subband-domain universal line modeling for robust power system …cj... · 2019-02-13 ·...

SUBBAND-DOMAIN UNIVERSAL LINE MODELING FOR

ROBUST POWER SYSTEM TRANSIENT SIMULATION

A Dissertation Presented

by

Paraskevas Argyropoulos

to

The Department of Electrical and Computer Engineering

In partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Electrical Engineering

Northeastern University

Boston, Massachusetts

August 2018

ii

ABSTRACT

Currently available transient simulation packages which include frequency dependent

transmission line (FDTL) elements rely on the use of either the J. Marti method or the Universal

Line Modeling (ULM) method. The advantage of the J. Marti method is that the N-phase FDTL

can be modeled by N-independent single-phase FDTL circuits which is very robust for transposed

or nearly transposed FDTLs. The disadvantages of this method become evident as one deviates

from the transposed-line assumption such as in the case of asymmetric-lines or underground cables.

The Universal Line Modeling (ULM) method is a generalization of the J. Marti method and opts to

model the FDTL directly in the phase-domain. This method is capable of modeling arbitrary

FDTLs, not limited to the transposed and nearly transposed cases as opposed to the J. Marti method.

The disadvantages of this method however are increased computational complexity as well as

numerical considerations due to high-order rational function approximation (RFA) modeling.

This dissertation introduces a procedure for modelling a network of frequency-dependent

transmission lines (FDTL) by means of a perfect-reconstruction filter-bank (PR-FB), which allows

us to decompose the transient simulation problem into independent narrow-band (subband)

problems. This allows us to: (i) obtain insights regarding the behavior of the transients within

frequency-bands of interest (ii) use low order approximations to reduce the complexity of the

overall system, (iii) improve numerical stability, (iv) leverage parallel processing capability of

modern computers to increase simulation speed, and (v) employ distributed computing frameworks

such as Hadoop MapReduce to increase simulation speed. We use several examples to demonstrate

the utility of our subband-ULM method, including a single-phase FDTL, a three-phase FDTL, and

a single-phase FDTL network based on the IEEE 5-bus grid.

iii

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................................................ v

LIST OF TABLES .............................................................................................................. xi

CHAPTER 1 Introduction .................................................................................................. 1

1.1 Review of Circuit Theory Fundamentals ............................................................... 4 1.2 Distributed Computing Fundamentals ................................................................... 5 1.3 FDTL Modeling using Rational Function Approximations .................................... 7 1.4 FDTL Modeling using Subband Rational Function Approximations ...................... 9 1.5 Subband Transient Simulation Results .................................................................. 13

CHAPTER 2 Frequency Dependent Transmission Line (FDTL) Modeling Fundamentals... 15

2.1 Two-Port Representation ...................................................................................... 18 Special Case: Single-Phase FDTL ........................................................................ 22

2.2 Rational Function Approximations (RFAs) ........................................................... 25 Modeling the Weighting Function: The J. Marti Model and the Modal

Transform .................................................................................................... 27 Modeling the Weighting Function: The Universal Line Modeling (ULM)

method ......................................................................................................... 28 Special Case: Single-Phase Transmission Line ..................................................... 33

2.3 Discretization of the Continuous-Time Model ....................................................... 35

CHAPTER 3 QMF Filter Banks and Phasor Banks............................................................. 38

3.1 Perfect Reconstruction Tree-Structured QMF Filter Banks .................................... 38 3.2 Phasor Banks ........................................................................................................ 43

Measures of Reconstruction Numerical Robustness .............................................. 46 Optimally Robust FIR Filter ................................................................................. 48 Optimally Robust IIR Filter.................................................................................. 49 Special Cases ....................................................................................................... 52

CHAPTER 4 Subband Vector Fitting ................................................................................. 53

4.1 Case #1: Arbitrary Transfer Function (Toy Problem) ............................................ 54 4.2 Case #2: Single-Phase FDTL ................................................................................ 63 4.3 Case #3: Three-Phase FDTL ................................................................................. 64

CHAPTER 5 Transient Simulation and Accuracy of Combined Model ............................... 70

5.1 Case #1: Arbitrary Transfer Function (Toy Problem) ............................................ 70 Transient Simulation Results ................................................................................ 74

5.2 Case #2: Single-Phase FDTL ................................................................................ 77 5.3 Case #3: Three-Phase FDTL ................................................................................. 84

iv

CHAPTER 6 Scalable, Multiphase, Discrete Simulation of Frequency Dependent

Transmission Line (FDTL) Networks .......................................................................... 86

6.1 Introduction: Transfer Function based simulation vs. Y-matrix approach............... 86 6.2 Y-matrix equivalent circuits of fundamental components ...................................... 87

Multiphase Resistor (P-phase) .............................................................................. 88 Multiphase Capacitor (P-phase) ........................................................................... 90 Multiphase Inductor (P-phase) ............................................................................. 92 Simulation Examples and Unit tests ..................................................................... 94

6.3 Y-matrix equivalent circuit of an arbitrary characteristic impedance (P-phase) ...... 96 Simulation Examples and Unit tests ..................................................................... 101

6.4 Recursive computation of an arbitrary propagation function (P-phase) .................. 103 Simulation Examples and Unit tests ..................................................................... 105

6.5 Y-matrix equivalent circuits of single-phase FDTL ............................................... 107 6.6 Y-matrix equivalent circuits of multi-phase FDTL ................................................ 111 6.7 Arbitrary FDTL Network Simulation Procedure .................................................... 117 6.8 Y-matrix based FDTL simulation results ............................................................... 121

Simulation #1: Single-Phase FDTL (2-nodes) ...................................................... 121 Simulation #2: IEEE 5-Bus FDTL system (single-phase)...................................... 124 Simulation #3: Three-Phase FDTL (2-nodes) ....................................................... 130

CHAPTER 7 Concluding Remarks..................................................................................... 132

7.1 Summary of dissertation contributions .................................................................. 132 7.2 Further Research Topics ....................................................................................... 134

APPENDIX I Modal Transform in a Transposed Transmission Line .................................. 135

APPENDIX II Relations between Real and Imaginary Parts of a Causal Transfer

Function ...................................................................................................................... 139

REFERENCES ................................................................................................................... 142

v

LIST OF FIGURES

Figure 1.1: Block diagram of a Hadoop MapReduce job. ..................................................... 6

Figure 1.2: Subband transient analysis by means of a perfect reconstruction filter bank. ...... 6

Figure 1.3: Single-phase (N=1) transmission line viewed as a two-port network. ................. 7

Figure 1.4: Frequency Dependent (FD) single-phase characteristic impedance Z0(jω): (i) magnitude response (top), (ii) phase response (bottom). .......................................... 8

Figure 1.5: Frequency Dependent (FD) single-phase weighting function A(jω):

(i) magnitude response (top), (ii) phase response (bottom). .......................................... 8

Figure 1.6: Two Channel PR QMF Filter-Bank. .................................................................. 9

Figure 1.7: Logarithmic frequency tree-structured Analysis Filter Bank. ............................. 10

Figure 1.8: Multi-band Logarithmic frequency tree-structured Analysis Filter Bank. ........... 10

Figure 1.9: Ten-channel logarithmic frequency Daubechies-12 (D12) analysis filter bank magnitude responses. ................................................................................................... 11

Figure 1.10: High order Z0(s) (blue) vs. low order Z0,j(s) subband approximations (dashed

colored segments). ....................................................................................................... 12

Figure 1.11: High order A(s) (blue) vs. low order Aj(s) subband approximations (dashed


Figure 1.12: (i) Broadband-source excitation voltage (solid-black), and (ii) 10-Subband

decomposition of (i) (solid-colored) ............................................................................. 13

Figure 1.13: Subband circuit responses viewed as independent network solutions

(transient simulations) ................................................................................................. 14

Figure 1.14: Sinusoidal energization of open-circuited line (peak voltage at t=0) based on the: (i) full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(s), Aj(s)} using the

PR filter bank (red). The input voltage is shown in black. ............................................ 14

Figure 2.1: N-Phase transmission line circuit model. ........................................................... 15

Figure 2.2: N-Phase transmission line viewed as a two-port network. .................................. 18

Figure 2.3: SimPowerSystems power_lineparam 3-phase (default) tower configuration. ...... 19

Figure 2.4: Frequency Dependent (FD) 3-phase characteristic admittance Y0(jω)

magnitude responses. ................................................................................................... 20

vi

Figure 2.5: Frequency Dependent (FD) 3-phase characteristic admittance Y0(jω) phase

responses. .................................................................................................................... 20

Figure 2.6: Frequency Dependent (FD) 3-phase weighting function eigenvalues

λk{A1(jω)}: (i) magnitude responses (left), (ii) phase responses (right). ........................ 21

Figure 2.7: Single-phase (N=1) transmission line viewed as a two-port network. ................. 22

Figure 2.8: Frequency Dependent (FD) single-phase characteristic impedance Z0(jω): (i) magnitude response (top), (ii) phase response (bottom). .......................................... 23

Figure 2.9: Frequency Dependent (FD) single-phase weighting function A(jω): (i)

magnitude response (top), (ii) phase response (bottom). ............................................... 24

Figure 2.10: Frequency Dependent (FD) idempotent matrix M1(jω) magnitude responses .... 29

Figure 2.11: Frequency Dependent (FD) idempotent matrix M1(jω) phase responses ........... 30

Figure 2.12: Frequency Dependent (FD) idempotent matrix M2(jω) magnitude responses .... 30

Figure 2.13: Frequency Dependent (FD) idempotent matrix M2(jω) phase responses ........... 31

Figure 2.14: Frequency Dependent (FD) idempotent matrix M3(jω) magnitude responses. ... 31

Figure 2.15: Frequency Dependent (FD) idempotent matrix M3(jω) phase responses. .......... 32

Figure 3.1: Two Channel PR QMF Filter-Bank. .................................................................. 38

Figure 3.2: Logarithmic frequency tree-structured Analysis Filter Bank. ............................. 40

Figure 3.3: PR Filter Bank viewed as a combination of forward and inverse DWT. ............. 40

Figure 3.4: Ten-channel logarithmic frequency Daubechies-2 (Haar/D2) analysis filter bank magnitude responses. .......................................................................................... 41

Figure 3.5: Ten-channel logarithmic frequency Daubechies-6 (D6) analysis filter bank




Figure 3.7: Ten-channel logarithmic frequency FIR2CHPR-based (N=99) analysis filter

bank magnitude responses. .......................................................................................... 43

Figure 3.8: Filter-bank interpretation of dynamic phasors .................................................... 44

Figure 3.9: Simplified perturbation model for a PR filter-bank. ........................................... 46

Figure 3.10: FIR vs IIR magnitude responses of the prototype analysis filter H(z). .............. 49

vii

Figure 4.1: Transfer function data: (i) magnitude response (top), (ii) phase response

(bottom). ..................................................................................................................... 54

Figure 4.2: High order Z(s) (blue) vs. low order Zj(z) subband approximations (colored

segments). ................................................................................................................... 57

Figure 4.3: High order Z(s) (blue) vs. low order Z0(z) subband-0 approximation (red).

Zoomed-in responses. .................................................................................................. 57



Figure 4.5: High order Z(s) (blue) vs. low order Z2(z) subband-2 approximation (red). Zoomed-in responses. .................................................................................................. 58









Figure 4.10: High order Z(s) (blue) vs. low order Z7(z) subband-7 approximation (red). Zoomed-in responses. .................................................................................................. 61





Figure 4.13: High order Z0(s) (blue) vs. low order Z0,j(z) subband approximations (dashed


Figure 4.14: High order A(s) (blue) vs. low order Aj(z) subband approximations (dashed


Figure 4.15: Subband FD admittance matrix Y0,j(jω) magnitude responses. ......................... 65

Figure 4.16: Subband FD admittance matrix Y0,j(jω) phase responses. ................................. 65

Figure 4.17: Subband FD weighting function eigenvalues λk,j{A1(jω)} (i) magnitude

responses (left), (ii) phase responses (right). ................................................................ 66

Figure 4.18: Subband FD idempotent matrix M1,j(jω) magnitude responses. ........................ 66

viii

Figure 4.19: Subband FD idempotent matrix M1,j(jω) phase responses................................. 67


Figure 4.21: Subband FD idempotent matrix M2,j(jω) 𝑝hase responses. ............................... 68


Figure 4.23: Subband FD idempotent matrix M3,j(jω) phase responses................................. 69

Figure 5.1: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order subband RFAs Zj(z) and the choice of a D2/Haar filter bank (red). ...................... 71

Figure 5.2: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low

order subband RFAs Zj(z) and the choice of a D6 filter bank (red). .............................. 71

Figure 5.3: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low

order subband RFAs Zj(z) and the choice of a D12 filter bank (red). ............................ 72

Figure 5.4: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order subband RFAs Zj(z) and the choice of a FIR2CHPR-based (N=99) filter bank

(red). ........................................................................................................................... 72

Figure 5.5: Transient responses based on the: (i) full order model Z(s) (blue), (ii) low

order Zj(z) models using the D2/Haar filter bank (red). ................................................ 74


order Zj(z) models using the D6 filter bank (red). ......................................................... 75

Figure 5.7: Transient responses based on the: (i) full order model Z(s) (blue), (ii) low order Zj(z) models using the D12 filter bank (red). ....................................................... 75


order Zj(z) models using the FIR2CHPR (N=99) filter bank (red). ............................... 76

Figure 5.9: High order Z0(s) (blue) vs. low order Z0,j(z) subband approximations (dashed colored segments). ....................................................................................................... 79

Figure 5.10: High order A(s) (blue) vs. low order Aj(z) subband approximations (dashed



decomposition of (i) (solid-colored) ............................................................................. 81

Figure 5.12: Subband circuit responses viewed as independent network solutions (transient simulations) [open-ended line] ..................................................................... 82

Figure 5.13: Subband circuit responses viewed as independent network solutions

(transient simulations) [short-ended line] ..................................................................... 82

ix

Figure 5.14: Sinusoidal energization of open-circuited line (peak voltage at t=0) based on

the: (i) full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(z), Aj(z)} using the PR filter bank (red). The input voltage is shown in black. ............................................ 83

Figure 5.15: Sinusoidal energization of short-circuited line (peak voltage at t=0) based on

the: (i) full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(z), Aj(z)} using the

PR filter bank (red). ..................................................................................................... 83

Figure 5.16: Sinusoidal energization of open-circuited line (peak voltage at t=0) based on

the: (i) full order model {Y0(s), λk{A1(s)}, Mk(s)} (blue), (ii) low order {Y0,j(z),

λk,j{A1(z)}, Mk,j(z)} using the PR filter bank (red). The input voltage is shown in black. .......................................................................................................................... 84

Figure 5.17: Sinusoidal energization of short-circuited line (peak voltage at t=0) based on

the: (i) full order model {Y0(s), λk{A1(s)}, Mk(s)} (blue), (ii) low order {Y0,j(z), λk,j{A1(z)}, Mk,j(z)} using the PR filter bank (red). ....................................................... 85

Figure 6.1: Continuous-time representation of resistor in the time-domain (left) and

complex-frequency (Laplace) domain (right). .............................................................. 88

Figure 6.2: Time-domain, Continuous-time representation of resistor (left). Discrete-time representation of resistor (right). .................................................................................. 89

Figure 6.3: Continuous-time representation of capacitor in the time-domain (left) and

complex-frequency (Laplace) domain (right) (initial condition included). .................... 90

Figure 6.4: Time-domain, Continuous-time representation of capacitor (left). Discrete-

time representation of capacitor (right). ....................................................................... 92

Figure 6.5: Continuous-time representation of inductor in the time-domain (left) and complex-frequency (Laplace) domain (right) (initial condition included). .................... 92

Figure 6.6: Time-domain, Continuous-time representation of inductor (left). Discrete-

time representation of inductor (right). ......................................................................... 94

Figure 6.7: Time-domain, Continuous-time representation of test circuit ............................. 94

Figure 6.8: Time-domain, Discrete-time representation of test circuit .................................. 95

Figure 6.9: Analytical (continuous-time) response (solid-blue) vs. discrete-time simulator

(solid-red) ................................................................................................................... 96

Figure 6.10: Continuous-time characteristic impedance (top) and Foster-form (cascade)

RC-implementation (bottom) ....................................................................................... 97

Figure 6.11: Continuous-time characteristic impedance (left). Discrete-time equivalent

circuit (right) ............................................................................................................... 98

Figure 6.12: Continuous-time simulator (solid-red) vs. discrete-time simulator (solid-

blue) for a characteristic impedance response............................................................... 102

x

Figure 6.13: Analytical (continuous-time) response (solid-red) vs. discrete-time simulator

(solid-blue) for a propagation function response........................................................... 106

Figure 6.14: Continuous-time two-port model for single-phase FDTL (complex-

frequency domain) ....................................................................................................... 107

Figure 6.15: Discrete-time equivalent circuit for single-phase FDTL ................................... 109

Figure 6.16: Continuous-time two-port model for multi-phase FDTL (complex-frequency domain) ....................................................................................................................... 111

Figure 6.17: Discrete-time equivalent circuit for multi-phase FDTL .................................... 114

Figure 6.18: Two terminal, continuous-time, single-phase FDTL network ........................... 121

Figure 6.19: Two terminal, discrete-time equivalent, single-phase FDTL network ............... 121

Figure 6.20: Receiving-end voltage: simulation based on the: (i) full order model

broadband (blue), (ii) low order subband model (red). ................................................. 122

Figure 6.21: Receiving-end current: simulation based on the: (i) full order model

broadband (blue), (ii) low order subband model (red)................................................... 123

Figure 6.22: IEEE 5-Bus network topology ......................................................................... 124

Figure 6.23: IEEE 5-Bus network topology with FDTL annotations .................................... 125

Figure 6.24: IEEE 5-Bus, node-3 voltage: (a) broadband simulation (blue), (b) low-order

subband simulation (red) ............................................................................................. 128

Figure 6.25: IEEE 5-Bus, node-4 voltage: (a) broadband simulation (blue), (b) low-order subband simulation (red) ............................................................................................. 128

Figure 6.26: IEEE 5-Bus, node-3 current: (a) broadband simulation (blue), (b) low-order

subband simulation (red) ............................................................................................. 129

Figure 6.27: Two terminal, continuous-time, 3-phase FDTL network .................................. 130

Figure 6.28: Two terminal, discrete-time equivalent, 3-phase FDTL network ...................... 130

Figure 6.29: Three-phase receiving-end voltage: simulation based on the: (i) full order

model broadband (blue), (ii) low order subband model (red). ...................................... 131

Figure 6.30: Three-phase receiving-end current: simulation based on the: (i) full order

model broadband (blue), (ii) low order subband model (red). ...................................... 131

xi

LIST OF TABLES

Table 2.1: Vector fitting based RFAs Z0(s) and A(s) ............................................................ 33

Table 2.2: Absolute value of the poles of Z0(z) and A(z)...................................................... 37

Table 4.1: Subband frequency range decomposition. ........................................................... 53

Table 4.2: Vector fitting based high-order RFA Z(s)............................................................ 55

Table 4.3: Absolute value of the poles of Z(z). .................................................................... 56

Table 5.1: Subband frequency range decomposition. ........................................................... 79

CHAPTER 1

Introduction

Accurate power system fault location studies such as ones based on travelling wave

methods [1] typically require electromagnetic transient program (EMTP) simulations over a very

wide range of frequencies (up to 1MHz) [2]. Obtaining a broadband, rational function

approximation (RFA) that describes each power system component (such as the characteristic

impedance 𝑍(𝑠) and propagation/weighting function 𝐴(𝑠) of a frequency dependent transmission

line) is essential for accurate transient simulation [3, 30]. Continuous-time (CT) RFA methods such

as vector fitting (VF) [4] applied on a broadband system [0Hz(dc)-1Mhz] result in a stable but high

order approximant 𝑍(𝑠). Discretization of the high-order 𝑍(𝑠) results in a discrete-time (DT)

transfer function 𝑍(𝑧), and is prone to numerical instability due to pole-clustering around the unit

circle [5].

Currently available transient simulation packages which include frequency dependent

transmission line (FDTL) elements rely on the use of either the J. Marti method [3] or the Universal

Line Modeling (ULM) method [7, 8]. The J. Marti method employs a broadband, high-order RFA

on the N-decoupled modes (N-eigenvalues) of an N-phase transmission line. This is achieved by

decomposing the N-phase transmission line into N-independent circuits by means of a fixed-modal

transform. The advantage of this method is that the N-phase FDTL can be modeled by

N-independent single-phase FDTL circuits. The solution of each circuit is finally recombined by

means of a fixed eigen-vector matrix also known as the Clarke transform. This approach is very

robust for transposed or nearly transposed FDTLs where the assumption of a fixed eigenvector-

matrix is valid. The disadvantages of this method become evident as one deviates from the

transposed-line assumption such as in the case of asymmetric-lines or underground cables. In such

2

cases the eigen-vector matrix becomes frequency-dependent and needs to be taken under

consideration using the RFA modeling process [3]. The Universal Line Modeling (ULM) method

is a generalization of the J. Marti method and opts to model the FDTL directly in the phase-domain.

This method is capable of modeling arbitrary FDTLs, not limited to the transposed and nearly

transposed cases as opposed to the J. Marti method. The disadvantages of this method however are

increased computational complexity due to the broadband, high-order RFAs associated with

modeling the FDTL directly in the phase-domain [7, 8] as well as numerical considerations due to

high-order RFA modeling.

This work introduces a filter-bank technique which allows us to decompose the transient

simulation problem into independent narrow-band (subband) problems. The fundamental idea

behind our proposed approach is that every linear, time-invariant (LTI) system can be mapped into

a collection of uncoupled subband versions of the same problem by means of a frequency-selective

Analysis Filter-bank (AFB). By decomposing the broadband problem into subbands we are able to

reduce the bandwidth required for modeling the FDTL in each subband which allows us to use low

order RFAs. This in turn results in more robust discretization. Moreover, since the subbands are

uncoupled, we are able to process each one of them independently by means of distributed/parallel

processing. The use of subband modeling allows us to: (i) obtain insights regarding the behavior of

the transients within frequency-bands of interest (ii) use low order approximations to reduce the

complexity of the overall system, (iii) improve numerical stability, (iv) leverage parallel processing

capability of modern computers to increase simulation speed, and (v) employ distributed computing

frameworks such as Hadoop MapReduce to increase simulation speed [6]. In summary, the

proposed method constitutes a subband alternative to the distributed parameter FDTL using an

electromagnetic transient simulation program (EMTP), and is very suitable for simulating typical

transient power-system phenomena such as: (i) line energization, (ii) capacitor-switching, and (iii)

fault simulation [1, 2].

3

This dissertation is organized as follows: Chapter 2 provides the fundamentals of modeling

a multi-conductor (𝑁 −phase) frequency dependent transmission line (FDTL) by means of high-

order RFAs [4] based on: (i) the modal decomposition [3], and (ii) the Universal Line Modeling

(ULM) method [7] using the idempotent (spectral) decomposition [8]. Moreover, a discussion

regarding discretization and numerical stability considerations is provided. In Chapter 3 we

introduce the fundamentals of perfect reconstruction filter banks (PR-FBs) and multirate

processing, which are essential for constructing (accurate) subband low order RFAs. In particular,

two classes of PR filter banks are presented: (i) logarithmic-frequency, tree-structured quadrature

mirror filter (QMF) banks [9], and (ii) Phasor Banks [10-12]. In Chapter 4 we use the tools

developed in Chapter 3 to decompose the single high-order RFA broadband simulation problem

into several subbands and apply low-order RFAs in each one of them. This is carried out for both

single-phase and three-phase FDTL data based on Chapter 2. In Chapter 5 we study how a particular

choice of PR-FB affects the accuracy of transient simulation by computing the frequency response

of the “combined” transfer function. This is because in Chapter 4 we implicitly assumed that the

prototype analysis filter 𝐻0(𝑧) is an ideal lowpass filter. Deviations from this assumption are

investigated in this chapter. Finally, in Chapter 6 we introduce the discrete-time, 𝑌-matrix state-

space based approach for simulating networks consisting of one (or more) FDTLs. This approach

is very suitable for simulating an arbitrarily large network without introducing significant

complexity in deriving the model (scalability). The 𝑌-matrix state-space based approach is used to

simulate several FDTL networks in order to establish proof of principle and feasibility of the

proposed subband-ULM method.

4

1.1 Review of Circuit Theory Fundamentals

The fundamental idea behind the proposed work lies within the concept of superposition

which holds true for a linear, time-invariant (LTI) system such as a transmission line [13]. This

concept is used throughout circuit analysis. Some variations of this fundamental principle include:

(i) Sinusoidal steady-state analysis using complex-numbers as a special case of the Fourier

transform for purely sinusoidal excitations [14].

(ii) Transient response analysis using the Laplace transform as a generalization of (i) [14].

(iii) 𝑁 −phase FDTL transient modeling using the Laplace transform and the modal transform

(Eigen-decomposition of the 𝑁 −phases) as a generalization of (ii) [3] .

(iv) 𝑁 −phase FDTL transient modeling directly in the phase domain by means of the Laplace

transform and the (idempotent) spectral decomposition of the phases as a generalization of

(iii) [7].

In this work we generalize approach (iv) by using a subband, 𝑁 −phase modeling

technique directly in the phase domain by means of the Laplace transform and the (idempotent)

spectral decomposition of the phases. The subband analysis is achieved by means of a perfect

reconstruction filter bank which decomposes all currents/voltages (or equivalently the

excitations/sources) into narrow band subcomponents [9]. An oversimplification of this approach

which can be helpful to understand the fundamental principle can be viewed by exploring approach

(i) in the case where the source consists of multiple purely sinusoidal excitations. In this case

several independent circuits can be constructed, each one of them characterized by its own set of

complex numbers, and be solved independently. All results are then combined (reconstructed) in

the time-domain after an inverse Fourier/Phasor transform is applied on each sub-circuit.

5

1.2 Distributed Computing Fundamentals

A distributed computation platform such as Hadoop consists fundamentally of two

subcomponents: (a) The Hadoop distributed file system (HDFS), and (b) the MapReduce

framework. MapReduce consists of the following methods/processes:

(i) Map: which according to the definition, takes an input list of ‘keys’ and ‘values’ and

produces another list of ‘key’ and ‘values’. It is a transformation (set of rules) which takes

as an input large amounts of data (or equivalently a large problem) and partitions it into

several smaller jobs/problems [6].

(ii) Reduce: Which according to the definition, takes an input list of ‘keys’ and ‘values’ and

produces a list of ‘values’ alone. It is a transformation (set of rules) which handles all the

results from intermediate jobs (after the ‘map’ stage) and combines them back together

(reconstruction). Fig. 1.1, shows the block diagram of a MapReduce job [6].

(iii) Finally, this framework consists of more details (such as data/job replication within

clusters) in order to achieve high reliability in case one of the clusters/jobs fails to complete

[6].

From Fig. 1.1 we observe that the structure of a MapReduce process is analogous to the concept

of applying a subband decomposition by means of a perfect reconstruction filter bank (Fig. 1.2).

In particular, the analogies for modeling an FDTL are the following: (i) “MAP” = “Analysis filter

bank”, (ii) “REDUCE” = “Synthesis filter bank”, and (iii) “JOBS” = “subband transient sub-

simulations”.

6

Figure 1.1: Block diagram of a Hadoop MapReduce job.

Figure 1.2: Subband transient analysis by means of a perfect reconstruction filter bank.

7

1.3 FDTL Modeling using Rational Function Approximations

An FDTL is uniquely characterized by its: (i) characteristic impedance 𝑍0(𝑠) ∈ ℂ [𝛺],

weighting function 𝐴(𝑠) ∈ ℂ [1], and (iii) length ℓ ∈ ℝ [𝑘𝑚]. This allows us to construct a two-

port representation for the FDTL which is suitable for time-domain transient simulation by means

of a circuit solver (Chapter 2) (Fig. 1.3).

Figure 1.3: Single-phase (N=1) transmission line viewed as a two-port network.

