095-10 doktorat lewandowski · in a signiﬁcant improvement of the vna measurement accuracy. we...

WARSAW UNIVERSITYOF TECHNOLOGY

Faculty of Electronics and InformationSystems

Ph.D. ThesisArkadiusz Lewandowski

Multi-frequency approach to vector-network-analyzerscattering-parameter measurements

SupervisorProfessor Janusz Dobrowolski, Ph.D., D.Sc.

Warsaw, 2010

Abstract

Vector network analyzer (VNA) is the basic measurement instrument used in the char-acterization of microwave and millimeter-wave electronic circuits and systems. Much efforthas been put throughout the past three decades in improving the designs of VNA instru-mentation and in establishing the principles of VNA calibration and uncertainty analysisof VNA measurements. Modern VNAs are a culmination of this long standing research,and are sophisticated, mature and reliable measurement instruments, commonly employedin the industry and laboratories.

Recently, however, several new trends in the vector network-analysis started to emerge.These new trends result from an increased interest in the application of millimeter- andsub-millimeter-wave signals (frequencies up to 1 THz), rapid development of the nanotech-nology, requiring characterization of structures with very large impedances (on the order of100 kΩ), and an increased demand for large-signal characterization of microwave circuits.These new trends result, on one hand, in new concepts in the design of the VNA instrumen-tation, such as special VNA extension units, allowing the conventional VNAs to operateup to 500 GHz, microwave scanning microscopes, or nonlinear vector network analyzers(NVNA). On the other hand, these trends lead to new challenging demands regarding themeasurement accuracy and its reliable and complete evaluation.

The multi-frequency approach introduced in this work addresses this last issue. Theprinciple of this approach is to account for the relationships between scattering parametermeasurements at different frequencies. We show that this new approach allows to reduce byseveral times the impact of errors in the description of calibration standards, resulting thusin a significant improvement of the VNA measurement accuracy. We further demonstratethat the multi-frequency approach to the description of VNA instrumentation errors yieldsbetter understanding of their physical origins, leading to their compact description basedon the stochastic modeling. We finally show that the multi-frequency representation ofthe uncertainty in VNA scattering-parameter measurements is essential when using thesemeasurements in the calibration of time-domain measurement systems, such as high-speedsampling oscilloscopes, or nonlinear vector network analyzers.

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StreszczenieWektorowy analizator obwodów (ang. Vector Network Analyzer - VNA) jest podsta-

wowym urządzeniem wykorzystywanym do charakteryzowania układów i systemów elek-tronicznych wielkiej częstotliwości (w. cz). Konstrukcja współczesnych analizatorów, jaki metody wykorzystywane w ich kalibracji oraz analizie niepewności pomiaru, są owocemwieloletnich prac badawczych oraz intensywnego rozwoju technologicznego. W konsekwen-cji nowoczesne wektorowe analizatory obwodów odznaczają się niezwykle zaawansowanymii dojrzałymi rozwiązaniami technicznymi oraz są z powodzeniem wykorzystywane w co-dziennej praktyce zarówno laboratoriów pomiarowych jak i przemysłu układów w. cz.

W wektorowej analizie obwodów pojawiły się w ostatnim czasie nowe kierunki roz-woju, wynikające z rosnącego zainteresowania wykorzystaniem sygnałów w zakresie falmilimetrowych i submilimetrowych (częstotliwości blisko 1 THz), rozszerzania się zakresuimpedancji mierzonych struktur w. cz. (impedancje rzędu 100 kΩ), związanego z intensyw-nym rozwojem nanotechnologii, oraz z zapotrzebowania na charakteryzowanie wielkosy-gnałowych własności układów w. cz. Te nowe zastosowania wektorowej analizy obwodówprowadzą, z jednej strony, do nowych rozwiązań konstrukcyjnych, jak na przykład głowicepowielająco-mieszające rozszerzające zakres pracy typowych analizatorów do częstotliwościrzędu 500 GHz, mikrofalowe mikroskopy skaningowe, czy też wielkosygnałowe wektoroweanalizatory obwodów (ang. Nonlinear Vector Network Analyzer-NVNA). Z drugiej strony,stawiają one zupełnie nowe wyzwania, jeżeli chodzi o dokładność pomiaru, oraz jej wiary-godne oszacowanie.

Przedstawione w niniejszej pracy nowatorskie wieloczęstotliwościowe podejście do po-miaru parametrów rozproszenia za pomoca wektorowego analizatora obwodów jest próbąodpowiedzi na te nowe wyzwania. Jego istotą jest uwzględnienie relacji między pomiaramiparametrów rozproszenia na różnych częstotliwościach. W pracy wykazano, że to noweujęcie pozwala kilkukrotnie zmniejszyć wpływ błędów wynikających z niedokładnego opisuwzorców kalibracyjnych, a tym samym znacząco zwiększyć dokładność pomiaru. Pokazanorównież, że wieloczęstotliwościowy opisu błędów losowych w pomiarach analizatorem wek-torowym pozwala lepiej wyjaśnić ich fizyczne przyczyny, prowadząc do prostego i spójnegoopisu tych błędów opartego na modelowaniu stochastyczym. W końcu, w pracy wykazano,że uogólniony wieloczęstotliwościowy opis niepewności pomiaru parametrów rozproszeniajest niezbędny, gdy wykorzystuje się te pomiary w kalibracji urządzeń działających w dzie-dzinie czasu, takich jak szybkie oscyloskopy próbkujące, albo wielkosygnałowe wektoroweanalizatory obwodów.

v

Wenn [meine] Arbeit einen Werthat, so besteht er [...] darin, dass inihr Gedanken ausgedrückt sind,und dieser Wert wird umso größersein, je besser die Gedankenausgedrückt sind.

Ludwig Wittgenstein

Meine Resultate kenne ich längst,ich weiß nur noch nicht,wie ich zu ihnen gelangen soll.

Carl Friedrich Gauss

vii

Acknowledgment

This research project would not have been possible without the support of many peo-ple and institutions. First of all, I would like to thank to my advisor Dr.Dylan Williamsfrom the National Institute of Standards and Technology (NIST), Boulder, USA, for manyfruitful discussions and his continuous support during my five years long stay at NIST. Iwish also to express gratitude to my supervisor at the Warsaw University of Technology,Prof. Janusz Dobrowolski for his constant help and patience during the long period in whichthis work was written. My gratitude is also due to Dr.Wojciech Wiatr for his encourage-ment and many invaluable advices without which this work have not been accomplished.

I would like also to acknowledge Denis LeGolvan of NIST, Boulder, USA, for introduc-ing me into the world of coaxial connectors, and for his enormous help with the measure-ments. I would also like to convey thanks to Grzegorz Kędzierski and Karol Korszeń ofthe National Institue of Telecommunications, Warsaw, Poland, for performing the Type-Nmeasurements described in this work.

Special thanks is also due to all of my colleges in the Electromagnetics Division, NIST,Boulder, and at the Institute of Electronic Systems, Warsaw, Poland, for their constantsupport throughout the entire time in which this project was carried out.

My deepest gratitude is also due to my family for their love, patience, and understandingwithout which finishing this work would not have been possible.

Last, but not least, I would like to acknowledge the Polish Ministry of Science andHigher Education for the grant N N517 4394 33 from which this work was partially funded.

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Contents

Abstract iii

Streszczenie v

Acknowledgment ix

Nomenclature xx

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Objective and scope of this work . . . . . . . . . . . . . . . . . . . . . . . 5

2 Principles of VNA S-parameter measurements 72.1 Definition of S -parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Waveguide voltage, current and characteristic impedance . . . . . . 92.1.2 Wave amplitudes and scattering parameters . . . . . . . . . . . . . 112.1.3 Practical implications . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 VNA S -parameter measurement . . . . . . . . . . . . . . . . . . . . . . . 172.3 Two-port VNA mathematical models . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Linear time-invariant two-port VNA . . . . . . . . . . . . . . . . . 202.3.2 Modeling VNA nonstationarity . . . . . . . . . . . . . . . . . . . . 28

2.4 Two-port VNA calibration techniques . . . . . . . . . . . . . . . . . . . . . 302.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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CONTENTS

3 Overview of uncertainty analysis for VNA S-parameter measurements 433.1 Sources of error in corrected VNA S -parameter measurements . . . . . . . 443.2 Statistical description of S -parameter measurement errors . . . . . . . . . . 45

3.2.1 Statistical model for S -parameter measurement . . . . . . . . . . . 453.2.2 Error description for a single S -parameter . . . . . . . . . . . . . . 463.2.3 Error description for a matrix of S -parameters . . . . . . . . . . . . 49

3.3 Statistical models for errors in corrected VNA S -parameter measurements 503.3.1 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 Random errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Representation of errors in corrected VNA S -parameter measurements . . . 533.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.2 Errors in VNA calibration coefficients . . . . . . . . . . . . . . . . . 543.4.3 Errors in VNA raw measurements . . . . . . . . . . . . . . . . . . . 58

3.5 Approximate uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . 613.5.1 Ripple analysis techniques . . . . . . . . . . . . . . . . . . . . . . . 623.5.2 Calibration comparison method . . . . . . . . . . . . . . . . . . . . 623.5.3 Statistical residual analysis . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Complete uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . 643.6.1 Linear error propagation . . . . . . . . . . . . . . . . . . . . . . . . 653.6.2 Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Multi-frequency description of S-parameter measurement errors 674.1 Statistical model for the multi-frequency S-parameter measurement . . . . 684.2 The notion of a physical error mechanism . . . . . . . . . . . . . . . . . . . 684.3 Statistical properties of the multi-frequency measurement error . . . . . . . 71

4.3.1 Probability distribution function . . . . . . . . . . . . . . . . . . . . 714.3.2 Uncertainty reporting . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.3 Multi-frequency covariance-matrix structure . . . . . . . . . . . . . 75

4.4 Physical error mechanisms in VNA S -parameter measurements . . . . . . . 754.4.1 Calibration standard errors . . . . . . . . . . . . . . . . . . . . . . 754.4.2 VNA instrumentation errors . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Practical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5.1 Time-domain waveform correction . . . . . . . . . . . . . . . . . . . 784.5.2 Device modeling based on S -parameter measurements . . . . . . . . 834.5.3 Error-mechanism-based VNA calibration . . . . . . . . . . . . . . . 84

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4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Generalized multi-frequency VNA calibration 895.1 Formulation of the VNA calibration problem . . . . . . . . . . . . . . . . . 905.2 Coaxial multi-line VNA calibration . . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 Classical multi-line VNA calibration . . . . . . . . . . . . . . . . . 935.2.2 Coaxial air-dielectric line as a calibration standard . . . . . . . . . 935.2.3 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Errors in the coaxial multi-line VNA calibration . . . . . . . . . . . . . . . 955.3.1 Variation of connector-interface electrical parameters . . . . . . . . 955.3.2 Variation of line’s characteristic impedance and propagation constant 1005.3.3 Line length error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3.4 Reflect asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Calibration standard models . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.4.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4.2 Reflect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.4.3 Thru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Solution uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5.1 Mathematical framework . . . . . . . . . . . . . . . . . . . . . . . . 1115.5.2 Relationships between estimated parameters . . . . . . . . . . . . . 1125.5.3 Optimal constraint choice . . . . . . . . . . . . . . . . . . . . . . . 114

5.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.7 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.8 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.8.2 Type-N coaxial connector . . . . . . . . . . . . . . . . . . . . . . . 1195.8.3 1.85 mm coaxial connector . . . . . . . . . . . . . . . . . . . . . . . 126

5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Multi-frequency stochastic modeling of VNA nonstationarity errors 1376.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2 Generic physical model for the VNA nonstationarity . . . . . . . . . . . . . 1396.3 Stochastic model for connector nonrepeatability and cable instability . . . 144

6.3.1 Statistical properties of circuit parameters . . . . . . . . . . . . . . 1446.3.2 Estimation of the covariance matrix of circuit parameters . . . . . . 145

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CONTENTS

6.3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.4 Stochastic model for VNA test-set drift . . . . . . . . . . . . . . . . . . . . 157

6.4.1 Drift as the multidimensional random walk . . . . . . . . . . . . . . 1576.4.2 Estimation of the process covariance matrix . . . . . . . . . . . . . 1596.4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7 Conclusions 167

A Real-valued representation of complex vectors and matrices 173

B Maximum likelihood approach to system identification 175B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175B.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.2.1 Errors in system responses . . . . . . . . . . . . . . . . . . . . . . . 177B.2.2 Errors in system responses and excitations . . . . . . . . . . . . . . 182

B.3 Covariance matrix of the estimates . . . . . . . . . . . . . . . . . . . . . . 186B.3.1 Errors in system responses . . . . . . . . . . . . . . . . . . . . . . . 186B.3.2 Errors in system responses and excitations . . . . . . . . . . . . . . 188

B.4 Numerical solution techniques . . . . . . . . . . . . . . . . . . . . . . . . . 189B.4.1 Errors in system responses . . . . . . . . . . . . . . . . . . . . . . . 189B.4.2 Errors in system responses and excitations . . . . . . . . . . . . . . 191

B.5 Solution uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191B.6 Systems with complex-valued inputs and outputs . . . . . . . . . . . . . . 191

C Air-dielectric coaxial transmission line 195C.1 Infinite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 195C.2 Finite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

D Center-conductor gap impedance 199D.1 Infinite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 199D.2 Finite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201D.3 Finger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

E Slightly nonuniform coaxial transmission line 205

F Small changes of two-port’s scattering parameters 209

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G Estimation of VNA nonstationarity model parameters 213

H Vector stochastic Wiener process 217

Bibliography 221

xv

CONTENTS

xvi

Nomenclature

a, b, c vectorsA, B, C matrices{ai}N

i=1 set of vectors a1, . . . , aN

AT transpose of the matrix AAH conjugate transpose (Hermitian transpose) of the matrix AA ⊗ B Kronecker product of the matrices A and Bvec(A) vector representation [a11, a21, a31, . . . , a12, a22, a32, . . .]T of the matrix ARex real part of x

Im x imaginary part of x

a real-valued representation [Re a1, Im a1,Re a2, Im a2, . . .]T of the complex-valuedvector a

x estimate of x

x true value of x

x measurement of x

Δx error in the measurement of x

E(x) expectation value of x

Var(x) variance of x

Cov(x, y) covariance of x and y

α attenuation constant, i.e., Re γ

β phase constant, i.e., Im γ

β sought parameters of the VNA and calibration standardsβ0 free-space phase constantc vector of calibration-standard unknown parametersC capacitancec speed of light in vacuumc0 vector of calibration-standard known parameters

xvii

CONTENTS

C ′ normalized capacitanceD diameter of the outer conductor in the coaxial transmission lined diameter of the inner conductor in the coaxial transmission linedp center-conductor-pin diameter in the coaxial transmission lineDUT device under teste eccentricity of the inner conductor in the coaxial transmission lineEDF forward directivityEDF reverse directivityEDF forward trackingEDF reverse trackingEDF forward source matchEDF reverse source matchε0 dielectric permittivity of vacuumεr relative dielectric permittivityη physical error mechanismf frequency; probability density functiong center-conductor-gap width in the coaxial transmission lineγ complex propagation constantΓ reflection coefficientIn identity matrix of size n × n

J Jacobian matrixK number of frequenciesk frequency indexL inductanceL′ normalized inductanceL′

g normalized gap inductance per-unit-lengthl transmission line lengthli inner conductor lengthlo outer conductor lengthΔl misalignment of outer and inner conductor symmetry axesΔlα loss correction factorM number of mechanismsμ0 magnetic permeability of vacuumN number of calibration standards

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CONTENTS

n calibration standard indexω angular frequencyp vector of VNA calibration coefficientsPDF probability density functionr vector of residualsR resistanceR′ normalized resistanceρ0 characteristic-impedance correction factorrDF effective forward directivityrDF effective reverse directivityrDF effective forward trackingrDF effective reverse trackingrDF effective forward source matchrDF effective reverse source matchS scattering matrixs vector representation vec(s) of the scattering matrix S; vector of calibration-

standard S-parameter-definitionssm vector of raw measurements of calibration standardsΣ covariance matrixσ conductivity, standard deviationSOLT short open load throughSOLT Singular Value DecompositionT transmission matrixTRL through reflect linev phase velocityVNA vector network analyzerV weight matrix in the VNA calibrationw width of the in-cut between the connector socket fingersY admittanceZ impedanceZ0 characteristic impedance for the TEM mode in a lossy coaxial transmission lineZ00 characteristic impedance for the TEM mode in a lossless coaxial transmission

lineY ′ normalized admittance

xix

NOMENCLATURE

Z ′ normalized impedanceZref reference impedance

xx

Chapter 1

Introduction

One day when Pooh Bear had nothing else todo, he thought he would do something [...].

A. A. Milne, “House at the Pooh corner”

1.1 Motivation

Vector network analyzer (VNA) is the basic measurement instrument for characteriza-tion of microwave and millimeter-wave electronic circuits. The VNA measures scatteringparameters (S-parameters) which constitute a complete description of small-signal deter-ministic properties of an electronic circuit [1]. This measurement is typically performed ina broad frequency range, starting from tens of kHz and reaching even hundreds of GHz [2–5]. The VNA measured S-parameters, along with noise parameters, are then traditionallyused in the design and testing of both single components and complex systems working atmicrowave and millimeter-wave frequencies.

Much effort has been put throughout the past 30 years in establishing the principles ofvector-network analysis and improving the VNA instrumentation. A good review of thisresearch can be found in [6–8]. Modern VNAs, such as [3–5], are a culmination of this longstanding research and are very mature and reliable measurement instruments, commonlyemployed in the industry and laboratories.

Recently, however, several new trends in the vector network-analysis started to emerge.These trends push the boundaries of the conventional VNAs with respect to the maxi-

1

1. INTRODUCTION

mum measurement frequency, impedance level of the device under test (DUT), and theassumption of the DUT linearity.

The trend to extend the frequency range of modern VNAs results from an increasedinterest in the use of signals with frequencies in the millimeter- and sub-millimeter-waverange. Examples are high-capacity data transmission systems [9], millimeter-wave radarsystem [10, 11], radiotelescopes [12, 13], or the broad range of terahertz applications [14].Efforts to extend the maximum VNA measurement frequency have recently brought aboutspecial VNA extension units, allowing the conventional VNAs to operate up to 500 GHzwith rectangular waveguide connectors [2]. Extension up to 1 THz is likely to happen inthe nearest future [15, 16].

Accurate VNA S-parameter measurement at millimeter- and sub-millimeter-wave fre-quencies, however, is very challenging due to some specific error sources negligible at lowerfrequencies. Due to small wavelength, these measurements require the use of waveguideswith aperture size below 1 mm in order to avoid overmoding. While it is possible to man-ufacture such waveguides with quite a high precision, some irregularities, such as roundingof the waveguide corners or erosion of the leading edges of the waveguide apertures, areunavoidable and may lead to significant systematic errors in the VNA calibration [17].Furthermore, the connection of two waveguide flanges with such small apertures requiresvery precise alignment. Although some special alignment solutions, involving the use ofmultiple alignment pins, have been devised, the random errors due to flange misalignmentcan still significantly deteriorate the measurement accuracy [18]. Finally, the noise fluctu-ations of the VNA test-signals are another important source of errors in VNA S-parametermeasurement at sub-millimeter-wave frequencies. These fluctuations, caused by the ther-mal and phase noise originating in the frequency multiplication and sub-harmonic mixingcircuitry of the VNA extension units, significantly reduce the dynamic range and increasethe short-term instability (also referred to as the “trace jitter” [19]), as compared withVNAs operating at lower frequencies.

While the VNA calibration techniques used at lower frequencies can be adapted towork with millimeter and sub-millimeter-wave VNAs, due to those specific errors, theiraccuracy is often not satisfactory [18, 20]. Thus, new more accurate calibration techniques,less sensitive to those specific error sources, need to be devised.

Similar challenges regarding the measurement accuracy are encountered in VNA S-pa-rameter measurements of devices whose impedance differs significantly from the typicalVNA impedance level of 50 Ω. Examples of such devices are nanotubes, nanowires or

2

1.1. MOTIVATION

metamaterials, which may exhibit impedances in the range of tens and hundreds of kΩs,or fractions of an Ω [21]. Since the VNA test-ports are typically built based on 50 Ω-transmission-line components, the energy coupling to devices with impedance level closeto 50 Ω is very good. Thus, such devices are measured with the highest accuracy. However,when the DUT impedance is much smaller or much larger than this value, only a littlesignal is coupled and most of the signal is reflected back to the VNA. While some specialVNA architectures (e.g. [22]), aiming at increasing the VNA receiver resolution, may helpto improve the measurement accuracy in such cases, new VNA calibration techniques, lesssensitive to the VNA measurement errors (e.g., [23]), are needed.

The last trend in the modern vector-network-analysis results from an increased demandfor the characterization of active high-frequency circuits in the large-signal regime. Accu-rate large-signal characterization of such circuits is essential in the design and testing ofvarious applications. Examples are portable data transmission systems where the highpower efficiency (hence, long battery life) needs to be combined with a minimal nonlin-ear distortion to the transmitted signal, or active high-frequency circuits such as signalgenerators, mixers or frequency multipliers, that are by nature operating in the nonlinearregime.

Characterization of large-signal properties of high-frequency circuits poses multiple dif-ficult problems. It requires specialized instrumentation, such as nonlinear vector-network-analyzer (NVNA), also referred to as large-signal network analyzer (LSNA) [24, 25]. TheNVNA characterizes the nonlinear DUT properties in terms of either voltages and currents,or wave quantities, such as X-parameters [26, 27] or S-functions [25, 28]. The character-ization is performed at the principal frequency and its harmonics. The measurement isthen either directly used, for example, in the circuit simulator, or converted into the timedomain in order to analyze the shape of the voltage or current waveforms.

Accurate NVNA measurements require a specialized calibration procedure. This pro-cedure, apart from the traditional linear VNA calibration, involves also power and phasecalibration. The power calibration is required so as to enable the measurement of absolutequantities (voltages, currents or wave quantities). The phase calibration is necessary be-cause in the NVNA measurement one is not only interested in magnitudes and phases ofvoltages and currents (or wave quantities) at each frequency, but also in the phase relation-ships between those quantities. These relationships are essential, for example, when recon-structing the time-domain voltage and current waveforms from the NVNA measurements.As a result, the accuracy assessment of NVNA measurements requires new uncertainty

3

1. INTRODUCTION

analysis approaches that account not only for uncertainties at a single frequency, but alsofor the statistical correlations between uncertainties at different frequencies.

Consequently, the new trends in the vector network analysis discussed above, whilestimulating the development of new hardware solutions, lead also to new, more stringentdemands as to the measurement accuracy and its reliable and complete evaluation. Themulti-frequency approach presented in this work addresses this issue.

1.2 Previous research

Enhancement of VNA measurement accuracy and its more reliable and complete eval-uation have always been stimulating the development of VNA measurement techniques. Adetailed review of this development can be found in [6–8, 19]. Here we shall indicate themost important turning points in this development, which will allow us to better under-stand the origins of the multi-frequency approach proposed in this work.

The first turning point was the invention of the self-calibration methods. The ideaof self-calibration in the two-port VNA calibration problem had first been employed inEngen’s TRL method [29] and was then generalized by Eul and Schiek [30]. The conceptof self-calibration in one-port VNA calibration methods appears also in papers by Wiatr[31–33] and Bianco [34]. The principle of self-calibration is to use calibration standardsthat are only partially known and to determine their complete S-parameter descriptionalong with the VNA calibration coefficients. For example, in the TRL method, the trans-mission line is used with known length and unknown propagation constant. Consequently,the contribution of systematic errors in calibration standard definition can be reduced,since instead of the specific numerical values of calibration standard parameters, which areinevitably subject to measurement errors, the information as to the relationships betweenthese parameters is used.

Another turning point was the application of statistical methods in VNA calibrationproblem. This approach was initiated in the case of one-port VNA calibration in [31, 32, 35]and in the case of two-port VNA calibration methods in [36]. The application of statisticalmethods in VNA calibration is based on the use of redundant calibration standards andstatistical processing of the resulting overdetermined set of equations. Consequently, thecontribution of random measurement errors can be significantly reduced.

Another important paradigm change in the development of VNA calibration methodswas initiated in [37] and [38]. In these references, for the first time, the relationships

4

1.3. OBJECTIVE AND SCOPE OF THIS WORK

between the calibration standard parameters at different frequencies are exploited in theVNA calibration. The experimental result presented in [37] and [38] indicate that thisimproves the accuracy and reliability of the VNA calibration.

The multi-frequency description of S-parameter measurement errors naturally comple-ments a VNA calibration approach that accounts for the relationships between S-parametermeasurements at different frequencies. Such a description was introduced in [39], and isbased on the covariance-matrix representation which had already been used in the uncer-tainty analysis of single-frequency S-parameter measurements in dual six-port measure-ment systems [40, 41], and then recently rediscovered in the context of uncertainty eval-uation in VNA S-parameter measurements [42, 43]. The generalized covariance-matrixdescription proposed in [39] uses additional terms in the covariance matrix in order toaccount for statistical correlations between uncertainties at different frequencies. Thesecorrelations have been shown to be essential when applying the VNA S-parameter mea-surements in the calibration of time-domain measurement systems [39, 44].

1.3 Objective and scope of this work

As pointed out above, accounting for the relationships between VNA S-parameter mea-surements at different frequencies can be beneficial in terms of increased measurement ac-curacy (see [37, 38]) and its more complete evaluation (see [39, 44]). The objective of thiswork is to generalize these results by developing a comprehensive multi-frequency approachto VNA S-parameter measurements.

We shall attain this objective in two step. In the first step, we will develop a mathe-matical description of the relationships between VNA measurement at different frequencieswhich unifies the descriptions used in the calibration approaches of [37, 38] and in the un-certainty analysis of [39]. With the use of this generalized description, in the second step,we will investigate the benefits which could be gained by accounting for those relationshipsat various stages of the VNA measurement procedure. A particular emphasis will be puthere on the VNA calibration.

This organization of this work is as follows. In the introductory part (Chapter 2 andChapter 3) we review the foundations of VNA S-parameter measurements and uncertaintyanalysis. This part serves as the theoretical background for the discussion presented in themain part of this work.

The main part of this work consists of three chapters. In Chapter 4 we develop a uniform

5

1. INTRODUCTION

framework for the representation of relationships between VNA S-parameter measurementsat different frequencies. We further review the practical applications for which accountingfor these relationships is important. These applications include the correction of time-domain measurements, measurement-based device modeling and the VNA calibration onwhich we focus in the this work. We show that a statistically sound description of the VNAcalibration problem should be done in terms of the error mechanisms underlying the cali-bration standard and VNA instrumentation errors. As a consequence, the VNA calibrationshould be performed jointly at all measurement frequencies so as to account for the simul-taneous contribution of those error mechanisms to S-parameter measurements at differentfrequencies. We refer to this approach as the error-mechanism-based VNA calibration andin the remainder of this work we develop the necessary tools for the implementation ofsuch a calibration approach. These tools include the generalized multi-frequency VNAcalibration (see Chapter 5) and the framework for error-mechanism-based description ofthe VNA nonstationarity errors (see Chapter 6).

In the last part of this work (see Chapter 7) we present conclusions and discuss possibledirections of further research.

6

Chapter 2

Principles of VNA S-parametermeasurements

All models are wrong, some are useful.

George Box

In this chapter, we review the principles of S-parameter measurements with the vectornetwork analyzer (VNA). We begin with a brief review of the S-parameter definition.Following on that, we discuss the two-port VNA S-parameter measurements, and analyzethe imperfections of a typical two-port VNA measurement setup. The errors caused bythese imperfections are systematic as they are very stable in the course of typical VNAmeasurement. Therefore, they can be characterized in a calibration procedure and thenremoved from the actual S-parameter measurements in the correction procedure. Bothprocedures assume a mathematical model of these VNA. In the calibration procedure, aset of devices with some known characteristics is measured and the parameters of theVNA model are determined. Then, in the correction procedure, the model obtained in thecalibration is used to correct for the imperfections of the VNA setup. We discuss differentmathematical VNA models and VNA calibration techniques in the last two sections of thischapter.

7

2. PRINCIPLES OF VNA S -PARAMETER MEASUREMENTS

2.1 Definition of S-parameters

Scattering parameters form a description of an electronic circuit in terms of complexamplitudes1 of electromagnetic waves interacting with the circuit. This description isused in the situation when the dimensions of the circuit are become comparable with thewavelength, which typically takes place at microwave and millimeter-wave frequencies. Inthis situation, the physical phenomena occurring in the circuit are of a wave nature, andthe conventional circuit description in terms of terminal voltages and currents looses itsphysical correspondence.

Scattering parameter description uses the concept of the circuit port instead of thecircuit terminal. A circuit port is defined as section of a uniform arbitrary waveguidethrough which the electromagnetic wave may enter and exit the electronic circuit. Inorder to ensure the uniqueness of the scattering parameter description, we require thatthese ports are electromagnetically separated and that a given set of circuit ports (withwaveguide modes propagating through them) encompasses all of the possible means bywhich electromagnetic energy can enter and leave the circuit. This means that we need toaccount not only for all of the physical ports through which the electromagnetic waves areinteracting with the circuit, but also for all of the waveguide modes propagating throughthe circuit ports. In order to simplify the scattering parameter description, we typicallyassume single mode propagation through circuit ports, however, extension to the case ofmultiple modes is possible.

Scattering parameters describe the relationships between the amplitudes of the wavespropagating through the circuit ports. These amplitudes are defined with the use of asimple normalization such that the wave with a unit root-mean-square amplitude, in theabsence of the wave propagating in the opposite direction, carries unit power. Althoughthe principle of this normalization is very simple, its systematic derivation requires someconsideration. In the following, we briefly review the origins of this normalization (formore details refer to, e.g., [1, 45]). We first introduce the concepts of waveguide voltage,current and characteristic impedance. These concepts, although not required for the def-inition of normalized waves and, consequently, of S-parameters, allow one the relate thescattering parameter description to the methods of the transmission line theory. We thenpresent two different types of normalized waves. The first type, referred to as the travelingwaves, originates in the physics of wave propagation in the waveguide. Thus, properties of

1We assume time-harmonic dependence of the fields.

8

2.1. DEFINITION OF S -PARAMETERS

traveling waves closely reflect the physical properties of the actual waves propagating inthe waveguide. In some cases, however, these properties lead to results that are surprisingin the context of the transmission line theory. Consequently, another type of normalizedwaves, referred to as pseudo-waves, is introduced, which leads to more intuitive results inthe framework of transmission line theory, at a cost, however, of “not as close” correspon-dence to the physics of wave propagation in the waveguide. We conclude with a discussionof practical implications of the different normalization schemes for scattering parametermeasurements.

2.1.1 Waveguide voltage, current and characteristic impedance

Electromagnetic waves traveling in a waveguide are described in terms of modes whichare solutions to the Maxwell equations in the waveguide cross-section. For time-harmonicfield dependence, these solutions can be characterized by the normalized transverse electricand transverse magnetic field distributions, et(x, y) and ht(x, y), respectively, and thecomplex-valued propagation constant γ. In the case of lossless transmission lines, the fielddistributions are real-valued, and the propagation constant is imaginary γ = jβ. In thecase of transmission lines with losses, the field distributions are in general complex-valuedand the propagation constant has also a real part, that is γ = α + jβ.

With the use of the normalized field distributions and the propagation constant, we canwrite the complex peak amplitudes of the fields at any point in the waveguide (propagationoccurs along the z axis) in a normalized way as

Et(x, y, z) = C+et(x, y)e−γz + C−et(x, y)e+γz = V (z)V0

et(x, y), (2.1)

Ht(x, y, z) = C+ht(x, y)e−γz + C−ht(x, y)e+γz = I(z)I0

ht(x, y) (2.2)

where V0 and I0 are normalization constants with the dimension of voltage and current,respectively, C+ and C− are unitless constants specifying the amplitude of the forwardand backward propagating wave at z = 0, respectively, and V (z) and I(z) are defined aswaveguide voltage and current.

The unitless constants C+ and C− depend on the normalization used for et(x, y) andht(x, y). The waveguide voltage and current, however, are independent of this normaliza-tion due to the use of normalization constants V 0 and I0. These constants have units ofvoltage and current, respectively, hence in the following we refer to them as the normal-

9


ization voltage and current, respectively.The choice of normalization voltage V0 and current I0 is not arbitrary and we require

thatP 0 =

12V0I∗

0 =12

∫S

et × h∗t dS (2.3)

where the superscript ∗ indicates the complex conjugate, and S denotes the cross-sectionof the waveguide. From (2.3) it follows that the net power flow in the waveguide is

P (z) = 12

∫S

Et(x, y, z) × H∗t (x, y, z)dS = 1

2V (z)I∗(z)∫

S et × h∗t dS

V0I∗0

= 12V (z)I∗(z). (2.4)

Consequently, by imposing the condition (2.3), we require that the net power flowingthrough the cross-section of the waveguide can be determined applying conventional circuit-theory definition to the waveguide voltage and current. Note that the magnitude of P0

depends on the normalizations used for et(x, y) and ht(x, y), however, its phase is inde-pendent of this normalizations and is an inherent property of the mode.

Due to the power constraint (2.3), only one of the constants V 0 and I0 can be arbitrarilychosen. For example, for the voltage V 0, we may choose to use the path integral alongsome arbitrarily chosen path P in the cross-section of the waveguide

V0 = −∫

Pet(x, y)dl, (2.5)

with an obvious constraint V0 �= 0, and then determine I0 from (2.3) (“voltage-power”normalization). Alternatively, we can also fix the current I0 based the loop integral arounda closed loop L in the cross-section of the waveguide

I0 =∮

Lht(x, y)dl, (2.6)

with a similar constraint I0 �= 0 and then determine V0 from the constraint (2.3) (“current-power” normalization). Other normalizations are also possible, e.g., [1] or [46].

Based on the definition of the normalization voltage V0 and current I0, we define thecharacteristic impedance of the mode

Z0 =V 0

I0. (2.7)

The magnitude of Z0 depends, in general, on the field normalizations and the chosenstrategy for setting the constants V0 and I0. The phase of Z0 can be easily determined in

10


terms of the mode power P0. Indeed, after some simple transformations, we obtain

ImZ0

ReZ0= ImP0

ReP0(2.8)

Phase of Z0 is therefore the inherent property of the mode and does not depend on thefield normalizations and the constants V0 and I0.

For TEM modes, the integral (2.5) depends only on the end points of the path and it isnatural to choose these point to lie on different conductors. Definition (2.7) becomes thenthe conventional definition of the characteristic impedance for TEM modes.

The above definition of the waveguide voltage, current and characteristic impedanceare the fundamental concepts of the transmission line theory. This theory extends theconventional circuit theory by allowing voltages and currents to depend also on the location.The main tool of this theory is a set of differential equations, referred to as Telegraphicequations, which describe the wave propagation in terms time- and location-dependentvoltages and currents. For details refer to, e.g., [1] or [47].

2.1.2 Wave amplitudes and scattering parameters

So far, we have presented two different means of representing fields in the waveguide.The voltage-current description is independent of the field normalizations used in et(x, y)and ht(x, y) and allows us to use the methods of the transmission line theory. However,this description does not represent well the underlying physics of wave propagation phe-nomenon which is best both analyzed and experimentally observed in terms of forward andbackward propagating waves rather than voltages and currents. The other description wediscussed, resulting directly from the solution of Maxwell equations, uses unitless constantsC+ and C−. These constants have straightforward physical interpretation and describe theamplitudes of the forward and backward propagating wave. However, they are difficult toboth interpret and measure sue to the dependence on the normalization of the mode fieldset(x, y) and ht(x, y).

Solution to that problems is the description in terms of traveling-wave amplitudes. Thisdescription arises from the following normalization of the constants C+ and C−

a0(z) =√2ReP0C+e−γz, and b0(z) =

√2ReP0C−e+γz. (2.9)

By use of this normalization, we readily obtain the fields in the waveguide as

11


Et(x, y, z) = C+et(x, y)e−γz + C−et(x, y)e+γz = a0(z) + b0(z)√2ReP0

et(x, y), (2.10)

Ht(x, y, z) = C+ht(x, y)e−γz + C−ht(x, y)e+γz = a0(z) − b0(z)√2ReP0

ht(x, y), (2.11)

where the quantities a0(z) and b0(z) are referred to as the forward and backward travelingwave amplitudes.

The adjective “traveling” indicates that amplitudes a0(z) and b0(z) describe the actualforward and backward propagating waves in the waveguide. Similarly to waveguide voltageand current, the traveling wave amplitudes are independent of the normalization used inthe modal fields et(x, y) and ht(x, y). The traveling wave amplitudes have also a closecorrespondence to the real power carried by the propagating mode. Indeed, we can showthat, in the absence of the backward propagating wave, the forward wave carries the realpower ReP (z) = 1

2Re∫

S Et(x, y, z)×H∗t (x, y, z)dS = 1

2 |a0|2. Hence, a traveling wave with aunit root-mean-square amplitude,

∣∣∣ a0√2

∣∣∣ = 1 carries a unit real power. Similar relationshipsholds for the backward propagating wave.

It is important to note that the definition of traveling wave amplitudes does not usethe waveguide voltage, current, nor the characteristic impedance. These amplitudes aretherefore directly related to the modal solutions of Maxwell equations in the waveguide.However, the traveling wave amplitudes a0(z) and b0(z) can easily be related to the voltage-current description. Indeed, by comparing (2.1) and (2.2) with (2.10) and (2.11), we obtain

a0(z) =1√2

√ReP0

V0[V (z) + I(z)Z0] , (2.12)

b0(z) =1√2

√ReP0

V0[V (z) − I(z)Z0] , (2.13)

and thus

V (z) = 1√2

V0√ReP0

[a0(z) + b0(z)] , (2.14)

I(z) = 1√2

I0√ReP0

[a0(z) − b0(z)] . (2.15)

The direct correspondence of the traveling wave amplitudes to the forward and back-ward propagating modes leads to some surprising results in the case when there is a phase

12


shift between the electric and magnetic fields, that is, when argP0 �= 0. This phase shift isa consequence of power loss in the waveguide, which is commonly encountered in practicedue to finite conductivity of real conductors and losses in dielectrics.

When the waveguide is lossy, the forward and backward propagating modes are notorthogonal. Therefore, the real power flowing through a given cross section is not equalto the sum of powers carried by the modes, which is usually assumed in the classicaltransmission line theory [48, 49]. This result can easily be confirmed by writing the realpower flowing through the waveguide cross-section with the use of the traveling waveamplitudes

ReP (z) = 12 |a0|2 − 1

2 |b0|2 + Im(a0b∗0)Imp0

Rep0= 12 |a0|2 − 1

2 |b0|2 + Im(a0b∗0)ImZ0

ReZ0. (2.16)

Indeed, for argP0 �= 0 we obtain an additional term related to the phase of P0. Note that,according to (2.8), this phase is a property of the mode and does not depend on the choiceof field normalizations. It is related to the characteristic impedance of the mode through(2.8), hence argP0 �= 0 implies that characteristic impedance Z0 is complex.

Another property, surprising in the context of transmission line theory, that results fromthe loss in the waveguide, is that the ratio of real powers incident at and reflected froman discontinuity in the waveguide is not equal to |Γ|2 where Γ is the reflection coefficientΓ = b0/a0 [48]. Therefore, in some cases, magnitude of Γ may exceed one which is alsounusual for the classical transmission-line theory. This result can also be easily obtainedwith the use of traveling wave amplitudes [45].

Therefore, for practical reasons, it is sometimes desirable to have an alternative normal-ization which would lead to more intuitive results in the context of the transmission-linetheory. Also, when characteristic impedance Z0 exhibits a significant frequency depen-dence, it is more convenient to have a fixed relationship between the wave amplitudes andwaveguide voltages and currents that be independent of the frequency dependence of Z0.A normalization that has these properties is proposed in [45] and has form

a(z) = |V0|V0

√ReZref

2|Zref | [V (z) + I(z)Zref ] , (2.17)

b(z) = |V0|V0

√ReZref

2|Zref | [V (z) − I(z)Zref ] , (2.18)

13


where Zref is an arbitrary parameter with ReZref ≥ 0, and a(z) and b(z) are referred toas the pseudo-wave amplitudes. This normalization has the property that the real powerflow in the waveguide is described by the expression as (2.16), however with Z0 replaced byZref . Consequently, for a real Zref , the additional term in expression (2.16) vanishes andwe obtain a description that is more intuitive in the context of transmission line theory.It can be also shown that for Zref = Z0, the pseudo-wave amplitudes (2.17) and (2.18)become traveling-wave amplitudes (2.12) and (2.13).

It is important to note that the pseudo-wave amplitudes do not directly correspondto the traveling wave amplitudes. Indeed, we can readily show that a(z) and b(z) arelinear combinations of a0(z) and b0(z). Therefore pseudo wave amplitudes are more of amathematical artifact than a physical representation of wave propagation in the waveguide.This results in a well know property that for an infinite waveguide stimulated by a travelingwave with |a0(z)| �= 0 we have |b0(z)| = 0, however, after conversion to the pseudo-waveamplitudes, we obtain |b(z)| �= 0 [45]. This again confirms that the pseudo-wave amplitudesdo not reflect the physics of wave propagation in the waveguide.

It is sometimes desirable to convert from on set of pseudo-wave amplitudes, a(z) andb(z), with a reference impedance Zref to another set, a(z)′ and b(z)′ with a differentreference impedance Z ′

ref . The relationship between the two sets can be easily determinedfrom (2.17) and (2.18) as [45]

⎡⎣ a(z)′

b(z)′

⎤⎦ = N

⎡⎣ a(z)

b(z)

⎤⎦ , (2.19)

where

N =√√√√1 − jImZref/ReZref

1 − jImZ ′ref/ReZ ′

ref

1√1 − Γ2

⎡⎣ 1 −Γ

−Γ 1

⎤⎦ , (2.20)

andΓ =

Z ′ref − Zref

Z ′ref + Zref

. (2.21)

It can be shown that for real reference impedances Zref and Z ′ref , matrix (2.20) becomes

the transmission matrix of an ideal impedance transformer [1].

Having discussed different definitions of wave amplitudes, we finally introduce the scat-tering parameters. For a circuit with N ports, we group the wave amplitudes (traveling-

14


wave amplitudes of pseudo-wave amplitudes) at the circuit ports into two vectors

a =

⎡⎢⎢⎢⎣

a1...

aN

⎤⎥⎥⎥⎦ , b =

⎡⎢⎢⎢⎣

b1...

bN

⎤⎥⎥⎥⎦ , (2.22)

and define a linear relationship between the two vectors with a matrix S

b = Sa. (2.23)

When matrix (2.23) is defined in terms of traveling-wave amplitudes, we refer to its elementsas scattering parameters. In the case of pseudo-wave amplitude, we refer to the elementsof S as pseudo-scattering parameters. However, since the pseudo-wave amplitudes becometraveling wave amplitudes for Zref = Z0, we often talk briefly about scattering parametersdefined with reference to a certain impedance Zref .

In the case of two-port devices2, it is sometimes more convenient to represent therelationship between the wave amplitudes with the use of transmission matrix defined as

⎡⎣ b1

a1

⎤⎦ = T

⎡⎣ a2

b2

⎤⎦ . (2.24)

This description has the useful property that the transmission matrix of the cascade con-nection of two-port networks described with transmission matrices Ti, for i = 1, . . . , N , isgiven by a product

T =N∏

i=1Ti. (2.25)

In the common case of a two-port device, we give the relationship between the tworepresentations explicitly as they are used very often throughout this work. For a two-portnetwork with the scattering parameters given by

S =⎡⎣ S11 S12

S21 S22

⎤⎦ , (2.26)

2The transmission matrix representation can easily be extend to the case of multiport devices with aneven number of ports, see [47].

15


and the transmission parameters

T =⎡⎣ T11 T12

T21 T22

⎤⎦ , (2.27)

the relationships between the elements of (2.26) and (2.27) is [1]

T = 1S21

⎡⎣ S12S21 − S11S22 S11

−S22 1

⎤⎦ , (2.28)

and

S = 1T22

⎡⎣ T12 T11T22 − T12T21

1 −T21

⎤⎦ . (2.29)

2.1.3 Practical implications

In practice, the general definition of S-parameters presented in the previous section,can often be simplified. In most practical cases, the normalization voltage V 0 is real3,hence we have |V0|/V0 = 1. We obtain then the following relations between the waveguidevoltages and currents, and the traveling-wave amplitudes

a0(z) =√ReZ0

2|Z0| [V (z) + I(z)Z0] , (2.30)

b0(z) =√ReZ0

2|Z0| [V (z) − I(z)Z0] , (2.31)

while for the pseudo-wave amplitudes we have

a(z) =

√ReZref

2|Zref | [V (z) + I(z)Zref ] , (2.32)

b(z) =

√ReZref

2|Zref | [V (z) − I(z)Zref ] , (2.33)

3Voltage V 0 becomes complex if the plane of a constant phase velocity it not perpendicular to thedirection of propagation. This occurs in waveguides with dielectrics that are anisotropic or inhomogeneousin the waveguide cross-section.

