THE UNIVERSITY OF MICHIGAN OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR AN ANALYSIS OF THE BRETT ULTRA-WIDEBAND VIDEO AMPLIFIER Technical Report No. 122 2899-55-T Cooley Electronics Laboratory Department of Electrical Engineering By: D. K. Adams Approved by: B. F. Barton Project 2899 Task Order No. EDG-4 Contract No. DA-36-039 sc-78283 Signal Corps, Department of the Army Department of the Army Project No. 3A99-06-001-01 July 1961

ABSTRACT A detailed analysis is made of the process of phase modulation by a nonlinear reactance element. This analysis is applied to a video amplifier circuit developed by Mr. H. Brett of the U. S. Army Signal Research and Development Laboratory and shows this circuit to be capable of ultra-wideband amplification. The basic mechanism for this amplification is discussed in terms of several analytical viewpoints, which are ultimately shown to be consistent. Formulas for all basic circuit quantities are derived, including gain, bandwidth, input and output impedance, and noise figure. This analysis is used to predict optimum design techniques when nonideal components are incorporated in the Brett circuit. An illustrative design is given for a video amplifier with 20 db gain, 100 Mc bandwidth (0 to 100 Mc), and a noise figure of less than 0.1 db. ii

TABLE OF CONTENTS Page ABSTRACT.............................................. ii LIST OF ILLUSTRATIONS............................................ iv LIST OF SYMBOLS............................................... v 1. INTRODUCTION............................................... 1 2. ANGLE MODULATION WITH TIME-VARYING CAPACITANCE................ 5 2.1 A General Formulation of Angle Modulation with Time-Varying Capacitance........................... 7 2.2 Solution When V = O, In > 1............................ 10 2.3 Phase Detection..................................... 13 3. A MORE GENERAL FORMULATION FOR ANALYSIS OF FIG. 2............ 17 3.1 Formulation for a Time-Varying Capacitor at f........... 22 3.2 Formulation for a Time-Varying Capacitance atf......... 23 3.3 Summary of Section 3.................................... 23 4. CIRCUIT PROPERTIES OF THE BRETT PARAMETRIC VIDEO AMPLIFIER................................................... 24 4.1 Total Input Admittances Present to the Ideal Sources in Figs. 8(a) and 8(b).......................... 26 4.2 Transducer Gain Between ow and..................... 28 4.3 Gain-Bandwidth Products for Convtrsion Between a, and aq............................................ 28 4.4 Numerical example......................... 31 5. DESIGN CONSIDERATIONS FOR OPTIMUM AMPLIFIER PERFORMANCE....... 34 5.1 Physical Limitations on............................... 34 5.2 Operation With State-of-the-Art Limitations on Varactor Elements....................................... 57 5.3 Noise Figures................................... 41 6. CONCLUSION.................................................... 48 APPENDIX I....................................................... 50 APPENDIX II............................................. 51 LIST OF REFERENCES............................................... 55 DISTRIBUTION LIST....5............ 54 iii

LIST OF ILLUSTRATIONS Figure Title Page 1 Power distribution for a typical doublesideband reactive mixer. 2 2 Block diagram of the Brett Parametric Video Amplifier. 4 3 A circuit for angle modulation by a timevarying capacitance. 5 4 Modified phase detector to replace hybrid junction in Fig. 2. 16 5 Anticipated spectral distribution for circuit in Fig. 2, with small video signals. 18 6 Assumed characteristic for total varactor capacitance vs. voltage. 19 7 A lumped parameter equivalent circuit for Fig. 2. 25 8(a) Video equivalent of N in Fig. 7. 27 8(b) High frequency equivalent of N in Fig. 7. 27 9 Upper and lower bounds on conversion bandwidth vs. the sum of the circuit bandwidths, normalized to As Ap 30 10 Gain characteristic of example video parametric amplifier, pump frequency 5000 Mc. 33 11 Estimates of (a)MAX * 36 12 Pump tank with varactor loss R and external load GL. 38 13 Noise contributions from pump circuit conductance. 44 iv

LIST OF SYMBOLS Defined or first Frequency Variables: used on page f applied (video) signal frequency 6 cs 22xf 6 y coSt 8 S p w, (W^)% 51 f0 local oscillator or pump frequency 6 co 2fo 6 0 0 x o t 8 0 q ) ( - 8 )2 51 ~ (Wo 51 f f0 + nf; n = O,, + l 2,... 7 ~n ~ ~n 7 z f5/f 10 cf~b ~ angular frequency where varactor Q is unity 37 Current Variables: i(t) total current flowing through the varactor, or through any of its representations 8 i (t) waveform of applied pump current 8 I, I complex Fourier amplitudes of components of i(t) at f and f, respectively 6,8 ISo' Ino complex Fourier amplitudes of applied current source at f and f respectively n6,8 ic(t) waveform of current through C 8 iL(t) waveform of current through L 9 isc(t) waveform of short circuit current 18 v

LIST OF SYMBOLS (Cont.) Defined or first Voltage Variables: used on page V(t) waveform of total varactor voltage 18 v(t) time-varying portion of V(t) 18 V dc portion of V(t) 18 v(t) waveform of voltage across c (t) 6 V, V complex Fourier amplitudes of coms n ponents of V(t) at f and f, respectively 6,7 angular deviation of phase modulation 6 m, V phase and amplitude, respectively 11 0 0 of V 0 S see (20b) 11 B, 0 amplitude and phase, respectively, of correlating signal for phase detection 13 A(t), cp(t) envelope and phase, respectively, of V(t) 15 M(t), r (t) envelope and phase, respectively, of composite signal (sum of (25) and (26) 15 J nth order Bessel function of 1st n kind 7 Time-Varying Capacitance Variables: c (t) waveform of capacitance varying harmonically at f 5 Cs complex Fourier amplitude of c (t) 6,8 c (t) waveform of capacitance varying harmonically at f 25 C complex Fourier amplitude of c (t) 25 vi

LIST OF SYMBOLS (Cont.) Defined or first Circuit Elements and Parameters: used on page C total shunt capacitance in pump circuit 6 L total shunt inductance in pump circuit 6 L, CB high pass filter in pump circuit 25 L, C low pass filter in video circuit 25 V V G total shunt conductance appearing in pump circuit 6 CT(V), Q(V) capacitance and change characteristics, respectively, of a varactor 18,21 m, Vc, Q' parameters characterizing Q(V) 34 C(V) h, h2 first and second order Taylor coefficients of CT(V) 20 n11 12 first and second order Taylor coefficients of Q(V) 20 G internal conductance of video b source 25 Gn effective shunt conductance of varactor 37 G effective shunt conductance reS~L ~ flected into pump circuit by external circuit 58 B susceptance of pump circuit at f 9 n n Y admittance of pump circuit at f 9 R varactor series resistance 37 Q varactor quality factor 10 Y, Y total input admittance at f and SN T n f, respectively 26 vii

LIST OF SYMBOLS (Cont.) Defined or first Circuit Elements and Parameters (Cont.) used on page Y transfer admittance from f to inn n f; I = integer 43 a pumping parameter (sec 51c) 29 r C /C 10 Power and Gain Variables: Psb total sideband power delivered to G 12 P total sideband power delivered by video source 12 (TG)s transducer gain from fs to f 28 (TG)5 2 (TG)s 28 (TG)o (TG)s evaluated fs-0 29 (GB) voltage-gain-times-bandwidth product 31 Bandwidth Variables: Pb ~ total system bandwidth in radians/sec 29 PS bandwidth of input video circuit in radians/sec 29 P half-bandwidth of pump circuit in radians/sec 29 $P portion of P due to varactor loss 58 PL portion of Po due to output load 38 BoN noise bandwidth of pump circuit in cps 45 BN amplifier noise bandwidth in cps 42 Noise Variables: F noise figure 42 N excess noise power originating in video band 42 viii

