ERRATA Technical Report No. 134 Cooley Electronics Laboratory Page 15, Equation 2. 7 n in numerator of first term should be m. Page 15, line following Equation 2. 7 (2. 4b) should be (2. 4). Page 95, Section 4. 3 Should be labeled Section 6. 3. Page 95, third line following Equation 6. 49 Fig. 4. 3 should be Fig. 6. 3. Page 125, Fig. 8. 5 Ordinate should be labeled t. Page 145, Fig. 9. 5 Ordinate should be labeled GAIN (DB). Page 155, Fig. 9. 7 Abscissa should be labeled w /J1. 56 3wd Page 198, Footnote 1 Should refer to footnote on page 56. Page 199, Fig. 10. 6 Ordinate should be labeled T/TD. 3g BO qnatucm 4o50. ShrouIA )a a 9' 1a"rgrg du + r 1 Lr Pag 15 thif p graph All refern ces to "noise ftgure" Ghould be more p ifically, to "nois temperat o" The noie te atse ie in the Table a rel. &aive to Td.

Technical Report No. 134 4853-7-T A STUDY OF DOUBLE-SIDEBAND REACTIVE MIXERS by H'e D. K. Adams Approved by: /_ _y__ _ _ B. F. Barton for COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor Contract No. DA-36-039 sc-89227 Signal Corps, Department of the Army Department of the Army Project No. 3A99-06-001-01 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan December 1962

ACKNOWLEDGEMENTS The author wishes to thank the members of his doctoral committee for their comments and technical guidance. Special appreciation is due Professor C. B. Sharpe for his encouragement and council. Further, the author wishes to thank Mr. Wilbur J. Nelson for his patient help with the experimental portions of this work, and to Mr. Herb Brett of the U. S. Army Signal Corps for his suggestion that the phase modulation amplifier problem be treated. Special gratitude is owed to Mr. Robert E. Graham, Mrs. Lillian M. Thurston, and to Miss Ann M. Rentschler for preparing this manuscript for publication. Finally, the author is appreciative of the support the U. S. Army Signal Corps has given this work.

TABLE OF CONTENTS Page ACKNOWLEDGEMENTSii LIST OF ILLUSTRATIONS v LIST OF SYMBOLS ix LIST OF APPENDICES xv ABSTRACT xvi CHAPTER I: INTRODUCTION 1 1.1 Statement of the Problem 5 1.2 Review of the Literature 8 1. 3 Topics of Investigation 10 CHAPTER II: MATHEMATICAL FORMULATION 13 2.1 Selection of Variables 13 2.2 General Energy Relations 14 2. 3 Formulation for First-Order Mixing 16 2. 4 Methods of Solving the Set (2. 13) 19 CHAPTER III: INTRODUCTION TO SINGLE-SIDEBAND REACTIVE MIXING 29 3. 1 Application of General Energy Relations 30 3. 2 An Alternate Viewpoint 32 3. 3 Circuit Effects Due to First-Order Interactions 34 3. 4 Derivation of Circuit Quantities 36 CHAPTER IV: THE UPPER-SIDEBAND CONVERTER 45 4.1 Bandwidth of the Upper-Sideband Converter 50 4. 2 Degrading Influences on Upper-Sideband Conversion 52 4.3 Down-Conversion from Cou to Cor 60 4. 4 Conclusion 61 CHAPTER V: THE LOWER SIDEBAND CONVERTER 63 5. 1 Bandwidth and Sensitivity of the Lower-Sideband Converter 67 5. 2 Degrading Influences on Lower-Sideband Conversion 69 5. 3 Parametric Amplification 74 5. 4 The Degenerate Parametric Amplifier 77 5. 5 Conclusion 78 CHAPTER VI: STABILITY OF NETWORKS WITH PERIODICALLY VARYING REACTIVE ELEMENTS 79 6.1 Solutions of the Set (6. 3) 80 6. 2 Illustrative Calculation of Stability from Sinusoidal Steady-State Response 84 6.3 Conclusion 95 iii

TABLE OF CONTENTS (Cont.) Page CHAPTER VII: GENERAL CONSIDERATIONS OF REACTIVE MIXING INVOLVING SYMMETRICAL SIDEBAND PAIRS 97 7. 1 Application of the General Energy Relations to Double-Sideband Reactive Mixers 98 7.2 Application of the General Energy Relations to Phase Modulation 102 7. 3 A Generalization of the Double-Sideband Mixer 104 7. 4 Conclusion 105 CHAPTER VIII: DOUBLE-SIDEBAND SINGLE-PUMP MIXING WITH INDEPENDENT SIDEBAND LOADING 107 8. 1 Gain and Bandwidth for Double-Sideband Conversion from w to u 111 r u 8. 2 Noise Figure for Conversion from or to cou 119 8. 3 Down-Conversion from wou to Cr 129 8. 4 Conclusion 135 CHAPTER IX: DOUBLE-SIDEBAND MIXING WITH DEPENDENT SIDEBAND LOADING 137 9. 1 Circuit Properties 143 9.2 Design Considerations 159 9.3 Conclusion 170 CHAPTER X: DOUBLE SIDEBAND MIXING WITH HARMONIC PUMPING 173 10. 1 Basic Analysis of Harmonic Pumping 174 10.2 General Physical Interpretations 176 10.3 Circuit Properties of Fig. 10. 3 182 10. 4 Harmonic Pumping with Dependent Sideband Loading 198 10. 5 Conclusion 203 CHAPTER XI: EXPERIMENTAL WORK AND CONCLUSIONS 205 11. 1 An Ultra-Wideband Video Amplifier 205 11. 2 Experiments with Double-Pump Circuits 212 11.3 Conclusion 216 REFERENCES 225 iv

LIST OF ILLUSTRATIONS Figure Page 1. 1 Energy conversion by a time-varying capacitor 2 1. 2 Amplification by mixing with a time-varying capacitor of period w/co 3 1. 3 The basic representation of a reactive mixer 4 1. 4 A basic double-sideband spectrum 6 1. 5 An equivalent circuit for Fig. 1.3 7 2. 1 An equivalent circuit for (2. 18c) - (2. 18e) 23 2. 2 An equivalent circuit for (2. 18a), (2. 18c) and (2. 18d) 26 3. 1 Single-sideband conversion on a quantum basis 33 3. 2 Circuits for single-sideband conversion 35 3.3 A further reduction of Fig. 3. 2b into a linear, time-varying model 36 3. 4 A two-terminal-pair representation for conversion from ol to w2 showing certain critical external parameters 37 3. 5 The input noise sources associated with gs and g! 41 3. 6 A general set of noise sources that may contribute to the noise figure for conversion from wo. to W2 42 3.7 The equivalent circuit for a varactor 43 4. 1 A symmetrical transformer and its dual in T equivalent form 47 4. 2 A functional equivalent circuit for upper-sideband conversion 48 4. 3 A comparison of maximally flat and equal ripple pass bands for equal pumping coefficients 53 4. 4 A special case of Fig. 3. 4 appropriate for uppersideband conversion 53 v

LIST OF ILLUSTRATIONS (Cont.) Figure Page 4. 5 Plots of conversion gain and noise temperature vs. frequency ratio, showing the influence of varactor loss 55 5. 1 A functional equivalent circuit for lower-sideband conversion 64 5. 2 The equivalent circuit of a lower-sideband converter after Yp is transformed to Wr 66 5. 3 Power gain by reflection through parametric amplification 75 5. 4 A parametric amplifier with a circulator for improved performance 76 5. 5 Parametric amplifier noise temperature vs. frequency ratio, showing the optimum predicted by (5. 35b) 77 6. 1 An appropriate linear model for first-order interactions in reactive mixers. 80 6. 2 A typical root locus for the circuit in Fig. 6. 1 showing how the characteristic roots may shift when C(t) is applied 84 6. 3 A canonical realization of the lower-sideband mixer using a time-varying capacitor 86 6. 4 The poles and zeros of the circuit in Fig. 6. 3 86 7. 1 Sketches of (7. 2) showing regions of positive and negative conversion gains. 100 7. 2 Sketches of (7. 3) showing regions of positive and negative conversion gain 101 7. 3 An illustration of phase modulation by reactive mixing 102 7. 4 A typical phase modulation spectrum and the associated stability criterion based on (7. 7) 103 8. 1 A functional equivalent circuit of double-sideband conversion with a single pumping frequency 108 8. 2 The input conductance at resonance vs. yp, showing regions of apparent instability 110 8. 3 Curves showing improved upper-sideband upconversion due to the introduction of the lower sideband 116 vi

LIST OF ILLUSTRATIONS (Cont.) Figure Page 8. 4 Gain improvement and corresponding gainbandwidth product for upper-sideband conversion in the presence of the lower sideband 120 8. 5 Solutions of (8. 54) 125 8. 6 Optimum noise temperatures corresponding to infinite conversion gain between cor and wu 126 8. 7 Effects of the lower sideband on Tru and Gru in the matched output case 128 8. 8 Optimum noise temperatures for down-conversion from cu to cor 133 9. 1 A limiting case of double-sideband mixing that occurs for cr/p << 1 1 38 9. 2 A basic technique for isolating the sidebands from the pump in Fig. 9. 1 139 9. 3 A detection system for recovering an amplified version of the modulating signal 143 9. 4 The bandwidth of Gr vs. the sum of the signal bandwidths, normalized to Jr3 146 9. 5 Gain characteristic of the example amplifier in Section 9. 1.2 149 9.6 Estimates of C, yp, and kp2 by Fourier analysis 151 9.7 A comparison of optimum VGr and voltage gain for fixed bandwidth and varactor resistance vs. pump frequency 155 9. 8 Noise contributions from pump circuit conductance 157 9. 9 Two examples of adding a carrier for phase detection 162 9. 10 A means for direct quadrature detection using the slope of the pump tank 163 9. 11 A balanced modulator configuration for improving the pump power efficiency in the case of slope detection 165 9. 12 A realization of the detection scheme in Fig. 9. 9b 166 9. 13 F(X) in (9. 38) 170 10. 1 Transmission line representation of the three possible conversions between wcr, ca, and cu 178 vii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 10. 2 The three basic configurations for unilateral conversion 181 10. 3 Equivalent circuit for matched up-conversion from w tow,u for T = T = T 184 r u r u o 10. 4 Variation of the normalized input (or output) admittance with the gain parameter 7, for various output (or input) loads 189 10. 5 A comparison of upper-sideband noise temperatures for single- and double-pump up-converters 196 10. 6 A comparison of double- and single-pump downconversion noise temperatures 199 10. 7 Harmonic pumping in a phase modulation amplifier 201 11. 1 A "breadboard" model of the phase modulation amplifier 206 11. 2 A close up of the varactor mount in Fig. 11. 1 207 11. 3 Theoretical circuit properties of a phase shift amplifier vs. pump frequency 209 11. 4 Signal waveforms from phase modulation amplifier in Fig. 11. 1 209 11. 5 Voltage gain vs. pump power for 9. 5 kMc pump 210 11. 6 Measured pass bands of the varactor mount and the detector 211 11. 7 A low frequency version of the double-pump reactive mixer 213 11. 8 Schematic for Fig. 11. 7 214 11. 9 The response to a swept signal at wr terminals in Fig. 11.8 215 11. 10 Stable response curves for Fig. 11. 8, similar to those in Fig. 11.9 215 viii

LIST OF SYMBOLS SYMBOL MEANING DEFINED BY, OR FIRST USED IN co angular frequency Fig. 1.2. co midband frequency footnote 1, Ch. I Wcp, r two reference frequencies (1. 1) mco mco + no (1.1) mn p r m,n subscript integers (1.1) m", m', n' dummy subscript integers (3.1) Table of Special Subscripts Subscript Equivalent m and n m n p 1 0 r 0 1 Fig. 1. 4 u 1 1 JQ e 1 -1 2p 2 0 x (Ch. II) co t (2. lc) y (Ch. II) co t (2. lc) i(t) instantaneous current Fig. 1.3 v(t) instantaneous voltage Fig. 1.3 q(t) instantaneous charge (2.3) 1E(t) instantaneous source current Fig. 1. 3 q(v) charge-voltage law of nonlinear capacitor (2. 2) I Fourier coefficient of i(t) at co (2. lb) V Fourier coefficient of v(t) at cm (2. la) mn mn Qmn Fourier coefficient of q(t) at omn (2. 4) 1m Fourier coefficient of 11(t) at comn Fig. 1. 3 mn mn Wmn average power entering mixing element at wc (2. 5a) ix

LIST OF SYMBOLS (Cont.) SYMBOL MEANING DEFINED BY, OR SYMBOLMEANING FIRST USED IN V dc voltage (2.11) Q dc charge (2.11) al, a2 Taylor series coefficients in charge-voltage representation (2. 11) i (t) instantaneous short circuit current (2. 13) sc v'(t), v"(t) special voltage waveforms (2. 19) c'(t) special capacitance waveform (2. 20) Y(c) total admittance (2. 17a) Y(o) external admittance (not parasitic) Fig. 1. 3 Y'(o) parasitic admittance (2. 17a) g, g, g' conductive parts of total, external, and parasitic admittances, respectively Fig. 3. 4 b, b, b' susceptive parts of total, external, and parasitic admittances, respectively Fig. 3. 4 Y mn Y Y' Fourier components of total, external, and mn' mn mn parasitic admittance respectively, at w (2. 17a), (1. 2) mn C average capacity of pumped, nonlinear capacitor (2. 14) Yin. g. input admittance and conductance, respectively Fig. 3. 4 y Fourier component of time-varying capacity - at c (2. 24b) Y7 Fourier component of time-varying capacity p at 2w (2.24c) vY Fourier component of time-varying capacity at cr (2.30b) C total shunt capacity at mn (4. 14) mn mn [YA], [Yg], admittance matrices (2. 32b) [P], [V], [I] matrices (2. 32c) [I], [Ymn] matrices (2. 33b) gS' gL source and load conductance, respectively Fig. 3. 4 a, A parameters (2. 10) x

LIST OF SYMBOLS (Cont.) SYMBOL MEANING DEFINED BY, OR FIRST USED IN;h8^ ~ ~(Planck's constant)/2v (3. 8) N events/second (3. 8) Gij transducer gain from )i to cj (3. 15b) Gi Gij, neglecting parasitic loss (3. 16b) G.. midband gain (3.17) B bandwidth in cycles/second (3. 17) 3 bandwidth in radians/second (3. 18) k Boltzmann's constant Fig. 3.5 T noise temperature Fig. 3.5 T 2900 K Fig. 3. 5 0 F.. noise figure from conversion from ai to )j (3. 19) Tij noise temperature for conversion from wi to Wj (3. 20b) T noise temperature of parasitic conductance Fig. 3. 6 TL noise temperature of load conductance Fig. 3. 6 T noise temperature of dummy load (8. 39) Td noise temperature of varactor diode (4. 26) t TL/Tdd (5.31b). ~k s =t, if W& is the output frequency s Tf/Tdd (8.40) rd parasitic series resistance of varactor Fig. 3. 7 gd effective parasitic shunt conductance of varactor Fig. 3.7 Qd Q of varactor (3.28) gdr' gd' gdu gd r at wry, w, wu, respectively (3 33) Wd frequency where varactor Q equals one (neglecting parasitic inductance) (3. 29) Wd WdY/C footnote 1, Ch. IV L self inductance Fig. 4.1 M mutual inductance Fig. 4.1 xi

LIST OF SYMBOLS (Cont.) SYMBOL MEANING DEFINED BY, OR SYMBOLMEANING FIRST USED IN k pumping coefficient at cop (4. 15) Pu a parameter relating effect of Cu on r (4. 18c) Pk a parameter relating the effect of co on co (5. 8) K a parameter for measuring the effect of varactor loss (4. 22) Rr R (c r/c),2 (cu/co)2 (4.32), (4.38) r u r d' d x, y, z efficiency factors (4.33) PA power available (5.7) S sensitivity (5.18) an, b coefficients (6.1) 0ij(t), 0i(t) periodic coefficients (6.3) x. dependent variable in a set of linear differential equations (6.3) M~iJ~ ~ characteristic root (6. 4) /zk kth characteristic root (6. 5) /IL, /kL, ITk zeroth, first, and second-order approximations to /k (6.33) Y., Yik periodic functions (6.4) Ok' rthk parameters (6.7), (6.39)?q7 ~detector efficiency factor (9.20) Al, A determinants (6.9) D, Ds determinants (6. llb) Ns a parameter (6. 12) nm Bs an initial condition (6.13) Ps a periodic function (6. 13) Pr' Pp poles of a network (6. 19) o, 12 real and imaginary part of network poles (6. 19) Z1, Z2 zeros of a network (6. 19) k1, k2 real parts of network zeros (6. 19) xii

LIST OF SYMBOLS (Cont.) TSYMBOL MEANINGT^ DEFINED BY, OR ~~~~~~~~~SYMBOL__ __MEANING__ _FIRST USED IN e, f parameters (6.24) L, C, R inductance, capacitance, and resistance of network Fig. 6.3 a (Ch. VI) a perturbation parameter (6. 27b) a a noise figure parameter (A-5) a.. a matrix element (6.38) aj, alj zeroth and first approximations to a.i (6.34) s (Ch. VI) a parameter (6.28) s TL/T d (8.40) s a critical value of s (Ch. VI) (6. 36) c.. a matrix element (6.38) ij A system determinant (8. 6) a (Ch. VIII) a parameter (8.49) N(Ch. VIII) a parameter (8.73) Or', p bandwidth at rc, wp respectively (9. 1), (9.21) M (Ch. IX) index of phase modulation (9. 4b) GPM transducer gain by phase modulation (9. 19) 7r1 detector efficiency factor (9. 20) Gr 272G (9. 20) r PM T effective noise temperature of pump tank (9. 58) T effective noise temperature of external circuit losses in pump tank (9. 58) diN incremental noise current (9. 59) P total noise power reaching external load (9. 64) B, 0B amplitude and phase of inserted carrier (9. 67) N, 4~ amplitude and phase of resultant carrier (9. 68a) A, v, /i (Ch. IX) parameters (9. 78b) 5 phase parameter (10. 1) xiii

LIST OF SYMBOLS (Cont.) SYMBOL MEANING DEFINED BY, OR ~~~~~~~~~SYMBOL__ __MEANING__ _FIRST USED IN T' T', T U normalized admittances of wr co, W U, respectively' (10. 1) Tr, T, normalized conductance at r, w W W u r'respectively (10. lb) T", T"' T" normalized susceptance at cO, w, W respectively (10. 1) A normalized system determinant (10. 2a) Tr. T, U. normalized input admittance at cr, cu, Co, in in in respectively (10. 4) Y12 image admittance at wc, due to conversion to w2 (10.5) 12 normalized image admittance (10. 6) 012 propagation constant between c) and c2 (10. 6) To sin (1 + j cos 1 (10. 14a) k2p coefficient of harmonic pumping (10. 23) p (Ch. X) IY2p/yp (10. 43b) p reflection coefficient (11. 1) p, q, u (App. B) parameters (B-l) xiv

LIST OF APPENDICES Page Appendix A The Derivation of Optimum Noise Figure Formulas 219 Appendix B The Derivation of Phase Modulation Noise Figure 223 xv

ABSTRACT Reactive mixing can be accomplished by any nonlinear, or time-varying, energy storage element. In double-sideband reactive mixing, interactions take place between information carrying signals at only three frequencies. Such interactions require the presence of pumping or local oscillator signals at appropriate frequencies. The basic mixing spectrum treated in this study employs wr, cw, and wo as information carrying signals, and op and 2co as pumping frequencies; where wc = cw + Wr and cow = c - o. In the text, these p p p rp p r pumping frequencies are shown to provide all possible first-order interactions between Wo, coW and c. Also, the interactions in the latter set are shown to occur in an infinite number of other signal sets. Therefore, the double-sideband reactive mixers problem is generic to many reactive mixer configurations of practical interest. The purpose of this study is to extend the understanding of reactive mixers through an analysis of the double-sideband configuration. Two basic cases are treated, one with a single pump at cp and the other with double pumping at wo and 2wo. The useful engineering properties of each system are evalup p ated, and compared with the better known properties of single-sideband reactive mixers. Consideration is given to such circuit properties as conversion gain, stability, bandwidth, and noise figure. In many cases, significant improvements are noted, which are explained by a new technique of reactive mixer modeling. In the single-pump case, single-sideband conversion between cow and wo offers the greatest gain-bandwidth product, but conversion between co and wo offers the greatest gain. When combining these cases, it is shown that the presence of w can produce a finite increase in gain between or and u, without reducing the original gain-bandwidth product. Alternately, an arbitrary increase in gain is possible with a 30 percent reduction in gainbandwidth product. By the same technique, improvements are also noted in the conversion noise figure between co and c, particularly in the down-conversion mode. xvi

In the double-pumping case, an interesting class of unilateral conversions are obtained. One of these offers arbitrarily large conversion gains between cr and Wu, with gain-independent input and output terminal impedances. This mode of operation is capable of yielding larger gain-bandwidth products, greater circuit stability, and significant improvements in down-conversion noise figure. Perhaps the most significant practical outcome of this study is the development of a stable, ultra-wideband, video amplifier by double-sideband reactive mixing. Gains of 20 db and bandwidth from dc to 100 Me are predicted for a single mixer element. Experimental realizations of this system are shown and discussed. It is also demonstrated that this is a form of parametric amplification by phase modulation. Another theoretical problem that has been closely related to this study is the stability of periodically-varying networks. The interrelation of these problems has led to a new method of predicting the stability of linear systems with periodically varying parameters. This method requires a knowledge of sinusoidal steady-state response, but it does not entail a solution of the system differential equations. Therefore, this method of determining stability is less tedious than the normal method of calculating characteristic roots. It also provides a new approach for approximating the characteristic roots of a periodically-varying system. xvii

CHAPTER I INTRODUCTION The general properties of reactive mixers have been illustrated in classical problems dating back 100 years or more to the work of Faraday, Melde, and Lord Rayleigh (Ref. 1 ). Although these early illustrations of reactive mixing were nonelectrical by nature, they demonstrate a principle that can be readily depicted by a simple electrical example. Consider the parallel plate capacitor in Fig. 1. 1. When charge exists on the plates, work will be done when the plate separation is increased, but if the charge is then removed, the plates can be returned to their original position without involving further work. This process can be made continuous by adding an inductor to form an oscillating circuit. The capacitor is then alternately charged and discharged twice each cycle, so by separating the capacitor plates suddenly when it is charged and returning them when it is discharged, energy will be continually added to the oscillating system. If a resistor is now added that exactly dissipates the supplied mechanical energy, steady ac power will be delivered to the resistor at half the frequency of the moving capacitor plates. But if the resistor is further reduced until the oscillation stops, it can then be established that an applied signal will be amplified if its frequency is near the oscillation frequency. In this case, the signal mixes with the time-varying capacitor and produces a difference frequency or lower sideband near the oscillation frequency, as depicted in Fig. 1. 2. The added presence of a lower sideband allows the applied signal to be amplified regardless of the timing (or phase) of the mechanical work. A more comprehensive explanation of this effect will follow from a general study of reactive mixing. Mixing is a general phenomena that arises whenever there is coupling between sinusoidal variations with different frequencies. In the most common instance, mixing is caused by nonlinear couplings; but it can also be associated with linear, time-varying 1

2 o (t) c(t) v( () t v((t) Fig. 1. 1. Energy conversion by a time-varying capacitor. couplings if the coupling element itself constitutes one of the two required variations. The previous example is illustrative of the latter case. Only in linear time-invariant couplings does mixing fail to occur, but such couplings exist only as an idealization, so it can be concluded that frequency mixing is prevalent in all physical systems. The basic process of frequency mixing can be described as follows. If two signals with angular frequencies or and wp are coupled by a mixing element, the resultant steady-state signal will tend to contain all positive frequencies in the set )m = m c + n); m,n = 0, + 1, + 2,..., (1.1) which set is fundamental to all steady-state mixing processes. An additional characteristic of mixing is the Fourier distribution of signal among (1. 1), but this distribution is strongly dependent upon the type of mixing element. There are two extreme element types that are useful for characterizing mixing distributions. One is the storage element, which is commonly called a reactance, and the other is the dissipative element, which can be characterized by a resistor. A marked difference in mixing properties is noted when resistive and reactive elements are compared. This difference arises even though it is always a nonlinear or time-varying coupling that produces mixing in any element type. Since the mixing properties of resistive elements are well-known, the contrasting properties of reactive mixers seem quite unusual. In fact, when compared with the resistive mixer, the reactive mixer is an intriguing physical system. Therefore, an investigation of the unusual properties of reactive mixers has been chosen aasathe general purpose of this analysis.

3 I IX C ^ ^C(t) 0 12Ww 2 -o_ TUNED TO Co (o OSCILLATION \ \ FREQUENCY Fig. 1. 2. Amplification by mixing with a timevarying capacitor at period I/O. Once the mixing element itself is restricted to a certain class, either resistive or reactive, the auxiliary circuitry surrounding this element gives rise to the next order of influence on its mixing properties. Such external circuitry will be assumed to be linear and time-invariant, so it can be represented at the terminals of the mixing element by a Norton equivalent admittance, Y(w). For the present, let it be considered that mixing is due to nonlinearity. The basic signals at w01 and 10 (i. e., at wr and w ) can then be applied by current generators with complex amplitudes 1101 and II10, respectively. However, the tendency for currents to flow through the mixing element at all frequencies in (1. 1) is unchanged when external generators are applied at any, or all, of the wm. Therefore, a general model for the nonlinear reactive mixer is provided by the circuit in Fig. 1. 3. Later, it will be shown that linear reactive mixers can also be treated by Fig. 1. 3, as a special case. The components I in the general source waveform 11(t) can exist at each frem n quency in the set wmn so a great variety of source waveforms are possible. The procedure of this investigation will be to determine the steady-state v(t) and i(t) that result from specific ]I(t) waveforms. In particular, source waveforms will be considered that are made up of certain combinations of sinusoidal signals in the set 1I. In each case, the discrete set cWm mn mn will include all involved frequencies, so it will be convenient to define Y = Y(mn) (1.2) mn mn

4 SOURCE I(t) r —- l- 0 i(t) Y(w) —-w LINEAR ) v(t) TIME-INVARIENT "min |loi o 10 FILTER _______ —------— f j_____{_____-o NONLINEAR REACTIVE ELEMENT Fig. 1. 3. The basic representation of a reactive mixer. Although v(t) and i(t) tend to contain all positive cmn, the steady-state power flow will be restricted to a finite band in all cases of interest here. The filter characteristic Y will thus be chosen to short circuit, or otherwise suppress, all but a finite set of fremn quencies. This step calls for unrealizable Y(c,), but realizable Y(w) will be shown to yield satisfactory approximations in many cases, partly because wr and wp will be assigned different roles in the mixing process. The frequency w will be treated as the reference frequency for a signal being processed by the mixer, while w and its harmonics mp will be considered local oscillators, p P or "pumps," for supplying RF power to the mixer. In some configurations, the signal will actually be applied at or, in which case wr can be properly termed the signal frequency. In other cases, the applied signal will have one of the "sideband" frequencies mwp + nwr (n A 0), which explains why co is given the more general term, the signal reference frequency. When n A 0, the set mwp + nwr will be termed the set of signal frequencies. The latter set is to be contrasted with the set of pump frequencies which are characterized by n = 0. When reactive mixers of higher complexity are employed in a system, it should be noted that a higher over-all system complexity does not necessarily result. For example, the reactive mixer configurations in common usage today frequently require auxiliary equipment, such as circulators, to yield useful characteristics. Also, the broadbanding techniques that have been developed for simple reactive mixers call for a higher complexity of nonmixing (i. e., linear) circuitry. This analysis will show, however, that high spectral complexity can yield the same improvements. In addition, reactive mixers offer circuit effects that are not easily duplicated with ordinary circuit components. Therefore, multiple-sideband reactive mixers are of practical engineering interest.

5 1. 1 Statement of the Problem The main interest in reactive mixers to date has been in cases where the signal components are linearly related. This situation tends to prevail when the applied signal amplitude is small when compared to the pump amplitudes, and in this case several important circuit effects have been successfully realized. Foremost of these is conversion gain between two or more signal frequencies. In addition, such conversion gain is frequently accompanied by low noise figures and by gain-bandwidth products that tend to be proportional to pump frequency. These attributes are readily demonstrated in reactive mixers employing simple spectral configurations, which immediately prompts such questions as, "What benefits can be derived from reactive mixers employing more complicated sideband configurations?", and "What analytical procedures are required to explain and to evaluate these higher-order systems?" This analysis will attempt to answer these questions. Particular attention will be given to the spectral configurations that occur when Y(cw) suppresses all but three frequencies in the signal frequency set, and all but two of the pump frequencies. These double-sideband cases will be termed symmetrical or unsymmetrical according to the symmetry of the signal frequencies, exclusive of co r, with respect to op. For example, a case of symmetrical, double-sideband mixing occurs when the mixing products of co and or are restricted to the pair p + co. However, the term symmetrical, as used here, does not distinguish the frequency at which signal is applied. For instance, in the symmetrical example just cited, the applied signal may be at r c - c a, or co + co. Two reasons can be given for restricting the emphasis in this analysis to threesignal frequency (or double-sideband) cases. These are: (1) In concentrating on single-sideband resistive mixers, previous treatments have shown that individual cases yield distinctive circuit properties. Therefore, circuit properties of greater scope can be anticipated when the effects of two individual sidebands are combined. (2) The increased complexity of multiple-sideband mixers greatly incumbers detailed analysis. Therefore, when consideration is limited to double-sideband circuits, a reasonable compromise is made between preserving

6 SPECIAL NOTATION 0 Wr W Wp u W2p FREQ-" C00 CO WI Wi WI W10 W0-0oo @01 @i1,-1 l10 cl C20 GENERAL NOTATION Fig. 1. 4. A basic double-sideband spectrum. Since this analysis will be primarily concerned with such a spectrum, the special notation above will frequently be substituted for the general notation defined in (1. 1). analytical simplicity and obtaining an increased perception of the inherent properties of reactive mixers. Whenever interest is restricted to a particular set of frequencies, it becomes convenient to introduce an alternate frequency notation. The special set of frequencies in Fig. 1. 4 will receive considerable attention here, so much of the formulation to follow will be based on the special notation in this figure. The assumption that all power flow is restricted to a few frequencies is an obvious idealization. It can always be employed heuristically, but for practical reasons an immediate appraisal of this assumption is required. Certainly real filtering techniques cannot prevent some power flow outside a desired set of frequencies. However, a peculiar aspect of reactive mixers influences this tendency. It will be shown that power flow between two frequencies chosen at random can occur in either direction. Therefore, with a large number of extraneous couplings, cancellation will tend to occur. On the other hand, a large number of extraneous couplings of approximately the same magnitude is unlikely, since multiple mixing is generally required for coupling to frequencies well removed from the desired signal set. Multiple mixing tends to produce weak coupling, particularly when small signal amplitudes are involved. Therefore, in any given circuit, it can be expected that only a few extraneous couplings will create problems. In this case, unrealizable Y(w), that are assumed to suppress all but a certain set of sidebands, can be successfully approximated. One technique

7 01 10 - mn i - I I? 1I - - I0 I — - m OI 10 0 ] [mn Fig. 1. 5. An equivalent circuit for Fig. 1. 3. It is equivalent to Fig. 1. 3 when none of the Omn are suppressed, and is a convenient idealization when Y(w) is assumed to suppress all but a finite number of the cmn. is to design Y(w) that support the desired sidebands and suppress any unwanted sidebands with large couplings. The asymptotic behavior of such Y(w) would then be relied upon to suppress the more weakly coupled sidebands. Filter networks of the type just described tend to yield relatively narrow bandwidths for the desired sidebands. It will be convenient to idealize this effect, by postulating a set of ideal filters with the following property: each filter is an open circuit in one narrow band of frequencies, centered at Wm n and a short circuit elsewhere. The modification made by introducing these ideal filters is shown in Fig. 1. 5, where each filter is represented by a box and is labeled with its open circuit center frequency. This equivalent of Fig. 1. 3 is nonrealizable, but because of its computational value it will be the basis of much of the analysis to follow. Now, for example, the circuit admittance influencing each frequency band appears at a separate terminal pair, and hence can be independently varied. In practical design problems, an optimum set of terminal admittances Y can m n be obtained more easily from the model in Fig. 1. 5, than from Fig. 1. 3. So to realize this design in an actual circuit (e. g., Fig. 1. 3), the linear network Y(w) would have to be chosen to approximate the desired Y when w w, and to become a short circuit elsewhere. mn mn mn This suggests that each Y. must resemble a parallel resonant circuit with fairly high Q. mn When this is the case, a good first approximation to Fig. 1. 3 will result if the ideal filters in The symbol w will be used to denote a midband value. For example, wmn denotes the midband value of those frequencies generally identified as Wmn.

8 Fig. 1. 5 are simply removed. In dealing with Fig. 1. 5, it will be demonstrated that conversion gain is possible between a variety of terminal pairs. Reactive mixer conversion gain also occurs with far more extensive characteristics than are possible in linear, bilateral, multiport networks. To aid in cataloging these effects, the following basic circuit quantities will be of interest: (1) Transducer conversion gain from Wm to co, mn m'n' (2) Conversion bandwidth (3) Noise figure (4) Sensitivity to parameter variation Each of these quantities will depend on (1) the number and character of the unsuppressed signal sidebands, (2) the terminal admittances Y at each unsuppressed signal m n frequency, and (3) degrading influences, such as incidental loss in the mixing element. The basic techniques for making these circuit evaluations will be reviewed in Chapter II. 1. 2 Review of the Literature One of the first practical applications of reactive mixers was by E. F. W. Alexanderson (Ref. 2), who, in 1916, used a saturable inductor to modulate a high-power alternator for a carrier telephone system. In this way, Alexanderson demonstrated how a nonlinear reactance can implement the control of high-level carrier power by low-level signal power. In turn, this suggested to Alexanderson, and to others, that immediate detection of the carrier could recover an amplified version of the applied signal. However, due to the rapid advance of other means of amplification, principally vacuum tubes, the latter step appears to have been postponed for many years. Both the Alexanderson modulator and the simple circuit in Fig. 1. 2 illustrate how the energy of an oscillating system (i. e., a signal) can be increased by supplying energy at another frequency (i. e., the carrier), which can be alternately viewed as frequency mixing. Hence the term reactive mixing could have been appropriately applied long before it was

9 actually introduced. In fact, the general topic of reactive mixers was obscure in technical literature for the next thirty years. Then, mainly by accident, certain diodes that were developed for mixers in radar receivers were found to yield gain. Being foreign to resistive mixers, this phenomenon was attributed to a voltage-dependent capacitance in the barrier junction. Shortly afterward, H. C. Torry (Ref. 3) developed a basic description of nonlinear capacitance converters, and showed theoretically how these effects were possible. The next five or ten years saw parallel developments in theory (Refs. 4, 5, and 6) and in experimental verifications (Ref. 7), although certain experimental deficiencies were yet to be overcome. Theory was predicting low-noise conversion with gain, but the measured noise figures were not particularly impressive. Then, in 1954, the work of Uhler (Ref. 8) and Bakanowski (Ref. 9) at Bell Telephone Laboratories led to the development of low-loss silicon diodes now known as varactors or vericaps. The quality of these components prompted more extensive theoretical studies (Refs. 10, 11, 12, 13, and 14) which by 1958 led to the realization of low-noise microwave mixers with gain (Ref. 15). Only one mixing mode was seriously investigated at this time. It was a case of single-sideband mixing that manifested negative resistance, and it relied only on conversion to the lower sideband, but this technique made possible such improvement in microwave amplification that all efforts suddenly concentrated on this configuration. In time, however, some academic interest was shown in the less spectacular upper-sideband converter (Ref. 16). The first specific suggestions regarding multiple-sideband reactive mixing appeared in qualitative papers by Hogan, Jepsen, and Vartanian (Ref. 17) and by Hsiung (Ref. 18). However, the first detailed discussion was by the present writer while speaking at the 1959 PGMTT National Symposium. This material was later published (Refs. 19 and 20). The author's first efforts in multiple-sideband reactive mixing tacitly assumed that each sideband was independently loaded. Later, prompted by the interesting experimental work of Mr. H. Brett of the U. S. Army Signal Corps (Ref. 21), the author treated doublesideband mixing under the constraint of dependent sideband loading (Ref. 22). The result has been a uniting of the modern theory of multiple-sideband reactive mixers with earlier work, such as the Alexanderson modulator. The constraint of dependent sideband loading is dis

10 cussed in Chapter IX, and it appears at the present time to be the most promising practical application of this study. 1. 3 Topics of Investigation As previously stated, the goal of this analysis is to extend our understanding of the circuit properties of reactive mixers through a study of the double-sideband configuration. In pursuing this goal, progress has been made in the following areas. 1. A detailed analysis has been made of two symmetrical double-sideband reactive mixers. One employs a single pumping frequency co and involves the complete set wr, p r cop, o, and c. The other employs double pumping, at wp and o2p, and so involves the set or, cop, cpo o p 2p' r' p 2p' Wi and c. u 2. The cases in (1) have been treated under each of the following circumstances. a. When the sidebands are widely spaced in frequency so they can be easily separated and individually processed. b. When the sidebands are closely spaced in frequency so that individual processing is impractical. 3. The useful engineering properties of each system in (1) and (2) have been evaluated and compared with the better known properties of single-sideband reactive mixers. This comparison has demonstrated that double-sideband reactive mixers offer advantages with respect to gain, bandwidth, and noise figure. 4. Experimental observations have been made in support of the conclusions in (3). In particular, an ultra-wideband, lownoise video amplifier has been developed from the analysis in category (2b) above.

