L ICE N T IAT E T H E S I S
ISSN 1402-1757
ISBN 978-91-7583-561-7 (print)
ISBN 978-91-7583-562-4 (pdf)
Luleå University of Technology 2016
Maria De Lauretis Transmission line theory for cable modeling: a delay-rational model based on Green’s functions
Department of Computer Science, Electrical and Space Engineering
EISLAB
Transmission line theory for cable
modeling: a delay-rational model
based on Green’s functions
Maria De Lauretis
Industrial Electronics
Transmission line theory for cable
modeling: a delay-rational model based on
Green’s functions
Maria De Lauretis
Dept. of Computer Science, Electrical and Space Engineering
Luleå University of Technology
Luleå, Sweden
Supervisors:
Jonas Ekman, Giulio Antonini
ii
To my parents
iii
iv
A BSTRACT
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 improve
electromagnetic modeling techniques for the study of conducted electromagnetic emissions
in variable-frequency drives, with the aim of enhancing their reliability in energy production
plants. Pulse-width-modulated voltage converters are used to feed an AC motor, and they are
considered to be the primary reason for high-frequency effects in both the motor and the supply
grid. In particular, high-frequency currents, known as common mode currents, flow between
all energized components and the ground and travel via low-resistance and low-inductance
interconnects such as the power cable between the inverter and the motor.
Electrically long power cables are commonly used in VFD installations, and require particular attention. Accurate models can be obtained using the theory of multiconductor transmission lines. In the case of nonlinear terminations, such as an inverter, only time-domain analysis
is possible. In recent years, several techniques have been proposed. Some of these techniques
include the lumped-element equivalent circuit method, the method of characteristics (MoC)
and its generalizations, and the Padé approach. In this context, a modeling technique based
on Green’s functions has been proposed. The input/output impedance matrix is expressed as
a 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 necessary to model the dynamics of the system, especially when electrically long interconnects
are considered. To overcome this limitation, we have proposed the Delay-Rational Green’sFunction-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 through
hyperbolic functions. The delay extraction enables the use of a reduced number of poles and
improves the accuracy of the model in general, avoiding any ringing effects in the time-domain
response. The primary advantage of the proposed method compared with other well-known
techniques lies in the delayed state-space representation. The obtained model can be computed
regardless of the terminations and/or sources, and the terminal conditions can be immediately
and essentially incorporated.
The next step will be to simulate the entire inverter-cable-motor system. The partial element
equivalent circuit (PEEC) technique will be used to model the interconnects as well as the
discontinuities in the power cable that can be caused, for example, by switch disconnectors.
The theoretical results will be verified against experimental measurements. The final objective
is to provide new techniques for modeling the electrodynamics of variable-frequency drives to
allow their complete EMC assessment as early as the design stage and to enable the planning
of corrective actions in advance.
v
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C ONTENTS
Part I
1
Acronyms
3
C HAPTER 1 – T HESIS INTRODUCTION
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
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8
9
C HAPTER 2 – VARIABLE - FREQUENCY DRIVES
11
2.1 Background on variable-frequency drives . . . . . . . . . . . . . . . . . . . . 11
2.2 Pulse-width-modulated waveform and harmonic distortion . . . . . . . . . . . 12
2.3 The role of simulations and measurements . . . . . . . . . . . . . . . . . . . . 14
C HAPTER 3 – C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
3.1 Background on multiconductor transmission lines . . . . . . . . . . . . . . . .
3.2 Main shortcomings of the present models . . . . . . . . . . . . . . . . . . . .
3.3 Transmission line theory for the study of common mode currents in cables . . .
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23
C HAPTER 4 – T HE PROPOSED DELAY- RATIONAL MODEL
4.1 Green’s functions and boundary problems: a brief background . . . . . . . . .
4.2 Delay extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 The delay-rational state-space form . . . . . . . . . . . . . . . . . . . . . . . .
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32
C HAPTER 5 – R ESEARCH CONTRIBUTIONS
5.1 Paper A . . . . . . . . . . . . . .
5.2 Paper B . . . . . . . . . . . . . .
5.3 Paper C . . . . . . . . . . . . . .
5.4 Paper D . . . . . . . . . . . . . .
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C HAPTER 6 – C ONCLUSIONS AND FUTURE WORK
41
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
R EFERENCES
43
vii
Part II
49
PAPER A
1
Introduction . . . . . . . . . . . . . . . . .
2
Transmission Line Spectral Model . . . . .
3
Rational macromodel . . . . . . . . . . . .
4
Delayed Lossless Transmission Line Model
5
Delayed Lossy Transmission Line Model .
6
Numerical Experiments . . . . . . . . . . .
7
Conclusions . . . . . . . . . . . . . . . . .
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PAPER B
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Review of the spectral model for Multiconductor Transmission Lines
3
Delayed Model of Lossless MTL . . . . . . . . . . . . . . . . . . .
4
The Delay-Rational Model for a lossy MTL . . . . . . . . . . . . .
5
Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . .
6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PAPER C
1
Introduction . . . . . . . . . . . . .
2
Green’s function based methods . .
3
Proposed solution for cable bundles
4
Numerical Experiments . . . . . . .
5
Conclusions . . . . . . . . . . . . .
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PAPER D
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Green’s function-based method background . . . . . . . . . . . . . . . .
3
Delay-Rational Green’s Method for MTL with frequency-dependent p.u.l.
rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Delay-Rational model in the time-domain . . . . . . . . . . . . . . . . .
5
Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
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109
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pa. . . 117
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ACKNOWLEDGMENTS
I acknowledge Svenska Kraftnät (Swedish national grid) for providing funding for this research. Also, I would like to acknowledge the people who have supported this research from
both a professional and personal point of view. In particular, I would like to acknowledge
my supervisors, Jonas Ekman and Giulio Antonini, for their constant support during these
two years. They have helped me to organize my PhD studies and to develop new research
ideas, always with enthusiasm and effort. Andreas Nilsson, for his irreplaceable assistance in
laboratory activities and his enormous patience and sarcastic humor. Johan Borg, for all the
valuable discussions that we had. Joakim Nilsson and Marcus Lindner, my two colleagues on
this project: I am glad to have such colleagues as you, always open to discussions and collaborations. 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 my
stay at the L-paviljon enjoyable. Andreas Lindner, for helping me with my labs. Dariusz Kominiak, for his collaborative attitude. Marcus Lindner, for being a valuable discussion partner
and, also, a valuable husband!
In general, I would like to thank all the nice people and friends who have taken the time to
talk with me, either for pleasure or for working reasons. Thank you all!
Luleå, May 2016
Maria De Lauretis
ix
x
Part I
1
2
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
EMI
electromagnetic compatibility.
electromagnetic interference.
IEC
international electrotechnical commission.
MoC
MOR
MTL
method of characteristics.
model order reduction.
multiconductor transmission line.
PEEC
PQ
PWM
partial element equivalent circuit.
power quality.
pulse-width modulation.
VFD
variable-frequency drive.
3
4
List of Figures in Part I
2.1
2.2
2.3
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
4.5
4.6
General schematic of a variable-frequency drive. . . . . . . . . . . . . . . . . 12
Simplified schematic of a pulse-width-modulated (PWM) voltage source inverter. 12
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 grounding
connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Per-unit-length parameters for a one-conductor transmission line. . . . . . . . . 18
Multiconductor transmission line represented as a 2N-port system, with a common reference conductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Common and differential current components for a typical four-conductor shielded
cable, treated as a multiconductor transmission line. . . . . . . . . . . . . . . . 23
Common mode current analysis of a 4-conductor shielded cable connected between the inverter and motor, with the shield connected on both sides. . . . . . 24
Original Green’s-function-based method. . . . . . . . . . . . . . . . . . . . . 29
Complex conjugate pole locations computed using a modal rational approximation approach for a 6-conductor ribbon cable with frequency-independent
p.u.l. parameters, from Paper C. . . . . . . . . . . . . . . . . . . . . . . . . . 31
Complex conjugate pole families in the complex plane for a 2-conductor line
with frequency-dependent parameters, from Paper D. . . . . . . . . . . . . . . 31
Complex conjugate pole locations for a 9-conductor cable bundle, from Paper
C. The blue circles represent the asymptotic real parts for each family, α̂k . . . . 32
Block diagram for the algorithm used to determine the m̂ mode, from Paper C. . 33
Block diagram of the state-space model (4.19). . . . . . . . . . . . . . . . . . 34
5
6
C HAPTER 1
Thesis introduction
1.1
Motivation
Energy production plants are responsible for the generation of energy that is distributed throughout the national electric grid. Typical examples are nuclear, hydroelectric, and fossil-fuel power
plants. Motor-driven systems are widely used in the industry; in these systems, the motor is, in
most cases, an electric motor, and typical examples include pumps, fans, and compressors. The
advent of power electronics in the early 1960s dramatically changed the way in which motors
are controlled and, ultimately, designed [1]. Previously, motors were designed with electromechanical controls, which were neither flexible nor easy to change. Power electronics allowed
the control to be shifted from the motor itself to external (cheaper) circuitry, which eventually
became programmable. Motors previously used only for constant-speed applications, such as
the well-known squirrel cage motor, found new life in the 1970s, when electronic inverters enabled control of their speed. Research progressed in the following directions: control switched
from analog to digital, digital processors were adopted, and real-time modeling and simulations drove a continuous increase in motor-drive performances. Today, the focus is on energy
efficiency and EU directives, and power electronics play a large role [2]. In fact, power modules 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 as
no surprise that power electronics have found constant and increasing application in the industrial environment. The system constituted by a power module and an electric motor is generally
called a variable-frequency drive (VFD) or an adjustable-speed drive (ASD).
However, unexpected problems have been arising, such as the premature deterioration of
motor bearings [3, 4] and increasing electromagnetic interference phenomena [5, 6]. From
recent reports and investigations [7, 8, 9], it has emerged that variable-frequency drives are
playing an increasing role in determining the reliability of energy production plants. They may
severely impact the quality of the energy supplied to and obtained from utilities, eventually
leading 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
T HESIS INTRODUCTION
of these events is generally referred to as a power quality (PQ) problem [10]. Disturbances
in the electric grid can be due to several causes, such as short circuits, abruptly changing
loads, or seasonal problems such as thunderstorms and lightning. According to [11], lightning
strikes cause at least 60% of all dips and short interruptions. It has been found that VFDs
are particularly sensitive to disturbances in the power grid [7]. Moreover, VFDs have been
recognized 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 electronic
components.
Since 1906, the International Electrotechnical Commission (IEC) has been engaged in the
preparation and publication of international standards for all electrical, electronic and related
technologies. The purpose of these standards is to define tests and criteria to predict and avoid
potential problems in a product. The IEC-61800 standard, Part 3, is entitled “Adjustable speed
electrical power drive systems”, and it specifies the electromagnetic compatibility (EMC) product standard and test methods for variable-frequency drives. Clear indications are given for the
assessment of the immunity of a VFD to voltage deviation and harmonic distortion as well as its
harmonic emissions. However, even if a product is compliant with the standard, other factors
could degrade its EMC performance. The installation of a system and its earth connections, for
example, play a large role [12], because they may seriously impact and amplify problems with
electromagnetic interference. Also, changes in the EMC behavior of a VFD because of fault
conditions are not considered.
In the context of EMC, two main types of disturbances can be identified: radiated disturbances and conducted disturbances. As the names suggest, the former are emitted or transmitted 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 emissions
are considered together with power quality as related topics [10]. Conducted emission tests
allow the measurement of the amount of frequency noise that is injected into the grid by the
equipment. In view of the above, the project “Improvement of variable-frequency drives in energy production plants” has been initiated, of which the sub-project “Simulations of conducted
disturbances” serves as the central topic of the PhD studies presented in this thesis.
1.2
Thesis outline
The thesis outline is as follows. A general overview of VFDs, with particular emphasis on
the electromagnetic compatibility aspects, is given in Chapter 2. In fact, conducted emissions
cause electromagnetic interference (EMI) in the system, leading to harmonic currents that can
penetrate back to the utility supply. Because of the high switching frequencies imposed by the
pulse-width-modulated control signal, the cables behave not only as interconnections but as
electric system components with specific properties, and the theory of multiconductor transmission lines can be used to study and model their behavior. In Chapter 3, a general introduction to multiconductor transmission line (MTL) theory is provided. In particular, the wellstudied solution in the frequency domain is briefly recalled. Some of the most well-known
techniques for time-domain simulations are summarized, such as the lumped-element equivalent circuit method, the method of characteristics (MoC) and its generalizations, and matrix-
1.3. R ESEARCH QUESTIONS
9
rational-approximation-based techniques. In Chapter 4, we introduce the Green’s-functionbased method that is at the root of the previously published papers pertaining to the research
presented herein. The proposed delayed model exploits delay extraction, which leads to the
so-called Delay-Rational Green’s-Function-based Method, or DeRaG. In general, the main
advantage of the proposed method compared with other well-known techniques lies in the delayed state-space representation. The model can be computed regardless of the terminations
and/or sources, and the terminal conditions can be immediately and essentially incorporated. In
Chapter 5, the research contributions are summarized and a short description of the four papers
included 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 this
purpose, a motor kit has already been tested, but the preliminary results are not covered in this
thesis. Additionally, the interconnections will be modeled using the partial element equivalent
circuit (PEEC) technique. The ability to build a reliable mathematical model will allow the
proposal of filtering techniques and corrective actions in case of fault conditions, with the primary purpose of avoiding the tripping of variable-frequency drives in energy production plants.
Possibly, in situ measurements may be performed.
1.3
Research questions
In the study of the electromagnetic interference in VFD, accurate time domain models need
to be adopted. In fact, the drive is characterized by nonlinear components that can only be
described in the time domain. Typically, most of the focus is on the power module and on the
motor time-domain models. Even though it is well recognized that a proper model of the cable
is crucial for the accuracy of the system simulation, see for example [5] and the considerations
in [13], it is not uncommon to see in literature the use of modeling techniques, such as the
lumped circuit model approximation [9], which have known limitations, as it will be explained
in Chapter 3. This is somehow surprising, because a considerable amount of research has been
devoted 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 so
far have shortcomings that prevent their implementation in commonly used circuit simulator
environments, such as SPICE-like transient simulators, where the nonlinear components can be
easily and accurately represented. The most common limitations of existing models for MTLs
modeling are:
• passivity is not guaranteed by-construction. Non-passive models may result in unstable
models 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 does
not account for the frequency-dependent nature of the cable may lead to totally incorrect
results 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 be
cumbersome to obtain. An exception is [19] where a minimal code is provided. How-
10
T HESIS INTRODUCTION
ever, the cable is studied under specific geometric assumptions. Also, complex models
normally require complex circuit descriptions, which can severely impact the overall
simulation time.
The method proposed in [20] for multiconductor transmission line modeling has been presented with the main object to provide a general, accurate and passive model for MTLs. The
model can include frequency-dependent parameters without resorting to the convolution product and admits a state-space form realization in the time domain. One of the major advantage
of the model relies on the state-space form derived. In fact, linear state-space systems have
many advantages, such as:
• their theory is well understood and established, and properties such as stability and passivity 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-known
circuit simulator solvers such as SPICE can be used.
The work done so far has been focused on improving the model presented in [20], in order
to overcome the main shortcomings, such as: the high number of poles necessary to gain a
satisfactory accuracy; the oscillations in the time domain due to the limited bandwidth of the
model; 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 order
to 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, and
the final state-space representation?
3. Is it possible to provide a straightforward circuit representation where only standard
circuit elements are used?
In the work presented to date, the first two questions have been answered, even though a formal
verification 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 high
accuracy by virtue of the delay extraction technique, and the new delayed state-space system
is suitable for inclusion in SPICE-like transient simulators. This last step is of fundamental
importance for efficiently simulating the “drive-cable-motor” system. The ability to simulate
a complete system, possibly also considering switch disconnectors, for EMC purposes would
allow a deep understanding of the system and a complete EMC system assessment as early as
the design stage, with the potential to save time and money.
C HAPTER 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 or
graduate textbook [1, 22]. In this chapter, we will emphasize the concepts that are critical from
an 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 means
for 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 and
of 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 general
schematic of a VFD is presented in Fig. 2.1. The main components, from left to right, are as
follows:
• 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 and
frequency;
• the power cable, often referred as the power interface, which connects the power module
to the electric motor;
• the controller, which provides a pulse-width modulation (PWM) signal to the inverter
based 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
Electric Drive
DC link
Rectifier
Inverter
Electric motor
ϭ'
M
Load
'ϭ
Supply
Cable
Controller
Input
command
Figure 2.1: General schematic of a variable-frequency drive.
Diode Rectifier
Supply
DC link
Inverter
Motor
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 the
DC link, which smooths the voltage. Note that for drives rated at over 2.2 kW, the DC current
will be also smoothed by an inductor built into the DC link. The PWM voltage source inverter
will 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 distortion
The switching nature of the converter circuits results in waveforms that contain not only the
fundamental component but also unwanted harmonic voltages. Harmonics in power systems
2.2. P ULSE - WIDTH - MODULATED WAVEFORM AND HARMONIC DISTORTION
13
result in increased heating in equipment and conductors, misfiring in variable-speed drives, and
torque pulsations in motors. Low-order harmonics may cause an unwanted torque response
from the motor, whereas high-order harmonics can lead to acoustic noise if they excite a mechanical resonance [1, 24]. The switching nature of the PWM waveform has been recognized
as 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 from
an electrical point of view, manifesting in unwanted interaction with the power grid. The use of
high-frequency switching power semiconductors causes rapid voltage variations, and stray currents flow between the inverter and the motor and between the supply and the inverter, mainly
through parasitic capacitances. Unwanted currents that travel through wired connections and
ground paths are known as conducted disturbances.
2.2.1
Impact of long inverter-motor cables
A PWM waveform exhibits high rates of change in voltage dV / dt, which cause a transiently
uneven voltage distribution across the motor windings and short-duration voltage overshoots
because 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. At
the terminal of the drive, dV / dt will be greater than 5 kV /µs. At such high rates of change in
voltage, a cable behaves as a transmission line, the theory governing which is the subject of the
next chapter. Generally speaking, when a cable behaves as a transmission line, it is no longer
merely a connection but rather acts as an electric circuit component in its own right. From
a practical perspective, this means that when the voltage edge reaches the motor terminals, a
reflection occurs because the motor impedance is higher than the surge (or characteristic) cable
impedance (impedance mismatch). The motor terminal voltage experiences an overshoot of
theoretically up to twice the step voltage, which can represent a problem for a well-insulated
motor. Additionally, at each pulse edge, the drive must provide a pulse of current to charge
the capacitance of the converter-motor cable, and in extreme cases, this current may exceed
the rated current of the motor, which determines the rating of the required drive. This problem
is intensified in drive systems with rated voltages greater than 690 V and in medium-voltage
drives, where dV / dt filters are included between the inverter and motor [1]. Cables are not
responsible only for over-voltage problems. They can also impact the overall electromagnetic
behavior 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 up
to megahertz. High levels of voltage steepness dV / dt generate high-frequency (HF) stray
currents in parasitic capacitance. EMI currents travel through cable as waves and are subject
to multiple reflections. As a result, conducted EMI arises in both the power mains and ground
system. These HF stray currents are divided into two components, based on their circulation
paths:
• 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
Parasitic capacitances
Parasitic capacitances
Utility
supply
Motor
Grounding connections impedances
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 supply
through parasitic capacitances and grounding connections.
