Multiconductor transmission lines wideband modeling: A delay-rational Green’s-function-based method

(1)

Multiconductor transmission lines wideband modeling

A delay-rational Green’s-function-based method

Maria De Lauretis

Industrial electronics

Department of Computer Science, Electrical and Space Engineering

ISSN 1402-1544 ISBN 978-91-7790-214-0 (print)

ISBN 978-91-7790-215-7 (pdf) Luleå University of Technology 2018

DOCTORA L T H E S I S

Maria De Lauretis Multiconductor transmission lines wideband modeling

(2)

(3)

Multiconductor transmission lines wideband modeling

A delay-rational Green’s-function-based method

Maria De Lauretis

Dept. of Computer Science, Electrical and Space Engineering Lule˚a University of Technology

Lule˚a, Sweden

Supervisors:

Jonas Ekman, Giulio Antonini

(4)

ISSN 1402-1544

ISBN 978-91-7790-214-0 (print) ISBN 978-91-7790-215-7 (pdf) Luleå 2018

www.ltu.se

(5)

To Carlotta and Marcus

“[...] the tools we are trying to use and the language or notation we are using to express or record our thoughts, are the major factors determining what we can think or express at all!”, Dijkstra, Edsger W. “The humble programmer.” (1972)

iii

(6)

(7)

A BSTRACT

The performance of variable-frequency drives (VFDs) commonly used in energy production plants can be severely affected by electromagnetic (EM) noise in the form of conducted disturbances. A VFD is composed of an inverter, a motor, and a connecting power cable. The insulated-gate bipolar transistor (IGBT) technology and the pulse-width modulation (PWM) technique, used in the inverter, amplified the role of the power cable, which experiences the so-called “high-frequency” or “transmission line” effects, such as reflections, crosstalk, and distortion. Therefore, a complete EM assessment of a VFD requires an accurate and computationally efficient mathematical model of the cable, which can be studied as a multiconductor transmission line (MTL). Accordingly, we developed the “delay-rational Green’s-function- based” (DeRaG) model that should overcome the main limitations of the existing methods in the literature. In the DeRaG model, the impedance (or admittance) matrix is the sum of a rational series and a so-called hyperbolic part realized by hyperbolic functions. The rational series consists of poles and residues and can be truncated to a suitable size by a delay extraction technique. The hyperbolic part retains the primary information of the high-frequency behaviors, such as attenuation and propagation delays, of a line; thus, the DeRaG model is a wideband model. The DeRaG model is independent of the terminations and sources of the line and enables a delayed state-space representation; it can also account for EM interference. Nevertheless, an EM assessment of a complex system can be performed only using a calculator and proper software. Most of the advanced models for MTLs have been adapted for SPICE-like transient solvers. However, power electronics applications are commonly simulated by using software packages such as Simulink that are optimized for system-level simulations. We thus proposed the implementation of the DeRaG model both in SPICE and in Simulink to embrace a larger group of users and applications. The Simulink implementation was notably proven to be extremely simple and easy to describe. In addition, we focused on the hyperbolic part to qualitatively assess the behavior of an MTL. Our investigation resulted in an outstanding outcome; namely, we provided the distortionless condition for MTLs, whereas the distortionless condition was previously defined only for single-conductor transmission lines as the well-known Heaviside condition. In conclusion, the DeRaG model is a wideband model for the EM analysis of generic transmission lines that is suitable for system-level simulations required in power electronics applications and offers new insights into the physics of the system.

v

(8)

(9)

C ONTENTS

Part I 1

Acronyms 3

CHAPTER1 – THESISINTRODUCTION 7

1.1 Motivation . . . . 7

1.2 Research questions . . . . 9

1.3 Thesis outline . . . . 9

CHAPTER2 – VARIABLE FREQUENCY DRIVES ANDEMC 11 2.1 The PWM technique . . . . 11

2.2 Common problems with VFDs . . . 13

CHAPTER3 – THEDERAGMODEL 17 3.1 MTLs: core concepts . . . 18

3.2 Overview of MTL models . . . 20

3.3 The DeRaG model . . . 23

3.4 The DeRaG model for EMI . . . 30

CHAPTER4 – COMPUTER SIMULATION 35 4.1 The implementation in PSPICE . . . 36

4.2 The implementation in Simulink . . . . 37

4.3 Conclusion . . . 39

CHAPTER5 – DISTORTIONLESS BEHAVIOR 41 5.1 Single-conductor case . . . . 41

5.2 MTL case . . . 43

5.3 Conclusions . . . 45

CHAPTER6 – RESEARCH CONTRIBUTIONS 47 6.1 Paper A. DeRaG for two-conductor TLs . . . . 47

6.2 Paper B. DeRaG for MTLs with frequency-independent p.u.l. parameters . . . 47

6.3 Paper C. DeRaG model for cable bundles . . . 49

6.4 Paper D. DeRaG model for f-pul . . . 49

6.5 Paper E. Realization of the DeRaG model in SPICE . . . 50

6.6 Paper F. DeRaG and the Heaviside condition . . . 50

6.7 Paper G. DeRaG for EMI . . . 50

6.8 Paper H. DeRaG and the distortionless conditions for MTLs . . . . 51

6.9 Paper I. A Simulink implementation of DeRaG . . . . 51 vii

(10)

