Inductance Simulation for Microelectronics and Transistorized Loy
INDUCTANCE SIMULATION FOR MICROELECTRONICS AND
TRANSISTORIZED LOY-FREQUENCY ACTIVE GIRATORS
by
KENNETH RAOUL MORIN
B.Sc., Queen's University, Kingston, Ontario, 1961
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED SCIENCE r > .
in the Department
of
Electrical Engineering
¥e accept this thesis as conforming to the
required standard
THE UNIVERSITY OF BRITISH COLUMBIA
October, 1963 In presenting this thesis in partial fulfilment of
the requirements for an advanced degree at the University of
British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that per• mission for extensive copying of this, thesis for scholarly purposes may be granted by the Head of my Department or by his representatives,, "it is understood that copying, or publi•
cation of this thesis for financial gain shall not be allowed without my written permission.
Department of Electrical Engineering
The University of British Columbia,. Vancouver 8, Canada.
Date October 25, 1963 ABSTRACT
An inductance can be simulated for microelectronics applications using semiconductor elements (e.g., the "inductance diode"), using circuits containing amplifiers, or using gyrators.
The last two methods are considered in this thesis.
Several "amplifier methods" have appeared in the literature; these methods are classified into integrating- or differentiatiiig-
type circuits, and a differentiating-type circuit is proposed which is believed to be new.
Gyrator realization methods are tabulated and compared.
An "active gyrator" ("AG") is proposed as a circuit element (it has unequal gyration resistances). The AG behaves much like a
gyrator; it can be used to simulate inductance, and an analysis
shows that it can be used to make isolators and circulators with a power gain.
Methods of realizing an AG with amplifiers are investigated,
and an analysis leads to seven 2-amplifier circuits. One of
these AG circuits appears "best" for inductance simulation, and
this one is investigated experimentally using a transistor
circuit.
An extensive bibliography of the inductance simulation and
gyrator literature is presented. ACKNOWLEDGEMENT
The author is indebted to Dr. M.P. Beddoes, the supervising professor of this project, for his help and guidance throughout the course of the project. Thanks are also given to those prof• essors and graduate students who provided helpful discussion, and in particular, to CR. James, for his thorough proof-reading.
Acknowledgement is gratefully g^iven to the National Research
Council of Canada for the assistance received through a Bursary and a Studentship held by the author during his studies.
The work described in this thesis was supported by the Nat-. ional Research Council under Grant BT-68.
ix TABLE OP CONTENTS
ACKNOWLEDGEMENT ix
1. INTRODUCTION ...... c 1
2. INDUCTANCE SIMULATION 5
2.1 Methods for Eliminating, Miniaturizing, and Simulat• ing Inductors . 5
2.2 Classification of the Various "Amplifier Methods" of Inductance Simulation 7
.1 Integrating circuit ...... »•«.<>.. 7
.2 Differentiating circuit 12
2.3 Relationship Between the Various "Amplifier Methods" of Inductance Simulation,and the AG ...... 15
3. GYRATOR AND ACTIVE GYRATOR THEORY AND APPLICATIONS ..... 17
3.1 Survey of Gyrator Literature .«...... <> . 17
3.2 Gyrator Symbols 20
3.3 Definition of an Active Gyrator 24
3.4 The Activity of the Active Gyrator 25
3.5 Applications of the AG 26
.1 Simulation of inductance 26
.2 The AG used as an isolator 27
.3 The AG used as a circulator 29
a) Introduction 29
b) Conditions for circulation , 30
c) Matching 34
d) Power Gains 36 Page
4. CIRCUITS FOB REALIZING ACTIVE GTRATORS 39
4.1 Introduction .*•...... «««•••..« 39
4.2 Effects of the 2 Types of Circuit Components on the
I Matrix . 41
.1 Input conductance ...... 41
.2 Output connections 42
• 3 Negative resistance ...... 43
4.3 Design of a 3-^Amplifier AG 43
4.4 Clue for Designing 2-Amplifier AGs 46 4.5 Classification and Preliminary Screening of 2-Amplifier AGs ...... 48
4.6 Results of the Analyses of the 16 Configurations ... 49
5. EXPERIMENTAL RESULTS 52
5.1 Allowance for the Finite Output Impedance of the Amplifiers 52
5.2 Circuit Diagrams for the Prototype AG 55
5.3 Experimental Results 57
.1 Preliminary adjustment of the AG ...... 57
a) Adjustment for Y-^ & 0 ...... 58
b) Adjustment for Y22 ~ 0 59
.2 Inversion of resistance .....•...«...... •••««•.«•« 60
.3 Simulation of inductance ...... 60
6. CONCLUSION 64
7. BIBLIOGRAPHY ..... 66
7.1 Subject Index to Bibliography ...... 66
7.2 References 68 Page
APPENDIX 81
A.l The 4 Possible output Connections for Each
Configuration 81
k»2 A Systematic Method of Analysis ...... 82
.1 The first step 84
.2 The second step ...... 87 LIST OF ILLUSTRATIONS
Figure Page
1.1 - Definition of an ideal gyrator ...... «... • 1
2.1 - (a) The simple R-C integrator; (b) the operational amplifier integrator 8
2.2 - '•. Simulated inductance using a simple R-C integrating circuit and a pentode tube (grid-leak resistor omitted) .•.«.•.*...... •«.•.«•»• 8
2.3 - Stern's circuit for a simulated inductance .««•««••. 9
2.4 - Basic circuit used by Holbrook and McKeown. (a)
actual circuit; (b) approximate equivalent circuit 10
2.5 - Midgley and Stewart's simulated inductance circuit 12
2.6 - (a) The simple R-C differentiator, and (b) the operational amplifier differentiator, shown with a small resistor R^ connected (to produce v^ ' from "the current x ^) +oeoo<>eoo**w*0*Qoooo»ot>***»+4ri*
2.8 - An inductance simulation circuit which uses the operational amplifier differentiator of Figure 2.6(b) 14
2.9 - Classification of simulated inductances which use amplifier me"fcho.cLs +ooco*e*ioooa9ot>» + Qo«o9904i** + «<> + 3»1 —• Representative cross section of a field effect tetrode ^ ^ ^ ^ ^ « ^ ^0 « ^ a m o • ^ o ^ o o c« c « o o 0 a « « « » » 0 • * ^ ^ ^ ^ ^ ^ ^ ^ 20 3.2 - Comparison of gyrator realization methods .«...„..«« 21 3.3 - Tellegen's symbol for the ideal gyrator. Sometimes one of the semicircles is omitted (e.g., st57c) «... 22 3.4 - Gyrator symbol proposed by Feldkeller ...... 22 3.5 - Shekel's symbol for the 3-terminal gyrator •»«.«.«.. 23 3.6 - Hogan's symbol for the (microwave) gyrator ...... 24 3.7 — Circuit symbol and equations for the AG ...... 25 3.8 - The AG as an isolator: (a) parallel connected^ and (b) series vi Figure Page 3.9 - Equivalent circuit for the isolator of Figure 3.8(a) 3.10 - A 3-port network made from a 3-terminal device ..... 29 3«11 - The two basic circulator configurations: (a) voltage sources, and (b) current sources ..... » ...... •• «.« 31 3012 •=- The first step in deriving the matrix defined in 3.13 - (a) and (b) Input admittances to the circulator shown in Figure 3.11(b); (c) and (d) values of input admittance obtained when an AG is used. The direction of circulation is indicated by the arrow.. 35 3.14 - An application of a circulator. The circle represents a 3-port circulator ...... <» o...... 37 3.15 - The product Gm "GT ^, as a function of G_/G_ ... 38 ia->c c—>b n m 4.1 - The two components which will be used to build AGss (a) conductance, and (b) ideali'zed voltage amplifier, which has A realj positive or negative 39 4 = 2 •=- Skeleton of the. AG- starting point for each AG design 40 4.3 =• Successive steps taken in the analysis of AG 4.4 - Illustrating the effect of output connections ...... 42 4.5 - Negative resistances using voltage amplifiers .«»«•« 43 4.6 - The start of D 4 o 7 3 ™*CHT1J) X i f X G I* AG!" • ^aa«««-«0*«*»*0e*a«0a««0o4>aA0 4»«a«^i0*« 43 4.8 - Obtaining the voltages +V^+V"2 from the port voltages 46 4.9 => Changing the skeleton of an AG circuit ...... »oo.«. 47 4.10 - Summary of 2-amplifier AG circuits ...... a... 50 5.1 = The output connections for an amplifier with finite output conductance, G : (a) actual circuit; (b) approximate -equivalent when G^rvG^«Go ...... 52 5.2 - Design sheet for configuration number 8, using the approximation given in Figure 5.1(b) ...... 54 5.3 - Circuit diagram of the Q amplifier ...... • 55 vii Figure Page 5.4 - Circuit diagram of the P. amplifier ...... »«•».. 56 5.5 — Schematic diagram of the AG prototype ...... 56 5.6 - Artwork for the printed circuit board (actual size) 57 5.7 - The AG used to invert an impedance Z-^: experimental apparatus 58 5.8 - Experimental results, inversion of resistance ...... 61 5.9 - Experimental results, simulation of inductance; (a) low frequency measurements, Z- ; (b) high frequency measurements, 62 5.10 - Approximate equivalent circuit for the simulated inductance (losses are neglected) 63 A.l - Design sheet used for the investigation of AG circuits 83 A.2 - Design sheet for configuration number 8, D-l- ...... 85 A.3 - Equivalent circuit for finding-1;the "intrinsic terms" • for the configuration D-l- 86 A.4 - Equivalent circuits for finding some of the "amplifier terms" for the configuration D-1-. T matrices are given ...... 87 A.5 - Design sheet for configuration number 5, D+1+ ...... 90 A.6 - Design sheet for configuration number 6, D+l- ...... 91 A.7 - Design sheet for configuration number 11, D-2+ ..... 92 viii 1. INTRODUCTION There is currently much interest in the making of tiny electronic assemblies. Both resistors and capacitors have quite successfully been miniaturized; however, inductors have not been miniaturized so successfully. Initially, an attempt was made to develop miniature inductances by circuit techniques; but as the work progressed, emphasis shifted to the synthesizing of^ gyrator circuits. Techniques have been described which eliminate the need for inductors by redesign of circuits. Other approaches deal with making miniature inductors* for instance by winding fine wire on a tiny ferrite core. Still other approaches involve the simulation of inductance. One approach for simulating an inductance uses a gyrator — a 4-^terminal, antireciprocal device which obeys the equations shown in Figure 1.1. One property of the gyrator is that of impedance 0 -R h 0 R 1 LV2. - 2 Figure 1*1 Definition of an Ideal Gyrator, inversion. If an impedance ZT is connected from c to d in Figure Li 1.1, then V_ = ™I_.ZT , and the input impedance between a and b is seen to be -HI 2 ..(1*1) in ,11 zT In particular, if = j^jj"» "then Zin = ja> CR , and a simulated 2 inductance of value L - CR results. However, gyrators are commercially available only for microwave frequencies* although several experimental low-frequency gyrators have been made, for instance by using the Hall effect, or coupled electromechanical transducers. In view of the current interest in solid—state techniques, a gyrator made using trans• istors would be of value. One achievement reported in this thesis is the design of a gyrator-like device (made with transistors) which is suitable for inductance simulation. The impedance matrix of this device is practically identical with that shown in Figure 1*1, except that the R's are unequal; i.e., it obeys the equation 0 -R n (1.2) R 0 L m This equation reduces to that of a gyrator if = = R. This device inverts an impedance ZT into R R /Z,, and has other * L m n L? * properties similar to a. gyrator; however, it is an active device,' and on this account it is called an "active gyrator", or "AG" for * This is shown in Section.3*4. ,<,. sho r t. . 3 Although many papers on inductance simulation and gyrators have been published, as yet no comprehensive bibliography has appeared for either field. It is hoped that the extensive biblio• graphy contained in this thesis will fill this need. Chapter 2, on inductance simulation, contains a survey of the literature in this field. Attention is focussed on those methods of inductance simulation which use voltage amplifiers, and these methods are classified according to their basic principle of operation. Further, it is shown that they all behave like AGs. Chapter 3, on gyrators and active gyrators, begins with a survey of the gyrator literature; the various gyrator symbols are. then given. The rest of the chapter deals with the circuit properties of active gyrators: the AG is defined, and then some of its properties and applications are given. In Chapter 4, on circuits for realizing AGs, a method is developed for obtaining AG circuits which contain only voltage amplifiers and resistors. It is shown that a 3-amplifier AG can be designed intuitively. The main problem is to design a 2- amplifier AG, and a rather lengthy analysis produces 7 such designs. Of these 7 designs, 4 have been published previously, and 3 are believed to be new. In fact, one of the 3 new circuits had been designed in the early stages of this project, and was an incentive for carrying out the present analysis. This particular circuit also appears to be the best of the 7 for inductance simulation. Chapter 5 contains experimental results. In order to test the validity of the theoretical design, a prototype model was made of the AG chosen as most suitable for inductance simulation. The sample measurements made on this model confirmed the validity of 4 the design. Chapter 6 contains the conclusion, and Gjiapter 7 contains the bibliography. This bibliography is a major part of the thesis, and consequently it is included in the main body of the thesis. It is indexed according to subject for handy reference . • 5 2, INDUCTANCE SIMULATION The first section of this chapter is a survey of various proposals for eliminating, miniaturizing, and simulating inductors. Then certain of these simulation methods are considered in more detail, and it is shown that they are all related to one another, and also to the AG. 2.1 Methods for Eliminating, Miniaturizing, and Simulating Inductors. Some approaches to the problems of avoiding the use of inductors, or alternatively to miniaturize or simulate them, are considered in the general works on microelectronics (e.g. du6l, ho58, ho62., as listed in the bibliography, Chapter 7). In some cases, it may be possible to avoid the use of inductors altogether, for example by substituting R-C circuits in place of inductive circuits (co58), or by using ceramic transformers, etc. (e!58, lu58, ma61a, ma61b). For power supply smoothing coil applications, it is not necessary actually to have an inductor, for any device with a low d.c. impedance and a high a.c. impedance can be used} a current limiter fills these specifications nicely (du61, pp. 47-48, la62, wa59). In other cases, it is possible to alter conventional techniques and produce tiny inductors. The most straight-forward approach involves the development of special tiny ferrite cores, on which very fine wire is wound (du6l, pp. 201-03, and 248-52, st57a, st57b? ). Inductances of the order of a millihenry, and * Some of the references given in the bibliography were un• available to the author, so their classification is uncertain. 