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
Physical Description:
viii, 412 leaves : ill. ; 29 cm.
Bockelman, David E., 1967-
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Subjects / Keywords:
Microwave circuits -- Design and construction   ( lcsh )
Radio circuits -- Design and construction   ( lcsh )
Electrical and Computer Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Electrical and Computer Engineering -- UF   ( lcsh )
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1997.
Includes bibliographical references (leaves 403-411).
Statement of Responsibility:
by David E. Bockelman.
General Note:
General Note:

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University of Florida
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Copyright 1997




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-


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 author's

wife, Erika. Her unquestioning commitment was the light which has led the way to this



ACKN OW LED GM ENTS ................................................. .......................... ....... .... ii

TA BLE O F CO N TEN TS................................ ...................................................... iii

A B ST R A C T ............................................. .............................................................. vii


1 IN T R O D U C TIO N ................................................................ .............................

2 PRIOR THEORIES AND TECHNIQUES .................... ........ ............. 6

2.1. Fundamental Theories of Analysis.............................. .....................7
2.1.1. Coupled Transm mission Line Pairs ..................... ...........................7
2.1.2. Analog Methods.............................................9
2.1.3. Linear Network Representations ............................. ............ 11 Analog Network Parameters......................................12 RF Network Parameters.................. ....................... 15
2.2. M easurement Techniques ................................... .....................20
2.2.1. Single Mode Analog Measurements....................................21
2.2.2. Single Mode RF and Microwave Measurements............................22 Scalar Power Measurements Including Baluns .................22 Scattering Parameters with Baluns ..................................24
2.3. Summary of Past Theory and Techniques ........................................26


3.1. Mode Specific Scattering Parameters in Differential Circuits....................27
3.1.1. Fundamental Definitions for Differential Circuits..........................28 M odal Voltage and Currents........................... ........... 30 Coupled M ixed-M ode Signals ......................................... 32 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


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. D detailed Operation........................................ ...................... 72
4.2.4. Control Software........................ ..... .......................77
4.3. On-W afer M easurem ents............................................. ..................... 79


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 M odel Derivation ........................ ...................... 101
5.2.3. Order of Uncertainty Calculations........................... ........... 106
5.3. Conclusions on Accuracy ................................... .................... 107


6.1. Types of VNA M easurement Errors.................................................... 108
6.2. Primary PMVNA Calibration ....................................... 110
6.2.1. Raw Perform ance............................... ......... ... 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-W afer Calibration Standards .................................................134
6.3. Phase Offset Pre-Calibration ....................................... 139
6.3.1. Phase O ffset Standards ............................................................. 140 First Principles ...................................... ......... ............ 14 1 Offset M odel ......................................... 142 M odified T-M atrix Solution ........................................ 144
6.3.2. Phase Offset Of An Unknown DUT............................................ 150 Variable Offset M odel ..................... ...................... 150 Using Multiple Offset Standards ...................................151 Calculating the Offset of an Arbitrary DUT .................152 Diagonalized Form ..................... ......... ............. 153
6.4. Calibration Procedure ...................................... ....................... 154

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. G general ..........................................................291
11.1.2. Port-Symmetric Reciprocal Devices............................... .....293
11.2. B balanced D evices............................... ..........................................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 .................................... ............... 13

12 C O N C LU SIO N S ........................................ ...........................................318


A ANALOG HALF-CIRCUIT TECHNIQUES.................... .......................323

B ANALOG MEASUREMENT TECHNIQUES ................... .......... ......... 327



E DESCRIPTION OF HP8510 VNA SUB-SYSTEMS ........................................ 339

F DETAILS OF HP8517 TEST-SET MODIFICATIONS.................................348

G PMVNA CONTROL SOFTWARE ........................................................357

H MULTI-PORT T-MATRIX DEFINITION...................................................383

I ERROR TERMS OF PMVNA AND FOUR-PORT VNA................................387


LIST OF REFERENCES ................. ...........................................................403

BIOGRAPHICAL SKETCH .............. ............................................................412

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



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-

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.


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 =vl v2 as

-T T
V1 ii3 "3
+ +
+ Device +
Port 1 Avi Under Test Av2 Port 2
+ W+
v2 2 i4 V4

Figure 1-1. Schematic of two-port differential circuit.

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 differen-

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

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


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


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.


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.

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.

