Transmission line theory for cable modeling: a delay-rational...

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Department of Computer Science, Electrical and Space EngineeringEISLAB Transmission line theory for cable

modeling: a delay-rational model based on Green’s functions

ISSN 1402-1757ISBN 978-91-7583-561-7 (print)ISBN 978-91-7583-562-4 (pdf)

Luleå University of Technology 2016

Maria D

e Lauretis Transm

ission line theory for cable modeling: a delay-rational m

odel based on Green’s functions

Maria De Lauretis

Industrial Electronics

Transmission line theory for cablemodeling: a delay-rational model based on

Green’s functions

Maria De Lauretis

Dept. of Computer Science, Electrical and Space EngineeringLulea University of Technology

Lulea, Sweden

Supervisors:

Jonas Ekman, Giulio Antonini

To my parents

iii

ABSTRACT

At present, induction motors are controlled via the so-called variable-frequency drives (VFD)that allow to control the speed for the motors. The purpose of this PhD thesis is to improveelectromagnetic modeling techniques for the study of conducted electromagnetic emissionsin variable-frequency drives, with the aim of enhancing their reliability in energy productionplants. Pulse-width-modulated voltage converters are used to feed an AC motor, and they areconsidered to be the primary reason for high-frequency effects in both the motor and the supplygrid. In particular, high-frequency currents, known as common mode currents, flow betweenall energized components and the ground and travel via low-resistance and low-inductanceinterconnects such as the power cable between the inverter and the motor.

Electrically long power cables are commonly used in VFD installations, and require partic-ular attention. Accurate models can be obtained using the theory of multiconductor transmis-sion lines. In the case of nonlinear terminations, such as an inverter, only time-domain analysisis possible. In recent years, several techniques have been proposed. Some of these techniquesinclude the lumped-element equivalent circuit method, the method of characteristics (MoC)and its generalizations, and the Pade approach. In this context, a modeling technique basedon Green’s functions has been proposed. The input/output impedance matrix is expressed asa rational series, whose poles and their residues are identified by solving algebraic equations.The primary disadvantage of this method lies in the large number of poles that is typically nec-essary to model the dynamics of the system, especially when electrically long interconnectsare considered. To overcome this limitation, we have proposed the Delay-Rational Green’s-Function-based Method, abbreviated as DeRaG. In this method, the line delay is extracted and,by virtue of suitable mathematical manipulation of the rational series, is incorporated throughhyperbolic functions. The delay extraction enables the use of a reduced number of poles andimproves the accuracy of the model in general, avoiding any ringing effects in the time-domainresponse. The primary advantage of the proposed method compared with other well-knowntechniques lies in the delayed state-space representation. The obtained model can be computedregardless of the terminations and/or sources, and the terminal conditions can be immediatelyand essentially incorporated.

The next step will be to simulate the entire inverter-cable-motor system. The partial elementequivalent circuit (PEEC) technique will be used to model the interconnects as well as thediscontinuities in the power cable that can be caused, for example, by switch disconnectors.The theoretical results will be verified against experimental measurements. The final objectiveis to provide new techniques for modeling the electrodynamics of variable-frequency drives toallow their complete EMC assessment as early as the design stage and to enable the planningof corrective actions in advance.

v

CONTENTS

Part I 1

Acronyms 3

CHAPTER 1 – THESIS INTRODUCTION 71.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER 2 – VARIABLE-FREQUENCY DRIVES 112.1 Background on variable-frequency drives . . . . . . . . . . . . . . . . . . . . 112.2 Pulse-width-modulated waveform and harmonic distortion . . . . . . . . . . . 122.3 The role of simulations and measurements . . . . . . . . . . . . . . . . . . . . 14

CHAPTER 3 – CABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY 153.1 Background on multiconductor transmission lines . . . . . . . . . . . . . . . . 153.2 Main shortcomings of the present models . . . . . . . . . . . . . . . . . . . . 223.3 Transmission line theory for the study of common mode currents in cables . . . 23

CHAPTER 4 – THE PROPOSED DELAY-RATIONAL MODEL 254.1 Green’s functions and boundary problems: a brief background . . . . . . . . . 254.2 Delay extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 The delay-rational state-space form . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER 5 – RESEARCH CONTRIBUTIONS 375.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 Paper D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

CHAPTER 6 – CONCLUSIONS AND FUTURE WORK 416.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

REFERENCES 43

vii

Part II 49PAPER A 51

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 Transmission Line Spectral Model . . . . . . . . . . . . . . . . . . . . . . . . 543 Rational macromodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Delayed Lossless Transmission Line Model . . . . . . . . . . . . . . . . . . . 585 Delayed Lossy Transmission Line Model . . . . . . . . . . . . . . . . . . . . 606 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

PAPER B 691 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 Review of the spectral model for Multiconductor Transmission Lines . . . . . 733 Delayed Model of Lossless MTL . . . . . . . . . . . . . . . . . . . . . . . . . 754 The Delay-Rational Model for a lossy MTL . . . . . . . . . . . . . . . . . . . 805 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

PAPER C 971 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992 Green’s function based methods . . . . . . . . . . . . . . . . . . . . . . . . . 1003 Proposed solution for cable bundles . . . . . . . . . . . . . . . . . . . . . . . 1024 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

PAPER D 1091 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112 Green’s function-based method background . . . . . . . . . . . . . . . . . . . 1123 Delay-Rational Green’s Method for MTL with frequency-dependent p.u.l. pa-

rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174 Delay-Rational model in the time-domain . . . . . . . . . . . . . . . . . . . . 1215 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

viii

ACKNOWLEDGMENTS

I acknowledge Svenska Kraftnat (Swedish national grid) for providing funding for this re-search. Also, I would like to acknowledge the people who have supported this research fromboth a professional and personal point of view. In particular, I would like to acknowledgemy supervisors, Jonas Ekman and Giulio Antonini, for their constant support during thesetwo years. They have helped me to organize my PhD studies and to develop new researchideas, always with enthusiasm and effort. Andreas Nilsson, for his irreplaceable assistance inlaboratory activities and his enormous patience and sarcastic humor. Johan Borg, for all thevaluable discussions that we had. Joakim Nilsson and Marcus Lindner, my two colleagues onthis project: I am glad to have such colleagues as you, always open to discussions and collabo-rations. Elena Miroshnikova, for all the nice chocolate and mathematically based study groups.Basel Kikhia, for being the best flatmate and the best “faky” ever. All the people who made mystay at the L-paviljon enjoyable. Andreas Lindner, for helping me with my labs. Dariusz Ko-miniak, for his collaborative attitude. Marcus Lindner, for being a valuable discussion partnerand, also, a valuable husband!

In general, I would like to thank all the nice people and friends who have taken the time totalk with me, either for pleasure or for working reasons. Thank you all!

Lulea, May 2016Maria De Lauretis

ix

Part I

1

Acronyms

ASD adjustable-speed drive.

CM common mode.

DEPACT delay extraction-based passive compact transmission line (algorithm).DeRaG delay-rational Green’s-Function-based method.DM differential mode.

EMC electromagnetic compatibility.EMI electromagnetic interference.

IEC international electrotechnical commission.

MoC method of characteristics.MOR model order reduction.MTL multiconductor transmission line.

PEEC partial element equivalent circuit.PQ power quality.PWM pulse-width modulation.

VFD variable-frequency drive.

3

List of Figures in Part I

2.1 General schematic of a variable-frequency drive. . . . . . . . . . . . . . . . . 122.2 Simplified schematic of a pulse-width-modulated (PWM) voltage source inverter. 122.3 Paths of common mode currents. The red arrows represent the common mode

currents, which travel between the energized component of the ASD, the motor,the ground, and finally the supply through parasitic capacitances and groundingconnections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Per-unit-length parameters for a one-conductor transmission line. . . . . . . . . 183.2 Multiconductor transmission line represented as a 2N-port system, with a com-

mon reference conductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Common and differential current components for a typical four-conductor shielded

cable, treated as a multiconductor transmission line. . . . . . . . . . . . . . . . 233.4 Common mode current analysis of a 4-conductor shielded cable connected be-

tween the inverter and motor, with the shield connected on both sides. . . . . . 244.1 Original Green’s-function-based method. . . . . . . . . . . . . . . . . . . . . 294.2 Complex conjugate pole locations computed using a modal rational approxi-

mation approach for a 6-conductor ribbon cable with frequency-independentp.u.l. parameters, from Paper C. . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Complex conjugate pole families in the complex plane for a 2-conductor linewith frequency-dependent parameters, from Paper D. . . . . . . . . . . . . . . 31

4.4 Complex conjugate pole locations for a 9-conductor cable bundle, from PaperC. The blue circles represent the asymptotic real parts for each family, αk. . . . 32

4.5 Block diagram for the algorithm used to determine the m mode, from Paper C. . 334.6 Block diagram of the state-space model (4.19). . . . . . . . . . . . . . . . . . 34

5

CHAPTER 1

Thesis introduction

1.1 Motivation

Energy production plants are responsible for the generation of energy that is distributed through-out the national electric grid. Typical examples are nuclear, hydroelectric, and fossil-fuel powerplants. Motor-driven systems are widely used in the industry; in these systems, the motor is, inmost cases, an electric motor, and typical examples include pumps, fans, and compressors. Theadvent of power electronics in the early 1960s dramatically changed the way in which motorsare controlled and, ultimately, designed [1]. Previously, motors were designed with electrome-chanical controls, which were neither flexible nor easy to change. Power electronics allowedthe control to be shifted from the motor itself to external (cheaper) circuitry, which eventuallybecame programmable. Motors previously used only for constant-speed applications, such asthe well-known squirrel cage motor, found new life in the 1970s, when electronic inverters en-abled control of their speed. Research progressed in the following directions: control switchedfrom analog to digital, digital processors were adopted, and real-time modeling and simula-tions drove a continuous increase in motor-drive performances. Today, the focus is on energyefficiency and EU directives, and power electronics play a large role [2]. In fact, power mod-ules allow efficient control of the speed of a motor and, as a consequence, of its power output,thereby improving the overall energy efficiency. For the aforementioned reasons, it comes asno surprise that power electronics have found constant and increasing application in the indus-trial environment. The system constituted by a power module and an electric motor is generallycalled a variable-frequency drive (VFD) or an adjustable-speed drive (ASD).

However, unexpected problems have been arising, such as the premature deterioration ofmotor bearings [3, 4] and increasing electromagnetic interference phenomena [5, 6]. Fromrecent reports and investigations [7, 8, 9], it has emerged that variable-frequency drives areplaying an increasing role in determining the reliability of energy production plants. They mayseverely impact the quality of the energy supplied to and obtained from utilities, eventuallyleading to catastrophic failures in a domino effect that is, at present, not entirely understood.When discontinuities or even failures occur in the transmission of energy, the underlying cause

7

8 THESIS INTRODUCTION

of these events is generally referred to as a power quality (PQ) problem [10]. Disturbancesin the electric grid can be due to several causes, such as short circuits, abruptly changingloads, or seasonal problems such as thunderstorms and lightning. According to [11], lightningstrikes cause at least 60% of all dips and short interruptions. It has been found that VFDsare particularly sensitive to disturbances in the power grid [7]. Moreover, VFDs have beenrecognized as one of the sources of a type of disturbance known as “harmonic distortion”,which is primarily caused by electromagnetic interference (EMI) generated in power electroniccomponents.

Since 1906, the International Electrotechnical Commission (IEC) has been engaged in thepreparation and publication of international standards for all electrical, electronic and relatedtechnologies. The purpose of these standards is to define tests and criteria to predict and avoidpotential problems in a product. The IEC-61800 standard, Part 3, is entitled “Adjustable speedelectrical power drive systems”, and it specifies the electromagnetic compatibility (EMC) prod-uct standard and test methods for variable-frequency drives. Clear indications are given for theassessment of the immunity of a VFD to voltage deviation and harmonic distortion as well as itsharmonic emissions. However, even if a product is compliant with the standard, other factorscould degrade its EMC performance. The installation of a system and its earth connections, forexample, play a large role [12], because they may seriously impact and amplify problems withelectromagnetic interference. Also, changes in the EMC behavior of a VFD because of faultconditions are not considered.

In the context of EMC, two main types of disturbances can be identified: radiated distur-bances and conducted disturbances. As the names suggest, the former are emitted or transmit-ted between two devices through the air, whereas the latter are conducted via conductive paths,such as cabling, earthing, and metal frames. Note that at present, only conducted emissionsare considered together with power quality as related topics [10]. Conducted emission testsallow the measurement of the amount of frequency noise that is injected into the grid by theequipment. In view of the above, the project “Improvement of variable-frequency drives in en-ergy production plants” has been initiated, of which the sub-project “Simulations of conducteddisturbances” serves as the central topic of the PhD studies presented in this thesis.

1.2 Thesis outlineThe thesis outline is as follows. A general overview of VFDs, with particular emphasis onthe electromagnetic compatibility aspects, is given in Chapter 2. In fact, conducted emissionscause electromagnetic interference (EMI) in the system, leading to harmonic currents that canpenetrate back to the utility supply. Because of the high switching frequencies imposed by thepulse-width-modulated control signal, the cables behave not only as interconnections but aselectric system components with specific properties, and the theory of multiconductor trans-mission lines can be used to study and model their behavior. In Chapter 3, a general intro-duction to multiconductor transmission line (MTL) theory is provided. In particular, the well-studied solution in the frequency domain is briefly recalled. Some of the most well-knowntechniques for time-domain simulations are summarized, such as the lumped-element equiva-lent circuit method, the method of characteristics (MoC) and its generalizations, and matrix-

1.3. RESEARCH QUESTIONS 9

rational-approximation-based techniques. In Chapter 4, we introduce the Green’s-function-based method that is at the root of the previously published papers pertaining to the researchpresented herein. The proposed delayed model exploits delay extraction, which leads to theso-called Delay-Rational Green’s-Function-based Method, or DeRaG. In general, the mainadvantage of the proposed method compared with other well-known techniques lies in the de-layed state-space representation. The model can be computed regardless of the terminationsand/or sources, and the terminal conditions can be immediately and essentially incorporated. InChapter 5, the research contributions are summarized and a short description of the four papersincluded in this thesis is given. In Chapter 6, we present plans for future work. In particular,the second part of these PhD studies will be devoted to the simulation of the overall system,for the verification of the theoretical results against experimental measurements. To serve thispurpose, a motor kit has already been tested, but the preliminary results are not covered in thisthesis. Additionally, the interconnections will be modeled using the partial element equivalentcircuit (PEEC) technique. The ability to build a reliable mathematical model will allow theproposal of filtering techniques and corrective actions in case of fault conditions, with the pri-mary purpose of avoiding the tripping of variable-frequency drives in energy production plants.Possibly, in situ measurements may be performed.

1.3 Research questionsIn the study of the electromagnetic interference in VFD, accurate time domain models needto be adopted. In fact, the drive is characterized by nonlinear components that can only bedescribed in the time domain. Typically, most of the focus is on the power module and on themotor time-domain models. Even though it is well recognized that a proper model of the cableis crucial for the accuracy of the system simulation, see for example [5] and the considerationsin [13], it is not uncommon to see in literature the use of modeling techniques, such as thelumped circuit model approximation [9], which have known limitations, as it will be explainedin Chapter 3. This is somehow surprising, because a considerable amount of research has beendevoted to cables, studied as multiconductor transmission lines (MTLs) in the time domain[14, 15, 16, 17, 18]. On the other side, most of the MTL modeling techniques proposed sofar have shortcomings that prevent their implementation in commonly used circuit simulatorenvironments, such as SPICE-like transient simulators, where the nonlinear components can beeasily and accurately represented. The most common limitations of existing models for MTLsmodeling are:

• passivity is not guaranteed by-construction. Non-passive models may result in unstablemodels when connected to external terminations, thus leading to misleading results;

• the frequency-dependent nature of the cable is not easy to capture. A model that doesnot account for the frequency-dependent nature of the cable may lead to totally incorrectresults because the dissipative and dispersive effects are not considered; and

• the circuit counterpart of the model is normally not explicitly given as a code and can becumbersome to obtain. An exception is [19] where a minimal code is provided. How-

10 THESIS INTRODUCTION

ever, the cable is studied under specific geometric assumptions. Also, complex modelsnormally require complex circuit descriptions, which can severely impact the overallsimulation time.

The method proposed in [20] for multiconductor transmission line modeling has been pre-sented with the main object to provide a general, accurate and passive model for MTLs. Themodel can include frequency-dependent parameters without resorting to the convolution prod-uct and admits a state-space form realization in the time domain. One of the major advantageof the model relies on the state-space form derived. In fact, linear state-space systems havemany advantages, such as:

• their theory is well understood and established, and properties such as stability and pas-sivity are relatively easy to check and, if necessary, to enforce;

• their dimension can easily be compressed by virtue of model order reduction (MOR)techniques, the use of which is constantly increasing in the research community; and

• they admit an immediate translation in terms of circuit components [21], and well-knowncircuit simulator solvers such as SPICE can be used.

The work done so far has been focused on improving the model presented in [20], in orderto overcome the main shortcomings, such as: the high number of poles necessary to gain asatisfactory accuracy; the oscillations in the time domain due to the limited bandwidth of themodel; and the SPICE representation, which is only mentioned, but not given.

This PhD study started with the following research questions:

1. Is it possible to implement delay-extraction techniques for the method in [20], in orderto reduce the number of poles used, while increasing the accuracy?

2. Is it possible to include the delay without compromising the passivity of the model, andthe final state-space representation?

3. Is it possible to provide a straightforward circuit representation where only standardcircuit elements are used?

In the work presented to date, the first two questions have been answered, even though a formalverification of the passivity needs to be addressed. The third question has also been answered,and a forthcoming paper is in preparation. The new proposed model (DeRaG) has a highaccuracy by virtue of the delay extraction technique, and the new delayed state-space systemis suitable for inclusion in SPICE-like transient simulators. This last step is of fundamentalimportance for efficiently simulating the “drive-cable-motor” system. The ability to simulatea complete system, possibly also considering switch disconnectors, for EMC purposes wouldallow a deep understanding of the system and a complete EMC system assessment as early asthe design stage, with the potential to save time and money.

CHAPTER 2

Variable-frequency drives

2.1 Background on variable-frequency drives

A general overview of converters and power electronics can be found in any undergraduate orgraduate textbook [1, 22]. In this chapter, we will emphasize the concepts that are critical froman EMC perspective.

Electric drives are used to control the speed, and thus the torque, of electric motors. In [23],the definition of a VFD is given as “An electric drive designed to provide easily operable meansfor speed adjustment of the motor, within a specified speed range. See also: electric drive”,and an electric drive is defined as “A system consisting of one or several electric motors andof the entire electric control equipment designed to govern the performance of these motors.The control equipment may or may not include various rotating electric machines”. A generalschematic of a VFD is presented in Fig. 2.1. The main components, from left to right, are asfollows:

• the rectifier, which converts the AC voltage from the supply to a DC voltage;

• the DC link, which is primarily used to eliminate ripples in the DC voltage;

• the inverter, which converts the DC voltage to a AC voltage of suitable amplitude andfrequency;

• the power cable, often referred as the power interface, which connects the power moduleto the electric motor;

• the controller, which provides a pulse-width modulation (PWM) signal to the inverterbased on the input from the motor’s sensors and from the user; and

• the electric motor, which is typically an induction motor.

11

12 VARIABLE-FREQUENCY DRIVES

MSupply

Rectifier DC link Electric motor

Controller

Input

command

Cable

Load

Electric Drive

Inverter

Figure 2.1: General schematic of a variable-frequency drive.

MotorSupply

Diode Rectifier DC link Inverter

Figure 2.2: Simplified schematic of a pulse-width-modulated (PWM) voltage source inverter.

An essential circuit for a typical three-phase AC variable-frequency drive is shown in Fig. 2.2.The input from the AC utility supply is rectified into a DC voltage across the capacitor in theDC link, which smooths the voltage. Note that for drives rated at over 2.2 kW, the DC currentwill be also smoothed by an inductor built into the DC link. The PWM voltage source inverterwill then chop the DC voltage into an AC voltage of the desired amplitude and frequency,which will feed the electric motor.

2.2 Pulse-width-modulated waveform and harmonic distor-tion

The switching nature of the converter circuits results in waveforms that contain not only thefundamental component but also unwanted harmonic voltages. Harmonics in power systems

2.2. PULSE-WIDTH-MODULATED WAVEFORM AND HARMONIC DISTORTION 13

result in increased heating in equipment and conductors, misfiring in variable-speed drives, andtorque pulsations in motors. Low-order harmonics may cause an unwanted torque responsefrom the motor, whereas high-order harmonics can lead to acoustic noise if they excite a me-chanical resonance [1, 24]. The switching nature of the PWM waveform has been recognizedas the culprit for most problems observed in variable-frequency drives [1, 9]. In particular,serious damage can arise both from a mechanical point of view, in the motor itself, and froman electrical point of view, manifesting in unwanted interaction with the power grid. The use ofhigh-frequency switching power semiconductors causes rapid voltage variations, and stray cur-rents flow between the inverter and the motor and between the supply and the inverter, mainlythrough parasitic capacitances. Unwanted currents that travel through wired connections andground paths are known as conducted disturbances.