16

2.2. VNA S -PARAMETER MEASUREMENT

and thus

V (z) = 1√ReZref

|Zref | [a0(z) + b0(z)] , (2.34)

I(z) = 1√ReZref

|Zref |Zref

[a0(z) − b0(z)] . (2.35)

For a real reference impedance, we further obtain the familiar expression known from thecircuit theory [50–52]

a(z) = 12√

Zref

[V (z) + I(z)Zref ] , (2.36)

b(z) = 12√

Zref

[V (z) − I(z)Zref ] , (2.37)

and

V (z) =√

Zref [a0(z) + b0(z)] , (2.38)

I(z) = 1√Zref

[a0(z) − b0(z)] . (2.39)

In the context of the VNA S-parameter measurements, it is important to note that theVNA measures S-parameters with respect to some unknown reference impedance. Hencean important aspect of the VNA calibration is the determination of this impedance. Thiswill be discussed in more detail in Paragraph 2.4.2-C.

2.2 VNA S-parameter measurement

In the previous section, we demonstrated that the scattering parameters, as captured inS-matrix defined by (2.23), describe relationships between normalized guided electromag-netic waves incident at and reflected from the ports of an electronic circuit. This suggestsan intuitive method for their measurement, namely through an observation of these wavesin some controlled conditions, such as when only one of the device ports is excited at atime. This observation should disturb the waves as little as possible (this is analogous tothe condition in low-frequency oscilloscope measurements that the probe has high input

17


��

��

��

Fig. 2.1: VNA block diagram (switch is shown in the “forward” position).

impedance such that it does not disturb voltages and currents in the circuit), and in orderto avoid any interferences, waves emerging from the device-under-test (DUT) should beabsorbed at some place far enough from the DUT (this is analogous to the measurementof impedance or admittance parameters when we require, respectively, low impedance orhigh impedance termination of the circuit terminals).

This simple idea is the operational principle of the vector network analyzer (VNA).A simplified diagram of a typical VNA, dedicated to the measurement of devices withtwo or less ports (in short, a two-port VNA), is shown in Fig. 2.1. On either side of theDUT, there is a set of two directional couplers with detectors. We refer to each set asa reflectometer. The function of the reflectometer is to measure the complex amplitudesof the wave incident at and reflected from the DUT. This process is realized by couplingpart of each wave out the detection circuit and converting it to a lower frequency at whichthe analog to digital (A/D) converters can be used. We show the detection circuit as asingle mixer excited by the local oscillator (LO), however, in the reality the signal mayundergo multiple frequency conversions. The A/D conversion may take place either atsome intermediate frequency (IF) or in the baseband.

18

2.2. VNA S -PARAMETER MEASUREMENT

In the “forward” position of the switch (this is the position shown in Fig. 2.1), signalfrom the RF source is sent to the first port of the device under test while the other portis terminated with a matched termination. The function of this termination is to absorbthe wave emerging from the non-excited port of the DUT. Complex voltages a1m andb1m, which are approximately proportional to the amplitudes of the waves incident at andreflected from port one, a1 and b1, respectively, are then measured in the left reflectometer.Similarly, signal b2m, approximately proportional to the wave b2 transmitted through theDUT, is measured in the right reflectometer. From these measurements, we approximatescattering parameters S11 and S22 of the DUT as

S11m = b1m

a1m

, and S21m = b2m

a1m

, (2.40)

respectively. A similar description holds for the “reverse” position of the switch and weobtain the following approximations of S22 and S12

S22m = b2m

a2m

, and S12m = b1m

a2m

. (2.41)

We refer these approximations as “raw”, measured or uncorrected S-parameters.

The practical implementation of the VNA is far more complicated then the diagramin Fig. 2.1. For detailed discussion of different architectures see for example [53]. Themain objective of the VNA construction is to provide wideband operation (for examplefrom 70 kHz up to 70GHz [4]) while maintaining the error of approximations (2.40) and(2.41) reasonably small. This objective is very hard to attain in practice, therefore theerrors of approximations are (2.40) and (2.41) usually not acceptable, even for approximateassessment of the DUT S-parameters.

These errors result from various imperfections of the VNA construction. The mostimportant ones are the finite directivity of directional couplers, impedance mismatches inthe VNA (such as between the generator and the adjacent coupler, or between the othercoupler and the matched termination), discontinuities in the transmission lines guiding themeasured signals, phase shift and attenuation introduced by these lines, parasitic couplingbetween the VNA ports (e.g. through the LO circuitry), and differences between the loadimpedance in the forward and reverse position of the switch.

19


2.3 Two-port VNA mathematical models

In this section, we discuss mathematical models for two-port VNA measurements.These models describe the relationship between the measured and actual S-parametersof a DUT as a function of a set of model parameters describing the systematic errorsintroduced by the VNA. These models form the foundation for the correction of VNAmeasurements and the development of VNA calibration algorithms.

The basic premise in the formulation of VNA models is that the relationships betweenthe measured and actual wave amplitudes are linear and time-invariant, thus the VNAis assumed to be a linear time-invariant (LTI) system. We begin our discussion with thedescription of VNA models based on this premise. Then we discuss situations when thetime-invariance assumption is violated. We do not discuss the case when the VNA becomesnon-linear as it is beyond the scope of this work.

2.3.1 Linear time-invariant two-port VNA

��

Fig. 2.2: 16-term model of a two-port VNA.

A. 16-term model. The most general model of alinear time-invariant two-port VNA is the 16-termmodel [6, 54–56]. This model, shown schematicallyin Fig. 2.2, results directly from the VNA diagramin Fig. 2.1. In the 16-term model model, the linearrelationship between the measured and actual wavesas a four-port liner network, denoted with E Fig. 2.2,and defined as

⎡⎢⎢⎢⎢⎢⎢⎣

b1m

b2m

a1

a2

⎤⎥⎥⎥⎥⎥⎥⎦= Se

⎡⎢⎢⎢⎢⎢⎢⎣

a1m

a2m

b1

b2

⎤⎥⎥⎥⎥⎥⎥⎦

, where Se =⎡⎣ S11e S12e

S21e S22e

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

e11 e12 e13 e14

e21 e22 e23 e24

e31 e32 e33 e34

e41 e42 e43 e44

⎤⎥⎥⎥⎥⎥⎥⎦

. (2.42)

Model parameters contained in the matrix Se encompass all possible transmission andreflection paths in the VNA. Referring to Fig. 2.2 and Fig. 2.1, we note that

• the diagonal terms of S11e, S12e, S21e, and S22e describe the systematic error intro-duced by the VNA reflectometers themselves;

20

2.3. TWO-PORT VNA MATHEMATICAL MODELS

• anti-diagonal terms of S11e (that is e12 and e21) correspond to the internal cross-talk between the VNA reflectometers; this cross-talk results from the finite isolationbetween different ports of the switch;

• the anti-diagonal terms of Se (that is e14, e23, e32 and e41) correspond to the couplingbetween VNA reflectometers, for example, through the IF circuitry; in modern VNAsthis coupling is negligible, therefore one typically assumes e14 = e23 = e32 = e41 = 0;

• the anti-diagonal terms of S22e (that is e34 and e43) correspond to the external cross-talk between the VNA reflectometers, that is, to the direct cross-talk between theVNA measurement ports; when measuring open-waveguide structures (e.g., mis-crostrip lines or coplanar waveguides), this cross-talk can be significant, however,in the case of enclosed waveguides (e.g., coaxial lines or rectangular waveguides) thiscross-talk does not occur.

We determine the relationship between the raw and actual S-parameters in the 16-termmodel by applying their definitions

⎡⎣ b1m

b2m

⎤⎦ = Sm

⎡⎣ a1m

a2m

⎤⎦ , and

⎡⎣ b1

b2

⎤⎦ = S

⎡⎣ a1

a2

⎤⎦ , (2.43)

respectively, to (2.42) and solving the resulting set of linear equations. This yields

Sm = S11e + S12eS (I − S22eS)−1 S21e = S11e + S12e

(S−1 − S22e

)−1S21e, (2.44)

andS =

[S21e (Sm − S11e)−1 S12e + S22e

]−1. (2.45)

By exploiting the structure of (2.44) and (2.45), we note that, although the model (2.2)has 16 terms, only 15 terms need to be known to solve (2.44) and (2.45). Indeed, if wemultiply all elements of S21e by an arbitrary constant and divide all elements S12e by thesame constant, relationships (2.44) and (2.45) do not change. Therefore one of the elementsin S12e or S21e can be arbitrarily chosen, for example fixed to one.

We further observe that relationships between the actual and measured S-parameters,S and Sm, respectively, given by (2.44) and (2.45), are nonlinear functions of the modelparameters contained in the matrix Se, defined in (2.42). Therefore, the 16-term model isoften expressed in an alternative form which uses a different set of parameters for which

21


the relationships between S and Sm become linear. This alternative form is defined by[6, 55, 56]

⎡⎢⎢⎢⎢⎢⎢⎣

b1m

b2m

a1m

a2m

⎤⎥⎥⎥⎥⎥⎥⎦= Te

⎡⎢⎢⎢⎢⎢⎢⎣

b1

b2

a1

a2

⎤⎥⎥⎥⎥⎥⎥⎦

, where Te =⎡⎣ T11e T12e

T21e T22e

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

t11 t12 t13 t14

t21 t22 t23 t24

t31 t32 t33 t34

t41 t42 t43 t44

⎤⎥⎥⎥⎥⎥⎥⎦

. (2.46)

After applying (2.43) to (2.46) and solving the resulting set of linear equations we obtain

T11eS + T12e − SmT21eS − SmT22e = 0, (2.47)

where the model parameters contained in the submatrices of Te can be expressed in termsof the original parameters of the 16-term model as

T11e = S12e − S11eS−121eS22e, (2.48)

T12e = S11eS−121e, (2.49)

T21e = −S−121eS22e, (2.50)

T22e = S21e. (2.51)

The relationship (2.47) can further be brought to a very convenient form with the use ofmatrix vectorization operator [57]. For a matrix X represented as X=[x1, . . . , xN ], wherex1, . . . , xN are the columns of X, the vectorization operator is defined as [57]

vec(X) =

⎡⎢⎢⎢⎣

x1...

xN

⎤⎥⎥⎥⎦ . (2.52)

Consequently, the vector (2.52) consists of stacked columns of matrix X. Applying thisoperator to (2.47) and with the use of the identity vec (ABC) =

(CT ⊗ A

)vec (B), where

⊗ is the Kronecker product (see [57]), we obtain

(ST ⊗ I2×2

)t11e + t12e −

(ST ⊗ Sm

)t21e − (I2×2 ⊗ Sm) t22e = 0, (2.53)

22


where tije = vec (Tije), for i, j = 1, 2. This can be further transformed to

[ST ⊗ I2 I4 −ST ⊗ Sm −I2 ⊗ Sm

]⎡⎢⎢⎢⎢⎢⎢⎣

t11e

t12e

t21e

t22e

⎤⎥⎥⎥⎥⎥⎥⎦= 0, (2.54)

where I2 and I4 are identify matrices of size 2× 2 and 4× 4, respectively. Equation (2.54)forms a foundation for the 16-term VNA model identification [55, 56].

B. 8-term model. For modern VNAs, the internal cross-talk and coupling are usuallyvery small. Also, when performing VNA measurements with closed waveguides, such ascoaxial transmission line or rectangular waveguide, the external cross-talk is negligible4. In this case, the VNA reflectometers are electrically separated and the model (2.42)simplifies to the 8-term model [6, 29], referred to also as error-box model (see Fig. 2.3). Inthe 8-term model, the VNA reflectometers are represented as two linear two-port networksA and B, referred to as the error boxes. We shall first formulate definitions of thesenetworks, following a similar convention to that used in model (2.46), and then showanother formulation which stems from the basic form (2.42) of the 16-term model.

As the networks A and B are electrically separated, we can rewrite (2.46) as two setsof independent equations

⎡⎣ b1m

a1m

⎤⎦ = TA

⎡⎣ b1

a1

⎤⎦ , and

⎡⎣ a2

b2

⎤⎦ = TB

⎡⎣ a2m

b2m

⎤⎦ , (2.55)

and represent the measured and actual S-parameters, Sm and S, as the transmissionparameters, Tm and T, respectively, defined by

⎡⎣ b1

a1

⎤⎦ = T

⎡⎣ a2

bb

⎤⎦ , and

⎡⎣ a1m

b1m

⎤⎦ = Tm

⎡⎣ a2m

b2m

⎤⎦ , (2.56)

which immediately yieldsTm = TATTB, (2.57)

4In the case of VNA measurements involving open waveguides, such as in the case of on-wafer mea-surements or measurements employing fixtures with microstrip lines, the external cross-talk may becomesignificant.

23


��

Fig. 2.3: 8-term model of a two-port VNA.

and consequentlyT = T−1

A TmT−1B . (2.58)

Equations (2.57) and (2.58) constitute probably the most commonly used formulationof the 8-term model. Following a similar reasoning as in the case of the 16-term model, wecan show that out of total 8 complex terms in TA and TB, only 7 are necessary to describethe relationship between Tm and T. Therefore one of the terms in the 8-term model canbe arbitrarily chosen, for example fixed to one.

Parameters of the eight term model can be chosen in different ways. A simple choiceis to directly use the coefficient of matrices TA and TB. Another choice is to relate theparameters of the eight term model to the actual sources if systematic error in the VNAmeasurement. This can be done with the use of the flow graph in Fig. 2.4 [58]. Theterms in the graph correspond to the different sources of systematic errors in the VNAreflectometers. The second letter in the subscript, “F” or “R”, denotes position of theswitch, “forward” or “reverse”, respectively, in which the signal source is connected to thereflectometer. The individual terms correspond to the systematic errors resulting from

• finite directivity of the reflectometers (EDF and EDR),

• mismatch at the reflectometer input (ESF and ESR),

• reflection tracking of the reflectometers (ERF and ERR).

The additional terms α and β describe the asymmetry in the parameters of both reflec-tometers. However, since the 8-term model has 7 independent terms, only the ratio α/β

appears in the equations (2.57) and (2.58). Therefore the flow graph for the 8-term modelcan also be represented in an alternative form, by lumping the non-reciprocity of both errorboxes into one of them. In Fig. 2.5, we show such an alternative representation where thenon-reciprocity is lumped into the error box representing port two of the VNA. Similargraph can also be obtained by lumping the nonreciprocity into error box representing portone of the VNA.

24


Fig. 2.4: Flow graph for the 8-term VNA model.

Fig. 2.5: Alternative form of the flow graph for the 8-term VNA model.

With the use of the terms shown in Fig. 2.4, we can rewrite (2.57) as

Tm = 1Et

EATEB, (2.59)

where the transmission matrices Tm and T can be derived from the measured and actualDUT S-parameters with the use of (2.28) , while

EA =⎡⎣ ERF − EDF ESF EDF

−ESF 1

⎤⎦ , EB =

⎡⎣ ERR − EDRESR ESR

−EDR 1

⎤⎦ , (2.60)

andEt =

α

βERR. (2.61)

The appealing simplicity of (2.57) and (2.58) allows to describe the VNA calibrationproblem in a very concise and elegant way which has lead to numerous interesting results(see for example [30, 36]). However, formulation (2.57) and (2.58), has also an importantdisadvantage. After examining (2.28), we note that transmission matrix T cannot bedefined for a DUT that does not have a forward transmission, that is, when S21 = 0.Indeed, in such a case, pairs of variables a1, b1 and a2, b2 are unrelated and the transmissionmatrix T does not exist. Hence, matrix formulation (2.57) and (2.58) cannot provide auniform description for the VNA measurements of both two-port and one-port devices.

In order to describe the measurement of a one-port device with the use of matrices TA

25


and TB, we apply the definitions of reflection coefficient measurement on port A and B

ΓmA = b1m

a1m

, and ΓmB = b2m

a2m

(2.62)

to obtainΓmA = EDF + ERFΓA

1 − ESFΓA

, and ΓmB = EDR + ERRΓB

1 − ESRΓB

, (2.63)

where ΓA and ΓB are the reflection coefficients of the one-port device connected to theVNA port A and B, respectively. Equations (2.63) can be easily inverted to obtain thecorrection formulas.

An alternative formulation of the eight-term model that provides a uniform descriptionof both one-port and two-port measurements, can be derived from the basic form (2.42)of the 16-term model. Taking into account that the VNA reflectometers are electricallyseparated and including the definitions in Fig. 2.4, we can write submatrices of Se as

Se =⎡⎣ S11e S12e

S21e S22e

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

EDFβ

αERF

EDR ERR

α

βESF

1 ESR

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (2.64)

Applying then (2.64) to formulas (2.44) and (2.45), we obtain a uniform description ofboth one-port and two-port measurements

S11m = EDF + S11ERF

D− ERF ESRΔS

D, (2.65)

S22m = EDR + S22ERR

D− ERRESFΔS

D, (2.66)

S21m = S21α

β

ERR

D, (2.67)

S12m = S12β

α

ERF

D, (2.68)

whereΔS = S11S22 − S21S12, (2.69)

andD = 1 − ESF S11 − ESRS22 + ESF ESRΔS. (2.70)

26


C. Model parametrization choice. We showed in the previous section that the 16-termand the 8-term model can be represented in terms of different sets of parameters. Althoughthis different parametrizations are equivalent, special attention needs to be paid to theirproperties in the context of their application in VNA calibration algorithms. Importantproperty in this context is the uniqueness of the parametrization. Parametrizations havingthis property allow one to avoid the so called root choice problem in analytical VNA cali-bration methods, and improves the robustness of the iterative VNA calibration techniques.We shall discuss this issue in more detail for the 8-term model.

The primary parametrization we use for the 8-term model, which we refer to as thebase parametrization, results directly from (2.64). In this parametrization, we write thevector of VNA-model parameters as

p =[EDF , ESF , ERF , EDR, ESR, ERR,

α

β

]T

. (2.71)

By expanding equations (2.44) and (2.45) in terms of parameters in p, we can easily showthat these parameters describe a unique solution to (2.44) and (2.45). In other words, ifsome p solves equations (2.44) and (2.45), there is no other p′ �= p that also solves theseequations.

Parametrization (2.71), however, is not the common one encountered in the literature.Two other parametrizations that are often used are the reciprocal parametrization (see [36,59]) and the transmission parametrization (see [29, 30]). In the reciprocal parametrization,write the vector of VNA parameters is written as

pR =[EDF , ESF ,

√ERF , EDR, ESR,

√ERR,

α

β

√ERR

ERF

]T

=

= [EDF , ESF , etF , EDR, ESR, etR, k]T . (2.72)

This parametrization has a very convenient property that the joint effect of the nonreciproc-ity of both VNA error boxes is lumped into a single non-reciprocity factor k = α

β

√ERR

ERF.

Thus the VNA error boxes A and B are represented as reciprocal two-port linear networkswith S21A = S12A = etF and S21B = S12B = etR, respectively. For reciprocal error boxes(that is when α = ERF /α and β = ERR/β), we have k = 1, otherwise |k| �= 1. Conse-quently, adding a reciprocal linear network between the VNA error box and the DUT doesnot change k. This is not the case of the base parametrization, for which adding such a

27


linear network affects, in general, all of the parameters contained in (2.71).Parametrization (2.72) is, however, not unique. We can demonstrate that by investi-

gating the conversion of the base parametrization (2.71) into (2.72). To this end, we writeetF and etR as

etF = sRF

√ERF

+, and etR = sRR

√ERR

+, (2.73)

where√

x+indicates one of the square roots of the complex number x (e.g., with the positive

real part), and sRF = ±1 and sRR = ±1. Based on that, we rewrite (2.72) as

pR(sRF , sRR) =⎡⎣EDF , ESF , sRF

√ERF

+, EDR, ESR, sRR

√ERR

+,sRR

sRF

α

β

√ERR

+

√ERF

+

⎤⎦

T

. (2.74)

Consequently, we see that there are four different vectors pR(sRF , sRR) which lead to thesame vector p, depending on which square root we choose.

The transmission parametrization is based on the transmission matrix representation(2.59) of the 8-term model. In this parametrization, the vector of VNA parameters iswritten as

pT =[EDF , ESF , EDF ESF − ERF , EDR, ESR, EDRESR − ERR,

α

βERR

]T

=

= [EDF , ESF ,ΔF , EDR, ESR,ΔR, r]T . (2.75)

We can easily show that (2.75) can be uniquely derived from the base parametrization(2.71).

2.3.2 Modeling VNA nonstationarity

The 16-term and eight term model presented in the previous section capture the pri-mary systematic errors introduced in the VNA measurements. These models assume thatthe VNA does not change with time, which is essential for reliable identification of andcorrection for VNA systematic errors.

In practice, however, this assumption is not met. The primary reason for the VNAnonstationarity is the very manner in which the VNA operates, that is, by repeatedlyswitching the source generator and the matched termination between the two reflectometers(see Fig. 2.1). Due to the asymmetry of the switch, the VNA reflectometers see a slightlydifferent impedance of the source generator and of the matched termination in each position

28


(a) (b)

Fig. 2.6: Modeling changes of the matched termination impedance: (a) forward measure-ment, (b) reverse measurement.

of the switch. The difference between the source generator impedance in the two differentpositions of the switch does not lead to an error since the reflectometer parameters, ascaptured in the 16-term or eight term model, do not depend on this impedance [60].However, the change of the matched termination impedance leads to systematic errors[6, 58].

Other sources of the VNA nonstationarity are the nonrepeatability of the switch, thetest-set drift, imperfect connector repeatability and errors due to cable flexure. Apart fromthe test-set drift which is strongly dependent on the temperature and humidity changes,these errors are of a random nature and without any further knowledge about their char-acter they cannot be corrected for. We discuss those sources of VNA nonstationarity inmore detail in Subsection 3.3.2.

As to the changes of the matched-termination impedance, there exist two commonapproaches for modeling their impact of VNA-model parameters. In either approach, adifferent models is used for the VNA operating in the forward and reverse direction. Inthe 12-term model [6], additional terms are used to described the effect of the matched-termination variation whose values, however, are not directly related to the value of thematched-termination impedance seen through the switch. Another approach, which isillustrated in Fig. 2.6, extends the VNA models presented in the previous section by addingsome additional terms related directly to the impedance of the matched termination. Inthe forward and reverse positions of the switch, the matched termination presents differentreflection coefficient, ΓF and ΓR, respectively. This reflection coefficients are referred to asthe switch terms. Consequently, the measured parameters can be expressed as a functionof the time-invariant VNA measurement Sm and the switch terms. In particular, when the

29


switch is in the forward direction, the VNA measures

SF11m = b1F m

a1F m

and SF21m = b2F m

a1F m

, (2.76)

and when the switch is in the reverse direction, the VNA measures

SR22m = b2Rm

a2Rm

and SR12m = b1Rm

a2Rm

. (2.77)

With the use of these measurements, one can solve the linear equations resulting from theflow graphs in Fig. 2.6 to obtain [58]

Sm = 11 − SR

12mSF21mΓRΓF

⎡⎣ SF

11m − SR12mSF

21mΓF SR12m − SF

11mSR12mΓR

SF21m − SR

22mSF21mΓF SR

22m − SR12mSF

21mΓR

⎤⎦ . (2.78)

Matrix Sm obtained in that way is then used with the time-invariant VNA models presentedin Section 2.3.1.

The reflection coefficients ΓF and ΓR are typically measured in a separate step [58].The VNA ports are then directly connected so that the matched termination is excitedfrom the opposite port. This impedance is typically very stable [58], therefore repeatedmeasurement of the reflection coefficients ΓF and ΓR are not necessary.

2.4 Two-port VNA calibration techniques

VNA calibration procedures have been extensively studied in the literature and thereexists several good and detailed reviews, such as [6], or more recently [8]. Based on thosereviews, one concludes that these procedures differ from each other in many aspects, suchas the number and the type of calibration standards used, model of the VNA systematicerrors, mathematical formulation of the calibration problem, or the numerical methodused to solve it. Thus, VNA calibration procedures seem to form an inhomogeneous realmwith the only commonality being the high complexity of the mathematical description.Therefore, in this overview, instead of going deep into mathematical details, we shall takea higher-level look at the VNA calibration procedures and try to draw some similaritiesbetween them.

The VNA can be thought of as a system whose input and output are the actual scatter-

30

2.4. TWO-PORT VNA CALIBRATION TECHNIQUES

ing parameters of a device and their raw measurement, respectively (see Fig. 2.7). Denotingthe vector representation of the measurement and the actual S-parameters of the deviceunder test (DUT) with the vectors sm and s, respectively, and the VNA model parameters(calibration coefficients) with the vector p, we may model the VNA operation at a givenfrequency as a vector function

sm = f (s, p) , (2.79)

where the particular form of this function depends on the VNA mathematical model andthe parametrization choice (see Section 2.3).

Fig. 2.7: Representation of theVNA operation.

The objective of the VNA calibration procedure isnow to determine the parameters p of the VNA based onthe measurement of a number of devices—referred to ascalibration standards—with some known characteristics.The VNA calibration problem falls therefore into the gen-eral class of system identification problems [61, 62], alsoreferred to as inverse problems [63] or nonlinear regression problems [64, 65].

��

Fig. 2.8: VNA measurement of a calibration stan-dard.

In order to formulate this problemin a more precise way, consider Fig. 2.8.Let sm

n and sn denote the measurementand the actual parameters of the n-thcalibration standard, for n = 1, . . . , N ,where N is the number of calibrationstandards. Let further the actual S-parameters of the n-th calibration standard be represented by the function

sn = gn (c0n, cn) , (2.80)

where c0n and cn are known and unknown parameters of the n-th calibration standard,respectively. The particular form of the function gn depends on the physical model of thecalibration standard, and the choice of the known and unknown parameters.

The VNA calibration procedure can now be thought of as a system identification prob-lem in which one determines the vector p of VNA parameters and the unknown calibra-tion standard parameters {cn}N

n=1 based on the definitions {c0n}Nn=1 and raw measurements

{smn }N

n=1 of the calibration standards. This description will serve as a basis for the overviewof VNA calibration techniques. In this overview, we consider three different aspects of VNA

31


calibration techniques: mathematical formulation of VNA calibration problem, types ofcalibration standards, and numerical methods used to solve the calibration problem.

2.4.1 Formulation

A. Deterministic and statistical approach. In order to solve the VNA calibrationproblem, we may use either a deterministic or a statistical approach. The deterministicapproach is based on the following assumptions:

• the measurements and definitions of the calibration standards are accurate,

• the number of independent relationships describing calibration standard measure-ments is equal to the total number of estimated parameters (i.e. parameters con-tained in the vector p and the set of vectors {cn}N

n=1).

Consequently, in the deterministic approach, the parameters p and {cn}Nn=1 are obtained

by solving a set of—in general non-linear—equations. This equations are formed by (2.79)and (2.80), written for all of the calibration standards, that is,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

sm1 = f (g1(c01, c1), p)

...sm

N = f (gN(c0N , cN), p). (2.81)

Since the number of equations is assumed to be equal to the number of sought parameters,the calibration problem has either a unique or a finite number of solutions. We generallystrive to pose the VNA calibration problem such that it has a unique solution, however,due to its non-linear character, some ambiguity often cannot be avoided (e.g., the so-called“root-choice” problems [29, 30]).

The deterministic approach had long been the primary direction along which the devel-opment of VNA calibration procedures was taking place. The flag examples of determin-istic VNA calibration procedures are the Short-Open-Load (SOL) procedure for one-portVNAs, and the Short-Open-Load-Thru (SOLT)5 procedure for two-port VNAs [66]. Inboth procedures, “Short”, “Open” and “Load” stand for a one-port device realized as atransmission line section terminated with a short circuit, an open circuit, and a matched

5In the SOLT procedure, the number of equations is actually larger then the number of sought VNAparameters. These equations, however, are solved using deterministic methods (see [6, 66]), by neglectingsome pieces of information.

32


Fig. 2.9: VNA measurement of a calibration standard with errors.

termination, respectively. The “Thru” denotes either the direct or indirect (e.g., with aknown transmission line or an adapter) connection of the VNA ports.

The main advantage of the deterministic approach is that is allows to better understandthe relationships between the solution to the calibration problem and the number andtype of calibration standards. The rigorous analysis of equations (2.81) has therefore leadto many interesting results, such as the self-calibration methods (see [29, 30, 67, 68]).The Thru-Reflect-Line (TRL) procedure [29, 67] and the unknown-thru procedure [69],discussed in more detail in Paragraph 2.4.2-A, are the most representative examples ofthese methods.

The main disadvantage of the deterministic calibration methods is their sensitivity tomeasurement errors. Indeed, since the definitions and measurements of calibration stan-dards are assumed to be error free, any error in those values is directly mapped into anerror of the VNA calibration coefficients. Therefore, when calibrating the VNA with deter-ministic procedures, it is a good practice to perform an additional verification procedure, inorder to ensure that no serious measurement error occurred during the calibration. In theverification procedure, one either remeasures the calibration standards and compares theircorrected measurements with the definitions (e.g., in order to exclude a serious connectorrepeatability error), or, preferably, measures a set of check standards whose S-parametersdiffer significantly from calibration-standard S-parameters. By comparing the measure-ment of these standards with their definitions, both systematic and random errors in theVNA calibration procedure can be detected.

Statistical approach the VNA calibration problem allows to avoid the above problems.This approach is based on the assumptions that:

• the measurements and definitions of the calibration standards are corrupted witherrors (see Fig. 2.9),

• the number of independent relationships describing calibration standard measure-ments is larger than the total number of estimated parameters (i.e. parameters

33


contained in vector p and the set of vectors {cn}Nn=1).

Consequently, in the statistical approach, we obtain an overdetermined (redundant) set ofequations describing the calibration standard measurements, that is,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

sm1 = f (g1(c01, c1) + Δs1, p) + Δsm

1...

smN = f (gN(c0N , cN) + ΔsN , p) + Δsm

N

, (2.82)

where Δsmn and Δsn denote the error in the measurement and definition of the n-th cali-

bration standard, respectively. Since the measurements and definitions of the calibrationstandards are both corrupted with errors, a deterministic solution to this set of equationsdoes not exist (see Section B.2.2). In other words, we cannot find a set of VNA modelparameters p and unknown parameters of the standards {cn}N

n=1 that would simultane-ously solve all of the calibration equations, under the assumption that Δsm

n = Δsn = 0,for n = 1, . . . , N . Hence, instead of looking for a deterministic solution, we need to lookfor a solution that is in some sense optimal.

In the system identification, the process of finding such an optimal solution if referredto as the estimation of system parameters, and the optimal solution is referred to as theestimator of system parameters [61, 62, 64, 65]. The criteria for the solution optimality areformulated based on the statistical model of the measurement errors. The most commonlyused estimation technique is based on the maximum likelihood criterion [64, 65]. Thefundamental concepts of this technique in the context of the system identification problemsare reviewed in Appendix B. Examples of other estimation techniques are the method ofmoments or the Bayesian estimation [64, 65, 70].

The key concept in the estimation based on the maximum likelihood criterion is thelikelihood function. This function is defined based on the statistical properties of themeasurement errors (see Appendix B). In the context of the VNA calibration, two commoncases are when either only the calibration standard definitions or both the calibrationstandard definitions and the raw measurements are affected by errors. In both cases, thesought parameters of the VNA and calibration standards can be represented as a vector

β =

⎡⎢⎢⎢⎢⎢⎢⎣

pc1...

cN

⎤⎥⎥⎥⎥⎥⎥⎦

, (2.83)

34


where p denotes the VNA calibration coefficients, and {cn}Nn=1 are the vectors of unknown

parameters of the calibration standards.

The case when only the calibration standard definitions are affected by errors is oftenencountered in practice. Indeed, most modern VNAs have a very low noise floor, and offeradditionally various means of reducing the raw measurement noise, such as averaging ornarrowing the IF bandwidth. Consequently, the dominant error sources are the connectornonrepeatability, cable instability and the test-set drift. Although these error sources areconsidered as instrumentation errors, their impact can be thought of as a disturbanceadded to the calibration standard definitions. Considering that, we can rewrite (2.82) as

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f−1 (sm1 , p) = g1(c01, c1) + Δs1

...f−1 (sm

N , p) = gN(c0N , cN) + ΔsN

. (2.84)

These equations describe a variation of the system (B.6), in which both the system outputsand inputs depend on the estimated parameters. We can still, however, explicitly definethe residual errors as

⎧⎪⎪⎪⎨⎪⎪⎪⎩

r1 (β) = f−1 (sm1 , p) − g1(c01, c1)...

rN (β) = f−1 (smN , p) − gN(c0N , cN)

. (2.85)

Assuming now that the errors Δsn, for n = 1, . . . N , have a normally probability densityfunction (PDF), we can write the estimator of system parameters (see Section B.2.1) as

β = argminβ

N∑n=1

rn(β)T Σ−1Δsn

rn(β)=argminβ

r (β)T Σ−1Δs r (β) , (2.86)

where

r (β) =

⎡⎢⎢⎢⎣

r1 (β)...

rN (β)

⎤⎥⎥⎥⎦ , ΣΔs =

⎡⎢⎢⎢⎣

ΣΔs1. . .

ΣΔsN

⎤⎥⎥⎥⎦ , (2.87)

the matrix ΣΔsnis the covariance matrix of the errors in calibration standard definitions,

and the underline denotes the convention for real-valued representation of complex vectorsdescribed in Appendix A. We often assume that the covariance matrix is known up to ascaling factor, that is, ΣΔs = σ2VΔs, where VΔs is a known matrix and σ2 is the unknown

35


residual variance (square of the residual standard deviation). The estimate β can then beobtained by replacing ΣΔs with VΔs in (2.86) (see Section B.2.1).

In the case when also the raw measurement errors need to be accounted for, the VNAcalibration problem is solved with the errors-in-independent-variables approach (see Sub-section B.2.2). The equations describing the measurements take on the form which is avariation of (B.5), that is,

⎧⎪⎪⎪⎨⎪⎪⎪⎩

f−1 (sm1 − Δsm

1 , p) = g1(c01, c1) + Δs1...

f−1 (smN − Δsm

N , p) = gN(c0N , cN) + ΔsN

. (2.88)

Along with the parameters (2.83), we estimate then also errors in raw measurements, thusthe vector of sought parameters becomes

θ =

⎡⎢⎢⎢⎢⎢⎢⎣

β

Δsm1...

ΔsmN

⎤⎥⎥⎥⎥⎥⎥⎦

, (2.89)

and the residuals are⎧⎪⎪⎪⎨⎪⎪⎪⎩

r1 (θ) = f−1 (sm1 − Δsm

1 , p) − g1(c01, c1)...

rN (θ) = f−1 (smN − Δsm

N , p) − gN(c0N , cN). (2.90)

Following (B.51), we can then write the maximum likelihood estimate of (2.89) as

φ = argminφ

N∑n=1

rn(θ)T Σ−1Δsn

rn(θ) +N∑

n=1Δsm

n Σ−1Δsm

nΔsm

n =

= argminφ

r (θ)T Σ−1Δs r (θ) + (Δsm)T Σ−1

ΔsmΔsm, (2.91)

where

r (θ) =

⎡⎢⎢⎢⎣

r1 (θ)...

rN (θ)

⎤⎥⎥⎥⎦ , Δsm =

⎡⎢⎢⎢⎣Δsm

1...

ΔsmN

⎤⎥⎥⎥⎦ , ΣΔsm =

⎡⎢⎢⎢⎣

ΣΔsm1

. . .ΣΔsm

N

⎤⎥⎥⎥⎦ . (2.92)

36


Again, we often assume that both covariance matrices are known up to a scaling factor,that is, ΣΔs = σ2VΔs and ΣΔsm = σ2VΔsm , where VΔs and VΔsm are known matricesand σ2 is the unknown residual variance (square of the residual standard deviation). Thesolution is then obtained by replacing ΣΔs with VΔs and ΣΔsm with VΔsm in (2.91) (seeSubsection B.2.2).

The statistical approach to the VNA calibration has many advantages. The most im-portant one is that the use of redundant calibration standards reduces the uncertainty dueto random VNA measurement errors. Thus, unlike the deterministic approaches, statisticalVNA calibrations methods are less sensitive to serious measurement errors. Moreover, sucherrors can easily be detected through the residual analysis. The principle of this analysisis to determine how well the optimal solution to the VNA calibration problem describesthe individual measurements of calibration standards. This is discussed in more detail inSection 3.5.3.

The drawback of statistical approaches is that in general they require more computerresources and are more time consuming than deterministic approaches. This drawback,however, looses its significance considering the constantly increasing computational powerof personal computers.

Development of statistical approach to the VNA calibration has been initiated by someearly papers on the calibration of one-port VNAs [33, 71, 72] and six-port reflectometers[35, 73]. Newer contributions on the statistical calibration of one-port VNAs include [38,74–77]. In all of these approaches, apart from [75], the least-squares formulation (2.86) isused.

Statistical approach in the two-port VNA calibrations was started in [36, 55]. In theexcellent paper of Marks [36] a rigorous uncertainty analysis of the TRL calibration isshown, and based on that analysis a statistical technique for including measurements ofmultiple transmission lines with different lengths is proposed. Newer publications, suchas [56, 59], generalize this approach to the case of arbitrary two-port VNA calibrationalgorithms. The approach of [59] uses the general formulation (2.91).

B. Single-frequency and multi-frequency approach. A typical formulation of theVNA calibration problem, as described in the previous section, is developed at a singlefrequency. Such a formulation, however, becomes inadequate when some of the parametersestimated in the VNA calibration are frequency independent. Examples include the slidingload calibration of [37] with unknown positions of the sliding load, or the multi-reflect

37


through calibration of [38, 78] in which the lengths of the offset terminations are unknown.In these approaches, the multi-frequency formulation of the VNA calibration problem isemployed. In the following, we demonstrate the principle of this formulation for the generalmodel (2.88).

Let the model parameters be defined by the vector

λ =

⎡⎢⎢⎢⎢⎢⎢⎣

θf1...

θfK

α

⎤⎥⎥⎥⎥⎥⎥⎦

, (2.93)

where θfkare the parameters (2.89) at the frequency fk, for k = 1, . . . , K, and α is a

vector of frequency independent model parameters (those parameters are removed from{θfk

}Kk=1). Assuming that the errors are normally distributed, we estimate the parameters

(2.93) by considering the a likelihood function defined jointly at all frequencies. In the casewhen the statistical correlations between the errors at different frequencies are neglected,this leads to the minimization of a weighted sum of squared residuals at all measurementfrequencies (see B.2.2)

λ = argminλ

K∑k=1

rfk(θk, α)T Σ−1

Δsfk

rfk(θk, α) +

K∑k=1

(Δsm

fk

)TΣ−1

Δsmfk

Δsmfk

, (2.94)

where the single frequency covariance matrices ΣΔsfkand ΣΔsm

fkare defined in (2.87) and

(2.92), respectively. If the errors in raw measurement are neglected, (2.94) reduces furtherto

λ = argminλ

K∑k=1

rfk(θk, α)T Σ−1

Δsfk

rfk(θk, α) , (2.95)

which is the formulation whose slightly modified versions are employed in [37, 38]. If thereare no frequency independent model parameters in the vector α, solving (2.94) and (2.95) isobviously equivalent to solving the problem (2.91) and (2.86), respectively, independentlyat each frequency.

Reference [37] is, to the author’s knowledge, the earliest contribution on the statisticalVNA calibration employing the multi-frequency formulation. In this work, a sliding loadcalibration (see [6, 32]) is considered for which the positions of the load are unknown.These positions are then determined along with the VNA calibration coefficients based on

38


the model for the frequency-dependence of the sliding-load’s phase. Thus, the accuracyand robustness of the calibration is improved as the positions of the load need not to beprecisely controlled.

Reference [78] presents for the first time, to the author’s knowledge, a complete VNAcalibration based on the multi-frequency formulation. In this work, the multiple offset-short calibration (see [32, 34, 79]) is described in which the lengths of the offset shorts areunknown. Based on the model for the frequency-dependence of the offset-short reflectioncoefficient, these lengths are then determined along with the VNA calibration coefficients.Thus, the calibration accuracy is significantly increased as the systematic errors in thelength determination of the offset shorts are corrected for.

2.4.2 Standards

A. Partially unknown standards. The principle of the VNA calibration is to de-termine the VNA model parameters based on measurements of a number of calibrationstandards with known characteristics. The term “known characteristics” is intuitivelyunderstood as known numerically, and this understanding underlies the VNA calibrationmethods based on fully known standards, such as the SOLT method [6]. However, this termcan also be understood as known in terms of some relationships governing the standardscharacteristics. This alternative way of defining calibration standards is the foundation ofVNA calibration methods based on the self-calibration of standards.

The principle of self-calibration is to use the relationships between calibration-standardS-parameters as an additional piece of information in the VNA calibration. Then, as aside result of the VNA calibration procedure, some missing parameters of the calibrationstandards, encapsulated in the set {cn}N

n=1, are determined. From these parameters thenumerical value of these characteristics, as defined by {sn}N

n=1, can be further derived.A flag VNA calibration method employing the principle of self-calibration is the TRL

method [29, 67]. In this method, three different types of standards are used: a fully knowndirect thru connection, a transmission line with known length and unknown propagationconstant, and an unknown reflective standard for which we only assume that it is presentsidentical reflection coefficient on both VNA ports. Then, in the course of calibration, alongwith VNA model parameters, the unknown propagation constant of the line standard andthe unknown reflection coefficient of the reflect standard are determined. As a result, withthe use of TRL calibration, one can calibrate the two-port VNA to measure S-parame-ters with reference to an unknown characteristic impedance of the transmission line. This

39


impedance is then determined based on the propagation constant and line’s cross-sectionalparameters (see Paragraph 2.4.2-C).

A dual calibration with respect to the multiline TRL method is the multi-reflect thrucalibration of [38], in which at least three offset terminations and a thru connection areused. In this calibration the thru connection is also assumed to be fully known. As for theoffset terminations, we assume that their lengths are known and that these sections areformed out of a transmission line with the same propagation constant and characteristicimpedance. The reflection coefficient of the termination at the end of each offset section isassumed to be unknown but identical for all offset sections. Similarly to the TRL method,in the course of calibration, along with VNA model parameters, the unknown propagationconstant of the offset sections and the the reflection coefficient of the termination aredetermined. Also, an additional step is needed to determine the characteristic impedanceof the offset sections.

A generalization of the self-calibration principle for the two-port VNA calibration meth-ods is discussed in [30], where other self-calibration methods, such as the “Line”-”Reflect”-”Match” (TRM) are proposed. In this method a fully known matched-termination standardis used instead of a line standard. The extension of the TRM method is the LRM methods(thru replaced with a transmission line) and the LRRM method (some parameters of thematch standard are determined in the process of self-calibration) [80]. Also a very inter-esting implementation of the self-calibration principle is the “unknown” thru method [69]which uses as a calibration standard a reciprocal but otherwise unknown two-port, referredto as the “unknown” thru.

The main advantage of the self-calibration methods is that they require less informationabout the calibration standards. Also, since this information is expressed in terms of somephysical relationships between the standards characteristics instead of numerical values,self calibration methods are, in general, less prone to measurement errors than the methodsemploying fully defined calibration standards. Consequently, self-calibration methods incombination with the statistical approach (e.g. [36–38]), lead to the most accurate VNAcalibration techniques.

B. Primary and transfer calibration standards. Primary calibration standards aredefined through their physical models and some fundamental parameters, such as geo-metrical dimensions. Transfer calibration standards are defined directly in terms of theirS-parameters which are obtained either from measurements or electromagnetic simulations.

40


Examples of primary calibration standards are the transmission line in the TRL calibra-tion [29, 67] or an offset termination in the multi-reflect thru calibration [38]. Anotherinteresting primary standard is the direct thru connection of VNA ports. This standard isonly applicable in connectorized measurements, such as with the rectangular waveguides orcoaxial transmission lines. An example of a transfer standard is the matched terminationfor which we usually do not have a physical model.

C. Calibration reference impedance. The objective of VNA calibration is to deter-mine VNA model parameters such that the corrected DUT S-parameters are expressedwith respect to some known reference impedance Zref , usually 50 Ω. In the VNA cali-bration methods which use fully defined calibration standards, this reference impedanceis set by the reference impedance with respect to which S-parameters of the calibrationstandards are given. Clearly, this reference impedance needs to be identical for all of thecalibration standards, otherwise a systematic error is introduced.

However, when using self-calibration methods, this reference impedance is often un-known. This is the case of self-calibration methods which use sections of a transmissionlines with an unknown characteristic impedance Z0 as a part of calibration standards (e.g.,multiline TRL of [36] and multi-reflect thru method of [38]). We can easily show thatthe VNA calibrated with these methods measures DUT S-parameters with respect to anunknown characteristic impedance Z0 of the transmission line. Therefore an additionalprocedure is required for determining Z0 so that DUT S-parameters can be expressed withrespect to a typical reference impedance, such as 50 Ω.

Much has been written on the issue of measuring the characteristic impedance of trans-mission lines, in particular in the case of on-wafer calibrations [81, 82]. For low-loss trans-mission lines, which we deal with here, there exist two simple methods. In these methods,one assumes the line has small dielectric losses and its capacitance per unit length doesnot significantly change with frequency. The characteristic impedance can then be approx-imately written as [81]

Z0 =γ

jωC0, (2.96)

where γ is the propagation constant of the lines determined in the calibration, and C0

is the quasi-static capacitance per-unit-length of the lines. This capacitance can thenbe determined either based on dimensional parameters of the line, such as in the case ofprecision coaxial air-dielectric lines [83], or based on the reflection coefficients measurementof a resistor with known DC resistance [82]. With the use of the relationship (2.19), the

41


reference impedance of corrected DUT S-parameters can then be reset from Z0 to a desiredvalue.

