![7.9 Resistor–Inductor–Capacitor Circuit 7.25
7.10 Response of RL Circuit to Various Functions 7.31
7.11 Response of RC Circuit to Various Functions 7.39
Exercises 7.49
Objective-Type Questions 7.52
Answers to Objective-Type Questions 7.53
8. NETWORK FUNCTIONS 8.1
8.1 Introduction 8.1
8.2 Driving-Point Functions 8.1
8.3 Transfer Functions 8.2
8.4 Analysis of Ladder Networks 8.5
8.5 Analysis of Non-Ladder Networks 8.15
8.6 Poles and Zeros of Network Functions 8.20
8.7 Restrictions on Pole and Zero Locations for Driving-Point Functions [Common Factors in N(s)
and D(s) Cancelled] 8.21
8.8 Restrictions on Pole and Zero Locations for Transfer Functions [Common Factors in N(s) and
D(s) Cancelled] 8.21
8.9 Time-Domain Behaviour from the Pole-Zero Plot 8.39
8.10 Graphical Method for Determination of Residue 8.42
Exercises 8.50
Objective-Type Questions 8.53
Answers to Objective-Type Questions 8.55
9. NETWORK SYNTHESIS 9.1
9.1 Introduction 9.1
9.2 Hurwitz Polynomials 9.1
9.3 Positive Real Functions 9.16
9.4 Elementary Synthesis Concepts 9.24
9.5 Realisation of LC Functions 9.30
9.6 Realisation of RC Functions 9.47
9.7 Realisation of RL Functions 9.63
Exercises 9.72
Objective-Type Questions 9.74
Answers to Objective-Type Questions 9.76
10. TWO-PORT NETWORKS 10.1
10.1 Introduction 10.1
10.2 Open-Circuit Impedance Parameters (Z Parameters) 10.2
10.3 Short-Circuit Admittance Parameters (Y Parameters) 10.8
10.4 Transmission Parameters (ABCD Parameters) 10.18
10.5 Inverse Transmission Parameters (A�B�C�D� Parameters) 10.24
10.6 Hybrid Parameters (h Parameters) 10.28
10.7 Inverse Hybrid Parameters (g Parameters) 10.33
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![Exercises 1.21
Simplifying the network (Fig. 1.47),
`
4 Ω
3 Ω
1 Ω
2 Ω 6 Ω Va
+
−
Va
+
−
18 V 3 Ω
1.33 Ω
1 Ω
6 Ω
18 V
4.5 A
4.5 A
(a) (b)
3 Ω
1.33 Ω 1 Ω
3 Ω
6 Ω
2.33 Ω
6 Ω
Va
Va
18 V
5.985 V
18 V 5.985 V
+
−
(c) (d)
Fig. 1.47
Applying KCL at the node,
V V V
a a a
−
+
−
+ =
18
3
5 985
2 33 6
0
.
.
Va = 9 23
. V
Exercises
1.1 Use source transformation to simplify the
network until two elements remain to the left
of terminals A and B.
20 mA
6 kΩ
2 kΩ 3 kΩ 12 kΩ
3.5 kΩ
A
B
Fig. 1.48
[88.42 V, 7.92 k W]
1.2 Determine the voltage Vx
in the network of
Fig. 1.49 by source-shifting technique.
2 V
3 Ω
2 Ω 1 Ω
2 Ω 5 Ω
Vx
Fig. 1.49
[1.129 V]](https://image.slidesharecdn.com/52608-250403080045-82d2fc77/85/Circuit-Theory-and-Transmission-Lines-2nd-Edition-Ravish-R-Singh-43-320.jpg)
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Circuit Theory and Transmission Lines 2nd Edition Ravish R. Singh
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- 5. Circuit Theory and Transmission Lines Second Edition
- 6. About the Author Ravish R Singh is presently Academic Advisor at Thakur Educational Trust, Mumbai. He obtained a BE degree from University of Mumbai in 1991, an MTech degree from IIT Bombay in 2001, and a PhD degree from Faculty of Technology, University of Mumbai, in 2013. He has published several books with McGraw Hill Education (India) on varied subjects like Engineering Mathematics (I and II), Applied Mathematics, Electrical Engineering, Electrical and Electronics Engineering, etc., for all-India curricula as well as regional curricula of some universities like Gujarat Technological University, Mumbai University, Pune University, Jawaharlal Nehru Technological University, Anna University, Uttarakhand Technical University, and Dr A P J Abdul Kalam Technical University. Dr Singh is a member of IEEE, ISTE, and IETE, and has published research papers in national and international journals. His fields of interest include Circuits, Signals and Systems, and Engineering Mathematics.
- 7. Ravish R Singh Academic Advisor Thakur Educational Trust Mumbai, Maharashtra Circuit Theory and Transmission Lines Second Edition McGraw Hill Education (India) Private Limited NEW DELHI McGraw Hill Education Offices New Delhi New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto
- 8. McGraw Hill Education (India) Private Limited Published by the McGraw Hill Education (India) Private Limited P-24, Green Park Extension, New Delhi 110 016. Circuit Theory and Transmission Lines, 2e Copyright© 2016, 2014, by McGraw Hill Education (India) Private Limited. No part of this publication may be reproduced or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior written permission of the publishers. The program listings (if any) may be entered, stored and executed in a computer system, but they may not be reproduced for publication. This edition can be exported from India only by the publishers, McGraw Hill Education (India) Private Limited. ISBN (13): 978-93-5260-270-4 ISBN (10): 93-5260-270-6 Managing Director: Kaushik Bellani Director—Products (Higher Education and Professional): Vibha Mahajan Manager—Product Development: Koyel Ghosh Specialist—Product Development: Piyali Chatterjee Head––Production (Higher Education and Professional): Satinder S Baveja Senior Production Executive: Jagriti Kundu AGM—Product Management (Higher Education and Professional): Shalini Jha Manager—Product Management: Ritwick Dutta General Manager—Production: Rajender P Ghansela Manager—Production: Reji Kumar Information contained in this work has been obtained by McGraw Hill Education (India), from sources believed to be reliable. However, neither McGraw Hill Education (India) nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill Education (India) nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw Hill Education (India) and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at Text-o-Graphics, B-1/56, Aravali Apartment, Sector-34, Noida 201 301, and printed at Cover Printer: Visit us at: www.mheducation.co.in
- 9. Dedicated to My Father Late Shri Ramsagar Singh and My Mother Late Shrimati Premsheela Singh
- 11. Contents Preface xiii 1. BASIC NETWORK CONCEPTS 1.1 1.1 Introduction 1.1 1.2 Resistance 1.1 1.3 Inductance 1.2 1.4 Capacitance 1.3 1.5 Sources 1.4 1.6 Some Definitions 1.6 1.7 Series and Parallel Combination of Resistors 1.7 1.8 Series and Parallel Combination of Inductors 1.9 1.9 Series and Parallel Combination of Capacitors 1.10 1.10 Star-Delta Transformation 1.10 1.11 Source Transformation 1.13 1.12 Source Shifting 1.19 Exercises 1.21 Objective-Type Questions 1.22 Answers to Objective-Type Questions 1.23 2. ELEMENTARY NETWORK THEOREMS 2.1 2.1 Introduction 2.1 2.2 Kirchhoff’s Laws 2.1 2.3 Mesh Analysis 2.2 2.4 Supermesh Analysis 2.17 2.5 Node Analysis 2.26 2.6 Supernode Analysis 2.46 Exercises 2.52 Objective-Type Questions 2.56 Answers to Objective-Type Questions 2.57 3. NETWORK THEOREMS (APPLICATION TO DC NETWORKS) 3.1 3.1 Introduction 3.1 3.2 Superposition Theorem 3.1 3.3 Thevenin’s Theorem 3.30 3.4 Norton’s Theorem 3.64 3.5 Maximum Power Transfer Theorem 3.91 3.6 Millman’s Theorem 3.112 Exercises 3.117 Objective-Type Questions 3.122 Answers to Objective-Type Questions 3.123
- 12. 4. COUPLED CIRCUITS 4.1 4.1 Introduction 4.1 4.2 Self-Inductance 4.1 4.3 Mutual Inductance 4.2 4.4 Coefficient of Coupling (k) 4.2 4.5 Inductances in Series 4.3 4.6 Inductances in Parallel 4.4 4.7 Dot Convention 4.9 4.8 Coupled Circuits 4.15 4.9 Conductively Coupled Equivalent Circuits 4.37 4.10 Tuned Circuits 4.40 Exercises 4.47 Objective-Type Questions 4.49 Answers to Objective-Type Questions 4.50 5. RESONANCE 5.1 5.1 Introduction 5.1 5.2 Series Resonance 5.1 5.3 Parallel Resonance 5.18 5.4 Comparison of Series and Parallel Resonant Circuits 5.21 Exercises 5.38 Objective-Type Questions 5.39 Answers to Objective-Type Questions 5.40 6. TRANSIENT ANALYSIS 6.1 6.1 Introduction 6.1 6.2 Initial Conditions 6.1 6.3 Resistor–Inductor Circuit 6.27 6.4 Resistor–Capacitor Circuit 6.49 6.5 Resistor–Inductor–Capacitor Circuit 6.66 Exercises 6.79 Objective-Type Questions 6.82 Answers to Objective-Type Questions 6.85 7. LAPLACE TRANSFORM AND ITS APPLICATION 7.1 7.1 Introduction 7.1 7.2 Laplace Transformation 7.1 7.3 Laplace Transforms of Some Important Functions 7.2 7.4 Properties of Laplace Transform 7.4 7.5 Inverse Laplace Transform 7.7 7.6 The Transformed Circuit 7.12 7.7 Resistor–Inductor Circuit 7.13 7.8 Resistor–Capacitor Circuit 7.19 viii�Contents
