FUNDAMENTALS OF ELECTRIC CIRCUIT THEORY

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FUNDAMENTALS OF ELECTRIC CIRCUIT THEORY For B.E. / B.Tech. Students of Electronics, Electrical, Computer, Control, Instrumentation, Communication Engineering and Information Technology.

2 FUNDAMENTALS OF ELECTRIC CIRCUIT THEORY For B.E. / B.Tech. Students of Electronics, Electrical, Computer, Control, Instrumentation, Communication Engineering and Information Technology. Book is also useful for B.Sc. and M.Sc. Students of Physics D. CHATTOPADHYAY M.Tech., Ph.D., D.Sc., P.R.S. (Mouat medalist) Former Professor and Head, Institute of Radio Physics and Electronics Calcutta University P. C. RAKSHIT M.Tech., Ph.D. Former Professor and Head, Institute of Radio Physics and Electronics Calcutta University S. CHAND & COMPANY LTD. (AN ISO 9001 : 2000 COMPANY) RAM NAGAR, NEW DELHI

3 S. CHAND & COMPANY LTD. (An ISO 9001 : 2000 Company) Head Office: 7361, RAM NAGAR, NEW DELHI Phone: , , Fax: Shop at: schandgroup.com; info@schandgroup.com Branches : AHMEDABAD : 1st Floor, Heritage, Near Gujarat Vidhyapeeth, Ashram Road, Ahmedabad , Ph: , , ahmedabad@schandgroup.com BENGALURU : No. 6, Ahuja Chambers, 1st Cross, Kumara Krupa Road, Bengaluru , Ph: , , bangalore@schandgroup.com BHOPAL : Bajaj Tower, Plot No. 243, Lala Lajpat Rai Colony, Raisen Road, Bhopal , Ph: bhopal@schandgroup.com CHANDIGARH : S.C.O , First Floor, Sector - 22-C (Near Aroma Hotel), Chandigarh , Ph: , , chandigarh@schandgroup.com CHENNAI : 152, Anna Salai, Chennai , Ph: , , chennai@schandgroup.com COIMBATORE : Plot No. 5, Rajalakshmi Nagar, Peelamedu, Coimbatore , (M) , coimbatore@schandgroup.com (Marketing Office) CUTTACK : 1st Floor, Bhartia Tower, Badambadi, Cuttack , Ph: ; , cuttack@schandgroup.com DEHRADUN : 1st Floor, 20, New Road, Near Dwarka Store, Dehradun , Ph: , , dehradun@schandgroup.com GUWAHATI : Pan Bazar, Guwahati , Ph: , guwahati@schandgroup.com HYDERABAD : Padma Plaza, H.No , Opp. Ratna College, Narayanaguda, Hyderabad , Ph: , , hyderabad@schandgroup.com JAIPUR : A-14, Janta Store Shopping Complex, University Marg, Bapu Nagar, Jaipur , Ph: , jaipur@schandgroup.com JALANDHAR : Mai Hiran Gate, Jalandhar , Ph: , , jalandhar@schandgroup.com JAMMU : 67/B, B-Block, Gandhi Nagar, Jammu , (M) (Marketing Office) KOCHI : Kachapilly Square, Mullassery Canal Road, Ernakulam, Kochi , Ph: , cochin@schandgroup.com KOLKATA : 285/J, Bipin Bihari Ganguli Street, Kolkata , Ph: , , kolkata@schandgroup.com LUCKNOW : Mahabeer Market, 25 Gwynne Road, Aminabad, Lucknow , Ph: , , lucknow@schandgroup.com MUMBAI : Blackie House, 103/5, Walchand Hirachand Marg, Opp. G.P.O., Mumbai , Ph: , , mumbai@schandgroup.com NAGPUR : Karnal Bag, Model Mill Chowk, Umrer Road, Nagpur , Ph: , nagpur@schandgroup.com PATNA : 104, Citicentre Ashok, Govind Mitra Road, Patna , Ph: , , patna@schandgroup.com PUNE : 291/1, Ganesh Gayatri Complex, 1st Floor, Somwarpeth, Near Jain Mandir, Pune , Ph: , pune@schandgroup.com (Marketing Office) RAIPUR : Kailash Residency, Plot No. 4B, Bottle House Road, Shankar Nagar, Raipur , Ph: , raipur@schandgroup.com (Marketing Office) RANCHI : Flat No. 104, Sri Draupadi Smriti Apartments, East of Jaipal Singh Stadium, Neel Ratan Street, Upper Bazar, Ranchi , Ph: , ranchi@schandgroup.com (Marketing Office) SILIGURI : 122, Raja Ram Mohan Roy Road, East Vivekanandapally, P.O., Siliguri , Dist., Jalpaiguri, (W.B.) Ph (Marketing Office) VISAKHAPATNAM: Plot No. 7, 1st Floor, Allipuram Extension, Opp. Radhakrishna Towers, Seethammadhara North Extn., Visakhapatnam , (M) , visakhapatnam@schandgroup.com (Marketing Office) 1987, D. Chattopadhyay & P.C. Rakshit All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Publishers. First Edition 1987 Subsequent Editions and Reprints 1991, 2000, 2003, 2004, 2006, 2009 Ninth Revised Edition 2011 ISBN : Code : 10B 108 PRINTED IN INDIA By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi and published by S. Chand & Company Ltd., 7361, Ram Nagar, New Delhi