We are predominantly interested in the common mode currents. In fact, the common mode
currents flow into the ground and can return to the supply utility; also, hazardous common
mode voltages can arise between system components and the ground. In Fig. 2.3, the paths
of the common mode currents are highlighted. Note that the total common mode current is
the sum of contributions from the cable and from the motor. In this thesis, we focus on cable
modeling [13].
2.3
The role of simulations and measurements
The primary motivation for mathematical models, and simulations in general, is that they allow a system to be prepared, checked and tested before its actual production. In fact, if the
conducted 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 electromagnetic compatibility, the cables and their impact on the overall system performance must
be 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 as
multiconductor transmission lines.
C HAPTER 3
Cables and multiconductor
transmission line theory
3.1
Background on multiconductor transmission lines
Generally 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 applications, they can be treated as transmission lines when the propagation delay TD of a traveling
signal 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 transmission line (MTL) theory [29]. The reader is referred to [29] for a comprehensive study of
multiconductor transmission lines and to [30] for a general overview of simulations of interconnects.
In a variable-frequency drive, the high rates of change in voltage due to PWM control influence the reliability of the system. In particular, the cable between the inverter and the motor
is 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 variablefrequency drive, it is therefore necessary to model the cable with the same attention and accuracy devoted to the inverter and the motor. In the following, we will provide a general overview
of MTL theory, in which more than two conductors (ground included) are considered. Note
that 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 the
time domain. In the frequency domain, we typically assume a sinusoidal excitation source, and
steady-state conditions can be studied. By contrast, time-domain analysis allows the consideration of sources with any arbitrary time variation, and both transient and steady-state solutions
15
16
C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
are considered. In power systems, it is important to be able to predict transient behavior because, as noted in Chapter 1, transient disturbances such as voltage dips or surges may cause a
variable-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, where
this wavelength “...represents the distance that a single-frequency, sinusoidal electromagnetic
wave must travel in order to change phase by 360◦ ” [31]. In the frequency domain, given a
frequency of interest f , the wavelength is defined as
λ=
v
,
f
(3.2)
where v is the velocity of the wave in the medium (v w 2.99 × 108 m/s in vacuum). If the largest
dimension of the MTL is electrically small ( λ ), then we can apply the lumped-circuit theory.
`
. The definition of
As a rule of thumb, a line is considered “electrically short” when λ < 10
an “electrically short” line becomes less clear in time-domain analysis because each signal
contains a continuum of sinusoidal frequency components. The problem is the identification
of the maximum frequency of interest. Typically, conditions are imposed on the pulse rise and
fall 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 electrically
short 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 necessary. Conventional circuit elements can be classified based on the number of dimensions that
are 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 uniform 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 elements
and can be described in terms of resistance, capacitance, inductance and conductance. The
same 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 into
so-called modes (partial fields): each mode exhibits a certain pattern, and the sum of all modes
gives the actual field distribution. The transverse electromagnetic (TEM) mode, in which the
field vectors are orthogonal to each other and to the propagation direction, is particularly important. 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 conductors
with a constant cross section. For realistic conductors, with losses and terminations, other field
modes are excited; at low frequencies, these modes are small, and the conductor is said to be
operating in a quasi-TEM mode.
Given a three-dimensional space xyz, the transmission line is regarded as a distributedparameter structure along the z axis, and the lumped-circuit analysis technique is extended to
structures that are electrically long in this dimension. The common approach to analyzing multiconductor transmission lines is to assume a quasi-TEM mode of propagation for the electric
and 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 inhomogeneous surrounding medium, are considered to be negligible.
Under the quasi-TEM mode assumption, Maxwell’s equations can be recast in the form of the
well-known telegrapher’s equations (3.4) [33]. These equations are a set of 2N coupled firstorder partial differential equations (PDEs) that describe the voltages and currents of a generic
transmission line structure. In the case of a multiconductor transmission line, the resistance
R, capacitance C, inductance L and conductance G are represented by full matrices. If the
matrices satisfy R = G = 0, then we are considering a lossless transmission line; otherwise,
the line is lossy.
∂
∂
V(z,t) = −RI(z,t) − L I(z,t) ,
∂z
∂t
∂
∂
I(z,t) = −GV(z,t) − C V(z,t) .
∂z
∂t
(3.4a)
(3.4b)
For eq. (3.4) to be used, the MTL must be discretized in the z direction, in infinitesimal sections
of 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 transmission line. Note that for a uniform line, the per-unit-length parameter matrices R, L, G and
18
C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
R∆z
i(z,t)
v(z,t)
L∆z
C∆z
z
i(z + ∆z,t)
v(z + ∆z,t)
G∆z
z+∆z
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 which
case 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 particularly
important at high frequencies [29]. They can be obtained in two main ways: by obtaining
a 2-D solution of Maxwell’s equations [34] or through measurements of the port response at
discrete frequency points [27]. The matrices R, L, G and C are typically full, positive-definite
symmetric matrices of order N, where N + 1 is the number of conductors with the reference
conductor included [29].
In the Laplace domain, the telegrapher’s equations read as ordinary differential equations
(ODEs), as follows:
d
V (z, s) = − [R + sL] I (z, s) = −Z0 (s)I (z, s) ,
dz
d
I (z, s) = − [G + sC] V (z, s) = −Y0 (s)V (z, s) .
dz
(3.5a)
(3.5b)
Z0 (s) and Y0 (s) are the N × N symmetric matrices of the per-unit-length impedance and admittance, respectively. V (z, s) and I (z, s) are N × 1 column vectors; they represent the voltage and
current vectors, respectively, which depend on the Laplace variable s and the position z along
the line.
Port models
Models in which the terminal currents and voltages are related are well suited for inclusion
in SPICE-like simulators. A generic MTL can be represented as a 2N-port system, as shown
in Fig. 3.2, where only the input (z = 0) and the output (z = `) of the transmission line are
considered. The most common solutions that are suitable for the 2N-port representation are
1. the modal solution and
2. the matrix-exponential solution.
3.1. BACKGROUND ON MULTICONDUCTOR TRANSMISSION LINES
V1(s)
V2(s)
MTL
+
+
VN+1(s)
V N+2(s)
͘͘͘
-
IN+2(s)
I2(s)
+
͘͘͘
VN(s)
IN+1(s)
I1(s)
+
19
IN(s)
I2N(s)
V 2N(s)
+
+
z=0
z= l
-
Figure 3.2: Multiconductor transmission line represented as a 2N-port system, with a common reference
conductor.
The solution can then be expressed in terms of ABCD parameters, Y parameters (the admittance representation) or Z parameters (the impedance representation) [29]. The representation
used in this thesis is the impedance representation, which reads in general as shown in eq.
(3.6):
V0 (s)
Z11 (s) Z12 (s) I0 (s)
I0 (s)
=
= Z(s)
,
(3.6)
V` (s)
Z12 (s) Z11 (s) I` (s)
I` (s)
where Z(s) is the open-end port impedance matrix, which represents the transfer function between the terminal voltages and currents.
The identification of the time delay of electrically long lines is crucial to ensure that a model
is efficient in terms of accuracy and memory occupation. In fact, if the delay is not properly
incorporated into the model, then a large number of unknowns will be necessary to achieve
acceptable accuracy. In the following, we will discuss the most common macromodeling techniques, and we will see that line delay identification is the most common means of improving
both the accuracy and size of a model.
An important property of a model is its passivity. In fact, stable but non-passive models are
not useful for transient simulations because passivity violations can lead to spurious oscillation
and incorrect simulations. Moreover, stable but non-passive models that are interconnected can
lead to a non-stable system, whereas interconnected passive models result in an overall passive
system. Note that passivity implies stability, but the converse does not hold.
3.1.1
Suitable models for time-domain simulations
Standard 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 equations
into ODEs. Moreover, although partial differential equations are better solved in the frequency
domain, 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
20
C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
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 review
the 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 electrically
short segments of length ∆z, each modeled with lumped parameters. The obvious primary disadvantage of this method is that it leads to large circuit matrices for electrically long lines, for
which the propagation delay must be considered. Additionally, the line delay is not explicitly
included but rather merely approximated, leading to an intrinsic violation of the causality condition for transmission lines [35]. To account for frequency-dependent parameters, additional
passive LR or LRC sections must be used to approximate the distributed loss [9, 36], thereby
increasing the size of the model used and limiting the numerical efficiency of time-domain
simulations. If the frequency dependence of the parameters is not correctly captured, then the
simulation 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 nonlinear and/or dynamic loads. It yields an exact solution for a lossless line and is the method
implemented in most computer programs for circuit simulations, such as SPICE. The extraction of the one-way delay TD = v` allows the voltage and current at one end of the line to
be specified using the voltage and current at the other end, by virtue of the time delay. The
transmission line is represented by a set of admittances/impedances and delayed sources. For
multiconductor transmission lines, the generalized method of characteristics has been developed [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 the
per-unit-length impedance Z0 and admittance Y0 . The modal voltages and modal currents thus
obtained are represented by N separate, uncoupled, two-conductor lossy lines. With the mode
characteristic 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 as
follows:
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)
3.1. BACKGROUND ON MULTICONDUCTOR TRANSMISSION LINES
21
where eΓmp
(s) is an N × N diagonal matrix that contains the N modal propagation constants
Γm,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 into
the time domain, and several techniques have been developed to overcome this difficulty, such
as the Padé rational approximation. As observed in [35], a direct rational approximation of
the propagation operator can be difficult, mainly because the matrix accounts for the line delay
and the line attenuation/dispersion, and a delay extraction technique is therefore proposed.
When frequency-dependent per-unit-length parameters are considered, the modal line delays
are defined in terms of the frequency asymptotic values of the capacitance and inductance, as
follows:
p
(3.9)
Tk = ` Λk ,
where the Λk are the eigenvalues of the matrix product C∞ L∞ ; C∞ and L∞ are the asymptotic
values of the frequency-dependent capacitance and inductance matrices. The main difficulty
in this approach lies in how these delays are extracted from the propagation operator. The
particular delay extraction technique used should be based on the process that is applied to the
delayless transfer function. This approach is not passive by construction. Recently, sufficient
conditions 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 written
as
∂ V(`, s)
Z` V(0, s)
=e
,
(3.10)
I(0, s)
∂ z I(`, s)
where
0 −a
Z=
−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 to
perform a matrix rational approximation (MRA) of this function [41, 42, 43]. The exponential function can be approximated as a ratio of polynomial matrices expressed in terms of
closed-form Padé 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 order
of the approximation. To overcome this difficulty, a new technique has recently been proposed,
named the Delay-Extraction-based Passive Compact Transmission-Line Macromodeling Algorithm (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 are
then necessary to model the delays, and the decoupling is achieved via transformation matrices.
This model is passive by construction.
22
C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
Other well-known techniques
Other well-known techniques are
• TOPline [14], which combines matrix rational approximation with modal delay extraction;
• 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 models
Lumped approaches are, in general, not advisable because they do not produce accurate broadband models. The delay must be taken into account with additional RLC networks, which, in
turn, lead to a large equivalent circuit, and the simulation in the time domain becomes cumbersome.
In general, the MoC model does not preserve passivity by construction, although sufficient
conditions can ensure the passivity of the model. Additionally, it requires approximation of the
exponential form.
Methods that generate a rational model of an MTL, such as the Padé rational approximation
and ladder network models, use expansion points to identify the poles, and their accuracy
improves as the order of the approximation increases. This can lead to large macromodels that
are 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 used
in 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-difference
approximation method; previously, it used U-elements, which have recently been replaced with
W-elements. In [46], it is suggested that circuit elements based on the MoC or matrix rational
approximation are more efficient then the U-elements. Regarding W-elements, nothing has
been said about the passivity of the model obtained [47].
3.3. T RANSMISSION LINE THEORY FOR THE STUDY OF COMMON MODE CURRENTS IN
23
CABLES
Iu1(s)
Iv1(s)
Iw1(s)
Id1 Ic
Ic
Id1
Id2
Ic
Ic
Id2
Id3
Ic
Ic
Id3
Ic
Ig1(s)
Is(s)
MTL
Ic
Iu2(s)
Iv2(s)
Iw2(s)
Ig2(s)
4Ic
4Ic
z=0
z= l
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 mode
currents in cables
The typical cables used for three-phase induction motors are coaxial shielded cables. They
generally contain three conductors (one for each phase) and the ground conductor. They are
typically shielded, with the shield connected directly to the chassis of the motor. As stated
in Chapter 1, we are primarily interested in studying the common mode currents that travel
through such cables. It is therefore worthwhile to clarify how an MTL model can be used
for this purpose. In general, differential mode currents Id are distinguished from common
mode currents Ic based on their circulation paths [25]. A differential mode current is typically
equal to the signal or power current, and such currents are not present in either the shield or
the 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 common
mode currents: the longitudinal conversion loss of the cable, which causes some of the power
currents to leak out through stray coupling, and the noise voltage between the connection point
of the cable and the ground reference of the circuit.
The shield conductor is treated as the reference conductor. For the TEM mode assumption
to 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 and
common mode currents in an MTL model. Referring to Fig. 3.3, the phase currents are denoted
by Iu1 , Iv1 , and Iw1 on the AC drive side and by Iu2 , Iv2 , and Iw2 on the motor side for consistency
with the nomenclature that is used in practice to name the connections. The ground currents
are denoted by Ig1 and Ig2 . In each phase conductor, the current is the sum of a differential
mode current and a common mode current. Referring to the inverter side, we can write
Iu1 = Id1 + Ic ,
(3.12a)
24
C ABLES AND MULTICONDUCTOR TRANSMISSION LINE THEORY
Inverter
Iu1
Id1 Ic
Ic
Id1
Iu2
Iv1
Id2
Ic
Ic
Id2
Iv2
Iw1
Id3
Ic
Ic
Id3
Iw2
3Ic
MTL
Ic
Ic
Motor
3Ic
4Ic
4Ic
Figure 3.4: Common mode current analysis of a 4-conductor shielded cable connected between the
inverter 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 Ic
represents the common mode component. In theory, in a perfectly balanced system, no current
should flow in the ground conductor. Because of asymmetries in the motor or in the cable
itself, however, unwanted currents do indeed flow in the ground conductor. Note that in the
simulations, only the phase conductors are excited. The total common mode current in the
shield conductor, named Is , can be written as
Is = Iu1 + Iv1 + Iw1 + Ig1 = Id1 + Id2 + Id3 +4Ic .
{z
}
|
(3.13a)
=0
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 contribution
can 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 and
motor. The total common mode current that interacts with the inverter and motor is 3Ic , as seen
in Fig. 3.4, where the inverter and motor are represented as loads.
C HAPTER 4
The proposed delay-rational model
4.1
Green’s functions and boundary problems: a brief background
The method developed in the studies presented in this thesis is called the Delay-Rational
Green’s-Function-based Method, or DeRaG. It stems from the work presented in [20]. The
previous chapter summarized various methods for studying MTLs. They are predominantly
based on either the matrix-exponential solution or the modal decomposition or on a combination of the two (DEPACT). In [20], a different perspective is adopted. Specifically, transmission
lines are treated as a special case of planar circuits [32], in which only one dimension is nonnegligible with respect to the wavelength. A rational macromodel is derived, which is suitable
for lossy and dispersive multiconductor transmission lines. The port voltages are expressed, in
the Laplace domain, in terms of port currents, which are considered as forced sources at the
terminals 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 of
the 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 force
sources. The telegrapher’s equations read as follows:
∂
V(z, s) = −Z0 (s)I(z, s) ,
∂z
∂
I(z, s) = −Y0 (s)V(z, s) + Is (z, s) .
∂z
(4.2a)
(4.2b)
By considering the sources directly in the equations, we obtain homogeneous boundary conditions. If Neumann-type boundary conditions are used, then the values of the derivative on the
25
26
T HE PROPOSED DELAY- RATIONAL MODEL
boundaries are specified, as follows:
∂
∂
V(z, s)|z=0 = V(z, s)|z=` = 0 .
∂z
∂z
(4.3)
It is then possible to differentiate the voltage equation (4.2a) again with respect to z to allow
the substitution of the current equation (4.2b), yielding
∂2
V(z, s) − γ 2 (s)V(z, s) = −Z0 (s)Is (z, s) ,
2
∂z
(4.4)
where γ 2 = Z0 (s)Y0 (s) is the propagation operator. The set of equations (4.3) and (4.4) represents a Sturm-Liouville problem, whose solution can be found using the Green’s function
approach. All details can be found in [20] and the references therein. In [20], it is shown that
the solution to the problem is provided by the following (dyadic) Green’s function:
mπ mπ mπ 2 −1
2
2
1
· Am cos
z cos
z0 ,
G(z, z , s) = − ∑ γ (s) +
`
`
`
m=0
0
∞
where 1 is the identity matrix and the coefficients Am are defined as
q
 1, m=0
Am = q `
 2 , m = 1, · · · , ∞ .
(4.5)
(4.6)
`
An intuitive explanation of the Green’s function in (4.5) is that it gives the effect at the spatial
point z when an excitation source is applied at point z0 . The port voltages at z = 0, ` can be
expressed 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)(−Z0 (s)I0 (s)) + G(0, `, s)(−Z0 (s)I` (s)) ,
(4.7a)
V` (s) = G(`, 0, s)(−Z0 (s)I0 (s)) + G(`, `, s)(−Z0 (s)I` (s)) .
(4.7b)
Eq. (4.7) can be written in matrix form, yielding the impedance representation for a generic
multiconductor transmission line as follows:
V0 (s)
Z11 (s) Z12 (s) I0 (s)
I0 (s)
=
= Z(s)
,
(4.8)
V` (s)
Z12 (s) Z11 (s) I` (s)
I` (s)
where Z(s) is the open-end port impedance matrix, which is block symmetric, where each
block has dimensions of N × N. Note that the block entries of Z(s) are indeed the Green’s
functions of eq. (4.5), where z and z0 have been suitably updated, and they read as follows:
mπ 2 −1
2
1
· A2m Z0 (s) ,
Z11 (s) = Z22 (s) = ∑ γ (s) +
`
m=0
+∞
(4.9a)
4.1. G REEN ’ S FUNCTIONS AND BOUNDARY PROBLEMS :
A BRIEF BACKGROUND
mπ 2 −1
2
Z12 (s) = Z21 (s) = ∑ γ (s) +
1
· A2m Z0 (s) cos (mπ) .