- 13. 7.9 Resistor–Inductor–Capacitor Circuit 7.25 7.10 Response of RL Circuit to Various Functions 7.31 7.11 Response of RC Circuit to Various Functions 7.39 Exercises 7.49 Objective-Type Questions 7.52 Answers to Objective-Type Questions 7.53 8. NETWORK FUNCTIONS 8.1 8.1 Introduction 8.1 8.2 Driving-Point Functions 8.1 8.3 Transfer Functions 8.2 8.4 Analysis of Ladder Networks 8.5 8.5 Analysis of Non-Ladder Networks 8.15 8.6 Poles and Zeros of Network Functions 8.20 8.7 Restrictions on Pole and Zero Locations for Driving-Point Functions [Common Factors in N(s) and D(s) Cancelled] 8.21 8.8 Restrictions on Pole and Zero Locations for Transfer Functions [Common Factors in N(s) and D(s) Cancelled] 8.21 8.9 Time-Domain Behaviour from the Pole-Zero Plot 8.39 8.10 Graphical Method for Determination of Residue 8.42 Exercises 8.50 Objective-Type Questions 8.53 Answers to Objective-Type Questions 8.55 9. NETWORK SYNTHESIS 9.1 9.1 Introduction 9.1 9.2 Hurwitz Polynomials 9.1 9.3 Positive Real Functions 9.16 9.4 Elementary Synthesis Concepts 9.24 9.5 Realisation of LC Functions 9.30 9.6 Realisation of RC Functions 9.47 9.7 Realisation of RL Functions 9.63 Exercises 9.72 Objective-Type Questions 9.74 Answers to Objective-Type Questions 9.76 10. TWO-PORT NETWORKS 10.1 10.1 Introduction 10.1 10.2 Open-Circuit Impedance Parameters (Z Parameters) 10.2 10.3 Short-Circuit Admittance Parameters (Y Parameters) 10.8 10.4 Transmission Parameters (ABCD Parameters) 10.18 10.5 Inverse Transmission Parameters (A�B�C�D� Parameters) 10.24 10.6 Hybrid Parameters (h Parameters) 10.28 10.7 Inverse Hybrid Parameters (g Parameters) 10.33 ���������ix
- 14. 10.8 Inter-relationships between the Parameters 10.37 10.9 Interconnection of Two-Port Networks 10.63 10.10 T-Network 10.79 10.11 Pi (� )-Network 10.79 10.12 Lattice Networks 10.84 10.13 Terminated Two-Port Networks 10.87 Exercises 10.97 Objective-Type Questions 10.100 Answers to Objective-Type Questions 10.103 11. FILTERS AND ATTENUATORS 11.1 11.1 Introduction 11.1 11.2 Classification of Filters 11.1 11.3 T-Network 11.1 11.4 � Network 11.4 11.5 Characteristic of Filters 11.6 11.6 Constant-k Low Pass Filter 11.7 11.7 Constant-k High-pass Filter 11.14 11.8 Band-pass Filter 11.18 11.9 Band-stop Filter 11.22 11.10 m-Derived Filters 11.25 11.11 m-Derived Low-Pass Filter 11.28 11.12 m-Derived High-Pass Filter 11.31 11.13 Terminating Half Sections 11.34 11.14 Composite Filter 11.37 11.15 Attenuator 11.40 11.16 Lattice Attenuator 11.41 11.17 T-Type Attenuator 11.42 11.18 �-Type Attenuator 11.45 11.19 Ladder-Type Attenuator 11.47 Exercises 11.48 Objective-Type Questions 11.49 Answers to Objective-Type Questions 11.50 12. TRANSMISSION LINES 12.1 12.1 Introduction 12.1 12.2 Types of Transmission Lines 12.1 12.3 Primary Constants of a Transmission Line 12.2 12.4 Equivalent Circuit Representation of Transmission Line 12.2 12.5 Secondary Constants of a Transmission Line 12.3 12.6 Infinite Line 12.10 12.7 Terminated Transmission Lines 12.11 12.8 General Solution of a Transmission Line 12.11 12.9 Reflection 12.15 x�Contents
- 15. 12.10 Reflection Coefficient 12.15 12.11 Standing Waves 12.16 12.12 Standing Wave Ratio 12.16 12.13 Scattering parameters or S-parameters 12.18 12.14 Impedance Smith Chart 12.19 12.15 Impedance Matching Using Smith Chart 12.36 Exercises 12.42 Objective-Type Questions 12.42 Answers to Objective-Type Questions 12.43 Solved Question Paper December 2015 (EXC 304) SQP.1 Solved Question Paper December 2015 (ETC 304) SQP.1 Solved Question Paper May 2016 (EXC 304) SQP.1 Solved Question Paper May 2016 (ETC 304) SQP.1 Index I.1 ���������xi
- 17. Overview Circuit Theory and Transmission Lines is an important subject for third-semester students of Electronics Engineering and Electronics & Telecommunication Engineering. With lucid and brief theory, this textbook provides thorough understanding of the topics of this subject. Following a problem-solving approach and discussing both analysis and synthesis of networks, it offers good coverage of dc circuits, network theorems, two-port networks, network synthesis, and transmission lines. Generally, numerical problems are expected in university examinations in this subject. The weightage given to problems in examinations is more than 70–80%. Questions from important topics of this subject are part of competitive examinations such as IAS, IES, etc. Hence, numerous solved examples and exercise problems are included in each chapter of this book to help students develop and master problem-solving skills required to ace any examination with confidence. Objective-type questions from various competitive examinations are also included at the end of each chapter for easy revision of core concepts. Salient Features Up-to-date and full coverage of the latest revised syllabus of University of Mumbai � Covers both analysis and synthesis of networks � Uses problem-solving approach to explain topics � Lucid coverage of network theorems, transient analysis, two-port networks, network synthesis � Separate chapter on transmission lines � Extensively supported by illustrations � Solution of 2015 and 2016 University of Mumbai question papers is provided at the end of the � book. Examination-oriented excellent pedagogy: � � Illustrations: 1500+ � Solved Examples within chapters 570 � Unsolved Problems: 185 � Objective Type Questions: 135 Chapter Organisation This text is organised into 12 chapters. Chapter 1 covers basic circuit elements and basic laws comprising of networks. Further, dc network theorems are elucidated in Chapters 2 and 3. Chapters 4 and 5 discuss coupled circuits and resonance, respectively. Chapters 6 and 7 discuss transient analysis in time domain and frequency domain, respectively. Chapters 8 and 9 cover network functions and network synthesis. Chapter 10 elucidates two-port networks. Chapter 11 describes filters and attenuators. Lastly, transmission lines and radio frequency is covered in Chapter 12. Acknowledgements My acknowledgements would be incomplete without a mention of the contribution of my family members. I feel indebted to my father and mother for their lifelong inspiration. I also send a heartfelt thanks to my wife Nitu; son Aman; and daughter Aditri, for always motivating and supporting me during the preparation of the Preface
- 18. xiv�Preface project. I appreciate the support extended by the team at McGraw Hill Education (India), especially Koyel Ghosh, Piyali Chatterjee, Satinder Singh Baveja, Anuj Shrivastava and Jagriti Kundu during the editorial, copyediting and production stages of this book. I am grateful to the reviewers mentioned below for taking out time to review certain chapters of the book and sharing their valuable suggestions: Amit Bagade Ramrao Adik Institute of Technology, Nerul, Navi Mumbai, Maharashtra Gajraj Singh Ramrao Adik Institute of Technology, Nerul, Navi Mumbai, Maharashtra Reena Sonkusare Sardar Patel Institute of Technology, Mumbai, Maharashtra Suggestions for improvements will always be welcome. Ravish R Singh Publisher’s Note Remember to write to us. We look forward to receiving your feedback, comments, and ideas to enhance the quality of this book. You can reach us at info.india@mheducation.com. Please mention the title and authors’ name as the subject. In case you spot piracy of this book, please do let us know.
