Waveguide characterization methodology on lossy silicon substrates by Pablo Federico Gabriel Silvoni

POLITECNICO DI TORINO SCUOLA DI DOTTORATO Dottorato in Dispositivi Elettronici â&#x20AC;&#x201C; XVII ciclo

Tesi di Dottorato

Waveguide Characterization Methodology on Lossy Silicon Substrates A theoretical and heuristic study

Pablo Silvoni

Tutore Prof. Giovanni Ghione

Coordinatore del corso di dottorato 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 째

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

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 and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Firmado digitalmente por Pablo Silvoni Nombre de reconocimiento (DN): cn=Pablo Silvoni, c=AR, email=pablosilvoni@yahoo.com Ubicación: Mar del Plata Fecha: 2014.05.05 12:43:55 -03'00'

Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED.

iii

To my Love and Inspiration: my dear wife Adriana and our three ”Rolling Stones”.

Table of Contents Table of Contents

List of Figures

viii

Abstract

Dedication

xii

Acknowledgements

xiii

1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transmission Line and Waveguide Theory 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 TEM mode propagation theory review . . . . . . . . . . . . . 2.2.1 Transmission Line description from Maxwell equations 2.2.2 Telegrapherâ&#x20AC;&#x2122;s equations and equivalent circuit model . 2.3 Multiconductor transmission line modelling . . . . . . . . . . . 2.4 Multimode description of MTL equations . . . . . . . . . . . . 2.5 Limitations of the quasi-TEM assumptions . . . . . . . . . . . 3 RF 3.1 3.2 3.3 3.4

Instruments and Tools Introduction . . . . . . . . . . . . . . . . . . . . . . . . Characterization of linear networks . . . . . . . . . . . Characterization problem in microwaves and millimeter Scattering parameters theory review . . . . . . . . . . .

. . . . . . . . waves . . . .

. . . . . . . . . . .

1 1 2 3

. . . . . . .

4 4 7 10 19 23 31 40

. . . .

46 46 47 50 54

3.5

. . . . . . . . .

60 60 61 62 64 66 67 70 74

4 Microwave and Millimiter Wave Measurement Techniques 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 VNA Calibration process . . . . . . . . . . . . . . . . . . . . . 4.3 Non Redundant Methods . . . . . . . . . . . . . . . . . . . . . 4.3.1 SOLT Calibration Technique . . . . . . . . . . . . . . . 4.3.2 QSOLT Calibration Technique . . . . . . . . . . . . . . 4.4 Self Calibration or Redundant Methods . . . . . . . . . . . . . 4.4.1 TRL technique . . . . . . . . . . . . . . . . . . . . . . 4.4.2 RSOL (UTHRU) technique . . . . . . . . . . . . . . .

. . . . . . . .

78 78 78 80 80 83 87 87 95

5 Calibration & Measurement Tool 5.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.2 MATLAB Calibration & Measurement Tool . . 5.3 Calibration & Measurement program . . . . . . 5.3.1 Switch Correction algorithm . . . . . . . 5.3.2 TRL algorithm and DUT deembedding . 5.3.3 Uploading and calibrated measurements 5.4 Coaxial Experimental Results . . . . . . . . . .

. . . . . . .

98 98 99 102 102 105 107 111

. . . . . . .

116 116 118 123 126 134 142 146

3.6

The Vector Network Analyzer . . . . . . . . . . . . 3.5.1 VNA General Description . . . . . . . . . . 3.5.2 Signal Source . . . . . . . . . . . . . . . . . 3.5.3 Test Set . . . . . . . . . . . . . . . . . . . . 3.5.4 Command Unit . . . . . . . . . . . . . . . . Systematic error removal and VNA calibration . . . 3.6.1 Measurement Errors . . . . . . . . . . . . . 3.6.2 Twelve Terms Error Model . . . . . . . . . . 3.6.3 Error Box Model (Eight-Term Error Model)

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

6 Networks characterization and parameter extraction 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Transmission line characterization methods . . . . . . . 6.2.1 Circuit parameters extraction from S-Matrix . . 6.2.2 On Wafer measurements and characterization . 6.3 MTL characterization methods . . . . . . . . . . . . . 6.3.1 MTL parameters extraction from S-Matrix . . . 6.3.2 MTL simulation and experimental results . . . .

. . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . .

7 Conclusions 151 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 154 Out of Context... ?

155

157

Bibliography

158

vii

List of Figures 2.1

Electromagnetic field structure of a TEM mode of propagation

. . .

2.2

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

. . . .

2.3

Derivation contours of the first transmission line equation

. . . . . .

2.4

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

2.5

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

2.6

Transmission Line equivalent lumped circuit model . . . . . . . . . .

2.7

Multiconductor Transmission Line system . . . . . . . . . . . . . . .

2.8

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

2.9

Conductor equivalent circuit of a MTL for two modes of propagation

3.1

Two Port Network Transmission line model

. . . . . . . . . . . . . .

3.2

Power Waves and Reference Planes interpretation . . . . . . . . . . .

3.3

Equivalent circuit of a linear generator . . . . . . . . . . . . . . . . .

3.4

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

3.5

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

3.6

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

3.7

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

3.8

Twelve Terms Error Model Forward Set . . . . . . . . . . . . . . . .

3.9

Twelve Terms Error Model Reverse Set . . . . . . . . . . . . . . . . .

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

3.11 An interpretation of the Error Box Model

. . . . . . . . . . . . . . .

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

4.1

viii

4.2

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

4.3

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

4.4

D.U.T. Setup Measurement Fixture . . . . . . . . . . . . . . . . . . .

5.1

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

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

. . . . . 103

. . . . . . . . . . . . . . . . . . . . . . . 115

5.10 LINE ηef f 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

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

6.7

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

6.8

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

6.9

Refraction index ηef f

. . . . . . . . . . . . . . . . . . 130

. . . . . . . . . . . . . . 132

. . . . . . . . . . . . . . . . . . . . . . . . . . 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

6.16 Modal attenuation constant dB/cm . . . . . . . . . . . . . . . . . . . 149 6.17 Modal Refractive index dB/cm . . . . . . . . . . . . . . . . . . . . . . 149 6.18 Modal Cross Power Îśnm merit coefficient index . . . . . . . . . . . . . 150

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. Validity and limitations were stated and discussed. An overview of the state of the art of the experimental characterization for linear networks, microwave and millimeter measurement 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. Experimental 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 parameters was implemented and experimental and simulated results were compared and discussed. Multi-transmission line characterization methods were discussed and compared. 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.

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.

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Most of all Iâ&#x20AC;&#x2122;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 Febbruary 1st, 2005

Pablo Silvoni

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 highspeed 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 development of more accurate models of devices and inter-chip interconnections because

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

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 Userâ&#x20AC;&#x2122;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 Waveguide Theory 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 Telegrapherâ&#x20AC;&#x2122;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 engineering. 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

First, the key assumptions of these theoretical descriptions will be aimed at understanding 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 structureâ&#x20AC;&#x2122;s largest dimension is electrically small, i.e., much less than a wavelength, we can approximately lump the EM effects into circuit elements as in lumpedcircuit theory and define alternative variables of interest such as voltages and currents. 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 represent a complete analysis. Other aspects, such as imperfect line conductors, also

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 Maxwellâ&#x20AC;&#x2122;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 experimentally be measured as the scattering matrix, and all other two-port descriptions defined for TEM structures, will be considered.

