"Gain, Sensitivity and Stability of Linear Two Port

"GAIN, SENSITIVITY AND STABILITY :.. OF LINEAR TWO PORT AMPLIFIERS"

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

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

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

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...... • ...221 7.9 DEVICE SPECIFICATION 221 7.10 FURTHER RESEARCH PROBLEMS 221 (1)Investigation into Gain-Bandwidth Limitation 221 (2)Investigation into Synthesis of Amplifiers Using Minimum Components 222 (3)Investigation into Relations between Physical Parameters and U and A 222 (4)Development of New Measurement Techniques 223 (5)Investigation into Use of Distributed Components 223 (6)Investigation into the Two-Port Oscillator 223

PAGE APPENDICES 225 Appendix 1 : Relation between k and Amplifier Operating Condition 225 Appendix 2 : Relation between n and Amplifier Operating Condition 227 Appendix 3 : Common Feedback Arrangements 229 Appendix 4 : Feedback and Gain Charts 234 Appendix 5 : Effect of Feedback on Spread in Non-Reciprocity Measure 236 Appendix 6 : Effect of Embedding on Spread in Measure of Goodness 239 Appendix 7 : "Maximum Efficiency" of General Two-Port 244 REFERENCES 246

LIST OF PRINCIPAL SYMBOLS SYMBOL FIRST USED DEFINITION ON PAGE A complex measure of non-reciprocity 38 G conjugate-matched power gain 19 g hybrid two-port matrix parameters 17 Gt transducer power gain 19 h hybrid two-port matrix parameters 17 I current into two-port 16 k Stern's absolute stability factor 21 k. i (411P22)/(L -I- M) 21 L magnitude of 10 -1210-21 19 M real part of P12P21 19 N imaginary part of P12P21 19 n P1P2/L 22 ni P11P22/L 22 p general two-port matrix parameters 19 sensitivity figure SF 107 inherent stability factor of two-port Si 19 U measure of goodness or Mason's lossless unilateral power gain 23 ✓ voltage at a port of two-port 16 y admittance, two-port matrix parameters 17 z impedance, two-port matrix parameters 17 8 skew factor 22 li inherent stability factor 19 X1/A, inverse measure of non-reciprocity 44 A' U/A, normalised inverse measure of non-reciprocity 45 P real part of "p" 19 PS + P11' real part of self immittance P1 at port 1 21 P22' real part of self-immittance P2 PL -I- at port 2 21 12

SYMBOL FIRST USED ON PAGE imaginary part of "p" 19 a1 aS all' imaginary part of self- immittance at port 1 68 a + 0 imaginary part of self- cr 2 L 22' immittance at tort 2 68 0 angle. (1/A) or if X 108

LIST OF PRINCIPAL SUBSCRIPTS FIRST USED SYMBOL SUBSCRIefS REFER TO ON PAGE

cy cylindrical spread 116 eq equivalent padded (matched) amplifier 65 F feedback 229 I imaginary part 42 i original two-port 19 L load 19 O spherical spread 118 P1 total port immittance at port 1 68 P2 total port immittance at port 2 68 real part 40 S source 19 U spread in U or 1/U 112 X spread in 1/A or X 109 1 port 1 of two-port 21 2 port 2 of two-port 21 13

LOCATION OF 2IGURES

FIGURE PAGE FIGURE PAGE 1.1 27 4.6 135 1.2 28 4.7 136 1.3 29 4.8 137 1.4 30 4.9 138 1.5 31 4.10 139 1.6 32 4.11 140 1>7 33 4.12 141 1.8 34 4.13 142 2.1 56 4.11+ 11+3 2.2 57 4.15 141+ 2.3 58 4.16 145 2.4 59 4.17 146 2.5 6o 4.18 147 2.6 61 4.19 148 2.7 62 5.1 175 2.8 63 5.2 176 3.1 95 5.3 177 3.2 96 5.4 178 3.3 97 5.5 179 3.4 98 5.6 i8o 3.5 99 5.7 Ai 3.6 100 5.8 181 3.7 101 5.9 182 3.8 102 5.10 183 3.9 103 5.11 183 4.1 130 5.12 184 4.2 131 5.13 185 4.3 132 5.14 186 4.4 133 5.15 187 4.5 134 5.3.6 188

FIGURE PAGE FIGURE PAGE 5.17 188 A.1.1 225 5.18 189 A.3.1 232 6.1 204 A.3.2 233 6.2 205 A.3.3 233 6.3 206 A.5.1 236 6.4 207 A.6.1 243 6.5 208 6.6 209 6.7 210 6.8 211 6.9 211 6.10 211 6.11 212 6.12 212 6.13 213 6.14 214 6.15 215 6.16 216

LOCATION OF TABLES

TABLE PAGE 1.1 35 1.2 36 15 CHAPTER

INTRODUCTION

1.1 INTRODUCTORY BACKGROUND

1.1.1 Device Characterisation

Active devices are frequently characterised by equivalent circuits, which often need to be quite complex, in order to represent the devices accurately. For example, the transistor may be adequately represented by a simple resistive T -equivalent circult 1at low frequencies; but at high frequencies, suitable equivalent circuits2'3 may have to be very complicated. Thus, with the mesa structure' an equivalent circuit of figure 1.1 may be necessary. Using this equivalent circuit, a simple calculation, say, of an amplifier between 75-ohm source and load impedances at a single frequency, can involve quite unwieldy algebraic expressions. An alternative form of device characterisation, which avoids the complexity of an equivalent circuit, is the specification of relations between terminal voltages and currents (as functions of frequency or d.c. bias if required). This concept amounts to treating a device as a "black box," avoiding explicit reference to the intricate contents, but centring attention wholly on the external electrical behaviour. In the above example of the transistor, the "black box" representation will be that shown in figure 1.2, with p = y, z, h or g, depending on the terminal quantities chosen as dependent or independent variables. This summed up5'6 in table 1.1, and described in the immediately following sub-section .

1.1.2 Two Port Networks and Matrix Analysis The treatment of a device as a "black box" springs from a much more general background, rich in mathematical tools and analytical techniques5-8. This is in the treatment of a large class of networks, each with a specific input port and a specific output port, as two-port 16 networks. Coming within this classification are a large number of electronic units, including attenuators, transmission lines, wave - 10,11 guides, filters, amplifiers, transducers and even mixers (if the local oscillator is regarded as part of the two-port). The signal transmission properties are determined only by the relations between terminal voltages and currents, namely : V1, the input voltage I1, the input current V2, the output voltage I the output current 2' Any two of these may be chosen as independent variables, and there are 6 ways of doing this, as listed in table 1.1, with corresponding two-port parameters. The sign convention used is indicated in figure 1.2.

Any set of two-port parameters is sufficient to specify the electrical characteristic of a two-port network, whether it contains an active device or not; any one set is convertible to another set via the conversion9 table 1.2. The choice of a convenient parameter set depends largely on the embedding network chosen and the inter- connection planned; a judicious choice can result in much saving of labour in mathematical manipulations. For example, in the design of 12 13 an amplifier, it has been shown 2 advantageous to choose a matrix set that corresponds to the mode of termination -- i.e. "matrix termination" -- as illustrated in figure 1.3. Also, when a two-port is to be connected to another two-port, a wise choice of parameter set can result in simple additions of matrix parameters. This is known as "matrix embedding" and is illustrated in figure 1.4.

1.1.3 Three Terminal Devices as Two-Port Networks Many two-port networks are made up of one or more 3-terminal devices, like the junction transistor, used in one of its configurations, for example, common-base or common-emitter configuration. To express the properties of a 3-terminal device in terms independent of its orientation14, a 3 X 3 matrix can be used, as shown in figure 1.5. Any particular two-port can then be obtained from one of the above 17

matrices by eliminating a row and a corresponding column. The suggested sign changes are such that the resulting two-port has the same sign convention as that in figure 1.2.

1.2 EMPHASIS In this and the following chapters, emphasis is placed, firstly, on the application of a 3-terminal active device in a linear two-port amplifier with useful power gain. As a reference device, the junction transistor is chosen and assumed to operate in a small-signal mode. This choice is made largely because of such properties of the transistor as its finite input impedance and non-unilateral nature. However, the general results obtained should be equally applicable to other two-ports, including other active devices, like the vacuum tube and the field-effect transistor.

Secondly, where matrix analysis is concerned, reference will be made only to y, z, h and g matrices, which are of more general use. These matrices form a group of their own, in the sense that elements in the principal diagonal are immittances, and off-diagonal elements are concerned with the transfer properties of the two-port. Thirdly, where reference is made to amplifier gain, this is to be taken to mean the transducer power gain, unless otherwise stated. The transducer power gain is defined as the ratio of power output to the load, to the maximum power available from the source.

Finally, in subsequent sections of this chapter, previous works which have more direct bearing upon the present one will be discussed in greater detail than others; and these results will be freely referred to later on.

1.3 HISTORICAL NOTES 1.3.1 Two-port Networks The generalised treatment of two-port networks by means of matrix 6 algebra began with Strecker and Feldtkeller in 1929, when these were called "vierpols" or "four-pole networks." Many refinements followed, 18 notably those by Guillemin5 and Baerwald7, with applications mainly to passive bilateral networks, such as transmission lines and filters. By this time, two-port networks were also known as "quadripoles," "four- 15 terminal networks" and "two-terminal pair networks." In 1948, Peterson first extended the theory to include applications to active non-bilateral devices, and used it to find simple equivalent circuits for vacuum-tubes 10,11 operating at u.h.f. A little later, in 1949, Wheeler and Dettinger in investigating the efficiency of a superheterodyne converter, suggested the term "two-port" as being more appropriate, particularly in microwave applications. They even recorded having received "a favourable reaction from the several engineers to whom it was presented." However, nowadays, two-port networks still continue to be known by all the above names even in microwave applications16.

1.3.2 Unilaterisation of Amplifiers The invention of the transistor gave a great impetus to the development of two-port amplifier theory. This is because the "inherent internal feedback" within the transistor gives rise to interdependence between input and output circuits, and amplifier designs using equivalent circuits become more involved. To ensure stability and simplify design techniques, early investigations17-21 were made towards unilateralising transistor amplifiers, by both matrix and equivalent circuit approaches. 20'2223 However, it soon became apparent that neutralisation or uni - 19 laterisation was not synonymous with stability, and "the primary performance consideration of the amplifier" did not depend upon unilateral or bilateral property.

1.3.3 Conjugate-Matched Amplifiers With the acceptance of non-unilateral amplifiers, amplifier design by two-port approach began making rapid progress. The inherently stable two-port -- i.e. stable under all conditions of passive terminations -- was first investigated by Roberts24, who derived power gain expressions for the conjugate-matched two-port ("Ultimate Gain") as well as the arbitrarily terminated two-port ("Actual Gain"). Later, 19

these expressions were rediscovered and re-expressed in various forms 25 26 27 by several authors, including Linvill and Schimpf , Karp , Gartner , 12 28 29 Venkateswaran 6 Boothroyd , Lathi and Rollett . In addition, Venka- 12 teswaran & Boothroyd have shown, firstly, that the transducer gain expression for the arbitrarily terminated two-port maintains the same form in all matrix environments. In terms of the general matrix p = p + jail this is;

G 4PSPLIP21 2 1 1 t (P11 Ps)(P22 PL) Pl2P21 I 1 2 Secondly, the conjugate-matched power gain has a unique value not dependent on matrix environment, and is given by:

P21 1 P12 Si

where Si Ali + /YL2i - 1 1 2

411P22 - M and i L

with L = 1P12P211 ' and M + X = P12P21.

The terminations for conjugate-matching are given by;

PS = RP11 4- j( -ail + ,No ) 2.22 13 N PL = Rp22 + j(-a22 + 411) 1 M N2 where R = (1 - P11P22 Lo2 02 - F11F22

These same results have, subsequently, been also obtained by 28 29 Lathi and Rollett . 20

1.3.4 Stability Measures for Conjugate-Matched Amplifiers

For conjugate-matching to be possible, Si in equation 1.2 has to be greater than 1, which means that 'ili must also be greater than 1. An equivalent form of stability criterion has been derived earlier by Llewellyn3o. However, the introduction of Si and ~i has given rise 31 to many stability factors being defined , primarily as amplifier design parameters. Thus, S is called the "invariant stability factor" by 32 i Venkateswaran , andshown to be the modulus of the internal loop loss. The reciprocal of Si is named "the maximum efficiency of the bilateral -- 28 "invariant stability figure" -- in Lathi's work , and as K in a paper by Aurell34. The reciprocal ofvti -- designated by C -- is called "critical factor" by Linvill & Schimpf25.

1.3.5 Padded Amplifiers leiEm. 1. is less than unity, the two-port becomes inherently unstable -- i.e. it will oscillate for certain values of terminations. 13 Venkateswaran , Scanlan and Singleton35, among others, have studied the stabilisation of such a two-port by the addition of damping elements at the input and/or output ports. Analyses have been made by including 13 these damping elements as part of a new, stable two-port. Venkateswaran , in particular, has used this means of changing stli to obtain a cascaded amplifier with specified power gain. The padding (or damping) elements may be added in a variety of ways, corresponding to the various modes of terminations, as shown in figure 1.6. Having been padded in one mode, an amplifier can then be terminated in another mode, if it so proves convenient.

1.3.6 Mismatched Amplifiers An alternative way of obtaining stable power gain from an inherently unstable two-port is by employing sufficiently mismatched terminations. This has been the subject of several investigations12' 21,25,28 The technique amounts to choosing the real parts of source

and load immittances p and p respectively -- to satisfy Stern's L stability criterion36: 2p1p2 k = > 1 L + M

where and p2 = P1 = Pll PS P22 PL Corresponding to the tuned condition, the source and load reactances 36 are respectively:

0 a S p1 15 + —0 a L 22 P2 where o is a solution of the cubic equation: 3 M N X +h(1 + 0 16 P1P2 PiP2 12,13 Venkateswaran 6 Boothroyd have made an extensive study of the mismatched amplifier. They have shown, firstly, that although in general there can be 3 real solutions to the cubic equation 1.6, in practice, particularly when the mismatch is large, there is usually only one solution. In terms of X , the transducer gain equation 1.1 o becomes:

12 4 . G PsPL IP211 2 2r 2 )2 4. (2x N )2-j 17 P1P2 xo P1P2 o P1P2 Secondly, if p11 and p22 are positive, the transducer gain has a maximum value for a given value of k, namely :

4(1 - 4,/ki/k)2 G 1)21 2L tymax,k r 1)12 k(L + ij,72- 2M \2 4. (2x 2N N 2 ....1.8 C( o TiTEITY ` o k(L+M) j This occurs when pl = •p11 19 and P2 = 17ci *P22

Finally, expression 1.8 may be more conveniently written in terms of "the performance factor" n = p1p2/L

4(1 - \rni/n )2 G 1:321 t, ,n = 1---i 1 10 12 n(1 - 00 - 1,)7-+ (no -4)2 where n = o 0 i ill'22/L 28 Lathi and Spence37 , assuming that p11 and p22 are positive, have also expressed the general mismatched power gain in terms of departure from the condition for maximum mismatched power gain. This is:

p 4(E, fni/h)(1/5 - Vni/n) G 21 1 1 11 t NN2 p12 n (1- - -f)z 2 + (2X0 - r71: where Ps = (15 In/ni - 1)1311 1 12 PL = 111/ni 1)1322

1.3.7 Stability Measures for Mismatched Amplifiers

Just as Si and 11. have been used in the case of conjugate-matched 12,13,28,35,36 amplifiers, so, here k has been adopted by several authors as a stability design parameter for mismatched amplifiers. However, Venkateswaran e. Boothroyd have found that k is not a convenient parameter because it can assume different values in different matrix sets, even for a given amplifier and its terminations. In its place, the performance factor n has been suggested, for the mismatched power gain then tends to be independent of matrix environment when the mismatch is large.

