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Tables of Contents for Transmission Lines and Lumped Circuits
Chapter/Section Title
Page #
Page Count
Foreword
xix
Preface
xxi
Introduction
1
2
A Survey of the Problem
3
2
The Idea of the Book
5
10
Transmission Line Equations and Properties
15
34
Transmission Line Model
16
5
Two-Conductor Transmission Line Equations
21
5
Ideal Transmission Lines
21
1
Lossy Transmission Lines
22
1
Nonuniform Transmission Lines
23
1
Transmission Lines with Distributed Sources
23
1
Transmission Lines with Frequency-Dependent Parameters
24
2
Multiconductor Transmission Line Equations
26
6
Ideal Multiconductor Transmission Lines
26
2
Lossy Multiconductor Transmission Lines
28
2
Multiconductor Transmission Lines with Distributed Sources
30
1
Multiconductor Transmission Lines with Frequency-Dependent Parameters
30
2
Poynthing's Theorem for Lines with Frequency Independent Parameters
32
2
Two-Conductor Transmission Lines
32
1
Multiconductor Transmission Lines
33
1
Uniqueness of the Solution of Transmission Lines Equations
34
3
Two-Conductor Transmission Lines
34
2
Multiconductor Transmission Lines
36
1
Poynting's Theorem for Lines in the Frequency Domain
37
2
Two-Conductor Transmission Lines
37
1
Multiconductor Transmission Lines
38
1
Uniqueness of the Solution of Transmission Lines Equations with Frequency-Dependent Parameters
39
2
Two-Conductor Transmission Lines
39
2
Multiconductor Transmission Lines
41
1
Transmission Line Equations in the Laplacde Domain
41
1
Reciprocity Theorems for Two-Conductor Transmission Lines
42
2
The First Form of the Reciprocity Theorem
42
2
The Second Form of the Reciprocity Theorem
44
1
The Third Form of the Reciprocity Theorem
44
1
Reciprocity Theorems for Multiconductor Transmission Lines
44
5
The First Form of the Reciprocity Theorem
45
2
The Second Form of the Reciprocity Theorem
47
1
The Third Form of the Reciprocity Theorem
47
1
Reciprocity Theorem for a Semi-infinite Transmission Lines
47
2
Ideal Two-Conductor Transmission Lines Connected to Lumped Circuits
49
44
d'Alembert Solution of Two-Conductor Transmission Lines Equations
50
3
Some Elementary Networks
53
11
An Infinite Line
53
1
A Semi-infinite Line Connected to an Ideal Current Source
54
2
A Semi-infinite Line Connected to a Linear Resistor; Reflection Coefficient
56
2
A Semi-infinite Line Connected to a Linear Capacitor
58
2
A Semi-infinite Line Connected to a Nonlinear Resistor
60
3
A Semi-infinite Line Connected to a Nonlinear Resistor in Parallel with a Linear Capacitor
63
1
Natural Frequencies of a Finite Length Transmission Lines Connected to Short Circuits
64
2
Two-Conductor Transmission Lines as Two-Ports
66
5
State Variables of the Line
67
1
Transmission Line Behaviour at the Ends
67
4
The Input-Output Description
71
1
The Input-State-Output Description, and Equivalent Circuits of Thevenin and Norton Type
72
3
Lines Connected to Linear Lumped Circuits
75
9
State Equations in Normal Form
77
1
Natural Frequencies of the Network
78
1
Solution in the Laplace Domain
79
5
A Glimpse at a Transmission Lines Connected to a Nonlinear One-Port: State Equations in Normal Form
84
4
A Line Connected to a Nonlinear Resistor
85
1
A Line Connected to a Nonlinear Resistor in Parallel with a Linear Capacitor
86
2
Ideal Two-Conductor Transmission Lines with Distributed Sources
88
5
A Particular Solution of the Line Equations With Distributed Sources
