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THE THEORY, MEASUREMENT, AND APPLICATIONS OF MODE SPECIFIC SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION By DAVID E. BOCKELMAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997 Copyright 1997 By DAVID E. BOCKELMAN ACKNOWLEDGMENTS The author would like to acknowledge the significant support of Motorola Radio Products Group Applied Research, without which this work would not have been possible. Many members of the staff of Applied Research gave support, advice and assistance which has been received with gratitude. The author would especially like to thank Charles Backof, VicePresident and Director of Research, Radio Products Group, who urged the pursuit of this work, and Dr. WeiYean Hwong, Principal Member of the Technical Staff, who gave his time and direction. The author is indebted to Robert Stengel, Member of the Technical Staff, who provided motivation for this work and guidance through its comple tion. Furthermore, the author would like to thank Professor William R. Eisenstadt, who demonstrated his generosity by giving essential support in technical and personal matters. The author would also like to thank the members of his advisory committee for their sup port and direction, who were critical elements in the partnership between the University, Motorola, and the student. Also appreciated by the author is the help of the staff of the University of Florida Microelectronics Lab, and the help of many others who can not be listed here. Of all who gave their support and assistance, none was as critical as the author's wife, Erika. Her unquestioning commitment was the light which has led the way to this conclusion. TABLE OF CONTENTS ACKN OW LED GM ENTS ................................................. .......................... ....... .... ii TA BLE O F CO N TEN TS................................ ...................................................... iii A B ST R A C T ............................................. .............................................................. vii CHAPTERS 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 2.1.3.1. Analog Network Parameters......................................12 2.1.3.2. 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 2.2.2.1. Scalar Power Measurements Including Baluns .................22 2.2.2.2. Scattering Parameters with Baluns ..................................24 2.3. Summary of Past Theory and Techniques ........................................26 3 FUNDAMENTAL THEORY OF MODE SPECIFIC SPARAMETERS............27 3.1. Mode Specific Scattering Parameters in Differential Circuits....................27 3.1.1. Fundamental Definitions for Differential Circuits..........................28 3.1.1.1. M odal Voltage and Currents........................... ........... 30 3.1.1.2. Coupled M ixedM ode Signals ......................................... 32 3.1.1.3. MixedMode Scattering Parameters ................................37 3.1.2. Choice of Reference Impedances for Multiple Modes...................39 3.1.3. Relationship of MixedMode and Standard SParameters ...............42 3.1.4. Interpretations of MultiMode 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 CONSTRUCTION OF THE PUREMODE VECTOR NETWORK ANALYZER. 61 4.1. Basic Operation of the PMVNA................... ..........................62 4.1.1. Fundamental Concepts.............................. .................62 4.1.2. General PMVNA TestSet Architecture........................................63 4.2. Implementation of a Practical PMVNA...................... ........................65 4.2.1. System Level Description............................ ....................66 4.2.2. TestSet Construction ........................................ ...................69 4.2.3. D detailed Operation........................................ ...................... 72 4.2.4. Control Software........................ ..... .......................77 4.3. OnW afer M easurem ents............................................. ..................... 79 5 ACCURACY OF THE PUREMODE VECTOR NETWORK ANALYZER.....82 5.1. ProbetoProbe 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 CALIBRATION OF THE PUREMODE VECTOR NETWORK ANALYZER .... 108 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 NonPure Mode Generation.......................124 6.2.5. Solution of the Calibration Problem............................................ 128 6.2.6. Coaxial Calibration Standards ........................ ... ................... 132 6.2.7. OnW afer Calibration Standards .................................................134 6.3. Phase Offset PreCalibration ....................................... 139 6.3.1. Phase O ffset Standards ............................................................. 140 6.3.1.1. First Principles ...................................... ......... ............ 14 1 6.3.1.2. Offset M odel ......................................... 142 6.3.1.3. M odified TM atrix Solution ........................................ 144 6.3.2. Phase Offset Of An Unknown DUT............................................ 150 6.3.2.1. Variable Offset M odel ..................... ...................... 150 6.3.2.2. Using Multiple Offset Standards ...................................151 6.3.2.3. Calculating the Offset of an Arbitrary DUT .................152 6.3.2.4. Diagonalized Form ..................... ......... ............. 153 6.4. Calibration Procedure ...................................... ....................... 154 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 THINFILM METALONCERAMIC 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 StepUp 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. SingleEnded 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. SingleEnded 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. PadtoPad Crosstalk.................... ........ .....................275 11 PROPERTIES OF MIXEDMODE SPARAMETERS ....................................291 11.1. Symmetry of Reciprocal Devices ..................... ...... ...................... 291 11.1.1. G general ..........................................................291 11.1.2. PortSymmetric Reciprocal Devices............................... .....293 11.2. B balanced D evices............................... ..........................................294 11.3. Indefinite MixedMode SParameters ............................................ ....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 APPENDICES A ANALOG HALFCIRCUIT TECHNIQUES.................... .......................323 B ANALOG MEASUREMENT TECHNIQUES ................... .......... ......... 327 C TRANSMISSION OF MODES FROM COUPLED TO UNCOUPLED LINES..... 330 D SIMULATED SPARAMETERS OF DIFFERENTIAL AMPLIFIER..............334 E DESCRIPTION OF HP8510 VNA SUBSYSTEMS ........................................ 339 F DETAILS OF HP8517 TESTSET MODIFICATIONS.................................348 G PMVNA CONTROL SOFTWARE ........................................................357 H MULTIPORT TMATRIX DEFINITION...................................................383 I ERROR TERMS OF PMVNA AND FOURPORT VNA................................387 J DEMONSTRATION OF COEFFICIENT MATRIX RANK ...........................390 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 THE THEORY, MEASUREMENT, AND APPLICATIONS OF MODE SPECIFIC SCATTERING PARAMETERS WITH MULTIPLE MODES OF PROPAGATION by David E. Bockelman May 1997 Chairman: William R. Eisenstadt Major Department: Electrical and Computer Engineering Modespecific scattering parameters (sparameters) are defined from fundamental concepts. Such sparameters 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 sparame ters: (1) pure differential mode sparameters with a differentialmode input and output, (2) pure commonmode sparameters with a commonmode input and output, (3) modecon version sparameters with a differentialmode input and a commonmode output, and (4) modeconversion sparameters with a commonmode input and a differentialmode out put. All of these sets of modespecific sparameters are shown to be useful in analysis of a differential circuit. A specialized system, called the puremode vector network analyzer (PMVNA), is developed for the measurement of the modespecific sparameters 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 fourport analyzer. The modespecific sparameter 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 modespecific sparameters. Several structures on a silicon integrated cir cuit (IC) are measured. The effects of differential topology on circuittocircuit coupling are quantified. Basic design methods are advanced for the design of high frequency dif ferential circuits. CHAPTER 1 INTRODUCTION In many applications, devices and circuits have been designed for only a single mode of operation. In the most general sense, a mode is a particular electromagnetic field configuration for a given device or circuit. In the case of one or two conductors, the modes are usually frequency dependent, so the existence of simultaneous modes can be avoided by proper selection of operating frequencies (or by proper physical design for a given frequency). However, with three or more conductors, there will usually exist multi ple modes even in static cases. In such situations, the simultaneous existence of two or more modes can be difficult to avoid. Differential circuits are a particular class of circuits of historic importance with three conductors. Sometimes called balanced circuits, the primary operation of differen tial circuits is to respond to the difference between two signals, such as Av =vl v2 as T T V1 ii3 "3 + + + Device + Port 1 Avi Under Test Av2 Port 2 (DUT) + W+ v2 2 i4 V4 Figure 11. Schematic of twoport differential circuit. shown in Figure 11. 