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 propagating 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 subscript to denote transverse. It is assumed that the medium is homogeneous, linear and isotropic and characterized by the scalar parameters of electric permitivity ², magnetic permeability µ and conductivity σ. Then Maxwell’s equations become: − → → − ∂Ht ∇ × Et = µ ∂t

(2.2.1)

− → − → − → ∂Ht ∇ × Ht = σEt − ² ∂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:

∂ ∂ + yˆ · ∂x ∂y ∂ = zˆ · ∂z

∇t = xˆ · ∇z

being xˆ, yˆ and zˆ 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: → − − → ∂Et ∂Ht zˆ × = −µ ∂z ∂t

(2.2.3)

− → → − − → ∂Ht ∂Ht zˆ × = σEt − ² ∂z ∂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 → − ∇t · H t = 0

∇2t Φ(x, y) = 0 ∇2t Ψ(x, y) = 0

(2.2.7)

Equations (2.2.7) show that these scalar potential functions satisfy Laplace’s equation in any transverse plane as they do for static fields. This permits the unique

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: Z

V (z, t) = −

− → ~ E t · dl

(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

− → ~ H t · dl

I(z, t) = −

(2.2.9)

These results can be applied to a TEM field structure propagating along a uniform transmission line with two parallel conductors to obtain the transmission line equations as shown in the following section.

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, ² and µ. Maxwell’s equations in integral form are:

− → ~ ∂ E · dl = −µ ∂t c

− → ~ H · dl = c

− → H · d~s

(2.2.10)

− → ∂ J · d~s + ² ∂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:

∂ (Ex dx + Ey dy) = −µ ∂t c

ZZ (Hx dx + Hy dy) = c

ZZ Hz dxdy = 0

(2.2.12)

∂ Jz dxdy + ² ∂t s

ZZ Ez dxdy

(2.2.13)

ZZ =

Jsz dxdy s

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, independent 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: Z

V (z, t) = −

− → ~ E t · dl

(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

− → ~ H t · dl

I(z, t) = −

(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

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:

Z 1

− → Ez · d~l +

3 2

− → Et · d~l +

4 3

− → Ez · d~l +

1 4

− → ∂ Et · d~l = −µ ∂t

− → Ht · d~s

(2.2.16)

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

Figure 2.3: Derivation contours of the first transmission line equation assumption of Ez = 0 gives: Z

V (z + ∆z, t) = − Z2 1 V (z, t) = −

− → Et (x, y, z + ∆z, t) · d~l − → Et (x, y, z, t) · d~l

Therefore (2.2.16) becomes: ∂ V (z + ∆z, t) − V (z, t) = −µ ∂t

− → Ht · d~s s

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

∂ ∂ 1 V (z, t) = −µ lim ∂z ∂t ∆z→0 ∆z

− → Ht · d~s s

(2.2.17)

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 inductance L for a ∆z section is: φ L= = −µ I

. − → Ht · d~s I s

Now it can be defined a per-unit-length (pul) inductance, L, at any cross section of the uniform line as: L L = lim = −µ ∆z→0 ∆z

. − → Ht · ~ndl

− → 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: ∂ ∂ V (z, t) = −L I(z, t) ∂z ∂t

(2.2.19)

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: ZZ

− → ∂ J · d~s = − Qenc ∂t Sv

(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: ZZ

− → J · d~s

ZZ +

− → ∂ J · d~s = − Qenc ∂t Se

(2.2.21)

The terms in last equation become: ZZ

→ − J · d~s = I(z + ∆z, t) − I(z, t);

(2.2.22)

→ − 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 capacitance C. From Gauss’ law, the total charge enclosed by a closed surface Sv is: ZZ

− → E · d~s

Qenc = Sv

(2.2.24)

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

C =

then, by substituting (2.2.24) and observing Fig. 2.4, the per-unit-length capacitance C is defined as: C C = lim = − ∆z→0 ∆z

. − → Et · ~nd~l

− → Et · d~l

(2.2.25)

Similarly, the conductance between the two conductors for a ∆z section can be defined as: ZZ

. − → J · d~s V (z, t)

G =

(2.2.26)

Then, from (2.2.23) we can define per-unit-length conductance G as: G = −σ G = lim ∆z→0 ∆z

. − → Et · ~nd~l ct

− → 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: ∂ ∂ I(z, t) = −GV (z, t) − C V (z, t) ∂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).

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 directed 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 component 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 assumption 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 homogeneous surrounding medium invalidates the basic assumption of a TEM field

structure because different portions of this medium are characterized by different dielectric constants ²i and magnetic permeabilities µi . Then, the phase velocities vphi √ (vphi = 1/ ²i µi ) 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. Nevertheless, 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 ²ef f is defined so that if the transmission line conductors are immersed in a homogeneous dielectric having this ²ef f , 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.

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 electrically 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 substructure 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 circuits 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:

∂ ∂ V (z, t) = −RI(z, t) − L I(z, t) ∂z ∂t ∂ ∂ I(z, t) = −GV (z, t) − C V (z, t) ∂z ∂t

(2.2.29) (2.2.30)

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]:

∂ V (z, ω) = −(R + jωL)I(z, ω) ∂z ∂ I(z, ω) = −(G + jω C)V (z, ω) ∂z

(2.2.31) (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=−

R + jωL

G + jω C

# (2.2.34)

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

V (z, ω)

" = Ev · exp(Λ.z) ·

I(z, ω)

V0+

# (2.2.35)

V0−

Constants V0+ and V0− are determined by the environment conditions. Ev is the eigenvector matrix and Λ is the eigenvalue matrix of the system given by: " Ev =

Yc −Yc

# ,

and

Λ=

−γ 0 0

# (2.2.36)

The variable γ is the propagation constant and Zc = Yc−1 is the characteristic impedance of the system expressed as functions of the pul parameters R, L, G and C and given by:

γ=

(R + jωL) · (G + jω C)

(2.2.37)

s Zc =

(R + jωL) (G + jω C)

(2.2.38)

These are the fundamental parameters that describe the behavior of the transmission line. The propagation constant γ can be expressed as a complex number as follows:

γ = α + jβ

(2.2.39)

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) = V0+ · e−γ.z + V0− · eγ.z = V + (z) + V − (z)

(2.2.40)

V0+ −γ.z V0− γ.z ·e − · e = I + (z) + I − (z) Zc Zc

(2.2.41)

I(z, ω) =

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 measurements 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.

2.3

Multiconductor transmission line modelling

The results obtained in the previous chapter for the general properties of a two conductors 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 section 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 Telegrapherâ&#x20AC;&#x2122;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 Îłmi and modal characteristic impedance Zmi , can not be measured directly. The more common description of the system starts by applying Kirchoffâ&#x20AC;&#x2122;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 P law gives I0 = nk=1 Ik .