`
m=0
27
+∞
(4.9b)
The summations are infinite, as expected for a delayed system. Eq. (4.9) is general, because no
conditions are imposed on the per-unit-length impedance or admittance matrices. The model
is, therefore, naturally suitable for both frequency-independent and frequency-dependent p.u.l.
parameters. To address the latter case, Z0 (s) and Y0 (s) are written in a rational form in terms
of poles and residues, for example, using the vector fitting technique of [48], as follows:
RZ
,
q=1 s − pq,Z
PZ
Z0 (s) = R0∞ + sL0∞ + ∑
0
Y
(s) = G0∞ + sC0∞ +
PY
RY
∑ s − pq,Y ,
(4.10a)
(4.10b)
q=1
where the R are the residual matrices, the p are the poles, and PZ and PY are the numbers of
poles used in the rational approximation; typically, a small number is required (1-3). The R0∞ ,
G0∞ , L0∞ and C0∞ terms describe the asymptotic behavior of the line at infinite frequency. In
the case of frequency-independent parameters, PZ = PY = 0. Eq. (4.10) can be expressed in
rational polynomial form as the ratios between the polynomial matrices B p (s) and D p (s) and
the polynomials A p (s) and C p (s), as follows:
Z0 (s) =
b0 sPZ +1 + b1 sPZ + · · · bPZ +1 B p (s)
,
=
a0 sPZ + a1 sPZ −1 + · · · aPZ
A p (s)
(4.11a)
Y0 (s) =
d0 sPY +1 + d1 sPY + · · · dPY +1 D p (s)
.
=
c0 sPY + c1 sPY −1 + · · · cPY
C p (s)
(4.11b)
Note that the polynomials A p (s) and C p (s) can be made strictly Hurwitz by construction, thus
imposing that the roots lie only in the left half-plane, which is an essential requirement for
stability. Regarding the passivity, if vector fitting is used, it applies an a posteriori passivity
enforcement. The impedance port matrix Z(s) can be recast as
∞
1
(−1)m
−1 2
,
(4.12)
Z(s) = ∑ Fm (s) Am B p (s)C p (s)
m
(−1)
1
m=0
2
where Fm (s)−1 = B p (s)D p (s) + A p (s)C p (s) mπ
1. The poles of the Z(s) matrix are eval`
uated as the determinants of the polynomial matrices Fm (s), for m = 0, · · · , ∞. Note that all
sub-blocks of the port impedance matrix share the same poles and that the residues are all
identical up to a multiplication factor of (−1)m .
In [49], we clarified the total number of poles computed per mode in the frequency-independent case. In particular, the m = 0 summation mode has N associated poles, whereas higher
modes have 2N poles (N complex conjugate pairs), where N is the number of conductors. It
is also worthwhile to clarify the number of poles per mode in the frequency-dependent case.
28
T HE PROPOSED DELAY- RATIONAL MODEL
We can follow the same reasoning given for the frequency-dependent case. In particular, for
the m = 0 mode, the number of poles is equal to (PY + 1)N. For the m > 0 modes, the number
of poles is equal to (PZ + PY + 2)N; this last general result has been explained in [20]. It can
be proven that a rational representation is well suited to be translated into a state-space system
using known macromodel synthesis techniques [30], as follows:
ẋ (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 is
crucial for the implementation of the model in a SPICE-like simulator. The main bottleneck
of the original approach presented in [20] was the large number of modes, and thus poles, that
was 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 of
residues and poles, a pole pruning method was proposed. Specifically, the dominant poles
and residues, those whose magnitudes cannot be neglected, are identified [18]. Pole pruning
is particularly necessary in the frequency-dependent case because the fitting procedure may
generate additional poles. The procedure presented in [20] is briefly summarized in the diagram
in Fig. 4.1. Under the assumption that Z0 and Y0 are passive, it was proven in [18] that this
Rational Green’s-function-based method is passive by construction.
4.2
Delay extraction
The primary disadvantage of the Rational Green’s-function-based method is the large number
of modes required. This is a common problem in rational techniques and is commonly resolved
using delay extraction techniques [35, 43]. In the work presented
√to date, we have used explicit
line delay extraction. The two-way line lossless delay TD = 2` L0 C0 can be extracted in two
main ways:
• By considering the imaginary part of the complex conjugate poles. This approach has
been explained in Papers A and B for the frequency-independent case and in Paper D for
the frequency-dependent one.
• By considering the eigenvalues of the matrix product L0 C0 , following an approach similar
to that proposed in [35]. This approach was used in Paper C.
When frequency-dependent p.u.l. parameters are considered, the last available frequency sample is taken. The delay can be incorporated into the model once the periodicity of the transfer
function 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 a
Fourier series allows the block impedances to be rewritten in terms of a Dirac comb, into which
the 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 expression
translates back into hyperbolic functions.
4.2. D ELAY EXTRACTION
29
Longitunal impedance Z’ and transversal admittance Y’
Available at discrete frequency samples
ࢅԢ ሺ‫ݏ‬ሻ ൌ ࡳሺ‫ݏ‬ሻ ൅ ‫࡯ݏ‬ሺ‫ݏ‬ሻ
ࢆԢ ሺ‫ݏ‬ሻ ൌ ࡾሺ‫ݏ‬ሻ ൅ ‫ࡸݏ‬ሺ‫ݏ‬ሻ
Rational polynomial forms of Z’ and Y’
ࢆԢሺ‫ݏ‬ሻ ൌ
࡮‫ ݌‬ሺ‫ݏ‬ሻ
‫ ݌ܣ‬ሺ‫ݏ‬ሻ
ࢅԢሺ‫ݏ‬ሻ ൌ
ࡰ‫ ݌‬ሺ‫ݏ‬ሻ
‫ ݌ܥ‬ሺ‫ݏ‬ሻ
Green’s function based method
Poles and residues form
࡮‫ ݌‬ሺ‫ݏ‬ሻ ࡰ‫ ݌‬ሺ‫ݏ‬ሻ
࡮‫ ݌‬ሺ‫ݏ‬ሻ
݉ߨ ʹ
ͳ
ࢆሺ‫ݏ‬ሻ ൌ ෍ ቈ
൅ ቀ ቁ ࢁ቉ ‫݉ʹܣ‬
൤
ሺ
ሻ݉
ሺ
ሻ
ሺ
ሻ
ሺ
ሻ
െͳ
‫ݏ ݌ܥ ݏ ݌ܣ‬
݈
‫ݏ ݌ܣ‬
൅λ
݉ ൌͲ
ࡾ࢔
ሺെͳሻ݉
൨ൌ ෍
‫ ݏ‬െ ‫݊݌‬
ͳ
൅λ
݊ൌͳ
Poles and residues pruning
Dominant poles and residues selected
݊݀
ࢆሺ‫ݏ‬ሻ ൌ ෍
݊ൌͳ
ࡾ࢔
‫ ݏ‬െ ‫݊݌‬
Figure 4.1: Original Green’s-function-based method.
To achieve these results, we analyzed the behavior of the poles and residues in the lossless 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].
30
T HE PROPOSED DELAY- RATIONAL MODEL
The second method must be used in the case of large cable bundles because the computation of
the poles becomes highly demanding of computation time and prone to numerical inaccuracies
when 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 the
same properties. The number of families is equal to the number of conductors. The time delay
for each conductor can be properly included by observing that given the complex conjugate
poles 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 certain
mode m̂. Denoting by β̂k,m the equispaced fictitious imaginary parts for m = 1, · · · , +∞, we
can write β̂k,m = mβ̂k,1 . It is worth noting that the imaginary parts of these fictitious poles
correspond to the lossless ones, such that
β̂k,1 =
2π
,
Tk
(4.15)
where the Tk are the delays associated with each conductor. With a proper mathematical formulation, 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, represented by the index k = 1, · · · , N, as seen in Fig. 4.2, where the 6 families of complex conjugate
poles 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 is
adopted, the real parts of the poles for the different families become asymptotically constant
after a certain mode m̂, as seen in Fig. 4.3 for a 2-conductor TL with frequency-dependent
parameters, from Paper D, and in Fig. 4.4 for a 9-conductor frequency-independent cable bundle, from Paper C. In fact, for large cable bundles, numerical inaccuracies can arise, and more
careful identification of the pole families is necessary. The asymptotic value of the real part for
family k is denoted by α̂k .
Regarding the residues, they are also clustered into families and exhibit asymptotic behavior; in particular, their phases tend asymptotically to zero, and the real part remains constant
among the families, that is,
Rk,n = R̂k
k = 1, · · · , N ,
(4.16)
where the index k spans all families.
The m̂ 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 when
the VF approach is used for the pole computations), over the real part of the pole families, as
shown in the diagram in Fig. 4.5.
Given a lossy MTL, we have proven that the block impedance matrices can be written, in
the Laplace domain, as follows:
Np
N R̂ 11 T
Tk
Rn11
k k
+∑
coth (s − α̂k )
,
(4.17a)
Z11 (s) = ∑
2
2
n=1 s − pn
k=1
| {z }
Rational delayless
4.2. D ELAY EXTRACTION
1
31
×1011
poles vf
ℑ
0.5
0
-0.5
-1
-4
-3.5
-3
-2.5
-2
-1.5
-1
×104
ℜ
Figure 4.2: Complex conjugate pole locations computed using a modal rational approximation approach for a 6-conductor ribbon cable with frequency-independent p.u.l. parameters, from Paper C.
2
×1012
Family 1
Family 2
ℑ (p)
1
0
-1
-2
-2.5
-2
-1.5
-1
ℜ (p)
-0.5
0
×10
9
Figure 4.3: Complex conjugate pole families in the complex plane for a 2-conductor line with frequencydependent parameters, from Paper D.
32
T HE PROPOSED DELAY- RATIONAL MODEL
×108
poles vf
p̂
1.5
1
ℑ
0.5
0
-0.5
-1
-1.5
-2
-7
-6
-5
-4
-3
ℜ
-2
×104
Figure 4.4: Complex conjugate pole locations for a 9-conductor cable bundle, from Paper C. The blue
circles represent the asymptotic real parts for each family, α̂k .
Np
N R̂ 12 T
Rn12
Tk
k k
Z12 (s) = ∑
csch (s − α̂k )
,
+∑
s
−
p
2
2
n
n=1
k=1
| {z }
(4.17b)
Rational delayless
where N p is the total number of poles/residues used, the pn are real or complex conjugate poles,
ij
and the Rn are the corresponding residue matrices for Zi j (s), i, j = 1, 2. The α̂k denote the
constant asymptotic real parts of the poles associated with each conductor (indexed by k), and
ij
the R̂k 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 delayless part of the system is stable, because all poles lie on the left side of the complex plane, that
is, α̂k = −|α̂k |.
4.3
The delay-rational state-space form
The impedances are written as a sum of rational functions, and this allows direct translation
into the time domain using known macromodel synthesis techniques [21, 30]. The resulting
model reads as follows:
ẋ (t) = Ax (t) + Bi (t) ,
(4.19a)
4.3. T HE DELAY- RATIONAL STATE - SPACE FORM
33
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000000000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000000000
N_mod = M;
STOP = FALSE;
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00000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
END
yes
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000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000
m == N_mod
m = m + 1;
no
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0૚૚
0 0 0 0 0 0 0 0 0 0 0݉
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00000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
% Port impedance evaluation for the mode m
ࢆ ȁ
ൌ ൤ࢆሺ࢙ሻ ࢅሺ࢙ሻ ൅ ቀ
݉ߨ
ቁ ࡵ൨
݈
‫ࢆ ܣ‬ሺ࢙ሻ
% Vector fitting in order to identify poles and
residues
VF
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00‫ࡾס‬
0 0 0 0 0 0 0 0 0 0 0 0݇ǡ݉
0 0 0 0 0 0 0ෝ
0 0 0 0 0 0 0െ
0 0 0 0 0 0 0 0 0‫ࡾס‬
0 0 0 0 0 0 0 0 0 0 0 0 0 0݇ǡ݉
0 0 0 0 0 0 0ෝ0 0 0 0 െͳ
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ቤ00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ቤ000 000 000 000 000 000൏
0 0 0 0 0 0 0 0 0‫݈݋ݐ‬
0 0 0 0 0 0 0 0 0 0 0ܴ݁‫ݏ‬
0000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00‫ࡾס‬
00 00 00 00 00 00 00 00 00 00 00 00݇ǡ݉
00 00 00 00 00 00 00ෝ
00 00 00 00 െͳ
00 00 00 00 00 00 00 00 00 00 00െ
00 00 00 00 00 00 00 00 00‫ࡾס‬
00 00 00 00 00 00 00 00 00 00 00 00 00 00݇ǡ݉
00 00 00 00 00 00 00ෝ
00 00 00 00 െʹ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ߙ
00 00 00 00 00 00݇ǡ݉
00 00 00 00 00 00 00ෝ
00 00 00 00 0000 00 00 00െ
00 00 00 00 00 00 00 00 00ߙ
00 00 00 00 00 ݇ǡ݉
00 00 00 00 00 00 00 00ෝ
00 00 00 00 െͳ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ቤ00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ቤ00 00 00 00 00 ൐
0 0 0 0 0 0 0 0 0 0‫݈݋ݐ‬
0 0 0 0 0 0 0 0 0 0 0 0ܴ݁
00000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ߙ
0 0 0 0 0 0 0 ݇ǡ݉
0 0 0 0 0 0 0 0ෝ
0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
no
yes
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 0000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00݇
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0݇
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
% Evaluation of the asymptotic values for
poles and residues, and ݉
ෝ
෡ ǡ ߙො ǡ ݉
ࡾ
ෝ
Figure 4.5: Block diagram for the algorithm used to determine the m̂ mode, from Paper C.
34
T HE PROPOSED DELAY- RATIONAL MODEL
D0
i(t)
B
+
Ẋ(t)
x(t)
ʃ
v(t)
C
+
A
vd(t)
D(t)
Figure 4.6: Block diagram of the state-space model (4.19).
v (t) = Cx (t) + D0 i (t) + vd (t).
(4.19b)
The system is presented in diagrammatic form in Fig. 4.6. The vector ẋ(t) is a suitable statespace variable.
The vectors i(t) and v (t) are the terminal currents and voltages, respectively, associated with
the 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 delayedattenuated contributions of the input currents. We recall the following:
• A is constructed using the poles of the rational delayless part; each pole is repeated on
the 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
N R̂k11 T2k
0
D0 = ∑
.
0
R̂k11 T2k
k=1
(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
N D11 (t) D12 (t)
D(t) = ∑
,
(4.21)
D12 (t) D11 (t)
k=1
with
D11 (t) = R̂k11 Tk e−|α̂k |t
+∞
∑ δ (t − mTk ) ,
m=1
(4.22a)
4.3. T HE DELAY- RATIONAL STATE - SPACE FORM
D12 (t) = R̂k12 Tk e−|α̂k |t
35
+∞
Tk
∑ δ t − mTk − 2 .
m=0
(4.22b)
The vd (t) term can be regarded as the delayed impulse response of the system, that is, the
output in response to a series of Dirac delta inputs. We can recognize a discrete convolution in
the expression vd (t) = D(t) ⊗ i(t).
The SPICE representation can be obtained by using current-controlled voltage sources and
standard circuit elements for the delayless part and, for the delay part, using lossless transmission lines to reproduce the lossless delay and current-controlled voltage sources to reproduce
the attenuation.
To implement the system (4.19) in MATLAB, a backward Euler integration solver has
been used, with a fixed time step of dt. We note that because the main goal is to implement
the system (4.19) in SPICE, no particular attention has been paid to the MATLAB solver itself
because SPICE will use an adaptive time-step method, for which the user can set the maximum
time-step size.
Note that the resulting model can be easily incorporated into recently developed modelreduction techniques [51], which would allow the state-space size to be further reduced, thereby
speeding up the time-domain simulations.
36
T HE PROPOSED DELAY- RATIONAL MODEL
C HAPTER 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 A
Title: 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 Compatibility (EMC Europe), Gothenburg, Sweden, 2014.
Summary: This paper represents the first attempt to extract the line delay and to incorporate
it into the Rational Green’s-function-based model. The simple case of a two-conductor transmission line (reference included) is considered, with frequency-independent per-unit-length
parameters. From a study of the poles, we determine that the real parts of the complex conjugate poles do not vary among the m > 0 modes. In the lossless case, the imaginary part is
linear with respect to the lossless line delay. In the lossy case, the imaginary parts become
equally spaced after a certain mode m. The analogy between the time-domain expression of
the block impedances and a Fourier series allows the block impedances to be written in the
form of a Dirac comb, into which the line delay TD can be easily incorporated. The extension
to the lossy case and the state-space representation are merely outlined.
Contribution: The author developed the mathematical formulation, performed the tests and
wrote the majority of the paper. The author presented the paper at the conference, with followup discussions.
5.2
Paper B
Title: A Delay-Rational Model of Lossy Multiconductor Transmission Lines with Frequency
Independent 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 multiconductor transmission lines. In particular, it is found that poles and residues can be grouped
into families that share the same properties: the same (constant) real part and the same asymptotic behavior of the imaginary part. From a given summation mode m̂, the poles and residues
exhibit periodic properties, which allow the explicit extraction and incorporation of the time
delay. Conditions for identifying m̂ are given. A delay-rational state-space model in the time
domain is presented, in which delayed Dirac combs are incorporated.
Contribution: The author developed the mathematical formulation, performed the tests and
wrote the majority of the paper. The author was responsible for the submission and revision
processes, responding to the reviewers and applying the suggested modifications.
5.3
Paper C
Title: 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 Advanced Applications (ICEAA), Turin, Italy, 2015.
Summary: In this paper, the authors address the problem of computing the poles in the case of
large cable bundles, and a method for reducing the order of the model is proposed. The wellknown vector fitting algorithm is used for this purpose, and the inversion of the polynomial
matrix 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. The
model 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 D
Title: 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 Compatibility (EMC), Dresden, Germany, 2015.
Summary: The delay-rational model is extended to include MTLs with frequency-dependent
per-unit-length parameters. The p.u.l. longitudinal impedance Z0 (s) and transverse admittance
Y0 (s) are considered to be available at N f discrete frequency samples. A rational polynomial
form is obtained using the vector fitting technique. Redundant poles and residues are discarded
through pole/residue pruning. Unlike in the frequency-independent case, the real part of the
poles 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 results
found for the case of frequency-independent p.u.l. parameters to be extended to the frequencydependent case.
5.4. PAPER D
39
Contribution: The author participated in the elaboration of the ideas, the development of the
mathematical formulation and the performance of the tests. The author wrote the majority of
the text and presented the paper at the conference, with follow-up discussions.
40
C HAPTER 6
Conclusions and future work
6.1
Conclusions
A new model for multiconductor transmission lines has been presented, called DeRaG. This
model is based on a previous work of [20], where the solution for the MTL equations is presented, in an original way, in terms of Green’s functions. The computed poles and residues can
be regarded as the exact poles and residues of the line, in contrast with other rational approximation techniques that estimate the poles, and whose accuracy depends on the approximation
order, 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 to
reduce the number of poles used, while increasing the accuracy?