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 differentialmode and the com monmode. 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 doublebalanced 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 circuittocircuit isolation. This characteristic has been exploited for many years, most notably in telephone systems in the form of twistedpair wire transmission lines [5]. The higher isolation of differential circuits (with respect to singleended 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 circuittocircuit isolation is critical. As a result, differential circuits are being applied where only singleended circuits have tradi tionally been used. Second, the differential circuit has increased dynamic range when compared to a ground referenced, or singleended, 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 singleended 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 selfconsistent, 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 (sparameters) have been developed for characterization and analysis at these frequencies, but have been applied primarily to singleended circuits. A modification of existing sparameter tech niques is needed for accurate measurement, analysis and design of differential circuits at microwave frequencies. This work extends the definitions of sparameters to modespe cific representations, where the sparameters 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 commonmode voltages and currents are shown for the first time to be nonorthogonal, 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 sparameters 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 multimode sparameters is constructed, and the inherent accuracy advantages of the system are estab lished. Fundamental work in multiport network analyzer calibration proceeds beyond any previously published work, and a verification procedure establishes the accuracy of the calibration. Measurements with the multimode network analyzer includes the first of integrated differential circuits. Extensions of sparameter design techniques to multi mode circuits are presented that will formalize the design and analysis of RF differential circuits. This dissertation is organized in the following manner. In Chapter 2, techniques for analysis and measurement of differential circuits, prior to this work, are discussed. Chapter 3 presents original work of extending scattering parameter theory to differential circuits. A new measurement system for the measurement of modespecific sparameters 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 modespe cific sparameter concepts. Chapter 8 applies the new modespecific concepts to power 5 splitters and combiners. Several thinfilm metal differential structures, fabricated on alu minum oxide, are studied in Chapter 9. Circuittocircuit 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 sparameters and provides basic analysis and design tools for use with RF differential circuits. Chapter 12 concludes this disserta tion with a summary, some discussions, and remarks on future research. CHAPTER 2 PRIOR THEORIES AND TECHNIQUES This chapter serves as a summary of past theoretical and experimental techniques that are applied to differential circuits. The focus of the chapter is RF and microwave dif ferential circuits. However, lower frequency work has had a profound effect on the sub ject, so the examination will include relevant analog techniques. In the area of theoretical analysis, the subjects presented include multimode 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 commonmode voltages and currents are shown to be nonorthogonal, 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 21. Electric field distributions in planar coupled transmission lines. a) Oddmode electric field; b) Evenmode 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 21. 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 21(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 7tmodes, respectively, and the symmetry in the field distribution was lost. With the loss of the even and oddmodes, the direct analogy to differential and com monmodes 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 multimode network analysis is built. Scattering parameters are a relative measure of a network's response, so any modespe cific sparameters 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 nearaudio 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 commonmode voltages. Referring to Figure 11, the differentialmode voltage at port one is defined as Vdl 1 2 (21) and the commonmode voltage at port one is defined as Vcl 2 (22) The differential current into port one is 1 i2 'dl 2 (23) and the commonmode current into port one is i1 + i2 icl = 2 (24) with similar definitions at port two. These definitions have lead to voltage gain concepts Vd2 Vc2 A d2= Ac = (25) 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 monmode halfcircuits. 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 halfcircuit 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 halfcircuit 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 differentialmode and a commonmode output signal are produced, then some conversion from differential to commonmode 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 commonmode rejection ration (CMRR). A critical parameter of differential amplifier design, CMRR quantifies the ability of an amplifier to amplify differential signals and reject commonmode 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 timeinvariant (LTI) network representation is a basic and useful circuit analysis technique which is widely applied to twoport and threeterminal circuits of both analog and RF applications [22]. Network representations are distinctly suitable for I S11 + Vl I 2 + V2 12 lk Yk I + Linear Time Invariant nPort Network Ik+1 .. 0 k+1 + k+J Vk+ l N (k+l)' Ik+2 ^k+2 + Ik2 +2 c (k+2)' In n + n  n Figure 22. Notation of an nport linear timeinvariant 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 cuits. A circuit, or network, with n pairs of terminals which are used as input/output con nections is known as an nport network. The notation conventions for an nport network are shown in Figure 22. 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 (sparameters), are based on functions of voltage and cur rent. 2.1.3.1. Analog Network Parameters Network representations can be applied to differential circuits in at least two ways. One possible application of network theory is to interpret each input and output terminal of the differential circuit as a port with the return path grounded. This approach is quite common, and will be referred to as the standard approach to network representation. With this approach, all of the inputs and outputs of the differential circuit are ground referenced (singleended). 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 fourport network, shown in Figure 23. Here the port voltages are related to the differ ential and commonmode voltages [12, 19, 20] by Vdl Vd2 v1 = 2+cl V3= 2+Vc2 (26) Vdl Vd2 2 =2 +Vcl 2 =2 + c2 The port currents can similarly be related to differential and commonmode currents i1 = idl +icl 3 = d2 + ic2 (27) 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 commonmode 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 23. 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 nonintuitive. 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 (21) to (24), 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 lownoise systems, as discussed in Chapter 1. For example the zparameters 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 TwoPort cl Network c2 + + Vc 1 c2' Figure 24. Modal notation of an twoport differential network. This network description can be interpreted directly in terms of differential and commonmode responses. The network diagram can be modified to reflect the explicit modes, as shown in Figure 24. This approach will be called the modal network represen tation. Note that a twoport differential circuit is represented again by a fourport net work; in general, an nport differential circuit will have a 2nport network. The separation of the differential and commonmode ports in the network representation is a useful con ceptual tool. The modal network representation presented thus far is useful in the analysis of analog differential circuits. However, the application of this technique to RF and microwave circuits is of limited use as will be discussed in the next section. 2.1.3.2. RF Network Parameters Powerbased 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 sparameters is ease in measurement. In distinction to voltagecur rent derived parameters, sparameters are measured with ports terminated in the character istic impedance. This has meaningful practical implications, since shortcircuits and opencircuits 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 25, satisfy the set of differential equa tions 2 d 2 2 SV(x) V(x) = 0 dx 2 (29) d 2 (x) y 1(x) = 0 dx where y is the propagation constant. The general solution of (29) is V(x)= AeYx+ Be^x S A yx B yx (210) 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 (211) Given the phasor notation V(x)=AeA' 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 as v(x) = i(x) = I(x) o (212) V (x) V (x) a(x)= V( b(x) =  With these definitions, (210) becomes v(x) = a(x) + b(x) (213) 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 sparameters. + I I V I x=l x=0 Figure 25. Terminated transmission line. When applied to an nport network, such as in Figure 22, the a and b waves result in a sparameter description Ibt Si s12 ... s,, a, b2 S21 S22 ... 