Communications S&S

8/13/2019 Communications S&S

1/277

LEVEL 3 3015 COMMUNICATIONS, SIGNALS

AND SYSTEMS 2003

Peter H. Cole

January 5, 2004


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Contents

1 INTRODUCTION 11.1 Course Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 Access to Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Essential Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Making Use of the Lectures . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Other Goodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 REVISION 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.2 Quantities derived by differentiation . . . . . . . . . . . . . . . . . 32.2.3 Frequency and angular frequency . . . . . . . . . . . . . . . . . . . 4

2.3 RC and RL Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.1 Things not possible . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 Establishing a capacitor voltage . . . . . . . . . . . . . . . . . . . . 42.3.3 Discharging a capacitor . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.4 Establishing an inductor current . . . . . . . . . . . . . . . . . . . . 52.3.5 De-energising an inductor . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Phasor Analysis for Linear a.c. Circuits . . . . . . . . . . . . . . . . . . . . 72.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.2 Particular cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.3 Peak value and r.m.s. value phasors . . . . . . . . . . . . . . . . . . 82.4.4 Properties of a sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.5 Unusual waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.6 Resistance and resistivity . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Characterisation of N-Port Networks . . . . . . . . . . . . . . . . . . . . . 92.5.1 Definitions of port variables . . . . . . . . . . . . . . . . . . . . . . 92.5.2 Definition of impedance matrix . . . . . . . . . . . . . . . . . . . . 102.5.3 Definition of admittance matrix . . . . . . . . . . . . . . . . . . . . 112.5.4 Relations between impedance and admittance parameters . . . . . . 12

2.5.5 Pathological cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.6 The reciprocity theorem . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Thevenins and Nortons Theorems . . . . . . . . . . . . . . . . . . . . . . 12

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2.6.1 Thevenin and Norton networks . . . . . . . . . . . . . . . . . . . . 122.6.2 Determination of Thevenin and Norton parameters . . . . . . . . . 142.6.3 Pathological cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6.4 Networks with non-linear elements . . . . . . . . . . . . . . . . . . 162.6.5 Frequency domain equivalent circuits . . . . . . . . . . . . . . . . . 162.6.6 Warning against undue generalisation . . . . . . . . . . . . . . . . . 18

2.7 Tuned Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7.1 Ideal series resonant circuit . . . . . . . . . . . . . . . . . . . . . . 192.7.2 Ideal parallel resonant circuit. . . . . . . . . . . . . . . . . . . . . . 202.7.3 Practical parallel resonant circuit . . . . . . . . . . . . . . . . . . . 222.7.4 Common features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7.5 Practical parallel to ideal parallel circuit transformation . . . . . . 24

2.8 Maximum Power Transfer Theorem . . . . . . . . . . . . . . . . . . . . . . 24

2.8.1 Available source power . . . . . . . . . . . . . . . . . . . . . . . . . 252.8.2 Logarithmic expression of power ratios . . . . . . . . . . . . . . . . 252.8.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.8.4 Insertion gain and insertion loss . . . . . . . . . . . . . . . . . . . . 26

3 INTRODUCTION TO PROBABILITY 273.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Belonging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.3 Subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.4 Set equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.5 Universal set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.6 Null set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.7 Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.8 Set operations and concepts . . . . . . . . . . . . . . . . . . . . . . 293.2.9 Mutual exclusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.10 Collective exhaustion . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.11 Theorems on sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Applying set theory to probability . . . . . . . . . . . . . . . . . . . . . . . 333.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.4 Further comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.6 Fundamental remark . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.7 Theorem on event spaces . . . . . . . . . . . . . . . . . . . . . . . . 343.3.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.9 A modest proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Probability Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Probability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5.1 Probabilities with the union of disjoint sets . . . . . . . . . . . . . . 363.5.2 Probabilities with the union of mutually exclusive sets . . . . . . . 37


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3.5.3 Probabilities of outcomes combined to form an event . . . . . . . . 373.5.4 Theorem on equally likely outcomes . . . . . . . . . . . . . . . . . . 373.5.5 Theorem on no outcome . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.6 Theorem on complementary outcomes . . . . . . . . . . . . . . . . 373.5.7 Theorem on arbitrary sets . . . . . . . . . . . . . . . . . . . . . . . 373.5.8 Theorem on partially ordered sets . . . . . . . . . . . . . . . . . . . 373.5.9 Theorem on adding probabilities of intersections . . . . . . . . . . . 373.5.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.3 Fundamental formula for conditional probability . . . . . . . . . . . 393.6.4 Law of total probability . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7 Bayes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.9 Sequential Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.9.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.9.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 Counting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.1 Fundamental Principle of Counting . . . . . . . . . . . . . . . . . . 423.10.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.3 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.10.4 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.11 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 RANDOM VARIABLES 454.1 Review of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.3 Bayes rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.1.4 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . 464.2.3 Probability Density Function . . . . . . . . . . . . . . . . . . . . . 474.2.4 Joint Distribution and Density Functions . . . . . . . . . . . . . . . 484.2.5 Conditional Density Function . . . . . . . . . . . . . . . . . . . . . 49

4.3 Expectation and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.3 Central moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 The Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.1 One Dimensional Gaussian Distribution . . . . . . . . . . . . . . . 514.4.2 Two Dimensional Gaussian Distribution . . . . . . . . . . . . . . . 52


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4.5 Transformation of Random Variables . . . . . . . . . . . . . . . . . . . . . 534.5.1 Single Variable Transformations . . . . . . . . . . . . . . . . . . . . 534.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5.3 Two-variable transformations . . . . . . . . . . . . . . . . . . . . . 544.5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5.5 Application to Gaussian variables . . . . . . . . . . . . . . . . . . . 56

4.6 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Second Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . 584.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.7.2 Cumulants of the Gaussian distribution . . . . . . . . . . . . . . . . 584.7.3 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 RANDOM PROCESSES 615.1 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 First and Second Order Statistics . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.2 Second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Mean, Autocorrelation & Autocovariance . . . . . . . . . . . . . . . . . . . 635.3.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.3 Autocovariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.4 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.5 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4.2 How the argument will unfold . . . . . . . . . . . . . . . . . . . . . 645.4.3 The Poisson point process . . . . . . . . . . . . . . . . . . . . . . . 665.4.4 TheX process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4.5 Additive properties of intervals . . . . . . . . . . . . . . . . . . . . 695.4.6 TheY process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4.7 TheP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4.8 The Poisson impulse process . . . . . . . . . . . . . . . . . . . . . . 725.4.9 Return to the X process . . . . . . . . . . . . . . . . . . . . . . . . 735.4.10 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4.11 Wiener-Levy Process . . . . . . . . . . . . . . . . . . . . . . . . . . 755.4.12 The reflection principle . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Sine wave and Random Noise . . . . . . . . . . . . . . . . . . . . . . . . . 775.6 Cross-correlation and Covariance . . . . . . . . . . . . . . . . . . . . . . . 79