In the new proposed model, called DeRaG, the properties of the poles and residues have
been 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 lossless
modal delays are used in order to reduce the number of poles, while increasing the accuracy
of the model itself. In fact, ringing effects in the time domain are eliminated because all the
bandwidth is correctly characterized. The delays can be computed either from the poles or
from 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 the
final state-space representation?
We have shown that the delays are explicitly incorporated into the model through a proper
mathematical formulation. The additional terms that account for the delays do not compromise
the simplicity of the model and a new delayed state-space system is obtained. The primary
advantage of the proposed method with respect to other well-known techniques relies on the
delayed 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 optimization achieved using standard MOR algorithms such as PRIMA [51]. A formal verification
of 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 circuit
elements are used?
The circuit implementation of a model in state-space form is straightforward to obtain,
R
thereby allowing the use of well-known circuit simulation tools such as PSPICE A/D
. In
fact, the delayless part in the frequency domain translates in a standard macromodel in the time
domain. A standard macromodel can be represented in PSPICE with standard circuit components, namely resistances, inductances and current-controlled sources, following the guidelines
provided in [21, 30]. The delay part, instead, can be accounted using standard lossless transmission lines, whose model in PSPICE is exact. Current-controlled voltage sources can be
used to account for the attenuation.
The resulting final circuit model can be identified and translated into PSPICE, making the
model accessible to a broader set of users who typically employ circuit tools for their analyses. The model will have the same structure both for the frequency dependent and frequency
independent cases, because no convolution products are involved. The terminations can be
incorporated in PSPICE in a straightforward way.
6.2
Future work
Some 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 model
will be explained in details in forthcoming papers.
Two additional models need to be developed to allow the analysis of a VFD system: one for
the AC drive, and one for the motor. These models must be suitable for addressing EMI conducted emissions. Once a complete detailed model “drive-cable-motor” is obtained, the impact
of 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 be
modeled. Additionally, switch disconnectors are typically used in the industry, and these components must be properly considered because they can also impact the system performance.
Disconnectors and interconnections within the converter, which are represented by imperfect
conductors with irregular geometries, require full-wave models. To this end, we will use the
partial element equivalent circuit (PEEC) approach to provide an electromagnetic model of the
interconnects.
The final phase of this PhD study will be devoted to measurements. The models obtained
through simulation must be verified against experimentally measurement results.
For this purpose, laboratory activities in the EMC laboratory of LTU and pre-compliance
testing of VFDs are being conducted.
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Part II
49
50
PAPER A
Delayed Impedance Models of
Two-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 Europe), Gothenburg, Sweden, 2014.
c 2014, IEEE, Reprinted with permission.
51
52
Delayed Impedance Models of Two-Conductor Transmission
Lines
Maria De Lauretis, Jonas Ekman and Giulio Antonini
Abstract
This paper presents a new delayed model of two-conductor transmission lines with frequencyindependent per-unit-length parameters. In particular, the line delay extraction problem is considered. By use of a dyadic Green’s function macromodel method, the rational form of the
open-end impedance matrix allows an easy identification of poles and residues, and a new
technique for the extraction of the line delay in an analytical way is gained, without any impact
on the complexity of the line macromodel itself. By use of Laplace and Fourier transforms, the
transfer function is expressed in terms of the Dirac comb. The delay is then easily identified
and directly incorporated into the system impulse response. Giving a current-controlled representation, the port voltages are evaluated. Thanks to the formulation of the transfer function by
use of the Dirac comb, the convolution product is avoided, gaining accuracy and time-saving
from a computational point of view. Numerical results confirm the validity of the proposed
delay-extraction technique. The basic ideas for the extension of the proposed technique to the
lossy case are outlined.
1
Introduction
Nowadays, interconnects play an increasing important role in high-speed VLSI chips and multichip modules. The system performance may be severely deteriorate by the interconnect effects such as signal delay, reflection and dispersion, or crosstalk[1, 2]. The literature devoted
to transmission line and interconnect modeling is vast (see [3, 4] and references therein). The
transmission line equations can be formulated in both the frequency and time domains, this
last important for power systems to predict the transient behaviour of long power lines and
cables; on the other hand, the knowledge of the frequency content, or spectrum, of a signal is
essential in the EMC context, for example in order to check if the system satisfies regulatory
limits, or how it interacts with other electronic systems [5, 2]. A still open issue in this field
is the effective modeling of long transmission lines (e.g. power/signal cables) since standard
techniques require a large number of unknowns not being able to incorporate the propagation
delay [1, 3, 6].
Over the years, some techniques have proposed delay extraction-based macromodeling algorithms both for time-domain and tabulated frequency data [7, 8]. A well-known method for
the analysis and the delay extraction is the Method of Characteristic (MoC)[9], and its generalizations [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 frequency points by use of modal analysis. Consequential, irrational functions arise in frequency
domain, and the time domain formulation requires some form of approximation of the inverse
Laplace transform; the numerical efficiency in terms of execution time is heavily dependent
on the particular implementation [12]. Another well studied technique is the one based on
Padé macromodeling approach [13, 14], where the delays are model implicitly. The reader is
referred 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 of
delay-rational macromodels of long interconnects. More specifically, stemming from the spectral representation of TLs described in [15], the pole analysis is performed. Dealing with linear
and time-invariant system, the use of Fourier and Laplace transforms [16] allow to express the
impulse 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 “inputstate 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 on
the Green’s function impedance formulation and only basic Fourier and Laplace transforms are
involved. It is to be pointed out that, since Dirac functions are used, the convolution product
used in the current-controlled representation is avoided, yielding to faster simulation and less
prone 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’s
method for the analysis of the TLs. Section III presents the pole/residue analysis of the rational
macromodel. In Section IV, the delayed lossless transmission line model is presented, with the
impedance-based representation by using of the Dirac comb. In Section V, the extension of the
proposed approach to lossy TLs is outlined. Sections VI and VII presents numerical results
and 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 electromagnetic
field mode propagates, read [18]
d
V (z, s) = − (R + sL) I (z, s) = −Z(s)I (z, s)
dz
d
I (z, s) = − (G + sC)V (z, s) + Is (z, s) = −Y (s)V (z, s) + Is (z, s)
dz
(1a)
(1b)
with R, L, C and G denoting per-unit-length (p.u.l.) parameters [18, 19], here assumed frequencyindependent. Is (z, s) represents a p.u.l. current source located at abscissa z. In [15] it has been
2. T RANSMISSION L INE S PECTRAL M ODEL
55
I 0 (s)
I l (s)
+
+
V 0 (s)
V l (s)
-
z
Figure 1: Two conductor transmission line.
demonstrated that, given the Green’s function as
0
∞
G(z, z ) = − ∑
nπ 0
nπ
` z cos ` z
2
γ 2 (s) + nπ
`
cos
A2n
n=0
(2)
where
q
 1, n=0
An = q `
 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) =
∑
A2n Z(s)
n=0 γ 2 (s) +
(5a)
nπ 2
`
A2n Z(s) cos (nπ)
∑
nπ 2
n=0 γ 2 (s) + `
∞
Z12 (s) = Z21 (s) =
(5b)
which is well suited to be translated into a state space realization because the impedances are
written as a sum of rational functions. Impedances (5) admit a closed form representation in
the Laplace domain which reads
Z11 (s) = Z22 (s) = η(s) coth (γ(s)`)
(6a)
Z12 (s) = Z21 (s) = η(s) csch (γ(s)`)
p
where γ(s) = (R + sL) (G + sC) is the propagation constant, and η(s) =
characteristics impedance of the line.
(6b)
q
(R+sL)
(G+sC)
is the
56
PAPER A
3
Rational macromodel
The poles of the transmission line can be evaluated as the poles of impedance Z(s) and the
zeros of the polynomial at the denominator of Z matrix entries
2
γ (s) +
nπ 2
`
=0
(7)
which can be re-written as a trinomial term [16]
RC + LG
2
LC s + s
+
LC
RG + nπ
`
LC
2 !!
= 0.
(8)
It is convenient to re-write the term in brackets in the well known canonical form [16]
RC + LG RG + nπ
`
+
s +s
LC
LC
2
2
= s2 + 2ζn ωn s + ωn 2
(9)
where:
• ωn is the resonance angular frequency (10). For the sake of simplicity, for n = 1 we call
ω1 = ω .
s
2 2
RG + n `π2
ωn =
(10)
LC
• ζn is the damping factor (11). For the sake of simplicity, for n = 1 we call ζ1 = ζ .
v
RC + LG u
u 1
ζn =
(11)
t
n2 π 2
2
LC RG +
`2
The complex conjugate poles of a trinomial term can be expressed as sn(1,2) = αn ± jβn (the
sub-index n will be omitted for n = 1). It can be simply proven that αn and βn can be written
in terms of ω and ζ , as will be shown in the next section, where the trinomial term is analyzed
for n = 0 and n > 0 respectively.
3.1
Poles analysis
• n=0
At the denominator, the poles s1 = − RL and s2 = − CG are real and stable (Re < 0). Considering
the numerator (R + sL) = L s + RL , a pole-zero cancellation occurs, this last operation allowed
for the stability of the pole. A binomial term then results.
3. R ATIONAL MACROMODEL
57
• n>0
For n = 1 and s(1,2) = α ± jβ we have
p
(
ω = α2 + β 2
ζ = √ −α
2
2
(
α = −ζ ω
p
β = ω 1−ζ2.
α +β
(12)
It is useful to express αn and βn (for n > 1) as a function of ω and ζ (obtained for n = 1), as
done in (13). This result is particularly significant since it proves that the real part of the poles
does not change with n, thus all the complex conjugate poles share the same real part.
αn = −ζ ω = α
q
βn = ω Q2n − ζ 2
v
u
u 1 + nπ 2 1
`
RG
t
Qn =
π 2 1
1 + ` RG
(13a)
(13b)
(13c)
Finally, we obtain a simpler expression for Z11 (s) and Z12 (s)
Z11 (s) =
2 +∞
R + sL
1
+
∑
`LC n=1 (s − α)2 + βn2
`C s + CG
Z12 (s) =
2 +∞
R + sL
1
+
(−1)n .
∑
G
`LC n=1 (s − α)2 + βn2
`C s + C
(14a)
(14b)
Following this approach, the corresponding residues can be computed splitting the term in the
summation as
k1,n βn
k2 (s − α)
+
(15)
2
2
(s − α) + βn (s − α)2 + βn2
with the residues being
(
k1,n =
1
βn (R + αL)
k2 = L .
(16)
By using the inverse Laplace transform, the corresponding expressions in time domain are
"
#
1 −Gt
2 αt +∞
z11 (t) =
e C +
e ∑ (k1,n sin(βnt) + k2 cos(βnt)) Θ(t)
`C
`LC n=1
"
#
1 −Gt
2 αt +∞
z12 (t) =
e C +
e ∑ (k1,n sin(βnt) + k2 cos(βnt)) (−1)n Θ(t)
`C
`LC n=1
where Θ(t) is the Heaviside step function.
58
PAPER A
4
Delayed Lossless Transmission Line Model
In the lossless case, being R = G = 0 ,
• for n = 0 the binomial term is reduced to
1
√ .
• for n > 0, α = 0 and βn = nπ
`
LC
1
`Cs ,
means there is a pole in zero;
The period of transient phenomena is
√
T = 2` LC .
(18)
Hence, the fundamental frequency is f0 = 2`√1LC . Note that this result is in agreement with the
well known theoretical one for the delay time of lossy TLs[5]
√
Td = ` LC .
In fact, following our notation, the delay time is equal to
The poles result to be purely imaginary as
(19)
T
2
.
sn(1,2) = ± jβn = ± j2πn f0 .
(20)
The Laplace self and transfer impedances transform read as
Z11 (s) =
1
2 +∞
s
+
∑
s`C `C n=1 s2 + βn2
(21a)
Z12 (s) =
1
2 +∞
s
+
(−1)n
∑
2
s`C `C n=1 s + βn2
(21b)
whose inverse is given by
z11 (t) =
z12 (t) =
!
1
2 +∞
+
∑ cos(2πn f0t) Θ(t)
`C `C n=1
!
+∞
1
2
+
cos(2πn f0t)(−1)n Θ(t) .
∑
`C `C n=1
(22a)
(22b)
4.1
Impedance-based input-output representation by using Dirac comb
• Self impedance z11 (t)
4. D ELAYED L OSSLESS T RANSMISSION L INE M ODEL
59
It can be observed that (22a) represents an unilateral Fourier series, that is generally expressed
as
+∞
t
s(t) = s0 + 2 ∑ |sn |cos 2πn + θn
(23)
T
n=1
1
and θn = 0. By using the exponential form of the Fourier
with identical coefficients equal to `C
series, the continuous Fourier transform reads as
!
+∞
Z11 ( f ) =
∑
sn δ ( f − n f0 ) ⊗ F (Θ(t))
(24)
n=−∞
where F denotes the Fourier transform and ⊗ denotes the convolution product. It is straightforward to obtain the time domain expression in terms of Dirac comb, as
z11 (t) =
where η =
q
L
C
+∞
T +∞
δ (t − nT )Θ(t) = 2η ∑ δ (t − nT )Θ(t)
∑
`C n=−∞
n=−∞
(25)
for the lossless case.
• Transfer impedance z12 (t)
Equation (22b) can also be regarded as a unilateral Fourier transform. It is worth to note
that (−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) =
1
2 [cos(2πn f 0t + πn) + cos(2πn f 0t − πn)] . Since in general A cos(α + π) = A cos(α − π) , the
term in the summation can be written as cos(2πn f0t − πn) . Finally, the z12 (t) expression can
be 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 Dirac
comb is (26), with T2 the looked delay.
"
#
+∞
T
z12 (t) = 2η ∑ δ t − nT −
Θ(t) .
(26)
2
n=−∞
4.2
Lossless case: time domain realization
Dirac 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 the
Dirac comb expression read as
+∞
z11 (t) ⊗ i1 (t) = ηi1 (t) + 2η
∑ i1(t − nT )
n=1
(28a)
60
PAPER A
+∞
T
z12 (t) ⊗ i2 (t) = 2η ∑ i2 t − nT −
2
n=0
(28b)
The same applies to port 2. As expected, both the contributions to port voltage v1 (t) are a
scaled and delayed replica of the port currents i1 (t) and i2 (t). For lossless two-conductor TLs
this 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
!
#
2 +∞
1
+
z11 (t) ⊗ i1 (t) =
∑ cos(2πn f0t) Θ(t) ⊗ i1(t)
`C `C n=1
"
!
#
1
2 +∞
z12 (t) ⊗ i2 (t) =
+
∑ cos(2πn f0t)(−1)n Θ(t) ⊗ i2(t) .
`C `C n=1
"
As it can be easily observed, this last expression is more involved than the one with the Dirac
comb formulation, and it requires the explicit computation of the convolution product.
5
Delayed Lossy Transmission Line Model
The lossy case can be regarded as a perturbed version of the lossless one. As in the lossless
case, the complex pole pairs share the same negative real part and this suggests the idea to use
the Dirac comb. This step is not straightforward because in the lossy case the imaginary parts
are not equally spaced as it happens in the lossless case and, thus, no periodic function can be
directly recovered. The basic steps in order to achieve this goal are described below. In the
following 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 as
βnLL , and βnLY for the lossy case as in (13b), the index n = ñ p is found such that the
following conditions hold
(
p
βnLY = ω Q2n − ζ 2 , n = 1, · · · , ñ p − 1
(30)
βnLY = βnLL ,
n = ñ p , · · · , ∞ .
Similar considerations hold for the residual: from the expression (16), using the subindex LL for the general lossless case, and LY for the lossy case, we have that k2 = L is
unchanged, while for k1,n we have
(
1
k1,nLY = βnLY
(R + αL), n = 1, · · · , ñ p − 1
(31)
k1,nLY = 0 ,
n = ñ p , · · · , ∞ .
6. N UMERICAL E XPERIMENTS
61
2. The summation in (14a) is then split in accordance with the previous observations, and
some 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 final
impedances for the lossy case
#
"ñ −1
ñ p −1
p
1
L(s − α)
2
R + sL
+
Z11 (s) =
∑ (s − α)2 + β 2 − ∑ (s − α)2 + β 2 +
`LC n=1
`C s + CG
n=1
nLY
nLL
T
1
+ η coth (s − α)
−
2
`C(s − α)
"ñ −1
#
ñ p −1
p
(R + sL)(−1)n
L(s − α)(−1)n
1
2
+
Z12 (s) =
∑ (s − α)2 + β 2 − ∑ (s − α)2 + β 2 +
`LC n=1
`C s + CG
n=1
nLY
nLL
T
1
+ η csch (s − α)
.
−
2
`C(s − α)
(32)
(33)
The rational contributions in (32) and (33) admit a standard state-space representation with notdelayed port currents as inputs and port voltages as outputs, the hyperbolic terms correspond
to a Dirac comb in the time domain. Hence, the global hybrid state-space Dirac comb model,
in the time domain, becomes
ẋ (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 lossy
case and its extension to multiconductor transmission lines will be presented in forthcoming
papers, along with the complete derivation of both the frequency and the time domain models,
here only outlined for lack of space.
6
6.1
Numerical Experiments
Lossless case
Let us consider a two-conductor lossless transmission line of length ` = 0.5 m, opened at both
ends. 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, rise
time and fall time equal to 200 ps, and unitary magnitude. When using (22), the summation
needs to be truncated to a suitable N [15]. It is worth noticing that fast simulations require to
keep minimum the value of N; on the other hand, N has to be large enough to preserve the
62
PAPER A
120
v (t) Dirac comb
11
100
v (t) Pole/residue
11
v11(t)
80
60
40
20
0
−20
0
50
100
150
Time [ns]
200
250
Figure 2: Port voltage v11 (t),(N = 30, ` = 0.5 m).
accuracy. When using the Dirac comb expression by (28), the infinite summation is truncated
based on the selected time slot (see also [17]).
The results obtained by using the standard (cosine) formulation in Section 4.2 and the new
Dirac 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” denotes
the new proposed approach. Using N = 30 , namely 60 complex conjugate poles and one real
pole, 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 obviously
it is not visible in the Dirac comb result, as highlighted in Fig. 3a. Secondly, the Dirac comb
formulation allows to correctly capture the rise time, whilst the standard (cosine) formulation
does 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 is
affected 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 .
6. N UMERICAL E XPERIMENTS
63
v11(t) Dirac comb
110
v11(t) Pole/residue
v11(t)
105
100
95
90
8.5
9
9.5
10
Time [ns]
(a) Ringing effect
100
v11(t)
80
60
40
20
v11(t) Dirac comb
0
v11(t) Pole/residue
8.2
8.4
8.6
8.8
Time [ns]
9
9.2
(b) Rise time
Figure 3: Details for port voltage v11 (t),(N = 30, ` = 0.5 m).