11 5. A fundamental relationship has been obtained between symmetrical double-sideband mixers and a class of unsymmetrical double-sideband mixers. Since this relationship enables the properties of the latter to be derived from the former, it adds to the generality of the results cited above. 6. A new representation has been discovered for single-sideband, reactive mixers which displays their properties through easily interpreted equivalent circuits. This representation can also be applied to double-sideband configurations, so it should aid even the mature investigator to more fully understand the surprising properties of double-sideband reactive mixers. 7. A new method for predicting the stability of a linear system with periodically-varying components has been derived. This method requires a knowledge of sinusoidal steady-state response, but it does not entail a solution of the system differential equations. Therefore, this method yields a determination of stability that is less tedious than the normal method of calculating characteristic roots. In addition, however, it provides a new approach for approximating the characteristic roots of a periodically varying system. The results cited above will be described in detail in subsequent chapters. Chapter III discusses the general principles of reactive mixers and outlines methods for calculating their basic circuit properties. Chapters IV and V treat the two fundamental singlesideband cases, whose properties are well-known, but whose study serves as a useful basis for later comparisons. Chapter VI, which treats stability, stands somewhat alone in this discussion, but the results there support the main contributions of the remaining chapters. Chapters VII, VIII, and IX discuss the single-pump, double-sideband cases, and Chapter X discusses the double-pump case. The final chapter reviews the experimental program that has accompanied the preparation of this dissertation, and also summarizes the conclusions of this study.

CHAPTER II MATHEMATICAL FORMULATION The basic circuit model in Fig. 1. 3 depicts a nonlinear system of arbitrary order, and hence its complete analysis is impossible with present mathematical methods. While encouraging progress continues to be made in obtaining such a general analysis (see Ref. 23, for example), the treatments of reactive mixers that have borne the greatest engineering fruits to date have been of the Fourier type. Hence they have dealt exclusively with steady-state behavior. But a shortcoming arises when steady-state methods are applied to the circuits represented by Fig. 1. 3, because the latter are potentially unstable. Nevertheless, it will be shown (in Chapter VI) that steady-state analysis alone can be used to accurately predict regions of instability in many practical cases. Hence, these steady-state methods are capable of determining their own region of validity. 2. 1 Selection of Variables When v(t) and i(t) have steady-state waveforms, they can be expressed by 1 j(mx+ny) v(t) - E E V ej(mx+ ny) (2. la) 2 mn m=-oc n=-oc 1i 0C C j(mx + ny) i(t): - E E I n (2. lb) 2 mn m=-ooc n=-oc where x = t y = t p r V = V* I I* (2. c) mn -m, -n mn -m, -n The assumption will be made that the mixing element in Fig. 1. 3 is capacitative, and that it is characterized by a charge-voltage curve of the form 13

14 q = q(v) (2.2) Of course, to be consistent with the assumption that the mixing element is lossless, q(v) must be single valued (e. g., free of hysteresis), but in any case the charge waveform will have the following form, 00 0C j(mx + ny) q(t) = Qmn (mxny) (2.3) m=-oc n=-oc and then I = j(mwo + n ) Q+ (2. 4) mn p r mn (24) In principle, I can also be related to the set of V by using (2. 2) to eliminate Qm' mn n mn However, this step can rarely be accomplished exactly. An alternate technique is to approximate (2. 2) by the first few terms of a Taylor series, which is particularly useful when formulating a small signal analysis. In fact, this approach will be the basis of the analysis to follow, but first an important general consequence of (2. 4) will be derived. 2. 2 General Energy Relations The average power entering the mixing element in Fig. 1. 3 at frequency wm mn is expressed by1 Wmn () Re(V I (2. 5a) mn 4 mn mn where by (2. Ic) W = W (2. 5b) mn -m, -n But the lossless character of this element dictates that cc cc z Z W = 0 (2.6) mn m=-oc n=-oo Read Re as "the real part of..."

15 The previous sum can be written equivalently c C W \ mn Z E (mi + no ) = 0 P r mc + nCo m=-oc n=- P r which, with the aid of (2. 5b), can be separated into the following two terms c o nW c oc nW mn mn 2; i — mnn 1 + 2~ E Z - = 0 (2.7) P I n m=m) + noo r moo - noo m=O n=-c p r n=O m=- p r Now, by substituting (2. 4b) into (2. 5a), it can also be noted that W mn _ (1) Re(-j V Q (2. 8) mow + no mn mn p r It was first observed by J. M. Manley and H. E. Rowe (Ref. 24) that (2. 8) is independent of wp and co if Qn and V are associated with a pure (but nonlinear) capacip r mn mn tor. That is, if co and wr are changed at will the tuning of Y(w) can always be altered so that (2. 8) remains constant. Consequently, each term in (2. 7) must vanish independently, which yields oc cc mW E - mn 0 (2.9a) Imuo + nco m=0 n=-o p r oc cc nW _ mn = (2.9b) mo + no n=O m=-oc p r These equations are referred to as the Manley-Rowe general energy relations because they describe the equilibrium energy flow in an arbitrary reactive mixer. Although Manley and Rowe originally derived (2. 9a) and (2. 9b) by direct calculation (Ref. 25), their alternate proof which was quoted above shows the physical significance of these relations more clearly. It will be demonstrated that the generality of the Manley-Rowe relations occurs at the expense of information content, and for this reason a more detailed analysis of Fig. 1. 3 remains to be made.

16 2. 3 Formulation for First-Order Mixing Only two facets of input information were used in the derivation of the general energy relations: (a) the mixing element is lossless (b) the mixing element is capacitative Because of the lack of detail in these assumptions, the Manley-Rowe relations provide only a limited description of reactive mixers. To make further progress, such as finding the actual voltage and current waveforms in Fig. 1. 3, would require a further specification of q(v). In practice, the q(v) characteristic can be obtained either empirically or by applying basic physical principles to a model of the particular reactive element. However, even with q(v) and v(t) specified, mathematical limitations often hinder ones attaining useful expressions for the Qmn and hence for i(t). For example, Sensiper and Weglein (Ref. 26) derived the pump coefficients Qo for the typical q(v) characteristic of a reverse-biased semiconductor diode, namely q(v) = Q(1 - v/A) (2.10) O < c< 1 They assumed a single pump generator for which v(t) = V + V cos ow t, where V is the dc p P bias, and then obtained the Fourier coefficients Qm in closed form. However, the resulting expressions involved gamma functions of Vp/V that were quite complex. A more general (but less exact) approach has been to represent q(v) by a Taylor series q(v) = Q + al((v - V) +a.. (2. 11) which has a twofold advantage. An inductive mixing element is also characterized by (2. 9), but (2. 8) would be replaced by its dual in this case.

17 a. The coefficients, aj, can be derived from any differentiable q(v) characteristic. b. The series can be truncated after a finite number of terms to permit a compromise between ease of analysis and accuracy. Although a disadvantage arises when (2. 11) converges slowly, the flexibility of the Taylor series representation warrants its use in determining the first-order mixing properties of Fig. 1. 3. To this end, (2. 11) will now be applied to a significant special case. As outlined in Section 1. 1, the situation that will receive primary attention in this analysis is the one where real power flow is limited to two pump frequencies, w and p',and three signal frequencies cor, co, and cu. The spectrum of these frequencies was depicted in Fig. 1. 4 and its occurrence can be insured if Y is either a short or an open m n circuit to all other frequencies (i. e., Yn = o or 0, respectively). Choosing the case where Y shorts all unwanted frequencies, the voltage waveform will be m n jv(t) - V Re F x ~ 2jx jy V j(x+y) j(x-y) () - V uRe V Ve + V + V (2. 2a) 1 (1jX x ( 62jxz -2jx =2 [(Vp E + VpE + ( i V2p ). * (2. 12b) The current waveform contains the same frequencies, and short circuited components in addition, so if the latter are collectively termed the short circuit current ic(t), the total current waveform can be written i~> ~~~LJY 2jx jy I (X+Y) j(x-y i(t) = ic (t) +Re [IpeI 2 + e E + ri( ). (2. 12c) The advantage of Fourier representation now appears because the unshorted components of i(t) and v(t) are related by the equations below, which have been obtained from (2. 4), (2. 11), and (2. 12).

18 I - j p a V + 2 (V2 V* + V+ V ] (2.13a) p I 2 V 2p p + (2. 3) FI 2j ~~~ a 2 1 2 p = j a V + Vu V) (2. 13b) 2I 2w 1 2p 2 2 p 2 2 I j a V +2 (V V + Vp V ] (2.13c) a Il = [alV 2 (V V V2+ V V (2. 13d) 1 j= a~ "V + (V Vu + V V * (2. 13e) k k 1 f 2 p r 2p The special case described above is representative of first-order interaction in all reactive mixers. For example, the following observations can be made on the basis of the set (2. 13), and can be applied to reactive mixers in general. (1) If all voltages except one, say Vmn, were short circuited, then a single sinusoidal voltage will appear across the mixing element and (2. 13) will reduce to I = j 1aV Thus, an isolated mn mn 1 mn signal will experience an effective capacity C a dq(v) (2. 14) v = V (2) All first-order interaction (or mixing) is assoicated with the coefficient a2. (3) Given any set of allowed voltage components, the expression for I can be written by inspection, bem n cause I depends only upon frequency pairs that add m n algebraically to mwo + no. For example, the contribution to I due to first-order mixing will be mn given by I ej(mx + ny) mn ji(m + nw) a, V j(m'x + n'Y) V ij[(m-m')x + (n-n)y] (2 15) r m-m', n-n'' l=-~ JC

19 where V V*. Typically only a few terms -m,-n mn in (2. 15) will be nonvanishing. 2. 4 Methods of Solving the Set (2. 13) The source waveform in Fig. 1. 3 is expressed by the Fourier series OC OC jmXy II(t) = Re 1I j(mx+ny) (2. 16a) m n m=-c n=-oc so 11(t) can be related to i(t), component by component, through the Thevenin-Norton theorem In = Imn + Y mnVmn (2. 16b) mn mn mn mn which was previously illustrated in Fig. 1. 5. However, the mixer current I contains a m n component due to the average capacity C, which can be conveniently lumped with Y to m n yield an effective terminal admittance Y. In fact, this procedure can be generalized by m n introducing the notation Y = Y + Y' (2. 17a) mn mn mn where Y denotes all useful external admittance at w while Y' denotes all parasitic mn mn mn or generally unwanted admittance. Up to this point, only one contribution to Y' has been m n noted, and this is the static susceptance of the mixer element, so using the modified frequency notation in Fig. 1. 4, (2. 17a) becomes in this case Y = Y + jwr C r r Jr Y = Y +jWc C. (2.17b) The influence of the mixing element upon the components of source current may now be formulated by substituting (2. 16) and (2. 17) into (2. 13), which yields j o at 1 =Y V w2 (V V* + V V* + Vp V) (2. 18a) p p P 2 2p p ur Hr

20 II Y V ~ jw2 (1V2 +V V ) (2.18b) 2p 2p2p + 2 2 p ) ( ) jcora2 1 = Y Vr + (V* V + V V) (2. 18c) r r r 2 p u p JCua2 jw a II = Y ( V + V V ) (2. 18) u u u 2 p r 2p ) Ja2: VY +- 2 (V V* + V2pV*). (2. 18e) Because the Imn are known, the most general application of the set (2. 18) [and hence of the set (2. 13)] will be to require a solution for the V. However, the inversion of (2. 18) is complicated by its nonlinear nature. There are several alternate representations of Fig. 1. 3 that help to organize the mathematical steps involved in the inversion of (2. 18). They also facilitate the physical interpretation of reactive mixer properties. If the nonlinear element is replaced by a timevarying element with a waveform that follows only a portion of the applied signals, the remaining signals can be related entirely by linear mixing action. To demonstrate, let a nonlinear capacitor have the charge-voltage characteristic in (2. 11), and let its voltage waveform be separated into two components, namely v(t) - V = v'(t) + v"(t) (2. 19) If it is assumed that the instantaneous capacity follows only the portion v'(t), then c'(t) = a1 + a2 v'(t) (2. 20) and if the remaining voltage v"(t) is now applied across this time-varying capacitance, the resulting current will be d i(t) = - [c'(t) v"(t)] (2. 21) The relevance of this approach to the inversion or solution of the set (2. 18) will now be demonstrated in two separate cases.

21 2. 4. 1 Mixing by Time-Varying Capacitance at wp and w2p. To formulate the first case, the voltage, waveform in (2. 2) will be partitioned in the following manner. 3 v'(t) = Re[V jx + Vp2jx] (2. 22a) p p 2p v"(t) = Re[Vjy + V j( ) + V j(x-y) (2. 22b) p r u Now, to follow the procedure outline in the previous section, v' (t) will be assumed to control the nonlinear capacitance. The resulting capacitance waveform will then be c'(t) = C + 2 Re[ype jx + 2j (2.23) where, by (2. 20), C = a1 (2.24a) yp = a2Vp/2 (2.24b) p 2 p'2p = a2V2p/2 (2. 24c) Next, the voltage v'(t) in (2. 22b) will be applied across this capacitance, so the resulting current waveform will be i"(t) = i" (t) + Re[I Ejy + I ej(x+y) + Iej(X-Y) (2.25) SCp When (2. 21) is applied, the voltage coefficients in (2. 25) are found to be related to the 4 current coefficients in (2. 22) by the following linear matrix equation. The subscript (p) is introduced here to distinguish this voltage partitioning from a similar one in the next section. Equation 2. 26 is linear unless wr = wp/2. r p/2

22 jw C jw y jwi y V r v r r p r p r uI = Ijuyp jwuC jwyuYp v u up u u'2p u IQ -jwc i p -jw 2p - jw C V (2. 26) It can now be observed that (2. 26) is equivalent to (2. 13c) through (2. 13e), if the identification in (2. 24) is made. By employing the ideal filters introduced in Section 1. 1, the set (2. 26) can be associated with the linear equivalent circuit in Fig. 2. 1. Incidentally, this figure also illustrates how the average capacity of a mixing element can be lumped with each terminal admittance, as expressed by the modified terminal admittances relation in (2. 17) Y = Y + jcw C mn mn mn When the sources in Fig. 2. 1 are included with (2. 26), the equivalent of (2. 18c) through (2. 18e) results, namely: Il Y jW c*'j y V r r r r p r = jw y Y j wy V u up u u 2p u pI-j -jw Y V* (2.27) Therefore, one method of solution for (2. 18) is to invert (2. 27) and to substitute the results into (2. 18a) and (2. 18b). In this way, the relations between V and lIp, and between V2p and I[2p are established, so yp and y2p in (2. 24) can be eliminated. This establishes that the analysis in Fig. 1. 3, subject to the frequency constraint in Fig. 1. 4, can be accomplished directly from an analysis of the linear model in Fig. 2. 1. It should be noted that Fig. 2. 1 has been the basis of previous reactive mixer analyses, but that this model is normally arrived at by direct small signal argument. In

23 11 ct FILTER u r | FILTER FILTER Yr Yu t, Y2p Fig. 2. 1. An equivalent circuit for equations (2. 18c) —(2. 18e). Each filter is an open circuit near the labeled frequency and a short circuit elsewhere. such an argument, all signal voltages V (n / 0) are assumed small in comparison with the m n pump voltages VO; and then, simply by virtue of size, it is assumed that the reactive element follows the pump waveform entirely, and that small signals will experience a linear reactance element varying harmonically at the pumping frequencies. These small-signal assertions are shown to be valid (for first-order interactions) by observing the relationship between (2. 27) and the last three equations in (2, 18). Thus, small signal arguments view all interactions as pump acting on signal. In fact, (2. 18a) and (2. 18b), which describe the first-order influence of the signal frequencies on the pump, were omitted in deriving (2. 27). In the next section, an alternate model for small signal analysis will be formulated that emphasizes the converse case. In fact, the effect of signal on pump will be shown to be equally as basic as that described by (2. 27). 2. 4. 2 Mixing by Time-Varying Capacitance at Wr and p. In the previous section a symmetrical, double-sideband reactive mixer was represented in terms of timevarying capacitance at co and w2p. This model provided a linear representation for Fig. p2p

24 1. 3. In addition, it was demonstrated that the signal frequencies voltages, Vr, V, and V are related entirely by the mixing action of the linear capacitance waveform in (2. 23). A second model will now be derived that describes the mixing in (2. 27) through a different set of voltage components. It is equivalent to the first model in the sense that it leads to an equivalent mathematical solution of (2. 13) by a similar method. However, this alternate representation has distinguishing characteristics that offer new physical interpretation of the same mixing process. To formulate this case, the voltage waveform in (2. 2) will be reapportioned as follows: v'(t) = Re [vriY + V 2j (2. 28a) v"(t) = Re [v i V e + V e (2. 28b) Again, following the method in Section 2. 3, the nonlinear capacitor is assumed to follow v (t). The resulting capacitance waveform is r c (t) =C + 2 Re y ey + e 2jx (2. 29) r r p where C = a (2. 30a) r = a2V /2 (2.30b) v!p = a2 V2/2 (2. 30c) If the remaining voltage v"(t) is applied across c' (t), the resulting current will be r r i"'(t) = i (t) + Re E + + + I e (2. 31) r sc p( u E30) rL r = a Applying (2. 20) again, the unshorted current and voltage components are found to be related as follows:

25 I- j C jw Yr ~ O O jw 2 V* I wYc jw C jo + 0 jwc 0 V Ip = i pr + p p2p p I 0 jw uYr jC C jW 0 0 V* (2. 32a) u ur u u 2p u This rather unusual matrix equation has the form [I] = [YA] [V] + [YB] [V*] (2. 32b) which will be termed semi-linear, because it fails to pass one of the two basic tests for linearity: superposition and scaling. The relation above is linear with respect to superposition, but nonlinear with respect to scaling, which can be demonstrated if [V] is multiplied by a nonreal, diagonal matrix [P]. The result is not equivalent to multiplying [I] by [P], since [P] [I] = [YA [P] [V] + [YB] [P] [V*] [YA] [P] [V] + [YB] [P*] [V*] Of course it is only the second term (2. 32) that violates this test for linearity, so when y2p is zero (2. 32) will be linear. Later it will be demonstrated that semi-linearity arises in this case because the frequency of y2p is precisely twice that of V. In (2. 32), the surviving linear quality of superposition simplifies the direct solution of this equation. The equivalent circuit in Fig. 2. 2 can be formed from (2. 32), if the semi-linear term is lumped into controlled sources at each terminal and if yr is taken to be real. 5 When the external sources in Fig. 2. 2 are included with (2. 32), the latter equation becomes Choosing the phase of one component, say yr, is not an added restriction because it is equivalent to choosing the time origin.

26 A C }FILTER Kw 21 -- t jWQcVQ I__ Iu nP (t) P I IVp FILTER FILTER u'^ " (t^], -T i i I c 3 1 j P (2 pVp ir ju Y2p V Fig. 2. 2. An equivalent circuit for equations (2. 18a), (2. 18c), and (2. 18d). The ideal filters have the same significance as in Fig. 2. 1. I ] j&Yr 0 V 0 0 jWp'W V* I 1 YI i'kr ~ v ~ ~ 2p v jw Y jWf'Y jw 0 0 Vr V p pYr wp Spr p I | py2p V p| I 0u j uY Y V 0 0 V* (2. 33a) u u'Yr u u u which has the form [] = ([YA] + [Ymn]) [V] + [Yb] [V*] (2. 33b) When the identification in (2. 30) is made, it will be recognized that (2. 33) is identical to (2. 18a), (2. 18d), and (2. 18e). Thus, the set (2. 18) can be solved by substituting the solution of (2. 33) into the remaining two equations (2. 18b) and (2. 18c). The most significant feature of this second formulation is that the linear timevarying element follows wform the a that is normally of smallest amplitude, but this model leads to the same solution as that in Section 2. 4. Thus, the intuitive notion of basing an analysis on the apparently dominant influence of large pump waveforms is valid, but not

27 fundamental. The significance of this fact is best emphasized by suppressing the pump harmonic V2p, which causes y2p to vanish. In this case, (2. 18b) can be dropped and the production of sideband frequencies by mixing becomes limited to the terms a2VpVr/2 or a V V*, 2, which appear in (2. 18d) and (2. 18e) respectively. Because of the symmetry of these terms, it is equally valid to view mixing as a capacitance a V p2 interacting with a 2 p voltage Vr (or V*); or as a capacitance a2V (or a2V*) interacting with the voltage V. So r r r p r although small signal considerations suggest mixing by pump acting on signal, the reverse mixing effect of signal affecting pump has identical characteristics. Also, either of these viewpoints is sufficient to solve the set in (2. 18). In later chapters, these formulations will be used to provide separate explanations of certain basic reactive mixer properties.

CHAPTER III INTRODUCTION TO SINGLE-SIDEBAND REACTIVE MIXERS Reactive mixing generally involves a multitude of sidebands, but in the past single-sideband cases have received the greatest attention. The term single-sideband will be applied when the signal set is restricted to two frequencies coupled by first-order interactions. It is soon apparent that an infinite number of such single-sideband pairs exist. However, they are all typified by two representative cases which involve r as one frequency, and either w = wc - Or or o =ow + o, for the other. p r' u p r To prove the latter statement let the general signal frequency pair mn and c) n' be coupled by a pump frequency oW,,0. When this coupling occurs by first-order interaction, the frequency indices must be related on the basis of (2. 15), which yields m" = m - m' n' = n (3. 1) The single-sideband cases involving r, Wp, and cp + cr have the following values: m' = 0, m" = m = 1, and n' = n = + 1; but all other cases can now be reduced to these by defining new pump and signal reference frequencies as follows: m"O = 10 = p co n = 01 + r (3. 2) Then, by (3. 1) mn 10-o 01o ) + cp = 10 1 p29 29

30 which shows that mixing within the pairs (or' co) and (or, wu) is representative of all single-sideband mixing. For most of this chapter, and for the next two chapters, Y(co) will be chosen to short circuit all frequencies except or, p, and either o, or coa. The basic mixing properties of these sets are well known, but they will be reviewed in preparation for later chapters. Also in the course of this review, several novel approaches to reactive mixer analysis will be introduced, as well as some previously unpublished aspects of single-sideband conversion. 3. 1 Application of the General Energy Relations For the restricted set of frequencies just specified, the general energy relations in (2. 9) reduce to W W W p + - + = O (3.2a) r u U - p u f + - 0 (3.2b) co co co k r u 2 These equations will be applied to the two typical single-sideband configurations, which will be distinguished as Case 3. 1. 1, when W = 0, and Case 3. 1. 2 when Wu = 0. In applying the general energy relations to these cases, it should be noted that Wn has been defined as the average power entering the reactive mixing element at frequency c mn. Therefore, when signal power is converted from one frequency to another, the associated conversion gain will be the negative ratio of the corresponding W mn 3. 1. 1 W, = 0. In this case, (3. 2b) predicts the following power gain for conversion form c) to co. r u -W co u U (3. 3) W co r r It is certainly noteworthy that this gain is greater than one. As a partial explanation of this result, the quotient of the two relations in (3. 2) yields W ac - P (3.4) W co r r

31 which shows, in this special case, that the pump and signal sources contribute power to the mixing process in proportion to their frequency. Therefore, if co > COr the pump will conp tribute more power to the convertedoutput at ow than the signal. On the other hand, if the signal is applied at wo and down-converted (demodulated) to c r, the corresponding gain will be the reciprocal of (3. 3), which is less than unity. Nevertheless, signal power that appears to be lost during down-conversion is not dissipated by the mixing element, which is lossless. Instead, it is reflected at the pump frequency. This point will become clearer after considering the lower-sideband converter. 3. 1. 3 W = 0. Conversion from co to co has several intriguing aspects. u r First, because the frequencies cr and c both cover the range from zero to cp, they are indistinguishable in the following sense: when two frequencies sum to op it is immaterial which is termed cw and which co. In either case, the conversion gain from one to the other is numerically equal to the ratio of output to input frequency, just as in the previous case. But by contrast, the conversion gain for lower-sideband reactive mixing is shown in (3. 2b) to be the negative of the frequency ratio, namely w co _ = _ _ (3.5) W co r r This aspect of reactive mixers is unheard of in resistive mixers, and so it requires further interpretation. Note that W cannot be positive in the absence of a source at wo, and also that WV must be nonpositive by (3. 5). Therefore, an applied signal at or must experience power reflections exceeding 100 percent, which further implies that the lower-sideband converter is potentially unstable. In support of the latter statement, steady-state analysis (in Chapter V) will show how this form of single-sideband converter is characterized by input and output admittances with negative real parts. Further support can be found by applying (3. 2a) to the present case, which yields W oL W (3.6) P P Therefore W is positive and exceeds the converted output power at ca In accord with the statements made about (3. 5), pump power, in excess of that converted to Lo, must be con

32 verted to wr and become radiated as a reflected signal. Returning now to the case in 3. 1. 1 that involved an applied signal at w, the following variable changes show why signal power was absorbed by the pump: CL = CO) u p p = - ) = CI C- (3.7) r p k Hence, as far as cw is concerned, the pump w is a lower-sideband frequency. Large signals -- _ - u- - p at wU could conceivably cause the circuit to oscillate at wp u p 3. 2 An Alternate Viewpoint (Ref. 27) Let us recall that the general energy relations are based on two properties of lossless mixing elements: (a) that energy is conserved, and (b) that the average power flow at wm is proportional to this frequency. These relations were derived by circuit analysis. mn Speaking in more general terms, however, the general energy relations deal with sinusoidal electromagnetic waves at each of the w mn and wave motion is subject to a far greater variety of descriptions than is provided by circuit theory. For example, the quantum theory predicts that wave motion can be regarded as a flow of photons of energy ao. Therefore, mn conversion from Aw to the upper sideband cu can be regarded as signal photons mixing with pump photons to produce upper-sideband photons. This situation is depicted in Fig. 3. la. If there are N such events per second, the average output power will be W = N ow (3.8) u u but the average signal input power will be W Nwo (3.9) The conversion gain will now be u u u _ (3. 10) r r 1The symboll denotes h/2v, where h is the Planck constant.

33 tWr 2hwr cp (a) (b) ywu )l) Fig. 3. 1. Single sideband conversion on a quantum basis: (a) upper-sideband case; (b) lower sideband case. which is precisely the ratio predicted by the Manley-Rowe relations in (3. 3). It is instructive to also consider lower-sideband conversion on a quantum basis. As depicted in Fig. 3. ib, the mixing of one signal photon with one pump photon must produce one difference frequency photon and two reflected signal photons. Therefore, the net input signal power will be W = N w - 2Ns = - N o (3.11) r r r r The output power is -W' = N{ (3. 12) so the power gain will be R w W _ (3.13) Mr o r r which is in agreement with (3. 5). Evidently, the reactive mixer can serve as a model for certain quantum processes. Hence, Fig. 1. 3 could be regarded as an equivalent circuit for quantum devices that employ conversion between discrete energy levels. A good example of such a device would be the solid-state maser. It is suggested, therefore, that the basic conclusions of this analysis are pertinent to masers and other devices that are normally considered to be beyond the scope of circuit analysis.

34 3. 3 Circuit Effects Due to First-Order Interactions Several interesting properties of single-sideband reactive mixers have been described by the general energy relations. These properties have shown a sharp contrast between the two typical cases, where only the upper or the lower sideband is present. However, numerous questions remain to be answered, such as: "What must be done to inject signal power into a reactive mixer, in order to realize the potential gain?" and "When will reactive mixer conversion gain be stable?" etc. It will be shown that these further details depend upon the actual law of the nonlinear element. For this reason, the mathematical formulation in Section 2. 2 will be the medium employed for further analysis, so let us now consider some typical single-sideband circuits. The circuit in Fig. 3. 2a is a simple realization of the basic single-sideband mixer, and it has the general form of Fig. 1. 3. The nonlinear capacitor is assumed to be biased with average capacity C, according to (2. 14), while the linear portion is made up of three parallel tanks in series. These tanks are assumed to be tuned so that when a fixed capacitor of value C is placed between terminals a-b, it will resonate the resulting linear network at w, co and w; where W denotes either ow or o Normally, a different tank will dominate in each of these resonances, so the voltage v(t) will appear primarily across one tank at each resonance frequency, which explains why the applied sources in Fig. 3. 2a have been associated with individual tank circuits. This type of filtering is common in mixers, and motivates the use of idealized models like Fig. 1. 5. For example, an ideal form of this type is shown in Fig. 3. 2b. The latter will be equivalent to Fig. 3. 2a, in the bands of interest, if the network associated with each filter in Fig. 3. 2b is chosen to be the appropriate Norton equivalent of Fig. 3. 2a. For example, Y must be the Norton equivalent admittance at terminals a-b, exclusive of C, for frequencies near o r, etc. Once again, the advantage provided by ideal filters is to separate the circuit admittances influencing each frequency band so they can be independently varied. The one main approximation incurred by these ideal filters, is that circuit effects originating beyond the desired bands are ignored. Figure 3. 2b will now be analyzed to determine the first order interactions between r', and we or wo. Following the method of Section 2. 4, the pump and nonlinear capaci

35 (a) T~V' v(t)- t a b (b) II P Y- p x - or u FILTE *r o T r ]i j -x; [ 7 r r r FILTER FILTER X Cr x ]r Yx realization and (b) is an idealized form of (a) having the form of Fig. 1. 5. In (a), each frequency appears in varying degrees at all terminals. In (b), the ideal filters cause each frequency to appear at only one terminal. tor will be replaced by the time-varying capacitor at wap, which produces a second equivalent circuit, namely the linear model in Fig. 3. 3. This final linear model will be recognized as a special case of Fig. 2. 2. It is obtained when 2p is set equal to zero, and when one of the two 2p sidebands is shorted. When similarly modified, the matrix equation in (2. 27) describes Fig. 3. 3, and the two basic single-sideband cases can be treated separately by alternately setting V l and V to zero. model in Fig. 3. 3 can be summarized as follows. First, all extraneous mixing products have been suppressed by assuming ideal filters. Secondly, the mixing element has been linearized

36 | x: or u jr( {{r ^ LrV W W c Lx{'X x x x.r ___r__ CX YC LX Yr Yx Yr Yx Fig. 3. 3. A further reduction of Fig. 3. 2b into a linear, time-varying model. The ideal filters allow C to be shown at both terminals. by substituting a time-varying capacitor. To be valid, these steps require small applied signals, and that the individual resonances in Fig. 3. 2a have fairly high Q's. Further evidence on the accuracy of these assumptions is given in Chapter VI. 3. 4 Derivation of Circuit Quantities The relationships established in the preceding sections are a convenient basis for systematic calculation of the fundamental circuit quantities listed in Section 1. 1. Consider first the conversion gain between two members of the signal set, say o1 and w2. As far as these two frequencies are concerned, the entire mixer reduces to the two terminal network in Fig. 3. 4, which will be convenient for introducing those external or parasitic parameters with important influence on the circuit properties of reactive mixers. The noteworthy parts of Fig. 3. 4 are: 1. A source at co. with short circuit current IS and 1 S internal conductance gs, which is normally the only contributor to g1. 2. A load conductance gL at co2, which is normally the only contributor to g2. 3. Tuning susceptances bl and b2 at c1 and wo2. 4. Parasitic admittances Y' = gl + jb' and Y'2 g9 + jb2 which may arise from cables, filters, or from parasitic losses in the mixer itself.

37 SOURCE TUNING PARASITIC PARASITIC TUNING LOAD I~~s ~~~~S ~ lbV2 L i L MIXER in YI YI I YI Y2 Y, I Y2'1 I 2 W. I W2 (INPUT) (OUTPUT) Fig. 3. 4. A two terminal pair representation for conversion from c 1 to c02 showing certain critical external parameters. The practical evaluation of mixer properties is simplified by lumping all admittances into general terminal admittances as Y1 = Y1 + Y = + jbl Y2 = Y + =g2+jb2 (3. 14) Once done, a particular mixer can be evaluated first as a function of Y1 and Y2. Then the external parameters can be separated out, so their influence on a particular circuit property can be determined. This procedure will be outlined next. 3. 4. 1 Conversion Gain. The most convenient measure of conversion gain is transducer gain, which is defined as follows: average power delivered to YL (at 0o2) G __________(3. 15a) 12 =average power available from the source (at (1) In the notation in Fig. 3. 4, 4gL V 2 1G -= _YggL 21 (3. 15b) in where Ylin denotes the total input admittance experienced by the ideal source IxS at the a1. terminals. Both Yin and V2/V1 can be calculated directly from appropriate mixer relations. -'*in z1

38 Comparing reactive mixers on the basis of transducer gain, rather than on pure power gain, has merit because different reactive mixers operate at radically different impedance levels. Hence the full realization of a potential power gain may call for unrealistic source impedances, but the optimum gain for a given source impedance is the maximum transducer gain. The effect of parasitic loss Dn transducer gain can be easily introduced. Calling G12 the modified transducer gain obtained by considering Y1 and Y2 the source and load admittances, respectively, then G12 can be expressed in terms of G12, by the following rewording of (3. 15a): available source power from 1S and (gS + g1) power to g G S _ 1 x L 12 available source power power to (g + g) from IfS and gs power to (gL + g2) available source power from IS and (gS + gp) Therefore ( gq\ ( g L 2 12 g= gs (g gL + G12 (= G (3. 16b) The latter equation is useful because G12 is independent of how the terminal admittances are partitioned between desired and parasitic components. Hence, (3. 16b) is a fundamental form that tends to remain fixed during various optimization procedures. 3. 4. 2 Noise Figure. In reactive mixers, as with most systems, both internal and external noise sources must be considered. For the mixers treated here, the external noise will be assumed to be of thermal origin, being due to resistance at the mixer terminals.2 The internal sources of noise will tend to be more varied, but the predominant internal noise in most reactive mixers is the thermal noise associated with parasitic losses. Therefore, two basic assumptions will be made for this noise analysis: (1) that all noise is thermal, and (2) that all noise sources are uncorrelated. 2Other nonthermal noise sources can often be included by appropriately raising the thermal noise temperature T. This step is useful when conversion bandwidths are narrow.

39 The conversion noise figure of the network in Fig. 3. 4 can be expressed in terms of the thermal noise of a conductance at standard temperature T = 2900K. The available noise power from such a conductance is kT B12, where k is the Boltzman constant and B12 is the noise bandwidth in cps for conversion from w1 to w2. The latter is normally defined by the expression B 1 f G(o) dc (3. 17) 27G o where G(c) is the system gain and G its midband value. For a mixer, however, this definition must be modified because the range of each sideband is limited by the pump frequency. Therefore, if the conversion gain G12 is assumed to diminish rapidly outside its conversion pass band (of width 12 rad/sec), the following definition can be adopted: 1 F o1+n3 12 = _ G12(Wl) dc; n> 1 (3. 18) no1 12 A tacit assumption in (3. 18) is that i3 12 will be essentially independent of n, if n is somewhat larger than one. As a practical example, let G(c)/ G correspond to a simple resonant circuit with 3 db bandwidth 3. Then (3. 17) yields a noise bandwidth IrT//2 rad/sec or /3/4 cps. In the corresponding case for a mixer, the pass band of G12 () would resemble a tuned circuit with bandwidth 12' and (3. 18) will yield B12 very nearly equal to 3 12/4 cps. It will be convenient to assign the noise temperature T to the source conductance gS in Fig. 3. 4. Then the definition of noise figure can be stated as follows total noise power at o2 F (3. 19) 12 noise power at c2 due to gS alone Since the output noise power at o2 is composed of excess noise contributions as well as uncorrelated noise from gS alone, (3. 21) can also be written in the form

40 excess noise power at c2 F 1 + -(3.20a) 12 noise power at c2 due to gS alone The quantity F12-1 is often termed the excess noise figure. A second alternate measure of excess noise can be introduced by writing (3. 20a) as F2 = 1 + T2 (3.20b) 1~2 T 0 The quantity T12 is called the effective input noise temperature. If gS had the noise temperature T12 it alone would yield the same excess output noise as all the actual noise sources put together. One advantage of measuring excess noise in terms of noise temperature is the elimination of the arbitrary reference temperature T. As a simple illustration of (3. 20), excess noise due to a parasitic input conductance g' will be considered, so g' will be assigned the noise temperature T' as noted in Fig. 3. 5. The presence of g' reduces the reference noise power available to the converter to the value kToB 12gS kTB9 l2g (3.21) + g S g1 Similarly, the excess noise power available to the converter due to g' is kT B12g kT1B 12g1 (3.22) 9 + gI Sg1 So if these are the only noise sources available to the output load, the noise figure becomes g.T 1+~ - (3.23) gsT0 There are several other forms of excess noise that will arise in this analysis, and these are depicted in Fig. 3. 6. It is again assumed that conversion from col to w2 is desired, but now an auxiliary sideband is introduced at w0c. This might be done for example to enhance the desired conversion in some way. The auxiliary sideband is terminated by a conductance g3 with a noise temperature T3: and the midband transducer gain and noise band

41 s,T Vo g;,T' GENERAL MIXER 44kTo B g /4 k,'B ____ Fig. 3. 5. The input noise sources associated with gS and g'. width for conversion between w3 and c2 are denoted by G32 and B32, respectively. The contribution by g3 to the excess noise figure F 12- is then G32B32T3 (3.24) 12 12 o A third source of excess noise illustrated by Fig. 3. 6 is load noise. Normally load noise is not counted as excess noise, but in systems with negative output conductance it must be included. Let the load conductance gL and the parasitic conductance g2 in Fig. 3. 6 have noise temperatures TL and T', respectively, and then their available noise powers will then be kTLB22 and kT2 B22 respectively, where B22 is the noise bandwidth of the transducer gain by reflection at co2. The latter quantity is defined as follows: power delivered to gL at w2 G (3. 95) G22 -power available from a source at wo with internal conductance gL Therefore the contribution to excess noise figure by the external noise sources at w2 is B22 G221 TL +22 (3.26) 12 12 o G12TfogL -1 The factor G22-1 is incorporated in the first term in (3. 26), instead of G22, so TL will not be held against a converter with a matched output. If G12 is large, (3. 26) will often be negligible in comparison with other noise figure contributions, unless G22, TL, or T' are also particularly large. The latter circum

42 24 kT3 83293 93,T3 1 3 T) 1 GLT L B2T?g T1 V4kTo B,1 2S 4kT, I2 G2 94kT; B2 Fig. 3. 6. A general set of noise sources that may contribute to the noise figure for conversion from o1 to c2. stance occurs in certain reactive mixers, however, as later chapters will indicate. A final expression for the noise figure of Fig. 3. 6 can now be formed by combining (3. 23), (3. 24), and (3. 26) which yields g1 T G32B32T3 B22 G22 TL G Tg F = 1 + 32 3 22 22 - 22 2 (3. 27) 12 gsT0o G12B12To B2 12 (3 T27) 3. 4. 3 The Varactor as a Mixer Element. An often used reactive mixer component is the reverse biased semiconductor diode, which is customarily termed a varactor. Varactors have had their greatest success at microwave frequencies, where the only serious degrading influence is their series resistance. For this reason, Fig. 3. 7(a) is a useful varactor equivalent circuit. Only the varactor capacity C varies with bias. The alternate representation in Fig. 3. 7b is more convenient for the present analysis, even though it varies with frequency. The parasitic loss gd(w) can be included directly with the parasitic admittance Y'. It is convenient to express gd(c) in terms of the Q of the varactor. In Fig. 3. 7a Qd = 1/oCrd (3.28) and if Qd2 1, the effective shunt capacity in Fig. 3. 7b is constant and equal to C, the series capacity, while the effective shunt conductance becomes

43 rd dC IC (a) (b) Fig. 3. 7. The equivalent circuit of a varactor in (a) series form, and (b) in shunt form. gd c 2C2rd - rd (3.29) d rd cod where cd is the frequency at which Qd = 1. When varactors are used, gd and C are the most fundamental contributors to the parasitic admittances Y' and Y'. In this analysis, varactors will be used to model the relations derived, and no other forms of parasitic loss will be considered. However, it is important to note that gd varies with frequency, so a different parasitic loss will appear at each mixer terminal. o 2 C wiC Atc,: w - - - = (3.30a) At c;: gd = o = gdl (3. 30a) d 2C At 2: d d gd2(3.30b) Therefore, when varactor loss is the only parasitic loss considered g = gdl g gd2 (3.31) 3. 4. 4 Comments on Notation. The general properties of Fig. 3. 4 will be applied to a number of reactive mixer configurations by identifying the frequencies ol and c2 with various signal frequencies such as c0r w, wc, etc. The total admittance at these

44 frequencies has been expressed by Y = Y + Y! (3.32a) r r r Y = Y + Y? (3. 32b) — etc — where Y' is the parasitic admittance at wc and Y is the chosen terminal admittance. r r r Considering all parasitic loss to be due to a varactor, then co 2C Y = g' + jb' = + j C (3.33a) r dr cdr jo rC d wo2C Y' = g + jb - C (3. 33b) The basic contributions to useful terminal admittance will be source, load, and tuning admittances. When a source of internal conductance gs and a tuning inductance L = I/ r2C are applied at w, then j- 2C r Y g - (3. 34a) r S Wd and Y;I = gS + gdr f jW~~i w s w+ Y =r g + g JrC ) (3. 34b) r g dr r which is the admittance function of a simple resonant circuit. It is typical of the admittance functions that will be employed throughout this analysis. The preferable term for Yr is chosen terminal admittance rather than the desired terminal admittance, because Y will often contain components that compensate for Y'. In the absence of Y', these added components would also be undesirable.