5.6.1 Real processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6.2 Complex processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.7 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.7.1 Definition: Stationary in the strict sense . . . . . . . . . . . . . . . 805.7.2 First order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 805.7.3 Second order statistics . . . . . . . . . . . . . . . . . . . . . . . . . 80


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5.7.4 Definition: Stationary in the wide sense . . . . . . . . . . . . . . . . 815.8 Transformation of Random Processes . . . . . . . . . . . . . . . . . . . . . 81

5.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.8.2 Memoryless Transformations . . . . . . . . . . . . . . . . . . . . . . 815.8.3 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.8.4 Example: Differentiation operator . . . . . . . . . . . . . . . . . . . 845.8.5 Further example: Poisson impulse process . . . . . . . . . . . . . . 845.8.6 Simplified results for stationary processes . . . . . . . . . . . . . . . 85

5.9 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.9.2 Time averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.9.3 Time and ensemble statistics . . . . . . . . . . . . . . . . . . . . . . 865.9.4 Definition of an ergodic system . . . . . . . . . . . . . . . . . . . . 86

5.9.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 POWER SPECTRA OF STATIONARY PROCESSES 916.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.1 Recall of definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1.2 Properties of correlation functions . . . . . . . . . . . . . . . . . . . 916.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.1 Example: Rectangular pulse . . . . . . . . . . . . . . . . . . . . . . 936.2.2 Example: Exponential pulse . . . . . . . . . . . . . . . . . . . . . . 95

6.2.3 Example: Triangular pulse . . . . . . . . . . . . . . . . . . . . . . . 956.2.4 Hermitian property . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.5 The Rayleigh energy theorem . . . . . . . . . . . . . . . . . . . . . 966.2.6 Properties of Fourier transforms . . . . . . . . . . . . . . . . . . . . 966.2.7 Example: Double exponential pulse . . . . . . . . . . . . . . . . . . 976.2.8 Another example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.1 Example: Application to periodic signals . . . . . . . . . . . . . . . 99

6.4 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.5 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.5.2 Wiener Khinchine theorem . . . . . . . . . . . . . . . . . . . . . . . 1016.5.3 Application of the theorem . . . . . . . . . . . . . . . . . . . . . . . 101

6.6 Cross Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . 1026.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.6.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.7 Linear Time Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . 1036.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.8 Cyclostationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.8.2 Example: sine wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.8.3 Example: random binary signal . . . . . . . . . . . . . . . . . . . . 106


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9.3.5 Phase characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.3.6 Pole-zero config u r a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4

9.4 Chebychev Type I Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379.4.3 Pole positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

9.5 Chebychev Type II Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.5.2 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.5.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.6 All Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.6.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.6.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449.7 Bessel-Thompson Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449.7.1 Defining property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449.7.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459.7.3 Graphical expression of responses . . . . . . . . . . . . . . . . . . . 145

9.8 Elliptic Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489.8.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489.8.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489.8.4 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.8.5 Magnitude response . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.8.6 Phase response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509.8.7 Pulse response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.9 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

10 FILTER DESIGN AND TRANSFORMATIONS 15310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

10.1.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15310.1.2 Notation for unit angular frequency . . . . . . . . . . . . . . . . . . 153

10.2 Low Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15310.2.1 Design steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

10.2.2 Frequency transformation of specifications . . . . . . . . . . . . . . 15410.2.3 Design of normalised filter . . . . . . . . . . . . . . . . . . . . . . . 15410.2.4 Scaling elements for frequency . . . . . . . . . . . . . . . . . . . . . 15510.2.5 Scaling elements for frequency and impedance . . . . . . . . . . . . 15510.2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.3 The High Pass Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 15610.3.1 Structure of low pass filter . . . . . . . . . . . . . . . . . . . . . . . 15610.3.2 Structure of a high pass filter . . . . . . . . . . . . . . . . . . . . . 15610.3.3 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15610.3.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10.3.5 Design steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15810.4 The Band Pass Transformation . . . . . . . . . . . . . . . . . . . . . . . . 162


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10.4.1 Structure of a band pass filter . . . . . . . . . . . . . . . . . . . . . 16210.4.2 Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16210.4.3 Design equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16410.5 The Band Stop Transformation . . . . . . . . . . . . . . . . . . . . . . . . 16610.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

11 REALISATION OF PASSIVE FILTER CIRCUITS 16711.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16711.2 Some important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

11.2.1 Aspects of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16711.2.2 Available source power . . . . . . . . . . . . . . . . . . . . . . . . . 16711.2.3 Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 168

11.3 Lumped L-C Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17011.3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17011.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17111.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17211.3.4 Generalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

11.4 Admittance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17311.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17411.4.2 Example: Third Order Butterworth Filter . . . . . . . . . . . . . . 175

11.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12 ACTIVE FILTERS 177

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17712.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17712.1.2 Realisation strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 177

12.2 First Order Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . 17712.2.1 Passive circuits realisations . . . . . . . . . . . . . . . . . . . . . . . 17712.2.2 Effect of circuit loading . . . . . . . . . . . . . . . . . . . . . . . . . 17812.2.3 Inverting amplifier c ircuits . . . . . . . . . . . . . . . . . . . . . . . 17912.2.4 Non-inverting amplifier circuits . . . . . . . . . . . . . . . . . . . . 18012.2.5 More complex functions . . . . . . . . . . . . . . . . . . . . . . . . 181

12.3 Second Order Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . 181

12.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18112.3.2 Low pass Sallen and Key circuit . . . . . . . . . . . . . . . . . . . . 18112.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18312.3.4 Band pass Sallen and Key circuit . . . . . . . . . . . . . . . . . . . 18312.3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18412.3.6 Inverting Band Pass Circuit . . . . . . . . . . . . . . . . . . . . . . 18412.3.7 Biquadratic Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 18512.3.8 Case 1: z< o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18512.3.9 Case 2:

z>

o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

12.4 Switched Capacitor Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

12.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18712.4.2 Low pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18912.4.3 Circuit properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189


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12.5 Frequency Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19012.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

A REFERENCES 191

B FORMULAE AND TABLES 193B.1 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193B.2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193B.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194B.4 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

B.4.1 The direct transform . . . . . . . . . . . . . . . . . . . . . . . . . . 194B.4.2 The reverse transform . . . . . . . . . . . . . . . . . . . . . . . . . 195B.4.3 Theorems and transforms . . . . . . . . . . . . . . . . . . . . . . . 195

C PROOFS 201C.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201C.2 Bounds on the Autocorrelation Function . . . . . . . . . . . . . . . . . . . 201C.3 The Wiener Khinchine Theorem . . . . . . . . . . . . . . . . . . . . . . . . 202C.4 Analysis of the Bessel-Thompson Filter . . . . . . . . . . . . . . . . . . . . 204