CHAPTER7 – CONCLUSIONS AND FUTURE WORK 53

7.1 Conclusions . . . 53

7.2 The answers to the research questions . . . 53

7.3 Future work . . . 54

REFERENCES 57 Part II 63 PAPERA 65 1 Introduction . . . . 67

2 Transmission Line Spectral Model . . . 68

3 Rational macromodel . . . 70

4 Delayed Lossless Transmission Line Model . . . 72

5 Delayed Lossy Transmission Line Model . . . 74

6 Numerical Experiments . . . 75

7 Conclusions . . . 78

PAPERB 81 1 Introduction . . . 83

2 Review of the spectral model for Multiconductor Transmission Lines . . . 85

3 Delayed Model of Lossless MTL . . . . 87

4 The Delay-Rational Model for a lossy MTL . . . 92

PAPERC 107 1 Introduction . . . 109

2 Green’s function based methods . . . 110

3 Proposed solution for cable bundles . . . 112

PAPERD 119 1 Introduction . . . 121

2 Green’s function-based method background . . . 122

3 Delay-Rational Green’s Method for MTL with frequency-dependent p.u.l. parameters . . . 127

4 Delay-Rational model in the time-domain . . . 131

5 Numerical Results . . . 132

6 Conclusions . . . 133 viii

(11)

PAPERE 137

1 Introduction . . . 139

2 Background on the impedance formulation obtained using Green’s functions . . 141

3 The Delay-Rational state-space form . . . 143

4 Equivalent circuit for the rational delayless part . . . 145

5 SPICE implementation of the delayed term vd(t) . . . 147

6 Numerical Examples . . . 152

PAPERF 163 1 Introduction . . . 165

2 Background . . . 166

3 Hyperbolic functions and Heaviside condition . . . 168

PAPERG 173 1 Introduction . . . 175

2 The frequency domain model: background formulations . . . 177

3 Asymptotic analysis of the plane-wave-to-MTL coupling model . . . 180

4 The DeRaG-EMI model: a delayed-rational model based on Green’s functions for plane wave to MTL coupling . . . 188

5 The DeRaG-EMI model in the time domain . . . 190

6 Numerical results . . . 193

P^APERH 203 1 Introduction . . . 205

2 Review of the Delay-Rational Green’s Function Method . . . 207

3 Hyperbolic functions and Heaviside condition . . . 209

PAPERI 225 1 Introduction . . . 227

2 The DeRaG model for MTLs . . . 228

3 The block model in Simulink . . . 231

ix

(12)

(13)

A CKNOWLEDGMENTS

First, I want to acknowledge Svenska Kraftn¨at (National Energy Grid of Sweden) for providing the funding for this research. I would like to thank my supervisors Jonas Ekman and Giulio Antonini; they supported me both technically and personally, and I feel honored (and lucky) to have had their support during this Ph.D. I would like to thank Andreas Nilsson for his invaluable assistance with the laboratory activities and for teaching me all that I know (or should know) in an EMC lab, and I am grateful to ˚Ake Wisten for sharing all of his knowledge and experience with me. My journey would not have been the same without sharing my thoughts and pain with my colleagues. I would like to personally thank Elena, Sergio, Sandeep, Darius, Jan, Denis, Andreas (Hartman), Gulnara, and Hasan. I would also like to thank the control group, especially Miguel Casta˜no, who invited me to talk during one of their meetings as well as Khalid Atta for the interest he showed in my research topic. Many thanks to Basel, who never left me from the moment I entered Sweden, and to many special friends, including Rania, Fabiola, Tobias, Anindita, and Arianna. Thanks are also due to my family in Italy, who always supported my professional and personal life decisions. Last but not least, I would like to thank my husband Marcus for always being by my side and my daughter Carlotta, who continuously reminds me that there is an entire world to discover in addition to books and circuits.

Lule˚a, November 2018 Maria De Lauretis

xi

(14)

(15)

Part I

1

(16)

(17)

Acronyms

CM common mode.

DeRaG delay-rational Green’s-function-based (method).

EM electromagnetic.

EMC electromagnetic compatibility.

EMI electromagnetic interference.

f-pul frequency-dependent per-unit-length parameters.

FDTD finite-difference time-domain.

IGBT insulated-gate bipolar transistor.

IOC instantaneous overcurrent.

MIMO multi-input and multi-output.

MoC method of characteristics.

MOR model-order reduction.

MRA matrix rational approximation.

p.u.l. per-unit-length.

PWM pulse-width modulation.

RaG rational Green’s-function-based (method).

TEM transverse electromagnetic.

TL transmission line.

ULM universal line model.

VF vector fitting.

VFD variable-frequency drive.