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- (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 2-1(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 7t-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 network's response, so any mode-spe-

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 -1 2 (2-1)

and the common-mode voltage at port one is defined as

Vcl 2 (2-2)

The differential current into port one is

1 i2
'dl 2 (2-3)

and the common-mode current into port one is

i1 + i2
icl -= 2 (2-4)

with similar definitions at port two. These definitions have lead to voltage gain concepts

Vd2 Vc2
A d2= Ac = (2-5)
A dl =Vcl

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, 211.

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.

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

Vl I

V2 12


Yk I

Time Invariant

.. 0 k+1
k+J Vk+ l
N -(k+l)'

Ik2 +2
c (k+2)'

n -- n

Figure 2-2. Notation of an n-port linear time-invariant network.

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-


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. 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

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
v1 = -2+cl V3= -2+Vc2
Vdl Vd2
2 =--2 +Vcl 2 =-2 + c2

The port currents can similarly be related to differential and common-mode currents

i1 = idl +icl 3 = d2 + ic2
i2 = -idl+icl i4 = -id2+ic2

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

+ +
Vl Differential v3

2 Amplifier i
2 2 4
2 -p-- a. o 4

Figure 2-3. Network representation of differential amplifier.

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

dl Zdl,d Zdlcl Zdl,d2 Zdl,c2 dl
vcl Zcl,dl Zcl,cl zcl,d2 Zcl,c2
Vd2 Zd2,di Zd2,cl Zd2,d2 Zd2,c2 1d2

c2 zc2,d Zc2, cl Z,2,d2 Zc2c2 c2

dl 'dl d2
+ +
Vdl i Vd2
dl Linear I d2'
Time Invariant
l icl Differential Two-Port
cl Network c2
+ +

Vc 1 c2'

Figure 2-4. Modal notation of an two-port differential network.

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 2n-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. 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

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-


d 2 2
SV(x)- V(x) = 0
2 (2-9)
d 2
--(x)- y 1(x) = 0

where y is the propagation constant. The general solution of (2-9) is

V(x)= Ae-Yx+ Be^x
S A -yx B yx (2-10)
I(x) = e e
Zo -Zo

where A and B are complex constants and Zo 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

7= J(R+j.oL)(G+jcoC) Z0 = +j (2-11)

Given the phasor notation V(x)=Ae-A' and V(x)=BeYx, and by limiting the transmission

line to be lossless (i.e. Z0 = Re {Z0), then the important normalized quantities are defined


v(x) = i(x) = I(x) o

V (x) V (x)
a(x)= -V( b(x) = -

With these definitions, (2-10) becomes

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

x=-l x=0

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

Ibt Si s12 ... s,, a,

b2 S21 S22 ... 2,n a2 (2-14)

bn sil sn2 ... Snn an

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

transmission lines. The definition is based on a generalized power wave at the n-th port

[23 25]

a = [v +i Z ]

b = -- [V -in Z *]
n 2 e(Z) n n n

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 n-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

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 V1 and V2 and source currents 71

and 72, respectively. Assume the sources are harmonically time varying (so Vi and 7i 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

P = Re(VI1*) P2 = Re(V212 ) (2-16)

and the total power in both sources is

P, = P +P2 (2-17)

By definitions (2-1) through (2-4), the differential and common-mode voltage and current

can be expressed as

Vd V V2 d 2 (11 -12)
Vc =- (VI +V2) Ic = (11 +12)

The power in each mode is then

d = Re(Vd) Pc = Re(Vcl ) (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)

1 T* ( *]
P = 2[Re(V1/* )+Re(V2 2*)-Re(VI 2*)-Re(V2 *)]
Pc = 4[Re(VIII*)+ Re(V2 2*)+ Re(VI 2*)+ Re(V2 2*)]

and the sum of mode power is

Pd+P = [[3Re(V *) + 3Re(V2 2*) -Re(VI1 2)-Re(V2l/*)] (2-21)

Expanding the sum of the source powers in (2-17)

PT = P +P2 = Re(V *) + Re(V212*) Pd + P (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 orthogonalityy) 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

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

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 1800 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. 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

can be swept at a fixed frequency, resulting in a output power versus input power charac-


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 1800 3 dB hybrid couplers, have magnitude and phase imbal-

ance in the splitting of a signal. Ideally, a 1800 3 dB hybrid coupler would take a single

input signal and split it into two equal amplitude signals with 1800 phase difference. With

Hybrid Hybrid

Figure 2-6. RF scalar power measurement of differential circuit.