2,n a2 (214) bn sil sn2 ... Snn an or simply b = Sa where the bar over a lowercase variable represents a column vector [23]. The definition of sparameters can be generalized to include complex characteris tic impedances. This generalization also removes the dependency of the sparameter on transmission lines. The definition is based on a generalized power wave at the nth port [23 25] a = [v +i Z ] (215) b =  [V in Z *] n 2 e(Z) n n n The sparameter matrix equation (214) remains the same. Scattering parameters have not been widely applied to the analysis or measure ment of differential circuits. Sparameters 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 sparameters is not difficult. In fact, with the standard network representation discussed earlier, a nport differential circuit can be described with a 2nby2n sparameter 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 commonmode responses. For a illustration of the difficulties of interpreting the standard fourport sparameters of an RF differential amplifier, see Appendix D. The above limitations could be removed by extending sparameter theory to a modal network representation. This extension has not been completed prior to this work, and this dissertation later presents the extension. A straightforward extension of sparameter theory to a modal network represen tation would apply the traditional definitions of modal voltages and currents in (21) through (24) to the generalized power wave definitions of (215). However, the voltage and current definitions of (21) through (24) are not an acceptable basis for a power wave network representation. Straightforward 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 uppercase variable) but have no specific phase relation. The power delivered by the two sources is P = Re(VI1*) P2 = Re(V212 ) (216) and the total power in both sources is P, = P +P2 (217) By definitions (21) through (24), the differential and commonmode voltage and current can be expressed as Vd V V2 d 2 (11 12) (218) Vc = (VI +V2) Ic = (11 +12) The power in each mode is then d = Re(Vd) Pc = Re(Vcl ) (219) 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 (219) 1 T* ( *] P = 2[Re(V1/* )+Re(V2 2*)Re(VI 2*)Re(V2 *)] S[(220) 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/*)] (221) Expanding the sum of the source powers in (217) PT = P +P2 = Re(V *) + Re(V212*) Pd + P (222) which clearly shows that the voltage and current definitions of (21) through (24) are not directly usable in power calculations. The voltage and current definitions of (21) through (24) can be used for power and powerwave calculations if care is taken to account for the nonorthogonal 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 sparameter network representations for differential circuits prior to this work. The attempts at applying sparameters to RF differential circuits have relied upon intuitive notions of differential sparameters [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 selfconsistent 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 nearaudio 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 singleended 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, centertapped 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 nonideal performance. Differentialmode RF measurements can be made with the use of 1800 power split ters/combiners, and commonmode RF measurements can be made with 0 power split ters/combiners. Like the analog measurements, these RF/microwave measurements assume singlemode inputs and output, and are called single mode measurements. 2.2.2.1. Scalar Power Measurements Including Baluns One widely used type of RF measurement of differential circuits is a scalar power measurement. This measurement provides the magnitude of the power gain. The mea surement may take the form of a constant amplitude input signal swept across frequency, resulting in a gain versus frequency characteristic. Alternatively, the input power level can be swept at a fixed frequency, resulting in a output power versus input power charac teristic. Regardless of the specific measurement, scalar power measurements have the same basic instrumentation. The signal source is an RF signal generator, the measurement instrument is a power meter or a spectrum analyzer, and RF baluns must be used. A typi cal measurement system is shown in Figure 26. 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 nonideal performance of the baluns. The nonideal 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 26. 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 monmode input. The differential and commonmode responses cannot be distinguished by the instruments, and the overall measurement accuracy is reduced. These effects are examined in detail in Chapter 8. 2.2.2.2. Scattering Parameters with Baluns A less prevalent, but important, technique for RF/microwave differential circuits is single mode (differential) sparameter measurement [26]. This approach, as implied by its name, attempts to measure sparameters 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 sparameters. The measurement system includes a standard twoport vector network analyzer (VNA) which automatically measures the sparameters of a twoport device and a pair of 1800 3dB power splitters/combiners. This approach has also been applied to onwafer measurements of differential circuits [26]. The schematic of the sys tem is shown in Figure 27. Figure 27. SParameter 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 sparameters. However, the sparameter approach represents an important extension of measurement techniques. In contrast to scalar measurements, sparameters 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 commonmode scattering parameters. Calibration of the measurement system, a necessity for all accurate VNA measurements, is also undefined. Although lim ited in accuracy due to the cited problems, a calibration for this system could be derived from the theory presented later in this work. 26 2.3. Summary of Past Theory and Techniques Clearly, an opportunity exists to extend the accuracy of analysis, design, and mea surement of differential circuits into the RF and microwave frequencies. By combining the core principles of differential circuits traditionally belonging to the analog domain with established RF techniques like scattering parameters, a strong contribution to both fields is achieved. In the next chapter, the concepts of multimode analog differential cir cuits are extended into a rigorous theory for the analysis, measurement and design of RF differential circuits. CHAPTER 3 FUNDAMENTAL THEORY OF MODE SPECIFIC SPARAMETERS 3.1. Mode Specific Scattering Parameters in Differential Circuits A severe limitation in differentialmode/commonmode circuit characterization is a lack of applicable power wave and sparameter theory in terms of these two modes. There is no previously reported way to describe sparameters based on mixed differential mode/commonmode 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 sparameters to describe differentialmode 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 ABCDparameter descriptions of TEM circuits. One notable exception to the Z/Y/ABCDparameter 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 multimode power waves and sparameters. Portions of this work have been published in summary form [30]. The details of the development of multimode sparameters, 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 twoport RF/ microwave dif ferential system is shown in Figure 31. Essential features of the microwave differential circuit in Figure 31 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 coupledpair (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, MixedMode Port I x=X1 MixedMode Port 2 x=x2 I V2 4 Figure 31. Schematic of RF differential twoport 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 quasiTEM, fashion with a welldefined characteristic impedance and propaga tion constant. Coupled line pairs, as in Figure 31, allow propagating differential signals (the quantities of interest) to exist. The differential circuit discussion in this chapter will be limited to the twoport case, but the generalized theory for nport circuits can be readily derived from this work. Most practical implementations of Figure 31 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 commonmode propa gation. Conceptually, the commonmode 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 commonmode and differentialmode 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, differentialmode, and commonmode) on a coupled transmission line will be referred to in this work as mixedmode propagation, from which mixedmode sparameters 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 selfconsistent fash ion. A differential signal propagates between the lines of the coupledpair (as opposed to propagating between one line and ground), and a commonmode 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 commonmode and differ entialmode 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, differentialmode, and commonmode) on a coupled transmission line will be referred to as mixedmode propagation, from which mixedmode sparameters will be defined. 