6 small enough to mount on a tiny "micromodule11 substrate have been produced by this method* For very small values of inductance (of the order of a microhenry) printed circuit techniques can be used to print a spiral, two-dimensional inductor (br55). Using thia- film techniques, spiral inductors with 1 mil line-width can be fabricated on a ferrite substrate, yielding inductances of several hundred microhenries (du61, p* 202, to62)i Finally, there are several methods for actually simulating an inductance* An AG and a capacitor can be used, as was discussed in Chapter 1, and thus any method of constructing an AG is also a method of simulating an inductance. (Several circuits for realizing the AG are considered in the next chapter). Semiconductor elements have been used for inductance simulation; these elements use one or several solid-state effects to produce inductive behaviour* Examples are the "inductance diode"* and the inductive transistor* Dill (di6l) gives a com• prehensive survey of these methods. He concludes that the / inductive transistor, with which it "can be possible to get inductances in the henry region with reasonably high Q", is "vpry promising", although it is plagued with temperature and stability problems* and requires a high power dissipation. Refer to the bibliography, page 66 f for further references* There are several methods of inductance simulation which might collectively be called "amplifier methods" (fu60, ho—y mc63j, mi60, st58, to53* wa47), since they all employ some sort of amplifying means. These circuits all contain a capacitor as a major component. These circuits are considered in much more detail in the rest of this chapter, where it is shown that they can all be derived from either an integrating or a differentiating circuit. In addition, it is shown that they all behave as AG circuits (terminated in the capacitor mentioned above), albeit only under the same conditions that are necessary for inductance simulation. 2.2 Classification of the Various "Amplifier Methods" of Inductance Simulation. The various amplifier methods of inductance simulation all employ the same basic principle: an amplifier is used in con• junction with the -90° phase angle of a capacitive impedance to produce the +90° phase angle of an inductor. The capacitor appears in either an integrating or a differentiating circuit. 2.2.1 Integrating circuit. The circuit equation for an inductor may be written • • •(2.1 ) Given an empty, two-terminal, black box with the voltage v^ applied between the terminals, what can be put inside the box in order that the input current be given by (.2.1)? If an integrating circuit is connected to the two terminals, one can obtain a voltage 8 If it now can be arranged for the input current to be proportional to this v , then the black box will behave like an inductance, o Common ways of integrating are with either a simple R-C •if integrator or an operational amplifier integrator, as in Figure 2.1. 4 Ri»fc — o—w 1 R 1 \L 2 v C (a) (b) Figure 2.1 - (a) The simple R-C integrator} (b) the oper- ational amplifier integrator. For the simple integrator of Figure 2.1(a) some sort of amplifying device will be needed. A pentode tube may be used (hu42, wa47), connected as shown in Figure 2.2. Bl O- A/y 11 a + A t c ap 1 b h O Figure 2.2 - Simulated inductance using a simple R-C integrating circuit and a pentode tube (grid-leak resistor omitted). * A simple L-R integrator might be used as well, but this would be defeating the purpose. If the integrating circuit draws negligible current, and the blocking capacitor C, is large, then the input impedance to the R1C circuit is approximately that of an inductance, L = » ^m Fulenwider (fu60) gives an improved version of this circuit* which contains 3 pentodes and 4 triodes. Instead of using a simple R-C integrating circuit, however, he uses a triode isolation amplifier and a pentode loaded with a capacitor; the other extra tubes cancel the remaining losses, to produce a Q limited only by stability considerations. The particular circuit which he gives has "a frequency range of 60 to 2000 cps, with an equivalent magni• tude range of 28 to 2800 henries." Stern (st58) completes the circuit of Figure 2.1(a) with a transistor. The base of the transistor is connected to term• inal 2, and the emitter is connected to terminal 3 through a relatively high resistance (so as to not load the capacitor C too heavily), thereby forcing the base current to be proportional to Vq. Finally, the collector is connected to terminal 1 (see Figure 2.3) so that, if the integrating circuit draws negligible current, then the input current is the collector current, ^ pi - dt, -O O— + a (1) Rl -vW (2) R, ' V =: V cap b o .2 O- -O O—1 (3) Figure 2.3 - Stern *s.circuit for a simulated inductance. 10 and a simulated inductance results. In practice, a single trans• istor seems insufficient to provide a reasonably high Q. A one- transistor circuit was assembled in this lab, and produced a Q of only about 2. Dill (di6l) reports achieving inductances as high as 10 henries with this circuit, but always with a Q less than 10. An interesting realization of the integrating circuit of Figure 2.1(a) has been studied from a practical point of v.iew . by Holbrook and McKeown (ho—> mc63). The basic circuit which they use (suggested by Bogert (bo55)) is shown in Figure 2.4. This circuit consists of a tube (or common-emitter transistor) with a capacitive load (C^), and a resistive feedback path (R^)* The R-C integrator is formed by r and C^, as shown in Figure 2.4(b). If I^yfl-^ (this requires g^p^l/R^t then the input impedance to this circuit is given by R (1+jtfCr ) Z. ^ 2_ , in (u+l)+j«Cr P R(bias) R, p <- •AAr •AAr- + :•. l V. in + (a). •(b)- ^ Figure 2.4 - Basic circuit used by Holbrook and McKeown: (a) actual circuit; (b) approximate equivalent circuit. and if 1 « then Z. ^jttj C ^ ). \ gm/ It is seen that the input signal is amplified before it is integrated; this arrangement might, result in.a low noise level. Against this, the amplifier must be capable .of handling, a large, signal. The single-stage triode and transistor circuits studied by Holbrook and McKeown exhibited a maximum'Q of less than 2, while a pentode circuit had a maximum Q of about 5. Two and 3-trans- istor circuits had maximum Q's of about 5 and 8 respectively, while a 3-transistor circuit, employing 2 of the transistors to produce a negative resistance, is reported to produce "high values of Q factor, up to the point of instability and oscillation." Most of the measurements were taken at 1000 c.p.s.1, with values of C^ ranging from 0.01 to 1.0 |xf.; the resulting inductances were of the order of 1 henry. For the operational amplifier integrator of Figure 2.1(b), one need merely connect a resistance from the output terminal 2 to the.input terminal 1, as shown in Figure 2.5. The input impedance will be .predominantly- inductive, if (i ): the ; integrating j . , : circuit draws negligible current- •(•I^vV^ ^'" i'.^)V''&rid-:(dl.)vy^<^:y^i;.* This circuit was used by Midgley and Stewart (mi6.0), who. * simulated inductances of the order of 100 henries, at •power ••''.' frequencies. Even if the operational amplifier has an infinite 12 Figure 2*5 - Midgley and Stewart's simulated inductance circuit. gain, the simulated inductance is. shunted by. a resistance 5 a second operational amplifier can be used to' cancel:this, resistance^ How• ever, if the gain of the first operational amplifier is finite, a series resistance also appears; one can compensate for this by using positive feedback within.the operational amplifier:to increase its gain. '. •>.• •.•'• 2.2.2 Differentiating circuit. , ' , \ The circuit equation for an inductor may be written di-j-'' , V-j^ = Ii —4 .0 * ( 2. 2 ) Given an empty, two-terminal,, black box with the current i-^ injected into one terminal and out the other, what can be put inside the box in order that the input voltage be given by(2i2)? If i-^ flows through a small series resistor connected 13 inside the black box, a voltage v ' = R^i-j^ will- appear across R^. A differentiating circuit can then be used to develop a voltage dv, ' di v = K —^- = KR, — 0 dt L dt If now K'Vq (where K'vJ^j^v^' ) is made to appear in series with R^ across the input terminals of the box, the terminal impedance will be inductive. Common ways of differentiating use either a simple R=C differentiator or an operational amplifier differentiator, (Figure 2.6). It is assumed that the differentiating circuits Figure 2.6 - (a) The simple R-C differentiatorj and (b) the operational amplifier differentiator, shown with a small resistor R^ connected (to produce v^' from the current i-j ). provide little loading on R^. / The simple R-C differentiator was used by Towner (to63) for his artificial inductor.' He solved the problem of connecting Vq (amplified) in series with R^, by using what Millman (mi58) calls a "bootstrap" circuit, as shown in Figure 2.7. The two amplifiers shown diagrammatically, and the R-C differentiator V between them, produce the amplified Vq voltage ( JA-J | -^-2! o^' Figure 2.7 - Diagram of Towner's artificial inductor. which is the input to the bootstrap (triode) circuit. The out• put of the bootstrap circuit appears acrossH^, and is thus in series with R^, as required. Towner has "easily obtained* an inductance of 10,000 henries with such a circuit, and although the frequency range was limited to below 500 cps by the particular components used, he says the upper limit can be raised indefinitely. If a high—Q inductance is desired, a resistor can be used to produce positive feedback from the output of the second amplifier to the input of the first. No published circuit based upon the operational amplifier differentiator of Figure 2«6(b) was.found, but the circuit shown in Figure 2.8 might be used. An - ^(CRlV- Figure 2.8 - An inductance simulation circuit which uses the operational amplifier differentiator of Figure 2.6(b). 15 The results of this section are summarized in a table, Figure 2.9. Classif• Integrating circuit Differentiating circuit ication Operational Operational Simple R-C Simple R-C Amplifier Amplifier -Hund, Figure 2,2, -Midgley and Proposed (hu42, wa47). -Towner, -Morin, -Fulenwider, Stewart, Figure 2.7, Figure 2.8, (fu60). Figure 2.5, (to53). (this by -Stern, (mi60). thesis). Figure 2.3, (st58). Figure 2.9 - Classification of simulated inductances which use amplifier methods. 2.3 Relationship Between the Various "Amplifier Methods" of Inductance Simulation, and the AG. All the circuits discussed so far contain a capacitor (the capacitor used in the integrating or differentiating circuit) plus some "external circuitry". Since a capacitor plus this external circuitry produces anc.inductance, and a capacitor.plus an AG produces an inductance, it may well be asked if the external circuitry is not in fact an AG. It is easily shown that the answer to this question is affirmative, but with some reser• vations, because' the conditions which are necessary to realize an inductance are also necessary in order that the "external circuitry" behaves like an AG. For example, consider the circuit of Figure 2.3. If this circuit is thought of as a 4-terminal device (terminals a,b,c,d) terminated at the output side (terminals c,d) by the capacitor C, then the current i through the capacitor is just the C Etp current i0 at the output port, , 16 i~ = i ^ - D~" (because. v1»v0). ...