2.4.3 Solution

VNA calibration methods can also be classified with respect to the mathematical meth-ods used to solve the VNA calibration problem. All of the deterministic methods and someof the statistical methods (such as [36]) use a close-form analytical solutions. Most of thestatistical methods, such as [33, 37, 38, 56, 59], use iterative algorithms which are usuallydifferent variations of the nonlinear least-squares minimization.

The main disadvantage of the iterative solution of VNA calibration problem, as withall optimization-based methods, is that it might not find a global minimum of the costfunction. Therefore special attention needs to be paid to a robust problem parametrization(see Section 2.3.1-C), and a good choice of the starting point.

42

Chapter 3

Overview of uncertainty analysis forVNA S-parameter measurements

The word ’chance’ then expresses only ourignorance of the causes of the phenomenathat we observe to occur and to succeedone another in no apparent order. Proba-bility is relative in part to this ignorance,and in part to our knowledge.

Pierre Simon de Laplace

Although the VNA calibration and correction procedure allows to remove the majorsystematic errors in the VNA measurement, corrected S-parameter measurements are stilldeteriorated with some residual measurement errors. These errors are caused by the im-perfection of the VNA calibration, and by errors in raw VNA S-parameter measurements(see Section 3.1).

The measurement error is in general quantified with the measurement uncertainty. Inthe case of scalar measurements, there exist well established methods for representationand evaluation of the measurement uncertainty [84, 85]. However, these methods cannotbe directly applied to complex- and matrix-valued measurand, such as the S-parameters.Therefore, multiple conventions for representing the measurement uncertainty in S-param-eters, extending the methods presented in [84, 85], are used. We review these conventionsin Section 3.2.

43

3. OVERVIEW OF UNCERTAINTY ANALYSIS FOR VNA MEASUREMENTS

When evaluating the measurement uncertainty in corrected VNA S-parameter mea-surements, we typically proceed in three steps. We first develop statistical models forphenomena responsible for the imperfection of the VNA calibration and for the errors inraw VNA measurements (see Section 3.3). With the use of different uncertainty analysismethods (see Section 3.5 and Section 3.6), these errors are the mapped into equivalenterrors in the VNA calibration coefficients. These equivalent errors, typically representedwith VNA residual-error model (see Section 3.4), are then propagated into the resultingerrors in corrected VNA S-parameter measurements.

3.1 Sources of error in corrected VNA S-parametermeasurements

The VNA calibration procedure allows to characterize and subsequently remove thesystematic errors of the VNA test-set. The resulting corrected VNA S-parameter mea-surements are, however, still subject to some measurement errors. These errors resultsfrom inaccurate calibration-standard definitions and from VNA instrumentation errorsthat occur during the measurement of both the calibration standards and the device undertest (DUT).

Errors in the definitions of calibration standards can be classified into two groups:numerical errors and inconsistency errors. Numerical errors are pertinent in the case ofcalibration standards that are treated as fully known. These errors result from the limitedaccuracy with which the values of some parameters of the calibration standard were deter-mined. Examples here are uncertainties in the length measurement of transmission linesin the TRL calibration, or the non-zero reflection coefficient of the matched terminationstandard in the SOLT calibration

Inconsistency errors stem from the violation of some assumptions made as to the charac-teristics of one or more calibration standards. These errors occur in the case of calibrationstandards to which we apply the self-calibration principle (see Paragraph 2.4.2-A), thatis, calibration standards that are partially unknown. Examples here are differences in thecharacteristic impedance of the lines or a variation in the connector interface discontinuityin the coaxial multi-line TRL calibration.

VNA instrumentation errors can be divided into two groups: nonstationarity errorsand receiver errors. The nonstationarity errors manifest themselves as changes of VNAelectrical parameters with time. These changes are caused by the VNA test-set drift

44

3.2. STATISTICAL DESCRIPTION OF S -PARAMETER MEASUREMENT ERRORS

(primarily due to temperature and humidity changes), bending of the connecting cables,and connector nonrepeatability.

The VNA receiver errors result from the noise and nonlinear effects in the measurementof the waves a1m, b1m, a2m, and b2m (see Fig. 2.1). The noise in these measurements isusually classified into low-level noise and high-level noise (also referred to as the tracejitter) [19]. The low-level noise results from the wideband thermal noise in the VNAreceivers. The high-level noise is caused by the narrowband thermal noise in the IF partof the receiver, and by the phase-noise of the LO.

The nonlinearity errors are caused by the nonlinearities of the VNA receivers. In mod-ern VNAs, the impact of these nonlinearities is minimized through some corrections andadaptive signal level adjustment. Therefore, in typical operating conditions, the nonlin-earity errors can be neglected [19, 86].

3.2 Statistical description of S-parameter measure-ment errors

Scattering-parameters measurement forms a complex-valued matrix. Uncertainty eval-uation for a measurement represented in such a form is very inconvenient, since standardstatistical methods operate on real-valued vector random variables [87]. Therefore, in thecontext of the uncertainty analysis, we typically represent S-parameter measurements asa real-valued vectors and describe their uncertainty with a covariance matrix [40, 42]. Inthe following, we review this representation starting with a general statistical model forS-parameter measurements, and following with the details of this model for the case ofmeasurement of a single S-parameter and a matrix of S-parameters.

3.2.1 Statistical model for S-parameter measurement

Our statistical model for S-parameter measurement is based on the measurement modelfor vector-valued measurands (see [88]). We represent the S-parameter measurement asvector by simply stacking the S-parameters in some arbitrary order. One preferred choiceof this order follows from the vectorization operator (2.52), and will be discussed in moredetail in Subsection 3.2.3.

We define a single frequency measurement s of S-parameters of N -port device as a sum

45


of the true value s of the S-parameters and the measurement error Δs

s = s +Δs, (3.1)

where the underline denotes the real-valued vector representation of a complex-valuedvector defined by (A.2). We assume further that the true value s is constant, and that themeasurement error Δs is a vector of 2N2 random variables

Δs = [Δs1, . . . ,Δs2N2 ]T . (3.2)

The statistical properties of the joint probability density function (PDF) fΔs(x) of themeasurement error Δs are discussed in the following sections. For now we only assumethat the measurement error has zero expectation value, that is, E (Δs) = 0, and henceE (s) = s.

3.2.2 Error description for a single S-parameter

For a single scattering parameter Sij, the model (3.1) reduces to

Sij = Sij +ΔSij, (3.3)

where

ΔSij =⎡⎣ ReΔSij

ImΔSij

⎤⎦ , (3.4)

is the real/imaginary part representation of the measurement error. This representationis the most fundamental description of errors in single S-parameter measurements. Itusually very well corresponds to the underlying physical origins of the measurement errorand leads to an intuitive statistical description [42]. However, this representation doesnot have a straightforward physical interpretation. Indeed, when interpreting S-parametermeasurement, instead of the real and imaginary part, we rather talk about the magnitudeand phase which correspond to the simple physical concepts of the attenuation and phaseshift. Therefore, the uncertainty in S-parameters is often alternatively represented interms of errors in magnitude and phase. However, the unapparently complicated statisticalproperties of this representation may lead to incorrect uncertainty estimates [42]. Wediscuss these properties below and show that the in-phase/quadrature error representationoffers a combination of the statistical simplicity of the real/imaginary part representation

46

3.2. STATISTICAL DESCRIPTION OF S-PARAMETER MEASUREMENT ERRORS

and straightforward physical interpretation of the magnitude/phase representation.

��

Fig. 3.1: Error representationfor complex measurands.

The statistical properties of all the representation canbe explained based on the graph shown in the Fig. 3.1. Inthis graph we show a complex-valued measurand z thatis perturbed with a measurement error Δz. We assumethat Δz has a two-dimensional PDF defined in termsof its real and imaginary part with EΔz = 0, and thecovariance matrix is

ΣRI =⎡⎣ Var (ReΔz) Cov (ReΔz, ImΔz)Cov (ReΔz, ImΔz) Var (ImΔz)

⎤⎦ .

(3.5)As a result of the perturbation Δz, we obtain a new valuez′ = z +Δz. A change in the magnitude and phase of z

can be easily derived as

Δ|z| = |z′| − |z| =∣∣∣|z| +ΔzI + jΔzQ

∣∣∣− |z|, (3.6)

andΔθ = arg z′ − arg z = arctan ΔzQ

|z| +ΔzI

, (3.7)

where ΔzI and ΔzQ and are the in-phase and quadrature error components, respectively,defined as

ΔzI = cos θReΔz + sin θ ImΔz, (3.8)

andΔzQ = − sin θReΔz + cos θ ImΔz. (3.9)

We see that the transformation between the real/imaginary and magnitude/phase repre-sentation is in general nonlinear. In order to better understand statistical properties of thistransformation, we shall investigate two limit cases, that is, when |Δz| � |z| and when|Δz| ≈ |z|.

The case when |Δz| � |z| corresponds the situation when a measurement of a largereflection coefficient (e.g., short or open), or a large transmission (e.g., line or thru) issubject to small measurement errors. Equations (3.6) and (3.7) can then approximatelybe written as

Δ|z| ≈ ΔzI , (3.10)

47


andΔθ ≈ ΔzQ

|z| . (3.11)

In this situation, the relationship between all the error representations is linear. We canthen write the covariance matrix of vector ΔzMA = [Δ|z|,Δθ]T as

ΣMA = M(|z|)R(θ)ΣRIR(θ)T M(|z|)T , (3.12)

where

R(θ) =⎡⎣ cos θ sin θ

− sin θ cos θ

⎤⎦ , (3.13)

and

M(|z|) =⎡⎣ 1

1/|z|

⎤⎦ . (3.14)

The case |Δz| ≈ |z| corresponds to the situation when measuring devices with a smallreflection coefficient (e.g. matched termination), or small transmission (e.g. attenuator)with large measurement errors. In this case, equations (3.6) and (3.7) take on a highlynonlinear form. The statistical distribution of Δ|z| and Δθ has then complicated properties,hence statistical inference based on Δ|z| and Δθ for |Δz| ≈ |z| requires special care [42].To illustrate that, consider the case when |z| ≈ 0. We obtain then

Δ|z| = |z′| − |z| ≈ |ΔzI + jΔzQ| , (3.15)

andΔθ = arg z′ − arg z ≈ arctan ΔzQ

ΔzI

. (3.16)

Now, if ΣRI corresponds to a Gaussian distribution with E (Δz) = 0, then Δ|z| has aRayleigh distribution [42]. This distribution is defined only for positive values, therefore weobtain E (Δ|z|) > 0 for E (Δz) = 0. Consequently, for example, when averaging magnitudeof matched termination measurements pertubed with noise, we would counterintuitivelyobtain a non-zero value.

As an alternative to the magnitude/phase representation, the in-phase/quadrature rep-resentation has been proposed [89]. This representation is defined by a fixed linear trans-formation (3.8) and (3.9) of the real/imaginary representation. The covariance matrix for

48

3.2. STATISTICAL DESCRIPTION OF S-PARAMETER MEASUREMENT ERRORS

vector ΔzIQ = [ΔzI , ΔzQ]T can be written as

ΣIQ = R(θ)ΣRIR(θ)T . (3.17)

As equations (3.10) and (3.11) demonstrate, for |Δz| � |z| the in-phase/quadrature repre-sentation has a similar interpretation as the magnitude/phase representation. The in-phasecomponent corresponds then directly to the magnitude change, and the quadrature com-ponent is the length of an arc corresponding the Δθ rotation of vector with magnitude|z|. As opposed to the magnitude/phase representation, however, the in-phase/quadraturerepresentation maintains its linear relationship to the real/imaginary representation for|Δz| ≈ |z|.

3.2.3 Error description for a matrix of S-parameters

In the case of matrix of S-parameters, the measurement uncertainty is represented witha covariance matrix containing 2N2 real elements where N is the number of devices ports.This matrix consists of the variances of the uncertainty components (i.e. real and imaginarypart, magnitude and phase, or in-phase and quadrature part) of individual S-parametersand covariances of all possible pairs of these components [40–42].

The particular structure of this covariance matrix depends on the ordering of real andimaginary part of S-parameters in the vector representation of S-parameters. The mostcommon convention, which we adopted in this work, is based on the vectorization operatorvec (·) [57]. To illustrate this convention, we apply the definition (2.52) of this operator toa two-port scattering matrix S of size 2× 2, to obtain a vector with four complex elements

s = vec (S) = vec

⎛⎝⎡⎣ S11 S12

S21 S22

⎤⎦⎞⎠ =

⎡⎢⎢⎢⎢⎢⎢⎣

S11

S21

S12

S22

⎤⎥⎥⎥⎥⎥⎥⎦

. (3.18)

The real-valued representation of complex variables in s (such as with the real/imaginary ormagnitude/phase components) can be obtained in multiple ways. One common convention,defined by (A.2), expands each element of the complex vector into its real and imaginarypart, that is, for (3.18) we obtain

s =[ReS11 ImS11 ReS21 ImS21 · · · ReS22 ImS22

]T. (3.19)

49


With the use of (3.19), we can apply the definition of the covariance matrix (see [87]) tothe measurement error Δs in the vector s, obtaining

ΣΔs =

⎡⎢⎢⎢⎢⎢⎢⎣

Var (ReΔS11) Cov (ReΔS11, ImΔS11) · · · Cov (ReΔS11, ImΔS22)Cov (ImΔS11,ReΔS11) Var (ImΔS11) · · · Cov (ImΔS11, ImΔS22)

... ... . . . ...Cov (ImΔS22,ReΔS11) Cov (ImΔS22, ImΔS11) · · · Var (ImΔS22)

⎤⎥⎥⎥⎥⎥⎥⎦

.

(3.20)Conversion to the in-phase/quadrature representation can easily be obtained as

ΣIQΔs =

⎡⎢⎢⎢⎢⎢⎢⎣

R(θ11)R(θ21)

R(θ12)R(θ22)

⎤⎥⎥⎥⎥⎥⎥⎦

ΣΔs

⎡⎢⎢⎢⎢⎢⎢⎣

R(θ11)R(θ21)

R(θ12)R(θ22)

⎤⎥⎥⎥⎥⎥⎥⎦

T

,

(3.21)where matrices R(θij) are defined by (3.13) and θij = argSij.

3.3 Statistical models for errors in corrected VNA S-parameter measurements

In this section we briefly review statistical models used to describe errors in VNAS-parameter measurements. Such models are used in the statistical methods for VNAcalibration to determine the appropriate weighting functions for the residual errors (seeParagraph 2.4.1-A) and are a prerequisite for the evaluation of uncertainties in correctedVNA S-parameter measurements.

In Section 3.1, we divided the errors in corrected VNA S-parameter measurementsaccording to their origin, that is, into the calibration standard errors and VNA instru-mentation errors. In the context of the statistical description of these errors, however,it is more convenient to talk about the systematic and random errors. Systematic errorsdo not change between experiments and result from the limited accuracy with which weone can determine parameters of the measurement set up. Random errors change betweenexperiments and result from random variations of some parameters of the measurementset up. In the case of VNA S-parameter measurements, systematic errors are primar-ily related to inaccuracy of the calibration-standard definitions or their inconsistency (see

50

3.3. STATISTICS OF ERRORS IN CORRECTED VNA MEASUREMENTS

Subsection 3.1). The random errors result mainly from VNA instrumentation errors, suchas connector nonrepeatability, cable flexure, and the test-set drift 1. In the evaluationof random errors, we typically use statistical methods based on repeated measurements(Type A uncertainty evaluation [84]) to determine these properties, while in the case ofsystematic errors, other methods are used (Type B uncertainty evaluation [84]).

3.3.1 Systematic errors

As already mentioned, systematic errors in corrected VNA S-parameter measurementsare primarily related to inaccuracies of the calibration-standard definitions. In the case ofprimary calibration standards (see Paragraph 2.4.2-B), their definitions are typically devel-oped based on the physical description of the calibration standards and their dimensionaland material parameters [90]. The uncertainties of these parameters are then propagatedinto the uncertainties of the S-parameter definitions with the use of either the linear errorpropagation (e.g. [36, 41] ) or the Monte Carlo approach (e.g. [77]). In the case of transferstandards, the systematic errors in their S-parameter definitions are derived directly fromthe uncertainties of the measurement technique used to characterize them. Often times, inparticular when specifying the accuracy of the calibration standards, these uncertaintiesare expressed in a simplified fashion, for example, by use of the magnitude/phase uncer-tainty representation, by neglecting the statistical correlations between uncertainties, or byapproximating their statistical properties with the circular-normal PDF (see Section B.6).

3.3.2 Random errors

A. Connector repeatability and cable instability. The impact of connector nonre-peatability and cable instability errors is usually described jointly with a single statisticalmodels. This model typically assumes the circular-normal PDF and statistical indepen-dence of the errors at different frequencies [56, 71, 76]. It is also typically assumed thatthe connector repeatability and cable instability errors affect independently each of thescattering parameters [56].

This typical approach, however, has no justification in the physics of the connectorrepeatability and cable instability errors. Reference [91] shows that the connector re-peatability errors exhibit a very regular frequency dependence, therefore errors at different

1Position of the center conductor in coaxial air-lines, or the plunger reflection coefficient in the slidingload and sliding short are exceptions here: these errors are random error but affect the calibration standarddefinition.

51


frequencies cannot be assumed statistically independent. Reference [92] develops a sim-ple equivalent circuit describing the connector repeatability errors in terms the electricalparameters of the connector discontinuity. Subsequently, it shows that the connector re-peatability errors strongly depend on the reflection coefficient the DUT. Reference [93]demonstrates also that the PDF of the connector repeatability errors, as determined fromrepeated reflection-coefficient measurements, has typically a PDF with one of the variancesmuch smaller then the other. Consequently, some calibration approaches (e.g., [75]) usedrepeated measurements to determine the full covariance matrix of the connector repeata-bility errors.

Another description of the connector repeatability errors, applicable primarily when de-termining the weights in the statistical VNA calibration methods, has been proposed in [36].This approach approximates the connector interface variation with a linear two-port withrandom S-parameters described by the circular-normal PDF. The connector repeatabilityerrors in the actual measurements of a two-port device are then determined by attachingsuch a connector-error two-port at either end of the device. Thus, the resulting error inthe DUT S-parameters due to the connector nonrepeatability depends also on the DUTS-parameters themselves, which is often observed in the actual measurements [91]. Conse-quently, the approach of [36] allows also to capture the relationships between the connectorrepeatability errors in different calibration standards, and thus to more accurately specifythe weighting matrices in the statistical calibration methods (see Paragraph 2.4.1-A).

B. Test-set drift. The impact of the VNA test-set drift has rarely been treated inthe literature. The approach of Reference [94] (see Subsection 3.5.2) is often used toapproximately quantify the impact of the VNA test-set drift. Another method, suggestedby Reference [95], uses repeated measurements of different verification standards. Basedon those measurements, the worst case estimate of errors due to the test-set drift is derived.

C. Receiver noise and nonlinearity. Similarly to the VNA test-set drift, the VNAreceiver errors are rarely considered. Reference [56] suggests that the VNA receiver noisecan be approximated with the circular-normal PDF which seems to well correspond to theactual operation of the VNA receivers. Reference [56] proposes also a simple model for thedependence of the the VNA receiver errors on the raw S-parameters. This dependence isthe consequence of the fact, that the raw S-parameters are obtained as ratios of complexvoltages measured by the VNA receivers. Following that reasoning, Reference [96] suggests

52

3.3. REPRESENTATION OF ERRORS IN CORRECTED VNA MEASUREMENTS

that the VNA receiver errors should be characterized directly in terms of VNA receivervoltages instead of raw S-parameter measurements.

Regarding the receiver nonlinearity errors, Reference [86], based on an experimentalstudy, develops an approximate method to predict the impact of receiver nonlinearities.It also points out that special attention needs to be paid to the linear operation of VNAreceivers as the nonlinear effects can nonintuitively alter the corrected S-parameter mea-surements.

3.4 Representation of errors in corrected VNA S-parameter measurements

In this section, we present a mathematical model for errors in corrected VNA S-param-eter measurements. This model consists of two components: description of errors causedby the imperfection of the VNA calibration (errors in VNA calibration coefficients) anddescription of errors due to noise and nonlinearities of VNA receivers (errors in raw VNAmeasurements). We begin with an overview of how the errors discussed in Section 3.1 affectthe VNA S-parameter measurements (see Subsection 3.4.1). Following on that, we presenta detailed analysis of how errors in VNA calibration coefficients (Subsection 3.4.2) and inVNA raw measurements (Subsection 3.4.3) propagate into the corrected VNA S-parametermeasurements.

3.4.1 Overview

We illustrate sources of error in corrected VNA S-parameter measurements in Fig. 3.2.In this figure we show the flow graph of the 8-term VNA model (compare Fig. 2.5) withadditional terms related to errors discussed in Section 3.1. We assume that errors relatedto the switch operation (see Subsection 2.3.2) and cross-talk between VNA ports are eithernot present or have been perfectly corrected for. This is a viable approximation, since theerrors introduced by the switch are typically very stable (see Subsection 2.3.2), while thecross talk between the VNA ports is neglegible in modern VNAs (see Paragraph 2.3.1-B).

In Fig. 3.2 the error sources are represented with two sets of parameters. The firstset describes the errors in the determination of VNA calibration coefficients and can be

53


Fig. 3.2: Basic error model for corrected two-port VNA S-parameter measurements.

written as a vector of perturbations

Δp =[ΔEDF ,ΔESF ,ΔERF ,ΔEDR,ΔESR,ΔERR,Δ

(α

β

)]T

, (3.22)

added to the nominal set of VNA calibration coefficients (2.71). The second set describeserrors in VNA raw measurements and is represented as an additional matrix

η =⎡⎣ η11 η12

η21 η22

⎤⎦ , (3.23)

added to the raw measurement (2.44).

3.4.2 Errors in VNA calibration coefficients

For the eight term model (see Subsection 2.3.1-B) with the basic parametrization de-fined by (2.71), we can determine the effect of these errors by analyzing the sensitivity of(2.45) to changes described by (3.22). This is, however, inconvenient, as it requires takingderivatives of (2.45) with respect to submatrices defined by (2.64). Therefore we follow adifferent approach, based on the residual error-box representation (see [6]).

In this representation, one uses a set of additional linear two-port networks, referredto as residual error-boxes, that are attached to the actual error boxes representing VNAcalibration coefficients. The residual error-boxes can be thought of as a set of VNA cali-bration coefficients that one would obtain in an error-free calibration, performed after theactual calibration. Thus, we referr to the parameters of the residual error-boxes as theeffective VNA parameters.

With the use of residual error-boxes, one can then easily determine the errors in bothcorrected VNA S-parameter measurement and VNA calibration coefficients, by attaching

54


(a) (b)

Fig. 3.3: Error models for corrected one-port VNA S-parameter measurements: (a) basicmodel, (b) residual error-box representation.

the residual error-boxes to the network representing the DUT or the actual VNA error-boxes, respectively. We first discuss the residual-error box representation for the case ofone-port VNA measurements and then show its extension to the the two-port case.

A. One-port measurement. Flow graph for a one-port VNA with errors in calibrationcoefficients and raw measurement and the equivalent residual error-box representation areshown in Fig. 3.3. Since the reflection coefficient measurement depends only on the productof forward and reverse transmission through the error box (ERF and ERR in Fig. 2.4), welump this product into a single parameter ER.

In the flow graph in Fig. 3.3a, parameters ΔED, ΔER, and ΔES denote errors in thedetermination of VNA calibration coefficients and η corresponds to the overall error inone-port-VNA raw measurement. The residual error-box representation in Fig. 3.3b allowsto easily determine the error in corrected reflection coefficient measurements as

ΔΓ = rD + (1 + rR)2 Γ1 − rSΓ

− Γ ≈ rD + 2rRΓ + rSΓ2. (3.24)

The parameters rD, rS, and rR are referred to as the effective directivity, effectivesource-match, and effective tracking of the calibrated one-port VNA. In order to determinethese parameters, we compare the two flow graphs in Fig. 3.3. We assume the that errorsΔED, ΔER, and ΔES are small which, after applying the standard methods for flow-graph

55


Fig. 3.4: Residual error-box representation of the error model for corrected two-port VNAS-parameter measurement.

analysis, leads to a first-order approximation

ΔED ≈ ERrD, (3.25)ΔES ≈ rS + 2ESrR + E2

SrD, (3.26)ΔER ≈ 2ERrR + 2ERESrD, (3.27)

which after some straightforward manipulations yields

rD ≈ ΔED

ER

, (3.28)

rR ≈ 12ΔER

ER

− ES

ER

ΔED, (3.29)

rS ≈ ΔES − ES

ER

ΔER + E2S

ER

ΔED. (3.30)

In the case when VNA systematic errors are small, that is, |ES| ≈ 0 and |ES/ER| ≈ 0,we can further simplify these expressions to obtain

rD ≈ ΔED

ER

, (3.31)

rR ≈ 12ΔER

ER

, (3.32)

rS ≈ ΔES. (3.33)

This last set of expression shows that in the case when the VNA systematic errors aresmall, the changes in VNA calibration coefficients can be represented as an additionallinear reciprocal two-port, whose parameters depends only on those changes and the singleVNA calibration coefficient ER.

56


B. Two-port measurement. Flow graph of the residual error-box representation ofthe error model from Fig. 3.2 is shown in Fig. 3.4. This representation is defined bya forward (parameters rDF , rSF , rRF ) and reverse (parameters rDR, rSR, rRR, and rK)residual error-box. The parameters rDF , rSF , rRF are referred to as the forward effectivedirectivity, forward effective source-match, and forward effective tracking, respectively,while the parameters rDR, rSR, rRR, are the reverse effective directivity, reverse effectivesource-match, and reverse effective tracking, respectively. The parameter rK is referredto as the effective non-reciprocity factor rK . We decided to lump the this factor into thereverse residual error-box, however, it can be also added to the forward residual error-box.

Based on the flow graph in Fig. 3.4 and under the assumption that the errors are small,we can determine errors in corrected S-parameter measurements as

ΔS11 ≈ rDF + 2rRF S11 + rSF S211 + rSRS21S12, (3.34)

ΔS22 ≈ rDR + 2rRRS22 + rSRS222 + rSF S21S12, (3.35)

ΔS21 ≈ (rRF + rRR + rK + rSF S11 + rSRS22)S21, (3.36)ΔS12 ≈ (rRF + rRR − rK + rSF S11 + rSRS22)S12. (3.37)

In order to obtain parameters of the residual error-boxes, we compare the flow graphsin Fig. 3.2 and Fig. 3.4. We assume the that errors in (3.22) are small which, after applyingthe standard methods for flow-graph analysis, leads to first-order approximations

rDF ≈ ΔEDF

ERF

, (3.38)

rRF ≈ 12ΔERF

ERF

− ESF

ERF

ΔEDF , (3.39)

rSF ≈ ΔESF − ESF

ERF

ΔERF + E2SF

ERF

ΔEDF , (3.40)

and

rDR ≈ ΔEDR

ERR

, (3.41)

rRR ≈ 12ΔERR

ERR

− ESR

ERR

ΔEDR, (3.42)

rSR ≈ ΔESR − ESR

ERR

ΔERR + E2SR

ERR

ΔEDR, (3.43)

57


with

rK = β

αΔ(

α

β

)+ 12ΔERR

ERR

+ ESR

ERR

ΔEDR. (3.44)

In the case when VNA systematic errors are relatively small, that is, |ESF | ≈ 0, |ESR| ≈0, |ESF /ERF | ≈ 0, and ESR/ERR| ≈ 0, we can further simplify these expressions to obtain

rDF ≈ ΔEDF

ERF

, (3.45)

rRF ≈ 12ΔERF

ERF

, (3.46)

rSF ≈ ΔESF . (3.47)

and

rDR ≈ ΔEDR

ERR

, (3.48)

rRR ≈ 12ΔERR

ERR

, (3.49)

rSR ≈ ΔESR, (3.50)

with

rK = β

αΔ(

α

β

)+ 12ΔERR

ERR

. (3.51)

Similarly to the one-port VNA case, this last set of expression shows that when the VNAsystematic errors are small, changes in VNA calibration coefficients can be represented asadditional linear reciprocal two-ports, whose parameters depend only on those changes andon the calibration coefficients ERF and ERR respectively.

3.4.3 Errors in VNA raw measurements

The description of VNA receiver errors is based on a set of perturbations added to theraw VNA S-parameter measurement. These perturbations form a joint description of VNAreceiver noise and nonlinearities. These errors cannot be accounted for with the residualerror-box representation, hence we need to directly determine derivatives of (2.45). To thisend, we first represent the differentials in DUT S-parameters and their measurement, ΔSand ΔSm, respectively, in a vector form by use of the vectorization operator [57], defined

58


by (2.52). We then obtain

vec (ΔS) =

⎡⎢⎢⎢⎢⎢⎢⎣

ΔS11

ΔS21

ΔS12

ΔS22

⎤⎥⎥⎥⎥⎥⎥⎦

, and vec (ΔSm) =

⎡⎢⎢⎢⎢⎢⎢⎣

η11

η21

η12

η22

⎤⎥⎥⎥⎥⎥⎥⎦

. (3.52)

By applying the rules of matrix calculus we then obtain⎡⎢⎢⎢⎢⎢⎢⎣

ΔS11

ΔS21

ΔS12

ΔS22

⎤⎥⎥⎥⎥⎥⎥⎦= J

⎡⎢⎢⎢⎢⎢⎢⎣

η11

η21

η12

η22

⎤⎥⎥⎥⎥⎥⎥⎦

, (3.53)

where

J = ∂vec (S)∂vec (Sm)T =

(ST ⊗ S

) (ST

12e ⊗ S21e

) [(Sm − S11e)−T ⊗ (Sm − S11e)−1

], (3.54)

where ⊗ denotes the Kronecker product [57] and A−T is a short form for (A−1)T =(AT)−1

.We can further simplify the second term in (3.54) by inserting (2.44), that is,

(Sm − S11e)−1 = S−121e

(S−1 − S22e

)S−1

12e, (3.55)

and(Sm − S11e)−T = S−T

12e

(S−T − ST

22e

)S−T

21e. (3.56)

With the use of identities (AB)⊗ (CD) = (A ⊗ C) (B ⊗ D) and (A ⊗ B)−1 = A−1 ⊗ B−1

we can further simplify this term

(Sm − S11e)−T ⊗ (Sm − S11e)−1 ==(S−T

12e ⊗ S−121e

) {[(S−T − ST

22e

)S−T

21e

]⊗[(

S−1 − S22e

)S−1

12e

]}=

=(ST

12e ⊗ S21e

)−1 [(S−T − ST

22e

)⊗(S−1 − S22e

)] (ST

21e ⊗ S12e

)−1, (3.57)

which after insertion into (3.54) gives

J =(ST ⊗ S

) [(S−T − ST

22e

)⊗(S−1 − S22e

)] (ST

21e ⊗ S12e

)−1, (3.58)

59


and after expansion simplifies to

J =[I4 −

(ST ST

22e

)⊗ I2 − I2 ⊗ (SS22e) +

(ST ⊗ S

) (ST

22e ⊗ S22e

)] (ST

21e ⊗ S12e

)−1,

(3.59)

where I2 and I4 are identity matrice of size 2 × 2 and 4 × 4, respectively.

Expression (3.59) is general and can be applied to both the 16-term and the 8-termmodel. In the case of the 8-term model, under the assumption that the VNA has smallreflections, that is, |ESF | ≈ 0, |ESR| ≈ 0, |ESF /ERF | ≈ 0, and |ESR/ERR| ≈ 0 and thatthe DUT has small transmissions (S21 ≈ 0 and S12 ≈ 0), we obtain

ΔS11 ≈ η111 − 2ESF S11 + E2

SF S211

ERF

, (3.60)

ΔS22 ≈ η221 − 2ESRS11 + E2

SRS211

ERR

, (3.61)

ΔS21 ≈ η21β

α

1 − ESF S11 − ESRS22 + ESF ESRS11S22

ERR

, (3.62)

ΔS12 ≈ η12α

β

1 − ESF S11 − ESRS22 + ESF ESRS11S22

ERF

. (3.63)

Comparison with (3.38) through (3.43) shows that errors in reflection coefficient measure-ment due to errors η11 and η22 in VNA raw measurements can then be described with anequivalent effective directivity equal to the VNA receiver error, that is, rDF = η11 andrDR = η22.

In the case when the DUT is well matched (S11 ≈ 0 and S22 ≈ 0) and the VNA hassmall reflections, we obtain

ΔS11 ≈ η11

ERF

, (3.64)

ΔS22 ≈ η22

ERR

, (3.65)

ΔS21 ≈ β

α

η21

ERR

, (3.66)

ΔS12 ≈ α

β

η12

ERF

. (3.67)

In this case, we see that the resulting error in reflection coefficient measurement can beseen as VNA receiver errors η11 and η22 scaled by inverse of the round-trip transmissionthough the relevant errors boxes, ERF and ERR. The error in transmission measurements

60

3.5. APPROXIMATE UNCERTAINTY EVALUATION

can be interpreted as VNA receiver error η21 and η12 scaled by the inverse of the first-orderoverall transmission through both error boxes.

3.5 Approximate uncertainty evaluation

The approximate uncertainty analysis techniques allow to determine parameters of theresidual error-box model (see Subsection 3.4.2) for a previously calibrated VNA. Thesetechniques use either an experimental or analytical approach to estimate how the actualrandom and systematic errors in the calibration standards and VNA raw measurementstransform into the parameters of the residual error-box model. Thus, approximate uncer-tainty evaluation techniques cannot be used to perform a complete uncertainty analysis.In such a analysis, instead of one particular realization of systematic and random errors,we are rather interested in the statistical distribution of measurement errors, induced bymultiple realizations of underlying errors in the calibration standards and the raw VNAmeasurement. In other words, we are interested in a distribution of these errors that wouldbe observe in a hypothetical experiment when a large number of VNA calibrations wereperformed with different realizations of the same set of calibration standards. Therefore,the approximate uncertainty evaluation techniques, although offering a very useful infor-mation about the error of one particular VNA calibration, cannot provide the informationabout the statistical distribution of this error. They cannot also be used to estimate theuncertainty due the errors occurring during corrected VNA measurements, such as theerrors due to imperfect DUT connector repeatability or cable flexure.

The other disadvantage of the approximate techniques is that are not traceable. Trace-ability requires in general that the uncertainties determined in an uncertainty analysis ofa measurement procedure be traceable back to uncertainties of some fundamental physicalstandards, such as length or impedance standards [97, 98]. In other words, it requiresthat these measurement uncertainties be expressed as a function of the uncertainties of thefundamental standards and the model of the measurement procedure. For example, in thecase of transmission lines used as calibration standards in the TRL calibration [29], theuncertainty in the length measurement of this lines should be traceable to the uncertaintyof some fundamental length standards. The residual error-box parameters determined withthe approximate techniques originate either from a direct measurement or mathematicaltreatment of the calibration problem and cannot be traced back to the uncertainties ofsome fundamental standards, such as length or impedance standard.

61


3.5.1 Ripple analysis techniques

Ripple analysis technique is probably the earliest method for an approximate evaluationof VNA calibration accuracy [99]. It is primarily used in the analysis for one-port VNAcalibrations [95]. Extensions to the two-port case were also proposed [100], however, wewill not discuss them here.

Ripple analysis technique is an experimental method. It uses a long section of a uniformtransmission line, terminated either with a well-matched or a highly-reflective termination.A calibrated measurement of such a verification device reveals a standing-wave pattern(“ripples”), parameters of which can be transformed into the equivalent parameters ofthe residual error-box model [95]. This transformation can be done in different ways. Inthe simplest case from the ripple amplitude one can determine the magnitude of residualdirectivity (line terminated with a matched termination) and residual source match (lineterminated with a highly reflective termination). More complex analysis techniques allowto determine the complete residual error-box model [101, 102].

3.5.2 Calibration comparison method

Calibration comparison method [94] was originally developed as a tool for the uncer-tainty evaluation in on-wafer VNA calibrations. Subsequently, however, it has also beenapplied to the uncertainty evaluation of VNA calibration in other environments (e.g.,[103]).The principle of this method is to analyze the difference between two calibrations—a bench-mark calibration and a calibration corrupted with errors—and then estimate the resultingworst-case error in the DUT measurement.

The difference between the two calibration in this method is expressed with the useof the residual error-box representation (see Subsection 3.4.2). The worst-case estimate oferror in the DUT S-parameters is then determined as follows. Consider a DUT describedwith S-parameters Sij, for i, j = 1, 2 and cascaded with the residual error-boxes to obtainerror-corrupted S-parameters Sij, for i, j = 1, 2. We define the worst-case error as [94]

ε = max|Sij |≤1i,j=1,2

|Sij − Sij|. (3.68)

Assuming that the difference between the calibrations is small, first order approximationfor (3.68) can be derived [94].

62

3.5. APPROXIMATE UNCERTAINTY EVALUATION

The worst case estimate (3.68) can be thought of as single-number measure of thedeviation captured in the residual error-boxes. It has been reported, however, that thisestimate is too conservative when applied to the measurement of actual passive DUTs [104].This can be attributed to the fact, the range of DUTs for which the worst case error isdetermined, specified by the constraints in (3.68), includes also non-physical devices, e.g.,when S11 = S21 = 1.

3.5.3 Statistical residual analysis

The principle of the statistical residual analysis is to assess how well the measurementmodel—that is the VNA calibration coefficients along with the definitions of the calibrationstandards—describes the actual measurements of the calibration standards.The inadequacyof the measurement model is then transformed into the equivalent uncertainties of the VNAcalibration coefficients. These uncertainties can futher be used to predict the accuracy ofcorrected VNA S-parameter measurements.

The inadequacy of the VNA measurement model can only be observed in statisticalVNA calibration methods, that is, when a redundant number of calibration standardsis measured (see Paragraph 2.4.1-A). The most common formulation of statistical VNAcalibration is based on the least-squares estimation. In the formulation (2.86) with ΣΔs =σ2VΔs, the inadequacy of the VNA measurement model (2.84) is quantified with the sum

S (β) = r (β)T V−1Δs r (β) , (3.69)

where r (β) are the residual errors of the VNA calibration model, given by (2.85), β areVNA calibration model parameters defined by (2.83), ΣΔs = σ2VΔs is the covariancematrix of errors in calibration standard definitions given in (2.87), VΔs is a known matrix,and σ2 is the unknown scaling factor, referred to as the residual variance. The estimateβ of VNA calibration model parameters (which contains also the unknown parameters ofcalibration standards if the calibration method employs the principle of self-calibration;see Paragraph 2.4.2-A) is obtained from (2.86) and the estimate of the residual variance isdetermined as (see Subsection B.2.1)

σ2 = 1ν

S(β), (3.70)

where ν is the number of degrees of freedom, that is, the number of equations less the

63


number of estimated parameters. With the use of above definition, we can derive theestimate of covariance matrix of errors in β caused by the inadequacy of the measurementmodel. This estimate can be written as (see see Subsection B.3.1)

Σβ = σ2[Jr(β)T Σ−1

r Jr(β)]−1

, (3.71)

and Jr(β) is the Jacobian of r (β) calculated at β, that is

Jr(β) =∂r (β)∂βT

∣∣∣∣∣β=β

. (3.72)

Statistical properties of errors in VNA calibration coefficients, as captured in covariancematrix (3.71), can be further transformed into the residual error-box representation withthe use of relationships derived in 3.4.2. With this representation we can then estimatethe resulting error in corrected VNA S-parameter measurements.

Statistical residual analysis is the most rigorous technique to estimate the contributionof systematic and random errors in a particular set of calibration standards and raw VNAmeasurements. It is typically built into the calibration software based on statistical meth-ods [56, 59], which allows to quickly verify the quality of the VNA calibration. However,similarly to other approximate methods, statistical residual analysis does not allow forcomplete uncertainty evaluation and is not traceable.

3.6 Complete uncertainty evaluation

Complete uncertainty analysis approaches are based on a rigorous analysis of how thecalibration standard errors and VNA instrumentation errors propagate into the correctedVNA measurement. These approaches can be divided into two groups: techniques basedon the linear error propagation [19, 73, 105–109], and techniques employing the numericalMonte Carlo simulation [110, 111]. These two groups of techniques are also referred to asthe uncertainty propagation and distribution propagation, respectively. This nomenclaturereflects the fact that in the linear error propagation one analyzes only the first two momentsof the input and output error distributions, while in the Monte Carlo simulation the entireprobability density function of the input and output error is determined.

64

3.6. COMPLETE UNCERTAINTY EVALUATION

3.6.1 Linear error propagation

The techniques based on linear error propagation usually employ analytical relation-ship between the errors in corrected VNA measurements and the underlying calibration-standard errors and VNA instrumentation errors. These relationships are based on thefirst-order Taylor expansion of the function describing the VNA measurement process.This expansion has usually a complicated form, closely bound to a specific VNA calibra-tion procedure and a VNA model. Therefore, whenever the model of the VNA measure-ment changes, a significant effort needs to be put into deriving these relationships anew(see [36, 108, 109, 112, 113]). This makes these techniques very cumbersome in practicalapplications.

In order to determine the uncertainties in the definitions of the calibration standards,these techniques usually employ a physical model of the standard (see [6, 90]). Based onthat model and uncertainties of its parameters, such as lengths, diameters, or materialparameters, the uncertainty in the S-parameters of the standards is then calculated. Thisprovides a traceability path for the resulting uncertainties [97]. In order to quantify theVNA instrumentation errors, usually a statistical analysis based on a large number ofrepeated measurements is used [41, 95, 114].

Probably the most advanced uncertainty analysis approach based on the analyticaltechniques has been developed at NIST [40, 41]. This approach has been developed for theTRL calibration [29] in the coaxial 7 mm standard applied to a 2 − 18 GHz dual six-portmeasurement system [115]. Apart from developing analytical expressions for the VNAmeasurement uncertainties, this approach features a very rigorous statistical treatment ofthe measurement results based on the covariance matrix description.

3.6.2 Monte-Carlo simulation

Monte Carlo simulation is a well-established numerical technique for uncertainty es-timation [116]. The principle of this technique is to treat the measurement problem asan arbitrary function transforming input quantities (in the case of VNA measurements—calibration standard definitions, their raw measurements, and the raw measurements ofthe DUT) into output quantities (in the case of VNA measurements—corrected DUT S-parameters). The input quantities are then randomly varied according to the statisticaldistribution of their errors, and the respective responses of the output quantities are col-lected. Based on that, statistical distribution of the output quantities is determined, from

65


which different uncertainty measures can be derived.The key advantages of numerical approaches based on the Monte Carlo simulation

are the statistical rigorousness and flexibility. Indeed, in the Monte Carlo simulation noassumptions need to be made as to the linearity of error propagation, and an arbitrarymeasurement model (in the case of VNA measurements—combination of the VNA modeland VNA calibration procedure) can easily be analyzed. This makes the Monte Carloapproach particularly attractive for complicated problems for which deriving analyticalequations would be infeasible.

The main disadvantage of the Monte Carlo simulation is that it is very time consumingand requires huge computer resources. The time consumption becomes particularly criticalwhen estimating covariance matrices, as the number of Monte Carlo iterations needs tosignificantly exceed the number of estimated parameters [87, 116]. Therefore, the MonteCarlo approaches are often used to verify uncertainty analyzes based on the linear errorpropagation.

66

Chapter 4

Multi-frequency description ofS-parameter measurement errors

Logic, after all, is a trick devised by the humanmind to solve certain types of problems.

Richard Bellman

In this chapter, we introduce the multi-frequency description of errors in VNA S-parameter measurements. This description generalizes the conventional single-frequencycovariance matrix representation (see Subsection 3.2.3) by accounting for the statisticalcorrelations between errors at different frequencies. These statistical correlations resultfrom the fact that the VNA S-parameter measurement errors at different frequencies havesome common physical causes.

The discussion we present in this chapter has two objectives. First of all, we shalldevelop the mathematical and physical foundations for the multi-frequency description oferrors in S-parameter measurements. Secondly, we shall identify the implications of suchan extended description for practical applications. A more detailed analysis of some ofthose applications will then be given in the remainder of this work.

We start out by introducing the statistical model for the multi-frequency VNA S-param-eter measurement. We then review the notion of the physical error mechanism and discussstatistical properties of the S-parameter measurement error defined jointly for multiplefrequencies. Subsequently, we briefly review the aspects of uncertainty reporting specificto the multi-frequency error description. Following on that, we discuss the physical error

67

4. MULTI-FREQUENCY ERROR DESCRIPTION IN VNA MEASUREMENTS

mechanisms in the VNA S-parameter measurements and finally we analyze the practicalimplications of the generalized multi-frequency error description we introduced.

4.1 Statistical model for the multi-frequency S-parameter measurement

Our statistical model for the multi-frequency S-parameter measurement is an extensionof the model introduced in Subsection 3.2.1. Following (3.1), we define measurement s ofS-parameters of N -port device for a set of frequencies {fk}K

k=1 as a sum of the true values of the S-parameters and the measurement error Δs, that is,

s = s +Δs, (4.1)

where

s =

⎡⎢⎢⎢⎣

s1...

sK

⎤⎥⎥⎥⎦ , (4.2)

the underline denotes the real-valued vector representation of a complex-valued vectordefined by (A.2), and sk, for k = 1, . . . , K is the real-valued vector representation ofS-parameters at the frequency fk. The vector s has thus Q = 2N2K elements.

