doktorat Lewandowski

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WARSAW UNIVERSITY

OF TECHNOLOGY

Faculty of Electronics and InformationSystems

Ph.D. Thesis

Arkadiusz Lewandowski

Multi-frequency approach to vector-network-analyzer

scattering-parameter measurements

Supervisor

Professor Janusz Dobrowolski, Ph.D., D.Sc.

Warsaw, 2010


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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 effort

has 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 analysis

of VNA measurements. Modern VNAs are a culmination of this long standing research,

and are sophisticated, mature and reliable measurement instruments, commonly employed

in 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- and

sub-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 of

100 k), 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 operate

up 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. The

principle of this approach is to account for the relationships between scattering parameter

measurements at different frequencies. We show that this new approach allows to reduce by

several times the impact of errors in the description of calibration standards, resulting thus

in a significant improvement of the VNA measurement accuracy. We further demonstrate

that the multi-frequency approach to the description of VNA instrumentation errors yields

better understanding of their physical origins, leading to their compact description based

on the stochastic modeling. We finally show that the multi-frequency representation of

the uncertainty in VNA scattering-parameter measurements is essential when using these

measurements in the calibration of time-domain measurement systems, such as high-speed

sampling oscilloscopes, or nonlinear vector network analyzers.

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Streszczenie

Wektorowy analizator obwodw (ang. Vector Network Analyzer - VNA) jest podsta-wowym urzdzeniem wykorzystywanym do charakteryzowania ukadw i systemw elek-

tronicznych wielkiej czstotliwoci (w. cz). Konstrukcja wspczesnych analizatorw, jak

i metody wykorzystywane w ich kalibracji oraz analizie niepewnoci pomiaru, s owocem

wieloletnich prac badawczych oraz intensywnego rozwoju technologicznego. W konsekwen-

cji nowoczesne wektorowe analizatory obwodw odznaczaj si niezwykle zaawansowanymi

i dojrzaymi rozwizaniami technicznymi oraz s z powodzeniem wykorzystywane w co-

dziennej praktyce zarwno laboratoriw pomiarowych jak i przemysu ukadw w. cz.

W wektorowej analizie obwodw pojawiy si w ostatnim czasie nowe kierunki roz-

woju, wynikajce z rosncego zainteresowania wykorzystaniem sygnaw w zakresie fal

milimetrowych i submilimetrowych (czstotliwoci blisko 1 THz), rozszerzania si zakresu

impedancji mierzonych struktur w. cz. (impedancje rzdu 100 k), zwizanego z intensyw-

nym rozwojem nanotechnologii, oraz z zapotrzebowania na charakteryzowanie wielkosy-

gnaowych wasnoci ukadw w. cz. Te nowe zastosowania wektorowej analizy obwodw

prowadz, z jednej strony, do nowych rozwiza konstrukcyjnych, jak na przykad gowice

powielajco-mieszajce rozszerzajce zakres pracy typowych analizatorw do czstotliwoci

rzdu 500 GHz, mikrofalowe mikroskopy skaningowe, czy te wielkosygnaowe wektorowe

analizatory obwodw (ang. Nonlinear Vector Network Analyzer-NVNA). Z drugiej strony,stawiaj one zupenie nowe wyzwania, jeeli chodzi o dokadno pomiaru, oraz jej wiary-

godne oszacowanie.

Przedstawione w niniejszej pracy nowatorskie wieloczstotliwociowe podejcie do po-

miaru parametrw rozproszenia za pomoca wektorowego analizatora obwodw jest prb

odpowiedzi na te nowe wyzwania. Jego istot jest uwzgldnienie relacji midzy pomiarami

parametrw rozproszenia na rnych czstotliwociach. W pracy wykazano, e to nowe

ujcie pozwala kilkukrotnie zmniejszy wpyw bdw wynikajcych z niedokadnego opisu

wzorcw kalibracyjnych, a tym samym znaczco zwikszy dokadno pomiaru. Pokazanorwnie, e wieloczstotliwociowy opisu bdw losowych w pomiarach analizatorem wek-

torowym pozwala lepiej wyjani ich fizyczne przyczyny, prowadzc do prostego i spjnego

opisu tych bdw opartego na modelowaniu stochastyczym. W kocu, w pracy wykazano,

e uoglniony wieloczstotliwociowy opis niepewnoci pomiaru parametrw rozproszenia

jest niezbdny, gdy wykorzystuje si te pomiary w kalibracji urzdze dziaajcych w dzie-

dzinie czasu, takich jak szybkie oscyloskopy prbkujce, albo wielkosygnaowe wektorowe

analizatory obwodw.

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Wenn [meine] Arbeit einen Wert

hat, so besteht er [...] darin, dass in

ihr Gedanken ausgedrckt sind,

und dieser Wert wird umso grer

sein, je besser die Gedanken

ausgedrckt sind.

Ludwig Wittgenstein

Meine Resultate kenne ich lngst,

ich wei nur noch nicht,

wie ich zu ihnen gelangen soll.

Carl Friedrich Gauss

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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 Williams

from 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. I

wish 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 which

this 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 Kdzierski and Karol Korsze of

the National Institue of Telecommunications, Warsaw, Poland, for performing the Type-N

measurements 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 constant

support 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 understanding

without which finishing this work would not have been possible.

