Electric Circuits & Networks

Cover
Contents
Preface
List of Reviewers
Part One: Basic Concepts
	Chapter 1: Circuit Variables and Circuit Elements
		Introduction
		1.1 Electromotive Force, Potential and Voltage
			1.1.1 Force Between two Moving Point Charges and Retardation Effect
			1.1.2 Electric Potential and Voltage
			1.1.3 Electromotive Force and Terminal Voltage of a Steady Source
		1.2 A Voltage Source with a Resistance Connected at its Terminals
			1.2.1 Steady-State Charge Distribution in the System
			1.2.2 Drift Velocity and Current Density
			1.2.3 Current Intensity
			1.2.4 Conduction and Energy Transfer Process
			1.2.5 Two-terminal Resistance Element
			1.2.6 A Time-varying Voltage Source with Resistance Across it
		1.3 Two-terminal Capacitance
		1.4 Two-terminal Inductance
			1.4.1 Induced Electromotive Force and its Location in a Circuit
			1.4.2 Relation Between Induced Electromotive Force and Current
			1.4.3 Faraday’s Law and Induced Electromotive Force
			1.4.4 The Issue of a Unique Voltage Across a two-terminal Element
			1.4.5 The two-terminal Inductance
		1.5 Ideal Independent two-terminal Electrical Sources
			1.5.1 Ideal Independent Voltage Source
			1.5.2 Ideal Independent Current Source
			1.5.3 Ideal Short-circuit Element and Ideal Open-circuit Element
		1.6 Power and Energy Relations for two-terminal Elements
			1.6.1 Passive Sign Convention
			1.6.2 Power and Energy in two-terminal Elements
		1.7 Classification of Two-terminal Elements
			1.7.1 Lumped and Distributed Elements
			1.7.2 Linear and Non-linear Elements
			1.7.3 Bilateral and Non-bilateral Elements
			1.7.4 Passive and Active Elements
			1.7.5 Time-invariant and Time-variant Elements
		1.8 Multi-terminal Circuit Elements
			1.8.1 Mutual Inductance Element
			1.8.2 Why Should M12 be Equal to M21?
			1.8.3 Ideal Dependent Sources
		1.9 Summary
		1.10 Problems
	Chapter 2: Basic Circuit Laws
		Introduction
		2.1 Kirchhoff's Voltage Law (KVL)
		2.2 Kirchhoff's Current LaW (KCL)
		2.3 Interconnections of Ideal Sources
		2.4 Analysis of a Single-Loop Circuit
		2.5 Analysis of a Single-Node-Pair Circuit
		2.6 Analysis of Multi-Loop, Multi-Node Circuits
		2.7 Summary
		2.8 Problems
	Chapter 3: Single Element Circuits
		Introduction
		3.1 The Resistor
			3.1.1 Series Connection of Resistors
			3.1.2 Parallel Connection of Resistors
		3.2 The Inductor
			3.2.1 Instantaneous Inductor Current versus Instantaneous Inductor Voltage
			3.2.2 Change in Inductor Current Function versus Area under Voltage Function
			3.2.3 Average Applied Voltage for a Given Change in Inductor Current
			3.2.4 Instantaneous Change in Inductor Current
			3.2.5 Inductor with Alternating Voltage Across it
			3.2.6 Inductor with Exponential and Sinusoidal Voltage Input
			3.2.7 Linearity of Inductor
			3.2.8 Energy Storage in an Inductor
		3.3 Series Connection of Inductors
			3.3.1 Series Connection of Inductors with Same Initial Current
			3.3.2 Series Connection with Unequal Initial Currents
		3.4 Parallel Connection of Inductors
			3.4.1 Parallel Connection of Initially Relaxed Inductors
			3.4.2 Parallel Connection of Inductors with Initial Energy