LIST OF SYMBOLS (Cont.) Defined or first Noise Variables (Cont.) used on page N excess output noise power originating in pump band 42 K Boltzmann' s constant 42 T room temperature 42 T' noise temperature of G 45 TD, TL noise temperature of G and GL, respectively 43 dig increment of RMS noise current in interval do 43 PG total noise power delivered to G 43 ix

COOLEY ELECTRONICS LABORATORY TECHNICAL REPORT NO. 122 AN ANALYSIS OF THE BRETT ULTRA-WIDEBAND VIDEO AMPLIFIER 1. INTRODUCTION The recent revival of engineering interest in reactive mixers, particularly in applications calling for low-noise energy conversion and amplification, has led to a wide variety of reactive-mixer circuits. This apparent versatility in reactive mixer design, when compared to that in conventional resistive mixers, can be attributed largely to the radical difference that occurs between the reactive-mixer sideband and its image. The nature of this dissimilarity is demonstrated in Fig. 1, which shows a typical frequency distribution of power for a double-sideband reactive mixer. The following features of Fig. 1 are noteworthy (Refs. 1 and 2): (1) The output power in either or both of the sidebands can exceed the input signal power. (2) The amounts of output power from these sidebands are typically unequal. (3) The reflected power at the signal frequency may greatly exceed the input power. Notes (1) and (2) above have been the basis of recent developments in low-noise microwave modulators and converters, while Note (5) is the basis of parametric amplification. In each of these applications, however, the emphasis to date has been on single-sideband circuits. This specialization has been motivated, at least in part, by the following factors:

(1) The greatest need for improved low-noise energy conversion has existed at microwave frequencies, but in this application practical factors also limit the local oscillator (i.e., the power source or pump) frequency to the microwave region. Therefore, the sidebands tend to have large frequency separations, and single-sideband operation is natural. (2) For single-sideband converters operating on microwave signals, lower-sideband energy conversion tends to yield greater gain for a given noise figure (Ref. 5). (3) If the output power is derived from only one sideband, any coupling to other sidebands -tends to degrade circuit performance by introducing excess noise. POWER LOCAL OSC. IN INPUT (PUMP) SIGNAL f \ o I r f FREQUENCY-* POWER OUT Fig. 1. Power distribution for a typical double-sideband reactive mixer. It should be recognized that the factors cited above have historical importance but are insufficient to justify an exclusive interest in single-sideband reactive mixers. In fact, work at this laboratory has previously shown that double-sideband circuits can be designed to yield improved converter performance, even at microwave frequencies (Ref. 4). 2

Thus multiple-frequency reactive mixers are of justifiable engineering interest. The purpose of this report is to extend the theory of reactive mixers to a particular multiple-sideband pattern associated with angular modulation. This analysis has been motivated by interesting experimental work conducted by Mr. H. Brett of the U. S. Army Signal Research and Development Laboratory (Ref. 5). The theory derived will be applied specifically to the Brett mixer-amplifier circuit shown in Fig. 2, which will be referred to hereafter as a parametric video amplifier.+ In Fig. 2, a local oscillator signal at a frequency near 1500 Me is applied to arm 1 of a hybrid junction and splits between arms 3 and 4. The signal emerging from arm 3 mixes with an applied video signal through the action of a nonlinear capacitor (varactor). This process will be shown to reflect back to arm 3 of the hybrid junction what is essentially a phase-modulated signal. At the same time the signal emerging at port 4 is reflected by an adjustable short. The two reflected signals now add in the hybrid, such that in-phase frequency components pass out arm 1, while out-of-phase components emerge from arm 2. Similarly, quadrature components divide equally in power between the two ports. Brett reports that the detected and filtered signal on arm 2 is an amplified version of the original video signal which appears to contain little excess noise. Noise figures of less than 2 db and video bandwidths that are roughly 1 percent of the pump frequency have been estimated. The goal of this analysis, then, is to attempt to explain the origin of these results and to suggest possible methods for extending the performance of Brett's circuit. +In this report, the term video will be applied to the extended frequency range from dc to several hundred megacycles. 5

I PUMP IN PHASE MOOULATED ^^ ~PUMP, —-i i- VIDEO IN ADJUSTABLE HYBRID A A W PAS SHORT 2 JUNCTION FILTER FILTER AMPLITUDEI VARACTOR MODULATED | VAR PUMP -ENVELOPE DETECTOR VIDEO OUT Fig. 2. Block diagram of the Brett Parametric Video Amplifier. Since low-noise video amplification appears to be a property of the Brett circuit the analysis offered in this report will be concerned with small video signals. While a small signal analysis has limited applicability, it simplifies the mathematics in that the amplification process itself can be assumed linear. However, there appear to be at least two different ways for formulating this small-signal treatment. Perhaps the more conventional of these is to argue that the microwave signal, which is the power source (or pump) for video ampliv fication, is much larger than the video signal, and therefore the varactor-plus-pump can be replaced by a time-varying capacitor at pump frequency. On the other hand, Brett has suggested that if the varactorplus-video-signal is replaced by a time-varying capacitor at video frequency, the video signal can be imagined to angle modulate the pump. This report will demonstrate that these two approaches are consisternt 4

with each other. Individually, however, they do tend to emphasize different aspects of parametric video amplification. Since the Brett concept of angle modulation by the video signal is the less conventional of the two, it will be treated first. 2. ANGLE MODUIATION WITH TIME-VARYING CAPACITANCE Traditionally, angle modulation is achieved either by varying the frequency of an oscillator (FM); or by splitting the output of an oscillator into two components, whose relative phase is shifted before they are recombined into a phase-modulated signal (Armstrong PM). In both of these systems the fundamental source of angle modulation is a circuit capable of presenting a varying reactance to the signal being modulated. In general this varying reactance is either an active circuit or is so closely allied with one that the energy conversion properties of the varying reactance are easily overlooked. For this reason, first attention will be devoted to an investigation of angle modulation by the time variation of a passive reactive component. In particular, the circuit in Fig. 3 will be considered and will be shown to illustrate a useful case of multiple-frequency reactive mixing. To relate Fig. 5 to the Brett circuit in Fig. 2, it is tentatively assumed that cs(t) T iO(t) L \G IC C1(t) v) Fig. 5. A circuit for angle modulation by a time varying capacitance. 5

simulates everything to the right of the band-pass filter, the latter being represented by L and C. In Section 3, a more rigorous justification will be given for applying the analysis of this section to the Brett circuit. In Fig. 3, a current at fixed frequency f is applied to the L-C circuit which is tuned to f, and is shunted by a capacitor varying harmonically in time at frequency f. (Note: This circuit is to be distinguished from that of a conventional FM modulator in which f is associated with a "loose", or tunable, oscillator). In the steady state, assuming io(t) = Ioos oCt (la) c (t) = CS sin w t (lb) this circuit will tend to produce a phase-modulated tank voltage that may be approximated by v(t) = Ioocos (wot + g sin wst) (2a) where = 2.fo:' s = 25fs (2b) The quantity p is customarily termed the angular deviation, and quasistatic considerations suggest that:, C ~ tan1 o cp G tan 1 os (2c) The preceding expression for v(t) and zp have been cautiously stated, and indeed they will be shown to be inaccurate. For example, one obvious oversight in (la) is incidental amplitude modulation. Other 6