Last, but not least, I would like to acknowledge the Polish Ministry of Science and

Higher 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 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Previous research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Objective and scope of this work . . . . . . . . . . . . . . . . . . . . . . . 5

2 Principles of VNA S-parameter measurements 7

2.1 Definition ofS-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Waveguide voltage, current and characteristic impedance . . . . . . 9

2.1.2 Wave amplitudes and scattering parameters . . . . . . . . . . . . . 11

2.1.3 Practical implications . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 VNAS-parameter measurement . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Two-port VNA mathematical models . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Linear time-invariant two-port VNA . . . . . . . . . . . . . . . . . 20

2.3.2 Modeling VNA nonstationarity . . . . . . . . . . . . . . . . . . . . 28

2.4 Two-port VNA calibration techniques . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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CONTENTS

3 Overview of uncertainty analysis for VNA S-parameter measurements 43

3.1 Sources of error in corrected VNAS-parameter measurements . . . . . . . 44

3.2 Statistical description ofS-parameter measurement errors . . . . . . . . . . 453.2.1 Statistical model for S-parameter measurement . . . . . . . . . . . 45

3.2.2 Error description for a singleS-parameter . . . . . . . . . . . . . . 46

3.2.3 Error description for a matrix ofS-parameters . . . . . . . . . . . . 49

3.3 Statistical models for errors in corrected VNAS-parameter measurements 50

3.3.1 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2 Random errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Representation of errors in corrected VNAS-parameter measurements . . . 53

3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.2 Errors in VNA calibration coefficients . . . . . . . . . . . . . . . . . 54

3.4.3 Errors in VNA raw measurements . . . . . . . . . . . . . . . . . . . 58

3.5 Approximate uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . 61

3.5.1 Ripple analysis techniques . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.2 Calibration comparison method . . . . . . . . . . . . . . . . . . . . 62

3.5.3 Statistical residual analysis . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Complete uncertainty evaluation . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6.1 Linear error propagation . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.2 Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Multi-frequency description ofS-parameter measurement errors 67

4.1 Statistical model for the multi-frequencyS-parameter measurement . . . . 68

4.2 The notion of a physical error mechanism . . . . . . . . . . . . . . . . . . . 68

4.3 Statistical properties of the multi-frequency measurement error . . . . . . . 71

4.3.1 Probability distribution function . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Uncertainty reporting . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 Multi-frequency covariance-matrix structure . . . . . . . . . . . . . 754.4 Physical error mechanisms in VNAS-parameter measurements . . . . . . . 75

4.4.1 Calibration standard errors . . . . . . . . . . . . . . . . . . . . . . 75

4.4.2 VNA instrumentation errors . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Practical implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5.1 Time-domain waveform correction . . . . . . . . . . . . . . . . . . . 78

4.5.2 Device modeling based on S-parameter measurements . . . . . . . . 83

4.5.3 Error-mechanism-based VNA calibration . . . . . . . . . . . . . . . 84

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CONTENTS

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

5 Generalized multi-frequency VNA calibration 895.1 Formulation of the VNA calibration problem . . . . . . . . . . . . . . . . . 90

5.2 Coaxial multi-line VNA calibration . . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 Classical multi-line VNA calibration . . . . . . . . . . . . . . . . . 93

5.2.2 Coaxial air-dielectric line as a calibration standard . . . . . . . . . 93

5.2.3 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Errors in the coaxial multi-line VNA calibration . . . . . . . . . . . . . . . 95

5.3.1 Variation of connector-interface electrical parameters . . . . . . . . 95

5.3.2 Variation of lines characteristic impedance and propagation constant 100

5.3.3 Line length error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3.4 Reflect asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Calibration standard models . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.2 Reflect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4.3 Thru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Solution uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5.1 Mathematical framework . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5.2 Relationships between estimated parameters . . . . . . . . . . . . . 112

5.5.3 Optimal constraint choice . . . . . . . . . . . . . . . . . . . . . . . 114

5.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.7 Residual analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.8.2 Type-N coaxial connector . . . . . . . . . . . . . . . . . . . . . . . 119

5.8.3 1.85 mm coaxial connector . . . . . . . . . . . . . . . . . . . . . . . 126

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

6 Multi-frequency stochastic modeling of VNA nonstationarity errors 137

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Generic physical model for the VNA nonstationarity . . . . . . . . . . . . . 139

6.3 Stochastic model for connector nonrepeatability and cable instability . . . 144

6.3.1 Statistical properties of circuit parameters . . . . . . . . . . . . . . 144

6.3.2 Estimation of the covariance matrix of circuit parameters . . . . . . 145

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CONTENTS

6.3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4 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 . . . . . . . . . . . . . 159

6.4.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.2.1 Errors in system responses . . . . . . . . . . . . . . . . . . . . . . . 177

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

B.3 Covariance matrix of the estimates . . . . . . . . . . . . . . . . . . . . . . 186



B.4 Numerical solution techniques . . . . . . . . . . . . . . . . . . . . . . . . . 189



B.5 Solution uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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

C Air-dielectric coaxial transmission line 195

C.1 Infinite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

C.2 Finite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

D Center-conductor gap impedance 199

D.1 Infinite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

D.2 Finite metal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

D.3 Finger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

E Slightly nonuniform coaxial transmission line 205

F Small changes of two-ports scattering parameters 209

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CONTENTS

G Estimation of VNA nonstationarity model parameters 213

H Vector stochastic Wiener process 217

Bibliography 221

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CONTENTS

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Nomenclature

a, b, c vectors

A,B

,C

matrices{ai}Ni=1 set of vectors a1, . . . , aNAT transpose of the matrix A

AH conjugate transpose (Hermitian transpose) of the matrix A

A B Kronecker product of the matrices Aand Bvec(A) vector representation [a11, a21, a31, . . . , a12, a22, a32, . . .]