2.2.1 Impact of long inverter-motor cablesA PWM waveform exhibits high rates of change in voltage dV/dt, which cause a transientlyuneven voltage distribution across the motor windings and short-duration voltage overshootsbecause of reflection effects in the motor cable. We consider the same example provided in [1],which is a practical and realistic case. The cited reference considers a 400 V power converter,with a DC link voltage of approximately 540 V and a voltage switching time of 100 ns. Atthe terminal of the drive, dV/dt will be greater than 5 kV/µs. At such high rates of change involtage, a cable behaves as a transmission line, the theory governing which is the subject of thenext chapter. Generally speaking, when a cable behaves as a transmission line, it is no longermerely a connection but rather acts as an electric circuit component in its own right. Froma practical perspective, this means that when the voltage edge reaches the motor terminals, areflection occurs because the motor impedance is higher than the surge (or characteristic) cableimpedance (impedance mismatch). The motor terminal voltage experiences an overshoot oftheoretically up to twice the step voltage, which can represent a problem for a well-insulatedmotor. Additionally, at each pulse edge, the drive must provide a pulse of current to chargethe capacitance of the converter-motor cable, and in extreme cases, this current may exceedthe rated current of the motor, which determines the rating of the required drive. This problemis intensified in drive systems with rated voltages greater than 690 V and in medium-voltagedrives, where dV/dt filters are included between the inverter and motor [1]. Cables are notresponsible only for over-voltage problems. They can also impact the overall electromagneticbehavior of an ASD [5]. Steep voltage and current slopes cause electromagnetic interference(EMI) in terms of both common mode (CM) and differential mode (DM) emissions [25, 26,27]. In the standards, the conducted EMI frequency range extends from a few kilohertz upto megahertz. High levels of voltage steepness dV/dt generate high-frequency (HF) straycurrents in parasitic capacitance. EMI currents travel through cable as waves and are subjectto multiple reflections. As a result, conducted EMI arises in both the power mains and groundsystem. These HF stray currents are divided into two components, based on their circulationpaths:

• differential mode currents, which flow between power lines, and

• common mode currents, which flow between all energized components and the ground.

14 VARIABLE-FREQUENCY DRIVES

MotorUtility

supply

Grounding connections impedances

Parasitic capacitancesParasitic capacitances

Figure 2.3: Paths of common mode currents. The red arrows represent the common mode currents,which travel between the energized component of the ASD, the motor, the ground, and finally the supplythrough parasitic capacitances and grounding connections.

We are predominantly interested in the common mode currents. In fact, the common modecurrents flow into the ground and can return to the supply utility; also, hazardous commonmode voltages can arise between system components and the ground. In Fig. 2.3, the pathsof the common mode currents are highlighted. Note that the total common mode current isthe sum of contributions from the cable and from the motor. In this thesis, we focus on cablemodeling [13].

2.3 The role of simulations and measurementsThe primary motivation for mathematical models, and simulations in general, is that they al-low a system to be prepared, checked and tested before its actual production. In fact, if theconducted EMI level in a VFD can be properly predicted, then corrective actions can be taken,such as, for example, the design of an appropriate EMI filter [9, 13, 28]. The interconnections,such as the cables, form an essential part of an electrical system. To assess a product for elec-tromagnetic compatibility, the cables and their impact on the overall system performance mustbe properly understood. Corrective actions can then be taken as early as the design stage [12].In this context, we start by proposing a new model for cables, in which they are viewed asmulticonductor transmission lines.

CHAPTER 3

Cables and multiconductortransmission line theory

3.1 Background on multiconductor transmission linesGenerally speaking, a transmission line is a structure that can guide electromagnetic (EM)waves between two or more points. Regarding the common cables found in industrial appli-cations, they can be treated as transmission lines when the propagation delay TD of a travelingsignal is large compared with the physical length ` of the cable, that is,

TD ` . (3.1)

Under certain conditions, a cable can be mathematically modeled using multiconductor trans-mission line (MTL) theory [29]. The reader is referred to [29] for a comprehensive study ofmulticonductor transmission lines and to [30] for a general overview of simulations of inter-connects.

In a variable-frequency drive, the high rates of change in voltage due to PWM control in-fluence the reliability of the system. In particular, the cable between the inverter and the motoris not merely an interconnection but rather behaves also as a load, and it displays resistive,capacitive and inductive effects. Various transmission line effects arise, such as reflections,overshoot, undershoot and crosstalk. To perform an accurate EMC assessment of a variable-frequency drive, it is therefore necessary to model the cable with the same attention and accu-racy devoted to the inverter and the motor. In the following, we will provide a general overviewof MTL theory, in which more than two conductors (ground included) are considered. Notethat all results can be easily adapted to the scalar (one-conductor transmission line) case.

The solutions for the MTL equations can be obtained in either the frequency domain or thetime domain. In the frequency domain, we typically assume a sinusoidal excitation source, andsteady-state conditions can be studied. By contrast, time-domain analysis allows the consider-ation of sources with any arbitrary time variation, and both transient and steady-state solutions

15

16 CABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY

are considered. In power systems, it is important to be able to predict transient behavior be-cause, as noted in Chapter 1, transient disturbances such as voltage dips or surges may cause avariable-frequency drive to trip.

Electrical dimension

A crucial concept, not only for transmission lines but also for electromagnetic compatibility,is the electrical dimension of a line. The electrical dimension is given as a wavelength, wherethis wavelength “...represents the distance that a single-frequency, sinusoidal electromagneticwave must travel in order to change phase by 360” [31]. In the frequency domain, given afrequency of interest f , the wavelength is defined as

λ =vf, (3.2)

where v is the velocity of the wave in the medium (vw 2.99×108 m/s in vacuum). If the largestdimension of the MTL is electrically small ( λ ), then we can apply the lumped-circuit theory.As a rule of thumb, a line is considered “electrically short” when λ < `

10 . The definition ofan “electrically short” line becomes less clear in time-domain analysis because each signalcontains a continuum of sinusoidal frequency components. The problem is the identificationof the maximum frequency of interest. Typically, conditions are imposed on the pulse rise andfall times, tr and t f , respectively. Under the assumption that tr = t f , a practical rule of thumb is[30]

fmax =0.35

tr. (3.3)

A faster signal transition time implies a smaller λ . The classification of a line as electricallyshort or long determines the model to be used: in the first case, a lumped model is sufficient(standard circuit theory), whereas in the second case, distributed or full-wave models are nec-essary. Conventional circuit elements can be classified based on the number of dimensions thatare comparable to the operating wavelength [32]:

• zero-dimensional, or lumped circuits;

• one-dimensional, or uniform transmission lines;

• two-dimensional, or planar circuits; and

• three-dimensional, or waveguides.

Electrically short lines belong to the first category. In the following, the main results for uni-form transmission lines are discussed.

3.1. BACKGROUND ON MULTICONDUCTOR TRANSMISSION LINES 17

The quasi-TEM mode and the telegrapher’s equations

In electrically short lines, the electromagnetic field effects can be lumped into circuit elementsand can be described in terms of resistance, capacitance, inductance and conductance. Thesame cannot be done for electrically long lines; another approach must be adopted.

A certain class of electromagnetic problems can be solved by decomposing the fields intoso-called modes (partial fields): each mode exhibits a certain pattern, and the sum of all modesgives the actual field distribution. The transverse electromagnetic (TEM) mode, in which thefield vectors are orthogonal to each other and to the propagation direction, is particularly im-portant. This mode occurs in many configurations, either alone or as the dominant mode [33].Transmission lines are such configurations: only the TEM modes are required in the ideal case,in which the transmission lines consist of two or more infinitely long and lossless conductorswith a constant cross section. For realistic conductors, with losses and terminations, other fieldmodes are excited; at low frequencies, these modes are small, and the conductor is said to beoperating in a quasi-TEM mode.

Given a three-dimensional space xyz, the transmission line is regarded as a distributed-parameter structure along the z axis, and the lumped-circuit analysis technique is extended tostructures that are electrically long in this dimension. The common approach to analyzing mul-ticonductor transmission lines is to assume a quasi-TEM mode of propagation for the electricand magnetic fields.

The quasi-TEM mode assumption is valid under the following conditions:

1. the cross-sectional dimensions, such as conductor separations, must be electrically small,and

2. any non-ideal effects of the line, such as imperfect line conductors and/or an inhomoge-neous surrounding medium, are considered to be negligible.

Under the quasi-TEM mode assumption, Maxwell’s equations can be recast in the form of thewell-known telegrapher’s equations (3.4) [33]. These equations are a set of 2N coupled first-order partial differential equations (PDEs) that describe the voltages and currents of a generictransmission line structure. In the case of a multiconductor transmission line, the resistanceR, capacitance C, inductance L and conductance G are represented by full matrices. If thematrices satisfy R = G = 0, then we are considering a lossless transmission line; otherwise,the line is lossy.

∂

∂ zV(z, t) =−RI(z, t)−L

∂

∂ tI(z, t) , (3.4a)

∂

∂ zI(z, t) =−GV(z, t)−C

∂

∂ tV(z, t) . (3.4b)

For eq. (3.4) to be used, the MTL must be discretized in the z direction, in infinitesimal sectionsof length ∆z. Each ∆z section can be described in terms of the so-called per-unit-length (p.u.l.)parameters R, L, G and C, as shown in Fig. 3.1 for the simple case of a one-conductor trans-mission line. Note that for a uniform line, the per-unit-length parameter matrices R, L, G and


R∆z L∆z

G∆zC∆z

z+∆zz

v(z + ∆z,t)v(z,t)

i(z,t) i(z + ∆z,t)

Figure 3.1: Per-unit-length parameters for a one-conductor transmission line.

C are independent of z. However, these matrices typically depend on the frequency, in whichcase we refer to them as frequency-dependent p.u.l. parameters. Frequency-dependent p.u.l.parameters account for skin and proximity effects (variations in the resistance and inductance)and dielectric losses (variations in the conductance and capacitance), which are particularlyimportant at high frequencies [29]. They can be obtained in two main ways: by obtaininga 2-D solution of Maxwell’s equations [34] or through measurements of the port response atdiscrete frequency points [27]. The matrices R, L, G and C are typically full, positive-definitesymmetric matrices of order N, where N + 1 is the number of conductors with the referenceconductor included [29].

In the Laplace domain, the telegrapher’s equations read as ordinary differential equations(ODEs), as follows:

ddz

V(z,s) =− [R+ sL]I(z,s) =−Z′(s)I(z,s) , (3.5a)

ddz

I(z,s) =− [G+ sC]V(z,s) =−Y′(s)V(z,s) . (3.5b)

Z′(s) and Y′(s) are the N×N symmetric matrices of the per-unit-length impedance and admit-tance, respectively. V(z,s) and I(z,s) are N×1 column vectors; they represent the voltage andcurrent vectors, respectively, which depend on the Laplace variable s and the position z alongthe line.

Port models

Models in which the terminal currents and voltages are related are well suited for inclusionin SPICE-like simulators. A generic MTL can be represented as a 2N-port system, as shownin Fig. 3.2, where only the input (z = 0) and the output (z = `) of the transmission line areconsidered. The most common solutions that are suitable for the 2N-port representation are

1. the modal solution and

2. the matrix-exponential solution.


V1(s)

V2(s)

VN(s)

VN+1(s)

VN+2(s)

V2N(s)

- -

+

+

+ +

+

+

MTL

z=0 z= l

I1(s)

I2(s)

IN(s)

IN+1(s)

IN+2(s)

I2N(s)

Figure 3.2: Multiconductor transmission line represented as a 2N-port system, with a common referenceconductor.

The solution can then be expressed in terms of ABCD parameters, Y parameters (the admit-tance representation) or Z parameters (the impedance representation) [29]. The representationused in this thesis is the impedance representation, which reads in general as shown in eq.(3.6): [

V0(s)V`(s)

]=

[Z11(s) Z12(s)Z12(s) Z11(s)

][I0(s)I`(s)

]= Z(s)

[I0(s)I`(s)

], (3.6)

where Z(s) is the open-end port impedance matrix, which represents the transfer function be-tween the terminal voltages and currents.

The identification of the time delay of electrically long lines is crucial to ensure that a modelis efficient in terms of accuracy and memory occupation. In fact, if the delay is not properlyincorporated into the model, then a large number of unknowns will be necessary to achieveacceptable accuracy. In the following, we will discuss the most common macromodeling tech-niques, and we will see that line delay identification is the most common means of improvingboth the accuracy and size of a model.

An important property of a model is its passivity. In fact, stable but non-passive models arenot useful for transient simulations because passivity violations can lead to spurious oscillationand incorrect simulations. Moreover, stable but non-passive models that are interconnected canlead to a non-stable system, whereas interconnected passive models result in an overall passivesystem. Note that passivity implies stability, but the converse does not hold.

3.1.1 Suitable models for time-domain simulationsStandard circuit simulators, such as SPICE, can solve nonlinear ordinary differential equations(ODEs) but typically not PDEs. It is therefore necessary to convert the telegrapher’s equationsinto ODEs. Moreover, although partial differential equations are better solved in the frequencydomain, only time-domain analysis is possible in the presence of nonlinear terminations [30].This problem is known as the problem of mixed time/frequency-domain representation. Direct


integration of the solution obtained in the frequency domain is generally not recommended,and several numerical techniques have been developed for this purpose. We will briefly reviewthe most common ones.

Lumped segmentation techniques

The idea of lumped segmentation techniques is to approximate the telegrapher’s equations (3.4)using lumped equivalent circuits. The line is divided into an appropriate number of electricallyshort segments of length ∆z, each modeled with lumped parameters. The obvious primary dis-advantage of this method is that it leads to large circuit matrices for electrically long lines, forwhich the propagation delay must be considered. Additionally, the line delay is not explicitlyincluded but rather merely approximated, leading to an intrinsic violation of the causality con-dition for transmission lines [35]. To account for frequency-dependent parameters, additionalpassive LR or LRC sections must be used to approximate the distributed loss [9, 36], therebyincreasing the size of the model used and limiting the numerical efficiency of time-domainsimulations. If the frequency dependence of the parameters is not correctly captured, then thesimulation of the common mode impedance becomes inaccurate, especially at high frequency[13].

Method of characteristics

The method of characteristics (MoC) is a numerical method that is suitable in the case non-linear and/or dynamic loads. It yields an exact solution for a lossless line and is the methodimplemented in most computer programs for circuit simulations, such as SPICE. The extrac-tion of the one-way delay TD = `

v allows the voltage and current at one end of the line tobe specified using the voltage and current at the other end, by virtue of the time delay. Thetransmission line is represented by a set of admittances/impedances and delayed sources. Formulticonductor transmission lines, the generalized method of characteristics has been devel-oped [37, 38, 39]. This method is based on the modal decomposition of voltages and currents,which is accomplished through similarity transformations that allow the diagonalization of theper-unit-length impedance Z′ and admittance Y′ . The modal voltages and modal currents thusobtained are represented by N separate, uncoupled, two-conductor lossy lines. With the modecharacteristic impedances defined as ZCm(s), the modal solution reads as follows:

Vm(0,s) =ZCm(s)Im(0,s)+E0(s) , (3.7a)

Vm(`,s) =ZCm(s)Im(`,s)+E`(s) , (3.7b)

where the subscript m stands for “modal”. The E are voltage-controlled sources, defined asfollows:

E0(s) = eΓm(s)` [Vm(`,s)−ZCm(s)Im(`,s)] , (3.8a)

E`(s) = eΓm(s)` [Vm(0,s)−ZCm(s)Im(0,s)] , (3.8b)


where eΓm(s) is an N ×N diagonal matrix that contains the N modal propagation constantsΓm,i(s) =

√Zmi(s)Ymi(s) on the main diagonal. The term eΓm(s)` is called the “propagation

operator”. For lossy transmission lines, it is not possible to directly translate this model intothe time domain, and several techniques have been developed to overcome this difficulty, suchas the Pade rational approximation. As observed in [35], a direct rational approximation ofthe propagation operator can be difficult, mainly because the matrix accounts for the line delayand the line attenuation/dispersion, and a delay extraction technique is therefore proposed.When frequency-dependent per-unit-length parameters are considered, the modal line delaysare defined in terms of the frequency asymptotic values of the capacitance and inductance, asfollows:

Tk = `√

Λk , (3.9)

where the Λk are the eigenvalues of the matrix product C∞L∞; C∞ and L∞ are the asymptoticvalues of the frequency-dependent capacitance and inductance matrices. The main difficultyin this approach lies in how these delays are extracted from the propagation operator. Theparticular delay extraction technique used should be based on the process that is applied to thedelayless transfer function. This approach is not passive by construction. Recently, sufficientconditions that guarantee the passivity of the MoC by construction have been specified in [40].

Matrix rational approximation

Using the terminal conditions, it can be proven that the telegrapher’s equations can be writtenas

∂

∂ z

[V(`,s)I(`,s)

]= eZ`

[V(0,s)I(0,s)

], (3.10)

where

Z =

[0 −a−b 0

]a = R(s)+ sL(s) b = G(s)+ sC(s) . (3.11)

The exponential function has no direct translation in the time domain. Algorithms exist toperform a matrix rational approximation (MRA) of this function [41, 42, 43]. The exponen-tial function can be approximated as a ratio of polynomial matrices expressed in terms ofclosed-form Pade rational function, and thus, ordinary differential equations can be obtained.Frequency-dependent p.u.l. parameters are included by expressing them as rational functions.The model becomes inefficient when long lossy lines are considered because of the high orderof the approximation. To overcome this difficulty, a new technique has recently been proposed,named the Delay-Extraction-based Passive Compact Transmission-Line Macromodeling Algo-rithm (DEPACT) [43]. Delay extraction is performed directly on the matrix-exponential form,and the order of the approximation can be kept low. Lossless decoupled transmission lines arethen necessary to model the delays, and the decoupling is achieved via transformation matrices.This model is passive by construction.


Other well-known techniques

Other well-known techniques are

• TOPline [14], which combines matrix rational approximation with modal delay extrac-tion;

• vector-fitting-based techniques [15, 44];

• the universal line model [16, 45]; and

• the finite-difference time-domain (FDTD) method [17, 29].

A comparative study between the MoC and rational approximation approaches can be found in[46].

3.2 Main shortcomings of the present modelsLumped approaches are, in general, not advisable because they do not produce accurate broad-band models. The delay must be taken into account with additional RLC networks, which, inturn, lead to a large equivalent circuit, and the simulation in the time domain becomes cumber-some.

In general, the MoC model does not preserve passivity by construction, although sufficientconditions can ensure the passivity of the model. Additionally, it requires approximation of theexponential form.

Methods that generate a rational model of an MTL, such as the Pade rational approximationand ladder network models, use expansion points to identify the poles, and their accuracyimproves as the order of the approximation increases. This can lead to large macromodels thatare inefficient in time-domain simulations.

In summary, the research community is still seeking a model with the following properties:

• passive by construction,

• suitable for the case of frequency-dependent p.u.l. parameters,

• easy to embed in a SPICE-like simulator,

• independent of the specific termination and sources, and

• accurate.

It is worth noting that typically, the PSPICE built-in model for transmission lines is not usedin the simulation of EMI emissions because its parameters cannot be dependent on frequency,and this limitation would lead to inaccurate simulations [9]. HSPICE uses the finite-differenceapproximation method; previously, it used U-elements, which have recently been replaced withW-elements. In [46], it is suggested that circuit elements based on the MoC or matrix rationalapproximation are more efficient then the U-elements. Regarding W-elements, nothing hasbeen said about the passivity of the model obtained [47].

3.3. TRANSMISSION LINE THEORY FOR THE STUDY OF COMMON MODE CURRENTS IN

CABLES 23

MTL

z=0 z= l

Iu1(s)

Iv1(s)

Iw1(s)

Iu2(s)

Iv2(s)

Iw2(s)

Id1

Id2

Id3

Id1

Id2

Id3

Ic

Ic

Ic

Ic

Ic

Ic

Is(s)

Ig1(s) Ig2(s)Ic Ic

4Ic 4Ic

Figure 3.3: Common and differential current components for a typical four-conductor shielded cable,treated as a multiconductor transmission line.

3.3 Transmission line theory for the study of common modecurrents in cables

The typical cables used for three-phase induction motors are coaxial shielded cables. Theygenerally contain three conductors (one for each phase) and the ground conductor. They aretypically shielded, with the shield connected directly to the chassis of the motor. As statedin Chapter 1, we are primarily interested in studying the common mode currents that travelthrough such cables. It is therefore worthwhile to clarify how an MTL model can be usedfor this purpose. In general, differential mode currents Id are distinguished from commonmode currents Ic based on their circulation paths [25]. A differential mode current is typicallyequal to the signal or power current, and such currents are not present in either the shield orthe ground. Common mode currents flow equally, in the same direction, in all conductors,including the shield and the ground. It is possible to distinguish two major sources of commonmode currents: the longitudinal conversion loss of the cable, which causes some of the powercurrents to leak out through stray coupling, and the noise voltage between the connection pointof the cable and the ground reference of the circuit.

The shield conductor is treated as the reference conductor. For the TEM mode assumptionto be valid, the total sum of the currents in all conductors, including the reference conductor,must be zero. It is therefore natural to ask how we can distinguish between differential andcommon mode currents in an MTL model. Referring to Fig. 3.3, the phase currents are denotedby Iu1, Iv1, and Iw1 on the AC drive side and by Iu2, Iv2, and Iw2 on the motor side for consistencywith the nomenclature that is used in practice to name the connections. The ground currentsare denoted by Ig1 and Ig2. In each phase conductor, the current is the sum of a differentialmode current and a common mode current. Referring to the inverter side, we can write

Iu1 = Id1 + Ic , (3.12a)


MTL

Iu1

Iv1

Iw1

Iu2

Iv2

Iw2

Id1

Id2

Id3

Id1

Id2

Id3

Ic

Ic

Ic

Ic

Ic

Ic

Ic Ic

4Ic 4Ic

3Ic 3Ic

Inverter Motor

Figure 3.4: Common mode current analysis of a 4-conductor shielded cable connected between theinverter and motor, with the shield connected on both sides.

Iv1 = Id2 + Ic , (3.12b)

Iw1 = Id3 + Ic , (3.12c)

Ig1 = Ic (3.12d)

where the Idi for i = 1,2,3 are the differential currents associated with each conductor and Icrepresents the common mode component. In theory, in a perfectly balanced system, no currentshould flow in the ground conductor. Because of asymmetries in the motor or in the cableitself, however, unwanted currents do indeed flow in the ground conductor. Note that in thesimulations, only the phase conductors are excited. The total common mode current in theshield conductor, named Is, can be written as

Is = Iu1 + Iv1 + Iw1 + Ig1 = Id1 + Id2 + Id3︸︷︷︸=0

+4Ic . (3.13a)

Because of the assumption that differential mode currents do not flow in the shield conductor,their sum is equal to zero. Then, given the input port currents, the common mode contributioncan be computed as

Ic =Iu1 + Iv1 + Iw1 + Ig1

4. (3.14)

Note that the shield and the ground are typically connected on both sides, to the inverter andmotor. The total common mode current that interacts with the inverter and motor is 3Ic, as seenin Fig. 3.4, where the inverter and motor are represented as loads.

