TRANSIENT ANALYSIS OF SPATIALLY DISTRIBUTED …OZKAR, METE. Transient Analysis of Spatially Distributed Microwave Circuits Using Con-volution and State Variables. (Under the direction

TRANSIENT ANALYSIS OF SPATIALLYDISTRIBUTED MICROWAVE CIRCUITS USING

CONVOLUTION AND STATE VARIABLES

by

METE OZKAR

A thesis submitted to the Graduate Faculty ofNorth Carolina State University

in partial fulfillment of therequirements for the Degree of

Master of Science

ELECTRICAL ENGINEERING

Raleigh

1998

APPROVED BY:

Chair of Advisory Committee

Abstract

OZKAR, METE. Transient Analysis of Spatially Distributed Microwave Circuits Using Con-volution and State Variables. (Under the direction of Michael B. Steer.)

A convolution-based transient analysis is developed. The implementation uses state vari-ables and the separation of the circuit into linear and nonlinear subcircuits. The linear part isformulated in the frequency domain according to the modified nodal admittance matrix formu-lation. This frequency domain matrix representation is then transformed into a time domainimpedance matrix through the inverse Fourier Transform technique. Some methods such asaugmentation and phase-shifting to bandlimit the frequency response are presented. The non-linear equation is solved in the time domain by using a nonlinear equation solver and discreteconvolution techniques. The analysis is used to model a soliton line circuit.

Dedication

This thesis is dedicated to my parents and to my sister whose endless patience and under-standing gave me motivation for my studies.

Biographical Summary

Mete Ozkar was born in Turkey on May 13, 1974. He received the B.S. degree in ElectricalEngineering at the Middle East Technical University. While pursuing the B.S. degree he workedas a co-op for Aerodata, Braunschweig Germany in the summer of 1995. He was admitted intothe Masters Program at North Carolina State University in Fall 1996. While working toward hisM.S. degree he held a Research Assistantship with the Electronics Research Laboratory in theDepartment of Electrical and Computer Engineering. His research interests include analog andRF circuit design, microwave systems, and computer aided simulation of nonlinear circuits. Heis a member of the Institute of Electrical and Electronic Engineers and the Microwave Theoryand Techniques Society.

Acknowledgements

I wish to express my appreciation to my advisor Dr. Michael Steer for his continuous supportduring this lengthy work. It has been an honor working with him in his quasi-optical group. Iwould also like to thank Dr. Griff Bilbro and Dr. James Mink for serving on my committee.

I would like to express my appreciation to my graduate colleagues and everyone in ourquasi-optical research group for their help. First to Mr. Carlos Christoffersen for helping mein understanding TRANSIM and programming. To Mr. Mostafa Abdulla, Mr. Ahmed Khalil,Mr. Chris Hicks, Dr. Todd Nuteson, and Dr. Huan-sheng Hwang for answering my theoraticalquestions. To Mr. Baribrata Biswas, Mr. Usman Azeez Mughal and Mr. Satoshi Nakazawafor giving up their valuable time to help me in many different ways.

My special thanks go to all of my other friends for their moral support and encouragement.And finally, I would like to express my gratitude to my past and present instructors who

taught me and who provided me useful information about electrical engineering.

Contents

List of Tables vii

List of Figures viii

List of Symbols x

1 Introduction 11.1 Motivations and Objectives of This Study . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Literature Review 32.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Aysmptotic Waveform Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 AWE method in detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Steady-State Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 The Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Harmonic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.3 Sources of Error and Limitations . . . . . . . . . . . . . . . . . . . . . . 9

3 Convolution-Based Transient Analysis 103.1 Equation Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Methods used in the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Packages used in the implementation . . . . . . . . . . . . . . . . . . . . 163.3.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Thresholding, truncation and DC normalization . . . . . . . . . . . . . . 18

4 Results and Discussions 194.1 Soliton Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Nonlinear Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Simulation Results For the Soliton Line . . . . . . . . . . . . . . . . . . . 20

4.2 A Simple Diode Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Analysis of running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

vi

5 Conclusions and Future Research 315.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A Netlist Examples and Terminology 36A.1 The netlist for the soliton line with 47 diodes . . . . . . . . . . . . . . . . . . . . 36A.2 The netlist for a simple diode circuit . . . . . . . . . . . . . . . . . . . . . . . . 42A.3 Netlist Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

B Element Models 44B.1 Transmission Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.1.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.1.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.2 Diode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.2.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

C Additions to TRANSIM 47

List of Tables

3.1 Software packages used in TRANSIM . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Execution times for different number of sampling points . . . . . . . . . . . . . . 30

C.1 Added Transim file names and descriptions . . . . . . . . . . . . . . . . . . . . . 47C.2 List of added functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

viii

List of Figures

2.1 The worst case example for the infinite impulse response . . . . . . . . . . . . . 4

3.1 The partitioning of the circuit into linear and nonlinear subcircuits . . . . . . . 103.2 Some examples of resistive augmentation and compensation circuits . . . . . . . 133.3 A quasi-optical cavity oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 An example of the calculation of the impulse response using FFT . . . . . . . . 143.5 An example of the calculation of the impulse response using FFT . . . . . . . . 153.6 General approach to the simulation using the convolution based transient analysis 163.7 The flow diagram of the analysis code . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 The soliton line model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 The augmentation networks used in the soliton line model . . . . . . . . . . . . 204.3 The first diode current for the 47 diode soliton line . . . . . . . . . . . . . . . . 204.4 The last diode current for the 47 diode soliton line . . . . . . . . . . . . . . . . . 214.5 The first diode state variable for the 47 diode soliton line . . . . . . . . . . . . . 214.6 The last diode state variable for the 47 diode soliton line . . . . . . . . . . . . . 224.7 The source voltage for the 47 diode soliton line . . . . . . . . . . . . . . . . . . . 224.8 The source current for the 47 diode soliton line . . . . . . . . . . . . . . . . . . 234.9 The direct impulse resonse as seen from the voltage source terminals in the 47

diode soliton line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.10 The enlarged version of Figure 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 244.11 The cross impulse resonse as seen from the voltage source terminals in the 47

diode soliton line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.12 The real part of the direct impulse resonse in frequency domain as seen from the

voltage source in the 47 diode soliton line . . . . . . . . . . . . . . . . . . . . . . 254.13 The imaginary part of the direct impulse resonse in frequency domain as seen

from the voltage source terminals in the 47 diode soliton line . . . . . . . . . . . 254.14 The real part of the cross impulse resonse in frequency domain as seen from the

voltage source in the 47 diode soliton line . . . . . . . . . . . . . . . . . . . . . . 264.15 The imaginary part of the cross impulse resonse in frequency domain as seen

from the voltage source terminals in the 47 diode soliton line . . . . . . . . . . . 264.16 A comparision between the lossy and no loss case in terms of the state variables 274.17 A comparision between the lossy and no loss case in terms of the currents . . . . 274.18 The diode state variable for the simple circuit . . . . . . . . . . . . . . . . . . . 284.19 The diode current for the simple circuit . . . . . . . . . . . . . . . . . . . . . . . 284.20 The effect of phase-shift on the simple diode circuit solution . . . . . . . . . . . 294.21 Experimental data for the soliton line simulated . . . . . . . . . . . . . . . . . . 30

B.1 The diode model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ix

List of Symbols

f – Frequency.fm – Boundary frequency.t – Time.FFT – Fast Fourier transform.AWE – Asymptotic Waveform Evaluation.HB – Harmonic Balance.X – State variable vector in frequency domain.X – Time domain state variables in vector form.X – A subset of X .VL – The linear voltage vector.VLi – The linear voltage at the ith nonlinear node.INL – The nonlinear current vector.iNLj – jth element of INL.τ – Phase shift constant.Ft() – Error function vector at a time point.Ft,i() – ith element of the error function vector at a time point.ω – Angular frequency.W – Matrix that rearranges the nonlinear currents.T – Transformation matrix.Msv – State variable impedance matrix.mij – ijth element of Msv.Ssv – State variable source vector.Ssvi – ith element of Ssv.Sfixed – Ideal source vector.ns – Total number of state variables in the circuit.nt – Variable for discrete time.NT – Number of total time points.nτ – Convolution variable for discrete time.Y (f) – Admittance matrix representation in frequency domain.Y (f) – Y (f) with phase shift.F – Reference Frequency in the transmission line model.R – Resistance parameter of the coaxial transmission line.G – Conductance parameter of the coaxial transmission line.C – Capacitance parameter of the coaxial transmission line.L – Inductance parameter of the coaxial transmission line.l – Physical length of the transmission line.dB – Decibel.s – Seconds.

c – Speed of light.k – Dielectric contant.A(f) – Attenuation as a function of frequency.tan δl – Loss tangent of the coaxial transmission line.γ – Complex phase constant.ID – Total diode current.Ic – Capacitor current in the diode model.Id – Conduction current in the diode model.vj – Voltage as a state variable in the diode model.NLTL – Nonlinear transmission line.CPW – Coplanar waveguide.Im – Imaginary part of a complex number.∆f – The smallest frequency increment.

Chapter 1

Introduction

1.1 Motivations and Objectives of This Study

Transient analysis of microwave circuits has been challenging because of convergence problems,aliasing issues, the lengthy convolution and other numerical errors introduced by methods,such as Newton’s method, when solving the nonlinear system. Harmonic balance plays a moreimportant role in microwave circuit modeling than transient analysis, but nevertheless, transientanalysis becomes important when analyzing oscillations and chaotic behaviour in microwavecircuits [3].