D ADVICE ON STUDYING FOR THE EXAMINATION 209

E NOTES ON STUDENT ERRORS 213E.1 Common Errors in Quiz 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

F STANDARD INTERNATIONAL TERMINOLOGY AND UNITS 217F.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217F.2 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217F.3 The Base Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218F.4 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220F.5 The Fundamental Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 220F.6 List of Standard Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

G STOP PRESS 225G.1 Thevenins and Nortons Theorems . . . . . . . . . . . . . . . . . . . . . . 225

G.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225G.1.2 Thevenins theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 225G.1.3 Nortons theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225G.1.4 Relations between the circuits . . . . . . . . . . . . . . . . . . . . . 226G.1.5 Commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226G.1.6 Cautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

H SOME NOTES ON PHASOR ANALYSIS 229H.1 Ob jective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229H.2 A Simple Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

H.3 Analysis by Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 229H.4 Disadvantageous Features . . . . . . . . . . . . . . . . . . . . . . . . . . . 230H.5 Representation of Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . 231


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H.6 Analysis Leading to Algebraic Equations . . . . . . . . . . . . . . . . . . . 232H.7 Application to Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . 232H.8 The Physical and Mathematical Systems . . . . . . . . . . . . . . . . . . . 232

H.8.1 The physical system . . . . . . . . . . . . . . . . . . . . . . . . . . 233H.8.2 The mathematical system . . . . . . . . . . . . . . . . . . . . . . . 233H.8.3 Deletion of the time function . . . . . . . . . . . . . . . . . . . . . . 233

I MODULATION SYSTEMS 237I.1 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237I.2 Introduction and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 237I.3 Modulation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

I.3.1 Definition of modulation . . . . . . . . . . . . . . . . . . . . . . . . 237I.3.2 Classes of modulation . . . . . . . . . . . . . . . . . . . . . . . . . 238

I.4 Bandpass Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . . . 238I.5 Linear modulation systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

I.5.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 238I.5.2 AM circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240I.5.3 Double sideband suppressed carrier modulation . . . . . . . . . . . 240I.5.4 Single sideband modulation . . . . . . . . . . . . . . . . . . . . . . 241I.5.5 Vestigial sideband modulation . . . . . . . . . . . . . . . . . . . . . 242I.5.6 Detection systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

I.6 Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244I.6.1 AM superheterodyne receiver . . . . . . . . . . . . . . . . . . . . . 244

I.7 Angle modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

I.7.1 Phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246I.7.2 Frequency modulation . . . . . . . . . . . . . . . . . . . . . . . . . 246I.7.3 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247I.7.4 Bandwidth requirements . . . . . . . . . . . . . . . . . . . . . . . . 249I.7.5 Detectors of angle modulation . . . . . . . . . . . . . . . . . . . . . 250

I.8 Performance of modulation systems . . . . . . . . . . . . . . . . . . . . . . 250I.8.1 A general communication system . . . . . . . . . . . . . . . . . . . 251I.8.2 Illustration of modulation effects . . . . . . . . . . . . . . . . . . . 252I.8.3 PM post detection noise spectrum . . . . . . . . . . . . . . . . . . . 254I.8.4 FM post detection noise spectrum . . . . . . . . . . . . . . . . . . . 255

I.8.5 Pre-emphasis and de-emphasis in frequency modulation . . . . . . . 255I.8.6 Parameters for broadcast FM . . . . . . . . . . . . . . . . . . . . . 256

I.9 Destination signal to noise ratios . . . . . . . . . . . . . . . . . . . . . . . 256I.9.1 FM threshold effect and mutilation . . . . . . . . . . . . . . . . . . 256I.9.2 FM threshold extension (advanced topic to be omitted) . . . . . . . 256

I.10 Comparison of CW modulation systems . . . . . . . . . . . . . . . . . . . . 257


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2.1 Charging a capacitor though a resistor. . . . . . . . . . . . . . . . . . . . . 42.2 Waveforms in charging a capacitor through a resistor. . . . . . . . . . . . . 42.3 Discharging a capacitor though a resistor. . . . . . . . . . . . . . . . . . . 52.4 Waveforms in discharging a capacitor through a resistor. . . . . . . . . . . 5

2.5 Energising an inductor though a resistor. . . . . . . . . . . . . . . . . . . . 52.6 Waveforms in energising an inductor through a resistor. . . . . . . . . . . . 62.7 De-energising an inductor though a resistor. . . . . . . . . . . . . . . . . . 62.8 Waveforms in de-energizing an inductor through a resistor. . . . . . . . . . 62.9 Polar representation of a complex number. . . . . . . . . . . . . . . . . . . 72.10 An N-port network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.11 A nice linear one port network. . . . . . . . . . . . . . . . . . . . . . . . . 132.12 A linear terminal voltage current relation. . . . . . . . . . . . . . . . . . . 132.13 Basic forms of Thevenin and Norton circuits. . . . . . . . . . . . . . . . . . 142.14 A network with no Norton equivalent. . . . . . . . . . . . . . . . . . . . . . 15

2.15 A network with no Thevenin equivalent. . . . . . . . . . . . . . . . . . . . 152.16 Simplification of a Thevenin network. . . . . . . . . . . . . . . . . . . . . . 162.17 A network with a non-linear element. . . . . . . . . . . . . . . . . . . . . . 162.18 Another network with a non-linear element. . . . . . . . . . . . . . . . . . 172.19 A further network with a non-linear element. . . . . . . . . . . . . . . . . . 172.20 Thevenin representation of an automobile battery. . . . . . . . . . . . . . . 182.21 Norton representation of the same battery. . . . . . . . . . . . . . . . . . . 182.22 Ideal series resonant circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 192.23 Variation of series circuit current vs frequency. . . . . . . . . . . . . . . . . 202.24 Ideal parallel resonant circuit . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.25 Variation of parallel circuit voltage vs frequency . . . . . . . . . . . . . . . 212.26 Practical parallel resonant circuit. . . . . . . . . . . . . . . . . . . . . . . . 222.27 Practical parallel to ideal parallel circuit transformation . . . . . . . . . . . 242.28 Context for maximum power transfer theorem. . . . . . . . . . . . . . . . . 252.29 Context for definition of insertion gain and insertion loss. . . . . . . . . . . 26

3.1 Venn diagram representingAcontained within B. . . . . . . . . . . . . . . 283.2 Venn diagram representing the union ofA and B. . . . . . . . . . . . . . . 293.3 Venn diagram representing the intersection ofAandB. . . . . . . . . . . . 303.4 Venn diagram representing the complement ofA. . . . . . . . . . . . . . . 30

