1 "GAIN, SENSITIVITY AND STABILITY :.. OF LINEAR TWO PORT AMPLIFIERS" by Anek Singhakowinta B.Sc. (Eng. ) $ I. a Thesis submitted for the Degree of DOCTOR OF PHILOSOPHY of the UNIVERSITY OF LONDON Department of Electrical Engineering, Imperial College of Science & Technology, June 1964 2. ABSTRACT In this thesis matched and mismatched two-port amplifiers with general feedback are analysed, compared and discussed in terms of the basic parameters of the two-ports. It is shown that the matched amplifier is generally superior to the mismatched amplifier, both in gain-sensitivity performance and in ease of design. The power gain of the matched amplifier can be expressed simply in terns of only two basic parameters, which are independently variable; one of these is real and the other complex. The relation between the power gain and the parameters is displayed geometrically in the forms of the Inverse Gain Space and the Inverse Gain Chart. Both of these geometrical representations find useful applications in amplifier design, particularly since an area or volume of spread in these representations tends to remain invariant in size as it is moved about by feedback. The gain-sensitivity performance of the matched amplifier is investigated in detail for various types of spreads of basic parameters. Sensitivity figures are defined to facilitate consideration of amplifier sensitivity performance, and it is found that gain-sensitivity figure products tend to remain constant in the stable region except for the area close to the border of marginal stability. Since basic parameters are so useful in amplifier consideration, new methods for measuring and determining them are described and verified experimentally. These measurements can be done using both lumped and distributed measuring circuits. Finally, new techniques for synthesizing amplifiers to have given sensitivity performance and frequency response are described. These techniques are illustrated by experiments using physical transistors. 3 ACKNOWLEDGENINTS The author wishes to express his deep gratitude to his supervisor, Dr. A.R. Boothroyd, for his guidance, encouragement and advice throughout the course of this research. He is also greatly indebted to Mr. R.A. King for many stimulating discussions and valuable criticisms, and to Dr. R. Spence for continued interest and helpful suggestions. Thanks are also extended to his colleagues in the Transistor Laboratory, in particular to R.G. Harrison, A.S. Oberai, J.V. Hanson, R. Johnston and V. Roengpithya for helpful discussions and assistance in the preparation of this thesis. Finally, financial support provided by International Computers and Tabulators, Ltd, in the form of the Hollerith Research Studentship is gratefulacknowledged. L. TABLE OF CONTENTS PAGE TITLE 1 ABSTRACT 2 ACKNOWLEDGMENTS 3 TABLE OF CONTENTS 4 LIST OF PRINCIPAL SYMBOLS 11 LIST OF PRINCIPAL SUBSCRIPTS 12 LOCATION OF FIGURES 13 LOCATION OF TABLES 14 CHAPTER 1 : INTRODUCTION 15 1.1 INTRODUCTORY BACKGROUND 15 1.1.1 Device Characterisation 15 1.1.2 Two-Port Networks and Matrix Analysis 15 1.1.3 Three-Terminal Devices as Two-Port Networks 16 1.2 EMPHASIS 17 1.3 HISTORICAL NOTES 17 1.3.1 Two-Port