A motivation for this work was implementing the convolution-based transient analysis ap-proach so that it can be used for any kind of linear/nonlinear circuit in microwaves.

Another motivation was to improve the pre-existing convolution-based transient analysisdeveloped by Basel [28]. The previous transient analysis had convergence problems in thenonlinear system solution for the types of microwave circuits (spatial power combiners andsoliton lines) being modeled.

In implementing the convolution analysis it was desirable to make use of the frequency do-main formulation of the linear subcircuit that is used in harmonic balance analysis in TRAN-SIM. In this way, the linear formulation is kept the same for both analysis types.

The most important achievements of this work are:

• Solving the problem of convergence by integrating a standard nonlinear equation solverinto the analysis code.

• Reducing the memory requirements in great amount by avoiding unnecessary convolutionloops and by using thresholding and truncation techniques that were developed previouslyby Basel [28].

• Using state variables. This allows the use of parametrized device models including non-linear behavioral models.

1.2 Thesis Overview

Chapter 2 presents a review of the previously published work done on transient analysis. Asymp-totic waveform evaluation is discussed in detail and advantages/disadvantages compared toconvolution analysis techniques using the inverse Fourier Transform or the inverse LaplaceTransform are reviewed.

1

CHAPTER 1. INTRODUCTION 2

Chapter 3 covers the formulation and implementation of the convolution-based transientanalysis in detail.

In Chapter 4 the modeling of a soliton line and some simulation results are presented.In Chapter 5 a summary of the thesis is given along with conclusions and suggestions for

future work in this topic.

Chapter 2

Literature Review

Several approaches are used in the transient analysis of circuits. One approach is to modelevery circuit element in the time domain using differential equations. In the well known SPICEimplementation, every element has its companion model which consists of linear elements.The values for these elements are chosen according to the differential equation that describesthe particular element. The solution is found by using numerical integration and iterationtechniques applied at each time step. An outline for the SPICE transient analysis is given byZuberek and et al. [42]. The stability properties of the integration methods in SPICE transientanalysis are investigated by Gubian and Zanella [36]. The simulation errors introduced by theSPICE transient analysis is discussed by Brambilla and DÀmore [34]. Brambilla and DÀmoreshow some unexpected results given by the trapezoidal integration algorithms in SPICE. Theyargue that this type of effect is present in every simulated circuit and that they are in mostcases hidden by similar behavior of the circuit under analysis. According to the authors, thiskind of effect results in an erroneous transient which is only due to the trapezoidal algorithmand not to the circuit which is working at its correct steady state.

2.1 Convolution

In microwave circuits, where the elements are described better in the frequency domain due tothe frequency dependent parameters, it is usually difficult to obtain a time domain model forsuch elements as dispersive transmission lines. Also, the data obtained through measurementsis usually in the frequency domain. Another approach is to model the linear elements in thefrequency domain and then to shift into the time domain using a Laplace Transform, or aFourier Transform. Convolution can then be performed in the time domain to get the responseof the circuit. In this method the impulse response of the linear circuit is convolved with thetime domain representation of the nonlinear circuit.

2.1.1 Fourier Transform

The Discrete Fourier Transform inversion is generally done using the Fast Fourier Transform(FFT) algorithms. The nature of the Fourier Transform requires periodic functions as its input.Unfortunately, the frequency models for elements are usually non-periodic. The FFT algorithmstill treats these functions as periodic, by considering the non-periodic function as one period ofanother periodic function. Aliasing effects occur in the inverse FFT (IFFT) unless the functionis bandlimited and it forms a smooth periodic waveform. Also, the continuous frequency domainfunction is required to be sampled at the Nyquist rate to avoid aliasing.

3

CHAPTER 2. LITERATURE REVIEW 4

The worst case scenerio in this method is shown in Figure 2.1. Since both the transmissionline and the source are ideal, there is no loss in the circuit. The impulse wave travels endlesslywithout being attenuated. This causes the time domain response to be of infinite time duration,which is impossible to calculate with the FFT algorithms. Even for moderately lossy lines, theduration of the response can exceed many transit times of the transmission line network andthe inverse FFT techniques become inefficient.

Since the frequency domain function cannot be sampled for all frequencies, a boundaryfrequency has to be chosen. It is very important that the function is bandlimited to this chosenfrequency, otherwise aliasing effects will be observed.

Brazil has solved the problems of aliasing in his “causal convolution” method [1, 2]. He usestechniques such as conversion to an all-transmission line network, conversion to non-minimum-phase system function and time-domain windowing. Time-domain windowing removes theeffects of non-causality which appear as a tail on the discrete impulse response. Conversion tonon-minimum-phase system function becomes important when discontinuities at the boundaryfrequency are present. By applying a phase shift the imaginary part is pulled down to zero.Although the magnitude part of the phase-shifted function might have discontinuities in itshigher order derivatives at the boundary frequency, this technique is quite powerful when usedproperly. This technique will be discussed further in Chapter 3. According to Brazil, thecapacitors and inductors can be made open and short circuits respectively at the boundaryfrequency. Again, this results in a naturally periodic frequency function with no discontinuities.

Basel, on the other hand used resistive augmentation networks to bandlimit the time domainresponse in the simulation of interconnection networks [28]. As it can be understood from Figure2.1, convolution methods have problems dealing with low loss or lossless lines. This situationcan lead to an infinite impulse response for the line and a corresponding infinite computationtime to do the convolution. Basel used parasitic elements as natural filters in addition tothe resistive augmentation networks to achieve frequency bandlimiting and so to overcome theproblem described above.

Figure 2.1: The worst case example for the infinite impulse response

2.1.2 Laplace Transform

The Laplace inversion is done by numerical inversion techniques. This method does not havealiasing problems since it does not have to assume that the function is periodic. The inverseLaplace transform exists for both periodic and non-periodic functions. One does not have toworry about the causality problem for double-sided Laplace Transforms, either. Unlike FFTmethods, the desired part of the circuit response can be achieved without doing tedious andunrelated calculations for the other parts of the response. For example, if the circuit response


for time t is required, this response can be calculated without the knowledge of the responseat other values of time. Laplace inversion techniques suffer from the series approximation sothe discontinuities cannot be seen exactly in the time domain. Another source of error fornumerical inversion comes from the nonlinear iterations involved.

The well known inversion integral for double-sided Laplace transform is:

g(t) =1

2πj

∫ e−j∞

e+∞G(s)estds =

1

2πj

∫ e−j∞

e+∞G(z/t)ezdz (2.1)

where z = st. Approximating the function ez in (2.1) by a Pade rational function,

HN,M(z) =PN (z)

QM(z)=

N∑i=0

(M +N − i)!(Ni

)zi

M∑i=0

(−1)i(M +N − i)!(Mi

)zi

(2.2)

can be obtained. Now, an approximated function g for g(t) can be obtained by inserting (2.2)into (2.1):

g(t) =1

2πjt

∫ e′+j∞

e′−j∞

G(z/t)HN,M(z)dz (2.3)

The above integral can be evaluated using the residue theorem by choosing an infinite arcas the closed contour [21]. By choosing N < M and closing the path of integration around thepoles in the right half plane, time response g is obtained as:

g(t) = −1

t

M∑i=1

Kig(zi/t), t > 0 (2.4)

where Ki is the residue at the pole zi of HN,M(z).

2.2 Aysmptotic Waveform Evaluation

Asymptotic waveform evaluation (AWE) is an efficient and general technique for simulatingand modelling large linear(ized) circuits. This method approximates the transient portion ofthe response by matching the initial boundary conditions and the first 2q − 1 moments ofthe exact response to a lower q-pole model. The word asymptotic is used because as theorder of this approximation increases the response of the system approaches the actual solutionasymptotically.

The two major inspirations for AWE came from the work of Elmore [13] who estimated thedelays in RC-tree networks and the work of McCormick and Allen [14] who showed that theinterconnect circuit moments can be used to achieve lower order circuit models. Elmore delaywas the first moment-method that was applied to the interconnect problem.

Two steps are involved in the implementation of AWE algorithms. The first step is tocalculate the moments and the second step is to match these moments to a lower order frequencydomain function.


2.2.1 AWE method in detail

A time domain solution of a differential equation or a network is assumed with the initialconditions. The time domain solution is transformed into Laplace domain, usually by numericinversion techniques or convolution. Nakhla [4] provides a different fast method which doesnot use neither convolution nor inversion techniques. He actually calculates the solution of thenonlinear circuit at a time point in terms of the previous solution rather than in terms of asolution that depends on all of the past solutions.

Let the solution of the system be x(t). The Laplace transform of this function, X(s) iscalculated by the one sided Laplace transformation equation as follows:

X(s) =∫ ∞

0x(t)e−stdt (2.5)

Using the Taylor series approximation about s = 0 (which becomes the Maclaurin Series) thefollowing is obtained:

X(s) =∫ ∞

0x(t)[1− st +

1

2s2t2 − ...]dt (2.6)

=∞∑k=0

1

k!(−s)k

∫ ∞0

tkx(t)dt

The series coefficients are:

mk =(−1)k

k!

∫ ∞0

tkx(t)dt (2.7)

These coefficients are also called the time moments which can be computed for a lumped, linear,time-invariant circuit using the initial conditions. The following matrix equation holds true fora lumped, linear, time-invariant circuit irregardless of the equation formulation such as nodalformulation, MNA formulation, and Tableau Analysis [20].

Mx(t) +Ndx

dt= b(t) (2.8)

The vector x(t) is the solution of the system, b(t) is the vector formed by the independentsources and M and N are formed by the resistors, capacitors and inductors.