CHAPTER 4

The proposed delay-rational model

4.1 Green’s functions and boundary problems: a brief back-ground

The method developed in the studies presented in this thesis is called the Delay-RationalGreen’s-Function-based Method, or DeRaG. It stems from the work presented in [20]. Theprevious chapter summarized various methods for studying MTLs. They are predominantlybased on either the matrix-exponential solution or the modal decomposition or on a combina-tion of the two (DEPACT). In [20], a different perspective is adopted. Specifically, transmissionlines are treated as a special case of planar circuits [32], in which only one dimension is non-negligible with respect to the wavelength. A rational macromodel is derived, which is suitablefor lossy and dispersive multiconductor transmission lines. The port voltages are expressed, inthe Laplace domain, in terms of port currents, which are considered as forced sources at theterminals z = 0, ` as follows:

Is(z,s) = I0(s)δ (z)+ I`(s)δ (z− `) , (4.1)

where δ (z) is the Dirac delta symbol and I0 and I` are the port currents at the two extremities ofthe line. Note that with this approach, the resulting model is described in terms of Z parameters.The model can be described in terms of Y parameters by treating the port voltages as forcesources. The telegrapher’s equations read as follows:

∂

∂ zV(z,s) =−Z′(s)I(z,s) , (4.2a)

∂

∂ zI(z,s) =−Y′(s)V(z,s)+ Is(z,s) . (4.2b)

By considering the sources directly in the equations, we obtain homogeneous boundary condi-tions. If Neumann-type boundary conditions are used, then the values of the derivative on the

25

26 THE PROPOSED DELAY-RATIONAL MODEL

boundaries are specified, as follows:

∂

∂ zV(z,s)|z=0 =

∂

∂ zV(z,s)|z=` = 0 . (4.3)

It is then possible to differentiate the voltage equation (4.2a) again with respect to z to allowthe substitution of the current equation (4.2b), yielding

∂ 2

∂ z2 V(z,s)− γ2(s)V(z,s) =−Z′(s)Is(z,s) , (4.4)

where γ2 = Z′(s)Y′(s) is the propagation operator. The set of equations (4.3) and (4.4) rep-resents a Sturm-Liouville problem, whose solution can be found using the Green’s functionapproach. All details can be found in [20] and the references therein. In [20], it is shown thatthe solution to the problem is provided by the following (dyadic) Green’s function:

G(z,z′,s) =−∞

∑m=0

[γ

2(s)+(mπ

`1)2]−1

·A2m cos

(mπ

`z)

cos(mπ

`z′), (4.5)

where 1 is the identity matrix and the coefficients Am are defined as

Am =

√

1` , m = 0√2` , m = 1, · · · ,∞ .

(4.6)

An intuitive explanation of the Green’s function in (4.5) is that it gives the effect at the spatialpoint z when an excitation source is applied at point z′. The port voltages at z = 0, ` can beexpressed in terms of the point currents in (4.1) by using the Green’s function given in (4.5),as follows:

V0(s) = G(0,0,s)(−Z′(s)I0(s))+G(0, `,s)(−Z′(s)I`(s)) , (4.7a)

V`(s) = G(`,0,s)(−Z′(s)I0(s))+G(`,`,s)(−Z′(s)I`(s)) . (4.7b)

Eq. (4.7) can be written in matrix form, yielding the impedance representation for a genericmulticonductor transmission line as follows:[

V0(s)V`(s)

]=

[Z11(s) Z12(s)Z12(s) Z11(s)

][I0(s)I`(s)

]= Z(s)

[I0(s)I`(s)

], (4.8)

where Z(s) is the open-end port impedance matrix, which is block symmetric, where eachblock has dimensions of N ×N. Note that the block entries of Z(s) are indeed the Green’sfunctions of eq. (4.5), where z and z′ have been suitably updated, and they read as follows:

Z11(s) = Z22(s) =+∞

∑m=0

[γ

2(s)+(mπ

`

)21

]−1

·A2mZ′(s) , (4.9a)

4.1. GREEN’S FUNCTIONS AND BOUNDARY PROBLEMS: A BRIEF BACKGROUND 27

Z12(s) = Z21(s) =+∞

∑m=0

[γ

2(s)+(mπ

`

)21

]−1

·A2mZ′(s)cos(mπ) . (4.9b)

The summations are infinite, as expected for a delayed system. Eq. (4.9) is general, because noconditions are imposed on the per-unit-length impedance or admittance matrices. The modelis, therefore, naturally suitable for both frequency-independent and frequency-dependent p.u.l.parameters. To address the latter case, Z′(s) and Y′(s) are written in a rational form in termsof poles and residues, for example, using the vector fitting technique of [48], as follows:

Z′(s) = R′∞ + sL′∞ +PZ

∑q=1

RZ

s− pq,Z, (4.10a)

Y′(s) = G′∞ + sC′∞ +PY

∑q=1

RY

s− pq,Y, (4.10b)

where the R are the residual matrices, the p are the poles, and PZ and PY are the numbers ofpoles used in the rational approximation; typically, a small number is required (1-3). The R′∞,G′∞, L′∞ and C′∞ terms describe the asymptotic behavior of the line at infinite frequency. Inthe case of frequency-independent parameters, PZ = PY = 0. Eq. (4.10) can be expressed inrational polynomial form as the ratios between the polynomial matrices Bp(s) and Dp(s) andthe polynomials Ap(s) and Cp(s), as follows:

Z′(s) =b0sPZ+1 +b1sPZ + · · ·bPZ+1

a0sPZ +a1sPZ−1 + · · ·aPZ

=Bp(s)Ap(s)

, (4.11a)

Y′(s) =d0sPY+1 +d1sPY + · · ·dPY+1

c0sPY + c1sPY−1 + · · ·cPY

=Dp(s)Cp(s)

. (4.11b)

Note that the polynomials Ap(s) and Cp(s) can be made strictly Hurwitz by construction, thusimposing that the roots lie only in the left half-plane, which is an essential requirement forstability. Regarding the passivity, if vector fitting is used, it applies an a posteriori passivityenforcement. The impedance port matrix Z(s) can be recast as

Z(s) =∞

∑m=0

Fm(s)−1A2mBp(s)Cp(s)

[1 (−1)m

(−1)m 1

], (4.12)

where Fm(s)−1 = Bp(s)Dp(s)+Ap(s)Cp(s)(mπ

`

)21. The poles of the Z(s) matrix are eval-

uated as the determinants of the polynomial matrices Fm(s), for m = 0, · · · ,∞. Note that allsub-blocks of the port impedance matrix share the same poles and that the residues are allidentical up to a multiplication factor of (−1)m.

In [49], we clarified the total number of poles computed per mode in the frequency-indepen-dent case. In particular, the m = 0 summation mode has N associated poles, whereas highermodes have 2N poles (N complex conjugate pairs), where N is the number of conductors. Itis also worthwhile to clarify the number of poles per mode in the frequency-dependent case.


We can follow the same reasoning given for the frequency-dependent case. In particular, forthe m = 0 mode, the number of poles is equal to (PY +1)N. For the m > 0 modes, the numberof poles is equal to (PZ +PY + 2)N; this last general result has been explained in [20]. It canbe proven that a rational representation is well suited to be translated into a state-space systemusing known macromodel synthesis techniques [30], as follows:

x(t) = Ax(t)+Bi(t) , (4.13)

v(t) = Cx(t) ,

which admits a direct translation in terms of circuit elements [30, 21]. This last property iscrucial for the implementation of the model in a SPICE-like simulator. The main bottleneckof the original approach presented in [20] was the large number of modes, and thus poles, thatwas necessary to achieve good accuracy. In fact, no automatic stopping criteria were given,and it was necessary to use a sufficiently high number of modes. To reduce the number ofresidues and poles, a pole pruning method was proposed. Specifically, the dominant polesand residues, those whose magnitudes cannot be neglected, are identified [18]. Pole pruningis particularly necessary in the frequency-dependent case because the fitting procedure maygenerate additional poles. The procedure presented in [20] is briefly summarized in the diagramin Fig. 4.1. Under the assumption that Z′ and Y′ are passive, it was proven in [18] that thisRational Green’s-function-based method is passive by construction.

4.2 Delay extractionThe primary disadvantage of the Rational Green’s-function-based method is the large numberof modes required. This is a common problem in rational techniques and is commonly resolvedusing delay extraction techniques [35, 43]. In the work presented to date, we have used explicitline delay extraction. The two-way line lossless delay TD = 2`

√L′C′ can be extracted in two

main ways:

• By considering the imaginary part of the complex conjugate poles. This approach hasbeen explained in Papers A and B for the frequency-independent case and in Paper D forthe frequency-dependent one.

• By considering the eigenvalues of the matrix product L′C′, following an approach similarto that proposed in [35]. This approach was used in Paper C.

When frequency-dependent p.u.l. parameters are considered, the last available frequency sam-ple is taken. The delay can be incorporated into the model once the periodicity of the transferfunction has been determined. In the lossless case, this periodicity is easily identified.

Basically, the analogy between the time-domain expression of the block impedances and aFourier series allows the block impedances to be rewritten in terms of a Dirac comb, into whichthe line delay TD can be easily incorporated. The Dirac comb expression is in the time domain.From the time domain, we can return to the Laplace model, and the Dirac comb expressiontranslates back into hyperbolic functions.

4.2. DELAY EXTRACTION 29

Rational polynomial forms of Z’ and Y’

Green’s function based methodPoles and residues form

Poles and residues pruningDominant poles and residues selected

Longitunal impedance Z’ and transversal admittance Y’

Available at discrete frequency samples

Figure 4.1: Original Green’s-function-based method.

To achieve these results, we analyzed the behavior of the poles and residues in the loss-less and lossy cases, for the cases of frequency-independent and frequency-dependent p.u.l.parameters.

Analysis of poles and residues

In our studies, two main methods have been used to compute poles and residues:

• the inversion of the polynomial matrix Fm(s) [20] and

• the vector fitting approach [50].


The second method must be used in the case of large cable bundles because the computation ofthe poles becomes highly demanding of computation time and prone to numerical inaccuracieswhen the number of conductors is large (more than 5 conductors).

In general, we observe that the poles and residues can be grouped into families that share thesame properties. The number of families is equal to the number of conductors. The time delayfor each conductor can be properly included by observing that given the complex conjugatepoles for the m = 1, · · · ,∞ modes, namely,

pk,m(1,2) = αk± jβk,m , (4.14)

where the index k spans the N families, the imaginary parts become equispaced after a certainmode m. Denoting by βk,m the equispaced fictitious imaginary parts for m = 1, · · · ,+∞, wecan write βk,m = mβk,1. It is worth noting that the imaginary parts of these fictitious polescorrespond to the lossless ones, such that

βk,1 =2π

Tk, (4.15)

where the Tk are the delays associated with each conductor. With a proper mathematical for-mulation, the observed asymptotic behavior can be expressed in terms of hyperbolic functions.Note that the real parts of the complex conjugate poles are shared between the families, repre-sented by the index k = 1, · · · ,N, as seen in Fig. 4.2, where the 6 families of complex conjugatepoles for a 6-conductor ribbon cable are clearly evident. This example is taken from Paper C.In the case of frequency-dependent parameters, or when the vector fitting (VF) technique isadopted, the real parts of the poles for the different families become asymptotically constantafter a certain mode m, as seen in Fig. 4.3 for a 2-conductor TL with frequency-dependentparameters, from Paper D, and in Fig. 4.4 for a 9-conductor frequency-independent cable bun-dle, from Paper C. In fact, for large cable bundles, numerical inaccuracies can arise, and morecareful identification of the pole families is necessary. The asymptotic value of the real part forfamily k is denoted by αk.

Regarding the residues, they are also clustered into families and exhibit asymptotic behav-ior; in particular, their phases tend asymptotically to zero, and the real part remains constantamong the families, that is,

Rk,n = Rk k = 1, · · · ,N , (4.16)

where the index k spans all families.The m summation mode can be found by using proper tolerances over the phase of the

residues and, when necessary (in the case of frequency-dependent p.u.l. parameters or whenthe VF approach is used for the pole computations), over the real part of the pole families, asshown in the diagram in Fig. 4.5.

Given a lossy MTL, we have proven that the block impedance matrices can be written, inthe Laplace domain, as follows:

Z11(s) =Np

∑n=1

R11n

s− pn︸︷︷︸Rational delayless

+N

∑k=1

R11k Tk

2coth

((s− αk)

Tk

2

), (4.17a)

4.2. DELAY EXTRACTION 31

ℜ ×104

-4 -3.5 -3 -2.5 -2 -1.5 -1

ℑ

×1011

-1

-0.5

0

0.5

1poles vf

Figure 4.2: Complex conjugate pole locations computed using a modal rational approximation ap-proach for a 6-conductor ribbon cable with frequency-independent p.u.l. parameters, from Paper C.

ℜ (p) ×109

-2.5 -2 -1.5 -1 -0.5 0

ℑ (

p)

×1012

-2

-1

0

1

2Family 1

Family 2

Figure 4.3: Complex conjugate pole families in the complex plane for a 2-conductor line with frequency-dependent parameters, from Paper D.


ℜ ×104

-7 -6 -5 -4 -3 -2

ℑ×10

8

-2

-1.5

-1

-0.5

0

0.5

1

1.5

poles vf

p

Figure 4.4: Complex conjugate pole locations for a 9-conductor cable bundle, from Paper C. The bluecircles represent the asymptotic real parts for each family, αk.

Z12(s) =Np

∑n=1

R12n

s− pn︸︷︷︸Rational delayless

+N

∑k=1

R12k Tk

2csch

((s− αk)

Tk

2

), (4.17b)

where Np is the total number of poles/residues used, the pn are real or complex conjugate poles,and the Ri j

n are the corresponding residue matrices for Zi j(s), i, j = 1,2. The αk denote theconstant asymptotic real parts of the poles associated with each conductor (indexed by k), andthe Ri j

k are the corresponding asymptotic residues. The Tk represent the modal lossless delays,which can be evaluated using the well-known formula

Tk = 2` ·Λ(CL), (4.18a)

where Λ(CL) denotes the eigenvalues of the matrix product CL. Note that the rational delay-less part of the system is stable, because all poles lie on the left side of the complex plane, thatis, αk =−|αk|.

4.3 The delay-rational state-space formThe impedances are written as a sum of rational functions, and this allows direct translationinto the time domain using known macromodel synthesis techniques [21, 30]. The resultingmodel reads as follows:

x(t) = Ax(t)+Bi(t) , (4.19a)

4.3. THE DELAY-RATIONAL STATE-SPACE FORM 33

N_mod = M;

STOP = FALSE;

m == N_mod

% Port impedance evaluation for the mode m

% Vector fitting in order to identify poles and

residues

VF

END

m = m + 1;

yes

no

% Evaluation of the asymptotic values for

poles and residues, and

yes

no

Figure 4.5: Block diagram for the algorithm used to determine the m mode, from Paper C.


i(t)ʃ

D0

A

B C+

D(t)

+

v(t)

vd(t)

x(t)Ẋ(t)

Figure 4.6: Block diagram of the state-space model (4.19).

v(t) = Cx(t)+D0i(t)+vd(t). (4.19b)

The system is presented in diagrammatic form in Fig. 4.6. The vector x(t) is a suitable state-space variable.The vectors i(t) and v(t) are the terminal currents and voltages, respectively, associated withthe linear network.

The matrices A, B, C, and D0 account for the rational delayless part of the impedance,whereas the term vd(t) corresponds to the hyperbolic functions and accounts for the delayed-attenuated contributions of the input currents. We recall the following:

• A is constructed using the poles of the rational delayless part; each pole is repeated onthe diagonal a number of times equal to the number of ports.

• B is a selection matrix.

• C consists of the residues of the rational delayless part.

• D0 is the so-called direct-link matrix.

All of these matrices are sparse. In particular, D0 is block diagonal and is defined as

D0 =N

∑k=1

[R11

kTk2 0

0 R11k

Tk2

]. (4.20)

The primary novelty of the system defined in (4.19) resides in the term vd(t) = D(t)⊗ i(t),where D(t) is a symmetric matrix defined as

D(t) =N

∑k=1

[D11(t) D12(t)D12(t) D11(t)

], (4.21)

with

D11(t) = R11k Tke−|αk|t

+∞

∑m=1

δ (t−mTk) , (4.22a)

4.3. THE DELAY-RATIONAL STATE-SPACE FORM 35

D12(t) = R12k Tke−|αk|t

+∞

∑m=0

δ

(t−mTk−

Tk

2

). (4.22b)

The vd(t) term can be regarded as the delayed impulse response of the system, that is, theoutput in response to a series of Dirac delta inputs. We can recognize a discrete convolution inthe expression vd(t) = D(t)⊗ i(t).

The SPICE representation can be obtained by using current-controlled voltage sources andstandard circuit elements for the delayless part and, for the delay part, using lossless transmis-sion lines to reproduce the lossless delay and current-controlled voltage sources to reproducethe attenuation.

To implement the system (4.19) in MATLAB, a backward Euler integration solver hasbeen used, with a fixed time step of dt. We note that because the main goal is to implementthe system (4.19) in SPICE, no particular attention has been paid to the MATLAB solver itselfbecause SPICE will use an adaptive time-step method, for which the user can set the maximumtime-step size.

Note that the resulting model can be easily incorporated into recently developed model-reduction techniques [51], which would allow the state-space size to be further reduced, therebyspeeding up the time-domain simulations.

CHAPTER 5

Research contributions

The papers included in this thesis are listed below in chronological order of publication.My contributions to each paper are highlighted.

5.1 Paper ATitle: Delayed Impedance Models of Two-Conductor Transmission Lines.Authors: Maria De Lauretis, Jonas Ekman and Giulio Antonini.Published in: Proceedings of IEEE, International Symposium on Electromagnetic Compati-bility (EMC Europe), Gothenburg, Sweden, 2014.Summary: This paper represents the first attempt to extract the line delay and to incorporateit into the Rational Green’s-function-based model. The simple case of a two-conductor trans-mission line (reference included) is considered, with frequency-independent per-unit-lengthparameters. From a study of the poles, we determine that the real parts of the complex con-jugate poles do not vary among the m > 0 modes. In the lossless case, the imaginary part islinear with respect to the lossless line delay. In the lossy case, the imaginary parts becomeequally spaced after a certain mode m. The analogy between the time-domain expression ofthe block impedances and a Fourier series allows the block impedances to be written in theform of a Dirac comb, into which the line delay TD can be easily incorporated. The extensionto the lossy case and the state-space representation are merely outlined.Contribution: The author developed the mathematical formulation, performed the tests andwrote the majority of the paper. The author presented the paper at the conference, with follow-up discussions.

5.2 Paper BTitle: A Delay-Rational Model of Lossy Multiconductor Transmission Lines with FrequencyIndependent Per-Unit-Length Parameters.Authors: Maria De Lauretis, Jonas Ekman and Giulio Antonini.

37

38

Published in: IEEE Transactions on Electromagnetic Compatibility, 2015.Summary: The results obtained for the scalar case in the first paper are extended to multi-conductor transmission lines. In particular, it is found that poles and residues can be groupedinto families that share the same properties: the same (constant) real part and the same asymp-totic behavior of the imaginary part. From a given summation mode m, the poles and residuesexhibit periodic properties, which allow the explicit extraction and incorporation of the timedelay. Conditions for identifying m are given. A delay-rational state-space model in the timedomain is presented, in which delayed Dirac combs are incorporated.Contribution: The author developed the mathematical formulation, performed the tests andwrote the majority of the paper. The author was responsible for the submission and revisionprocesses, responding to the reviewers and applying the suggested modifications.

5.3 Paper CTitle: Enhanced Delay-Rational Green’s Method for Cable Time Domain Analysis.Authors: Maria De Lauretis, Jonas Ekman, Giulio Antonini and Daniele Romano.Published in: Proceedings of IEEE, International Conference on Electromagnetics in Ad-vanced Applications (ICEAA), Turin, Italy, 2015.Summary: In this paper, the authors address the problem of computing the poles in the case oflarge cable bundles, and a method for reducing the order of the model is proposed. The well-known vector fitting algorithm is used for this purpose, and the inversion of the polynomialmatrix is avoided. The properties previously exploited in Paper B are adapted to the new case.We observe that an additional criterion must be imposed on the real parts of the poles. Themodel order reduction is performed over the delayless part of the open-end port impedance.Tests are performed on a 9-conductor cable bundle with a length of `= 20 m.Contribution: The author participated in the elaboration of the ideas and performed the tests.

5.4 Paper DTitle: Delay-Rational Model of Lossy and Dispersive Multiconductor Transmission Lines.Authors: Maria De Lauretis, Jonas Ekman, Giulio Antonini.Published in: Proceedings of IEEE, International Symposium on Electromagnetic Compati-bility (EMC), Dresden, Germany, 2015.Summary: The delay-rational model is extended to include MTLs with frequency-dependentper-unit-length parameters. The p.u.l. longitudinal impedance Z′(s) and transverse admittanceY′(s) are considered to be available at N f discrete frequency samples. A rational polynomialform is obtained using the vector fitting technique. Redundant poles and residues are discardedthrough pole/residue pruning. Unlike in the frequency-independent case, the real part of thepoles becomes constant among the different pole families after a certain summation mode.Suitable tolerances over the real part of the poles and the phase of the residues allow the resultsfound for the case of frequency-independent p.u.l. parameters to be extended to the frequency-dependent case.

5.4. PAPER D 39

Contribution: The author participated in the elaboration of the ideas, the development of themathematical formulation and the performance of the tests. The author wrote the majority ofthe text and presented the paper at the conference, with follow-up discussions.

CHAPTER 6

Conclusions and future work

6.1 ConclusionsA new model for multiconductor transmission lines has been presented, called DeRaG. Thismodel is based on a previous work of [20], where the solution for the MTL equations is pre-sented, in an original way, in terms of Green’s functions. The computed poles and residues canbe regarded as the exact poles and residues of the line, in contrast with other rational approxi-mation techniques that estimate the poles, and whose accuracy depends on the approximationorder, as in matrix rational approximation techniques.

This PhD study started with the objective to overcome the main limitations of the method.The first research question was:

Is it possible to implement delay-extraction techniques for the method in [20], in order toreduce the number of poles used, while increasing the accuracy?

