Technical Report ECOM-01870-189 November 1967 IMPEDANCE CHARACTERISTICS OF PUMPED VARACTORS C. E. L. Technical Report No. 7695-189 Contract No. DA28-043-AMC-01870(E) DA Project No. 1 PO 21101 A042. 01. 02 Prepared by David E. Oliver COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan for U. S. Army Electronics Command, Fort Monmouth, N. J. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL-S.

ABSTRACT Since the publication in 1956 by Manley and Rowe of their now famous study of power flow in nonlinear reactive media, a considerable amount of work has been reported on the analysis of pumped varactor diode circuits. However all these studies have been limited in generality by two simplifying assumptions: the limitation of the pump voltage to low amplitudes, and the restriction of signal power flow to only a few sidebands. The first of these assumptions excludes the possibility of harmonic generation by the nonlinear diode, while the second presupposes the existence of ideal filters in the linear circuitry. This study is concerned with an analysis of pumped varactor diode circuits without these two restrictions. Because the governing nonlinear equations are extremely complex and are insoluble using algebraic techniques, recourse was made to numerical techniques utilizing the digital computer. The input impedances of several basic pumped varactor diode circuits were analyzed theoretically by determining a suitable model for each circuit and then predicting its iii

impedance characteristics. The results showed good correlation with experimental measurements, indicating the adequacy of the circuit model, and the validity of the analysis methods. In developing an accurate circuit model, a number of interesting and useful results were discovered. It was shown that several circuit parameters, which had never before been considered important, significantly affect circuit behavior. These parameters include: the exact nature of the diode depletion layer capacity-voltage characteristic; the circuit termination at the pump harmonic frequencies and their sidebands; the minority carrier lifetime of the semiconductor material; and the character of the electric fields in the diode junction. The relationships between the exact expressions derived here and those obtained using the assumptions of previous workers were explored and clarified. A practical technique was developed for experimentally measuring the pump voltage amplitude across the diode junction. This method is useful for all save the smallest pump voltage magnitudes and does not depend upon the presence of rectified current flow through the diode. The most significant outcome of this study is the development of a circuit model which can accurately characterize a varactor diode circuit. This model together with mathematical techniques developed here makes possible the accurate analysis of this type of circuit. iv

TABLE OF CONTENTS Page ABSTRACT iii LIST OF TABLES vii LIST OF SYMBOLS viii LIST OF ILLUSTRATIONS xix LIST OF APPENDICES xxii I. INTRODUC TION 1 1. 1 Statement of Problem 1 1. 2 Review of Literature 4 1.3 Topics of Investigation 6 1. 4 Thesis Organization 11 II. FUNDAMENTAL PROPERTIES OF PUMPED VARAC TORS 13 2. 1 Introduction 13 2.2 Depletion Layer Capacity 14 2. 3 Diffusion Capacity 21 2.4 Varactor Equivalent Circuit 32 2.5 Manley-Rowe Relations 35 III. HARMONIC BALANCE RELATIONS 40 3. 1 Introduction 40 3. 2 Elastance Expansion 41 3. 3 Capacitance Expansion 49 3. 4 Short Circuit and Open Circuit Assumptions 51 3.5 Summary 65 IV. STABILITY 66 4. 1 Introduction 66 4. 2 The Ferroresonant Effect 67 4.3 Other Instabilities 80 v

TABLE OF CONTENTS (Cont.) Page V. DEPLETION LAYER CAPACITY 84 5. 1 Introduction 84 5. 2 Pump Circuit Configuration 87 5. 3 An Iterative Technique Applicable to Single Tuned Circuits 98 5. 4 An Iterative Technique Applicable to Distributed Circuits 106 5.5 Small Signal Equations 112 5.6 Summary 117 VI. DIFFUSION EFFECTS 119 6. 1 Introduction 119 6. 2 Pump Circuit Representation 121 6.3 Iterative Techniques 127 6.4 Small Signal Equations 135 6.5 Summary 145 VII. EXPERIMENTAL WORK AND CONCLUSIONS 146 7. 1 Introduction 146 7. 2 Experimental Methods 151 7. 3 Experimental and Comparative Theoretical Results 156 7. 4 Conclusions 192 REF ERENCES 245 DISTRIBUTION LIST 250 vi

LIST OF TABLES Table Title Page 2. 1 Comparison of capacities. 31 3. 1 Frequency notation. 46 6. 1 Iteration sequence. 133 7. 1 Lumped and diffusion capacity circuit parameters. 155 7.2 Distributed circuit parameters 156 B. 1 Fourier coefficients for N = 1. 0. 215 B. 2 Fourier coefficients for N = 2. 0. 216 B. 3 Fourier coefficients for N = 3. 0. 217 B. 4 Fourier coefficients for N = 4. 0. 218 B. 5 Fourier coefficients for N = -1. 0. 219 B. 6 Fourier coefficients for N = -2.0. 220 B. 7 Fourier coefficients for N = -3. 0. 221 B. 8 Fourier coefficients for N = - 4. 0. 222 E. 1 Varactor measurements. 234 E. 2 Measured parameters of diode PC0622C. 243 vii

LIST OF SYMBOLS Defined by or Symbol Meaning first used in w angular frequency Section 1. 1 op angular frequency of the pump Section 1. 1 W1 angular frequency of the applied input signal Section 1. 1 ~, n, m dummy subscript integers Section 1. 1 mn m + nI Eq. 1. 1 mn p 1 qt total charge stored in varactor junction Eq. 2. 1 Vt total voltage across varactor junction Eq. 2. 1 C varactor junction capacity Eq. 2. 1 Ca varactor junction capacity at zero bias Eq. 2. 1 0 semiconductor contact potential Eq. 2. 1 M non-integer exponent Eq. 2. 1 f function of one variable Eq. 2. 2 QC constant of integration Eq. 2. 3 VBias bias voltage across diode Eq. 2. 4 VN normalized time varying voltage component across abrupt junction diode Eq. 2. 4 QBias DC charge stored in diode junction Eq. 2. 5 QN normalized time varying charge component in abrupt junction diode Eq. 2. 5 K1, K2 normalization in constants Eq. 2. 5 S elastance of diode Ea. 2.9 viii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in S5 normalized small signal elastance at the bias point Eq. 2.9 CN normalized capacity of abrupt junction diode Eq. 2. 13 SN normalized elastance of abrupt junction diode Eq. 2. 14 VL normalized time varying voltage across linear junction diode Eq. 2. 16 QL normalized time varying charge in linear junction diode Eq. 2. 16 CL normalized capacity of linear diode Eq. 2.17 SL normalized elastance of linear diode Eq. 2. 18 x distance from junction Section 2. 3 t time (real) Section 2. 3 q magnitude of charge on an electron Section 2. 3 p thermal-equilibrium hole density in n- region Section 2. 3 T lifetime of holes in the base Section 2. 3 j current density due to holes Section 2.3 p P(x, t) excess hole density in n- region Section 2. 3 D diffusion constant for holes Eq. 2.22 E magnitude of retarding electric field Eq. 2.22 Ax mobility of holes Eq. 2.22 K Boltzmann constant Eq. 2.23 T junction temperature Eq. 2.23 ix

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in Total total current through diode which is attributable to diffusion Eq. 2.25 Ip current transported by motion of holes Eq. 2. 25 eI current transported by motion of.electrons Eq. 2.25 e 2. 718281827 Eq. 2.27 u(a, component of hole density of frequency w and x=0 Eq. 2.28 00o DC component of hole density at x=0 Eq. 2.28 j f-T Eq. 2.29 I diffusion current through diode at frequency w Eq. 2.30 electric field term Eq. 2.30 V W voltage component at frequency co Eq. 2.32 Y junction admittance due to diffusion Eq. 2. 33 P normalized hole density Eq. 2.33 TDC direct current through junction Eq. 2.34 Ls diode series inductance Section 2. 4 Cp diode package capacitance Section 2. 4 Rs diode series resistance Section 2. 4 B. imaginary part of junction admittance Section 2. 4 G. real part of junction admittance Section 2. 4 P power entering the reactance at c Eq. 2.35 x

