POLITECNICO DI TORINO

SCUOLA DI DOTTORATODottorato in Dispositivi Elettronici – XVII ciclo

Tesi di Dottorato

Waveguide CharacterizationMethodology on Lossy Silicon

SubstratesA theoretical and heuristic study

Pablo Silvoni

Tutore Coordinatore del corso di dottoratoProf. Giovanni Ghione Prof. Carlo Naldi

14 Febbraio 2005

WAVEGUIDE CHARACTERIZATION METHODOLOGY

ON LOSSY SILICON SUBSTRATES

By

Pablo Silvoni

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

AT

POLITECNICO DI TORINO

TURIN, ITALY

JANUARY 2005

c© Copyright by Pablo Silvoni, 2005

POLITECNICO DI TORINO

DEPARTMENT OF

ELECTRONICS

The undersigned hereby certify that they have read and recommend

to the Faculty of Graduate Studies for acceptance a thesis entitled

“Waveguide Characterization Methodology on Lossy Silicon

Substrates” by Pablo Silvoni in partial fulfillment of the requirements

for the degree of Doctor of Philosophy.

Dated: January 2005

External Examiner:Prof. Marco Pirola

Research Supervisor:Prof. Giovanni Ghione

Examing Committee:Prof. Ermanno Di Zitti

Prof. Heinrich Chirstoph Neitzert

ii

POLITECNICO DI TORINO

Date: January 2005

Author: Pablo Silvoni

Title: Waveguide Characterization Methodology on

Lossy Silicon Substrates

Department: Electronics

Degree: Ph.D. Convocation: 14th February Year: 2005

Permission is herewith granted to Politecnico di Torino to circulate andto have copied for non-commercial purposes, at its discretion, the above titleupon the request of individuals or institutions.

Signature of Author

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iii

To my Love and Inspiration: my dear wife Adriana and

our three ”Rolling Stones”.

iv

Table of Contents

Table of Contents v

List of Figures viii

Abstract xi

Dedication xii

Acknowledgements xiii

1 Introduction 1

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

1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Transmission Line and Waveguide Theory 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 TEM mode propagation theory review . . . . . . . . . . . . . . . . . 7

2.2.1 Transmission Line description from Maxwell equations . . . . 10

2.2.2 Telegrapher’s equations and equivalent circuit model . . . . . 19

2.3 Multiconductor transmission line modelling . . . . . . . . . . . . . . . 23

2.4 Multimode description of MTL equations . . . . . . . . . . . . . . . . 31

2.5 Limitations of the quasi-TEM assumptions . . . . . . . . . . . . . . . 40

3 RF Instruments and Tools 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Characterization of linear networks . . . . . . . . . . . . . . . . . . . 47

3.3 Characterization problem in microwaves and millimeter waves . . . . 50

3.4 Scattering parameters theory review . . . . . . . . . . . . . . . . . . . 54

v

3.5 The Vector Network Analyzer . . . . . . . . . . . . . . . . . . . . . . 60

3.5.1 VNA General Description . . . . . . . . . . . . . . . . . . . . 60

3.5.2 Signal Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.3 Test Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.4 Command Unit . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Systematic error removal and VNA calibration . . . . . . . . . . . . . 66

3.6.1 Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.2 Twelve Terms Error Model . . . . . . . . . . . . . . . . . . . . 70

3.6.3 Error Box Model (Eight-Term Error Model) . . . . . . . . . . 74

4 Microwave and Millimiter Wave Measurement Techniques 78

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 VNA Calibration process . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Non Redundant Methods . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.1 SOLT Calibration Technique . . . . . . . . . . . . . . . . . . . 80

4.3.2 QSOLT Calibration Technique . . . . . . . . . . . . . . . . . . 83

4.4 Self Calibration or Redundant Methods . . . . . . . . . . . . . . . . . 87

4.4.1 TRL technique . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4.2 RSOL (UTHRU) technique . . . . . . . . . . . . . . . . . . . 95

5 Calibration & Measurement Tool 98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 MATLAB Calibration & Measurement Tool . . . . . . . . . . . . . . 99

5.3 Calibration & Measurement program . . . . . . . . . . . . . . . . . . 102

5.3.1 Switch Correction algorithm . . . . . . . . . . . . . . . . . . . 102

5.3.2 TRL algorithm and DUT deembedding . . . . . . . . . . . . . 105

5.3.3 Uploading and calibrated measurements . . . . . . . . . . . . 107

5.4 Coaxial Experimental Results . . . . . . . . . . . . . . . . . . . . . . 111

6 Networks characterization and parameter extraction 116

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2 Transmission line characterization methods . . . . . . . . . . . . . . . 118

6.2.1 Circuit parameters extraction from S-Matrix . . . . . . . . . . 123

6.2.2 On Wafer measurements and characterization . . . . . . . . . 126

6.3 MTL characterization methods . . . . . . . . . . . . . . . . . . . . . 134

6.3.1 MTL parameters extraction from S-Matrix . . . . . . . . . . . 142

6.3.2 MTL simulation and experimental results . . . . . . . . . . . . 146

vi

7 Conclusions 151

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

154

Out of Context... ? 155

A 157

Bibliography 158

vii

List of Figures

2.1 Electromagnetic field structure of a TEM mode of propagation . . . 9

2.2 Two conductor line: (a) Current and Voltage (b) TEM fields . . . . 12

2.3 Derivation contours of the first transmission line equation . . . . . . 13

2.4 Derivation contours of the second transmission line equation . . . . . 14

2.5 Effect of conductor losses, non-TEM field structure . . . . . . . . . . 17

2.6 Transmission Line equivalent lumped circuit model . . . . . . . . . . 20

2.7 Multiconductor Transmission Line system . . . . . . . . . . . . . . . 24

2.8 Modal equivalent circuit of a MTL for two modes of propagation . . . 32

2.9 Conductor equivalent circuit of a MTL for two modes of propagation 38

3.1 Two Port Network Transmission line model . . . . . . . . . . . . . . 48

3.2 Power Waves and Reference Planes interpretation . . . . . . . . . . . 55

3.3 Equivalent circuit of a linear generator . . . . . . . . . . . . . . . . . 57

3.4 HP8510 Block Diagram (Agilent Technologies 2001) . . . . . . . . . . 61

3.5 HP8511 S-Parameter Test Set (Agilent Technologies 2001) . . . . . . 63

3.6 HP8511A Frequency Converter (Agilent Technologies 2001) . . . . . . 64

3.7 HP8510 DSP Block Diagram (Agilent Technologies 2001) . . . . . . . 65

3.8 Twelve Terms Error Model Forward Set . . . . . . . . . . . . . . . . 73

3.9 Twelve Terms Error Model Reverse Set . . . . . . . . . . . . . . . . . 73

3.10 Ideal Free Error VNA and Error Boxes . . . . . . . . . . . . . . . . . 75

3.11 An interpretation of the Error Box Model . . . . . . . . . . . . . . . 76

4.1 1 - Port Error Model (Port 1) . . . . . . . . . . . . . . . . . . . . . . 81

viii

4.2 Ideal VNA and Error Box (Port 1) . . . . . . . . . . . . . . . . . . . 84

4.3 Thru - Line Setup Measurement Reference Planes . . . . . . . . . . . 93

4.4 D.U.T. Setup Measurement Fixture . . . . . . . . . . . . . . . . . . . 94

5.1 R. Marks Error-Box Error Model of a Three-Sampler VNA . . . . . 103

5.2 Measurement System for two 2-Port networks . . . . . . . . . . . . . 104

5.3 Error Model of a Four Sampler VNA . . . . . . . . . . . . . . . . . . 108

5.4 Twelve Terms Error Model - Forward and Backward sets . . . . . . . 109

5.5 S11 Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.6 S11 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.7 S21 Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.8 S21 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.9 LINE Attenuation constant . . . . . . . . . . . . . . . . . . . . . . . 115

5.10 LINE ηeff coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1 CPW stratified dielectric structure . . . . . . . . . . . . . . . . . . . 129

6.2 per-unit-length Inductance nHy/cm . . . . . . . . . . . . . . . . . . . 130

6.3 per-unit-length Capacitance pF/cm . . . . . . . . . . . . . . . . . . 130

6.4 per-unit-length Resistance Ω/cm . . . . . . . . . . . . . . . . . . . . 131

6.5 per-unit-length Conductance S/cm . . . . . . . . . . . . . . . . . . . 131

6.6 Module of the Characteristic Impedance Zc . . . . . . . . . . . . . . 132

6.7 Phase of the Characteristic Impedance Zc . . . . . . . . . . . . . . . 132

6.8 Attenuation dB/cm . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.9 Refraction index ηeff . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.10 MTL T-circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.11 MTL Π-circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.12 Asymmetric Coupled Microstrip Line . . . . . . . . . . . . . . . . . . 146

6.13 per-unit-length R(f) Ω/cm matrix . . . . . . . . . . . . . . . . . . . . 147

6.14 per-unit-length L(f) nHy/cm matrix . . . . . . . . . . . . . . . . . . 147

6.15 per-unit-length C(f) pF/cm matrix . . . . . . . . . . . . . . . . . . . 148

ix

6.16 Modal attenuation constant dB/cm . . . . . . . . . . . . . . . . . . . 149

6.17 Modal Refractive index dB/cm . . . . . . . . . . . . . . . . . . . . . . 149

6.18 Modal Cross Power ζnm merit coefficient index . . . . . . . . . . . . . 150

x

Abstract

PABLO F. G. SILVONI. Waveguide Characterization Methodology on Lossy Silicon

Substrates. Advisor: Prof. Dr. Giovanni Ghione.

A theoretical review of transmission line and waveguide theory was given. Valid-

ity and limitations were stated and discussed. An overview of the state of the art of

the experimental characterization for linear networks, microwave and millimeter mea-

surement instruments, tools and error models were described and discussed. Vector

network analyzer calibration techniques were presented and discussed. A calibration

and measurement tool was developed based on the TRL calibration technique. Ex-

perimental results based on TRL and SOLT calibration techniques were made and

compared on coaxial media. Single transmission line characterization methods were

discussed and compared. A characterization method based on scattering parame-

ters was implemented and experimental and simulated results were compared and

discussed. Multi-transmission line characterization methods were discussed and com-

pared. A multi-transmission line characterization method, based on the scattering

matrix without optimization was implemented. Experimental results were compared

with an experiment selected in the scientific literature.

xi

Dedication

This thesis is dedicated to Dr. Daniel Avalos, Full Professor of Physics of the Facultad

de Ingeniera de la Universidad Nacional de Mar del Plata, Argentina; who shared with

us his love for experiments and the pleasure for the interpretation of reality considering

Physics a great adventure of the thought. His passion, fantasy and creativity to explain

the phenomena, his faith in his students and his sense of humor, were the force and

motivation for a lot of his students, friends and children; reminding us the fact that

Imagination is more important than knowledge as uncle Albert taught us almost like

a belief.

My recognition forever.

xii

Acknowledgements

I’d like to express my sincere gratitude to Prof. Giovanni Ghione, my thesis advisor,

for his support and guidance, help and continuous encouragement during my graduate

studies and research work.

I’d also like to express my sincere appreciation and gratitude to Prof. Marco

Pirola for the constant intellectual help and patience along the research work. This

thesis would not have been possible without his continuous help.

I like to thank to Prof. Dr. Gianpaolo Bava for all his encouragement, intellectual

help, fine sense of humor and patience in our discussions. I would like to thank the

whole Gruppo di Microonde Politecnico di Torino for all its great cooperation and

participation, especially to Dr. Michelle Goano, Dr. Franco Fiori, Dr. Valeria

Teppatti, Prof. Andrea Ferrero, Prof. Pisani and Mr. Renzo Maccelloni.

I’d like to thank to my great friend Carlos Issazadeh and his family who always

believed, and helped me in the moments when I forgot to believe in myself.

I’d like to thank to my good friends Jorge Finocchietto for all his faith and love,

who constantly supported me; and Stefan Tannenbaum who helped, supported and

encouraged me with patience all the time.

I’d like to thank to my cousin Marcelo, Patricio Valdivia and his wife Marytas,

Martin Fernandez and his wife Silvia, Roberto Kiessling and Pedro Kolodka for their

love, faith and sincere friendship; and to all my friends for their love and support.

I’d like to thank to nonno Giovanni, for all his love.

I’d like to thank to my aunt and godmother Mirta for all his all his love.

I’d like to thank to my mother and father, my brother Ricardo and my sisters

Maria Gabriella, Annamaria and Luisa, my uncle Mirta and all my family who have

always stood by me and made it possible for me to pursue graduate studies and for

giving love, spiritual and material support to my wife and children all this time.

xiii

xiv

Most of all I’d like to thank to my lovely and dear wife Adriana, my son Juan

Salvador, and my daughters Constanza Guadalupe and Maria Jose who have always

supported me with love, faith and patient. They are my source of inspiration and

motivation.

I like to thank the Lord for all his Blessings.

To all of you, many thanks

Turin, Italy Pablo Silvoni

Febbruary 1st, 2005

Chapter 1

Introduction

1.1 Motivation

Active research in silicon technology has become more specialized in subfields related

to RF and high-speed applications derived from the complexity and sophistication of

embedded systems and integrated circuits, paradigms for the present state of the art

of the information technology.

In the last decade, the global expansion of mobile telecommunications and high-

speed electronic applications has stimulated both, basic and applied research on the

main issues critical to the implementation of these sophisticated devices.

In particular such embedded systems, containing complex functions, are integrated

and interconnected within complex communication structures; and the high circuit

density together with the ever increasing operating frequency requires to deal with

the problem of Electro Magnetic Interference.

EMI phenomena need to be taken into account for design, requiring the devel-

opment of more accurate models of devices and inter-chip interconnections because

1

2

interconnection of high-speed systems has become a critical issue since they are af-

fected by EMI phenomena as crosstalk, time delay and distortion [2][21].

The performance of system and on-chip interconnections has become crucial for

high-speed and high frequency applications [12] and CAD tools need to use accurate

models based on EM propagation theory, which must be accurately developed and

validated.

From these considerations comes the motivation for the present work, conceived as

a framework of ideas integrated into a methodology for characterizing high frequency

waveguides on silicon substrates.

This methodology was intended to be based both on a theoretical and experimental

study which can be extended to waveguides on different substrate materials.

1.2 Thesis Overview

The present work is the result of an applied research program which seeks primarily

for a fundamental understanding of the phenomena under investigation and at the

same time looks for possible applications [44].

The present work was divided to seven chapters, being the present the first chapter.

Chapter 2 is a review of the transmission lines theory, its validity, assumptions

and limitations.

Chapter 3 is an overview of the state of the art of the experimental characterization

for linear networks, it describes the vector network analyzer VNA and presents an

introduction to the microwave and millimeter measurement problem together with a

description of the error models.

Chapter 4 presents the more modern microwave and millimeter measurement and

3

calibration techniques, giving an extensive description of the TRL technique which is

extensively used for planar waveguide characterizations.

Chapter 5 describes a calibration and measurement tool specially developed for

the present work. Experimental results are given and discussed.

Chapter 6 presents single and multi-transmission line characterization methods

based on experimental measurements of scattering parameters. Experimental results

and simulations comparison are given and discussed.

Chapter 7 contains the Conclusions.

An Appendix contains a User’s Guide of the calibration and measurement tool

developed ad hoc for the present work.

1.3 Original Contributions

A Calibration and Measurement tool was developed in MATLAB environment based

on the TRL algorithm. This tool uses the capacity of the VNA HP8510C to be

connected to a remote computer through an IEEE 488.2 interface. Different features

and experimental results are described in Chapter 5.

Chapter 2

Transmission Line and WaveguideTheory

2.1 Introduction

In this chapter, a review of the relevant theoretical topics of linear transmission lines

and multiconductor transmission line (MTL) models will be presented with a rigorous

physics description using Maxwell equations. Then, the Telegrapher’s equation will be

developed as a distributed-parameter, lumped-circuit description that is the common

model used in engineering. Limitations of the descriptions will be presented and

discussed.

Transmission line structures serve to guide electromagnetic (EM) waves between

two points. The analysis of transmission lines consisting of two parallel conductors of

uniform cross section is a fundamental and well understood subject in electrical engi-

neering. However, the analysis of similar lines consisting of more than two conductors

is somewhat more difficult than the analysis of two-conductor lines. Matrix methods

and notation provide a straightforward extension of most of aspects of two-conductor

to multiconductor transmission lines.

4

5

First, the key assumptions of these theoretical descriptions will be aimed at under-

standing their restrictions on the applicability of the representations and the validity

of the results obtained.

Electromagnetic fields are, actually, distributed continuously throughout space.

If a structure’s largest dimension is electrically small, i.e., much less than a wave-

length, we can approximately lump the EM effects into circuit elements as in lumped-

circuit theory and define alternative variables of interest such as voltages and cur-

rents. The transmission-line formulation views the line as a distributed-parameter

structure along the propagation axis and thereby extends the lumped-circuit analysis

techniques to structures that are electrically large in this dimension. However, the

cross-sectional dimensions, e.g., conductor separations, must be electrically small in

order for the analysis to yield valid results.

The fundamental assumption for all transmission-line formulations and analysis,

whether for a two-conductor or a MTL, is that the field structure surrounding the

conductors obeys to a Transverse Electro Magnetic or TEM structure. A TEM field

structure is one in which the electric and magnetic fields in the space surrounding the

line conductors are transverse or perpendicular to the line axis which will be chosen

to be the z axis of a rectangular coordinate system. The waves on such lines are said

to propagate in the TEM mode.

Transmission-line structures having electrically large cross-sectional dimensions

have, in addition to the TEM mode of propagation, other higher-order modes of

propagation. An analysis of these structures using the transmission line equation

formulation would then only predicts the TEM mode component and does not rep-

resent a complete analysis. Other aspects, such as imperfect line conductors, also

6

may invalidate the TEM mode transmission-line equation description. In addition,

an assumption inherent in the MTL equation formulation is that the sum of the line

currents at any cross section of the line is zero; and it is assumed that a conductor,

the reference conductor, is the return for all the line currents. This last assumption

may not be true and there may be other non-TEM currents in existence on the line

conductors due to EMI and/or asymmetries in physical terminal excitation.

A complete solution of the transmission-line and MTL structures, which does

not presuppose only the TEM mode, can be obtained with Full-Wave solutions of

Maxwell’s equations, techniques that require numerical methods and are outside of

the scope of this work.

In my approach only the analytical solutions for TEM and quasi-TEM modes of

propagation that are consistent with the characterization parameters which can ex-

perimentally be measured as the scattering matrix, and all other two-port descriptions

defined for TEM structures, will be considered.

7

2.2 TEM mode propagation theory review

First to examine the classical formulae and parameters of Transmission Line and

waveguide theory, the results of the TEM propagation mode will be reviewed.

The fundamental assumption in any transmission line formulation is that the

electric field intensity vector−→E (x, y, z, t) and the magnetic field intensity vector

−→H (x, y, z, t) satisfy the transverse electromagnetic (TEM) field structure, and they

lie in a plane (the x-y plane) transverse or perpendicular to the line axis (the z axis).

Considering a rectangular coordinate system as shown in Fig. 2.1 where a propa-

gating TEM wave in which field vectors are assumed to lie in a plane transversal

to the propagation direction is illustrated, we denote the field vectors with a t sub-

script to denote transverse. It is assumed that the medium is homogeneous, linear

and isotropic and characterized by the scalar parameters of electric permitivity ε,

magnetic permeability µ and conductivity σ. Then Maxwell’s equations become:

∇×−→E t = µ∂−→H t

∂t(2.2.1)

∇×−→H t = σ−→E t − ε

∂−→H t

∂t(2.2.2)

The ∇ operator can be broken into two components, one component, ∇z, in the

z direction and one component, ∇t, in the transverse plane as ∇ = ∇t +∇z, where:

∇t = x · ∂

∂x+ y · ∂

∂y

∇z = z · ∂

∂z

8

being x, y and z the unit vectors in the rectangular directions. Separating (2.2.1)

and (2.2.2) by equating the field components in the z direction and in the transverse

plane gives:

z × ∂−→E t

∂z= −µ

∂−→H t

∂t(2.2.3)

z × ∂−→H t

∂z= σ

−→E t − ε

∂−→H t

∂t(2.2.4)

∇t ×−→E t = 0

∇z ×−→H t = 0(2.2.5)

Equations (2.2.5) are identical to those for static fields. As a consequence, the

electric and magnetic fields of a TEM field distribution satisfy a static distribution

in the transverse plane. Then, each of the transverse field vectors can be defined as

the gradients of some auxiliary scalar fields or potential functions Φ and Ψ, such as:

−→E t = g(z, t) · ∇Φ(x, y)−→H t = f(z, t) · ∇Ψ(x, y)

(2.2.6)

And by applying Gauss’s laws:

∇t · −→E t = 0 ∇2t Φ(x, y) = 0

∇t · −→H t = 0 ∇2t Ψ(x, y) = 0

(2.2.7)

Equations (2.2.7) show that these scalar potential functions satisfy Laplace’s equa-

tion in any transverse plane as they do for static fields. This permits the unique

9

Figure 2.1: Electromagnetic field structure of a TEM mode of propagation

definition of voltage between two points in a transverse plane as the line integral of

the transverse electric field between those two points:

V (z, t) = −∫ 2

1

−→E t · d~l (2.2.8)

Similarly the last equation of (2.2.7) shows that we may uniquely define current in

the z direction as the line integral of the transverse magnetic field around any closed

contour lying solely in the transverse plane:

I(z, t) = −∮

ct

−→H t · d~l (2.2.9)

These results can be applied to a TEM field structure propagating along a uni-

form transmission line with two parallel conductors to obtain the transmission line

equations as shown in the following section.