3.5 Venn diagram representing the difference betweenA and B. . . . . . . . . 313.6 Venn diagram representing mutually exclusive sets. . . . . . . . . . . . . . 323.7 Venn diagram representing collectively exhaustive sets. . . . . . . . . . . . 32

xi


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3.8 Tree diagram for resistor production. . . . . . . . . . . . . . . . . . . . . . 42

4.1 Distribution functions of continuous and discrete variables. . . . . . . . . . 47

4.2 Probability density function of a continuous variable. . . . . . . . . . . . . 484.3 Probability density function of a discrete variable. . . . . . . . . . . . . . . 494.4 Gaussian probability density function. . . . . . . . . . . . . . . . . . . . . 514.5 The Q function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.6 Functional relation between random variables. . . . . . . . . . . . . . . . . 534.7 General functional relation. . . . . . . . . . . . . . . . . . . . . . . . . . . 544.8 Cartesian to polar conversion. . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Conceptual representation of a random process. . . . . . . . . . . . . . . . 615.2 Probability density function of a random process. . . . . . . . . . . . . . . 625.3 Conditional probability density function. . . . . . . . . . . . . . . . . . . . 63

5.4 Poisson point process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Probabilities of number of points in an interval. . . . . . . . . . . . . . . . 675.6 Probability density function of time between successive pulses. . . . . . . . 685.7 X variable for Poisson point process. . . . . . . . . . . . . . . . . . . . . . 695.8 Y variable for Poisson point process. . . . . . . . . . . . . . . . . . . . . . 705.9 Overlapping and non-overlapping intervals. . . . . . . . . . . . . . . . . . . 705.10 Correlation function of a Poisson pulse process. . . . . . . . . . . . . . . . 735.11 Poisson impulse process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.12 Random walk process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.13 Probabilities in a random walk. . . . . . . . . . . . . . . . . . . . . . . . . 75

5.14 The reflection principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.15 Phasor diagram of sine wave and random noise. . . . . . . . . . . . . . . . 785.16 Probability density functions of r and. . . . . . . . . . . . . . . . . . . . 795.17 Illustration of a linear system. . . . . . . . . . . . . . . . . . . . . . . . . . 825.18 Examples of random processes. . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1 Correlation functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Autocorrelation function of a sine wave. . . . . . . . . . . . . . . . . . . . 936.3 Rectangular pulse and its Fourier transform. . . . . . . . . . . . . . . . . . 946.4 The sinc function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.5 Fourier transform of exponential pulse. . . . . . . . . . . . . . . . . . . . . 956.6 Fourier transform of triangular pulse. . . . . . . . . . . . . . . . . . . . . . 966.7 Fourier transform of double exponential pulse. . . . . . . . . . . . . . . . . 986.8 Fourier transform of a constant by limiting process. . . . . . . . . . . . . . 996.9 Fourier transform of a periodic signal. . . . . . . . . . . . . . . . . . . . . . 1006.10 Convolution of two time functions. . . . . . . . . . . . . . . . . . . . . . . 1016.11 Autocorrelation and power spectrum of Poisson pulse process. . . . . . . . 1026.12 Linear time invariant system. . . . . . . . . . . . . . . . . . . . . . . . . . 1036.13 Meaning of power spectral density. . . . . . . . . . . . . . . . . . . . . . . 1056.14 Random binary waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.15 Autocorrelation function of random binary waveform. . . . . . . . . . . . . 107

6.16 Average autocorrelation and power spectrum. . . . . . . . . . . . . . . . . 108

7.1 Power spectrum and autocorrelation function of white noise. . . . . . . . . 111


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LIST OF FIGURES xiii

7.2 Bandlimited white noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.3 White and narrowband noise. . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4 Field containment structure with single port access. . . . . . . . . . . . . . 113

7.5 Thermal noise equivalent circuits. . . . . . . . . . . . . . . . . . . . . . . . 1147.6 Power spectrum of shot noise. . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.1 Context for definition of transfer function. . . . . . . . . . . . . . . . . . . 1178.2 Poles and zeros ofH(s)H(s) in the s plane. . . . . . . . . . . . . . . . . 1198.3 Poles and zeros of examplefilter. . . . . . . . . . . . . . . . . . . . . . . . 1208.4 Zero selection for minimum and non-minimum phase filters. . . . . . . . . 1218.5 Definition of various ideal filter responses. . . . . . . . . . . . . . . . . . . 1228.6 Definition of parameters of realistic low pass filter response. . . . . . . . . 1238.7 Example circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.8 Normalised circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.9 Impedance scaled circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.10 Response of original circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.11 Response of frequency scaled circuit. . . . . . . . . . . . . . . . . . . . . . 1268.12 Impedance and frequency scaled circuit. . . . . . . . . . . . . . . . . . . . 1278.13 Ideal low pass filter response. . . . . . . . . . . . . . . . . . . . . . . . . . 1278.14 Specification of required response. . . . . . . . . . . . . . . . . . . . . . . . 128

9.1 Amplitude responses of Butterworth fil ters. . . . . . . . . . . . . . . . . . . 1329.2 Phase responses of Butterworthfilters. . . . . . . . . . . . . . . . . . . . . 1349.3 Pulse response of ann= 3 Butterworth fil ters. . . . . . . . . . . . . . . . . 135

9.4 Poles of Butterworth filter forn = 5. . . . . . . . . . . . . . . . . . . . . . 1369.5 Magnitude responses of Chebychev Ifilters. . . . . . . . . . . . . . . . . . 1389.6 Phase responses of Chebychev I filters. . . . . . . . . . . . . . . . . . . . . 1399.7 Pulse response of ann= 3 Chebychev I filter. . . . . . . . . . . . . . . . . 1409.8 Poles of ann = 5 Chebychev I fil t e r s . . . . . . . . . . . . . . . . . . . . . . 1 4 29.9 Magnitude responses of Chebychev II fil ters. . . . . . . . . . . . . . . . . . 1439.10 Poles and Zeros of an all pass fil t e r . . . . . . . . . . . . . . . . . . . . . . . 1 4 49.11 Frequency responses of Bessel-Thompson fil ters. . . . . . . . . . . . . . . . 1459.12 Phase responses of Bessel-Thompson filters. . . . . . . . . . . . . . . . . . 1469.13 Pulse response ofn = 3 Bessel-Thompson filter. . . . . . . . . . . . . . . . 147

9.14 Magnitude responses of elliptic filters. . . . . . . . . . . . . . . . . . . . . . 1499.15 Phase responses of elliptic filters. . . . . . . . . . . . . . . . . . . . . . . . 1509.16 Pulse responses of an n= 3 elliptic fil t e r . . . . . . . . . . . . . . . . . . . . 1 5 1