LIST OF SYMBOLS (Cont. ) Defined by or Symbol Meaning first used in Pi power entering the reactance at the pump Eq. 2.36 u Wp + W1, upper sideband frequency Eq. 2. 36 P u power entering the reactance of the upper sideband Eq. 2.36 co coWp - W1' lower sideband frequency Eq. 2. 38 PQ power entering the reactance of the lower sideband Eq. 2.38 nu power entering the reactance at nop + w1 Eq. 2. 40 Pnk power entering the reactance at nwp - wc1 Eq. 2.40 BQp pump component of charge Eq. 3. 2 q s signal component of charge Eq. 3. 2 V s signal voltage across junction Eq. 3.3 Sn complex Fourier coefficient of elastance at frequency nop Eq. 3.6 is signal current through varactor Eq. 3.8 nw co1 component of signal current at p frequency nwop co1 Eq. 3.8 Vnw ~1o component of signal voltage at p frequency nwp ~ w Eq. 3. 12 On special notation for signal frequency components Table 3. 1 I n signal current component at frequency Con Eq. 3. 14 Vn signal voltage component at frequency Cn Eq. 3. 14 xi

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in [S] depletion layer impedance matrix Eq. 3. 15 [i ] signal current through depletion layer capacity Eq. 3. 15 [v ] signal voltage across junction Eq. 3. 15 Zmm m, n element in [S] Eq. 3. 16 Vp pump component of varactor voltage Eq. 3. 21 C complex Fourier coefficient of depletion layer capacity at frequency nwp Eq. 3.21 [C] pumped diode depletion layer admittance matrix Eq. 3.22 Ymn m, n element in [C] Eq. 3.23 [Y'(n)] n-frequency admittance matrix Eq. 3. 29 amn submatrix of [Y'(n)] Eq. 3.30 9n submatrix of [Y'(n)] 1 Eq. 3. 31 n ymn m n element of the a22 matrix Eq. 3. 32 7y pumping "hardness" coefficient Eq. 3. 34 Q M normalized charge Section 4. 2 VM normalized time varying component of voltage Section 4. 2 Co; small signal resonant frequency Section 4. 2 a amplitude of sinusoidal waveform Section 4. 2 T phase angle Eq. 4. 8 V normalized voltage across tuned circuit Eq. 4. 8 xii

LIST OF SYMBOLS (Cont. ) Defined by or Symbol Meaning first used in IIN current source Eq. 4.7 Q quality factor of circuit Eq. 4.7 L inductance Eq. 4.7 a small parameter Eq. 4.7 ac change in capacity Eq. 4. 11 6 frequency deviation from resonance Eq. 4. 16 E.in amplitude of voltage source Eq. 4. 17 Et perturbation of qt Eq. 4. 19 q0 amplitude of sinusoidal charge Eq. 4.20 v, C, I normalization constants Eq. 4. 22 QA(n) coefficient of cos nopt Eq. 5. 1 QB(n) coefficient of sin nw pt Eq. 5. 1 s complex Laplace parameter Eq. 5. 3 Z Thevenin equivalent impedance Section 5. 2 L1 primary inductance Section 5. 2 L2 secondary inductance Section 5. 2 M1 mutual inductance between L1 and L2 Section 5. 2 Cs parasitic coil capacity Section 5.2 G inductor loss Section 5.2 ZL impedance of ideal inductor Section 5. 2 X reactance of shorted transmission line Section 5.2 xiii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in R resistance of shorted transmission line Section 5. 2 wL resonant frequency of transmission line Section 5. 2 Z0 characteristic impedance of line Section 5. 2 a' and b' inner and outer diameters of coaxial line Section 5. 2 a conductivity Section 5. 2 V LIN voltage across linear circuit Eq. 5. 8 EA cosine component of E.in Eq. 5.9 A in EB sine component of E.in Eq. 5.9 R(n) resistance at nth harmonic Eq. 5. 11 X(n) reactance at nth harmonic Eq. 5. 11 I. current source Eq. 5. 15 in Qm estimate for Qafter m iterations Eq. 5. 19 Em error after m iterations Eq. 5.20 QAm estimate for QA after m iterations Eq. 5. 30 QBm estimate for QB after m iterations Eq. 5. 30 eA(n)m coefficient of cos nw pt in series for Em Eq. 5.31 B(n)m coefficient of sin n opt in series for Em Eq. 5.31 ZLR impedance of lossy inductor Eq. 5.37 Zcoil impedance of inductor with parasitic capacity Eq. 5.39 xiv

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in W c coil self resonant frequency Eq. 5.41 56 mequation error after m iterations Section 5. 4 e ' correction term in mth iteration Eq. 5. 12 m [Zs] linear signal impedance matrix Section 5. 5 Zn nth element in [Z ] Section 5. 5 [Vs] input signal matrix Section 5. 5 Vi component of input signal at co Eq. 5.55 'N Norton current source Eq. 6. 1 IN ILIN current through linear network Eq. 6. 1 'd diffusion current through diode Eq. 6. 1 Ip current through barrier layer capacity Eq. 6. 1 pA cosine component of IN Eq. 6.2 A cosine component of IN Eq. 6. 2 Y admittance of linear circuit Eq. 6. 3 G(n) conductance at nth harmonic Eq. 6. 4 B(n) susceptance at nth harmonic Eq. 6.4 VA cosine component of Vp Eq. 6. 11 VB sine component of Vp Eq. 6. 11 EN voltage source Section 6. 3 e error in initial approximation to solution Section 6. 3 V' estimate for Vp after m iterations Eq. 6. 12 XV

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in VAi cosine component of V' Eq. 6. 12 Am m v sine component of V' Eq. 6. 12 VBm m Fm, G constants in mth iteration Eq. 6. 15 m' m [Is] input signal current matrix Eq. 6. 19 [I1s,] signal current through linear circuit Eq. 6. 19 [Ids] signal current through diffusion capacity Eq. 6. 19 [YS] linear admittance matrix Eq. 6. 22 Y nth element in Y Eq. 6.23 n s Pp carrier density due to pump Eq. 6. 28,n excess hole density component at x=O, at frequency nwp Eq. 6.28 P carrier density due to signal Eq. 6.29 [Yd] diffusion admittance matrix Eq. 6. 34 [Yt] total admittance Eq. 6.35 M2 mutual inductance Section 7. 2 L3 inductance Section 7.2 CBI' CB2' Section 7- 2 CB1, CB2, blocking capacitor B3, CB4 Cfl, Cf2, Cf3 feed through capacitor Section 7. 2 CR1, CR2 varactor diode Section 7. 2 F frequency term in matrix Eq. A. 3 xvi

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in B cofactor of / Eq. A. 4 nl,n mn F' frequency term in matrix Eq. A. 5 G.in input admittance to ladder network Eq. A. 7 S M elastance of M law diode Eq. B. 4 normalized time varying charge in M law diode Eq. B. 5 K3 normalization constant Eq. B. 5 N general non- integer exponent Eq. B. 7 K' normalized elastance or capacitance Eq. B. 7 Kn normalized coefficient Eq. B. 10 G' real part of -1/jnw Z(nwp) Fig. C. 1 n p p Y~ imaginary part of - l/jnwp Z(nco) Fig. CO 1 n p p A(n)m coefficient of cos nwpt Eqo Do 2 5B(n)m coefficient of sin nw pt Eqo D. 2 E (n) coefficient of cos co t Eq, D. 3 e j(n)m coefficient of sin wp t Eq, Do 3 B M. p R1n) real part of -41 + Z(nwp)] Eq. D. 4 (n) Xn) imaginary part of -[1 + Z(nwp)] Eqo D. 4 r, s integer dummy indices Eqo D. 7 IF forward current Eqo E. 2 R resistor in pulse circuit Eq. E. 2 EF pulse generator output before switching Eq. E 2 xvii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in ER pulse generator output after switching Eq. E. 2 VF diode voltage before switching Eq. E. 2 VR diode voltage after switching Eq. E. 3 T length of reverse current pulse Eq. E. 3 IR reverse current Eq. E. 4 xviii

LIST OF ILLUSTRATIONS Figure Title Page 1. 1 The basic representation of a pumped varactor circuit. 8 1.2 Pumped varactor circuit consisting of lumped elements. 8 1. 3 Pumped varactor circuit consisting of a transmission line circuit. 9 2. 1 Typical varactor DC characteristics 15 2. 2 Typical varactor capacitance characteristics 17 2.3 Varactor equivalent circuit. 33 2. 4 Uhlir varactor model. 34 4. 1 Parallel RLC network excited by a current source. 68 4.2 Fractional increase in average capacity necessary to produce ferroresonant effect. 74 4.3 Frequency response curves for parallel resonant circuit containing a varactor diode. 75 4.4 Theoretical results showing region of ferroresonant instability. 77 4. 5 Experimental results showing region of ferroresonant instability. 78 4. 6 Series resonant circuit containing a varactor diode. 79 5. 1 Thevenin equivalent pump circuit. 86 xix