- 19. (As per latest revised syllabus of University of Mumbai) This text is useful for Electronics Engineering Circuit Theory—EXC304 Module 1:Analysis of Electrical Circuits 1.1 Analysis of DC Circuits: Analysis of circuits with and without controlled sources using generalized loop, node matrix, superposition, Thevenin, Norton, Millman theorems 1.2 Analysis of Coupled Circuits: Self and mutual inductances, coefficient of coupling, dot convention, equivalent circuit, solution using loop analysis 1.3 Series and Parallel Resonance Circuits: Selectivity, bandwidth, quality factor GO TO: CHAPTER 1. BASIC NETWORK CONCEPTS CHAPTER 2. ELEMENTARY NETWORK THEOREMS CHAPTER 3. NETWORK THEOREMS (APPLICATION TO dc NETWORKS) CHAPTER 4. COUPLED CIRCUITS CHAPTER 5. RESONANCE Module 2: Time and Frequency Domain Analysis 2.1 Time Domain Analysis of R-L and R-C Circuits: Forced and natural response, time constant, initial and final values Solution Using First Order Equation for Standard Input Signals: Transient and steady state time re- sponse, solution using universal formula 2.2 Time Domain Analysis of R-L-C Circuits: Forced and natural response, effect of damping Solution Using Second Order Equation for Standard Input Signals: Transient and steady state time response 2.3 Frequency Domain Analysis of RLC Circuits: S-domain representation, applications of Laplace trans- form in solving electrical networks, driving point and transfer function, poles and zeros, calculation of residues by analytical and graphical method, frequency response GO TO: CHAPTER 6. TRANSIENT ANALYSIS CHAPTER 7. LAPLACE TRANSFORM AND ITS APPLICATION CHAPTER 8. NETWORK FUNCTIONS Roadmap to the Syllabus
- 20. Module 3: Synthesis of RLC Circuits 3.1 Positive Real Functions: Concept of positive real function, testing for Hurwitz polynomials, testing for necessary and sufficient conditions for positive real functions 3.2 Synthesis of RC, RL, LC Circuits: Concepts of synthesis of RC, RL, LC driving point functions GO TO: CHAPTER 9. NETWORK SYNTHESIS Module 4: Two-Port Networks 4.1 Parameters: Open circuit, short circuit, transmission and hybrid parameters, relationships among pa- rameters, reciprocity and symmetry conditions 4.2 Series/Parallel Connection: T and Pi representations, interconnection of two-port networks GO TO: CHAPTER 10. TWO-PORT NETWORKS Module 5: Filters and Attenuators 5.1 Basic Filter Circuits: Low pass, high pass, band pass and band stop filters, transfer function, frequency response, cutoff frequency, bandwidth, quality factor, attenuation constant, phase shift, characteristic impedance 5.2 Concept of Design and Analysis of Filters: Constant K, M derived and composite filters 5.3 Attenuators: Basic concepts, classification, attenuation in dB, K factor (impedance factor) and design concepts GO TO: CHAPTER 11. FILTERS AND ATTENUATORS Module 6: Transmission Lines 6.1 Power Frequency Lines: Representation, losses and efficiency in power lines, effect of length, calcula- tion of inductance and capacitance 6.2 Radio Frequency Lines: Representation, propagation constant, attenuation constant, phase constant, group velocity, input impedance, characteristic impedance, reflection coefficient, standing wave ratio, VSWR, ISWR, S-parameters 6.3 Smith Chart: Impedance locus diagram, impedance matching GO TO: CHAPTER 12. TRANSMISSION LINES xvi�Roadmap to the Syllabus
- 21. This text is useful for Electronics and Telecommunication Engineering Circuits and Transmission Lines—ETC304 Module 1: Electrical Circuit Analysis 1.1 Analysis of DC Circuits: Analysis of circuits with and without controlled sources using generalized loop and node matrix methods and source transformation, superposition, Thevenin, Norton, Millman theorems 1.2 Magnetic Circuits: Self and mutual inductances, coefficient of coupling, dot convention, equivalent circuit, solution using loop analysis 1.3 Tuned Coupled Circuits: Analysis of tuned coupled circuits GO TO: CHAPTER 1. BASIC NETWORK CONCEPTS CHAPTER 2. ELEMENTARY NETWORK THEOREMS CHAPTER 3. NETWORK THEOREMS (APPLICATION TO dc NETWORKS) CHAPTER 4. COUPLED CIRCUITS Module 2: Time and Frequency Domain Analysis 2.1 Time Domain Analysis of R-L and R-C Circuits: Forced and natural response, time constant, initial and final values Solution Using First Order Equation for Standard Input Signals: Transient and steady state time re- sponse, solution using universal formula 2.2 Time Domain Analysis of R-L-C Circuits: Forced and natural response, effect of damping Solution Using Second Order Equation for Standard Input Signals: Transient and steady state time response 2.3 Frequency Domain Analysis of RLC Circuits: S-domain representation, applications of Laplace trans- form in solving electrical networks, driving point and transfer function, poles and zeros, calculation of residues by analytical and graphical method, analysis of ladder and lattice network Response to Standard Signals: Transient and steady state time response of R-L-C circuits GO TO: CHAPTER 6. TRANSIENT ANALYSIS CHAPTER 7. LAPLACE TRANSFORM AND ITS APPLICATION CHAPTER 8. NETWORK FUNCTIONS ������������������������xvii
- 22. Module 3: Synthesis of RLC circuits 3.1 Positive Real Functions: Concept of positive real function, testing for Hurwitz polynomials, testing for necessary and sufficient conditions for positive real functions 3.2 Synthesis of RC, RL, LC and RLC Circuits: Properties and synthesis of RC, RL, LC driving point functions GO TO: CHAPTER 9. NETWORK SYNTHESIS Module 4: Two-Port Circuits 4.1 Parameters: Open circuits, short circuit, transmission and hybrid parameters, relationship among pa- rameters, reciprocity and symmetry conditions 4.2 Interconnections of two-port circuits, T & � representation 4.3 Terminated two-port circuits GO TO: CHAPTER 10. TWO-PORT NETWORKS Module 5: Radio Frequency Transmission Lines 5.1 Transmission Line Representation: T and � representations, terminated transmission line, infinite line 5.2 Parameters of Radio Frequency Lines: Propagation constant, attenuation constant, phase constant, group velocity, input impedance, characteristic impedance, reflection coefficient, standing wave ratio, VSWR, ISWR, S-parameters 5.3 Smith Chart: Impedance locus diagram, impedance matching GO TO: CHAPTER 12. TRANSMISSION LINES xviii�Roadmap to the Syllabus
- 23. 1 Basic Network Concepts 1.1 IntroductIon We know that like charges repel each other whereas unlike charges attract each other. To overcome this force of attraction, a certain amount of work or energy is required. When the charges are separated, it is said that a potential difference exists and the work or energy per unit charge utilised in this process is known as voltage or potential difference. The phenomenon of transfer of charge from one point to another is termed current. Current (I) is defined as the rate of flow of electrons in a conductor. It is measured by the number of electrons that flow in unit time. Energy is the total work done in the electric circuit. The rate at which the work is done in an electric circuit is called electric power. Energy is measured in joules (J) and power in watts (W). 1.2 resIstance Resistance is the property of a material due to which it opposes the flow of electric current through it. Certain materials offer very little opposition to the flow of electric current and are called conductors, e.g., metals, acids and salt solutions. Certain materials offer very high resistance to the flow of electric current and are called insulators, e.g., mica, glass, rubber, Bakelite, etc. The practical unit of resistance is ohm and is represented by the symbol W. A conductor is said to have resistance of one ohm if a potential difference of one volt across its terminals causes a current of one ampere to flow through it. The resistance of a conductor depends on the following factors. (i) It is directly proportional to its length. (ii) It is inversely proportional to the area of cross section of the conductor. (iii) It depends on the nature of the material. (iv) It also depends on the temperature of the conductor. Hence, R l A R l A ∝ = ρ where l is length of the conductor, A is the cross-sectional area and r is a constant known as specific resistance or resistivity of the material.