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 1800. 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. 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 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 1800 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.

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.


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.


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/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 suitable for characterization of a generic differential circuit.

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 (Avl # 0,

Port I

Port 2

I V2 4

Figure 3-1. Schematic of RF differential two-port network.

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. Avl = 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

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. 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

Vd(x)E 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

id(x) 2(l i2) (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

vc(x)2 (vI+ 2) (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

ic(x) -= i+2 (3-4)

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

P1 = Re(vi,*) P2 = Re(v2i2*) (3-5)

and the total power in both sources is

PT = PI +P2 (3-6)

The power in each mode is

Pd = Re(vdid*) P = Re(vcic*) (3-7)

By definitions (3-1) to (3-4)

Pd = [Re(vlil*)+Re(v2i2*)-Re(vii2*)-Re(v2il*)]
Pc = [Re(vlil*)+Re(v2i2*)+Re(v i2*)+Re(v2il*)]

and the sum of mode power is

Pd+Pc = [2Re(vlil*)+2Re(v2i2*)] = Re(vlil*)+ Re(v2i2*) (3-9)

Expanding the sum of the source powers in (3-6)

PT = PI+P2 = Re(vii*)+Re(v2i2*) (3-10)

Therefore the sum of the modal power is equal to the total power

Pd+ P =P + P2 = PT (3-11)

and energy is conserved by the definitions of common and differential-mode voltages and

currents. 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


+ i4
'3 14
34 .... x=L
Line B Port 2
Port 2

Figure 3-2. Schematic of terminated asymmetric coupled-pair transmission line.

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]

d = (ZI + Zm'2)

x =-(z2i2 +Zmi)
dT = -(YlV +mV2)

di = -(Y2V2 + YV)

where zl and z2 are self-impedances per unit length; yl and Y2 are admittances per unit

length; and Zm and ym are mutual impedance and admittance per unit length, respectively.

Also, a harmonic time dependence (i.e. e'"') is assumed.

The solution to the set of equations (3-12) as published by Tripathi [8] is given as

V1 = Ale-*c+A2eYcx+A3e-Y X+A4ey7x

v2= AIRce-cx+ A2RceY c+A3Re-7~x+ A4RreT'Y

A -cx A2 cx A A4 7x (3-13)
I = --e -Z--e + -e --e
cl c2 tl x2
A Rc -ycx A2Rc "cx A3Rx -e x A4R3 x
12 e Zc2 e + Zt- e e
Z Z Z 2

where A and A3 represent the phasor coefficients for the forward (positive x) propagating

c and n-modes, respectively, and A2, and A4 represent the phasor coefficients for the

reverse (negative x) propagating c and i-modes, respectively. The characteristic imped-

ance of the c-modes are represented by Zli and Zc2 for lines A and B, respectively, and the

characteristic impedance of the e-modes are represented by Zli and Z2 for lines A and B,

respectively. Additionally, Rc = v2/v1 for ~--yc, Rn = V2/V1 for y-y, and

2 ZI + Y2Z2
2 +-- YmZ (3-14)

C, (1 1 2 2+-4(zim 2 m 2m
+ (yIZl -Y2Z2)2 + 4(Zlym + Y2Zm)(Z2Ym + Y

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 R, = -1,

and the c and the i-modes become the even and odd modes, respectively, as first used by

Cohn [6]. For notational purposes, we shall use the substitutions c -- e and xi -- o for

even-mode and odd-mode, respectively. With these substitutions, the mode characteristic

impedances and propagation constants become

Zcl = Zc2 = Ze
Zl = Z2 = Zo (3-15)

Expressing (3-13) in the symmetric case
Expressing (3-13) in the symmetric case

v, = Ale-Ye+A2eYeX +A3e-ox+A4eYoX

v2 = Ale ex+A2eex- A3e-Y A4e7ox
A1 -ex A2 Yex A3 -Yo A4 x (3-16)
I = e -Ze + e -Z e
Ze Ze o o
A -ex A2 ex A3 A4 Yox
i2= -e-yex 2 eyX 3 e-Yo + 4 e
Ze Ze e Z +

