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The theory, measurement, and applications of mode specific scattering parameters with multiple modes of propagation

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The theory, measurement, and applications of mode specific scattering parameters with multiple modes of propagation
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Bockelman, David E., 1967-
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viii, 412 leaves : ill. ; 29 cm.

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Calibration ( jstor )
Crosstalk ( jstor )
Differential circuits ( jstor )
Electric potential ( jstor )
Mathematical vectors ( jstor )
Microwaves ( jstor )
Power gain ( jstor )
Signals ( jstor )
Software ( jstor )
Transmission lines ( jstor )
Dissertations, Academic -- Electrical and Computer Engineering -- UF ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1997.
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Includes bibliographical references (leaves 403-411).
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Typescript.
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Vita.
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by David E. Bockelman.

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THE THEORY, MEASUREMENT, AND APPLICATIONS OF MODE SPECIFIC
SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION











By

DAVID E. BOCKELMAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1997




THE THEORY, MEASUREMENT, AND APPLICATIONS OF MODE SPECIFIC
SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION
By
DAVID E. BOCKELMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1997


Copyright 1997
By
DAVID E. BOCKELMAN


ACKNOWLEDGMENTS
The author would like to acknowledge the significant support of Motorola Radio
Products Group Applied Research, without which this work would not have been possible.
Many members of the staff of Applied Research gave support, advice and assistance
which has been received with gratitude. The author would especially like to thank Charles
Backof, Vice-President and Director of Research, Radio Products Group, who urged the
pursuit of this work, and Dr. Wei-Yean Hwong, Principal Member of the Technical Staff,
who gave his time and direction. The author is indebted to Robert Stengel, Member of the
Technical Staff, who provided motivation for this work and guidance through its comple
tion.
Furthermore, the author would like to thank Professor William R. Eisenstadt, who
demonstrated his generosity by giving essential support in technical and personal matters.
The author would also like to thank the members of his advisory committee for their sup
port and direction, who were critical elements in the partnership between the University,
Motorola, and the student. Also appreciated by the author is the help of the staff of the
University of Florida Microelectronics Lab, and the help of many others who can not be
listed here.
Of all who gave their support and assistance, none was as critical as the authors
wife, Erika. Her unquestioning commitment was the light which has led the way to this
conclusion.


TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
TABLE OF CONTENTS iii
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
2 PRIOR THEORIES AND TECHNIQUES 6
2.1. Fundamental Theories of Analysis 7
2.1.1. Coupled Transmission Line Pairs 7
2.1.2. Analog Methods 9
2.1.3. Linear Network Representations 11
2.1.3.1. Analog Network Parameters 12
2.1.3.2. RF Network Parameters 15
2.2. Measurement Techniques 20
2.2.1. Single Mode Analog Measurements 21
2.2.2. Single Mode RF and Microwave Measurements 22
2.2.2.1. Scalar Power Measurements Including Baiuns 22
2.2.2.2. Scattering Parameters with Baiuns 24
2.3. Summary of Past Theory and Techniques 26
3 FUNDAMENTAL THEORY OF MODE SPECIFIC S-PARAMETERS 27
3.1. Mode Specific Scattering Parameters in Differential Circuits 27
3.1.1. Fundamental Definitions for Differential Circuits 28
3.1.1.1. Modal Voltage and Currents 30
3.1.1.2. Coupled Mixed-Mode Signals 32
3.1.1.3. Mixed-Mode Scattering Parameters 37
3.1.2. Choice of Reference Impedances for Multiple Modes 39
3.1.3. Relationship of Mixed-Mode and Standard S-Parameters 42
3.1.4. Interpretations of Multi-Mode Scattering Parameters 46
3.2. Generalizations of Mode Specific Scattering Parameters 53
3.2.1. Other modes 53
3.2.2. Eigen modes 57
iii


4 CONSTRUCTION OF THE PURE-MODE VECTOR NETWORK ANALYZER.
61
4.1. Basic Operation of the PMVNA 62
4.1.1. Fundamental Concepts 62
4.1.2. General PMVNA Test-Set Architecture 63
4.2. Implementation of a Practical PMVNA 65
4.2.1. System Level Description 66
4.2.2. Test-Set Construction 69
4.2.3. Detailed Operation 72
4.2.4. Control Software 77
4.3. On-Wafer Measurements 79
5 ACCURACY OF THE PURE-MODE VECTOR NETWORK ANALYZER 82
5.1. Probe-to-Probe Crosstalk 83
5.1.1. Simulated Probe Crosstalk 84
5.1.2. Measured Probe Crosstalk 88
5.2. Uncertainty Calculations 90
5.2.1. Discussion of Accuracies 99
5.2.2. Uncertainty Model Derivation 101
5.2.3. Order of Uncertainty Calculations 106
5.3. Conclusions on Accuracy 107
6 CALIBRATION OF THE PURE-MODE VECTOR NETWORK ANALYZER ....
108
6.1. Types of VNA Measurement Errors 108
6.2. Primary PMVNA Calibration 110
6.2.1. Raw Performance 110
6.2.2. PMVNA Error Model 115
6.2.3. Development of Calibration Equation 121
6.2.4. Switching Errors and Non-Pure Mode Generation 124
6.2.5. Solution of the Calibration Problem 128
6.2.6. Coaxial Calibration Standards 132
6.2.7. On-Wafer Calibration Standards 134
6.3. Phase Offset Pre-Calibration 139
6.3.1. Phase Offset Standards 140
6.3.1.1. First Principles 141
6.3.1.2. Offset Model 142
6.3.1.3. Modified T-Matrix Solution 144
6.3.2. Phase Offset Of An Unknown DUT 150
6.3.2.1. Variable Offset Model 150
6.3.2.2. Using Multiple Offset Standards 151
6.3.2.3. Calculating the Offset of an Arbitrary DUT 152
6.3.2.4. Diagonalized Form 153
6.4. Calibration Procedure 154
iv


7 VERIFICATION OF THE PMVNA 156
8 POWER SPLITTER AND COMBINER ANALYSIS 167
8.1. Splitters 169
8.2. Combiners 178
8.3. Extensions to Arbitrary Phase 180
9 THIN-FILM METAL-ON-CERAMIC STRUCTURES 183
9.1. Differential Transmission Lines 184
9.1.1. Uniform Differential Transmission Line 184
9.1.2. Balanced Step Differential Transmission Line 189
9.1.3. Unbalanced Step-Up Differential Transmission Line 194
9.2. Comparison Between Measurements and Simulations 199
9.2.1. Unbalanced Step Differential Transmission Line 200
9.2.2. Balanced Step Differential Transmission Line 206
9.3. Crosstalk Between Differential Transmission Lines 212
9.3.1. Balanced Differential Transmission Lines 214
9.3.2. Unbalanced Differential Transmission Lines 230
10 PASSIVE INTEGRATED CIRCUIT STRUCTURES 239
10.1. Transmission Lines without Metal Ground Planes 243
10.1.1. Single-Ended Transmission Lines 243
10.1.2. Simple Uniform Differential Transmission Line 248
10.2. Transmission Lines with Ground Metal Ground Planes 252
10.2.1. Single-Ended Transmission Lines 253
10.2.2. Uniform Differential Transmission Lines 254
10.3. Unbalanced Differential Transmission Lines 259
10.4. Vertical Differential Transmission Lines 265
10.5. Pad-to-Pad Crosstalk 275
11 PROPERTIES OF MIXED-MODE S-PARAMETERS 291
11.1. Symmetry of Reciprocal Devices 291
11.1.1. General 291
11.1.2. Port-Symmetric Reciprocal Devices 293
11.2. Balanced Devices 294
11.3. Indefinite Mixed-Mode S-Parameters 297
11.4. Device Mode Specific Gains of Ideally Balanced Differential Circuit 304
11.4.1. Transducer Power Gains 305
11.4.2. Maximum Power Gains 308
11.4.3. Power Gain Circles 313
12CONCLUSIONS
318


APPENDICES
A ANALOG HALF-CIRCUIT TECHNIQUES 323
B ANALOG MEASUREMENT TECHNIQUES 327
C TRANSMISSION OF MODES FROM COUPLED TO UNCOUPLED LINES
330
D SIMULATED S-PARAMETERS OF DIFFERENTIAL AMPLIFIER 334
E DESCRIPTION OF HP8510 VNA SUB-SYSTEMS 339
F DETAILS OF HP8517 TEST-SET MODIFICATIONS 348
G PM VNA CONTROL SOFTWARE 357
H MULTI-PORT T-MATRIX DEFINITION 383
I ERROR TERMS OF PMVNA AND FOUR-PORT VNA 387
J DEMONSTRATION OF COEFFICIENT MATRIX RANK 390
LIST OF REFERENCES 403
BIOGRAPHICAL SKETCH 412
vi


Abstract of Dissertation Presented to the Graduate School of the University of Florida in
Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
THE THEORY, MEASUREMENT, AND APPLICATIONS OF MODE SPECIFIC
SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION
by
David E. Bockelman
May 1997
Chairman: William R. Eisenstadt
Major Department: Electrical and Computer Engineering
Mode-specific scattering parameters (s-parameters) are defined from fundamental
concepts. Such s-parameters directly express the response of a device in its intended
modes of operation. The development is specifically applied to high frequency differen
tial circuits. Differential circuits are shown to be characterized by four sets of s-parame
ters: (1) pure differential mode s-parameters with a differential-mode input and output, (2)
pure common-mode s-parameters with a common-mode input and output, (3) mode-con
version s-parameters with a differential-mode input and a common-mode output, and (4)
mode-conversion s-parameters with a common-mode input and a differential-mode out
put. All of these sets of mode-specific s-parameters are shown to be useful in analysis of a
differential circuit.
A specialized system, called the pure-mode vector network analyzer (PMVNA), is
developed for the measurement of the mode-specific s-parameters of a high frequency dif
ferential circuit. The calibration of this analyzer is developed and implemented. Verifica-
vii


tion establishes error correction accuracy. The PMVNA is shown to have accuracy
advantages for the measurement of differential circuits when compared to a traditional
four-port analyzer.
The mode-specific s-parameter concepts are applied to several practical high fre
quency differential circuits. Power splitters and combiners are analyzed with these con
cepts. Traditional specifications of phase and magnitude imbalance are shown to
correspond to spurious mode responses. Differential transmission line structures, imple
mented on ceramic substrates, are examined. The effects of imbalance and symmetry are
analyzed with mode-specific s-parameters. Several structures on a silicon integrated cir
cuit (IC) are measured. The effects of differential topology on circuit-to-circuit coupling
are quantified. Basic design methods are advanced for the design of high frequency dif
ferential circuits.
viii


CHAPTER 1
INTRODUCTION
In many applications, devices and circuits have been designed for only a single
mode of operation. In the most general sense, a mode is a particular electromagnetic field
configuration for a given device or circuit. In the case of one or two conductors, the
modes are usually frequency dependent, so the existence of simultaneous modes can be
avoided by proper selection of operating frequencies (or by proper physical design for a
given frequency). However, with three or more conductors, there will usually exist multi
ple modes even in static cases. In such situations, the simultaneous existence of two or
more modes can be difficult to avoid.
Differential circuits are a particular class of circuits of historic importance with
three conductors. Sometimes called balanced circuits, the primary operation of differen
tial circuits is to respond to the difference between two signals, such as Av]=Vj v2 as
-T
vi 1
+
+
Port 1 Avt
+
v2 2
-I
Figure 1-1. Schematic of two-port differential circuit.
T.
i


2
shown in Figure 1 -1. The two conductors can also have a common voltage (or a current
flow) with respect to a third conductor, namely ground. As a result, two modes of opera
tion are generally possible with differential circuits: the differential-mode and the com
mon-mode. Furthermore, both modes can exist simultaneously in general.
There are many applications of differential circuits. Twisted pair transmission
lines, operational amplifiers, baluns, coupled transmission lines, power splitters and com
biners are all examples of differential circuits [1 3], More recent applications include
radio frequency (RF) low noise amplifiers (LNA) with differential inputs and outputs, as
well as double-balanced mixers such as Gilbert cell mixers [4].
RF differential circuit applications have become common as the commercial
demand for radio systems has grown. Two characteristics of differential circuits make
them particularly attractive for RF applications. The first advantage of the differential cir
cuit is circuit-to-circuit isolation. This characteristic has been exploited for many years,
most notably in telephone systems in the form of twisted-pair wire transmission lines [5].
The higher isolation of differential circuits (with respect to single-ended circuits) is due to
the nullification of any noise common to both constituent signals in the differential signal,
i.e. (v+n) {-v+n) = 2v where n represents an interfering signal. This isolation increase is
important to integrated circuit (IC) implementations. As integration density increases, and
operation frequencies increase, improving circuit-to-circuit isolation is critical. As a
result, differential circuits are being applied where only single-ended circuits have tradi
tionally been used.
Second, the differential circuit has increased dynamic range when compared to a
ground referenced, or single-ended, circuit. With a given voltage swing v, a pure difieren-


3
tial signal will be doubled, i.e. v (-v) = 2v. This increased dynamic range is particularly
important as the supply voltages decrease in modern radio systems. This decreasing sup
ply voltage has made single-ended implementations of receiver functions difficult, since
maximum signal swing in a circuit is typically less than the supply voltage. By imple
menting radio functions with differential circuits, the available signal swing, and hence
the dynamic range, can be increased while retaining a low supply voltage.
The emphasis of traditional RF and microwave techniques has been to avoid the
simultaneous existence of multiple modes. As a result, there is a lack of self-consistent,
rigorous theory that is applicable to the measurement, analysis and design of RF and
microwave differential circuits. Typically, differential circuits are designed and analyzed
with traditional analog techniques, which employ lumped element assumptions. RF and
microwave differential circuits contain distributed circuit elements, and require distributed
circuit analysis and testing. Furthermore, traditional methods of testing differential cir
cuits have required the application and measurement of voltages and currents, which is
difficult at RF and microwave frequencies. Scattering parameters (s-parameters) have
been developed for characterization and analysis at these frequencies, but have been
applied primarily to single-ended circuits. A modification of existing s-parameter tech
niques is needed for accurate measurement, analysis and design of differential circuits at
microwave frequencies. This work extends the definitions of s-parameters to mode-spe
cific representations, where the s-parameters are defined in terms of the natural modes of
operation of a circuit.
This dissertation presents original work in the following areas. The traditionally
accepted definitions of differential and common-mode voltages and currents are shown


4
for the first time to be non-orthogonal, and therefore unacceptable for direct application of
these definitions for power calculations. New orthogonal definitions for these voltages
and currents are presented, and shown to be appropriate for power calculations. Multiple
mode s-parameters are developed which for the first time completely describes the linear
behavior of an RF differential circuit. These concepts are verified through simulations of
RF differential circuitry. The first network analyzer for the measurement of multi-mode
s-parameters is constructed, and the inherent accuracy advantages of the system are estab
lished. Fundamental work in multi-port network analyzer calibration proceeds beyond
any previously published work, and a verification procedure establishes the accuracy of
the calibration. Measurements with the multi-mode network analyzer includes the first of
integrated differential circuits. Extensions of s-parameter design techniques to multi-
mode circuits are presented that will formalize the design and analysis of RF differential
circuits.
This dissertation is organized in the following manner. In Chapter 2, techniques
for analysis and measurement of differential circuits, prior to this work, are discussed.
Chapter 3 presents original work of extending scattering parameter theory to differential
circuits. A new measurement system for the measurement of mode-specific s-parameters
is introduced in Chapter 4. Chapter 5 examines the accuracy, and establishes the intrinsic
accuracy advantages, of this system for the measurement of differential circuits. The cali
bration theory and implementation for the new measurement system is developed in
Chapter 6. In Chapter 7, the results of accuracy verification of this new system are pre
sented. The remaining chapters of the dissertation focus on applications of the mode-spe
cific s-parameter concepts. Chapter 8 applies the new mode-specific concepts to power


5
splitters and combiners. Several thin-film metal differential structures, fabricated on alu
minum oxide, are studied in Chapter 9. Circuit-to-circuit crosstalk for IC structures on sil
icon is studied in Chapter 10, and conclusions are made about practical implementation of
ICs. Chapter 11 discusses properties of the new s-parameters and provides basic analysis
and design tools for use with RF differential circuits. Chapter 12 concludes this disserta
tion with a summary, some discussions, and remarks on future research.


CHAPTER 2
PRIOR THEORIES AND TECHNIQUES
This chapter serves as a summary of past theoretical and experimental techniques
that are applied to differential circuits. The focus of the chapter is RF and microwave dif
ferential circuits. However, lower frequency work has had a profound effect on the sub
ject, so the examination will include relevant analog techniques.
In the area of theoretical analysis, the subjects presented include multi-mode trans
verse electromagnetic (TEM) structures such as coupled transmission lines. The analog
methods that have provided the basic concepts of differential circuit analysis are summa
rized. Network representation of a differential circuit is reviewed, and its application to an
RF differential circuit is explored. The traditionally accepted definitions of differential
and common-mode voltages and currents are shown to be non-orthogonal, i. e., as a sys
tem, the definitions do not conserve energy.
The measurement techniques of RF differential circuits are then summarized.
Analog differential circuit measurement techniques are briefly examined as background.
All widely practiced measurement approaches for RF and microwave differential circuits
are presented in general, and are shown to provide inadequate characterization of the
device under test.
6


7
2.1. Fundamental Theories of Analysis
The topics presented in the following sections represent some of the most promi
nent concepts in differential circuit analysis. The subjects are coupled transmission lines,
analog methods, and network representations, and each topic holds a unique concept
which shapes later theoretical developments.
2.1.1. Coupled Transmission Line Pairs
In a survey of prior work in the RF and microwave fields, one early area of work is
found to share many concepts with differential circuits in general. The work done in cou
pled transmission lines, and their applications, describe multiple mode behavior that is
analogous to differential circuit modes. The importance of this transmission line work in
this context is the treatment of the simultaneous existence of two modes of propagation.
The coupled transmission line pair theories have their foundations in electromag
netic field descriptions [6, 7]. Systematic treatment of the coupled pair transmission line
begins with the examination of the two fundamental TEM modes. Planar coupled trans
mission lines such as stripline are of particular importance due to fabrication advantages.
As a result, much early work considers planar structures. With such structures, the two
fundamental TEM modes are called odd and even for their respective field symmetry, with
(a)
(b)
Figure 2-1. Electric field distributions in planar coupled transmission lines,
a) Odd-mode electric field; b) Even-mode electric field.


8
the terminology apparently first introduced by Cohn [6], Representations of the electric
field distributions for the two fundamental TEM modes are shown in Figure 2-1. In
Figure 2-1 (a) the signal conductors are at equal but opposite potentials and carry equal but
opposite currents, and hence this is called the odd mode. In Figure 2-l(b) the signal con
ductors are the same potential and carry equal currents; this is called the even mode.
Early work was limited to only physically symmetric structures [6], and the terms
even and odd apply only with such limitations. Tripathi later extended the theory to
include any coupled pair transmission line structure [8], With this extension, the two
modes became c and 7i-modes, respectively, and the symmetry in the field distribution was
lost. With the loss of the even and odd-modes, the direct analogy to differential and com
mon-modes becomes less clear.
Although important for the treatment of simultaneous modes, the coupled trans
mission line work is limited to transmission line applications. The theory is cast in terms
of characteristic impedances, propagation constants, etc., and is not directly applicable to
a general differential circuit. Previous work has been specific to descriptions of coupled
transmission lines [6, 8-15] and shielded balanced transmission lines. In the literature,
the coupled transmission work has been most commonly applied to directional couplers
[2, 16 18] rather than to differential circuits in general. All of the referenced work deals
with specific TEM structures, and is not suitable for characterization of a generic differen
tial circuit.
Despite the narrow application of prior work, the theory of coupled transmission
lines provides the foundation on which general multi-mode network analysis is built.
Scattering parameters are a relative measure of a networks response, so any mode-spe-


9
cific s-parameters must be defined with respect to some reference impedance. In
Chapter 3, the theory of coupled transmission lines will be used to rigorously define the
reference impedance for the different modes.
2.1.2. Analog Methods
Traditional analog methods play a central role in the prior work with differential
circuits. The work, which employs lumped element assumptions, is used primarily for
audio or near-audio frequencies. Of course, most any general analog circuit techniques
can be applied to an analog differential circuit, but some specialized concepts are of par
ticular importance.
Analog theories have provided the fundamental definitions of differential and
common-mode voltages. Referring to Figure 1-1, the differential-mode voltage at port
one is defined as
vdl s V1 v2 i2'1
and the common-mode voltage at port one is defined as
(2-2)
The differential current into port one is
(2-3)
and the common-mode current into port one is
(2-4)


10
with similar definitions at port two. These definitions have lead to voltage gain concepts
These definitions are widely accepted as evidenced by examples found in the texts by
Gray and Meyer [4], Middlebrook [19] and Giacoletto [20], as well as other recent works
[12,21],
Of particular interest in analog techniques is the method of differential and com
mon-mode half-circuits. This technique allows circuit analysis simplification by consider
ing separately the response of the circuit to a pure differential signal and a pure common
mode signal [4], For an instructive example of the application of half-circuit methods, see
Appendix A. These analog techniques are useful in gaining insight into differential ampli
fier operation. However, these techniques have some limitations. First, these equivalent
circuit approaches can become intractable as high frequency effects are included. Also,
they are inherently lumped element approaches, and are not easily adapted to include dis
tributed circuit elements that become important at RF and higher frequencies.
The most fundamental limitation is that the half-circuit techniques are applied only
to symmetric differential circuits. It has been shown [19], that perfectly symmetric (or
balanced) differential circuits exhibit no mode conversion. By limiting the analysis to
symmetric circuits, or by neglecting any asymmetry, the phenomenon of mode conver
sion is completely ignored. Mode conversion occurs when a stimulus of a pure mode cre
ates an output of more than one mode. For example, if a pure differential signal drives an
amplifier, and both a differential-mode and a common-mode output signal are produced,
then some conversion from differential to common-mode has occurred.


11
Mode conversion is an important phenomenon for RF and microwave differential
circuits, and in amplifiers in particular. It has also been shown [19] that mode conversion
will affect the maximum achievable common-mode rejection ration (CMRR). A critical
parameter of differential amplifier design, CMRR quantifies the ability of an amplifier to
amplify differential signals and reject common-mode signals. Understanding, predicting,
and measuring the phenomenon of mode conversion can be important to the performance
of RF differential circuits.
2.1.3. Linear Network Representations
Linear time-invariant (LTI) network representation is a basic and useful circuit
analysis technique which is widely applied to two-port and three-terminal circuits of both
analog and RF applications [22]. Network representations are distinctly suitable for
1.
V ~k+l
IM ^
g1 > (k+iy
h
Linear
Time Invariant
V -M.
T0- ^
h Vk+2
T2 -3 (k+2)

,, 4
Network
4
4
k'0- ^
+
/
-
Figure 2-2.
Notation of an rc-port linear time-invariant network.


12
descriptions of distributed element circuits as encountered in RF and microwave applica
tions. It is of use then to examine the application of LTI network theory to differential cir
cuits.
A circuit, or network, with n pairs of terminals which are used as input/output con
nections is known as an n-port network. The notation conventions for an n-port network
are shown in Figure 2-2. The dashed line connecting the 1 terminal to ground indicates
that some or all of the return terminals can be grounded. The behavior of the network is
described by a set of equations that are expressed in terms of the defined voltages and cur
rents (or quantities that are related), which can be written in matrix form. The matrix
description leads to a convenient set of parameters for a circuit. Some commonly used
parameters are impedance Z, admittance Y, hybrid h, and chaining ABCD. These param
eters are all based on voltage and current descriptions of the network. Other parameters,
such as scattering parameters (s-parameters), are based on functions of voltage and cur
rent.
2.1.3.1. Analog Network Parameters
Network representations can be applied to differential circuits in at least two ways.
One possible application of network theory is to interpret each input and output terminal
of the differential circuit as a port with the return path grounded. This approach is quite
common, and will be referred to as the standard approach to network representation. With
this approach, all of the inputs and outputs of the differential circuit are ground referenced
(single-ended). In this case, the network will always have 2n ports, where n is number of
differential inputs and outputs. For example, a differential amplifier can be represented as


13
a four-port network, shown in Figure 2-3. Here the port voltages are related to the differ
ential and common-mode voltages [12, 19, 20] by
Ml
'l = -Y + Vc\
vdl
M2
v3 = -T + Vc2
(2-6)
2 + vcl v'
M2
2 2 + vc2
The port currents can similarly be related to differential and common-mode currents
3
¡.11 + I
d2 _r c2
- hn + L
(2-7)
l = 'dl+icl
*2 = ~ *dl + !cl 4 d2 'r c2
By defining the port voltages and currents as such, the network description of the differen
tial circuit can be completed in terms of any useful parameters.
There is a critical limitation with this particular approach to network representa
tion of differential circuits. Since all port voltages and currents are functions of both dif
ferential and common-mode quantities, all the resulting network parameters will also be
combinations of both mode responses. The response of the circuit to a specific mode is
not obvious from inspection of the parameters. This commingling of the modal responses
Figure 2-3. Network representation of differential amplifier.


14
is a distinct disadvantage, since the implicit purpose of a differential circuit is to provide a
certain response to a differential stimulus. So, although the standard approach to network
representation is a sufficient description of differential circuits, it is non-intuitive. There
fore, a second approach to network descriptions of differential circuits will be described.
The second type of network representation of differential circuits describes the cir
cuit explicitly in terms of modal responses. By using modal definitions of voltage and
current as given in (2-1) to (2-4), a network description can be defined. First, the inputs
and outputs must be paired appropriately. For example, ports one and two can be paired to
create a differential port. This pairing of signals is extremely useful in low-noise systems,
as discussed in Chapter 1. For example the z-parameters of a differential amplifier can be
defined [20] as
vdl
zdl,dl
zdl,cl zdl,d2 zdl,c2
dl
vcl
=
zcl,dl
Zcl,cl zcl,d2 zcl,c2
cl
<
Cl
NJ
zd2,dl
zd2,cl zd2,d2 zd2,c2
ld2
>2
zc2,dl
zc2,cl zc2,d2 zc2,c2_
lc2_
Figure 2-4. Modal notation of an two-port differential network.


15
This network description can be interpreted directly in terms of differential and
common-mode responses. The network diagram can be modified to reflect the explicit
modes, as shown in Figure 2-4. This approach will be called the modal network represen
tation. Note that a two-port differential circuit is represented again by a four-port net
work; in general, an n-port differential circuit will have a 2-port network. The separation
of the differential and common-mode ports in the network representation is a useful con
ceptual tool. The modal network representation presented thus far is useful in the analysis
of analog differential circuits. However, the application of this technique to RF and
microwave circuits is of limited use as will be discussed in the next section.
2.1.3.2. RF Network Parameters
Power-based scattering parameters are widely used in RF and microwave fields to
represent circuits and devices with distributed elements. As its name implies, scattering
parameters represent a scattering or separation of a signal by a device under test. The scat
tered signals are the reflected and transmitted electromagnetic waves that are produced
when a device is stimulated with an incident wave. Scattered wave descriptions of net
works are very important when operation frequencies are high enough such that circuit
elements become a significant fraction of a wavelength (approximately one tenth of a
wavelength). Scattering parameters originate in transmission line concepts. As such, they
are always defined with respect to a characteristic impedance, or reference impedance.
The primary benefit of s-parameters is ease in measurement. In distinction to voltage-cur-
rent derived parameters, s-parameters are measured with ports terminated in the character
istic impedance. This has meaningful practical implications, since short-circuits and


16
open-circuits are extremely difficult to realize at RF and microwave frequencies due to
distributed element effects.
Scattering parameters will first be developed in terms of transmission line quanti
ties, to provide insight into their wave aspect. Following this definition, the generalized
definition will be given.
The following development is summarized from Gonzalez [23], The voltage and
current along a transmission line, such as in Figure 2-5, satisfy the set of differential equa
tions
-^jV(x)-y2V(x) = 0
dx
d2 2
-VW-vm = 0
dx
where y is the propagation constant. The general solution of (2-9) is
(2-9)
V(x) = Ae yx + Beyx
(2-10)
where A and B are complex constants and Z0 is the characteristic impedance. The propa
gation constant and the characteristic impedance can be expressed in terms of the parame
ters R, G, L, C which are the resistance, conductance, inductance, and capacitance per unit
length of the transmission line
y = +ja>L)(G +ycoC) ZQ = (2-11)
Given the phasor notation V+(x)=Ae~'fx and V(x)=Be,x, and by limiting the transmission
line to be lossless (i.e. Zq = Re(Zfj}), then the important normalized quantities are defined
as


17
a(x) =
i(x) = I(x)JZQ
With these definitions, (2-10) becomes
(2-12)
v(x) = a(x) + b(x)
i(x) = a(x)-b(x) ^ ^
The a and b waves are the incident and reflected/transmitted normalized power waves, and
they are the primary quantities of s-parameters.
I
I
-U
I
x=k)
Figure 2-5, Terminated transmission line.
When applied to an n-port network, such as in Figure 2-2, the a and b waves result
in a s-parameter description
b\
'll
*12 '
*1 n
b2
=
S21
*22
*2 n
bn
fnl
*n 2
V
(2-14)
or simply b = Sa where the bar over a lower-case variable represents a column vector [23],
The definition of s-parameters can be generalized to include complex characteris
tic impedances. This generalization also removes the dependency of the s-parameter on


18
transmission lines. The definition is based on a generalized power wave at the n-tU port
[23 25]
a = , [v +i Z ]
n n n n
b = [ v i Z ]
n 2 ^R¡(F) n nn
(2-15)
The s-parameter matrix equation (2-14) remains the same.
Scattering parameters have not been widely applied to the analysis or measure
ment of differential circuits. S-parameters would provide the same benefits to RF differ
ential circuits as they do for other RF and microwave circuits. Conceptually, the
representation of differential circuits with s-parameters is not difficult. In fact, with the
standard network representation discussed earlier, a -port differential circuit can be
described with a 2n-by-2n s-parameter matrix, without any additional consideration.
However, this approach has the same disadvantages as previously described, namely, the
parameters do not provide useful indications of the differential and common-mode
responses. For a illustration of the difficulties of interpreting the standard four-port
s-parameters of an RF differential amplifier, see Appendix D.
The above limitations could be removed by extending s-parameter theory to a
modal network representation. This extension has not been completed prior to this work,
and this dissertation later presents the extension.
A straight-forward extension of s-parameter theory to a modal network represen
tation would apply the traditional definitions of modal voltages and currents in (2-1)
through (2-4) to the generalized power wave definitions of (2-15). However, the voltage
and current definitions of (2-1) through (2-4) are not an acceptable basis for a power wave


19
network representation. Straight-forward application of these definitions results in quanti
ties that do not conserve energy. The difficulties with power calculations using these
quantities can be demonstrated with a simple example.
Suppose two sources of power have potentials and V2 and source currents 7j
and 72, respectively. Assume the sources are harmonically time varying (so V¡ and 7; are
phasors, as indicated by the bar over the upper-case variable) but have no specific phase
relation. The power delivered by the two sources is
/>, = Re^/j*) P2 = R z(V2T*) (2-16)
and the total power in both sources is
PT=Pl+P2 (2-17)
By definitions (2-1) through (2-4), the differential and common-mode voltage and current
can be expressed as
\ 2>
VC = \(V¡ + V2) 7"c = \(h+T2)
(2-18)
The power in each mode is then
pd = Re(Vd*> Po = Re(Vc*> (2-19)
If the modal definitions are consistent, then the total power of the modes must be equal to
the total power of the sources by the conservation of energy. Expanding (2-19)
Pd = jIRefV^VReV^VReV^V RefVy,*)]
Pc = |[Re(^*) + Re(^*) + Re(V^7¡*) + Re(V^7;*)]
(2-20)
and the sum of mode power is


20
Pd + Pc = pReVj/jVSReV^VReVjVReVi*)] (2-2D
Expanding the sum of the source powers in (2-17)
PT = P]+P2 = Re(Vj/^*) + Re(V^7¡*) *Pd+Pc (2-22)
which clearly shows that the voltage and current definitions of (2-1) through (2-4) are not
directly usable in power calculations.
The voltage and current definitions of (2-1) through (2-4) can be used for power
and power-wave calculations if care is taken to account for the non-orthogonal nature of
the system. However, it is much more convenient to define new mode voltages and cur
rents that are orthogonal. These new definitions are given in Chapter 3, Section 3.1.1
Despite the advantages of scattering parameters, there is no acceptable theoretical
treatment of s-parameter network representations for differential circuits prior to this
work. The attempts at applying s-parameters to RF differential circuits have relied upon
intuitive notions of differential s-parameters [26, 21]. As such, the prior incomplete theo
ries have not treated several fundamental principles that a rigorous theory requires. The
principles in question are conversation of energy (orthogonality) in the modes, precise
definitions of reference impedances for all modes, rigorous definitions of all pure modal
responses, and self-consistent definitions of conversion responses between modes.
2.2. Measurement Techniques
When examining prior work on circuits with multiple simultaneous modes of
propagation, consideration must be given to accepted measurement techniques. The state
of the theoretical development, and of the organization of the field as a whole, can be


21
observed in the completeness and accuracy (or the lack thereof) of generally accepted
measurement techniques.
Again, the scope of this survey of prior work will be limited to subjects related to
differential circuits. The topics presented in the following sections represent some of the
most widely practiced measurement techniques for differential circuits. The subjects are
divided between the analog techniques, RF/microwave scalar approaches, and scattering
parameter measurements. The treatment of the measurement techniques is not intended to
be exhaustive, but it is representative of the common types of measurements presently
applied to differential circuits.
2.2.1. Single Mode Analog Measurements
Analog measurements of differential circuits are typically direct measurements of
voltages and currents, which are primarily limited to audio or near-audio frequencies (i.e.
typical analog frequencies) [27]. The reason for this limitation is due to distributed nature
of circuits as frequencies approach RF. With distributed circuits, there will be transmis
sion line effects. With transmission line effects, the voltage and current will be functions
of the position along the line. Furthermore, parasitic capacitances and inductances
become significant at these frequencies, and effect the performance of the DUT. As a
result, it is difficult to make unambiguous measurements of voltage and current at RF and
higher frequencies.
Differential analog measurements typically employ single-ended to differential
converters (called baluns) to stimulate and measure the DUT in an ideally pure differential
mode. However, these converters are not ideal, and they affect the accuracy of the mea
surements. Most significantly, the measurements inevitably include the effects of these


22
converters, and little can be done to remove them. For a more detailed description of typ
ical analog differential measurements, see Appendix B.
2.2.2. Single Mode RF and Microwave Measurements
When a differential circuit operates in the RF/microwave frequencies, voltage and
current measurements are no longer practical. Instead, the appropriate measurements deal
with transmission of waves and power. Some of the most common and important RF
measurements of differential circuits are presented below.
The primary difficulty with RF differential measurements, like analog measure
ments, is the generation and reception of differential signals. Also like analog measure
ments, RF measurements require baluns. For RF, center-tapped transformers are available
that can operate to 1 GHz [28]. For higher frequencies, power splitter/combiners, such as
hybrid couplers, are generally used. The one consequential difference from the analog
baluns is that the RF/microwave baluns have more non-ideal performance.
Differential-mode RF measurements can be made with the use of 180 power split
ters/combiners, and common-mode RF measurements can be made with 0 power split
ters/combiners. Like the analog measurements, these RF/microwave measurements
assume single-mode inputs and output, and are called single mode measurements.
2.2.2.1. Scalar Power Measurements Including Baluns
One widely used type of RF measurement of differential circuits is a scalar power
measurement. This measurement provides the magnitude of the power gain. The mea
surement may take the form of a constant amplitude input signal swept across frequency,
resulting in a gain versus frequency characteristic. Alternatively, the input power level


23
can be swept at a fixed frequency, resulting in a output power versus input power charac
teristic.
Regardless of the specific measurement, scalar power measurements have the
same basic instrumentation. The signal source is an RF signal generator, the measurement
instrument is a power meter or a spectrum analyzer, and RF baluns must be used. A typi
cal measurement system is shown in Figure 2-6.
Like the analog measurements, the scalar RF power measurements include the
effects of the baluns. The effects of the baluns are even more difficult to remove at RF
frequencies than at analog. This difficulty is due to the increased non-ideal performance
of the baluns. The non-ideal performance is typically specified in terms of loss, magni
tude imbalance, and phase imbalance. The effect of the baluns on the accuracy of the
measurement can be examined qualitatively.
RF baluns, such as 180 3 dB hybrid couplers, have magnitude and phase imbal
ance in the splitting of a signal. Ideally, a 180 3 dB hybrid coupler would take a single
input signal and split it into two equal amplitude signals with 180 phase difference. With
Hybrid Hybrid
Figure 2-6. RF scalar power measurement of differential circuit.


24
an ideal splitter a pure differential mode signal could be constructed. However, the mag
nitude and phase imbalance means that the two outputs of the splitter are not exactly equal
amplitude, and the phase difference is not 180. As a result, a pure differential signal is
not produced by a real power splitter, and a test circuit is only driven in an approximately
single mode fashion. The magnitude and phase imbalance also affect the combination of
two signals. In essence, the imbalance causes a spurious response to a common mode
input. The combined effect of the imbalances in the power splitter and combiner is to
measure a commingled response of the circuit to both a large differential and small com
mon-mode input. The differential and common-mode responses cannot be distinguished
by the instruments, and the overall measurement accuracy is reduced. These effects are
examined in detail in Chapter 8.
2.2.2.2. Scattering Parameters with Baiuns
A less prevalent, but important, technique for RF/microwave differential circuits is
single mode (differential) s-parameter measurement [26]. This approach, as implied by its
name, attempts to measure s-parameters of a circuit with input signals and output signals
of a single (differential) mode.
Like other single mode measurements, this technique employs baluns. The most
common application of this method is the measurement of the differential response of a
circuit with s-parameters. The measurement system includes a standard two-port vector
network analyzer (VNA) which automatically measures the s-parameters of a two-port
device and a pair of 180 3dB power splitters/combiners. This approach has also been
applied to on-wafer measurements of differential circuits [26], The schematic of the sys
tem is shown in Figure 2-7.


25
Figure 2-7. S-Parameter measurement of differential circuit.
This measurement technique suffers from the same problems as the scalar RF mea
surements. The magnitude and phase imbalance in the splitters/combiners and the neglect
of mode conversion will all produce errors in the measured s-parameters. However, the
s-parameter approach represents an important extension of measurement techniques. In
contrast to scalar measurements, s-parameters are by their nature vector quantities, and
hence they represent both magnitude and phase measurements.
Another limitation of this technique as reported is the lack of rigorous definition of
differential and common-mode scattering parameters. Calibration of the measurement
system, a necessity for all accurate VNA measurements, is also undefined. Although lim
ited in accuracy due to the cited problems, a calibration for this system could be derived
from the theory presented later in this work.


26
2.3. Summary of Past Theory and Techniques
Clearly, an opportunity exists to extend the accuracy of analysis, design, and mea
surement of differential circuits into the RF and microwave frequencies. By combining
the core principles of differential circuits traditionally belonging to the analog domain
with established RF techniques like scattering parameters, a strong contribution to both
fields is achieved. In the next chapter, the concepts of multi-mode analog differential cir
cuits are extended into a rigorous theory for the analysis, measurement and design of RF
differential circuits.


CHAPTER 3
FUNDAMENTAL THEORY OF MODE SPECIFIC S-PARAMETERS
3.1. Mode Specific Scattering Parameters in Differential Circuits
A severe limitation in differential-mode/common-mode circuit characterization is
a lack of applicable power wave and s-parameter theory in terms of these two modes.
There is no previously reported way to describe s-parameters based on mixed differential-
mode/common-mode propagation. Previous work most closely related to this work has
been specific to descriptions of coupled transmission lines [8 15] and shielded balanced
transmission lines. Work by the National Bureau of Standards on balanced transmission
lines used s-parameters to describe differential-mode propagation, but neglected common
mode propagation and any mode conversions [21], In the literature, the coupled transmis
sion work has been most commonly applied to directional couplers [2, 16-18] with Cohn
and Levy [3] providing a historical perspective on the role of coupled transmission lines in
directional coupler development. Past work on coupled transmission lines has largely
focused on voltage/current relationships and Z, Y, and ABCD-parameter descriptions of
TEM circuits. One notable exception to the Z/K/ABCD-parameter approach is work by
Krage and Haddad [29] which employs traditional normalized power waves to describe
coupler behavior. However, all of the referenced prior work deals with specific TEM
structures, and is not suitable for characterization of a generic differential circuit.
27


28
The following sections contain original work in the definitions of multi-mode
power waves and s-parameters. Portions of this work have been published in summary
form [30], The details of the development of multi-mode s-parameters, and new related
material, are contained in the remainder of the chapter.
3.1.1. Fundamental Definitions for Differential Circuits
In a practical RF/ microwave implementation, a differential circuit is based on
pairs of coupled transmission lines. A schematic of a typical two-port RF/ microwave dif
ferential system is shown in Figure 3-1. Essential features of the microwave differential
circuit in Figure 3-1 are the coupled pair transmission line on the input and output of the
DUT. As described in Chapter 2, this coupled line structure allows the propagation of two
TEM modes.
It is conceptually beneficial to define a signal that propagates between the lines of
the coupled-pair (as opposed to propagating between one line and ground). Such signals
are known as differential signals, and can be described by a difference of voltage (Av[ 0,
Mixed-Mode Mixed-Mode
Figure 3-1. Schematic of RF differential two-port network.


29
Av2 0) and current flow between the individual lines in a pair. By such a definition, the
signal is not referenced to a ground potential, but rather the signal on one line of the cou
pled pair is referenced to the other. Further, this differential signal should propagate in a
TEM, or quasi-TEM, fashion with a well-defined characteristic impedance and propaga
tion constant. Coupled line pairs, as in Figure 3-1, allow propagating differential signals
(the quantities of interest) to exist. The differential circuit discussion in this chapter will
be limited to the two-port case, but the generalized theory for n-port circuits can be readily
derived from this work.
Most practical implementations of Figure 3-1 will incorporate a ground plane, or
some other global reference conductor, either intentionally or unintentionally. This
ground plane allows another mode of propagation to exist, namely common-mode propa
gation. Conceptually, the common-mode wave applies equal signals with respect to
ground at each of the individual lines in a coupled pair, such that the differential voltage is
zero (i.e. Av[ = Av2 = 0). The ability of the microwave differential circuit to propagate
both common-mode and differential-mode signals requires any complete theoretical treat
ment to include characterization of all simultaneously propagating modes. For conve
nience, the simultaneous propagation of two or more modes (namely, differential-mode,
and common-mode) on a coupled transmission line will be referred to in this work as
mixed-mode propagation, from which mixed-mode s-parameters will be defined.
To begin the development of a rigorous theory of mixed differential and common
mode normalized power waves, the two modes must be defined in a self-consistent fash
ion. A differential signal propagates between the lines of the coupled-pair (as opposed to
propagating between one line and ground), and a common-mode signal propagates with


30
equal signals with respect to ground at each of the individual lines in a coupled pair. The
ability of the microwave differential circuit to propagate both common-mode and differ
ential-mode signals requires any complete theoretical treatment to include characterization
of all simultaneously propagating modes. For convenience, the simultaneous propagation
of two or more modes (namely, differential-mode, and common-mode) on a coupled
transmission line will be referred to as mixed-mode propagation, from which mixed-mode
s-parameters will be defined.
3.1.1.1. Modal Voltage and Currents
At this point, it is important to define the differential and common-mode voltages
and currents to develop a self-consistent set of mixed-mode s-parameters. Referring to
Figure 3-1, define the differential-mode voltage at a point, x, to be the difference of
between voltages on node one and node two
vdW = vi ~v2 (3-1)
This standard definition establishes a signal that is no longer referenced to ground. In a
differential circuit, one would expect equal current magnitudes to enter the positive input
terminal as leaves the negative input terminal. Therefore, the differential-mode current is
defined as one-half the difference between currents entering nodes one and two
(3-2)
These definitions differ from previously published definitions by Zysman and Johnson
[12] due to change in references. The common-mode voltage in a differential circuit is
typically the average voltage at a port. Hence, common-mode voltage is one half the sum
of the voltages on nodes one and two


31
vc = 2(vl +v2)
(3-3)
The common-mode current at a port is simply the total current flowing into the port.
Therefore, define the common-mode current as the sum of the currents entering nodes one
and two
(3-4)
ic(x) = i{ + i2
Note that the differential current includes the return current, and the return current for the
common-mode signal flows through the ground plane. For this reason, the differential
mode current is halved where the common-mode current is not. This definition of com
mon-mode current differs from the traditionally accepted definition [4, 12, 19 21],
Definitions in (3-1) to (3-4) are self-consistent with the differential power deliv
ered to a differential load. This can be shown by demonstrating that these definitions con
serve the total energy in the modes. The power at each terminal (x = 0 for example) can be
expressed as
P, = Re(v [ i, *) P2 = Re(v2!2*)
(3-5)
and the total power in both sources is
(3-6)
The power in each mode is
(3-7)
By definitions (3-1) to (3-4)
Pi ~ 2[Re(vll*) + Re(v22*)-Re(vi2*)-Re(v2 Pc = jIRe^'^ + ReOYy'O + ReVjiyO + ReOY'j*)]
(3-8)


32
and the sum of mode power is
Pd + Pc = ^[2Re(v,i1*) + 2Re(v2i2*)] = RefVji',*) + Re(v2¡2*) (3-9)
Expanding the sum of the source powers in (3-6)
PT = P \ + P2 = Re(V|t'j*) + Re(v2¡2*) (3-10)
Therefore the sum of the modal power is equal to the total power
Pd + Pc = Pl+P2 = PT (3-11)
and energy is conserved by the definitions of common and differential-mode voltages and
currents.
3.1.1.2. Coupled Mixed-Mode Signals
To begin the presentation of mixed-mode s-parameters, a general asymmetric cou
pled transmission line pair over a ground plane will be analyzed. This analysis yields mul
tiple propagating modes all referenced to ground. These modes will be used to express the
x = L
Port 2
Figure 3-2. Schematic of terminated asymmetric coupled-pair transmission line.


33
desired differential signal between the lines of the coupled-pair, as well as the common
signal referenced to ground. Figure 3-2 is a diagram of such a coupled-pair transmission
line, with all pertinent voltages and currents denoted. Also shown in Figure 3-2 is a repre
sentation of a termination for the coupled-pair line. Subject to the simplifying assump
tions, the mathematical results of this chapter are applicable to any pair of conductors with
a nearby ground conductor.
Referring again to Figure 3-2, the behavior of the coupled-line pair can be
described by [8]
dv j
Tx = -(Zll+Zm'2)
dv2
Tx =-(z2'2 + Vl)
(3-12)
di\
Tx = -0'iv1 +3'mv2)
Tx = +
where zj and z2 are self-impedances per unit length; yy andy2 are admittances per unit
length; and zra and ym are mutual impedance and admittance per unit length, respectively.
Also, a harmonic time dependence (i.e. e/<0<) is assumed.
The solution to the set of equations (3-12) as published by Tripathi [8] is given as
v1=A,e-V + VV + A3e-V + vV
= A xRce^cX + A2RceV + A3RKe~V + A^e
1
^ ll
II
-V A\V
1 z ,
cl
ZC2 Z*l Z*2
A\Rc -ycx
y "Y y AaR n
2 cJcx + 3 yV 4 *
- ~ e
zc 1
ry C 1 rj C n C-
Zc2 Z7il Ztc2
(3-13)


34
where A j, and A3 represent the phasor coefficients for the forward (positive x) propagating
c and Tt-modes, respectively, and A2, and A4 represent the phasor coefficients for the
reverse (negative x) propagating c and jt-modes, respectively. The characteristic imped
ance of the c-modes are represented by Zc] and Zc2 for lines A and B, respectively, and the
characteristic impedance of the 7t-modes are represented by Zi and Zn2 for lines A and B,
respectively. Additionally, Rc = v2/v¡ for y=+yc, RK = v2/v\ for y=yn, and
y-
c,n
y ,zi+>2z2
+v.
(3-14)
>Jiy izi y^1+^z\ym++?
Each voltage/current pair at each node represent a single propagating signal referenced to
the ground potential. These signals will be called nodal waves.
A practical simplification in the development of mixed-mode s-parameter theory is
to assume symmetric coupled pairs (i.e. lines A and B have equal width) as reference
transmission lines. This assumption allows simple mathematical formulations of mixed
mode s-parameters. Furthermore, this assumption is not overly limiting, since reference
lines may be made arbitrarily short. For symmetrical lines, in (3-13) Rc = 1 and Rn = -1,
and the c and the 7t-modes become the even and odd modes, respectively, as first used by
Cohn [6], For notational puiposes, we shall use the substitutions c > e and 7t o for
even-mode and odd-mode, respectively. With these substitutions, the mode characteristic
impedances and propagation constants become
(3-15)
yc = ye yn = ya
Expressing (3-13) in the symmetric case


35
v,=Aie-V+vV+A3e-V+vV
V2 = Aie-V + VV_A3e-V_VV
i, = li e-v_l?ev+£e-v_liev
A1 -7A A2 yx A3 -Y0x ^4 y x
¡2 = e ~Z^e e ~Z~e U + Z~e
(3-16)
As before, these voltage/current pairs are nodal waves at each terminal that are referenced
to ground.
Expressing the differential and common-mode values (3-1) through (3-4) in terms
of the line voltages and currents (3-16)
vdw = 2(a3*-V+VV)
Ux) = %~V-A*eyoX
-y y y x
vcM = A,e ,e' +A2e,e
ic(x) =
(3-17)
Recall that A, and A2 are the forward and reverse phasor coefficient for the even-mode
propagation, and A3 and A4 are the forward and reverse phasor coefficient for the odd
mode propagation. If a short hand notation is introduced, a better understanding of these
definitions can be had. Let
v+(x)=A3e
v+(x)=A,e
,-v
v-W=A4eV
o
rV
vJ=A2eV
CW-.V
(3-18)


36
Then (3-16) becomes
v, = v*(x) + v>) + v>) + vjx)
v2 = v*(x) + Ve(x) v+o(x) V0(x)
¡I = igW !c(x) + 10W ia(x)
2 = Cm !ew 0w+i0w
and (3-17) becomes
vdW = 2(v*(jc) + v~(jc) )
+ v>)-v>)
dW = 'W-'oW = z
vcM = ^W + veW
+ V(W v(jc)
*CW = 2(ie(x)-ie(x)) = 2 =
(3-19)
(3-20)
Note that, in general, Za Ze.
Characteristic impedances of each mode can be defined as the ratio of the voltage
to current of the appropriate modes at any point, x, along the line. These impedances can
be expressed in terms of the even and odd-mode (ground referenced) characteristic imped
ances
vJW 2v+(x)
d ¡JW v+o(x)/Zo
z = v+(x) = Ze
c (2v+)/Zc 2
(3-21)
(3-22)


37
These relations between the even/odd mode characteristic impedances and the differential/
common mode characteristic impedances are consistent with the matched load termina
tions discussed in the literature [9, 10].
3.1.1.3. Mixed-Mode Scattering Parameters
Now that voltages, currents, and characteristic impedances have been defined for
both differential and common modes, the normalized power waves can be developed. By
the definition for a generalized power wave at the n-th port [23, 24]
a = , [v + i Z ]
" 2jmz n n n
b = =L=[v i Z *]
n 2jR^) n nn
(3-23)
where an is the normalized wave propagating in the forward (positive x) direction; bn is
the normalized wave propagating in the reverse (negative x) direction; and Z is the char
acteristic impedance of the port. With the above definitions, the differential normalized
waves become, at port one
adlsfld(*i> = 2/Re(zd)[vd(j) + blsblx0 = Jg=[Vdto-'dZd*]
Similarly, define the common-mode normalized waves, at port one, as
= Yjmfvx)+ix)Zc]
hclUW = ^^[vcW- (3-24)
(3-25)


38
Analogous definitions at port two can easily be found by setting x = x2.
Imposing the condition of low-loss transmission lines on the coupled-pair of
Figure 3-1, the characteristic impedances are approximately purely real [23]. Under this
restriction, Z¡¡ ~ Re/ZJ = R and Zc ~ Re(ZJ = Rc. With this assumption, the normalized
wave equations at port one can be simplified
dl
[vdl + 'dlRdll
fcdl = JR^¡[V dlW-'dlW^dll
(3-26)
acl = ^bct + 'cW^]
bc\ = J^=[v c(x)-ic(x)Rc]
(3-27)
With the normalized power waves defined, the development of mixed-mode
s-parameters is straight forward. The definition of generalized s-parameters [24, 23] is
b = Sa (3-28)
where the bar over the lower-case letters denote an n-dimensional column vector and the
bold upper-case letter an n-by-n matrix. Given a coupled-line two-port like Figure 3-1, or
any arbitrary mixed-mode two-port, the generalized mixed-mode s-parameters can be
described by
*dl = ^ddl^dl +iddl2ad2 + idcllacl +idcl2ac2
bd2 = ^ddl^dl+idd22ad2 + idc21acl + ^022^02
, (3-29)
"cl icdlladl +icdl2ad2 + iccllacl +iccl2ac2
bc2 = icd21adl +'scd22ad2 + icc21acl +icc22ac2


39
Each parameter has the notation
smom^pp. ~ 'v(output-inode)(mput-mode)ioutput-port)(input-port) (3-30)
to indicate the modes and ports of the signal path which the parameter represents. The dif
ferential and common-modes are denoted by a subscript d and c, respectively, and the
ports are denoted by there port number, in this case, one and two. The set of equations in
(3-29) can be expressed as a partitioned matrix
'dl
d2
c2
(3-31)
The following names are used: Sdd are the differential s-parameters, Scc the com
mon-mode s-parameters, and and Scd the mode-conversion or cross-mode s-parame-
ters. In particular, Sdc describes the conversion of common-mode waves into differential
mode waves, and Scd describes the conversion of differential-mode waves into common
mode waves. These four partitions are analogues to four transfer gains (Acc, Add, Acd,
Adc) introduced by Middlebrook [19]. These mixed-mode two-port s-parameters can be
shown graphically (see Figure 3-3) as a traditional four-port. It must be remembered,
however, that the ports are conceptual tools only, and not physically separate ports.
3.1.2. Choice of Reference Impedances for Multiple Modes
If one is to make a general purpose RF measurement port, the values of character
istic port impedances must be chosen. It is useful to require the even and odd-mode char
acteristic impedances of the measurement system to be equal, thus reducing the number of
different valued matched terminations required. In contrast, it is difficult to fabricate


40
Physical
P^rt 1
Mixed-Mode
Two-Port
Physical
Port 2
I
Figure 3-3. Signal flow diagram of mixed-mode two-port network.
accurate lumped termination standards for coupled lines where Ze does not equal Z0. If
the characteristic impedances of the lines are defined to be equal (say, 502), then a further
simplification of the above expressions can be accomplished with the substitution Ze = Z0
= Zq where in the low-loss case Zq ~ Re(Zo¡ = R0.
By choosing equal even and odd-mode characteristic impedances, one is selecting
a special case of coupled transmission line behavior, as described in (3-12). Enforcing
equal even and odd-mode characteristic impedances is equivalent to the conditions of
uncoupled transmission lines. As has been shown in the literature [9], the condition Ze =
Za results in the mutual impedances and admittances being zero (zm=0, ym=0). Under
these conditions, the describing differential equations of the transmission line system


41
(3-12) clearly become uncoupled, resulting in two independent transmission line solu
tions. Although very specific, this is a valid solution to (3-12), and all results up to this
point are also valid under the special case of equal even and odd-mode characteristic
impedances. Therefore, we choose the reference lines of the mixed-mode s-parameters to
be uncoupled transmission lines. The key to this choice is that these uncoupled reference
lines can be easily interfaced with a coupled line system, as discussed below.
To interpret the meaning of uncoupled reference transmission lines, consider a sys
tem of transmission lines: one coupled pair and one uncoupled pair connected in series. If
even and odd (or c and n) modes are both propagating (forward and reverse) on the cou
pled pair, then it can be shown that the waves propagating on each of the uncoupled trans
mission lines are linear combinations of the waves propagating on the coupled system (see
Appendix C). Furthermore, the differential and common-mode normalized waves of the
coupled pair system can be reconstructed from the normalized waves at a point on the
uncoupled line pairs (see Appendix C). This point of reconstruction is arbitrary, and one
may choose the point to be the interface between the coupled system and the uncoupled
reference lines.
It it interesting to note that an alternative requirement can be found through which
the nodal and mixed-mode waves can be related. One could require the differential-mode
and common-mode characteristic impedances to be equal (i.e. Zd = Zc = Z0). The rela
tionship between mixed-mode and standard s-parameters (discussed in the next section)
will change, however. This alternate requirement may have value in some cases, but the
original requirement (Ze = Z0 = Zg) best relates mixed-mode s-parameters to standard s-
parameters.


42
3.1.3. Relationship of Mixed-Mode and Standard S-Parameters
The most straightforward means of implementing a mixed-mode s-parameter mea
surement system is to directly apply differential and common-mode waves while measur
ing the resulting differential and common-mode waves. Unfortunately, the generation and
measurement of these modes of propagation is not easily achievable with standard vector
network analyzers (VNA). However, under certain conditions, one can relate the total
nodal waves (each representing two modes of propagation) to the desired differential and
common-mode waves. These nodal waves are readily generated and measured with stan
dard VNAs, and with consideration, the differential and common-mode waves, and hence
the mixed-mode s-parameters, can be calculated. Therefore, the relationships between the
normalized mixed-mode waves (adl, bA,, acl, bc\, etc.) and the nodal waves (a¡, b¡, a2, b2,
etc.) will be derived, and the necessary conditions for these relationships to exist will be
found.
To begin the development of the relationship between the nodal and mixed-mode
normalized power waves, the normalized differential-mode incident wave at mixed-mode
port one, adl,will be expressed in terms of the normalized single-ended (nodal) power
waves at port one, a\, and at port two, a2. First, the normalized nodal waves of the cou
pled lines at the interface are defined, with Z0 ~ R0, as
(3-32)


43
where a¡ and b¡ are the normalized forward and reverse propagating nodal waves at node i,
respectively, and i 6 {1,2,3,4). Next, the definition of the normalized differential-mode
incident wave at mixed-mode port one, a<¡¡, will be repeated
dl
2^
[vdlW + !dlWdl]
(3-33)
Recalling that the differential voltage and current at port one are defined through (3-1) and
(3-2) as
vdlW = vi(x)-v2(x)
1 (3-34)
¡diW = ^(iW-^W)
and that the differential reference characteristic impedance is defined in (3-21), with the
substitution Ze = Za = Zg ~ R0, as
dl = 2R0 (3-35)
then (3-33) can be re-written as
fldl = 77s=[vdiW + idiWdi]
J= |-y= IV!W v2(x) + R0(it(x)~ i2(x))]
{^[ViW + 0,lW]- [V2W + V2W]
(3-36)
By applying the definition of normalized waves at port one and two (3-32), then (3-36)
becomes simply


44
3dl = J(ala2)
(3-37)
This equation has a meaningful analogy with the differential voltage and current defini
tions. Similarly, the differential and common-mode waves a port one are
dl =
b&\ J2^b 1 b^
Similarly, for port two
cl + a^
bcl = jl(fcl+2)
(3-38)
d2 = J^a 3"4)
bd2 =
c2 = J(il3 + a4)
bc2 = -(*3 + *4>
J2
(3-39)
Equations (3-38) and (3-39) represent important relationships from which mixed
mode s-parameters can be determined with a practical measurement system.
By using the definition of s-parameters [23] for a four port network together with
the relations in (3-38) and (3-39), a transformation between mixed-mode and standard
s-parameters can be found. The transformation can be developed by considering the rela
tionships between the standard and mixed-mode incident waves, a, which can be written
ad\
1-10 0
l
ad2
1
0 0 1-1
a 2
cl
~ J2
110 0
3
ac2
0 0 1 1
4
or, compactly
- mm
a
Ma
std
(3-40)
(3-41)


45
where amm and a1*111 are the mixed-mode -waves vectors, respectively, and
1-10 0
0 0 1-1
J2 1 1 0 0
(3-42)
0 0 11
Similarly, for the response waves, b, it is found
(3-43)
Applying the generalized definition of s-parameters from (3-28), it can be shown
(3-44)
where Smm are the mixed-mode s-parameters, Sstd are the standard four-port s-parameters.
The transformation in (3-44) gives additional insight into the nature of mixed
mode s-parameters. The transformation is a similarity transformation, which indicates
that a change of basis has occurred between standard and mixed-mode s-parameters. Con
ceptually, the nodal currents and voltages correspond to the basis of standard four-port
s-parameters, and the modal currents and voltages of (3-1) to (3-4) correspond to the basis
of mixed-mode s-parameters. (precisely what is meant by a basis of an s-parameter repre
sentation will be explored in Section 3.2).
The transformation (3-44) also gives information into the nature of the chosen
mode-specific a- and b-waves. It is easily demonstrated that the operator M has the prop
erty M'1 = Mt (where the superscript T indicates the matrix transpose operator). This
indicates that the M operator is a unitary (also called orthonormal) operator [31]. This can
be easily demonstrated by applying the definition of a unitary operator
M(M*)
T
(3-45)


46
where indicates the complex conjugate. A unitary transformation is one that transforms
one orthonormal bases to another orthonormal bases. If it is accepted (until Section 3.2,
where it can be established) that standard four-port s-parameters are operators in an
orthonormal basis, then it follows from (3-45) that the definitions of the differential and
common-mode normalized power waves must also represent an orthonormal basis. This
is yet another indication that the mode currents and voltages in (3-1) to (3-4) provide a
self-consistent framework for power calculations.
Further, it indicates clearly that the two sets of s-parameters are different represen
tations of the same device, and that, ideally, the two representations contain the same
information about the device. However, it will be shown in Section 5.2 that transforma
tion according to (3-44) of measured data from practical measurement systems (with mea
surement errors) can lead to significant errors in the transformed data.
3.1.4. Interpretations of Multi-Mode Scattering Parameters
Equations (3-26) and (3-27) form the basis of an ideal mixed-mode s-parameter
measurement system. These equations can be implemented into a microwave simulator,
and can provide a quick and simple method of illustrating the usefulness of mixed-mode s-
parameters.
The circuit in Figure 3-4 was implemented into Hewlett-Packards Microwave
Design System (MDS) [32], The phase difference, 0, between the two sources was set to
0 for the common-mode and common-to-differential-mode forward s-parameters. For
the forward differential-mode and differential-to-common-mode s-parameters, the phase
difference was set to 180. In each case, the nodal waves were calculated from (3-26) and
(3-27), and the s-parameters were calculated with the appropriate ratios. The reverse s-


47
Ang=0
Mag=l V
-^-
+ 0
Length=l inch
h=25 mils
er=9.6
4{7)-^r-
v2
Width= 100pm
Space= 100pm
Width= 100pm
v3
v4 zo
Ang=0
Mag=l V
Port 1
Port 2
A^y
*4
Figure 3-4. Schematic of mixed-mode simulation of symmetric coupled-pair line.
parameters were calculated by driving mixed-mode port two of the DUT, with 50Q loads
at port one.
The first example of mixed-mode s-parameters uses a DUT of a pair of coupled
microstrip transmission lines, with symmetric (i.e. equal width) top conductors. This
symmetric coupled-pair, and the accompanying circuitry, is shown in Figure 3-4. Each
runner width is 100pm with an edge-to-edge spacing of 100pm. The substrate is 25 mil
thick alumina with a relative permittivity of 9.6 with a loss tangent of 0.001, and the metal
conductivity is that of copper, ~5.8xl07 S/m. A one-inch section of this line was simu
lated in MDS as described above, and the mixed-mode s-parameters at 5 GHz are
^dd'fdc
Scd'Scc
-0.001 Z-141 0.972Z9.530 0 0
0.972^9.53 0.001^-141 0 0
0 0 0.341 Z-60.4 0.915Z-26.40
0 0 0.915Z-26.40 0.341Z-60.40
(3-46)


48
As expected, each partitioned sub-matrix demonstrates the properties of a reciprocal, pas
sive and (port) symmetric DUT. The differential s-parameters, S^, show the coupled
pair possesses an odd-mode characteristic impedance of 50Q (100Q differential imped
ance), and has low-loss propagation in the differential mode. The common-mode
s-parameters, Scc, show the coupled pair possesses an even-mode characteristic imped
ance other than 50£1 Actually, the even-mode impedance of the pair is 1402 (702 com
mon-mode impedance). Note the cross-mode s-parameters are zero for the symmetric
coupled pair indicating no conversion between propagation modes.
The second example is similar to the first, except the coupled microstrip transmis
sion lines are asymmetric (i.e. unequal widths). This asymmetric coupled-pair, and the
accompanying circuitry, is shown in Figure 3-5. One top conductor width is 100pm, and
the second is 170pm, with an edge-to-edge spacing of 65pm. Again, the substrate is 25
mil thick alumina with a relative permittivity of 9.6 with a loss tangent of 0.001, and the
metal conductivity is that of copper. A one-inch section of this line was simulated in
MDS at 5 GHz, and the mixed-mode s-parameters are
Ang=0
Mag=l V
Length=l inch
h=25 mil
er=9.6
V|
Ang=0
Mag=l V
v2
Width= 100pm
Space=65pm
Width= 170pm
v4
Port 1
Port 2
Zo
Wv-
4
Figure 3-5. Schematic of mixed-mode simulation of asymmetric coupled-pair line.