- 24. 1.2 Circuit Theory and Transmission Lines 1. Power Dissipated in a Resistor We know that v = R i When current flows through any resistor, power is absorbed by the resistor which is given by p = v i The power dissipated in the resistor is converted to heat which is given by E vi dt Rii dt i Rt t t = = = ∫ ∫ 0 0 2 1.3 Inductance Inductance is the property of a coil that opposes any change in the amount of current flowing through it. If the current in the coil is increasing, the self-induced emf is set up in such a direction so as to oppose the rise of current. Similarly, if the current in the coil is decreasing, the self-induced emf will be in the same direction as the applied voltage. Inductance is defined as the ratio of flux linkage to the current flowing through the coil. The practical unit of inductance is henry and is represented by the symbol H. A coil is said to have an inductance of one henry if a current of one ampere when flowing through it produces flux linkages of one weber-turn in it. The inductance of an inductor depends on the following factors. (i) It is directly proportional to the square of the number of turns. (ii) It is directly proportional to the area of cross section. (iii) It is inversely proportional to the length. (iv) It depends on the absolute permeability of the magnetic material. Hence, L N A l L N A l ∝ 2 2 = µ where l is the mean length, A is the cross-sectional area and m is the absolute permeability of the magnetic material. Current–Voltage Relationships in an Inductor 1. We know that v L di dt = Expressing inductor current as a function of voltage, di L v dt = 1 Integrating both the sides, di l L v dt i t L v dt i t i i t t = = + ∫ ∫ ∫ 0 0 0 1 0 ( ) ( ) ( ) ( )
- 25. 1.4 Capacitance 1.3 The quantity i(0) denotes the initial current through the inductor. When there is no initial current through the inductor, i t L v dt t ( ) = ∫ 1 0 Energy Stored in an Inductor 2. Consider a coil of inductance L carrying a changing current I. When the current is changed from zero to a maximum value I, every change is opposed by the self-induced emf produced. To overcome this opposition, some energy is needed and this energy is stored in the magnetic field. The voltage v is given by v L di dt = Energy supplied to the inductor during interval dt is given by dE vi dt L di dt i dt Li dt = = = Hence, total energy supplied to the inductor when current is increased from 0 to I amperes is E dE Li di L I I I = = = ∫ ∫ 0 2 0 1 2 1.4 capacItance Capacitance is the property of a capacitor to store an electric charge when its plates are at different potentials. If Q coulombs of charge is given to one of the plates of a capacitor and if a potential difference of V volts is applied between the two plates then its capacitance is given by C Q V = The practical unit of capacitance is farad and is represented by the symbol F. A capacitor is said to have capacitance of one farad if a charge of one coulomb is required to establish a potential difference of one volt between its plates. The capacitance of a capacitor depends on the following factors. (i) It is directly proportional to the area of the plates. (ii) It is inversely proportional to the distance between two plates. (iii) It depends on the absolute permittivity of the medium between the plates. Hence, C A d C A d ∝ = ε where d is the distance between two plates, A is the cross-sectional area of the plates and e is absolute permittivity of the medium between the plates.
- 26. 1.4 Circuit Theory and Transmission Lines Current–Voltage Relationships in a Capacitor 1. The charge on a capacitor is given by q = Cv where q denotes the charge and v is the potential difference across the plates at any instant. We know that i dq dt d dt Cv C dv dt = = = Expressing capacitor voltage as a function of current, dv C i dt = 1 Integrating both the sides, dv C i dt v t C i dt v v v t t t ( ) ( ) ( ) ( ) 0 0 0 1 1 0 ∫ ∫ ∫ = = + The quantity v (0) denotes the initial voltage across the capacitor. When there is no initial voltage on the capacitor, v t C i dt t ( ) = ∫ 1 0 Energy Stored in a Capacitor 2. Let a capacitor of capacitance C farads be charged from a source of V volts. Then current i is given by i C dv dt = Energy supplied to the capacitor during interval dt is given by dE vi dt vC dv dt dt = = Hence, total energy supplied to the capacitor when potential difference is increased from 0 to V volts is E dE C v dv CV V V = = = ∫ ∫ 0 0 2 1 2 1.5 sources Source is a basic network element which supplies energy to the networks. There are two classes of sources, namely, 1. Independent sources 2. Dependent sources
- 27. 1.5 Sources 1.5 1.5.1 Independent Sources Output characteristics of an independent source are not dependent on any network variable such as a current or voltage. Its characteristics, however, may be time-varying. There are two types of independent sources: 1. Independent voltage source 2. Independent current source Independent Voltage Source 1. An independent voltage source is a two-terminal network element that establishes a specified voltage across its terminals. The value of this voltage at any instant is independent of the value or direction of the current that flows through it. The symbols for such voltage sources are shown in Fig. 1.1. The terminal voltage may be a constant, or it may be some specified function of time. 2. Independent Current Source An independent current source is a two-terminal network element which produces a specified current. The value and direction of this current at any instant is independent of the value or direction of the voltage that appears across the terminals of the source. The symbols for such current sources are shown in Fig. 1.2. The output current may be a constant or it may be a function of time. 1.5.2 dependent Sources If the voltage or current of a source depends in turn upon some other voltage or current, it is called as dependent or controlled source. The dependent sources are of four kinds, depending on whether the control variable is voltage or current and the controlled source is a voltage source or current source. 1. Voltage-ControlledVoltageSource(VCVS) A voltage-controlled voltage source is a four-terminal networkcomponentthatestablishesavoltagevcd between two points c and d in the circuit that is proportional to a voltage vab between two points a and b. The symbol for such a source is shown in Fig. 1.3. The (+) and (−) sign inside the diamond of the component symbol identifies the component as a voltage source. vcd = m vab The voltage vcd depends upon the control voltage vab and the constant m, a dimensionless constant called voltage gain. 2. Voltage-Controlled Current Source (VCCS) A voltage-controlled current source is a four-terminal network component that establishes a current icd in a branch of the circuit that is proportional to the voltage vab between two points a and b. The symbol for such a source is shown in Fig. 1.4. v(t) (b) V (a) Fig. 1.1 Symbols for independent voltage source (b) I (a) i(t) Fig. 1.2 Symbols for independent current source a c vab b − + + − vcd d + − mvab Fig. 1.3 Symbol for VCVS a c vab b − + vcd icd d + − gmvab Fig. 1.4 Symbol for VCCS
- 28. 1.6 Circuit Theory and Transmission Lines The arrow inside the diamond of the component symbol identifies the component as a current source. icd = gm vab The current icd depends only on the control voltage vab and the constant gm , called the transconductance or mutual conductance. The constant gm has dimension of ampere per volt or siemens (S). 3. Current-ControlledVoltageSource(CCVS) A current-controlled voltage source is a four-terminal networkcomponentthatestablishesavoltagevcd between two points c and d in the circuit that is proportional to the current iab in some branch of the circuit. The symbol for such a source is shown in Fig. 1.5. vcd = r iab The voltage vcd depends only on the control current iab and the constant r called the transresistance or mutual resistance. The constant r has dimension of volt per ampere or ohm (W). 4. Current-Controlled Current Source (CCCS) A current-controlled current source is a four-terminal network component that establishes a current icd in one branch of a circuit that is proportional to the current iab in some branch of the network. The symbol for such a source is shown in Fig. 1.6. icd = b iab The current icd depends only on the control current iab and the dimensionless constant b, called the current gain. 1.6 some defInItIons 1. Network and Circuit The interconnec- tion of two or more circuit elements (viz., volt- agesources,resistors,inductorsandcapacitors) is called an electric network. If the network contains at least one closed path, it is called an electric circuit. Every circuit is a network, but all networks are not circuits. Figure 1.7(a) shows a network which is not a circuit and Fig. 1.7(b) shows a network which is a circuit. 2. Linear and Non-linear Elements If the resistance, inductance or capacitance offered by an element does not change linearly with the change in applied voltage or circuit current, the element is termed as linear element. Such an element shows a linear relation between voltage and current as shown in Fig. 1.8. Ordinary resistors, capacitors and inductors are examples of linear elements. a c iab b − + + − vcd d + − r iab Fig. 1.5 Symbol for CCVS a c b − + icd iab d + − b iab Fig. 1.6 Symbol for CCCS R L C (a) V (b) C R L Fig. 1.7 (a) Network which is not a circuit (b) Network which is a circuit