3.1.1.1. Modal Voltage and Currents At this point, it is important to define the differential and commonmode voltages and currents to develop a selfconsistent set of mixedmode sparameters. Referring to Figure 31, define the differentialmode voltage at a point, x, to be the difference of between voltages on node one and node two Vd(x)E V2 (31) 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 differentialmode current is defined as onehalf the difference between currents entering nodes one and two 1 id(x) 2(l i2) (32) These definitions differ from previously published definitions by Zysman and Johnson [12] due to change in references. The commonmode voltage in a differential circuit is typically the average voltage at a port. Hence, commonmode voltage is one half the sum of the voltages on nodes one and two 1 vc(x)2 (vI+ 2) (33) The commonmode current at a port is simply the total current flowing into the port. Therefore, define the commonmode current as the sum of the currents entering nodes one and two ic(x) = i+2 (34) Note that the differential current includes the return current, and the return current for the commonmode signal flows through the ground plane. For this reason, the differential mode current is halved where the commonmode current is not. This definition of com monmode current differs from the traditionally accepted definition [4, 12, 19 21]. Definitions in (31) to (34) are selfconsistent 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*) (35) and the total power in both sources is PT = PI +P2 (36) The power in each mode is Pd = Re(vdid*) P = Re(vcic*) (37) By definitions (31) to (34) 1 Pd = [Re(vlil*)+Re(v2i2*)Re(vii2*)Re(v2il*)] (38) Pc = [Re(vlil*)+Re(v2i2*)+Re(v i2*)+Re(v2il*)] and the sum of mode power is 1 Pd+Pc = [2Re(vlil*)+2Re(v2i2*)] = Re(vlil*)+ Re(v2i2*) (39) Expanding the sum of the source powers in (36) PT = PI+P2 = Re(vii*)+Re(v2i2*) (310) Therefore the sum of the modal power is equal to the total power Pd+ P =P + P2 = PT (311) and energy is conserved by the definitions of common and differentialmode voltages and currents. 3.1.1.2. Coupled MixedMode Signals To begin the presentation of mixedmode sparameters, 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 Termination + i4 '3 14 34 .... x=L Line B Port 2 Port 2 Figure 32. Schematic of terminated asymmetric coupledpair transmission line. desired differential signal between the lines of the coupledpair, as well as the common signal referenced to ground. Figure 32 is a diagram of such a coupledpair transmission line, with all pertinent voltages and currents denoted. Also shown in Figure 32 is a repre sentation of a termination for the coupledpair 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 32, the behavior of the coupledline pair can be described by [8] dv, d = (ZI + Zm'2) dv2 x =(z2i2 +Zmi) (312) di1 dT = (YlV +mV2) di2 di = (Y2V2 + YV) where zl and z2 are selfimpedances 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 (312) as published by Tripathi [8] is given as V1 = Ale*c+A2eYcx+A3eY X+A4ey7x v2= AIRcecx+ A2RceY c+A3Re7~x+ A4RreT'Y A cx A2 cx A A4 7x (313) I = e Ze + 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 nmodes, respectively, and A2, and A4 represent the phasor coefficients for the reverse (negative x) propagating c and imodes, respectively. The characteristic imped ance of the cmodes are represented by Zli and Zc2 for lines A and B, respectively, and the characteristic impedance of the emodes are represented by Zli and Z2 for lines A and B, respectively. Additionally, Rc = v2/v1 for ~yc, Rn = V2/V1 for yy, and 2 ZI + Y2Z2 2 + YmZ (314) 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 mixedmode sparameter 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 sparameters. Furthermore, this assumption is not overly limiting, since reference lines may be made arbitrarily short. For symmetrical lines, in (313) Rc = 1 and R, = 1, and the c and the imodes 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 evenmode and oddmode, respectively. With these substitutions, the mode characteristic impedances and propagation constants become Zcl = Zc2 = Ze Zl = Z2 = Zo (315) Expressing (313) in the symmetric case Expressing (313) in the symmetric case v, = AleYe+A2eYeX +A3eox+A4eYoX v2 = Ale ex+A2eex A3eY A4e7ox A1 ex A2 Yex A3 Yo A4 x (316) I = e Ze + e Z e Ze Ze o o A ex A2 ex A3 A4 Yox i2= eyex 2 eyX 3 eYo + 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 commonmode values (31) through (34) in terms of the line voltages and currents (316) vd(x) = 2(A3e7 ox +A4" A3 _oX A4 oX 0 0 id(x) = Ze Z eY (317) 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 evenmode 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 (318) v(x)=Ai' v(x) =A2e e(x)e tex Ze e e  Then (316) becomes v v(x) + ve(X) + v(x)+ V(x) 1 e(X) + o(X v2 = e(x)+e(x) vo(x) vo(x) (319) ( = x) ie(X) + (x) i(x) +e + 2 = (x) e() o(x) + io(x) and (317) becomes Vd(X) = 2(vo() + vo(x)) + vo(x) Vo(x) id(x) = io(X) i oW () Z Z (320) v(x) = ,(x) + v() ) + (x) e(x) ic(x) = 2(i(x) ie()) = 2 Ze 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 oddmode (ground referenced) characteristic imped ances v'(x) 2v+(x) Zd i ()/Z 2 (321) v+(x) v+(x) Z Z (322) 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]. 3.1.1.3. MixedMode 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 nth port [23, 24] a [v +i Z ] n 2 () n n n (323) 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= =X1 bdI d = d( id(X)Z(324) bdl bd(Xl) = [Vd(X) x=xl Similarly, define the commonmode normalized waves, at port one, as acl ac(xl) (= ) + ic(x)Zc] 2 Ike(Z) c (325) 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 lowloss transmission lines on the coupledpair of Figure 31, 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 dl x= x 1 (326) bdl = [Vdl(x) dl(x)Rdl] 1x dl 2 ic I x R] x=xl acl = [V(c(x)+ RI 1 (327) X = X1 1 = i c X[v c(x) c] With the normalized power waves defined, the development of mixedmode sparameters is straight forward. The definition of generalized sparameters [24, 23] is b = Sa (328) where the bar over the lowercase letters denote an ndimensional column vector and the bold uppercase letter an nbyn matrix. Given a coupledline twoport like Figure 31, or any arbitrary mixedmode twoport, the generalized mixedmode sparameters can be described by bdl = Sddlladl+ Sddl2ad2+ SdcIlacl+Sdcl2ac2 bd2 = Sdd21adl+ Sdd22ad2 + Sdc21acl + dc22ac2 (329) bcl = Scdlladl+scdl2ad2+scclIacl +Sccl2ac2 bc2 = cd2adl + Scd22ad2+ Scc2acl+ Scc22ac2 Each parameter has the notation momiPPi = S(outputmode)(inputmode)(outputport)(inputport) (330) to indicate the modes and ports of the signal path which the parameter represents. The dif ferential and commonmodes 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 (329) can be expressed as a partitioned matrix bdl "dl bd2 = ddSd ad2 (331) bcl LScdSc acI bc2 ac2 The following names are used: Sdd are the differential sparameters, Scc the com monmode sparameters, and Sdc and Scd the modeconversion or crossmode sparame ters. In particular, Sdc describes the conversion of commonmode waves into differential mode waves, and Scd describes the conversion of differentialmode waves into common mode waves. These four partitions are analogues to four transfer gains (Acc, Add, Acd, Adc) introduced by Middlebrook [19]. These mixedmode twoport sparameters can be shown graphically (see Figure 33) as a traditional fourport. 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 oddmode 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 MixedMode Physical Pqrt TwoPort Port 2 I I I 1 Sdd21 1 adbd2 bdl Sddl Sddl2 / Sdd22 1d2 a c c cc2 bc2 .. c  I Scc Sec2 I bcl. 1 S1Sccl 2 ac2 I I c I I Figure 33. Signal flow diagram of mixedmode twoport 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 lowloss case Zo = Re[Zo)} Ro. By choosing equal even and oddmode characteristic impedances, one is selecting a special case of coupled transmission line behavior, as described in (312). Enforcing equal even and oddmode 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 (312) clearly become uncoupled, resulting in two independent transmission line solu tions. Although very specific, this is a valid solution to (312), and all results up to this point are also valid under the special case of equal even and oddmode characteristic impedances. Therefore, we choose the reference lines of the mixedmode sparameters 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 commonmode 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 mixedmode waves can be related. One could require the differentialmode and commonmode characteristic impedances to be equal (i.e. Zd = Zc = Zo). The rela tionship between mixedmode and standard sparameters (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 mixedmode sparameters to standard s parameters. 3.1.3. Relationship of MixedMode and Standard SParameters The most straightforward means of implementing a mixedmode sparameter mea surement system is to directly apply differential and commonmode waves while measur ing the resulting differential and commonmode 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 commonmode waves. These nodal waves are readily generated and measured with stan dard VNAs, and with consideration, the differential and commonmode waves, and hence the mixedmode sparameters, can be calculated. Therefore, the relationships between the normalized mixedmode 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 found. To begin the development of the relationship between the nodal and mixedmode normalized power waves, the normalized differentialmode incident wave at mixedmode port one, adl,will be expressed in terms of the normalized singleended 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 1 ai= i [vi+iiRO] 2(330 (332) b = [viiiRO] 24o 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 differentialmode incident wave at mixedmode port one, adl, will be repeated adl = [dl(x)+idl(x)Rdl] (333) d = dl x X1 Recalling that the differential voltage and current at port one are defined through (31) and (32) as vdl(x) = v(x)v2(x) 1 (334) idl(X) = (il(x)i2(x)) and that the differential reference characteristic impedance is defined in (321), with the substitution Ze = Zo = Zo = Ro, as Rdl = 2Ro (335) then (333) can be rewritten as adl = J [d Ix) +dl(x)Rdl ] (336) x x = {o[(X) V2(x) + Ro(i1(x) i2(x))] x=xt = { o[v(x)+Roil(x)] o2(x)+Roi2(x)]x x=x By applying the definition of normalized waves at port one and two (332), then (336) becomes simply adl = (a a2) (337) This equation has a meaningful analogy with the differential