3

(18)

(19)

List of Figures in Part I

1.1 General schematic of a variable-frequency drive (VFD). . . . 8

2.1 General schematic of a variable-frequency drive (VFD). . . 12

2.2 Three-phase inverter . . . 12

2.3 Simulink model of a basic 3-phase inverter. . . 13

2.4 Voltages measured for the inverter in Fig. 2.3. . . 14

3.1 MTL of N+1 conductors described in terms of input and output ports. . . 18

3.2 Elementary cell with p.u.l. parameters for a one-conductor transmission line. . 19

3.3 Real and imaginary parts of R¹¹_1,1for the 3-conductor TL studied in [1], ordered by the summation mode. . . . 27

3.4 Complex poles for the 3-conductor TL studied in [1]. . . 28

3.5 Definitions of the parameters characterizing the incident field as a uniform plane wave. . . . . 31

3.6 MTL illuminated by an EM plane wave. . . 32

4.1 Block model of the equations in Simulink (continuous-time model). . . 38

4.2 Implementation of the matrix D(t) in Simulink. . . 39

4.3 Dirac comb implementation in Simulink of the block D¹¹(t) in fig. 4.2 . . . 40

6.1 Maps of the papers that compose this thesis organized in chronological order based on the year of publication. . . 48

5

(20)

(21)

C HAPTER 1 Thesis Introduction

1.1 Motivation

Energy production plants rely on electrical machines to convert either electrical energy into mechanical energy (motors) or mechanical energy into electrical energy (generators). Among such machines, induction (asynchronous) machines are the most widespread. Induction motors can be controlled by power converter circuits that are based on the electronic switching of power transistors. The system composed of electric motor(s) and electric control equipment is called a variable-frequency drive (VFD), as depicted in Fig. 1.1. The inverter can convert the utility supply voltage and frequency into the desired values, thereby achieving control over the rotational speed of the motor. The advent of the use of insulated-gate bipolar transistors (IGBTs) in inverters dramatically increased the control precision but also created unexpected problems for the motor [2,3], for the cables [4,5], and for the inverter itself [6]. The most common failures related to the cable were reported as cable failures (e.g., due to corona discharge), unexpected drive overcurrent alarms, and susceptible external circuit malfunctions. In [7], the authors divide the main shortcomings of the VFDs used in energy production plants throughout Sweden into the following three categories:

• software problems: the configurations of different types of software and many drives are challenging;

• hardware problems: the hardware lifespan is short (compared with other industrial equipment); and

• electromagnetic (EM) problems: problems with the harmonics generated at the input stage of the drive, the premature failure of the motor, and the susceptibility of the drive to interference from the net (e.g., during thunderstorm season).

The “Improvement of variable-frequency drives in energy production plants” project conducted at Lule˚a University of Technology (LTU) addresses each of these three topics separately as subprojects. This thesis focuses on EM problems, and the corresponding subproject is called

7

(22)

M

Supply

Rectifier DC link Electric motor

Controller

Input command

Cable

Load Electric Drive

Inverter

Figure 1.1: General schematic of a variable-frequency drive (VFD).

“Simulations of conducted disturbances”, partially addressed in [8]. Conducted disturbances or emissions are, as the name suggests, related to the noise current that is channeled via conductive paths in the system, such as cabling, earthing systems, and metal frames. Noise current, which is not functional to the system, constitutes the root cause of various EM problems related to the premature deterioration of motor bearings and motor insulation, the deterioration of the cable performance, the occurrence of ground loops, and the increasing of EM interference phenomena in the power net [9]. Accordingly, a complete EM assessment of a VFD requires accurate high-frequency models for the inverter, the motor, and the power cable; among these components, we focus our attention on the power cable. The power cable experiences the so-called “high-frequency” or “transmission line” effects, such as reflections, distortion and (more generally) EM problems. The power cable can be susceptible to external EM noise, but it can also carry EM noise, possibly amplifying its effects as in the case of reflections (see Chapter 2). The mathematical model of the power cable should be able to account for all of these effects, which can be accurately described by multiconductor transmission line (MTL) theory [10]. However, whereas the solution of the MTL equations is straightforward in the frequency domain, the solution becomes less obvious in the time domain. Working in the time domain is unavoidable in the presence of nonlinear devices such as power transistors, which can be described only in the time domain [11]. There are several MTL models in the literature that can be used for time-domain simulations; we review the most important models in Chapter 3, and we refer interested readers to the relevant literature [10, 11]. Each model has its own advantages and disadvantages and can be more or less suitable depending on the specific application. For EM analysis, the model in the time domain should be accurate, passive [12], able to account for the frequency-dependent nature of the cable, able to include EM interference, and readily implementable in commonly used software simulation tools.

(23)

1.2. RESEARCH QUESTIONS 9

1.2 Research questions

The research questions beneath the derivation of the DeRaG model have been provided in the licentiate thesis [8]. The research questions that motivate this thesis are listed here as follows:

1. Is the DeRaG model capable of including the effects of EM interference in terms of plane-wave coupling?

2. Is the DeRaG model suitable for system-level simulations as required in power electronics applications?

3. Could the hyperbolic part of the model be used to assess the behavior of an MTL qualitatively?

We answer each question in Chapter 7, which summarizes our research outlined in Part I of this thesis.