1.3.8 Activitx

A two-port can give useful power gain only when it is active. The most general condition for a network to be active has been determined 38 by Raisbeck . For a two-port network, an equivalent condition has 22 39 been derived by Mason ' , namely that the lossless unilateral power gain U should be greater than unity. In terms of y or z matrix, this is: 23

, 1 2 I 21 12 U P - P > 1 4(P11q2 - PiAl) where p' = p' + j a' = y or z matrix.

Mason has shown further that the value of U is invariant to changes in configuration and lossless embedding. U then becomes a measure of goodness of a 3-terminal device, and has been exploited 41 by several authors, including Weinreich40, Zuleeg , Stats et al42 , and Page and Boothroyd43, in investigations on "transit-time" oscillators, maximum frequency of oscillation, and maximum loads for oscillators. The concept of U, moreover, has been extended to n-port networks by Meadows and Dasher44.

1.3.9 Reciprocity Active devices are usually non-reciprocal. Since analytical groundwork has been laid largely for reciprocal networks, many authors have considered devices being split up into bilateral and unilateral parts. Thus Haus45 has considered a general two-port as being made up of a bilateral part in cascade with a unilateral one. Shekel46, on the other hand, has shown that a general 3-terminal device is made up of a 3-terminal gyrator (which is an extension of Tellegen's idea47) embedded in a delta of bilateral elements, which include a negative • resistor if the device is active.

1.4 FORMULATION OF PROBLEM There are many features not altogether satisfactory in previous works on two-port amplifiers, particularly those concerning power gain, sensitivity and stability performance. These features are discussed below. Investigations into them have led to the present work.

1.4.1 Sensitivity Active devices cannot usually be manufactured to as close a tolerance as desired and certainly not as accurately as passive elements. An important consideration in the design of an amplifier, therefore, is 21. its sensitivity to changes in device parameters. With the general two- port, no satisfactory analysis has yet been made of this parameter sensitivity or of the general ways to control it. Many authors, like Iinvill and Gibbons°, Meritt49 and Lathi28 have based their analyses either on specific equivalent circuits, concentrating attention on specific elements in the circuits, and on voltage, current and immittance transfer functions, instead of the more basic power gain functions. Further, all these works have been generally restricted to specific two-ports and specific parameter sets, with no indication as to the superiority of one parameter set over another, or whether an improved performance could be obtained with feedback and/or change of device configuration.

1.4.2 Stability and Sensitivity

Stability factors have often been quite arbitrarily defined. In particular, k and n have been derived under conditions which are not necessary tuned, as shown in Appendices 1 and 2. This being so, any relation between stability factors and sensitivity performance can only be a vague one, assessable only from the general experience that, for given parameter variations, the larger stability factors are, the less power gain changes tend to be.

Common stability factors like k, n and Si can, in fact, be shown to give only poor indication of amplifier sensitivity performance. Thus, the conjugate-matched amplifier in figure 1.7 has fixed terminations; yet, the values of k and n differ considerably in different matrix sets, whereas the sensitivity performance of an amplifier should be independent of the methods used in computation. Similarly, in figure 1.8 are two amplifiers with the same values of Si and power gain; yet, when their parameters are varied by the same amounts, the changes in power gain are vastly different. 25

1.4.3 Gain-Sensitivity Capability

It is common knowledge that some power gain has generally to be exchanged for improved sensitivity performance. What is not clear from previous works is the rate of exchange, whether it is variable, and if so how, particularly with feedback and/or change of configuration. Knowledge of this exchange rate is important in the synthesis of amplifiers such that the spread in power gain will be within given limits.

1.4.4 Negative Real Parts of Self-Immittances

Previous amplifier analyses have generally been restricted to two-ports with both p11 and p22 positive. Though these conditions are necessary for conjugate-matching to be possible, they are not so when two-ports are used in padded and mismatched amplifiers. Further, 50 in practice, p22 at least can be naturally negative or made so.

1.4.5 Simplification of Design Procedures

Amplifier design techniques used by previous authors generally involve a great number of parameters and laborious calculations, particularly when an optimum matrix set or an optimum embedding network is to be chosen. In fact, tedious computations are necessary even to determine whether given amplifier specifications can be met, using certain devices. It is clear that a much simpler design technique is desirable, particularly when designing optimum amplifiers for given sensitivity performance.

1.4.6 Present Work

In the present work, investigations are made into both the matched and the mismatched techniques of obtaining stable power gain from a device. The former includes the padded technique as a special case. The study of amplifier sensitivity performance is made the backbone of the work, but where possible, the relation between stability factors and sensitivity performance is indicated. It is realised that matrix parameters are merely tools of analysis, and the sensitivity 26 problem is, therefore, examined in the light of more basic parameters. Emphasis throughout this thesis is on 3-terminal devices used in two-port amplifiers, and analyses are made quite general, to cover cases of feedback and change of configuration at the same time. The sensitivity consideration here is restricted to that of power gain of single-frequency, tuned amplifiers, designed from the mean values of device parameters. The emphasis is on tuned condition; and this excludes the change of power gain with temperature and similar effects, which depend largely on practical details like biasing arrangements and tolerances of passive elements. In this thesis, the matched amplifier technique is shown to be superior to the mismatched technique in simplicity of design, cascading and in sensitivity performance. The use of basic parameters in both these cases makes possible general analyses covering cases of two-ports with negative p11 and/or p22. It also results in considerable simplifications, particularly in the case of the matched amplifier, where only 2 basic parameters are found to be necessary for amplifier design. The use of basic parameters leads, first of all, to a novel approach in the design of a wide-band amplifier. Secondly, it simplifies sensitivity consideration, and leads to the discovery of definite relationships between power gain and sensitivity performance. The establishment of these relationships makes possible a simple design of matched amplifiers for given sensitivity performance and optimum power gain.

1,5 ORIGINALITY Except where reference is made to the works of others, the research work reported herein has been carried out independently by the author, and the conclusions reached are original. 27

11 CSec

r ce r I L sc1 I c E C C.

ad. t sc2

C Sbe -I I-- Ctc3 bb' c I tc2

1—C Sbc

FIGURE 1.1

AN EQUIVALENT CIRCUIT FOR TRANSISTOR WITH MESA STRUCTURE 28

0 11 sv-----./12

p21 - 22

Positive directions of currents and voltages are indicated by arrows. Dotted lines show the use of a 3—terminal device as a two—port.

FIGURE 1.2

A TWO—PORT NETWORK 29

FIGURE 1.3

MATRIX TERMINATIONS

Y11 912 V1 Ps 'a L Y21 922 V2 —1— .0

I Z11 Z12 1 a z 1 Z21 22 2 1

h h 11 121 PL h h V 21 22 2

e 1 gll g12 V

PS JaL g21 g22 12 T —1—

I lT Yl1N+ YllP Y12N+ Yl2P V1T 1 21- I 0 2T 3r2ie Y21P Y22N+ Y22P V2T 2T

V + z + z 1T Z11N 11P Z12N 12P V z + z 2T Z21N+ Z21P 22N 22P2T / • . 1T

V h + h h + h 1T 11N 11P 12N 12P 1T 1 h + h h + h 2T 21N 21P 22N 22P V2T v1 T I 6 }V2T

• g + g • V 11N 11P g1 2N+• g 1 2P 1T I g21N+ g21P g22N+• g 22P 2T

12T a a 'V1T a11N a 12N 11P 12P V2T P V V1TI N .1 2T 1 a a -I 1T a21N a22N 21P 22P 2T

T► 2T / . • / V, b b b b V Ze.T 1N 12N 11P 12P 1T T1 V 1 P N I 2 I = b b b b -I I 2T 21N 22N 21P 22P IT

FIGURE 1.4

MATRIX EMBEDDING 31

v V z I 1 yll 912 913 i a aa ab z a 1 = V V z z I 2 921 922 923 2 b ba zbb bc b 1 V V z z z 1 3 Y31 Y32 Y33 3 c ca cb cc c \ -% .s.

common chan(re common change eliminate t, eliminate terminal sign of terminal sign of

1 row 1, col 1 nil 1 row 1, col 1 Vb, I b ,z zbc cb 2 row 2, col 2 nil 2 row 2, col 2 V,1 c c z lz ca ac 3 row 3, col 3 nil 3 row 3, col 3 V , l a a z zabt ba

FIGURE 1.5

THREE—TERMINAL DEVICE AS TWO—PORT

O original P" y—mode of padding P" two—port

padded two—port

P" original z—mode of padding two—port

padded 'two—port

original h—mode of padding two—port

padded two—port

original g—mode of padding p two—port 0

padded two—port p" are lossy padding elements.

FIGURE 1.6

MATRIX PADDING -j 51 2 1

69;18

MATCHED 2 - PORT

DATA OF MATCHED TWO -PORE

Feequenoy f 2, 3 Nsis Lomas.. Unilateral Gain U = 42.3 Measure of Non-reciprocity A = -10.3 ••$9.6 Inherent Stability Factor S = 1.5 Transducer Gain Gt = 66.7 or 18.2 dB

COMPUTED 'VALUES OF "h" AND "n"

MATRIX SE "k" • “ 9 1.510 1.484. h 2.259 0.983 1.510 1.4.75 g 2.772 • 0.897

FIGURE 1 .7 INCONSL9fINCr OF "k" AHD "n" AS SENSITIVITY MEASURES A = 100 e j63 A ==125. 100 II = 62.5 II G0 50 G = 50 Si= 2 Si = 2

PsfPL = Matched Terminations U = Lossless Unilateral. Gain A = Complex Measure of Non-reciprocity Gc Conjugate Matched Gain Si = Inherent Stability Factor

IF Ws OF BOTH TWO-PORTS INCREASE BY 20 44

G6 becomes 76.5 _._ - Gc becomes 56.3 ' • IF LA's OF BOTH TWO-PORTS INCREASE BY 10°

Go becomes 67 ______as becomes 58.7

FIGURE 1.8.. . INCONSISTENCY OF "Si" AS SENSITIVITY MEASURE

TABLE 1.1

TWO-PORT MATRIX CHARACTERISATION

independent p matrix equation variables becomes • I1 Y11 Y12 V1 Vl' V2 y = • 12 Y91 Y22 V2 .. . \ / •-•. e

.... Z z V 1 11 12 I1 1l' 19 z = z V2 21 Z22 I2 -.. / ..

-- --, ,..• -- / V1 e• h h11 12 I 1 I1, V2 h = ' V , I, h21 h22 \ / \ / \ '.'

4, I N e a ' 'y 1 gll -12 1 V1,I 2 g = V2 g21 g22 12 N / N. /\

• ..- ,.. ,f '.. a a V V 1 11 12 2 V2, -I2 " a = I a -1 1 a21 22 ... 2 /

• S. , ... , -> V b 2 11 b12 V1 V1/ -I1 b = b -I 12 21 b22 1 . 5. i • 1

TABLE 1.2 : MATRIX INTERRELATIONS

FROM1-4- lz j [hi Ni [a] [11 TO [Y] z22 -z12 1 -h12 A7 g12 a22 -pa bll -1 Az A z hil hi_ Yll Y12 i -22 g22 a12 a12 012 b12 [Y] -z21 z11 h21 ph -g21 1 -1 all -Ab b22 Y22 Y21 Az Az hll h11 322 322 a12 a12 b12 b12

Y22 -Y12 ph h12 1 312 all pa b22 1 7, 11 z12 h h fr 7 a a b b A y A y 22 22 '11 '11 21 21 21 21 Ez1 -Y21 Yll -h21 1 321 AFT 1 a22 A b bll h22 Ay A y z21 z22 h22 gll 311 a21 a21 b21 b21

-Y A z12 g22 -312 a12 Aa b12 1 1 12 z h h — — 11 12 b Y11 Yll z22 z22 Ag Ag a22 a22 b11 11 [hi -g21 g11 -1 a21 -Ab b21 Y21 Ay -z21 1 h h21 22 Yll Yll z22 z22 A,, Ag a22 a22 bll bll a -p b Ay Y12 1 z1 2 h22 -h12 21 a 21 -1 311 312 b22 b22 ph all all Y22 Y22 z11 zll ..-4"L h [g ] -y z h21 h11 - 21 1 21 Az - 1 a12 A b b12 321 g22 a b b Y22 Y22 z11 zli ph ph ll all 22 22 o b -Y22 -1 z11 Az -ph -h11 1 g22 a 22 12 11 a12 A A y21 y21 z21 z21 h21 h21 321 g21 b b [al , z -h ' -Ay -Y11 1 22 22 -1 g11 L. D21 bll TT eT ----- a21 a22 A A b Y21 y21 z21 z21 1121 21 -21 321 b

Y11 -1 z22 Az 1 hll -A, -S22 a22 a12 b b pa 11 12 Y12 Y12 z12 z12 h12 h12 g12 312 Aa [b] h L -g a a -A y -Y22 1 z11 22 h -'11 -1 21 11 b b21 22 pa Y12 y12 z12 z12 h12 h12 g12 g12 pa

A 9 = 911999 P12P21 Prom "Principles of Transistor Circuits," by R.P. Shea, ref. 9 37

CHAPTER 2

ANALYSIS OF THE MATCHED AYTLIFIER

2.1 INTRODUCTION When a two-port is inherently stable or is made so, the maximum power gain is obtained by conjugate-matching it. "This maximum 12 13 power gain is independent ' of the mode of termination -- i.e., y, z, h or g mode shown in figure 1.3. For analytical purposes, it is convenient to classify all two-ports with matched terminations as matched amplifiers. This is possible by including feedback networks and/or padding elements as part of the two-ports. In previous works, the matched amplifier has been analysed in terms of L. complex parameters, and laborious calculations have had to be made, first to decide if the amplifier is inherently stable, and then, to compute the power gain. After computing the power gain, there still remains a considerable doubt if this is the best performance -- considering both gain and sensitivity -- obtainable from the device (or devices) that goes to make the two-port. For example, it is not readily known if (further) feedback, padding or both should be applied to improve the performance of the amplifier. A more general approach to amplifier analysis is, therefore, desirable, from which the optimum operating point, embedding and terminations may be calculated without much difficulty. To do this, clearly, one has to work in terms of some basic properties of the two-port, whose variations with padding and/or feedback may be closely correlated to changes in power gain and sensitivity performance. 22 Two such important parameters are, firstly, Mason's U, which is closely related to the activity of a device, and, secondly, the amount of non-reciprocity in a two-port, which allows more power to pass in one direction than in the other. In this chapter, the relation between these two parameters and power gain is established, leading to a novel approach in designing matched amplifiers and in cascading them.