voltage and current defini tions. Similarly, the differential and commonmode waves a port one are I 1 adl = (ala2) acl =J (al+a2) (338) bdl = (bb2) bcl = (b +b2) Similarly, for port two 1 1 ad2 = (a3 a4) ac2 = (a3 4) (339) bd2 = (b3b4) bc2= 2(b33+b4) Equations (338) and (339) represent important relationships from which mixed mode sparameters can be determined with a practical measurement system. By using the definition of sparameters [23] for a four port network together with the relations in (338) and (339), a transformation between mixedmode and standard sparameters can be found. The transformation can be developed by considering the rela tionships between the standard and mixedmode incident waves, a, which can be written adl 1 1 0 0 a0 ad2 1 0 1 1 a2 (340) ac 1 1 0 0 a3 c2 0 0 1 4 or, compactly a mm= Ma std a = Ma (341) where a"mm and std are the mixedmode awaves vectors, respectively, and 1 1 0 0 1 0 0 1 1342) M = (342) = 2 1 1 0 0 0 01 Similarly, for the response waves, b, it is found Smm= Mb std (343) Applying the generalized definition of sparameters from (328), it can be shown Smm = MSstd M (344) where Smm are the mixedmode sparameters, Sstd are the standard fourport sparameters. The transformation in (344) gives additional insight into the nature of mixed mode sparameters. The transformation is a similarity transformation, which indicates that a change of basis has occurred between standard and mixedmode sparameters. Con ceptually, the nodal currents and voltages correspond to the basis of standard fourport sparameters, and the modal currents and voltages of (31) to (34) correspond to the basis of mixedmode sparameters. (precisely what is meant by a basis of an sparameter repre sentation will be explored in Section 3.2). The transformation (344) also gives information into the nature of the chosen modespecific a and bwaves. 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 (345) 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 fourport sparameters are operators in an orthonormal basis, then it follows from (345) that the definitions of the differential and commonmode normalized power waves must also represent an orthonormal basis. This is yet another indication that the mode currents and voltages in (31) to (34) provide a selfconsistent framework for power calculations. Further, it indicates clearly that the two sets of sparameters 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 (344) 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 MultiMode Scattering Parameters Equations (326) and (327) form the basis of an ideal mixedmode sparameter measurement system. These equations can be implemented into a microwave simulator, and can provide a quick and simple method of illustrating the usefulness of mixedmode s parameters. The circuit in Figure 34 was implemented into HewlettPackard's Microwave Design System (MDS) [32]. The phase difference, 0, between the two sources was set to 0 for the commonmode and commontodifferentialmode forward sparameters. For the forward differentialmode and differentialtocommonmode sparameters, the phase difference was set to 1800. In each case, the nodal waves were calculated from (326) and (327), and the sparameters were calculated with the appropriate ratios. The reverse s Ang=9 Mag=l V Port 1 Port 2 Figure 34. Schematic of mixedmode simulation of symmetric coupledpair line. parameters were calculated by driving mixedmode port two of the DUT, with 500 loads at port one. The first example of mixedmode sparameters uses a DUT of a pair of coupled microstrip transmission lines, with symmetric (i.e. equal width) top conductors. This symmetric coupledpair, and the accompanying circuitry, is shown in Figure 34. Each runner width is 100im with an edgetoedge 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 oneinch section of this line was simu lated in MDS as described above, and the mixedmode sparameters at 5 GHz are I i = (346) S'cd, See 0.001 Z1410 0.972Z9.530 0 0 0.972Z9.530 0.001Z141 0 0 0 0 0.341 Z60.40 0.915Z26.40 0 0 0.915/26.40 0.341 60.4 As expected, each partitioned submatrix demonstrates the properties of a reciprocal, pas sive and (port) symmetric DUT. The differential sparameters, Sdd, show the coupled pair possesses an oddmode characteristic impedance of 502 (1000 differential imped ance), and has lowloss propagation in the differential mode. The commonmode sparameters, Sce, show the coupled pair possesses an evenmode characteristic imped ance other than 50,. Actually, the evenmode impedance of the pair is 140L (70Q com monmode impedance). Note the crossmode sparameters 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 coupledpair, and the accompanying circuitry, is shown in Figure 35. One top conductor width is 100lm, and the second is 170gm, with an edgetoedge 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 oneinch section of this line was simulated in MDS at 5 GHz, and the mixedmode sparameters are i2 Ang= Port 1 Port 2 Mag=l V Figure 35. Schematic of mixedmode simulation of asymmetric coupledpair line. [ddlc Sd (347) I cd 0.003Z1750 0.956Z1.8190 0.005Z1770 0.031Z80.70 0.956/1.819 0.003Z1750 0.031 80.70 0.005Z1770 0.005Z1770 0.031 Z80.70 0.502Z48.00 0.844Z40.20 0.031Z80.70 0.005Z1770 0.844Z40.2o 0.502Z48.00 As in the first example, each partitioned submatrix demonstrates the properties of a reciprocal, passive and (port) symmetric DUT. Also like the first example, the differen tial sparameters show the coupled pair possesses an oddmode characteristic impedance of nearly 500 (actually 49Q), and has lowloss propagation in the differential mode. The commonmode sparameters show the coupled pair has a greater degree of mismatch than the first example (the evenmode impedance is 152Q in this case). The most important difference between the two examples is seen in the cross mode sparameters. The data in (347) shows significant conversion between propagation modes, particularly in transmission parameters Sdc21 and Scd21. Note these two submatri ces are equal indicating equal conversion from differential to commonmode and from common to differentialmode. These nonzero sparameters 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 commonmode waves exist. Some of the energy of the differential wave is converted to a commonmode propa gation, and the total energy is preserved (except for losses in the metal and dielectric). . r A G I h I 21. Figure 36. Simulated magnitude in dB of Sdd21 and Scc21 versus frequency for asym metric coupledpair transmission line 1.0 Freq (GHz) 21. Figure 37. Simulated magnitude in dB of Sddll and Scc,1 versus frequency for asym metric coupledpair line. ' 1.0 p 21.0 oq Freq (GHz) 0  d o 1.0 Freq (GHz) Figure 38. Simulated magnitude in dB of Scd21 versus frequency for asymmetric cou pledpair line. o 1.0 Freq (GHz) 21.0 Figure 39. Simulated magnitude in dB of Scdl versus frequency for asymmetric cou pledpair line. This example circuit was simulated across frequency, and the magnitudes of selected mixedmode sparameters are plotted in Figures 36, 37, 38 and 39. Figure 36 shows both Sdd21 and Scc21 in dB from 1 GHz to 21 GHz. The ripple pattern across fre quency in the commonmode transmission (Scc2i) indicates an impedance mismatch at the ports for commonmode propagation. At the higher frequencies of the plot, the finite con ductivity of the conductors is evident as average loss increases. The differentialmode 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 37, 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 differentialmode, and hence Sdd21 is reduced. Here, differential energy is converted to both commonmode transmission Scd21 and commonmode reflection Scd ll Figure 38 shows the crossmode transmission Scd21 in dB, and Figure 39 shows the crossmode reflection Scdl in dB. The minima in the dif ferentialmode 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 missmatch can be shown to be approximately Loss(dB)=101og[l (ISddlI2+ Scd2112+ lScd112)] (348) 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 mixedmode sparameters 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 (31) to (34) represent only one possible definition of modes. There are infinitely many such definitions with a fourport 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 sparameter matrix as a linear operator. Traditionally, an sparameter matrix is interpreted from a physical view, where the elements of the matrix represent the gain coefficients of a certain inputtooutput path. The operator interpretation views the sparameter matrix as an oper ator that maps one ndimensional vector space into an mdimensional space [31] (with typ ical devices, m and n are equal). With such an interpretation, it will be shown that the transformation to another mode definition can be regarded simply as a transformation of coordinates. al b, b at b, 3 b2 a2 a2 DUT b4 b3 DUT a3 b2 4 b4 a4 (a) (b) Figure 310. Two views of a fourport sparameter matrix. a) The physical view. b) The linear operator view. To illustrate the operator view of sparameters, consider the fourport example in Figure 310. Define basis vectors corresponding to each physical port S01 0 0 1* 0 2 3 4 = (349) One can clearly see that these vectors are linearly independent, that is clP + 2P2 + c33 + c4P4 0 (350) 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 = (351) This means that the system of basis vectors is orthonormal. Continuing, an arbitrary set of input signals becomes a = al, ++ a2 + a33 + a44 (352) and the output signals are b = bl1P + b2P2 + b33 + b4P4 (353) With the basis definitions of (349), the coordinates of the input and output signals are a2 b=b2 a = b = (354) a3 b3 a4 b4 The traditional sparameter 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/commonmode definitions of (31) to (34) they are P1' = Pdl P2 = Pci 3 = Pd2 4 = P(355) 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' (356) 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' (357) has the coordinates in the new basis a a2 a' = (358) a3 a4 By expressing the input vector in the original basis (352) in terms of the new basis vectors via (356), 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 (360) 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/commonmode definitions, it can be shown that (360) becomes mm std a = Ma (361) As illustrated in (353), the input and output vector spaces share the same basis vectors, so the output in the new basis becomes b = Xb (362) or, for differential/commonmodes Smm= Mb std (363) The linear operator representing the DUT can be translated between bases by S' = XSX1 (364) In general, if both sets of bases are orthonormal, as defined in (351), 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 sparameters 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 eigenmodes. Eigenvalues arise from the diagonalization of a matrix, and the matrix of eigenvectors become the transformation matrix. Symbolically, A = TST (365) where A = diag(X ,,..., X,) (366) where 1i are the eigenvalues of S, and T is a matrix whose columns are composed of the eigenvectors of S [33]. In linear system analysis, eigenvalues represent the natural frequencies of a sys tem. When described in state space notation, the statefeedback matrix, A, determines these natural frequencies. The natural frequencies, or eigenvalues, are the solutions to IXIAl = 0 (367) Corresponding to each eigenvalue, ki, there is a eigenvector, e such that (XIA)ei = 0 (368) Physically, the eigenvalues are the complex frequencies at which the system will have (unforced) oscillations, and the eigenvectors are the amplitude coefficients of each of the state variables under the conditions of oscillation. In contrast, the eigenvalues and vectors of an sparameter matrix do not represent system oscillations. For an sparameter operator, the eigenvectors 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 eigenvectors of a operator, S, will be called canonical modes. The eigenvalues represent the DUT response in terms of the canonical modes. In general, an nport 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 oneport (multimode) 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 eigenvalues of a sparameter 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 eigenvalues of a matrix, S, remain unchanged by a change of basis (i. e. a similarity transformation as in (365)). The eigenvalues, therefore, 1. The definitions of mixedmode sparameters presented in Section 3.1.1 define nodall) ports one and two as mixedmode port one, and so on. However, any other combination of two ports could have also been chosen as a mixedmode port. are immutable properties of an sparameter matrix, and the canonical modes of a device are properties of the device. Eigenvectors 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 eigenvalues across all such bases indi cates the all representations of a device are leaving the essence of the device unchanged. Mixedmode sparameters are indeed an equivalent representation of a standard fourport sparameter matrix. Not every device has a canonical representation. A matrix, S, is diagonalizable if and only if S has n linearly independent eigenvectors. It can be shown [33] that S has n linearly independent eigenvectors ifS has n distinct eigenvalues (the converse is not true, however). Therefore, if all eigenvalues 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 sparameter matrix does not have n linearly independent eigenvectors, then it is possible to find n independent generalized eigenvectors. Under these conditions, the new operator matrix is not diagonalizable, but generally in Jordan form. A Jordan form matrix has some nonzero offdiagonal elements. Such a device requiring a Jordan form representation will exhibit modeconversion between some of its canonical modes. Despite this limitation, the Jordan form representation of an sparameter operator can have some utility in calculations. Not every nondiagonalizable matrix has a Jordan form representation. In such cases, other decomposition methods are available, such as LDUfactorization [33]. These decompositions cause representations that are as close as possible to a diagonal form. 60 This work can be extended to include these other representations of an sparameter opera tor. With the fundamental theory of mixedmode sparameters developed, the applica tion of these concepts to practical circuits can begin. The first step in this progression is to measure the mixedmode sparameters of an RF differential circuit. These new sparame ters require the design and construction of a specialized measurement system. The devel opment of this new system is the subject of the next chapter. CHAPTER 4 CONSTRUCTION OF THE PUREMODE VECTOR NETWORK ANALYZER As a result of the limitations of measuring RF differential circuits and devices with a singlemode system, as discussed in Chapter 2, a custom vector network analyzer (VNA) has been designed to measure mixedmode sparameters in the most direct and accurate fashion. The existence of a transformation between standard and mixedmode sparame 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 fourport VNA. A traditional VNA would measure standard sparameters by stimulating each terminal of the differen tial circuit individually, and these sparameters would then be transformed to mixedmode sparameters for analysis. Alternatively, the mixedmode sparameters of the differential circuit can be measured directly by stimulating each mode individually. A pure differen tialmode stimulus could be produced, and the differential and commonmode responses of the DUT could be measured, thus providing a direct measurement of mixedmode sparameters. A network analyzer that directly measures mixedmode sparameters will be referred to as a puremode vector network analyzer (PMVNA) due to its generation and measurement of pure single mode signals. The two approaches do not yield equally accurate mixedmode sparameters of differential devices, however. It is shown in Chapter 5 that the PMVNA has an accuracy advantage over a traditional fourport VNA while measuring a differential circuit. Mixed mode sparameters generated by transforming standard sparameters measured by a tradi tional fourport VNA exhibit higher levels of uncertainty in a differential device measure ment than those measured by a PMVNA. This accuracy advantage of a puremode 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 sparameter measurement system is to directly apply differential and commonmode waves while measuring the resulting differential and commonmode 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 (338) and (339), 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 commonmode waves, and hence the mixedmode sparameters, can be calculated. Equations (338) and (339) represent important relationships from which a PMVNA can be constructed with components of standard singleended VNAs. To under stand the utility of the above relationships, consider Figure 41, 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 commonmode or differentialmode forward sparameters, 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., Ang=O Mag=l V Figure 41. Conceptual diagram of puremode measurement system. through the generalized definitions given in (332). From these nodal waves, the differen tial and commonmode normalized waves, and, hence, the mixedmode sparameters, can be calculated. Physically, the various ratios of nodal waves, a1, bl, a2, b2, etc., are mea sured, and from theses ratios the mixedmode sparameters are found. 4.1.2. General PMVNA TestSet Architecture The physical implementation of a mixedmode sparameter measurement system can be achieved with extensions of standard VNA techniques. The differential stimulus of a coupled twoport 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 (338) and (339), can be also completed through a 1800 splitter/combiner. The commonmode stimulus of a coupled twoport 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 commonmode reflected and transmitted waves also completed through a 00 splitter/combiner. A VNA testset is the portion of the test system that generates the normalized power waves, a and b. A typical testset uses directional couplers to separate the forward and reverse waves. A testset also samples the stimulus signal, either with a directional coupler or a power splitter. The testset generally downmixes 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 testset. A testset also provides RF switches to allow automated measurement of all sparameters of the DUT with a single connection. A basic puremode testset is shown in part in Figure 42. The figure includes mechanisms by which all of the mixedmode 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 differentialmode 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 monmode RF stimulus signal. Switches SW2 and SW3, which operate in concert, pro vide the means to stimulate either mixedmode port one or two. Directional couplers DI, D2, D3, and D4 separate all forward and reverse signals at each singleended port (i.e. nodal waves). These nodal waves are combined, in accordance to (338) and (339), 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 commonmode normalized power waves (adl, ac, bdl, bct, etc.). Mixed Mode Port 2 "...... plitter/ Variable I Mixed Combiner Attenuator Mode adi al (x4) I Port 1 44  ^ I 3 act H2 bdl  a bt o3 H3 Figure 42. RF Section of basic testset of PMVNA. From the appropriate ratios of these power waves, the mixedmode sparameters can be calculated. 