In the Laplace domain (2.8) becomes (assuming zero initial conditions)

(M + sN)X(s) = B(s) (2.9)

If one is interested in the impulse response of the circuit, B(s) becomes B0. Assuming thatX(s) is the impulse response and using a Taylor series approximation

(M + sN)(X0 + sX1 + s2X2 + ....) = B0 (2.10)

is obtained. When X(s) is expanded at some point(s) other than zero this leads to frequencyshifting and multipoint AWE methods [19].

The following equations can be obtained by equating the coefficients of the terms with thesame order of s.

MX0 = B0 (2.11)

MXk = −NXk−1 for k > 0. (2.12)


These two equations can be solved recursively and by only using one LU-factorization [15]. The2q moments will be the solutions of these equations. The first moments are calculated by a dcanalysis. That is the capacitors are replaced with open circuits and inductors are replaced withshort circuits. The calculated voltages across the capacitors and calculated currents throughthe short circuited inductors will be the first set of moments. The next set of moments are thencalculated using these dc values [16].

Now, having calculated the moments, the frequency domain function can be re-written usingthe moment expression given by (2.7):

X(s) =∫ ∞

0x(t)dt− s

∫ ∞0

x(t)dt+1

2s2∫ ∞

0x(t)dt+ ... (2.13)

= m0 +m1s+m2s2 + ...

As mentioned previously, these moments are matched to a lower order frequency domain func-tion by using various techniques (such as Pade approximation). The reduced-order modelfunction with q poles has the form:

X(s) =bq−1sq−1 + ...+ b1s+ b0

aqsq + ...+ a1s+ 1(2.14)

Equating (2.13) and (2.14) and solving the linear system of equations, the coefficients am andbm can be obtained. Having obtained the coefficients, the roots of the denominator which arethe poles of the reduced-order model can be determined. Knowing the poles of the approximatefunction, the partial fraction expansion can be applied to obtain the function in the followingform:

X =k1

s− p1+

k2

s− p2+ ...+

kqs− pq

=q∑l=1

kls− pl

(2.15)

=∑l=1

qkl/pl

1− s/pl

Now that the impulse response of the circuit is known, the response of the circuit to any kind ofinput can be determined. For example one can calculate the step response just by multiplying(2.15) by 1/s and evaluating the inverse Laplace transform numerically.

2.2.2 Limitations

There are two major approaches for the moment-matching [6]. The first approach, calledpure moment matching is represented by the continued-fraction method of Chen and Shieh[7] and the state-variable formulation of Bosley and Lees [8]. The second method, known asstable moment matching, is represented by the Routh approximation [9] and Pade-Hurwitzapproximation. Other existing reduction methods are mentioned by Bosley and Lees [10, 8].

The pure moment-matching method is known to suffer from instability and sensitivity prob-lems [10, 11]. Independent of the original system being stable or instable, the reduced modelcan be instable. The Routh and Pade-Hurwitz methods solve the problem first by approxi-mating the denominator of the frequency domain representation of the system with a lowerdegree Hurwitz polynomial to achieve stability [12] and then matching certain moments by as-signing proper coefficients to the numerator polynomial. Although these methods create stable


approximates, the response of these approximates are usually not close to the original systemresponse.

LU factorization and the dc analysis involved in the AWE methods are usually the biggestissues in determining efficiency. According to [15] the LU decomposition is O(n3) for an arbi-trary circuit matrix. However, this reduces to O(n) for tree structured circuits when Sparsematrix packages are used. Sparse matrix packages can be used to speed up the LU factoriza-tion to O(n1.4−1.7) for typical matrices. AWE is really powerful when analyzing the interconnectstructures since interconnects are large tree structures. AWE is much faster in the interconnectcircuit analysis than transient analysis, because only a few dc analyses are required to computeall the moments and matching the moments to the actual poles and residues is efficient for asmall number of poles. Another issue in the limitations of AWE methods is that resistor loopsare undesirable in AWE method, because their currents are not known a priori.

An implementation of the AWE algorithm, RICE [16] approaches the above problems bymaking use of path tracing, compaction and factorization techniques. AWEswit [22] is anothersimulator which solves for the state variables in each of the continuous-time circuits that re-sult from switching. PL-AWE is a general purpose circuit simulation program that has beendeveloped especially for the analysis of VLSI circuits [17]. It uses different approaches to theAWE implementation to make the performance very good, but it has not been optimized forexecution efficiency.

AWE methods do not apply to nonlinear circuits naturally. For this reason, the nonlinearcircuits have to be linearized. One way to do this is to use piecewise linearization techniques[17]. The multi-port characterization of the networks (macromodelling) have been applied tononlinear transient analysis [4, 18].

Special considerations have to be taken into account during the pole-zero computation oflarge size microwave circuits and circuits with distributed elements . AWE technique can onlyextract the low frequency poles because the moments carry information about the low-frequencycharacteristics of the circuit. Different techniques (such as frequency hopping technique, theusage of Markov parameters to improve the accuracy of the transient response near t = 0) havebeen proposed [19].

2.3 Steady-State Methods

In microwave applications, steady-state periodic or quasi-periodic response of circuits is usuallyof more interest than transient response. If transient analysis is used to obtain the steady-stateresponse of the circuit, one has to wait until the transients die out completely in order to see thesteady-state response. The transients in microwave frequencies might take a long time due tothe large time constants and hence the transient analysis becomes very inefficient to calculatethe steady-state response. This is where harmonic balance (HB) and shooting methods becomeimportant.

2.3.1 The Shooting Method

The idea in shooting methods is to skip the transients by adjusting the initial conditions.The resulting nonlinear system of algebraic equations can be solved using nonlinear iterationmethods. However, the convergence of these nonlinear iterations depend on the estimatedinitial conditions. Transient analysis is the only rigorous way of obtaining a good estimate forthe required initial conditions so that the transients are bypassed. Also, transient analysis isessential if transient information such as turn-on time or stability are required.


2.3.2 Harmonic Balance

Harmonic balance analysis is the most powerful analysis technique if one is interested in steady-state response of a circuit. The conventional approach begins by partitioning the circuit intolinear and nonlinear subcircuits. The linear subcircuit is solved in the frequency domain whilethe nonlinear subcircuit is solved in the time domain. The formulation starts by calculatingthe error function of either the voltage waveforms or the current waveforms at the commonnodes between the two subcircuits. The resulting nonlinear equation can then be solved usingnonlinear iterative techniques.

2.3.3 Sources of Error and Limitations

Three sources of error are of concern in harmonic balance [23]. The first two result fromtruncating the harmonics considered to be some finite number, and the third results from thenonlinear iteration involved to solve the nonlinear system of equations. The calculation of theinverse Jacobian in this nonlinear iteration solution seems to be the main source of cost in thecomputation [26].

Even though the actual goal of the shooting method was to get away from the lengthyand costly calculation of transients, transient analysis cannot be avoided completely. Anotherdrawback of the shooting method is again the dense Jacobian matrix that arises in the nonlinearequation formulation [20]. Due to the lengthy inversion of this dense Jacobian matrix, themethod is inefficient.

Chapter 3

Convolution-Based Transient Analysis

The modified nodal admittance (MNA) matrix is currently used in TRANSIM for the frequencydomain formulation of the linear subcircuit of the network as shown in Figure 3.1. This new ap-proach to transient analysis combines this frequency domain formulation with the state variableformulation.

3.1 Equation Formulation

The idea of partitioning a circuit into linear and nonlinear parts was introduced by Nakhla andVlach [5] for harmonic balance analysis. This idea has been present in the circuit simulatorused here, called TRANSIM and has been used by both harmonic balance and transient anal-ysis codes of the simulator. The frequency domain formulation for the linear subcircuit waspreviously implemented by Christoffersen [27] and some of the equations are repeated here forconvenience.

NETWORK

LINEAR

Device

Nonlinear

Device

NonlinearDevice

NonlinearIL2

IL1

IL3

ILm LnI

ILl

LkI

V1

V

V

2

3

VV

V

V

l

k

m n

Y(f)

Figure 3.1: The partitioning of the circuit into linear and nonlinear subcircuits

The vector of port voltages at the linear circuit in frequency domain is:

VL(X, f) = Ssv(f) +Msv(f) INL(X, f) (3.1)

where VL(X, f) is the linear voltage vector and X is the state variable vector with each element

10

CHAPTER 3. CONVOLUTION-BASED TRANSIENT ANALYSIS 11

containing all the frequency components of the same state variable. Therefore X is given by:

X =

x1,f0 x1,f1 x1,f2 · · · x1,fm

x2,f0 x2,f1 x2,f2 · · · x2,fm...

xns,f0 xns,f1 xns,f2 · · · xns,fm

(3.2)

Expanding the matrix multiplication in equation (3.1), each element of the voltage vectorVL(X, f) can be written as:

VLi(X, f) = Ssvi +ns∑j=1

mij(f) iNLj(X, f) (3.3)

where mij is the ijth element of the matrix Msv and iNLj is the jth element of the nonlinearcurrent matrix INL. Equation (3.3) can be rewritten in time domain by replacing the multipli-cations with the convolution operation and replacing the frequency domain variables with theirtime domain equivalents.

VLi(X, t) =ns∑j=1

mij(t) ∗ iNLj(X, t) + Ssvi(t) (3.4)

Expanding the convolution operation we get:

VLi(X, t) =ns∑j=1

∫ t

−∞mij(t− τ ) iNLj(X, τ ) dτ (3.5)

In order to do the convolution on the computer, the time domain equation needs to be indiscrete form.