		3.5 The Capacitor
		3.6 Series Connection of Capacitors
			3.6.1 Series Connection of Capacitors with Zero Initial Energy
			3.6.2 Series Connection of Capacitors with Non-zero Initial Energy
		3.7 Parallel Connection of Capacitors
		3.8 Summary
		3.9 Questions
		3.10 Problems
Part Two: Analysis of Memoryless Circuits
	Chapter 4: Nodal Analysis and Mesh Analysis of Memoryless Circuits
		Introduction
		4.1 The Circuit Analysis Problem
		4.2 Nodal Analysis of Circuits Containing Resistors with Independent Current Sources
		4.3 Nodal Analysis of Circuits Containing Independent Voltage Sources
		4.4 Source Transformation Theorem and its Use in Nodal Analysis
			4.4.1 Source Transformation Theorem
			4.4.2 Applying Source Transformation Theorem in Nodal Analysis of Circuits
		4.5 Nodal Analysis of Circuits Containin Dependent Current Sources
		4.6 Nodal Analysis of Circuits Containing Dependent Voltage Sources
		4.7 Mesh Analysis of Circuits with Resistors and Independent Voltage Sources
			4.7.1 Principle of Mesh Analysis
			4.7.2 Is Mesh Current Measurable?
		4.8 Mesh Analysis of Circuits with Independent Current Sources
		4.9 Mesh Analysis of Circuits Containing Dependent Sources
		4.10 Summary
		4.11 Problems
	Chapter 5: Circuit Theorems
		Introduction
		5.1 Linearity of a Circuit and Superposition Theorem
			5.1.1 Linearity of a Circuit
		5.2 Star-Delta Transformation Theorem
		5.3 Substitution Theorem
		5.4 Compensation Theorem
		5.5 Thevenin’s Theorem and Norton’s Theorem
		5.6 Determination of Equivalents for Circuits with Dependent Sources
		5.7 Reciprocity Theorem
		5.8 Maximum Power Transfer Theorem
		5.9 Millman’s Theorem
		5.10 Summary
		5.11 Problems
	Chapter 6: The Operational Amplifier as a Circuit Element
		Introduction
		6.1 Ideal Amplifiers and their Features
			6.1.1 Ground in Electronic Amplifiers
		6.2 The Role of DC Power Supply in Amplifiers
			6.2.1 Linear Amplification in Electronic Amplifiers
			6.2.2 Large-signal Operation of Amplifiers
			6.2.3 Output Limits in Amplifiers
		6.3 The Operational Amplifier
			6.3.1 The Practical Operational Amplifier
		6.4 Negative Feedback in Operational Amplifier Circuits
		6.5 The Principles of ‘Virtual Short’ and ‘Zero Input Current’
		6.6 Analysis of Operational Amplifier Circuits using the IOA Model
			6.6.1 The Non-Inverting Amplifier Circuit
			6.6.2 The Voltage Follower Circuit
			6.6.3 The Inverting Amplifier Circuit
			6.6.4 The Inverting Summer
			6.6.5 The Non-Inverting Summer Amplifier
			6.6.6 The Subtractor Circuit
			6.6.7 The Instrumentation Amplifier
			6.6.8 Voltage to Current Converters
		6.7 Offset Model for an Operational Amplifier
		6.8 Effect of Non-Ideal Properties of Opamp on Circuit Performance
		6.9 Summary
		6.10 Questions
		6.11 Problems
Part Three: Sinusoidal Steady-State in Dynamic Circuits
	Chapter 7: Power and Energy in Periodic Waveforms
		Introduction
		7.1 Why Sinusoids?
		7.2 The Sinusoidal Source Function
			7.2.1 Amplitude, Period, Cyclic Frequency and Angular Frequency
			7.2.2 Phase of a Sinusoidal Waveform
			7.2.3 Phase Difference Between Two Sinusoids
			7.2.4 Lag or Lead?
			7.2.5 Phase Lag/Lead Versus Time Delay/Advance