In the new proposed model, called DeRaG, the properties of the poles and residues havebeen studied and exploited. In particular, the poles and residues are grouped into families,from which the modal lossless delays can be identified and properly incorporated. The losslessmodal delays are used in order to reduce the number of poles, while increasing the accuracyof the model itself. In fact, ringing effects in the time domain are eliminated because all thebandwidth is correctly characterized. The delays can be computed either from the poles orfrom standard eigenvalue decomposition of the matrix product CL.

The second research question was:

Is it possible to include the delay without compromising the passivity of the model, and thefinal state-space representation?

We have shown that the delays are explicitly incorporated into the model through a propermathematical formulation. The additional terms that account for the delays do not compromisethe simplicity of the model and a new delayed state-space system is obtained. The primaryadvantage of the proposed method with respect to other well-known techniques relies on thedelayed state-space representation, whose size can be further compressed by virtue of model

41

42

order reduction (MOR) techniques. Preliminary tests have already confirmed the size opti-mization achieved using standard MOR algorithms such as PRIMA [51]. A formal verificationof the passivity for the DeRaG model has not been addressed yet.

The third research question was:

Is it possible to provide a straightforward circuit representation where only standard circuitelements are used?

The circuit implementation of a model in state-space form is straightforward to obtain,thereby allowing the use of well-known circuit simulation tools such as PSPICE A/D R©. Infact, the delayless part in the frequency domain translates in a standard macromodel in the timedomain. A standard macromodel can be represented in PSPICE with standard circuit compo-nents, namely resistances, inductances and current-controlled sources, following the guidelinesprovided in [21, 30]. The delay part, instead, can be accounted using standard lossless trans-mission lines, whose model in PSPICE is exact. Current-controlled voltage sources can beused to account for the attenuation.

The resulting final circuit model can be identified and translated into PSPICE, making themodel accessible to a broader set of users who typically employ circuit tools for their analy-ses. The model will have the same structure both for the frequency dependent and frequencyindependent cases, because no convolution products are involved. The terminations can beincorporated in PSPICE in a straightforward way.

6.2 Future workSome future work remains to be done for the Rational-Delayed Green’s-Function-based Method(DeRaG). Although the passivity of the model should be preserved after the delay extraction,a formal verification must be performed. Also, the exact circuit implementation of the modelwill be explained in details in forthcoming papers.

Two additional models need to be developed to allow the analysis of a VFD system: one forthe AC drive, and one for the motor. These models must be suitable for addressing EMI con-ducted emissions. Once a complete detailed model “drive-cable-motor” is obtained, the impactof ground connection typologies can be analyzed and classified based on their performances.Solutions for conducted emissions in form of EMI filters will be studied.

Moreover, the interconnections between the cable and the system components must bemodeled. Additionally, switch disconnectors are typically used in the industry, and these com-ponents must be properly considered because they can also impact the system performance.Disconnectors and interconnections within the converter, which are represented by imperfectconductors with irregular geometries, require full-wave models. To this end, we will use thepartial element equivalent circuit (PEEC) approach to provide an electromagnetic model of theinterconnects.

The final phase of this PhD study will be devoted to measurements. The models obtainedthrough simulation must be verified against experimentally measurement results.

For this purpose, laboratory activities in the EMC laboratory of LTU and pre-compliancetesting of VFDs are being conducted.

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[49] M. De Lauretis, G. Antonini, and J. Ekman, “A delay-rational model of lossy multicon-ductor transmission lines with frequency-independent per-unit-length parameters,” IEEETransactions on Electromagnetic Compatibility, vol. 57, no. 5, pp. 1235–1245, 2015.

[50] M. De Lauretis, J. Ekman, G. Antonini, and D. Romano, “Enhanced delay-rationalGreen’s method for cable time domain analysis,” in 2015 International Conference onElectromagnetics in Advanced Applications (ICEAA), Sept 2015, pp. 1228–1231.

[51] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: passive reduced-order intercon-nect macromodeling algorithm,” in Proceedings of the 1997 IEEE/ACM internationalconference on Computer-aided design. IEEE Computer Society, 1997, pp. 58–65.

48 REFERENCES

Part II

49

PAPER A

Delayed Impedance Models ofTwo-Conductor Transmission

Lines

Authors:Maria De Lauretis, Jonas Ekman and Giulio Antonini

Reformatted version of paper originally published in:Proceedings of IEEE, International Symposium on Electromagnetic Compatibility (EMC Eu-rope), Gothenburg, Sweden, 2014.

c© 2014, IEEE, Reprinted with permission.

51

Delayed Impedance Models of Two-Conductor TransmissionLines

Maria De Lauretis, Jonas Ekman and Giulio Antonini

Abstract

This paper presents a new delayed model of two-conductor transmission lines with frequency-independent per-unit-length parameters. In particular, the line delay extraction problem is con-sidered. By use of a dyadic Green’s function macromodel method, the rational form of theopen-end impedance matrix allows an easy identification of poles and residues, and a newtechnique for the extraction of the line delay in an analytical way is gained, without any impacton the complexity of the line macromodel itself. By use of Laplace and Fourier transforms, thetransfer function is expressed in terms of the Dirac comb. The delay is then easily identifiedand directly incorporated into the system impulse response. Giving a current-controlled repre-sentation, the port voltages are evaluated. Thanks to the formulation of the transfer function byuse of the Dirac comb, the convolution product is avoided, gaining accuracy and time-savingfrom a computational point of view. Numerical results confirm the validity of the proposeddelay-extraction technique. The basic ideas for the extension of the proposed technique to thelossy case are outlined.

1 Introduction

Nowadays, interconnects play an increasing important role in high-speed VLSI chips and mul-tichip modules. The system performance may be severely deteriorate by the interconnect ef-fects such as signal delay, reflection and dispersion, or crosstalk[1, 2]. The literature devotedto transmission line and interconnect modeling is vast (see [3, 4] and references therein). Thetransmission line equations can be formulated in both the frequency and time domains, thislast important for power systems to predict the transient behaviour of long power lines andcables; on the other hand, the knowledge of the frequency content, or spectrum, of a signal isessential in the EMC context, for example in order to check if the system satisfies regulatorylimits, or how it interacts with other electronic systems [5, 2]. A still open issue in this fieldis the effective modeling of long transmission lines (e.g. power/signal cables) since standardtechniques require a large number of unknowns not being able to incorporate the propagationdelay [1, 3, 6].

Over the years, some techniques have proposed delay extraction-based macromodeling al-gorithms both for time-domain and tabulated frequency data [7, 8]. A well-known method forthe analysis and the delay extraction is the Method of Characteristic (MoC)[9], and its general-izations [10, 11, 12]. The MoC has the main idea to represent the TL with a set of admittance

53

54 PAPER A

and delayed sources as terminal behaviour, and the delay is explicitly extracted over fixed fre-quency points by use of modal analysis. Consequential, irrational functions arise in frequencydomain, and the time domain formulation requires some form of approximation of the inverseLaplace transform; the numerical efficiency in terms of execution time is heavily dependenton the particular implementation [12]. Another well studied technique is the one based onPade macromodeling approach [13, 14], where the delays are model implicitly. The reader isreferred to [2] for a comparative analysis between these two techniques.

In this context, the aim of this paper is to provide a new approach for the identification ofdelay-rational macromodels of long interconnects. More specifically, stemming from the spec-tral representation of TLs described in [15], the pole analysis is performed. Dealing with linearand time-invariant system, the use of Fourier and Laplace transforms [16] allow to express theimpulse response of the system in terms of the Dirac comb, which clearly identifies the delay,thus resulting to be well suited for efficient time domain analysis.

A similar impulse response formulation is actually not completely new in literature. In[17], the lossless case analysis is divided in “input-state-output representation” and “input-state representation”, this last making use of an impulse response in terms of the Dirac comb.In [17], Bessel functions are basically used, whereas in our approach the derivation is based onthe Green’s function impedance formulation and only basic Fourier and Laplace transforms areinvolved. It is to be pointed out that, since Dirac functions are used, the convolution productused in the current-controlled representation is avoided, yielding to faster simulation and lessprone to numerical errors, as it will be shown in Section 6.

This paper is organized as follows. Section II presents a short overview about the Green’smethod for the analysis of the TLs. Section III presents the pole/residue analysis of the rationalmacromodel. In Section IV, the delayed lossless transmission line model is presented, with theimpedance-based representation by using of the Dirac comb. In Section V, the extension of theproposed approach to lossy TLs is outlined. Sections VI and VII presents numerical resultsand conclusions, respectively.

2 Transmission Line Spectral Model

A two-conductor transmission line, as sketched in Fig. 1, is considered.At the generic abscissa z the propagation equations, known as Telegrapher’s equations, in

the Laplace domain and under the hypothesis that only the quasi-transverse electromagneticfield mode propagates, read [18]

ddz

V (z,s) =−(R+ sL) I (z,s) =−Z(s)I (z,s) (1a)

ddz

I (z,s) =−(G+ sC)V (z,s)+ Is(z,s) =−Y (s)V (z,s)+ Is (z,s) (1b)

with R, L, C and G denoting per-unit-length (p.u.l.) parameters [18, 19], here assumed frequency-independent. Is(z,s) represents a p.u.l. current source located at abscissa z. In [15] it has been

2. TRANSMISSION LINE SPECTRAL MODEL 55

V 0 ( s ) V

l ( s )

+

-

+

-

I 0 ( s ) I

l ( s )

z

Figure 1: Two conductor transmission line.

demonstrated that, given the Green’s function as

G(z,z′) = −∞

∑n=0

A2n

cos(nπ

` z)

cos(nπ

` z′)

γ2(s)+(nπ

`

)2 (2)

where

An =

√

1` , n = 0√2` , n = 1, · · · ,∞

(3)

the general solution for the end voltages of the transmission line in terms of port currents is

V0(s) = Z11(s)I0(s)+Z12(s)I`(s) (4a)

V`(s) = Z21(s)I0(s)+Z22(s)I`(s) (4b)

where

Z11(s) = Z22(s) =∞

∑n=0

A2nZ(s)

γ2(s)+(nπ

`

)2 (5a)

Z12(s) = Z21(s) =∞

∑n=0

A2nZ(s)cos(nπ)

γ2(s)+(nπ

`

)2 (5b)

which is well suited to be translated into a state space realization because the impedances arewritten as a sum of rational functions. Impedances (5) admit a closed form representation inthe Laplace domain which reads

Z11(s) = Z22(s) = η(s)coth(γ(s)`) (6a)

Z12(s) = Z21(s) = η(s)csch(γ(s)`) (6b)

where γ(s) =√(R+ sL)(G+ sC) is the propagation constant, and η(s) =

√(R+sL)(G+sC) is the

characteristics impedance of the line.

56 PAPER A

3 Rational macromodelThe poles of the transmission line can be evaluated as the poles of impedance Z(s) and thezeros of the polynomial at the denominator of Z matrix entries

γ2(s)+

(nπ

`

)2= 0 (7)

which can be re-written as a trinomial term [16]

LC

(s2 + s

(RC+LG

LC

)+

(RG+

(nπ

`

)2

LC

))= 0 . (8)

It is convenient to re-write the term in brackets in the well known canonical form [16]

s2 + sRC+LG

LC+

RG+(nπ

`

)2

LC= s2 +2ζnωns+ωn

2 (9)

where:

• ωn is the resonance angular frequency (10). For the sake of simplicity, for n = 1 we callω1 = ω .

ωn =

√RG+ n2π2

`2

LC(10)

• ζn is the damping factor (11). For the sake of simplicity, for n = 1 we call ζ1 = ζ .

ζn =RC+LG

2

√√√√ 1

LC(

RG+ n2π2

`2

) (11)

The complex conjugate poles of a trinomial term can be expressed as sn(1,2) = αn± jβn (thesub-index n will be omitted for n = 1). It can be simply proven that αn and βn can be writtenin terms of ω and ζ , as will be shown in the next section, where the trinomial term is analyzedfor n = 0 and n > 0 respectively.

3.1 Poles analysis• n = 0

At the denominator, the poles s1 =−RL and s2 =−G

C are real and stable (Re < 0). Consideringthe numerator (R+sL) = L

(s+ R

L

), a pole-zero cancellation occurs, this last operation allowed

for the stability of the pole. A binomial term then results.

3. RATIONAL MACROMODEL 57

• n > 0

For n = 1 and s(1,2) = α± jβ we have

ω =

√α2 +β 2

ζ = −α√α2+β 2

α =−ζ ω

β = ω√

1−ζ 2 .(12)

It is useful to express αn and βn (for n > 1) as a function of ω and ζ (obtained for n = 1), asdone in (13). This result is particularly significant since it proves that the real part of the polesdoes not change with n, thus all the complex conjugate poles share the same real part.

αn = −ζ ω = α (13a)

βn = ω

√Q2

n−ζ 2 (13b)

Qn =

√√√√1+(nπ

`

)2 1RG

1+(

π

`

)2 1RG

(13c)

Finally, we obtain a simpler expression for Z11(s) and Z12(s)

Z11(s) =1

`C(s+ G

C

) + 2`LC

+∞

∑n=1

R+ sL

(s−α)2 +β 2n

(14a)

Z12(s) =1

`C(s+ G

C

) + 2`LC

+∞

∑n=1

R+ sL

(s−α)2 +β 2n(−1)n .

(14b)

Following this approach, the corresponding residues can be computed splitting the term in thesummation as

k1,nβn

(s−α)2 +β 2n+

k2(s−α)

(s−α)2 +β 2n

(15)

with the residues being k1,n =

1βn(R+αL)

k2 = L .(16)

By using the inverse Laplace transform, the corresponding expressions in time domain are

z11(t) =

[1`C

e−GC t +

2`LC

eαt+∞

∑n=1

(k1,n sin(βnt)+ k2 cos(βnt))

]Θ(t)

z12(t) =

[1`C

e−GC t +

2`LC

eαt+∞

∑n=1

(k1,n sin(βnt)+ k2 cos(βnt))(−1)n

]Θ(t)

where Θ(t) is the Heaviside step function.

58 PAPER A

4 Delayed Lossless Transmission Line ModelIn the lossless case, being R = G = 0,

• for n = 0 the binomial term is reduced to 1`Cs , means there is a pole in zero;

• for n > 0, α = 0 and βn =(nπ

`

) 1√LC

.

The period of transient phenomena is

T = 2`√

LC . (18)

Hence, the fundamental frequency is f0 =1

2`√

LC. Note that this result is in agreement with the

well known theoretical one for the delay time of lossy TLs[5]

Td = `√

LC . (19)

In fact, following our notation, the delay time is equal to T2 .

The poles result to be purely imaginary as

sn(1,2) =± jβn =± j2πn f0 . (20)

The Laplace self and transfer impedances transform read as

Z11(s) =1

s`C+

2`C

+∞

∑n=1

ss2 +β 2

n(21a)

Z12(s) =1

s`C+

2`C

+∞

∑n=1

ss2 +β 2

n(−1)n (21b)

whose inverse is given by

z11(t) =

(1`C

+2`C

+∞

∑n=1

cos(2πn f0t)

)Θ(t) (22a)

z12(t) =

(1`C

+2`C

+∞

∑n=1

cos(2πn f0t)(−1)n

)Θ(t) .

(22b)

4.1 Impedance-based input-output representation by using Dirac comb• Self impedance z11(t)

4. DELAYED LOSSLESS TRANSMISSION LINE MODEL 59

It can be observed that (22a) represents an unilateral Fourier series, that is generally expressedas

s(t) = s0 +2+∞

∑n=1|sn|cos

(2πn

tT+θn

)(23)

with identical coefficients equal to 1`C and θn = 0. By using the exponential form of the Fourier

series, the continuous Fourier transform reads as

Z11( f ) =

(+∞

∑n=−∞

snδ ( f −n f0)

)⊗F (Θ(t)) (24)

where F denotes the Fourier transform and ⊗ denotes the convolution product. It is straight-forward to obtain the time domain expression in terms of Dirac comb, as

z11(t) =T`C

+∞

∑n=−∞

δ (t−nT )Θ(t) = 2η

+∞

∑n=−∞

δ (t−nT )Θ(t) (25)

where η =√

LC for the lossless case.

• Transfer impedance z12(t)

Equation (22b) can also be regarded as a unilateral Fourier transform. It is worth to notethat (−1)n = e jπn = cos(nπ) . The term of the summation becomes cos(2πn f0t)cos(nπ) .By applying the well known result of the Werner formula, we get cos(2πn f0t)cos(nπ) =12 [cos(2πn f0t +πn)+ cos(2πn f0t−πn)] . Since in general Acos(α +π) = Acos(α−π) , theterm in the summation can be written as cos(2πn f0t−πn) . Finally, the z12(t) expression canbe written like an unilateral Fourier series with phase shift θn =−πn .

Following the same steps as before, the time domain expression of z12(t) in terms of Diraccomb is (26), with T

2 the looked delay.

z12(t) = 2η

[+∞

∑n=−∞

δ

(t−nT − T

2

)]Θ(t) . (26)

4.2 Lossless case: time domain realizationDirac comb formulation

Considering the current control representation [5][17], the port voltage v1(t) can be written as

v1(t) = z11(t)⊗ i1(t)+ z12(t)⊗ i2(t) (27)

Having the time domain expressions for z11(t) and z12(t), the two convolutions by using theDirac comb expression read as

z11(t)⊗ i1(t) = η i1(t)+2η

+∞

∑n=1

i1(t−nT ) (28a)

60 PAPER A

z12(t)⊗ i2(t) = 2η

+∞

∑n=0

i2

(t−nT − T

2

)(28b)

The same applies to port 2. As expected, both the contributions to port voltage v1(t) are ascaled and delayed replica of the port currents i1(t) and i2(t). For lossless two-conductor TLsthis result can also be obtained from the multi-reflections method [20].

Standard (cosine) formulation

The convolution by using the standard (cosine) z expressions (17a) and (17a) yields

z11(t)⊗ i1(t) =

[(1`C

+2`C

+∞

∑n=1

cos(2πn f0t)

)Θ(t)

]⊗ i1(t)

z12(t)⊗ i2(t) =

[(1`C

+2`C

+∞

∑n=1

cos(2πn f0t)(−1)n

)Θ(t)

]⊗ i2(t) .

As it can be easily observed, this last expression is more involved than the one with the Diraccomb formulation, and it requires the explicit computation of the convolution product.

5 Delayed Lossy Transmission Line ModelThe lossy case can be regarded as a perturbed version of the lossless one. As in the losslesscase, the complex pole pairs share the same negative real part and this suggests the idea to usethe Dirac comb. This step is not straightforward because in the lossy case the imaginary partsare not equally spaced as it happens in the lossless case and, thus, no periodic function can bedirectly recovered. The basic steps in order to achieve this goal are described below. In thefollowing we will refer to Z11(s) since the derivation for Z12(s) is similar.

1. The general expression (13b) of βn is analysed. Referring to βn for the lossless case asβnLL, and βnLY for the lossy case as in (13b), the index n = np is found such that thefollowing conditions hold

βnLY = ω√

Q2n−ζ 2, n = 1, · · · , np−1

βnLY = βnLL , n = np, · · · ,∞ .(30)

Similar considerations hold for the residual: from the expression (16), using the sub-index LL for the general lossless case, and LY for the lossy case, we have that k2 = L isunchanged, while for k1,n we have

k1,nLY = 1βnLY

(R+αL), n = 1, · · · , np−1

k1,nLY = 0 , n = np, · · · ,∞ .(31)

6. NUMERICAL EXPERIMENTS 61

2. The summation in (14a) is then split in accordance with the previous observations, andsome manipulations are made to restore the Dirac comb.

We do not report here all the derivations for lack of space, we hereby present only the finalimpedances for the lossy case

Z11(s) =1

`C(s+ G

C

) + 2`LC

[np−1

∑n=1

R+ sL(s−α)2 +β 2

nLY−

np−1

∑n=1

L(s−α)

(s−α)2 +β 2nLL

]+

+η coth((s−α)

T2

)− 1

`C(s−α)(32)

Z12(s) =1

`C(s+ G

C

) + 2`LC

[np−1

∑n=1

(R+ sL)(−1)n

(s−α)2 +β 2nLY−

np−1

∑n=1

L(s−α)(−1)n

(s−α)2 +β 2nLL

]+

+η csch((s−α)

T2

)− 1

`C(s−α). (33)

The rational contributions in (32) and (33) admit a standard state-space representation with not-delayed port currents as inputs and port voltages as outputs, the hyperbolic terms correspondto a Dirac comb in the time domain. Hence, the global hybrid state-space Dirac comb model,in the time domain, becomes

x(t) = Ax(t)+Bi(t) (34a)

v(t) = Cx(t)+D(τ)i(t) (34b)

where D(τ) acts as delay operator on the port currents i(t). A more detailed study of the lossycase and its extension to multiconductor transmission lines will be presented in forthcomingpapers, along with the complete derivation of both the frequency and the time domain models,here only outlined for lack of space.

6 Numerical Experiments

6.1 Lossless caseLet us consider a two-conductor lossless transmission line of length `= 0.5 m, opened at bothends. The p.u.l. parameters are

L = 336 nH/m C = 129 pF/m

with R and G both set equal to zero. Port 1 is excited by a pulse current i1(t) of width 1 ns, risetime and fall time equal to 200 ps, and unitary magnitude. When using (22), the summationneeds to be truncated to a suitable N [15]. It is worth noticing that fast simulations require tokeep minimum the value of N; on the other hand, N has to be large enough to preserve the

62 PAPER A

0 50 100 150 200 250−20

0

20

40

60

80

100

120

Time [ns]

v1

1(t

)

v11

(t) Dirac comb

v11

(t) Pole/residue

Figure 2: Port voltage v11(t),(N = 30, `= 0.5 m).

accuracy. When using the Dirac comb expression by (28), the infinite summation is truncatedbased on the selected time slot (see also [17]).