10

2.2.1 Transmission Line description from Maxwell equations

By considering a two-conductor transmission line as shown in Fig. 2.2 the following

properties are assumed: a. the conductors are parallel to each other and the z axis, b.

the conductors have uniform cross sections along the line axis and c. the conductors

are perfect (conductor resistivity ρ = 0). The first two properties define a uniform

line. The medium surrounding the conductors is assumed to be lossy (σmedium 6= 0)

and is homogeneous in σ, ε and µ. Maxwell’s equations in integral form are:

∮

c

−→E · d~l = −µ

∂

∂t

∫∫

s

−→H · d~s (2.2.10)

∮

c

−→H · d~l =

∫∫

s

−→J · d~s + ε

∂

∂t

∫∫

s

−→E · d~s (2.2.11)

Open surface s is enclosed by the closed contour c. The quantity−→J is a current

density in A/m and contains conduction current,−→Jc = σ

−→E , as well as any source

current,−→Js , as

−→J =

−→Jc +

−→Js .

By assuming the TEM field structure about the conductors in any cross-sectional

plane as indicated in Fig. 2.2 (b), we can choose the contour c to lie solely in the

cross-sectional plane between the two conductors into the xy plane, and the surface

s enclosed to be a flat surface in the transverse xy plane. By the TEM assumptions,

there are no z -directed fields so that Hz = 0. Similarly by the TEM assumptions,

(Ez = 0), and there is no z -directed displacement current, only z -directed source

currents, Jsz. Then equations (2.2.10) and (2.2.11) become:

11

∮

c

(Exdx + Eydy) = −µ∂

∂t

∫∫

s

Hzdxdy = 0 (2.2.12)

∮

c

(Hxdx + Hydy) =

∫∫

s

Jzdxdy + ε∂

∂t

∫∫

s

Ezdxdy (2.2.13)

=

∫∫

s

Jszdxdy

Equations (2.2.12) and (2.2.13) are identical to those for static time variation as

was pointed in equation (2.2.7) in the TEM assumptions. Therefore, from equation

(2.2.12) it may be uniquely defined the voltage between the two conductors, indepen-

dent of path, so long as we take the path to lie in a transverse plane as indicated in

Fig. 2.2 (b), in the same way as was pointed in equation (2.2.8) as:

V (z, t) = −∫ 1

0

−→E t · d~l (2.2.14)

Similarly, equation (2.2.13) allows the unique definition of the current by choosing

a closed contour in the transverse plane encircling one of the conductors as indicated

in Fig. 2.3, in the same way as was pointed in equation (2.2.9) as:

I(z, t) = −∮

ct

−→H t · d~l (2.2.15)

This current defined by (2.2.15) lies solely on the surface of the perfect conductor.

If both conductors are enclosed with the same contour it can be shown that the net

12

Figure 2.2: Two conductor line: (a) Current and Voltage (b) TEM fields

current is zero, being the current in any cross section on the lower conductor equal

and opposite to the current on the upper conductor.

Now the transmission-line equations can be derived in terms of the voltage and

current defined above. First, the open surface s is considered , enclosed by the contour

cl as is shown in Fig. 2.3. Integrating Faraday’s Law given in equation (2.2.10) around

this contour gives:

∫ 2

1

−→Ez · d~l +

∫ 3

2

−→Et · d~l +

∫ 4

3

−→Ez · d~l +

∫ 1

4

−→Et · d~l = −µ

∂

∂t

∫∫

s

−→Ht · d~s (2.2.16)

By defining the voltages between the two conductors as in (2.2.14) with the TEM

13

Figure 2.3: Derivation contours of the first transmission line equation

assumption of Ez = 0 gives:

V (z + ∆z, t) = −∫ 3

2

−→Et(x, y, z + ∆z, t) · d~l

V (z, t) = −∫ 1

4

−→Et(x, y, z, t) · d~l

Therefore (2.2.16) becomes:

V (z + ∆z, t)− V (z, t) = −µ∂

∂t

∫∫

s

−→Ht · d~s

Rewriting this and taking the limit as ∆z → 0 gives:

∂

∂zV (z, t) = −µ

∂

∂tlim

∆z→0

1

∆z

∫∫

s

−→Ht · d~s (2.2.17)

14

Figure 2.4: Derivation contours of the second transmission line equation

The right hand of (2.2.17) can be interpreted as an inductance of the loop formed

between the two conductors. Being φ = the magnetic flux, by definition the induc-

tance L for a ∆z section is:

L =φ

I= −µ

∫∫

s

−→Ht · d~s

/I

Now it can be defined a per-unit-length (pul) inductance, L, at any cross section

of the uniform line as:

L = lim∆z→0

L

∆z= −µ

∫ 1

2

−→Ht · ~ndl

/ ∮

ct

−→Ht · d~l (2.2.18)

where ~n is the unit vector perpendicular to the open surface s. Combining this

with (2.2.17), the first transmission line equation is obtained:

∂

∂zV (z, t) = −L ∂

∂tI(z, t) (2.2.19)

15

To derive the second transmission line equation, we recall the continuity equation

which states that the net outflow of current from a closed surface Sv equals the time

rate of decrease of the charge enclosed Qenc by that surface:

∫∫

Sv

−→J · d~s = − ∂

∂tQenc (2.2.20)

Integrating the continuity equation over the closed surface Sv of length ∆z that

encloses each conductor as is shown in Fig. 2.4 gives:

∫∫

So

−→J · d~s +

∫∫

Se

−→J · d~s = − ∂

∂tQenc (2.2.21)

The terms in last equation become:

∫∫

Se

−→J · d~s = I(z + ∆z, t)− I(z, t); (2.2.22)

∫∫

So

−→J · d~s = σ

∫∫

So

−→Et · d~s (2.2.23)

The right-hand side of (2.2.21) can be defined in terms of a per-unit-length capac-

itance C. From Gauss’ law, the total charge enclosed by a closed surface Sv is:

Qenc =

∫∫

Sv

−→E · d~s (2.2.24)

16

The capacitance between two conductors for a ∆z section is:

C =Qenc

V

then, by substituting (2.2.24) and observing Fig. 2.4, the per-unit-length capaci-

tance C is defined as:

C = lim∆z→0

C

∆z= −

∮

ct

−→Et · ~nd~l

/ ∫ 1

0

−→Et · d~l (2.2.25)

Similarly, the conductance between the two conductors for a ∆z section can be

defined as:

G =

∫∫

So

−→J · d~s

/V (z, t) (2.2.26)

Then, from (2.2.23) we can define per-unit-length conductance G as:

G = lim∆z→0

G

∆z= −σ

∮

ct

−→Et · ~nd~l

/∫ 1

0

−→Et · d~l (2.2.27)

Finally, substituting (2.2.22), (2.2.25) and (2.2.27) into (2.2.21) gives the second

transmission-line equation:

∂

∂zI(z, t) = −GV (z, t)− C ∂

∂tV (z, t) (2.2.28)

Equations (2.2.19) and (2.2.28) are the transmission-line equations represented as

a coupled set of first-order, partial differential equations in the line voltage, V (z, t),

and line current I(z, t).

17

Figure 2.5: Effect of conductor losses, non-TEM field structure

Most of the previous derivations assumed perfect conductors. Unlike losses in the

surrounding medium, lossy conductors invalidate the TEM field structure assumption.

As is shown in Fig. 2.5, the line current flowing through the imperfect line conductor

generates a nonzero electric field along the conductor surface, Ez(z, t), which is di-

rected in the z direction violating the basic assumption of the TEM field structure in

the surrounding medium. The total electric field is the sum of the transverse compo-

nent Et(z, t) and this z directed component Ez(z, t). However, if the conductor losses

are small, this resulting field structure is almost TEM. This is the quasi-TEM as-

sumption and, although the transmission line equations are no longer valid, they are

nevertheless assumed to represent the situation for small losses through the inclusion

of the per-unit-resistance parameter R.

Another limitation of the transmission-line equations description is that non ho-

mogeneous surrounding medium invalidates the basic assumption of a TEM field

18

structure because different portions of this medium are characterized by different di-

electric constants εi and magnetic permeabilities µi. Then, the phase velocities vphi

(vphi= 1/

√εiµi) of TEM waves in these regions will be different; when it is required

for a TEM field structure to have only one propagation velocity in the medium. Nev-

ertheless, the transmission-line equations are solved by assuming to represent the

situation so long as these velocities are not substantially different, referred to as the

quasi-TEM assumption. To describe this situation of a non homogeneous medium, an

effective dielectric constant εeff is defined so that if the transmission line conductors

are immersed in a homogeneous dielectric having this εeff , the propagation velocities

and all other attributes of the solutions for the original non homogeneous medium

and for this one will be the same.

In order to solve the transmission-line equations by obtaining a closed analytical

solution for them, the above quasi-TEM assumptions will be taken into account by

adding the conductor losses to the model in a heuristic or engineering approach.

This is the classical form of Telegrapher’s equations with all the above described

parameters involved in a distributed-parameter, lumped circuit as will be seen in the

next paragraphs.

19

2.2.2 Telegrapher’s equations and equivalent circuit model

The previous two derivations of the transmission-line equations were rigorous. In

order to add the conductor losses to the transmission-line model, a quasi-TEM field

structure will be assumed, and the usual derivation of a distributed-parameter, lumped

equivalent circuit model will be developed. The concept stems from the fact that

lumped-circuit concepts are only valid for structures whose largest dimension is elec-

trically small, i.e., much less than a wavelength, at the frequency excitation.

If a structural dimension is electrically large, we may break the transmission line

into the union of electrically small substructures and can then represent each sub-

structure with a lumped circuit model. In order to apply this to a transmission-line,

an equivalent lumped circuit model is considered in Fig. 2.6. In this figure, the

transmission line is subdivided into infinitesimal pieces of incremental lumped cir-

cuits composed by the per-unit-length (pul) parameters R, L, G and C, embedded

and connected within little cross sections of ∆z length. In the approach, these R, L,

G and C pul parameters will be assumed to be constant with frequency to obtain the

solution of Telegrapher’s equations. This assumption will be reexamined later when

the propagation of EM waves with microwave and millimeter wavelengths into silicon

waveguides (microstrip lines, CPWs, etc) will be considered.

From Fig. 2.6 it is straightforward to derive the circuital equations that describe

the lumped circuit model [41] known as the Telegrapher’s equations :

∂

∂zV (z, t) = −RI(z, t)− L ∂

∂tI(z, t) (2.2.29)

∂

∂zI(z, t) = −GV (z, t)− C ∂

∂tV (z, t) (2.2.30)

20

Figure 2.6: Transmission Line equivalent lumped circuit model

To solve this equation system the harmonic voltage and current with the phasor

representation in the frequency domain as V (z, ω) and I(z, ω) will be considered.

Then the above equation system can be rewritten as [22]:

∂

∂zV (z, ω) = −(R+ jωL)I(z, ω) (2.2.31)

∂

∂zI(z, ω) = −(G + jω C)V (z, ω) (2.2.32)

The above equation system can be expressed in a more concise form as:

∂

∂z

[V (z, ω)

I(z, ω)

]= M ·

[V (z, ω)

I(z, ω)

](2.2.33)

where M is the system equation associated matrix given by:

M = −[

0 R+ jωLG + jω C 0

](2.2.34)

21

The solution of the Telegrapher’s equations, expressed as in (2.2.33) is given by:

[V (z, ω)

I(z, ω)

]= Ev · exp(Λ.z) ·

[V +

0

V −0

](2.2.35)

Constants V +0 and V −

0 are determined by the environment conditions. Ev is the

eigenvector matrix and Λ is the eigenvalue matrix of the system given by:

Ev =

[1 1

Yc −Yc

], and Λ =

[−γ 0

0 γ

](2.2.36)

The variable γ is the propagation constant and Zc = Y −1c is the characteristic

impedance of the system expressed as functions of the pul parameters R, L, G and Cand given by:

γ =√

(R+ jωL) · (G + jω C) (2.2.37)

Zc =

√(R+ jωL)

(G + jω C)(2.2.38)

These are the fundamental parameters that describe the behavior of the trans-

mission line. The propagation constant γ can be expressed as a complex number as

follows:

γ = α + jβ (2.2.39)

22

where α represents the attenuation constant that takes into account the power

attenuation of the EM wave along the line, and β is the phase constant that represents

the behavior of the phase velocity vph = ω/β of the EM wave along the line.

Then, modal voltage V (z, ω) and modal current I(z, ω) along the line can be

expressed as the well known expressions based on the travelling waves or forward

intensities V +, I+ and the backward intensities V −, I− as:

V (z, ω) = V +0 · e−γ.z + V −

0 · eγ.z = V +(z) + V −(z) (2.2.40)

I(z, ω) =V +

0

Zc

· e−γ.z − V −0

Zc

· eγ.z = I+(z) + I−(z) (2.2.41)

From the above considerations it is clear that the pul parameters R, L, G and

C fully characterize a transmission line, and they are analytically related with the

EM wave parameters γ = α + jβ and Zc. As will be shown in next chapters these

parameters can be obtained indirectly through two-ports scattering parameters mea-

surements and using matrix calculations.

In next paragraphs an extension of this theory will be applied to a multiconductor

transmission line system and matrix equations will be obtained and solved in analogy

with the Telegrapher’s equations.

23

2.3 Multiconductor transmission line modelling

The results obtained in the previous chapter for the general properties of a two con-

ductors transmission line will be extended to multiconductor transmission lines or

MTLs. As seen, the TEM field structure and associated mode of propagation is

the fundamental, underlying assumption in the representation of a transmission-line

structure with the transmission-line equations. The class of lines will be restricted

to those that are uniform lines consisting of (n + 1) conductors of uniform cross sec-

tion that are parallel to each other. The same quasi TEM assumptions used for the

two conductors transmission line will be used to derive the MTLs equations from

an equivalent circuit that takes into account lossy conductors and inhomogeneous

surrounding medium. The MTL equations will have an identical form to the Teleg-

rapher’s equations by using a matrix notation.

A full development of the MTL equations will be not given, but only the useful

results related to the experimental characterization will be shown. For developments

are refer to works of K. D. Marx [39], C. Paul [41] and Marks and Williams [8].

Taking into account Fig. 2.7, a set of n conductors (1,2,..i,..,j...n) and a reference

conductor 0 define a multiconductor transmission line system. Maxwell equations

applied to these systems demonstrate that different i modes propagate along the z

axis, but their modal parameters, the modal propagation constant γmiand modal

characteristic impedance Zmi, can not be measured directly. The more common

description of the system starts by applying Kirchoff’s laws to the different ith circuits

given by the ith conductors and the reference conductor. It is assumed that the

reference conductor collects all the n conductor currents and applying the 2nd Kirchoff

law gives I0 =∑n

k=1 Ik.

24

Figure 2.7: Multiconductor Transmission Line system

This model represents very well the cases of coupled microstriplines, coupled

CPW transmission lines, and PCB traces on different substrates and will be assumed

throughout this work.

In order to apply the circuit theory to the model shown in Fig. 2.7, the different pul

matrices parameters R, L, C and G,that describe the mutual interactions between

the different conductors, will be defined. From the model in Fig. 2.7 the voltage V

and current I vectors are defined as:

V(z, t) =

V1(z, t)...

Vi(z, t)...

Vn(z, t)

I(z, t) =

I1(z, t)...

Ii(z, t)...

In(z, t)

(2.3.1)

25

By using the assumption of a quasi-TEM field structure, small losses in lossy

conductors can be described by a pul conductor resistance Ri for any ith conductor.

Then, by taking into account the model in Fig. 2.7, the per-unit-length resistance

matrix R is defined as:

R =

(R1 +R0) R0 · · · R0

R0 (R2 +R0) · · · R0

......

. . ....

R0 R0 · · · (Rn +R0)

(2.3.2)

The Ψ vector contains the total magnetic flux per unit length, ψi, which penetrates

the ith circuit defined between the ith conductor and the reference conductor; and is

related with the I current vector and the per-unit-length inductance matrix L as

follows:

Ψ =

ψ1

...

ψi

...

ψn

= L · I (2.3.3)

where the per-unit-length inductance matrix L contains the individual per-unit-

length self-inductances, Lii, of the circuits and the per-unit-length mutual-inductances

between the circuits, Lij:

26

L =

L11 L12 · · · L1n

L21 L22 · · · L2n

......

. . ....

Ln1 Ln2 · · · Lnn

(2.3.4)

With similar considerations to the two conductors transmission line, the transverse

conduction current flowing between conductors can be considered by defining per-

unit-length conductances, Gij, between each pair of conductors. Then a per-unit-length

conductance matrix, G, that represents the conduction current flowing between the

conductors in the transverse plane, can be defined as:

G =

∑nk=1 G1k −G12 · · · −G1n

−G21

∑nk=1 G2k · · · −G2n

......

. . ....

−Gn1 −G2n · · · ∑nk=1 Gnk

(2.3.5)

Similarly, the per-unit-length charge can be defined in terms of the per-unit-length

capacitances, Cij, between each pair of conductors. Then, the displacement current

flowing between the conductors in the transverse plane is represented by the per-unit-

length capacitance matrix C defined as:

C =

∑nk=1 C1k −C12 · · · −C1n

−C21

∑nk=1 C2k · · · −C2n

......

. . ....

−Cn1 −C2n · · · ∑nk=1 Cnk

(2.3.6)

27

If the total charge per unit of line length on the ith conductor is denoted as Qi,

the fundamental definition of C is given by:

Q =

Q1

...

Qi

...

Qn

= C ·V (2.3.7)

The above per-unit-length parameter matrices contain all the cross-sectional di-

mension information that fully characterizes and distinguishes one MTL structure

from another. Then, a set of 2n, coupled, first-order, partial differential equations,

the MTL equations , can be derived by analogy to the two conductors transmission

line as a generalization of Telegrapher’s equations in matrix notation as follows:

∂

∂zV(z, t) = −RI(z, t)− L

∂

∂tI(z, t) (2.3.8)

∂

∂zI(z, t) = −GV(z, t)−C

∂

∂tV(z, t) (2.3.9)

To find the solutions of the above MTL equations , the frequency-domain repre-

sentation where the excitation sources are sine waves in steady state will be considered

and the line voltages and currents will be denoted in their phasor form as:

Vi(z, t) = <Vi(z)ejωt (2.3.10)

Ii(z, t) = <Ii(z)ejωt (2.3.11)

28

where <· denotes the real part of the enclosed complex quantity and all complex

or phasor quantities will be denoted with the ˜ over the quantity. Then, substituting

the phasor forms in equations (2.3.8) and (2.3.9), the MTL equations for harmonic

steady state excitation are given by:

∂

∂zV(z) = −Z · I(z) (2.3.12)

∂

∂zI(z) = −Y · V(z) (2.3.13)

where the per-unit-length impedance and admittance matrices (or per-unit-length

conductor impedance and admittance matrices) Z and Y are:

Z = R + jωL

Y = G + jωC(2.3.14)

In taking time derivatives to give the equations (2.3.12) and (2.3.13), it was as-

sumed that the per-unit-length parameter matrices R, L, C and G are time inde-

pendent, i.e. the cross sectional dimensions and surrounding media properties do not

change with time.