CHAPTER IV THE UPPER-SIDEBAND CONVERTER Conversion between wr and w, for the case where other signal frequencies are not present, will now be treated. An appropriate linear model for this problem is Fig. 3. 3, with a =. In turn it is described by (2. 27), with Ve =0, which leads to the following basic relation. If Y jci)*jw V r Yr r'p r Iu jWu y Y V (4. 1) u up u u While a resemblance will be noted between (4. 1) and the admittance characterization of a passive, two-node, linear, bilateral circuit, there are two important differences that should be observed: (1) the node voltages in (4. 1) are at different frequencies, and (2) its matrix is nonreciprocal. It can now be anticipated that the upper sideband converter will have some of the same properties as a passive, two-node, linear, bilateral network; but contrasting properties as well. Fortunately, these two property categories can be isolated in a manner having definite pedagogical value by converting (4. 1) into the so-called ABCD representation given below: V A B V r u Ir C D I (4.2) 45

46 Before converting (4. 1) to (4. 2), it is convenient to separate the internal and external current components as follows [ Y 0 V I I jK C jw y V r r r r r r r p r LuH Y PVI jwCj C V (4. 3) u u u u u up I u u One additional modification in (4. 3) is that p has been made real, which in no way restricts the validity of this relation, since the phase of one Fourier coefficient is always arbitrary. In fact, choosing yp real is equivalent to choosing a particular time origin. When the individual relations in (4. 3) are transformed into ABCD notation, they yield V 1 0 V lr r K] Y 1 K i I (4.4a) V - C/Yp -j/wuyp v Ir - jco (Y2 C 2 )/Y C p I (4. 4b) V 1 0 V u u I -Yu 1 l (4. 4c) u u u The (b) relation above contains all mixing effects. It can be further factored as follows. Vr -C/'p -j/or7p 1 0 V -jo(y 2 -_C2) Ir _- __P'-P C/y 0 cO /w I] (4.5) L Iy - r ruj Iij The significance of the first term in (4. 5) is easily recognized by comparing with the ABCD representation of the symmetrical transformer. For the transformer in

47 II 1I2 II- P I2 t t A d M _ I| C1~ r i p t2 (a) (b) Fig. 4. 1. A symmetrical transformer (a) and its dual (b) in 7r equivalent form. Fig. 4. la, one can write -L/M -j/wM 12 K -(M2 -L2) VJc(M - LI) L/M L V (4.6) which is the dual of the first factor in (4. 5). Therefore, the latter must represent two coupled capacitors with self-capacity C and mutual capacity yp. Admittedly, mutually coupled capacitors are seldom encountered in practice, but their properties are easily visualized from the well-known properties of their dual, the inductive transformer; or from their 7r equivalent, which is shown in Fig. 4. lb. The second factor in (4. 5) represents a broadband frequency converter operating between wr and w. It has unity voltage gain, nonreciprocal current gain, and nonreciprocal power gain. The latter two gains equal the output to input frequency ratio, so this converter has exactly the power gain predicted by the general energy relations of Manley and Rowe (see Eq. 3. 3). Therefore, it is appropriately termed an ideal, upper-sideband, Manley-Rowe converter. A further property of this ideal converter is admittance transformation, in which it obeys the relation Y( ) # r ( ) (4.7) V cL= Yu(Ou) u

48 Xp — Wrp -- - WU * C C MANLEY- ROWE;rt~ C T ~ CONVERTER L —-—.'.o DUAL OF FIG. 4.1 a Fig. 4. 2. A functional equivalent circuit for upper sideband conversion. where Y(w ) is the reflected admittance at or, and Y (w ) is the external admittance at the wc terminals. u From the interpretation of (4. 4b) just given, the set of relations in (4. 4) can be collectively represented by the circuit in Fig. 4. 2, which is equivalent to Fig. 3. 3. The advantage of Fig. 4. 2 is its clear comparison of a first-order mixer operating between a r and Lo, and a passive, two-node linear, bilateral network. Effects in the latter category will arise only from the external admittances and the mutually-coupled capacitor. The remaining effects, of mixing and nonreciprocal power gain, all take place in the Manley-Rowe converter. Therefore, the presence of the latter in Fig. 4. 2 can be attributed entirely to asymmetry in (4. 1). Since the maximum power gain of a transformer is unity, the maximum power gain in Fig. 4. 2 must be cu/W r, the gain of the Manley-Rowe converter. To actually achieve this gain, the capacitive transformer must match the load and source admittances. This matching process can be traced out by first noting that the load admittance at cu is transformed to ro by the converter, which reflects the load (4. 7) into the secondary of the capacitive transformer. In turn, the transformer reflects loads between primary and secondary according to the rule: 2 2 admittance reflected] total admittanc (4. 8) into primaryrtotal admittance into primaryin s Therefore, maximum power gain secondary Therefore, maximum power gain between uc and co occurs for

49 2 2 * - jwC = r p (4.9a) r' Y + jC C cw u r U 2 CL o y 2 r u p (4.9b) Y + j C The technique of analyzing the upper-sideband converter by dissection has successfully illustrated the origin of its basic mixing properties. However, this procedure could hardly be called an expedient analytical tool. A more direct analysis would treat the circuit properties of (4. 1) as it stands. If the real parts of Y and Y are regarded as the source and load conductance respectively, then the transducer gain defined in (3. 15) can be found directly from (4. 1). Setting Iu = 0, the voltage gain is V jwO) u u p (4. 10a) V Y r u while the input admittance seen by Id is r u Y = Y + r (4. 10b) r. r Y in u Therefore, by (3. 15b), 42 2 4 gr g u co y G | u u p (4. 11) ru IYrYu+ Wr 0u Yp2 2 which can be shown to be maximum under the condition (4. 9b). The maximum transducer gain then equals the maximum power gain, as it obviously should since (4. 9b) is the condition for admittance matching. Without the representation in Fig. 4. 2 to serve as a guide, the significance of (4. 11) would be more obscure. Therefore, Fig. 4. 2, and other dissected circuits that will be introduced in later chapters, will continue to be convenient for interpretation, but direct matrix analysis will be the primary method followed henceforth.

50 4. 1 Bandwidth of the Upper-Sideband Converter The bandwidth of the converter under discussion can be attribtued to the capacitive transformer in Fig. 4. 2. Being a transformer, its properties depend upon the "coefficient of coupling" p/C, which can be more appropriately termed the coefficient of pumping. Remember that y is proportional to pump voltage, according to (2. 23b). By transformer analogy, conversion pass bands that are either under, critical, or over-coupled can be anticipated for appropriate choices of pump level. Several other well known results from transformer theory will also be drawn upon (Ref. 28). A given transformer bandwidth will require the least coefficient of coupling if its primary and secondary are synchronously tuned. In applying this result to the present discussion, maximum conversion bandwidth can be anticipated, for a given y /C ratio, if Y and Yr resonate C at W and wr respectively. If Y = g + jb and Y = gr + jbr, resonance corresponds to b + c C = 0 r r b + C = 0 (4. 12) u u The condition for maximum power gain (4. 9b) then becomes r co yp r up (4. 13) r 9 gu It can also be written 2 r - P - k 2 (4. 14) -- -C C p co W r u r u where 3r and 3u are the individual bandwidths (in rad./sec) at or and wc (in the absence of pumping), and where C and C are the total shunt capacities affecting the or and wu resonances. Generally, C and C exceed C due to stray or filter capacity in Y and Y, and since bandwidth and conductance level are more easily measured that total shunt capacity, C and C will be defined by the relations Cr = gu/3, and Cu = gu/3, where g and g are

51 the individual conductance levels in the absence of pumping. It is now evident that -_ = kp (4.15) should be considered the pumping coefficient rather than yp/C. This substitution will be introduced henceforth by assuming C = C C r u Returning now to (4. 14), the latter can be seen to predict the smallest pumping coefficient for obtaining a given terminal bandwidth with maximum conversion gain. A second result from transformer theory can now be applied directly to the upper-sideband mixer. For a given coefficient of coupling, maximum transformer bandwidth results if the primary and secondary bandwidths are equal (Ref. 28). In the present analysis the bandwidth of the secondary depends upon the secondary admittance, which is given by +1(Y + j C) = -Y (4. 16) co U u co u u u Over reasonably small bandwidths, o /cu can be considered constant, and then the secondary bandwidth equals 0U, the bandwidth of Y. The condition for optimum conversion bandwidth is now [r = 3', but this condition can be satisfied with a variety of pass bands. As an example, Table 4. 1 gives specifications for both the equal ripple and the maximally flat cases, which can be obtained directly from the corresponding transformer relations. A conclusion to be noted in Table 4. 1 is that a given pumping coefficient will yield the greatest bandwidth in a maximally flat characteristic. For example, the ratio of bandwidth to pumping coefficient is 1/12 smaller for the equal ripple pass band, than for the maximally flat conversion pass band, with the same pumping coefficient. A comparison of the latter two cases is also shown in Fig. 4. 3.

52 Pass Band Bandwidth Pumping /G | 3 Coefficient k P Maximally Flat 2(B. W) (B. W) F- 2 k ou (critically coupled) Lrwu P Equal Ripple 2. 2[2(B. W)] 2. 4 (B. W) u] [2 k ] rwu [2 kp u] Table 4. 1. Realizable pass band characteristics for upper-sideband conversion. (B. W) denotes the bandwidth of the individual terminal admittance, which are assumed to be equal. Although these conclusions were obtained from simple transformer theory, by identifying yp/ JC C with the coefficient of coupling, a practical difference must be noted. p r u Pumping is a dynamic quantity, and any increase in the coefficient of pumping will generally call for more pump oscillator power. However, coupling is a static quantity, which can usually be incrased by a geometrical adjustment. Therefore, one should remember that conversion pass bands other than maximally flat demand premium pump power, and this must be paid for continually. An alternate way of summarizing the performance of mixers is through gainbandwidth product, which will be defined as (transducer gain)2x (bandwidth). This quantity appears in Table 4. 1 for the case of upper-sideband conversion. Sometimes gain-bandwidth evaluation is made on the basis of voltage gain, but since (voltage gain) > (transducer gain)2 the latter yields a more conservative estimate of gain-bandwidth product. In reactive mixers it is important to observe that gain-bandwidth product is proportional to pump frequency (at least for ideal cases), while in conventional mixers it is independent of frequency. 4. 2 Degrading Influences on Upper-Sideband Conversion The most fundamental limitation on reactive mixer performance is loss in the mixer element. It has been previously mentioned that the varactors are the most commonly used reactive mixer components, and their losses have been represented by series resistance in Fig. 3. 7a, or by equivalent shunt conductance in Fig. 3. 7b. The latter representation can be used with the circuit in Fig. 3. 4 to appropriately describe an upper-sideband converter with a nonideal varactor. In this case, cv1 and v2 will be identified with or and wu

53 -— 100~% UP-CONVERSION FREQUENCY RESPONSE: MAXIMALLY FLAT EQUAL RIPPLE — 70% I I FREQUENCY INDIVIDUAL TERMINAL CHARACTERISTICS. It-100% REQUIRED FOR: MAXIMALLY FLAT CURVE ABOVE- // \ EQUAL RIPPLE CURVE ABOVE - 70% J FREQUENCY Fig. 4. 3. A comparison of maximally flat and equal ripple Fig. 4. 3. A comparison of maximally flat and equal ripple pass bands for equal pumping coefficients. Yp,y w 9 br 9gdr C r c 9du bu 9L Yrin r Y YrY rfn Yr Yru Y Fig. 4. 4. A special case of Fig. 3. 4 appropriate for upper-sideband conversion.

54 respectively, which yields the circuit in Fig. 4. 4. Using the approximation in (3. 29), the parasitic conductances in Fig. 4. 4 have values co 2c gdr =ad (4. 17a) dr cod o 2C u ddu - (4. 17b) du = o 4. 2. 1 Effects of Varactor Loss in Gain. With the source at cw and the load at cu, as shown, the effect of parasitic loss on gain can be evaluated by (3. 16). In making this calculation, the condition that each terminal is independently tuned will be assumed, since it has been previously shown to yield optimum conversion bandwidths. The modified midband gain G in (3. 16) is equivalent to (4. 11), which with all susceptance tuned out can ru be written cou rgg cru n G r 4 u 2 2-^uIJyp_2 (4. 18a) ru wr L(grgU + cor up ) r u rup u = 4Pu u"- (4.18b) r (1 + Pu) 2 where o) o 2 r u (4. 18c) Now, using (4. 18b) in (3. 16), yields the true midband gain Now, using (4. 18b) in (3. 16), yields the true midband gain - / gSgL 4pU ru r r \Yu (1 + Pu) ( Of course, in the limit of zero varactor loss, (4. 19) reduces to (4. 18b) which is a maximum when pu = 1, corresponding to (4. 13). An interesting aspect of reactive mixer analysis now arises. When the varactor is ideal, conversion gain for a fixed wr increases indefinitely with cw, as shown by r p

55 1.0 / / / / / CONVERSION GAIN IDEAL VARACTOR — / LOSSY VARACTOR S / (SCALE ON LEFT) 0.1 ____/1 / R1. ~~R e1.0. / / | NOISE TEMPERATURE / / (SCALE ON RIGHT).01 0.1 1.0 10 -Rr W r Fig. 4. 5. Plots of conversion gain and noise temperature vs. frequency ratio showing the influence of varactor loss. the dotted curve in Fig. 4. 5. In the presence of degrading series resistance, the effective shunt loss at wu increases as the square of the pump frequency. Therefore, for a given r (signal frequency), Wd (varactor cut-off frequency), and pumping coefficient, ever diminishing returns will accompany increases in pump frequency. For a fixed r and w, it has been shown by Jones and Honda (Ref. 29) and also by Kurokawa and Uenohara (Ref. 30) that (4. 19) has a maximum value given by

56 (ru)m(ax ) (-+ r (4. 20) \ max r subject to the condition gs gL K-l - = K, or p (4.21) gd PPu u K+1 dr du where 2 K = 1 +r u (4. 22) gdrgdu The influence of frequencies variations can now be determined by substituting (4. 17) into (4. 21), which yields 2 2 K2 = 1 + p (4. 23a) WrUcC d)2 1+ (4.23b) wherel Co = cwdy /C. Therefore, parasitic varactor loss causes K and p to fall off monotonically with op (or with cw ), such that the maximum realizable gain of the up converter is p u l im d li G - (4.24) c0 - oc ru 2cor P The plot of optimum gain vs. pump frequency, shown in Fig. 4. 5, suggests that pump frequencies higher than (cow))2/ offer little added advantage. It is interesting to now interpret the conditions for maximum gain, as stated in (4. 22). The reflected input conductance at or is Pgr = P(g + gdr) K i gS g - gr (4. 25a) 1The quantity ypQ/wcc = yp wd/Coo is often called the dynamic figure of merit for a varactor, so ado is the frequency where this figure of merit becomes unity.

57 Similarly, the reflected input conductance at w is U Pugu L - gdu (4. 25b) Therefore, when the parasitic losses gdr and gdu are included, the total input and output conductance seen by the source and load are gS and gL respectively. Hence optimum gain again corresponds to a matched condition, but now the source and load are matched to the degraded mixer terminals. 4. 2. 2 Noise Figure. A second measure of the degradation induced by varactors loss is noise figure. Varactor loss makes a contribution to the up converter noise figure in both the r and the wU bands, but these contributions can be conveniently associated with the parasitic conductances gdr and gdu in Fig. 4. 4. Since these conductances are assumed to originate solely in varactor loss, as expressed by (4. 17), they can be assigned a common noise temperature Td. Therefore, applying the noise figure formula in (3. 27) (without the T3 term), yields ___ TL ~ uu du 1 + -- +(4.26) ru To gS G Gr u Td Gu gL where G is the transducer gain by reflection at c. By way of review, Gu is the midband power delivered to gL relative to the power available from a source at w o with internal conductance gL [see (3. 25)]. GU depends on the input admittance at w, and by a calculation similar to that yielding (3. 15b), the total input admittance at WU is o) co)' 2 Y Y +r up (4.27a) u. u Y in r At midband Y =g g(l+ ) (4.27b) u. u. u u in in and using the source Iu shown in Fig. 4. 4, the gain GU is easily shown to be 4gL2 2 gu u G u -4- ( —-p) (4.28) U U2 l+ P

58 Therefore uu = r gLgr 1 (4.29) Gru WUu gugs. and T d gdr rgr 1 ~F ~ ~ru 1 + [ - + (rr 1)T~j (4.30) ru T [gS \ugSPu ru d When (3. 27) is satisfied, so G is a maximum, (3. 32a) yields ru (ru)Max Gain 1 To K (4.31) 0 \Gru which agrees with the results previously obtained in Ref. 29 and Ref. 30. An interesting property of (4. 31) is its lack of dependence on T2 which arises because the load gL is matched. A generalization of this result is treated in Section 9. 1. Since K is a monotonically decreasing function of pump frequency, according to (4. 23b), F increases monotonically with pump frequency and approaches the value Td 1 + T- (4 r + 1) (4. 32a) 0 o where R = (r/d)2 (4. 32b) More convenient for plotting, however, is the equivalent noise temperature T = (F - 1)Td, which is shown in the lower portion of Fig. 4. 5. The curves of T show that it can be less ru than Td, if r < 0. 1 approximately. The minimum noise temperature occurs when wu /Ir =1, but this is an unlikely operating point because then G < 1. Therefore to obtain high gains ru with low noise figures, R must be much less than 1, and 0. 1 < Rr w/w < 1. r r u r The parameter R is a convenient basis for normalization because it only depends on input frequency, varactor quality, and the degree of pumping. Each of these quantities is usually specified in the early phases of a design problem. Also, the dynamic

59 2 figure of merit is equal to 1/R. r 4. 2. 3 Efficiency Factors. The degradation due to varactor loss can be alternately expressed in terms of input and output efficiency factors. These will be introduced at this time because they will be more convenient than the parameter K in later analysis. Let r gr x = -(4.33a) gr by the efficiency factor at r, and gu y: (4. 33b) OU by the similar quantity at U. Then, for the case of conversion from ow to wu u r u gs Lg - - = ~s y = gu (4.34) gr gu which yields the following restatements of the conditions for optimum G: ru x = y (4.35a) G -u P (4.35b) ru cor u Pu = 2x - 1 = 1 + 2 u R 1 + r (4.35c) )u (-x)a)r r(4. 35 au R(1 - x) ~r ~(2x - 1) Rr (4.35d) rp r (Fru)max gain T x (-) ( ) The minimum of the latter expression occurs for r 2x- 1 (4.36) 2See footnote, Page 56.

60 which calls for wu/r = 1, as previously discussed. u r 4. 3 Down-Conversion From w to cw u r The Manley-Rowe converter in Fig. 4. 2 is nearly unilateral, because its gain is less than unity for down-conversion from wu to w. While this case is of less practical interest than the up-converter, an analysis of the down-converter follows directly from the preceding analysis, simply by interchanging the subscripts r and u everywhere. This observation follows directly from the symmetry of (4. 1), when yp is real. In other words, the response to lI is the same mathematical form as that to I,r if r and u are interchanged. Therefore, by (4. 20) -F,,,,, K - 1 \ r cr (Gur)max K+1 I Pu 4u since K and p are symmetric in r and u. As expected, (Gur)max is less than unity. The curves in Fig. 4. 5 can also be applied to this case by interchanging r and u. This includes replacing R = (Wr/W )2 by r r d Ru =(u) = R (4.38) u w'?r r so the abcissa in Fig. 4. 5 becomes (c /wu) R. A numerical comparison of up- and downconversion is shown in Table 4. 2. R Abcissa in Gain Noise Fig. 4. 5 Temperature Up-Conversion R 0.01 R = 1 5. 5 0. 35T r r w d r Down-Conversion R =1 R 0.1 0.055 5.8 T u u w d u Table 4. 2. A comparison of up- and downconversion for U /( = 10. u r

61 In summary, the ratio of down-conversion gain to up-conversion gain is approximately (wcr/u)2, which rarely exceeds 0. 1. The corresponding ratio of noise temperatures is approximately w wor which generally exceeds three. Therefore, the down-conversion mode offers low gains and high noise figures, making it rather unattractive. 4. 4 Conclusion In describing the basic properties of upper sideband conversion, the purpose of the present chapter has been fourfold: (1) to illustrate the variety of effects that relatively simple relations, such as (4. 1), can yield; (2) to note how the basic circuit quantities outlined in Section 3. 4 can be derived in a particular case; (3) to show a physical significance for the major results derived; and (4) to prepare for comparison with other systems that will be analyzed later.

CHAPTER V THE LOWER-SIDEBAND CONVERTER Perhaps the most fascinating aspect of reactive mixing is the radical difference that occurs between upper and lower sidebands. Having just studied the properties of the upper sideband, the contrasting properties of the lower sideband will now be developed. Following the procedure of Chapter IV, a basic relation for lower-sideband mixing can be derived from (2. 27) by setting V = 0, which yields r Yr jrYp Vr 11* -jc7 Y*~ V* (5. 1) The similarity between (4. 1) and (5. 1) is quite apparent, if yp is again made real. In fact, by taking the conjugate of the second line of (4. 1), and by replacing the subscript u by C, one obtains (5. 1) directly. The value of this resemblance is that many results from the study of (4. 1) can be carried over directly to the solution of (5. 1). For example, by making the substitution noted above, (5. 1) can immediately be written in the informative ABCD notation of (4. 4) and (4. 5). This step yields the following interpretive relation. V 1 0 C/Y j/wY 1 0 1 0 V -jOr(yp2 C2) - Co) I\ r Y 1 0I^1 rp C 0 -Y, 1 11* r r Y C/Yp -Y terminal capacitive ideal terminal admittance transformer converter admittance (5.2) 63

64 C b' c I Ir yp -.6 __~~_ —rf M Vrr 1 1 I SIDEBAND V' C c MANLEY-ROWE CONVERTER CAPACITIVE 0 TRANSFORMER b c Fig. 5. 1. A functional equivalent circuit for lower sideband conversion. Since each factor above can be ascribed to one stage in a cascade, they together produce the equivalent circuit in Fig. 5. 1. In resemblance to Fig. 4. 2, Fig. 5. 1 contains a capacitive transformer and an ideal frequency converter with unity voltage gain and nonreciprocal current and power gain. In contrast, however, the current and power gains of the latter are the negative of the output to input frequency ratio. Therefore, on the basis of (3. 5), this converter should be termed an ideal, lower-sideband, Manley-Rowe converter. A peculiar aspect of Fig. 5. 1 is that all currents and voltages to the right of the line cc' are complex conjugates of the corresponding currents and voltages in an actual lower-sideband mixer. The effect of this alteration can be discovered by applying a load Y = ga + jb, (5.3) at the wa terminals. If Y is assumed to be passively realizable, it will satisfy the conditions ab g 0'; ac > 0 (5.4a) which the conjugate of Y would fail to pass. For it, ab < 0 (5.4b) acdo * * and hence YQ is unrealizable. However, being in an equivalent circuit, the effect of Y at

65 the co terminals of Fig. 5. 1 will be the same as the effect of Y in the actual mixer. * When the load Yp is transformed through the Manley-Rowe converter in Fig. 5. 1, it appears in the secondary of the capacitive transformer in the form r r Y C Y (-ge + jb) (5 5) which is illustrated in Fig. 5. 2. A significant feature of the admittance (5. 5) is its negative real part and its nonmonotonic susceptance. Hence, some rather unusual circuit effects can be expected from lower-sideband mixing, but with the aid of the equivalent circuit just generated these effects can by systematically determined. The approach will be no more complicated than extending the known properties of the simple capacitive transformer to include unrealizable loads, like (5. 5). For example, combining (5. 5) and (4. 8) shows the total input admittance at co to be r CL) co 2 Y Y r Wk Wp (5.6) rin r Y Other results of this type will now be developed. One unusual feature of the circuit in Fig. 5. 2 is power gain (by reflection) in the primary, which occurs in addition to the transmission power gain from primary to secondary. To demonstrate these effects, let L and C in Fig. 5. 2 be resonant at r, and let the primary conductance gr be partitioned into a load and a source conductance, gL and g., respectively. The available source power is then 4P |r(5.7) - 4gS The load power depends on the reflected primary admittance, which at midband is - 2 y 2 y 2 p _p r y p = g = - Pgr (5.8) Here, pg is analogous to p in (4. 18c). The power delivered to gL is

66 o bb 1r gs 9L L jb -g I I gr Y rg I I Y a b Fig. 5. 2. The equivalent circuit of a lower-sideband converter after Y is transformed to co. PL r r(11p 4 g) L = P 9 2 (5. 9) L r L gr (1-01)2 so it is evident that PL can greatly exceed the available source power. In fact, PL becomes infinitely large as P - 1 (5. 10) The power amplification in this case can be attributed to an enhancement of the input voltage by negative conductance reflected into the primary. The power converted from co to w can also greatly exceed the available r 1 source power, but then the load gL will be coupled to w c instead of to o (i. e., g = gL) In Fig. 5. 2, the power reaching the secondary of the capacitive transformer is P r ) r(1-Pf) cor ) (gr (1-p )2 A (5. 11) so the midband transducer gain from wr to wo becomes 4pk / \ gS G _j P) ( ) S (5. 12) (lpk)2 gr Similarly, from (5. 9), the midband transducer gain by reflection is - 4 gSgL / 1 \2 G" g - (5. 13) rr _O~a

67 which reduces to Grr ( (5. 14) if the first factor is optimized by letting gS = g = 1/2 g. Both G and G can be arbirr rk trarily large, by allowing p to approach one. Considering (5. 8) further, it is seen that the condition p - 1 produces a cancellation of the external terminal conductance at cr (or at wc ) by the reflected conductance at w (and at w ). So as p passes through one, the total input conductance passes from positive to negative, which raises the question of stability. A rigorous treatment of stability is made in Chapter VI, where it is demonstrated that the circuit under discussion is stable, to first approximation, if Pk < 1 (5. 15) This agrees with the intuitive prediction based on (5. 10). oth Gr and G depend upon pQ in the same way, but gr and g will differ in these two cases because only one contains the load gL. Also, a slight distinction is to be noted between the various reflection gains that have been introduced so far. For G in uu (4. 28) the source and load conductance are identical, but for G in (5. 13) the source and load conductance are separate components at the same terminal pair. Since a simple modification [based on (3. 16)] can reverse the sense of either of these quantities, context will henceforth be relied upon for making these cases distinct. 5. 1 Bandwidth and Sensitivity of the Lower-Sideband Converter There are several auxiliary parameters that shed further light on the circuit properties of the lower-sideband converter. One is bandwidth, and here again the transformer rule applies: the greatest conversion bandwidth occurs when the individual circuit bandwidths are equal. For large gains, where p -1 0, the conversion bandwidth is (Ref. 31) _ _ 2 7 (se-pc. (5. 16) 13r{ 2C r I

68 Thus for high gains the gain-bandwidth product is Gr C = Qkp (5. 17) where k is the pumping coefficient. In comparing with Table 4. 1, which gives the gain-bandwidth product for upper-sideband conversion with the same wp and kp, Table 4. 1 is seen to exceed (5. 17) by a factor of more than ~. So in spite of the fact that lower-sideband conversion offers much larger gains, its gain-bandwidth product is less, which speaks for the superior bandwidth of the upper-sideband converter. For the latter, wc increases when wo increases u r and both circuits become detuned in the same direction. However, transformer action tends to cancel these two detuning effects. The result can be critical or over-coupled characteristics that are extremely broadband. For the lower-sideband converter, w decreases as ow increases and the detuning effects are additive. Hence, narrow, undercoupled pass bands result, whose width decreases with increasing gain. A detailed study of the bandwidth of lower-sideband converters has been made by Lombardo and Sard (Ref. 32 ). By direct analysis, they confirm the rule of equal terminal bandwidths which was introduced here by equivalent circuit arguments. They also confirm the optimum gain-bandwidth product in (5. 17) for the case where parasitic loss is negligible. However, they predict an important correction when the effective shunt capacity at wr and wd exceeds the varactor capacity C. In this case, C in (5. 17) should be replaced by /CrC~, where C and C equal C plus the stray or filter capacity at c or or respectively. A similar observation was made regarding the pumping coefficient in Section 4. 1. A second parameter of interest in lower-sideband conversion is sensitivity, which can be explained as follows. If a circuit characteristic Z depends on a variable X, the sensitivity of Z to X is defined as S a log Z _ XaZ X aolog X Zax (1 In the case of lower-sideband conversion, the transucer gain is sensitive to variations in

69 the load or source conductance. According to (Ref. 31) S — P (5. 19) while the sensitivity to pumping (y ) is YP g p Therefore, one serious drawback to lower-sideband conversion is the high sensitivity that accompanies large gains. Particularly disturbing is the large sensitivity to external source and load admittance variations since these admittances are not well stabilized in many applications. In the ensuing analysis, however, multiple-sideband schemes will be considered which offer considerably reduced sensitivities to the external admittance, as well as other benefits. 5. 2 Degrading Influences on Lower-Sideband Conversion The basic form of degradation to be considered is varactor loss, which influences both gain and noise figure. For upper-sideband converters, varactor loss not only reduces the gain from its frequency dependent value, but it also limits the usefulness of raising pamp frequency. For lower-sideband converters, which offer infinite gain, varactor loss will be seen to increase gain sensitivity and to have a more pronounced effect on noise figure. The evaluation of these quantities will now be reviewed. 5. 2. 1 Effect of Varactor Loss on Gain. Using the basic form in (3. 16) together with the lossless gain relation in (5. 12), lower-sideband conversion gain in the presence of varactor loss becomes Gr( = (lpp4p (5. 2 la) where gr = gs + gdr Q = gL + gdQ (5.21b)

70 and where gdr = r C/wd g = C/d (5.22) Therefore, for finite gains, some increase in p is required to overcome the loss factor (gSgL/grg ), but losses also influence pe. Following the procedure outlined above, we find that gL g /p2 gL g S~ _- r ( L Yr (5. 23) ggr gLS ggr k where pe is pi in the absence of loss. Therefore, due to varactor loss, two increments of increase in p, are required to restore Gr. As a result, the sensitivity in (5. 19) increases by rgg S S (5.24) g \/gSgL g where S is the sensitivity due to gS and gL alone. 5. 2. 2 Noise Figure. There are two basic contributions to excess noise figure in lower-sideband converters. The first is thermal noise from the parasitic input and output losses gdr and gdl which will be assumed to have noise temperature Td. The second is thermal noise from the output load gL, which will again be assigned the noise temperature TL. The effect of these two contributions can be determined from the general noise figure relation in (3. 27), but first the gain ratio (GQ /Gr ) and the noise bandwidth ratio (B /Br ) must be calculated. Here G e denotes the midband transducer gain by reflection at cow which is defined in (3. 25). However, the proper expression for Gp, corrected for varactor loss, can be obtained from (5. 13) by a derivation similar to that yielding Grk in (5. 21). The result is GQf ( ( ) (g1} (5.25)

71 so the gain ratio is GpQ 1 ~r grgL:-L (5.26) GrQ Pp cO gSgQ For large gains, where p 1, the bandwidths of G p p and G are both given by (5. 16), so (3. 27) beomes gdrT d Wr g T d +gT gr F = 1+ + + (5.27a) gs T c gT - ~ "So W L b^'o S gdrTd r grT =1 +T (5. 27b) gSTo S gS o where gdk Td + gLTL T = (5. 27c) gdP + gL is the effective noise temperature of all conductance at the wp terminals. Because the parasitic conductances are frequency sensitive, further study of Frp becomes an optimization problem, which can be summarized as follows. First it is desirable for gL to be much larger than gdk to efficiently utilize the large conversion gains that are possible when p 1. Or, stated in other words, the efficiency factors gL z = = (5. 28a) gj o0 W2C 1+ Wd gs gS 1 x = =- (5. 28b) gr o 2C 1 +r Cod gS should be close to unity. Since wr and the varactor are often specified, x depends basically on the source conductance gs. However, z depends basically on uop and falls off at 12 db per

72 octave when co > cdgL/C. Below the latter frequency, F tends to decrease with w due to the factor owr/jo, in the second term. So extremely large oQ/'Wr ratios will be useful if gL in (5. 28a) is large. On the other hand, the pumping coefficient (p /C) should be considered a constant, so gS must decrease as gL increases to keep p - 1. This decreases the input efficiency and raises the first term in (5. 27). Therefore, the second term in (5. 27) will dominate for small co, while the first term is dominant for large WQ. This suggests optimum values for co /co and x, for a given pumping coefficient (kp), varactor cut-off frequency (wd), signal center frequency (car), and output efficiency (z). In Appendix A, these optimum values are shown to be ( - R F1 1 + 1 + (5.29a) (k opt 1-z / Rr[l+z (t-l)] R R - __ r i r l r -l if r << 1 (5.29b) (i-z)(l+z)(t-1)' 1and (x) = 1 — i _(5. 30a) opt 11 + R [l+Z (t-l)] R _ 1 _; if - << 1 (5.30b) 1/ -z Rr[l+z (t-1)] where R =(- r( C ) (: r (5. 31a) r od d ~pp co / TL t = T (5.31b) d See footnote on p. 56 for a discussion of R and c,. r

73 The optimum noise figure in this case is (Frapt 1 + 2R 1+z(- 1 -z (532a) Rr[1l+z (t-1)] tz R 1 + 2/Rr(l+ ); if r << 1 (5.32b) r+ 1-z 1 lZ 1-z The corresponding source and load conductances that must accompany this optimum noise figure are given by / gW +\ 1Wrp d R( + + t+z (5.33a) ( C /opt (( ) d) (5. 33b) A universal plot of (F ) opt is found in Appendix A. Now to illustrate these noise figure relations, consider the following example. (1) A signal at 3 kMc is to be converted to Co with maximum gain and minimum noise figure. (2) An output efficiency z = 0. 5 will be tolerated. (3) A varactor with cut-off frequency Wd = 27r (100 kMc) is available, and it can be pumped so y p/C = 0. 3. (4) TL= Td = T. Then t = 1, z = 0. 5, and R = 0. 01, which yields: by (5. 29) (r = 0.02 1+ J~ )= 1/6.1 (5. 34a) by() pt by (5. 32) /F,p = 1 + 0 04 51 = 1.29 (5. 34b)

74 by (5. 33) (.03) J51 =.21 (5. 34c) WrC opt ( gCo = 0. 18 (5. 34d) V\" /opt The last two values correspond to a terminal Q of 5 at each terminal, but gd, = gL in this case, so QQ is actually 2. 5. (The input loss gdr = gS/6 does not affect Qr appreciably. ) However, for optimum gain-bandwidth product the terminal bandwidths must be equal, which corresponds to wu Q = w Qr when both terminal capacities are assumed equal. Hence, if Qr =5, then an optimum gain-bandwidth product occurs for Qf = 30, but an optimum noise figure requires Qk = 2. 5. A proportionately narrower input bandwidth is required for optimum noise figures because input noise receives more amplification than output noise, if o < w, according to (5. 26). A brief comparison with upper-sideband noise figure will now be constructive. Using the same varactor, the same signal frequency and R = 0. 01, Fig. 4. 5 yields Fu = 1. 38 for the same pump frequency which is a somewhat larger value. However, if ru wc is reduced until wo occurs at the frequency where co was previously, then Fr = 1. 28 which is essentially the same as (5. 34b). 5. 3 Parametric Amplification The application of lower-sideband conversion that has received the greatest prior attention is power gain by reflection. It provides a means for one-port amplification that is commonly termed parametric amplification. Therefore, the portion of Fig. 5. 1 extending to the right of the line aa' constitutes a parametric amplifier, and has the driving point characteristic shown in Fig. 5. 3a. The basic circuit configuration for parametric amplification is shown in Fig. 5. 3b, and a typical gain characteristic of this circuit appears in Fig. 5. 3c. One of the drawbacks to the configuration in Fig. 5. 3b is that both the source and the load intercept the reflected power. A common technique for overcoming this limitation, at least at microwave frequencies, is by using a circulator to isolate the source, the load, and the negative conductance port. Basically a circulator is a lossless device that permits nonreciprocal coupling between each pair of its three or more ports. In Fig. 5. 4

75 a c REFL. (a) I 9f S ~/TUNING INDUCTANCE (b) POWER (C) TO gL (c) ( p \2 PUMP ON (i- p')2 pO ----— L^ ^PUMP OFF (p =0O) Wr r Fig. 5. 3. Power gain by reflection through parametric amplification. (a) shows the negative conductance generated by lower-sideband mixing, (b) shows a simple circuit for realizing reflector gain, and (c) shows a typical gain response. a three-port circulator is shown which operates as follows: an incident signal at port A is coupled only to port B, where it impinges upon a parametric amplifier. The latter in turn reflects an amplified version of the signal back into port B, where it now passes entirely to port C. If gL matches the circulator at port C, none of the reflected power will reach the source at port A, and the desired separation of source and load is achieved. Usually the characteristic admittance is the same for each circulator port. Call this Y. Then if gS = g Y the conductance seen by the parametric amplifier is no 0 0 longer g + g = 2Y, but now g - Y. Therefore, in the presence of a circulator, the critical parameter p in (5. 13) doubles its value. Alternately, a 30 percent reduction in yp 2 can be achieved through the use of a circulator. 2In addition, for equal values of p, the circulator provides up to 3 db more gain (because the amplified signal is isolated from the source). A circulator also reduces the sensitivity of power gain to gL and gs. Parasitic losses are neglected in these comparisons.