- 29. 1.7 Series and Parallel Combinations of Resistors 1.7 A non-linear circuit element is one in which the current does not change linearly with the change in applied voltage. A semiconductor diode operating in the curved region of characteristics as shown in Fig. 1.8 is common example of non-linear element. Other examples of non-linear elements are voltage- dependent resistor (VDR), voltage-dependent capacitor (varactor), temperature-dependent resistor (thermistor), light- dependent resistor (LDR), etc. Linear elements obey Ohm’s law whereas non-linear elements do not obey Ohm’s law. 3. Active and Passive Elements An element which is a source of electrical signal or which is capable of increasing the level of signal energy is termed as active element. Batteries, BJTs, FETs or OP-AMPs are treated as active elements because these can be used for the amplification or generation of signals. All other circuit elements, such as resistors, capacitors, inductors, VDR, LDR, thermistors, etc., are termed passive elements. The behaviour of active elements cannot be described by Ohm’s law. 4. Unilateral and Bilateral Elements If the magnitude of current flowing through a circuit element is affected when the polarity of the applied voltage is changed, the element is termed unilateral element. Consider the example of a semiconductor diode. Current flows through the diode only in one direction. Hence, it is called an unilateral element. Next, consider the example of a resistor. When the voltage is applied, current starts to flow. If we change the polarity of the applied voltage, the direction of the current is changed but its magnitude is not affected. Such an element is called a bilateral element. 5. Lumped and Distributed Elements A lumped element is the element which is separated physically, like resistors, inductors and capacitors. Distributed elements are those which are not separable for analysis purposes. Examples of distributed elements are transmission lines in which the resistance, inductance and capacitance are distributed along its length. 6. Active and Passive Networks A network which contains at least one active element such as an independent voltage or current source is an active network. A network which does not contain any active element is a passive network. 7. Time-invariant and Time-variant Networks A network is said to be time-invariant or fixed if its input–output relationship does not change with time. In other words, a network is said to time-invariant, if for any time shift in input, an identical time-shift occurs for output. In time-variant networks, the input–output relationship changes with time. 1.7 serIes and parallel combInatIons of resIstors Let R R R 1 2 3 , and be the resistances of three resistors connected in series across a dc voltage source V as shown in Fig. 1.9. Let V V V 1 2 3 , and be the voltages across resistances R R R 1 2 3 , and respectively. In series combination, the same current flows through each resistor but voltage across each resistor is different. I V 0 Linear Elem ent Non-Linear Elem ent Fig. 1.8 V-I characteristics of linear and non-linear elements R1 I V R2 R3 V3 V2 V1 Fig. 1.9 Series combination of resistors
- 30. 1.8 Circuit Theory and Transmission Lines V V V V R I R I R I R I R R R R T T = + + = + + = + + 1 2 3 1 2 3 1 2 3 Hence, when a number of resistors are connected in series, the equivalent resistance is the sum of all the individual resistance. Voltage Division and Power in a Series Circuit 1. I V R R R V R I R R R R V V R I R R R R V = + + = = + + = = + + 1 2 3 1 1 1 1 2 3 2 2 2 1 2 3 V R I R R R R V 3 3 3 1 2 3 = = + + Total power P P P P T = + + 1 2 3 = + + = + + I R I R I R V R V R V R 2 1 2 2 2 3 1 2 1 2 2 2 3 2 3 Figure 1.10 shows three resistors connected in parallel across a dc voltage source V. Let I I I 1 2 3 , and be the current flowing through resistors R R R 1 2 3 , and respectively. In parallel combination, the voltage across each resistor is same but current through each resistor is different. I I I I V R V R V R V R R R R R R R R R R R R R T T T = + + = + + = + + = + 1 2 3 1 2 3 1 2 3 1 2 3 2 3 3 1 1 1 1 1 + + R R 1 2 Hence, when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of reciprocals of individual resistances. Current Division and Power in a Parallel Circuit 2. V R I R I R I R I I V R R I R R R R R R R R R I T T = = = = = = = + + 1 1 2 2 3 3 1 1 1 2 3 1 2 2 3 3 1 I V R R I R R R R R R R R R I I V R R I R R R R R R R T T 2 2 2 1 3 1 2 2 3 3 1 3 3 3 1 2 1 2 2 3 = = = + + = = = + + + R R I 3 1 R1 R2 R3 I1 I2 I3 I V Fig. 1.10 Parallel combination of resistors
- 31. 1.8 Series and Parallel Combination of Inductors 1.9 Total power P P P P I R I R I R T = + + = + + 1 2 3 1 2 1 2 2 2 3 2 3 = + + V R V R V R 2 1 2 2 3 3 Note: For two branch circuits, R R R R R T = + 1 2 1 2 V R I R I R I I V R R I R R R R I I V R R I R R R R I T T T = = = = = = + = = = + 1 1 2 2 1 1 1 2 1 2 2 2 2 1 1 2 1.8 serIes and parallel combInatIon of Inductors Let L L L 1 2 3 , and be the inductances of three inductors connected in series across an ac voltage source v as shown in Fig. 1.11. Let v v v 1 2 3 , and be the voltages across inductances L L L 1 2 3 , and respectively. In series combination, the same current flows through each inductor but the voltage across each inductor is different. v v v v L di dt L di dt L di dt L di dt L L L L T T = + + = + + = + + 1 1 3 1 2 3 1 2 3 Hence, when a number of inductors are connected in series, the equivalent inductance is the sum of all the individual inductances. Figure 1.12 shows three inductors connected in parallel across an ac voltage source v. Let i i i 1 2 3 , and be the current through each inductance L L L 1 2 3 , and respectively. In parallel combination, the voltage across each inductor is same but the current through each inductor is different. i i i i L v dt L v dt L v dt L v dt T = + + = + + ∫ ∫ ∫ ∫ 1 2 3 1 2 3 1 1 1 1 1 1 1 1 1 2 3 1 2 3 1 2 2 3 3 1 L L L L L L L L L L L L L L T T = + + = + + Hence, when a number of inductors are connected in parallel, the reciprocal of the equivalent inductance is equal to the sum of reciprocals of individual inductances. v L1 v1 v2 i L2 L3 v3 Fig. 1.11 Series connection of inductors L1 L2 L3 i1 i2 i3 i v Fig. 1.12 Parallel connection of inductors
- 32. 1.10 Circuit Theory and Transmission Lines 1.9 serIes and parallel combInatIon of capacItors Let C C C 1 2 3 , and be the capacitances of three capacitors connected in series across an ac voltage source v as shown in Fig 1.13. Let v v v 1 2 3 , and be the voltages across capacitances C C C 1 2 3 , and respectively. In series combination, the charge on each capacitor is same but voltage across each capacitor is different. v v v v C i dt C i dt C i dt C i dt T = + + = + + ∫ ∫ ∫ ∫ 1 2 3 1 2 3 1 1 1 1 1 1 1 1 1 2 3 C C C C T = + + C C C C C C C C C C T = + + 1 2 3 1 2 2 3 3 1 Hence, when a number of capacitors are connected in series, the reciprocal of the equivalent capacitance is equal to the sum of reciprocals of individual capacitances. 1. Voltage Division in a Series Circuit Q C V C V C V C V V Q C C V C C C C C C C C C V T T = = = = = = = + + 1 1 2 2 3 3 1 1 1 2 3 1 2 2 3 3 1 V Q C C V C C C C C C C C C V V Q C C V C C C C C C C T T 2 2 2 1 3 1 2 2 3 3 1 3 3 3 1 2 1 2 2 3 = = = + + = = = + + + C C V 3 1 Figure 1.14 shows three capacitors connected in parallel across an ac voltage source v. Let i i i 1 2 3 , and be the current through each capacitance C C C 1 2 3 , and respectively. In parallel combination, the voltage across each capacitor is same but current through each capacitor is different. i i i i C dv dt C dv dt C dv dt C dv dt C C C C T T = + + = + + = + + 1 2 3 1 2 3 1 2 3 Hence, when a number of capacitors are connected in parallel, the equivalent capacitance is the sum of all the individual capacitance. 1.10 star-delta transformatIon Whenacircuitcannotbesimplifiedbynormalseries–parallelreductiontechnique,thestar-deltatransformation can be used. Figure 1.15 (a) shows three resistors RA , RB and RC connected in delta. Figure 1.15 (b) shows three resistors R1 , R2 and R3 connected in star. v1 C1 C2 C3 v2 v v3 Fig. 1.13 Series combination of capacitors C1 C2 C3 i1 i2 i3 i v Fig. 1.14 Parallel combination of capacitors
- 33. 1.10 Star-delta Transformation 1.11 1 2 3 1 RB R1 R2 R3 RC RA 2 3 (a) (b) Fig. 1.15 (a) Delta network (b) Star network These two networks will be electrically equivalent if the resistance as measured between any pair of terminals is the same in both the arrangements. 1.10.1 delta to Star Transformation Referring to delta network shown in Fig. 1.15 (a), the resistance between terminals 1 and 2 = R R R R R R R R R C A B C A B A B C ( ) ( ) + = + + + Referring to the star network shown in Fig. 1.15 (b), the resistance between terminals 1 and 2 = R R 1 2 + . Since the two networks are electrically equivalent, R R R R R R R R C A B A B C 1 2 + = + + + ( ) ...(1.1) Similarly, R R R R R R R R A B C A B C 2 3 + = + + + ( ) ...(1.2) and R R R R R R R R B A C A B C 3 1 + = + + + ( ) ...(1.3) Subtracting Eq. (1.2) from Eq. (1.1), R R R R R R R R R B C A B A B C 1 3 − = − + + ...(1.4) Adding Eq. (1.4) and Eq. (1.3), R R R R R R B C A B C 1 = + + Similarly, R R R R R R A C A B C 2 = + + R R R R R R A B A B C 3 = + + Thus, star resistor connected to a terminal is equal to the product of the two delta resistors connected to the same terminal divided by the sum of the delta resistors.