more subtle discrepancies in (2a) and (2c) will be demonstrated. It is instructive to consider the flow of energy in Fig. 3. Since charge is present on the time-varying capacitor, the latter can be expected to do work upon this charge. At the same time, modulation of the tank circuit voltage tends to draw extra power from the current source io(t), so in this phase-modulation process energy may enter the circuit at both input frequencies. Thus, it can be anticipated that a suitable demodulator attached across the tank could recover an amplified version of the original f signal. The amplification in this case is s due both to the power contributed by io(t) to the modulation process and to the lossless character of the time-varying element. In the following paragraphs a quantitative evaluation of this effect will be presented. 2.1 A General Formulation of Angle Modulation with Time-Varying Capacitance One well-known characteristic of (la) is that it contains spectral components at all frequencies in the set f = f + nfs, where n = 0, + 1, + 2,.... This fact follows from the relation: 00 cos (wot + p sin ost) = Z Jn(c) cos (n)t (3) nL= -oo where wn = 2ifn and J(cp) is the nth-order Bessel function with the property that J n(c) = (-1)n J n(). In the steady state, v(t) in Fig. 3 will have the same spectral frequencies as (3), although the actual Fourier coefficients of v(t) will differ from those in (3) since v(t) is not simply sinusoidally phase-modulated. Therefore the following steady-state representation of v(t) will be taken. 00 v(t) = Re VneJ(x + An) 4) n= -oo 7

where x = 2nfot and y = 2jf t. From Fig. 3, this voltage must satisfy 0 s the equation iO(t) = ic(t) + iL(t) + Gv(t) + i(t) (5) where io(t) = Re(IoeJX) (6a) c (t) = Re(C ejY) (6b) i(t) dt [c(t) v(t)] (6c) Note that the coefficients C and I 0 have now been made complex so the phases of i (t) and cs(t) are arbitrary and are no longer restricted to those indicated in (lc). The subscript on c,(t) has been introduced to distinguish this time-varying capacitor from another that will be introduced in Section 3. As a consequency of (5), i(t) will have the same spectral frequencies as v(t), or: i(t) = Re L I ej(x + (7) 11=- -oo Now, by employing (6c), these spectral components can be precisely related to the components of v(t), which yields the following basic result. = 1. (cv +C* (8) n 2 n sn-l + n+l Here, * denotes a complex conjugate. A final step prior to calculating the V through (5) is to note from Fig. 3 that n i(t) = Re j CVne(x + ny) (9a) 8= n (9a)

oV ~ej(x + ny) iL(t) Re - n (9b) Then using (4) and (9), (5) yields = GV + j (Co s 1 ( la) 0 = YV + jn (CsVn + Cv ); n 0 (lob) n n 2 n s n-1 s n+l where (since a 2L = 1) Yn= G + jB (lla) B W nC L 2noSC (11b) n n c L s n Therefore, given Ioo and C, the resulting v(t) can be found if (lOb) can be solved for the V. It is instructive to write (10) in matrix form, as follows: 0 JcoCC s 2 3_ V? I 2 2j s 12) -1.. V2 o Y W S -1 O 1 JSs 1 V o -3LC Y -jcC* V o%22s 2, 2' 2 s 2 A complete solution for v(t) entails inverting the infinite matrix above which poses a difficult mathematical problem. However, approximate solutions have been obtained that are sufficiently enlightening for the present application. If it is assumed that v(t) is approximately representable by 9

(2), the magnitude of cp will never exceed x/2, so only the first three or four terms in (3) will ever be significant. Furthermore, if this circuit is employed as a phase modulator for small signals, the smallsignal condition ICs/C << 1 can be imposed. In this case, C)u Ic C G -S Q 1-_j1 (13) G C so the small-signal condition is more precisely given by IC# I<~< (14) When (14) is satisfied, all Fourier coefficients of v(t) should decrease rapidly with |n|. This assumption can be checked by neglecting 1ll V n in (12) except those for which In| < 2 and solving (10) for the remaining sidebands. In support of the above statements, this calculation yields V2 2 Cs (15) V C Consequently, by imposing (14), all V except those with n = 0, + 1 will n be ignored for the present to focus attention on the small-signal properties of Fig. 3. 2.2 Solution When Vn = O0, n| > 1 Using the approximate value for Bn given in (lib), (10) yields: V1 -jrQ(l + z) (l6a) V;- 2~(1 + 2jz&Q)(16a) V-1 -jr*(1 - z) V 2(1 - 2jzQ)16 [ 2 (1 + 4zQ. )] (16c) 0

where z ='s/o, and r= Cs/C. An interesting feature revealed by (16) is that the two sidebands have unequal amplitudes, these being in proportion to their respective frequencies: V 1v l+ (17) One might be tempted to attribute this result to the approximations made in ignoring higher-order sidebands, but (11) is a true limiting value independent of both r and Q. Further inspection of (12) reveals that sideband asymmetry is a general characteristic of this circuit, since it continues to occur as additional sideband pairs are introduced. The significance of this asymmetry will now be demonstrated. By neglecting all but the first-order sidebands in (4), the total tank voltage in Fig. 5 becomes V ~V v(t) = Re tVoe 1 + V ej + e (18) Since the origin of time in (18) is arbitrary, it will be convenient to choose the phase of r equal to tan-1 2zQ, which simplifies (16). Equation 18 now becomes v(t) = Vi i(l + 2zS sin y)2 + (2S cos y)2 } cos (x + p(t) + po] (19a) where: $p(t) = tan-1 2S cos y (19b) 1 + 2zS sin y V = V eJ o 20a) 0 o s = 20b) The effect of sideband asymmetry is now revealed through terms in (19a) containing the frequency ratio z, since these terms would not occur in 11

the typical phasor description of angle modulation, such as that used to obtain (2). In (19a) and (19b), the term 2zS sin y contributes to both phase and amplitude distortion. Although both of these effects are very small under the present assumptions, it is instructive to inquire about their origin. The total power delivered by the two sidebands is sb 2 2 Psb - 2(Lv12 + v-112) G 2 V 2 (1 + z2) G (21) 0 The total power supplied to the sidebands by io(t) is (by Eq. 16c) G S2 V (22) 0 So the power supplied by the time-varying capacitor must be the difference between (22) and (21), or 2-2 2 3 P = S V z G (23) S 0 which increases proportionally to the square of the signal frequency. Therefore, the sideband asymmetry noted in (17) must be due to the energy contribution from the time-varying capacitor c (t). The power gain between s this time-varying capacitor and the sidebands is s _ lz (2)2 (22 s ( ) ( ) ) (2) which is consistent with the fundamental power relations of Manley and Rowe (Ref. 1). Now, if the sideband power in (21) could be completely converted back to fs, (24) would constitute an overall amplifier power gain, which stresses the importance of this relation. 12

2.3 Phase Detection Since the sideband power in (21) is associated with a signal that is essentially phase-modulated, the question arises as to methods of suitable sideband power detection. Rather than employing a typical discriminator circuit, which inherently suppresses the lower video frequencies, direct phase detection appears to be the most efficient approach. Conventionally, the first step in direct phase detection is to add a quadrature carrier component (the pump component in this case) to the phase-modulated signal, which tends to convert the original phase modulation to a proportional amplitude modulation. The second step is envelope detection to recover the original phase information. To demonstrate this principle, (19a) will be rewritten as v(t) = Vo [1 + A(t)] cos [x + p(t) + (Po], (25) 0 and the signal B sin (x + e) (26) will be added to (25) to produce the sum M(t) cos [Uot + *(t)] where M2(t) = VO[l + A(t)]2 + B2 + 2B V 1 + A(t)] sin [e - cP(t)] (28a),(t)= t l + A(t)) sin (p(t) + po) - B cos 0 (28b) Vo (1 + A(t)) cos (cp(t) +.p) + B sin e 13