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)

vd(x) = 2(A3e7 ox +A4"
A3 _oX A4 oX

0 0
id(x) = Z-e Z- eY
vc(x) = Ale-'e +A2eYex
[A -' ex A2 \ex
ic(x) = 2~ ee ye ex
Recall that A and A2 are the forward and reverse phasor coefficient for the evenmode
Recall that A I 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

A3 -.X A4 yx
v+(x) -A3e x vo(x) A4eYx i(x) e io( x) e
S YeX ex A A2 Yex (3-18)
v(x)=-Ai-' v(x) =A2e e(x)-e tex Ze e
e -

Then (3-16) becomes

v v(x) + ve(X) + v(x)+ V-(x)
1 e(X) + o(X
v2 = e(x)+e(x)- vo(x)- vo(x)
( = x)- ie(X) + (x) i(x)
+e +
2 = (x) e() o(x) + io(x)

and (3-17) becomes

Vd(X) = 2(vo() + vo(x))

+ vo(x) Vo(x)
id(x) = io(X) i- oW () Z
v(x) = ,(x) + v() )

+ (x) e(x)
ic(x) = 2(i(x) ie()) = 2

Note that, in general, Zo 4 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-


v'(x) 2v+(x)
Zd i ()/Z 2 (3-21)

v+(x) v+(x) Z
Z (3-22)
ZC (x (2v+(x) )l/Ze 2

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]. 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 ]
n 2 () n n n
bn=2 [vn -inZn*]

where a, 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

'l I ad(x1) 1 d(x) + id(X)Zd
adl=-ad(XI) = [Vd(X=

bdI d = d( id(X)Z(3-24)
bdl bd(Xl) = [Vd(X)-

Similarly, define the common-mode normalized waves, at port one, as

acl -ac(xl) (= ) + ic(x)Zc]
2 Ike(Z)

c (3-25)
bcl bc(xl) =[V(x)-ic(x)Z]
= x

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, Zd = Re(Zdj = Rd and Zc = ReZcJ = Rc. With this assumption, the normalized

wave equations at port one can be simplified

adl = [Vdl(x)+idl(x)Rdl
x= x
1 (3-26)
bdl = [Vdl(x) -dl(x)Rdl]

dl 2 ic I x R]

acl = --[V(c(x)+ RI

1 (3-27)
X = X1
1 = i c X-[v c(x)- c]

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

bdl = Sddlladl+ Sddl2ad2+ SdcIlacl+Sdcl2ac2
bd2 = Sdd21adl+ Sdd22ad2 + Sdc21acl + dc22ac2
bcl = Scdlladl+scdl2ad2+scclIacl +Sccl2ac2
bc2 = cd2adl + Scd22ad2+ Scc2acl+ Scc22ac2

Each parameter has the notation

momiPPi = S(output-mode)(input-mode)(output-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

bdl "dl
bd2 = ddSd ad2 (3-31)

bcl LScdSc acI
bc2 ac2

The following names are used: Sdd are the differential s-parameters, Scc the com-

mon-mode s-parameters, and Sdc 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

Physical Mixed-Mode Physical
Pqrt Two-Port Port 2
I 1 Sdd21 1

bdl Sddl Sddl2 / Sdd22 1d2

a c c cc2 bc2
.. c -
I Scc Sec2 I
bcl. 1 S1Sccl 2 ac2
I I c

Figure 3-3. Signal flow diagram of mixed-mode two-port network.

accurate lumped termination standards for coupled lines where Ze does not equal Zo. If

the characteristic impedances of the lines are defined to be equal (say, 500), then a further

simplification of the above expressions can be accomplished with the substitution Ze = Zo

= Zo where in the low-loss case Zo = Re[Zo)} Ro.

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 Z =

Zo results in the mutual impedances and admittances being zero (zm=0, y,=0). Under

these conditions, the describing differential equations of the transmission line system

(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 7t) 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 = Zo). 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 = Z, = Zo) best relates mixed-mode s-parameters to standard s-


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, bdl, acl, bl, etc.) and the nodal waves (a1, bl, a2, b2,

etc.) will be derived, and the necessary conditions for these relationships to exist will be


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 nodall) power

waves at port one, al, and at port two, a2. First, the normalized nodal waves of the cou-

pled lines at the interface are defined, with Zo = Ro, as

ai= -i [vi+iiRO]
b = [vi-iiRO]

where ai and bi are the normalized forward and reverse propagating nodal waves at node i,

respectively, and i E { 1,2,3,4}. Next, the definition of the normalized differential-mode

incident wave at mixed-mode port one, adl, will be repeated

adl = [dl(x)+idl(x)Rdl] (3-33)
d = dl
x X1

Recalling that the differential voltage and current at port one are defined through (3-1) and