2.2 COMPLEX MEASURE OF NON-RECIPROCITY

Non-reciprocity is an essential feature of a stable and useful two-port amplifier. This is evident from the fact that if there is the same power gain from port 1 (input) to port 2 (output), as from port 2 to port 1, instability will occur. Therefore, a thorough investigation into the dependence of amplifier operation on the amount of non-reciprocity there is in a two-port should shed considerable light on the capability of the two-port itself.

For generality, a measure of non-reciprocity is defined so as to admit of a complex value. Thus, in figure 2.1 (a), a voltage source is applied to port 1 of the two-port, and a short-circuit connected to port 2. Let the current in the short-circuit be 12. When the voltage source and the short-circuit are interchanged, as in figure 2.1 (b), let the new current in the short-circuit be Il. The complex measure of non-reciprocity (A) is then defined as: 12 A = I1 A may be evaluated in terms of two-port matrix parameters. Consider, for example, the y-matrix:

= Y117.1 Y12V2

22 = y21V1 y22V2 Corresponding to figure 2.1 (a), if V1 = V, and V2 = 0,

12 = y21V When the voltage source and the short-circuit are interchanged, V1 = 0, V2 = V, then

I = y V 1 12 Therefore, A = I2/I1 = Y21/912. If use is made of the conversion table 1.2, it can be shown that z h Y21 21 A = = = 21 = g21 2 1(a) h Y12 Z12 12 g12

which may be expressed more generally as:

p21 A = * 2 1(b) P12 where + sign applies when p = y, z, and - sign applies when p = h, g.

2.3 MEASURE OF GOODNESS Nason22'39 and other authors40-43 have shown that there are many interesting properties of U, the lossless unilateral power gain of a two-port. Firstly, U is unchanged by lossless embedding, which includes change of device configuration as a special case. Secondly, a device is active or passive depending on whether U is greater or less (but not negative) than unity. With U greater than unity, Mason has shown that a device can always be made into a useful two-port amplifier. If U is negative, on the other hand) a device can be used chiefly as a one-port negative resistance amplifier.

It is clear that U is an important measure of goodness of a device, determining its usefulness in a two-port amplifier, in all device configurations. As our interest lies chiefly in two-port amplifiers with useful power gain, we shall restrict our attention to devices with U > 1.

U has, in the past, been written in a different form for y and z matrices, from that for h and g matrices. Thus, for p' = p' + j a' = y, z matrices:

13 U = 21 - p12 2 4(13111°22 - P12P21) while, for p" = pfl a" = h, g matrices,

2 nft n" '21 '12 U 411.1P22 a3.2°. 21) • These two forms of U may be expressed in terms of the general matrix p= y, z) h or g as: 40

1- - 2 E121 U 1 1'21 2 2 2Re(Ap -D11-e!2Tz ; D ) whereAp= -11-22 - P12P21 and asterisk (*) signifies complex conjugate quantity. The negative sign applies for p = y, z and the positive sign for p = h, g.

It is not often realised that, although algebraically U assumes different forms, the numerical value remains the same in all matrices. This may be checked by changing the expression for U from one matrix environment to another, using the conversion table 1.2. The constancy of U is also obvious from a conclusion by Venkateswaran & Boothroyd12 that, the conjugate-matched power gain is independent of matrix environment, for U also represents the conjugate-matched power gain of a two-port that has been unilateralised by lossless network.

2.4 BASIC POWER GAIN RELATION It is evidently advantageous to incorporate U and A into power gain analysis. A good starting point appears to be in the examination of expression 2.2 for U. The alternative (;) signs seem to be due to the alternative signs for A (see equation 2.1 (b)). It is obviously useful to introduce A into the expression for U. Dividing the numerator and the denominator of equation 2.2 by 2 11 P12[1 = Pl2PI2' we get:

± P21 11 2 U P12 2R 6°P P1142 e(I ) 2Re( * P12 12 P12 where the top sign applies for p = y, z, and the bottom sign for p = h,g. A = t (P21/P12) may now be substituted into the above equation, giving: 1 2 I A - U = 2 3 2Re(AP 4. P111322) 2AR P12 1 1°21 where subscript R indicates the real part.

4.1

But, from equation 1.2,

AP + P11 P *22 2P11P22 - M P21 2Re( ) = 2. - 2111AI L p P12 12.12 ' 12

Therefore, equation 2.3 becomes:

U 2(101 - AR) 24

It is interesting to note that, if U is positive, there is a IninirmunlimittothevalueofYi.in11 all possible device configurations and embedding networks. Thus, from equation 2.4, if U :.0,

n AR > i pT 0 AR or 111 > I The most negative value of AR/IA j is -1. Therefore, -1 2 5

Assuming that the two-port of interest has stable matched power gain, G, further simplifications may be made to equation 2.4. From equation 1.2 G 2rt Si1, + 'AI S. G + 1 W by substitution into equation 2.4 A - 2 U

G 2 (Al2 - 2AR.G G

or A 1 26 A - G

The matched power gain has now been expressed very concisely in terms of two basic parameters, and two only. These are (a) the measure of goodness U of the device (or combination of devices) that goes into the two-port, (b) the complex measure of non-reciprocity A of the two-port.

2.5 GEOMETRICAL REPRESENTATIONS

2.5.1 Normal Gain Chart

The basic gain equation 2.6 is in a form which can be displayed graphically with advantage: there are few parameters and the relation between them is simple. In fact, for a fixed value of U, equation 2.6 is simply an equation of a family of circles, one for each value of G. This becomes obvious if equation 2.6 is re-written in an alternative form:

) 2 2 G(U - 1) GU(G- i 2 LAR + G - U -E. AI 27 (G U) where subscripts R and I indicate real and imaginary parts respectively. On a constant-U plane, a gain circle is characterised by: 1 centre co-ordinates: T--U1) 0 2 8 radius : \raj. l(Gal :.36)

It is clear that a useful chart may be obtained from equation 2.7 if the value of U is fixed. However, it is more convenient to obtain a universal chart, where all parameters are normalised with respect to U. This chart may then be interpreted according to the value of U concerned. If all parameters in equation 2.6 are normalised by U, the result is: 1/UI 1 29 IrG/U A/U - G/U If U >> 1, or IA I >> 1, equation 2.9 reduces to: 1 2 10 Irvu - Git is now a function of APT only, and lines of constant G/U can be plotted on MU-plane, to form a two-dimensional chart, with Re(/U) 43 and Im(/U) as the Cartesian co-ordinates. This plot is still a plot of a family of circles, with a gain circle being characterised by:

(normalised) centre co-ordinates: (G/U - 1 , 211 G/U (normalised) radius : U' q/u - 1

Circles of constant G/U are usually not full circles: they are limited by the requirement that, for stability of operation, Si = 'A VG > 1.

Thus, the result is the Normal Gain Chart of figure 2.2, with separate regions of stable operation and unstable operation. Here, lines of constant (normalised) power gain, G/U, are all incomplete circles, tangential to the line of marginal stability, corresponding to Si = 1. An equation for this marginal stability line may be obtained by equating IA1 = G in equation 2.10, giving:

1 - cos /A = IAOU 2 12 which is a simple cardioid.

In figure 2.2, lines of constant stability factors Si = 2 and Si = 4 have also been drawn. These lines are less like cardioids. In fact, the larger Si is, the more circular the corresponding line becomes. Equations for these lines may be obtained from equation 2.4, which gives, for U>> 1 or lAl>> 1,

ni - cos 4= 1.4y2u 2 13 where, it will be remembered that = + 1/Si)

The Normal Gain Chart is rather limited in its application to amplifier design, particularly because the unstable region occupies rather a prominent place in the foreground. 'introvert it could be-used, for oxampla,in assessing the general effect of feedback and padding, as shown in sub-section 2.6.4. "Further, the Chart should prove to be of great use in oscillator design, which is not within the scope of the present work.

2.5.2 Inverse Gain Chart

In amplifier design, the interest lies chiefly in the stable region. It is, therefore, expedient to invert the Normal Gain Chart, bringing the stable region into the foreground, and pushing the unstable region into the background. Since inversion is a conformal transform, the lines of constant (normalised) gain, G/U, remain circles in the Inverse Gain Chart, which is effectively a plot of constant G/U lines on the complex plane U/A. This is shown in figure 2.3 for U >> 1 or IAA >>1. This chart bears a marked similarity to those obtained by 34 Aurell , Scanlan and Singleton35. However, it is actually very different from them in that it is in terms of useful, basic parameters invariant to a change in matrix environment, and not restricted to 51 a particular device configuration or a particular matrix set .

The Inverse Gain Chart may be plotted directly from the gain equation 2.6. Thus, in terms of X= 1/A, the latter may be written: _ x 2 lk(a) 1[17 (T- _ GXI

U - 1 2 2 1[1.1 G - 1 or + % 2 2 114.(b) 11 UG - 1 I G UG - 1 from which, a circle In %-plane is given by U - 1 centre co-ordinates: , UG - 1 215 radius G UG - 1

Assuming that U >>1, or lAl>> 1, expressions 2.15 in normalised forms -- normalised by 1/U -- become:

(normalised) centre co-ordinates: 01 216 } (normalised) radius :

1+5

The Inverse Gain Chart is much simpler geometrically than the Normal Gain Chart. Thus, constant (normalised) gain lines remain circles, as has been noticed. In addition, the line of marginal stability (Si = 1) on the Inverse Gain Chart is now a simple parabola, and lines of larger stability factors (Si > 1) are simple ellipses.

As in the case of the Normal Gain Chart, an equation for the line of marginal stability may be obtained for the Inverse Gain Chart by substituting the condition Si = IA = 1 into equation 2.14 (a) or (b). This gives, for U >> 1 orlAl>> 1 2 X' R + — 2 17 where X' is the normalised form of X, and is given by X' = UX. Equation 2.17 will be recognised as that of a parabola, with focus at the origin, apex at (.1/4, 0) and semi-latus rectum = 1/2.

Similarly, equations for lines of larger stability factors may be expressed conveniently in terms of Ili as: IX' 1 2('12i - 1)} 2 X12 ,2 1 = 1 2 18 4(12i - 1) 4(12 - 1)2

Equation 2.18 is that of an elipse, with

semi-major axis

semi-minor axis = 2 2 - 1

I 1 and centre of ellipse at 0} 2(111 - 1)

In amplifier design, the Inverse Gain Chart usually proves more useful than the Normal Gain Chart, and is referred to more 46 frequently in the following pages. This is because:

(a) The geometry of the Inverse Gain Chart is simpler, as has been pointed out.

(b)The most useful part of the stable region is in the foreground of this chart.

(c)The size of spread in X or 1/A, as shown in Appendix 5, tends to remain unchanged by feedback under certain circumstances, which are not uncommon in practice. This facilitates sensitivity consideration (see Chapter 4), for movement on the Chart and size of spread may be considered separately.

(d)The effect of feedback on X is more easily evaluated than on A. On the Inverse Gain Chart, feedback tends to make the point of operation move on a straight line, as shown in Appendix 4. On the Normal Gain Chart, on the other hand, the movement due to feedback tends to be circular, and more difficult to follow.

2.5.3 Gain Sp.9.,ce. In many applications, it is necessary to consider changes in U in conjunction with changes in A, for example, when studying the effect of lossy feedback. Under such circumstances, it is still possible to use two-dimensional charts, by allowing for variation in th6 normalising factor U. However, an alternative geometrical representation, with both U and A (or their functions) as variables, can be very advantageous. This may be done by developing either the Normal Gain Chart or the Inverse Gain Chart, removing the normalising factor and introducing it as another variable. Thus we have a 3-dimensional representation, which can take 2 convenient alternative forms, namely: (a)Normal Gain Space, with the 3 Cartesian axes corresponding to AR, AI and U; (b)Inverse Gain Space, with the 3 Cartesian axes corresponding to AR, X I and 1/U. 47

The second alternative is more suitable for amplifier design, since, as shown in the previous section, the corresponding Inverse Gain Chart is more useful than the Normal Gain Chart for this purpose. The choice of 1/11 is also more appropriate than U, because, as shown in Appendix 6: (a)a spread in 1/U -- namelyL(1/U) -- is more readily calculated than a spread in U;

(b)LS(1/U) tends to remain constant under some circumstances as U is modified, say, by lossy padding. This constancy of size of spread again facilitates sensitivity consideration, as shown in Chapter 4. A model of the Inverse Gain Space has been made, consisting of planes of constant values of 1/U, interconnected by surfaces of constant gain. A photograph of this is shown in figure 2.4-, and a sketch, for greater clarity, in figure 2.5. In the photograph is also clearly visible a surface of constant value ofSi , forming a cone, a cross-section of which is an ellipse at a plane of constant 1/U.

2.6 PROPERTIES OF BASIC POWER GAIN RELATION It has been shown that the matched power gain of a two-port is uniquely related to only two basic parameters, U and A : U being the measure of goodness, and A the measure of non-reciprocity. There are many interesting features associated with this relation, leading to useful applications and a better understanding of amplifier performance. These features will be discussed below, some in detail, while others only briefly to begin with, to be enlarged upon in later chapters.

2.6.1 Activity and Power Gain

A two-port (and a device in general) can be active only when IF> 1. The larger U is, the more active the two-port becomes. This increasing activity with increasing U may be interpreted in 2 ways: 4.8

(1)If the two-port is made to oscillate, the larger U is, the larger the maximum load it can generally tolerate43. (2)If the two-port is made into a matched amplifier, a given power gain can be realised with greater ease if U is larger; i.e., the gain-sensitivity capability of the two-port is increased. This fact is obvious from the Inverse Gain Chart, for as the power gain of a two-port is raised with respect to its U, the operating point on the chart tends to move into a region with denser gain lines. Therefore, for the same power gain, an amplifier with a larger U can operate in an area less sensitive to parameter changes than one with a smaller U. This gain-sensitivity aspect will be enlarged upon in Chapter 4.. A device ceases to be active when U< 1. When U = 1, the maximum power gain obtainable is unity, independent of what A is. This fact 13 has been noticed by Venkateswaran , after some algebraic manipulations. However, this is straightaway obvious from the basic power gain equation 2.6, where, if the power gain is unity, U has to be unity, and vice versa.

2.6.2 Maximum Matched Gain For a given value of U, there is a maximum value of matched power gain obtainable by any means whatsoever. For U >>1, this is

G x 219 ma = 4U as evident on the Inverse Gain Chart. This occurs when 0 1800, and lAi= G, at the boundary between stable and unstable regions. A more precise expression for the maximum matched gain, valid for any value of U, is obtainable by substituting the same conditions, i.e•, kLI = G and 0 = 1800 into equation 2.6, giving:

2 20 Gmax = (ar - 1) 2 -1 A transducer gain greater than this may be obtained in a mismatched amplifier. However, in practice, it is not desirable to operate an amplifier, matched or mismatched, with power gain approaching 4U, since slight changes in biasing conditions or device parameters 4,9 can give rise to instability. In the case of the matched amplifier this poor sensitivity performance is evident from the Inverse Gain Chart, since the point of operation then approaches the boundary between stable and unstable regions. Moreover, the matching terminal immittances 13 in this case also tend to zero , as is clear from equation 1.3, Chapter 1.