The phase modulation cp(t) now appears on the amplitude envelope. For the case under consideration, where the maximum value of cp(t)l is small, the original amplitude modulation, A(t), is negligible and cp(t) makes its greatest contribution to (28a) if (@ - (po) = O,,... In this case, (28a) reduces to M2(t) w Vo + B2 + 2 B V cp(t) (29a) \ B- 2 B q /t - (V2 + B 2) + 2) (29b) But since lcp(t)l << 1 +B V B0 (t) M(t) V 2 + B2 1+.( (30) v%' L Vo2 + B and the resulting envelope waveform approximates B V_ c(t),,, (31) V.2 +B2 which approaches V p(t) for large B. If B = V, this envelope wave0 form becomes 0.707 V0 cp(t). For small phase deviations (19b) yields?p(t) s 2S cos y (32) so the available power from, say, the positive envelope is s2 V 2 B2 G 2 -B2 (33) V + B o while the total sideband power is given by (21). Therefore the fraction of sideband power converted to envelope power, through the addition of a quadrature signal to the original phase modulated signal, is approximately 14

B2 2- - ) - 2- (34) (V2 + B2) (1 + z2) which approaches unity for large B and small z. When coupling losses are included with (34), the fraction of sideband power actually available for envelope detection is obtained. This fraction will be referred to as the modulation conversion efficiency. The latter must then be multiplied by the envelope detection efficiency to obtain the total sideband to signal demodulation efficiency. Returning now to Fig. 2, it can be anticipated that the signal returning to port 5 will contain angle modulation. If so, this modulation can be efficiently detected by positioning the short on arm 4 to reflect a quadrature pump signal back into the junction, where it will add to the signal from arm 3. This sum will then pass through an envelope detector on arm 2. Thus, the basic role of the hybrid junction is to provide conversion from angle to amplitude modulation. For a symmetrical hybrid, the carriers on arms 3 and 4 are equal (i.e., B = V ) so (34) assumes the value one-half. However, another one-half factor must be introduced because half the sideband power incident on the hybrid from arm 5 goes out arm 1. Therefore, the modulation conversion efficiency of the hybrid junction system in Fig. 2 is approximately 25 percent (-6 db). To this must eventually be added the envelope detection loss to arrive at the total demodulation efficiency. A modification of the hybrid junction phase detector that would avoid much of this 6 db modulation conversion loss is shown in Fig. 4. Assume for illustration that the pump source is increased by 3 db so a -6 db directional coupler will couple out a signal at the original pump level (recall that a hybrid junction acts as a -3db coupler). This 15

PUMP SOURCE DIRECTIONAL I COUPLER Ndb QUADRATURE PUMP ( A (SEE FIG.2) PUMP QUAD PUMP CIRCULAT WITH VIDEO P M. ENVELOPE: DETECTOR' AMPLIFIED VIDEO OUT Fig. 4. Modified phase detector to replace hybrid junction in Fig. 2. signal passes directly into the phase modulator through a circulator, which then channels the entire reflected wave into the envelope detector. The pump source is also applied directly to the envelope detector, so ideally B2 now exceeds V2 by 6 db in (34), and (34) becomes approximately -1 db. With this technique, the circulator eliminates the original 3 db insertion loss of the hybrid. and the directional coupler eliminates 2 db of the loss predicted by (34), but requires 3 db more pump power. The directional coupler provides an additional advantage of a jr/2 phase shift, 16

so the two pump components will arrive at the envelope detector in quadrature, if the line lengths are equal. 3. A MORE GENiERAL FORIIUAJLTTIO FOR ANALYS IS OF FIG. 2 The following question arises in att-empting to apply the analysis of the previous section to Fig. 2, "Is it valid to neglect nonlinear effects introduced by the large pump signal and then to assume that the varactor follows a small video signal? " To investigate this question, the general behavior of the varactor in Fig. 2 will be considered briefly. Because the varactor capacitance depends upon its terminal voltage, application of the pump (fo) and the video signal (f ) will tend to produce pump harmonics (mf), video harmonics (nf ), and mixing products mf + nf_, where m and n are integers. If low-noise, low-level amplification is of interest, only small video signals need be considered and video harmonics should be of negligible importance. Similarly, pump harmonics can be assumed to be suppressed if the varactor is tuned to the pump frequency, so the major voltage components experienced by the varactor will be at f, f, and f = f + nf, for those values of n that correspond to o s n o s sidebands within the pump circuit pass band (Fig. 5). To determine the validity of the assumptions just stated, and of the previous representation of the varactor as a time-varying capacitance at f, the following procedure wsill be carried out with the aid of Fig. 6. (1) The analysis of Section 2 will be assumed to accurately represent the circuit in Fig. 2. Consequently, all sidebands except those at f will be neglected by temporarily assuming them to be short circuited, as shown in Fig. 6a. 17

/ PUMP PASS BAND / / \ VIDEO PASS BAND / / \ s f-f f-I f- f2 fP FREQ Fig. 5. Anticipated spectral distribution for circuit in Fig. 2, with small video signals. (2) The total varactor voltage waveform in Fig. 6a will be denoted by V(t) = V + v(t) (36a) where V is the dc bias voltage and v(t) = Re [vseiY + V0 ej + vlej(X+) + V 1e(x Y] (36b) The varactor current, i(t), will have components at the same frequencies as v(t), and short-circuited components isc (t) in addition. Therefore, i(t) = i c(t) + ReIse + Ioe + ej( + I(37) SCt) i(t) ReIe+ Ioe +e(X-)] (37) (3) The total varactor capacitance, CT(V), which is defined as the ratio of its total instantaneous charge, Q(V), to total instantaneous voltage, V(t), will be represented by the curve in Fig. 6b. The varactor will be assumed to be biased at a value C by the dc voltage V, and its characteristic in the vicinity of C will be represented by 18

i(t) SHORT CIRCUIT V(t) r CT(V) EXCEPT AT f AND fl (a) CT(V) C - -,\,SLOPE h, V t) V _(t) t (b) Fig. 6. Assumed characteristic for total varactor capacitance vs. voltage. V is bias voltage. 19

CT(V) = C + hi (V-V) + (V-V) (3) T,(V) 1 2 2. (4) By using this varactor characteristic and the fundamental relation in (6c), basic equations will be derived relating v(t) and the unshorted components of i(t). These equations will be shown to be consistent with those derived in Section 2, and will be used to further evaluate the circuit properties of the Brett amplifier. Also, they will be used to derive the components of short-circuit current that were introduced in step (1). Conditions will then be obtained to show when the short circuits in Fig. 6a can be removed without seriously changing i(t) and v(t). To proceed withthese steps, the instantaneous capacitance of the varactor will be approximated by CT(t) = C + h1 v(t) + 2 h2 vt) (39) Although this relation only approximately represents a varactor, it is sufficiently accurate to illustrate the basic mixing process that occurs in Fig. 2. Using this approximation, the basic law for varactor current, in (6c), becomes i(t) =d [[C + hl(t) + 2 h2v2t)][V + v(t) (40) Now, substituting (36b) in (40), and comparing the result with (37), yields the following set of fundamental relations: i = _Js [Vs + 1 + 2 VOV 1] (41a) is + 2 I12Vl X 0 X I = -W-i [V 1 + VOVS] (41b) 20

1= jiW [TVL + 2 TA2 Vs] (41c) I, = Jco [yo + 2 2VV + 2V- (V-d where: = + h V; h = 2h + h2 V (41e) These equations describe a mixing process that can be traced through, starting with I and Io. The first terms in (41a) and (41d) show that s 0 I and I experience an effective varactor capacitance equal to T1, which gives rise to voltage components Vs and V. In turn, these voltages pros o0 duce sideband currents through the mixing terms 1 2Vo*V in (4lb) and 2 2os 12VoVs in (41c). These sideband currents also experience the effective varactor capacitance ql, producing sideband voltages that mix to produce reflected currents at f and f, according to the last two terms in (41a) 5 o and (41d). Equations (4l) demonstrate that Tl is the effective varactor capacitance, since this value of capacitance is experienced when a (small) sinusoidal signal, plus bias, is applied to a varactor (i.e., when Vs and V1 are zero). Therefore, published curves of varactor capacity vs. bias, that are measured by RF techniques, are actually plots of TI and have slope q2 It is instructive to note that the set (41) would also be obtained if the varactor were represented by the following charge-voltage characteristic: Q(V) = C V + %1(V-V-) 4+ 2(V-V)2 (42) Again, T1 and q2 are related to the coefficients in (38) by (41e). 21

The relations in (41) can now be used to answer the question posed at the beginning of this section, which asked if it was valid to assume that the varactor follows only one of two applied signals, as in Fig. 2. This question will be answered by showing that the set of nonlinear equations in (41) can be solved by converting it to one of two linear sets; each of which applies when the varactor follows only one signal, either the signal at fs or the signal at fo. Therefore, these two cases result from replacing the varactor by a time-varying capacitor at either f or f. It will be shown below that these two representations s 0 are related by the symmetry that occurs with respect to V (or V*) and O Vs in the mixing terms in (41b) and (41c). 3.1 Formulation for a Time-Varying Capacitor at f This representation arises because (41b) through (41d) can also be derived from (6c) if, before substituting in (6c), Vs is deleted from (36b), and (39) is replaced by the following harmonically-varying capacitance ) = + ) T 2Re V seJ t] Physically this situation is identical to that treated in Section 2, and it can be made mathematically equivalent by letting C = nI + stray capacity (43a) s = 2Vs (43b) This equivalence is seen by comparing (41b) through (4ld) with (8) or (10). The significance of (43b) is consistent with intuition, since it shows that CT(t) follows the video voltage, in proportion to the slope (I2) of 22

the effective capacitance characteristic. Now, since (43) converts (41b) through (41d) into a linear set whose solution was given in Section 2, this solution can now be substituted into (41a) to relate Cs (or Vs) to the video current Is, which completes the solution of the set (41). However, the actual implementation of this step will be postponed until Section 4, 3.2 Formulation for a Time-Varying Capacitance at f 0 This representation arises in a similar manner to the previous one, although it is derived from the first three equations of (41), instead of the last three. If CT(t), in (39), is replaced by the following harmonically-varying capacitance (at f ) 1 + o(t) = Tl + 12Re Voe ] (44a) and if V is deleted from (36b), then (6c) will again yield (41a) through (41c) if C = lV (44b) The latter relation now converts ('la) through (41c) into a linear set, whose solution can be substituted into (41d) to again complete the solution of the set (41). 3.3 Summary of Section 3 It can now be concluded, within the assumptions stated at the start of this section, that the varactor in Fig. 2 can be considered to follow either the small video signal or the large pump signal, since these approaches lead to the same solution for the set (41). This property is the result of the symmetry of th.e mixing terms in (4lb) and (41c), which 23

can be illustrated as follows. Note that mixing occurs in (4lb) through the term Tr2VoVs, but this term can be treated equally well as: (1) a capacitor r2 V mixing with V*, or (2) as a capacitor'r V* mixing with 2 S 0 2 0 Vs; and either case leads to a systematic solution of (41). The final step of verifying the assumptions at the beginning of this section is mainly of mathematical interest, so it is treated in Appendix 1. Since Appendix 1 makes use of relations that are derived in Section 4, it should be noted for now that the above procedure is valid if IVsl << IVl. 4. CIRCUIT PROPERTIES OF THE BRETT PARAMETRIC VIDEO AMPLIFIER Through the introduction of (4la), the analysis in Section 3 has provided a method for using the results of Section 2 to determine more of the properties of Fig. 2. However, the alternate approach outlined in Section 3.2 is more conventionally used for deriving the properties of reactive mixer circuits (Refs. 2 and 4). Therefore, for ease of comparison w7ith other works in this area, the method of replacing the varactor by the time-varying capacitor in (44a) will be used to relate (4l) to Fig. {2 and to thereby obtain the basic properties of this circuit. At the same time, the effective varactor capacitance r1 will be denoted by C to be consistent with Fig. 3. To go to this second analysis, a more precise specification of the circuit in Fig. 2 is necessary. The analysis of the previous section employed lumped circuit concepts, and therefore the circuit in Fig. 7 provides a useful eauivalent for that in Fig. 2. In Fig. 7, a hybrid coil has been substituted for the hybrid junction, and simple filters separate the video and pump, signals that appear across the 24

N C L 3 ^ c /HP LrK I I r-t G L~ — i0 ioVARACTOR G^ LC csot) Cv Gs CISO) PHASE MOD. PUMP OUT t4 Fig. 7. A lumped parameter equivalent circuit for Fig. 2. L >> L and C ~> C are assumed. aractor. The tuned circuit on arm 4 represents the adjustable short in the transmission line version (Fig. 2). It will be recalled that this short provides a variable pump component that can be added to the phase-modulated signal on arm 3 to accomplish phase detection. At video frequencies, I is the video source, and the varactor so network N in Fig. 7 reduces to Fig. 8a. At the sideband frequencies (fl) N reduces to Fig. 8b (Note: the sideband current sources that have been included in Fig. 8b will be treated as noise generators in Section 5.3 so they will be carried along in the present analysis.) By Figs. 8a and 8b, Io = I + (Gs + JsC + C CB]) Vs I + YsV (45a) I10 = I + Yl; Ilo = I- + Y_1V1 (4b) where Y1 and Y-1 are defined in (lla) and (llb) and are again assumed to 25

have the value Y = Y1 G + 2jw C. (46) Note: This assumed symmetry of the pump circuit admittance will be shown to have an important bearing on the operation of the Brett circuit, Now, by combining (41a) through (41c) with (45), the following fundamental admittance relation is obtained (Refs. 2 and 4). y J1 jc* 1 so s 2 0so 2 s lo 1 J Y 0 V1 ~10 = 2j(w1)Co Y1 1 * 1 * * -0l | 2 ( )CO 0 Y Vl (47) -10 2 J (-l)Co -1 Since the approximation cited in (46) will be assumed in applying (47), Y 1 can be replaced by Y1. It should be noted that (47) plays the same role as (12) did in Section 2, and therefore it will yield the basic circuit properties of Fig. 7, which are given in the following sections (see Refs. 2 and 4 for typical derivations of these quantities). It should also be noted that the procedure outlined in Section 3.1 leads to identical circuit properties. 4.1 Total Input Admittances Presented to the Ideal Sources In Figs. 8a and 8b._,, _ fJ\C 12 At ow: = Y Y + 2 (48a) -s S^ s 2Y1 C |Co IcC12 Y1 At c,:= Y 1 + _.s. 1 (48b) 6IN 4 Y Y- ~.-1CC|2 26

N c 3 | Fc(t)==CB = C =CV G f VSIGNAL e ~ ~ ~~~ — (ISO) Fig. 8(a). Video equivalent of N in Fig. 7. 103F loQ t r_ IG IL c c- o Fig. -8(b). High frequency equivalent of N in Fig. 7. 27 //