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:  V1 (z, t)   ..   .      V(z, t) =  Vi (z, t)     . .   .   Vn (z, t) 

 I1 (z, t)   ..   .      I(z, t) =  Ii (z, t)     . .   .   In (z, t) 

(2.3.1)

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 .. .

···

(R2 + R0 ) · · · .. .. . .

R0 .. .

···

     

(2.3.2)

(Rn + R0 )

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:  ψ1  .   ..      =L·I Ψ= ψ i    .   ..    

(2.3.3)

ψn

where the per-unit-length inductance matrix L contains the individual per-unitlength self-inductances, Lii , of the circuits and the per-unit-length mutual-inductances between the circuits, Lij :



 L11 L12 · · ·

L1n

  L  21 L22 · · · L2n L= . .. .. ..  .. . . .  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 perunit-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:  P n G1k −G12 ··· −G1n  k=1 P n  −G −G2n 21  k=1 G2k · · · G= . . .. ... .. ..  .  Pn −Gn1 −G2n ··· k=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-unitlength capacitance matrix C defined as:  P n C1k −C12 ··· −C1n  k=1 P n  −C −C2n 21  k=1 C2k · · · C= . . .. ... .. ..  .  Pn −C2n ··· −Cn1 k=1 Cnk

      

(2.3.6)

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:  Q1  .   ..       Q =  Qi  =C·V  .   ..    

(2.3.7)

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

∂ ∂ V(z, t) = −RI(z, t) − L I(z, t) ∂z ∂t ∂ ∂ I(z, t) = −GV(z, t) − C V(z, t) ∂z ∂t

(2.3.8) (2.3.9)

To find the solutions of the above MTL equations, the frequency-domain representation 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) = <{V˜i (z)ejωt }

(2.3.10)

Ii (z, t) = <{I˜i (z)ejωt }

(2.3.11)

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:

∂ ˜ V(z) = −Z · ˜I(z) ∂z ∂˜ ˜ I(z) = −Y · V(z) ∂z

(2.3.12) (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 assumed that the per-unit-length parameter matrices R, L, C and G are time independent, 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: ∂ ˜ ˜ X(z) = A(z) · X(z) ∂z

(2.3.15)

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

˜ V(z) ˜I(z)

" A(z) =

−Z

−Y

# (2.3.16)

Then, by using the results of state-variable equations a 2n × 2n state-transitionˆ matrix, Φ(z), can be defined with the following properties:

ˆ Φ(0) = I2n

being

I2n

the 2n × 2n identity matrix

z2 2 z3 3 z I2n + A+ A + A +· · · 1! 2! 3!

ˆ −1 (z) = Φ(−z) ˆ Φ ˆ Φ(z)

exp{Az}

(2.3.17)

The general solution is found straightforwardly from the state-variable equations solutions assuming that the initial states are zero and giving: ˜ ˆ − z0 ) · X(z ˜ 0) X(z) = Φ(z

(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)

" ˆ = Φ(z) ·

˜ V(0) ˜I(0)

" =

ˆ 11 (z) Φ ˆ 21 (z) Φ

ˆ 12 (z) Φ ˆ 22 (z) Φ

# " ·

˜ V(0) ˜I(0)

# (2.3.19)

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

ˆ In order to solve (2.4.1) by finding the chain parameter matrix Φ(z), the uncoupled 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 deˆ 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.

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 twodimensional 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 simplest 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 Vm Im where Vm and Im are the modal voltage

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

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 equivalentcircuit 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

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 Im 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 ˜ ˜ V(z) = ZY · V(z) ∂z 2 ∂2 ˜ I(z) = YZ · ˜I(z) ∂z 2

(2.4.1) (2.4.2)

˜ and ˜I are the column vectors of conductor voltages and currents. where the V These magnitudes can be defined as to be arbitrary invertible linear transformations ˜ m and ˜Im as: of the modal voltages and currents V ˜ = Mv · V ˜m V ˜I = Mi · ˜Im

(2.4.3)

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

˜ and ˜I, and the between the actual phasor line or conductor voltages and currents, V ˜ m and ˜Im . modal voltages and currents V In order to be valid, these n × n transformation matrices must be nonsingular, −1 must exist in order to go between both sets and the inverse matrices M−1 v and Mi

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

˜m V ˜Im

" =

−M−1 v ZMi

−M−1 i YMv

# " ·

˜m V ˜Im

# (2.4.4)

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

the per-unit-length modal impedance and admittance matrices Zm and Ym can be defined as:     Zm = M−1 ZM =  i v  



Zm1

0 .. .

Zm2 .. .

···

Ym1

  0  −1 Ym = Mi YMv =  .  ..  0

0 Ym2 ... ···

··· ...

0 .. .

Zmn

··· ...

0 .. .

...

Ymn

      

(2.4.5)

      

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

(2.4.6)

∂2 ˜ ˜ m (z) = γ 2 · V ˜ m (z) Vm (z) = Zm Ym · V 2 ∂z ∂2 ˜ Im (z) = Ym Zm · ˜Im (z) = γ 2 · ˜Im (z) ∂z 2

(2.4.7) (2.4.8)

with the modal propagation constant matrix γ defined as:     2 γ = Zm Ym = Ym Zm =   

γ12 0 .. .

··· .. . γ22 ... ...

···

 0 ..  .    0   2 γn

(2.4.9)

Now, a straightforward solution for the modal uncoupled equations (2.4.7) and (2.4.8) is:

˜ m (z) = V ˜ + e−γz + V ˜ − eγz V m m

(2.4.10)

˜Im (z) = ˜I+ e−γz − ˜I− eγz m m

(2.4.11)

where the matrix exponentials e±γz are defined as:  e±γ.z

   =  

e±γ1 z

0 .. .

e±γ2 z .. .

···

··· ...

0 .. .

e±γn z

      

(2.4.12)

where each mode is described by a couple of travelling waves, the vectors modal ˜ − , ˜I− . ˜ + , ˜I+ and the vectors modal backward intensities V forward intensities V m m m m Then, substituting the similarity transformations Mv and Mi into equations (2.3.12) and (2.4.2) implies:

∂2 ˜ ˜ V(z) = Mv γ 2 M−1 v V(z) ∂z 2 ∂2 ˜ ˜ I(z) = Mi γ 2 M−1 i I(z) ∂z 2

(2.4.13) (2.4.14)

where the matrices ZY and YZ are related to γ 2 by the similarity transformations and per-unit-length modal impedance and admittance matrices Zm and Ym as follows:

−1 2 ZY = Mv Zm Ym M−1 v = Mv γ Mv

(2.4.15)

−1 2 YZ = Mi Ym Zm M−1 i = Mi γ Mi

(2.4.16)

Thus all four matrices have the identical eigenvalues γ 2 . The similarity transformation 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)

where the superscript

indicates the Hermitian adjoint (conjugate transpose)

and I is the identity matrix. Then, with this assumption Mi = M−1 = M, the v characteristic impedance matrix ZC can be defined as: ZC = Y−1 MγM−1 = ZMγ −1 M−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: ˜ ˜ + e−γz + V ˜ − eγz ) V(z) = ZC M · (V m m ˜I(z) = M · (˜I+ e−γz − ˜I− eγz ) m m