		7.3 Instantaneous Power in Periodic Waveforms
		7.4 Average Power in Periodic Waveforms
		7.5 Effective Value (RMS Value) of Periodic Waveforms
		7.6 The Power Superposition Principle
			7.6.1 RMS Value of a Composite Waveform
		7.7 Summary
		7.8 Question
		7.9 Problems
	Chapter 8: The Sinusoidal Steady-State Response
		Introduction
		8.1 Transient State and Steady-State in Circuits
			8.1.1 Governing Differential Equation of Circuits – Examples
			8.1.2 Solution of the Circuit Differential Equation
			8.1.3 Complete Response with Sinusoidal Excitation
		8.2 The Complex Exponential Forcing Function
			8.2.1 Sinusoidal Steady-State Response from Response to
			8.2.2 Steady-State Solution to and the Operator
		8.3 Sinusoidal Steady-State Response using Complex Exponential Input
		8.4 The Phasor Concept
			8.4.1 Kirchhoff’s Laws in terms of Complex Amplitudes
			8.4.2 Element Relations in terms of Complex Amplitudes
			8.4.3 The Phasor
		8.5 Transforming a Circuit into A Phasor Equivalent Circuit
			8.5.1 Phasor Impedance, Phasor Admittance and Phasor Equivalent Circuit
		8.6 Sinusoidal Steady-State Response from Phasor Equivalent Circuit
			8.6.1 Comparison Between Memoryless Circuits and Phasor Equivalent Circuits
			8.6.2 Nodal Analysis and Mesh Analysis of Phasor Equivalent Circuits – Examples
		8.7 Circuit Theorems in Sinusoidal Steady-State Analysis
			8.7.1 Maximum Power Transfer Theorem for Sinusoidal Steady-State Condition
		8.8 Phasor Diagrams
		8.9 Apparent Power, Active Power, Reactive Power and Power Factor
			8.9.1 Active and Reactive Components of Current Phasor
			8.9.2 Reactive Power and the Power Triangle
		8.10 Complex Power under Sinusoidal Steady-State Condition
		8.11 Sinusoidal Steady-State in Circuits with Coupled Coils
			8.11.1 Dot Polarity Convention
			8.11.2 Maximum Value of Mutual Inductance and Coupling Coefficient
			8.11.3 A Two-Winding Transformer – Equivalent Models
			8.11.4 The Perfectly Coupled Transformer and The Ideal Transformer
		8.12 Summary
		8.13 Questions
		8.14 Problems
	Chapter 9: Sinusoidal Steady-State in Three-Phase Circuits
		Introduction
		9.1 Three-Phase System versus Single-Phase System
		9.2 Three-Phase Sources and Three-Phase Power
			9.2.1 The Y-connected Source
			9.2.2 The Δ-connected Source
		9.3 Analysis of Balanced Three-Phase Circuits
			9.3.1 Equivalence Between a Y-connected Source and a Δ-connected Source
			9.3.2 Equivalence Between a Y-connected Load and a Δ-connected Load
			9.3.3 The Single-Phase Equivalent Circuit for a BalancedThree-Phase Circuit
		9.4 Analysis of Unbalanced Three-Phase Circuits
			9.4.1 Unbalanced Y–Y Circuit
			9.4.2 Circulating Current in Unbalanced Δ-connected Sources
		9.5 Symmetrical Components
			9.5.1 Three-Phase Circuits with Unbalanced Sources and Balanced Loads
			9.5.2 The Zero Sequence Component
			9.5.3 Active Power in Sequence Components
			9.5.4 Three-Phase Circuits with Balanced Sources and Unbalanced Loads
		9.6 Summary
		9.7 Questions
		9.8 Problems
Part Four: Time-Domain Analysis of Dynamic Circuits
	Chapter 10: Simple RL Circuits in Time-Domain
		Introduction
		10.1 The Series RL Circuit
