AN d s y N thesis
s. k. Bhattacharya
Director (Academics)
Shaheed Udham Singh Engineering College Mohali, Punjab India
Assistant Professor
Department of Electrical Engineering
BBBS Engineering College
Fatehgarh Sahib, Punjab India
Manpreet singh
Delhi • Chennai
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To my mother Late Smti. Shaila Bala Bhattacharya
—S. K. Bhattacharya
To my mother Smti. Paramjeet Kaur
—Manpreet Singh
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1. Basic Concepts
1.1 Introduction 1
1.2 Voltage, Current and Resistance 2
1.3 Ohm’s Law 2
1.4 Electrical Power and Energy 3
1.5 Series and Parallel Connections of Resistors 3
1.5.1 Series Connection of Resistors 3
1.5.2 Parallel Connection of Resistors 4
1.5.3 Series–Parallel Circuits 5
1.5.4 Ladder Network 5
1.6 Basic Circuit Elements 9
1.6.1 Resistors 9
1.6.2 Inductors—Self-Inductance and Mutual Inductance 10
1.6.3 Capacitors 14
1.7 Inductors and Capacitors in DC Circuits 17
1.8 DC Network Terminologies and Circuit Fundamentals 17
1.8.1 Network Terminologies 18
1.8.2 Voltage and Current Sources 19
1.8.3 Source Transformation 21 Review Questions 23
2. kirchhoff’s laws, Mesh and Nodal Analysis
2.1 Kirchhoff’s Laws 26
2.1.1 Kirchhoff’s Current Law 26
2.1.2 Kirchhoff’s Voltage Law 29
2.2 Mesh Analysis 30
2.3 Nodal Analysis 43
2.4 Super Nodal Analysis 53
2.5 Super Mesh Analysis 54
2.6 Methods of Solving Complex Network Problems 56
2.6.1 Numerical Problems Based on Kirchhoff’s Laws 56
26
2.6.2 Numerical Problems Based on Mesh and Nodal Analysis 60
Review Questions 82
Multiple Choice Questions 86 Answers 89
3. steady state Analysis of AC Circuits 90
3.1 AC Voltage Applied Across a Resistor 90
3.2 AC Voltage Applied Across an Inductor 92
3.3 AC Voltage Applied Across a Capacitor 95
3.4 R–L Series Circuit 99
3.5 Apparent Power, Real Power and Reactive Power 101
3.6 Power in R–L Series Circuit 101
3.7 Power Triangle of R–L Series Circuit 102
3.8 R–C Series Circuit 103
3.8.1 Power and Power Triangle of R–C Series Circuit 105
3.9 R–L–C Series Circuit 105
3.10 AC Parallel Circuits 108
3.10.1 Phasor or Vector Method of Solving Circuit Problems 108
3.10.2 Admittance Method of Solving Circuit Problems 110
3.10.3 Use of Phasor Algebra in Solving Circuit Problem 115
3.11 AC Series–Parallel Circuits 133
Review Questions 139
Multiple Choice Questions 143 Answers 146
4. R–L–C Circuits and resonance 147
4.1 R–L–C Series Circuit with Variable Frequency Input Voltage 147
4.2 Series Resonance 148
4.2.1 Effect of Variation of Frequency on Current and Voltage Drops 149
4.2.2 Effect of Variation of Frequency on Impedance and Power Factor 150
4.3 Applications of R–L–C Circuits 151
4.3.1 Band-pass Filter 151
4.3.2 Band-stop Filter 154
4.4 Parallel Resonance 155
4.4.1 Ideal Tank Circuit 155
4.4.2 Non-ideal Tank Circuit 156
4.4.3 Resonant Frequency 156
4.5 Parallel Resonant Filters 156
4.5.1 Band-pass Filter 157
4.5.2 Band-stop Filter 157
4.6 Applications of Resonant Circuits 157
4.6.1 Tuned Amplifier 157
4.6.2 Input to Receiver from an Antenna 158
4.6.3 Other Applications 158
4.6.4 Locus Diagram 167
Review Questions 169
5. Network theorems and Applications 172
5.1 Introduction 172
5.2 Superposition Theorem 173
5.3 Thevenin’s Theorem 174