(2.4.19) (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: √

and

YZ = MγM−1

³√ ´−1 √ ZC = Y−1 YZ = Z YZ

(2.4.21)

(2.4.22)

The above definitions are used into (2.4.19) and (2.4.20) and its results are substiˆ ij (z) tuted in (2.3.18). Then, using the properties of the exponential matrices, the Φ ˆ terms of the chain parameter matrix Φ(z), that represent the general solution of the MTL equations, are given by: " ˆ Φ(z) =

ˆ 11 (z) Φ ˆ 21 (z) Φ

ˆ 12 (z) Φ ˆ 22 (z) Φ

´ ³√ ´  YZ −ZC sinh YZ ´ ³√ ´  (2.4.23) ³√ = −1 −ZC sinh YZ cosh YZ 

cosh

³√

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 equivalent 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 ˜ mk are V˜mk and I˜mk and modal electric fields and magnetic field strengths E˜mk and H given by: X V˜mk

(z) · E˜mk (x, y) ˜0k V k X I˜mk ˜ mk (x, y) Ht (x, y, z) = (z) · H I˜0k Et (x, y, z) =

(2.4.24) (2.4.25)

where the normalizing voltage V˜0k and current I˜0k are restricted by: Z P0k =

∗ V˜0k I˜0k

≡ S

∗ ˜ mk E˜mk × H · 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 ∗ . The characteristic impedance of the kth mode forward direction is given by V˜mk I˜mk ∗ is ZCk ≡ V˜0k /I˜0k = |V˜0k |2 /P0k = P0k /|I˜0k |2 ; its magnitude is fixed by the choice of

|V˜0k | or |I˜0k | while its phase is fixed by (2.4.26). With this definition, ZCk corresponds 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: 

  Zm1 · · · 0 γ1 Z C 1 · · · 0  .   . . .. .. .. . ..  =  .. . . Zm = γZC =  .  .   .. .. . Zmn . γn ZCn 0 0

   

(2.4.27)

In the following paragraphs the topics that represent limitations of quasi-TEM assumptions for the MTLs and the evaluation of the model in the case of lossy structures will be discussed.

2.5

Limitations of the quasi-TEM assumptions

One of the more important facts that defines the TEM structures for TEM propagating fields, is the assumption that the different modes propagating along the structure are TEM or quasi-TEM and orthogonal. In high speed electronic circuits different 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 =

±γn z c± (Etn ± Ezn z) ne

(2.5.1)

±γn z c± (±Htn + Hzn z) ne

(2.5.2)

H =

X n

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: Z X γn z −γn z γm z −γm z ∗ P = E × H∗ · zdS = (c+ + c− )(c+ + c− ) Pnm ne ne me me

(2.5.3)

Z and

Pnm =

∗ · zdS Etn × Htm

(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.

Lossless modes are power orthogonal when they are not degenerate; that is, their 2 modal cross powers Pnm are zero when γn2 6= γm . Most equivalent circuit descriptions

for MTLs assume power orthogonal modes, that are congruent with quasi-TEM assumptions. 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 degeneracies 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 influence 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 =

Pnm Pmn Pnn Pmm

(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

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 =

where the superscript

[b(λm )T a(λn )][b(λn )T a(λm )] [b(λn )T a(λn )][b(λm )T a(λm )]

(2.5.6)

signifies Hermitian adjoint (conjugate transpose) and

a(λm ) and a(λn ) are the eigenvectors of β = YZ with eigenvalues λn = γn2 and 2 λ m = γm , 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 symmetric, then β = αT , where the superscript

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 powernormalized 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 powernormalization and the modal cross powers on the equivalent circuit parameters, that is consistent with the modal representation of the present work.

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: Z P=

X V˜mj (z) I˜∗ (z) Z mk ˜ ∗ · zdS E × H · zdS = E˜mk × H mk ∗ ˜ ˜ V0j I0k S j,k ∗

(2.5.7)

˜ mk the modal electric fields and magnetic field strengths. This being E˜mk and H can be put into the more compact form:

T ˜m P = ˜Im · X · V

(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 · = ∗ V˜0j I˜0k

Z S

˜ ∗ · zdS E˜mk × H mk

(2.5.9)

This cross-power matrix X takes into account the influence of all modes propagating 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 result of the power-normalization (2.4.26). When the a conductor equivalent circuit representation is given, the (2.5.8) can be written as:

−1 ˜ T P = ˜I (M−1 i ) · X · Mv V

(2.5.10)

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

=⇒

T ˜ P = ˜I · 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 = · V˜0j

Z E˜mk · dl

∀j

=⇒

V˜0j =

E · dl

(2.5.12)

T T T Then Mi would be given by Mi = (XM−1 v ) = (Mv )X . Another choice could

be used by defining first the Mi by fixing the conductor currents, and then Mv would be determined from Mv = (MTi )−1 X. 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

symmetric with the requirement that Mtv Mi becomes diagonal implying that Z = Zt and Y = Yt (the superscript

indicates transpose matrix) [16]. The requirement

that Mtv Mi diagonal is not always compatible with the condition MTi Mv = X as discussed in [8]. But with high lossy lines this orthogonality is lose and X 6= I. Then the product Mtv Mi 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 Îśnm 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 experimental 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 described putting emphasis on the microwave metrology problematic. Measurement error models, their physical causes and removal procedures will be presented and discussed.

3.2

Characterization of linear networks

The theory of transmission lines is called distributed circuit analysis, and it is intermediate 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. Inductors can only store magnetic energy, whereas capacitors only store electric energy. • Application of the Maxwell equation (∇ · J = −∂ρ/∂t = 0) for charge conser¯ P vation at nodes gives Kirchoff’s current law: iκ = −∂q/∂t¯nodes = 0.

Figure 3.1: Two Port Network Transmission line model • Application of the Maxwell equation (∇ × E = −∂B/∂t = 0) at loops gives ¯ P 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: # # " # " " V1 cosh(γ`) Zc · sinh(γ`) V2 (3.2.1) · = cosh(γ`) I1 Zc−1 · sinh(γ`) I2

Then it is possible, by doing linear transformations, to describe the linear transmission lines or waveguides with their impedance and/or admittance matrices.

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

# " ·

(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 independent. 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.

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 Telegrapherâ&#x20AC;&#x2122;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

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 ² 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 invalidate 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 represent 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 inhomogeneous 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 ²1 , µo and ²2 , µo . The classical way of characterizing this

situation is to obtain an effective dielectric constant ²ef f , defined such that if the line conductors are immersed in a homogeneous dielectric with a dielectric constant ²ef f , 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 equation 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 electrically long, i.e., much greater than a wavelength λ, and they are properly handled by lumped-circuit characterization. However structures whose cross-sectional dimensions 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 simultaneously 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

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 â&#x20AC;?rough estimateâ&#x20AC;? 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 VNAâ&#x20AC;&#x2122;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.

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 conventional 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.