T of the matrix A

Re x real part ofx

Im x imaginary part ofx

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

vector a

x estimate ofx

x true value ofx

x measurement ofx

x error in the measurement ofx

E(x) expectation value ofx

Var(x) variance of x

Cov(x, y) covariance ofxand y

attenuation constant,i.e., Re phase constant,i.e., Im

sought parameters of the VNA and calibration standards

0 free-space phase constant

c vector of calibration-standard unknown parameters

C capacitance

c speed of light in vacuum

c0 vector of calibration-standard known parameters

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CONTENTS

C normalized capacitance

D diameter of the outer conductor in the coaxial transmission line

d diameter of the inner conductor in the coaxial transmission linedp center-conductor-pin diameter in the coaxial transmission line

DUT device under test

e eccentricity of the inner conductor in the coaxial transmission line

EDF forward directivity

EDF reverse directivity

EDF forward tracking

EDF reverse tracking

EDF forward source matchEDF reverse source match

0 dielectric permittivity of vacuum

r relative dielectric permittivity

physical error mechanism

f frequency; probability density function

g center-conductor-gap width in the coaxial transmission line

complex propagation constant

reflection coefficient

In identity matrix of sizen nJ Jacobian matrix

K number of frequencies

k frequency index

L inductance

L normalized inductance

Lg normalized gap inductance per-unit-length

l transmission line length

li inner conductor length

lo outer conductor length

l misalignment of outer and inner conductor symmetry axes

l loss correction factor

M number of mechanisms

0 magnetic permeability of vacuum

N number of calibration standards

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CONTENTS

n calibration standard index

angular frequency

p vector of VNA calibration coefficientsPDF probability density function

r vector of residuals

R resistance

R normalized resistance

0 characteristic-impedance correction factor

rDF effective forward directivity

rDF effective reverse directivity

rDF effective forward trackingrDF effective reverse tracking

rDF effective forward source match

rDF effective reverse source match

S scattering matrix

s vector representation vec(s) of the scattering matrix S; vector of calibration-

standard S-parameter-definitions

sm vector of raw measurements of calibration standards

covariance matrix

conductivity, standard deviation

SOLT short open load through

SOLT Singular Value Decomposition

T transmission matrix

TRL through reflect line

v phase velocity

VNA vector network analyzer

V weight matrix in the VNA calibration

w width of the in-cut between the connector socket fingers

Y admittance

Z impedance

Z0 characteristic impedance for the TEM mode in a lossy coaxial transmission line

Z00 characteristic impedance for the TEM mode in a lossless coaxial transmission

line

Y normalized admittance

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NOMENCLATURE

Z normalized impedance

Zref reference impedance

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Chapter 1

Introduction

One day when Pooh Bear had nothing else to

do, 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 scattering

parameters (S-parameters) which constitute a complete description of small-signal deter-

ministic properties of an electronic circuit [1]. This measurement is typically performed in

a 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 traditionally

used in the design and testing of both single components and complex systems working at

microwave and millimeter-wave frequencies.Much effort has been put throughout the past 30 years in establishing the principles of

vector-network analysis and improving the VNA instrumentation. A good review of this

research can be found in [68]. Modern VNAs, such as [35], are a culmination of this long

standing research and are very mature and reliable measurement instruments, commonly

employed 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-

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1. INTRODUCTION

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

assumption of the DUT linearity.

The trend to extend the frequency range of modern VNAs results from an increased

interest in the use of signals with frequencies in the millimeter- and sub-millimeter-wave

range. Examples are high-capacity data transmission systems [9], millimeter-wave radar

system [10, 11], radiotelescopes [12, 13], or the broad range of terahertz applications [14].

Efforts to extend the maximum VNA measurement frequency have recently brought about

special VNA extension units, allowing the conventional VNAs to operate up to 500 GHz

with rectangular waveguide connectors [2]. Extension up to 1 THz is likely to happen in

the 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 lower

frequencies. Due to small wavelength, these measurements require the use of waveguides

with 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 rounding

of the waveguide corners or erosion of the leading edges of the waveguide apertures, are

unavoidable and may lead to significant systematic errors in the VNA calibration [17].

Furthermore, the connection of two waveguide flanges with such small apertures requires

very precise alignment. Although some special alignment solutions, involving the use ofmultiple alignment pins, have been devised, the random errors due to flange misalignment

can 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-parameter

measurement at sub-millimeter-wave frequencies. These fluctuations, caused by the ther-

mal and phase noise originating in the frequency multiplication and sub-harmonic mixing

circuitry of the VNA extension units, significantly reduce the dynamic range and increase

the short-term instability (also referred to as the trace jitter [19]), as compared with

VNAs operating at lower frequencies.While the VNA calibration techniques used at lower frequencies can be adapted to

work with millimeter and sub-millimeter-wave VNAs, due to those specific errors, their

accuracy 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 VNAS-pa-

rameter measurements of devices whose impedance differs significantly from the typical

VNA impedance level of 50 . Examples of such devices are nanotubes, nanowires or

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1.1. MOTIVATION

metamaterials, which may exhibit impedances in the range of tens and hundreds of ks,

or fractions of an [21]. Since the VNA test-ports are typically built based on 50 -

transmission-line components, the energy coupling to devices with impedance level closeto 50 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 little

signal is coupled and most of the signal is reflected back to the VNA. While some special

VNA architectures (e.g. [22]), aiming at increasing the VNA receiver resolution, may help

to improve the measurement accuracy in such cases, new VNA calibration techniques, less

sensitive to the VNA measurement errors (e.g., [23]), are needed.

The last trend in the modern vector-network-analysis results from an increased demand

for 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 of

various applications. Examples are portable data transmission systems where the high

power 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 signal

generators, mixers or frequency multipliers, that are by nature operating in the nonlinear

regime.

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]. The

NVNA 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 is

then either directly used, for example, in the circuit simulator, or converted into the time

domain 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 absolute

quantities (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 of

voltages 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

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1. INTRODUCTION

analysis approaches that account not only for uncertainties at a single frequency, but also

for the statistical correlations between uncertainties at different frequencies.