5.3.1 Procedure for Applying Thevenin’s Theorem 175
5.4 Norton’s Theorem 179
5.5 Millman’s Theorem 181
5.6 Maximum Power Transfer Theorem 182
5.7 Maximum Power Transfer Theorem for Complex Impedance Circuits 185
5.8 Reciprocity Theorem 186
5.9 Tellegen’s Theorem 188
5.10 Compensation Theorem 189
5.11 Star Delta Transformation 190
5.11.1 Transforming Relations from Delta to Star 190
5.11.2 Transforming Relations from Star to Delta 191
5.12 Numericals on Network Theorems 195
Review Questions 223
6. transient response of Circuits
Using differential equations 226
6.1 Transient Condition in Networks 226
6.2 Transient Response of R–L Series Circuits Having DC Excitation 227
6.2.1 Rise of Current Through R–L Series Circuit 227
6.2.2 Time Constant of R–L Series Circuit 230
6.2.3 Decay of Current Through R–L Series Circuit 230
Contents
6.3 Transient Response in R–C Series Circuits Having DC Excitation 239
6.3.1 Case I: Capacitor is Getting Charged 239
6.3.2 Case II: Discharging of Capacitor 243
6.4 Transient Response of R–L–C Series Circuits Having DC Excitation 249
6.5 Sinusoidal Response of R–L Circuits 253
6.6 Sinusoidal Response of R–C Circuits 258
6.7 Sinusoidal Response of R–L–C Circuits 262
Review Questions 268
7. laplace transform 271
7.1 Concept of Laplace Transform 271
7.2 Laplace Transform of Standard Functions 272
7.3 Laplace Transform Problems Based on Standard Formula 280
7.4 Properties of Laplace Transform 286
7.4.1 Property 1: First Shifting Property 286
7.4.2 Property 2: Multiplication by t n 287
7.4.3 Property 3: Division by ‘t’ 288
7.4.4 Property 4 290
7.5 Summary of Useful Properties of Laplace Transform 291
7.6 Initial Value Theorem 292
7.7 Final Value Theorem 295
7.8 Inverse Laplace Transform 299
7.9 Convolution Theorem 310 Review Questions 315 Multiple Choice Questions 317 Answers 319
8. transient response of Circuits Using laplace transform 320
8.1 Steps to Find Transient Response Using Laplace Transform 320
8.2 Circuit Elements in the s-Domain 321
8.2.1 Resistor in the s-Domain 321
8.2.2 Inductor in s-Domain 321
8.2.3 Capacitor in s-Domain 321
8.3 DC Response of R–C Series Circuit 322
8.4 DC Response of R–L Series Circuit 324
8.5 DC Response of an R–L–C Series Circuit 326
8.6 Sinusoidal Response of R–L Series Circuit 329
8.7 Sinusoidal Response of R–C Series Circuit 333
Review Questions 355
9. three-Phase systems and Circuits
9.1 Introduction 358
9.2 Advantages of Three-Phase Systems 359
9.3 Generation of Three-Phase Voltages 360
9.3.1 Equation of Three-phase Voltages 360
9.3.2 Balanced Three-phase System 362
9.4 Terms Used in Three-Phase Systems and Circuits 363
9.5 Three-Phase Winding Connections 363
9.5.1 Star Connection 363
9.5.2 Delta Connection 364
9.5.3 Relationship of Line and Phase Voltages and Currents in a Star-connected System 365
9.5.4 Relationship of Line and Phase Voltages and Currents in a Delta-connected System 366
9.6 Active and Reactive Power 368
9.7 Comparison Between Star Connection and Delta Connection 369
9.8 Measurment of Power in Three-Phase Circuits 376
9.8.1 One-wattmeter Method 376
9.8.2 Two-wattmeter Method 377
9.8.3 Three-wattmeter Method 379
9.8.4 Star to delta and Delta to Star Transformation 384
9.9 More Numericals Basesd on Three-Phase Balanced Load 387
10.