2.6.3 Power Gain Capability of a 3-Terminal Device

Active devices used in amplifiers are usually 3-terminal devices. When these are characterised by two-port parameters or equivalent circuits, it is not always clear which device configuration gives optimum performance. It is usually necessary to examine every configuration individually before a choice can be made. Further, it is never clear if feedback (and how much of it) will give additional improvements.

In the gain relation of equation 2.6, and its geometrical representations, on the other hand, we have an interrelation between the performance of a device in all configurations. A change of device configuration amounts to only a change in the complex measure of non- reciprocity. Furthermore, this change of configuration may be classified as a special case of lossless transformer feedback, as shown in Appendix 3. Therefore, in more general terms, the gain relation 2.6 describes the performance of a device in a general passive embedding network. If the embedding network is lossless, only A is altered; if it is lossy, both U and A may be affected. The process of getting the optimum performance out of a device, therefore, boils down to choosing an appropriate operating point, say, on the Inverse Gain Chart, and designing an optimum eiibedding network (which may be a simple change of device configuration) to achieve this operating point. The optimum embedding network may be decided on the basis of bandwidth performance, or minimum components required, or even ease of construction.

2.6.4 Effect of Stabilisation

Amplifier stabilisation has usually been taken to mean satis- faction of certain stability criterions, like Stern's stability factor 50

k, and Venkateswaran's inherent stability factor Si, both of which must be greater than unity for stability (see Chapter 1). These requirements may be met in several ways, for example by port padding, feedback or both. The end results are, however, generally not the same, even though the final values of, say Si and k may be made the same. This has been shown in section 1.4.2.

A more satisfactory interpretation of stabilisation can be obtained from the power gain equation 2.6 and the Inverse Gain Chart; for here it does not mean merely moving away from the unstable region, but rather moving into a stable region of acceptable performance. The Inverse Gain Chart of figure 2.3 also makes it clear why amplifiers with the same value of S. need not have the same sensitivity performance; for, constant Si lines on this chart pass through regions of varying densities of constant gain lines -- i.e. varying sensitivity performance. SinceS.is closely related12'13 to k in the case of the matched amplifier, it is not surprising that amplifiers designed for a fixed value of k by different techniques can have quite different sensitivity performance. The process of stabilisation can be seen rather strikingly on the Normal Gain Chart. An example is shown in figure 2.6. Initially the two-port is inherently unstable, with the point of operation at o in the unstable region. If damping elements are applied at one or both ports, only the value of U is changed, but A remains the same. If the change in U is sufficient, the new point of operation may be at a, in the stable region. Conversely, if the point of operation a is chosen (say, for satisfactory sensitivity performance), the change in U required can be computed from the difference in length between Io and Ia. Alternatively, if the required point of operation is at c, lossless feedback can first be applied, to shift point o to b (via a circular arc -- see Appendix 5); then port padding may be used to move the point of operation from b to c. 51

2.7 APPLICATIONS OF BASIC POWER GAIN RELATION

2.7.1 Device Specification The usefulness of U and A, and their simple relation to amplifier performance, strongly suggest their use in the specification of a two- port, particularly when it is intended for amplifier application. This system of specification offers many advantages over that by means of an equivalent circuit, or a set of two-port parameters, namely:

(1) The specification of U and A involves the minimum possible parameters. This is advantageous from the point of view of both the designer and the manufacturer, particularly when parameter spreads are to be specified: it is simpler to calculate gain spread from spreads in 2 parameters than to do so from spreads in all elements of a two-port matrix set or of an equivalent circuit. (2)The power gain of a two-port is readily computed from U and A, through the use of either equation 2.6 or a gain chart. This is in contrast to the complicated gain calculation from a set of two-port parameters or an equivalent circuit. (3)From the knowledge of U and A of a two-port, and the Inverse Gain Chart, the designer can rapidly estimate the sensitivity performance of the matched amplifier made from the two-port, and also know the gain- sensitivity capability of the device(s) in the two-port. Thus, this method specification is far superior to the not-uncommon method of specifying that a device has a given power gain at a given frequency, say, 10 db at 100 Mc/s; for the power gain could then have been boosted by positive feedback so that the sensitivity performance is quite unacceptable.

(4.) Both U and A can be calculated without much difficulty from a set of matrix parameters. They can also be measured and computed more directly, as shown in Chapter 5. (5) Given the values of U and A and their spreads, it is possible to choose an operating point on a gain chart, such that the sensitivity 52 performance is acceptable. If a typical set of two-port parameters is also given, an embedding network may be designed to obtain this required point of operation. Examples of this design technique are given in Chapter 6. Note that here the points of operation are not restricted to those available only by simple change of device configuration.

From the above, it appears that the specification of U and A forms an ideal system of two-port characterisation, if an amplifier can be constructed from the knowledge of U and A only. That means that the amplifier has to be built without prior knowledge of terminal matching immittances, which is possible if both the real and imaginary parts of these immittances are tunable to realise the maximum power gain. Such amplifier circuits are, in fact, possible and quite simple, as shown in figures 2.7 (a) and (b), where lumped and distributed tunings are employed respectively. In the case of lumped tuning, only variable capacitances are shown, but some of these may in fact by inductances. In these circuits, the real parts of matching immittances are tuned by adjusting Cl and C4, and the imaginary parts by C2 and C3.

The presence of C and C should not mar the bandwidth per- l 4 formance. A judicious choice of these, and source and load impedances can result in reasonable frequency response. An example of such a circuit and its frequency response are shown in figure 2.8. In any case, if the value of U is determined by the method indicated in Chapter 5, the precise values of matching immittances become automatically known; then, there will be no necessity for Cl and c4, except to take up spreads in input and output immittances of the two-port.

2.7.2 Study and Control of•Sensitivity The usefulness of active devices depends not only on the gain capability of a "typical" device, but also on parameter spreads. For example, if parameter spreads are large, the average gain of a device may have to be reduced considerably to obtain satisfactory overall performance. 53

An important item of data to a designer is, therefore, the magnitude of parameter spread. When a two-port is characterised by U and A, spreads in only these two parameters need be specified, and gain spread is then uniquely determined. This may be seen rather pictures- quely from the Inverse Gain Space, where spreads in U and A give rise to a volume of spread, intersecting a number of constant gain surfaces. The values of these gain surfaces then determine the size of gain spread. If the volume of spread is unaffected by feedback (see Appendices 5 and 6), one can visualise the effect of feedback as the movement of this volume to a suitable region of acceptable sensitivity performance, i.e., where only a few gain surfaces are cut by the volume of spread.

The concept of U and A, therefore, gives rise to a great deal of simplification in the study and control of sensitivity. This is in contrast to the usual but complicated way of assessing sensitivity performance from spreads in matrix parameters or equivalent circuit elements. As the subject of sensitivity is a very important one, this will be thoroughly investigated in a later chapter.

2.7.3 Simplification of Circuit Design Present analysis and synthesis of active circuits are often greatly encumbered by numerous, sometimes varying parameters. A great deal of simplification can arise, even if for part of the work, if one can think in terms of fewer parameters, which may be groups of other parameters. From this viewpoint, U and A are groups of parameters with special significance as far as the gain property of a two-port is concerned.

The advantages accruing from the use of U and A have been mentioned in previous sections. These include simplification of amplifier design and of assessing and controlling its gain-sensitivity performance. Because U and A may be varied independently -- U by lossy port padding and A by lossless embedding -- the designer can exploit the full capability of a device, instead of being limited 54 to using definite device configurations. In this connection, it should bb noted that previous works on two-port amplifiers have ignored cases when the real parts of self immittances, , are negative. p11 and p22 As mentioned previously, this situation can occur naturally, as shown by Pritchard50. The use of U and A in amplifier design, however, completely avoids this limitation. A useful amplifier can be made as long as U of the device is greater than unity. One further important use can be made of the independent variations of U and A : this is in the design of a wide-band amplifier. Since U of a two-port usually varies with frequency, the design amounts to constraining A so that the points of operation stay on a constant gain surface in the Inverse Gain Space (see figures 2.4 and 2.5) over the desired frequency range. The design can be conveniently divided into 2 independent stages. The first stage involves the synthesis of a lossless feedback network to constrain A as described above. The second stage is to synthesize lossless interstage networks to match the two-port (plus feedback network) to source and load immittances. A practical example of this is described in Chapter 6.

2.8 Cascading Consideration It frequently happens that sufficient power gain, with reasonable sensitivity performance, cannot be obtained from a single device. Then use may be made of a compound structure of devices, for example, a cascode connection or a Darlington pair connection52. These compound structures can usually be treated as simple two-ports, -- each with an input port and an output port -- and the theory set out above will apply. An alternative method of obtaining sufficient power gain is to cascade amplifier stages. With matched amplifiers, if both the real and the imaginary parts of port immittances are tuned, cascading of stages presents little difficulty. This is particularly true if each amplifier stage is tuned separately between source and detector having the same impedance, say 50 ohms. When properly tuned, the optimum

55 input and output impedances of a stage will be exactly 50 ohms, and cascading will produce no additional deterioration in sensitivity performance. The overall power gain is also given simply by the product of power gain of individual stages.

Suppose there are n amplifiers, each consisting of m stages, with overall power gains given by GT,, GT2, ...GTn. If the power gains of individual stages for the jth amplifier are (i + ogji),

Cg 6gj2), CE + 05gjm), then

2 21 GTj = Cg + 8g jl )(i + gj2) Cg + ogom) where g is the mean power gain of a stage, averaged over all n amplifiers, and Og the departure of gain from the mean value.

Equation 2.21 may be written approximately as:

GTj = Ci)11 + (Ogil + ogj2 + 222 •

Thus, the spread in gain of the jth amplifier from the mean value mg is due solely to the spreads in gain of individual stages. Therefore, the final gain spread of overall amplifiers is predictable accurately. 56

(a)

TWO - PORT

(b)

1 2 COMPLEX MEASURE OF NON-RECIPROCITY A =

FIGURE 2.1

DEFINITION OF COMP= MEASURE OF NON-RECIPROCITY G/U 1.4 WU s1.0 1.6 0.8 1.8 0.6 WI

FIGURE 2.2

NORMAL GAIN CHART

_(4 •

• • • 0 2 . •

•

FIGURE 2, 3 INVERSE GAIN- CHART

.••• 0.0

1....1.•..1 I I I I I I I I •• 0 .2 .3 .4 .6 .7 .8 .9 1.0 SCALE 59

surface of constant S. 1

-or firs

FIGURE 2.4

MODEL OF INVERSE GAIN SPACE 60

FIGUE 2.5

SKETCH OF INVi:RSE GAIN SPACE FIGURE2.6 gFFECT OF PADDING WITH AND WITHOUT FEEDBACK

1 • iiperating point ftn(gdur) q (padding only) operating point (feedback & padding

ReA/U)

= initial operating point (unstable) effect of padding only effect of feedback and padding 6z

C4 Cl 50a LOAD 50A. SOURCE

"/4,

'fee V cc FIGURE 2.7(0) MATCHING WITH LUMPED TUNING

50n 5on. SOURCE LOAD 2- PORT

stub stub C3 stub Cl stub C4 C2 FIGURE 2.7(b)

MATCHING WITH DISTRIBUTED TUNING

0-750

ILI200 pf 0-30pf 1ti ci 74= 0 4- 7 C 75 Q G. 7 5 Q LOAD SOURCE

4 : 1 (a)

AMPLIFIER WITH REAL AND IMAGINARY PARTS OF PORT IMMITTANCES TUNABLE

(b) FREQUENCY RESPONSE OF CIRCUIT IN (a)

FIGURE 2.8 64

CHAPTER 3

ANALYSIS OF THE MISMATCHED AMPLIFIER

3.1 INTRODUCTION

In previous works on two-port amplifiers, the two-port is often taken as it comes. If the two-port is inherently stable, or made so by simple port padding, it is used in a matched amplifier. If it is inherently unstable, it is used with mismatched terminations to get stable power gain. Feedback is seldom employed, because the general effect of feedback is difficult to analyse in terms of two-port matrix elements, particularly when the question of sensitivity is to be considered.

When a two-port is mismatched, the mismatching is usually confined to the real parts of port immittances, in order to obtain a reasonable bandpass characteristic. The imaginary parts are usually tuned to maximise power gain at the required central frequency. In this case, for stability, the real parts of source and load immittances, Ps and pL, are chosen to satisfy Stern's stability criterion36: 2 P1P2 L +M >1. Previous analyses of mismatched amplifiers have been built largely around k and its near relative, n, as design parameters. This is because of the general experience that the larger k and n are made, the better the sensitivity performance tends to be. Actually, however, the connection between k and n with sensitivity performance is rather vague, as shown in sub-section 1.4.2. Further, as Venkateswaran and Boothroyd12 have shown, for a given value of k or n, there can be as many as 3 sets of values of source and load immittances; and each set corresponds to a different amplifier sensitivity performance. k and n are, therefore, not quite convenient design parameters to use. Moreover, they do not provide a convenient basis on which to compare matched and mismatched amplifiers, and give little clue as to possible improvements by combining mismatching with feedback. 65

In this chapter, an analysis is made of the mismatched amplifier, first in terms of the two-port matrix parameters and a stability factor, which is closely linked to the inherent stability factor of the matched amplifier. This is done partly for convenience in later comparisons. The analysis is then re-expressed, as far as possible, in terms of basic parameters, and made independent of matrix environment and device configuration. The use of basic parameters makes possible a general study of the mismatched amplifier with feedback, just as in the case of the matched amplifier. The results of the single-stage analysis are then made a basis for the discussion of the problems involved in cascading mismatched amplifier stages, particularly that of the change in gain-sensitivity performance due to cascading.

3.2 MISMATCHED TWO-PORT AND EQUIVALENT PADDED TWO-PORT

3.2.1 Basic Conceit

The mismatched two-port may be looked upon as a development of the padded two-port. An example of this is shown in figure 3.1. The "original" two-port in (a) is padded by z' and zL, and then matched by terminations (z5 - zA) and (zi, - zL). If zs and zL are removed from the two-port and added to the source and load terminations, the result is the mismatched amplifier of (b).

3.2.2 Equivalent Stability Factor The above development shows that the original two-port sees the same "terminations" in both the equivalent padded amplifier and the mismatched amplifier. Therefore, as far as these terminations are concerned, the two amplifiers may be characterised by the same stability factor. Thus, if the inherent stability factor of the equivalent eq i is, in equation 1.2, Chapter 1 -- then we can say that the stability factor of the mismatched amplifier is also Seq. The subscript "eq" has been used instead of "i" to distinguish the equivalent padded two-port from the original two-port. From equation 1.2, therefore, one can write the equivalent 66 stability factor as:

I 3.1 Seq = '14 eq eq-

411eqP22eq M where L 3.2

Since M and L are unaffected by port padding, they have been left unscripted.