The resulting MTL equations (2.3.12) and (2.3.13) can be put in a more compact

form similar to the state-variable equations :

∂

∂zX(z) = A(z) · X(z) (2.3.15)

29

being X(z) a 2n× 1 vector and A(z) a 2n× 2n matrix and where:

X(z) =

[V(z)

I(z)

]A(z) =

[0 −Z

−Y 0

](2.3.16)

Then, by using the results of state-variable equations a 2n × 2n state-transition-

matrix, Φ(z), can be defined with the following properties:

Φ(0) = I2n being I2n the 2n× 2n identity matrix

Φ−1(z) = Φ(−z)

Φ(z) = expAz = I2n+z

1!A+

z2

2!A2+

z3

3!A3+· · · (2.3.17)

The general solution is found straightforwardly from the state-variable equations

solutions assuming that the initial states are zero and giving:

X(z) = Φ(z − z0) · X(z0) (2.3.18)

then, choosing z0 = 0 and equating (2.3.16) and (2.3.17), the general solution

of MTL equations is given by:

[V(z)

I(z)

]= Φ(z) ·

[V(0)

I(0)

]=

[Φ11(z) Φ12(z)

Φ21(z) Φ22(z)

]·[

V(0)

I(0)

](2.3.19)

where the Φij(z) are n× n submatrices of the chain parameter matrix Φ(z).

30

In order to solve (2.4.1) by finding the chain parameter matrix Φ(z), the uncou-

pled form of the MTL equations, where the different modes of propagation can be

defined and analytical solutions are found, will be used. The method to be used is

a similarity transformation [41] and is the most frequently used technique for de-

termining the chain parameter matrix Φ(z). In the following paragraphs a simple

form of this representation will be presented as an uncoupled multimode description

[8]. Relationships between the modal and conductor parameters will be given as an

equivalent description of the same MTL structure.

31

2.4 Multimode description of MTL equations

The validity and usefulness of the modal or multimode description will be discussed.

Maxwell’s equations are separable in the longitudinal and transverse directions of

uniform waveguides and transmission lines. This leads to a natural description of

the electromagnetic fields within the line in terms of the eigenfunctions of the two-

dimensional eigenvalue problem. These eigenfunctions form a discrete set of forward

and backward modes which propagate independently with an exponential dependence

along their lengths.

This modal description has a natural equivalent-circuit representation even in

presence of small losses by applying quasi-TEM assumptions. In this representation,

each unidirectional mode is described by a modal voltage and current that propagate

independently of those associated with the other modes of the line; this is the sim-

plest equivalent-circuit representation of a lossy multimode transmission line from a

physical point of view. An example of a MTL modal equivalent-circuit model for two

modes of propagation is given in Fig. 2.8.

The modal description of Ref. [34] is close to the low frequency theory, in which

the complex power Pm is given by VmI∗m where Vm and I∗m are the modal voltage

and current respectively. This allows the construction of a low-frequency equivalent-

circuit analogy and the straightforward application of the methods of nodal analysis.

To create the analogy, reference planes are specified to be far enough away from the

ends of the lines interconnecting the circuit elements to ensure that only a single

mode is present there. Then a node is assigned to each of these modes, setting the

nodal voltages and currents equal to the modal voltages and currents. Brews [5] [6]

proposed a normalization that ensures that the power in the actual circuit corresponds

32

Figure 2.8: Modal equivalent circuit of a MTL for two modes of propagation

to that in the equivalent-circuit analogy, which fixes the relationship between the

modal voltages and currents. Typically the modal voltage is defined to correspond

to the actual voltage between conductor pairs across which the circuit elements are

attached to, and the modal current is determined from the power constraint.

Models of the embedded circuit elements can be further simplified in the equivalent-

circuit analogy by representing them as an interior circuit connected to lines with

lengths equal to those physically connected to the element. This approach allows for

simple lumped-element circuit models for the interior circuits that correspond closely

to those predicted from physical models.

When multiple modes of propagation are excited in a transmission line, the total

voltage across a given conductor pair will in general be a linear combination of all

the modal voltages and currents. As a result, the voltage across even the simplest

of circuit elements will not correspond to any one of the modal voltages but to a

linear combination of all of them. Then the modal voltages and currents, which

33

are associated with the modes rather than with the connection points of the circuit

elements, do not correspond to those across the device terminals.

These considerations are important because in the similarity transformations not

all available descriptions have physical sense. Theerefore, the power considerations

are necessary to give the opportune constraints for the problem to be solved.

The power normalization given by [8] is constructed so that the product of the

modal voltage Vm and current I∗m give the modal power Pm carried by a single mode

in the absence of other modes in the structure; and it permits a useful modal or

multimode description of the multiconductor transmission line.

The MTL equations (2.3.12) and (2.3.13), can be placed in the form of uncoupled,

second-order ordinary differential equations by differentiating both with respect to z

and substituting as:

∂2

∂z2V(z) = ZY · V(z) (2.4.1)

∂2

∂z2I(z) = YZ · I(z) (2.4.2)

where the V and I are the column vectors of conductor voltages and currents.

These magnitudes can be defined as to be arbitrary invertible linear transformations

of the modal voltages and currents Vm and Im as:

V = Mv · Vm

I = Mi · Im

(2.4.3)

being the n × n complex matrices Mv and Mi the similarity transformations

34

between the actual phasor line or conductor voltages and currents, V and I, and the

modal voltages and currents Vm and Im.

In order to be valid, these n × n transformation matrices must be nonsingular,

and the inverse matrices M−1v and M−1

i must exist in order to go between both sets

of variables. Substituting (2.4.3) into (2.3.16) gives:

∂

∂z

[Vm

Im

]=

[0 −M−1

v ZMi

−M−1i YMv 0

]·[

Vm

Im

](2.4.4)

If it is possible to obtain Mv and Mi so that M−1v ZMi and M−1

i YMv are diagonal,

the per-unit-length modal impedance and admittance matrices Zm and Ym can be

defined as:

Zm = M−1v ZMi =

Zm1 0 · · · 0

0 Zm2. . .

......

. . . . . . 0

0 · · · 0 Zmn

(2.4.5)

Ym = M−1i YMv =

Ym1 0 · · · 0

0 Ym2. . .

......

. . . . . . 0

0 · · · 0 Ymn

(2.4.6)

Substituting the above definitions into equations (2.4.1) and (2.4.2) gives:

35

∂2

∂z2Vm(z) = ZmYm · Vm(z) = γ2 · Vm(z) (2.4.7)

∂2

∂z2Im(z) = YmZm · Im(z) = γ2 · Im(z) (2.4.8)

with the modal propagation constant matrix γ defined as:

γ2 = ZmYm = YmZm =

γ21 0 · · · 0

0 γ22

. . ....

.... . . . . . 0

0 · · · 0 γ2n

(2.4.9)

Now, a straightforward solution for the modal uncoupled equations (2.4.7) and

(2.4.8) is:

Vm(z) = V+

me−γz + V−meγz (2.4.10)

Im(z) = I+

me−γz − I−meγz (2.4.11)

where the matrix exponentials e±γz are defined as:

e±γ.z =

e±γ1z 0 · · · 0

0 e±γ2z . . ....

.... . . . . . 0

0 · · · 0 e±γnz

(2.4.12)

36

where each mode is described by a couple of travelling waves, the vectors modal

forward intensities V+

m, I+

m and the vectors modal backward intensities V−m, I

−m.

Then, substituting the similarity transformations Mv and Mi into equations

(2.3.12) and (2.4.2) implies:

∂2

∂z2V(z) = Mvγ

2M−1v V(z) (2.4.13)

∂2

∂z2I(z) = Miγ

2M−1i I(z) (2.4.14)

where the matrices ZY and YZ are related to γ2 by the similarity transformations

and per-unit-length modal impedance and admittance matrices Zm and Ym as follows:

ZY = MvZmYmM−1v = Mvγ

2M−1v (2.4.15)

YZ = MiYmZmM−1i = Miγ

2M−1i (2.4.16)

Thus all four matrices have the identical eigenvalues γ2. The similarity transfor-

mation matrices Mv and Mi diagonalizes ZY and YZ respectively. Therefore, there

is a direct relationship between the MTL conductor model and the modal description

if the proper similarity transformation matrices are encountered.

In the case of quasi-TEM assumptions with small losses, the Z and Y matrices

are intended to be symmetric for reciprocal structures, and is demonstrated that [41]:

MTi ·Mv = I (2.4.17)

37

where the superscript T indicates the Hermitian adjoint (conjugate transpose)

and I is the identity matrix. Then, with this assumption Mi = M−1v = M, the

characteristic impedance matrix ZC can be defined as:

ZC = Y−1MγM−1 = ZMγ−1M−1 (2.4.18)

Using the above ZC definition and the similarity transformation relationships

(2.4.3), a general solution for the uncoupled MTL equations (2.4.1) and (2.4.2) can

be written in terms of the modal MTL solution as:

V(z) = ZCM · (V+

me−γz + V−meγz) (2.4.19)

I(z) = M · (I+

me−γz − I−meγz) (2.4.20)

Finally, by equating and combining equations (2.4.15), (2.4.16) and (2.4.18), the

following matrices can be defined:

√YZ = MγM−1 (2.4.21)

and

ZC = Y−1√

YZ = Z(√

YZ)−1

(2.4.22)

The above definitions are used into (2.4.19) and (2.4.20) and its results are substi-

tuted in (2.3.18). Then, using the properties of the exponential matrices, the Φij(z)

terms of the chain parameter matrix Φ(z), that represent the general solution of

the MTL equations, are given by:

Φ(z) =

[Φ11(z) Φ12(z)

Φ21(z) Φ22(z)

]=

cosh

(√YZ

)−ZC sinh

(√YZ

)

−Z−1C sinh

(√YZ

)cosh

(√YZ

) (2.4.23)

38

Figure 2.9: Conductor equivalent circuit of a MTL for two modes of propagation

An equivalent-circuit model for MTLs can be constructed based on the assumption

that Z and Y matrices are symmetric, and the modes of propagation are orthogonal

into a reciprocal structure [16]. An example of this representation for two modes of

propagation is given in Fig. 2.9.

An important remark will be given for the definition of the characteristic impedance

matrix ZC where the power normalization given in [8] is assumed. In this modal equiv-

alent circuit representation, the total transverse electric field Et and magnetic field

strength Ht in the MTLs due to the excited modes with modal voltages and currents

Vmk and Imk and modal electric fields and magnetic field strengths Emk and Hmk are

given by:

Et(x, y, z) =∑

k

Vmk

V0k

(z) · Emk(x, y) (2.4.24)

Ht(x, y, z) =∑

k

Imk

I0k

(z) · Hmk(x, y) (2.4.25)

39

where the normalizing voltage V0k and current I0k are restricted by:

P0k = V0kI∗0k ≡

∫

S

Emk × H∗mk · zdS (2.4.26)

where <(P0k) ≥ 0. This normalizes the modal voltages and currents so that when

only the kth mode is present, the complex power carried by the kth mode alone in the

forward direction is given by VmkI∗mk. The characteristic impedance of the kth mode

is ZCk≡ V0k/I0k = |V0k|2/P ∗

0k = P0k/|I0k|2; its magnitude is fixed by the choice of

|V0k| or |I0k| while its phase is fixed by (2.4.26).

With this definition, ZCkcorresponds to the ratio of the modal voltage to the modal

current in the line when only the kth mode is present. Then, a direct relationship

between the modal impedance matrix Zm and the characteristic impedance matrix ZC

is given by:

Zm = γZC =

Zm1 · · · 0...

. . ....

0. . . Zmn

=

γ1ZC1 · · · 0...

. . ....

0. . . γnZCn

(2.4.27)

In the following paragraphs the topics that represent limitations of quasi-TEM as-

sumptions for the MTLs and the evaluation of the model in the case of lossy structures

will be discussed.

40

2.5 Limitations of the quasi-TEM assumptions

One of the more important facts that defines the TEM structures for TEM propagat-

ing fields, is the assumption that the different modes propagating along the structure

are TEM or quasi-TEM and orthogonal. In high speed electronic circuits differ-

ent MTLs with lossy conductors that violate these assumptions are used; then, the

different modes propagating along the structure are composed by a set of TEM or

quasi-TEM orthogonal modes and a set of interdependent or coupled modes [7].

The total electric electric field E and magnetic field H in a closed, uniform and

isotropic MTL along z axis can be expressed as:

E =∑

n

c±n e±γnz(Etn ± Eznz) (2.5.1)

H =∑

n

c±n e±γnz(±Htn + Hznz) (2.5.2)

where c±n are the forward and reverse excitation coefficients of the nth mode, γn is

the nth modal propagation constant, its transversal modal electric and magnetic fields

Etn and Htn respectively, and its longitudinal modal electric and magnetic fields Ezn

and Hzn are only functions of the transverse coordinates x and y.

When only a finite number of the discrete modes are excited in the line, the total

complex power P is:

P =

∫E×H∗ · zdS =

∑nm

(c+n eγnz + c−n e−γnz)(c+

meγmz + c−me−γmz)∗Pnm (2.5.3)

and Pnm =

∫Etn ×H∗

tm · zdS (2.5.4)

where the sum is taken over all the excited modes, and the integrals are performed

over the transmission-line cross section. The power Pnm is called for n 6= m the modal

cross power.

41

Lossless modes are power orthogonal when they are not degenerate; that is, their

modal cross powers Pnm are zero when γ2n 6= γ2

m. Most equivalent circuit descriptions

for MTLs assume power orthogonal modes, that are congruent with quasi-TEM as-

sumptions. In this case the total power in the line can be calculated as a simple sum

of the powers carried by each pair of forward and backward modes, assuming that the

modal symmetries eliminate the possibility of existence for modal cross powers.

In the case of highly lossy lines, typical of modern circuits, losses develop degen-

eracies that permit the existence of modal cross powers Pnm and the total power in

the line can no longer be calculated as a simple sum of the powers carried by each

pair of forward and backward modes. Fache and De Zutter [17] have constructed an

equivalent circuit theory based on power-normalized conductor voltages and currents

that accounts rigorously for modal cross powers even when losses are large. The influ-

ence of modal cross powers for dominant quasi-TEM modes of asymmetrical coupled

transmission lines are large at useful frequencies and need to be taken into account

in thermal noise calculations as is remarked by [51].

In Ref. [50] the mechanisms and conditions that give rise to large modal cross

powers are discussed, and to evaluate the influence of modal cross powers in lossy

transmission lines, a merit coefficient ζnm was defined to quantify their significance:

ζnm =PnmPmn

PnnPmm

(2.5.5)

The Pnm fix relations between the modal and the power-normalized conductor

voltages and currents of [17] and can be determined from products of the matrices

relating those quantities. The unitless coefficient ζnm can be determined solely from

42

the power-normalized per unit length conductor impedance matrices Z = R + jωZ,

and admittance matrices Y = G + jωC of the MTL without the detailed knowledge

of how the modal and circuit quantities in the theory are normalized. Then, the

quantity ζnm is found from Z and Y by:

ζnm =[b(λm)T a(λn)][b(λn)T a(λm)]

[b(λn)T a(λn)][b(λm)T a(λm)](2.5.6)

where the superscript T signifies Hermitian adjoint (conjugate transpose) and

a(λm) and a(λn) are the eigenvectors of β = YZ with eigenvalues λn = γ2n and

λm = γ2m, and b(λm) and b(λn) are the eigenvectors of α = ZY with eigenvalues λn

and λm.

When the per-unit-length impedance and admittance matrices Z and Y are sym-

metric, then β = αT , where the superscript T signifies Hermitian adjoint (conjugate

transpose). This implies that b(λm)T a(λn) = b(λm)T a(λn) = 0, and it can be seen

from (2.5.6) that ζnm = 0 whenever the eigenvectors of α and β can be taken real.

The influence of limitations of quasi-TEM assumptions in lossy MTLs are taken

into account by evaluating the influence of the modal cross powers on the power-

normalized equivalent circuit parameters as is demonstrated by Williams et altri [8].

As is shown in [8], the power-normalization affects the definition of the similarity

transformation matrices Mv and Mi, by giving a condition for them that is directly

related with the modal cross powers.

A brief discussion follows to remark the effects of the above mentioned power-

normalization and the modal cross powers on the equivalent circuit parameters, that

is consistent with the modal representation of the present work.

43

The complex power P transmitted across a reference plane is given by the integral

of the Poynting vector over the MTL cross section S as:

P =

∫E×H∗ · zdS =

∑

j,k

Vmj(z)

V0j

I∗mk(z)

I∗0k

∫

S

Emk × H∗mk · zdS (2.5.7)

being Emk and Hmk the modal electric fields and magnetic field strengths. This

can be put into the more compact form:

P = IT

m ·X · Vm (2.5.8)

where the superscript T indicates the Hermitian adjoint (conjugate transpose) and

the cross-power matrix X is defined with its elements as:

Xkj =1

V0j I∗0k

·∫

S

Emk × H∗mk · zdS (2.5.9)

This cross-power matrix X takes into account the influence of all modes propa-

gating along the MTL, the orthogonal modes and the coupled modes, and as is shown

in [50], the off-diagonal elements of this matrix are often large in lossy quasi-TEM

MTLs near modal degeneracies. The diagonal elements of X are unitary as a re-

sult of the power-normalization (2.4.26). When the a conductor equivalent circuit

representation is given, the (2.5.8) can be written as:

P = IT(M−1

i )T ·X ·M−1v V (2.5.10)

44

In order to assign a node to each pair of conductor voltages and currents in the

conductor representation as shown in Fig. 2.9, the above power expression (2.5.10)

can be simplified by imposing the following restriction:

MTi Mv = X =⇒ P = I

T · V (2.5.11)

This gives a useful representation because it mimics that of the low-frequency

nodal equivalent-circuit theory where a node can be assigned to each pair of conductor

voltages and currents by finding that the power P flowing into any circuit element

corresponds exactly to that in the equivalent-circuit analogy.

The restriction (2.5.11) leaves the determination of either Mv or Mi open (but

not both). To determinate the similarity transformation matrices, the conductor

voltages can be fixed, for example, to correspond to the integral of the total electric

fields E along any given path lk between the conductors to which circuit elements are

connected by choosing the elements of Mv with:

Mvkj=−1

V0j

·∫

lk

Emk · dl ∀j =⇒ V0j =

∫

lk

E · dl (2.5.12)

Then Mi would be given by Mi = (XM−1v )T = (MT

v )XT . Another choice could

be used by defining first the Mi by fixing the conductor currents, and then Mv would

be determined from Mv = (MTi )−1X.

For lossless MTLs the matrix X is equal to the identity I and only orthogonal

modes are present and the pul conductor impedance and admittance Z and Y are

45

symmetric with the requirement that MtvMi becomes diagonal implying that Z = Zt

and Y = Yt (the superscript t indicates transpose matrix) [16]. The requirement

that MtvMi diagonal is not always compatible with the condition MT

i Mv = X as

discussed in [8]. But with high lossy lines this orthogonality is lose and X 6= I. Then

the product MtvMi is not diagonal in this case, and the pul conductor impedance and

admittance Z and Y are no longer symmetric.

All the above discussion remarks the fact of the influence of losses in a quasi-TEM

representation, and deviations of the model are taken into account. The modal and

conductor equivalent circuit representations depends on the modal cross powers that

are not present in the original MTL equations.

As a conclusion, these models can be corrected by using a proper definition of

the similarity transformation matrices Mv or Mi that takes into account modal cross

powers by assuming a cross-power matrix X. The above described phenomena can

be estimated by using a merit coefficient ζnm that can be calculated through the

measured pul conductor impedance and admittance Z and Y.

This important result provides a powerful instrument to evaluate the high lossy

lines behaviors and their divergencies from quasi-TEM assumptions through experi-

mental measurements.

Chapter 3

RF Instruments and Tools

3.1 Introduction

In this chapter a review of the characterization principles of two-port linear networks,

power waves and Scattering parameters representation will be given, together with a

description of the instruments and tools available to make microwave measurements.

The convenience of lumped circuits characterization compared to classical circuit

theory and their extension to transmission lines will be briefly presented. As will be

seen, for high frequency, many assumptions of lumped circuit theory are no longer

valid and another kind of representation needs to be used. The assumptions for the

scattering parameters representation will be presented for metrology.

Finally a description of the VNA Vector Network Analyzer system will be de-

scribed putting emphasis on the microwave metrology problematic. Measurement

error models, their physical causes and removal procedures will be presented and

discussed.

46

47

3.2 Characterization of linear networks

The theory of transmission lines is called distributed circuit analysis, and it is inter-

mediate between the low-frequency extreme of lumped circuits and the most general

field equations. Lumped circuit theory is associated with the following assumptions

and approximations:

• Physical size of the circuit is assumed to be much smaller than the wavelength

of the signals that exist therein (size of circuit is assumed < λ/8 )

• Practically there is no time delay between both voltages and currents at different

parts of the network. The applied voltage at one port is sensed immediately at

any other port.