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 5. 2 Practical pumped varactor circuits. 8$ 5.3 Pump circuit configuration used in Chapter V. 95 5.4 Equivalent circuit for an inductor with parasitic capacity. 104 5. 5 Signal circuit configuration used in Chapter V. 114 6. 1 Varactor circuit used to investigate diffusion effects. 120 6. 2 Norton equivalent pump circuit. 123 6. 3 Pump circuit configuration used in Chapter VI. 123 6. 4 Exponential function and linear approximations. 132 6. 5 Signal circuit configuration used in Chapter VI. 134 7. 1 Block diagram of measurement apparatus. 147 70 2 Circuit used for measurement on lumped circuit and diffusion circuit. 143 70 3 Circuit used for measurement on distributed circuit. 149 7. 4 Pumping hardness as a function of resonant frequency shift. 150 7. 5 Diode voltage waveforms. 157 7.6 Total current waveforms. 158 7. 7 Partial current waveformns. 159 7. 8 Harmonic elastance coefficients for a lumped circuit. 163 70 9 Harmonic elastance coefficients for a lumped circuit with distributed capacity. 164 XX

LIST OF ILLUSTRATIONS (Cont.) Figure Title Page 7. 10 Harmonic elastance coefficients for a distributed circuit. 165 7. 11 Harmonic elastance coefficients for an actual diode. 166 7. 12 Comparison of harmonic elastance coefficients for different circuit conditions. 167 7. 13 Input admittance of the lumped circuit. 170 7. 14 Comparison of methods for computing the input admittance of the lumped circuit. 174 7. 15 Input admittance of the distributed circuit. 176 7. 16 Comparison of methods for computing the input admittance of the lumped circuit. 178 7. 17 Effect of harmonics upon the input admittance. 181 7. 18 Input admittance of the diffusion circuit. 184 7. 19 Effect of diode model upon the input admittance. 189 A. 1 Resistive ladder network. 203 C. i The computer flow diagram to solve the pump circuit equation for single tuning. 225 D. 1 The computer flow diagram to solve the distributed circuit equation. 233 E. 1 Measured depletion layer capacity. 235 E. 2 Measured stored charge. 236 E. 3 Test circuit for pulse measurements. 238 E. 4 Minority carrier density in base of diode. 239 Eo 5 Normalized switching time for diode with charge storage. 244 xxi

LIST OF APPENDICES Page APPENDIX A: INVERSION OF MATRICES USED IN CHAPTER III 199 APPENDIX B: VOLTAGE AND CURRENT PUMPING 208 APPENDIX C: THE COMPUTER FLOW DIAGRAM TO SOLVE THE PUMP CIRCUIT EQUATION FOR SINGLE TUNING 223 APPENDIX D: THE COMPUTER FLOW DIAGRAM TO SOLVE THE DISTRIBUTED CIRCUIT EQUATION 226 APPENDIX E: MEASUREMENT OF VARACTOR DIODES 234 xxii

CHAPTER I INTRODUC TION 1. 1 Statement of the Problem This thesis is concerned with analysis of the input impedance characteristics of a pumped varactor diode circuit. This analysis, which includes both theoretical and experimental studies, avoids simplifying assumptions which would restrict the accuracy and generality of the results. Although the response of pumped varactor diode circuits to small input signals has been studied extensively, most of this effort has been concentrated upon low-noise amplifying devices. Reflection type parametric amplifiers have received particularly wide-spread attention, although both parametric up-converters and phase-shift amplifiers have also been discussed frequently. Despite the multitude of studies reported on parametric circuit development, no general study of the behavior of these circuits has been reported. Certain assumptions are inherent in all the published analyses and cause appreciable error in many cases. This study examines the response of some simple pumped varactor circuits using a method free from these assumptions. Consequently, the analysis presented here is more accurate than any previously reported study. It is capable of accurately predicting pumped diode circuit behavior, even under conditions of hard pumping where the applied pump power is of such magnitude that the diode conducts over part of the pump cycle.

The operation and terminology of pumped varactor circuits are conveniently reviewed by considering Fig. 1. 1 which shows all the basic elements present in a pumped varactor circuit in block diagram form. These elements are 1) A varactor diode. This is a semiconductor PN junction designed for operation under reverse-biased conditions such that the principal impedance component comes from the nonlinear voltage-dependent depletion layer capacitance. 2) A linear network. This network may include lumped or distributed circuit components. It contains a load resistor which absorbs the output signal, as well as the impedances associated with the external sources. It may also contain filters designed to separate the various frequency components generated in the parametric action. 3) A current or voltage source at frequency wp. The function of the source is similar to that of the local-oscillator in a mixer circuit. In parametric circuits it is usually designated as the pump. 4) A current or voltage source at frequency w1) This source provides the signal to be processed through amplification or frequency conversion. The pump at frequency co and the input signal at frequency -1 are coupled parametrically by the nonlinearity of the varactor 2

capacitance. In general, the resulting signal contains all positive frequencies in the set = m + n ~ n, m ~ 1, +2, Wm =1' O,~1,~2,... (1. 1) In practice, the amplitude of the input signal is small compared with the pump amplitude to minimize the nonlinearity of the signal processing. The frequency of the input signal, wl, is usually smaller than the pump frequency w p. The frequencies are then restricted to the set Wm = mWp + w (1. 2) where as before m = 0, 1, 2,... but for linear processing to occur, f can have only three values = 0, -1, +1 The components at frequencies where f = 0 are the pump and its harmonics. The largest and most important of these is the pump fundamental (m = 1, - 0). First order approximations, which are valid at small pump amplitudes, ignore all other components and treat both pump voltage (or current) and the resulting variations in diode capacity (or elastance) as sinusoidal. For "hard" or large amplitude 3

pumping, the sinusoidal approximations are often not valid, and more accurate analytical methods are required. Components at frequencies where Q = -1, or +1 have the same information content as the input signal. Frequencies in this set are denoted as signal frequencies. The output signal may be taken at any of these signal frequencies. However, it is usually taken at a,, p +,1' or wp - W 1. 1. 2 Review of Literature The literature on pumped circuits principles dates back more than a century to Faraday and Rayleigh, who showed that oscillation can be produced in a mechanical resonant system by supplying energy or "pumping" at a frequency twice that of the fundamental resonant frequency of the system (Ref. 1). Prior to World War I, the principle was expanded to include frequency ratios other than two-to-one and was applied to electrical networks. In 1916, Alexanderson built a "Magnetic Amplifier" (Ref. 2). Although he was primarily interested in modulation, or up-conversion, he did present data showing that negative resistance and parametric gain could exist. The next really significant advance occurred during World War II when it was discovered by Torrey and Whitmer (Ref. 3) that certain microwave diodes had a nonlinear contact capacitance and could be used for amplification. Shortly thereafter, van der Ziel recognized the potential for building a low-noise amplifier (Ref. 4). fiowever. it was 4

not possible to realize this potential until 1954 when Uhlir and his coworkers at Bell Telephone Laboratories developed low-loss silicon nonlinear capacitors (varactors) (Refs. 5 and 6). Since that time, developments have been rapid, (Ref. 7) The Manley-Rowe relations which describe the power flow at various frequencies in an arbitrary nonlinear reactance were published in 1956, (Refs. 8 and 9). In April 1957, Suhl (Ref. 10 ) proposed construction of a parametric amplifier using a nonlinear ferrite material, and in July of that year experimental realization was reported (Ref. 11). Bloom and Chang (Ref. 12) in their paper of Dec. 1957 considered theoretically the performance of parametric amplifiers using nonlinear inductors, and the following year both Heffner and Wade (Ref.13) and Rowe (Ref. 14) published analyses of parametric amplifiers utilizing nonlinear capacitors. All these analyses assumed that the nonlinear element was varying sinusoidally at the pump frequency (cop) They also assumed that only two voltages were present: one at the original frequency (w1); and the other at the difference frequency or lower sideband (p - 1 ) Leenov (Ref. 15)in a study of varactor up-converters published in 1958 used a somewhat different set of assumptions. He assumed sinusoidal elastance (elastance is the reciprocal of capacitance) and the presence of only two currents at frequencies L1 and w + 1. Three years later, Robinson (Ref. 16) and Penfield and Rafuse (Ref. 17)

