Transmission Lines and Lumped Circuits: Fundamentals and Applications
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Table of contents
- Contentsvii
- Forewordxix
- Prefacexxi
- Introduction1
- Chapter 1. Transmission Line Equations and Properties15
- 1.1 Transmission Line Model16
- 1.2 Two-Conductor Transmission Line Equations21
- 1.3 Multiconductor Transmission Line Equations26
- 1.4 Poynting's Theorem for Lines with Frequency Independent Parameters32
- 1.5 Uniqueness of the Solution of Transmission Line Equations34
- 1.6 Poynting's Theorem for Lines in the Frequency Domain37
- 1.7 Uniqueness of the Solution of Transmission Line Equations with Frequency-Dependent Parameters39
- 1.8 Transmission Line Equations in the Laplacde Domain41
- 1.9 Reciprocity Theorems for Two-Conductor Transmission Lines42
- 1.10 Reciprocity Theorems for Multiconductor Transmission Lines44
- Chapter 2. Ideal Two-Conductor Transmission Lines Connected to Lumped Circuits49
- 2.1 d'Alembert Solution of Two-Conductor Transmission Line Equations50
- 2.2 Some Elementary Networks53
- 2.3 Natural Frequencies of a Finite Length Transmission Line Connected to Short Circuits64
- 2.4 Two-Conductor Transmission Lines as Two-Ports66
- 2.5 The Input-Output Description71
- 2.6 The Input-State-Output Description, and Equivalent Circuits of Thévenin and Norton Type72
- 2.7 Lines Connected to Linear Lumped Circuits75
- 2.8 A Glimpse at a Transmission Line Connected to a Nonlinear One-Port: State Equations in Normal Fo84
- 2.9 Ideal Two-Conductor Transmission Lines with Distributed Sources88
- Chapter 3. Ideal Multiconductor Transmission Lines93
- 3.1 d’Alembert Solution for Ideal Multiconductor Transmission Lines93
- 3.2 Infinite Multiconductor Transmission Lines103
- 3.3 Semi-infinite Multiconductor Transmission Lines and Equivalent Circuits104
- 3.4 Ideal Multiconductor Transmission Lines as Multiports106
- 3.5 The Input-State-Output Description and the Equivalent Circuits of Thévenin and Norton Type112
- 3.6 Multiconductor Lines with Homogeneous Dielectric115
- 3.7 Multiconductor Transmission Line Connected to Linear Resistive Multiports117
- 3.8 A Particular Solution of the Ideal Multiconductor Transmission Line Equations with Distributed S121
- 3.9 Properties of the Characteristic Conductance Matrix Gc and Resistance Matrix Rc125
- Chapter 4. Lossy Two-Conductor Transmission Lines129
- 4.1 Lossy Transmission Lines are Dispersive130
- 4.2 Solution of the Lossy Transmission Line Equations in the Laplace Domain132
- 4.3 The Propagation Along a Lossy Transmission Line136
- 4.4 Semi-infinite Lossy Line Connected to an Ideal Current Source141
- 4.5 Representation of Lossy Two-Conductor Lines as Two-Ports148
- 4.6 The Input-State-Output Description154
- 4.7 Input-Output Descriptions in Explicit Form160
- 4.8 A Lossy Transmission Line Connecting Two Linear Resistive One-Ports168
- 4.9 The Matching Problem for Lossy Lines172
- 4.10 Lossy Transmission Lines with Distributed Sources174
- 4.11 Characterization of the Terminal Behavior of the Line Through the Scattering Parameters178
- Chapter 5. Lossy Two-Conductor Transmission Lines with Frequency-Dependent Parameters181
- 5.1 Introduction181
- 5.2 Frequency Behavior of the Per-Unit-Length Admittance Y(184
- 5.3 Frequency Behavior of the Per-Unit-Length Impedance Z(s)193
- 5.4 Properties of the Describing Functions P(s), Zc(s), and Yc(s)200
- 5.5 Qualitative Behavior of the Impulse Responses p(t), Zc(t), and Yc(s)206
- Chapter 6. Lossy Multiconductor Transmission Lines215
- 6.1 Introduction215
- 6.2 Lossy Multiconductor Lines Exhibiting a Structural Symmetry217
- 6.3 Lossy Multiconductor Line Equations in the Laplace Domain219
- 6.4 Lossy Multiconductor Transmission Line as Multiports in the Laplace Domain224
- 6.5 The Input-State-Output Description and the Equivalent Representations of Thévenin and Norton Ty228
- 6.6 Input-Output Descriptions in Explicit Form231
- 6.7 The Problem of the Inverse Laplace Transform of the Matrix Operators P(s), Zc(s), and Yc(s)235
- 6.8 Study of the Asymptotic Behavior of the Matrix Operator A(s) Through the Rayleigh-Schrödinger M240
- 6.9 Asymptotic Expressions for the Matrix Operators A(s) and Tv(s)246
- 6.10 Evaluation of the Impulse Responses for Lossy Multiconductor Lines with Frequency-Independent P247
- 6.11 Evaluation of the Impulse Responses of Lossy Multiconductor Lines with Frequency-Dependent Para254
- Chapter 7. Nonuniform Transmission Lines265
- 7.1 Introduction265
- 7.2 Equations for Nonuniform Lossless Transmission Lines268
- 7.3 Analytical Solutions for Lines with Transversally Homogeneous Dielectric and Particular Profiles271
- 7.4 Representation of Nonuniform Transmission Lines as Two-Ports in the Laplace Domain279
- 7.5 The Equivalent Circuit of Thévenin Type in the Time Domain285
- 7.6 The Solution of the Line Equations for a Generic Profile of L(x) and C(x)295
- Chapter 8. Transmission Line Equations in Characteristic Form305
- 8.1 Introduction305
- 8.2 A First-order Wave Equation in Characteristic Form and the Characteristic Curves306
- 8.3 The Characteristic Form Equations for Lines with Frequency Independent Parameters312