88
3
Characterization as Two-Ports
91
2
Ideal Multiconductor Transmission Lines
93
36
d'Alembert Solution for Ideal Multiconductor Transmission Lines
93
10
Properties and Diagonalization of the Matrices LC and CL
95
4
Characteristic Resistance and Conductance Matrices
99
2
Natural Modes of Propagation
101
2
Infinite Multiconductor Transmission Lines
103
1
Semi-infinite Multiconductor Transmission Lines and Equivalent Circuits
104
2
Ideal Multiconductor Transmission Lines as Multiports
106
6
Characterization of the Transmission Line Behavior at the Ends
108
4
The Input-State-Output Description and the Equivalent Circuits of Thevenin and Norton Type
112
3
Multiconductor Lines with Homogeneous Dielectric
115
2
Characterization of the Transmission Line Behavior at the Ends
116
1
Multiconductor Transmission Line Connected to Linear Resistive Multiports
117
4
Natural Frequencies of the Network
120
1
A Particular Solution of the Ideal Multiconductor Transmission Line Equations with Distributed Sources
121
4
Transversally Homogeneous Lines
124
1
Properties of the Characteristic Conductance Matrix Gc and Resistance Matrix Rc
125
4
Lossy Two-Conductor Transmission Lines
129
52
Lossy Transmission Lines are Dispersive
130
2
The Heaviside Condition
131
1
Solution of the Lossy Transmission Line Equations in the Laplace Domain
132
4
The Propagation Along a Lossy Transmission Line
136
5
Semi-infinite Lossy Line Connected to an Ideal Current Source
141
7
Properties of the Characteristic Impedance Zc(s) and of the Impulse Response Zc(t)
142
3
A Fast Convolution Algorithm
145
3
Representation of Lossy Two-Conductor Lines as Two-Ports
148
6
Terminal Behaviour of the Line in the Laplace Domain: the Describing Functions Zc(s) and P(s)
149
3
Properties of the Global Propagation Operator P(s) and of the Impulse Response p(t)
152
2
The Input-State-Output Description
154
6
Laplace Domain Equivalent Circuits of Thevenin and Norton Type
154
2
Time Domain Thevenin Description
156
2
Time Domain Norton Description
158
2
Input-Output Descriptions in Explicit Form
160
8
The Impedance Matrix
160
4
The Admittance Matrix
164
1
The Hybrid Matrices
165
1
The Transmission Matrices
166
2
A Lossy Transmission Line Connecting Two Linear Resistive One-Ports
168
4
Numerical Solution of Volterra Integral Equations of the Second Kind
171
1
The Matching Problem for Lossy Lines
172
2
Lossy Transmission Lines with Distributed Sources
174
4
Particular Solution of Line Equations with Distributed Sources
175
2
Two-port Characterisation
177
1
Characterization of the Terminal Behavior of the Line Through the Scattering Parameters
178
3
Lossy Two-Conductor Transmission Lines with Frequency-Dependent Parameters
181
34
Introduction
181
3
Frequency Behavior of the Per-Unit-Length Admittance Y(s)
184
9
Homogeneous Embedding Medium
187
3
Nonhomogeneous Embedding Medium
190
2
Asymptotic Expression of Y(s)
192
1
Frequency Behavior of the Per-Unit-Length Impedance Z(s)
193
7
Skin Effect
196
2
Anomalous Skin Effect
198
1
Superconducting Transmission Lines
199
1
A Single Wire Above a Finite Conductivity Ground plane
200
1
Properties of the Describing Functions P(s), Zc(s), and Yc(s)
200
6
Asymptotic Expression of Zc
203
1
Asymptotic Expression of P
204
1
Asymptotic Behaviour of Yc
205
1
Qualitative Behavior of the Impulse Responses p(t), zc(t), and Yc(s)
206
9
Inverse Laplace Transform of Functions Behaving as s-1 for s → ∞