𝐼𝑘(𝑠) =𝑉𝑘(𝑠)

𝑍0(𝑠)− [𝑉𝑚(𝑠)

𝑍0(𝑠)+ 𝐼𝑚(𝑠)] 𝐴(𝑠) (1.1𝑎)

𝐼𝑚(𝑠) =𝑉𝑚(𝑠)

𝑍0(𝑠)− [𝑉𝑘(𝑠)

𝑍0(𝑠)+ 𝐼𝑘(𝑠)] 𝐴(𝑠) (1.1𝑏)

Typical FD data for 𝑍0(𝑗𝜔) and 𝐴(𝑗𝜔) for a given length “ℓ” are shown in Figs. 1.4, 1.5 (solid

blue lines). In order to facilitate accurate numerical simulation of an FDTL, the FD data need to be

fitted by means of a rational function approximation (RFA) [3, 4]. A prominent numerical

technique for obtaining such continuous-time RFAs is based on vector fitting [4]. The solid-red

lines in Figs. 1.4, 1.5 correspond to the result of high-order RFAs by means of vector fitting.

Finally, in order to support discrete-time simulation using a fixed-step solver the continuous-time

RFAs are converted to discrete-time RFAs by means of a generalized bilinear transform

(Chapter 2). This approach results in high-order RFAs which lead to increased computational

complexity and are prone to numerical instability after the discretization step is applied.

8

Figure 1.4: Frequency Dependent (FD) single-phase characteristic impedance Z0(jω):

(i) magnitude response (top), (ii) phase response (bottom).

Figure 1.5: Frequency Dependent (FD) single-phase weighting function A(jω): (i) magnitude response (top), (ii) phase response (bottom).

9

1.4 FDTL Modeling using Subband Rational Function Approximations

In order to construct low-order RFAs (Chapter 4) that match the original (broadband)

FDTL data very closely, we propose to employ the concept of a Perfect Reconstruction Filter Bank

(PR-FB) (Chapter 3). A 2-channel perfect reconstruction filter bank is characterized by: (i) the

analysis stage filters {𝐻0(𝑧), 𝐻1(𝑧)}, (ii) the frequency partition (subband domain), and (iii) the

synthesis stage filters {𝐺0(𝑧), 𝐺1(𝑧)} (Fig. 1.6) [9].

Figure 1.6: Two Channel PR QMF Filter-Bank.

Perfect reconstruction (PR) means that

𝑦[𝑛] = 𝑥[𝑛 − 𝑛0] (1 .2)

for some (integer) delay 𝑛0.

A convenient way to support more than two-subbands is realized by expanding the analysis

filters into scales as shown in Fig. 1.7. This approach also results in a logarithmic frequency

structure in the subbands. The equivalent of a multi-subband PR-FB using the expansion of

Fig. 1.7 is shown in Fig. 1.8 [9]. Our objective is to fit a low order rational function model in each

subband so that it closely resembles the high order frequency responses 𝑍0(𝑗𝜔), 𝐴(𝑗𝜔).

10

Figure 1.7: Logarithmic frequency tree-structured Analysis Filter Bank.

Figure 1.8: Multi-band Logarithmic frequency tree-structured Analysis Filter Bank.

Experimenting with different perfect reconstruction filter banks suggests that choosing a

prototype analysis filter 𝐻0(𝑧) which corresponds to a good lowpass filter (in terms of passband

ripple and stopband attenuation) is a desired characteristic for decoupling the broadband simulation

into narrowband sub-simulations (Chapter 5).

Typical frequency responses of the analysis filters (AF) in each band are shown in Fig. 1.9.

11


magnitude responses.

The resulting low-order (second-order) transfer functions { ��0,𝑗(𝑧), ��𝑗(𝑧) ; 𝑗 = 0,1, … ,9 }

based on vector fitting and the ten-channel (𝑀 = 10) PR-FB of Fig. 1.9 are all stable. Details

regarding our proposed subband FDTL modeling technique are provided in Chapter 4.

Figs. 1.10, 1.11 show plots of the high-order {��0(𝑠), ��(𝑠)} vs. the low-order discrete-time

approximations {��0,𝑗(𝑧), ��𝑗(𝑧)}. The bandwidth of each subband is defined by the x-coordinates

corresponding to two consecutive solid black dots in Figs. 1.10, 1.11.

12

Figure 1.10: High order Z0(s) (blue) vs. low order Z0,j(s) subband approximations (dashed colored segments).

Figure 1.11: High order A(s) (blue) vs. low order Aj(s) subband approximations (dashed colored

segments).

13

1.5 Subband Transient Simulation Results

Each subband circuit characterized by { ��0,𝑗(𝑧), ��𝑗(𝑧) ; 𝑗 = 0,1, … ,9 } can be solved

independently by means of parallel/distributed processing. The resulting subband transient

simulations can: (i) be viewed individually in order to obtain insights regarding the effect of a

particular transient within a frequency band of interest, and (ii) be reconstructed through the

synthesis filter-bank (Fig. 1.8) in order to obtain the equivalent broadband transient simulation

result. Details regarding our proposed subband FDTL transient simulation are provided in

Chapter 4. Fig. 1.12 shows the broadband voltage-source and its 10-subband decomposition used

to excite an open-circuited line. Fig. 1.13 shows the independent network solutions (transient

responses) using the subband-sources of Fig. 1.12 and the low order RFAs of Figs. 1.10, 1.11.

Finally, Fig. 1.14 compares the transient response (receiving-end voltage of an open-circuited line),

due to sinusoidal voltage energization (peak voltage at 𝑡 = 0), of the original CT high-order system

{��0(𝑠), ��(𝑠)} to the low-order PR filter bank proposed approach after the synthesis filter bank

(reconstruction stage). Transient simulation of the {��0(𝑠), ��(𝑠)} based system was carried out using

a variable-step solver, while simulation of the {��0,𝑗(𝑧), ��𝑗(𝑧)} based PR filter bank system was

done using a fixed step 𝛥𝑡 = 1𝜇𝑠 solver. The two responses match indeed very closely.


decomposition of (i) (solid-colored)

14

Figure 1.13: Subband circuit responses viewed as independent network solutions (transient

simulations)

Figure 1.14: Sinusoidal energization of open-circuited line (peak voltage at t=0) based on the: (i)

full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(s), Aj(s)} using the PR filter bank (red).

The input voltage is shown in black.

CHAPTER 2

Frequency Dependent Transmission Line (FDTL) Modeling Fundamentals

A N-phase frequency dependent transmission line (FDTL) is fully characterized by the

following per unit length parameters: (i) resistance 𝑹 ∶= 𝑹(𝑗𝜔) = 𝑹𝑇(𝑗𝜔) ∈ ℝ𝑁×𝑁[𝛺/𝑘𝑚], (ii)

inductance 𝑳 ∶= 𝑳(𝑗𝜔) = 𝑳𝑇(𝑗𝜔) ∈ ℝ𝑁×𝑁[𝐻/𝑘𝑚], (iii) capacitance 𝑪 ∶= 𝑪(𝑗𝜔) = 𝑪𝑇(𝑗𝜔) ∈

ℝ𝑁×𝑁[𝐹/𝑘𝑚], (iv) conductance 𝑮 ∶= 𝑮(𝑗𝜔) = 𝑮𝑇(𝑗𝜔) ∈ ℝ𝑁×𝑁[𝑆/𝑘𝑚], and (v) transmission line

length ℓ ∈ ℝ [𝑘𝑚]. Moreover, we can define: 𝒁(𝑗𝜔) = 𝒁𝑇(𝑗𝜔) = [𝑹(𝑗𝜔) + 𝑗𝜔𝑳(𝑗𝜔)] ∈

ℂ𝑁×𝑁[𝛺/𝑘𝑚] and 𝒀(𝑗𝜔) = 𝒀𝑇(𝑗𝜔) = [𝑮(𝑗𝜔) + 𝑗𝜔𝑪(𝑗𝜔)] ∈ ℂ𝑁×𝑁[𝑆/𝑘𝑚] so that 𝒁 ∙ 𝒀 = (𝒀 ∙

𝒁)𝑇 ∈ ℂ𝑁×𝑁[𝑘𝑚−2] and 𝒇(𝒁 ∙ 𝒀) = [𝒇(𝒀 ∙ 𝒁)]𝑇 ∈ ℂ𝑁×𝑁[𝑓(𝑘𝑚−2)] where 𝒇(. ) represents a

function of a matrix [7]. Parameters (i-iv) are positive definite matrices and functions of the analog

angular frequency (𝜔 ∈ ℝ [𝑟𝑎𝑑/𝑠𝑒𝑐], 𝜔 = 2𝜋𝐹, 𝐹 ∈ ℝ [𝐻𝑧]). The diagonal elements correspond

to “self” values and the off-diagonal elements correspond to cross-coupling values based on the

conductor type and the tower geometry [13]. Given these parameters and the fundamental circuit

model of a transmission line of Fig. 2.1 we can obtain the telegraph equations [13].

Figure 2.1: N-Phase transmission line circuit model.

16

(𝐾𝑉𝐿): − 𝒗(𝑥, 𝑡) + 𝛥𝑥 ∙ 𝑹 ∙ 𝒊(𝑥, 𝑡) + 𝛥𝑥 ∙ 𝑳 ∙𝜕

𝜕𝑡𝒊(𝑥, 𝑡) + 𝒗(𝑥 + 𝛥𝑥, 𝑡) = 𝟎 (2.1𝑎)

(𝐾𝐶𝐿) : − 𝒊(𝑥, 𝑡) + 𝛥𝑥 ∙ 𝑪 ∙𝜕

𝜕𝑡𝒗(𝑥 + 𝛥𝑥, 𝑡) + 𝛥𝑥 ∙ 𝑮 ∙ 𝒗(𝑥 + 𝛥𝑥, 𝑡) + 𝒊(𝑥 + 𝛥𝑥, 𝑡) = 𝟎 (2.1𝑏)

where 𝒗(𝑥, 𝑡) ∈ ℝ𝑁[𝑉], 𝒊(𝑥, 𝑡) ∈ ℝ𝑁[𝐴] are the voltage and current N-phase vectors respectively.

By rearranging the terms in (2.1) and applying the definition of the derivative

( lim𝛥𝑥→0

𝒇(𝑥+𝛥𝑥,𝑡)−𝒇(𝑥,𝑡)

𝛥𝑥≜

𝜕

𝑑𝑥𝒇(𝑥, 𝑡)) we obtain:

𝜕

𝜕𝑥𝒗(𝑥, 𝑡) = − [𝑹 ∙ 𝒊(𝑥, 𝑡) + 𝑳 ∙

𝜕

𝜕𝑡𝒊(𝑥, 𝑡)] (2.2𝑎)

𝜕

𝜕𝑥𝒊(𝑥, 𝑡) = − [𝑮 ∙ 𝒗(𝑥, 𝑡) + 𝑪 ∙

𝜕

𝜕𝑡𝒗(𝑥, 𝑡)] (2.2𝑏)

Finally, the Laplace transform ( 𝐹(𝑠, 𝑥) ≜ 𝐿{𝑓(𝑡, 𝑥)} ) applied on (2.2) yields the telegraph

equations:

𝜕

𝜕𝑥𝑽 = −𝒁 ∙ 𝑰 (2.3𝑎)

𝜕

𝜕𝑥𝑰 = −𝒀 ∙ 𝑽 (2.3𝑏)

where 𝑽 ∶= 𝑽(𝑥, 𝑠) ∈ ℂ𝑁[𝑉] is the Laplace transform of the N-phase voltage vector, 𝑰 ∶= 𝑰(𝑥, 𝑠) ∈

ℂ𝑁[𝐴] is the Laplace transform of the N-phase current vector, 𝒁 = 𝒁𝑇 ∶= 𝒁(𝑠) = 𝑹 + 𝑠 ∙ 𝑳 ∈

ℂ𝑁×𝑁[𝛺/𝑘𝑚] is the per unit length impedance matrix and 𝒀 = 𝒀𝑇 ∶= 𝒀(𝑠) = 𝑮+ 𝑠 ∙ 𝑪 ∈

ℂ𝑁×𝑁[𝑆/𝑘𝑚] is the per unit length admittance matrix.

The (matrix-vector) wave equations for the voltages and currents can be obtained from (2.3) by

differentiation in terms of “𝑥”:

𝜕2

𝜕𝑥2𝑽 − 𝒁 ∙ 𝒀 ∙ 𝑽 = 𝟎 (2.4𝑎)

𝜕2

𝜕𝑥2𝑰 − 𝒀 ∙ 𝒁 ∙ 𝑰 = 𝟎 (2.4𝑏)

17

The solutions of the (matrix-vector) wave equations are:

𝑽 = 𝒆−√𝒁∙𝒀∙𝑥 ∙ 𝑽1 + 𝒆+√𝒁∙𝒀∙𝑥 ∙ 𝑽2 (2.5𝑎)

𝑰 = 𝒆−√𝒀∙𝒁∙𝑥 ∙ 𝑰1 + 𝒆+√𝒀∙𝒁∙𝑥 ∙ 𝑰2 (2.5𝑏)

Where 𝑽1, 𝑽2, 𝑰1 , 𝑰2 are constants which depend on boundary conditions, 𝒁 ∙ 𝒀 = [𝒀 ∙ 𝒁]𝑇 ∈

ℂ𝑁×𝑁[𝑘𝑚−2] , and {𝒆±√𝒁∙𝒀∙𝑥 , 𝒆±√𝒀∙𝒁∙𝑥} ∈ ℂ𝑁×𝑁[1] are matrix-exponential functions [15, 16].

Given the eigenvalue-eigenvector decomposition of a matrix "𝑨 ∈ ℂ𝑁×𝑁": 𝑨 = 𝑴 ∙ 𝜦 ∙ 𝑴−1 where

"𝑴 ∈ ℂ𝑁×𝑁" is the eigenvector matrix and "𝜦 ∈ ℂ𝑁×𝑁" is the diagonal eigenvalue matrix, a

function of a matrix 𝒇(𝑨) ∈ ℂ𝑁×𝑁 (such as the matrix exponential) is defined as:

𝒇(𝑨) = 𝑴 ∙ 𝒇(𝜦) ∙ 𝑴−1, where 𝒇(𝜦) corresponds to element-wise operations along the diagonal

elements. Furthermore, 𝒇1(𝑨) ∙ 𝒇2(𝑨) = 𝒇2(𝑨) ∙ 𝒇1(𝑨) [16].

By combining the solutions of the wave equations (2.5) and the telegraph equations (2.3) we have:

𝜕

𝜕𝑥[𝒆−√𝒁∙𝒀∙𝑥 ∙ 𝑽1 + 𝒆

+√𝒁∙𝒀∙𝑥 ∙ 𝑽2] = −𝒁 ∙ 𝑰 ⇒

𝑰 = 𝒀0 ∙ [𝒆−√𝒁∙𝒀∙𝑥 ∙ 𝑽1 − 𝒆

+√𝒁∙𝒀∙𝑥 ∙ 𝑽2] (2.6𝑎)

𝜕

𝜕𝑥[𝒆−√𝒀∙𝒁∙𝑥 ∙ 𝑰1 + 𝒆

+√𝒀∙𝒁∙𝑥 ∙ 𝑰2] = −𝒀 ∙ 𝑽 ⇒

𝑽 = 𝒁0 ∙ [𝒆−√𝒀∙𝒁∙𝑥 ∙ 𝑰1 + 𝒆

+√𝒀∙𝒁∙𝑥 ∙ 𝑰2] (2.6𝑏)

where 𝒁0 ∈ ℂ𝑁×𝑁[𝛺] is the characteristic impedance matrix and 𝒀0 ∈ ℂ

𝑁×𝑁[𝑆] is the characteristic

admittance matrix with properties:

𝒀0 ≜ 𝒁−1 ∙ √𝒁 ∙ 𝒀 = 𝒁0

−1 = [√𝒀 ∙ 𝒁]−1∙ 𝒀 = √𝒀 ∙ 𝒁 ∙ 𝒁−1 (2.7𝑎)

𝒁0 ≜ 𝒀−1 ∙ √𝒀 ∙ 𝒁 = 𝒀0

−1 = [√𝒁 ∙ 𝒀]−1∙ 𝒁 = √𝒁 ∙ 𝒀 ∙ 𝒀−1 (2.7𝑏)

18

2.1 Two-Port Representation

A two-port representation of the FDTL can be obtained by setting the following boundary

conditions: sending-end voltage/current 𝑽(𝑥 = 0, 𝑠) = 𝑽𝑘, 𝑰(𝑥 = 0, 𝑠) = 𝑰𝑘 and receiving-end

voltage/current 𝑽(𝑥 = ℓ, 𝑠) = 𝑽𝑚 , 𝑰(𝑥 = ℓ, 𝑠) = −𝑰𝑚 (Fig. 2.2).

Figure 2.2: N-Phase transmission line viewed as a two-port network.

This allows us to solve the boundary value problem of (2.5) in terms of the unknown constants

𝑽1, 𝑽2, 𝑰1, 𝑰2.

𝑽𝑘 = 𝒁0 ∙ 𝑰𝑘 + 𝑨2 ∙ [𝒁0 ∙ 𝑰𝑚 + 𝑽𝑚 ] (2.8𝑎)

𝑽𝑚 = 𝒁0 ∙ 𝑰𝑚 + 𝑨2 ∙ [𝒁0 ∙ 𝑰𝑘 + 𝑽𝑘 ] (2.8𝑏)

𝑰𝑘 = 𝒀0 ∙ 𝑽𝑘 −𝑨1 ∙ [𝒀0 ∙ 𝑽𝑚 + 𝑰𝑚] (2.8𝑐)

𝑰𝑚 = 𝒀0 ∙ 𝑽𝑚 −𝑨1 ∙ [𝒀0 ∙ 𝑽𝑘 + 𝑰𝑘 ] (2.8𝑑)

𝒁0 = 𝒁0𝑇 = 𝒀0

−1 = √𝒁 ∙ 𝒀 ∙ 𝒀−1 (2.8𝑒)

𝒀0 = 𝒀0𝑇 = 𝒁0

−1 = √𝒀 ∙ 𝒁 ∙ 𝒁−1 (2.8𝑓)

𝑨1 = 𝑨2𝑇 = 𝑒−ℓ∙√𝒀∙𝒁 = 𝒀0 ∙ 𝑨2 ∙ 𝒁0 (2.8𝑔)

𝑨2 = 𝑨1𝑇 = 𝑒−ℓ∙√𝒁∙𝒀 = 𝒁0 ∙ 𝑨1 ∙ 𝒀0 (2.8ℎ)

19

where 𝑰𝑘, 𝑰𝑚, 𝑽𝑘, 𝑽𝑚 ∈ ℂ𝑁𝑥1 are vectors and functions of the Laplace variable 𝑠, 𝒁0(𝑠) ∈

ℂ𝑁𝑥𝑁 [𝛺] is the characteristic impedance matrix, 𝒀0(𝑠) ∈ ℂ𝑁𝑥𝑁 [𝑆] is the characteristic admittance

matrix, and 𝑨1(𝑠) = 𝑨2𝑇(𝑠) ∈ ℂ𝑁𝑥𝑁 [1] is the weighting (propagation) function matrix.

The frequency responses can be obtained by setting 𝑠 = 𝑗𝜔 in (2.8).

For one of the simulation examples studied in this work and without loss of generality, we

consider a three-phase (𝑁 = 3) FDTL. In particular, MATLAB’s SimPowerSystems

POWER_LINEPARAM function was used to generate the (slightly non-transposed) FD data

characterized by the default tower configuration (DefaultLineParameters.mat), with 𝐺(𝑗𝜔) = 30 ∙

𝑰 [𝑛𝑆/𝑘𝑚] ∀𝜔 ∈ ℝ where 𝑰 is the NxN identity matrix and ℓ = 161𝑘𝑚.

The magnitude and phase responses of 𝒀0(𝑗𝜔) based on the (default) tower configuration

(Fig. 2.3) are shown with solid-blue in Figs. 2.4, 2.5. Only the lower triangular portion of the matrix

is shown since the upper part can be obtained through the symmetry property 𝒀0(𝑗𝜔) = 𝒀0𝑇(𝑗𝜔).

The magnitude and phase responses of the eigenvalues associated with 𝑨1(𝑗𝜔) = 𝑨2𝑇(𝑗𝜔), that is,

𝜆𝑘{𝑨1(𝑗𝜔)} = 𝜆𝑘{𝑨2(𝑗𝜔)} ; 𝑘 = 1,2, . . , 𝑁 based on this tower configuration are shown with solid-

blue in Fig. 2.6.

Figure 2.3: SimPowerSystems power_lineparam 3-phase (default) tower configuration.

20

Figure 2.4: Frequency Dependent (FD) 3-phase characteristic admittance Y0(jω) magnitude

responses.

Figure 2.5: Frequency Dependent (FD) 3-phase characteristic admittance Y0(jω) phase responses.

21

Figure 2.6: Frequency Dependent (FD) 3-phase weighting function eigenvalues λk{A1(jω)}: (i)

magnitude responses (left), (ii) phase responses (right).

As an illustration, consider two simple examples of a boundary conditions:

Open-ended / open-circuit Line: 𝑽𝑘 = 𝑬𝑠 (source voltage), 𝑰𝑚 = 𝟎 where (2.8) simplify to

𝑽𝑘 = 𝑬𝑠 (2.9𝑎)

𝑽𝑚 = 𝑨2 ∙ [2 ∙ 𝑬𝑠 −𝑨2 ∙ 𝑽𝑚 ] (2.9𝑏)

𝑰𝑘 = 𝒀0 ∙ [𝑬𝑠 −𝑨2 ∙ 𝑽𝑚 ] (2.9𝑐)

𝑰𝑚 = 𝟎 (2.9𝑑)

22

Short-ended / short-circuit Line: 𝑽𝑘 = 𝑬𝑠 (source voltage), 𝑽𝑚 = 𝟎 where (2.8) simplify to

𝑽𝑘 = 𝑬𝑠 (2.10𝑎)

𝑽𝑚 = 𝟎 (2.10𝑏)

𝑰𝑘 = [𝑰 + 𝑨12 ] ∙ 𝒀0 ∙ 𝑬𝑠 + 𝑨1

2 ∙ 𝑰𝑘 (2.10𝑐)

𝑰𝑚 = −𝑨1 ∙ [𝒀0 ∙ 𝑬𝑠 + 𝑰𝑘 ] (2.10𝑑)

Where 𝑰 ∈ ℝ𝑁×𝑁 is the identity matrix.

Special Case: Single-Phase FDTL

For the special case corresponding to a single-phase FDTL (𝑁 = 1) the matrix-vector two-

port relations of (2.8) reduce to the following (scalar) expressions (Fig. 2.7):

Figure 2.7: Single-phase (N=1) transmission line viewed as a two-port network.


𝑍0(𝑠)− [𝑉𝑚(𝑠)

𝑍0(𝑠)+ 𝐼𝑚(𝑠)] 𝐴(𝑠) (2.11𝑎)


𝑍0(𝑠)− [𝑉𝑘(𝑠)

𝑍0(𝑠)+ 𝐼𝑘(𝑠)] 𝐴(𝑠) (2.11𝑏)

𝑍0(𝑠) = √𝑅(𝑠) + 𝑠𝐿(𝑠)

𝐺(𝑠) + 𝑠𝐶(𝑠) (2.11𝑐)

𝛾(𝑠) = √(𝑅(𝑠) + 𝑠𝐿(𝑠))(𝐺(𝑠) + 𝑠𝐶(𝑠)) (2.11𝑑)

23

𝐴(𝑠) = 𝑒−𝛾(𝑠)ℓ (2.11𝑒)

where 𝑍0(𝑠) ∈ ℂ [𝛺] is the characteristic impedance, 𝛾(𝑠) ∈ ℂ [1/𝑘𝑚] is the propagation function

and 𝐴(𝑠) ∈ ℂ [1] is the weighting function.

MATLAB’s SimPowerSystems POWER_LINEPARAM function was used to generate the

FD data based on the default tower configuration with one conductor, with 𝐶(𝑗𝜔) =

10.713 𝑛𝐹/𝑘𝑚, 𝐺(𝑗𝜔) = 30 𝑛𝑆/𝑘𝑚 ∀𝜔 ∈ ℝ and ℓ = 161𝑘𝑚.

The frequency responses of 𝑍0(𝑠) and 𝐴(𝑠) based on this configuration can be obtained by

setting 𝑠 = 𝑗𝜔 in (2.11) and are shown in Figs. 2.8, 2.9 respectively.

Figure 2.8: Frequency Dependent (FD) single-phase characteristic impedance Z0(jω):

(i) magnitude response (top), (ii) phase response (bottom).

24

Figure 2.9: Frequency Dependent (FD) single-phase weighting function A(jω): (i) magnitude

response (top), (ii) phase response (bottom).

As an illustration, consider two simple examples of a boundary conditions:

Open-ended / open-circuit Line: 𝑉𝑘 = 𝐸𝑠 (source voltage), 𝐼𝑚 = 0 where (2.9) simplify to

𝑉𝑘 = 𝐸𝑠 (2.12𝑎)

𝑉𝑚 = 2 ∙𝐴

1 + 𝐴2∙ 𝐸𝑠 (2.12𝑏)

𝐼𝑘 = 1 − 𝐴2

1 + 𝐴2∙𝐸𝑠𝑍0 (2.12𝑐)

𝐼𝑚 = 0 (2.12𝑑)

Short-ended / short-circuit Line: 𝑉𝑘 = 𝐸𝑠 (source voltage), 𝑉𝑚 = 0 where (2.10) simplify to

𝑉𝑘 = 𝐸𝑠 (2.13𝑎)

𝑉𝑚 = 0 (2.13𝑏)

25

𝐼𝑘 = 1 + 𝐴2

1 − 𝐴2∙𝐸𝑠𝑍0 (2.13𝑐)

𝐼𝑚 = −2𝐴

1 − 𝐴2∙𝐸𝑠𝑍0 (2.13𝑑)

2.2 Rational Function Approximations (RFAs)

A constant-parameter (CP) method is a fast but crude approach of modeling a transmission

line: It uses a single frequency value 𝜔0 to represent the entire bandwidth of interest, that is,

𝒀0(𝑗𝜔) = 𝒀0(𝑗𝜔0) ∀𝜔 ∈ ℝ, and 𝑨1(𝑗𝜔) = 𝑨1(𝑗𝜔0) ∀𝜔 ∈ ℝ, where 𝜔0 = 2𝜋𝐹0 and 𝐹0 =

{50, 60} 𝐻𝑧. The CP method tends to severely overestimate the high-frequency transients [3].

A far more accurate technique relies on the use of rational functions ��0(𝑠), ��1(𝑠) that

closely match 𝒀0(𝑠) and 𝑨1(𝑠) respectively. A rational function {𝐺(𝑠) ≜ 𝐾∏ (𝑠−𝜁𝑖)𝑀𝑖=1

∏ (𝑠−𝑝𝑖)𝑁𝑖=1

, 𝑁 < 𝑀} is

uniquely determined by its gain "𝐾", poles {𝑝𝑖 ; 𝑖 = 1,… ,𝑁} and zeros {𝜁𝑖 ; 𝑖 = 1,… ,𝑀} and is

required in order to represent a realizable system [17]. A prominent numerical technique for

obtaining RFAs is based on vector fitting [4, 18]. Vector fitting may be applied on a frequency

dependent matrix (such as the one describing 𝒀0(𝑗𝜔)) in order to obtain a transfer function matrix

characterized by a common set of poles but different zero sets for each element of the matrix [4].

Using MATLAB’s RATIONALFIT function we obtain such high-order RFAs. The solid-blue lines

in Figs. 2.4-2.6 correspond to the measurement data based on (2.8), Fig. 2.3 and the solid-red lines

in Figs. 2.4-2.6 correspond to the result of high-order RFAs by means of vector fitting. The resulting

RFAs ��0(𝑠), ��𝑘{𝑨1(𝑗𝜔)} ; 𝑘 = 1,2, . . , 𝑁 are minimum phase, that is, 𝑅𝑒{𝑝𝑖} < 0, 𝑅𝑒{𝜁𝑖} < 0 ∀𝑖

and of order 16, 8 respectively.