- 8.4 The Characteristic Form Equations for Lines with Frequency-Dependent Parameters323
- 8.5 Characteristic Equations for Multiconductor Lines329
- 8.6 Stepwise Integration of the Transmission Line Equations in Characteristic Form332
- Chapter 9. Lumped Nonlinear Networks Interconnected by Transmission Lines337
- 9.1 Introduction337
- 9.2 Time Domain Formulation of the Network Equations338
- 9.3 A Glimpse at the Uniqueness Problem for Ideal Two-Conductor Transmission Lines341
- 9.4 A Glimpse at the Uniqueness Problem for Imperfect Two-Conductor Transmission Lines: Associated R349
- 9.5 A Glimpse at the Numerical Solution for Imperfect Two-Conductor Transmission Lines: Associated D354
- 9.6 Well-Posedness of the Network Equations363
- 9.7 Numerical Solution of the Network Equations368
- 9.8 Lumped Circuits Connected Through Multiconductor Transmission Lines371
- Chapter 10 Qualitative Analysis of an Ideal Two-Conductor Line Connected to Nonlinear Resistors: Per377
- 10.1 Introduction377
- 10.2 State Equations in Normal Form for an Ideal Two-Conductor Line Connected to Nonlinear Resistors380
- 10.3 A Glimpse at the Scalar Maps384
- 10.4 Passivity, Eventual Passivity, and Local Passivity395
- 10.5 Some General Properties of the Dynamics398
- 10.6 Qualitative Behavior of the Solution for Locally Active Resistors: a Glimpse at the Bifurcation404
- 10.7 A Glimpse at the Behaviour of Noninvertible Maps: Chaotic Dynamics410
- 10.8 Lossy Transmission Lines429
- Appendix A. Some Useful Notes on the Matrix Operators435
- A1 Preliminary Definitions435
- A2 The Eigenvalue Problem Au = λu436
- A3 The Generalized Eigenvalue Problem Au = λBu439
- A4 Function of a Matrix Operator440
- A5 Perturbation of a Matrix Operator: Asymptotic Behavior of the Eigenvalues441
- Appendix B. Some Useful Notes on the Laplace Transformation445
- B1 General Considerations445
- B2 Asymptotic Behavior of the Object Function for t→∞450
- Appendix C. Some a-priori Estimates453
- C1 a-priori Estimates for the Solution of Equation (9.8)453
- C2 a-priori Estimates for the Solution of Equation (9.18)454
- Appendix D. Tables of Equivalent Representations of Transmission Lines457
- References463
- Index471
The theory of transmission lines is a classical topic of electrical engineering. Recently this topic has received renewed attention and has been a focus of considerable research. This is because the transmisson line theory has found new and important applications in the area of high-speed VLSI interconnects, while it has retained its significance in the area of power transmission. In many applications, transmission lines are connected to nonlinear circuits. For instance, interconnects of high-speed VLSI chips can be modelled as transmission lines loaded with nonlinear elements. These nonlinearities may lead to many new effects such as instability, chaos, generation of higher order harmonics, etc. The mathematical models of transmission lines with nonlinear loads consist of the linear partial differential equations describing the current and voltage dynamics along the lines together with the nonlinear boundary conditions imposed by the nonlinear loads connected to the lines. These nonlinear boundary conditions make the mathematical treatment very difficult. For this reason, the analysis of transmission lines with nonlinear loads has not been addressed adequately in the existing literature. The unique and distinct feature of the proposed book is that it will present systematic, comprehensive, and in-depth analysis of transmission lines with nonlinear loads.
* A unified approach for the analysis of networks composed of distributed and lumped circuits
* A simple, concise and completely general way to present the wave propagation on transmission lines, including a thorough study of the line equations in characteristic form
* Frequency and time domain multiport representations of any linear transmission line
* A detailed analysis of the influence on the line characterization of the frequency and space dependence of the line parameters
* A rigorous study of the properties of the analytical and numerical solutions of the network equations
* The associated discrete circuits and the associated resisitive circuits of transmission lines
* Periodic solutions, bifurcations and chaos in transmission lines connected to noninear lumped circuits
* A unified approach for the analysis of networks composed of distributed and lumped circuits
* A simple, concise and completely general way to present the wave propagation on transmission lines, including a thorough study of the line equations in characteristic form
* Frequency and time domain multiport representations of any linear transmission line
* A detailed analysis of the influence on the line characterization of the frequency and space dependence of the line parameters
* A rigorous study of the properties of the analytical and numerical solutions of the network equations
* The associated discrete circuits and the associated resisitive circuits of transmission lines
* Periodic solutions, bifurcations and chaos in transmission lines connected to noninear lumped circuits