(2.3) z cap it^ JL Also , i, ^ i (because i-,» i~) 1 c l ^ 2 ...(2.4) Equations (2.3) and (2.4) are, taken together, in the form of equation (1.2). Stern's circuit, minus his capacitor, therefore acts like an AG. Similar results can be obtained for the other inductance simulation circuits. In all cases, the conditions for the "external circuitry" to behave like an AG are the same as the con• ditions for obtaining an inductance. 17 3. GYRATOR AND ACTIVE GYRATOR THEORY AND APPLICATIONS The first section is a survey of the gyrator literature. In the 2nd section, the gyrator symbols in common use are given. Then, the more general "active gyrator" (AG) is defined, and some of its properties are shown. Finally, several applications Of the AG are given - inductance simulation, use as an isolator, and use as a circulator. 3.1 Survey of Gyrator Literature. In 1948, Tellegen (te48a) proposed a new, anti-reciprocal network element, which he called the "gyrator". The gyrator is an idealized element, which has the impedance matrix 0 -R R 0 as was shown in Figure 1*1. The gyrator produces zero phase shift for transmission in the forward direction, and a 180° phase shift for transmission in the reverse direction. Among other things, it can be used to simulate inductance, and to make isolators and circulators. Two years before, McMillan had published a paper (mc46) in which he described a 4-terminal "anti-reciprocal box". This device was really a non-ideal gyrator, for it differed from the gyrator only in that the diagonal elements in its impedance matrix were not zero. 18 The problem of realizing ajj'gyrator has received consider• able attention, since, unlike the resistor, capacitor, inductor, and transformer, the gyrator is not something which can easily be made in the workshop. Concurrently, much theoretical work has been done, and much has been written on network synthesis using gyrators. The problem of realization has been approached in many different ways: electromechanical, Hall effect, ferrite, tube, transistor, field-effect, and parametric devices have all been used. McMillan's "anti-reciprocal box" (mc46), consisting of 2 coupled electromechanical transducers, is an approximate realiz• ation of a gyrator, although the two self-impedance terms are not zero. It has been represented (b!53) by an equivalent circuit which contains a gyrator. Several papers have been published in the last few years on piezoelectric-piezomagrietic gyrators (7 references ). These coupled transducers have also been used to make isolators (9 references) and circulators (si62b). Another electromechanical device, the generalized machine, has also been used as a gyrator (pe 58) . McMillan (mc47) suggested using the Hall effect to obtain an "anti-reciprocal box" (a non-ideal gyrator), and several papers have since appeared on Hall effect gyrators, isolators, and circulators (9 references). Gyrators have been realized most successfully for the micro• wave range of frequencies. A microwave gyrator consists of a piece of ferrite material inserted inside a waveguide in a biasing These references are listed under the appropriate heading in the bibliography, chapter 7. 19 magnetic field. The first successful microwave gyrator was reported by Hogan (ho52), who has subsequently given an excellent summary of microwave gyrators, etc. (ho56). Many other papers on microwave gyrators have been published, some of which are included in the bibliography (about 20 references). The special ferrites issue of the Proceedings of the IRE, October, 1956, contains many references* Comparatively few papers have been written oh tube and transistor gyrators. Shekel (sh53) has proposed a 3-terminal gyrator (the 4th terminal is eliminated by connecting b and d together in Figure 1.1,, page 1 ), and shown that any non-re• ciprocal 3-terminal element, such as a vacuum tube or transistor, may be represented by a 3-terminal gyrator in parallel with a triangle of admittances; these admittances may be "stripped" by admittances of the opposite sign to leave the gyrator (this process will involve at least one negative resistance if the non- reciprocal element is active •( sh'54) )". As an example of this method, Shekel shows how a gyrator can be built using a vacuum ..' tube and 3 conductances, 2 of which are negative. Shekel's cir• cuit (sh53, Fig. 4) is essentially the same as one given by Bogert (bo55, Fig. 3(d)), who uses an amplifier to provide the two negative conductances, although the equivalence is not at all obvious. In all, Bogert gives 4 gyrator circuits, each of which contains 2 low-gain amplifiers. Nonnenmacher (no54) has proposed a circuit containing 2 low-gain amplifiers and 2 transformers, which can be used, among other things, to make a gyrator. de Pian (pi62) mentions Shekel's method of synthesizing a gyrator with a vacuum tube. 20 A field effect tetrode is a 4-terminal semiconductor device (et62, st59a, st59b, st6l), with the representative cross-section shown in Figure 3.1. Under proper biasing conditions, it behaves a, b, c, and d are ohmic contacts. Figure 3.1 - Representative cross section of a field effect tetrode. like a gyrator. Prototype field-effect tetrodes have been made with difficulty, but it is hoped that their fabrication will be easier using epitaxial growth techniques (et62). Parametric devices have been described (ka60, ko6l) which can be adjusted to perform as gyrators. The foregoing methods of realizing gyrators are compared in the table of Figure 3.2. This table is intettded only as a rough guide. 3.2 Gyrator Symbols. In this section, the "ideal gyrator" as proposed by Tel- legen is considered. Its equations may be written: 0 -R ... (3.1a1) V R 0 I 21 Me^^F eature ^Frequency How Close Accessory Other Features thod o*f\ ^ j Range to Ideal? Equipment Realiz ation>J Required Figure 3.2 - Comparison of gyrator realization methods. 22 or ^1 V " 0 G ..•(3.1b) X _-G 0_ _2. -V2. .where R = l/G* . a. (3.1c) R is called the "gyration resistance". The. symbol proposed by Tellegen for the ideal gyrator is given in Figure 3.3. + A O- IV a + b ) c d O- -O Figure 3.3 Tellegents symbol for the ideal gyrator. Sometimes one of the semicircles is omitted (e st57c). An alternate symbol, which clearly shows the anti- reciprocal nature of the gyrator, has been proposed by Feld- keller (fe54). This symbol is shown in Figure 3.4. Signals a O- -5K b O -j f— O d Figure 3.4 - Gyrator symbol proposed by Feldkeller. 23 travelling from right to left suffer a phase inversion, while those travelling in the opposite direction do not. In practice, gyrators are often built and/or used with ter minals b and d common (Figure 3.3), that is, as 3-terminal dev• ices. For such a 3-terminal gyrator, the indefinite admittance matrix can be written (li6l) from (3.1b): 0 G -G* -G 0 G . *>« . (3.2) indefinite G -G 0 Depending upon which terminal is grounded, crossing out of the corresponding row and column yields the usual 2X2 admittance matrix. No matter which terminal is grounded, the circuit has the same matrix (with a possible change of sign, corresponding to an interchange of input and output terminals), as pointed out by Shekel (sh53). In view of this property, Shekel proposed the symbol shown in Figure 3*5 for the 3-terminal gyrator. The Figure 3.5 - Shekel's symbol for the 3-terminal gyrator. direction of the arrow indicates that each Y..^ term is either +G or -G, according to whether j—>k is in the same or opposite direction as the arrow respectively (j,k, = 1,2, or 3, and j ^ k). 24 A fourth symbol was proposed by Hogan (ho52) for his micro• wave gyrator. This symbol, shown in Figure 3.6, emphasizes the • % Figure 3.6 - Hogan's symbol for the (microwave) gyrator. 180° phase shift in one direction; the interpretation is that the element produces a 180° phase shift for transmission in the direction of the arrow, and a zero phase shift in the opposite direction. 3.3 Definition of an Active Gyrator. A "generalized gyrator" has been considered by Nonnenmacher (no54), who replaced the gyration resistances R by arbitrary complex impedances, Zffl and Z .. The case where these Imgedances are pure resistances, Rm and Rn, is of particular interest for inductance simulation: v "o -R ~ " l~ n . . . (3.3) R 0 I 2 . 2. J _ m Because of its active property (see section 3.4), this device will be called an "active gyrator", or "AG" for short. Setting G = l/R , and G = l/R , ...(3.4) m • m' n n' 25 the Y-matrix equation for the AG can be written, •..(3.5) Feldkeller's gyrator symbol (Figure 3.4) will be used for the AG, since it gives more information than Tellegen's symbol, and since Shekel's symbol is clearly unsuited to the AG. See Figure 3.7. Figure 3.7 - Circuit symbol and equations for the AG. 3.4 The Activity of the Active Gyrator. With regard to such things as phase inversion, interchanging ports, cascading 2 units, land impedance inversion, the AG has properties closely resembling those of the gyrator. Consequently, these topics are not considered further. In this section, the activity of the AG is considered. 7* This fact is easily seen by comparing the indefinite Y-matrix for the AG with that for the gyrator (equation (3.2)). 26 The following conditions are used for the passivity of a linear 2-terminal-pair network (see, for example, mc46 or bo57): Rll> 0 R22^" 0 . . . (3 . 6a,b , 4R11B22 r- (Z12+Z2|)(Z1* + Z21) > 0, where the impedances refer to the Z matrix. Comparing (3.6) with (3.3) , it is seen that the conditions of (3.6a) and(3.6b) are satisfied with the equal sign, and the condition of (3.6c) becomes -(-R + R )2>- 0 . n m ^ Since R ^ R , the AG is not passive; i.e. it is active. 3.5 Applications of the AG. Three applications are considered in this sections 1) simulation of inductance, 2) use as an isolator, and 3) . use as a circulator. 3.5.1 Simulation of inductance. If an AG is terminated by a capacitance, Z^ = l/(jwC), then the input impedance (see Chapter l) is given by Z. = R R /ZT = 0 j«(R R C), m m n' L m n ' ' ; 27 which represents a simulated inductance, L = R.R C« «•.(3»7) m- n This result is of. much practical interest in microelectronics, for if a tiny AG can be built, then a tiny inductance can be obtained by merely adding a capacitor. In addition to being very small, such a simulated inductor woul shielding problems would be minimized. 3.5.2 The AG used as an isolator. It is shown here that the AG can be uarad to make an isolator, and this isolator can have a power gain greater than unityJ if G > G . n ' m The element in the Y matrix of an ideal isolator must be zero (to provide isolation in the reverse direction), while the Y21 element must be non-zero (to provide transmission in the forward direction). The Y^ an^ ^22 e-'-emen"'>s are then the input admittances at the two ports, since Tinl = Yll " T12Y21//^22 + TL^ = Y^^ since Y^ — 0» Y nus and similarly ^-^n2 ~ 22' ^ these elements should provide a match with the input and output lines. Two ways of making an isolator from an AG and a resistor are shown in Figure 3.8. 28 „,„ "X . —-—_ • 0 -R ' n . R m 0-J R I (b) R - R G (G - G) V, R V n m (G+G ) G V R + R R n L 2J m Figure 3.8 (b) series connected. In FigurFi pree 3.8(a), the Y.^ element will be zero if Gm= G . The m resultant Y matrix is 0 Y =• -(G + G ) G : n For a perfect match, G, should equal the characteristic admit- " * tance of the line in which the isolator is inserted. The equivalent circuit of this isolator is shown in Figure 3.9, and the power gain (matched) is (G + G )' out ...(3.8) P. in 4G2 This power gain will be greater than unity if G> i.e., if G > G . n ^ m It is assumed that the input and output lines have the same characteristic admittance. 29 1 r- I O r G. G .= G m ,2 „ N2(G + GJ' |V/G P , = VJ" G = in out I 2' 4G Figure 3.9 - Equivalent circuit for the isolator of Figure 3..8(a). For the configuration of Figure 3.8(b), R = R for perfect n isolation, and the (matched) power gain is P , (R + R ) out m' ...(3.9) P. 4R£ m 3.5.3 The AG used as a circulator. It is shown here that an AG can be used as a circulator if the1 terminating impedances are properly chosen. Such a circulator exhibits a power gain, which varies with the ratio GM/G . a ) Introduction A 3-port network can be made from a 3-terminal device, as \ shown in Figure 3.10. If the 3-terminal device and th6 termin• ations at each port are suitably chosen, such a network may 30 Port b Port a Port c Figure 3c10 - A 3-port network made from a 3-terminal device. behave like a circulator (si62b). Such a circulator would be useful, for instance, for simultaneously sending and receiving audio-frequency signals over a single pair of wires, or for replacing the hybrid coils in two-way, two-wire amplifiers. b) Conditions for circulation The ports shown in Figure 3..10 may be driven by either voltage or current sources, as shown in Figure 3.11. For each case, a given direction of circulation is achieved if certain elements of the associated 3x3 matrix (defined separately for each figure) are set to zero. These matrix elements can be made zero by suitable choice of the terminating (source) immitbances. These immittances are of course independent of whether voltage or current sources are used, so the analysis need be carried out for only one of Figures 3.11(a) and (b). Since the associated matrix of Figure 3.11(b) can be derived more simply, this configuration will be analysed. 