358
9.10 Method of Solving Problems on Unbalanced Load 397 Review Questions 401
Multiple Choice Questions 405 Answers 407
Network Functions − s-domain Analysis of Circuits 408
10.1 Introduction 408
10.1.1 Terminals and Ports 408
10.1.2 Concept of Complex Frequency 409
10.2 Transformed Impedances in s-Domain 411
10.2.1 Resistance 411
10.2.2 Inductance 412
10.2.3 Capacitance 412
10.3 One-Port Network 413
10.3.1 Driving Point Impedance and Admittance Functions 413
10.4 Two-Port Network 420
10.4.1 Network Functions of a Two-port Network 421
10.5 Transfer Function 422
10.6 Network Function in Generalised Form 423
10.7 Poles and Zeros of Network Functions 424
10.7.1 Poles of a Network Function 424
10.7.2 Zeros of a Network Function 424
10.8 Pole–Zero Diagram 424
10.9 Time-Domain Response from Pole–Zero Plot 427
10.10 More Examples on Network Function 437
10.11 Poles and Zeros of Network Functions and Their Significance 449
10.12 Stability Criterion for an Active Network 450
10.13 Examples Based on Pole–Zero Plot 452
Review Questions 458
11. two-port Network Parameters 460
11.1 Introduction 460
11.2 Two-port Network Parameters 461
11.2.1 Open-circuit Impedance-parameters or Z-parameters 461
11.2.2 Short-circuit Admittance Parameters 462
11.2.3 Relationship Between Impedance and Admittance Matrix 464
11.2.4 Hybrid or h-parameters 466
11.2.5 Inverse Hybrid or g-parameters 467
11.2.6 Transmission Parameters 468
11.2.7 Inverse Transmission Parameters 469
11.3 Correlation of Two-Port Network Parameters 477
11.3.1 Conversion of Y-parameters to Z-parameters 477
11.3.2 Conversion of A, B, C and D or t-parameters to h-parameters 478
11.3.3 Conversion of h-parameters to Y-parameters 479
11.4 Two-Port Reciprocal and Symmetrical Networks 479
11.4.1 Reciprocal Two-port Network 479
11.4.2 Symmetrical Two-port Networks 480
11.5 Terminated Two-Port Network 480
11.6 Interconnected Two-Port Network 482
11.7 T-Circuit Representation of Two-Port Network 483
11.8 p -Circuit Representation of Two-Port Network 484
11.9 Image Impedance 484
11.10 More Solved Numericals 485
Review Questions 495
12. Network synthesis and realisability 499
12.1 Introduction 499
12.2 Hurwitz Conditions for Stability 500
12.3 Properties of Positive Real Functions 506
12.4 Synthesis of Networks by Foster’s and Cauer’s Methods 513
12.5 Foster and Cauer Forms 513
12.5.1 Synthesis of R–C Network 513
12.5.2 Properties of the R–C Impedance or R–L Admittance Function 513
12.5.3 Foster Form-I of R–C Network 513
12.5.4 Foster Form-II of R–C Network 514
12.5.5 Cauer Forms of R–C Network 515
12.5.6 Synthesis of R–L Network 515
12.5.7 Properties of R–L Impedance Function/R–C Admittance Function 515
12.5.8 Foster Form-I of R–L Network 516
12.5.9 Foster Form-II of R–L Network 516
12.5.10 Cauer Form-I of R–L Network 517
12.5.11 Cauer Form-II R–L Network 517
12.5.12 Synthesis of L–C Networks 517
12.5.13 Properties of L–C Immittance 517
12.5.14 Foster Form-I of L–C Network 517
12.5.15 Foster Form-II of L–C Network 518
12.5.16 Cauer Form-I of L–C Network 518
12.5.17 Cauer Form-II of L–C Network 519
12.6 More Numericals on Synthesis of L–C Network 548
Review Questions 576
Multiple Choice Questions 582 Answers 584
13. Filters and Attenuators
13.1 Introduction 585
13.1.1 Measurement in Decibels 587
13.2 Types of Filters 587
13.3 Classification of Passive Filters 588
13.3.1 Low-Pass Filters 588
13.3.2 High-Pass Filters 588
13.3.3 Band-Pass Filters 588
13.3.4 Band-Stop or Band-Elimination Filter 589
13.4 Parameters of a Filter 589
13.4.1 Propagation Constant (g ) 590
13.4.2 Attenuation Constant 590
13.4.3 Phase Shift Constant (b ) 591
13.4.4 Characteristic Impedance (Z0 ) 591
13.5 Filter Networks 591
13.5.1 Formation of Symmetrical T-Network 591
13.5.2 Formation of Symmetrical p-Network 592
13.5.3 Ladder Network 592
13.6 Analysis of Filter Networks 593
13.6.1 Symmetrical T-Network 593
13.6.2 Analysis of p-Network 601
13.6.3 Summary of Parameters of Filter Network 604
13.7 Classification of Filters 604