We assume further that the true value s is constant, and that the measurement errorΔs is a vector of Q random variables

Δs = [Δs1, . . . ,ΔsQ]T . (4.3)

The statistical properties of the joint PDF fΔs(x) of the measurement error Δs are dis-cussed in the following sections. For now, we only assume that the measurement error haszero expectation value, that is, E (Δs) = 0, thus E (s) = s.

4.2 The notion of a physical error mechanism

The notion of physical error mechanism is essential for our multi-frequency descriptionof errors in S-parameter measurements. It reflects the intuition that the overall measure-ment error is caused by some common fundamental error mechanisms. The fundamental

68

4.2. THE NOTION OF A PHYSICAL ERROR MECHANISM

character of these mechanisms manifests itself in the fact that they give the simplest phys-ical explanation of the underlying causes of the measurement error. For example, in thecase of VNA S-parameter measurements, the fundamental error mechanisms correspondto both systematic measurement errors (e.g., uncertainties on dimensional and materialparameters of the calibration standards) and random measurement errors (e.g., bendingof the cables, misalignment of inner and outer conductors or displacements of the innerconductor fingers in coaxial connectors, VNA test-set drift, or VNA receiver noise and non-linearities). Consequently, we see that the fundamental error mechanisms characterizingthe causes of the measurement error, have physical character, and are statistically uncor-related and frequency independent. In the following, we referr to the error mechanismsfulfilling these three conditions as the physical error mechanisms.

We define a single physical error mechanism as a scalar random variable ξ that describesthe variability of an underlying physical parameter characterizing the mechanism, and afunction

Δs = m (ξ) , (4.4)

which represents a physical model that relates the parameter ξ and the correspondingerror Δs in the S-parameter measurement. Since the physical error mechanism characterizechanges, we assume that E (ξ) = 0, m (0) = 0. We further define the variance Var (ξ) = σ2

ξ .Typically, the random variable ξ has a Gaussian or uniform probability density function.

The function m (ξ) is in general nonlinear. However, assuming that the variance σ2ξ is

small, we can approximate m (ξ) with the first order Taylor expansion around ξ = 0, thatis,

Δs ≈ jξ, (4.5)

wherej = ∂m (ξ)

∂ξ

∣∣∣∣∣ξ=0

. (4.6)

The vector (4.6) can be thought of as a nominal response of the measurement s to theerror mechanism ξ.

We further represent a set of M physical error mechanisms with the vector

ξ = [ξ1, . . . , ξM ]T , (4.7)

69


the covariance matrix

Σξ =

⎡⎢⎢⎢⎣

σ2ξ1

. . .σ2

ξM

⎤⎥⎥⎥⎦ , (4.8)

and the Jacobian matrixM = [j1, . . . , jM ] . (4.9)

The matrix (4.8) is diagonal due to the assumption of statistical independence of themechanisms in the vector (4.7). With the use (4.9), we can eventually approximate theerror in S-parameter measurement as

Δs = Mξ. (4.10)

The properties of the matrix M require some additional remarks. Based on the dis-cussion at the beginning of this chapter, a natural assumption would be that the columnsof the matrix M are linearly independent, or in other words, that we can distinguish be-tween the contributions of different physical error mechanisms to the measurement error.In practice, however, we often encounter the situation when different physical mechanismlead to indistinguishable contributions. For example, in the case of a coaxial transmissionline, we cannot distinguish in the frequency-dependent S-parameters of the line betweenthe effect of the same relative change in the inner and outer conductor diameter.

When the contributions of some of the physical error mechanisms to the measurementerror Δs are indistinguishable, the columns of the matrix M become linearly dependent,thus, the matrix M is rank deficient. In order to avoid that, we introduce the concept ofelectrically-equivalent physical error mechanisms. We write ξ = Qξ′, where ξ′ is the set ofelectrically-equivalent physical error mechanisms, and define the measurement error as

Δs = MQξ′, (4.11)

where we require the matrix MQ to be full rank. Columns of the matrix Q constitute thebasis for the range of the matrix M and can be easily determined based on the physicalconsiderations. For example, coming back to the example of the coaxial transmissionline, the effect of a change in the inner and outer conductor diameter can be equivalentlycaptured in terms of a change of the characteristic impedance of the line.

In the following, unless otherwise noted, we assume that the physical error mechanisms

70

4.2. STATISTICS OF THE MULTI-FREQUENCY MEASUREMENT ERROR

describing the measurement error are distinguishable. We always denote the error mech-anisms with the vector ξ which, depending on the context, refers either to the actual orelectrically-equivalent physical error mechanisms.

4.3 Statistical properties of the multi-frequency mea-surement error

We now investigate the statistical properties of the measurement error as defined by(4.10). We first rewrite (4.10) with the use of real-valued vectors. To this end, we use theconventions (A.2) and (A.6) to write

Δs = Mξ, (4.12)

and consequently, we write the covariance matrix of (4.12) as

ΣΔs = E[Mξ (Mξ)T

]= MΣξMT , (4.13)

where the matrix ΣΔs has form

ΣΔs =

⎡⎢⎢⎢⎢⎢⎢⎣

Var(Δs1) Cov(Δs1,Δs2) · · · Cov(Δs1,ΔsQ)Cov(Δs2,Δs1) Var(Δs2) · · · Cov(Δs2,ΔsQ)

... ... . . . ...Cov(ΔsQ,Δs1) Cov(ΔsQ,Δs2) · · · Var(ΔsQ)

⎤⎥⎥⎥⎥⎥⎥⎦

, (4.14)

where Cov(Δsi,Δsj) = Cov(Δsj,Δi), for i �= j and i, j = 1, . . . , Q. Now, we note that thelength Q of the measurement error Δs, in most cases, is much smaller than the numberM of physical error mechanisms. Consequently, although the matrix (4.8) has size Q × Q,its rank is only M . Hence, the matrix (4.8) is singular. In the following we discussthe consequences of this fact related to the form of the PDF for Δs and reporting theuncertainties described with ΣΔs, and finally we take a closer look at the structure of thematrix ΣΔs.

4.3.1 Probability distribution function

Assuming that the measurement error has the normal probability distribution, we can-not write the PDF of Δs in the typical form (see [87]) due to singularity of ΣΔs. The

71


probability distribution function for a normal vector random variable ζ with a mean μ

and a singular covariance matrix Σζ has the form [117]

fζ (x) =1

(2π)M2

M∏i=1

λ12i

e− 12 (x−μ)T Σ−

ζ(x−μ)T

, (4.15)

where {λm}Mm=1 are the positive eigenvalues of Σζ and Σ−

ζ denotes any generalized inverse ofthe matrix Σζ . The distribution (4.15) is also referred to as the singular normal distribution[118, 119]. The generalized inverse A− of a matrix A is defined as a matrix fulfilling thecondition (see [120])

AA−A = A. (4.16)

A well known generalized matrix inverse is the Moore-Penrose inverse A+ , also referredto as the matrix pseudo-inverse, which, additionally to (4.16), fulfills the conditions

(AA+

)T= AA+, (4.17)(

A+A)T

= A+A, (4.18)

A+AA+ = A. (4.19)

Unlike the generalized inverse defined by (4.16), the Moore-Penrose inverse is uniquelydefined for a given matrix A [120]. For a full rank matrix A we can easily show thatA+ = A−1 by simply inserting A−1 into the conditions (4.16) through (4.19) 1.

In the case of the random variable Δs defined by (4.12), the probability distributionfunction takes on the form

fΔs (x) =1

(2π)M2 det(Σξ)

12 det(MT M) 1

2e− 1

2(M−x)T Σ−1ξ

M−x. (4.20)

where M− is any generalized inverse of M. By comparing (4.20) and (4.15) we see, that inorder to justify the form (4.20), we have to prove two conjectures: that the product ∏M

i=1 λi

of the positive eigenvalues of ΣΔs is det (Σξ) det(MT M

), and that any generalized inverse

of the matrix ΣΔs can be written as Σ−Δs =

(M−)T

Σ−1ξ M− .

1Another, more intuitive, definition of the Moore-Penrose inverse is A+ = arg minX ‖AX − I‖F , where‖·‖F is the Frobenius matrix norm (square root of the sum of all of the matrix elements squared) and I isthe identity matrix [121].

72

4.3. STATISTICS OF THE MULTI-FREQUENCY MEASUREMENT ERROR

Consider the singular-value decomposition of the matrix M, that is [121],

M = USVT , (4.21)

where U ∈ RQ×Q and V ∈ R

M×M are orthogonal square matrices, and S ∈ RQ×M is a

rectangular matrix containing on the diagonal the M singular values of the matrix M, thatis,

S =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

s1. . .

sM

Ø

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.22)

where Ø is an empty matrix. The singular values are non-zeros as the matrix M is assumedto be full rank [121]. We can easily show that the eigenvalues of the matrix MT M areequal to squared singular values in S, consequently det

(MT M

)= det

(ST S

). Inserting

(4.21) into (4.13) and postmultiplying by US we obtain

ΣΔsUS = US(VT ΣξVST S

)= USP, (4.23)

where P = VT ΣξVST S. Now, consider the spectral decomposition of the matrix P

P = XΛX−1, (4.24)

where X−1 = XT since the matrix P is symmetric. Inserting (4.24) into (4.23) and post-multiplying by X we obtain

ΣΔsUSX = USXΛ, (4.25)

which expresses the condition for the eigenvalue decomposition (with only positive eigenval-ues) of ΣΔs: the eigenvalues are contained in the diagonal matrix Λ and the correspondingeigenvectors are the columns of the product USX. The eigenvalues of P are all positiveand real since P is full rank and symmetric. These eigenvalues constitute all of the pos-itive eigenvalues of ΣΔs since rankΣΔs = rankP = M . The product of the eigenvaluescontained in Λ is ∏M

m=1 λi = det (P) = det(VT ΣξV

)det

(ST S

)= det (Σξ) det

(MT M

),

which proves the first conjecture.

As to the second conjecture, based on the definition of the generalized inverse (4.16),

73


we need to show that

MΣξ

(M−M

)TΣ−1

ξ M−MΣξMT = MΣξMT , (4.26)

for any generalized inverse M−. This requires that M−M = IM for any generalized inverseM− which is only true if the matrix M is full rank. Indeed, taking the singular valuedecomposition of the matrix M, we can easily show that

M− = VS−UT (4.27)

fulfills the condition (4.16). With the use of (4.27), we can write

M−M = VS−SVT . (4.28)

Now, considering the diagonal structure of the matrix S, as shown in (4.22), and accountingfor the assumption that singular values are non-zero (i.e., the matrix M is full rank), wesee that the condition (4.16) is equivalent to S−S = IM , thus M−M = IM .

4.3.2 Uncertainty reporting

Second important consequence of the singularity of the matrix ΣΔs concerns estimatingconfidence regions. The typical methods for the confidence region estimation (e.g., [88])require the inversion of the covariance matrix, hence cannot be used with ΣΔs. Conse-quently, some other methods for reporting measurement uncertainty need to be devised,based on projecting the uncertainty regions in the space of the physical error mechanismsonto the space of the actual measurement errors (e.g. see [122, 123]). This is, however,beyond the scope of this work.

In practice, however, we are often interested in the uncertainties of only single elementsof the vector Δs. Examples include the real and imaginary part of an S-parameter ata particular frequency. The confidence region then reduces to a typical one-dimensionalinterval and can be determined with the use of classical uncertainty evaluation methods(see [88]).

74

4.3. PHYSICAL ERROR MECHANISMS IN VNA MEASUREMENTS

4.3.3 Multi-frequency covariance-matrix structure

We shall now look in more detail at the structure of (4.14). Matrix (4.14) containsthe variances and covariances for all components of the vector (4.3), that is, for real andimaginary parts of all of the S-parameters of the N -port device for frequencies fk, fork = 1, . . . , K. We can rewrite it as a block matrix

ΣΔs =

⎡⎢⎢⎢⎢⎢⎢⎣

Σf1,f1 Σf1,f2 . . . Σf1,fK

Σf2,f1 Σf2,f2 . . . Σf2,fK

... ... . . . ...ΣfK ,f1 ΣfK ,f2 . . . ΣfK ,fK

⎤⎥⎥⎥⎥⎥⎥⎦

, (4.29)

where all of the sub-matrices have size 2N2 × 2N2. The diagonal sub-matrices Σfk,fkare

covariance-matrices for the single-frequency measurement errors Δsk, and have the samestructure as the single-frequency covariance matrix defined in Subsection 3.2.3. In thefollowing, we referr to those matrices in short as Σfk

. The off-diagonal sub-matrices Σfk,fl

define the correlations between measurement error in S-parameters at different frequencies,that is, for vectors Δsk and Δsl, for k �= l. Note that if there is no statistical correlationsbetween S-parameter measurement errors at different frequencies, the off-diagonal matricesΣfk,fl

have all entries equal zero.

4.4 Physical error mechanisms in VNA S-parametermeasurements

The source of errors in corrected VNA S-parameter measurements and their statisticalmodels have been discussed in Section 3.1 and Section 3.3, respectively. In the following,we shall take another look at these errors, however, from the perspective of the underlyingphysical error mechanisms.

4.4.1 Calibration standard errors

The error mechanisms in the calibration standards are identified based on physicalmodels of the standards. These models express S-parameters of calibration standards interms of some fundamental dimensional and material parameters. For example, in thecase of the coaxial transmission lines, these parameters are the lengths and diameters of

75


the conductors. The uncertainties of the measurements of these fundamental parametersbecome then the estimates for standard deviations of the error mechanisms ξ, while therelationships resulting from the calibration standard physical models give the functionsm (ξ). We discuss this in more detail in Chapter 5.

4.4.2 VNA instrumentation errors

A. Nonstationarity. A physical-mechanism-based description of the VNA nonstation-arity errors poses a complex problem. In practice, it is difficult to characterize the VNAnonstationarity errors with analytical models derived from fundamental mechanical andelectrical parameters of the VNA. For example, in the case of connector repeatability er-rors, this would require careful mechanical characterization of all of the possible mechanicaldisplacements in the connector interface, and then electro-mechanical modeling of their in-fluence on the interface S-parameters. Although such an approach has been applied tosimplified connector models (e.g. see [124–126]), it is extremely difficult to model real con-nector structures. The situation is even more difficult with the cable instability or VNAtest-set drift—analytical modeling of electrical parameters of such complex structures isbeyond our capacity.

On the other hand, as demonstrated in [91], the connector repeatability errors exhibita very regular frequency dependence which can be modeled with a simple lumped-elementequivalent circuit. This observation is a foundation of our approach to the descriptionof the VNA nonstationarity which is presented in Chapter 6. We show there that theVNA nonstationarity errors can be characterized in terms of a very small set of frequencyindependent error mechanisms ξ and fixed functions capturing the frequency dependenceof this disturbance errors. Consequently, we represent the VNA nonstationarity errors as

Δsk = m (sk, fk, ξ) (4.30)

which, under the assumption that errors are small, can be approximately written as

Δsk ≈ M (sk, fk) ξ, (4.31)

where the matrix M (sk, fk) is defined by (4.9).

76

4.4. PHYSICAL ERROR MECHANISMS IN VNA MEASUREMENTS

B. Receiver noise. As discussed in Section 3.1, the VNA receiver noise is caused by thethermal noise and phase noise in the receivers. Since the VNA performs the measurementat one frequency at-a-time, the noise realizations at different measurement frequenciescan be considered statistically independent2. Consequently, the VNA receiver noise canbe described with a separate set of physical error mechanisms ξm (fk) at each frequencyfk. The statistical properties of these mechanisms depend on the raw S-parameters (orVNA receiver voltages), which need to be accounted for in the physical model of the errormechanism. Consequently, at a given measurement frequency fk, we postulate the followingdescription of raw S-parameter measurement error

Δsmk = mm (sm

k , fk, ξmk ) (4.32)

which, under the assumption that error are small, can approximately be written as

Δsmk ≈ Mm (sm

k , fk) ξmk , (4.33)

where the matrix M (smk , fk) is defined by (4.9).

C. Receiver nonlinearity. The nonlinearity of VNA receivers introduces errors of asystematic nature as their impact is clearly repeatable for the same measurement condi-tions. Following a similar reasoning as for the VNA receiver noise, we can assume thereare no statistical correlations between the VNA receiver nonlinearity errors at different fre-quencies. Provided that a model for the nonlinear behavior of VNA receivers is available,these errors could be quantified and eventually corrected for. A preliminary analysis of[86] could be a starting point for the development of such a model. However, as the VNAreceiver nonlinearity errors are negligible under typical VNA operating conditions, we willnot consider them in more detail in this work.

2The exception here is the measurement in the so-called “ramp” mode, available in some VNAs. Inthis mode, the source is swept continuously without stopping at each measurement frequency. Hence,we can expect the noise for adjacent measurement frequencies to be statistically correlated. However, inthe laboratory-precision VNA measurements, we avoid operation in the “ramp” mode due to the loss ofaccuracy caused by the increased measurement speed.

77


4.5 Practical implications

In this section, we discuss the practical implications of the multi-frequency approach tothe description of S-parameter measurement errors. These implications include the error-mechanism-based approach to the VNA calibration, uncertainty analysis in time-domainwaveform correction, and device modeling based on S-parameter measurements.

4.5.1 Time-domain waveform correction

VNA S-parameter measurements are often used in the calibration of time-domain mea-surement systems such as high speed oscilloscopes [127] or electro-optic sampling systems[44]. In the calibration process of such systems, systematic errors of some of the systemcomponents, such as adapters, pulse sources, oscilloscopes, or on-wafer probes are charac-terized with S-parameters. These errors are then removed from the actual time-domainmeasurement in a correction procedure. In the following, we present the details of time-domain waveform correction and demonstrate the importance of the multi-frequency errordescription is the uncertainty analysis of this procedure. The exposition presented belowgeneralizes the discussion presented in [39, 128]. Reference [39] shows also experimentalresult that illustrate the role of the multi-frequency error description in the uncertaintyanalysis of the time-domain waveform correction.

A. Introduction. A typical situation of time-domain waveform correction is shown inFig. 4.1. A waveform source, for example, a pulse generator, is connected through atwo-port device to a load, for example, an oscilloscope. In the frequency domain, thesource is characterized with the wave amplitude bS (jω) and the input reflection coefficientΓS (jω). The corresponding voltage delivered by the source to a matched terminationcan be determined from (2.36) and (2.37) as VS (jω) = bS (jω)

√Zref (jω) where Zref (jω)

is the real reference impedance, typically 50Ω. The load is characterized with a loadreflection coefficient ΓL (jω). We measure the voltage VL (jω) at the load which is relatedto the forward wave amplitude aL (jω) through VL (jω) = aL (jω)

√Zref (jω) 3. Finally,

the two-port device is characterized with the scattering parameters.In the time-domain waveform correction problem, we are interested in determining the

time-domain waveform vS (t) based on the measurement of other time-domain waveform3In the following, for the sake of notational simplicity, we skip the explicit indication of the frequency

dependence.

78

4.5. PRACTICAL IMPLICATIONS

Fig. 4.1: Time-domain waveform source connected to a load through a two-port device.

vL (t), and the measurements of the two-port S-parameters and the reflection coefficientsof the source and load, ΓS and ΓL, respectively. Based on the analysis of the flow graphin Fig. 4.1, one can easily show that the relationships between these parameters in thefrequency domain has the form [128]

VL = HVS, (4.34)

where the factor H is defined by

H = 1 − ΓSS11 − ΓLS22 − ΓLΓS [S21S12 − S11S22]S21

. (4.35)

In the case without the two-port device (that is S11 = S22 = 0 and S21 = S12 = 1), thisfactor reduces

H = 1 − ΓLΓS, (4.36)

which corresponds to mismatch correction.In order to convert (4.34) into the time domain, we first introduce some notational

conventions. We represent the frequency-domain measurements of VL and VS, and thecorrection factor H as the vectors

v′L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

VL [0]...

VL [N ]VL [N ]∗

...VL [1]∗

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, v′S =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

VS [0]...

VS [N ]VS [N ]∗

...VS [1]∗

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, and h′ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

H [0]...

H [N ]H [N ]∗

...H [1]∗

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.37)

where VL [n] = VL(jnΔω), VS [n] = VS(jnΔω), and H [n] = H(jnΔω), for n = 0, . . . , N ,Δω = 2πΔf it the angular frequency step, the star denotes the complex conjugate, and

79


N + 1 is the number of positive frequencies (including the DC). The conversion into thetime-domain requires the knowledge of the spectrum for both the positive and negative fre-quencies. However, since all of the waveforms considered here are real-valued, the spectrumfor negative frequencies is the complex conjugate of the spectrum for positive frequencies[129]. This is explicitly shown in the vectors in (4.37) which have M = 2N + 1 elements.

We further represent the corresponding time-domain waveforms as vectors

vL =

⎡⎢⎢⎢⎣

vL [0]...

vL [M − 1]

⎤⎥⎥⎥⎦ , vS =

⎡⎢⎢⎢⎣

vS [0]...

vS [M − 1]

⎤⎥⎥⎥⎦ , and h =

⎡⎢⎢⎢⎣

h [0]...

h [M − 1]

⎤⎥⎥⎥⎦ , (4.38)

where vL [n] = vL (nΔt), vS [n] = vS (nΔt) , and h [n] = h (nΔt) , for n = 0, . . . , M −1, andΔt = 2π

MΔω.

The relationship between the frequency-domain spectrum and time-domain waveformis now determined by the Fourier series. For a frequency domain spectrum X (jnΔω) andthe time domain waveform x (nΔt), this relationship has the form [129]

x (nΔt) =∞∑

k=−∞X (jnΔω) ejnkΔωΔt. (4.39)

In the case of truncated frequency data, as described by the vectors (4.37), equation (4.39)reduces to the inverse discrete Fourier transform [129]

x [n] =M∑

k=0X [k] ejnk 2π

M , (4.40)

which is typically evaluated with the use of the Fast Fourier Transform (FFT) algorithm[129].

Now, in order to write the relationship between the time-domain waveforms (4.38), weapply (4.40) to (4.34). With the use of the convolution theorem, we obtain [129]

vL [n] =M−1∑k=0

h [k] vS [(n − k) modM ] =M−1∑k=0

vs [k]h [(n − k) modM ] (4.41)

which describes the circular convolution of discrete series vS [n] and h [n], and the operatorxmod y is the remainder of the division of two integer numbers x and y.

80


B. Matrix formulation. The equation (4.41) can be used directly to perform the time-domain correction based on the measured waveform samples vs [n] and the time-domainrepresentation h [n] of the frequency-domain correction factor (4.35), as obtained through(4.40). However, in order to perform the uncertainty analysis of this procedure, it is moreconvenient to write (4.41) in a matrix notation as

vL = T (vS)h, (4.42)

where

T (vS) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

vS [0] vS [M − 1] vS [M − 2] . . . vS [1]vS [1] vS [0] vS [M − 1] . . . vS [2]vS [2] vS [1] vS [0] . . . vS [3]... ... ... . . . ...

vS [M − 1] vS [M − 2] vS [M − 3] . . . vS [0]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.43)

is the Toeplitz matrix (see [121]) constructed from the vector vS. In a similar manner, wecan obtain the relationship

vL = T (h)vS, (4.44)

where the Toeplitz matrix T (h) is defined analogously to (4.43).We can further represent the inverse discrete Fourier transform (4.40) with the use of

matrix notation to obtainh = Fh′ = FCMh′, (4.45)

where F is the Vandermonde matrix [121]

F =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 . . . 11 w w2 . . . wM−1

1 w2 w4 . . . w2(M−1)

... ... ... . . . ...1 wM−1 w2(M−1) . . . w(M−1)(M−1)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.46)

w = ej 2πM , CM is defined by (A.3), and the underline denotes the real-valued convention

for representation of complex vector defined by (A.2). Now, combining (4.42) and (4.45),we eventually obtain

vL = T (vS)FCMh′. (4.47)

81


C. Uncertainty analysis. Equation (4.47) is an important results. It describes explic-itly the relationship between the samples vL [n] of the de-embedded waveform and samplesH [n] of the correction factor spectrum. The correction factor spectrum is a function ofVNA S-parameter measurements of the two-port device and the source and load reflec-tion coefficients. Uncertainty of this factor can be easily derived from the uncertaintiesof S-parameter measurements with the use of linear error propagation. We denote thisuncertainty with a multi-frequency covariance matrix Σh′ .

Now, the uncertainty of the waveform samples vL [n], denoted with a covariance matrixΣvL

, can be determined directly from (4.47). However, we derive this covariance matrixin two steps in order to better demonstrate the role of statistical correlation betweenuncertainties at different frequencies in Σh′ . In the first step, we write the relationshipbetween uncertainties in vL [n] and h [h]. This relationship can be easily derived from(4.42) as [87]

ΣvL= T (vS)ΣhT (vS)T . (4.48)

The matrix T (vS) depends on the waveform vS [n]. In the simplest case of vS [n] = δ [n],this matrix becomes diagonal and then the uncertainty for each sample of vL [n] dependsonly on the uncertainty of h [n] at the same time point. In a general case, however, whenvS [n] contains non-zero elements for n > 0, the uncertainty for each sample of vL [n]depends on the uncertainties and statistical correlations between them for all samples inh [n].

Now, the covariance matrix Σh can be derived from (4.45) as

Σh = FCMΣh′ (FCM)T . (4.49)

The matrix F, as defined by (4.46), is dense. The matrix CM is block diagonal and rep-resents only the transformation between the two different representations of the waveformspectrum. Consequently, the product FCM is also a dense matrix. This property has animportant consequence. Examining closer (4.49), we note that the uncertainty of each timedomain sample h [n], thus also of each time domain sample vL [n], depends not only onthe single frequency variances in Σh′ but also on all of the statistical correlations betweenthem, including the correlations between uncertainties at different frequencies. Experi-mental results illustrating the importance of these correlations for the correctness of theuncertainty analysis are given in [39].

82


4.5.2 Device modeling based on S-parameter measurements

In the device modeling based on S-parameter measurements, we identify the parametersof an electrical model of the device based on the measurement of the device’s S-parameters.Typically, these measurements are performed at multiple frequencies. Examples includemodeling of active devices, such as microwave transistors, and passive devices, such astransmission line discontinuities.

The approaches used in the measurement-based device modeling fall into the categoryof the system identification methods. The most common approach used in the systemidentification is based on the maximum-likelihood estimation principle. In Appendix B,we review the main aspects of the use of this principle in the system identification problems.In this section, we briefly demonstrate how to account for the multi-freqency descriptionof errors in S-parameter measurements in the maximum-likelihood system identification.

Consider a nonlinear system identification problem of the class described in Sec-tion B.2.1. The equations describing the measurements are given by

⎧⎪⎪⎪⎨⎪⎪⎪⎩

s1 = f (x1, θ) + Δs1...

sN = f (xN , θ) + ΔsN

, (4.50)

where {xn}Nn=1 are the known system inputs, {sn}N

n=1 are multi-frequency S-parametermeasurements in a vector form, and {Δsn}N

n=1 are the measurement errors. We furtherassume that the measurement errors are normally distributed with E (Δsi) = 0, and thecovariance matrix E

(ΔsnΔsT

n

)= ΣΔsn

which may be singular.

Following the methodology presented in Appendix B and by use of the definition (4.20),we can write the log-likelihood function as

lnL (θ) = −12 ln (2π)

N∑n=1

Mn − 12

N∑n=1

Mn∑m=1

ln λn,m − 12

N∑n=1

[M−

n rsn(θ)]T

Σ−1ξn

M−n rsn

(θ),

(4.51)with the residuals defined by

⎧⎪⎪⎪⎨⎪⎪⎪⎩

rs1 (θ) = s1 − f (x1, θ)...

rsN(θ) = sN − f (xN , θ)

, (4.52)

83


which readily leads to the following least-squares optimization problem

θ = argminθ

N∑n=1

[M−

n rsn(θ)]T

Σ−1ξn

M−n rsn

(θ), (4.53)

where M−n is an arbitrary generalized inverse of the matrix Mn. Now, consider the use

of the Moore-Penrose generalized inverse M+n of the matrix Mn. Since matrices Mn, for

n = 1, . . . , N are full rank, we have M+n =

(MT

n Mn

)−1MT

n (see [121]), which yields

θ = argminθ

N∑n=1

[(MT

n Mn

)−1MT

n rsn(θ)]T

Σ−1ξn

(MT

n Mn

)−1MT

n rsn(θ). (4.54)

Now, this result is different from the typical maximum-likelihood estimators discussedin Appendix B which simply use the sum of squared residuals weighted by the inverse of thecovariance matrix. The term

(MT

n Mn

)−1MT

n rsn(θ) in (4.54) is the orthogonal projection

of the residual error rsn(θ) onto the column space of the matrix Mn. In other words, this

term contains the coefficients of the representation of the frequency-dependent residualerror in terms of the error mechanisms captured in the covariance matrix ΣΔsn

. Hence, theestimator θ minimizes the weighted sum of residuals expressed in terms of error mechanismsrather then frequency-dependent S-parameters. This result can be easily extend to morecomplex system identification models discussed in Appendix B.

4.5.3 Error-mechanism-based VNA calibration

In the error-mechanism-based VNA calibration, we recognize the fact that the measure-ment errors are caused by a set of physical error mechanisms. Consequently, the incon-sistency of the calibration equations describing the measurement of a redundant numberof calibration standards is completely characterized by a realization of these mechanisms,instead of some derived statistics expressed in terms of S-parameters (see Section 3.3).This is shown in Fig. 4.2 which summarizes the influence of different classes of VNA errormechanisms discussed in Section 4.4 on the measurements of calibration standards. Withthe use of the Fig. 4.2, we can rewrite the equations (2.82) describing the measurementsof calibration standards as

84


Fig. 4.2: Physical error mechanisms in the VNA calibration.

⎧⎪⎪⎪⎨⎪⎪⎪⎩

smnk = sm

nk + mmn (ξm

nk, smnk)

smnk = f (snk + mp

n (ξpn, snk, fk) , pk)

snk = gn (c0n + ξcn, cnk, fk)

, for n = 1, . . . , N, and k = 1, . . . , K. (4.55)

where N is the number of calibration standards, K is the number of frequencies, theindices n and k, refer to the calibration standard number and the measurement frequencyfk, respectively, and ξc

n, ξpn and ξm

nk are the physical error mechanisms, to be explainedbelow. The unknown calibration parameters are defined by

θ =

⎡⎢⎢⎢⎣

β1...

βK

⎤⎥⎥⎥⎦ , (4.56)

where βk denotes the solution vector to the single-frequency VNA calibration problem atthe frequency fk, for k = 1, . . . , K, given by (2.83) and repeated here for the sake of clarity

βk =

⎡⎢⎢⎢⎢⎢⎢⎣

pk

c1k

...cNk

⎤⎥⎥⎥⎥⎥⎥⎦

. (4.57)

85


Errors in the calibration standard definitions and their raw measurements, as defined inParagraph 2.4.1-A, can be expressed in terms of the physical error mechanism as

Δsnk = mpn (ξp

n, snk, fk) , and Δsmnk = mm

n (ξmnk, sm

nk) . (4.58)

The physical error mechanisms are defined as a vector

ξ =

⎡⎢⎢⎢⎣

ξc

ξp

ξm

⎤⎥⎥⎥⎦ , (4.59)

where the subvectors ξc, ξp, and ξm contain the physical error mechanisms correspondingto the calibration standard errors, VNA nonstationarity errors, and VNA receiver errors,respectively, and are defined as

ξc =

⎡⎢⎢⎢⎣

ξc1...

ξcN

⎤⎥⎥⎥⎦ , ξp =

⎡⎢⎢⎢⎣

ξp1...

ξpN

⎤⎥⎥⎥⎦ , and ξm =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ξm11...

ξm1K

ξmN1...

ξmNK

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (4.60)

The joint PDF of the error mechanism is fξ (ξ). As the mechanisms are assumed to bestatistically independent (see Section 4.2), we have E (ξ) = 0 and the covariance matrixof ξ is given by (4.8). We further explicitly recognize the fact that the physical errorsmechanisms modeling the VNA receiver errors at different frequencies are statisticallyindependent (see Paragraph 4.4.2-B). Consequently, the vector ξm is made up of a set ofmechanisms for each measurement frequency.

Now, in order to formulate the maximum-likelihood solution of the set of equations(4.55), we need to define the residual errors (see Appendix B). The residual errors areestimates of the error mechanisms, defined by (4.59) and (4.60), and are thus defined as

r =

⎡⎢⎢⎢⎣

rc

rp

rm

⎤⎥⎥⎥⎦ , (4.61)

86

4.6. SUMMARY

and

rc =

⎡⎢⎢⎢⎣

rc1...

rcN

⎤⎥⎥⎥⎦ , rp =

⎡⎢⎢⎢⎣

rp1...

rpN

⎤⎥⎥⎥⎦ , and rm =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

rm11...

rm1K

rmN1...

rmNK

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (4.62)

Now, the residuals need to satisfy the equations (4.55), from which we obtain the relation-ships

⎧⎪⎪⎪⎨⎪⎪⎪⎩

smnk = sm

nk + mmn (rm

nk, smnk)

smnk = f (snk + mp

n (rpn, snk, fk) , pk)

snk = gn (c0n + rcn, cnk, fk)

, for n = 1, . . . , N, and k = 1, . . . , K.. (4.63)

With these definitions, we can write the maximum likelihood estimate of the unknownresidual errors r and the calibration parameters θ as

argmaxθ,r

fξ (r) , (4.64)

under the equality constraints (4.63).

The remainder of this work (Chapter 5 and Chapter 6) is devoted to developing the nec-essary means for implementing the VNA calibration based on the formulation (4.64).

4.6 Summary

In this chapter, we developed the theoretical and physical foundations of the multi-frequncy description of errors in S-parameter measurements. The key concepts we intro-duced are the physical error mechanism and the multi-frequency S-parameter measurementerror. As pointed out in [39], the main mathematical difficulty with the multi-frequencydescription of errors in S-parameter measurements is the singularity of the multi-frequencycovariance matrix. This singularity makes it impossible to use the traditional form of themultivariate normal PDF. We showed that this PDF can be defined by use of the physicalerror mechanisms rather then directly in terms of the S-parameter measurement them-

87


selves.We further discussed the practical implications of the multi-frequency description of

S-parameter measurement errors for calibrated time-domain measurements. Examples ofsuch measurements are oscilloscope measurement with the correction for the impedancemistmatch or for the adapter parameters. We performed a detailed uncertainty analysis of ageneralized form of a time-domain-waveform correction procedures that employs S-parame-ter measurements. This analysis revealed that the statistical correlations between S-param-eter measurement errors at different frequencies are essential for the correct evaluation ofuncertainties in such waveform correction procedures. Measurement examples illustratingthe importance these correlations are given in [39].

Subsequently, we analyzed the use of the multi-frequency description of errors in S-pa-rameter measurements in the device modeling. We focused on the most common approachbased on the maximum-likelihood estimation. Our analysis yielded an intersting resultsshowing that when using the multi-frequency error description in the device modeling, themisfit between the model and the measurement (quantified with residuals) needs to bewritten in terms of the physical error mechanisms rather the S-parameters.

Finally, we showed that the statistically sound description of the VNA calibrationproblem should be done in terms of the error mechanisms underlying the calibration stan-dard and VNA instrumentation errors. As a consequence, the VNA calibration should beperformed jointly at all measurement frequencies so as to account for the simultaneouscontribution of the error mechanisms to S-parameter measurements at different frequen-cies. We referr to this approach as the error-mechanism-based VNA calibration and in theremainder of this work we develop some necessary tools for the implementation of such acalibration approach. These tools include the generalized multi-frequency VNA calibration(see Chapter 5) and the error-mechanism-based description of the VNA nonstationarityerrors (see Chapter 6) which are the primary source of VNA instrumentation errors.

88

Chapter 5

Generalized multi-frequency VNAcalibration

Pooh looked at his two paws. He knew one ofthem was right, and he knew that when you haddecided which one of them was the right, thenthe other one was left. But he never could re-member how to begin.

A. A. Milne, “House at the Pooh corner”

In this chapter, we develop a generalized multi-frequency VNA calibration approachand demonstrate it in the context of the multi-line TRL calibration method [36] in thecoaxial environment. In the multi-frequency approach to the VNA calibration (see Para-graph 2.4.1-B), the relationships between the calibration standard S-parameters at differ-ent frequencies are exploited as an additional piece of information in the VNA calibration.Here we generalize this way of looking at the VNA calibration problem by using the con-cept of the physical error mechanism (see Section 4.2) to characterize these relationships.We focus only on the mechanisms affecting the calibration standard definitions and use anapproximate description of the VNA instrumentation errors (see Section 3.3.2). However,by use of the framework for the description of the VNA nonstationarity errors presented inChapter 6, the results presented here can be extended to establish a calibration approachbased entirely on the concept of the physical error mechanism (see Subsection 4.5.3).

We start out with the generalized multi-frequency formulation of the VNA calibrationproblem and then proceed with a brief overview the coaxial multi-line TRL calibration.

89

5. GENERALIZED MULTI-FREQUENCY VNA CALIBRATION

Following on that, we discuss sources of errors in calibration standards used in the coax-ial multi-line VNA calibration and based on that we establish the calibration standardmodels. Compared to the classical multi-line TRL calibration, these models contain someadditional parameters which characterize the impact of the physical error mechanisms onthe calibration-standard S-parameters. These parameters, such as inner and outer con-ductor length errors or inner conductor displacement, are then identified along with theVNA calibration coefficients in the course of generalized multi-frequency VNA calibration.In the last section, we present experimental verification of our calibration approach formulti-line TRL calibration with Type-N and 1.85 mm coaxial transmission lines.

5.1 Formulation of the VNA calibration problem

Our generalized multi-frequency VNA calibration approach is a simplified variant of theerror-mechanisms-based calibration discussed in Subsection 4.5.3. The simplifications wemake concern the description of the VNA instrumentation errors, and statistical propertiesof the physical error mechanisms affecting the calibration standards.

As for the VNA instrumentation errors, we pointed out in Subsection 4.4.2 that iden-tification of the physical error mechanisms responsible for these errors is fairly difficult.Therefore, simplified statistical description of the VNA instrumentation error is often used,such as the models discussed in Section 3.3. In our calibration approach, we adapt thesemodels to the multi-frequency formulation of the VNA calibration. In particular, we neglectthe VNA receiver errors, assume that the VNA nonstationarity errors at a single frequencyfk are normally distributed, and neglect the statistical correlations between those errors atdifferent frequencies. Following the notation in Fig. 4.2, we write the probability densityfunction (PDF) of the VNA nonstationarity errors as

fξsk(Δsk) = (2π)−NW/2 |ΣΔsk

|−1/2 exp(

−12ΔsT

k Σ−1Δsk

Δsk

), (5.1)

where N is the number of calibration standards, and W is the number of real-valuedmeasurements obtained for each calibration standard. For the VNA with P ports, wehave W = 2P 2. We further assume that the covariance matrix ΣΔsk

is known up to amultiplicative factor, that is,

ΣΔsk= σ2

kVΔsk, (5.2)

where VΔskis a known matrix and σ2

k is the residual variance (square of the residual

90

5.1. FORMULATION OF THE VNA CALIBRATION PROBLEM

standard deviation).

Concerning the calibration standard errors, we employ the physical error mechanismdescription discussed in Subsection 4.4.1. We assume that the physical error mechanismsuniform PDF, that is, within the range of the parameters ξc we have the PDF fξc(ξc) = C,where C is a constant. Consequently, we can write the original problem (4.64) as

arg maxθ, rc, rp

1, . . . , rpK

σ21, . . . , σ2

K

CK∏

k=1fξs

k(Δrk|σ2

k), (5.3)

under the equality constraints⎧⎨⎩ sm

nk = f (snk + rsnk, pk)

snk = gn (c0n + rcn, cn, fk)

, for n = 1, . . . , N, and k = 1, . . . , K. (5.4)

Taking now the natural logarithm of the likelihood function, we can write this optimizationproblem equivalently as

arg maxθ, rc, rs

1, . . . , rsK

σ21, . . . , σ2

K

lnL(θ, rc, rs1, . . . , rs

K , σ21, . . . , σ2

K) (5.5)

under the equality constraints (5.4), where the log-likelihood function, after dropping theconstant terms, is given by

lnL(θ, rc, rs1, . . . , rs

K , σ21, . . . , σ2

K) = −NW

2

K∑k=1

ln σ2k − 1

2

K∑k=1

1σ2

k

rsTk V−1

Δskrs

k. (5.6)

We further incorporate the constraints (5.4) into the log-likelihood function, which even-tually yields the following problem

arg maxθ, rc, σ2

1, . . . , σ2K

lnL(θ, rc, σ21, . . . , σ2

K), (5.7)

91


where the likelihood function is given by

lnL(θ, rc, σ21, . . . , σ2

K) = −NW

2

K∑k=1

ln σ2k − 1

2

K∑k=1

1σ2

k

rsk(θ, rc)T V−1

Δskrs

k(θ, rc), (5.8)

the underline denotes the convention (A.2), and the residual vector is defined as

rsk(θ, rc) =

⎡⎢⎢⎢⎣

rs1k(θ, rc)

...rs

Nk(θ, rc)

⎤⎥⎥⎥⎦ , (5.9)

and

rsnk(θ, rc) = f−1 (sm

nk, pk) − gn (c0n + rcn, cn, fk) , for n = 1, . . . , N, and k = 1, . . . , K.

(5.10)With the formulation (5.7), we generalize the multi-frequency approach to VNA cali-

bration initiated in [37, 38] (see Paragraph (2.4.1-B)). Our formulation (5.7) differs fromthe methods presented in References [37, 38] in that it does not require the complete knowl-edge of statistical properties of the errors Δsk. Instead, only the relationship between theseerrors, as captured by the matrix VΔsk

, need to be known, and the absolute value of theseerrors is determined from the residual variances σ2

k. In this manner, we follow the typicalapproach used in the single-frequency statistical VNA calibration (see [58, 59] and thediscussion in Paragraph 2.82).

Furthermore, the formulation (5.7) along with the numerical technique described in(5.6), is generic and can be easily adapted to any type of VNA calibration. This, again,differs our approach from the methods presented in References [37, 38] which are dedicatedto a particular VNA calibration technique.

5.2 Coaxial multi-line VNA calibration

In this section, we briefly review the coaxial multi-line TRL calibration. This calibrationserves as a workhorse for the demonstration of the generalized multi-frequency approach toVNA calibration. We begin with a brief overview of the classical multi-line TRL methodof [36], and then discuss the implementation of the multi-line TRL method in the coaxialenvironment. Following on that, we briefly review some computational aspects of themulti-line TRL method.

92

5.2. COAXIAL MULTI-LINE VNA CALIBRATION

5.2.1 Classical multi-line VNA calibration

The multi-line TRL calibration method [36] is an extension of the classical TRL method[29, 67]. In the TRL method, the following calibration standards are used:

• a section of a transmission line with known length and unknown characteristicimpedance and propagation constant,

• a reflect standard which is assumed to have the same but otherwise unknown reflec-tion coefficient on both VNA ports,

• a thru standard which is typically realized as a direct connection of the VNA ports1.

In the course of the calibration, the unknown propagation constant of the line is determinedalong with the VNA calibration coefficients. Based on this propagation constant and thecapacitance per unit length of the transmission line, one can then determine the charac-teristic impedance of the line, and thus reset the reference impedance of the calibration toan arbitrary value, typically 50 Ω (see Paragraph 2.4.2-C).

Also, the unknown reflection coefficient of the reflect standard is determined along withthe VNA calibration coefficients. The value of this reflection coefficient is not necessaryto complete the calibration, however, it can be used to verify the calibration correctness,provided that an á priori estimate of this reflection coefficient is available.

In the multi-line TRL method, instead of a single transmission line, a set of multipletransmission lines with different lengths is used. This allows to overcome the inherentbandwidth limitation of the classical TRL method [29] and provides higher accuracy due toredundant measurements [36]. Also, the use of redundant number of calibration standardsallows to employ the residual analysis (see Subsection 3.5.3) to detect random connectionerrors or calibration-standard-definition errors.

5.2.2 Coaxial air-dielectric line as a calibration standard

The implementation of the multi-line TRL calibration in the coaxial environment isbased on transmission lines realized as air-dielectric lines (commonly referred to as airlines)and the reflect standard realized as an offset termination, that is, a termination with a

1If such a connection cannot be realized (e.g., in on-wafer environments, or a non-insertable configu-ration of VNA test ports in coaxial connector environment), a short section of transmission line with adifferent length than the base line is used instead. This modification of the TRL method is referred to asLRL method [130].

93


section of an airline. The center conductor in airlines is not supported by plastic beads,which are used in other coaxial devices, such as adapters or attenuators. The lack of beadsallows to better predict the S-parameters of the air-lines, because electrical parametersof the beads cannot be well predicted nor controlled [131]. Connecting the air-lines withan unsupported center conductor is, however, difficult and requires great operator skill,particularly at smaller connector sizes.

As for the offset terminations, their center conductor is supported by the terminationitself, therefore connecting them is much simpler and less prone to accidental damaging.Consequently, calibration methods based on multiple reflect standards, such as the offset-short calibration (see [33, 38, 79, 132]), are often preferred at smaller connector sizes [133].