* (DCDJc2IY At W-1 S ol (48c) _ _ _ _ -i * ) IN - 4 YsY + C l2 Perhaps one surprising result in (48a) is that the reflected input admittance at ws (the second tern in 48a) has a positive real part. The derivation of (48a) shows that this result is a consequence of the slight difference between the amplitudes of V1 and V 1, as first noted in (17). This difference arises in spite of the pass band symmetry assumed in (46). A second feature of (48) to be noted is that Y1 / Y-1 even though Y = Y*. This result has bearing on the overall circuit-noise figure. 4.2 Transducer Gain Between C and w s +1 The transducer gain (TG) from w to c is defined as the outsn s n put power at O divided by the available source power at C. By (47), n s GaG W 21C012 Ws @o: (TG)51 + 22s - 1212 (49a) G G C IC IC I. 2 o -91 - (TG) s -1 lo o (49b) -S sl IYsY1 + s21Co1212 4.3 Gain-Banfiwidth Products for Conversion Between Cs and +1 Because V1 and V 1 combine with the pump to produce an angle modulated signal, the total power in V1 and V 1 will eventually determine the video power available from the final envelope detector. Therefore, the sum of the two transducer gains in (49) is of basic interest, namely (TG)s = (TG)l + (TG),-1 GG ( 2 2)a2 ( s IYY'1 + 1 2 lC 1212 28

which will be considered a function of s. In evaluating (45) and other related quantities, it will be convenient to introduce the following auxiliary parameters: G S s, = C - C + C = video circuit bandwidth (radians) (51a) V B o = 2C = one-half pump circuit bandwidth (radians) (5lb) Ico12 = tC 4 + C + C+ ) (51c) Equation (50) now becomes (TG)s b (52) z z z + (w-) LK + j +l( Therefore, maximum gain occurs at CD = 0 and has the value (TG) - (53) - s o The bandwidth p of (TG), is- also of interest, and to calculate this quantity it will be assumed that s << C at the -3 db points. In this case =I- t -IKI + K2 + 2 (5+ 4a) where w-~l ^(^1 -1 0 a < 1 Inspection of these relations shows that p can be approximated by,IF O152 29

This approximation is very accurate for << 1 or >> 1 Po lo and incurs a maximum error of about 30 percent for Ps = and a o 0. S 0 Exact solutions of (54a) and (54b) are plotted in Fig. 9 for a = 0 and = 1. For other values of a, the curves for P lie between those plotted; so (55), which corresponds to the a = 1 curve in Fig. 9, can be taken as a practical lower bound on the bandwidth of (TG)s..70. \.60.50 0.40.30 - \LOpCUS OF,20 EQ. 55 2 3 4 5 6 (aS+ i/ --- Fig. 9. Upper and lower bounds on conversion bandwidth vs. the sum of the circuit bandwidths, normalized to ~j~i~. 30

Now, combining (53) with (55) yields the following lower bound for the gain-bandwidth product of the circuit in Fig. 7. (GB) = >. (56) 0s The expression on the right has a maximum value when s = Po' or (GB)MAX > fp / (57) which emphasizes the value of a high pump frequency. 4.4 Numerical Example Because of the abundance of circuit variables introduced thus far, a numerical example will perhaps be appreciated. Assume that a video amplifier is to be designed with the following characteristics: Maximum Gain (dc) = 20 db Bandwidth = 100 Me Pump Frequency = 5000 Mc If U = P these requirements yield (10)(100 MC 1 (by Eq. 57) a < [(10)(00 Mc) = 0.02 (by Eq. 55) = 2 (100 Me) = 200 M However, with s3 chosen equal to p and a very much less than one, the calculations above contain nearly the full 30 percent error in (55). The exact solution of (54) in this case yields f = 0.65 s (58a) 31

so (56) becomes (GB) =f << 1 (58b) Hence, the conditions in this example can be more accurately satisfied with (by 58b) a = 0.0084 PS PO (by 58a) = 4 Mc 2-i: 2-ic It will be shown later that the value 0.0084 is well within the physical limitations on a. To complete the example under discussion, when ps = o the s 0 video input admittance can be written 2 2 N = 1+ (.) G5+j' \/ 2 G8 -i s % Y1 + G 1+ j S By this relation, it is evident that Y is very nearly equal to Y as IN s long as c << 1, which means that the circuit in this example operates essentially as a voltage amplifier. The fact that very little video power is accepted at the input terminals also explains the difference between the power gains in (50) and (24). The former depends upon the video power available at the input while the latter depends upon video power delivered to the input. A final calculation of interest is the total transducer gain, which (by 52) is given very nearly by (G ~ 100 (TG) - - - 32

This quantity is plotted in Fig. 10 and shows a 12 db/octave roll off at high frequency and a half-power bandwidth of very nearly 100 Mc. 0 0' 10' 100 1000 f IN MC - Fig. 10. Gain characteristic of example video parametric amplifier, pump frequency 5000 Mc. 33

5. DESIGN CONSIDERATIONS FOR OPTIMUM AMPLIFIER PERFORMANCE 5.1 Physical Limitations on a Since gain-bandwidth product (by 56) is proportional to>/, it is important to consider possible limitations on this quantity. By definition (in 51c), a is one-fourth of the ratio of the dynamic varactor capacitance squared (C2) to the product of the total fixed capacitance shunting C at f and f. Thus maximum ao occurs when the varactor experi0 S 0 ences minimum stray, filter, or bypass capacitance at each operating frequency, or /C \2 -LX= (C) (59) and by (44a) and (44b) AX i" [2 tin ) (60) It should be noted that (60) is strictly valid only for /V < 1, or when the effective varactor capacity rL can be assumed to vary linearly about its value at V. However, graphical analysis of typical varactor characteristics will show that (60) does approximate (59) over a significant range of V/V. It is well known that the charge-voltage characteristic of back biased semiconductor diodes can be approximated by the relation Q(V) = Q 1+ -; < m < 1 (61) where V is the junction contact potential and normally V < V. ThereC C fore, by (42), = (A ) Q(V)(l + V/ )34

= ( A) N (l + V/v -1 and so (60) becomes AX (1 + V/v) c c which reduces to ( 72 (V 2 if V, V V. Since V = V marks the threshold of excessive varactor conduction, the upper limit on a that is predicted by (60) will be taken to m - 1)/2, which has the value 0.06 for the typical case of m = 1/2. However, for V # V, the accuracy of (60) deteriorates, and an exact Fourier analysis is required to evaluate (59). This has been done in Ref. 6, and the result for m = 1/2 is shown in Fig. 11. Since the latter curve predicts a c 0.1 for V0/V = 0.9, the value 0.1 will be taken as a safe upper limit on a, in the sense that a = 0.1 should be approachable with large pump signals and minimum stray or auxiliary capacity across the varactor. Filter or bypass capacity that mlay shunt the varactor on one or more of the operating frequencies will reduce both c and the circuit bandwidth, but these reductions will produce opposing effects on the transducer gain (by 52). To illustrate this point, consider a case where the input signal is available from a 100 ohm source and the effective varactor capacitance (C) is 1 pf, so (neglecting any bypass capacity) Ps/2g = 1600 Mc. If the desired amplifier bandwidth is significantly smaller than 1600 Me, the extra input circuit bandwidth in this case would unnecessarily reduce the transducer gain; but if p is reduced by 35 35

0.12 -- 0.10 O.0 - BY FOURIER ANALYSIS (AFTER FIG. 5.3 IN REF. 6) o X 0.06 0.04 X 0 //B BY (60) 0.020o n 0.4 0.6 0.8 1.0 Fig. 11. Estimates of (a)MAX. 36

adding extra capacity in the video circuit, a will be reduced simultaneously to nullify the possible advantage to be derived from the former reduction. Therefore, to make optimum use of the varactor, video bandwidths must be controlled by inductive source transformations. To continue the illustration above, a stray capacity of 1 pf will be allowed for, making C + C = 2 pf, and a broadband 2:1 step-up transformer will be assumed to change the source impedance to 400 ohms. Then Ps/2nt is reduced to 200 Me, with an accompanying transducer gain improvement of 6 db (Note: the reduction in P accounts for 9 db, while the reduction in x accounts for -3 db). In the Brett amplifier, the importance of a high source impedance arises because the video input impedance is large as long as f << f. Thus, as a wideband amplifier, S O the Brett circuit is particularly useful with high impedance sources such as solid state particle detectors or various photoelectric devices. 3.2 Operation With State-of-the-Art Limitations on Varactor Elements The equation for maximum gain bandwidth product (b7) suggests that f should be as large as possible, but the high frequency losses in 0 actual varactors make this choice incompatible with the other optimizing condition that p = p. The high frequency loss in a varactor is due to a fairly constant series resistance R, and therefore the varactor will load the pump tank (see Fig. 12) with an effective shunt conductance GD z ciRC2 = (62) where i is the 2, of the varactor at wo, and oD is the frequency where AD = 1 (commonly termed the varactor cut-off eziequelcy). The approximation in (62) is valid if 2 >a 1. 37

In Fig. 12, GL represents the effective shunt conductance of the external circuit, so the total tank conductance G is the sum GL + GD. Regarding Fig. 2, GL represents the conductance seen looking into arm 3 of the hybrid junction, which in turn depends upon the internal conductance of the pump generator and the external load conductance reflected through the envelope detector. If these are both matched to the hybrid, GL will be the characteristic admittance of the hybrid junction. In the L case of the modified pump circuit in Fig. 4, the nonreciprocal properties of the circulator cause GL to depend only upon the envelope detector and its output load. Henceforth, GL will be assumed to included all exL ternal loading on the pump tank, but its precise value will depend upon the particular pump circuit configuration employed. EQU I VALENT VARACTOR I FC RI C R -,-1- - C I, I II I, — _... I- l I GL GD + GL GL PUMP TANK Fig. 12. Pump tank with varactor loss R and external'load GL. With varactor loss, the half-bandwidth of the pump circuit is o 2C = + L (63a) 38

where GD G GD 2 2L (63b) AD = 2 -; = 2 (6 For a given pump frequency, it is evident that the minimum value of Po is PD, which can also be expressed as 1 2 ^o)N = Pg =~~ 2(64) (oMIN D - (6 24) Similarly, the expression in (53) for low frequency gain must be changed to account for varactor loss. It will be sufficient to do this by considering the transducer gain between source and GL rather than G, or 42 / GL \ (TG) - (GL ) (6a) and by (64) and (63b) ~V,'PL PD (TG)o = D [ D (65b) Since (65b) is maximum for PL = D, varactor loss will be best compensated for by choosing G = GD, which yields (TG)o = (66) s In practice, (66) must now be reduced by the fraction of GL that constitutes the actual output load, but this step will not alter the significance of this relation. From (66), it is now desirable to minimize Ps. By (55) and (63), the bandwidth (3) can be approximated by the relation 1 1 1 Ps PD + PL Therefore, if fo is high enough to make ED >> _s' so that B t Is' than the desired amplifier bandwidth will be obtained with maximum gain. 39

In summary, varactor losses influence the choice of pump frequency. To design an amplifier with a given bandwidth h, the pump frequency should be chosen so wc2 >> ai) in order to make SD >>' Then the varactor should be loaded at pump frequency such that GD = GL. By D L this technique, the following optimum low-frequency gain can be obtained: (TG)o - (67) max The maximum value of voltage-gain-times-bandwidth is now (GB)M = (2)l ) y ^ (68) By comparing (68) with (57) the effect of varactor losses is quite evident. If W = a, then a must be at least a2 times larger with varactor loss than without to yield the same gain-bandwidth product. However, little advantage results from choosing (a) in excess of 3, which limits the usefulness of arbitrarily raising the pump frequency. Returning to the example at the end of Section 4, the effects of varactor loss will now be illustrated by assuming a varactor with cutoff frequency %b = 2J(105 Mc). To retain a 20-db gain and 100-Mc bandwidth with minimum pumping (i.e., minimum a) it is necessary that w > > Q =: 2n(3.3.103 Me) Choosing wp = 2 (10l Mc) yields (by 64) PD = 2((500 Mc) Letting GL = GD, so ~o = D + pL = 2t(lOOO1 Mc), 40

then by (66) 2t 1 1 1 - o100 Me 1000 Mc 110 Me s and by (66) = (100)(110 Me) = 0.05 2.105 Mc which is some five times larger than the a predicted in the absence of varactor loss. (Note: in addition, the previous calculation assumed a pump frequency only half as large.) Since the value 0.05 is half the safe upper limit assumed for a, it will be concluded that 20 db gain is an upper limit for a Brett amplifier with a 100 Mc bandwidth and employing a varactor with 10l cut off frequency. 5.3 Noise Figures Because the active element in the Brett amplifier is reactive (i.e., a pumped, nonlinear capacitance), it might be anticipated that negligible excess noise would be associated with it. However, in other types of reactive mixers (e.g., parametric amplifiers) thermal noise associated with parasitic resistance in the varactor element provides the lower limit for excess noise. In addition, negative resistance tends to be reflected at various sideband frequencies, which causes the reactive mixer to accentuate external circuit noise. It is important to examine the Brett circuit for these types of noise mechanisms. Considering the diagram in Fig. 5, it is evident that excess noise can arise in both the video band and the pump pass band. In the latter, thermal noise will come from the varactor loss (GD) and the load (GL). Normally load noise would not be held against the amplifier, but 41

analysis will show that the pump accentuates load noise in this circuit, so the load is a source of excess noise. In the video band, excess noise will come primarily from parasitic losses in cables and transformers, etc., since varactor loss can be neglected at the rather low signal frequencies involved (i.e., s < < p). Therefore, no excess noise will be attributed p to the varactor in the video band#. With these assumptions, the noise figure of the Brett amplifier can be expressed by N + N F += ^ (G (69) N 0 where N = output noise power due to parasitic losses in video input circuit N = excess output noise power due to pump tank losses and the output load T = room temperature 3B = amplifier noise bandwidth in cps N If G is the effective parasitic conductance of the video input circuit, and G is the video source conductance (G = G + G ), then so S so S N G 5 S KBN (TG) Go 0 s0 if G' and G are at room temperature. By careful design, Gs/Gso should s so s so be small, but the actual value of G could depend on numerous external S factors. No attempt will be made to evaluate N numerically. The more fundamental noise source is N, which depends upon the varactor. To calculate N0, first consider an increment of the pump cir0' Although the preceding analysis has been applied for 0 < f_ < hundreds of megacycles, the use of input transformers will generally limit the lower value of fs (to say 0.1 Mc), so nonthermal, low-frequency noise mechanisms will be neglected in this report. 42

cuit pass band lying between c, and +, + dc, as diagramed in Fig. 13. Due to thermal noise arising from the pump conductance G, each such increment of the pump pass band will experience an incident RMS noise current 44KT Gdco di - v ~ (70) where TLGL + TD% T = - _ GL +GD is the noise temperature of the parallel combination of GL and GD, assuming the individual noise temperatures are TL and TD, respectively. This incident noise current will contribute to output noise in two ways, by noise reflected in each dcu increment, and by noise converted from the image s increment lying between C1 and -1 - dw. To evaluate these two noise contributions, which will be assumed to be uncorrelated, let Y1 input admittance at cl IN Y = input admittance at U(1 IN Y 1= transfer admittance from w1 to c Y =1 = transfer admittance from Ic1 to -1 The total output noise power in the increment dus in Fig. 13 now becomes p(u) dO - i 12 IYiiN (1) By assuming w( constant and treating w(D as the variable, (71) expresses the noise power distribution over the right half of Fig. 13. 43