13.8 Constant K-Type or Prototype Filters 605
13.8.1 Constant K-type Low-Pass Filters (LPF) 605
13.8.2 Constant K-type High-Pass-Filters (HPF) 615
13.8.3 Comparison of Constant K-Type LPF and HPF 619
13.8.4 Constant K-type Band-Pass Filter 624
13.8.5 Constant K-type Band-Stop/Band-Elimination Filter 637
13.8.6 Comparison of Constant K-type Filters 645
13.8.7 Limitations of Constant K-type Filters 649
13.9 m-Derived Filters 649
13.9.1 m-Derived T-section 650
13.9.2 m-Derived p-section 651
13.9.3 m-Derived Low-Pass Filter 653
13.9.4 Summary of m-Derived Low-Pass Filter 657
13.9.5 m-Derived High-Pass Filter 659
13.9.6 Summary of m-Derived HPF 662
13.9.7 Comparison of m-Derived LPF and HPF 663
13.9.8 m-Derived Band-Pass Filter 665
13.9.9 m-Derived Band-Stop Filter 667
13.10 Composite Filters 668
13.10.1 Composite Low-Pass Filter 669
13.10.2 Composite High-Pass Filter 670
13.11 Additional Solved Numericals on Filters 671
13.11.1 Problems on m-Derived Low-pass Filters 671
13.11.2 Problems on m-Derived High-pass Filters 674
13.11.3 Problems on Composite Filters 675
13.12 Attenuators 681
13.12.1 Introduction 681
13.12.2 T-type Attenuator 682
13.12.3 p-type Attenuator 685
13.12.4 Lattice Attenuator 688
13.12.5 Bridged T-type Attenuator 691
13.13 More Solved Problems on Filters and Attenuators 693
Review Questions 702
Multiple Choice Questions 705 Answers 707
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Preface
It is with great pleasure we present you with this comprehensive book, Network Analysis and Synthesis, which has been developed over a period of time. The content of this book has been decided on the basis of analysis of the syllabus prescribed by all the leading Indian universities on the subject of ‘Network Analysis and Synthesis’ or ‘Circuits and Networks’. Although, a range of books are available on this subject, this book is expected to satisfy the needs of students of different learning abilities. Presentations of topics have been made so simple that even an average student will be able to follow this book almost independently.
For each chapter, the objectives have been well-stated so as to guide students in selfevaluation after studying each chapter. After studying the entire book, it is expected that students will be able to analyse, design, and synthesise electrical circuits and networks. Although, the students might be familiar with some basic concepts and principles dealt with in this book, a review of understanding of these basic concepts will facilitate understanding of the advanced topics. We believe that basic knowledge of differential and integral calculus is essential to understand this book.
PedAgogy
1. Defining the Learning Objectives Course Content provides simply an outline of the topics. It is passive and does not convey specifically what is to be learnt. Once the objectives are stated, the students know what to learn and the teachers also know how to teach and how to evaluate learning.
In this book, learning objectives have been stated to help both students and teachers.
2. Delivery of Concepts and Principles All the chapters have been defined by some key features: simple language, examples to rule, known to unknown, simple to complex and concrete to abstract. These presentations have been made in an interactive way.
3. Developing Higher Order Intellectual Abilities Ability to recall and reproduce is the lowest level of intellectual attainment. Emphasis has been given to develop in students the higher order intellectual abilities including abilities of application, analysis and synthesis which together may be defined as problem solving ability.
4. Creating Ample Opportunities for Practice
A large number of solved examples in each chapter demonstrate the many ways to solve the problems. This is followed by review questions which is composed of ample number of exercise problems with answers so that students can practice and gain confidence. Exercises have been graded from simple to complex to make learning motivating.