4 PREFACE TO THE NINTH EDITION In this edition, changes have been made at some places in the text, and several new problems have been included for the benefit of the students. The users of the book are sincerely thanked for their support. PREFACE TO THE FIRST EDITION AUTHORS This book arises primarily out of the course given by the authors at the Institute of Radio Physics and Electronics, Calcutta University. It is meant to be a textbook for the beginners in Electronics, Electrical, Computer, Control, and Communication Engineering. Parts of the book may also be used with advantage by the undergraduate and the postgraduate students of Physics. Normally, a beginner has to depend for the theory of linear circuits on a number of books, all of which are not readily available. Hence there is definitely a need for a standard textbook which covers in a single volume the material needed for a beginner in a comprehensive way. The present book is an attempt in this direction. The contents of this book are suitable for the two-semester course on circuit theory. Emphasis has been, as the title indicates, on the fundamentals, keeping in view the requirements of the beginners. Some modern aspects of circuit theory have also been included for the benefit of students. We have tried to present the subject matter in a clear and concise manner with numerous diagrams and examples. In each chapter, a variety of problems has been worked out by way of illustrations. A large number of problems has also been set as exercises at the end of each chapter so that the students may find an opportunity to apply the principles and techniques they have learnt in the text. Short review questions and miscellaneous problems are given at the end of the book. The problems given in the book induce in the students a habit of thinking much needed in their professional career in future. Every effort has been made to give as much insight into circuit theory as possible within a time a student can reasonably devote to the subject in his busy curriculum. We wish to thank the Institution of Electronics and Telecommunication Engineers, the Institution of Engineers (India), and Union Public Service Commission for their permission to use problems from past examination papers. Institute of Radio Physics and Electronics 92, Acharya Prafulla Chandra Road Calcutta , India. AUTHORS

5 GUSTAV ROBERT KIRCHHOFF ( ) The German physicist, Gustav Robert Kirchhoff, was born at Königsberg in March, He was educated at the University of Königsberg and was appointed as an extraordinary professor of physics at Breslau in He became a professor of physics at Heidelberg in 1854 and shifted to Berlin in Kirchhoff made significant contributions in conduction and radiation of heat, thermodynamics of solutions, vaporization and chemical reactions, crystalline reflections and refractions, spectral analysis etc. In electricity, he developed two useful laws Kirchhoff s voltage and current laws - which provided the basis for electric circuit theory. A major part of his work is described in his Vorlesungen iiber mathematische physik. Kirchhoff died in Berlin in October, 1887.