The results obtained by using the standard (cosine) formulation in Section 4.2 and the newDirac comb formulation in Section 4.2 are compared. For convenience, in the pictures the label“Pole/residue” denotes the standard (cosine) formulation, and the label “Dirac comb” denotesthe new proposed approach. Using N = 30, namely 60 complex conjugate poles and one realpole, evident differences arise, as we can see from the waveforms of v11(t) in Fig. 2. Firstly,some ringing [5] affects v11(t) expressed by the standard (cosine) formulation, while obviouslyit is not visible in the Dirac comb result, as highlighted in Fig. 3a. Secondly, the Dirac combformulation allows to correctly capture the rise time, whilst the standard (cosine) formulationdoes not, as highlighted in Fig. 3b.

As second example, a 5 m long TL is considered, with the same p.u.l. parameters as before.Using N = 300 in the standard (cosine) formulation is not enough since, as before, the result isaffected by ringing, as clearly seen in Fig. 4 for the port voltage v21(t) when port 1 is excited.Instead, the proposed approach based on the Dirac comb gives satisfactory results.

6.2 Lossy case

For the lossy case, the same example as before is considered, with ` = 0.5m and N = 30,adding the p.u.l. resistance and conductance

R = 25.2 Ω/m G = 0.05 S/m .


8.5 9 9.5 10

90

95

100

105

110

Time [ns]

v1

1(t

)

v11

(t) Dirac comb

v11

(t) Pole/residue

(a) Ringing effect

8.2 8.4 8.6 8.8 9 9.2

0

20

40

60

80

100

Time [ns]

v1

1(t

)

v11

(t) Dirac comb

v11

(t) Pole/residue

(b) Rise time

Figure 3: Details for port voltage v11(t),(N = 30, `= 0.5 m).

64 PAPER A

34 35 36 37

0

2

4

6

8

10

12

Time [ns]

v2

1(t

)

v21

(t) Dirac comb

v21

(t) Pole/residue

Figure 4: Port voltage v21(t),(N = 300, `= 5 m).

Let us denote with:

• Z12LY -delayed the new mixed rational-delayed formulation (33);

• Z12LY -poles/residues the standard rational formulation (5b).

The relative error of these two approaches compared to the analytical formula (6) is computedand shown in Fig. 5. As clearly visible, the new formulation (33) provides, at high frequen-cies, better results than the poles/residues one (5b), because it incorporates the high frequencybehavior through the Dirac comb.

7 ConclusionsIn this paper, a mixed rational-delayed model of two-conductor transmission lines is devel-oped. To this aim, the dyadic Green’s function-based approach, with frequency-independentparameters, has been assumed as background. The investigation of the poles for lossless andlossy TLs reveals that in both cases the real part of the complex conjugate pairs is identical.Furthermore, the analysis of the imaginary part of poles allows to identify the fundamentalfrequency for the transient phenomena. Hence, through a combination of basic Fourier andLaplace transforms, the impulsive response of the lossless case is obtained in terms of Diraccomb, whose periodicity is related to the extracted delay of the line. The lossy case is easilyrecovered from the lossless one using a perturbation approach which naturally leads to a mixedrational-delayed transmission line model. The major advantage of the proposed method with

REFERENCES 65

0 2 4 6 8 10

x 109

10−6

10−4

10−2

100

102

Freq [Hz]

Re

lative

err

or

Z12LY

delayed

Z12LY

poles/residues

Figure 5: Relative error of Z21(s) using (33) and (5b), compared with the rigorous solution (6b).

respect to full rational models is represented by the limited number of poles/residues, whichreduces the complexity of the state-space model, since the Dirac comb already takes the delayinto account. The numerical experiments have been demonstrated that the proposed methodoutperforms the full rational model while providing a good accuracy when compared with therigorous solution in the frequency domain.

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66 PAPER A

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[13] A. Dounavis, X. Li, M. S. Nakhla, R. Achar, “Passive closed-form transmission-linemodel for general-purpose circuit simulators,” IEEE Transactions on Microwave Theoryand Techniques, vol. 47, no. 12, pp. 2450–2459, Dec. 1999.

[14] A. Dounavis, E. Gad, R. Achar, M. S. Nakhla, “Passive model reduction of multiportdistributed interconnects,” IEEE Transactions on Microwave Theory and Techniques,vol. 48, no. 12, pp. 2325–2334, Dec. 2000.

[15] G. Antonini, “A dyadic Green’s function based method for the transient analysis of lossyand dispersive multiconductor transmission lines,” IEEE Transactions on Microwave The-ory and Techniques, vol. 56, no. 4, pp. 880–895, Apr. 2008.

[16] R. Beerends, Fourier and Laplace Transforms, ser. Fourier andLaplace Transforms. Cambridge University Press, 2003. [Online]. Available:http://books.google.se/books?id=frT5 rfyO4IC

[17] G. Miano and A. Maffucci, Transmission Lines and Lumped Circuits. Academic Press,2001.

[18] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York, NY: John Wiley& Sons, 1992.

67

[19] J. Brandao Faria, Multiconductor Transmission-Line Structures: Modal Analysis Tech-nique. New York, NY: John Wiley & Sons, 1993.

[20] D. Pozar, Microwave Engineering, 3Rd Ed. Wiley India Pvt. Limited, 2009. [Online].Available: http://books.google.se/books?id=UZgvwJ3Eex8C

PAPER B

A Delay-Rational Model of LossyMulticonductor Transmission Lines

with Frequency IndependentPer-Unit-Length Parameters


Reformatted version of paper originally published in:IEEE Transactions on Electromagnetic Compatibility, 2015.


69

A Delay-Rational Model of Lossy MulticonductorTransmission Lines with Frequency Independent

Per-Unit-Length Parameters


Abstract

Cables, printed circuit boards, and VLSI interconnects are commonly modeled as multicon-ductor transmission lines. Models of electrically long transmission lines are memory and timeconsuming. In this paper, a robust and efficient algorithm for the generation of a delay-basedmodel is presented. The impedance representation via the open-end matrix Z is analyzed. Inparticular, the rational formulation of Z in terms of poles and residues is exploited for bothlossless and lossy cases. The delays of the lines are identified, and explicitly incorporated intothe model. A model order reduction of the system is automatically performed, since only alimited number of poles and residues are included in the rational part of the model, whereasthe high frequency behavior is captured by means of closed expressions that account for thedelays. The proposed method is applied to two relevant examples and validated through thecomparison with reference methods. The time domain solver is found to be more accurate andsignificantly faster than the one obtained from a pure-rational model.

1 IntroductionThe increasing clock speeds and the complexity of circuits made signal integrity and elec-tromagnetic compatibility topics of real importance for circuit and system designers. In thisframework, interconnects at various level (PCBs, chip) play an important role, since their per-formances may be severely deteriorated by high frequency effects such as signal delay, reflec-tion, dispersion, or crosstalk[1]. Interconnects can be seen as transmission lines with multiplesignal lines, and the well known multiconductor transmission line (MTL) theory can be applied(see [1, 2, 3] and references therein).

The transmission line equations can be formulated in both the frequency and time domains.While the analytical solution of a MTL in the frequency domain is completely assessed, thetransient analysis still deserves some attention, especially for electrically long transmissionlines. In fact, non-linear devices acting as terminations, such as drives and receivers in PCBsor non-linear protection devices in power MTLs, can only be represented in the time domain.It is then mandatory to model MTLs in time-domain when non-linear terminations are used.The classic frequency analysis of MTLs followed by the inverse fast Fourier transform is notsuitable to this aim due to the presence of the non-linearities.

In order to make time domain simulations reliable and fast from a computational time point

71

72 PAPER B

of view, several techniques have been proposed for an electrically long MTL. For example,there is the well-known lumped-element equivalent circuit method. This is a segmentationtechnique, whose main drawback is that a large number of unknowns is needed, mainly becausethe propagation delays need to be approximated by a large number of lumped elements [1, 2, 4].In order to overcome this limitation, other approaches have been proposed, most of them basedon delay extraction algorithms, leading to macromodels for both the time-domain and tabulatedfrequency data [5, 6].

A well-known method is the Method of Characteristic (MoC) [7], and its generalizations[8, 9, 10]. The MoC represents the MTL with a set of admittances and delayed sources, wherethe delay of the line is explicitly extracted over fixed frequency points by use of modal analy-sis. As a consequence, irrational functions arise in the frequency domain, and the time domainformulation requires some approximation of the inverse Laplace transform; the numerical effi-ciency in terms of execution time is heavily dependent on the particular implementation [10].

Also very popular, especially in the power community, and similar to the MoC technique,is the universal line model (ULM) [11] in which the propagation matrix H is first fitted in themodal domain. The resulting poles and time delays are then used for fitting H in the phasedomain, under the assumption that all poles contribute to all elements of H . The unknownresidues are then calculated by solving an overdetermined linear equation as a least squaresproblem. An improved version of this approach has been recently presented in [12].

Another well studied technique is the one based on the Pade approach [13, 14], where thedelays are modeled implicitly. The reader is referred to [15] for a comparative analysis betweenMoC and Pade techniques. A different approach is based on the idea to accelerate the transientanalysis of a MTL using waveform relaxation techniques [16, 17, 18, 19].

A rigorous rational model of a MTL has been proposed in [20], where the input/outputimpedance matrix is expressed in terms of the dyadic Green’s function of the 1-D propagationproblem. A rational series form of this last one is proposed, where the poles are identified bysolving algebraic equations. Although general, such an approach may lead to large state-spacemodels when an electrically long MTL is considered, since a significant number of poles isnecessary to gain a good accuracy, especially in terms of ringing effects in the time domain. Inthis context, the aim of this paper is to overcome such limitation. A new approach for the iden-tification of delay-rational macro-models of electrically long interconnects is presented, wherea reduced number of poles and, consequently, a reduced state space size in the time domainare gained. It is not just a matter of size, in fact a higher accuracy over the all frequency rangeis gained, and no ringing effects are present in the time domain response. From a computa-tional point of view, the algorithm is also faster. Stemming from the spectral representation ofa MTL as described in [20], the pole analysis is first performed. Assuming a MTL as a linearand time-invariant system, the use of Fourier and Laplace transforms [21] allows to expressthe impulse response of the system in terms of Dirac combs, which clearly identify the delays,thus resulting to be well suited for efficient time domain analysis.

The paper is organized as follows. Section 2 presents a short review of the Green’s functionbased method for the analysis of MTL; some peculiar properties of poles and residues arepointed out. In Section 3, the delay-rational impedance model of a lossless MTL is presented.The extension of the proposed method for a lossy MTL is outlined in Section 4. Pertinent

2. REVIEW OF THE SPECTRAL MODEL FOR MULTICONDUCTOR TRANSMISSION LINES 73

V1(s)

V2(s)

VN(s)

VN+1(s)

VN+2(s)

V2N(s)

- -

+

+

+ +

+

+

MTL

z= 0 z= l

I1(s)

I2(s)

IN(s)

IN+1(s)

IN+2(s)

I2N(s)

Figure 1: Multiconductor transmission line represented as 2N port system, with a common referenceconductor.

numerical results are presented in Section 5 and the conclusions are drawn in Section 6.

2 Review of the spectral model for Multiconductor Trans-mission Lines

A multiconductor transmission line, as sketched in Fig. 1, is considered. At the generic ab-scissa z the propagation equations, known as Telegrapher’s equations, in the Laplace domainand under the hypothesis that only the quasi-transverse electromagnetic field mode propagates,read [1]

ddz

V(z,s) = − [R+ sL]I(z,s)

= −Z′(s)I(z,s) (1a)

ddz

I(z,s) = − [G+ sC]V(z,s)+ Is(z,s)

= −Y′(s)V(z,s)+ Is (z,s) (1b)

where R, L, C and G are frequency-independent per-unit-length (p.u.l.) matrices and are non-negative definite symmetric of order N, being N + 1 the number of the conductors with thereference conductor included [1, 22]. Z′(s) and Y′(s) are the p.u.l. longitudinal impedance andadmittances matrices. Is(z,s) represents a p.u.l. current source located at abscissa z. V(z,s) andI(z,s) represent the voltage and current vectors depending on Laplace variable s and positionz along the line.

In [20], it has been proved that the general solution for the end voltages of a transmission

74 PAPER B

line of length ` in terms of port currents is[V0(s)V`(s)

]=

[Z11(s) Z12(s)Z12(s) Z11(s)

][I0(s)I`(s)

]= Z(s)

[I0(s)I`(s)

](2)

V0(s) and I0(s) are the voltage and the current port vectors respectively, of dimensionN× 1, related to the input ports at z = 0. V`(s) and I`(s) are the voltage and the current portvectors respectively, of dimension N× 1, related to the output ports at z = `. Z(s) is the portimpedance matrix, that is block symmetrical, each block of dimension N×N, defined as

Z11(s) = Z22(s) (3a)

=+∞

∑n=0

[γ

2(s)+(nπ

`

)2I

]−1

·A2nZ′(s)

Z12(s) = Z21(s) (3b)

=+∞

∑n=0

[γ

2(s)+(nπ

`

)2I

]−1

·A2nZ′(s)cos(nπ) .

In (3) γ2(s)=Z′(s)Y′(s)= (R+ sL)·(G+ sC), A20 = 1/` for n= 0, A2

n = 2/` for n= 1, · · · ,+∞,and I is the identity matrix. By using the Green’s function method developed in [20], the Z(s)can be expressed as an infinite sum of poles and residues as in (4). In fact, for each mode m, anumber n = 1, ..., np of poles and residues can be evaluated, leading to a final number of polesequal to np = ∑

Mm=0 np,m.

Z(s) =np

∑n=1

Rn

s− pn. (4)

It is worth noting that, in order to simplify the notation, it is not specified whether the poles arereal or complex conjugate.

A first way to use the rational impedance model (4) is to express it in the time-domain; theresulting transient impedance z(t) is then convoluted with the port currents i(t) of the MTL[23].

Since the impedances are written as a sum of rational functions, this model is well suitedto be translated into a state space realization as

x(t) = Ax(t)+Bi(t) (5)

v(t) = Cx(t) .

Typically, the poles are used to compute the matrix A, the residues for the matrix C, whilethe matrix B maps the inputs to state space equations. These equations can be then coupledwith those describing the terminations, that can be either linear or non-linear. More detailsabout the state-space model construction can be found in [3].

3. DELAYED MODEL OF LOSSLESS MTL 75

The main drawback of this pure rational, yet rigorous, approach relies on the large numberof modes (theoretically +∞) needed to accurately capture the frequency response of electricallylong lines. In the present work, a new formulation of the impedance matrix is given, that allowsto properly model the MTL by using delays and a reduced number of modes and, thus, of poles.In general, we refer to “delay” as the traveling time a pulse spends from the input port, to theoutput port, then again to the input port: summarizing, is the “input port-output port-input port”time. Through an explicit extraction of the delays, a delay-rational model is obtained, wherethe size of the rational part is dramatically reduced (thanks to the reduction of the numberof poles), still keeping accuracy in the time range of interest. Some useful observations arenecessary in order to develop the delay-rational model:

• in the lossless case, the transient impedances are expected to be periodic functions [1];

• the residues of a lossless MTL can be clustered into N groups, or families, sharing thesame magnitude, and the corresponding poles can be identified. Within each group, theyare equally spaced along the imaginary axis, confirming the expected periodicity;

• when computing the poles and residues of a lossy MTL with frequency independent p.u.l.parameters using the rational form (3), from n = 1, · · · ,+∞ (the zero mode n = 0 of thesummation will be treated as a special case), the poles and residues gather again in fami-lies, sharing the same real part and magnitude, respectively. Even thought the periodicityis not so clear to identify, it can be indeed restored thanks to a proper mathematicalformulation.

3 Delayed Model of Lossless MTLIn the lossless case, R = G = 0, where 0 is the null matrix. It is necessary to treat separatelythe mode zero from the higher order ones.

• n = 0

Recalling that A20 = 1/` , the mode zero contribution to the impedances (3) can be written as

Zi j(s) |n=0 =1s`

C−1 (6)

for i, j = 1,2.The pole in zero has always multiplicity 1, that is crucial to guarantee the asymptotic sta-

bility.

• n 6= 0

Given a N-conductor transmission line, we have that each mode n 6= 0 generates N poles [20].For each pole pk,n the corresponding matrix of residues Rk,n is computed, with k = 1, · · · ,Nkeeping track of the family. Hence, each impedance Zi j(s) can be represented in terms of polesand residues as

76 PAPER B

Z11(s) =C−1

s`+

N

∑k=1

(+∞

∑n=1

(Rk,n

s− pk,n+

R∗k,ns− p∗k,n

))(7a)

Z12(s) =C−1

s`+

N

∑k=1

(+∞

∑n=1

(Rk,n

s− pk,n+

R∗k,ns− p∗k,n

)(−1)n

)(7b)

where ∗ denotes the complex conjugate of the complex numbers.From numerical results, we could observe that:

• all the residues are purely real;

• the residues of the k-th pole can be split into N groups, or families. Within each group,they remain constant in magnitude through the mode n. Since they are all real, we canwrite

Rk,n = Rk,1 , n = 1, · · · ,+∞ and k = 1, · · · ,N (8)

• all the poles pk,n are purely imaginary pairs, then pk,n = jβk,n, for n = 1, · · ·+∞;

• the residues clustering suggests a corresponding clustering of poles into N families. Infact, the poles belonging to each family are equispaced along the imaginary axis, mean-ing βk,n = nβk,1, for k = 1, · · · ,N and n = 1, · · · ,+∞, where k spans over the differentfamilies, as it can be seen in the example in Fig. 2. This observation suggests the ideathat poles and residues of each family describe a periodic function, as it will be bettershown in the following;

• the fundamental frequency fk,1 for each family is

fk,1 =βk,1

2π(9)

• higher order harmonic frequencies are simply given as fk,n = n fk,1, for n = 1, · · · ,+∞ ;

• the pole in the origin for n = 0 is equally shared between all families. From numericalresults, we could observe that (6) can be written as

C−1

`=

N

∑k=1

Rk,1 . (10)

Accordingly to the above observations, since (10) is related to the n = 0 mode, we pose Rk,0 =Rk,1, with Rk,0 the residue associated to the zero mode. A more compact expression is thengained


ℜ(pk,n

)

-1 -0.5 0 0.5 1

ℑ(p

k,n

)

×109

-6

-4

-2

0

2

4

6

Family1

Family2

Figure 2: Poles locus in the complex plane for a two-conductor lossless transmission line. For the sakeof clarity, the number of modes is n = 10. Note that the pole in the origin is not considered, and 4 poles(in complex conjugate pairs) are detected for each family.

Z11(s) =N

∑k=1

(Rk,0

s+

+∞

∑n=1

(Rk,0

s− pk,n+

R∗k,0s− p∗k,n

))(11a)

Z12(s) =N

∑k=1

(Rk,0

s+

+∞

∑n=1

(Rk,0

s− pk,n+

R∗k,0s− p∗k,n

)(−1)n

). (11b)

In the following, the self and the mutual impedances will be separately treated, for the sakeof clarity.

Self-impedance

Since the sub-blocks of Z11(s) are all positive and real, the expression for Z11(s) can be furthersimplified as

Z11(s) =N

∑k=1

(Rk,0

s+2Rk,0

+∞

∑n=1

s

s2 +(nβk,1

)2

)(12)

z11(t) =N

∑k=1

(Rk,0 +2Rk,0

+∞

∑n=1

cos(nβk,1t

))Θ(t) . (13)

78 PAPER B

s(t) = s0 +∑+∞

n=1 2 |sn|cos(

2πn tT0+θ

)KSF

S( f ) = ∑+∞n=−∞ snδ ( f −n f0)KS

F−1

s(t) = T0 ∑

+∞n=−∞ snδ (t−nT )

Figure 3: Equivalences via Fourier transform.

Mutual impedance

For the Z12(s), the following expression is gained

Z12(s) =N

∑k=1

(Rk,0

s+2Rk,0

+∞

∑n=1

ss2 +(nβk,1)2 (−1)n

). (14)

Because of the term (−1)n = cos(nπ), a phase shift equal to −nπ is present in the transientimpedance z12(t)

z12(t) =N

∑k=1

(Rk,0 +2Rk,0

+∞

∑n=1

cos(nβk,1t−nπ

))Θ(t) . (15)

Eq. (13) and (15) confirm that transient impedances zi j(t), for i, j = 1,2, are the sum of Nperiodic functions whose period is

Tk =1

fk,1=

2π

βk,1, k = 1, · · · ,N . (16)

3.1 Impedance-based I-O representation by Dirac combThe general series form of the impedances requires a high number of modes in order to havesufficient accuracy. It leads to large state-space models, and it requires cumbersome convolu-tion products in the time domain. Taking advantage of the delays explicit extraction techniquehere developed, the next step will be to re-write the impedances in terms of the Dirac combs. Inthe MTL lossless case, each impedance will be represented by the sum of N Dirac comb trains,each of them having a different frequency of repetition. From a system theory point of view,this is equivalent to have N delayed systems in parallel, each contributing with a time delayequal to Tk,k = 1, · · · ,N. In the following, we will always exploit the equivalences collected inFig. 3. The self and mutual impedances will be treated separately.


Self-impedance

Let us consider the expression in (12). Exploiting the analogy with the Fourier series as in Fig.3, it is straightforward to obtain the time domain expression in terms of Dirac comb as

z11(t) =N

∑k=1

(Rk,0Tk

+∞

∑n=−∞

δ (t−nTk)

)Θ(t) (17)

where Θ(t) is the Heaviside step. The Heaviside step function can be written in terms of thesign function sgn(t) as

Θ(t) =12+

12

sgn(t) . (18)

Hence, the transient impedance z11(t) will read as

z11(t) =N

∑k=1

Rk,0Tk

(δ (t)

2+

+∞

∑n=1

δ (t−nTk)

). (19)

The Laplace transform of (19) is then performed as

L (z11(t)) =N

∑k=1

Rk,0Tk

(12+

+∞

∑n=1

e−snTk

)

=N

∑k=1

Rk,0Tk

(

es Tk2 + e−s Tk

2

)2(

es Tk2 − e−s Tk

2

)

=N

∑k=1

Rk,0Tk

2

cosh(

sTk2

)sinh

(sTk

2

) (20)

leading to the final expression for Z11(s)

Z11(s) =N

∑k=1

Rk,0Tk

2coth

(sTk

2

). (21)

It is worth noting that, in the single-conductor case, the residual is equal to 1/`C and T = Tk[24]. As expected, the expression (21) is the natural extension of the single-conductor model[24] to the MTL case.