Consequently, the new trends in the vector network analysis discussed above, while

stimulating the development of new hardware solutions, lead also to new, more stringent

demands as to the measurement accuracy and its reliable and complete evaluation. The

multi-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. A

detailed review of this development can be found in [68, 19]. Here we shall indicate the

most 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 idea

of self-calibration in the two-port VNA calibration problem had first been employed in

Engens TRL method [29] and was then generalized by Eul and Schiek [30]. The concept

of self-calibration in one-port VNA calibration methods appears also in papers by Wiatr

[3133] and Bianco [34]. The principle of self-calibration is to use calibration standards

that are only partially known and to determine their complete S-parameter description

along 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 are

inevitably subject to measurement errors, the information as to the relationships between

these 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 statistical

methods in VNA calibration is based on the use of redundant calibration standards and

statistical processing of the resulting overdetermined set of equations. Consequently, the

contribution of random measurement errors can be significantly reduced.

Another important paradigm change in the development of VNA calibration methods

was initiated in [37] and [38]. In these references, for the first time, the relationships

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1.3. OBJECTIVE AND SCOPE OF THIS WORK

between the calibration standard parameters at different frequencies are exploited in the

VNA calibration. The experimental result presented in [37] and [38] indicate that this

improves the accuracy and reliability of the VNA calibration.

The multi-frequency description ofS-parameter measurement errors naturally comple-

ments a VNA calibration approach that accounts for the relationships betweenS-parameter

measurements at different frequencies. Such a description was introduced in [39], and is

based 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-matrix

description proposed in [39] uses additional terms in the covariance matrix in order toaccount for statistical correlations between uncertainties at different frequencies. These

correlations 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 VNAS-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 this

work is to generalize these results by developing a comprehensive multi-frequency approach

to 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 frequencies

which 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 put

here on the VNA calibration.

This organization of this work is as follows. In the introductory part (Chapter 2 and

Chapter 3) we review the foundations of VNA S-parameter measurements and uncertainty

analysis. This part serves as the theoretical background for the discussion presented in the

main part of this work.

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

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1. INTRODUCTION

framework for the representation of relationships between VNA S-parameter measurements

at different frequencies. We further review the practical applications for which accounting

for these relationships is important. These applications include the correction of time-domain measurements, measurement-based device modeling and the VNA calibration on

which we focus in the this work. We show that a statistically sound description of the VNA

calibration problem should be done in terms of the error mechanisms underlying the cali-

bration standard and VNA instrumentation errors. As a consequence, the VNA calibration

should 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 different

frequencies. We refer to this approach as the error-mechanism-based VNA calibration and

in the remainder of this work we develop the necessary tools for the implementation ofsuch a calibration approach. These tools include the generalized multi-frequency VNA

calibration (see Chapter 5) and the framework for error-mechanism-based description of

the VNA nonstationarity errors (see Chapter 6).

In the last part of this work (see Chapter 7) we present conclusions and discuss possible

directions of further research.

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Chapter 2

Principles of VNA S-parameter

measurements

All models are wrong, some are useful.

George Box

In this chapter, we review the principles ofS-parameter measurements with the vector

network 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 analyze

the imperfections of a typical two-port VNA measurement setup. The errors caused by

these imperfections are systematic as they are very stable in the course of typical VNA

measurement. Therefore, they can be characterized in a calibrationprocedure and thenremoved from the actual S-parameter measurements in the correction procedure. Both

procedures assume a mathematical model of these VNA. In the calibration procedure, a

set of devices with some known characteristics is measured and the parameters of the

VNA model are determined. Then, in the correction procedure, the model obtained in the

calibration is used to correct for the imperfections of the VNA setup. We discuss different

mathematical VNA models and VNA calibration techniques in the last two sections of this

chapter.

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2. PRINCIPLES OF VNA S-PARAMETER MEASUREMENTS

2.1 Definition ofS-parameters

Scattering parameters form a description of an electronic circuit in terms of complex

amplitudes1 of electromagnetic waves interacting with the circuit. This description is

used in the situation when the dimensions of the circuit are become comparable with the

wavelength, which typically takes place at microwave and millimeter-wave frequencies. In

this situation, the physical phenomena occurring in the circuit are of a wave nature, and

the conventional circuit description in terms of terminal voltages and currents looses its

physical correspondence.

Scattering parameter description uses the concept of the circuit port instead of the

circuit terminal. A circuit port is defined as section of a uniform arbitrary waveguidethrough which the electromagnetic wave may enter and exit the electronic circuit. In

order to ensure the uniqueness of the scattering parameter description, we require that

these ports are electromagnetically separated and that a given set of circuit ports (with

waveguide modes propagating through them) encompasses all of the possible means by

which electromagnetic energy can enter and leave the circuit. This means that we need to

account not only for all of the physical ports through which the electromagnetic waves are

interacting with the circuit, but also for all of the waveguide modes propagating through

the circuit ports. In order to simplify the scattering parameter description, we typicallyassume single mode propagation through circuit ports, however, extension to the case of

multiple modes is possible.

Scattering parameters describe the relationships between the amplitudes of the waves

propagating through the circuit ports. These amplitudes are defined with the use of a

simple normalization such that the wave with a unit root-mean-square amplitude, in the

absence of the wave propagating in the opposite direction, carries unit power. Although

the principle of this normalization is very simple, its systematic derivation requires some

consideration. In the following, we briefly review the origins of this normalization (for

more 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, ofS-parameters, allow one the relate the

scattering parameter description to the methods of the transmission line theory. We then

present two different types of normalized waves. The first type, referred to as the traveling

waves, originates in the physics of wave propagation in the waveguide. Thus, properties of

1We assume time-harmonic dependence of the fields.