5. Self Feedback in Achieving the Learning Objectives Self feedback provides students the opportunity to evaluate their learning abilities. For this, a large number of short answer type and multiple choice type review questions have been provided in each chapter.
CoNteNts ANd CoverAge
This book is divided into 13 chapters. The content and coverage of chapters are as follows:
Chapter 1 reviews the basic concepts related to electrical circuits, series parallel connections, functions of circuit elements, property of storing energy by inductors and capacitors, voltage and current sources and source transformation.
Chapter 2 explains applications of Kirchhoff’s current and voltage laws in mesh and nodal analysis of circuits.
Chapter 3 presents steady state analysis of R–L, R–C and R–L–C series and parallel circuits with plenty of solved numericals.
Chapter 4 establishes the condition for series and parallel resonance. The chapter also deals with the significance of bandwidth and quality factor. Also explains how series and parallel resonant circuits are used in the field of electronics.
Chapter 5 defines and explains the network theorems like Superposition theorem, Thevenin’s theorem, Millman’s theorem, Maximum Power transfer theorem, etc. The chapter solves a large number of complex network problems using network theorems. Also shows methods of simplifying networks using star-delta transformation technique.
Chapter 6 explains the transient condition that may occur in electrical networks. The chapter derives expressions for current and voltage under transient condition in R–L, R–C and R–L–C series circuits. It also carries out transient analysis of R–L, R–C, and R–L–C series circuits with sinusoidal inputs.
Chapter 7 provides explanation of Laplace transform and its application in solving circuit problems.
Chapter 8 lists the steps to find transient response of electrical networks using Laplace transform. Determines transient response of electrical circuits by Laplace transform method.
Chapter 9 explains three-phase balanced and unbalanced systems. The chapter shows how to calculate power and power factor of balanced three-phase loads. It solves numerical problems related to balanced and unbalanced star and delta connected loads.
Chapter 10 explains the concept of complex frequency. The chapter shows how to convert circuit parameters from time domain to s-domain. The chapter explains the method of finding transfer function of electrical networks. It also explains the concept of zeros and poles of a transfer function. Shows how to write the characteristic equation and then apply Routh’s stability criterion.
Chapter 11 defines the two-port network parameters and shows how to represent them in the form of matrix equation. Calculates Z-parameters and Y-parameters. Defines A, B, C, D transmission parameters. The chapter also shows how to calculate A, B, C, D parameters of a given networks. Also calculates Z-parameters and Y-parameters of interconnected two-port networks.
Preface xix
Chapter 12 explains the concept of network Synthesis. The chapter shows how to synthesize networks by Foster and Cauer methods.
Chapter 13 explains the basic function of a filter circuit. Draws and explains basic filter networks in different sections. The chapter shows method of analysis of K-type and m-derived filters. Composite filters using constant K-type and m-derived filters have also been discussed in this chapter.
ACkNowledgeMeNts
This work is the outcome of years of experience in teaching ‘Network Analysis and Synthesis’ and other related subjects to the students of electrical and electronics engineering. We would like to thank all those who provided feedback in the form of their learning difficulties when we were teaching the subject. Thanks are also due to the faculty of various engineering colleges/ universities, for reviewing the manuscript and giving valuable suggestions.
We appreciate the excellent editorial work done by the team members of Pearson Education. Lastly, we would like to convey our special thanks to Mrs Sumita Bhattacharya and Mrs Shivdeep Kaur for their patience and encouragement which helped us to complete this big task.
S. K. Bhattacharya Manpreet Singh
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Basic Concepts 1
Chapter objeCtives
After carefully studying this chapter, you should be able to do the following:
Explain the concept of voltage, current, and resistance.
State and explain Ohm’s law.
Distinguish between electrical power and energy.
Distinguish between series and parallel connections of resistances.
Calculate branch currents and potential drops in parallel and series–parallel circuits.
Reduce a ladder network into a single equivalent resistance.
Explain the basic function of circuit elements like resistance, inductance, and capacitance.
Distinguish between self-inductance and mutual inductance and establish their relationship.