3.1 Y " \ 12 / .A/ a' -T21 22j V a C/C (a) 1** Y , Y V V " Z Z " I a aa ab £ ^ab a aa ac a Y Y xY V Z,, Z X ba bb l ^ba bc b V b bb Y Y . Y V V Z Z I L cj cb c cb J- ca L cJ - ca L cj Figure 3.11 - The two basic circulator configurations: (a) volt• age sources, and (b) current sources. First, the conditions for circulation will be stated. For circulation in the direction a-H>b-^-c, the associated matrix of Figure 3.11(b) must have the form " V ~ Z 0 a aa ac a V = J 0 ...(3.10) b Jba bb V 0 Z L c _ cb CC-! For circulation in the opposite direction, c •a, the associated matrix must have the form V Z I ab a aa 0 a V 0 Z.. .. . (3.11) b be h bb V Z 0 Z I _ c _ ca cc J c The associated matrix defined in Figure 3.1l(b) will now be derived, in terms of the Y parameters defined in the same figure. The circuit of Figure 3.11(b) can be formed by connecting two 32 circuits in parallel, as shown in Figure 3.12. The resultant in parallel .j with Y, , +Y +T. Y, 0-T. 11 •12 a b b 11 a b 12b -Y, Y, +Y Y Y Y +Y +Y •21 '22. b b c_ L 21- b 22 b c Figure 3.12 - The first step in deriving the matrix defined in Figure 3.11(b). Y matrix can be inverted. Y00 + Y, + Y Y - Y 22 b c xb x12 ...(3.12) AY Y - Y Y,, + Y + Y, 11 a b b 21 where AY = (Y^ + J& + Xb) (T22 .+ Yfc + Yc)-(Y21 - tb)(T12 - Tb). Hence the indefinite Z matrix (li6l) is easily written, Y +T +T (Y +Y +Y "va 22 b c - 12 22 c) a 1 vV -(Y21+Y22+Yc) Yn+Y22+Y12+Y21+Ya+Yc T +I +I b ~ AY < 11 12 a> V x . c. 21 b ...(3.13) 33 and this is in fact the matrix defined in Figure 3.11(b). Now, comparing (3.10) with (3.13), the conditions for circulation in the direction a—»b—^c are YL - Y Conditions for circulation b ~ x21 in direction a—*>b—>c. Yc = - Comparing (3.11) with (3.13)» the conditions for circulation in the direction c —>b—>a are + Ta = "(Til V Y - Y Conditions for circulation Xb _ x12 in direction c—*b —»a. Yc = " ..(3.15) For the AG, T-, = Y00 =0, Y,0 = G , and T_N •= -G ; there- 11-22 12 m 21 n fore, (3.14) and (3.15) become -G a m Y, =. -G Conditions for circulation b n in direction a—>b—^c with AG, Y = -G , c m' ...(3.14a) 34 and Y = G a n Y. = G b m Conditions for circulation in direction c—»>b —^a with AG. c n ...(3.15a) For circulation in the direction a-^b—>c with the AG, negative source immittances are required. This result might be useful when it is desired to use a circulator in conjunction with an amplifier having a negative input and/or output impedance. For . circulation in the direction c—s»b—*-a, the required source immittances are all positive. Suppose such a circulator is operating with the.source immittances given by (3.15a).. Are the 3 ports simultaneously matched? Can a net power gain be achieved? c) Matching The self terms in (3.13) are the open-circuit input impedances (including the source immittances) at the respective ports in Figure 3.11(b);. Therefore, the admittance seen by each source is given by the reciprocal of the respective self term minus the source admittance. This result is summarized in Figures 3.13(a) and (b), in •which the source admittances have been chosen according to (3.14) and (3.15), thereby producing ; 1 circulation in the directions af»b-»c and c->-b —:>a respectively. The results obtained when an AG is used are given in (c) and (d) of the same figure; note that, in this c 35 Figure 3.13 - (a) aad (b) Input admittances to the circulator shown in Figure 3*11 (b); (c) and (d») values of input adm• ittance obtained when an AG is used. The direction of circul• ation is indicated by the arrow. match cannot be obtained at any port unless Gffl - Gn (i.e., the AG becomes a gyrator), in which case all 3 ports are matched simultaneously. ' ... • In general, however, the 3 ports are all mismatched. It might be argued that reflections caused by these mismatches would destroy the circulation properties; however, unless the sources are connected by very long lines, or by delay lines, reflections should be negligible at the low frequencies (long wavelengths) of interest. Of course* matching transformers cannot be u3ed^ for then the effective terminating immittances would no longer 36 satisfy the conditions for circulation. One effect of the mismatches would be to cause the power transfers to be less than maximum; the question of power gain is considered next. d) Power gains For circulation in the direction c—*»b—> a using an AG, (3.13) becomes -1 o a a 0 -l ... (3.13a) G + G m n L-i 0 1 . I . Hence, if a current I is applied at port a (while I, = I -6), a the power delivered at port c is given by 2 P _ G . ...(3.16a) c y U m+ \ tn / n Similarly, driving ports c and b, respectively, -I, G™> and Po ...(3.16b,c G + 0 1I m' a G + G n* m n/ m n Now, the powers available from each of the sources (P ) av are [ 2 I 2 bl 1 ; P c B/V 4G .•av. 4G av 4G a n m n ..(3.17) 37 Therefore, the transducer power gains (G^) in the direction of circulation are 4G G m n Gm = G Tc->b Tb_>a (G + G )2 m n' . . . (3.18) 4G 2. n G (G + G )' a—^c m n Note that GT = GT — 1, and c —^ b b —a ...(3.19) > Gn > Grp = 1 as Q = 1. a—>-c *^ m ^ Thus.; a power gain can be realized from port a to port c if G"^Gm« How much of this power gain can be utilized? A typical application of the circulator is shown diagrammatically in Figure 3.14. A measure of the power gain produced by the load AAAAA Port b Port a Signato l in / ~ZT , \ , • . •. - • Port c. transmission line Figure 3.14 - An application of a circulator., The circle represents a 3-port circulator. circulator is the product G^ . G^ . This product is : a — > c c --s»b 38 given by 16G G ' b c m n 'T .* "G-Tn . ~ P P a—>-c c—>b av av (G + G )' a c m n' ....(.3.20) This function varies with G /.G^. as. shown in Figure 3.15. The function is greater than 1 for 1 «C Gn/Gm<^11.5, reaching a maximum of about 1.69 when G /G ~3.0. n m 10 11.5 15 Figure 3.15 - The product G^, *» G™ as a function of a—*-c • c—^-b G /G . n m Another measure of the circulator's power gain is given by the product G^ .... G,jv.^, . Grj^ , .This product varies with a—>- c c —>b b —>a with G /G in a manner similar to that shown in Figure 3.15. It n m 6 is greater than 1 for 1 <, G /G ^4.24, and reaches a maximum of 1.39 when G /G =2. n m In summary, the active property of the AG may be t^ken ad• vantage of to realize a circulator with a net gain greater than unity. 39 4. CIRCUITS FOR REALIZING ACTIVE GYRATORS In the literature, circuits have been described (bo55) which are essentially In AG form (i.e., which obey 3.5)). These circuits are made up of only two types of elements, the ideal conductance, and the idealized voltage amplifier, as shown in Figure 4.1. An approach based on these two elements will be used Output G -AAA" (a) Figure 4.1 - The two components which will be used to build AGs: (a).conductance, and (b) idealized voltage ampli• fier, which has A real, positive or negative. to design several AG Circuits. 4.1 Introduction. As was mentioned above, only the two types of elements, shown in Figure 4.1 will be used to realize AGs. Starting with nothing but the external AG terminals a,b,c,d, as shown in Figure 4.2, amplifiers and conductances will be connected inside An analogous investigation could be made using idealized current amplifiers. 40 -O Y 1 c a "Black Box" V2 b (empty) d T Y O -O }2. . 2l'. 22, 2j Figure 4.2 - Skeleton of the AG - starting point for each AG design. the black box, one by one: each such addition will alter the Y matrix*, and the aim will be to add components so that the Y matrix becomes that of an AG, ' o G " V m ...(3.5) -G 0 V L n 2J The foregoing idea underlies the whole of this chapter. The successive steps are shd|m. in a block diagram, Figure 4.3. * Section • 4.2 4.3 4 .4 4. 5 Appendix 4. 6 The two How to Design Clue 16 con• Analy• Be suits compon• use of a for figur• sis of of the ents of these 3-amp• de sign• ations the 16 analy• Figure circuit lifier ing a Ifor config• sis , 4.1 comp• AG 2-amp• 2-amp- urat• and onents lifier lifier ions dis• AG AGs cussion Useful One circuit in its particularly own : useful for right inductance simulation Figure 4.3 - Successive, steps taken in the analysis of AG circuits* * Whenever the Y matrix is mentioned in this chapter, reference is made to the Y matrix for the terminals a,b,c,d, as defined m Figure 4.2. 41 4*2 Effects of the 2 Types of Circuit Components on the T Matrix. The input of a particular amplifier will be connected bet• ween 2 of the terminals a,b,c»dj separate conductors will then be connected between the output terminal, and some of the 4 terminals a,b,c,d, each such conductor forming what will be referred to as an "output connection". For each amplifier, both the input con• ductance and these "output connections" will have an effect upon the X matrix, and these two effects will be considered separately* 4*2,1 Input conductance. Suppose only the input terminals of several amplifiers have been connected to certain pairs of the 4 terminals a,b,c,d* The Y matrix may then be expressed in terms of the input conduct• ances of the amplifiers. All the self-admittance elements will necessarily be of positive sign and the transfer admittance elements will necessarily be of negative sign. Since both self- admittances must be zero (equation (3.5)), any such self-admit• tance element will have to be cancelled somehow by a negative conductance (see the next paragraph). Considering the transfer admittance matrix elements now-, ^2 must be positive, and ^21 negative; thus any (negative) transfer admittances caused by input conductances will be of proper sign for,-Y^l* but not for Y.^; consequently, might not have to be altered, but will. : _ . * • • • ; Some interesting exceptions would arise if amplifiers with neg• ative input conductance were used* 42 4.2.2 Output connections* Consider the amplifier and conductance connected as in Figure 4.4. If PT were connected to Pl, a current I' = V^(l-A)G would flow through G, and if (l-A) «^0, the effect would be that Figure 4.4 - Illustrating the effect of output connections, of a negative conductance connected from a to b. This type of connection affects a self-admittance matrix element (Y-^-for. the connection shown in Figure 4.4), and could be used to make a self-admittance zero, as mentioned in section 4.2.1 above. On the other hand, if P' were connected to P2, a current I1 = V^G - V^AG would flow through G. The first term on the r.h.s (a self term) is just the current which would flow if G were connected from c to d, and therefore its. effect is similar to that of an amplifier input admittance, as discussed in section 4.2.1 above. The second term (a transfer term) can be varied both in magnitude and sign by choice of A, and thus, a connection of this type affords control of a transfer admittance matrix element by choice of A. 43 4.2.3 Negative resistance The point regarding negative conductance (or resistance) will be considered in more detail. Negative resistances (e.g., st60; see bibliography, page 68) are of two basic types, which are usually called the shunt, or short-circuit stable type, and the series, or open-circuit stable type. A shunt—type negative resistance of value -G- will remain stable for any passive termination G + jB which has G ^> Gv A series-type negative resistance of value ''-B will remain stable for any passive termination R + jX, for which R Rq. The amplifier of Figure 4.1(b) can be used to produce either a shunt or a series—type negative resistance, and circuits for both types are shown in Figure 4.5. R B]_+ (l-A)ji G G, + U-A)G10 a) Shunt or short- b) Series, or open- circuit stable type. circuit stable type. Figure 4.5 - Negative resistances using voltage amplifiers. 4.3 Design of a 3-Amplifier AG. The ideas discussed in section 4.2.1 will be used to design an AG. Two amplifiers may be connected as shdwn in Figure 4.6 to 44 a Y 11 12 Y Y 21 22 Figure 4.6 - The start of an AG circuit. provide control over the transfer admittances. As the circuit stands, the Y matrix is -A2G5 Gl + G5 "A1G4 G2 + G4 Since this Y matrix must have the form 0 G m -G 0 _ n A2 must be negative, and A^ must be positive. Next, the two self- admittance terms must be made zero, by providing a negative conductance (shunt-type) from a to b (to cancel G^ + G,-), and another from c to d (to cancel G2+ G^). Amplifier A^ can be used for the former (since a positive—gain amplifier is needed), 45 and a 3rd amplifier connected from c to d (using the configuration of Figure 4.5(a)) for the latter. The final circuit is shown in Figure 4.7. Figure 4.7 - 3-amplifier AG. This AG has several useful features. One important feature is that all 3 amplifiers have a common ground point, so only one power supply is needed. In addition, this common ground point is connected to the common negative input-output terminal of the AG. (Several 2-amplifier AGs will be designed later, but none of them will have this property.) Gm and Gn can be independently controlled for this AG, so G ^ G can be obtained. ' m < n In the development above, it was assumed that the ampli :'- fiers all had zero output impedance, In a practical circuit, this will not be true. In particular, if amplifier 1 has a finite output impedance, the analysis will be complicated . The problem of designing a 2-amplifier AG is Considered next. This point is considered further in section 5.1. 