Networks 17 1.3.2 Unilaterisation of Amplifiers 18 1.3.3 Conjugate-Matched Amplifiers 18 1.3.4 Stability Measures for Conjugate-Matched Amplifiers 20 1.3.5 Padded Amplifiers 20 1.3.6 Mismatched Amplifiers 20 1.3.7 Stability Measures for Mismatched Amplifiers 22 1.3.8 Activity 22 1.3.9 Reciprocity 23 1.4 FORMULATION OF PROBLEM 23 1.4.1 Sensitivity 23 1.4.2 Stability and Sensitivity 24 1.4.3 Gain-Sensitivity Capability 25 1.4.4 Negative Real Parts of Self-Immittances 25 1.4.5 Simplification of Design Procedures 25 1.4.6 Present Work 25 5 PAGE 1.5 ORIGINALITY 26 CHAPTLH 2 : ANALYSIS OF THE MATCHED AMPLIFIER 37 2.1 INTRODUCTION ....... 37 2.2 COMPLEX MEASURE OF NON-RECIPROCITY 38 2.3 MEASURE OF GOODNESS 39 2.4 BASIC POWER GAIN RELATION 40 2.5 GEOMLTPICAL REPRESENTATIONS 42 2.5.1 Normal Gain Chart 1+2 2.5.2 Inverse Gain Chart 44 2.5.3 Gain Space 46 2.6 PROPERTIES OF BASIC POWER GAIN RELATION 47 2.6.1 Activity and Power Gain 47 2.6.2 :ffaximum Matched Gain 48 2.6.3 Power Gain Capability of a 3-Terminal Device 1+9 2.6.4 Effect of Stabilisation 49 2.7 APPLICATIONS OF BASIC POWER GAIN RELATION 51 2.7.1 Device Specification 51 2.7.2 Study and Control of Sensitivity 52 2.7.3 Simplification of Circuit Design 53 2.8 CASCADING CONSIDERATION 54 CHAPTER 3 : ANALYSIS OF THE MISMATCHED AMPLIFIER 64 3.1 INTRODUCTION 64 3.2 MISMATCHED TWO PORT AND EQUIVALENT PADDED TWO-PORT 65 3.2.1 Basic Concept 65 3.2.2 Equivalent Stability Factor 65 3.2.3 Relation between Mismatched Amplifier and Equivalent Padded Amplifier 66 3.3 ANALYSIS IN TERMS OF TWO-PORT PARAMETERS AND EQUIVALENT STABILITY FACTOR 68 3.3.1 Evaluation of Equivalent Padding Elements 68 3.3.2 Evaluation of Equivalent Stability Factor 69 3.3.3 Skew Factor 8 70 6 PAGE 3.3.4 Mismatched Gain Expressions 72 3.3.5 Some Useful Expressions 73 3.4 PRACTICAL CONSIDERATION 74 3.5 MISMATCHED GAIN AND BASIC PARAMETERS 75 3.5.1 Use of Basic Parameters 75 3.5.2 Mismatched Gain and Matrix Environment 76 3.5.3 Mismatched Amplifier with Feedback 77 3.5.4 Relative Magnitude of Ueq •78 3.6 GEOMETRICAL REPRESENTATION ..78 3.7 PROPERTIES OF MISMATCHED GAIN RELATIONS 80 3.7.1 Power Gain at Large Mismatch 80 3.7.2 Maximum Mismatched Gain 81 3.7.3 Regions of Positive and Negative n. 81 3.8 DESIGN CONSIDERATIONS 82 3.8.1 M/L and Power Gain 82 3.8.2 N/L and Sensitivity 83 3.8.3 Choice of Skew Factor 45 85 3.8.4 General Sensitivity Consideration 87 3.8.5 Near-Optimal Mismatched Amplifier Design 88 3.9 CASCADING CONSIDERATION 90 3.9.1 Relation betireen Stage Gain and Overall Gain 90 3.9.2 Relation between Transducer Gain and Actual Gain 91 3.9.3 Design of Interstage Transformers 93 3.9.4 Sensitivity Consideration when Cascading .93 CHAPTER 4 : GAIN-SENSITIVITY CONSIDERATION 104 4.1 INTRODUCTION 104 4.2 DESIGN REQUIREMENTS 105 4.3 CHOICE OF SENSITIVITY DEFINITION 106 4.4 SPREAD IN MEASURE OF NON-RECIPROCITY a07 4.4.1 Maximum Gain Deviation due to Circular Spread in X 107 4.4.2 Sensitivity and Stability 109 4.4.3 Sensitivity Figure for Spread in X .110 4.4.4 Gain-Sensitivity Figure Product for Spread in X 111 7 PAGE 4.5 SPREAD IN MEASURE OF GOODNESS 112 4.5.1 Maximum Gain Deviation due toil.