Since the system is assumed to be causal, iNLj = iNLj(0) = 0 for t < 0. Therefore

VLi(X, nt) =ns∑j=1

nt−1∑nτ=0

mij(nt − nτ ) iNLj(X, nτ)

+NT∑

nτ=nt+1

mij(nτ ) iNLj(0)

+ Ssvi(nt)

=ns∑j=1

nt−1∑nτ=0

mij(nt − nτ ) iNLj(X, nτ)+ Ssvi(nt) (3.6)

and the error function at the ith nonlinear node is defined as

Ft,i(X) = VLi(X, nt)− VNLi(X, nt) (3.7)

In vector form, the error function of the complete system is

Ft(X) =

ns∑j=1

nt−1∑nτ=0

mij(nt − nτ )iNLj(nτ ) + Ssv1(nt)− VNL1

ns∑j=1

nt−1∑nτ=0

mij(nt − nτ )iNLj(nτ ) + Ssv2(nt)− VNL2

...ns∑j=1

nt−1∑nτ=0

mij(nt − nτ )iNLj(nτ ) + Ssvns (nt)− VNLns

ns×1

(3.8)


3.2 Methods used in the analysis

Before converting the frequency domain MNA representation of the linear network into thestate variable time domain impedance matrix the frequency response needs to be bandlimitedso that the function forms a periodic waveform. This will reduce the aliasing errors duringthe FFT operation. The idea behind this method implemented here was borrowed from [2].The last frequency component of the imaginary part is pulled down to zero by multiplying thewhole frequency vector by ejwτ which corresponds to a time domain shift. Fortunately, formost cases the time shift required for this correction is much smaller than the fixed time step(which is calculated from the given information in the netlist) used in TRANSIM. Let Y (f) bethe frequency domain function which is assumed to be periodic since inverse FFT algorithmis about to be applied. Here is the required operation to pull the value of ImY (f) at theboundary frequency (The highest frequency considered in the frequency formulation) to zero:

Y (f) = Y (f)e−j2πfτ (3.9)

This is a linear phase shift operation since each frequency is shifted by the same amount. Onehas to be careful about a few points when using (3.9). The value of τ being greater thansome value will cause the exponential function to be periodic and this will introduce an extraperiodicity in every period of the multiplied function. The discrete version of (3.9) is

H(n∆f) = Y (n∆f)e−j2πn∆fτ (3.10)

where n is the total number of dicrete frequencies. It is easy to see that if n∆fτ is greaterthan 1, the multiplying exponential begins to repeat itself. Therefore, the condition on τ canbe found as:

n∆fτ ≤ 1 (3.11)

τ ≤ 1

n∆f(3.12)

Another caution with the use of phase shift described by Brazil is related to the discontinu-ities in the higher derivatives of the magnitude part of the resulting function at the boundaryfrequency [1]. But when used properly the effect of phase shifting on reducing discontinuitiesat the boundary frequencies can be highly beneficial.

If the real part of the frequency vector is not bandlimited, a resistive augmentation circuitcan be used to overcome this problem. This type of augmentation circuit gave good results forthe simulation of interconnects for Basel’s work [40]. Some examples of resistive augmenationcircuits are shown in Figure 3.2. In these circuits R1 is usually chosen as positive and R2 ischosen as negative.

An example for both addding a resistive augmentation circuit and phase shifting is seen inFigure 3.4 and 3.5. The frequency domain y11 parameter of the cavity oscillator shown in Figure3.3 (obtained from a code written by Nuteson [35]) was transformed into time domain usingIFFT command in MATLAB. Before the transformation, it was made sure that the frequencydomain function was bandlimited. The procedure described above was applied to achieve this.Resistor values for the augmentation circuit were chosen as 50 Ω and 30 Ω for this case.

As it can be seen, time domain data has a tiny imaginary part (due to the FFT algorithmin MATLAB) and a bandlimited real part. The modifications done in the frequency domaincan now be removed since the resistors are not frequency dependent and the phase shift justcorresponds to a time shift. If this is not done, the correct impulse response is not obtained since


R1

R1

R1

R1-

R1-

R1-

Augmentation Network Compensation Network



R - R22

Figure 3.2: Some examples of resistive augmentation and compensation circuits

S1

PARTIALLYTRANSMITTINGREFLECTOR

-

z = 0

z = D

S2

ay

az

ACTIVEANTENNAARRAY

PLANARREFLECTOR

z = d

2ay

Figure 3.3: A quasi-optical cavity oscillator


periodicity condition is the assumption when one takes IFFT or FFT of a function. Althoughit is hard to see in the figure, the original impulse response does not completely die out but itactually moves away from zero due to the aliasing effects. This would not be a realistic impulseresponse and would yield errors in all other responses obtained from convolving this impulseresponse with the other input function.

Figure 3.6 summarizes the methods described above. It shows the most general approachto the simulation of a circuit using the implemented convolution based transient analysis. Asmentioned previously, the LC network provides natural filtering and the phase-shift and theresistive networks bandlimit the frequency and the time domain responses of the linear network.

Cavity Oscillator Impulse Response

Frequency Domain Response

Real Imaginary

−20 −15 −10 −5 0 5 10 15 200.0166

0.0168

0.017

0.0172

0.0174

0.0176y11

frequency (GHz)

Rea

l par

t of y

11

−20 −15 −10 −5 0 5 10 15 20−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−5 y11

frequency (GHz)

Imag

inar

y pa

rt o

f y11

Impulse Response

Real Imaginary

0 100 200 300 400 500 600 700 800−5

0

5

10

15

20x 10

−3 Impulse Response

discrete time

Rea

l par

t

0 100 200 300 400 500 600 700 800−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5 Impulse Response

discrete time

Imag

inar

y pa

rt

Figure 3.4: An example of the calculation of the impulse response using FFT

3.3 Implementation

The flow diagram for the implemented transient analysis is shown in Figure 3.7. After the netlistis parsed and the frequency domain admittance representation of the linear network is formed,the so-called compressed MNA matrix is obtained. The following operation is performed when


Cavity Oscillator Impulse Response

Step Response

Real Imaginary

0 100 200 300 400 500 600 700 800−5

0

5

10

15

20x 10

−3 Step Response

discrete time

Rea

l par

t

0 100 200 300 400 500 600 700 800−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−5 Step Response

discrete time

Imag

inar

y pa

rt

Original Impulse Response

Real Imaginary

0 100 200 300 400 500 600 700 800 900−4

−2

0

2

4

6

8

10

12

14x 10

−3 Original Impulse Response

discrete time

Rea

l par

t

0 100 200 300 400 500 600 700 800 900−0.01

−0.005

0

0.005

0.01

0.015Original Impulse Response

discrete time

Imag

inar

y pa

rt

Final Step Response (Real): 0.0000000

Final Step Response (Imaginary): -0.0000000

Final-10 Step Response (Real): 0.0000002

Final-10 Step Response (Imaginary): -0.0000011

Figure 3.5: An example of the calculation of the impulse response using FFT


DEVICE

NONLINEAR

R1-

R

1R-

- R

1

R2

-

-

R- 1

2- R

2

-

R2

NONLINEAR

DEVICE

C

C

L

L

L

C

C

L

LINEAR

NETWORK

CompensationNetwork

jwe

AugmentationNetwork

R1

R1

R1

R1

R2

R2

R2

R2

FilteringNetwork

Figure 3.6: General approach to the simulation using the convolution based transient analysis

obtaining the compressed MNA matrix:

Msv(f) = TM−1W (3.13)

Similarly, the compressed source matrix is

Ssv(f) = TM−1Sfixed (3.14)

Again, the meanings of T and W matrices are discussed in Christoffersen’s work [25]. Tand W are used to get the correct linear matrix to be used in the state variable formulationfrom the general representation in frequency domain. Msv(f) is an impedance matrix ratherthan an admittance matrix, because the error function compares the voltages at the nonlinearterminals. In general, ideal sources were of no concern during this work even though they wereimplemented.

3.3.1 Packages used in the implementation

In developing the frequency model, the Sparse 1.2 package [30] was used. This is a packagethat solves linear equations which have sparse matrices.

The nonlinear error equation formulated in (3.8) is solved using the NNES package [31].At each time point the error function is calculated using the state variable vector that has aninitial value. This initial value is set to zero at zero time, but at other times different thanzero it is set to the previous time solution to improve the convergence and to decrease thenumber of iterations. The NNES routine calls the element routines in order to calculate theerror function. Element routines return the nonlinear currents and voltages given the statevariable vector. Each call to the nonlinear element routines from NNES corresponds to oneiteration. The iterations continue until the error function value is close to zero. Each statevariable vector calculated as a solution of the error function equation is stored at each timepoint. Once the end of the time points is reached, the results can be displayed.

The software packages used in TRANSIM are listed in Table 3.1.


MNA

Obtain

Parse the

Netlist

Create the compressed

Source and MNA

matrices

Time Domain

Conversion (FFT)

NNES

Routines

Element

End of

time points?

OUTPUT

Transient Analysis

Figure 3.7: The flow diagram of the analysis code

Table 3.1: Software packages used in TRANSIMSparse Sparse matrix libraryNNES Nonlinear equation systems solver

ADOL-C Automatic differentiation packageGnuplot Plotting utility


3.3.2 Convolution

A lot has been said and published about convolution so far. The idea of convolving the impulseresponse of the linear circuit with the nonlinear outputs or any input function is not new intransient analysis. Gordon et al. have implemented a similar approach for a time domain sim-ulation of multiconductor transmission lines [24]. However, they indicate that the convolutionintegral, which becomes a convolution sum for the computer simulations, is O(n2) when it isimplemented (Here, n is the total number of discrete time points used). That is why theyuse an exponential series approximation for the convolution sum. The implementation of theconvolution sum in the transient analysis described in this thesis uses less CPU time by simplyeliminating the “all-zero” frequency vectors. This greatly reduces the time and memory spentfor the convolution sum. However, it is still true that the CPU time to do convolution willincrease as the loop advances through the time steps. This causes the analysis to graduallyslow down as it progresses. This is unavoidable if one wants to calculate an exact convolutionsum without any approximations. Chiprout and Nakhla [4] have avoided the use of convolutionin the AWE method that they developed by expressing the current time point in terms of theprevious time point only.