Figure 3.2: Power Waves and Reference Planes interpretation To clear the concept a brief discussion follows. By considering a n-port electromagnetic 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: Vi + Zi Ii ai = p 2 | <(Zi ) |

Vi − Zi∗ Ii bi = p 2 | <(Zi ) |

(3.4.1)

and by inverting the system, 1

Vi = p

| <(Zi ) |

· (Zi∗ ai + Zi bi )

Ii = p

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

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 = Ziâ&#x2C6;&#x2014; 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 â&#x2C6;&#x2019; Zi Ii . 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)

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: <{Vi Ii∗ } = |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= Zi∗ 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 0

reflected powers because there is a linear relation between ai s and bi s 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

in terms of V and I as follows: b = F (V + G∗T · I)

a = F (V + G · I), Where

∗T

(3.4.6)

denotes conjugate transposed matrix and: ³ p ´ 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)−1 F −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 = <{Vi Ii∗ } = |Vi+ |2 /Zci − |Vi− |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: Vi+

ai = p

| <(Zci ) |

bi = p

Vi− | <(Zci ) |

(3.4.11)

the total power into each ith port junction becomes: Pi = |ai |2 â&#x2C6;&#x2019; |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.

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 instrumentâ&#x20AC;&#x2122;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 transmitted 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

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.

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. Discrete frequencies measured range from 51 up to 801 samples in a single sweep. Frequency 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.

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 oneport 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.

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 receives the four 20 Mhz channels from the Test Set unit, and provides another heterodyne 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

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 microprocessor. 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.

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 cannot easily be connected directly to VNA-supporting coaxial or waveguide media. Often, 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 becomes 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.

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 contributions, 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

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 standards. 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: â&#x20AC;˘ Error Model Definition â&#x20AC;˘ Calibration Process â&#x20AC;˘ 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.

In the frequency domain, all known calibration techniques are based on the insertion 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 calibration 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.

3.6.2

Twelve Terms Error Model

Historically the Twelve Terms Error Model was developed from the causes of measurement 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 transmission 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.

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 developed 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 ET F and Reverse Transmission Tracking ET R . Twelve Terms Error Model Forward Set is shown in Fig. 3.8 and Reverse Set is shown in Fig. 3.9.

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:  · i¸ h¡ i ¡ S −E ¢ h ¡ S −E ¢ ¢ ¡ S −E ¢  S21M −EXF  11M DF . 1+ 22M DR .E 12M XR .E  . − SR LF ERF ERR ET F ET R     S = 11A  (Deno)       h ¡  ¢¡ ¢i ¡ S −E ¢   S22M −EDR XF  1+ . E −E . 21M SR LF  ERR ET F    (Deno)  S21A = h ¡  ¢¡ ¢i ¡ S −E ¢   S11M −EDF XR  1+ . ESF −ELR . 12M  ERF ET R   S12A =  (Deno)        ·  i¸ h¡ i  ¡ S −E ¢ h ¡ S −E ¢ ¢ ¡ S −E ¢  S21M −EXF  22M DR . 1+ 11M DF .E 12M XR .E − .  SF LR ERR ERF ET F ET R    S22A = (Deno) · Deno = 1 +

¸· ¸ ´ ³S − EDF ´ 22M − EDR .ESF . 1 + .ESR − ERF ERR · ¸ ´ ³S ´ ³S 12M − EXR 21M − EXF . .ELF .ELR − 1+ ET F ET R

³S

(3.6.1)

11M

(3.6.2)

Figure 3.8: Twelve Terms Error Model Forward Set

Figure 3.9: Twelve Terms Error Model Reverse Set

3.6.3

Error Box Model (Eight-Term Error Model)

The most modern formulation of measurement errors is the physical model of systematic errors. The concept is based in a Ideal Free Error VNA, connected to the D.U.T through two â&#x20AC;?black boxesâ&#x20AC;?, 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 :

Figure 3.10: Ideal Free Error VNA and Error Boxes

b1m

# = EA 路

a1 "

b2m

a2m b2

" ,

= EB 路

a1m

with

EA =

# with

EB =

# (3.6.3)

e10 e11 A A "

e00 e01 A A

e00 e01 B B

# (3.6.4)

e10 e11 B B

In matrix notation, relationships at each port can be written as follows: "

b1m a1m

" = Ta 路

b1 a1

" and

a2m b2m

" = Tb 路

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

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:

1 Ta = 10 · eA 1 Tb = 10 · eB

−∆A e00 A −e11 A

−e11 B

e00 B

−∆B

≡

1 · Xa e10 A

11 01 10 with ∆A = e00 A · eA − eA · eA

(3.6.6)

≡

1 · Xb e10 B

11 01 10 with ∆B = e00 B · eB − eB · eB

(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: ¡ ¢−1 Tm = Ta · Td · Tb

(3.6.8)

Thus the deembedding formula that gives Td , the D.U.T. cascading matrix , is obtained just by inverting (3.6.8) as is shown: ¡ ¢−1 ¡ ¢−1 Td = Ta · Tm · Tb = α−1 · Xa · Tm · Xb

(3.6.9)

with

α=

e10 B e10 A

(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 rewriting terms of the error box matrices and the deembedding formula as follows: " Ta = p · Xa = p ·

k p k p

·a b 1

¡ ¢−1 Td = α−1 · Xa · Tm · Xb

" ,

Tb = w · Xb = w ·

with

1 f

α=

u w u w

p w

# ·g

(3.6.11)

(3.6.12)

Chapter 4 Microwave and Millimiter Wave Measurement 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. Limitations of the different techniques will conduce to use them in diverse environments (coaxial, microstrip lines, etc.). The deembedding process as the major characterization 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.

Calibration techniques in frequency domain are based on the insertion of standard 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 remote 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 standard 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

[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 procedure 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)

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 ET F for the forward case, and doing the same with the source at Port 2 ET R is obtained. Measured and actual transmission coefficients are equated by:

S21M = S21A · ET F

S12M = S12A · ET R

(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 exhibit 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. Alternatively, commercial VNAs describe transmission lines in terms of a delay coefficient with a small resistive loss component. The open standard exhibits further imperfections and is often described in terms of a frequency-dependent fringing capacitance expressed as a polynomial expansion. Standard models need to be provided by calibration 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.