4.2. Implementation of a Practical PMVNA Rather than build an entire PMVNA from elementary components (such as direc tional couplers and mixers), a more practical approach has been followed by modifying a standard VNA. As will be discussed below, a PMVNA can be constructed in a straight forward manner by adapting a modular HewlettPackard 8510 VNA system. First, a sys temlevel description of the PMVNA, as implemented for this work, will be given. Fol lowing this, a detailed description of the PMVNA testset 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 HewlettPackard 8510C VNA sys tem. The complete block diagram of the implemented system is shown in Figure 43. The basic idea behind the implemented PMVNA is to use the subsystems of a standard 8510 (each contained as a single piece of test equipment) in a nonstandard configuration with little or no modification to the individual subsystems. The subsystems (85101, 85102, 8517, 85651, etc.) are shown in Figure 43. For a description of these subsystems and the standard 8510 configuration, see Appendix E. Basically, the PMVNA is an 8510 VNA with two testsets, where both testsets are used simultaneously. The implementation of a PMVNA with an HP8510 VNA requires the addition of a second 8517 testset 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 testsets. The option is actually an additional circuit board for switching IF signals which is installed in one of the two testsets. 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 testset to be accomplished simply by changing the address of the active testset (contained in a register in the 85101) to the address of the desired testset. The address of the active testset can be set through standard general purpose interface bus GPEB 85101 I.E Bus 85102 3488A opt 014 u TS A Enable d 0 GP IBSystem TS B Enable I.F B u s I n u RF Input RF Input 8517A 8517A TestSet A I i t h TestSet B (1) (2) (2) (1) (4) 85651 S MixedMode MixedMode Port l Port 2 Figure 43. PMVNA system block diagram. (GPIB) commands. The availability of the testset selection function to GPIB commands enables highlevel control of the subsystems in the PMVNA. The PMVNA system also requires some minor modifications to the control hard ware of the testsets. As developed by HP, Option 001 allows the selection of one active testset, and the deactivation of all other testsets. This deactivation includes the moving of the RF port selection switch (internal to the testset) to a terminated position, so that no RF signal is present at the ports of the deselected testsets. Also upon deactivation, the variable attenuators in a testset (used to control the incident power on a DUT) are reset to 0 dB. The suppressing of the RF signal from the inactive testsets and the change of attenuation setting are unwanted side effects. The modification to both testsets is needed to allow RF to continue at the ports of inactive testsets and to keep the attenuator settings unchanged. The modification requires a minor change to the testset 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 testset. When the signal, called testset 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 testset modifications, see Appendix F. These hardware changes are implemented to block unwanted system commands from the 8510 operating system as the active testset is changed. An alternative to these hardware modifications is to change the operating system. Such a change, to allow switching between testsets 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 HewlettPackard. 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 testsets simultaneously. The use of a 1800/00 splitter allows for the generation of both differential and commonmode 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 testset 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. TestSet Construction One of the most useful aspects of a PMVNA implemented as shown in Figure 43 is the straightforward manner by which the differential and commonmode normalized power waves can be derived from the nodal power waves. Referring first to the basic PMVNA testset of Figure 42, 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 (nonideal) components, and so is subject to errors (see Chapter 8). A more practical and accurate method of constructing the differential and commonmode responses is through digital calculation of (338) and (339). This technique exploits the architecture of the standard 8510, which downmixes and digitizes the normalized power waves. Once the nodal waves are digitized, the differential and commonmode 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 twoport testsets. The connection of this simplified PMVNA testset is shown in Figure 44, which includes two standard (singleended) 8517A sparameter testsets. These testsets 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 testsets are needed. T. IT Dit ctors Hr 4 11 I ll To IF Detectors (HP85101) Figure 44. RF section of simplified PMVNA testset. A significant advantage of this testset configuration is its symmetry. If the two testsets 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 differentialmode, for example, both testsets have the same switch configuration. The two RF paths that comprise the dif ferential signal (one through testset A, the other through testset 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 modeconversion 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 testset configuration has one significant disadvantage, namely, the use of two independent voltage controlled ocsillators (VCOs). Referring to Figure 44, one can see that each testset contains a VCO. During measurements, this VCO is phase locked to the RF input signal of the testset (for details, see Appendix E). This VCO gen erates a signal the drives all four downmixers in the testset. As all mixers in a single testset are driven by the same VCO, the phase relationship between the downmixed a and b signals remains the same as it was at RF However, as the PMVNA switches between the two testsets, the phase relationship between the VCOs of the two testsets is unknown. As a result, the straightforward application of the measured power wave data will result in significant errors. This disadvantage can be removed, however, through a precalibration 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 nonideal mode generation, as will be shown in Chapter 8. This nonideal 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 mixedmode sparameters 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 highlevel 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 mixedmode sparameters. 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 subsystems that is even more fundamental. These lowlevel events, such as the lock ing of the main phaselock 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) mixedmode sparameters of a DUT. Optionally, this operation can produce standard fourport raw sparameters directly (in contrast to transformation of mixedmode sparameters). 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 commonmode responses of a DUT to both a differential and a commonmode stimulus. Referring to the flow diagram in Figure 45, the PMVNA first measures the DUT with a differential stimu lus, which is accomplished by setting SW1 to position one (see Figure 43). 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 downmixers: 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 differentialforward 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 downmixers. This con figuration of the PMVNA is called the differentialreverse mode (DR). Figure 45. Flow chart of PMVNA measurement. Next, the PMVNA measures the DUT with a commonmode stimulus, which is accomplished by setting SW1 to position two (Figure 43). The forward measurements are repeated in the same way as with the differential stimulus described above. This con figuration is called the commonforward mode (CF). Similarly, the reverse measurements are repeated with the commonmode stimulus; this configuration of the PMVNA is called the commonreverse mode (CR). The calculation of the mixedmode 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 45, the arrangement of the a data is illustrated, where DF DR CF CR a al a 1 ai DF DR CF CR DF a2 DR a2 CF a2 CR a2 a = a = a = a = (41) DF DR CF CR a3 a3 a3 a3 DF DR CF CR 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 (42) 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 Bmatrices, generated phasecorrected versions, A, and Bc, respectively. The mixedmode normalized power waves are now calculated in matrix form Amm= MAstd Bmm= MBtd (43) where the matrix M is the similarity operator described in (342). The designation of ele ments of the mixedmode power wave matrix, Amm is also composed of column vectors, one for each PMVNA configuration Amm = mmDF mmDR mmCF mmCR] (44) and where DF adl DF mmDF ad2 a =F (45) act DF ac2 with the subscript d referring to the differentialmode quantity, c to the commonmode, and the subscript numbers referring to the mixedmode port numbers (in contrast to the singleended node numbers). The remaining vectors of (44) are defined in the same fash ion. Likewise, the mixedmode Bmatrix, Bmm, can be defined. The calculation of raw mixedmode sparameters is examined in detail next. After calculation of the mixedmode normalized power matrices Amm and Bmm, the raw mixed mode sparameter 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 fourport sparameters. The standard sparameters can be calculated through the similarity transformation of (344), but they can also be calculated directly from raw A and Bmatrices. With this method, the standard A and Bmatrices of (43) are used to directly calculate the standard sparameters std Std std S = B td(AStd) (47) The accuracy of these standard sparameters 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 fourport 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 nondifferential DUT have higher residual errors than measurements of the same device from a standard fourport 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 sourcecode 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 46. 