• Since the largest dimension of the circuit is much smaller than the wavelength,

radiation is negligible.

• Energy stored between currents and charges at different points in the circuit

(stray inductance and capacitance) is assumed to be very small with respect to

the energy in the truly lumped elements. The stored energy in the region around

an element is predominantly electric or magnetic, and it changes from one form

being dominant to the other when the device goes through self-resonance. In-

ductors can only store magnetic energy, whereas capacitors only store electric

energy.

• Application of the Maxwell equation (∇ · J = −∂ρ/∂t = 0) for charge conser-

vation at nodes gives Kirchoff’s current law:∑

iκ = −∂q/∂t∣∣nodes

= 0.

48

Figure 3.1: Two Port Network Transmission line model

• Application of the Maxwell equation (∇ × E = −∂B/∂t = 0) at loops gives

Kirchoff’s voltage law:∑

vκ = −∂φ/∂t∣∣loops

= 0.

Under the above assumptions the transmission lines can be modelled as Two Port

Black boxes applying the circuit theory to describe its behavior. Normally the two

port matrices are used to characterize the transmission lines electrically. Typically

the Transmission T and ABCD matrices are used to express the transmission line

parameters as function of the propagation constant γ, the characteristic impedance

Zc and the physical length ` as is shown in Fig. 3.1 and the following equation:

[V2

I2

]=

[cosh(γ`) Zc · sinh(γ`)

Z−1c · sinh(γ`) cosh(γ`)

]·[

V1

I1

](3.2.1)

Then it is possible, by doing linear transformations, to describe the linear trans-

mission lines or waveguides with their impedance and/or admittance matrices.

49

The single mode solution of Telegrapher’s equation is considered in Fig. 3.1. From

this figure we can relate the modal waveguide voltage V and modal waveguide current

I with the travelling waves or forward intensities V +, I+ and the backward intensities

V −, I− (by assuming that z = 0 leftwards and increases rightwards) as follows:

V (z) = V +e−γz + V −eγz and I(z) = V +/Zc · e−γz − V −/Zc · eγz (3.2.2)

with the straightforward relationships

V1 = V (0) I1 = I(0)

V2 = V (`) I2 = I(`)

Then, the voltage, current and impedance magnitudes can be used to describe

their physical behavior. The Z impedance matrix describes the behavior of the linear

network by relating the two port magnitudes:

[V1

V2

]=

[Z11 Z12

Z21 Z22

]·[

I1

I2

](3.2.3)

In electric circuits the two port voltages and currents can normally to be measured

and characterization is a straightforward task in linear lumped circuit measurements.

The Telegrapher’s parameters R,G, L and C per-unit length are frequency indepen-

dent. Thus all linear theory can be applied to this mathematical description.

As will be seen in the next section, these parameters are not more constant when

frequency increases, mainly because the conductor losses due to the skin effect and

dielectric losses increase in high frequency. This linear approach will be valid in

microwave and millimeter frequencies only if the narrow frequency band is studied,

since this approach is not more valid in the wide frequency band where the wavelength

λ is comparable to the physical dimensions of waveguides.

50

3.3 Characterization problem in microwaves and

millimeter waves

Classical waveguide circuit theory proposes an analogy between an arbitrary linear

waveguide circuit and a linear lumped electrical circuit. The lumped electrical circuit

is described by an impedance matrix, which relates the normal electrical currents and

voltages at each of its terminals, or ports. The waveguide circuit theory likewise

defines an impedance matrix relating the waveguide voltage and waveguide current

at each port. In both cases, the characterization of a network is reduced to the

characterization of its circuit components.

The general conditions satisfied by the impedance matrix are different in the two

cases. The waveguide voltage and current are highly dependent on definition and

normalization, in contrast to linear electrical circuits. The waveguide circuits are

described by travelling waves, not as lumped electrical ones.

As described in the last chapter, classical waveguide circuit theory is based on

defined waveguide voltage and waveguide current ; indeed these definitions rely upon

the electromagnetic analysis of a single and uniform waveguide [7]. Solutions of

Telegrapher’s linear equation are the eigenfunctions of the electromagnetic boundary

conditions. These eigenfunctions correspond to waveguide modes which propagate in

either direction with an exponential dependence on the axial coordinate.

A basic assumption of waveguide theory circuit is that at each port, a pair of

identical waveguides must be joined without discontinuity and must transmit only

a single mode, or a finite number of modes. When limited to a single mode, the

field distribution is completely described by these complex numbers indicating the

51

complex intensity of these two opposite travelling waves [34]. The waveguide voltage

and current related to the electric and magnetic fields−→E and

−→H of the mode, are

linear combinations of the two travelling waves. This linear relationship is function

of the characteristic impedance of the mode. Telegrapher’s equation is derived under

the following assumptions [41]:

• The conductors are perfect (resistivity % = 0) with uniform cross sections along

the line axis.

• The dielectric medium is homogeneous and isotropic characterized by the same

electrical permitivity ε and magnetic permeability µ along the line axis.

conductor losses and inhomogeneity are not taken into account and only TEM

mode waves are propagated.

The above assumptions are not more valid in the case of a lossy conductor that in-

validate the TEM field structure assumption because the line current flowing through

an imperfect line conductor generates a non zero electric field along the conductor

surface. However if the conductor losses are small, this resulting field structure is

almost TEM. This is referred to as the quasi-TEM assumption and, although the

transmission-line equations are no longer valid, they are nevertheless assumed to rep-

resent the situation for small losses through the inclusion of the per-unit-resistance

parameter R.

Another situation that invalidates the TEM assumptions is the presence of inho-

mogeneous cross section media around conductors, as in microstrip lines and coplanar

waveguides. The field velocity of propagation will be different in two different media

characterized respectively by ε1, µo and ε2, µo. The classical way of characterizing this

52

situation is to obtain an effective dielectric constant εeff , defined such that if the line

conductors are immersed in a homogeneous dielectric with a dielectric constant εeff ,

the velocities of propagation and all other attributes of the solutions for the original

inhomogeneous case will be the same.

Other implicit assumptions in the TEM characterization transmission-line equa-

tion are taken into account for its derivation. The distribution of lumped elements

along the line with infinitesimal section length means that the line lengths are elec-

trically long, i.e., much greater than a wavelength λ, and they are properly handled

by lumped-circuit characterization. However structures whose cross-sectional dimen-

sions are electrically large at the frequency of excitation will have, in addition to the

TEM field modes, other higher-order TE and TM modes of propagation simultane-

ously with the TEM mode. Therefore, the solution of Telegrapher’s equation does not

give the complete solution in the range of frequencies where these non-TEM modes

coexist on the line.

As an example of the complexity of the problem of propagation in high frequency

and inhomogeneous media the classical study on propagation in microstrip line given

by Hasegawa [24] is mentioned, where the existence of three different fundamental

modes of propagation as a function of the product of Si substrate resistivity and

the frequency is demonstrated. These three modes can be classified as the dielectric

quasi-TEM mode, the skin-effect mode and the slow-wave mode. In this work not only

is the influence of Si substrate resistivity on the mode propagation demonstrated, but

also heuristic proof of the frequency-substrate resistivity product influence on the

propagation modes.

All the assumptions and examples above mentioned show that there is no general

53

theory that applies and describes propagation, and therefore the characterization of

electromagnetic waves on lossy and inhomogeneous waveguides, that are the major

involved phenomena in microwave and millimeter range of frequencies. Thus the

treatment and modelling are based on an engineering approach assuming heuristic

arguments founded on low-frequency circuit theory that only serves to have a ”rough

estimate” of the actual behavior of microwave waveguides in the conditions indicated.

From the experimental point of view it is not possible to measure the modal

waveguide intensities in the microwave and millimeter range of frequencies. The

travelling waves are associated with the concept of voltage and current along the line

and generally are not available for measurements. As will be seen in the next section,

another linear combination of the waveguide intensities, the power waves, will be

used because they are easily able measurable by commercial VNA’s. Through these

power waves the waveguide or transmission line can be characterized directly from

the measurements of the Scattering matrix parameters that are linearly related to

them. In next section the major results of Scattering matrix theory will be presented

and the concepts used for metrology.

54

3.4 Scattering parameters theory review

The waveguide voltages and currents are never well defined in waveguides. These

magnitudes represent the vector sum of all mode contributions into the waveguide

interconnections and further they are very difficult or impossible to measure by con-

ventional measurement instruments. Historically it was more common to measure

power relationships in microwave fields. Although is not very intuitive, the travelling

waves concept is more closely related to the voltage or current along the line than to

the power in a stationary state.

If a circuit which terminates a line at the far end does not match the characteristic

impedance of the line, even if the circuit has no source at all, we have to consider

two travelling waves in opposite directions along the line. This makes the power

calculation twice as complicated. For this reason, when the main interest is in the

power relation between various circuits in which the sources are uncorrelated, the

travelling waves are not considered as the best independent variables to use for the

analysis.

A different concept of waves was introduced by Kurokawa [28], the power waves.

This new approach is theoretically equivalent to the characterization with Z or Y

matrices but it is more convenient because:

• the voltage and currents can not be directly measured in high frequency

• the power waves can be measured by VNA’s

• for wide bandwidths it is easier to obtain matched loads than open or short

circuits that are necessary to define the Z or Y matrices.

55

Figure 3.2: Power Waves and Reference Planes interpretation

To clear the concept a brief discussion follows. By considering a n-port electro-

magnetic structure as shown in Fig. 3.2 the power waves ai and bi at the different

reference planes in each port junction i of the structure can be defined as follows:

ai =Vi + ZiIi

2√| <(Zi) |

bi =Vi − Z∗

i Ii

2√| <(Zi) |

(3.4.1)

and by inverting the system,

Vi =1√

| <(Zi) |· (Z∗

i ai + Zibi) Ii =1√

| <(Zi) |· (ai − bi) (3.4.2)

where Vi and Ii are the voltage and the current flowing into the ith port of a

junction and Zi is the impedance looking out from the ith port. The positive real

value is chosen for the square root in the denominators.

In this definition each port is described by a reference plane transverse to a uniform

56

lossless waveguide leading to the junction and this reference plane is located at a

sufficient distance from the junction for far-field conditions to apply. Thus there is a

single mode propagated across each reference plane. The reference planes need to be

sufficiently far from any cross-section change of the uniform waveguide for evanescent

modes excited by the change to have decreased to negligible proportions at the plane.

The propagated mode need not be the same at all ports so different interfaces can be

described in these terms.

The physical meaning of power waves is related to the exchangeable power of a

generator. For this purpose, let us consider the equivalent circuit of a linear generator

as shown in Fig. 3.3, in which Zi is the internal impedance and E0 is the open

circuit voltage of the generator or Thevenin voltage. By applying the maximum

power transference theorem, the maximum power PL into the load ZL is given when

ZL = Z∗i then:

PL|max = Pa =|E0|2

4<(Zi)with <(Zi) > 0 (3.4.3)

The maximum power is called the available power of the generator, and is the

maximum power that the generator can supply to the load. If <(Zi) < 0 the eq.

(3.4.3) represents the exchangeable power which is finite but is not equal to the

maximum power. The voltage at the generator terminals is given by Vi = E0 − ZiIi.

Replacing it in the definition of ai from the power waves given by (3.4.1) and taking

the square of the magnitude, we have:

|ai|2 =|E0|2

4<(Zi)(3.4.4)

57

Figure 3.3: Equivalent circuit of a linear generator

which means the same as the available power of the generator, and if E0 = 0 then

ai = 0. If we develop the expression |ai|2 − |bi|2 in the definition (3.4.1) we have:

<ViI∗i = |ai|2 − |bi|2 (3.4.5)

The left-hand side of (3.4.5) represents the actual power transferred from the

generator to the load in the case where <(Zi) > 0. The generator sends the power

|ai|2 towards a load, regardless of the load impedance. If the load is not matched,

then ZL 6= Z∗i and a part of the incident power is given by |bi|2 so that the net power

absorbed by the load is equal to |ai|2−|bi|2. Associated to these incident and reflected

powers, are waves ai and bi respectively. These waves are considered the incident and

reflected powers because there is a linear relation between a′is and b

′is and this can

be used to advantageously define the Scattering parameters.

To define the Scattering matrix of a linear n-port network we consider the vectors

a, b, V and I at the ith port of the network respectively. Then a and b can be written

58

in terms of V and I as follows:

a = F (V + G · I), b = F (V + G∗T · I) (3.4.6)

Where ∗T denotes conjugate transposed matrix and:

F = diag(1/

√| <(Zi) |

), G = diag(Zi) (3.4.7)

Using the linear relation between V and I given by V = Z.I where Z is the

impedance matrix, the linear combination between a and b, the Scattering matrix S

is given by:

b = S · a (3.4.8)

and then, by equating (3.4.6),(3.4.7) and (3.4.8) we obtain the following relations

for S and Z matrices:

S = F (Z −G∗T )(Z + G)−1F−1

Z = F−1(I − S)−1(SG + G∗T )F(3.4.9)

Although the above definitions are related to the Zi impedance, this impedance

can be arbitrary, giving a different linear combination between a and b for different

impedances Zi. Therefore the Scattering matrix S is referred to a Zref reference

impedance by changing Zi by Zref in definition (3.4.1).

If we take an n-port structure as transmission line with characteristic impedance

Zci, from equation (3.2.2) we can express the power in the ith line as:

Pi = <ViI∗i = |V +

i |2/Zci − |V −i |2/Zci (3.4.10)

and by taking Zref = Zci in each ith port junction we can associate the power

waves with travelling waves in each ith junction as follows:

ai =V +

i√| <(Zci) |

bi =V −

i√| <(Zci) |

(3.4.11)

59

the total power into each ith port junction becomes:

Pi = |ai|2 − |bi|2 (3.4.12)

The equivalence between the Z and S matrices that fully characterize a n-port

electromagnetic structure was developed by starting from a power point of view.

A more modern definition of Scattering parameters is given by Marks and Williams

[34]. This work demonstrates that power waves as defined by (3.2.2) have no physical

meaning which the travelling waves do and they are only mathematical artifacts. The

power waves are equivalent to travelling waves only when the reference impedance Zref

is equal to the characteristic impedance Zi of the mode in each port.

When Zi varies greatly with frequency, as is often the case in lossy lines, the

resulting measurements using Zref = Zi may be difficult to interpret and a Zref = cte

is preferred. If Zref is chosen to be real, the power is given by (3.4.12). This is the

normal definition that is used in commercial VNAs, and as will be seen in the next

chapter, Zref is defined by a calibration process.

60

3.5 The Vector Network Analyzer

Vector Network Analyzer (VNA) system is intended as a complex electronic apparatus

that consents acquisition, management and presentation of data related to microwave

structures. Magnitude and phase characteristics of networks and components such as

filters, amplifiers, attenuators, and antennas are measured by VNAs.

Measurement of four scattering parameters is made by separating the incident

and reflected waves at two ports, Port 1 and Port 2, and then converting them to low

frequency to be sampled by the microprocessor system. Sampling and treatment of

the information is performed by an internal microprocessor system in the instrument’s

control unit that obtains data presentation and computes the numeric process. VNA

may be commanded by a remote computer through an external GPIB databus for

automatic instrumentation.

3.5.1 VNA General Description

A fully integrated Vector Analyzer System HP8510C [47] as used in this work is

composed by:

Source provides the RF signal. It is a Synthesized Sweeper.

Test Set separates the signal produced by the source into an incident signal, sent

to the device-under-test (DUT), and a reference signal against which the trans-

mitted and reflected signals are later compared. The test set also routes the

transmitted and reflected signals from the DUT to the receiver (IF/detector).

Command Unit includes the Display/Processor and the IF/Detector or Receiver.

The Receiver, together with the Display/Processor, processes the signals. Using

61

Figure 3.4: HP8510 Block Diagram (Agilent Technologies 2001)

its integral microprocessor, it performs accuracy enhancement and displays the

results in a variety of formats.

Peripherals system components that include peripheral devices such as a printer, a

plotter, and a disc drive. Measurement results and other kinds of information

can be sent to peripherals.

A Vector Network Analyzer simplified block diagram is presented in Fig. 3.4

3.5.2 Signal Source

In a measurement, the signal source is swept from the lower measurement frequency

to the higher measurement frequency using a linear ramp controlled by the VNA.

62

This sweep is called a ramp sweep and it gives the fastest update of the measurement

display. In step-sweep mode, the source is phase-locked at each discrete measurement

frequency controlled by the VNA. Useful bandwidths from 45 Mhz to 26 Ghz are

available and they can be enhanced to 40 Ghz.

Sweeper scan range and frequency can be selected by the VNA system bus. Dis-

crete frequencies measured range from 51 up to 801 samples in a single sweep. Fre-

quency resolution goes from 1 Hz at low frequency to 4 Hz at 26 Ghz. The sweeper

has an internal frequency reference and can be used an external reference as well.

Frequency resolution depends on the frequency reference used by.

3.5.3 Test Set

Test Set is the key component of the system and is designed to avoid the need to

reverse connections to the DUT when a reversed signal flow is required. Each test set

provides the following features:

• Input and output ports for connecting the device to test

• Signal separation for sampling the reference signal and test signals

• Test signal frequency to 20 MHz conversion

These functions can be integrated in a unique device, the S-parameter test set

that is connected to the Command Unit by an internal bus and to the Synthesized

Sweeper by coaxial connectors. Internally it has direct couplers, deviators, etc, that

are necessary to measure the scattering parameters and a frequency converter to

provide the signal frequency conversion to 20 Mhz.

63

Figure 3.5: HP8511 S-Parameter Test Set (Agilent Technologies 2001)

S-parameter test sets as shown in Fig. 3.5 provide automatic selection of S11,

S12, S21, and S22. The stimulus is automatically switched for forward and reverse

measurements. This capability allows for fully error-corrected measurements on one-

port devices and two-port devices without needing to manually reverse the DUT.

By taking the ratio after electronic switching, switching path repeatability errors are

eliminated. The bias input and sense connections provided allow the testing of active

devices.

Internal 10 dB steps attenuators (from 0 dB to 90 dB) are available to control the

incident stimulus level at the DUT input, without causing a change in the reference

signal level. As deviator a diode network that gives a good isolation (= 80 dB) and

providing fast switching is used. Bias Tee are provided for DC Biasing in active

devices.

64

Figure 3.6: HP8511A Frequency Converter (Agilent Technologies 2001)

A frequency converter block diagram is shown in Fig. 3.6 and its conversion

procedure is performed by four diode samplers excited contemporarily by an impulse

burst whose frequency is selected to convert to 20 Mhz all amplitude and phase

characteristics of the four RF signals. One of the four signals becomes the reference

channel and the other three signals are phaselocked to it.

3.5.4 Command Unit

The Command Unit automatically manages all functions of the instrument. It re-

ceives the four 20 Mhz channels from the Test Set unit, and provides another hetero-

dyne down-conversion to 100 Khz for application in the detection and data processing

elements of the receiver.

Magnitude and phase relationships between the input signals are maintained

throughout the frequency conversion and detection stages, because the frequency

65

Figure 3.7: HP8510 DSP Block Diagram (Agilent Technologies 2001)

conversions are phase-coherent and the IF signal paths are carefully matched. Each

synchronous detector develops the real (X) and imaginary (Y) values of the reference,

or test signal, by comparing the input with an internally generated 100 kHz sine wave.

This method practically eliminates measurement uncertainty errors resulting from

drift offsets, and circularity. Each X,Y data pair is sequentially converted to digital

values and read by the central processing unit CPU. Accuracy of sampled data is

given by a 19 bit analog to digital conversion.

Digital data processing is performed by the CPU and a Math dedicated micropro-

cessor. Multiple operations, analysis, and data display presentation can be produced.

When error correction is selected, the raw data and error coefficients from the selected

calibration coefficient set are used in appropriate computations by a dedicated vector

math processor.

66

Corrected data are represented in time domain by converting from the frequency

domain to time domain using the inverse Fourier Chirp-Z transform technique. A

dedicated display processor asynchronously converts the formatted data for viewing

at a flicker-free rate on the vector-writing display. A block diagram of the VNA

Digital Signal Processing DSP is shown in Fig. 3.7.

3.6 Systematic error removal and VNA calibration

Vector Network Analyzers (VNA) find very wide application as primary tools in

measuring and characterizing circuits, devices and components. At higher frequencies

measurements pose significantly more difficulties in calibrating the instrumentation

to yield accurate results with respect to a known or desired electrical reference plane.

Characterization of many microwave components is difficult since the devices can-

not easily be connected directly to VNA-supporting coaxial or waveguide media. Of-

ten, the device under test (DUT) is fabricated in a non coaxial or waveguide medium

and thus requires fixturing and additional cabling to enable an electrical connection

to the VNA.