In the following it will be assumed that Seq andlea are always greater than Si andYt.ti respectively. This is generally true in practice.

3.2.3 Relation between Mismatched Amplifier and Equivalent Padded Amplifier

The transducer gain of the mismatched amplifier is clearly related to that of the equivalent padded amplifier. Consider the amplifiers in figure 3.1. The transducer gain of the mismatched amplifier is:

P P' P PL L L A G t P 3.3 A PA. PL where, writing R as the real part of z generally,

PL power output to RL P' = power output to (RL

P = power available from source with resistance R A PA power available from source with resistance (Rs - RA)

Note that the power gain of either the matched or the mismatched amplifier, when tuned, is unaffected if, instead of zs, zL, zA and zij, one writes Rs, RL, RA and RI,. This is equivalent to removing the reactive parts of these impedances and adding them all to the "original" two-port. If this is done, equation 3.3 may be written simply as: P' SRL - R t PA' - VOL - 3.4 6 67

But, PL/`PA is the matched power gain G of the equivalent padded eq amplifier, and, from equation 1.2, Chapter 1, may be written as:

G PL eq T7 Seq Therefore, for the mismatched amplifier, RsRL G w t Seq. "(Rs - RS)(RL RL) 3 5(a) Equation 3.5 (a) is true in all matrix environrIonts (though it may assume different values), and may be written in terms of the general matrix p = p + ja as: A I G PSPL 3 5 ( b ) t = Seq ( Ps - P)(101, - Pi) transducer gain transducer gain i.e. of = of equivalent X (factor) ...3.5(c) mismatched amplifier) padded amplifier where the factor can have different values in different matrix environments. In the following sections, the above approach to mismatched amplifier analysis will be adopted and further extended. This is because the approach possesses many advantages namely: first, since no prior assumption regarding the sign of p11 and p22 has been made, they can be either positive or negative; this is in contrast to previous analyses, when only positive signs have been accepted. Second, the use of Seq to denote the stability performance of a mismatched amplifier means that for the amplifier to be stable, S must be greater than unity; and this eq is true in all matrix sets and device configurations. Note the improve- ment this has over the use of n (see Appendix 2). Third, the analysis links the performance of the mismatched amplifier to the equivalent padded amplifier. Thus, to some extent, use can be made of the well- established matched amplifier analysis in Chapter 2 in the design of the mismatched amplifier. 68

3.3 ANALYSIS IN TERMS OF TWO-PORT PARAMETERS AND EQUIVALENT STABILITY FACTOR

3.3.1 Evaluation of Equivalent Padding Elements

In its present form, equation 3.5 (b) is not very useful in amplifier design, particularly since it contains the equivalent padding elements, pA and pi,' which are not easily measurable. A modification is necessary; and to do this, it is noticed that the total port immittances of the mismatched amplifier and the equivalent padded amplifier are the same. These are:

input PP1 = PP1 j GPI = Pin + PS 3 6 output = PP2 = PP2 j (11'2 = Pout + PL where pin = pin +ja in = input immittance, and Pout = pout j POUt = output immittance. Since the equivalent padded amplifier is a matched amplifier,

1 21)P1 = (PS - PO = PA + Pin 1 3.7 = 2PP2 = (PL - PO Pi + Pout Therefore, knowing ps and pli p q and pi, may be evaluated from ppl and pp2, which, for tuned terminations, are given by: M + X N M+ jiN 0 = PS + Pll - Re(777) = pi - PP1 P2 P2(1 + x2o) 3.8 M + AN Re( M iN PP2 = PL P22 - p1(1 + jX0)) = P2 2 P1(1 xo) where use has been made of equation 1.5, which may also be written as:

p1 = P1 j al = p1(1 j7to) 15 P2 = p2 4- j a2 = P2(1 ixo) From equations 3.7 and 3.8, therefore,

M + X N PS Pll o 2 2p2(1 + X0) 3.9 M + X N PL -• P22 o 2 2p1(1 + x2)

3.3.2 Evaluation of Equivalent Stability Factor

The substitution of equations 3.9 into equation 3.5 (b) results in a mismatched power gain expression involving two interdependent parameters S and X . The interdependence is clear from the fact eq o that p p is determined by a choice of S , and from equation 1.6, 1 2 eq X is a function of p1p2. Therefore, further simplification can be o obtained by eliminating one of these parameters from the power gain expression, preferably,X since S is obviously a more convenient o eq design parameter. The elimination is done by first expressing Seq in terms of X as follows: o From equations 3.9, the self immittances of the equivalent padded two-port are: 1 M + X N 1 _ p o Plleq = Pll +• PS - 2 + 2p2(1 + X2) 310 M + X N P2 0 P22eq = P22 +• PI, = 2 + 41(1 + X0)

, gives: The substitution of equations 3.10 into equation 3.2, for r‘ eq M + X N o ) 2 n(1 + X2 + nL 3.11 - eg. r 2(1 + X20)2

From the cubic equation 1.6, however, if X0 is a solution, M + X N 2 2 M o (x + 1)(X + 3 12 0 nL

Therefore, equation 3.11 can be further simplified to give: 70

n +X2 4. -21)2 _ M 3.13 '1 eq = o nL L Equation 3.13 contains n and X, which are related by equation 1.6, i.e. N - X M n 0 1 3.14 L .X0(1 + 0) Substituting equation 3.14 into equation 3.13, therefore, gives: ,2 2 2 (N - 1 0) + L X o 315 2LX0(N - X0M) S may be evaluated using equation 3.1. This gives, after From 1eq eq simplification: N - MXo S 3.16 eq LX.o

N or 3.17 o M + LS eq

3.3.3 Skew Factor 8 The simple nature of equation 3.17 makes it possible to eliminate as an intermediate design parameter. Thus, equation 3.17 the use of X0 may be substituted into equations 3.9, and then into the mismatched power gain equation 3.5 (b). The result, however, is still not altogether satisfactory, since ps and pi, can still be chosen in a variety of ways , giving rise to different values of power for a given value of Seq alone is not sufficient; p gain. Hence, the specification of Seq and p have to be specified as well. For pl, and p22 both positive, h 37 Lathi and Spence have defined PS and pL as: (o _ PS = i 1)1)11 n 3.18

PL (8 n. - 1)P22 p22/L. Here ps and PL have been defined in terms of whereni = 1)11 departure from the reference values corresponding to 8 = 1, when the

. 12 28 37 mismatched power gain is maximum ' . This maximum power gain condition is often termed the "optimum" mismatched condition. The definitions for PS and pL in equations 3.18 may be extended to cover cases when pli and/or p22 are negative. Thus,

but p22 < 0, and pi, may be defined as: f" P11> C/' PS

PS = (, _1.11 1)p11 3.19 ,r-E7 , PL = `8 -n. i` v22'

Similarly 0, 0, then, if Pll< P22>

PS = (8 \7-111. 1)( P11) 3.20 fl,rid- PL 1)P22

and, if Pll <0' P22‹ 0,

PS = (8417n1. + 1)(1311) 321

PL 1)(-P22)

The reference values for ps and pi, in equations 3.19 - 3.21, -- i.e., when 8 = 1 -- do not correspond to the condition of maximum mismatched power gain, which may be infinite for a given value of Seq as shown in sub-section 3.8.3. In particular, in equation 3.21, if 8 = 1, it may be shown that the power gain is actually minimum. Equations 3.18 to 3.21 contain n, which may be usefully replaced by S n eq, a more convenient design parameter. The relation between and S is obtained by substituting equation 3.17 into equation 3.14, eq giving, after simplification: 72

(M/L + se )2 n 2(rieq L) 3.22 Therefore, equations 3.18 - 3.21 may be re-written as: 8(M/L + Seq) P11> ° ‘ PS P11 [q2n i(rleq + M/L) 3.23 (1/8)(M/L + Seq) P22> 0 f ( PL P22 L2ni(Ileq + M/L)

8(M/L + Seq) Pll > 0} P11 PS (Nleq + M/L) c-2ni 3 24 (1/8)(M/LA + sea)4. P22 < 0 P22 ) PL + M/L) (- 8(m/L + Seq) Pll < 0 PS 1 41-2ni(n e + m/L) 325 (1/8)( L + sea) 122P > 0 PL L-2nOleci + m/L) P22

8(MA + Seq) P11 < Ps + 1](- P11) -2nieeq + M/L) 3.26 (1/8)(NA + } + ( P22 Ph [12n i(neq + -] `-P221

3.3.4 Mismatched Gain Expressions To evaluate the general mismatched gain equation 3.5 (b), it remains to determine the product (PS pA)(pl, PL). From equations 3.7 - 3.8 this is, after simplification : 1 _ 1_1, ,2 _11% 2 (PS - PA)(PL = PP1PP2 /4.o41 1` 3.27 By eliminating X0 and n, using equations 3.17 and 3.22, equation 3.27 is reduced to:

2 2 (Seq - 1) (Ps - P0(PL - . 2 06„) 3 28 43eq `Ileq Using equations 3.23 - 3.26 and equation 3.28, the mismatched gain expressions under various conditions may now be evaluated. Thus: for p11> 0, P22 >0 41 AI S eq 18014 ,) m G 42ni leciiry g(-17+Seq.) 0, q-p-f)M% . . . 3.29 t (s2 1\2 L eq ( ` eq _

for P11> °' P22 <0 {oaks Gt = I Alseq2 2 I L -4112ni(vieq+11-1) —+S ) +11-2ni(leq-itt)1 ...3.30 (Sea-1) eq) il1(141, eq

0, for Pll< P22 >0 : 24.1AISeci tem Gt = 4-2ni(YlecA-14-28(—+ S i(neq4)} 1.13.31 (s2 _1\2 s eq) LM eq) -11-2n ` eq

for P11< °' P22 <0 1A IS G eq 8(1 ) + f2nineq+t)} t 2 2 `L4seq ) + 4/2ni(leq4;4{1(t+S11 eq ( ...3.32 (seq-1)

3.3.5 Some Useful Expressions In equations 3.23 to 3.26, the definitions of ps and pi, have been made dependent on the signs of Pll P22* These equations and may be expressed more generally as:

8(M/L + S ) PS Pll = P1 f2nil (neg.eq + MA) IP111 3.33 (1/6)(M/L + s ) PL ' P22 P2 7A / 1 2n l (leg + M/L)e.1P221 i 74

which are valid for all signs of and p22 p11 . Using these equations, together with equations 3.23 to 3.26, many useful expressions may be considerably simplified, and put into forms independent of the signs of oP1111 and p22. Thus, input and output total port immittances, defined by equations 3.8, may be written as: 2 S eq - 1 PP1 = 8IP111 S edl2nil (neg + M,/L) 334- S2 - 1 eq PP2 11p .ISegV12nil (ri eq + M/L)

Similarly, equations 3.10 for self-immittances of the equivalent padded two-port become:

Plleq 3.35 1 leg + P22eq = -8" 11'221\1 2n . I 24 The input and output immittances of the mismatched amplifier can also be simplified to:

81P111(1/Seq 14/1) Pin pll 1f12 it(leg M/L) 3.36

IP221(1/seq + M/L) Pout P22 - 6,112ni1(ne4 M/L)

3.4. PRACTICAL CONSIDERATION

In practice, p11 is usually positive, while p22 can be either positive or negative. In subsequent sections, therefore, p11 is taken tobealwayspositive,unlessotherwisestated.IfiL is negative, the assumption will be that p22 is negative. 75

3.5 MISMATCHED GAIN AND BASIC PARANETERS

3.5.1 Use of Basic Parameters

In previous sections, the mismatched amplifier has been analysed as a modification of the equivalent padded amplifier. In particular, the stability factor of the latter, namely S , has been used to denote eq the stability character of the former. This "equivalent" concept can be extended, by using the basic parameter U of the equivalent padded eq . two-port as an additional characterising parameter, instead of Seq V1ith the use of Ue , it is also convenient to replace ni by U., the 9. measure of goodness of the original two-port. The replacement of ni and S (as well as vl eq l eg ) may be done by the use of equations 1.2, 2.4 and 3.1, which give:

n. 1 i II) 2 - 11 + AR + 2 2Uil A I CAI L 3.37

3.1 Seq = leq 41eq2 - 1 - 11 2 AR 3.38 Ileq = 2Ue IA.I

The last equation has been obtained by subscripting all parameters in equation 2.4 by "eq" to denote application to the equivalent padded amplifier. Note that A = A. = A, since port padding does not modify eq -s — the measure of non-reciprocity. The substitution of equations 3.37, 3.1 and 3.38 into equations 3.29 to 3.32 results in mismatched gain expressions which are usually very complicated. For example, 22-7 0 and 6 = 1, then, if pli> 0, p from equation 3.29, the mismatched gain is:

A - 11 I 2 AR - 11 2 2Uectl.A1 + I Al + 2UegILLI + I Gt X A 112 ARIA 112 12 )2 2 - 1 - U4204 +11.1 2Ue0 1.1 1411

M IA-112 AR i 22 2 A-11 M ±R1 1. 1-112 ARMr12 + AR X L2U0441 +111- 2Uegil +/Al i1-1 i 2UifAlIAI+ + L Weygilli + L [ 3 39 Note that the complexity of equation 3.39 -- and this is for a relatively simple instance -- is in sharp contrast to the simplicity of equation 2.6 for the matched amplifier.

3.5.2 Mismatched Gain and Matrix Environment Although the use of basic parameters results in complicated gain expressions, it reveals certain properties of the mismatched amplifier, which are otherwise obscure. One of these is the relation between mismatched power gain and matrix environment. Thus, of the 5 parameters used in the preceding section, /+ have values independent of matrix environment. These are U. and A, which are the basic parameters of the original two-port, and U and 8, whose values are fixed eq by design. Therefore, NA is the only parameter with its value depending on the choice of matrix environment. Since different values of M/L give rise to different values of mismatched power gain, amplifier operation in one matrix environment can result in more power gain than another. Amplifier operation in different matrix environments, of course, corresponds to the use of different modes of termination, as shown in figure 1.3.

The dependence of M/L on the choice of matrix environment can be simply removed, for M/L is variable by padding one or both ports of a two-port with reactive elements in such a way that they are not tuned out. An example is shown in figure 3.2, where a reactive element fie is placed across the input port, while series tuning arrangement (i.e., 77 z or h mode of termination) is used there. Thus "g" is not tuned out, and any value of M/L for the overall two-port can be obtained. In this way, the maximum power gain for a given value of Seq, if this is desired, may be obtained with any desired mode of termination. The addition of untuned reactive padding elements like "g" in theaboveexampleaffectsonlyMA.Theotherparameters,U.1,A, U eq and 8, are unchanged, because U. is invariant to lossless embedding, and A, being a ratio of transfer functions, is unchanged by port padding in general. U eq and 6 are, of course, fixed by design.