206
1
Qualitative Behavior of the Impulse Responses zc and yc
207
1
Qualitative Behavior of the Impulse Response p
208
2
A Numerical Experiment
210
5
Lossy Multiconductor Transmission Lines
215
50
Introduction
215
2
Lossy Multiconductor Lines Exhibiting a Structural Symmetry
217
2
Lossy Multiconductor Line Equations in the Laplace Domain
219
5
The Eigenvalues and Eigenvectors of the Matrices Λ = ZY/s2 and Π = YZ/s2
219
2
The General Solution
221
3
Lossy Multiconductor Transmission Line as Multiports in the Laplace Domain
224
4
The Input-State-Output Description and the Equivalent Representations of Thevenin and Norton Type
228
3
Laplace Domain
228
2
Time Domain
230
1
Input-Output Descriptions in Explicit Form
231
4
The Impedance Matrix
231
1
The Admittance Matrix
232
1
The Hybrid Matrices
233
1
The Transmission Matrices
234
1
The Problem of the Inverse Laplace Transform of the Matrix Operators P(s), Zc(s), and Yc(s)
235
5
Study of the Asymptotic Behavior of the Matrix Operator Λ(s) Through the Rayleigh-Schrodinger Method
240
6
The Eigenvalues of Λ(0) = LC Are Nondegenerate
242
1
The Eigenvalues of Λ(0) = LC Are Degenerate
243
2
A Particular Case of Degeneracy: Lines with Transverse Homogeneous Dielectric
245
1
Asymptotic Expressions for the Matrix Operators A(s) and Tv(s)
246
1
Evaluation of the Impulse Responses for Lossy Multiconductor Lines with Frequency-Independent Parameters
247
7
Asymptotic Expressions for the Describing Functions P(s), Zc(s), and Yc(s)
248
2
Evaluation of the Principal Parts of the Impulse Responses p(t), zc(t), and yc(t)
250
2
An Application to a Three-Conductor Line
252
2
Evaluation of the Impulse Responses of Lossy Multiconductor Lines with Frequency-Dependent Parameters
254
11
Multiconductor Lines with Skin Effect
254
6
Application to a Three-Conductor Line
260
2
Per-Unit-Length Impedance and Admittance Matrices with Arbitrary Frequency Dependence
262
3
Nonuniform Transmission Lines
265
40
Introduction
265
4
Equations for Nonuniform Lossless Transmission Lines
The Dyson Series
269
2
Analytical Solutions for Lines with Transversally Homogeneous Dielectric and Particular Profiles of L(x)
271
8
Exponential Profile
271
2
Linear Profile
273
1
Gaussian Profile
274
3
General Solution of the Traveling Wave Type
277
2
Representation of Nonuniform Transmission Lines as Two-Ports in the Laplace Domain
279
6
Terminal Behavior of the Line
280
3
The Input-State-Output Description and an Equivalent Circuit of Thevenin Type
283
1
Input-Output Descriptions in Explicit Form
284
1
The Equivalent Circuit of Thevenin Type in the Time Domain
285
10
Asymptotic Behavior of the Describing Functions
286
2
Impulse Responses
288
7
The Solution of the Line Equations for a Generic Profile of L(x) and C(x)
295
10
Study of the Asymptotic Behavior of the Solution Through the Liouville-Green Transformation
295
3
Semianalytical Evaluation of Traveling Wave Solutions Based on the WKB Method
298
7
Transmission Line Equations in Characteristic Form
305
32
Introduction
305
1
A First-order Wave Equation in Characteristic Form and the Characteristic Curves
306
6
The Domain of Dependence of the Solution
309
1
The Transport of the Irregularities
309
3
The Characteristic Form Equations for Lines with Frequency Independent Parameters
312
11
The Domain of Dependence of the Solution
317
4
The Transport of the Discontinuities
321
1
The Riemann Variables