6 CONTENTS Chapter Pages 1. BASIC CIRCUIT ELEMENTS Introduction (1) 1.2. Classification of Elements (2) 1.3. Voltage and Current Sources (2) 1.4. Nullators and Norators (5) 1.5. AC Responses of Circuit Parameters (6) (a) Resistance (7); (b) Inductance (7); (c) Capacitance (8); 1.6. Two Important Conservation Laws (9) 1.7. Effective or Root-Mean-Square (RMS) Values (10) 1.8. Worked-out Problems (11) Questions (13). 2. COMPLEX NUMBERS IN CIRCUIT ANALYSIS The Sinusoid (16) 2.2. Phasor Algebra (18) 2.3. Application to Circuits (19) (a) R L. Circuit (19); (b) R C. Circuit (21); (c) R L C. Circuit (23) 2.4. Average Power and Reactive Power (24) 2.5. Complex Frequency (26) 2.6. Worked-out Problems (27) Questions (32). 3. SOME IMPORTANT CONCEPTS Kirchhoff s Laws (34) 3.2. Series and Parallel Connection of Impedances (35) 3.3. Admittance (36) 3.4. The Quality Factor or Q (37) Q of an inductor (37) Q of a Capacitor (38) 3.5. B and G in terms of Q (39) 3.6. Duality (40) 3.7. Transformation of Voltage and Current Sources (40) (a) Reduction of a number of voltage (or current) sources to a single voltage (or current) source (40) (b) Equivalence of voltage and current sources (41); Voltage source with no series impedance (42) Current source with no parallel impedance (42) Reduction of complicated connection of source branches (42) 3.8. Worked-out Problems (43) Questions (49). 4. RESONANCE Series Resonance (52) Sharpness of resonance (53) Occurrence of maximum voltage across L and C (55) Locus diagrams (55) Reactance and susceptance sketches (57) 4.2. Parallel Resonance (58) Simpler parallel resonant circuit (59) Currents in parallel resonant circuit (60) Reactance, susceptance and impedance sketches (61) 4.3. Universal Resonance Curve (61) 4.4. Selectivity of Resonant Circuits (63) (a) Series circuit (63); (b) Parallel circuit (64) 4.5. Self Resonance of Coils (64) 4.6. Poles and Zeros : Circuit Behaviour from their Location (65) 4.7. Worked-out Problems (67) Questions (74). 5. NETWORK ANALYSIS Network and Its Topological Description (79) 5.2. Network Variables (80) Mesh Method of Analysis (81) Node Method of Analysis (83) 5.5. Some Comments (84) 5.6. Matrix Representation (86) Dual and Inverse Networks (87) 5.8. Driving Point and Transfer Impedances (88) State Variable Analysis (90) Worked-out Problems (93) Questions (102). 6. ELEMENTS OF GRAPH THEORY AND NETWORK APPLICATIONS Introduction (108) 6.2. A Few Important Terms (108) 6.3. Incidence Matrix (108) (v)

7 6.4. Tie-Set Matrix (110) 6.5. CutSet Matrix (112) 6.6. Generalised Network Equations (114) 6.7. Worked-Out Problems (115) Questions (123). 7. NETWORK CONVERSIONS AND NETWORK THEOREMS Two-Port Networks and their Reduction (125) 7.2. T- Transformation (127) 7.3. The Bridged-T Network (128) 7.4. The Lattice Network (129) 7.5. Superposition Theorem (130) 7.6. Millman s Theorem (130) 7.7. Thevenin s Theorem (131) 7.8. Norton s Theorem (132) 7.9. Reciprocity Theorem (133) Substitution and Compensation Theorems (134) Maximum Power Transfer Theorem (135) Matching and Sharpness of Resonance of a Parallel Resonant Circuit Connected to a Voltage Source of Finite Internal Resistance (137) Tellegen s Theorem (138) Worked-out Problems (139) Questions (151). 8. INDUCTIVELY COUPLED CIRCUITS Introduction (156) Mutual Inductance and its Evaluation (156) 8.3. Identification of Relative Polarities (158) 8.4. Equivalent Inductance of Conductively Connected Mutually Coupled Coils (158) 8.5. Transformer (160) 8.6. Transformer Adjustment for Maximum Power Transfer to the Load (163) 8.7. Singly Tuned Transformer (163) 8.8. Doubly Tuned Transformer (165) 8.9. Worked-out Problems (171) Questions (182). 9. SYNTHESIS OF TWO-TERMINAL REACTIVE NETWORKS Introduction (187) 9.2. Foster s Reactance Theorem (187) 9.3. Separation Property of Poles and Zeros (188) 9.4. Network Realisation of Reactance Function (189) 9.5. Different Forms of Reactance Function (190) 9.6. Driving-Point Admittance (Second Foster Network) (191) 9.7. Analytic Form for an Arbitrary Two-Terminal Network (192) 9.8. Canonic Networks (193) 9.9. Continued Fraction Networks or Cauer Networks (194) Worked-out Problems (196) Questions (201). 10. SYNTHESIS OF TWO-TERMINAL R-L AND R-C NETWORKS Introduction (204) Two-Terminal R-L Network (204) (a) Driving-point impedance (204); (b) Driving-point admittance (207); (c) Networks derived from continued fraction expansion of Z D (s) (208); Two-Terminal R-C. Network (209) (a) R-C impedance (209); (b) R-C admittance (211) (c) R-C continued fraction network (212) Instructions for synthesis of R-L and R-C. impedances (213) Positive Real Functions (212) Properties of positive real functions (213) Testing Hurwitz polynomials (213) Testing Re[F(s)] 0 for PRF (215) Synthesis from Positive Real Functions (217) Worked-out Problems (217) Questions (225). 11. TWO-PORT NETWORKS Introduction (228) Open-Circuit and Short-Circuit Parameters (228) Transmission Parameters (231) Hybrid or Mixed Parameters (h-parameters) and g-parameters (232) Matrix forms of Input-Output Relations (235) Interconnection of Two Four-Terminal Networks (235) (i) Connection in cascade (235) (ii) Connection in parallel (236) (iii) Connection in series (236) Validity tests for the matrix method (237) Characteristics of Unsymmetrical Networks (239) Image impedances (239) Expressions for Zi 1 and Zi 2 (240) Propagation function (241) Characteristics of Symmetrical Networks (242) Propagation function (242) Iterative Impedances (243) Lattice Network (244) Balanced and Unbal- (vi)