			10.1.1 The Series RL Circuit Equations
			10.1.2 Need for Initial Condition Specification
			10.1.3 Sufficiency of Initial Condition
		10.2 Series RL Circuit with Unit Step Input – Qualitative Analysis
			10.2.1 From t = 0– to t = 0+
			10.2.2 Inductor Current Growth Process
		10.3 Series RL Circuit with Unit Step Input – Power Series Solution
			10.3.1 Series RL Circuit Current as a Power Series
		10.4 Step Response of an RL Circuit by Solving Differential Equation
			10.4.1 Interpreting the Input Forcing Functions in Circuit Differential Equations
			10.4.2 Solving the Series RL Circuit Equation by Integrating Factor Method
			10.4.3 Complementary Function and Particular Integral
		10.5 Features of RL Circuit Step Response
			10.5.1 Step Response Waveforms in Series RL Circuit
			10.5.2 The Time Constant ‘τ’ of a Series RL Circuit
			10.5.3 Rise Time and Fall Time in First Order Circuits
			10.5.4 Effect of Non-Zero Initial Condition on Step Response of RL Circuit
			10.5.5 Free Response of Series RL Circuit
		10.6 Steady-State Response and Forced Response
			10.6.1 The DC Steady-State
			10.6.2 The Sinusoidal Steady-State
			10.6.3 The Periodic Steady-State
		10.7 Linearity and Superposition Principle in Dynamic Circuits
		10.8 Unit Impulse Response of Series RL Circuit
			10.8.1 Unit Impulse Response of RL Circuit with Non-Zero Initial Current
			10.8.2 Zero-State Response for Other Inputs from Zero-State Impulse Response
		10.9 Series RL Circuit with Exponential Inputs
			10.9.1 Zero-State Response for a Real Exponential Input
			10.9.2 Zero-State Response for Sinusoidal Input
		10.10 General Analysis Procedure for Single Time Constant RL Circuits
		10.11 Summary
		10.12 Questions
		10.13 Problems
	Chapter 11: RC and RLC Circuits in Time-Domain
		Introduction
		11.1 RC Circuit Equations
		11.2 Zero-Input Response of RC Circuit
		11.3 Zero-State Response of RC Circuits for Various Inputs
			11.3.1 Impulse Response of First-Order RC Circuits
			11.3.2 Step Response of First-Order RC Circuits
			11.3.3 Ramp Response of Series RC Circuit
			11.3.4 Series RC Circuit with Real Exponential Input
			11.3.5 Zero-State Response of Parallel RC Circuit for Sinusoidal Input
		11.4 Periodic Steady-State in a Series RC Circuit
		11.5 Sinusoidal Steady-State Frequency Response of First-Order RC Circuits
			11.5.1 The Use of Frequency Response
			11.5.2 Frequency Response and Linear Distortion
			11.5.3 Jean Baptiste Joseph Fourier and Frequency Response
			11.5.4 First-Order RC Circuits as Averaging Circuits
			11.5.5 Capacitor as a Signal-Coupling Element
			11.5.6 Parallel RC Circuit for Signal Bypassing
		11.6 The Series RLC Circuit – Zero-Input Response
			11.6.1 Source-free Response of Series RLC Circuit
			11.6.2 The Series LC Circuit – A Special Case
			11.6.3 The Series LC Circuit with Small Damping – Another Special Case
			11.6.4 Standard Formats for Second-order Circuit Zero-input Response
		11.7 Impulse Response of Series RLC Circuit
		11.8 Step Response of Series RLC Circuit
		11.9 Standard Time-Domain Specifications for Second-Order Circuits
		11.10 Examples on Impulse and Step Response of Series RLC Circuits