49
0.003 Z-175
0.956Z1.819
0.005 Z-177
0.031Z80.70
0.956^1.819
0.003 Z-175
| 0.031 Z80.7
0.005 Z-177
0.005 Z-177
0.031Z80.70
1 0.502Z48.00
0.844Z-40.2
0.031Z80.70
0.005Z-1770
0.844Z-40.20
0.502Z48.0
As in the first example, each partitioned sub-matrix demonstrates the properties of
a reciprocal, passive and (port) symmetric DUT, Also like the first example, the differen
tial s-parameters show the coupled pair possesses an odd-mode characteristic impedance
of nearly 50Q (actually 49£2), and has low-loss propagation in the differential mode. The
common-mode s-parameters show the coupled pair has a greater degree of mismatch than
the first example (the even-mode impedance is 152S2 in this case).
The most important difference between the two examples is seen in the cross
mode s-parameters. The data in (3-47) shows significant conversion between propagation
modes, particularly in transmission parameters Sdc2i and Scd2j. Note these two sub-matri
ces are equal indicating equal conversion from differential to common-mode and from
common to differential-mode. These non-zero s-parameters can be interpreted conceptu
ally in the following way. In the case of Scd2i, a Pure differential mode wave is impinging
on port 1 of the DUT. However, at port 2, both differential and common-mode waves
exist. Some of the energy of the differential wave is converted to a common-mode propa
gation, and the total energy is preserved (except for losses in the metal and dielectric).


50
Figure 3-6. Simulated magnitude in dB of S21 and ^cc21 versus frequency for asym
metric coupled-pair transmission line
Figure 3-7. Simulated magnitude in dB of Sddl ( and Sccl t versus frequency for asym
metric coupled-pair line.


51
Figure 3-8. Simulated magnitude in dB of SC(¡21 versus frequency for asymmetric cou
pled-pair line.
Figure 3-9. Simulated magnitude in dB of Sc(m versus frequency for asymmetric cou
pled-pair line.


52
This example circuit was simulated across frequency, and the magnitudes of
selected mixed-mode s-parameters are plotted in Figures 3-6,3-7, 3-8 and 3-9. Figure 3-6
shows both Sdd2i and Scc2i in dB from 1 GHz to 21 GHz. The ripple pattern across fre
quency in the common-mode transmission (Scc2i) indicates an impedance mismatch at the
ports for common-mode propagation. At the higher frequencies of the plot, the finite con
ductivity of the conductors is evident as average loss increases. The differential-mode
transmission (Sdd2i) shows smaller ripples (0.2 dB maximum), indicating smaller mis
match, and also shows lower average loss. However, the losses due to the reflections at
the ports do not account for all of the ripple in the differential transmission. As can be
seen in Figure 3-7, the return loss for the differential mode is greater than 20 dB, which
can account for approximately 0.04 dB of worst case loss (over ohmic losses). Mode con
version accounts for the remaining reduction in the differential-mode, and hence Sdd2i is
reduced. Here, differential energy is converted to both common-mode transmission Scd2i
and common-mode reflection Scdll. Figure 3-8 shows the cross-mode transmission Scd2i
in dB, and Figure 3-9 shows the cross-mode reflection Scd], in dB. The minima in the dif
ferential-mode transmission Sdd2i correspond to a worst case point in the relative phases
of Sdd2i, Scd2i, and 5cdj j. In a low loss transmission line case, the insertion loss due to
mode conversion and miss-match can be shown to be approximately
Loss(dB) =-101og [l -(|Sddn|2 + |Scd2||2 + |Scdu|2)] (3-48)
This is consistent with the increasing ripple in Sdd2i with increasing frequency since the
mode conversion (Scd2i and Scd| [) increases with frequency.
The use of mixed-mode s-parameters can be further illustrated with an example of
a differential amplifier. Such an example is found in Appendix D.


53
3.2. Generalizations of Mode Specific Scattering Parameters
3.2.1. Other modes
The voltages and currents of (3-1) to (3-4) represent only one possible definition of
modes. There are infinitely many such definitions with a four-port network, although the
chosen set has important practical value. Furthermore, a network with more ports can
support more modes of propagation. It is useful to generalize the proceeding work to
include all possible mode definitions as it leads to insight into the nature of the mixed
mode definitions presented.
To begin the generalization, it is helpful to establish the concept of an s-parameter
matrix as a linear operator. Traditionally, an s-parameter matrix is interpreted from a
physical view, where the elements of the matrix represent the gain coefficients of a certain
input-to-output path. The operator interpretation views the s-parameter matrix as an oper
ator that maps one n-dimensional vector space into an m-dimensional space [31] (with typ
ical devices, m and n are equal). With such an interpretation, it will be shown that the
transformation to another mode definition can be regarded simply as a transformation of
coordinates.
(a) (b)
Figure 3-10. Two views of a four-port s-parameter matrix.
a) The physical view, b) The linear operator view.


54
To illustrate the operator view of s-parameters, consider the four-port example in
Figure 3-10. Define basis vectors corresponding to each physical port
1
0
0
0
P\s
0
Pl=
1
0
p^
0
0
0
1
0
0
0
0_
1
One can clearly see that these vectors are linearly independent, that is
c + c2p2 + c3p3 + c4p4 0 (3-50)
for all possible complex scalars {Cj, c2, c3, c4} 3 C, where C is the set of all complex
numbers. Furthermore, this set of basis vectors {pyPyPyP^) have a zero scalar prod
uct, that is
prpj II
i j
i = j
(3-51)
This means that the system of basis vectors is orthonormal. Continuing, an arbitrary set of
input signals becomes
a = a xp ] + a2p2 + a3p3 + aAp4 (3-52)
and the output signals are
b = bipi+b2p2 + b3p3 + h4p4 (3-53)
With the basis definitions of (3-49), the coordinates of the input and output signals are
b =
(3-54)


55
The traditional s-parameter matrix equation, b = Sa, can now be said to express a linear
operator, S, mapping an input space to an output space. It is important to note that both
the input and output spaces have the same basis vectors.
Now, considering the same example, define a new set of basis vectors,
{p ]', p2', py p4'}. These new basis vectors can describe any arbitrary mode definitions.
In the case of the differential/common-mode definitions of (3-1) to (3-4) they are
(3-55)
where the new subscripts are used to clearly indicate that the new basis does not corre
spond to physical network ports. Assuming linear relationships between the old and new
bases, they can be generally related
P\ = xuPi' + x\2P2+xnP3+xt4P4
p2 = x2lp]'+x22p2+x23p2+x24p4
p3 = *31P|'+*32P2' + *333'+*34P4'
p4 = *4 lP!1 + *4202' + *4303' + *4404'
(3-56)
An input signal vector in the new basis
a
flj'Pj' + ci2P2 + + a4'P4'
(3-57)
has the coordinates in the new basis
a
a
(3-58)


56
By expressing the input vector in the original basis (3-52) in terms of the new basis vectors
via (3-56), and then by equating the coefficients of the basis vectors, it can be shown that
al
*11 *12 *13 *14
a2
*21 *22 *23 *24
a3
*31 *32 *33 *34
aA
*41 *42 *43 *44
which can be simply expressed as
(3-59)
a' = Xa (3-60)
where A" is a transformation of coordinates matrix. Therefore, the translation between dif
ferent mode definitions is simply a transformation of coordinates. In the case of the differ
ential/common-mode definitions, it can be shown that (3-60) becomes
mm std
= Ma
(3-61)
As illustrated in (3-53), the input and output vector spaces share the same basis vectors, so
the output in the new basis becomes
V = Xb (3-62)
or, for differential/common-modes
r mm std
b = Mb (3-63)
The linear operator representing the DUT can be translated between bases by
S' = XSAf'1 (3-64)
In general, if both sets of bases are orthonormal, as defined in (3-51), then the
T
transformation matrix, X, will always be unitary, that is AT(AT*) = /. Conversely, if a
defined transformation matrix is unitary, then both systems of basis vectors are orthonor-


57
mal [31]. With the concept of s-parameters as linear operators, one can define any number
of new and potentially useful modes of propagation.
3.2.2. Rigen modes
One particularly interesting new mode definition arising from the operator view of
networks is the concept of eigen-modes. Eigen-values arise from the diagonalization of a
matrix, and the matrix of eigen-vectors become the transformation matrix. Symbolically,
A = TlST (3-65)
where
A = diagO.,, ...,Xn) (3-66)
where X¡ are the eigen-values of S, and T is a matrix whose columns are composed of the
eigen-vectors of S [33].
In linear system analysis, eigen-values represent the natural frequencies of a sys
tem. When described in state space notation, the state-feedback matrix, A, determines
these natural frequencies. The natural frequencies, or eigen-values, are the solutions to
|X/-A| = 0 (3-67)
Corresponding to each eigen-value, X¡, there is a eigen-vector, e¡, such that
(\I-A)ei = 6 (3-68)
Physically, the eigen-values are the complex frequencies at which the system will have
(unforced) oscillations, and the eigen-vectors are the amplitude coefficients of each of the
state variables under the conditions of oscillation.
In contrast, the eigen-values and vectors of an s-parameter matrix do not represent
system oscillations. For an s-parameter operator, the eigen-vectors represent the coeffi-


58
cients of a transformation to a new basis. The new basis further represents new modes of
propagation. This new basis is special, in that it transforms the operator, S, into a diagonal
matrix. For this reason, the modes corresponding to the eigen-vectors of a operator, S,
will be called canonical modes. The eigen-values represent the DUT response in terms of
the canonical modes.
In general, an n-port device will have n canonical modes. When stimulated by one
of the canonical modes, the device will generate a response proportional to only the mode
by which it was stimulated. There is only one port definition possible for canonical
modes. Each canonical mode is formed from a linear combination of signals at all of the
physical ports. There are n possible canonical modes of propagation supported by a
device with n physical ports. This removes any ambiguity that exists in the port number
ing convention1.
The canonical representation of a device allows for very simple calculations of
responses. Since the canonical form of a device is a one-port (multi-mode) network, the
response of the device to a canonical mode input is simply a reflection of the same canon
ical mode. The canonical mode reflection has a scaling, or gain, factor that is conceptually
equivalent to the traditional definition of reflection coefficients. The eigen-values of a
s-parameter matrix are the canonical reflection coefficients. Furthermore, a given device
generates no conversion between its canonical modes. As a result, the canonical represen
tation can be interpreted as the natural modes of a device.
It is interesting to note that eigen-values of a matrix, S, remain unchanged by a
change of basis (i. e. a similarity transformation as in (3-65)). The eigen-values, therefore,
1. The definitions of mixed-mode s-parameters presented in Section 3.1.1 define
(nodal) ports one and two as mixed-mode port one, and so on. However, any other
combination of two ports could have also been chosen as a mixed-mode port.


59
are immutable properties of an s-parameter matrix, and the canonical modes of a device
are properties of the device. Eigen-vectors are not unique, since they need only to be inde
pendent. As stated earlier, infinitely many modes (not independent) can be defined for a
given network. However, the consistency of the eigen-values across all such bases indi
cates the all representations of a device are leaving the essence of the device unchanged.
Mixed-mode s-parameters are indeed an equivalent representation of a standard four-port
s-parameter matrix.
Not every device has a canonical representation. A matrix, S, is diagonalizable if
and only if S has n linearly independent eigen-vectors. It can be shown [33] that S has n
linearly independent eigen-vectors if S has n distinct eigen-values (the converse is not true,
however). Therefore, if all eigen-values are different, then one can be assured the device
has a canonical representation. If some values are repeated, then the existence of a canon
ical representation depends on S.
If an s-parameter matrix does not have n linearly independent eigen-vectors, then it
is possible to find n independent generalized eigen-vectors. Under these conditions, the
new operator matrix is not diagonalizable, but generally in Jordan form. A Jordan form
matrix has some non-zero off-diagonal elements. Such a device requiring a Jordan form
representation will exhibit mode-conversion between some of its canonical modes.
Despite this limitation, the Jordan form representation of an s-parameter operator can have
some utility in calculations.
Not every non-diagonalizable matrix has a Jordan form representation. In such
cases, other decomposition methods are available, such as LDU-factorization [33], These
decompositions cause representations that are as close as possible to a diagonal form.


60
This work can be extended to include these other representations of an s-parameter opera
tor.
With the fundamental theory of mixed-mode s-parameters developed, the applica
tion of these concepts to practical circuits can begin. The first step in this progression is to
measure the mixed-mode s-parameters of an RF differential circuit. These new s-parame-
ters require the design and construction of a specialized measurement system. The devel
opment of this new system is the subject of the next chapter.


CHAPTER 4
CONSTRUCTION OF THE PURE-MODE VECTOR NETWORK ANALYZER
As a result of the limitations of measuring RF differential circuits and devices with
a single-mode system, as discussed in Chapter 2, a custom vector network analyzer (VNA)
has been designed to measure mixed-mode s-parameters in the most direct and accurate
fashion. The existence of a transformation between standard and mixed-mode s-parame-
ters, discussed in Section 3.1.3, suggests two possible approaches to the measurement of
differential circuits. One approach is the use of a traditional four-port VNA. A traditional
VNA would measure standard s-parameters by stimulating each terminal of the differen
tial circuit individually, and these s-parameters would then be transformed to mixed-mode
s-parameters for analysis. Alternatively, the mixed-mode s-parameters of the differential
circuit can be measured directly by stimulating each mode individually. A pure differen
tial-mode stimulus could be produced, and the differential- and common-mode responses
of the DUT could be measured, thus providing a direct measurement of mixed-mode
s-parameters. A network analyzer that directly measures mixed-mode s-parameters will
be referred to as a pure-mode vector network analyzer (PMVNA) due to its generation and
measurement of pure single mode signals.
The two approaches do not yield equally accurate mixed-mode s-parameters of
differential devices, however. It is shown in Chapter 5 that the PMVNA has an accuracy
advantage over a traditional four-port VNA while measuring a differential circuit. Mixed
mode s-parameters generated by transforming standard s-parameters measured by a tradi-
61


62
tional four-port VNA exhibit higher levels of uncertainty in a differential device measure
ment than those measured by a PMVNA. This accuracy advantage of a pure-mode
measurement system provides motivation for the development of a specialized measure
ment system for differential circuits. Portions of this chapter have been published in sum
mary form [34],
4.1. Basic Operation of the PMVNA
4.1.1. Fundamental Concepts
As discussed above, the most straightforward means of implementing a mixed
mode s-parameter measurement system is to directly apply differential and common-mode
waves while measuring the resulting differential and common-mode waves. Unfortu
nately, the generation and measurement of these modes of propagation is not easily
achievable with standard vector network analyzers (VNA). However, as shown in (3-38)
and (3-39), one can relate the total nodal waves to the desired differential and common
mode waves. These nodal waves are readily generated and measured with standard
VNAs, and with consideration, the differential and common-mode waves, and hence the
mixed-mode s-parameters, can be calculated.
Equations (3-38) and (3-39) represent important relationships from which a
PMVNA can be constructed with components of standard single-ended VNAs. To under
stand the utility of the above relationships, consider Figure 4-1, which is a conceptual
model for a PMVNA system. By adjusting the phase difference, 0, between the two
sources to 0 or 180 one can determine the common-mode or differential-mode forward
s-parameters, respectively. Conceptually, the measured quantities are the voltages and
currents. These values can be related to the normalized nodal waves, a,, bt, a2, b2, etc.,


63
Ang=0
Figure 4-1. Conceptual diagram of pure-mode measurement system.
through the generalized definitions given in (3-32). From these nodal waves, the differen
tial and common-mode normalized waves, and, hence, the mixed-mode s-parameters, can
be calculated. Physically, the various ratios of nodal waves, aj, fej, i>2 sured, and from theses ratios the mixed-mode s-parameters are found.
4.1.2. General PMVNA Test-Set Architecture
The physical implementation of a mixed-mode s-parameter measurement system
can be achieved with extensions of standard VNA techniques. The differential stimulus of
a coupled two-port requires the input waves at the reference plane to be 180 apart. One
possible way this can be achieved through a single signal source is with the use of a 180
3dB hybrid splitter/combiner. The construction of the differential reflected and transmit
ted waves, via (3-38) and (3-39), can be also completed through a 180 splitter/combiner.
The common-mode stimulus of a coupled two-port requires the input waves at the refer
ence plane to be 0 apart. This can also be achieved through a single signal source with
the use of a 0 3dB hybrid splitter/combiner, with the construction of the common-mode
reflected and transmitted waves also completed through a 0 splitter/combiner.


64
A VNA test-set is the portion of the test system that generates the normalized
power waves, a and b. A typical test-set uses directional couplers to separate the forward
and reverse waves. A test-set also samples the stimulus signal, either with a directional
coupler or a power splitter. The test-set generally down-mixes all signal to an intermedi
ate frequency (IF), so that all RF functions of the VNA (other than the RF signal source)
are contained within the test-set. A test-set also provides RF switches to allow automated
measurement of all s-parameters of the DUT with a single connection.
A basic pure-mode test-set is shown in part in Figure 4-2. The figure includes
mechanisms by which all of the mixed-mode wave components are generated. Not shown
are the down mixers and the rest of the VNA system, which are discussed in Section 4.2.1
and Appendix E. When switch one (denoted as SW1) is in position one, the 3dB hybrid
coupler, HI, splits the RF signal into two signals with nominally equal amplitudes and
180 phase difference, thus generating the differential-mode RF stimulus signal. Note
that all switches have their unused ports terminated in 50£2 loads in all cases. By placing
SW 1 in position two, the coupler, H1, again splits the RF signal into two signals, in this
case with nominally equal amplitudes and 0 phase difference, thus generating the com
mon-mode RF stimulus signal. Switches SW2 and SW3, which operate in concert, pro
vide the means to stimulate either mixed-mode port one or two. Directional couplers D1,
D2, D3, and D4 separate all forward and reverse signals at each single-ended port (i.e.
nodal waves). These nodal waves are combined, in accordance to (3-38) and (3-39), in
3dB hybrid couplers H2, H3, H4, H5, each providing a (nominal) sum and difference
between the corresponding nodal waves. The output of these couplers are proportional to
the differential and common-mode normalized power waves (ad], ac], dl, fcc], etc.).


65
Figure 4-2. RF Section of basic test-set of PMVNA.
From the appropriate ratios of these power waves, the mixed-mode s-parameters can be
calculated.
4.2. Implementation of a Practical PMVNA
Rather than build an entire PMVNA from elementary components (such as direc
tional couplers and mixers), a more practical approach has been followed by modifying a


66
standard VNA. As will be discussed below, a PMVNA can be constructed in a straight
forward manner by adapting a modular Hewlett-Packard 8510 VNA system. First, a sys
tem-level description of the PMVNA, as implemented for this work, will be given. Fol
lowing this, a detailed description of the PMVNA test-set will be given. Next, the
operation of the PMVNA will be detailed, and the control software will then be described.
4,2.1. System Level Description
The construction of the PMVNA is based the Hewlett-Packard 8510C VNA sys
tem. The complete block diagram of the implemented system is shown in Figure 4-3. The
basic idea behind the implemented PMVNA is to use the sub-systems of a standard 8510
(each contained as a single piece of test equipment) in a non-standard configuration with
little or no modification to the individual sub-systems. The sub-systems (85101, 85102,
8517, 85651, etc.) are shown in Figure 4-3. For a description of these sub-systems and the
standard 8510 configuration, see Appendix E.
Basically, the PMVNA is an 8510 VNA with two test-sets, where both test-sets are
used simultaneously. The implementation of a PMVNA with an HP8510 VNA requires
the addition of a second 8517 test-set to supply all required RF hardware. Some additional
control hardware, and some minor modifications to the 8517 test sets are also needed, as
will be described below.
The flexibility of the 8510 VNA system greatly facilitates the implementation of a
PMVNA. One important feature of the 8510 is exploited in order to reduce the complex
ity of the control software and hardware in the adaptation to the PMVNA. The feature,
known as Option 001, allows selection between multiple test-sets. The option is actually
an additional circuit board for switching IF signals which is installed in one of the two


67
test-sets. The board works in coordination with features of the 8510 operating system
(standard firmware of the 8510). The operating system of the standard 8510 allows the
selection of a test-set to be accomplished simply by changing the address of the active
test-set (contained in a register in the 85101) to the address of the desired test-set. The
address of the active test-set can be set through standard general purpose interface bus
Figure 4-3. PMVNA system block diagram.


68
(GPIB) commands. The availability of the test-set selection function to GPIB commands
enables high-level control of the sub-systems in the PMVNA.
The PMVNA system also requires some minor modifications to the control hard
ware of the test-sets. As developed by HP, Option 001 allows the selection of one active
test-set, and the deactivation of all other test-sets. This deactivation includes the moving
of the RF port selection switch (internal to the test-set) to a terminated position, so that no
RF signal is present at the ports of the deselected test-sets. Also upon deactivation, the
variable attenuators in a test-set (used to control the incident power on a DUT) are re-set
to 0 dB. The suppressing of the RF signal from the inactive test-sets and the change of
attenuation setting are unwanted side effects. The modification to both test-sets is needed
to allow RF to continue at the ports of inactive test-sets and to keep the attenuator settings
unchanged. The modification requires a minor change to the test-set digital control hard
ware to allow the masking of commands to change the position of the RF port selector
switch or attenuators. The masking of system commands is achieved through a single dig
ital control signal for each test-set. When the signal, called test-set enable, is asserted, the
test set can receive system commands effecting RF switch and attenuators; otherwise,
these system commands are blocked (other system commands are unaffected by the modi
fication). With these changes, the option 001 can now be used to multiplex the two test-
sets while maintaining an uninterrupted RF signal at the ports. For complete details of the
test-set modifications, see Appendix F.
These hardware changes are implemented to block unwanted system commands
from the 8510 operating system as the active test-set is changed. An alternative to these
hardware modifications is to change the operating system. Such a change, to allow


69
switching between test-sets without changing the RF switch position or attenuator set
tings, is certainly possible, and quite attractive since it would eliminate the need for any
modification of the test equipment of the 8510 system. The option of modifying the 8510
operation system is unavailable, however, as it is proprietary property of Hewlett-Packard.
Due to the unavailability of the operating system software, the hardware modifications
have been performed.
A single 3dB hybrid 180/0 splitter/combiner is added to the standard 8510 con
figuration. This splitter generates the two RF signals needed to operate both test-sets
simultaneously. The use of a 180/0 splitter allows for the generation of both differential
and common-mode stimuli. An RF switch is required to select between the two modes,
and a driver for the switch is required to allow automatic control. The switch driver and
test-set enable control lines are interfaced to a GPIB controllable digital switches (3488A
with option 014). With this switch controller, the PMVNA can be completely automated.
4.2.2. Test-Set Construction
One of the most useful aspects of a PMVNA implemented as shown in Figure 4-3
is the straight-forward manner by which the differential and common-mode normalized
power waves can be derived from the nodal power waves. Referring first to the basic
PMVNA test-set of Figure 4-2, one can see that the calculation of the modal normalized
waves is accomplished through four 180/0 splitter/combiners. The calculation is done
at RF with real (non-ideal) components, and so is subject to errors (see Chapter 8). A
more practical and accurate method of constructing the differential and common-mode
responses is through digital calculation of (3-38) and (3-39). This technique exploits the
architecture of the standard 8510, which down-mixes and digitizes the normalized power


70
waves. Once the nodal waves are digitized, the differential and common-mode normal
ized power waves can be simply calculated in the control software.
In the PMVNA implemented for this work, the calculations of the normalized
power waves are accomplished by using two standard two-port test-sets. The connection
of this simplified PMVNA test-set is shown in Figure 4-4, which includes two standard
(single-ended) 8517A s-parameter test-sets. These test-sets have all required RF circuitry
to separate the different waves, and all of the down converter circuitry. No modifications
to the RF portions of these test-sets are needed.
RF Source
To IF Detectors
(HP85101)
Figure 4-4. RF section of simplified PMVNA test-set.


71
A significant advantage of this test-set configuration is its symmetry. If the two
test-sets are the same model (as they are for this work) then the RF paths of the PMVNA
are well balanced. When the PMVNA is set to forward differential-mode, for example,
both test-sets have the same switch configuration. The two RF paths that comprise the dif
ferential signal (one through test-set A, the other through test-set B) are identical (within
manufacturing tolerances), and thus the phase and magnitude balance between the two
paths is good. If the paths are poorly balanced, then high levels of mode-conversion will
be generated in the PMVNA. Good balance is required to have sufficient raw dynamic
range for accurate measurements. Again, the amount of tolerable imbalance must be
determined by experience. The raw performance of the implemented PMVNA is exam
ined in Section 6.2.1.
This simplified test-set configuration has one significant disadvantage, namely, the
use of two independent voltage controlled ocsillators (VCOs). Referring to Figure 4-4,
one can see that each test-set contains a VCO. During measurements, this VCO is phase-
locked to the RF input signal of the test-set (for details, see Appendix E). This VCO gen
erates a signal the drives all four down-mixers in the test-set. As all mixers in a single
test-set are driven by the same VCO, the phase relationship between the down-mixed a
and b signals remains the same as it was at RF. Flowever, as the PMVNA switches
between the two test-sets, the phase relationship between the VCOs of the two test-sets is
unknown. As a result, the straight-forward application of the measured power wave data
will result in significant errors. This disadvantage can be removed, however, through a
pre-calibration process that characterizes the phase offset between the two VCOs. This
process is detailed in Section 6.3


72
Only the 180/0 splitter/combiner and the RF switch for the source (SW1) limit
the bandwidth of the simplified PMVNA. The 8517A test sets operate from 45 MHz to 50
GHz, and with relaxed requirements on the 180/0 splitter/combiner, accurate measure
ments are possible from about 100 MHz to above 25 GHz with one hybrid [35]. A second
hybrid allows accurate measurements from 45 MHz to above 5 GHz [36]. The factor lim
iting the frequency range of any splitter is the amount of imbalance that is tolerable in a
PMVNA system. This imbalance leads to non-ideal mode generation, as will be shown in
Chapter 8. This non-ideal mode generation can be tolerated and corrected through cali
bration (see Chapter 6) but only to a point. At some level of imbalance, the corrected
dynamic range of one or more of the mixed-mode s-parameters becomes unacceptable.
The frequency at which the level of imbalance is unacceptable generally occurs beyond
the specified operation frequencies of the splitter (splitter frequency specifications are
linked to specified levels of phase and magnitude imbalance), but the exact level of tolera
ble imbalance is usually found through experience.
With this PMVNA configuration, all mode responses, including mode conversion,
can be measured. With all responses available, very accurate, repeatable calibrations and
measurements are possible. Additionally, with the use of standard, readily available mea
surement equipment, the PMVNA can be easily and economically duplicated.
4.2.3. Detailed Operation
This section details the theory of operation of the PMVNA. The operation is pre
sented as a sequence of high-level events that affect the measurement of a DUT by the
analyzer. This discussion is meant to clarify the way raw data is collected and manipu
lated in the measurement of raw mixed-mode s-parameters. In general, each event


73
described in this section is comprised of many more elementary events which are not
described here. The referenced elementary events are performed by the control software
of the PMVNA which has been developed solely for this work. For more details on the
PMVNA control software, see Section 4.2.4. Furthermore, there is a level of operation of
the sub-systems that is even more fundamental. These low-level events, such as the lock
ing of the main phase-lock loop, are accomplished by the operating system of the 8510
system, and are transparent to the PMVNA control software. This most basic level of
operation is not described here, but can be found in 8510 documentation [43],
This section details only the measurement operation of the PMVNA. This opera
tion is the foundation of the general operation of the PMVNA, and the output of this oper
ation is raw (uncorrected) mixed-mode s-parameters of a DUT. Optionally, this operation
can produce standard four-port raw s-parameters directly (in contrast to transformation of
mixed-mode s-parameters). The calibration and subsequent error correction procedures,
and all other functions of the PMVNA, are detailed in Section 4.2.4.
The basic operation of the PMVNA measures the differential and common-mode
responses of a DUT to both a differential and a common-mode stimulus. Referring to the
flow diagram in Figure 4-5, the PMVNA first measures the DUT with a differential stimu
lus, which is accomplished by setting SW1 to position one (see Figure 4-3). Forward
operation of the DUT is measured by setting the RF port selection switches of both test-
sets into forward position. This drives PMVNA ports (nodes) one and two with a nominal
180 phase difference. Normalized waves are measured at all down-mixers: aj, bi, a2, b2,
a3, fe4, a4, 4 (the reasons for measuring all possible normalized waves, even those that are
apparently unneeded, are to correct for RF switch imperfections; see Section 6.2.4). This


74
configuration of the PMVNA is called the differential-forward mode (DF). Next, reverse
operation of the DUT is measured by setting the RF port selection switches of both test-
sets into reverse position. This drives PMVNA ports three and four with a nominal 180
phase difference. Again, normalized waves are measured at all down-mixers. This con
figuration of the PMVNA is called the differential-reverse mode (DR).
Figure 4-5. Flow chart of PMVNA measurement.


75
Next, the PMVNA measures the DUT with a common-mode stimulus, which is
accomplished by setting SW1 to position two (Figure 4-3). The forward measurements
are repeated in the same way as with the differential stimulus described above. This con
figuration is called the common-forward mode (CF). Similarly, the reverse measurements
are repeated with the common-mode stimulus; this configuration of the PMVNA is called
the common-reverse mode (CR).
The calculation of the mixed-mode normalized power waves is as follows. After
all data from a DUT measurement has been collected, the raw a and b data are arranged
into column vectors, where each vector corresponds to a single measurement mode (DF,
DR, CF, CR), and the a and b data are collected into two corresponding vectors. In
Figure 4-5, the arrangement of the a data is illustrated, where
DF
DR
CF
CR
a\
al
ai
a\
DF
DR
CF
CR
-DF
*2
-DR
a2
-CF
a2
-CR
a2
a =
DF
a =
DR
a =
CF
a =
CR
3
a3
a3
a3
DF
DR
CF
CR
a4 .
a4 .
a4 _
4
and where the subscripts one through four correspond to the PMVNA port (node) num-
DF DR CF CR
bers. Similarly, the b data are arranged into vectors b b b and b The data
are the placed in two matrices
astd ["-DF -DR -CF -Cr] Rs,d = [-DF -DR 7CF rCRl
[a a a a j [bbbb
(4-2)
where the superscript std indicates that the matrices are nodal data rather than mixed
mode data. The phase offset correction process, which will be described in detail in
Section 6.3, is applied to the A and B-matrices, generated phase-corrected versions, A,


76
and Bc, respectively. The mixed-mode normalized power waves are now calculated in
matrix form
,mm ...Std mm ..-std
A = MA B = MB (4-3)
where the matrix M is the similarity operator described in (3-42). The designation of ele
ments of the mixed-mode power wave matrix, Amm is also composed of column vectors,
one for each PMVNA configuration
and where
mm 14' _mint)R -mmCF
a a
-mmCRj
(4-4)
(4-5)
DF
d2
DF
cl
DF
c2
with the subscript d referring to the differential-mode quantity, c to the common-mode,
and the subscript numbers referring to the mixed-mode port numbers (in contrast to the
single-ended node numbers). The remaining vectors of (4-4) are defined in the same fash
ion. Likewise, the mixed-mode B-matrix, Bmm, can be defined.
The calculation of raw mixed-mode s-parameters is examined in detail next. After
calculation of the mixed-mode normalized power matrices Amm and Bmm, the raw mixed
mode s-parameter matrix, Smra, can be simply calculated
piiim nmm mm,
S = n (A )
(4-6)


77
One of the added benefits of the PMVNA is that it can also be used to measure
standard four-port s-parameters. The standard s-parameters can be calculated through the
similarity transformation of (3-44), but they can also be calculated directly from raw A and
B-matrices. With this method, the standard A and B-matrices of (4-3) are used to directly
calculate the standard s-parameters
Sstd = Bstd(Astd)_1 (4-7)
The accuracy of these standard s-parameters must be considered carefully, how
ever, As is shown in Chapter 5, the PMVNA has lower residual errors when measuring a
differential device. By similar arguments, it can be shown that a standard four-port VNA
(where only one test port is stimulated at a time) will have lower residual errors when
measuring a device that exhibits no differential behavior. Stated another way, the four-
port measurements of the PMVNA of a non-differential DUT have higher residual errors
than measurements of the same device from a standard four-port VNA.
4.2.4. Control Software
The control software of the PMVNA was implemented in LabVIEW. LabVIEW is
a graphical instrument control language which is well suited for the automation of the
PMVNA [37]. The control software has many functions (1) general measurement control
(2) VNA operation settings such as measurement frequencies, attenuation settings, etc.,
(3) PMVNA calibration, (4) general user interface, (5) data display, and (6) data input/out-
put (I/O) in files.
The control software of the PMVNA represents a significant development effort.
This software is highly specialized, and has been developed solely for this work. The pro
gram is graphically developed, so that wiring diagrams take the place of traditional


78
source-code listings. The control software represents more than 11.5 Mbytes of code, so
including all diagrams is prohibitive. Instead, flow diagrams are presented to indicate the
substance of the software.
This section reviews the control software at the highest level of functionality. For
detailed descriptions of the various functions, see Appendix G. The basic flow of the soft
ware is indicated in Figure 4-6. The first step in using the PMVNA is to set the basic oper
ating parameters of the analyzer. This includes the frequencies of measurement, the
attenuator settings for all ports, the number of averages, RF source power level, and so on.
Figure 4-6.
Top-level flow chart of PMVNA control software.


79
Next, a phase offset pre-calibration must be completed. This characterizes and allows for
the correction of the phase offset between the two VCO signals in the test sets. For theo
retical details on this calibration step, see Section 6.3. The primary calibration character
izes linear time-invariant errors in the PMVNA, allowing for error correction of measured
data. The theoretical development of the PMVNA calibration is given in Chapter 6. The
next step in the software flow is DUT measurement. This includes measurements as
detailed in Section 4.2.3, and error correction of the measured DUT mixed-mode s-param-
eters. The final two steps in the software flow are optional, but are almost always used.
The first of these is data display, which allows the user to examine the raw or corrected
DUT and calibration standard s-parameters in a variety of formats. The last step is file 10
which allows the user to save any of measured data to a file in CITI format [38]. Also, the
software allows the user to re-calculate any portion calibration and error correction algo
rithms, which is used mainly for de-bugging purposes.
4.3. On-Wafer Measurements
The PMVNA can make measurements of devices with coaxial connectors, or
devices that are meant to be probed at the wafer level. Wafer-level measurements, or on-
wafer measurements, require special RF wafer probes to make good performance RF con
nections to integrated devices that are typically quite small (on the order of 300pm on a
side). For the PMVNA, careful attention must be given to the signal launch from the
probe tip to the wafer surface. As shown in Appendix C, the mixed-mode s-parameters of
an arbitrary differential DUT can be accurately measured with uncoupled reference trans
mission lines (or ports), independent of any coupled modes of propagation that may exist
in the DUT. This is achieved through the decomposition of any coupled-mode signals into


80
G
S,
G
S2
G
Figure 4-7. GGB dual-RF wafer probe, top view (not to scale).
uncoupled modes, which results in mixed-mode s-parameters that are normalized to the
reference impedance of the uncoupled lines. Accordingly, the wafer probes that interface
with a differential DUT can be composed of isolated single-ended probes.
In order to maintain a smooth transition to any coupled-modes, two single-ended
probes are paired into a single mixed-mode probe. Each mixed-mode probe provides two
RF measurement ports that are in reasonably close proximity, but are ideally uncoupled.
Hence, a mixed-mode probe footprint of GSjGS2G is adopted. The PMVNA system, as
implemented for this work, is fitted with a pair of 150|im pitch dual-RF probes manufac
tured by GGB Industries [39]. A dual-RF probe is illustrated in Figure 4-7, with a detail
showing the probe contact configuration.
Wafer probes require special calibration standards. These standards are meant to
be contacted directly by the probe, so that the calibration reference planes are at the probe


81
tips. These wafer probe-able standards are widely available for two-port VNAs. How
ever, the unique nature of the PMVNA required custom wafer-probe standards to be
designed and manufactured. These standards are discussed in detail in Section 6.2.7 of
Chapter 6.
With the construction and operation of the PMVNA detailed, the measurement
accuracy remains to be assessed. An important aspect of the PMVNA is its accuracy in
the measurement of differential devices, relative to that of a more traditional VNA. This
is the central issue that will be examined in the next chapter.


CHAPTER 5
ACCURACY OF THE PURE-MODE VECTOR NETWORK ANALYZER
As indicated in Section 3.1.3, mixed-mode s-parameters and standard four-port
s-parameters are related by a linear similarity transform. This relationship suggests that a
traditional four-port VNA (where only one measurement port is stimulated at a time)
could be used to measure a differential DUT, and the resulting four-port s-parameters
could be transformed to mixed-mode s-parameters for easy analysis. Instead, a special
ized VNA has be constructed to directly measure mixed-mode s-parameters. These two
approaches do not yield equally accurate mixed-mode s-parameters of differential devices,
however. The PMVNA will be shown to be more accurate than a traditional four-port
VNA while measuring a differential circuit. Mixed-mode s-parameters generated by
transforming standard s-parameters measured by a traditional four-port VNA exhibit
higher levels of uncertainty than those measured by a PMVNA. In particular, the uncer
tainties of transformed mode-conversion parameters, Sdc and Scd, can be significantly
larger than the actual device parameter magnitudes. The accuracy advantage of a pure
mode measurement system provides motivation for the development of this specialized
measurement system for differential circuits.
In order to better understand the benefits and limitations of the PMVNA, the mea
surement accuracy of the system will be examined. The goal of this chapter is to quantify
the error in mixed-mode s-parameters of differential devices as measured by a PMVNA.
Since it has been earlier established that a linear transform exists between mixed-mode
82


83
s-parameters and standard s-parameters, a traditional four-port vector network analyzer
(FPVNA) can theoretically be used to measure a differential device. Here, a traditional
four-port network analyzer refers to a network analyzer that stimulates each port individu
ally while un-stimulated ports are terminated with a matched load. If a FPVNA is to be
considered for measurement of differential devices, it is important to understand the
errors that result by transforming standard s-parameters into mixed-mode s-parameters.
The accuracy of both systems must be compared to understand the advantages and disad
vantages of each. To quantize the errors in both a PMVNA and a FPVNA, the analysis is
divided into two important areas of consideration: probe-to-probe crosstalk and maximum
measurement uncertainty. It will be shown that the PMVNA has a higher dynamic range
than the FPVNA due to the 1/d3 and 1/d (d is distance) dependence of probe crosstalk,
respectively. It will also be shown that the uncertainty of mode-conversion parameters is
significantly lower for the PMVNA than for the FPVNA.
5.1. Probe-to-Probe Crosstalk
For a wafer-probe measurement system, the uncorrected probe-to-probe crosstalk
is an important specification. This crosstalk can limit the dynamic range of the measure
ment system, making high dynamic range measurements impractical. An important
example of such a high dynamic range measurement is the reverse isolation of an inte
grated RF amplifier. The unacceptable probe crosstalk of single-ended two-port VNA
provided some of the original motivation for the development of the PMVNA. The differ
ential mode of operation of the PMVNA is expected to have reduced probe crosstalk, due
to the natural common-mode signal rejection characteristic of a differential circuit. This
reduced crosstalk would allow higher dynamic range measurements than FPVNA. For


84
these reasons, the raw probe-to-probe crosstalk of the PMVNA and a traditional four-port
VNA are first quantified. The examination of the crosstalk levels is based on electromag
netic simulations of probe tips. Measured probe-to-probe crosstalk is also provided as fur
ther evidence of the higher dynamic range of the PMVNA.
5.1.1. Simulated Probe Crosstalk
The mixed-mode probe is simulated as a ground-signal i-ground-signal2-ground
(GS1GS2G) probe, as described in Section 4.3. The crosstalk of the four-port system is
represented through simulations of ground-signal-ground (GSG) probes. The use of the
two-port single-ended probes allows a consistent comparison between the crosstalk levels.
For simulation, the probes are modeled as 50pm wide by 100pm long metal strips
arranged in a 150pm pitch configuration, as shown in Figure 5-1. The strips are situated
(a)
(b)
G
51
G
52
G
G
S
G
t
4
I ^ 100pm
jOOpm
I^JPOp
150pm
50pm
150pm
50pm
Figure 5-1. Probe crosstalk simulation layout.
a) Mixed-mode probe layout, b) Single-ended probe layout.


85
on the surface of a 25mil substrate, and for purposes of this demonstration, the substrate
relative dielectric constant has been chosen to be one. Under the substrate is an ideal
ground plane. The probes are simulated in opposing pairs where the distance between
probes is specified. The electromagnetic simulator used is Hewlett-Packards Momen
tum, which is a method-of-moments simulator [40]. Multiple simulations of both the
mixed-mode and single-ended structures have been executed over a range of distances
between the probes tips.
The results of the multiple simulations are shown in Figure 5-2 to Figure 5-5. A
direct comparison of the crosstalk in the differential mode of the PMVNA to that of the
single-ended VNA is shown in Figure 5-2 as a function of probe separation at 1.0 GHz.
The simulations show that the single-ended crosstalk maintains an approximate I/d char
acteristic, whereas the differential crosstalk behaves as I/d3. This different dependence on
probe separation provides significant decrease in crosstalk for the differential mode with
respect to the single-ended operation, and hence provides for greater dynamic range in the
corresponding measurement. Also shown in Figure 5-2 is the common-mode crosstalk of
the PMVNA. The common-mode shows nearly the same level of crosstalk as the single-
ended system, as expected. This indicates that the common-mode measurements will
have approximately the dynamic range as traditional single-ended measurements. This
plot illustrates the dynamic range advantages of differential measurements over single-
ended measurements. Figure 5-3 shows a comparison of the crosstalk of the PMVNA to
that of the single-ended VNA 10.0 GHz. Figure 5-4 and Figure 5-5 show crosstalk as a
function of frequency for single-ended and differential probes, respectively.


86
The previous figures assume perfect phase and magnitude balance in the PMVNA
system. However, all real systems will have some degree of imbalance, degrading the
modal purity of any stimulus signal. The effects of imbalances on probe-to-probe
crosstalk can be quantified with the use to the same electromagnetic simulations. For
example, a 5 phase imbalance from the ideal 180 differential results in a probe crosstalk
level of-106 dB at 1.0 GHz and 1500pm separation, which reduces the dynamic range
improvement over single-ended to approximately 34 dB. The phase imbalance of the
present PMVNA is less than 5 from 1 to over 5 GHz with very small magnitude imbal
ance.
V100 Separation (pm) 10K
Figure 5-2. Simulated probe crosstalk vs. separation distance at 1.0 GHz.


87
V100 Separation (|xm) 10K
Figure 5-3. Simulated probe crosstalk vs. separation distance at 10 GHz.
150|im
300pm
500pm
700pm
1000pm
1500pm
Figure 5-4. Simulated single-ended probe crosstalk vs. frequency for several probe
separations.


88
150pm
300pm
500pm
700pm
1000pm
1500pm
T 1.0 freq(GHz) 10.0
Figure 5-5. Simulated differential probe crosstalk vs. frequency for several probe sepa
rations.
5.1.2. Measured Probe Crosstalk
Measured probe-to-probe crosstalk for the PMVNA is shown in Figure 5-6 to
Figure 5-8. This data was collected with GGB 150 pm-pitch dual RF probes (as discussed
in Section 4.3), where the probe tips were suspended in air approximately 10 cm above a
ground plane. Figure 5-6 shows the measured and simulated differential and common
mode crosstalk as a function of probe separation at 1.0 GHz. Figure 5-7 shows the same at
10.0 GHz. Figure 5-8 shows the measured differential crosstalk versus frequency for sev
eral probe separations.
From these figures, one can see that the measured crosstalk, regardless of mode, is
generally higher than that of the simulated structures. The source of the difference is most
likely due to the structural differences between the simulated structures and the actual
probes. Despite the differences in the absolute level of crosstalk, the measured data shows
similar trends versus probe separation. The measured data shows a 30 dB difference
between the differential and common-mode crosstalk at 1.0 GHz and 1500 pm separation,


89
and 22 dB at 10.0 GHz (compared to simulated 40 dB and 30 dB, respectively). This dif
ference in the crosstalk of the modes clearly indicates a higher dynamic range for the dif
ferential-mode in the PMVNA.
Figure 5-6. Measured probe crosstalk vs. separation distance at 1.0 GHz.
Figure 5-7. Measured probe crosstalk vs. separation distance at 10 GHz.


90
V 1.0 freq(GHz) 10
150pm
300pm
500pm
700pm
1000pm
1500pm
Figure 5-8. Measured differential probe crosstalk vs. frequency for several probe sepa
rations.
5.2. Uncertainty Calculations
A generally accepted quantification of error in VNA measurements is the maxi
mum uncertainties in the magnitude and phase of a set of s-parameters [43]. This section
seeks to quantify the error in a mixed-mode measurement, and compare that to the error in
a standard four-port measurement.
All measurements have errors, and these (unknown) errors add uncertainty to the
measurements. This uncertainty limits how accurately a DUT can be measured. VNA
errors can be separated into raw and residual errors. Both types of errors can be further
sorted into systematic (repeatable) and non-systematic (non-repeatable) errors. For a
complete description of VNA errors, see Chapter 6. Residual errors are the errors that
remain after calibration. During calibration, standards with known characteristics are


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THE THEORY MEASUREMENT AND APPLICATIONS OF MODE SPECIFIC SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION By DAVID E. BOCKELMAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997

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Copyright 199 7 By DA YID E. BOCKELMAN

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ACKNOWLEDGMENTS The author would like to acknow l edge the significant support of Motorola Radio Products Group Applied Research, without which this work would not have been possible. Many members of the staff of Applied Research gave support, advice and assistance which has been received with gratitude. The author would especially like to thank Charle s Backof, Vice-President and Director of Research, Radio Products Group, who urged the pursuit of this work and Dr. Wei-Yean Hwang Principal Member of the Technical Staff who gave his time and direction. The author i s indebted to Robert Stengel, Member of the Technical Staff who provided motivation for this work and guidance through it s completion Furthermore the author would like to thank Professor Wil l iam R. Eisenstadt who demonstrated hi s generosity by giving essential support in technical and personal matters. The author would also like to thank the members of his advisory committee for their s up port and direction, who were critical elements in the partner s hip between the University, Motorola, and the s tudent. Also appreciated by the author is the help of the staff of the University of Florida Microelectronics Lab and the help of many others who can not be listed here. Of a l l who gave their s upport and assistance none was as critical as the author's wife, Erika. Her unquestioning commitment was the light which has led the way to this conclusion 11

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TABLE OF CONTENTS AC.KNOWLEDGMENTS .......... ... .. .... .. .. . . .. .......... . . .. .......................................... .. .... 11 TABLE OF CONTENTS .................. ..... ...... .................. .......................................... ..... 111 ABSTRACT . ............. ..... .. .. .... ..... .......... .... . .... .. .. ....... ............................... ..... .... ... Vll CHAPTERS l IN'"TR OD U CTI ON ... ...... ... ..................................... .. .. . ....................................... 1 2 PRIOR THEORIES AND TECHNIQUES ... .. ..... .. .. .......................................... 6 2.1. Fundamental Theories of Analy s i s . ............ .......... ............ ..................... ..... 7 2 1 1 Coupled Transmis s ion Line Pairs .. ..... ... .. .... .... .... ............. .... ...... 7 2 1 .2. Analog Method s .............................................. .... .... . .. . ..... ...... ....... 9 2.1.3. Linear Network R e pr ese ntations ........... . .... ............ ... ..................... 11 2.1. 3 .1. Analog Network Parameter s ... .... ..................................... 12 2.1.3.2 RF Network Parameters .... .. ........................... . .. ........ ..... . 15 2.2. Mea s urement Te c hnique s ............................................................................. 20 2.2.1. Single Mode Analog Mea s urement s ....... .... .. ............ .. ................... 21 2.2.2. Single Mod e RF and Mi c rowave Measurement s .. .. ... ........ . .... ........ 22 2.2.2. 1. Scalar Power Measurement s Including Baluns .. .. .. ... ...... 22 2.2.2.2. Scatterin g Parameter s with Balun s .................................... 24 2. 3. Summary of Pa s t Th eo ry and Techniques .................................................... 26 3 FUNDAMENTAL THEORY OF MODE SPECIFIC S-PARAMETERS ............ 27 3.1. Mode Specific Scattering Parameters in Differential Circuits .... ....... ....... .. 27 3.1.1. Fundamental Definition s for Differential Circuit s .......... ..... .... ........ 28 3 1.1.1. Modal Volta ge and Currents ............... .. .. .......... ............... 30 3.1.1 2. Coupled Mixed-Mode Signals ........................................... 32 3.1.1.3. Mixed Mod e Scattering Parameters .................................. 37 3.1.2. Choice of R efe renc e Imp e dance s for Multiple Modes ..................... 39 3 1. 3. Relation s hip of Mixed Mode and Standard S-Parameter s ............... 42 3.1.4. Interpretation s of Multi Mode Scattering Parameters .. ....... .. .......... 46 3.2. Generalization s of Mode Spe c ifi c Scattering Parameter s ... ......................... 53 3.2.1. Other mode s .................................................... ............. .................... 53 3.2.2. Eigen modes ................................................................ .... ............... 57 111

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4 C ONSTR UC TION O F THE PURE MOD E VE CTOR NE TWORK AN AL Y ZE R 6 1 4.1. B as i c Op e r a tion of th e PMVNA .. .... .. . .. .. .. . . .. . .. ............ .. .... .. . ... ....... 62 4 1 1 F undam e n ta l Co n ce pt s ........ .... .. .. .. . .. .... .. ......... ......... .. . ...... .... ... 62 4. 1 .2. Ge n e r a l PMVNA T estS e t Ar c hit ect ur e .. ...... .... .. . ....... ............... 63 4.2. Im p l eme ntati o n of a P rac ti ca l PMVNA .............................. .. .... .. ............... 65 4 .2. 1 S ys t e m Leve l D esc rip t i o n ...................... .. .. . .... ....... ..................... 66 4.2 2. TestS e t Co n st ru ct i o n ... ...... .... . .......... .................... .. .... ... .......... 69 4.2.3. D e t a il ed O p e r at i o n ............... . ........................... .. .... .... .. .. .... ...... 72 4 .2.4. Co n t r o l S o ft ware ....................... ..... ................................................... 77 4.3. On-W afe r M eas ur eme n ts .... .. .. .. ...................... .... ....... . .. ....................... 79 5 A CC URA CY O F THE P U RE MOD E VE C TOR NETWORK ANALY ZE R ..... 82 5. 1. Pr o b et oPr o b e C r ossta lk ....................... .. ........ .... ..................................... 83 5 1 1 S imulat e d Pr o b e C r oss t a lk .......................... . .. .. ... .. ... . .... ....... ...... 8 4 5 1 .2. M e a s ur e d Pr o b e Cr oss t a l k .................... ............ . .. .. . ..... .. ........... 88 5 2 U n cer t ai nt y C alc ul at i o n s .......... ................ ......... .. .. .. .. ......... . ........ ......... 9 0 5 .2. 1 Di sc u ss i o n of A cc ur ac i es ............................................... .................. 99 5 .2.2. U n ce rt ai nt y M o d e l D e ri va ti o n ........ .. ..... ............ .... .... .............. 10 1 5.2.3. Ord e r o f U n ce rt a int y C a l c ul a tion s ... . .. . .. . .. .. . . .. . .. . .. .. .. ....... 1 06 5 3. C o n c l us i o n s o n A cc ur acy .................................... ...................................... 107 6 C ALIBRATION OF THE PURE MODE VECTOR NETWORK ANAL YZE R .... 10 8 6 .1. Ty p es of V N A M eas u reme n t E rr ors ........................................................... 108 6.2. P r im ary PMV N A Ca l i br at i o n ..................................................................... 110 6 2. 1 R aw P e r fo rm a n ce ... . . .......... .. ......... .. .. ... .. .. .... .... ... . .. .. ....... ... 110 6 .2.2. PMVN A E rr o r M o d e l ........... .. ....... ............. ......... .. . ..... ......... ... 1 15 6.2 3. D eve l o p me nt of Ca libr a ti o n E qu atio n ............................................ 12 1 6 2.4. Sw it c hin g E rr ors a nd N o n Pur e Mod e G e n e r a ti o n .. .... .. .. ..... ..... 124 6.2.5. So lu t i o n of th e C a libr a ti o n Pr o bl e m .... ..... .. ........................... .... 128 6 .2.6. Co a x ial Ca librati o n S t and a rd s ....................................................... 132 6.2.7. On W afe r Ca libr a ti o n St a n da rd s . .............. ...... .... .... .. .... ..... ...... 134 6.3. Pha se Offse t Pr e-Ca li bra ti o n ...................... ..... ..... ................. . .. ..... ...... 1 39 6.3. 1 Ph ase O ffset St a n da rd s . ..... . .. .. .. .. .. .. . . . .. .... .... .... ........... ...... 140 6.3 1 1 Fir s t Prin c ipl es .. .. .. . . .. .. .. . .. .. ....... ......... .................. 14 1 6.3 1 .2. O ffse t M ode l .............. ....................... .......... ..... ............ 1 42 6.3 .1.3. M o di f i e d T M a tr ix S o luti o n ....... .................................... 1 44 6.3.2. Pha se O ffse t O f An U nkn ow n D U T .............. .......... ..................... 1 50 6.3 2. 1 V a ri a bl e Of fset M o d e l . ..... ..... . ............ ........................ 150 6. 3. 2.2. U s in g Multip l e O ff se t St a nd a rd s ..................................... 15 1 6.3.2.3. Ca l c ul a tin g th e O ffse t o f a n Arbi t r a r y D UT . .................. 152 6.3.2.4. D iago n a l ized F o rm .. ..... .... ..... ...................................... 153 6.4. Ca libr atio n P roced ur e .. .. ............. .. .. . ........ .. .. .... .. ......... .. ................. 1 54 I V

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7 VERIFICATION OF THE PMVNA ..... ................................ ....... .................... 156 8 POWER SPL I TTER AND COMBINER ANALYSIS .. ............................ ......... 167 8.1. Splitter s . ... .. ........ ...... . .................................. .. ..... . .... . .. . ........ ..... .. .. 169 8 2. Combiner s ................................ ........ .. .. ....................................................... 178 8.3. Exten s ions to Arbitrary Phase ................................. ................. .... ............ 180 9 THIN-FILM MET AL ON CERAMIC STRUCTURES .. .... .. ......... . .. ... .. ....... 183 9 1 Differential Transmission Line s ..... .. ................ .. ....... .... .... .................. 184 9 .1.1. U nifor1n Differential Transmi ss ion Line ........................................ 184 9. 1.2. Balanced Step Differential Transmi ss ion Line ............................... 189 9.1.3. Unba l anced Step -Up Differential Tran s mis s ion Line ........... ......... 194 9 .2. Comparison Between Mea s ur e ments and Simu l ations ...... ........ ......... ....... 199 9 .2.1. Unbalanced Step Differential Tran s mi ss ion Line ........................... 200 9 .2.2. Balanced Step Differential Transmi ss ion Line ......... .. . .. ............. 206 9. 3. Crosstalk Between Differential Transmi ss ion Lines .... ............ .. .... ...... .. .. 212 9.3.1. Balanced Diff e rential Transmi ss ion Line s ..................................... 214 9.3.2. Unbalanced Differential Tran s mi ss ion Line s ................................. 230 l O PASSIVE INTEGRATED CIRCUIT STRUCTURES ..................... .. ..... ........ 239 10.1. Tran s mi ssi on Line s without Metal Ground Planes ...... ... ..... ... ....... . .... ...... 243 10 1 1 Single -E nded Transmi ss ion Lines . .. ...... .. .. . .... .... .................... 243 10.1.2 Simple Unifor1r1 Differential Tran s m i ssion Line ............................ 248 10 .2. Tran s mission Lines with Ground Metal Ground Plane s ....................... .. ... 2 52 10.2.1 Single-Ended Tran s m i ss ion Lines ... ........................................ . ... 253 10.2. 2. Uniform Differential Transmission Line s ....................................... 25 4 10.3. Unbalanced Differential Transmi ss ion Line s .. ............... . ....... ................. 259 10.4. Vertical Differential Tran s mi ss ion Line s .............................. ..... .............. 265 10.5 Pad -toPad Crosstalk ................................................................................... 27 5 11 PROPERTIES OF MIXED-MODE S PARAMETERS ...... .... . ....................... 291 11.1. Symmetry of Reciprocal Devi ces ... .. .......................................... .. ............ 291 11.1.1 General ................................ ............................................. ..... ......... 291 11.1 .2. Port-Symmetri c Reciprocal Devices .... .. .. .............................. .. ... 293 11.2. Balanced Devices ......................... .. ..... ..... . .. .. .. . .... ............ .......... ... ...... 29 4 11.3. Indefinite Mixed-Mode S-Parameters .......... ....... ......... ....... ..................... 297 11.4. Device Mode Specific Gains of Ideally Balanced Differential Circuit ...... 304 11 .4. 1 Transducer Power Gain s ...................................... . ........................ 30 5 11.4.2. Maximum Power Gains ....................... . .. .. . .................................. 308 11.4. 3. Power Gain Circles ....... ..................... . ..... . . ..... . ........................ 313 12 CONCLUSIONS ....................................................... ...... ..................................... 318 V