1.3 Thesis outline

This thesis is divided into two parts. Part I serves as an introduction to the work performed during the Ph.D., and Part II includes all of the accepted and published papers that are considered major contributions to this thesis. Part I is divided into six chapters, with this chapter as the first. In Chapter 2, we provide a general overview of VFDs with an emphasis on the EM aspects related to the power cable. In Chapter 3, we review the MTL theory and describe the DeRaG model as derived in [13–17], which are papers A, B, C, D and G, respectively, in Part II. In particular, the DeRaG model is suitable for transmission lines with frequency-independent [13, 14] (papers A and B) and dependent [15, 16] (papers C and D) per-unit-length parameters, and it can be extended to describe the plane-wave coupling problem [17] (paper G). In Chapter 4, we summarize the PSPICE and the Simulink implementations, as described in [1] and [18] (papers E and I in Part II), respectively. In Chapter 5, we present the mathematical formulation that provides the distortionless condition for MTLs; we start from a single-conductor transmission line, as detailed in [19] (paper F), and extend the result to MTLs, as detailed in [20] (paper H).

We provide a summary of the content and of my personal contributions to each paper in Chapter 6. Finally, the conclusions and future work are presented in Chapter 7.

(24)

(25)

C HAPTER 2 Variable frequency drives and EMC

In the following sections, we review the operating principles of a generic VFD as depicted in Fig. 2.1, placing emphasis on the parts that are related to electromagnetic compatibility (EMC) aspects.

2.1 The PWM technique

The inverter provides the motor with an AC waveform of the desired voltage and frequency. The average value of the output voltage from the inverter is controlled by the pulse-width modulation (PWM) technique, the nomenclature of which reveals its principle of operation: this technique allows controlling the fundamental component of the output voltage magnitude from the inverter by generating pulses of the desired width and spacing. Several PWM inverters are available depending on the application and the control applied [21]. The three-phase PWM depicted in Fig. 2.2 is found in most applications. We can easily identify three “legs” composed of two power transistors, each with free-wheel diodes. This type of inverter contains one carrier wave signal and three sinusoidal modulating signals. The carrier and modulating signals are compared, and a pulse is generated whenever the magnitude of the sinusoidal signal is smaller or larger than the carrier signal. Then, the pulse is realized by switching the power transistors, which are generally IGBTs. The switching frequency is the rate at which a transistor is turned on and off, and it depends on the result of the comparison. The generic leg voltage (VAo, VBo, or VCoin Fig. 2.2) consists of rectangular pulses, the widths of which are modulated, thereby allowing the fundamental component of the output voltage magnitude to be controlled. Note that the voltage waveform of the inverter is not sinusoidal; however, the motor behavior depends mainly on the fundamental component of the applied voltage, which is sinusoidal [21]. To understand this concept more easily, in Fig. 2.3, we show a basic implementation of a 3-phase inverter in Simulink (MATLAB suite) composed of a PWM generator, a universal bridge, and a 3-phase dummy inductive load. Figure 2.4 shows the measured voltages with respect to the phase A. Typical switching frequencies range from a few KHz to a few MHz (20 kHz - 2 MHz).

Consequently, the pulses are trapezoidal signals with a small rising time, where the rise time of a pulse is the time that the pulse spends rising from 10 % to 90 % of the peak voltage. In

11

(26)

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

Figure 2.2: Three-phase inverter

the context of a VFD, a very fast-rising pulse is a pulse with a rise time of 0.1 µs or shorter.

An increased switching frequency reduces the sizes of components, such as the inductors, transformers, resistors and capacitors, and diminishes the space requirements; an increase in the switching frequency also enables a faster transient load response time and lower output ripple. However, among other effects, higher frequencies represent the cause of electromagnetic interference (EMI) problems, as summarized in the next section.

(27)

2.2. COMMON PROBLEMS WITHVFD^S 13

Figure 2.3: Simulink model of a basic 3-phase inverter.

2.2 Common problems with VFDs

The fast switching of a power transistor causes high^dV_dt values at the rising and falling edges of the inverter output waveforms. For example, a 750 V DC bus can be varied within approximately 75 to 300 ns with values of^dV_dt that can exceed 10 kV/µs. On the motor side, the most common reported failures are premature bearing deterioration [22] and motor winding insulation failure.

The latter is caused by several factors, and is determined mainly by the peak value of the voltage at the motor terminal that can reach two or even three times the DC bus voltage. In the next section, we focus on the role of the power cable.

2.2.1 The power cable

The power cable between the inverter and the motor has several impacts on the drive applications, as is well documented in [4]. With the advent of IGBT technology, well-known transmission line effects such as reflected waves have been amplified coincident with the appearance of numerous new effects, such as the occurrence of instantaneous overcurrent (IOC) trips, the failure of susceptible external circuits, and the degradation of the cable performance. The reason for these effects originates in the concept of the electrically long lines. An electrically long line is a line that behaves as a circuital component in itself and is able to store energy, distort the signal, and cause overshoots, undershoots, reflections, and (more generally speaking) EM

(28)

Figure 2.4: Voltages measured for the inverter in Fig. 2.3.

problems. In fact, electrically long lines experience so-called “transmission line effects” or

“high speed effects” [10]. A generic line or transmission line (TL) can be defined as electrically long or short based on the wavelength of the traveling signal. The definitions of the wavelength and the electrically long line follow.

