fundamentals of - kopykitab

FUNDAMENTALS OFELECTRIC CIRCUIT

THEORYFor B.E. / B.Tech. Students of Electronics, Electrical,

Computer, Control, Instrumentation, CommunicationEngineering and Information Technology.

Book is also useful for B.Sc. and M.Sc. Students of Physics

D. CHATTOPADHYAYM.Tech., Ph.D., D.Sc., P.R.S. (Mouat medalist)

Former Professor and Head,Institute of Radio Physics and Electronics

Calcutta University

P. C. RAKSHITM.Tech., Ph.D.

Former Professor and Head,Institute of Radio Physics and Electronics

Calcutta University

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© 1987, D. Chattopadhyay & P.C. RakshitAll rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording or otherwise, without the prior permission of the Publishers.

First Edition 1987Subsequent Editions and Reprints 1991, 2000, 2003, 2004, 2006, 2009Ninth Revised Edition 2011

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By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055and published by S. Chand & Company Ltd., 7361, Ram Nagar, New Delhi -110 055.


PREFACE TO THE NINTH EDITIONIn 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.

AUTHORS

PREFACE TO THE FIRST EDITIONThis 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 inElectronics, Electrical, Computer, Control, and Communication Engineering. Parts of the bookmay 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 textbookwhich 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. Emphasishas been, as the title indicates, on the fundamentals, keeping in view the requirements of thebeginners. Some modern aspects of circuit theory have also been included for the benefit ofstudents.

We have tried to present the subject matter in a clear and concise manner with numerousdiagrams and examples. In each chapter, a variety of problems has been worked out by way ofillustrations. A large number of problems has also been set as exercises at the end of eachchapter so that the students may find an opportunity to apply the principles and techniques theyhave learnt in the text. Short review questions and miscellaneous problems are given at the endof the book. The problems given in the book induce in the students a habit of thinking muchneeded in their professional career in future. Every effort has been made to give as much insightinto circuit theory as possible within a time a student can reasonably devote to the subject in hisbusy curriculum.

We wish to thank the Institution of Electronics and Telecommunication Engineers, theInstitution of Engineers (India), and Union Public Service Commission for their permission touse problems from past examination papers.

Institute of Radio Physics and Electronics AUTHORS92, Acharya Prafulla Chandra RoadCalcutta-700 009, India.


GUSTAV ROBERT KIRCHHOFF(1824 – 1887)

The German physicist, Gustav Robert Kirchhoff, was born at Königsberg inMarch, 1824. He was educated at the University of Königsberg and wasappointed as an extraordinary professor of physics at Breslau in 1850. Hebecame a professor of physics at Heidelberg in 1854 and shifted to Berlin in1875. Kirchhoff made significant contributions in conduction and radiationof heat, thermodynamics of solutions, vaporization and chemical reactions,crystalline reflections and refractions, spectral analysis etc. In electricity, hedeveloped two useful laws — Kirchhoff’s voltage and current laws - whichprovided the basis for electric circuit theory. A major part of his work isdescribed in his Vorlesungen iiber mathematische physik. Kirchhoff died inBerlin in October, 1887.


CONTENTSChapter Pages

1. BASIC CIRCUIT ELEMENTS 1—151.1. 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 Conserva-tion 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 16—332.1. 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 Powerand Reactive Power (24) 2.5. Complex Frequency (26) 2.6. Worked-out Problems (27)Questions (32).

3. SOME IMPORTANT CONCEPTS 34—513.1. 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 Capaci-tor (38) 3.5. B and G in terms of Q (39) 3.6. Duality (40) 3.7. Transformation of Volt-age and Current Sources (40) (a) Reduction of a number of voltage (or current) sourcesto a single voltage (or current) source (40) (b) Equivalence of voltage and currentsources (41); Voltage source with no series impedance (42) Current source with noparallel impedance (42) Reduction of complicated connection of source branches (42)3.8. Worked-out Problems (43) Questions (49).