The point at which the DUT connects with the measurement system is defined as

the DUT reference plane and is the point where it is desired that measurements be

referenced. However, any measurement includes not only the DUT, but contributions

from the fixture and cables as well.

By increasing frequency, the electrical contribution of the fixture and cables be-

comes increasingly significant. In addition, practical limitations of the VNA in the

form of limited dynamic range, isolation, imperfect source/load match, and other

imperfections contribute to systematic errors of measurements.

67

A perfect measurement system would have infinite dynamic range, isolation, and

directivity characteristics, no impedance mismatches in any part of the test setup,

and flat frequency response. In practice, this ”perfect” network analyzer is achieved

by measuring the magnitude and phase of known standard devices, using this data

in conjunction with a model of the measurement system to determine error contri-

butions, then measuring a test device and using vector mathematics to compute the

actual response by removing the error terms.

The dynamic range and accuracy of the measurement is then limited by system

noise and the accuracy to which the characteristics of the calibration standards are

known. The following paragraphs describe the source of measurement errors, error

model definitions and error correction.

3.6.1 Measurement Errors

Network analysis measurement errors can be separated into three categories:

• Systematic Errors

• Random Errors

• Drift Errors

Drift errors can be compensated by an accurate project of the electronic and

mechanical parts of the systems and are minimized by a warm up period before to

start a measurement. Random errors are non-repeatable measurement variations

due to factors like system noise, connector repeatability, temperature variations, and

other environment and physical changes in the test setup between the calibration and

the measurement. These errors cannot be modelled and measured with an acceptable

68

degree or certainty, they are unpredictable and therefore cannot be removed from the

measurement, and produce a cumulative ambiguity in the measured data.

Systematic errors are repeatable and arise from imperfections within the VNA.

They include mismatch and leakage terms in the test setup, isolation characteristics

between the reference and test signal paths, and system frequency response. These

errors are the most significant at RF and microwave frequencies and they can be

largely removed by a calibration process. Causes of these errors are very complex and

they will be not discussed here. A full treatment of them is given in [43][47].

Such errors are quantified by measuring characteristics of known devices or stan-

dards. Hence systematic errors can be removed from the resulting measurement. The

choice of calibration standards is not necessarily unique. Selection of a suitable set of

standards is often based on such factors as ease of fabrication in a particular medium,

repeatability, and the accuracy with which the characteristics or the standard can be

determined.

The Systematic error correction process can be divided in:

• Error Model Definition

• Calibration Process

• Measurement of DUT and Error Correction or Deembedding

Error Models can be defined by their causes within the measurement instrument

or through a black box approach. The calibration process involves the actions needed

to identify correctly the error model parameters. Calibration is fully dependent on

the error model and on the number of parameters to be identified.

69

In the frequency domain, all known calibration techniques are based on the inser-

tion of standards or devices with well known electrical behavior on the place of DUT.

Measurement of standards gives the calculation of error model parameters. These

coefficients can be stored into a computer memory or into the VNA firmware to be

used to correct the DUT raw measurements mathematically within a deembedding

process.

In the following paragraphs the two most important approaches of Error Model

definitions will be discussed [43]. Historically the Twelve Terms Error Model is the

best known and it is the Error Model used internally by the VNA. The Error Box

Model approach was developed over the last two decades and it gives a more physical

meaning for the deembedding process, it also permits new and more accurate cali-

bration techniques to be followed by a computer outside the VNA. The last model

presented was adopted in this work and will be discussed with more attention. First

the Twelve Terms Error Model will be presented.

70

3.6.2 Twelve Terms Error Model

Historically the Twelve Terms Error Model was developed from the causes of mea-

surement uncertainties. They can be classified in the following categories:

• Directivity.

• Source Match.

• Load Match.

• Isolation.

• Tracking.

Directivity error is mainly due to the inability of the signal separation device

to absolutely separate incident and reflected waves. Residual reflection effects of

test cables and adapters give their contribution too in this uncertainty. Reflection

measurements are most affected by this error.

Source Match error is given by the inability of the source to maintain absolute

constant power at the test device input and by cable and adapter mismatches and

losses. This error is dependent on the relationship between input impedance of the

device under test DUT and the equivalent match of the source. It affects both trans-

mission and reflection measurements.

Load Match error is due to the effects of impedance mismatches between DUT

output port and the VNA test input. It is dependent on the relationship between the

output impedance of the DUT and the effective match of the VNA return port. It

affects both transmission and reflection measurements.

71

Isolation error is due to crosstalk of the reference and test signal paths, and

signal leakage within both RF and IF sections of the receiver. It affects high loss

transmission measurements.

Tracking error is the vector sum of frequency response, signal separation device,

test cables and adapters, and variations in frequency response between the reference

and test signal paths. It affects both transmission and reflection measurements.

The VNA provides different possibilities to measure these errors and they are de-

veloped in the literature [47][43]. The Full 2-Port Error Model or Twelve Terms Error

Model that provides full directivity, source match, load match , isolation and tracking

error correction for transmission and reflection measurements will be presented.

This model provides measurement accuracy for two-port devices requiring the

measurement of all four S parameters of the two-port device. There are two sets of

error terms, forward and reverse, with each set consisting of six error terms. Error

terms are the following:

• Forward Directivity EDF and Reverse Directivity EDR

• Forward Source Match ESF and Reverse Source Match ESR.

• Forward Load Match ELF and Reverse Load Match ELR.

• Forward Isolation EXF and Reverse Isolation EXR.

• Forward Reflection Tracking ERF and Reverse Reflection Tracking ERR.

• Forward Transmission Tracking ETF and Reverse Transmission Tracking ETR.

Twelve Terms Error Model Forward Set is shown in Fig. 3.8 and Reverse Set is

shown in Fig. 3.9.

72

SijA represent the actual DUT S-parameters and SijM are the measured S-

parameters . After a Calibration process the twelve error terms are calculated and

actual DUT parameters are given by the Error Correction Deembedding equations :

S11A =

[(S11M−EDF

ERF

).

[1+

(S22M−EDR

ERR

).ESR

]]−[(

S21M−EXFETF

).(

S12M−EXRETR

).ELF

]

(Deno)

S21A =

[1+

(S22M−EDR

ERR

).(

ESR−ELF

)].(

S21M−EXFETF

)(Deno)

S12A =

[1+

(S11M−EDF

ERF

).(

ESF−ELR

)].(

S12M−EXRETR

)(Deno)

S22A =

[(S22M−EDR

ERR

).

[1+

(S11M−EDF

ERF

).ESF

]]−[(

S21M−EXFETF

).(

S12M−EXRETR

).ELR

]

(Deno)

(3.6.1)

Deno =

[1 +

(S11M − EDF

ERF

).ESF

].

[1 +

(S22M − EDR

ERR

).ESR

]−

−[1 +

(S21M − EXF

ETF

).(S12M − EXR

ETR

).ELF .ELR

](3.6.2)

73

Figure 3.8: Twelve Terms Error Model Forward Set

Figure 3.9: Twelve Terms Error Model Reverse Set

74

3.6.3 Error Box Model (Eight-Term Error Model)

The most modern formulation of measurement errors is the physical model of system-

atic errors. The concept is based in a Ideal Free Error VNA, connected to the D.U.T

through two ”black boxes”, the Error Boxes A and B where all measurement errors

are concentrated. This concept permits a more systemic vision and error treatment

by becoming independent of their actual causes.

An Ideal Free Error VNA and two fictitious networks named Error Boxes define

the measurement system as shown in Fig. 3.10. The Error Boxes A and B take

into account the systematic error for the two ports in the measurements. Port A

and Port B represent the measurement reference planes, the error boxes contain the

contribution of the systematic errors and Port 1 and Port 2 represent the ideal error

free ports of the network analyzer.

Two basic hypotesis are assumed to define the Error Box Model : the isolation of

the ports and the linearity of the relation between the waves at each port. Isolation

of the ports is intended that the measured waves at each port depend only upon

the real waves at the same port, hence the signal path between the measured waves

a1m, b1m, a2m and b2m lays only inside the D.U.T., and not inside the test set (see

Fig. 3.11). This is quite reasonable and offers a dramatic simplification for the

calibration process. The majority of the calibrations algorithms known are based

on this assumption. Linearity allows to describe the model with standard two-port

parameters by indicating a straightforward relation between all magnitudes.

Based on the Error Box Model shown in Fig. 3.11, a mathematical description

that uses matrix notation is given. They can be defined as the error box S-matrices

EA and EB :

75

Figure 3.10: Ideal Free Error VNA and Error Boxes

[b1m

a1

]= EA ·

[a1m

b1

], with EA =

[e00

A e01A

e10A e11

A

](3.6.3)

[b2m

a2

]= EB ·

[a2m

b2

], with EB =

[e00

B e01B

e10B e11

B

](3.6.4)

In matrix notation, relationships at each port can be written as follows:

[b1m

a1m

]= Ta ·

[b1

a1

]and

[a2m

b2m

]= Tb ·

[a2

b2

](3.6.5)

The Ta is the Error Box A cascading matrix from left to right following the

signal path from Port 1 to Port 2, and Tb is the Error Box B transmission matrix

76

Figure 3.11: An interpretation of the Error Box Model

from right to left from Port 2 to D.U.T. Relationships between error box S-matrices

parameters and cascading and transmission matrices are given by:

Ta =1

e10A

·[−∆A e00

A

−e11A 1

]≡ 1

e10A

·Xa with ∆A = e00A · e11

A − e01A · e10

A (3.6.6)

Tb =1

e10B

·[

1 −e11B

e00B −∆B

]≡ 1

e10B

·Xb with ∆B = e00B · e11

B − e01B · e10

B (3.6.7)

The relationship between D.U.T. and error box parameters is given by equating

the measured and actual power waves through the matrix description of the model

as is shown in Fig. 3.11. The chain of matrices Tm represents the raw measurement

and is given by the following equation:

Tm = Ta · Td ·(Tb

)−1(3.6.8)

77

Thus the deembedding formula that gives Td, the D.U.T. cascading matrix , is obtained

just by inverting (3.6.8) as is shown:

Td =(Ta

)−1 · Tm · Tb = α−1 · (Xa

)−1 · Tm ·Xb (3.6.9)

with

α =e10

B

e10A

(3.6.10)

As can be seen from the above measurement system definition, the eight error terms

are totally defined by the parameters of the Error boxes A and B.

A different notation, as was presented by Ferrero in [19][20], will be used in this

work to describe error boxes in calibration algorithms. It is presented here by rewrit-

ing terms of the error box matrices and the deembedding formula as follows:

Ta = p ·Xa = p ·[

kp· a b

kp

1

], Tb = w ·Xb = w ·

[1 u

w

f uw· g

](3.6.11)

Td = α−1 · (Xa

)−1 · Tm ·Xb with α =p

w(3.6.12)

Chapter 4

Microwave and Millimiter WaveMeasurement Techniques

4.1 Introduction

In this chapter a calibration process will be defined and the more relevant calibration

techniques will be presented and discussed. To understand the calibration problem,

different techniques based in the Error Models definitions will be discussed. Limi-

tations of the different techniques will conduce to use them in diverse environments

(coaxial, microstrip lines, etc.). The deembedding process as the major characteriza-

tion procedure after a calibrated measurement will be presented in all cases.

4.2 VNA Calibration process

VNA Calibration process is intended as the actions needed to determine correctly

the numerical values of all the error model parameters at each frequency of interest.

This process is fully dependent upon the Error Model and the number of parameters

to identify.

78

79

Calibration techniques in frequency domain are based on the insertion of stan-

dard devices, with well known electrical characteristics, at the place of the D.U.T.

The measurement of these standard devices permits the identification of Error Model

parameters. These coefficients can be stored in the instrument’s memory or a re-

mote computer to be used to correct raw measurements through vector mathematics.

Modern VNAs are able to correct raw measurements in real time with a calibration

technique that is in the instrument’s firmware. Practical procedures are explained in

the HP 8510C Programmer’s Handbook [47].

Calibration techniques can be divided in two categories:

• Non redundant methods

• Redundant methods or self calibration

Non redundant methods are used where uncertainties about standard devices are

not admitted. These methods are based on the connection of well known standard

device fabricated specifically and grouped into Calibration Kits. There are different

Calibration Kits with standards as Short, Open, Thru, Line and Match; fabricated

in different technologies that are used in VNAs as coaxial, microstrip line, etc. The

best known non redundant method is SOLT and it is implemented in the commercial

VNA’s firmware.

Self calibration is based on system redundancy where not all parameters of stan-

dard devices need to be known because the number of independent measurements is

greater than number of parameters to be identified. Some electrical characteristics

of standard devices are found from the solution of the calibration process. Different

methods where developed, the most important is the TRL invented by Bianco et al

80

[4], with developments added by Engen and Hoer [11], Speciale [45],[46] and others;

the LRM developed by Eul and Schieck [13], and the modern UTHRU by Ferrero and

Pisani [19]. The following paragraphs will describe the more important calibration

techniques, their field of use and differences between them in terms of accuracy.

4.3 Non Redundant Methods

4.3.1 SOLT Calibration Technique

SOLT (Short-Open-Load-Through) is the earlier calibration technique and it is a pro-

cedure to calculate the Twelve Terms Error Model. Although fabrication techniques

favor SOLT standards in coaxial, it is difficult to implement them precisely in other

media such as microstrip and coplanar. So this calibration technique is suited to be

used with coaxial media. Known standards are short, open, load and through . There

are two kinds of measurements to determine the error terms: 1 - Port or reflection

measurement, and 2 - Port or transmission measurement.

In 1-Port measurements at Port 1 and Port 2 the Directivity, Source Match and

Reflection Tracking errors of backward and forward error models can be determined.

Standards used are a Short, an Open and a Matched Load. If D.U.T. is connected

to Port 1 EDF , ESF and ERF can be determined, instead if it is connected to Port

2 EDR, ESR and ERR can be determined. In Fig. 4.1 the 1 - Port Error model is

shown.

In the above model S11M is the measured reflection coefficient and S11A is the

actual one at Port 1. The relationship between them is given by Mason’s Rules as:

S11M = EDF +S11A · ERF

1− ESF · S11A

(4.3.1)

81

Figure 4.1: 1 - Port Error Model (Port 1)

By connecting standards with reflection coefficients as:

• known Short

• known Open

• known Load

it is possible to obtain a 3 equation system from (4.3.1) and to calculate EDF ,

ESF and ERF . Connecting the standards to Port 2 we have a similar 1 - Port model

as it is shown in Fig. 4.1. It is possible to calculate the error terms EDR, ESR and

ERR with the same assumptions as in Port 1 by the following equation:

S22M = EDR +S22A · ERR

1− ESR · S22A

(4.3.2)

In a 2 - Port measurement, connecting the source at Port 1 and the standard

through (Thru) between the two ports it is possible to determine ETF for the forward

case, and doing the same with the source at Port 2 ETR is obtained. Measured and

actual transmission coefficients are equated by:

82

S21M = S21A · ETF S12M = S12A · ETR (4.3.3)

Isolation terms EXF and EXR are measured by connecting as terminations two

loads at two ports and by placing them at the points at which the D.U.T. will be

connected. Then, with a transmission configuration, the isolation error coefficients

are measured. These terms are the part of incident wave that appears at the receiver

detectors without actually passing through the D.U.T.

Ideal standards with reflection coefficients like Γshort = −1, Γopen = 1 and Γload =

0, and transmission coefficients S21thru = 1 are impossible to achieve. Specially with

increasing frequency it is impossible to fabricate lossless standards and they will ex-

hibit differences from ideal behavior. Effects such as a nonzero length of transmission

line associated with each standard are acknowledged. If the electrical length of the

transmission line associated with the standards is short, losses become small and

attenuation α can be neglected without a significant degradation accuracy. Alterna-

tively, commercial VNAs describe transmission lines in terms of a delay coefficient

with a small resistive loss component. The open standard exhibits further imperfec-

tions and is often described in terms of a frequency-dependent fringing capacitance

expressed as a polynomial expansion. Standard models need to be provided by cali-

bration kits manufacturers.

SOLT Calibration accuracy is rigidly connected to standards behavior. Systematic

errors are removed by deembedding using equation (3.2.1) from the Twelve Terms

Error Model . Uncertainty of measurement is given by a residual systematic error

as non-ideal switching repeatability (switching error), non-infinite dynamic range,

cables stability and by casual errors.

83

4.3.2 QSOLT Calibration Technique

An improvement for the SOLT calibration technique was invented by Pisani and Fer-

rero [18], the QSOLT. This new procedure permits to take only a 1-Port measurement

by compared with the two 1-Port measurements taken in SOLT. A global accuracy

improvement is achieved by reducing the total number of necessary standards. Influ-

ence of uncertainties in standard model definitions can be reduced, by reaching more

repeatable and precise measurements. This technique is a procedure to calculate the

Error Box Model terms. By using the model shown in the Fig. 3.11 and rewriting

equations (3.6.6) and (3.6.7) in a convenient way, the mathematical description of

this solution is given by:

Ta = e01A · 1

t11

·[−∆A e00

A

−e11A 1

]≡ 1

e10A

·Xa (4.3.4)

Tb = e01A · 1

t12

·[

1 −e11B

e00B −∆B

]≡ 1

e10B

·Xb (4.3.5)

with ∆A = e00A · e11

A − e01A · e10

A and ∆B = e00B · e11

B − e01B · e10

B

Where the T Matrix coefficients are expressed as follows:

t11 = e01A e10

A , t12 = e01A e10

B , t21 = e10A e01

B , t22 = e01B e10

B (4.3.6)

with t22 = e01B · e10

B = t21 · t12 · t−111 (4.3.7)

84

Figure 4.2: Ideal VNA and Error Box (Port 1)

Considering a 1-Port measurement as in SOLT but only in one port, Port 1 (Port

2), as indicated in Fig. 4.2, the two ports scheme is reduced to an ideal VNA followed

by an Error Box EA. It is demonstrated [18] that it is not necessary to know all four

Error Box parameters but only three: e00A , e11

A and the product t11 = e10A · e01

A . The

following relationship is given between the measured Γm and the actual Γa standard

reflection coefficients:

Γm = e00A +

e10A · e01

A · Γa

1− e11A · Γa

(4.3.8)

then, by connecting three known standards: short, open and load as in SOLT, it

is possible to have 3 independent equations and to calculate the desired error terms

e00A , e11

A and t11.

QSOLT measures a standard Thru in a 2-Port measurement with a known Tat

transmission matrix. By replacing expressions (4.3.4) and (4.3.5) into equation (3.6.8),

the relationship between the measured (subindex tm) and known Thru matrices with

Error Box terms are found to be:

Ttm = Xa · Tat ·X−1b (4.3.9)

85

then, because Xa was fully defined by the 1-Port measurement, the Error Box

transmission matrix Xb is determined by inverting (4.3.9) as:

Xb = T−1tm ·Xa · Tat (4.3.10)

The Xb Error Box terms are calculated with the following formulae:

e00B = X21

b · (X11b )−1

e11B = −X12

b · (X11b )−1

t22 = det(Xb) · (X11b )−2

t12 = (X11b )−1

t21 = t11 · det(Xb) · (X11b )−1

(4.3.11)

If an ”ideal” Thru (quasi ideal for typical applications as S21thru = S21thru ≈ 1) is

used as two-port device, the following equations apply:

S11tm = e00A + (t11 · e11

B ) · (1− e11A · e11

B )−1

S21tm = t21 · (1− e11A · e11

B )−1

S12tm = t12 · (1− e11A · e11

B )−1

S22tm = e00B + (t22 · e11

A ) · (1− e11A · e11

B )−1

(4.3.12)

Equating (4.3.7) with the above equation system (4.3.12) the Xb Error Box coef-

ficients are encountered:

e11B = (S11tm − e00

A ) · [t11 + e11A · (S11tm − e00

A )]−1

t21 = S21tm · (1− e11A · e11

B )

t12 = S12tm · (1− e11A · e11

B )

t22 = S21tm · S12tm · (1− e11A · e11

B )2 · t−111

e00B = S22tm − t22 · e11

A · (1− e11A · e11

B )−1

(4.3.13)

86

The QSOLT improvement is the reduction of the number of standards to be con-

nected from 7 to 4 without the need to take a Port 2 (Port 1) reflection measurement,

achieving more accuracy and reducing influence of uncertainties. This technique is not

implemented in the VNA firmware and needs to be performed on a remote computer.