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APPENDICES A ANALOG HALF CIRCUIT TECHN I Q U ES . .. .. . .. . . .. .... .. .... .. . . .. . . . .. ... ... 32 3 B ANALOG MEASUREMENT TECHNIQUES ... .... . .. . .. . .. .. ... . .. . .. . .. .. ..... .... 327 C TRANSMISSION OF MODES FROM COUPLED TO UNCOUPLED LINES ..... 3 30 D SIMULAT E D S PARAMETERS OF DIFFERENTIAL AMPLIFIER .. .. .. .. .... 33 4 E DESCRIPTION OF HP8510 VNA SUB SYSTEMS .. ................. . . . .. .. ......... 33 9 F DETAILS O F HP851 7 TEST SET M ODIFICAT I ONS .... .. .. . . . .. . .. .. ... ..... 3 48 G PMVNA CONT R OL SOFfW ARE .. ... ....... .. .. .. . .. .. .. ... .. .. .. . .... .. .. . ..... .. .... 3 57 H MULTI-PORT T MATRIX DEF'INI I ION . . .. .. .... .. .... . . .. .. . . ... . . .. .. ..... ..... 3 8 3 I E RR O R TERMS OF PMVNA AND FOUR-PO R T VNA . . .......... . . . .. .. ... ... 3 87 J DEMONSTRAT I ON OF COEFFICIENT MAT R IX RANK . .... .... . . .. . ...... .... 3 90 LIST OF REFERENCES . . .. .. . ............ .. .... .. . .. . . . ... . ....... . . .. .. . . .. . . . .. .. ..... ... ... 40 3 BIOGRAPHICAL SKETCH .......... ... .. .. .. . .. . .. ........ . .. .. . .. . .. .. .. . . .... ....... . .... .. .. . ..... 4 1 2 Vl

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirement s for the Degree of Doctor of Philo so ph y THE THEORY, MEASUREMENT AND APPLICATIONS OF MODE SPECIFIC SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION by Da v id E. Bockelman May 1997 Chairman: William R. Eisenstadt Major Department : E lectrical and Computer Engineering Mode -s pecific sca ttering parameter s (s -parameter s) are defined from fundamental co ncept s. Such s-parameters directly ex pre ss the re s ponse of a device in it s intend ed mode s of operation. The development i s s pecifically applied to high frequency differen tial circuits. Differential circuits are shown to be characterized by four se t s of sparam ter s : ( 1 ) pure differential modes-parameters with a differential mode input and output, (2) pure common-modes-parameters with a common-mode input and output, (3) mode -con version sparameter s with a diff e renti a l-mode input and a common-mode output, and (4) 1node-conversion s -parameter s with a common-mode input and a differential mod e out put. All of the se sets of mode -s p ec ifics parameter s are s hown to be u se ful in analysis of a differential circuit. A specia lized sys tem called the pure mode v ec tor network analyzer ( PMVNA ), i s developed for the mea s urement of the modes pecific sparameters of a high frequency dif ferential circuit. Th e calibration of thi s analyzer is developed and implemented Verif icaVll

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tion establishes error correction accuracy. The PMVNA is shown to have accuracy advantages for the measurement of differential circuits when compared to a traditional four-port analyzer. The mode-specifics-parameter concepts are applied to several practical high fre quency differential circuits. Power splitters and combiners are analyzed with these con cepts Traditional specifications of phase and magnitude imbalance are shown to correspond to spurious mode responses. Differential transmission line structures, imple mented on ceramic substrates, are examined. The effects of imbalance and symmetry are analyzed with mode-specific s-parameters Several structures on a silicon integrated cir cuit (IC) are measured. The effects of differential topology on circuit-to-circuit coupling are quantified. Basic design method s are advanced for the design of high frequency dif ferential circuits. Vlll

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CHAPTER 1 INTRODUCTION In ma n y applicat i o n s, device s a nd circ uit s have been designed for only a si ngle mode of operation. In the most general se n se, a mode is a particular electromagnetic field configuration for a given device or circuit. In the case of one or two conductors, the modes are usually frequency dependent, so the existence of si multaneous modes can be avoided by proper selectio n of operating frequencies ( or by prope r physical design for a given frequency). However, with three or more conductors there will usually exist multi ple modes even in static cases In such s ituation s, the s imult aneo u s existence of two or more modes can be diffict1 l t to avoid. Differential c ircuits are a particular c la ss of circuits of historic importance with three conductors. Sometimes called balanced circuits, the primary operation of differen tial circuits is to respond to the difference between two s ignals, s uch as ~v 1 =v 1 v 2 as T T VJ z 1 l3 V3 + + + Devi ce + Port 1 ~Vl Under Test ~V2 Port 2 ( DUT ) + + V2 l 2 V4 l4 1_ 1_ Figure 1 1. Schematic of two-port diffe r ential c ircuit. 1

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2 shown in Figure 1-1. The two conductors can also have a common voltage ( or a current flow) with respect to a third conductor, namely grot1nd. As a result two modes of opera tion are generally possible with differential circuits: the differential-mode and the com mon-mode. Furthermore, both modes can exist simultaneously in general. There are many application s of differential circuits. Twi s ted pair transmission lines, operational amplifiers, balun s, coupled tran s mission lines power s plitter s and com biners are all examples of differential circuits [1 3]. More recent applications include radio frequency ( RF ) low noise amplifiers (LNA) with differential inputs and output s, as well as double-balanced mixers such as Gilbert cell mixers [4] RF differential circuit application s have become common as the commercial demand for radio systems has grown. Two characteristics of differential circuits make them particularly attractive for RF app l ications. The first advantage of the differential cir cuit is circuit-to-circuit isolation Thi s characteristic has been exploited for many years most notably in telephone systems in the for1n of twi s ted-pair wire transmi ss ion line s [5]. The higher isolation of differential circuits ( with respect to single-ended circuits) i s due to the nullification of any noise common to both constituent signals in the differential signal, i.e. (v+n) (-v+n) = 2v where n represents an interfering signal This isolation increase is important to integ1ated ci1cuit ( IC) implementations As integration den s ity increa ses, and operation frequencies increase, improving circuit-to-circt1it isolation is critical. As a result differential circuits are being applied where only single-ended circuits have tradi tionally been u se d. Second the differential circuit ha s increased dynamic range when compared to a ground referenced, or single-ended, circuit. With a given vo l tage swing v a pure differen

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3 tial s ignal will be doubled i.e. v (-v) = 2v. Thi s increa se d dynamic range is particularly important as the supply voltages decr e ase in modern radio systems. This decrea si ng sup ply voltage has mad e s ingle -e nded implementation s of receiver functions difficult, since maximum s ignal sw ing in a circuit i s typically les s than the s upply voltage. By imple menting radio functions with differential circuits, the available s ignal s wing, and hence the dynamic ran ge, ca n be increa se d while retaining a low supply voltage. The emphasis of traditional RF and microwave techniques has been to avoid the s imultaneou s existence of multiple mode s. A s a result there i s a lack of se lf-con s i ste nt rigorou s theory that is applicable to the measurement analysis and design of RF a nd microwave differential circuits. Typically, differential circuits are designed and analyzed with traditional analog techniques, which employ lumped element assumptions. RF and 1nicrowave differential circuits contain distributed circuit elements, and require di s tributed circuit analysis and testing. Furthe1more, traditional methods of testing differenti a l cir c uit s have required the application and m eas urement of voltages and currents, which i s difficult at RF and microwave fr e qu e ncie s. S ca tt er in g parameters (s-pa ramet e r s) ha ve been developed for c haracterization and analysis at these frequencies, but have been applied primarily to si n g le -e nded c ir c uit s. A modification of existing sparameter te ch niqu es i s needed for accurate mea s urement, analysis and design of differential circuits at microwave frequencies. Thi s work extends the definitions of sparameters to mode -s p cific repre se ntation s, where the sparameter s are defined in term s of the natural mode s of operation of a circuit. This di sse rtation presents original work in the following areas. The traditionally accepted d efi nition s of differential and common-mode voltages and c urrent s are s h own

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4 for the first time to be non-orthogonal, and the ref ore unacceptable for direct application of these definitions for power calculations. New orthogonal definitions for these voltages and currents are pre se nted, and shown to be appropriate for power calculations. Multiple mode sparameter s are developed which for the first time completely describes the linear behavior of an RF differential circuit. These concepts are verified through simulations of RF differential circuitry. The fir s t network analyzer for the measurement of multi mode sparameters is constructed, and the inherent acct1racy advantages of the system are estab li s hed Fundamental work in multi-port network analyzer calibration proceeds beyond any previously published work, and a verification procedure establishes the accuracy of the calibration. Measurements with the multi-mode network analyzer includes the first of integrated differential circuits. Extensions of s-parameter design techniques to multi mode circuits are pre se nted that will formalize the design and analysis of RF differential circuits. This dissertation is organized in the following manner. In Chapter 2, technique s for analysis and mea s urement of differential circuits, prior to this work, are discu sse d. Chapter 3 presents original work of extending scattering parameter theory to differential circuits. A new measurement sys tem for the measurement of mode-specifics-parameters is introduced in Chapter 4. Chapter 5 examines the accuracy, and establishes the intrinsic accuracy advantages, of this system for the measurement of differential circuits. The cali bration theory and implementation for the new meast1rement system is developed in Chapter 6. In Chapter 7, the re s ults of accuracy verification of thi s new sys tem are pre sented. The remaining chapters of the dis se rtation focu s on applications of the modespe cific sparameter concepts. Chapter 8 applies the new mode-specific concepts to power

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5 splitters and combiners. Several thin-film metal differential structures, fabricated on alu minum oxide, are studied in Chapter 9. Circuit-to-circuit crosstalk for IC s tructur es on si l icon is s tudied in Chapter 10 and conclusions are made about practical implementation of ICs. Chapter 11 discusses properties of the new s-pa rameter s and provides basic analysis and design tool s for use with RF differential circuits. Chapter 12 concludes this disserta tion with a summary, some discu ss ions and remark s on future research.

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CHAPTER2 PRIOR THEORIES AND IBCHNIQUES This chapter se rves as a s ummary of pa s t theoretical and experimental techniqu es that are applied to differential circuits. The focus of the chapter is RF and microwave dif ferential circuits. However lower frequency work has had a profound effect on the s ub ject so the examination will include relevant analog techniques. In the area of theoretical analysis, the subjects presented include multi -mode tran verse electromagnetic ( TEM ) st 1ucture s such as coupled tran s mi ss ion line s. The analog methods that have provided the ba s ic concepts of differential circuit analysis are summa rized. Network repre se ntation of a differential circuit is reviewed and its application to an RF differential circuit is explored. The traditionally accepted definitions of differential and commo n-mode voltages and currents are s hown to be non-orthogonal i. e., as a sys tem, the definitions do not conserve energy. The mea s urement technique s of RF differential circuits are then summarized. Analog differential circuit mea s urement techniques are briefly examined as background. All widely practiced measurement approaches for RF and microwave differential ci rcuit s are presented in general, and are s hown to provide inadequate characterization of the device under te st. 6

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7 2.1. Fundamental Theories of Analysis The topics presented in the following sections represent some of the most promi nent concepts in differential circuit analysi s. The s ubjects are coupled transmission line s, analog method s, and network repre se ntations and each topic hold s a unique concept which s hape s later theoretical developments 2. 1.1. Coupled Tran s mission Line Pairs In a s urv ey of prior work in the RF and microwave field s, one early are a of work i s found to s hare many concepts with differential circuits in general. The work done in cou pled tran s mi ss ion line s, and their application s, de sc ribe multiple mode behavior that i s analogou s to differential circuit mode s. The importance of this transmi ss ion line work in thi s context is the treatment of the s imultaneous existence of two modes of propagation The coupled transmission line pair theorie s have their foundation s in electromag netic field de sc ription s (6 7]. Sy s tematic treatment of the coupled pair tran s mi ss ion line begin s with the examination of the two fundamental TEM mode s Planar coupled tran mi ss ion lin es s u c h as s triplin e are of particular importance due to fabrication advantages. A s a r es ult, much early work co n side r s planar s tructure s. With s uch s tru c ture s, the two fundamental TEM mod es are called odd and even for their re s pective field symmetry, with Figure 2-1. (a) ( b ) Electric field di s tribution s in planar coupled tran s mi ss ion line s. a) Odd -mo de e l ec tric field ; b ) Even-mode electric field.

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8 the terminology apparently fir s t introduced by Cohn [6] Representations of the electric fi e ld di s tribution s for th e two fundamental TEM modes a re s hown in Figure 21 In Fi g ure 2 -1 (a) the s i g nal conductors are at equal bt1t oppo s ite potential s and carry e qual but opposite c urrent s, a nd hen ce thi s i s called the odd mode. In Figure 2 -1 ( b ) the s ignal con ductors are the sa m e pot e ntial and carry eq ual currents; thi s i s called the even mode. Early work was limited to only phy s ically sy mmetric s tructures [6] and the terms even and odd apply only with s uch limitation s. Tripathi later extended the theory to includ e any coupled pair tran s mi ss ion lin e st ructur e [8]. With thi s extension the two modes became c and 1t modes re s pectively and the symmetry in the field di s tribution was lo s t With the lo ss of the ev en and odd-modes, the direct analogy to differential and com mon-mode s become s l ess clear. Althou g h imp o rtant for the tr ea tm e nt of s imultan eo u s mod es, the co upled tr a n mis s ion line work i s limit e d to tr a nsmi ss ion lin e applications. The theory i s cast in term s of characteristic imp e dance s, pr o pa ga tion co n s t a nt s, etc., and i s not directly applicable to a ge n eral differential c ir c uit Previou s work ha s be e n s p ec ific to de sc ription s of co upl ed tran s mi ss ion line s [6 8 15] and shielded balan ce d tran s mi ss ion lines. In the literature the coupled tran s mi ss ion work ha s been most commonly applied to directional couplers [2, 16 1 8 ] r a th e r than to diff ere nti al circuits in ge neral. All of the referenc e d work de a l s with specific TEM st ructure s, a nd is not s uitable for characterization of a ge n er ic differen tial circuit. De s pit e the narrow application of prior work, th e theory of co upled tr a n s mi ssion Jine s provides the foundation on which ge neral multi-mod e network analysis i s built Scattering parameter s are a relative mea s ure of a network 's re s pon se, s o any mode -spe

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9 cific s-parameters must be defmed with respect to some reference impedance. In Chapter 3, the theory of coupled transmission lines will be used to rigorously define the reference impedance for the different modes. 2.1.2. Analog Methods Traditional analog methods play a central ro l e in the prior work with differential circuits. The work, which employs lumped element assumptions, is used primarily for audio or near-audio frequencies. Of course, most any general analog circuit techniques can be applied to an analog differential circuit, but some specialized concepts are of particular importance. Analog theories have provided the fundamental definitions of differential and common-mode voltages. Referring to Figure 1-1, the differential-mode voltage at port one is defined as (2-1) and the common-mode voltage at port one is defined as (2-2) The differential cunent into port one is (2-3) and the common-mode current into port one is (2-4)

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10 with similar definition s at port two These definition s have lead to voltage gain concepts (25 ) These definitions are widely accepted as evidenced by examples found in the texts by Gray and Meyer [4] Middlebrook [19] and Giacoletto [20] as well as other recent works [ 12 21 ]. Of particular intere st in analog technique s is the method of differential and com mon-mode half -c ircuits. This technique allows circuit analysis simplification by consider ing se parately the response of the circuit to a pure differential signal and a pure common mode signal [ 4]. For an instructive example of the application of half-circuit method s, see Appendix A. The se analog techniques are useful in gaining in sig ht into differential ampli fier operation. How eve r the se techniques have some limitations. First, these equivalent c ircuit approaches can be co me intractable as high frequency effects are included. Al so, they are inherently lumped element approaches, and are not easily adapted to include dis tributed circuit elements that become important at RF and higher frequencies. The mo s t fundamental limitation i s that the half-circuit techniques are applied only to sy mmetric differential circuits. It ha s been s hown [ 19] that perfectly sy mmetric ( or balanced) differential circuits exhibit no mode conversion. By limiting the analysis to sy mmetric circuits, or by neglecting any asymmetry, the phenomenon of mode conver s ion is completely i gnored. Mode conversion occurs when a stimulus of a pure mode cre ates an output of more than one mode For example, if a pure differential signal drive s an amplifier, and both a differential-mode and a common-mode output signal are produced then some conversion from differential to common-mode has occt1rred.

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11 Mode conversion is an important phenomenon for RF and microwave differenti a l circuits, and in amplifiers in particular. It ha s also been s hown [19] that mode conversion will affect the maximum achievable co mmon -1node rejection ration (C MRR). A critical parameter of differ ential amplifier design, CMRR quantifie s th e ability of an amplifier to amplify differential signa ls and reject common-mode sig nals. Understanding, predi cting, and measuring the phenomenon of mode conversion can be important to the performance of RF differential circuits. 2.1.3. Linear Network Repre se ntation s Linear time-invariant (LT I ) network repr esen tation is a basic and useful circuit analysis te chniq ue which is widely applied to two-port and three-terrninal circuits of both analog and RF applications [ 22] Network representations are distinctly suitable for 1 /1 lk+l k+J + + Vi 11 I Vk+l ----1' k+l (k+l)' 2 1 2 l k+2 k+2 + Linear + V2 I Vk+2 1 2 Time Invariant 2' k+2 (k+2)' n-Port Network k l k 1, i n + + Vk l k 1, i vn k' n Figure 2-2. Notation of an n-port linear time-invariant network.

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12 descriptions of distributed element circuits as enco11ntered in RF and microwave applica tions. It is of use then to examine the application of LTI network theory to differenti al cir cuits. A circuit, or network, with n pair s of terminals which are used as input/output con nection s is known as an n-port network. The notation conventions for an n-port network are shown in Figure 2-2. The da s hed line connecting the 1' ternlinal to ground indicate s that so me or all of the return terminals can be grounded. The behavior of the network is described by a set of eq uation s that are expressed in terrr1s of the defined voltages and cur rent s ( or quantitie s that are related ), which can be written in matrix form. The matrix description leads to a convenient set of parameters for a circuit. Some commonly used parameters are impedance Z admittance Y, hybrid h and chaining ABCD. These param eters are all ba sed on voltage and current descriptions of the network. Other parameter s, s uch as sca ttering parameters (s-pa rameter s), are based on functions of voltage and cur rent. 2.1.3.1. Analog Network Parameter s Network representations can be applied to differential circuits in at least two ways. On~ po ss ible application of network theory i s to interpret each input and output tern1inal of the differential circuit as a port with the return path grounded. This approach i s quite common, and will be referred to as the s tandard approach to network representation With this approach, all of the inputs and outputs of the differential circuit are ground referenced (s ingle -en ded ). In this case, the network will always have 2n port s, where n is number of differential input s and outputs. For example, a differential amplifier can be repre sen ted as

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13 a four-port network shown in Figure 2-3. Here the port voltages are related to the differ ential and common-mode voltages [ 12, 19, 20] by vdl vd2 Vl + vcl V3 + vc2 2 2 vd l vd2 v2 + vcl V2 + vc2 2 2 The port currents can s imilarly be related to differential and common-mode currents i I = idl + icl i2 = -idl + icl i 3 = id2 + ic1 i4 = id2 + i c2 (2-6) (2-7) By defining the port voltages and currents as such, the network description of the diff e ren tial ci1cuit can be completed in terms of any useful parameters. There i s a critical limit ation with this particular approach to network representa tion of differential circuits. Since all port voltages and currents are functions of both dif ferential and common-mode quantities, al l the resulting network parameter s will also be combinations of both mode responses The response of the circuit to a specific mode i s not obvious from inspection of the parameters. This comming lin g of the modal respon ses I l 1 Z 3 3 + + Vl Differential V3 l_ Amplifier l_ l2 l4 2 4 + + '2 V4 l_ Four-Port LTI Network l_ (sma lls ignal ) Figure 2-3. Network representation of differential amplifier.

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14 is a distinct disadvantage since the implicit purpose of a differential circuit is to provide a certain response to a differential stimulus So, although the standard approach to network representation is a sufficient description of differential circuits, it is non-intuitive. There fore, a second approach to network descriptions of differential circuits will be described The second type of network representation of differential circuits describe s the c ir cuit explicitly in terrns of modal responses. By using modal definitions of voltage and current as given in ( 2 -1) to (2-4 ) a network description can be defined. First, the input s and outputs must be paired appropriately For example ports one and two can be paired to create a differential port This pairing of signals is extremely useful in low-noise systems as discussed in Chapter I. For example the z-parameters of a differential amplifier can be defined [20] as vdl z dl d l Z ctl ,c l Z ctl d 2 z dl ,c 2 ld1 v c l Zc l dl 2 c l ,c l Zc l d 2 Zc 1 ,c2 z c l ( 2 8 ) vd 2 Z ct 2 d 1 2 ct 2 c 1 Z ct 2 d2 Z ct 2 c2 ' ' zd2 v c2 2 c2, d I 2 c2,c l Z c 2, d 2 2 c 2 ,c2 l c 2 dl lctl lct d2 + + VctJ Vct 2 dl Linear d2' Time Invariant Differential Two-Port cl 1 c 1 c2 Network + + Vc l V c2 cl c 2 Figure 2-4. Modal notation of an two-port differential network.

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15 This network description can be interpreted directly in terms of differential and common-mode responses. The network dia g ram can be modified to reflect the explicit modes, as s hown in Figure 2-4. This approach will be called the modal network represen tation. Note that a two-port differential circuit i s represented again by a four-port net work; in general, an n-port differential circuit will have a 2n-port network. The separation of th e differential and common-mode ports in the network repre se ntation is a u sefu l con ceptual tool The modal network representation pre se nted thus fat i s useful in the analysis of analog differential circuits. Howev e r the application of this technique to RF and microwave circuits i s of limited u se as will be di scussed in the next section. 2.1.3.2. RF Network Para1neters Power ba sed sca tterin g parameters are widely u sed in RF and mi c row ave fields to repre se nt circuits and device s with distributed elements. A s its name implie s, scattering parameters repre se nt a scattering or se paration of a signal by a device unde1 test. The scat tered s ignals are the reflected and tr a n s mitted electromagnetic waves that are produced when a device i s s timulated with an incident wave. Scattered wave de s criptions of net works are very important when operation frequencies are high enough s uch that circu it elements become a sign ificant fraction of a wavelength (a pproximately one tenth of a wavelength ). Scattering parameter s originate in transmission line concepts. A s such, they are always defined with r espect to a characteristic impedance, or reference impedanc e. The primary benefit of sparam ete r s is ease in mea s urement In distinction to voltage-cur rent derived parameters, sparameter s are measured with port s terminated in the character istic impedance. Thi s ha s meaningful practical implications s ince short-circuits and

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16 open-circuits are extremely difficult to realize at RF and microwave frequencies due to distributed element effects. S ca ttering parameters will first be developed in ter1n s of tran s mi ss ion line quanti ties, to provide insight into their wave aspect. Following this definition, the generalized definition will be given. The following development is s ummari ze d from Gonzalez [23]. The voltage and current along a transmission line s uch as in Figure 2-5, sa tisfy the set of differential equations d 2 2 2 V(x) y V(x) = 0 dx d 2 2 2 / (x) y l (x) = 0 dx where y i s the propagation constant The general solution of (2-9) is yx "(X V(x) = Ae +Be / (x) A -yx B yx = -e --e Zo Zo (2-9) (2-10) where A and B are complex constants and z 0 is the c haracteristic impedance. Th e propa gation constant and the characteristic impedance can be expressed in terms of the parame ters R, G, L, C which are the resistance conductance, inductance, and capacitance per unit length of the transmi ss ion line y = J(R+JwL)(G+jwC) R + jroL G + jroC (2-11) Given the phaser notation v+(x)=Ae-yx and V (x) =Be 'YX and by limitin g th e transmission line to be lossle ss ( i.e. Z 0 = Re{ Z 0 } ), then the important nor1naJized quantitie s are defined as

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17 v(x) V(x) i(x) = I(x ) Jzo Jzo v+(x) a(x) b(x) V (x) Jzo Jzo With these definitions, (210 ) becomes v(x) = a(x) + b(x) i(x) = a(x) b(x) (2-12) (2-13) The a and b waves are the incident and reflected/tran smit ted normalized power waves, and they are the primary quantities of s-para meters. + V I I I x=-1 Figure 2-5. Terminated transmission line. I x=O When applied to an n-port network, s uch as in Figure 2-2, the a and b waves result in as-parameter description bl sl 1 s12 sln al b1 s21 s22 s2n a2 (2-14) b,i sn l sn2 snn an or simply 75 = Sa where the bar over a lower-ca se variable repre se nt s a column vector [23]. The definition of s-parameters can be generalized to include complex characteris tic impedances. This ge neralization al so removes the dependency of the s-parameter on

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18 transmission lines. The definition is based on a generalized power wave at the n-th port [23 25] 1 [ z ] a = -=== V +l n 2 JRe(Z n) n n n b 1 [ z ] =--;:===V l n 2/e(Zn) n n n (2-15) The s-parameter matrix equation ( 2-14 ) remain s the s ame. Scattering parameters have not been widely applied to the analysi s or measure ment of differential circuits. S-parameters would provide the same benefits to RF differ ential circuits as they do for other RF and microwave circuit s. Conceptually the repre se ntation of differential circuits withs-parameters is not difficult. In fact, with the s tandard network representation di s cu sse d earlier, an-port differential circuit can be de sc ribed with a 2n by -2 n s -parameter matri x, without any a dditional consideration. However, thi s appro ac h ha s the s ame di s advanta ges as previously de sc ribed, namely the parameter s do not provide useful indications of the differential and common-mode re s pon s e s. For a illu s tration of the difficultie s of interpreting the s tandard four-port sparameters of an RF differential amplifier, see Appendix D The above limitations co uld be removed by extending sparameter theory to a modal network repre se ntation. This extension ha s not been completed prior to thi s work and tl1i s dissertation later pre se nt s the extension. A s traight -fo rward extension of sparameter theory to a modal n e twork repre se n tation would apply the traditional definition s of modal voltages and currents in (2-1) through (24 ) to the generalized power wave definition s of (215 ). However, the voltage and current definition s of (21 ) through (24 ) are not an acceptable ba s is for a pow e r wa ve

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19 netw o rk repre se ntation Straight-forward applic a tion of these definitions re s ult s in quanti tie s that do not conserve energy Th e difficultie s with power calculations usin g these quantitie s ca n b e d emo n s tr a ted with a s impl e example. Suppo se tw o so urce s of pow e r h ave potential s V 1 and V 2 and so ur ce c urrent s 7 1 and 7 2 re s pectively Assume the sources are harmonically time varying (so ii; and I i are pha so r s, as indicated by the bar over the upper -case variable) but have no s pe cific phase relation The power delivered by the two sources is (2-16) and the tot a l power in both so urc es i s (2-17) By definitions (2-1) through (2-4), the diff ere nti a l and common-mode voltage a nd current can be exp re sse d as Vd = VI V2 l Ve = 2CV1 + V2) The pow e r in eac h m o de i s then p d = R e(~ 1d *) 1 Id = 2(11 / 2) 1 = C 1 2(/1 + 1 2) (2-18) (2-19) If th e modal definition s are consistent, then the total power of the mode s mu s t be e qual to th e total power of the so urc es by the conservation of energy. Expanding (219 ) 1 --* --* --* --* Pd= 2 [Re(V 1 1 1 ) +Re (V 2 I 2 )Re ( V 1 I 2 )Re ( V 2 / 1 ) ] I --* --* --* --* P c = 4 [Re ( V 1 I 1 ) + Re (V 2 / 2 ) + Re ( V 1 / 2 ) + Re (V 2 J 1 ) ] and the sum of mode power i s (2 -20 )

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20 (2 2 1 ) Expanding the sum of the source powers in (2-17) (2-22) which clearly shows that the voltage and current definitions of (2-1) through (2-4) are not directly u sa ble in power calculations. The voltage and current definitions of (21) through (2-4) can be used for power and power-wave calculations if care is taken to account for the non -ort hogonal nature of the syste m. However it i s much more convenient to define new mode voltages and currents that are orthogonal. These new definitions are given in Chapter 3, Section 3.1.1 Despite the advantages of scattering parameters there is no acceptable theoretical treatment of sparameter network repre se ntations for differential circuits prior to this work. The attempts at applying sparameter s to RF differential circuits have relied upon intuitive notions of differential sparameter s [26 21 ]. A s such, the prior incomplete theo ries have not treated seve ral fundamental principle s that a rigorous theory requires. The principles in question are conversation of energy (or thogonality ) in the mode s, pre cise definition s of reference impedance s for all modes rigorou s definitions of all pure modal re spo nses and self-consistent definitions of conversion responses between modes. 2.2. Measurement Techniques When examining prior work on circuits with multiple simultaneous modes of propagation consideration mu st be given to accepted measurement techniques. The state of the theoretical development and of the organization of the field as a whole, can be

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21 observed in the completeness and accuracy ( or the lack thereof) of generally accepted mea s urem e nt techniqu es. Again the scope of thi s s ur vey of prior work will be limit e d to subjects r elated to differ e ntial circuits. Th e topic s pr ese nt e d in the following sections repre se nt so m e of the mo s t wid e l y practiced mea s ur e ment techniques for differential circuits. Th e s ubje c t s are divided between the analog technique s, RF / microwave sca lar approaches, an d scat terin g parameter mea s urement s. The treatment of the mea s urement techniques i s not int e nded to b e exhaustive, but it is representative of the common types of mea s urement s pre se ntly applied to differential circ uit s. 2.2. 1 Single Mod e Analog Me as urement s Analo g m eas ur e ment s of differential circuits are typi cal l y dire c t m eas urement s of vo lt ages and c t1rr e n ts, which are primarily limit ed to audio or n ea r -a udio freqt1encies ( i .e. typical analog frequ e ncie s) [27] The reason for thi s limitation i s due to di st ribut ed natur e of circuits as frequencies approach RF. With di s tributed circuits, there will be tran smis sio n line effects With transmi ss ion line effects, the voltage and current will b e functions of the position along t he line. Furthermore, parasitic ca pacitance s and induct a nce s b ecome significa nt at these frequencies, and effec t the performance of the DUT As a r es ult it i s difficult t o make un am bi g uou s measurements of voltage and cu rrent at RF and hi gher frequencies. Differ e ntial analog mea s ur e m e nt s typically e mploy si ngle-ended to differential co nverter s (ca lled baluns) to s timulat e and m eas ure the DUT in an ideally pur e differential mode. However the se co nvert ers are n o t id ea l, a nd they affect th e accuracy of th e mea s urements Mo s t s ignificantly, the mea s urement s inevitably include the effects of these

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22 converters, and littl e can be done to remove them. For a more detailed description of typ ical ana l og differential measurements, see Appendix B. 2.2.2. Single Mode RF a nd Microwave Measurements When a differential circuit operates in the RF/microwave frequencies, voltage and current measurements are no l onger practical. Instead, the appropriate measurements deal with transmission of waves and power. Some of the most common and important RF measurements of differential circuits are presented below. The primary difficulty with RF differential mea s urement s l ike analog measure ments, is the generation and reception of differential signa l s Also like analog measure ments, RF measurements require baluns. For RF, center-tapped transformers are available that can operate to 1 GHz [ 28]. For higher frequencies, power splitter/combiners, such as hybrid cot1plers, are genera ll y used. The one consequential difference from the analog baluns is that the RF/microwave balun s have more non-ideal performance. Differential-mode RF measurements can be made with the use of 180 power split ters/combiner s and common-mode RF measurements can be made with 0 power split ter s /co1nbiners. Like the analog mea s urements, these RF/microwave measurements assume single-mode inputs and output, and are called single mode measurements. 2.2.2.1. Scalar Power Measurements Including Baluns One widely used type of RF measurement of differential circuits is a scalar power measurement. This measurement provides the magnitude of the power gain. The mea surement may take the form of a constant amplitude input signal swept across frequency resulting in a gain versus frequency characteri s tic. Alternatively, the input power level

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23 can be swept at a fixed frequency, re s ulting in a output power versus input power characteristic Regardless of the specific mea s urement sca lar power measurement s have the sa me basic instrumentation. The signal so urce i s an RF signal generator, the mea s urement instrument is a power meter or a spectru m analyzer, and RF baluns must be u sed. A typi cal measurement system is s hown in Figure 2-6. Like the analog mea s urement s, the scalar RF power measurement s include the effects of the balun s. The effects of the baluns are even more difficult to remove at RF frequencies than at analog. This difficulty is due to the increased non ideal performance of the baluns. The non ideal perfo1mance is typically specified in terms of loss, magni tude imbalance and phase imbalance. The effect of the baluns on the accuracy of the measurement can be examined qualitatively RF balun s, such as 180 3 dB hybrid couplers, have magnitude and phase imbal ance in th e splitting of a signal Ideally, a 180 3 dB hybrid coupler would take a si ngl e input s ignal and sp lit it into two equal amplitude s ignal s with 180 pha se difference With Signal Generator Figure 2-6. Hybrid Cou ]er o o 180 Differential Circuit Hybrid Cou ler o o 180 RF scalar power measurement of differential circuit. Spectrum Anal zer

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24 an ideal splitter a pure differential mode signal cou ld be constructed. However, the mag nitude and phase imbalance means that the two outputs of the splitter are not exactly equal amplitude and the phase difference is not 180 As a result a pure differential signal i s not produced by a real power splitter and a test circ11it is only driven in an approximately single mode fashion. The magnitude and phase imbalance also affect the combination of two signals. In essence the imbalance causes a spurious response to a common mode input. The combined effect of the imbalances in the power splitter and combiner is to measure a commingled response of the circuit to both a l arge differential and small com mon-mode input. The differential and common-mode responses cannot be distinguished by the instruments, and the overall measurement accuracy is reduced. These effects are examined in detail in Chapter 8. 2 2.2.2. Scattering Parameters with Baluns A less prevalent, but important, technique for RF/microwave differential circuits is single mode (differential) s-parameter measurement [26]. This approach, as implied by its name, attempts to measure s-parameters of a circuit with input signa l s and output signals of a single ( differential) mode. Like other single mode measurements, this technique employs baluns. The most common application of this method is the measurement of the differential response of a circuit with s-parameters. The measurement system includes a standard two-port vector network analyzer (VNA) which automatically measures the s parameters of a two-port device and a pair of 180 3dB power splitters/combiners. This approach has also been applied to on-wafer measurements of differential circuits [26]. The schematic of the sys tem is shown in Figure 27.

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Figure 2-7. Hybrid Coupler o o 180 25 port 1 Network Analyzer port2 Differ e ntial Circuit Hybrid Coupler o o 180 S Par a m e t e r m eas ur e m e nt of differential circuit. Thi s measurement technique s uff e rs from the same problem s as the scalar RF mea s urement s. The ma g nitude and phas e imbalan ce in the splitters/combiners and the ne g l ect of mode conversion will all produc e errors in the mea s ured s-parameters. Howev e r the sp a ramet er approach represents an important extension of measurement technique s. In co ntra s t t o sca lar m easu rement s, s-para m e t ers a r e by their nature vector quantities, a nd h e nc e th ey represent both magnitude and ph ase m eas ur e m e nt s Another limitation of thi s technique as reported i s th e lack of rigorous definition of di ffe rential and common-mode sca tterin g parameter s. Calibration of the mea s urement syste m a nece ss ity for all accurate VNA measurement s, is al so undefmed. Althou g h lim ited in a cc uracy due to the cited problem s, a ca libration for this system could be derived from the theory pr esente d l a t er in thi s work.

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26 2.3. Summary of Past Theory and Techniques Clearly, an opportunity exists to extend the accuracy of analysis, design, and mea surement of differential circuits into the RF and microwave frequencies. By combining the core principles of differential circuits traditionally belonging to the analog domain with established RF techniques like scattering parameters, a strong contribution to both fields is achieved. In the next chapter, the concepts of multi-mode analog differential cir cuits are extended into a rigorous theory for the analysis, measurement and design of RF differential circuits.

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CHAPTER3 FUNDAMENTAL THEORY OF MODE SPECIFICS-PARAMETERS 3.1. Mode Specific Scattering Parameters in Differential Circuits A seve re limitation in differential-mode/common-mode circuit characterization is a lack of applicable power wave and s-parameter theory in terms of these two modes. There is no previou sly reported way to describe sparameter s based on mixed differenti al mode/common-mode propagation. Previou s work most closely related to this work ha s been s pe c ific to descriptions of coupled tran smiss ion line s (8 15] and shielded balanced transmission lines. Work by the National Bureau of Standards on balanced transmission lines used sparameter s to describe differential-mode propagation, but neglected common mode propagation and any mode conversions [21]. In the literature, the coupled transmis sion work has been most commonly applied to directional couplers (2, 16 18] with Cohn and Levy [3] providing a hi storic al perspective on the role of coupled tran smiss ion lines in directional coupler development. Pa st work on coupled transmission line s has largely focused on voltage/current relationships and Z, Y, and ABCD-parameter description s of TEM circuits One notable exception to the Z/Y/ABCD-parameter approach is work by Krage and Haddad (29] which employs traditional normalized power waves to describe coupler behavior. However all of the referenced prior work deals with specific TEM structures, and is not s uitable for characterization of a generic differential circuit. 27

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28 The following s ections contain original work in the defmition s of multi-mode power waves and sparameters Portions of this work have been published in summary form [30]. The detail s of the development of multi-mode sparameter s, and new related material, are c ontained in the remainder of the chapter. 3.1 1. Fundamental D ef inition s for Differential Circuit s In a practi ca l RF / microwav e implementation a differential circuit i s ba sed on pair s of coupled tran s mis s ion line s. A sc hematic of a typical two port RF/ microwave dif ferential sys tem is s hown in Figure 31 Es se ntial features of the microwave diff e rential circuit in Figure 3-1 are the coupled pair tran s mi ss ion line on the input and output of th e DUT As described in Chapter 2, thi s coupled line structure allows the propagation of two TEM mode s. It i s conceptually b e neficial to define a signal that propagates between the line s of the coupled-pair (as opposed to propag a ting between one line and ground). Such s ignal s are known as diff e r e ntial s ignal s, and can be described by a differ e nce o f voltage (~v 1 -:/= 0, Mixed Mode Port 1 coupled line s coupled X = X1 (Z n1 Z ct, 'Y1tt, 'Yc1) line s (Z1t2,Zc2, 'Y1t2 ,'Yc2) I .l\.f\f'i 1 ( 1 ) }+~ --9111111! ..... \ --~ I V1 1 \ I l I ...L. f : I l 2 \ 1 (2) J--1 ~\ : ___ ___. I + \J I V2 I ...L. DUT Figure 3-1. Schematic of RF differential two port network . I .. l ' Mix e d-Mode Port 2 X = X2 I -"\J/V\. l 3 + (3) V3 I I ...L. 1 ..rlrv\_ l4 ( 4 ) + I V4 I ...L.

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29 Llv 2 -:/= 0) and current flow between the individual line s in a pair By such a definition, the signal is not 1 ferenced to a ground potential, but rather the s ignal on one line of the cou pled pair i s referenced to the other. Further, thi s differential s ignal should propagate in a TEM or qua s i -TE M, fashion with a well-defined characteristic impedance and propaga tion constant. Coupled line pairs as in Figure 3-1, allow propagating differential signals ( the quantities of intere s t ) to exist The differential circuit di sc ussion in this chapter will be limited to the two-port case, but the generalized theory for n-port circuits can be readily derived from thi s work. Mo s t practical implementation s of Figure 3-1 will incorporate a ground plane or sorne other global r efe rence conductor, either intentionaJJy or unintentionally Thi s ground plane allows another mode of propa ga tion to exist, namely common-mode propa gation. Conceptually the common-mode wave applies equal signals with respect to gro und at each of the individual line s in a coupled pair s uch that the differential voltage is zero (i.e Ll v 1 = Ll v 2 = 0 ) The ability of the microwave differential circuit to propagate both common-mode and differential mode s ignals require s any complete theoretical tr eat ment to include characterization of aJl s imultan eo u s ly propagating mode s. For conve nience the s imultaneou s propa g ation of two or more mode s ( namely differential-mode, and common-mode) on a coupled transmi ss ion line will be referred to in thi s work as mixed-mode propagation from which mixed mode sparameter s will be defined To begin the d eve lopment of a rigorous theory of mixed differential and common mode norrr1alized power waves, the two mode s mu s t be defined in a se lf-consistent fash ion A differential signal propagates between the lines of the coupled-pair ( a s opposed t o propa ga tin g between one line and g round ), and a common-mode s ignal propagate s with

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30 equal signals with respect to ground at each of the individual lines in a coupled pair. The ability of the microwave differential circuit to propagate both common-mode and differ ential-mode signals requires any complete theoretical treatment to include characterization of all simultaneously propagating modes. For convenience, the simultaneous propagation of two or more mode s ( namely differential-mode and common-mode) on a coupled transmission line will be referred to as mixed mode propagation, from which mixed-mode sparameter s will be defined. 3.1.1.1 Modal Voltage and Currents At this point, it is important to define the differential and common-mode voltages and currents to develop a se lf-consistent set of mixed-mode sparameter s Referring to Figure 3-1, define the differential-mode voltage at a point x, to be the difference of be twee n voltages on node one and node two (3-1) This sta ndard definition establishes a signal that i s no longer referenced to ground. In a differential c ircuit, one would expect equal cu1rent magnitude s to enter the po s itive input terminal as leave s the negative input terminal. Therefore, the differential-mode current is defined as one-half the difference between currents entering nodes one and two (3-2) These definitions differ from previou sly published definitions by Zysman a nd Johnson [ 12] due to change in references The common-mode voltage in a differential circuit is typically the average voltage at a port. Hence common-mode voltage is one half the sum of the voltages on node s one and two

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31 (3-3) The common-mode c urrent at a port i s sim ply the total current flowing into the port. Therefore define the common-mode current as the sum of the currents entering nodes one and two i C (X) = i 1 + i 2 (3-4) Note that the differential current include s the return current, and the return c urrent for the common-mode signal flows through the ground plane For this rea so n the differential mode current i s hal ve d where the common-mode c u1 r e nt i s not This definition of common-mode cu1Tent differs from the traditionally accepted d e finition [4 12 19 21]. Definition s in (31 ) to (34 ) are se lf-con s i s tent with the differential power deliv ered to a differential load. Thi s can be shown by demon s trating that the se definition s con se rve the total energy in the modes The power at each terminal (x = 0 for example) can be expressed a s a nd the total power in both so ur ces i s The power in each m ode i s By defmitions (3 -1 ) to (3 -4) 1 Pd= 2 [ R e(v 1 i 1 *) +Re (v 2 i 2 *)Re (v 1 i 2 *)Re (v 2 i 1 *)] p = C 1 2 [R e(v 1 i 1 *) + Re (v 2 i 2 *) + Re (v 1 i 2 *) + Re (v 2 i 1 *) ] (35 ) (3-6) (3-7) (3-8)

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32 and the sum of mode power i s Expanding the s um of the source power s in (3-6) (3-10) Therefore the s um of the modal power i s equal to the total power (3-1 1 ) and energy is conserved by the definition s of common and differential-mode voltages and cu r rents. 3.1.1.2. Coupled Mixed-Mode Signals To begin the presentation of mixed mode sparameters a general asymmetric cou pled tran s mi ss ion lin e pair over a ground plane will be analyzed. Thi s analysis yields mul tiple propagating mode s all referenced to grot1nd. The se modes will be u sed to express the Figure 3-2. Termination Line B Lin e A + V 4 ....... x=O Port 1 x =L Port 2 Schematic of terminated asymmetric coupled-pair transmission line

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33 desired differential s ignal between the line s of the coupled-pair, as well as the common signal referenced to ground. Figure 3-2 is a diagram of s uch a coupled-pair transmis s ion line, with all pertinent voltages and currents denoted. Also s hown in Figure 3-2 is a repre se ntation of a termination for the coupled-pair line Subject to the s implifying assump tions, the mathematical re s ult s of thi s chapter are applicable to any pair of conductors with a nearby ground conductor. Referring again to Figure 3-2, the behavior of the coupled-line pair can be described by [8] dv 1 . dx = -( z l z 1 + z m z 2) dv 2 . dx = -(Z2l2 + Zm' l) di 1 dx = -(y 1 v l + Y rri v2) di 2 dx (3-12) where z 1 and z 2 are self-impedances per unit length; y 1 and y 2 are admittances per unit length ; and Zm and y 01 are mutual impedance and admittance per unit length respectively Also, a harmonic time dependence ( i.e. jrot) i s assumed. The solution to the se t of equations (312 ) as publi s hed by Tripathi [8] is given as y cx Y cx YrcX Yrcx v 1 = A 1 e +A 2 e +A 3 e +A 4 e y c x Y cX Yrcx YrcX v 2 = A 1 Rce +A 2 R ce +A 3 Rrc e +A 4 Rrc e A1 Y X A2 "( x A 3 Y X A4 Yrcx (3-13) l l e C e c + e 1t zcl e 2 c2 z1tl 2 rc2 A I R e "{ x A2Rc Y x A3R1t Y X A4R7t y X l = e C e C + e 7t e 7t 2 c2 2 rcl 2 z c l z1t2

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34 where A 1 and A 3 represent the phasor coefficients for the forward (positive x) propagating c and re-modes, respectively, and A 2 and A 4 represent the phasor coefficients for the reverse (negative x) propagating c and n mode s, respectively. The characteristic imped ance of the c-modes are represented by Zc 1 and Zc 2 for lines A and B, respectively, and the characteristic impedance of the re-mode s are represented by Z1tl and Z7t 2 for lines A and B, Y 1 Z 1 + Y2Z2 'Y; ,1t = 2 + y mZ,n (3-14) ~J(Y1Z1 -Y2Z2) 2 + 4 ( Z 1 Ym + Y2Zm )(Z2Ym + Y Each voltage/current pair at each node represent a single propagating signal referenced to the ground potential. These s ignals will be called nodal waves. A practical simplification in the development of mixed-modes-parameter theory is to assume symmetric coupled pairs (i.e. lines A and B have equal width) as reference transmission lines. This assumption allows simple mathematical formulations of mixed mode s-parameters. Furthermore, this assumption is not overly limiting since reference line s may be made arbitrarily short. For symmetrical lines in (3-13) R e = l and R1t = -1 and the c and the re-modes beco1ne the even and odd modes respectively, as first used by Cohn [6]. For notational purpose s, we shall use the substitutions c e and o for even-mode and odd-mode, re s pectively With these substitutions, the mode characteristic impedances and propagation constants become = Z 2 = Z C e =Z 2 =Z 1t 0 (3-15) Expressing (3-13) in the symmetric case

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35 Y X y X -Y X y X vi = A1e e +A2e e +A3e o +A 4e o yex 'Y eX Y oX YoX v 2 = A 1 e +A 2 e -A 3 e -A 4 e A 1 Y x A2 "{ X A 3 Y X A4 Y X il = -e e -e e + -e o -e o z e ze zo zo A1 -Y x A2 y x A3 y x A4 y x i2 = -e e -e e -e o + -e o z e ze z o zo (3-16) As before, these voltage/current pairs are nodal waves at each terminal that are referenced to ground. Expressing the differential and common-mode va lu es ( 3-1) th rough (3-4) in terms of the line voltages and currents (3-16) (3-17) Recall that A 1 and A 2 are the forward and reverse phaser coefficient for the even-mode propagation, and A 3 and A 4 are the forward and reverse phaser coefficient for the odd mode propagation. If a short hand notation is introduced, a better understanding of these definitions can be had Let A A4 y x Y X yx +c ) 3 -YoX z 0 x = z e i;(x) = Z e 0 v;(x) = A 3 e 0 v;(x) =A 4 e 0 0 0 Al -Y X A2 y x (3-18) Y X yx v;(x) = A 1 e e ve(x)=A 2 e e i;(x) = z e e i ;(x) = z e e e e

PAGE 45

Then ( 3-16) become s and (317 ) become s 36 + + v 2 = ve(x) + v e(x)-v 0 (x)-v 0 (x) i 1 = i ;(x) i e(x) + i ;(x) i 0 (x) i 2 = i; (x) i ~(x) i; (x) + i 0 (x) Note that, in general Z 0 :/=. Z e (319 ) ( 3-20) Characteristic impedance s of each mode ca n be defined as the ratio of the voltage to c urrent of the appropriate mode s at any point, x, along the lin e. These impedance s can be exp re sse d in te1m s of the eve n and odd-mode (g round referenced ) characteristic impedances vd(x) z =-= d ij (x) 2v ;(x) v;(x)IZ 0 = 2Z 0 v;(x) z e = ---(2v;(x))/ Z e 2 (3 21) (3-22)

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37 The se relation s betw ee n the even/odd mode characteristic impedances and the differenti al/ common mode characteristic impedances are consistent with the matched load ter1r1ination s di sc u sse d in th e literature [9 1 O]. 3 .1 1 .3. Mixed Mod e Scattering Param e t e r s Now that voltage s, currents, and characteristic impedance s have been defined for both differential and common mode s, the normalized power waves can be developed By the definition for a generalized power wave at then-th port [23, 24] 1 [ z ] a = --=== v +z n 2J'Re(Zn) n n n b = l [ v i Z ] n 2 J'Re(Z n) n n n (3-23) where an i s the norm a lized wave propagating in the forward ( po s itive x) dir ec tion ; bn i s the normalized wave propagating in the rever se ( negative x) direction ; and Zn is the char acteri s tic impedance of the port With the above definition s, the differential normalized waves be co me at port one (3 24) Similarly define the co mmon mod e n or m a lized waves at port one, as (3-25)

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38 Analogous definition s at port two can easily be found by s etting x = x 2 Imposing the condition of low-loss transmission lines on the coupled-pair of Figure 3-1, the characteristic impedances are approximately purely real [23]. Under thi s restriction, Za Re{Za} = Ra and Z c RefZ c J = R e With this assumption, the normalized wave equations at port one can be s implified (3-26) acl 2 R C X = X I (3-27) b c l 2 R C x = x 1 With the normalized power waves defined the development of mixed-mode sparameters is straight forward The definition of generalized sparameters (24, 23] i s b = Sa ( 3-28 ) where the bar over the lower-case letter s denote an n-dimensional column vector and the bold upper -c ase letter an n-by-n matrix Given a coupled-line two-port like Figure 3-1, or any arbitrary mixed-mode two-port the generalized mixed-mode s-parameters can be described by bdl = sddll adl + sdd12ad2 + sd c l I acl + sdc12ac2 b d2 = s dd 21 adl + sdd 22a d 2 + sdc21 acl + sdc22ac2 b cl = scdl 1 adl + scd12ad2 + Seel 1 acl + sccl2ac2 b c2 = scd21 adl + scd22ad2 + s cc2 1 a c l + scc22ac2 (3 29)

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39 Each parameter has the notation s r n m p o. = s (o utput m o d e)( input m o d e)(o utput p o rt )( input p o rt ) 0 l O' l ( 33 0 ) to indicate the mode s and port s of the signal path which the parameter represents. The dif ferentia] and common-modes are denoted by a sub s cript d and c re s pectively, and the port s are denoted by there port number in this ca s e one and two. The set of equation s in ( 3 29 ) can be expre ss ed as a partitioned matrix bdl ad l bd 2 I SddlSd c ad 2 (3-31 ) -,b c l S c d S ec a c l b c2 a c2 The following names are used: Sctd are the differential s -parameters S ec the com mon-modes-parameter s and Sd c and S e ct the mode-conversion or cro ss -mode s-parame ter s In particular, Sct c de s cribe s the conversion of common-mode waves into differential mode waves and S c d de s cribe s the conversion of differential mode waves into common mode wave s These four partition s are analogues to fou1 tran s fer gain s (A ce Add, Acct Ad e) introduced by Middlebrook [19]. These mixed-mode two-ports-parameters can be shown graphically (see Figure 3-3 ) as a traditional four-port. It must be remembered, however, that the ports are conceptual tools only, and not phy s ically separate port s 3.1.2 Choice of Reference Impedances for Multiple Modes If one is to make a general purpo s e RF mea s urement port, the value s of character i s tic port imp e dance s mu s t be cho s en It is useful to require the even and odd-mode char acteri s tic impedance s of the measurement sy s tem to be equa l thus reducing the number of different valued matched termination s requir e d. In contrast, it i s difficult to fabricate

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40 Physical Mixed-Mode Physical Pqrt 1 Port 2 Two-Port I I I I 1 Sdd21 1 I bd 2 adl I Sddl Sdd22 I bdl I Sdd1 2 l ad2 I 1 <.I' 1 I C'e7 I I ot-1< I 1 S cc21 l I ac hc2 Seel I S cc22 I Sc c12 1 1 I ac2 I I I I Figure 3-3. Sign a l flow diagram of mixed-mode two-po rt network. accurate lump ed terrnination standards for co upl ed lin es where Ze does not equal Z 0 I f the c h aracteristic impeda n ces of the lin es are defined to be equal (say, 50 .Q ), then a f u rther s implification of the above expressions can be accompli s hed with the s ub stitution Ze = Z 0 = z 0 where in the low-loss case Z 0 = R e[Z 0 } = R 0 B y choosing equal even and oddm ode c h aracteristic impedances, one is se lecting a specia l case of coupled transmission lin e behavior, as described in (3-12). Enforcing equal even and odd-mode c h aracteristic impedance s i s eq ui va l ent to the conditions of uncoupled tran smissio n lin es. As ha s been shown in the literature [9], the condition Ze = Z 0 re s ults in the mutual impedances and a dmitt ances being zero (zm =O Ym=O). Under these co nditi ons, the descri bin g differential equa ti ons of the transmission lin e system

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41 (3-12) clearly become uncoupled, resulting in two independent transmission line solu tions. Although very specific this is a valid solution to (3 -12), and all results up to this point are also valid under the special case of equal even and odd-mode characteristic impedances. Therefore, we choose the reference lines of the mixed-mode s-parameters to be uncoupled transmission lines. The key to this choice is that these uncoupled reference lines can be easily interfaced with a coupled line system, as discu sse d below To interpret the meaning of uncoupled reference transmission lines consider a sys tem of transmission line s: one coupled pair and one uncoupled pair connected in s eries If even and odd (or c and 7t) modes are both propagating (fo rward and reverse) on the cou pled pair then it can be s hown that the waves propagating on each of the uncoupled trans mission line s are linear combinations of the waves propagating on the coupled syste m (see Appendix C). Furthermore, the differential and common-mode normalized waves of the coupled pair sys tem can be reconstructed from the normalized waves at a point on the uncoupled line pairs (see Appendix C). Thi s point of reconstruction is arbitrary, and one may choose the point to be the interface between the coupled sys tem and the uncoupled refer e nce line s. It it interesting to note that an alternative requirement can be found through which the nodal and mixed -mo de waves can be related. One could require the differential-mode and common-mode characteristic impedances to be equal (i.e. Zct = Z c = Z 0 ). The rela tion s hip between mixed mode and s tandard sparameters ( discussed in the next section) will change, however This alternate requirement may have value in some cases, but the original requirement (Ze = Z 0 = Z 0 ) best relate s mixed-mode s-parameters to standard parameters

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42 3.1 3. R e l ations hip of Mixed-Mode and Standard S-Parameters The most straightforward means of implementing a mixed-mode s-parameter mea sureme nt system is to directly apply differential and common-mode waves while measur ing the resulting differential and common-mode waves. Unfortunately, the generation and measurement of these modes of propagation is not easily achievable with standard vector network analyzers (VNA) However under certain conditions, one can relate the total nodal waves (eac h representing two modes of propagation) to the desired differential and common-mode waves. These nodal waves are readily generated and mea s ured with stan dard VNAs, and with consideration, the differential and common-mode waves, and hence the mixed-modes-parameters, can be calculated. Therefore, the relationships between the normalized mixed-mode waves (act 1, hct 1 ac 1 b c 1 etc.) and the nodal waves (a 1 b 1 a 2 b 2 etc.) will be derived, and the necessary conditions for these relationships to exist will be found. To begin the development of the relationship between the nodal and mixed-mode normalized power waves, the normalized differential-mode incident wave at mixed-mode port one, ad I, will be expressed in ter1ns of the normalized si ngle-ended (noda l ) power waves at port one, a 1 and at port two, a 2 First, the normalized nodal waves of the cou pled lines at the interface are defined with z 0 R 0 as (3-32)

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43 where a i and b i are the normalized forward and rever s e propagating nodal wave s at node i, re s pectively, and i E { 1 2 3 4}. Next the definition of the normalized differential-mode incident wave at mix e d-mode port one ad 1 will be repeated ( 3 -3 3 ) R e callin g th a t the diff e rential volta g e and current at port one are defined through ( 3-1 ) and (32 ) a s v d 1 (x) = v 1 (x)-v 2 (x) 1 id 1 (x) = 2 ( i 1 (x)i 2 (x)) (3 34) and that the differential reference characteri s tic impedance i s defined in ( 3-21 ), with th e s ub s titution Z e = Z 0 = Z 0 ::::;:; R o, as (3-3 5 ) then ( 3 -3 3 ) can be re written as 1 ( 33 6 ) By applyin g the definition of normali z ed waves at port one and two ( 3-32 ) then ( 3-36 ) b e come s s imply

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44 (3-37) This equation ha s a meaningful analogy with the differential voltage and current defini tions. Similarly, the differential and common-mode waves a port one are 1 1 adl Ji(al a2) acl Ji(al + a2) (3-38) I l bc1 bdl = Ji(b1-b2) Ji(b1 +b 2) Similarly for port two 1 ad2 = Ji(a3 a4) I b d2 = Ji(b3 -b4) ac2 = (3-39) Equations (3-38) and (3-39) represent important relation ships from which mixed mode s-parameters can be deterrnined with a practical measurement system. By using the definition of s-parameters [23] for a four port network together with the relations in (3-38) and (3-39), a transfor1r1ation between mixed-mode and standard s-parame ter s can be found The transformation can be developed by considering the rela tionships between the sta ndard and mixed-mode incident waves, a, which can be written adl 1 1 0 0 al ad2 1 0 0 1 -1 a2 (3-40) Ji 1 1 0 0 acl a3 a c2 0 0 1 1 a4 or, compactly mm st d a = Ma (3-41)

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45 where amm and cz8td are the mixed-mode a-waves vectors, respectively, and M= 1 -1 0 0 1 0 0 Ji 1 1 1 -1 0 0 0 0 1 1 Similarly, for the response waves, b, it is found b mm= Mb std Applying the generalized definition of s-parameters from (3-28), it can be shown (3-42) (3-43) (3-44) where smrn are the mixed-modes-parameters, s st d are the standard four-ports-parameters. The transformation in (3-44) gives additional insight into the nature of mixed mode s-parameters. The transformation is a similarity transformation, which indicates that a change of basis has occurred between standard and mixed-mode s-parameters. Con ceptually, the nodal currents and voltages correspond to the basis of standard four-port s-parameters, and the modal currents and voltages of (3-1) to (3-4) correspond to the basis of mixed-mode s-parameters. (precisely what is meant by a basis of an s-parameter repre sentation will be explored in Section 3.2). The transformation (3-44) also gives information into the nature of the chosen mode-specific aand b-waves. It is easily demonstrated that the operator M has the prop erty M -I = MT (where the superscript T indicates the matrix transpose operator). This indicates that the M operator is a unitary (also called orthonormal) operator (31]. This can be easily demonstrated by applying the definition of a unitary operator M(M*)T = I (3-45)

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46 where indicates the complex conjugate. A unitary transformation is one that transforn1s one orthonormal bases to another orthonormal bases. If it is accepted (until Section 3.2 where it can be established) that standard four-port s-parameters are operators in an or t honormal basis, then it fallows from (3-45) that the definitions of the differential and common-mode nor1naJized power waves must also represent an orthonor1r1al basis. This is yet another indication that the mode currents and voltages in (3-1) to (3-4) provide a se l f-consistent framework for power calculations. Further, it indicates clearly that the two sets of s-parameters are different represen tations of the same device, and that, ideal l y, the two representations contain the same information about the device. However, it will be shown in Section 5.2 that transfor1na tion according to (3-44) of measured data from practical measurement systems (with mea surement errors) can lead to significant errors in the transformed data. 3.1.4. Interpretations of Multi-Mode Scattering Parameters Equations (3-26) and (3-27) fortn the basis of an ideal mixed-mode s-parameter measurement system These equations can be imple1nented into a microwave simulator and can provide a quick and simple method of illustrating the usefulness of mixed-modes parameters The circuit in Figure 3-4 was imp l emented into Hewlett-Packard's Microwave Design System (MDS) [32]. The phase difference, 0, between the two sources was set to 0 for the common-mode and common-to-differential-mode forward s-parameters. For the forward differential-mode and differential-to-common-mode s-parameters, the phase difference was set to 180 In each case, the nodal waves were calculated from (3-26) and (3-27), and the s-parameters were calculated with the appropriate ratios. The reverses

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Ang=0 Mag=l V + Zo Ang=0 Mag=l V VJ V 2 47 Length= 1 inch h=25 mils er=9.6 V 3 Width=lOO m Zo , , , , , , , l Space=lOOm I I 1 : Zo V4 Width=lOO m Port 1 Port 2 Z4 Figure 3-4. Schematic of mixed-mode simulation of symmetric coupled pair line parameter s were calculated by driving mixed-mode port two of the DUT, with 50Q load s at port one The first example of mixed-mode s-parameters uses a DUT of a pair of coupled micro s trip transmission lines, with symmetric ( i.e. equal width ) top conductor s This s ymmetric coupled pair and the accompanying circuitry, is shown in Figure 3-4. Each runner width is 1 OOm with an edge-to-edge spacing of 1 OOm. The s ubstrate is 25 mil thick alumina with a relative permittivity of 9.6 with a loss tangent of 0.001, and the metal conductivity i s that of copper ~ 5 8x10 7 Sim. A one inch s ection of this line wa s s imu lated in MDS as de s cribed above and the mixed mod esparameter s at 5 GH z are ( 3 46 ) 0.001 L-141 0.972L9.53 0 0 0.972L9 53 0.001 L141 0 0 0 0 0 341 L 60 4 0 915 L26.4 0 0 0 915 L26 4 0 341 L-60 4

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48 As expected, each partitioned sub-matrix demonstrates the properties of a reciprocal, pa s sive and (port) symmetric DUT. The differential s-parameters, Sctct show the coupled pair possesses an odd-mode characteristic impedance of 50.Q ( 100.Q differential imped ance), and has lo w-loss propagation in the differential mode. The common-mode s-parameters, S ec, show the coupled pair possesses an even-mode characteristic imped ance other than 50.Q. Actually, the even-mode impedance of the pair is 140.Q (70.Q com mon-mode impedance). Note the cross-modes-parameters are zero for the symmetric coupled pair indicating no conversion between propagation modes. The second example i s s imilar to the first, except the coupled microstrip transmis sion lines are asymmetric (i.e. unequal widths). This asymmetric coupled-pair, and the accompanying circuitry, is shown in Figure 3-5. One top conductor width is IOOm, and the second is 170m with an edge-to-edge spacing of 65m. Again, the substrate i s 25 mil thick alumina with a relative permittivity of 9.6 with a loss tangent of 0.001, and the metal conductivity is that of copper. A one-inch section of this line was simulated in MDS at 5 GHz, and the mixed-modes-parameters are Ang=0 Length= 1 inch h=25 mil Mag=l V er=9.6 s V1 V3 Width=IOO m + Zo Zo : : I , : j Space=65m I + Zo Zo 2 V2 V4 Width= 170m lz Ang=0 l4 Port 1 Port 2 Mag=l V Figure 3-5. Schematic of mixed-mode simulation of asymmetric coupled-pair line

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49 0.003L-175 0.956Ll.819 I O.OOSL-177 0.031L80.7 0 .956L l .8 19 0 003 L175 I 0.031L80.7 O.OOSL 177 -----------------------0 00SL 177 0 0 .0 31 L80 .7 I 0.502L48.0 0.844L-40 2 I 0.031 L80.7 O.OOSL-177 I 0 .844L-40.2 0.502L48.0 (3-47) As in the first exa mple each partitioned submatrix demonstrates the properties of a reciprocal, passive and (po rt ) symmetric DUT. Also like the first example, the differen tial s-pa rameter s s how the coupled pair possesses an odd-mode characteristic impedance of nearly 50Q (actually 49Q) and has low-los s propagation in the differential mode. The common-mode s-parame ters show the coupled pair has a greater degree of mismatch than the frrst example (the even-mode impedance is l 52Q in this case). The most important difference between the two examples is seen in the cross mode s-parameters. The data in ( 3-47 ) shows signif icant conversion between propagation mod es, particularly in transmi ss ion parameters Sct c 2 1 and S c ct 2 1 Note these two sub-matri ces are eq ual indicating equal co nver s ion from differential to common-mode and from commo n to differential-mode. These nonzero s-paramete rs can be interpreted conceptu ally in the following way. In the case of Scct 21 a pure differential mode wave is impinging on port 1 of the DUT However, at port 2, both differential and common-mode waves exist. Some of the energy of the differential wave is converted to a common-mode propa gation, and the total energy i s preserved ( except for losses in the metal and dielectric ).