3.5.3 Mismatched Amplifier with Feedback A further use of basic parameters in mismatched gain expressions is in the study of the combined effect of feedback and mismatch. Here, the use of two-port matrix parameters is rather limited, since feedback generally affects all the parameters at once. Arduous computations are necessary to evaluate the change in amplifier performance, and each individual type and amount of feedback need be considered separately; no general study is possible.

The use of U1., Ueq and A, on the other hand, makes it possible to examine the general effect of feedback combined with mismatch, particularly if the mismatched amplifier is viewed as a development of the equivalent padded amplifier. Thus, the mismatched amplifier can be studied on a plane of constant U in the gain space of the eq matched amplifier. Since the mismatched amplifier is stable only if the equivalent padded amplifier is stable, the stable region on U eq plane is the same for both amplifiers. Therefore, use can be made of the gain charts in Chapter 2, with suitable modifications to constant gain lines, to study the mismatched amplifier with feedback (see section 3.6) As an example, the mismatched amplifier can be considered as formed from a two-port that is first embedded in a lossless network. This moves the point of operation on U1 plane to a required place on the same plane. Next, padding is applied, moving the point of operation 78

from U.-plane to Ueq -plane. The resulting padded amplifier is then converted to the mismatched amplifier, by the process described in section 3.2. From this it is clear that, if the mismatched amplifier is to have satisfactory sensitivity performance, the equivalent padded amplifier itself must have satisfactory sensitivity performance. It would not do, for example, to have the equivalent padded amplifier operating on the border of marginal stability. This design technique is treated further in section 3.8.6. A rather over-simplified picture has been presented above. In practice, the situation is much more complicated. Thus, lossless feedback will modify not only A, but also M,/L, unless untuned compensating reactive padding is employed. The modification of A and MYL entails corresponding changes in source and load immittances, in order to maintain the desired values of 8 and U . These problems eq are treated further in subsequent sections.

Relative Magnitude of U 3.5.4 eq to be greater or less Theoretically, it is possible for Ueq than IL, depending, among other things, on whether the equivalent padding elements z' and z' in figure 3.1 (a) have negative or positive real less than parts. In practice, however, it is more usual to make Ueq U. for better sensitivity performance. Therefore, in the following, , unless otherwise it will be generally assumed that Ueq is less than Ui stated.

3.6 GEOMETRICAL REPRESENTATION The geometrical representation of power gain relation has been shown to be very advantageous in the case of the matched amplifier, especially when dealing with the sensitivity problem, and when considering the effect of feedback in general. This is because there are so few variables to deal with, and they all have special significance in relation to amplifier performance. 79

In the case of the mismatched amplifier, the power gain is expressible in terms of 5 parameters, U., U , A, 8 and NA; of these, eq — A is complex. Therefore, a complete geometrical representation will require 6 dimensions in all, and will be far too complicated to be of practical use.

A simple and useful geometrical representation is possible, if the mismatched power gain is plotted as a. function of a few of these parameters at a time, the remaining parameters being kept fixed. Some such useful representations are plots of constant G /U. lines on .1 t ./Ueq - plane and Ue/APlane -- forming Normal and Inverse Mismatched Gain Charts respectively -- with Ueei, 8 and M/L kept constant. Note that different charts are also obtained here by using different values of U eq/U., 8 and M/L. Since the mismatched amplifier is stable only if the equivalent padded amplifier is stable, it is clear that the Normal and the Inverse Gain Charts for the mismatched amplifier have the same stable regions as the Normal and the Inverse Gain Charts for the matched amplifier.

To facilitate analysis in terms of the equivalent padded amplifier, the mismatched gain charts corresponding to the Inverse Gain Chart of the matched amplifier are preferred. Examples of these charts are shown in figures 3.3 to 3.6, which have been plotted numerically using gain equations 3.29 and 3.30, in conjunction with equations 3.37 and 3.38. It is assumed in these plots that A`>>1 or U so that e normalised plots are possible to begin with. The different figures have been obtained by varying the parameters Ueei, 8 and M/L one at a time, to show the effects of their variations. The use of gain equations 3.29 and 3.30 implies the assumption that p11 is always positive and that a negative value of ni is caused by a negative value of p22 only. In all these figures, lines of constant stability factors Seq have been omitted to keep the figures simple. These lines occupy exactly the same positions as lines of constant Si do in figure 2.3, the Inverse Gain Chart for the matched amplifier. 80

Note that, where the variations of "fixed" parameters, Ueq /U., 6 and 4/1., are of interest, these can be studied by a superimposition of such charts as those in figures 3.3 to 3.6. It may, of cource, be necessary to superimpose several charts together in order to allow for sufficient variations of these parameters.

3.7 PROPIIRTIES OF MISMATCHED GAIN RELATIONS

3.7.1 Power Gain at Lame Mismatch It has been seen that mismatched gain equations are generally very complicated. There are some special circumstances when these equations may be considerably simplified, and one of these is when the mismatch is large. Defining large mismatch as S :;:?-1, equations eq 3.29 to 3.32 may be reduced to 1 G = 1+1.111 s - 3.40 t eq if 8 is not far from unity. Note that the power gain of the mismatched amplifier is, in this instance, 4 times that of the equivalent padded amplifier with the same value of S . The mismatched amplifier, therefore, appears eq to be superior to the equivalent padded amplifier. However, when sensitivity performance is taken into account, the converse can be true. From section 3.2, it is clear that spreads in the real parts of input and output immittances, pin and Pout' will give rise to spreads in equivalent padding elements, 6 and PL. These spreads, in their turn, cause spreads in U and, therefore, in power gain of the mismatched eq amplifier. In the matched amplifier, on the other hand, the real parts of source and load immittances, ps and pi,' are tunable to take up spreads in pin and and these do not give rise to gain spread. It is Pout' possible, therefore, that if spreads in pin and o are large, the Pout sensitivity performance of the mismatched amplifier can be considerably , poorer than that of the padded amplifier with the same value of Seq even when Se q]> 1. 81

3.7.2 Maximum Mismatched Gain

The mismatched power gain can be boundless, a fact which is not obvious from the gain charts in figures 3.3 to 3.6, because these have been plotted with L restrained, except in one instance, to a value of -1, and 8 to values of 0 and 1.

That mismatched power gain may be boundless is obvious from equations 3.29 to 3.32. As S eq tends to unity, the denominators of these gain expressions tend to zero. Therefore, unless the numerators also tend to zero, which is possible if M/L = -1, the corresponding values of power gain will also tend to be boundless.

As Seq tends to unity, however, the equivalent padded amplifier will be operating near the border of marginal stability -- a region of poor sensitivity performance. Therefore, the mismatched amplifier will also be in a poor sensitivity region. The sensitivity performance of the mismatched amplifier will be considered further in sub-section 3.8.4..

3.7.3 Regions of Negative and Positive ni

In the stable region on Ueq/A-plane, there are usually 2 distinct regions corresponding to n. >0 and ni

IM AR M _ zui + ITT + L - 0 311 Note that equation 3.41 is independent of 8. In normalised form it may be written as: I22s. 9 0 34.2 2U. A R + VA1

In the particular case when N/L = -1, equation 3.42 reduces to: 82

lisa\ 2 Ilear229. req.\2 \ A U71 A iR 4. 2 .1 3.43 which is the equation of a parabola. It is interesting to note that ifilecU. 1 -- i.e., the mismatched amplifier becomes a matched amplifier-- equation 3.43 becomes identical with equation 2.17, which is the equation for the contour of marginal stability on the Inverse Gain Chart in Chapter 2.

The discontinuity of constant gain lines at ni = 0 is due to the definition of the skew factor in equations 3.23 and 3.24. There, if p22 tends to zero, p2 also tends to zero. Therefore, to maintain a given value of S (and hence n = p1p2 eq /L, which is related to Seq by equation 3.22), pi has to tend to infinity, which is not a realisable situation.

3.8 DESIGN CONSIDERATIONS It has been shown that the mismatched amplifier, made from a two-port with initial measure of goodness Ui, may be considered to have been developed from the equivalent padded amplifier, with U eq and A as basic parameters. Given these values of basic parameters, the mismatched amplifier may still have different values of power gain, depending on the values of M/L and 6 used. In this section, the influence of these two parameters are examined in greater detail, leading to a possible design of the mismatched amplifier for given sensitivity performance.

3.8.1 M/L and Power Gain

From section 3.5.2 it is clear that an appropriate choice of M/I, can result in more power gain being obtained for a given value of S eq. This value of M/L may be realised by choosing a correct mode of termination, or by using untuned reactive padding.

It is usually possible to select a value of MA such that the power gain is maximum, using charts like those in figures 3.3 to 3.6. However, under some special circumstances, this value of M/L may be 83

determined relatively simply by algebraic means. For example, if 1 =(ii ivt/L)> 0 and 6 = 1, -- a situation not uncommon in practice then the power gain equation 3.29 may be written as:

2 Gt 4jAi. (F) 3.44 ' (Seq - 1)

where F = (M/L Seq) - ali M/L)(1eq M/L) 3.45

To maximise G with respect to M/L, it is sufficient to maximise F t with respect to the same variable, since S and A are independent of eq M/L. Since and %I are also independent of M/L, equation 3.45 may l eq be differentiated directly with respect to M/L, giving:

Oli + M/L) + Oleg + M/L) dF 1 - 3.46 d(Wh) 2 Ir(i i + M/L)(leci + M/E7

Now, (li + M/L) is positive by assumption, and (rieg + L) is also positive, because 1eq >1 for stability and the maximum negative value of M/L = Re(13121321)41)121)211 is -1. Therefore equation 3.46 may be written as: 1 dF ) - Ole + m/10 n 2 - 347 d7/T) = 2)f(li + m/L)(leci + M/L) which is always a negative quantity. Therefore, the maximum value of F, and hence of power gain, are obtained when M/L is most negative, namely M/L = -1. In general, the value of M/L giving maximum power gain, for a given value of S , depends on what S, and are. This is eq eq 1 obvious from figure 3.5, where at point P, for example, more power gain is obtained when M/L = +0.2 than when M/L = -1.

3.8.2 M/L and Sensitivity

Although a proper choice of M/L can give rise to more power gain for a given value of S , the amplifier performance under this eq 814. condition is not necessarily optimum. In the following, it is shown that there is generally no correlation between the value of M/L and the sensitivity performance of the mismatched amplifier. Consider, for example, the two-port in figure 3.7 (a), where the stray capacitance Cs has a large spread. If y or h mode of termination is employed (figure 3.7 (b) ), the spread in Cs is taken up by port tuning, and there is no spread in power gain. On the other hand, if z or g mode of termination is employed (figure 3.7 (c) ), Cs gives rise to a spread in L as well as spreads in input and output immittances, and the spread in power gain can be large. Now, the mean value of M/L in y or h mode may be greater or less than in z or g mode. Therefore, it is clear that more or less power gain can be obtained with better or worse sensitivity performance, depending on the presence of such elements as Cs. This means that, there is no unique relation between power gain and sensitivity, which is invariant to changes in matrix environment. The above conclusion also applies when reactive padding is used to modify WL. Consider he same two-port in figure 3.7 (a), where C has a large spread. It has been shown that a choice of y or h mode of termination can lead to these spreads being tuned out. Suppose that one of these modes of termination is employed, but M/L is modified (e.g. to raise the power gain) by adding a series reactance between points 2 and 2" (figure 3.7 (d) ). It is not difficult to see, then, that some of the spread in C may not be tuned out. If, on the other s hand, this reactance is placed at the other port (figure 3.7 (e) ), the spread in Cs can be tuned out, and more power gain can result without impairing the sensitivity performance. In practice, the nature of spread may be more complicated. For example, there can be a spread in Cs as well as one in lead inductance at the same port. In this case, it is not clear what mode of termination and what value of M/L are best to choose for maximum gain and optimum sensitivity performance.

3.8.3 Choice of Skew Factor 6 Skew factor 6 cannot always be chosen for maximum power gain, since it can be infinite and/or p22 are negative. Consider, ifPll for example, equation 3.30 for pli> 0, p92

41A1Seci G ) M/L t .2 8(NA2 + S ) V:egS1.i l + (Seq - 1) eq

and the power gain increases proportionally with 8. With a large increase in power gain, the sensitivity performance tends to get poorer. This is seen from equation 3.24; a large value of 8 implies that p2 tends to zero, and a small change in p22 can cause large changes in Seq and 8, giving rise to a large change in power gain. On the other hand, 8 can usually be chosen for optimum sensitivity performance. Consider the general power gain equation 3.5 (b), which is:

IAL PsPL 35 (b) Gt = s eq (PS - P0(PL - PI) Suppose that source and load immittances ps and pl., are fixed. A proper and (ps - pA)(pl, choice of 8 can result in minimum spreads of Seq and hence of power gain.

Consider first the spread in (p3 - p;)(pl, - pt). It is convenient to denote the spread in a quantity q by 41q, which is then treated as a small quantity, whose magnitude only is of interest. Using equations 3.6 and 3.7, and the fact that ps and pig are fixed quantities, one can write

4f( Ps PO(PL P6 t(Pp14Pout Pp2APin) 348

By substituting for ppl, (31,2, pin o from equations 3.34 and and P out 3.36, the right-hand side of equation 3.48 becomes: 86

A 113221 1 , 7: 5*PP1"422 '-g Pp2'6'Pli 6.1PA 349 where pia and 42 are total port immittances when 5 = 1, and 1/S eq + M/L x = 3 50 /(2nif (leg + M/L)

As a first approximation, it is assumed that there is no spread in 8. Expression 3.49 can then be written as:

-21-(S. P1'1.11322 +462.411) + terms independent of 8 3 51 If this expression is now differentiated with respect to S and the result equated to zero, the value of 8 for minimum spread (checked by a second differentiation) in - - (p5 pL) is given by: 2 all/p1"1 4P11411I 8 3 52 422142 AP22/P22I where the modulus signs have been used, since only the magnitudes are of interest.

It can be shown that the value of 8 given by equation 3.52 also gives rise to minimum spread in S eq, and, therefore, in A q, to which Seq is related by equation 3.1. Thus, consider equation 3.2 for %leg:

= 4.1121.22_2.1- M 'leq L 32 a choice of 8 can affect only the spread in(I o llee22eq). Therefore, to minimise the spread in S eq, it is sufficient to minimise the spread in (P11eqP22eq). From equations 3.10, 3.7 and 3.6, 1 1 '8'(P11eqP22eq) = Plleq41(P22 PL 2PP2) P22eq6(P11+ PS - 1Pp1) which, by substituting for pla and pp2, becomes:

PL_1 ) PS 1 1 I1(PlleqP22eq) = Plleg'6(P22 2 gout' P22eci&P11+ 2 - 7Pin/ ..-A "Az

Using equations 3.35 and 3.36, and the fact that ps and pi, have no spreads, the right-hand side of equation 3.53 may be written as: 87

P22 1P22) 1 A l Pll 81P111 2— + 2 .x) 8* Pllee ( P22 - — + F322eq." P 11 2 3.54 where pileci and when 6 = 1 respectively. and P22eq are Plleq P22eq Assuming that 6 has no spread, expression 3.54 may be expanded to:

8 , P22 1 P11 'Plleq. + T P2, 2eq . A2 + terms independent of S. If this expression is now differentiated with respect to 5, and the result equated to zero, the optimum value of 6 will be found to be given again by equation 3.52. Hence, if 5 is chosen to be that given by equation 3.52, the spread in power gain will be minimum, to a first approximation. Note that when the spread in nfeci is minimised, the spread in Ueq is also minimised, since r t eq and Ueq are related via equation 3.38.