Definition 2.2.1 The wavelength λ is defined as λ = v

f , (2.1)

where v is the phase speed of the wave and f is the frequency.

Definition 2.2.2 An electrically long line is a line whose physical length ` (assumed to be the largest dimension of the line) is larger than the wavelength λ of the traveling signal:

λ ` . (2.2)

(29)

2.2. COMMON PROBLEMS WITHVFD^S 15

For VFD applications, a practical formula is given in [4] as λ = c

√ εr

1 fu

, (2.3)

where c is the speed of light, εris the dielectric constant of the cable insulation, and fuis the equivalent pulse frequency defined as

fu= 1 π τr

, (2.4)

where τris the rise time of the pulse generated by the inverter. As a rule of thumb, a line is electrically long if ` > ₁₀^λ. Let us consider a very fast rising pulse of 0.1 µs for bundled PVC wires with εr= 5.5. The wavelength of the signal is 40 m, a tenth of which is 4 m. This means that a PVC cable that is longer than 4 m will experience high-frequency effects. In industrial plants, it is not uncommon to encounter cables with a length that spans hundreds of meters;

therefore, these cables will behave as TLs rather than mere interconnections. This constitutes the main reason why cables in VFDs must be studied with the same level of accuracy previously devoted only to vias and traces of printed circuit boards. In fact, cables within VFDs experience the same problems but on a larger (physical) scale in terms of the voltage and current levels.

We review the main mathematical models used for MTLs in Chapter 3, where we introduce the model that we developed throughout the papers listed in Part II. In the following, we summarize the main effects of electrically long power cables in VFDs.

Reflected waves are caused by an impedance mismatch between the characteristic impedances of the motor and the cable. Consequently, the motor can experience a voltage peak at its termi- nals. In practical applications such as VFDs, the critical length of the cable is ^λ₄. In fact, at^λ₄, the motor experiences the theoretically higher amplitude (2 times the DC bus voltage) of the peak voltage at the terminal connections due to reflected waves [4]. Due to high PWM carrier switching frequencies, the critical cable length can become quite short, and a power cable as short as 10 m can already be considered critical. Accordingly, longer cable lengths and higher PWM switching frequencies have been proven to increase the reflected wave motor voltage by up to 3 times [23–25]. IOC trips can be caused by the capacitive coupling of two or more conductors in a cable [26]. Therefore, recalling that the current passing through a capacitor is proportional to the variation in the voltage, higher ^dV_dt values clearly lead to a higher current, thereby giving rise to current spikes that can approach overcurrent trip levels in a VFD.

Finally, the cable plays a large role in the occurrence of EMI noise [5, 27]. Common mode (CM) noise is a form of conducted EMI noise [10]. The high equivalent frequency (2.4) of the pulses causes unwanted current to travel within the parasitic capacitances of the system between the inverter and the ground, the cable and the ground, and the motor and the ground [28].

These capacitances offer a low-impedance path for high-frequency stray currents caused by the steepness of the voltage of the pulses. A major problem is represented by the current in the ground plane because the ground reference potential is no longer zero. CM currents may also radiate, thereby causing radiated EMI that could affect susceptible circuitry. To correctly capture all of the CM currents in a simulation environment, we need high-frequency models for the inverter, the motor, and the power cable. In this thesis, we focus on the power cable, and we provide a new mathematical model that can be used for an EM assessment of VFDs.

(30)

(31)

C HAPTER 3 The DeRaG model

Nomenclature

` physical length of the transmission line [m]

N number of conductors

z axis of the line in a rectangular coordinate system p.u.l., pul, or superscript⁰ per-unit-length quantities

f-pul frequency-dependent per-unit-length parameters

R⁰, L⁰, G⁰, C⁰ p.u.l. resistance [Ω/m], inductance [H/m], conductance [S/m] and capacitance [F/m] matrices

ΓΓΓ(s) propagation constant matrix

Z impedance matrix [Ω]

V voltage port vector [V]

I current port vector [A]

R residue matrix [ω/Hz]

p= α + jβ complex pole [Hz]

Am positive coefficients equal top1/` for m = 0 and p2/` for m> 0

Φ matrix of hyperbolic functions

TD lossless (modal) propagation delay vector [s]

ˆ asymptotic quantities

λ wavelength [m]

c₀ speed of light [m/s]

17

(32)

In this chapter, we introduce some of the main concepts and relevant literature about MTLs, and we present the DeRaG model, which we developed for the study and analysis of MTLs.