Mutual impedance

the transient mutual impedance z12(t) reads as in (15). Following similar steps as for the selfimpedance, the time domain expression for z12(t) is obtained in term of Dirac combs

z12(t) =N

∑k=1

(Rk,0Tk

+∞

∑n=0

δ

(t−nTk−

Tk

2

)). (22)

80 PAPER B

The Laplace transform of (22) is then performed

L (z12(t)) =N

∑k=1

Rk,0Tk

(+∞

∑n=0

e−snt−s Tk2

)=

N

∑k=1

Rk,0Tk

2

1

sinh(

sTk2

) (23)

gaining the final expression for Z12(s)

Z12(s) =N

∑k=1

Rk,0Tk

2csch

(sTk

2

). (24)

The expression (24) is the natural extension of the single-conductor model [24] to the MTLcase.

It is also worth noting that, in both the self and mutual impedance (21) and (24), the quan-tity Rk,0Tk/2 acts as a generalized characteristic impedance for the k-th family of poles andresidues. In the single conductor TL case outlined in [24], it was found that η = T/(2`C).Having in mind that, in the single conductor TL case, there is only one family of poles andresidues, and all residues are equal to Rk = 1/`C, it is easy to verify that Rk,0Tk/2 is actually acharacteristic impedance.

4 The Delay-Rational Model for a lossy MTLThe lossy case is considered. As done before, it is convenient to treat separately the n = 0mode and the higher order modes n > 0.

• n = 0

The same considerations hold for both the self and the mutual impedance. From the expression(3) we have

Zi j(s) |n=0 =A20 (G+ sC)−1 =

A20Ad j (G+ sC)

det (G+ sC)(25)

for i, j = 1,2. The zeros of det (G+ sC) correspond to N negative real poles pk,0, with k =1, · · · ,N. Denoting Rk,0 the associated residues, the expression is rewritten as

Zi j(s) |n=0 =N

∑k=1

Rk,0(s− pk,0

) . (26)

The constant A20 is already included in Rk,0.

• n > 0

In the lossy case, the poles pk,n are complex conjugate pairs but it is still possible to subdividethem into N groups. The corresponding residues Rk,n are also complex conjugate. Fromnumerical computation, it is possible to observe that the poles share the same real part αk,nwithin each k-th family through the n modes. Then, αk,n = αk,1 for k = 1, · · · ,N, as in theexample in Fig. 4.

4. THE DELAY-RATIONAL MODEL FOR A LOSSY MTL 81

ℜ (pk,n

) ×105

-5 -4.5 -4 -3.5 -3 -2.5

ℑ(p

k,n

)

×109

-6

-4

-2

0

2

4

6

Family1

Family2

Figure 4: Poles locus in the complex place for a two-conductor lossy transmission line. For the sake ofclarity, the number of modes is n = 10. Note that the N = 2 real poles for n = 0 are not plotted, and 4poles (in complex conjugate pairs) are detected for each family.

• n≥ m, n 6= 0

Some important properties regarding poles and residues are observed for modes of index n> m,n 6= 0, where the choice of m is the subject of Section 4.1:

• Rk,n = Rk is a constant value, for k = 1, · · · ,N, as in the example in Fig. 5. Morespecifically, the imaginary part of Rk,n tends to zero for n→ +∞, meaning the phasetends to zero, as in Fig. 6. Denoting by Rk the real part of Rk,n, from a certain mode m,the following approximation can be used Rk,n = Rk, for n = m, · · · ,+∞;

• for each k-th family, the imaginary parts βk,n of the poles become equispaced from acertain mode m, then for n = m, · · · ,+∞. Denoting by βk,n the fictitious imaginary partsequispaced for n = 1, · · · ,+∞, we can write βk,n = nβk,1, n = 1, · · · ,+∞. It is worthnoting that the imaginary parts of these fictitious poles correspond to the lossless ones.

These properties hold also for a single conductor TLs in [24]. Hence, for n > m, poles andresidues exhibit the same periodic property as for the lossless case. However, in order to recoverthe periodicity, the full set of poles and residues needs to be reconstruct for n = 0, · · · ,+∞. Tothis aim, fictitious poles and residues are defined as

pk,0 = αk,1 (27a)

Rk,0 = Rk (27b)

82 PAPER B

Mode

0 2 4 6 8 10

|Res

11| [Ω

· s

]×10

9

6

8

10

12

14

Family1

Family2

Figure 5: Magnitude of the residue Rk,n of Z11(s), for a two-conductor lossy transmission line. For thesake of clarity, the number of modes is n = 10.

Mode

0 2 4 6 8 10

Res

11[r

ad]

×10-4

0

1

2

3

4

5

Family1

Family2

Figure 6: Phase of the residue Rk,n of Z11(s), for a two-conductor lossy transmission line. For the sakeof clarity, the number of modes is n = 10.


pk,n(1,2) = αk,1± jnβk,1 (27c)

Rk,n = Rk (27d)

for k = 1, · · · ,N. The expressions of the impedances Zi j(s), for i, j = 1,2 in the Laplace domainare

Z11(s) =N

∑k=1

(Rk,0

s− pk,0+

m−1

∑n=1

(Rk,n

s− pk,n+

R∗k,ns− p∗k,n

)+

+∞

∑n=m

(Rk

s− pk,n+

Rk

s− p∗k,n

))(28a)

Z12(s) =N

∑k=1

(Rk,0

s− pk,0+

m−1

∑n=1

((Rk,n

s− pk,n+

R∗k,ns− p∗k,n

)+

+∞

∑n=m

(Rk

s− pk,n+

Rk

s− p∗k,n

))(−1)n

)(28b)

that in the time domain are

z11(t) =N

∑k=1

(Rk,0epk,0t +2eαk,1tRk

+∞

∑n=m

cos(nβk,1t)

+ eαk,1tm−1

∑n=1

2∣∣Rk,n

∣∣cos(βk,nt +∠Rk,n)

)Θ(t) (29a)

z12(t) =N

∑k=1

(Rk,0epk,0t +2eαk,1tRk

+∞

∑n=m

cos(nβk,1t−nπ)

+ eαk,1tm−1

∑n=1

2∣∣Rk,n

∣∣cos(βk,nt +∠Rk,n−nπ)

)Θ(t) . (29b)

The transient impedances (29) implicitly contain the time-delays. In the following, the delayswill be explicitly extracted. For the sake of clarity, self and mutual impedances Z11(s) andZ12(s) respectively, are treated separately.

4.1 Selection of m

In order to assign a value for m, the following criterion is adopted. As it was already ob-served, the phase of the residues tends to zero for n → +∞. It means, the difference inphase

(∠Rk,n+1−∠Rk,n

)→ 0 for n = 0, · · · ,+∞. We observe that

(∠Rk,n+1−∠Rk,n

)>(

∠Rk,n+2−∠Rk,n+1)

for n = 0, · · · ,+∞. Then, given a certain tolerance value tol, m is cho-sen such that

∠Rk,m−∠Rk,m−1

∠Rk,2−∠Rk,1< tol . (30)

84 PAPER B

4.2 Self-impedanceGiven the expression in (28a), the summation ∑

+∞

n=m is rewritten as

Z11(s)|n≥m =N

∑k=1

(2Rk

+∞

∑n=m

(s−αk,1)

(s−αk,1)2 +(nβk,1)2

). (31)

The rational contributions with fictitious poles and residues (27) are added and subtracted inorder to reconstitute the full summation

Z11(s)|n≥m =N

∑k=1

(Rk

s−αk,1+2Rk

+∞

∑n=1

(s−αk,1)

(s−αk,1)2 +(nβk,1)2

− Rk

s−αk,1−2Rk

m−1

∑n=1

(s−αk,1)

(s−αk,1)2 +(nβk,1)2

). (32)

The inverse Laplace transform of the first two terms of (32) is performed to exploit the equiv-alences in Fig. 3. A Fourier series expression in the time domain is obtained

L −1(•) =N

∑k=1

Rk

(eαk,1t +2eαk,1t

+∞

∑n=1

cos(nβk,1t)

)Θ(t) (33)

where, for the sake of simplicity, the argument of the Laplace inverse transform is denoted as•. It is possible to recognize (33) as the extension to the lossy case of (13) for the lossless case.The time domain Dirac comb is obtained

L −1(•) =N

∑k=1

Rk

(Tkeαk,1tδ (t)

2+Tkeαk,1t

+∞

∑n=1

δ (t−nTk)Θ(t)

). (34)

The Laplace transform of (34) reads as

L (•) =N

∑k=1

Rk

(Tk

2coth

((s−αk,1)

Tk

2

)). (35)

4.3 Mutual ImpedanceIn the Z12(s) case, the summation ∑

+∞

n=m can be rewritten as

Z12(s)|n≥m =N

∑k=1

(2Rk

+∞

∑n=m

(s−αk,1)(−1)n

(s−αk,1)2 +(nβk,1)2

)(36)

where the multiplicative term (−1)n is considered explicitly. Then, the same steps as for Z11(s)are performed. Hence, the rational contributions involving fictitious poles and residues (27) areadded and subtracted in order to reconstitute the full summation

Z12(s)|n≥m =N

∑k=1

(Rk

s−αk,1+2Rk

+∞

∑n=1

(s−αk,1)(−1)n

(s−αk,1)2 +(nβk,1)2


− Rk

s−αk,1−2Rk

m−1

∑n=1

(s−αk,1)(−1)n

(s−αk,1)2 +(nβk,1)2

). (37)

The inverse Laplace transform of the first two terms of (37) is performed to exploit the equiv-alences in Fig. 3. A Fourier series in the time domain is obtained, as shown in (38):

L −1 (•) =N

∑k=1

Rk

(eαk,1t +2eαk,1t

+∞

∑n=1

cos(nβk,1t−nπ)

)Θ(t) . (38)

Following the same steps as for Z11(s), it can be written in terms of the time domain Diraccomb as

L −1(•) =N

∑k=1

RkTkeαk,1t+∞

∑n=0

δ

(t−nTk−

Tk

2

). (39)

The Laplace transform of (39) reads

L (•) =N

∑k=1

Rk

(Tk

2csch

((s−αk)

Tk

2

)). (40)

Finally, it is possible to rewrite the general expressions (28) as

Z11(s) =N

∑k=1

(Rk,0

s− pk,0+

m−1

∑n=1

(Rk,n

s− pk,n+

R∗k,ns− p∗k,n

)

−m−1

∑n=1

(Rk

(s− pk,n)+

Rk

(s− p∗k,n)

)

− Rk

(s−αk,1)+

RkTk

2coth

((s−αk,1)

Tk

2

))(41a)

Z12(s) =N

∑k=1

(Rk,0

s− pk,0+

m−1

∑n=1

(Rk,n

s− pk,n+

R∗k,ns− p∗k,n

)

−m−1

∑n=1

(Rk

(s− pk,n)+

Rk

(s− p∗k,n)

)(−1)n

− Rk

(s−αk,1)+

RkTk

2csch

((s−αk,1)

Tk

2

)). (41b)

In both the impedances in (41), the first four terms admit a standard state-space realizationwhile the hyperbolic terms correspond to time domain Dirac combs. Indeed, in the time do-main, these two expressions read as

z11(t) =N

∑k=1

(Rk,0epk,0t

Θ(t)+m−1

∑n=1

2∣∣Rk,n

∣∣eαk,nt cos(βk,nt +∠Rk,n

)Θ(t)

86 PAPER B

−2Rkeαk,1tm−1

∑n=1

cos(nβk,1t)Θ(t)− Rkeαk,1tΘ(t)

+ Rk

(Tkeαk,1tδ (t)

2+Tkeαk,1t

+∞

∑n=1

δ (t−nTk)

))(42a)

z12(t) =N

∑k=1

(Rk,0epk,0t

Θ(t)+m−1

∑n=1

2∣∣Rk,n

∣∣eαk,nt cos(βk,nt +∠Rk,n−nπ

)Θ(t)

−2Rkeαk,1tm−1

∑n=1

cos(nβk,1t−nπ)Θ(t)− Rkeαk,1tΘ(t)

+RkTkeαk,1t+∞

∑n=0

δ

(t−nTk−

Tk

2

)). (42b)

4.4 State space modelThe rational contributions in (41) admit a standard state-space representation with port currentsi(t) as inputs, and port voltages v(t) as outputs. The state space matrix A has dimension nss

nss = (2(N +2N(m−1)))Np (43)

where N denotes the number of conductors and Np = 2N denotes the number of ports. The portcurrent vector i(t) has dimension Np, then the input space is ℜq×1 with q = Np, and it can bewritten as

i(t) = [i1(t)|i2(t)]T (44a)

with i1(t) the currents related to the input ports, and i2(t) the currents related to the outputports; the apex T denotes the transposition. The output vector is given by the voltages at theinput and the output ports, thus the output space is ℜq×1 with q = Np ,

v(t) = [v1(t)|v2(t)]T . (45a)

The summation of hyperbolic terms in the frequency domain corresponds to a summation ofDirac combs in the time domain. Hence, the global delay-rational model in the time domaincan be written as

x(t) = Ax(t)+Bi(t) (46a)

v(t) = Cx(t)+D0i(t)+ id(t) (46b)

where id(t) = D(t)⊗ i(t), with D(t) described hereinafter. Compared to a standard state spacemodel as in (5), two matrices, D0 and id(t), are used. The first one is defined as

D0 =N

∑k=1

[Rk

Tk2 0

0 RkTk2

]. (47)


Note that the convolution δ (t)⊗ i(t) = i(t), and then it is omitted in the state-space model for-mulation. The second matrix id(t) = D(t)⊗ i(t) represents the delayed currents contributions.In particular, D(t) acts as a delay operator on the port currents i(t), then it is symmetric withthe form

D(t) =N

∑k=1

[D11 D12D12 D11

](48)

with

D11 = RkTkeαk,1t+∞

∑n=1

δ (t−nTk) (49a)

D12 = RkTkeαk,1t+∞

∑n=0

δ

(t−nTk−

Tk

2

)(49b)

where the infinite summation will be truncated to n = max(t)Tk

, where t is the time window underanalysis. Note that the convolution product D(t)⊗ i(t) returns only an attenuated and translatedversion of the currents, so no convolutions products are actually performed in the time-domainsolver.

4.5 DiscussionIt is worth pointing out the main advantages of this method when compared to the existingones.

1. Both the standard rational and the delay-rational models do not require any discretizationof the line as they are based on the Green’s function for MTL;

2. the computation of poles and residues is performed separately for each mode. Hence, wedo not search all the poles at a time as fitting techniques usually do and, as a consequence,the proposed method is less prone to numerical round-off errors;

3. once the number m of modes to be included in the delay-rational model is identified,a reduced order model is automatically computed, since m m for electrically longtransmission lines;

4. the proposed technique provides a significant insight into the theory of MTL since itnaturally leads to identify the delays, without resorting to any modal decomposition nornumerical estimation, apart from the computation of poles for each mode.

5 Numerical ExperimentsThe proposed delay-rational method is validated through two test cases. The results, in the fre-quency and time domain, are compared with the one obtained by the rational model from the

88 PAPER B

dyadic Green’s function method [20]. In the following, we will refer to the existing dyadicGreen’s function method as “Rational”, and the proposed delayed-rational one as “Delay-rational”. For the simulations, the Matlab R2014b [25] software is used, on a computer running64-bit Windows 7 O.S., with Intel Core Xeon @2.27 GHz (2 processors) and 48 GB of RAM.For a intuitive comparison, the weighted RMS-error respect to the impedance computed as perthe MTL classical theory [1] is defined as

Err =

√√√√ 14N2Ns

2N

∑i=1

2N

∑j=1

Ns

∑k=1

∣∣∣∣Zi j,r(k)−Zi j(k)Zi j,r(k)

∣∣∣∣2 (50)

where Zi j,r is the i, j-th entry of the impedance matrix by MTL classical theory, assumed asreference; Zi j denotes the i, j-th entry of the impedance matrix obtained by the rational modelor the delay-rational one; Ns is the number of frequency samples. The generic 2N-port repre-sentation in Fig. 1 is adopted.

5.1 Example 1: four-conductor transmission lineA four-conductor lossy TL of length ` = 0.2 m is considered, with p.u.l. parameters reportedbelow [26]. The conductor 1 (as per Fig. 1) is excited by a smooth pulse voltage of amplitude1 V, rise and fall times tr = t f = 0.5 ns and pulse-width τ = 2 ns. All the ports are terminatedon 250 Ω resistances.

L =

1 0.11 0.03 0

0.11 1 0.11 0.030.03 0.11 1 0.11

0 0.03 0.11 1

µH/m (51)

R =

3.5 0.35 0.035 0

0.35 3.5 0.35 0.0350.035 0.35 3.5 0.35

0 0.035 0.35 3.5

Ω/m (52)

G =

10 −1 −0.1 0−1 10 −1 −0.1−0.1 −1 10 −1

0 −0.1 −1 10

mS/m (53)

C =

1.5 −0.07 −0.03 0−0.07 1.5 −0.07 −0.03−0.03 −0.07 1.5 −0.07

0 −0.03 −0.07 1.5

nF/m . (54)


The number of modes is set as m = 150 in the standard rational formulation (3). In the pro-posed delay-rational (41), the value of m = 7 is automatically computed, based on the accuracydesired. The poles in the complex plane are plotted in Fig. 7. In the picture, are highlighted thepoles generated by the n = 0 mode and by the n > 0 modes, which are gathered in four families(equal to the number of conductors), sharing the same negative real parts. The four time-delaysare τ1 = 14.67 ns, τ2 = 15.07 ns, τ3 = 15.72 ns, τ1 = 16.19 ns. The resulting state-space hassize 9632 and 832 for the standard rational and the new delay-rational approaches, respectively.Fig. 8 shows the voltages in time domain for the ports 1 and 4. The two methods are comparedalso with the IFFT standard solution, that uses diagonalization and modal solution evaluation.No significant differences are observed. As it can be seen from Table 1, the proposed delay-rational technique is faster and more accurate than the rational standard one, and provides highaccuracy with a reduced number of modes, over the entire frequency range.

−7.5 −7 −6.5 −6 −5.5 −5 −4.5

x 106

−4

−3

−2

−1

0

1

2

3

4x 10

10

Re

Ima

g

n=0 poles

n>0 poles

Figure 7: Poles location for a four-conductor transmission line, example 1 in Section 5.1.

5.2 Example 2: electrically long cable

A `= 10 m long cable is considered. Its p.u.l. parameters are [27]

L = 2.756914 µH/m , R = 1.975946Ω/m

G = 8.719413 µS/m , C = 0.827367nF/m .(55)

90 PAPER B

0 50 100 150 200−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Time [ns]

V1 [

V]

Rational

Delay−Rational

IFFT

0 50 100 150 200−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time [ns]

V4 [

V]

Rational

Delay−Rational

IFFT

Figure 8: Voltages at ports 1 and 4 for a four-conductor transmission line, example 1 in Section 5.1.

6. CONCLUSIONS 91

Table 1: Comparative results for the two numerical examples.

Example Method Time domain solver [s] Weighted RMS-error

Example 1Rational (m = 150) 14704.75 21.2829

Delay-Rational (m = 7) 1474.45 0.0401

Example 2Rational (m = 800) 12459.14 66.1996

Delay-Rational (m = 7) 991.62 0.0370

The input port is excited by a smooth pulse voltage of amplitude 1 V, rise and fall times tr =t f = 3 ns, and pulse-width τ = 15 ns. It is terminated at both ends on Rt = 500Ω resistances.The standard rational approach requires 800 modes to provide a reasonable accuracy, whilethe new delay-rational method requires only 7 modes to achieve even a better accuracy, assummarized in Table 1.

The resulting state spaces for the delay-rational approach and the rational standard one areof size 52 and 3202, respectively. The time-delay is τ = 955 ns, in agreement with the wellknown result τ = 2`

√LC (it is the time required for the wave to go back and forth, then a length

2` needs to be considered). The port voltages are in Fig. 9. From Table 1, it can be claimedagain that the proposed method proves to be more accurate and significantly faster than thestandard rational one.

6 Conclusions

A new delay-rational model has been proposed for lossless and lossy MTL with frequency-independent per-unit-length parameters. It is an improved version of the rational method basedon the dyadic Green’s function of 1-D propagation along the line. The basic observation isthat the residues of a lossless N-conductor MTL can be gathered in N families sharing thesame magnitude, with the corresponding poles being equally spaced along the imaginary axis.When a lossy MTL is considered, poles and residues can be gathered as well, since they sharethe same negative real part (poles) and magnitude (residues) in the families, with the poles againequally spaced along the imaginary axis. This property does not hold for the low frequencypoles and the corresponding residues. Thus, mathematical manipulations are made in orderto restore the periodicity of the poles. A delay within each family can be easily identified,thus achieving a delay-rational impedance model with a compact state-space realization. Themain enhancement of the proposed method relies on the explicit delays extraction, that allowsto reduce the size of the rational macromodel. The high-frequency behavior of impedancesis preserved by using closed form expressions in the frequency domain whose time-domaincounterparts are Dirac combs, that explicitly take the delays into account. Numerical resultsconfirm the advantages of the new technique in terms of compactness of the models state-space,accuracy and speed-up, when compared to the standard rational approach. The extension ofthe method to the frequency-dependent per-unit-length parameters as well as the passivity ofthe model will be described in forthcoming reports.

92 PAPER B

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

0.12

Time [ns]

V1 [

V]

Rational

Delay−Rational

0 500 1000 1500 2000 2500 3000−0.05

0

0.05

0.1

0.15

0.2

Time [ns]

V2 [

V]

Rational

Delay−Rational

Figure 9: Port voltages for a cable of length `= 10 m, example 2 in Section 5.2.

REFERENCES 93

AcknowledgmentThe authors acknowledge Svenska Kraftnat (Swedish national grid) for providing funding forthis research.

References[1] C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed. New York, NY:

John Wiley & Sons, 2008.



[4] P. Maffezzoni, A. Brambilla, “Modelling Delay and Crosstalk in VLSI Interconnect forElectrical Simulation,” Electron. Letters, vol. 36, no. 10, pp. 862–864, May 2000.