3.8.4 General Sensitivity Consideration In the case of the matched amplifier, the closeness of gain surfaces in a 3-dimensional gain space is sufficient indication of amplifier sensitivity performance. With the mismatched amplifier, however, there are more parameters to handle, and the study of sensitivity becomes more involved. A gain space here has 6 dimensions; and to assess gain spread one has not only to consider the closeness of gain surfaces in this space, but also to know both the mode of termination and the type of reactive padding being used. The latter, as has been shown, have a bearing on the sizes of parameter spreads. Assuming that the maximum spreads in various parameters are known, a simple assessment of sensitivity performance may be made through the use of charts like those in figures 3.3 to 3.6. These charts have been intentionally plotted with mean values of parameters given by U.I/Ueq = 5, 8 = 1 and M/I, = -1. The various charts show the effects of parameter deviations from these mean values. The total gain spread may be estimated by adding up the maximum changes in power gain due to various parameter spreads. This is likely to be a pessimistic estimate, since a particular device, in a group of n devices, is not 88 likely to have all its parameter values simultaneously differing from the mean values most unfavourably.

The chart method of sensitivity assessment is simpler than the alternative method of differentiating appropriate gain equations with respect to various basic parameters and then evaluating maximum gain changes due to parameter spreads. This is because gain equations in terms of basic parameters are very complicated, even in the simple example shown in equation 3.34, and the differentiated functions will be even more so. Numerical evaluation of these functions will prove more tedious than an evaluation from charts. The use of charts has one further advantage in that it can be obvious at a glance if the sensitivity performance is likely to be most unsatisfactory, for example, when the point of operation is in a region where the density of gain lines is great.

Finally, it is usually true that when A (ort9 is small or zero, changes in power gain with changes in parameter values are comparatively small. This is illustrated in figure 3.8, where the sensitivity performance at two points on the charts in figures 3.3 . Point P is to 3.6 is compared for variations of (5, N/I, and U./Ueq near the margin of stability and Q is at/X.. = 0. Note that the sensitivity performance at Q is much better than that at P.

3.8.5 Near-Optimal Mismatched Amplifier Design Strictly, the optimal design of a mismatched amplifier amounts to examining the gain-sensitivity capability at all points of operation in a 6-dimensional gain space, and selecting the best one. In this selection, it is necessary to take account of the fact that the volume of spread in the gain space depends also on matrix environment and on the use of untuned reactive padding elements. This design procedure is far too impractical, even when the sensitivity performance is assessed by the approximate method using charts, which is described in the preceding section. 89

A more practical, but approximate, design procedure is to treat the mismatched amplifier to be designed, first as the equivalent padded amplifier, with parameters A and U , and to allow for changes in power eq gain and power gain spread when the padded amplifier is "converted" and A is made such into the mismatched amplifier. The choice of Ueq that the gain-sensitivity performance of the equivalent padded amplifier is optimum. This selection may be made by the method shown in Chapter 4, assuming, in particular, that the spread in is the same as that 1/11eq in 1/U..

When U eq and A are known, Seq may be found from equation 3.38. The knowledge of S eq will facilitate the calculations of other parameters, as well as the terminating immittances. The next step is to "convert" the equivalent padded amplifier into the mismatched amplifier. The "conversion" amounts to selecting suitable values of M/L and 8. As shown in section 3.8.1, MA can usually be chosen for maximum power gain by using charts like those in figures 3.3 to 3.6, or in simple circumstances, by algebraic means. However, in choosing the optimum values of Aand U that LL.= 0, in which eq, it often happens case, a change of M/L produces a negligible change in power gain anyway. Thus, it is frequently more useful to select terminations such that the spread in M/L is minimum, and this involves an examination of the physical nature of spreads (see sub-section 3.8.2). Finally, there is the choice of 5. With 8 can usually be chosen for maximum power gain as well. However, as in the case of selecting M/L, the power gain is then not very sensitive to changes in 8, and it is advisable to select 8 for minimum spread in power gain. That is, 8 is chosen according to equation 3.52. With all 5 parameters now known, the terminations may be computed: (i)the real parts of the terminations, ps and pL, are found from equations 3.23 to 3.26; (ii)the imaginary parts, a s and. 010 are evaluated by substituting the value of X from equation 3.17 into equation 1.5. 90

It should be emphasized that this design technique does not necessarily produce optimum mismatched amplifiers from a given batch of devices. In fact, the final power gain and power gain spread cannot be known accurately at the beginning of the design procedure. This is so, particularly, because of the assumption made at the beginning, thatthespreaaimOjecl isequalto-thespreadinl;whereas,X in fact, the spread in .5, which 1/Ueq depends upon the choice of 11/L and is made towards the end of the design.

3.9 CASCADING CONSIDERATION As mentioned in section 2.8, when a single-stage amplifier will not give sufficient power gain with satisfactory sensitivity performance, one solution is to cascade amplifier stages. In this section, we consider the problems involved in cascading mismatched amplifiers. It is not intended to make an exhaustive analysis of the multi-stage mismatched amplifier, but rather to deduce its performance from that of the constituent stages.

3.9.1 Relation between Stage Gain and Overall Gain Consider an n-stage mismatched amplifier. The overall transducer gain G is given by: T P G L = power supplied to load T power available from source PA P P ol o2 Poi PL *P 3 55 PA °Pol o2 Po(n-1) where Po. = power output from jth stage, and PA. = power available from is equal to power source.Sincepoweroutputfromjthstage,POjl input into (j+l)th stage, Pi(j+1), equation 3.55 may be written as: P P p P ol o2 'o3 L 356 TP.A 12 13 1,n transducer gain Product of actual of X gains of first stage remaining stages

91 where the actual gain of jth stage is defined as Poi/Pij. Several interesting points arise from considering equation 3.56. Firstly, when single-stage mismatched amplifiers are cascaded, the overall power gain is not given by the product of transducer gains of the individual stages, as is the case with matched amplifiers. Secondly, since input and output immittances of single-stage mismatched amplifiers can differ considerably from load and source immittances, cascading can often be done only by using interstage transformers, if too much power gain is not to be lost. Finally, the interstage transformers limit the degree of skewness to definite values, depending on the mode of termination. In fact, Boothroyd53 has shown that skewness factor S = 1 can be realised only when h and g matrices are used. This will be discussed further later on.

3.9.2 Relation between Transducer Gain and Actual Gain of a mismatched It is convenient to write the actual gain Ga stage in terms of the transducer gain G of that stage. Thus: t P P F o o A Ga • IT P.1 15.-A • 1 which Boothroyd53 has shown may be written as: 2 PP1 Ga 357 Gt 4.pspin Consider a long chain of identical mismatched amplifier stages, with identical interstage transformers of ratio T : 1, as shown in figure 3.9. The relations between the real parts of source and output immittances, and between the real parts of load and input immittances are fixed by the transformer ratio; these are z matrix: 1 and 2 3.58(a) PS = T2'Pout PL = T pin 1 y matrix: and PL 3.58(b) PS = T2'Pout T2 92 h matrix: 1 1 2' 0 and. pi, 3.58(o) PS T Fout 1 g matrix: Ps T. and. PL 3 58(d) Pout

By substituting these relations into equation 3.57, the following power gain relations are obtained:

for y and z matrices

G = ll— = G , the power gain of the equivalent 1— S1 eq a eq padded amplifier

for h and g matrices le3.59 PL PL = 111.117, = G • Ga oq Pout eq Pout Therefore, with y and z modes of terminations, a mismatched amplifier stage is not superior to the equivalent padded amplifier stage in power gain. This is because the skew factor 8 is limited to unfavourable values, given by:

1111'22 T for y matrix: 8 = P11 360

for z matrix: 8 = AirP1221 equations 3.60 may be derived by substituting the immittance relations 3.58 into equations 3.34 With h and g modes of termination, on the other hand, the immittance relations 3.58 give:

Pin Pout 3 61 PS PL which, from equations 3.34 and equations 3.23 to 3.26, is satisfied by skew factor 8 = 1. Here, the power gain obtained can be greater (but not necessarily so) than that of the equivalent padded amplifier. 93

3.9.3 Design of Interstage Transformers If the value of S for a mismatched stage is fixed, the trans- eq former turn-ratio T may be evaluated from immittance relations 3.58, in conjunction with equations 3.23 to 3.26 and equations 3.36. For example, for z matrix, relations 3.58 give:

PS 1 2 3.62 Pout = T If Pll >0, P22> 0, PS and Pout may be substituted from equations 3.23 and 3.36 respectively, giving, after some simplification, using the value of 8 from equations 3.60:

2(Y7 + m/L) = (1 + 1t fZE v7,7 Pll leg T Pll I This may be solved for T, giving: -2(leci + M/L) + V*2(lea + WL) - 4ni P22 11 T 91 3.63 Pll 2)57; In a similar way, expressions for T may be found for other modes of and termination, and other values (including negative values) of Pll P22*

3.9.4 Sensitivity Consideration when Cascading The sensitivity performance of a multi-stage mismatched amplifier generally cannot be assessed from the performance of individual stages) which have been designed using fixed source and load immittances. This is because a stage in the middle of a chain of stages has the output of the preceding stage as its source, and the input immittance of the following stage as its load. Therefore, to assess the sensitivity performance of the overall amplifier, the additional gain spread due to spreads in source and load immittances of a stage need also be taken into account. Note that a spread in, say, load immittance pi, of a particular stage tends to give rise to an additional spread in input immittance of that stage, since this may be written as: 914.

Pl2P21 Pin =-12111 P22 PL As this input immittance serves as the load for the preceding stage, it is clear that the spread in load immittance can grow in magnitude up the chain of stages. In designing a multi-stage mismatched amplifier, therefore, the eventual number of stages to be used need be often kept in mind when considering allowable sensitivity performance for a single stage.

The deterioration of gain-sensitivity performance by cascading can make the multi-stage mismatched amplifier inferior to the amplifier made by cascading equivalent padded amplifier stages. This is particularly so when y and z modes of termination are used, because, as shown in equation 3.59, the power gain of a mismatched stage is then equal to the power gain of the equivalent padded stage, while the sensitivity performance of the former can be worse, due to spreads in source and load immittances. Moreover, whereas the multi-stage matched amplifier may be tuned by tuning individual stages separately) the multi-stage mismatched amplifier requires rather a complicated tuning arrangement. This is because the source and load immittances of a particular stage in the latter amplifier are determined by the tuning of adjacent stages. If there are many stages, therefore, there is a possibility that the mismatched amplifier may not be tuned properly, causing even more gain spread.

It appears, then, that when the possibility of cascading is to be considered, the matched amplifier technique is far superior to the mismatched technique, both in ease of design and in performance. Further comparisons between these two alternative techniques are made in the. following chapter. 95

EQUIVALENT PADDED TWO—PORT (MATCHED)

Z I z' z' S

z zll 12 z — z' L L

z z 21 22

z. z in Out!

equivalent padding impedances zsP L (zs— = equivalent source impedance (z1,— = equivalent load impedance

(a)

EQUIVALENT PADDED AMPLIFIER

z 11 z12 z L ,• z z 21 22

(b) MISMATCHED AMPLIFIER

FIGURE 3.1

RELATION BETWEEN MISMATCHED AMPLIFIER AND EQUIVALENT PADDED AMPLIFIER 96

j X RS S OC-44 AA Ni• 0 COMMON—BASE I = 1 mA e pL VCB = —10 V O f = 3 Me/s

IOVERALL TWO—PORT

g m-mhos 6. *0.4

1_1.0.2

- 0.2 10.2 (noz - 0:4 T 0.4

(ii)z 0.6

- 0.6 0.13

±-1,2

- 0.8

FIGURE 3.2 EFFECT OF UNTUNED REACTIVE PADDING ON M/I,

X`I" ;' • • ' • / g Geq/Ue • • / = 0.8 • / • / V', = 1.0 • • •• • • 4 = 1.2 • aT 1-- 1.4— • / _ - • • kr = 1.6 , - = 1.8 • , , 0- = 2.0 7 / I :1 ,•

I? ; , ...... "...... '...... -. 1;

l 1 .....°' i 1 1 , ....1. I 1-.. I 0. . I 4 i ....„. 1 • ' I (.9 I 1 1 Q . 1 1 I I 1 i I I X' 1 I I I R . I 1 I III i .... I ....,..„ I I , ...... i % % t ••• I I I 1 .... I .S4,...... 1 I \*tit \t 1 1 '''...... % IA ...... ‘ t\ 't % ..

, ...... ‘ Gt/Ui = 0.7 1 . = 0.6 • . II

.... ‘ • • • • = 0.5 • „‘ ‘‘ % • = 0.4 • • • • • = 0.3 • • • •, \ s • .. • = 0.2 • • • • ss %• \ s% • s • • • • • • • • ss • 1/4.0

A & . I . •.. . • ..., , ... • ,, • . • , % • , • _ ,.. _ . ------. (6=1/2) -..,, ., --- . • , 1„,. -- 0 -- G.t/Ui = 01 .. " ,. • -- .1--- - . ••. ___ .1' = 0.3 4111Pr ___.-- - / s• _ ----- .. = OA .- s , t (1,-..\ , ,,•,,\ , = 0.5 4* 4 ' s. s, ' s tO --N. , / 74W. ,, . ,' . ... ‘.....,

, I If I' ... 1 _ : 1. .X / • 07 , I i I . It 1 S.' 1 I I ..... ; I • ..l • t i I .. 1 : ; A• i / 0 ,.. . I : ' 4. 1.' I Q I tD , ; : ! • V , ., 6 , s,-- ,% ,, _ s...,,.. , X ,. R . . , k-.., .... ‘ ‘ .