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Figure 3 6. Figure 3 7 0 I 0 \. 0 0 ...... I 1.0 50 \ Freq ( GHz ) \ 0 V) 0 V) I 2 1 0 Simulated m a gnit u d e in dB of S ct ct 2 1 and S ee 2 1 v er s us fr e quenc y for a s ym metri c coupled pair tran s mi ss ion lin e 0 0 0 0 V) I 1 0 r, I r r (\ ... Fr e q ( GH z) I "" "' (\ ,... "' l 0 0 u V'.) u 0 0 V) I 21 0 Simul a ted magn i tude in dB of Sctdl l and S ee l 1 ver s u s fr e quency for asym metri c c oupled-pair lin e

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0 0 /' I 0 0 V) I 1.0 / t I I V 51 I\ I\ (' /f I \1 'v I / \ I .I I V V Freq (G H z) 21.0 Figure 3-8 Simulated magnitude in dB of Scct 21 versus frequency for asymmetric cou pled-pair line 0 0 ./", I ] t I I /I I ., I t I \j 0 0 1.0 Freq (GHz) 21.0 Figure 3-9. S i mulated magnitude in dB of Sect 11 versus frequency for asymmetric cou pled-pair line.

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52 This example circuit was simulated across frequency, and the magnitudes of selected mixed-modes-parameters are plotted in Figt1res 3-6, 3-7, 3-8 and 3-9. Figure 3-6 shows both Sctd 2 1 and Scc 21 in dB from 1 GHz to 21 GHz. The ripple pattern across fre quency in the co mmon-mode transmission ( S cc 21 ) indicates an impedance mi s match at the ports for common-mode propagation. At the higher f1equencies of the plot the finite con dt1ctivity of the conductors is evident as average los s increase s. The differential-mode transmission (Sdd 21 ) s hows s maller 1ipple s (0.2 dB maximum ), indicating smaller mis match and also s how s lower average loss However, the losse s due to the reflections at the ports do not account for all of the ripple in the differential transmi ss ion. As can be see n in Figure 3-7, th e return lo ss for the differential mode is greater than 20 dB which can account for approximately 0.04 dB of wor s t case lo ss ( over ohmic lo ss e s). Mode con version accounts for the remaining reduction in the differential-mode, and hence Sdd 21 i s r e duc e d. Here differential energy i s converted to both common-mode transmission S cd 2 1 and common-mode reflection Scdl 1 Figure 3-8 shows the cross-mode transmission Scd 21 in dB and Figure 3-9 shows the cross-mode reflection Scdl 1 in dB. The minima in the dif ferential-mode transmission Sdd 2 1 co n es pond to a wor st case point in the relative phases of Sctct21, Scd21, and Scdl l In a low loss transmi ss ion line case, the insertion lo ss due to mode conversion and mi ssmat c h can be s hown to be approximately Loss (dB) ~lOlog 1 -( sddll 2 + IScd21 2 + scdll 2) (3 -48 ) Th i s i s consistent with the increasing ripple in Sdct 21 with increasing frequency since the mode conversion (Scct 21 and Scdl 1 ) increase s with frequency. The use of mix e d-modes-parameter s can be further illu s trated with an example of a d if ferential amplifier Such an example i s found in Appendix D

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53 3.2. Generalizations of Mode Specific Scatterin~ Parameters 3.2.1. Other modes The voltages and currents of (31) to (3-4) represent only one possible defmition of mode s. There are infinitely many such definition s with a four-port network, although the chosen set has important practical value. Furtherrnore a network with more ports can support more mode s of propagation. It i s useful to generalize the proceeding work to include all possible mode definition s as it lead s to in s ight into the nature of the mixed mode definition s pre se nted. To begin the generalization, it is helpful to establish the concept of an s-parameter matrix as a linear operator. Traditionally, an s-parameter matrix is interpreted from a phy s ical view, where the elements of the matrix represent the gain coefficients of a certain input to-output path. The operator interpretation views the s-parameter matrix as an oper ator that map s one n-dimensional vector s pace into an m dimensional space [31] (with typ ical devices, m and n are equal). With s uch an interpretation, it will be shown that the transformation to another mode definition can be regarded simply as a transformation of coordinates. al b3 b1 a1 b1 a3 b2 a2 a2 DUT b4 b3 DUT a3 b2 a4 b4 a4 (a) (b) Figure 3-10. Two views of a four-port sparameter matrix. a) The physical view. b ) The linear operator view.

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54 To illustrate the operator view of s -parameters consider the four-port example in Figure 3 10 Define basis vector s corre s ponding to each phy s ical port 1 0 0 0 0 1 0 0 ( 3 49 ) p = p = p = p = 1 0 2 0 3 I 40 0 0 0 1 One can clearly s ee that the s e vector s are linearly independent, that i s ( 3-50 ) for all po ss ible compl e x s calar s { c 1 c 2 c 3 c 4 } 3 C where C i s the s et of all complex numbers Furthermore thi s set of ba s i s vector s {p 1 p 2 p 3 p 4 } have a zero s calar prod uct that i s " p . p = l ) J 0 i j l = J ( 3 51 ) Thi s mean s that the s y s tem of basis ve c tor s is orthonormal Continuing an arbitrary s et of input signal s become s A A a = a 1P1 + a 2 P 2 + a 3 P 3 + a4P 4 ( 3-5 2) and the output s i g nal s ar e b = h1P1 + b 2 ft 2 + b 3 P 3 + bJ; 4 ( 3-53 ) With the ba s i s definitions of ( 3 49 ), the coordinates of the input and output signal s are al bl a2 b 2 b = ( 3 54 ) a = a 3 b 3 a 4 b4

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55 The traditional s-parameter matrix equation, b = Sa, can now be said to express a linear operator, S, mapping an input space to an output space. It is important to note that both the input and output spaces have the s ame basis vectors. Now, considering the sa me example, define a new set of basis vectors, {p 1 ', p 2 ', p 3 ', p 4 '} These new basis vectors can de sc ribe any arbitrary mode definition s. In the case of the differential/common-mode definitions of (3-1) to (3-4) they are (3 -55) where the new subscripts are used to clearly indicate that the new basis does not corre spond to physical network ports. Assuming linear relationships between the old and new bases they can be generally related A "f Al A l Af P2 = X21P1 + x22P2 +x 23P3 +x 24P4 P3 = X31P I 1 + X32P2' + X33P3 1 + X3~ 4 1 (3-56) An input signal vector in the new ba s is t '"' ,,.., ,,.., ,,.., a = al P 1 + a2P2 + a3 P 3 + a4 P 4 (3-57) has the coordinates in the new basi s al a' 2 a' (3-58) I a3 a 4

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56 By expressing the input vector in the origina] basis (3-52) in terms of the new basis vectors via (3-56), and then by equating the coefficients of the basis vectors, it can be shown that al I Xl l X12 X13 X14 al a' X21 X22 X23 X24 a2 2 (3-59) a' X31 X32 X33 X34 a3 3 a' 4 X41 X42 X43 X44 a4 which can be simply expressed as a' = Xa (3-60) where Xis a transformation of coordinates matrix. Therefore, the translation between dif fe r ent mode definitions is simply a transformation of coordinates. In the case of the differ ential/common-mode definitions, it can be shown that (3-60) becomes mm std a = Ma (3-61) As illustrated in (3-53), the input and output vector spaces share the same basis vectors, so the output in the new basis becomes b' = Xb (3-62) or, for differential/common-modes b mm= Mb std (3-63) The linear operator representing the DUT can be tra11slated between bases by s = xsx1 (3-64) In general, if both sets of bases are orthonormal, as defined in (3-51 ), then the transformation matrix, X, will always be unitary, that is X(X*) T = I. Conversely, if a defined transformation matrix is unitary, then both systems of basis vectors are orthonor

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57 mal [31]. With the concept of s-parameters as linear operators, one can define any number of new and potentially useful modes of propagation. 3 2.2. Eigen modes One particularly intere s ting new mode definition arising from the operator view of networks is the concept of eigen-modes. Eigen-values arise from the diagonalization of a matrix and the matrix of eigen-vectors become the transformation matrix. Symbolically, (3-6 5 ) where (3 -66 ) where Ai are the eigen-values of S and Tis a matrix whose columns are composed of the eigen-vectors of S [33] In linear system analy s i s, eigen-values represent the natural frequencies of a sys tem. When described in state space notation, the state-feedback matrix, A determines the se natural frequencies The natural frequencies, or eigen-values, are the solutions to IA/ Al = 0 (3-67) Corresponding to each eigen-value, Ai, there i s a eigen-vector, e i, such that ( A ./ A ) e = 0 l l (3-68) Physically the eigen -va lues are the complex frequencies at which the system will have (unforced) oscillations, and the eigen-vectors are the amplitude coefficients of each of the state variables under the conditions of oscillation. In contrast, the eigen-values and vectors of ans-parameter matrix do not represent system oscillations. For ans-parameter operator, the eigen-vectors represent the coeffij

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58 cients of a transforrnatjon to a new basis. The new basis fu1ther represents new modes of propagation. This new basis is special, in that it transforms the operator, S, into a diagonal matrix. For this reason, the modes corresponding to the eigen-vectors of a operator, S, will be called canonical modes. The eigen-values represent the DUT response in terms of the canonical modes In general, an n-port device will haven canonical modes. When stimulated by one of the canonical modes, the device will generate a response proportional to only the mode by which it was stimulated. There is only one port definition possible for canonical modes. Each canonical mode is formed from a linear combination of signals at all of the physical ports There are n possible canonical modes of propagation supported by a device with n physical ports. This removes any ambiguity that exists in the port number ing convention 1 The canonical representation of a device allows for very simple calculations of responses. Since the canonical forn1 of a device is a one-port (multi-mode) network, the response of the device to a canonical mode input is simply a reflection of the same canon ical mode. The canonical mode reflection has a scaling, or gain, factor that is conceptually equivalent to the traditional definition of reflection coefficients. The eigen-values of a s-parameter matrix are the canonical reflection coefficients. Furthermore, a given device generates no conver s ion between its canonical modes. As a result, the canonical represen tation can be interpreted as the natural modes of a device. It is interesting to note that eigen-values of a matrix, S, remain unchanged by a change of basis (i.e. a similarity transformation as in (3-65)) The eigen-values, therefore, I The definitions of mixed-mode s -parameters presented in Section 3.1.1 define (nodal) ports one and two as mixed-mode port one, and so on. However, any other combination of two ports could have also been chosen as a mixed-mode port.

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59 are immutable properties of ans-parameter matrix, and the canonical modes of a device are properties of the device. Eigen-vectors are not t1nique, since they need only to be inde pendent. As stated earlier, infinitely many modes (not independent) can be defined for a given network. However, the consistency of the eigen-values across all such bases indi cates the all representations of a device are leaving the essence of the device unchanged. Mixed-mode s-parameters are indeed an equivalent representation of a standard four-port s-parameter matrix. Not every device has a canonical representation. A matrix, S, is diagonalizable if and only if S has n linearly independent eigen-vectors. It can be shown [33] that S has n linearly independent eigen-vectors if S has n distinct eigen-values (the converse is not true, however). Therefore, if all eigen-values are different, then one can be assured the device has a canonical representation If some values are repeated, then the existence of a canon ical representation depends on S If an s-parameter matrix does not haven linearly independent eigen-vectors, then it is possible to find n independent generalized eigen-vectors. Under these conditions, the new operator matrix is not diagonalizable, but generally in Jordan forrr1. A Jordan form matrix has some non-zero off-diagonal elements. Such a device requiring a Jordan forrn representation will exhibit mode-conversion between some of its canonical modes. Despite this limitation, the Jordan forrn representation of ans-parameter operator can have some utility in calculations. Not every non-diagonalizable matrix has a Jordan forrn representation. In such cases, other decomposition methods are available, such as LOU-factorization [33]. These decompositions cause representations that are as close as possible to a diagonal form.

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60 This work can be extended to include these other representations of ans-parameter opera tor. With the fundamental theory of mixed-modes-parameters developed the applica tion of these concepts to practical circuits can begin. The first step in this progression is to measure the mixed-modes-parameters of an RF differential circuit These new s-parame ters require the design and construction of a specialized measurement system. The devel opment of this new system is the subject of the next chapter.

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CHAPTER4 CONSTRUCTION OF THE PURE-MODE VECTOR NETWORK ANALYZER As a result of the limitations of measuring RF differential circuits and devices with a single-mode system, as discussed in Chapter 2, a custom vector network analyzer ( VNA ) has been designed to measure mixed-mode s-parameters in the most direct and accurate fashion. The existence of a transformation between standard and mixed mode s-p arame ters, discussed in Se c tion 3.1.3, suggests two possible approaches to the measurement of differential circuits. One approach i s the use of a traditional four-port VNA. A traditional VNA would measure standard s-parameters by stimulating each terminal of the differen tial circuit individually, and theses-parameters would then be transformed to mixed-mode s-parameters for analysis. Alternatively, the mixed-modes-parameters of the differential circuit can be measured directly by stimulating each mode individually. A pure differen tial-mode stimult1s could be produced, and the differentialand common-mode response s of the DUT could be measured, thus providing a direct measurement of mixed-mode sparameter s. A network analyzer that directly measure s mixed-mode sparameter s will be refened to as a pure-mode vector network analyzer (PMVNA) due to its generation and measurement of pure single mode signals. The two approaches do not yield equally accurate mixed-modes-parameters of differential device s, however. It is shown in Chapter 5 that the PMVNA has an accuracy advantage over a traditional four-port VNA while measuring a differential circuit. Mixed mode sparameters generated by transforming standards-parameters measured by a tradi61

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62 tional four-port VNA exhibit higher levels of t1ncertainty in a differential device measure ment than those measured by a PMVNA. This accuracy advantage of a pure-mode measurement system provides motivation for the development of a specialized measure ment system for differential circuits. Portions of this chapter have been published in sum mary form [34]. 4.1. Basic Operation of the PMVNA 4.1.1. Fundamental Concepts As discussed above, the most straightforward means of implementing a mixed mode s-parameter measurement system is to directly apply differential and common-mode waves while measuring the resulting differential and common-mode waves. Unfortu nately, the generation and measurement of these modes of propagation is not easily achievable with standard vector network analyzers (VNA). However, as shown in (3-38) and (3-39), one can relate the total nodal waves to the desired differential and common mode waves. These nodal waves are readi ly generated and measured with standard VNAs, and with consideration, the differential and common-mode waves, and hence the mixed-modes-parameters, can be calcu l ated. Equations (3-38) and (3-39) represent impo1tant relationships from which a PMVNA can be constructed with compo nent s of standard single-ended VNAs. To under stand the utility of the above relationships, consider Figure 4-1, which is a conceptual model for a PMVNA system. By adjusting the phase difference, 0, between the two sources to 0 or 180 one can determine the common-mode or differential-mode forward s-parameters, respectively. Conceptually, the measured quantities are the voltages and currents. These values can be related to the normalized nodal waves, a 1 b 1 a 2 b 2 etc.,

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Ang=0 Mag=l V s + Ang=0 Mag=l V i1 Zo 63 V1 pt+ Zo DUT Zo Figure 4-1. Conceptual diagram of pure-mode measurement system. through the generalized definitions given in (3-32). From these nodal waves, the differen tial and common-mode normalized waves, and, hence the mixed-modes-parameters, can be calculated Phy sica lly the various ratios of nodal waves, a 1 b 1 a 2 b 2 etc., are mea sured, and from the ses ratios the mixed mode s-parameters are found. 4.1.2 General PMVNA Test-Set Architecture The physical implementation of a mixed-modes-parameter measurement system can be achieved with extensions of sta ndard VNA techniques. The differential stimulus of a coupled two-port requires the input waves at the reference plane to be 180 apart. One possible way thi s can be achieved through a single signal source is with the u se of a 180 3dB hybrid splitter/combiner. The construction of the differential reflected and transmit ted waves, via (3-38) and (3-39), can be also completed through a 180 splitter/combiner. The common-mode stim ulus of a coupled two-port req11ire s the input waves at the refer ence plane to be 0 apart. This can also be achieved through a s ingle signal source with the use of a 0 3dB hybrid splitter/combiner, with the construction of the common-mode reflected and transmitted waves also completed through a 0 splitter/combiner.

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64 A VNA testse t is the portion of the test system that generates the nonnalized power waves, a and b. A typical test-set uses directional couplers to separate the forward and reverse waves. A test -s et also samples the stimulus signal, either with a directional coupler or a power sp litter The testset generally down-mixes all signal to an intermedi ate frequency (IF), so that all RF functions of the VNA ( other than the RF signal so urce ) are contained within the testse t. A test-set also provides RF switches to allow automated measurement of all sparameters of the DUT with a single connection. A basic pure mode test-set is shown in part in Figure 4-2 The figure includes mechanisms by which all of the mixed-mode wave components are generated. Not shown are the down mixers and the rest of the VNA system, which are discussed in Section 4.2.1 and Appendix E. When switch one (denoted as SWI) is in position one, the 3dB hybrid coupler, HI, splits the RF signal into two signals with nominally equal amplitudes and 180 phase difference, thus generating the differential-mode RF stimulus signal Note that all switches have their unused ports terminated in 50Q loads in all cases By placing SWI in position two, the coupler, Hl again splits the RF signal into two signals, in thi s case with nominally equal amplitudes and 0 phase difference, thus generating the com mon-mode RF stimulus signal. Switches SW2 and SW3, which operate in concert, pro vide the means to stimulate either mixed-mode port one or two Directional couplers Dl, D2, D3, and D4 separate all forward and reverse signals at each single-ended port (i.e. nodal waves). These nodal waves are combined, in accordance to (3-3 8) and (3-39), in 3dB hybrid couplers H2, H3 H4, H5 each providing a (nominal) sum and difference between the corresponding nodal waves. The output of these couplers are proportional to the differential and common-mode normalized power waves (act 1 ac 1 hct 1 hc 1 etc.).

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To D ow n Mixe r s b c2 bd2 a c2 ad 2 3 dB Hybr i d Sp l itfe1/ Combi n er 0 -c H 0 :J 0 0 <] 00 ,...... 0 H 0 ... 0 <] 0 00 S W 2 3 dB H(eb r id 2 Split er/ ... ... RF Source Comb in er l 0 /' "' 2 1.r[ H 0 0'v I 1: 0 I .... .. <] 0 I"' "L!: ..... \... I 00 u,...... SW3 SWl HI 2 . '"' .. 1 II To Down 3 dB H(ebr i d Mixers Sp li t er/ Combiner adl 0 H 0 ac1 0 <] 0 00 ... bdl 0 H 0 c l 0 0 <] 00 6 5 H4 HS I .i.ij --.,.-, / _,., .-,.I ) I -NII I ... ,, -rr-r/ Variable Atte n uator ( x4) H2 H 3 Figure 4-2. RF Section of bas i c tests et of PMVNA. : ) >< t : D1 Mi M xed ode rt 2 Po \ : J>< t : D4 Mi xed ode ort I M p From the appropriate ratios of the se power waves, the mixed-modes-parameter s can be calct1lated. 4.2. Implementation of a Practica l PMVNA Rather than build an entire PMVNA f r om e l ementary compo n ents (such as direc tional couplers and mixers), a more practica l app r oac h h as bee n fol l owed by modifyi n g a

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66 standard VNA. As will be discussed below, a PMVNA can be constructed in a s traight forward manner by adapting a modular Hewlett-Packard 8510 VNA system. First, a sys tem-level description of the PMVNA as implemented for this work, will be given. Fol lowing this a detailed description of the PMVNA test-set will be given. Next, the operation of the PMVNA will be detailed, and the control so ftware will then be described. 4.2 1. System Level Description The construction of the PMVNA is ba se d the Hewlett-Packard 8510C VNA sys tem. The complete block diagram of the implemented sys tem is shown in Figure 4-3. The basic idea behind the implemented PMVNA i s to 11 se the sub-systems of a s tandard 8510 ( each contained as a si ngle piece of test equipment) in a nons tandard configuration with little or no modification to the individual sub-systems. The s ub-sy s tems (8 5101 85102, 8517, 85651 etc.) are s hown in Figure 4-3 For a description of these sub-systems and the s tandard 8510 configuration, see Appendix E. Ba s ically the PMVNA i s an 8510 VNA with two testse ts where both te s tse ts are u sed simultaneously. The implementation of a PMVNA with an HP8510 VNA require s the addition of a second 8517 te s t -se t to supply all required RF hardware Some additional control hardware, and s ome minor modification s to the 8517 test sets are also needed, as will be described below The flexibility of the 8510 VNA sys tem greatly facilitate s the implementati o n of a PMVNA. One important feature of the 8510 i s exploited in order to reduce the complex ity of the control softwa re and hardware in the adaptatjon to the PMVNA. The feature known as Option 001, a llows selection between multiple test -se ts. The option is actually an additional circuit board for switching IF s ignal s which is in s talled in one of the two

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67 t es ts et s. The board work s in c oordin a tion with features of the 8510 operatin g s y s t e m (s tandard f1ID1ware o f th e 8510 ). Th e operating sy s tem o f the s tandard 8510 allow s th e se l e ction of a te s tse t to be accompli s hed s imply by changing the addre ss of the a c tiv e te s t -s et ( c ontained in a register in the 85101 ) to th e addres s of the desired tests et The addre ss of the ac tiv e te s ts et c an b e se t throu g h s tandard g eneral purpo s e int e rf a ce bu s ,( ~ )( 85101 GPIB I F. Bu s 8510 2 TS A Enable I I 7 3488A opt014 I ,------------+-" I,_____, GPIB Sy s tem TS B En a ble ( 1 ) opt 001 ( 1 ) L .J 85 1 7 A T e s tSet A --(2) LJ (/.) 180 0 ..... u 3 dB SP. l it t e 'a) Com1:5ine r (/.) L I J I J : = s w it c h N ,dri ve r L .J --------~ 85651 (2) ( 1 ) L ..J RF Input 851 7 A T es t-Set B (3) ( 2 ) L .J ( 4 ) Mi x ed-Mode Port 1 Mixed Mod e Port 2 Fi g ur e 4 3 PMVNA s y s t e m block dia g r a m.

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68 (GPIB) commands. The availability of the test-set selection function to GPIB commands enables high-level control of the sub-systems in the PMVNA. The PMVNA system also requires some minor modifications to the control hard ware of the test-sets. As developed by HP, Option 001 allows the selection of one active te s t-set, and the deactivation of all other te s t-sets. This deactivation includes the moving of the RF port selection switch ( internal to the test-set ) to a terminated po s ition, so that no RF signal is present at the ports of the deselected test-sets. A l so upon deactivation the variable attenuators in a test-set (used to control the incident power on a DUT) are re-set to O dB. The suppressing of the RF signal from the inactive test-sets and the change of attenuation setting are 11nwanted side effects. The modification to both test-sets is needed to allow RF to continue at the ports of inactive te s t-sets and to keep the attenuator settings unchanged The modification requires a minor change to the test-set digital control hard ware to allow the masking of command s to change the position of the RF port selector sw itch or attenuator s The masking of sys tem commands is achieved through a single dig ital control signal for each test-set. When the signal, called test-set enable, is asserted, the test set can receive sys tem commands effecting RF switch and attenuators; otherwise, these system commands are blocked ( other system commands are unaffected by the modi fication) With these changes, the option 00 I can now be used to multiplex the two test sets while maintaining an uninterrupted RF signal at the ports For complete detail s of the te s t-set modification s, see Appendix F. These hardware changes are implemented to block unwanted system commands from the 85 IO operating system as the active test-set is changed. An alternative to these hardware modifications i s to change the operating system. Such a change to allow

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69 switching between tests et s without changing the RF switch position or attenuator s et tin gs, i s ce rtainly po ss ible and quite attractive s ince it would eliminate the need for any modification of the te s t equipment of th e 8510 system. The option of modifying the 8510 operation system i s unavailable, however, a s it is proprietary property of Hewlett-Packard Due to the unavailability of the operating s y s tem so ftware the hardware modification s have been performed A single 3dB hybrid 180 0 s plitter/combiner is added to the s tandard 8510 con figuration. This s plitter generates the two RF s i g nal s needed to operate both te s t -se t s s imultaneou s ly The u se of a 180 /0 sp litter allows for the generation of both diff e rential and co mmon mod e st imuli. An RF s witch i s r e quired to se l e ct b e tween the two mode s, and a driver for the sw itch i s required to allow automatic control. The s witch driver and test -se t enable control line s are interfaced to a GPIB controllable digital switches (3 488A with option 014 ) With thi s switch controller, the PMVNA can be completely automated. 4 2.2. Test Set Construction One of the mo s t u se ful aspects of a PMVNA implemented a s s hown in Figure 4-3 i s the s traight-forward manner by which the differential and common-mode normalized power wave s can be d e rived from the nodal power waves Referring first to the ba s ic PMVNA te s ts et of Fi g ure 4-2, one ca n see that the calculation of the modal normalized waves i s a c c ompli s h ed through four 180 / 0 splitter/combiners. The calculation i s done at RF with real ( non ideal ) component s, and s o i s s ubject to errors (see Chapt e r 8 ). A more practical and accurate method of constructing the differential and common-mode re spo n ses i s through digital calculation of (3-38) and (339 ). Thi s technique exploits the ar c hit ec h1re of the s tandard 8510, which down mixe s and digitize s the normalized power

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70 waves. Once the nodal waves are digitized, t h e differen t ial and common-mode normal ized power waves can be simply calculated in t h e control software. In the PMVNA implemented for thi s work, the calcu l ations of the nor111alized power waves are accomplished by using two standard two-port test-sets. The connection of this simplified PMVNA test-set is shown in Figu r e 4-4, which includes two standard (single-ended) 8517 As-parameter test-sets. These test-sets have all required RF circuitry to separate the different waves, and al l of t h e dow n converter circuitry No modification s to the RF portions of these test-sets are needed. To IF D etec t o r s ( HP85101 ) __ ___ ___ __,-~ b2~""""''~' ----.i===::;;;'.i;'I II Y' ;><"' I a2 ,";,, u~J--+----, S W ~ B ,...., RF in :, --;,y~ p ort 2 HP8517A T es t -Set B l\ cn tl l --1vco......_ a I )C''i----+---' 3 dB H ybrid 'u_ D Ort l II RF Source Splitter/ bl ~~. ><: c .. i1hiner '-----t.r-<.u25Yr -E= ~ J, 2 _-+,, +-t'" t> r-.. r .... U .. V 1-1-h ,, ,Iii \. ,IL '1 w ,._ M I ;;; Jo 0 0 .r 0 SWl a3 I b2 I a2 ,c, ....... ____. __ SWA RF in cntl l --1v co......_ ; al ~,- --+-__. >< "" port 2 HP8 5 1 7A Test-Set A port I ---------fi-.,..::....: b 1 ;___-<:~?r-----t ;><===:!::ti To IF Detectors ( HP85101 ) Figure 4-4. RF section of simplified PMVNA test-set port 4 Mix e d M o de P o rt 2 p o rt 3 port 2 '------1 Mixed-Mode P o rt 1 port 4

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71 A signif icant adva ntag e of thi s test-set configuration is its symmet ry. If the two te st-se t s are the same model (as they are for this work) then the RF paths of the PMVNA are well balanced. When the PMVNA is se t to forward differential-mode, for example, both te s tsets have the sa me s witch configuration. The two RF paths that comprise th e dif ferential s ignal (o ne through te s tset A, the other through te s t -se t B ) are identical (wit hin manufacturing tolerances ), and thus the pha se and magnitude balance between the two path s is good. If the paths are poorly balanced then high level s of mode -co nver s i o n will be ge n era t e d in the PMVNA Good balance i s r e quired to hav e s ufficient raw dynamic ran ge for accurate measurements. Again, the amount of tolerable imbalance mu s t be determined by experience. The raw performan ce of the implemented PMVNA i s exa m ined in Section 6.2.1 Thi s simplified testse t configuration has one significant disadvantage namely the u se of two independ e nt voltage co ntrolled ocsillators ( VCO s). Referrin g to Figure 4-4, one ca n see that each te s t -set co ntain s a VCO. Durin g mea s urements thi s VCO i s pha se lo cke d to the RF input sig nal of the test-set (fo r d e tail s, see Appendix E). This VCO ge n era te s a s ignal the drive s all four down-mixers in the te s t -se t. As all mixer s in a si n g le test-set are driven b y the s ame VCO the pha se r e lation s hip between the down-mixed a and b s ignal s remain s the sa m e as it was at RF How eve r as the PMVNA swi tch es betw ee n the two t est-se t s, the pha se relation s hip betwee11 the VCO s of the two te s t -se t s is unknown. As a result the straight-forward application of the measured power wave data will r es ult in s ignifi ca nt errors. Thi s di sa dvanta ge ca n be removed however through a pr e-ca libration pro cess that characterizes the pha se offset between the two VCOs. Thi s pro cess i s detailed in Section 6.3

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72 Only the 180/0 splitter/combiner and the RF switch for the source (SWl) limit the bandwidth of the simplified PMVNA. The 8517 A test sets operate from 45 MHz to 50 GHz, and with relaxed requirements on the 180 /0 splitter/com bin er, accurate measure ments are possible from about 100 MHz to above 25 GHz with one hybrid [35]. A second hybrid allows accurate meast1rements from 45 MHz to above 5 GHz (36]. The factor lim iting the frequency range of any splitter is the amount of imbalance that is tolerable in a PMVNA system. This imbalance leads to non-ideal mode generation, as will be shown in Chapter 8. This non-ideal mode generation can be tolerated and corrected through cali bration (see Chapter 6) but only to a point. At some level of imbalance, the corrected dynamic range of one or more of the mixed-mode s-parameters becomes unacceptable. The frequency at which the level of imbalance is unacceptable generally occurs beyond the specified operation frequencies of the splitter (splitter frequency specifications are linked to specified levels of phase and magnitude imbalance), but the exact level of tolera ble imbalance is usually found through experience. With this PMVNA configuration, all mode responses, including mode conversion, can be measured. With all responses available, very accurate, repeatable calibrations and measurements are possible. Additionally, with the use of standard, readily available mea surement equipment, the PMVNA can be easily and economically duplicated. 4.2.3. Detailed Operation This section details the theory of operation of the PMVNA. The operation is pre sented as a sequence of high-level events that affect the measurement of a DUT by the analyzer. This discussion is meant to clarify the way raw data is collected and manipu lated in the measurement of raw mixed-mode s-parameters. In general each event

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73 described in this section is comprised of many more elementary events which are not described here. The referenced elementary events are perforrned by the control software of the PMVNA which has been developed solely for this work. For more details on the PMVNA control software, see Section 4.2.4. Furthermore there is a level of operation of the sub-systems that is even more fundamental These low-level events, such as the lock ing of the main phase-lock loop are accomplished by the operating system of the 8510 system, and are transparent to the PMVNA control software This most basic level of operation is not de scri bed here, but can be found in 8510 documentation [43]. This section details only the mea s urement operation of the PMVNA Thi s opera tion i s the foundation of the general operation of the PMVNA, and the output of thi s oper ation i s raw ( uncorrected) mixed-mode s-pararne ters of a DUT. Optionally, thi s operation can produce s tandard four-port raw sparameter s diIectly (i n contrast to tran sfo 1mation of mixed-modes-parameters). The ca libration and s ubsequent error correction procedures, and all other functions of the PMVNA are detailed in Section 4.2.4. The basic operation of the PMVNA measures the differential and common-mode responses of a DUT to both a differential and a common-mode stimulus Referring to the flow diagram in Figure 4-5, the PMVNA first measures the DUT with a differential stimu lus, which is accomplished by setting SWl to position one (see Figure 4-3 ). Forward operation of the DUT is measured by set ting the RF port se lection sw itches of both test sets into forward position. Thi s drives PMVNA ports (nodes) one and two with a nominal 180 phase difference Normalized waves are measured at all down-mixers: a 1 b 1 a 2 b 2 a3, b4, a4, b4 (the reasons for meas11ring all po ss ible nor1nalized waves, even those that are apparently unneeded are to correct for RF swi tch imperfections; see Section 6.2.4). This

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74 configuration of the PMVNA is called the differential-forward mode (DF). Next, reverse operation of the DUT is measured by setting the RF port selection switches of both test sets into reverse position. This drives PMVNA ports three and four with a nominal 180 phase difference. Again, normalized waves are measured at all down-mixers. This con figuration of the PMVNA is called the differential-reverse mode (DR) Set drive to 180 ( differential) Set drive to 0 (common-mode) ... Set test-sets to forward operation Set test-sets to forward operation Measure test-set A (a 1 b 1 a 3 b3) Measure test-sets A &B, build aCF Measure test-set B (a2, b2, a4, b4) Set test-sets to reverse operation Build column vector aDF Measure test-sets A &B, build aCR .. ... Set test-sets to reverse operation Bu i ldA std B std ... Measure test-sets A &B, build aDR Correct phase offset: Ac, B c .. ... Calculate A mm Bmm ... Calculate mixed-modes-parameters Figure 4-5. Flow chart of PMVNA measurement.

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75 Next, the PMVNA measures the DUT with a common-mode stimulus, which is accomplished by setting SWI to position two (Figure 4-3). The forward measurements are repeated in the same way as with the differential stimulus described above. This con figuration is called the common-forward mode (CF). Similarly, the reverse measurements are repeated with the common-mode stimulus; this configuration of the PMVNA is called the common-reverse mode (CR). The calculation of the mixed-mode nor1nalized power waves is as follows. After all data from a DUT measurement has been collected, the raw a and b data are arranged into column vectors, where each vector corresponds to a single measurement mode (DF, DR, CF, CR), and the a and b data are collected into two corresponding vectors. In Figure 4 5, the arrangement of the a data is illustrated where DF DR CF CR al al al al DF DR CF CR -DF a2 -DR a2 -CF a2 -CR a2 a a a a (4-1) DF DR CF CR a 3 a3 a3 a3 DF DR CF CR a4 a4 a4 a4 and where the subscripts one through four correspond to the PMVNA port (node) num-DF -DR -CF -CR bers. Similarly, the b data are arranged into vector s b b b and b The data are the placed in two matrices A s t d = -DF -DR -CF -C R a a a a B st d DF -DR -CF -CR b b b b (4-2) where the superscript std indicates that the matrices are nodal data rather than mixed mode data. The phase offset correction process, which will be described in detail in Section 6.3, is applied to the A and B-matrices, generated phase-corrected versions, A c

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76 and B e, respectively The mixed-mode norrnalized power waves are now calculated in matrix forn1 Amm = MA s td (4-3) where the matrix Mi s the similarity operator described in (3-42). The designation of ele ment s of the mixed mode power wave matrix A mm is also composed of column vector s, one for each PMVNA configuration A mm = mmDF mrnDR -mmCF mmCR a a a a ( 4 4 ) and where DF adl OF -mmDF ad2 ( 4-5 ) a DF a cI DF a c2 with th e s ub s cript d referring to the differential mode quantity, c to the common mode and the subscript numbers referring to the mixed-mode port number s ( in contrast to the s ingle-ended node nu1nbers ) The remaining vectors of ( 4-4 ) are defined in the s ame fa s ion. Likewise, the mixed-mode B matrix Bmm can be defined. The calculation of raw mixed-mode s parameters is examined in detail next. After calculation of the mixed-mode normalized power matrice s A mm and Bmm the raw mixed mode s parameter matrix smrn can be simply calculated 1 Smm = B mm ( A mm) (4 6 )

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77 One of the added benefits of the PMVNA is that it can also be used to measure s tandard four-port sparameter s. The standards-parameter s can be calculated through the s imi1arity transformation of ( 3-44 ), but they can also be calculated directly from raw A and B matrices. With thi s method, the standard A and B-matrices of ( 4-3 ) are u s ed to directly calculate the standard s -parameters ( 47) The accuracy of these s tandard s -parameters must be considered carefully how ever. As i s s hown in Chapter 5, the PMVNA has lower residual errors when measuring a differential device. By similar argument s it can be shown that a standard four port VNA ( where only one te s t port i s stimulated at a time ) will have lower re s idual errors when mea s uring a device that exhibit s no differenti a l behavior. Stated another way the four port mea s urement s o f the PMVNA of a non differential DUT have higher residt1al error s than measurement s of the same devi c e from a s tandard four-port VNA. 4.2 4 Control Software The control so ftware of the PMVNA was implemented in Lab VIEW. Lab VIEW i s a graphical instrument control language which is well suited for the automation of the PMVNA [37]. The c ontrol software ha s many function s ( 1) general measurement control ( 2 ) VNA operation s etting s such as mea s urement frequencies, attenuation s ettings, etc. ( 3 ) PMVNA calibrati o n ( 4 ) general u s er interface ( 5 ) data di s play and (6 ) data inpuUout put ( I/0 ) in file s The control software of the PMVNA repre s ent s a significant development effort Thi s s oftware i s highly s peciali z ed and ha s been developed s olely for this work. The pro gram is graphically d e veloped s o that wiring diagram s take the place of traditional

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78 source-code listing s. The control software represents more than 11.5 Mbytes of code so including all diagram s i s prohibitive Instead flow diagrams are presented to indicate the s ub s tance of the software. Thi s s ection reviews the control software at the highest level of functionality. For de t ailed description s of the variou s function s, see Appendix G The basic flow of the soft ware is indicated in Figure 4-6 The first step in using the PMVNA is to set the basic oper ating parameter s of the analyzer Thi s includes the frequencies of measurement the attenuator se ttings for all port s, the number of average s, RF s ource power level, and so on. Wait for Corrunand Re -C alculate I I I I SetupPMVNA I I I I I I + Pha se Offset Pre-Calibration I I I I I Primary Calibration I I DUT Measurement Display Re s ults File I/0 Exit Figure 4-6 Top -leve l flow chart of PMVNA control s oftware

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79 Next, a phase off s et pre-calibration must be completed. This characterize s and allows for the correction of the phase offset between the two VCO signals in the test sets. For theo retical details on thi s c alibration s tep s ee Section 6 3. The primary calibration character izes linear time-invariant error s in the PMVNA allowing for error correction of measur e d data The theoretical development of the PMVNA calibration i s given in Chapter 6 Th e next s tep in the s oftware flow is DUT m e a s urement. This includes mea s urement s a s detailed in Section 4 .2. 3 and en or correction of the measured DUT mixed-mode s-param eter s The final two s teps in the s oftware flow are optional but are almost always used The first of these is data display, which allows the user to examine the raw or corrected DUT and calibration s tandard s parameters in a variety of formats The last s tep is file IO which allows the user to save any of measured data to a ftle in CITI format [38]. Al s o, the s oftware allow s the u s er to re-calculate any portion calibration and error correction algo rithms which i s used mainly for de bugging purpo s e s. 4 3 On Wafer Measurement s The PMVNA can make measurement s of devices with coaxial connectors, or device s that are meant to be probed at the wafer level Wafer level mea s urements or on wafer measurement s, require s pecial RF wafer probes to make good performance RF con nection s to integrated devices that are typically quite small ( on the order of 300m on a side ) For the PMVNA careful attention must be given to the signal launch from the probe tip to the wafer s urface As shown in Appendix C the mixed-mode s -parameter s o f an arbitrary differential DUT can be accurately measured with uncoupled ref e rence tran s mi ss ion line s ( or port s), independent of any coupled mode s of propagation that may exi s t in the DUT This i s achieved through the decomposition of any coupled mode signal s into

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0 0 0 Figure 4-7. 80 6 6 GGB dual-RF wafer probe, top view (not to scale). uncoupled modes, which results in mixed-mode s-parameters that are normalized to the reference impedance of the uncoupled lines. Accordingly the wafer probes that interface with a differential DUT can be composed of isolated single-ended probes. In order to maintain a s mooth transition to any coupled-modes, two single-ended probes are paired into a single mixed-mode probe. Each mixed-mode probe provide s two RF measurement ports that are in reasonably clo se proxinuty, but are ideally uncoupled Hence a mixed-mode probe footprint of GS 1 GS 2 G is adopted. The PMVNA system, a s implemented for thi s work, i s fitted with a pair of 150m pitch dual-RF probes manufac tured by GGB Indu stries [39] A dual-RF probe is illustrated in Figure 4-7 with a detail showing the probe contact configuration. Wafer probe s reqt1ire special calibration standards. These standards are meant to be contacted directly by the probe, so that the calibration reference planes are at the probe

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81 tips. These wafer probe-able standards are widely available for two-port VNAs. How ever, the unique nature of the PMVNA required custom wafer-probe standards to be designed and manufactured. These standards are discussed in detail in Section 6.2.7 of Chapter 6. With the construction and operation of the PMVNA detailed, the measurement accuracy remains to be assessed. An important aspect of the PMVNA is its accuracy in the measurement of differential devices, relative to that of a more traditional VNA. This is t he central issue that will be examined in the next chapter.

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CHAPTERS ACCURACY OF THE PURE-MODE VECTOR NETWORK ANALYZER As indicated in Section 3.1.3, mixed-modes-parameters and standard four-port s-parameters are related by a linear similarity transform. This relationship suggests that a traditional four-port VNA (where only one measurement port is stimulated at a time) could be used to measure a differential DUT, and the resulting four-ports-parameters could be transformed to mixed-mode s-parameters for easy analysis. Instead, a special ized VNA has be constructed to directly measure mixed-mode s-parameters. These two approaches do not yield equally accurate mixed-modes-parameters of differential devices, however. The PMVNA will be shown to be more accurate than a traditional four-port VNA while measuring a differential circuit. Mixed-modes-parameters generated by transforrning standard s-parameters measured by a traditional four-port VNA exhibit higher levels of uncertainty than those measured by a PMVNA. In particular, the uncer tainties of transfor1r1ed mode-conversion parameters, Sdc and Scd, can be significantly larger than the actual device parameter magnitudes The accuracy advantage of a pure mode measurement system provides motivation for the development of this specialized measurement system for differential circuits. In order to better understand the benefits and limitations of the PMVNA, the mea surement accuracy of the system will be examined The goal of this chapter is to quantify the error in mixed-mode s-parameters of differential devices as measured by a PMVNA. Since it has been earlier established that a linear transform exists between mixed-mode 82

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83 s-parameters and standard s-parameters, a traditional four-port vector network analyzer (FPVNA) can theoretically be used to measure a differential device. Here, a traditional four-port network analyzer refers to a network analyzer that stimulates each port individu ally while un-stimulated ports are terminated with a matched load If a FPVNA is to be considered for measurement of differential devices, it is important to understand the errors that result by transforming standard s-parameters into mixed-mode s-parameters. The accuracy of both systems must be compared to understand the advantages and disad vantages of each. To quantize the errors in both a PMVNA and a FPVNA, the analysis is divided into two important areas of consideration: probe-to-probe crosstalk and maximum measurement uncertainty. It will be shown that the PMVNA has a higher dynamic range than the FPVNA due to the l!d 3 and lid (dis distance) dependence of probe crosstalk, respectively. It will also be shown that the uncertainty of mode-conversion parameters is significantly lower for the PMVNA than for the FPVNA. 5.1 Probe-to-Probe Crosstalk For a wafer-probe measurement system, the uncorrected probe-to-probe crosstalk is an important specification. This crosstalk can limit the dynamic range of the measure ment system, making high dynamic range measurements impractical. An important example of such a high dynamic range measurement is the reverse isolation of an inte grated RF amplifier. The unacceptable probe crosstalk of single-ended two-port VNA provided some of the original motivation for the development of the PMVNA. The differ ential mode of operation of the PMVNA is expected to have reduced probe crosstalk, due to the natural common-mode signal rejection characteristic of a differential circuit. This reduced crosstalk would allow higher dynamic range measurements than FPVNA. For

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84 these reasons, the raw probe-to-probe crosstalk of the PMVNA and a traditional four-port VNA are first quantified. The examination of the crosstalk levels is based on electromag netic simulations of probe tips. Measured probe-to-probe crosstalk is also provided as fur ther evidence of the higher dynamic range of the PMVNA. 5.1.1. Simulated Probe Crosstalk The mixed-mode probe is simulated as a ground-signal 1 -ground-signal 2 -ground (GS 1 GS 2 G) probe as described in Section 4.3. The crosstalk of the four-port system is represented through s imulations of ground-signal-ground (GSG) probes. The use of the two-port single-ended probes allows a consistent comparison between the crosstalk levels For simulation, the probes are modeled as 50m wide by 1 OOm long metal strips arranged in a 150m pitch configuration as shown in Figure 5-1 The strips are situated d lOOm G Ill t Ill S1 (a) Ill 150m G Ill f 50m S 2 1111 G d lOOm G 1111 1111 t ( b ) s 1111 1111 G 1111 150m 1111 50m Figure 5-1. Probe crosstalk simulation layout a ) Mixed-mode probe layout. b ) Single-ended probe layout

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85 on the swface of a 25mil substrate, and for purposes of this demonstration, the substrate relative dielectric constant has been chosen to be one. Under the substrate is an ideal ground plane. The probes are simulated in opposing pairs where the distance between probes is specified. The electromagnetic simulator used is Hewlett-Packard's Momen tum, which is a method-of-moments simulator [40]. Multiple simulations of both the mixed-mode and single-ended structures have been executed over a range of distances between the probes tips. The results of the multiple simulations are shown in Figure 5-2 to Figure 5-5. A direct comparison of the crosstalk in the differential mode of the PMVNA to that of the single-ended VNA is shown in Figure 5-2 as a function of probe separation at 1.0 GHz. The simulations show that the single-ended crosstalk maintains an approximate lid char acteristic, whereas the differential crosstalk behaves as lld 3 This different dependence on probe separation provides significant decrease in crosstalk for the differential mode with respect to the single-ended operation, and hence provides for greater dynamic range in the corresponding measurement. Also shown in Figure 5-2 is the common-mode crosstalk of the PMVNA. The common-mode shows nearly the same level of crosstalk as the single ended system, as expected. This indicates that the common-mode measurements will have approximately the dynamic range as traditional single-ended measurements. This plot illustrates the dynamic range advantages of differential measurements over single ended measurements. Figure 5-3 shows a comparison of the crosstalk of the PMVNA to that of the single-ended VNA 10.0 GHz. Figure 5-4 and Figure 5-5 show crosstalk as a function of frequency for single-ended and differential probes, respectively.

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86 The previous figu r es assume perfect phase and magnitude balance in the PMVNA system. However, all rea l systems will have some degree of imbalance, degrading the modal purity of any stimu l us signal. The effects of imbalances on probe-to-probe crosstalk can be quantified with the use to the same electromagnetic simulations. For examp l e a 5 phase imbalance from the ideal 180 differential results in a probe crosstalk level of -106 dB at 1.0 GHz and 1500m separation, which reduces the dynamic range improvement over single-ended to approximately 34 dB. The phase imbalance of the present PMVNA is less than 5 from 1 to over 5 GHz with very small magnitude imbalance. ---'6-S cc 2 1 simulated ~o"Sdd21 simulated 0 0 I / S2 1 (single-ended) simulated 0 0 N 100 I I f-;J "1 l "' I"I/ A ''" ... .. "' j y' \ ~ 45dB I'-... l :l. I\ h / Separation (m) Figure 5-2 Simulated probe crosstalk vs separation distance at 1.0 GHz. lOK

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----,,_6_~Scc 2 1 simulated ----1Qr-S dct 21 simulated 0 'o:::t I I/ S2 1 (s ingle -e nded) simulated 0 100 I 13 I A l "' "" 87 ,. z. .. I ""'~ .. i\ )1 I'/ I'\ Separation (m) Figure 5-3. Simulated probe crosstalk vs. separation distance at 10 GHz 0 V) I I'\. V / I r I J 0 1 0 /2 / V I 1 I 1 ......1 "'_.......-: ,....-/ .,,,,' I/" / frequency (GHz) 10.0 ,, lOK 150m 300m 500m 700m lOOOm 1500m Figure 5-4. Simulated single-ended probe crosstalk vs. frequency for several probe separations

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Figure 5-5. 0 I 0 N I I l I I I _/ .....-0/ ,,.. I'] / .....-:r / .,r :J / ...-., / 88 -__,,,,,1 1 I 1 1 1 1 150m 300m 500m 700m IOOOm 1500m 1.0 freq (GHz) 10.0 Simulated differential probe crosstalk vs. frequency for several probe sepa rations. 5 1.2 Measured Probe Crosstalk Measured probe-to-probe crosstalk for the PMVNA is shown in Figure 5-6 to Figure 5-8. This data was collected with GGB 150 m-pitch dual RF probes (as discussed in Section 4.3), where the probe tips were suspended in air approximately 10 cm above a ground plane. Figure 5-6 shows the measured and simulated differential and common mode crosstalk as a function of probe separation at 1 0 GHz. Figure 5-7 shows the same at 10.0 GHz Figure 5-8 shows the measured differential crosstalk versus freqt1ency for sev eral probe separations. From these figures, one can see that the measured crosstalk, regardless of mode, is generally higher than that of the simulated structures The source of the difference is most likely due to the structural differences between the simulated structures and the actual probes. Despite the differences in the absolute level of crosstalk, the measured data shows similar trends versus probe separation. The measured data shows a 30 dB difference between the differential and common-mode crosstalk at 1.0 GHz and 1500 m separation,

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89 and 22 dB at 10.0 GH z (c ompared to s imulated 40 dB and 30 dB respectively ). Thi s dif ference in the crosstalk of the mod es clear l y indicate s a higher dynamic range for the dif ferential-mode in th e PMVNA. Sc c 2 1 mea s ur ed S dd 21 m eas ur ed 6. S cc 21 simulated 0 Sctd21 simulated 0 I 0 ~100 I I G.. I r.l l l "' '\ V '\ ...... I N... ... ..... i\ \ \ \ ') ") I\. h' / 1"1 S epara tion ( ) Figure 5-6. Measured prob e crossta lk vs. separatio n di sta n ce at 1.0 GHz. S cc 2 1 measured Sdd21 me as u red 6. S cc 2 1 sim ulat ed 0 Sdd21 sim ul ated 0 I \. V 0 C'l ,_ 100 I r:I. r.l l "' '\ V i'... 71' '\... "' I\ ""ft 1. \ l ' ' l;,I J't] / Separation ( ) Figure 57. Me as ur ed probe cross t alk vs. separat ion distance at 10 GHz. lOK lOK

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0 'tj1 0 N I I ( I I I If i_......---r / r / /\ "; \ IV / / V VV 'I I r-' V 90 ....... / ,.. / r\ / ..... I \ J \ V V V I ...... . ,v J -J IJ J l J 150 m 300 m 500 m 700 m lOOOm 1500m 1.0 freq (GHz) 10 Figure 5-8. Measured differe nt ia l probe crosstalk vs. frequency fo r several probe sepa ration s. 5.2. Uncertainty Calc ul ations A genera l ly accepted quantificatio n of error in VNA measurements i s the maxi mum uncertai n t i es in the magnitude and p h ase of a set of s-parameters [43). This sect i on seeks to q u antify the error i n a mixed-mode measurement, and compare t h at to t h e error in a standard fo u r-port meas u rement All measurements h ave errors a n d these (unknown) errors add uncertainty to the measurements. This uncertainty l i mits how acc u rate l y a DUT can be meast1red. VNA errors can be separated into raw and residual errors. Both types of errors can be further sorted into systematic (repea t able) and non-systematic ( non-repeatab l e) errors. For a comp l ete descr i ption of VNA errors, see Chapter 6. R es i d u a l errors are th e errors that remain after calibration. D u r i ng ca l i b ratio n sta n da r ds wit h k n own c h aracteristics are

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91 measured by the VNA, and the systematic errors of the VNA are quantified. Any mea surement of a DUT can be corrected by mathematically removing the effects of the sys tematic errors (PMVNA calibration and error correction are examined in detail in Chapter 6). However, this conection process is not completely accurate. Limitations on how accurately the standards are known and non-systematic e1Tors (in calibration and DUT data) cause the correction to be imperfect. Measurement accuracy is specified as a certain level of maximum possible magni tude and phase error for a given DUT measurement, which is called maximum measure ment uncertainty. The numerical values for these specifications arise from the combination of three elements: ( 1) detailed mechanical tolerances of the calibration stan dards (from the manufacturer of the standards) which lead to uncertainties in the standards electrical response, (2) raw instrumentation measurement errors, and (3) the precise VNA calibration process used [ 41]. All of these elements contribute to the actual measurement error. To make the accuracy specifications independent from the DUT, the sources of error, from the three areas listed, are stated as a set of equivalent residual errors. These residual errors are based on an assumed error model. Since the actual error, produced by the three factors above, cannot be directly known, the residual error terms are expressed as maximum magnitudes. It is asst1med that these residual errors can combine in a way to produce the maximum error in the corrected DUT s-parameters. To ensure that a VNA is producing measurements within the accuracy limits set by the residual error terms, a verification process is typically employed [ 41]. This process involves measuring a set of verification standards (different than those used in calibra

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92 tion), and comparing the corrected s-parameters to NIST traceable measurements of the same verification standards ( previou s ly generated through a meteorology lab ). These traceable measurement al s o have an associated maximt1m uncertainty which has been cal culated from the s ame three factors li s ted above ( these uncertaintie s are typically smaller than tho se of a typical VNA due to extreme tolerances used in developing meteorology s tandard s) [41] Th e difference between the mea s ured sparameters and the traceable s parameters repre se nts the total measurement uncertainty. The effects of the uncertainty in the traceable data can be removed from the measurement data (in a worst-ca s e fa s hion ), leaving the VNA mea s urement uncertainty From the established residual error s, an allowable maximum mea s urement uncertainty ca n be calculated for each of the verifica tion s tandard s, and the VNA measurement uncertainty must be less than this value to be considered to be operating within s p ec ifi c ation s [43] For sparamet e r mea s ur e ment s, uncertainty is u s ually expressed as a maximum magnitude uncertainty and a maximum pha se un c ertainty for each paramet e r. Maximum uncertainty i s calculated by finding the pha ses for all residual errors that cause the maxi murn error in the corrected data either in magnitude or pha se Furthermore, the uncertain tie s of a measurement are highly dependent on the actual sparameter s of the DUT Becau s e of this maximum uncertaintie s are given with respect to a particulars-paramet e r value or s pecific DUT. Thi s s tudy of uncertainty focu ses on gaining a r e asonable e s timation of the maxi mum uncertainty of the PMVNA while mea s uring a differential device The nece ss ity of th e es timation results from the lack of well established residual error s for the PMVNA, and the lack of appropriate mixed mode verification s tandard s through which measure

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93 ment uncertainties can be found. Another goal of this study is to estimate the maximum uncertainty of a FPVNA while measuring the same differential device. In particular, it is important to quantize the maximum uncertainty of transforming four-port s-parameters with uncertainty into mixed-mode s-parameters. The calculation of maximum uncertainty is based upon a residual error model. The error model used for calculation of t1ncertainty of the FPVNA is shown in Figure 5-9. The figure shows the equivalent (composite) representation of several errors. The devel opment of this model required several assumptions, but it is extended directly from accepted two-port residual error models. The basic assumptions of the model are dis cussed below, but a detail description is available in Section 5 .2.2. The basic assumptions used in the FPVNA are: (1) perfect port-to-port isolation. This is done primarily to simplify calculations. The effects of probe-to-probe crosstalk may limit accuracy of some parameters, but is considered in the earlier sections. (2) No drift errors are included, to simplify calculations. This error typically has only a minor effect. (3) No cable variation errors are considered, to simplify calculations. This error typically has only a minor effect. (4) No connector repeatability errors are considered, to simplify calculations. Again, this e1Tor typically has only a minor effect. (5) Forward and reverse error quantities are considered to be equal. This is done to simplify calculations, and from typical two-port residual errors, this is a good approximation. ( 6) Any source or load variations due to switching are geometrically averaged. Again, this is done to sim plify calculations. This allows the development of a single error model instead off our, while allowing reasonable estimation of the effects of switching errors. (7) All residual error values are based on typical 851 OC residual errors, TOSL sliding load calibration,

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94 with I 024 averages [ 43] (se e Table 5 1 ). The residual error model of the PMVNA is s hown in Figure 5-10 It i s essentially identical to that of the FPVNA. The values of the maximum uncertainties hav e b ee n found u s in g a pr ocess of numeri ca l ca lculation of many random tri a ls where all variables are considered to have a un i form pr o bability distribution ( thi s approach i s generally called Monte -Ca rlo analysis). This analysis ha s be e n done with a M athemat i ca program Sin ce th e error in a ny o n e pararneter i s in ge neral a function of all DUT s-p arameter s and all error parameter s, the direct so lution of the worst-case error i s extremely difficult without s implifyin g assump tions Th e num e rical technique allows the estimation of the wor s t -case errors without makin g any further sim plifying assumptions. The following ste p s are taken in the mont car lo ca l culation: ( 1 ) a set of actual DUT s-parameters are pre s umed. (2) The pha se angles of th e error terms are randoml y se t with a uniform distribution over 0 to 360 and (3) the re s ulting '' me as ur e d '' s-parameters are calculated by a embedding process (t he inverse function of error co rrection ). ( 4) The error betw ee n the magnitude s of the actual and '' mea s ured '' spar a m e t e r s are calculated, likewi se for errors in pha ses. ( 5 ) Many ran dom tri a l s are run and the maximum errors in ma g nitud es and phases are co llecte d for eac h parameter. Numerical calculations are ba se d on the sparameters of an example differential RF ainplifier as the D UT. The sparameter s, given in ( 5-1 ), are the 1.0 GHz mixed mod e sp a rameter s of an RF pre -a mp The values of the error term s u s ed in the calculation ar e g iv en in Tabl e 5 1 For this s tudy the number of random trials i s one million

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Port 1 Port 2 E' ct2 Port 3 E' ct 3 Port4 E ct 4 1 E's1 E r 1 E p 2 E's2 E 'r2 E ~ p 3 E's3 E 'r3 E's 4 E 'r4 95 Assumptions : Forward and rever s e error s equal P erfect port isolat i on No drift variation s No connector repeatability eff ec t s DUT Sour c e/Load variation s g eometrically averaged No c able variat i on s 851 OC typical re s idual err o r s : 2 4mm s lot l e ss c onnector s TOSL, s liding load Fi g ure 5 9 Equj va lent four port un ce rta i nty model Table 5 1. Typical 851 OC re s idual error ma g nitude s. Fr e q Ed E r E s EP Ent Ab En f ( GHz ) ( dB ) ( dB ) ( dB ) ( dB ) ( dB ) ( dB ) ( dB ) 045-2 -41 90 0 00796 41. 9 0 0.00796 0 0009 0.0010 99 3

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DM 1 E ct1 DM2 E'ct 2 CM 1 E'ct3 CM2 E ct4 E's1 E'r1 E'p 2 E's2 E'r2 E'p3 E's3 E'r 3 E'p4 E's4 96 Assumptions: Forward and reverse errors equal Perfect port isolation No drift variations No connector repeatability effects DUT Source/Load variations geometrically averaged No cable variations 851 OC typical residual errors: 2.4mm slot-less connectors TOSL, sliding load Figure 5-10 Equivalent mixed-mode uncertainty model 0 329 0.0007 0.0080 0.0019 S dd S de 1.24 0 973 0 0140 0.0190 S S 0.0132 0.0013 0 286 0 0165 cd cc (5-1) 0.0351 0 0183 1 02 0.962 Calculations of maximum magnitude an d phase uncertainties of the PMVNA and the FPVNA are presented below. In the following expressions, the s uperscript ''mm'' indi cates the parameters are in mixed-mode format; the superscript 'std '' indicates standard

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97 s-parameters. The vertical bars indicate the values are magnitude terms, and the angle sign, L, represents the phase of the parameter. The symbol L1 01 ax indicate s the maximum calculated error in magnitude or phase as appropriate. As the number of random trials increase, the maximum ca l culated error will approach the maximum uncertainty For this work, it is assumed that the number of trials is sufficient so that the maximum calculated error is approximately equal to the maximum uncertainty, so the terms will be used inter changeably. The maximum magnitude uncertainty in the mixed-mode s-parameters of the amplifier, as measured by the PMVNA, are given in (5-2). -3 6 5 -5 9.35 x l0 9.24 x l0 7.23xl0 2.52xl0 -2 2 -4 -4 l .47xl0 l 79xl0 4.06xl0 3.58xl0 4 5 -3 -4 l.OOxlO l 84x10 9.45 x l0 2 04 x 10 ( 5 -2) 4 4 -2 -2 7 05 x 10 3.38xl0 l .25xl O l .78x l 0 It can be seen that the uncertainties are roughly proportional to the magnitude of the corresponding s-parameter. In all cases, the maximum uncertainties are more than one order of magnitude s maller than the magnitude of the parameter illustrating good overall accuracy. Thi s result i s analogous to the behavior of two-port uncertaintie s The values of maximum magnitude uncertainty of mixed-mode s-parameters tran sf ormed from the FPVNA s-parameters is given in ( 5 3). This calculation has been accomplished during the monte-carlo calculation by first transforming the ''measured'' fou r -ports-parameter s of each random trial into mixed-modes-parameters, and then accu mulating the maximum error s with respect to the actual DUT mixed-mode s-parameters. Thi s order of calculation is indicated by explicitly showing the transformation (MS M 1 )

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98 before the uncertainty operator An alternate order of calculation i s discu ss ed in Section 5. 2 .3 at the e nd of thi s chapt e r -3 5 3 5 9.38 x 10 2.13 x 10 9.24 x 10 7.82 x 10 -2 -2 2 2 l.51 x 10 1.76 x 10 1.23 x 10 1.7l x 10 3 4 -3 4 9.30 x 10 1.48 x 10 9.15 x 10 1.96 x l0 2 2 -2 -2 1.40 x 10 1.69 x 10 l.19 x 10 1.7l x 10 ( 53) In general th e uncertaintie s of the tran s f orrned four port s-parameter s are great er than those measured directly by the PMVNA In the highlighted mode conversion s tenn s ( Sd c and S c d ) the unc e rtainties have increased si g nificantly over those of the PMVNA so that the uncertaintie s a re approximately the s am e magnitude a s the corresponding sparameter. In the s e mode-conversion s terms, the overall error is dominated by the lar e s t error in the s tandard s-parameter terms. Consider the conversion parameter s c d 2 1 ( 5-4 ) In a typical differenti al device ( 5-5 ) s o that the 11ncertainty in the tran s forr r 1 e d parameter is approximately st d 1 ~ I oc I Lim ax MSD U TM c d 2 1 ~ Lim a x S 3 1 S 3 1 ( 5-6 ) With s imilar approximation s, the differ e ntial gain of a device is ( 57) s ~ 1 ~ I oc I Lim ax MSD U ~ d d 2 1 ~ Lim ax S 3 1 S 3 1 ( 5-8 )

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99 The ref ore, the errors in the mode-conversion parameter will be in proportion to the gain of the device, rather than in proportion to the magnitude of the actual mode-conversion The maximum phase uncertaintie s of the mixed-mode s-parameters of the differ ential amplifier can also be calculated. The phase uncertainty of the directly measured mixed-mode parameters is shown in (5-9), and that of the transformed four-port data is given in (510 ), both in degrees. 1.63 0.756 0.516 0.755 A L Smm 0.679 1.05 1 .67 1.08 umax DUT = 0.433 0.816 1.89 0.707 1 .15 1.06 0.705 1.06 1.63 1.75 180 2.34 0.696 1.04 61.7 63.8 44.6 6.59 1 83 0.680 23.5 67.2 0.672 1 .02 ( 59) (5-10) The pha se errors are of similar size, except in the mode-conver s ion parameters. The large phase errors in the conversion parameters of the transformed four-port data is directly related to the large magnitude errors in the same parameters. 5.2.1. Discussion of Accuracies This study of uncertainties has demonstrated that the PMVNA has a higher accu racy than the FPVNA when measuring a differential device. When a differential device is measured by a PMVNA, the pure-mode parameters (Sdd and S ec) have about one-half of the magnitude error of the corresponding parameters of transformed four-port s-parame ter s measured by a FPVNA. The mode-conversion parameters ( Sct c and S ect) as measured by a PMVNA can ha ve s ub sta ntially lower error than those measured by a FPVNA.