46 4.4 Clue for Designing 2-Amplifier AGs. In this section* justification is given for considering only circuits in which b is connected directly to d; therefore all the circuits will have the same "skeleton", consisting of the 3 terminals a,c, and b-d. In the 3-amplifier circuit of Figure 4.7 > amplifier A-^ amplifies , and the other two amplify V^. Gan one °^ these amplifiers be eliminated? For instance, if an amplifier were connected to amplify V^-V^, perhaps it could perform the functions of both A^, and one of the other two. It is easy to show that this is in fact the case, and both A^ and A2 can be replaced by a single +'ve gain amplifier amplifying - V.^ (this circuit is analysed in Figure 4.16), Here, then, is a clue: use one amplifier to amplify +V^+V2. Will this clue lead to still more 2-amplifier AG circuits? The voltages iv'-j.+v^ can easily be obtained directly from the 4 terminals a,b,c,d, as shown in Figure 4.8. In each of the 4 cases shown, two terminals* one from each port, are connected +y V -V, 1 '2 a o =0 c (a) v. (b) b o» Od b O' \> d a a Figure 4.8 - Obtaining the voltages +Vr^±\1^ from the port voltages. 47 together, and the required voltage is taken between the other two terminals. However, the 4 methods are really equivalent for, given an AG built on one of the skeletons of Figures 4.8(b) to (d), the position of.its terminals can easily be changed so the skeleton becomes that of Figure 4.8(a), while the form of the Y matrix remains unchanged . An example is given in Figure 4.9. 1 > d O a • 2 > vi c b O -O G 0 G o m n -G 0 -G 0 n am Figure 4.9 - Changing the skeleton of an AG circuit. Consequently, the complete set of AG combinations possible with the skeleton of Figure 4.8(a) >; will include all the possible AG circuits for any of the other skeletons. Therefore, only one The manipulation proceeds in 2 steps. First, reverse the polarity at one port, whence the Y matrix 0 G becomes '0 -G Second, m m -G 0 G 0 n n either (a) reverse the polarity at the other port, or else (b) interchange ports, whence the Y matrix becomes either 0 G or -G 0 n '0 G respectively, and in either case is of the original form. n •G 0 m 48 skeleton will be considered, say that of Figure 4.8(a). This is # the consequence of our clue . 4.5 Classification and Preliminary Screening of . 2-Amplifier AGs. In this . section, all the possible circuits which can be built on the skeleton of Figure 4.8(a) using 2 amplifiers are classified, and it is shown that only 16 of the 24 possible types need be considered. The circuits will be classified by the way the amplifier inputs are connected. How many useful ways can the inputs of two amplifiers be connected to the 3 terminals of Figure 4.8(a)? Disregarding the polarity of the inputs for the moment, a con• nection between a and c will be designated as "D" (for difference), between a and b-d as "1", and between c and b-d as "2". Then there are 6 possible ways of connecting the 2 amplifiers: DD, Dl, D2, 12, 11, and .22. The latter 2 a,re eliminated, for they allow only (one of) Y-^ or ^22 ^° ^e se^ ^° zero• In order to take polarity into account, a, a, and c will be considered as the positive terminals for the D, 1, and 2 connections respectively. Then the input connections, of the two amplifiers may be completely described by 4 characters, for example "D+1-". .There are therefore 4X4 = 16 possible ways of connecting the amplifier inputs which might yield an AG. The problem of systematically analysing these. 16 configurations is considered in detail in the Appendix, and the results of this analysis are contained in the next section. The skeleton was obtained on the basis of using a difference amplifier; circuits will also be considered which do not have; this property (the 1+2+ configurations). : :.. , 49 4.6 Results of the Analyses of the 16 Configurations. The results of the analyses carried out in the Appendix are summarized in a table, Figure 4.10. From the 1st column of entries, note that 7 of the 16 configurations yield an AG, numbers 5, 7, 8, 11, 12, 14, and 16. It is easy to show that the last 4 are equivalent to circuits given by Bogert (bo55). From the 2nd column, note that only one of the 7 AGs, configuration riuniber 8, has a common ground for both amplifiers. This is a desirable feature in a practical circuit, for otherwise 2 separate power supplies would be needed. From the 3rd column of entries, note that configuration num• ber 8 requires one + 've and one —'ve gain amplifier, while the. other 6 configurations require both amplifiers to have a +'ve gain. In practice, a -t-'ve gain amplifier may require 2 stages of amp• lification, while a -'ve gain amplifier requires only 1, and thus only 3 stages of amplification are required for configuration number 8, while the others require 4. i From the 4th column of entries, note that most of the feasible configurations require only one output connection per amplifier, except for number 8, which requires 2 for the +'ve gain amplifier. In a practical circuit, the amplifiers will have a finite output resistance. If only one output connection is required, this finite output resistance may be used as (part of) the connection resis• tance. For configuration number 8, though, the equations will be complicated if the + 've gain amplifier has a finite output resistance; and if this output resistance is too high, perhaps the configuration will not be feasible. This problem is considered in 50 Amplifier Is the Common How Is only Can the Num• In*put Con• Config• Ground many one Res• Gyration Comments ber figuration uration for + 've istor Conduct• Feasible? both Amps. per Apip. ances be Amps.? Reqd,? Reqd.? made =? 1 D+D+ NO 2 D+D- NO 3 D-D+ \ 4 D-D- NO une Daiance 5 D+1 + YES NO 2 YES YES condition is that G~ = Gr- 6 D+l- NO Same as for 7 D-1 + YES NO 2 YES YES number 5 This MJ is best, 8 D-l- YES YES 1 NO NO for inductance simulation 9 D+2+ NO 10 D+2- NO Bogert gives 11 D-2+ YES NO 2 YES YES thia_ circuit (bo55,Fig.3(c)) .; Same as for 12 D-2- YES NO 2 YES YES numb e r 11 13 1+2+ NO Bogert gives 14 1+2- YES NO 2 YES YES this circuit Cbo5;5,Eig.Md)) 15 1-2+ NO Same as for 16 1-2- YES NO • 2 YES YES number 14 Figure 4.10 - Summary of 2-amplifier AG circuits. 51 section 5.1. From the 5th column of entries, note that with all the configurations but number 8j the gyration conductances can be made equal. Configuration number 8, however, will not yield a gyrator. Which of the 7 circuits is best for inductance simulation? Configuration number 8 has the advantages of requiring only 1 power supply, and only one +rve gain amplifier. The only dis• advantage of configuration number 8 is that one of the ampli• fiers has 2 output connections; it is shown in section 5.1 that, for the case where the output resistance of this amplifier is small, the configuration still yields an AG. Therefore, number 8 seems to be the best of the 7 configurations . In fact, on the basis of these considerations, number 8 is just as good as the 3-amplifier AG of section 4.3. The circuits of configuration numbers 5, 7, and 8 are all believed to be new. However, since the application of primary interest is inductance simulation, and since number 8 seems the most advantageous for this purpose, numbers 5 and 7 will not be pursued any further. In Chapter 5, configuration number 8 is considered in more detail, and a few experimental measurements which were obtained with this circuit are given. Questions of high Q, stability, and noise performance (the first two are interdependent) have not entered into this decision. It is interesting to note that the circuit of configuration number 8 was designed before the present analysis was carried out. 52 5, EXPERIMENTAL RESULTS In this chapter* a practical circuit for AG configuration number 8 (Figure A.£ ) is investigated. In the first section, the analysis of this configuration is repeated, this time for the case where the amplifiers have a finite output resistance. The second section contains the: actual circuit diagram for the prototype which was built* In the third section, the results of some experimental measurements are given. . These results are sufficient to confirm the theory. 5.1 Allowance for the Finite Output Impedance of the Amplifiers. In the analysis of configuration number 8 presented in the Appendix, it was assumed that the amplifiers had zero output impedance. In practice, this is not true; this point was dis• cussed in section 4*6* Let the P amplifier.(see Appendix) have a finite output conductance, Gq. Then the output connections for the amplifier are as shown in Figure 5*1(a). The approximateV-equivalent of this network is shown in Figure 5.1(b), for the case where G ' ^G-G./G Go G, PI ° , 4 0 P2 p cMv -A/V-o -A/V 4. P 6 Figure 5.1 - The output connections for an amplifier with finite output conductance, G0: (a) actual circuit; (b) approximate ^—equivalent when G.j'V G^« G0 . 53 G3^ G4«V ..(5.1) The effect of the finite output . impedance, then, is to introduce a conductance G ' ^ G^G./G between Pl and P2 (c and d). o 3 4/o In Figure 5.2, configuration number 8 is analysed for the case where the P amplifier has a finite output conductance Gq . The approximation of Figure 5.1(b) is used. It is seen that, with Gq included, the configuration still yields an AG. It is easy to show that for this AG ; in fact, equation (2 ) of Figure 5.2 can be written T12 = I T2l| + (G2+ G4+ Gor)* •••(5.2) This section is concluded with a sample design, based on the analysis given in Figure 5.2. Suppose G = G, = G~ = 200 umho, o 1 2 ' A1 = 6, and A2 - "10- Then equations (la) and (2 ) in Figure 5.2 become, respectively, (5000)G3G4 +0.0002 + G& - 5G3 =0, and ...(5.3) 0.0002 - (5000)G3G4 + 10G6 - 5G4 =0. Since there are 2 equations in 3 unknowns, one of the unknowns Note that the AG terminals can easily be re-named so that this inequality is reversed. See footnote in section 4,4, page 47. 54 + a 'P-Pl •V P-P2 Vifl-AiVfl&Ai Q-Qi Q-Q2 P2.QI AG Intr— Terms Produced by Amps. insic admit• Subtotals term, P Amplifier fifir tance term intr PI P2 Ql v 02 via in= o (0 Affec- 0 via o Y12= + va Ml via T21=. - Affect -Si %M 0 (3) I ? viv,a 2 *X 0 0 T22= 0 o 0 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (ll) ^ t'sth-z output <**t/> Pt«UPZ u>*'ll U us For- y^2T0^ & 6* a?€c/. T^-e ce>* J rife* Ti^paSeJ /"f -hd^C/l^i)^, Cl) T~/4.e sn£>~tot<*/s Tk Figure 5.2 - Design sheet for configuration number 8, using the approximation given in Figure 5.1(b). 55 may be chosen at will, say G^ = 50 u-mho, in order to comply with (5.l). Solution of equations (5.3) yields G^ = 91 jumho, Gg = 27*. 4 p,mho . In the next section a prototype is described which was built using these values as a design centre. 5.2 Circuit Diagrams for the Prototype AG. A one—transistor, common-emitter amplifier was designed to produce a negative gain of about 10. The circuit diagram is shown in Figure 5*3* ^4.7K Figure 5»3 - Circuit diagram of the Q amplifier. A two—transistor, common-emitter, R-C coupled amplifier was designed to produce a positive gain of about 6. The circuit 56 ~ +6, at OUT f•= 1000c.p.s. O p Zout - 5K Figure 5*4 - Circuit diagram of the P amplifier. diagram is shown in Figure 5.4. The emitter resistors in both circuits provide negative current feedback, and lower the output resistance. These 2 amplifiers were connected in the D-l- configuration (number 8.) as shown in Figure 5.5. Design centre values calculated E 04 3 200K AH capacitors are 10 u-f tantalum electrolytic. Figure 5.5 - Schematic diagram of the AG prototype. in the sample design of section 5.1 were used, and variable resistors were used for G-^ and G^ to permit adjustment* The circuit was built on a printed circuit board, Figure 5*6* 57 Figure 5.6 - Artwork for the printed circuit board (actual size). For convenience* identical coupling capacitors were used through• out. 5.3 Experimental Results. Two sets of measurements were, taken with the prototype AG described in the preceeding section. In the first set, the AG was used to invert a resistance, for comparison with similar results given by Bogert (bo55). In the second set, it was used to simulate an inductance. The experimental apparatus is shown in Figure 5.7. Z^n was measured with a General Radio X-Y bridge, type 1603A. 5.3.1 Preliminary adjustment of the AG. Recall from section 5*2 that G_ and G, were made variable 58 Heathkit Audio Oscillator Model AG.