again assumed sinusoidal elastance and the existence of only a limited number of currents. They advocated analyses which limited the number of currents and treated the elastance variation as sinusoidal rather than the methods used in previous analyses which restricted the number of voltage components and treated capacitance variation as sinusoidal. In a paper on parametric amplifier noise, Kurokawa and Uenohara (Ref. 18) discussed both the case in which two currents (at 1 and wp + w 1) are allowed to flow as well as the case in which two voltages (also at w and p + w 1) are permitted to exist. Adams (Ref. 19) in a 1962 paper allowed both sideband currents to flow along with the input signal current: wl, wp + w 1, and wp - w. He also relaxed the capacitance waveform assumption somewhat by considering a second harmonic component in addition to the fundamental. Korpel and Ramasaway's (Refs. 20, 21) 1965 study of pumped varactor input conductance was premised on essentially the same assumptions as that of Adams (Ref. 19), except that they recognized that the average capacity of a varactor is dependent upon pump amplitude. 1. 3 Topics of Investigation The objective of this study is the accurate analysis of pumped varactor diode circuit behavior. Detailed study of a general circuit, such as that shown in Fig. 1. 1, does not yield results which are directly applicable to circuit design. To make the problem tractable, while 6

retaining some generality, it was decided to study the circuits in Figs. 1. 2 and 1. 3. These two circuits are simple, but they contain most features found in more complex, practical circuits. The results of the study of these basic circuits are directly applicable to the analysis of more complex networks. The device shown in Fig. 1. 2 is a very simple pumped varactor circuit. The circuit is assumed to be resonant near the pump frequency, so that a large voltage is produced across the diode with a small expenditure of power. Most varactor diode pump circuits utilize resonant tuning to minimize the pump power requirement. The pump circuit resonance may also be adjusted for tuning at, or near, a sideband frequency. If the lower sideband (wp - w I) is near resonance, and the upper sideband is outside the passband, the device is a simple)negative-resistance parametric amplifier. The device can also be made to behave as a simple positive- resistance varactor up-converter by placing the upper sideband near resonance, and the lower sideband below the passband. If the sidebands are tuned symmetrically, the device behaves as an elementary double- sideband or phase-modulation circuit. The simple circuit in Fig. 1. 2, thus can represent a large class of useful circuits. The distributed circuit shown in Fig. 1. 3 is somewhat more complex. Most pumped varactor circuits are actually built using distributed rather than lumped elements, and this simple transmission line circuit was chosen to represent this class of devices. The lumped 7

Varactor Signal Linear Pump Source 1 Network Source Output Fig. 1. 1. The basic representation of a pumped varactor circuit Z. in Varactor Pump Source Mu tul Bias Voltage M Inductance Mutual Inductance Fig, 1. 2. Pumped varactor circuit consisting of lumped elements

in Transmission Line L Varactor Short Short Circuit Bias Voltage r Coupling Loop Pump (p Source Fig. 1. 3. Pumped varactor circuit consisting of a transmission line circuit The input impedance for both circuits 1. 2 and 1, 3 is defined as the ratio of signal voltage (at frequency wl) to signal current. Currents and voltages are present at other frequencies (w1, wp+ 1 Z p+ 1'. o. etc. ), and they affect the impedance significantly.

circuit (Fig. 1. 2) has only one resonance (or passband), while the distributed circuit (Fig. 1. 3) has an infinity of resonances. This results in marked differences in the circuit behavior. The theoretical method used to determine the behavior of both circuits will include four steps: (1) The pump voltage waveform will be found by solution of a nonlinear differential equation. Once the voltage across the diode is known, its small signal capacity can be computed easily at every instant of time during the pump cycle. Under these circumstances it is unnecessary to assume that the diode capacitance (or elastance) is sinusoidal, since this quantity can be computed as accurately as desired. (2) The capacitance (or elastance) found in (1) will be expressed as a Fourier series, with the fundamental at the pump frequency, (co) and additional terms at harmonics of the pump. (3) The Fourier coefficients found in (1) and (2) will be inserted into the harmonic balance equations. These describe the mixing of the pump and its harmonics with the input signal (at frequency w1). (4) The set of harmonic balance equations will be solved, and the impedance at the input signal frequency will be found. 10

The whole process will be systematized so that all four steps of the computation can be carried out on the digital computer. 1o 4 Thesis Organization Chapter II is a discussion of background material that is necessary for the remainder of this study. Two important but unrelated topics are discussed: (a) the properties of varactor diodes, and (b) the Manley-Rowe general energy relations, Chapter III discusses the harmonic balance equations. These are a general formulation of mixing in a lossless, nonlinear reactance. A discussion of the assumptions made by others and the relationships between these assumptions and the exact expressions is included. Chapter IV is a short discussion of pump stability in varactor circuits. The conditions under which the circuit is unstable are determined, and compared with experimental results. The comparison indicates that the simplified method of analysis which is developed in this chapter is adequate for at least a quantitive understanding. Chapter V is a theoretical study of pumped diode circuits (Figs. 1. 2 and 1. 3) under conditions of small signal pumping. The nonlinear differential equation describing pump circuit behavior is set up and a method for solving it is presented. However, the solution is not in closed form, necessitating use of numerical methods for a specified set of input conditions. 11

Chapter VI is a theoretical study of diffusion or conduction effects in the circuit of Fig. 1, 2. The results are applicable even under conditions of very hard pumping, where the diode is reversebiased with a DC voltage, but is driven into forward conduction during a portion of the pump cycle by the large AC pump voltage, Chapter VII contains all the numerical results from the studies of Chapters V and VI, as well as the results of a rather extensive experimental investigation, The experimental and numerical results are then compared and discussed. This chapter closes with the conclusions of the study, 12

CHAPTER II FUNDAMENTAL PROPERTIES OF PUMPED VARACTORS 2. 1 Introduction The performance of a pumped varactor circuit is intimately dependent upon the characteristics of the diode. The diode capacityvoltage relationship, the diode resistance, the carrier lifetime, etc., all significantly influence circuit behavior. Thus, it is appropriate to review some fundamental properties of varactor diodes prior to setting up and solving the nonlinear equations describing the circuit behavior. In the next two sections, 2.2 and 2. 3, the basic equations describing varactor diodes are first set down and then put in special forms which will be utilized in later chapters. A circuit model which accurately characterizes the varactor is presented in Section 2. 3, and is used throughout the remainder of this paper. Some additional background material of a different nature is discussed in Section 2. 4, where the Manley-Rowe equations are presented. These are the general energy relations which describe the lossless varactor mixing process. Unfortunately, they can provide only a limited description of circuit performance, and hence are not used quantitatively in this paper. They can provide some important insights into circuit behavior, however; some of this insight will be particularly useful in Chapter VI, where the numerical results of the computer 13

programs and the experimental results are set down and interpreted. 2. 2 Barrier-Layer Capacity Modern varactor diodes are made from p-n semiconductor junctions, which exhibit a nonlinear capacitance when biased in the voltage range between forward conduction and reverse breakdown. A varactor diode is built to utilize this nonlinear reactive effect while minimizing the bulk series resistance of the semiconductor material. Figure 2. 1 shows the DC voltage current characteristics of a typical microwave varactor. The current in the forward direction increases exponentially, doubling in magnitude every 18 my. In the reverse direction, breakdown typically occurs between -5 and - 150 volts, depending upon the method of manufacture. In between, the conduction current is negligible and the device shows a voltage dependent capacitance called the barrier, or depletion layer capacity. The physical origins of this capacity are thoroughly discussed by several authors (Refs. 22, 23, 24, 25, 26), whose results are only summarized here. Under reverse bias,the barrier layer capacity is the principal factor in determining the capacity of the varactor diode. At forward bias, however, the diffusion capacity associated with current flow may become important (Ref. 27). The barrier layer capacity of a p-n junction is independent of frequency up to at least 20 GHz (Ref. 28). It can be expressed approximately as: 14

(ICamps) +40 - Current +30 +20 + 10 - - Voltage -8 -6 -4 -2 +2 +4 +6 -10 -20 -30 -40 Fig. 2. 1. Typical varactor DC characteristics 15

dqt A C t A C (2. 1) dvt 1 M where vt = applied voltage C = junction capacity at zero bias a 0 = contact potential qt = charge stored in junction and M= 2 for an abrupt junction device M= 3 for a graded junction device The value of 0 varies from 0. 5 to 1. 2 volts depending upon the semiconductor material and the manufacturing process. The exponent is a function only of the junction configuration. Actual devices can often be closely approximated by using a value between 0. 25 and 0. 50 for the exponent, and then adjusting the value of 0 for the best curve fit. Figure 2. 2 shows the measured capacitance-voltage curve of a typical alloy junction varactor along with a very close approximation of the form (2. 1). In Chapters V and VI, where the circuit equations are set up and then solved, it is necessary to have an expression of the following from which relates voltage to charge: 16