- 34. 1.12 Circuit Theory and Transmission Lines 1.10.2 Star to delta Transformation Multiplying the above equations, R R R R R R R R A B C A B C 1 2 2 2 = + + ( ) ...(1.5) R R R R R R R R A B C A B C 2 3 2 2 = + + ( ) ...(1.6) R R R R R R R R A B C A B C 3 1 2 2 = + + ( ) ...(1.7) Adding Eqs (1.5), (1.6) and (1.7), R R R R R R R R R R R R R R R R R R R R R A B C A B C A B C A B C A B C 1 2 2 3 3 1 2 + + = + + + + = + + ( ) ( ) = = = = R R R R R R A B C 1 2 3 Hence, R R R R R R R R R R R R R R R R R R R R R R R A B = + + = + + = + + = + + 1 2 2 3 3 1 1 2 3 2 3 1 1 2 2 3 3 1 2 1 3 R R R R R R R R R R R R R R R R R C 3 1 2 1 2 2 3 3 1 3 1 2 1 2 3 = + + = + + Thus, delta resistor connected between the two terminals is the sum of two star resistors connected to the same terminals plus the product of the two resistors divided by the remaining third star resistor. Note: (1) When three equal resistors are connected in delta (Fig. 1.16), the equivalent star resistance is given by R R R R R R R Y = + + = ∆ ∆ ∆ ∆ ∆ ∆ 3 R RY ∆ = 3 A A B B C C R∆ RY RY RY R∆ R∆ Fig. 1.16 Equivalent star resistance for three equal delta resistors (2) Star-delta transformation can also be applied to network containing inductors and capacitors.
- 35. 1.11 Source Transformation 1.13 1.11 source transformatIon A voltage source with a series resistor can be converted into a equivalent current source with a parallel resistor. Conversely, a current source with a parallel resistor can be converted into a voltage source with a series resistor as shown in Fig. 1.17. V V R R R (a) (b) ⇔ I = Fig. 1.17 Source transformation Source transformation can be applied to dependent sources as well. The controlling variable, however must not be tampered with any way since the operation of the controlled sources depends on it. example 1.1 Replace the given network of Fig. 1.18 with a single current source and a resistor. 10 A 6 Ω 5 Ω 20 V A B Fig. 1.18 Solution Since the resistor of 5 W is connected in parallel with the voltage source of 20 V it becomes redundant. Converting parallel combination of current source and resistor into equivalent voltage source and resistor (Fig. 1.19), By source transformation (Fig. 1.20), 13.33 A A B 6 Ω Fig. 1.20 example 1.2 Reduce the network shown in Fig. 1.21 into a single source and a single resistor be- tween terminals A and B. 6 Ω 6 Ω 80 V 60 V 20 V A B B A Fig. 1.19
- 36. 1.14 Circuit Theory and Transmission Lines 1 A A B 6 V 4 V 3 V 2 Ω 2 Ω 1 Ω 3 Ω Fig. 1.21 Solution Converting all voltage sources into equivalent current sources (Fig. 1.22), 1 A 2 A 3 A 2 A 2 Ω 2 Ω 1 Ω 3 Ω A B Fig. 1.22 Adding the current sources and simplifying the network (Fig. 1.23), 3 A 1 A 1 Ω 0.75 Ω A B Fig. 1.23 Converting the current sources into equivalent voltage sources (Fig. 1.24), 1 Ω 0.75 Ω 1.75 Ω 0.75 V 3.75 V 3 V A B B A Fig. 1.24 example 1.3 Replace the circuit between A and B in Fig. 1.25 with a voltage source in series with a single resistor.
- 37. 1.11 Source Transformation 1.15 50 Ω 30 Ω 3 A 5 Ω 6 Ω 20 V A B Fig. 1.25 Solution Converting the series combination of voltage source of 20 V and a resistor of 5 W into equivalent parallel combination of current source and resistor (Fig. 1.26), 3 A 4 A 30 Ω 50 Ω 5 Ω 6 Ω A B Fig. 1.26 Adding the two current sources and simplifying the circuit (Fig. 1.27), 7 A A B 30 || 50 || 5 || 6 = 2.38 Ω Fig. 1.27 By source transformation (Fig. 1.28), 16.67 V 2.38 Ω A B Fig. 1.28 example 1.4 Find the power delivered by the 50 V source in the network of Fig. 1.29. 5 Ω 2 Ω 50 V 3 Ω 10 V 10 A Fig. 1.29
- 38. 1.16 Circuit Theory and Transmission Lines Solution Converting the series combination of voltage source of 10 V and resistor of 3 W into equivalent current source and resistor (Fig. 1.30), 2 Ω 3 Ω 5 Ω 10 A 50 V 3.33 A Fig. 1.30 Adding the two current sources and simplifying the network (Fig. 1.31), 5 Ω 1.2 Ω 50 V 13.33 A Fig. 1.31 By source transformation (Fig. 1.32), 5 Ω 1.2 Ω 50 V 16 V I Fig. 1.32 I = − + = 50 16 5 1 2 5 48 . . A Power delivered by the 50 V source = 50 × 5.48 = 274 W example 1.5 Find the current in the 4 W resistor shown in network of Fig. 1.33. 2 Ω 4 Ω 6 V 2 A 5 A Fig. 1.33 Solution Converting the parallel combination of the current source of 5 A and the resistor of 2 W into an equivalent series combination of voltage source and resistor (Fig. 1.34),
- 39. 1.11 Source Transformation 1.17 2 Ω 4 Ω 2 A 6 V 10 V Fig. 1.34 Adding two voltage sources (Fig. 1.35), 2 Ω 4 Ω 4 V 2 A Fig. 1.35 Again by source transformation (Fig. 1.36), 2 Ω 4 Ω 2 A 2 A Fig. 1.36 Adding two current sources (Fig. 1.37), 2 Ω 4 Ω 4 A Fig. 1.37 By current-division rule, I4 4 2 2 4 1 33 W = × + = . A example 1.6 Find the voltage across the 4 W resistor shown in network of Fig. 1.38. 3 Ω 6 Ω 4 Ω 2 Ω 1 Ω 6 V 3 A Fig. 1.38
- 40. 1.18 Circuit Theory and Transmission Lines Solution Converting the series combination of the voltage source of 6 V and the resistor of 3 W into equivalent current source and resistor (Fig. 1.39), 2 A 3 A 3 Ω 6 Ω 4 Ω 1 Ω 2 Ω Fig. 1.39 By series–parallel reduction technique (Fig. 1.40), 2 A 3 A 2 Ω 4 Ω 1 Ω 2 Ω Fig. 1.40 By source transformation (Fig. 1.41), 2 Ω 2 Ω 1 Ω 4 Ω 4 V 3 A (a) 4 Ω 1 Ω 4 Ω 4 V 3 A (b) 1 A 3 A (c) 4 Ω 4 Ω 1 Ω (d) 4 A 4 Ω 4 Ω 1 Ω (e) 4 Ω 1 Ω 4 Ω 16 V I Fig. 1.41 I = + + = 16 4 1 4 1 78 . A Voltage across the 4 W resistor = = × = 4 4 1 78 7 12 I . . V
- 41. 1.12 Source Shifting 1.19 example 1.7 Find the voltage at Node 2 of the network shown in Fig. 1.42. 50 Ω 100 Ω 100 Ω + − 15 V I 1 2 10 I Fig. 1.42 Solution We cannot change the network between nodes 1 and 2 since the controlling current I, for the controlled source, is in the resistor between these nodes. Applying source transformation to series combination of controlled source and the 100 W resistor (Fig. 1.43), 50 Ω 50 Ω 50 Ω 100 Ω 100 Ω 50 Ω + − 15 V 15 V 15 V 5 I 1 1 2 2 50 Ω 1 2 0.1 I 0.1 I I I I Fig. 1.43 Applying KVL to the mesh, 15 50 50 5 0 15 105 0 143 − − − = = = I I I I . A Voltage at Node 2 = − = − × = 15 50 15 50 0 143 7 86 I . . V 1.12 source sHIftInG Source shifting is the simplification technique used when there is no resistor in series with a voltage source or a resistor in parallel with a current source.