3.3.3 Thresholding, truncation and DC normalization

Another way of reducing the time required for convolution is to use thresholding. Thresholdingis a way of improving the speed by sacrificing little accuracy. Also, truncation can be done ifall the points after some point are known to be zero. DC normalization is a correction for thealiasing errors introduced by thresholding. For more on DC normalization and thresholding,the reader is referred to Basel’s dissertation [28].

Chapter 4

Results and Discussions

4.1 Soliton Line

4.1.1 Nonlinear Transmission Lines

Nonlinear transmission lines (NLTLs) find applications in a variety of high speed, wide band-width systems including picosecond resolution sampling circuits, laser and switching diodedrivers, test waveform generators, and mm-wave sources [38]. They have three fundamentalcharacteristics: nonlinearity, dispersion and dissipation. The actual NLTLs consist of coplanarwaveguides (CPWs) periodically loaded with reverse biased Schottky dioes. In an ideal NLTL,the balance between the nonlinearity of the loaded nonlinear elements and the dispersion ofthe periodic structure can result in the formation of a stable solitary wave called a soliton [39].The nonlinearity in NLTLs is due to the voltage dependent capacitance of the diodes and thedissipation is due to the conductor losses in the CPWs.

4.1.2 Modeling

In this work, the NLTLs described above were modelled by regular low-loss coaxial transmissionlines and microwave diodes [32]. Skin effect was taken into account in the modelling of thetransmission lines. Details about the modelling issues can be found in Appendix B. The circuitused as a model for NLTLs (shown in Figure 4.1) can either be excited by sinusoids with a DCbias or by a pulse.

The augmentation network used is shown in Figure 4.2. Without the presence of thisaugmentation network, convergence is not achieved. The resistive part of the augmentationnetwork is compensated in the nonlinear element routines. In this case R was chosen as 100Ω and L was chosen as 1.0 nH. Convergence was not achieved for smaller values of L for theparticular choice of R. .

~-

+

Figure 4.1: The soliton line model

19

CHAPTER 4. RESULTS AND DISCUSSIONS 20

R

R

L

L

~+

-

Augmentation networks

Figure 4.2: The augmentation networks used in the soliton line model

4.1.3 Simulation Results For the Soliton Line

Figures 4.3-4.15 show some simulation results for a soliton line excited by a pulse as shown inFigure 4.7. The netlist for this soliton line with 47 diodes is shown in Appendix A.1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time (ns)

Cur

rent

(A

)

The current of the first diode

Figure 4.3: The first diode current for the 47 diode soliton line

The time domain impulse responses are all limited in time as seen from Figure 4.9 and 4.11.This means that the aliasing effects were minimized during the inverse FFT operation.

It has to be indicated here that the L value chosen (1 nH) was quite large and this affectsthe behavior of the original circuit. Finding a way to obtain the optimum set of values for Rand L would solve this problem.

A comparision between the simulations of the 4 diode soliton line with lossy and no losstransmission lines is shown in Figure 4.16 and 4.17. The losses were chosen small in order tomake the transmission line model accurate. The excitation voltage consisted of a -6 V DC anda sinusoid with 6 V magnitude.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Time (ns)

Cur

rent

(A

)

The current of the last diode

Figure 4.4: The last diode current for the 47 diode soliton line

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−12

−10

−8

−6

−4

−2

0

2

Time (ns)

Vol

tage

(V

)

The state variable of the first diode

Figure 4.5: The first diode state variable for the 47 diode soliton line


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5

−4

−3

−2

−1

0

1

Time (ns)

Vol

tage

(V

)

The state variable of the last diode

Figure 4.6: The last diode state variable for the 47 diode soliton line

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−12

−10

−8

−6

−4

−2

0

Time (ns)

Vol

tage

(V

)

The source voltage

Figure 4.7: The source voltage for the 47 diode soliton line


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (ns)

Cur

rent

(A

)

The source current

Figure 4.8: The source current for the 47 diode soliton line

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

0

20

40

60

80

100

Time (ns)

Impu

lse

(Ohm

)

The direct Impulse response as seen from the voltage source

Figure 4.9: The direct impulse resonse as seen from the voltage source terminals in the 47 diodesoliton line


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−20

0

20

40

60

80

100

Time (ns)

Impu

lse

(Ohm

)

The direct Impulse response as seen from the voltage source (Enlarged)

Figure 4.10: The enlarged version of Figure 4.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (ns)

Impu

lse

(Ohm

)

The cross impulse response as seen from the voltage source

Figure 4.11: The cross impulse resonse as seen from the voltage source terminals in the 47 diodesoliton line


0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

70

80

90

100

Frequency (GHz)

Rea

l im

puls

e re

spon

se (

Ohm

)

The frequency domain direct impulse response

Figure 4.12: The real part of the direct impulse resonse in frequency domain as seen from thevoltage source in the 47 diode soliton line

0 50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

35

40

45

Frequency (GHz)

Imag

inar

y im

puls

e re

spon

se (

Ohm

)

The frequency domain direct impulse response

Figure 4.13: The imaginary part of the direct impulse resonse in frequency domain as seen fromthe voltage source terminals in the 47 diode soliton line


0 50 100 150 200 250 300 350 400 450 500−4

−2

0

2

4

6

8

10

12

Frequency (GHz)

Rea

l im

puls

e re

spon

se (

Ohm

)

The frequency domain cross impulse response

Figure 4.14: The real part of the cross impulse resonse in frequency domain as seen from thevoltage source in the 47 diode soliton line

0 50 100 150 200 250 300 350 400 450 500−10

−8

−6

−4

−2

0

2

4

Frequency (GHz)

Imag

inar

y im

puls

e re

spon

se (

Ohm

)

The frequency domain cross impulse response

Figure 4.15: The imaginary part of the cross impulse resonse in frequency domain as seen fromthe voltage source terminals in the 47 diode soliton line


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−12

−10

−8

−6

−4

−2

0

2

Time (ns)

Vol

tage

(V

)

The state variable of the last diode for the 4−diode soliton line

Lossy case No loss case

Figure 4.16: A comparision between the lossy and no loss case in terms of the state variables

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time (ns)

Cur

rent

(A

)

The current through the last diode for the 4−diode soliton line

Lossy case No loss case

Figure 4.17: A comparision between the lossy and no loss case in terms of the currents


4.2 A Simple Diode Circuit

The results shown in Figure 4.18 and 4.19 are the outputs of the netlist in Appendix A.2. Onceagain it is important to have the capacitor and the inductor after the voltage source to providenatural filtering. The careful choice of these inductance and capacitance values bandlimits thefrequency domain response of the circuit as seen from the voltage source. Also, phase-shiftingplays an important role in this circuit. Without phase-shifting the correct voltage levels arenot obtained. The results are compared with the HB analysis simulation results in Figure 4.18and 4.19. The plots for HB show only one period of the solution, since it is known that thesolution is periodic.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−3

−2

−1

0

1

2

time(ns)

volta

ge(V

)

The diode state variable

Transient Analysis HB Analysis

Figure 4.18: The diode state variable for the simple circuit

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time (ns)

Cur

rent

(A

)

The diode current

Transient Analysis HB Analysis

Figure 4.19: The diode current for the simple circuit

It is important to realize that the compared voltages may not always be the same, becausetransient analysis outputs the state variable and the voltage obtained from HB analysis is


the actual voltage across the diode terminals. In this particular case, the series resistanceof the model (which is shown in Appendix B) is 10 Ω, and hence the state variable and thevoltage across the diode element are not the same. The difference becomes noticable when theconduction current in the model dominates the current flowing through the diode. At all othertimes when the low capacitor current is dominant and the diode is reverse biased, the voltagesare almost the same.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2

Time (ns)

Vol

tage

(V

)

The state variable for two different netlists

Without phase−shift With phase−shift

Figure 4.20: The effect of phase-shift on the simple diode circuit solution

Figure 4.20 shows the effect of phase-shifting for the transient analysis of this simple diodecircuit. It is clear that the voltage levels are not right and there is a lot of aliasing effectsseen in the output waveform of the diode state variable if the phase-shift is turned off. This isan example of a circuit where phase-shift works well, however one should always consider thelimitations of this method as mentioned in Chapter 3. The time shift between the two results inthis case is not due to the phase-shift operation. Since the voltage across the diode is expectedto increase to a positive value due to the excitation source, the waveform observed for the casewithout phase-shift is completely wrong.

4.3 Analysis of running time

Table 4.1 shows the running time for different parts of the transient analysis. The circuit usedhere is described by the netlist in Appendix A.2. As the number of sampling points is increasedthe IFFT time increases, as expected. If the number of sampling points are N , the computationtime for the IFFT is known to be proportional to N log N [37]. Although the IFFT time shownis the total execution time of the related routine, rough calculations can still be made to seethat the above fact holds for this case, too. In order to minimize the effects of the calculationsother than the IFFT in the code, it is best to consider cases for the sampling points of 8192and 16384. Using the proportionality, one expects a factor of 2.1538 and the measured IFFTtime is indeed in agreement with this.