26

It is important to note at this point that ��𝑘{𝑨1(𝑗𝜔)} cannot be directly obtained from

𝜆𝑘{𝑨1(𝑗𝜔)} because the latter contains an unknown delay term 𝜏𝑘 ∈ ℝ [𝑠𝑒𝑐] which is heavily

influenced by the transmission line length ℓ. This delay can be estimated using Bode’s gain-phase

relation [19] (Appendix II). This relation states that the phase-response of 𝜆𝑘{𝑨1(𝑗𝜔)} although

known from the data, can be uniquely obtained from its magnitude response at a single analog

frequency 𝜔0 ∈ ℝ [𝑟𝑎𝑑

𝑠𝑒𝑐] using [19]

𝑀𝑘(𝑗𝜔0) =𝜋

2∫

𝑑𝑊𝑘𝑑𝜈

𝑓(𝜈)𝑑𝜈+∞

−∞

(2.14𝑎)

𝑓(𝜈) =2

𝜋2ln (𝑐𝑜𝑡ℎ (

|𝜈|

2)) (2.14𝑏)

𝑊𝑘(𝜈) = 𝑙𝑛|𝜆𝑘{𝑨1(𝑢)}| (2.14𝑐)

𝑢 = 𝜔0 ∙ 𝑒𝜈 (2.14𝑑)

where 𝑀𝑘(𝑗𝜔0) ∈ ℝ ; 𝑘 = 1,2, … ,𝑁. Once the phases are computed via (2.14) the delays can be

estimated using

��𝑘 =𝑀𝑘(𝑗𝜔0) − 𝑝ℎ𝑎𝑠𝑒{ 𝜆𝑘{𝑨1(𝑗𝜔0)} }

𝜔0 (2.15)

Numerical evaluation of (2.14) and substitution in (2.15) for the 3-phase data given and for several

frequencies 𝜔0 results consistently in ��1 = 540.35𝜇𝑠𝑒𝑐, ��2 = 538.50𝜇𝑠𝑒𝑐, and ��3 = 538.92𝜇𝑠𝑒𝑐

(Fig. 2.6). Once the delays ��𝑘 are computed we can obtain accurate estimates ��𝑘{𝑨1(𝑠)} of

𝜆𝑘{𝑨1(𝑠)} using

��𝑘{𝑨1(𝑠)} = ��𝑘{𝑨1(𝑠)} ∙ 𝑒−𝑠��𝑘 (2.16)

The phase responses shown in Fig. 2.6 (right) are a comparison between ��𝑘{𝑨1(𝑠)} (solid-

red) and 𝜆𝑘{𝑨1(𝑠)} ∙ 𝑒+𝑠��𝑘 (solid-blue) 𝑘 = 1,2, … , 𝑁, that is, the delays ��𝑘 extracted from the

data. We observe that the phase responses match indeed very closely except at the regions where

the magnitude responses are equal to zero in which case the phase has no meaning.

27

Modeling the Weighting Function: The J. Marti Model and the Modal Transform

Although detailed approximations ��𝑘{𝑨1(𝑠)} of 𝜆𝑘{𝑨1(𝑠)} using Bode’s formula and

vector fitting were obtained, the problem of approximating 𝑨1(𝑠) has not yet been addressed. The

modal decomposition method [3] assumes that the eigenvectors associated with 𝑨1(𝑠) are

frequency-independent and are given by the Clarke (matrix) transform 𝑽 (Appendix I). For this

example:

𝑨1(𝑠) = 𝑽 ∙ 𝜦(𝑠) ∙ 𝑸 (2.17𝑎)

𝜦(𝑠) = 𝑑𝑖𝑎𝑔{𝜆𝑘{𝑨1(𝑠)} ; 𝑘 = 1,2,3} (2.17𝑏)

𝑽 = 𝑸−1 (2.17𝑐)

𝑽 = [1 1 11 −2 11 1 −2

] (2.17𝑑)

𝑸 =1

3[1 1 11 −1 01 0 −1

] (2.17𝑒)

This allows us to solve three independent scalar (single-phase) transmission-line problems

in the eigenvalue (modal) domain and construct the three-phase solution using the constant 𝑽,𝑸

matrices [3]. Moreover, since all transmission lines in a large network use the same eigenvector

matrices 𝑽,𝑸 the entire network can be decomposed to three independent networks each one of

them consisting of single-phase transmission-lines. Although this model is attractive due to its

simplicity, it is accurate only for fully transposed (symmetric) lines. The more asymmetric the line

is, the more inaccurate the model becomes, especially in the study of underground cables [7]. An

extension of the modal transform relies on the use of frequency-dependent eigenvectors

𝑽(𝑗𝜔), 𝑸(𝑗𝜔) but fails to extend to a network consisting of multiple transmission lines because

𝑽(𝑗𝜔), 𝑸(𝑗𝜔) differ (in principle) for each transmission-line and thus we cannot transform an entire

network to a modal domain.

28

Modeling the Weighting Function: The Universal Line Modeling (ULM) method

The Universal Line Modeling (ULM) method [7] is based on the idempotent (spectral)

decomposition [8] and allows us to solve a network of transmission lines directly in the (matrix-

vector) phase-domain. The idempotent (spectral) decomposition of 𝑨1(𝑠) in terms of its

eigenvector matrices 𝑽(𝑠), 𝑸(𝑠) is given by:

𝑨1(𝑠) = ∑𝑴𝑘(𝑠) ∙

𝑁

𝑘=1

𝜆𝑘{𝑨1(𝑠)} (2.18𝑎)

𝑴𝑘(𝑠) = 𝒗𝑘(𝑠) ∙ 𝒒𝑘𝑇 (𝑠) (2.18𝑏)

𝑽(𝑠) = [𝒗1(𝑠), 𝒗2(𝑠), … , 𝒗𝑁(𝑠)] ∈ ℂ𝑁𝑥𝑁 (2.18𝑐)

𝑸(𝑠) = [𝒒1(𝑠), 𝒒2(𝑠), … , 𝒒𝑁(𝑠)] ∈ ℂ𝑁𝑥𝑁 (2.18𝑑)

where 𝒗𝑘(𝑠), 𝒒𝑘𝑇 (𝑠) ∈ ℂ𝑁𝑥1, are the eigenvectors of 𝑨1(𝑠) and 𝑴𝑘(𝑠) ∈ ℂ

𝑁𝑥𝑁 are the (rank-1)

idempotent matrices. Unlike the infinite choices of eigenvectors for a given matrix, the idempotent

(spectral) decomposition is unique [8]. The ULM method relies on (frequency-dependent) RFAs

that approximate: (i) the eigenvalues 𝜆𝑘{𝑨1(𝑗𝜔)}, and (ii) the idempotent matrices 𝑴𝑘(𝑗𝜔). For

the 3-phase example studied in this work (𝑁 = 3) and thus 3 such RFAs are required. RFAs for

{𝑴𝑘(𝑗𝜔); 𝑘 = 1,2,3} are obtained the same way as RFAs for 𝒀0(𝑗𝜔). The magnitude and phase

responses of {𝑴𝑘(𝑗𝜔); 𝑘 = 1,2,3} based on the tower configuration of Fig. 2.3 are shown with

solid-blue in Figs. 2.10-2.15.

In particular, the solid-blue lines in Figs. 2.10-2.15 correspond to the measurement data

based on (2.8, 2.18) and the solid-red lines in Figs. 2.10-2.15 correspond to the result of high-order

RFAs by means of vector fitting. The resulting RFAs ��1(𝑠), ��2(𝑠), ��3(𝑠) are minimum phase,

that is, 𝑅𝑒{𝑝𝑖} < 0, 𝑅𝑒{𝜁𝑖} < 0 ∀𝑖 and of order equal to 13, 13, 1 respectively. Finally, the

idempotent matrices corresponding to the Clarke transform (2.17) are shown in Figs. 2.10-2.15

using dashed-black lines. From Figs. 2.10-2.15 we observe that: (i) the line is non-transposed since

29

the data-specific idempotent matrices (solid-blue) differ from the Clarke-based matrices (dashed-

black), and (ii) the high-order RFAs (solid-red) match the data-specific idempotent matrices (solid-

blue) very closely.

Figure 2.10: Frequency Dependent (FD) idempotent matrix M1(jω) magnitude responses

30

Figure 2.11: Frequency Dependent (FD) idempotent matrix M1(jω) phase responses

Figure 2.12: Frequency Dependent (FD) idempotent matrix M2(jω) magnitude responses

31

Figure 2.13: Frequency Dependent (FD) idempotent matrix M2(jω) phase responses

Figure 2.14: Frequency Dependent (FD) idempotent matrix M3(jω) magnitude responses.

32

Figure 2.15: Frequency Dependent (FD) idempotent matrix M3(jω) phase responses.

Finally, an idempotent matrix-based RFA ��1(𝑠) corresponding to 𝑨1(𝑠) is given by:

��1(𝑠) = ∑��𝑘(𝑠) ∙

𝑁

𝑘=1

��𝑘{𝑨1(𝑠)} (2.19𝑎)

��𝑘{𝑨1(𝑠)} = ��𝑘{𝑨1(𝑠)} ∙ 𝑒−𝑠��𝑘 (2.19𝑏)

This allows to replace {𝒀0(𝑠), 𝑨1(𝑠)} with {��0(𝑠), ��1(𝑠)} in (2.8) and express the problem in terms

of RFAs in the time domain by means of the inverse Laplace transform:

𝒗𝑘(𝑡) = ��0(𝑡) ∗ 𝒊𝑘(𝑡) + ��2(𝑡 − ��𝑘) ∗ [��0(𝑡) ∗ 𝒊𝑚(𝑡) + 𝒗𝑚(𝑡) ] (2.20𝑎)

𝒗𝑚(𝑡) = ��0(𝑡) ∗ 𝒊𝑚(𝑡) + ��2(𝑡 − ��𝑘) ∗ [��0(𝑡) ∗ 𝒊𝑘(𝑡) + 𝒗𝑘(𝑡)] (2.20𝑏)

𝒊𝑘(𝑡) = ��0(𝑡) ∗ 𝒗𝑘(𝑡) − ��1(𝑡 − ��𝑘) ∗ [��0(𝑡) ∗ 𝒗𝑚(𝑡) + 𝒊𝑚(𝑡)] (2.20𝑐)

𝒊𝑚(𝑡) = ��0(𝑡) ∗ 𝒗𝑚(𝑡) − ��1(𝑡 − ��𝑘) ∗ [��0(𝑡) ∗ 𝒗𝑘(𝑡) + 𝒊𝑘(𝑡)] (2.20𝑑)

where lower-case denotes the inverse Laplace transform of the corresponding upper-case (matric-

vector) function and “*” denotes linear convolution.

33

Special Case: Single-Phase Transmission Line

For the single-phase (𝑁 = 1) studied in this work the characteristic impedance and

weighting function (2.19) are scalar functions and there is no requirement for a spectral

decomposition to eigenvalues and idempotent matrices. Using MATLAB’s RATIONALFIT

function we obtain such scalar high-order RFAs (Figs. 2.8, 2.9). The solid blue lines in Figs. 2.8,

2.9 correspond to the measurement data based on (2.11) and the dashed red lines in Figs. 2.8, 2.9

correspond to the result of high-order RFAs by means of vector fitting. The poles and zeros of the

RFAs are shown in Table 2.1.

Table 2.1: Vector fitting based RFAs Z0(s) and A(s)

��0(𝑠) ��(𝑠)

Zeros

𝜁𝑖(1)

Poles

𝑝𝑖(1)

Zeros

𝜁𝑖(2)

Poles

𝑝𝑖(2)

-1.23e+6 -1.75e+9 -1.28e+5 + j6.33e+5 -1.04e+5 + j2.02e+5

-1.49e+5 -1.19e+6 -1.28e+5 - j6.33e+5 -1.04e+5 - j2.02e+5

-2.29e+4 -1.43e+5 -2.04e+5 + j1.77e+5 -8.05e+4 + j5.09e+4

-3.76e+3 -2.17e+4 2.04e+5 - j1.77e+5 -8.05e+4 - j5.09e+4

-633 -3.56e+3 -1.47e+4 -2.73e+4

-106 -600 -4.24e+3 -1.15e+4

-18.25 -101 -863 -3.81e+3

-4.74 -16.57 -86.25 -837

-0.3791 -3.55 -85.1

-0.39

From Table 2.1 we observe that: (i) both RFAs are minimum phase, that is, 𝑅𝑒{𝑝𝑖(1,2)

} < 0,

𝑅𝑒{𝜁𝑖(1,2)

} < 0 ∀𝑖, (ii) the orders of ��0(𝑠) and ��(𝑠) are 10, 9 respectively. Finally the RFAs ��0(𝑠)

and ��(𝑠) are given by

��0(𝑠) =𝐵1(𝑠)

𝐴1(𝑠)=∏ (𝑠 − 𝜁𝑖

(1))9𝑖=1

∏ (𝑠 − 𝑝𝑖(1))10

𝑖=1

(2.21𝑎)

��(𝑠) =𝐵2(𝑠)

𝐴2(𝑠)=∏ (𝑠 − 𝜁𝑖

(2))8𝑖=1

∏ (𝑠 − 𝑝𝑖(2))9

𝑖=1

(2.21𝑏)

34

Using Bode’s gain-phase relations (2.14, 2.15) the estimated delay term 𝜏 ∈ ℝ [𝑠𝑒𝑐] associated

with 𝐴(𝑠) is �� = 539.93𝜇𝑠𝑒𝑐 (Fig. 2.9). Once �� is computed we can obtain an accurate estimate

��(𝑠) of 𝐴(𝑠) using

��(𝑠) = ��(𝑠)𝑒−𝑠�� (2.22)

The phase response shown in Fig. 2.9 is a comparison between ��(𝑠) and 𝐴(𝑠)𝑒+𝑠��, that is,

the delay �� extracted from the data. We observe that the phase responses match indeed very closely

except at the region where the magnitude response is equal to zero in which case the phase has no

meaning. Finally, the matrix-vector equations (2.20) reduce to

𝑖𝑘(𝑡) = 𝑣𝑘(𝑡) ∗ ��0(𝑡) − [𝑣𝑚(𝑡) ∗ ��0(𝑡) + 𝑖𝑚(𝑡)] ∗ ��(𝑡 − ��) (2.23𝑎)

𝑖𝑚(𝑡) = 𝑣𝑚(𝑡) ∗ ��0(𝑡) − [𝑣𝑘(𝑡) ∗ ��0(𝑡) + 𝑖𝑘(𝑡)] ∗ ��(𝑡 − ��) (2.23𝑏)

��0(𝑠) =1

��0(𝑠) (2.23𝑐)

��(𝑠) = ��(𝑠) = 𝑒+𝑠�� (2.23𝑑)

where lower-case denotes the inverse Laplace transform of the corresponding upper-case function

and “*” denotes linear convolution

35

2.3 Discretization of the Continuous-Time Model

In this section we shall establish a connection between the concept of recursive convolution

[3, 20] and the well-known bilinear transform [5, 17]. A popular technique for evaluating the

convolution integrals in (2.20, 2.23) using a fixed-step (𝛥𝑡) discrete simulator is based on the

following idea [3, 20]. Consider a delay-transfer function 𝐻(𝑠) with simple (multiplicity=1),

possibly complex, stable poles (𝑅𝑒{𝑝𝑗} < 0), and 𝑁 < 𝑀. Its rational function expansion is given

by

𝐻(𝑠) ≜𝑆(𝑠)

𝐹(𝑠)=∑𝐻𝑗(𝑠)

𝑁

𝑗=1

=∑𝑟𝑗

𝑠 − 𝑝𝑗

𝑁

𝑗=1

𝑒−𝑠𝜏 (2.24)

𝑠𝑗(𝑡) ≜ 𝑓(𝑡) ∗ ℎ𝑗(𝑡) = 𝑟𝑗∫ 𝑓(𝑡 − 𝑢)𝑒𝑝𝑗(𝑢−𝜏)𝑑𝑢

+∞

𝜏

(2.25)

A recursive relation between 𝑠𝑗(𝑡) and 𝑠𝑗(𝑡 − 𝛥𝑡) of the form

𝑠𝑗(𝑡) = 𝑐0𝑠𝑗(𝑡 − 𝛥𝑡) + 𝑐1𝑓(𝑡 − 𝜏) + 𝑐2𝑓(𝑡 − 𝜏 − 𝛥𝑡) (2.26𝑎)

𝑐0 = 𝑒𝑝𝑗𝛥𝑡 (2.26𝑏)

𝑐1 = −1

𝑝𝑗−1 − 𝑒𝑝𝑗𝛥𝑡

𝑝𝑗2𝛥𝑡

(2.26𝑐)

𝑐2 =𝑒𝑝𝑗𝛥𝑡

𝑝𝑗+1− 𝑒𝑝𝑗𝛥𝑡

𝑝𝑗2𝛥𝑡

(2.26𝑑)

can be established by: (i) applying integration by parts, (ii) setting 𝑑𝑓(𝑡−𝑢)

𝑑𝑢≈𝑓(𝑡−𝜏−𝛥𝑡)−𝑓(𝑡−𝜏)

𝛥𝑡 [20],

and (iii) ignoring the scaling factor 𝑟𝑗. Standard digital signal processing (DSP) notation allows us

to rewrite (2.26) as

𝑠𝑗[𝑛] = 𝑐0𝑠𝑗[𝑛 − 1] + 𝑐1𝑓[𝑛 − 𝐿] + 𝑐2𝑓[𝑛 − 1 − 𝐿] (2.27𝑎)

36

𝐿 = ⌊𝜏

𝛥𝑡⌋ (2.27𝑏)

Where ⌊. ⌋ denotes the “floor” operation. Taking the z-transform of (2.27) leads to a

discrete-time transfer function of the form:

𝐻𝑗(𝑧) ≜𝑆𝑗(𝑧)

𝐹(𝑧)=𝑐1 + 𝑐2𝑧

−1

1 − 𝑐0𝑧−1𝑧−𝐿 (2.28)

After ignoring the delay term 𝑧−𝐿 , it can be shown that the mapping 𝐻𝑗(𝑠) ↔ 𝐻𝑗(𝑧) is a

generalized bilinear transform [5] of the form 𝑠 =𝐴+𝐵𝑧−1

𝐶+𝐷𝑧−1 with 𝐴 = 1 + 𝑝𝑗𝑐1, 𝐵 = 𝑝𝑗𝑐2 − 𝑐0, 𝐶 =

𝑐1, 𝐷 = 𝑐2. Furthermore, assuming that the product 𝑝𝑗𝛥𝑡 is small enough so that the approximation

𝑒𝑝𝑗𝛥𝑡 ≈ 1+ 𝑝𝑗𝛥𝑡 is valid, reveals that the mapping 𝐻𝑗(𝑠) ↔𝐻𝑗(𝑧) (2.26-2.28) corresponds to the

well-known forward rectangular bilinear transform 𝑠 =1−𝑧−1

𝛥𝑡∙𝑧−1 [15]. In this work, discretization of

𝐻𝑗(𝑠) is carried out using the “prewarped Tustin” bilinear transform, namely [15]

𝐻𝑗(𝑠)

𝑠=𝜔0,𝑗

𝑡𝑎𝑛(𝜔0,𝑗𝛥𝑡

2)

∙ 1−𝑧−1

1+𝑧−1

→ 𝐻𝑗(𝑧) (2.29)

where 𝜔0,𝑗 is the analog angular frequency (Chapter 4). This allows us to leverage the stability and

frequency representation properties of the classical Tustin transform while maintaining accuracy at

a particular frequency of interest 𝜔0,𝑗 via prewarping [5]. Accuracy around a particular frequency

is a very desirable characteristic when we decompose a broadband simulation to several

narrowband sub simulations (Chapter 3).

Finally, although the Tustin transform guarantees stability in the discrete time domain,

we may still encounter numerical instability due to pole clustering (numerical multiplicity) around

the unit circle. This can be illustrated by means of the following example. Consider 𝐻(𝑠) =1

𝑠−𝑝

and the mapping (2.29) with 𝜔0,𝑗 = 0, so that 𝐻(𝑧) =

𝛥𝑡

2−𝑝𝛥𝑡(1+𝑧−1)

1−(2+𝑝𝛥𝑡

2−𝑝𝛥𝑡)𝑧−1

. The pole of 𝐻(𝑧) is located at

37

𝑧 =2 + 𝑝𝛥𝑡

2 − 𝑝𝛥𝑡 (2.30)

If 𝛥𝑡 ≪ 𝑝 then 𝑧 → 1, resulting in a marginally-stable discrete-time system 𝐻(𝑧). Now let

us consider a system 𝐻(𝑠) with three poles {𝑝1, 𝑝2, 𝑝3} so that 𝑝3 ≫ 𝑝1 and 𝑝3 ≫ 𝑝2. The condition

number 𝐾 ≜𝑚𝑎𝑥|𝑅𝑒{𝑝𝑖}|

𝑚𝑖𝑛|𝑅𝑒{𝑝𝑖}| is large and the step size 𝛥𝑡 should be of the order of

1

|𝑅𝑒{𝑝3}| for accurate

simulation [15, 20]. This results in {𝑝1𝛥𝑡, 𝑝2𝛥𝑡} → 0 so that two poles {𝑧1, 𝑧2} → 1, resulting in

numerical instability [5, 17].

Table 2.2 shows the absolute value of the poles of the single-phase discretized, high-order

transfer functions ��0(𝑧), ��(𝑧) clearly illustrating the numerical instability (shaded entries).

Similarly, for the 3-phase example studied in this work, a few of the absolute value of the poles of

the discretized, high-order transfer functions ��0(𝑧), { ��𝑘{𝑨1(𝑧)} , ��𝑘(𝑧) ; 𝑘 = 1,2,3 } are

unstable (greater than one) illustrating the numerical instability. In Chapter 4 we improve numerical

stability by employing narrowband (subband) processing with low order RFA models.

Table 2.2: Absolute value of the poles of Z0(z) and A(z)

ABS( Poles of

��0(𝑧) )

ABS( Poles of

��(𝑧) )

0.9977 0.9022

1.0062 0.9022

1.0062 1.0058

1.0000 0.9976

1.0000 0.9976

0.9918 0.9807

0.9918 0.9754

0.9784 0.9227

0.8668 0.9227

0.2523

CHAPTER 3

QMF Filter Banks and Phasor Banks

In order to construct low-order RFAs (Chapter 4) that match the original (broadband)

FDTL data very closely, the concept of a Perfect Reconstruction Filter Bank (PR-FB) shall be

employed. In particular we shall study the structure of two PR-FB classes: (i) Logarithmic-tree

quadrature mirror filter banks (QMF) [9], and (ii) Phasor banks [10-12].

3.1 Perfect Reconstruction Tree-Structured QMF Filter Banks

A 2-channel perfect reconstruction filter bank is characterized by: (i) the analysis stage

filters {𝐻0(𝑧), 𝐻1(𝑧)}, (ii) the frequency partition (subband domain), and (iii) the synthesis stage

filters {𝐺0(𝑧), 𝐺1(𝑧)} (Fig. 3.1) [9].

Figure 3.1: Two Channel PR QMF Filter-Bank.

39

Perfect reconstruction (PR) means that

𝑦[𝑛] = 𝑥[𝑛 − 𝑛0] (3 .1)

for some (integer) delay 𝑛0. A convenient, and frequently used, set of choices for 𝐻0(𝑧), 𝐻1(𝑧),

𝐺0(𝑧), 𝐺1(𝑧) with the PR property is provided by the family of FIR para-unitary quadrature mirror

filters (QMF). This family is completely specified by the constraints

1) 𝐻0(𝑧) ∶ ℎ𝑎𝑠 𝑑𝑒𝑔𝑟𝑒𝑒 𝑁 = 𝑜𝑑𝑑 (3.2)

2) 𝐻1(𝑧) = 𝑐 ∙ 𝐻0#(−𝑧) , |𝑐| = 1 (3.3)

3) |𝐻0(𝑒𝑗𝜔)|

2+ |𝐻0(𝑒

−𝑗𝜔)|2= 2 (Power complementary property) (3.4)

For the synthesis Filters:

4) 𝐺0(𝑧) = 𝐻0#(𝑧) (3.5)

5) 𝐺1(𝑧) = 𝐻1#(𝑧) (3.6)

Where "#" denotes conjugate reversal of a polynomial, namely

𝐻0 (𝑧) ≜ ℎ0 (0) + ℎ0 (1)𝑧−1 +⋯+ ℎ0 (𝑁)𝑧

−𝑁 ⇒

𝐻0#(𝑧) ≜ ℎ0

∗(𝑁) + ℎ0∗(𝑁 − 1)𝑧−1 +⋯+ ℎ0

∗(1)𝑧−(𝑁−1) + ℎ0∗(0)𝑧−𝑁 (3.7)

The QMF constraints (3.2-3.6) reduce the design task to finding a (prototype) FIR low-pass filter

𝐻0(𝑧) that satisfies (3.3), since 𝐻0(𝑧), 𝐻1(𝑧), 𝐺0(𝑧), 𝐺1(𝑧) are all expressed in terms of this filter.

Logarithmic frequency structure in the subbands is realized by expanding the analysis

filters into scales as shown in Fig. 3.2. The Discrete Wavelet Transform (DWT) constitutes a

special case within the family of tree-structured filter banks [9]. The equivalent of the

forward/inverse DWT represented as a filter bank is shown in Fig. 3.3 [9]. Our objective is to fit

low order rational function models in each subband so that it closely resembles the high order

frequency response 𝒀0(𝑗𝜔) , { 𝜆𝑘{𝑨1(𝑗𝜔)} ,𝑴𝑘(𝑗𝜔); 𝑘 = 1,2,3 } of Chapter 2.

40

Figure 3.2: Logarithmic frequency tree-structured Analysis Filter Bank.

Figure 3.3: PR Filter Bank viewed as a combination of forward and inverse DWT.

Experimenting with different perfect reconstruction filter banks suggests that choosing a

prototype analysis filter 𝐻0(𝑧) which corresponds to a good lowpass filter (in terms of passband

ripple and stopband attenuation) is a desired characteristic for decoupling the broadband simulation

into narrowband sub-simulations (Chapter 5). All filter banks used in this work are members of the

Quadrature-Mirror Filter (QMF) bank family, so that only the (prototype analysis) filter 𝐻0(𝑧)

needs to be specified. The other filters {𝐻1(𝑧), 𝐺0(𝑧), 𝐺1(𝑧)} are determined by the QMF design

specification (3.2-3.6). In particular, {𝐻0(𝑧), 𝐻1(𝑧), 𝐺0(𝑧), 𝐺1(𝑧)} were constructed using: (a) a

Daubechies-2 (Haar) FB, (b) a Daubechies-6 FB, (c) a Daubechies-12 FB, and (d) using

41

MATLAB’s FIR2CHPR with order 𝑁 = 99 and passband-edge frequency 𝑓𝑝 = 0.45 [9, 17]. The

frequency responses of the analysis filters (AF) in each band are shown in Figs. 3.4-3.7.

Experimental results suggest that the order of 𝐻0(𝑧) can be reduced down to 𝑁 = 6 − 12,

while still maintaining good frequency separation. Moreover, members of the Daubechies family

(i.e., D-12 FB) constitute good choices for achieving frequency separation in the subbands due to

the MAXFLAT property, that is, a flat passband and monotonic rejection in the stopband [9, 17].

Figure 3.4: Ten-channel logarithmic frequency Daubechies-2 (Haar/D2) analysis filter bank

magnitude responses.

42

Figure 3.5: Ten-channel logarithmic frequency Daubechies-6 (D6) analysis filter bank magnitude

responses.

Figure 3.6: Ten-channel logarithmic frequency Daubechies-12 (D12) analysis filter bank magnitude responses.

43

Figure 3.7: Ten-channel logarithmic frequency FIR2CHPR-based (N=99) analysis filter bank magnitude responses.