Coaxial air lines are manufactured in either insertable or noninsertable configuration.In the insertable configuration, line ends have connectors with opposite sexes, while in thenoninsertable configuration, the line connectors are of the same sex, typically male. Thenoninsertable configuration is preferred as far as the calibration accuracy is concerned sincethe same reflect standard can be used on both VNA ports; thus the error due to its asym-metry is reduced (see Subsection 5.3.4). However, two-port DUTs are typically insertable,therefore an additional adapter needs to be first characterized, in order to measure suchDUTs. The insertable configuration eliminates this problem, however, at the cost of tworeflect standards with different connector sexes which inevitably have slightly different re-flection coefficients. In this work, we focus on the coaxial multi-line TRL calibration inthe insertable calibration.

5.2.3 Computational aspects

The original implementation of the multi-line TRL method [36] is based on an approx-imate error analysis of the classical TRL method. This analysis yields the uncertaintiesfor individual TRL calibrations that can be constructed out of a given set of transmissionlines. Based on these uncertainties, at each frequency, a set of optimal TRL calibrationsis constructed whose results are then statistically averaged.

The uncertainty analysis of the classical TRL method by [36] is based on the linearerror propagation, hence the statistical averaging is performed with the use of linear least-squares techniques. Consequently, the resulting algorithm is very robust as it does notrequire any starting point. However, it does not provide a true least-squares solution tothe calibration problem due to some simplifying assumptions made in the error analysis.Also, the resulting algorithm is quite complicated. Therefore, some other implementations

94

5.3. ERRORS IN THE COAXIAL MULTI-LINE VNA CALIBRATION

of the multiline TRL method have been proposed [56, 59] which use a true least-squaresformulation of the calibration problem and employ nonliner optimization techniques tosolve it. These approaches are conceptually much simpler than that of [36], however, theyare more time consuming and require a good starting point.

5.3 Errors in the coaxial multi-line VNA calibration

In this section, we describe the sources of errors in the implementation of the multi-lineTRL calibration in the coaxial environment. We discuss the physical origins of these errorsand develop models that characterize their impact on calibration-standard S-parameterdefinitions. The discussion in this section serves as background for the development ofmodels for the coaxial calibration standards (see Subsection 5.4) that employ the conceptof the physical error mechanism (see Section 4.2).

5.3.1 Variation of connector-interface electrical parameters

��

� ��

��

Fig. 5.1: Connector discontinuity and the electri-cal reference plane.

One assumption commonly made inthe VNA measurements, and in partic-ular in the multi-line TRL calibration,is that the connector interface param-eters, that is, electrical parameters ofthe discontinuity at the VNA test port,are identical for all calibration stan-dards and DUTs. This is illustratedin Fig. 5.1. Connector discontinuity D

physically occupies some space aroundthe connector mating plane. Defini-tions of the calibration standards, however, do not account for this discontinuity andare established with respect to the electrical reference plane. Consequently, as long as thediscontinuity D is the same for all measured device, it becomes part of the measurementsystem and the electrical reference plane becomes the calibration reference plane.

In practice, however, this assumption is often not met. As a result, some common partof the connector interface discontinuity gets lumped in to the VNA calibration coefficients,however, the differences with respect to that common part contribute to the measurementerror. The main reason for the variation of the connector interface electrical parameters,

95


apart from imperfect mechanical repeatability of the connection, are the width variationof the center conductor gap and variation of the socket fingers bending due to dimensionaltolerances on the connector pin and thickness of socket fingers, and variation of the centerconductor eccentricity.

��

��

�

Fig. 5.2: Simplified schematic of the coaxialconnector interface (not in scale).

The origin of these two errors can bebetter understood based on the Fig. 5.2.This figure shows a simplified longitudi-nal cross section of the slotted pin-socket(sexed) connector interface [134]. The con-nection between the inner conductors inthis interface occurs through the fingers ofthe socket (usually four or six fingers) thatcontact the pin. These fingers are flexibleso as to reliably contact pins that have dif-ferent diameters or are slightly eccentricity.Consequently, for the same socket, the con-

figuration of its fingers vary when connecting different pins. Also, dimensional toleranceson the thickness of the socket fingers lead to the variation of the electrical parameter ofthe interface.

As to the center conductor gap, its presence is necessary so as to allow the connectionbetween the pin and the socket. Consequently, both female and male center conductors areset back by a small distance of around few tens of micrometers. This distance is referred toas the pin depth. Although the pin depth limits can be well controlled in the manufacturingprocess, its particular value for a given device cannot be predicted. The pin depth mayalso vary due to multiple reconnections or mishandling of the connectors2.

In the case of the air-dielectric coaxial transmission lines the situation is even morecomplicated. The center conductor of such lines is unsupported, thus the center conductorgap is different each time the line is connected. This variation can be minimized througha proper operator technique, however, it cannot be completely eliminated.

A. Center conductor gap. The gap forms a radial in-cut in the center conductor whosewidth g is much smaller than the depth, determined by the inner conductor and pin

2Calibration standard manufactures typically provide the users with special gages allowing for a periodiccontrol of the pin-depth and thus eliminating devices with protruding center conductors.

96


diameters, d and dp, respectively (see Fig. D.1). Dimensions of this in-cut for typicalconnector sizes can be determined based on the values given in Tab. 5.1.

Connector Type-N 3.5mm 2.4mm 1.85mmd [μm] 3040 1520 1042 804dp [μm] 1651 927 511 511

gmax [μm] 13 13 13 13

Tab. 5.1: Inner conductor and pin diameter, d and dp, re-spectively, and maximal width of the gap gmax for typicalcoaxial-connector types [135].

Electrical properties of thecenter conductor gap can bemodeled with a short sec-tion of a radial waveguide ter-minated with an ideal shortcircuit [92, 136]. In Ap-pendix D, we show that theinput impedance of such asection can be approximatedas

Z = jωμ0

2πg ln(

πd − Nw

πdp − Nw

)+ (1 + j)

√ωμ0

2σ1πln(

πd − Nw

πdp − Nw

). (5.11)

where N is the number of fingers, w is the width of the space between the fingers, andσ is the metal conductivity. Equation (5.11) shows that the gap impedance is a sum ofan inductive component, depending on gap width g and a surface-impedance componentwhich is independent of g. This surface-impedance component results from the skin-deptheffect in the gap walls and the connector pin. Since the ratio of the gap width to thegap depth is small, that is, 2g

d−dp� 1, we neglect the skin-depth effect in the connector

pin. Reference [136] gives, without derivation, a similar expression for the gap inductancewhich, however, does not account for losses. Also the formula given in [92] for the 7mmconnector can be derived from (5.11) by substituting proper dimensions for d and dp (see[135]), N = 0, and neglecting the conductor loss.

Scattering parameters of the series impedance representing the center conductor gapcan be written as [1]

Sgap =1

Z ′ + 2

⎡⎣ Z ′ 2

2 Z ′

⎤⎦ , (5.12)

whereZ ′ = Z

Z0, (5.13)

is the normalized impedance, and Z0 is the characteristic impedance of the coaxial trans-mission line given by (C.7). Typically, the correction due to conductor losses is very small,

97


that is, vRc

2ωZ00� 1, hence we can approximate (5.13) as

Z ′ = Z

Z00[1 + (1 − j) vRc

2ωZ00

] ≈ Z

Z00= jωL′ + (1 + j)R′

√ω

ω0, (5.14)

where the normalized gap inductance L′ and surface resistance R′ are functions of the gapgeometry and are given by

L′ = gμ0

2πZ00ln(

πd − Nw

πdp − Nw

), (5.15)

andR′ = 1

Z00

√ω0μ0

2σ1πln(

πd − Nw

πdp − Nw

)(5.16)

where ω0 is a fixed reference frequency, typically ω0 = 2π109rad/s. Consequently, to firstorder, the frequency dependence of the normalized gap impedance does not depend onthe frequency variation of the characteristic impedance Z0. Taking further into accountthat the gap impedance is small, that is |Z ′| � 1, we can eventually approximate itsS-parameters as

Sgap ≈⎡⎣ 1

2Z ′ 1 − 12Z ′

1 − 12Z ′ 1

2Z ′

⎤⎦ . (5.17)

In order to quantitatively illustrate the effect of the gap, we calculated the magnitudeof S11 in (5.17) based on model (5.11). We used the the dimensions corresponding to the1.85 mm coaxial standard taken from Tab. 5.1 and assumed the conductivity of copper(σ = 5.7 · 107S/m). For this connector we have four fingers, that is, N = 4, and fromobservations under the microscope we estimated w ≈ 100μm. Inserting those values into(5.15) and (5.16), we obtain the normalized inductance per micrometer of gap’s width

L′1.85 mm = 2.28 · 10−15[1/μm], (5.18)

and the normalized surface resistance at 1GHz

R′1.85 mm@1GHz = 1 · 10−4. (5.19)

Magnitude of the gap’s reflection coefficient for the frequency range 0−67GHz is shown inFig. 5.3. We see that this reflection coefficient increases linearly with frequency. The maxi-mum value we expect based on the model (5.11) for the gap width within the specifications

98


given in Tab. 5.1 is around 0.006 which corresponds to −44 dB. The surface-impedancecomponent of this reflection coefficient is of order 4 · 10−4 at 67GHz, hence it can beneglected compared to the inductive component described by (5.15).

� ��

��

��

��

��

��

��

Fig. 5.3: Magnitude of the gap reflection coef-ficient for the 1.85mm coaxial connector stan-dard for some typical gap widths.

Measurements and electromagnetic sim-ulations, in general, confirm that the pri-mary electrical properties of the gap canbe modeled as a series inductance [124,126, 135]. In measurement results reportedin [135], reflection coefficients larger byroughly up to 30% than the prediction ofthe model (5.11) are obtained. However,no uncertainties for those measurements arereported and some obvious systematic er-rors (ripples) can be observed in those mea-surements. Reference [124] simulates a sim-plified model of the gap which does not in-clude the fingers. Reflection coefficients of the gap obtained in [124] for 3.5mm, 2.4mm,and 1.0mm coaxial connectors agree well with the model (5.11) when setting N = 0.

� ��

��

��

��

Fig. 5.4: Magnitude of the gap reflection coef-ficient at 67 GHz for some typical gap widths,based on the model (5.11) and electromag-netic simulations [126].

Reference [126] simulates a more com-plex model of the gap which accounts forthe socket fingers and other geometricaldetails of the gap structure. In Fig. 5.4we show a comparison of results reportedin [126] with predictions of the analyticalmodel (5.11) at the frequency 67GHz, forsome typical gap widths. We see that re-flection coefficients reported in [126] are ingeneral much larger than those predicted bymodel (5.11). This is due to the fact thatanalysis of [126] accounts also for the re-flection caused by the socket fingers. Thisreflection is independent of the gap width,and from Fig. 5.4 it can be estimated to be around 0.01 at 67GHz. If we add this reflec-tion to the prediction of (5.11), both sets of results agree very well. This confirms that

99


the analytical model (5.11) is capable of estimating the reflection coefficient caused by thegap itself.

Reference [126] reports also that for extremely small gaps, a resonance behavior canbe observed. We believe, however, that such a situation can be avoided through a properoperator technique.

B. Connector socket fingers. The errors due to nonreproducibility of the connector-socket finger bending have not received much attention in the literature. Reference [135]compares results of various electromagnetic analyzes with the prediction of analytical mod-els. From these results, one concludes that these errors may be comparable with the errorsdue to nonreproducibility of the center conductor gap. However, no further details aregiven in Reference [135] as to the description of the socket finger used in the electromag-netic analyzes nor the assumptions made in the analytical models. In this work, we decidedto neglect this source of errors.

5.3.2 Variation of line’s characteristic impedance and propaga-tion constant

In the multiline TRL calibration method, we assume the all of the lines have the samecharacteristic impedance Z0 and propagation constant γ. Nominal S-parameters of a singleline are therefore given by

S =⎡⎣ 0 e−γl

e−γl 0

⎤⎦ , (5.20)

where γ is the complex propagation constant and l is the line length. Scattering parametersin (5.20) are normalized with reference to the characteristic impedance Z0 of the line, thusthe reflection coefficients are equal to zero.

In practice, however, we observe a slightly different value of Z0 and γ for each line. Thisvariation results from manufacturing tolerances of line diameters and metal conductivity,and changes in the eccentricity due to different alignment of each line’s center conductorwith respect to the test-ports. Another important factor is the nonuniformity of line diam-eters, and variation of the eccentricity and conductivity along the line. This nonuniformityis primarily attributed to the mechanical vibrations of the tools used to manufacture theinner and outer conductor.

In the following, we analyze deviations in S-parameters of a line caused by the abovefactors. We first consider the case of a line with uniform but perturbed diameters and

100


conductivity. We then proceed with the analysis of a slightly nonuniform line and showthat such a line can be approximately treated as a uniform line with some equivalentperturbations of the propagation constant and characteristic impedance.

A. Uniform line. Characteristic impedance and propagation constant are linked withthe diameters, eccentricity and metal conductivity through the relationship (C.7) given inAppendix C. Since the dependence on the eccentricity e is quadratic, small changes in thisparameter can be neglected. As for the other parameters, we can readily determine therelationship between changes in the inner and outer diameter, Δd and ΔD, respectively,and the metal conductivity Δσ and the resulting changes ΔZ0 and Δγ in the characteristicimpedance and propagation constant, respectively, as

ΔZ0

Z0≈ ΔZ0

Z00≈ ΔZ00

Z00+ (1 − j) α

β0

ΔRc

Rc

≈ ΔZ00

Z00, (5.21)

andΔγ ≈ (1 + j)α

(ΔRc

Rc

− ΔZ00

Z00

), (5.22)

whereΔZ00

Z00= 1ln D

d

(ΔD

D− Δd

d

), (5.23)

andΔRc

Rc

= −12Δσ

σ− d

D + d

ΔD

D− D

D + d

Δd

d, (5.24)

and α = Reγ is given by (C.9), Z00 is the lossless characteristic impedance of the line givenby (C.1), and Rc is the resistance per unit length of the transmission line given by (C.10).In (5.21) we neglected the contribution of errors in Rc to the change of characteristicimpedance, since for low loss coaxial airlines α

β0�1.

Equation (5.21) indicates that the relative changes in the characteristic impedanceare to first order frequency independent. On the other hand, from (5.22), we see thatchanges of the propagation constant follow the frequency dependence of the real part ofthe propagation constant.

The impact of errors in the line’s characteristic impedance on its S-parameters canbe determined with the use of the relationship (2.19). Assuming that the characteristicimpedance changes from Z0 to Z0 = Z0 + ΔZ0, scattering parameters of the two portnetwork described by (2.19), which correspond to a circuit description of a transformer,

101


can be written as [45]

Str =⎡⎣ Γ Q

√1 − Γ2

Q−1√1 − Γ2 −Γ

⎤⎦ , (5.25)

whereΓ = Z0 − Z0

Z0 + Z0, (5.26)

and

Q =

√√√√1 − jImZ0/ReZ0

1 − jImZ0/ReZ0. (5.27)

is a factor accounting for the complex-valued character of the characteristic impedance.For the coaxial transmission line with small conductor losses, that is, when α

β0� 1, we can

write the first-order approximations

Γ ≈ 12ΔZ00

Z00, (5.28)

andQ ≈ 1 − j

12

α

β0

(ΔRc

Rc

− ΔZ00

Z00

)≈ 1. (5.29)

Assuming further that the changes of the characteristic impedance are small, we can finallyapproximate (5.25) as

Str ≈⎡⎣ 1

2ΔZ00Z00

11 −1

2ΔZ00Z00

⎤⎦ . (5.30)

By adding the network (5.30) on either end of the line described by (5.20) and account-ing for the error in propagation constant (5.22), we obtain a first-order approximation ofline’s S-parameters

S11 = S22 ≈ 12ΔZ00

Z00

(1 − e−2γL

), (5.31)

S21 = S12 ≈ e−γl

[1 − (1 + j)α

(ΔRc

Rc

− ΔZ00

Z00

)l

]. (5.32)

In Tab. 5.2 we summarize the worst case values of ΔZ00Z00

and ΔRc

Rcdue to typical variations

of diameters obtained from [135]. Based on that, we estimated that for the 1.85mmstandard the maximum error in magnitude of the reflection coefficient due to dimensionalvariation is around 0.0023, that is −53 dB. As to the transmission coefficient, we estimatethat for a line with conductivity of copper (σ = 5.7 · 107S/m), at 67GHz, an error of

102


1% in Rc or Z00 leads to 0.4% of error in α which corresponds to a relative error inthe transmission coefficients of 0.006% per 1 cm of line length. Consequently, we seethat dimensional variations affect mainly the reflection coefficients, while the transmissioncoefficients are much less sensitive to the tolerances on line’s diameters.

Connector Type-N 3.5mm 2.4mm 1.85mmD [μm] 7000 3500 2400 1850d [μm] 3040 1520 1042 804

ΔD [μm] 5 2.5 2.5 2.5Δd [μm] 2.6 2 2 2(

ΔZ00Z00

)MAX

[%] 0.19 0.24 0.35 0.46(ΔRc

Rc

)MAX

[%] 0.08 0.11 0.17 0.21

Tab. 5.2: Diameters and their tolerance for typical connec-tor types [135], and the resulting maximum relative changesin the lossless characteristic impedance Z00 and resistanceper-unit-length Rc.

Regarding the error inmetal conductivity, the situ-ation is more complex. Inorder to lower the resistiv-ity, air transmission lines aretypically plated with silveror gold. The thickness ofthe plating layers and theirroughness are in general notwell controlled [136, 137],hence the effective conductiv-ity of the plating may undergomuch larger relative varia-tions than dimensional pa-rameters. From (5.24), we see that 1% of error in conductivity leads to 0.003% of errorin transmission coefficients per 1 cm of line length. Therefore variation of the conductivitymay be the most important contributor to the error in the propagation constant.

B. Slightly nonuniform line. Another factor causing the variation of the characteris-tic impedance is the nonuniformity of the diameters, conductivity and eccentricity alongthe line. In Appendix E, we show that S-parameters of a slightly nonuniform coaxialtransmission line with small conductor losses can be written as

S11 ≈ γ

l∫0

ΔZ00 (x)Z00

e−2γxdx, (5.33)

S22 ≈ γe−2γl

l∫0

ΔZ00 (x)Z00

e2γxdx, (5.34)

and

103


S21 = S12 = e−γl [1 − (1 + j)αΔlα] , (5.35)

where

Δlα =l∫

0

[ΔRc (x)

Rc

− ΔZ00 (x)Z00

]dx, (5.36)

In the derivation made in Appendix E, we neglect the effect of conductivity variation onthe reflection coefficients, and the eccentricity variation as they lead to second order terms.

The frequency-independent parameter Δlα characterizes the change in line losses due tothe nonuniformity of the diameters, conductivity and eccentricity along the line. Rewriting(5.35) in a slightly different form

S21 = S12 ≈ e−(1+j)α(l+Δlα)e−jβl, (5.37)

we see that this parameter can be interpreted as an equivalent “lengthening” of the linefor the losses.

The change in the reflection coefficients, as given by (5.33) and (5.34), cannot in generalbe characterized by a single frequency independent parameter. Furthermore, the frequencydependence of (5.33) and (5.34) is determined be the actual profile of the diameters. Since,in general, we do not know this profile, we cannot also predict these dependence.

In the context of multiline TRL calibration, a slightly nonuniform line can be seen as auniform line with an equivalent perturbation of the propagation constant and characteristicimpedance and some residual reflections resulting from the actual nonuniformity profile.This can be prooven in the following manner. In the calibration algorithm, we assume thatthe line is uniform, and optimize the VNA calibration coefficients and unknown parametersof the line so as to minimize the misfit between the line model prediction and the line’sS-parameters determined from raw measurements. In the case of a nonuniform line, thismisfit is thus characterized by the integrals in (5.33) and (5.34). We can show that theseintegrals become minimal when we subtract the mean value of the impedance profile, thatis,

104


S11 ≈ 12ΔZ00

Z00

(1 − e−2γl

)+ γ

l∫0

[ΔZ00 (x)

Z00− ΔZ00

Z00

]e−2γxdx, (5.38)

S22 ≈ 12ΔZ00

Z00

(1 − e−2γl

)+ γe−2γL

l∫0

[ΔZ00 (x)

Z00− ΔZ00

Z00

]e2γxdx, (5.39)

whereΔZ00

Z00= 1

l

l∫0

ΔZ00 (x)Z00

dx. (5.40)

Consequently, a nonuniform line is seen in the multiline TRL calibration as a uniform linewith the characteristic impedance error defined by (5.40), error in the propagation constantdescribed by (5.35), and residual reflections given by the integrals in (5.38) and (5.39).

5.3.3 Line length error

The only numerical parameters passed to the multiline TRL calibration method are theline lengths. For the coaxial airlines, these lengths are typically defined as the lengths of theouter conductor and are measured with mechanical blocks [90]. The value stemming fromthis measurement will be in general different than the actual length of the line due to themeasurement error and the compression of the line in the actual measurement environmentdue to the torque applied to the connector interface [138].

Scattering parameters of a line with perturbed length can be readily obtained from(5.20) as

S11 = S22 = 0, (5.41)

and

S21 = S12 ≈ e−γl (1 − γΔl) . (5.42)

From (5.42), we can derive the upper bound for relative errors in the transmission coeffi-cients as

∣∣∣∣∣ΔS21

S21

∣∣∣∣∣ =∣∣∣∣∣ΔS12

S12

∣∣∣∣∣ ≤ |β0|Δl. (5.43)

105


For 1.85mm precision air lines, typically the error in length determination is smallerthan 5μm. This leads to the maximum relative error in transmission coefficients of 0.7%at 67GHz.

5.3.4 Reflect asymmetry

In the classical TRL [29] and multi-line TRL [36] calibrations, it is assumed that thereflect standard presents the same reflection coefficient on both VNA ports. Thus, thisstandard can be seen as a symmetrical two-port network without transmission.

Difference in the reflection coefficient of the reflect standard presented on both VNAports, which we refer to as the asymmetry of the reflect standard, leads to errors in boththe classical TRL [29] and the multiline TRL [36] calibrations. In the case of the coaxialimplementation of the multiline VNA calibration, sources of the reflect asymmetry aredifferent in the noninsertable and insertable configurations. In the case of the noninsertableconfiguration, we use the same reflect standards on both VNA ports. Consequently, theasymmetry results only from the nonrepeatability of the connector interface parameterswhich is of a random nature. However, in the case of the of the insertable configurationthe situation is different. In this configuration, we use in fact two reflect standards withdifferent connector sexes. Although these standards are designed so as to exhibit the samereflection coefficient, the nonreproducibility of the connector interface (see Subsection 5.3.1)and the manufacturing tolerances always lead to some small difference in the reflectioncoefficient of both devices.

In the case of a single reflect standard, its asymmetry gets lumped into the VNAcalibration coefficients and thus cannot be detected. This asymmetry, however, could bedetected with the use of multiple reflect standards with different reflection coefficients. Theresulting calibration would use both multiple lines and multiple pairs of reflect standards.Implementation of such a calibration method, however, is beyond the scope of this work.

5.4 Calibration standard models

In this section, we present extendend models for the coaxial calibration standards usedin the multi-line VNA calibration. These models account for the error mechanisms dis-cussed in Section 5.3. Contribution of these mechanisms is described with additionalfrequency-independent parameters which can be thought of as the electrically-equivalent

106

5.4. CALIBRATION STANDARD MODELS

��

��

��

Fig. 5.5: Coaxial airline in the multi-frequency multiline TRL calibration: (a) overview,(b) electrical model.

error mechanisms (see Section 4.2). These parameters are then identified, along with theVNA calibration coefficients, in the course of the multi-frequency VNA calibration.

5.4.1 Lines

As discussed earlier on, a set of coaxial airline exhibits characteristics that slightly differfrom the nominal description (5.20). We illustrate that in Fig. 5.5a where we show differentphysical error mechanisms affecting a real coaxial airline. These mechanisms include (seeSection 5.3):

• errors in the lengths of the inner and outer conductor, Δlin and Δlon, with respectto the nominal lengths, li0n and lo0n, respectively;

107


• nonreproducibility of the center conductor gap on both line ends due to the randomdisplacement Δln of the horizontal symmetry axis of the inner conductor with respectto the horizontal symmetry axis of the outer conductor;

• nonuniformity and nonreproducibility of the diameters of the inner and outer con-ductor, ΔDn(x) and Δdn(x), defined with respect to the nominal diameters, d0 andD0, respectively;

• nonuniformity and nonreproducibility of the conductor losses, characterized by theconductivity error Δσ(x) defined with respect to the nominal conductivity σ0.

The impact of these physical error mechanisms on the line S-parameters can be equivalentlydescribed with the circuit shown in Fig. 5.5b. This circuit consists of:

• an ideal transmission line with the length li0n +Δli0n, propagation constant γ +(1+j)αΔlαn and the nominal characteristic impedance Z0; the frequency-independentparameter Δlαn, referred as the loss correction factor, is defined by (5.36), and char-acterizes the impact of diameter errors and conductivity error, and the nonuniformityof the line on the propagation constant;

• two transformers describing the change of the line’s characteristic impedance withrespect to the nominal impedance Z0; these transformers are characterized by thefrequency-independent characteristic impedance correction ΔZ00n, defined by (5.40),which captures the impact of diameter errors and the nonuniformity on the line’scharacteristic impedance; in the calibration algorithm we use the normalized charac-teristic impedance correction ρ0n = ΔZ00n/Zref , where Zref = 50Ω.

• two inductances modeling the nonreproducibility of the center conductor gap; theseinductances are determined from the model (5.15) and are function of the inner andouter conductor length errors, Δlin and Δlon, respectively, and the center conductordisplacement Δln.

Assuming that the errors are small, scattering parameters of the line standard shown inFig. 5.5b can to first order be written as

SLn = SL0n +⎡⎣ wT

11ξLne−γli0nwT

12ξLn

e−γli0nwT21ξLn

wT22ξLn

⎤⎦ , (5.44)

108

5.4. CALIBRATION STANDARD MODELS

where the electrically-equivalent error mechanisms (see Section 4.2) are given by

ξTLn

= [Δlαn, ρ0n,Δln,Δlin,Δlon] , (5.45)

the frequency-dependent weighting functions, resulting from models developed in in Sub-sections 5.3.1 through 5.3.3, are captured in vectors

w11 =12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

01 − e−2γli0n

jωL′g(1 − e−2γli0n)

−j 12ωL′

g(1 + e−2γli0n)j 1

2ωL′g(1 + e−2γli0n)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, w22 =12

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

01 − e−2γli0n

−jωL′g(1 − e−2γli0n)

−j 12ωL′

g(1 + e−2γli0n)j 1

2ωL′g(1 + e−2γli0n)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (5.46)

and

w21 = w12 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− (1 + j)α

00

j 12ωL′

g − γ

−j 12ωL′

g

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (5.47)

and SL0n are the nominal S-parameters of the air-line. These parameters are determinedunder the assumptions that ξLn

= 0, that is, for a uniform airline with the nominalconductivity σ0 and characteristic impedance Z0, the nominal diameters, d0 and D0, thenominal conductor lengths, li0n and lo0n, and equal center-conductor-gap widths on bothline ends. Consequently, nominal S-parameters of a coaxial airline can be written as

SL0n =⎡⎣ j 1

2ωL′g

(1 + e−2γli0n

)g0 e−γli0n

(1 − jωL′

gg0)

e−γli0n

(1 − jωL′

gg0)

j 12ωL′

g

(1 + e−2γli0n

)g0

⎤⎦ , (5.48)

where L′g is the normalized inductance per-unit-length of the center conductor gap, defined

in (5.15), and g0n = (lo0n − li0n)/2 is the nominal center conductor gap.

109


Consequently, a set of NL real lines is characterized by the following parameters

ξΔlα = [Δlα1, . . . ,ΔlαNL]T (5.49)

ξρ0 = [ρ01, . . . , ρ0NL]T , (5.50)

ξΔl = [Δl1, . . . ,ΔlNL]T (5.51)

ξΔli= [Δli1, . . . ,ΔliNL

]T , (5.52)ξΔlo = [Δlo1, . . . ,ΔloNL

]T . (5.53)

which are determined, along with the VNA calibration coefficients, in the course of themulti-frequency VNA calibration. However, not all of the parameters in vectors (5.49)through (5.53) can be uniquely determined. This will be discussed in Section 5.5.

5.4.2 Reflect

A nominal description of the reflect standard in the classical multi-line TRL calibrationis given by

SR =⎡⎣ ΓR 0

0 ΓR

⎤⎦ , (5.54)

where ΓR is the unknown reflection coefficient of the reflect, determined during the calibra-tion. In our approach we do not account for the possible asymmetry of the reflect standard(see Subsection 5.3.4) and use the nominal description (5.54).

5.4.3 Thru

In the noninsertable calibration, the thru standard is realized as a line section anddescribed by the model from Subsection 5.4.1. In the case of the insertable configuration,considered in this work, a direct thru connection of VNA ports is typically used. Weassume that its S-parameters are

ST =⎡⎣ 0 11 0

⎤⎦ . (5.55)

110

5.5. SOLUTION UNIQUENESS

5.5 Solution uniqueness

In this section, we discuss this issue of assuring the uniqueness of the solution to thecalibration problem (5.7). We first review the mathematical conditions for the solutionuniqueness and show that these conditions are related to the linear independence of theestimated parameters in the vicinity of the solution. We then analyze the relationshipsbetween the parameters estimated in the multi-line VNA calibration and identify thosethat are linearly dependent. Following on that, we propose linear constraints for theseparameters so as to provide their unique identification. These constraints are relatedto some intuitive statistical properties of the error mechanisms affecting the calibrationstandards.

5.5.1 Mathematical framework

Conditions for the solution uniqueness in the maximum-likelihood system identificationare reviewed in Section B.5. We show there that the solution is unique if the Jacobianmatrix of the residuals determined at the solution is full rank. In the case of the multi-frequency VNA calibration problem (2.94), with residuals defined by (5.10), this Jacobianmatrix is given by3

J =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

∂rs1(θ, rc)∂θT

∂rs1(θ, rc)

∂(rc)T

... ...∂rs

K(θ, rc)∂θT

∂rsK(θ, rc)∂(rc)T

⎤⎥⎥⎥⎥⎥⎥⎥⎦

. (5.56)

Now, direct investigation of the rank of the matrix (5.56) is complicated, hence wefollow here a different approach. We first note the the matrix (5.56) has more rows thencolumns as there are more equations than estimated parameters . Thus, if the matrix J isrank deficient, some of its columns are linear combinations of other columns. This in turnreflects the fact that there exist some relationships between the estimated parameters θ

and rc. Hence, when verifying the uniqueness of the solution to (5.7), instead of analyzingthe rank of (5.56), we look for relationships between the identified parameters.

The existence of such relationships indicates, that the description of the estimationproblem uses more parameters than the actual number of degrees of freedom. The most

3The dependence of the likelihood function on the residual variances σ21 , . . . σ2

K has been eliminated byuse of the concentrated likelihood approach (see Subsection 5.6).

111


common reason for such a situation, apart from the error in the mathematical descriptionof the problem, is the inherent unidentifiability of some parameters from a given set ofmeasurements. An example of such a situation is the identification of the 8-term VNAmodel (see Section 2.3.1-B). Although the physical description of this model uses eightparameters, only seven independent terms can be identified in the calibration since some ofthe physical model parameters appear always as products in the equations (2.65) through(2.68). Consequently, we need to either accept this ambiguity (which does not introduceany error in the case of the 8-term VNA model, since both the calibration and correctionequations use only seven independent terms describing the VNA), or make some additionalassumptions as to the relationships between parameters that cannot be identified.

One way to write such assumptions is through some additional constraints imposed onthe optimization problem (5.7). In the simplest case, these constraints take on the form oflinear equality constraints, that is,

Aθ = 0, (5.57)

where matrix A consists of M linearly independent rows and rank (A) = M where M

is the dimension of the null space of the Jacobian matrix J. The constrained optimiza-tion problem formed by (5.7) and (5.57) can then be solved through the direct variableelimination [139].

5.5.2 Relationships between estimated parameters

The parameters estimated in the multi-frequency multiline TRL calibration are con-tained in the vectors θ and rc. The elements of the vector rc are the estimates of theairline parameters grouped in the vectors (5.49) through (5.53), while the vector θ, definedby (2.89), contains the VNA calibration coefficients along with the unknown propagationconstant of the lines and the unknown reflection coefficient of the reflect at the frequencyfk, for k = 1, . . . , K.

The potential non-uniqueness of the solution to (5.7), given by estimates of θ andrc, may result either from relationships between the frequency independent parametersgrouped in the vector rc or relationships between those parameters and the parametersgrouped in the vector θ. Indeed, the columns of the Jacobian matrix of the residualfunction (5.10), corresponding to the parameters θ, are obviously linearly independent,since the VNA model does not account for the relationships between VNA calibration

112

5.5. SOLUTION UNIQUENESS

coefficients at different frequencies.Now, in the classical multiline TRL calibration, the identification of the propagation

constant requires the knowledge of the line lengths. In the course of the TRL calibrationthe electrical length of the line is identified which, based on the physical line length, allowsto determine the propagation constant. In the multi-line TRL calibration, the algorithmfor calculating the propagation constant is more complicated and uses t averaging of thepropagation constants estimated in individual TRL calibrations. As a consequence, theunique identification of both line length corrections captured in the vector (5.52) and (5.53),and the propagation constant is impossible. Indeed, we can easily show that for a givensolution to the calibration problem (5.7), another solution with the same maximum valueof the likelihood function can be derived by simply adding an arbitrary length to all of theelements of the vectors (5.52) and (5.53). This does not affect the center conductor gapand the error resulting from this additional length gets compensated by the adjustmentthe complex propagation constant γ determined in the calibration. Consequently, we needto make some assumption as to the length corrections (5.52) and (5.53) in the form of theconstraints ⎡

⎣ aΔli

aΔlo

⎤⎦

T ⎡⎣ pΔli

pΔl0

⎤⎦ = 0. (5.58)

A similar reasoning leads to the conclusion that another assumption in the form

aTΔlαpΔlα = 0, (5.59)

needs to be made as to the loss correction factors in the vector (5.51). Indeed, a constantvalue Δlα added to all of the elements of the vector (5.51) gets also compensated by theadjustment of the propagation constant.

Regarding the correction of the characteristic impedance captured in the vector (5.50),we also note that the unique identification of those parameters and the VNA model param-eters is not possible. Indeed, for a given solution to the calibration problem (5.7), addingan arbitrary impedance to the parameters in the vector (5.50) leads to another solutionwith the same value of the likelihood function. The transmission parameters of the lines in(5.44) do not change due to lack of dependence on ρ0n. The resulting error in the reflectioncoefficients of the lines can be compensated with additional transformers with oppositeimpedance ratios added to the VNA error boxes. These transformers do not change themeasurement of the through standard and get absorbed in the parameters of the reflectstandard (and into the calibration coefficient determined based on the measurement of the

113


reflect standard). Hence, in order to uniquely identify the elements of the vector (5.50), weneed to make an assumption as to the values of these elements. We write this assumptionin the form of the constraint

aTρ0pρ0 = 0. (5.60)

Finally, as for the width of the center conductor gap, we first note that this widthis determined by the parameters contained in the vectors (5.51), (5.52) and (5.53). Wealready noticed that some assumptions need to be made regarding the parameters (5.52)and (5.53). As far as the the parameters (5.51) are concerned, consider perturbing themby Δl′. This is equivalent to adding the inductance L′

gΔl′ to the inductances on the lineend which is attached to VNA port 1, and subtracting this same inductance from theinductances on the line end which is attached to VNA port 2 (see Fig. 5.5b). We caneasily show that this leads to another solution of the VNA calibration problem (5.7) withthe same value of the likelihood function. Indeed, the transmission coefficient of the linesin (5.44) remain the same as they do not depend on the parameter Δln. The resultingerror in the reflection coefficients of the lines is further compensated by adding impedances−jωL′

gΔl′ and jωL′gΔl′ in series to the VNA ports. These impedances do not change the

measurement of the through standard, and get absorbed in the parameters of the reflectstandard. Consequently, we see the parameters in the vector (5.49) cannot be uniquelydetermined and we also need to make an assumption as to their values. We write thisassumption in the form

aTΔlpΔl = 0. (5.61)

5.5.3 Optimal constraint choice

The assumptions given by equations (5.58) through (5.61) can be made in various ways.The simplest choice is to select one of the lines as a reference and assume that it is notaffected by the errors captured in the vectors (5.49) through (5.53). Through this choice,however, one assumes that the dimensional measurements for the reference line are perfect.This is obviously not true, and introduces an additional error.

The optimal choice for the constraints (5.58) through (5.61) can be deduced based on thestatistical properties of the errors captured in the vectors (5.49) through (5.53). We assumethat the dimensional measurements and manufacturing tolerances are not corrupted with asystematic error and the variance of each error type is the same for all lines. Furthermore,it is plausible to assume that the VNA ports are equally likely to pull the center conductor

114

5.6. NUMERICAL SOLUTION

towards itself, therefore the expectation value of the displacement Δl should be zero.Hence, we can write assumptions that express these observations as

aTΔli

= [1, . . . , 1] , (5.62)aT

Δl0 = [1, . . . , 1] , (5.63)

and

aTΔlα = [1, . . . , 1] , (5.64)aT

ρ0 = [1, . . . , 1] , (5.65)aT

Δl = [1, . . . , 1] . (5.66)

These assumptions are met with a much smaller error than when taking one of the lines asa reference. Indeed, we can show that the standard deviation of this error will be smallerby 1√

Nwhere N is the number of lines.

5.6 Numerical solution

In this section, we review the key points of the algorithm for solving the optimizationproblem (5.7). This problem has a large number of optimized variables, hence its directsolution is infeasible. Therefore we apply different techniques to reduce the dimensionalityof (5.7).

We first separate the determination of the residual variances σ21, . . . , σ2

K and the pa-rameters θ and rc by use of the stagewise minimization approach, referred to as the con-centrated likelihood approach (see Section B.2.1). To this end, we determine the valuesσ2

1(θ, rc), . . . σ2K(θ, rc) of the residual variances which maximize the likelihood function

(5.8) for a given value of the parameters θ and rc. This values are then inserted into (5.8)to yield the concentrated likelihood function which is further maximized with the use ofnumerical techniques.

In order to determine σ21(θ, rc), . . . σ2

K(θ, rc), we take the derivative of (5.8) with respectto σ2

k to obtain

∂ lnL(θ, rc, σ21, . . . , σ2

K)∂σ2

k

= −NW

21σ2

k

+ 12

1(σ2

k)2 rs

k(θ, rc)T V−1Δsk

rsk(θ, rc), (5.67)

115


which after setting to zero yields

σ2k(θ, rc) = 1

NWrs

k(θ, rc)T V−1Δsk

rsk(θ, rc). (5.68)

Inserting these solutions into the original likelihood function (5.8) and dropping the con-stant terms, gives the concentrated likelihood function

lnL(θ, rc) = −NW

2

K∑k=1

ln rsk(θ, rc)T V−1

Δskrs

k(θ, rc). (5.69)

Consequently, we can rewrite (5.7) as a two step optimization process. In the first step,we obtain the estimates θ and rc from

arg maxθ, rc

lnL(θ, rc), (5.70)

and then calculate the maximum-likelihood estimates of residual variances as

σ2MLE,k = σ2

k(θ, rc). (5.71)

We can further show, that the estimates σ2MLE,k are biased. The unbiased estimates of the

residual variances can be written as

σ2k =

NW

NW − R/Kσ2

MLE,k, (5.72)

where R is the length of the vector rc and K is the number of frequencies.

Now, the maximization problem (5.70) can be equivalently written as

arg minθ, rc

K∑k=1

ln rsk(θ, rc)T V−1

Δskrs

k(θ, rc). (5.73)

In order to solve this problem, we use an iterative numerical approach based on theLevenberg-Marquardt algorithm (see SubsectionB.4.1-B). We can show that after expand-ing the residuals into the Taylor series and neglecting the nonlinear terms, the equationsfor the corrections to the estimates θ(q) and rc(q) (without the Levenberg-Marquardt reg-

116

5.6. NUMERICAL SOLUTION

ularization) are given by

X(q)T V −1X(q)Δφ(q) = −X(q)T V −1r(q), (5.74)

where

Δφ(q) =⎡⎣ Δθ(q)

Δrc(q)

⎤⎦ , (5.75)

X(q) = D−1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂rs1(θ, rc)∂θT

∣∣∣∣∣θ=θ(q),rc=rc(q)

∂rs1(θ, rc)

∂(rc)T


... ...∂rs

K(θ, rc)∂θT


∂rsK(θ, rc)∂(rc)T


⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (5.76)

and

r(q) = D−1

⎡⎢⎢⎢⎣

rs1(θ(q), rc(q))

...rs

K(θ(q), rc(q))

⎤⎥⎥⎥⎦ , (5.77)

and

V =

⎡⎢⎢⎢⎣

VΔs1. . .

VΔsK

⎤⎥⎥⎥⎦ , (5.78)

and

D =

⎡⎢⎢⎢⎣

rs1(θ(q), rc(q))T V−1

Δs1rs

1(θ(q), rc(q)). . .

rsK(θ(q), rc(q))T V−1

ΔsKrs

K(θ(q), rc(q))

⎤⎥⎥⎥⎦⊗ INW ,

(5.79)where the underline denotes the convention (A.2), INW is the identity matrix of size NW ×NW , and ⊗ denotes the Kronecker product [57]. In order to solve the normal equations(5.74), we further exploit the sparsity of the matrix X(q) to reduce the dimensionality ofthe problem (5.74). We use here a similar approach to [140].

117


5.7 Residual analysis

In order to assess uncertainties of the the estimates θ and rc, we perform the residualanalysis (see Subsection 3.5.3). We can show that the unbiased estimate of the covariancematrix of errors in these estimates is given by (see Section B.3)

ΣΔφ = 1NW − R/K

(XT V −1X

)−1, (5.80)

where the matrix X is given by (5.76) evaluated at θ and rc. When evaluating (5.80), weare often interested in only covariance matrices of rc or parts of the vector θ correspondingto one frequency. This submatrices of ΣΔφ can be easily determined by accounting for thesparsity of the matrix X and applying the formulas for the inversion of block matrices (see[121]).

5.8 Experimental results

In order to verify our multi-frequency multiline TRL algorithm, we compare it withthe classical multiline TRL method [36]. To this end, for a given set of check devices andmultiline-TRL calibration standards, we measure their S-parameters as raw quantities(i.e., without applying any error correction in the VNA). Subsequently, we process themeasurement data off-line with the two calibration algorithms and compare the results.In this section, we present results of this verification for the multiline TRL calibration inthe Type-N and 1.85 mm coaxial connector standard. Other results, obtained for coaxialtransmissions lines with the LPC-7 connector standard, can be found in [141].

5.8.1 Overview

We implemented our multi-frequency multiline TRL algorithm in the MATLAB envi-ronment [142]. We used the 8-term VNA model (see Paragraph 2.3.1-B) with the baseparametrization (2.71). We determined the switch terms (see Subsection 2.3.2) from themeasurement of the through connection and removed them from the raw measurements byuse of the relationship (2.78). In the absence of any detailed information as to the natureof the VNA instrumentation errors, we assumed that these errors have a circular-normal

118

5.8. EXPERIMENTAL RESULTS

Fig. 5.6: Measurement setup.

PDF (see Paragraph B.6) and are uncorrelated. Thus, the matrix VΔsk, as introduced in

(5.2), is the identity matrix.The classical multiline TRL calibration was performed with the use of the software

package MultiCAL [143, 144]. As a result of the calibration, this package yields a set ofVNA calibration coefficients which we then use in the MATLAB environment to correctthe DUT measurements. The MultiCAL package uses the 8-term VNA model with thereciprocal parametrization (2.72). Thus, the VNA calibration coefficients obtained fromthe MultiCAL package were converted to the base parametrization (2.71). The switchterm correction in the MultiCAL package is performed with the methods described inSubsection 2.3.2.

5.8.2 Type-N coaxial connector

The measurement setup is schematically depicted in Fig. 5.6. Measurements wereperformed with the Agilent E8363 VNA in the frequency range 0.1− 18 GHz (359 points).In order to reduce the impact of VNA receiver errors, the IF bandwidth was reduced downto 50 Hz. In the setup two flexible 3.5mm cables with additional 3.5mm-Type-N adapterswere used.

As calibration standards we used six transmissions lines, a pair of reflect standards, anda direct through connection of the VNA ports. The lengths of the airline outer conductorsspecified along with the measurement error, as specified by the manufacturer [145], aregiven in Tab. 5.3. The lengths of the inner conductors were not specified by the manufac-turer, thus we assumed them to be unknown and modified the condition (5.58) to account

119


only for the outer-conductor length corrections.As the reflect standard we used a pair of female and male offset shorts from the SOLT

calibration kit [146]. According to the manufacturer, the two offset shorts were designedso as to have the same electrical length, corresponding to the phase shift of a 9.9 mm longcoaxial airline.

In Tab. 5.3 we put the corrected lengths of the inner and outer conductor of the airlinesdetermined in our multi-frequency multiline TRL calibration. The corrected lengths of theouter conductor differ from the nominal ones by up to 33μm, and thus are well within the±100μm measurement uncertainty bounds specified by the manufacturer.