64
PAPER A
v21(t) Dirac comb
12
v21(t) Pole/residue
10
v21(t)
8
6
4
2
0
34
35
36
Time [ns]
37
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 computed
and shown in Fig. 5. As clearly visible, the new formulation (33) provides, at high frequencies, better results than the poles/residues one (5b), because it incorporates the high frequency
behavior through the Dirac comb.
7
Conclusions
In this paper, a mixed rational-delayed model of two-conductor transmission lines is developed. To this aim, the dyadic Green’s function-based approach, with frequency-independent
parameters, has been assumed as background. The investigation of the poles for lossless and
lossy 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 fundamental
frequency for the transient phenomena. Hence, through a combination of basic Fourier and
Laplace transforms, the impulsive response of the lossless case is obtained in terms of Dirac
comb, whose periodicity is related to the extracted delay of the line. The lossy case is easily
recovered from the lossless one using a perturbation approach which naturally leads to a mixed
rational-delayed transmission line model. The major advantage of the proposed method with
R EFERENCES
65
2
10
Z12LY delayed
Z12LY poles/residues
0
Relative error
10
−2
10
−4
10
−6
10
0
2
4
6
Freq [Hz]
8
10
9
x 10
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, which
reduces the complexity of the state-space model, since the Dirac comb already takes the delay
into account. The numerical experiments have been demonstrated that the proposed method
outperforms the full rational model while providing a good accuracy when compared with the
rigorous solution in the frequency domain.
References
[1] C. R. Paul, Analysis of multiconductor transmission lines. Second Edition, IEEE Press,
Hoboken, NJ: John Wiley Interscience, 2008.
[2] I. M. Elfadel, H-M. Huang, A. E. Ruehli, A. Dounavis, M. S. Nakhla, “A comparative
study of two transient analysis algorithms for lossy transmission lines with frequencydependent data,” in Digest of Electr. Perf. Electronic Packaging, Oct. 2001, pp. 255 –
258.
[3] R. E. Collin, Field Theory of Guided Waves.
IEEE Press, New York, 1991.
[4] R. Achar, M. Nakhla, “Simulation of high-speed interconnects,” Proceedings of the IEEE,
vol. 89, no. 5, pp. 693–728, May 2001.
[5] C. R. Paul, Introduction to Electromagnetic Compatibility.
Interscience, 2006.
Hoboken, NJ: John Wiley
66
PAPER A
[6] P. Maffezzoni, A. Brambilla, “Modelling Delay and Crosstalk in VLSI Interconnect for
Electrical Simulation,” Electron. Letters, vol. 36, no. 10, pp. 862–864, May 2000.
[7] A. Charest, M. Nakhla, R. Achar, D. Saraswat, N. Soveiko, and I. Erdin, “Time domain
delay extraction-based macromodeling algorithm for long-delay networks,” IEEE Transactions on Advanced Packaging, vol. 33, no. 1, pp. 219–235, Feb 2010.
[8] P. Triverio, S. Grivet-Talocia, and A. Chinea, “Identification of highly efficient delayrational macromodels of long interconnects from tabulated frequency data,” IEEE Transactions on Microwave Theory and Techniques, vol. 58, no. 3, pp. 566–577, March 2010.
[9] F. H. Branin, “Transient analysis of lossless transmission lines,” Proceedings of the IEEE,
vol. 55, no. 11, pp. 2012–2013, Nov. 1967.
[10] A. J. Gruodis, C. S. Chang, “Coupled lossy transmission line characterization and simulation,” IBM Journal of Research and Development, vol. 25, no. 1, pp. 25–41, 1981.
[11] S. Lin and E. S. Kuh, “Transient simulation of lossy interconnects based on recursive
convolution formulation,” IEEE Transactions on Circuits and Systems, I, vol. 39, no. 11,
pp. 879–892, Nov. 1992.
[12] S. Grivet-Talocia, H.-M. Huang, A. E. Ruehli, F. Canavero and I. M. (Abe) Elfadel, “Transient analysis of lossy transmission lines: an efficient approach based on the method of
characteristics,” IEEE Transactions on Advanced Packaging, vol. 27, no. 1, pp. 45–56,
Feb. 2004.
[13] A. Dounavis, X. Li, M. S. Nakhla, R. Achar, “Passive closed-form transmission-line
model for general-purpose circuit simulators,” IEEE Transactions on Microwave Theory
and Techniques, vol. 47, no. 12, pp. 2450–2459, Dec. 1999.
[14] A. Dounavis, E. Gad, R. Achar, M. S. Nakhla, “Passive model reduction of multiport
distributed 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 lossy
and dispersive multiconductor transmission lines,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 4, pp. 880–895, Apr. 2008.
[16] R. Beerends,
Fourier and Laplace Transforms,
ser. Fourier and
Laplace 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.
2001.
Academic Press,
[18] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York, NY: John Wiley
& Sons, 1992.
67
[19] J. Brandão Faria, Multiconductor Transmission-Line Structures: Modal Analysis Technique. 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
68
PAPER B
A Delay-Rational Model of Lossy
Multiconductor Transmission Lines
with Frequency Independent
Per-Unit-Length Parameters
Authors:
Maria De Lauretis, Jonas Ekman and Giulio Antonini
Reformatted version of paper originally published in:
IEEE Transactions on Electromagnetic Compatibility, 2015.
c 2015, IEEE, Reprinted with permission.
69
70
A Delay-Rational Model of Lossy Multiconductor
Transmission Lines with Frequency Independent
Per-Unit-Length Parameters
Maria De Lauretis, Jonas Ekman and Giulio Antonini
Abstract
Cables, printed circuit boards, and VLSI interconnects are commonly modeled as multiconductor transmission lines. Models of electrically long transmission lines are memory and time
consuming. In this paper, a robust and efficient algorithm for the generation of a delay-based
model is presented. The impedance representation via the open-end matrix Z is analyzed. In
particular, the rational formulation of Z in terms of poles and residues is exploited for both
lossless and lossy cases. The delays of the lines are identified, and explicitly incorporated into
the model. A model order reduction of the system is automatically performed, since only a
limited number of poles and residues are included in the rational part of the model, whereas
the high frequency behavior is captured by means of closed expressions that account for the
delays. The proposed method is applied to two relevant examples and validated through the
comparison with reference methods. The time domain solver is found to be more accurate and
significantly faster than the one obtained from a pure-rational model.
1
Introduction
The increasing clock speeds and the complexity of circuits made signal integrity and electromagnetic compatibility topics of real importance for circuit and system designers. In this
framework, interconnects at various level (PCBs, chip) play an important role, since their performances may be severely deteriorated by high frequency effects such as signal delay, reflection, dispersion, or crosstalk[1]. Interconnects can be seen as transmission lines with multiple
signal 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, the
transient analysis still deserves some attention, especially for electrically long transmission
lines. In fact, non-linear devices acting as terminations, such as drives and receivers in PCBs
or 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 not
suitable 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 segmentation
technique, whose main drawback is that a large number of unknowns is needed, mainly because
the 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 based
on delay extraction algorithms, leading to macromodels for both the time-domain and tabulated
frequency 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, where
the delay of the line is explicitly extracted over fixed frequency points by use of modal analysis. As a consequence, irrational functions arise in the frequency domain, and the time domain
formulation requires some approximation of the inverse Laplace transform; the numerical efficiency 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 the
modal domain. The resulting poles and time delays are then used for fitting H in the phase
domain, under the assumption that all poles contribute to all elements of H . The unknown
residues are then calculated by solving an overdetermined linear equation as a least squares
problem. An improved version of this approach has been recently presented in [12].
Another well studied technique is the one based on the Padé approach [13, 14], where the
delays are modeled implicitly. The reader is referred to [15] for a comparative analysis between
MoC and Padé techniques. A different approach is based on the idea to accelerate the transient
analysis 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/output
impedance matrix is expressed in terms of the dyadic Green’s function of the 1-D propagation
problem. A rational series form of this last one is proposed, where the poles are identified by
solving algebraic equations. Although general, such an approach may lead to large state-space
models when an electrically long MTL is considered, since a significant number of poles is
necessary to gain a good accuracy, especially in terms of ringing effects in the time domain. In
this context, the aim of this paper is to overcome such limitation. A new approach for the identification of delay-rational macro-models of electrically long interconnects is presented, where
a reduced number of poles and, consequently, a reduced state space size in the time domain
are gained. It is not just a matter of size, in fact a higher accuracy over the all frequency range
is gained, and no ringing effects are present in the time domain response. From a computational point of view, the algorithm is also faster. Stemming from the spectral representation of
a MTL as described in [20], the pole analysis is first performed. Assuming a MTL as a linear
and time-invariant system, the use of Fourier and Laplace transforms [21] allows to express
the 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 function
based method for the analysis of MTL; some peculiar properties of poles and residues are
pointed 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. R EVIEW OF THE SPECTRAL MODEL FOR M ULTICONDUCTOR T RANSMISSION L INES 73
V1(s) +
V2(s) +
-
I2(s)
IN+2(s)
MTL
+
+
VN+1(s)
VN+2(s)
͘͘͘
͘͘͘
VN(s) +
IN+1(s)
I1(s)
IN(s)
I2N(s)
+
z= 0
V2N(s)
z= l
-
Figure 1: Multiconductor transmission line represented as 2N port system, with a common reference
conductor.
numerical results are presented in Section 5 and the conclusions are drawn in Section 6.
2
Review of the spectral model for Multiconductor Transmission Lines
A multiconductor 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 electromagnetic field mode propagates,
read [1]
d
V (z, s) = − [R + sL] I (z, s)
dz
= −Z0 (s)I (z, s)
(1a)
d
I (z, s) = − [G + sC] V (z, s) + Is (z, s)
dz
= −Y0 (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 nonnegative definite symmetric of order N, being N + 1 the number of the conductors with the
reference conductor included [1, 22]. Z0 (s) and Y0 (s) are the p.u.l. longitudinal impedance and
admittances matrices. Is (z, s) represents a p.u.l. current source located at abscissa z. V (z, s) and
I (z, s) represent the voltage and current vectors depending on Laplace variable s and position
z 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
I0 (s)
V0 (s)
Z11 (s) Z12 (s) I0 (s)
=
= Z(s)
I` (s)
V` (s)
Z12 (s) Z11 (s) I` (s)
(2)
V0 (s) and I0 (s) are the voltage and the current port vectors respectively, of dimension
N × 1, related to the input ports at z = 0. V` (s) and I` (s) are the voltage and the current port
vectors respectively, of dimension N × 1, related to the output ports at z = `. Z(s) is the port
impedance matrix, that is block symmetrical, each block of dimension N × N, defined as
Z11 (s) = Z22 (s)
nπ 2 −1
+∞ 2
= ∑ γ (s) +
I
· A2n Z0 (s)
`
n=0
(3a)
Z12 (s) = Z21 (s)
nπ 2 −1
+∞ 2
I
· A2n Z0 (s) cos (nπ) .
= ∑ γ (s) +
`
n=0
(3b)
In (3) γ 2 (s) = Z0 (s)Y0 (s) = (R + sL)·(G + sC), A20 = 1/` for n = 0, A2n = 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, a
number n = 1, ..., ñ p of poles and residues can be evaluated, leading to a final number of poles
equal to n p = ∑M
m=0 ñ p,m .
np
Z(s) =
Rn
∑ s − pn .
(4)
n=1
It is worth noting that, in order to simplify the notation, it is not specified whether the poles are
real or complex conjugate.
A first way to use the rational impedance model (4) is to express it in the time-domain; the
resulting 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 suited
to be translated into a state space realization as
ẋ (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, while
the matrix B maps the inputs to state space equations. These equations can be then coupled
with those describing the terminations, that can be either linear or non-linear. More details
about the state-space model construction can be found in [3].
3. D ELAYED M ODEL OF L OSSLESS MTL
75
The main drawback of this pure rational, yet rigorous, approach relies on the large number
of modes (theoretically +∞) needed to accurately capture the frequency response of electrically
long lines. In the present work, a new formulation of the impedance matrix is given, that allows
to 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 the
output 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, where
the size of the rational part is dramatically reduced (thanks to the reduction of the number
of poles), still keeping accuracy in the time range of interest. Some useful observations are
necessary 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 the
same magnitude, and the corresponding poles can be identified. Within each group, they
are 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 the
summation will be treated as a special case), the poles and residues gather again in families, sharing the same real part and magnitude, respectively. Even thought the periodicity
is not so clear to identify, it can be indeed restored thanks to a proper mathematical
formulation.
3
Delayed Model of Lossless MTL
In the lossless case, R = G = 0, where 0 is the null matrix. It is necessary to treat separately
the 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 =
1 −1
C
s`
(6)
for i, j = 1, 2.
The pole in zero has always multiplicity 1, that is crucial to guarantee the asymptotic stability.
• 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, · · · , N
keeping track of the family. Hence, each impedance Zi j (s) can be represented in terms of poles
and residues as
76
PAPER B
N
C−1
Z11 (s) =
+∑
s`
k=1
N
C−1
Z12 (s) =
+∑
s`
k=1
+∞
∑
n=1
+∞
∑
n=1
∗
Rk,n
Rk,n
+
s − pk,n s − p∗k,n
!!
∗
Rk,n
Rk,n
+
s − pk,n s − p∗k,n
!
(7a)
!
(−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 can
write
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. In
fact, the poles belonging to each family are equispaced along the imaginary axis, meaning βk,n = nβk,1 , for k = 1, · · · , N and n = 1, · · · , +∞, where k spans over the different
families, as it can be seen in the example in Fig. 2. This observation suggests the idea
that poles and residues of each family describe a periodic function, as it will be better
shown 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 numerical
results, we could observe that (6) can be written as
N
C−1
= ∑ Rk,1 .
`
k=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 then
gained
3. D ELAYED M ODEL OF L OSSLESS MTL
6
77
×109
Family1
Family2
4
ℑ(p k,n)
2
0
-2
-4
-6
-1
-0.5
0
0.5
1
ℜ(p k,n)
Figure 2: Poles locus in the complex plane for a two-conductor lossless transmission line. For the sake
of 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.
N
Z11 (s) = ∑
k=1
N
Z12 (s) = ∑
k=1
Rk,0 +∞
+∑
s
n=1
∗
Rk,0
Rk,0
+
s − pk,n s − p∗k,n
!!
Rk,0 +∞
+∑
s
n=1
∗
Rk,0
Rk,0
+
s − pk,n s − p∗k,n
!
(11a)
!
(−1)n
.
(11b)
In the following, the self and the mutual impedances will be separately treated, for the sake
of clarity.
Self-impedance
Since the sub-blocks of Z11 (s) are all positive and real, the expression for Z11 (s) can be further
simplified as
!
N
+∞
Rk,0
s
Z11 (s) = ∑
(12)
+ 2Rk,0 ∑
2
s
n=1 s2 + nβk,1
k=1
!
+∞
N
(13)
z11 (t) = ∑ Rk,0 + 2Rk,0 ∑ cos nβk,1t Θ(t) .
k=1
n=1
78
PAPER B
t
|s
|
2
cos
2πn
s(t) = s0 + ∑+∞
+
θ
n
n=1
T0
KS
F
S( f ) = ∑+∞
n=−∞ sn δ ( f − n f 0 )
KS
F −1
s(t) = T0 ∑+∞
n=−∞ sn δ (t − nT )
Figure 3: Equivalences via Fourier transform.
Mutual impedance
For the Z12 (s), the following expression is gained
N
Z12 (s) = ∑
k=1
+∞
Rk,0
s
+ 2Rk,0 ∑ 2
(−1)n
2
s
s
+
(nβ
)
k,1
n=1
!
.
(14)
Because of the term (−1)n = cos(nπ), a phase shift equal to −nπ is present in the transient
impedance z12 (t)
N
z12 (t) = ∑
Rk,0 + 2Rk,0
+∞
∑ cos
nβk,1t − nπ
!
Θ(t) .
(15)
n=1
k=1
Eq. (13) and (15) confirm that transient impedances zi j (t), for i, j = 1, 2, are the sum of N
periodic functions whose period is
Tk =
3.1
1
fk,1
=
2π
,
βk,1
k = 1, · · · , N .
(16)
Impedance-based I-O representation by Dirac comb
The general series form of the impedances requires a high number of modes in order to have
sufficient accuracy. It leads to large state-space models, and it requires cumbersome convolution products in the time domain. Taking advantage of the delays explicit extraction technique
here developed, the next step will be to re-write the impedances in terms of the Dirac combs. In
the 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 delay
equal to Tk , k = 1, · · · , N. In the following, we will always exploit the equivalences collected in
Fig. 3. The self and mutual impedances will be treated separately.
3. D ELAYED M ODEL OF L OSSLESS MTL
79
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
!
N
z11 (t) = ∑
Rk,0 Tk
+∞
∑
δ (t − nTk ) Θ(t)
(17)
n=−∞
k=1
where Θ(t) is the Heaviside step. The Heaviside step function can be written in terms of the
sign function sgn(t) as
Θ(t) =
1 1
+ sgn(t) .
2 2
(18)
Hence, the transient impedance z11 (t) will read as
N
z11 (t) = ∑ Rk,0 Tk
k=1
!
δ (t) +∞
+ ∑ δ (t − nTk ) .
2
n=1
(19)
The Laplace transform of (19) is then performed as
!
1 +∞ −snTk
L (z11 (t)) = ∑ Rk,0 Tk
+∑e
2
n=1
k=1

 Tk
Tk
N
es 2 + e−s 2

= ∑ Rk,0 Tk  Tk
Tk
k=1
2 es 2 − e−s 2


N R T
cosh s T2k
k,0 k 

=∑
2
sinh s Tk
k=1
N
(20)
2
leading to the final expression for Z11 (s)
Rk,0 Tk
Tk
Z11 (s) = ∑
coth s
.
2
2
k=1
N
(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 self
impedance, the time domain expression for z12 (t) is obtained in term of Dirac combs
!
N
+∞ Tk
z12 (t) = ∑ Rk,0 Tk ∑ δ t − nTk −
.
(22)
2
n=0
k=1
80
PAPER B
The Laplace transform of (22) is then performed
L (z12 (t)) =
+∞
N
Tk
!
∑ Rk,0Tk ∑ e−snt−s 2
k=1
n=0
=


Rk,0 Tk 
1

2
sinh s Tk
k=1
N
∑
(23)
2
gaining the final expression for Z12 (s)
Rk,0 Tk
Tk
csch s
.
Z12 (s) = ∑
2
2
k=1
N
(24)
The expression (24) is the natural extension of the single-conductor model [24] to the MTL
case.
It is also worth noting that, in both the self and mutual impedance (21) and (24), the quantity Rk,0 Tk /2 acts as a generalized characteristic impedance for the k-th family of poles and
residues. 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 and
residues, and all residues are equal to Rk = 1/`C, it is easy to verify that Rk,0 Tk /2 is actually a
characteristic impedance.