(3-2) as

vdl(x) = v(x)-v2(x)
1 (3-34)
idl(X) = (il(x)-i2(x))

and that the differential reference characteristic impedance is defined in (3-21), with the

substitution Ze = Zo = Zo = Ro, as

Rdl = 2Ro (3-35)

then (3-33) can be re-written as

adl = J [d Ix) +dl(x)Rdl ] (3-36)
x x

= {-o[(X)- V2(x) + Ro(i1(x) -i2(x))]


= -{ o[v(x)+Roil(x)] --o2(x)+Roi2(x)]x

By applying the definition of normalized waves at port one and two (3-32), then (3-36)

becomes simply

adl = (a -a2) (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

I 1
adl = (al-a2) acl =J (al+a2)
bdl = (b-b2) bcl = (b +b2)

Similarly, for port two

1 1
ad2 = (a3 a4) ac2 = (a3 4)
bd2 = (b3b4) bc2= 2(b33+b4)

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

adl 1 -1 0 0 a0
ad2 1 0 1 1 a2 (3-40)
ac 1 1 0 0 a3
c2 0 0 1 4

or, compactly

a mm= Ma std
a = Ma


where a"mm and std are the mixed-mode a-waves vectors, respectively, and

1 -1 0 0
1 0 0 1 -1342)
M = (3-42)
= 2 1 1 0 0
0 01

Similarly, for the response waves, b, it is found

Smm= Mb std (3-43)

Applying the generalized definition of s-parameters from (3-28), it can be shown

Smm = MSstd M- (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 Tindicates 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


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-


The circuit in Figure 3-4 was implemented 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 1800. 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-

Mag=l V

Port 1 Port 2

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 500 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 100im 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

I i- = (3-46)
S'cd, See

0.001 Z-1410 0.972Z9.530 0 0
0.972Z9.530 0.001Z-141 0 0
0 0 0.341 Z-60.40 0.915Z-26.40
0 0 0.915/-26.40 0.341 -60.4

As expected, each partitioned sub-matrix demonstrates the properties of a reciprocal, pas-

sive and (port) symmetric DUT. The differential s-parameters, Sdd, show the coupled

pair possesses an odd-mode characteristic impedance of 502 (1000 differential imped-

ance), and has low-loss propagation in the differential mode. The common-mode

s-parameters, Sce, show the coupled pair possesses an even-mode characteristic imped-

ance other than 50,. Actually, the even-mode impedance of the pair is 140L (70Q 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 100lm, and

the second is 170gm, with an edge-to-edge spacing of 65gm. 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=- Port 1 Port 2
Mag=l V

Figure 3-5. Schematic of mixed-mode simulation of asymmetric coupled-pair line.

[ddlc Sd (3-47)
I cd

0.003Z-1750 0.956Z1.8190 0.005Z-1770 0.031Z80.70-
0.956/1.819 0.003Z-1750 0.031 80.70 0.005Z-1770
0.005Z-1770 0.031 Z80.70 0.502Z48.00 0.844Z-40.20
0.031Z80.70 0.005Z-1770 0.844Z-40.2o 0.502Z48.00

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 500 (actually 49Q), 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 152Q 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 Sdc21 and Scd21. 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 Scd21, 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).

. r A G I h I 21.

Figure 3-6. Simulated magnitude in dB of Sdd21 and Scc21 versus frequency for asym-
metric coupled-pair transmission line

1.0 Freq (GHz) 21.

Figure 3-7. Simulated magnitude in dB of Sddll and Scc,1 versus frequency for asym-
metric coupled-pair line.

' 1.0




Freq (GHz)

0 -


Freq (GHz)

Figure 3-8. Simulated magnitude in dB of Scd21 versus frequency for asymmetric cou-
pled-pair line.