87

4.4 Self Calibration or Redundant Methods

4.4.1 TRL technique

TRL (Thru Reflect Line) was invented by Bianco et al [4] and developed by Engen

and Hoer [11] as an improvement of TSD [45]. This technique is used to calculate the

terms of the Error Box model as was presented in Fig. 3.11. This solution is based

upon the measurement of a device in each of the two ports and two bilateral devices

connected between the ports:

• Thru: a piece of line with known length and characteristic impedance connected

to the two ports. Typically a zero length thru with an identity transmission

matrix is assumed.

• Reflect : a load (typically a piece of line opened or shorted) from which it is

only necessary to know the sign (phase) of its reflection coefficient within the

measurement frequency bandwidth. This device is alternatively connected to

Port 1 and Port 2.

• Line: a piece of line with the same characteristic impedance as the Thru but

with different length.

The goal of this solution is that it doesn’t rely on fully known standards and

it uses only three simple connections to completely characterize the error model.

The major problem in non-coaxial media is to separate the transmission medium

effects from the device characteristics. The accuracy of this measurement depends

on the quality of calibration standards. TRL calibration accuracy relies only on

the characteristic impedance of a short transmission line, and for this reason this

88

technique can be applied in dispersive media such as microstrip, coplanar, waveguide,

etc. TRL currently provides the highest accuracy in coaxial measurements available

today. The key advantages by using transmission lines as reference standards are:

a. transmission lines are among the simplest elements to realize in many non-

coaxial media, b. the impedance of transmission lines can be accurately determined

from physical dimensions and materials. Finally the TRL Calibration is the unique

technique that gives the propagation constant γ as a direct result of it. This is the

reason why is widely used to determine transmission line parameters.

Mathematics associated with this solution is based on matrix transmission repre-

sentation as was pointed out in formulae (3.6.6), (3.6.7), (3.6.11) and (3.6.12).

By measuring the Thru and the Line in 2-Port measurements and using (3.6.7),

we obtain:

TmT = Ta · TT ·(Tb

)−1(4.4.1)

TmL = Ta · TL ·(Tb

)−1(4.4.2)

where TmT and TT are the measured and actual Thru transmission matrices; and TmL

and TL the measured and actual Line transmission matrices

By properly equating (4.4.1) and (4.4.2) we have:

RM = TmL.(TmT )−1

= Ta.TL.(Tb)−1[Ta.TT .(Tb)

−1]−1

= Ta.TL.(TT )−1.(Ta)−1

= Ta.RT .(Ta)−1

(4.4.3)

89

andRN = (TmT )−1.TmL

= [Ta.TT .(Tb)−1]−1Ta.TL.(Tb)

−1

= Tb.(TT )−1.TL.(Tb)−1

= Tb.RS.(Tb)−1

(4.4.4)

Matrices RM and RT have the same eigenvalues as RN and RS given by the

following eigenvalue matrix:

Λ =

[λ1 0

0 λ2

](4.4.5)

The RM , RT , RN , RS eigenvector matrices are given by M , T , N and S respec-

tively, then it follows:

RM = M.Λ.M−1 = Ta.T.Λ.(T−1a .T−1) with Ta = M · T−1 (4.4.6)

RN = N.Λ.N−1 = Tb.S.Λ.(T−1b .S−1) with Tb = N · T−1 (4.4.7)

The Line transmission matrix with a length ` and propagation constant γ is given

by:

TL =

[e−γ.` 0

0 e+γ.`

](4.4.8)

By replacing the actual Thru and Line transmission matrices TL and TT in equa-

tion (4.4.3) we have:

RT = TL.T−1T = T.Λ.T−1 =

[e−γ.∆` 0

0 e+γ.∆`

]= Λ (4.4.9)

90

with ∆` = `line − `thru

Similar reasoning applies to eq. (4.4.4) with the same result for RN . Since matrices

RT = RS = Λ are diagonal their eigenvector matrices are equal to identity matrix

T = S = I, then:

Ta = M = p ·Xa = p ·[

a · k/p b

k/p 1

](4.4.10)

Tb = N = w ·Xb = w ·[

1 u/w

f g · u/w

](4.4.11)

The columns of Ta and Tb are the eigenvectors of RM and RN respectively. The

entities a, b, f and g are elements of the normalized eigenvectors. By a knowledge of

the length ∆` and from (4.4.9), the eigenvalues of RM and RN are given by [20]:

λ1 = e−γ.∆` λ2 = eγ.∆` (4.4.12)

being solutions of the characteristic equation of RM (RN):

λ1,2 =1

2·[RM11 + RM22 ±

√4.RM12RM21 + (RM11 −RM22)2

](4.4.13)

The normalized eigenvectors of RM and RN are computed as [30]:

a =RM12

λ1 −RM11

b =RM12

λ2 −RM11

(4.4.14)

91

f =λ1 −RN11

RN12

g =λ2 −RN11

RN12

(4.4.15)

From the measurement of the Reflect Γa at Port 1 (Γm1) and Port 2 (Γm2) the

following relationships are provided:

Γm1 =b + a · Γa · k/p

1 + Γa · k/pΓm2 =

f + g · Γa · u/w

1 + Γa · u/w(4.4.16)

The measured Thru input reflection coefficient SmT11 gives the following equation:

SmT11 =b− a · k/p · u/w

1− k/p · u/w(4.4.17)

By combining equations (4.4.16) and (4.4.17) the TRL algorithm calculates the

actual reflection coefficient Γa of the Reflect as follows:

Γa = ±√

(b− Γm1).(f − Γm2).(SmT11 − a)

(a− Γm1).(g − Γm2).(SmT11 − b)(4.4.18)

Reflect cannot be matched (Γa 6= 0). To solve the sign ambiguity the algorithm

needs a rough knowledge of the reflection phase.

By replacing eq. (4.4.18) in eqs. (4.4.16) and (4.4.17) the following coefficients

are obtained:

k

p=

Γm1 − b

(a− Γm1).Γa

u

w=

Γm2 − f

(g − Γm2).Γa

(4.4.19)

92

The multiplying factors p and w need not to be calculated but only their ratio

α = p/w. This property is clear by combining eqs. (4.4.10) and (4.4.11) into the raw

measurement fundamental equation (3.6.7) obtaining:

Tm = α.Xa.Td.(Xb)−1 =

p

w·[

a · k/p b

k/p 1

]· Td ·

[1 u/w

f g · u/w

]−1

(4.4.20)

From the Thru measurement, the transmission coefficient SmT21 is obtained and

the α coefficient is given by:

α =u/w · (g − f)

(1− u/w · k/p) · SmT21

(4.4.21)

As subproducts of the TRL Calibration the propagation constant γ of the Line

and the actual reflection coefficient Γa of the Reflect are calculated.

There are important features to consider with this technique:

• The reference plane is put in the middle of the Thru.

• The reference impedance of the measurement system is defined by the charac-

teristic impedance of the Line.

• The TRL has frequency limitations and it needs multiple lines to cover a broad-

band. It is necessary that ∆` = `line−`thru 6= n·λ/2 because at these frequencies

the algorithm doesn’t work and produces ill conditioned matrices.

93

Figure 4.3: Thru - Line Setup Measurement Reference Planes

To make a TRL Calibration it is necessary to take into account some practical

considerations:

• The electrical length of the Line section should be λ/4 or 90 in the middle

of the measurement span frequency and a phase difference between 20 and

160 along the same span assures that, the TRL algorithm is in a convergency

bandwidth, sufficiently far from the 6= n · λ/2 frequencies.

• TRL is frequency limited to bandwidths no larger than 8:1. For wider band-

widths, ulterior lines are employed to split the band.

• To measure the Line its position needs to be centered with respect to the center

of the Thru and reference planes will be re-positioned as shown in Fig. 4.3.

• Within a planar measurement with an accurate fixture setup is required to have

the proper position of microprobes with respect to the devices. To assure that

the reference planes will be just besides the edge faces of the D.U.T. a piece of

94

Figure 4.4: D.U.T. Setup Measurement Fixture

Thru with a length of 1/2.`thru has to be added to both sides of the D.U.T.

centering it as shown in Fig. 4.4.

• If the Thru is not ideal then matrix T 6= I. If T matrix is diagonal the

consequence is a different reference plane than ideal. This it is taken into ac-

count with the considerations shown in Fig. 4.4. If T matrix is complete, then

Line and Thru have different characteristic impedances and the reference

impedance of the system will be different from the Line. Heuristic consider-

ations are made to solve this situation by taking a compromise value of the

reference impedance as the geometric mean of the Thru and Line character-

istic impedances Zref ≈√

Zthru · Zline .

95

4.4.2 RSOL (UTHRU) technique

This technique developed by Pisani and Ferrero [19] is an innovative self calibration

solution where the greatest obstacle in modern techniques like TRL or LRM that is

the full knowledge of at least one two-port network, the Thru standard is surpassed.

In many applications this Thru standard can not be completely known. An example

of this is the case where it is not possible to connect directly the two probes, then

it is necessary to have as short as possible Thru that guarantees low losses and easy

modelling. An example of this is the case of two port on-wafer devices with unaligned

ports or having a 90 angle between them as shown in Fig. ??, a very important

situation in today’s actual RF ICs.

RSOL (reciprocal - short - open - load) technique doesn’t requires any particular

Thru knowledge. This procedure is based on the two ports Error Box model where

any reciprocal two-port can be used as Thru . The unique requirement of the Thru

standard is reciprocity and a rough knowledge of its transmission coefficient S21 phase

shift.

Associated mathematics with this solution is given by Error box model equations

(3.6.6), (3.6.7), (3.6.9) and (3.6.10) that are rewritten here for the sake of simplicity:

Ta =1

e10A

·[−∆A e00

A

−e11A 1

]≡ 1

e10A

·Xa with ∆A = e00A · e11

A − e01A · e10

A

Tb =1

e10B

·[

1 −e11B

e00B −∆B

]≡ 1

e10B

·Xb with ∆B = e00B · e11

B − e01B · e10

B

Td =(Ta

)−1.Tm.Tb = α−1.

(Xa

)−1.Tm.Xb with α =

e10B

e10A

96

As in the SOLT calibration technique it is necessary to take two 1-Port Mea-

surements to obtain the error coefficients of Xa and Xb matrices. The relationship

between the measured Γm and the actual Γa standard reflection coefficients at Port

1 is the following:

Γm = e00A +

e10A · e01

A · Γa

1− e00A · Γa

(4.4.22)

and by connecting three known standards: short, open and load , it is possible to

have 3 independent equations and to calculate the desired error terms e00A , e11

A and

the product e10A · e01

A . The same reasoning applied at Port 2 gives the error terms e00B ,

e11B and the product e10

B · e01B . With these error terms it is straightforward to obtain

∆A and ∆B.

Finally, the coefficient α is obtained by connecting a reciprocal unknown two-port

network between the ports. By applying the reciprocity properties, the transmission

matrix of a reciprocal unknown Thru has an unitary determinant. From (3.6.5), it

follows:

det(Tm) = α2 · det(XA) · det(XB)−1 (4.4.23)

therefore,

α = ±√

det(Tm) · det(XB)

det(XA)(4.4.24)

The sign ambiguity is solved as follows. Let

Y = (XA)−1 · Tm ·XB (4.4.25)

97

which is fully known from the above measurements. Then, by applying (3.6.5) the

Thru S21 scattering parameter is given by:

S21thru =α

Y22

(4.4.26)

From the above equation, a rough knowledge of the Thru S21 phase shift is all

that is necessary to solve the α sign ambiguity.

This solution allows to calibrate the two ports although they have identical sex

connectors or different port transitions as coaxial in Port 1 and Port 2 directly an

on-wafer probe, without complicated models for the transitions or elaborated deem-

bedding procedures. Accuracy of this technique is comparable to modern LRM

technique as proven by Pisani and Ferrero [19].

Chapter 5

Calibration & Measurement Tool

5.1 Introduction

As an original contribution, a Calibration and Measurement Tool based on the

TRL algorithm was developed. This tool uses the capacity of the VNA HP8510C to

be connected to a remote computer through an IEEE 488.2 interface. The program

was developed in MATLAB code and it runs in different platforms giving a versatile

use. Interesting features were implemented into this tool. Full TRL calibrations can

be performed through the use of an easy-to-use GUI designed to this effect. Deem-

bedding and plot of results are available for the user. Further, it is possible to perform

the Uploading of Twelve Error coefficients in the VNA. This feature allows a unique

calibration in a remote computer and store it into the measurement instrument, giv-

ing a powerful utility for repetitive measurements. In the following paragraphs a

description of the tool is provided. An example of calibration is presented and is

compared with other calibration techniques. Original equations for the equivalence

between the Twelve Error coefficients and the Error box model are presented for the

first time in literature.

98

99

5.2 MATLAB Calibration & Measurement Tool

This tool exploits the MATLAB Instrument Toolbox by connecting the computer to

a remote measurement instrument through a GPIB card and an IEEE 488.2 bus for

virtual instrumentation. This feature permits to develop a code program in a easy

way through the only configuration of the computer card by the user, without taking

into account low level signals.

A GUI (General User Interface) was implemented to achieve an easy interaction

with the user. All features of the software are performed by interaction with the

GUI and proper callback functions, giving a structured and efficient code. The code

program uses these functions to subdivide tasks in simple routines that pass inputs

and results as function arguments. In the Appendix A a User Guide is provided where

all user actions are fully explained. This particular tool was developed by dividing

the main routines in two functional blocks:

• Environment Values

• Calibration and Measurement

The Environment Values is a block that permits the user to configure a particu-

lar calibration and measurement. The user can define these environment values by

writing the start and stop frequencies, number of samples, source RF power and the

average factor. The average factor is defined because the tool uses the Step Mode of

the VNA by phase locking single sample frequencies and averaging single frequency

measurement. By pressing a button all user’s values are automatically communicated

to the instrument.

100

Calibration and Measurement is the heart of the program and is divided in three

functional parts:

• TRL Calibration

• DUT and Deembedding

• Uploading and calibrated measurement

The TRL Calibration is performed by the measurement of the known Reflect

standard at Port 1 and Port 2 and the LINE and THRU standards. In this block

the user gives the software a rough knowledge of the phase of the Reflect to be used

in the TRL algorithm. When the four standard measurements have been made, the

TRL algorithm is implemented calculating the Error Box parameters.

Once TRL standards have been measured, a DUT measurement of raw data

can be taken. After this, automatically DUT Corrected data are calculated by the

Deembedding procedure as was explained for the TRL algorithm in the last chapter.

Lastly can be performed the uploading of the twelve error terms to the measure-

ment instrument, using an internal routine that calculates the equivalence between

the Error Box model and Twelve Terms that is in the VNA. This equivalence was

developed explicitly for the first time in this work. Once the uploading is achieved, a

calibrated measurement can be performed by using the uploaded twelve terms coef-

ficients.

All standard, DUT, DUT Corrected data and calibrated measurements, are stored

into files in Touchstone format and their names can be changed by the user through

the GUI. The tool permits easy calibration and measurement to be performed as well

101

as deembeded data for characterization. By applying TRL Calibration, the propa-

gation constant γ and the actual Reflect standard Γa are measured. The following

paragraph describes the implementation of the TRL algorithm in the tool as well as

the equivalence between Error Box model and Twelve Terms , with the calculated

terms to be uploaded.

102

5.3 Calibration & Measurement program

The calibration & measurement tool implements the TRL algorithm for calibration.

Formulae used for this algorithm are given. Using the Error Box model as shown in

Fig. 3.10 and Fig. 3.11, a description of the algorithm will be given.

Classical error model representations as given in Marks’ work [36] take into ac-

count unbalanced and imperfect switching by two switch terms, that represent the

reflection coefficients ΓF and ΓR of the port termination in the forward and backward

stimulation configurations as shown in Fig. 5.1.

They represent the switch error contribution (this model is only presented for

convenience and its parameters will be not explained. A total equivalence with our

representation stems from X = Ta, Y = Tb and T = Td. The α and β coefficients are

constants that comprise a different presentation of the same model).

In our work these reflection coefficients are omitted because the Switch Correction

algorithm that permits to minimize (and practically eliminated) the switching error

was implemented.

Implementing the Switch Correction algorithm simplifies the Error model and the

switching error contribution is eliminated. To explain the algorithm’s implementation

a brief explanation of the Switch Correction algorithm as implemented in our program

will be given.

5.3.1 Switch Correction algorithm

The TRL Calibration and DUT measurements are made by applying the Switch

Correction algorithm that calculates the scattering parameters by measuring the 4

power waves.

103

Figure 5.1: R. Marks Error-Box Error Model of a Three-Sampler VNA

The RF source signal is injected at Port 1 and Port 2 alternatively. It allows to

minimize the isolation error, assuming a zero value for the Error Box Model calcu-

lation. The algorithm is applied to a Four-Sampler VNA. When the signal source is

applied to Port 1, as can be seen Fig. 5.2, the relationship between power waves and

the measured scattering matrix [Sm] is given by:

[b′1m

b′2m

]=

[Sm11 Sm12

Sm21 Sm22

]·[

a′1m

a′2m

](5.3.1)

where the′supraindex is a remark for power waves measured with the signal source

applied at Port 1. Then, applying the signal source to Port 2, a second measurement

of the power waves is made and the relationship between these power waves becomes:

[b′′1m

b′′2m

]=

[Sm11 Sm12

Sm21 Sm22

]·[

a′′1m

a′′2m

](5.3.2)

104

Figure 5.2: Measurement System for two 2-Port networks

where the′′

supraindex is a remark for power waves measured with the signal

source applied at Port 2.

The measured scattering parameters matrix [Sm] is now found by combining

(5.3.1) and (5.3.2) as follows:

[Sm11 Sm12

Sm21 Sm22

]=

[b′m1 b

′m2

b′′m1 b

′′m2

]·[

a′m1 a

′m2

a′′m1 a

′′m2

]−1

(5.3.3)

This procedure is followed for all 2-Port devices to be measured, giving the actual

measured S-parameters with the switch error corrected by (5.3.3), balancing the two

ports switching.

105

5.3.2 TRL algorithm and DUT deembedding

The TRL Calibration algorithm is implemented by measuring the 1-Port Reflect at

Port 1 and Port 2, and by the two port measurements of LINE and THRU. All the

measurements performed by the tool are made using the Switch Correction algorithm.

For each frequency sample the following steps are performed.

First the TmT and TmT matrices (eqs. 4.4.1 and 4.4.2) are calculated by trans-

forming the THRU and LINE measured S-matrices to cascade T matrices. Then RM

and RN are obtained as:

RM = TmL.(TmT )−1 RN = (TmT )−1.TmL (5.3.4)

By using the MATLAB function eig, the eigenvectors matrices M and N respec-

tively of RM and RN are calculated and given as:

[M ] = eig(RM) =

[M11 M12

M21 M22

][N ] = eig(RN) =

[N11 N12

N21 N22

](5.3.5)

By using the conclusions of (4.4.9) where matrices RT = RS are diagonal, the

coefficients of Error Box model are given by:

Ta = M = p ·Xa =

[ka pb

k p

]Tb = N = w ·Xb =

[w u

wf ug

](5.3.6)

106

And the a, b, f and g coefficients are calculated as follows:

a =M11

M21

b =M12

M22

f =N21

N11

g =N22

N12

(5.3.7)

From the measurement of the Reflect at Port 1 (Γm1) and Port 2 (Γm2), and the

measured Thru input reflection coefficient SmT11; the actual Reflect Γa is calculated

by solving (5.3.8) as:

Γa = ±√

(b− Γm1).(f − Γm2).(SmT11 − a)

(a− Γm1).(g − Γm2).(SmT11 − b)(5.3.8)

Using the above result and data, the coefficients k/p and u/w are calculated by

the algorithm as:

k

p=

Γm1 − b

(a− Γm1).Γa

u

w=

Γm2 − f

(g − Γm2).Γa

(5.3.9)

Finally from the above results and the measured Thru transmission coefficient

SmT21, the α coefficient is calculated as:

α =u/w · (g − f)(

1− u/w · k/p) · SmT21

(5.3.10)

The above calculations provide all the Error Box model coefficients that are nec-

essary to get the corrected data from the DUT raw data through the deembedding

process.

107

The DUT is measured in the same way as the other two port devices. Once

this measurement is achieved, the software has all the necessary data to perform

the deembedding calculation for the actual DUT data. With the Error Box model

coefficients and the DUT raw data, the deembedding formula is calculated by the

program as:

Td = α−1 · (Xa

)−1 · Tm ·Xb (5.3.11)

5.3.3 Uploading and calibrated measurements

This utility is useful to perform repeated measurements with the same calibration.