-8 Detector: General Tektronix Radi o Type 502A X-Y Bridge Oscilloscope Type 1603A O Figure 5.7 - The AG used to invert an impedance Z^: experimental apparatus. to permit adjustment. These two variable conductances can be adjusted to set Y^ — 0 and ^ 0* a) Adjustment for — 0 Terminals c and d are short-circuited. Under this condition, Y. in 'v2.=.o so Y-j^ can be measured directly. Y^ can be set arbitrarily close to zero by adjusting G^ (see equation (2 ), Figure 5.2). Since the circuit would be potentially unstable if Y^ were -'ve, Gg is adjusted to make Y^ slightly +'ve, say Y. - Y,, =5 umho in 11 This value of Y,, will provide an (arbitrary) margin of stability. 59 The adjustment was carried out by setting the dials of "the X-T bridge to 5 + JO u.mhos, and then adjusting G^ to balance the bridge. b) Adjustment for Y22>'0 Terminals c and d are.openrcircuited. Under this condition, z. •• • 22 Jin T11Y22 + JY12||Y2l| But Y^^ ~ 0 because of the adjustment made in ,a) above, and there• fore Y 2 ^ 22 in lIi2| Pail Y22 can be set arbitrarily close to zero by adjusting G-j (see equation (la) in Figure 5.2). Again, a margin of stability can be achieved by making Y22 slightly positive; in practice, this is easily done by adjusting G^ to make Z^n slightly positive, say Z. : = 5 Jl. in This adjustment was carried out by setting the dials of the X-Y bridge to 5 + JO ohms, and then adjusting G^ to balance the bridge. In practice, a slight sliding balance is observed between adjustments a) and b), so the preliminary adjustment is completed by repeating a) and b). This sliding balance is not evident in equations (la) and (2 ) of Figure 5.2, because of the approximation (Figure 5.1(b)) which was made-. 60 5.3.2 Inversion of resistance. A calibrated Heathkit resistance substitution box was used for ZT in Figure 5.7. The resistance was varied from 22 ohms to Li 10 megohms in 18 steps, and Z^n was measured for each setting. Three such sets of readings were taken, at frequencies of 200 c.p.s., 1 Kc., and 5 Kc. The measured values were converted to polar form, and are plotted in Figure 5.8 (the curve given by Bogert (bo55, taken at 400 c.p.s.) is included in the figure for comparison).. At 1000 c.p.s., the inversion is excellent for values of ranging from about 500 ohms to 100 kilohms (about 2\ decades), with a phase angle of less than 1° over this entire range. At both 200 c.p.s. and 5 Kc, the phase response is much poorer. 5.3.3 Simulation of inductance. A 0.1 ufd., 75v., ceramic disk capacitor was used for Z-^ in Figure 5.7. was measured as the frequency was varied from 20 c.p.s. to 20 Kc. The measured values, plotted in Figure 5.9, indicate that the simulated inductance has the approximate equi• valent circuit shown in Figure 5.10. The series capacitance in this equivalent circuit represents the coupling capacitor at the input to the Q amplifier (C in Figure 5.5), while the parallel s capacitance is probably the result of stray capacitances and phase shift effects within the AG circuit. At 1000 c.p.s., the Q of the inductor was about 16. Note, however, that this Q can be increased by merely adjusting G^ and GA, If the Q is made too large, however, drift in the amplifiers -100K -X r-l° (O) (phase angle of Z±n) +5° (A) t-lOK- ^-10° ( + ) f =0.2 Kc. A -5° •( + ) 10 o _>>N 1 lZinl -ft- f "= 1.0 Kc . o f = 5.0 Kc. + (Bogert1s curve) ,-5° (A) f = 0.4 Kc. biK- Curve given by Bogert (bo55) ••(+>•••-.•• (shifted 1 decade to the right) s^--io° (A) iov hioo +5° (O) 10M 100 IK R ,_n_. ' : I0K 100K > IM » I i t i n> I i I i \ > t i i > J il "Q i i i i i > il 1 O > I I I M'tl +io° Id). Figure 5.8 - Experimental results, inversion of resistance. 62 I.. ^ u.mho, 20 c.p.s. •300 (a) 0.1 Kc. Figure 5.9 - Experimental results, simulation of inductance: (a) low frequency measurements, Z±N', (b) high frequency measurements, Yi; n 63 ~1".C- ' — 15u.f. Series resonance. s at f ^ 37 c.p.s. 600pf. Parallel resonance at f ^ 6Kc . O— J Figure 5.10 - Approximate equivalent circuit for the simulated inductance (losses are neglected). may cause the Q to become -'ve, thereby producing potential instability. Although no explicit drift measurements were made, the initial and final readings in one set of measurements ind• icated that had1 drifted from about +0.5 umho to -l.0u.mho, over a period of a few hours (Y-^ an£l |^2l| ^ 300 urriho). On the basis of the results given above in sections 5.3.2 and 5.3.3, one can conclude that the prototype AG performs as expected from the theoretical design. 64 6. CONCLUSION The topics of inductance simulation and gyrator realization have been dealt with in this thesis* An extensive survey has been given of the inductance simulation literature. It has been shown that several seemingly unrelated "amplifier methods" of inductance simulation are really closely related. An extensive survey Of the gyrator literature has been given The "active gyrator" (A|J) has been defined, and several of its properties and applications given* One property of the AG is that of inductance simulation. An active gyrator can also be used to realize an isolator or a circulator with a power gain. An analysis has been presented which results in 7 circuit configurations for the realization of the AG. One of these cir• cuits is the best for inductance simulation. Of the other cir• cuits^ 4 have been published previously, arid 2 are believed to be new. The "best"circuit is also believed to be new. Experimental results obtained with the best AG circuit for inductance simulation have been given. These results confirm the validity of the design. The main contributions of this thesis are: 1) classification of the "amplifier methods" of ind• uctance simulation (Chapter 2), 2) proposal of ari operational amplifier method of inductance simulation which is believed to be new (Figure 2.8), 3) comparison of gyrator realization methods (Figure 3 65 proposal of the AG (section 3.3), analysis of isolators and circulators made with AGs, showing the power gains available (section 3*5)9 design of an AG circuit which turns out to be "best" for inductance simulation (section 4.6), presentation of an analysis which results in several. circuits for the realization of the AG (Chapter 4, and the Appendix), experimental testing of one of these AG circuits (Chapter 5), and compilation of an extensive bibliography of the inductance simulation and gyrator literature (Chapter 7)« 66 7. BIBLIOGRAPHY The bibliography which was compiled during the course of this project contains well over 100 references. It is believed that a bibliography of this extent has not appeared elsewhere. Each reference is indexed with a 4-character code; for example, "do63". The first two characters are the first two letters of the surname of the principal author, while the last two characters indicate directly the year of publication. For instance, a paper written by J. Doe, and published in 1963 would be indexed under "do63"«. If several papers fall under the same index code, a 5th character is added to differentiate them; for example, do63a, do63b, etc. It is hoped that this bibliography will serve as a useful guide to anyone seeking information regarding inductance sim• ulation and gyrators. 7.1 Subject Index to Bibliography. INDUCTANCE, etc* Conventional inductors: ho49, hu25, po37. Microminiature inductors: br55, du61, ho58, ho62, st57a, st57b, to62. Simulated inductance: Semiconductor devices : di61, di62a, di62b, du63 , ei52, fi59, ga61, go57, gu56, hu60r ka55« ko54, ko55, la60, la6l, ni58, ni60, no63, on56, sc60, sc62, sp58. 67 Other: bo55, bo56, co58, et62, fu60, ho—, hu42(?-), mc63, mi55, mi60, st58, to53, wa47. Current limiters: du61, la62, wa59. Ceramic IF transformers, etc.: el58, lu58, ma61a, ma61b. Increasing Q factor: sa6l. GYRATORS, etc. Theoretical: Network theory: au55, ca55a, ca55b, ca56, fr58, on6la, oo54, pr57, te48b, te4'9a, te49b, te50, te51, te55, tw55, wa51. Others: ca45, fa58, na6l, pi62, sh53, sh54, te48ar (original paper), te52, te56. Practical (physical realization): Electromechanical: Coupled transducers: bl53, br44, cl61, cu—, ga52, ge61, ha54, mc46, mi47, on61b, on62, si6l, si62a. " " as isolator: ga52, ga54, ga59, mc46, mc47, on62, si61,'si62a. " " as circulator: si62b. Generalized' machine: pe58. Hall Effect: ar60, be63, ch58, gr58, gr59a,•gr59b, hu61, ma53, mc47, se52, si62a, wi54. Ferrite (microwave frequencies): al57, be50, bl56, bl57, ca54, ca56, fo55, go53, he56, ho52, ho53, ho56, jo59, ka53, ow56, ^ re54, sc61, so60. " as isolator: be57, jo59. " as circulator: ho52, ho56. Tube and transistor: bo55, bo56, no54, pi62, sh53. Field effect tetrode: et62, st59a, st59b, st6l. Parametric: ka60, ko61. 68 OTHER Network properties! bo57, sh54. Circulators: pe62, st57c, tr56. Negative resistance: bo45, cr31, fa52, gi45, he35, ro54, st60, uz63. 7.2 References, al57 Allin, P.E.V., Ferrites at Microwaves, Electronic Eng., Vol. 29, pp. 292-96; June, 1957. ar60 Arlt, G., Halleffekt-Vierpole mit Hohem ¥irkungsgrad, Solid-State Electronics, Vol. 1, pp. 75-84; March, 1960. au5 5 Aurell, C.G., Representation of General Linear 4-Terminal Network and Some of its Properties, Ericsson Technics, Vol. 11, pp. 155-79; 1955. be50 Beljers, H.G., and Snoek, J.L. , Gyromagnetic Phenomena Occurring Within Ferrites, Philips Tech. Rev., Vol. 11, pp. 315-22; May, 1950. be 57 ' Beljers, H.G., Application of Ferroxcube to Unidirectional Waveguides, Philips Tech. Rev., Vol. 18, p. 158; 1957.. be63 Beckman Instruments, Inc., Fullerton, Calif., Hall Effect Manual, (pamphlet, with extensive bibliography); 1963. bl53 Black, L.J. and Scott, H.J.,- Measurements on Nonreciprocity in Electromechanical Systems, Journ. Acoustical Soc. Am., Vol. 25, pp. 1137-40; Nov., 1953. bl56 Bloembergen, N., Magnetic Resonance in Ferrites, PIRE, Vol. 44, pp. 1259-69; Oct., 1956, (Bibliography of 63 art., see #»s 6, 8, 10 (biblio), 43) bl57 Blackman, L.C.F., Low Loss Magnesium Manganese Ferrites, J. Electronics, Vol. 2, pp. 451-56; March, 1957. 69 Bode, H.W., Network Analysis and Feedback Amplifier Design (book), Van Nostrand Co., New York, pp. 185-88; 1945V Bogert, B.P., Some Gyrator and Impedance Inverter Circuits, PIRE, Vol. 43, pp. 793-96; July, 1955. Bogert, B.P., Impedance Inverters, U.S.Pat. 2,757,345, July 31, 1956. Bolinder, E.F., Survey of Some Properties of Linear Networksj IRE Trans., Vol. CT-4, pp. 70-18; Sept., 1957. Braun, K., Title unavailable, Telegr. Fernspr. Technik, Vol. 33, pp. 85- $ 1944. Bryan, H.E., Printed Inductors and Capacitors, Tele-Tech and Electronic Industries, Vol. 14, p. 68; Dec, 1955. Casmir, H.B.G., On Onsager's Principle of Microscopic Reversibility, Rev. Mod'. Phys., Vol. 17, pp. 343-50; 1945.: Carlin, H.J., Principles of Gyrator Networks, Proc. Symp. Modern Advances in Microwave Techniques, Polytechnic Institute of Brooklyn, Nov. 8-10, 1954, pp. 175-204. Carlin, H. J.,, Synthesis of ,Non-Reciprocal Networks, Presented at Symposium on Modern Network Synthesis, Polytechnic Institute of Brooklyn, April 13—15$ 1955. Carlin, H.J., On Physical Realizability of Linear Non-Reciproeal Networks, PIRE, Vol. 43, pp. 608-16; May, 1955. Carlin, H.J., Non-Reciprocal Network Theory Applied to Ferrite Microwave Devices, (Convention on Ferrites, Oct.-Nov., 1956), IEE Proc, Vol. 104, part B, supp. #6, pp. 316-19; 1957. Champness, C.H., Hall Effect and Some of its Possible Applications, IEE Can. Convention Bee, 1958, pp. 265-69. 70 cl6l Clevite Corp., Low Frequency Gyrator Development,, (report), Nov., 1961, 44 pages. co58 Cooperman, J.I., and Franklin, P.J., Some Circuit Techniques to Eliminate Large-Volume Components; a Literature Survey, Microminiaturization of Electronic Assemblies (book), Hayden Book Co., Inc., N.Y., 1958; pp. 193-212. Also see Electronic Design, vol. 7, pp. 57-61; March, 1959. cr31 Crisson, G., Negative Impedances and the Twin 21-Type Repeater, BSTJ, Vol.. 10, pp. 485-513; July, 1931. cu— Curran, D.R., Germano, CP., Silverman, J.H., and Dishotels, ¥.J., Low-Frequency Gyrator Development, Armed Services Technical Information Agency AD 269-374. di61 Dill, H.G., Inductive Semiconductor Elements and Their Application in Bandpass Amplifiers, IRE Trans., Vol. MIL-5, pp. 239-50; July, 1961. di62a Dill, H.G., Avalanche Pulse Generators, Semiconductor Products, Vol. 5, Feb., 1962, pp. 23-30. di62b Dill, E.G., Semiconductor Inductive Elements, Semiconductor Products, Vol. 5, April, 1962, pp. 30-33; and May, 1962, pp. 28-31. do36 Doherty, W.H., A New High Efficiency Power Amplifier for Modulated Waves, PIRE, Vol. 24, pp. 1163-82; Sept., 1936 .(Also seem51,BP. 399-404.) du6l Dummer, G.W.A., and Granville, J.W., Miniature and Microminiature Electronics (book), John Wiley & Sons, New York, 1961, 310 pages, ($7.50). du63 Dutta Roy, S.C., The Inductive Transistor, IRE Trans., Vol. CT-10, pp. 113-15; March, 1963. ei52 Einsele, T., Uber die Traegheit des Flussleitwarts von Germaniumdioden, Z. Angew, Phys., Vol. 4, pp. 183-87; May, 1952. el58 Elders, D., and Gikow, E., Ceramic IF Filters Match Transistors, Electronics, Apr. 25, 1958, pp. 59-61. Also see 1957 Electronic Components Conference, Chicago, May 2, 1957. 