100 16 10 --- C 1 asymptote - 1v12 t...C 1 1 theoretical abrupt junction 1 (1 —)2 - _ Actual diode PC 1430 D 10 4-} T y c' U -O. 1 -1 -1O -100 Bias Voltage Fig.:.. Ty.:i var ractor capacitance characteristics

Vt = f(qt) (2. 2) In the abrupt junction case, a relation of this form can be obtained from Eq. 2. 1. The first step is to find the charge stored in the junction. This is done simply by integrating Eq. 2. 1. 1 / 2 t a2C0( t ) Q (2.3) where Qc is a constant of integration. It is convenient here to introduce new variables. Let both the total voltage and charge be the sum of two terms. The first term is dependent only upon DC bias; the second is a function of the tinme varying signals. K1 and K2 are constants which will be used to scale both VN and QN' and thus obtain the simplest possible form for expression (2.2). Vt Vgias + K1VN(t) (2. 4) qt - QBias + K2QN(t) (2. 5) Combining Eqs. 2. 3, 2. 4 and 2. 5 /+ias K1QN(t) V -2 Ca6) QBias + KQN(t) = C Qc (2.6) Qc can now be chosen so that QN(t) is zero when VN(t) is zero. 13

Q 2 Biasl- BCias (2.7) ce ~qBias ~.a 0 - ~ ) ~'~} Equation 2. 6 can be combined with Eq. 2. 7, solved for VN(t) and rewritten -(0 - VBia) K QN2(t) K 2QN(t) VN(t) = (Q as-Q - (QBias Qc) (2.8) 1 L Bias c The small signal elastance at the bias point is found from Eq. 2.8. A d VN(t) 2(0 - VBias) K2 __= _ Bis 2(2.9) s d QN(t) (QB as -QC K Bias c 1 limit QN- 0 Expressions 2. 1, 2. 8, and 2. 9 can now be scaled by normalizing both signal and impedance to convenient values: (1) Normalize the impedance level by setting the small signal elastance at the bias point equal to one daraf. Ss =1 (2. 10) (2) Normalize the signal level by setting the "total DC bias" equal to VBias- K 1 (2. 11) 19

Then from Eq. 2. 9, Eq. 2. 10, andEq. 2. 11, K2 can be written as: -(QBias- Qc) K2 2 The normalized equation for voltage (Eq. 2. 8) now becomes Q (t)2 VN(t) = QN(t) + 4 (2. 12) This is the desired expression having the form of Eq. 2. 2. In normalized form, the equation for capacitance (Eq. 2. 1) becomes CN(t) = 1 (2. 13) [1 + VN(t)12 The small signal elastance can be computed from Eq. 2. 12 d VN(t) QN(t) (2. 14) SN(t) = dQN(t) = 1+ 2 (2. 14) In Eqs. 2. 12 through 2. 14, the variables are constrained by the following limits: QN(t) > -2 VN(t) > -1 (2. 15) SN(t) > 0 20

The same procedure can, of course, also be carried through for the diode having a linear variation of the majority carrier concentration in the junction area. VL(t) - 1+ [1+~ QL(t)] (2. 16) CL(t) = 1 (2. 17) [1 + VL(t)] SL(t)= [1 3 QL(t)] (2. 18) The normalization here is carried out exactly as in the abrupt junction case (Eqs. 2. 10 and 2. 11). The limits, however, are different. QL(t) > - VL(t) > - 1 (2. 19) SL(t) > 0 2. 3 Diffusion Capacity Diffusion capacity arises from conduction processes in a p-n junction, which occur through charge carriers (electrons and holes) being injected across the junction, and diffusing into the region on the opposite side. Once across the junction they continue to exist as minority carriers having a finite lifetime, and traveling with a finite velocity. 21

The circuit effect of this process is a frequency-dependent capacity with a large associated loss. For mixing processes, this compares unfavorably with the low loss, frequency independent capacity formed by the p-n junction depletion layer. This section includes a short discussion of some p-n junction physics, for, unlike the depletion layer capacity which is treated in several references, there appears to be no adequate discussion of the mixing processes occurring in diffusion capacity. To compute the diffusion capacitance, it is necessary to make several idealizing assumptions concerning the nature of the p-n junction. The assumptions made in this paper can be summarized as follows (Refs. 25, 27, 29, 30, 31). 1) The problem can be reduced to one dimensional variations along the x-axis. This assumption introduces little error, for in all practical configurations, the radii of the junction and the area of the junction are both much greater than the junction thickness. 2) The device can be represented by a step junction, in which the doping changes discontinuously at the junction. A depletion region which is swept free of charge carriers is formed around this junction. The potential across the depletion region is the sum of the applied voltage plus the contact potential. Although this assumption is an obvious over-simplification, exact computations for some special cases show it to be surprisingly good (Refs. 32, 26). 22

3) The energy of the charge carriers can be described by Boltzmann statistics. This condition is applicable for varactor diodes constructed of silicon and germanium and for most GaAs diodes. However, it may break down for GaP. 4) The p- region is so heavily doped that the electron current can be neglected. Forward current then is carried entirely by holes, which leave the p-region to become minority carriers in the nregion. Reverse current is due to holes originating in the n- region, which are swept across the junction into the p-region. This assumption is consistent with current device fabrication techniques for silicon devices. 5) A retarding electric field which is assumed to be constant with distance exists in the n- region. This assumption is clearly at odds with the step junction approximation. It represents only a first order approximation to the actual fields in the junction. 6) The n- region is long compared with the diffusion length of holes in the material. This is a valid assumption for most devices, but is only approximately satisfied for some low loss epitaxial devices which have very thin n-type layers. 7) The hole current is constant throughout the depletion layer. Both transit time across the depletion layer and hole recombination in this layer are neglected. This assumption is almost always valid for germanium PN junction, but is less accurate for silicon. 23

8) The minority-carrier densities are small compared with majority carrier densities. This is the "low level injection" assumption (Ref. 29). It allows both the carrier lifetime and mobility to be treated as constants. This assumption is always valid for low current levels, but breaks down under conditions of high currents and high resistivitity n-type materials. Under these assumptions, the continuity equation is valid for holes in the n- region. - a P(x, t) P(x, t) - = q + q (2.20) ax at + whe re j (x t) is the current density due to holes (p-type carriers) q is the magnitude of the charge on an electron P(x, t) is the excess hole density above pn?n is the normal thermal-equilibrium density of holes in the n- region T is the lifetime of holes in the n-type material. Since the carrier density cannot be less than zero, P(x, t) has a lower limit given by P(x,t) > -Pn (2.21) The transport equation for holes in the presence of an electric field also is valid in this situation. 24

*-j (x,t) = qD axt) + q E i P(x, t) (2.22) where D is the diffusion constant for holes in the n-region E is the magnitude of the retarding field.L is the mobility of holes in the n- region The diffusion constant and the mobility are related by the so- called Einstein relationship IL q (2.23) D KT Differentiating Eq. 2. 22 with respect to x and combining it with the continuity equation (2. 20) yields an equation which describes hole flow in the n- region. D (x, t) a P(x, t) P(x t) a P(xt) Since this is a linear differential equation, superposition and a variety of transform techniques are applicable. In general, the current through any cross section of the diode is carried by both holes and electrons, that is, I = I +1 (2. 25) Total P e 25

However, one of the assumptions made here (assumption number 4) is that the current across the junction itself is carried entirely by holes. Then, taking the coordinate system origin at the junction, the total current is given by I(t) = I = -q DA aP(t) - qAEL P(o, t) (2. 26) Total a x=O x=O where A is the area of the junction. The final important relationship to be introduced in this section comes from Boltzmann statistics. It gives the instantaneous density of excess carriers at the junction in terms of the instantaneous voltage across the junction (Ref. 33). P(o,t) P (eKTvt -t (2. 27) This equation, in contrast to the equations describing hole flow in the n- region, is nonlinear. To compute the diffusion capacity of a PN junction at a general frequency w, assume that the excess hole density at the junction has a component aW, at that frequency plus a constant term, (o 26