But from (47) and (48), *^ ~ Y Y Y + 2(2I2 1 1 ( -) Y- (-W) 1: s 1y s 2((IN IN Y Ys - s(o -Ws) C (72a) + C 2 -Y1 Y~ Y1 + 2~s2 11 iI Y, ) (-W) s 2 (72b) Ws (Wo'+ W ) cl1 and so (71) is valid over the left half of Fig. 13 when aS is negative. dwi: wI Wo WI Fig. 13. Noise contributions from pump circuit conductance. The total noise power delivered to G, PG, can be found by integrating (71) for - < s < m. If c<< ~o and << 1, it is shown in Appendix 2 that PG can be expressed approximately by 00I 2~~~~~/ CI D s \o2 PG Ss ds t) L + 2( (73) \l+p 1.- PP + W'S B-+ P, The effects of pumping on load noise are now evident through the term in (73) containing a. Since a vanishes in the absence of pumping, this term 44

will be the excess load noise that must be held against the amplifier. The quantitative evaluation of (73) is carried out in Appendix 2 for the condition where 0 >> s. This case has been chosen because it was previously shown to yield maximum gain in the presence of varactor loss. Denoting the total noise power reaching the external load (GL) by PG L (73) now yields L G 2P\ W22 GL G ) [4KT (2 (2)( 2) (74) Therefore N0 = (GL GD T B(B) () where ON = 2iT is the noise bandwidth of the pump circuit in cps, as can be identified by comparing the first term in (74) with (69). Again concentrating on the adjustments associated with optimum gain, as discussed in Section 5.2, let GL = G so o = /a (by 63), then No = (4KT'BoN) - o) (75a) o 2 or, by (66), and noting that (s s 3, 1 - (TG)0 /( 0 = 2 (4T'BN) (75b) Therefore, by (69), the basic contribution of the varactor to the amplifier noise figure will be 45

F + 1 + o T' (76) In the practical example in Section 5.2, design considerations were given for obtaining a 1000-Mc gain-bandwidth product, using a varactor with a 105-Mc cut-off frequency. This design called for a = 0.05, oP; 10 B, and (TG)o = 100; so, by (75b), the load noise power is approximately doubled through the action of the pump. However, if T' = T, this excess power contributes less than 0.1 db to the circuit noise figure in (76). This extremely low varactor noise figure occurs primarily because of the choice of the operating frequencies. The close spacing and equal loading of the output sidebands greatly suppresses the undesirable, negative resistance effects due to the lower sideband effects that are well known in conventional single-sideband parametric amplifiers. The moderate enhancement of output load noise that is predicted by (75b) contributes very minutely to noise figure, since excess output noise contributes to F in proportion to the inverse of (TG)o, and (TG)o is large by virtue of aD the high - ratio. If measured noise figures exceed the numerical values predicted above, experiments will have to determine the source of additional noise contributions. For example, concern must be given for the effective load noise temperature T'. While the varactor can be assumed to be at room temperature (TD = T), the load temperature TL may be much larger, since pump oscillator noise will contribute to TL. If this effect is large, pump oscillator padding will be required to reduce TL. Another contribution to TL will be the load temperature itself. If this is large compared to room temperature, it can be reduced with the aid of a nonreciprocal device such as an isolator or circulator. Finally, 46

the magnitude of input circuit noise must be estimated in an actual circuit, so it can be included in (76) according to (69). But these extraneous effects are largely under the control of the circuit designer, so very low noise figures can generally be expected from the Brett circuit. 47

6. CONCLUSION In spite of the complexity of the analysis in this report, the Brett amplifier is a simple, and effective, circuit. It is capable of gain-bandwidth products that surpass those of the best vacuum tubes and most transistors. In addition, the Brett circuit is capable of noise figures that are significantly lower than are provided by these more conventional components. It has been determined that the Brett circuit operates by allowing an input (video) signal to phase-modulate a carrier (pump) by means of a nonlinear reactive element (varactor). Amplification is possible because the pump and signal, respectively, contribute power to the modulation process approximately according to the ratio pump frequency \2 s ignal frequency/ An amplified version of the input signal can thus be obtained by detecting the modulated carrier with an efficient phase detector. An additional advantage of the Brett circuit is its positive input resistance at the video frequency. For an ideal circuit, with fixed pump frequency, optimum gain occurs when the bandwidths of the pump and video circuits are in the ratio 2:1, respectively. However, for optimum transducer gains it is essential to control these bandwidths with minimum shunt capacity. For this reason, input transformers are useful for controlling the video bandwidths. When varactor losses are considered, a useful upper limit in pump frequency occurs because the above bandwidth rule cannot be satisIn this connection, several simple, ultra-wideband transformers have recently been constructed by this laboratory and will be the subject of a future report. 48

fied at high pump frequencies. To obtain optimum gain in this case, the pump frequency must be raised until the pump bandwidth is much larger than the video bandwidth, so the amplifier bandwidth can be determined entirely by the video circuit. Since varactor losses will also provide the ultimate limitation on noise figures, a noise analysis has been made for the case where varactor loss is significant. Noise figure formulas were derived, and an illustrative calculation was made for an amplifier with a 1000-Mc gainbandwidth product. It was shown that a varactor with a 105-Mc cut-off frequency would contribute less than 0.1 db to the total amplifier noise figure. These results are generally consistent with the experimental results reported to date by Brett and others (private communications). 49

APPENDIX I The analysis of Section 3 assumes that all extraneous sidebands are short-circuited. However, this assumption is unnecessary if IVsl<<Ilv To prove this statement, the voltage components calculated by (47), due to Iso, can be used to find the components of short-circuit current in (41). For example, if I2s is the complex Fourier component of short-circuit current at 2o, then by (40) s I2 = 2jwc V1 V*1 and by (45a) and(48a) sO = Vs L N s iL J Since the factor in the first bracket is independent of V, the shortcircuit current available to the video circuit, at the video harmonic, will be small compared to the applied video current when |V I <<v I. as similar conclusion can be reached for the other components of short-circuit current in (41). Therefore, for small signals, the assumptions stated at the beginning of Section 3 are valid. 50

APPEMDIX II The derivation of (73) and (74) proceeds as follows. Letting P = - q = --; 4~ = +- (77) then (72a) and (72b) become 2 L1 G2 ( + S p2) -.p2 + Cp2 + ( p IN o 1 - p2 - ~ p(q -p)a u + j pu Y!.-1 2 = G + 2) + C + (8b) Po I P(q + p) a When these equations are combined to form (71), terms of odd symmetry in cO (or x) can be dropped, since the integral of (71) is desired. The S noise power delivered to G, PG, now becomes J P(Os)d ) 2 p - (1 + p2) ) [1 - p2(1 ) ]2 + (pu)2 + ( p )2(p2 + 2q2) ) ft [1 — p2 (1 C-a)]2 + (pu)2 and if a << 1 and p2 << q2, this relation can be reduced to 51

p j + 22 (79) PG a2, [ (1 - p2)2 + (pu)2) o which is equivalent to (73). When (79) is integrated term by term, the first term yields (Dwight 120.01) 2n J 00) 2 )(ftV ) (80) l+ ~ In the second term, (1 - p2)2 + (pu)2 ( (p2 + )(p2 + U2 -2) if u2> 4. Therefore, for the case where 3 >> ps, so that u2, 1, the second term in (80) becomes approximately (Dwight 120.2). ( w)f t1O (I) dp = (t)(caq) (I) (81) Equation 74 can now be obtained directly from (80) and (81). It should be noted that the approximations made in obtaining (81) introduce positive errors, so (74) is a slightly conservative estimate of output noise power. 52

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. 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. 3. K. Kurrokawa and M. Uenohara, "Minimum Noise Figure of the VariableCapacitance Amplifier," BSTJ, Vol. XL, No. 3, May 1961, pp. 695-722. 4. D. K. Adams, "An Analysis of Four-Frequency Nonlinear Reactance Circuits," IRE Transactions on Microwave Theory and Techniques, Vol. MIT-8, No. 3, May 1960, pp. 274-283. (See also, D. K. Adams,"Some Considerations of Four-Frequency Nonlinear Reactance Circuits," Cooley Electronics Laboratory Technical Report No. 96, The University of Michigan, September 1959.) 5. H. Brett, P. A. Brand, W. G. Matthei, "A Varactor Diode Parametric Standing-Wave Amplifier," Proc. IRE, Vol. 49, No. 2, February 1961, pp. 509-510. 6. L. Blackwell and K. Kotzebue, "Semiconductor-Diode Parametric Amplifiers," Prentice-Hall, Inc., 1961. 53

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