Another point of interest in Table 4.1 is the execution time for the iteration routines. For512 sampling points the time required to solve the nonlinear problem is almost twice the timerequired for the 1024 sampling points. This is due to the fact that more frequency domaininformation is available for the higher number of sampling points. Convergence gets more


40 50 60 70 80 90 100−15

−10

−5

0Experimental result for the soliton line simulated

Time (ps)

Vol

tage

(V

)

Figure 4.21: Experimental data for the soliton line simulated

difficult as interpolation needs to be done for the lower number of sampling points. For thecircuits simulated so far, 1024 was found to be the optimum number of sampling points. Usingmore points than the optimum number of sampling points does not effect the execution timetoo much for the nonlinear iterations.

Table 4.1: Execution times for different number of sampling points

Sampling points IFFT time(s) Total time(s) Iteration time(s)512 0.08 33.40 33.06

1024 0.15 16.90 16.182048 0.29 17.27 15.914096 0.66 18.82 15.988192 1.66 22.12 16.00

16384 3.54 31.96 15.90

4.4 Experimental Data

Experimental data presented in Figure 4.21 was obtained from Case [38]. The voltage waveformshown is the output voltage at the last diode of the circuit. In this case the NLTL was drivenby a 27 dBm sine wave with -3.0 V DC bias. The impulse of the waveform shown is 11.4 Vp−pand has a duration of 5.1 ps.

Chapter 5

Conclusions and Future Research

5.1 Conclusions

A convolution based transient analysis with state variables was developed for the simulation ofspatially distributed microwave circuits. The important aspects implemented here are

• Bandlimited frequency representations for the linear subcircuit is the most importantissue in the convolution analysis implemented in this work.

• Augmentation networks are important in bandlimiting the frequency domain signals, inminimizing the aliasing effects in the inverse FFT and in limiting the time domain impulseresponses.

• Natural filtering by the use of small capacitances and inductances provides bandlimiting.

• Careful use of phase-shifting can avoid the aliasing problems in the inverse FFT.

• The use of well-chosen state variables makes it easier to implement the nonlinear elementequations and improves the robustness of the analysis.

• The separation of the circuit into linear and nonlinear parts can still be used in transientanalysis with the linear part implemented in frequency domain and nonlinear part im-plemented in time domain. This provides an advantage because many nonlinear devicessuch as sources can best be described in time domain. Devices with admittance parameterdescriptions in frequency can be included in the linear part.

5.2 Future Research

Several improvements can be added to the newly developed transient analysis in TRANSIM.Of these improvements the most urgent one is to provide the compatibility with ADOL-C [29].ADOL-C is a software package written in C and C++ that performs automatic differentiation.This is urgent because it will definitely help the convergence issues in the nonlinear solution,since the derivatives calculated by ADOL-C are exact. Currently, the differences are used toapproximate the required derivatives in the nonlinear equation solver or in the element routines.ADOL-C can be used in both nonlinear element models and in the calculation of the Jacobianmatrix in the solution of the nonlinear equation system in order to reduce the nonlinear solutiontime. ADOL-C is currently being used in the state variable harmonic balance analysis [25].

31

CHAPTER 5. CONCLUSIONS AND FUTURE RESEARCH 32

Christoffersen showed that the convergence rate was greater using automatic differentiationthan using differences in the case of harmonic balance [26].

Another improvement that can be made is in outputing the results. Currently, only the statevariables and currents at the nonlinear terminals can be viewed at the output. A large convo-lution operation is needed to calculate every voltage at every node in the circuit. Frequencydomain calculation is also possible provided that the aliasing issues discussed throughout thethesis are solved.

Considering thermal issues and having the temperature parameters included in the transientanalysis (or in TRANSIM in general) is another subject for future work.

Bibliography

[1] T. J. Brazil Causal, “Causal Convolution-A New Method for the Transient Analysis ofLinear Systems at Microwave Frequencies,” IEEE Transactions on Microwave Theory andTechniques, Vol. 43, no.2, pp. 315-23, February 1995.

[2] T. J. Brazil, “A New Method For The Transient Simulation of Causal Linear SystemsDescribed In The Frequency Domain,” IEEE MTT-S International Microwave SymposiumDigest, pp. 1485-88, 1992.

[3] M. B. Steer, “Simulation of Nonlinear Microwave Circuits - an Historical Perspective andComparisons,” IEEE MTT-S International Microwave Symposium Digest, Vol. 2, pp. 599-602, 1991.

[4] E. Chiprout and M. Nakhla. “Fast nonlinear waveform estimation for large disributednetworks,” IEEE MTT-S International Microwave Symposium Digest, Vol. 3, pp. 1341-1344, June 1992.

[5] M. S. Nakhla and J. Vlach, “A piecewise harmonic balance tecnique for determination ofperiodic response of nonlinear systems,” IEEE Transactions on Circuits and Systems, Vol.CAS-23, pp. 85-91, February 1976.

[6] P. K. Chan, Comments on “Aysmptotic Waveform Evaluation for Timing Analysis,” IEEETransactions on CAD, Vol. 10, pp. 1078-79, August 1991.

[7] C. Chen and L. Shieh, “A novel approach to linear model simplification,” Int.J. Contr.,Vol. 8, pp. 561-570, 1968.

[8] M. Bosley and F. Lees, “A survey of simple transfer-function derivations from high-orderstate-variable models,” Automat., Vol. 8, pp. 765-775, 1972.

[9] M. F. Hutton and B. Friedland, “Routh approximations for reducing order of linear, time-invariant systems,” IEEE Trans. Autom. Contr., Vol. AC-20, pp. 329-337, June 1975.

[10] S. Kung and D. Lin, Recent progress in linear system model -reduction via Hankel ma-trix approximation, Techical Report, Univ. Southern California, Department of ElectricalEngineering, 1980.

[11] R. F. Brown, “Model stability in use of moments to estimeate pulse transfer functions,”Electron Lett., Vol. 7, pp. 587-589, September 1971.

[12] Y. Shamash, “Stable reduced-order models using Pade-type approximations,” IEEE Trans.Autom. Contr., Vol. AC-19, pp. 615-616, October 1974.

33

BIBLIOGRAPHY 34

[13] W. C. Elmore, “The transient response of damped linear networks with particular regardto wideband amplifiers,” Journal of Applied Physics, Vol. 19, no.1, pp. 55-63, 1948.

[14] “Modeling and Simulation of VLSI Interconnections with Moments,” Ph.D Dissertation,Massachusetts Institute of Technology, June 1989.

[15] V. Raghavan, R. A. Rohrer, L. T. Pillage, J. Y. Lee, J. E. Bracken and M. M. Alaybeyi,“AWE Inspired,” Proceedings of the Custom Integrated Circuits Conference, pp. 18.1.1-18.1.8, 1993.

[16] C. L. Ratzlaff and L. T. Pillage, “RICE: Rapid Interconnect Circuit Evaluation usingAWE,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,Vol. 13, pp 763-76, June 1994.

[17] C. T. Dikmen, M. M. Alaybeyi, S. Topcu, A. Atalar, E. Sezer, M. A. Tan, and R. A.Rohrer, “Piecewise Linear Asymptotic Waveform Evaluation For Transient Simulationof Electronic Circuits,” Proceedings of IEEE International Symposium on Circuits andSystems, pp. 854-57, June 1991.

[18] R. J. Trihy and R. A. Rohrer, “AWE macromodels for nonlinear circuits,” Proceedings ofthe 36th Midwest Symposium on Circuits and Systems, Vol. 1., pp. 633-636, August 1993.

[19] M. Celik, O. Ocali, M. A. Tan, and A. Atalar, “Pole-zero computation in microwave circuitsusing multipoint Pade approximation, IEEE Transactions on Circuits and Systems,” Vol.42, pp. 6-13, January 1995.

[20] P. J. C. Rodrigues, Computer-aided analysis of nonlinear microwave circuits, Artech House,1998.

[21] R. V. Churchill and J. W. Brown, Complex Variables and Applications, 5th edition, Mc-Graw Hill, 1990.

[22] R. J. Trihy and R. A. Rohrer, “AWEswit: A Switched Capacitor Circuit Simulator,” IEEEJournal of Solid State Circuits, Vol. 29, pp. 217-25, March 1994.

[23] K. S. Kundert, J. K. White and A. Sangiovanni-Vincentelli, Steady-state methods for sim-ulating analog and microwave circuits, Boston, Dordrecht, Kluwer Academic Publishers,1990.

[24] C. Gordon, T. Blazeck and R. Mittra, “Time Domain Simulation of Multiconductor Trans-mission Lines with Frequency-Dependent Losses,” IEEE Transactions on Computer AidedDesign of Integrated Circuits and Systems, Vol. 11, pp. 1372-87, November 1992.

[25] C. E. Christoffersen, State Variable Harmonic Balance Simulation of a Quasi-optical PowerCombining System, M.S. Thesis, Department of Electrical and Computer Engineering,North Carolina State University, Raleigh, North Carolina, U.S.A., 1998.

[26] C. E. Christoffersen, Private Communication.

[27] C. E. Christoffersen, M. B. Steer and M. A. Summers, “Harmonic Balance Analysis ForSystems With Circuit-Field Interactions”, IEEE MTT-S International Microwave Sympo-sium, June 1998.

BIBLIOGRAPHY 35

[28] M. S. Basel, Simulation of High Speed Digital Circuit Interconnection Networks, Ph.DDissertation, Department of Electrical and Computer Engineering, North Carolina StateUniversity, Raleigh, North Carolina, U.S.A., 1993.