3.2 Phasor Banks

The classical dynamic phasor representation is based on the so-called Short-Time Fourier

Trasform (STFT): we associate a Fourier series representation with the (continuous-time) signal

segment {𝑥(𝑠); 𝑡 − 𝑇 < 𝑠 ≤ 𝑡} where we consider "t" as a parameter, viz.

𝑥(𝑠) = ∑ 𝑋𝑘(𝑡)𝑒𝑗𝑘𝜔𝑠

∞

𝑘=−∞

, 𝑡 − 𝑇 < 𝑠 ≤ 𝑡 (3.8𝑎)

𝑋𝑘(𝑡) =1

𝑇∫ 𝑥(𝑠)𝑒−𝑗𝑘𝜔𝑑𝑠𝑡

𝑡−𝑇

(3.8𝑏)

44

Here 𝜔 =2𝜋

𝑇, where 𝑇 is the duration of a cycle (or the period in a steady state).

Phasor Banks provide a generalization of the well-known discrete Gabor transform [9, 21,

22], which can be (equivalently) viewed as a perfect reconstruction M-channel uniform DFT filter

bank (Fig. 3.8). We define a Phasor Bank as a filter bank that satisfies the following conditions:

Figure 3.8: Filter-bank interpretation of dynamic phasors

(i) Causal & stable:

Phasor Banks are targeted for applications that require dynamic processing, namely

processing carried out online in real time, using digital signal processing hardware/software, one

sample at a time. As a consequence, all processing must be causal and stable.

(ii) No Decimation:

In contrast to filter banks designed for coding applications, which employ down-sampling

to achieve the smallest possible data rate subject to the constraint of perfect reconstruction, we opt

to avoid down-sampling altogether, so that 𝐷 = 1 in Fig. 3.8. The resulting single-rate processing

scheme allows for a closer relation between our discrete-time generalized Dynamic Phasors and

the continuous-time systems from which our signals of interest are acquired, as well as greater

flexibility in the selection of the Phasor Bank synthesis stage [10]. This type of filter-bank is known

in the literature as fully oversampled [23].

45

(iii) Uniform DFT-like structure:

Both analysis and synthesis filters have a uniform DFT-like structure, that is, they can be

expressed in terms of a prototype analysis filter 𝐻(𝑧) (PAF) and a prototype synthesis filter 𝐺(𝑧)

(PSF) respectively, viz.

𝐻𝑘(𝑧) = 𝐻(𝑒−𝑗𝑘𝜔0𝑧) (3.9𝑎)

𝐺𝑘(𝑧) = 𝑒−𝑗𝑘𝜔0𝛥𝐺(𝑒−𝑗𝑘𝜔0𝑧) (3.9𝑏)

where 0 ≤ 𝑘 ≤ 𝑀 − 1, 𝑀 represents the number of channels, 𝜔0 ≜2𝜋

𝑀, and Δ is the controllable

reconstruction delay (Fig. 3.8). When Δ = 0 the expressions (3.9) coincide with the definition of a

uniform DFT filter bank.

(iv) Unconstrained Analysis Filter:

The prototype analysis filter 𝐻(𝑧) of the Phasor Bank is unconstrained and thus can be

designed to meet the specifications of a particular application of interest. For example, when used

for dynamic frequency analysis in electric power systems, the analysis stage should be highly

frequency selective. This reduces the leakage between frequency bands and allows unambiguous

association of each Dynamic Phasor 𝒳𝑘(∙) with a single harmonic (Fig. 3.8).

(v) Controllable 𝛥 −Delay Perfect Reconstruction (PR):

Perfect reconstruction indicates that 𝑦[𝑛] = 𝑥[𝑛 − 𝛥], where 𝛥 is the reconstruction delay

(Fig. 3.8). Control applications usually require zero reconstruction delay, which cannot be

achieved, for instance, with paraunitary QMF filter banks. The reconstruction delay 𝛥 of our Phasor

Banks is a design parameter, with any desired (integer) value in the range [0,𝑀 − 1], and is entirely

independent of the analysis stage of the phasor bank. The perfect reconstruction requirement

𝑦[𝑛] = 𝑥[𝑛 − 𝛥] for a phasor-bank with given causal 𝐻(𝑧), 𝐺(𝑧) reduces to [10].

46

𝑓[𝑛] = {

1

𝑀 𝑛 = 𝛥

0 𝑛 = 𝛥 + 𝑙𝑀, . 𝑙 ≥ 1

(3.10)

where 𝐹(𝑧) ≜ 𝐻(𝑧)𝐺(𝑧), and 𝑓[𝑛] represents the impulse response of 𝐹(𝑧) [10]. Thus, 𝐹(𝑧) is a

delayed version of a Nyquist filter [9]. Since 𝐻(𝑧) is unconstrained, 𝐺(𝑧) is responsible for

satisfying (3.10). Because we do not down-sample the phasor-domain coefficients, every 𝐻(𝑧) has

(infinitely) many matching choices of 𝐺(𝑧) that satisfy the PR condition (3.10). This added freedom

of choice allows us to customize the selection of 𝐺(𝑧) with respect to other metrics of performance,

such as reconstruction numerical robustness.

Measures of Reconstruction Numerical Robustness

Numerical robustness (a.k.a "stability of reconstruction") refers to the effects of

perturbations/errors in 𝒳𝑘[𝑛] on the reconstructed signal 𝑦[𝑛] [24], (Fig. 3.9).

Figure 3.9: Simplified perturbation model for a PR filter-bank.

Using vector-matrix notation, let 𝑨, 𝑺 represent the input-output signal space mappings induced by

the analysis (AFB) and synthesis (SFB) filters, respectively (Fig. 3.9). Thus, we have

𝜂𝑟𝑜𝑏(𝑨, 𝑺) = ‖𝜟𝐱‖ ‖𝐱‖⁄

‖𝜟𝒛‖ ‖𝒛‖ ⁄≤ ‖𝑨‖ ∙ ‖𝑺‖ (3.11)

47

which is true for every norm, where 𝒛 represents the collection of all 𝒳𝑘 's (Fig. 3.9). Including the

requirement of perfect reconstruction (i.e., ∙ 𝑨 = 𝑰 ) we conclude that ‖𝑨‖ ∙ ‖𝑺‖ ≥ 1, so that

numerical robustness of filter banks is synonymous with reducing the product ‖𝑨‖ ∙ ‖𝑺‖. The

minimum value of this product ( ‖𝑨‖ ∙ ‖𝑺‖ = 1 ) is achieved by using: (i) the canonical left inverse

( 𝑺 = 𝑨+ ), and (ii) making 𝑨 unitary [9]. This optimal solution is not causal, and attempting to

make it causal results in significant reconstruction delay. We shall focus here on the Euclidean

vector norm, so that

𝜂𝑟𝑜𝑏(𝑨, 𝑺) =

‖𝛥𝐱‖2 ‖𝐱‖2

√ ∑ ‖𝛥𝒳𝑘‖2

2 𝑀−1𝑘=0

∑ ‖𝒳𝑘‖22𝑀−1

𝑘=0

(3.12)

Notice that the same robustness index values are obtained when we replace the Euclidean

norm in (3.12) by the "RMS - norm" ‖𝐱‖𝑅𝑀𝑆 ≜ lim𝑁→∞

√ 1

𝑁∑ |𝑥[𝑛]|2𝑁−1𝑛=0 , which is more appropriate

for persistent (finite power) signals. This allows us to model the perturbations 𝛥𝒳𝑘(∙) as

independent white noise processes, all with the same variance, and the input signal 𝑥[𝑛] as a

stationary ergodic process.

Using these assumptions along with causality, stability and the property (3.9), the

robustness index (for 𝑁 → ∞) is minimized by selecting 𝐺(𝑧) as the solution of the linearly

constrained quadratic optimization problem:

min𝒈 1

2‖𝒈‖2

2 (3.13𝑎)

subject to

𝓗𝒈 = (1

𝑀)𝒆0 (3.13𝑏)

48

where 𝓗 is a lower-triangular (block-Toeplitz) matrix, viz.,

𝓗 = [

ℎ𝛥 ⋯ ℎ0 0 ⋯ ⋯ ⋯ ⋯ℎ𝛥+𝑀 ⋯ ⋯ ⋯ ℎ0 0 ⋯ ⋯ℎ𝛥+2𝑀 ⋯ ⋯ ⋯ ℎ𝑀 ⋯ ℎ0 0⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

] (3.14𝑎)

𝒆0 ≜ [ 1 0…0 ]𝑇 (3.14𝑏)

The optimally robust solution of (3.13) is given by

[𝑔[0]

𝑔[1]⋮

]

𝑜𝑝𝑡

= (1

𝑀)𝓗†𝒆0 (3.15)

where 𝓗† denotes the generalized (Moore-Penrose) inverse of this infinite matrix. The resulting

optimal impulse response 𝒈𝑜𝑝𝑡 has infinite length, since 𝓗† and 𝒆0 have infinite dimensions. The

optimal IIR solution can be approximated, to any desired level of accuracy, by truncating 𝓗, 𝒆0 to

finite size, which results in an FIR approximation for 𝒈𝑜𝑝𝑡 .

Optimally Robust FIR Filter

The optimally robust solution (3.15) often turns out to be FIR. For instance, when 𝐻(𝑧) is

FIR with 𝑑𝑒𝑔{𝐻(𝑧)} ≤ 𝑀 −1, the expression (3.15) reduces to

𝐺𝑜𝑝𝑡(𝑧) =1

𝑀∑ |ℎ[𝑙]|2𝛥𝑙=0

∑ℎ∗[𝛥 − 𝑙]𝑧−𝑙𝛥

𝑙=0

(3.16)

where the asterisk (*) denotes complex conjugation. Thus, the constraint 𝑑𝑒𝑔{𝐻(𝑧)} ≤ 𝑀 − 1

induces a similar constraint for the (optimally-robust) 𝐺(𝑧). The family of phasor banks with FIR

𝐻(𝑧) and 𝐺(𝑧) of degree 𝑀 − 1 or less is known as Windowed-FIR phasor banks [10]. Since the

frequency selectivity of such an 𝐻(𝑧) is quite limited (Fig. 3.10), we turn our attention in the

following section to phasor banks that employ an IIR prototype analysis filter 𝐻(𝑧).

49

Figure 3.10: FIR vs IIR magnitude responses of the prototype analysis filter H(z).

Optimally Robust IIR Filter

In view of the need to truncate the expression (3.15) to finite dimensions, we consider in

this section optimized choices of the prototype synthesis filter 𝐺(𝑧) that are FIR of a prescribed

length. In particular, when the prototype analysis filter 𝐻(𝑧) satisfies a mild regularity condition,

the optimal FIR 𝐺(𝑧) can be expressed, without approximation, in terms of finite dimensional

matrices.

Theorem 3.1: (Optimal FIR 𝐺(𝑧))

Consider a given IIR prototype analysis filter 𝐻(𝑧) = 𝑏(𝑧) 𝑎(𝑧)⁄ with 𝑑𝑒𝑔{𝑏(𝑧)} ≤ 𝑑𝑒𝑔{𝑎(𝑧)} ≤

𝑀 − 1 , so that 𝑎(𝑧) ≜ ∑ 𝑎[𝑖]𝑧−𝑖𝑀−1𝑖=0 and 𝑏(𝑧) ≜ ∑ 𝑏[𝑖]𝑧−𝑖𝑀−1

𝑖=0 . Let {𝑝𝑖 ; 1 ≤ 𝑖 ≤ 𝑀 − 1} denote

the poles of this filter (i.e., the roots of 𝑎(𝑧)), and assume that

50

𝑝𝑖𝑀 ≠ 𝑝𝑗

𝑀 for all 𝑖 ≠ 𝑗 (3.17)

Then, the optimally-robust FIR prototype synthesis filter 𝐺𝑜𝑝𝑡(𝑧) of a prescribed length 𝐿𝑔 is given

by 𝐺𝑜𝑝𝑡(𝑧) = 𝛾𝑜𝑝𝑡(𝑧)𝑎(𝑧) where 𝛾𝑜𝑝𝑡(𝑧) is an FIR filter of length 𝐿𝛾 ≜ 𝐿𝑔 − 𝐿𝑎 + 1 whose

impulse response is

𝜞𝑜𝑝𝑡 = 1

𝑀(𝑫𝑇𝑫)−

12 (𝑩(𝑫𝑇𝑫)−

𝑇2)†

𝒆0 (3.18𝑎)

or, equivalently,

𝜞𝑜𝑝𝑡 = 1

𝑀(𝑫𝑇𝑫)−1𝑩𝑇[ 𝑩(𝑫𝑇𝑫)−1𝑩𝑇 ]−1𝒆0 (3.18𝑏)

with 𝒆0 ≜ [1 0…0]𝑇 , and

𝑫 ≜

[ 1 0 ⋯ ⋯ ⋯𝑎1 1 0 ⋯ ⋯⋮ 𝑎1 1 0 ⋯

𝑎𝐿𝑎−1 ⋮ ⋱ ⋯ ⋯

0 𝑎𝐿𝑎−1 ⋱ ⋯ ⋯

⋮ 0 ⋱ ⋯ ⋯⋮ ⋱ ⋱ ⋱ ⋱]

(3.18𝑐)

𝑩 ≜ [𝑏𝛥 ⋯ 𝑏0 ⋯ ⋯ ⋯ ⋯ ⋯0 ⋯ ⋯ 𝑏𝑀−1 ⋯ 𝑏0 0 ⋯⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

] (3.18𝑑)

Both 𝑩 and 𝑫 are finite dimensional, viz., [𝑩] = 𝐾 × 𝐿𝛾, [𝑫] = 𝐿𝑔 × 𝐿𝛾 , where 𝐾 ≜ 1 +

floor (𝐿𝛾+𝐿𝑏−𝛥−2

𝑀). We use the notation 𝐿𝑔 to denote the length of the impulse response of the FIR

filter 𝐺(𝑧), i.e., 𝐿𝑔 ≜ 𝑑𝑒𝑔{ 𝐺(𝑧) } + 1 and similarly for 𝐿𝛾 , 𝐿𝑎 , 𝐿𝑏 . Also, (𝑫𝑇𝑫)1/2 is the lower

triangular Cholesky factor of the symmetric positive-definite matrix 𝑫𝑇𝑫 ∎

Theorem 3.1 guarantees that 𝐹(𝑧) = 𝐻(𝑧)𝐺(𝑧) is FIR, so that the perfect reconstruction

condition (3.10) gives rise to a finite set of linear equations whose optimal solution is given by

(3.18). Typically, 𝐻(𝑧) is the product of a classic filter technique (such as Butterworth, Chebyshev,

51

etc.) in which condition (3.17) is rarely violated (Fig. 3.10). The following theorem shows that the

PR condition can sometimes be satisfied with a specially structured IIR 𝐹(𝑧).

Theorem 3.2: (Optimal IIR 𝐺(𝑧))

Consider a given IIR prototype analysis filter 𝐻(𝑧) =𝑏(𝑧)

𝑎(𝑧) where the poles of 𝑎(𝑧) satisfy (3.17),

and a prototype synthesis filter of the form 𝐺(𝑧) =𝑐(𝑧)

𝑝0(𝑧). A necessary condition for 𝐹(𝑧) ≜

𝐻(𝑧)𝐺(𝑧) to satisfy the PR condition (3.10) is that 𝑐(𝑧) = 𝛾(𝑧)𝑎(𝑧) for some polynomial 𝛾(𝑧),

and that all roots of 𝑝0(𝑧) violate (3.17), namely ∀𝑝𝑖 ∃𝑝𝑗, 𝑗 ≠ 𝑖, such that 𝑝𝑖𝑀 = 𝑝𝑗

𝑀. With this

condition satisfied, there are multiple choices of 𝛾(𝑧) that result in Perfect Reconstruction (PR).

∎

One simple choice of 𝑝0(𝑧) consists of a collection of complex pole pairs

{ (𝑝𝑗, 𝑝𝑗∗) ; 0 ≤ 𝑗 ≤ 𝐿 − 1 } such that 𝑝𝑗 = 𝑟𝑗𝑒

𝑗𝜋

𝑀. Thus 𝑝0(𝑧) ≜ ∏ {1 − 2𝑟𝑗 𝑐𝑜𝑠 (𝜋

𝑀)𝑧−1 +𝐿−1

𝑗=0

𝑟𝑗2𝑧−2}. The resulting optimization problem is now non-linear since we are optimizing for both

𝛾(𝑧) and the special pole magnitudes 𝑟𝑗 . The optimally-robust IIR prototype synthesis filter with

𝛾(𝑧) of a given length 𝐿𝛾 and 𝑝0(𝑧) as in Theorem-3.2, with prescribed pole magnitudes 𝒓 ≜

{𝑟0, … , 𝑟𝐿−1} is given by 𝐺𝑜𝑝𝑡(𝑧) =𝛾𝑜𝑝𝑡(𝑧)𝑎(𝑧)

𝑝0(𝑧) so that 𝐹(𝑧) ≜ 𝐻(𝑧)𝐺(𝑧) =

𝑏(𝑧)𝛾𝑜𝑝𝑡(𝑧)

𝑝0(𝑧), as required

by Theorem-3.2. The optimal choice of 𝛾𝑜𝑝𝑡(𝑧) coefficients depends on 𝒓 and is given by

𝜞𝑜𝑝𝑡(𝒓) =1

𝑀(𝑫1

𝑇𝑫1)−12 (𝑩1(𝑫1

𝑇𝑫1)−𝑇2)†

𝒆0 (3.19)

where 𝑫1 ∶= 𝑫1(𝒓) = 𝑷0−1𝑫, 𝑩1 ∶= 𝑩1(𝒓) = 𝑩𝑷0

−1 and 𝑷0 ∶= 𝑷0(𝒓) is the lower-triangular

Toeplitz matrix associated with 𝑝0(𝑧). Also, (𝑫1𝑇𝑫1)

1/2 is the lower triangular Cholesky factor of

the symmetric positive-definite matrix 𝑫1𝑇𝑫1. Since 𝑷0 is a square matrix of infinite size, this

approach requires truncation as opposed to the method based on (3.18).

52

Special Cases

(i) The optimized Windowed-FIR Phasor Bank [10] is a special case of Theorem-3.1. It is obtained

by setting 𝑎(𝑧) = 1 so that 𝐺𝑜𝑝𝑡(𝑧) = 𝛾𝑜𝑝𝑡(𝑧) and 𝑫 = 𝑰 is an identity matrix. Since the rows of

𝑩 are mutually orthogonal, 𝑩𝑩𝑇 is a diagonal matrix.

(ii) The optimized Constrained FIR (CFIR) solution (3.18) can be obtained from the Constrained

IIR (CIIR) solution (3.19) by setting 𝑝0(𝑧) = 1 and 𝑃0 = 𝐼, that is, there are no special poles of

𝐺(𝑧) that violate (3.17).

CHAPTER 4

Subband Vector Fitting

In this chapter we opt to use the tools presented in Chapter 3 in order to decompose the

single high-order RFA broadband simulation problem into several subbands and apply low-order

RFAs in each one of them. In particular, we shall study three examples: (i) Generic transfer function

(toy-problem) in order to establish proof of principle, (ii) single-phase FDTL based on the data of

Chapter 2, and (iii) 3-phase FDTL based on the data of Chapter 2. For all examples we shall use

the same logarithmic-tree QMF perfect reconstruction filter-bank characterized by 10-channels

(𝑀 = 10). The frequency ranges [𝑓1(𝑗), 𝑓2(𝑗)] ∈ ℝ [𝐻𝑧] of each subband are described by (4.1)

(Table 4.1)

[𝑓1(𝑗), 𝑓2(𝑗)] ; 𝑓1

(𝑗)= {

2𝑗−1

𝛥𝑡 ∙ 2𝑀 , 𝑗 > 0

0 , 𝑗 = 0 , 𝑓2

(𝑗)= 𝑓2

(𝑗+1) (4.1)

Table 4.1: Subband frequency range decomposition.

Band

𝑗

𝑓1(𝑗) [𝐻𝑧] 𝑓0

(𝑗) [𝐻𝑧] 𝑓2

(𝑗) [𝐻𝑧]

0 0 488.28 976.56

1 976.56 1464.84 1953.16

2 1953.16 2929.69 3906.25

3 3906.25 5859.38 7812.50

4 7812.50 11718.80 15625.00

5 15625.00 23437.50 31250.00

6 31250.00 46875.00 62500.00

7 62500.00 93750.00 125000.00

8 125000.00 187500.00 250000.00

9 250000.00 375000.00 500000.00

54

Moreover, discretization 𝐻𝑗(𝑠) → 𝐻𝑗(𝑧) is carried out for each band by means of the

prewarped Tustin bilinear transform (Chapter 2) with 𝜔0,𝑗 = 𝜋(𝑓1(𝑗)+ 𝑓2

(𝑗)) ∀ 𝑗 > 0 and

𝜔0,𝑗 = 0 for 𝑗 = 0 in order to match the DC frequency. The simulation step was chosen to be

𝛥𝑡 = 1𝜇𝑠 as outlined in Chapter. 2.

4.1 Case #1: Arbitrary Transfer Function (Toy Problem)

For the purposes of our simplistic simulation example and without loss of generality let us

consider a lowpass, broadband continuous-time (CT) 𝑍(𝑠) with bandwidth up to 500kHz. The solid

blue lines in Fig. 4.1 correspond to frequency dependent data and the solid red lines in Fig. 4.1

correspond to the result of a 15th-order RFA ��(𝑠) by means of vector fitting. The poles and zeros

of ��(𝑠) are shown in Table 4.2.

Figure 4.1: Transfer function data: (i) magnitude response (top), (ii) phase response (bottom).

55

Table 4.2: Vector fitting based high-order RFA Z(s).

��(𝑠) : ⍺ = 1

Poles

𝑝𝑖

Zeros

𝜁𝑖

-88188.65 -95985.9154

-87222.21 -92092.7167

-72457.11 -85505.6159

-66808.09 -75656.8273

-65817.73 -72372.4254

-64029.30 -68686.0463

-47584.31 -48051.4785

-47469.10 -47983.1591

-46480.94 -39780.6546

-36650.95 -37503.705

-32483.34 -32836.0047

-28677.03 -16752.5817

-7272.89 -11264.5052

-3452.15 -7502.9093

-2777.26 -7302.6899

From Table 4.2 we observe that: (i) the RFA is minimum phase, that is, 𝑅𝑒{𝑝𝑖} < 0, 𝑅𝑒{𝜁𝑖} < 0

∀𝑖, and (ii) the order of ��(𝑠) is 15. Finally the RFA ��(𝑠) is given by

��(𝑠) =𝐵(𝑠)

𝐴(𝑠)= ⍺

∏ (𝑠 − 𝜁𝑖)15𝑖=1

∏ (𝑠 − 𝑝𝑖)15𝑖=1

(4.2)

Table 4.3 shows the absolute value of the poles of the discretized, high-order transfer

function ��(𝑧), clearly illustrating the numerical instability (yellow-shaded entries).

56

Table 4.3: Absolute value of the poles of Z(z).

ABS( Poles of ��(𝑧) )

1.1294

1.1294

1.0975

1.0975

1.0414

1.0414

0.9733

0.9733

0.9060

0.9060

0.8505

0.8505

0.8141

0.8141

0.8014

The resulting second-order transfer functions { ��𝑗(𝑧) ; 𝑗 = 0,1, … ,9 } based on vector

fitting and the ten-channel (𝑀 = 10) PR-FB are all stable. Figs. 4.2-4.12 show plots of the high-

order ��(𝑠) vs. the low-order discrete-time subband approximations ��𝑗(𝑧) = 𝐾𝑗𝐵𝑗(𝑧)

𝐴𝑗(𝑧). The

bandwidth of each subband is defined by the x-coordinates corresponding to two consecutive solid

black dots in Fig. 4.2. We observe that construction of the subband approximations ��𝑗(𝑧) depends

on the number of channels (𝑀 = 10), 𝛥𝑡 and 𝑍(𝑠) but not the particular choice of a PR-FB

(Figs. 3.4-3.7). The effect of the latter in terms of accuracy in the transient simulation is studied in

more detail in Chapter. 5. Figs. 4.3-4.12 correspond to zoomed-in versions of Fig. 4.2 and are

intended to show the vector fitting accuracy of the low-order subband models within the bandwidth

determined by the x-coordinates enclosed by the two solid black dots.

57

Figure 4.2: High order Z(s) (blue) vs. low order Zj(z) subband approximations (colored segments).

Figure 4.3: High order Z(s) (blue) vs. low order Z0(z) subband-0 approximation (red). Zoomed-in

responses.

58

Figure 4.4: High order Z(s) (blue) vs. low order Z1(z) subband-1 approximation (red). Zoomed-in responses.


responses.

59


responses.


responses.

60



responses.

61


Figure 4.11: High order Z(s) (blue) vs. low order Z8(z) subband-8 approximation (red). Zoomed-

in responses.

62


Finally, the transfer functions { ��𝑗(𝑧) ; 𝑗 = 0,1, . . ,9 } are given by

��0(𝑧) = 0.0187271 + 0.010618𝑧−1 − 0.98938𝑧−2

1 − 1.975𝑧−1 + 0.97509𝑧−2 (4.3𝑎)

��1(𝑧) = 0.092861 + 0.0097279𝑧−1 − 0.99027𝑧−2

1 − 1.8604𝑧−1 + 0.86056𝑧−2 (4.3𝑏)

��2(𝑧) = 0.602331 + 0.0092821𝑧−1 − 0.99072𝑧−2

1 − 1.0702𝑧−1 + 0.070684𝑧−2 (4.3𝑐)

��3(𝑧) = 0.216811 + 0.0079067𝑧−1 − 0.99209𝑧−2

1 − 1.6827𝑧−1 + 0.68302𝑧−2 (4.3𝑑)

��4(𝑧) = 0.381511 + 0.054585𝑧−1 − 0.94542𝑧−2

1 − 1.2989𝑧−1 + 0.32249𝑧−2 (4.3𝑒)

��5(𝑧) = 0.9081 + 0.087394𝑧−1 − 0.91261𝑧−2

1 − 0.18089𝑧−1−0.72794𝑧−2 (4.3𝑓)

��6(𝑧) = 1.01291 + 0.092274𝑧−1 − 0.90773𝑧−2

1 + 0.038487𝑧−1 − 0.93188𝑧−2 (4.3𝑔)

��7(𝑧) = 1.02051 + 0.095003𝑧−1 − 0.905𝑧−2

1 + 0.054273𝑧−1 − 0.94398𝑧−2 (4.3ℎ)

63

��8(𝑧) = 1.0231 + 0.10431𝑧−1 − 0.89569𝑧−2

1 + 0.060605𝑧−1 − 0.9393𝑧−2 (4.3𝑖)

��9(𝑧) = 1.04061 + 0.1809𝑧−1 − 0.8191𝑧−2

1 + 0.10698𝑧−1 − 0.89301𝑧−2 (4.3𝑗)

4.2 Case #2: Single-Phase FDTL

The resulting second-order transfer functions { ��0,𝑗(𝑧), ��𝑗(𝑧) ; 𝑗 = 0,1, … ,9 } based on

vector fitting and the ten-channel (𝑀 = 10) PR-FB are all stable. Figs. 4.13-4.14 show plots of the

high-order {��0(𝑠), ��(𝑠)} vs. the low-order discrete-time approximations {��0,𝑗(𝑧), ��𝑗(𝑧)}. The


black dots in Figs. 4.13, 4.14.

Figure 4.13: High order Z0(s) (blue) vs. low order Z0,j(z) subband approximations (dashed colored

segments).

64

Figure 4.14: High order A(s) (blue) vs. low order Aj(z) subband approximations (dashed colored segments).

4.3 Case #3: Three-Phase FDTL

The resulting second-order transfer functions ��0,𝑗(𝑧), { ��𝑘,𝑗{𝑨1(𝑧)} , ��𝑘,𝑗(𝑧) ; 𝑘 =

1, 2, 3 ; 𝑗 = 0,1, … ,9 } based on vector fitting and the ten-channel (𝑀 = 10) PR-FB are all stable.