[5] A. Charest, M. Nakhla, R. Achar, D. Saraswat, N. Soveiko, and I. Erdin, “Time domaindelay extraction-based macromodeling algorithm for long-delay networks,” IEEE Trans-actions on Advanced Packaging, vol. 33, no. 1, pp. 219–235, Feb 2010.

[6] P. Triverio, S. Grivet-Talocia, and A. Chinea, “Identification of highly efficient delay-rational macromodels of long interconnects from tabulated frequency data,” IEEE Trans-actions on Microwave Theory and Techniques, vol. 58, no. 3, pp. 566–577, March 2010.





[11] A. Morched, B. Gustavsen, and M. Tartibi, “A universal model for accurate calculation ofelectromagnetic transients on overhead lines and underground cables,” IEEE Transactionson Power Delivery, vol. 14, no. 3, pp. 1032–1038, Jul 1999.

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[12] T. Noda, “Application of frequency-partitioning fitting to the phase-domain frequency-dependent modeling of overhead transmission lines,” IEEE Transactions on Power De-livery, vol. PP, no. 99, pp. 1–1, to be published 2014.



[15] I. M. Elfadel, H-M. Huang, A. E. Ruehli, A. Dounavis, M. S. Nakhla, “A comparativestudy of two transient analysis algorithms for lossy transmission lines with frequency-dependent data,” in Digest of Electr. Perf. Electronic Packaging, Oct. 2001, pp. 255 –258.

[16] F. Y. Chang, “The generalized method of characteristics for waveform relaxation analy-sis of lossy coupled transmission lines,” IEEE Transactions on Microwave Theory andTechniques, vol. 37, no. 12, pp. 2028–2038, Dec. 1989.

[17] N. J. Nakhla, A. E. Ruehli, M. S. Nakhla and R. Achar, “Simulation of coupled inter-connects using waveform relaxation and transverse partitioning,” IEEE Transactions onAdvanced Packaging, vol. 29, no. 1, pp. 78–87, 2006.

[18] N. J. Nakhla, A. E. Ruehli, M. S. Nakhla, R. Achar and C. Chen, “Waveform relaxationtechniques for simulation of coupled interconnects with frequency-dependent parame-ters,” IEEE Transactions on Advanced Packaging, vol. 30, no. 2, pp. 257–269, 2007.

[19] J. Guo, Y.-Z. Xie, K.-J. Li, and F. Canavero, “Convergence analysis of the distributedanalytical representation and iterative technique (DARIT-Field) for the field coupling tomulticonductor transmission lines,” IEEE Transactions on Electromagnetic Compatibil-ity, to appear, 2014.


[21] R. Beerends, Fourier and Laplace Transforms, ser. Fourier and Laplace Transforms.Cambridge University Press, 2003.


[23] G. Miano and A. Maffucci, Transmission Lines and Lumped Circuits. Academic Press,2001.

95

[24] M. De Lauretis, J. Ekman, and G. Antonini, “Delayed impedance models of two-conductor transmission lines,” in Electromagnetic Compatibility (EMC Europe), 2014International Symposium on, Sept 2014, pp. 670–675.

[25] The MathWorks Inc., MatLab v8.4.0.150421 (R2014b), Natick, Massachusetts, 2014.

[26] A. Saini, M. Nakhla, and R. Achar, “Generalized time-domain adjoint sensitivity analysisof distributed MTL networks,” IEEE Transactions on Microwave Theory and Techniques,vol. 60, no. 11, pp. 3359–3368, Nov 2012.

[27] A. Ruehli, A. Cangellaris, and H.-M. Huang, “Three test problems for the comparison oflossy transmission line algorithms,” in Electrical Performance of Electronic Packaging,2002, Oct 2002, pp. 347–350.

PAPER C

Enhanced Delay-Rational Green’sMethod for Cable Time Domain

Analysis

Authors:Maria De Lauretis, Jonas Ekman, Giulio Antonini and Daniele Romano

Reformatted version of paper originally published in:Proceedings of IEEE, International Conference on Electromagnetics in Advanced Applications(ICEAA), Turin, Italy, 2015.


97

Enhanced Delay-Rational Green’s Method for Cable TimeDomain Analysis

Maria De Lauretis, Jonas Ekman, Giulio Antonini and Daniele Romano

Abstract

State-space models of multiconductor transmission lines can be generated by means of theGreen’s function based method which allows to write the open-end impedance in a rationalform as an infinite sum of “modal impedances”. It can be then embedded in a circuit sim-ulation environment for efficient time domain analysis. The previous rational approach hasbeen improved through a proper mathematical formulation, that makes use of explicit delayextraction and pole/residue asymptotic behavior. Nevertheless, the computation of the polesbecomes computationally expensive when the number of conductors increases, since the zerosof high order polynomials have to be evaluated. A rational fitting over the “modal impedances”is proposed, which allows a fast identification of the poles that, together with the delays, modelthe high frequency behavior of the cable in terms of standard hyperbolic functions. The low-frequency behavior is captured by a reduced size state-space model, via rational fitting. Nu-merical results confirm the accuracy of the proposed modeling approach for electrically longcables, with a large number of conductors.

1 IntroductionThe increasing in the switching frequencies in power electronic applications is the main re-sponsible for noise and electromagnetic interference (EMI), that are carried mainly throughthe cable, especially in terms of conducted emission noise [1]. While the well-establishedmulticonductor transmission line (MTL) theory [2] is used for cable frequency analysis, theresearch community is still focused on efficient time-domain models which can be more easilyintegrated in circuit simulators, such as SPICE [3], where non-linear power electronics devicesare efficiently modeled. The Green’s function based method proposed in [4] relies on the eval-uation of the exact poles and residues of the open-end impedance matrix, expressed as a sumof infinite rational transfer functions, denoted as “modal impedances”. A time-domain state-space model is easily gained. Although general, this approach may lead to large state-spacemodels when an electrically long cable is considered, since a significant number of poles isrequired to achieve a good accuracy. In [5, 6, 7], a delayed impedance model has been pro-posed to overcome such limitation, by using an explicit delay extraction and the analysis ofthe pole/residue asymptotic behavior. Nevertheless, when the number of conductors increases,such an approach may suffer in terms of performances, since the computation of the poles re-quires the inversion of a polynomial matrix, that is computationally expensive. In this paper,in order to speed-up the poles identification, the vector fitting algorithm [8] is applied to each

99

100 PAPER C

“modal impedance”; in fact, it allows an easier computation of the asymptotic poles/residuesthat, together with the delays, are used to model the high frequency behavior of the cable interms of standard hyperbolic functions. The low-frequency behavior of the cable is captured bya reduced size state-space model which can be easily extracted by resorting to a rational fitting.The final delay-rational model allows computing the time-domain response of a generic cablebundle by solving a set of delay differential equations with delayed sources. Simulation resultsare presented in order to validate the proposed approach.

2 Green’s function based methodsThe open-end impedance representation of a MTL of length ` reads [2][

V0(s)V`(s)

]=

[Z11(s) Z12(s)Z12(s) Z11(s)

]︸︷︷︸

Z(s)

[I0(s)I`(s)

](1)

where Z(s) is block symmetrical, with each block of dimension N×N, being N the number ofconductors.

2.1 Standard Green’s function based methodThe dyadic Green’s function method for the analysis of N-conductor MTLs, treated as a 2Nport system, has been proposed in [4], where the impedance matrix Z(s) is expressed as aninfinite sum of matrices of rational functions as follows

Z11(s) =+∞

∑m=0

Fm(s)−1(s)A2mZ′(s) (2a)

Z12(s) =+∞

∑m=0

(−1)mFm(s)−1(s)A2mZ′(s) (2b)

Fm(s) = Z′(s)Y′(s)+(mπ

`

)2I (2c)

with Z′(s) = R′+ sL′ the p.u.l. longitudinal impedance and Y′(s) = G′+ sC′ the p.u.l. trans-verse admittance. The computation of poles and residues requires the inversion of the polyno-mial matrix Fm(s) defined in (2). All the sub-blocks of Z(s) share the same poles which canbe computed as the zeros of the determinant of Fm(s). We remind that the number of polesis N for the zero mode m = 0, and 2N poles for higher order modes m > 0. Once the polespn,n = 1, · · · ,+∞ are identified, the residues Rn can be computed [4]. Finally, the open-endimpedance matrix Z(s) is expressed as follows

Z(s) =∞

∑n=1

Rn

s− pn. (3)

2. GREEN’S FUNCTION BASED METHODS 101

This rational model is well suited to be translated into a state-space form [9], but two maindrawbacks can be identified:

1. electrically long lines may require a large number of modes to achieve a good accuracy,resulting in a large state-space model;

2. the computation of poles for each mode m requires the inversion of the polynomial matrixFm(s), which becomes computationally expensive for a large number of conductors.

2.2 Delayed Green’s function based methodThe first drawback has been addressed in [5, 6] for the frequency-independent p.u.l. parametercase, and in [7] for the frequency-dependent one. The number of modes can be drasticallyreduced by explicit delay extraction, along with the poles/residues asymptotic behavior. Morespecifically, it has been observed that poles obtained by the standard Green’s function basedapproach are clustered in N groups, called “families”, identified by their real part. Withineach family, the poles become equispaced along the imaginary axis when the phase of thecorresponding residues tends to zero. A truncation mode m can be found accordingly to thefollowing two criteria:

• the pole families are identified by their real part;

• the residue families are identified by their phase.

The two conditions that need to be satisfied are∠Rk,m−∠Rk,m−1

∠Rk,2−∠Rk,1< tolR (4a)

αk,m−αk,m−1

αk,m> tolα . (4b)

where tolR and tolα are suitable tolerance values.Once the families have been properly identified, the infinite summation can be then truncatedusing the following hyperbolic functions

Zh11(s) = ∑Nk=1

RkTk2 coth

((s− αk)

Tk2

)(5)

Zh12(s) = ∑Nk=1

RkTk2 csch

((s− αk)

Tk2

). (6)

These expressions can be translated into the time domain in terms of attenuated Dirac combs.Poles and residues that do not belong to the identified families, pn and Rn respectively,

along with auxiliary poles and residues, pn and Rn respectively, as detailed in [6], contributeto the rational part of the model as follows

Zi j(s) =np

∑n=1

Ri j,n

s− pn−

m−1

∑m=0

(N

∑k=1

Ri j,k

s− pk,m

)(7)

where i, j = 1,2 identify the blocks in (2) and np identifies the number of poles for m = 0 tom−1.

102 PAPER C

3 Proposed solution for cable bundles

As pointed out before, the computation of poles becomes computationally time demandingand also prone to numerical inaccuracies when the number of conductors increases, as in thecase of cable bundles. This section aims to present a strategy to overcome this limitation. Thepoles computation can be more efficiently performed by using the vector fitting (VF) algorithm[8, 10]. In fact, the direct application of the VF to the open-end impedance matrix samples overthe frequency range of interest would not be efficient either, because it would require to searchall the poles at once, resulting in a more difficult identification of the pole families, crucial inorder to restore the hyperbolic functions.

3.1 Rational approximation of “modal impedances”

Instead, taking advantage of the modal decomposition provided by the Green’s function basedmethod (2), the rational fitting procedure can be adopted to approximate each “modal impedan-ce”, completely avoiding the inversion of the polynomial matrix in (2), thus leading to theexplicit computation of poles and residues. Furthermore, the number of poles to be used in therational fitting of each “modal impedance” is known. In the frequency independent per-unit-length parameter case, they are N for the mode zero m = 0, and 2N for m > 0. A schematic ofthe general algorithm is proposed in (1). Because of the symmetric block structure of Z(s) , it iseasy to observe that all the sub-blocks share the same poles, and that the residues are the sameexcept for the multiplication for (−1)m for the off-diagonal blocks. This allows evaluating onlythe modal frequency response for the N ×N block Z11|m. From the frequency independentcase theory [6], it is known that the 2N poles of each mode m divide into N families, eachfamily sharing the same real part αk, at least from a certain mode order m on. When poles arecomputed using a rational approximation as VF, the poles can have a different behavior, and itis indeed expected since we are relying on a numeric technique. Nevertheless, the same mainproperties are retained. In particular, the poles gather again into families in number equal tothe number of conductors. In Fig. 2, the poles for a N = 6 lossy ribbon cable are plotted inthe complex plane, and indeed, 6 families can be easily recognized. The family identificationis performed as outlined in Section 2.2.

3.2 Time delay extraction

As observed in previous reports, the time-delay used in the final delay-state-space model canbe computed by analyzing the lossless line. It is important to observe that the delays usedin the frequency independent case are indeed coincident with the modal delays defined asTk = 2`

√Λk, where Λ are the eigenvalues of the product matrix C′L′ [11], and k = 1, . . . ,N. It

is worth to notice that we always use, in our formulation, the two-way delay. The βk,1 can becomputed as βk,1 =

2π

Tk.

3. PROPOSED SOLUTION FOR CABLE BUNDLES 103

N_mod = M;

STOP = FALSE;

m == N_mode

% Port impedance evaluation for the mode m

% Vector fitting in order to identify poles and

residues

VF

END

m = m + 1;

yes

no

% Evaluation of the asymptotic values for

poles and residues, and

yes

no

Figure 1: Block diagram for the proposed algorithm.

104 PAPER C

ℜ ×104

-4 -3.5 -3 -2.5 -2 -1.5 -1

ℑ×10

11

-1

-0.5

0

0.5

1poles vf

Figure 2: Poles location computed using a modal rational approximation approach, for a 6 conductorribbon cable.

3.3 State-space modelIn order to generate the minimal size state-space model, it can be observed that the rationalexpression in (7) has one main drawback: even if a small value of m− 1 can be found, thenumber of poles is double, since two summation in m− 1 have to be used. We notice thatthe rational part (7) actually represents a delay-less impedance. For this reason, it can befitted using a reduced number of poles. The vector fitting algorithm can be used again to thisaim. The double summation (7) would require Np = 2(N + 2N(m− 1)) poles which can besignificantly reduced by the direct application of the vector fitting algorithm to the delay-lessimpedance, computed as

Z11−dl(s) = Z11(s)−N

∑k=1

RkTk

2coth

((s− αk)

Tk

2

)(8a)

Z12−dl(s) = Z12(s)−N

∑k=1

RkTk

2csch

((s− αk)

Tk

2

). (8b)

4 Numerical ExperimentsIn order to evaluate the accuracy of the proposed approach, the cable bundle with 9 conductorsis considered [12], of length ` = 20 m. The p.u.l. parameters are those reported in [12]. Thefirst conductor is excited by a smooth pulse voltage (amplitude 1 v, rise and fall time 20 ns,


pulsewidth 50 ns and initial delay 30 ns); all the ports are terminated on 50-Ω resistances. Theproblem has been analyzed by using the standard theory of MTL and by using the proposedenhancement of the Delay-Green’s function approach (Delay-GVF). All the code has beendeveloped in MATLAB [13]. For the Delay-GVF method, the number of modes m = 7 isautomatically chosen by the algorithm. The poles computed for the delay extraction (chosenby the vector fitting procedure), in the complex plane, are shown in Fig. 3 where the 9 familiesof poles are clearly identified.

ℜ ×104

-7 -6 -5 -4 -3 -2

ℑ

×108

-2

-1.5

-1

-0.5

0

0.5

1

1.5

poles vf

p

Figure 3: Poles location for the 9 conductor cable bundle.

Furthermore, the low frequency behavior of the cable has been captured using only 20 poles,resulting in a state-space model of order 360. Since in this case m= 7 has been found, using theprevious approach to set the state-space model would have led to a total number of poles equalto Pn = 2(N +2N(m−1)) = 234, with the final state-space model of size 2 ·N ·Pn = 4212.

The transient voltages at port 1 and 14 are shown in Fig. 4 where IFFT denote the solutionobtained by using the standard theory of MTL in the frequency domain, transformed in thetime domain by using the Inverse Fast Fourier Transform algorithm. Finally, the average norm2 error between the solution computed by the standard theory of MTL and the Delay-GVFmethod is 1.08 ·10−3, where the error, for a single frequency sample, is computed as

||Z−ZGreen−DV F ||2||Z||2

(9)

where Z and ZGreen−DV F denote the impedance computed by using the standard theory of MTLand by using the Green-DVF method, respectively.

106 PAPER C

0 200 400 600 800 1000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [ns]

V1 [

V]

Delay−GVF

ifft

0 200 400 600 800 1000−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [ns]

V1

4 [

V]

Delay−GVF

ifft

Figure 4: Transient voltages at port 1 and 14 for the 9 conductor cable bundle.

5 Conclusions

In this paper, an enhancement of the delay Green’s function based method for the analysisof electrically long cables is presented. The identification of the asymptotic poles/residuesbehavior, along with the explicit delay extraction, allow the decomposition of the open-end

REFERENCES 107

matrix in a rational and a hyperbolic part, leading to a state-space model and delayed sources,respectively. The hyperbolic part extraction requires the computation of poles, which waspreviously performed by computing the determinant of a polynomial matrix of order 2N, whitN the number of conductors. This method becomes intractable when the number of conductorexceeds 6. This limitation has been overcome by applying a rational fitting procedure to the”modal impedances” used in the Green’s function based approach. The vector fitting algorithmallows a fast computation of poles and residues and, as a consequence, a fast identificationof the hyperbolic functions used to model the high frequency part of the spectrum The lowfrequency part can be efficiently represented using again the vector fitting algorithm, with asignificant compression of the state-space model. A pertinent example has been presented, thatproves the capability of the proposed method to efficiently model electrically long cables witha large number of conductors.

AcknowledgmentThe authors acknowledge Svenska Kraftnat (Swedish national grid) for providing funding forthis research.

References[1] S. Skibin, B. Wunsch, I. Stevanovic, and B. Gustavsen, “High frequency cable models

for system level simulations in power electronics applications,” in Electromagnetic Com-patibility (EMC EUROPE), 2012 International Symposium on, Sept 2012, pp. 1–6.

[2] C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed. New York, NY:John Wiley & Sons, 2008.

[3] L. W. Nagel, “SPICE: A computer program to simulate semiconductor circuits,” Univer-sity of California, Berkeley, Electr. Res. Lab. Report ERL M520, May 1975.


[5] M. De Lauretis, G. Antonini, and J. Ekman, “Delayed impedance models of two-conductor transmission lines,” in Proc. Int. Symp. Electromagn. Compat., Sep. 2014,,Sept 2014, pp. 670–675.

[6] ——, “A delay-rational model of lossy multiconductor transmission lines with frequency-independent per-unit-length parameters,” IEEE Trans. Electromagn. Compat., vol. PP,no. 99, pp. 1–11, 2015.

[7] ——, “Delay-rational model of lossy and dispersive multiconductor transmission lines,”in Proc. Int. Symp. Electromagn. Compat., Aug. 2015, accepted.

108

[8] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responsesby vector fitting,” IEEE Transactions on Power Apparatus and Systems, vol. 14, no. 3,pp. 1052–1061, Jul. 1999.


[10] B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans.Power Del., vol. 21, no. 3, pp. 1587–1592, 2006.


[12] G. Andrieu and A. Reineix, “The “equivalent cable bundle method”: A solution to modelcable bundles in presence of complex ground structures,” in Electromagnetic Compatibil-ity (EMC EUROPE), 2013 International Symposium on, Sept 2013, pp. 259–263.

[13] “Matlab User’s Guide,,” The Mathworks, Inc., Natick, 2001.

PAPER D

Delay-Rational Model of Lossyand Dispersive Multiconductor

Transmission Lines


Reformatted version of paper originally published in:Proceedings of IEEE, International Symposium on Electromagnetic Compatibility (EMC),Dresden, Germany, 2015.


109

Delay-Rational Model of Lossy and DispersiveMulticonductor Transmission Lines


Abstract

In this paper, the transient analysis of lossy and dispersive multiconductor transmission linesis considered. The existing Green’s function-based method is extended to explicitly includedelays extraction, thus leading to a significant compressed time-domain state-space model. Theproposed method is mainly based on poles and residues asymptotic analysis and lossless delaysextraction. The resulting hybrid state-space model incorporates Dirac-combs in the input andresults into a reduced number of state variables. A test case has been considered in order toclearly demonstrate the effectiveness of the proposed methodology. The results are comparedwith the original rational Green’s function method, and with the standard inverse fast Fouriertransform technique.

1 Introduction

Transmission-line structures allow to guide electromagnetic waves between two or more points.Multiconductor transmission lines (MTLs) consist of more than two conductors (reference in-cluded), and are normally found in both signal and power applications (interconnects in printedcircuit boards (PCB) or chips, power transmission lines, cables). Multiconductor transmissionlines are typically modeled using Telegrapher’s equations assuming a quasi-TEM field structure(see [1, 2, 3] and references therein). The modeling of MTLs in the frequency domain is well-established in literature, while the research is still focused on efficient time-domain models,especially in case of electrically long MTLs. In fact, the presence of non-linear terminationssuch as drivers and receivers in PCBs or non-linear protection devices in power transmissionlines requires time-domain models, since the use of frequency-domain ones is not effective.Efficient transient MTLs models are then strictly needed. Standard approaches for transientanalysis adopt the Fast Fourier Transform (FFT), or the numerical inverse Laplace transform(NILT). These are normally computationally expensive; for example, the numerical convolu-tion product in the time domain with non-linear terminations is a well-known computationalbottleneck. Several other methods nowadays available relay in rational function approxima-tions, that lead to rational models in terms of poles and residues. For example, techniquesbased on the Pade approach [4, 5], or on the well-known Method of Characteristic [6] and itsgeneralizations [7, 8, 9]. Rational models may not be efficient, however, for electrically longMTLs, mainly due to the huge number of poles required to properly represent the propaga-tion line delays. A rigorous rational model of a MTL has been proposed in [10], where the

111

112 PAPER D

impedance port matrix is expressed in terms of the dyadic Green’s function of the 1-D prop-agation problem. A rational series form is presented, where poles are identified by solvingalgebraic equations and the corresponding residues are easily computed. Although general,such an approach may lead to state-space models with high model-order when an electricallylong MTL is considered, since a significant number of modes, and thus poles, is necessary toachieve a good accuracy, especially in terms of ringing effects in the time domain. In orderto overcome this drawback, the method in [10] has been improved for frequency-independentper-unit-length (p.u.l.) parameters single conductor TLs by the authors in [11], thanks to theexplicit delay extraction. The delay-rational model obtained has a reduced size and better ac-curacy in the time domain. Recently, the method has been further extended to MTLs withfrequency-independent p.u.l. parameters in [12]. The aim of the present paper is the general-ization of the model in [12], proposed for frequency-independent p.u.l. parameters MTLs, toMTLs with frequency-dependent per-unit-length parameters. Hence, a delay-rational Green’smethod is proposed, that consists of the following main steps:

• poles and residues computation of the impedance matrix as described in [10] (Section2);

• analysis of poles and residues in order to identify and exploit periodical behaviors (Sec-tions 3.1 and 3.2);

• poles and residues pruning in order to retain only the dominant ones (Sec. 3.4);

• delayed state-space model (Sec. 4).