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100 A practical example of where such errors can be important is in the calculation of the common-mode rejection of a radio receiver. In a differential-in-differential-out ampli fier, the common-mode rejection ratio is essentially the mode-conversion parameters. In a case of a strong common-mode signal, errors in the mode-conversion parameters can lead to errors in the calculation of the amount of spurious differential response of the amplifier. Consider a O dBm common-mode signal at the input of the amplifier in (5-1). Assuming 50Q terminations for ease, the actual differential signal at the output of the amp is 20Log(sdc 21 ) = -37.1 dBm. The signal level predicted by the PMVNA measurements would be, at worst, -36.2 dBm. In contrast, the signal predicted by the FPVNA could be between -31.6 dBm and -55.4 dBm. The mode-conversion parameters represent important behavior in the analysis and design of differential circuits. A fundamental advantage of differential circuits is increased noise immunity when compared to single-ended circuits. Here, noise immunity means the rejection of common-mode signals of all type. Typical examples of such com mon-mode noise include interfering signals, known as electromagnetic interference (EMI), from clocks, VCOs, etc. Also, even-order distortion products are rejected in an ideal differential circuit. The ability of a real circuit to achieve the rejection of these sig nals is directly linked to the degree of balance in the differential circuit. The mode-con version parameters are direct measurements of the degree of imbalance in a differential circuit Thus, errors in the measured mode-conversion limit the ability to analyze and ulti mately realize the advantages of differential topologies.

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101 5.2.2 Uncertainty Model Derivation The uncertainty error models of Figure 5-9 and Figure 5-10 are developed by extending the basic concepts of the well established two-port uncertainty model The two port model is typically developed in terms of separate forward and reverse model s. These models are shown, in s implified form, in Figure 5-11 neglecting drift, cables, and connec tors The four-port models have simply applied two different two-port models The major differences are two fold. First, the forward and reverse models have been combined into a single effective error model. Second, the two different two-port error models that com po se the four-port model can not be considered to be independent In other words, the increa se in the number of port s requires additional error terms so that each port has a rel tive error path with respect to all other ports In general, with the assumption of perfect port isolation, an n-port VNA will reqt1ire n 2 1 error paths and one unity-valued path. The models in Figure 5-11 show several cascaded blocks each with multiple error ter 1 r1s These blocks are reduced to a single block representing the composite error paths containing the appropriate error terms. For simpler calculation, the composite error block is represented in ter1n s of chaining scattering parameters known as t-parameters (see Appendix H). The error block, T i s partitioned into four-by-four sub-matrices T Ell T T TE2 2 The ''measured'' s-parameters, including errors, can be calculated by (5-11) (5-12) where Sa is that actuals-parameter matrix of the DUT. For a complete discussion of the concepts leading to this equation, see Section 6.2

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102 I I I I I I I Noise & I I I Residual : Noi se & I Re s idu a l I Dynami c I I DUT I Dynamic I Error Error Ac c uracy I Accuracy I I I I I I I I I I 1 +Etn t 1 +A b2 I 1 1 S 2 1 J+Eft I I I I I I I l+E fnt I Efd Ef s I S11 S22 I Efl I I l+Abl I I Efnf I I l+Err I S12 I I I I I I I I I I I I I I I (a) Emf I I I I I I I I I I I I I Emf I S21 l + Err I I I I I I J+Emt J+Ab 2 I Er s I S11 S22 I Er s Erd I I I I I I I I I I J+A b 1 l+Emt l+Ert S12 1 1 I I I I I I I I Accuracy I I I I Accuracy I 1 Error DUT I Error I I I D y namic I Dynamic 1 Residual Residual I Noise & I I I Noise & I I I I I I I (b) Figure 5 11 Two-port uncertainty model. a) Forward uncertainty mode l b ) Rever se uncertainty model. The above deve l opment i s applicab l e to both standard and mixed modes-parameters The uncertainty models of the PMVNA and FPVNA are essentially the same, but the corre spo nding s-parameter ex pre ss ion s are s li g htl y different due to their re s pective definitions. However is can be s hown (see Appendix I), that under the condition of the no-leakage

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103 model ( i.e perfect isolation) the appropriate conversions of the two different s-parameter matrices results in identical t-parameter expressions, that is Tmm E no l eakage T std E no l eakage (5-13) As a result, the remaining development applies equally to the FPVNA and the PMVNA uncertainty models The partition s of the T matrix can now be related to the residual error terms in Figure 5-11. For perfect port isolation, the partitions T Eij, are diagonal matrices, and have the form T Ell = Diag ( Xli ) T EI2 = Diag (X2i) T 2 1 = Diag (X 3 i) T E22 = Diag(X4i ) iE {1,2,3,4} ( 5-14 ) where the s ubscript i indicates a te1m associated with a particular measurement port, and X ii are intermediate variables. The se terms are related to the error term s by Xli = E' E'diE 1 s i E'di X2 = fl E'pi l E' pl ( 5 -1 5) E' s i x 3 i x4i 1 E'pi E'pi The primed quantities are further expanded as (5-16) The port s are a ss umed to have the s ame re s idua] error magnitudes so the error terms are identical for all port s i. The noise and dynamic range errors, D d, are expanded as

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104 (5-17) As stated earlier the effects switching between ports have been estimated to pro vide a single error model for the entire VNA. The estimation is accomplished by geomet rically averaging the quantities that change as the port switches change. Since the FPVNA and PMVNA uncertainty models are derived from two-port VNA models, the switching estimation is done with respect to the two-port model of Figure 5-11 For example, the source impedance error of the FPVNA i s assumed to be equal for all ports and constant, regardless of which port i s being s timulated The value of this error is estimated as the geometric mean of the source match error and the load match error of a two-port VNA. This same estimation is used for re s ponse/tracking enors and dynamic range errors Er = JEfrEft (5-18) Furthermore, the numerical values used are assumed to be equal in forward and reverse directions E* = E* = E* for ward reverse (5-19) where represents any error term designator ( e. g. d, s, r) A s discu sse d ea rlier in this section, additional error terms a1e required to describe the rel a tive error between ports of the FPVNA and PMVNA In absence of actual four port residual errors, it i s assumed that the port-to-port error will be similar in magnitude to the average response error ( 5-20 ) The error matrix TE, ha s fifteen non-unity composite error terms. However, it con t ains thirty-one independent error variables (Ep Ed, etc.). To reduce the number of

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105 independent variables, the composite error term s in TE are considered as the variables for the monte carlo analysis. The maximum magnitudes of the composite error terms must be u se d s ince the individual error term s are independent, and they are fot1nd to be jEdj Es IDmax IX1; 5 ( I +IE, ) Dmax \ + J IEPI Edi Dm ax Es X2i 1 IEPI X3j 1 IEPI (5-21) (X4 1 = 1 ) with (5-22) The fact that the error terms are magnitudes only i s emphasized by explicitly s howing magnitude bar s on all error terms The random variables of the composite error term s are constructed from the mag nitude s of ( 5 -2 1 ) where the phase of the error is the randomly varied quantity. where X indicated a random variable. j~2 X2i = X 2 i e 1 (X41 = 1) ( 5 -23) From the composite error ter1n s in ( 5-23 ), the maximum uncertainties can be cal culated. Each trial of the monte -car lo calculation se t the random phases and the error matrix TE i s calculated by ( 5-14 ). From this, the '' mea s ured'' sparameters are calculated from ( 5-12 ), and the magnitude errors are found by ( 5-24 )

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106 and the phase error can be calculated by [43] (5-25) 5.2.3. Order of Uncertainty Calculations The order of ca l c ul at ion has b ee n careful ly considered, as there is an alternative approach. The maximum errors could first be accumulated in the four-ports-parameters, and then the maximum errors of each mixed-mode parameter could then be calculated. Thi s la s t method requires that the uncertaintie s of th e approp1iate standards-parameter be combined in a way to maximize the r es ulting mixed-mode uncertainty. This approach, although s impler to ca lculate, i s not u se d as the primary method of calculation becau se it contains two level s of error maximization, and might unn ecessa rily inflate the un certain ties of the transformed s-para meter s. In practice the method of calculation makes little difference, as seen in (5-26) Ll M s td M 1 = max L'.imax S DUT (5-26) 2 4 2 4 1.03x10 2.38x10 l 03 x 10 2.38xl0 2 2 -2 -2 1.75x10 2.0lxlO l.75 x 10 2.0lxlO -2 -4 2 4 l.03xl0 2.38x10 1.03 x 10 2.38xl0 2 2 2 2 l .75x 10 2.0lxlO l.75 x 10 2.0lxlO The si milarity in the un ce rtaintie s of ( 5-6 ) and (5-26) indicates that the tran sform provides no advantage in reducing uncertainties. A particular random trial may produce correlated errors in the four-port s-parameters, and these correlations may reduce errors in

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107 the transformed s-parameters. However, another random trial exists that, when the result ing four-ports-parameters are transformed, again produces an error that is as large as approach indicated in ( 5-26). 5.3. Conclusions on Accuracy This study of uncertainties has demonstrated that the PMVNA has a higher accu racy than the FPVNA when measuring a differential device. The PMVNA has the advan tage of the natural noise suppression of differential circuits. This results in greatly enhanced dynamic range in differential measurement with respect to traditional VNA measurements. Consideration of residual error in an practical VNA system has shown tha t when measuring a differential device, the pure-mode parameters from the PMVNA have about one-half of the magnitude error of the corresponding parameters of trans formed four-ports-parameters measured by a FPVNA. Most importantly, the mode-con version parameters as measured by a PMVNA can have substantially lower error than those measured by a FPVNA. These findings indicate that the PMVNA has a clear accu racy advantage in the measurement of differential devices. With the advantages of the PMVNA clear, the calibration and error correction of the PMVNA is next required for the measurement of RF differential circuits. The accu racy conclusions of this chapter are based, in part, on a assumed level of calibration accu racy. Care must be taken to ensure the completeness of the PMVNA calibration to ensure the accuracy advantages are maintained.

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CHAPTER6 CALIBRATION OF THE PURE-MODE VECTOR NETWORK ANALYZER 6.1. Types of VNA Measurement Errors All VNA measu1 ements have errors which can be grouped into several major cate gories. The most common groupings are: linear systematic errors, linear non-systematic (non-repeatable) errors, non-linear errors, source frequency errors. A further source of error, which is important, but not relevant to this discussion, is human operator error. Some of these error s are further sub-divided such as non-systematic errors divided into random and drift error s A representation of the classes of errors is shown in Figure 6-1 Operator Non Linear Raw Source Freq. Raw Residual Sy s tematic Raw Re s idual Linear Random Raw Residual NonSy s tematic Drift Raw Residual Figure 6-1. Types of VNA measurement errors 108

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109 Some errors will not be considered here. In particular, non-linear errors, and so urce frequency errors wi ll not be treated. Non-line ar errors include errors such as non linear performance in the analog-to-digital converters in the VNA [41] and undesired non linear behavior in down mixer s. Source frequency e1Tors include absolute and r elat iv e ( dri ft) errors in the frequency of the RF signal generated in the VNA so urce Both of the se typ es of e rror s can have se riou s impact on mea s urem e nt accuracy. However the se errors are essentially se t by the test equipment architecture, and are generally not improvable by the equipment user. For this rea so n these errors will be assumed as re s idual uncorrect a ble errors, and not treated in the calibration proce ss Sy ste matic errors include all sta tic ( r e peatable ) errors, and non-sy stemat i c errors include noi se, drift, a nd other time variant errors. Theoretically, the effects of all static linear systema ti c errors ca n b e math e matically removed if the errors are known This pro cess, called calibration, involves m eas uring certain well known devi ces, called standards, with the non -i deal VNA The se mea s urement s, in combination with the known re s pon ses of th e standards, can be used to solve for all systematic errors. After calibration, the sys tematic VNA errors ca n be removed from the mea s urement s of any unknown device; this is called error correction. The ca libration of the PMVNA is sepa rated into two major divi s ion s: the primar y calibration and the pha se offset pre-calibration The primary calibration characterize s lin ear syste matic errors as described above. This i s done in with the traditional method of me as uring a se t of well known standards. The phase offset pre-calibration i s the proce ss by which the pha se r e lationship between the VCO s of the two test-set s i s characterized. This process is much different than the primary calibration in that very little need s to be

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110 known about the standards assoc iated with the pre-calibration. After a full description of the se calibration steps, the step-by-step calibration proce ss implemented for this work will be detailed. Thi s final sec tion of thi s chapter will explain the mechanics by which all of the previou s theoreti ca l development of the chapter is used to make a practical, accurate calibration of the PMVNA. 6.2. Pri mary PMVNA Calibration 6.2.1. Raw Performance The raw performance of any VNA i s important as it will effect the ultimat e cali brated mea s urement accuracy. A s an indication of the raw accuracy the mea s ured raw data of two device s are presented. The first pre se nted device is the match, where matched 50.Q termination s are placed on all ports The second device is a pair of through connec tion s. Together, the se devices give an indication of the raw dynamic range of the PMVNA. For in s tanc e, the ratio of the raw tran s mission of the through connection to the re s idual tran s mi ss ion of the match gives a measure of the raw dynamic range of that par ticular parameter Other mea s ures of raw performance might be propo s ed such as non matched impedance levels but it i s felt that no other definition has a particular advantage. Further1nore the raw dynamic range doe s not indicated directly the dynamic ran ge of th e corrected VNA The cotTected dynamic range of the VNA will typically be s ignifi cantly greater than the raw dynamic range. The exact level of enhancement by calibration depends on the accuracy of the calibration, the stability of the VNA, and other conditions. However the raw performance of the VNA will ultimately influence the corrected accu racy of the VNA.

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0 itrttU ,--.. y "O "O Cl) ......., /:0 "'O 0 S1 I freq. (GHz) 21 0 ,--.. N :g Cl) ......., /:0 "O 0 ":' 1 freq. (GHz) 21 0 IA /J. io\ .lU f V 0 ,--.. N "'O "O Cl) ......., ~1 I 0 ,--.. j N\ N 11 N "O "O Cl) ......., /:0 "O 0 0 --;1 0 "O u Cl) ......., 0 1 freq. (GHz) 21 1 0 0 freq. (GHz) 21 ij V I freq. (GHz) 21 .,-freq. (GHz) 21 ~ VI N 0 I 1 u Cl) ......., /:0 "O 8 freq. (G Hz ) 21 1 freq. (G H z) 21 111 0 0 ,..... ,--.. ... J\ Ill A I\ ,, viiB r-,. YI N ,..... I u u k>'"O "O Cl) Cl) ...._,, ......., /:0 "'O 0 S1 I freq. (GHz) 21 ~1 I freq. (GHz) 21 0 0 ,--.. r--,. ~LA f>ltA i/J ,..... r' N i VI N N u (.) '"d "O Cl) Cl) ......., ......., /:0 /:0 "O "'O 0 0 0 ":' 1 freq. (GHz) 2 1 --;1 freq. (GHz) 21 0 0 ,..... rl,; ,--.. 'Y I r,,' r1~1 ~ N u u u u Cl) Cl) ...._,, ......., /:0 /:0 "'O "O 8 0 I 1 freq. (GHz) 21 I 1 freq. (GHz) 21 0 0 I/ p N' ~ J -\4P ..... N N u u u u u Cl) Cl) ......., ......., f /:0 "O 8 0 I I freq. (GHz) 21 I l freq. (G H z) 21 Figure 6-2. R aw measured mixed-mode s-parameters of a pair of through connections (coaxia l ) between ports one and three, and ports two and four. Note that sca l es of the parameters are different. The uncorrected through co nn ection data are shown in Figure 6-2. The through is accomplished by connecting the mixed-mode ports to get h er with coaxial cables. (Specifi cal l y, ports one and three are co nn ected toget her and ports two and four are connected, where the port numbers are indicated in Figure 3-1) The figure shows 1S 21 1 and IS 11 1 in dB for differential-to-differential ( dd), common-mode-to-diffe r ential (de), differential-to

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0 .........-...---.---,-.,.--.---r-r--r-, N i--+--+-+-i--+-+-t---+-1 I -0-----1--t--+-+-t-i---t-~-,--, 1---ab freq. (G Hz ) 21 u,--+-1--1-1-+--+-1---+-4 Cl) "--'1---+--+-+-t-i--+-~l--+--+-+-1-1-+--+-t"'O 0 -0 -0 0 0 L '\..J .nu \. V, 1 I freq. (G H z) 21 0 .--.--,--,.......,................--...,........--.--, N 1--+--+-+-t-i--+--+-+-+-i I U> ~-l--+--4-1---1---+-+---l---oil Cl) "--' l--+--+-+-1--1--+--+-+l--+--l--+--4-1---1---+-+--+--l "'O -----0 N I 1 freq. (GHz) 21 I freq. (G H z) 2 1 ,.........--,--,---,-.,..-,--.,........--r-, I l--t--t--t--t--t--t-1---+-1 Nt--t--t--t--t--t--t--t---+-1 '8--+--+-->--t-....-.---.-.. Cl) "--' 1---1--+-+-t--t--t--t---+-1 t--t--t--t--t--t--t-1---+-1 0 N 0 1> Cl) "--' "'O 8 ttll ~, ~ii( I "" r I I freq. (G H z) 21 1 freq. (GHz) 2 1 112 0 ...-........................ .--.--...--.-~ N i---t--+-+-t--t--t-1--r-1 I -0 ~l--+--+-t--1-i--+--+-+ 1--t--+-+-t-i---t-~ "'O 1-+-+-+-+-l-hr+-lr--t.:,t freq. (G H z) 21 Cl) "--' t--1---t--t-+--+-+--+-+--+-, l--+---+--+-1--+-+--+-+--+-, "'O 8~~-0 -......,....... ........................ -,--, N i--+--+-+-+--+-+--+-+ 0 -0 0 0 ;J ~1 H'r~ ; r freq. (GHz) 21 0 .--.-....--.----.--............ ..,........--.--, N i--+--+-+-i--+-+--+-+-+-i I .-+----t--t--+--+-+--+-+--t--'I Cl) '-'1--+---+--+-1--+-+--+-+--t--, 1--t-----+--t--+--+-+--+-+ "'O 0 !-+-+it-~~ N .__.___.___.._.__-..... .......... __.___. I 1 freq. (GHz) 21 1 freq. (GHz) 21 0 ,--,-..,..............,.........--r-...,........--.--, --------1--1-+-+-+-t-+--+-+---+-t C"ll-..-+-+-1--t-+--+-+--+-, t,,~+-+-+-t-+-+-+--+-4 Cl) "--' 1--1-+-+-+-t-+--+-+---+-t 1--1-+-+-+-t-+--+-+---+-t 0 :8 'l. I' \ V f I freq. (G Hz ) 21 l freq. (GHz) 21 Figt1re 6-3. Raw measured mixed-1node s-parameters of matched loads (coaxial) at all ports. Note that sca le s of the parameters are different. common-mode (cd) and common-mode-to-common-mode (cc) responses. The raw data shows a return loss of about 20 dB from 1 GHz to 21 GHz for the dd and cc respon ses, which i s commensurate with the raw perfor mance of a s tandard 8510 VNA. The raw dd and cc transmission show typical 8510 performance. The cd and de transmission is l ess than -20 dB which indicates a rea so nably low level of imbalance in the raw system. The uncorrected mixed-modes-parameter s of mea s ured matched l oads is s h own in Figure 6-3.

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113 0 0 ;... I V ffl1.,. ,, freq.(GHz) 2 1 freq ( GH z) 21 1 freq. (GHz ) 21 ...--.. ,......,1-t-+-+-t-t--+-+-+--+--I N t---1---t-t---t-t--+--t-t---;--<1 l-+--+-+---+--1t-t-+-t----t-i "O ---t-T---t--t---1--1 0 ~ :r---r-r'""1r---1--"t"-r-;-t-;7 Cl) ~+-+--+-t-t-+-t----+-1 ..._., t-t--t--t--t-1r--t-"t"-r......,......, "O t--+-+-+---+--1t-t--t--t-......,......, 0 N a __. .. j 0 0 0\ 0\ 0 ..... .t' ~ ~ II" 0 ~l\l "' C"1 8 1 freq ( GH z ) 21 1 freq. (GH z) 21 I 0 N "
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11 4 e r e n ce, the raw fo u r-ports-pa r a m eter da t a of th e t hr o u g h s a nd m a t c h es a r e s h own i n F ig ur e 64 a nd F i g u re 6-5, r es p ec ti ve l y. Th ese s h o w that th e tr a n s f o rm e d fo u rp o rt d ata a l so h ave goo d d y n amic r a n ge. Th ese a nd ot h e r raw m eas ur e m e nt s in dica t e good raw sys t e m perfo r1 nance, whic h s h o ul d r es ul t is goo d ca librat e d p e rf or m a n ce 0 0 II '.(' n .r :w: OL ""i\f .~ I \ . V "fr e q. ( G H z ) 21 0 0 J. r A i \J "" Ii ,. '1..11 "8 '7 1 8 freq ( G H z ) 2 1 '7 1 0 N I 0 N I ~ "I.J ~ "" / m 0 0 N N IV 1 1 freq. ( G H z ) 2 1 'l 0 N I 0 N I 'V I '{\ hit [ r freq ( GHz ) 21 v1l I freq. (G H z) 21 ~'V ,~ ~ ti 0 N I 0 "' ~ 1 I I 0 N u '7 1 0 0 0 freq. (G H z ) 21 1 1 0 0 N l ,.., \ I ~ l . ~ ~ Wf;t t '+if .) 0 0 ].. N 0 I 1 freq. ( G H z ) 2 1 1 freq. ( G H z) 21 1 ft \ 'I' freq ( GHz ) 21 I ( I I I ~-. ru, -~ .. ~ ~ T freq ( GHz ) 2 1 \ ", J II rt. I I I .., V 0 N I ~ / ~ \ l i (\"'1 .. 1 0 ~ 1 I 0 N I ,r 0 N '7 1 0 freq ( G H z ) 21 L , f r eq (G H z ) 2 1 0 0 w 1 111,t '1W 1. 1 1 (\ .. . freq. ( GHz ) 2 1 1 freq ( GH z) 2 1 I r Y'f~ T 0 0 0 ~., freq. ( G H z ) 21 1 1 . 1A J /' ,. l/ ,~ freq ( GH z ) 2 1 F i gu r e 6-5 R aw meas ur e d f o ur -po r ts-pa r amete r s of m a t c h e d l oa d s ( co a x.ia l ) at a ll po rt s Note th a t sca l es of t h e parame t e r s a r e di ffe r e n t.

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115 6.2 2 PMVNA EtTor Model The calibration and correction of linear systematic errors must be pursued only in the context of an assumed error model The error model dictate s the which errors of a VNA will be admitted to consideration and correction For many years, the generally accepted approach to a VNA error model ha s been to propose an equivalent error model. In such a model, the total effect of all linear systematic errors (there may be many hun dred s of sources of error in a VNA ) i s combined into a simpler set of equivalent error term s [42]. This equivalent error model may have only a few enor terms but, if properly constructed, it exactly reproduces the errors in the VNA. An analogy exists between thi s equivalent error model and a Thevenin 's equivalent network. Just as a simple two param eter Thevenin 's equivalent exists for a linear circuit [22] regardles s of its complexity, so doe s a s imple equivalent error model exists for a complex VNA system with many sources of linear errors. A more direct argument can be drawn from signal flow graph reduction techniques, where the flow graph of the entire VNA, with all errors included, ca n be reduced to a small set of equivalent signal paths [23] A large body of work has been publish in the literature over the past thirty years in the area of VNA calibration The error model s propo se d have tended to become more general, but the question of the number of appropriate error ter1ns and the number and types of calibration standards has been the subject of considerab l e debate The de facto s tandard in VNA calibration for the past fifteen years is the so-called twelve-ter1r1 error model for the two port VNA The rea so n for its prominence is primarily due to its use in the HP8510 VNA. This error model [ 43], shown for reference in Figure 6-6, i s actually two independent half enor model s, one for forward operation of the test se t the other for

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116 I I 1 I S21 I b2 a1 (a) I S11 S22 I Efl E fd Efs I I b1 Efr S12 I I I DUT I I DUT I I S21 I Er, b2 E,l I S11 S22 I Er s E,d (b) I I b1 a2 E,-r S12 1 I I I I Erx Figure 6-6. Twelve-tertn two-port VNA error model. a) Forward error model. b) Reverse error model. reverse. This error model i s typically evaluated through a calibration procedure known as Through Open-Short-Load (T OSL ) 1 It can also be adapted to more sophisticated calibra tion techniques [44] such as Through-LineR e flect ( TRL ) [45] and Line-Reflect-Match (LRM) [ 46]. Although not considered optima l due to the requirement of four standards, the twelve-term error mode l i s well adapted to treating test-set swi tching enors, s ince each half model applied to only one port selection switch position (see Section 6 2.4 for more detail s on sw itching errors). One limitation of the twe lv e-te rt n error mode l is it treatment of port-to-port crosstalk The error model assumes that only two error terms, Efx and E, x, 1 Or by so me other permutation of the standard name s, s u c h as SOLT.

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117 are required to characterize crossta l k under all test con dition s. This assumption has been s hown to be inaccurate under certain conditions [47], and mu st be recognized as a limita t i on of thi s error model. The calibration and correction of the PMVNA begins by adopting an error model that depart s from the tradi ti onal twelve -te rms model. Due to its unique architecture, the PMVNA required a very general error model for accurate calibration development. The model u se d i s the generalized error model for an n-port net work analyzer, including all port-to port leakage errors, which has been intr oduced by Speciale [4 8]. The error model s h own in Figure 6-7, employs a s ingle error network with 2n ports, and the error network i s an eq ui va l ent repre se ntation of a ll linear syste matic errors The error n etwork is expressed with chaining scattering parameters called t-parameters. These t-parameters are mathematically related to s-parameters, and they h ave the prop erty w h ere the re s ulting t-parameters of a se rie s of cascaded networks is equal to the matrix product of the t-parameters of the individual network s [23] (simi lar to ABCD-parameters ). The t-paramR eal VNA with Errors Ideal VNA Error Network DUT n-port 1 1 n+l 1 n-port 2 2 n+2 2 3 3 n+3 3 TEI I TE1 2 S m S a TE2 2 TE2l n-by-n n-b y-n .... n-1 n-1 2n-by-2n 2n-1 n-1 n n 2n n Figure 6-7. Generalized error model for then port VNA.

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118 eters are typically defined only for two port networks so a definition of a 2n-port t-matrix is found in Appendix H. All linear systematic errors in the PMVNA can be represented with this general ized error model where n i s equal to four. In th e case of the PMVNA the error network is a mixed-mode repre se ntation as defined in Figure 3-1. A s imil arity transform relates the enor matrices of the two-port PMVNA and the sta ndard four-port VNA (see Appendix I), thu s the applicability of the error model of Fig ur e 67 is assured. Due to this transforma tion, the calibration theory of the PMVNA parallels that of a standard four-port VNA. Each s ignal path in the error network repre se nt s an unknown error term The error model of Figure 6-7 include s all possible error term s, including all port-to-port leakage paths. For an-port VNA, there are 4n(n-1) l eakage term s out of a total of (2n) 2 error terms. It is important to address the PMVNA calibration problem in the most general Real VNA (w ith Errors) Ideal VNA Error Box DUT r----------, n -po rt 1 n port 1 I 1 n+l I (n even) 2 I 2 n+2 I 2 (n even) I I I I I I n/2 I n/2 n+n/2 I n/2 S ,n I I S a I I n-by-n n by-n I I n/2+1 n/2 +1 n+n/2+1 n/2+1 I I n/2+2 n/2 + 2 n+n/2+2 n/2+2 I I I I I I n n 2n n I I L __________ ..J Figure 6-8. Half-leakage error model for genera l VNA.

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119 te rm s, but there are some simplifications of the erro r model that are also of interest where some of the leakage paths are neglected One useful simplification sp li ts the measurement ports into two groups, and neglects all error te11r1 s connecting the two groups while preserving all error ter1r1s of each group, as shown in Figure 6-8. This unique s implifi cation arises because a sing l e mixed mode port is comprised of two sing l e-ended ports This simp lifi cation particu l arly app l ies when the PMVNA is used in wafer-level measurements. Typically, the port-to-port leak age in a wafer-probe syste m must be neglected, de s pite the fact the leakage can be sign ifi cant, since the l eakage is a strong funct i on of probe placement and is not a static error. (A ttempting to correct for leakage between probes can l ead to significant errors in cor rec te d measurements if the relative po s ition of the probes are moved afte r calibration ) As implemented in C hap ter 4, this error model is app l icab le as each PMVNA wafer probe Real VNA ( with Errors) Ideal VNA Error Box DUT r----------, n-port 1 n-port 1 I 1 n+l I (n even) 2 (n even) 2 I 2 n+2 I I I I I I I 3 3 n+3 3 I S m 4 4 n+4 I 4 Sa I I n-b y-n n-by-n I I n/2+1 I I I I I I n-1 n-1 2n-l n-1 I 2n I n n n I I L __________ J Figure 6-9 Pair leakage error mod e l for general VNA.

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120 (35] is a static component with two single-ended probes. This error model, which will be called the half-leakage model, is a special case of the full model. With the half-leakage model there is a total of 2n 2 error terms ( note that n must be even for this model to apply). Another useful simplification splits the measurement ports into pairs, and neglects all error terms connecting the two group s while preserving all enor terms of each pair a s shown in Figure 6-9 This simplification i s similar to the half-iso l ation model, but is more appropriate for mixed-mode measurements with more than two mixed-mode ports This simplifications similarly applies when the PMVNA is used in wafer-level measurements This error model, which wi l l be called the pair-leakage model, is a special case of the full model. With the pair-leakage model, there is a total of 8n error terms (note that n must again be even for this model to apply). For the two mixed-mode ports of the PMVNA, the pair-leakage and the ha l f-leakage models are identical Real VNA (with Errors ) Idea l VNA Error Network DUT r----------, n port 1 I I n-port 1 n+J 1 (n even) I I I I 2 I 2 I n+2 2 I I I I Sm I I Sa n-b yn n-by n I I I I I I n I I n 2n I I n L ____ ___ _ .J Figure 6 10. No leakage error model for general VNA.

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121 The second simplification of the error model neglects all leakage between ports. This greatly reduces the number of error terrns in the model leading to a simpler calibra tion problem. This simplified error model shown in Figure 6 10, will be called the no leakage model. The no-leakage model can be applied when the leakage levels in the mea surement s ystem are insignificant ( compared to the DUT), such as a system using coaxial connector measurement interface s Such system s have significantly less crosstalk between ports than wafer probe interfaces. The no-leakage model must also be used if none of the leakage errors are static, such as with independently moving wafer probes. The no-leakage model also a special case of the full model, leaves a total of 4n error ter1ns. 6.2.3. Development of Calibration Equation The development of the calibration procedure for the PMVNA continues with the development of the fundamental relation that describes the calibration problem This so called calibration equation describes the relationship between measured s-pararneters, actual s-parameters, and error terms. A general formulation of the calibration has been introduced by Speciale [48], and this section will initially follow this published work. The development will start with variable definitions as illustrated in Figure 6-11: Sm are the measured s-parameter s (with linear sy s tematic errors) TE (SE ) are the t-parameters parameters) of the error network, and Sa are the actual ( error-less) s-parameters of a DUT. Note that the normalized waves have been expressed as n-dimensional vectors, as

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122 de sc ribed in Appendi x H. As further e xplained in Appendix H, the matrix equation for T E may be expanded in t e 1 rn s of the partition s b1 = T E ll a2 + T E 12 b 2 ( 6-1 ) a1 = T E2 1a 2 +T E22 b 2 where T E iJ are the four n-by n partition s of T and a i and b i are t h e n-dimen s iona l a and b-wa v e ve c t or s, re s p ec ti v ely Similar l y, b 1 a = S aa i a but from Figure 6-11 it c an be s een tha t a 1 a = b 2 and b 1 a = a 2 s o a 2 = S a b 2 By s ubstitutin g thi s l a s t expre ss ion into ( 6-1 ), on e find s b1 = ( T E 11 8 a +T E 1 2) b 2 ( 6 -2) ( 6 -3) Th e la s t equ a tion can be re-arranged to find ( 6 -4) I n Err o r n a1 Network b 2 aal I I S EH TE I I n port I T E 11 T E 1 2 I DUT I I I n T E 22 T E2 n I b1 a 2 b a t Fi g ure 6 11 Ve c t o r eq u iva l ent of ge n e rali ze d error mode l for th e n port VNA.

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123 Combining (6-4) and (6-2) However, it is true that b 1 = S,na 1 so, by observation, 1 S ,n = (TE 11 Sa + TE 12) ( T E2 l Sa + T E22) By a series of matrix multiplications and inversions, (6-6) can be expressed as (6-5) (6-6) (6-7) This equation is the fundamental relationship call the calibration equation, on which the PMVNA calibration will be developed. From the calibration equation the basic concept of VNA calibrations can be readily observed. By measurement (Sm) of known devices called standards (Sax), the unknown error ter1ns (TE) can be mathematically found. In general, VNA calibrations require the application of multiple calibration standards. Each standard is measured by the VNA (Smi), and the actual s-parameters of the standards are assumed to be known (Saxi for the i-th standard). For each standard, the calibration equation (6-7) applies. In terms of an n-port VNA, the error network is represented by a 2n-by-2n unknown network, and each s-parameter matrix is an n-by-n matrix. The matrix equation (6-7) can be expanded, and the rest1lting scalar equations are linear in the elements of TE The set of all scalar eqt1a tions can then be re-written as (6-8) where tE is a column vector comprised of the elements of TE [46, 47] Given that m differ ent calibration standards are applied, the coefficient matrix,AE, has dimensions (m n 2 )-by

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124 (2n) 2 and tE has dimensions (2n) 2 -by-l. The mathematical solution of (6-8) now becomes the key step in the PMVNA calibration process. 6.2.4. Switching Errors and Non-Pure Mode Generation The development of the calibration equation ( 67) is predicated on the assumption that the error model remains static throughout the calibration process and through any sub sequent measurements. Referring to Figure 4-4, one can see an RF switch to set the stim ulus mode, as well as two HP8517 test sets, where each test set uses an RF switch to set forward or reverse operation [43]. By means of these three switches, the PMVNA has four distinct modes of operation: differential forward, differential reverse, common-mode for ward, and common-mode reverse. The changing of the switch positions violates the pri mary assumption of the error model, however. By changing the switch positions, the error model also changes, and is no longer static. While in a single switch state, the error model is static, so the errors caused by switches are called quasi-static. These quasi-static errors must be effectively removed before the error model from the previous section can be applied to a calibration. Another issue with the measurement of raw mixed-modes-parameters with the PMVNA is imperfections in the generation of a pure-mode stimulus. As shown in Figure 4-5, the PMVNA generates the differential and common-mode stimuli from a 0 / 180 hybrid power splitter. It has been shown that any imbalances in the splitter, together with any phase and magnitude imbalance in the paths of the HP8517 test sets, will gener ate a spurious mode simultaneously with the desired mode (see Chapter 8). These imbal ances can cause spurious modes of significant amplitude. If the mode imperfections are neglected, and the measured response of a device to be attributed to the nominal mode

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125 only, then significant inconsistencies can occur in the raw mixed-mode s-parameters. For accurate calculation of raw mixed-mode s-parameter s any imperfections in the stimulus must be characterized Additionally the changes in s witch position s cause changes in the amount of imbalance in the stimuli. Again, the s e switch effects also violate the static requirement of the error model The removal of the s witchin g effect s and the s timulu s imbalance can be achieved through the application of all eight samplers in the PMVNA This approach is an exten sion of two-port VNA techniques [44] The traditional model for the effects of imperfect s witches i s shown in Figure 6 12 for a two-port VNA. The model as s umes that all system atic errors have been repre s ented in the error network so the directional couplers can be considered to be ideal, or error-free. The switch is typically s aid to have some non matched terminating impedance cau s ing it to be non-ideal. By measuring a 1 a 2 b 1 and b 2 with the four down-mixers of the test set s at both switch position s, the raw s-parame ter s can be calculated Traditionally this approach of using all four down-mixer mea s ure"'I J I I .. 'I 1 J Ideal 2-port En or Directional "v I u Network DUT / Ill tll Coupler s 2 Imperfe c t 2 :l>
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126 ments is only used during certain s tep s in a TRL or LRM calibration. After the calibration is complete, the TRL/LRM error coefficients are tran s lated to the traditional twelve-port error model [ 49] of Figure 6-6. With thi s approach, only three down-mixers are used dur ing DUT measureme11ts (a 1 b 1 b 2 for forward and a 2 b 1 b 2 for reverse). The PMVNA as illustrated in Figure 4-3, i s equipped with two HP8517 A te s t sets, each having four down-mixers for a total of eight down-mixer s. A generalized model for all switching errors and imbalance errors is shown in Figure 6 13. This model can be u sed to model any s ystematic errors that change a s a function of switch se lection. By using mea s urements at all sa mpler s for each switch position the effects of both switching errors and mode imbalance can be removed from the measured s-parameters. Collecting all a and b data into matrices A= DF DR CF CR al al al al DF DR CF CR a2 a2 a2 a2 DF DR CF CR a3 a3 a3 a3 DF DR CF CR a4 a4 a4 a4 B= bDF bDR bCF bCR 1 1 1 1 bDF bDR bCF bCR 2 2 2 2 bDF bDR bCF bCR 3 3 3 3 (6 -9 ) bDF bDR bCF bCR 4 4 4 4 where the s uper sc ript indicates differential ( D ) or common-mode (C) drive and forward (F) or rever se (R) drive, and the s ub sc ripts indicate port number (1, 2, 3, 4). Expressed with the vector notation of Appendix H, (6-9) become s A = -op DR -cp -C R a a a a B DF -DR -CF -C R b b b b (6-10) With these matrices the raws-parameters can be calculated as S = BA 1 m (6-11)

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127 Equation ( 6 11 ) i s the s olution to a sys tem of sixteen equations with the sixteen raw s-parameters as unknown s, throu g h which the raw sparameter s of the DUT are sepa rated from the effect s of the imperfe c t s witches and i1nperfect s timulu s generation. By applying this approach to every calibration s tandard measurement a single s tatic error model can be applied, and the calibration equation ( 6 -7) can be used Additiona l ly, all s ub se quent DUT m eas urements are al so made using equations (6 9 ) and ( 6 11 ) b1 bn S' sw Switch S-parameters (i-th position) l l a asrc,l a asrc,n l a src Source Input Signa l s (i-th positio n ) Error Network a1 an+J a1 an a2n a,i b2,1 1-__,,. ----1 b n Figure 6-13. General sw itching error model signal flow g raph DUT S a

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128 6.2.5 Solution of the Calibration Problem The construction of the ca1ibration equation (6-8), and its subsequent solution, are the heart of calibration process. The conditions under which a solution to (6-8) can be found will first be presented in terms of a general n-port problem. These conclusions will then be applied to th e ca libration of the PMVNA. There has b een a variety of s tatement s m a de in the literature about the minimum number of s tandard s required for a calibration of a VNA In the original presentation of the general n-port error model, it is sa id that only three n port standards are required to solve all error terrns including all leakage terrns [ 48], but is later changed to five [50]. In 1991, the minimum number is s tated as four s tandards required to solve the general error model with (a pparently ) all leaka ge path s on a two-port VNA [47] M a ny other calibra tion te c hniques for VNAs with two three and four ports have been published in the la st decade with a the number of s t a ndard s u se varying from three to ten [51 53] Thi s sec tion attempts to re so l ve the ambiguity s urroundin g the number of standards requir e d for a so luti o n to the calibration e qu atio n For purpo ses of this examination, a so lution i s valid for calibration only if it i s unique within one arbitrary scalar. That i s, if ts i s a valid so lution vector of (6 -8 ), then the only other solution vectors that exist are a.t s, where a is any complex sca lar In other words, th e Null-space of AE must be of dimen sio n one [31] For ease, thi s type of so lution will be ca lled an ordinary solution. Furtherrnore this section will consider only the gen eral enor model of Fi gu re 6 -7, with all leaka ge path s in c luded, and the three s pe cia l cases illu s trated in Figure 6-8 to Figure 6-10. For an n-port calibration, a single sta ndard will be co n s id e red to always haven-ports, r ega rdle ss of actual co n st ruction of the phy s ic a l sta n

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129 dard. For example, an n-port match standard may be constructed of a group of n indepen dent one-port match loads, but for purposes of discussion the group will be considered as a single standard. Due to the special cases of the etTor models, calibration standards will be considered to be either a reflection standard or a full standard. A reflection standard is defined as a group of n one-port reflection standards (such as then-port match example above); a full standard is defined to have transmission between at least two ports. For the general error model, both types of standards are treated the same, but in the case of the half-leakage, pair-leakage or no-leakage models, the reflection standards generate fewer sets of measurement data (hence, fewer equations) than the full standards. The determination of the number of required standards for ca l ibration is based on consideration of the rank of the coefficient matrix, AE. For an ordinary solution to (6-8), matrix AE must have a rank of exactly (2n) 2 1. Recall that with m standards, AE has dimensions (m n 2 )-by-(2n) 2 For the full error model, each standard generates n 2 equa tions. It has been found that AE will have rank of exactly (2n) 2 1 only with five or more n-port standards. This means that 5n 2 equations are generated for the solution of 4n 2 error terms, so the system of equations (6-8) is over-determined. (The over-determined nature of the system of equations can be used to red11ce the number of known s-parameters of Sax, which can give significant accuracy advantages.) Furthermore, with four standards the matrix AE is square (same number of equations and error terrns), but the rank of AE is 4n 2 -n. Similar conclusions can be made about the special cases of half-leakage, pair-leak age and no-leakage mode l s All conclusions are summarized in Table 6-1.

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130 Table 6-1. Calibration summary. Eq n s fro m S t ds M i n. Std w/ Square A E Min. Stds Error for Model Number of Ordinary Type Unknowns Fu l l Reflect No. Std Rank(A) Sol u tion Full-leakage 4n 2 n2 n2 4 4n 2 n 5 Half l eakage 2n 2 n2 n 2 /2 2 2n 2 n 3 ( n even) Pair l eakage 8n n2 2n depends depends depend s ( n even) onn on n on n No-leakage 4n n 2 n depends depends depends onn onn on n Conclusions about the pair-leakage and no-leakage model are not as general, since the minimum number of required standards depends on the number of ports, n. For exam ple, with the no leakage model, if n is four (as for the PMVNA) at least two full standards are required for rank of 4n 1 = 15 This is again over-determined, with one full standard giving a square AE matrix with rank of fourteen. In contrast, if n equals two, at least three standards are required (for example: LRM [ 46] and TRL [ 45]). These properties have been found through the use of numerical s imulation s. The simulations have been performed with a program written in Mathematica [54]. The pro gram allows generation of a coefficient matrix, AE based on postulated error terms and calibration standards, for any number of ports, standards, and leakage model. With ran domly generated error terms and calibration s tandards (fo r n=2, 3, 4 and 5 ), the previous conclusions about n-port calibrations have been found inductively. For more information about this process, see Appendix J.

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131 Of course, the rank of the coefficient matrix, A, is affected by the type of each standard used in calibration. It has been found that at least one of the standards must be constructed so that at least n-1 non-zero transfer functions exist between its ports In other words, such a standard interconnects all of the VNA's measurement ports simultaneously. Hence this standard will be called a generalized through standard. It is important to note that the generalized through does not necessarily provide low loss interconnection between the ports. Without a generalized through standard as one of the minimal set of standards, the rank of the coefficient matrix will not be sufficient for an ordinary solution to the calibration equation. As its name implies, the generalized through is a generalization of the through standard of two-port VNA calibrations For a two-port VNA, the through provides trans fer between all ports However with more than two ports, the generalized through stan dard is less familiar It can be shown that one or more two-port through used as a single n-port standard, is not sufficient for an ordinary solution to (6-8). Furthermore, use of 1 2 3 n-1 n R 3 Rn I Rn+I Rn Figure 6-14. Preferred embodiment of n-port generalized through standard.

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132 multiple pairs of throughs, each connecting different ports, will not provide sufficient rank in AE for an ordinary solution. With three or more ports, the implementation of a generalized through standard is no longer straight forward. The suggested implementation of a complete n-port standard is shown in Figure 6-14. This network provides the required connections between all measurement ports. This type of network will be called a star network. The star network is particularly practical to implement for wafer-probe standards (See Section 6.2.7). By isolating the ground connection of the network, the value of each resistor can be measured directly with a two ter1ninal Ohmmeter. This property is important during fabrication of star standards when trimming individual resistors is necessary. 6.2.6. Coaxial Calibration Standards The PMVNA calibration has been implemented with the application of the above conclusions about general VNA calibrations. With n being four, the PMVNA calibration standards can be defined in physical terms. For this section, calibration standards with 3.5mm coaxial connectors are used. Five standards are combined to create the calibration kit: (1) a four-port match (four 50Q loads), (2) a four-port short (four offset shorts), (3) a four-port open (four offset opens), (4) a pair of zero-length through lines (by connecting test cable of ports one to three and two to four), and (5) a four-port resistive star network as the generalized through. The star network is constructed from two resistive power dividers [55] connected as shown in Figure 6-15. To use the star network as a calibration standard, its s-parameters must be accu rately known. The star network, as shown in Figure 6-15, has been characterized through a series of two-ports-parameter measurements, with the unused ports terminated with

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133 50.Q loads. By makin g s ix two port measurements all of the four-ports-parameters of the star network can be found. For this proces s, the VNA ( operated as a standard two-port VNA ) i s calibrated with the Hewlett-Packard 85052B precision 3.5mm calibration kit [56] using the Through-Short-Open-Load (T OSL ) technique with sliding loads. When making the two-port measurements of the s tar network, the remaining ports are terminated with the 50.Q loads from the 85052B kit and the se are assumed to be perfect matched load s. These loads have return loss of no less than 35dB, and this assumption limits the accuracy of the four-ports-parameters of the star network There are published methods that can remove this assumption [57], but for this work the assumption of perfect matched loads is reasonable Traditionally calibration standards are modeled by s imple equivalent circuits with a sma ll set of parameter s which allow calculation of the standards' s-parameters at any 3.5mm (F) 166 ( 1 ) /i '.Q a r\0 \0 3.5mm (F) 3.5mm (M) (3) b'\Q; \.b Lucas Weinschel Model 1580 adapter HP8 3059A 3.5m m (M) t o 3.5mm (M) 3 .5 mm ( ~F~ --. (2) 3.5 mm (F) a r--\0 \0----Lucas Weinschel Model 1580 ( 4 ) 3.5mm (M) Figure 6-15. Schematic of coaxial s tar network s tandard.

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134 freq11ency. The star network has more complex behavior than traditional standards, and accurate modeling would require a complex equivalent circuit with many parameter s. To avoid this difficulty, interpolation between measured s-parameters i s used to generate the star network's s-parameters at an arbitrary frequency point. The interpolation is done through a tenth order polynomial fit, over a range of nine frequency points about the de s ired frequency for the real and imaginary parts of each s-parameter. This interpolation process is implemented in Lab VIEw, and represents a useful general capability for cali bration s that allows a simple means to incorporate non-traditional standards. The remaining s tandards are treated with traditional models [43]. The calibration parameters of the shorts and opens u se d for the PMVNA calibration are provided by the manufacturer [58 59] With the published parameters theoretical s-parameters (Sax) of the short and open can be calculated. The through-lines are assumed perfect zero-length throughs, and the S ax is defined accordingly. The S ax of the star network is generated through interpolation as described above. The 50Q load Sa x is also generated through interpolation of measurements of the load standards done with sliding load calibrations. 6.2.7. On Wafer Calibration Standard s Calibration standards for the PMVNA have also been constructed for on-wafer measurements. These standards have been designed so the plane of calibration is the wafer probe tips. The standards are thin film metal-on-ceramic structures, and two ver s ion s have been designed and fabricated, one for 150m pitch probes, and another for 500m pitch probes. Both have been fabricated with thin-film gold (4m thick+/0.25m) on a poli s hed al11mina substrate (Er~ 9.9, tan8 0.001). The resi s tive layer is a nickel-chrome ( NiCr) metal approximately 400A thick, with a sheet re s istance of about 40

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135 Q/square. After fabrication, the resistors were tuned to the desired DC values ( +/0.1 % ) with a laser. All photo-masks required for fabrication have been generated in the Univer sity of Florida Microelectronics Laboratory, and all fabrication has been done in the Motorola Thin-Film Research Laboratory in Plantation, Florida. In accordance with Section 6.2.5, five types of standards have been fabricated: (1) a four-port match (four 50.Q loads), (2) a four-port short (four offset shorts), (3) a four port open (four offset opens), (4) a pair of zero-length through lines (by connecting test cable of ports one to three and two to four), and (5) a four-port resistive star network as the generalized through. The layouts of the 150m pitch standards are shown in Figure 6-16 to Figure 6-20. GND Resistor (50.Q trimmed) '-----~ 1 Mixed-Mode Port 1 GND 2 GND GND 3 Mixed-Mode GND Port2 4 GND Figure 6-16. Physical layout of l 50m pitch four-port match standard.

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GND 1 Mixed Mode Port 1 GND 2 GND 136 GND 3 Mixed-Mode GND Port 2 4 GND Figure 6-17. Phy s ical layout of l 50m pitch four port short s tandard. GND 1 Mixed M o de Port 1 GND 2 GND GND 3 Mixed Mode GND Port2 4 GND Figure 6-18 Physi c al layout of l SOm pit c h four port open s tandard

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GND 1 Mixed-Mode Port 1 GND 2 GND 137 GND 3 Mixed-Mode GND Port 2 4 GND Figure 6-19. Physical layout of 150m pitch pair-of-throughs standard. GND 1 Mixed-Mode Port 1 GND 2 GND GND 3 Mixed-Mode GND Port2 4 GND Figure 6-20. Phy s ical layout of l 50m pitch four-port star s tandard.

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138 For both sets of on-wafer calibration standards, the theoretical s-parameters (Sax) are calculated by electromagnetic simulation. The simulator used is Hewlett-Packard 's Mom en tum [40]. The simulations assumed a substrate with a relative dielectric constant of 9 .9 and a lo ss tangent of 0.001. The gold was assumed to be 4m thick with a conductiv ity of 5.8x10 7 Sim The resistive layer s were asst1med to be infmitely thin resistor mate rial with a defined sheet resi s tance of the nominal design value of 50.Q/square After simulation, the resulting s-parameters have been used to generate the theoretical s-param eters of the calibration standards, S ax, through the interpolation process described in Section 6 2 6 The star standard of Figure 6-20 has s pecial usefulne ss in the calibration of a PMVNA This standard can help enhance the accuracy of the se n s itive mode-conver s ion parameters. Theoretically, the accuracy of a calibration i s limited by the accuracy to which th e sta ndards are known. However, due to noise, repeatability l imitations, and numerical and/or mea s urement dynamic range limitations, the accuracy of a calibration is greatest in the neighborhood of the sta ndard s In other words, a corrected DUT measure ment will have greatest accuracy if its respon se is nearly that of at least one of the stan dards. PMVNA can measure the conversion between modes, such as differential-to common-mode conversion. Different devices can have dramatically different level s of mode conversion, making accurate error conection difficult for the mode conversion respon ses.

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1 2 I I Mixed-Mode Port 1 139 R1 R3 R2 R4 Rs Figure 6-21. Po ss ible embodiment of mode conversion standard. I I 3 I I 4 I Mixed-Mode Port 2 The basic advantage of the sta r standar d i s that it allows the design of calibration stan dard s with specified amount of mode-conver sio n The basic network for this standard is the four-port resi stive s tar and it s circuit diagram is given in Figure 6-21. By choosing the appropriate values of the re s i sto rs in the stat network the mode conversion pruameters of the network can be varied over a wide range Practical imple mentation s of this standa rd have been able to reliably achieve mode -co nversion magni tudes from -6 dB to less than -80 dB 6.3. Pha se Off set Pre-Calibration A unique aspect of the PMVNA as implemented for this work is the unknown pha se relationship between the VCO s in the two te s tse ts, as de sc ribed in Section 4.2.2. The pha se offset between the test -se t s is not a sys tematic error in the strict se n se b eca u se it changes as th e RF switc h po s ition s change. Furthermore, direct application of switching error model s, s uch as Figure 6-13, are in s ufficient to overcome the problem of an

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140 unknown phase relation between the VCOs. Use of (6-9) and (611) with the raw a and b data from implement e d PMVNA re s ult s in incorrect calculation of the raw s-pa rameter s. This is due to the fact that there is an unknown phase offset between the data from test-set A and te s t -se t B for eac h set of RF sw itch po s ition s. However the phase offset relation s hip can be characterized and corrected. The process operates on the raw a and b data from both test-sets, and i s independent of the pri mary calibration process Furthermore, the phase offset co1Tection must be applied to al] a and b data in the primary calibration. For these reason s, the phase offset calibration pro cess is called a pre-calibration procedure. The phase offset between the test sets i s actually a function of s-parameters of the DUT. As a result, the phase offset pre -cal ibration is broken into two steps. The first step calculates the pha se offset for each of several so called offset standards. The second step uses the se results to calculate the parameters that will describe the pha se offset of an arbi trary DUT. 6.3.1. Pha se Off se t Standards A phase offset standard is a device that is measured to allow calculation of the te s set phase offsets. It i s significantly different from the type of standards u se d in the pri mary calibration, in that its actual s-parameters do not need to be known. The offset stan dard doe s have some restrictions in its general characteristics, as will be discussed Conceptually, a offset standard is a device that allows one test-set to measure the response of the other test-set.