[29] A. Griewank, D. Juedes, J. Utke, “Adol-C: A Package for the Automatic Differentiationof Algorithms Written in C/C++,” Version 1.7, September 1996.

[30] K. S. Kundert, A. Songiovanni-Vincentelli, Sparse User’s Guide - A Sparse Linear Equa-tion Solver, Department of Engineering and Computer Sciences, University of California,Berkeley, California, U.S.A, Version 1.3a, April 1988.

[31] R. S. Bain, NNES User’s Manual, 1993.

[32] Compact Software, Microwave Harmonica Elements Library , 1994.

[33] M. A. Summers, Simulation of a quasi-optical grid amplifier, M.S. Thesis, Departmentof Electrical and Computer Engineering, North Carolina State University, Raleigh, NorthCarolina, U.S.A., 1998.

[34] A. Brambilla and D. DÀmore, “The Simulation Errors Introduced by the SPICE TransientAnalysis,” IEEE Transactions on Circuits and Systems, Vol. 40, pp 57-60, January 1993.

[35] T. W. Nuteson, Electromagnetic Modeling of Quasi-Optical Power Combiners, Ph.D Dis-sertation, Department of Electrical and Computer Engineering, North Carolina State Uni-versity, Raleigh, North Carolina, U.S.A., 1996.

[36] P. Gubian, M. Zanella, “Stability properities of integration methods in SPICE transientanalysis,” Proceedings IEEE International Symposium on Circuits and Systems, Vol. 5, pp.2701-4, 1991.

[37] A. V. Oppenheim, R. W. Schafer, Dicrete-Time Signal Processing, Prentice-Hall, Inc.,1989.

[38] M. G. Case, Nonlinear Transmission Lines For Picosecond Pulse, Impulse and Millimeter-wave Harmonic Generation, Ph.D Dissertation, Department of Electrical and ComputerEngineering, University of California, Santa Barbara, California, U.S.A., 1993.

[39] H. Shi, C. W. Domier, N. C. Luhmann, “A monolithic nonlinear transmission line systemfor the experimental study of lattice solutions,” Journal of Applied Physics, Vol 4., pp.2558-64, August 1995.

[40] M. S. Basel, M. B. Steer, P. D. Franzon, “Simulation of High Speed Interconnects Using aConvolution-Based Hierarchical Packaging Simulator,” IEEE Transactions on components,packaging, and manufacturing technology, Vol. 18 no. 2, pp. 74-82, February 1995.

[41] Series IV Manual, Circuit Network Items, p. 19-9, July 1995.

[42] W. M. Zuberek, A. Konczykowska, H. Wang, “Distributed Transmission Lines and Time-Domain Analysis in SPICE-like Circuit Simulators,” IEEE International Symposium onCircuits and Systems, pp. 709-12, 1989.

Appendix A

Netlist Examples and Terminology

This appendix contains the netlists that were used to produce the results shown in the thesis.The voltage source in the netlist in section A.1 is the Thevenin equivalent of the original voltagesource with the augmentation resistor. In order to get the actual value across the terminalsof this voltage source, one should divide the voltage levels in the parameter description of thevoltage source by 8.

A.1 The netlist for the soliton line with 47 diodes

svtr_47cellpulse.net.options type = "svtr"*.options dcNormal = "on".options delay = "on".options tr_debug = "on".options keep_svM = "on"

.options ind = 1.0nH

.options rs = 1.0

.options bv = -30.0

.options fs = 1.0e10

.options tl = 0.0001

.options zc = 75.0

.tran 0 1000.0e-12 0.

.options spts = 2048

.options type = "svtr" sfrq = 500.e9

.options LPFOrder = 3 impulselength = 1

.options ytthresthru = 0.00 ytthrescross = 0.0

.options impulsescale = 1.0

.options tolerance = 1.0e-6 maxNoOfIterates = 100

.options LPFCornerFrequency = 500e9

*vsin:1 203 0 rs = 300.0 mag = 48 dc = -48 freq = 9e9vpulse:1 203 0 rs = 300.0 tr = 100.0e-12 clockspeed = 4.0e9+ tf = 100e-12 high = -100.0 low = 0.0 starttime = 0.0+ sequence = "1000000000000"res:s3 0 203 r=100ind:1 203 201 l=ind

diode:d1 401 0 js=1.e-12 alfa=38.696 e=10 ct0=1.08e-15

36

APPENDIX A. NETLIST EXAMPLES AND TERMINOLOGY 37

+ r0 = rs fi=0.643 gama=0.451 jb=0. vb= bv area=271.64ind:d1 1 401 l=indres:d11 401 0 r=100

diode:d2 402 0 js=1.e-12 alfa=38.696 e=10 ct0=1.08e-15+ r0 = rs fi=0.643 gama=0.451 jb=0. vb= bv area=258.63ind:d2 2 402 l=indres:d21 402 0 r=100

diode:d3 403 0 js=1.e-12 alfa=38.696 e=10 ct0=1.08e-15+ r0 = rs fi=0.643 gama=0.451 jb=1.e-5 vb=bv area=246.24ind:d3 3 403 l=indres:d31 403 0 r=100









diode:d12 412 0 js=1.e-12 alfa=38.696 e=10 ct0=1.08e-15+ r0 = rs fi=0.643 gama=0.451 jb=1.e-5 vb=bv area=158.3ind:d12 12 412 l=ind


res:d121 412 0 r=100
























diode:d35 435 0 js=1.e-12 alfa=38.696 e=10 ct0=1.08e-15+ r0 = rs fi=0.643 gama=0.451 jb=1.e-5 vb=bv area=51.19


ind:d35 35 435 l=indres:d351 435 0 r=100














*tlinp:t0 201 1 z0mag=zc length=501.29e-6 k=7 tand=tl f=Fstlinp:t1 1 2 z0mag=zc length=978.57e-6 k=7 tand=tl f=Fstlinp:t2 2 3 z0mag=zc length=931.69e-6 k=7 tand=tl f=Fstlinp:t3 3 4 z0mag=zc length=887.06e-6 k=7 tand=tl f=Fstlinp:t4 4 5 z0mag=zc length=844.57e-6 k=7 tand=tl f=Fstlinp:t5 5 6 z0mag=zc length=804.11e-6 k=7 tand=tl f=Fstlinp:t6 6 7 z0mag=zc length=765.59e-6 k=7 tand=tl f=Fstlinp:t7 7 8 z0mag=zc length=728.92e-6 k=7 tand=tl f=Fstlinp:t8 8 9 z0mag=zc length=694.00e-6 k=7 tand=tl f=Fstlinp:t9 9 10 z0mag=zc length=660.75e-6 k=7 tand=tl f=Fstlinp:t10 10 11 z0mag=zc length=629.10e-6 k=7 tand=tl f=Fstlinp:t11 11 12 z0mag=zc length=598.97e-6 k=7 tand=tl f=Fstlinp:t12 12 13 z0mag=zc length=570.27e-6 k=7 tand=tl f=Fstlinp:t13 13 14 z0mag=zc length=542.96e-6 k=7 tand=tl f=Fstlinp:t14 14 15 z0mag=zc length=516.95e-6 k=7 tand=tl f=Fstlinp:t15 15 16 z0mag=zc length=492.18e-6 k=7 tand=tl f=Fstlinp:t16 16 17 z0mag=zc length=468.61e-6 k=7 tand=tl f=Fstlinp:t17 17 18 z0mag=zc length=446.16e-6 k=7 tand=tl f=Fstlinp:t18 18 19 z0mag=zc length=424.79e-6 k=7 tand=tl f=Fstlinp:t19 19 20 z0mag=zc length=404.44e-6 k=7 tand=tl f=Fstlinp:t20 20 21 z0mag=zc length=385.06e-6 k=7 tand=tl f=Fstlinp:t21 21 22 z0mag=zc length=366.62e-6 k=7 tand=tl f=Fstlinp:t22 22 23 z0mag=zc length=349.05e-6 k=7 tand=tl f=Fstlinp:t23 23 24 z0mag=zc length=332.33e-6 k=7 tand=tl f=Fstlinp:t24 24 25 z0mag=zc length=316.41e-6 k=7 tand=tl f=Fstlinp:t25 25 26 z0mag=zc length=301.26e-6 k=7 tand=tl f=Fstlinp:t26 26 27 z0mag=zc length=286.83e-6 k=7 tand=tl f=Fstlinp:t27 27 28 z0mag=zc length=273.09e-6 k=7 tand=tl f=Fstlinp:t28 28 29 z0mag=zc length=260.00e-6 k=7 tand=tl f=Fstlinp:t29 29 30 z0mag=zc length=247.55e-6 k=7 tand=tl f=Fstlinp:t30 30 31 z0mag=zc length=235.69e-6 k=7 tand=tl f=Fstlinp:t31 31 32 z0mag=zc length=224.40e-6 k=7 tand=tl f=Fstlinp:t32 32 33 z0mag=zc length=213.65e-6 k=7 tand=tl f=Fstlinp:t33 33 34 z0mag=zc length=203.42e-6 k=7 tand=tl f=Fstlinp:t34 34 35 z0mag=zc length=193.67e-6 k=7 tand=tl f=Fstlinp:t35 35 36 z0mag=zc length=184.39e-6 k=7 tand=tl f=Fstlinp:t36 36 37 z0mag=zc length=175.56e-6 k=7 tand=tl f=Fstlinp:t37 37 38 z0mag=zc length=167.15e-6 k=7 tand=tl f=Fstlinp:t38 38 39 z0mag=zc length=159.14e-6 k=7 tand=tl f=Fstlinp:t39 39 40 z0mag=zc length=151.52e-6 k=7 tand=tl f=Fstlinp:t40 40 41 z0mag=zc length=144.26e-6 k=7 tand=tl f=Fstlinp:t41 41 42 z0mag=zc length=137.35e-6 k=7 tand=tl f=Fstlinp:t42 42 43 z0mag=zc length=130.77e-6 k=7 tand=tl f=Fstlinp:t43 43 44 z0mag=zc length=124.51e-6 k=7 tand=tl f=Fstlinp:t44 44 45 z0mag=zc length=118.54e-6 k=7 tand=tl f=Fstlinp:t45 45 46 z0mag=zc length=112.86e-6 k=7 tand=tl f=Fstlinp:t46 46 47 z0mag=zc length=107.46e-6 k=7 tand=tl f=Fstlinp:t47 47 48 z0mag=zc length=102.31e-6 k=7 tand=tl f=Fs**res:rl 48 0 r=75. nonlinear=1 poly 1 2 0 0. 1.


res:rl 48 0 r=75.