3.1 MTLs: core concepts

MTLs are of increasing importance because they permeate each electric and electronic field, such as the interconnections in printed circuit boards (PCBs) and the power transmission lines employed for energy distribution. MTLs can be described by N parallel conductors of length

` that are able to transmit electrical signals. Figure 3.1 shows an MTL as a system with input and output ports where the voltages and currents are functions of both the position and the time. The (N+1)th conductor is the reference conductor for the port voltages, and it is the return conductor for the currents. Classic transmission line (TL) theory assumes that the EM field

I

₁

(0,t) I

₁

(`,t)

V

₁

(0,t) I

₂

(0,t) I

₂

(`,t) V

₁

(`,t)

V

₂

(0,t) V

₂

(`,t)

V

_N

(0,t) V

_N

(`,t)

I

_N

(0,t) I

_N

(`,t)

z x

`

Figure 3.1: MTL of N+1 conductors described in terms of input and output ports.

surrounding the conductors has a transverse electromagnetic (TEM) structure. In a TEM field structure, the electric and magnetic fields are orthogonal to each other and to the line axis, and they propagate along the line axis as waves. Under the TEM mode hypothesis, MTLs can be studied by considering distributed lumped elements along the line axis. In particular, the line is divided into electrically small cells along its length `; each cell of length ∆z is described in terms of per-unit-length (p.u.l.) parameters, as depicted in Fig. 3.2. For a generic MTL, we have four p.u.l. parameter matrices, namely, the resistance (R⁰), inductance (L⁰), capacitance (C⁰) and conductance (G⁰) matrices, which can be either dependent on or independent of the

(33)

3.1. MTL^S:CORE CONCEPTS 19

frequency. The superscript⁰is used to be consistent with the notation found in the literature.

The matrix R⁰= 0 for perfect conductors, whereas G⁰= 0 if the surrounding medium is lossless.

If the conductors and the medium are both lossless, then R⁰= G⁰= 0, which is referred to as the lossless case; otherwise, we will discuss the lossy case. The TEM mode is invalidated by

I₁(z,t) R⁰∆z L⁰∆z

C⁰∆z

I₁(z + ∆z,t)

G⁰∆z

I₀(z,t) I₀(z + ∆z,t)

V₁(z,t) V₁(z + ∆z,t)

∆z

Figure 3.2: Elementary cell with p.u.l. parameters for a one-conductor transmission line.

imperfect line conductors and/or an inhomogeneous surrounding medium. Additionally, the cross-sectional dimension of the line must be electrically small. When nonideal effects can be neglected, we can discuss the quasi-TEM mode assumption [10]. The DeRaG model is valid under the quasi-TEM mode assumption, and the TEM mode is assumed to be the only propagating mode. The well-known telegrapher’s equations describe the voltages and currents at the ends of a TL under the quasi-TEM mode assumption. In the time domain, they read as partial differential equations (PDEs):

∂

∂ zV(z,t) = −Z(t)⁰∗ I(z,t) , (3.1a)

∂

∂ zI(z,t) = −Y(t)⁰∗ V(z,t) , (3.1b) where ∗ denotes the convolution product and Z⁰(t) and Y⁰(t) are the p.u.l. impedance and admittance matrices, respectively, in the time domain [10]. PDEs are not easily integrable, and their direct solution is a research topic in itself. Moreover, standard circuit simulator tools, such as SPICE, can solve ordinary differential equations (ODEs) but not PDEs. The preferred approach is to study the MTL equations in the Laplace domain, where they read as ODEs:

d

dzV (z, s) = −R⁰(s) + sL⁰(s) I (z, s) = −Z⁰(s)I (z, s) , (3.2a)

(34)

d

dzI (z, s) = −G⁰(s) + sC⁰(s) V (z, s) = −Y⁰(s)V (z, s) , (3.2b) where s ∈ C is the complex variable of the Laplace transform and Z⁰(s) and Y⁰(s) are the N × N symmetric matrices of the p.u.l. impedance and admittance, respectively. The p.u.l. matrices are dependent on s if they are frequency dependent p.u.l. parameters (f-pul); otherwise, they are frequency independent. V (z, s) and I (z, s) are N × 1 column vectors representing the voltage and current vectors, respectively. A standard approach is to solve (3.2) and convert the solution in the time domain. In fact, finding the solution in the time domain is necessary in all cases of nonlinear terminations, such as power transistors in inverters, which can be described and solved only in the time domain.

High-frequency models

In the context of MTLs, a high frequency model denotes a model that is suitable for electrically long TLs, as defined in the previous chapter. Other than basic lumped models for electrically short lines, all MTL models are high-frequency models. Along an electrically long line, a traveling signal experiences a propagation time delay TDthat depends on both the length of the line and the phase wave velocity. For single-conductor TLs, the time delay is the time that the signal spends traveling from the input to the output of the line (one-way time delay), and it is defined as

TD= `√

C⁰L⁰. (3.3)

For MTLs, the time delay vector can be defined as the modal lossless time delay as [29]

TD= `p

Λ(C⁰L⁰) , (3.4)

where Λ(C⁰L⁰) denotes the eigenvalues of the matrix product C⁰L⁰. If C⁰(s) and/or L⁰(s) is frequency dependent, then the asymptotic value for s −→ ∞ is considered. TDis an N × 1 column vector in which each element represents the delay of the corresponding conductor. Note that TLs are causal [30]; therefore, the output signal can appear only after the input signal has traveled along the TL to generate a (one-way) time delay. This observation, which may sound trivial, is actually crucial for MTL models because a model that does not properly account for time delay(s) is intrinsically non-causal.

3.2 Overview of MTL models

In the licentiate thesis [8], we discussed the advantages and disadvantages of commonly used MTL models that exist in the literature. In this section, we briefly review the existing techniques for TLs. The reader is referred to [11] for a complete overview of high-speed interconnect models. We limit our discussion to port models that use p.u.l. parameters. Port models, such as impedance/admittance models or ABCD models [10], relate voltages and currents at each end of the line. Some well-known models are described as follows:

(35)

3.2. O^{VERVIEW OF}MTL^MODELS 21

• Lumped segmentation techniques.