8 anced Networks (245) Bartlett s Bisection Theorem (245) Proof of the theorem (246) Determination of Z 0 and of T, and Bridged-T networks (246) Conversion of Lattice into Equivalent T and Networks (248) Nonreciprocal Two-port Network (249) Terminated Two-port Circuits (251) Special Two-port Networks (252) Worked-out Problems (255) Questions (267). 12. ATTENUATORS, INSERTION LOSS, AND EQUALISERS Introduction (273) Units of attenuation (273) Lattice Attenuator (274) T-Type Attenuator (275) Type Attenuator (276) Bridged-T Attenuator (276) Asymmetric Attinuators (277) (a) L-Type Attenuator (277) (b) Asymmetric T and Attenuators (278) Ladder Type Attenuator (279) Balanced Attenuator (280) Insertion Loss (280) Amplitude Equalisers (282) Workedout Problems (285) Questions (291). 13. CLASSICAL FILTERS Definition and Classification (293) Filter Configurations and Characteristics (293) Mechanism of Filter Action (295) Low-Pass Filter (296) (i) Possible Lattice Section (296) (ii) Design of prototype sections (296) High-Pass Filters (299) (i) Lattice configuration (299) (ii) Constant-k prototype sections (300) m-derived Structure (302) m-derived T-section (303) Impedance match with m- derived half sections (L-sections) (305) m-derived p-section (308) Composite Filter (309) Band-Pass Constant-k Filter (309) m-derived band-pass filter (312) Band Elimination Filter (314) Doubly Tuned Transformer as a Band-pass Filter (314) Modern Filter Design Concept : Approximation Methods (316) Worked-out Problems (320) Questions (324). 14. ACTIVE FILTERS Introduction (326) Basic RC Filters (327) Low-Pass Active Filter (329) High-Pass Active Filter (332) Active Band-Pass Filter (333) Active Band-Reject (Notch) Filter (335) Switched-Capacitor Filters (336) Worked-out Problems (339) Questions (341). 15. LAPLACE TRANSFORMATION : TIME DOMAIN RESPONSE OF CIRCUITS Introduction (342) The Laplace Transformation (342) Evaluation of Some Laplace Transforms (343) A Few Important Theorems (347) Application to Networks : Simple Cases (349) 1. Series R.L circuit subjected to a step input voltage (349) 2. Series R.C. circuit subjected to a step input voltage (351) 3. Pulse response of series R-C network (353) 4. Exponentially damped voltage applied to a series R-L circuit (355) 5. Series R-C circuit with a sinusoidal driving voltage (356) 6. Series RLC circuit with a step input voltage (357) Characteristic Equation and Natural Response (360) Time Domain Response from Pole-Zero Plots (361) An AC Switching Transient (362) More Complicated Networks (364) Circuit Elements in s-domain (364) s-domain Circuit Analysis (365) Impulse Response and Transfer Function (365) The Convolution Integral and Its Application to Circuits (368) Computer-Aided Circuit Analysis (371) Workedout Problems (371) Questions (388). (vii)