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2.1. DEFINITION OFS-PARAMETERS

traveling waves closely reflect the physical properties of the actual waves propagating in

the waveguide. In some cases, however, these properties lead to results that are surprising

in 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 in

the 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 discussion

of practical implications of the different normalization schemes for scattering parameter

measurements.

2.1.1 Waveguide voltage, current and characteristic impedance

Electromagnetic waves traveling in a waveguide are described in terms ofmodeswhich

are solutions to the Maxwell equations in the waveguide cross-section. For time-harmonic

field dependence, these solutions can be characterized by the normalized transverse electric

and transverse magnetic field distributions, et(x, y) and ht(x, y), respectively, and the

complex-valued propagation constant. In the case of lossless transmission lines, the field

distributions are real-valued, and the propagation constant is imaginary = j. In the

case of transmission lines with losses, the field distributions are in general complex-valued

and the propagation constant has also a real part, that is =+j.

With the use of the normalized field distributions and the propagation constant, we can

write the complex peak amplitudes of the fields at any point in the waveguide (propagation

occurs along the zaxis) in a normalized way as

Et(x,y,z) =C+et(x, y)e

z +Cet(x, y)e+z =V(z)

V0et(x, y), (2.1)

Ht(x,y,z) =C+ht(x, y)e

z +Cht(x, y)e+z =I(z)

I0ht(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 forward

and backward propagating wave at z= 0, respectively, and V(z) and I(z) are defined as

waveguide voltageand current.

The unitless constants C+ and C depend on the normalization used for et(x, y) and

ht(x, y). The waveguide voltage and current, however, are independent of this normaliza-

tion due to the use of normalization constants V0 and I0. These constants have units of

voltage and current, respectively, hence in the following we refer to them as the normal-

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ization voltage and current, respectively.

The choice of normalization voltage V0 and current I0 is not arbitrary and we require

that

P0 =1

2V0I

0 =

1

2

S

et htdS (2.3)

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

P(z) =1

2

S

Et(x,y,z) Ht (x,y,z)dS=1

2V(z)I(z)

Set htdS

V0I0=

1

2V(z)I(z). (2.4)

Consequently, by imposing the condition (2.3), we require that the net power flowing

through 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 constantsV0and I0can be arbitrarily

chosen. For example, for the voltage V0, we may choose to use the path integral along

some arbitrarily chosen path Pin the cross-section of the waveguide

V0= P et(x, y)dl, (2.5)with an obvious constraint V0= 0, and then determine I0 from (2.3) (voltage-powernormalization). Alternatively, we can also fix the currentI0based the loop integral around

a closed loop L in the cross-section of the waveguide

I0 =L

ht(x, y)dl, (2.6)

with a similar constraintI0= 0 and then determineV0 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 the

characteristic impedanceof the mode

Z0=V0

I0. (2.7)

The magnitude of Z0 depends, in general, on the field normalizations and the chosen

strategy for setting the constants V0 and I0. The phase ofZ0 can be easily determined in

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

ImZ0ReZ0 =

ImP0ReP0 (2.8)

Phase ofZ0 is therefore the inherent property of the mode and does not depend on the

field 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 is

natural to choose these point to lie on different conductors. Definition (2.7) becomes then

the conventional definition of the characteristic impedance for TEM modes.

The above definition of the waveguide voltage, current and characteristic impedance

are 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 Telegraphic

equations, which describe the wave propagation in terms time- and location-dependent

voltages 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 and

backward propagating waves rather than voltages and currents. The other description we

discussed, resulting directly from the solution of Maxwell equations, uses unitless constants

C+ andC. These constants have straightforward physical interpretation and describe the

amplitudes of the forward and backward propagating wave. However, they are difficult to

both interpret and measure sue to the dependence on the normalization of the mode fields

et(x, y) and ht(x, y).

Solution to that problems is the description in terms oftraveling-wave amplitudes. This

description arises from the following normalization of the constants C+ and C

a0(z) =

2ReP0C+ez , and b0(z) =

2ReP0C

e+z . (2.9)

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

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shift between the electric and magnetic fields, that is, when arg P0= 0. This phase shift isa consequence of power loss in the waveguide, which is commonly encountered in practice

due to finite conductivity of real conductors and losses in dielectrics.When the waveguide is lossy, the forward and backward propagating modes are not

orthogonal. Therefore, the real power flowing through a given cross section is not equal

to the sum of powers carried by the modes, which is usually assumed in the classical

transmission line theory [48, 49]. This result can easily be confirmed by writing the real

power flowing through the waveguide cross-section with the use of the traveling wave

amplitudes

ReP(z) =1

2|a0|2 1

2|b0|2 + Im(a0b0) Imp0Rep0 =

1

2|a0|2 1

2|b0|2 + Im(a0b0) ImZ0ReZ0 . (2.16)

Indeed, for arg P0= 0 we obtain an additional term related to the phase ofP0. Note that,according to (2.8), this phase is a property of the mode and does not depend on the choice

of field normalizations. It is related to the characteristic impedance of the mode through

(2.8), hence arg P0= 0 implies that characteristic impedance Z0 is complex.Another property, surprising in the context of transmission line theory, that results from

the loss in the waveguide, is that the ratio of real powers incident at and reflected from

an discontinuity in the waveguide is not equal to||2 where is the reflection coefficient = b0/a0 [48]. Therefore, in some cases, magnitude of may exceed one which is also

unusual for the classical transmission-line theory. This result can also be easily obtained

with 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-line

theory. Also, when characteristic impedance Z0 exhibits a significant frequency depen-

dence, it is more convenient to have a fixed relationship between the wave amplitudes and

waveguide voltages and currents that be independent of the frequency dependence ofZ0.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)