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 46. Toplevel flow chart of PMVNA control software. Next, a phase offset precalibration 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 timeinvariant 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 mixedmode sparam 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 sparameters 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 recalculate any portion calibration and error correction algo rithms, which is used mainly for debugging purposes. 4.3. OnWafer Measurements The PMVNA can make measurements of devices with coaxial connectors, or devices that are meant to be probed at the wafer level. Waferlevel 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 mixedmode sparameters 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 coupledmode signals into S, 0 G 0 S2 G Figure 47. GGB dualRF wafer probe, top view (not to scale). uncoupled modes, which results in mixedmode sparameters 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 singleended probes. In order to maintain a smooth transition to any coupledmodes, two singleended probes are paired into a single mixedmode probe. Each mixedmode probe provides two RF measurement ports that are in reasonably close proximity, but are ideally uncoupled. Hence, a mixedmode probe footprint of GSIGS2G is adopted. The PMVNA system, as implemented for this work, is fitted with a pair of 150glm pitch dualRF probes manufac tured by GGB Industries [39]. A dualRF probe is illustrated in Figure 47, with a detail showing the probe contact configuration. Wafer probes require special calibration standards. These standards are meant to be contacted directly by the probe, so that the calibration reference planes are at the probe 81 tips. These wafer probeable standards are widely available for twoport VNAs. How ever, the unique nature of the PMVNA required custom waferprobe standards to be designed and manufactured. These standards are discussed in detail in Section 6.2.7 of Chapter 6. With the construction and operation of the PMVNA detailed, the measurement accuracy remains to be assessed. An important aspect of the PMVNA is its accuracy in the measurement of differential devices, relative to that of a more traditional VNA. This is the central issue that will be examined in the next chapter. CHAPTER 5 ACCURACY OF THE PUREMODE VECTOR NETWORK ANALYZER As indicated in Section 3.1.3, mixedmode sparameters and standard fourport sparameters are related by a linear similarity transform. This relationship suggests that a traditional fourport VNA (where only one measurement port is stimulated at a time) could be used to measure a differential DUT, and the resulting fourport sparameters could be transformed to mixedmode sparameters for easy analysis. Instead, a special ized VNA has be constructed to directly measure mixedmode sparameters. These two approaches do not yield equally accurate mixedmode sparameters of differential devices, however. The PMVNA will be shown to be more accurate than a traditional fourport VNA while measuring a differential circuit. Mixedmode sparameters generated by transforming standard sparameters measured by a traditional fourport VNA exhibit higher levels of uncertainty than those measured by a PMVNA. In particular, the uncer tainties of transformed modeconversion 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 mixedmode sparameters of differential devices as measured by a PMVNA. Since it has been earlier established that a linear transform exists between mixedmode sparameters and standard sparameters, a traditional fourport vector network analyzer (FPVNA) can theoretically be used to measure a differential device. Here, a traditional fourport network analyzer refers to a network analyzer that stimulates each port individu ally while unstimulated 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 sparameters into mixedmode sparameters. 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: probetoprobe 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 modeconversion parameters is significantly lower for the PMVNA than for the FPVNA. 5.1. ProbetoProbe Crosstalk For a waferprobe measurement system, the uncorrected probetoprobe 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 singleended twoport 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 commonmode signal rejection characteristic of a differential circuit. This reduced crosstalk would allow higher dynamic range measurements than FPVNA. For these reasons, the raw probetoprobe crosstalk of the PMVNA and a traditional fourport VNA are first quantified. The examination of the crosstalk levels is based on electromag netic simulations of probe tips. Measured probetoprobe crosstalk is also provided as fur ther evidence of the higher dynamic range of the PMVNA. 5.1.1. Simulated Probe Crosstalk The mixedmode probe is simulated as a groundsignallgroundsignal2ground (GSIGS2G) probe, as described in Section 4.3. The crosstalk of the fourport system is represented through simulations of groundsignalground (GSG) probes. The use of the twoport singleended 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 51. The strips are situated G S, 1 m (a) G 1501m S2 m m 50gm Gd m m LOOm G (b) S 5Q 50gm Figure 51. Probe crosstalk simulation layout. a) Mixedmode probe layout, b) Singleended 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 HewlettPackard's Momen tum, which is a methodofmoments simulator [40]. Multiple simulations of both the mixedmode and singleended structures have been executed over a range of distances between the probes tips. The results of the multiple simulations are shown in Figure 52 to Figure 55. A direct comparison of the crosstalk in the differential mode of the PMVNA to that of the singleended VNA is shown in Figure 52 as a function of probe separation at 1.0 GHz. The simulations show that the singleended 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 singleended operation, and hence provides for greater dynamic range in the corresponding measurement. Also shown in Figure 52 is the commonmode crosstalk of the PMVNA. The commonmode shows nearly the same level of crosstalk as the single ended system, as expected. This indicates that the commonmode measurements will have approximately the dynamic range as traditional singleended measurements. This plot illustrates the dynamic range advantages of differential measurements over single ended measurements. Figure 53 shows a comparison of the crosstalk of the PMVNA to that of the singleended VNA 10.0 GHz. Figure 54 and Figure 55 show crosstalk as a function of frequency for singleended 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 probetoprobe 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 singleended 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 ance. o 0 U A Scc21 simulated 0 Sdd21 simulated  S21 (singleended) simulated o 45dB ___ __ _ i 100 Separation (Igm) Figure 52. Simulated probe crosstalk vs. separation distance at 1.0 GHz. 0  S .,2 1 sim u lated  Sdd2l1 simulated  SS21 (singleended) simulated 100 Separation (lm) Figure 53. Simulated probe crosstalk vs. separation distance at 10 GHz. S1.0 frequency (GHz) Figure 54. Simulated singleended probe crosstalk vs. frequency for several probe separations. F. ... .... .. . I I I i I I I I 150gm  300tm 500Am I 700gm I 10001m i 1500gm j ) 150gm ..  300gmO 5001jm r   700gm _____ 1000jm  15000m 7 1.0 freq (GHz) 10.0 Figure 55. Simulated differential probe crosstalk vs. frequency for several probe sepa rations. 5.1.2. Measured Probe Crosstalk Measured probetoprobe crosstalk for the PMVNA is shown in Figure 56 to Figure 58. This data was collected with GGB 150 impitch 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 56 shows the measured and simulated differential and common mode crosstalk as a function of probe separation at 1.0 GHz. Figure 57 shows the same at 10.0 GHz. Figure 58 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 commonmode 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 ferentialmode in the PMVNA.  Scc21 measured  Sdd21 measured  Scc21 simulated 0 Sdd21 simulated C 100 Separation (im) Figure 56. 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 57. Measured probe crosstalk vs. separation distance at 10 GHz. '0 90 __ i 150gm 300gm .  500gm II I' J "^'^1 looolm  1000gm v _V v 1500gm S1.0 freq (GHz) 10 Figure 58. Measured differential probe crosstalk vs. frequency for several probe sepa rations. 5.2. Uncertainty Calculations A generally accepted quantification of error in VNA measurements is the maxi mum uncertainties in the magnitude and phase of a set of sparameters [43]. This section seeks to quantify the error in a mixedmode measurement, and compare that to the error in a standard fourport 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 nonsystematic (nonrepeatable) 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 nonsystematic 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 error. To make the accuracy specifications independent from the DUT, the sources of error, from the three areas listed, are stated as a set of equivalent residual errors. These residual errors are based on an assumed error model. Since the actual error, produced by the three factors above, cannot be directly known, the residual error terms are expressed as maximum magnitudes. It is assumed that these residual errors can combine in a way to produce the maximum error in the corrected DUT sparameters. 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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