In Tab. 5.3 we also give the standard deviation of the corrected lengths determinedin the residual analysis (see Section 5.7). This standard deviation can be thought of asthe precision with which the corrected line lengths are determined4, and for all of thelines is 0.2μm, which indicates that our calibration method is capable of a very precisedetermination of those lengths.

In Tab. 5.4 we put together the center conductor displacements Δl, corrections to thecharacteristic impedance ΔZ0 = ρ0Zref , and the loss correction factors Δlα determinedin our multi-frequency multiline TRL calibration, along with the standard uncertaintiesdetermined in the residual analysis. We see that these uncertainties are typically on theorder of several percent which shows that our calibration algorithm can very preciselydetermine those corrections. The center conductor displacements are less then 35 μm.The characteristic impedance corrections are less then 0.1 % and are smaller then theworst-case value of 0.19 %, given in Tab. 5.2 and derived from the specifications [135].Finally, the relative loss corrections (i.e., Δlα/li0), except for the line L2, are less than10 %. For the line L2 this correction is on the order of 20%.

In Fig. 5.7 we compare the real and imaginary part of the propagation constant deter-mined with the use of the classical multiline TRL calibration (solid gray), and with the useof our multi-frequency multiline TRL calibration (solid black). We see that our method,by using more detailed models for the coaxial airlines, leads to a slightly different estimateof both the real and imaginary part of the propagation constant, as compared with theclassical multiline TRL calibration. The relative difference in the phase constant is largerthan in the attenuation constant, which indicates that the errors of a reactive character(i.e., nonrepeatability of the connector interface, nonuniformity of the diameters) have a

4In order to determine the accuracy of these corrections, we would have to perform full uncertaintyanalysis, accounting for the error of the constraints discussed in Subsection 5.5.3. This will not be discussedhere.

120


Line

Outer conductor Inner conductor Measurementuncertainty

Residuallength length standard

Measured Corrected Corrected uncertainty[mm] [mm] [mm] [mm] [μm]

L1 29.97 30.001 29.978 ±0.1 0.2L2 49.94 49.958 49.933 ±0.1 0.2L3 59.93 59.934 59.913 ±0.1 0.2L4 74.93 74.913 74.904 ±0.1 0.2L5 99.88 99.876 99.856 ±0.1 0.2L6 149.83 149.797 149.772 ±0.1 0.2

Tab. 5.3: Outer conductor lengths with measurement uncertainties, as specified by themanufacturer [145], and outer and inner conductor lengths obtained in our multi-frequencymultiline TRL calibration, along with standard uncertainty from the residual analysis.

Line

Δl ΔZ0 Δlα

Value Standard Value Standard Value Standarduncertainty uncertainty uncertainty

[μm] [μm] [mΩ] [mΩ] [mm] [mm]L1 −8.7 0.1 6 1 −2.3 0.2L2 −12.7 0.1 −51 1 9.4 0.2L3 35.1 0.1 45 1 −3.0 0.2L4 −6.1 0.1 −7 1 0.9 0.2L5 −5.8 0.1 −20 1 −1.8 0.2L6 −1.8 0.1 27 1 −3.2 0.2

Tab. 5.4: Estimates of the center conductor displacement Δl, characteristic impedancecorrection ΔZ0, and loss correction factor Δlα obtained in our multi-frequency multilineTRL calibration, along with standard uncertainties from the residual analysis.

dominant influence on the determination of the propagation constant.

In order to assess the accuracy of our multi-frequency multiline TRL calibration, wecompared the residual standard deviation obtained in our approach (see Section 5.1) andin the classical multiline TRL method (see Paragraph 2.4.1-A). This comparison is shownin Fig. 5.8 as a function of frequency. We see that the residual standard deviation yieldedby our methods is at least four times smaller than for the classical multiline TRL method.This indicates that our description of the calibration standards (i.e., definitions of thecalibration standards along with the VNA model) fits better the actual measurements ofthe standards than the classical multiline TRL method.

121


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Fig. 5.7: Propagation constant for the multiline TRL calibration in the Type-N coaxialconnector standard: (a) imaginary part normalized to the free-space propagation constantβ0, (b) real part; as obtained from the classical multiline TRL (solid gray) [36], and fromour multi-frequency multiline TRL calibration (solid black).

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Fig. 5.8: Residual standard deviation for themutiline TRL calibration in the TypeN coax-ial connector standard, as obtained from theclassical multiline TRL (solid gray) [36], andfrom our multi-frequency multiline TRL cal-ibration (solid black).

We further performed the residual anal-ysis and calculated the residual standarddeviations of the VNA calibration coeffi-cients (see Section 5.7). In Figure 5.9 weshow the residual standard deviation for theeffective source match, effective directiv-ity and effective tracking, respectively (seeParagraph 3.4.2-B), as obtained from theclassical multiline TRL calibration (solidgray) and from our multi-frequency multi-line TRL calibration (solid black). We ob-serve that our method consistently yieldsmuch smaller uncertainties of VNA calibra-tion coefficients than the classical multilineTRL method. Our uncertainties are typi-cally four times smaller than in the classical

multiline TRL, and at lower frequencies (i.e., below 2GHz) the uncertainties we provideare even of an order of magnitude smaller than in the classical multiline TRL method.

In order to verify that the VNA calibrated with our approach yields correct measure-ments, we measured some check devices: a pair of offset-open terminations, a pair ofmatched terminations, and a 10 dB attenuator. In Fig. 5.10 and Fig. 5.11, we show mag-

122


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Fig. 5.9: Residual standard deviation of the effective directivity of (a) the VNA port 1 and(b) the VNA port 2, effective source match of (c) the VNA port 1 and (d) the VNA port2, and effective tracking of (a) the VNA port 1 and (b) the VNA port 2, as obtained fromthe classical multiline TRL (solid gray) [36], and from our multi-frequency multiline TRLcalibration (solid black).

123


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Fig. 5.10: Reflection coefficient magnitude for a coaxial Type-N (a) female and (b) maleoffset-open termination, along with the residual standard uncertainty (c,d), as obtainedfrom the classical multiline TRL (solid gray) [36], and from our multi-frequency multilineTRL calibration (solid black).

nitude and phase (with the linear part removed), respectively, of the corrected reflectioncoefficient of the male and female offset-open termination, as determined with the classi-cal multiline TRL calibration (solid gray), and with the use of our method (solid black),along with the residual standard uncertainties. We see that the agreement between bothmethods is very good, and our method yields residual standard uncertainties that are atleast four time smaller than for the classical multiline TRL method.

We also note the for both methods the reflection coefficient magnitude at some frequen-cies exceeds one. This non-physical result can be attributed to the asymmetry of the reflectstandard which may have a significant impact on measurements of highly reflective devices[109]. This asymmetry cannot be accounted for in the approximate uncertainty assessmentprovided by the residual analysis, since the errors in the reflect standard get completely

124


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Fig. 5.11: Reflection coefficient phase (with linear part removed) for a coaxial Type-N (a)female and (b) male offset-open termination, along with the residual standard uncertainty(c,d), as obtained from the classical multiline TRL (solid gray) [36], and from our multi-frequency multiline TRL calibration (solid black).

absorbed into the VNA calibration coefficients, and thus do not manifest themselves as themisfit between the calibration standard measurements and their model.

In Fig. 5.12, we show the reflection coefficient magnitude of a male and female matched-termination, along with the in-phase residual standard uncertainties (see Section 3.2), asdetermined with the classical multiline TRL calibration (solid gray), and with the use ofour method (solid black). We again see a very good agreement between the two methodswith much smaller uncertainties (at least four times) provided by our method.

In Fig. 5.13 we show the magnitude and phase (with the linear part removed) of thetransmission through the 10 dB attenuator, as determined with the classical multiline TRLcalibration (solid gray), and with the use of our method (solid black), along with resid-ual standard uncertainties. While the agreement between the two calibration methods is

125


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Fig. 5.12: Reflection coefficient for a coaxial Type-N (a) female and (b) male matchedtermination, along with the in-phase residual standard uncertainty (c,d), as obtained fromthe classical multiline TRL (solid gray) [36], and from our multi-frequency multiline TRLcalibration (solid black).

excellent, our method again yields much smaller uncertainties.

5.8.3 1.85 mm coaxial connector

The measurement setup is schematically depicted in Fig. 5.14. Measurements wereperformed with the Agilent E83631 VNA in the frequency range 0.2−67 GHz (335 points).In order to reduce the impact of VNA receiver errors, the IF bandwidth was lowered downto 100 Hz, and an average of four sweeps was taken. We used only one cable, connectedto the VNA port 2, in order to minimize errors due to the cable instability. Both theVNA port 1 and the cable end on the DUT side were equipped with additional adaptersin order to minimize the wear on the VNA-port and cable connectors as those elementscannot easily be replaced. These adapters were carefully selected to have a small pin depth

126


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Fig. 5.13: Transmission for a coaxial TypeN 10 dB attenuator: (a) magnitude, (b) phase(with the linear part removed), along with the (c) magnitude and (d) phase residual stan-dard uncertainty, as obtained from the classical multiline TRL (solid gray) [36], and fromour multi-frequency multiline TRL calibration (solid black).

(≤ 10 μm), in order to reduce the range within which the center conductor gap may vary.

The calibration kit consisted of five transmissions lines, a pair of reflect standards, anda direct through connection of the VNA ports. The lengths of the air-line outer and innerconductors were measured with a mechanical block and are given, along with the standarduncertainty of the measurement, in Tab. 5.5. As the reflect standard we used a pair offemale and male offset shorts. According to the manufacturer, the two offset shorts weredesigned so as to have the same electrical length, corresponding to the phase shift of a5.4 mm long coaxial airline.

In Tab. 5.5 we compare the nominal lengths of the inner and outer conductor of theairlines with the corrected lengths obtained from our multi-frequency multiline TRL cal-ibration. The corrected lengths differ from the nominal lengths by up to 13μm which is

127


Fig. 5.14: Measurement setup.

within the 99% confidence interval (±15μm), determined from the dimensional measure-ment uncertainty.

In Tab. 5.5 we also show the standard deviation of the corrected lengths determined inthe residual analysis. This standard deviation quantifies the precision with which we candetermine the corrected line lengths, and for all of the lines is 0.2μm, which indicates thatwe can very precisely determine those lengths.

In Tab. 5.6 we put together the center conductor displacements Δl, corrections to thecharacteristic impedance ΔZ0 = ρ0Zref , and the loss correction factors Δlα determinedin our multi-frequency multiline TRL calibration, along with the standard uncertaintiesfrom the residual analysis. We see that the precision with which we determine most of thecorrections is very high, typically of on the order of a few percent. The center conductordisplacements are very small and less than 4 μm. The characteristic impedance correctionsare less then 0.3 % and are smaller then the value of 0.46 %, given in Tab. 5.2 and derivedfrom the specifications [135]. Finally, the relative loss corrections are less than 15 %.

Comparing to the results for the Type-N connector given in Tab. 5.4, we note the thecharacteristic impedance and loss corrections for the 1.85mm coaxial airlines are in generallarger. This may be attributed to the larger impact of manufacturing tolerances for smallerconnector sizes.

In Fig. 5.15 we compare the real and imaginary part of the propagation constant de-termined in the classical multiline TRL calibration (solid gray), and with the use of ourmulti-frequency multiline TRL method (solid black). Similarly to results for the Type-Nconnectors, we see that our calibration algorithm yields slightly different estimate of boththe real and imaginary part of the propagation constant, as compared with the classical

128


Line

Outer conductor Inner conductor Measurement Residuallength length standard standard

Measured Corrected Measured Corrected uncertainty uncertainty[mm] [mm] [mm] [mm] [μm] [μm]

L1 14.999 15.012 14.991 14.990 5 0.2L2 16.337 16.344 16.331 16.326 5 0.2L3 18.573 18.580 18.572 18.567 5 0.2L4 23.066 23.076 23.060 23.051 5 0.2L5 29.981 29.982 29.977 29.967 5 0.2

Tab. 5.5: Outer and inner conductor lengths, as measured with mechanical blocks anddetermined in our multi-frequency multiline TRL calibration, along with the measurementstandard uncertainty and standard uncertainty from the residual analysis.

Line

Δl ΔZ0 Δlα

Value Standard Value Standard Value Standarduncertainty uncertainty uncertainty

[μm] [μm] [mΩ] [mΩ] [mm] [mm]L1 −1.7 0.1 78 2 2.25 0.05L2 2.4 0.1 27 2 1.90 0.05L3 −4.0 0.1 −167 2 −2.70 0.05L4 3.5 0.1 −1 2 0.60 0.05L5 −0.2 0.1 63 2 −2.05 0.05

Tab. 5.6: Estimates of the center conductor displacement Δl, characteristic impedancecorrection ΔZ0, and loss correction factor Δlα obtained in our multi-frequency multilineTRL calibration, along with standard uncertainties from the residual analysis.

multiline TRL calibration. The relative differences in the real part are larger than for themultiline TRL calibration in the coaxial Type-N calibration standard. We believe that thismay be attributed to a larger impact of surface roughness on the loss variation for smallerconnector diameters and at higher frequencies.

In Fig. 5.16 we show a plot of the residual standard deviation as a function of frequency,as obtained in the classical multiline TRL (solid gray) and in our multi-frequency multilineTRL method (solid black). We see that our method delivers the residual standard deviationwhich for most frequencies is at least two times smaller than in the classical multilineTRL method. This indicates that our description of the calibration standards (i.e., thedefinitions of the calibration standards along with the VNA model) fits better the actualmeasurements than the classical multiline TRL method. It is also important to note thatthe improvement in the residual standard deviation is not as large as for the multiline

129


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TRL calibration in the coaxial Type-N calibration standard. We think that this can beexplained with an increased impact of connector nonrepeatability errors (i.e., bending onthe connector socket fingers, misalignment of the conductors) for smaller connector sizes.

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In Figure 5.17 we compare the residualstandard deviation of the effective sourcematch, effective directivity and the effectivetracking, respectively, as obtained from theclassical multiline TRL calibration (solidgray) and from our multi-frequency multi-line TRL calibration (solid black). We ob-serve that our method consistently yieldsmuch smaller uncertainties in the VNA cal-ibration coefficients than the classical mul-tiline TRL method. Our uncertainties aretypically two times smaller than from theclassical multiline TRL, and at lower fre-quencies (i.e., below 1GHz), the uncertain-ties we provide are by up to four timessmaller than in the classical multiline TRL

method.

130


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131


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Fig. 5.18: Reflection coefficient magnitude for a 1.85mm coaxial (a) female and (b) maleflat-short termination, along with the residual standard uncertainty (c,d), as obtained fromthe classical multiline TRL (solid gray) [36], and from our muli-frequency multiline TRLcalibration (solid black).

In order to verify that the VNA calibrated with our approach yields correct measure-ments, we measured three verification standards: a pair of flat-short terminations, a pairof matched terminations, and a 10 dB attenuator. In Fig. 5.18 and Fig. 5.19 we show themagnitude and phase, respectively, of the corrected reflection coefficient of the male and fe-male flat-short termination, along with the residual standard uncertainties, as determinedwith the classical multiline TRL calibration (solid gray), and with the use of our method(solid black). We see that the agreement between both methods is very good, however,our method yields residual standard uncertainties that are at least two time smaller thanfor the classical multiline TRL method.

We also note the for both methods the reflection coefficient magnitude at some fre-quencies is larger than one. Similarly to the measurement of the Type-N offset-open

132


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Fig. 5.19: Reflection coefficient phase (with linear part removed) for a 1.85mm coaxial (a)female and (b) male flat-short termination, along with the residual standard uncertainty(c,d), as obtained from the classical multiline TRL (solid gray) [36], and from our muli-frequency multiline TRL calibration (solid black).

terminations, this non-physical result may be attributed to the asymmetry of the reflectstandard. Another reason for this behavior is that when connecting the flat-short termina-tion, a different connector interface is formed than for the calibration standards, as thereis no center conductor gap on the DUT side. This difference in the connector interfacecontributes to a systematic error in the measurement of the flat short.

In Fig. 5.20 we show the reflection coefficient magnitude of a male and female matched-termination, along with the in-phase residual standard uncertainty (see Section 3.2), asdetermined with the classical multiline TRL calibration (solid gray), and with the useof our multi-frequency multiline TRL method (solid black). We again see a very goodagreement between the two methods with much smaller uncertainties (at least four times)provided by our method.

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In Fig. 5.21 we show the magnitude and phase (with the linear part removed) of thetransmission through the 10 dB attenuator, as determined with the classical multiline TRLcalibration (solid gray), and with the use of our method (solid black), along with resid-ual standard uncertainties. While the agreement between the two calibration methods isexcellent, our methods again yields much smaller uncertainties.

5.9 Summary

We presented a generalization of the multi-frequency approach to VNA calibration (seeParagraph 2.4.1-B). The principal idea of our generalization is to use the concept of thephysical error mechanism (see Section 4.2) to capture the relationships between calibration

134

5.9. SUMMARY

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standard S-parameters at different frequencies. We further developed a generic numericalframework for the solution of such a multi-frequency VNA calibration problem and demon-strated it in the context of the multiline TRL calibration in the coaxial environment. Theexperimental results for the multiline TRL calibration with Type-N and 1.85mm coaxialtransmission lines clearly show that by accounting for the relationships between calibrationstandard S-parameters at different frequencies, we can significantly (by a few times) reducethe uncertainty in the corrected VNA S-parameter measurements.

The approach we presented is the first step towards the development of the VNAcalibration approach based entirely on the description of measurement errors in terms of thephysical error mechanisms (see Subsection 4.5.3). In order to implement such an approach,a description of VNA instrumentation errors in terms of physical error mechanisms is

135


needed. In the following chapter, we propose a description of the VNA nonstationarityerrors, which are the primary source of VNA instrumentation errors, employing the conceptof the physical error mechanism.

136

Chapter 6

Multi-frequency stochastic modelingof VNA nonstationarity errors

Entities should not be multipliedbeyond necessity.

William of Ockham

In this chapter, we present a framework for a multi-frequency description of VNA non-stationarity errors. These errors, caused by the temperature and humidity drift, cableinstability, and connector nonrepeatability, are the primary source of the VNA instru-mentation errors (see Section 3.1). Our framework is based on a stochastic model whoseparameters are identified from repeated S-parameter measurements. The core of this modelis a generic description of the VNA nonstationarity which uses a set of lumped elementsadded to the two-port network capturing the VNA calibration coefficients. The stochasticmodel is then constructed by allowing the parameters of the lumped elements to ran-domly vary. As a result, we describe the complicated frequency-dependence of the VNAnonstationarity errors with the use of a very small set of frequency-independent randomvariables.

6.1 Introduction

In Chapter 4, we introduced the multi-frequency description of errors in S-parame-ter measurements and discussed the origins of the statistical correlations between theseerrors at different frequencies. We showed that these correlations result from the fact the

137

6. MULTI-FREQUENCY STOCHASTIC MODELING OF VNA NONSTATIONARITY

S-parameter measurement errors are caused by some common fundamental physical errormechanisms (see Subsection 4.2).

We further pointed out that the number of these mechanism is usually much smallerthan the number of variables captured in the S-parameter measurement, thus the informa-tion contained in the covariance matrix for the multi-frequency S-parameter-measurementerror is redundant. Consequently, when developing the multi-frequency error description, itis more efficient to follow the bottom-up approach, that is, to first identify and characterizethe fundamental error mechanisms, and from that to construct the covariance matrix.

This bottom-up approach is easily followed in the case of calibration standard errors.Indeed, these errors result from some well-defined physical error mechanisms, related pri-marily to the tolerances on the dimensional and material parameters of the standards.Consequently, physical models for errors in S-parameters of calibration standards followdirectly from the physical models of the standards themselves.

In the case of VNA nonstationarity errors, however, application of the bottom-up ap-proach is much more difficult. The nonstationarity of VNA electrical parameters, mani-festing itself as the connector nonrepeatability, cable instability, and the test-set drift, iscaused by a number of different complicated physical mechanism and is commonly regardedas difficult to both understand and model [19]. In the case of the connector nonrepeata-bility, some interesting efforts have been made towards understanding of these mechanismbased on electromagnetic modeling of simplified connector structures [125, 126]. Theseefforts provide valuable guidelines as to the connector interface design, however, it seemsthat the complexity of real connector and cable structures is far beyond the capacity ofdirect electromagnetic modeling.

Also, some attempts have been made in the literature to statistically model the drift ofmeasurement instruments [147–149]. These models, however, developed for simple scalar-measurement instruments, such as power meters, are not easily extendable to complicatedmeasurement systems such as the VNA.

Because of this complexity, VNA instrumentation errors are traditionally characterizedwith the use of either an approximate statistical analysis or some heuristic methods. Forexample, in the case of connector nonrepeatability or cable instability, the standard uncer-tainty of these errors is assessed based on repeated measurements of devices with differentS-parameters, such as matched and highly reflective terminations [93, 95, 150, 151]. Inthe case of the VNA test-set drift, the heuristic method of [94] is often used (see Subsec-tion 3.5.2). In this method, the difference between two sets of VNA calibration coefficients,

138

6.2. GENERIC PHYSICAL MODEL FOR THE VNA NONSTATIONARITY

obtained before and after an experiment, is used as an estimate of the VNA test-set driftduring the experiment.

The above methods, however, have two important drawbacks. First of all, these meth-ods do not allow to reliably estimate the statistical correlations between measurementerrors at different frequencies. Indeed, due to the large size of the covariance matrix forthe multi-frequency S-parameter measurement error, direct estimation of these correlationsbased on repeated measurements would require an impractically large number of measure-ments [87]. Another important drawback of these methods, except for [94], is that insteadof considering the statistical properties of the VNA nonstationarity itself, they analyze onlythe impact of this nonstationarity on the S-parameter measurement of some individual de-vices. As a results, the impact of the VNA nonstationarity errors on the measurements ofdevices with other S-parameters is difficult to predicted.

Hence, our description of VNA nonstationarity errors is different. Instead of focusingon the impact of the VNA nonstationarity on measurements, we begin with a genericphysical model for the VNA nonstationarity itself, proposed in [152] and discussed inmore detail in Section 6.2. This model describes the VNA nonstationarity with a set oflumped elements added to the two-port network describing the VNA calibration coefficients(commonly referred to as the VNA error box—see Paragraph 2.3.1-B). Based on thatdescription, we build a stochastic model for the VNA nonstationarity by allowing thefrequency-independent parameters of the lumped elements to randomly vary. We thenidentify statistical properties of these parameters based on repeated measurements of asingle highly-reflective offset termination. As a results, our model can be used to predictthe complicated frequency-depedendence of VNA nonstationarity errors for devices witharbitrary S-parameters.

6.2 Generic physical model for the VNA nonstation-arity

VNA nonstationarity errors often exhibit a “regular” frequency-dependence [91, 93, 150,151, 153]. Also, a common experience of a microwave metrologist is that any instability inthe VNA setup, for example, due to cable bending or a loose connection, leads to periodicripples in corrected S-parameter measurements.

This type of a regular frequency-dependent behavior can easily be explained from aphysical point of view. Consider a small discontinuity that occurs within the VNA setup

139


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after the calibration has been performed. Correction with the original VNA calibrationcoefficients does not account for this new discontinuity, hence the reflections off this dis-continuity affect the corrected VNA S-parameter measurements. Now, if one measuresa device that is not perfectly matched, will ripples are observed which correspond to thedistance between this new discontinuity and the plane at which the reflection occurs in thedevice.

Reference [91] attempts to quantitatively describe this behavior in the case of connectorrepeatability errors. It uses a connector-interface equivalent circuit of [92] which is formedby a series inductance, a series resistance, and a shunt capacitance inserted between theDUT and the VNA. The elements of this circuit model the effects responsible for theconnector repeatability errors, such as the misalignment the center and outer conductors(the shunt capacitance) and variation of the joint impedance (the series inductance andresistance). A simple procedure for the identification of model parameters is then proposedwhich is based on repeated measurement of a highly-reflective offset termination.

The model for VNA nonstationarity errors we propose here (see [152]) generalizes theapproach of [91, 92]. We observe that in many cases a single discontinuity cannot explainthe observed ripple behavior in corrected S-parameter measurements. This is especiallytrue for connector repeatability errors for which changes of connector interface often occurnot only at the connector joint plane. Examples are bending of the center conductor fingersor flexing of the beads supporting the center conductor due to the mechanical strain appliedto this conductor.

Our equivalent circuit accounts therefore an arbitrary but otherwise fixed number ofdiscontinuities, located at different distances from the VNA reference plane. This circuit

140


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Fig. 6.2: Lumped-element equivalent circuit for a single perturbation.

is shown in Fig. 6.1. It describes a small change of one-port-VNA calibration coefficientswith a set of N + 1 small perturbations. These perturbations are added at different fixeddistances dn, for n = 0, . . . , N , to the linear two-port network representing the VNAcalibration coefficients. For the perturbation P0 which occurs at the VNA reference planewe have d0 = 0.

We describe each of the perturbations with a lumped-element circuit shown in Fig. 6.2.Compared to [91, 92], our circuit has an additional transformer which accounts for changesof the characteristic impedance. Such changes may result, for example, from eccentric-ity of the inner conductor, variation of the connector socket diameter, or changes of thetransmission line diameters due to the temperature drift. We further describe the seriesimpedance perturbation as

Zn = Rn + jXn = RDC,n + (1 + j)RRF 0,n

√f

f0+ jωLn, (6.1)

where f0 is a fixed reference frequency. Such a description allows to account for both thechange of DC resistance and the change in the skin-depth resistance and inductance.

The circuit in Fig. 6.2 is fairly general and accounts for typical electrical characteristicsthat a small discontinuity in a coaxial line might exhibit. In the case when additionalinformation on the electrical character of the perturbations is available, this circuit can beaccordingly modified. However, in this work, we do not strive to obtain nor to account forsuch information. Instead, we show that a statistically and physically sound descriptionof the VNA nonstationarity errors can be developed with the use of the general modelpresented in Fig. 6.1, and thus without any need to directly model the design details ofthe connector interface, cable, or the VNA itself.

We represent the joint contribution of the perturbations Pn, for n = 0, . . . , N to the

141


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error in corrected DUT S-parameter with an equivalent two-port network E inserted be-tween the VNA and the DUT (see Fig. 6.3). In order to determine parameters of thisnetwork, we assume that the perturbations are small, which allows us to neglect multiplereflections and superimpose the contributions of individual perturbations. The scatteringmatrix of this equivalent network is given in compact form as (see Appendix F)

S =

⎡⎢⎢⎣

N∑n=0

wR1(f)pR,nejβ0dn 1 − wT (f)T pT

1 − wT (f)pT

N∑n=0

wR2(f)pR,nejβ0dn

⎤⎥⎥⎦ , (6.2)

where real-valued model parameters are defined by

pT = [N∑

n=0R′

DC,n

N∑n=0

R′RF 0,n

N∑n=0

(L′n + C ′

n) ]T , (6.3)

pR,n =[

R′DC,n R′

RF 0,n L′n − C ′

n Z ′0,n

]T, (6.4)

and

R′DC,n =

RDC,n

Zref

, (6.5)

R′RF 0,n =

RRF 0,n

Zref

, (6.6)

L′n =

Ln

Zref

, (6.7)

C ′n = CnZref , (6.8)

Z ′0,n =

Z0,n

Zref

, (6.9)

142


where Zref is the reference impedance, typically 50 Ω. The quantities wT (f), wR1(f), andwR2(f) are the frequency dependent weighting functions defined as

wT (f) =

⎡⎢⎢⎢⎣

1√ff0

j2πf

⎤⎥⎥⎥⎦ , (6.10)

and

wR1(f) =

⎡⎢⎢⎢⎢⎢⎢⎣

1√ff0

j2πf

−1

⎤⎥⎥⎥⎥⎥⎥⎦

, wR2(f) =

⎡⎢⎢⎢⎢⎢⎢⎣

1√ff0

j2πf

1

⎤⎥⎥⎥⎥⎥⎥⎦

, (6.11)

where β0 = ω/c is the propagation constant in the air, c is the speed of light in vacuum.

We neglect the conductor loss in the transmission lines since the distances dn are small.Also, we assume that Z0,n is real since small dimensional changes are, to first order, affectingonly the magnitude of line’s characteristic impedance (see Subsection 5.3.2).

The overall error due to the perturbations Pn, as described by (6.2), has importantproperties. To first order, this error is independent of the VNA calibration coefficientsand is determined only by the electrical parameters of the perturbations, pT and the pR,n,and the distances dn of these perturbations from the VNA reference plane. Consequently,equation (6.2) can be applied to describe the connector repeatability and cable instabilityerrors which are independent of the individual VNA setup and are rather a property of aparticular connector interface or cable.

The distances dn, for n = 0, . . . , N are fixed and do not undergo any statistical variation.For example, for the connector repeatability errors, these distances correspond to somedesign-fixed dimensions (e.g. position of the bead, length of connector socket fingers,length of the connector pin) of the connector interface. The same reasoning applies to thecable instability errors or the test-set drift errors. Hence, the only parameters that varyare pT and the pR,n, which we thus group into a single vector

p =

⎡⎢⎢⎢⎢⎢⎢⎣

pT

pR,0...

pR,N

⎤⎥⎥⎥⎥⎥⎥⎦

. (6.12)

143


The models for VNA nonstationarity errors we describe in the following sections are thenconstructed by assuming different statistical properties of the parameters (6.12).

Finally, it is important to note that the frequency dependence of (6.2) is set by fixedweighting functions (6.10) and (6.11), the distances dn, and a small set of frequency inde-pendent parameters (6.12). Consequently, once the distances dn and statistical propertiesof (6.12) are known, the impact of the VNA nonstationarity can easily be determined anyset of measurement frequencies.

6.3 Stochastic model for connector nonrepeatabilityand cable instability

In this section, we describe the stochastic models for the connector nonrepeatability andcable instability. We describe these models in the same section because the assumptionsunderlying them are identical. However, it is important to note that both models areidentified and used independently of each other. In these models, we use the description(6.2) for VNA nonstationarity with the parameters (6.12) replaced by random variables.In the following, we first discuss the assumptions made as to the statistical properties ofthese variables. We then proceed with the procedure for identification of those properties,and finally present experimental verification of our models. The discussion we present hereis an extension of [154].

6.3.1 Statistical properties of circuit parameters

We capture the electrical parameters of the connector interface or the cable whichundergo random variations when reconnecting the interface in the vector (6.12). Sincethese parameters describe the changes of some characteristics, we assume that E (p) = 0.We further assume that the probability density function (PDF) of the vector p is normal.Consequently, the statistical properties of the vector p are completely described with thecovariance matrix

Σp = E(ppT

). (6.13)

The assumption that the vector p has a normal PDF can easily be justified fromthe physical point of view. Parameters in the vector p represent the impact of mechanicalchanges of the connector interface or the cable in terms of some electrical parameters. Sincethese changes are small, parameters in the vector p are linear combinations of changes in

144

6.3. STOCHASTIC MODEL FOR CONNECTOR AND CABLE ERRORS

some geometrical parameters of the connector interface or the cable, such as displacementsof the conductors, bending of the socket fingers, flexing of the center conductor beads orchange of the cable length. It is reasonable to assume that all of those mechanical changesare normally distributed, hence the vector p has also a normal probability distributionfunction.

6.3.2 Estimation of the covariance matrix of circuit parameters

We estimate the covariance matrix (6.13) from repeated corrected VNA-measurementsof an offset termination. In the case of the connector-nonrepeatability model, we per-form repeated measurement of the termination on the VNA port without the cable whilereconnecting and rotating the device before each measurements. When identifying thecable-instability model, we perform measurements of the termination on the VNA portextended with the cable, while randomly bending the cable before each measurement. Pre-ferred devices for the model identification are highly-reflective offset-terminations, due tothe larger impact of VNA nonstationarity on the their measurements [91]. However, ourestimation method can be used for offset terminations with arbitrary non-zero reflectioncoefficients.

Let Γk, denote the unknown true value of the one-port device reflection coefficient atthe frequency fk, for k = 1, . . . , K, and let Γmk be the m-th corrected measurement of thisreflection coefficient in a series of repeated measurements, where m = 1, . . . , M . We canmodel this measurement by transforming Γk through the linear two-port network given by(6.2). Considering only first order terms we obtain the approximation

Γmk = Γk + xk(d, Γk)T pm + εmk, (6.14)

where the parameters pm characterize the change of VNA calibration coefficients in them-th measurement, εmk is the VNA measurement noise,

xk(d, Γk) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−2ΓkwT (fk)e−j2β0d0wR1(fk) + ej2β0d0Γ2

kwR2(fk)e−j2β0d1wR1(fk) + ej2β0d1Γ2

kwR2(fk)...

e−j2β0dN wR1(fk) + ej2β0dN Γ2kwR2(fk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (6.15)

145


andd = [d0, . . . , dN ]T . (6.16)

Our goal now is to determine the estimates pm of the connector interface parameters pm,the estimate Σp of their covariance matrix, the estimates dn of the perturbation locationsdn, for n = 0, . . . , N , and the estimates Γk of the unknown reflection coefficient Γk.

Generic solution to that problem requires the use of statistical methods for the analysismixed-effect models for repeated measurements [155]. In these methods, the likelihoodfunction for all of the estimated parameters is written and the sought parameters are de-termined by maximizing this function. In the case of the nonlinear model (6.14), this leadsto a difficult constrained nonlinear optimization problem (the estimate of the covariancematrix (6.13) needs to be positive definite), therefore in the following we proceed in asimplified manner.

We first note that, since the VNA nonstationarity errors are small, the estimate Γk ofthe reflection coefficient at the frequency fk is well approximated by the mean value

Γk = Γk =1

M

M∑m=1

Γmk. (6.17)

After inserting this estimate into (6.14), we only have to determine pm, dn, and the covari-ance matrix Σp. We further assume that this matrix can be approximated as the samplecovariance of the estimates pm, that is [87]

Σp = 1M − 1

M∑m=1

pmpTm. (6.18)

The error of this approximation will depend on the accuracy with which we determinethe estimates pm. This accuracy depends on the variance of the VNA measurement noiseεmk, the number of measurement frequencies, and the error of the model (6.2). In orderto increase this accuracy, we perform the measurements for a large number of frequencies,and reduce the VNA receiver errors by narrowing the IF bandwidth and averaging multipleresults.

Equipped with the above assumptions, we can pose the original problem as the task ofdetermining pm, for m = 1 . . . , M , and dn, for n = 0, . . . , N under the condition (6.17).To this end, we write the measurements (6.14) in a more compact form as

ym = y + X (d)pm + εm, (6.19)

146


where

ym =

⎡⎢⎢⎢⎣Γm1...

ΓmK

⎤⎥⎥⎥⎦ , y =

⎡⎢⎢⎢⎣Γ1...ΓK

⎤⎥⎥⎥⎦ , εm =

⎡⎢⎢⎢⎣

εm1...

εmK

⎤⎥⎥⎥⎦ , (6.20)

and

X (d) =

⎡⎢⎢⎢⎣

x1(d,Γ1)T

...xK(d,ΓK)T

⎤⎥⎥⎥⎦ . (6.21)

We further assume that the VNA receiver errors have the same circular-normal PDF at eachfrequency (see Subsection B.6), and that the errors at different frequencies are statisticallyuncorrelated (see Subsection 4.4.2). Thus, the VNA measurement noise εmk has also thecircular-normal PDF whose variance can be determined from (3.60) as

E(εmkε∗mk) =

∣∣∣∣∣∣1 − 2ESkΓk + E2

SkΓ2k

ERk

∣∣∣∣∣∣2

E(ηη∗) = V 2εk

σ2η, (6.22)

where ESk and ERk are the source match and tracking at the frequency fk, determined inthe VNA calibration preceeding the experiment, and E(ηη∗) = σ2

η is the unknown varianceof the VNA receiver errors. Consequently, we can write the covariance matrix for thevector εm as

E(εmεHm) = σ2

η

⎡⎢⎢⎢⎣

V 2ε1

. . .V 2

εK

⎤⎥⎥⎥⎦ = σ2

ηVε. (6.23)

Now, we use the maximum likelihood approach (see Appendix B) and determine the pm

and the dn as the solution to the nonlinear least squares problem

θ = arg minθ

M∑m=1

rm (d, pm)H V−1ε rm (d, pm) , (6.24)

where the parameter vector is

θ =

⎡⎢⎢⎢⎢⎢⎢⎣

p1...

pM

d

⎤⎥⎥⎥⎥⎥⎥⎦

, (6.25)

147


and the residuals arerm (d, pm) = ym − y − X (d)pm. (6.26)

We find the solution to the problem (6.24) with the use of the Levenberg-Marquardtalgorithm (see Subsection B.4.1-B). We exploit the linear dependence of (6.26) on theparameters pm to reduce the dimensionality of the optimization problem. Details aregiven in Appendix G.

Once the estimates pm are determined, we calculate their sample covariance matrix(6.31). We further reduce the dimensionality of this covariance matrix with the use of theprincipal component analysis [87]. That is, we determine the approximation

Σp ≈ UΣξUT , (6.27)

where Σξ is a diagonal matrix of size M × M with non-zero diagonal elements, referred toas principal components, U is a full column-rank rectangular matrix of size P × M withorthogonal columns, P is the length of the vector p, and M < P is the number of principalcomponents. We choose the number of principal components in the approximation (6.27)so as to capture 99% of the total variance contained in the original matrix (6.31), whichcorresponds to the error of this approximation (in the sense of the Frobenius matrix norm[121]) less than 1%. This results typically in two or three principal components.

The principal components obtained through the approximation (6.27) can be consideredas the electrically-equivalent physical error mechanisms (see Section 4.2) characterizing theconnector interface or cable instability errors. Indeed, by rescaling the weighting functions(6.10) and (6.11) by the columns of the matrix U, we can completely characterize theconnector nonrepeatability or cable instability, as described by the matrix (6.2), with a setof M independent variables with variances determined by the diagonal elements of Σξ.

6.3.3 Experiments

A. Connector repeatability errors. We illustrate our approach with experimentalresults for the 1.85 mm coaxial-connector interface. We used two measurement setupswhich are schematically shown in Fig. 6.4. In each case, measurements were performedwith the Agilent E83631 VNA in the frequency range 0.2−67 GHz and with the frequencyspacing of 50 MHz. In order to reduce the influence of the VNA receiver errors, we narrowedthe IF bandwidth to 10 Hz and averaged four sweeps. This resulted in a relatively longmeasurement time of around 2min for a single reflection-coefficient measurement. Both the

148


(a) (b)

Fig. 6.4: Measurement setup for (a) female and (b) male one-port DUTs.

VNA port 1 and the cable end on the DUT side were equipped with additional adapters,in order to minimize the wear on the VNA-port and cable connectors as those elementscannot easily be replaced.

In the setup shown in Fig. 6.4a, we first calibrated the VNA with the SOLT calibrationemploying four offset shorts with different lengths, an offset open, a matched terminationand a direct thru connection. Then we performed repeated measurements of differentone-port female DUTs on the VNA port 1. Before each measurement, the DUT wasdisconnected, rotated by 45◦, and connected again. A similar experiment was performedin the setup show in Fig. 6.4b, however, after the calibration, we measured different one-port male DUTs.

Fig. 6.5 shows the in-phase and quadrature component (see Subsection 3.2.2) of thestandard deviation of 16 repeated reflection-coefficient measurements of 5.4 mm and7.6 mm long, female and male offset shorts, and 5.4 mm long female and male offset opens,along with the standard deviation predicted from our stochastic model. In the model,we used three perturbations, corresponding to the nonrepeatability of the connector joint(perturbation located at the connector joint plane), flexing of the bead supporting thecenter conductor (perturbation located within the adapter, 3 mm away from the connectorjoint plane), and bending of the connector socket fingers (perturbation located within theDUT for female DUTs, and within the adapter for male DUTs, in each case 1.1 mm awayfrom the connector joint plane). We estimated the locations of the perturbations basedon observations under the microscope and the specifications [135], and kept them fixed inthe analysis.

The stochastic models we used employed two random variables in the case of the 5.4 mm

149


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150


long offset shorts, three random variables in the case of the 7.6 mm long short, two randomvariables for the 5.4 mm long female offset open, and only one random variable for the5.4 mm long male offset open. We chose the number of variables such that the error of theapproaximation (6.27) is less than 1%. We observed that for each termination, when usingonly one random variable, the error of the approximation (6.27) was less then 10% (in thecase of the 5.4 mm long male offset open it was alread less than 1%), and increasing thenumber of variables yielded only a small improvement.

In Fig. 6.5 we see that the overall agreement between the stochastic model predictionand the measurements for the quadrature (phase) errors is very good except for a smalldiscrepancy in the frequency range below 4 GHz. We think that the discrepancy is causedby an increased VNA test-set drift in this frequency range (the 16 repeated measurementstook over 30min), since we observe it in the measurements of all of the devices.

The agreement for the in-phase (magnitude) errors is not as good as for the quadratureerrors and, in general, our model underestimates the in-phase errors. We think that thisis due to the VNA receiver errors which are not accounted for by our model. On theother hand, the in-phase errors are of an order of magnitude smaller than the quadratureerrors, and therefore less important. Consequently, our stochastic model is capable ofrepresenting the dominant component of the connector repeatability errors, as observed inthe measurements of the three different offset terminations.

We further note that the connector repeatability errors for the 5.4 mm long male openare much larger than for other terminations. We think that this might be caused by anexcessive eccentricity of the center conductor or a loose connection between the adapter andthe VNA ports during the measurment of the open. In order to verify those hypotheses,we plan on inspecting the offset open under a microscope, and if necessary, repeating theexperiment.

In the measurments of all of the terminations, except for the 5.4 mm long female open,we observe that the connector repeatability errors are comparable for each pair of maleand female offset terminations with the same length. This can be justified by the factthat both male and female terminations have the same connector interface, and that themechnical forces put on this interface (e.g., due to the misalignment of the inner and outerconductors) are approximately the same since both termiantions have the same length.

We also note that the connector nonrepeatability errors are smaller for longer offset-terminations. We think that this can be explained by the fact that for longer terminationsthe displacement of the inner conductor at the connector joint causes a more gradual

151


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Fig. 6.6: Connector repeatability errors in 16 repeated reflection coefficient measurementsof 1.85mm coaxial one-port devices along with the 95% uncertainty bounds predicted fromthe stochastic model developed from the measurements of a 5.4mm long female offset-short: (a) in-phase error for a 6.3 mm long female offset-short, (b) quadrature error for the6.3 mm long female offset-short, (c) in-phase error for a 5.4 mm long female offset-open,(d) quadrature error for the 5.4 mm long female offset-open, (e) real part of the error fora matched termination, (f) imaginary part of the error for the matched termination.

152


bending of the center conductor along the termination.In the next step, we investigated whether the model we developed based on the mea-

surements of an offset short can be used to predict the connector repeatability errors in themeasurements of other one-port devices. Fig. 6.6 compares the distribution of 16 repeatedmeasurements of a 6.3 mm long offset short, a 5.4 mm long offset open, and a matchedtermination, respectively, with the 95% uncertainty bounds predicted from the stochasticmodel developed based on the measurements of the 5.4 mm long offset short. In Fig. 6.6aand Fig. 6.6b, we compare the in-phase and quadrature component of the deviation of eachmeasurement from the mean, while in Fig. 6.6c we show the real and imaginary part of thisdeviation. In the case of the 6.3 mm long offset short and the 5.4 mm long offset open, ourstochastic model accurately predicts the uncertainty bounds for the quadrature component.For the in-phase component, we underestimate the observed variability which, however, isof an order of magnitude smaller then the variability in the quadrature component.

In the case of the matched termination our prediction of the uncertainty bounds is notas good as for the offset terminations. We think that this may be attributed to the designof the matched termination. This termination is constructed with the use of a planarresistor inserted into the coaxial transmission line at some distance from the connectorjoint. The center conductor in such a construction is less rigid than in an offset short,hence the mechanical strain put on this conductor affects not only the connector interface,but also the planar structure forming the matched termination.

B. Cable instability errors. In order to verify our model, we made an experimentwith a high-quality 66 cm long microwave cable. The measurements were performed in thesetup shown in Fig. 6.4b, with the same VNA settings as in the experiment described in theprevious paragraph. We performed repeated reflection-coefficient measurements of differentone-port devices attached to VNA port 2 while randomly bending the cable between themeasurements. We kept the cable positions within the range typically observed in practice.

We determined the initial estimates of the perturbation locations for the optimizationprocedure (6.24) based on a time-domain representation of the measurments. We obtainedthis represenation by transforming each reflection coefficient measurement into the timedomain (see [156, 157]), and then taking the standard deviation of the resulting waveformsat each time point. The maxima of such a representation correspond therefore to thelocations within the cable (and the adapter attached to it) at which changes occur whilethe cable is being bent.

153

6. MULTI-FREQUENCY STOCHASTIC MODELING OF VNA NONSTATIONARITY��

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Fig. 6.7: Standard deviation of the time-domain representation of 16 repeated reflectioncoefficient measurements of a 5.4mm long 1.85mm coaxial male offset-short, with randomcable flexure: (a) from −1.4m to the VNA reference plane (b) zoom into the range from−80mm to the VNA reference plane.

Fig. 6.7 shows the time-domain represenation we obtained for 16 repeated measurementsof a 5.4 mm long 1.85 mm coaxial male offset short. The time points were then transformedinto equivalent distances by assuming that the wave is propagating in the free space. TheVNA reference plane is located at the distance 0 and negative distances correspond to thelocations away from the VNA reference plane towards the VNA. In Fig. 6.7a we see thatthe standard deviation has the largest values at around −50 mm, −700 mm, and −800 mm.These distances correspond to both ends of the cable1. The changes around 50 mm aredominant, therefore we decided to disregard the perturbations at −700 mm and −800 mm.