(1/t) 112 4.5.2 Sensitivity and Stability 113 4.5.3 Sensitivity Figure for Spread in (1/U) 113 4.5.4 Gain-Sensitivity Figure Product for Spread in 1/u 114 4.6 VOLUME OF SPREAD IN INVERSE GAIN SPACE 115 4.6.1 Cylindrical Volume of Spread 116 4.6.2 Spherical Volume of Spread 117 4.7 CHOICE OF EMBEDDING NETWORKS FOR CONTROLLING SENSITIVITY 119 4.7.1 Lossless and Lossy Embeddings 119 4.7.2 Consideration of Gain-Sensitivity Figure Products 119 4.7.3 Detailed Comparison between Lossy and Lossless Techniques 120 (a)Spread in X i21 (b)Spread in (1/U) 123 (c)Cylindrical Spread 124 (d)Spherical Spread 125 4.8 CHOICE OF DEVICE CONFIGURATION 127 CHAPTER 5 : MEASURE/ENT OF BASIC PARAMETERS 149 5.1 INTRODUCTION 149 5.2 DIRECT MEASUREMENT OF A USING LUMPED BRIDGE CIRCUITS 150 5.2.1 Basic Transformer Ratio-Arm Bridge 151 5.2.2 Adaptation of Transformer Ratio-Arm Bridge 151 (a)Principle of Operation 151 (b)Experimental Circuits 152 (c)Experimental Results 153 (d)Alternative Arrangements 153 5.2.3 New Bridge Circuit 154 (a)Principle of Operation (b)Practical Bridge l55 (c)Bridge Calibration 157 (d)Measurement Procedure 8 PAGE (e) Experimental Results 159 5.3 DIRECT MEASUREMENT OF A USING DISTRIBUTED CIRCUIT 160 5.3.1 Basic G-R Transfer-Function Bridge ...... 160 5.3.2 Modification of G-R Transfer-Function Bridge 161 (a)Principle of Operation 161 (b)Bridge Set-Up Procedure 162 (c)Experimental Results 164 5.4. MEASURUTENT OF X - 1/A 164 5.5 MEASUREMENT OF U 165 505.1 Introduction 165 5.5.2 Principle of Determination of U 166 5.5.3 Determination of 1/S1 167 Method A • 167 Construction A-1 -- Zawels' Construction 167 Construction A-2 -- A Modification of Zawels' Construction 168 Construction A-3 -- Further Modification of Zawels' Construction 168 Additional Construction 169 Method B 170 Construction B-1 -- Wheeler's Construction 170 Construction B-2 -- Improved Construction 171 5.6 EVALUATION OF U 171 5.6.1 Geometrical Construction 171 5.6.2 Evaluation of U Using a Chart 173 5.6.3 Experimental Results 173 CHAPTER 6 : DESIGN EXAMPLES 0 • • 0 • 0 ......... 0 • ........ ....190 6.1 INTRODUCTION 190 6.2 SYNTHESIS OF AMPLIFIERS FOR GIVEN SENSITIVITY PERFORMANCE ....190 6.2.1 Preliminary Data 190 6.2.2 Design Problem 191 6.2.3 Amplifier Design by Lossless, Matched Technique 192 9 PAGE 6.2.4 Comparison with Amplifiers Designed by Lossy, Matched Technique 194 6.2.5 Comparison with Amplifiers Designed by Mismatched Technique 196 6.2.6 Discussion on Synthesis of Amplifiers .197 6.3 NEV TFCHNIQUE FOR SYNTHESIZING WIDE-BAND AMPLIFIERS 197 6.3.1 Design Principle 197 6.3.2 Design Specifications 198 6.3.3 Preliminary Data 199 6.3.4 Design of Feedback Circuit 200 6.3.5 Design of Matching Networks 201 6.3.6 Experimental Results 202 6.3.7 Discussion on New Technique for Synthesizing Wide-Band Amplifiers 203 CHAPTER 7 : CONCLUSIONS AND GENERAL DISCUSSION •••opocaot•0•13••21.7 7.1 GENERAL CONCLUSION 217 7.2 THE MATCHED AMPLIFIER 217 7.3 THE MISMATCHED AMPLIFIER 218 7.4 COMPARISON BETWEEN THE MATCHED AND THE MISMATCHED AMPLIFIERS-318 7.5 GAIN-SENSITIVITY RELATIONS .319 7.6 MEASUREMENT OF BASIC PARAMETERS 220 7.7 SYNTHESIS OF AMPLIFIERS FOR GIVEN SENSITIVITY PERFORMANCE 220 7.8 SYNTHESIS OF WIDE-BAND AMPLIFIERS ......-..o..........•