Electrical Network Theory

1. Fundamental Concepts
 1.1 Introduction
 1.2 Elementary Matrix Algebra
  Basic Operations
  Types of Matrices
  Determinants
  The Inverse of a Matrix
  Pivotal Condensation
  Linear Equations
  Characteristic Equation
  Similarity
  Sylvester's Inequality
  Norm of a Vector
 1.3 Notation and References
 1.4 Network Classification
  Linearity
  Time-Invariance
  Passivity
  Reciprocity
 1.5 Network Components
  The Transformer
  The Gyrator
  Independent Sources
  Controlled or Dependent Sources
  The Negative Converter
 Problems

2. Graph Theory and Network Equations
 2.1 Introductory Concepts
  Kirchhoff's Laws
  Loop Equations
  Node Equations
  State Equations—A Mixed Set
  Solutions of Equations
 2.2 Linear Graphs
  Introductory Definitions
  The Incidence Matrix
  The Loop Matrix
  Relationships between Submatrices of A and B
  Cut-sets and the Cut-set Matrix
  Planar Graphs
 2.3 Basic Laws of Electric Networks
  Kirchhoff's Current Law
  Kirchhoff's Voltage Law
  The Branch Relations
 2.4 Loop, Node, and Node-Pair Equations
  Loop Equations
  Node Equations
  Node-pair Equations
 2.5 Duality
 2.6 Nonreciprocal and Active Networks
 2.7 Mixed-Variable Equations
 Problems

3. Network Functions
 3.1 Driving-Point and Transfer Functions
  Driving-Point Functions
  Transfer Functions
 3.2 Multiterminal Networks
 3.3 Two-Port Networks
  Open-circuit and Short-circuit Parameters
  Hybrid Parameters
  Chain Parameters
  Transmission Zeros
 3.4 Interconnection of Two-Port Networks
  Cascade Connection
  Parallel and Series Connections
  Permissibility of Interconnection
 3.5 Multiport Networks
 3.6 The Indefinite Admittance Matrix
  Connecting Two Terminals Together
  Suppressing Terminals
  Networks in Parallel
  The Cofactors of the Determinant of Yi
 3.7 The Indefinite Impedance Matrix
 3.8 Topological Formulas for Network Functions
  Determinant of the Node Admittance Matrix
  Symmetrical Cofactors of the Node Admittance Matrix
  Unsymmetrical Cofactors of the Node Admittance Matrix
  The Loop Impedance Matrix and its Cofactors
  Two-port Parameters
 Problems

4. State Equations
 4.1 Order of Complexity of a Network
 4.2 Basic Considerations in Writing State Equations
 4.3 Time-Domain Solutions of the State Equations
  Solution of Homogeneous Equation
  Alternate Method of Solution
  Matrix Exponential
 4.4 Functions of a Matrix
  The Cayley-Hamilton Theorem and its Consequences
  Distinct Eigenvalues
  Multiple Eigenvalues
  Constituent Matrices
  The Resolvent Matrix
  The Resolvent Matrix Algorithm
  Resolving Polynomials
 4.5 Systematic Formulation of the State Equations
  Topological Considerations
  Eliminating Unwanted Variables
  Time-invariant Networks
  RLC Networks
  Parameter Matrices for RLC Networks
  Considerations in Handling Controlled Sources
 4.6 Multiport Formulation of State Equations
  Output Equations
 Problems

5. Integral Solutions
 5.1 Convolution Theorem
 5.2 Impulse Response
  Transfer Function Nonzero at Infinity
  Alternative Derivation of Convolution Integral
 5.3 Step Response
 5.4 Superposition Principle
  Superposition in Terms of Impulses
  Superposition in Terms of Steps
 5.5 Numerical Solution
  Multi-input, Multi-output Networks
  State Response
  Propagating Errors
 5.6 Numerical Evaluation of eAT
  Computational Errors
  Errors in Free-state Response
  Errors in Controlled-state Response
 Problems

6. Representations of Network Functions
 6.1 Poles, Zeros, and Natural Frequencies
  Locations of Poles
  Even and Odd Parts of a Function
  Magnitude and Angle of a Function
  The Delay Function
 6.2 Minimum-phase Functions
  All-pass and Minimum-phase Functions
  Net Change in Angle
  Hurwitz Polynomials
 6.3 Minimum-phase and Non-minimum-phase Networks
  Ladder Networks
  Constant-Resistance Networks
 6.4 Determining a Network Function from its Magnitude
  Maximally Flat Response
  Chebyshev Response
 6.5 Calculation of a Network Function from a Given Angle
 6.6 Calculation of Network Function from a Given Real Part
  The Bode Method
  The Gewertz Method
  The Miyata Method
 6.7 Integral Relationships between Real and Imaginary Parts
  Reactance and Resistance-Integral Theorems
  Limitations on Constrained Networks
  Alternative Form of Relationships
  Relations Obtained with Different Weighting Functions
 6.8 Frequency and Time-Response Relationships
  Step Response
  Impulse Response
 Problems