71 et62 Etter, P.J., and Wilson, B.L.H., Inductance Prom a Field-Effect Tetrode, PIRE, Vol. 50, pp. 1828-29; Aug., 1962. fa52 Farley, B.G., Dynamics of Transistor Negative-Resistance Circuits, PIRE, Vol. 40, pp. 1497-1508; Nov., 1952. fa58 Fabrikov, V.A., Vozmozhnost Usileniya Giromagnitnoi Snedoi Moshchnosti Slabogo Moduliruyushchego Signala, Radio-Tekhnika i Electronika, Vol. 3, pp. 190-97; Feb.,.1958. Published in English: Possibility of Power Amplification of Weakly Modulated Signals by Gyromagnetic Medium, Radio Eng. and Electronics, Vol. 3, pp. 265-75; 1958. fi59 Firle, T.E., and Hayes, O.E., Some Reactive Effects in Forward Biased Junctions, IRE Trans., Vol. ED-6, pp. 330-34; July, 1959. fo55 Fox, A.G., Miller, S.E., and Weiss, M.T., Behaviour and Applications of Ferrites in Microwave Region, BSTJ, Vol. 34, pp. 5-103; Jan., 1955. fr58 Frank, V., and Hojgaard-Jensen, H., Note on Reciprocity Theorem for Electrical Systems, Applied Science Research, Sec. 8, Vol. 7, pp. 145-49; 1958. fu60 Fulenwider, J.E., High Q Inductance Simulation, PIRE, Vol. 48, pp. 954-55; May, 1960. ga52 Gamo, H., Four Terminal Networks Violating Reciprocal Theorem and One-Way Systems, Proc. 26th Joint Conf. of Elec. Eng'g, Soc, 1952, Paper # 9-1. ga54 Gamo, H., Electromechanical One-Way Systems, J. Acoust. Soc. (Japan), Vol. 10, pp. 65-76; June, 1954. ga59 Gamo, H.^ On Passive One-Way Systems, Trans. Int. Symp. on Cct. & Inf. Theory, 1959, pp. 283-98. ga6l Gartner, WW., and Schuller, M., • Three-Layer Negative Resistance and Inductive Semiconductor Diodes, PIRE, Vol. 49, pp. 754-67; Apr., 1961. ge6l Germano, CP., and Curran, D.R., Low Frequency Gyrators, Proc. Electrnc. Compts. Conf., May, 1961, pp. 24-1 to 24-13. 72 gi45 Ginzton, E.L., Stabilized Negative Impedances, Electronics, Vol. 18, July, Aug., and Sept., 1944, pp. 140-50, 138-48, and 140-44 respectively. go53 Goldstein, L., and Lampert, M.A., A New Linear Passive Non-Reciprocal Microwave Cct. Component, PIRE, Vol. 41, pp. 295-6; 1953. go57 Good, E.F., A fwo-Phase Low Frequency Oscillator, Electronic Engineering, Vol. 29, pp. 164-69 and,210-l3; April & May, 1957. gr58 Grubbs, ¥.J., The Hall-Effect Circulator - a Passive ... Device, IRE Wescon, 1958, pt. 3, pp. 83-93. gr59a Grubbs, W.J., Hall Effect Devices, BSTJ, Vol. 38, pp. 853-76; May, 1959. gr59b Grubbs, W.J., Hall Effect Circulator, PIRE, Vol.. 47, p. 528; 1959. gu56 Guggenbuehl, W., Theoretische Ueberlegungen zuj|, Physikalischen Begruendung des Ersatzschaltbildes von Halbleiterdioden bei Hohen Stromdichten, Arch. Elektrotech. Ubertragung, Vol. 10, pp. 483—85; Nov., 1956. .ha!4 Haus, H.A., Equivalent Circuit for Passive Non-Reciprocal Network, J. Appl. Phys., Vol. 25, pp. 1500-02; Dec, 1954. he35 Herold, F.W., Negative Resistance and Devices for Obtaining it, PIRE, Vol. 23, pp. 1201-23; Oct., 1935. he56 Heller, G.S., Ferrites as Microwave Circuit Elements, PIRE, Vol. 44, pp. 1386-93; Oct., 1956. ho49 Howe, G.W.O., The Q Factor of Single-Layer Coils, Wireless Eng., Vol. 26, pp. 179- ; June, 1949. ho52 Hogan, C.L., Ferromagnetic Faraday Effect at Microwave Frequencies and: its Applications - Microwave Gyrator, BSTJ, Vol. 31, pp. 1-31; Jan., 1952., See also Wireless Engr., Vol. 29, pp. 171-73; July, 1952. 73 ho53 Hogan, CL. , The Ferromagnetic Faraday Effect at Microwave Frequencies and its Applications, Rev. Mod. Phys., Vol. 25, pp. 253-263; Jan., 1953. ho56 Hogan, CL., The Elements of Nonreciprocal Microwave Devices, PIRE, Vol. 44, pp. 1345-68; Oct., 1956. Paper presented at Symposium on Microwave Props. &' Appls. of Ferrites, Harvard Univ., Cambridge, Mass., April 2-4. 1956. (See p. 1345, rh, top, re impedance inversion) ho58 Horsey, E.F., and Shergalis, L.D. (editors), Microminiaturizationcof Electronic Assemblies, Hayden Book Co., Inc., N.Y., 1958. ho62 Horsey, E.F. and Franklin, , Status of Microminiaturization, IRE Trans., Vol. CP-9, pp. 10- ; Mch., 1962. ho— Holbrook, G.W., and McKeown, D.L., Simulation of Inductance by Active Circuit Elements, To be published. hu25 Hund, A., and DeGroot, H.B., Radio Frequency Resistance and Inductance of Coils Used in Broadcast Reception, Technologic Papers of the Bureau of Standards, Washing• ton, D.C., No. 298; Oct. 22, 1925. hu4 2 Hund, A., Frequency Modulation (book), McGraw-Hill, 1942, pp. 15 5-74. hu60 Hughes Aircraft Co., Semiconductor Division., Newport Beach, California, Molecular Bandpass Amplifier, Interim Sci. Rept., Nos. IF and 2, Contract AF33(616) - 7252; July 15 and Oct. 15, I960, respectively. hu61 Hubbard, C.H., LoSasso, L.A., and Rousso, E., Microwave Isolator Combines Hall Effect and Tunnel Diodes, Electronics, June 16, 1961, pp. 56-57. jo59 Jones, EwM.T., Matthaei, G.L., and Cohn, S..B,, Non-Reciprocal, TEM-Mode Structures for "Wide-Band Gyrator and Isolator Applications, IRE Trans., Vol. MTT-7, pp. 453-60; Oct., 1959. ka53 Kales, M.L., Chait, H.N., and Sakiotis, N.G., A Non-Reciprocal Microwave Component, J. Appl. Phys., Vol. 24, p. 816; 1953. 74 Kanai, Y., On the Inductive Part in the a.c Characteristics of the Semiconductor Diodes, J. Phys. Soc. Japan, Vol. 10, pp. 719-20; 1955. Kamal, A.K., A Parametric Device as a Non-Reciprocal Element, PIRE, Vol. 48, pp. 1424-30; Aug., 1960. Kohn, G., and Nonnenmacher, V., Inductives Verhalten von p-n Uebergaengen in Flussrichtung Arch. Elektrotech. Ubertragung, Vol. 8, pp. 561-64; Dec, 1954. * Kohn, G., Die Beruecksichtigung des Uebergangsgebietes Zwischen Pluss und Sperrgebiet im Ersatzschaltbild fuer Traege' Germaniumdioden, Arch. Elektrotech. Ubertragung, Vol. 9, pp. 241-45; May, 1955. Korpel, A., and Desmares, P., Experiments "with Non-Reciprocal Parametric Devices, PIRE, Vol. 49, p. 1582; Oct., 1961. Ladany, I., An Analysis of Inertial Inductance in a Junction Diode IRE Trans., Vol. ED-7, pp. 303-10; Oct., I960. Ladany, I., and Kearney, R.J., A High-Q Tuned Circuit Using a Solid-State Inductance, J. of Electronics and Control, Vol. 10, pp. 241-43; Mch., 1961. Lawrence, H., A Diffused Field-Effect Current Limiter, IRE Trans., Vol. ED-9, pp. 82-87; Jan., 1962. Linvill, J.G., and Gibbons, J.F., Transistors and Active Circuits (book), McGraw-Hill, 1961, pp.. 21-5-18. Lungo, A., and Henderson, K., Application of -Piezoelectric Resonators to Modern Bandpass Amplifiers, IRE Natl. Conv. Rec , 1958, part 6, pp. 235-42. Mason, W.P., Hewitt, W.H., and Wick, R.F., Hall Effect Modulators and. "Gyrators" Employing Magnetic Field Independent Orientations in Germanium, J. Appl. Phys., Vol. 24, pp. 166-75; Feb., 1953.. Macario, R.C.V., Design Data for Bandpass Ladder Filters Employing Ceramic Resonators, Electronic Engineering, Vol. 33, pp. 171-77; Mch., 1961. 75 ma6lb Macario, R.CJ. , Ceramic IP Transformers, Wireless World, Vol. 67, pp. 253-56; May, 1961.. mc46 McMillan, E.N., Violation of the Reciprocity Theorem in Linear Passive Electromechanical Systems, J. Acoust. Soc. Am., Vol. 18, pp. 344-47; Oct., 1946. mc47 McMillan, E.N,, Letter to Editor, J. Acoust. Soc. Am., Vol. 19, p. 922; 1947. (See same journal, Trent, and Jefferson, pp. 502-03). mc63 McKeown, D.L., Simulation of Inductance by Active Circuit Elements, (?), Thesis, Nova Scotia Technical College, 1963. mi47 Miles, J.W., Coordinates and the Reciprocity Theorem in Electromech• anical systems, J. Acoust. Soc. Am., Vol. 19, pp. 910-13; Sept., 1947. mi55 Miller, S..L. , and Ebers,J.J., Alloyed Junction Avalanche Transistors, BSTJ, Vol. 34, pp. 883-902; Sept., 1955. mi58 Millman, J., Vacuum Tube and Semiconductor Electronics, (book), McGraw-Hill, 1958, p. 584. mi60 Midgley, D«, and Stewart, J.M., Simulation of Inductance by an Integrating Circuit, Electrical Review, Vol .• 164, pp. 281-85; Feb. 12, 1960. na61 Nair, K.K,, and Swamy, M.N.S., On Gyrator Networks, J. Institution Telecomm. Engrs. India, Vol. 7, pp. 111- 16; May, 1961. ni58 Nishizawa, J., Iwasa, S., and Watanabe, Y., Simplified Theory on Inductive Impedance of p-n Junction, Repts. Res. Inst. Elec. Commun., Tohoku Univ., Vol. 10, pp. 45-57; July, 1958. ni60 Nishizawa, J., and Watanabe, I., Semiconductor Inductance Diode, Proc. Solid State Circuits Conf., Philadelphiaj Pa., Feb. 10-12, 1960. no54 Nonnenmacher, W., and Schreiber, F., Der Zweidrahtverstarker als Gyrator Vierpol zur Herstellung Ungewohnlic'Aer Scheinwiderstunde, Frequenz, Vol. 8,;pp. 201-04; July, 1954. 76 nb63 Nordman, J.E., and Greiner, R.A., The Small-Signal Inductive Effect in a Long P-I-N Diode, IEEE Trans., Vol. ED-10, pp. 171-77; May, 1963. on56 Onoe, M«, and Ushirokawa, A., Inductive a.c* Admittance of Junction Transistor, PIRE, Vol*. 44,- p. 1475; Oct., 1956. on61a Onodera, R., Topological Synthesis of Non-Reciprocal Resistance Net• works, Proc* IEE, pt* C. Vol. 108, pp. 325-34; Sept., 1961. on61b Onoe, M., and Sawabe, M., A Piezoelectric-Piezomagnetic Gyrator, f Proc. Spring Mtg. Acoust. Soc. Japan, May, 1961, Paper on62 Onoe, M., and Sawabe, M., A Piezoelectric-Piezomagnetic Gyrator, PIRE, Vol. 50, pp. 1967-73; Sept., 1962. oo54 Oono, Y., and Yasuura, K., (something about network theory) Ann. Te'lecomm., Vol. 9, pp. 73 and 109; 1954. ow56 Owens, CD., Modern Magnetic Perrites and Their Engineering Appli• cations, IRE Trans, Vol. CP-3, pp. 54-62; Sept., 1956. pe58 Penhune, J., Operation of the Generalized Machine as a Gyrator, Mimeographed article, M.I.T., June, 1958. pe62 Penfield, P.* Jr., A Classification of Lossless 3-Ports, IRE Trans., Vol. CT-9, pp. 215-223; Sept., 1962. pi62 dePian, L., Linear Active Network Theory (book), Prentice-Hall, 1962, pp. 236-48. po37 Pollack, D., The Design of Inductances for Frequencies Between 4 and 25 Mc, Trans. AIEE, Vol. 56, pp. 1169-76; Sept., 1937. pr57 Prudhon, M., Gyrateurs et Systemes a Sens Unique, ra63 Cables & Transmission, Vol. 11, pp. 66-73; Jan., 1957. Rand, A.* Inductor Size vs. Q: A Dimensional Analysis, IEEE Trans., Vol CP-10, pp. 31-35; March, 1963. 77 re54 Redfield, A.G., Electrodynamic Perturbation Theorem, with Application to Non-Reciprocal Systems, J. Appl. Phys., Vol. 25, pp. 1021-24; Aug., 1954. ro54 Roddam, T., Negative Resistance, Wireless World, Vol. 60, pp. 335-38; JUly, 1954. sa61 Sabih, D., Increasing the Q of a Filter Through the Use of Tunnel Diodes, Solid State Journal, Vol. 2, June, 1961, pp. 30-33. sc60 Schuller, M., and Gartner, W.W., Inductive Elements For Solid-State Circuits, Electronics, April 22, 1960, pp. 60-61. sc61 Schlicke, H.M., Essentials of Dielectromagnetic Engineering (book), Wiley, 1961, pp. 135, 142, and 202-15. sc62 Scharfetter, D.L., and Jordan, A.G., Reactive Effects in Semiconductor Filaments Due to Conductivity Modulation, and an Extension of the Theory of the Double-Base Diode, . IRE Trans., Vol. ED-9, pp. 461-73; Nov., 1962. se52 Semmelman, CL. , Nonreciprocal Transmitting Devices, U.S. Patent #2,774,890, filed 1952; granted 1956. sh53 Shekel, J., The Gyrator as 3-Terminal Element, PIRE, Vol. 41, pp. 1014-16; Aug., 1953. sh54 Shekel, J., Reciprocity Relations in Active 3-Terminal Elements, PIRE, Vol. 42, pp. 1268-70; Aug., 1954. si6l Silverman, J.H. , Schoeffler, J.D., and Curran, D.R., Passive Electromechanical Gyrators and Isolators, Proc. Natl. Elec. Conf., Oct., 1961, pp. 521-29. si62a Silverman, J.H., Nonreciprocal Behaviour in Passive Systems, IRE Trans., Vol. CP-9, pp. 77-81; June, 1962. si62b Silverman, J.H., The Electromechanical Circulator, IRE Trans., Vol. CP-9, pp. 81-85; June, 1962. 78 Soohoo, R.F., Theory and Application of Ferrites (book), Prentice Hall, 1960, pp. 2, 9, and 159-61. Spenke, E., Das Induktive Verhalten von p-n Gleichrichtern bei Starken Durchlassbelastungen, Z. Angew. Phys., Vol. 10, pp. 65-88; Feb., 1958. Stone, H.A. Jr., Micro-Miniature Inductors, Electronic Design, July 1, 1957, p. 44. Stone, H.A. Jr., Component Development for: Microminiaturization, IRE Components Convention, U.S.A., 1957. Storer, J.E., Passive Network Synthesis (book), McGraw-Hill, 1957, pp. 239-41. Stern, R.H., Transistor-Simulated Reactances, Electronic Design, March 5, 1958, pp. 24-27. Stone, H.A. Jr., Theory and Use of' Field-Effect Tetrodes, Electronics, May 15, 1959, pp. 66-68. Stone, H.A. Jr., The Field-Effect Tetrode, 1959 IRE Natl. Conv. Rec, pt. 3, pp. 3-8. Strauss, L., Vave Generation and Shaping (book), McGraw-Hill, 1960, pp. 315-56. Stone, H*A* Jr., and Warner, R.M., The Field-Effect Tetrode, PIRE, Vol. 49, pp. 1170-83; July, 1961. Tellegen, B.D.H., The Gyrator, a New Electric Network Element, Philips Res. Repts., Vol. 3, pp. 81-101; April, 1948. Tellegen, B.D.H., Synthesis of Passive Resistanceless 4-Poles That May Violate Reciprocity Relation, Philips Res. Repts., Vol. 3, pp. 321-37; Oct., 1948. Tellegen, B.D.H., Synthesis of Passive 2-Poles by Means of Networks Cont• aining Gyrators, Philips Res. Repts., Vol. 4, pp. 31-37; Feb., 1949. 