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