7. Fundamentals of Network Synthesis
 7.1 Transformation of Matrices
  Elementary Transformations
  Equivalent Matrices
  Similarity Transformation
  Congruent Transformation
 7.2 Quadratic and Hermitian Forms
  Definitions
  Transformation of a Quadratic Form
  Definite and Semi Definite Forms
  Hermitian Forms
 7.3 Energy Functions
  Passive, Reciprocal Networks
  The Impedance Function
  Condition on Angle
 7.4 Positive Real Functions
  Necessary and Sufficient Conditions
  The Angle Property of Positive Real Functions
  Bounded Real Functions
  The Real Part Function
 7.5 Reactance Functions
  Realization of Reactance Functions
  Ladder-Form of Network
  Hurwitz Polynomials and Reactance Functions
 7.6 Impedances and Admittances of RC Networks
  Ladder-Network Realization
  Resistance-Inductance Networks
 7.7 Two-Port Parameters
  Resistance-Capacitance Two-Ports
 7.8 Lossless Two-Port Terminated in a Resistance
 7.9 Passive and Active RC Two-Ports
  Cascade Connection
  Cascading a Negative Converter
  Parallel Connection
  The RC-Amplifier Configuration
 Problems

8. The Scattering Parameters
 8.1 The Scattering Relations of a One-Port
  Normalized Variables—Real Normalization
  Augmented Network
  Reflection Coefficient for Time-Invariant, Passive, Reciprocal Network
  Power Relations
 8.2 Multiport Scattering Relations
  The Scattering Matrix
  Relationship To Impedance and Admittance Matrices
  Normalization and the Augmented Multiport
 8.3 The Scattering Matrix and Power Transfer
  Interpretation of Scattering Parameters
 8.4 Properties of the Scattering Matrix
  Two-Port Network Properties
  An Application—Filtering or Equalizing
  Limitations Introduced by Parasitic Capacitance
 8.5 Complex Normalization
  Frequency-Independent Normalization
  Negative-Resistance Amplifier
 Problems

9. Signal-Flow Graphs and Feedback
 9.1 An Operational Diagram
 9.2 Signal-Flow Graphs
  Graph Properties
  Inverting a Graph
  Reduction of a Graph
  Reduction to an Essential Graph
  Graph-Gain Formula
  Drawing the Signal-Flow Graph of a Network
 9.3 Feedback
  Return Ratio and Return Difference
  Sensitivity
 9.4 Stability
  Routh Criterion
  Hurwitz Criterion
  Liénard-Chip art Criterion
 9.5 The Nyquist Criterion
  Discussion of Assumptions
  Nyquist Theorem
 Problems

10. Linear Time-Varying and Nonlinear Networks
 10.1 State Equation Formulation for Time-Varying Networks
  Reduction to Normal Form
  The Components of the State Vector
 10.2 State-Equation Solution for Time-Varying Networks
  A Special Case of the Homogeneous Equation Solution
  Existence and Uniqueness of Solution of the Homogeneous Equation
  Solution of State Equation—Existence and Uniqueness
  Periodic Networks
 10.3 Properties of the State-Equation Solution
  The Gronwall Lemma
  Asymptotic Properties Relative to a Time-Invariant Reference
  Asymptotic Properties Relative to a Periodic Reference
  Asymptotic Properties Relative to a General Time-Varying Reference
 10.4 Formulation of State Equation for Nonlinear Networks
  Topological Formulation
  Output Equation
 10.5 Solution of State Equation for Nonlinear Networks
  Existence and Uniqueness
  Properties of the Solution
 10.6 Numerical Solution
  Newton's Backward-Difference Formula
  Open Formulas
  Closed Formulas
  Euler's Method
  The Modified Euler Method
  The Adams Method
  Modified Adams Method
  Milne Method
  Predictor-Corrector Methods
  Runge-Kutta Method
  Errors
 10.7 Liapunov Stability
  Stability Definitions
  Stability Theorems
  Instability Theorem
  Liapunov Function Construction
 Problems

Appendix 1 Generalized Functions
 A1.1 Convolution Quotients and Generalized Functions
 A1.2 Algebra of Generalized Functions
  Convolution Quotient of Generalized Functions
 A1.3 Particular Generalized Functions
  Certain Continuous Functions
  Locally Integrable Functions
 A1.4 Generalized Functions as Operators
  The Impulse Function
 A1.5 Integrodifferential Equations
 A1.6 Laplace Transform of a Generalized Function

Appendix 2 Theory of Functions of a Complex Variable
 A2.1 Analytic Functions
 A2.2 Mapping
 A2.3 Integration
  Cauchy's Integral Theorem
  Cauchy's Integral Formula
  Maximum Modulus Theorem and Schwartz's Lemma
 A2.4 Infinite Series
  Taylor Series
  Laurent Series
  Functions Defined by Series
 A2.5 Multivalued Functions
  The Logarithm Function
  Branch Points, Cuts, and Riemann Surfaces
  Classification of Multivalued Functions
 A2.6 The Residue Theorem
  Evaluating Definite Integrals
  Jordan's Lemma
  Principle of the Argument
 A2.7 Partial-Fraction Expansions
 A2.8 Analytic Continuation

Appendix 3 Theory of Laplace Transformations
 A3.1 Laplace Transforms: Definition and Convergence Properties
 A3.2 Analytic Properties of the Laplace Transform
 A3.3 Operations on the Determining and Generating Functions
  Real and Complex Convolution
  Differentiation and Integration
  Initial-Value and Final-Value Theorems
  Shifting
 A3.4 The Complex Inversion Integral

Bibliography
 1. Mathematical Background
  Complex Variable Theory
  Computer Programming
  Differential Equations
  Laplace Transform Theory
  Matrix Algebra
  Numerical Analysis
 2. Network Topology and Topological Formulas
 3. Loop, Node-Pair, Mixed-Variable Equations
 4. Network Functions and Their Properties
 5. State Equations
 6. Network Response and Time-Frequency Relationships
 7. Network Synthesis
 8. Scattering Parameters
 9. Signal-Flow Graphs
 10. Sensitivity 947
 11. Stability
 12. Time-Varying and Nonlinear Network Analysis