P(o,t) = Re a cJt + cr (2. 28) The only restrictions on ac is that -Iar i + oa > -P wo o - n The behavior of holes in the base region can now be expressed as a boundary value problem. The coordinate system is set up so that the base region extends indefinitely in the positive direction starting at x = O. D a P(x, t) aP(xt) P(x, t) E a P(x t) (2.29 ax = T ax P(o,t) - Re ca jWto + } P(cc, t) = l(t) = -qDA aP(x,t) -q AEpI P(o, t) Total x=O The solution to this boundary value problem can be found using conventional techniques. Since the differential equation for the excess hole density is linear, the component of current at frequency w is dependent only upon c~r and is independent of any other signal component which may be present. Using the set of equations (2. 29), the component of current at frequency co is found to be 27

qA(1+jwT) ac I = (2.3C) 2 KT 1 + + (1+jT)( where 4 K2 T2 DT q2 E2 The voltage across the diode can be obtained from the Boltzmann relation, (Eq. 2. 27). vt Rear + a ejwt |v (t) - e K f K n + (2. 31) Pn If a is assumed to be small, this can be simplified and the voltage component at w identified. V = KT ( j (2. 32) co\ Cq o pr n If ao contains any time varying components. the situation is somewhat more complex due to frequency mixing. This case is discussed in Chapter VI. The admittance of the junction is found from Eqs. 2. 30 and 2. 32. Diffusion capacity is defined as the imaginary part of this admittance divided by co. 28

I= (KT) qA(1+ jW)(~O+ Pn) --- - (2.33) or q V qP (1+ jWT)KT Bias KT Pn(l+ j(T) e Y where q A pn n T qE 2 KT Under these conditions, the DC current through the junction is p KT Bias 1 DC - (2 3 where VBias is the DC voltage, and a0 is the excess carrier density associated with this voltage. It is interesting to compare Eq. 2. 33 with that for depletion layer capacity (2. 1). Both depletion layer and diffusion capacity are functions only of the instantaneous diode voltage. Depletion layer capacity, however, is frequency independent and lossless, while diffusion capacity is not frequency independent and has a large loss associated with it at all frequencies.

Some further comparison of diffusion and depletion layer capacity can be made with a numerical example. The data presented here was obtained experimentally on a high quality diode having a measured capacity-voltage relationship of the form in Eq. 2. 1. The measurement techniques, together with a more complete set of data are presented in Appendix E. The results are merely summarized here. Carrier lifetime T = 0.5,isec "Normalized carrier density" Pn = 10 12 amps Electric field term = 0. 1 Junction field E = 600 volts/cm Capacity at zero voltage C' = 33. 1 pf a Contact potential 0 = 1 volt The following table (Table 2. 1) was computed by using this data in Eqs. 2. 1 and 2. 33. It illustrates clearly why the diffusion capacity can be neglected at reverse-bias voltages, and why it is important under forward-bias conditions. 30

Diffusion Capacity Bias Depletion- layer Voltage DC Current Capacity @ Mc @ 1000 Me + 0. 5 0. 24 ma 47. 0 pf 4500 pf 390 pf + 0. 4 4. 45 /i amps 42. 9 pf 89 pf 7. 1 pf 0. 2 0. 0015 Iiamps 37. 1 pf 0. 028 Pf 2. 4 x 10- pf 0 0 33. 1 pf 9. 6 x 10 6 pf 0. 79 x 10- 6 pf 4.0 -- 14. 8 f — -10.0 -- 10. pf Table 2. 1. Comparison of Capacities

The actual reverse current of the diode is several orders of magnitude larger than the values calculated above using Pn. This additional current is carried by holes and electrons which are generated thermally within the depletion region, and are then immediately swept out by the electric field. Diffusion capacity changes by a factor of about 10 for every 60 mv change in junction potential. 2.4 Varactor Equivalent Circuit Varactor operation can be described by the equivalent circuit in Fig. 2. 3 (Refs. 5, 17, 34, 35, 36). The diode package is characterized by the package capacity Cp, and a series inductance Ls. The impedances of this equivalent circuit cannot be attributed to individual effects, but must be considered as a lumped equivalent circuit. The values in the equivalent circuit hold only at one frequency, and at other frequencies the parameters have to be re-evaluated. The junction itself is usually formed on a base of n-type material. To achieve a reasonable reverse breakdown voltage, this material must have a high resistivity (Ref. 37). The resulting "bulk" resistivity is the major source of loss in varactor diodes. Usually, it is independent of both voltage and frequency. T-However, some high quality epitaxial varactors are formred on very thin base layers (Ref. 38). Changes in voltage, which change the width of the depletion layer,cause an appreciable percent change in base width for these diodes. 32

r - - - - - - - - Semiconductoor I ulk Junction I Resistance AdmittanceI Ls I Rs jB ISI I I Fig. 2.3. Varactor Equivalent Circuit 33

R C(v) s t) o- - A A —o Fig. 2.4. Uhlir varactor model 34

The total admittance of the junction consists of the admittance of the barrier layer capacity in parallel with the admittance due to diffusion effects. The nature of these admittances are discussed in Sections 2. 2 and 2. 3. Measurements by Eng (Refs. 24, 39) indicate that for some devices skin effects on the semiconductor surface are important. To account for this he proposes that a linear network consisting of two resistors and one capacitor be placed across the junction. The reactances of the package and of the skin effect are small and can often be neglected, but in any case, they are linear and can be lumped into the external circuit. If, in addition, R can be considered independent of voltage, the diode model reduces to the socalled Uhlir Model, Fig. 2.4 (Ret. 5). This model usually includes only the effects of barrier layer capacity, but may be easily extended to include diffusion effects as well. It will be used throughout the remainder of this paper. 2.5 Manley-Rowe Relations The relationships which describe the real power flow through a nonlinear capacitor were first derived by Manley and Rowe (Refs. 8, 9): cc o0 nm P V V mn /Z. / dmw + nam m.a=0 n=-oc p 1 (2. 35) cc cc nP ~n v.mn = m=- Al m o + nco m=-cc n=0 p 1 35

where P ll is the power entering the reactance at frequency mra + no1 Usually power entering is taken as positive, while power leaving is taken as negative. These relationships are applicable to a lossless capacitance and give the inherent limitations governing power flow at a set of frequencies. 2. 5. 1 Single Sideband Up-converters. The circuit which allows power to flow at only three frequencies w 1' sap. and w + w 1 is pI p 1 termed an upper sideband up-converter. Power is applied to the circuit at the signal and the pump. while the output is taken at their sum (upper sideband). The Manley-Rowe relationships (Eq. 2. 35) in this case reduce to P' P p+ ~ = O P u (2. 36) P P 1 u - -_ +0 1 u where o= co + e1, i e., the upper sideband. P' is the power supu p p plied by the pump, and P1 is the power supplied by signal source. A little algebra reduces these to 36

u 1 p Up p' 1 Ip P P (2. 37) 1 P p co u u P1 C1 The first of these equations (2. 37) is simply conservation of energy; the power out at upper sideband is the sum of power in at signal and pump. The second shows that the two sources contribute to the converted power in the ratio of their frequencies. The third equation predicts that the conversion gain from 1 to Wu is proportional to the frequency ratio. The circuit which utilizes the lower sideband, i. e., W = p - c 1 in addition to the signal and pump is called a parametric amplifier. For these frequencies, Eq. 2. 35 reduces to p p 1 WIQ (2. 38) P1 P_ -- 0 1 c.Q These equations reduce immediately to Pi -1= - (e (2.39) 37

Power is not supplied at Pp (i. e., Pa < 0); and therefore, P1 < 0. This means more power is leaving the circuit at w 1 than is supplied. The resultant circuit effect is a negative resistance at w 1 and at Lo. This is in contrast to the upper sideband up-converter in which the signal source supplies a net positive power to the circuit, which results in a positive resistance. 2.5. 2 Multiple Sideband Up-converter. In the most general linear case, power is allowed to flow at the set of frequencies (1. 2). The second of the Manley-Rowe relations (2. 35) then reduces to P1 P Pu P2f P2u P3 PL u 2u P2u 3 _______ + + P3u + +... = 0 (2. 40) p 1 Here Pne and PnU denote the power flow at the lower and upper sidebands of the nth pump harmonic (i. e., at nwp -w and nwp + 1) The power flow at the signal frequency is P2 P2 P3f P1 L1l + 2w -w1 3w -W 1 u p 1 p 1 38

Now P P < 0 c Pu PP P fNo 2Q f ' u 2u' 3u' ' 0 because power is supplied to the circuit only at 1 and w p. Thus, power output at any of the upper sidebands (P, P2u' P3u ) causes power to flow into the circuit at 1,; but power output at any of the lower sidebands (Ply P21f' P3'.. ) causes power to flow out of the circuit at w 1 The resultant circuit effect is that the lower sidebands tend to cause negative resistance at wl, while the upper sidebands tend to cause positive resistance at w1. By virtue of the frequency ratios, the power at the upper and lower sidebands of the pump fundamental usually determine the sign of this resistance, but the sidebands of the pump harmonics can sometimes be significant.