• • N • • , .. 4 , ..• •• • • N • ' • , ' ' •••••• • • / • • " '''',." • • • , '-... s • . • , , ... • • , ' • • i . "...., • -• . • - -. -- ' ... -.,,...,. • , • • _ . • ( 6 =1) ,, J____, • , • G . = 03 FIGURE 3.4 t/U1 - = 0.6 P ,- , =0.5 ,1 MISMATCHED GAIN CHART WITH =0.4 , = 0.3 .. U /U = 5, M/L = —1 ,r i eq , .a2 / and 6 = 1 ...(dotted lines) ,., 6 = 1/2...(solid lines) /' Lc, cc

• • • (M/L=0.2) • • • • Gt/Ui .4 0.3 • = 0.4 • • = 0.5 • 0.6 • 5, _ ..._9)_ , 0.7 — S _ o j2; --- / = 0.8 ...... e."" e• e e + -- / = 0.9 - • •/ • = 1.0 • • • • • • • • I ^ ....-N I

6 N I I' e•-•, I II C; I ....., .--.‘ N 0 -n tiJ ci, 9 6 ; 1/ , ,,,r n.= 0 It I (.9 ...., 0 ...... ' St .4 (M/ L=0.2 .... , ..... ,...... ,I 61 Z

• • S

• (MIL=-1.0) Gt/Ui= 0.7 FIGURE 3.5 = 0.6 = 0.5 MISMATCHED GAIN CHART WITH = 0.4 U./Ueq = 5, Et = 1 = 0.3 1 = 0.2 and M/L = 0.2 (solid lines) = —1 (dotted lines)

X i (Ui/Ueq 3) \ • • Gt/Ui = 03 • = 0.4 • 0 • • = 0.5 r• 3 ) (qUeck 0.6 = 0.7 (0 ilUeq' /

,/ I / / ., / , / 'Y c„," ..., / o st 1, di ,43 Q st,t, 0. -I ---.. cu ...... J i ...... -:. 0'7 j 04- ...... ,•...... I

FIGURE 3.6 MISMATCHED GAIN CHART WITH = -1.0, 8 = 1 and U.1/Ueq = 3 (solid lines) U./U = 5 (dotted lines) (Uittieci=5) 1 eq Gt/Ui = 0.7 = 0,6 = 0.5 = 0,4 ---- -. 1-)----- ,_ / = 0.3 / = 0.2 / . / . / o;--, 0 101

TO TWO-PORT WITH SPREAD IN C SOURCE s

(a)

SPREAD IN CS IS TUNED OUT

(b)

2 SPREAD IN C TO s SOURCE IS NOT TUNED OUT

( c)

SPREAD IN C IS NOT s TUNED OUT COMPLETELY

(d)

TO SPREAD IN C s SOURCE L CAN BE TUNED OUT

( 0)

FIGURE 3.7

TUNING OUT OF SPREAD IN C s 102

M/L 8 = 1 Mean values of parameters: U.a./Ueq = 5, . -1,

Points on chart P Q

near margin X'I = 0 Locations of stability

Mean values of power —1 gain corresponding to, GM . 0.3 G/U = 0.3 mean parameter values' (fig. 3.3)

If 8 becomes 1/2 G/U decreases G/U decreases (fig. 3.1.)4 to 0.05 to 0.26

If M/L becomes 0.2 j G/U increases G/U decreases (fig. 3.5) I to 1.0 to 0.28

becomes G/U increases G/U increases If U./U1 eq 3 (fig. 3.6) to 0.52 to 0.43

FIGURE 3.8

COMPARISON OF SENSITIVITY PERFORMANCE AT TWO POINTS ON GAIN CHARTS 103

T:1 T:1 T:1

TWO-PORT TWO-PORT

PL Pout Pin

p p + j0

T = int erstage transformer

FIGURE 3.9

CASCADED ALIPLIFIFIR 104

CHAPTER 4

GAIN-SENSITIVITY CONSIDERATION

4.1 INTRODUCTION In chapters 2 and 3, the two alternative techniques of obtaining stable power gain, namely the matched and the mismatched techniques, have been analysed and discussed. We have seen that, if the power gain required from a two-port is greater than 4 times the measure of goodness of the two-port, i.e., greater than 4U, then only the mismatched technique can be adopted. If the power gain is less than 4U, either technique may be used. We have also seen that, in general, the greater the power gain obtained from a two-pOrt amplifier is, the poorer is its sensitivity performance.

In practice, active devices like the transistor cannot be made to a close tolerance. A spread of 200%to 300% in some parameters of the transistor is not uncommon. For instance, the value of the internal base resistance r bb' of 0C-44, as quoted by the manufacturer, can range from 40 ohms to 250 ohms. An important consideration in the design of amplifiers is, therefore, that of sensitivity, or, more precisely, that of gain-sensitivity capability. Given a batch of devices, it is not sufficient just to make them into simple amplifiers with maximum power gains, for the sensitivity performance can then be very poor; nor is it satisfactory merely to damp the amplifiers with passive elements to get the required sensitivity performance, for the resulting power gain can be very low. On the other hand, it is not desirable to use a very complicated design procedure involving complicated networks, in order to obtain amplifiers with optimum power gains for given sensitivity performance, or optimum sensitivity performance for given power gains.

In this chapter, an investigation is made into the simplest but most effective way of obtaining optimum gain-sensitivity performance from a batch of devices. Emphasis will be placed on designing amplifiers for a given value of gain spread. The investigation involves a 105 preliminary search for the most promising design technique to adopt, and finally a detailed examination of the gain-sensitivity capability of the devices when this technique is used. It will be assumed that the power gain required from a device is not excessive, i.e. certainly less than 4.U. If more power gain than this is required, it will be assumed that compounds of devices are used (resulting in two-ports with larger values of U), or that amplifiers are cascaded together.

4.2 DESIGN REQUIREMENTS

With the power gain demanded from a device being less than 4U, there are two alternative design techniques to choose from, namely, the matched and the mismatched techniques. In making this choice, particularly in the design of amplifiers for given sensitivity performance, there are a number of criteria to consider, namely:

(a) The design procedure should require the minimum number of parameters to be specified, measured and/or calculated.

(b)The procedure should be quite simple; but

(c)it should produce maximum power gain for given sensitivity performance.

(d)The resulting amplifier circuit and tuning procedure should be simple.

(e)Cascading procedure should be simple, and the sensitivity performance after cascading should be predictable from that of the individual stages.

In figure 4.1 are summarised the merits and demerits of the matched and the mismatched techniques. From this, and the analyses and discussions in the preceding chapters, it is obvious that the above criteria are best satisfied by the matched amplifier design technique. This is particularly so when the possibility of cascading has to be taken into account.

In the following) therefore, the sensitivity performance of the matched amplifier is examined in greater detail, with a view to 106

establishing the relationships between power gain and sensitivity performance with various types of spreads. An investigation is also carried out to determine the conditions under which lossy embedding may be used with advantage in controlling amplifier sensitivity. Throughout this consideration, except where otherwise indicated)an assumption will be made that sizes of spreads in (1/0 and X (= 1/A) are unchanged by embedding a two-port in a passive network -- an assumption not unrealistic, in view of the examples in Appendices 5 and 6. In any case, if the sizes of spreads do change with embedding, the results of the following analyses can usually be re-interpreted to allow for the changes.

4.3 CHOICE OF SENSITIVITY DEFINITION Several definitions of sensitivity have been made in the literature. For example, if G, the quantity of interest (e.g. power gain), is a function of p with a spread Op, then, denoting the spread in G by AG, the more commonly used definitions are: y S - 41 1 Aln G

Lan G pGfG 42 S2 Ain p AID/10

S Ain G AVG 43 3 AP A P Definition 4.1 has been used by Bode54, while definition 4.2 is preferred by Truxal55 , Mason- 56 and several subsequent authors. Definition 4.3 is given by Hakimi and Cruz57, in conjunction with systems with multiple parameter variations. There are two criteria in the choice of an appropriate sensitivity definition. Firstly, when a number is assigned to it, say, S = 6, it should be immediately evident what the gain spread is, given a value of parameter spread. In this respect, all the above definitions are satisfactory. Secondly, the sensitivity definition chosen should have a simple bearing on an invariant property of the devices, if possible. This may be in the nature of: 107

(a)A simple relation between feedback and gain spread or parameter spread. The feedback may be in a particular category, e.g. lossless feedback.

(b)A unique relation between gain and sensitivity, from which it is readily calculated how one quantity need be modified to improve the other.

With the matched amplifier, it has been mentioned that spreads A(1/U) and dX. tend to remain constant under certain circumstances, while the values of U and X themselves are modified. It is clear, therefore, that for the present purpose, definition 4.3 or a function of it, is most suitable, since 4p then tends to remain constant, and a modification of S can be related directly to a change inA.G/G. 3 S , as it is defined, may be appropriately termed "sensitivity", 3 since the smaller it is, the better is the sensitivity performance. In amplifier design, however, it is usually more convenient to associate improved performance with an increasing quantity. The simplest function of S3 with this property is, of course, the reciprocal of S3, or 1/S3, which may be called "sensitivity figure." That is,

sensitivity figure = S 4. 4 F y G - AG---r-APG Good sensitivity performance now corresponds to large values of S. This definition of S is particularly useful in studying the gain- F sensitivity capability of devices, since, as gain and sensitivity performance are somewhat interchangeable quantities, the product of gain and sensitivity figure should prove a valuable quantity. This, in fact, is the case, as will be shown in subsequent sections.

4.4. SPREAD IN MEASURE OF NON-RECIPROCITY

4.4.1 Maximum Gain Deviation due to Circular Spread in X

Consider a circular spread of radius ,X on X-plane, as shown in figure 4.2. Let the centre of the circle correspond to the location of the average device, with mean parameters equal to 1/U and X, and

108 let the average gain be G. It is proposed to find the maximum gain deviationiG, from the average gain G, due to this spread. The maximum gain deviation obviously occurs on the circumference of the spread circle. Let the average device be denoted by a vector X, and any device on the circumference by hl. Then, the change from —X to -Xa is given approximately by: 8X = 2iX cos (Q - 0) 45 4+ sin (G - 0) where X= I XI and 0 = arg X.

Now, the basic gain relation 2.6 for the matched amplifier may be written as:

4 6 U - 6:6-442 - 4.3")cos 0 +2 g)2 = 0

Let G, X and 0 change by 61, SX and 80 respectively. Then, it may be shown that

- 61 1(1 - 1) + (1 - *)cos - a0(1 #)sin 0 al X GX U 47 .1\`1. .. cos 0 21X 2U - ( Ui and 60 may be substituted into equation 4.7 from equations 4.5, giving, after simplification:

Ax (- 1).cos(Q - 0) + (1 - Ti.).cos(Q - 20) a; = 4.8 G 2 f - T).cos GX 12UXX 0

To find the maximum gain deviation in the spread circle, equation 4.8 is partially differentiated with respect to A and then equated to zero, giving

- 1)sin 0 + (1 - -Psin 20 tan = _ 1)..s 0 (1 - cos 20

109

By substituting the value of g in equation 4.9 into equation 4.8, the latter becomes:

Gx ... 1)1.2+ (1 -/-t 1)(1 - -137- )cos (AGI 4.10 G/X X 2 GX 12UX (1 - 1)cos 0 where AG has been used instead of 817, to indicate maximum gain spread, and (AG/G) has been subscripted by X to indicate gain spread due to a circular spread on X.plane.

Equation 4.10 may be considerably simplified, if U>> 1 and 2 ,2 X << 1 (i.e. LAI>>1), to:

1*2 + 1 (U 1)cos 0 (AGA AX11{4,1 GX 411 GX cos 0

From equation 4.6, if U >> 1, G may be written in terms of X' = UT,

[ f(2 x121 412 x12 ('2- R) 4- Xi)2

Hence, G may be eliminated from the right-hand side of equation 4.11, giving:

xt2 X.R) 114 XL)2 4.13 * Ll xi)2 xI2 ‘2

All parameters on the right-hand side of equation 4.13 have been normalised by 1/11, including the radius of the spread circle, which becomes1170.

444.2 Sensitivity and Stability Equation 4.11 may also be written in terms of inherent stability factor Si, by eliminating G and X, with the use of equations 1.2 and 2.13. The result is:

110

s - 2S c0 s + 1 dx 1 2 2 4)4 ° X S. -

It is interesting to note that in equation 4.14 we have an interrelation between S. and the sensitivity performance with respect to a spread in X = 1/A. In general, Si in itself is insufficient for sensitivity indication; arg X (= 0) is required as well. Given Si only, however, the maximum and minimum limits of gain spread may be determined. Thus, by setting cos 0 in equation 4.14 to +1 and -1, it is found that:

X 1 lies between 2.4 and 2.Q .S 1 .S

4.4.3 Sensitivity Figure for Spread in X

Using the definition in equation 4.4, the sensitivity figure

SFX for spread in X may be evaluated from equation 4.15, which gives: xi)2 xI2 ( 2 SX. 415 (k.G/G)x 0.1 il .. A1,1 4. x,2 xRi ‘R 2 AR1) 2 - where the .1, sign has been omitted, since only the magnitude is of interest.

Equation 4.15 contains both normalised and unnormalised terms. For convenience in graphical display, a normalised sensitivity figure S'' is defined such that: F xliz)2 xt2 Si - FX (AG/G) 4.16 X + 7ki) _ ift(2 xiz)2 702

S'Sri is thus a function of .'only. If lines of constant S' are plotted FX on A'-plane, the result is the chart of figure 4.3. A use of this chart is in the design of amplifiers, such that gain spread due to a spread in X is controlled to a given value. Thus, suppose a batch of devices are given, with no spread in U, but with a circular spread 111 in X, of radius 4A. If the maximum value of (AGA) acceptable is X specified, then the appropriate value of SF% or Sk can be calculated, and the corresponding line of constant Sh located on the chart. Loss- less feedback, suggested in Appendices 3 and 4 can next be applied to obtain an operating point on this line of constant Sh. It is clear that, given a value of Sbx, the maximum power gain is obtained when X' is real, i.e., along M- axis in figure 4.3. Hence, an optimum amplifier should have its operating point on this axis. An example of amplifier design involving sensitivity control similar to this is given in Chapter 6.

4.4.4 Gain-Sensitivity Figure Product for Spread in X

It is evident from figure )+.3 that an increase in power gain is generally accompanied by a decrease in sensitivity figure. This suggests the existence of an "exchange rate" between gain and sensitivity performance, which is best stuaied by forming a product of gain and sensitivity figure. Using equations4.12 and 4.16, this is:

,h1 7 )2 - x ,21...4.17 G.SFX = -56SAX 4r16 xA)2- xt2H4 In figure 4.4 G.SFx is plotted on X.'-plane. Note that lines of constant are quite far apart in the interior of the stable GoSFX region, but they become very close together, just next to the contour of marginal stability, Si = 1. This means that the product G.SFx tends to remain constant over the most useful part of the stable region, but decreases rapidly to zero at the contour of marginal stability. This fact is further demonstrated in figures 4.5 (a) and (b), which show the values of G.SFX at various sections across the Inverse Gain Chart. In both these figures, the curves remain nearly constant, except at the very edges -- i.e., near the contour of marginal stability -- where they drop precipitously to zero. From figure 4.4, it is obvious that, given a value of X', the value of G.SFx is largest when X' = Xi -- i.e., on Taxis. This may be ascertained by differentiating equation 4.17 (squared, for convenience) 112 with respect to angle 0, where X'cos 0 = X" i.e., R'

2 + x02 2 -ca2(G-.Sipx) — sin 0[1 + X

X [2(1 XA) -'J(2 + 1A)2 - X'2 4.18

Stationary values of G.SFx, therefore, occur when 0 = 0 or 7t i. e. , along XL-axis. From figures 4.4 and 4..5, it is clear that the value of 0 G.SFX is larger when = 0 than when 0 = 7t. By substituting 0 = 0 into equation 4.17, and differentiating with respect to XA, it is found that the maximum value of G.SFx occurs at XA = m, with

(G.S FX) max = 1

4.5