.out plot 0 0 impulse in "impulse.out"

.out plot 0 0 fimpulse real in "fimpulse_real.out"

.out plot 0 0 fimpulse imag in "fimpulse_imag.out"

.out plot 0 1 impulse in "impulse01.out"

.out plot 0 1 fimpulse real in "fimpulse_real01.out"

.out plot 0 1 fimpulse imag in "fimpulse_imag01.out"

*.out plot node "vsin:1" 0 itx in "source_current".out plot node "vpulse:1" 0 itx in "source_current"*.out plot node "vsin:1" 0 xt in "source_voltage".out plot node "vpulse:1" 0 xt in "source_voltage".out plot node "diode:d1" 0 itx in "diode1.current".out plot node "diode:d47" 0 itx in "diode47.current".out plot node "diode:d1" 0 xt in "diode1.state".out plot node "diode:d47" 0 xt in "diode47.state"

.end

A.2 The netlist for a simple diode circuit

svtr_diode.net*.options dcNormal = "on".options delay = "on".options keep_svM = "on".options tr_debug = "on"*.options verb_level = 3

vsin:1 12 0 rs = 10. mag = 10 dc = 0 freq = 10e8*vpulse:1 12 0 rs = 50. tr = .1e-9 clockspeed = 5e8*+ tf = .1e-9 high = 30 low = 0.0 starttime = 0*+ sequence = "1010" x = 0 y = 0ind:1 12 11 l=1e-9*ind:1 12 11 l=10e-9*cap:1 11 0 c=5e-12cap:1 11 0 c=1e-12res:1 0 1 r =50res:2 0 3 r = 50res:3 11 1 r = 50

diode:D1 1 3 js=5.1e-14 alfa=38.696 e=10 ct0=1.32767e-15+ vb=-1.0e50+ fi=0.8 gama=0.810205 jb=1e-5 area=1 afac=38.696 r0=10.

.tran 0 5.0e-9 0.

.options spts = 1024

.options type = "svtr" sfrq = 100.e9

.options LPFOrder = 3 impulselength = 1.0

.options ytthresthru = 0.00 ytthrescross = 0.00

.options impulsescale = 1.0

.options tolerance = 1.0e-6 maxNoOfIterates = 100


.options LPFCornerFrequency = 100e9

.out plot node "vsin:1" 0 itx in "current_s.out"*.out plot node "vpulse:1" 0 itx in "current_s.out".out plot node "vsin:1" 0 xt in "voltage_s.out"*.out plot node "vpulse:1" 0 xt in "voltage_s.out".out plot node "diode:D1" 0 itx in "current_d.out".out plot node "diode:D1" 0 xt in "voltage_d.out".out plot 0 0 impulse in "impulse1.out".out plot 1 1 impulse in "impulse2.out".out plot 0 0 fimpulse mag in "fimpulse1.out".out plot 1 1 fimpulse mag in "fimpulse2.out".end

A.3 Netlist Terminology

The netlists in A.1 and A.2 can be run in TRANSIM to produce results. More informationabout TRANSIM can be found in other sources [25, 33, 28]. The following features were addedduring this work:

spts Number of sampling points in frequency.

delay Sets the phase correction on or off.

keep svM If set to on, it keeps the frequency domain circuit impedance matrix.

svtr State-variable transient analysis is run if type is set to this.

sfrq Sampling frequency.

impulse Time domain impulse response for the corresponding state variables.

fimpulse Frequency domain impulse response for the corresponding state variables.

xt Extracts the requested state variable for the display.

itx Extracts the requested nonlinear current for the display.

Appendix B

Element Models

B.1 Transmission Line Model

The transmission line model used in the simulations is a regular coaxial line with small losses,therefore perturbation method was used to arrive at the equations. Skin effect is taken intoaccount by a frequency scaling term [41]. That is, loss increases as the frequency is increased.This model was implemented in frequency domain, so it will be included in the linear subcircuitduring the analysis. Hence, it can also be used in harmonic balance analysis code of TRANSIM.

B.1.1 Parameters

z0mag Characteristic impedance in resistance unitslength Physical length of the transmission line in metersk Dielectric constanttand Loss tangentf Reference frequency (Equivalent to F below) in Hza Attenuation constant in dB per meters

B.1.2 Equations

A(f) = A (forF = 0) (B.1)

A(f) = A(F )

√f

F(forF 6= 0) (B.2)

C =

√k

Z0 c(B.3)

L = Z20 C (B.4)

R = 2A(f)Z0 (B.5)G = tan δl × ω C (B.6)

Characteristic impedance is calculated according to the formula

Z0 =

√R + jωL

G + jωC(B.7)

44

APPENDIX B. ELEMENT MODELS 45

and the attenuation is calculated according to the formula

γ =√

(R + jωL) (G + jωC) (B.8)

These are converted to the required y-parameter representation by using the following formulae.

y11 =1

Z0 tanh(γl)(B.9)

y12 = − 1

Z0 sinh(γl)(B.10)

y22 = y11 (B.11)y21 = y12 (B.12)

B.2 Diode Model

The diode model used in the simulations consists of a series reistor, a nonlinear capacitor, anda regular diode. This model was implemented as a nonlinear device in the analysis.

B.2.1 Parameters

js Saturation current in amperesalfa Slope factor of the conduction currente Power-law parameter of breakdown currentct0 Zero-bias depletion capacitancer0 Bias-dependent part of series resistance in forward-biasfi Built-in barrier potentialgama Capacitance power-law parameterjb Breakdown saturation currentvb Breakdown voltagearea Area multiplier

Rs

C j

I d

ID

I c

Vj

Figure B.1: The diode model

B.2.2 Equations

The model in Figure B.1 is used. The state variable is chosen as vj. The currents are calculatedaccording to the equations provided in Compact Software Elements Library [32]. Then, the

APPENDIX B. ELEMENT MODELS 46

capacitor current is given by

Ic = Cdvjdt

(B.13)

and the total current is given by

ID = Ic + Id

= Cdvjdt

+ Id (B.14)

As it can be seen, the derivative of the state variable is required. In this model, the derivativeswere calculated using the differences, which do not actually yield the exact derivatives.

The parametrized model [25] was also implemented and it yielded the same results for thesame circuits.

Appendix C

Additions to TRANSIM

In this appendix the additions to the TRANSIM code during this work are presented in tab-ular form. More information on the TRANSIM code can be found in the work of Basel andChristoffersen [28, 25].

Table C.1: Added Transim file names and descriptionssvtr misc.c Routines for reading the options in the netlistsvtr phys.c Physical routines

svtr func ev.c Function evaluation routinesan main svtr.c Main routinesvtr interface.c Interface routines between NNES & TRANSIM

svtr freq.c Frequency vector set up routine

47

APPENDIX C. ADDITIONS TO TRANSIM 48

Table C.2: List of added functionsFunction Name File Name DescriptionGet svtroptions svtr misc.c It reads and stores the options.Free svtroptions svtr misc.c It frees the memory used to store

the options.svtr Get States svtr phys.c It finds the number of states re-

quired and creates the nonlinear el-ement list.

svtr Create T svtr phys.c It allocates and fills the T matrixsvtr Destroy T svtr phys.c It frees the T matrix and the non-

linear element list.svtr Get U and I svtr phys.c It gets the nonlinear voltages and

currents for the given states.svtr output svtr phys.c It outputs the results to the graph

structures.svtr Create ws svtr phys.c It initializes the structures for the

evaluation routine.svtr Destroy ws svtr phys.c It frees the memory created by

svtr Create ws.Create svtrdata svtr func ev.c It allocates the memory for svtrdata

structure.New svM svtr func ev.c It allocates space for the time do-

main matrix information structure.Free svM svtr func ev.c It frees the memory created by

New svM.Destroy svtrdata svtr func ev.c It frees the memory created by Cre-

ate svtrdata.svtr Filter svtr func ev.c It filters the given data.svtr IFFT svtr func ev.c It takes the IFFT of the given data.

svtr Func ev svtr func ev.c It calculates the error function.svtr solve svtr func ev.c It returns the solution (the state

variables) in vector form.an main svtr an main svtr.c This is the main routine of the tran-

sient analysis.svtr solve int svtr interface.c It makes the call to the FORTRAN

interface.Create svtr freq svtr freq.c It sets up the frequency vector.

Documents

TRANSIENT ANALYSIS OF SPATIALLY DISTRIBUTED …OZKAR, METE. Transient Analysis of Spatially Distributed Microwave Circuits Using Con-volution and State Variables. (Under the direction