321
2
The Characteristic Form Equations for Lines with Frequency-Dependent Parameters
323
6
Characteristic Equations for Multiconductor Lines
329
3
Stepwise Integration of the Transmission Line Equations in Characteristic Form
332
5
Lumped Nonlinear Networks Interconnected by Transmission Lines
337
40
Introduction
337
1
Time Domain Formulation of the Network Equations
338
3
A Glimpse at the Uniqueness Problem for Ideal Two-Conductor Transmission Lines
341
8
An Ill-Posed Problem
344
1
A Circuit with an Additional Parasitic Reactance
345
4
A Glimpse at the Uniqueness Problem for Imperfect Two-Conductor Transmission Lines: Associated Resistive Circuit
349
5
The Volterra Integral Equation of the Second Kind in Normal Form
349
1
Uniqueness Condition for Imperfect Lines
350
2
A Circuit with an Additional Parasitic Reactance
352
2
A Glimpse at the Numerical Solution for Imperfect Two-Conductor Transmission Lines: Associated Discrete Circuit
354
9
Effects of an Additional Parasitic Reactance
358
2
Numerical Solution of an Ill-Posed Circuit by Artificially Enforcing the Continuity of the Terminal Voltage
360
3
Well-Posedness of the Network Equations
363
5
Numerical Solution of the Network Equations
368
3
Lumped Circuits Connected Through Multiconductor Transmission Lines
371
6
Associated Resistive Circuit
372
1
Lines Connecting Resistive Multiports
372
3
Associated Discrete Circuit
375
2
Qualitative Analysis of an Ideal Two-Conductor Line Connected to Nonlinear Resistors: Periodic Solutions, Bifurcations and Chaos
377
58
Introduction
377
3
State Equations in Normal Form for an Ideal Two-Conductor Line Connected to Nonlinear Resistors: Formulation in Terms of a Scalar Map un + 1 = f(un)
380
4
A Glimpse at the Scalar Maps
384
11
Stair-Step Diagrams
386
1
Linear Map
387
2
Affine Linear Map
389
1
Bounded Solutions and Stability
390
2
Steady-State Solution
392
3
Passivity, Eventual Passivity, and Local Passivity
395
3
Some General Properties of the Dynamics
398
6
Study of the Boundedness of the Solution Through the Liapunov Function Method
399
1
Local Behavior of the Map un + 1 = f(un)
400
2
Qualitative Behavior of the Solution for Locally Passive Resistors
402
2
Qualitative Behavior of the Solution for Locally Active Resistors: A Glimpse at the Bifurcations
404
6
Pitchwork Bifurcation of a Nonhyperbolic Fixed Point
407
1
Period-Doubling Bifurcation of a Nonhyperbolic Fixed Point
408
2
A Glimpse at the Behaviour of Noninvertible Maps: Chaotic Dynamics
410
19
A Tent Map
410
6
The Lyapunov Exponent and Chaotic Transients
416
2
A Unimodal Map
418
6
The Logistic Map
424
5
The Spatio-Temporal Chaos
429
1
Lossy Transmission Lines
429
6
Appendix A Some Useful Notes on the Matrix Operators
435
10
A1 Preliminary Definitions
435
1
A2 The Eigenvalue Problem Au = λu
436
3
A3 The Generalized Eigenvalue Problem Au = λBu
439
1
A4 Function of a Matrix Operator
440
1
A5 Perturbation of a Matrix Operator: Asymptotic Behavior of the Eigenvalues
441
4
Appendix B Some Useful Notes on the Laplace Transformation
445
8
B1 General Considerations
445
5
B2 Asymptotic Behavior of the Object Function for t → ∞
450
3
Appendix C Some a-priori Estimates
453
4
C1 a-priori Estimates for the Solution of Equation (9.8)
453
1
C2 a-priori Estimates for the Solution of Equation (9.18)
454
3
Appendix D Tables of Equivalent Representations of Transmission Lines
457
6
References
463
8
Index
471