1.1 INTRODUCTION
Determine the factors on which resistance, inductance, and capacitance depend.
Explain how an inductor and a capacitor can work as energy-storing devices.
Explain the role of magnetic core material in an inductor.
Explain the role of dielectric material in a capacitor.
Distinguish between series and parallel connections of resistors and capacitors.
Be conversant with the network terminologies.
Distinguish between a voltage source and a current source.
Convert a voltage source into a current source and vice-versa.
In this chapter, we shall review the basic concepts related to electrical networks. The students must have studied these concepts in their earlier classes. However, a brief review will help them to understand the contents of chapters that would follow. We will review the basic concepts of voltage, current, resistance, Ohm’s law, electrical power and electrical energy, series and parallel connections of resistors and division of currents and voltages, and circuit parameters such as resistor, inductor and capacitor.
1.2 VOLTAGE, CURRENT AND RESISTANCE
When two oppositely charged bodies are connected by a wire, flow of electrons occurs from one body to the other. This movement of electrons constitutes a flow of electric current. The wire that allows the flow of electrons is called a conductor.
Electromotive force is an electric pressure that causes current to flow through a conducting wire. A conducting wire of large cross-sectional area allows more current to flow than a conductor with small cross-sectional area. Therefore, a conductor with a large crosssectional area offers less resistance to flow of current than a conductor with a small crosssectional area. Therefore, we can say that resistance is the opposition to the flow of current through a conductor.
Voltage or Electro motive force (EMF) or potential difference is an electric pressure that causes the flow of current, that is, the movement of electrons through a conductor. The opposition offered by the conductor to the flow of current is called its resistance.
The direction of electron flow is from the negatively charged body to the positively charged body. However, current flow is considered from the positive to the negative terminal.
The current that flows in one direction is called direct current (DC). In alternating current (AC), the flow of charge reverses directions alternately every fraction of a second (every halfcycle). The cycle repeats over and over again.
1.3 OHM’S LAW
In an electric circuit, current, voltage and resistance are related by an important law, called the Ohm’s law. Ohm’s law is expressed as follows:
where I is in amperes, V is in volts and R is in ohms.
Ohm’s law states that the current flowing through a resistance is directly proportional to the potential difference between its ends and inversely proportional to the value of the resistance, provided the temperature remains constant.
By applying Ohm’s law, the current flowing through a resistor can be calculated by measuring the voltage drop across it, provided the value of the resistance is known.
Conductance (G) is the inverse of resistance (R). In terms of conductance, Ohm’s law can be represented as follows:
The unit of G is siemens and that of R is ohms.
1.4 ELECTRICAL POWER AND ENERGY
When electric current flows through a conducting material, work is done in moving the electrons. Further, heat is dissipated as a result of the work done.
Work is also done when electricity is converted into light, sound and heat as in an electric lamp, a speaker and an electric heater, respectively.
Power (P) is defined as the time rate of doing work (W ). The power supplied to any electrical equipment is as follows: P = V I watts
By Ohm’s law, V = IR
Therefore,
Energy consumed is expressed as watt-hour or kilowatt-hour (kWh). Energy consumed or work done is as follows:
Energy = Power × Time
W = P × t Wh or kWh; The unit of work is joule. If unit of power is watt and unit of time is second, 1 watt-second = 1 joule; or, joules/second = watt (1.4)
Calorie is the unit of energy 1 calorie = 4.2 joules
We can convert electrical energy in kWh into calories as 1 36 42 5 kWh1 100060 60 watt-second or joules 1 kWh 10 86 =× ×× = × = 0 010 calories × 3
Thus, we have explained the relationship between work, power, and energy.
1.5 SERIES AND PARALLEL CONNECTIONS OF RESISTORS
A number of resistors need to be connected in series, parallel, or series-parallel in practical circuits. It is often required to determine their equivalent resistance and also calculate current and voltage distribution in these resistances. The steps are explained in the following sections.
1.5.1 Series Connection of Resistors
When a number of resistors are connected in series, there is only one path through which the current can flow. The magnitude of current is the same in all parts or components of a series circuit. When a number of resistances are connected in series, their equivalent resistance is given as follows:
