2899-9-T ENGN UMR0011 DEPARTMENT OF ELECTRICAL ENGINEERING 0ri^~ I~ELECTRON IC DEFENSE GROUP': a~~~~~0 Technical Report No. 96 Some Cnsderaonsiertn of Four-Frequency Nonlinear Reactance Circuits By: D.K. ADAMS Approved by: H. W > Under Contract With: CONTRACT NO. DA-36-039. sc-78283, DErT. OF ARMY PROJ. NO. 3-99-04-106, PLACED BY:,SIGNAL CORPS ENGINEERING LABORATORIES, FORT MONMOUTH, N. J. Administered by: September 1959 ==~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR SOME CONSIDERATIONS OF FOUR-FREQUENCY NONLINEAR REACTANCE CIRCUITS Technical Report No. 96 Electronic Defense Group Department of Electrical Engineering koBy: D. K. Adams Approved by: Charles B. Sharpe Project 2899 TASK ORDER NO. EDG-4 CONTRACT NO. DA-36-039 sc-78283 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-106 September 1959 THE UNIVERSITY OF MICHIGAN ENGINEERING LIBRARY

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv ACKNOWLEDGEMENT v ABSTRACT vi 1. INTRODUCTION 1 2. GENERAL ENERGY CONSIDERATIONS 3 3. SMALL-SIGNAL THEORY 10 4. CONCLUSION 31 APPENDIX 33 REFERENCES 37 DISTRIBUTION LIST 38

LIST OF ILLUSTRATIONS Figure Page 1. General circuit model. 2 2. General characteristics of power conversion gain from CI to D2 and w5. 6 3. General characteristics of power conversion gain from o2 to e1 and w. 8 4. A small-signal model. 11 5. Representation of a modified external admittance notation. 14 6. A locus of external circuit adjustments for parametric amplification at C2. 22 7. A cascade of upper-sideband up-converters of the type described in Case 2. 28 8. A positive-input-conductance, nonlinear reactance amplifier theoretically capable of unlimited gain. 29 iv

ACKNOWLEDGEMENT The author wishes to express his appreciation to Professor C. B. Sharpe and to Mr. R. T. Denton for frequent helpful discussions during the course of this research.

ABSTRACT Several advantages of multiple-frequency nonlinear reactance circuits are described in this paper. In particular, a circuit is considered in which a nonlinear reactance couples four basic frequencies: CO), (D1, c2o, and C3 which are so related that w2 =- Oc + ol and w3 alO - 0l. Here,,ca is taken to be the power source or pump. It is found to be desirable to allow for the possible presence of the pump harmonic, 2rto, and individual cases are characterized accordingly as au2 is present or not. The major results are: (1) Unlimited amplification gain is theoretically possible at frequencies higher than the pump, by reflecting negative input resistance at wp2, but without relying on any effects due to the pump harmonic. (2) Unlimited up- or down-conversion gains between ml and )2 are theoretically possible in the presence of the pump harmonic, but without reflecting negative input or output resistance. (3) Unlimited amplification gain is theoretically possible at frequencies both lower and higher than the pump fundamental, without reflecting negative input resistance. vi

SOME CONSIDERATIONS OF FOUR-FREQUENCY NONLINEAR REACTANCE CIRCUITS 1. INTRODUCTION In recent years considerable attention has been given to the unusual properties of nonlinear reactance modulators. The basic attribute of these circuits that has been exploited thus far is frequency-conversion with power gain. Since nonlinear reactance conversion gain can be obtained in the presence of a signal (or reference frequency), a local oscillator (or "pump"), and just one of their sidebands, emphasis to date has been on three-frequency circuits. It has been demonstrated, however, that remarkably different modulator characteristics arise, depending upon the particular choice of sideband (Ref. 1). In the upper-sideband case, the magnitude of conversion gain is limited to the ratio of the output frequency to the input frequency. In the lower-sideband case, arbitrary conversion gain is possible through regenerative action, which reflects negative resistance into both the input and output terminals. Such negative resistance also enhances the input signal; this-effect is known as parametric amplification. In view of the striking differences between the two cases cited, the simultaneous presence of both sidebands causes the anticipation of interesting effects. Several of these effects will be investigated in this paper, and the following major results will be demonstrated: (1) Parametric amplification is possible at frequencies higher than the pump frequency with no reliance upon effects due to pump harmonics. (2) In the presence of the first pump harmonic, arbitrary upor down-conversion gain is possible between the reference frequency and the

-wo,2Wo I W2l 2 W 3 T IFI FI LERTER T F I LTER T F I LTER 1 rte t W, W, W, W, I w I - 2 1 Io I10 120 I30 Fig. 1. A general circuit model for the nonlinear reactance problem under discussion.

upper-sideband with-no reflection of negative resistance at either frequency. Furthermore, the reflected input and output resistances can be made relatively insensitive to gain. With nonlinear reactance elements possessing strong nonlinearities and good high-frequency response, the value of the first result lies primarily in its contribution to the general understanding of nonlinear reactance circuits. However, the second result has considerable potential value in spite of its dependence on the presence of the first pump harmonic. As a consequence of statement (2), up- or down-converters of arbitrary gain can be cascaded. In the special case in which an up-converter of the type mentioned above is followed by a similar down-converter, a nonlinear reactance amplifier can be constructed which does not rely on the reflection of negative resistance at the input terminals. Therefore, unlike conventional parametric amplifiers and negative-input-resistance converters, the full potential of this improved converter circuit can be obtained without the use of circulators. 2. GENERAL ENERGY CONSIDERATIONS The basic problem to be considered in the following analysis is that of a lossless, nonlinear reactance in the presence of signals at 0D, l' and at their upper and lower sidebands: u2 = 0 + wl, and ='0 - ~1. In general, the signal at w0 will be considered as the power source or pump and will be assumed to have arbitrarily-large available power. It is practical to allow also for the possible presence of the pump harmonic, 2w0, since it will be shown to provide valuable effects in some cases. A circuit model for this problem is suggested in Fig. 1, in which a nonlinear capacitor has been chosen to represent the nonlinear -53

reactive element, and ideal filters have been assumed (1) to short circuit all frequencies except those chosen above, and (2) to allow these frequencies to be coupled only by the nonlinear capacitor. While the assumption of ideal filters seems to be very restricting, the external circuits will ultimately be constrained to be resonant with fairly high Q's. Thus, the ideal filters can eventually be removed without seriously affecting circuit operation. Although small-signal analysis will be the primary medium employed, a general foundation for the results previously cited is suggested by the general properties of nonlinear reactances. It has been shown by Manley and Rowe (Ref. 1) that the following general statements can be made about lossless, nonlinear reactances in the presence of sinusoidal signals with frequencies from the set Dw n + + r, where mn =, 2,.. mn - o-l o- o n, rz 0 _ mn W O E ~ (la) 00 00 Z _ n-w- = O mn (lb) n=-O m=-oo () Here, W is the average power entering the nonlinear reactance at o m mn mri ana W = Wm n The only restrictions on Eqs. I are that 1/o be mn -m, -n irrational, and that the nonlinear reactive element have a single-valued characteristic. For application to the problem of interest here, Eqs. 1 reduce to Wo + W20 w W...- + — + = o (2a) W1 W 2 W -- + --- = 0 (2b) C:1 32 c~3 -4

where the double subscript notation has been dropped, except in the case of average power at the first pump harmonic, W20. It is important to note that the artifice of restricting the Manley-Rowe relations to a particular set of desired frequencies does not necessarily mean that a predicted result will be generally realizable without the presence of certain of the eliminated frequencies. For example, if the external circuitry reactively terminates a particular frequency, it would be eliminated from Eqs. 1. However, through internal conversion this frequency may be a hidden mechanism behind effects that seem, from Eqs. 1, to be independent of such frequency. This situation arises in this problem in connection with the first pump harmonic. Equation la suggests nothing about the relative importance of the pump and its harmonic, and, in fact, falsely suggests that they are interchangeable. Since it is the reactive power of the pump harmonic that is important in this case, W20 will be ignored henceforth. Further study of Eqs. 2 will be divided into two parts. Consider first the application of signals at ol and (0, with 2 and o3 seeing passive loads only. Thus, W2 and W are nonpositive and Eq. 2b yields the following expressions for up-conversion power gains from w to w2 or (w3 W2 G 2 1 (3a) P12 1 W1 3l 2 WW3 P1 Wl aWl 1 -W G -. - = W(3b) 1W2-3 -5

wo 0 /G (WO WI W3 W0 W2 ) LW2 W __.0 _ / WP: W2 WO2 WI Fig. 2. Sketches of Eqs. (3) showing the regions of positive and negative conversion gains from m1 to o2 and 3 vs. the ratio of output powers at 03 and w2' Pump power is applied at o, and Wo/W is the ratio of pump to signal power. 6

W- 0), W(Wu, 1 + WO =. 3 (3C) w32 W23 To illustrate the remarks to follow, sketches of Eqs. 3 are shown in Fig. 2. In the limit where w3 is short circuited (W3 = 0), G has the well-known value O l/1* However, G increases as (W2W /WDW2) is increased, and arbitrarily-large positive values of G can be achieved P12 in the vicinity of (W 3JW23A) < 1. This suggests that large conversion gain is possible without reflecting negative input conductance at CDl For (W 2W/W2 ) > 1, G and G are negative. Hence, this condition defines a general region of potential instability in which parametric amplification is possible at a1' It is important to note that negative conversion gains do not necessarily imply the reflection of negative conductance at the input frequency terminals, although this has been the case in previous nonlinear reactance circuits. Equations 1 would also predict negative conversion gains if the input signal were completely absorbed at one terminal pair, amplified, and then expelled at the same frequency (along with other conversion frequencies) at another terminal pair. The realization of this effect (without using circulators) is one goal of this paper. The second case of interest is with signals applied at c0 and C2, and with only passive loads for ul and c3,* In this case Eqs. 2 yield 2 1 5 G = _1 _ 1 G - C- _W (4a) 2l1 W2 aw2 1 W1 1w3 -7

II'I I I w~ I CA.3 Wo 2 -W WP2_ / JWo W2 | - / Wi ^Gp2-27~~I >~Wo Fig. 3. Sketches of Eqs. (4) showing the regions of positive and negative conversion gains from C2 to w and c3 vs. the ratio of output powers at 03 and w2 Pump power is applied at o0, and W /W is the ratio of pump to signal power. 8

w3 1 G ~-..... L — -.. (4b) P23 W2 O 1- WP Wo 2 - 1 O -- W3 (4c) W2 C2 W wcn 1W3 which are sketched in Fig. 3. For (W3 1/Wlw3) < 1/2, G21 and G2 are positive, but WO/W2 is negative. Therefore, the pump circuit is unstable in this region and the signal at a2 will act as the power source. For (ao/a2) < (w3 /w1 3) < 1, Gp is greater than unity. Therefore, this 21 is a region of arbitrary down-conversion gain with positive input conductance. For (W /W ) > 1; G and WW2 are negative. Thus, 5 L5211 a region of potential instability exists at.,, which suggests that parametric amplification is possible at this frequency.l Again it should be noted that Eqs. 1 in no way guarantee that just any nonlinear reactance will yield the results above. For a particular nonlinear reactance, Eqs. 1 may have only the trivial solution, W = 0. mn For example, it can be noted from Fig. 3 that parametric amplification at o2 is predicted even in the limit where a), is short circuited (W1 = O). However, with the nonlinear reactance model employed in small-signal analysis (a model which satisfies the Manley-Rowe equations), it can be 1. The possibility of parametric amplification at o2, using wl and 03 as so-called "idler" frequencies, was originally proposed by Hogan et al (Ref. 2). Since this work seems to be based on purely qualitative arguments, parametric amplification at a>2 will be treated in more detail here. It will be shown that rather unobvious circuit adjustments are required to realize this effect at a small-signal level. 2. This special case has been previously treated through small-signal analysis by Bloom and Chang (Ref. 3). -9

shown that the pump harmonic (a0O) is necessary for parametric amplification at 2, if W1 = 0.. SMALL-SIGNAL TEORY Although the Manley-Rowe equations provide a basis for predicting general properties of nonlinear reactance circuits, they yield no specific information on impedance levels, bandwidth, or the importance of extraneous frequencies. Such information requires further knowledge of the element in question plus mathematical techniques for handling the associated nonlinear circuits. Fortunately, several approximate characterizations of nonlinear elements lend themselves to standard analytical methods. In general, these involve some form of smallsignal analysis, the most basic of which neglects responses of the nonlinear element to all signals except the power source and its harmonics. Small signals at all other frequencies then see essentially a periodically varying circuit element at fundamental frequency w0 (Refs. 4, 5). In this sectbn, a nonlinear capacitor will be assumed to be driven by a power source at w0. To employ the small-signal model outlined above, the power source and nonlinear capacitor will be immediately replaced by a periodically varying capacitor for which the following Fourier representation applies. n=-oo *x 1 where C C.n Now, assuming the external circuitry essentially short-circuits all small-signal rrequencies except al, w2 and (D, the 1. * indicates complex conjugate. 410

+130 g3 Y3,, I3f, Y3,*'3 b3 YIN,, IN, Y2,r2 _4.~4 I 1,or1 __ Id 2 0 ^3 t__________ I [ FILTER [ [ normalized admittance, according to the definitions in E. (12). ~FILTER FILTER > C (Wo) Fig. 4. A small-signal model, based on Eq. (7), for the nonlinear capacitance circuit shown in Fig. 1. Here, 7 represents a normalized admittance, according to the definitions in Eq. (12).

small-signal voltage ana current across the time-varying capacitor will be given by So~J t.'jo t V(t) = 2 (Ve n + Ve ) (6a) n=l n 35 jt * -jt (6b) I(t) = ~ (I e n + Ie ) +(terms representing n=l short circuit currents at other frequencies) In Eq. 6b, it is necessary to consider only terms in l' D2' and w3, since these are the only current components accessible to external terminals. By combining Eqs. 5 and 6, the following admittance relation is found between the currents and voltages across the time-varying capacitor at each of the small-signal frequencies (Ref. 4). II mJalCo 3JlCl lCl V1 I2 J2C 1 2Co 2C2 V2 (7) ~3 Cl -JW3C2 J3Co V 3 31 3 3o V It is important to note, due to the terms in Co in Eq. 7, that smallsignal effects on Cw1, 2 ana CD will generally include effects of both the pump and its first harmonic. Thus, for a more general formulation of this problem, terms in C2 will be included. Equation 7 suggests the linear circuit in Fig. 4 as a smallsignal model for this problem. The external admittances present in Fig. 4 are now easily included with Eq. 7 to yield the complete admittance relationship shown below. -12

o10 Yl+jlCo V lC J I 1v o20 2CI Y2+2Co j2C2 V2 (8) ISO -Sw C -jw5 C2 Y<jW C V 30 3 1 3 2 y L3 3 In general, two basic phenomena will be investigated with the aid of Eq. 8: (1) power conversion from wm to (D; and (2) parametric M ~n amplification at wD; where m, n = 1, 2 or 2, 1, respectively. Referring m to Fig. 5, if Gt is the ratio of the output power at Dn to the availmn able generator power from a source at c o then 4gs (gmIN - gm) Gt = -I.mx G (9) tmn YmN I2 Pmn where Y = g + jb is the driving point admittance at terminals ^MI MIN MIN (m), gs is the internal admittance of the input generator, and Gp is given by Eqjs. 3a or 4a. Since G can be alternately expressed by pmn n n V 2 g G n= m (10) Pmn m rIN m Eq. 9 becomes 4gsgn Vn( | | (11) Gt iNI V Ultimately, conditions will be sought under which Gt is arbitrarily mn large. As will be shown shortly, this condition usually arises only when IY | approaches zero. Since the vanishing of Ym is also a sufficient condition for parametric amplification at (Dm, first attention will be directed toward this quantity. -13

m IN Wm Gpm n NONLINEAR ImoO | 0 9s I9m-9s t bm REACTANCE bn, CONVERTER I 1_.__ __II Ym SOURCE Fig. 5* The modified external admittance notation that is assumed whenever external sources are present, as in the derivation of Eq. (9).

Considerable simplification of the ensuing analysis results if the following normalized admittances are introduced. Y +ja 1 C21 I_______ 1IN 2 7 = 1 -; ( 2a ) Y + C 2 (12b) 2 2 C271iN 21 Y Y —JW C 0NiN 73 = 3c... (12c) 3 I 2 - W32 The notations 7k = k + Jik and 7k = N + JP (k =, 2, and 5) 5 ak "kIN IN IN will also be adopted. The input admittances now take the following normalized forms. r3 - 72 + 2jcp, = Y7 + y y (13a) -IN 7 -2 Y, - 71 + 2jCp Y = y2 +,. - (13b) 7, + 72- 2jcp 7 3 = 7, 2 2jp (13c) 5IN 7172 Here, 2 2 (14) IC C21 and hence is a function of only the relative phases or C. and C2. Equations 13 are assumed to apply simultaneously, ancd express both the normalized input admittances presented to an ideal generator at any -15

terminal, or the normalized output admittances presented to an ideal voltmeter at any terminal. Therefore, whenever real sources and meters are applied, their internal admittances will be assumed to be lumped with each Yk or 7k (k = 1, 2, 3), as in Fig. 5. The form of each of Eqs. 13 readily identifies the physical significance of each term. The first term in each expression for 7 is kIN the normalized load admittance at terminals k (k = 1, 2, 3), while the second term is the corresponding reflected admittance at terminals k. Inspection of Eqs. 13 reveals that many sets of passive loads exist for which one or more of the 7k have negative real parts. This situation IN will be interpreted as complete instability, and hence Eqs. 13 will be considered physically significant only when the real parts of yl,' Y2 IN IN and 73 are simultaneously positive. In other words, a negative reflected'IN conductance at any terminal pair will be considered useful only when the external conductance at that terminal has the greater magnitude of the two. Since circuit adjustments are desired that yield large values of Gt in Eq. 11, the zeros of Ymi and the poles of |Vn / Vmi will be of immediate interest. It follows, from the inversion of Eq. 8, that the poles of IVn / Vm| occur when 17m7 - 11 = 0 (m = 1,. 2). However, by Eqs. 12 and 13, this condition usually yields a pole for Ym (m = 1, 2). When this is the case, then, the zeros of YIN (and hence mN ) are the only infinite gain points. The only exception to this observation occurs when Eqs. 13a and 13b are made independent of 73, which is treated in the section marked Case 2. -16

Case 1. IC2 = 0 The first application of Eqs. 13 will be to the true four. frequency problem in which all effects of the pump harmonics are assumed negligible. This corresponds to letting IC21 approach zero, whereupon Eqs. 13 reduce to the following limiting forms: Y3 - 72 71 = 1 +.- - (15a) l i Y2Y3 73?IN 2 1 75Y2 (15c) y = y - 35N 3 7172+1 Note that C,21 cancels identically from each of the above expressions. Therefore, it will be required that Eqs. 15, and those to follow in this case, have solutions other than the trivial one, IC21 = 0. Since Eqs. 15a and 15b cannot be made independent of 7 in this case, only arbitrarily small values of 17m | (m = 1 or 2) will IN allow unlimited conversion gains from co1 to W2, or vice versa. Therefore, when IC21 = 0, the possible advantages of four-frequency circuits over conventional three-frequency circuits are: (1) the reflection of useful negative conductance at ul in the presence of w2, and (2) the reflection of useful negative conductance at D 2. Since the latter result is the more novel of the two, in that it provides parametric amplification at a frequency higher than the true pump frequency (i.e., %>o, since IC2| = 0), primary consideration will be given to statement (2). This study will -17

be done under two constraints: (1) a, a, and a must be positive IN IN TIN for stability; and (2) 1' 2, and 3 must be small (preferably zero) IN IN ^IN at the center operating frequency. The latter condition makes all three terminals nearly resonant, which is desirable as a practical means of eliminating the ideal filters that are tacit to this analysis. Only approximate resonance is required at the external terminals in this case, because it will be shown that some detuning is necessary for parametric amplification at 2 to occur. For purposes of introduction, a special case of four-frequency operation will be described first. One technique for resonating the external terminals is to make Ad = 2 = P = 0, which means physically that external reactances are chosen to resonate C independently at each terminal. This choice of external reactance, which has been generally satisfactory in conventional three-frequency circuits, is attractive from the practical point of view because it allows circuit adjustments in the "cold" condition (i.e., with the pump turned off). However, this tuning procedure will be shown to be unproductive in this case. Case la.!C2 0; = 1 = = 0. Under these conditions, Eqs. 15 become a-a a = a1 + a(16a) 1IN 1 32 a a2 =a2 +-aa-1 (16b) 2IN + 3all-18

Ta =ac a-y2+ 1 (c(16c) O3IN 3 Ca +l Inspection of Eqs. 16 shows the condition for stability to be a1a5>1. Therefore, no useful region of negative reflected conductance exists at w2 and the minimum value of aN is limited by circuit stability to 1/a2. 2 IN Thus, parametric amplification at 2 is not possible in this case, and the presence of u2 limits the gain by parametric amplification at wD1 It also follows that little practical improvement of conversion gain results over that possible in conventional three-frequency circuits. Thus, from the point of view of gain, little reward is offered by fourfrequency circuits whose external terminals are independently resonant. It will now be shown, through a more general investigation of Eqs. 15, that the desired negative reflected conductances at ol and (2 can be obtained with more arbitrary tuning techniques. These techniques are based on the following general theorems, which pertain to the fourfrequency time-varying reactance circuit under consideration here. Theorem 1: If IC2 = 0 and al, a2i aN, then P2 0, IN IN IN IN or is a necessary condition for parametric amplifiIN IN cation at (2. Proof: See Appendix. Theorem 2: If I |C1 0, the necessary and sufficient conditions for aIN a2IN' IN > 0 and N = = 0 (i.e., a stable circuit with resonant terminals at ml and w3) are: P3 = 0 (17a) 62 = 1172l 2 (17b) -19

~-a P~~2 a, - T> [T a lesser of (1, 2) Proof: See Appendix. It should be noted that Theorem 2 can be applied to conventional parametric amplifiers in the case where the upper sideband (X2) is not completely shorted, but where the pump harmonic is supressed. In this case, Eq. 17b indicates that "cold" tuning is not sufficient and some retuning must accompany changes in pump level. In view of Theorem 1, parametric amplification a 2 will be approached by only requiring the terminals at w2 to be resonant (2IN 0), for then IY2 | approaches zero as a2 approaches zero. The additional IN IN assumption that P = 0 will be shown below to lead to useful solutions. Case lb: IC2 = 0; 3 = 00; 1 0$ 2 O0. Equations 15 now yield 2 < a- < (18) 1721nd (2) the refl as the condition for: (1) stability at! ana O, and. (2) the reflection of negative conauctance at w2. It Eq. lI is satisfied, the condition ror resonance at co2 then becomes: 2 12 1,!2 1 (a- ) + (1- 2 = ( (19) 3 2 2 If Eq. 19 is satisfied, the condition for stability at m2 is P2 < BPl 2 (20) 1. If P3 = 0, it can be shown, as a3 decreases, that )3 never becomes unstable before w1 becomes unstable. Thus Eq. 18 only explicitly defines the region of stability at o1. -20

These conditions are made clearer in Fig. 6, which is a plot of Eq. 19 for a1a3 < 1. Here, p1 is considered to be the independent variable determining the magnitude of negative reflected conductance at w2. Point A is the threshold of parametric amplification at 2,' point B marks the threshold of circuit instability at 1l (Eq. 18), while infinite gain at 2 corresponds to the point C (Eq. 20). Therefore, useful parametric amplification at w2 may be limited to the finite gains corresponding to the range between points A and B on the tuning curve in Fig. 6. This limitation on gain can be 2 2 overcome by choosing a2 = 2 which makes the points B and C coincide. In this case, the negative conductances reflected at both ol and w2 allow arbitrary gain at either frequency. It can be shown that a similar result is possible if the terminals at 1 are constrained to be resonant while those at w2 are detuned. Thus, in the notation of Fig. 4, sufficient conditions for arbitrary gain by parametric amplification at w2 (and at a1), with IC21 = 0, are: b3 + 03C = O (21a) 5 5~0 2 b2 (21b) 92 2(f b b 1< 2 < 1 (21c) 2 1l2 Icll2 [ l2 W [W C w22 2b] 2 (21d) culI 2 + 2 2b 2b It is important to note that conditions 20 or 21c satisfy Theorem 2 only at the point of infinite gain (5152 = 1). Thus, as previously mentioned, the terminals at 1l and (3 are not resonant in this case and suffer detuning at center frequency by an amount given by: -21

aF —3 3 2 l^2 2 ^ 0 I C / / INCREASING I\ ~I / / GAIN AT W2 -al22 -- INSTABILITY AT w, INFINITE GAIN AT W2 Fig. 6. The locus of external circuit adjustments for tuning and parametric amplification at c2, as described in Case lb.

1011 b = b (22a) 1 N 1 2b2 IN 2 ~~b'2b2c?13|Cl 2(2blb2 - (D)2l2 12 (22b) 3IN (2g b2 + 1|21CJ12)2 + (2b b2- al21cll2)2 Equations 22 indicate that the amount of detuning at cu1 and (3 is very small when the gains at c)2 (or at (1) are large. Therefore, if 1C2| t 0, the operation described in Fig. 6 (with points B and C coincident) allows: (1) arbitrary gain by parametric amplification at either Cl or >2; and (2) arbitrary up- or down-conversion gain between ml and 2'. Both these effects arise through the reflection of negative conductance at the input terminals and are accompanied by some detuning of the terminals either at ol or m2. Case 2: IC21 0O It has been shown previously that the design of idealized, three- or four-frequency, nonlinear reactance converters for large, smallsignal, conversion gains (i.e., small-signal gains considerably greater than the ratio of output frequency to input frequency) must invariably rely on the reflection of negative resistance at the input terminals. Consequently, such devices frequently employ circulators as auxiliary equipment in an effort to reduce noise figures and to increase gain-bandwidth products. It will now be shown, when |C 2| 0, that arbitrarily large conversion gains are possible between l1 and w2 without reflecting negative resistance at either frequency. Thus, the importance of circulators in high-gain, nonlinear reactance converters can be eliminated. -23

The technique for achieving the result cited above is to choose external circuit components that allow Vn / Vm| to be arbitrarily large (see Eq. 11) without producing a similar effect on Y (m, n = 1, 2 or mIN 2, 1, respectively). It has been previously remarked that this condition can be achieved by making Eqs. 13a and 13b independent of 73, which can be accomplished only if Y1= 2 = cp + jcp (23) Obviously, only the positive square root is permitted here, as will be the case henceforth unless (+) signs are specifically indicated. Substitution of Eq. 23 into Eqs. 13a and 13b yields 7 = y = 2 1cp (24a) "IN 2IN Thus, when Eq. 25 is satisfied, the terminals at l1 and w2 are conjugately matched for all 7y. Similarly, Eq. 13c reduces to 7 7 3 -cp + jcp (24b) 3IN which indicates that a constant negative conductance is reflected at c3 Consequently, z3 > vl-AP must be required for stability. Although 7rm (m = 1, 2) has been made independent of y3, it IN can be shown that Gt still has a useful pole as a function of 73. By mn solving Eq. 8 for IVn / V m, and using Eqs. 12, Eq. 11 yields the following gain expressions: lu (a 3 [ + -(i3 + (2 +p)(a) G1 2 -, -- -- (25a) 12 2 (a - v )2 + (3 + p)2 j 1 I-24- +

Here, the (+) signs arise because cp specifies only the magnitude of the relative phase angle between C1 and C2 and not its sign. In each of Eqs. 2 Lj* 25 the upper sign corresponds to Im (C2C2) > 0 while the lower sign * 2 corresponds to Im (CC2 )<0. Thus, for a given value of cp, there are two possible up-conversion gains and two possible down-conversion gains. In the case of negative square roots, both Gt and Gt reduce to the ratio 12 21 of output to input frequency, independently of y3; while with the positive square roots both Gt and Gt become arbitrarily large as 73 approaches r- "12 21 the value l-c + jcp (assuming, for stability, that it does so with a3 > f l- ). Within a single converter stage, however, Eqs. 25 can not be simultaneously satisfied with positive square roots, which yields the useful result that each converter stage is semi-directional. The directivity is dependent on both the forward gain and the relative magnitudes of the frequencies employed. Perhaps the most practical operating condition is with the terminals at w3 resonant (i.e., 3 = -cp), whereupon Eqs. 25 reduce to: a + - 22 GD [ < r(26a) 21 e2 a Gt21 = w[]2 (26b) In the positive square root cases, the conversion gains now depend entirely on a, which means physically that variation of the external circuit Q at o3 controls conversion gain without changing the input or output admittance at any terminal pair. -25

The fact that two completely different conversion characteristics arise according to the sign of Im (C1C2) is somewhat surprising, since the input and output admittances at all three terminals are independent of this sign. From Figs. 2 and 3 it is evident that the fixed gain cases of Eqs. 25 and 26 must correspond to W3 = 0, which can be verified by solving Eq. 8 for V3. With sources first at C1 and then at 2 P) V3 is found to vanish independently of 73 whenever the negative square roots occur in Gt or Gt, respectively. However, in the positive square root cases, 12 21 V3 increases with Gt2 and Gt, for sources at lw and C respectively, which is again in agreement with Figs. 2 and 3. It is interesting to also assume a source at o3 and to evaluate the quantities V1 / V3 and V2 / V. It can be shown that couples only to the wl terminals when Gt employs the positive square root, and 12 that w couples only to the co terminals with the positive square root in G. Thusthe sign of Im (C2C2) determines the coupling between wl, c2, and o3 once the source frequency is specified. In each case, coupling to the external terminals takes place for only two of the three possible frequency pairs. This qualitatively accounts for the two types of up- and down-conversion found in this case. In summary, for a given magnitude of phase angle between the pump and its harmonic, the results above require the following external circuit adjustments: C0 C2 At 0wl b + OlCo; (27a) 1 10 Q1 V1 = f7i (2a 1 Co At U2: b + Co = P (27b) -26

At w 3: b3+ +3CO; Q3 < Q2 (27c) It can be seen from Eqs. 27 that each external terminal will be detuned in the absence of the pump, unless cp =. Also, the necessary external Q's increase as (p approaches unity. Therefore the condition cp = 0 appears to be the preferable operating condition, since it calls for minimum Q circuits and permits circuit tuning prior to the application of the pump signals. Since the results of this section have led to a small-signal converter with: (1) arbitrary power gain, and (2) conjugately matched input and output terminals, several applications immediately suggest themselves. The first application, which is shown in Fig. 7, is to cascaded up-converters. Here, the gain of each stage can be individually controlled at the respective lower sidebands without requiring any adjustments of the interstage networks. Since each stage of this cascade is an up-converter, it is necessary that Im (C2C2) > 0 for each stage, which makes the maximum gain in the reverse direction the ratio of input frequency to output freauency. A similar result follows if the sign of Im (CC2) is reversed in each stage, to make the cascade a high gain down-converter. A modification of this application is shown in Fig. 8, where an up-converter and a down-converter are placed "back-to-back," such that the input and output frequencies are the same. In this configuration the only basic difference between the two stages is in the sign of Im (C2C2). Therefore, this circuit yields a nonlinear reactance amplifier with: (1) positive input and output conductance, (2) arbitrary forward gain, and (3) a maximum reverse gain of unity. It is interesting to apply the Manley-Rowe equations to the circuit in Fig. 8. Since only l1 and o3 appear at the external terminals, -27

I WLo-WI Wo-Wo- I II I I, +o < n w. 2 I. <. 2W1 O+ I 2- WO T 20 Fig. 7. A cascade of upper-sideband up-converters of the type described in Case 2. The reflected conductances, which are indicated in each box with appropriate signs, are gain-insensitive. Each stage is theoretically capable of unlimited gain.

Wo'WI WO'WI WI I IV WOi WI~~~~~ I IV~ IIW~l __ Ln _IW 4 + IO,22Wo Wo, 2wo Fig. 8. A positive-input-conductance, nonlinear reactance amplifier formed by cascading an up-converter and a down-converter of the type described in Case 2. The only basic difference between these two stages is in the sign of the phase angle between D and 2w

this circuit is described by the same special case of Eqs. 1 as is the conventional nonlinear reactance amplifier, where (by Eq. 3b, with W = 0) W (28) G = - = - (28) P13 W1 1 In this case, however, W1 has components at two separate terminals and hence can be negative even though the input is conjugately matched, Although the circuits described above may have desirable gain and control aspects, their noise characteristics will serve as the ultimate test of their value. The primary sources of noise are associated with the external resistances and with internal resistance that is inherently present in any real nonlinear reactance element. In practice, the noise temperature of the output load will generally be fixed and may be large, which is one reason for the use of circulators in conventional negativeinput resistance amplifiers and converters. However, with the possibility for conjugate-match offered in this case, output noise that is fed back to the input (due to the semi-directional nature of the circuits under consideration) can be largely absorbed. In addition, the decoupling of sources at c, from the output terminals suggests that the noise contribution of g3 should be small. Thus, the so-called idler noise that is present in conventional nonlinear reactance circuits should be considerably reduced in this circuit. It can be proposed, therefore, even without the use of circulators, that the noise figure of this circuit should be improved over that of conventional negative-input-resistance amplifiers and converters, even when these latter circuits employ circulators. However, a more precise -30

estimate of noise figure has not been made at this time, because the original formulation of this problem (i.e., Eq. 7) does not permit the accurate inclusion of losses internal to the nonlinear reactance element. 4. CONCLUSION A detailed analysis of Eq. 8, or equivalently Fig. 4, has demonstrated several useful extensions of conventional nonlinear reactance circuits. It has been shown that parametric amplification at a frequency higher than the pump can, from the point of view of the nonlinear reactance element, be the same order of effect as conventional parametric amplification at a frequency lower than the pump. Also, a nonlinear reactance circuit has been developed that is capable of unlimited up- or down-conversion gain without reflecting negative resistance at either the input or output terminals. The latter result suggests a means by which a nonlinear reactance amplifier can be obtained which does not require a circulator for optimum performance. A more detailed summary of the results of this analysis is given in Table I. Within the limits of approximation provided by the circuit model in Fig. 4, Table I contains a compilation of the more important conclusions regarding three- and four-frequency, nonlinear reactance circuits. DThe term "four-frequency" has been employed throughout this paper even though a fifth frequency, in the form of the pump harmonic, has frequently been included. This choice of nomenclature has been made because only one of the two pump components need be applied externally and hence only four frequencies need be observed at external terminals. -51

TABLE I A SUMMARY OF THE MAJOR EFFECTS OBTAINABLE FROM THE NONLINEAR REACTANCE CIRCUIT MODEL OF FIG. 4. In entries where one or more frequencies or time-varying capacity components are omitted, they are assumed to be short circuited. Key: C.G. = conversion gain; P.A. = gain by parametric amplification; I.R. = input resistance; and T = tuning conditions. Frequencies Pump Components (a) C1 only (b) C2 only (c) C1 and C2 (1) 1 YW2 C.G.: Limited No effects Same as (la) P.A.: None I.R.: Positive T: Gain independent (2) 0W3 0 |C.G.: Unlimited No effects Same as (2a) P.A.: Unlimited I.R.: Negative T: Gain independent (35) W2' W3 No effects Same as (2a) Same as (2a) (4) e1' )(2,'3 C.G.: Unlimited Same as (3b) C.G.: Unlimited P.A.: Unlimited P.A.: Unlimited I.R.: Positive I.R.: -Positive at CD and a 2 T: -Gain dependent -Negative at -Tuning impossible ate all three frequencies 3 2 T: Gain independent 32