4. RESONANCE 52—784.1. Series Resonance (52) Sharpness of resonance (53) Occurrence of maximum volt-age 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 parallelresonant circuit (60) Reactance, susceptance and impedance sketches (61) 4.3. Univer-sal 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 79—1075.1. Network and Its Topological Description (79) 5.2. Network Variables (80).5.3. Mesh Method of Analysis (81). 5.4. Node Method of Analysis (83) 5.5. SomeComments (84) 5.6. Matrix Representation (86). 5.7. Dual and Inverse Networks (87)5.8. Driving Point and Transfer Impedances (88). 5.9. State Variable Analysis (90)5.10. Worked-out Problems (93) Questions (102).

6. ELEMENTS OF GRAPH THEORY AND NETWORK APPLICATIONS 108—1246.1. Introduction (108) 6.2. A Few Important Terms (108) 6.3. Incidence Matrix (108)

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6.4. Tie-Set Matrix (110) 6.5. CutSet Matrix (112) 6.6. Generalised Network Equa-tions (114) 6.7. Worked-Out Problems (115) Questions (123).

7. NETWORK CONVERSIONS AND NETWORK THEOREMS 125—1557.1. 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 Theo-rem (130) 7.6. Millman’s Theorem (130) 7.7. Thevenin’s Theorem (131) 7.8. Norton’sTheorem (132) 7.9. Reciprocity Theorem (133) 7.10. Substitution and CompensationTheorems (134) 7.11. Maximum Power Transfer Theorem (135) 7.12. Matching andSharpness of Resonance of a Parallel Resonant Circuit Connected to a Voltage Sourceof Finite Internal Resistance (137) 7.13. Tellegen’s Theorem (138) 7.14. Worked-outProblems (139) Questions (151).

8. INDUCTIVELY COUPLED CIRCUITS 156—1868.1. Introduction (156). 8.2. Mutual Inductance and its Evaluation (156) 8.3. Identifi-cation of Relative Polarities (158) 8.4. Equivalent Inductance of Conductively Con-nected Mutually Coupled Coils (158) 8.5. Transformer (160) 8.6. Transformer Adjust-ment 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) Ques-tions (182).

9. SYNTHESIS OF TWO-TERMINAL REACTIVE NETWORKS 187—2039.1. Introduction (187) 9.2. Foster’s Reactance Theorem (187) 9.3. Separation Prop-erty 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 (Sec-ond 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 Net-works (194) 9.10. Worked-out Problems (196) Questions (201).

10. SYNTHESIS OF TWO-TERMINAL R-L AND R-C NETWORKS 204—22710.1. Introduction (204) 10.2. Two-Terminal R-L Network (204) (a) Driving-pointimpedance (204); (b) Driving-point admittance (207); (c) Networks derived from con-tinued fraction expansion of ZD (s) (208); 10.3. 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) 10.4. Positive RealFunctions (212) Properties of positive real functions (213) Testing Hurwitz polynomi-als (213) Testing Re[F(s)] 0 for PRF (215) 10.5. Synthesis from Positive Real Func-tions (217) 10.6. Worked-out Problems (217) Questions (225).

11. TWO-PORT NETWORKS 228—27211.1. Introduction (228) 11.2. Open-Circuit and Short-Circuit Parameters (228)11.3. Transmission Parameters (231) 11.4. Hybrid or Mixed Parameters (h-parameters)and g-Parameters (232) 11.5. Matrix forms of Input-Output Relations (235) 11.6. Inter-connection 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 ma-trix method (237) 11.7. Characteristics of Unsymmetrical Networks (239) Image im-pedances (239) Expressions for Zi1 and Zi2 (240) Propagation function (241)11.8. Characteristics of Symmetrical Networks (242) Propagation function (242) 11.9.Iterative Impedances (243) 11.10. Lattice Network (244) 11.11. Balanced and Unbal-


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anced Networks (245) 11.12. Bartlett’s Bisection Theorem (245) Proof of the theorem(246) 11.13. Determination of Z0 and of T, and Bridged-T networks (246)11.14. Conversion of Lattice into Equivalent T and Networks (248) 11.15.