Figs. 4.15-4.23 show plots of the high-order {��0(𝑠), ��𝑘{𝑨1(𝑠)}, ��𝑘(𝑠)} (solid-blue) vs. the low-

order discrete-time approximations {��0,𝑗(𝑧), ��𝑘,𝑗{𝑨1(𝑧)}, ��𝑘,𝑗(𝑧)} (dashed-colored). The


black dots in Figs. 4.15-4.23.

65

Figure 4.15: Subband FD admittance matrix Y0,j(jω) magnitude responses.

Figure 4.16: Subband FD admittance matrix Y0,j(jω) phase responses.

66

Figure 4.17: Subband FD weighting function eigenvalues λk,j{A1(jω)} (i) magnitude responses

(left), (ii) phase responses (right).

Figure 4.18: Subband FD idempotent matrix M1,j(jω) magnitude responses.

67

Figure 4.19: Subband FD idempotent matrix M1,j(jω) phase responses.


68

Figure 4.21: Subband FD idempotent matrix M2,j(jω) 𝑝hase responses.


69

Figure 4.23: Subband FD idempotent matrix M3,j(jω) phase responses.

CHAPTER 5

Transient Simulation and Accuracy of Combined Model

The result of low order RFAs in each subband compared to the original high-order RFA

was studied in detail in Chapter 4 using three examples. Although we were able to obtain a very

close match between the RFAs, we did not consider the effect of choosing a particular analysis

filter bank (Figs. 3.4-3.7). In fact, we implicitly assumed that the prototype analysis filter 𝐻0(𝑧) is

an ideal lowpass filter. In this chapter we study how a particular choice of PR-FB affects the

accuracy of transient simulation by computing the frequency response of the “combined” transfer

function ��(𝑧), namely, the frequency response of the low-order RFAs ��𝑗(𝑧) including the PR-FB

characterized by 𝑀 = 10 channels and a particular choice of {𝐻0(𝑧), 𝐻1(𝑧), 𝐺0(𝑧), 𝐺1(𝑧)}

(Fig. 3.3). In particular, we shall study the effect of a particular choice of PR-FB for the same three

examples as in Chapter 4, namely: (i) Generic transfer function (toy-problem) in order to establish

proof of principle, (ii) single-phase FDTL based on the data of Chapter. 2, and (iii) 3-phase FDTL

based on the data of Chapter. 2.

5.1 Case #1: Arbitrary Transfer Function (Toy Problem)

For this example we compute the “combined” transfer function ��(𝑧), namely, the

frequency response of the low-order RFAs ��𝑗(𝑧) including the PR-FB characterized by 𝑀 = 10

channels and a particular choice of {𝐻0(𝑧), 𝐻1(𝑧), 𝐺0(𝑧), 𝐺1(𝑧)} (Fig. 3.3) for the 4-different

choices of prototype-analysis filters 𝐻0(𝑧) as seen in Figs. 3.4-3.7. The resulting “combined”

transfer functions ��(𝑧) for these four combinations of 𝐻0(𝑧) are shown below (Figs. 5.1-5.4)

71

Figure 5.1: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order subband RFAs Zj(z) and the choice of a D2/Haar filter bank (red).

Figure 5.2: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order subband RFAs Zj(z) and the choice of a D6 filter bank (red).

72

Figure 5.3: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order subband RFAs Zj(z) and the choice of a D12 filter bank (red).

Figure 5.4: High order Z(s) (blue) vs. “combined” transfer function Z(z) based on the low order

subband RFAs Zj(z) and the choice of a FIR2CHPR-based (N=99) filter bank (red).

73

From Figs. 3.4-3.7, 5.1-5.4 we observe that: (a) increasing the length of the prototype

analysis filter 𝐻0(𝑧) improves frequency selectivity for each channel; (b) the plots of ��(𝑧)

(Figs. 5.1-5.4) suggest that a filter length of 6 − 12 results in a good match, and further increase of

filter length offers negligible improvement.

So far we have established two design parameters. The first design parameter is the number

of channels (𝑀 = 10) which is related to the order of the subband RFAs. For instance, more

channels would allow us to reduce the orders of ��𝑗(𝑧) down to one, while fewer channels would

make the second order approximations chosen for ��𝑗(𝑧) inaccurate with the extreme case being a

single channel (𝑀 = 1; no subbands) approximation, where the entire 15-order model would be

required. The second design parameter is related to the frequency selectivity of the prototype

analysis filter 𝐻0(𝑧) associated with the perfect reconstruction filter bank chosen (Fig. 3.1). The

effect of a particular PR-FB choice can be studied by computing the “combined” transfer function

��(𝑧) between the input 𝑥[𝑛] and the output 𝑦[𝑛] (Fig. 3.3). This captures: (a) the low-order RFAs

that were computed in Chapter 4, and (b) the choice of the PR-FB itself (ex: D2, D6, D12, etc.).

Figs. 5.1-5.4 show the frequency responses of the “combined” transfer functions. The solid blue

lines in Figs. 5.1-5.4 represent the frequency responses of the high-order CT RFA ��(𝑠) and the

solid red lines represent the “combined” frequency responses based on the low-order subband RFAs

��𝑗(𝑧) for different PR-FB choices. To be more specific, Fig. 5.1 shows that a D2-PR-FB constitutes

a very poor choice since the magnitude mismatch between RFAs is large over the entire frequency

range. This can be attributed to the poor frequency selectivity of the D2-PR-FB. From Fig. 5.2 we

observe that a D6-PR-FB constitutes a major improvement over a D2-PR-FB in both magnitude

and phase over the entire frequency range. From Fig. 5.3 we observe that a D12-PR-FB offers a

slight improvement over a D6-PR-FB, especially in the phase response for frequencies greater than

10kHz. Finally, from Fig. 5.4 we observe that an 𝑁 = 99 order FIR2CHPR-based PR-FB offers

negligible improvement over a D12-PR-FB. This analysis suggests that while frequency

74

separation/selectivity of the PR-FB is an important characteristic for accurate transient simulation

it does not have to be perfect. A D6-PR-FB or a D12-PR-FB are sufficient choices (Figs. 3.5, 3.6).

Transient Simulation Results

Figs. 5.5-5.8 compare the transient responses, due to a periodic square wave input, of the

original CT high-order system ��(𝑠) to the low-order PR filter bank approach proposed, for different

choices of analysis filters, that is, D2, D6, D12, and FIR2CHPR 𝑁 = 99 respectively. Transient

simulation of the ��(𝑠) based system was carried out using a variable-step solver, while simulation

of the ��𝑗(𝑧) based PR filter bank system was done using a fixed step 𝛥𝑡 = 1𝜇𝑠 solver. From

Fig. 5.5 we observe a large mismatch in the responses, while Figs. 5.6-5.8 reveal a close match

between the two responses as predicted from our earlier analysis of the “combined” transfer

function.

Figure 5.5: Transient responses based on the: (i) full order model Z(s) (blue), (ii) low order Zj(z)

models using the D2/Haar filter bank (red).

75

Figure 5.6: Transient responses based on the: (i) full order model Z(s) (blue), (ii) low order Zj(z) models using the D6 filter bank (red).


models using the D12 filter bank (red).

76


models using the FIR2CHPR (N=99) filter bank (red).

77

5.2 Case #2: Single-Phase FDTL

We may use the tools presented in Chapters 2, 3, 4 in order to simulate a single-phase

FDTL using our proposed subband-ULM approach. The inputs to our design procedure are: (i) the

FDTL per-unit-length parameters and tower geometry configuration (Sec. 2.1), and (ii) the driving

inputs (sources) and terminal conditions of the FDTL network (ex. Open-ended and short-ended

buses) (Sec. 2.1).

In particular the steps associated with our subband-ULM procedure are the following:

Extracting the delay component of the propagation/weighting function 𝐴(𝑠) using Bode’s gain-

phase relation (Sec. 2.2). In our example the estimated delay is equal to 𝜏 = 539.93𝜇𝑠.

Choosing the perfect reconstruction filter-bank (PR-FB) that is appropriate for the FDTL data

of interest. In this example we employ a logarithmic-frequency, tree-structured, QMF perfect

reconstruction filter-bank characterized by a Daubechies-6 (D6) prototype analysis filter

𝐻0(𝑧), and 10-channels (𝑀 = 10) (Chapter 3).

Obtaining a rational function approximation (RFA) for the (continuous-time) characteristic

impedance 𝑍(𝑠), and the propagation/weighting function 𝐴(𝑠) (after we extract the delay) for

each subband (Sec. 2.2). In our example we used second-order RFA for each subband.

Performing the RFAs in the subbands allows us to: (i) obtain insights regarding the behavior

of the transients within frequency-bands of interest (ii) use low order approximations to reduce

the complexity of the overall system, (iii) improve numerical stability, (iv) leverage parallel

processing capability of modern computers to increase simulation speed, and (v) employ

distributed computing frameworks such as Hadoop MapReduce to increase simulation speed.

Mapping each subband, low-order RFA pair {𝑍𝑗(𝑠), 𝐴𝑗(𝑠)} into a discrete-time equivalent

{𝑍𝑗(𝑧), 𝐴𝑗(𝑧)} using the prewarped Tustin bilinear transform (Sec. 2.3), applied to a partial

fraction expansion of {𝑍𝑗(𝑠), 𝐴𝑗(𝑠)}.

78

Obtaining a difference equation which represents the FDTL configuration of interest. In

particular we study the network equations of an open-ended and short-ended FDTL for each

subband (Sec. 2.1).

Decomposing the driving input-signals (sources) into its subband components.

Solving each subband network in terms of the desired voltages and currents. These are the

receiving-end subband voltages and receiving-end subband currents for the open-ended and

short-ended configurations respectively.

Reconstructing each subband voltage/current of interest using the synthesis filter-bank.

In summary, we employ a logarithmic-frequency, tree-structured, QMF perfect

reconstruction filter-bank characterized by 10-channels (𝑀 = 10). The frequency ranges

[𝑓1(𝑗), 𝑓2(𝑗)] ∈ ℝ [𝐻𝑧] of each subband are shown below (Table 5.1)

The discretization 𝐻𝑗(𝑠) → 𝐻𝑗(𝑧) is carried out for each band by means of the prewarped

Tustin bilinear transform (Chapter 2) with 𝜔0,𝑗 = 𝜋 (𝑓1(𝑗)+ 𝑓2

(𝑗)) ∀ 𝑗 > 0 and 𝜔0,𝑗 = 0 for 𝑗 = 0

in order to match the DC frequency. The simulation step was chosen to be 𝛥𝑡 = 1𝜇𝑠 as outlined in

Chapter 2.

𝑓1(𝑗)= {

2𝑗−1

𝛥𝑡 ∙ 2𝑀 , 𝑗 > 0

0 , 𝑗 = 0 , 𝑓2

(𝑗)= 𝑓1

(𝑗+1) (5.1)

79

Table 5.1: Subband frequency range decomposition.

Band

𝑗

𝑓1(𝑗) [𝐻𝑧] 𝑓0

(𝑗) [𝐻𝑧] 𝑓2

(𝑗) [𝐻𝑧]

0 0 488.28 976.56

1 976.56 1464.84 1953.16

2 1953.16 2929.69 3906.25

3 3906.25 5859.38 7812.50

4 7812.50 11718.80 15625.00

5 15625.00 23437.50 31250.00

6 31250.00 46875.00 62500.00

7 62500.00 93750.00 125000.00

8 125000.00 187500.00 250000.00

9 250000.00 375000.00 500000.00

Figure 5.9: High order Z0(s) (blue) vs. low order Z0,j(z) subband approximations (dashed colored

segments).

80

Figure 5.10: High order A(s) (blue) vs. low order Aj(z) subband approximations (dashed colored

segments).

The resulting second-order transfer functions { ��0,𝑗(𝑧), ��𝑗(𝑧) ; 𝑗 = 0,1, … ,9 } based on

vector fitting and the ten-channel (𝑀 = 10) PR-FB are all stable. Figs. 5.9, 5.10 show plots of the

high-order {��0(𝑠), ��(𝑠)} vs. the low-order discrete-time approximations {��0,𝑗(𝑧), ��𝑗(𝑧)}. The


black dots in Figs. 5.9, 5.10.

For the purposes of our single-phase, FDTL transient simulation example we consider the

cases of an open-circuited line given by 𝑉𝑚 = 2 ∙𝐴

1+𝐴2∙ 𝐸𝑠 (2.12b), and a short-circuited line given

by 𝐼𝑚 = −2𝐴

1−𝐴2∙𝐸𝑠

𝑍0 (2.13d). The problem with this approach is that it is not scalable for larger

networks. We address this issue in Chapter 6 by means of introducing a “master circuit” and a “sub-

circuit” update equation.

81

Finally, we compare the transient simulation results of our proposed subband-ULM method

for a single-phase FDTL. Fig. 5.11 shows the broadband voltage-source and its 10-subband

decomposition used to excite an open-circuited and a short-circuited line. Figs. 5.12, 5.13 show the

independent network solutions (transient responses) using the subband-sources of Fig. 5.12 and the

low order RFAs of Figs. 5.9, 5.10 for an open-ended and short-ended line respectively. Figs. 5.14,

5.15 compare the transient response, due to sinusoidal voltage energization (peak voltage at 𝑡 = 0),

of the original CT high-order system {��0(𝑠), ��(𝑠)} to the low-order PR filter bank proposed

approach after the synthesis filter bank (reconstruction stage). Transient simulation of the

{��0(𝑠), ��(𝑠)} based system was carried out using a variable-step solver, while simulation of the

{��0,𝑗(𝑧), ��𝑗(𝑧)} based PR filter bank system was done using a fixed step 𝛥𝑡 = 1𝜇𝑠 solver. The two

responses match indeed very closely. Finally, Fig. 5.14 represents the receiving-end voltage of an

open-circuited line given by (2.12b) while Fig. 5.15 represents the receiving-end current of a short-

circuited line given by (2.13d).


decomposition of (i) (solid-colored)

82


simulations) [open-ended line]


simulations) [short-ended line]

83


full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(z), Aj(z)} using the PR filter bank (red).

The input voltage is shown in black.

Figure 5.15: Sinusoidal energization of short-circuited line (peak voltage at t=0) based on the: (i)

full order model {Z0(s), A(s)} (blue), (ii) low order {Z0,j(z), Aj(z)} using the PR filter bank (red).

84

5.3 Case #3: Three-Phase FDTL

Figs. 5.16, 5.17 compare the transient response, due to sinusoidal voltage energization

(peak voltage at 𝑡 = 0), of the original CT high-order system {��0(𝑠), ��𝑘{𝑨1(𝑠)}, ��𝑘(𝑠)} to the

low-order PR filter bank approach proposed. Transient simulation of the

{��0(𝑠), ��𝑘{𝑨1(𝑠)}, ��𝑘(𝑠)} based system was carried out using a variable-step solver, while

simulation of the {��0,𝑗(𝑧), ��𝑘,𝑗{𝑨1(𝑧)}, ��𝑘,𝑗(𝑧)} based PR filter bank system was done using a

fixed step 𝛥𝑡 = 1𝜇𝑠 solver. The two responses match indeed very closely. Finally, Fig. 5.16

represents the receiving-end voltage of an open-circuited line given by (2.9b) while Fig. 5.17

represents the receiving-end current of a short-circuited line given by (2.10d).


full order model {Y0(s), λk{A1(s)}, Mk(s)} (blue), (ii) low order {Y0,j(z), λk,j{A1(z)}, Mk,j(z)} using the PR filter bank (red). The input voltage is shown in black.

85

Figure 5.17: Sinusoidal energization of short-circuited line (peak voltage at t=0) based on the: (i) full order model {Y0(s), λk{A1(s)}, Mk(s)} (blue), (ii) low order {Y0,j(z), λk,j{A1(z)}, Mk,j(z)}

using the PR filter bank (red).

CHAPTER 6

Scalable, Multiphase, Discrete Simulation of Frequency Dependent

Transmission Line (FDTL) Networks

6.1 Introduction: Transfer Function based simulation vs. Y-matrix approach

In this chapter we introduce the discrete-time, 𝑌-matrix state-space based approach for

simulating networks consisting of one (or more) FDTLs. This is an alternative to the transfer-

function based approach discussed in Chapter 5 and (2.9, 2.10). Although both methods are

functionally equivalent, the 𝑌-matrix, state-space approach is better suited for scalability, that is,

simulating an arbitrarily large network without introducing significant complexity in deriving the

model. The objective is to start with continuous-time, time-domain expressions for electrical

components and derive a discrete-time equivalent circuit which is characterized by the following,

discrete-time, state-space equations:

𝒀[𝑛] ∙ 𝒗[𝑛] = 𝒊[𝑛] + 𝒊𝒉𝒊𝒔𝒕[𝑛 − 1] (6.1𝑎)

𝒊𝒉𝒊𝒔𝒕[𝑛] = 𝑓( 𝒗[𝑛] , 𝒊𝒉𝒊𝒔𝒕[𝑛 − 1] ) (6.1𝑏)

Where 𝒀[𝑛] ∈ ℂ𝑁𝑥𝑁 represents the known admittance matrix, 𝒊[𝑛] ∈ ℂ𝑁𝑥1 represents the known

input current-sources, 𝒊𝒉𝒊𝒔𝒕[𝑛 − 1] ∈ ℂ𝑁𝑥1 represents a known history/memory term and 𝒗[𝑛] ∈

ℂ𝑁𝑥1 is the unknown voltage vector.

At every simulation-step we solve (6.1a) for 𝒗[𝑛] and update the history/memory term of

(6.1b). The memory term of (6.1b) depends on the voltage solution of (6.1a) and its previous value.

It is essentially a first-order (discrete state-space) difference equation. An equivalent circuit which

can be described by (6.1) forms the basis of a scalable discrete-time simulator for an arbitrarily

large electrical network consisting of FDTLs and other fundamental components. We shall refer to

87

(6.1a) as the “master circuit” equation since the resulting voltage vector 𝒗[𝑛] corresponds to a

physical network (measurable) quantity. Finally, we shall refer to (6.1b) as the “sub-circuit” update

equation since the resulting current/voltage quantities correspond to auxiliary variables that are not

part of the physical network. These auxiliary variables correspond to non-measurable

current/voltage quantities that were introduced as part of the modeling process.

6.2 Y-matrix equivalent circuits of fundamental components

In this section we present the derivation of a discrete, multiphase (𝑃-phase), 𝑌-matrix

equivalent circuit for the three fundamental circuit components. Namely, multiphase (a) resistor,

(b) capacitor, (c) inductor. These fundamental components shall be reused when building the

𝑌-matrix based FDTL model. Derivation of the discrete, 𝑌-matrix based equivalent circuit may be

realized using the generalized bilinear transform. The generalized bilinear transform allows us to

construct a general model which can later be reduced to equivalent circuits that have better accuracy

around a particular frequency (ex: DC-focused using Tustin transform, or arbitrary frequency

focused which are more suitable for subband approximations). Finally, the recursive convolution

approximation will be presented as a special case of the generalized bilinear transform. This is

essentially a pole-dependent bilinear transformation which is used in FDTL modeling (Chapter 2).

We shall use the following notation thought the chapter:

(1) Lower-case, bold quantities represent vectors.

(2) Upper-case, bold quantities represent matrices.

(3) 𝐿{. } Represents the Laplace transform and “𝑠” is the Laplace operator

(4) 𝑠 = 𝑓(𝑧) Represents the z-domain representation of a continuous-time 𝑠-domain transfer

function. Such mapping could be a generalized bilinear transform, namely, 𝑠 = 𝑓(𝑧) =

𝑏0+𝑏1∙𝑧−1

𝑏2+𝑏3∙𝑧−1

88

(5) 𝑍−1{. } Represents the inverse 𝑍-transform.

(6) [ 𝑓(. ) ] Represents the physical units of quantity 𝑓(. ).

(7) 𝑓(0−) Represents the initial condition for quantify 𝑓(. ).

(8) 𝑇𝑠 Represents the fixed-step simulation size. [ 𝑇𝑠 ] = 𝑠𝑒𝑐 , 𝑇𝑠 =1

𝐹𝑠 .

(9) 𝛿[𝑛] Represents the discrete-time (Kronecker) delta function. 𝛿[𝑛] = {1, 𝑛 = 00, 𝑛 ≠ 0

Our objective is to obtain recursive relations of the form (6.1) which are suitable for scalable

simulation of arbitrarily large networks.

Multiphase Resistor (P-phase)

The following figure (Fig. 6.1), represents the continuous-time circuit schematic of a 𝑃-phase

resistor

Figure 6.1: Continuous-time representation of resistor in the time-domain (left) and complex-

frequency (Laplace) domain (right).

Where

Voltage: 𝒗𝑅(𝑡) ∈ ℝ𝑃×1, [𝒗𝑅(𝑡)] = 𝑉

Current: 𝒊𝑅(𝑡) ∈ ℝ𝑃×1, [𝒊𝑅(𝑡)] = 𝐴

Resistance: 𝑹 ∈ ℝ𝑃×𝑃, [𝑹] = Ω

89

Discretization:

𝒗𝑅(𝑡) = 𝑹 ∙ 𝒊𝑅(𝑡)𝐿{.}⇒ 𝒗𝑅(𝑠) = 𝑹 ∙ 𝒊𝑅(𝑠)

𝑠=𝑓(𝑧)⇒

𝒗𝑅(𝑧) = 𝑹 ∙ 𝒊𝑅(𝑧)𝑍−1{.}⇒ 𝒗𝑅[𝑛] = 𝑹 ∙ 𝒊𝑅[𝑛] ⇒

𝒊𝑅[𝑛] = 𝒀𝑅 ∙ 𝒗𝑅[𝑛] (6.2𝑎)

𝒀𝑅 = 𝑹−1 (6.2𝑏)

Where:

𝒗𝑅[𝑛] ∈ ℝ𝑃×1 , 𝒊𝑅[𝑛] ∈ ℝ

𝑃×1 , 𝒀𝑅 ∈ ℝ𝑃×𝑃 , [ 𝒀𝑅 ] = 𝑆

We observe that (6.2) is in the desired form (6.1) where 𝒊𝒉𝒊𝒔𝒕[𝑛] = 𝒊𝒉𝒊𝒔𝒕[𝑛 − 1] = 𝟎. This

is expected since resistors are memoryless elements. Finally, 𝒀𝑅 = 𝑹−1 need only to be computed

once, since it is not a function of time “𝑛”.

The following figure (Fig. 6.2) represents the discrete-time equivalent circuit for an 𝑃-

phase resistor.

Figure 6.2: Time-domain, Continuous-time representation of resistor (left). Discrete-time

representation of resistor (right).

90

Multiphase Capacitor (P-phase)

The following figure (Fig. 6.3), represents the continuous-time circuit schematic of an 𝑃-phase

capacitor

Figure 6.3: Continuous-time representation of capacitor in the time-domain (left) and complex-

frequency (Laplace) domain (right) (initial condition included).

Where

Voltage: 𝒗𝑐(𝑡) ∈ ℝ𝑃×1, [𝒗𝑐(𝑡)] = 𝑉

Current: 𝒊𝑐(𝑡) ∈ ℝ𝑃×1, [𝒊𝑐(𝑡)] = 𝐴

Capacitance: 𝑪 ∈ ℝ𝑃×𝑃, [𝑪] = F

Discretization:

𝒊𝑐(𝑡) = 𝑪 ∙𝑑

𝑑𝑡𝒗𝑐(𝑡)

𝐿{.}⇒

𝒊𝑐(𝑠) = 𝑪 ∙ ( 𝑠 ∙ 𝒗𝑐(𝑠) − 𝒗𝑐(0−) ) ⇒

𝒊𝑐(𝑠) = 𝑠𝑪 ∙ 𝒗𝑐(𝑠) − 𝑪 ∙ 𝒗𝑐(0−)

𝑠=𝑓(𝑧)⇒

Let 𝑠 = 𝑓(𝑧) =𝑏0+𝑏1∙𝑧

−1

𝑏2+𝑏3∙𝑧−1 be the generalized bilinear transform, then:

𝒊𝑐(𝑧) =𝑏0 + 𝑏1 ∙ 𝑧

−1

𝑏2 + 𝑏3 ∙ 𝑧−1∙ 𝑪 ∙ 𝒗𝑐(𝑧) − 𝑪 ∙ 𝒗𝑐(0

−) ⇒

𝑏2 ∙ 𝒊𝑐(𝑧) + 𝑏3 ∙ 𝑧−1 ∙ 𝒊𝑐(𝑧) = 𝑏0 ∙ 𝑪 ∙ 𝒗𝑐(𝑧) + 𝑏1 ∙ 𝑧

−1 ∙ 𝑪 ∙ 𝒗𝑐(𝑧)

−𝑏2 ∙ 𝑪 ∙ 𝒗𝑐(0−) − 𝑏3 ∙ 𝑧

−1 ∙ 𝒗𝑐(0−) ⇒

91

𝒊𝑐(𝑧) =𝑏0𝑏2∙ 𝑪 ∙ 𝒗𝑐(𝑧) − 𝑧

−1 ∙ [ 𝑏3𝑏2∙ 𝒊𝑐(𝑧) −

𝑏1𝑏2∙ 𝑪 ∙ 𝒗𝑐(𝑧) +

𝑏3𝑏2∙ 𝑪 ∙ 𝒗𝑐(0

−)] − 𝑪 ∙ 𝒗𝑐(0−)

𝑍−1{.}⇒

𝒊𝑐[𝑛] = 𝒀𝐶 ∙ 𝒗𝑐[𝑛] − 𝒉𝐶[𝑛 − 1] − 𝒊𝑐(𝐼𝐶)[𝑛] (6.3𝑎)

𝒀𝐶 =𝑏0𝑏2∙ 𝑪 (6.3𝑏)

𝒊𝑐(𝐼𝐶)[𝑛] =

1

𝑇𝑠∙ 𝑪 ∙ 𝒗𝑐(0

−) ∙ 𝛿[𝑛] = 𝐼0(𝐶)∙ 𝛿[𝑛] (6.3𝑐)

𝒉𝐶[𝑛 − 1] =𝑏3𝑏2∙ 𝒊𝑐[𝑛 − 1] −

𝑏1𝑏2∙ 𝑪 ∙ 𝒗𝑐[𝑛 − 1] +

𝑏3𝑏2∙1


−) ∙ 𝛿[𝑛 − 1] (6.3𝑑)

Now 𝒉𝐶[𝑛] may be expressed as a function of 𝒉𝐶[𝑛 − 1] as follows:

𝒉𝐶[𝑛] = (𝑏0𝑏3 − 𝑏1𝑏2𝑏0𝑏2

) ∙ 𝒀𝐶 ∙ 𝒗𝑐[𝑛] −𝑏3𝑏2∙ 𝒉𝐶[𝑛 − 1] (6.3𝑒)

The discrete-time equivalent circuit of the 𝑃-phase capacitor may be summarized by (6.4) and

Fig. 6.4:

𝒊𝑐[𝑛] = 𝒀𝐶 ∙ 𝒗𝑐[𝑛] − 𝒉𝐶[𝑛 − 1] − 𝒊𝑐(𝐼𝐶)[𝑛] (6.4𝑎)

𝒀𝐶 =𝑏0𝑏2∙ 𝑪 (6.4𝑏)

𝒊𝑐(𝐼𝐶)[𝑛] =

1


−) ∙ 𝛿[𝑛] = 𝐼0(𝐶)∙ 𝛿[𝑛] (6.4𝑐)

𝒉𝐶[𝑛] = (𝑏0𝑏3 − 𝑏1𝑏2𝑏0𝑏2

) ∙ 𝒀𝐶 ∙ 𝒗𝑐[𝑛] −𝑏3𝑏2∙ 𝒉𝐶[𝑛 − 1] (6.4𝑑)

𝒉𝐶[−1] = 0 (6.4𝑒)

Where:

𝒗𝑐[𝑛] ∈ ℝ𝑃×1 , 𝒊𝑐[𝑛] ∈ ℝ

𝑃×1 , 𝒊𝑐(𝐼𝐶)[𝑛] ∈ ℝ𝑃×1 , 𝒉𝐶[𝑛] ∈ ℝ

𝑃×1 , 𝒀𝐶 ∈ ℝ𝑃×𝑃 , [ 𝒀𝐶 ] = 𝑆.