CHAPTER III HARMONIC BALANCF RELATIONS 3. 1 Introduction The objective of this chapter is to present the Harmonic Balance Relations (Ref. 40). These are a general formulation for parametric circuit analysis. In Section 3. 2, a set of linear equations relating diode voltage to currents is developed by expaniding the diode depletion layer elastance in a Fourier series. Iin this set of equations, diode currents are the independent variables. The dual set of equations involving diode depletion layer capacitance. and ai indeperndent set of voltages is presented in Section 3. 3. The relationships developed in these two sections form an essential step in the comlputation of the input imnpedance; and they will be used in both Chapters V and VI. The two sets of equationxs are compared in Section 3. 4. It is shown that the two sets of equations are intimately related to the two types of assunmptions which have been made previously by workers in this field., and the relationship between these two types of assumptions are explored and clarified.This provides a convenient framework in which to examine the exact analysis presented in succeeding chapters.

3. 2 Flastance Expansion The sniall signal (differential) elastance of a varactor diode can be written as a function of the total charge stored in the junction. such that dv 1 - t (3. 1) S(qt) C dq (3.1) kt tdq where S(qt) represents the small-signal elastance and C(qt) the smallsignal capacitance; vt is the total potential across the junction and qt is the charge. This definition of elastance is consistent with the definition of varactor incremental elastance as used by Penfield and Rafuse (Ref. 17). The potential across the jun-ction can also be expressed as a function of charge. If the charge co.nsists of a large component Qp due to the "pump" source plus a "small signal" charge qs, vt can be written in the form of a Taylor series expansion t;S dvv 1 2+ ~, 2 _ + v(1) = +t(Q l +q - V(Q ) + d2t t.Idqt 3 qt =p qt Qp where Now let - (Q i)tv 41

The first term is simply the voltage due to the pump in the absence of any signal. The remainder of the expansion is an expression for the signal voltage vs dv dvvt2 vqdq s d.+ (3. 3) dqt dqQ qt=Qp qt=Q qt=Qp When qs is small Land v(qt) is sufficiently smooth], the first term of the expression is dominant, and the relationship between the signal voltage and signal charge may be approximated by the linear equation. dvt v s t O q (3.4) qt-Qp whenever qs p< Qp This expression can be rewritten by using the definition of elastance in Eq. 3. 1 vS S(Q) qs (3. 5) If the charge due to the pump is periodic with frequency Wp, a Fourier expansion may be written for the depletion layer elastance 42

o00 j nwO t S(Q S e P (3.6) n — - where Sn = S* because S(Qp) is a real quantity. n -n p Now diode small-signal voltages and currents must be found. These must satisfy Eqs. 3. 5 and 3. 6 and be consistent with the constraints imposed by whatever linear network which is connected to the terminals of the diode. It is easy to show that if the external signal source is sinusoidal at frequency wcl, these conditions are satisfied when the signal current consists of components at the positive frequencies in the set. np + w1 n= 01,... (3. 7) That is c j(n.+w 1)t + c j(nwo -w 1)t sp1 n- - (n+o n~=-ocm (npw W1) p (3. 8) The charge must be of the form 43

j(nop+C 1) t q s i(+c ( pw1) n= p 1 j+c (nw -) )t p1 (3.9) +jI 1 L(n-W )o t Equations 3. 5, 3.6 and 3.9 can be combined to obtain the signal voltage, vs, across the diode. S I e +cC m (nw ~tW e, = -c P P m. 1:(=-c cp 1 ~j+ SI (3. 10) i j(nwe - ) m, n=-oc p 1 It is clear from Eq. 3. 10 that vS contains the same set of frequencies as is, that is, all the frequencies in the set (3. 7). In particular there are no voltages (and thus no currents) at any frequencies in the set nwI + 2w P- n= 0 1, 2... nw~ + 3Lo (3.11) p- 1 The presence of these conponents would not be consistent 44

with the linear equation (3. 4). However, if the signal amplitude becomes large, the higher order terms in Eq. 3. 3 cannot be neglected and the additional components appear. From a practical point of view, their existence means that the signal is being processed nonlinearly. They can always be rendered insignificant simply by reducing the signal amplitude. No signal frequencies other than those in (3. 7) can exist if linearity is to hold. Equation 3. 10 can be rearranged and written in the form j(nw +o 1) t +O S n-n (na +co e1) m, n- - c p 1 j(nco -c ) t S I e P +oc n-n (mwco -co1) /X....j(mw -a ) Im, n=-oc p 1 The individual terms in this equation can now be identified, and the signal voltage can be written as +oc j(no +co 1)t v V = e + Y V e n=-cc p 1n=-c p p 12) where the voltage coefficients can be identified as (Ref. 40) 45

+Oc n-m (mw +1) (p+ 1) m=-cC p 1 (3. 13) VOC Sn- m I(M cop1 (nw +cc (nl (w -o ) (Wp r1) m=- ( nwp 1 Because both v and i are real quantities IS=V S p 1 p and (nw +~w) (-nw -w ) It is convenient here to introduce some special notation. Each signal frequency in the set (3. 7) can be assigned a subscript according to the scheme of Table 3. 1. Special Notation for Signal Frequency Conmponents Frequency w1 W-W 1 W +W1 2p- 2Wp w1 Special notation l w 2 o 3 co4 5 Table 3. 1. Frequency Notation Lower sidebands are assigned even subscripts, while upper sidebands are denoted by odd subscripts. The relationship between positive frequency components and those at negative frequency can now be written as 46

I = I* n -n V = V n -n The coefficients of Eq. 3. 13 can be expressed conveniently in terms of this special notation. The results for the first few coefficients are written out here: SO T1 S1 I_2 S_1I3 S2 -4 S_2 15 V. + + + + 1 Jw1 -jW2 jW3 -j 4 jW5 Sil-I S 12 S2 I-3 S VI_ 1C 2 3 + 1 4 V = + + + + 2 -j 1 2 -jo 3 jo4 S ll S2 I-2 S0 I3 S3 I-4 V3= +. + + +.. 3 jwo -jw jo3 -jLL4 1 2 3 4 S2 I-1 V... (3. 14) 4 -jw1 The complete set of signal voltage components may be found in this manner. They may easily be expressed in matrix form. The signal voltage, vs, can now be written as a column matrix, which is the product of a square impedance matrix [S] which represents pumped depletion layer impedance and a column matrix [i, which represents the signal currents. 47

To obtain a consistant set of signal currents, take the complex conjugate of all the even rows in Eq. 3. 14. The resulting matrix is written in terms of V and I for n odd and V and I for n n n -n -n even. The first few terms of the matrix are written out in Eq. 3. 15. S S S S 0 1 -1 2. V co j -j jIc W 1 c2 S3 W4 S_! S S S V I..-0 -J 2 -2 1 2 3 4 - S1 S2 S2 S3 V3 - J ~.1 J2 -3 4 3 S S S S V2 3 3 4 V-4 l i j i ' S3 W1 2 3 4 S2 S3 S1x S 2 3 ( 4 z 5&j (m n) -6 L1 (3. 15) The elements of the impedance matrix are of the form mn (1) Wn (3. 16) 48

where k(m, n) is an integer and is given by (m n) = (1)n /2n- 1+()n )m 2m- 1+(- 1)n ) 4 4 3.3 Capacitance Expansion The harmonic balance equations in Section 3. 2 were derived by defining elastance as dv S(qt) d= q (3. 17) and by expressing the voltage across the diode junction as a function of the charge stored in the junction vt = vt(qt). (3. 18) The computations were carried up to a point where a signal voltage matrix was equated to the product of an impedance matrix and a current matrix (Eq. 3. 15). In this section, an alternate approach is presented. The charge stored in the junction is expressed as a function of the voltage across the junction. qt qt(vt) (3. 19) Capacity can be defined as 49

dqt(vt) C(vt) dv (3. 20) This definition of capacitance is consistent with the definition used by Rowe (Ref. 14), and Heffner's definition of "incremental capacity" (Ref. 41). Heffner (Ref. 41) contains a complete discussion on the definition of capacity. For a periodic pump voltage, the depletion layer capacitance can be expressed in a Fourier series C(V) C ejnpt (3. 21) p n The arguments from this point on are the dual of the arguments in Section 3. 2, and the computations are carried on in exactly the same manner. The results can be expressed in matrix form. The signal currents are written as column matrix is]. This is the product of an admittance matrix, [C], which represents depletion layer admittance, and a column matrix, [vs, which represents the signal voltages. The first few terms of the resulting expression are written out in Eq. 3.22. 50