79 te49b Tellegen, B.D.H., Complementary Note on the Synthesis of Passive Resistance- less 4-Poles, , Philips Res. Rept., Vol. 4, pp. 366-69; Oct., 1949. te50 Tellegen, B.D.H., and Klauss, E., The Parameters of a 4-Pole That:*May Violate the Recip• rocity Relation, Philips Res. Repts., Vol. 5, pp. 81-86; Apr., 1950. te51 Tellegen, B.D.H., and Klauss, E., Resonant Circuits Coupled by a Passive 4-Pole that may Violate the Reciprocity Relation, Philips Res. Repts., Vol. 6, pp. 86-95; Apr., 1951. te52 Tellegen, B.D.H.(?), Gyrator, Wireless Engineer, Vol. 29, pp. 143-45; June, 1952. te55 Tehon, S., The Gyrator as a Circuit Element, Proc. Symp. on Circuit Analysis, Univ. of Illinois, 1955. te56 Tellegen, B.D.H., Gyrator, Electric Network Element, Philips Tech. Rev., Vol. -18, pp. 120-24; 1956-57. to53 Towner, G.H., . Artificial Inductor, U.S. Patent # 2,800,586, filed July 31, 1953; granted. July 23, 1957. to62 Tomaino , -M..P. , Modern Electronic Components, Electronics, May 11, 1962, pp. 51-74. tr56 Treuhaft, M.A., Network Properties of Circulators Based on the Scattering Concept, // PIRE, Vol. 44, pp. 1394-1402; Oct., 1956. ^ IRE Trans., Vol. CT-3, pp. 127-3 5; June, 1956. Also see^ tw55 Twiss, R.Q., Nyquist's and Thevenin's Theorems Generalized for Non- Reciprocal Linear Networks, J. Appl. Phys., Vol. 26, pp. 599-602; May, 1955. uz63 Uzunoglu,. V. , A New Look at Negative Resistance Devices, Electronics, Vol. 36, March 8, 1963, pp. 35-39. wa47 War Training Staff of the Cruft Lab., Harvard Univ., Cambridge, Mass., Electronic Circuits and Tubes (book), McGraw-Hill, 1947, pp. 666-67. 80 wa51 Wallace, B,L. Jr., and Baisbeck, G. , Duality as a Guide in Transistor Circuit Design, BSTJ, Vol. 30, pp. 381-417; Apr., 1951 (pp. 394-95 in particular). va59 Warner, R.M. Jr., Jackson, W.H., Doucette, E.I., and Stone, H.A. Jr., A Semiconductor Current Limiter, PIRE, Vol. 47, pp. 44-56; Jan., 1959. wi54 Wick, B.P., Solution of the Field Problem of the Germanium Gyrator, J. Appl. Phys., Vol. 25, pp. 741-56; June, 1954. 31 APPENDIX •< ANALYSIS OF THE 16 CONFIGURATIONS OP) SECTION 4.5 In the search for 2-amplifier AG circuits, it vas decided (section 4.4) to consider only circuits which have the skeleton of Figure 4.8(a), (terminals b and d are connected). A pre• liminary screening (section 4.5) narrowed the multitude of poss• ible circuit configurations to 16* Each such configuration represents a different connection of the input terminals of the two amplifiers. The problem at hand is to determine which configurations yield AG circuits* It is first shown that only 2 output connections need be considered for each amplifier. Thus, since each configuration contains 2 amplifiers, the problem reduces to choosing some or all of the 4 possible output connections. A systematic method is given for making this choice. A.l The 4 Possible Output Connections for Each Configuration. For each given configuration, the input terminal connections are fixed for the 2 amplifiers. It remains to connect the output terminal of each amplifier, through separate conductances, to 1 or more points in the circuit. Only the 3 points a, c, and b-d will be considered as candidates for such connections. The ground of each amplifier will already be connected to one of these 3 points, and connection of the output to this point as well would produce a loop current which would merely load the amplifier. 82 Thus, only the remaining 2 points need be considered as candid• ates for output connections. One of these 2 will already be connected to the input of the amplifier; this point will be called the "1" point, and the other the "2" point, for the amp• lifier in question. The output terminals of the 2 amplifiers will be labelled P and Q, and accordingly the 1 and 2 points for the P amplifier will be called the Pi and P2 points, etc. (see Figure 4.4). Therefore, for each of the 16 given con• figurations, an attempt will be made to design an AG by choice of some or all of the 4 output terminal connections P to PI, P to P2, Q to Ql, and Q to Q2. A.2 A Systematic Method of Analysis. The systematic analysis of these circuits is facilitated by the use of a table, as shown in Figure A.l. The table cont• ains 4 rows (labelled 1 to 4 along the right hand side) and 11 columns (labelled 1 to 11 along the bottom), of which the first 2 and the 8th one are already filled in. Each row relates to a particular Y matrix element, and both the matrix element concerned, and its desired value (equation 3.5)) are indicated in the 8th column. For each given configuration, the analysis is done in 2 steps. The first step is to fill in the diagram and voltages shown above the table, and columns 3 to 7 of the table. This is a straightforward process. The second step is to try and 83 a o c + O k 'P-Pl V P-P2 rQ-Qi b o- r •o d Q-Q2 AG Intr• Terms Produced by Amps, insic admit• i Subtotals term, P Amolifier Q Ampl] f i fir tance Y term Pl intr P2 : oi ••••' 02 via Affec" 'u- 0 (I) Ii7 via Va via Aff ec-t v, T21= " (3) I ? via I22= 0 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (ll) Figure A.l - Design sheet used for the investigation of AG circuits. 84 design an AG, with the help of columns 3 to 7 of the table (columns 9 to 11 are used to keep track of the progress made). The procedure is illustrated by an example. A.2.1 The first step. The first step is carried out as follows (refer to Figure A.2) : 1) Draw in the amplifiers on the "skeleton" provided (terminals a, c, and b-d), with their inputs connected according to the particular configuration being analysed. 2) Label the various P and Q points on this diagram, as discussed in section A.1 above. 3) Write down the expressions for the 4 voltages Vp_p-^, etc., in the space provided. 4) Fill in the blanks in columns 3 to 7 of the table. This process is explained in detail in the next few paragraphs. Columns 3 to 7 of the table are designed to facilitate the selection of some or all of the 4 output connections P to PI, etc., in an attempt to obtain a Y matrix having the form of (3.5). They are completed in such a manner that the matrix element Y-^J f°r example, can be found by merely adding together ro w (superposition theorem) the entries in the Y^2 ' which cor• respond to the particular output connections used. The terms in column 3 represent the contribution to the Y matrix made by the input conductances, G-^ and G2> and are there• in : :—: ~~ ! • Whenever the Y matrix is mentioned in this appendix,: •. refer• ence is intended to the Y matrix for the terminals a,b,c,d, as defined in Figure 4.2. 85 + a Vl0-A,HVa(Al-l) i I P-Pl V.O-A,) +V2A, P-P2 Q-Qi b 6 P2, Ql O d Q-Q2 AG Intr• Terms Produced by Amps. insic admit• Subtotals term, P Amu'lifier Q Ampl] fier tance term xintr PI P2 01 * 02 via Affec- (aw Tn=° o (0 I ? via i12= + 1 o 0 v2 0 via Affeci -6, 0 0 (3) I ? viv,a 0 T22= 0 o 0 0 (1) V(2z ) (3) (4) (5) (6) (7) (8) (9) (10) (11) for V22= 0^ PI must b*. uSZ *7""/?€ for V2y -Ve j <#2 mus^ ^je«/, *^oiS^ —* Ve . /^r Yn~Oj p2. must £e US€ refn^y '*•» co/u*,* /Ij reputeah A Figure A.2 -. Design sheet for configuration number. 8, D-1-. 86 fore called "intrinsic terms". They are found by removing everything but G^ and G^ from the circuit skeleton, as shown for the D-l- configuration in Figure A.3. G G Gl + 2 ~ 1 -G, G, I 2j Figure A.3 - Equivalent circuit for finding the "intrinsic terms" for the configuration D-l-. The terms in columns 4 to 7, the "amplifier terms", repres• ent the contributions to the Y matrix which will be made by each output connection (if it is used). These terms can be deter• mined, for each output connection, by drawing an equivalent cir• cuit similar to that of Figure A.3. The equivalent circuits for the two P connections in the D-l- configuration shown in Figure A.2 are given below in Figure A.4, along with the terms they produce. The terms can also be obtained directly from the voltages Vp_p^> etc., with the help of columns 1 and 2 of the table. For instance, the P2 connection would cause a current to flow from a to b-d, and thus affect only 1^ (see column 1 of the table). This current would be equal to G V 4 P-P2 = Wl-V + V2G4AX, The intrinsic terms will be the same, of course, for all of the 4 configurations D+1+. 87 O c r+ VP-P2 i- —Od CKjU-A^-- G3(A1-1) "G^l-A^ Vl G3(AX-1) G3(l-A1) 0 0 Figure A.4 - Equivalent circuits for finding some of the "amp• lifier terms" for the configuration D-1-. Y mat• rices are given. in the same direction as 1^, and thus its effect on 1^ via (see column 2 of the table) would yield the admittance term G^(l-A^), and its effect on 1^ via would yield the term G^A^. These admittance terms are entered in the proper spaces in the table (column 5 in this case). When columns 3 to 7 have been filled, the first step is completed. A. 2.2 The second step. The second step is intelligently to select which output connections to use. Refer to the table of Figure A.2. In the row for Y^-^, the intrinsic term is (always) +'ve, and therefore a -'ve amplifier term is needed to make Y^ = 0; any of the 4 output connections could be used for this purpose. In the row for Y^2> the intrinsic term is (always) -'ve, and therefore a +'ve amplifier term is needed to make Y, - = +G : either con- 1 12 m' nection Pl or P2 could be used for this purpose. In the row for 88 ^2±9 ^e intrinsic term is (always) -'ve, and thus no change is needed to have Y_, = -G (=-G.. here) , so this row need not be 21 n 1 ' row ne considered for the time being. Finally, for the Y22 > ^ intrinsic term is (always) +'ve, and therefore a -'ve amplifier term is needed to make ^2 = ^' only "the PI connection can provide such a term. Now, after looking at all 4 rows, what can be concluded? Obviously, the PI connection must be used, for it is the only one which can be used to set ^22= 0. The elements of the Y matrix when G^ is connected from P to PI are found by adding the terms in columns 3 and 4. Of course, the condition Y22 = Gx + G3(l-A1) = 0 ...(l) must be imposed; this is equation (l) in Figure A.2. A^ must be +'ve to satisfy this condition. The elements of the resultant Y matrix are written in column 9 (columns 9 to 12 are provided for this purpose). Now the entries in column 9 must be compared with those in the columns for the still unused output connections P2, Ql, and Q2. This comparison is similar to that made in the paragraph above, except that this time the row for neea- not be consid• ered. It is seen that connection Q2 must be used, for it is the only one which can be used to make Y^^ -'ve. Note that this connection will also affect Y22, so the condition of equation (l) must be altered to read Y22 = Gl + G6 + G3(l-Al) = 0...... (la) 89 The condition for Y to be negative is 21 Y = -G1 + G3(A1-1) + G6(A2-l) 21 • 9 • (2) n A2 must be -'ve to satisfy this condition. The, resultant Y matrix is written in column 10.. Another comparison is made, this time between column 10, and the columns for the unused output connections, P2 and Ql. It is seen that connection P2 must be used, both to, set - 0, and to make Y _ +'ve. The condition imposed is T = G + G + (3) ll 2 n G4(l-Ax) ;= 0. The resultant Y matrix is shown in column 11. . Since this matrix has the form of (3.5), the analysis stops here. Conclusion: the given configuration yields an AG. The foregoing analysis is described concisely in the space below the table in Figure A.2. The configuration just analysed actually yielded an AG; however, some of the configurations will not (e.g., the D+l- configuration, analysed in Figure A.6). The analysis sheets for several other configurations are given in Figures A.5 to A.7. _9_0_ Pi,ai a 0— V p P1 =M(Ari)+v2(i-A,) V. V V V V2-Q2 =V,A - 2 P2 o d 2 AG Intr• Terms Produced by Amps. insic admit• Subtotals term, P Amplifier Q Amp!] f i pr tance term y Pl P2 Ql • 02 intr via =o in 0 Affec- = Vi 0 (0 via 0 o 0. v2 via o Affec-t v, - /^o-r -Ve; For- y/f~o, ^ Figure A, 5 -. Design sheet for configuration number 5, D+1-+. 91 Pi Q2.' 'P-Pl V P-P2 Q-Qi b 6 -O d Q-Q2 P2,QI AG Intr• Terms Produced by Amps. insic admit• Subtotals term, P Amulifier 0 A.mpl If i er tance term intr PI P2 •. Ql - 02 via G5(l-A2\ (0 Affec- &,+<*, 0 via T 0 0 i.; A-* v2 via 64 A, 0 Y Affect v, 21= " (3) I ? via 0 6yH) 0 ^6 *22= (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) For- ya , P/ J>€ OfS fee Q2. t^^st he. otS^d. '* 9\2. Must oe. — Ve. 7^e so,{> to~tci/s are. For- Yzz:~Oj 5e£ 7^€re^r« <^/-e cahf^rci"Hoy\ does /\/bT~ yfe/d Qh f\G0 Figure A.6 - Design sheet for configuration number 6, D+1-, 92 QZ iP PI, Ql , a Q rP-Pi P-P2 b o- r P2 ^ Q-Q2 AG Intr• Terms Produced by Amps. insic admit• Subtotals term, P Amplifier Q A.mpl -if i Pr tance y term Pl P2 02 mtr Ql # via T 0 ll= & 0 Affec- >V. < (0 I ? via 1 Va 0 T12= + via T = -Cr, (3) Affec-t v, 0 21 " I ? via i22= o 2 0 0 0 o 0 v2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) /Sr y,/-0, e/"Her- PI cr PZ m«S"6 6e use,*/. If PI rSdSeJ} 67 e re^/retj cawJ^r^ Kmfas V^Oar u/^ll. •/laar« r/*ice d crkhvc c^nec^ 1V1C affect yZf y*~P£with used* 7fi9. coh-c/rfian imposed TS /hrs c/rctf/t? /y e^t/Vdle*?^' 6* if^e -oLive* &y Figure A.7 - Design sheet for configuration number 11, D-2+.