It permits the user to do a calibration on a remote computer and to upload the

calculated coefficients to the VNA memory. This feature calculates the equivalent

Twelve terms of the VNA model from the Error Box model coefficients. In our work

an equivalence between the two models was implemented and explicit expressions of

Twelve Terms Error Model are given for the fist time in literature.

The equivalence is based on the Error Model of a four sampler VNA developed

by Marks [36], shown in Fig. 5.2, and another equivalence given in [3]. In this model

the Error Boxes are given by X and Y as cascade matrices respectively, the actual

DUT as the T matrix and the measured raw data as Tm. By combining and equating

properly the presented formulae in this model, the following equation results:

Tm = β/α · 1

ERR

[ERF − EDF .ESF EDF

−ESF 1

]T

[ERR − EDR.ESR ESR

−EDR 1

](5.3.12)

108

Figure 5.3: Error Model of a Four Sampler VNA

where the α coefficient is totally different from the other one given in the above

equation (5.3.11). To show the equivalence between this model with coefficients of

the Twelve Terms Error Model expressed, we first rearrange the equation (4.4.20)

properly and we set Td = T . Then, an equivalent equation to (5.3.12) is found, using

uniquely Error Box model coefficients are written:

Tm =p/w

u/w · (g − f)·[

a · k/p b

k/p 1

]· T ·

[−g · u/w u/w

−f 1

](5.3.13)

Properly equating the terms expressed in (5.3.12) and (5.3.13) we find an equiva-

lence for the first six terms expressed as follows:

EDF = b

EDR = f

ESF = −k/p

ESR = −u/w

ERF = k/p · (a− b)

ERR = u/w · (g − f)

(5.3.14)

To find the equivalence of the last terms from the Twelve Terms Error Model we

109

Figure 5.4: Twelve Terms Error Model - Forward and Backward sets

use the formulae extracted from the model shown in Fig. 5.3 and given by R. Marks

in [36] as follows:

β/α =ETR

ERF + EDF · (ELR − ESF )(5.3.15)

α/β =ETF

ERF + EDR · (ELF − ESR)(5.3.16)

By replacing the results of (5.3.14)in (5.3.15) and (5.3.16) and equating properly

we find that:

ETR = p/w · [k/p · a + b · ELR] (5.3.17)

ETF = (p/w)−1 · [u/w · g + f · ELF ] (5.3.18)

110

With the assumptions made in [36] that switch coefficients do not have any im-

portant influence, (ΓF = ΓR = 0) (fact that is reasonable in our case because the

Switch Correction algorithm was applied to all the two port measurements), we find

the following equivalences:

ELF = ESR and ELR = ESF (5.3.19)

Another important assumption used in all Error Box model formulations is that

the isolation of the error boxes, and thus the forward and reverse isolation terms on

the Twelve Terms Error Model, are assumed to be null EXF = EXR = 0.

Replacing the terms of (5.3.19) in (5.3.17) and (5.3.18) and by equating we find

the last equivalences for ETR and ETF :

ELF = −u/w

ELR = −k/p

ETF = (p/w)−1 · u/w · (g − f)

ETR = (p/w) · k/p · (a− b)

EXF = 0

EXR = 0

(5.3.20)

From the above expressions (5.3.14) and (5.3.20), the software calculates the

Twelve Terms Error Model from the Error Box model coefficients presented in our

work. After that, they can be uploaded into the memory of the instrument by the user

to perform automated calibrated measurements. Therefore with a single calibration,

it is possible to perform repeated calibrated measurements using this utility and the

deembedding process is performed automatically by the VNA using the Twelve Terms

calculated by the user calibration algorithm.

111

5.4 Coaxial Experimental Results

TRL Calibration and a DUT measurement with a Coaxial Kit were performed and

compared with another on board SOLT Calibration as an example of the automated

features that the software brings.

The selected DUT was a precision 6 dB SMA Coaxial Attenuator. A 30 mm length

Rigid Coaxial SMA connector was used as the LINE. As REFLECT the OPEN Loads

of a Mauryr Coaxial Calibration Kit were used.

With another feature of the program, the attenuation constant α of the LINE and

the ηeff = c/vph coefficient were calculated. Plots of the different magnitudes of Raw

data, Corrected DUT data and the actual Reflect coefficient Γa are provided.

From the plot of the DUT Reflection Coefficient S11 the DUT corrected data from

the TRL Calibration performed by the tool can be seen in a smooth trace. Around

this plot there is the trace (with ”ripple” wave form) of the DUT corrected data given

by the calibration performed with the uploaded 12 error terms calculated by the tool.

The other two calibration performed by the VNA firmware, the on board SOLT have

more irregular traces.

The graph highlights that the phase of DUT transmission coefficient S21 corrected

by the TRL Calibration performed has a linear behavior along the entire bandwidth

as opposed to the same coefficient S21 performed with a SOLT on board calibration

(performed with the same standards as the tool TRL calibration) that has phase skips

in the band.

112

1 1.5 2 2.5 3 3.5 4−80

−70

−60

−50

−40

−30

−20

−10

0S11 (dB) Module

Frequency Ghz

Raw DataTRL on PCUploaded 12 TermsSOLT on Board

Figure 5.5: S11 Module

1 1.5 2 2.5 3 3.5 4−200

−150

−100

−50

0

50

100

150

200S11 Angle

Frequency Ghz

Raw DataTRL on PCUploaded 12 TermsSOLT on Board

Figure 5.6: S11 Phase

113

1 1.5 2 2.5 3 3.5 4−10

−9

−8

−7

−6

−5

−4

−3

−2S21 (dB) Module

Frequency Ghz

Raw DataTRL on PCUploaded 12 TermsSOLT on Board

Figure 5.7: S21 Module

1 1.5 2 2.5 3 3.5 4−160

−140

−120

−100

−80

−60

−40

−20S21 Angle

Frequency Ghz

Raw DataTRL on PCUploaded 12 TermsSOLT on Board

Figure 5.8: S21 Phase

114

The LINE parameters like the attenuation constant α and the refractive index

for the phase velocity ηeff = c/vph are calculated from the measured propagation

constant γ and from the length difference ∆` = `line− `thru. The algorithm calculates

the eigenvalue matrix Λ of matrix (5.3.4) RM = TmL.(TmT )−1, rewritten in the same

way as equation (4.4.9) by doing:

RM = TmL.(TmT )−1 = T.Λ.T−1 =

[e−γ.∆` 0

0 e+γ.∆`

]=

[λ1 0

0 λ2

]= Λ (5.4.1)

By equating the eigenvalues and length difference ∆` properly, we take the mean

value of the attenuation constant α and the refractive index ηeff parameters, that are

calculated by the tool as follows:

〈α〉 = 1/2 · ln |λ1|+ ln |λ2|∆`

(5.4.2)

with

〈ηeff〉 =1

2πf∆`

∣∣∣ arctan[(=(λ1)/<(λ1)

]∣∣∣ +∣∣∣ arctan

[(=(λ2)/<(λ2)

]∣∣∣

(5.4.3)

and

εr = η2eff (5.4.4)

From the above measurement results, the calculated LINE parameters α and ηeff

are shown in Fig. 5.9 and Fig. 5.10:

115

1 1.5 2 2.5 3 3.5 40.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02TRL Measurements α (dB/cm) vs Freq

α (d

B/c

m)

Frequency Ghz

Figure 5.9: LINE Attenuation constant

1 1.5 2 2.5 3 3.5 41.001

1.0015

1.002

1.0025

1.003

1.0035

1.004

1.0045

1.005

1.0055

1.006

TRL Measurements ηeff

vs Freq

η eff

Frequency Ghz

Figure 5.10: LINE ηeff coefficient

Chapter 6

Networks characterization andparameter extraction

6.1 Introduction

In this chapter the approaches used to characterize experimentally single two con-

ductors transmission lines and MTLs through the measurement of the Scattering

parameters will be given.

First the relevant methodologies used to extract the different transmission line

parameters R, L, G and C from direct measurements will be discussed. Then an ac-

curate extraction method from Scattering parameters matrix that takes into account

frequency dependency of R(f), L(f), G(f) and C(f) will be presented. An exam-

ple of measurement and extraction will be discussed and compared with theoretical

predictions of a full wave simulation.

Finally different methodologies for the extraction of the multi transmission line

parameter matrices R, L, C and G from Scattering matrix will be presented and the

results of an example will be discussed. Drawbacks and limitations will be highlighted

and discussed.

116

117

The parameter extraction methodologies included in this chapter are directly con-

nected with the useful implementation of different measurements of scattering pa-

rameters with the Measurement & Calibration Tool presented in last chapter.

It is remarked that a useful close set of measurements can be taken by a powerful

tool, and transmission line parameters can be fully and accurately characterized by

the measurements of the Scattering matrix with a single VNA instrument.

118

6.2 Transmission line characterization methods

Different methodologies are used to characterize a transmission line by direct mea-

surements. The more common experimental procedures [40] [26] [32] [33] and their

limitations will be presented. Then, a more accurate methodology [9] [10] that sur-

passes the classical methods’ limitations will be explained and an implementation

through a measurement of Scattering parameters will be discussed.

For any transmission line mode, the per-unit-length circuit parameters R, L, Gand C are defined in terms of the characteristic impedance Zc and the propagation

constant γ by:

γ

Zc

= G + jωC (6.2.1)

γZc = R+ jωL (6.2.2)

Then, if the characteristic impedance Zc and the propagation constant γ are

known, the per-unit-length circuit parameters R, L, G and C are given by:

R = <γZcL = =γZc/ωG = <γ/ZcC = =γ/Zc/ω

(6.2.3)

The problem consists in determining the propagation constant γ and the char-

acteristic impedance Zc through experimental methodologies, and then to solve the

equation system (6.2.3).

The measurement of the propagation constant γ is an easy task using the TRL

calibration, a subproduct of this procedure. Accurate results are given by this method

119

and it is used as the standard for its determination. Instead, one of the more prob-

lematic parameters to be measured is the characteristic impedance Zc and it only can

be estimated.

An approach based on the TRL calibration methodology that permits to estimate

the characteristic impedance Zc was given by J. Kasten et al [26]. This procedure

argues that Zc can be determined from a measurement of the propagation constant

γ and knowledge of the ”free-space capacitance”. The idea is attractive since γ is

readily determined using the TRL calibration.

The method supposes lossless conductors (R¿ ωL), then:

Zc ≈√LC =

1

vphC =1

cC0`(6.2.4)

where vph is the phase velocity, c is the free-space light velocity, C0 is the free-space

per-unit-length capacitance and ` is the transmission line structure length.

The drawback of this methodoloy is that it fails in low frequencies, therefore the

estimation of Zc by this method can be problematic.

Another procedure proposed by Marks and Williams [32] explores the possibility

of an alternative indirect prediction of Zc trough the measurement of γ by TRL

calibration. The method, while approximate, was demonstrated quite precise for

quasi-TEM lines with low substrate losses [31].

This analysis supposes that when the substrate loss is low and the transverse

currents in the conductors are weak, as is typically true at very high frequencies,

then G is negligible (G ¿ ωC). With this approximation the (6.2.1) becomes:

120

γ

Zc

= G + jωC ≈ jωC (6.2.5)

and

Zc ≈ γ

jωC (6.2.6)

In order to predict the value of the characteristic impedance Zc, this method

proposes an experimental measurement of the propagation constant γ and the pul

capacitance C.

There are different methodologies to measure the pul capacitance C and their

goal is the accuracy and complexity of the measurement. Approximate procedures

were presented in [33] with a reasonable complexity. The first one is based on the

measurement of the per-unit-length dc resistance Rdc, an easily measurable quantity.

The procedure takes the imaginary part of the product of (6.2.1) and (6.2.2):

RC + LG = <(

γ2

jω

)(6.2.7)

In the case of low losses substrates G is small at microwaves frequencies and

LG ¿ RC. If R is approximately equal to the per-unit-length DC resistance Rdc,

then equation (6.2.7) becomes:

C ≈ 1

Rdc

<(

γ2

jω

)(6.2.8)

121

These approximate values are expected to deviate significantly from the actual

value except at low frequencies, where the current in the conductors is highly uniform

and the approximation R ≈ Rdc is valid. For this reason, a least squares fit of a

quadratic to the approximation of C is used to extrapolate to DC.

To achieve realistic results in low frequencies, another measurement is proposed

in the same work [33] where a small lumped resistor is measured at low frequencies

giving:

Zc1 + Γload

1− Γload

= Zload ≈ Rload,dc (6.2.9)

where Rload,dc is the dc resistance of the lumped load and Γload is its complex

measured reflection coefficient. Substituting (6.2.9) in (6.2.1) gives:

C[1− j(G/ωC)] ≈ γ

jωRload,dc

1 + Γload

1− Γload

(6.2.10)

In Ref. [33], a least-squares to fit a quadratic to the measured values of C was

used to extrapolate the approximate values of C to dc. Approximate values of G/ω

are also obtained with this technique. Limitations of this technique are that it is

only applicable to quasi-TEM lines but not necessarily to other types of waveguides

mainly in the case of lossy substrates where the approximation G ¿ ωC is not valid.

Added to this, the approximations and complexity of measurements allow for further

errors.

A different approach, based on a single measurement of the Scattering parameters

is shown and used in the following paragraphs of this work. This technique does not

122

assume approximations and uses the full information given by the S-matrix.

123

6.2.1 Circuit parameters extraction from S-Matrix

The above traditional approaches used to extract the per-unit-length circuit parame-

ters R, L, G and C assume resistance and capacitance constant with frequency. These

assumptions are inaccurate when high frequency transmission parameters need to be

extracted because they strongly depend on the frequency.

A different methodology based on the direct extraction of the Telegrapher’s equa-

tion per-unit-length circuit parameters R, L, G and C from S-parameter measurements

was proposed by W. Eisenstadt [9] [10].

This procedure characterizes interconnections and transmission lines using stan-

dard on-chip microwave probing directly from S-parameter measurements. Standard

automated microwave test equipment can be used to obtain results.

The theoretical basis of the method is Telegrapher’s equation taking into account

the frequency dependency of the per-unit-length circuit parameters R(f), L(f), G(f)

and C(f).

The S-parameter responses measured from a lossy unmatched transmission line

with length `, propagation constant γ, characteristic impedance Zc and a controlled

reference impedance Zref are [27]:

[S] =1

DS

[(Z2

c − Z2ref ) sinh(γ`) 2ZcZref

2ZcZref (Z2c − Z2

ref ) sinh(γ`)

](6.2.11)

where

DS = 2ZcZref cosh(γ`) + (Z2c + Z2

ref ) sinh(γ`)

The above matrix is assumed symmetrical and contains two independent linear

equations. This S-parameter matrix is converted to ABCD parameter matrix as:

124

[ABCD] =

[cosh(γ`) Zc sinh(γ`)

Zc sinh(γ`) cosh(γ`)

](6.2.12)

and the relationship between the S-parameters and the ABCD matrix is [7]:

A = (1 + S11 − S22 −∆S)/(2S21)

B = (1 + S11 + S22 + ∆S)Zref/(2S21)

C = (1− S11 − S22 + ∆S)/(2S21Zref )

D = (1− S11 + S22 −∆S)/(2S21)

(6.2.13)

where

∆S = S11S22 − S21S12

Combining equations (6.2.11) to (6.2.13) yields [9]:

e−γ` =

1− S2

11 + S221

(2S21)2±K

−1

(6.2.14)

where

K =

(S2

11 − S221 + 1)2 − (2S11)

2

(2S21)2

1/2

(6.2.15)

and

Z2c = Z2

ref

(1 + S11)2 − S2

21

(1− S11)2 − S221

(6.2.16)

125

Once γ(f) and Zc(f) are determined from (6.2.12)and (6.2.14), Telegrapher’s equa-

tions model per-unit-length circuit parameters R(f), L(f), G(f) and C(f) are given

by:

R(f) = <γZcL(f) = =γZc/ωG(f) = <γ/ZcC(f) = =γ/Zc/ω

(6.2.17)

The procedure converges very well for small length ` segments of transmission

line, being the convergency bandwidths limited by this length `. It is shown that the

procedure is independent of the calibration technique used to extract the calibrated

Scattering matrix parameters.

This procedure was used in our work to extract the per-unit-length circuit param-

eters from the S-parameters matrix measured with a VNA HP8510C of a two port

CPW structure and the results where compared with a Full Wave EM simulation to

validate the experimental performance of the method.

Results of the parameter extraction and calibrated Scattering matrix are given,

and compared with the simulated CPW structure data.

126

6.2.2 On Wafer measurements and characterization

Modern VNAs can easily make accurate measurements in situations where calibration

standards can be connected to the test ports. There are, however, many devices that

cannot be connected directly to the test port of a VNA and require a fixture system or

on-wafer probe to complete the bridge between the DUT and the test instrumentation.

The use of test fixtures presents problems and additional errors are introduced in the

measurement process.

Mainly network analysis, in the general situation, is used to characterize the linear

behavior of a device. The data resulting from the measurements will not be truly

accurate because of imperfections in the instrument and in the hardware used to

connect the device. As was seen in previous chapters, random errors, including drift,

noise and repeatability are difficult to handle but systematic errors can be addressed

by means of calibration techniques.

Some of the problems specific to the fixtured measurements include connection

repeatability and difficulty in providing reference standards. In addition, the nature

of the transmission medium may include dispersion, losses and other problems which

make it difficult to establish a reliable, known characteristic impedance.

A number of factors need to be considered to measure with a microwave test fixture

[42]:

• Compatibility: Many devices have performances which are strongly depen-

dent on the environment in which they are embedded and it is therefore neces-

sary to provide an environment similar to that used in the application. This is

met by arranging for a similar physical geometry in the measurement environ-

ment, ensuring that the field configuration in the vicinity of the device closely

127

matches that of the application and is more likely to give useful data. The

fixture is optimized for the range of impedances being measured and this may

require that the fixture transforms the measurement environment impedance.

• Calibration: The success for fixture design is the calibration technique to be

used. The very nature of a test fixture is such that conventional calibration

techniques are unsuitable because the device to be tested does not have ports

terminated in precision connectors. There are two distinct approaches for de-

embedding device measurements from those of a fixture.

The first method consists in calibrating the VNA system at reference planes of

the device by employing calibration components which replace the DUT within

the fixture. The method is very simple in principle and relies only on the quality

of the calibration components, the repeatability of the fixture and the validity

of the calibration algorithms. In this case all the discontinuities, losses, etc. are

all included in the Error models of the fixture.

The second method uses a model for the fixture and with de-embeds the device.

Such a model may be as simple as a length of transmission line at the test port

or include complications due to multiple discontinuities, losses, etc. There are

many combined possibilities involving calibration at accessible reference planes

which are as close as possible to the device in conjunction with a model with

the minimum complexity. The majority of these imperfections are not included

in the Error models and need to be added to the total fixture to implement the

de-embedding process.

In our measurement the first method was used, then all the imperfections between

128

probe tips and contacts with the transmission line measured where included in the

Error Boxes of the fixture’s Error model.

Measurements were made with reference planes coinciding with the position of the

probe tips in contact with the DUT. Then, differences between measurement values

and simulation values can be attributed to the extraction process methodology used

and/or the accuracy of the simulated model, but no to the imperfections of the fixture.

The extraction methodology [9] presented in the last section, was validated by a

measurement of a Coplanar Waveguide CPW with stratified dielectric that was made

by implementing the calibration techniques and measurement tools presented in the

last chapters. Results were compared with a Full Wave EM simulation [14][15] of the

CPW structure. In Fig. 6.1 the front view of the tested CPW structure1 is shown. A

sample of this CPW structure of a length of 2.585 mm was simulated and measured

within a bandwidth from 1 to 6 Ghz.

As can be seen from Figs. 6.2 and 6.3, the extracted per-unit-length circuit pa-

rameters L(f) and C(f) are in good agreement with the FW simulation model’s

parameters. A disagreement is shown in Figs. 6.4 and 6.5 for the per-unit-length

circuit parameters R(f) and G(f). For the parameter R(f), the simulated model

predicts a lower influence of the skin effect on the structure behavior. The difference

can be explained by the assumption that in the measurement the microwave measure-

ment fixture probe tips were not deembedded, giving an additional contribution for

dispersion losses. The simulated dielectric losses, present in G(f), are greater than

the measured data. Causes for this behavior can be attributed to the assumption of

a highly lossy dielectric synthesized Debye model [1] for the complex permittivity ε.

1CPW structure data were provided by Prof. Franco Fiori of the EMI Microwave Group atPolitecnico di Torino, Italy

129

Figure 6.1: CPW stratified dielectric structure

An excellent match is achieved between the simulated and measured characteristic

impedance Zc as shown in Figs. 6.6 and 6.7, when the differences are attributed to

the microwave measurement fixturing, where the de-embedding process did not include

the probe tips interfaces.