76 gL i ^ REFLECTED C ^< ^ ^POWER. INCIDENT Y X/ ^ POWER / 9 S PARAMETRIC AMPLIFIER r Fig. 5. 4. A parametric amplifier with a circulator for improved performance. In the high gain condition (pQ = 1), it can be shown that the noise figure of a parametric amplifier with a circulator is also given by (5. 27). Hence, the subscripts r and f on F in (5. 27) can be ignored for the present. For parametric amplification, the load gL at c can be zero since the actual load is attached the circulator at co. At one time it was felt that additional loading at oQ could reduce the noise figure by being cooled, since this would reduce the effective lower-sideband noise temperature appearing in (5. 27c). However, to keep p l 1, an added conductance at ci calls for a reduction in the source conductance gs, which in turn causes an unfavorable increase in g r/gs in (5. 27). This dr S conclusion can also be noted from (5. 32a) where variations in gk are equivalent to variations 3 in z, and (5. 32) is actually a minimum when z equals zero. Therefore, the optimum noise figure of a parametric amplifier with circulator is (F \ = 1 + 2Rr 1 +-r (5.35a) \ ~opt r ~r and it occurs for Ft R i + tJi+ (5. 35b) o i< r d ur This is true even if TL < TD. Further discussion of this point is found in Appendix A.

77 1.2 \ /R0.1 1.0: R=0.01 0.80.6 - 0.4 - 0.2o I I., I. I I 1.0 10 100 (_r Figo 5. 5. Parametric amplifier noise temperature vs. frequency ratio, showing the optimum predicted by (5. 35b). Consider now an example similar to the one in the previous section, where Wd 100 kMc, wr = 3 kMc, y /C = 0. 3, and Td T. In this case, a parametric amplifier r p d o at 3 kMc has a minimum noise figure of 1. 2, which is smaller than the corresponding value in (5. 34b). The smaller loading at aw in the parametric amplifier accounts for its lower noise figure. A sketch of parametric amplifier noise temperature for arbitrary cow/co ratios is shown in Fig. 5. 5. It illustrates the existence of an optimum pumping frequency, but it can be noted that a rather broad minimum occurs. 5. 4 The Degenerate Parametric Amplifier A special case of parametric amplification occurs when wr = o = wp/2. In this case, the basic analytical form in (5. 1) reduces to a single equation I = Y V + jyV* (5.36a) r r r r'Yp r whose solution is Y *E jW y I * r r (5. 36b) r I 12 -_ 21 P 2 Again, very large gains can occur, but now they are sensitive to the relative phase of the

78 pump,pand the source II. At midband, with Y = gr, maximum gain occurs for: - "'p r r r gr corYpI (5. 37a) gr = cr l r pi phase y - 2(phase I ) = n7/2 (5. 37b) p r However zero gain occurs is phase yp - 2(phase I ) = -7r/2 (5. 38) Therefore, if a degenerate parametric amplifier breaks into oscillation, the phase relation in (5. 37b) can be anticipated. To date, this form of parametric amplification has been regarded mainly as a novelty. However, a practical application of this principle will be discussed in Chapter X. b. 5 Conclusions All effects in this chapter have been the result of a regenerative coupling between or and the lower sideband, which has manifested itself as negative resistence. This effect has given rise to large conversion gains, but with small bandwidths and high sensitivities to parameter variations. In succeeding chapters, other frequency components will be introduced that will greatly extend the potentialities of single-sideband reactive mixing. Henceforth, Chapters IV and V will serve as convenient bases for comparing the newly found properties of double-sideband reactive mixers.

CHAPTER VI STABILITY OF NETWORKS WITH PERIODICALLY VARYING REACTIVE ELEMENTS It has been demonstrated in Chapter II that an analysis of the general reactive mixer in Fig. 1. 3 can be accomplished by treating an alternate network; one that is linear, but time-varying. For example, the time-varying network in Fig. 2. 1 was shown to be appropriate for treating the spectrum in Fig. 1. 4. It thereby illustrated a linearizing procedure in which the pump and nonlinear element are replaced by a linear element that varies periodically at the fundamental pump frequency. Since this substitution has been shown to be valid for all first-order mixing, the circuit in Fig. 6. 1 constitutes a first-order linear model for all reactive mixers. The sinusoidal steady-state properties of two special cases of Fig. 6. 1 were evaluated in Chapters IV and V. Several of these properties were suggestive of instability, which is discomforting since rigorous statements about stability normally originate outside the steady-state realm. The present chapter will consider, more rigorously, the stability of the circuit in Fig. 6. 1. In particular, the relationship between true instability and that predicted by sinusoidal steady-state analysis will be considered, and it will be shown that knowledge of the sinusoidal steady-state response is sufficient to distinguish between stable and unstable solutions. In Fig. 6. 1, the time-varying capacitance is assumed to have period 2r/wp, and the source is assumed to be sinusoidal with period 27/wo. Since the admittance of the time-varying portion of Fig. 6. 1 can be written in the form N Z a (jw)n Y(jw) = M - (6. 1) Z bm(j)m m =0 79

80 Y (jw) LINEAR TIME-INVARENT jwt NETWORK C C(t)-C Fig. 6. 1. An appropriate linear model for first-order interactions in reactive mixers. C denotes average value of C(t). the entire circuit is described by the following linear differential equation: dLI dm N dnv(t) Z b d- [QC(t) - C] v(t) + E a- = b (j)m ejt (6.2) m=l ndt= n d m=O e However, (6. 2) can also be written as a set of M first-order differential equations, of the form: dx. N dt +'ij(t)xi = 1 * (6. 3) — I+ 2 0 (tMx 0. (i = 1, 2,... M) (6.3) j =1 In this set, the x. are functions of v(t) or its derivatives, the 0ij are periodic coefficients with period 27/cop, and the i. have the period 2r/co. Given a typical circuit of the type shown in Fig. 6. 1, there are several alternate methods by which sets in the form of (6. 1) can be obtained. These all take the form of (6. 3), but in each case the individual 0ij depend upon how the x.i are chosen with respect to the circuit variables. In any case, the same stability characteristics will be predicted, so it is immaterial for now just how the set (6. 3) is chosen. In Section 6. 2, an example is given that shows how the set (6. 3) can be obtained in a particular case. 6. 1 Solutions of the Set (6. 3) From the well-known theory of linear differential equations with periodic coefficients (Ref. 33), the homogeneous portion of the set (6. 3) has solutions of the form x. = et y (6.4) 1 i

81 where the yi are periodic in t with period 2r/wp. In general, there are M values of /i (although they need not all be distinct) that characterize (6. 4), namely U = /,k' (k = 1, 2,..., M) (6.5) Since the yi are periodic, the Alk must determine the stability of this system and a necessary and sufficient condition for stable solutions is that Re(lk)< 0 (k = 1, 2,..., M) (6.6) If the /k are distinct, there are M linearly independent solutions of the form (6. 4) and the general solution of the homogeneous portion of (6. 3) is M Ukt x. = e yik(t) k (6. 7) i where the 7k are, for the moment, arbitrary constants. The solution of the inhomogeneous set (6. 3) can now be found by the method of variation of parameters, which regards (6. 6) as a transformation of the set xi to the set 7 k. When (6. 7) is substituted into (6. 3), the 7k are found to satisfy the relation llkt d7k Z e y. (6.8) k e Yik dt i (6.8) Therefore, the solution of the inhomogeneous set (6. 3) can be obtained by replacing the constant 77k in (6. 7) by functions of time that satisfy (6. 8). Solving this latter set for the derivatives of 77s, yields d7r dt = (s = 1, 2,..., M) (6.9) where A is the determinant of the coefficients of drs/dt, A = e k (6. 10)kt ~= Yijek (6. 10)

82 and A is the determinant A with its sth column replaced by the 0.. Therefore s 1 Yll.' 81''' Y1M Y21.' 0 2.'. Y2M -II t s e dt YM1- 0M YMM ___ (6.1 a) dt Yik D - t S S = e (6. lib) Since the yik all have period 27i/wp, and the i have period 2r/0, the ratio D /D will contain only those frequencies in the set m = 0, 1, + 2,... W = mw + non mn n = 0, + 2,... Therefore, D _ j t s s mn N m e (6. 12) ~D mn mn and tD -i t = BS + e dt (6. 13a) 0 (jC) -/L )t Owmn As )t = B + Ns N e 1 (6. 13b) s mn - 11 + jO -I- + mom mn s mn -As t = B + P (t)e - P (0) (6.13c) where B is the initial condition and P (t) varies at the frequencies mn. The final solution of the set (6. 3) is now found to be

83 ILkt X. = Z e Yik(t) [Bk - Pk(0)] + Yik(t) P(t) (6. 14) k k The second term in (6. 14) is the steady-state response. The first term is the transient response, and hence it is the part of the solution that is normally used to determine stability. A question of interest, however, is whether the interrelation between these two solutions is such that either can predict stability. To pursue this question, consider a typical set of characteristic roots for the circuit in Fig. 6. 1, as shown in Fig. 6. 2. In the absence of pumping (i. e., when C(t) - C is 0), these roots are simply the poles of impedance of the static network in Fig. 6. 1. However, when the pump is applied, each root tends to move and may cross the imaginary axis. Assume that no root crosses the imaginary axis prior to s' which will be assumed to cross at jw. Also, let o in (6. 11) be selected so that one of the Wc has the value ). Then, by writing s = - + jW, 6 = 0 will mark the boundary between mn s stability and instability. In this case, the component of Ps(t) [in (6. 13)] at ws becomes 5 St -p t N e [Ps(t)]mn e - (6. 15) which approaches infinity as 5 approaches 0. However, -s nm lim F -plt 1 __ 0 Ps(t) Se - P (0) t (6. 16) which is the familiar form of solution at resonance. Thus, when 6 = 0, the total solution remains bounded for all finite time, but its growth marks an instability threshold. However, the steady-state term in (6. 14) itself becomes infinite when a characteristic root crosses the imaginary axis. Therefore, if one can obtain an accurate estimate of the sinusoidal steadystate response of a periodically varying system (hopefully even without solving the differential equations), the singular points of this estimate will determine the thresholds of instability. This result is well known in constant parameter linear systems, where the characteristic roots themselves can be calculated by a direct and simple algebraic method. In systems with time varying parameters, however, or even in those with periodically varying parameters, approximate (and often tedius) methods are required to locate the characteristic roots (see

84 jWx X X x) X v Fig. 6. 2. A typical root locus for the circuit in Fig. 6. 1 showing how the characteristic roots may shift when C(t) is applied. Section 6. 2 for a typical calculation). Therefore, the use of sinusoidal steady-state solutions to determine stability promises to be a significant aid to the study of systems with periodically varying parameters. In Chapter II, the steady-state performance of several periodically varying networks were represented by characteristic matrix equations [i. e., (2. 27) and (2. 33)], so the singular points of these representations occur when the inverse fails to exist. For linear equations such as (2. 27), the vanishing of the system determinant marks the threshold of instability. For semilinear equations, such as (2. 23b), the same principle can be applied. For example, by rewriting (2. 33b) in the form [Y] [I] - [YB] [11*1 [YA] [YA] - [ [Y]}[ V] the vanishing of the determinant of the real matrix [YA] [YA] - [Y] [Y] is seen to mark the threshold of instability. 6. 2 Illustrative Calculation of Stability from Sinusoidal Steady-State Response A steady-state analysis of the elementary lower-sideband reactive mixer was formulated in Chapter V. The basic assumption in this analysis, as depicted by the ideal circuit in Fig. 3. 3, is that all frequencies except or and wc are short circuited. The remaining steady-state voltages, V and V, can then be determined by (5. 1). According to the rule established in the previous section, instability can be expected when V or V ber{

85 come unbounded, which occurs when the system determinant of (5. 1) vanishes, Y Y* - cow y2 0 (6. 17) r a r p An important special case occurs when the wr and ow loads are synchronously tuned, so Wr = Op- W. Then (6. 17) will vanish for the smallest possible value of y, namely that given by: gg2 c c 2 = 0 (6. 18) This condition was previously noted in (5. 10) to yield infinite conversion gain. An important question that we now ask is, "How well does (6. 18) predict instability in a practical circuit?" To answer this question, a practical circuit will be tested by this method. 6. 2. 1 Synthesis of a Prototype Circuit. The simplest type of circuit for approximating a lower-sideband converter is the four-pole network (and attached timevarying capacitor) shown in Fig. 6. 3. For convenience, the frequency of the time-varying capacitor has been normalized to two rad/sec, so the basic period of this system is m seconds. The poles and zeros of impedance of the static portion of this network are shown in Fig. 6. 4, where the pole locations are denoted by P -r + PQr P -uk + jQ~ P -CT - jQ p^ = -9 - j2 (6.19a) r r r2 and the zero locations are denoted by Z = -k2 Z -k1 +j Z = -k - j (6.19b) The choice (j) for the imaginary component of a is arbitrary and is made only for convenience. If it is assumed that

86 IL, L2 TR21 T - 2y COS2t RI R2 Fig. 6. 3. A canonical realization of the lower-sideband mixer using a time-varying capacitor. -2j -KI I X 0 —-Ko -Ocr — 2j -2j Fig. 6. 4. The poles and zeros of the circuit in Fig. 6. 3. |r a > 1> |, >> 1 (6.20) the resonant frequencies of the static network will be very nearly equal to Qr and 2, and the condition for synchronous tuning will become simply 2Q + Q2 = 2 (6.21) r from to = 2 with i frequecies Figure 6. 3 will then be viewed as a mixer from wc to wc = 2-co with midband frequencies r 1~ Wr' ihmdadrqece

87 r and w that equal 2Q and Q2, respectively. In view of the assumption in (6. 20), the reactive portions of Fig. 6. 3 will be synthesized first, and the resistors R1 and R2 will be added by perturbation. The synthesis yields the following element values 1 L2C = - (6.22a) 2 Q2 2~ 2 r k LC~ 1 L1 (1 (6. 22b) C c = (l-Q 1 2 )(Q 2 1) (6. 22c) When losses are introduced, by adding R, and R2 as shown in Fig. 6. 3, the poles and zeros move off the jw axis. The real parts of the zero locations are then related to R1 and R2 by R R I 2 k, =, k = (6.23) 1 L! 2 while the real parts of the pole locations satsify the relations k ea + fau 2 e + f (6.24a) k, ea + fa 2 e+ f (6. 24b) 2 e+f where 2z Q 2 e = -2 f = 2 (6. 24c) 1-2Q 1-2r The principle relating Fig. 6. 3 and Fig. 3. 3 can be stated as follows: Fig. 6, 3 has been designed to support frequencies in the vicinity of w = Qr and o = Q2, and to suppress (as effectively as possible) other mixing products. For approximate steady-state analysis, Fig. 3. 3 would then be chosen to duplicate Fig. 6. 3 in the vicinity of wr = 2r and wc = 2, and to give perfect suppression elsewhere. Therefore, if Y(w) is the admittance of the static

88 portion of Fig. 6. 3, then Fig. 3. 3 would use the quantities Y = Y(r ) and Y = Y(w). In the present calculation, in which (6. 18) is being tested, only the real part of Y(o) is needed which can be written k (1-S 2 )(Q2 -1) k 2 2 2 1! — - + (6.25b) C 1 2 2 k +(co - 2 L2 CL) The approximation above is valid if the poles and zeros are reasonably well separated. This will normally be the case in a practical converter. Within the wr and w) bands, (6. 25b) will be considered to have the constant values g = g(2r ) and g = g(Q ). Using these relations in (6. 18) yields the following steady-state prediction for the instability threshold. (6. 26a) which can be simplified to read (/ \2 -4(e-f) ra (r Q 26 P, r r e (6. 26b):(~)~ ef It will now be shown that (6. 26) accurately predicts the pump level for which the characteristic roots of Fig. 6. 3 cross the imaginary axis. 6. 2. 2 Analysis of Fig. 6. 3. By writing the mesh equations, the differential equations for the circuit in Fig. 6. 3 can be obtained as shown below. Q 1 +Q2 Q1+ klQ1 + Q1 + L1C (1 - a cos 2t) = 0 Qi+Q2 Q2 + kQ2 + L2C (1 - a cos 2t) = 0 (6.27a) where Q1 = S iidt Q2 = S i2dt

89 Here the assumption has been made that 1 1 C + 2 lyp[ os 2t - C (1-a cos 2t) C + 2 I'Yp Ics 2t C a 2 -C (6.27b) Now let us consider a change in (6. 27a) which will be explained later; namely to replace k1 and k2 by the quantities sak2 and sak2, respectively. We note that the condition s = a restores the original set of equations in (6. 25). These equations can now be put into normal form [i. e., that of (6. 3)] by making the following substitutions. x1 = Q1 x3 = Q2 x2 = Q1 X4 = Q2 (6.27c) which yields 1 0 1 0 0 x Ix2 a21 -sak a23 X2 X3 0 0 0 1 x x 4 a41 0 a43 -sak2 X4 (6.28) where 21 = -(Qr2 2 2 + 2 -(1 - Q2 )( 2 - 1) cos 2t = a21 + a al a23 = -(1 - Q2r2)(Q2 - 1)(1 - a cos 2t) = a3 +f a 3 a41 = -r2 3 2 (1 c 2t) = a43 = -Qr2 z2 (1 - a cos 2t) = a43 + a a43 (6.29)

90 Equation (6. 28) can be abbreviated as follows, x. ikxk a k (aik + ak)k (i = 1,,4) (6. 30) k k ik To test the stability condition predicted in (6. 18), the characteristic roots of the system in (6. 28) will be determined by perturbation methods. Realizing the form of the solution stated in (6. 4) the substitution will be made x. eltyi (6.31) where it can be asserted that yi is a function with period iT. Equation (6. 30) now becomes yi + Iyi = Z aikYk (i = 1, 2, 3, 4) (6.32) k 6. 2. 3 The Perturbation Method. In the present problem there are four characteristic roots, so M = /k where k = 1, 2, 3, 4. To obtain these roots by perturbation methods, let /k = k + a k + a 2 ilk +.(6. 33a) Corresponding to each root there will occur a periodic function Yik satisfying (6. 31). Therefore, let Yik =ik + a y ik 2 +' (6.33b) where each term in (6. 33b) is required to be periodic. Substituting (6. 33a) and (6. 33b) into (6. 37), and collecting powers of a, yields the following schedule: 0 00 O 0 y + [y - O a~ y? I -y +O a! y0 0(6.34b) 0 0 T 0 0' + ~ik - L'vk = - (6. 34b) Yik k ik aiYk + k Yi+ IkYik Cv aivk /kYik+ (al-k)Y k (6. 34c) V I Since the first equation above (6. 33a) pertains to an analysis of Fig. 6. 3 with R1=R2=yp =0, the solutions Yik are readily found. This unperturbed solution can then be applied as a forcing

91 function in (6. 33b). The latter equation has constant coefficients, so it can be readily solved for Yik. However, only those terms in Yik with period mT are of interest, so it will be found that such terms exist only for certain values of uk. Thus the desired quantities: /ik and periodic ik' are obtained. In turn, these can be applied to (6. 34c) to yield still higher approximations. A noteworthy feature of this method is that any one root can be approximated to an arbitrary degree, without carrying along any approximations to the other roots. 6. 2. 3. 1 The Unperturbed Solution. Since the unperturbed solution is used to calculate higher approximations to the desired solution, it is expedient for Y'k to be as simple as possible. Therefore, the resistors R1 and R2 will be added as perturbations, and the unperturbed roots will be located in the simplest possible position, namely on the imaginary axis. Therefore, the 1lk will be assigned as follows. /j 2 -j= j 3 = j = -j2 (6.35) It may be imagined that introducing two perturbing effects simultaneously would reduce the rate of convergence of the entire perturbation process. However, just the opposite is true in the present case, because the effect of y is to move the characteristic roots into the unstable half-plane [as defined in (6. 6)], while R1 and R2 tend to displace each root into the stable half-plane. Thus, for a given value of yp, there exists some value of s [in (6. 28)] that causes the least change in the characteristic roots. When s is in this neighborhood, the most rapid convergence of (6. 33a) will occur. In the present analysis, the case of interest occurs when the characteristic roots in Fig. 6. 3 have just crossed the imaginary axis. When the corresponding situation occurs in (6. 28), the real part of the perturbed characteristic roots will vanish. This situation must occur for a certain value of s, say sc. Therefore the threshold of instability in Fig. 6. 3 will be marked by SYP = c (6.36) C 2 Because synchronous tuning has been assumed, it can be anticipated that there will be little root motion in the imaginary direction. Therefore, the value of s should lead to rapid convergence of (6. 33a).

92 In the special case of constant coefficients, the yik in the general solution (6. 7) are constants, so the unperturbed solution of (6. 28) becomes it x. = ~ c. e (6. 37) However, one of the constants associated with each root is arbitrary, so let c1 = 1 for f = 1, 2, 3, and 4. The remaining cik can be found by substituting (6. 37) into the unperturbed portion of (6. 28), which yields jQ -jig j' -jig i~r r i Ici. e e f f jQne -jire jQ f -jQ f (6. 38) To simplify the perturbation method of solution, the dependent variable will now be transformed from x. to yi, according to (6. 31). This transformation is made separately for each root /k, which yields 0 o (/ - k)t ~ik E c ~ e= 2 kt (6. 39) ik i= Pk where the c. are as defined in (6. 38) and the 7rk are undetermined constants that will be used to distinguish those parts of (6. 39) with period 7Ti Since this perturbation allows each /k root locus to be separately evaluated, the root /l (with unperturbed value jQ ) will be considered first. In this case - r v = 1: - = = 3: -- \i = 4j(3-:) v = 2: /I 1 v -2jQ r = 4- /: = -J(Q +tr) (6.40) so two terms in (6. 39) have period m. Therefore, letting ~ = 7~ 0, the unperturbed so two terms in (6. 39) have period 7T. Therefore, letting /12 = 2/13 = 0, the unperturbed solution becomes o o o -2jt (6. 41) Yi1 = ll 11 + ci4741 e

93 where 7 11 and 7741 remain to be determined. 6. 2. 3. 2 First-Order Perturbations. The unperturbed solutions in (6. 40) can now be substituted into (6. 34b) to obtain the first-order solutions Yi'. Note that the forms of the Yi!, and of all higher order solutions, are quite similar because the homogeneous part of each equation in (6. 34) has the characteristic roots given in (6. 40). Therefore, the general solution of the unforced portion of (6. 34b) is O O (/11- 1k)t Yk = ci.k e When the forcing function in (6. 34b) is considered, solution by the method of variation of parameters yields the following first-order differential equation for the 77 O O vk ~k dP ek (6. 42) dt Ci e Here, Dk is the determinant of c. with the Ath column replaced by the column O O ik = L (aiVY k) + /kYik (6. 43) Therefore, D k has period 7m. The solution of (6. 42) has the same form as (6. 13a), with the index n absent. Therefore, O O (47mj - i~ + /k)t'k = B' + Z Nkm Re m o o(6.44) rk Pk km. o o' A m 47rmj - it + /k However, as previously illustrated in (6. 16), r7k exhibits a resonance type solution if 0 0 exp(-/u - Ak)t has period T. Therefore, two forms of (6. 44) must be distinguished O o -( )t ^k = ^k -+ P k(t) e k() (6.45a)

94 O o b) lp-/k = 2nj: N' t 7k M= k + — 2- + Q(t)(6. 45b) where B'k is constant and Pk(t) and Ql'(t) have period i. Once the solutions for r/'k are w k dP ( ktk obtained, they are to be substituted into (6. 41) to obtain the y'k, but the latter are required to have period 77. Therefore, the following conditions are necessary to insure that the yik have this period [in (6. 45a)] B' - P'(0) = 0 (6. 46a) N' [in (6. 45b)] 2 0 (6. 46b) The condition in (6. 46a) can always be met because B' is simply a constant of integration. vk However, (6. 46b) requires that (6. 42) have no constant terms when v and k are such that (,j - jk) is an integral multiple of 2j. This constraint will determine /k' O o When k = 1, the values of 0i - lk are given in (6. 40). There are two values of v for which the resonance solution in (6. 45b) can occur, namely, v = 1 and v = 4. Therefore, from (6. 42), /ik must be chosen so Dl and D 1e jt have no constant terms. Setting these constant terms equal to zero yields two simultaneous equations in /il and ir41 /1i. Eliminating 41/7711 yields the following expression for /i as a function of s. s(k k) s(k f k k e 2 1 2(1-^) i 2(1 —) 1 1 (6.47) 16 l( -^)(1 f 16 QQe - e) (1 - e) r e f The effect of pumping on pi. can now be found by setting s = a, as noted on p. 89. In general, values of /i with both positive and negative real parts will result, but the case of interest is for s = s [see (6. 36)], where the real part of /i vanishes. Assuming c 1

95 that the imaginary part of /i also vanishes when s = s, then s occurs for \ii = 0 in (6. 47) which yields s 2 16(e-f)2 (6. 48) c 16(e-f)2 ar r rC r where (6. 24) has been used to simplify this result. The fact that sc2 above is positive and real, shows the validity of the assumption that Im (ii) =0. Now, using (6. 36), the stability condition becomes (yp2 ^-4(e-f)2 a Cr r )C < __ r- r (6. 49) ef which agrees exactly with (6. 26). Therefore, direct calculation of the characteristic roots of Fig. 6. 3 has verified the much simpler stability prediction in (6. 18), although the latter was derived from an idealized representation of Fig. 4. 3 by steady-state analysis. The main approximations that have been obtained in this first-order verification are these in (6. 20) and (6. 25b). If they were not satisfied a new formulation of the steady-state analysis could be made to account for these cases. 4. 3 Conclusion In this chapter the following theorem has been proven: The formal steady state response of a periodically varying system becomes unbounded if and only if one or more of the characteristic roots of the system is purely imaginary. The purpose of subsequent analysis has then been shown how approximate steady-state solutions can be used with this theorem to yield useful and accurate stability predictions. A secondary purpose has also been accomplished in showing the laborous procedure involved in obtaining the characteristic roots of a periodically varying system. In this way the practicality of the above stated theorem has been demonstrated.

CHAPTER VII GENERAL CONSIDERATIONS OF REACTIVE MIXING INVOLVING SYMMETRICAL SIDEBAND PAIRS The preceding chapters, summarizing single-sideband reactive mixers, have demonstrated two extremes in circuit properties. In the first case considered, only the upper sideband was present. A completely stable conversion gain was demonstrated, which was limited in magnitude to the ratio of output to input frequency. Lower-sideband mixing, on the other hand, demonstrated arbitrarily large conversion gains, but at the expense of an ever increasing threat of instability. Large, lower-sideband conversion gains are also accompanied by large input reflections. While the latter are useful, as a means of one-port (parametric) amplification, they are difficult to control and to exploit fully. Furthermore, single-sideband parametric amplification is limited to frequencies below the pump frequency. In view of the striking differences between these two single-sideband cases (both with respect to their useful features and their limitations), the presence of sideband pairs causes the anticipation of interesting effects. The main effects that will be discussed are enumerated below: 1. Parametric amplification at frequencies higher than pump frequency, without relying upon effects from pump harmonics. 2. Conversion gains between wr and w u' well in excess of the output to input frequency ratio, and either with or without negative input resistance. 3. Amplification by phase modulation and detection. In each of these cases, the added influence of the pump harmonic 2cw will be considered. In most instances, it will be shown to provide still further improvement in mixer performance. The detailed realization of these effects will be the subject of later chapters, but the present chapter will discuss them from the point of view of the general energy relations. It will be shown that the latter predict each of the effects noted above. 97

98 7.1 Application of the General Energy Relations to Double-Sideband Reactive Mixers When applied to double-sideband reactive mixers, the Manley-Rowe general energy equations in (2. 9) reduce to the set in (3. 2). But when the pump harmonic is added, they become W +W W W p 2p u p 2 + u + =0 (7. la) p u & W W W r u. 0 r + - = (7. lb) r u ~ Before going on, it is important to consider the artifice of restricting the Manley-Rowe relations to a particular set of frequencies. We note that the realization of a predicted energy distribution, without the influence of certain of the eliminated frequencies, is not guaranteed. For example, if the external circuitry reactively terminated a particular frequency, it would be eliminated from (7. 1), but this frequency may still contribute internally to the conversion process. It may be a hidden mechanism behind effects that seem from (7.1) to be independent of this frequency. In other words, the Manley-Rowe equations only relate frequencies where real power flows. Situations of this nature will be illustrated herein, particularly in connection with the first pump harmonic. For example, (7. la) suggests nothing about the relative importance of the pump and its harmonic. It even suggests that they are interchangeable, which is rarely true. A further distinction between these two frequencies must come through a knowledge of their reactive power components. Therefore, the term W2p in (7. la) can be ignored for the present, if it is realized that reactive power at 2co (or other suppressed frequencies) may still be required to realize the effects predicted by (7.1). Further study of (7. 1) will be divided into two parts. Consider first the application of signals at wa and op, with co and ou experiencing passively realizable loads only (in particular, loads whose resistive components are positive). Thus, W and W are' u nonpositive and (7. Ib) yields the following expressions for the up-conversion power gain from Co to Co or to o. r 9' u

99 W co u u pr = - w = ) ( X ) (7. 2a) ru ( W - Xf r r Pu - ( (7. 2c) shown in Fig. 7. 1. In the limit where cw is short circuited (W = O0), PrU has the previously discussed value w u /w. However, P increases as X is increased, and arbitrarilylarge positive values of PrU can be achieved in the vicinity of X u < 1. This suggests that large conversion gain is possible without reflecting negative input conductance at ar. For Xf > 1, PrU and Prf are negative. Hence, this condition defines a general region of potential instability in which parametric amplification is possible at or. 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 was the case in the circuits discussed previously. The Manley-Rowe equations 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 analysis. The second case of interest is with signals applied at wp and w, and where cor and co are terminated only in passive loads. In this case, with Xpr = Wow r/W ra, (7. 1) yields the following expressions, which are also plotted in Fig. 7. 2. W I (r r Pur w " = IX-) (7. 3a) u u u u Pr

100 Wp I I I I W Wr W/ /r A Wp u W) Pru Wr (r - WI conversion gains from or to ou and co s. the ratio of output ^^ ~~~~~~~Wu Wu Pru powers at wj and wo. Pump power is applied at w and /WW is the ratio of pump to signal power. For X. < 1/2, P and P are positive, but W /W is negative. Therefore, the pump circuit is unstable in this region and the signal at cu will act as the power source. For oV p/ov < (Xfr) < 1, P is greater than unity, so this is a region of arbitrary down-conp'~~~~~~~ ur ~ version gain with positive input conductance. For (Xre ) > 1; Posit and are negative For X ) r 1/2' PUr and p/c are positive, but Wp/W is negative. Therefore, the pump veus, a region wi potentive inpustaity stsat, wich suggests that paramet 1; r u ar ai Thus, a region of potential instability exists at w, which suggests that parametric amplification is possible at this frequency. 1See Section 3.1. 2.

101 o I I i I Jr aIJ1 Wp Cu a) Wp / Pu. Pur Wr WI~ WJu ^___________w Wp w /- ----— 1 —-- powers at w I and wc. Pump power is applied at op, and Wp/W is the ratio of pump to signal power. Again, it should be noted that the Manley-Rowe equations do not guarantee that just any reactive element will yield the results above. For a particular element, only the trivial solution, W = 0, may be possible. For example, it can be noted from Fig. 7. 2 mn that parametric amplification at wU is predicted even in the limit where wr is short circuited (Wr =0). But, applying the time-varying reactance model in Fig. 2.1, as described by (2. 27) [a model that satisfies the Manley-Rowe equations in (7. 1)], it can be shown that Wu is completely decoupled when wr is suppressed, unless the pump harmonic y2p is present. r

102 +7)(n w I r^ ~V(t) PHASE Y P t T |t L (MODULATED) Fig. 7.3. An illustration of phase modulation by reactive mixing. 7. 2 Application of the General Energy Relations to Phase Modulation An important characteristic of reactive mixers is their tendency, under certain circumstances, to produce phase modulation. This effect is illustrated in Fig. 7.3, where a signal at co varies the tuning of a resonant circuit and hence varies its phase shift. A true phase modulation spectrum, as shown in Fig. 7. 4, is symmetrical in the sense in which the term is used here. (That is, the distribution of sidebands by frequency is symmetrical with respect to cp.) However, the spectrum in Fig. 7. 4 has still a higher degree of symmetry, since W = W mn m, -n It will now be of interest to apply the Manley-Rowe equations to this configuration. This step is facilitated if (2. 9b) is first written in the form W oc n nW -r + z zmn + 0 -m =0 (7.4) r n=0 m=l r p r If (2. 5b) is applied to the last term, (2. 9b) can be further modified to read: W oo cC nW nW r mn, -n - + z m n m- = 0 (7.5) c mop + nco mc - nco r n=0 m=1 mp r p r But, in Fig. 7. 4, m = and W = W, so in this case mn m,-n W oo 2n2 co W r r mn (7.6) or n=1 wo 2 +(no )2 p r Each sideband in Fig. 7. 4 is passively terminated, so the right hand side of (7. 6) must be negative, and hence W must also be negative. Therefore, the phenomenon of phase modulation by reactive elements is potentially unstable. Instability in this spectrum arises be

103 - STABILITY _ ^* ~CRITERION UNSTABLE 0 Wr W,-n )p In ( STABLE Fig. 7. 4. A typical phase modulation spectrum and the associated stability criterion based on (7. 7). cause the lower sideband is dominant in each sideband pair in (7. 5). Further inspection of (7. 5) shows that W is negative as long as W mwo + no n < p r (7.7) W nw - nw m, -n p r This relation defines the line shown dotted in Fig. 7. 4, which is an envelope that separates stable and unstable spectral distributions. For example, if a line is drawn through the peaks of the power spectral components of any symmetrical pair of sidebands, then its slope, relative to the dotted line in Fig. 7. 4, determines the influence of this pair on stability. If the slope is greater than the reference line, the pair will contribute positively to Wr and to stability. With lesser slope, the same sidebands contribute negatively to W and hence contribute to instability. It will be of interest to now consider pump power. If the decomposition employed in (7. 4) and (7. 5) is applied to (2. 9a), the following relation for pump power is obtained W so oo W W P + Z n mn + m+, n 0 (7.8) p n=1 m=1 mP r m nwrj Applying this relation to the spectrum in Fig. 7. 4, where m=1, yields W oo 2 W p p - i n (7.9) p n=1 2 + (nc) )2 p r If this result is now compared with (7. 6), a given sideband pair will be seen to receive pump

104 and signal power in the ratio 0) 2 P (7.10) 2 2 nw r Therefore, nearly all sideband power is contributed by the pump, in the ratio of pump to signal frequency squared. This will be an important result later on. 7. 3 A Generalization of the Double-Sideband Mixer The reader will eventually see that the remainder of this study is essentially an investigation of the 3 X 3 matrix in (2. 27). While this matrix was derived for the symmetrical mixing scheme in Fig. 1. 4, it is of interest to note that an infinity of unsymmetrical mixer schemes are equally well described by (2. 26) or (2. 27). As proof, consider a general time-varying capacitor with components ymp at all pump harmonics nop, and consider further that the signal set includes any of the frequencies mw ~ wO. It can be shown p r that the relation corresponding to (2. 26) in this case is the following: -I4,/j c4,- C Yp Y2p Y3p 4p - - - 1 /jco-IJ3 C Y Y2 Y Vyp c C lY -coyy -y p 2p 7p3p 4p - v - * * * ** r/J p =2,-i -2p1p Y2p Ip )22p p 3p- - I /jiu -l Yp 3p p p C p p - V I21/j21 = - _ 4P 73 P j2'2p 7p 74p - r r -'4p~'3pJ~' p p4p 72p p2p C- V I jw - Yp C- Vu I31//j31 - - Y4p Y3p 2p p C - V31 It can be observed that the center 3 X 3 submatrix in (7.11) is identical to (2. 26). Also, all other 3 X 3 submatrices along the main diagonal are equivalent to (2. 26),

105 although each involves a different set of signal frequencies. Therefore, all signal frequency subsets that are defined by -(Wp -ow) ~ ncp - ~ nw r + ncp (7.12) w + Co nc p r p are identical. Consequently, all properties in later chapters that are attributed to or, C' and wu can also be realized with the mixing spectra in the table below, providing the indicated correspondence is followed. Reference Set w Co o.1 r u 2co - c co -co co All lower p r p r r All lower sideband 3cw - w 2co -o co - co equivalents l P r p r p r co co + co 2c + c All upper r p r p r sideband co + o+ 2cw + co 3Co + co equivalents P r r r Table of equivalent signal spectra. 7. 4 Conclusion The general energy relations have suggested some interesting effects associated with double-sideband reactive mixing. In particular, it has been seen that extracting power at the lower sideband, wc,will increase the power transfer between or and cou. Also, these same interactions can be attained with an infinite number of sets of three signal frequencies. The following chapters will now treat these individual cases by means of appropriate small signal analysis. In this way, the many improved circuit effects associated with double-sideband reactive mixing will be discovered.

CHAPTER VIII DOUBLE-SIDEBAND, SINGLE-PUMP MIXING WITH INDEPENDENT SIDEBAND LOADING A convenient first step towards investigating the advantages of double-sideband reactive mixers is provided by a simple combination of the two typical single-sideband cases. The single-pump frequency wp will be retained, but the signal frequency set will be extended to include co, WC, and cw simultaneously. The formulation for first-order mixing in this case is essentially in (2. 27). The only modification required to make (2. 27) applicable now is to set y2p to zero which yields: II Y jcwy jw or7 V r r rp rp r IL jw y Y 0 V u up u u -J *~ 0 Ye V* (8. 1) Again, yp will be assumed real. The effects incurred by neglecting y2p will be explained in Chapter 10. An appropriate equivalent circuit for (8. 1) is Fig. 2. 1 (with y2p = 0), but in employing this figure cow and Cu will be assumed to be widely spaced in frequency. Ideal filtering can then be approximated reasonably well, and the sidebands can then be independently loaded. Hence Y and Y can be regarded as arbitrary, positive-real admittances. An alternate equivalent circuit can be formed by combining the equivalent circuits in Figs. 4. 2 and 5. 1, as shown in Fig. 8. 1. Here, the r terminals are simultaneously coupled to the ow and the wu terminals, but the latter two frequencies have no direct coupling. By recalling the separate functions of the individual sidebands, the following operation of Fig. 8. 1 can be visualized. The load at Co reflects positive conductance 107

108 Wr WAu UPPER YPo II A I Vr 1 C SIDEBAND MANLEY-ROWE VA Y U A CONVERTER * Fig. 8. 1. A functional equivalent circuit of double-sideband conversion with a single pumping frequency. into the wc terminals, while the load at wQ reflects negative conductance. The latter tends to enhance the input signal by parametric amplification. Therefore, any signal power that is convertedto co will be amplified before conversion. As a result, the transducer gain from wr to wc can greatly exceed the ratio of these frequencies. Ironically then, the extraction of power at a increases the power conversion to cw. Figure 8. 1 also suggests enhanced down-conversion gain from co to o, because the down-converted signal will receive parametric amplification before emerging from the cor terminals. However, this action at or radically changes the input admittance at o, even to the extent of reflecting negative conductance at this frequency. When this is the case, direct parametric amplification will occur at w u — a frequency higher than the true pump frequency. Note that this accomplishment is quite different from "parametric amplification by low frequency pumping," as reported by Bloom and Chang (Refs. 34, 35, 36), who used harmonic generation to make a low frequency pump produce capacitance variations at the normal pumping frequency. Basically their system is a conventional lower-sideband converter. The question arises as to what benefits wo might bestow on conversion from c) to w. In short, this case appears to offer no advantage because the single-sideband case has already demonstrated arbitrary conversion gain between or and wo and the prer

109 sence of wc simply introduces excess loading. Of course, in certain applications this loading may be useful for controlling stability, but the effect of wo on cW appears to be less deserving of attention than the previously noted effects of wQ on c. Therefore, the exploit of the present chapter will be limited to effects that can be derived at the wr and wo terminals. Using the results of Chapters IV and V, the input admittances at each terminal in Fig. 8. 1 can be written by inspection. At w r the two reflected admittances appear in parallel, so by (4. lOb) and (5. 6) one obtains 2 2 Y Y + rW u p Wr _ p (8. 2) r. r Y Y in u To perform a similar calculation at w, the load at o will first be reflected to wr according to (5. 6) and then the parallel combination thus formed with Y will be reflected to w. Upon adding this reflected admittance to Y, one obtains Oj Co y 2 U. u 2 YH = \ - _ r usp _ (8 3) in W os p Yr Y Similarly, at cow 10 W oy 2 tY Y* - - r p (8. 4) k 2 in W cWyp y + r up r \Y u It is generally desirable to have all frequencies synchronously tuned. In this case, (8. 2) through (8. 4) reduce to WrW u Wp r p (8.5a) pr p- + -= - I (8. 5a) rin r gu g( W )'y 2 g r =g + r (8. 5b) in w y, -,9WrW - (p. (8.5c) in rao oQ y g - g gir -g. which are plotted, vs. yp, in Fig. 8. 2.

110 9 rin u / ONLY/ /u DOMINANT (a) 9r \ WOi DOMINANT gu in ONL ~/ONLY (b) l)I DOMINANT gQ in (C) g9 --,- A I f~Wu DOMINANT F ONLY A | \ Or DOMINANT Fig. 8. 2. The input conductance at resonance vs. yn, showing regions of apparent instability. The important question of stability arises again in connection with Fig. 8. 2. It can be seen that regions of negative input conductance appear at each of the individual terminals, but for different values of yp. For example, gun becomes negative for a smaller value of yp than g i or gri,, and so the stable regions of Fig. 8. 2 are a matter of concern. P "in in In a preliminary study of this problem by the author (Ref. 19), the range of stable operation was assumed to be determined by the negative conductance region that first appeared as yp increases. However, this assertion is inconsistent with the more rigorous criterion outlined in Chapter VI. For example, further examination of Fig. 8. 2 shows that

1ll the input conductance at the individual terminal pairs can become negative in two ways, either by going through zero or through infinity. According to the criterion just mentioned, only the latter instance is actually unstable. Thus the possible range of stable loadings in Fig. 8. 2 is greater than once expected. A more specific statement about the stability of double-sideband, single-pump mixers can be made directly from (8. 1). It has been established that a sinusoidal, steadystate response must become infinite at thresholds of instability. Therefore, to the extent that (8. 1) predicts the steady-state response in an actual mixer, the vanishing of a system determinant (A) will mark the instability thresholds. From (8. 1), Y Y + co y2 ( 8.6) r Y u rp u( u or, by (8.2), A = Y* Y Y (8.7) u r. in which vanishes only when Yr. vanishes. The fact that the input admittance at wr can be inr used to predict stability is important because there the influence of the individual sidebands can easily be distinguished through the representation in Fig. 5. 2. In essence, the lowersideband is the source of all instability, so it can be said that w2 is always regeneratively coupled to wr. By contrast, the upper-sideband is degeneratively coupled to w, and a proper balance of these two feedback mechanisms will produce an interesting variety of circuit effects. 8. 1 Gain and Bandwidth for Double-Sideband Conversion from w to co r u It has been noted that the transducer gain from wr to wu increases significantly with the dissipation of power at wo. To quantitatively evaluate this effect, the transducer gain from or to ou can be calculated from (8. 1) and (3. 15b) which yields G = _ 4 g g r Wu _ (8. 8) ru uc I _ w rY Y ~CWwUy -w wOy 2Y /y* 2 r\ r u up r 2p u/ / The most fundamental application of (8. 8) occurs when the individual terminals are syn

112 chronously tuned. Then, if each terminal admittance is equivalent to a simple resonant circuit, the latter can be approximated by Y r gr(1 2j ) (8. 9a) r - g (l + 26 ^ ) (8.9b) u ye g ( 2jb) (8. 9c) where cr = or +, and f3, 3 and 0u are the bandwidths in angular frequency of the co, wo, and wu terminals, respectively. The approximations in (8. 11) are useful because the most significant circuit behavior occurs for values of 25 that are less than the individual bandwidths. In this case, the terminal admittances are well represented by (8. 9). Substituting (8. 9) in (8. 8) yields the following expression for first-order dependence of transducer gain upon 5. W u3(gSgL P Gru(6) / \ u (8.10) Or. +r 6, +- u + I1 + 2j5//32 Two additional approximations made in writing (8. 10) are that 2 2 to 9 y 7 C c y _ru p r up (8. 1a) 2 2 -re p re ep4 p (8. llb) gr gk gr gThe midband value of (8. 10) is -\ Sr (grgu u G r (8. 12) ru (1+ p - p)2 which becomes arbitrarily large as pQ - 1 + pu

113 In (8. 11) p depends on the input and output conductance, while p is rather arbitrary. Therefore it is instructive to regard (8. 12) as a function of Pu' with p as a parameter. It is apparent from (8. 12), that an optimum value of p exists. To further evaluate (8. 12), it will be rewritten as follows Gu Nxy(l-x)(1-y) G - (8. 13) ru r [1-P1 + N(1-x)(1-y)] 2 2 N - r up (8. 14a) gdrgdu (I- )(1-Y) x = y = (8. 14b) gr gu The quantities in (8. 14b) are the same input and output efficiency factors that were previously introduced in Section 4. 2. 3. For a given varactor, a given p, and for fixed frequencies, N will be constant in (8. 13) so G can be optimized with respect to x and y. The result is expressed ru below in several alternate forms if p < 1:( p _ (r) P(1- ( 5) which occurs for x = y (8. 16a) Pu = (2x-1)(1-p) (8. 16b) ifp > 1: (G r) = (8. 17) opt which occurs for Pu = P^-l (8. 18)

114 Therefore, the optimum gain depends on p. If p is varied, GU increases monotonically until (8. 18) is satisfied. But if pu is varied, the optimum gain can be either finite or infinite depending on p. To interpret the optimum gains above, note that (8. 16) reduces to the previous result [in (4. 35)] when pQ = 0. But in the latter case, optimum gain corresponded to a matched source and load. Therefore, it will be of interest to see if (8. 16) retains this special significance in its more general application. By (8. 5a) gr. = g+ gdr + gr (Pu- (8.19a) in gs + gS - 1- (8. 19b) but the first term denotes the source itself. Therefore, the source will be matched when p - pk = 2x- 1 (8. 19c) Similarly, by (8. 5b), the load will be matched when (P) gu + gL (8.20a) or 2y-1 (8.20b) 1-P Since (8. 20b) is identical to (8. 16), it is evident that (8. 19b) and (8. 20b) cannot be satisfied simultaneously unless p 0. Therefore, in the presence of the lower sideband, but with pk < 1, optimum gain from cr to aw corresponds to the output load being matched, but not the source. Under these conditions the source is mismatched according to (8. 19b), which yields gr. = gg (1 - 2po) (8. 20c) in The second term here is the conductance seen by the source, which cannot exceed gs. Thus

115 the relationship between p and pf in (8. 15) yields arbitrarily large gains with: (1) a matched output, and (2) an optimum compensation for varactor loss. However, as the gain becomes infinite in this case. p must become vanishingly small, which puts a severe restriction on the output load. WVhen p > 1, alternate advantages accure because infinite gain can result for nonzero values of p. No optimization with respect to x or y occurs in this case because infinite gain is theoretically possible for any x or y. 8. 1. 1 Gain Improvement When p < 1. The optimum gain in (8. 15) can be regarded as the simplest extension of the pure upper-sideband converter, Therefore, to obtain a better feeling for the influence of the lower sideband, the dependence of (Gru) ru opt upon frequency will be calculated for this case. Combining the relations in (4. 18c), (5. 8), and (8. 14 through 8. 16), the following normalized gain expression can be obtained (,ru) > r_ /2x-1 ~u /"rY2 (GruI0ot ( r)' —2) (8.21) where = * ^ - ( dd 2)(l-x) Ur. ( (2x —l)(1-p) (8.22) Eliminating x from these two relations yields the curves in Fig. 8. 3. These curves show that gain increases with p even in the presence of varactor loss. However, a further limitation arises because pC depends on the varactor loss gd at o). The maximum value of pp occurs when g = gd, which yields co cr yp Pxmax - rp - (1-x) N (8. 23a) (P )max -gr gdW u Or, by (8. 14) and (8. 16), (P max (2 ) [-( K) (8. 23b) Therefore ~(p,)2x - 1 ^P&max = -- — (8.24a) u(1-x) + (2x-l) 9O

116 I.0 /.0 / LOCUS OF MAXIMUM OBTAINABLE GAIN FOR LOSSY VARACTOR / /I / / >__.~__. ~__.......IDEAL VARACTOR', (/ 00 _ __ LOSSY VARACTOR.01 //I I.01 O.2 1.0 10 Wrr W'd' Fig. 8. 3. Curves showing improved upper sideband up-conversion due to the introduction of the lower sideband. The parameter pp is a measure of the amount of parametric amplification due to the lower sideband. When pk = 1, the lower sideband is coupled just as it would be in a simple lower-sideband converter with infinite gain. vwhich is less than 2x-1 u (Ppmax x if - 1 (8. 24b) ( max x t Substituting the latter value into (8. 21) yields (w 2 2 2 i(0)G ~ (^'.(5} l —( (8.25) (ru max Krr x ) (.2-p5) which means that the limiting case of p = 1 yields a maximum gain that is only four times larger than the maximum gain possible when p = 0. However, the larger gain occurs at a lower pump frequency, which offers additional advantages.

117 Therefore, for a given wu/ w ratio, varactor loss will limit p. in (8. 16b) to u r some value less than one. This limit is such that (8. 15) can be improved by a factor of at most 4 (6 db) over the optimum value for p = 0. The maximum gain for a given p, based on (8. 24b), is also shown in Fig. 8. 3. It should be noted that the error of approximation in (8. 24c) is such that the curves in Fig. 8. 3 are a slight overestimate of Gr 8. 1. 2 Gain Improvement When p > 1. When p > 1, the condition in (8. 18) will yield infinite gain. In this case, the double-sideband converter has essentially the same gain and bandwidth properties as the simple lower-sideband converter. The lower sideband now completely dominates in the circuit, so only the output frequency is different, being wc instead of wc. The gain-bandwidth product is given by (8. 37), which differs from (5. 17) only by the factor w u/w. Not to be overlooked, however, is that now nearly identical outputs are available at two separate frequencies. 8. 1. 3 Bandwidth of G. For a further description of double-sideband conru version, the bandwidth of (8. 12) will be calculated. First, it will be noted that the input and output bandwidths must be equal to obtain maximum conversion bandwidth. Then it will be reasoned that if an additional signal frequency produces desirable effects, its presence will be desired over the entire conversion band. Therefore, a case of general interest occurs when the terminal bandwidths are equal, namely when 3r = d 3 = Mu =. In this case, (8. 10) reduces to /0 \ 4xyp ~~~G (4X5) = = —----- (8.26) ru cr [1 +p p + 2 +4(22)2 (8. 26) The bandwidth of (8. 21) is found to be 1 2A (A - 2) + 14 - ~4A 4- A2 2 (8.28) 251 [z A AA + ]21 (8.28) A if A << 1 (8.29) where: A = 1 + Pu - pp. (8.30)

118 Thus, as A - 0, G - x, while 61 - 0. The significance of these limits can be convenru 2 iently demonstrated in terms of the gain-bandwidth product, which has the value 2 ru cA2A (u) (A-2) + 4-4A2A j 26; ^ J~ 2r3 Xu A2~ - [ (8.31) It appears from (8. 31) that p should be large, but p is not independent of f3. By rewriting p as follows 2'c co y r u p r u p (8.32) u 3r u Cr Cu and using f3 = i =,3, then r u i / w= (p >y ) (8.33) which is a constant for fixed frequencies. Therefore, (8. 31) is not proportional to jU as it appears. Using (8. 33) in (8. 31), the latter becomes A-2 + 44-4A + 2A2 26, Gr = 2y o (8.34) - u u C C A2 Vr u Tne expression in square brackets in (8. 31) has maximum value 2 which occurs for A = 2. For this value, a maximally flat bandpass results with a bandwidth J/23. The same result was noted for the pure upper-sideband converter (p = 0), in Table 3. 1. When the lower sideband is introduced, so p increases from zero, the optimum bandwidth associated with A = 2 can be retained if Pu also increases. Therefore, one provision offered by introducing the lower sideband is the means of trading gain for bandwidth, while retaining an optimum gain-bandwidth product. The latter quantity can be found by setting Pu 1 + P

119 which makes A = 2. Then by (8. 34) 2 = WGp xo y (8.35) ~2' r u which is optimum for a given pair of input and output efficiency factors x and y. Comparing (8. 34) with (8. 16) shows that optimum gain-bandwidth and the (finite) optimum gain occur simultaneously only if p = 0. When (8. 16) is satisfied A = 2x(1-p ) < 2 (8.36) which approaches zero for large gains. However, as A - 0, the gain-bandwidth product in (8. 34) approaches 2 2X 2 2 r = p / C (8. 37) which is 30 percent less than the optimum value in (8. 35). The complete dependence of gainbandwidth on A is shown in Fig. 8. 4. Also shown in this figure is a plot, vs. pQ, of the gain relative to that at optimum gain-bandwidth. Since the minimum value of this curve corresponds to A = 1 + Pu, this minimum value will be greater than zero db if u < 1. The optimum gain-bandwidth product could not be realized in this case. Of the two optimum gain cases cited in (8. 16) and (8. 18), the latter still seems preferable. Both yield slightly less than optimum gain-bandwidth product, but (8. 18) yields infinite gain with more flexible conditions. 8. 2 Noise Figure for Conversion From cor to wo __ __r u The up-conversion noise figure in the presence of w o can be calculated and optimized directly from (3. 27). However, two basic cases should be distinguished. The first is with the load matched, which has been shown to yield only finite gains in practical application. The second is with infinite gain, where both the source and load experience negative reflected conductance. In these two cases, (3. 27) becomes

120 20 GAIN IMPROVEMENT RELATIVE TO MATCHED ft \GA N _ \ I m \I T 10 DB GAIN IMPROVEMENT RELATIVE TO P = 0 100% RELATIVE GAIN-BANDWIDTH \ I v | PRODUCT 50% A = I + - P — - 0' 2 A=I*+P-P - 3 Fig. 8. 4. Gain improvement and corresponding gain-bandwidth product for upper-sideband conversion in the presence of the lower sideband. G T cr uu d gdu with output -- -i,6. ~ T matched gdr Td G T ru To g mtched - 1 + + -+ (8. 38) ru g Gru T ~ G T g (8.38) ru = gSo T G Tu u with infinite g ain Gru To gain The evaluation of F will be simplified by employing the efficiency factors ru gS gL g x = y = z = (8. 39) gr gU D

121 and the noise temperature ratios T T L S t T (8.40) d d Here g~ denotes an external load at co with noise temperature T. Since g is a dummy load, it will be adjusted to optimize Fr The various factors in (8. 38) can be expressed in terms of the normalized quantities in (8. 39) and (8. 40). For example, the gain ratios can be found by calculations similar to those leading to (8. 12). This yields Gku Pkor Gr xw (8. 41a) ru k uu Y wOr( l-po)2 u^a~u -_ — ~ r _ - 0(8. 41b) Gru x Pu Wu Perhaps it should be noted that z does not appear in (8. 41a) because, at a dummy terminal like wn, g is considered to be the source conductance, and not g. The noise temperature ratios are given by m- gda Td + Pk T - ( ) T - = 1i + z(s-1) (8. 42a) T-d (gd + gL) Td T gdu Td+gL TL -u _ - T+g T = i + y(t-1) (8.42b) -d gdu + gL d which can be used to evaluate the bracketed terms in (8. 38). For the two cases, it follows that Td gdu T d matched output rT TgL ( y ) case (8. 43a) T0gL TT case u - Td + infinite gain TTe rf- Ttesmfo 8 38 io t) ife gi c an la (8. 43b) Tor ty case Therefore, the same form of (8. 38) is appropriate for the infinite gain case and for the

122 matched output case. Setting t = 0, one obtains the second from the first. The following single expression for F now results. ru Fr 1 d l-x c r P [l+ z(s-1)] r(1-p)2[1+y(t-1)] (8.44) F = 1 + pT - (8.45) ru T- x cited. 8.2.1 Noise Figure for Infinite Gain. For the infinite gain case, the appropriate condition is I + Pu - Pk = 0 (8. 45) which can be expressed in terms of the same fundamental quantities that appear in (8. 44). With the aid of the identity in (8. 14a), together with (3. 29), pu and p2 are given by - r(l-x) (I-y) p = (8.46a) ur r(-X) (1-z) p (8.46b) c w;Rr The quantity Rr in (8. 46) appeared previously in (4. 32b). It has the value R -r PC (8. 47) r: d kCd The high gain condition in (8. 45) now becomes r _ r (w) (1-y + L, Uc Q Z (8. 48) w - (1-x)(1-z) + -) (848) but = o + 2c. Therefore, letting wcr/ = a, (8. 48) becomes a - (x-z) r + - (1 +2a) (8. 49)

123 Before attempting to solve this equation exactly it should be noted that ow and wu must be much larger than wr if small noise figures are to be obtained. Also, if the noise figures are not small, their optimization will not be of critical interest. Therefore, asserting that (a) is small, the following approximate solution for (8. 49) is obtained R r a (1-x)(y-z) if a << 1 (8.50) Since the denominator of (8. 50) must be positive and cannot exceed one, two conditions are necessary for (8. 50) to be valid, namely R << 1 and z < y. However only the first is an approximation in this case, since z < y is necessary for p > 1. By (8. 46b), an alternate form for (8. 50) is Pk Rr a (i-x)(1-z) and therefore 1- z 7r(8.51a) P~ y-z Pu W (li-y) y-z (8. 51b) Pu cWu (1-z) y-z The importance of (8. 50) and (8. 51) is that p does not depend explicitly on x when (a) is small. Therefore the noise figure in (8. 44) can be written -ru T x(1-x) where a = R {l + y(t-1) (1-y) 1 + z(s-1) (-z) (8. 52b) r \ (y-z)2 (y-z) J Since (8. 52a) has the form of (A-5), its optimum value will be given by (A-7a) whenever p, and hence a, are independent of x. It is evident from Fig. A-I, that a should be as small as possible. Considering first the factor

124 1 + z (s-1) 1-z its minimum value occurs for z = 0, which means g = 0. Hence any excess loading at wo is harmful. When p is given by (8. 51), as it will be in most cases of interest, z = 0 also minimizes this factor. Therefore, the minimum value of the second term in (8. 52b) is (1/y)2 which occurs for z = 0. In this case, =R [1+y(t-1) + 1 (8. 53) r I 1-Y y2 The latter quantity will be minimum with respect to y if y3 2 _ _ -_ <2(8. 54) (1_y)2 t The solution of this equation is shown in Fig. 8. 5. Also shown is the corresponding value of a/R. These values can be used in (A-7b) to find the effective noise temperature of the converter. =eru = 2[1 + (8. 55) Td However, it is convenient to relate this optimum noise temperature to the frequency ratio u 1 + - (8. 56) c r a Or, by (8. 50) u r A = [2R + y(1-x) (8. 57) Eliminating a and R between (8. 57), (8. 55), and Fig. 8. 5 yields the minimum noise temperature and optimum frequency ratio curves in Fig. 8. 6. For o / or > 2, the accuracy of the approximation in Fig. 8. 6 increases rapidly. As an illustration in the use of these curves, when wco/w is large ur

125 100 IO0 0.1 0.3.5.7 1.0 Fig. 8. 5. Solutions of (8. 54). (Tru) 2,/ (8.58) Trd /opt When t = 1, Fig. 8. 5 yields y = 0. 67 and a = 5R, so ru (T) 2J5RW (8.59) Td /opt (8. 59) This result can be compared with the corresponding lower sideband converter. For the same output efficiency factor, z = 0. 67, (5. 32b) yields

126 100 0 1o \\ 10 \ / \WU Wr 1.0 OPTIMUM NOISE TEMPERATURES 0.1 I I.01 0.1 1.0 - 10 Rr Fu~ Fig. 8. 6. Optimum noise temperatures corresponding to infinite conversion gain between w and co. ~Trf~r u T 2t r (8 60) T u /opt which is some 20 percent less for equal values of R. r On the other hand, comparing with the pure upper-sideband converter, Fig. 4. 5 shows T ru. 34 d for R 0.01, = 10 r Wr G = 6 r ru For the same R, (8. 59) yields T u = 0.43 Td

127 but here G is infinite. So this particular technique for adding the lower sideband greatly enhances G with only a small increase in noise temperature. ru There is another interesting difference that can be noted between the pure lower-sideband converter and the present case. The noise figure of the former decreases uniformly with the output efficiency factor, while the latter calls for a particular output efficiency according to Fig. 8. 5. In practice, however, this distinction would probably disappear. In discussion conversion between wr and op in Chapter V, no effect from cw was assumed but with a real varactor, pure lower-sideband conversion is impossible. Considering only the varactor loss gdu, an increase in noise temperature would occur that would require a specific choice of output efficiency for minimization. Of course, it may be found that reactive detuning at wc reduces the effect of grU, but in any case cu will make some contribution to T such that the final comparison of lower and upper sideband conversion will yield nearly identical noise temperatures. Therefore, the most outstanding difference between these two cases is not their noise temperature, but rather their ease of realization. A precise adjustment at wp is required to yield infinite gain from or to ow, while that from wr to wc is relatively insensitive to the suppression of o. 8. 2. 2 The Noise Figure, Subject to (8. 16). The noise figure relation in (8. 44) can be optimized only when some constraint relates p and p. In (8. 16b), a constraint is given yielding optimum gain with a matched output, namely P, = (2x- 1)(1 -p) In this case, it has already been noted that the noise figure is given by (8. 44) if t = 0. Therefore, Td -x WP p[l+z(s-l)] r F = 1 +- - + — + r (8.61) ru T- x oox XPu Wu But R r r xpU U x(l-x) (8. 62a) xu u = x

128 1.0 100 / 0.75 CONVERSION / P' 0.5 GAINS / / / / / / -,- / /Io =t I a/ I \p i=0. 75' |I'/./ \\ NOISE I ^\c pi-0.5 TEMPERATURE I, \' R=O'l -- 0.01C~ SC\ pO_ \ —— ^R < 0.01 PAs075 - P =0.5 ~~.~~=~~~ R 0.01.001 I I 0.I.01 0.1 1.0 10 R (U- ) Fig. 8. 7, Effects of the lower sideband on T and G ru ru in the matched output case. and cr po k r / 1 \ co~PA = ^ _A /-L _\(8. 62b) \ u/\ /U 1-2 u ru which facilitates the evaluation of (8. 61)o Curves of TrU are shown in Fig. 8. 7. These were obtained by eliminating x between (8. 61) and (8. 62). An interesting aspect of these curves is that x increases with p for fixed wo co, and ow'. In (8. 61) this causes the first term to decrease with p and the second two terms to increase with p. As a result, improved noise figures occur over certain ranges of the abcissa. For example, if R = 0. 01 and ou/Wr = 16, Tr = 66 for p = but T =.57 for p = 0 5. Yet, in the latter case, G is some three times larger. The noise temperature curves have been superimposed in the gain curves to show their correspondence. Because a maximum value of (wUc/o) R exists whenever ur r

129 pk / 0, the range of R is limited. For p = 0. 5, Rr < 0. 2 is necessary. However, for some range below this value of R, an optimum noise temperature exists, that depends on Pk 8.3 Down-Conversion from co to cW u r In an earlier discussion, in Section 4. 3, down-conversion from co to co was u r shown to be impractical, but the lower sideband was absent in that treatment. The lower sideband will now be introduced and it will be shown to greatly enhance down-conversion gain. The first step is to calculate G from (8. 1). With the aid of (3. 15b), one obtains ur co. 4 a2 G - r gSL ru Yp 1 (8. 63) ur Wu YrYu + cor 2p 2 (u - co Yu/Y *) By comparison with (8. 8), it is interesting to observe that fru G = Wr- G (8. 64) Ur \ co/ ru The proportionality just noted means that both up-and down-conversion have the same bandwidth [see (8. 27)]. Consequently, the down-conversion gain-bandwidth product is cor/w smaller than that for up-conversion. Being less than one, this ratio is one of the disadvantages plagueing the down-conversion mode. The result in (8. 64) applies even with oQ shorted, and it was obtained for this case earlier. However, a point of interest arises because the derivation in Section 4. 3 made use of a symmetry that does not exist when c is present. Without cow, the basic equation for relating or and ou is (4. 1), which is symmetrical in these variables, but the more general expression is (8. 1), which is not generally symmetrical in wr and co. Therefore, though conversion gain obeys the scaling relationship in (8. 64), other properties of up-and down-conversion may be found to differ by more than a simple scaling. Using the notation of (8. 11), (8. 63) can be written Wr r 4 xyu ur - -L- pu j (6 G (8. 65a) UI~ OU 1(1 + pu - PQ)N

130 where again gr gu gl x =, y = and z = (8. 65b) gr gu gl However, the source and load are now interchanged (relative to their positions for up conversion) so it follows that gr = gL and gu = gS, while gf is a dummy load. As anticipated, G is no longer limited to the frequency ratio w /Wu, but ur r/ u can be arbitrarily large if p - 1 + Pu. Therefore, by introducing the lower sideband, one limitation of simple down-conversion is overcome, namely that of gain being less than unity. However, high gain again occurs at the expense of negative reflected conductance at each terminal, so no improvement in gain-bandwidth product results. Another problem that previously characterized down-conversion was a high noise figure, so let us now consider the influence of co on the down-conversion noise figure. It will be sufficient to consider the noise figure associated with infinite gain, which (using the form of 2. 48) can be expressed as follows: gdu Td fGr Z Grr Tr gr u =1 + + + T (8.66) ur go T G T G Tcr S 0 ur o ur o0 L Here T is the effective noise temperature of the wr terminals, and by (8. 40) it follows that Tr = [l+x(t-1)] Td T = [1 +z(s-1)] Td (8.67) Again using (8. 1) it can be shown that C) r xp k G =4: -4 p r (8.68a) Gr o 1 + PL( U - P) G = (8. 68b) rr (1 + Pu -pf)2 Therefore (8. 66) becomes Fur = 1 + Td y + u + ( ] (8.69)

131 Two important aspects of (8. 69) are the frequency ratios wc /wQ and w u/Or' which both exceed one. Because of these ratios, it will be shown that Fr is invariably larger than Fr ru Further examination of Fr is aided by the following identities, which are obtained from (8. 46) 1-z 0u U (<^- e(8. 70a) wo R u u puCTr 1 (8. 70b) The quantity R is the same as previously defined in (4. 38), namely R = (-) R = ( (8. 70c) R is useful for normalizing down-conversion noise figures because it only depends on: (1) input frequency, (2) the varactor, and (3) the pump level. Using the identities in (8. 70), F in (8. 69) now becomes ur F To+ - Y y(-y+ (8. 71a) ur T ly y(l-y) where R 2 R u u a = (1 - z)[1 +z(s-l)] + 1 + x(t —1)] (8. 71b) As written, (8. 71a) has the form treated in Appendix A (with the variable y substituted for x), but there a is assumed to be independent of y. Now, a depends on y, because z, y, and wu/c)r are related by the high gain constraint 1 + Pu - p = 0, which [by (8. 46)] can be written in the following form: W w \ / R \ (1-z) = () (i X + (1 - y) (8. 72) Therefore, the optimization of (8. 71a) will differ from that of other noise figures, because the variation of a with y must be accounted for. It will be expedient to assume that Ru, x, and t are specified, so the second

132 term in (8. 71b) will be constant, that is R 1- [1 + x(t-1)] = N (8. 73) First consideration will then be given to the case where all noise sources are at room temperature. Using (8. 72) in (80 71b), the latter becomes: s=t=l wo R / a = (l-y) lx (8.74a) and F To 2 a 2( l-x) which is minimum for y 1 - ------- (8. 75) ~opt 1 2(1-x) + 1 + R u It is interesting to note that Yopt is independent of frequency ratio in this case, and therefore the optimum Fur decreases with W p/! V. Hence, for a given wu, the lowest noise figures will result if wr/Wu <K 1. In this case, the effective noise temperature depends on R /(1-x) as shown in Fig. 8. 8. An interesting difference now arises over the previous noise figures considered, in that idler cooling can reduce the noise figure. To demonstrate, consider an extreme case where the external load at we is cooled to absolute zero (i. e., s = 0), and then s=0, t=1 wc R a = (a+ 1-y)2 + N; a = r r (8.76a) w 1-x u from which Fr = 1 + Td [F N( + a -] ur - T0 T Ly\~a/ 0j (817Gb)

133 100 30 - t 20 zj 0o ~.o^~ _______~Ru I-X 1.00.1I 0.01.02.03.05.07 0.1.2.3.5.7 1.0 2 3 5 7 10 R u I-X Fig. 8. 8. Optimum noise temperatures for down-conversion from ou to or, with and without idler load cooling (at woe), and with a room temperature load. The basic requirement for these optimum noise temperatures is Wr << Wu. The latter is minimum for opt 1 J 1- - 1 (8.77) Yopt 1+ 1 2(1+a) N + a2 Again the optimum noise figure increases monotonically with c)r/ WU, through the factor a. If a < 1, then s =0: (F ) pt 1 + 2+ Y - t2 (8. 78) which is plotted in Fig 8. 8.opt which is plotted in Fig. 8. 8.

134 For other values of s and t, curves similar to those in Fig. 8. 8 can be derived. In general, raising t moves all curves to the right, while s influences the lower portions of each curve. Lower sideband cooling can improve the down-conversion noise figure because lower sideband noise is less amplified than output noise (at w ). The ratio of these two contributions is P e r//, which is less then one if cor/'w is small. Therefore, the optimization obtained here requires relatively heavy loading at ow and wou, but light loading at the output frequency cr. Cooling the load at Co will then reduce the noise figure. In practice, the input and idler frequency loadings are adjusted for high gain, while the output is loosely coupled to keep the noise figure down. This procedure would require that the output be followed by a power amplifier with high input impedance. It is interesting to compare the optimum down-conversion noise figures with and without w, the latter being taken from Fig. 4. 5. For the case where R = 0. 1 and cu/w = 10 are specified, the results are shown in the following table: u r an0 | eow unshorted; x = 0. 1 shorted i(Fig. 4. 5) gi at room temp. gk cooled to 0~K T = 1.1 T 2.2 T =1.2 ur ur ur The noise figures that accompany the introduction of w are somewhat larger than those without wc, but it should be remembered that the former are accompanied by huge gains while the gain is less than one with CL absent. In addition, cooling the ow termination can reduce the difference between these two noise figures to something quite negligible. Only one other technique has been considered in which infinite down-conversion gains are possible. This would be in the pure lower-sideband converter where wr > co. The noise figure in this case is given by (5. 27a) which has the optimum value in (5. 32a). However, the latter optimum calls for the particular cor /Wl ratio in (5. 29a). This ratio exceeds one only if Rr/(1-z) > 0. 2. Therefore, the optimization leading to (5. 32a) is not too practical for down conversion, since the latter often involves specified input and output frequencies. In this case, a definite minimum noise figure exists, which is

135 Tdw r F > 1 Td + (8. 79) Therefore, if the input frequency (wr) to output frequency (w ) ratio is 10 to 1, as in the previous illustration, F will equal 11. It can be noted then that double sideband operation offers significant improvements in the down-conversion mode. It is an interesting aspect of this down-conversion analysis that high w /Wr ratios, are required, which means that wu, w' and wp will become very closely spaced. Consequently, all three frequencies may fall within the same resonance. This would be a practical mode of operation, since it has been shown that w and wQ require approximately the same loading when minimum noise figures are obtained. Further discussion of dependent sideband loading will be found in Chapter IX. 8. 4 Conclusions It has been shown that double-sideband reactive mixers have some advantages over single-sideband reactive mixers, two forms of up-conversion gain increase are possible, but both are accompanied by negative reflected conductance at the input. One form offers infinite gain and negative output conductance, while the other offers finite gains with a matched output. In the second case, an improvement in noise figure is possible, but in the first case a slight noise figure degradation occurs. In both cases, the gain-bandwidth product may reduce by about 30 percent with the gain increase, but it is possible to increase the gain by about 6 db without reducing gain-bandwidth product. One general advantage that can be cited is the opportunity to control gain between w and ow by varying the loading at a dummy frequency cw. A significant improvement in down-conversion noise figure has been noted. Noise figures of slightly over 3 db are possible, with essentially infinite down-conversion gain, and for a wide selection of operating frequencies.

CHAPTER IX DOUBLE-SIDEBAND MIXING WITH DEPENDENT SIDEBAND LOADING In the configurations considered in previous chapters, up-conversion was found to be the most useful reactive mixer mode. An optimum sideband-to-signal frequency ratio was found to characterize this mode, but it varied according to varactor loss. For low signal frequencies or high varactor quality, the optimum frequency ratio is very large, with the result that the pump and sideband frequencies become very closely spaced. But an alternate limitation on frequency ratio then arises due to the difficulty in filtering. As an extreme case, the sidebands may fall within the pump resonance as illustrated in Fig. 9. 1. When this occurs, Yp and Y are no longer independent quantities, and the character of double-sideband conversion changes radically. At first glance, the configuration in Fig. 9. 1 appears useless since sideband power would tend to be absorbed in pump internal conductance. However, as shown in Fig. 9. 2a, the sidebands and the reflected pump signal can be separated from the pump generator with the aid of a circulator. The total reflected energy that emerges from the circulator thus constitutes a modulated signal. The useful properties of this signal will now be investigated. For frequencies near wp, Fig. 9. 2a is equivalent to the tuned circuit in Fig. 9. 2b, which will be assumed resonant at wo for the case of basic interest. Then wo and wu p u will be equally detuned so that Y Y (9. la) u gp + j rp (9. lb) where p = gp + gdp 137

138 MIXING SPECTRUM 0 WOr WJ W p Wu --- Fig. 9. 1. A limiting case of double-sideband mixing that occurs for c/ p < < 1. It is assumed in writing (9. 1) that gp is the characteristic admittance of the circulator, so the varactor will see g at the circulator terminals. The first step in evaluating this special case will be to determine the type of modulation produced by the wo and wu sidebands. In general, the total modulated waveform can be written in the form V (t) = R e jP + V - jt V+ -je (9.2) ^*" {p p where the sideband voltages can now be found from (8. 1) and (9. 1). By direct calculation, the latter yields -ji y V Vu p r (9. 3a) u Y u -jw YV V = Cp r (9. 3b) Y u But, recalling from (2. 24b) that yp = a2Vp/2, the sideband voltages can be alternately expressed in the form V V u — j M u = -jwM V -jolM* (9. 4a) P P where a2V OM r J M M = r IM e (9.4b) u

139 lip <VARACTOR 9gp (a) gp > C dp r (b) Fig. 9. 2. (a) A basic technique for isolating the sidebands from the pump in Fig. 9. 1; (b) the high and low frequency equivalent circuits of (a). Substituting (9. 4) in (9. 2) now yields V i(t) = |VpI [(1 + 2wr|M|Isin 0)2 + (2w p|IMcos )2] cos(Wpt + 0 + 0p) (9. 5a) where 0 w t + 0 (9. 5b) 0 = -tan 1 1 M+ 2rM sin 0 (9. 5c) 1_1 + wrjMj sin & Vp = |Vple P. (9. 5d) Since M is proportional to the signal voltage, a small signal assumption can be used to make |MI arbitrarily small. Also, w >- Wr will be assumed, and together these assumptions render the fundamental nature of (9. 5a) and (9. 5c) observable by reducing them to the following simplified expressions: 0 -2owp MIcos(wrt+ 0M) (9. 6a) Vm(t) |VpICOS[Opt + 0 - 2Wp |M|coS(Wrt + 0M)]. (9. 6b)

140 Three observations can now be made: (1) The first-order effect of the signal at cr is to phase or angle modulate the pump voltage. (2) The angular deviation of this modulation is a2V 2 r 2 2wiM |M 2 | — u (9.7a) p (9.7b) P P where 2yr = a V is the amplitude of equivalent time varying r 2 r r capacity at co. [See (2. 30b).] (3) The maximum angular deviation occurs as wc approaches zero. Returning to the expressions for the sideband voltages in (9. 3), the origin of phase modulation can be attributed to the common factor j. However, the fact that angular deviation is proportional to the signal voltage through the term yr suggests a physical explanation for this effect. The signal voltage, however small, will vary the capacity of the nonlinear element; and if this element is part of a resonant circuit tuned to p, these capacity variations will phase modulate the pump voltage. It would seem perhaps that this effect should be masked by nonlinear effects from the larger pump signal, but the analysis in Chapter II refutes this notion. As stated in Section 2. 4, first-order mixing can be alternately regarded as pump acting on signal, or signal acting on pump. For double-sideband configurations, the latter case is depicted by (2. 33a) with y2p = 0, and since the pump is the only source in this case, (2. 33a) becomes 0 Y2 jco r 0 " V l j~pY,= jp Yp cr V 0 0 j yo Y Vj (9.8) which attributes all modulation to time-varying capacitance at r.

141 The formulations in (9. 8) and (8. 1) are effectively equivalent because V y rV y r(9.9) V y P P but they express the mixing process in different terms. The relative advantage of these two methods depends upon how the sidebands are used to form the output. For example, in the analysis in Chapter VIII, the output signal was a single-sideband filtered from the pump, so no advantage would result from retaining the pump in electrical form in this case. However, when the pump and sidebands together form the useful output, as in the present chapter, the formulation in Section 2. 4 is more convenient. When the constraint in (9. 1) is included in (9. 8), the sideband voltages are identical to those in (9. 4) as expected. In addition, however, (9. 8) yields the effect of signal on pump directly, through the relation p1 + 2W.2IM2 (1 t d (9. 10) in Vp p P II The corresponding relation from (8. 1) is? 2 y /2 Y Y + p (9. 11) r. r Y in u which is a special case of (8. 2). It indicates the effect of pump on signal under the conditions in (9. 1). In (9. 10), the term 2w M2 g (gM(1 P = P P refl (9.12a) is the reflected conductance at p due to the modulating signal at wr, while the corresponding reflected conductance in (9. 11) is 2w2 y V 12b) (g) r g - 22 g (912b) refl IYu2 P r V Therefore, in this mixing process the relative power contributions by the pump and signal are given by the ratio

142 pump power Vp (prefl 3) signal power V (gr)refl In view of this enormous potential gain, it can be anticipated that immediate phase detection of the waveform in (9. 6b) will recover a well-amplified version of the original signal. This point will be discussed further in Section 9. 1, but first another curious aspect of this phasemodulation process should be noted. In (9. 4), the two sidebands have slightly different amplitudes such that V U 2o V 1 + r (9. 14) VP cooWp Since this asymmetry vanishes as wr approaches zero, it is apparently responsible for the terms in (9. 5) that involve wr. One such term appears as incidental AM and the other appear: as phase distortion. Both of these distortions are negligible when c << c. The sideband asymmetry noted in (9. 14) has its greatest influence at w. In Section 7. 2, the Manley-Rowe equations were applied to phase-like modulation and there the tendency toward instability was found to be very sensitive to sideband asymmetry. In the present case, the ratio of the two sideband powers is W~ (= w)2, (9. 15) which satisfies the stability condition in (7. 7). In fact, using (7. 5) and (9. 15) together, one obtains the following sideband-to-signal power ratio (W W/) wc2 co2 2 u p r p W 2 Y"2 2 (Wu'+ (9. 16) Wr r r which is just the negative of the unstable ratio corresponding to pure phase modulation. Therefore, while the sideband asymmetry noted in (9. 14) has a more or less negligible influence on distortion, it has a useful influence on stability. Further evidence of this appears in the derivation of Yr. in (9. 11), where the sideband asymmetry is seen to be just sufficient rin for the positive reflected conductance from the upper sideband to just overcome the negative conductance from the lower sideband. The net reflected conductance in (9. 11) is positive, but small.

143 0 — (aI! TO POINT A PUMP / PHA IN FIG. 9.2 IN /PHASE MODULATED - PUMP L_-~ PM TO AM CONV. PHASE AM DETECTOR DETECTION Wr Fig. 9. 3. A detection system for recovering an amplified version of the modulating signal. 9. 1 Circuit Properties It is now evident that a phase-modulated signal emerges from the circulator in Fig. 9. 2. Hopefully, an amplified version of the modulating signal can be recovered by placing a phase detector at its output port, which is illustrated in Fig. 9. 3. Since (9. 16) suggests power gains in proportion to pump frequency squared for this case, it differs considerably from the cases previously studied, where power gain was proportional to only the first power of pump frequency. In deriving (9. 16), the sideband powers were added as if they were incoherent signals. This step is satisfactory for merely estimating the power gain, but once it is recognized that the sidebands add coherently to form a phase-modulated signal, a more precise measure of gain is required. Helpful in this respect is the classical view that the two sideband voltages can be regarded as vectors rotating with respect of V at angular frequency w r. The phase envelope is then formed by the total projection of V and Ve in quadrature with V so the peak of the phase envelope will occur whenever the sidebands are in phase. Hence, a transducer gain for phase modulation can be defined by (Maximum quadrature component) 2 i of Vu + Vk with respect to Vp GpM = 2L Au l____ P / g p (9. 17a) PM 2 (Available source power) g (9. 17a) at o r Or, in the manner of (3. 15b), P 4 gr g M V pn \ 17b) c - ~r p P /Maximum imaginary component/ (9. PM jYrin 2 Vr of (Vu +V)/Vp

144 The phase of (V + V )/V, as found from (9. 4), is somewhat arbitrary since the phase of V is arbitrary. However its maximum possible imaginary component is (wU + ) IM I =2w 2 IMI (9. 18) which occurs when the instantaneous phases of V and Y are equal. As could be expected, (9. 18) equals the angular deviation in (9. 7a). Using (9. 4b), (9. 9), and (9. 11) in (9. 17b), the conversion gain by phase modulation becomes 16 grgp c 2 (9 19 PM YrYu+ 2wr 2 plP |2 Of course this gain will be useful only if the phase modulation sidebands can be converted back to wr efficiently, which calls for a good phase detector. But it will be shown that phase detection invariably introduces loss, so if i72 is the fraction of phase envelope power that is actually converted back to wr, the over-all transducer gain will then be r = 2GpM (9.20) The numerical value of 72 will be discussed later, but for now it can be stated that 0. 1 < 72 <. 5. A more detailed discussion of phase detection efficiency will be taken up in Section 9. 3. For a given source and load conductance, both (9. 19) and (9. 20) are maximum for wo = 0. Therefore, an untuned input will be assumed having the form Y gr ( j r (9. 21) \ r where or = g /Cr is the bandwidth and Cr is the total input capacity at or (i. e., C = C + stray). A low pass amplifier characteristic now results with the following amplifier characteristic /pr/3p/ o ^p / G (o ) = 872kP2 (j ) (1+ 2r) (9.22) PZPP~rr r 2 (po) n 2 1+ (k 2 - r(ii Y vr'p/ \r p

145 where k 2 is the pumping coefficient and in this case p k 2 = 2 //CrC p'p /r p There is a resemblance between the transducer gains in (9. 19) and (9. 22) and the power gain in (9. 16), since both are proportional to W 2, but the former two remain finite p as w r approaches zero. This difference occurs because the reflected input conductance at w r vanishes with w, so no signal power can enter the converter when Wr =0. Still, the maximum transducer gain occurs when wr = 0 and has the value r2 k 2 2 _ 8i k wo G P- r p (9.23) 9. 1. 1 Bandwidth and Gain-Bandwidth Product. To calculate the bandwidth (3) of the gain in (9. 20) it will be assumed that wr is much less than cw within the pass band /3 (i. e., <3 < c ). Direct calculation then yields 1/2 3 /3r/3p -D+ V2D: +1 (9. 24a) r Vr p 1-k 2 where D r -1 —2 ((r +' ) -1 (9. 24b) 2(l -kp2) p ^r Inspection of these relations shows that 8[^ +^'1 (9. 25a) ~ 0 if /3p - >3 > 0 (9. 25b) The maximum error in (9. 25a) is less than 30 percent, and it occurs when /3 = and k 2 0. r p p Exact solution of (9. 26) are plotted in Fig. 9. 4 for k 2= 0 and 1. For other values of k 2 p p lies between the two curves plotted. Since the two curves in Fig. 9. 4 are separated by no more than 30 percent, it can be concluded that the conversion bandwidth is roughly equal to the lesser of the two termin

146.70.60 t C I.50.40.30 - A 0.20 - 2 3 4 5 6 Pr +'P Fig. 9. 4. The bandwidth of Gr vs. the sum of the circuit bandwidths, normalized to r3 p. The two curves shown are upper and lower bounds. al bandwidths. This result can be explained by considering the input and output admittances. The former was given in (9. 11) and the latter can be found by substituting (9. 1) into (8. 3) and (8. 4), which yields ) co u I Y Y = Y + W ur ul 2 u (9. 26a) uin u YrYu- cowrw)ypl2 Y - WrW yp u (9.26b) in Yr + uwrculyp l From the point of view of bandwidththe importance of these admittance relations is the ratio of the second term to the first, which shows the ratio of reflected to fixed admittance. In (9. 11), the reflected admittance at or has the components: ) [) (gr +' r (gib) =13r/3p;+ jr/ g + 3 b.(9.27) Note that p is actually half the bandwidth of the pump tank, but it is referred to here as the pump bandwidth.

147 The key factor here is k 2 which, in Section 9. 1. 3, will be shown to be much less than one. Therefore (g ) and (b r) are much smaller than g and b respectively, and the bandrefl r refl r' width of the co terminals (gr/Cr) changes very little when the pump is applied. Investigation of the sideband admittance in (9. 26a) and (9. 26b) shows the same general trend, that Y Y (9. 28a) u. u in Yc Ye. (9. 28b) in However, it should be noted that Y / Y* even though Y. In fact (9. 26) shows u. k. U in in Y (w) =Y* (-w) (9.29) U. r r in in Some confusion may arise about the significance of Yu and Y, when they are in in part of the same resonance, but they apply on different sides of resonance, so they can be regarded as separate quantities. As one consequence of (9. 29), frequencies above wp experience positive reflected conductance while those below see negative conductance. Nevertheless, these reflected conductances are numerically small compared to gp, so their asymmetry is relatively unimportant. Now that the bandwidth has been determined, a related quantity of basic importance is gain-bandwidth product. By (9. 23) G = 2 -Jw - - * (9. 30) r fi =13rA/cp It is of interest to optimize (9. 30) but a number of practical factors can influence the result. For example, if the varactor is lossless, -3 can be chosen independently of c. Then by p P Fig. 9. 4, f/ J3 /3 is maximum when 3r = 3p which leads to the relation /Gr d / 1.83 ow k 7 (1-0. 283 k +...); k < 1. (9.31) V ro PP P' p Since this gain-bandwidth product is proportional to w), the lossless case emphasizes the

148 value of a high pump frequency. However, varactor loss eventually limits the usefulness of raising the pump frequency, as will be shown in Section 9. 1. 4. 9. 1. 2 A Numerical Example. To illustrate the great promise afforded by this double-sideband configuration, a numerical example will now be considered. Since the gainbandwidth product in (9. 29) is proportional to pump frequency, this suggests that the highest possible pump frequency should be selected. Today, it is well within the state-of-the-art for varactors to be pumped at 10 kMc, which means that a gain-bandwidth product of 1000 Mc is possible if kpk can be made made as large as 0. 1/1. 83 = 0. 055. Later it will be shown that qkp = 0. 055 is easily realized, so let us continue the present example by designing an amplifier with the following characteristics: Midband Gain, G = 20 db (9. 32a) Bandwidth _ = 100 Me (dc to 100 Me) (9. 32b) Together these correspond to a gain-bandwidth product of 1000 Mc, which in the present case requires /7k = 0.055. (9. 33) p Now, assuming /r = /p as suggested above, and k 2 << 1 as will be justified in the next section, Fig. 9. 4 yields /3 = 3 = _0_4 = 1.56 /3 = 27 (156 Mc) (9.34) 3s p 0.64 for the pump and signal bandwidths. As expected, neither the pump nor signal terminal admittances are much affected by pumping. For example, applying (9. 27) to the present case gives k 2 Y __ _, 0 0552 r (Y)efl Y ( ) 2 - r 012 Y (9.35) rrefl- 2 r 07 2 r where 772 = 1/8 has been assumed. Therefore, the input admittance is essentially that of the varactor itself, which is purely capacitive for frequencies in the cLr band. It is evident that double-sideband reactive mixing provides a technique for dc amplification with extremely wide bandwidths, from dc to UHF. Although no standard termi

149 20.17 10-\ I0 (rL) IN MC r2 7r Fig. 9. 5. Gain characteristic of the example amplifier in Section 9. 1. 2. nology exists for this broad range of frequencies, it will be termed an extended video range and the configuration under discussion will henceforth be termed an ultra-wideband video amplifier. The frequency response of the amplifier in this example is shown in Fig. 9. 5. 9. 1. 3 Physical Limitations on k 2. By (9. 31), the gain-bandwidth product is p proportional to kp, so it is important to consider possible limitations on this quantity. According to the definition in (9. 32) kp2 = p y2/ rCp (9. 36) where C and C are the total circuit capacities at wo and co respectively. Therefore, in the r p r p notation of the general Taylor expansion in (2. 11), Cp = al + stray or filter capacity a V and y = 2, P for which p< 2p. (9 37) p2 2a1~j.P I'(9. 37)

150 However, this relation is valid only over a range where the varactor capacity varies linearly about the bias point, and typically this is true only when V - V <, where V is the bias P 2' voltage. As previously noted in (2. 10), varactors are characterized by relations in the form q(v) = Q(1 - v/A)1-; 0 < a < 1 (9. 38) In this case, (9. 37) becomes /|V \-2 2 /V2 k 2 < (o)2 ( A ( (2) ( ) (9.39) where the latter approximation is valid when A, the contact potential, is small compared to |VpI and V. An upper limit on |Vp| also arises because a varactor is basically a diode and excessive conduction takes place when Vp = V + A V. Therefore, the upper limit on k 2 as predicted by (9. 39), is (a/2). In a typical case, a = 1/2, and then k 2 7 0. 06. However, for |Vp| V, the accuracy of (9. 39) deteriorates because the formulation behind this relation is only concerned with first-order interactions. When Vp is large, higher-order interactions also contribute to the circuit effects being observed, but this can be partially accounted for by replacing yp by the Fourier component of time-varying capacity at w9. p 27T, ]j2d t p? = SO d e d(pt) (9. 40a) where: jw t v = V + Re(V c P) 0 p This expression for yp is proportional to V for small V /V, but deviates from proportionality P P p as the latter increases. Similarly, with large pumping signals the average varactor capacity deviates from the bias value, and is given by the relation C 2 s [(dv) (p] *d 2 (9.40b) 0 L''vo

151 1.4 1.20 2_C 0 0.2 0.4 0.6 0.8.2 0.4 - 0.08 0.3- / 0.06 0.2 -n / 0.04 0.1, BY (C) 9.39) 0.02 0 0.2 0.4 0.6 0.8 1.0 1.2 The integrals in (9. 40) have been evaluated in Ref. 37 and the results are shown in Fig. 9. 6 for a = 1/2. Also plotted is the upper limit on kp2, namely (yp/2C)2 Based on Fig. 9. 6, the value 0. 1 will be taken as a practical upper limit on k 2 in the sense that k2 = 0. 1 should be approachable with large pump signals and minimum stray or auxiliary capacity across the varactor. Of course, filter of bypass capacity that may shunt the varactor at one or more of the operating frequencies will reduce both k 2 and the circuit bandwidth, and these reductions produce opposite effects on the gain G. 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) r/27T = 1600 Mc. If the desired amplifier bandwidth is significantly smaller than 1600 Me, the extra input circuit bandwidth in this case would unnecessarily reduce G; but if /3 is reduced by adding extra capacity in the video r r~~

152 circuit, k 2 would reduce simultaneously to nullify the possible advantage to be derived from p this 3r reduction. Therefore, to make optimum use of a varactor for phase modulation, the video bandwidths must be controlled by inductive source transformations. To continue the illustrations above, a stray capacity of 1 pf will be allowed for, making C = 2 pf, and a broadband 2:1 step-up transformer will be assumed to change the source impedance to 400 ohms. Then 3 /27T is reduced to 200 Me, with an accompanying transducer gain improvement of 6 db. (Note: the reduction in 3r accounts for 9 db, while the reduction in k 2 accounts for -3 db.) In this case, the importance of a high source impedance arises p because the video input impedance is large, as a result of wr being much less than w. Thus, phase modulation is particularly useful for wideband amplification with high impedance sources such as solid-state particle detectors or various photoelectric devices. 9. 1. 4 The Effect of Varactor Loss. As shown in (3. 30), varactor loss can be represented by a frequency-sensitive shunt conductance. At w(j it has the value a) "C 9 P (9. 41) dp Wd and therefore g gp + gdp (9.42) where g is the sum of all external conductance. As previously noted g would typically be the characteristic admittance of the circulator. The half-bandwidth of the pump tank now becomes 9p+ g dp - ^ ~- d^20- _ = + (9.43a) Op ~2C p +dcp where g g P ~~~~~~~~~~~(9. 43b) p 2C' dp 2C (943) Therefore, the minimum possible value of f3 is (/3) ~~/3 = p ( mm) dp 2W (9. 44) wmin th a which occurs in the absence of external loading.

153 The expression in (9. 23) for midband gain can be similarly modified to exclude sideband power lost in the varactor, which yields 872k 2 2 g 12 G = g L- +dp j (9. 45a) r p 9p + gdp or by (9. 43) 167/k pad p dp G = P d. (9.45b) r fr r (3p + /dp)2 Since the last expression is maximum for p = dp' varactor loss is best compensated for by choosing gdp = gp, which matches the varactor to the circulator and yields 4ij2k 2d G r - (9.46) /r Without a circulator, gp = gs gL (9. 47) and then (9. 46) must be reduced by the fraction of gp that constitutes the output load, namely g p /g. However, this step will not alter the significance of (9. 46). A final step in optimizing G is to minimize Ir in (9. 46). Using the approximation in (9. 25a), - 1 i 1' rr + p 3 + (9.48) 0 Or TP Os Op+ dp it suggests that cp be increased until /3dp and hence /3p, are much greater than r3. Then r3 will have the minimum value: and the corresponding optimum dc gain will be 4172k 2w (") d G( )m p d- - (9. 49) Gr max P which occurs for C 2 > > o,. The maximum gain-bandwidth product is now = 2 (9 50)

154 By comparing (9. 50) with (9. 31), the effect of varactor loss is quite evident. For example, if w = N T)Wd, then?7k must be at least N times larger with varactor loss then without to yield the same gain-bandwidth product. Considering again the example in Section 9. 1. 2, a varactor with C d = 27(105 Me) will be assumed. To retain 100 Me bandwidth with minimum pumping, let Cp = 3 /JC = 27(104 M). (9. 51) Then by (9. 44) dp = 27(500 Me) (9. 52) and if j3p = /dp' then p = 23dp = 27T(1000 Me). (9. 53) By (9. 48) /3 100 Mc - 10 M - = i (9. 54),3 100 Me 1000 Me 110 Mc So by (9. 46) G =4q2k 2 (10%) k 2 = 0. 1 P = 50 (17 db) for 2 = 1/8. (9. 55) Therefore, a gain-bandwidth product of 700 Me seems to be the upper limit when varactor loss is considered in this case. Still, it must be noted that the optimum gain-bandwidth product in (9. 50) is proportional to /f-. Therefore, if the bandwidth is extended to 200 Mc, G /3 will equal 1000 Mc for the same C)d, 2, and k 2 but the gain in this case is = 25 = 14 db. Also, it will be shown that 7'2 can be as large as 1/2, which would add 6 db r to (9. 55). A summary of how optimum gain varies with pump frequency is shown in Fig. 9. 7. It assumes the case where a definite bandwidth /3 is required. When the pump frequency is low, the condition t3dp = /3 yields insufficient pump bandwidth, so 3p must be increased. yilsisfiin up bnwdhso1p

155 100 P3p 10 NORMALIZED 1.0 _ TRANSDUCER \ GAIN NORMALIZED \ ~/ ~VOLTAGE GAIN \ 0.1.0 1I-II Fig. 9. 7. A comparison of optimum /Gr and voltage gain for fixed bandwidth and varactor resistance vs. pump frequency. Gr is normalized to 4kpnwd and the voltage gain is normalized to 2(yp/C) wd/l. 56 The optimum choice in this case is = p + 3d = 1. 56 (9. 56) as noted in (9. 34). However, as wc increases, 3dp increases in proportion to wc 2 and p p reaches the value (1. 5613/2) when cwp = 56cd. The latter is the highest pump frequency at which equal pump and signal bandwidths are optimum. For still higher pump frequencies, dp = Up must be satisfied for optimum efficiency, but then /p increases in proportion to c 2 p

156 By (9. 18), the midband voltage gain is voltage gain (9 57) ~p which is an increasing function of cp as long as ~p is constant. However, for wp > 1. 563wod, 3p increases faster than wp which reduces the voltage gain. At the same time, g increases p p p in proportion of w 2, so the output power remains constant for p > /1. 56/3c even though the voltage falls off. Therefore, the main advantage to be gained from increasing wp beyond the point where varactor loss can account for half the pump bandwidth is that fr can be reduced from 1. 563 to 3. In this region G increases asymptotically toward the value 4kp od' 9. 1. 5 Noise Figure. A calculation of the noise figure associated with the phase modulation gain GpM, or the detected gain Gr, follows the general procedure outlined in Section 3. 4. 2, which requires the specification of each external noise source. The main type of noise source that can be anticipated is that due to parasitic loss, which would be thermal in origin. In the co band, excess noise will originate primarily in cables and transformers, etc., since varactor loss can be neglected in the video range (e. g., below 100 Me, microwave varactors have Q's greater than 1000). It is actually in the pump tank that the varactor makes a fundamental noise contribution particularly when varactor loss is a significant fraction of the total pump tank loss. Therefore excess noise figures due to thermal noise sources in the pump tank will now be calculated, and the result used as a guide for determining when other, more arbitrary noise contributions, are significant. The effective noise temperature in the pump tank is T d gp pp(9. 58) gp + gpd where Td and T are the noise temperatures of the varactor and the auxiliary pump circuitry, respectively. Therefore, considering the pump pass band as drawn in Fig. 9. 8, each increment dw will experience an (incoherent) noise source 4kT g dco N 2\ - 2 (9.59)

157 di di I I j'^^ \ — ~^^ —dww coA oou r Up u Fig. 9. 8. Noise contributions from pump circuit conductance. Consider first the increment lying between ow and wU + dw. Two output noise contributions will arise in this band; one due to noise reflected from diN and the other due to noise converted from the image band which lies between wo and wc - dw. To evaluate these two noise contributions let Y = total tank admittance at w u. u in Y. = total tank admittance at o in Y k = transfer admittance from w to ow Y = transfer admittance from wo to w. -.~u u' The total output noise power in the original increment now becomes Idi_ gp IdiN 2pg P(Cor) dc = 2- + (9. 60) I uinl IYu I which expresses the output noise power distribution over the right half of Fig. 9. 8. But, as previously noted in (9. 29) Y (w) = Y* (- ) (9.61) u. r Y r. in in Similarly, it can be shown that Yu(or) = YQu(-r) (9.62)

158 so (9. 55) is equally valid over the left half of Fig. 9. 8 by letting cor be negative. The total noise power delivered to gp, which will be called Pp, is found by integrating (9. 60) over the complete range -oc < co < oc. If or << op and k 2 < 1, it is shown in Appendix B that P can be expressed approximately by 4kCC ~~2T I ____~~2 f dwCL T, pj1 ij 1 2/Lk 2 rc 2(9.63) _ P 1 oc 4k 2 T 21 Or P t [< Lv 1 dI+ /9p / ) 1:)63 \I C ~ +t rt I rr + c3p2 The effects of pumping on load noise are now evident through the term in P containing kp2 which vanishes in the absence of pumping. Hence this excess noise must be held against the amplifier even though it is partly due to the output load. The quantitative evaluation of (9. 63) is carried out in Appendix B for the condition %p > /3 rbecause it includes each of the optimum gain cases that have been treated so far. Denoting the total noise power reaching the external conductance gp by Pp, then I- - 4 2 gP 4kT 2 P p cp p = TP 4kTp P) (2) (9.64) P g p 2 4/p 4'r/p and the excess noise power reaching g. is I' excess k 32 r] (P) = P (4kTpBpn) P ) ) /p (9. 65a) p/excess g \ pP \32/\73 / 2 where BPn =() (P (9. 65b) is the noise bandwidth of the pump circuit in cps. The noise figure due to varactor loss can now be found by referring (9. 65a) to the input in the manner of (2. 39), which yields

159 Pp excess F = + (9. 66a) 4k T B ~)2 o pnT =-1 + 32 3r T (9. 66b) Therefore, for the high gain case where k = 0. 1, p = 3 andT = T, p p' r p o F = 1.003 = 0.01 db which is negligably small. It can therefore be concluded that sources of noise other than the varactor itself will determine the noise figure of the configuration under study. Measured noise figures will undoubtedly exceed the numerical values predicted above, so experiments will have to determine the source of additional noise contributions, although several can be anticipated. For example, the varactor can be assumed to be at room temperature (Td = T ), but the load temperature T may be much larger, particularly if pump oscillator noise contributes to T. If this effect is large, pump oscillator padding will be required to reduce T. Other forms of load noise should be small if a circulator is employed. Finally, the magnitude of input circuit noise must be estimated in an actual circuit, so it can be included in (9. 66). However, these extraneous effects are largely under the control of the circuit designer so very low noise figures can generally be expected. 9. 2 Design Considerations The principle of amplification by phase modulation is quite simple, but the optimum usage of this principle will involve a number of further considerations. As depicted in Fig. 9. 3 this process contains three basic steps: 1. phase modulation (PM) 2. PM to AM conversion 3. AM detection. The last two steps together constitute phase detection but they have been separated to indicate the important fact that all phase detectors are really amplitude detectors preceded by phase to amplitude conversion networks.

160 So far, only step 1 has been treated in detail, but the factor t7 has been reserved for the gain reductions due to steps 2 and 3. The fact that loss must be incurred in steps 2 and 3 is fundamentally true, even though detectors can be built employing either nonlinear reactive or nonlinear resistive elements. Reactive demodulators must be ruled out because they incur losses proportional to the ratio of output to input frequency squared, or just the inverse of the modulation gain [see (7. 6)]. Resistive demodulators also incur loss, but more or less independently of frequency. Therefore, demodulation must be by resistive elements, with care being taken to minimize loss. The statement that reactive mixer demodulators have high loss does not contradict the conclusions of Chapters V and VIII. There, high-gain down-converters were described, but because they employed indpendently loaded sidebands they would be unsuitable for video demodulation. Nevertheless, such converters could be used for secondary amplification prior to demodulation. Phase detectors operate by adding a second signal to a phase modulated signal in such a way that phase variations become at least partially converted to amplitude variations. The second signal may be generated actively or passively through filter action, but in either case a conventional AM detector does the final demodulation. For small angular deviations, the added signal will properly contain a quadrature carrier component. Therefore, taking the case of (9. 6a), where 2wc M is small, the p general signal B sin(p t + 0B) (9. 67) will be added to (9. 6b) to yield the sum N(t) cos [p t + 4(t)] (9. 68a) where N2 = IVp 2 B2 + 2B[Vpl sin[0B-0p-0(t)] (9. 68b) 1(tV sin[e(t) + p] - B cos (9.68c) Vpcos[0(t) + 0p] + B sin 0B

161 Two significant cases can be isolated from (9. 68) and these are illustrated in Fig. 9. 9. The first is where the two carriers are precisely in quadrature, which yields an enhanced carrier and corresponds to (0B-p) = 0. The second is for (0B-p) 0, which yields a partially suppressed carrier. However, each detection system will be further characterized by the degree of isolation that can be achieved between the varactor, the pump, and the detector. Isolation between the pump and detector is particularly desirable when the reference signal for phase detection is also taken from the pump generator, since the reference signal needs to be pure in frequency (and phase). There are two basic isolation techniques that can be considered. One is isolation by active circuitry through secondary amplification. The other is isolation by passive circuitry such as circulators. Of course the former gives an added advantage of increased gain, but it calls for an extra active stage which will be considered undesirable for the present analysis. Instead, our interest will be restricted to techniques in which the pump oscillator is the only active component, and secondary amplification will be regarded only as a potential means for extending each method discussed. 9. 2. 1 Quadrature or Slope Detection (0B = 0P). For this case, (9. 68b) yields 1/2 F 2 w BIV M cos () N(t) = V -2 + Bz L2 p~ + ( tjP (9. 69a) 2co) B V pMI cos 0(t) V 2 + V 1 2 P + B2 1 (9. 69b) IVpi + B while by (9. 68c), /(t) = tan1 - 20 Mj cos 0(t) (970) so the modulation now appears partly in the envelope and partly in the phase is depicted in Fig. 9. 9a. Since (2wop M ) is the original angular deviation, the index for amplitude variations in (9. 69b) is ( B2Vl 2 (angular deviation) (9.71) VP B/

162 ---- -RESULTANT AMPLITUDE / DEVIATION I / I / I / I / I (a) B / / RESULTANT / CARRIER /,~/ _I/-INITIAL ANGULAR I Vp I DEVIATION RESULTANT AMPLITUDE (/ DEVIATION ~(b) B 4. /RESULTANT - ^j 1- CARRIER _< _,_-~INITIAL ANGULAR " - ~ v^r" --- 1' DEVIATION Fig. 9. 9. Two examples of adding a carrier for phase detection. Case (a) corresponds to slope detection, while Case (b) is basically a suppressed carrier system. Hence the maximum AM index occurs for B = IVp and equals half the angular deviation, which means that only one-fourth of the coherent sideband power is made available for detection. Of course the latter fraction can be increased to unity if B >> Vp, but this would represent an extravagant usage of pump power. It follows from (9. 69b) that the greatest efficiency from the standpoint of pump power also occurs for B = IVp, and thus just doubles the original pump power requirement. Hence, for quadrature detection systems, the maximum available AM sideband power is 1/8 (-9 db) that possible as pure PM, for a given pump power. However, other detection systems will be shown to give better performance when they are usable. There are several ways in which the quadrature signal in Fig. 9. 9a can be generated, but each of these methods are equivalent to splitting the original pump signal and

163 PEp |]L g p VC Hp gp tP (a) (b) J o Il gp T jP V i, (c) Vp in (b) Vp in (c) 22 (d) Fig. 9. 10. A means for direct quadrature detection using the slope of the pump tank: (a) shows the basic pump tank; (b) is the same tank at resonance; (c) the same tank at its half power point, and (d) shows the corresponding pump phasors. applying part to the varactor and part to the detector. To this end, let us consider the circuits in Fig. 9. 10. It is evident that adding a susceptance jg to the conductance g introduces a quadrature component of voltage. However, this configuration is equivalent to operating a tuned circuit at its half power point. Thus, phase detection by the addition of a quadrature pump signal is equivalent to the well-known technique of slope detection. Slope detection can be accomplished directly if the pump circuit is tuned as shown in Fig. 9. lOc. This appears to be the most effective detection method whenever detector isolation is impractical. However, detuning the pump tank does cause a small reaction at the video input, which will be discussed in Section 9. 2. 4. Figure 9. 10 further illustrates the 9 db PM to AM conversion efficiency previously cited. Tuning to the half power point reduces |V by 1/ /, but the sidebands are

164 doubly effected; first because they are proportional to Vpl and second because they are detuned, which yields a net sideband reduction of (1/F)22 or -6 db. Finally, only half the sideband power appears in the AM envelope which introduces another -3 db loss, making (-9 db) in all. Some improvement in the above system can be obtained using the balanced modulator configuration shown in Fig. 9. 11, since it gives a better utilization of the extra pump power demanded by slope or quadrature detection. Two varactors are used and they are tuned so the upper half-power point of one coincides with the lower half-power point of the other. Hence, the pump sees a purely resistive load to which it can be matched, but it produces a dc envelope on each varactor equal to IVp //2-. When a modulating signal is applied simultaneously in each circuit, a balanced output results from the two detectors, and the resulting AM envelope in each detector is the same as that for a single detector in an unbalanced system. Therefore, for a given pump power, the maximum AM sideband power from a balanced modulator is 1/4 (-6 db) that possible as pure PM. Note, however, that the 3 db improvement in going from an unbalanced to a balanced system is due to pump matching and not to better handling of the sidebands. Hence 72 is the same for both cases. 9. 2. 2 Detection of Double-Sideband, Suppressed Carrier Signals (0B-p > 0). A still more efficient detection method is depicted in Fig. 9. 12, which shows the sidebands being completely converted to AM by employing a second signal with both quadrature and outof-phase components. The out-of-phase component will just cancel the carrier if B cos 0 ___ _ __/ 7p V sin-0B (9. 72) I pl B Hence this will be termed a suppressed carrier method. For convenience, 0 will be set equal to zero and then (9. 69) and (9. 72) yield N = Vp ctn 0B [1 + 20(t) tan B] 1/2 (9. 73a) IVpl ctn 0p [1 + (t) tan 0B. (9. 73b) In this case, the AM index equals (tan 0B) times the angular deviation, which gives 100 percent PM to AM efficiency for 0B = T/4.

165 DETECTOR 1 X; 22 VARACTOR IUp 4Vp -- AMPLIFIED I-'=- SIGNAL OUT SIGNAL (BALANCED). /AMPLIFIED (a) A TUNING TANK TANK V A Y B (b) (c) Fig. 9. 11. A balanced modulator configuration, (a) for improving the pump power efficiency in the case of slope detection. The effect of a signal, as shown in (b), is to detune both tanks in the same direction. Figure (c) compares the individual tank voltages to the total pump voltage. A suppressed carrier modulation is easily obtained when the varactor is matched to the pump, since the sidebands are all that emerge from the varactor in this case and a circulator can be used to isolate the sidebands from the pump generator. Since this is just the case depicted in Fig. 9. 3, the latter with the circulator gp matched to the varactor loss gdp is equivalent to Fig. 9. 9b with B = - Vp. Efficient detection can now be accomplished by adding a second carrier at the output port that is in quadrature with the original pump signal. However, this added signal can now be relatively small, perhaps an order of magnitude greater than the sidebands themselves, since it only needs to be large enough for linear detection. Figure 9. 12 summarized this detection process, which for the same pump power yields nearly the same AM sideband power as can be obtained in pure PM. To illustrate this conclusion, consider a case where 100 milliwatts of pump power yields a phase modulation gain of 100. Then 10 microwatts of available signal power would produce 0. 25 milliwatts in each sideband and a total coherent PM power of 1 milliwatt. But with a circulator this modulation can be completely converted

166 PUMP IN DIRECTIONAL COUPLER VARACTOR DOUBLE i Ur SIDEBAND. 2 t SUPPRESSED 1T < CARRIER B < ADDER ENV DET cr (AMPLIFIED) Fig. 9. 12. A realization of the detection scheme in Fig. 9. 9b. to AM if a second signal, at about a 10-milliwatt level, is supplied. 9. 2. 3 Upper Limits on 72. Returning now to the factor 72, it should be noted that only the sideband power losses incurred during detection are included in 72 and not pump power demands. The factor r7 expresses the product (PM to AM efficiency) times (AM detection efficiency) and the preceding discussion has shown that the first factor varies from 1/8 to 1, depending on technique. The second factor 77, the detection efficiency, is even less tangible because it depends upon detector diode quality. In fact, one is suprised to find that detection efficiency is relatively undiscussed in standard engineering texts. Although matched detectors are easily made at microwave frequencies, they can be relatively inefficient unless care is taken. At lower frequencies 50 percent efficiencies can be easily obtained, so there is reason to expect efficiencies of this order at microwave frequencies too, but further study of this question is still in progress. Therefore, by using a circulator, so the sole contribution to 772 will be from envelope detection, and using efficient diodes, r72 can be as large as 1/2 in this case. Without a circulator, or other means of isolation, direct slope detection should be used which yields 772 =(1/4)(1/2) = 1/8.

167 9. 2. 3 The Effect of Direct Slope Detection on Circuit Properties. The pump tank can be tuned for direct amplitude modulation by placing wop at one of the half-power frequencies, but then a new constraint must replace (9. 1). Using upper and lower signs (~) to denote tuning to the upper and lower half-power points respectively, then Y - p [I + j(l + co1p)] Y gp[1 + j(1 or/3p)] (9. 74) The approximations above are excellent for w r Q p3p, which will be an ample range if Qp 3. The effect of slope detection is demonstrated most clearly by the input admittance at or In this case, (8. 2) yields -+ - 1_ r __ o o | Vp 2 3p 23 L/1^ Y =Y+ r (9.75) r. r g L +)2% co'(4 1 + which has several interesting properties. Considering first the case where r is small, the main effect of pumping is the reflection of an input capacity T+e wl. (9. 76) gr Therefore, when the pump is tuned to the upper half-power point, the reflected input susceptance tends to cancel the nominal input susceptance wr C. Just the reverse occurs at the lower sideband. One other effect is a reflected conductance co 20co I Vp + Plp (9. 77) gp which is negative when tuning is to the upper half-power point. Together these effects yield a useful variety of circuit properties.