		11.11 Frequency Response of Series RLC Circuit
			11.11.1 Sinusoidal Forced-Response from Differential Equation
			11.11.2 Frequency Response from Phasor Equivalent Circuit
			11.11.3 Qualitative Discussion on Frequency Response of Series RLC Circuit
			11.11.4 A More Detailed Look at the Band-pass Output of Series RLC Circuit
			11.11.5 Quality Factor of Inductor and Capacitor
		11.12 The Parallel RLC Circuit
			11.12.1 Zero-Input Response and Zero-State Response of Parallel RLC Circuit
			11.12.2 Sinusoidal Steady-State Frequency Response of Parallel RLC Circuit
		11.13 Summary
		11.14 Questions
		11.15 Problems
	Chapter 12: Higher Order Circuits in Time-Domain
		Introduction
		12.1 Analysis of Multi-Mesh and Multi-Node Dynamic Circuits
		12.2 Generalisations for an nth Order Linear Time-Invariant Circuit
		12.3 Time-Domain Convolution Integral
			12.3.1 Zero-State Response to Narrow Rectangular Pulse Input
			12.3.2 Expansion of an Arbitrary Input Function in Terms of Impulse Functions
			12.3.3 The Convolution Integral
			12.3.4 Graphical Interpretation of Convolution in Time-Domain
			12.3.5 Frequency Response Function from Convolution Integral
			12.3.6 A Circuit with Multiple Sources – Applying Convolution Integral
			12.3.7 Zero-Input Response by Convolution Integral
		12.4 Summary
		12.5 Questions
		12.6 Problems
Part Five: Frequency-Domain Analysis of Dynamic Circuits
	Chapter 13: Dynamic Circuits with Periodic Inputs – Analysis by Fourier Series
		Introduction
		13.1 Periodic Waveforms in Circuit Analysis
		13.2 The Exponential Fourier Series
		13.3 Trigonometric Fourier Series
		13.4 Conditions for Existence of Fourier Series
		13.5 Waveform Symmetry and Fourier Series Coefficients
		13.6 Properties of Fourier Series and Some Examples
		13.7 Discrete Magnitude and Phase Spectrum
		13.8 Rate of Decay of Harmonic Amplitude
		13.9 Analysis of Periodic Steady-State Using Fourier Series
		13.10 Normalised Power in a Periodic Waveform and Parseval’s Theorem
		13.11 Power and Power Factor in AC System with Distorted Waveforms
		13.12 Summary
		13.13 Questions
		13.14 Problems
	Chapter 14: Dynamic Circuits with Aperiodic Inputs - Analysis by Fourier Transforms
		Introduction
		14.1 Aperiodic Waveforms
			14.1.1 Finite-Duration Aperiodic Signal As One Period of a Periodic Waveform
		14.2 Fourier Transform of an Aperiodic Waveform
			14.2.1 Fourier Transform of a Finite-Duration Aperiodic Waveform
			14.2.2 Fourier Transform of Infinite-Duration Aperiodic Waveforms
			14.2.3 Interpretation of Fourier Transforms
		14.3 Convergence of Fourier transforms
			14.3.1 Uniqueness of Fourier Transform Pair
		14.4 Some Basic Properties of Fourier transforms
			14.4.1 Linearity of Fourier Transform
			14.4.2 Duality in Fourier Transform
			14.4.3 Time Reversal Property
			14.4.4 Time Shifting Property
		14.5 Symmetry Properties of Fourier transforms
			14.5.1 Conjugate Symmetry Property
			14.5.2 Fourier Transform of an Even Time-Function
			14.5.3 Fourier Transform of an Odd Time-Function
			14.5.4 Fourier Transforms of Even Part and Odd Part of a Real Time-Function
			14.5.5 v(0) and V(j0)