We observe that (6.4) is in the form of (6.1) with: 𝒀[𝑛] = 𝒀𝐶 , [𝑛] = 𝒗𝑐[𝑛] , 𝒊[𝑛] = 𝒊𝑐[𝑛] +

𝒊𝑐(𝐼𝐶)[𝑛] , 𝒊𝒉𝒊𝒔𝒕[𝑛] = 𝒉𝐶[𝑛] , and (6.4d) corresponds to (6.1b).

92

Figure 6.4: Time-domain, Continuous-time representation of capacitor (left). Discrete-time

representation of capacitor (right).

Multiphase Inductor (P-phase)

The following figure (Fig. 6.5), represents the continuous-time circuit schematic of an 𝑃-phase

inductor

Figure 6.5: Continuous-time representation of inductor in the time-domain (left) and complex-

frequency (Laplace) domain (right) (initial condition included).

Where

Voltage: 𝒗𝐿(𝑡) ∈ ℝ𝑃×1, [𝒗𝐿(𝑡)] = 𝑉

Current: 𝒊𝐿(𝑡) ∈ ℝ𝑃×1, [𝒊𝐿(𝑡)] = 𝐴

Inductance: 𝑳 ∈ ℝ𝑃×𝑃, [𝑳] = H

Discretization:

𝒗𝐿(𝑡) = 𝑳 ∙𝑑

𝑑𝑡𝒊𝐿(𝑡)

𝐿{.}⇒

93

𝒗𝐿(𝑠) = 𝑳 ∙ ( 𝑠 ∙ 𝒊𝐿(𝑠) − 𝒊𝐿(0−) ) ⇒

𝒗𝐿(𝑠) = 𝑠𝑳 ∙ 𝒊𝐿(𝑠) − 𝑳 ∙ 𝒊𝐿(0−)

𝑠=𝑓(𝑧)⇒

Let 𝑠 = 𝑓(𝑧) =𝑏0+𝑏1∙𝑧

−1

𝑏2+𝑏3∙𝑧−1 be the generalized bilinear transform, then:

𝒗𝐿(𝑧) =𝑏0 + 𝑏1 ∙ 𝑧

−1

𝑏2 + 𝑏3 ∙ 𝑧−1∙ 𝑳 ∙ 𝒊𝐿(𝑧) − 𝑳 ∙ 𝒊𝐿(0

−) ⇒

𝒊𝐿(𝑧) =𝑏2𝑏0∙ 𝑳−1 ∙ 𝒗𝐿(𝑧) + 𝑧

−1 ∙ [ 𝑏3𝑏0∙ 𝑳−1 ∙ 𝒗𝐿(𝑧) −

𝑏1𝑏0∙ 𝒊𝐿(𝑧) +

𝑏3𝑏0∙ 𝒊𝐿(0

−)] +𝑏2𝑏0∙ 𝒊𝐿(0

−)𝑍−1{.}⇒

𝒊𝐿[𝑛] = 𝒀𝐿 ∙ 𝒗𝐿[𝑛] + 𝒉𝐿[𝑛 − 1] + 𝒊𝐿(𝐼𝐶)[𝑛] (6.5𝑎)

𝒀𝐿 =𝑏2𝑏0∙ 𝑳−1 (6.5𝑏)

𝒊𝐿(𝐼𝐶)[𝑛] =

1

𝑇𝑠∙𝑏2𝑏0∙ 𝒊𝐿(0

−) ∙ 𝛿[𝑛] = 𝐼0(𝐿)∙ 𝛿[𝑛] (6.5𝑐)

𝒉𝐿[𝑛 − 1] =𝑏3𝑏2∙ 𝑳−1 ∙ 𝒗𝐿[𝑛 − 1] −

𝑏1𝑏2∙ 𝒊𝐿[𝑛 − 1] +

1


−) ∙ 𝛿[𝑛 − 1] (6.5𝑑)

Now 𝒉𝐿[𝑛] may be expressed as a function of 𝒉𝐿[𝑛 − 1] as follows:

𝒉𝐿[𝑛] = (𝑏0𝑏3 − 𝑏1𝑏2𝑏0𝑏3

) ∙ 𝒀𝐿 ∙ 𝒗𝐿[𝑛] −𝑏1𝑏0∙ 𝒉𝐿[𝑛 − 1] + (

𝑏0𝑏3 − 𝑏1𝑏2

𝑏02 ) ∙ 𝒊𝐿(0

−) ∙ 𝛿[𝑛] (6.5𝑒)

Finally, the discrete-time equivalent circuit of the 𝑃-phase inductor may be summarized by (6.6)

and Fig. 6.6:

𝒊𝐿[𝑛] = 𝒀𝐿 ∙ 𝒗𝐿[𝑛] + 𝒉𝐿[𝑛 − 1] + 𝒊𝐿(𝐼𝐶)[𝑛] (6.6𝑎)

𝒀𝐿 =𝑏2𝑏0∙ 𝑳−1 (6.6𝑏)

𝒊𝐿(𝐼𝐶)[𝑛] =

1


−) ∙ 𝛿[𝑛] = 𝐼0(𝐿)∙ 𝛿[𝑛] (6.6𝑐)

𝒉𝐿[𝑛] = (𝑏0𝑏3 − 𝑏1𝑏2𝑏0𝑏3

) ∙ 𝒀𝐿 ∙ 𝒗𝐿[𝑛] −𝑏1𝑏0∙ 𝒉𝐿[𝑛 − 1] + (

𝑏0𝑏3 − 𝑏1𝑏2

𝑏02 ) ∙

1

𝑇𝑠∙ 𝒊𝐿(0

−) ∙ 𝛿[𝑛] (6.6𝑑)

𝒉𝐿[−1] = 0 (6.6𝑒)

94

Where: 𝒗𝐿[𝑛] ∈ ℝ𝑃×1 , 𝒊𝐿[𝑛] ∈ ℝ

𝑃×1 , 𝒊𝐿(𝐼𝐶)[𝑛] ∈ ℝ𝑃×1 , 𝒉𝐿[𝑛] ∈ ℝ

𝑃×1 , 𝒀𝐿 ∈ ℝ𝑃×𝑃 , [ 𝒀𝐿 ] = 𝑆.

We observe that (6.6) is in the form of (6.1) with: 𝒀[𝑛] = 𝒀𝐿 , [𝑛] = 𝒗𝐿[𝑛] , 𝒊[𝑛] = 𝒊𝐿[𝑛] +

𝒊𝐿(𝐼𝐶)[𝑛] , 𝒊𝒉𝒊𝒔𝒕[𝑛] = 𝒉𝐿[𝑛] , and (6.6d) corresponds to (6.1b).

Figure 6.6: Time-domain, Continuous-time representation of inductor (left). Discrete-time

representation of inductor (right).

Simulation Examples and Unit tests

In this section we validate the discrete-time equivalent circuits of (6.2, 6.4, 6.6) by comparing the

simulated output to known analytical responses.

For the following circuit (Fig. 6.7)

Figure 6.7: Time-domain, Continuous-time representation of test circuit

95

With parameters: = 2Ω , 𝐿 = 1𝐻 , 𝐶 = 1𝐹 , 𝑣𝑐(0−) = 10𝑉 , 𝑖𝐿(0

−) = 5𝐴

The analytical solution for 𝑖𝐿(𝑡) is:

𝑖𝐿(𝑡) = 5 ∙ 𝑒−𝑡 ∙ (1 + 𝑡) ∙ 𝑢(𝑡) , where 𝑢(𝑡) = {

1, 𝑡 ≥ 00, 𝑡 < 0

The discrete-time equivalent circuit based on (6.2, 6.4, 6.6) is shown below (Fig. 6.8)

Figure 6.8: Time-domain, Discrete-time representation of test circuit

The circuit of Fig. 6.8 has two (master) nodes {𝐴, 𝐵} thus the simulation 𝑌-matrix based equations

are:

[𝑌𝑅 + 𝑌𝐶 −𝑌𝑅−𝑌𝑅 𝑌𝑅 + 𝑌𝐿

] ∙ [𝑣𝑐[𝑛]

𝑣𝐿[𝑛]] = [

00] + [

ℎ𝑐[𝑛 − 1]

−ℎ𝐿[𝑛 − 1]] + [

𝒊𝑐(𝐼𝐶)[𝑛]

−𝒊𝐿(𝐼𝐶)[𝑛]

]

𝑖𝐿[𝑛] = 𝑌𝑅 ∙ (𝑣𝑐[𝑛] − 𝑣𝐿[𝑛])

Where all parameters are according to definitions (6.2, 6.4, 6.6).

Finally, Fig. 6.9 compares the analytical response of this circuit (solid-blue) to the discrete-

time simulator (solid-red). The responses indeed match very closely, thus validating the discrete-

time models developed in (6.2, 6.4, 6.6)

96

Figure 6.9: Analytical (continuous-time) response (solid-blue) vs. discrete-time simulator (solid-red)

6.3 Y-matrix equivalent circuit of an arbitrary characteristic impedance (P-phase)

In this section we present the derivation of a discrete, multiphase, 𝑌-matrix equivalent

circuit for a characteristic impedance. The characteristic impedance is a rational function and the

first essential component of an FDTL.

Characteristic Impedance (𝑃-phase)

Voltage: 𝒗(𝑠) ∈ ℂ𝑃×1, [𝒗(𝑠)] = 𝑉

Current: 𝒊(𝑠) ∈ ℂ𝑃×1, [𝒊(𝑠)] = 𝐴

Impedance: 𝒁(𝑠) ∈ ℂ𝑃×𝑃, [𝒁(𝑠)] = Ω

97

The 𝑃-phase characteristic impedance element from Fig. 6.10 can be described by the following

transfer-function matrix expression

𝒗(𝑠) = 𝒁(𝑠) ∙ 𝒊(𝑠) (6.7)

where we assume 𝒁(𝑠) to be a proper rational function with a partial fraction expansion as follows:

𝒁(𝑠) = ∑1

𝑠 − 𝑝𝑛∙ 𝑲𝑛

𝑁

𝑛=1

, 𝑲𝑛 ∈ ℂ𝑃×𝑃 (6.8)

Combining (6.7) and (6.8) yields:

𝒗(𝑠) = ∑𝒗𝑛(𝑠)

𝑁

𝑛=1

(6.9𝑎)

𝒗𝑛(𝑠) =1

𝑠 − 𝑝𝑛∙ 𝑲𝑛 ∙ 𝒊(𝑠) (6.9𝑏)

Eq. (6.9) may be realized by means of a cascaded (Foster-form) RC network as shown below

(Fig. 6.10)

Figure 6.10: Continuous-time characteristic impedance (top) and Foster-form (cascade) RC-

implementation (bottom)

98

From the (Foster-form) RC network of (Fig. 6.10) we may derive the following. We assume

zero initial conditions

𝒊(𝑠) = 𝒊𝑅𝑛(𝑠) + 𝒊𝐶𝑛(𝑠) ⇒ (𝑹𝑛−1 + 𝑠𝑪𝑛) ∙ 𝒗𝑛(𝑠) ⇒

(𝑠𝐼 + 𝑪𝑛−1 ∙ 𝑹𝑛

−1) ∙ 𝒗𝑛(𝑠) = 𝑪𝑛−1 ∙ 𝒊(𝑠) (6.10)

Equating (6.9) and (6.10) yields:

𝑪𝑛 = 𝑲𝑛−1 (6.11𝑎)

𝑹𝑛−1 = −𝑝𝑛 ∙ 𝑲𝑛

−1 (6.11𝑏)

Eq.(6.11) describes the equivalent (Foster-form) RC network parameters given a

characteristic impedance of the form (6.8). In order to obtain a discrete-time equivalent circuit we

substitute each 𝑹𝑛 and 𝑪𝑛 component from the (Foster-form) RC network with its discrete-time

equivalent component as derived in (6.2, 6.4) and figures (6.2, 6.4). Substitution and application of

fundamental circuit theory properties (such as source transformation, etc.) allows us to obtain the

following simplified discrete-time equivalent circuit which is very suitable for scalable simulation

(Fig. 6.11). This can be summarized by the following equations:

Figure 6.11: Continuous-time characteristic impedance (left). Discrete-time equivalent circuit (right)

99

Start with a given characteristic impedance matrix 𝒁(𝑠) = 𝒁𝑇(𝑠) ∈ ℂ𝑃×𝑃

𝒁(𝑠) = ∑1


𝑁

𝑛=1

(6.12𝑎)

And a choice of a generalized bilinear transform 𝑠 = 𝑓(𝑧)

𝑠 = 𝑓(𝑧) =𝑏0 + 𝑏1 ∙ 𝑧

−1

𝑏2 + 𝑏3 ∙ 𝑧−1 (6.12𝑏)

The discrete equivalent circuit parameters are:

𝒀𝑛 = (𝑏0𝑏2− 𝑝𝑛) ∙ 𝑲𝑛

−1 (6.12𝑐)

𝒀𝑇𝑂𝑇 = (∑𝒀𝑛−1

𝑁

𝑛=1

)

−1

(6.12𝑑)

𝒊[𝑛] = 𝒀𝑇𝑂𝑇 ∙ 𝒗[𝑛] − 𝒉𝑇𝑂𝑇[𝑛 − 1] (6.12𝑒)

Where we solve (6.12e) for 𝒗[𝑛] and the update term 𝒉𝑇𝑂𝑇[𝑛] is computed by the following steps:

[ 𝒀1 + 𝒀2 −𝒀2 𝟎 … 𝟎−𝒀2 𝒀2 + 𝒀3 −𝒀3𝟎⋮𝟎

−𝒀3 𝒀3 + 𝒀4 ⋱ −𝒀𝑁−1−𝒀𝑁−1 𝒀𝑁−1 + 𝒀𝑁]

∙

[ 𝒗2[𝑛]

𝒗3[𝑛]⋮⋮

𝒗𝑁[𝑛]]

=

[ 𝒀1 ∙ 𝒗[𝑛]𝟎⋮⋮𝟎 ]

+

[ 𝒉𝐶2 [𝑛 − 1] − 𝒉𝐶1[𝑛 − 1]

𝒉𝐶3 [𝑛 − 1] − 𝒉𝐶2[𝑛 − 1]

⋮⋮

𝒉𝐶𝑁[𝑛 − 1] − 𝒉𝐶𝑁−1[𝑛 − 1]]

(6.12𝑓)

A short-hand for (6.12f) may be given by:

�� ∙ ��[𝑛] = ��[𝑛] + ��[𝑛 − 1] (6.12𝑔)

Where: �� ∈ ℂ𝑃∙(𝑁−1)×𝑃∙(𝑁−1) is a symmetric block-tridiagonal matrix, with 𝑃 × 𝑃 blocks, and

��[𝑛] ∈ ℂ𝑃∙(𝑁−1)×1 , ��[𝑛] ∈ ℂ𝑃∙(𝑁−1)×1 , ��[𝑛] ∈ ℂ𝑃∙(𝑁−1)×1.

100

The objective is to solve for the “sub-circuit” voltage vector ��[𝑛] given the “master” voltage value

𝒗[𝑛] obtained from (6.12e).

The “sub-circuit” history terms ��[𝑛] are updated as follows:

𝒉𝐶𝑛[𝑛] =1

𝑏2∙ [𝑏0𝑏3 − 𝑏1𝑏2

𝑏2] ∙ 𝑲𝑛

−1 ∙ ( 𝒗𝑛[𝑛] − 𝒗𝑛+1[𝑛] ) −𝑏3𝑏2∙ 𝒉𝐶𝑛[𝑛 − 1] (6.12ℎ)

∀𝑛 ∈ {1,2, … ,𝑁 − 1}

𝒉𝐶𝑁[𝑛] =1

𝑏2∙ [𝑏0𝑏3 − 𝑏1𝑏2

𝑏2] ∙ 𝑲𝑁

−1 ∙ ( 𝒗𝑁[𝑛] − 0 ) −𝑏3𝑏2∙ 𝒉𝐶𝑁[𝑛 − 1] (6.12𝑖)

Again, we shall refer to (6.12f, 6.12g, 6.12h, 6.12i) as the “sub-circuit” update equations

because they correspond to the capacitive terms inside the (Foster-form) RC network and are

“hidden” from any other interconnections that may occur with other characteristic impedance

elements.

Finally, the “master circuit” update equation is:

𝒉𝑇𝑂𝑇[𝑛] = 𝒀𝑇𝑂𝑇 ∙ ∑ 𝒀𝑛−1 ∙ 𝒉𝐶𝑛[𝑛]

𝑁

𝑛=1

(6.12𝑗)

In summary: The master circuit equation (6.12e) corresponds to (6.1a) and sub-circuit equation

(6.12j) corresponds to (6.1b) in terms of our scalable network representation.

101


In this section we validate the discrete-time equivalent circuits of (6.12) by comparing the simulated

output to known analytical responses.

The responses shown in Fig. 6.12 are obtained as follows:

Input Source Description: 3-phase input current source 𝒊(𝑡) = [𝑖1(𝑡), 𝑖2(𝑡), 𝑖3(𝑡)]𝑇 with

components:

𝑖1(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1) + 𝑢(𝑡 − 2) − 𝑢(𝑡 − 3) + 𝑢(𝑡 − 4),

𝑖2(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 2) + 𝑢(𝑡 − 4),

𝑖3(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 4) where 𝑢(𝑡) = {1, 𝑡 > 00, 𝑡 < 0

is the continuous-time step function.

System Description: A (continuous-time) 3 × 3 characteristic impedance transfer function

matrix with individual rational transfer functions of order 16.

The voltage response shown in solid-red “𝑣𝑐” corresponds to the output of a continuous-time

based simulator (using MATLAB’s “lsim” function). In particular, this is the second phase of

the 3-three phase response voltage vector. Typical solvers for continuous-time simulation are

“Runge-Kutta” methods of order “𝑁” (RK-N).

The voltage response shown in solid-blue “𝑣𝑑” correspond to the output of the scalable,

discrete-time simulator based on (6.12). In particular, this is the second phase of the 3-three

phase response voltage vector.

Error metric: ‖𝑣𝑐−𝑣𝑑‖∞‖𝑣𝑐‖∞

. We observe that the two responses match very closely (within 2.3%)

using this relative-error metric.

102

Figure 6.12: Continuous-time simulator (solid-red) vs. discrete-time simulator (solid-blue) for a characteristic impedance response.

Fig. 6.12 compares the continuous-time simulator-based response of this circuit (solid-blue) to the

discrete-time simulator described by (6.12) (solid-red). The responses indeed match very closely

(within 2.3%), thus validating the discrete-time models developed in (6.12).

103

6.4 Recursive computation of an arbitrary propagation function (P-phase)

In this section we introduce a recursive expression for computing the output of a

propagation function. The propagation function is a delay-rational function and the second essential

component of an FDTL.

Propagation Matrix Function (𝑃-phase)

Here we consider the dimensionless propagation matrix function: 𝑨(𝑠) ∈ ℂ𝑃×𝑃, [𝑨(𝑠)] = 1.

This typically corresponds to a delay-transfer function of the form:

𝑨(𝑠) = 𝑯(𝑠) ∙ 𝑒−𝑠∙𝜏 (6.13)

so that an input “𝒙(𝑠)” , output “𝒚(𝑠)” relationship can be written as:

𝒚(𝑠) = 𝑨(𝑠) ∙ 𝒙(𝑠) = 𝑯(𝑠) ∙ 𝒙(𝑠) ∙ 𝑒−𝑠∙𝜏 (6.14)

The function “𝑒−𝑠∙𝜏” corresponds to a pure delay term by “𝜏” seconds and 𝑯(𝑠) may be

expressed by the following partial fraction expansion:

𝑯(𝑠) = ∑1


𝑁

𝑛=1

(6.15)

Consequently,

𝒚(𝑠) = ∑𝑲𝑛 ∙ 𝒚𝑛(𝑠)

𝑁

𝑛=1

(6.16𝑎)

𝒚𝑛(𝑠) =1

𝑠 − 𝑝𝑛∙ ��(𝑠) (6.16𝑏)

��(𝑠) = 𝒙(𝑠) ∙ 𝑒−𝑠∙𝜏 (6.16𝑐)

Discretization of (6.16) may be achieved (for example) by means of “recursive

convolution” as discussed in Chapter 2. “Recursive convolution” can be alternatively viewed as a

special, pole-dependent bilinear transform of the form:

𝑠 = 𝑓(𝑧; 𝑛) =𝑏0(𝑛)+ 𝑏1

(𝑛)∙ 𝑧−1

𝑏2(𝑛)+ 𝑏3

(𝑛)∙ 𝑧−1

(6.17𝑎)

104

𝑏2(𝑛)= 1 (6.17𝑏)

𝑏0(𝑛)=𝑝𝑛 ∙ 𝑐1

(𝑛)+ 1

𝑐1(𝑛)

(6.17𝑐)

𝑏1(𝑛)=−(𝑐0

(𝑛)− 𝑝𝑛 ∙ 𝑐2

(𝑛))

𝑐1(𝑛)

(6.17𝑑)

𝑏3(𝑛)=𝑐2(𝑛)

𝑐1(𝑛) (6.17𝑒)

𝑐0(𝑛)= 𝑒𝑝𝑛∙𝑇𝑠 (6.17𝑓)

𝑐1(𝑛)= −

1

𝑝𝑛−(1 − 𝑒𝑝𝑛∙𝑇𝑠)

𝑝𝑛2 ∙ 𝑇𝑠

(6.17𝑔)

𝑐2(𝑛)=𝑒𝑝𝑛∙𝑇𝑠

𝑝𝑛+(1 − 𝑒𝑝𝑛∙𝑇𝑠)


(6.17ℎ)

And the recursive expression for computing the output 𝒚[𝑛] is:

𝒚𝑛[𝑛] = 𝑐0(𝑛) ∙ 𝒚𝑛[𝑛 − 1] + 𝑐1

(𝑛) ∙ 𝒙[𝑛 − 𝑇𝐴𝑈] + 𝑐2(𝑛) ∙ 𝒙[𝑛 − 𝑇𝐴𝑈 − 1] (6.17𝑖)

𝒚[𝑛] = ∑𝑲𝑛 ∙ 𝒚𝑛[𝑛]

𝑁

𝑛=1

(6.17𝑗)

𝑇𝐴𝑈 = 𝑓𝑙𝑜𝑜𝑟 (𝜏

𝑇𝑠 ) (6.17𝑘)

Thus, in order to summarize, given an input-output relationship of the form:

𝒚(𝑠) = 𝑨(𝑠) ∙ 𝒙(𝑠) = 𝑯(𝑠) ∙ 𝒙(𝑠) ∙ 𝑒−𝑠∙𝜏 (6.18𝑎)

Where “𝑒−𝑠∙𝜏” corresponds to a pure delay term by “𝜏” seconds and 𝑯(𝑠) may be expressed by the

following partial fraction expansion:

𝑯(𝑠) = ∑1


𝑁

𝑛=1

(6.18𝑏)

105

The discrete-time recursive convolution-based relation is given by (Chapter 2):

𝑐0(𝑛)= 𝑒𝑝𝑛∙𝑇𝑠 (6.18𝑐)

𝑐1(𝑛)= −

1

𝑝𝑛−(1 − 𝑒𝑝𝑛∙𝑇𝑠)


(6.18𝑑)

𝑐2(𝑛)=𝑒𝑝𝑛∙𝑇𝑠

𝑝𝑛+(1 − 𝑒𝑝𝑛∙𝑇𝑠)


(6.18𝑒)

𝒚𝑛[𝑛] = 𝑐0(𝑛)∙ 𝒚𝑛[𝑛 − 1] + 𝑐1

(𝑛)∙ 𝒙[𝑛 − 𝑇𝐴𝑈] + 𝑐2

(𝑛)∙ 𝒙[𝑛 − 𝑇𝐴𝑈 − 1] (6.18𝑓)

𝒚[𝑛] = ∑𝑲𝑛 ∙ 𝒚𝑛[𝑛]

𝑁

𝑛=1

(6.18𝑔)

𝑇𝐴𝑈 = 𝑓𝑙𝑜𝑜𝑟 (𝜏

𝑇𝑠 ) (6.18ℎ)

From (6.18g) we observe that the current value of 𝒚[𝑛] depends only on history terms, since 𝒚𝑛[𝑛]

is a function of 𝒚𝑛[𝑛 − 1] , 𝒙[𝑛 − 𝑇𝐴𝑈] , 𝒙[𝑛 − 𝑇𝐴𝑈 − 1] (6.18f). These equations are

incorporated in the FDTL model (Section 6.5).


In this section we validate the discrete-time equivalent expressions of (6.18) by comparing the

simulated output to known analytical responses.

The responses shown in Fig. 6.13 are obtained as follows:

Input Source Description: 3-phase voltage source 𝒗(𝑡) = [𝑣1(𝑡), 𝑣2(𝑡), 𝑣3(𝑡)]𝑇 with

components:

𝑣1(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1) + 𝑢(𝑡 − 2) − 𝑢(𝑡 − 3) + 𝑢(𝑡 − 4),

𝑣2(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 2) + 𝑢(𝑡 − 4),

𝑣3(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 4)

where “𝑢(𝑡)” is the continuous-time step function, namely, 𝑢(𝑡) = {1, 𝑡 > 00, 𝑡 < 0

106

System Description: A (continuous-time) 3 × 3 propagation transfer function matrix with

individual rational transfer functions of order 16.

The voltage response shown in solid-red “𝑣𝑐” corresponds to the output of a continuous-time

based simulator. In particular, this is the first phase of the 3-three phase response voltage vector.

The voltage response shown in solid-blue “𝑣𝑑” correspond to the output of the scalable,

discrete-time simulator based on (6.18). In particular, this is the first phase of the 3-three phase

response voltage vector.

Error metric: ‖𝑣𝑐−𝑣𝑑‖∞‖𝑣𝑐‖∞

. We observe that the two responses match very closely (within 2.4%)

using this relative-error metric.

Fig. 6.13 compares the continuous-time simulator based response of this circuit (solid-

blue) to the discrete-time simulator described by (6.18) (solid-red). The responses indeed match

very closely (within 2.4%), thus validating the discrete-time models developed in (6.18).

Figure 6.13: Analytical (continuous-time) response (solid-red) vs. discrete-time simulator (solid-

blue) for a propagation function response.

107

6.5 Y-matrix equivalent circuits of single-phase FDTL

In this section we present the derivation of a discrete, single-phase, Y-matrix equivalent

circuit for the single-phase FDTL using the fundamental components derived in sections 6.2-6.4.

Single-Phase FDTL:

The objective is to obtain a discrete-time equivalent circuit for a single-phase FDTL that is suitable

for scalable simulation in networks consisting of multiple FDTLs. Fig. 6.14 shows the continuous-

time model of a single-phase FDTL. The FDTL is fully characterized by: (i) the characteristic

impedance 𝑍0(𝑠) , (ii) the propagation function 𝐴(𝑠), and (iii) it’s length “ ℓ ”. Both quantities

{ 𝑍0(𝑠) , 𝐴(𝑠) } are scalars for a single-phase FDTL. From Fig. 6.14 we may obtain the following

relations for the two ports.