Nonreciprocal Two-port Network (249) 11.16. Terminated Two-port Circuits (251)11.17. Special Two-port Networks (252) 11.18. Worked-out Problems (255) Questions(267).

12. ATTENUATORS, INSERTION LOSS, AND EQUALISERS 273—29212.1. Introduction (273) Units of attenuation (273) 12.2. Lattice Attenuator (274) 12.3.T-Type Attenuator (275) 12.4. -Type Attenuator (276) 12.5. Bridged-T Attenuator(276) 12.6. Asymmetric Attinuators (277) (a) L-Type Attenuator (277) (b) AsymmetricT and Attenuators (278) 12.7. Ladder Type Attenuator (279) 12.8. Balanced Attenu-ator (280) 12.9. Insertion Loss (280) 12.10. Amplitude Equalisers (282) 12.11. Worked-out Problems (285) Questions (291).

13. CLASSICAL FILTERS 293—32513.1. Definition and Classification (293) 13.2. Filter Configurations and Characteris-tics (293) 13.3. Mechanism of Filter Action (295) 13.4. Low-Pass Filter (296) (i) Pos-sible Lattice Section (296) (ii) Design of prototype sections (296) 13.5. High-PassFilters (299) (i) Lattice configuration (299) (ii) Constant-k prototype sections (300)13.6. m-Derived Structure (302) m-derived T-section (303) Impedance match with m-derived half sections (L-sections) (305) m-derived p-section (308) 13.7. CompositeFilter (309) 13.8. Band-Pass Constant-k Filter (309) m-derived band-pass filter (312)13.9. Band Elimination Filter (314) 13.10. Doubly Tuned Transformer as a Band-passFilter (314) 13.11. Modern Filter Design Concept : Approximation Methods (316)13.12. Worked-out Problems (320) Questions (324).

14. ACTIVE FILTERS 326—34114.1. Introduction (326) 14.2. Basic RC Filters (327) 14.3. Low-Pass Active Filter(329) 14.4. High-Pass Active Filter (332) 14.5. Active Band-Pass Filter (333) 14.6.Active Band-Reject (Notch) Filter (335) 14.7. Switched-Capacitor Filters (336) 14.8.Worked-out Problems (339) Questions (341).

15. LAPLACE TRANSFORMATION : TIME DOMAIN RESPONSEOF CIRCUITS 342—39615.1. Introduction (342) 15.2. The Laplace Transformation (342) 15.3. Evaluation ofSome Laplace Transforms (343) 15.4. A Few Important Theorems (347) 15.5. Appli-cation to Networks : Simple Cases (349) 1. Series R.L circuit subjected to a step inputvoltage (349) 2. Series R.C. circuit subjected to a step input voltage (351) 3. Pulseresponse of series R-C network (353) 4. Exponentially damped voltage applied to aseries 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) 15.6. Characteristic Equation andNatural Response (360) Time Domain Response from Pole-Zero Plots (361) 15.7. AnAC Switching Transient (362) 15.8. More Complicated Networks (364) 15.9. CircuitElements in s-domain (364) 15.10. s-Domain Circuit Analysis (365) 15.11. ImpulseResponse and Transfer Function (365) 15.12. The Convolution Integral and Its Appli-cation to Circuits (368) 15.13. Computer-Aided Circuit Analysis (371) 15.14. Worked-out Problems (371) Questions (388).


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16. FOURIER ANALYSIS : STEADY STATE RESPONSE 397—42016.1. Introduction (397) 16.2. Fourier Series (397) 16.3. Determination of FourierCoefficients (398) 16.4. Waveform Symmetry and Fourier Coefficients (400) 16.5.The Effective Value of a Periodic Function (403) 16.6. Exponential Form of FourierSeries (404) 16.7. Steady-State Response of Networks to Periodic Signals (405) 16.8.Average Power Calculation for Periodic Signals (405) 16.9. Mean-Square Error (407)16.10. Worked-out Problems (408) Questions (416).