		14.6 Time-Scaling Property and Fourier transform of Impulse Function
			14.6.1 Compressing a Triangular Pulse in Time-Domain with its Area Content Constant
		14.7 Fourier Transforms of Periodic Waveforms
		14.8 Fourier Transforms of Some Semi-Infinite Duration Waveforms
			14.8.1 Fourier Transform of e–α t u(t)
			14.8.2 Fourier Transform of Signum Function
			14.8.3 Fourier Transform of Unit Step Function
			14.8.4 Fourier Transform of Functions of the Form
		14.9 Zero-State Response by Frequency-Domain Analysis
			14.9.1 Why Should the System Function and Fourier Transform of Impulse Response be the Same?
		14.10 The System Function and Signal Distortion
			14.10.1 The Signal Transmission Context
			14.10.2 Linear Distortion in Signal Transmission Context
			14.10.3 Pulse Distortion in First Order Channels
		14.11 Parseval’s Relation for a Finite-Energy Waveform
		14.12 Summary
		14.13 Questions
		14.14 Problems
	Chapter 15: Analysis of Dynamic Circuits by Laplace Transforms
		Introduction
		15.1 Circuit Response to Complex Exponential Input
		15.2 Expansion of a Signal in Terms of Complex Exponential Functions
			15.2.1 Interpretation of Laplace Transform
		15.3 Laplace Transforms of Some Common Right-Sided Functions
		15.4 The s-Domain System Function H(S)
		15.5 Poles and Zeros of System Function and Excitation Function
		15.6 Method of Partial Fractions for Inverting Laplace Transforms
		15.7 Some Theorems on Laplace Transforms
			15.7.1 Time-shifting Theorem
			15.7.2 Frequency-shifting Theorem
			15.7.3 Time-Differentiation Theorem
			15.7.4 Time-integration Theorem
			15.7.5 s-Domain-Differentiation Theorem
			15.7.6 s-Domain-Integration Theorem
			15.7.7 Convolution Theorem
			15.7.8 Initial Value Theorem
			15.7.9 Final Value Theorem
		15.8 Solution of Differential Equations by Laplace Transforms
		15.9 The s-Domain Equivalent Circuit
			15.9.1 s-Domain Equivalents of Circuit Elements
		15.10 Total Response of Circuits Using s-Domain Equivalent Circuit
		15.11 Network Functions and Pole-Zero Plots
			15.11.1 Driving-point Functions and Transfer Functions
			15.11.2 The Three Interpretations for a Network Function H(s)
			15.11.3 Poles and Zeros of H(s) and Natural Frequencies of the Circuit
			15.11.4 Specifying a Network Function
		15.12 Impulse Response of Network Functions from Pole-Zero Plots
		15.13 Sinusoidal Steady-State Frequency Response from Pole-Zero Plots
			15.13.1 Three Interpretations for H
			15.13.2 Frequency Response from Pole-Zero Plot
		15.14 Analysis of Coupled Coils Using Laplace Transforms
			15.14.1 Input Impedance Function and Transfer Function of a Two-Winding Transformer
			15.14.2 Flux Expulsion by a Shorted Coil
			15.14.2 Breaking the Primary Current in a Transformer
		15.15 Summary
		15.16 Problems
Part Six: Introduction to Network Analysis
	Chapter 16: Two-Port Networks and Passive Filters
		Introduction
		16.1 Describing Equations and Parameter Sets for Two-Port Networks
			16.1.1 Short-Circuit Admittance Parameters for a Two-Port Network
			16.1.2 Open-Circuit Impedance Parameters for a Two-Port Network
			16.1.3 Hybrid Parameters and Inverse-Hybrid Parameters for a Two-Port Network
		16.2 Equivalent Circuits for a Two-Port Network
		16.3 Transmission Parameters (ABCD Parameters) of a Two-Port Network
		16.4 Inter-relationships between Various Parameter Sets
		16.5 Interconnections of Two-Port Networks
		16.6 Reciprocity and Symmetry in Two-Port Networks