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where Zrefis 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 power

flow in the waveguide is described by the expression as (2.16), however withZ0replaced byZref. Consequently, for a realZref, the additional term in expression (2.16) vanishes and

we 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 correspond

to the traveling wave amplitudes. Indeed, we can readily show that a(z) and b(z) are

linear combinations ofa0(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 traveling

wave 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) and

b(z), with a reference impedance Zref to another set, a(z) and b(z) with a different

reference impedanceZref. The relationship between the two sets can be easily determined

from (2.17) and (2.18) as [45]

a(z)

b(z)

= N

a(z)

b(z)

, (2.19)

where

N=

1 jImZref/ReZref1 jImZref/ReZref

11

2

1

1

, (2.20)

and

=Zref ZrefZref+Zref

. (2.21)

It can be shown that for real reference impedances Zref and Zref, 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 withNports, we group the wave amplitudes (traveling-

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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 elements

as scattering parameters. In the case of pseudo-wave amplitude, we refer to the elementsofS aspseudo-scattering parameters. However, since the pseudo-wave amplitudes become

traveling wave amplitudes forZref=Z0, we often talk briefly about scattering parameters

defined with reference to a certain impedance Zref.

In the case of two-port devices2, it is sometimes more convenient to represent the

relationship between the wave amplitudes with the use oftransmission matrixdefined 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 , is

given by a product

T=Ni=1

Ti. (2.25)

In the common case of a two-port device, we give the relationship between the two

representations explicitly as they are used very often throughout this work. For a two-port

network 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].

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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= 1

S21

S12S21 S11S22 S11

S22 1

, (2.28)

and

S=

1

T22 T12 T11T22

T12T21

1 T21

. (2.29)

2.1.3 Practical implications

In practice, the general definition ofS-parameters presented in the previous section,

can often be simplified. In most practical cases, the normalization voltage V0 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 V0 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.

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2.2. VNAS-PARAMETER MEASUREMENT

and thus

V(z) = 1ReZref |Zref| [a0(z) +b0(z)] , (2.34)

I(z) = 1ReZref

|Zref|Zref

[a0(z) b0(z)] . (2.35)

For a real reference impedance, we further obtain the familiar expression known from the

circuit theory [5052]

a(z) = 1

2

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

b(z) = 1

2

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 the

VNA measures S-parameters with respect to some unknown reference impedance. Hence

an important aspect of the VNA calibration is the determination of this impedance. This

will 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 in

S-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 suggests

an intuitive method for their measurement, namely through an observation of these waves

in some controlled conditions, such as when only one of the device ports is excited at a

time. This observation should disturb the waves as little as possible (this is analogous to

the condition in low-frequency oscilloscope measurements that the probe has high input

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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 be

absorbed at some place far enough from the DUT (this is analogous to the measurement

of impedance or admittance parameters when we require, respectively, low impedance or

high 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 with

two or less ports (in short, a two-port VNA), is shown in Fig. 2.1. On either side of the

DUT, 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 amplitudes

of the wave incident at and reflected from the DUT. This process is realized by coupling

part of each wave out the detection circuit and converting it to a lower frequency at which

the analog to digital (A/D) converters can be used. We show the detection circuit as a

single mixer excited by the local oscillator (LO), however, in the reality the signal may

undergo multiple frequency conversions. The A/D conversion may take place either at

some intermediate frequency (IF) or in the baseband.

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2.2. VNAS-PARAMETER MEASUREMENT

In the forward position of the switch (this is the position shown in Fig. 2.1), signal

from the RF source is sent to the first port of the device under test while the other port

is 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 and

b1m, which are approximately proportional to the amplitudes of the waves incident at and

reflected from port one,a1 andb1, respectively, are then measured in the left reflectometer.

Similarly, signal b2m, approximately proportional to the wave b2 transmitted through the

DUT, is measured in the right reflectometer. From these measurements, we approximate

scattering parametersS11 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 we

obtain the following approximations ofS22 and S12

S22m= b2ma2m

, and S12m= b1ma2m

. (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 diagram

in Fig. 2.1. For detailed discussion of different architectures see for example [53]. The

main objective of the VNA construction is to provide wideband operation (for example

from 70 kHz up to 70 GHz [4]) while maintaining the error of approximations (2.40) and

(2.41) reasonably small. This objective is very hard to attain in practice, therefore the

errors of approximations are (2.40) and (2.41) usually not acceptable, even for approximate

assessment of the DUT S-parameters.

These errors result from various imperfections of the VNA construction. The most

important ones are the finite directivity of directional couplers, impedance mismatches in

the VNA (such as between the generator and the adjacent coupler, or between the other

coupler and the matched termination), discontinuities in the transmission lines guiding the

measured signals, phase shift and attenuation introduced by these lines, parasitic coupling

between the VNA ports (e.g. through the LO circuitry), and differences between the load

impedance in the forward and reverse position of the switch.

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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-parameters

of a DUT as a function of a set of model parameters describing the systematic errors

introduced by the VNA. These models form the foundation for the correction of VNA

measurements and the development of VNA calibration algorithms.