4
The Delay-Rational Model for a lossy MTL
The lossy case is considered. As done before, it is convenient to treat separately the n = 0
mode 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
A20 Ad 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
N
Zi j (s) |n=0 =
∑
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 subdivide
them into N groups. The corresponding residues Rk,n are also complex conjugate. From
numerical computation, it is possible to observe that the poles share the same real part αk,n
within each k-th family through the n modes. Then, αk,n = αk,1 for k = 1, · · · , N, as in the
example in Fig. 4.
4. T HE D ELAY-R ATIONAL M ODEL FOR A LOSSY MTL
6
81
×109
4
Family1
Family2
ℑ(p k,n)
2
0
-2
-4
-6
-5
-4.5
-4
-3.5
ℜ (p k,n)
-3
-2.5
×105
Figure 4: Poles locus in the complex place for a two-conductor lossy transmission line. For the sake of
clarity, the number of modes is n = 10. Note that the N = 2 real poles for n = 0 are not plotted, and 4
poles (in complex conjugate pairs) are detected for each family.
• n ≥ m̂, n 6= 0
Some important properties regarding poles and residues are observed for modes of index n > m̂,
n 6= 0, where the choice of m̂ is the subject of Section 4.1:
• Rk,n = R̂k is a constant value, for k = 1, · · · , N, as in the example in Fig. 5. More
specifically, the imaginary part of Rk,n tends to zero for n → +∞, meaning the phase
tends to zero, as in Fig. 6. Denoting by R̂k the real part of Rk,n , from a certain mode m̂,
the following approximation can be used Rk,n = R̂k , for n = m̂, · · · , +∞;
• for each k-th family, the imaginary parts βk,n of the poles become equispaced from a
certain mode m̂, then for n = m̂, · · · , +∞. Denoting by β̂k,n the fictitious imaginary parts
equispaced for n = 1, · · · , +∞, we can write β̂k,n = nβ̂k,1 , n = 1, · · · , +∞. It is worth
noting 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 > m̂, poles and
residues exhibit the same periodic property as for the lossless case. However, in order to recover
the periodicity, the full set of poles and residues needs to be reconstruct for n = 0, · · · , +∞. To
this aim, fictitious poles and residues are defined as
p̂k,0 = αk,1
R̂k,0 = R̂k
(27a)
(27b)
82
PAPER B
×109
14
Family1
Family2
|Res11 | [Ω · s]
12
10
8
6
0
2
4
6
8
10
Mode
Figure 5: Magnitude of the residue Rk,n of Z11 (s), for a two-conductor lossy transmission line. For the
sake of clarity, the number of modes is n = 10.
5
×10-4
Family1
Family2
Res11 [rad]
4
3
2
1
0
0
2
4
6
8
10
Mode
Figure 6: Phase of the residue Rk,n of Z11 (s), for a two-conductor lossy transmission line. For the sake
of clarity, the number of modes is n = 10.
4. T HE D ELAY-R ATIONAL M ODEL FOR A LOSSY MTL
p̂k,n(1,2) = αk,1 ± jnβ̂k,1
R̂k,n = R̂k
83
(27c)
(27d)
for k = 1, · · · , N. The expressions of the impedances Zi j (s), for i, j = 1, 2 in the Laplace domain
are
!
!!
∗
N
m̂−1
+∞
Rk,n
Rk,0
Rk,n
R̂k
R̂k
Z11 (s) = ∑
+∑
+
+∑
+
(28a)
s − p∗k,n
s − p̂∗k,n
n=1 s − pk,n
n=m̂ s − p̂k,n
k=1 s − pk,0
!
!!
!
∗
m̂−1
+∞
N
Rk,n
Rk,n
Rk,0
R̂k
R̂k
n
+∑
+
+
+∑
(−1)
Z12 (s) = ∑
s − pk,n s − p∗k,n
s − p̂∗k,n
n=1
n=m̂ s − p̂k,n
k=1 s − pk,0
(28b)
that in the time domain are
N
z11 (t) = ∑
Rk,0 e pk,0t + 2eαk,1t R̂k
+ eαk,1t
∑
∑ cos(nβ̂k,1t)
n=m̂
k=1
m̂−1
+∞
!
2 Rk,n cos(βk,nt + ∠Rk,n ) Θ(t)
(29a)
n=1
N
z12 (t) = ∑
k=1
m̂−1
+ eαk,1t
∑
Rk,0 e pk,0t + 2eαk,1t R̂k
+∞
∑ cos(nβ̂k,1t − nπ)
n=m̂
!
2 Rk,n cos(βk,nt + ∠Rk,n − nπ) Θ(t) .
(29b)
n=1
The transient impedances (29) implicitly contain the time-delays. In the following, the delays
will be explicitly extracted. For the sake of clarity, self and mutual impedances Z11 (s) and
Z12 (s) respectively, are treated separately.
4.1
Selection of m̂
In order to assign a value for m̂, the following criterion is adopted. As it was already observed, the phase of the residues tends to zero for n → +∞. It means, the difference in
phase ∠Rk,n+1 − ∠R
k,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, m̂ is chosen such that
∠Rk,m̂ − ∠Rk,m̂−1
< tol .
(30)
∠Rk,2 − ∠Rk,1
84
4.2
PAPER B
Self-impedance
Given the expression in (28a), the summation ∑+∞
n=m̂ is rewritten as
N
Z11 (s)|n≥m̂ =
∑
2R̂k
k=1
+∞
(s − αk,1 )
∑
!
.
2
2
n=m̂ (s − αk,1 ) + (nβ̂k,1 )
(31)
The rational contributions with fictitious poles and residues (27) are added and subtracted in
order to reconstitute the full summation
+∞
(s − αk,1 )
R̂k
+ 2R̂k ∑
2
2
n=1 (s − αk,1 ) + (nβ̂k,1 )
k=1 s − αk,1
!
m̂−1
(s − αk,1 )
R̂k
−
− 2R̂k ∑
.
2
2
s − αk,1
n=1 (s − αk,1 ) + (nβ̂k,1 )
N
Z11 (s)|n≥m̂ =
∑
(32)
The inverse Laplace transform of the first two terms of (32) is performed to exploit the equivalences in Fig. 3. A Fourier series expression in the time domain is obtained
!
L −1 (•) =
N
+∞
k=1
n=1
∑ R̂k eαk,1t + 2eαk,1t
∑ cos(nβ̂k,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
!
+∞
N
αk,1 t δ (t)
T
e
k
+ Tk eαk,1t ∑ δ (t − nTk ) Θ(t) .
L −1 (•) = ∑ R̂k
(34)
2
n=1
k=1
The Laplace transform of (34) reads as
N
L (•) =
∑ R̂k
k=1
4.3
Tk
Tk
coth (s − αk,1 )
.
2
2
(35)
Mutual Impedance
In the Z12 (s) case, the summation ∑+∞
n=m̂ can be rewritten as
N
Z12 (s)|n≥m̂ =
∑
k=1
2R̂k
+∞
∑
(s − αk,1 )(−1)n
!
2
2
n=m̂ (s − αk,1 ) + (nβ̂k,1 )
(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) are
added and subtracted in order to reconstitute the full summation
N
Z12 (s)|n≥m̂ =
∑
k=1
+∞
(s − αk,1 )(−1)n
R̂k
+ 2R̂k ∑
2
2
s − αk,1
n=1 (s − αk,1 ) + (nβ̂k,1 )
4. T HE D ELAY-R ATIONAL M ODEL FOR A LOSSY MTL
85
m̂−1
(s − αk,1 )(−1)n
R̂k
−
− 2R̂k ∑
2
2
s − αk,1
n=1 (s − αk,1 ) + (nβ̂k,1 )
!
.
(37)
The inverse Laplace transform of the first two terms of (37) is performed to exploit the equivalences in Fig. 3. A Fourier series in the time domain is obtained, as shown in (38):
!
L −1 (•) =
+∞
N
∑ R̂k
eαk,1t + 2eαk,1t
∑ cos(nβ̂k,1t − nπ)
Θ(t) .
(38)
n=1
k=1
Following the same steps as for Z11 (s), it can be written in terms of the time domain Dirac
comb as
N
+∞ Tk
−1
αk,1 t
L (•) = ∑ R̂k Tk e
(39)
∑ δ t − nTk − 2 .
n=0
k=1
The Laplace transform of (39) reads
N
L (•) = ∑ R̂k
k=1
Tk
Tk
csch (s − αk )
.
2
2
(40)
Finally, it is possible to rewrite the general expressions (28) as
!
∗
Rk,n
Rk,n
Z11 (s) = ∑
+
s − pk,n s − p∗k,n
k=1
!
m̂−1
R̂k
R̂k
−∑
+
(s − p̂∗k,n )
n=1 (s − p̂k,n )
!
R̂k Tk
R̂k
Tk
+
coth (s − αk,1 )
−
(s − αk,1 )
2
2
!
∗
N
m̂−1
Rk,n
Rk,0
Rk,n
Z12 (s) = ∑
+ ∑
+
s − p∗k,n
n=1 s − pk,n
k=1 s − pk,0
!
m̂−1
R̂k
R̂k
n
−∑
+
∗ ) (−1)
(s
−
p̂
)
(s
−
p̂
k,n
k,n
n=1
!
R̂k
R̂k Tk
Tk
−
+
csch (s − αk,1 )
.
(s − αk,1 )
2
2
N
m̂−1
Rk,0
+ ∑
s − pk,0 n=1
(41a)
(41b)
In both the impedances in (41), the first four terms admit a standard state-space realization
while the hyperbolic terms correspond to time domain Dirac combs. Indeed, in the time domain, these two expressions read as
N
z11 (t) =
∑
k=1
Rk,0 e pk,0t Θ(t) +
m̂−1
∑ 2 Rk,n eαk,nt cos
n=1
βk,nt + ∠Rk,n Θ(t)
86
PAPER B
− 2R̂k eαk,1t
m̂−1
∑ cos(nβ̂k,1t)Θ(t) − R̂k eαk,1t Θ(t)
n=1
+ R̂k
N
z12 (t) =
∑
!!
+∞
Tk eαk,1t δ (t)
+ Tk eαk,1t ∑ δ (t − nTk )
2
n=1
Rk,0 e
m̂−1
pk,0 t
Θ(t) +
∑ 2 Rk,n eαk,nt cos
βk,nt + ∠Rk,n − nπ Θ(t)
n=1
k=1
− 2R̂k eαk,1t
(42a)
m̂−1
∑ cos(nβ̂k,1t − nπ)Θ(t) − R̂k eαk,1t Θ(t)
n=1
!
T
k
+R̂k Tk eαk,1t ∑ δ t − nTk −
.
2
n=0
+∞
4.4
(42b)
State space model
The rational contributions in (41) admit a standard state-space representation with port currents
i(t) as inputs, and port voltages v(t) as outputs. The state space matrix A has dimension nss
nss = (2 (N + 2N(m̂ − 1))) N p
(43)
where N denotes the number of conductors and N p = 2N denotes the number of ports. The port
current vector i(t) has dimension N p , then the input space is ℜq×1 with q = N p , and it can be
written 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 output
ports; the apex T denotes the transposition. The output vector is given by the voltages at the
input and the output ports, thus the output space is ℜq×1 with q = N p ,
v(t) = [v1 (t)|v2 (t)]T .
(45a)
The summation of hyperbolic terms in the frequency domain corresponds to a summation of
Dirac combs in the time domain. Hence, the global delay-rational model in the time domain
can be written as
ẋ (t) = Ax (t) + Bi (t)
(46a)
v (t) = Cx (t) + D0 i (t) + id (t)
(46b)
where id (t) = D(t) ⊗ i (t), with D(t) described hereinafter. Compared to a standard state space
model as in (5), two matrices, D0 and id (t), are used. The first one is defined as
N 0
R̂k T2k
.
(47)
D0 = ∑
0
R̂k T2k
k=1
5. N UMERICAL E XPERIMENTS
87
Note that the convolution δ (t) ⊗ i(t) = i(t), and then it is omitted in the state-space model formulation. 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 with
the form
N D11 D12
D(t) = ∑
(48)
D12 D11
k=1
with
D11 = R̂k Tk eαk,1t
+∞
∑ δ (t − nTk )
(49a)
n=1
D12 = R̂k Tk e
αk,1 t
+∞
Tk
∑ δ t − nTk − 2
n=0
(49b)
where the infinite summation will be truncated to n = max(t)
Tk , where t is the time window under
analysis. Note that the convolution product D(t)⊗i (t) returns only an attenuated and translated
version of the currents, so no convolutions products are actually performed in the time-domain
solver.
4.5
Discussion
It is worth pointing out the main advantages of this method when compared to the existing
ones.
1. Both the standard rational and the delay-rational models do not require any discretization
of 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, we
do 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 m̂ of modes to be included in the delay-rational model is identified,
a reduced order model is automatically computed, since m̂ m for electrically long
transmission lines;
4. the proposed technique provides a significant insight into the theory of MTL since it
naturally leads to identify the delays, without resorting to any modal decomposition nor
numerical estimation, apart from the computation of poles for each mode.
5
Numerical Experiments
The proposed delay-rational method is validated through two test cases. The results, in the frequency 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 dyadic
Green’s function method as “Rational”, and the proposed delayed-rational one as “Delayrational”. For the simulations, the Matlab R2014b [25] software is used, on a computer running
64-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 per
the MTL classical theory [1] is defined as
v
u
u 1 2N 2N Ns Zi j,r (k) − Zi j (k) 2
Err = t 2 ∑ ∑ ∑ (50)
4N Ns
Zi j,r (k)
i=1 j=1 k=1
where Zi j,r is the i, j-th entry of the impedance matrix by MTL classical theory, assumed as
reference; Zi j denotes the i, j-th entry of the impedance matrix obtained by the rational model
or the delay-rational one; Ns is the number of frequency samples. The generic 2N-port representation in Fig. 1 is adopted.
5.1
Example 1: four-conductor transmission line
A four-conductor lossy TL of length ` = 0.2 m is considered, with p.u.l. parameters reported
below [26]. The conductor 1 (as per Fig. 1) is excited by a smooth pulse voltage of amplitude
1 V, rise and fall times tr = t f = 0.5 ns and pulse-width τ = 2 ns. All the ports are terminated
on 250 Ω resistances.


1 0.11 0.03 0
0.11 1 0.11 0.03

L= 
(51)
0.03 0.11 1 0.11 µH/m
0 0.03 0.11 1


3.5
0.35 0.035
0
 0.35
3.5
0.35 0.035
 Ω/m
R =
0.035 0.35
3.5
0.35 
0
0.035 0.35
3.5

10
−1 −0.1
0
 −1
10
−1 −0.1
 mS/m
G =
−0.1 −1
10
−1 
0
−0.1 −1
10
(52)


1.5 −0.07 −0.03
0
−0.07 1.5 −0.07 −0.03

C =
−0.03 −0.07 1.5 −0.07 nF/m .
0
−0.03 −0.07 1.5
(53)

(54)
5. N UMERICAL E XPERIMENTS
89
The number of modes is set as m = 150 in the standard rational formulation (3). In the proposed delay-rational (41), the value of m̂ = 7 is automatically computed, based on the accuracy
desired. The poles in the complex plane are plotted in Fig. 7. In the picture, are highlighted the
poles 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-delays
are τ1 = 14.67 ns, τ2 = 15.07 ns, τ3 = 15.72 ns, τ1 = 16.19 ns. The resulting state-space has
size 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 compared
also 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 delayrational technique is faster and more accurate than the rational standard one, and provides high
accuracy with a reduced number of modes, over the entire frequency range.
10
4
x 10
3
2
n=0 poles
Imag
1
0
−1
n>0 poles
−2
−3
−4
−7.5
−7
−6.5
−6
Re
−5.5
−5
−4.5
6
x 10
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 = 1.975946 Ω/m
(55)
G = 8.719413 µS/m , C = 0.827367 nF/m .
90
PAPER B
0.12
Rational
Delay−Rational
IFFT
0.1
V1 [V]
0.08
0.06
0.04
0.02
0
−0.02
0
50
100
Time [ns]
150
200
0.03
Rational
Delay−Rational
IFFT
0.02
V4 [V]
0.01
0
−0.01
−0.02
−0.03
0
50
100
Time [ns]
150
200
Figure 8: Voltages at ports 1 and 4 for a four-conductor transmission line, example 1 in Section 5.1.
6. C ONCLUSIONS
91
Table 1: Comparative results for the two numerical examples.
Example
Example 1
Example 2
Method
Time domain solver [s]
Weighted RMS-error
Rational (m = 150)
14704.75
21.2829
Delay-Rational (m̂ = 7)
1474.45
0.0401
Rational (m = 800)
12459.14
66.1996
991.62
0.0370
Delay-Rational (m̂ = 7)
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 = 500 Ω resistances.
The standard rational approach requires 800 modes to provide a reasonable accuracy, while
the new delay-rational method requires only 7 modes to achieve even a better accuracy, as
summarized in Table 1.
The resulting state spaces for the delay-rational approach and the rational standard one are
of size 52 and 3202,√respectively. The time-delay is τ = 955 ns, in agreement with the well
known 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 claimed
again that the proposed method proves to be more accurate and significantly faster than the
standard rational one.
6
Conclusions
A new delay-rational model has been proposed for lossless and lossy MTL with frequencyindependent per-unit-length parameters. It is an improved version of the rational method based
on the dyadic Green’s function of 1-D propagation along the line. The basic observation is
that the residues of a lossless N-conductor MTL can be gathered in N families sharing the
same 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 share
the same negative real part (poles) and magnitude (residues) in the families, with the poles again
equally spaced along the imaginary axis. This property does not hold for the low frequency
poles and the corresponding residues. Thus, mathematical manipulations are made in order
to 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. The
main enhancement of the proposed method relies on the explicit delays extraction, that allows
to reduce the size of the rational macromodel. The high-frequency behavior of impedances
is preserved by using closed form expressions in the frequency domain whose time-domain
counterparts are Dirac combs, that explicitly take the delays into account. Numerical results
confirm 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 of
the method to the frequency-dependent per-unit-length parameters as well as the passivity of
the model will be described in forthcoming reports.
92
PAPER B
0.12
Rational
Delay−Rational
0.1
V1 [V]
0.08
0.06
0.04
0.02
0
0
500
1000
1500
2000
Time [ns]
2500
3000
0.2
Rational
Delay−Rational
V2 [V]
0.15
0.1
0.05
0
−0.05
0
500
1000
1500
2000
Time [ns]
2500
3000
Figure 9: Port voltages for a cable of length ` = 10 m, example 2 in Section 5.2.
R EFERENCES
93
Acknowledgment
The authors acknowledge Svenska Kraftnät (Swedish national grid) for providing funding for
this research.