		16.7 Standard Symmetric T and Pi Equivalents
		16.8 Image Parameter Description of a Reciprocal Two-Port Network
			16.8.1 Image Parameters for a Symmetric Reciprocal Two-Port Network
			16.8.2 Image Parameters in terms of Open-Circuit and Short-Circuit Impedances
		16.9 Characteristic Impedance and Propagation Constant of Symmetric T and Pi Networks Under Sinusoidal Steady-State
			16.9.1 Attenuation Constant α and Phase Constant β
		16.10 Constant-k Low-pass Filter
			16.10.1 Ideal Low-pass Filter Versus Constant-k Low-pass Filter
			16.10.2 Prototype Low-pass Filter Design
		16.11 m-Derived Low-pass Filter Sections for Improved Attenuation
		16.12 m-Derived Half-Sections for Filter Termination
			16.12.1 m-Derived Half-Sections for Input Termination
			16.12.2 Half-Π Termination Sections for Π-Section Filters
		16.13 Constant-k and m-Derived High-Pass Filters
			16.13.1 Design Equations for Prototype High-Pass Filter
			16.13.2 m-Derived Sections for Infinite Attenuation
			16.13.3 Termination Sections for High-Pass Filter
		16.14 Constant-k Band-Pass Filter
			16.14.1 Design Equations of Prototype Band-Pass Filter
		16.15 Constant-k Band-Stop Filter
		16.16 Resistive Attenuators
			16.16.1 Attenuation provided by a Symmetric Resistive Attenuator
			16.16.2 The Symmetrical T-Section Attenuator
			16.16.3 The Symmetrical Π-Section Attenuator
			16.16.4 The Symmetrical Lattice-Section Attenuator
			16.16.5 The Symmetrical Bridged-T-Section Attenuator
			16.16.6 Asymmetrical T-Section and Π-Section Attenuators
		16.17 Summary
		16.18 Questions
		16.19 Problems
	Chapter 17: Introduction to Network Topology
		Introduction
		17.1 Linear Oriented Graphs
			17.1.1 Connected Graph, Subgraphs and Some Special Subgraphs
		17.2 The Incidence Matrix of a Linear Oriented Graph
			17.2.1 Path Matrix and its Relation to Incidence Matrix
		17.3 Kirchhoff’s Laws in Incidence Matrix Formulation
			17.3.1 KCL Equations from A Matrix
			17.3.2 KVL Equations and the A Matrix
		17.4 Nodal Analysis of Networks
			17.4.1 The Principle of v-Shift
			17.4.2 Nodal Analysis of Networks Containing Ideal Dependent Sources
		17.5 The Circuit Matrix of a Linear Oriented Graph
			17.5.1 The Fundamental Circuit Matrix Bf
			17.5.2 Relation between All Incidence Matrix Aa and All Circuit Matrix Ba
		17.6 Kirchhoff’s Laws in Fundamental Circuit Matrix Formulation
			17.6.1 Kirchhoff’s Voltage Law and the Bf Matrix
			17.6.2 Kirchhoff’s Current Law and the Bf Matrix
		17.7 Loop Analysis of Electrical Networks
			17.7.1 The Principle of i-Shift
			17.7.2 Loop Analysis of Networks Containing Ideal Dependent Sources
			17.7.3 Planar Graphs and Mesh Analysis
			17.7.4 Duality
		17.8 The Cut-Set Matrix of a Linear Oriented Graph
			17.8.1 Cut-sets
			17.8.2 The All Cut-set Matrix
			17.8.3 Orthogonality Relation Between Cut-set Matrix and Circuit Matrix
			17.8.4 The Fundamental Cut-set Matrix
			17.8.5 Relation Between Qf, A and Bf
		17.9 Kirchhoff’s Laws in Fundamental Cut-Set Formulation
			17.9.1 Kirchhoff’s Current Law and the Qf Matrix
			17.9.2 Kirchhoff’s Voltage Law and the Qf Matrix
		17.10 Node-Pair Analysis of Networks
			17.10.1 Node-Pair Analysis of Networks Containing Ideal Dependent Sources
		17.11 Analysis Using Generalised Branch Model
			17.11.1 Node Analysis
			17.11.2 Loop Analysis
			17.11.3 Node-pair Analysis
		17.12 Tellegen’s Theorem
		17.13 Summary
		17.14 Problems
Answers to Selected Problems
Index