1.0 Freq (GHz) 21.0
Figure 3-9. Simulated magnitude in dB of Scdl versus frequency for asymmetric cou-
pled-pair line.

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 Sdd21 and Scc21 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 (SddI2) 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 Sdd21 is

reduced. Here, differential energy is converted to both common-mode transmission Scd21

and common-mode reflection Scd ll Figure 3-8 shows the cross-mode transmission Scd21

in dB, and Figure 3-9 shows the cross-mode reflection Scdl in dB. The minima in the dif-

ferential-mode transmission Sdd21 correspond to a worst case point in the relative phases

of Sdd21, Scd21, and Scdl. 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 -(ISddlI2+ Scd2112+ lScd112)] (3-48)

This is consistent with the increasing ripple in Sdd21 with increasing frequency since the

mode conversion (Scd21 and Scdll) 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.

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


al b, b at
b, 3 b2 a2
a2 DUT b4 b3 DUT a3
b2 4 b4 a4
(a) (b)

Figure 3-10. Two views of a four-port s-parameter matrix.
a) The physical view. b) The linear operator view.

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

S01 0 0
1* 0 2- 3 4 -= (3-49)

One can clearly see that these vectors are linearly independent, that is

clP + 2P2 + c33 + c4P4 0 (3-50)

for all possible complex scalars {cl, c3, c4} 3 C, where C is the set of all complex

numbers. Furthermore, this set of basis vectors {p1'2',p3' P4} have a zero scalar prod-

uct, that is

0 i j
Pi ={ I = (3-51)

This means that the system of basis vectors is orthonormal. Continuing, an arbitrary set of

input signals becomes

a = al, ++ a2 + a33 + a44 (3-52)

and the output signals are

b = bl1P + b2P2 + b33 + b4P4 (3-53)

With the basis definitions of (3-49), the coordinates of the input and output signals are

a2 b=b2
a = b = (3-54)
a3 b3
a4 b4

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,

{p1 I2"' 3" 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

P1' = Pdl P2 = Pci 3 = Pd2 4 = P(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, = xl1I'+xl2P2' +xl33' +x14P4

2 = x21/1]' +x22P2' +x23P3' +244'
P3 = x3P1'+x32P2' +x33P3' +x34P4

4 = X41P1' + x42P2' + 43P3' + X44P4'

An input signal vector in the new basis

a' = al'Pl' +a2'2' +a3' + a44' (3-57)

has the coordinates in the new basis

a' = (3-58)

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

a, x11 xl2 x13 X14 a1

a2 x21 X22 X23 x4 2 2(-
a3' x31 x32 X33 34 a3

a4 X41 X42 X43 X44 a4

which can be simply expressed as

a' = Xa (3-60)

where X 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
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

Smm= Mb std (3-63)

The linear operator representing the DUT can be translated between bases by

S' = XSX-1 (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*) = I. Conversely, if a

defined transformation matrix is unitary, then both systems of basis vectors are orthonor-

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 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 = T-ST (3-65)


A = diag(X ,,..., X,) (3-66)

where 1i 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

IXI-Al = 0 (3-67)

Corresponding to each eigen-value, ki, there is a eigen-vector, e such that

(XI-A)ei = 0 (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-

clients 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 convention.

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
nodall) 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.

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 ifS 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.


This work can be extended to include these other representations of an s-parameter opera-


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.


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-

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, E, between the two

sources to 00 or 1800 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,, bl, a2, b2, etc.,

Mag=l V

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, a1, bl, a2, b2, etc., are mea-

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 1800 apart. One

possible way this can be achieved through a single signal source is with the use of a 1800

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 1800 splitter/combiner.

The common-mode stimulus of a coupled two-port requires the input waves at the refer-

ence plane to be 00 apart. This can also be achieved through a single signal source with

the use of a 00 3dB hybrid splitter/combiner, with the construction of the common-mode

reflected and transmitted waves also completed through a 00 splitter/combiner.

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, H1, splits the RF signal into two signals with nominally equal amplitudes and

1800 phase difference, thus generating the differential-mode RF stimulus signal. Note

that all switches have their unused ports terminated in 500 loads in all cases. By placing

SW1 in position two, the coupler, HI, again splits the RF signal into two signals, in this

case with nominally equal amplitudes and 00 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 DI,

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 (adl, ac, bdl, bct, etc.).

Port 2

"...... plitter/ Variable I Mixed-
-Combiner Attenuator Mode
adi al (x4) I Port 1
44 ----- ^ -------I 3
act H2

bdl -

bt o3 H3

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


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

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

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


I.E Bus
opt 014
u TS A Enable d 0
GP IB-System --TS B Enable

I.F B u s I n u
RF Input RF Input

8517A 8517A
Test-Set A I i t h Test-Set B
(1) (2) (2)

(1) (4)
85651 S

Mixed-Mode Mixed-Mode
Port l Port 2

Figure 4-3. PMVNA system block diagram.