4.3.2

QSOLT Calibration Technique

An improvement for the SOLT calibration technique was invented by Pisani and Ferrero [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. Influence 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: 1 Ta = e01 · A · t11

Tb =

with

e01 A

1 · · t12

−∆A e00 A −e11 A

−e11 B

e00 B

−∆B

11 01 10 ∆A = e00 A · eA − eA · eA

and

≡

1 · Xa e10 A

(4.3.4)

≡

1 · Xb e10 B

(4.3.5)

11 01 10 ∆B = e00 B · eB − eB · eB

Where the T Matrix coefficients are expressed as follows:

10 t11 = e01 A eA ,

with

10 t12 = e01 A eB ,

01 t21 = e10 A eB ,

10 t22 = e01 B eB

10 −1 t22 = e01 B · eB = t21 · t12 · t11

(4.3.6) (4.3.7)

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 11 10 01 Error Box parameters but only three: e00 A , eA and the product t11 = eA · eA . The

following relationship is given between the measured Γm and the actual Γa standard reflection coefficients: Γm = e00 A +

01 e10 A · eA · Γa 1 − e11 A · 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 11 e00 A , eA 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 · Xb−1

(4.3.9)

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: −1 Xb = Ttm · Xa · Tat

(4.3.10)

The Xb Error Box terms are calculated with the following formulae:  11 −1 21   e00 B = Xb · (Xb )     11 12 11 −1    eB = −Xb · (Xb ) t22 = det(Xb ) · (Xb11 )−2     t12 = (Xb11 )−1      t21 = t11 · det(Xb ) · (X 11 )−1 b

(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      S 21tm  S12tm      S 22tm

11 11 11 −1 = e00 A + (t11 · eB ) · (1 − eA · eB ) 11 −1 = t21 · (1 − e11 A · eB ) 11 −1 = t12 · (1 − e11 A · eB )

(4.3.12)

11 11 11 −1 = e00 B + (t22 · eA ) · (1 − eA · eB )

Equating (4.3.7) with the above equation system (4.3.12) the Xb Error Box coefficients are encountered:  £ ¤−1   e11 = (S11tm − e00 ) · t11 + e11 · (S11tm − e00 ) B A A A     11 11    t21 = S21tm · (1 − eA · eB ) 11 t12 = S12tm · (1 − e11 A · eB )    −1 11 2  t22 = S21tm · S12tm · (1 − e11  A · eB ) · t11     e00 = S22tm − t22 · e11 · (1 − e11 · e11 )−1 B A A B

(4.3.13)

The QSOLT improvement is the reduction of the number of standards to be connected 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.

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

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 representation 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: ¡ ¢−1 TmT = Ta · TT · Tb

(4.4.1)

¡ ¢−1 TmL = Ta · TL · Tb

(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)

and RN = (TmT )−1 .TmL = [Ta .TT .(Tb )−1 ]−1 Ta .TL .(Tb )−1

(4.4.4)

= Tb .(TT )−1 .TL .(Tb )−1 = Tb .RS .(Tb )−1

Matrices RM and RT have the same eigenvalues as RN and RS given by the following eigenvalue matrix:

" Λ=

λ1

λ2

# (4.4.5)

The RM , RT , RN , RS eigenvector matrices are given by M , T , N and S respectively, then it follows: RM = M.Λ.M −1 = Ta .T.Λ.(Ta−1 .T −1 )

with

Ta = M · T −1

(4.4.6)

RN = N.Λ.N −1 = Tb .S.Λ.(Tb−1 .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.`

e+γ.`

(4.4.8)

By replacing the actual Thru and Line transmission matrices TL and TT in equation (4.4.3) we have: " RT = TL .TT−1 = T.Λ.T −1 =

e−γ.∆`

e+γ.∆`

# =Λ

(4.4.9)

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 ·

k/p "

Tb = N = w · Xb = w ·

a · k/p b

# (4.4.10)

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

· ¸ p 1 = · RM 11 + RM 22 ± 4.RM 12 RM 21 + (RM 11 − RM 22 )2 2

(4.4.13)

The normalized eigenvectors of RM and RN are computed as [30]:

RM 12 λ1 − RM 11

RM 12 λ2 − RM 11

(4.4.14)

λ1 − RN 11 RN 12

λ2 − RN 11 RN 12

(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/p

Γm2 =

f + g · Γa · u/w 1 + Γa · u/w

(4.4.16)

The measured Thru input reflection coefficient SmT 11 gives the following equation:

SmT 11 =

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: s Γa = ±

(b − Γm1 ).(f − Γm2 ).(SmT 11 − a) (a − Γm1 ).(g − Γm2 ).(SmT 11 − 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: Γm1 − b k = p (a − Γm1 ).Γa

u Γm2 − f = w (g − Γm2 ).Γa

(4.4.19)

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

" · Td ·

1 u/w

#−1

f g · u/w

(4.4.20)

From the Thru measurement, the transmission coefficient SmT 21 is obtained and the α coefficient is given by:

α=

u/w · (g − f ) (1 − u/w · k/p) · SmT 21

(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 characteristic impedance of the Line. • The TRL has frequency limitations and it needs multiple lines to cover a broadband. It is necessary that ∆` = `line −`thru 6= n·λ/2 because at these frequencies the algorithm doesn’t work and produces ill conditioned matrices.

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 bandwidths, 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

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 account 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 considerations 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 .

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:

1 Ta = 10 · eA 1 Tb = 10 · eB

−∆A e00 A −e11 A

−e11 B

e00 B

−∆B

≡

1 · Xa e10 A

11 01 10 with ∆A = e00 A · eA − eA · eA

≡

1 · Xb e10 B

11 01 10 with ∆B = e00 B · eB − eB · eB

¡ ¢−1 ¡ ¢−1 Td = Ta .Tm .Tb = α−1 . Xa .Tm .Xb

with

α=

e10 B e10 A

As in the SOLT calibration technique it is necessary to take two 1-Port Measurements 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 = e00 A +

01 e10 A · eA · Γa 1 − e00 A · Γa

(4.4.22)

and by connecting three known standards: short, open and load , it is possible to 11 have 3 independent equations and to calculate the desired error terms e00 A , eA and 01 00 the product e10 A · eA . The same reasoning applied at Port 2 gives the error terms eB , 10 01 e11 B and the product eB · eB . 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, s 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)

which is fully known from the above measurements. Then, by applying (3.6.5) the Thru S21 scattering parameter is given by:

S21thru =

Îą 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 Îą 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 deembedding

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. Deembedding 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, giving 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.

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 particular 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 measurement 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 coefficients. 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 propagation constant Îł and the actual Reflect standard Î&#x201C;a 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 account 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 calculation. 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: "

b1m

" =

b2m

Sm11 Sm12 Sm21 Sm22

# " 路

a1m

# (5.3.1)

a2m

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: "

b1m 00

b2m

" =

Sm11 Sm12 Sm21 Sm22

# " 路

a1m 00

a2m

# (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

" =

bm1 bm2 bm1 bm2

# " Âˇ

am1 am2 am1 am2

#â&#x2C6;&#x2019;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 transforming 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 respectively of RM and RN are calculated and given as: " [M ] = eig(RM ) =

M11 M12 M21 M22

" [N ] = eig(RN ) =

N11 N12

# (5.3.5)

N21 N22

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

" Tb = N = w · Xb =

wf ug

# (5.3.6)

106

And the a, b, f and g coefficients are calculated as follows:

M11 M21

M12 M22

N21 N11

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 SmT 11 ; the actual Reflect Γa is calculated by solving (5.3.8) as: s Γa = ±

(b − Γm1 ).(f − Γm2 ).(SmT 11 − a) (a − Γm1 ).(g − Γm2 ).(SmT 11 − b)

(5.3.8)

Using the above result and data, the coefficients k/p and u/w are calculated by the algorithm as: k Γm1 − b = p (a − Γm1 ).Γa

u Γm2 − f = w (g − Γm2 ).Γa

(5.3.9)