10.1 Specification of the required response. . . . . . . . . . . . . . . . . . . . . . 15510.2 Filter design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15610.3 Typical low pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15610.4 Typical high pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15710.5 High pass response specifications. . . . . . . . . . . . . . . . . . . . . . . . 15710.6 Specifications of required response. . . . . . . . . . . . . . . . . . . . . . . 157

10.7 Filter design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.8 Coaxial line filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.9 Specification of required response. . . . . . . . . . . . . . . . . . . . . . . . 160


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xiv LIST OF FIGURES

10.10Normalised low pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16010.11The required filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16110.12Typical band pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.13Band pass transformation of elements. . . . . . . . . . . . . . . . . . . . . 16310.14Band pass transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.15Band pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11.1 Thevenin equivalent circuit of a source. . . . . . . . . . . . . . . . . . . . . 16811.2 Two-port network with interconnected transmission lines. . . . . . . . . . . 16911.3 Two port circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17011.4 Realisation of third order Butterworth filter. . . . . . . . . . . . . . . . . . 17211.5 Realisation of third order elliptic filter. . . . . . . . . . . . . . . . . . . . . 17311.6 A two port circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17311.7 An example circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17411.8 Realisation offilter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12.1 First order RC circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17812.2 First order RC circuit with buffer amplifier. . . . . . . . . . . . . . . . . . 17812.3 Inverting operational amplifier circuit. . . . . . . . . . . . . . . . . . . . . 17912.4 Non-inverting operational amplifier. . . . . . . . . . . . . . . . . . . . . . . 18012.5 Low pass Sallen and Key circuit. . . . . . . . . . . . . . . . . . . . . . . . 18212.6 Band pass Sallen and Key circuit. . . . . . . . . . . . . . . . . . . . . . . . 18312.7 Inverting band pass circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . 18512.8 Bi-quadratic band pass circuit. . . . . . . . . . . . . . . . . . . . . . . . . . 186

12.9 Switched capacitor circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18812.10Switch waveforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18812.11Switched capacitor low pass fil t e r . . . . . . . . . . . . . . . . . . . . . . . . 1 8 9

H.1 A simple circuit for analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 229H.2 Sign conventions for circuit variables. . . . . . . . . . . . . . . . . . . . . . 230H.3 Phasors, rotating arms and projections. . . . . . . . . . . . . . . . . . . . . 231

I.1 Fourier spectrum for a single tone message. . . . . . . . . . . . . . . . . . . 238I.2 Illustration of a simple amplitude modulated signal. . . . . . . . . . . . . . 239I.3 Fourier spectrum with amplitude modulation by a single tone. . . . . . . . 239

I.4 Amplitude modulation using a balanced mixer. . . . . . . . . . . . . . . . 240I.5 Circuit for envelope detection. . . . . . . . . . . . . . . . . . . . . . . . . . 240I.6 Illustration of a double sideband suppressed carrier modulated signal. . . . 241I.7 Single sideband modulation using sideband filter. . . . . . . . . . . . . . . 241I.8 Phase shift single sideband modulator. . . . . . . . . . . . . . . . . . . . . 242I.9 Illustration of vestigial sidebandfil ter ing. . . . . . . . . . . . . . . . . . . . 243I.10 Synchronous detection of a modulated signal. . . . . . . . . . . . . . . . . 243I.11 Block diagram of a superheterodyne receiver. . . . . . . . . . . . . . . . . . 244I.12 Passbands in a superheterodyne receiver. . . . . . . . . . . . . . . . . . . . 245I.13 Narrow band phase modulator using balanced mixer. . . . . . . . . . . . . 246

I.14 Bessel functions of various orders. . . . . . . . . . . . . . . . . . . . . . . . 248I.15 Amplitude spectrum of an angle modulated signal. . . . . . . . . . . . . . 248I.16 Magnitude spectrum of an angle modulated signal. . . . . . . . . . . . . . 248


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LIST OF FIGURES xv

I.17 Number of significant sidebands in a frequency modulated signal. . . . . . 249I.18 A balanced discriminator circuit. . . . . . . . . . . . . . . . . . . . . . . . 250I.19 Derivation of balanced FM discriminator transfer function. . . . . . . . . . 251

I.20 A general CW communication system. . . . . . . . . . . . . . . . . . . . . 251I.21 Signal and noise in various modulation systems. . . . . . . . . . . . . . . . 253I.22 PM post detection noise spectrum. . . . . . . . . . . . . . . . . . . . . . . 254I.23 FM post detection noise spectrum. . . . . . . . . . . . . . . . . . . . . . . 255I.24 FM noise performance as a function of gamma. . . . . . . . . . . . . . . . 257I.25 FMFB receiver block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 257


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Chapter 1

INTRODUCTION

1.1 Course Definition

The course has the formal title: 3015 COMMUNICATIONS, SIGNALS AND SYS-TEMS, and occupies approximately 22 lectures at Level 3. In addition, it is expectedthat there will be, in times set aside in the timetable for lectures, two brief examinations,which will supplement the main examination at the end of the semester.

1.2 Objectives

The course seeks to introduce

Basic probability theory and applications thereof.

The statistical properties of signals, the noise which can accompany those signals,and what we can do about it.

The behaviour and design of circuits for filtering signals.

Elementary communication theory and simple communication ciruits.

1.3 Acknowledgments

Chapter 3 of these notes has drawn heavily on the book listed as Reference1 3 in Appen-dix A. The notes also draw extensively on material originally written for other courses byDr. B. R. Davis. An emphasis on the concept of essential knowledge has been introducedby the current lecturer.

1.4 Access to Notes

The current version of the lecture notes, homework and tutorial problems, and somesolutions, will be progressively available at the current Lecturers web page which is at

http://www.eleceng.adelaide.edu.au/Personal/peter/peter.

1


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2 CHAPTER 1. INTRODUCTION

1.5 Bibliography

A list of useful reference books appears in Appendix A. Some of those books are concerned

with professional practice beyond the limits of this introductory course.

1.6 Essential Knowledge

An attempt will be made in this course to connect the material to that of other courses,and in particular to courses on basic circuit theory, and to descriptions in engineeringterms of practical hardware.

The presentation will also be based on the belief that the understanding of advancedwork is facilitated by having a firm and correct grasp of elementary and fundamentalmaterial.

These two factors will lead to the identification of a number of fundamental concepts,recorded in Chapter 2, of which all students should be able to demonstrate clear under-standing without recourse to notes.

A significant part of the assessment in the subject will be devoted to testing whethersuch skill has been acquired.

1.7 Making Use of the Lectures

Not all of the material to be presented in this course will be treated in detail in these notes.

Students are therefore advised to attend all lectures, and to come to lectures prepared totake notes on occasion.

1.8 Other Goodies

Advice on how to study for the examination is contained in Appendix D.