In Fig. 6.7b, we further expand the time-domain representation for the range from−80 mm to 0 mm. The largest variations are occuring in the range between −60 mmand −40 mm, and between −30 − 0 mm and 0 mm, that is, within the adapter attachedto the cable. The change withint the adapeter are smaller then the perturbations at−700 mm and −800 mm, hence we disregerd them in our model. As a result, we usetwo perturbations related to two peaks with maximal amplitude which occurr in the rangebetween −60 mm and −40 mm. Based on the Fig. 6.7a, we estimated the locations ofthese peaks to −53.7 mm and −49.2 mm, which were further refined in the estimationprocedure to −54.2 mm and −48.7 mm, respectively.

In Fig. 6.8 we show the in-phase and quadrature component of the standard deviation of

1Since the cable is filled with a dielectric, the difference between the location of the first and lastperturbation is not equal to the physical length of the cable.

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Fig. 6.8: Standard deviation of 16 repeated reflection coefficient measurements of a 5.4mmlong 1.85mm male coaxial offset-short with random cable flexure, represented as the in-phase (blue) and quadrature (red) component, along with the stochastic model prediction(black).

the repeated measurements for the offset short along with the standard deviation predictedfrom our stochastic model. The model we used employed only two random variablesdetermined from on the approximation (6.27). The agreement between our model and themeasurements is very good for both the in-phase and quadrature component. We notethat in-phase errors are comparable with the quadrature errors which indicates that theinstability of the cable affects both its phase shift and attenution.

We further used the stochastic model for cable instability errors developed for the5.4 mm long 1.85 mm coaxial male offset short to predict these errors in measurementsof other one-port devices, that is, a 7.6 mm long male offset-short, a 5.4 mm long maleoffset-open and a coaxial male matched termination. The results of those measurementsare shown in Fig. 6.9. In both figures, we show the deviations of the measurements fromtheir mean along with the 95% uncertainty bounds predicted from our stochastic model.In the case of the offset open, we show separately the in-phase and quadrature componentof the deviations, while in the case of the matched termination, we compare the real andand imaginary part of the deviations. In both cases, we see that the measurements liewithin the uncertainty bounds predicted from the model. The prediction of these boundsfor the matched termination is slightly worse than for the open. This may be attributedto the VNA receiver errors whose relative contribution is larger for devices with smallerreflection coefficients.

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Fig. 6.9: Cable instability errors in 16 repeated reflection coefficient measurements of1.85mm coaxial one-port devices along with the 95% uncertainty bounds predicted fromthe stochastic model developed from the measurements of a 5.4mm long male offset-short:(a) in-phase error for a 7.6 mm long male offset-short, (b) quadrature error for the 7.6 mmlong male offset-short, (c) in-phase error for a 5.4 mm long male offset-open, (d) quadra-ture error for the 5.4 mm long male offset-open, (e) real part of the error for a matchedtermination, (f) imaginary part of the error for the matched termination.

156

6.4. STOCHASTIC MODEL FOR VNA TEST-SET DRIFT

6.4 Stochastic model for VNA test-set drift

In the following, we discuss the stochastic model for the VNA test-set drift. In thismodel, we use the description (6.2) of VNA nonstationarity errors with the vector (6.12)replaced by a set of stochastic processes. We employ here the stochastic Wiener processwhich forms the mathematical description of the random walk pheonomenon. We firstdiscuss the properties of the scalar stochastic Wiener process, and then show how to adoptthis concept to a more complex situation of the VNA test-set drift. Then we discussthe estimation of the process covariance matrix and present experimental verification ourmodel.

6.4.1 Drift as the multidimensional random walk

The drift manifests itself as a measurement bias varying with time [147]. In the case ofVNA S-parameter measurements, this bias results from a continuous change of VNA elec-trical parameters in time due to aging, and temperature and humidity changes. Therefore,at any time point after the calibration, the VNA calibration coefficients slightly differ fromthose determined during the calibration procedure. We further observe that the maximumvalue of this difference increases with time. However, it is also possible that, after someperiod of time, the VNA calibration coefficients take on again the values close to thosedetermined during the calibration.

The above description of the VNA test-set drift process resembles closely the randommotion of a particle in a fluid, commonly referred to as the Brownian motion [158]. Atany time after the motion begins, the particle, due to a series of random collisions, is awayfrom the starting point. Although we expect the worst case value of the distance of theparticle from the starting point to increase with time, it is also possible the particle comesback to the vicinity of the starting point.

The mathematical description of the one-dimensional Brownian motion is given by thestochastic Wiener process [158]. Considering the similarities between the Brownian motionand the drift process, Reference [147] suggests applying the stochastic Wiener process tomodel the drift of a simple scalar measurement instrument, such as power meters. Theexperimental results presented in [147] confirm, that the statistical properties of the driftprocess agree well with those of the stochastic Wiener process.

In the context of scalar measurements, we can summarize the properties of the stochas-tic Wiener process w(t) as follows [147, 158]:

157


a) w(t) = 0, that is, at the time t = 0, right after the calibration has been performed,the measurement instrument has no bias;

b) w(t) ∼ N(0, αt), that is, w(t) has a normal PDF with E[w(t)] = 0 and E[w(t)2] = αt,where α > 0 is the process parameter;

c) w(t2)− w(t1) ∼ N(0, α(t2 − t1)), that is, differences in bias for non-overlapping timeperiods are statistically independent and have the normal PDF with zero expectationvalue and the variance proportional to the length of the time period;

d) E[w(t)|w(t1); t1 ≤ t] = w(t1), that is, if the measurement bias w(t1) at the time t1

were subtracted from the measurements for time t ≥ t1, the expectation value of theprocess would again be zero, as for t1 = 0.

In the case of the VNA test-set drift, we have to do with complex system characterized by alarge number of frequency-dependent parameters. However, as pointed out in Section 6.2,small changes of VNA calibration coefficients at multiple frequencies can be approximatedwith the model (6.2). This model employs a set of frequency-independent parameters cap-tured in the vector (6.12) and some fixed functions characterizing the frequency-dependenceof the change in the VNA calibration coefficients. Consequently, we can describe theVNA test-set drift by letting the parameters (6.12) vary according to the vector stochasticWiener process. This process has analogous properties to the scalar process, that is (seeAppendix H):

a) w(t) = 0,

b) w(t) ∼ N(0, Λt),

c) w(t2) − w(t1) ∼ N(0, Λ(t2 − t1)),

d) E[w(t)|w(t1); t1 ≤ t2] = w(t1),

where Λ is positive-definite symmetric full-rank matrix characterizing the process. Conse-quently, we construct the description of the VNA test-set drift with the model (6.2) wherestatistical properties of the model parameters p are described by the vector stochasticWiener process.

158


6.4.2 Estimation of the process covariance matrix

We estimate the process covariance matrix Λ based on repeated corrected VNA mea-surements of a one port device, performed over a long period of time. During the measure-ments, the VNA setup is kept unchanged, that is, the device is not reconnected and theposition of the cables is not changed. Similarly to the estimation of the connector interfacemodel, the preferred devices are highly reflective terminations, due to the larger impact ofthe changes in VNA calibration coefficients on the their measurements.

The measurements are described by a model similar to (6.14), that is

Γmk = Γ0k + xk(d, Γ0k)T pm + εmk, (6.28)

where pm is the change of VNA parameters in the m-th measurement with reference tothe state at the beginning of the experiment, Γ0k is the value of the reflection coefficientmeasured with the drift-free VNA, εmk is the VNA receiver noise, d is given by (6.16) and

xk(d, Γ0k) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−2Γm,kwT (fk)e−j2β0d0wR1(fk) + ej2β0d0Γ2

0kwR2(fk)e−j2β0d1wR1(fk) + ej2β0d1Γ2

0kwR2(fk)...

e−j2β0dN wR1(fk) + ej2β0dN Γ20kwR2(fk)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (6.29)

Our goal now is to determine the estimates pm of the VNA parameters changes pm, form = 1 . . . , M where M is the number of measurements, the estimate Λp of the processcovariance matrix, the estimates dn of the perturbation locations dn, for n = 0, . . . , N ,and the estimates Γ0k of the unknown reflection coefficient Γ0k, for frequencies fk, wherek = 1, . . . , K.

We use a similar approximate approach as in the case of the connector interface errors.We first note that the true value Γ0k of the reflection coefficient can be well approximatedby the measurement taken right after the calibration, that is,

Γ0k = Γ1k. (6.30)

After inserting this estimate into (6.28), we only need to determine pm , dn, and Λp. Wefurther approximate the estimate Λp with the use of the pm and based on (H.13). That

159


is,

Λp = 1(M − 1)ΔT

M−1∑m=1

ΔpmΔpTm, (6.31)

where Δpm = pm+1 −pm, for m = 1, . . . , M −1, and ΔT is the time increment between theconsecutive measurements. The error of this approximation will depend on the accuracywith which we determine the estimates pm. In order to increase the accuracy of theapproximation (6.31), we perform the measurements for a large number of frequencies,and reduce the VNA measurement noise by narrowing the IF bandwidth and averagingmultiple results.

By use of the above assumptions, we can now pose the original problem as the task ofdetermining pm, for m = 1 . . . , M , and dn, for n = 0, . . . , N under the condition (6.30).To this end we write the measurements (6.28) in a more compact form as

ym = y + X (d)pm + εm, (6.32)

where

ym =

⎡⎢⎢⎢⎣Γm,1...

Γm,K

⎤⎥⎥⎥⎦ , y =

⎡⎢⎢⎢⎣Γ11...

Γ1K

⎤⎥⎥⎥⎦ , εm =

⎡⎢⎢⎢⎣

εm1...

εmK

⎤⎥⎥⎥⎦ , (6.33)

and

X (d) =

⎡⎢⎢⎢⎣

x1 (d,Γ11)T

...xK (d,Γ1K)T

⎤⎥⎥⎥⎦ (6.34)

Now, we can determine the pm and the dn as the solution to a nonlinear least squaresproblem analogous to (6.24). We solve this problem with the same methods as in the caseof the connector interface errors (see Appendix G).

Once the estimates pm are determined, we calculate the estimate (6.31) of the processcovariance matrix. Similarly to the connector repeatability and cable instability model, wereduce the dimensionality of this covariance matrix with the use of the principal componentanalysis [87]. That is, we determine the approximation

Λp ≈ UΛξUT , (6.35)

where Λξ is a diagonal matrix of size M × M with non-zero diagonal elements, referred toas principal components, U is a full column-rank rectangular matrix of size P × M with

160


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Fig. 6.10: Standard deviation of the time-domain representation of changes in the reflectioncoefficient of a 5.4mm long 1.85mm male coaxial offset-short within 2 min long periods,as repeatedly measured over the period of 12 hours: (a) from −1.4m to the VNA referenceplane (b) zoom into the range from −100mm to the VNA reference plane.

orthogonal columns, P is the length of the vector p, and M is the number of principalcomponents. We choose the number of principal components in the approximation (6.35) soas to capture 99% of the total variance contained in the original matrix (6.31). This resultstypically in two or three principal components. Analogously to the connector interfacemodel, these principal components can be treated as the electrically-equivalent physicalerror mechanisms (see Section 4.2), characterizing the VNA test-set drift.

6.4.3 Experiments

We verified our approach by making repeated reflection-coefficient measurements ofdifferent one-port devices over a long period of time. The measurements were performedin the setup shown in Fig. 6.4b, with the same setting as for the connector repeatabilitymeasurements (see Paragraph 6.3.3-A). The one-port devices were measured on the VNAport 1.

Fig. 6.7 shows the time-domain representation of the results we obtained for a 5.4 mmlong 1.85 mm coaxial female offset short. The measurements were performed every 2 minover the period of 12 hours. For each pair of consecutive measurement, we calculated thechange in the reflection coefficients, transformed those changes into the time domain inthe same manner as in the case of the cable instability errors (see Paragraph 6.3.3-B), andthen calculated the standard deviation at each time point. The peaks in this representation

161


correspond therefore to locations within the VNA at which changes due to the drift areoccuring.

The standard deviations shown in Fig. 6.7 are much smaller (almost by an order ofmagnitude) than for the cable instability errors. This is because we are observing changesdue to the drift within short 2 min long periods of time. For the same reason, the impactof the VNA receiver errors is more visible in Fig. 6.7, and manifests itself by a high noisefloor at the level of around 2 × 10−6.

In Fig. 6.7a we see further, that the dominant changes are occuring in the range between−100mm and the VNA reference plane. This range is expanded in Fig. 6.7b, where wesee peaks in the range between −70mm and −50mm, and between −10mm and 0mm.Based on that, we decided to use in our model four perturbations. In Tab. 6.1 we puttogether the initial locations of those perturbation, approximated based on the Fig. 6.7b,along with the refined estimates determined in the modeling procedure.

Locations [mm]Initial 0 -4.5 -31.3 -63.8 -66.0

Estimated 0 -4.9 -30.9 -63.5 -66.3

Tab. 6.1: Locations of the perturbations in thestochastic model for the VNA test-set drift developedbased on the measurements of a 5.4 mm long 1.85 mmcoaxial female offset-short.

In Fig. 6.8 we present resultsof this modeling procedure. Inthis figure, we show the in-phaseand quadrature component of thestandard deviation of the changescalculated in the frequency do-main, along with the predictionof the stochastic model we devel-oped. The model we used em-ployed two random variables. The

agreement between the stochastic model prediction and measurement for the quadrature(phase) errors is very good except for the discrepancy in the frequency range below 4 GHz.We also noticed this discrepancy is in other experiments (see Subsection 6.3.3-A). We thinkthe this discrepancy is related to the fact that the VNA uses a different set of of couplersand mixers in below 4 GHz. Another confirmation for this hypothesis can be seen in thein-phase component of the standard deviation, determined based on the measurements,which abruptly increases below 4 GHz.

The agreement for the in-phase (magnitude) errors is not as good and, in general,our model underestimates the in-phase component of the standard deviation. We thinkthat this component is larger in the actual measurements due to the presence of the VNAreceiver noise not accounted for by our model. On the other hand, the in-phase errors

162


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Fig. 6.11: Standard deviation in the reflection coefficient of a 5.4mm long 1.85mm malecoaxial offset-short within 2 min long periods, as repeatedly measured over the period of12 hours, represented as the in-phase (blue) and quadrature (red) component, along withthe stochastic model prediction (black).

are much smaller and, therefore, less important than the quadrature errors. Consequently,our stochastic model is capable of adequately representing the VNA test-set drift errors asobserved in the measurement of the short.

We further used the stochastic model for VNA test-set-drift we developed for the 5.4 mmlong 1.85 mm coaxial female offset short to predict the errors related to the drift in othermeasurements performed on the same VNA. To this end, we remeasured the same short,and then measured also other one-port devices, that is, a 5.4 mm long 1.85 mm coaxial offsetfemale open and a coaxial female matched termination. In each case, the measurementwere performed every 1/2 hour for the period of 12 hours.

The results of those measurements are shown in Fig. 6.12. In all figures, we showthe deviations of the measurements from the initial measurement performed right afterthe calibration, along with the 95% uncertainty bounds predicted from our stochasticmodel. In the case of the offset open and offset short, we show separately the in-phase andquadrature component of the deviations, while in the case of the matched termination, wecompare the real and and imaginary part of the deviations.

In the case of the offset short, the variation of the quadrature component of the devi-ations is well predicted by the uncertainty bounds predicted from the model. As for theoffset open, our model slightly underestimates the deviations of the quadrature componentof the deviations. We think that this may be caused by a larger variation of the tempera-ture in the laboratory during the measurements performed for the open, as compared withthe variation that occurred during the measurements used to establish the model. Also,

163


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(b)Fig. 6.12: VNA test-set drift errors in repeated reflection-coefficient measurements of1.85mm coaxial one-port devices, taken every 1/2 hour over the period of 12 hours, alongwith the 95% uncertainty bounds predicted from the stochastic model developed based onthe measurements of the 5.4mm long female offset-short: (a) in-phase error for a 5.4mmfemale long offset-short, (b) quadrature error for the 5.4mm female long offset-short, (c)in-phase error for a 5.4mm female long offset-open, (d) quadrature error for the 5.4mmfemale long offset-open, (e) real part of the error for a matched termination, (b) imaginarypart of the error for the matched termination.

164

6.5. SUMMARY

in both cases, our model underestimates the in-phase component of the deviations, whichmay be again attributed to the VNA receiver errors which are not accounted for by ourmodel. This component, however, is much smaller than the quadrature component, con-sequently, our stochastic model is capable of adequately predicting the dominant elementof the VNA test-set drift errors, as observed in the measurements of the offset short andoffset open.

The prediction of the uncertainty bounds due to the VNA test-set drift for the matchedtermination is slightly worse than for other one-port standards. This may be again at-tributed to VNA receiver errors whose relative contribution is larger for devices with smallerreflection coefficients.

6.5 Summary

We proposed a new approach to the description of the VNA nonstationarity errors. Ourapproach is based on a stochastic model constructed by adding randomly varying lumpedelements to the two-port describing the VNA calibration coefficients. In the case of theconnector nonrepeatability and cable instability, we describe the statistical properties ofthose elements with a multivariate Gaussian PDF, while in the case of the VNA test-setdrift, we employ the vector stochastic Wiener process.

Parameters of the stochastic model are then identified based on a simple experimentsinvolving repeated measurements of a single highly-reflective offset terminations. Thesimplicity of this measurement allows us to independently characterize different VNA non-stationarity sources, that is connector nonrepeatability, cable instability, and the test-setdrift.

The preliminary experimental results demonstrate that our models are capable of mod-eling the complicated frequency-dependent characteristics of the VNA nonstationarity er-rors in the measurements of coaxial one-port devices by use of only a few statistically un-correlated and frequency inpdendent random variables. We further show that our models,once establihed, can fairly accurately predict the uncertainty bounds for VNA nonstation-arity errors in the future measurements of devices with arbitrary reflection coefficients.This indicates that the models we developed consistently describe the properties of a givenconnector interface, cable, or VNA, rather than just accurately reproducing the measure-ments obtained in one experiment. Hence, the random parameters that underlie our modelsare can be treated as the electrically-equivalent error mechanisms responsible for the VNA

165


nonstationarity errors (see Section 4.2).The stochastic models for we developed here have already been applied to assess the

uncertainties due the VNA nonstationarity errors in the uncertainty analysis for VNA mea-surements [39] which accounts for the statistical correlations between VNA measurementuncertainties at different frequencies. These models, once fully verified and extended tothe measurements two-port devices, can also be incorporated into the error-mechanism-based VNA calibration (see Subsection 4.5.3), in order to further increase the calibrationaccuracy.

166

Chapter 7

Conclusions

Good tests kill flawed theories;we remain alive to guess again.

Karl Popper

In this work, we developed theoretical foundations and presented the most importantpractical applications of the multi-frequency approach to VNA S-parameter measurements.The principle of this approach is to account for the relationships between S-parametermeasurements at different frequencies. Thus, the multi-frequency approach breaks withthe traditional paradigm in VNA measurements, according to which the measurementprocedure is carried out independently at each frequency under consideration.

The multi-frequency approach stems from the observation that S-parameter-measurement errors at different frequencies are related to each other. These relationshipscan be intuitively explained by the fact that all of the measurement errors, both at thesame and at different frequencies, have some common fundamental physical causes. Thesecauses correspond to errors in the definitions of calibration standards (e.g., tolerances oftheir dimensional and material parameters) and to errors in the VNA instrumentation(e.g., instability of the cables, misalignment of the inner and outer conductors or displace-ments of the inner conductor fingers in coaxial connectors, test-set drift, or receiver noiseand nonlinearities).

The simplest mathematical representation for the relationships between measurementerrors at different frequencies is given by the covariance matrix [39]. The diagonal termsof this matrix describe the variances of the measurement errors while the off-diagonalterms capture the statistical correlations between those errors, including the correlations

167

7. CONCLUSIONS

between errors at different frequencies. The vector underlying the covariance matrix iscomprised of the real and imaginary parts of the measurement errors at all frequencies.Thus, we refer to this vector as the multi-frequency S-parameter-measurement error. Sucha representation generalizes the approach, employed in some early contributions on six-portS-parameter measurement systems, and recently rediscovered for the VNA measurements[42], in which the single-frequency S-parameter measurement errors are also characterizedwith a covariance matrix.

The use of the covariance matrix for the multi-frequency S-parameter measurementerror, although seemingly intuitive, leads to some difficulties. In most practical cases, thecolumns and rows of this matrix are linearly dependent due to the fact that it charac-terizes the variability of a large number of random variables (real and imaginary partsof S-parameters at all frequencies), dependent on a much smaller number of other inde-pendent random variables, corresponding to the fundamental causes of the measurementerrors. Consequently, the covariance matrix is rank deficient and cannot be inverted, hencethe conventional form of the multivariate Gaussian PDF (see [87]) cannot be used withthe multi-frequency S-parameter-measurement error. As a result, applications where theknowledge of the PDF is required, such as statistical VNA calibration methods or statis-tical procedures for the measurement-based device modeling, cannot be directly extendedto use the covariance matrix for the multi-frequency S-parameter-measurement error.

Therefore, in order to remedy this mathematical difficulty, we followed a different ap-proach in this work. Instead of directly employing the covariance matrix of the multi-frequency S-parameter measurement error, we focused in our analysis on the fundamentalcauses of the measurement errors which determine the structure of this matrix. A math-ematical model for these causes is given by the notion of the physical error mechanism.We define a single physical error mechanism as a scalar random variable, correspond-ing to a physical parameter which is affected by errors, and a corresponding frequency-dependent function which characterizes the relationships between this parameter and themulti-frequency S-parameter measurement error. By applying this concept and with theuse of the generalized matrix-pseudo-inverse (see [117]), we then construct the multivariateGaussian PDF for the multi-frequency S-parameter-measurement error. This PDF turnsout to have an intuitive form in which the probability for given region in the domain of S-parameters is expressed in terms of the corresponding region in the domain of the physicalerror mechanisms. This result is an important and unique contribution of this work.

We further analyze how to adapt the VNA measurement procedure to account for a

168

more complete description of measurement errors offered by the concepts of the physicalerror mechanism and the multi-frequency S-parameter-measurement error. We focus onthe VNA calibration and on the use of the multi-frequency S-parameter-measurement errorin the uncertainty analysis and in the measurement-based device modeling.

We first reformulate the VNA calibration problem in terms of the physical error mecha-nisms underlying the measurement errors. As these mechanisms contribute simultaneouslyto the measurement errors at all frequencies, such a generalized VNA calibration problemneeds to be solved jointly at all measurement frequencies. We refer to this generalizedphysics-based formulation of the VNA calibration problem as the error-mechanism-basedVNA calibration. This novel formulation is a unique contribution of this work and, to theauthor’s knowledge, has not been considered in the literature before.

Practical implementation of the error-mechanisms-based VNA calibration leads to somedifficulties. It requires, on one hand, detailed modeling and characterization of all of thephysical error mechanisms responsible for the VNA measurement errors. On the otherhand, the error-mechanism-based VNA calibration results in an optimization task whichis difficult to solve due to its nonlinear and ill-posed character, and large scale. In themain part of this work, we propose solutions to those problems in the context of VNAcalibration in the coaxial environment.

The development of an error-mechanism-based description of VNA measurement errorsrequires modeling of the error mechanisms affecting the calibration standards and the VNAitself. In the case of the coaxial transmission lines, employed as calibration standards inthis work, we model those mechanism based on a detailed analysis of possible errors inthe definitions of the standards. This analysis results in extended models of the lines thataccount for the phenomena such as errors in the determination of the inner and outerconductor length, variation of conductor diameters and conductor loss among the lines,nonuniformity of the conductor diameters, and nonreproducibility of the center conductorgap.

Regarding the modeling of error mechanisms affecting the VNA itself, we focus onthe VNA nonstationarity errors, that is, the connector nonrepeatability, cable instabilityand test-set drift, which are the primary sources of VNA instrumentation errors. Wepropose a novel approach that uses a stochastic model whose parameters are identified fromrepeated S-parameter measurements. The core of this model is a generic description for thefrequency dependence of the VNA nonstationarity errors which uses a set of lumped elementperturbations, added at fixed distances to the two-port describing the VNA calibration

169

7. CONCLUSIONS

coefficients. The stochastic models for the VNA nonstationarity errors are then establishedby allowing the parameters of the lumped elements to vary randomly.

In the case of the connector repeatability and cable instability errors, we assume thatthe parameters of these elements vary according to the multivariate Gaussian PDF. Thisassumption has a simple physical justification. The lumped elements model the impactof changes in some dimensional parameters on the electrical properties of the connectorinterface or the cable, and we can reasonably assume those changes to be small and to followthe multivariate Gaussian PDF. The electrical parameters are therefore, approximately,linear combinations of the dimensional parameters, and have also the multivariate GaussianPDF.

As for the test-set drift, we follow the approach used in the drift modeling of scalarmeasurement instruments, and assume that it can be described as the random walk phe-nomenon, referred to also as the Brownian motion [147, 148]. The mathematical descriptionof this phenomenon in the scalar case is given by the stochastic Wiener process [158]. Wegeneralize this description to the multidimensional case and then apply it to model thevariability of the lumped elements in the perturbations.

We further present methods to identify parameters of the stochastic models. Thesemethods involve repeated reflection-coefficient measurements of only a single highly-reflective offset termination. Preliminary experimental results for the coaxial connectorstandard show that we are able to model and predict the complicated frequency de-pendence of the connector nonrepeatability, cable instability and test-set drift errors,in the measurements of one-port devices, by use of only a few statistically uncorrelatedand frequency-independent random variables. Establishing the validity bounds for thestochastic models we developed and their extension to the two-port measurements are thetopics of our ongoing research.

Our description of VNA nonstationarity errors is a significant and unique contribution ofthis work. To the author’s knowledge, such as simple and yet comprehensive model for thisimportant source of errors in VNA measurements has not been proposed in the literatureso far. The variables used in this description are frequency independent and statisticallyuncorrelated and can therefore be thought of as the electrically-equivalent error mechanismsresponsible for the VNA nonstationarity errors. Consequently, our description, once fullyestablished and verified, can be used in the error-mechanism-based VNA calibration tofurther improve the measurement accuracy. This is the topic of our further research.

Regarding the numerical algorithm for the error-mechanism-based VNA calibration, we

170

develop it in the context of the coaxial multiline TRL calibration [36]. In our implemen-tation, we account only for the error mechanisms affecting the calibration standards anduse a conventional description of the VNA instrumentation errors. The resulting algorithmis generic and can easily be adopted to other redundant calibration schemes, such thoseconsidered in [37, 38]. This novel algorithm is one of the main contributions of this work.

Two main problems we faced with the implementation of this algorithm are the largescale of the resulting nonlinear optimization task and its ill-posed character. The largescale of the optimization task results from the fact the VNA calibration coefficients aresought simultaneously at all measurement frequencies. Consequently, direct solution ofthe error-mechanism-based VNA calibration problem is very time consuming. Hence, wedeveloped a robust iterative numerical approach which exploits the relationships betweenthe sought parameters in order to reduce the dimensionality of the optimization problem.Our approach is based on a modified version of the classical Levenberg-Marquardt algo-rithm (see [159]) in which we account for the sparse structure of the Jacobians of the goalfunction in a similar manner to [140].

The ill-posed character of the error-mechanism-based VNA calibration, manifesting it-self with dificulties in obtaining a unique solution, results from the fact that some of theestimated parameters are related to each other. We analyze the origins of those relation-ships and devise a general methodology for assuring the identifiability of the solution. Thismethodology relies on restricting the space of possible solutions with a set of linear equalityconstraints, based on some intuitive statistical properties of the physical error mechanisms.

We test the resulting multi-frequency calibration algorithm for a coaxial multiline TRLcalibration with the 1.85mm and 7.0mm transmission-line standards. The extensive ex-perimental results we present indicate that by exploiting the relationships between errorsin calibration standard S-parameters at different frequencies, we can reduce the residualstandard uncertainty in corrected S-parameter measurements by a few times. This re-sult clearly demonstrates the benefit of the multi-frequency approach to VNA calibration.Adaptation of the error-mechanism-based VNA calibration to other calibration schemes,such as the offset-short calibration [79], and to measurement environments with otherwaveguiding structures, such as the planar (microstrip line, coplanar waveguide) or rect-angular waveguides, is the topic of our further research.

Equipped with the error-mechanism-based representation of multi-frequency S-param-eter-measurement errors, we further analyze the application of such a description in theuncertainty analysis and in the measurement-based device modeling. We first show that

171

7. CONCLUSIONS

the statistical correlations between measurement errors at different frequencies are essentialwhen evaluating the uncertainties in calibrated time-domain measurements that employthe VNA S-parameter measurements. Examples of such time-domain measurements arethe correction for the impedance mismatch or removing the errors due to an adapter inoscilloscope measurements. The importance of these correlations has already been pointedout in [44], and in this work we present a detailed derivation of the uncertainty analysisfor a generalized procedure for calibrated time-domain measurements.

We further reformulate the standard methods for measurement-based modeling of elec-tronic devices (e.g. transistors or transmission-line discontinuities), based on the maximumlikelihood principle, to account for the relationships between S-parameter measurement er-rors at different frequencies. The key conclusion of our analysis concerns the quantificationof the misfit between the measurement and the model. In standard methods, which donot account for the statistical correlations between S-parameter measurement errors atdifferent frequencies, this misfit is quantified in terms of S-parameters. We showed thatwhen accounting for these correlations, the misfit needs to be expressed in terms of thephysical error mechanisms. This new result is an important contribution of this work that,to the author’s knowledge, has not been published before.

Summarizing, the multi-frequency approach presented in this work, by accountingfor the relationships between measurements at different frequencies, introduces a newparadigm into the VNA S-parameter measurements. We show that this new paradigmleads to a significant improvement of the VNA S-parameter measurement accuracy andits more complete description. We hope that the multi-frequency approach to VNA S-parameter measurements may become an answer to the new demands put on the VNAmeasurements accuracy by the terahertz applications, nanotechnology, and nonliner mea-surements.

172

Appendix A

Real-valued representation ofcomplex vectors and matrices

In this appendix, we briefly review the convention used in this work for a real-valuedrepresentation of complex-valued vectors and matrices. Let the vector x ∈ C

M where

x = [x1, . . . , xM ]T . (A.1)

This convention (see also [40, 42, 160]) expands each element of the complex vector into itsreal and imaginary part, that is, the real-valued representation, denoted with underline, is

x = [Rex1, Imx1,Rex2, Imx2, . . . ,RexN , ImxN ]T . (A.2)

The inverse relationship between the two representations is readily written as

x = CMx =

⎡⎢⎢⎢⎢⎢⎢⎣

1 j

1 j. . .

1 j

⎤⎥⎥⎥⎥⎥⎥⎦

x, (A.3)

where the size of the matrix CM is M ×2M . Let further the vector y ∈ Cn, and the matrix

A ∈ Cn×m where

A =

⎡⎢⎢⎢⎣

a11 · · · a1M

... . . . ...aN1 · · · aNM

⎤⎥⎥⎥⎦ (A.4)

173

A. REAL-VALUED REPRESENTATION OF COMPLEX VECTORS AND MATRICES

and y = Ax. With the use of the convention (A.2), we can now write the relationshipbetween the two vector as

y = A x. (A.5)

We can easily show that

A =

⎡⎢⎢⎢⎣

R (a11) · · · R (a1M)... . . . ...

R (aN1) · · · R (aNM)

⎤⎥⎥⎥⎦ , (A.6)

where the operator R (x) is defined by

R (x) =⎡⎣ Rex −Imx

Imx Rex

⎤⎦ . (A.7)

174

Appendix B

Maximum likelihood approach tosystem identification

In this appendix, we discuss the key concepts of the application of the maximum-likelihood approach to the nonlinear system identification. Our discussion is based onReferences [64, 65].

B.1 Introduction

A generic nonlinear static system can be described with a multivariate vector-valuedfunction

y = f (x, θ) , (B.1)

where x ∈ RP is a vector of system inputs (excitations), y ∈ R

Q is a vector of systemoutputs (responses), θ ∈ R

R is a vector of system parameters. The objective of systemidentification is to determine the system parameters θ based on a set of measured responses{yn}N

n=1, obtained for a set of applied excitations {xn}Nn=1. In the error-free case, these

measurements are described by a consistent set of equations⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1 = f (x1, θ)...

yN = f (xN , θ), (B.2)

which, provided that NQ ≥ R, can readily be solved for θ by selecting any R differentequations out of (B.2).

175

B. MAXIMUM LIKELIHOOD APPROACH TO SYSTEM IDENTIFICATION

Fig. B.1: System with errors in responses and excitations.

In the reality, however, both system excitations and responses are affected by errors,that is, we observe the system response with some measurement error

yn = yn +Δyn, (B.3)

under a disturbed excitationxn = xn +Δxn, (B.4)

where the vectors Δyn and Δxn are the error in the measurement of the system responseyn and the error in the application of the system excitation xn, respectively, yn and yn

are the measured value and the unobservable true value of the system response yn, and xn

and xn are the unobservable true value and the estimated value of the system excitationxn, respectively. In the following, we refer to the errors Δyn and Δxn in short as themeasurement errors and excitation errors, respectively.

The measurements we obtain in the real case are thus described by a set of equations⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1 = f(x +Δx1, θ

)+Δy1

...yN = f

(xN +ΔxN , θ

)+ΔyN

. (B.5)

where the measurement and excitation errors, {Δyn}Nn=1 and {Δxn}N

n=1, respectively, areadditional unknowns, and θ denotes the unknown true value of the system parameters.This set of equations (B.5) cannot be uniquely solved for θ, {Δyn}N

n=1 and {Δxn}Nn=1,

as the number of unknowns is larger the the number of equations in (B.5). Hence, someadditional criteria need to be used to obtain a solution. In the following, we describe theapplication of the maximum-likelihood approach for solving the set (B.5).

The principal idea of the maximum-likelihood approach is to assign a metric to differentsolutions of (B.5), based on the statistical properties of the errors {Δyn}N

n=1 and {Δxn}Nn=1.

This metric is referred to as the likelihood function. Equations (B.5) are then solved byselecting the solution for which the likelihood function attains the maximum.

176

B.2. FORMULATION

In the following, we first consider different formulations of the maximum-likelihoodsolution to the problem (B.5), that result from different assumptions as to the measurementand excitation errors. Then we discuss the error analysis of the solution to (B.5) andbriefly review numerical techniques used for obtaining the maximum-likelihood estimators.Finally, we discuss the conditions for the identifiability of the solutions to (B.5) and discussthe extension of our results to the case of systems with complex-valued inputs and outputs.

B.2 Formulation

The two common cases we consider when solving the set (B.5) are the simplified casewhen the system excitations are assumed to be error-free, and the general case when boththe system responses and excitations are affected by errors. In the following, we firstdiscuss in detail the simplified case and then extend our results to the general case.

B.2.1 Errors in system responses

In the case when only system responses are affected by errors, the set (B.5) reduces to⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1 = f(x1, θ

)+Δy1

...yN = f

(xN , θ

)+ΔyN

, (B.6)

and we are seeking the vector θ and the set of vectors {Δyn}Nn=1 that satisfy the set

(B.6). In order to form the maximum-likelihood solution to this problem, we need to knowthe statistical properties of the measurement errors {Δyn}N

n=1 . For each measurementΔyn, these properties are captured in the probability density function fΔyn (Δyn). Fornormally distributed errors with E (Δyn) = 0 and with a full-rank covariance matrixE(ΔynΔyT

n

)= ΣΔyn , this PDF is given by

fΔyn (Δyn) = (2π)−Q/2 |ΣΔyn|−1/2 exp(

−12ΔyT

n Σ−1Δyn

Δyn

). (B.7)

Assuming further that errors for the consecutive measurements are statistically indepen-dent, that is, E

(ΔykΔyT

l

)= 0 for k �= l, we can write the joint PDF for the vector

Δy = [Δy1, . . . ,ΔyN ]T , (B.8)

177


as

fΔy (Δy) =N∏

n=1fΔyn (Δyn) = (2π)−NQ/2 |ΣΔy|−1/2 exp

(− 12ΔyT Σ−1

ΔyΔy)

, (B.9)

where the covariance matrix for the joint PDF is

ΣΔy =

⎡⎢⎢⎢⎣

ΣΔy1. . .

ΣΔyN

⎤⎥⎥⎥⎦ . (B.10)

Now, when formulating the maximum-likelihood solution to the system identificationproblem, two concepts of errors and residuals play an important role[64]. Both conceptsrefer to the variables used in the description of the system that are subject to some randomdisturbances. For a given variable x, the error is defined as the difference between theobserved value x and the true (unobservable) value x of this variable, that is, ex = x − x.The residual is an estimate rx of the error ex. The likelihood function is then formed bysubstituting the residuals into the joint PDF of the errors.

For the measurements (B.6) performed on the system shown in Fig. B.1, under theassumption of error- free excitations, the errors are defined by the vector (B.8). Theresiduals

ry = [ry1 , . . . , ryN]T (B.11)

are the estimates of the errors (B.8). For a given sample {yn}Nn=1 of system response mea-

surements and system excitations {xn}Nn=1, and for a given estimate of system parameters

θ, the residuals need to satisfy the equations (B.6), which yields⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1 = f (x1, θ) + ry1 ,...

yN = f (xN , θ) + ryN

. (B.12)

The likelihood function is now formed by substituting the residuals ry into the PDF of theerrors Δy, which immediately yields

L (θ, ry|x, y) = (2π)−NQ/2 |ΣΔy|−1/2 exp[

− 12ry

T Σ−1Δyry

]. (B.13)

wherey = [y1, . . . , yN ]T , (B.14)

178

B.2. FORMULATION

andx = [x1, . . . , xN ]T . (B.15)

In the following we drop the dependence on x and y in order to simplify the notation.

The maximum likelihood estimates of the system parameters and the errors, θ and Δyn

are then obtained by maximizing the likelihood function (B.13) accounting for the equalityconstraints (B.12), that is

argmaxθ,ry

L(θ, ry), (B.16)

under the constraints (B.12). (B.17)

By directly incorporating the constraints we obtain an unconstrained problem

θ = argmaxθ

L (θ) , (B.18)

whereL (θ) = (2π)−NQ/2 |ΣΔy|−1/2 exp

[− 12ry(θ)T Σ−1

Δyry(θ)], (B.19)

withry(θ) = [ry1(θ), . . . , ryN

(θ)]T , (B.20)

and ⎧⎪⎪⎪⎨⎪⎪⎪⎩

ry1 (θ) = y1 − f (x1, θ)...

ryN(θ) = yN − f (xN , θ)

. (B.21)

From θ we can obtain the maximum likelihood estimate Δyn of the measurement errorΔyn by evaluating (B.21) at θ, that is, Δyn = ryn(θ), for n = 1, . . . , N .

The maximum likelihood estimate (B.18) has an intuitive interpretation [64]. Let δΩdenote a region in the hyperplane around the vector of measurements y. Then the probabil-ity obtaining the measurements in a region δΩ around the actually observed measurementsis L (θ) δΩ. Consequently, for all possible values of θ, the probability of obtaining mea-surements within δΩ of the actual ones attains the maximum for the estimate (B.18).

Now, when determining the estimate (B.18), both numerically and analytically, it istypically much easier to maximize the logarithm of L (θ), referred to as the log-likelihood[64]. This readily leads to the same result owing to the monotonicity of the logarithm.

179


That is,θ = argmax

θL (θ) = argmax

θlnL (θ) . (B.22)

In the case of (B.13), we the log-likelihood function takes on the form

lnL (θ) = −12NQ ln (2π) − 1

2

N∑n=1

ln |ΣΔyn| − 12

N∑n=1

ryn(θ)T Σ−1

Δynryn(θ)=

= −12NQ ln (2π) − 1

2 ln |ΣΔy| − 12ry(θ)T Σ−1

Δyry. (B.23)

Now, when solving the problem (B.18), we encounter mainly two cases: when thecovariance matrices ΣΔyn of errors in the system responses are fully known and when thematrices ΣΔyn are known up to a scaling factor, that is,

ΣΔyn = σ2VΔyn , (B.24)

where the VΔyn are known matrices, and the scaling factor σ2 is referred to as the resid-ual variance (square of the residual standard deviation). In the case of the fully knowncovariance matrices ΣΔyn , maximizing the log-likelihood function (B.23) readily leads tothe minimization of a weighted sum of squared residuals, that is,

θ = argminθ

N∑n=1

ryn(θ)T Σ−1

Δynryn(θ)=argmin

θry(θ)T Σ−1

Δyry(θ), (B.25)

Thus, in this case, the maximum likelihood estimation is equivalent to the familiar weightedleast squares problem [64, 65].

In the case of when the covariance matrices are given by (B.24), the log-likelihoodfunction takes on the form

lnL(θ, σ2) = −12NQ ln (2π)−1

2

N∑n=1

ln |VΔyn|−12NQ ln σ2− 1

2σ2

N∑n=1

ryn(θ)T V−1

Δynryn(θ),

(B.26)

and, in order to obtain the estimates of θ and σ2, the function (B.28) needs to be maximizedin terms of both parameters. A common approach here is the stagewise maximization ofthe likelihood function, referred to as the concentrated likelihood approach [65]. In this

180

B.2. FORMULATION

approach, we first assume that θ is fixed and, from the condition for the stationary point

∂ lnL(θ, σ2)∂σ2 = −NQ

2σ2 + 12 (σ2)2

N∑n=1

ryn(θ)T V−1

Δynryn(θ) = 0,

we obtain the estimate σ2 (θ) as

σ2 (θ) = 1NQ

N∑n=1

ryn(θ)T V−1

Δynryn(θ). (B.27)

Now, after inserting this estimate into the original likelihood function (B.26), we obtain

lnL(θ, σ2 (θ)) = −12NQ[ln (2π) + 1 − ln (NQ)] − 1

2

N∑n=1

ln |VΔyn|+

− 12NQ ln

N∑n=1

ryn(θ)T V−1

Δynryn(θ), (B.28)

from which we obtain the estimate of θ as

θ = argmaxθ

lnL(θ, σ2 (θ)). (B.29)

Due to the monotonicity of the logarithm, this estimate (B.29) can be equivalently deter-mined as

θ = argminθ

N∑n=1

ryn(θ)T V−1

Δynryn(θ)=argmin

θry(θ)T V−1

Δyry(θ). (B.30)

where

VΔy =

⎡⎢⎢⎢⎣

VΔy1. . .

VΔyN

⎤⎥⎥⎥⎦ . (B.31)

The estimate θ inserted into (B.27) yields eventually the estimate of the residual variance

σ2MLE = σ2(θ) = 1

NQ

N∑n=1

ryn(θ)TV−1

Δynryn(θ) =

1NQ

ry(θ)TV−1

Δyry(θ). (B.32)

This estimate can be shown to be biased [64]. The unbiased estimate of the residual

181


variance is given by [64]σ2 = NQ

NQ − Rσ2

MLE, (B.33)

where R is the number of elements in the vector θ.

B.2.2 Errors in system responses and excitations

When both system responses and excitations are subject to disturbances, the solutionto (B.5) is constituted by not only the estimates of system parameters θ and the errors{Δyn}N

n=1 in the system responses, but also by the estimates of the errors {Δxn}Nn=1 in

the system excitations . The resulting problem and its variants are known in the literatureunder different names, such as: errors-in-independent-variables regression (EIV) [64, 65],orthogonal-distance (ODR) regression [140], total least-squares (TLS) regression [161], fullleast-squares regression [162], or generalized-distance regression [75]. In this work we followthe nomenclature of [64, 65].

In order to write the likelihood function for the problem (B.5), similarly to the problem(B.6), we need to know the statistical properties of the errors {Δyn}N

n=1 and {Δxn}Nn=1.

Regarding the errors in system responses we follow the assumptions made in Section B.2.1.As for the errors {Δxn}N

n=1 in the system excitations, for each excitation we describestatistical properties of these errors with the probability density function fΔxn (Δxn). Fornormally distributed errors with E (Δxn) = 0 and with a full-rank covariance matrixE(ΔxnΔxT

n

)= ΣΔxn this PDF can be written as

fΔxn (Δxn) = (2π)−P/2 |ΣΔxn|−1/2 exp(

− 12ΔxT

n Σ−1Δxn

Δxn

). (B.34)

Assuming further that these errors for the consecutive measurements are statistically in-dependent, that is E

(ΔxkΔxT

l

)= 0 for k �= l, we can write the joint PDF for the vector

Δx = [Δx1, . . . ,ΔxN ]T (B.35)

as

fΔx (Δx) = (2π)−NQ/2N∏

n=1|ΣΔxn|−1/2 exp

(− 12

N∑n=1

ΔxTn Σ−1

ΔxnΔxn

)=

= (2π)−NQ/2 |ΣΔx|−1/2 exp(

−12ΔxT Σ−1

ΔxΔx)

, (B.36)

182

B.2. FORMULATION

where the covariance matrix for the joint PDF is

ΣΔx =

⎡⎢⎢⎢⎣

ΣΔx1. . .

ΣΔxN

⎤⎥⎥⎥⎦ . (B.37)

We also assume that the errors in the excitations and errors in the responses are statisticallyindependent, that is that is E

(ΔxkΔyT

l

)= 0 for k, l = 1, . . . , N .