168 To study the input circuit properties in this case it is convenient to rewrite (9. 37) as follows: Y. L 71()2 +J ( 1 +2) (9. 78a) in1 + ( 1 + ( ) w k 2 = r/3p v = /3 = Pp (9. 78b) Perhaps the most interesting situation occurs at the upper half-power point, where increased bandwidth is possible. Study of this relation shows how input capacity is cancelled in proportion to the product piv. However, when X A 1, a peak negative conductance appears which threatens stability. Therefore, to insure stability, a compromise must be made between suppressing capacity and introducing negative conductance. The normal input bandwidth in the absence of pumping is X 1/ / =. Therefore, if v < 1/2, the input bandwidth could be doubled and still the sidebands would remain in a region of linear slope on the pump resonance. By some experimentation with (9. 18), it can be shown that iv = 1/2 approximately doubles the input bandwidth if v < 1/2, with a fairly safe stability margin. In contrast, if iv = 1, the input susceptance is nearly zero over the entire linear slope of the pump tank (O < X < 1), but negative reflected conductance will produce instability for 1/ i < X < 1. Therefore, operating at the upper half-power point provides a safe increase in bandwidth, up to a factor of about two, such that (/r)effective 2 r (9. 79) The asymmetry demonstrated in (9. 78), with respect to the side of resonance being used, can be attributed to sideband distortion. At the upper half-power point, the lower sideband is enhanced at the expense of the upper sideband, so the negative conductance from the lower sideband is no longer balanced out by positive conductance reflected by the upper sideband. Just the reverse situation occurs on the lower side of resonance. It is interesting to also examine the effect of sideband asymmetry on conversion gain. The gain relation corresponding to (9. 17) in the case of direct slope detection is

169 4 g rgp Vp p maximum real com- GAAM Yl 2 Vr Iponent of (V + V)/V 8/ r. r L u pj where voltage ratios needed above can be found by repeating the derivation of (9. 3), but using (9. 74) instead of (9. 1). Now V -jw y V -jwy * u u r l 1_ r uVp gp[l~+J(l )] Vp gp[l (l+A)] r(9. 81) V g[p -( X)] VP gp1+ j(1T X)] and hence rmaximum real component 1 (W 1p)/ F(A) (9. 83) of(Vu+V v )/v o 2 n where F(A) = f + 1 (9.83) 1 + (1+A)2 1 +(1+ )2 Since F(X) is symmetrical in X, it is independent of the choice of operating slope, and so the only asymmetry in GAM with respect to side-of-resonance will be that due to Yr. F(X) is plotted in Fig. 9. 13 showing that it is essentially constant for X < 2. 2. in Therefore, the effective half-bandwidth of the pump tank is (tp)effective 2 p2 p Because F(A) is essentially < F(0) always, the midband gain can be taken at X = 0, which yields 4 2 y 2 2k 2 W 2 M - p P P (9. 84) AM = ggp = /t3p r&p r p By comparison with (9. 23), GAM is 1/4 as large as GpM for the same yp, and hence 1/8 as large for the same pump power. While these conclusions regarding gain reduction are consistent with the observations in Section 9. 2. 1, the latter suggest nothing about the bandwidth improvement. Because both /3 and /3 are effectively doubled with direct slope detection, f3 and /3 need r p r p

170 1.5 1.0I.O 0.5 0 1.0 2.0 3.0 Fig. 9. 13. F(X) in (9. 38). be only half as large in (9. 84) as in (9. 23). Therefore, for a given bandwidth and pumping coefficient, GAM, and GpM can be equally large, but for different circuit adjustments. However, in spite of this theoretical conclusion, direct slope detection will be less preferable in practice because of its greater sensitivity to adjustment. Without good detector isolation, such as that achieved with a circulator, the tuning adjustments for gain and detection are dependent on one another, which makes them more critical. Also, with the appearance of negative resistance at the input terminals, stability will be more critically dependent on pump tuning with direct slope detection. 9. 3 Conclusions The form of double-sideband reactive mixing that has been treated in this chapter yields an immediate practical application, namely the construction of an ultra-wideband video amplifier. This technique uses a form of reactive mixing to phase modulate a pump or carrier. The, when the latter is detected, an amplified video signal will result. The treatment of this problem has considered first the modulation process, and secondly the detection process. The optimization of both has been discussed, with the result that gainbandwidth products of the order of 1000 Me are possible. Much experimental work has been done in connection with this principle, which the reader will find discussed in Chapter XI.

171 The application of varactors to wideband video amplifiers is original with the present writer, as far as he knows, but the principle used here can be traced back several decades (Ref. 2). Nearly all prior attention to this concept can be found in the field of magnetic amplifiers, but there, little attention is given to ultra-wideband systems, mainly due to the type of material used. It can be said that the advent of microwave varactors has lead to wideband generalizations of basic magnetic amplifier principles. However, the customary form of analysis in reactive mixer circuits tends to obscure these relationships. Hopefully, the present analysis has bridged this gap.

CHAPTER X DOUBLE-SIDEBAND MIXING WITH HARMONIC PUMPING Several improvements have been demonstrated by the comparison of doublesideband reactive mixing in Chapter VIII with single-sideband conversion. The most noteworthy improvements were with respect to the upper sideband, and were due to negative resistance being reflected from w o. Such negative resistance increased the conversion gain and also made possible parametric amplification at co. However, the scope of these improvements was limited because the negative resistance increased the sensitivity to external circuitry at the input and output terminals. It will now be shown that the above limitations can be further relaxed by adding the pump harmonic. One characteristic of the configuration treated in Chapter VIII is the lack of a direct coupling between co and ou. However this limitation can be overcome by adding the pump harmonic. With this added coupling, each pair of signal frequencies interacts in two ways. For example, a signal at cu is coupled directly to co by the pump fundamental. It is u r also indirectly coupled to or through ci, by first mixing with the pump harmonic to yield co,which in turn mixes with co to yield r. The scope of the present chapter will be to study the properties associated with this greater coupling flexibility. These properties can be divided into two categories according to whether the sidebands are independently or dependently loaded. In the former category, the following properties will be noted: (1) Unilateral up- or down-conversion between any two of the three signal frequencies co r co, and co (2) Simultaneous unilateral conversion from cr to co and from wc to c, and vice versa. (3) Arbitrarily large conversion gains between wr and Wo with gain independent matching at the input and output terminals. 173

174 (4) Gain-bandwidth products and noise figures that compare favorably with those in conventional reactive mixers. In the case of dependent sideband loading it will be shown that considerable improvement in gain-bandwidth product can result from harmonic pumping. It therefore provides a means for cascading phase modulation amplifiers, as described in Chapter IX, without using interstage phase detection. This point will be further expounded in Section 10. 4, but since the case of independent sideband loading is the more general of the two, it will be treated first, and than dependent loading taken as a special case. 10.1 Basic Analysis of Harmonic Pumping Mathematically, the first-order coupling of cr co co U c, and 2c is described by (2. 27), which is already familiar through the special cases treated in previous chapters. In fact, relative to the treatment in Chapter VIII, all that is necessary to restore (2. 27) to its most general form is to account for the pump harmonic by adding 2p. The formal solution of (2. 27) is straightforward, but there are ten or more independent parameters in this equation, so more than a formal solution is required to understand all of the effects involved. The equivalent circuit for this configuration is Fig. 1. 2. In analyzing (2. 27), no assumptions will be made regarding the relative phases of yp and y2p, so a phase parameter ~ will be introduced. It is convenient to define d by the relation 2 p 2p = e (10. la)!Tp Y2pl Another helpful variable change is to introduce the following set of normalized admittances r Y2p T = r - -, + j (10. lb) rp' = -u =' iU ( ~ 10.1c) Y T = _- - = T + jT" (10. Id) co i p u 2pI

175 They are proportional to the actual terminal admittances for constant pump level. In the present application of (10. 1), no restrictions on sideband loading are imposedso r''u' and 7 are independent parameters. As an illustration of their usefulness, the normalized admittances reduce the system determinant of (2. 27) to the following expression: a = cWrWt uo2p 2 p A (10.2a) r { u 2p 2p where r =, +7 -, r- + 2jcos0 (10.2b) This determinant is particularly important in applying the stability criterion of Chapter VI. The latter predicts that the region of validity of all steady-state solutions can be determined by the vanishing of A, and all solutions must eventually be tested in this way. For example, if a set of terminal admittances and pump levels are chosen to yield a certain circuit effect, then A must not vanish even as the pumps are brought up to the required level. In the absence of pumping A = Y Y (10. 2c) ru & The formal solution of (2. 27) is achieved through the inversion of its admittance matrix [Y]. Using the notation in (10. 2), it becomes r (-u -1 2p'u - 1 + JEi T -j! 72p' U k CWr Yp2 jWup j co [Y] -1 1 |rT — r T - r * j- (10.3) j corp ou 2p' i c f2p A Jryp 2pw c j 2pp -7 + ]C -7 + jE J 7T u r J r u jr~p u2p 2 &[2p1

176 The diagonal terms in (10. 3) are of particular interest because they are the input impedance as experienced by an- ideal source at each terminal in Fig. 2.1. In the cases treated previously, with' = 0, the various network effects have often been strongly dependent on / p negative terminal resistance, so it is important to see if this would again be the case. However, more convenient expressions result from considering the input admittance as given by the reciprocal of each diagonal element in (10. 3). The result is n 2p in in u ( 2 - T + 2jcos5 uY ur 2p'u;'. + r ~- c (10. 4a) r. r. r r - 1 in p n in u'T - 7 + 2jcos^ Y -T (10. 4b) in In in i -r + u -2jcos Y (-2p'C.; = 77 +1 (10. 4c) in in in ru Again the normalized notation of (10. 2) has been used, but the first term in each expression above can be identified with the terminal admittance in the absence of pumping, while the second term denotes the admittance reflected by pumping. The appearance of both the terminal and the reflected admittance in one expression makes this definition of input admittance particularly convenient for comparisons. 10. 2 General Physical Interpretations The physical significance of (10. 4) can be seen by calculating the normalized image admittance relating each pair of frequencies. If coI and c2 denote a general pair of frequencies, then the image admittance is 12 - s) 1(o(10. 5) Y12 = /Yl(S) Yl(o) where Yl(s) and Yl(o) denotes the reflected admittance at cl due to a short or open respectively at w2. By applying this concept to (10. 4), the following image admittances result: i2 between Y 7 r p Y = W and: ru ru' ur u i 2p ur r u!Y2p; (10.6a)

177 7 + 2j cos 5 ru ur (10. 6b) / - 2j cos rC {r 7 between * co and w Y u - W 2p Tp u Yu u 2pi'u{ (10. 6c) /T - 2j cos.u = ~ = -u u X r Since the expressions above are somewhat foreign looking, a useful interpretation is provided by transmission line theory, where image admittance is equivalent to characteristic admittance. This identification is convenient in the present case because the magnitudes of the normalized image admittances are independent of direction, and therefore the set (10. 6) gives rise to the informative representations in Fig. 10. 1. The second transmission line characteristic of interest is the propagation constant 0, which is generally defined by 012 -= n (V _Y ) (10.7) sion line portions of Fig. 10.1, which yields 0 -\y tne. Th i s + ief i e ru -jp -. e e (a); tiur v p s (b) 2p 77 (a);-12 2 (b)1 12p u l ) [ r fi 0E^ -- _D_ (c); r -^\ (d)I (10.8) ur 72p ~ ~ ~ ~ = a; e b

178 wr IypI2 rIY2pl Wu1Y2p1:I t Tru -u (a) VT r Vr T V r 1 ~ 0 -0~ IDEAL IDEAL IY2pl Wu1Y2pI t Tr t *o~ (b) Vr VA IDEAL --— IDEAL Wr I T p2 VI2pl WuIY2pI:__ f (c) V I Vt IDEAL ----- IDEAL Fig. 10.1. Transmission line representations of the three possible conversions between w, r', and. r U~~~~~~~~- 0~ -j'Y2p Tr -jE B0 u -P12p - Tr+ j t j 0 (e_; _ (f) (10. 8) 1-'2p1 r u'2p r & In studying these relations, we first note that 0 ij 0. (i, j=r, u) (10.9) and hence the transmission lines in Fig. 10. 1 are nonreciprocal. In fact, they can even be unilateral. To illustrate the latter point, the numerator of (10. 8a) can be written rk + (sin 5 + j cos ) (10. 10) and therefore if T8 = Isin 5f - j cos; m < 5 < 27r (10.11)

179 (10.10) will vanish. There is no transmission from Wr to c in this case. In contrast, the numerator of (10. 8b) can be written 7T + (-sin 5 + j cos 5) (10.12) which does not vanish but equals 21 sin ~[ when (10.11) is satisfied. However, the vanish0 0 ing of e and e reverses when 0 < < T. Hence, between each terminal pair in Fig. 2. 1, a state of unilateral conversion can be set up, whose conversion direction depends only on the relative phase of the pump and its harmonic. Further inspection of (10. 9) yields the following general conditions under which unilateral conversion is possible: between c and; if T = T (10.13a) r u o between c and Wo; if T = T (10. 13b) between W and w; if T = T (10.13c) where T = [sin I1 + j cos 5 (10.14a) IT I 1 Hence, unilateral conversion between each pair of frequencies depends only on the termination at the third frequency. The direction of unilateral conversion depends on the sign of sin. The characteristic admittances also take on interesting values under (10.13). These are noted below: if T- = T* Tu = -T T (10.15a) f 0 ru o ur if T= T0: = Tr -T r (10.15b) if T = T0: T = T0 =-T (10.15c) The sign of each characteristic admittance cannot be determined directly from (10. 6), but it must be found from the corresponding input admittance in (10. 4). For example, setting T = T* in (10. 4a) yields &9 =7

180 T - T r = + u0 (10. 16a) r. r * in 7 T -1 u o which seems to depend on T. However, by (10. 14b), O = 1/T-, so (10. 15) reduces to T = T - 7 (10.16b) r. r o In which agrees with (10.15a). It is reasonable that each input admittance would be independent of the output load in the unilateral condition, since unilatarity depends only on the loading at the third frequency in each case. Therefore, a result similar to (10.16b) is found for the other input admittances as summarized in Table 10.1. Direction of Zero Conditions Image Admittance Conversion Gain ___ (Normalized) Critical Sin Sin E Load ) - w <0 -T r u * o u r o r f < 0 * i SQ ~r > o T T u 0 [ -cW >0 -T r 0 w - ^ < 0 * u 0 T = T r 0 0W -w > -T u o Table 10. 1. A summary of conditions associated with unilateral conversion gains. It is important to observe that the critical parameter T0 is real when = 7/2, which corresponds to 2p being 900 out of phase with yp. Therefore, the original assumption of arbitrary pump phase is crucial to the main conclusions of this chapter. It is noteworthy that the least rewarding case occurs when thie pumps are in phase, although this case might have been chosen initially for analytical reasons. Had this case been assumed, the

181 To Wr Wu (a) (/~ AV Wu 0/ T r) + To (b) (c) Fig. 10. 2. The three basic configurations for unilateral conversion. Key: 7 < <1 < 27; 0 < 2 < r; > nonvanishing conversion;. —— 4 —- unilateral conversions, vanishing for 5 = k. unique features of harmonic pumping would have been missed. Further consequences of the choice of 4 will become evident later, but in general the more 4 differs from 7r/2, the harder it is, for practical reasons, to achieve unilateral conversion. Continuing our evaluation by transmission line terminology, two important facts can now be noted about (10. 8) and (10.15). (1) The transfer functions in (10. 8) dealing with conversions that originate at either wr or co have singular points when Tr = 1/Tu or 1/Tr, respectively, and hence large conversion gains can occur in these cases. (2) The transmission lines in Fig. 10. can have negative as well as positive characteristic admittances so reflection gains can be obtained at any frequency. These remarks are depicted in Fig. 10. 2, where three-port networks are used to illustrate the three conditions of unilateral conversion.

182 It is interesting to note the similarity between the (b) and (c) parts of Fig. 10. 2. In fact, the symmetry with respect to wr and wu has been apparent in each expression derived so far, particularly where the normalized factors are concerned. Therefore, unilateral conversion can be treated in two cases rather than three. The first is depicted by Fig. 10. 2a, which corresponds to (10. 13a). In this case, the coupling between wr and w resemble a conventional lower-sideband converter, but in addition c and w are regeneratively coupled so amplification by reflection can take place at w. However, the most unusual feature of this configuration is the nonreciprocal conversion between r and u' which provides a means for unilateral conversion with large forward gain. Also, the forward direction can be changed by varying the relative phase of the two pumps. The second case that will be separated from Fig. 10. 2 is a combination of the configurations in (b) and (c). If T = To, as depicted in (b), then TU = Tr produces a conjugate match at w c, which corresponds to (c). Therefore, both (b) and (c) can exist simultaneously, and the resulting configuration is shown in Fig. 10. 3. It yields two unilateral couplings, so the only transmission between wr and co in this case is by the direct path. Coupling through Aw goes only one way. As a result, the conjugate match at r and WU is independent of Tr, the normalized load at w. Since this second case (in Fig. 10. 3) is more novel than the first (in Fig. 10. 2a), it will receive the major attention of the remainder of this chapter. 10. 3 Circuit Properties of Fig. 10. 3 The basic properties that distinguish Fig. 10. 3 from other circuits are gain, bandwidth, stability, parameter sensitivity, and noise figure. These can all be obtained systematically from (10. 3), which will now be done in such a way that the results can be easily compared with those of the other configurations previously treated. 10. 3.1 Gain and Bandwidth. The first circuit property of interest will be the conversion gain between each pair of terminals. Although the general gain definition in (3.15a) will again be used, in the present analysis it is convenient to note that -1 Gil = 4 ggL[Y ]i; i, = r,, u (0.7) where [Y 1] ji denotes the diagonal element of (10. 3), and where gS and gL are the source

183 and load conductance at either terminal. Therefore, when the constraint 7 = 7 = T is inr u o troduced, the gain expressions in Table 10. 2 result. It is particularly interesting to note, due to the signs in Table 10. 2, that only the denominators vanish. In addition, the following basic properties can also be observed for this case. Gain Function Sin; > 0 Sin; < 0 W [(T +sin )2 ( +(T + cos )2] W u u ru o[(Tw -sin )2+(' + cos)2] r Wr w [(Tc - sin )2 + (Tr + cos ))2] ur w 4 co TT sin Gr C[(T -sin )2 +(-r + cos ~)2] 4 T'r sin T ur ~o0 u^r | or| we[(~T + sin t)2 + (T' + cos 5)2] G4 Wu sin 0 r i | [(T - sin 5)2 + (T + cos 2)2] 4w oT' sins I u u) 2 u[(~ e+ sin )2 + (rT + cos O)2] Table 10. 2. Conversion gains when y1 = y2 = y. 1. With T'' = cos 0, arbitrarily large gains can be obtained between each frequency pair as T - sin 0. (Note: Tr = T~ + jT-) 2. With sources at co or o,u the input terminals are conjugately matched independently of gain, and therefore input circulators are never needed. 3. Unilateral gain is possible between or and Co, and between Cu and co, for any value of forward gain.

184 T Wr T o __W ]Ir 0 T ^ ^ - T r ToO u Fig. 10. 3. Matched-up conversion from oor to cou, for Tr = T = To. Solid arrows depict the direction and relative magnitude of conversion gain for 0 < ~ < ir. All arrows reverse for 7T < ~ < 271. 4. Although conversion between wr and wu is not unilateral, the ratio of forward to backward gain is proportional to forward gain, and hence can be made arbitrarily large. The variable gain functions in Table 10. 2 are large only when their denominators are small, and hence when A z 0. Therefore, the bandwidths of each conversion depend solely on A, which is proportional to A. By (10. 2b) and (10. 4c) a = (T T +1) T m~~~~in ~~(10. 18a) which in the present case becomes A = (2T + 1) T (10.18b) in where the first factor is a constant. Therefore, the conversion bandwidth depends only on the bandwidth at the lower sideband terminals. To calculate this large-gain bandwidth, let T' = - cos 5 at midband (o ) and let wo = w (1 + 6), where 161 << 1. Then by (10. 4c) T' 2C 6 - cos l (10.19) 2Cf l'2p[ where Cp is the total fixed capacity at (w. The bandwidth f3 can then be determined from the value of 6 that yields

185 (T" + cos 5)2 = (T - sin 5)2 (10. 20) The result is W D'l( - Isin ~I) f3 = T - s2p in (10. 21) C 2 Y[2p cos For large gains, T-' Isin I, and therefore the gain-bandwidth product for conversion between cr and wc is given by (i,j = r or u) r u W j 2 wc i2p sin ( VGi j = - --- (10.22) C - 2 I2p cos 7f It is evident that 5 = + is optimum in this case, which can be explained by noting, in the basic condition T = T = T, that detuned terminals are required whenever ~ That is, in the absence of pumping, each terminal must be detuned in proportion to cos 0, so when the pumps are applied the reflected susceptance yields the required tuning. But, in addition to being inconvenient from a practical point of view, 5 = + 7 yields reduced bandwidth. In the limit of in-phase pumps (~ =0), the gain-bandwidth product vanishes. It is interesting to compare the gain-bandwidth product in (10. 22) with that of a conventional upper-sideband up-converter. Assuming the optimum case, where ~ = ~, and defining the coefficient of harmonic pumping as:?2p k (10. 23) 2p C then (10. 22) yields /oW 2W co3 3/2 G 2kpu 2k P (10. 24a) ru 2p or 2p 12 p while from Table 3. 1, in the conventional case, G 3 =2kw 2k w (10.24b) ru p u P p (10.24b) Therefore, by adding the pump harmonic, the gain-bandwidth product can increase by

186 k2 co (10. 24c) p r When C = ~, g = y 1 2/ ypl while gu = uo y [. Therefore, (10. 24c) calls for' g r p 2p u 2p gu > gr. This is quite different from the simple upper-sideband converter described by (10. 24b), which calls for g = g.r An important advantage in the double-pump case is that 3 is independent of the bandwidths at the r and wu terminals, which means greater flexibility in realizing the gain-bandwidth product in (10. 24a). 10. 3. 2 Stability. Since it has been noted that the high gains in Table 10. 2 are associated with A approaching zero, the question of stability arises. For Tr = Tr = To A can be regarded as a function of r, and it vanishes only when T = 7. Therefore, a sufficient condition for stability is 7- > I sin I. However, the normalization that has been introduced in the T parameters somewhat obscures the effect of pump variation on stability. Therefore let us consider the case where - =, such that T = T= 1 are appropriate 2 r u o conditions for realizing Fig. 10. 3. By (10. 2), Tr = T =1 yields ~ lpl 2 gr g =';gc 2 (10.25) r 2p u u 2p 172p where y and y2p denote the final operating values of p and y2p, respectively. That is, yp and 2p equal p and yp when the pumps are warmed up. Then at midband | 21 gf - 2 /fr2pp 2 Ax = co co cou[ I I l' - + lpy I -I r ( u 2p 2p- ( 2)} ~r<9wuiY~p " iY~plz Opp IYP t f p,2p I'p I (10. 26) which is positive for r/ = gk Io I y2pl > as long as Iy2pl does not exceed its specified value Y2p. Of course the latter condition only arises because the entire gain is due to a negative conductance proportional to y2pl, which is reflected at the w) terminals. Therefore, the basic configuration in Fig. 10. 3 yields arbitrarily large conversion gains, but with stability that is independent of how the pumps are brought up to their desired levels. 10. 3. 3 Sensitivity. Sensitivity is another circuit property that is closely related to stability. Referring to the discussion in Section 5.1, the sensitivity of a circuit

187 property Z to a parameter X is defined as the ratio of the percentage change in Z to the percentage change in X, when the latter is small. In the negative resistance amplifiers and converters treated prior to the present chapter, gain was very sensitive to important external parameters, such as load conductance, source conductance, and pump level. In the cases of present interest, however, the gain sensitivity vanishes both with respect to the load and to the source conductance. These and all other gain sensitivities are recorded in Table 10. 3 for the circuit configuration in Fig. 10. 3. The entries in Table 10. 3 have been found from the general gain formula in (10. 17) by using the following expression for the gain sensitivity from c. to cj, 4X0 F Z..r12 a(gigi) az.] Sx(G.ij) = G (X) Z2gR i. ) X X= (10.27) WX( - G^T)ij2 3X + 2gigiR e \ o where X denotes a parameter with normal value X. It is evident from Table 10.3 that ~ = ~ 7 is optimum from the standpoint of sensitivity, which is interesting because this same condition yields optimum gain-bandwidth product. For this optimum case, the only nonvanishing gain sensitivities are those with respect to g9 and y; but this is consistent with the gain formulas themselves, which depend explicitly on the gain control parameter T = (10.29) co [)'2p[ It is interesting to note that circuit properties other than gain can also have high sensitivities. However, these other properties may be sensitive to a different set of parameters than the gain. For example, the reflected input admittance is sensitive to the output load according to the relation (T aT-I u 7 )in 2 ST (Yi - Y) = T IT a'( ) (10. 30) u'in in u/ T =T U O Also, the sensitivity of the output admittance to the source obeys the same relation. However, the sensitivity of the reflected admittance at Col to the source and load admittance is zero.

188 Parameter Normal Value Gain Sensitivity X X co. to c). O 1 1 SX(Gij) I 1 arbitrary ctn l I Y0 arbitrary Gt ZP -\ I, i]sinc I | 0.U. gu iu Zp 1 is an a, te arb itraryi m grelative cha s sin ( I Y p 0 siniii i i pi y i br, bu, and b1 cos 0 *Since ~ is an angle, the sensitivity to ~ is measured with respect to absolute changes in ~ (say a/) rather than to relative changes in (A^ ia). Table 10. 3. Gain sensitivities for the circuit depicted by Fig. 10. 3. In the ideal case, where r = o, the input admittance is independent of rT, and hence of gain. But, if Tu T, the input admittance becomes quite sensitive to gain, as was shown in Fig. 10. 4 for the case where ~ = ~ 7-. Yet, the circuit action is such that admittance variations at either the input or the output have no first-order effects on gain. Nevertheless, it can be anticipated that admittance variations can have an influence on noise figure. If the input is improperly coupled, negative resistance will appear at the output and will amplify the output noise. Further discussion of this point will appear in the next section. Therefore, as far as sensitivity is concerned, the main advantage of double pumping is that no gain takes place at the input or output terminals. Consequently, the gain

189 TA, \ / ^Wr 3.0 X 09 A^T|l CCONVERTER 3.0 -/ 09 2,0 1.0 - ~~2'*,~0 - / ^ ^'(orT~2 t / ^-dINSTABILITY 02 0 2 3 Fig. 10. 4. Variation of the normalized input (or output) admittance with the gain parameter Tr, for various output (or input) loads. The ideal case is for Tr = 1. is not sensitive to imperfections in termination, even though some negative resistance may result from such. To illustrate these observations, consider up-conversion from wr to wc and let us assume that a 10 percent variation in source conductance is possible, but that the output load is constant. Thus, assuming ~ =, the proper output load will correspond to T' = 1 SO U 2 G = u - (10. 31) ru co IT - But when T' = 1 ~ 0. 1, the reflected output conductance at co can go negative if T- < 1. 1. r u U Therefore, by arbitrarily choosing 7r = 1. 1, G will equal (26 db + 10 log u/or) with negligible fluctuation. However, from Fig. 10. 4, the output conductance will fluctuate between an open circuit and about twice the load conductance. As a result, one obtains a high

190 gain circuit that remains very stable in spite of mismatch fluctuation. 10. 3..5 The Effects of Varactor Loss. According to the method outlined in Section 3. 4. 3, varactor loss can be introduced as frequency-dependent shunt conductance at each sideband terminal. As a result, the normalized terminal conductances have two components; one due to loss and one due to the externally applied conductance. For example, - T 2rd - (gr + grd) (10. 32) WriYp' where gr is the external source or load conductance at r', and grd is the loss conductance. Several interesting consequences of varactor loss will now be examined with the aid of this notation. Perhaps the most outstanding feature introduced by double pumping is unilateral conversion, but the latter requires a critical value for each normalized admittance, according to Table 10. 1. Therefore, when loss is introduced, unilateral conversion can only be retained by reducing the external conductances until Table 10. 1 is again satisfied, but then varactor loss appears as a mismatch. One can compensate for this mismatch by deviating somewhat from perfect unilateral conversion. For example, in the presence of loss, Fig. 10. 3 can yield a match at Wr and w;, or can yield perfect isolation between the pairs (co, r w) or (o, Ou), but not both. Also, the conditions for isolation are still gain independent, but not those for match. It will be shown, however, that the conditions for match approach those for isolation as the gain increases, and therefore the imperfections due to loss are not serious. To examine the conditions for simultaneous match with loss, consider the circuit in Fig. 10. 3, and let us assume all terminals synchronously tuned with T = /2. If x is the efficiency factor at or (x = Tr/r) then match will occur if r r r T~ - T'U r' = r +' - u (10. 33) r rd T -1 or T' - T' (2x- 1) r' = (10.34a) ir'- 1

191 Similarly, if y is the efficiency factor at cou, match will occur at wu if r-?r (2y - 1) T' = 1 (10. 34b) u T TI - 1 r Although it is rather tedious to solve equations (10.34) simultaneously, a significant special case can easily be treated. If' and T' are approximately unity when the co and wc termir u r u nals are matched, then (10. 34) can be approximated by 1+7' (2x - 1) (rT - 1) 1 (7' - 1) + 2(1 - x) (10. 35a) r 1- T u 1+T (2y - 1) (T - 1) = 1 (T - 1) + 2(1 - y) (10. 35b) u 1 -'- r Therefore, with high gain (i. e., with 7T - 1) r' 1 = 2(y- 1) ( ) (10. 36a) 1 \ T - 1 = 2(x- 1) (1, (10. 36b) u 1+' which both approach zero as T' goes to one. Therefore, varactor loss causes very little mismatch in the unilateral conversion mode, if the gain is high. The process described by (10. 35) can be visualized with the aid of Fig. 10. 4. If we start with T' = 1, and then introduce loss at co, the resulting increase in T' reflects a small negative conductance at cr, which reduces T' But at high gain, the latter reducr' r. in tion actually overcompensates for the loss conductance at or, so 7' can be reduced until T' r' u r is only slightly greater than 1. Similarly, 7T is reduced until the loss at co is cancelled, r u which occurs for T 1. In this case, the worst mismatch occurs at low gain, r and then the input reflection coefficient is (1 - x). However, the latter vanishes as the gain increases. 10. 3. 4. 1 Up-Conversion Noise Figure. The most significant effect of varactor loss in multiply-pumped circuits will undoubtedly be the influence on noise figure. However, it will be of interest to see what noise figure benefits accompany the rather unique

192 circuit properties that have come to light through double pumping. The basic circuit that will be treated for noise figure is Fig. 10. 3, so let us first consider the noise figure for up conversion from ro to w. The general noise figure expression in (3. 27) can again be applied, but it takes on a simpler form in this case. The third term in (3. 27) is the noise figure contribution by the output load, which would be due to an output mismatch. This term will be negligible if the output mismatch is small, which is the case in Fig. 10. 3. Ideally, the output of Fig. 10. 3 is perfectly matched, so mismatch only arises as a result of varactor loss. However, the influence of varactor loss has been shown to diminish when conversion gain increases. Therefore, a case of basic interest will be the noise figure for high gain under the conditions of unilateral conversion, namely -r = T r u o By (3. 27), the up-conversion noise figure can be written Td G- T F = 1 + ( + (10.37) ru x T G T o ru o where x is again the input efficiency factor (at wr in this case). Also, in writing (10. 37), the bandwidths of GU and Gru have been assumed equal according to the argument of Section 10. 1. By (10. 17) and (10. 3), the gain ratio in (10. 37) becomes Gu cr' Tr - je G- - ( r)O ( XT) -r_ j (10.38) ru f r Tk + je which depends on. Since optimum up-conversion bandwidth and sensitivity occurs under the following conditions = 7/2 (10. 39a) T = T = 1 (10. 39b) r u To = T' (i.e., real) (10. 39c) the latter will be assumed in calculating the noise'figure. Now (10. 38) becomes Gu C_ r _ 2 2 Gru - (

193 in which case Tdrl aco F - 1 (x) [ 1 + z(s - 1) (10. 40) ru T x X where T~ = Td[1 + z(s-1)] (10. 41a) z g = /g; s /Td (10. 41b) 47' a - (10.41c) (T + 1)2 The noise temperature expression in (10. 41a) is equivalent to (5. 27c), but it has been written in the form used in Appendix A. Further evaluation of (10. 40) is possible by using the constraints in (10. 37a). In particular, if varactor loss is the only parasitic noise source, then by (3. 30) co2 C 2 gr = xgr + w. (10.42) c y2p Hence x = 1 - Rr p (10. 43a) where Y2p p p (10. 43b) p r o C 2 R (= (10.43c) Similarly co 2C l = z g ~ = o 2p (10.44) c and therefore r) 1 (1045) Cv( p 1-z. (10. 45) ~-~: ) (

194 A final observation before calculating the noise figure is that a z 1 in (10. 41c) when T' z 1, and therefore choosing a = 1 corresponds to high gain. The noise figure for unilateral conversion with high gain now becomes F = 1 + d ( r) [+ + (1 1) (10.46) ru 1 2 1 + z The three parameters in (10. 46) have important physical significance, so they will be reviewed below: - = the dynamic quality factor of the varactor. It is a measure of R r varactor quality and the degree of pumping, and is the same factor that appeared in (4. 32b). p = the ratio of the two pump components. z = a measure of the external loading at the dummy terminal w. Therefore p and z are parameters that can be varied to optimize the noise figure. However, Rr is an independent parameter that all reactive mixers, using the same varactor and pump, will have in common. One other constraint that must be considered in optimizing (10. 46) is the output condition r = 1. When considered together with the high gain condition 1 = 1, one obtains the relation Cu i _-y (10. 47) co i-z where y is the output efficiency factor. Now it is clear that y must be less than z, so z cannot equal zero in this case. Now, to proceed with the minimization of (10. 46), let f 1 +z(s- 1) (10. 48) 1-z and then (P)opt = -f r f f (10. 49) opt r r

195 from which (F ) op= 1 + T r [Rf + 1+R f (10. 50a) (ruopt T rj O or (Tru) opt =2 Td/(popt (10. 50b) The smallest value of (10. 50) occurs when f has its minimum value of one. However, (f) must exceed one in all practical cases because (z) must exceed zero to satisfy (10. 47). Therefore to obtain low noise figures, the output efficiency (y) must be low so (z) can be small, which in turn makes (f) small in (10. 50). An additional help in this case would be idler cooling to reduce f by reducing s. It is interesting to observe that (10. 50) is identical to the formula for optimum noise figure of a lower-sideband converter, as given by (A-10) in Appendix A. Therefore, Fig. A. 1 with a = fR is a plot of the equivalent noise temperature in (10. 50b). Actually, it is not unreasonable to find that these two converters have identical optimum noise figures, since they have the same noise sources —namely loss at the wr and woQ terminals. However, the value of harmonic pumping is not reduced by this conclusion. The point of interest is that multiple pumping offers such circuit advantages as increased stability and increased gain-bandwidth product, without an increased noise figure. A further comparison of double-pumping noise figure can be based on the conventional upper-sideband converter (which also operates between c and wu ). By (4. 35f), the optimum noise figure of the latter is given by (10. 50) with f = 1, but the conversion gain is less than one in this case. Therefore, to obtain gain in the conventional upper-sideband upconverter, one must operate with less than optimum noise figure and hence must trade gain for noise figure. The corresponding result for the doubly-pumped converter is the necessity for trading output efficiency for noise figure. This point is summarized in Fig. 10. 5, which shows the absolutely minimum noise temperature of a double-pump converter and the corresponding noise temperature of a single-pump converter operating at the same frequency. Also shown is a curve of idler efficiency (z) for which both converters have the same noise figure. Therefore, by accepting a low output efficiency, one can obtain lower noise figures

196 100 70- 503020B. REALIZABLE NOISE 7sin - and double p TEMPERATURE ~ \^.^ L _ — A. MINIMUM THEORETICAL i__~^o~~~~~~~~~~~ ^NOISE TEMPERATURE.7 W.^ ^- (z) TO YIELD CURVE B IN DOUBLE PUMP CASE.01.02.03.05.07.1.2.3.5.7 1 2 3 5 7 10 r2 -- Fig. 10. 5. A comparison of upper-sideband noise temperatures for single- and double-pump up-converters. Curve A is a theoretical limit for two cases: (1) single pump with zero gain, and (2) double pump with zero efficiency (i. e., z = 0 at [wu/Cr] opt). Curve B is the noise temperature for: (1) the single-pump converter operating at the [cu/crl opt for the double-pump case, or (2) the double-pump converter with the z indicated. Therefore, with Curve B both converters have the same frequencies, the same noise temperature, and the same external stability, but the one with two pumps has essentially infinite gain. in a double-pump, rather than in the single-pump, upper-sideband converter. 10. 3. 4. 2 Down-Conversion Noise Temperatures. In the previous discussions of down-conversion, in Sections 4.3 and 8.3, the gain-bandwidth products and noise figures were considerably inferior to those of up-conversion. This result is basic to all reactive mixer configurations, but down-conversion is still of practical interest. There are many applications where even modest down-conversion gain-bandwidth products are useful, if their noise figures are reasonably low. Therefore, any method for reducing down-conversion noise figures is of interest, even if the same configuration yields better noise figures as an up-converter. One technique that falls in this category is harmonic pumping.