References
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[10] S. Grivet-Talocia, H.-M. Huang, A. E. Ruehli, F. Canavero and I. M. (Abe) Elfadel, “Transient analysis of lossy transmission lines: an efficient approach based on the method of
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[11] A. Morched, B. Gustavsen, and M. Tartibi, “A universal model for accurate calculation of
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[12] T. Noda, “Application of frequency-partitioning fitting to the phase-domain frequencydependent modeling of overhead transmission lines,” IEEE Transactions on Power Delivery, vol. PP, no. 99, pp. 1–1, to be published 2014.
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[14] A. Dounavis, E. Gad, R. Achar, M. S. Nakhla, “Passive model reduction of multiport
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[15] I. M. Elfadel, H-M. Huang, A. E. Ruehli, A. Dounavis, M. S. Nakhla, “A comparative
study of two transient analysis algorithms for lossy transmission lines with frequencydependent data,” in Digest of Electr. Perf. Electronic Packaging, Oct. 2001, pp. 255 –
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[16] F. Y. Chang, “The generalized method of characteristics for waveform relaxation analysis of lossy coupled transmission lines,” IEEE Transactions on Microwave Theory and
Techniques, 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 interconnects using waveform relaxation and transverse partitioning,” IEEE Transactions on
Advanced 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 relaxation
techniques for simulation of coupled interconnects with frequency-dependent parameters,” 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 distributed
analytical representation and iterative technique (DARIT-Field) for the field coupling to
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[20] G. Antonini, “A dyadic Green’s function based method for the transient analysis of lossy
and dispersive multiconductor transmission lines,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 4, pp. 880–895, Apr. 2008.
[21] R. Beerends, Fourier and Laplace Transforms, ser. Fourier and Laplace Transforms.
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[24] M. De Lauretis, J. Ekman, and G. Antonini, “Delayed impedance models of twoconductor transmission lines,” in Electromagnetic Compatibility (EMC Europe), 2014
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[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 analysis
of distributed MTL networks,” IEEE Transactions on Microwave Theory and Techniques,
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[27] A. Ruehli, A. Cangellaris, and H.-M. Huang, “Three test problems for the comparison of
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PAPER C
Enhanced Delay-Rational Green’s
Method 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.
c 2015, IEEE, Reprinted with permission.
97
98
Enhanced Delay-Rational Green’s Method for Cable Time
Domain 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 the
Green’s function based method which allows to write the open-end impedance in a rational
form as an infinite sum of “modal impedances”. It can be then embedded in a circuit simulation environment for efficient time domain analysis. The previous rational approach has
been improved through a proper mathematical formulation, that makes use of explicit delay
extraction and pole/residue asymptotic behavior. Nevertheless, the computation of the poles
becomes computationally expensive when the number of conductors increases, since the zeros
of 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, model
the high frequency behavior of the cable in terms of standard hyperbolic functions. The lowfrequency behavior is captured by a reduced size state-space model, via rational fitting. Numerical results confirm the accuracy of the proposed modeling approach for electrically long
cables, with a large number of conductors.
1
Introduction
The increasing in the switching frequencies in power electronic applications is the main responsible for noise and electromagnetic interference (EMI), that are carried mainly through
the cable, especially in terms of conducted emission noise [1]. While the well-established
multiconductor transmission line (MTL) theory [2] is used for cable frequency analysis, the
research community is still focused on efficient time-domain models which can be more easily
integrated in circuit simulators, such as SPICE [3], where non-linear power electronics devices
are efficiently modeled. The Green’s function based method proposed in [4] relies on the evaluation of the exact poles and residues of the open-end impedance matrix, expressed as a sum
of infinite rational transfer functions, denoted as “modal impedances”. A time-domain statespace model is easily gained. Although general, this approach may lead to large state-space
models when an electrically long cable is considered, since a significant number of poles is
required to achieve a good accuracy. In [5, 6, 7], a delayed impedance model has been proposed to overcome such limitation, by using an explicit delay extraction and the analysis of
the 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 requires 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/residues
that, together with the delays, are used to model the high frequency behavior of the cable in
terms of standard hyperbolic functions. The low-frequency behavior of the cable is captured by
a 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 cable
bundle by solving a set of delay differential equations with delayed sources. Simulation results
are presented in order to validate the proposed approach.
2
Green’s function based methods
The open-end impedance representation of a MTL of length ` reads [2]
V0 (s)
Z11 (s) Z12 (s) I0 (s)
=
V` (s)
Z12 (s) Z11 (s) I` (s)
|
{z
}
(1)
Z(s)
where Z(s) is block symmetrical, with each block of dimension N × N, being N the number of
conductors.
2.1
Standard Green’s function based method
The dyadic Green’s function method for the analysis of N-conductor MTLs, treated as a 2N
port system, has been proposed in [4], where the impedance matrix Z(s) is expressed as an
infinite sum of matrices of rational functions as follows
+∞
Z11 (s) =
∑ Fm(s)−1(s)A2mZ0(s)
(2a)
m=0
+∞
Z12 (s) =
∑ (−1)mFm(s)−1(s)A2mZ0(s)
(2b)
m=0
Fm (s) = Z0 (s)Y0 (s) +
mπ 2
`
I
(2c)
with Z0 (s) = R0 + sL0 the p.u.l. longitudinal impedance and Y0 (s) = G0 + sC0 the p.u.l. transverse admittance. The computation of poles and residues requires the inversion of the polynomial matrix Fm (s) defined in (2). All the sub-blocks of Z(s) share the same poles which can
be computed as the zeros of the determinant of Fm (s). We remind that the number of poles
is N for the zero mode m = 0, and 2N poles for higher order modes m > 0. Once the poles
pn , n = 1, · · · , +∞ are identified, the residues Rn can be computed [4]. Finally, the open-end
impedance matrix Z(s) is expressed as follows
∞
Z(s) =
Rn
∑ s − pn .
n=1
(3)
2. G REEN ’ S FUNCTION BASED METHODS
101
This rational model is well suited to be translated into a state-space form [9], but two main
drawbacks 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 matrix
Fm (s), which becomes computationally expensive for a large number of conductors.
2.2
Delayed Green’s function based method
The first drawback has been addressed in [5, 6] for the frequency-independent p.u.l. parameter
case, and in [7] for the frequency-dependent one. The number of modes can be drastically
reduced by explicit delay extraction, along with the poles/residues asymptotic behavior. More
specifically, it has been observed that poles obtained by the standard Green’s function based
approach are clustered in N groups, called “families”, identified by their real part. Within
each family, the poles become equispaced along the imaginary axis when the phase of the
corresponding residues tends to zero. A truncation mode m̂ can be found accordingly to the
following 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,m̂ − ∠Rk,m̂−1
< tolR
∠Rk,2 − ∠Rk,1
αk,m̂ − αk,m̂−1
> tolα .
αk,m̂
(4a)
(4b)
where tolR and tolα are suitable tolerance values.
Once the families have been properly identified, the infinite summation can be then truncated
using the following hyperbolic functions
R̂k T̂k
T̂k
Zh11 (s) = ∑N
coth
(s
−
α̂
)
(5)
k 2
k=1 2
R̂k T̂k
T̂k
Zh12 (s) = ∑N
csch
(s
−
α̂
)
(6)
k 2 .
k=1 2
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, p̂n and R̂n respectively, as detailed in [6], contribute
to the rational part of the model as follows
!
n̂ p
Ri j,n m̂−1 N R̂i j,k
Z̄i j (s) = ∑
−∑ ∑
(7)
n=1 s − pn
m=0 k=1 s − p̂k,m
where i, j = 1, 2 identify the blocks in (2) and n̂ p identifies the number of poles for m = 0 to
m̂ − 1.
102
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PAPER C
Proposed solution for cable bundles
As pointed out before, the computation of poles becomes computationally time demanding
and also prone to numerical inaccuracies when the number of conductors increases, as in the
case of cable bundles. This section aims to present a strategy to overcome this limitation. The
poles 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 over
the frequency range of interest would not be efficient either, because it would require to search
all the poles at once, resulting in a more difficult identification of the pole families, crucial in
order 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 based
method (2), the rational fitting procedure can be adopted to approximate each “modal impedance”, completely avoiding the inversion of the polynomial matrix in (2), thus leading to the
explicit computation of poles and residues. Furthermore, the number of poles to be used in the
rational fitting of each “modal impedance” is known. In the frequency independent per-unitlength parameter case, they are N for the mode zero m = 0, and 2N for m > 0. A schematic of
the general algorithm is proposed in (1). Because of the symmetric block structure of Z(s) , it is
easy to observe that all the sub-blocks share the same poles, and that the residues are the same
except for the multiplication for (−1)m for the off-diagonal blocks. This allows evaluating only
the modal frequency response for the N × N block Z11 |m . From the frequency independent
case theory [6], it is known that the 2N poles of each mode m divide into N families, each
family sharing the same real part αk , at least from a certain mode order m̂ on. When poles are
computed using a rational approximation as VF, the poles can have a different behavior, and it
is indeed expected since we are relying on a numeric technique. Nevertheless, the same main
properties are retained. In particular, the poles gather again into families in number equal to
the number of conductors. In Fig. 2, the poles for a N = 6 lossy ribbon cable are plotted in
the complex plane, and indeed, 6 families can be easily recognized. The family identification
is 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 can
be computed by analyzing the lossless line. It is important to observe that the delays used
in the frequency
independent case are indeed coincident with the modal delays defined as
√
T̂k = 2` Λk , where Λ are the eigenvalues of the product matrix C0 L0 [11], and k = 1, . . . , N. It
is worth to notice that we always use, in our formulation, the two-way delay. The β̂k,1 can be
.
computed as β̂k,1 = 2π
T̂
k
3. P ROPOSED SOLUTION FOR CABLE BUNDLES
103
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
N_mod = M;
STOP = FALSE;
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
END
yes
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
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00000000000000000000000000000000000000000000000000000000000000
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00000000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000
m == N_mode
m = m + 1;
no
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ʹ
0 0 0 0 0 0 0 0 0 0 0 0 0 0െͳ
0000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00Ԣ00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00Ԣ00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ʹ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00૚૚
00 00 00 00 00 00 00 00 00 00 00 ݉
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 ݉
00000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
% Port impedance evaluation for the mode m
݉ߨ
ࢆ ȁ ൌ ൤ࢆሺ࢙ሻ ࢅሺ࢙ሻ ൅ ቀ ቁ ࡵ൨
݈
‫ࢆ ܣ‬ሺ࢙ሻ
% Vector fitting in order to identify poles and
residues
VF
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
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000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00‫ࡾס‬
0 0 0 0 0 0 0 0 0 0 0 0݇ǡ݉
0 0 0 0 0 0 0ෝ
0 0 0 0 0 0 0െ
0 0 0 0 0 0 0 0 0‫ࡾס‬
0 0 0 0 0 0 0 0 0 0 0 0 0 0݇ǡ݉
0 0 0 0 0 0 0ෝ0 0 0 0 െͳ
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ቤ00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ቤ000 000 000 000 000 000൏
00 00 00 00 00 00 00 00 00‫݈݋ݐ‬
00 00 00 00 00 00 00 00 00 00 00 ܴ݁‫ݏ‬
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
‫ࡾס‬
െ
‫ࡾס‬
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00݇ǡ݉
0 0 0 0 0 0 0ෝ
0 0 0 0 െͳ
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0݇ǡ݉
0 0 0 0 0 0 0ෝ
0 0 0 0 0െʹ
00000000000000000000000000000000000000000000000000000000000000000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ߙ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00െ
0 0 0 0 0 0 0 0 0 00 00 00 00 00 ݇ǡ݉
00 00 00 00 00 00 00 00ෝ
00 00 00 00 00െͳ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
ෝ 0 0 0 0 0 0 0 0 0ߙ
݇ǡ݉
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00 00 00 00 00 00 00 00 00 00‫݈݋ݐ‬
00 00 00 00 00 00 00 00 00 00 00 00ܴ݁
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00ߙ
00 00 00 00 00 00 00 ݇ǡ݉
00 00 00 00 00 00 00 00ෝ
00 00 00 00 0000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
yes
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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00݇
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0݇
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
% Evaluation of the asymptotic values for
poles and residues, and ݉
ෝ
෡ ǡ ߙො ǡ ݉
ࡾ
ෝ
Figure 1: Block diagram for the proposed algorithm.
no
104
PAPER C
1
×1011
poles vf
ℑ
0.5
0
-0.5
-1
-4
-3.5
-3
-2.5
ℜ
-2
-1.5
-1
×10
4
Figure 2: Poles location computed using a modal rational approximation approach, for a 6 conductor
ribbon cable.
3.3
State-space model
In order to generate the minimal size state-space model, it can be observed that the rational
expression in (7) has one main drawback: even if a small value of m̂ − 1 can be found, the
number of poles is double, since two summation in m̂ − 1 have to be used. We notice that
the rational part (7) actually represents a delay-less impedance. For this reason, it can be
fitted using a reduced number of poles. The vector fitting algorithm can be used again to this
aim. The double summation (7) would require N p = 2(N + 2N(m̂ − 1)) poles which can be
significantly reduced by the direct application of the vector fitting algorithm to the delay-less
impedance, computed as
N
T̂k
R̂k T̂k
coth (s − α̂k )
Z11−dl (s) = Z11 (s) − ∑
2
k=1 2
(8a)
N
R̂k T̂k
T̂k
Z12−dl (s) = Z12 (s) − ∑
csch (s − α̂k )
.
2
k=1 2
4
(8b)
Numerical Experiments
In order to evaluate the accuracy of the proposed approach, the cable bundle with 9 conductors
is considered [12], of length ` = 20 m. The p.u.l. parameters are those reported in [12]. The
first conductor is excited by a smooth pulse voltage (amplitude 1 v, rise and fall time 20 ns,
4. N UMERICAL E XPERIMENTS
105
pulsewidth 50 ns and initial delay 30 ns); all the ports are terminated on 50-Ω resistances. The
problem has been analyzed by using the standard theory of MTL and by using the proposed
enhancement of the Delay-Green’s function approach (Delay-GVF). All the code has been
developed in MATLAB [13]. For the Delay-GVF method, the number of modes m̂ = 7 is
automatically chosen by the algorithm. The poles computed for the delay extraction (chosen
by the vector fitting procedure), in the complex plane, are shown in Fig. 3 where the 9 families
of poles are clearly identified.
×108
poles vf
p̂
1.5
1
ℑ
0.5
0
-0.5
-1
-1.5
-2
-7
-6
-5
-4
ℜ
-3
-2
×104
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 m̂ = 7 has been found, using the
previous approach to set the state-space model would have led to a total number of poles equal
to Pn = 2(N + 2N(m̂ − 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 solution
obtained by using the standard theory of MTL in the frequency domain, transformed in the
time domain by using the Inverse Fast Fourier Transform algorithm. Finally, the average norm
2 error between the solution computed by the standard theory of MTL and the Delay-GVF
method 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 MTL
and by using the Green-DVF method, respectively.
106
PAPER C
0.7
Delay−GVF
ifft
0.6
0.5
V1 [V]
0.4
0.3
0.2
0.1
0
−0.1
0
200
400
600
Time [ns]
800
1000
0.01
0
Delay−GVF
ifft
V14 [V]
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
0
200
400
600
Time [ns]
800
1000
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 analysis
of electrically long cables is presented. The identification of the asymptotic poles/residues
behavior, along with the explicit delay extraction, allow the decomposition of the open-end
R EFERENCES
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 was
previously performed by computing the determinant of a polynomial matrix of order 2N, whit
N the number of conductors. This method becomes intractable when the number of conductor
exceeds 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 algorithm
allows a fast computation of poles and residues and, as a consequence, a fast identification
of the hyperbolic functions used to model the high frequency part of the spectrum The low
frequency part can be efficiently represented using again the vector fitting algorithm, with a
significant compression of the state-space model. A pertinent example has been presented, that
proves the capability of the proposed method to efficiently model electrically long cables with
a large number of conductors.
Acknowledgment
The authors acknowledge Svenska Kraftnät (Swedish national grid) for providing funding for
this 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 Compatibility (EMC EUROPE), 2012 International Symposium on, Sept 2012, pp. 1–6.
[2] C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed.
John Wiley & Sons, 2008.
New York, NY:
[3] L. W. Nagel, “SPICE: A computer program to simulate semiconductor circuits,” University of California, Berkeley, Electr. Res. Lab. Report ERL M520, May 1975.
[4] G. Antonini, “A dyadic Green’s function based method for the transient analysis of lossy
and dispersive multiconductor transmission lines,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 4, pp. 880–895, Apr. 2008.
[5] M. De Lauretis, G. Antonini, and J. Ekman, “Delayed impedance models of twoconductor 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 frequencyindependent 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 responses
by vector fitting,” IEEE Transactions on Power Apparatus and Systems, vol. 14, no. 3,
pp. 1052–1061, Jul. 1999.
[9] R. Achar, M. Nakhla, “Simulation of high-speed interconnects,” Proceedings of the IEEE,
vol. 89, no. 5, pp. 693–728, May 2001.
[10] B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans.
Power Del., vol. 21, no. 3, pp. 1587–1592, 2006.
[11] S. Grivet-Talocia, H.-M. Huang, A. E. Ruehli, F. Canavero and I. M. (Abe) Elfadel, “Transient analysis of lossy transmission lines: an efficient approach based on the method of
characteristics,” IEEE Transactions on Advanced Packaging, vol. 27, no. 1, pp. 45–56,
Feb. 2004.
[12] G. Andrieu and A. Reineix, “The “equivalent cable bundle method”: A solution to model
cable bundles in presence of complex ground structures,” in Electromagnetic Compatibility (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 Lossy
and Dispersive Multiconductor
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),
Dresden, Germany, 2015.
c 2015, IEEE, Reprinted with permission.