Now, for the measurements (B.5) are defined by the vectors (B.8) and (B.36). As thoseerrors are statistically independent, their joint PDF is given by

f(Δx,Δy) = fΔx (Δx) fΔy (Δy) . (B.38)

The residualsrx = [rx1 , . . . , rxN

]T (B.39)

andry = [ry1 , . . . , ryN

]T (B.40)

are the estimates of the errors Δx and Δy, respectively. For a given sample {yn}Nn=1 of

system response measurements and system excitations {xn}Nn=1, and for a given estimate

of system parameters θ, the residuals need to satisfy the equations (B.5), which yields⎧⎪⎪⎪⎨⎪⎪⎪⎩

y1 = f (x + rx1 , θ) + ry1...

yN = f (xN + rxN, θ) + ryN

. (B.41)

The likelihood function is now formed by inserting the residuals into the joint PDF (B.38)of the errors. This yields

L (θ, rx, ry|x, y) =

= (2π)−NQ/2−NP/2 |ΣΔy|−1/2|ΣΔx|−1/2 exp[

− 12ry

T Σ−1Δyry − 1

2rxT Σ−1

Δxrx

], (B.42)

wherey = [y1, . . . , yN ]T , (B.43)

183


andx = [x1, . . . , xN ]T . (B.44)

In the following, we drop the dependence on x and y, in order to simplify the notation.

The maximum likelihood estimates of the system parameters θ, and the errors in re-sponses and excitations, Δyn and Δxn, respectively, is then obtained by maximizing thelikelihood function (B.42) while accounting for the equality constraints (B.41), that is,

arg maxθ,ry,rx

L(θ, ry, rx), (B.45)

under the constraints (B.41). (B.46)

We can further directly incorporate the constraints into (B.45), which yields an equivalentproblem

φ =⎡⎣ θ

Δx

⎤⎦ = argmax

φL(φ), (B.47)

where

φ =⎡⎣ θ

rx

⎤⎦ , (B.48)

and

L (φ) = (2π)−NQ/2−NP/2 |ΣΔy|−1/2|ΣΔx|−1/2

exp(

− 12ry(φ)T Σ−1

Δyry(φ) − 12rT

x Σ−1Δxrx

), (B.49)

with ⎧⎪⎪⎪⎨⎪⎪⎪⎩

ry1 (φ) = y1 − f (x1 + rx1 , θ)...

ryN(φ) = yN − f (xN + rxN

, θ). (B.50)

The maximum likelihood estimate of measurement errors Δyn is obtained by evaluating(B.50) at φ, that is, Δyn = ryn(φ), for n = 1, . . . , N .

Similarly to the problem (B.6), we can show that with fully known covariance matricesof errors in the system excitations and measurements, (B.48) is equivalent to the weighted

184

B.2. FORMULATION

least squares problem, that is,

φ = argminφ

N∑n=1

ryn(φ)T Σ−1

Δynryn(φ) +

N∑n=1

rTxn

Σ−1Δxn

rxn =

= argminφ

ry(φ)T Σ−1Δyry(φ)+rT

x Σ−1Δxrx, (B.51)

Another case we often consider in practice is when the covariance matrices are known upto a scaling factor, that is,

ΣΔyn = σ2VΔyn , and ΣΔxn = σ2VΔxn , (B.52)

whereVΔxn and VΔyn , for n = 1, . . . , N are known matrices, and σ2 is the unknown residualvariance. In this case, we can also perform a stagewise optimization of the log-likelihoodfunction to obtain (see [64])

φ = argminφ

N∑n=1

ryn(φ)T V−1

Δynryn(φ) +

N∑n=1

rTxn

Σ−1Δxn

rxn =

= argminφ

ry(φ)T V−1Δyry(φ) + rT

x Σ−1Δxrx, (B.53)

and

σ2MLE = 1

Nn

[ N∑n=1

ryn(φ)TV−1

Δynryn(φ) +

N∑n=1

rTxn

Σ−1Δxn

rxn

]=

= 1Nn

[ry(φ)

TV−1

Δyry(φ) + rTx Σ−1

Δxrx

]. (B.54)

Similarly to (B.32), the estimator (B.54) is biased. The unbiased estimator of the residualvariance is given by [64]

σ2 = NQ

NQ − NP − Rσ2

MLE. (B.55)

We shall now make one additional remark about the formulation (B.51) and (B.53).Both problems can be equivalently written in a form similar to (B.25) and (B.30), respec-tively. That is, in the case of (B.51), we can write

φ = argminφ

r (φ)T Σ−1r (φ) , (B.56)

185


where

r (φ) =⎡⎣ ry(φ)

rx

⎤⎦ , (B.57)

and the matrix Σ is given by

Σ =⎡⎣ ΣΔy

ΣΔx

⎤⎦ . (B.58)

In the case of the formulation (B.53), we replace Σ in (B.56) with

V =⎡⎣ VΔy

VΔx

⎤⎦ (B.59)

This formulation is helpful in the error analysis of the estimate φ and in the developmentof the numerical methods for solving (B.51) and (B.53).

B.3 Covariance matrix of the estimates

Once we have the estimate θ of the system parameters (and the estimates Δxn of thedisturbances in the system excitations), we are often interested in the uncertainty of theseestimates. These uncertainties are determined in the residual analysis and are obtainedbased on sensitivities of the maximum-likelihood solution θ to changes in system responsesand excitations.


We first analyze the problem (B.25) when the covariance matrices of errors in systemresponses are fully known, and the extend our results to the case (B.30) when the covariancematrices are given by (B.24).

In order to determine the sensitivity of the solution (B.25), we consider the perturbedsolution θ + Δθ due to the errors in system responses Δy′

n with E(Δy′n) = 0 and with a

known covariance matrix E(Δy′nΔy′T

n ) = ΣΔyn , for n = 1, . . . N . That is,

θ +Δθ = argminθ

N∑n=1

[yn +Δy′n − f (xn, θ)]T Σ−1

Δyn[yn +Δy′

n − f (xn, θ)] . (B.60)

By expanding the function (B.1) in the Taylor series around the unperturbed estimate θ

186

B.3. COVARIANCE MATRIX OF THE ESTIMATES

and neglecting the nonliner terms, we obtain

θ +Δθ = argminθ

N∑n=1

[ryn(θ) + Δy′

n + J(θ)(θ − θ)]T

Σ−1Δyn

[ryn(θ) + Δy′

n + J(θ)(θ − θ)]

, (B.61)

whereJ(θ)n =

∂ryn(θ)∂θT = −

[∂f (x1, θ)

∂θT

∣∣∣∣∣θ=θ

· · · ∂f (xM , θ)∂θT

∣∣∣∣∣θ=θ

], (B.62)

is the Jacobian matrix of the residuals. Now, expanding (B.61), removing the constantterms and accounting for the fact that from the stationarity of the unperturbed solutionwe have J(θ)T ryn(θ)=0, we readily obtain a linear least squares problem

Δθ = argminΔθ

N∑n=1

[Δy′

n + Jn(θ)Δθ]T

Σ−1Δyn

[Δy′

n + Jn(θ)Δθ]

, (B.63)

where Δθ = θ − θ, which has the solution

Δθ = −[J(θ)T Σ−1

ΔyJ(θ)]−1

J(θ)T Σ−1ΔyΔy′, (B.64)

where

J(θ) =

⎡⎢⎢⎢⎣

J(θ)1...

J(θ)N

⎤⎥⎥⎥⎦ , Δy′ =

⎡⎢⎢⎢⎣Δy′

1...

Δy′N

⎤⎥⎥⎥⎦ , and ΣΔy =

⎡⎢⎢⎢⎣

ΣΔy1. . .

ΣΔyN

⎤⎥⎥⎥⎦ . (B.65)

Now, writing the covariance matrix ΣΔθ of Δθ, we obtain

ΣΔθ = E(ΔθΔθT ) =

=[J(θ)T Σ−1

ΔyJ(θ)]−1

J(θ)T Σ−1ΔyE(Δy′Δy′T )Σ−1

ΔyJ(θ)[J(θ)T Σ−1

ΔyJ(θ)]−1

=

=[J(θ)T Σ−1

ΔyJ(θ)]−1

. (B.66)

In the case when the covariance matrices are known up to the scaling factor, we replace

187


ΣΔy in (B.66) with the estimate

ΣΔy = σ2

⎡⎢⎢⎢⎣

VΔy1. . .

VΔyN

⎤⎥⎥⎥⎦ = σ2VΔy, (B.67)

which yieldsΣΔθ = E(ΔθΔθT ) = σ2

[J(θ)T V−1

ΔyJ(θ)]−1

. (B.68)


In order to determine the covariance matrix of the estimates φ, we use the formulation(B.56) and follow the similar reasoning as above. For the problem (B.51) we then obtain

ΣΔφ = E(ΔφΔφT ) =[J(φ)T Σ−1J(φ)

]−1, (B.69)

while for the problem (B.53) we have

ΣΔφ = E(ΔφΔφT ) = σ2[J(φ)T V−1J(φ)

]−1, (B.70)

with

J(φ) =

⎡⎢⎢⎢⎢⎢⎢⎣

Jθ(φ)1 Jx(φ)1... ...

Jθ(φ)N Jx(φ)N

0 IP

⎤⎥⎥⎥⎥⎥⎥⎦

, (B.71)

whereJθ(φ)n =

∂rn(φ)∂θT

∣∣∣∣∣φ=φ

, Jx(φ)n =∂rn(φ)

∂xT

∣∣∣∣∣φ=φ

(B.72)

and

Σ =

⎡⎢⎢⎢⎣

Σ1. . .

ΣN

⎤⎥⎥⎥⎦ , and V =

⎡⎢⎢⎢⎣

V1. . .

VN

⎤⎥⎥⎥⎦ , (B.73)

and IP is the identity matrix of the size P × P .

188

B.4. NUMERICAL SOLUTION TECHNIQUES

B.4 Numerical solution techniques

Numerical techniques for solving the problems (B.25), (B.30) and (B.51), (B.53) are ingeneral iterative. We first discuss classical algorithms for solution of the problems (B.25)and (B.30), and then briefly review the possible strategies for solving (B.51) and (B.53).


The principle of the Gauss-Newton and Levenberg-Marquardt algorithm is to solve(B.25) in a sequence of quadratic approximations. In the q-th iteration of this sequence,the solution estimate is θ(q). From a quadratic approximation of the goal function aroundthis solution estimate, we obtain a correction Δθ(q) which is then used to update thesolution estimate with the formula

θ(q+1) = θ(q) + Δθ(q). (B.74)

This procedure is repeated until some convergence criteria are met. The two algorithmsdiffer in the way the correction Δθ(q) is computed.

A. Gauss-Newton algorithm. In the Gauss-Newton algorithm, the quadratic approxi-mation of the goal function in (B.25) is obtained by approximating (B.1) with its first-orderTaylor series expansion, that is,

f(xn, θ) ≈ f(xn, θ(q)) + J(q)f (xn)(θ − θ(q)), (B.75)

where the matrix J(q)f is the Jacobian of the function at θ(q), that is,

J(q)f (xn) =

∂f (xn, θ)∂θT

∣∣∣∣∣θ=θ(q)

. (B.76)

The correction Δθ(q) can then be obtained from a classical linear weighted least squaresproblem, that is,

Δθ(q) = arg minΔθ(q)

N∑n=1

[r(q)

yn− J(q)

f (xn)Δθ(q)]T

Σ−1Δyn

[r(q)

yn− J(q)

f (xn)Δθ(q)]

, (B.77)

189


wherer(q)

yn= yn − f(xn, θ(q)), for n = 1, . . . , N. (B.78)

Solution to (B.77) can then be obtained from the normal equations

X(q)T Σ−1ΔyX(q)Δθ(q) = X(q)T Σ−1

Δyr(q)y (B.79)

and

r(q)y =

⎡⎢⎢⎢⎣

r(q)y1...

r(q)yN

⎤⎥⎥⎥⎦ , ΣΔy =

⎡⎢⎢⎢⎣

ΣΔy1. . .

ΣΔyN

⎤⎥⎥⎥⎦ , and X(q) =

⎡⎢⎢⎢⎣

J(q)f (x1)...

J(q)f (xN)

⎤⎥⎥⎥⎦ . (B.80)

Solution to the normal equations (B.79) can be written as

Δθ(q) =(X(q)T Σ−1

ΔyX(q))−1

X(q)T Σ−1Δyr(q)

y , (B.81)

however, equation (B.81) is not recommended to compute the correction Δθ(q) due topoor numerical properties. The singular value decomposition (SVD) method (see [121]) istypically used to solve the normal equations (B.79).

B. Levenberg-Marquardt algorithm. The problem with the Gauss-Newton algorithmis that the updated solution computed from (B.74) and (B.82) may lay outside the regionin which the approximation (B.75) holds. The Levenberg-Marquardt algorithm (see [159])addresses this problem by introducing an adaptively controlled damping in the computationof the step Δθ(q). This damping is implemented through a following modification of thenormal equations (

X(q)T Σ−1ΔyX(q) + μA

)Δθ(q) = X(q)T Σ−1

Δyr(q)y , (B.82)

where μ is a damping factor and matrix A contains the main diagonal of X(q)T Σ−1ΔyX(q)Δθ(q).

The factor μ may be chosen according to different strategies (see [64, 65, 139, 159]). Thegeneral principle here is to start with a small number, such as μ = 10−3. We thencompute the step Δθ(q) and check if the goal function at θ(q+1) has decreased. If not,then the damping factor μ is increased (say by 10) and the step is recomputed. If yes,then the new solution θ(q+1) is accepted and the damping factor is decreased (say by10). In the recomputing of the step Δθ(q) for different values of the damping factor μ,the Sherman-Morrison-Woodbury identity for matrix inversion (see [121]) may be used to

190

B.5. SOLUTION UNIQUENESS

speed up the computations.


The numerical solution to the problems (B.51) and (B.53) is, in general, obtained bytransforming these problem to the formulation (B.56) and then by applying the methodsdescribed in Subsection B.4.1. The particular structure of the Jacobian matrix (B.76), asgiven by (B.71), can be exploited to reduce the dimensionality of the optimization problem[140].

B.5 Solution uniqueness

When solving the system identification problems, we are often interested in verifyingwhether the estimate of the system parameters we obtained is unique. We can show thatfor the solution to class of problem described in Section B.2.1 and Section B.2.2 to beunique, the Jacobian matrices (B.76) determined at the solution θ need to be full rank.Indeed, consider the normal equations (B.79) written at the solution, that is,

XT Σ−1ΔyXΔθ = XT Σ−1

Δyry, (B.83)

where we dropped the iteration index to signify that the residuals (B.78) and the Jacobian(B.76) are evaluated at the solution θ. At the solution, we have Δθ = 0 and consequentlyXT Σ−1

Δyry = 0. Now, if the matrix X is not full rank, the dimension of its null space islarger than one. Consequently, from the definition of the null space, there exist an infinitenumber of vectors z such that z �= 0 and Xz = 0. For a vector z ∈ null (X), we thenreadily have

XT Σ−1ΔyXz = XT Σ−1

Δyry = 0, (B.84)

hence θ∗ = θ + x is also a solution. As a result, instead of a unique solution, we have aspace of solutions defined by θ and the null space of X.

B.6 Systems with complex-valued inputs and outputs

In practice, we often encounter the case when the system has complex-valued inputsand/or outputs. In this case, we can easily adopt the methods described in this appendix

191


by representing the complex-valued vectors as real-valued vectors, comprised of real andimaginary parts of relevent complex quantities. Typical convention for such a representa-tion is reviewed in the Appendix A.

In the case, however, when the complex random variables characterizing the errorshave the circular-normal PDF, we can use a simpler approach [62, 163, 164]. A vector ofcomplex random variables x is said to have a circular PDF if [62]

E[(x − E (x)) (x − E (x))T

]= 0. (B.85)

A complex random variable with such a property is also sometimes referred to as a propercomplex random variable [163]. Expanding (B.85), we can easily show that it is equivalentto the following conditions

Cov (Rex,Rex) = Cov (Imx, Imx) , (B.86)Cov (Rex, Imx) = −Cov (Rex, Imx)T . (B.87)

In the case of a scalar random variable x, these conditions imply Var (Rex) = Var (Imx) =σ2

x and Cov (Rex, Im x) = 0, that is,

Σx =⎡⎣ σ2

x

σ2x

⎤⎦ , (B.88)

where the underline denotes the convention (A.2) for real-valued representation of complexrandom variables.

Now, let Cov (Rex,Rex) = Cov (Imx, Imx) = ΣRR and Cov (Rex, Imx) = ΣRI . Wecan then show that

E[(x − E (x)) (x − E (x))H

]= 2ΣRR + j2ΣRI = Σx, (B.89)

where the superscript H denotes the conjugate transpose (Hermitian transpose). Conse-quently, the statistical properties the vector of complex random variables with a circularPDF are characterized with a single complex matrix and can be obtained in the analogousfashion to the real-valued case. We can further show that

Σx =12Σx, (B.90)

192

B.6. SYSTEMS WITH COMPLEX-VALUED INPUTS AND OUTPUTS

where the underline denotes the convention (A.6) for the real valued representation ofcomplex matrices.

We assume now that the vector x has a normal PDF with a non-singular covariancematrix Σx, fulfilling the conditions (B.86) and (B.87). We can then show that its PDFcan be written with the use of (B.89) as [62]

fx (x) = fx (x) =1

(2π)N |Σx|e−(x−μ)Σ−1

x (x−μ) (B.91)

This property greatly simplifies the formulation and solution of least-squares problemsinvolving vectors of complex normal random variables, as the PDF given by B.91 has asimilar form to the PDF for real-valued normally distributed vectors. We can show thatfor a system with complex inputs and outputs the estimates of system parameters canbe obtained from equations given in Subsection B.2.1 and Subsection B.2.2, by employingthe complex covariance matrices B.89 and replacing the transpose T with the conjugatetranspose H.

193


194

Appendix C

Air-dielectric coaxial transmissionline

In this appendix we summarize characteristics of the TEM mode in the coaxial trans-mission lines with air dielectric. We first consider the case of line without conductor lossesand then discuss the case with small conductor losses.

C.1 Infinite metal conductivity

Let the diameters of the inner and outer conductor of the line d and D, respectively. Forboth conductors we assume the infinite conductivity. Let the center of the inner conductorbe offset from the center of the outer conductor by e. Characteristic impedance of theTEM mode in the line is given by [137]

Z00 =η

2π lnd2 + D2 − 4e2 +

√(D2 − d2 + 4e2)2 − (4De)2

2dD, (C.1)

whereη =

√μ0

εrε0, (C.2)

and μ0 = 4π10−7H/m and ε0 ≈ 8.854 10−7F/m1 are the magnetic permeability and dielec-tric permittivity of vacuum, respectively, and εr is the relative dielectric permittivity of

1Dielectric permittivity of vacuum is defined as ε0 =(μ0c2)−1, where c = 299792458 m/s is the speed

of light in vacuum.

195

C. AIR-DIELECTRIC COAXIAL TRANSMISSION LINE

the air. The propagation constant γ is imaginary and is given by

γ = jβ0 = jω√

εrε0μ0 = jω/v, (C.3)

and the phase velocity v isv = c√

εr

, (C.4)

where c = 1/√μ0ε0 is speed of light in vacuum.

For concentric conductors, that is, when e = 0, equation C.1 reduces to [165]

Z00 =η

2π ln D

d. (C.5)

For slightly eccentric lines, that is, when 2eD−d

≈ 0 , characteristic impedance C.1 can beapproximated as [131]

Z00 ≈ η

2π ln[

D

d

(1 − 4e2

D2 − d2

)]≈ η

2π

(ln D

d− 4e2

D2 − d2

). (C.6)

C.2 Finite metal conductivity

The characteristic impedance and propagation constant of the quasi-TEM mode in acoaxial transmission line with small conductor losses are given by [131, 165]

Z0 = Z00

√1 + (1 − j) vRc

ωZ00≈ Z00

[1 + (1 − j) vRc

2ωZ00

], (C.7)

and

γ = jβ0

√1 + (1 − j) vRc

ωZ00≈ jβ0

[1 + (1 − j) vRc

2ωZ00

]= α + j (β0 + α) , (C.8)

withα = β0

vRc

2ωZ00= Rc

2Z00, (C.9)

where Rc is the conductor resistance per-unit-length. For concentric conductors, this re-sistance is given by

Rc =1

σδ

( 1πD

+ 1πd

), (C.10)

196

C.2. FINITE METAL CONDUCTIVITY

and σ is the metal conductivity and δ is the skin depth given by

δ =√

2ωμ0σ

, (C.11)

which lead toZ0 ≈ Z00

[1 + (1 − j) v

2Z00

√μ0

2σω

( 1πD

+ 1πd

)], (C.12)

andγ ≈ jβ0

[1 + (1 − j) v

2Z00

√μ0

2σω

( 1πD

+ 1πd

)]. (C.13)

In the case of eccentric conductors, the surface resistance is given by [137]

Rc =1

σδ

(1

πρDD+ 1

πρdd

), (C.14)

whereρd =

D2 − d2 − 4e2√(D2 − (d + 2e)2

) (D2 − (d − 2e)2

) , (C.15)

andρD = D2 − d2 + 4e2√(

(D + 2e)2 − d2) (

(D − 2e)2 − d2) . (C.16)

For small values of eccentricity, that is, when 2eD−d

≈ 0, these expressions can be approxi-mated as

ρd ≈ 1 + 2d2e2

(D2 − d2)2 , (C.17)

andρD ≈ 1 + 2D2e2

(D2 − d2)2 . (C.18)

197

C. AIR-DIELECTRIC COAXIAL TRANSMISSION LINE

198

Appendix D

Center-conductor gap impedance

D.1 Infinite metal conductivity

Fig. D.1: Cross-section of the center conduc-tor gap (not in scale).

A simplified schematic of the center con-ductor gap is shown in Fig. D.1. For gapwidths g much smaller the depth of the gap(d − dp) /2, electrical properties of the cen-ter conductor gap can be modeled as thatof a radial waveguide stub [92, 136]. For thefundamental TEM mode in this waveguide,the admittance transformation is given by[48]

y(r) = − j + y(r0)ξ(β0r, β0r0)ct(β0r, β0r0)Ct(β0r, β0r0) + jy(r0)ξ(β0r, β0r0)

, (D.1)

whereZ0(r) =

1Y0(r)

= ηg

2πr, (D.2)

is the characteristic impedance at r, y(r) and y(r0) are the normalized input admittancesat r and r0, where r > r0, β0 is the propagation constant of the air defined by (C.3), η is

199

D. CENTER-CONDUCTOR GAP IMPEDANCE

the wave impedance of the air given by (C.2) and [48]

ct(x, y) = J1(x)N0(y) − N1(x)J0(y)J0(x)N0(y) − N0(x)J0(y)

, (D.3)

Ct(x, y) = J1(y)N0(x) − N1(y)J0(x)J1(x)N1(y) − N1(x)J1(y)

, (D.4)

ξ(x, y) = J0(x)N0(y) − N0(x)J0(y)J1(x)N1(y) − N1(x)J1(y)

, (D.5)

where Jm(t) is the Bessel function of the first kind of order m, and Nm(t) is the Besselfunction of the second kind (also referred to as the Neumann function [166]) of order m.In the case shown in Fig. D.1, this stub transforms a short circuit at r = dp/2, that is,y(dp/2) = ∞, to some admittance y(d/2) at r = d/2. Inserting y(dp/2) = ∞ into (D.17),we readily obtain

y(d/2) = Y (d/2)Y0(d/2) = jct(β0d/2, β0dp/2). (D.6)

For β0d/2 � 1 and β0dp/2 � 1, we can approximate (D.21) with the use of approximateforms of Bessel functions for t � 1 [1]

J0(t) ≈ 1, (D.7)

J1(t) ≈ t

2 , (D.8)

N0(t) ≈ 2πln t, (D.9)

N1(t) ≈ − 2π

1t, (D.10)

to obtainY (d/2)Y0(d/2) ≈ −j

1β0d/2 ln d

dp

, (D.11)

which, after inserting (D.2), gives the gap impedance

Z(d/2) ≈ jηg

πdβ0d/2 ln d

dp

= jμ0

2πωg ln d

dp

. (D.12)

200

D.2. FINITE METAL CONDUCTIVITY

D.2 Finite metal conductivity

For conductors with small losses, the propagation constant and characteristicimpedance become complex and can be approximated as

γ ≈ j (β0 + α) + α, (D.13)

andZ0(r) = η

g

2πr

[1 + (1 − j) α

β0

], (D.14)

whereα = Rc

2Z0, (D.15)

and Rc is the conductor resistance per unit length. For the radial waveguide, this resistancedepends on r and can be easily calculated as

Rc = Rs1

πr= 1

σδ

1πr

=√

ωμ0

2σ1

πr, (D.16)

where Rs is the conductor surface resistance, σ is the metal conductivity, and δ is the skindepth given by (C.11). Inserting (D.16) into (D.15) gives

α =√

ωμ0

2σ1ηg

.

In order to account for the losses in the admittance transformation, we need to solvethe Telegraphic equations for the radial line, accounting for the complex character of thepropagation constant. This solution can be shown to have the form

y(r) = −1 + y(r0)ξh(β0r, β0r0)cth(β0r, β0r0)Cth(β0r, β0r0) + y(r0)ξh(β0r, β0r0)

, (D.17)

where

cth(x, y) = I1(x)K0(y) + K1(x)I0(y)I0(y)K0(x) − K0(y)I0(x)

, (D.18)

Cth(x, y) = I1(y)K0(x) + K1(y)I0(x)I1(y)K1(x) − K1(y)I1(x)

, (D.19)

ξh(x, y) = I0(y)K0(x) − K0(y)I0(x)I1(y)K1(x) − K1(y)I1(x)

, (D.20)

201


and Im(t) is the modified Bessel function of the first kind of order m, and Km(t) is themodified Bessel function of the second kind of order m. Inserting y(dp/2) = ∞ into (D.17),we readily obtain

y(d/2) = Y (d/2)Y0(d/2) = cth(β0d/2, β0dp/2). (D.21)

For β0d/2 � 1 and β0dp/2 � 1, we can approximate (D.21) with the use of approximateforms of the modified Bessel functions for |z| � 1 [1] as

I0(z) ≈ 1, (D.22)

I1(z) ≈ z

2 , (D.23)

K0(z) ≈ − ln z, (D.24)

K1(z) ≈ 1z

, (D.25)

After inserting the above equations into (D.21), we obtain the first order approximation

Z(d/2) ≈ jωμ0

2πg ln d

dp

+ (1 + j)√

ωμ0

2σ1πln d

dp

. (D.26)

We note the surface impedance in (D.26) does not depend on the gap width. This is dueto the fact that this impedance results from the skin-depth effect in the gap walls. Indeed,the total surface impedance of the gap walls can be calculated as

Zs = 2d/2∫

dp/2

Rs

2πrdr = Rs

πln d

dp

= (1 + j)√

ωμ0

2σ1πln d

dp

. (D.27)

D.3 Finger effect

The fingers of the connector socket are formed by in-cuts along the center conductor.The in-cuts are very narrow, that is, we have w � πd, where w is the in-cut width andd is the diameter of the inner conductor. Due to the manufacturing method, the in-cutshave the same width at both outer and inner side of the fingers. We can approximatelysubstitute the in-cuts with magnetic walls and treat the center conductor gap as a set ofN sectorial radial waveguides, with the angle θ of each waveguide given approximately by

θ/2π ≈ 1d/2

πd − Nw

2πN= 1

N− w

πd≈ 1

N,

202

D.3. FINGER EFFECT

where N is the number of fingers. Each of the waveguides supports a TEM mode withthe characteristic impedance given by

Z0θ(r) =1

Y0θ(r)= η

g

θr= 2π

θZ0(r) ≈ 1

NZ0(r), (D.28)

and impedance transformation given by (D.21), with Y0 determined from (D.28). In orderto determine the diameters that need to be used in (D.21), we define the effective radiusfor a given arc length L and angle θ as

reff =L

θ. (D.29)

For the outer diameter, we have then

deff

2 = (πd − Nw) /N

2π/N= d

2 − Nw

2π , (D.30)

and similarly for the pin diameter

dp,eff

2 = dp

2 − Nw

2π . (D.31)

After inserting those effective diameters into (D.32), we obtain

Z(d/2) ≈ jωμ0

2πg ln πd − Nw

πdp − Nw+ (1 + j)

√ωμ0

2σ1πln πd − Nw

πdp − Nw. (D.32)

A similar expression for the effect of fingers is given in [136].

203


204

Appendix E

Slightly nonuniform coaxialtransmission line

Consider a nonuniform transmission line with length l and characterized by the char-acteristic impedance Z0(x) and complex propagation constant γ(x), for x ∈ [0, l]. In orderto determine S-parameters of such a line, we define the port one at x = 0 with refer-ence impedance Zref1 = Z(0), and the port two at x = l and with reference impedanceZref2 = Z0(l). The first order solution to S-parameters (accounting for the power normal-ization introduced in Chapter 2) of such a line is [167]

S11 = −l∫

0

N (x) e−2

x∫0

γ(x′)dx′

dx, (E.1)

S22 = −l∫

0

N ′ (x) e−2

x∫0

γ′(x′)dx′

dx, (E.2)

and

S21 = e−

l∫0

γ(x)dx

, (E.3)

S12 = e−

l∫0

γ(x)dx

(E.4)

205

E. SLIGHTLY NONUNIFORM COAXIAL TRANSMISSION LINE

where N ′ (x) = N (l − x), γ′ (x) = γ (l − x), and

N(x) = 12

1Z0(x)

dZ0(x)dx

. (E.5)

Writing the characteristic impedance as

Z0(x) = Z0 +ΔZ0(x), (E.6)

and assuming that |ΔZ0 (x) | � |Z0|, we can write approximately

N (x) ≈ 121

Z0

dΔZ0 (x)dx

= 12

d

dx

[ΔZ0 (x)

Z0

]. (E.7)

Writing further the propagation constant as

γ(x) = γ +Δγ(x), (E.8)

we obtainl∫

0

γ(x)dx = γl +l∫

0

Δγ(x)dx, (E.9)

which gives us an approximation for the transmission coefficients

S21 = S12 = e−γl

⎡⎣1 −

l∫0

Δγ(x)dx

⎤⎦ . (E.10)

Inserting further (E.7) in (E.1), we obtain an approximation

S11 = −l∫

0

N (x) e−2γx−2

x∫0

Δγ(x′)dx′

dx ≈ −l∫

0

N (x)⎡⎣1 − 2

x∫0

Δγ(x′)dx′⎤⎦ e−2γxdx ≈

≈ −l∫

0

N (x) e−2γxdx + 2l∫

0

N (x)⎡⎣ x∫

0

Δγ(x′)dx′⎤⎦ e−2γxdx. (E.11)

For a slightly nonuniform coaxial transmission line, we can write with the use of (5.21)

206

and (5.22)

N(x) ≈ 12

d

dx

[ΔZ00 (x)

Z00

], (E.12)

Δγ (x) ≈ (1 + j)α

[ΔRc (x)

Rc

− ΔZ00 (x)Z00

]. (E.13)

Inserting the last expression into the transmission coefficient equation, we readily obtain

S21 = S12 ≈ e−γl

⎡⎣1 − (1 + j)α

l∫0

[ΔRc (x)

Rc

− ΔZ00 (x)Z00

]dx

⎤⎦ , (E.14)

which leads to a phase shift of the transmission coefficients.As to the reflection coefficient the situation is more complicated. The first integral can

be simplified after integrating by parts to the form

−l∫

0

N (x) e−2γxdx = 12ΔZ00 (0)

Z00− 12ΔZ00 (0)

Z00e−2γl + γ

l∫0

ΔZ00 (x)Z00

e−2γxdx. (E.15)

Regarding the second integral, after integrating by parts, we obtain approximately

l∫0

N (x)⎡⎣ x∫

0

Δγ(x′)dx′⎤⎦ e−2γxdx ≈

≈ −l∫

0

ΔZ00 (x)Z00

⎧⎨⎩Δγ(x)e−2γx − γe−2γx

⎡⎣ x∫

0

Δγ(x′)dx′⎤⎦⎫⎬⎭ dx,

which to first order can be neglected. Hence we obtain the approximation for the reflectioncoefficient

S11 ≈ −12ΔZ00 (0)

Z00+ 12ΔZ00 (L)

Z00e−2γl + γ

l∫0

ΔZ00 (x)Z00

e−2γxdx. (E.16)

In a similar manner, we can derive

S22 ≈ −12ΔZ00 (l)

Z00+ 12ΔZ00 (0)

Z00e−2γl + γe−2γl

l∫0

ΔZ00 (x)Z00

e2γxdx. (E.17)

We further change the reference impedance at both ports to Z0, which finally leads to

207

E. SLIGHTLY NONUNIFORM COAXIAL TRANSMISSION LINE

approximations

S11 ≈ γ

l∫0

ΔZ00 (x)Z00

e−2γxdx, (E.18)

S22 ≈ γe−2γl

l∫0

ΔZ00 (x)Z00

e2γxdx, (E.19)

while S21 to first order remains the same and is determined by (E.14).We can now show that setting ΔZ00 (x) = ΔZ00 and ΔRc (x) = ΔRc reduces (E.18),

(E.19), and (E.14) to

S11 = S22 =ΔZ00

Z00

(1 − e−2γl

), (E.20)

S21 = S12 = e−γl

[1 − (1 + j)α

(ΔRc

Rc

− ΔZ00

Z00

)l

], (E.21)

which agrees with the description (5.32) of a slightly mismatched uniform transmissionline.

If we now assume that the line is lossless, that is, γ = jβ0, expressions (E.16), (E.17),and (E.14) reduce to

S21 = S12 ≈ e−jβ0l, (E.22)

and

S11 ≈ jβ01

ln Dd

l∫0

[ΔD (x)

D− Δd (x)

d

]e−j2β0xdx, (E.23)

S22 ≈ jβ0e−j2β0l 1ln D

d

l∫0

[ΔD (x)

D− Δd (x)

d

]ej2β0xdx. (E.24)

Similar results are reported in [168].

208

Appendix F

Small changes of two-port’sscattering parameters

In this appendix, we give a detailed derivation of the model (6.2). Consider a calibratedone-port VNA, with an error-box described by a transmission matrix T. Due to some ran-dom fluctuations, the VNA error-box changes to a new transmission matrix T′. We assumethat the change is due to a small perturbation that has occurred at some location insidethe VNA (see Fig. 6.1). We describe the perturbation with a transmission matrix ΔTn

(scattering matrix ΔSn), where n is the index of the perturbation. For small perturbationsCn, Ln, Rn, and Z0,n (see Fig. 6.2), we may write approximately

ΔTn ≈⎡⎣ 1 − εT,n εR1,n

−εR2,n 1 + εT,n

⎤⎦ , and ΔSn ≈

⎡⎣ εR1,n 1 + εT,n

1 + εT,n εR2,n

⎤⎦ , (F.1)

where εR1,n, εR2,n, and εT,n are small numbers defined as

εR1,n =12 (Z

′n − Y ′

n − ρn) , (F.2)

εR1,n =12 (Z

′n − Y ′

n + ρn) , (F.3)

εT,n =12 (Z

′n + Y ′

n) , (F.4)

209

F. SMALL CHANGES OF TWO-PORT’S SCATTERING PARAMETERS

with

Z ′n =

RDC,n + RRF 0,n

√ff0+ jωLn

Zref

= R′DC,n + R′

RF 0,n

√f

f0+ jωL′

n, (F.5)

Y ′n = jωCnZref = jωC ′

n, (F.6)

ρn =Z0,n

Zref

= Z ′0,n. (F.7)

In order to express T′ with ΔTn, we split the error box T into two parts. The partbetween the raw measurement plane and the plane, where the perturbation is located, isdescribed with a matrix T1. The remaining part of the error-box is represented by a matrixT2. Thus we can write

T = T1T2, and T′ = T1ΔTnT2. (F.8)

We then represent the error in corrected VNA S-parameter measurements due to the per-turbation ΔTn by cascading a transmission matrix ΔTe,n with the error-box transmissionmatrix T. Thus we can write

ΔTe,n = T−1T′ = (T1T2)−1 T1ΔTnT2 = T−12 ΔTnT2. (F.9)

In order to expand this last expression, we assume that the error-box can be seen as a setof electrically lumped discontinuities connected with low-loss transmission-line sections.Hence, we can write approximately

T2 =N∏

n=1T2,n, (F.10)

where

ΔT2,n ≈⎡⎣ (1 − δT,n) e−jβ0ln δR1,n

−δR2,n (1 + δT,n) ejβ0ln

⎤⎦ , (F.11)

and δR1,n, δR2,n, and δT,n are small numbers, jβ0 is the propagation constants, ln is thephysical length of the n-th section, and N is the number of sections. By inserting (F.10)and (F.11) in (F.9) and considering only first-order terms, we obtain

ΔTe,n ≈⎡⎣ 1 − εT,n εR1,ie

jβ0dn

−εR2,ne−jβ0dn 1 + εT,n

⎤⎦ , (F.12)

where dn =∑n

n′=1 ln′ is the distance of the perturbation from the reference plane. Finally,

210

since we assumed that the perturbation are small, we can neglect multiple reflections andwrite the overall error as the linear combination

ΔTe ≈

⎡⎢⎢⎣

1 − N∑n=0

εT,n

N∑n=0

εR1,nejβ0dn

− N∑n=0

εR2,ne−jβ0dn 1 +N∑

n=0εT,n

⎤⎥⎥⎦ , (F.13)

where N + 1 is the number of perturbations. We further expand (F.13) by insertingequations (F.2) through (F.7), which, after conversion to scattering parameters, yields theexpressions (6.2) through (6.11).

211

F. SMALL CHANGES OF TWO-PORT’S SCATTERING PARAMETERS

212

Appendix G

Estimation of VNA nonstationaritymodel parameters

We rewrite (6.24) in a more compact form as

θ = arg minθ

r (θ)H V−1r (θ) , (G.1)

where

r (θ) =

⎡⎢⎢⎢⎣

r1 (d, p1)...

rM (d, pM)

⎤⎥⎥⎥⎦ , (G.2)

and

V =

⎡⎢⎢⎢⎣

Vε

. . .Vε

⎤⎥⎥⎥⎦ , (G.3)

contains M copies of the matrix Vε on its diagonal. We solve the nonlinear optimizationproblem (G.1) with the use of the iterative numerical Levenberg-Marquardt algorithm. Ina single iteration, we determine a new estimate θ of sought parameters from the previousestimate θ0 as θ = θ0 + Δθ, where the correction Δθ is obtained as a solution of thenormal equations (

JHV−1J + λD)Δθ = JHV−1r (θ0) , (G.4)

213

G. ESTIMATION OF VNA NONSTATIONARITY MODEL PARAMETERS

where the Jacobian matrix J is defined as

J = −∂r (θ)∂θT

∣∣∣∣∣θ=θ0

, (G.5)

D is a fixed positive definite diagonal matrix (typically the matrix constructed out of themain diagonal of JHV−1J [64]), and λ is chosen according to the strategy presented in [159].In the following, we first consider the case when λ = 0 (the Gauss-Newton algorithm). Wethen show how to derive the solution to (G.4) with λ �= 0 based on the solution for λ = 0.

The Jacobian J can be written as a block matrix

J =

⎡⎢⎢⎢⎣

X (d0) Λ1 (d0). . . ...

X (d0) ΛM (d0)

⎤⎥⎥⎥⎦ , (G.6)

where X(d0) is given by (6.21) in the case of the connector interface model, and by (6.34)in the case of the test-set drift model, and

Λm (d0) =∂ (X (d)pm)

∂dT

∣∣∣∣∣d=d0

. (G.7)

Taking into account (6.21) and (6.15), we can show that

Λm (d0) = −j2β0

⎡⎢⎢⎢⎣

x′1 (d0,Γ1)T

...x′

K (d0,ΓK)T

⎤⎥⎥⎥⎦ = −j2β0X′ (d0) , (G.8)

where

x′k(d0,Γk) =

⎡⎢⎢⎢⎢⎣

(e−j2β0d1wR1(fk) − ej2β0d1Γ2

kwR2(fk))T

pR,1...(

e−j2β0dN wR1(fk) − ej2β0dNΓ2kwR2(fk)

)TpR,N

⎤⎥⎥⎥⎥⎦ , (G.9)

and Γk is given by Γk in the case of the connector interface model, and by Γ1k in the caseof the test-set drift model.

With the use of (G.6), we can rewrite the normal equations (G.4) for λ = 0 as a set of

214

equations (we omit the dependence on d0 for the sake of notational simplicity)

XHV−1ε XΔp1 + XHV−1

ε Λ1Δd = XHV−1ε r1

...XHV−1

ε XΔpM + XHV−1ε ΛMΔd = XHV−1

ε rM

, (G.10)

andM∑

m=1ΛH

mV−1ε XΔpm +

(M∑

m=1ΛH

mV−1ε Λm

)Δd =

M∑m=1

ΛHmV−1

ε rm. (G.11)

Solution to (G.10) and (G.11) can be obtained in two steps. We first assume that Δd isfixed, and write the solutions to (G.10) as

Δpm =(XHV−1

ε X)−1

XHV−1ε (rm − ΛmΔd) , (G.12)

for m = 1, . . . , M . These solutions are the inserted into (G.11), which yields(

M∑m=1

ΛHmHΛm

)Δd =

M∑m=1

ΛHmHrm, (G.13)

whereH = V−1

ε − V−1ε X

(XHV−1

ε X)−1

XHV−1ε . (G.14)

Consequently, instead of solving the original optimization problem (G.4), we solve theproblem (G.13) which has smaller dimensionality.

In order to account for λ �= 0, a simple modification need only be made to (G.10) and(G.11). Writing the matrix D as

D =

⎡⎢⎢⎢⎢⎢⎢⎣

DX. . .

DX

DΛ

⎤⎥⎥⎥⎥⎥⎥⎦

, (G.15)

where DX = diag(XHV−1

ε X), and DΛ = diag

(∑Mm=1 ΛH

mV−1ε Λm

). We can the rewrite

(G.13) and (G.14) as(

λDΛ +M∑

m=1ΛH

mH′Λm

)Δd =

M∑m=1

ΛHmH′rm, (G.16)

215

G. ESTIMATION OF VNA NONSTATIONARITY MODEL PARAMETERS

andH′ = V−1

ε − V−1ε X

(XHV−1

ε X + λDX)−1

XHV−1ε . (G.17)

216

Appendix H

Vector stochastic Wiener process

Consider an ensemble of N independent scalar stochastic Wiener processes

w (t) = [w1 (t) , . . . , wN (t)]T (H.1)

Each of these processes is characterized with the parameter αn > 0, for n = 1, . . . , N .Thus we can write

w(t) ∼ N(0, Dt) (H.2)

and

w(t2) − w(t1) ∼ N(0, D(t2 − t1)) (H.3)

where

D =

⎡⎢⎢⎢⎣

α1. . .

αN

⎤⎥⎥⎥⎦ . (H.4)

We now define a general N -dimensional Wiener process, that is, a process whose un-derlying one-dimensional processes can be statistically correlated. Let A be a full rankN × N matrix, and let ψ(t) be a new process defined as

ψ(t) = Aw(t). (H.5)

Since the transformation is linear, ψ(t) is also normally distributed. We calculate the first

217

H. VECTOR STOCHASTIC WIENER PROCESS

two moments of this distribution as

E [ψ(t)] = E [Aw(t)] = AE [w(t)] = 0, (H.6)

and

E[ψ(t)ψ(t)T

]= E

[Aw(t) (Aw(t))T

]= AE

[w(t)w(t)T

]AT = tADAT = tΛ, (H.7)

henceψ(t) ∼ N(0, tΛ). (H.8)

Using (H.8), we can readily show that

ψ(t2) − ψ(t1) ∼ N (0, Λ(t2 − t1)) . (H.9)

Matrix Σ = tΛ = tADAT is the covariance matrix of the new process. We can readilyshow that it is a valid covariance matrix, that is, a symmetric and positive semi-definitematrix. Indeed,

ΣT = t(ADAT

)T= t

(AT)T

DT AT = tADAT = Σ, (H.10)

and since for every x such that |x| > 0 we have

xT tDx ≥ 0, (H.11)

we also see that

xT tΛx = xT tADAT x =(AT x

)TtD(AT x

)= yT tDy ≥ 0, (H.12)

since |y| =∣∣∣AT x

∣∣∣ ≥ 0 for |x| ≥ 0.

In order to estimate the parameter Λ, we use the maximum likelihood estimationtheory. For a given realization y(t) for t = 0 < t1 < . . . < tN of the process ψ(t), we formthe differences Δyn = y(tn) − y(tn−1). It can be then easily shown that the maximum

218

likelihood estimator Λ of the process parameters Λ has form

Λ = 1N

N∑n=1

ΔynΔyTn

tn − tn−1. (H.13)

Estimator Λ is not biased. Indeed, for each n we have

Δyn

tn − tn−1∼ N(0, Λ), (H.14)

henceE ˆ(Λ) = 1

N

N∑n−1

E

{ΔynyT

n

tn − tn−1

}= 1

N

N∑n−1

Λ = Λ. (H.15)

219

H. VECTOR STOCHASTIC WIENER PROCESS

220

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