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Chapter 2

REVISION

2.1 Introduction

This chapter provides a listing of what is considered to be essential knowledge in thiscourse. The first assessment exercise will be set on what may be found here.

2.2 Units

2.2.1 SI Units

In this course and presumably other courses, quantities will be expressed using terminol-ogy and units defined in SI units and recommendations for the use of their multiples andof certain other units, International Standard ISO 1000 (1992), International Organisa-tion for Standardisation, Case postale 56, CH-1211, Genevre 20, Switzerland.

You are expected to know the names and SI units for the circuit theory quantitiescommonly denoted by R, X, Z, G, B, Y, L, and C, for the field quantities of voltage,current, and power.

While we are discussing terminology and units, we take the opportunity to remindourselves that the names and units of the electromagnetic field quantities denoted by E,H,D, BandJare as shown in the Table below.

E E Electric field intensity Vm1

H H Magnetic field intensity Am1

D D Electric flux density Cm2

B B Magnetic flux density Wbm2

J J Volume current density Am2

Table 2.1: Names and units for field quantities.

2.2.2 Quantities derived by differentiation

An important point to remember is that when one quantity is derived from another bydifferentiating with respect to a dimensioned variable, for example position or time, achange of dimensions and thus of units results.

3


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4 CHAPTER 2. REVISION

2.2.3 Frequency and angular frequency

It is also important to distinguish between the concepts of frequency and angular fre-

quency. Something called a frequency is measured in Hz. Normally such a quantity ifdenoted by the symbol f. Something called an angular frequency is measured in rad/s.Normally such a quantity if denoted by the symbol . The relation between the two is

= 2f (2.1)

It is very common that students lose marks by not observing the distinction.The quantity known as bandwidth, and denoted by BW, in also measured in Hz. A

related quantity, known as angular bandwidth, and greater than BW by a factor 2, ismeasured in rad/s. Not observing the distinction is a further common source of error.

2.3 RC and RL Circuits

2.3.1 Things not possible

It should be remembered that it is not possible to have a step change in capacitor voltage,not is it possible to have a step change in inductor current. We will focus below on whatis possible, and review some standard results, which should be committed to memory.

2.3.2 Establishing a capacitor voltage

Figure 2.1: Charging a capacitor though a resistor.

Figure 2.2: Waveforms in charging a capacitor through a resistor.


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2.3. RC AND RL CIRCUITS 5

2.3.3 Discharging a capacitor

Figure 2.3: Discharging a capacitor though a resistor.

Figure 2.4: Waveforms in discharging a capacitor through a resistor.

2.3.4 Establishing an inductor current

Figure 2.5: Energising an inductor though a resistor.

2.3.5 De-energising an inductor


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Figure 2.6: Waveforms in energising an inductor through a resistor.

Figure 2.7: De-energising an inductor though a resistor.

Figure 2.8: Waveforms in de-energizing an inductor through a resistor.


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2.4. PHASOR ANALYSIS FOR LINEAR A.C. CIRCUITS 7

2.4 Phasor Analysis for Linear a.c. Circuits

2.4.1 Notation

Much of the time in this course we will be dealing with variables which directly expressthe values of the physical quantities, such as, for example, voltage or current. If thosephysical quantities have a time variation, so do the variables of our equations.

In such a case, when the quantities represented are scalars, as in the example ofvoltage or current just mentioned, we use lower case Roman or sometimes Greek lettersto represent them. Sometimes the time variation is shown, and sometimes it is not, as forexample in the equation for a real time-varying voltage

v= v(t). (2.2)

In many cases it will be convenient to restrict the time variation of all physical quan-tities to be either constant or sinusoidal, or more explicitly to be ofcosine form.

In such cases, the behaviour of each time-varying quantity is known for all time if weknow thefrequency, theamplitudeand thephaseof the cosine function of time. In a singlecontext, all such quantities are assumed to have the same frequency, which is stated onceas a fixed part of that context, but the different variables representing different quantitiescan have various amplitudes and phases.

In this situation, it is convenient to introduce a variable called a complex phasorwhichwhile not itself being a function of time, does represent in the manner described belowa time-varying quantity. What we do is to introduce a complex number, constructed so

that the magnitudeof the complex number is the amplitudeof the cosine waveform, andthe angle of the complex number in its polar representation, as shown for example inFigure 2.9, is the phase angleof the cosine waveform.

Figure 2.9: Polar representation of a complex number.

Thus for the sinusoidally varying quantity

v= Vmcos(t +) (2.3)

which has an amplitude Vm and a phase angle , we construct using Vm and the

time invariant complex phasorV given by

V =Vmej (2.4)


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It may be noted that the relation between the time invariantcomplex phasor V andthe time-varyingvariable v(t) which it represents is

v(t) = Vejt (2.5)In words, this relation says that to recover the time function from the complex phasor,

we multiply the complex phasor by ejt and take the real part.

Agraphicalinterpretation of the mathematical operation just defined is that to recoverthe time function from the complex phasor, we can first represent as shown in Figure 2.9the phasor in the Argand diagram, and take its projectionon the horizontal axis as theexpression of the value of the time function at the time t= 0. To visualise the behaviourof the physical quantity as a function of time, we must construct a rotating arm whoseposition at t = 0 is that of the complex phasor, and rotate it in a counter-clockwisedirection on the Argand diagram at an angular frequency , starting at time t = 0 at theposition illustrated in Figure 2.9, and watch the values of the projection on the horizontalaxisof the rotating arm which results.

Notice in the above exposition that we have notsaid that the phasor rotates. To doso would contradict the definition of the phasor as a time invariant quantity. We havegiven a different name, namely rotating arm, to the thing that rotates.

In establishing the notation described above, we have been able, because it is available,to use different calligraphy, namely v (italic) and V (upright Roman), to distinguish thereal time-varying variables directly representing the physical quantities, and the time

invariant complex phasors indirectly representing them. The difference in notation ishelpful in avoiding misunderstandings.

You are expected to have a facility for conversion of time domain variables to frequencydomain variables and vice versa.

2.4.2 Particular cautions

There can be noj in a time domain expression.

There can be not in a frequency domain expression.

2.4.3 Peak value and r.m.s. value phasors

There are two common conventions in which sinusoidal quantities are expressed. In thefirst, known as the peak value phasor convention, the phasors have magnitudes that areequal to the peak values of the sinusoidal quantities expressed. Thus the relation betweenthe phasors and the real time varying quantities represented is given by

v(t) =

Vejt

(2.6)

In the second convention, known as the r.m.s. value phasor convention, the phasors

have magnitudes that are equal to the r.m.s. values of the sinusoidal quantities expressed.those r.m.s. values are less by a factor of

2 than the peak values. Thus the relationbetween the phasors and the real time varying quantities represented is given by


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2.5. CHARACTERISATION OF N-PORT NETWORKS 9

v(t) =

2Vejt

(2.7)

How can you tell which convention is in use? Well, in a perfect world, wheneveran r.m.s. phasor is used, it should have r.m.s. after it. Another thing which helpsis knowing that power engineers generally employ r.m.s. phasors, and communicationengineers generally employ peak value phasors. Do you seem to live in a perfect world?