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141 6.3.1.1 First Principles The entire phase offset pre-calibration is based upon two observations about the PMVNA as implemented in Figure 4-3 The first of these is that, for a particular PMVNA test-set RF switch configuration, the phase offset i s highly repeatable. This means that, for example, when PMVNA is switched into differential-forward (DF) mode, the phase offset between the VCO in test-set A and that of test-set Bi s always the same within a high degree of accuracy (as long as the DUT remains unchanged). The seco nd important observation is that the PMVNA test set has four additional RF switch configurations that have not been used to this point Referring to Figure 4 -4, one can see there are three independent RF switches (SW 1, SWA and SWB) each having two positions, giving a total of eight possible switch configurations. Four of these po s i tions represent the fundamental operation of the PMVNA ( i. e. DF DR, CF, and CR). The remaining four positions are not used in the normal PMVNA operation, and together the se positions are called seco ndary operation. Measuring a device (in thi s case a offset standard) in both the fundamental and secondary modes should give the same set of four-port (or mixed-mode) s-parameters, within the dynamic range of the PMVNA. Symbolically, 8 rn f = 8 n is (6-12) where Sn if i s the raw s-pa rameter s mea s ured under fundamental operation, and S ms is the raws-parameters measured under seco ndary operation. This equality holds only if the data from the eight down-mixers of the PMVNA have no relative pha se offset. Therefor e, equation (612) can be u se d to calculate any unknown phase offsets. This important approach, based on (612), shall be called the principle of equality of operation. It is

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142 important to note that (6 -12) contains only raw uncorrected s-parameters; this allows the offset pre-calibration to be perfor111ed indep e ndently of the primary calibration as the error term s of the PMVNA do not need to be known 6 3.1.2. Offset Mod e l The VCO in a 8517 test-set pha se -lock s to a particular phase point on the RF sig nal input (see Figure 4-4). For example, if the PMVNA i s in DF mode where port s one and three are driven 180 out of pha se, the VCO in test -se t A is phase-locked to a~F. The down-mixed a~F data will always have a pha se that corresponds to a particular phase of the RF a~F signal. For example, if the VCO is locked to a~F, then the 0 point of the down mixed data will alway s correspond to the sa me pha se point of the RF s ignal say 0 The actual phase points are arbitrary, but the correspondence always hold s. The two test sets will lo ck to their appropriate pha se points and their sa mpled data will indicate a fixed pha se difference between the two drive s ignal s (a~F and a~F for this example), regard le ss of the actual phase differen ce between them Before proceeding, a mathematical model of the pha se offset must be established The simplest way of describing such an offset i s by stating that, for a given measured a or b-wave the actual quantity is equal to the measured quantity multiplied by an unknown complex scalar. For example (6 -13 ) where the subscript c indicates the corrected quantity and m indicate s the measured quan tity. The model of (6-13) allows both a phase and magnitude offset, but actual offsets found from measured data (as will be de sc ribe later) have unity magnitude s. In general the A matrix become s

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A = C 143 DF DF DR DR CF CF CR CR al XI al XI a l Xi al xl DF DF DR DR CF CF CR CR a2 x2 a2 x2 a2 x2 a2 x2 DF DF DR DR CF CF CR CR a3 X3 a3 X3 a3 X3 a3 X3 DF DF DR DR CF CF CR CR a4 X4 a4 X4 a4 X4 a4 X4 (6-14) This can be simplified with the use of the Haddamard matrix product [61] (also known as the entry-wise product ) A = A X C m (6-15) where DF DR CF CR Xl XI Xi Xi DF DR CF CR X= X2 X2 X2 X2 DF DR CF CR (6-16) X3 X3 X3 X3 DF DR CF CR X4 X4 X4 X4 No assumptions have been made at this point as to which a orb data have actual phase off sets. If no offset exi s ts, the correspondi n g offset variab l e, x, will be unity Similar expres sions can be stated for the b data (6-17) Applying (6-15) and (617) to the equality of operation equation (6-12), and re-arranging one finds (618 ) where x 1 a nd X s are the phase off se t matrices for the fundamental and secondary modes of operation respectively. In general, the matrices are not equal

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144 By solving equation (6-18) for x 1 and X s, the phase offsets for each a and b-wave can be found. However, the solution of this equation is not easily found, as it is non-linear in the elements of x 1 and Xs. For this reason, the approach presented above has been mod ified to allow a simpler so lution to the problem This modified approach is presented in the next section. 6.3.1.3. Modified T-Matrix Solution The enabling simplification begins with the observation that a single test-set has no phase offset between its corresponding a and b data. For example, a 1 DF and a 3 DF have no phase offset since they are measured by the same test-set By using this fact, together with the concept oft-parameters, (6 -18 ) can be restated as a linear equation. Grouping port s one and three as a pair and two and four as another, a new T matrix can be defined in a similar fashion as in Appendix H. The new T-matrix can be developed by considering an arbitrary four-port matrix, where the aand b-waves are related by h1 sll s12 S13 S14 al b 2 s21 s22 s23 s24 a2 (6-19) b3 5 31 5 32 S33 S34 a3 b4 5 4 1 5 42 S43 S44 a4 The column of b-waves is the ''o utput '' vector, and the column of a-waves is the ''input'' vector. By re-arranging the aand b-waves, a new set of '' input '' and ''output'' vectors are defined as al h2 a3 b4 (6-20) u v' bl a2 b3 a4

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145 where the primed quantities denote the new port grouping (in contrast to that described in Appendix H which i s used throughout the primary calibration). The new T-matrix equation becomes v' = T 'u (6-21) The new T-matrix T ', can be expressed in ter1r1 s of the originals-parameters of the net work, but the actual s-parameters are not of interest. In a similar fashion as the A and B matrices, one can define or, more s imply DF DR CF CR al al al al DF DR CF CR a3 a3 a3 a3 bDF bDR bCF bCR 1 1 1 l bDF bDR bCF bCR 3 3 3 3 If If bDF bDR bCF bCR 2 2 2 2 bDF bDR bCF bCR 4 4 4 4 DF DR CF CR a2 a2 a2 a2 DF DR CF CR a4 a4 a4 a4 (6-22) If (6-23) If In these matrices, the subscript f denotes fundamental operation of the PMVNA. For sec ondary U' s and V s si milarly defined (but have different numerical values). The full matrix expressions of (6 -21 ) are (6-24) The phase offset in the meast1red a and b data can treated in the same fashion as the previ ous section. The phase-corrected a and b data can be expressed as

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U' c f = v cf = 146 DF DF DR DR CF CF CR CR al X 1 a 1 X 1 al X 1 a 1 X 1 DF DF DR DR CF CF CR CR a3 X3 a3 X3 a3 X3 a3 X3 b DF DF bDR DR bCF CF bCR CR 1 XI I XI 1 xl 1 xl b DF DF bDR DR bCF CF bCR CR 3 X3 3 X3 3 X3 3 X3 b DF DF bDR DR bCF CF bCR CR 2 X2 2 X2 2 X2 2 X2 b DF DF bDR DR bCF CF bCR CR 4 X4 4 X4 4 X4 4 X4 DF DF DR DR CF CF CR CR a2 x2 a2 x2 a2 x2 a2 x2 OF OF DR DR CF CF CR CR a4 X4 a4 X4 a4 X4 a4 X4 If If or more compactly U U' X' c f n1f lf V c f = V mf X12f where the X' If and X1 2 f are the new offset matrices, and are defined as OF DR CF CR OF DR CF CR X1 XI Xl Xl X2 X2 X2 X2 DF DR CF CR DF DR CF CR X'If = X3 X3 X3 X3 OF DR CF CR X4 X4 X4 X4 OF DR CF CR XI Xl XI Xl X2 X2 X2 X2 DF DR CF CR DF DR CF CR X3 X3 X3 X3 X4 X4 X4 X4 If (6-25) (6-26) (6-27) (6-28) If Notice that X' lf contains only phase terms from test-set A, and X1 2 f contains only phase terrris from test-set B. In a given configuration (e. g. fundamental DF), the relative phase offset in the a and b data from a single test-set will be zero. As a result, X1 lf and X' 2 f simplify to a single offset variable for each row

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X'1 f = DF DR CF CR Xl X 1 X 1 X I DF DR CF C R x 1 x 1 x 1 x 1 DF DR CF CR Xl X I Xl X l DF DR CF C R X I X l X l Xl 147 X' 2f = If DF DR CF CR X2 X2 X2 X2 DF DR CF CR X2 X2 X2 X2 DF DR CF CR (6-29) X2 X2 X2 X2 DF DR CF CR X2 X2 X2 X2 If Sin ce only the relativ e pha se offset b e tween th e two te s t -se t s i s important either test-set may be considered as the phase reference. Accordingly X' I f can be chosen arbitrarily The Haddamard identity matrix, J i s chosen, where all the elements are unity ( where U 'mf l = U' mf ). As a result U' cf = U 'mf (6-30) Dropping the numeric s ub sc ript and u s ing/ to denote fundamental operation, the remain ing off se t matrix i s defined as DF DR CF CR xf xf xf xf DF DR CF CR xf xf xf xf DF DR CF CR xf xf xf xf DF DR CF CR xf xi. xf xf Similarly for seco ndary operation U' cs = U' nis V cs = V, ns X' 2s where (6 -31 ) (6-32)

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148 DF DR CF CR XS XS XS XS DF DR CF CR X' 2s = x s x s X s x s DF DR CF CR = ' ' DF DR CF CR X s x s X s X s (6 -33 ) XS XS XS XS DF DR CF CR XS XS XS XS It must be r e membered that th e pha se offset for the fundamental operation is not generally equal to that of the seco ndary operation. Furtherrr1ore, the sw itch configuration de scr ip tor s (e.g., DF ) do not have the no11nal meaning in se condary operation. Combining (6-24) through (6 -33 ), one frnds -1 T 's = ( V ms X' 2s) U' ms The principle of equality of operation ( 6-12 ) can now be re s tated as or, with the unknown phase off se t s, X' 21 and X' 2 s, as (6-34) (6 35) (6 -36 ) Provided that T' and T' s exist, then ( 6-36) is linear in the elements of X' 21 and X' 2 s. Thi s equation ca n be expanded, and re -w ritten as where A x = 0 p p (6 -37) (6-38) and AP i s the coefficient matrix of ( 6 -36). Equation (6-37) can now b e solved with a numerical te c hnique known as s ingular value d ecom po si tion ( SVD ) [60] Thi s techniqu e

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149 finds a solution vector, ts, that minimizes the productAE.ts in a lea s t -sq uare s sense. The SVD approach i s u sefu l for the so lution of the calibration equation where real data is used, where the real data has both random errors due to noise, drift, etc. and residual systematic errors dt1e to imperfe c tly known standards. SVD i s also used in the sa me way in the pri mary calibration (see Section 6.4). Once a solution is found the offsets of the A and B matrix, defined in (616 ), can be directly found as X= 1 1 1 1 DF DR CF CR ~f xf xf ~f I I I 1 DF DR CF CR xf xf xf xf (6 -39) The offset-corrected A c and B e matrix can be found by (6 -15) and (6-17). For examples of actual off set data calculated from measured data, see Appendix L As indicated T 't of the offset standard must exist for equation ( 6-36) to have a so lution. This requirement ha s important implications on the general characteristic of off set standards. Referring to the matrix conversion betweens-parameters and t-parameter s, one can find that a T-matrix will exist only ifs ;! exist. This is true if S 21 :t O With the new definition of the T-matrix for the phase offset calibration, this restriction can be stated in term s of the offset s t a ndard' s s -parameters as (6-40) Phy s ically this restriction means that ''cross-over'' transmission, s 41 s 23 mu s t be distin guishable from ''straight-through'' transmission, s 21 s 43 For a reciprocal offset standard, the re s triction (6-40) means that either s 41 s 23 = 0 or s 21 s 43 = 0 Simply stated, an

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150 acceptable offset standard will connect ports one and two and ports three and four, or it will connect ports one and four and ports two and three, while all other transmission is essentially eliminated These two types of devices are illustrated in Figure 6-22. 6.3.2. Phase Offset Of An Unknown DUT As mentioned earlier, the phase offset between the two test-sets is actually a func tion of the s-parameters of the DUT. As a result the fundamental mechanics of this vari able offset must be described and characterized so that the phase offset of an arbitrary, unknown device can be calculated. 6.3 .2.1. Variable Offset Model The reason for the variation of the phase offset can be found by examination of Figure 6-13. This figure has been simplified, and presented in terms of vector a and waves in Figure 6-23. With the RF switches of the test-sets in the i-th configuration, the switches can be collectively described by an s-parameter matrix s:w. The systematic error s of the PMVNA are included with the s -parameters of the DUT in Sm With the i-th configuration, a particular stimulus condition is generated, a;rc. For the moment assuming that all a and b data is sampled without phase errors, it can be shown that TSA 1 3 TS A 1 TSB 2 4 TSB 2 (a ) Figure 6-22. Schematic of types of acceptable offset standards. a) Straight-through. b) Cross-over. 3 4 (b)

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151 4 4 4 bn am l Sm S S W 4 an bm Switch S parameters 4 ( i th po s iti o n ) S o ur c e l a s rc Input Signa l s ( i th po s iti o n ) Figure 6-23. Simplified genera l switching error model signal flow graph (6-41) (6-42) This clearly indicates that the actual a and b data vary with the DUT raw s-parameters. Referring to (6-41), one can see that the actual phase difference between the com ponents of the a-wave s is a function of the s -parameters of the DUT, Sm. Since the VCOs are always locked to the same phase point of the actual a-waves, the phase offset in the sampled a data is a l so a fu n ct i on of S m As a result the switc h s-parameter matrix, s:w, and the source vector a ; r c must be found for every switch position, i, to allow the calculation of the phase offset for an arbitrary DUT. 6 3 2.2. Using Multiple Offset Standards The ca l culation of s~w and a l r c is accomplis h ed through the use of multiple offset standard s For each offset standard the actual phase offset is calculated as described in

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152 Section 6.3.1.3. The raw a and b data for each offset standard is corrected with the corre sponding phase offset, giving the actual a and b data for each offset standard. Combining (6-41) and (6-42), it is found that (6-43) Grouping the actual a and b data from each offset standard into matrices for each switch configuration, (6-43) becomes S l B l Al = Al SW C + s r c C (6-44) This can be expanded into a non-homogeneous system of equations that are linear in the l l elements of S sw and asrc This can be expressed as (6-45) where A~ is the coefficient matrix for the i-th position, a~ is a column vector of the elet -z t -l ments of A c, and xT is a co l umn vector of the unknown elements of S s w and asr c This system of equations can be solved via SVD. Of course, this process must be done for all switch positions i = {DF, DR, CF, CR}. 6.3 2.3. Calculating the Offset of an Arbitrary DUT Now that the fundamental mechanism for the phase offset has been characterized, the actual phase offset for an arbitrary DUT can be calculated. For any device measure ment the raw a and b data, are collected into a column vector for each switch position The offset vector for this position, xi, can be calculated by the linear expression (6-46) After calculating x l for all switch positions, the offset matrix can be constructed by

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153 (6-47) From this offset corrected data, the raws-parameters of the DUT can be accurately calculated by (6-48) This process i s not restricted in the type of DUT that can be corrected The limita tions of the offset s tandards, namely that the T-matrix exists, does not apply to ( 6-46) 6.3.2.4. Diagonalized Form In practice equation (6-46) can be further simplified. This is due to the fact that s :w is a diagonal matrix within practical limits. That is where l . . S S W = DiagonalMatrix( rll r~ r~ r~ ) -i r = rl r i r' r' 1 2 3 4 T By recognizing this fact, (6-46) can be simplified to -z -i -i -i -i -i r X b m = X am a src With this simplification, the vector offset can be symbolically found as l asrc l X = --. -i -i i am-r m (6-49) (6-50) (6-51) (6-52) where the division is the Haddamard ( entry-wise ) division. Now, the offset matrix can be constructed according to (6 -47 ), and the raws-parameters are calculated by (6-48).

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154 6.4. Calibration Procedure The calibration process for the PMVNA is summarized as follows. First the phase offset pre-calibration is completed. Several offset standards are measured during this pro cess. These standards are generated with the use of a pair of resistive splitters [55] in com bination with various terminations. The test-port cables from ports one and four (one and two) are connected to one resistive splitter, and ports two and three (three and four) are connected to the other resistive splitter. The remaining ports of the resistive splitters are ter1ninated with a variety of one-port devices, such as opens and shorts. This combination of splitters and terminations provide a simple means of generating a large variety of offset standards. The offset pre-calibration is always executed at the coaxial interface, regard less of the type of primary calibration that will follow (i. e., on-wafer or coaxial). The off set pre-calibration is independent of the reference plane of the primary calibration. The primary calibration is accomplished next. Each calibration standard is con nected to the PMVNA, and all down-mixers are measured for each stimulus mode and direction. The raw A and B data matrices are constructed as shown in (6-9). The phase offset of each standard in calculated via (6-47), and the raw measured s-parameters (Sm.x) are calculated as shown in (6-48). The measured s-parameters and the corresponding the oretical s-parameters (Sax) of all standards are used to generate the coefficient matrix, AE, via equation (67). Finally, the calibration equation (6-8) is solved, and the error matrix, TES, is constructed. For the PMVNA, the solution of the calibration equation (6-8) is implemented with a numerical technique known as singular value decomposition (SVD) [60]. SVD is an technique that finds a solution vector, ts, that minimizes the product A Ets in a least

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155 squares sense. The SVD approach is useful for the solution of the calibration equation where real data is used, where the real data has both random errors due to noise drift, etc ., and residual systematic errors due to imperfectly known standards. The SVD solution provides the best least-square-error solution to the calibration data. Since the solution to the calibration problem by this method is actually an estimate of the actual error network, TE, the notation TES is adopted to make a clear distinction. For this work the SVD solu tion algorithm was integrated into Lab VIEW so calibration and correction could be accom plished within the control software. The basic Lab VIEW application does not include a SVD algorithm. As a result a custom C-code routine (from [60]) for SVD has been inte grated into Lab VIEW for this work After the error matrix, T Es, is found, any subsequent device measurements can be corrected through the application of (6-7). First the DUT offset corrected a and b data i s found via (6-47), and the raw DUT s-parameters are calculated by (6-48). Then, the error matrix TES is partitioned as di s cussed earlier, and then (6-7) is solved for S 0 (6-53) which now represents the corrected s parameters of the DUT. For the PMVNA, the s-parameters are expressed in terms of mixed-modes-parameters, but the corrected s-parameters can be transformed into standard four-ports-parameters if desired. For more details on the calibration software, see Appendix G.

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CHAPTER 7 VERIFICATION OF THE PMVNA To provide a verification of the accuracy of the PMVNA calibration it is required to measure some standard other than those used in calibration. By comparing the cor rected s-parameters from the PMVNA to the theoretical s-parameters of the verification s tandard, one can get a measure of the accuracy of the calibration. It is desirable to find verification s tandards where the theoretical s-parameters are as independent as po ss ible from the PMVNA mea s urement system (and calibration s tandards used). It is al so de sir able to use devices that are representative of typical differential device perforrnance. Although verification se t s are commercially available for two-port VNAs, there are no s uch differential verification s tandards (w ith NIST traceable measurements provided ). For this reason, the verification standards used here are provided by a Hewlett Packard 85057B ve1ification kit [62]. This kit contains four two-port standards, each accompanied with NIST traceable sparameter measurements ( these measurements have associated maximum uncertainties, also provided ). For verification of the PMVNA, vari ous combinations of two verification s tandards are measured, and the corrected measurements are compared to the provided s-parameters. While combinations of two-port devices do not represent a general differential device that the PMVNA is designed to mea sure, the 85057B kit provides a readily available means of accuracy verification. The 85057B verification kit contains a 20 dB attenuator, and 40 dB attenuator, a 50.Q air-dielectric tran s mission line, and a 25.Q air-dielectric transmi ss ion line. The stan156

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157 dards have 2.4mm coaxia l connectors, but the PMVNA uses 3.5mm connecto r s. These devices are used despite t hi s incompatibil ity s ince they are readily availab l e to the author. As a result of the connector incompatibility, adapters are used betw een the connectors of the PMVNA (rep r esenting the calibration reference plane) and the verification s tandards. The corrected s -pararneters of the verification device and a d apters are manipulated to de embed the adapters, after which the s -parameters can be compared directly to the provided verifications-parameters. The s-pararneters of the adapters n eeded for the de-embedding Figure 7-1. 0 ,...... ,...... C/.) 0 0 I ~f\l ,,,,. I I' I , / ,. Nl\f f\. I V r-.. A \ I 'J 0.25 freq. (GHz) 25.25 0 0 0 I ,.__ --, ~ 0.25 freq. (GHz) 25 25 0 0 ~\ y \I C"l 0 I 0.25 freq. (G H z) 25.25 0 0 0 I QV' -~ J / 1 1 J A. r I rl-0 \ I 0.25 freq. (GHz) 25.25 Measured s-parameters with adapters de-embedded ( bold) and verification s -p ararnete r s of the 50.Q air-die l ectric transmission line, connected between ports one and three while the 25 .Q ai r-di electric tran s mis s ion line is connected b etween port two and four.

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l/') 0 0 ~ ..._,, ...... Cl'.) <] l/') 0 0 I l/') 0 ...... N Cl'.)
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159 The verification measurements are made by th e PMVNA which has been cali brated using all five sta ndards de sc ribed ear li er, with perfect i so lation between the port assumed All measurements ( ca libr ation and verificat i on) are made with 1024 averages The first verification stan dard measurement is th e simu lt aneo u s measurement of the son air-dielectric tr ansmiss i on line a nd the 2s n a ir -die l ectric transmission line. The son tran smiss ion lin e i s connected between ports one and three while the 2s n transmission lin e i s co nn ected between port s two and four. The measured s-paramete r s of the son 0 0 0 ' f ' ,-... ,-... I V \ ..._,, '--" N Cl') Cl') 0 0 V) V) I 0.25 freq. (GHz) 25.25 I 0.25 freq. (GHz) 25.25 0 0 0 , I , ,-... V ,-... "' -..._; ..._,, N N N Cl') Cl') 0 0 V) V) I I 0 .2 5 freq. (GHz) 25.25 0.25 freq (GHz) 25.25 Figure 7-3. Measured s-parameters with adapters de-embedded (bold) and verification s-parameters ( light : present but not distingui s h able) of the 2s n air-dielec tric transmission line, connected between ports two and four whi l e the son air-dielectric transmission lin e i s connected between port one and three

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160 transmission line are shown in Figure 7-1 toget h er with the s-parameters provided with the verification kit The agreement between the two sets of data is quite good; the error between the sets of data is shown in Figure 7-2 The agreement of S 21 and S 12 is good in bo t h phase and magnitude. The error in the parameter magnitudes, with respect to the ver ifications-parameter s, is less than .05 dB over the entire measurement band, where the error is defined as~ Sij = 201og S~erif 20 l og sJe as Also shown in Figure 7-2, is the maximum uncertainty of the provided s-parameters of the verification standards. For mo s t lr) 0 0 ..... -., C/) <] lr) 0 0 I lr) 0 C"I C/) <] lr) 0 I / factory. uncerta1nt}11 ~-"" J\ .....i \ I A -' V" I y ~mag. 0.25 freq. (GHz) 25.25 factory . / uncertainty AJ, lk, .. ,tf_ 'J .. r't. ,, A ... ~ ... "' ... .. \,I ..... .... Ll mag. 0.25 freq. (GHz) 25.25 lr) 0 factory uncertaint) ~I'.' .. / .. .. .. ""' ... ,., \ ..... V C"I C/) <] ... '" .... t,j -. ,. ... .. .. .... 41 I ~mag. l I I lr) 0 1 0.25 freq. (GHz) 25 .2 5 lr) 0 0 C/) <] lr) 0 0 I ~. -factory / uncertainty ,, I .. ,, I \ "', Ll mai ;. 0.25 freq. (GHz) 25 25 Figure 7-4 Differences between mea s ured s-parameters and verification s -p arameters of Figure 7-3 (sol id ) and facto r y uncertainty of ver i fication s-parameters (dashed). Errors in S 11 and S 22 expressed as the difference of the linear magnitudes of the respective data Errors in S 12 and S 2 1 are expressed as the difference in dB of magnitudes in dB.

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161 of the measurement band, the errors fall within the uncertainty window of the verification data, indicating that the measurements are reasonab l y accurate The phase agreement of the transmission paramete r s is also very good but d u e to space li mitations, the phase of the s-parameters are not shown. The phase error for the transmission parameters less than verif m eas 2 where ~LSij = LSij LSij The agreement between S 11 and S221s not as good, but the reflections r ep r esented are very sma l l (-40 dB to -50 dB), so errors of these magn i tudes a r e not overly ob j ectio n ab l e Furt h ermore, t h e adapter de-embedding process wi l l effectively mask return l osses greater than that of the adapters, which is in the 40 dB 0 0 0 I '\'j" 0\ I '\'j" 0 N .. / v,r .... .. ....... V I....... I/ 0.25 freq. (GHz) 25.25 '' ' 0.25 freq. (GHz) 25.25 '\'j" 0\ I '\'j" 0 N I 0 0 0 I JJ~ '\ '-V ..__ ........ ~I fc: 0.25 freq. (GHz) 25.25 ,.,.. r .. l l~~w K A .. ........ ' \ 0.25 freq. (GHz) 25.25 Figure 7-5. Measured s-parameters with adapters de-embedded (bo l d) and verification s-parameters of t h e 20 dB atte nu ator, connected between ports o n e and three while the 40 dB Attenuator connected between po1t two and four.

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162 range. The error for these reflections are represented in Figure 7-2 in ter1r1s of a (non-rela tive) difference between the magnitudes of the measured and verification data, where verif meas h d h f h d h s .. = s.. s .. W en compare tot e uncertainty o t e vert 1cat1on ata, t e 11 11 11 errors of the reflection parameters are not excessive. Had the de-embedding process not been necessary, the errors would undoubtedly compared more favorably with the uncer tainty bounds. The phase errors of S 11 and S 22 (not shown) vary rapidly over the measure ment band. However the uncertainty of verification data for these parameters is Figure 7-6. tr) 0 0 Cl'.) <] tr) 0 0 I tr) 0 ...... N Cl'.) <] tr) 0 I factory ,. uncertainty_ J>.. -. 7 ,1 .... \/ .. :----~ L .... V,. I mag. 0.25 freq. (GHz) 25 25 imag. CJ J-' ,, ... . . --.. .. ~ .. uncertainty factory 0.25 freq. (GHz) 25 25 tr) 0 factory / uncerta1nt1 I! J ... .A .. .. ...i I Cl'.) <] --,_ .. .. mag. tr) 0 0 .2 5 freq. (GHz) 25.25 tr) 0 0 ..... '-" N N Cl'.) <] tr) 0 0 I -.,. -factory. / uncerta1nt, -J. Ii---'. fw .... .. w 'f mag. 0.25 freq. (GHz) 25.25 Differences between meast1red s-parameters and verifications-parameters of Figure 7-5 (solid) and factory uncertainty of verifications-parameters ( dashed ). Errors i n S l l and S 22 expressed as the difference of the linear magnitudes of the respective data. Errors in S 12 and S 21 are expressed as the difference in dB of magnitudes in dB.

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163 The diminishing magnitudes of S 11 and S 22 cause the phase error of these parameters to be of little significance. From the same PMVNA measurement, the s-parameters of the 25Q transmission line are also determined, and they are shown in Figure 7-3 together with the s-parameters provided with the verification kit Again the agreement is quite good, and the two sets of data overlay each other so well as to make the verification traces barely discernible. The enors are shown in Figure 7-4 in the same format as discussed above. The errors of S 11 and S 22 are less than about .04, which is good considering the large variation in the 0 ...... 0 0 I 0\ I J dJA. I / I,. ' I V I 0.25 freq (GHz) 25.25 ., t'-.. 0.25 freq. (GHz) 25.25 ...... 0 0 0 I "" "' 0.25 freq. (GHz) 25.25 .,, ,, -...._ ,.,, I ,/ I'\. / / / j V __, I 0.25 freq. (GHz) 25.25 Figure 77 Measured s-parameters with adapters de-embedded (bold) and verification s-parameters of the 40 dB attenuator, connected between ports two and four while the 20 dB Attenuator connected between port one and three.

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164 magnitudes of the parameters. The transmission parameter s, S 21 and S 1 2 which also vary s ignificantly over th e measurement, have less than .2dB magnitude error, wh i ch com pare s rea so nably to the uncertainties and no more than 5 phase error with respect to the sparameter s provid ed wit h the verification kit. The second verification standard mea s urement i s t h e simu lt aneo u s mea s urement of the 20 dB attenuator a nd the 40 dB attenuator. The 20 dB attenuator is co nnected between ports one and three of the PMVNA while the 40 dB attenuator i s connected Figure 7-8. Vi 0 0 ....... -. Vi 0 factory. / uncertam ty. r I j ~ mag I I :j '..... '--' 7\.. ,_. ~ .. I .l ,,_ ,_ ,,., \ ~., ""' I'\.. / U'J U'J
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165 between ports two and four. The measured s-parameters of the 20 dB attent1ator are shown in Figure 7-5 together with the s-parameters provided with the verification kit. The differences are again quite small, as shown in Figure 7-6. The reflection parameters S 11 and S 22 show similar behavior to that of the son transmission line, as is expected since the attenuator is a well matched device. The transmission parameters, S 21 and S 12 agree well with less than .1 dB magnitude error and less than 4 phase error. From the same PMVNA measurement, the measured s-parameters of the 40 dB attenuator are also found, and they are shown in Figure 77 together with the s-parameters provided with the verification kit. Again the agreement is quite good, as seen in the errors, shown in Figure 7-8. The reflection parameters S 11 and S 22 show similar behavior to other well matched devices. The errors of S 21 and S 12 are less than .1 dB in magnitude and less than 4 in the phase. The calibration of the PMVNA has been shown to be accurate in terms indepen dent of the PMVNA. Strictly speaking, the accuracy of the calibration has been estab lished for only the specific verification standards measured. These verification standards are meant to represent some extremes of possible DUT performance, so it is argued that the accuracy of the measurements of any DUT can be reasonably assured. The verifica tion standards as shown do not exercise all of the sixteen s-parameters measurable. How ever many other combinations of the same verification standards have been made. The combinations include connecting the standards between different ports, such as one to two or one to four, and many combinations of different standards, such as the son air-dielec tric transmission line with the 40 dB attenuator. These measurements have not been shown due to space limitations, but all compare to verification data with the same general

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166 level of accuracy. It is argued that the se many measurements verify the accuracy of all sixteens-parameters measured by the PMVNA. Even with the acceptable accuracy indicated, the actual accuracy of the PMVNA calibration is further argued to be higher than what is indicated through the discussed mea surements The de-embedding process of the adapters from the measured verification device s must be recognized as a significa tion so urce of error. The s-parameters of the adapters include any residual errors from the 3.5mm calibrations compounded with any errors in the 2.4mm ca libration process The residual error of the ba s ic PMVNA i s argued to be so mething less than what is indicated through this verification process. The errors shown here are felt to be over-estimations of the actual errors of the PMVNA calibration, but are useful as conservative measures of accuracy. The calibration of the PMVNA has been successfully completed in the theoretical framework of a general VNA calibration. The appropriateness of this approach ha s been established through theoretical arguments and validated through measured results. The requirements for a solution of a general calibration problem have been clarified, and new approaches to calibration standards and models have proved accurate.

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CHAPTERS POWER SPLITTER AND COMBINER ANALYSIS The fundamental concepts of mixed-mode scattering parameter s have been estab li s hed by the preceding chapters. The methods of measurement of mixed-mode s-parame ters has also been thoroughly explored, and a specialized measurement s ystem has been described The error correction and measurement accuracy of the mixed-mode measure ment system have been demonstrated. Now, the tools of mixed-modes-parameter theory and the mea s urement sys tem will be applied to the analysis of some important RF devices, s tructures and circuits. This chapter will focus on power splitters and combiners, and the mixed-modes-parameters will provide new insight into the performance of such compo nents. Chapter 9 and Chapter 10 will focus on the analysis of seve ral RF differential applications, and Chapter 11 will provide so me important mixed-mode design concepts. Power splitters and combiners are indispensable components in RF and microwave systems, being used in mixers, balanced amplifiers, baluns, phase shifters, and many other applications. Some of the more commonly t1sed splitters/combiners include 180 hybrid rings [66 70] and 90 branch-line couplers [71 73] There are many other varieties of these components such as tightly coupled microstrip [74 75]. Some recent development s have focused on use of uniplanar transmission line to simplify MMIC implementation s [76 79]. Recent MMIC applications include active splitters with an arbitrary phase rela tionship [80], baluns for double-balanced mixers [81], and linear vector modulators [82] 167

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168 The performance of power splitters and combiners, particularly phase and magni tude balance, can have a strong influence on the performance of some systems. Systems such as a balanced s-parameter measurement s ystem [83] rely on phase and magnitude relationship s in power splitters/combiners to make accurate measurements. However, practical splitters/combiners, such as 3dB hybrids, have varying amounts of phase and magnitude imbalance over their bandwidth which leads to system performance degrada tion or measurement errors. Typically splitter/combiner imbalance is specified across a bandwidth in terms of maximum magnitude and phase variation [66 82] Manufacturers of splitters/combiner s also specify imbalance with this method For example, a typical s pecification of a 1.0 GHz to 12.4 GHz 180 3 dB hybrid splitter is .8 dB amplitude imbalance and 10 phase imbalance [35] This chapter present s an analysi s of imbalances in power splitters and combiners in terms of differential and common modes. Portions of this work has been published by the author [84]. The analysis yields approximate expressions for the differential and com mon-mode nor1nalized waves as a function of magnitude and phase imbalance. These expressions demonstrate that splitter/combiner imbalance can be represented in terms of differential and common-mode responses, and provide insight into the nature of such responses. The combined differential and common-mode analysis represents a more com plete way of quantifying imbalance and new performance metrics are suggested to simul taneously characterize phase and magnitude imbalance. These metrics promote a fundamental understanding of the physical performance of non-ideal splitters and combin ers which is useful in the design and analysis of sensitive differential circuits and systems.

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169 8.1. Splitters Consider a splitter, such as a hybrid splitter, which has a single input and two out puts. The relationships between the two outputs can be described in several ways, but this analysis will use the total voltages and currents at the outputs in terms of differential and common-mode voltages and currents. The definitions of these quantities for a two output system are defined in Section 3.1.1, and are repeated below. (8-1) (8-2) (8-3) (8-4) where v 1 and v 2 are the voltages at outputs one and two, respectively, and i 1 and i 2 are the currents flowing into the outputs one and two, respectively. From these voltages and cur rents, the differential and common-mode normalized waves have been shown to be (8-5) (8-6) where b 1 and b 2 are nor1r1alized output waves at ports one and two, respectively. To continue the analysis a specific type of splitter/combiner will be used, say a 180 3 dB hybrid in a splitter configuration. A simplified signal flow graph of this device is shown in Figure 8-1 neglecting port mismatch and the output-to-output signal path

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a l 1 J2 170 a(l +~ )J(0+~+1t) b2 Figure 8-1. Simplified sig nal flow graph of 180 3dB sp litter with phase and amplitude imbalance. Included in this flow graph is the amplitude imbalance (~), phase imbalance ( 0 ), balanced loss (a) and balanced phase shift(). Note, the outputs of the splitter are ideally of equal magnitude and 180 out of phase. From (8-5) and Figure 8-1, the differential-mode norrnalized output can be found to be aa .ei ~ b d = ; ( l + ( l + ~) ei e) (8-7) where ai is the no1malized input wave. Similar l y, from (8 -6 ) the common-mode normal ized output can be found to be (8-8) If the pha se imbalanc e is small ( 101< < 1 ) then the complex exponential can be approxi mated by ei 8 === 1 + )0 (8-9) Applying this approximation to (8-7), the differential mode norrnalized output can be approximated as

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171 aa.e j bd:=::; ; (2+~+}0(1+~)) (8-10) which can be expressed as (8-1 1 ) which can be further approximated by 'Al+~ Aaa .el't' ________ j0 -aa el't' 2 2 (8-12) where the final approximation assumes the imbalances to be s mall (101<<1, ~<
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172 which in the case of a 180 splitter, can be approximated by CMRR~ (2 + ~) 2 + 0 2 ( 1 + ~) 2 2 + ~ ~ ~2+ 8 2( 1 +~)2 J~2+ 8 2 (8-14) (8-15) This quantity gives a single measure of the effects of both magnitude and phase imbalance in a splitter. The relation between CMRR and the traditional measures of imbalance can be examined by plotting contours of constant CMRR from (8-14) as a function of 0 and 1 +~, as shown in Figure 8-2 By plotting 0 and 1 +~ in rectangular coordinates, these contours form ellipses, and by choosing units of degrees and dB for phase and magnitude imbal ance, respectively, the specifications can be plotted on the same plane as horizontal and vetticaJ lines. The phase and magnitude imbalance specification lines define a rectangle in the imbalance plane. The lowest CMRR in this rectangle, the result of maximum phase and maximum magnitude imbalance occurring simultaneously, is indicated by the ellipse which intersects the corners of the rectangle. Another significant ellipse is the one which is entirely contained in the specifica tion rectangle. This ellipse represents the highest CMRR that can occur while one of the imbalances is at it maximum specified limit In Figure 8-2, such an ellipse is limited by the magnitude imbalance, indicating the worst-case best performance is limited by magni tude imbalance. The shaded regions in Figure 8-2 indicate performance that is within the traditional specification limits, but has relatively poor CMRR.

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173 Had the phase specifications been made s uch that their representative lines were also tangent to the sma ll est dashed ellipse, areas of lower CMRR would still exist in th e come rs of the re s ulting rectangle. The se region s il lu s trate t h e advantage of specifying minimum CMRR over maximum magnitude and phase imba l ance. Two splitters with the sa me magnitude and phase spec i fications may have different minimum CMRR. However, if minimum CMRR i s s pecified, no s uch ambiguity exists, and the CMRR ellipse indicate s the maximum simultaneous phase and magnitude imb a l ance. 15 I I I I 10 +10 0 (spec) I .,,,,,. 20 dB -....., I -----~---------~,----/ I I / /~ 20. \ dB (min imum ) I / I ____ 2s d~ I 26.7 dB / I ; 1 ....., I max \ I / 30d..__. \ I II/ 3d d..,..... I \ 5 ,;;;-I I 40 dB \ lo1--.-----+--+-+-+~-+-~~~--~~ I I J I \ ~-,1 / a, \ 1' ----J--/I / -5 -10 -15 -2 u l ', ./ 1 1 \ 0 ...... __ ., 0.. / \ ~ I I ~ / I l :g / "O In spec but oo / ci I P CMRR l o / 10 0 (spec) oo r + / ------~--------~------1 ..... _,, I ..... __ ----I I I 0 l+Ll(dB) I I 1 2 Figure 8-2. Loci of constant CMRR (dB) in the plane of phase imbalance, e ( degrees ), versus magnitude imbalance, 1 +L\ ( dB ). D as h ed line s indicate manufac turer' s s pecification s, and dashed el lip ses indicate worst-case and best-ca se CMRR interpretation of specifications Shaded regions indicate perfor mance wit hin specification with poor CMRR

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174 To illustrate the use of above analysis, the s-parameters of a 180 3dB hybrid split ter/combiner (Merrimac, part number HJM-4R-6.5G [35] ) were measured. The measure ments were made on a standard HP8510C two-port VNA with 3.5mm coaxial connectors, wi t h a sliding-load TOSL calibration. In this case, a ''round-robin'' method was u se d to determine the multi-ports-parameters of the sp litter. This method employs multiple two port measurements of the device with remaining ports of the device ter1ninated in preci sion son loads. A total of three two-port s-parameter sets were generated as the three ports were measured two at a time, with the remaining port terminated The errors due to any imperfect termination of the free port were considered to be negligibly small. The (a) (b) 0 C'l 0 C'l I C'l I I + 10 (spec) -----I ---t 1 10 (s pec) I +I I tI 0.045 freq (GHz) I I I +0 .8 dB (spec) I I I ....J. -... .L I I J' .". I\ I I\ ,., ,, I V V \.,j V V" IV', ,V"' "' V '\ .. ... -r-0.8dB (spec) I I I I I 0 045 freq (GHz) 20.045 / r ' ,... \j w 120.045 Figure 8-3. Measured imbalance of a 180 3dB hybrid power splitter with dashed lines indicating manufacturer's specifications. a) Measured magnitude imbalance in dB. b) Measured phase imbalance in degrees.

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175 overall three-port s-parameters of the splitter were constructed f r om the appropriate ele ments of the three two-port s-parameter se t s. A direct mea s urement of the output magnitude imbalance is s h own in Figure 8-3(a) This power sp li tter is s pecified to have no more than .8dB magnitude variation from 1 GHz to 12.4 GHz The measured phase imbalance is shown in Figure 8-3(b ). Over its bandwidth this sp litt er should have a maximum phase variation of As can be see n fro m these figures, the sp litter is within specificatio n. The differential and common-mode re s ponses of the spl itt er were calculated from the measured s-parameters. With the definition s S I i = b 1 /ai, and S 2 i = b 2 /ai, and the u se of (8-5) and (86 ), it is easily shown that 0 .... ..t::) 0 V) I 0 .04 5 ,v \ I Ii I V freq ( GHz ) ... V"'\ JI \ V I 1,JJ 20.045 0 ..t::) 0 ,..... I Figt1re 8-4. Rati o in dB of normalized common-mode-toinput waves and nortnalized differential-mode -toinput waves of a 180 3dB hybrid power spl i tter. Data is derived from mea s ured data. Note t h e different vertical scales.

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176 be 1 S = = h ( S 1 + S2 ) C ai ,_;2 l l (8-16) which are readily calculated from measured data. The magnitude of these differential and common-modes-parameters are shown in Figure 8-4. This figure clearly shows the gener ation of a common-mode signal at the output of the splitter. Outside of the splitter's intended bandwidth, the common-mode response becomes large in magnitude and the differential-mode response shows ripple in its magnitude. The CMRR of the splitter can also be easily calculated from measured s-parame ters. With the use of (8-14) and (8-16) 0 tr) u 0 I I I I I I I A I Jl I I I I I I -. I I ... I 0 045 ... freq (GHz) (8-17) I I I I I I ' I r I V\. I I 1 ] I I I I I 20.045 Figure 8-5. Common-mode rejection ratio (magnitude) in dB of a 180 3dB hybrid power splitter Data is derived from measured data

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177 The calculated CMRR magnitude is shown in Figure 8-5. In the specified bandwidth the s plitter ha s approximately 30 dB CMRR, decreasing rapidly out of band The minimum mea s ured CMRR is 27.2 dB. The measured imba la nce i s plotted on the imbalance plane of 8 and 1 +L\ in Figure 8-6 over the specif ication bandwidth of 1 0 GHz to 12.4 GHz. The data i s entirely contained within the 27 .2 dB e llip se, which falls within the limit s of the highe s t possible CMRR specification of 26 7 dB (see Figure 8 -2). 15--------------------I I .___...----,20 d ts-__ __ lO 1--+10 (s p ec) ___ __,..,:;;.__ .....,__ __ --,;:a,.... ____ I 5 I / / I ,,--...., C/) Q.) Q.) J 6'o 0 Q.) I \" "O '-.../ CD ,.....,_ u .... f l 5 ~ 1 0 (spec) ~ I -10 I 15 -2 1 25 1 d 27.2d B .,, / .... ..... ... ,, -~ -0 l+L\ ( dB) I I I ,,,, I I / / (.) G) I ~ I I ~ I 1 2 Figure 8-6. Mea sured imbalance of 180 3dB hybrid power sp litt er ( 1 .0 GHz to 12.4 GHz ) plotted in the plane of phase imbalance, 8 ( degrees), versus magni tude imbalance, 1 + L\ ( dB) with loci of constant CMRR (dB). Dashed line s indicate manufacturer 's specifications, and da s hed ellipse indicate s mini mum meas ur ed CMRR

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178 8.2. Combiners The effect of magnitude and phase imbalance on power combiners can also be ana lyzed in terms of differential and common-mode normalized waves. In a typical hybrid power splitter/combiner, two signals will be combined in two fashions s imultaneously, producing two output s. For example the 3 dB hybrid 180 s plitter/combiner will produce the s um of two input s ignals at one output and the difference of the s ignals at the seco nd output. The combiner can be analyzed as two s ingle output combiners, each delivering the s um or differen ce sig nal as appropriate. The ref ore, only a 180 combiner will be consid e red, with generalizations to any pha se of combining. The simplified signal flow graph of a 180 combiner is similar to that of the s plitter and is shown in Figure 87. A linear com bination of differential and common-mode wave s i s as s umed at each input (818 ) which can be found by so lvin g the forward wave relations s imilar to ( 8-5) and ( 8 6) for a 1 and a 2 Th e total output of the co mbiner is then a function of ad, ac and imbalan ces fl. and 0. The 180 combiner ha s an output 1 J2 a c 1 +l:l. )ei< 0 +<1> + n ) Figure 8 7. Simplified s ign a l flow graph of 180 3dB co mbin e r with phase and ampli tude imbalance.

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179 h1so = (8 -19) A rejection ratio can not be directly applied to combiners. However, a common-mode combiner re s ponse ratio (RCM co mb ) can be defined as RCM = h1so co mb ac (8-20) which can be shown to be approximately _aJ 2 2 2 RCM comb ~ 2~ + 0 ( 1 + ~) (8 -21) Equations (819 ) and ( 8-21) indicate that due to imbalance, the output of the combiner is clearly influenced by the presence of a common-mode signal at the inputs In addition to a response from an unde s ired mode, imbalance in the combiner also causes error in the combining of the desired mode. To illustrate this, consider the differ ential-mode response ratio ( RDM co mb ) of the 180 combiner defined as RDM = h1so co mb ad a = 0 C (8-22) Thi s illustrates the error in the combination of the de s ired mode is a function of imbalance It is interesting to note that the ratio of (8-22) and (8-21) is equal to the CMRR of the split ter in (815 ). The measured s-parameters of the example splitter of Section 8.1 can also be examined for combiner perfor1nance. The re s pon se ratios of ( 8-20) can be related to the measured s-parameters by

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180 where port three is the output and ports one and two are the inputs. Similarly the response ratios of (8-22) can be related to the measured s-parameters by 1 1 1 RDM comb = J2 S -S31 32 (8-23) (8-24) Plots of these ratios are not included since they are closely related the splitter perfor111ance shown in Figure 8-5 8.3. Extensions to Arbitrary Phase The above combined differential and common-mode analysis suggests an alter nate and useful, way of interpreting the imbalances in a splitter/combiner The measured sparameters of a practical power splitter further illustrate the combined-mode, or mixed mode, concepts. From the application of these concepts, CMRR i s found to quantify the effects of both magnitude and phase imbalance in power splitters. The above analysis considers only a 180 power splitter, but the analysis is easily extended to an arbitrary phase relation. For an arbitrary phase splitting the CMRR is exactly CMRR = 1 ( 1 + ~ ) ei < n + 8 ) 1 + ( 1 + ~) eJ < n + 8 ) (8-25) where Q is the phase of the split (e.g 0 180). With fust order approximations, (8-25) has been simplified for three common angles, and these results are summarized in Table 8-1. A study of errors re s ulting from the approximations (101 << 1, << 1) indicates both forms to be quite accurate over practical values of phase and magnitude imbalance. For a CMRR of 25 dB, both approximations have a maximum error of less than .2 dB.

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181 Note that when Q is not 180, CMRR does not have the traditional meaning or value, so that its name is misleading. For example, with Q = 0 CMRR is ideally zero since the dif ferential-mode output is ideally zero. For 0 splitters/combiners another metric is more conceptually natt1ral, namely differential-mode-rejection-ratio (DMRR) which can be defined as DMRR=l/CMRR. For 90 and other angles, neither DMRR nor CMRR is interpreted in a typical fashion. The use of the single metric CMRR for all phases of split ters is both simple and accurate, if its meaning is properly interpreted for each phase splitter. Table 8-1. Splitter CMRR for common values of Q Q CMRR (1 st approx .) CMRR (2 nd approx.) Ideal Value o o ~2+02(1 +~)2 J~2 + 02 0 (2 + ~)2 + 02( 1 + ~)2 2+~ 90 (1 +8(1 +~)) 2 +(1 +~) 2 1+~+8 1 A (l-8(1+~)) 2 +(1+~)2 A. 1+~-8 180 (2 + ~)2 + 82( 1 + ~)2 2+~ ~2+82(1 +~)2 J~2 + 92 00 A The effects of magnitude and phase imbalance in power combiners have also been described through a mixed-mode analysis. Response ratios such as RDM and RCM quan tify these effects. The combiner analysis considers only a 180 power combiner, but the analysis is easily extended to an arbitrary phase relation. For an arbitrary phase combin ing these responses are exactly

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182 RD M I 1 ( 1 + L\) ei < n + 0 ) 2 RCM = I 1 + ( 1 + L\) ei< n + 0 ) I 2 (8-26) To a first order approximation, (8-26) has been simplified for three common angles, and these results are summarized in Table 8-2. The impact on systems using power splitters and combiners can now be considered in terrns of differential and common-mode responses. The effects of an undesired signal mode, such as common-mode, can be investigated with simple network theory, and limits on such undesired signals can be set through the use of CMRR. Table 8-2. Combiner RDM and RCM for common values of Q. Q RDM RCM o o ~J1i2 + e2c 1 + L\)2 2 ~Jc2 + L\)2 + e2c 1 + L\)2 90 ~Jc1 + ec1 + L\)) 2 + c1 + ti) 2 ~J(l-0(1 +L\)) 2 +(1 +ti) 2 180 Jc2 + L\) 2 + e 2 c 1 + ti) 2 ~J1i2 + e2c 1 + ti)2 2

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CHAPTER9 THIN-FILM METAL-ON-CERAMIC STRUCTURES This chapter examines a series of RF differential struct11res that are fabricated on ceramic substrates. The structures are designed to provide examples of the use of mixed mode s-parameter concepts in the measurement and analysis of RF differential structures. The test results are from differential transmission structures. These structures provide important information about practical differential implementation of one of the most basic RF components, the transmission line. These experiments examine the perfortnance of various differential transmission line in terms of ( 1) mode-specific transmission behavior and (2) line-to-line crosstalk ( coupling). All structwes are fabricated on an IC scale and they are all designed to be directly probed with 150m pitch probes. The structures are thin-film metal-on-ceramic struc tures, fabricated with thin-film gold ( 4m thick .25m) on a polished alumina substrate (r:::: 9.9, tan8:::: 0.001). Resistors are fabricated with a resistive layer of a nickel-chrome 0 (NiCr) metal approximately 400A thick, with a sheet resistance of about 40 Q/square. In this case, the resistors were left un-tuned at their fabricated values. All photo-masks required for fabrication have been generated in the University of Florida Microelectronics Laboratory, and all fabrication has been done in the Motorola Thin-Film Research Labora tory in Plantation, Florida. All measurements presented in this chapter were made with 1024 averages and half-leakage correction using the methods presented in Chapter 6. 183

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184 9 .1. Differential Transmission Lines 9 .1.1. Uniform Differential Transmission Line The first s tructure presented is a simple coplanar wave guide (CPWG) transmis sion line pair. The constituent transmission lines are placed close together so that the pair can be considered a s ingle differential transmission line. Their closeness causes the indi vidt1al transmission lines to be coupled, so that they act as a single differential transmis sion line The transmi s sion line, shown in Figure 9-1, ha s a constant cro s s-section over its length, and is therefore called a uniform differential transmission line. Each constituent transmission line wa s designed to have a nominal 50.Q characteristic impedance 1 The signal conductors are 104m wide and the spaces between the edge of the signal conduc tors and the edge of the ground planes are 58m. The ground strip between the signal conductors is 80m wide. The s ignal conductor s are 4000m long, and the outer ground planes are 400m wide The s haded regions, 400m wide, represent strips of resistor material which have been added to reduce parasitic re s onances of the s tructure at high frequenc1es. MMl MM2 Figure 9-1. Layout of uniform differential transmi ss ion line with intermediate ground. 1 Th e de s ign o f ea c h line wa s a s ingl e s ignal co ndu c t o r d es ign negle c ting any co upling b e tw ee n the tw o adja ce nt transmis s i o n line s and th e e ff ec t s o f th e trun c ated gr o und plane s In oth e r w o rd s, the de s i g n a ss um e d un co upl e d tran s mi ss i o n lin es. Thu s th e e v e n and o dd m o de c har a c t e ri s ti c imp e d a n c e s are a ss um e d t o b e son during th e d es i g n pro ces s

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185 0 0 0 0 __/ ,,--.._ I ,,.r ,,--.._ N ,:) ,:) ,:) ,:) C'-l C'-l ...._,, ...._,, 0 0 0 I 0 25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 0 0 0 0 ,,--.._ ,,--.._ r:::: N N N ,:) ,:) -0 ,:) C'-l C'-l ...._,, ...._,, 0 0 0 I 0 .2 5 freq. (GHz) 20.25 I 0 25 freq. (G H z) 20.25 Figure 9-2. Measured p u re diffe r entia l s-parameters (DD) of structure of the uniforrn differentia l transmission line magnit u de in dB The measu r ed mixed-modes-parameters of the uniform differential transmission line are shown in Figure 9-2 to Figure 9-5. T h e magnitudes in dB of the pure differential mode responses (DD) are p l otted versus freq u ency i n Fig11re 9-2. From this figure, one can see t h at the structure behaves as a matched, low-loss transmission line to the differen tial mode signals. The struct u re has better than 20 dB differentia l return loss at 20 GHz, a n d l ess than 4 dB different i al insertion loss at 20 GHz. T h e actual differential character

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186 0 0 0 0 I "\. I/ i\ / I V --...... N .... ...... t.) t.) t.) u VJ VJ -._,; -._,; 0 0 0 I 0.25 freq (GHz) 20.25 I 0 .25 freq. ( GHz) 20.25 0 0 0 "' 0 r I '\. ., ... \ / I --...... N N N t.) t.) t.) u VJ VJ -._,; -._,; "'O 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-3. Mea s ured pure common-modes-parameters ( CC ) of structure of the uni form differential transmission line, magnitude in dB istic impedance is approximately 1010., which is s li ghtly more than twice the nominal sin gle-ended impedance of the individual line s The small deviation from the nominal single-ended impedance indicates the individual line s are loosely coup led The magnitudes in dB of the pure common-mode responses (C C ) are plotted ver s us frequency in Figure 9-3. These measurement s indicate that the s tructure i s good tran s mi ss ion line to the common-mode, but it is not as well matched as to the differential

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187 0 0 0 0 ,--... ,,-.... N (..) (..) "'O "'O vV) .\/\A V) ) '--" '--" ~ fv .t /\. I ll 'r-1 l,/\ J -i. ..JWn 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20 25 0 0 0 0 ,--... ,--... N N N (..) (..) "'O ,./ "'O t\, V) ,r V) _/' ~/ n '--" '--" i:Q "O i:Q "O }( t-,,,'" 0 0 0 0 I 0,25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-4. Measured common-to-differential mode-conversions-parameters (DC) of structure of the uniform d i fferential transmission li n e, magnitude in dB. mode. The periodic variations i n the commo n -mode retur n loss indicate an impedance miss-match, although sl i g h t. The actual common-mode charac t eristic impedance is approximately 290. Note, h owever, t h at the two reflectio n parameters are near l y indistin guishab l e, indicating a high level of port symmetry, as is expecte d from the l ayout. T h e magnitudes i n dB of the mode-conversion respo n ses (DC and CD) are p l otted versus frequency in Figure 9-4 and Figure 9-5. These plots show a reasonably low level of

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188 0 0 0 0 ------"Cl "Cl (.) u v-v:, .I A A l v:, .J ..._,, ..._,, f .M ... V V .. 1 f V -# tu. ,... ~,.. 1 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 0 0 0 0 _..._ (.) "Cl -0 1,/ (.) I 'V r\ / en / en ., ~..._,, .._, f f .J .. \J I rv 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20 25 Figure 9-5. Measured differential to-common mode-conversion s-parameters (CD) of structure of the unifo1111 differential transmission line, magnitude in dB mode-conversion in reflection (in the -40 dB to -50 dB range), but indicate more signifi cant level s of conversion in the transmis s i on parameters as frequency increases. These mode-conversions are most like l y due to the combined effect of probe-to-probe crosstalk and raw imba l ance in the PMVNA.

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189 9 1.2. Balanced Step Differential Transmission Line The second structure presented is a differential transmission line with a change in the width of both signal conductors. This change in the width is equal for both conduc tors, and is therefore called a balanced, or symmetric, step in width. Referring to Figure 9-6, the section of transmission line with the wide signal conductors is cascaded between two sections of narrower line. Both end sections are of the same dimensions as the nominal 50Q single-ended characteristic impedance ( conductors 104m wide, spaces 58m wide) as examined in Section 9.1.1. The middle section has conductors 140m wide and spaces 40m wide, and is 2000m long. Both end sections are lOOOm long, making the entire transmission line 4000m long. The measured pure differential-mode (DD) results from the balanced step-in-width transmission line are show in Figure 97, as magnitudes in dB. The differential return loss has a distinctive periodic variation with freqt1ency Furtherrnore, the transmission param eters also have a slight periodic variation. These plots, together with polar plots of the dif ferential data (not shown) indicate a step in the differential characteristic impedance. In other words, a differential-mode signal encounters a transmission line system with imped ance first of Z 1 then Z 2 then back to Z 1 This type of transmission line system is often MMI MM2 Figure 9-6. Layout of differential transmission line with symmetrical step in conductor widths.

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190 0 0 0 0 .... / / 'I ,,.....__ C'l -0 -0 -0 -0 Cl) Cl) ,.__,, '--" 0 0 0 .,..... I 0.25 I 20.25 freq. (GHz) 20.25 0.25 freq. (GHz) 0 0 0 0 / ;' i\... / I C'l N N -0 -0 -0 -0 Cl) Cl) '--" '--" 0 0 0 ,..... ...... I 0 25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-7. Measured pure differentia l s-parameters (DD) of strt1cture of the balanced step differentia l transmission l ine, magnitude in dB. cal l ed a Beatty standard [85 ] This behav i or of the differential transmission line of Figure 9-6 is not surpr i sing since t he cons t ituen t s i ngle-ended transmission lines each embody a Beatty standard Analysis shows the differentia l characteristic impedance of the center section to be 81.Q, w hil e both end sections are 10 In. The pure common-mode response (CC) of Figure 9-6 is shown in Figure 9-8 as magnitudes in dB. The periodic nature of the common mode response is more evident than that of the differentia l -mode. Clearly the common-mode response of the transmis

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191 0 0 0 0 "' r/ I" / \1 ( ,-.-. ,-.-. ('I t) t) t) t) V) V) 0 0 0 I 0.25 I freq. (GHz) 20.25 0.25 freq. (GHz) 20 25 0 0 0 0 r/ I" I'\. I J V \J ,-.-. ('I ('I (.) u (.) t) V) V) --"'tj 0 0 0 I 0.25 freq (GHz) 20 25 I 0.25 freq. (GHz) 20.25 Figure 9-8. Mea s ured pure common-modes-parameters (CC) of structure of balanced s tep differential transmission line magnitude in dB. sion line is also that of a Be ady standard. Analysis shows the common-mode characteris tic impedance of the center section to be 230., while both end sect i ons are 29Q F urth ermore, the return loss minima occt1rs at l ower frequencies in the balanced step sys tem than in the uniform transmission line. This downward shift indicates that the common-mode electrical length of the balanced step section is l onger than that of the an equal length of uniform tran sm ission line This difference is due to the quasi-TEM nature of

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192 0 0 0 0 ,...-._ ,...-._ N t.) t.) -0 -0 Vl Al ,_~ ... Vl ..,...__,, ...__,, j ..., 1 !1 -w~ '"O 0 0 0 0 I 0.25 freq. (GHz) I 20.25 0.25 freq. (GHz) 20.25 0 0 0 0 ,...-._ ,...-._ N N N A t.) t.) -0 V -0 l..' V "' v Vl r;J) ...__,, /" ...__,, ,,, ti w 0 0 0 0 I 0.25 freq. (GHz) 20 25 I 0.25 freq (GHz) 20.25 Figure 9-9. Mea sured common-to-differential mode-conversion s-paramete r s (DC) of str u cture of balanced step differential transmission line, magnitude in dB. CPWG, where the velocity of propagation is a function of th e transmission line cross-sec tion. It is also interesting to note that minima in the differential return loss occur at about twice the frequency of the corresponding common-mode minima. This indi cates t h at the overa ll structure bas a differential electrica l l e n gt h that is about half that of the common rnode.

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193 0 0 0 0 ,......._ ,......._ ...... N ...... ...... -0 All -0 C) C) v-' Cl) ,. Cl) --a --j I ~ ...__ ,.'"\! ' 0 0 0 0 I 0,25 I freq (GHz) 20.25 0.25 freq. (GHz) 20.25 0 0 0 0 ,......._ ,..... N N C) -0 j -0 ,J C) .... V ~ CJ') CJ') r --/'" --""f J I N 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0 25 freq. (GHz) 20.25 Figure 9-10. Measured d i fferentia l -to-common mode-conversio n s-parameters (CD) of structure of balanced step differential transmission line, magn i tude in dB. The measured mode-conversions r esponses (DC and CD) are shown in Figure 9-9 and Figt1re 9-10. Despite the step in the conductor widths, t h e mode-co n version levels are essent i a ll y the same as the unifortn trans mi ss i o n l ine of F i gure 9-1. I t can be observed that, at any d i stance along t h e line, the s i gnal conductors are of eq u a l width. This type of structure can be said to have electromagnetic fie l d symmetry (t h at is the even and odd mode exist in co n trast to the c and 1t-modes). As a resu l t of this symmet r y, the structure

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194 has a high degree of balance, that is, there is low levels of mode-conversion. The relation s hip between field symmetry and mode -co nversion will be further examined in the follow ing sections. 9.1.3. Unbalanced Step Up Differential Transmission Line Th e third struc ture presented ha s a change in the width of only one signa l conduc tor. A s show n in Figure 9-11, thi s s tructure is similar to that of the previou s sec tion, but in this case only one of the signal conductors have a step in width. Like the previous tran s mission line, the section of transmis s ion line with the wide signal conductors is cascaded between two sections of narrower line. Both end sec tions are of the sa me dimension s as the nominal 50Q single-ended characteristic impedance ( conductors 104m wide, spaces 58m wide ) as examined in Section 9 1 1 The middle sec tion ha s one signal conductor 140m wide and spaces 40m wide, but the other s ignal conductor is 104m wide with 58m wide spaces. Again the middle sec tion is 2000m long and both end se ction s are 1 OOOm long, making the entire tran s mi ss ion line 4000m long. Since the middle section doe s not have s ignal conductors of equal width, the dif ferential transmis s ion line is considered to be unbalanced. As discussed in Section 3.1.1, the structural asymmetry in the coupled tran s mission lines lead s to a lack of field symmeMMl MM2 Figure 9-11. Layout of differential transmission line with unbalanced step-up in conduc tor width.