The measured and simulated attenuation constant α shown in Fig. 6.8 are in

excellent agreement, where differences at high frequency becomes evident due to the

over valuated dielectric losses in the simulated model. The measured refractive index

ηeff presents a close behavior to the FW simulated model as is seen in Fig. 6.9.

130

1 2 3 4 5 60

2

4

6

8

10

Frequency, GHz

L, n

Hy/

cmMeasurementFW Simulation

Figure 6.2: per-unit-length Inductance nHy/cm

1 2 3 4 5 62

4

6

8

10

12

14

16

Frequency, GHz

C, p

F/c

m

MeasurementFW Simulation

Figure 6.3: per-unit-length Capacitance pF/cm

131

1 2 3 4 5 60

10

20

30

40

50

Frequency, GHz

R, Ω

/cm

MeasurementFW Simulation

Figure 6.4: per-unit-length Resistance Ω/cm

1 2 3 4 5 60

0.05

0.1

0.15

0.2

Frequency, GHz

G, S

/cm

MeasurementFW Simulation

Figure 6.5: per-unit-length Conductance S/cm

132

1 2 3 4 5 60

10

20

30

40

50

Frequency, GHz

Zc

Mod

ule,

ΩMeasurementFW Simulation

Figure 6.6: Module of the Characteristic Impedance Zc

1 2 3 4 5 6−60

−40

−20

0

20

40

60

Frequency, GHz

Zc

Ang

le, Ω

MeasurementFW Simulation

Figure 6.7: Phase of the Characteristic Impedance Zc

133

1 2 3 4 5 60

5

10

15

20

25

Frequency, GHz

α, d

B/c

mMeasurementFW Simulation

Figure 6.8: Attenuation dB/cm

1 2 3 4 5 61

2

3

4

5

6

7

8

9

10

Frequency, GHz

nef

f

MeasurementFW Simulation

Figure 6.9: Refraction index ηeff

134

6.3 MTL characterization methods

Various parameter extraction techniques for MTL structures were studied and valu-

ated. A brief discussion follows and finally an example in which an accurate technique

without optimization [38] is implemented will be presented.

Groudis and Chang [23] have previously developed a frequency domain method

to extract parameter matrices R, L, C and G from the two-port impedance Z and

admittance Y matrices. This method is based on a combination of the method of

characteristics and the decoupled mode transformation in frequency domain. In this

procedure, the solution of the MTL equations (2.3.12) and (2.3.13) is assumed as

follows:

V = A.exp(−Γz) + B · exp(Γz) (6.3.1)

Y−1C I = A.exp(−Γz)−B.exp(Γz) (6.3.2)

where

Γ = (ZY)1/2 = PγP−1 (6.3.3)

YC = Z−1Γ = YΓ−1 (6.3.4)

P is the eigenvector matrix of Γ. It is also the eigenvector matrix of the ZY prod-

uct, being γ the diagonal eigenvalue matrix of Γ. Applying properties the following

relationship is derived:

exp(−Γz) = Pexp(−γz)P−1 (6.3.5)

135

In the solution it is assumed that the matrix Γ = (ZY)−1 exists and that the

characteristic admittance matrix YC is symmetrical. Following the reasonings given

in [41], a MTL with n+1 conductors of length d can be treated as a 2n-port network,

having n ports on the input end (subindex i) and n ports on the output end (subindex

o). Then, it can be proven [7] that:

[Ii

Io

]=

[YC cothΓd −YC sinh−1 Γd

−YC sinh−1 Γd YC cothΓd

]·[

Vi

Vo

](6.3.6)

with a short-circuit admittance matrix Y2n of the 2n-port network given as:

Y2n =

[YA YB

YB YA

](6.3.7)

where

YA = YC cothΓd

YB = −YC sinh−1 Γd(6.3.8)

and

cothΓd = P(coth γd)P−1

sinh−1 Γd = P(sinh−1 γd)P−1(6.3.9)

And, the open-circuit impedance matrix Z2n is given by:

Z2n =

[ZA ZB

ZB ZA

](6.3.10)

where

ZA = (cothΓd)Y−1C

ZB = (sinh−1 Γd)Y−1C

(6.3.11)

136

Two methods to derive the YC and Γ matrices were proposed in [23]. The first one

is to be used when the transmission line attenuation is small, and it is not interesting

for lossy lines.

The second method is to be used with high attenuation or lossy lines. This case

may occur either because the line is sufficiently long, or because the frequency of

interest is so high that losses due to skin effect and proximity effect are significant.

It involves measurements at both input and output ends of the MTLs.

A brief discussion of this method is provided in the following. From equation

(6.3.7) we have:

−Y−1B YA = coshΓd (6.3.12)

Γd = P(cosh−1 Λ)P−1 (6.3.13)

where P is the eigenvector matrix of Γd and −Y−1B YA. Equations (6.3.3) and

(6.3.13) give:

γd = cosh−1 Λ (6.3.14)

and from (6.3.7) and (6.3.13) the characteristic admittance matrix YC is found

to be:

YC = −YB sinhΓd

= −YBP[sinh(cosh−1 Λ)]P−1(6.3.15)

This, from (6.3.14) the per-unit-length modal attenuation constant αm and per-

unit-length modal phase constant βm are obtained by dividing the real and the imag-

inary parts by d and ωd respectively.

137

Then, the per-unit-length parameters R, L, C and G can be derived by replacing

and equating into (6.3.4) the results Γ and YC respectively obtained from (6.3.13)

and (6.3.16), as follows:

Z = R + jωL = ΓY−1C

Y = G + jωC = YCΓ(6.3.16)

This method was tested by using a Full Wave EM simulation2 on a 4-Ports asym-

metric microstrip line and it was found not to achieve symmetry for the above per-

unit-length parameters R, L, C and G and to produce results without physical

meaning. Reasons for this are that the procedure presupposes symmetries for Z and

Y up to frequencies under the Ghz region, in which their authors have validated it.

As was explained in last chapters, these symmetries are intended to be broken in high

lossy media, as in high frequency, this the procedure needs to be modified to take

asymmetries into account.

A great limitation of the method is that it needs to be optimized by a proper

algorithm. Another drawback is that the convergency bandwidth is limited by the

heuristic rule of thumb `MTL ≤ λ/10, where `MTL is the MTL length in the prop-

agation direction and λ is the wavelength of the EM wave propagating along the

structure.

Another methodology was developed by Knockaert et al [29] to recover lossy MTL

parameters from Scattering matrix. The method is based on a generalization of the

simultaneous diagonalization technique by means of congruence transformations to

the general lossy reciprocal case.

2Simulation was made with the EM simulator EMSight of AWR, that includes a fast Full Waveelectromagnetic solver based in a modified Spectral-domain method of moments.

138

This procedure is based on the symmetry and reciprocity properties of the Z and

Y matrices, and the solution is obtained through the chain parameter matrix Φ(z)

defined in equations (2.3.16) and (2.3.17), that is rewritten as:

Φ(z) = exp

− z

(0 Z

Y 0

)=

[Ω α

β ΩT

]

=

[φ1

(z2ZY

) −zZφ2

(z2ZY

)

−zYφ2

(z2ZY

)φ1

(z2ZY

)] (6.3.17)

where the superscript T indicates Hermitian adjoint matrix (conjugate transpose)

and the entire functions φ1(z) and φ2(z) are defined as:

φ1(z) = cosh(√

z)

φ2(z) = sinh(√

z)/√

z(6.3.18)

Noting that α and β are symmetric, also Ωα and βΩ are symmetric. The authors

of [29] assume that Ω2 = In + αβ (with In is the n× n identity matrix) and that the

following relation needs to be achieved:

[Ω α

β ΩT

]·[

Ω −α

−β ΩT

]=

[In 0

0 In

]= I2n (6.3.19)

With this assumptions, the 2n× 2n Z2n-matrix description of a MTL is given by:

[Vi

Vo

]=

[Z2n

] ·[

Ii

Io

](6.3.20)

139

and based on the symmetry assumption:

Z2n =

[A B

B A

]=

[−β−1ΩT −β−1

α− Ωβ−1ΩT −β−1Ω−1

](6.3.21)

where A and B are symmetric n× n matrices. These relationships follow:

β = −B−1, Ω = AB−1, α = B − AB−1A (6.3.22)

With the matrices A and B given from the Z2n-matrix description, the Z and Y

matrices are recovered from the equations:

AB−1 = φ1

(z2ZY

)

B−1 = −zYφ2

(z2ZY

) (6.3.23)

By assuming that the eigenvalues of AB−1 and ZY are all distinct, the resulting

decompositions are given as:

AB−1 = PδzP−1, ZY = PδtP

−1 (6.3.24)

and the related simultaneous congruence decompositions given by

A = PδaPT , B = PδbP

T , Z = PδrPT , Y = P−T δgP

−1 (6.3.25)

Using the above relationships, the following equations for the δ(·) diagonal matrices

were derived [29]:

δaδ−1b = δz

δrδg = δt

δz = φ1(z2δt)

δ−1b = zδgφ2(z

2δt)

(6.3.26)

140

From the above equations, the MTL parameters can be obtained from the Z2n-

matrix. The following general formula is derived:

δt =1

z2[arg cosh(δz) + j2πδn]2 (6.3.27)

where δn is a diagonal matrix with integer entries that takes into account the

multiple branches of the inverse function of φ1(z) = cosh(√

z).

This algorithm intends to solve (6.3.27) specifying an index vector of integers, the

entries of the diagonal matrix δn, in order to retrieve the correct MTL parameters. It

has a direct connection with the MTL length z, since the method tries to find T from

a matrix exponential exp(zT). For the scalar case this creates phase related problems

to be solved to obtain an estimate for λ, given t = eλz. Approximations for this scalar

case are generalized to the matrix exponential and a general solution, that includes

the generation of the index vectors, was developed in [29].

The author of the present work has tested the mentioned algorithm in a 4-Ports

asymmetric microstrip line simulated with a Full Wave EM simulation3. Limitations

due to convergency problems were encountered in solving (6.3.27), where a difficulty

to reach diagonal matrices needs to be optimized in the original algorithm. In this

procedure, symmetries for Z and Y matrices were assumed, thus high lossy MTL

structures are not properly characterized.

Other approaches are proposed in the literature [25][38][48][49]. The procedure

presented by Arz et al [48] uses statistical measurement methodology based on Marks

algorithm [35] as an enhancement of the TRL algorithm.

3Simulation was made with the EM Full Wave simulator EMSight of AWR.

141

In the present work a procedure that doesn’t require optimization [37][38] and

that gives accurate results was implemented. A discussion of the method, examples

of characterization and limitations are given in the following paragraphs.

142

6.3.1 MTL parameters extraction from S-Matrix

A method for extracting the circuit models for MTLs from black-box parameters

was developed by Martens and Sercu [38]. If the number of conductors in a MTL is

large, the model will have many parameters to be extracted and non physical values

may be obtained or the extraction process does not converge as was seen in previous

paragraphs. Simultaneous optimization requires great computational effort and needs

error estimation routines.

The direct extraction method without optimization is valid for small MTL lengths

compared with wavelengths of propagating waves. If T or Π circuit models are pro-

posed, a direct relation is found between the Z and Y matrices and the circuit

parameters.

A brief explanation is given: an MTL with 2n access ports is considered, then if

two RL sections and one GC section (T-circuit model) or two GC sections and one RL

section (Π-circuit model) are sufficient to obtain an accurate model, no optimization

process is needed to determine the parameter values of the model [38]. They can be

directly calculated from the black-box Scattering parameters.

We consider a (n+1) conductors MTL, the 2n×2n S-matrix consists in four n×n

submatrices as follows [25]:

S =

[Sin,in Sin,out

Sout,in Sout,out

](6.3.28)

Then, we can find the impedance Z2n and admittance Y2n matrices from the

143

Scattering matrix as:

Z2n = Zref · [I+ S][I− S]−1

Y2n = Z−1ref · [I− S][I+ S]

(6.3.29)

where I is the 2n×2n identity matrix. To obtain the parameter values the 2n×2n

Z2n and Y2n are defined as:

Z2n =

[Zin,in Zin,out

Zout,in Zout,out

]and Y2n =

[Yin,in Yin,out

Yout,in Yout,out

](6.3.30)

Then the per-unit-length parameters R, L, C and G of the T-circuit model, as

shown in Fig. 6.10, are related to the above matrices as follows:

R(1) = <(Zin,in − Zin,out)

L(1) = =(Zin,in − Zin,out)/ω

C = =(Z−1in,out)/ω

G = <(Zin,out)

R(2) = <(Zout,out − Zin,out)

L(2) = =(Zout,out − Zin,out)/ω

for T circuit (6.3.31)

where the supraindex (1) indicates the input RL branch of the T-circuit model and

the supraindex (2) indicates the output T-circuit model RL branch.

144

Figure 6.10: MTL T-circuit model

In an analogue way, the per-unit-length parameters R, L, C and G of the Π-circuit

model, as shown in Fig. 6.11, can be found from the following relationships:

G(1) = <(Yin,in + Yout,in)

C(1) = =(Yin,in + Yout,in)/ω

R = <(−Y−1out,in)

L = =(−Y−1out,in)/ω

G(2) = <(Yout,out + Yout,in)

C(2) = =(Yout,out + Yout,in)/ω

for Π circuit (6.3.32)

where the supraindex (1) indicates the input GC branch of the Π-circuit model

and the supraindex (2) indicates the output Π-circuit model GC branch.

Although this methodology was originally developed for small high-speed IC inter-

connections, it was proven to work very well for MTLs with lengths `MTL ≤ λmin/20

being λmin the wavelength for the maximum frequency propagated along the line.

145

Figure 6.11: MTL Π-circuit model

The major advantage of the direct calculation method is that a very accurate

model is obtained quickly. A disadvantage is that the model is only valid for lengths

that are small with respect to the wavelength.

In the next paragraphs the experimental results of a simulation and characteri-

zation of a 4-Ports asymmetric coupled microstrip line structure using the present

methodology will be discussed.

146

6.3.2 MTL simulation and experimental results

A 4-Port asymmetric coupled microstrip line structure was tested through the method

discussed in the last section. It was assumed that the signal paths are connected

from the ground plane to the signal conductors, and coupled modes are intended to

be propagated through the line.

A geometry of the structure with length `MTL = 1 mm is shown in Fig. 6.12

where the 30 µm wide signal conductor on the left is separated from the 200 µm wide

signal conductor on the right by a 50 µm wide gap. The 100 µm thick substrate has

a relative dielectric constant of 12.9. The 0.5 µm thick signal conductors and 5 µm

thick ground plane have a conductivity σ = 3.602× 107 S/m.

Figure 6.12: Asymmetric Coupled Microstrip Line

A Full Wave EM simulation of the structure’s behavior was performed by using the

MWOffice c© EM simulator of AWRr, based on the modified Spectral-domain method

of moments in a range of frequency from DC to 5 Ghz. Through the S-matrix the

above characterization method [38] was used to extract the circuit parameters and

the following results were obtained:

147

1 2 3 4 50

5

10

15

20

Frequency, GHz

R, Ω

/cm

R11

R22

R12

R21

Figure 6.13: per-unit-length R(f) Ω/cm matrix

1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency, GHz

L, n

Hy/

cm

L11

L22

L12

L21

Figure 6.14: per-unit-length L(f) nHy/cm matrix

148

1 2 3 4 50

0.5

1

1.5

2

2.5

Frequency, GHz

C, p

F/c

m C

11

C22

−C12

−C21

Figure 6.15: per-unit-length C(f) pF/cm matrix

As can be seen the skin effect influences the R matrix values as is evident from

Fig. 6.13. A frequency dependent behavior or the L matrix values is observed and

can be seen in graphic Fig. 6.14.

In the graphic of Fig. 6.15, the C matrix values are plotted with a quasi constant

behavior the along frequency bandwidth. A similar experiment was presented in [8]

and compared with our results, giving a very good agreement between them. This

fact evidences the power of the characterization methodology proposed and tested,

as it is compared with other different approaches.

On the other hand, the diagonal matrices modal attenuations [α] and the modal

refractive indices [ηeff ] were extracted by equating the formula (6.3.3) as follows:

[α] = <(Γ), and [ηeff ] = c[β]

ω(6.3.33)

giving the results shown in figures 6.16 and 6.17:

149

1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Frequency, GHz

α, d

B/c

m

modal attenuation αm1

modal attenuation αm2

Figure 6.16: Modal attenuation constant dB/cm

1 2 3 4 51.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Frequency, GHz

ηef

f

modal refractive index ηeff1

modal refractive index ηeff2

Figure 6.17: Modal Refractive index dB/cm

150

Finally, the influence of the modal cross powers for coupled modes along the struc-

ture was validated by calculating the ζnm merit coefficient index [50] from the ex-

tracted Z and Y matrices. As can be seen from Fig. 6.18, a small 1.5 % influence

of the modal cross powers is noted near DC frequency values.

0 1 2 3 4 50

0.5

1

1.5

2

Frequency, GHz

ζ mn (

%)

Figure 6.18: Modal Cross Power ζnm merit coefficient index

Chapter 7

Conclusions

7.1 Summary

As stated in the Introduction, the present work was conceived as a framework of ideas

focused on a methodology for characterizing high frequency waveguides on silicon

substrates.

A general methodology for RF lossy lines characterization was implemented and

tested. This methodology is based on the transmission line theory and on heuristic

assumptions that take care of deviations for the classical model.

To achieve realistic results it was necessary to explore the limitations of the Teleg-

rapher’s equation in order to understand the important phenomena at high frequency.

Accurate models and suitable parameters were analyzed and presented. A large part

the information in the present work derives from a large number of sources and a

particular effort was made to organize it into a coherent framework of tools for this

particular field of knowledge.

A quantification of the classical model deviation is presented as a starting point

for the development of further models.

151

152

A fully automated VNA driver under MATLAB environment has been developed

and used to do experimental measurements. Different measurements cases were stud-

ied, compared and used.

Particular attention was given on the TRL calibration technique as the more

suitable technique for waveguide and transmission line characterization.

Different measurements were made with the automated measurement tool devel-

oped ad hoc for the present work. Good agreement with theoretical behaviors of

coaxial media have proven the correct functionality of this setup.

Different methods to extract useful electrical pul parameters of transmission lines

from scattering matrix were explored, studied and compared. An extraction proce-

dure for single lossy lines has been tested by using experimental measurement on

silicon substrate lines. Good agreement was found with full wave simulations as

theoretical reference, giving a confirmation of the correctness of the characterization

methodology adopted.

Accurate models and extraction procedures of multi-conductor transmission lines

were tested and compared with the selected literature results. Also in this case there

has been a good agreement between the results obtained by the proposed approach

and the published ones.

153

7.2 Future works

The extension of the MTL extraction technique using experimental data requires the

implementation of a multiport calibration technique.

Different ways for the optimization of MTL characterization methods for lossy

lines accounting for topology asymmetries need to be explored.

The development of a general model with parameters that take into account the

effects of coupled modes in MTL is referred in the present work and even if its validity

has to be experimentally addressed.

The reward of a thing well done is to have done it.

Ralph Waldo Emerson

154

Out of Context... ?

What is truth?

As a question out of context it seems a pretext for a possible answer.

The truth, intended as a knowledge who has a particular meaning for our brain;

is based on the interpretation of the reality through experience and it needs to be

revised in case of misinterpretation.

Interpretation of the facts of Nature, the phenomena, is the aim of modern science,

where the method of inquiring is the most important thing to be solved.

As a dialogue between man and nature, the inquiring predisposes to the answers...

From Galileo to the Information Age; the question was the ”nature of things and

its relationships”, where the method of interpretation was the key for the construction

of knowledge.

After the consequences on the nature by the arbitrary use of the method of in-

quiring, and after all the knowledge, the following questions need to be answered:

It is possible to change the method of inquiring?

How to do with the knowledge?.

Technologies, can only answer the question of the efficient use of ”knowledge”.

Then, the ”sense of the use of the knowledge” needs to be answered.

A philosophical answer for the sense of the use of the knowledge can be The Truth,

intended as a relationship between man and nature.

Science needs to ask itself some philosophical question... as for example: which

are the ultimate scopes and attitudes who command this relationship?

The whole dialogue implicates a relationship, then, a sense can be defined into the

context of a relationship, but the sense for a relationship is the relationship itself!

The author’s experience taught him that the ultimate sense of knowledge is Love,

intended as a relationship; then, as a consequence, he believes that empirically :

155

156

The Truth is in the Truth

Love is Truth

Appendix A

See User’s Guide Draft Version attached

157

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