Finally from the above results and the measured Thru transmission coefficient SmT 21 , the α coefficient is calculated as:

α= ¡

u/w · (g − f ) ¢ 1 − u/w · k/p · SmT 21

(5.3.10)

The above calculations provide all the Error Box model coefficients that are necessary 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: ¡ ¢−1 Td = α−1 · Xa · Tm · Xb

5.3.3

(5.3.11)

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:

1 Tm = β/α · ERR

ERF − EDF .ESF EDF −ESF

# " T

ERR − EDR .ESR ESR −EDR

# (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: p/w Tm = · u/w · (g − f )

a · k/p b k/p

" ·T ·

−g · u/w u/w −f

# (5.3.13)

Properly equating the terms expressed in (5.3.12) and (5.3.13) we find an equivalence for the first six terms expressed as follows:   EDF = b      EDR = f      E = −k/p SF  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 =

α/β =

ERF

ET R + EDF · (ELR − ESF )

(5.3.15)

ERF

ET F + 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: ET R = p/w · [k/p · a + b · ELR ]

(5.3.17)

ET F = (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 important 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 ET R and ET F :   ELF = −u/w      ELR = −k/p      E = (p/w)−1 · u/w · (g − f ) TF  ET R = (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 M auryr Coaxial Calibration Kit were used. With another feature of the program, the attenuation constant α of the LINE and the ηef f = 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

S11 (dB) Module 0

−10

−20

−30

−40

−50

−60 Raw Data TRL on PC Uploaded 12 Terms SOLT on Board

−70

−80

1.5

2.5 Frequency Ghz

3.5

Figure 5.5: S11 Module

S11 Angle 200

150

100

−50

−100

Raw Data TRL on PC Uploaded 12 Terms SOLT on Board

−150

−200

1.5

2.5 Frequency Ghz

Figure 5.6: S11 Phase

3.5

113

S21 (dB) Module −2

−3

−4

−5

−6

−7

−8

Raw Data TRL on PC Uploaded 12 Terms SOLT on Board

−9

−10

1.5

2.5 Frequency Ghz

3.5

Figure 5.7: S21 Module

S21 Angle −20

−40

−60

−80

−100

−120 Raw Data TRL on PC Uploaded 12 Terms SOLT on Board

−140

−160

1.5

2.5 Frequency Ghz

Figure 5.8: S21 Phase

114

The LINE parameters like the attenuation constant α and the refractive index for the phase velocity ηef f = 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.∆`

e+γ.∆`

" =

λ1

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 ηef f parameters, that are calculated by the tool as follows:

hαi = 1/2 ·

ln |λ1 | + ln |λ2 | ∆`

(5.4.2)

with ( ) ¯ ¯ ¯ ¯ £ ¤ £ ¤ 1 ¯ ¯ ¯ ¯ hηef f i = ¯ arctan (=(λ1 )/<(λ1 ) ¯ + ¯ arctan (=(λ2 )/<(λ2 ) ¯ 2πf ∆`

(5.4.3)

and

2 ²r = ηef f

(5.4.4)

From the above measurement results, the calculated LINE parameters α and ηef f are shown in Fig. 5.9 and Fig. 5.10:

115

TRL Measurements α (dB/cm) vs Freq 0.02

0.018

0.016

α (dB/cm)

0.014

0.012

0.01

0.008

0.006

0.004

0.002

1.5

2.5 Frequency Ghz

3.5

Figure 5.9: LINE Attenuation constant

TRL Measurements ηeff vs Freq 1.006

1.0055

1.005

1.0045

ηeff

1.004

1.0035

1.003

1.0025

1.002

1.0015

1.001

1.5

2.5 Frequency Ghz

3.5

Figure 5.10: LINE ηef f coefficient

Chapter 6 Networks characterization and parameter extraction 6.1

Introduction

In this chapter the approaches used to characterize experimentally single two conductors 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 accurate 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 example 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 connected with the useful implementation of different measurements of scattering parameters 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 measurements. The more common experimental procedures [40] [26] [32] [33] and their limitations will be presented. Then, a more accurate methodology [9] [10] that surpasses 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, G and C are defined in terms of the characteristic impedance Zc and the propagation constant γ by: γ = G + jωC Zc

(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 = <{γZc } L = ={γZc }/ω G = <{γ/Zc }

(6.2.3)

C = ={γ/Zc }/ω The problem consists in determining the propagation constant γ and the characteristic 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 problematic 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: r Zc ≈

L 1 1 = = C vph C 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 = G + jωC ≈ jωC Zc

(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): µ

γ2 RC + LG = < 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: µ 2¶ γ 1 < C≈ Rdc 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:

1 + Γload = Zload ≈ Rload,dc 1 − Γload

(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 1 + Γload jωRload,dc 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 parameters 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 equation 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 standard 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]: 1 [S] = DS

2 (Zc2 − Zref ) sinh(γ`)

2Zc Zref

2 (Zc2 − Zref ) sinh(γ`)

# (6.2.11)

where 2 DS = 2Zc Zref cosh(γ`) + (Zc2 + Zref ) 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`)

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)/(2S21 Zref )

(6.2.13)

D = (1 − S11 + S22 − ∆S)/(2S21 ) where ∆S = S11 S22 − S21 S12

Combining equations (6.2.11) to (6.2.13) yields [9]: ½ −γ`

2 2 1 − S11 + S21 ±K (2S21 )2

¾−1 (6.2.14)

where ( K=

2 2 (S11 − S21 + 1)2 − (2S11 )2 (2S21 )2

)1/2 (6.2.15)

and

2 Zc2 = Zref

2 (1 + S11 )2 − S21 2 (1 − S11 )2 − S21

(6.2.16)

125

Once γ(f ) and Zc (f ) are determined from (6.2.12)and (6.2.14), Telegrapher’s equations model per-unit-length circuit parameters R(f ), L(f ), G(f ) and C(f ) are given by: R(f ) = <{γZc } L(f ) = ={γZc }/ω G(f ) = <{γ/Zc }

(6.2.17)

C(f ) = ={γ/Zc }/ω 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 parameters 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]: â&#x20AC;˘ Compatibility: Many devices have performances which are strongly dependent on the environment in which they are embedded and it is therefore necessary to provide an environment similar to that used in the application. This is met by arranging for a similar physical geometry in the measurement environment, 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. â&#x20AC;˘ 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 deembedding 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 structure 1 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 parameters 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 measurement 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 ². 1

CPW structure data were provided by Prof. Franco Fiori of the EMI Microwave Group at Politecnico 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 Îą 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 Îˇef f presents a close behavior to the FW simulated model as is seen in Fig. 6.9.