109
110
Delay-Rational Model of Lossy and Dispersive
Multiconductor Transmission Lines
Maria De Lauretis, Jonas Ekman and Giulio Antonini
Abstract
In this paper, the transient analysis of lossy and dispersive multiconductor transmission lines
is considered. The existing Green’s function-based method is extended to explicitly include
delays extraction, thus leading to a significant compressed time-domain state-space model. The
proposed method is mainly based on poles and residues asymptotic analysis and lossless delays
extraction. The resulting hybrid state-space model incorporates Dirac-combs in the input and
results into a reduced number of state variables. A test case has been considered in order to
clearly demonstrate the effectiveness of the proposed methodology. The results are compared
with the original rational Green’s function method, and with the standard inverse fast Fourier
transform 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 included), and are normally found in both signal and power applications (interconnects in printed
circuit boards (PCB) or chips, power transmission lines, cables). Multiconductor transmission
lines 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 wellestablished 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 terminations
such as drivers and receivers in PCBs or non-linear protection devices in power transmission
lines 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 transient
analysis adopt the Fast Fourier Transform (FFT), or the numerical inverse Laplace transform
(NILT). These are normally computationally expensive; for example, the numerical convolution product in the time domain with non-linear terminations is a well-known computational
bottleneck. Several other methods nowadays available relay in rational function approximations, that lead to rational models in terms of poles and residues. For example, techniques
based on the Padé approach [4, 5], or on the well-known Method of Characteristic [6] and its
generalizations [7, 8, 9]. Rational models may not be efficient, however, for electrically long
MTLs, mainly due to the huge number of poles required to properly represent the propagation 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 propagation problem. A rational series form is presented, where poles are identified by solving
algebraic 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 electrically
long MTL is considered, since a significant number of modes, and thus poles, is necessary to
achieve a good accuracy, especially in terms of ringing effects in the time domain. In order
to overcome this drawback, the method in [10] has been improved for frequency-independent
per-unit-length (p.u.l.) parameters single conductor TLs by the authors in [11], thanks to the
explicit delay extraction. The delay-rational model obtained has a reduced size and better accuracy in the time domain. Recently, the method has been further extended to MTLs with
frequency-independent p.u.l. parameters in [12]. The aim of the present paper is the generalization of the model in [12], proposed for frequency-independent p.u.l. parameters MTLs, to
MTLs with frequency-dependent per-unit-length parameters. Hence, a delay-rational Green’s
method is proposed, that consists of the following main steps:
• poles and residues computation of the impedance matrix as described in [10] (Section
2);
• analysis of poles and residues in order to identify and exploit periodical behaviors (Sections 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 macromodel is obtained without resorting to moment-matching techniques.
2
Green’s function-based method background
A generic MTL can be represented as a multiport network [3] as in Fig. 1. 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 electromagnetic field mode propagates,
read [3]
∂z V(z, s) = −Z0 (s)I(s, z)
∂z I(z, s) = −Y0 (s)V(s, z) + Is (z, s)
(1a)
(1b)
where Z0 (s) = R(s) + sL(s) is the per-unit-length (p.u.l.) longitudinal impedance, Y0 (s) =
G(s) + sC(s) is the p.u.l. transverse admittance, with R(s), L(s), C(s) and G(s) the frequencydependent p.u.l. parameter matrices, non-negative definite symmetric of order N, being N + 1
the number of the conductors with the reference conductor included [3, 13]. Is (z, s) represents
a p.u.l. current source located at abscissa z. V (z, s) and I (z, s) represent the voltage and current
2. G REEN ’ S FUNCTION - BASED METHOD BACKGROUND
V1(s) +
V2(s) +
-
IN+2(s)
MTL
+
+
VN+1(s)
VN+2(s)
͘͘͘
͘͘͘
VN(s) +
IN+1(s)
I1(s)
I2(s)
113
IN(s)
I2N(s)
+
z= 0
V2N(s)
z= l
-
Figure 1: Multiconductor transmission line represented as 2N port system, with a common reference
conductor. 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 ` in
terms of port currents is given as
V0 (s)
Z11 (s) Z12 (s) I0 (s)
I0 (s)
=
= Z(s)
V` (s)
Z12 (s) Z11 (s) I` (s)
I` (s)
(2)
where Z(s) is the port impedance matrix, that is block symmetrical, each block of dimension
N × N. V0 (s) and I0 (s) are the voltage and the current port vectors respectively, of dimension
N × 1, related to the input ports at z = 0. V` (s) and I` (s) are the voltage and the current port
vectors 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 approximation, in a poles/residues form. In particular, a series form of the Dyadic Green’s Function is
given, 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 numerical computation, the infinite summation is truncated to M modes. Assuming Z0 (s) and
Y0 (s) available at N f discrete frequency samples sq = j2π fq , with q = 1, ..., N f , the vector
fitting technique [14] is applied, and a rational polynomial form is obtained. By using the
Green’s function method developed in [10], the Z(s) can be expressed as an infinite sum of
poles and residues. In fact, for each mode m, a number n = 1, ..., ñ p of poles and residues can
be evaluated, leading to a final number of poles equal to
M
np =
∑ ñ p,m .
(3)
m=0
The nd dominant poles p and residues R are then selected [10], leading to the final expression
nd
Z(s) =
Rn
∑ s − pn .
n=1
(4)
114
PAPER D
Longitunal impedance ǯ and transversal admittance ǯ
Available at discrete frequency samples
ࢆԢ ሺ‫ݏ‬ሻ ൌ ࡾሺ‫ݏ‬ሻ ൅ ‫ࡸݏ‬ሺ‫ݏ‬ሻ
ࢅԢ ሺ‫ݏ‬ሻ ൌ ࡳሺ‫ݏ‬ሻ ൅ ‫࡯ݏ‬ሺ‫ݏ‬ሻ
Rational polynomial forms of ǯ and ǯ
Vector fitting ሾͳͶሿ
ࢆԢሺ‫ݏ‬ሻ ൌ
࡮‫ ݌‬ሺ‫ݏ‬ሻ
‫ ݌ܣ‬ሺ‫ݏ‬ሻ
ࢅԢሺ‫ݏ‬ሻ ൌ
ࡰ‫ ݌‬ሺ‫ݏ‬ሻ
‫ ݌ܥ‬ሺ‫ݏ‬ሻ
Green’s function based method
Poles and residues form
࡮‫ ݌‬ሺ‫ݏ‬ሻ ࡰ‫ ݌‬ሺ‫ݏ‬ሻ
࡮‫ ݌‬ሺ‫ݏ‬ሻ
݉ߨ ʹ
ͳ
ࢆሺ‫ݏ‬ሻ ൌ ෍ ቈ
൅ ቀ ቁ ࢁ቉ ‫݉ʹܣ‬
൤
‫ ݌ܣ‬ሺ‫ݏ‬ሻ ‫ ݌ܥ‬ሺ‫ݏ‬ሻ
݈
‫ ݌ܣ‬ሺ‫ݏ‬ሻ ሺെͳሻ݉
൅λ
݉ ൌͲ
ሺെͳሻ݉
ࡾ࢔
൨ ൌ෍
ͳ
‫ ݏ‬െ ‫݊݌‬
൅λ
݊ൌͳ
Poles and residues pruning
Dominant poles and residues selected
݊݀
ࢆሺ‫ݏ‬ሻ ൌ ෍
݊ൌͳ
ࡾ࢔
‫ ݏ‬െ ‫݊݌‬
Figure 2: Green’s function-based method for MTLs with frequency-dependent per-unit-length parameters.
It is worth noting that, in order to simplify the notation, it is not specified whether the poles
are real or complex conjugate. The poles/residues representation is suited to be translated in a
state space form as described in [2]
ẋ (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 extraction
Despite the pole/residue pruning adopted, the number of summation modes required in order
to model the frequency response of electrically long lines can still be significantly large. This
is a problem especially for time domain simulations, since the model order of (5) is equal to
2Nn p , where n p is the number of poles, and 2N = N p is the number of ports. In order to
reduced the size of the model, a new delay-rational macromodel for frequency-independent
p.u.l. parameter N + 1 = 2 conductor TLs has been proposed in [11]. In this case, we have one
2. G REEN ’ S FUNCTION - BASED METHOD BACKGROUND
115
complex conjugate pair, for each mode m > 0, and one real pole for m = 0. The main advantage
relies 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 = ± j2π m
T , 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 m̂;
• the lossless delay can be used in order to gain a closed expression
in terms of hyperbolic
q
functions in the frequency domain as in (6), where η =
of poles
L
C,
and n̂ p is the total number
n̂ p
n̂ p
Rn
Rn
T
Z11 (s) = ∑
−∑
+ η coth (s − α)
2
n=1 s − pn
n=1 s − p̂n
(6a)
n̂ p
T
Rn
Rn
−∑
+ η csch (s − α)
;
Z12 (s) = ∑
2
n=1 s − p̂n
n=1 s − pn
(6b)
n̂ p
• it is possible to prove that the hyperbolic functions lead to a Dirac comb formulation in
the time domain [12, 11]
+∞
T
→ ηeαt δ (t) + 2ηeαt ∑ δ (t − mT )
η coth (s − α)
2
m=1
+∞ T
T
αt
→ 2ηe ∑ δ t − mT −
.
η csch (s − α)
2
2
m=0
(7)
(8)
The Dirac-comb formulation allows to avoid the convolution product. A delay-rational model
in the time domain is gained as
ẋ (t) = Ax (t) + Bi (t)
(9a)
v (t) = Cx (t) + D0 i (t) + id (t)
(9b)
where id (t) = D(t) ⊗ i (t). The D0 is defined as
η 0
D0 =
.
0 η
(10)
It is worth noting that the convolution δ (t) ⊗ i(t) = i(t), and then it is omitted in the state-space
model formulation. The 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 in the
form
D11 D12
D(t) =
(11)
D12 D11
116
PAPER D
with
+∞
D11 = 2ηeαt
∑ δ (t − mT )
(12a)
m=1
+∞
T
D12 = 2ηe ∑ δ t − mT −
2
m=0
αt
(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 translated
version of the currents, so no convolutions products are actually performed in the time-domain
solver.
Recently, the method has been extended by the authors to MTLs with frequency-independent
p.u.l. parameters [12]. The main steps are below summarized.
1. Given the p.u.l. longitudinal impedance Z0 and transverse admittance Y0 , the Green’s
function method is applied. Each sub-block Zi j for i, j = 1, 2 in (2) is expressed as sum
of poles and residues
∞
Zi, j =
Rn
∑ s − pn , for i, j = 1, 2 .
(13a)
n=1
2. The complex conjugate poles are divided into N families, each family identified from its
real part, as better visually explained in Fig. 3. Each mode m > 0 exhibits N complex
conjugate pair of poles. The mode m = 0 normally exhibits real poles in number equal
to N. For m ≥ m̂, the complex conjugate poles exhibit a periodic behavior inside each
family, meaning they are equi-spaced along the imaginary axis
p̂k,m(1,2) = αk ± jmβk , for m = m̂, ..., +∞.
(14a)
Their corresponding residue matrices R̂k are real and constant for each family, since their
phase tends asymptotically to zero.
3. Because of the periodicity for m = m̂, ..., +∞, the lossless delay Tk can be extracted for
each family of poles/residues, and incorporated in the frequency model through hyperbolic functions
!
n̂ p
m̂−1
N
N
R̂k
R̂k Tk
Rn
Tk
+∑
−∑ ∑
coth (s − αk )
(15a)
Z11 (s) = ∑
2
2
m=0 k=1 s − p̂k,m
n=1 s − pn
k=1
!
n̂ p
m̂−1
N
N
Rn
R̂k
R̂k Tk
Tk
Z12 (s) = ∑
−∑ ∑
+∑
csch (s − αk )
.
(15b)
2
n=1 s − pn
m=0 k=1 s − p̂k,m
k=1 2
It is worth observing that, in order to simplify the notation, it is not specified whether the poles
are real or complex conjugate. Note that (15) is the natural extension to the multiconductor
case of the expressions in (6). The state space time domain model reads as (9), with D0 and
D(t) opportunely redefined.
3. D ELAY-R ATIONAL G REEN ’ S M ETHOD FOR MTL WITH FREQUENCY- DEPENDENT
P. U . L . PARAMETERS
117
6
×109
4
Family1
Family2
ℑ(p k,n)
2
0
-2
-4
-6
-5
-4.5
-4
-3.5
ℜ (p k,n)
-3
-2.5
×105
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 frequencydependent p.u.l. parameters
The delay-rational model in [12] and summarized in the previous section is now extended
to MTLs with frequency-dependent p.u.l. parameters. Basically, starting with the original
approach 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 explain
the mathematical results. In particular, it is considered the N + 1 = 3 conductor transmission
line given as “Line 2” in [15], with frequency-dependent p.u.l. parameters as given in [15] and
length ` = 10 cm. The p.u.l. matrices R(s), L(s), G(s), and L(s) are available at N f discrete
frequency points sq = j2π fq , with q = 1, ..., N f . As summarized in Fig. 2, the vector fitting
technique is applied [16, 17], and Z0 (s) and Y0 (s) are expressed in a rational polynomial form.
Normally, a reduced number of poles is sufficient to capture their frequency behavior. Denoting
with Pz and Py the number of poles required for Z0 (s) and Y0 (s) respectively, Pz = Py = 3 are
used 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 impedance
matrix Z(s). The summation is truncated for m = M, with M normally high. Each mode has a
number of poles equal to (Pz + Py + 2)N [10]. For the Line 2 example, M = 400 has been used
to 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
2
×1012
Family 1
Family 2
ℑ (p)
1
0
-1
-2
-2.5
-2
-1.5
-1
-0.5
ℜ (p)
0
×10
9
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 are
analyzed, 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 analysis
We 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 a
static real part αk , but the real part changes with the summation mode αk,m . It then reaches an
asymptotic value after a certain mode m ≥ m̂
αk,m −→n≥m̂ α̂k .
(16)
For m = m̂, ..., +∞, the poles are approximated using
• the real asymptotic value α̂k ,
• 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 for
the frequency-dependent one the imaginary parts βk,m of the poles become equispaced for
3. D ELAY-R ATIONAL G REEN ’ S M ETHOD FOR MTL WITH FREQUENCY- DEPENDENT
P. U . L . PARAMETERS
119
3.75
×1010
Family 1
Family 2
ℜ Res11
3.7
3.65
3.6
3.55
0
100
200
300
400
Mode
Figure 5: Real part of the residue Rk,n , for the example Line 2 in [15].
m = m̂, ..., +∞. Denoting by β̂k,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 β̂k,m =
mβ̂k,1 . The lossless delay associated with each family can be then expressed as
T̂k =
2π
, for k = 1, ..., N .
(17)
β̂k,1
The asymptotic poles used will be
p̂k,m = α̂k ± jmβ̂k,1 .
3.2
(18)
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 approximated
with the asymptotic real value reached after a certain mode of the summation m̂, as shown in
Fig. 5.
3.3
Identification of truncation mode m̂
The value of m̂ can be selected considering both the real parts of the poles, and the phase of
the residues. Given a certain tolerance value tolRes , each element of Rk,m has to satisfy the
condition (19). Regarding the poles, given tolRe , the condition (20) has to be satisfied. The
120
PAPER D
6
×10-3
Family 1
Family 2
(Res11 )[rad]
4
2
0
-2
-4
0
100
200
300
400
Mode
Figure 6: Phase of the residue Rk,n , for the example Line 2 in [15], that is a p.u.l. frequency-dependent
p.u.l. parameters transmission line with N + 1 = 3 conductors.
value of m̂ is then chosen such that
and
∠Rk,m̂ − ∠Rk,m̂−1
< tolRes
∠Rk,2 − ∠Rk,1
(19)
αk,m̂ − αk,m̂−1
> tolRe .
αk,m̂
(20)
We note that, since the high frequency behavior will be covered thanks to the extracted lossless
delay, a low value of tolerances can be used. Note that, for m ≥ m̂, Rk,m = R̂k is a constant
value, for k = 1, ..., N. For the Line 2 example, we set the tolerances as tolRes = 10−1 and
tolRe = 10−3 , leading to a number of modes m̂ = 88, with a total number of poles equal to
1416.
3.4
Reduced-order model via poles/residues pruning
Even though the infinite summation is truncated to m̂ M, still a pole/residue pruning is
necessary, as already observed in [10], following the guidelines in [18]. After the poles/residues
pruning, for the Line 2 example, the number of poles decreases from 1416 to 354. It is worth
to note that the number of poles for each mode changes, and it is not known a priori. For a
intuitive comparison, the relative error can be observed: as it can be seen in Fig. 7, the accuracy
of the delay-rational proposed method is comparable with the previous rational one but it is
significantly less demanding in terms of memory storage and computational complexity.
4. D ELAY-R ATIONAL MODEL IN THE TIME - DOMAIN
Relative error for Z24
10
121
Z12 block
0
10-1
10-2
10-3
Delay-Rational
Rational
10-4
0
2
4
6
8
10
×109
Freq [Hz]
Figure 7: Relative error of the rational and the delay-rational methods, compared to the frequency
response 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-domain
The final Delay-Rational state space model reads as in (9), with D0 and D(t) redefined as
"
#
N
R̂k T̂2k
0
D0 = ∑
.
(21)
0
R̂k T̂2k
k=1
N
D(t) =
∑
k=1
D11 D12
D12 D11
(22)
with
+∞
D11 = R̂k T̂k eα̂k t
∑δ
t − mT̂k
(23a)
m=1
+∞
α̂k t
D12 = R̂k T̂k e
5
T̂k
∑ δ t − mT̂k − 2 .
m=0
(23b)
Numerical Results
The proposed method has been validated through simulations with Matlab R2015a [19], on a
computer 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
354
2816
484
Delay-Rational
0.4
Rational
Delay-Rational
IFFT
V1 [V]
0.3
0.2
0.1
0
-0.1
0
5
10
15
20
25
30
Time [ns]
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 voltage of amplitude 1 V, rise and fall times tr = t f = 50 ps and pulse-width τ = 2 ns. All the
ports 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 standard
Inverse-Fast-Fourier Transform (IFFT) technique are compared, and the computational times
between the Rational and the Delay-Rational methods are summarized in Tab. 1. The extracted
lossless delays are T1 = 1.5777 ns and T2 = 1.6405 ns. The port voltages V1 (input port for the
first conductor) and V4 (output port of the second conductor) are shown in Figs. 8 and 9. A
satisfactory agreement is achieved.
6
Conclusions
In this work, a delay-rational model of MTLs with frequency-dependent p.u.l. parameters has
been proposed. Starting with the Green’s function method, the approach is based on asymptotic
analysis of poles and residues, and lossless delays extraction. The reduced number of poles
required by the new formulation leads to a reduced model-order of the state-space form. The
R EFERENCES
123
0.05
V4 [V]
Rational
Delay-Rational
IFFT
0
-0.05
0
5
10
15
20
25
30
Time [ns]
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 compress
the size of the model when compared to the standard rational one, leading to a significant
speed-up in time-domain simulations.
Acknowledgments
The authors acknowledge Svenska Kraftnät (Swedish national grid) for funding this research.
References
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Errata Corrige
Chapter 3
1. 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 transmission lines when the propagation delay TD of a traveling signal is large compared with the
physical length ` of the cable, that is,
TD `
(3.10)
must be substituted with
Regarding 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 is
large compared with the period T of the signal, that is,
TD T
(3.10)
2. Page 16, in the sentence
`
. The definition of
As a rule of thumb, a line is considered “electrically short” when λ < 10
an “electrically short” line becomes less clear in time-domain analysis because each signal
contains 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 written
as
∂ V(`, s)
Z` V(0, s)
=e
,
(3.10)
I(0, s)
∂ z I(`, s)
where
0 −a
Z=
−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 line
equations can be written as
V(`, s)
Z` V(0, s)
=e
,
(3.10)
I(0, s)
−I(`, s)
where
0 −a
Z=
−b 0
a = R(s) + sL(s)
b = G(s) + sC(s) .
(3.11)