The line is modeled with a cascade of elementary cells, and the number of cells is chosen based on both the length of the line and the wavelength of the signal. A well- known example is the π-section method, where the elementary cells are arranged in a π configuration. However, there are well-known drawbacks: the number of cells significantly increases for electrically long lines, the accuracy is poor, and the model does not take into account the delay of the line, leading to an intrinsic violation of the causality condition for TLs. Moreover, these models can account for f-pul [31], but the size increases further as a result. Despite these drawbacks, lumped segmentation techniques are widely used for transient simulations of VFDs, as discussed in Chapter 5, because they are easy to implement.

• The method of characteristics (MoC) and its generalizations.

The MoC is a numerical method that was first proposed in [32]. The MoC is implemented in most computer programs for circuit simulators to solve lossless TLs, for which it provides an exact solution. Instead, the solution for a lossy TL is an approximation. In the MoC model, the solution to the telegrapher’s equations (3.2) is expressed in circuit equations (casted in a Norton-like form) as

I(0, s) = Yc(s)V(0, s) − J(0, s) , (3.5a) I(`, s) = Yc(s)V(`, s) − J(`, s) , (3.5b) where J are the currents of controlled current sources defined as

J(0, s) = H(s) [Yc(s)V(`, s) + I(`, s)] , (3.6a) J(`, s) = H(s) [Yc(s)V(0, s) + I(0, s)] , (3.6b) with

Γ Γ

Γ²(s) = Y(s)Z(s) , Yc(s) = ΓΓΓ⁻¹(s)Y(s) , (3.7)

H(s) = e^−`Γ^Γ^Γ(s), (3.8)

where ΓΓΓ²(s) is the squared propagation matrix, Yc(s) is the characteristic admittance matrix, and H(s) is the propagation operator, which accounts for time delays. For lossless lines, the propagation operator reduces to a matrix of pure delays. For lossy lines, the exponential term cannot be directly translated into the time domain. Several techniques have been applied to overcome this difficulty, such as the Pad´e rational approximation with delay extraction [29, 33]. The Bergeron’s MoC includes the losses in the lossless MoC formulation. In its original derivation [34], Bergeron’s MoC is used in well-known general-purpose computer programs, such as PSCAD/EMTDC and Simulink, for transient simulations of MTLs. Subsequently, the authors of [35] proposed an extension for the f-pul case.

(36)

• The finite-difference time-domain (FDTD) method [10].

The FDTD method discretizes the derivatives of the telegrapher’s equations with respect to both space and time [10]. In particular, the space intervals ∆z need to be electrically short. This implies that, given a TL, the number of unknowns changes if either the length

` changes or the λ of the signal changes, and can quickly expand. The stability of the FDTD method relies on the Courant condition on ∆t.

• The matrix rational approximation (MRA) method [36, 37].

The MRA technique approximates the MTL exponential transfer matrix of the solution of (3.2) expressed as

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

= e^−Φ`V(0, s) I(0, s)

, Φ =

0 Z⁰(s) Y⁰(s) 0

. (3.9)

However, the exponential term does not have a direct representation in the time domain.

The delay-extraction-based passive compact transmission line (DEPACT) algorithm in [38] employs delay extraction to limit the order of the approximation.

• Vector-fitting (VF)-based methods.

A VF based method takes the impedance or admittance matrix as input and outputs the transfer function expressed in a rational form with poles, residues and possibly a constant term in addition to the corresponding state space model [39]. The pole/residue representation reads as

Y(s) =

M

∑

m=1

R s− pm

+ D + sE , (3.10)

where R denotes the residue matrices and pmare the poles of the admittance matrix Y(s). The representation of a transfer function as written in (3.10) is called a rational representation, a pole/residue representation, or a spectral representation. However, the order of the approximation M can quickly become large for electrically long lines. The authors of [40] proposed a delayed vector fitting (DVF) algorithm, which includes the propagation delay of the line in the VF algorithm.

• The universal line model (ULM).

The ULM, which is based on the MoC, is very popular, particularly in the power commu- nity; it can be used to study TLs with f-pul [41, 42]. The idea of the ULM is similar to the VF approach. The propagation function is decomposed into modes and each mode is fitted with a set of poles and single time delays, followed by a final fitting in the phase domain with only residues as unknown quantities.

A common approach used to improve the accuracy of the model and the computational efficiency relies on extracting the line modal delays. However, if the delay is merely approximated, the model will be large (the number of unknowns will be large) and intrinsically band-limited (the approximation will be accurate only for a limited range of frequencies), and the TL-causality condition will be intrinsically violated.