The basic premise in the formulation of VNA models is that the relationships between

the measured and actual wave amplitudes are linear and time-invariant, thus the VNA

is assumed to be a linear time-invariant (LTI) system. We begin our discussion with the

description 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 becomes

non-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 a

linear time-invariant two-port VNA is the 16-term

model [6, 5456]. This model, shown schematically

in Fig. 2.2, results directly from the VNA diagram

in Fig. 2.1. In the 16-term model model, the linear

relationship between the measured and actual waves

as a four-port liner network, denoted with EFig. 2.2,

and defined as

b1m

b2ma1

a2

=Se

a1m

a2mb1

b2

, where Se=

S11e S12eS21e S22e

=

e11 e12 e13 e14

e21 e22 e23 e24e31 e32 e33 e34

e41 e42 e43 e44

. (2.42)

Model parameters contained in the matrix Se encompass all possible transmission and

reflection 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;

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2.3. TWO-PORT VNA MATHEMATICAL MODELS

anti-diagonal terms of S11e (that is e12 and e21) correspond to the internal cross-talkbetween the VNA reflectometers; this cross-talk results from the finite isolation

between different ports of the switch;

the anti-diagonal terms ofSe (that ise14,e23,e32 ande41) correspond to the couplingbetween VNA reflectometers, for example, through the IF circuitry; in modern VNAs

this coupling is negligible, therefore one typically assumes e14 = e23=e32=e41 = 0;

the anti-diagonal terms ofS22e (that ise34 ande43) correspond to the external cross-talkbetween the VNA reflectometers, that is, to the direct cross-talk between the

VNA 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) this

cross-talk does not occur.

We determine the relationship between the raw and actual S-parameters in the 16-term

model 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

S1 S22e1

S21e, (2.44)

and

S=

S21e(Sm S11e)1 S12e+ S22e1

. (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 ofS21e by an arbitrary constant and divide all elements S12e by the

same constant, relationships (2.44) and (2.45) do not change. Therefore one of the elements

in 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 model

parameters contained in the matrix Se, defined in (2.42). Therefore, the 16-term model is

often expressed in an alternative form which uses a different set of parameters for which

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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 ofTe can be expressed in terms

of the original parameters of the 16-term model as

T11e= S12e S11eS121eS22e, (2.48)T12e= S11eS

121e, (2.49)

T21e= S121eS22e, (2.50)T22e= S21e. (2.51)

The relationship (2.47) can further be brought to a very convenient form with the use of

matrix vectorization operator [57]. For a matrix X represented as X=[x1, . . . , xN], where

x1, . . . , xNare the columns ofX, 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 I22

t11e+ t12e

ST Sm

t21e (I22 Sm) t22e= 0, (2.53)

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

ST I2 I4 ST Sm I2 Sm

t11et12e

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 as

coaxial 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 aserror-box model(see Fig. 2.3). In

the 8-term model, the VNA reflectometers are represented as two linear two-port networks

A and B, referred to as the error boxes. We shall first formulate definitions of these

networks, following a similar convention to that used in model (2.46), and then show

another 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 sets

of 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 transmission

parameters, Tm and T, respectively, defined by

b1

a1

= T

a2

bb

, and

a1m

b1m

= Tm

a2m

b2m

, (2.56)

which immediately yields

Tm=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.

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

and consequently

T= T1A TmT1B . (2.58)

Equations (2.57) and (2.58) constitute probably the most commonly used formulation

of the 8-term model. Following a similar reasoning as in the case of the 16-term model, we

can show that out of total 8 complex terms in TA andTB, only 7 are necessary to describe

the relationship between Tm and T. Therefore one of the terms in the 8-term model can

be arbitrarily chosen, for example fixed to one.

Parameters of the eight term model can be chosen in different ways. A simple choice

is to directly use the coefficient of matrices TA and TB. Another choice is to relate the

parameters of the eight term model to the actual sources if systematic error in the VNA

measurement. This can be done with the use of the flow graph in Fig. 2.4 [58]. The

terms in the graph correspond to the different sources of systematic errors in the VNA

reflectometers. The second letter in the subscript, F or R, denotes position of the

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

appears in the equations (2.57) and (2.58). Therefore the flow graph for the 8-term model

can also be represented in an alternative form, by lumping the non-reciprocity of both error

boxes into oneof them. In Fig. 2.5, we show such an alternative representation where the

non-reciprocity is lumped into the error box representing port two of the VNA. Similar

graph can also be obtained by lumping the nonreciprocity into error box representing port

one of the VNA.

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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= 1

EtEATEB, (2.59)

where the transmission matrices Tm and T can be derived from the measured and actual

DUTS-parameters with the use of (2.28) , while

EA=

ERF EDFESF EDF

ESF 1

, EB=

ERR EDRESR ESR

EDR 1

, (2.60)

and

Et =

ERR. (2.61)

The appealing simplicity of (2.57) and (2.58) allows to describe the VNA calibration

problem 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 important

disadvantage. After examining (2.28), we note that transmission matrix T cannot be

defined for a DUT that does not have a forward transmission, that is, when S21 = 0.

Indeed, in such a case, pairs of variablesa1, b1and a2, b2are unrelated and the transmission

matrix T does not exist. Hence, matrix formulation (2.57) and (2.58) cannot provide a

uniform 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

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

mA = b1ma1m , and

mB = b2ma2m (2.62)

to obtain

mA=EDF+ ERFA1 ESFA , and mB =EDR+

ERRB1 ESRB , (2.63)

where A and B are the reflection coefficients of the one-port device connected to the

VNA port A and B, respectively. Equations (2.63) can be easily inverted to obtain the

correction formulas.