197 Following the procedure of the last section, the down-conversion noise figure becomes T d Gfr T f Fur =1 + (y ) T G T (10.51) o ur o By (10. 17) and (10.3) G Tu -je-j co Gur _ U ( ) (10. 52) Gr T + jej ( Y u so again 5 must be selected. In comparison with (10. 39), optimum down-conversion occurs for = -7r/2 (10. 53a) T = T = 1 (10. 53b) r u T real (= Tr). (10. 53c) In this case, (10. 52) becomes G co 2 T Gtr:u ( 2) Or - _( 2 2 (10. 54) ur j +1 and, using the notation in (10. 41), (10. 51) reduces to Td 1 aw F 1 + [1+ z(s- 1)] (10.55) ur T y ywc J For high gain, a - 1, and u 1 -y 9 1 -z Therefore, the minimum noise figure occurs for y; 1, which yields cw T u U (10.56) min 1Q o

198 Thus the minimum noise temperature equals the lower-sideband noise temperature T, and this occurs for or << u. In this case (10. 55) requires z - y z 1, which means heavy idler loading. However, the condition y = 1 is not always possible. By (10. 43a) x = 1 - /R p (10.57a) and by a similar derivation y = 1 - Ru/P (10. 57b) wherel /Cd C \2 / c \2 R ( ) =(u r) R. (10. 57c) c d Tp/ or\ r Therefore, eliminating p yields R/RR Ro R y 1- ru 1 - r u (10.58) 1 -ox coW(1 - x) u In a typical down-converter application, co and R would be specified. Therefore y will depend on the possible choices for co and x. The smaller the latter are, the greater y can be. This point is illustrated in Fig. 10.6, which shows the conversion noise figures that accompany harmonic pumping. It is apparent that the greater parameter flexibility offered by harmonic pumping is advantageous. For example, even if the figure of merit at cw (i. e., 1/ u) is small, u R u) is small, y can approximate one if p = y2p/ypl is large. Therefore, the pump harmonic (y2p) is useful for reducing the down-conversion noise figure, particularly if R is large. This result is also illustrated in Fig. 10. 6, which shows the high-gain down-conversion noise figure for both the double and single pump cases. The latter is taken from Fig. 8. 8. 10. 4 Harmonic Pumping with Dependent Sideband Loading A simple case of harmonic pumping occurs when oD and Co are dependently See (4. 38) and the footnote on page for further discussion of R.

199 5- SINGLE PUMP _ ur/wuul << I \ /^ /.01.02.03.05.07.1.2.3.5.7 1 2 3 5 7 10 ig(FROM. 0. 6 A comparison of double- and single-pump, down-conversion, noise temperatures. In both cases the lower-sideband is present to yield high gains. Its load is assumed to be at.7-,2 -'i.01.02.03.05.07.1.2.3.5.7 1 2 3 5 7 10 Ru/I-X F'ig. 10. 6. A comparison of double- and single-pump, down-conversion, noise temperatures. In both cases the lower-sideband is present to yield high gains. Its load is assumed to be at diode temperature. loaded; for example, by being closely spaced in the same resonance. This basic configuration was treated in Chapter IX for the case of a single pump, and was found to be a means of very wideband video amplification. The question that now arises is, "What further benefits could be obtained by adding the pump harmonic in this case?" A simple qualitative viewpoint of this system is suggested in Fig. 10. 7. All frequencies in the normal pump pass band are in a degenerate parametric amplifier contiguration with respect to the pump harmonic. Therefore, when wc and wc are created by the mixing of or and c, they will receive further amplification from ow2p However, wc and ow are phase coherent with respect to both op and 2p.'Therefore, it may be anticipated that an optimum gain improvement will require a proper phase between co and Wco There are two ways in hh th r an which this problem can be treated analytically. The most convenient, at this point, would be to use the formulation in (10. 3), which is based on a model that treats pumping signals as time-varying capacitors. But the equivalent model,

200 based on (2. 33a), considers instead the pump harmonic and the modulating signal as timevarying capacitors. The latter is a more descriptive model, because it yields directly the output waveform consisting of cp, wu, and co. However, the further pursuit of this second method will be postponed for a later study. While not as elegant, the original formulation gives the desired answers, and hence the second model will be used only as a means of supplementary interpretation. To proceed with the analysis of phase modulation gain with harmonic pumping, (10. 3) will first be solved for the sideband voltages due to a modulating current I. This step yields V T+ + (je-j) u = T- + (je-i) (10. 59a) r jwryp [rT T Tu + TU -(Tr - 2j cos )]; T- (jej ) f = U-(je — ) (10. 59b) r Wrp [7r7 Tju + T~ - U -(r - 2j cos )] where the terms in parenthesis vanish as \y2pl -0. Therefore, when these terms are deleted, the resulting expressions in (10. 59) must describe the phase modulation configuration of Chapter IX. Referring to (9. 17b), maximum phase modulation gain occurs when V and V are in quadrature with yp, the pump fundamental. Maximum gain also occurs when oU and co are nearly centered in the pump tank, which means or = 0. Then, all the T's are real, and the optimum value of ~ can be seen to be iT/2. It is of interest that this value of 5 corresponds to (5. 38), which places yp and y2p in the phase relationship for minimum gain by degenerate parametric amplification. Therefore, when the pump harmonic has maximum effectiveness with respect to ro, C, and cou it also has maximum stability with respect to the degenerate frequency p. Now using = v/2, the remaining quantities in (9. 17b) can be calculated, which results in the following expression for phase modulation gain: 1 o k2p/P 2 8 2k 1+ p / G - k_ jwP P 8t) k 2 k ( 1 + 1j v / )3 r p k2r r ~PM - 3r'3p 1+(k 2-1p -) ( w - tip2^"/ (1+jw/ (10.60a) P -~

201 Wr I WP 2Wp W I I W cu) 8I u FREQUENCY - (a) W,Wu DUE TO MIXING WrAND wp i / p TUNED TO Wp (b) 7ig. 10. 7. Harmonic pumping in a phase modulation amplifier; (a) shows the spectral configuration, and (b) pictures the normal pump pass band as a degenerate parametric amplifier. where Y Y = gp(1 + /p) (10.60b) Y = gr(l+jwc// ) (10.60c) r z r r 2 1 I' 2 I'2 k2 = CPC; k2 = 2 (10.60d) p C C 2p r p C This expression differs from (9. 22) only by the terms in k2p. But, as the latter quantity increases, the gain increases and becomes essentially infinite as ) k2 - /. To explore this case, let (wp 2p) 1 6 (10.61) and then if 6 is small, the midband gain becomes

202 16 C 2k 2 z -- _ p p 2 GPM - (V ) (10. 62) r p Now, the bandwidth will be quite small (i. e., /3 < /p or fr). Therefore, assuming =/3 r for illustration, the bandwidth becomes df~~~~~- -~ 2 (10. 63) r from which the gain-bandwidth product can be found to be GM = 2.82 o k (10.64) PM PP It is of interest that the latter is 60 percent larger than the corresponding result for the single-pump case [see (9. 31)]. This improvement in gain-bandwidth product is due to the factor wpk2p 1 + p (10. 65) in the numerator of (10. 60a), which doubles as k2p increases from zero to the high gain value in (10. 61). This factor of two is precisely the improvement that would result in the single-pump case if the pump fundamental were increased to 2w. Therefore, the main advantage of harmonic pumping in this application is as a convenient way of trading gain for bandwidth. Harmonic pumping produces a narrowing down of the output pass band at cp, which increases the apparent varactor Q. This multiplies the gain, but it decreases the bandwidth proportionately. It is interesting to also inquire about the effect of harmonic pumping on the input admittance. From (10. 4a) 2k) 2 in r gp(l+J r)(l+jp) 2 2 1- P p2p But, in the high gain case wo will be much less than /3 or /p in the overall pass band; so if

203 2k2 2/3p2 = 1 - 6, then 2w 2k 2 Y r Yr 1 + rp (1)j rin r P At the half power point, wr = 6 /3p/2, so 6k 2 Y - Y + P Y r. r 2 r in 6-0 which shows that the reflected admittance is insignificant over the conversion pass band with double pumping, just as it is with single pumping. Therefore the main advantage of double pumping in this case is greater circuit flexibility. While a single-pump converter can yield the same gain-bandwidth product, the circuit bandwidths, r and /3, must then be controlled by passive design, or by varactor fabrication. Often, these designs will call for unsuitable impedance levels and hence will require transformers. With double pumping, one can use natural impedance levels and then rely on the pump harmonic to sharpen the bandwidth and yield the optimum gain-bandwidth product. This could be an important advantage in many cases. 10. 5 Conclusion The introduction of harmonic pumping has been shown to yield an interesting set of circuit properties. Due to the phase coherence of the pump and its harmonic, unilateral conversions are possible. Also, a pseudo-circulation effect arises that improves circuit stability with respect to external parameter variations. One interesting configuration has shown arbitrarily large conversion gains between co and c, with matched input and output terminals. The gain depends on the load at wa, but the match is essentially independent of this load. The noise figure of this converter is as good as a conventional lower-sideband converter with infinite gain; yet, it is more stable than the latter. Also, it offers larger gain-bandwidth products. The application of harmonic pumping to down-conversion has also been successful, particularly from the point of view of noise figure. Noise figures that are significantly improved over those by other down-conversion methods have been calculated.

I

CHAPTER XI EXPERIMENTAL WORK AND CONCLUSIONS Throughout the course of this research considerable experimental work has been performed, but it has been exploratory in nature. While it is difficult to summarize this work precisely, two important experimental functions have been served, which are noted below: (1) In the course of each experiment, a number of more or less unexpected effects were observed. Often these effects called for critical circuit adjustments, particularly those dealing with stability, so precise data were difficult to take. Nevertheless, these effects were clear enough so that appropriate problems for theoretical study could be defined. As such, they served to show the importance of certain parameters that may otherwise have been neglected in analysis. (2) In one case, the preceding theory was used to design a working circuit. Using the principle discussed in Chapter IX, an ultra-wideband video amplifier was designed and constructed —with 20 db gain from dc to 100 Mc. A laboratory version of this amplifier is pictured in Fig. 11. 1. The results of these experimental programs will be the subject of the next two sections of this chapter. 11. 1 An Ultra-Wideband Video Amplifier The configuration of Fig. 11. 1 is a realization of the schematic shown in Fig. 9. 12. The circuit pictured is unnecessarily large, since it contains several nonessential 205

206 Fig. 11.1. A breadboard model of the phase modulation amplifier. Pump frequency 9. 4 kMc. Signal bandwidth 100 Mc. been realized in a much smaller package. The essentials of this circuit are the varactor mount and the circulator, which are shown more clearly in Fig. 11. 2. In the operation of actor. This part should experience a matched resonant circuit at the varactor. The other carrier insertion. Regarding the varactor, it was found that a commercially available varactor in mately a 10 percent frequency range. The optimum frequency for this effect was about 9. 4 kMc, but it varied somewhat with pump level. This matching effect can be attributed to parasidic lead inductance, which reduces the effective shunt resistance of the varactor. Therefore, waveguide, and by tuning the remaining varactor susceptance by means of a sliding short, the varactor will behave like a matched load at this frequency. Since pump power will be reflected pat wih saaialet tedtCIRCULATOphs siRr n atnutr i se o

,SS,8,:~SS~SS,~* SS S,:2 (a) Fig. 11. 2...A close up of the varactor mount in Fig. 1... (a) shows the varactor mount and circulator. (b) shows the varactor mount disassembled. only when a modulating signal is applied, the circulator acts like a balanced modulator in this case. A similar form of balanced modulator, although not as a gain-producing device, has been described recently by Mackey (Ref. 43). If the varactor in Fig. 11. 1 is matched and tuned in the absence of modulation, the reflection coefficient due to a modulation of the varactor capacity (AC) will be jw AC jw AC. 12 A o s 2Y (11. 1) (2Yo)2 + (w) C)2 o a......... t.. va.at o m ou................

208 If this reflected voltage can be converted entirely into useful output, the voltage gain will be PVp _ Q ac v Vr 2 C (11. 2) where the quantity in parenthesis can be of the order of 1/2 for a typical varactor. Therefore, the voltage gain can be as large as 1/4 the varactor Q. Similarly, the voltage gain times bandwidth will be PV Wi p acV lVr l = _4 (V C) V(11.3) r which shows the value of a high pump frequency. By pumping at 10, 000 Me for example, one can obtain gain-bandwidth products in excess of 1000 Mc, returning now to Fig. 11. 1, the reflected signal is passed through the circulator to the detector, where it adds to the second (inserted-carrier) component. The latter is adjusted in phase so the resultant signal across the detector is amplitude modulated. Detection then recovers the original modulating signal in an amplified form. It is of interest that (11. 2) is consistent with the sideband voltage expression in (9. 3). However, one factor not accounted for in Chapter IX is the parasitic series inductance of the varactor, which produces a series resonance at about 3 kMc in the varactors that we tested. Above series resonance, the varactor Q will increase with frequency, as indicated in Fig. 11. 3. Also, shown in Fig. 11. 3 are gain and bandwidth [based on (11. 2)] both above and below the frequency where the parasitic inductance is dominant. It can be seen, in spite of parasitic inductance, that the maximum gain-bandwidth product remains proportional to pump frequency even though the voltage gain itself falls off with pump frequency below selfresonance. The maximum bandwidth is limited by the R/L ratio of the varactor parasitics. Some typical experimental results are shown in Figs. 11. 4 through 11. 6. The first of these shows the response to a low-frequency triangular input signal. The detector output in the absence of the inserted carrier is the typical folded response of a balanced modulator. However, when the second carrier is inserted, the ac envelope "blooms," giving the response shown in Fig. 11. 4. Therefore, the inserted carrier performs two important roles. One is to restore fidelity to the detected output waveform and the other is to provide coherent

209 VOLTAGE GAIN GAIN BANDWIDTH BANDWIDTH SERIES Cup RESONANCE Fig. 11. 3. Theoretical circuit properties of a phase shift amplifier vs. pump frequency. At each frequency the varactor is assumed to be matched to a circulator. OUTPUT SIGNALS'WITH INSERTED a CARRIER INPUT SIGNAL WITHOUT INSERTED' CARRIER Fig. 11. 4. Signal waveforms from phase modulation amplifier in Fig. 11. 1. RF bias for the detector. The latter greatly enhances the sensitivity of the detector. Figure 11. 5 shows the voltage gain vs. pump power. For each change in pump power it is appropriate to readjust both the short behind the varactor and detector tuning. The gain shown in Fig. 11. 5 is essentially GpM, because the output detector was not loaded. Undoubtedly, the greatest weakness in this amplifier system is the output detector. To date, a 1N23B has been the most successful diode used, but it has a rather high output impedance. Also, its output impedance depends rather strongly on the RF matching used in the waveguide.

210 20 10DETECTOR PEAKED AT EACH POINT A- EACH / "-DETECTOR PEAKED AT 100 mw 5z 0 o x / CD 3-.~ I_ I I I I I I I I I I I I I I 2 3 5 7 10 20 30 50 70 100 PUMP POWER (mw) Fig. 11. 5. Voltage gain vs. pump power for 9. 5 kMc pump. It has been found that the voltage gain can be enhanced by matching the detector diode, but it also raises the impedance. Because the detector output impedance is high, it is hard to couple this output efficiently into a receiver or other measuring device. For example, a typical output impedance is 1000 ohms. In this case the output capacity should not exceed 1 uj/if if 100 Mc bandwidth is required. It is anticipated that future work will concentrate on improving the characteristics of the output detector. One possible approach is to pump at S-band, so a 1N21B diode could be used for detection. Experiments show a much lower output impedance for this diode. The data in Fig. 11. 5 was taken with a 10 kc signal. The X-band bandwidth of the varactor and detector mounts are shown in Fig. 11. 6 which shows that a video bandwidth of 300 Mc is possible if appropriate video input and output networks can be constructed. One possible solution to this problem is to use a high gain traveling wave tube for further X-band amplification prior to detection. The detector could be heavily loaded to yield the desired

211 1.0.// / ---- 100 mw INCIDENT z x/ H "LL 0.5 - 0 o -J / IL / \ I't_ x /\ w 8.7 9.0 9.5 0.0 10.5 FREQUENCY (kMc) 1.0 07-0. 0 0 I — 8.7 9.0 9.5 10.0 10.3 FREQUENCY (kMc) Fig. 11. 6. Measured pass bands of the varactor mount and the detector. The latter curve is taken from an oscilloscope trace and is for a 1N23B diode in a triple-tuned mount.

212 video bandwidth, even though its efficiency is low. The net result would be a wideband video amplifier system, with noise figure determined by the varactor stage, and a net gain that is essentially that of the traveling wave tube. These observations illustrate that much work remains to be done on this principle of amplification. Therefore, plans have been made to extend this study in the following directions: 1. To further evaluate the practical limitations on the basic version of this amplifier, by trying to improve detection efficiency and by comparing the performance obtained with different varactor types. 2. To consider multiple varactor structures for the purpose of greatly extending bandwidth. 3. To consider the use of predetection amplifiers, of a more conventional type, to obtain very wideband amplifier systems with low over-all noise figures. 11. 2 Experiments with Double-Pump Circuits The earliest experimental efforts were with the double-pump circuit described in Section 10. 3. A low frequency version of this circuit was built, which is shown in Fig. 11. 7. It has the schematic shown in Fig. 11. 8, which is basically a three-pole network containing two time-varying capacitors; one pumped at wp and one at 2ow. Each pump circuit is balanced so pump tuning can be done independently of signal tuning. The signal circuit was then designed to be simultaneously resonant at 2. 4 Me (= wr/27r), 5. 4 Me (= w /27), and 10. 2 Mc (= w /27T). Appropriately tuned networks were then coupled to the signal network to allow for independent sideband loading. This circuit did show the predicted gain improvement due to harmonic pumping. It also showed the predicted phase criticality between wo and 2o. However, it was hard to P P keep the three signal frequencies aligned well enough to obtain useful data. The alignment difficulty was further enhanced by the generation of negative resistance, which greatly narrows the conversion bandwidth. Therefore, it became apparent that a low-frequency version of this circuit was probably not a simplification. In fact, Y. Kaito of the Nippon Electric Company in

213 2.4-Mc SIGNAL...... - INPUT 7.8-Mc PUMP INPUT ~~~5.610.2 MP Fig. 11. 7. A low frequency version of the BRIDGE BRASS COIL SHIELDS LOWER SIDEBAND 5.44Mc UPPER SIDEBAND**~!0.2 Mc Fig. 11. 7. A low frequency version of the double-pump reactive mixer. Japan has recently employed this double-pumping technique at microwave frequencies (Ref. 39). He converted from 190 Mc to 1220 Mc with 13 db gain and 500 kc bandwidth. His pumping signals were 0. 5 Mw at 1030 Me and 4 Mw at 2060 Me, and to further support the theory in Section 10. 3, Kaito measured an input VSWR of two and an output VSWR of three, which shows a fair condition of match. The circuit in Fig. 11. 7 has had its greatest success in studying stability. In the initial experiments, only the pump fundamental was applied and then the loading at w - and wou was varied independently. In following this procedure, the regenerative and degenerative effects of the lower and upper sideband respectively were clearly seen. A typical result is

214 Wu (10.2 mc) I~ ~~~~~ i A 77 50 I,," )/ o 22.4 i * - It ^ oor (2.4mc) w _ ( 2( 15.4 mc) _p ^ /zpy 100 _ 14_-_-_-_0 172_ _______ PUMP - PUMP 1 -+BIAS p, A ^PC117 IAS Fig. 11. 8. Schematic for Fig. 11. 7. shown in Fig. 11. 9, which shows the effect of a swept signal applied to the w r coupling network. In the absence of pumping, the latter experiences a suck-out (curve A) due to the resonance of the secondary three-pole network. When the pump was applied, the suck-out was modified according to the relative loading at aU and w, as shown in curves B and C. The influence of cw was to reflect negative resistance that greatly sharpened the effect of the secondary. In fact, the primary current could be reduced to zero by loading the lower sideband lightly with respect to the upper sideband. This case corresponds to a net resistance of zero in the secondary. Further, pumping was found to reflect large negative resistances into the primary, but these were expected to cause instability according to the criterion adopted in the early investigations of this problem (Ref. 19). But this experiment made it apparent that some distinction had to be made between short and open circuit stability, which lead to the analysis in Chapter VI. From the

215 PRIMARY CURRENT NO PUMP - WITH 7.8 mc PUMP PRIMARY ONLY B. wQ SUPPRESSED A. SECONDARY TUNED C.wu SUPPRESSED TO 2.4 mc 2,4 mc Wr /27 -- Fig. 11. 9. The response to a swept signal at wr terminals in Fig. 11. 8. The regenerative effects of the lower sideband are clearly distinguishable from the regenerative effects of the upper sideband. latter, it could be concluded that the net series resistance in the primary of Fig. 11. 9 must go through zero, rather than infinity, to produce instability. This criterion is supported by the stable response curves shown in Fig. 11. 10, which show the net primary resistance going negative, first through infinity, and then through zero. Only the latter was found to be unstable. R< 0O R= OD R<<O W r r -- Fig. 11. 10. Stable response curves for Fig. 11. 8, similar to those in Fig. 11. 9. Instability occurs when R = 0, where R is the net primary resistance (initial plus reflected).

216 Although the added complexity of double-pump circuits with independent sideband loading make their application more tedious, they do offer greater flexibility. Still their use may have to wait for more demanding applications that will undoubtedly accompany future advances in the state-of-the-art. However, the relative simplicity of double pumping with dependent sideband loading, as depicted in Fig. 10. 7, may lead to an immediate application of double pumping. To date, no experiments have been conducted on this effect, but the theory offered in Section 10. 4 motivates such an effort. 11. 3 Conclusions The advent of time-varying circuit elements at microwave frequencies has added exciting new degrees of freedom to circuit theory. In this study, a thorough examination has been made of a representative system of this new work area. In particular, the circuit properties of a nonreciprocal three-port network have been analyzed. It is defined by the admittance matrix in (2. 27). The active elements in this network are two time-varying capacitors (i. e., pumps) at harmonically-related frequencies. In addition to the general case, several special cases have been treated. They have been obtained by restricting the original network to: one pump frequency, and two or three signal frequencies. The simplest case of reactive mixing occurs with one pump and two signal frequencies. Since the properties of the latter were well known prior to this study, they have been used as a basis of comparison for the more complex cases that are introduced in this study. The latter have shown advantages in the form of higher gain-bandwidth products, lower noise figures, and greater stability, but by incurring greater system complexity. While the above advantages do not occur simultaneously, it is encouraging that some configurations have shown improvements in one property, without being seriously degraded in others. A number of original contributions have been made in this study, which include the following: 1. Improvements have been made in the conversion between Cr and u, by introducing wQ, but by retaining a single pump frequency. These include: (a) A 6 db improvement in up-conversion gain, while retaining an optimum gain-bandwidth product.

217 (b) Infinite gain improvement, with a 30 percent reduction in gain-bandwidth product. (c) A small reduction in up-conversion noise figure that is further enhanced by cooling the lower sideband load. (d) A significant reduction in down-conversion noise figures. 2. When the pump harmonic is added, further improvements in conversion between ow and w have been noted. These r u include: (a) Infinite nonreciprocal gains, with matched inputs and outputs. (b) Less sensitivity to external parameter variations. (c) Increased gain-bandwidth products. (d) Optimum noise figures, which are the same as those for conventional infinite gain converters, but that can be improved by cooling the lower sideband load. (e) Greatly improved down-conversion noise figures with an asymptotic limit of 3 db. The latter is nearly realizable over a wide parameter range. 3. A new method of parametric video amplification has been treated that offers very wide bandwidths (dc to 100 Me) and low noise figures. WVhen realized with modern reactive mixer components (i. e., varactors), it yields an extremely useful amplifier that can operate in several ways. The method corresponding to phase modulation has been shown to be the most successful. 4. A simple method of estimating the stability of a periodically varying network has been derived. It depends only on a

218 knowledge of the steady-state response, and it is considerably easier than the conventional method of calculating characteristic roots. 5. Experimental observations have been made on all of the effects cited above, and each has been confirmed at least qualitatively. Regarding areas for further study, it should be noted that much work remains to be done in the analysis of time-varying systems. Even in relatively simple systems the number of sensitive parameters tends to be large, so many useful effects remain undiscovered. Hopefully, by building system complexity one layer at a time, the basic properties of timevarying networks will become more evident. Perhaps this will aid the development of a more complete theory. Also, much work remains in practical applications. It takes considerable time to learn techniques for working with nonlinear reactance elements, but once mastered these elements can often be used in relatively simple circuit configurations. Then, as each circuit is reduced to its simplest form, it becomes ready for the next level of sophistication.

APPENDIX A THE DERIVATION OF OPTIMUM NOISE FIGURE FORMULAS The basic noise figure formula for lower-sideband reactive mixers has been given in (5. 27). This formula can be written more simply as 1 + gdrd + r) ) (A-1) where T is the effective noise temperature of the total conductance at w [see (5. 27c)] Considerably study has been given to this relation, which applies both to conversion gain and to parametric amplification. Perhaps the earliest of these studies was by Knechtli and Weglein (Ref. 40), although it was followed closely by Kotzebue (Ref. 41). A more elegant, but somewhat later treatment, was by Greene and Sard (Ref. 42) who studied (A-1) as it applies to a one-port parametric amplifier with circulator (see Section 5. 3). Their approach is followed here. Using the notation gS gL X Z = gr gT then T - = 1 + z(t-1) 219

220 and (A-1) becomes F= 1 + { + [1 + z(t-1)i} - (A-2) An added constraint on (A-2) is that for large gains ( (r (1-x)(l-z) Pk -:=- -- _ - ~ 1 ^(A-3) Eliminating W r/ C from the last two relations yields Td1 Rr[l+z(t-1)] (A4) F = 1 + - d I 1 + (1-z)(-x) 1- (A-4) Two basic observations can now be made regarding (A-4): (1) Because this relation increases monotonically with z, any external loading at w (i. e., z f 0) will raise the noise figureyunless this extra loading has a zero degree noise temperature (t = 0). (2) There is an optimum input efficiency x for any R, t, and output efficiency z. Each optimum x corresponds to a unique w k/ co ratio, so F will improve with increasing op only up to a point. This point depends basically on varactor quality. The noise figure relation in (A-4) has a general form that appears frequently in reactive mixer analysis, which is F = 1+ (1 )-1 (A -5) T [ 1 + -1] The parameter a depends basically upon properties of the varactor, but other effects may be included. When (A-5) is minimized with respect to x, the optimum input efficient is found to be

221 opt = 1 + a - a + a2 1 - (A-6) ~~opt ~1 + which is plotted in Fig. A. 1. The corresponding optimum noise figure is (F)opt 1 + 2a T- [I + (A-7a) so the optimum noise temperature is (T)opt = 2 Td [ + H1a] (A-7b) which is also plotted in Fig. A. 1. From this figure it is evident that small a's are desirable. For example, T is less than T, if a is less than 0. 1. For lower-sideband mixers, whose noise figures are given by (A-4), the parameter a has the value r[ l+z(t- 1)] lY (A-8) a - - (A81-z Therefore, by (A-3), the optimum choice of frequencies is ~r c(r- - /-opt =(opt z) (A-9a) ( ) i[ + Rr[+z(t-l)] ] (A-9b) and by (A-7a) 2R [l+z(t-l)] (F) + r 1 + (A-10) (F)lower sideband 1-z R[l+z(t-l) (AThe previous studies cited (Refs. 40, 41, and 42) have considered (A-10) as it applies to the one-part parametric amplifier. A later study, by Karokawa and Uenohara (Ref. 30), has shown that (A-l) through (A-4) also apply to conversion from wr to W, so (A-10) is the optimized value of (A-4) in this case too. The latter optimization is further discussed in Section 5. 2. 2.

222 100 /10 / Rd OPT/.01 0.1 1.0 10 Fig. A. 1. Optimum noise temperatures for (A-5) vs. the general parameter a. The main difference that arises between the amplifier and the converter is that z is proportional to the output load in the converter and hence cannot be zero. However, for a given z, the optimum noise figure of the converter can be found as above, which yields the result in (5. 32). Therefore, the optimum converter noise figure exceeds that of an amplifier with the same input frequency wr. This fact should serve to correct some misleading statements in the literature. For example, on p. 721, Ref. 30 says that "... the amplifier and converter give the same noise figure..." A more accurate statement would be that they give the same noise figure formula (for high gains) but not with the same minimum obtainable value. The more general optimum noise figure relations in (A-7) will be shown to apply to many reactive mixer configurations. Therefore, they provide a convenient basis for comparing various reactive mixers, which is the purpose of this study.

APPENDIX B THE DERIVATION OF PHASE MODULATION NOISE FIGURE The derivation of (9. 63) and (9. 64) proceeds as follows. Letting w /3 r; q = r + /P (B-1) J 3 p J/rp rp Or then (9. 61) and (9. 62) become 2 2 ( \r2 1 -p2 +k 2 p2 + jpu 2'",~'= g p + p (B-2) iun = ginK )p 1 - p2 - 1/2 p(q- p)k2 + (B-2) //3^ \ 1 -p2 +k 2 p2 +jpu Y 2 = gP2 i 2 P (B-3) 2 KP 3P) 1/2 p(q + p)kp2 (B-3) When these equations are combined to form (9. 60), terms of odd symmetry in wr can be dropped, since the integral of (9. 60) is desired. The noise power delivered to gp now becomes oc 4kT co 3 fr dp P S P(w )dwr - tp -c c + r r p {[1 _p (1 _ p-)] 2 + (pU)2 +(k2 P2 (p~ + 22) (B-4) + p p x 2' 2 (B-4) [1 - P (1 - k 2 )]2 + (pU) 2 and if kp2 << 1 and pZ << q2, this relation can be reduced to 4kT oo J r p dp [ 1/2 (kp 2 pq)2 - 2p 7 f r 1 5) 223

224 which is equivalent to (9. 63). When (B-5) is integrated term by term, the first term yields (Dwight 120. 01) 4kT _ _ O_ __ dp 4kT 4kT 27dp _2 (I O') ( 2- T (r) ( (B-6) 2 7T O; r (iT " ) In the second term, (1-p2)2 +(pu)2 (p + -)(p2 +u2 -2) U if u2 > 4. Therefore, for the case where /3 >> p, so that u2 P >> 1, the second term in (B-6) becomes approximately (Dwight 120. 2). However, at the other extreme, if p/3 = r so u = 2 4kT) c 1/2 (k q)2p2 dp 4kT (4 ) /Q u (23 - (2,_P/ k 4 q2 (j) /3 (B-8) 2', Orop 2f r +p 2 ki2 p ~f q ( u Tr) which is the same as (B-7) with u = 2. Therefore (B-7) is valid for u > 2. Equation (9. 64) can now be obtained directly from (B-6) and (B-7). It should be noted that the approximations made in obtaining (B-7) introduce positive errors, so (9. 64) is a slightly conservative estimate of output noise power.

REFERENCES 1. W. W. Mumford, "Some Notes of the History of Parametric Transducers," Proc. IRE, Vol. 48, May 1960, pp. 848-853. 2. E. F. W. Alexanderson, "A Magnetic Amplifier for Radio Telephony, " Proc. IRE, Vol. 4, April 1916, pp. 101-149. 3. H. Torry and 0. A. Whitmer, Crystal Rectifiers, MIT Radiation Laboratory Series, McGraw-Hill Book Co., Inc., New York, New York, Vol. 15, 1948. 4. J. M. Manley and E. Peterson, "Negative Resistance Effects in Saturable Reactor Circuits," Trans. AIEE, Vol. 65, December 1946, pp. 870-881. 5. L. C. Peterson and F. B. Llewellyn, "The Performance and Measurements of Mixers in Terms of Linear Network Theory," Proc. IRE, Vol. 33, July 1945, pp. 458-476. 6. A. Van der Zeil, "On the Mixing Properties of Nonlinear Condensers, " Journal of Applied Physics, November 1948, pp. 999-1006. 7. R. V. Pound, Microwave Mixers, MIT Radiation Laboratory Series, McGraw-Hill Book Co., Inc., New York, New York, Vol. 16, 1948. 8. A Uhlir, Jr., Possible Uses of Nonlinear Capacitor Diodes, Bell Telephone Laboratories, Eighth Interim Report on Task 8, Signal Corps Contract No. DA-36-039-565589, July 1956. 9. A. E. Bakanowski, The Nonlinear Capacitor as a Mixer, Second Interim Report on Task 8, Crystal Rectifiers, Signal Corps Project 2-7-323A, December 1954. 10. A. C. Macpherson, "An Analysis of the Diode Mixer Consisting of Nonlinear Capacitance and Conductance and Ohmic Spreading Resistance, " IRE Trans. on Microwave Theory and Techniques, Vol. MIT-5, January 1957, pp. 43-51. 11. J. M. Manley and H. E. Rowe, "Some General Properties of Nonlinear Elements, Part I. General Energy Relations," Proc. IRE, Vol. 44, July 1956, pp. 904-913. 12. C. F. Edwards, "Frequency Conversion by Means of a Nonlinear Admittance, " Bell System Technical Journal, Vol. 35, November 1956, pp. 1403-1416. 13. S. Duinker, "General Properties of Frequency Converting Networks, " Phillips Research Reports, Vol. 13, February 1958, pp. 37-78, April 1958, pp. 101-148. 14. H. Heffner and G. Wade, "Noise, Gain and Bandwidth Characteristics of the Variable Parameter Amplifiers, " presented at IRE PGED meeting, Washington, D. C., October 31 to November 1, 1957. (Abstract, IRE Trans. on Electron Devices, Vol. ED-5, April 1958, p. 112.) 15. G. G. Herrmann, M. Uenohara, and A. Uhlir, Jr., "Noise Figure Measurements on Two Types of Variable Reactance Amplifiers Using Semiconductor Diodes, " Proc. IRE, Vol. 46, June 1958, pp. 1301-1303. 225

226 REFERENCES (Cont.) 16. D. Leenov, "Gain and Noise Figure of a Variable Capacitance Up-Converter, " Bell Systems Technical Journal, Vol. 37, July 1958, pp. 989-1008. 17. C. L. Hogan, R. L. Jepsen, and P. H. Vartanian, "New Type of Ferromagnetic Amplifier, " Journal of Applied Physics, Vol. 29, March 1958, pp. 422-423. 18. Hsu Hsiung, "Multiple Frequency Parametric Devices, " Digest of Technical Papers, 1959 Solid-State Circuits Conference, Philadelphia, Pennsylvania, February 1959. 19. 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, Ann Arbor, Michigan, September 1959.) 20. D. K. Adams, "Circuit Properties of a Double-Sideband, Doubly-Pumped Nonlinear Reactance Modulator," Proc. of National Electronics Conference, Vol. XVI, October 1960, pp. 480-486. 21. H. Brett, F. A. Brand, and W. G. Matthei, "A Varactor Diode Parametric StandingWave Amplifier, " Proc. IRE, Vol. 49, No. 2, February 1961, pp. 509-510. 22. D. K. Adams, An Analysis of the Brett Ultra-Wideband Video Amplifier, Cooley Electronics Laboratory Technical Report No. 122, The University of Michigan, Ann Arbor, Michigan, July 1961. 23. B. J. Leon, "A Frequency Domain Theory for Parametric Networks, " IRE Trans. on Circuit Theory, Vol. CT-7, September 1960, pp. 321-329. 24. J. M. Manley and H. E. Rowe, "General Energy Relations in Nonlinear Reactances, " Proc. IRE, Vol. 47, December 1959, pp. 2115-2116. 25. J. M. Manley and H. E. Rowe, "General Energy Relations in Nonlinear Reactances, " Proc. IRE, Vol. 44, July 1956, pp. 904-913. 26. S. Sensiper and R. D. Weglein, "Capacitance and Charge Coefficients for Parametric Diode Devices," Proc. IRE, Vol. 48, August 1960, pp. 1482-1483. 27. M. T. Weiss, "Quantum Derivation of Energy Relations Analogous to Those for NonLinear Reactances, " Proc. IRE, Vol. 45, July 1957, pp. 1012-1013. 28. F. E. Terman, Electronic and Radio Engineering, McGraw-Hill Book Co., Inc., New York, New York, 1955. 29. E. M. T. Jones and J. S. Honda, "A Low-Noise Up-Converter Parametric Amplifier," IRE Wescon Convention Record, August 18-21, 1959, pp. 99-107. 30. K. Kurokawa and M. Uenohara, "Minimum Noise Figure of the Variable-Capacitance Amplifier," The Bell System Technical Journal, Vol. XL, No. 3, May 1961. 31. H. E. Rowe, "Some General Properties of Nonlinear Elements, II. Small Signal Theory," Proc. IRE, Vol. 46, May 1958, pp. 850-860. 32. P. P. Lombardo and E. W. Sard, "Low-Noise Microwave Reactance Amplifiers with Large Gain-Bandwidth Products, " 1959 IRE Wescon Convention Record, August 18-21, 1959, pp. 83-98.

227 REFERENCES (Cont.) 33. F. R. Moulton, Differential Equations, Dover Publications, Inc., New York, New York, 1958. 34. S. Bloom and K. K. N. Chang, "Parametric Amplification Using Low Frequency Pumping, " Journal of Applied Physics, Vol. 29, March 1958, p. 594. 35. K. K. N. Chang and S. Bloom, "Parametric Amplifier Using Lower Frequency Pumping, " Proc. IRE, Vol. 46, July 1958, pp. 1383-1386. 36. K. K. N. Chang and S. Bloom, "A Parametric Amplifier Using Low Frequency Pumping, " 1958 Wescon Convention Record, Pt. 3, pp. 23-27. 37. L. A. Blackwell and K. L. Kotzebue, Semiconductor-Diode Parametric Amplifiers, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1961. 38. R. C. Mackey, "Some Characteristics of Microwave Balanced Modulators, " IRE Trans. on Microwave Theory and Techniques, Vol. MTT-10, March 1962, pp. 114-117. 39. Y. Kaito, "Two-Frequency Pumping Parametric Amplifier, " Convention Record of the Institute of Electrical Communication Engineers of Japan, May 1962. 40. R. C. Knechtli and R. D. Weglein, "Low-Noise Parametric Amplifiers, " Proc. IRE, Vol. 47, April 1959, pp. 584-585. 41. K. L. Kotzebue, "Optimum Noise Performance of Parametric Amplifiers, " Proc. IRE, Vol. 47, October 1959, pp. 1782-1783. 42. J. C. Greene and E. W. Sard, "Optimum Noise and Gain-Bandwidth Performance for a Practical One-Port Parametric Amplifier, " Proc. IRE, Vol. 48, September 1960, pp. 1583-1590.

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