9 16. FOURIER ANALYSIS : STEADY STATE RESPONSE Introduction (397) Fourier Series (397) Determination of Fourier Coefficients (398) Waveform Symmetry and Fourier Coefficients (400) The Effective Value of a Periodic Function (403) Exponential Form of Fourier Series (404) Steady-State Response of Networks to Periodic Signals (405) Average Power Calculation for Periodic Signals (405) Mean-Square Error (407) Worked-out Problems (408) Questions (416). 17. FREQUENCY DOMAIN DESCRIPTION : FOURIER TRANSFORM Introduction (421) The Fourier Transform (421) Fourier Transform of a Pulse (422) Fourier Integrals of Some Functions (425) Comparison of Fourier and Laplace Transforms (430) Some Interesting Features of Fourier Transforms (432) Appliction to Networks (434) Energy Calculations in Frequency Domain (435) Worked-out Problems (436) Questions (441). 18. DIGITAL FILTER Introduction (443) Digital Filter Structure (444) z-transform (446) Properties of z-transform (450) Digital Filter Design (453) (a) IIR Filter Design (453) (b) FIR Filter Design (456) Frequency Response (460) Digital Filter Realization Forms (462) Advantages of Digital Filters (465) Workedout Problems (465) Questions (472). 19. THREE-PHASE CIRCUITS Introduction (474) Three-phase Voltages (475) The Wye-Wye Circuit (477) The Wye-Delta Circuit (479) The Delta-Wye Circuit (480) The Delta-Delta Circuit (481) Power in Balanced Three-Phase Systems (481) Two wattmeter method (482) Worked-out Problems (483) Questions (487). 20. SPICE AND PSPICE : THE SOFTWARE PACKAGES FOR CIRCUIT ANALYSIS Introduction (489) Execution of PSpice (489) Resistors and Independent Sources (489) Resistive Circuits with voltage-dependent Sources (491) Resistive Circuits with Current-Dependent Sources (492) Transient Response of Circuits (493) A.C. Analysis of Circuits (495) Time-dependent Functions (498) Fourier Analysis (500) Concluding Remarks (504) Questions (504) SHORT REVIEW QUESTIONS (Objective Type) MISCELLANEOUS PROBLEMS INDEX (viii)

10 1 Basic Circuit Elements 1.1. INTRODUCTION In an electric circuit, the two terminals of a source of electrical energy are connected by wires (or conductors) to resistors, inductors, capacitors, other vacuum or solid state devices or their combinations. The interconnection is such that a closed path is available for the current to flow. In Fig. 1.1, a simple circuit is shown : the source of electrical energy is the generator G; the closed path for current is pqrsp. The three basic circuit elements are resistors, inductors and capacitors. These are actually devices having the properties of resistance, inductance and capacitance, respectively. These properties are also called circuit parameters. Physical definitions of these three circuit parameters are Fig A basic electric circuit. given below. The property that decides for a given current the rate at which electrical energy is transformed into heat or radiant energy is termed resistance. Self-inductance, or simply inductance, is the property that gives the voltage* induced in a circuit for a given rate of change of current in that circuit, and also gives the magnetic energy stored for a given current. Capacitance is that property that determines the charge on a system of conductors per unit voltage and also determines the electrical energy stored for a given charge. It must be emphasised that all practical circuit elements possess varying degrees of resistance, inductance and capacitance. But over a certain frequency range the desired circuit property predominates. Thus, the inductance of the leads of a commercial resistor may be ignored at low frequencies. At high frequencies, however, the inductance effect may have to be incorporated. Similarly, a coil having a low resistance to direct current may be taken to be a pure inductor at low frequencies. At high frequencies, however, the capacitive effect between the turns and to ground assumes more and more importance; also, the effective resistance is increased due to skin effect and dielectric losses. It is thus no longer justified to treat the coil as a pure inductor, and a more careful representation is necessary. In fact, owing to the parasitic effect of capacitance, the coil may assume a distributed configuration in which the current at any particular instant varies from point to point along the coil. Such complicated cases will not be dealt with here. We shall confine our attention only to lumped configurations in which the current in a series circuit will have the same value at every point at a given instant. Also, we shall frequently assume the existence of pure resistors, inductors and capacitors, each possessing only one of the three basic circuit properties. The purpose of doing so is to highlight the primary effects of resistance, inductance and capacitance. *Potential difference expressed in volt is called voltage. 1

Fundamental of Electric Circuit Theory 10% OFF Publisher : SChand Publications ISBN : 9788121900089 Author

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MATHEMATICAL PHYSICS MATHEMATICAL PHYSICS [For the Students of B.Sc. (Honours) and M.Sc. (Physics)] H.K. DASS M.Sc. Diploma in Specialist Studies (Mathematics) University of Hull England Assisted by Dr.