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195 0 0 0 0 ,,--... / ,,--... C"I "O "O "Cl "Cl ti) ti) '-" '-" 0 0 0 ...... ...... I 0.25 I freq. (GHz) 20.25 0.25 freq. (GHz) 20.25 0 0 0 0 ""'" ,J ,,--... ,,....._ / C"I C"I N "Cl "Cl "O "O ti) ti) '-" '-" t::Q "'d 0 0 0 ...... ...... I 0.25 freq (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-12. Measured pure differential s-para m eters (DD) of structure of the unbal anced step-up differential transmission lin e, magnitude in dB. try. Hence, the even and odd-modes are n o l onger defined, being replaced by the more genera l c and 7t mode s. The measured pure differential-mode (DD) responses are s h own in Figure 9 -1 2. The structure acts as a reasonably good differential-mode transmission line despite its unbalanced nature. The differential return l oss i s quite good (especially when compared to that of the uniform transmission line of Section 9 .1.1 ), but it does show some periodic

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196 0 0 0 0 ......... / / / \, / ,,-..... ,,-..... C"l ...... u u u u CJ) CJ) ...._,,, ...._,, "'O 0 0 0 I 0.25 freq. ( GHz ) 20 .2 5 I 0 25 freq. ( GHz ) 20.25 0 0 0 0 .,,. I /' / \ I/ .J V ,,-..... ,,-..... ...... C"l C"l C"l u u u u CJ) CJ) ...._,, ...._,,, "'O 0 0 0 I 0.25 freq. ( GHz ) 20.25 I 0 .2 5 freq. (G Hz ) 20.25 Fi g ure 9-13. Mea s ured pure common-modes-parameters (C C ) of s tructure of the unbal anced s t e p-up differential tran s mi ss ion line magnitud e in dB variations indicatin g a Beady like effect. Th e minima in the differential return lo ss of the unb a lanc e d structure are at s li g htl y hi g her frequencies than th e balanced s tep sys tem Thi s difference indicate s that the differential-mode electrical length of the unbalan c ed sect ion is less than that of th e balanced sec tion of th e sa m e phy s i ca l length R ecal l that th e differ ential and common-modes do not actually exist on a sy mmetrical s tructur es, but that

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197 0 0 0 0 /' ----I \ N .... ...... C) C) -0 'J'1 C/J C/J ..._,,, ~t\ J\,JA/ ., 0 0 0 0 I 0.25 I freq. (GHz) 20 25 0 .25 freq. (GHz) 20.25 0 0 0 0 V"'" ......... ----... / ...., N I '\ I N ,_/ N C) 0 -0 -0 C/J ..._,,, .. l/ C/J' M I ~ 0 0 0 0 I 0.25 freq ( GHz ) 20.25 I 0 ,2 5 freq. (GHz) 20.25 Figure 9-14. Mea s ured common-to-differential mode-conversion sparameters (DC) of s tructure of the unbalanced step-up differential transmission line magni tude in dB related differential and common mode re s pon ses can be defined ( or measured) by use of a zero-length symmetric reference transmission l ine (see Appendix C ). The measured pure common mode (CC) responses are shown in Figure 9-13. The common-mode re spo n se of the unbalanced s tep line i s very similar to that of the balanced step syste m The most notable difference s are in the reflection parameters, particularly at low frequencies. The common-mode return lo ss minima occur at very slightly higher fre

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198 0 0 0 0 .,,r -,, -...... I \ I N ...... ...... ,r -0 -0 (.) ., (.) /" Cf.) Cf.) '-' ....._,,, "O .N .J 1 0 0 0 0 I 0.25 I freq. ( GHz ) 20.25 0.25 freq ( GHz ) 20.2 5 0 0 0 0 ,,,,,........ --...... N I N N I (.) J -0 -0 (.) I Cf.) IJ Cf.) '--" "' A. V "O 0 0 0 0 I 0 .2 5 freq. (G H z) 2 0. 2 5 I 0 .2 5 freq (G H z) 20.2 5 Figure 9-15. Mea su red differential-to -co mmon mode -co nver s ion s -parameter s (C D ) of st ructure of the unbalanced s tep up differential tran s mi ss ion line magni tud e in dB. que11 c ie s in the unbalanced sys tem again indicating that the common mode electrical length of the unbalan ced section i s le ss than that of the balanced sec tion. The mea s ured mode-conversions r es p o n ses ( DC and CD) of th e unbalanced st ru tur e are s hown in Figure 9-14 a nd Figure 9-15. The lev e l of conversion i s hi g her in this s tructur e than in the unifor1r1 or balanced step tran s mission line s ( parti c ularly in the refl ec tion parameters). Th e mode -co nver s ion in r e flection i s particularly strong. The se reflec

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199 tion parameters exhibit behavior that is very similar to that of a traditional Beady standard. Furthermore, the mode-conversion reflections are very s imilar to the pure differential mode reflections However, the minima in the mode-conver s ion reflections occur at lower frequencies than the differential-mode. Thi s indicates an effective mode-conver s ion elec trical length that is higher than the differential-mode. AJternatively, the effective velocity of propagation of the mode-conversion is less than that of the differential-mode. Further more, the common-mode propagation velocity is about half of the differential mode. Hence, the mode -conve rsion equivalent propagation velocity is greater than the common mode but less than the differential mode This relationship is consistent with the concepts of mode -co nversion The mode -co nver s ion re s ponse can be considered to be one mode for some of it s propa gat ion, and the other mode for the remainder. As a result the effec tive propagation velocity of the mode-conversion response will be some sort of average of the velocities of the two pure modes Also note that the two directions of mode-conversion are equal for this structure. The differential-to-common-mode conversion (CD) is equal to the common-to-differen tial-mode conversions (DC). 9.2. Comparison Between Measurements and Simulations To illustrate the on-wafer measurement accuracy of the PMVNA, two simple dif ferential structures have been mea s ured and compared to simulated results. These struc tures have been chosen because they provide good examples of mixed mode behavior, such as mode conversion. Furtherrnore, these structures can be simulated with a reason ably high degree of confidence in the accuracy of the results. The results of this section have been publi s hed, along with a description of the PMVNA [34]

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200 9 2.1. Unbalanced Step Differential Transmission Line The first device, shown in Figure 9-16, is an unbalanced stepped impedance transmission line pair. In contrast to the structures presented in Section 9 .1, this device does not posses port symmetry. In other words, the ports are not interchangeable, as the equivalent impedance seen by mixed-mode port one is different than that of mixed-mode port two. As is Section 9.1 the structure is fabricated with thin-film gold (4m thick .25m) on a polished alumina substrate (Er~ 9.9, tan8 0.001). The structure has three sections of different impedances, where Lines A and C are pairs of nominally (single ended) 69Q transmission lines, and Line B provides a pair of transmission lines with dif ferent impedances (one 720., the other 40Q). The lengths of the three sections are shown in Figure 9-16, and are known 1 m. The vertical dimensions of the structure are shown in Figure 9-17. The fabrication tolerance on the spaces between the conductors is 1 m. The structure of Figure 9-16 has been simulated in Hewlett-Packard's Microwave Design System (MDS) [32]. The simulation employs the five conductor coupled microsMMl MM2 2000m 5000m 3000m Line A LineB Line C Figure 9-16. Layout of unbalanced step differential transmission line for comparison between measured and simulated results.

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201 j l 4 OOm 7 5 7 1 5m Om 5m OOm 7 7 5m Om 5m l 4 OOm I Line A, C (a) j j I LineB (b) 3 8 5 8 8 ~ l 1 r "'I l 3 90m 5m Om 5m Om 5m SOm 5m 90m Figure 9-17. Detail layout of Figure 9-16 showing critical dimensions. a) Detail of Lines A and C. b) Detail of Line B. trip transmission line model incorporated within MDS with the first, third and fifth con ductor grounded. See Figure 9-18 for the schematics of thi s circuits. The transitions between the sections of line are modeled with the MDS microstrip step-in-width model, but the coupling between each step i s not modeled. The simulation make s a quasi-static approximation, but includes metal losses (cr = 4. lx10 7 S/m) and dielectric losses. The nominal values for all dimension s and physical properties a1e used in all simulations, with the exception of the end sections (L ines A, C). The lengths of these sections are reduced by the distance the probes overlap the end of the structures. It is assumed that the overlap is 25m at each end, but the actual probe overlap can vary, causing discrepancies between measured and simulated data. The simulations generate standard s-parameters which are converted to mixed-modes-parameters through the transformation (3 -44 )

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202 The corrected measured and simulated mixed-modes parameter s of the asymmet rically stepped tran s mi ss ion line pair are plotted in Figure 9-19 through Figure 9-22. The agreement between the measured and simulated data i s quite good across the entire band width The mea s ured and s imulat e d data share many of the unu s ual fine feature s of th e re s ponses. For example the measured S cc l 2 has several abrupt increa ses in insertion lo ss at about 6 1 GHz 12 .2 GHz and 18 3 GHz~ likewise, the s imulated data shows similar response s, although at slightly lower frequencies. Additionally, the device demon s trate s a s trong level of mode-conver s ion ( Sct c S ect), and the agreement between mea s ured and s im ulated mode-conversion indicate s that the effect s of imbalance in the PMVNA are largely removed through calibration. W5 = 350 mW2 = 350 m W 5 = 340 m W 2 = 350 m W5 = 350 m I I I : I II 1----1 I : I I I t> : S4 = 7 5 m WI = 340 m : S4 = 85 m : W 1 = 340 m : S4 = 7 5 m : I I f I I I I I I I : W4 = 50 m : W 4 = 50 : : W4 = 50 : p o rt 1 ~.___ __ ____. I I I p o rt 3 ' : S 3 = 7 5 m : S 3 = 85 m : ' : S3 = 75 m : I I I I W 3 10 0 m W 2 100 W 3 80 m' ' W 2 100 W 3 100 m m I m ' I < ' I I r : S 2=7 5m : Wl = 1 50 :S2=35 : W1 = 150m :S2=7 5m : I I I I I I t I I I I I : W 2 = 50 : W2 = 50 m : W 2 = 1 50 qi W2 = 50 : W2 = 50 : p o rt 2 ~I 1----1 ,___~: I I I I : I I port 4 l S 1 = 7 5 m l W 1 = 15 0 m l S 1 = 3 5 m W 1 = 150 m S 1 = 7 5 m l I I I I I I /1 Wl = 350 ~ W2 = 350 m : Wl = 340 rti W 2 = 3 50 ni Wl = 350 rti ~---111 I : I j i I : I i i t> L = L 1 W l = 340 m L = L 2 L = L 3 1-HU = l.OE +3 m T = thi ck t = h e i g ht M SSUBS TRATE SUB S T = s 1 Omil ER= er _s ub MUR = 1 COND =cond ROUGH =Om TAND = t and e r _s ub = 9.9 tand = 0.001 thi ck= 4um co nd = 4. l e7 h e i g ht = 25 mil Ll = 2000 m L2= 500 0 m L 3 = 3000 m Figure 9-18. MDS schematic of unbalanced step differential tran s mission line of Figure 9 16 for comparison between measured and s imulated result s.

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203 Note that both the measured and simt1lated pure differential parameters (Sdd), common-mode parameters (Sec), and mode-conversion parameters (Sdc, S cd) all show approximately periodic variations across frequency. As discussed in Section 9 .1, these variations are analogous to the effects of a single transmission line with a step in imped ance. Hence, each partition of the mixed-modes-parameter matrix ( Sdd, S ec, Sdc, Scd ) can be interpreted as an effect i ve single transmission line with stepped impedance. 0 0 0 l/') j I I 0.1 0 0 IL... l/') I 0 1 I l l T. Measured V \.. ""'-., l Simulated freq. (GHz) 20 1 ts freq. (GHz) 20.1 0 0 l/') I 0 0 0 V) IL... ,. 0.1 I 0.1 freq. ( GHz) 20.1 1 7 V freq. ( GHz ) 20.1 Figure 9-19. Measured (heavy) and sim u lated (light) pure differentials-parameters (DD) of structure of the unbalanced step differential transmission line of Figure 9-16, magnitude in dB

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0 0 0 tr) I 0.1 0 0 .... tr) I O, l M_ i I T easured '"'" ~' ~im ul ated freq (GHz) I~ freq. (GHz) 204 ~"I'.. ~ l' 20.25 .,,. '\.w Jr 20.1 0 0 .... tr) I 0 0 0.1 -IN 8 ,......; I 0.1 "' )' freq. (GHz) 20.1 -.... '' r-,\ li freq. (GHz) 20.1 Figure 9-20. Measured (heavy) and simulated (light) pure common-modes-parameters (CC) of structure of the unbalanced step differential transmission line of Figure 9-16, magnitude in dB. By close examination of Figure 9-19 through Figure 9-22, it is found that only ten of the mixed-modes-parameters are clearly unique 2 This is a property of reciprocal devices. In contrast, the devices of Section 9 1 possessed only six unique mixed-modes parameters This difference is due to the fact that the devices of Section 9 .1 all possessed 2. Random noi s e and measurement error s make each parameter unique to s ome degree In the c ontext of thi s dis c u s sion, a unique parameter i s one that is substantially different than all other parameter s

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0 0 0 V) .II I 0.1 0 0 0 V) I 0.1 205 Measured .... .... .J 7 1 ~im u latec freq. (GHz) 20.1 ...... A ... I 7f tl I\ \ V, freq. (GHz) 20.1 0 0 c--; C. ,;;; 0 V) I 0.1 0 0 .... 0 V) I 0 1 .. v. ht t ""1, freq (GHz) 20.1 ., .... .. 4( ,,. , Ill lJ freq. (GHz) 20.1 Figure 9-21. Measured (heavy) and s imulated (light) common-to-differential mode-con version s-parameters (DC) of structure of the unba l anced s tep differential transmission line of Figure 9-16, magnitude in dB. port symmetry, where the device of Figure 9-16 does not In general, it can be shown that for a mixed-mode reciproca l device, smm = (Smm )T. Further consideration of the proper ties of mixed-mode s-p arameters will be made in Chapter 11.

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0 0 0 V) ,j I 0.1 0 0 0 V) I 0.1 I Measured ......,, .... ff I Ill Iii ,I I imulatec I freq. (GHz) 20.1 !Iii; ?r~ 1, .., M freq. (GHz) 20.1 206 0 0 0 V) I 0 0 0 V) 0.1 I 0.1 ;;;;;:: J' freq. (GHz) 20.1 ......... .... 1' //( ,. lll freq (GHz) 20.1 Figure 9-22. Measured ( hea vy) and sim ul ated ( light ) differential-to-common mode-con version s-parameters (CD) of structu re of the unbalanced step differential transmission lin e of Figure 9-16, magnitude in dB. 9,2.2. Balanced Step Differential Transmission Line The second device, s h own in Figure 9-23, is an balanced stepped impedance trans mission lin e pair. Like the previous struct ur e, this device does not possess port symmetry. The fabrication details for this device are the same as those of the previous device. The structure again has three sections of different imp eda n ces, where Lines D and Fare pairs of nominally (single-e nded ) 69Q transmission line s, and Line E provides a pair of trans

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207 MMl MM2 I --------------------------.r.------------------------------------------------------------------~~,$~--_,,, _______ ,, __ ,, ___________________ 2000m 5000m 3000m LineD LineE LineF Figur e 9-23. Layout of ba l anced step differential transmi ss ion line for comparison between measured and simulated re s ult s. mi ss ion lines with equal impedance s ( 40Q ) The vertical dimensions of the s tructure are s hown in Figure 9-24. The corre c ted mea s ured and simulated mixed-mode s-parameters of the balanced s tep tran s mission lin e pair are plotted in Figure 9-25 through Figure 9-28 Again, the agreement between measured and s imulated data i s good. Like the unbalanced s tructure the balanced structure demon s trates the periodic re s ponses analogou s to a single transmis sio n line with a s tep in impedance. Th e balanced s tru c ture s upport s only differential and c ommon mode and hence no significant mode-conversion occur s. Referring to Figure 9-27 and Figure 9-28, the measured mode-conversions-parameters are s mall and are mo s t likely generated by probe crosstalk and residual errors in the calibration. From the se two s tructur es, one can see that the measured mixed-mode sparam eter s compare well to the simulated parameter s Thi s i s indication that the PMVNA on waf er calibration i s accurate. Of course, the conclusion of accurate measurement results h e re re s t s on the assumption of s imulation accuracy The ac c uracy of the simulations can

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J l ' I 1 Line D F (a) 400 m 5 7 1 7 7 5m Om 5m OOm 5 m Om 5 m 400 m 208 390 m --+--J5 m 150m ----+--".15 m _--4-_,,80 m --+-__,5 m 11 150m 5m 390 m LineE (b) Figur e 9-24. Det ai l layout of Figure 9-23 s howing critical dimen s ion s. a) D eta il of Line s A and C. b ) Detail of Line B be assumed, at lea st for thi s comparative exercise, s in ce the models are ba se d on well established electromagnetic so lution methods. With the assumption of accurate sim ula tion results, the good correlation of the mea s ured data becomes only n ecessa ry but not s uffi cie nt, evidence of c alibration accuracy. Due to the la c k of traceable metrol ogy s tan dard s at the wafer l eve l ( particularly fo ur-port s tandard s), thi s sort of relative mea s ure s are the only means of assessing on-wafer m eas ur eme nt accuracy of the PMVNA

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0 0 ---0 -0 VJ ..._,,, 0 lf) j /J I 0 1 0 0 --C'-1 -0 -0 VJ ..._,,, lf) I 0 1 II"'" ..... / rt, \I j/ M eas ur e d imu la tec f r e q ( GH z) 2 0 1 ~ -, '\. A \ f r e q ( GH z) 2 0 1 20 9 0 0 lf) I 0 0 0 lf) L 0 1 I 0 1 ..... I\ F '\. vv f r e q. ( GHz ) 20. 1 /( r f req ( GH z) 20. 1 F i g ur e 9-2 5 M eas ur e d ( h eavy) and s imul a t e d (l i g ht ) pur e d iffer e ntial spar a m e t e r s ( DD ) of s tru c tur e of th e b a l a n ce d s t e p diff e r e ntial tr a n s mi ss ion lin e of F i gure 9-23, m a gnitud e in dB

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0 0 ~1mulated 0 V) I 0.1 0 0 k V) I 0.1 +SJ _,-J L i ~reasured freq. (G H z) 1\ y freq. (G Hz ) 210 ........ I20.25 l I' \ r\ 20.1 0 0 V) I 0 0 0 0 .0 1 r r' 1 0.1 fi!llii ' ' \ I'\ v ,t freq. (GHz) 20.1 )(' / 1 .I freq. (GHz) 20 .1 Figure 9-26. Measured (heavy) and simulated (l i ght) pure common-modes-parameters (CC) of structt1re of the balanced step differential transmission line of Figure 9-23, magnitude in dB.

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211 ~-+---+---+----+---+---+___ ..... ..... Ui---+---+---+----+---+---+___ "'O Cl) .._,,,~ easured ff3 1---+---+---+----+-t-+---+___ I----+---+---+---+-+--+---+ S imu ated ~ .di I 0.1 freq. (GHz) 20.1 ,---,-.,..--r---,--,:i:,---r---,-----,-----,.-, 0 ~ --------------t-----1 ,-1 N u "'01---+---+---+----+---+---+___ Cl) .._,,, 1)---1--+--I--+---+--+--+-+---+~ ff3 I---+--_-+---+__ __ ~= I 0.1 freq. (GHz) 20.1 ,,-.....1---+--+---+---+--+--+--+--+----t----4 N ...... u __________ -----i___ "'O Cl) .._,,, 1)---t---t---t--+--+--+-+--+--;~ ff3 ---------------t---0.1 freq. (GHz) 20.1 ~-+---+---+----+---+---+----1-----1---4----1 N u 1---+---+---+----+---+--+----1--+------1 "'O Cl) .._,,, 1)---t---t---t--+--+--+-+--+--;~ ff3 -----------0 Vi I 0.1 freq. (GHz) 20.1 Figure 9-27. Measured (heavy) and simu l ated ( l ight) common-to-differential mode-con version s-parameters (DC) of structure of the balanced step differential transmission line of Figure 9-23, magnitude in dB.

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,----,--,---,---,--......,....---r------r-----.---r---i 0 ,......._I---+---+---+---+---+____ .."01---+---+---+---+---+____ u Cl) ij9-Measured Simu ated 0 V) '0 1 0 0 0 V) 0.1 freq. (GHz) freq. (GHz) 20 1 ,r II 20.1 212 ~-1---~-t--t-----t----t-------t------t----l ..... "01---+---+--+---i---+---+---+---+---t---l u Cl) ...._,,, 9---+---+--+---i---+---+---+---+---t~ 0 V) 0 0.1 freq. (GHz) 20.1 ,---,---,----r-~-----,----,---,-----,---, 0 ~--+---+---+---+---+---+___ ('l ]1--1---~--------t----t-------t---+ CI) ...._,,, 9---+---+--+---i---+---+---+---+---t~ 0 V) 0.1 freq (GHz) 20.1 Figure 9-28. Measured (heavy) and simulated (light) differential-to-common mode-con version s-parameters (CD) of structure of the balanced step differential transmission l ine of Figure 9-23, magnitude in dB 9.3. Crosstalk Between Differential Transmission Lines One of the primary motivation s for the use of differential circuits is the inherent immunity of such circuits to interfering signals In particular, differential circuits exhibit reduced crosstalk between adjacent circt1its with respect to single-ended circuits. As illus trated in Section 5.1, the crossta l k between adjacent circuits is not eliminated by use of differential topologies; rather it is reduced.

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213 Unintentional transmission line coupling is a major concern in high density inte grated circuits. The application of differential transmission line should reduce this cou pling. However, it is important to quantify the levels of residual crosstalk between differential interconnections. Such knowledge is important for understanding the limita tions on differential circuit-to-circuit isolation at maximum integration levels, or alterna tively, the minimum separation between differential circuits for a desired level of isolation. The PMVNA provides a unique capability for the measurement and analysis of differential circuits. This capability can be directly applied to make accurate measurement of differential crosstalk Such measurements have not been possible before the advent of the PMVNA. As a result, analysis of differential crosstalk has been previously limited to indirect (and usually ambiguous) measurements, or to overly simplistic electromagnetic simulations. The PMVNA allows direct measurement of crosstalk of differential systems in a practical circuit implementation. Further1r1ore, the PMVNA also provides increased measurement dynamic range over traditional single-ended equipment, particularly for on wafer measurements. To begin the study of differential transmission line crosstalk, several experiments have been fabricated on ceramic substrates. These structures are generally larger than IC implementations, but they provide insight nevertheless. The experiments are primarily a series of pairs of adjacent differential transmission lines, with the separation between the differential transmission lines is varied. The experiments are divided into two groups: ( 1) pairs of balanced differential transmission lines, and (2) pairs of unbalanced transmission lines.

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214 9.3.1. Balanced Differential Transmission Lines These experiments are a series of pairs of differential transmission lines with vary ing separations between the two transmission lines. Each differential transmission line is terminated with a load of nominal values of 1 OOQ differentially and 25'2 common-mode. By terminating two adjacent differential transmission lines at opposite ends, the overall structure becomes a two-port mixed-mode device which can be measured with the PMVNA. The goal of these experiments is to quantify the level of crosstalk between bal anced differential transmission lines. It is a further goal to find the dependence on trans mission line separation tor crosstalk. This series of experiments are based on the differential transmission line shown in Figure 9-29 This structure is meant to approximate a practical differential transmission line in an IC implementation, where ground planes are not interposed between the two sig nal conductors. As can be seen in Figure 9-29, the signal conductors of the transmission line are of equal width, so the line is balanced. The signal conductors are designed to be 1 OOm wide with 1 OOm spaces between the signal conductors and grounds, which results in a differential characteristic impedance of approximately 100.U No attempt was made to set the common-mode characteristic impedance. The tapered separation of the Figure 9-29. Layout of simple balanced differential transmission line without interrnedi ate ground.

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215 0 0 0 0 ,,,,., ,,; ... / ./ / \ I ,,-... ,,-... ..... ,_ N ..... v ..... C/J C/J ..._,,, ..._,,, '"'O 0 0 0 ....... ....... I 0.25 freq. (GHz) I 20.25 0.25 freq. (GHz) 20.25 0 0 0 0 ., ,,; / / I ,,-... ,,-... ...... N N N "' C/J C/J ..._,,, ..._,,, '"'O 0 0 0 I 0 25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-30. Measured pure differentials-parameters (DD) of structure of the simple balanced differential transmission line, magnitude in dB. signal conductors at both ends of the transmission line is required to al l ows connection with l 50m pitch dual RF probes. The transmission line are 4000m long. The measured results of this transmission line are shown in Figure 9-30 through Figure 9-33. The pure differential-mode response, shown in Figure 9-30, indicates the structure is a reasonably good differential transmission line By the periodic variation of the differential return loss, it is clear than the characteristic impedance of the line is not

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216 0 0 0 0 ., ...... ... ,. / ., ' I I '\ I / ., i--.... _/ '\ u ,,-..., ...... u u u u VJ VJ ..._,, ..._,, 0 0 0 ,...., ,...., I 0 25 freq. (GHz) 20.25 I 0 25 f r eq (GHz) 20.25 0 0 0 0 / ., / "" .... .... ,' / I I \ I / '' w ,,-..., ,,-..., N N N u u u u VJ VJ ..._,, ..__, "1::) 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-31. Measured p u re common-modes-parameters (CC) of structure of the simple balanced differential transmission li n e, magn i tude in dB. p r ecise l y 100 .Q, b u t it is less than -20dB to above 20 GHz which is acceptable. The actual differentia l characteristic impedance is approximately 970. Also, the differential inser tion l oss is less than 0.5dB over the entire meas u rement ba n d The p ur e common-mode response i s shown in Figure 9-31. As can be seen, the structure does not provide a wel l -matched common-mode transmission line. This is not a concern, as practical IC imp l ementation of differential transmission line will generally

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0 0 -u -0 Cl) 0 0 \719 I 0.25 0 0 ,N u -0 ,A II. ... 0 0 I 0.25 ,) ... \J ..... ,, r~ 'YW, freq. (GHz) "" .. J', I.I .. I freq. (GHz) 217 IA ( 20.25 '>v,, 20.25 0 0 0 0 I 0 0 "1M1IJ_ .. /f l 0.25 ~1 ... \A .. freq. (GHz) "./', .. ,J' I "' f 0 0 I 0.25 V I freq. (GHz) ~, 'V 20.25 ,.Jf 20 25 Figure 9-32. Measured common-to-differential mode-conversion s-parameters (DC) of structure of the simple balanced differential transmission line, magnitude in dB exhibit common-mode characteristic impedances far from the match value of 25Q. The actua l common-mode characteristic impedances of the structure is 45Q. The mode-conversions responses are shown in Figure 9-32 and Figure 9-33. These measurements indicate that the structure is we l l balanced. The mode-conversion levels, both in transmission and reflection, are in the -40dB to -50dB range. These levels could result from probe-to-probe crosstalk or from residual errors in the calibration.

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218 0 0 0 0 --.,.-.._ ...... N ...... /' ...... -0 --g u ... '1 V \.. ,I V) jJ Vr ) V) j ,..,, .._.., ..._,, f I \, ti ftA ,..I\... I 7 \J' 1 L& )J Y"' I ,, 'I T 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 0 0 0 0 ----...... N N N / u -0 -0 u ~J J V) / V) J .. V"" .. ... ~I"' .._.., .. '---' i f 'IV' !I f \,J 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-33. Measured differential-to-common mode-co n versions-parameters (CD) of structure of the simple ba l anced differentia l transmission line, magnitude in dB. In the crosstalk experiments the transmission line of Figure 9 29 is resistively ter minated at one end. A test structure of such a device has been fabricated, as shown in Figure 9-34. The terminating resistors, made of thin NiCr, are design for nominal values of 1 OOQ for the different i a l mode and 25Q for the common-mode. The fabricated resis tors have not been tuned, so the actual values are about 10% low. The structure of Fig u re 9-34 has been measured as a mixed-mode one-port, and the results are plotted in Figure 9-35. From this measurement, one can see that the structure is

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219 terminated reasonably well in the differential-mode. This indicates that the actual values of the terminating resistor are close to there nominal values. The common-mode response indicate s a miss-matched transmission line, as expected. The mode-conversion responses show that the balance of the structure has been preserved. The crosstalk experiments are constructed through the use of two of the terminated transmission lines of Figure 9-34. The first of these experiments is shown in Figure 9-36. The two differential transmission lines are separated by 600m, measured from the center of the signal conductor space of the top transmission line the that of the bottom. Since it is desirable to make the results as general as possible, the transmission line separation is reported as a normalized value where the separation distance is divided by the signal con ductor width. This choice of normalization does not make the results completely indepen dent of the physical dimensions of the structures, however. For example, the substrate height, relative to the conductor width, will also influence the responses of these struc tures, particularly the common-mode re s ponses The separation ratio of Figure 9 36 is six. The remaining crosstalk experiments are shown in Figure 9-37 through Figure 9-39. The separation ratios of these structures are ten, fifteen, and twenty Figure 9-34. Layout of terminated simple balanced differential transmi s sion line with out intermediate ground.

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220 0 0 0 0 -.. -.. ..... ..... ..... ..... "'d 0 "'d "'d V Cl.) Cl.) "' . ....__,, ....__,, """ "" ~, If~ 0 0 0 0 ....... I 0 25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 0 0 0 0 / .... r,... .... / \ I ,/ -.. -.. ..... .--( ..... ,. .--( -0 u 0 'V 1 'I I u Cl.) f Cl.) ....__,, I ....__,, I I N 0 0 0 0 ....... I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-35. Measured one-port mixed-mode r eflections-parameters of structure of the terminated s imple balanced differential transmission line, magnitude in dB. The measured mixed-modes-parameters from these four s tructure s are presented in Figure 9-40 through Figure 9-43. The s-parameters of the four structures are overlaid in each of the figures. Figure 9-40 contains the pure differentia l -mode response of the four crosstalk experiments The differential transmission, Sdd 21 and Sddt 2 represents the crosstalk level between the transmission line s (in the differential-mode only). The differ ential crosstalk clearly indicates a dependence on the separat ion ratio, as expected, with

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221 Figure 9-36. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line (without intermediate ground) and transmission lines separated by 600m (D/W=6). Figure 9-37. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line (without intermediate ground) and transmission lines separated by I OOOm (D/W = I 0). the closest transmission lines having the highest levels of crosstalk. The differential reflection parameters, Sddl 1 and Sctct 22 show a very slight dependence on the separation ratio (note the difference in scales). The reflection response of the widest separation is essentially equal to that of the single terrninated transmis s ion line of Figure 9-34. The lack of a strong dependence of the reflection parameters of the separation indicates that, even with a small separation, the differential transmission lines are only weakly coupled.

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222 Figure 9-38. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line (without intermediate ground) and transmission lines separated by 1500m (D/W=15). Figure 9-39. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line (without intermediate ground) and transmission lines separated by 2000m (D/W =20).

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0 0 0 l/') I 0 25 0 0 / ~b ~/ 0 0 rr I I 0.25 223 I I D/W=6 1 0 15 20 f req (G Hz ) 20.25 DIW=6 1 0 01 5 20 \. I ~ "' 'l f req (G H z) 2 0 .25 0 0 0 0 ..... I 0 0 0 l/') Ill' 0.25 I 0.25 _/ / I T DIW =6 1 0 1 5 20 '~ --freq. (GHz) 20.25 DIW=6 _\. 1q ~ 15 2 ~ :::: f req (G Hz ) 20.25 Figure 9-40. M easured pur e differential s-parame t e r s ( DD ) of s tru ct ur es of Figure 9-36 to Figure 9-39, magnitude in dB. Note the different sca le s. It is important to not e that the transmission parameters repre se nt the crosstalk of the struc ture only while terminated with reference impedances. The effect of termination imped an ces on c rosstalk ca n be found by calculating the tran s ducer power gain ( or voltage ga in if desired) with the pure-differential s-paramete r s. Mor e detail s of this type of ca l c ulati o n will be g i ven in Chapter 11.

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224 0 0 0 0~ ..... :IL.. '\.. i,., h ,-... ,-... ... N /2~' (.) 8 (.) D!W=6, J; 1 7 ti'.) ti'.) -....._/ -....._/ 10 7 7 15 ) 20 0 0 0 0 .--1 ....-i I 0 25 I 20.25 freq. (GHz) 20 25 0.25 freq. (GHz) 0 0 l .. 0 0 .. ..... =II ,~ II Nt I ,-... ... I 'I ,-... ..... N N /;, N (.) (.) (.) (.) ti'.) D/W=6 ti'.) -....._/ 10 11 -....._/ 15 1 20 0 0 0 0 .--1 ....-i I 0.25 freq. ( GHz ) 20.25 I 0 .2 5 freq (GHz) 20.25 Figure 9-41 Measured pure common-mode s-parameters (CC) of stn1ctures of Figure 9-36 to Figure 9-39, magnitude in dB. The pure common-mode responses of the crosstalk experiments are s hown together in Figure 9-41. The common-mode transmission, Scc 21 and S cc i 2 represents the common-mode crosstalk between the transmission li nes with reference terminations In a practical differential circuit, the common-mode crosstalk physically represents the extent to which an unde s ired common-mode signal (s uch as noise from a power supply) can be coupled into an adjacent differential circuit (as a common-mode signal). Several observa tion s can be made about the common-mode crosstalk. First, it is general l y at a higher

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0 0 ...... Cl) ...__ 0 0 I 0,25 0 0 ,.-... .... /:Q "'O 0 0 I 0.25 225 ...... I freq. (GHz) 20.25 '~ I -/ I! I I /,' DIW=6 I / 10 '/ 15 l 20 f r eq. (GHz) 20.25 0 0 ,.-... C"l .. -01 ,I Cl) 0 0 I 0.25 0 0 C"J u -0 0 0 I 0.25 I, DIW=6 1// Io 15 I 20 freq. (GHz) -:..-\ freq. (GHz) I 20.25 ""' 20.25 Figure 9-42. Measured common-to-different i al mode-conversions-parameters (DC) of structures of Figure 9-36 to Fig ur e 9-39, magnitude in dB. level than the differential mode cross t a l k Second, the periodic nature of both the trans mission and reflection illustrate the dependence of crosstalk o n impedance l evels. Varia tio n s in magnitudes of common-mode reflections indi cate impedance levels along the transmission lin e are also changing. Corresponding to these impedance changes are also changes in the common-mode crosstalk.

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0 0 "8 0 0 0 25 0 0 0 0 r~ 11 ~ I 0.25 226 ===s V freq. (GHz) 20.25 i ..,,::i '/ I ,J JV /;, D/W=1t /// 15 '' 20 1 freq. (GHz) 20.25 0 0 ,-..... C"l ,.._ -0 u 0 0 I 0.25 0 0 C"l "8 0 0 I 0.25 I ,, iii// t J '/ /; I DIW=6. J; 10 1 15 1 20 freq. (GHz) 20.25 freq. (GHz) 20.25 Figure 9-43. Measured differentia l -to-common mode-conversions-parameters (CD) of stn1ctures of Figure 9-36 to Figure 9-39, magnitude in dB. The mode-conversion responses of the crosstalk experiments are plotted in Figure 9-42 and Figure 9-43. The levels of mode-conversion exhibited by the crosstalk experiments is significant l y higher than those of the single differential transmission line structure of Figure 9-29. This difference is due to an imba l ance that has been introduced in the crosstalk experiments. In the crosstalk experiment structures, the differential trans mission lines have a ground plane near only one of the signal condt1ctors. In contrast, the

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227 0 0 0 0 ,,-.... ...... ...... N c-,.: -a rJ) rJ) .._ .._ 1..--I,..,-" 0 0 0 0 I 6 D/W 20 I 6 D/W 20 0 0 0 0 ,,-.... ...... ..... N N (.) (.) (.) rJ) rJ) '--" '--" 0 0 0 0 I 6 D/W 20 '6 D/W 20 Fig u re 9-44. Measured mixed-mode transmission s-parameters (S 21 ) as a function of separation d i stance, at 10 GHz (heavy) and 1 .0 GHz (light), of structures of Figure 9-36 to Figure 9-39, magnitude in dB. single differential transmission l i ne in Figure 9-29 has ground pla n es equal distances from each signa l conductor. This oversight in the crosstalk experiment de s ign results in errone ously high mode-conversion levels As a result, it is difficult to make any quantitative conclusions about the effects of mode-conversion on crossta l k. Nevertheles s, it is instruc tive to consider the physica l meaning of transmissive mode-conversion in crosstalk situa tions. Consider the common-to-differential conversion, Sd c 2 1 This term represents the

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228 level at which a common-mode (undesired) signal on one transmission line is converted into a differential signal on an adjacent line This type of conversion is particularly unde sirable since the differential circuit is designed to re s pond to differential signals. When a common-mode interfering signal is c onverted into a differential-mode signal, the interfer ing signal i s effectively injected into the adjacent differential circuit This injection of the interfering signal n ega te s the isolation advantages of differential topologie s to so me extent. As it has be e n demon s trated, imbalance in a differential system generate s mode conversion. Therefore imbalance can greatly reduce the i so lation effectiveness of differ ential transmission lines To better understand the relationship of circuit separation to crosstalk strength the four unique transmi ss ion parameters of the crosstalk experiments, Sctd 2 l S cc2 l Sd c21 and Scct 21 ha ve been plotted as a function of the se paration ratio D/W in Figure 9-44. The se crosstalk parameter s have been plotted for I GHz and 10 GHz. In all but one case, the magnitudes of the crosstalk parameters decrea se monotonically with increasing sepa ration ratio as i s expected. The one exception is the differential crosstalk at 1 GHz which increases a s the separation increases By inspecting Figure 9-40, it i s observed a deep minimum occurs in Sctct 21 near 1 GHz. Thi s null is mo s t likely due to s ome unexpected parasitic behavior of the structures. Quantitative conclusions about crosstalk near the null are not reliable. In general, it can be s een that the differential crosstalk at higher frequencie s does decrea se as a function of separation. However the rate of decrea se with se parati o n is not at the rate one might expect, particularly in light of the measurements of Section 5.1. In Se c tion 5.1, it wa s s hown that probe to-probe crosstalk has a lld 3 dependence on se par a

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229 tion. The differential transmission line systems of this section show a much weaker dependence on separation. The cause of this difference is found in the means by which the coupling occurs. The probe to-probe coupling is a high impedance effect; that is, the cou pling mechanism is primarily the electric field. With the adjacent differential transmission lines the primary cot1pling mechanism is the magnetic field (low-impedance effects) Each differential transmission line creates a relative] y large loop through which magnetic fields pass. This magnetic field coupling, in combination with the residual electric field coupling, and the distributed nature of transmission lines, result in an overall crosstalk profile that is weaker than the expected l/d 3 dependence on separation. This behavior in differential transmi s sion lines has given rise to the so-called twisted-pair transmission line, which greatly reduces the magnetic coupling between adjacent differential transmis sion lines [5]. This suggests the need for analogous ''twisted'' differential transmission line structures for IC applications. Despite the magnetic coupling, the differential transmission lines still provide iso lation advantage over single-ended transmission line systems. By comparing the pure dif ferential-mode transmission to that of the common-mode (Scc 21 ), one can see the crosstalk advantages of the untwisted differential transmission system. The common-mode crosstalk is essentially the same as single-ended crosstalk Care must be taken in this comparison, however, since crosstalk in any interconnection system is a function of termi nation impedances. The behavior of Scc 21 versus separation shows that single-ended transmission line crosstalk decreases slower than differential crosstalk, so that at D/W =20, the differential system has a 30 dB advantage at 1 GHz and 20 dB at 1 O GHz. Therefore,

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230 it can be concluded generally that the untwisted differential transmission line systems have lower crosstalk than single-ended systems. 9.3.2. Unbalanced Differential Transmission Lines To study the effects of imbalance on differential transmission line crosstalk, a series of experiments have been designed with unbalanced differential transmission lines Like the previous section, these experiments are a series of pairs of differential transmis sion lines with varying separations between the two transmission lines. In this case, one of the transmission lines have and unbalanced step-up in signal conductor width The unbalanced transmission line in these experiments in shown in Figure 9-45. The wide sec tion is 125m wide, leaving a space between signal conductors of 87m. All other dimensions are the same as in the previous section. The measured mixed-mode s-parameters of the unbalanced step are shown in Figure 9-46 through Figure 9-49. From these figures, it can be seen that the step in width has strongly effected the pure-common mode and mode-conversion responses and the pure differential-mode response to a lesser extent The most pertinent changes are in the mode-conversion parameters which clearly show the imbalance of the transmission line. Figure 9-45. Layout of simple differential transmission line with an unbalanced step in conductor width, without intermediate ground.

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231 0 0 0 0 / I"\ V ,--.... V V ,--.... ...... N ...... "O "O "O "O Cl) Cl) '-"' '-"' f f 0 0 0 ....... ....... I 0.25 I freq. (GHz) 20.25 freq. (GHz) 20.25 0.25 0 0 0 0 V ,--.... -. v V ...., N N "O "O "O Cl) Cl) '-"' '-"' f f 0 0 0 ....... ...-I I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-46. Measured pure differentials-parameters (DD) of structure of the simple differential transmission line with an unbalanced step in conductor width, magnitude in dB. Comparing the mode-conversion responses of Figure 9-48 and Figure 9-49 to Figure 9-44, one can see the mode-conversion of the unbalanced transmission line are on the same order as those of the crosstalk experiments with balanced transmission lines in the previous section. The mode-conversion of the balanced transmission experiments is caused by unintentional imbalances in the layouts. The crosstalk experiments, shown in Figure 9-50 to Figure 9-53, have the same unintentional imbalances, which will cause

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232 0 0 0 0 I/ "" / .,,, I',.. ... I \ I I / .I '\. "' ,-.... ,-.... N u u (.) (.) fZJ en ....._,, ..._, 0 0 0 ..-I I I 0.25 freq. (GHz) 20 25 0 .2 5 freq. (GHz) 20.25 0 0 0 0 ./ / .,,, ,JO ... / / I I I \. .... ' )J ,-.... ,-.... N N N u u u u fZJ en ...._,,, ....._,, "'O 0 0 0 ..-I ..-I I 0 .2 5 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-47. Measured pure common-modes-parameters (CC) of structure of the simple differential transmission line with an unbalanced step in conductor width, magnitude in dB. similar levels of mode-conversion. This indicates that the crosstalk effects of the inten tional imbalance will be masked by those of the unintentional imbalances. Nevertheles s, the four unique transmission parameters of the crosstalk experi ments Sdd 2 J, S cc2 1 Sdc21 and Scd2J, have been plotted as a function of the separation ratio, D/W in Figure 9-54. Like the previous section, the crosstalk parameters have been plotted for 1 GHz and IO GHz The masking of the intention mode-conversion effects is evident

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233 0 0 0 0 ,,-..... .. ,,.-, / \ / ~ -...... /' 7 (.) ,, "'O I "'O r \ Cl'.) Cl'.) .,.. ..__ ..__ V 0 0 0 0 I 0.25 I 20.25 freq. (GHz) 20.25 0.25 freq. (GHz) 0 0 0 0 __.,. ....... ,,-.., ,,-.., V' / \ / C"I ---... (.) .,, "'O ., \ ,(' .., "'O I Cl'.) / Cl'.) ..__ "'C I/' 0 0 0 0 I 0,25 freq. (GHz) 20.25 I 0.25 freq (GHz) 20.25 Figure 9-48. Measured common-to-differential mode-conversions-parameters (DC) of structure of the simple differential transmission line with an unbalanced step in conductor width, magnitude in dB. by comparing the conversion parameters of Figure 9-54 to those of Figure 9-44. Despite the imbalance in the transmission line signal conductors, the mode-conversion levels in the crosstalk experiments are unchanged. Due to these difficulties no quantitative conclusions can be drawn about the dependence of mode-conversion crossta l k upon separation.

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234 0 0 0 0 I,"' i,r ,,-... ,,-... / \ I N _/ -0 -0 u u I .... Vt I (/). rJJ '-" I 0 0 0 0 I 0.25 freq. (GHz) 20.25 I 0.25 freq. (GHz) 20.25 0 0 0 0 ,_ i..--,,-... ,,-... N / "" 7 N N u """"" / -0 -0 I/" ti u \lJ rJJ / Cl) '-" I "'C 0 0 0 0 I 0.25 freq (GHz) 20.25 I 0.25 freq. (GHz) 20.25 Figure 9-49 Measured differential-to-common mode-conversions-parameters (CD) of structure of the simple differential transmission line with an unbalanced step in conductor width, magnitude in dB.

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235 Figure 9-50. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line with an unbalanced step in conductor width ( without intermediate ground) and transmission lines separated by 600m (D/W=6). Figure 9-51. Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line with an unbalanced step in conductor width ( without intermediate ground) and transmission lines separated by I OOOm (D/W=lO)

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236 Figure 9-52. Layout of differential to-differential crosstalk experiment with simple dif ferential transmission line with an unbalanced step in conductor width (without intermediate ground) and transmission lines separated by 1500m ( D/W=l5). Figure 9-53 Layout of differential-to-differential crosstalk experiment with simple dif ferential transmission line with an unbalanced step in conductor width (without intermediate ground) and transmission lines separated by 2000m (D/W=20).

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237 0 0 0 0 ,,-..._ ,,-..._ ,---( ,---( "'O "'O r/1 r/1 "-" "-" ,, V / / / 0 0 0 0 I I D/W 20 6 D/W 20 6 0 0 0 0 ,,-..._ ,---( ,---( (' N .., (.) (.) (.) r/1 r/1 -.._., "-" "'O 0 0 0 0 I 6 D/W 20 '6 D/W 20 Figure 9-54. Measured mixed-mode transmission s-parameters (S 21 ) of the simple differ ential transmission line with an unbalanced step in conductor width as a function of separation distance, at 10 GHz (heavy) and 1.0 GHz (light), of structures of Figure 9-50 to Figure 9-53, magnitude in dB. Regardless of the problems with the experiments of this section and Section 9.3.1, several important conclusions about differential transmission line crosstalk can be made First the PMVNA provides a means by which the crosstalk of differential systems can be directly and accurately measured. Second, with PMVNA measurements, Sdd 21 represents the differential crosstalk. This type of crosstalk is usually the most important in differen

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238 tial circuit applications Third S cc 2 1 is the transmission level of an undesired common mode signal from one differential transmission line to an adjacent line Fourth mode-con version crosstalk S c d 2 1 and Sd c 21 can significantly reduce the isolation effectiveness of differential system s This is done by converting an undesired common-mode signal from one differential tran s mission line to an undesired differential signal on an adjacent line ( in the case of Sd c 2 1 and the reverse in the case of S c d 2 1 ) Fifth, the transmission paramet e r s ( Sdd 2 l S cc 2 1 etc. ) are equal to the actual crosstalk in a system only when that s ystem i s terminated at the s ource and load with the reference impedances If the actual circuit s have other terrnination impedances the actual crosstalk can be calculated with the use of the mixed modes parameters.

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CHAPTERlO PASSIVE INTEGRATED CIRCUIT STRUCTURES This chapter presents a series of RF differential structures that are fabricated with a silicon IC process. The structures are designed to provide practical examples of RF IC interconnections for differential circuits. Moreover, this chapter provides original studies of IC interconnection crosstalk using mixed-mode s-parameter concepts in measurement and analysis. The majority of presented experiments are based on transmission lines structures, including traditional single-ended transmission lines and a variety of differen tial transmission lines. These experiments are designed primarily to assess the line-to-line coupling of such transmission lines on a silicon IC, but some consideration is also given the mode-specific transmission perforrr1ance of these structures. Finally, the coupling between probe pads will be studied a function of two dimensions. All structures were fabricated on a silicon substrate using the IC processing facili ties of the University of Florida Microelectronics Laboratory. All design, mask genera tion, and IC processing was done by the author. The IC process is shown schematically in cross-section of Figure 10-1. The final wafer has two metal layers, polysilicon diffused resistors, and an implanted substrate surface. The original substrate was three inches in diameter, and had a resistivity of approximately 100 Q-cm. Ion impla ntation (dose of 3.3 x 10 15 cm2 of boron at 30 KeV) decreased the surface resistivity to approximately 0.1 Q-cm. A first layer of oxide was thermally grown on the implanted surface. After all processing, this oxide was approximately 1 KA. Substrate contacts were defined by wet 239

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240 etching a photo-lithographically defined opening in the thermal oxide. A resistor layer was implemented with a layer of doped polysilicon (with a sheet resistance of approxi mately 50 Q/square ) A second layer of oxide, 5 KA thick, was deposited on top of the photo-defined polysilicon layer. The first metal layer was formed by electron beam evap0 oration of aluminum 5 KA thick, directly on to the photo-defined polysilicon layer. A 20KA ts KA sKA sKA 1 KA 0 4KA ox ox thermal OX n substrate (100 0-cm) Figure 10 1 Cross section of IC structure. (Not to scale) ___ poly resistor

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241 third layer of oxide, 15 KA thick, was deposited on top of the photo-defined first metal layer. Contacts to the first metal were also defined by wet etching photo-defined vias through the third oxide. The second metal layer was for1ned by electron beam evaporation of aluminum, 20 KA thick, on to the third oxide layer. It was assumed that sufficient ohmic metal-to-silicon and metal-to-metal contacts were established through physical contact of the materials during fabrication This assumption i s clearly not optimal but its adoption was dictated by processing limitations. The risk of poor inter-layer contacts was particularly high for the first-to-second-metal contact, where a thin oxide readily forms on the surface of bare aluminum when expo se d to air. These risks were accepted due to processing limitations, and efforts were made to reduce the oxidation of the aluminum (s uch as storing the wafers in a small air-tight con tainer between proce ss ing steps, and minimizing the time between critical processing ste p s). Despite the se efforts, the finished wafers exhibited poor contact resistance for both the first-to-second-metal contact and the first -me tal-to-poly s ilicon contact. The contact re s istance for a small via (2 0m by 20m) was on the order of 1 OOKQ. This high contact resi s tance rendered so me experiments unusable The problems with inter-layer contacts being foreseen, several experiments were de s igned to be useful even with poor contacts. These experiments are constructed either entirely from one metal layer or from two metal layers where interconnections are not required. These structures were successfully fabricated, and the results are pre se nted in the following sec tions. All mea s urements presented in this chapter were made with 256 averages and half-leakage correction.

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242 IO orroy_oS}mdiffm 1 arroy_<18)'mtv.fat2 111 0 o rroy_verteq ual arroy_verttopxZ o rray ve rttwf st ll :!::L========~!E ... ==========~~L=========~ .. ~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~ array_d1ffm 1 array_symm 1 array_twlst1000 orroy_offset1000 array_vertslot =========== :E:L====~!!::-====~~L====~ ~ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!~ oll'CPfJln~emz._m1ip1 DITO)'_drffm2...m 1gnd <:lrray lwi Bi:500 orroy of faet500 o rroy_verttwiBtalot ===== :E:L====~! ... E:====~!!:J..====~E::====~ orray_singlem1 array_osymm1 orray_twfst:200 orroy_offset200 g~-"'rttv,(stsloloff ----+=-J..----;:==!!-=====:::;-i ..pod_ to_ pod_lood pod...ta_pod...open I I jaton....cneyl ongn flat an g n tloi ju~! Figure 10-2. Map of wafer with groups of experiment shown as rectangles, and mea sured experiments indicated with heavy lines. The layout of the entire wafer is summarized in Figure 10-2. Each rectangle repre sents a series of experiments on a particular transmission line structure. The transmission line experiments consist primarily of a pair of terminated transmission lines with varying line-to-line separation. The layout cell name is shown in each rectangle. For example, ''array _diffml '' is the ser i es of experiments using simple un i form differential transmis

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243 sion lines formed in first metal. The heavy lines of some rectangles indicate that these experiments have been measured, and the results are summarized in this chapter. The structures presented in this chapter are grouped into several categories. The first presented are transmission line structures without a metal ground plane. This is f al lowed by a study of similar structures with ground metal ground planes. The effect of unbalanced transmission lines on crosstalk is next examined. A series of new structures, called vertical differential transmission lines is then presented. The chapter is concluded with a study of crosstalk between probe pads. 10.1. Transmission Lines without Metal Ground Planes This set of experiments examines the crosstalk performance of transmission lines when only the semi-conductor substrate is available for a ground plane. These structures are ter1ned ''without metal ground planes'' in contrast to the series of experiments in the following section (Section 10.2). This type of transmission line is very common in today's !Cs, where a single metal runner is used as an interconnection. With these struc tures, the semi-conductor substrate unavoidable becomes part of the transmission line. These experiments examine the basic RF transmission performance of such single-ended transmission line using the substrate as the ground plane, as well as the line-to-line crosstalk of such structures. These single-end structures are contrasted to differential transmission line st1uctures, also without a metal ground plane. 10.1.1. Single-Ended Transmission Lines The layouts of the experiments of single-ended transmission lines without metal ground planes are shown in Figure 10-3. Each transmission line is 10 mm long (not including the 100 m square probe pads), and the signal conductors are 20 m wide. For

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244 this experiment, the signal conductors are formed in first metal. The crosstalk experiments are for1ned from two terminated transmission line placed with a certain separation. The separation is measured from the center of one transmission line conductor to the cen ter of that of the adjacent transmission line. This center-to-center distance is divided by the conductor width to create a normalized separation distance (D/W). The tran smiss ion line s were intended to be terminated to the substrate via a 50.Q polysilicon resistor but contact problems render the terminations as open-circuits. This deviation from the design goal does not invalidate the experiments, however. Rather, the results of these experi ments are valid, but must be interpreted in the context of high impedance terminations For the crosstalk experiments, the separation ratio is varied through the series of structures, and the crosstalk between the transmission lines has been measured with the PMVNA. In this case, four-ports-parameters have been generated, and the data associ ated with the unused pads has been ignored. A single-ended transmission line ha s been provided with both ends connected to probe pads. The basic RF transmission perfor1nance D/W = 100 DIW= 50 D/W=20 D/W = 10 DIW=5 basic line Figure 10-3 Layout of te s t structures for single-ended transmission line s.

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245 can be estab l ished through direct measureme n t of t h e single-ended transmission line with out a me t al ground plane. T h e meas u red resu l ts of these s in g l e-ended experime n ts are summarized in Figure 10-4, Figure 10-5, and Fig u re 10-9. Figure 1 0-4 s h ows the magnitude in dB of the two-ports-parameters of the transmiss i on l ine from 45 MHz to 5 G H z. Figure 10-5 shows the s-parameters i n polar forrn, indicating linear magni tu des and phases as a function of 0 0 0 I.I') I 0.045 0 0 0 0 I 0.045 j freq (GHz) 5.045 ............ r--,....,_ ........ .......... ...._ freq. (GHz) 5.045 0 0 0 0 I 0,045 0 0 0 I.I') I 0.045 / ............ ............. ....... ............ f r eq. (GHz) 5.045 ., freq. (GHz) 5.045 Figure 10-4. Measured s-parameters of single-ended transmission line, magnitude in dB versus frequency.

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246 ,----:::;::::::::--1-:::::::::::----7 45MHz 0.045 freq. (GHz) 5 045 0.045 freq. ( GHz) 5.045 r---~::::::::==""-1----;::::::------, r.f) l--+--+---1----h~ 1---+-......+L-+--+--I 45MHz 0.045 freq. (GHz) 5.045 0 045 freq. (GHz) 5.045 Figure 10 5. Mea s ured s-parameters of single-ended and differential transmission line linear magnitudes and phase versus frequency in polar plots. frequency. From these two figures, it is apparent that this structure has very poor RF transmission qualities Poor substrate contact may be a contributing factor in the struc tt1re' s poor performance, but it is most likely primarily due to poor ground plane conduc tivity at RF. It is concluded that the use of the s ilicon substrate as a ground plane is the primary source of the poor performance. This conclusion is supported by the literature [87]. The structure of the single-ended transmi ss ion line with a doped silicon semi-con

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247 ducting substrate causes complex behavior as a function of substrate resistivity and fre quency. This structure supports three distinct modes: the skin-effect mode, the slow-wave mode, and dielectric quasi-TEM mode (87]. These mode do not necessarily have desir able transmission line performance. The modes can occur in various combinations as the frequency of operation changes for a given structure, thus causing unexpected and abrupt changes in the effective transmission line characteristic impedance, propagation velocity, and loss. As a result, single conductors over a silicon substrate do not generally make rea sonable RF transmission lines. The results of the line-to-line crosstalk for the single-ended transmission line are summarized in Figure 10-9 of the next section As can be seen, this single-ended structure exhibits very high levels of crosstalk. The crosstalk magnitude decreases about 15 dB over a decade of separation increase. This indicates that crosstalk between single-ended lines has dependence on separation approximately proportional to lid. D/W = 100 DIW=50 DIW=20 D/W= 10 DIW=5 ::::::::::::::::::::::::::::::::::::::::::::::::: basic line :1-===---~======-====--==--==---========e
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248 10.1.2. Simple Unifor1n Differential Transmission Line The layouts of the experiments of sim ple uniform differential transmission lines without metal ground planes are shown in Figure I 0-6. Like the prior structures, each transmission line is 10 mm long. The signal conductors are each 20 m wide with a 20 m space between conductors. The signa l conductors are formed in first metal, so there is no metal ground plane under the differential transmission lines. Additionally, the transmission line s have a constant cross-section over their length. The lack of a distinct 0 0 0 V) "" I 0.045 0 0 0 0 I rr '--.... 0.045 "" ' "' common "'-. mode I I I ~, differential freq. (GHz) 5.045 differential ......... r-,.....,. ...__ ..... A ... common/ mode .., freq. (GHz) 5.045 0 0 0 0 I 0 0 0 V) I --differential rr \..... ........... ........... ..... ..... Acommon I/ mode / / 0.045 freq. (G Hz ) 5.045 == I ,, I"' common "'-. mode ' I I I "-, differential 0.045 freq. (GHz) 5.045 Figure 107. Measured s -parameters of a simp le uniform differential transmission line.

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249 V) 4 5 MH z differential 0.045 freq. (GHz) 5.045 0.045 freq. (GHz) 5.045 common 4 5 MHz differential 0.045 freq. (GHz) 5.045 0.045 freq. ( GHz) 5.045 Figt1re 10-8. Measured s-parameters of a simp le uniform differential transmission line, magnitudes and phase ground plane, and the uniform structure of these lines warrant the name of ''simple uni form'' differential transmi ss ion line s. A directly mea s urable differential transmission line has also been provided in this se rie s of experiments. The response of thi s s tructure, as measured by the PMVNA is illu s trated in Figure s 10-7 and 10-8 Figure 10 7 s how s the magnitudes in dB of the pure differential-mode and pure common-modes-parameters. Figure 10-8 s hows the same

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250 parameters in polar form. The mode-conversion responses of the transmission line are not shown, but the conversions magnitudes are less than -20 dB, indicating a reasonable degree of balance. Both of these figures illustrate the basic advantage of differential trans mission lines as IC interconnections. The differential RF performance of the line despite the lack of a discernible ground plane, is acceptable. In this case, the differential charac teristic impedance is approximately 32Q The differential-mode also shows low disper sion and low loss (in comparison to the common-mode). In contrast, the common-mode behavior is poor and not useful in RF applications. The behavior of the common-mode is very similar to that of the single-ended transmission line, and is influenced by the same physical factors. In particular, the lack of a distinct ground plane has caused the common mode RF transmission perf or1nance to be poor. The crosstalk experiments are fo11ned from two terminated differential transmis sion lines placed with a certain separation. In this case, the separation is measured from the center of the space between the conductors of one differential transmission line to of that of the adjacent differential transmission line This center-to-center distance i s divided by the width of a single conductor to create the normalized separation distance (D/W). The transmission lines were intended to be tertninated with a l OOQ poly silicon re s i s tor but contact problems also rendered these as open-circuits Again, the results of these experiments must be interpreted in the context of high impedance terminations. The measured line-to-line crosstalk of the simple uniform differential transmission line is summarized in Figure 10-9. This figure shows the differential-mode and common mode crosstalk at 1 0 GHz as a function of the norrnalized line separation. Like the previ ous chapter, crosstalk is measured as the magnitude of the transmission between the adja

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251 cent line s (i.e. Sdct 21 Scc 21 etc.) Also shown in Figure 10-9 is the line-to-line crosstalk of the single-ended tran s mission line without a metal ground plane. This plot clearly shows the reduction of crosstalk in the differential-mode in comparison to the common-mode and the single-ended line. At larger separation ratios (greater than 20), the differential crosstalk decreases about 50 dB per decade of separation increase. This rate of decrease is indicates a functional dependence on separation nearly equal to the theoretical limit of l/d 3 The common-mode crosstalk, although not monotonic, decreases at a rate of approx imately 20 dB per decade of separation increase. The common-mode crosstalk is again similar to that of the single-ended transmission line. The differential transmission line, even without a distinct ground plane has signif icant crosstalk advantages for IC interconnections. At a separation ratio of one hundred, the differential mode crosstalk is nearly 60 dB less than the common-mode, and nearly 0 N I 0 N ,........i I I I t IY I 111111 :::::--..... sin2:le-ended ......... ...... ... -~ H'II .... ""Mo common ""mode \. \ \. differential \ ., -I se paration (D/W) 1000 Figure 10-9 Mea s ured line-to-line crosstalk, at 1.0 GHz, for single-ended and simple uniform differential transmission lines (both without ground planes) as a function of line separation.