- 42. 1.20 Circuit Theory and Transmission Lines example 1.8 Calculate the voltage across the 6 W resistor in the network of Fig. 1.44 using source- shifting technique. 3 Ω 4 Ω 2 Ω 6 Ω 1 Ω Va + − 2 1 3 4 18 V Fig. 1.44 Solution Adding a voltage source of 18 V to the network and connecting to Node 2 (Fig. 1.45), we have 3 Ω 4 Ω 1 Ω 2 Ω 6 Ω Va + − 2 1 3 4 18 V 18 V Fig. 1.45 Since nodes 1 and 2 are maintained at the same voltage by the sources, the connection between nodes 1 and 2 is removed. Now the two voltage sources have resistors in series and source transformation can be applied (Fig. 1.46). 3 Ω 4 Ω 1 Ω 2 Ω 6 Ω Va + − 18 V 18 V Fig. 1.46
- 43. Exercises 1.21 Simplifying the network (Fig. 1.47), ` 4 Ω 3 Ω 1 Ω 2 Ω 6 Ω Va + − Va + − 18 V 3 Ω 1.33 Ω 1 Ω 6 Ω 18 V 4.5 A 4.5 A (a) (b) 3 Ω 1.33 Ω 1 Ω 3 Ω 6 Ω 2.33 Ω 6 Ω Va Va 18 V 5.985 V 18 V 5.985 V + − (c) (d) Fig. 1.47 Applying KCL at the node, V V V a a a − + − + = 18 3 5 985 2 33 6 0 . . Va = 9 23 . V Exercises 1.1 Use source transformation to simplify the network until two elements remain to the left of terminals A and B. 20 mA 6 kΩ 2 kΩ 3 kΩ 12 kΩ 3.5 kΩ A B Fig. 1.48 [88.42 V, 7.92 k W] 1.2 Determine the voltage Vx in the network of Fig. 1.49 by source-shifting technique. 2 V 3 Ω 2 Ω 1 Ω 2 Ω 5 Ω Vx Fig. 1.49 [1.129 V]
- 44. 1.22 Circuit Theory and Transmission Lines Objective-Type Questions A network contains linear resistors and ideal 1.1 voltage sources. If values of all the resistors are doubled then the voltage across each resistor is (a) halved (b) doubled (c) increased by four times (d) not changed Four resistances 80 1.2 W, 50 W, 25 W, and R are connected in parallel. Current through 25 W resistor is 4 A. Total current of the supply is 10 A. The value of R will be (a) 66.66 W (b) 40.25 W (c) 36.36 W (d) 76.56 W Viewed from the terminal AB, the network 1.3 of Fig. 1.50 can be reduced to an equivalent networkofasinglevoltagesourceinserieswith a single resistor with the following parameters 10 V 5 V 4 Ω 10 Ω A B Fig. 1.50 (a) 5 V source in series with a 10 W resistor (b) 1 V source in series with a 2.4 W resistor (c) 15 V source in series with a 2.4 W resistor (d) 1 V source in series with a 10 W resistor A 10 V battery with an internal resistance of 1 1.4 W is connected across a nonlinear load whose V-I characteristic is given by 7 2 2 I V V = + . The current delivered by the battery is (a) 0 (b) 10 A (c) 5 A (d) 8 A If the length of a wire of resistance 1.5 R is uniformly stretched to n times its original value, its new resistance is (a) nR (b) R n (c) n2 R (d) R n2 All the resistances in Fig. 1.51 are 1 1.6 W each. The value of I will be I 1 V Fig. 1.51 (a) 1 15 A (b) 2 15 A (c) 4 15 A (d) 8 15 A The current waveform in a pure resistor at 10 1.7 W is shown in Fig. 1.52. Power dissipated in the resistor is i t 0 9 3 6 Fig. 1.52 (a) 7.29 W (b) 52.4 W (c) 135 W (d) 270 W Two wires 1.8 A and B of the same material and length L and 2L have radius r and 2r respectively. The ratio of their specific resistance will be (a) 1 : 1 (b) 1 : 2 (c) 1 : 4 (d) 1 : 8
- 45. Answers to Objective-Type Questions 1.23 Answers to Objective-Type Questions 1.1. (d) 1.2. (c) 1.3. (b) 1.4. (c) 1.5. (c) 1.6. (d) 1.7. (d) 1.8. (b)
- 47. 2 Elementary Network Theorems 2.1 IntroductIon In Chapter 1, we have studied basic network concepts. In network analysis, we have to find currents and voltages in various parts of networks. In this chapter, we will study elementary network theorems like Kirchhoff’s laws, mesh analysis and node analysis. These methods are applicable to all types of networks. The first step in analyzing networks is to apply Ohm’s law and Kirchhoff’s laws. The second step is the solving of these equations by mathematical tools. 2.2 KIrcHHoFF’S LAWS The entire study of electric network analysis is based mainly on Kirchhoff’s laws. But before discussing this, it is essential to familiarise ourselves with the following terms: Node A node is a junction where two or more network elements are connected together. Branch An element or number of elements connected between two nodes constitute a branch. Loop A loop is any closed part of the circuit. Mesh A mesh is the most elementary form of a loop and cannot be further divided into other loops. All meshes are loops but all loops are not meshes. 1. Kirchhoff’s Current Law (KCL) The algebraic sum of currents meeting at a junction or node in an electric circuit is zero. Consider five conductors, carrying currents I1, I2, I3, I4 and I5 meeting at a point O as shown in Fig. 2.1. Assuming the incoming currents to be positive and outgoing currents negative, we have I I I I I I I I I I I I I I I 1 2 3 4 5 1 2 3 4 5 1 3 5 2 4 0 0 + − + + − + = − + − + = + + = + ( ) ( ) Thus, the above law can also be stated as the sum of currents flowing towards any junction in an electric circuit is equal to the sum of the currents flowing away from that junction. 2. Kirchhoff’s Voltage Law (KVL) The algebraic sum of all the voltages in any closed circuit or mesh or loop is zero. If we start from any point in a closed circuit and go back to that point, after going round the circuit, there is no increase or decrease in potential at that point. This means that the sum of emfs and the sum of voltage drops or rises meeting on the way is zero. I1 I2 I3 I4 I5 O Fig. 2.1 Kirchhoff’s current law
- 48. 2.2 Circuit Theory and Transmission Lines 3. Determination of Sign A rise in potential can be assumed to be positive while a fall in potential can be considered negative. The reverse is also possible and both conventions will give the same result. (i) If we go from the positive terminal of the battery or source to the negative terminal, there is a fall in potential and so the emf should be assigned a negative sign(Fig. 2.2a). If we go from the negative terminal of the battery or source to the positive terminal, there is a rise in potential and so the emf should be given a positive sign(Fig. 2.2b). (a) Fall in potential (b) Rise in potential Fig. 2.2 Sign convention (ii) When current flows through a resistor, there is a voltage drop across it. If we go through the resistor in the same direction as the current, there is a fall in the potential and so the sign of this voltage drop is negative(Fig. 2.3a). If we go opposite to the direction of the current flow, there is a rise in potential and hence, this voltage drop should be given a positive sign(Fig. 2.3b). (b) Rise in potential (a) Fall in potential I I + + − − Fig. 2.3 Sign convention 2.3 MESH AnALYSIS A mesh is defined as a loop which does not contain any other loops within it. Mesh analysis is applicable only for planar networks. A network is said to be planar if it can be drawn on a plane surface without crossovers. In this method, the currents in different meshes are assigned continuous paths so that they do not split at a junction into branch currents. If a network has a large number of voltage sources, it is useful to use mesh analysis. Basically, this analysis consists of writing mesh equations by Kirchhoff’s voltage law in terms of unknown mesh currents. Steps to be Followed in Mesh Analysis Identify the mesh, assign a direction to it and assign an unknown current in each mesh. 1. Assign the polarities for voltage across the branches. 2. Apply KVL around the mesh and use Ohm’s law to 3. express the branch voltages in terms of unknown mesh currents and the resistance. Solve the simultaneous equations for unknown mesh 4. currents. Consider the network shown in Fig. 2.4 which has three meshes. Let the mesh currents for the three meshes be I1, I2, and I3 and all the three mesh currents may be assumed to flow in the clockwise direction. The choice of direction for any mesh current is arbitrary. Applying KVL to Mesh 1, V R I I R I I R R I R I R I V 1 1 1 2 2 1 3 1 2 1 1 2 2 3 1 0 − − − − = + − − = ( ) ( ) ( ) …(2.1) V1 V3 V2 R3 R1 R4 R2 R5 I3 I2 I1 Fig. 2.4 Circuit for mesh analysis