An alternative formulation of the eight-term model that provides a uniform description

of 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 electrically

separated and including the definitions in Fig. 2.4, we can write submatrices ofSe as

Se=

S11e S12e

S21e S22e

=

EDF

ERF

EDR ERR

ESF

1 ESR

. (2.64)

Applying then (2.64) to formulas (2.44) and (2.45), we obtain a uniform description of

both one-port and two-port measurements

S11m=EDF+S11ERF

D ERFESRS

D , (2.65)

S22m= EDR+S22ERR

D ERRESFS

D , (2.66)

S21m= S21

ERRD

, (2.67)

S12m=S12

ERFD

, (2.68)

where

S=S11S22 S21S12, (2.69)

and

D= 1 ESFS11 ESRS22+ESFESRS. (2.70)

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C. Model parametrization choice. We showed in the previous section that the 16-term

and the 8-term model can be represented in terms of different sets of parameters. Although

this different parametrizations are equivalent, special attention needs to be paid to theirproperties in the context of their application in VNA calibration algorithms. Important

property in this context is theuniquenessof the parametrization. Parametrizations having

this 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 the

base parametrization, results directly from (2.64). In this parametrization, we write the

vector of VNA-model parameters as

p=

EDF, ESF, ERF, EDR, ESR, ERR,

T. (2.71)

By expanding equations (2.44) and (2.45) in terms of parameters in p, we can easily show

that these parameters describe a unique solution to (2.44) and (2.45). In other words, if

some 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 thereciprocal parametrization(see [36,

59]) and thetransmission parametrization(see [29, 30]). In the reciprocal parametrization,

write the vector of VNA parameters is written as

pR=

EDF, ESF,

ERF, EDR, ESR,

ERR,

ERRERF

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 =

ERRERF

.

Thus the VNA error boxes A and B are represented as reciprocal two-port linear networks

with S21A = S12A = etF and S21B = S12B = etR, respectively. For reciprocal error boxes

(that is when = ERF/ and = ERR/), we have k = 1, otherwise|k| = 1. Conse-quently, adding a reciprocal linear network between the VNA error box and the DUT does

not change k. This is not the case of the base parametrization, for which adding such a

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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 write

etF and etR as

etF =sRF

ERF+

, and etR=sRR

ERR+

, (2.73)

where

x+

indicates one of the square roots of the complex numberx(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+

,sRRsRF

ERR

+

ERF

+

T

. (2.74)

Consequently, we see that there are four different vectors pR(sRF, sRR) which lead to the

same 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 is

written as

pT =

EDF, ESF, EDFESF ERF, EDR, ESR, EDRESR ERR,

ERR

T=

= [EDF, ESF, F, EDR, ESR, R, 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 and

correction for VNA systematic errors.

In practice, however, this assumption is not met. The primary reason for the VNA

nonstationarity is the very manner in which the VNA operates, that is, by repeatedly

switching 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 slightly

different impedance of the source generator and of the matched termination in each position

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(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 different

positions of the switch does not lead to an error since the reflectometer parameters, as

captured 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, the

test-set drift, imperfect connector repeatability and errors due to cable flexure. Apart from

the 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 in

more detail in Subsection 3.3.2.

As to the changes of the matched-termination impedance, there exist two common

approaches for modeling their impact of VNA-model parameters. In either approach, a

different models is used for the VNA operating in the forward and reverse direction. In

the 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 is

illustrated in Fig. 2.6, extends the VNA models presented in the previous section by adding

some additional terms related directly to the impedance of the matched termination. In

the forward and reverse positions of the switch, the matched termination presents different

reflection coefficient, Fand R, respectively. This reflection coefficients are referred to as

the switch terms. Consequently, the measured parameters can be expressed as a function

of the time-invariant VNA measurement Smand the switch terms. In particular, when the

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switch is in the forward direction, the VNA measures

SF11m= b1Fma1Fm andS

F21m= b2Fma1Fm , (2.76)

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

SR22m= b2Rma2Rm

andSR12m= b1Rma2Rm

. (2.77)

With the use of these measurements, one can solve the linear equations resulting from the

flow graphs in Fig. 2.6 to obtain [58]

Sm= 1

1 SR12mSF21mRF

SF11m SR12mSF21mF SR12m SF11mSR12mR

SF21m SR22mSF21mF SR22m SR12mSF21mR

. (2.78)

Matrix Smobtained in that way is then used with the time-invariant VNA models presented

in Section 2.3.1.

The reflection coefficients F and R are typically measured in a separate step [58].

The VNA ports are then directly connected so that the matched termination is excited

from the opposite port. This impedance is typically very stable [58], therefore repeated

measurement of the reflection coefficients Fand R are not necessary.

2.4 Two-port VNA calibration techniques

VNA calibration procedures have been extensively studied in the literature and there

exists several good and detailed reviews, such as [6], or more recently [8]. Based on those

reviews, one concludes that these procedures differ from each other in many aspects, such

as the number and the type of calibration standards used, model of the VNA systematicerrors, mathematical formulation of the calibration problem, or the numerical method

used to solve it. Thus, VNA calibration procedures seem to form an inhomogeneous realm

with the only commonality being the high complexity of the mathematical description.

Therefore, in this overview, instead of going deep into mathematical details, we shall take

a higher-level look at the VNA calibration procedures and try to draw some similarities

between them.

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

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2.4. TWO-PORT VNA CALIBRATION TECHNIQUES

ing parameters of a device and their raw measurement, respectively (see Fig. 2.7). Denoting

the vector representation of the measurement and the actual S-parameters of the device

under 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 given

frequency as a vector function

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

where the particular form of this function depends on the VNA mathematical model and

the parametrization choice (see Section 2.3).

Fig. 2.7: Representation of the

VNA operation.

The objective of the VNA calibration procedure is

now to determine the parameters p of the VNA based on

the measurement of a number of devicesreferred to as

calibration standardswith some known characteristics.

The VNA calibration problem falls therefore into the gen-

eral class of system identification problems [61, 62], also

referred 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 problem

in a more precise way, consider Fig. 2.8.

Letsmn andsndenote the measurement

and the actual parameters of the n-th

calibration standard, forn = 1, . . . , N ,

where N is the number of calibration

st

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doktorat Lewandowski