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252 70 dB less than that of the single-ended transmission line. The differential transmission line retains crosstalk advantages at small separations. Even at a separation ratio of five (which corresponds to 40 m edge-to-edge separation between adjacent transmission lines, equal to twice the conductor spacing of a differential transmission line), the differ ential-mode crosstalk is 20 dB less than the crosstalk between two single-ended transmis sion lines. This measured evidence is in contrast to the results of Section 9.3. The structures of Section 9.3 provided only slight reduction of crosstalk in the differential mode com pared to that of the common-mode. The reason for this discrepancy is two-fold. First, the structures of Section 9.3 have significant imbalances and parasitic responses so that effec tiveness of the differential structures in decreasing crosstalk is impaired. Second, the structures of Section 9.3 are limited to small separation ratios (six to twenty). Further more, the transmission lines of Section 9 .3 have cross-sectional dimensions five times larger than those of this chapter. The theoretical decrease in crosstalk for differential modes is based on approximations that the conductor widths and spaces of each differen tial transmission line are much smaller than the separation between adjacent lines. The larger dimensions of Section 9 .3 cause these approximations to be less accurate, and hence the increase the actual differential crosstalk. I 0.2. Transmission Lines with Ground Metal Ground Planes This set of experiments examines the crosstalk perfor111ance of integrated trans mission lines where a metal ground plane is used. These structures are the same as those in Section 10.1, except that first metal is used for a ground plane, and the transmission lines are forrned from second metal. This type of transmission line is not common in

PAGE 262

253 today 's IC s but could be easily implemented With the s e structures, the effect s of s emi conductor s ub s trate are sub s tantially removed from the transmission line performance. These experiments examine the basic RF transmission performance of such single-ended tran s mission line, a s well as the line to-line cros s talk of s uch structures These s ingle-end structure s are contra s ted to differential transmi ss ion line structure s, both with a metal ground plane. 10.2.1. Single Ended Transmission Line s The layout s of the experiment s of single-ended transmission lines with metal ground planes are s hown in Figure 10-10. Like the prior s tructures, each transmission line i s 10 mm long with s ignal conductor s 20 m wide. The signal conductors are formed in second metal and the ground plane under the transmi ss ion lines is formed in fir s t metal. Referring to Figure 10-1, the height between the signal conductor and the ground plane i s 0 approximately 15 KA. The cro s stalk experiments of this s ection are a function of the norJ t rl DIW= 100 DIW= 50 D/W=20 D/W= 10 DIW=5 basic line i = -----. ------. J ---;;; Figure 10-10 Lay o ut of te s t s tructure s for s ingle-ended transmi ss ion line with metal ground plane

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254 malized separation, measured the same as the single-ended transmission lines of Section 10 1 1. Similarly, these experiments are terminated with open-circuits. The measured results of the single-ended transmission lines with metal ground plane are shown in Figures 10-12 and 10-13 of the next section. With a metal ground plane, the single-ended transmission line has good RF performance. The characteristic impedance of this line is approximately 27Q, and is essentially constant across the mea surement band. In contrast, the line without a metal ground plane exhibits a characteristic impedance that varies rapidly over frequency. Furthermore, the losses of the single-ended line with the metal ground plane are significantly less than without a metal ground plane. The measured line-to-line crosstalk of the single-ended transmission lines with metal ground plane is summarized in Figure 10-14, also of the next section. The magni tude of the crosstalk between single-ended transmission lines is significantly reduced by the use of the metal ground plane. The ground plane reduces the crosstalk between adja cent transmission line by nearly 45 dB at small separations and more than 60 dB at large separations. Clearly, the use of metal ground planes with single-ended interconnections has significant advantages for RF transmission line performance and reduced crosstalk. 10.2.2. Uniform Differential Transmission Lines The layouts of the experiments of simple uniform differential transmission lines with metal ground planes are shown in Figure 10-11. Each transmission line is again 10 mm long. The signal conductors are each 20 m wide with a 20 m space between conductors, like the differential lines of Section 10.1.2. The signal conductors are for1ned in second metal, with a first metal ground plane beneath. Due to the constant cross-sec tion, these line are called uniform differential transmission lines.

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DIW= 100 DIW= 50 D/W=20 D/W = 10 DIW=5 basic line 255 Figure 10-11. Layout of test structures for uniforrn differential transmission line with metal ground plane. The measured response of the uniform differential transmission line with a metal ground plane is s ummarized in Figures 10 12 and 10-13 Figure 10-12 shows the magni tudes in dB of the pure differential-mode and pure common-modes-parameters. Figure 10-13 shows the same parameters in polar form. The mode-conversion re s pon ses of the transmission line again are not shown, bt1t the conversions magnitude s are less than -20 dB, indicating a rea so nable degree of balance. Also shown of these figures i s the response of the single-ended transmission line with a metal ground plane The differential and common-mode responses are virtually indistinguishable from one another and from the single-ended transmission line response. This similarity of the re spo n ses indicates that the signal conductors comprising the differential transmission line are essentially uncoupled With uncoupled conductors, the differential and common mode behavior degenerate into identical modes, which are equal to that of a single-ended

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256 transmission line (see Section 3.1.1). The physical construction of the differential line supports the conclt1sion that the signal conductors are uncoupled. The two conductors are 0 20 m apart, but they are only 15 KA ( 1.5 m) above the ground plane. Relative to the height above the ground plane, a conductor over ten times farther away has essentially no effect on another signal conductor 0 0 0 V) I 0.045 0 0 -differential ". common mode "single-ended freq. (GHz) 5.045 differential 11 / / common V mode / --, / single-ended v / 0 0 ....... I 0 045 freq. (GHz) 5.045 0 o I differential / / common / mode ._ single-ended / 0 0 ....... I 0.045 0 o 0 V) I A 0.045 "' freq (GHz) 5.045 differential common mode "' single-ended freq. (GHz) 5.045 Figure 10-12. Mea s ured s-parameters of single-ended and uniform differential transmis sion lines, both with metal ground planes.

PAGE 266

differential common~ mo~e sin le-ended 0.045 freq. (GHz) differential common mode ... ---sin le-ended 5.045 l--+---+---+--ll----311,--1--+---+-' t---i 0.045 freq. (GHz) 5.045 257 differential common mode ... .,,. f sin le-ended N 1---+--+--t--fl--3 ,--t--+--+~ :-t-----1 0.045 0.045 freq. (GHz) 5.045 differential common ~ode sin le-ended freq. (GHz) 5.045 Figure 10-13. Measured s-parameters of single-ended and differential transmission line both with ground planes, magnitude s and phase. The crosstalk experiments of this section are again function of the norrnalized sep aration. The separation between adjacent lines is measured by the ratio of center-to-center distance and a signal conductor width, which is the same as in Section 10.1 .2. Similarly, these experiments are terminated with open-circuits The line-to-line crosstalk of both the differential and common-modes are summarized in Figure 10-14. Like previous sect ions, the crosstalk is quantified by the transmission s-parameters, as measured by the PMVNA.

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0 N I 0 N ,...-1 I 1 258 I I,. ............. sin~le-ended w o :?:round common mode w/ 1 ;t~ 11 d y single-ended w/ 2:round i-.... I/ .... ,JR l'I diff erentia .0 w/ v 1 1 11 l n rl .. Nib separation (D/W) 1000 Figure 10-14. Measured line-to-line crossta l k, at 1 0 GHz, for sing l e-ended and differen tia l transmission lines with metal ground p l anes, as a function of line sepa ration 0 N I 0 N I 1 J.. I I IT I 11111111 common-mode ' w/o ground plan ,, e ..... differential I~ ' w/o ground planP I\J '\. A 111 11111 common-mode l ',I ... w I ground plan ~ I \ j{ e differential "'r, "' : V w/ ground olane separation (D/W) 1000 Figure 10-15. Measured line-to-line crossta l k, at 1 0 GHz, for differential transmission lines with and without metal ground planes, as a function of line separa tion.

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259 The ground plane under the differential transmission line has s ubstantially reduced th e amount of crosstalk for both differential and common-modes. Notice that the common mode cro ss talk is approximately 6 dB higher than the single ended crosstalk particularly as the s eparation increa ses This difference is in accordance with the approximate re s ult s expected with uncoupled lines. The overall decrea s e in cro s stalk is illustrated in Figure 10 15. For the differential line the differential-mode cros s talk is reduced between 20 dB to 40 dB by the presence of the ground plane. The common-mode crosstalk i s reduced 30 dB to 50 dB by the ground plane. Thi s reduction is a result of the small height of the s ignal conductors over the ground plane. In this case, the metal ground plane con fines electromagnetic field s, cau s ing s ignificantly less coupling ( crosstalk ) compared t o the s ame s tructure s without the ground plane Clearly the u s e of metal ground planes for RF IC interconnection s has several advantages over s tructures without ground planes. By e s tablishing a distinct ground plane single-ended interconnection s become good RF transmission lines. In IC imple mentations, the s eparation between metal layer s is typically small compared to s ignal con ductor width s Thi s s mall separation between ground planes and s ignal conductors ha s the effect of s ignificantly reducing the crosstalk between interconnections. The addition o f a ground plane also benefits the differential transmission line. The nearness of the ground plane further reduce s differential mode cro ss talk. With such differential structures, a very high degree of circuit-to-circuit i s olation can be achieved, even on high density IC s. 10 3. Unbalanced Differential Transmission Line s This set of experiments examine the effects of imbalance on crosstalk between dif ferential IC interconnections. These differential s tructures are the s ame as those in

PAGE 269

D/W = 100 DIW=50 DIW=20 D/W= 10 DIW=5 basic line 260 -----------------------=============: !;>~--===--= -----------------======-. t========-======================---= ....... -=====-======================C:.~ --------~ -~-~;).-=====-==============-=--===<~ Figure 10-16. Layout of test structures for unbalanced differential transmission lines. Section 10.1.2, except that imbalance is intentionally introduced. Like Section 10.1.2, these transmission lines have no metal ground plane. The layout s of the experiments of simple uniform differential transmission line s without metal ground planes are shown in Figure 10-16. Like the prior structures, each transmission line is 10 mm long The first metal signal conductors are each nominally 20 m wide with a 20 m space between con ductors. The imbalance is introduced by a step in the width of one of the signal conduc tors of the differential transmission lines. The width of this unbalanced conductor is 40 m wide with a length of 5 mm. The measured response of the unbalanced differential transmission is summarized in Figures 10-17 and 10-18. Figure 10-17 s how s the magnitudes in dB of selected pure differential-mode and pure common-mode s-parameters. Also shown in the figure are the same parameters for the (ba lanced ) simp le uniform differential line of Figure 10-6. Since

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261 the structures are port-symmetric, only S 11 and S 12 are shown. Figure 10-18 shows the magnitudes in dB of selected mode-conversion parameters of the same structures. Due to the reciprocal nature of the devices, only four mode-convers i on s-parameters are unique (see Chapter 11), so only Sctc is shown. From these figures, it is seen that the step-in-width has little effect on the differential lines. The pure-mode responses are very similar, with only the insertion loss of the unbalanced line increasing slightly (about I dB at 5 GHz). The imbalance has a more perceptible impact on the mode-conversion s-parameters. All 0 0 0 If) I r 0.045 0 0 0 If) I ; 0.045 un b 1 a ance d == f balanced freq (GHz) 5.045 unbalanced I'\. balanced freq. (GHz) 5.045 0 0 0 If) I t-.... 0.045 0 0 \~ balanced / = unbalanced / freq. (GHz) 5.045 '" balanced '\ "\.. "-... --unbalanced/ "' 0 If) I 0.045 freq. (GHz) \v~ ~ 5.045 Figure I 0-17. Measured differential and common-mode s-parameters of unbalanced and simple uniform (ba l anced) differential transmission lines

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0 .--..----.--.....---.---...---r---r--r--,---, 0 unbalanc e d balanced ---1,L----1-++-+--+1---1r----1 0 V) L,._..1,___i__--L.---L-~----'---L--'------'1 0.045 freq. (GHz) 0 0 unbalanced w( .JI I I bal anced / 0 V) I 0.045 ,/ freq. (G H z) 5.045 . ., l .. 5.045 262 0 .--.......--.---.----r--.......---.---r~---..---, 0 1---+--+--4--+----+unbalanced L~:t:l-C=t::t~b..L (.) -.._. I---I---L--4----1---4--1----t-; :...+-~---+--'--l -balanced -+-l,.---4---l---1----1 0 ";) L--.,l__....L,_..J.__l,_-4...---l...--1.---l.___JL--J 0.045 freq. (G Hz ) 5 045 0 .---.---.---.----r-----.---.--..--.----. 0 ___ ..._ unbalanced ---1----'-----'-----N .--jH--1---+-----+---+_:_:__:_:, (.) ~,._+-' -.._. l--+---1----1---4--1-..,,.q_-.J..----+--i---l __ balanced-A---1--.J..----+--1---1 0 ";) '----'----L--...l.----L-.._.a,.._--L..---.1...---L----10 045 freq. (GHz) 5 045 Figure 10 18. Mea su red mode-conversion sparamet ers of unbalanced and simple uni form ( balanced ) differential tran smissio n line s. mode conversions-parameters in creased about 10 dB with the unbalanced tran smiss i on lin e. The magnitudes of the m ode-conversion are relatively s mall (near -10 dB to -20 dB), but sti ll sign ifi cant. The crosstalk exper im e nt s with the unb a lan ced line s are function of the normal ized separatio n between two differential transmission lines. Only one of the lin es is unbal ance; the other is the same as the balanced simple uniform differential transmission lines of Section I 0.1.2. The separat ion b e tw ee n adjacen t line s is measured by the ratio of cen

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263 ter-to-center distance and a signal conductor width, which is the same as in Section 10.1.2. Simi l arly, these experiments are terminated with open-circuits. The line-to-line crosstalk of the pure modes is summarized in Figure 10-19. Included in this figure is the pure mode crosstalk of the simple uniform differential transmission lines of Section 10.1.2. From this figure, there is very little meaningful change in the magnitude of crosstalk between the balanced and unbalanced differential transmission lines. However, contrary to expecta tions, the common-mode crosstalk of the balanced line is slightly higher than the unbalanced line, over most of the range of line separations. This is due to different common mode losses for the two transmission lines. Referring to Figure 10-17, the common-mode insertion loss of the unbalanced line is greater than that of the balanced line. Because there is no significant difference in the return loss of the two lines, it can be concluded that 0 N I co 0 N ....... mmon-mode .. unbalanced ml unbalanced JI I'.. d common mo e balanced / differential balanced I/ V \. ,..._~ I 1.0 separation (D/W) 1000 Figure 10-19. Measured line-to-line crosstalk, at 1.0 GHz, for uniform and unbalanced and simple uniform (balanced) differential transmission lines, as a function of line separation.

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0 N I 0 N I 1 0 264 dB(Scdla: balance I I "' I I/ dB(Sdc12) ... balanced dB~Sd c 12J / unba ance / J}I I 11111111 :/ dB(Scct12) I / unbalanced,/ O& separation (D/W) 1000 Figure 10-20. Measured line-to-line mode-conversion crosstalk, at 1. 0 GHz, for unif orn1 and unbalanced and simple uniform (balanced) differential transmission lines as a function of line separation. the ohmic losses of the unbalanced line are higher than the balanced line. The common mode crosstalk of the unbalanced lines will naturally be lower due to these losses. The mode-conversion crosstalk of the unbalanced differential line is s ummarized in Figure 10-20 Again included for comparison is the mode-conversion crosstalk of the simple uniform differential transmission lines of Section 10.1.2. The mode-conversion crosstalk of the unbalanced transmission line is very near that of the balanced line. This figure confirn1s that, in this case, the intentionally introduced physical imbalances of the unbalanced differential transmission line do not cause a significant increase in mode-conversion

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265 10.4. Vertical Differential Transmission Lines This section examines the characteristics of an uncommon type of differential transmis s ion line for IC applications. This transmission line is formed by stacking the two signal conductors in the vertical direction, rather than in the horizontal direction as seen in previous sections. This vertical s tacking gives the structure the name of ''ve rtical '' differ ential tran s mission line. The vertical differential line i s essentially a broad-side coupled pair transmission lin e [14]. Thi s s tructure requires about one-third le ss surface area to construct co mpared to a typical horizontal differential tran s mission line which i s an advantage in RF IC applications. Specifically, this sections considers three varieties of vertical differential transmis s ion line s The first structure has equal width top and bottom conductors. The seco nd variation ha s a top conductor width twice that of bottom. The third type of vertical line al so has a double width top conductor, but the top conductor D/W= 100 DIW= 50 D/W=20 D/W = 10 DIW=5 ba s ic line Figure 10 -2 1. Lay o ut of te s t s tructure s for vertical differential tran s mi ss ion line with equal widths of top and bottom conductors.

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266 additionally has long open slots down the center. In this section, a set of experiments examines the RF transmission 1 ine performance, and the crosstalk behavior, of these verti cal differential transmission lines. The layout for the three types of vertical differential transmission lines are shown in Figures 10-21, 10-22, and 10-23. Figure 10-21 shows the layout of the vertical lines with equal top and bottom conductor widths. The layout of the vertical transmission lines with the top conductor width twice the bottom is shown in Figure 10-22. Figure 10-23 shows the layout of the final variety of vertical differential transmission line, where the double width top is slotted down its length. A detail of the layout of one slotted transmis sion line is shown in Figure 10-24. Like previous sections, these experiments include a series of pairs of terminated transmission lines for the measurement of crosstalk. These lines were intended to be D/W = 100 ~I)=====~~~~~~~---==-----DIW= 50 === ------== ==== D/W=20 D/W= 10 DIW=5 basic line !)I Figure 10-22. Layout of test structures for vertical differential transmission line with width of top conductor twice the width of the bottom conductor.

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267 -------------~ --~ ~ DIW= 100 DIW= 50 D/W = 20 D/W= 10 !) DIW=5 basic line Figure 10-23. Layout of te s t structures for vertical differential transmi s sion line with top conductor with slots and width of top conductor twice the width of the bot tom c onductor. Figure 10-24 Detail of layout of test structure s for vertical differential transmi ss ion line with top conductor with slot s and width of top conductor twice the width of the bottom conductor.

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268 ter1ninated with I oon resistors, but like previous experiments, processing problems have left the lines terminations open. In all cases, the separation between transmission lines is measured from the center of the bottom conductor of one line to the center of the bottom conductor of the adjacent line. This distance is divided by the width of the bottom con ductor to calculate the separation ratio (D/W). Also included is a structure for direct mea surement of the RF perf or1nance of each type of vertical differential transmission line. Each variation of the vertical transmission line is designed to address a certain per forrnance issue. The design process is illustrated by the cross-sectional sketch of the three vertical differential transmission lines in Figure 10-25 The simple vertical differential line is meant to maintain low crosstalk while increasing the integration density of differen tial IC interconnections. The vertical line with twice the top conductor width is designed to improve the balance of the vertical line. The top conductors of the simple vertical line (a) (b) (c ) Figure 10 25. Cross section of vertical differential transmission lines a) Equal width top and bottom conductors. b) Top width twice the bottom. c) Top with slot

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269 (a) have unequal parasitic capacitances to ground. The nearness of the bottom conductor to ground results in a larger capacitance than that of the top conductor. This unequal capacitance generates an unbalanced differential transmission line. By increasing the width (b), the capacitance of the top conductor to ground is increased. The design goal of this structure is to balance the parasitic capacitances of the two condt1ctors The actual width of the top conductor required for complete balance is a function of the location of the ground and the height between the conductors of the vertical transmission line. While the calculation of the width of the top conductor is straight-forward with a distinct ground plane, it is difficult when only the semi-conductor substrate is used under the vertical transmission line. As the IC process used for these experiments is limited to two metal layers, no metal ground plane could be used under the vertical lines. As a result, the design of the width of the top conductor is best accomplished through empirical methods. For this work, the width of the top conductor has been set at twice that of the bottom con ductor as a baseline for future work. For this work, the top conductor is made 40 m wide, and the bottom conductor is 20 m wide. The total transmission line length is 10mm. The design of the third variation of vertical differential transmission line is one with a slotted top conductor, as illustrated in Figure 10-25(c). This transmission line is designed to further improve the balance for the vertical transmission line. As a conse quence of its greater width, the top conductor of Figure 10-25(b) is expected to exhibit lower series resistance than the bottom conductor. This unequal resistance will again gen erate imbalance in the differential transmission line To compensate for this decreased resistance, a slot is formed in the top conductor. The slot is long and narrow, made down

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270 the center of the top conductor. Ideally, this slot should increase the series resistance of the top conductor, while maintaining the balanced capacitance to ground, while having minima l impact on the transmission performance of the structtire. The actual amount of increased balance, and the proper design of the slot, is again left to empirical methods. For this work, the slot is made 20 m wide in the center of the 40 m top conductor The length of each slot is 900 m, fallowed by a 100 m long section of transmission line 0 0 r I '/ / 0 N I I I 0.045 0 0 0 N I r 0.045 ., / ., ., ,,__ / / / too x2 / / / I I equal/ / I / slot ., freq. (GHz) I I equal / I I too x2 V -c. "' slot freq (GHz) 5.045 5 045 0 0 ... .... 7 7f / / / / too x2., / / equal 0 if) I 0.045 0 0 0 if) I 0.045 slot / / freq (GHz) slot ~ "'eaual '\' '' ... too x2/ freq. (GHz) / ~..,, 5.045 ,...,___ .......... V 5.045 Figure 10-26. Measured differential and common-modes-parameters of the three types of vertical differential transmission l ines.

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271 witho ut a slot This 1000 m cycle is repeated of the length of the line The bottom con ductor is again 20 m wide. The total transmission line length is 10 mm. The mixed-modes-parameters of these three vertical differential transmission lines are summarized in Figures 10-26 and 10-27. Figure 10-26 shows selected pure-mode responses of all three vertical lines. Due to port-symmetry, only S 11 and S 12 are shown for each mode. From this figure, it is clear that all three lines have acceptable differential transmission characteristics. In contrast, the common-mode characteristics of all three 0 ,-,-----.--~--.--------..---,--.---.----, 0 e ual I --slot u .__. to x2 0 ";) .__..__.......__,___.__ ___ __.____.____.___.____. 0.045 0 0 0 V) I 0 045 freq. (GHz) 5.045 s lot .,, I / ')( eaual ,~ ... ' """" .... ton x2 ,/ &freq. (G Hz ) 5.045 0 0 ,,--.... u -0 V'.) .__. 0 tr) I 0.045 slot / I s: / equal -.......... .. -., t '-.. ...... ton x2 ,/ freq. (GHz) 5 045 0 ,.--,-----,---,---,--------..---,--.---.----, 0 e ual I slot (.) -0 V'.) .__. to x2 0.045 freq. (GHz) 5.045 Figure 10-27. Measured mode-conver s ion s-parame ter s of the three types of vertical dif ferential transmission lines.

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272 vertical lines are poor. The common-mode behavior is due to the lack of a distinct ground plane, and is very similar to the results of Section I 0.1. The transmission line with equal width conductors has a differential characteristic impedance of about 20il For compari son, the simple uniform (horizontal ) differential transmission line has a differential char acteristic impedance of 32il The small height between the top and bottom conductors ( 15 KA) has decreased the differential characteristic impedance of the vertical transmis sion line. The transmission line with the double width top conductor has a differential char acteristic impedance of about 1 Sil The increase of the width of the top conductor has increased the capacitance to the bottom conductor, which decreases the differential char acteristic impedance of the structure. This decrease largely accounts for the higher inser tion loss of this structure with respect to the equal width vertical line. The increased insertion loss is primarily due to miss-match losses, as supported by the increased reflec tion magnitude. The transmi s sion line with the slotted top conductor has a differential characteris tic impedance of approximately 29il In addition to raising the series resistance, the slot decreased the capacitance between the top and bottom conductors. This decrease results in an increased differential characteristic impedance, which further results in lower inser tion loss and lower reflection magnitudes. The mode-conversion behavior of the three vertical transmission line is summa rized in Figure 10-27 Again only four mode-conversion parameters are shown due to symmetry considerations. The mode-conversion levels of all three vertical transmission line are relatively high when compared to the simple unifor1n differential line. Further

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273 more, the magnitudes of mode -co nversion in transmission apparently run counter to the intended behavior of the three transmission lines The slotted line has the highest mode conversion magnitudes, despite the most extensive efforts to balance the structure. The apparent increase in mode -co nversion is due to the different characteristic impedances of the three transmis sio n line s. The slotted line, with its differential characteristic impedance closest to 50Q, is better matched than the other vertical lines. The better match of the slot ted line increases the amount of signal that can be delivered to the load. This better match increases the magnitude of differential transmi ss ion sddl 2 as well as the mode-conversion transmission, sac 12 and sect 1 2 Inspection of Figure 10-27 reveals the relative strengths of the transmission mode-conversion correspond to the relative magnitudes of the differen tial characteristic impedances of the three vertical transmission lines With the change in 0 ('l I 0 ('I ....... I 1,0 I:~ I;~~~' j 1 \ \ rliff. ,. :nti~l \ cnmmon-mode equal I top x2 top x2 equal slot slot separation ( D/W ) 1000 Figure 10 -28 Mea s ured line-to-line crosstalk (p ure mode ), at 1.0 GHz, for the three ver tical differential transmission lines as a function of line separation.

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0 N I 0 ti 1.0 274 (', I'\ r-.. ; p rl I'\ I' separation (D/W) 1000 Figure 10-29. Measured line-to-line crosstalk (mode-conversion), at 1.0 GHz, for the three vertical differential transmission lines, as a function of line separa tion. Note different scale from previous figures. characteristic impedances considered, no significant difference can be seen in the imbal ance of the three vertical transmission lines. The crosstalk behavior of the three vertical transmission lines is summarized in Figure 10-28. The crosstalk magnitudes between vertical transmission lines are signifi cantly higher than in the horizontal differential lines of Section 10.1. This increase in crosstalk is due to the large magnitude of mode-conversion in the vertical transmission lines Despite the general increase in its magnitude, the differential-mode crosstalk char acteristic still retains an advantage over the common-mode crosstalk. The three variations of vertical lines exhibit similar levels of differential-mode crosstalk, but have consistent, but small, differences in common-mode crosstalk. The large imbalance in all three verti cal lines is illustrated in Figure 10-28. Because of the similarity of the responses, the indi

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275 vidual traces on the plot are not identified. This figure clearly shows that all three lines po ssess a large degree of imbalance. This imbalance is responsible for the increased line to-line crosstalk in these structures. It is concluded that the mode-conver s ion of the vertical transmission Jine s i s a result of IC processing i ss ue s. The series resistance of the first metal layer is higher than that of the second metal. For example, in terms of standards-parameters, at 1.0 GHz the vertical differential line with equal width conductors has about 7. 7 dB of insertion loss in the top conductor, but about 17 dB loss in the bottom conductor. Thi s difference is large enough to generate sign ificant mode-conversion The mode-conversion produced by the re s istance imbalance is much larger than the imbalances generated by unequal capacitance of the conductors to ground. As a result the effects of the balancing techniques used on the vertical Jines are masked by the dominating effect of the unbalanced metal re sistance. This type of problem will typically dominate most IC implementation of vertical differen tial transmission lines For the advantages of thi s struct ure to be realized, care must be taken to compensate the vertical transmission line for unequal resistance in the constituent metal layers. I 0.5 Pad-to Pad Crosstalk This set of experiments examines the crosstalk between probe pad s as a function distance. Pads are relatively large (compared to other integrated devices), typically sq uare metal structures u se d for both wafer-probe or wire-bond connections. In many applications, s ignal coupling through pads can be a sig nificant problem This is due to the fact that pads can be large with respect to other IC features, and can sometimes capaci tively couple signals into the s ubstrate at relatively high levels Pure differential and com

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276 moo-mode coupling between probe pads is examined, as well as mode-conversion coupling. Unlike the simple study of Section 5 1, this set of experiments quantify pad-to pad crosstalk for a practical silicon IC. Furthermore, these experiments map the pad-to pad crosstalk as a two-dimensional function of position of the pads. As discussed in Section 4.3, the PMVNA uses ground-signal 1 -ground-signal 2 ground (GSGSG) probes, so the pads for the experiments in this section (and throughout the chapter) are constructed accordingly The construction of GSGSG pads is illustrated in Figure 10-30. The GSGSG pads are placed in a grid to form the coupling experiments of this section, as shown in Figure 10-31. The rows of pads are spaced 500 m center-to center, and the columns are spaced 1000 m center-to-center The origin of the array is defined to be at the center set of pads in the bottom row. With the wafer probe of mixedlOOm ground signal pad pad E s 2nd metal ::::l ::::l 0 0 V) 0 1st metal poly signal v1as resistor pad ground substrate (a) pad (b) Figure 10-30. Probe pad construction. a) Top view of GSGSG pads. b) Cross-sectional view of process layers in signal and ground pads.

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277 mode port one fixed at the origin of the pad array the probe of mixed-mode port two ha s been placed at various pads in the array where the PMVNA has mea s ured the mixed mode s-parameters Like other experiment s, the variou s transmission s -parameter s quan tify cro ss talk between the pads. Treating the pad s a s sample points in a continuous domain contours of constant cro ss talk magnitude s can be constructed from the measur e d mixed-mode s-parameters The contour s of con s tant cro ss talk are two dimen s ional maps of the coupling between two differential s tructures This contour map of coup l ing can provide in s ight into how signal s from one circuit on an IC are coupled into others on the same IC The maps s how the crosstalk level of the differential-mode between two differential circuit s, as w e ll a s the common mode cro ss talk Additionally the maps can also s how the modec onver s ion c ro ss talk between two differential circuits a s a function of position Thi s two-dimen s ional mapping of the mode-conversion illu s trates how mode-conversion cro ss talk can occur between two otherw i se balanced differential circuits ..... 2500 m 2000 m 1 500 m I OOO m 500 m I I I I I N w U\ V\ w N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1= 1= 1= 3 8 3 0 8 3 Figure 10 31. Layout of test structure for pad-to-pad crosstalk.

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278 The measured crosstalk result s are summarized in Figures 10-32 to 10-35 These figures are contour plots of measured mixed-mode transmission. Figure 10-32 is the pure differential-mode crosstalk between pads at 1.0 GHz and 2.0 GHz. The mea s ured sdd 21 between each pair has been used to interpolate the contours of constant crosstalk over the surface of the IC. This figure shows the crosstalk decreases rapidly as the radial distance 2500 m ri, =--=--=--~ = ::~~===:=;~ ~, ~===r===-r ~~ ~~ 7 ~I ~:-==~~-=-=-~ 7 2000 m ---<= ~ 1 500 m l OOO m 500 m 0 2500 m 2000 m 1 500 m l OOO m 500 m .0 2500 m 2000 m l SOO m l OOO m SOO m I I I I I 1 1 -1 -1 07 _, I I I I 0 N w v,) N 0 0 0 0 0 0 0 8 8 8 0 0 0 8 0 0 0 0 0 0 ,:: 1= "s 1= 3 3 3 I 1 0 I 11 I I I I 1 5 I 11-1 I _, ps I 1 I 1 5 11 I I I -1 1 I 11 0 I I I I 0 N w 8 w N 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1= 1= 1= 3 3 3 I -I I 1 d ~ 80 -l o I I I O N w Figure 10-32 Constant crosstalk magnitudes contour s ( sdd 21 in dB) of pad-to-pad test structure. ( a) (b) (c) a) Crosstalk at 1.0 GHz. b) Crosstalk at 2.0 GHz. c) Analytical approxi mation of crosstalk

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279 from the origin increases. This indicates that the differential crosstalk decreases as a func tion between two differential circuits increase, as has been demonstrated earlier. Figure 10-33 is the pure common-mode crosstalk between pads at 1 0 GHz and 2.0 GHz. Similarly to the previous figure, the measured s cc 2 1 between each pair has been used to interpolate the contours of constant cros s talk over the surface of the IC Compared to the 2000 m 1500m lOOOm SOO m 0 I .:,. 0 0 0 I I w N 0 0 0 0 0 0 1= ,:::: 8 3 I f~-:_~~1:;~~:_1_;::->-I -I ;;:::::;;~:t_~:;::~ -1 ( a) -1 I 0 ..... N w .:,. 0 0 0 0 0 0 0 0 0 8 0 0 0 0 1= 1= 1= t3 3 3 2500m -I __ 1 -I 2000 m 1 500m .0 0 0 0 1= a I I w N 0 0 0 0 0 0 ,:::: 3 I 0 0 0 1= 3 0 0 0 0 N 0 f w 0 8 ,:: a ( b ) 2$00 m p-t-._--.._.......,.....--t----.--n,----,....----t,,-----------;-------.+----....--..------------2000 m 1$ 00m l OOOm r500 m I I I I ,, -==-~i=--~I I I I I 1 r I L L -.-c:::-:~~ I I I I --1 I I I I w 0 0 0 N 0 0 0 ,:: 8 0 0 0 0 ..... 0 8 -1-1 _J 1 -1 -6 I I I -1 I I w 0 8 1= 8 (c) Figure 10-33. Constant crosstalk magnitude s contours (scc 21 in dB) of pad-to-pad test structure a ) Cro ss talk at 1 .0 GHz. b) Crosstalk at 2.0 GHz c ) Analytical approxi mation of crosstalk

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280 differe n tial crosstalk map the common-mode crosstalk decreases slower than differential c ro ss ta l k with increa si ng radial distance. Figure 10-34 is the common-to-differential crosstalk between pad s at 1.0 GHz and 2.0 GH z In this case, the measured sdc 21 between each pair has been used to interpo l ate the contours of constant crossta l k over the s urface of the IC. Finally the differentia l to-common crosstalk i s shown in Figure 10 -34 at 1 0 GH z 2500m I 2000m 1 4 I 1 5 00m -1 IOOOm -1 I 1 2~ 500m -ltl I 0 ' I 0 N I.,.) 8 I.,.) N 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 1= 1= -,:::: 8 8 8 2 500m FT'"""=--==-= ::-i r=--=--=:5~r-=~~=-=c:----==-=p-r:='""'-=-"'F""=--=-'T"7f::::::-==-==~=ii-=~=--=-=r, 2000m 1 500 m I OOO m 500m 0 2000m 1500m lOOOm 500m I 0 I -:. 13 I -=108 1 1 3 I I fC,-~ 08 .,___, I I ' I I.,.) N 0 0 0 0 8 8 0 1= -,:::: 3 8 I 0 0 0 -,:::: a 0 I.,.) 0 8 -,:::: 3 I I I 95 I -----~~~ -----1 I I I I I 0 11 1 -1 -1 I 1L ,,,__ L -l -l 1 -1 I I I I I 8 I I I ----------1 I I I I I I I.,.) 0 8 I N 0 8 I 0 0 8 -,:::: 3 0 8 I.,.) 0 8 Figure 10 34 Constant crosstalk magnitudes contours (sdc 21 in dB ) of pad-to-pad te s t s tructure (a) ( b ) (c) a ) Cro ss talk at 1.0 GHz. b ) Crosstalk at 2.0 GHz c ) Analytical approxi mation of crosstalk.

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281 and 2 0 GHz In this case, the measured scd 21 between each pair ha s b ee n u se d to interpo late the contours of co n s tant crosstalk o ver the s urface of the IC. Both of the mode conversion crosstalk map s have minimal crosstalk along the y -axis 2000m 1500 m lOOOm SOOm J,. 0 0 0 1= 3 w 0 8 i= 3 N 8 i= a I 8 ,::: 3 0 0 8 i= a N 0 8 i= 3 0 0 0 i= 3 2500m rr=-=-=--=~T -=--:~ 7 j;jf=' == 9 M=-=::~y.~::-:c:::-":::ll = ~ ~=i;;;;~ ~--: =r=-=--=r".1 I 1 04 :fog 1 04 2000 m ________ !__ __ 1500m l OOO m 500m 0 2500m; 2000m: 1500,n ; LOOOm t 500m; 01 I ,------, I o ""'v--:f.J--7 r-/ ';:-"'I ..:-..-..--rrrrr' J,. ' I w N 0 0 0 0 0 0 0 0 0 0 0 0 i= ,::: i= ,::: a a 3 3 1 I-I I I I I I 1l-I I I I I I I I ' t.,.) N 0 0 0 0 0 0 0 8 0 0 0 i= i= 1= i= 3 3 a 8 0 0 I I i 1 ~-/-./ -la"'c-..... ~-=1 0~ 1 119 I 1 09 ...... 04 I I N w 0 0 8 0 0 0 0 0 0 0 0 s s s s ~ -; I I -1 _j 1 -1 I -so 5 I -9 I I I I I I N w 8 0 8 8 0 0 0 0 0 s s i= s a ( a ) (b) (c) Figure 10-35 Con s tant crosstalk magnitude s contours ( s cd 2 1 in dB ) of pad-to-pad te s t structure. a) Cro ss talk at 1.0 GH z b ) Cro ss talk at 2 0 GHz. c) Analytical approxi mation of crosstalk.

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282 With some approximations, the crosstalk behavior exhibited in these pad experi ments can be theoretically explained. The theoretical development of the pad crosstalk illuminates the basic principles leading to circuit-to-circuit crosstalk. By assuming that a probe pad is much smaller than a wavelength, the pad can then be approximated as a lumped element, neglecting distributed circuit effects. The structure can be further simpli fied by assuming the pads can be approximated by small conductive sphere at some poten tial. This sphere has a radius, a, where a is the smallest radius that completely encloses the pad (and a is much smaller than a wavelength). To greatly simplify the following expressions, the loading of the pads will be neglected. In other words, the characteristic source and load impedances of the PMVNA will be assumed to have no effect on the crosstalk. This is a course approximation, but it provides expression that are easily inter preted. Furthermore, the predicted crosstalk maintains the same qualitative behavior as the measured data. Neglecting the PMVNA ports impedances is the same as assuming the pads are part of a high impedance circuit, where current is negligible. By neglecting the PMVNA port impedances, the crosstalk can be approximated with a simple electrostatic approximation. With two pads, each approximated as a sphere, (x, y) / / / / / / / / / / (0,0) Figure 10-36. Model of single-ended circuit-to-circuit crosstalk

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283 the electric potential between the spheres can be easily calculated as a function of di s tance For example consider two single-ended pads as illustrated in Figure l 0 -36 One pad, radius a, is located at the origin, and the other pad, radius a, is located at a point in the .xyplane (x, y). The electric potential of a charged s phere is known to be [88] V(r ) Q s 1 (r> a) 41tr Q s 1 ( 10-1 ) (r a) V ( r ) 4na where r is radial distance from the center of the s phere If the sphere at the origin i s defined to have a potential of V 51 then the s urface charge can be found to be Q s = ( 41ta ) V s 1 s o that the potential of the sphere can be stated a s 1 V ( r ) = a V s Ir (r> a) The potential at a point in the x-y plane outside of the s phere can be expressed as 1 V (x, y) = a V s I J 2 2 X + y ( 102) ( 10-3 ) The radiu s of the second pad will be neglected, so that its potential is simply expre sse d by ( 10-3 ), so that Vi 1 (x, y) = V (x, y). The crosstalk between the two pad s will be defined as the ratio of the potential s of the pad s. Crosstalk = Vi 1 = --;:::=a== V s l Jx2 + y2 a r From ( 10-4 ), the 1/r proportionality of s ingle-ended crosstalk is readily ob se rved ( 10-4) This same approach can be appli e d to two s ets of differential probe pads and the pure differential, common-mode, and mode -c onversion crosstalk can be approximated Referring to Figure 10 -37, the pair of differential pads can be modeled by four small

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284 (x, y) (x-s/2, y) I/ f (x+s/2, y) I; /, I; /; I /; I // I I f'I I // I I//; I I/ I I 1/ I; Vs1 // V ___ __s 2 _ +s/2 (0,0) Figure 10-37. Model of differential circuit-to-circuit crosstalk. spheres. Two spheres on the x-axis represent the ''source'' pads. These pads, are at (-s/2, 0) and ( +s/2, 0) wheres is the pad-to-pad separation of the differential pad set, and these pads have electric potentials of V s l and V s 2 respectively The potentials at the other two pads, located at (x-s/2, y) and (x+s/2, y), are approximated by (10-5 ) aVsl aVs2 v 2(x y) = + -;:::=== l 2 2 J2 2 (x+s) +y x +y (10-6) Differential-mode operation of the source pads is set by driving the pads with equal but opposite potentials, V s 1 = -Vs 2 = VO Common-mode is set with equal potential sources, V s l = V s 2 = v O. The differential signal at the output pads is the difference of the potentials of the two pads

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285 (10-7) and the common-mode output is the average potential ( 10-8 ) The pure differential mode crosstalk i s defined a s Crdd = (10-9) The pure common-mode crosstalk i s defined as Cr ee = (10-10) The differential to-common-mode crosstalk is defined as Cr e d = ( 10-11 ) The common-to-differential-mode crosstalk is defined as Crd e = (10-12) Using equations ( 10-9) through ( 10-12) in combination with (10-5) and ( 10-6 ) the variou s types of cro ss talk can be approximated everywhere in the .xy-plane. The se expressions have been u se d to generate approximate contour maps of the pad-to-pad crosstalk. The s e maps are included in Figures 10-32 to 10-35. Comparing the measured pure differential-mode crosstalk in Figure 10-32, s trong similarities can be seen. Along

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286 both the x-axis and the y-axis, a rapid decline in the crosstalk with increasing distance can be seen. The theoretical relationship of the on-axis behavior can be readily derived from earlier approximation s. Setting the y ordinate to zero, ( 10-9) can be written 1 1 1 1 aV ----aV 0 ---0 x x-s x+s x Crdd = 2aV 0 2 -s 3 2 X -S X (10-13) As the distance between pad sets becomes much greater than the space between the indi vidual pads of a differential pair, x >> s, and ( 10-13) becomes 2 s Crdd 3 X ( 10-14) With these approximations the 1/d 3 characteristic of differential crosstalk becomes clear. However, the analysi s also indicates that the differential crosstalk will have this desired behavior only as the distance between circuits is large compared to the dimensions of the differential circuits. Returning to Figure 10-32, both measured and theoretical differential crosstalk decreases approximately proportional to the inverse of the cube of the distance between pads (or -60 dB per decade of distance). The theoretical contour map shows unexpected behavior along radii approximately from the y-axis. At these locations, the theoreti cal map show a series of deep nulls in the differential crosstalk. This effect is caused when the four paths (see Figure 10-37) are precisely balanced, forcing the coupled differ ential signal to vanish. This phenomenon illustrates the limitation of differential circuits in general. Differential circuits in general, can only approximately reject interfering sig nals This is due to the fact that interfering signals will be at slightly different strengths at each of the two conductors of a differential circuit. Only at the points where the interfer

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287 ing signal strength is precisely equal at the two conductors will the differential circuit completely reject the signal. This behavior is not clearly indicated in the measured data. The measurements are limited to a course grid, and rapidly changing features, such as the deep nulls, will likely be missed. Nevertheless, some evidence of the nulls can be seen in the measured data The data at 1.0 GHz has symmetrical minima at a pair of points near the expected locations. Furthermore, the contours near the y-axis are nearly triangular, which approximates the theoretical contours. In general, the theoretical calculations of the probe-to-probe crosstalk match the measured data, despite the crude approximations applied Comparing the measured pure common-mode crosstalk in Figure 10-33, strong similarities can be again be seen Both the measured and theoretical behavior are nearly independent of the angle of rotation in the .xy-plane. The common-mode crosstalk level is approximately proportional to the inverse of the radial distance between the pads In com mon-mode, the pairs of pads essentially behave as a single pads. As a result, the common mode coupling characteristic closely follows that of a single-ended pad, as described in (10-4). The approximate theoretical mode-conversion crosstalk maps are shown in Figure 10-34 and Figure 10-35. The measured mode-conversion responses are very simi lar for differential-to-common and common-to-differential. This is expected due to the port symmetry of the test structures. The theoretical maps show contours that are very similar to the measured data. The differences between the two theoretical mode-conver sion maps are due to neglecting the radius of one pair of pads. Nevertheless, the measured data corresponds well to the theoretical predictions. Both show the mode-conversion lev

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288 els decreasing very rapidly near the y-axis. The off-axis mode-conversion of the pads is due to the difference in path lengths between the four individual pads. Consider differen tial-to-common mode conversion, for example. The distance from one of the pads in a differential pair to a point on the y-axis equal to that of the other pad in the differential pair. Thus, a differential signal on the pair of pads at the origin creates a zero potential on the y-axis. If a pair of non-source pads are symmetrical about the y-axis, the average volt age coupled to these pads is also zero. The voltage difference between the pads is not zero, however, so the pure differential crosstalk is non-zero. As the output pads are moved off of the y-axis, the path lengths are increasingly asymmetric, leading to mode conversion The effect of this asymmetry is reduced as the radial distance between the pads increase, causing the difference in path lengths to be small relative to the overall length. Along the x-axis, it can be shown that the mode-conversion crosstalk is Crcd s = Crdc::::: 2 X (10-15) if x >> s Thus, mode-conversion displays a crosstalk characteristic that is proportional to the inverse of the square of distance (or -40 dB per decade of distance). This indicates that, in some cases, mode-conversion crosstalk can become stronger than differential crosstalk, even in differential circuits that are otherwise perfectly balanced. This conver sion crosstalk can be a serious limitation in the effectiveness of differential IC interconnects. These pad-to-pad experiments have provided the means for insight into circuit crosstalk on ICs. The coarse approximations lead to analytical relations that qualitatively follow the actual crosstalk. These relations represent the fundamental mechanisms of cir

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289 cuit crosstalk, but did not provide accurate quantitat i ve prediction of the strength of crosstalk. Many improvements in the accuracy of the expressions can be made by remov ing or improving the applied approximations. These improvements lead to increasingly complex analytic expressions, obscuring the meaning of the expressions. Accuracy of the analytic expressions can be significantly improved by including the effects of termination impedances. Other effects, such as inductive coupling can also be inc l uded. Ultimately, full electromagnetic analysis of the structures can be accomplished Regardless of the degree of completeness, the accuracy of theoretical crosstalk predictions can be improved. These predictions can be used as a tool in IC design to reduce crosstalk between circuits. In concluding this chapter, it is important to emphasize the enabling role that the PMVNA plays in the study of IC crosstalk, particularly for differential circuits. With this instrument, empirical validation of the basic crosstalk theory is achieved. Furthermore, the advantages and limitations of differential circuits on practical silicon ICs have been demonstrated. Several important conclusions can be made from this study. Even with minimal separation, the application of differential IC interconnections significantly reduces circuit-to-circuit crosstalk with respect to single-ended interconnections. The use of metal ground planes beneath interconnections greatly reduces crosstalk and improves the RF performance of interconnections, for both single-ended and differential topologies The limitations of differential circuits have been demonstrated also. The ability of differ ential circuits to reject crosstalk is generally dependent on the separation between adjacent circuits. The general rule of 1/d 3 applies only as separations are large with respect to the cross-sectional dimensions of the differential circuit At smaller separations, the crosstalk is stronger than that predicted by the l/d 3 rule. The most important limitation is perhaps

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290 the significant level of mode-conversion crosstalk that can exist between otherwise bal anced differential circuits, due simply to the relative spatial arrangement of the circuits. The next chapter will defined properties and methods to aid in the design of differ ential circuits. With the use of mixed-mode s-parameters, the design of differential cir cuits can be accomplished in a straight-forward manner. These tools will allow efficient differential IC design which, with the application of the finding of this chapter, will lead to significant reduction in circuit-to-circuit crosstalk.

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CHAPTER 11 PROPERTIES OF MIXED-MODES-PARAMETERS Mixed-mode s-parameters have been developed for the accurate measurement and analysis of differential RF devices and circuits. In particular, mixed-modes-parameters of two-port differential devices provide mode-specific interpretations that are analogous to traditional (single-ended) two-port device analysis However, some of the well-known properties of traditional s-parameters are yet be for1nulated in terms of mixed-modes parameter. This chapter will derive the mixed-mode counterparts of some of the most basic, and useful, properties of traditional s-parameters. In addition, some important new properties of mixed-modes-parameters will be discussed. The first section will examine symmetry properties of mixed-mode s-pararneters. The next section defines a balanced differential device, and considers its implications. The following section will define the indefinite mixed-modes-parameter matrix, and derive the properties associated with it. The final section of the chapter examines mode-specific gain calculations, and basic design methods. 11.1. Symmetry of Reciprocal Devices 11. l. l. General Provided that all ports are norrnalized by the same reference impedance, the tradi tional s-parameter matrix of an n-port device can be shown to be symmetric if the device is reciprocal [89]. For a reciprocal device, the standard s-pararneters have the property 291

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292 Sstd = (Sstd) T (11-1) Equivalently, the property can be expressed in terms of the elements of the s-parameter matrix as s .. = s ... By direct calculation of the mixed-mode s-parameters with t) )l (11-2) it can be shown that (11-3) In other words, a reciprocal device has a symmetric mixed-modes-parameter matrix. In terms of the mode-specific responses of a reciprocal device, it follows directly from ( 11-3) that T sec = sec (11-4) Thus, a reciprocal differential device has symmetric pure-mode s-parameter partitions. The reciprocal differential device has differential s-parameters that behave like a tradi tional reciprocal device. Similarly, common-modes-parameters also behave as reciprocal device. In contrast, the mode-conversion parameters do not behave as reciprocals devices in the traditional sense. Rather, a reciprocal differential device has complementary mode conversion. For the case of a two-port mixed-mode device, the differential-to-common mode forward transmission, scd 21 is equal to the common-to-differential reverse transmis sion, sdcl2 This does not imply that the differential-to-common mode forward transmis sion, scdl2, is equal to either of the previous terms. In this sense, the conversion process is fixed with respect to the ports, but reciprocal with respect to the modes.

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293 11.1.2. Port-Symmetric Reciprocal Devices An additional class of reciprocal devices has played an important role in previous chapters. This device is one that possess differential port symmetry. This type of symme try is not widely discussed in traditional two-port devices. However, the motivation for defining this type of symmetry arises from two-port applications. A traditional two-port network can be said to have port-symmetry when the ports can be interchanged without affecting the response. For a two-port device to have port symmetry, the reflection param eters of the ports must be equal, and the transmission parameters must also be equal. This type of symmetry is more restrictive than reciprocity since the reflection parameters are encompassed. Physically, a port-symmetric device is one where there is an axis of sym metry perpendicular to the direction of propagation (see Figure 11-1). port 1 port 1 port 2 I direction of propagation I I I port 2 I I axis of symmetry I I I I port 3 port 4 direction of propagation (a) (b) Figure 11-1. Port-symmetric devices a) Traditional two-port device. b) Mixed-mode two-port device.

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294 Extending the definition of port-symmetry to differential devices relies of the con cept of an axis of symmetry. In particular, when an axis of symmetry is perpendicular to the direction of propagation of the differential-mode, for example, the differential device can be said to possess port-symmetry This does not require that the two constituent sin gle-ended ports of a differential port be identical (see Figure 11-1 ). Under port-symmetry, it can be s hown that the mixed-mode s-parameters have some important properties Significantly, the mode-conversion partitions are equal (11-5) The device is also reciprocal so S cd = src still holds. This means the mode-conversion parameters behave as reciprocal devices. The condition of port-symmetry has further implication for mixed-mode s-parameters. Each of the four two-by-two partition s of the mixed-mode parameters contain only two unique parameters : one for transmission and one for reflection. That is 5 ddll 5 ddl2 5 dc11 5 dc l2 5 11 5 1 2 5 1 3 5 14 5 ddl2 5 ddll 5 dcl2 5 dcll H 5 12 5 22 5 14 5 24 5 cdll 5 cdl2 5 ccll 5 ccl2 5 13 5 14 5 11 5 12 5 cd 12 5 cdll 5 cc12 5 ccll 5 14 5 24 5 12 5 22 Hence, the partitions of the mixed-modes-parameters are each port-symmetric. 11.2. Balanced Devices (11-6) This type of device is defined in terrr1s of a mixed-mode representation. By defini tion, a balanced differential device has no mode-conversion (11-7) j

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295 This definition can apply to any type of differential device: passive, active, reciprocal, and so on Under balanced conditions, the device can be separated into two independent net works, one for the differential response and one for the common-mode response. These two network can be separated in both analysis and design. The condition of balance has some implications for the traditional s-parameters that are useful to consider. By applying the transformation (11-8) one finds sddll sddl2 0 0 s11 s12 s13 s14 sdd21 sdd22 0 0 s12 sll S14 S13 H (11-9) 0 0 Seel] sccl2 S3} S32 S33 S34 0 0 scc21 scc22 S32 S31 S34 S33 This gives some insight into the meaning of balance with respect to traditional parameters. In such ter1ns, a balanced device has identical signal paths on each ''side'' of the differen tial circuit. For example, the transmission from port one to port three ( on, say, the ''posi tive side'' of a differential circuit) is equal to the transmission from port two to port four ( on the ''negative side'' of the differential circuit). In this way balance in the device is achieved.

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296 direction of propagation I I port 1 port 3 (a) axis of symmetry I I I I port2 port 4 I I port 1 port 3 no axis of symmetry (b) port 2 port4 direction of propagation Figure 11-2. Balance in devices. a) Balanced device illustrating propagation symmetry. b) Unbalanced device without propagation symmetry. An alternative interpretation of circuit balance can be given in terms of symmetry. Examining the fortn of the traditional four-port s-parameters in ( 11-9), and comparing it to those of the port-symmetric device in ( 11-6), one can see that balance is related to symme try in direction of propagation. Specifical l y, balance in a device is possible when there i s an axis of symmetry parallel with direction of propagation. This type of symmetry, which can be called propagation symmetry, is illustrated in Figure 11-2. Such symmetry gener ally is physical (as in symmetry of physical dimensions), but must also be electrical response symmetry. For example, active differential circuits can be physically symmetric, but care must be taken in the balance of electrical operation of the active devices (bias, device area matching etc.) so that balanced operation is maintained.

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297 11.3. Indefinite Mixed-Mode S-Parameters Indefinite s-parameter matrices are typically used in active devices. In terms of traditional single-ended circuits, the indefinite s-parameter matrix relates the s-parameters of a transistor with one terminal grounded to the s-parameters of the same device with a different port grounded. For example, the s-parameters of a bipolar transistor in common base (CB) connection can be related to the s-parameters of the same device in common emitter (CE) connection. The indefinite s-parameters of a device are defined as those found when all terminals of the device are used as measurement ports. The name ''indefi nite'' is derived since no terminal is grounded as a definite reference [23]. Thus, a three terminal device, like a typical bipolar transistor, has a three-port indefinite s-parameter matrix. 2 1 3 5 11 5 12 5 13 5 21 5 22 5 2 3 5 31 5 32 5 33 s 11 s 12 s 3 5 21 5 22 s 3 s l S22 s23 S 1 5 32 S33 common-emitter (CE) common-collector (CC) common-ba s e ( CB) Figure 11 -3. Use of traditional indefinites-parameter matrix of a bipolar transistor.

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298 The indefinite s-parameter matrix is useful since the s-parameters of all possible referenced configurations of a device can be derived from it by observation. For example, let port one of the indefinite matrix corresponds to the base of a bipolar transistor, port two to the collector, and port three to the emitter. Then the s-parameters of the device in com mon-collector (CC) configuration can be immediately found by striking the third row and third column of the indefinite s-parameter matrix. The remaining parameters form the two-port s-parameters of the device in CC configuration. Direct measurement of the indefinite s-parameter matrix of a device requires a three-port VNA. (This is a possible application of the PMVNA.) However, the true utility of the indefinites-parameter matrix is that it can be readily found from the s-parameters of one of the definite configurations. The indefinite s-parameter matrix has the property that each row and column sum to one [23]. As a result, the indefinite s-parameter matrix can be found from any configuration. For example, the CE s-parameters of a device can be measured, and the indefinite s-parameter matrix calculated. The s-parameter of any other configuration can now be found. (This approach can have practical difficulties at RF. The parasitic effects of the grounding and biasing elements can make each of the configura tions differ. Care must be taken when applying the indefinites-parameter matrix method for RF operation.) The concepts of the indefinites-parameter matrix can be extended to differential circuits and mixed-mode s-parameters. The fundamental differential device must be defined, however. The basic unit in traditional indefinite s-parameter matrices is the sin gle three-terminal transistor. By analogy, a pair of matched transistors will be defined as the basic differential device. This basic device is illustrated in Figure 11-4, showing port

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299 numbering conventions to be used in the fallowing developments. From this basic differ ential device a large variety of practical differential circuits can be constructed. These differential circuits can have any of the mixed-mode ports grounded. For example grounding mixed-mode port three will result in a simple differential amplifier ( essentiall y, two single-ended CE amplifiers in parallel). By connecting the terminals of mixed-mode port three together and grounding them through a single re s istor, a more familiar diff eren tial amplifier is con s tructed By knowledge of the indefinites-parameter matrix of the basic differential pair the s-parameter s of these specific amplifier configurations can be easily found Now that the basic differential active device has been defined, it is important to find the relationships that govern its indefinite s-parameter matrix. With these estab lished the indefinite s -parameter matrix can be readily found from a single measurement of the basic differential pair. The relationships that are required can be derived from conMixed-Mode Port 1 Mixed-Mode Port 2 r -, I I I Q1 Q2 I I I I I L .J Mixed-Mode Port 3 Figure 11-4. Schem a tic of basic differential gain block for indefinite s-parameter defini tion.

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300 sideration of the direct measurement of the indefinites-parameter matrix. The schematic for such a measurement is shown in Figure 11-5. The currents into the ternlinals of the ports can be expressed as a sums of forward and reverse current waves .+ .+ l 11 l]l-lll l 12 ll2-zl2 .+ .+ (11-10) z2 l l2 l l21 l22 = 1 22 l22 .+ .+ l31 l31 l3 J 1 32 = l32 l32 This is always true, regardless of port tertninations or stimulus. With mixed-mode ports two and three ter1ninated in their characteristic impedances, as shown in Figure 11-5, cer tain terrns of ( 11-10) vatlish .+ 0 .+ 0 l2 l l22 = .+ .+ 0 0 Z31 '32 = Summing current at the ground node, it is found that l 11 V11 Zof2 Zof2 Zof2 Zof2 (11-11) (11-12) lt2 Figure 11-5. Schematic for calculating indefinite mixed-modes-parameter relations.

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301 which, when combined with ( 11-11 ), becomes When the pair is stimulated with a pure differential-mode signal at port one, .+ .+ lll = -ll2 which cau ses ( 11 13 ) to become However, from the fundamental definitions of common-mode currents of (3-4 ), icI = ill+ i1 2 i c2 = i21 + i2 2 i c3 = i31 + i 32 equation (11-15) becomes i cI + ic2 + i c3 = O Dividing by the differential input current, i; 1 (11-17) becomes By definition of s-parameters l cl l c2 l c3 -+-+= 0 .+ .+ .+ zdI ldl ldl l S J.k = _j_ .+ lk so equation ( 11-18) can be stated as S c dl l + S c d21 + S c d31 = O (11-13) (11-14) ( 11-15 ) ( 11-16) ( 11-17) ( 11-18 ) ( 11-19) ( 11-20) Similar results can be found when driving other ports. In general, it can be said that the sums over each column of S c d equals zero

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302 n S d .. = 0, Vi Cl) (11-21) j = I In the same fashion, it can be shown that the sum over each row of Scd equals zero n L scdij = 0, VJ i = I By summing the currents entering Q 1 it can be shown But from ( 11-20), this can be written as sddll + sdd21 + sdd31 = l So, in general, the sum over each column of Sdd equals one n L sddiJ = 1, Vi j = 1 Similarly, it can be shown that the sum over each row of Sdd equals one n L sddij = 1 VJ i = l ( 11-22) ( 11-23) ( 11-24) ( 11-25) (11-26) With similar arguments, can also be shown that the sum over each row of S ec equals one n L s ccij = 1, Vj i = 1 sum over each column of Se c equals one ( 11-27 )

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303 n S .. = 1 \::Ii CCL] ( 11-28 ) j = 1 sum over each row of S de equal s zero n L sdcij = O, \::/j ( 11-29 ) i = 1 sum over each column of Sct c equals zero n Sd . = 0 \::/ i C l] ( 11-30 ) j = 1 The important relation s hips governing the indefinite mixed-mode s-