As a result, delays are explicitly identified and a model-order reduction of the MTL macro-model is obtained without resorting to moment-matching techniques.

2 Green’s function-based method backgroundA generic MTL can be represented as a multiport network [3] as in Fig. 1. At the genericabscissa z the propagation equations, known as Telegrapher’s equations, in the Laplace domainand under the hypothesis that only the quasi-transverse electromagnetic field mode propagates,read [3]

∂zV(z,s) = −Z′(s)I(s,z) (1a)

∂zI(z,s) = −Y′(s)V(s,z)+ Is(z,s) (1b)

where Z′(s) = R(s) + sL(s) is the per-unit-length (p.u.l.) longitudinal impedance, Y′(s) =G(s)+sC(s) is the p.u.l. transverse admittance, with R(s), L(s), C(s) and G(s) the frequency-dependent p.u.l. parameter matrices, non-negative definite symmetric of order N, being N +1the number of the conductors with the reference conductor included [3, 13]. Is(z,s) representsa p.u.l. current source located at abscissa z. V(z,s) and I(z,s) represent the voltage and current

2. GREEN’S FUNCTION-BASED METHOD BACKGROUND 113

V1(s)

V2(s)

VN(s)

VN+1(s)

VN+2(s)

V2N(s)

- -

+

+

+ +

+

+

MTL

z= 0 z= l

I1(s)

I2(s)

IN(s)

IN+1(s)

IN+2(s)

I2N(s)

Figure 1: Multiconductor transmission line represented as 2N port system, with a common referenceconductor. Vi for i = 1, ...,2N are the voltages between each conductor and the ground.

vectors. In [10], the general solution for the end voltages of a transmission line of length ` interms of port currents is given as[

V0(s)V`(s)

]=

[Z11(s) Z12(s)Z12(s) Z11(s)

][I0(s)I`(s)

]= Z(s)

[I0(s)I`(s)

](2)

where Z(s) is the port impedance matrix, that is block symmetrical, each block of dimensionN×N. V0(s) and I0(s) are the voltage and the current port vectors respectively, of dimensionN× 1, related to the input ports at z = 0. V`(s) and I`(s) are the voltage and the current portvectors respectively, of dimension N×1, related to the output ports at z = `.

In [10], it has been proven that Z(s) can be expressed by means of a rational approxima-tion, in a poles/residues form. In particular, a series form of the Dyadic Green’s Function isgiven, and the port impedance can be expressed by means of eigenvalues and eigenfunctions,defined for a summation index, or summation mode, m = 0, ...,+∞. When performing a nu-merical computation, the infinite summation is truncated to M modes. Assuming Z′(s) andY′(s) available at N f discrete frequency samples sq = j2π fq, with q = 1, ...,N f , the vectorfitting technique [14] is applied, and a rational polynomial form is obtained. By using theGreen’s function method developed in [10], the Z(s) can be expressed as an infinite sum ofpoles and residues. In fact, for each mode m, a number n = 1, ..., np of poles and residues canbe evaluated, leading to a final number of poles equal to

np =M

∑m=0

np,m . (3)

The nd dominant poles p and residues R are then selected [10], leading to the final expression

Z(s) =nd

∑n=1

Rn

s− pn. (4)

114 PAPER D

Rational polynomial forms of and Vector fitting

Green’s function based methodPoles and residues form

Poles and residues pruningDominant poles and residues selected

Longitunal impedance and transversal admittance

Available at discrete frequency samples

Figure 2: Green’s function-based method for MTLs with frequency-dependent per-unit-length parame-ters.

It is worth noting that, in order to simplify the notation, it is not specified whether the polesare real or complex conjugate. The poles/residues representation is suited to be translated in astate space form as described in [2]

x(t) = Ax(t)+Bi(t) (5)

v(t) = Cx(t) .

The main steps of the results in [10] are proposed in a convenient schematic in Fig. 2.

2.1 Explicit delay extractionDespite the pole/residue pruning adopted, the number of summation modes required in orderto model the frequency response of electrically long lines can still be significantly large. Thisis a problem especially for time domain simulations, since the model order of (5) is equal to2Nnp, where np is the number of poles, and 2N = Np is the number of ports. In order toreduced the size of the model, a new delay-rational macromodel for frequency-independentp.u.l. parameter N +1 = 2 conductor TLs has been proposed in [11]. In this case, we have one

2. GREEN’S FUNCTION-BASED METHOD BACKGROUND 115

complex conjugate pair, for each mode m> 0, and one real pole for m= 0. The main advantagerelies in the explicit delay extraction. The procedure is briefly summarized in its main steps:

• in the lossless case, the periodicity of the purely imaginary poles pm = ± j2πmT , for

m = 1, ...,+∞ allows the explicit extraction of the time delay T ;

• in the lossy case, the complex conjugate poles pm(1,2) = α ± jβ , for m = 1, ...,+∞,become also periodic in their imaginary parts after a certain summation mode m;

• the lossless delay can be used in order to gain a closed expression in terms of hyperbolic

functions in the frequency domain as in (6), where η =√

LC , and np is the total number

of poles

Z11(s) =np

∑n=1

Rn

s− pn−

np

∑n=1

Rn

s− pn+η coth

((s−α)

T2

)(6a)

Z12(s) =np

∑n=1

Rn

s− pn−

np

∑n=1

Rn

s− pn+η csch

((s−α)

T2

); (6b)

• it is possible to prove that the hyperbolic functions lead to a Dirac comb formulation inthe time domain [12, 11]

η coth((s−α)

T2

)→ ηeαt

δ (t)+2ηeαt+∞

∑m=1

δ (t−mT ) (7)

η csch((s−α)

T2

)→ 2ηeαt

+∞

∑m=0

δ

(t−mT − T

2

). (8)

The Dirac-comb formulation allows to avoid the convolution product. A delay-rational modelin the time domain is gained as

x(t) = Ax(t)+Bi(t) (9a)

v(t) = Cx(t)+D0i(t)+ id(t) (9b)

where id(t) = D(t)⊗ i(t). The D0 is defined as

D0 =

[η 00 η

]. (10)

It is worth noting that the convolution δ (t)⊗ i(t) = i(t), and then it is omitted in the state-spacemodel formulation. The id(t) = D(t)⊗ i(t) represents the delayed currents contributions. Inparticular, D(t) acts as a delay operator on the port currents i(t), then it is symmetric in theform

D(t) =[

D11 D12D12 D11

](11)

116 PAPER D

with

D11 = 2ηeαt+∞

∑m=1

δ (t−mT ) (12a)

D12 = 2ηeαt+∞

∑m=0

δ

(t−mT − T

2

)(12b)

where the infinite summation will be truncated to Tw = max(t)T , whit t the time window under

analysis. Note that the convolution product D(t)⊗ i(t) returns only an attenuated and translatedversion of the currents, so no convolutions products are actually performed in the time-domainsolver.

Recently, the method has been extended by the authors to MTLs with frequency-independentp.u.l. parameters [12]. The main steps are below summarized.

1. Given the p.u.l. longitudinal impedance Z′ and transverse admittance Y′, the Green’sfunction method is applied. Each sub-block Zi j for i, j = 1,2 in (2) is expressed as sumof poles and residues

Zi, j =∞

∑n=1

Rn

s− pn, for i, j = 1,2 . (13a)

2. The complex conjugate poles are divided into N families, each family identified from itsreal part, as better visually explained in Fig. 3. Each mode m > 0 exhibits N complexconjugate pair of poles. The mode m = 0 normally exhibits real poles in number equalto N. For m ≥ m, the complex conjugate poles exhibit a periodic behavior inside eachfamily, meaning they are equi-spaced along the imaginary axis

pk,m(1,2) = αk± jmβk, for m = m, ...,+∞. (14a)

Their corresponding residue matrices Rk are real and constant for each family, since theirphase tends asymptotically to zero.

3. Because of the periodicity for m = m, ...,+∞, the lossless delay Tk can be extracted foreach family of poles/residues, and incorporated in the frequency model through hyper-bolic functions

Z11(s) =np

∑n=1

Rn

s− pn−

m−1

∑m=0

(N

∑k=1

Rk

s− pk,m

)+

N

∑k=1

RkTk

2coth

((s−αk)

Tk

2

)(15a)

Z12(s) =np

∑n=1

Rn

s− pn−

m−1

∑m=0

(N

∑k=1

Rk

s− pk,m

)+

N

∑k=1

RkTk

2csch

((s−αk)

Tk

2

). (15b)

It is worth observing that, in order to simplify the notation, it is not specified whether the polesare real or complex conjugate. Note that (15) is the natural extension to the multiconductorcase of the expressions in (6). The state space time domain model reads as (9), with D0 andD(t) opportunely redefined.

3. DELAY-RATIONAL GREEN’S METHOD FOR MTL WITH FREQUENCY-DEPENDENT

P.U.L. PARAMETERS 117

ℜ (pk,n

) ×105

-5 -4.5 -4 -3.5 -3 -2.5

ℑ(p

k,n

)

×109

-6

-4

-2

0

2

4

6

Family1

Family2

Figure 3: Poles locus in the complex place for a N = 2 conductor lossy frequency-independent p.u.l.parameters transmission line. N families are clearly identified, based on the real parts αk, for k =1, ...,N.

3 Delay-Rational Green’s Method for MTL with frequency-dependent p.u.l. parameters

The delay-rational model in [12] and summarized in the previous section is now extendedto MTLs with frequency-dependent p.u.l. parameters. Basically, starting with the originalapproach in [10], the considerations regarding the periodicity of poles and residues made in[11] and [12] have been opportunely extended and adapted for the frequency-dependent p.u.l.case. For the sake of simplicity, we will refer to a practical example in order to better explainthe mathematical results. In particular, it is considered the N + 1 = 3 conductor transmissionline given as “Line 2” in [15], with frequency-dependent p.u.l. parameters as given in [15] andlength ` = 10 cm. The p.u.l. matrices R(s), L(s), G(s), and L(s) are available at N f discretefrequency points sq = j2π fq, with q = 1, ...,N f . As summarized in Fig. 2, the vector fittingtechnique is applied [16, 17], and Z′(s) and Y′(s) are expressed in a rational polynomial form.Normally, a reduced number of poles is sufficient to capture their frequency behavior. Denotingwith Pz and Py the number of poles required for Z′(s) and Y′(s) respectively, Pz = Py = 3 areused for the Line 2 example. The Green’s method for MTLs with frequency-dependent p.u.l.parameters in [10] can be adopted, gaining a poles/residues expression for the port impedancematrix Z(s). The summation is truncated for m = M, with M normally high. Each mode has anumber of poles equal to (Pz +Py +2)N [10]. For the Line 2 example, M = 400 has been usedto gain a good accuracy in the frequency domain.

At this point, in the original rational Green’s method a model order reduction is performed

118 PAPER D

ℜ (p) ×109

-2.5 -2 -1.5 -1 -0.5 0

ℑ (

p)

×1012

-2

-1

0

1

2Family 1

Family 2

Figure 4: Complex conjugate poles families in the complex plane, for the example Line 2 in [15].

by aid of poles and residues pruning [10]. Conversely, in this work the poles and residues areanalyzed, with two main goals:

• reduce M by considering the asymptotic behavior of poles and residues;

• incorporate the lossless extracted delay for each conductor.

3.1 Poles asymptotic behavior analysisWe observe that, similarly to the frequency-independent p.u.l. parameters case studied in [12],the complex conjugate poles can be gathered in k families equal to the number of conductors,it means k = N, as it can be seen in Fig. 4. In this case though, the families do not share astatic real part αk, but the real part changes with the summation mode αk,m. It then reaches anasymptotic value after a certain mode m≥ m

αk,m −→n≥m αk . (16)

For m = m, ...,+∞, the poles are approximated using

• the real asymptotic value αk,

• the imaginary part of the poles for the transmission line made lossless, with the p.u.l.parameters evaluated for the last available frequency sample.

In fact, as observed in [12] for the frequency-independent p.u.l. parameters case, also forthe frequency-dependent one the imaginary parts βk,m of the poles become equispaced for

3. DELAY-RATIONAL GREEN’S METHOD FOR MTL WITH FREQUENCY-DEPENDENT

P.U.L. PARAMETERS 119

Mode

0 100 200 300 400

ℜ R

es

11

×1010

3.55

3.6

3.65

3.7

3.75Family 1

Family 2

Figure 5: Real part of the residue Rk,n, for the example Line 2 in [15].

m = m, ...,+∞. Denoting by βk,n the fictitious imaginary parts equispaced for m = 1, ...,+∞,the periodicity allows us to write the imaginary parts as a linear function in n, it means βk,m =

mβk,1. The lossless delay associated with each family can be then expressed as

Tk =2π

βk,1, for k = 1, ...,N . (17)

The asymptotic poles used will be

pk,m = αk± jmβk,1. (18)

3.2 Residues asymptotic behavior analysis

As in the frequency-independent case, the imaginary part of Rk,m tends to zero for m−→+∞,so does the phase, as in Fig. 6. For this reason, the residues for each k family are approximatedwith the asymptotic real value reached after a certain mode of the summation m, as shown inFig. 5.

3.3 Identification of truncation mode m

The value of m can be selected considering both the real parts of the poles, and the phase ofthe residues. Given a certain tolerance value tolRes, each element of Rk,m has to satisfy thecondition (19). Regarding the poles, given tolRe, the condition (20) has to be satisfied. The

120 PAPER D

Mode

0 100 200 300 400

(R

es

11)[

rad]

×10-3

-4

-2

0

2

4

6Family 1

Family 2

Figure 6: Phase of the residue Rk,n, for the example Line 2 in [15], that is a p.u.l. frequency-dependentp.u.l. parameters transmission line with N +1 = 3 conductors.

value of m is then chosen such that

∠Rk,m−∠Rk,m−1

∠Rk,2−∠Rk,1< tolRes (19)

andαk,m−αk,m−1

αk,m> tolRe. (20)

We note that, since the high frequency behavior will be covered thanks to the extracted losslessdelay, a low value of tolerances can be used. Note that, for m ≥ m, Rk,m = Rk is a constantvalue, for k = 1, ...,N. For the Line 2 example, we set the tolerances as tolRes = 10−1 andtolRe = 10−3, leading to a number of modes m = 88, with a total number of poles equal to1416.

3.4 Reduced-order model via poles/residues pruningEven though the infinite summation is truncated to m M, still a pole/residue pruning isnecessary, as already observed in [10], following the guidelines in [18]. After the poles/residuespruning, for the Line 2 example, the number of poles decreases from 1416 to 354. It is worthto note that the number of poles for each mode changes, and it is not known a priori. For aintuitive comparison, the relative error can be observed: as it can be seen in Fig. 7, the accuracyof the delay-rational proposed method is comparable with the previous rational one but it issignificantly less demanding in terms of memory storage and computational complexity.

4. DELAY-RATIONAL MODEL IN THE TIME-DOMAIN 121

Freq [Hz] ×109

0 2 4 6 8 10

Rela

tive e

rror

for

Z2

4

10-4

10-3

10-2

10-1

100

Z12block

Delay-Rational

Rational

Figure 7: Relative error of the rational and the delay-rational methods, compared to the frequencyresponse as per classical MTL theory. The Z24(s) element of the port impedance, related to its Z12(s)block, is considered.

4 Delay-Rational model in the time-domainThe final Delay-Rational state space model reads as in (9), with D0 and D(t) redefined as

D0 =N

∑k=1

[Rk

Tk2 0

0 RkTk2

]. (21)

D(t) =N

∑k=1

[D11 D12D12 D11

](22)

with

D11 = RkTkeαkt+∞

∑m=1

δ(t−mTk

)(23a)

D12 = RkTkeαkt+∞

∑m=0

δ

(t−mTk−

Tk

2

). (23b)

5 Numerical ResultsThe proposed method has been validated through simulations with Matlab R2015a [19], on acomputer equipped with 64-bit Windows 7 O.S., Intel Core Xeon @2.27 GHz (2 processors)

122 PAPER D

Table 1: Comparative results between the Rational and the Delay-Rational methods.

Method Number of poles Model-order (state-space form) cpu-time [s]

Rational 6408 25632 7484

Delay-Rational 354 2816 484

Time [ns]

0 5 10 15 20 25 30

V1 [V

]

-0.1

0

0.1

0.2

0.3

0.4

Rational

Delay-Rational

IFFT

Figure 8: Transient port voltage V1 (input port of the first conductor, as per notation in Fig. 1) with the“Rational”, “Delay-Rational”, and IFFT methods.

and 48 GB of RAM. The Line 2 example so far discussed is excited by a smooth pulse volt-age of amplitude 1 V, rise and fall times tr = t f = 50 ps and pulse-width τ = 2 ns. All theports are terminated on 250 Ω resistances. The standard Green’s function method, referred as“Rational”, the proposed extension that will be referred as “Delay-Rational”, and the standardInverse-Fast-Fourier Transform (IFFT) technique are compared, and the computational timesbetween the Rational and the Delay-Rational methods are summarized in Tab. 1. The extractedlossless delays are T1 = 1.5777 ns and T2 = 1.6405 ns. The port voltages V1 (input port for thefirst conductor) and V4 (output port of the second conductor) are shown in Figs. 8 and 9. Asatisfactory agreement is achieved.

6 Conclusions

In this work, a delay-rational model of MTLs with frequency-dependent p.u.l. parameters hasbeen proposed. Starting with the Green’s function method, the approach is based on asymptoticanalysis of poles and residues, and lossless delays extraction. The reduced number of polesrequired by the new formulation leads to a reduced model-order of the state-space form. The

REFERENCES 123

Time [ns]

0 5 10 15 20 25 30

V4 [V

]

-0.05

0

0.05

Rational

Delay-Rational

IFFT

Figure 9: Transient port voltage V4 (output port for the second conductor, as per notation in Fig. 1)with the “Rational”, “Delay-Rational”, and IFFT methods.

numerical example has shown the capability of the proposed method to significantly compressthe size of the model when compared to the standard rational one, leading to a significantspeed-up in time-domain simulations.

Acknowledgments

The authors acknowledge Svenska Kraftnat (Swedish national grid) for funding this research.

References



[3] C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed. New York, NY:John Wiley & Sons, 2008.


124 PAPER D







[11] M. De Lauretis, G. Antonini, and J. Ekman, “Delayed impedance models of two-conductor transmission lines,” in Electromagnetic Compatibility (EMC Europe), 2014International Symposium on, Sept 2014, pp. 670–675.

[12] ——, “A delay-rational model of lossy multiconductor transmission lines with frequencyindependent per-unit-length parameters,” IEEE Transactions on Electromagnetic Com-patibility, in press.


[14] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responsesby vector fitting,” IEEE Transactions on Power Apparatus and Systems, vol. 14, no. 3,pp. 1052–1061, Jul. 1999.

[15] A. Ruehli, A. Cangellaris, and H.-M. Huang, “Three test problems for the comparison oflossy transmission line algorithms,” in Electrical Performance of Electronic Packaging,2002, Oct 2002, pp. 347–350.

[16] B. Gustavsen, “Computer code for rational approximation of frequency dependent admit-tance matrix,” IEEE Transactions on Power Delivery, vol. 17, no. 4, pp. 97–104, Oct.2002.

REFERENCES 125

[17] ——, “Fast passivity enforcement for pole-residue models by perturbation of residuematrix eigenvalues,” IEEE Trans. Power Delivery, vol. 23, no. 4, pp. 2278–2285, Oct.2008.

[18] G. Antonini, “A new methodology for the transient analysis of lossy and dispersive multi-conductor transmission lines,” IEEE Transactions on Microwave Theory and Techniques,vol. 52, no. 9, pp. 2227–2239, Sep. 2004.

[19] The MathWorks Inc., MatLab Release R2015a, Natick, Massachusetts, 2015.

Errata Corrige

Chapter 31. Page 15, the second paragraph up to the formula (3.1) included

Regarding the common cables found in industrial applications, they can be treated as trans-mission lines when the propagation delay TD of a traveling signal is large compared with thephysical length ` of the cable, that is,

TD ` (3.10)

must be substituted withRegarding the common cables found in industrial applications, generally speaking they

can be treated as transmission lines when the propagation delay TD of a traveling signal islarge compared with the period T of the signal, that is,

TD T (3.10)

2. Page 16, in the sentence

As a rule of thumb, a line is considered “electrically short” when λ < `10 . The definition of

an “electrically short” line becomes less clear in time-domain analysis because each signalcontains a continuum of sinusoidal frequency components.

“electrically short” must be substituted with “electrically long”.

3. Page 19, the last sentence of the 3.1 section

Note that passivity implies stability, but the converse does not hold.

must be removed.

4. Page 21, the paragraph in the “Matrix rational approximation” subsection

Using the terminal conditions, it can be proven that the telegrapher’s equations can be writtenas

∂

∂ z

[V(`,s)I(`,s)

]= eZ`

[V(0,s)I(0,s)

], (3.10)

where

Z =

[0 −a−b 0

]a = R(s)+ sL(s) b = G(s)+ sC(s) . (3.11)

must be substituted with

Using the terminal conditions, it can be proven that the solution of the transmission lineequations can be written as [

V(`,s)−I(`,s)

]= eZ`

[V(0,s)I(0,s)

], (3.10)

where

Z =

[0 −a−b 0

]a = R(s)+ sL(s) b = G(s)+ sC(s) . (3.11)

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