130

10 Measurement FW Simulation

L, nHy/cm

0 1

3 4 Frequency, GHz

Figure 6.2: per-unit-length Inductance nHy/cm

16 14

Measurement FW Simulation

C, pF/cm

12 10 8 6 4 2 1

3 4 Frequency, GHz

Figure 6.3: per-unit-length Capacitance pF/cm

131

50 Measurement FW Simulation

R, Ω/cm

0 1

3 4 Frequency, GHz

Figure 6.4: per-unit-length Resistance Ω/cm

0.2

Measurement FW Simulation

G, S/cm

0.15

0.1

0.05

0 1

3 4 Frequency, GHz

Figure 6.5: per-unit-length Conductance S/cm

132

50 Measurement FW Simulation

Zc Module, Ω

0 1

3 4 Frequency, GHz

Figure 6.6: Module of the Characteristic Impedance Zc

60 Measurement FW Simulation

Zc Angle, Ω

40 20 0 −20 −40 −60 1

3 4 Frequency, GHz

Figure 6.7: Phase of the Characteristic Impedance Zc

133

α, dB/cm

Measurement FW Simulation

0 1

3 4 Frequency, GHz

Figure 6.8: Attenuation dB/cm

10 9 Measurement FW Simulation

neff

7 6 5 4 3 2 1 1

3 4 Frequency, GHz

Figure 6.9: Refraction index ηef f

134

6.3

MTL characterization methods

Various parameter extraction techniques for MTL structures were studied and valuated. 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: ˜ = A.exp(−Γz) + B · exp(Γz) V

(6.3.1)

˜ Y−1 C I = A.exp(−Γz) − B.exp(Γz)

(6.3.2)

Γ = (ZY)1/2 = PγP−1

(6.3.3)

YC = Z−1 Γ = YΓ−1

(6.3.4)

where

P is the eigenvector matrix of Γ. It is also the eigenvector matrix of the ZY product, 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 coth Γd

−YC sinh−1 Γd

YC coth Γd

# " ·

Vi Vo

# (6.3.6)

with a short-circuit admittance matrix Y2n of the 2n-port network given as: " Y2n =

YA YB

# (6.3.7)

YB YA

where YA =

YC coth Γd

YB = −YC sinh−1 Γd

(6.3.8)

and coth Γd

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 =

(coth Γd)Y−1 C

ZB = (sinh−1 Γd)Y−1 C

(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−1 B YA = cosh Γd Γd = P(cosh−1 Λ)P−1

(6.3.12) (6.3.13)

where P is the eigenvector matrix of Γd and −Y−1 B 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 sinh Γd = −YB P[sinh(cosh−1 Λ)]P−1

(6.3.15)

This, from (6.3.14) the per-unit-length modal attenuation constant αm and perunit-length modal phase constant βm are obtained by dividing the real and the imaginary 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−1 C Y = G + jωC = YC Γ

(6.3.16)

This method was tested by using a Full Wave EM simulation 2 on a 4-Ports asymmetric microstrip line and it was found not to achieve symmetry for the above perunit-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 `M T L ≤ λ/10, where `M T L is the MTL length in the propagation 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. 2

Simulation was made with the EM simulator EMSight of AWR, that includes a fast Full Wave electromagnetic 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 ˆ 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: ( ˆ Φ(z) = exp " =

Ã −z

0 Z

Ω

Y 0

β ΩT

¢ ¡ ¢ # φ1 z 2 ZY −zZφ2 z 2 ZY ¡ ¢ ¡ ¢ −zYφ2 z 2 ZY φ1 z 2 ZY ¡

(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: √ cosh( z) √ ±√ φ2 (z) = sinh( z) z φ1 (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

β ΩT

# " ·

Ω

−α

−β

ΩT

" =

# = 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 −1 ΩT

−β −1

α − Ωβ −1 ΩT −β −1 Ω−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 −1 A

(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 z 2 ZY ¡ ¢ B −1 = −zYφ2 z 2 ZY

(6.3.23)

By assuming that the eigenvalues of AB −1 and ZY are all distinct, the resulting decompositions are given as: AB −1 = P δz P −1 ,

ZY = P δt P −1

(6.3.24)

and the related simultaneous congruence decompositions given by A = P δa P T ,

B = P δb P T ,

Z = P δr P T ,

Y = P −T δg P −1

(6.3.25)

Using the above relationships, the following equations for the δ(·) diagonal matrices were derived [29]: δa δb−1 = δz δr δg = δt δz = φ1 (z 2 δt ) δb−1 = zδg φ2 (z 2 δt )

(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 [arg cosh(δz ) + j2πδn ]2 z2

(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(z T). 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 simulation 3 . 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. 3

Simulation was made with the EM Full Wave simulator EMSight of AWR.

141

In the present work a procedure that doesnâ&#x20AC;&#x2122;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 proposed, 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

(6.3.29)

−1 Y2n = Zref · [I − S][I + S]

where I is the 2n×2n identity matrix. To obtain the parameter values the 2n×2n Z2n and Y2n are defined as: " # Zin,in Zin,out Z2n = 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

= =(Z−1 in,out )/ω

= <(Zin,out )

f or

circuit

(6.3.31)

R(2) = <(Zout,out − Zin,out ) L(2)

= =(Zout,out − Zin,out )/ω

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−1 out,in )

= =(−Y−1 out,in )/ω

f or

Π circuit

(6.3.32)

G(2) = <(Yout,out + Yout,in ) C(2) = =(Yout,out + Yout,in )/ω where the supraindex and the supraindex

(2)

(1)

indicates the input GC branch of the Π-circuit model

indicates the output Π-circuit model GC branch.

Although this methodology was originally developed for small high-speed IC interconnections, it was proven to work very well for MTLs with lengths `M T L ≤ λmin /20 being λmin the wavelength for the maximum frequency propagated along the line.

145

Figure 6.11: MTL Î -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 characterization 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 `M T L = 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

R, Ω/cm

R11 15

R22 R

R21

0 1

2 3 Frequency, GHz

Figure 6.13: per-unit-length R(f ) Ω/cm matrix

L, nHy/cm

7 6

L11

L22 L12

L21

3 2 1 0

2 3 Frequency, GHz

Figure 6.14: per-unit-length L(f ) nHy/cm matrix

148

2.5

C, pF/cm

2 C

1.5

C22 −C

−C21

0.5

2 3 Frequency, GHz

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 [ηef f ] were extracted by equating the formula (6.3.3) as follows:

[α] = <(Γ),

and

[ηef f ] = c

giving the results shown in figures 6.16 and 6.17:

[β] ω

(6.3.33)

149

1.2 1 modal attenuation α

α, dB/cm

0.8

modal attenuation αm2 0.6 0.4 0.2 0

2 3 Frequency, GHz

Figure 6.16: Modal attenuation constant dB/cm

2.5 modal refractive index ηeff1

2.4

modal refractive index ηeff2

ηeff

2.3 2.2 2.1 2 1.9 1.8

2 3 Frequency, GHz

Figure 6.17: Modal Refractive index dB/cm

150

Finally, the influence of the modal cross powers for coupled modes along the structure was validated by calculating the ζnm merit coefficient index [50] from the extracted 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. 2

ζmn (%)

1.5

0.5

0 0

2 3 Frequency, GHz

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 Telegrapherâ&#x20AC;&#x2122;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 studied, 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 developed 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 procedure 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

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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 inquiring, 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

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The Truth is in the Truth Love is Truth

Appendix A See Userâ&#x20AC;&#x2122;s Guide Draft Version attached

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