APPENDIX Proofs of Theorems 1 and 2 can be obtained from Eqs. 15 by the following considerations. Recalling that 7K = Ki + jK (K = 1, 2, IN IN IN 3), the conditions for resonance at each pair of terminals in Fig. 4 (with 1C21 = ) become P 2 at: =Pl +. -- = 0 (29a) at (D: 2 = - - 3 = 0 (29b) 2 2INK 2'|7i7-l12 - pily12 at 5:. ffi =0 -_ 2 __ —- = 0 (29c) J 3IN 13 7l72+11 Similarly, the conditions for circuit stability are ai2 -! a at: + - - > 0 (30a) lIN + 17 2 11732 at ( 2 L'2 2 2 17i2a > 0 3b) a 24a 117212 at 03: a = a3 - 1 1212 > 0 (30c) 3 3IK 3 ly l +112 Therefore, Eqs. 30 are explicit expressions for the basic conditions under which Theorems 1 and 2 apply. The proof of Theorems 1 and 2 will now be divided into the 1Vollowing steps: -33

Step A. It is easily seen that there are two possible conaitions under which Eqs. 29a and 29c can be satisfied simultaneously; namely, 5 3= 0 (31a) or 72 1 ^ 2 s, 1.2 (5lb) 17172+11 However, for Eqs. 30a and 50c to be satisfied simultaneously it is necessary that 1 < 7 2 — __ -__-12 2- <:i (32) 17172+11 a l2+a11y2l Therefore, of the two possible conditions described in Eqs. 51, only Eq. 31a will yield both stability and resonance at %l and (3. Step B. If B 0, the condition for resonance at w and wo becomes PI7212 = 2' which, with the aid of Eq. 29b, reduces Eq. 30b to the following expression a a+a + 2IN2 2 2N 2f > 0 (33) ~2 =:2 -+ 2:2' P2 IN a' 2 However, for parametric amplification at w2, Eq. 33 must be satisfied in such a way that the second term takes on the negative sign. Clearly, this is not possible if 2 = 0. Consequently, if l = B = 0, then IN IN IN 2 N + 0 is necessary for parametric amplification at (2. 2 =0 can IN IN occur only if IN equals 2a2 or zero (i.e., only under conditions of 2IN - -54

conjugate match or infinite gain at w2). Step C. This step amounts simply to recalling that, under Case lb of the text, 2 = 0 and P = 0 were shown to yield P 1 0 and i3 + O in the 2IN 3 IN 3IN region of parametric amplification at 2'. Therefore, under the conditions stated in Theorem 1, P3 = 0 yields either I = 3 = 0 anca P2 0 (step B), or p = O 4i f 0 IN 3IN IN IN IN and p 3 0 (step C). Similarly, with 3 0, B and 3 cannot vanish IN IN IN simultaneously (step A). Therefore, stable parametric amplification at 2), with IC21 = 0, requires at least one of the three quantities, p1 i IN 23, or i, to not vanish, which proves Theorem 1. IN IN Now consider Theorem 2. By steps A and B above, the necessary and sufficient conditions for Dl and w3 to be both resonant and stable (see Footnote p. 20) are P 3= 0 (34a) P = l 72I (34b)?2 ss Pj/g'2 a a,- ^'> l2 (34c) 2 a3 1721 If Eqs. 34 are satisfied, Eq. 30b yields. 3 >2a2 2 ( al - a > 2a * [1 - (1-4a^a21] 2 <(35) to be necessary and sufficient for stability at U2'. However, by Eq. 34b 2 = 2 -35

whereupon Eq. 35 yields a r P21 1 > _ 2 _ 2 1 > - 2 lesser of (1, ) (37) 3 1 72 1 a2 J as the necessary and sufficient condition for stability at o2 Since Eq. 37 is also sufficient for stability at D1 and o3, the proof of Theorem 2 is completed. -36

REFERENCES 1. J. M. Manley and H. E. Rowe, "Some General Properties of Nonlinear Elements - Part I. General Energy Relations," Proc. IRE, Vol. 44, pp. 904-913, July 1956. 2. C. L. Hogan, R. L. Jepsen, and P. H. Vartanian, "New Type of Ferromagnetic Amplifier," J. Appl. Phys., Vol. 29, pp. 422-423, March 1958. 3. S. Bloom and K. K. No Chang, "Parametric Amplification Using Low Frequency Pumping," Proc. IRE, Vol. 46, pp. 1583 - 1387, July 1958. 4. H. E. Rowe, "Some General Properties of Nonlinear Elements - Part II. Small Signal Theory," Proc. IRE, Vol. 46, No. 5, May 1958, pp. 850-860. 5. D. Lennov, "Gain and Noise Figure of a Variable Capacitance Up-Converter," Bell Sys. Tech. J., Vol. 37, pp. 989-1008, July 1958. -37

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