Figure 6.14: Continuous-time two-port model for single-phase FDTL (complex-frequency

domain)

Sending-end Voltage: 𝑉𝑘(𝑠) ∈ ℂ, [𝑉𝑘(𝑠)] = 𝑉

Sending-end Current: 𝐼𝑘(𝑠) ∈ ℂ, [𝐼𝑘(𝑠)] = 𝐴

Receiving-end Voltage: 𝑉𝑚(𝑠) ∈ ℂ, [𝑉𝑚(𝑠)] = 𝑉

Receiving-end Current: 𝐼𝑚(𝑠) ∈ ℂ, [𝐼𝑚(𝑠)] = 𝐴

Characteristic Impedance: 𝑍0(𝑠) ∈ ℂ, [𝑍0(𝑠)] = Ω

Propagation Function: 𝐴(𝑠) ∈ ℂ, [𝐴(𝑠)] = 1

108

Sending-End Current “ 𝐼𝑘(𝑠) ” :


𝑍0(𝑠) − [ 𝑉𝑚(𝑠)

𝑍0(𝑠)+ 𝐼𝑚(𝑠) ] ∙ 𝐴(𝑠) (6.19)

Receiving-End Current “ 𝐼𝑚(𝑠) ” :


𝑍0(𝑠) − [ 𝑉𝑘(𝑠)

𝑍0(𝑠)+ 𝐼𝑘(𝑠) ] ∙ 𝐴(𝑠) (6.20)

We may define the following terms:

𝐻𝑚𝑘(𝑠) =𝑉𝑚(𝑠)

𝑍0(𝑠)+ 𝐼𝑚(𝑠) (6.21𝑎)

𝐻𝑘𝑚(𝑠) =𝑉𝑘(𝑠)

𝑍0(𝑠)+ 𝐼𝑘(𝑠) (6.21𝑏)

We assume that the propagation function “𝐴(𝑠)” can be expressed as a product of a proper

rational function and a pure delay term, namely

𝐴(𝑠) = 𝜆(𝑠) ∙ 𝑒−𝑠∙𝜏 (6.21𝑐)

𝜆(𝑠) = ∑𝑘𝑛(𝜆)

𝑠 − 𝑝𝑛(𝜆)

𝑁𝜆

𝑛=1

(6.21𝑑)

We also assume that the characteristic impedance “𝑍0(𝑠)” is a proper rational function with

𝑍0(𝑠) = ∑𝑘𝑛(𝑍0)

𝑠 − 𝑝𝑛(𝑍0)

𝑁𝑍0

𝑛=1

(6.21𝑒)

So that the sending and receiving end currents become:


𝑍0(𝑠) − 𝐻𝑚𝑘(𝑠) ∙ 𝜆(𝑠) ∙ 𝑒

−𝑠∙𝜏 (6.22𝑎)


𝑍0(𝑠) − 𝐻𝑘𝑚(𝑠) ∙ 𝜆(𝑠) ∙ 𝑒

−𝑠∙𝜏 (6.22𝑏)

109

Manipulating (6.21) and (6.22) yields the following continuous-time recursive relations:

𝐻𝑚𝑘(𝑠) = 2 ∙𝑉𝑚(𝑠)

𝑍0(𝑠)− 𝐻𝑘𝑚(𝑠) ∙ 𝜆(𝑠) ∙ 𝑒

−𝑠∙𝜏 (6.23𝑎)

𝐻𝑘𝑚(𝑠) = 2 ∙𝑉𝑘(𝑠)

𝑍0(𝑠)− 𝐻𝑘𝑚(𝑠) ∙ 𝜆(𝑠) ∙ 𝑒

−𝑠∙𝜏 (6.23𝑏)

If we define the following terms:

𝑦𝑚(𝑠) = 𝜆(𝑠) ∙ ( 𝐻𝑚𝑘(𝑠) ∙ 𝑒−𝑠∙𝜏 ) (6.24𝑎)

𝑦𝑘(𝑠) = 𝜆(𝑠) ∙ ( 𝐻𝑘𝑚(𝑠) ∙ 𝑒−𝑠∙𝜏 ) (6.24𝑏)

So that (6.22) becomes:


𝑍0(𝑠) − 𝑦𝑚(𝑠) (6.25𝑎)


𝑍0(𝑠) − 𝑦𝑘(𝑠) (6.25𝑏)

This may be viewed as superposition of a characteristic impedance-based term and a

recursive-convolution based-term. Discretization of (6.2-6.25) involves substitution of (6.12) and

(6.18) that were derived previously as building blocks.

Finally, the discrete-time equivalent circuit that is suitable for scalable simulation of

networks with several FDTLs is shown in below (Fig. 6.15). The discrete-time equations associated

with this network are summarized below:

Figure 6.15: Discrete-time equivalent circuit for single-phase FDTL

110

Compute { 𝑦(𝑚)[𝑛], 𝑦(𝑘)[𝑛] } by using (6.18) and 𝑇𝐴𝑈 = 𝑓𝑙𝑜𝑜𝑟 (𝜏

𝑇𝑠).

Then solve the following (6.26) for { 𝑣𝑘[𝑛], 𝑣𝑚[𝑛] }

𝑖𝑘[𝑛] = 𝑌𝑇𝑂𝑇 ∙ 𝑣𝑘[𝑛] − ℎ𝑇𝑂𝑇(𝑘) [𝑛 − 1] − 𝑦(𝑚)[𝑛] (6.26𝑎)

𝑖𝑚[𝑛] = 𝑌𝑇𝑂𝑇 ∙ 𝑣𝑚[𝑛] − ℎ𝑇𝑂𝑇(𝑚) [𝑛 − 1] − 𝑦(𝑘)[𝑛] (6.26𝑏)

ℎ𝑘𝑚[𝑛] = 2 ∙ 𝑌𝑇𝑂𝑇 ∙ 𝑣𝑘[𝑛] − 2 ∙ ℎ𝑇𝑂𝑇(𝑘) [𝑛 − 1] − 𝑦(𝑚)[𝑛] (6.26𝑐)

ℎ𝑚𝑘[𝑛] = 2 ∙ 𝑌𝑇𝑂𝑇 ∙ 𝑣𝑚[𝑛] − 2 ∙ ℎ𝑇𝑂𝑇(𝑚) [𝑛 − 1] − 𝑦(𝑘)[𝑛] (6.26𝑑)

Finally, update { ℎ𝑇𝑂𝑇(𝑘) [𝑛], ℎ𝑇𝑂𝑇

(𝑚) [𝑛] } using (6.12) and Fig. 6.11.

111

6.6 Y-matrix equivalent circuits of multi-phase FDTL

In this section we present the derivation of a discrete, multiphase, 𝑌-matrix equivalent

circuit for the single-phase FDTL using the fundamental components derived in sections 6.2-6.4.

Multi-Phase FDTL (𝑃-phases):

The objective is to obtain a discrete-time equivalent circuit for a 𝑃-phase FDTL that is suitable for

scalable simulation in networks consisting of multiple FDTLs. Fig. 6.16 shows the continuous-time

model of a single-phase FDTL. The FDTL is fully characterized by: (i) the characteristic admittance

matrix 𝒀0(𝑠) , (ii) the propagation function matrix 𝑨(𝑠), and (iii) it’s length “ ℓ ”. From Fig. 6.16

we may obtain the following relations for the two ports.

Figure 6.16: Continuous-time two-port model for multi-phase FDTL (complex-frequency

domain)

Sending-end Voltage: 𝒗𝑘(𝑠) ∈ ℂ𝑃×1, [𝒗𝑘(𝑠)] = 𝑉

Sending-end Current: 𝒊𝑘(𝑠) ∈ ℂ𝑃×1, [𝒊𝑘(𝑠)] = 𝐴

Receiving-end Voltage: 𝒗𝑚(𝑠) ∈ ℂ𝑃×1, [𝒗𝑚(𝑠)] = 𝑉

Receiving-end Current: 𝒊𝑚(𝑠) ∈ ℂ𝑃×1, [𝒊𝑚(𝑠)] = 𝐴

Characteristic Admittance: 𝒀0(𝑠) ∈ ℂ𝑃×𝑃 , [𝒀0(𝑠)] = S

Characteristic Impedance: 𝒁0(𝑠) ∈ ℂ𝑃×𝑃, [𝒁0(𝑠)] = Ω

Propagation Function: 𝑨1(𝑠) ∈ ℂ𝑃×𝑃, [𝑨1(𝑠)] = 1

112

Characteristic Impedance/Admittance matrix properties:

𝒀0(𝑠) = 𝒀0𝑇(𝑠) = 𝒁0

−1(𝑠) (6.27𝑎)

𝒁0(𝑠) = 𝒁0𝑇(𝑠) = 𝒀0

−1(𝑠) (6.27𝑏)

Propagation function matrix properties:

𝑨1(𝑠) = 𝑨2𝑇(𝑠) (6.27𝑐)

Where “𝑨1(𝑠)” and “𝑨2(𝑠)” are terms derived when expressing the FDTL two-port equations in

terms of current and voltage respectively.

Sending-End Current “ 𝒊𝑘(𝑠) ” :

𝒊𝑘(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) − 𝑨1(𝑠) ∙ [ 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) + 𝒊𝑚(𝑠) ] (6.28𝑎)

Receiving-End Current “ 𝒊𝑚(𝑠) ” :

𝒊𝑚(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) − 𝑨1(𝑠) ∙ [ 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) + 𝒊𝑘(𝑠) ] (6.28𝑏)

Eigen/Spectral Decomposition of the propagation function matrix “𝑨1(𝑠)”

𝑨1(𝑠) = 𝑽(𝑠) ∙ 𝚲(𝑠) ∙ 𝑸(𝑠) = ∑𝑴𝑝(𝑠) ∙ 𝜆𝑝(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝

𝑃

𝑝=1

(6.29)

Where:

𝚲(𝑠) ∈ ℂ𝑃×𝑃 is a diagonal matrix consisting of complex-frequency dependent eigenvalues

𝜆𝑝(𝑠) ∈ ℂ

𝐕(𝑠) ∈ ℂ𝑃×𝑃 , (𝑠) ∈ ℂ𝑃×𝑃 , 𝐐(𝑠) = 𝑽−1(𝑠) are the eigenvector matrices associated with 𝚲(𝑠),

with eigenvectors { 𝒗𝑝(𝑠) , 𝒒𝑝(𝑠) } ∈ ℂ𝑃×1 respectively.

𝑴𝑝(𝑠) = 𝒗𝑝(𝑠) ∙ 𝒒𝑝𝑇(𝑠) ∈ ℂ𝑃×𝑃 is the (rank-1) idempotent/spectral decomposition matrix

associated with 𝜆𝑝(𝑠).

𝑒−𝑠∙𝜏𝑝 is a pure-delay term that can be extracted from 𝜆𝑝(𝑠) by means of Bode’s gain-phase

relation.

113

Substituting (6.29) in (6.28) yields:

𝒊𝑘(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) −∑𝑴𝑝(𝑠) ∙ 𝜆𝑝(𝑠) ∙ [ 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) + 𝒊𝑚(𝑠) ] ∙ 𝑒−𝑠∙𝜏𝑝

𝑃

𝑝=1

(6.30𝑎)

𝒊𝑚(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) −∑𝑴𝑝(𝑠) ∙ 𝜆𝑝(𝑠) ∙ [ 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) + 𝒊𝑘(𝑠) ] ∙ 𝑒−𝑠∙𝜏𝑝

𝑃

𝑝=1

(6.30𝑏)

Where we can define the quantities { 𝒉𝑘𝑚(𝑠) , 𝒉𝑚𝑘(𝑠) } in the same way as the single-phase FDTL:

𝒉𝑚𝑘(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) + 𝒊𝑚(𝑠) (6.31𝑎)

𝒉𝑘𝑚(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) + 𝒊𝑘(𝑠) (6.31𝑏)

So that (6.30) becomes:

𝒊𝑘(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) −∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑚𝑘(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.32𝑎)

𝒊𝑚(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) −∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑘𝑚(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.32𝑏)

The recursive equations of { 𝒉𝑘𝑚(𝑠) , 𝒉𝑚𝑘(𝑠) } are:

𝒉𝑚𝑘(𝑠) = 2 ∙ 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) −∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑘𝑚(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.33𝑎)

𝒉𝑘𝑚(𝑠) = 2 ∙ 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) −∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑚𝑘(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.33𝑏)

We may now define the current sources associated with the recursive convolution terms:

𝒉𝐶𝑂𝑁𝑉(𝑚) (𝑠) = ∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑚𝑘(𝑠) ∙ 𝑒

−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.34𝑎)

𝒉𝐶𝑂𝑁𝑉(𝑘) (𝑠) = ∑𝑴𝑝(𝑠) ∙ [ 𝜆𝑝(𝑠) ∙ 𝒉𝑘𝑚(𝑠) ∙ 𝑒

−𝑠∙𝜏𝑝 ]

𝑃

𝑝=1

(6.34𝑏)

114

So that (6.32, 6.33) simplify to:

𝒊𝑘(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) − 𝒉𝐶𝑂𝑁𝑉(𝑚) (𝑠) (6.35𝑎)

𝒊𝑚(𝑠) = 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) − 𝒉𝐶𝑂𝑁𝑉(𝑘) (𝑠) (6.35𝑏)

𝒉𝑚𝑘(𝑠) = 2 ∙ 𝒀0(𝑠) ∙ 𝒗𝑚(𝑠) − 𝒉𝐶𝑂𝑁𝑉(𝑘) (𝑠) (6.35𝑐)

𝒉𝑘𝑚(𝑠) = 2 ∙ 𝒀0(𝑠) ∙ 𝒗𝑘(𝑠) − 𝒉𝐶𝑂𝑁𝑉(𝑚) (𝑠) (6.35𝑑)

We observe that (6.35) look very similar to the expressions derived for the single-phase

FDTL, namely, superposition of a characteristic impedance based term and a recursive-convolution

based-term. Discretization of (6.35) involves substitution of (6.12) and (6.18) that were derived

previously as building blocks.

Finally, the discrete-time equivalent circuit that is suitable for scalable simulation of

networks with several FDTLs is shown in below (Fig. 6.17). The discrete-time equations associated

with this 𝑃-phase FDTL network are summarized below:

Figure 6.17: Discrete-time equivalent circuit for multi-phase FDTL

Compute { 𝒉𝐶𝑂𝑁𝑉(𝑚) [𝑛] , 𝒉𝐶𝑂𝑁𝑉

(𝑘) [𝑛] } by using (6.34, 6.18) and 𝑇𝐴𝑈𝑝 = 𝑓𝑙𝑜𝑜𝑟 (𝜏𝑝

𝑇𝑠). This involves the

computation of “ 2 ∙ 𝑃 ” recursive convolutions for each sending and receiving nodes { 𝑘 , 𝑚 }.

Namely, each { 𝑴𝑝(𝑠) , 𝜆𝑝(𝑠) ; 𝑝 = 1,… , 𝑃} have their own partial-fraction expansions

{𝑐0(𝑀,𝑝;𝑛) , 𝑲𝑀,𝑝;𝑛 ; 𝑝 = 1,… , 𝑃} , {𝑐0

(𝜆,𝑝;𝑛) ,𝑲𝜆,𝑝;𝑛 ; 𝑝 = 1,… , 𝑃} respectively, and go through the

process of (6.18) independently to construct{ 𝒉𝐶𝑂𝑁𝑉(𝑚) [𝑛] , 𝒉𝐶𝑂𝑁𝑉

(𝑘) [𝑛] }.

115

For example, the computation of 𝒉𝐶𝑂𝑁𝑉(𝑚) [𝑛] is carried out by the following steps:

From (6.34a) we may define 𝜽𝑝(𝑠) ∶= [ 𝜆𝑝(𝑠) ∙ 𝒉𝑚𝑘(𝑠) ∙ 𝑒−𝑠∙𝜏𝑝 ]. This corresponds to a single

recursive convolution of the form (6.18f, 6.18g), namely

𝜽𝑝;𝑛[𝑛] = 𝑐0(𝜆,𝑝;𝑛)

∙ 𝜽𝑝;𝑛[𝑛 − 1] + 𝑐1(𝜆,𝑝;𝑛)

∙ 𝒉𝑚𝑘[𝑛 − 𝑇𝐴𝑈(𝑝)] + 𝑐2

(𝜆,𝑝;𝑛)∙ 𝒉𝑚𝑘[𝑛 − 𝑇𝐴𝑈

(𝑝) − 1]

𝜽𝑝[𝑛] = ∑𝑲𝜆,𝑝;𝑛 ∙ 𝜽𝑝;𝑛[𝑛]

𝑁

𝑛=1

And (6.34a):

𝒉𝐶𝑂𝑁𝑉(𝑚) (𝑠) = ∑ 𝑴𝑝(𝑠) ∙ 𝜽𝑝(𝑠)

𝑃𝑝=1 corresponds to reapplication of (6.18f, 6.18g) so that

𝒚𝑝;𝑛[𝑛] = 𝑐0(𝑀,𝑝;𝑛)

∙ 𝒚𝑝;𝑛[𝑛 − 1] + 𝑐1(𝑀,𝑝;𝑛)

∙ 𝜽𝑝[𝑛] + 𝑐2(𝜆,𝑝;𝑛)

∙ 𝜽𝑝[𝑛 − 1]

𝒚𝑝[𝑛] = ∑𝑲𝑀,𝑝;𝑛 ∙ 𝒚𝑛[𝑛]

𝑁

𝑛=1

Finally:

𝒉𝐶𝑂𝑁𝑉(𝑚) [𝑛] = ∑𝒚𝑝[𝑛]

𝑃

𝑝=1

We repeat the same process for 𝒉𝐶𝑂𝑁𝑉(𝑘) [𝑛].

Then solve the following (6.36) for { 𝒗𝑘[𝑛], 𝒗𝑚[𝑛] }

𝒊𝑘[𝑛] = 𝒀𝑇𝑂𝑇 ∙ 𝒗𝑘[𝑛] − 𝒉𝑇𝑂𝑇(𝑘) [𝑛 − 1] − 𝒉𝐶𝑂𝑁𝑉

(𝑚) [𝑛] (6.36𝑎)

𝒊𝑚[𝑛] = 𝒀𝑇𝑂𝑇 ∙ 𝒗𝑚[𝑛] − 𝒉𝑇𝑂𝑇(𝑚) [𝑛 − 1] − 𝒉𝐶𝑂𝑁𝑉

(𝑘) [𝑛] (6.36𝑏)

𝒉𝑘𝑚[𝑛] = 2 ∙ 𝒀𝑇𝑂𝑇 ∙ 𝒗𝑘[𝑛] − 2 ∙ 𝒉𝑇𝑂𝑇(𝑘) [𝑛 − 1] − 𝒉𝐶𝑂𝑁𝑉

(𝑚) [𝑛] (6.36𝑐)

𝒉𝑚𝑘[𝑛] = 2 ∙ 𝒀𝑇𝑂𝑇 ∙ 𝒗𝑚[𝑛] − 2 ∙ 𝒉𝑇𝑂𝑇(𝑚) [𝑛 − 1] − 𝒉𝐶𝑂𝑁𝑉

(𝑘) [𝑛] (6.36𝑑)

Then update { 𝒉𝑇𝑂𝑇(𝑘) [𝑛] , 𝒉𝑇𝑂𝑇

(𝑚) [𝑛] } by using (6.11) and Fig. 6.11.

116

Remarks:

(1) For a fully transposed FDTL the idempotent matrices 𝑴𝑝(𝑠) are frequency independent

(constants), namely: 𝑴𝑝(𝑠) = 𝑴𝑝 thus reducing the computational complexity of the

simulator.

(2) The 𝑃-phase relations of (6.36) reduce to the single-phase relations of (6.26) by setting 𝑃 = 1.

Equations (6.36) apply to both broadband and subband simulations. Each subband has its own

equivalent circuit as described by (6.36). The partial fraction expansions of the characteristic

admittance and propagation functions are the only quantities that differ per subband.

117

6.7 Arbitrary FDTL Network Simulation Procedure

In this section we provide an outline of the steps involved in constructing and simulating an

arbitrary network which consists of multiple FDTLs and other fundamental components (ex.

sources, loads).

Step #1: Master Circuit Y-Matrix Construction (N-Nodes)

Starting from the network schematic (ex. IEEE 5-Bus grid, etc.), obtain the 𝑌-matrix network

equations by visual inspection using the following steps:

(i) The 𝑌-matrix is diagonal with elements that correspond to the sum of all admittance terms

(sources, loads, and transmission lines) at each node {1,…,N}.

(ii) The source vector “𝑰𝑠[𝑛]” corresponds to the presence (or absence) of a source at each node.

(iii) The history terms “𝒉𝑇𝑂𝑇[𝑛 − 1], 𝒚[𝑛]” associated with the transmission lines correspond to

sums of connections that enter each node.

(iv) The history term “𝒉𝐿𝐶[𝑛 − 1]” corresponds to the sum of all capacitive/inductive loads that

are connected to each node.

The outlined procedure is “scalable” in the sense that steps (i-iv) are straight-forward to implement

for an arbitrarily large network by visual inspection.

Step #2: FDTL Sub-Circuit Construction

Starting from the tower geometry of each FDTL obtain (for each subband) the rational function

approximation (RFA) associated with each characteristic impedance 𝒁0(𝑠) and construct the

discrete-time (Foster-form) RC sub-circuit update equations based on (6.12) and the network

topology. This will produce sub-circuit update equations that resemble (6.1b) for the characteristic

impedance of each FDTL.

118

Step #3: FDTL Recursive Convolution Construction

Starting from the tower geometry of each FDTL obtain (for each subband) the rational function

approximation (RFA) associated with each propagation function 𝑨(𝑠) and construct the recursive

convolution update equations according to (6.18) and the network topology. This will produce

update equations that resemble (6.1b) for the propagation function of each FDTL.

Step #4: Source/Load Update Equation Construction

Construct the update equation for each individual capacitive or inductive source/load component,

based on (6.4, 6.6). This will produce update equations that resemble (6.1b) for each source/load

L, C individual component.

Step #5: Subband (Parallel) Computation Setup

Choose the perfect reconstruction filter-bank (PR-FB) parameters: (a) prototype analysis filter

𝐻0(𝑧), and (b) number of subbands “𝑀”. Then decompose each source to “𝑀” subband sources

using the analysis filter bank. Each subband source is to be processed in parallel using a distributed

computational framework (such as Hadoop, MapReduce). The computational savings due to this

step are of the order of “𝑀” due to downsampling/decimation.

Step #6: Subband Simulation Procedure

At this step all equations are in the standardized form of (6.1a, 6.1b) for the master circuit and the

sub-circuit respectively. For each parallel job (subband source), the following procedure is carried

out for each time instant “𝑛”:

(i) Solve the master circuit equation (6.1a) for 𝒗[𝑛].

(ii) Update the sub-circuit equations for each individual component: (a) Characteristic

impedance; (b) Propagation function; (c) Individual (lumped) L, C components

(iii) Go back to step (i) can compute for the next simulation step: 𝑛 → 𝑛 + 1.

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Step #7: Broadband Transient Response

In this step we collect all subband transient simulation responses and recombine them through the

synthesis filter-bank in order to construct the broadband transient responses (voltages, currents) of

interest.

Computational Complexity

For a 𝑃-phase FDTL network with “𝑁” nodes/buses, “𝐿” FDTLs and “𝑅𝑀𝐴𝑋” maximum-RFA

order, the computational complexity per time instant is given by the following two components:

The matrix associated with the “master circuit” is block-diagonal with block-size “𝑃” and

dimensions “𝑃 ∙ 𝑁 × 𝑃 ∙ 𝑁”. Thus, the complexity of solving the master circuit equation (6.1a)

is 𝑂(𝑁 ∙ 𝑃3).

The matrices associated with the “𝐿” characteristic impedance sub-circuits are 𝑃-block

tridiagonal with dimensions “𝑃 ∙ 𝑅𝑀𝐴𝑋 × 𝑃 ∙ 𝑅𝑀𝐴𝑋”. The complexity associated with solving

the sub-circuit time-update equation (6.1b) for a single FDTL is 𝑂(𝑅𝑀𝐴𝑋 ∙ 𝑃3) using Thomas’s

block-tridiagonal algorithm [25, 26]. Thus, for “𝐿” FDTLs the computational complexity is

𝑂(𝐿 ∙ 𝑅𝑀𝐴𝑋 ∙ 𝑃3).

Adding these two partial costs results in a combined cost of 𝑂(𝑃3 ∙ 𝑚𝑎𝑥{𝑁, 𝐿 ∙ 𝑅𝑀𝐴𝑋}). Finally,

the graph describing the network consists of a single connected component, so that 𝐿 ≥ 𝑁 − 1.

Thus the combined computational cost simplifies to 𝑂(𝑃3 ∙ 𝐿 ∙ 𝑅𝑀𝐴𝑋). We observe that the

combined cost is linear in the number of FDTLs, while “𝑃” and “𝑅𝑀𝐴𝑋” are independent of the size

of the network. Moreover, the level of connectivity of most electric power networks is such that

𝐿

𝑁≤ 2 (sparsely connected networks). In fact, for the IEEE 5-bus, 9-bus, 14-bus, 30-bus and

57-bus networks this ratio does not exceed 1.5. Thus, we conclude that the total cost of our subband

procedure is also linear in size of the network, which can be quantified either in terms of the number

of FDTLs, or in terms of the number of buses.

120

At each time-instant, the computational savings per subband are of the order of the

multiplicative factor “1

𝑟” (the smaller the better), where 𝑟 ≜

𝑅𝑀𝐴𝑋(𝑏𝑟𝑜𝑎𝑑𝑏𝑎𝑛𝑑)

𝑅𝑀𝐴𝑋(𝑠𝑢𝑏𝑏𝑎𝑛𝑑) . Subband transient

simulation allows us to use low-order RFAs, so that 𝑅𝑀𝐴𝑋(𝑠𝑢𝑏𝑏𝑎𝑛𝑑)

≪ 𝑅𝑀𝐴𝑋(𝑏𝑟𝑜𝑎𝑑𝑏𝑎𝑛𝑑)

and thus 𝑟 ≫ 1

allowing for significant savings in computation. Furthermore, 𝑀-subband transient simulation

allows us to employ parallel/distributed processing frameworks so that the desired broadband

transient responses (voltages, currents) are obtained “𝑀” times faster compared to a traditional

broadband (single-threaded) simulator.

Finally, we need to account for the overhead associated with setting-up a parallel job and

filtering through the analysis and synthesis filters. If the time duration of the simulation is

sufficiently large, then the overhead associated with setting up the parallel jobs and filtering through

the PR-FB becomes negligible.

In the next section (Sec. 6.8) we provide a collection of single-phase and three-phase FDTL

networks that illustrate the use of the general procedure outlined here.

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6.8 Y-matrix based FDTL simulation results

In this section we provide simulation results utilizing the single-phase and multi-phase FDTLs as

part of simple (2-node) and more complicated networks (more than two-nodes).

Simulation #1: Single-Phase FDTL (2-nodes)

We shall explore the following network (Fig. 6.18) which consists of (i) a single voltage source,

(ii) a single-phase FDTL, (iii) a load termination. The purpose of this example is to verify that we

can obtain the same results as with the non-scalable, analytical (transfer function) based approach

of (2.12, 2.13).

Figure 6.18: Two terminal, continuous-time, single-phase FDTL network

Figure 6.19: Two terminal, discrete-time equivalent, single-phase FDTL network

122

After transforming the voltage source to a current source, and substituting the discrete-time

equivalents for each component, we have the circuit shown in Fig. 6.19.

This circuit is characterized by the two master nodes {𝑘,𝑚} and following network

equations:

[𝑌𝑠 + 𝑌𝑇𝑂𝑇 0

0 𝑌𝐿 + 𝑌𝑇𝑂𝑇] ∙ [

𝑣𝑘[𝑛]

𝑣𝑚[𝑚] ] = [

ℎ𝑇𝑂𝑇(𝑘) [𝑛 − 1] + 𝑦(𝑚)[𝑛]

ℎ𝑇𝑂𝑇(𝑚) [𝑛 − 1] + 𝑦(𝑘)[𝑛]

]

If 𝑌𝐿 → 0 we have an open-ended line. The following figure (Fig. 6.20) matches the results from

(2.12).

Figure 6.20: Receiving-end voltage: simulation based on the: (i) full order model broadband

(blue), (ii) low order subband model (red).

If 𝑌𝐿 → +∞ we have a short-ended line. The following figure (Fig. 6.21) matches the results from

(2.13).