17. FREQUENCY DOMAIN DESCRIPTION : FOURIER TRANSFORM 421—44217.1. Introduction (421) 17.2. The Fourier Transform (421) 17.3. Fourier Transform ofa Pulse (422) 17.4. Fourier Integrals of Some Functions (425) 17.5. Comparison ofFourier and Laplace Transforms (430) 17.6. Some Interesting Features of Fourier Trans-forms (432) 17.7. Appliction to Networks (434) 17.8. Energy Calculations in FrequencyDomain (435) 17.9. Worked-out Problems (436) Questions (441).

18. DIGITAL FILTER 443—47318.1. Introduction (443) 18.2. Digital Filter Structure (444) 18.3. z-Transform (446)18.4. Properties of z-Transform (450) 18.5. Digital Filter Design (453) (a) IIR FilterDesign (453) (b) FIR Filter Design (456) 18.6. Frequency Response (460) 18.7. Digi-tal Filter Realization Forms (462) 18.8. Advantages of Digital Filters (465) 18.9. Worked-out Problems (465) Questions (472).

19. THREE-PHASE CIRCUITS 474—48719.1. Introduction (474) 19.2. Three-phase Voltages (475) 19.3. The Wye-Wye Circuit(477) 19.4. The Wye-Delta Circuit (479) 19.5. The Delta-Wye Circuit (480) 19.6. TheDelta-Delta Circuit (481) 19.7. Power in Balanced Three-Phase Systems (481) Twowattmeter method (482) 19.8. Worked-out Problems (483) Questions (487).

20. SPICE AND PSPICE : THE SOFTWARE PACKAGESFOR CIRCUIT ANALYSIS 488—50320.1. Introduction (489) 20.2. Execution of PSpice (489) 20.3. Resistors and Indepen-dent Sources (489) 20.4. Resistive Circuits with voltage-dependent Sources (491) 20.5.Resistive Circuits with Current-Dependent Sources (492) 20.6. Transient Response ofCircuits (493) 20.7. A.C. Analysis of Circuits (495) 20.8. Time-dependent Functions(498) 20.9. Fourier Analysis (500) 20.10. Concluding Remarks (504) Questions (504)

SHORT REVIEW QUESTIONS (Objective Type) 504—509

MISCELLANEOUS PROBLEMS 509—516

INDEX 517—520


1.1. INTRODUCTIONIn 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 theircombinations. The interconnection is such that a closed path is available for the current to flow.In Fig. 1.1, a simple circuit is shown : thesource of electrical energy is the generatorG; the closed path for current is pqrsp.

The three basic circuit elements areresistors, inductors and capacitors. Theseare actually devices having the propertiesof resistance, inductance and capacitance,respectively. These properties are alsocalled circuit parameters. Physical defini-tions of these three circuit parameters aregiven below.

The property that decides for a given current the rate at which electrical energy is transformedinto heat or radiant energy is termed resistance. Self-inductance, or simply inductance, is theproperty that gives the voltage* induced in a circuit for a given rate of change of current in thatcircuit, and also gives the magnetic energy stored for a given current. Capacitance is that propertythat determines the charge on a system of conductors per unit voltage and also determines theelectrical 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 propertypredominates. Thus, the inductance of the leads of a commercial resistor may be ignored at lowfrequencies. 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 atlow frequencies. At high frequencies, however, the capacitive effect between the turns and toground assumes more and more importance; also, the effective resistance is increased due to skineffect and dielectric losses. It is thus no longer justified to treat the coil as a pure inductor, anda more careful representation is necessary. In fact, owing to the parasitic effect of capacitance, thecoil may assume a distributed configuration in which the current at any particular instant variesfrom point to point along the coil. Such complicated cases will not be dealt with here. We shallconfine our attention only to lumped configurations in which the current in a series circuit will havethe same value at every point at a given instant. Also, we shall frequently assume the existence ofpure 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

Fig. 1.1. A basic electric circuit.

1 Basic Circuit Elements


Fundamental of Electric Circuit Theory

Publisher : SChand Publications ISBN : 9788121900089Author : Dr DChattopadhyaya And Dr PC Rakshit

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