(37)

3.3. T^HED^ER^AG^MODEL 23

3.3 The DeRaG model

The DeRaG model originates from the literature [43], where it was presented as a rational Green’s-function-based (RaG) method without the time-delay extraction. During this Ph.D., we extracted and explicitly incorporated time delays into the model, as fully described in the papers listed in Part II. The DeRaG model is computationally more efficient than the RaG model and is wideband by virtue of the abovementioned time-delay extraction. The DeRaG model can handle both frequency-dependent and frequency-independent p.u.l. parameters. In this section, we briefly review the DeRaG model in the frequency domain and in the time domain in the impedance formulation. The admittance formulation is also outlined.

3.3.1 The DeRaG model in the frequency domain

The impedance matrix representation reads as

V(0, s) V(`, s)

= Z¹¹(s) Z¹²(s) Z²¹(s) Z²²(s)

| {z }

Z(s)

I(0, s) I(`, s)

, (3.11)

where V(0, s) and I(0, s) are the N × 1 voltage and current vectors, respectively, related to the input ports, V(`, s) and I(`, s) are the N × 1 voltage and current vectors, respectively, related to the output ports, and Z(s) is the 2N × 2N block matrix-valued function between the input (currents) and the output (voltages) and is symmetric, where each block Z^{i j}(s) is an N × N matrix for i, j = 1, 2. In [43], the author proves that the impedance matrix representation reads as

Z(s) =

+∞

∑

m=0

Z¹¹_m(s) (−1)^mZ¹¹_m(s) (−1)^mZ¹¹_m(s) Z¹¹_m(s)

, with (3.12a)

Z¹¹_m(s) =

ΓΓ Γ(s)²+

mπ

`

2

1

−1

A²_mZ⁰(s) . (3.12b)

ΓΓ

Γ²(s) = Z⁰(s)Y⁰(s) is the N × N square propagation matrix, 1 represents the N × N identity matrix, and

Am= p1/`, m = 0

p2/`, m > 0 (3.13)

are positive coefficients. The expression in (3.12) is valid for both frequency-dependent and frequency-independent p.u.l. parameters. To simplify the notation, we start with the frequency- independent case. The impedance in (3.12) can be rewritten in a rational form as an infinite sum of the poles and a residue block matrix as

Z(s) =

+∞

∑

m=0 N

∑

k=1

Rmk

s− pmk

+ Rmk

s− ¯pmk

!

. (3.14)

(38)

Specifically, each summation mode m generates a certain number of residues and poles (N for m= 0 and 2N for m > 0 because poles and residues exist in complex conjugate pairs). The notation adopted in (3.14) is suitable for all modes because the zero mode generates only real poles and residues; therefore, the conjugate part will be equal to zero. The poles pmkare defined as

pmk=

α0k, m= 0

αmk± jβmk, m > 0, (3.15)

αmk∈ (−∞, 0], β_mk∈ (0, +∞), where the real parts are zero in the lossless case. The generic k-residue matrix Rmkin (3.12) corresponding to a mode m reads as

Rmk=R¹¹_mk R¹²_mk R²¹_mk R²²_mk

=

R¹¹_mk (−1)^mR¹¹_mk (−1)^mR¹¹_mk R¹¹_mk

. (3.16)

The residues are real for m = 0 and complex for m > 0. The matrix blocks R^{i j}_mkfor i, j = 1, 2 have dimensions of N × N, and they are the same for i, j = 1, 2, except they are multiplied by (−1)^m, which is derived from (3.12), for the nondiagonal blocks (i 6= j).

The f-pul case

The frequency-dependent case is handled by using VF [39] that expresses the p.u.l. impedance Z⁰(s) and admittance Y⁰(s) as

Z⁰(s) = H⁰_Z+ sE⁰_Z+

P_Z⁰

∑

q=1

R⁰_Z s− pqZ⁰

, and Y⁰(s) = H_Y⁰ + sE_Y⁰ +

P_Y⁰

∑

q=1

R⁰_Y s− pqY⁰

, (3.17a)

where R_Z,Yare the residue matrices of Z⁰(s) and Y⁰(s) as indicated by the subscript, p denotes the poles, P_Z⁰ and P_Y⁰ are the numbers of poles used in the rational approximations, and H⁰_Z,Yand E⁰_Z,Yare real, possibly zero, matrices [44]. We remark that it is not possible to know a priori the optimal number of poles required; thus, each case must be treated separately. The eq. (3.17) can be expressed in a rational polynomial form as the ratios between the polynomial matrices Bp(s) and Dp(s) and the polynomials Ap(s) and Cp(s):

Z⁰(s) =b₀s^P^Z⁺¹+ b₁s^P^Z+ · · · bPZ+1

a₀s^P^Z+ a₁s^P^Z⁻¹+ · · · aPZ

=Bp(s)

Ap(s), (3.18a)

Y⁰(s) =d0s^P^Y⁺¹+ d₁s^P^Y+ ·· · dPY+1

c0s^P^Y+ c₁s^P^Y⁻¹+ · · · cP_Y

=Dp(s)

Cp(s). (3.18b)

The block impedances in (3.12) are computed in the same way by considering the new polynomial expressions in (3.18), as detailed in [43]. In particular, the block matrix in (3.12b) is

Z¹¹_m(s) = Ep(s)⁻¹A²_mBp(s)Cp(s) with Ep(s) = Bp(s)Dp(s) + Ap(s)Cp(s)

mπ

`

2

1 . (3.19)