123

Figure 6.21: Receiving-end current: simulation based on the: (i) full order model broadband (blue), (ii) low order subband model (red).

The simulation results agree with Chapter 5. The main difference is that we employ the

(scalable) 𝑌-matrix network approach instead of the analytical (transfer function based) relations

of (2.12, 2.13).

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Simulation #2: IEEE 5-Bus FDTL system (single-phase)

We shall explore the following network which corresponds to the IEEE 5-Bus system topology

(Fig. 6.22).

Figure 6.22: IEEE 5-Bus network topology

The red-numbers enclosed in a square indicate the master node numbers associated with this IEEE

5-Bus system.

In addition, we may annotate the FDTLs as follows (Fig. 6.23) in order to avoid producing

a detailed equivalent circuit as the one presented in Fig. 6.19.

(i) Blue numbers enclosed by a circle indicate the FDTL number.

(ii) {𝑘,𝑚} correspond to the sending-end and receiving-end nodes used for each FDTL.

125

Figure 6.23: IEEE 5-Bus network topology with FDTL annotations

Obtaining analytical relations of the form (2.12, 2.13) for this network is a non-trivial task. Using

the scalable 𝑌-matrix based approach developed in this chapter results in the following relations

(which may be obtained by visual inspection of Fig. 6.23)

𝒀 ∙ 𝒗[𝑛] = 𝑰𝑠[𝑛] + 𝒉𝑇𝑂𝑇[𝑛 − 1] + 𝒚[𝑛] + 𝒉𝐿𝐶[𝑛 − 1] (6.37𝑎)

Where 𝑰𝑠[𝑛] = [ 𝐼𝑠1 , 𝐼𝑠2 , 𝐼𝑠3 , 𝐼𝑠4 , 𝐼𝑠5 ]𝑇 𝜖 ℝ5×1 is the input current-source vector of this topology,

with

𝐼𝑠1 = 𝑌𝐺1 ∙ 𝑉𝐺1[𝑛] , 𝐼𝑠2 = 𝑌𝐺2 ∙ 𝑉𝐺2[𝑛] , 𝐼𝑠3 = 𝐼𝑠4 = 𝐼𝑠5 = 0.

The (time-independent) admittance matrix associated with this network topology is:

𝒀 = 𝑑𝑖𝑎𝑔{𝑌1, 𝑌2, 𝑌3, 𝑌4 , 𝑌5} 𝜖 ℝ5×5 (6.37𝑏)

With:

𝑌1 = 𝑌𝑇𝑂𝑇(1)+ 𝑌𝑇𝑂𝑇

(2)+ 𝑌𝐺1 (6.37𝑐)


(3)+ 𝑌𝑇𝑂𝑇


(5)+ 𝑌𝐺2 + 𝑌𝐿2 (6.37𝑑)



(6)+ 𝑌𝐿3 (6.37𝑒)



(7)+ 𝑌𝐿4 (6.37𝑓)


(7)+ 𝑌𝐿5 (6.37𝑔)

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The (Foster-form) RC sub-circuit update equation vector associated with the characteristic

impedances of the FDTLs based on (6.12) is shown below (6.38):

𝒉𝑇𝑂𝑇[𝑛 − 1] = [ ℎ𝑇1 , ℎ𝑇2 , ℎ𝑇3 , ℎ𝑇4 , ℎ𝑇5 ]𝑇 𝜖 ℝ5×1 (6.38𝑎)

ℎ𝑇𝑗 ∶= ℎ𝑇𝑗[𝑛 − 1] , { 𝑗 = 1,2,3,4,5 } (6.38𝑏)

ℎ𝑇1 = ℎ𝑇𝑂𝑇(𝑘1)[𝑛 − 1] + ℎ𝑇𝑂𝑇

(𝑘2)[𝑛 − 1] (6.38𝑐)

ℎ𝑇2 = ℎ𝑇𝑂𝑇(𝑚1)[𝑛 − 1] + ℎ𝑇𝑂𝑇

(𝑘3)[𝑛 − 1] + ℎ𝑇𝑂𝑇(𝑘4)[𝑛 − 1] + ℎ𝑇𝑂𝑇

(𝑘5)[𝑛 − 1] (6.38𝑑)


(𝑚3)[𝑛 − 1] + ℎ𝑇𝑂𝑇(𝑘6)[𝑛 − 1] (6.38𝑒)


(𝑚6)[𝑛 − 1] + ℎ𝑇𝑂𝑇(𝑘7)[𝑛 − 1] (6.38𝑓)


(𝑚7)[𝑛 − 1] (6.38𝑔)

The recursive convolution update equation vector associated with the propagation functions of the

FDTLs based on (6.18) is shown below (6.39):

𝒚[𝑛] = [ 𝑦𝑇1 , 𝑦𝑇2 , 𝑦𝑇3 , 𝑦𝑇4 , 𝑦𝑇5 ]𝑇 𝜖 ℝ5×1 (6.39𝑎)

𝑦𝑇𝑗 ∶= 𝑦𝑇𝑗[𝑛] , { 𝑗 = 1,2,3,4,5 } (6.39𝑏)

𝑦𝑇1 = 𝑦(𝑚1)[𝑛] + 𝑦(𝑚2)[𝑛] (6.39𝑐)

𝑦𝑇2 = 𝑦(𝑘1)[𝑛] + 𝑦(𝑚3)[𝑛] + 𝑦(𝑚4)[𝑛] + 𝑦(𝑚5)[𝑛] (6.39𝑑)

𝑦𝑇3 = 𝑦(𝑘2)[𝑛] + 𝑦(𝑘3)[𝑛] + 𝑦(𝑚6)[𝑛] (6.39𝑒)

𝑦𝑇4 = 𝑦(𝑘4)[𝑛] + 𝑦(𝑘6)[𝑛] + 𝑦(𝑚7)[𝑛] (6.39𝑓)

𝑦𝑇5 = 𝑦(𝑘5)[𝑛] + 𝑦(𝑘7)[𝑛] (6.39𝑔)

The update equation that corresponds to individual capacitive or inductive source/load

components based on (6.4, 6.6) is shown below (6.40). The elements of (6.40) are all zero because

we consider purely resistive (memoryless) terminations for this example.

𝒉𝐿𝐶[𝑛 − 1] = [ 0 , 0 , 0 , 0 , 0 ]𝑇 𝜖 ℝ5×1 (6.40)

127

In summary, (6.37a) corresponds to the desired (scalable) form of (6.1a) and (6.38-6.40)

along with (6.12h-j, 6.18, 6.4d, 6.6d) correspond to the desired (scalable) form of (6.1b).

The order of updating the equations associated with this network are as follows (Section 6.7):

(i) Construct the 𝑌-matrix (once) according to (6.37b-g).

(ii) Solve (6.37a) for 𝒗[𝑛]. This is the “master circuit” equation as defined in (6.1a).

(iii) Update the characteristic impedance terms of the FDTL according to (6.38) and (6.12).

These equations depend on 𝒗[𝑛]. These are “sub-circuit” equations as defined in (6.1b).

(iv) Update the propagation matrix terms of the FDTL according to (6.39) and (6.18). These

equations depend on 𝒗[𝑛]. These are “sub-circuit” equations as defined in (6.1b).

(v) Update the capacitive/inductive load terms according to (6.40) and (6.4, 6.6). These

equations depend on 𝒗[𝑛]. These are “sub-circuit” equations as defined in (6.1b).

(vi) Go back to step (ii) “master-circuit” and repeat for each simulation step.

The outlined procedure is “scalable” in the sense that the “master-circuit” equation is straight-

forward to obtain for an arbitrarily large network by visual inspection. The “sub-circuit” equations

are decoupled from the “master-circuit” in the sense that they only depend on their corresponding

node-voltage and previous states (6.1b).

We observe that the broadband-based simulation matches the one based on subband vector fitting

(Figs. 6.24-6.26).

128

Figure 6.24: IEEE 5-Bus, node-3 voltage: (a) broadband simulation (blue), (b) low-order subband

simulation (red)

Figure 6.25: IEEE 5-Bus, node-4 voltage: (a) broadband simulation (blue), (b) low-order subband

simulation (red)

129

Figure 6.26: IEEE 5-Bus, node-3 current: (a) broadband simulation (blue), (b) low-order subband

simulation (red)

130

Simulation #3: Three-Phase FDTL (2-nodes)

We shall explore the following network (Fig. 6.27) which consists of (i) a single 3-phase voltage

source, (ii) a 3-phase FDTL, (iii) a 3-phase load termination. The purpose of this example is to

verify that we can obtain the same results as with the non-scalable, analytical (transfer function)

based approach of (2.9, 2.10).

Figure 6.27: Two terminal, continuous-time, 3-phase FDTL network

After transforming the voltage source to a current source, and substituting the discrete-time

equivalents for each component, we have the following circuit (Fig. 6.28).

Figure 6.28: Two terminal, discrete-time equivalent, 3-phase FDTL network

This circuit is characterized by the two master nodes {𝑘,𝑚} and following network equations:

[𝒀𝑠 + 𝒀𝑇𝑂𝑇 0

0 𝒀𝐿 + 𝒀𝑇𝑂𝑇] ∙ [

𝒗𝑘[𝑛]

𝒗𝑚[𝑚] ] = [

𝒉𝑇𝑂𝑇(𝑘) [𝑛 − 1] + 𝒉𝐶𝑂𝑁𝑉

(𝑚) [𝑛 − 1]

𝒉𝑇𝑂𝑇(𝑚) [𝑛 − 1] + 𝒉𝐶𝑂𝑁𝑉

(𝑘) [𝑛 − 1]]

131

If 𝒀𝐿 → 𝟎 we have an open-ended line. The following figure (Fig. 6.29) matches the results from

(2.9).

Figure 6.29: Three-phase receiving-end voltage: simulation based on the: (i) full order model broadband (blue), (ii) low order subband model (red).

If 𝒀𝐿 → +∞ we have a short-ended line. The following figure (Fig. 6.30) matches the results from

(2.10).

Figure 6.30: Three-phase receiving-end current: simulation based on the: (i) full order model

broadband (blue), (ii) low order subband model (red).

CHAPTER 7

Concluding Remarks

7.1 Summary of dissertation contributions

The main contributions of this dissertation are summarized below. Publications directly

related to this dissertation are [11, 12, 27-29].

Decomposition of the broadband transient simulation problem into a collection of

decoupled narrow-band problems. Since each subband can be simulated independently of

the rest, the simulation can be carried out in parallel, possibly using distributed

computational frameworks.

Independent analysis of transient responses at each subband. We may isolate a specific

narrow-band voltage/current transient response and independently study its contribution to

the broadband simulation. This may allow us to classify certain transient simulations

according to their frequency content.

Reduction of the RFA order in each subband, due to the narrow subband bandwidth. This

results in: (a) computational cost reduction, as compared to the standard (broadband)

simulation approach, (b) numerical stability robustness due to the improved (not as tightly

clustered) pole pattern of the subband RFA.

Introduction of the prototype analysis filter frequency response (𝐻0(𝑧) in the wavelet

version) as a customizable mean of controlling the quality of simulation and the choice of

RFA order. This constitutes a tradeoff between simulation performance and computational

cost.

133

Efficient discretization of the continuous-time FDTL model (Sec. 2.3). In particular,

discretization of the subband RFAs is carried out using the “prewarped Tustin” bilinear

transform (2.29), which allows us to leverage the stability and frequency representation

properties of the classical (DC-centered) Tustin transform while maintaining accuracy at a

particular frequency of interest 𝜔0 via prewarping. Accuracy around a particular frequency

𝜔0 ≠ 0 is a very desirable characteristic when we decompose a broadband simulation to

several narrowband sub simulations.

Derivation of a 𝑌-matrix equivalent circuit (complete with recursive time-update

equations) for the characteristic impedance and (delay-compensated) propagation function

of a polyphase, non-transposed FDTL.

Derivation of a 𝑌-matrix equivalent circuit (complete with recursive time-update

equations) for polyphase, (lumped) passive components (resistor, capacitor, inductor).

Development of a complete subband-domain electromagnetic transient simulation program

(SD-EMTP) for arbitrarily sized (and structured) network of buses, FDTLs, generators and

loads as demonstrated using the IEEE 5-Bus network topology (Sec. 6.8).

Demonstration of the efficiency of SD-EMTP by means of computational savings per-

subband and utilization of parallel/distributed computational frameworks (Sec. 6.7).

Demonstration of linear-scalability of the SD-EMTP to grids of any size and structure.

Construction of a novel efficient (uniform subband) filter bank, as a customizable

alternative of the paraunitary QMF filter bank (Phasor banks, Sec. 3.2).

134

7.2 Further Research Topics

As part of the future/further research section we propose to investigate the following:

Model subband network/topology changes using the method of equivalent current

injection.

Complete validation of our subband procedure on a “larger-scale” 3-phase (or multi-phase)

network subband simulation (similar to the IEEE 5-Bus system of Chapter 6).

Provide detailed computational cost comparison between the broadband and subband

transient simulations.

Analyze/optimize selection of the prototype analysis filter 𝐻0(𝑧).

Analyze/optimize RFA order selection for each subband.

Examine the effect of network non-linearities.

APPENDIX I

Modal Transform in a Transposed Transmission Line

A transposed transmission line has characteristics that remain invariant under flipping of

any two phases. This means that its 𝒁(𝑠) and 𝒀(𝑠) matrices have the general form

[𝑎 𝑏 𝑏𝑏 𝑎 𝑏𝑏 𝑏 𝑎

] ≜ (𝑎 − 𝑏) ∙ 𝑰 + 𝑏 ∙ [111] ∙ [1 1 1] (𝐴1.1)

where 𝑰 ∈ ℝ3×3 is the identity matrix and their product has the same form.

Since this 3 × 3 matrix is circulant, its eigenvalues and eigenvectors can be determined from the

properties of such matrices [16]. However, the eigenvectors are highly non-unique. For instance:

In addition to the columns of a 3 × 3 DFT matrix, we can also choose the eigenvectors from the

Clarke transform, and we can even have choices that contain only 1, 0, and -1 elements.

In general, the eigenvalue/eigenvector equation which corresponds to (A1.1) is:

(𝑎 − 𝑏) ∙ 𝒙 + 𝑏 ∙ [111] ∙ [1 1 1] ∙ 𝒙 = 𝜆 ∙ 𝒙 (𝐴1.2)

or equivalently:

(𝜆 + 𝑏 − 𝑎) ∙ 𝒙 = 𝑏 ∙ [111] ∙ {[1 1 1] ∙ 𝒙} (𝐴1.3)

This equation has two distinct solutions:

𝜆 + 𝑏 − 𝑎 ≠ 0 ⇒ 𝜆 = 𝑎 − 𝑏 + 𝑁 ∙ 𝑛 ↔ 𝑥~ [111] (𝐴1.4𝑎)

𝜆 = 𝑎 − 𝑏 ↔ 𝑥 ⊥ [111] (𝐴1.4𝑏)

where 𝑁 = 3. Thus we have a single eigenvalue defined by (A1.4a), and (𝑁 − 1) identical

eigenvalues (𝑎 − 𝑏) whose corresponding eigenvectors reside in the orthogonal complement of

136

𝑠𝑝𝑎𝑛 { [111] }. The choices include: (i) the DFT matrix:

1

√3∙ [

1 1 1

1 𝑒−𝑗2𝜋

3 𝑒−𝑗2𝜋

3

1 𝑒−𝑗2𝜋

3 𝑒−𝑗2𝜋

3

], (ii) the Clarke

matrix (transposed and rearranged): √2

3∙

[ √2

21 0

√2

2−1

2

√3

2

√2

2−1

2−√3

2 ]

, and (iii) many others.

In general the matrix constructed from the eigenvectors has the form: [𝑎 𝑎1 𝑏1𝑎 𝑎2 𝑏2𝑎 𝑎3 𝑏3

] where:

(i) 𝑎1 + 𝑎2 + 𝑎3 = 0, and (ii) 𝑏1 + 𝑏2 + 𝑏3 = 0 are needed to satisfy the constraint in (A1.4b).

Thus we still have 5 −degrees of freedom and a great amount of flexibility.

If we restrict the eigenvector matrix to be real-valued and orthogonal, then 𝑎 = ±1

√3 and

we get a set of 5 −constraints, namely:

𝑎1 + 𝑎2 + 𝑎3 = 0 (𝐴1.5𝑎)

𝑏1 + 𝑏2 + 𝑏3 = 0 (𝐴1.5𝑏)

𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 = 0 (𝐴1.5𝑐)

𝑎12 + 𝑎2

2 + 𝑎32 = 1 (𝐴1.5𝑑)

𝑏12 + 𝑏2

2 + 𝑏32 = 1 (𝐴1.5𝑒)

so that we have only 1 −degree of freedom left (5 −equations in 6 −unknowns). If we set, for

instance 𝑎1 = 0 (or 𝑎2 = 0) we get the Clarke transform. ∎

In general, we can choose to express all variables in terms of 𝑎1. Thus 𝑎3 = −(𝑎1 + 𝑎2),

𝑏3 = −(𝑏1 + 𝑏2) and so 𝑎1𝑏1 + 𝑎2𝑏2 + (𝑎1 + 𝑎2)(𝑏1 + 𝑏2) = 0 from which we can extract 𝑏2,

viz., (2𝑎2 + 𝑎1)𝑏2 = −(2𝑎1 + 𝑎2)𝑏1 or

𝑏2 = −𝜉 ∙ 𝑏1 (𝐴1.6𝑎)

𝜉 ≜2𝑎1 + 𝑎22𝑎2 + 𝑎1

(𝐴1.6𝑏)

137

Now 𝑏12 + (𝜉 ∙ 𝑏1)

2 + [(1 − 𝜉) ∙ 𝑏1]2 = 1 or 2 ∙ 𝑏1

2 ∙ (1 − 𝜉 + 𝜉2) = 1⇒

𝑏1 = ±1

√2 ∙ (1 − 𝜉 + 𝜉2) (𝐴1.7)

Finally, 𝑎12 + 𝑎2

2 + (𝑎1 + 𝑎2)2 = 1 results in the quadratic equation (in terms of 𝑎2):

2 ∙ 𝑎12 + 2 ∙ 𝑎1 ∙ 𝑎2 + 2 ∙ 𝑎2

2 = 1 ⇒

𝑎22 + 𝑎1 ∙ 𝑎2 + (𝑎1

2 −1

2) = 0 ⇒

𝑎2 =1

2∙ [−𝑎1 ±√𝑎1

2 − 4 ∙ (𝑎12 −

1

2)] ⇒

𝑎2 =1

2∙ [−𝑎1 ±√2− 3 ∙ 𝑎1

2] (𝐴1.8)

Since we have restricted our discussion to real-valued eigenvectors, we must have:

|𝑎1| ≤ √2

3 (𝐴1.9)

This assumption also guarantees that 𝜉 =2𝑎1+𝑎2

2𝑎2+𝑎1 is well defined.

Indeed: 2 ∙ 𝑎2 + 𝑎1 = [−𝑎1 ±√2 − 3 ∙ 𝑎12] + 𝑎1 = ±√2− 3 ∙ 𝑎1

2 ≠ 0 for |𝑎1| < √2

3 so that the

option of setting |𝑎1| = √2

3 is excluded.

In summary,

|𝑎1| < √2

3 (𝐴1.10𝑎)

𝑎2 =1

2∙ [−𝑎1 ± √2− 3 ∙ 𝑎1

2] (𝐴1.10𝑏)

𝑎3 = −(𝑎1 + 𝑎2) (𝐴1.10𝑐)

𝑏1 = ±1

√2 ∙ (1 − 𝜉 + 𝜉2) (𝐴1.10𝑑)

138

𝜉 =2𝑎1 + 𝑎22𝑎2 + 𝑎1

(𝐴1.10𝑒)

𝑏2 = −𝜉 ∙ 𝑏1 (𝐴1.10𝑓)

𝑏3 = −(𝑏1 + 𝑏2) = (𝜉 − 1) ∙ 𝑏1 (𝐴1.10𝑔)

If we now set: 𝑎1 = 0, we get:

𝑎2 = ±√2

2 , 𝑎3 = −𝑎2 , 𝜉 = +

1

2 , 𝑏2 = −

1

2∙ 𝑏1 , 𝑏1 = ±√

2

3 , 𝑏3 = −

1

2∙ 𝑏1

The choices of signs give us 8 −posibilities which are all the Clarke transform, namely:

[±1 0 00 ±1 00 0 ±1

] ∙

[

1

√30 √

2

3

1

√3

√2

2−1

2∙ √2

3

1

√3−√2

2−1

2∙ √2

3

]

(𝐴1.11)

In addition we can exchange the roles of the 2nd and 3rd columns, and we can permute the order of

all 3 −rows: all these are essentially the same transformation. ∎

APPENDIX II

Relations between Real and Imaginary Parts of a Causal Transfer Function

Let ℎ(𝑡) be a real-valued, causal and stable impulse response of an LTI system, with

transfer function 𝐻(𝑠) so that it is analytic in 𝑅𝑒{𝑠} > −𝑎 for some 𝑎 > 0. Define the symmetric

and skew-symmetric components of ℎ(𝑡), i.e.,

ℎ𝑒(𝑡) ≜1

2∙ [ℎ(𝑡) + ℎ(−𝑡)] (𝐴2.1𝑎)

ℎ𝑜(𝑡) ≜1

2∙ [ℎ(𝑡) − ℎ(−𝑡)] (𝐴2.1𝑏)

where ℎ𝑒(0) = ℎ(0) and ℎ0(0) = 0. The Fourier transform of (A2.1) yields:

𝐻𝑒(𝑗𝜔) =1

2∙ [ 𝐻(𝑗𝜔) + 𝐻∗(𝑗𝜔) ] = 𝑅𝑒{ 𝐻(𝑗𝜔) } (𝐴2.2𝑎)

𝐻𝑜(𝑗𝜔) =1

2∙ [ 𝐻(𝑗𝜔) − 𝐻∗(𝑗𝜔) ] = 𝑗 ∙ 𝐼𝑚{ 𝐻(𝑗𝜔) } (𝐴2.2𝑏)

where “*” denotes the complex conjugate and ∫ ℎ(−𝑡)𝑒−𝑗𝜔𝑡𝑑𝑡+∞

−∞= ∫ ℎ(𝑢)𝑒𝑗𝜔𝑢𝑑𝑢

+∞

−∞= 𝐻∗(𝑗𝜔)

for a real-valued function ℎ(𝑡). Since ℎ(𝑡) is one sided, we have

𝑠𝑖𝑔𝑛(𝑡) ∙ ℎ(𝑡) = {ℎ(𝑡), 𝑡 > 0 0, 𝑡 ≤ 0

(𝐴2.3)

because, by definition, 𝑠𝑖𝑔𝑛(0) = 0. Moreover:

𝑠𝑖𝑔𝑛(𝑡) ∙ ℎ(−𝑡) = { 0, 𝑡 ≥ 0 −ℎ(−𝑡), 𝑡 < 0

(𝐴2.4)

In other words 𝑠𝑖𝑔𝑛(𝑡) ∙ ℎ(𝑡) = ℎ(𝑡) and 𝑠𝑖𝑔𝑛(𝑡) ∙ ℎ(−𝑡) = −ℎ(−𝑡) except for 𝑡 = 0.

Consequently, for all 𝑡,

𝑠𝑖𝑔𝑛(𝑡) ∙ ℎ𝑒(𝑡) =1

2∙ [ ℎ(𝑡) − ℎ(−𝑡) ] = ℎ𝑜(𝑡) (𝐴2.5)

140

Mapping this into the frequency domain results in

ℱ{ 𝑠𝑖𝑔𝑛(𝑡) } =2

𝑗𝜔≜ 𝐺(𝑗𝜔) (𝐴2.6)

and

𝐻𝑜(𝑗𝜔) = 𝐺(𝑗𝜔) ∗ 𝐻𝑒(𝑗𝜔) = ∫2

𝑗(𝜔 − 𝑢)

+∞

−∞

∙ 𝐻𝑒(𝑗𝜔)𝑑𝑢

2𝜋 (𝐴2.7)

where “*” denotes linear convolution. This provides us with a relation between 𝑅𝑒{ 𝐻(𝑗𝜔) } and

𝐼𝑚{ 𝐻(𝑗𝜔) } viz.,

𝐼𝑚{ 𝐻(𝑗𝜔) } = −1

𝜋∫

1

𝜔 − 𝑢∙ 𝑅𝑒{ 𝐻(𝑗𝑢) }𝑑𝑢

+∞

−∞

(𝐴2.8)

Because of the singularity at 𝑢 = 𝜔 this integral has to be interpreted as the Cauchy

principal value. The relation described by (A2.8) can be converted to integration in [0, +∞) instead

of (−∞,+∞). Notice that

∫1


0

−∞

= ∫1

𝜔 + 𝑣∙ 𝑅𝑒{ 𝐻(−𝑗𝑣) }𝑑𝑣

+∞

0

(𝐴2.9)

But, since ℎ(𝑡) is real, 𝑅𝑒{ 𝐻(−𝑗𝑣)} = 𝑅𝑒{ 𝐻(𝑗𝑣)} so that

𝐼𝑚{ 𝐻(𝑗𝜔) } = −1

𝜋[∫

1


+∞

0

+∫1

𝜔 + 𝑢∙ 𝑅𝑒{ 𝐻(𝑗𝑢) }𝑑𝑢

+∞

0

] ⇒

𝐼𝑚{ 𝐻(𝑗𝜔) } =2𝜔

𝜋∫

1

𝑢2 −𝜔2∙ 𝑅𝑒{ 𝐻(𝑗𝑢) }𝑑𝑢

+∞

0

(𝐴2.10)

Now, suppose that 𝐻(𝑠) = 𝑒𝛾(𝑠) is a causal and stable transfer function. This means that

its region of convergence (ROC) is 𝑅𝑒{𝑠} > −𝑎 for some 𝑎 > 0. Under the assumption that 𝛾(𝑠)

can be viewed as the transfer function of a causal stable system we can apply (A2.10) to 𝛾(𝑗𝜔).

141

Thus, let 𝛾(𝑗𝜔) = 𝛼(𝑗𝜔) + 𝑗 ∙ 𝛽(𝑗𝜔), so that 𝛽(𝑗𝜔) =2𝜔

𝜋∫

1

𝑢2−𝜔2∙ 𝛼(𝑗𝑢)𝑑𝑢

+∞

0, where

𝛼(𝑗𝜔) = 𝑙𝑛| 𝐻(𝑗𝜔) |.

Integration by parts, and changing the integration variable to 𝑣 ≜ 𝑙𝑛 (𝑢

𝜔) results in

𝛽(𝑗𝜔) =1

𝜋∫𝑑𝑀

𝑑𝑣∙ 𝑙𝑛 (𝑐𝑜𝑡ℎ (

|𝑣|

2))𝑑𝑣

+∞

−∞

(𝐴2.11)

where 𝑀(𝑣) = 𝛼(𝑢)|𝑢=𝜔∙𝑒𝑣 = 𝛼(𝜔 ∙ 𝑒𝑣).

In summary for any stable minimum-phase system 𝐺(𝑗𝜔), the phase ∠𝐺(𝑗𝜔) is uniquely

related to the magnitude |𝐺(𝑗𝜔)| by:

∠𝐺(𝑗𝜔) =1

𝜋∫𝑑𝑀

𝑑𝑣∙ 𝑊(𝑣)𝑑𝑣

+∞

−∞

(𝐴2.12𝑎)

𝑀(𝑣) = 𝑙𝑛 ( |𝐺(𝑢)||𝑢=𝜔∙𝑒𝑣

) = 𝑙𝑛( |𝐺(𝜔 ∙ 𝑒𝑣)| ) (𝐴2.12𝑏)

𝑊(𝑣) = 𝑙𝑛(𝑐𝑜𝑡ℎ (|𝑣|

2)) (𝐴2.12𝑐)

𝑣 ≜ 𝑙𝑛 (𝑢

𝜔) (𝐴2.12𝑑)

This result is known as Bode’s Gain-Phase Relation. ∎

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