2.4.4 Properties of a sinusoid

For the sinusoidal voltage given in peak value phasor terms by

V = 10j (2.8)

you should be able to persuade yourself that the maximum value is 10 V, the minimumvalue is -10 V, the average value is 0 V, the value at time t = 0 is 0 V, and the r.m.s.value is approximately 7.07 V.

When a sinusoidal current flows in a resistor, the power dissipated is time-varying,but is at all points of time either positive or zero. It oscillates, at a rate of twice thefrequency of the sinusoidal current, between a value of zero and twice the average value.The average value of the power dissipated is the same as would be dissipated by d.c.current equal to the r.m.s. value of the sinusoid.

2.4.5 Unusual waveforms

In calculating r.m.s. values for unusual waveforms, please remember that r.m.s means the(root (of the mean (of the square))).

2.4.6 Resistance and resistivity

You should be familiar with the formula

R= l

A (2.9)

for the resistanceR of a rectangular bar of material of length l, resistivity and crosssectional areaA, and be able to work practical exercises thereon.

2.5 Characterisation of N-Port Networks

2.5.1 Definitions of port variables

For characterisation of an N-port network such as is illustrated in Figure 2.10, we mustfirst define senses for the voltage and currents at each of the ports, a portbeing a pair ofterminals.

The convention universally chosen is that the current Iis defined as that which entersa selected terminal of the pair, while the voltageVis simultaneously defined as the voltageof the selected terminal with respect to the other. It is also assumed for the purpose of


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Figure 2.10: An N-port network.

characterisation that if current I enters the selected terminal, a current I emerges fromthe other terminal.

This condition might or might not be a natural property of the network, but it canbe certainly be imposed from outside by ensuring that each of the signal generators orpassive impedances which are connected externally to the ports are floating i.e. haveno interconnections with one another. In line with this picture offloating external portconnections, we make no inquiry as to what voltages may exist between one pair ofterminals considered as a port and another pair of terminals considered as another port.

The lack of interest in this matter can be excused in some cases because the external

connections are truly fl

oating, in which case the number of independent currents whichcan flow between the 2N terminals is limited to just N, and can be excused in someother cases because there is at each port a common ground connection, in which casethe number of independent voltages which can exist between the 2Nterminals is limitedto just N. In some other cases in which neither of these situations applies, we must beaware that the viewpoint we are adopting is not capable of representing the full range ofbehaviour of a network with 2Nterminals. Whatever be the case, we are going to persistwith our notion that the variables shown in Figure 2.10 are sufficient for our purpose.

2.5.2 Definition of impedance matrix

With the variables so defined, we can take the view that we could in an empirical senseimpose, by means of ideal current generators at each port, whatever currents we please,


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2.5. CHARACTERISATION OF N-PORT NETWORKS 11

and leave it to the network to determine the voltages at the ports, which voltages wecould then measure. If we assume that the network contains only linear passive circuitelements, we can conclude from this view that the network is describable by the set of

equations below, in which theN Narray of parameters is called the impedance matrixfor the network.

V1V2...

VN

=

Z11 Z12 ... Z 1NZ21 Z22 ... Z 2N

... ...

. . . ...

ZN1 ZN2 ... Z NN

I1I2...

IN

(2.10)

It is clear from the above equation that the element Zijmay be determined by injectinga current into portj , ensuring that no currents are injected into the other ports by leavingthem open circuit, and studying the voltage developed at port i, taking care that the

process of observing that voltage is conducted via an ideal voltmeter, i.e. one whichdraws no current from the port.

In the light of this observation, the set of impedance parameters of a network aresometimes given the alternative name of the open circuit parametersof that network.

Of course the ideal voltmeter contemplated in the paragraph above does not exist,and some care must be taken to correct for the finite impedance of real voltmeters usedin the observation.

2.5.3 Definition of admittance matrix

An alternative view which we could in an empirical sense take is that we could impose,by means of ideal voltage generators at each port, whatever port voltages we please, andleave it to the network to determine the currents which then flow at the ports, whichcurrents we could then measure. If we continue to assume that the network containsonly linear passive circuit elements, we can conclude from this view that the network isdescribable by the set of equations below, in which the NN array of parameters iscalled the admittance matrixfor the network.

I1I2...IN

=

Y11 Y12 ... Y 1NY21 Y22 ... Y 2N

..

.

...

. ..

...YN1 YN2 ... Y N N

V1V2...VN

(2.11)

It is clear from the above equation that the elementYij may be determined by placinga known voltage across port j, ensuring that no voltages appear across the other portsby placing short circuits across them, and studying the current flowing in the referencedirection at port i when a short circuit is placed across that port, taking care that theprocess of observing that current is conducted via an ideal ammeter, i.e. one which placesno burden on the port, i.e. one which has no series resistance.

In the light of this observation, the set of admittance parameters of a network aresometimes given the alternative name of the short circuit parametersof that network.

Of course the ideal ammeter contemplated in the paragraph above does not exist, andsome care must be taken to correct for the non-zero burden of real ammeters used in theobservation.


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2.5.4 Relations between impedance and admittance parameters

It is easy to show that when both sets of parameters exist, the impedance matrix and the

admittance matrix are inverses of one another.

2.5.5 Pathological cases

While it is true that most networks have both impedance and admittance matrices, thereare special cases in which one or other of the matrices (or even both) may not exist. Forexample a network with an internal short circuit across one port may have an impedancematrix, but cannot have an admittance matrix, and a network with one port internallyopen circuited and not otherwise internally connected may have an admittance matrix,but cannot have an impedance matrix.

2.5.6 The reciprocity theorem

A fundamental theorem which may be applied in the network theory context to linear pas-sive networks and in the electromagnetic theory context to screened enclosures containingmedia which exhibit appropriate behaviour, is the reciprocity theorem, which states, in thenetwork theory context, that under appropriate conditions the impedance and admittancematrices are symmetric.

A set of sufficient conditions for the reciprocity theorem to apply is that the net-work may contain resistors, inductors and capacitors, but contains no voltage or current

generators.The most enlightening proof of the reciprocity theorem is that provided for the fullygeneral version of the theorem which is established in the electromagnetic theory contextat Level 4 of the course. That fully general treatment illuminates both the assumptionsunderlying lumped element network theory and the restrictions on material constitutiveparameters which are required for the theorem to be valid.