(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

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 1800/00 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 1800/00 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 1800/00 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

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.

T.- IT Dit ctors
Hr- 4 11 I ll

To IF Detectors

Figure 4-4. RF section of simplified PMVNA test-set.

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 However, 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

Only the 1800/00 splitter/combiner and the RF switch for the source (SWI) limit

the bandwidth of the simplified PMVNA. The 8517A test sets operate from 45 MHz to 50

GHz, and with relaxed requirements on the 1800/00 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

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

1800 phase difference. Normalized waves are measured at all down-mixers: al, bl, a2, b2,

a3, b4, a4, b4 (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

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 1800

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.

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

a al a 1 ai
-DF a2 -DR a2 -CF a2 -CR a2
a = a = a = a = (4-1)
a3 a3 a3 a3
a4 a4 0 4 a4

and where the subscripts one through four correspond to the PMVNA port (node) num-

bers. Similarly, the b data are arranged into vectors DF, bDR CF and CR The data

are the placed in two matrices

Astd [DF aDR aCF aCR] std = DF DR CF CR (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,

and Bc, respectively. The mixed-mode normalized power waves are now calculated in

matrix form

Amm= MAstd Bmm= MBtd (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

Amm = mmDF -mmDR -mmCF -mmCR] (4-4)

and where

-mmDF ad2
a =F (4-5)

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, Smm, can be simply calculated

Smm = Bmm(Amm -1
S = B (A )

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

std Std std
S = B td(AStd)- (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

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.

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 1O

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 300im 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

0 G

0 S2


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 GSIGS2G is adopted. The PMVNA system, as

implemented for this work, is fitted with a pair of 150glm 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


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.


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

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 l/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

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-signall-ground-signal2-ground

(GSIGS2G) 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 100mI long metal strips

arranged in a 150gm pitch configuration, as shown in Figure 5-1. The strips are situated

S, 1- m-
(a) G 1501m

S2 m m 50gm

Gd m m LOOm

(b) S 5Q


Figure 5-1. Probe crosstalk simulation layout.
a) Mixed-mode probe layout, b) Single-ended probe layout.

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-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 1/d char-

acteristic, whereas the differential crosstalk behaves as 1/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.

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 50 phase imbalance from the ideal 1800 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 50 from I to over 5 GHz with very small magnitude imbal-



--A- Scc21 simulated
-0-- Sdd21 simulated
- S21 (single-ended)


___ __ _

100 Separation (Igm)

Figure 5-2. Simulated probe crosstalk vs. separation distance at 1.0 GHz.


----- S .,2 1 sim u lated
-- Sdd2l1 simulated
---- SS21 (single-ended)

100 Separation (lm)

Figure 5-3. Simulated probe crosstalk vs. separation distance at 10 GHz.


frequency (GHz)

Figure 5-4. Simulated single-ended probe crosstalk vs. frequency for several probe

F. ...-

.... .. -.

I I I i I I I I

| 300tm
I 700gm
I 10001m
i 1500gm



..---- ----- 300gmO
r ------- -- 700gm

_____- 1000jm
-- 15000m

7 1.0 freq (GHz) 10.0

Figure 5-5. Simulated differential probe crosstalk vs. frequency for several probe sepa-

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 im-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 gm separation,

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.

--- Scc21 measured
-- Sdd21 measured
- Scc21 simulated
--0-- Sdd21 simulated


100 Separation (im)
Figure 5-6. Measured probe crosstalk vs. separation distance at 1.0 GHz.

-- Scc21 measured
- Sdd21 measured
A--- Scc21 simulated
-0- Sdd21 simulated

Il I 1 1 Illi k 1 I I I Ill II
7 100 Separation (gIm) 10K
Figure 5-7. Measured probe crosstalk vs. separation distance at 10 GHz.



__ --i 150gm
.-- -- 500gm

II I' J- "^-'^1 looo-lm

v _V v 1500gm

S1.0 freq (GHz) 10

Figure 5-8. Measured differential probe crosstalk vs. frequency for several probe sepa-

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

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 correction process is not completely accurate. Limitations on

how accurately the standards are known and non-systematic errors (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


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 assumed 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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