- 49. 2.3 Mesh Analysis 2.3 Applying KVL to Mesh 2, V R I R I I R I I R I R R R I R I V 2 3 2 4 2 3 1 2 1 1 1 1 3 4 2 4 3 2 0 − − − − − = − + + + − = ( ) ( ) ( ) …(2.2) Applying KVL to Mesh 3, − − − − − + = − − + + + = R I I R I I R I V R I R I R R R I V 2 3 1 4 3 2 5 3 3 2 1 4 2 2 4 5 3 3 0 ( ) ( ) ( ) …(2.3) Writing Eqs (2.1),(2.2), and(2.3) in matrix form, R R R R R R R R R R R R R R I I I 1 2 1 2 1 1 3 4 4 2 4 2 4 5 1 2 3 + − − − + + − − − + + = V V V 1 2 3 In general, R R R R R R R R R I I I V V V 11 12 13 21 22 23 31 32 33 1 2 3 1 2 3 = where, R11 = Self-resistance or sum of all the resistance of mesh 1 R12 = R21 = Mutual resistance or sum of all the resistances common to meshes 1 and 2 R13 = R31 = Mutual resistance or sum of all the resistances common to meshes 1 and 3 R22 = Self-resistance or sum of all the resistance of mesh 2 R23 = R32 = Mutual resistance or sum of all the resistances common to meshes 2 and 3 R33 = Self-resistance or sum of all the resistance of mesh 3 If the directions of the currents passing through the common resistance are the same, the mutual resistance will have a positive sign, and if the direction of the currents passing through common resistance are opposite then the mutual resistance will have a negative sign. If each mesh current is assumed to flow in the clockwise direction then all self-resistances will always be positive and all mutual resistances will always be negative. The voltages V1, V2 and V3 represent the algebraic sum of all the voltages in meshes 1, 2 and 3 respectively. While going along the current, if we go from negative terminal of the battery to the positive terminal then its emf is taken as positive. Otherwise, it is taken as negative. Example 2.1 Find the current through the 5 W resistor is shown in Fig. 2.5. 10 V 5 V 20 V 1 Ω 2 Ω 3 Ω 6 Ω 4 Ω 5 Ω Fig. 2.5 Solution Assigning clockwise currents in three meshes as shown in Fig. 2.6. Applying KVL to Mesh 1, 10 1 3 6 0 10 3 6 10 1 2 1 3 1 2 3 − − − − − = − − = I I I I I I I I 1 ( ) ( ) …(i)
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- 54. The Project Gutenberg eBook of The Automatic Toy Works
- 55. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: The Automatic Toy Works Author: New York Automatic Toy Works Release date: January 7, 2018 [eBook #56336] Most recently updated: July 20, 2020 Language: English Credits: Produced by MWS and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) *** START OF THE PROJECT GUTENBERG EBOOK THE AUTOMATIC TOY WORKS ***
- 56. Transcriber’s Note: A warning to the reader. The toys depicted in this catalogue are, for the most part, outdated and offensive racial stereotypes. THE AUTOMATIC TOY WORKS THE AUTOMATIC TOY WORKS MANUFACTURERS OF THE BEST NOVELTIES IN Mechanical and other Toys UNDER LETTERS PATENT. No. 20 College Place, (CORNER PARK PLACE,) NEW YORK CITY, U. S. A. Lockwood & Crawford, Stationers, 59 Pine Street, New York.
- 57. TO THE TRADE. SPRING, 1882. e present this revised and enlarged edition of our Illustrated Catalogue to the Trade, confidently believing that it will be of service in the selection of Mechanical Toys and Novelties which have gained so great favor and are now so popular in this country and abroad. In style, finish and dress of our figures, we have made very many improvements and have added several new kinds to our list. In the partiality which has been universally shown in favor of our Toys, and the constantly increasing demand for them, we find an incentive to more than sustain their reputation in the future. The Toys represented in this Catalogue are all mechanical and are set in motion, on being wound up, by patent movements. Each Toy is packed in a substantial wooden box, and will be sent, postpaid, to any address on receipt of price, where our goods are not found with dealers. Discount rates sent to the trade on application. Automatic Toy Works, 20 College Place, NEW YORK CITY, U. S. A.
- 59. The Mechanical Bear. This wonderful toy imitates the movements of a bear, by means of clock-work, in the most life-like manner. The bear rises up on its hind legs, turns its head from side to side, growls, moves its paws, and snaps its jaws together. At intervals it gets down on its fore-paws and goes through similar motions. It runs a long time, and while going it is difficult to believe it is not alive. It is elegantly made, and covered with fine fur. The mechanism is so strong and
- 60. perfect, that only the greatest abuse can put it out of order. It amuses old as well as young, and is exceedingly attractive for a show window. It is conceded to be one of the most ingenious toys ever invented, on account of its variety of motion and resemblance to nature. Made in black and white fur. Price, $4.25.
- 61. THE Mechanical Sewing-Machine Girl THE DELIGHT OF ALL GIRLS.—A CHARMING TOY, AND BEAUTIFULLY FINISHED A little girl is seated at a cabinet sewing-machine. On winding up the mechanism her feet begin to work the
- 62. treadle, and the sewing-machine begins to sew rapidly; she leans forward, puts the work in position, watches it, occasionally rising up and bringing the work up to examine it. These movements are repeated for a long time. The little figure is elegantly dressed in the latest fashion. It combines the attractiveness of a beautiful French doll with the interest of life-like motion. Price, $3.50.
- 63. Old Uncle Tom, THE COLORED FIDDLER. We consider this toy one of the most comically quaint of anything yet made. When seen in motion, laughter is irresistible. The old fellow commences the performance by slowly rocking backward and forward, as if debating what he should play, then suddenly he strikes his “favorite,” and rolling his head from side to side, fiddles in an ecstacy of
- 64. enjoyment. Funny as it is, there is something almost pathetic in it, too. This toy is well and carefully made, and with ordinary care will last for years. Price, $2.50.
- 65. The Celebrated Negro Preacher. He stands behind a desk, and slowly straightening himself up, turns his head from side to side and gestures vigorously with his arm. As he warms to his work, he leans forward over the pulpit, and shakes his head and hand at the audience, and vigorously thumps the desk. The motions are so life-like and comical that one almost
- 66. believes that he is actually speaking. The face and dress alone provoke irresistible laughter. He preaches as long as any preacher ought to, and stops when he gets through. Price, $2.50.
- 67. Our New Clergyman. BRUDDER GARDNER. The description on the opposite page applies to this brudder also. Price, $2.50.
- 69. Old Aunt Chloe, THE NEGRO WASHERWOMAN. Old Aunt Chloe demonstrates that happiness may be found in a wash tub as well as in a palace. She is faithful at her toil, and we commend her to our young ladies as an artist of no mean pretentions, after whom they may pattern if they choose to revive and become proficient in one of the lost accomplishments.
- 70. Price, $2.50.
- 71. The Old Nurse. This mechanical toy is made to imitate an old negro nurse with a child. Her motions are as natural as life. She holds the child in her hands and when the mechanism is started, (by being wound) she leans backward and forward tossing the child up and down in a most surprising manner. This is a very pleasing toy for children and is very popular.
- 72. Price, $2.50.
- 73. Fing Wing, A MELICAN MAN. This image, with its shaven head, long queue and quaint looking dress, gives a striking and life-like picture of a Chinese Laundryman. When at work, he bends over the tub, and rubs the garment which he holds in his hands with a naturalness so perfect he might easily be mistaken for a real Celestial.
- 74. Price, $2.50.
- 75. Ah-Sin. “THE HEATHEN CHINESE.” This piece is similar to the Laundryman represented on the opposite page. It shows Ah-Sin with a smoothing iron, putting the polish upon a gentleman’s linen. The mechanism of these novelties is so perfectly made, that only the greatest abuse can put them out of order.
- 76. Price, $2.50.
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