2.6 Thevenins and Nortons Theorems

2.6.1 Thevenin and Norton networks

An illustration of a fairly general linear network is provided in Figure 2.11 below. Thenetwork has many internal connections, but only a single pair of external terminals onwhich we will shortly focus attention. We expect that a set of liner equations, derivingfrom the equations describing each element, will determine the behaviour of the circuit,and in particular the relation between the current and voltage at the external terminals.

It is important to note that this expectation of a linear relation between the currentand voltage at the external terminals derives from the linearity of the circuit elements eminternal to the network, and does not depend upon whether we plan to connect eitherlinear elements or non-linear elements externally.

The linear relation between external terminal current and voltage can be expressedgraphically as in Figure 2.12 below. The figure reminds us that the most general linearrelation can be described in terms of two parameters, which might be chosen as the slope


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2.6. THEVENINS AND NORTONS THEOREMS 13

Figure 2.11: A nice linear one port network.

and the intercept on the vertical axis, or might alternatively be chosen as the interceptson the two axes.

Figure 2.12: A linear terminal voltage current relation.

We now ask the question of whether the circuit, viewed solely in terms of the relationbetween its external terminal voltage and current, can be represented by some simplernetwork in which there are fewer elements. The obvious answer is that it can, and in morethat one way. The two networks shown in Figure 2.13 are each capable of representinga relation such as shown in Figure 2.12. The resistance appearing in both circuits is the

negative of the slope in Figure 2.12, while the voltage and current generators have valuesequal to the intercepts on the axes in Figure 2.12.These circuits are called respectively the Thevenin and Norton equivalent circuits of


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2.6.3 Pathological cases

Not all networks have both Thevenin and Norton circuits. In fact an attempt to construct

a Norton equivalent for the ideal voltage source shown in Figure 2.14 leads to an infi

nitecurrent in parallel with a zero resistance, and no sensible calculations can be performedwith that result. We say the Norton circuit does not exist.

Figure 2.14: A network with no Norton equivalent.

Similarly an attempt to construct a Thevenin equivalent for the ideal current sourceshown in Figure 2.15 leads to an infinite voltage in series with an infinite resistance, andno sensible calculations can be performed with that result. We say the Thevenin circuit

does not exist.

Figure 2.15: A network with no Thevenin equivalent.

Some networks lend themselves to immediate simplification in the construction of aThevenin or Norton circuit. For example an ideal voltage source in parallel with some

other component, as shown in Figure 2.16, produces exactly the same effect as that idealvoltage source alone, so the other component may be immediately omitted, ie replacedby an open circuit.


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Figure 2.16: Simplification of a Thevenin network.

A similar simplification occurs when an ideal current source is in series with someother component. A little thought reveals that the other component may be omittedfrom the circuit, ie replaced by a short circuit.

2.6.4 Networks with non-linear elements

Figure 2.17: A network with a non-linear element.

Although the presence of a nonlinear element such as is shown in Figure 2.17 belowwill produce a nonlinear relation between terminal current and voltage, and thus precludethe existence of a Thevenin or Norton circuit, there are pathological cases, such as areillustrated in Figure fig:netw:anne or Figure fig:netw:fnne, in which a non-linear elementis present in such a way as to produce no non-linearity of the external terminal voltagecurrent relation, and a Thevenin or Norton circuit may still exist.

2.6.5 Frequency domain equivalent circuits

Although we have conducted the above discussion in the implied context of d.c. networks,

the results are also valid in the context of sinusoidally varying voltages, represented bycomplex phasors, in networks consisting of complex impedances. That this is true followsfrom the homomorphormism of the equations describing the above two contexts.


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Figure 2.18: Another network with a non-linear element.

Figure 2.19: A further network with a non-linear element.


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2.6.6 Warning against undue generalisation

It is most important to note that the Thevenin and Norton equivalent circuits do not

represent the details of the internalbehaviour of the network. They do not indicate whatcomponents are actually inside the network, nor do they give details of properties otherthan the relation between external terminal voltage and current, for example the internalpower dissipation.

A good example of the above statement is provided by the representation of an au-tomobile battery, which has an open circuit voltage of about 12 V d.c. and which dropsonly slightly when quite large currents are drawn from it. A Thevenin equivalent circuitfor a typical battery is shown in Figure 2.20.

12 V

0.01 WA

B

Figure 2.20: Thevenin representation of an automobile battery.

From the terminal voltage and current point of view the battery may equally well berepresented by the Norton equivalent circuit of Figure 2.21. Either circuit will give correctresults for the terminal voltage and current when any particular load, linear or non-linear,is connected to the terminals.

A

B

0.01 W1200 A

Figure 2.21: Norton representation of the same battery.


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Figure 2.23: Variation of series circuit current vs frequency.

It may be shown for a simple series resonant circuit that

Q=2f0L

r (2.15)

At resonance the voltage appearing across the inductor (or, alternatively, the capac-itor) is just Q times the source voltage. For that reason, Q is sometimes known as thevoltage magnification factor.

The bandwidth BW is defined as the difference (in Hz) between the frequencies atwhich the series circuit current is 1

2of the value at resonance. At those so-called half-

power pointsthe power dissipated in the circuit is just half the value at resonance. It maybe shown that the bandwidth and Q are related by

BW =f0Q

(2.16)

2.7.2 Ideal parallel resonant circuit.

A simple parallel resonant circuit excited by an ideal current generator is shown in Fig-ure 2.24 below. In that circuit the resonant frequency is again given by

f0 = 1

2

LC(2.17)

The impedance of the circuit is a maximum at resonance and is just equal to theresistanceR. This maximum impedance at resonance is known as the dynamic impedance.

For excitation with a constant current over a range of frequencies the variation of thevoltage appearing across the circuit versus frequency is as shown in Figure 2.25 below.

The quality factor Q is again defined as


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2.7. TUNED CIRCUITS 21

Figure 2.24: Ideal parallel resonant circuit

Figure 2.25: Variation of parallel circuit voltage vs frequency


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Q= 2(energy stored at resonance)

(energy dissipated per cycle) (2.18)

It may be shown for an ideal parallel resonant circuit that

Q= R

2f0L (2.19)

The bandwidthBW is defined as the difference (in Hz) between frequencies at whichthe parallel circuit voltage is 1

2of the value at resonance. At those so-calledhalf-power

points the power dissipated in the circuit is just half the value at resonance. It may beshown that the bandwidth and Q are related by

BW =f0Q

(2.20)

2.7.3 Practical parallel resonant circuit

A alternative parallel resonant circuit excited by an ideal current generator is shown inFigure 2.26 below. This circuit is known is the practical parallel resonant circuitbecausein a parallel resonant circuit, the significant losses generally occur within the inductor inthe manner indicated by the circuit.

Analysis of that circuit is algebraically more complex than for each of the two circuitsconsidered previously, and we in the below statements provide the results ofan approx-imate analysis which

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