11 jW1lCo jwlC1 j1 C2... V1 -2 -jW2C_1 - jw2C0 -jW C2 - j2C1 2 V-2 I3 j3C 1 jw3C2 jw3Co jw3C3... V3 I-4 -j4C_2 - jW4C_1 _-jw4C3 -j4C0... x V4 15 jw5C2 jw5C3 j5C 1 j5C4... V5 V5 I..... V -6.6 i ] =-[C xlv ~~~~~~~~~~~~s s ~~(3.22) The Y term of the matrix has the value nin ll j(-1)-~l Cf( n) (3.23) where (a, n) is an integer and, as before, can be expressed as (mnI)-= (_1)n [2n-1+(-1)] [2-1+1) (m,,~)~ (.U" [Ir~ijl] (44-1) m 4 3.4 Short Circuit and Open Circuit Assumptions Equations 3. 15 and 3. 22 are two representations of the most general case of small signal varactor mixing. They are, however, exceedingly large and unwieldy. Thus far, maost workers have chosen not to deal directly with themn, but rather to make simiplifying

assumptions. All the assumptions have one common goal: reduction of the infinite matrices to finite matrices. Many of the authors referred to in the literature review (Section 1. 2) (Refs- 13, 42, 18, 19, 20, 21, 43, 4, 14) choose to deal with the set of admittance equations (Eq. 3. 22). They reduce this matrix to a manageable size by making the so-called "short circuit" assumptions. The voltages are the independent quantities. The unwanted voltages are assumed to be shorted out by ideal filters. In addition. the diode is represented by a capacitance varying periodically. This is usually a sinusoid, although a few authors (Refs. 19, 20) have accounted for a second harmonic component in addition to the fundamental. Ona the other hand, some authors (Refs. 15, 16, 18, 17) deal with the impedance?matrix (Eq. 3. 15). To reduce the size of this matrix, they make a different set of assumptions, called the "open circuit" assumptions. Only a finite number of currents are permitted to flow through the diode junction. The currents are treated as the independent quantities and the voltages are the dependent quantities. Ideal filters prevent currents from flowing at the unwanted frequencies, i. e., open circuit them. In this case, the diode is represented as an elastance varying sinusoidally at the pump frequency. It has been argued that the open- circuit assumptions correspond more closely to the physical situation than do the shortcircuit assumptions, due to the presence of the diode lead inductance, 52

which has a reactance increasing with frequency. However it is possible for the circuit external to the diode to present a reactance which resonates in series with the diode lead inductance and thus provides a short-circuit termination. In practice,the presence of the diode series resistance usually results in some power dissipation at all sideband frequencies, and. neither assumption. is strictly valid. 3. 4. 1 Two-Freque-ncy Case. Most of the early parametric amplifier work was done by considering only two frequencies and using the "short circuit assumptions." In this case, signal voltages are present only at the input frequency, w1' and the lower sideband cc2. The voltages at other frequencies are shorted out, and while currents may flow at the shorted frequencies, they are unimportant because there is no power flow. In this case (3. 22) reduces to the two frequency short circuit equation. jw Co0 j cl C1 V1 ] x (3.24) '-2 IjC2C 1 -j cc 2 On the other hand, the "open circuit assumptions" can also be used. In the two-frequency case, signal currents exist only at frequencies w1 and cc2 and voltages at other frequencies are of no interest. These assumptions reduce Eq. 3. 15 to the two-frequency open circuit equation. 53

1 S V v co1 ~~~x ~(3.25) S _1 S0 V j v-2 | | 1 se -2 The relationship between the open and short circuit assumptions may easily be seen by inverting Eq. 3. 24 to obtain voltage in terms of current (Ref. 18). -j C _ -j C j 1 jC0 1 wl1(Co -C1C1) w2(C02-C1C_1) x (3. 26) JC_1 JC0 2V wI(Col2 C C (CO -C1C 1) Equation 3. 26 is identical with (3. 24) if one defines C A C1 S 0 and S = 0= 1 C2 22 CO - iC12 C0 C The open circuit and short circuit assumptions also give identical results for the other two frequency cases in which power is permitted to flow only at the input signal frequency, wco 1, and at the upper sideband frequency, wc3. 3. 4.2 Three-Frequency Case. The relationship between the open circuit and short circuit assumptions for three or more 54

frequencies is somewhat more complex. For the short circuit assumption assuming only three non-zero voltages, three non-trivial currents, and sinusoidal capacitance, Eq. 3. 22 reduces to I1 CO jC1 C j1C1 j o 1V1 121 = ] -jw 2C_ -j c2 C O x V_2 (3.27) 13 j 3 C1 0 j w3CO V3 From Eq. 3. 15 the open circuit assumption for the three frequency case reduces to SO S1 S_1 V ~ j i -j sI 1 W2 3 S S V2 |j j O x I 2 (3.28) V-2 ~~ -1 2 S1 S V 3-j 0 -j 1 3 Equation 3. 27 can be inverted to give voltages in terms of currents, but if this is done, the resulting equation is not the same as Eq. 3. 28. In particular, the terms on the diagonal have different forms. In Eq. 3. 28. these are just the impedances of ordinary frequency independent elastances. The elastances in the diagonal terms of the inverted Eq. 3. 27 lack the property of frequency independence. For the three-frequency case, the open and short circuit 55

assumptions yield different results. 3. 4. 3 n-Frequency Case. The open circuit and the short circuit assumptions also yield different results for the n frequency case (n > 2). It will be shown here that only in the limit as n - cc can the two assumptions become compatible. This will be demonstrated by inverting the n-frequency admittance matrix which is a squarf (nby n) matrix. It will be proved that only as this matrix becomes infinitely large does the inversion procedure result in an impedance matrix. The;iathematical techniques used in this section follow those of Desoer (Ref. 44). Desoer, however, was concerned with the errors produced by truncating an admittance matrix of the form (3. 22). In this section the relationship between the admittance formulation of reactive mixing and the impedance formulation will be explored. If the capacitance variation is sinusoidal, the general admittance matrix equation (3. 22) reduces to the simpler [ Y'(n)] matrix (Eq. 3. 29) in the n-frequency case. 56

jwIlC jW1C1 jWlC1 O 0 0 0 -jw2C1 -jW2C0 -jW2C1 ' 0 0 0 jw3C1 O jw C C jw3C 0 O O O 0 -2jw4C0 1 -jw4CO -0 C C (n)] j15C1 0, _ -jw0C 0 -jw6C0 0 -jW6C1 o O O O C jw7C1 0 jw7Co 0 jw7C O O O O 0 -jwC1 -jw8CO O O 0 I O O 0 jwgC1 0 jwgC 0 (3.29) I ~ ~ ~ ~ ~ -wc0-c

The [Y'(n)I matrix of i:q. 3. 29 is shown partitioned into submreatrices as follows: a 1 - 3 x 3 matrix a 12(n) - (n-3) x 3 matrix a21(n) - 3 x (n-3) matrix a2 2(n) - (n- 3) x (n - 3) mat rix c11 c C12(n) IY'(n)] =[7>_ - -( (3. 30) 21(n) } c22(n) Because a12(n) and a21(n) each have only two non-zero terms, the computation of the nine elements in the upper right hand corner of the [Y'(n)] matrix is relatively simple. After some manipulation the resulting inverse matrix can be written as ~ l(n)[Y'(n) ] _____ _ (3.31) where: 58

jClC1 jWoC1 jl1C_1 01(n) - cjw2C1O+ll(n)'2'4C1C 1 -12(n)2w 5C1C1 (3. 32) jw3C1 -21(n') w3w4C 1C 1 jw3Co + 22(n) c33w5C1C 1 O11(), 1312(n), 1321(n), and f22(n) are the upper four elements in the a22- 1 nmatrix. They are evaluated in Appendix A, and only the results are set down here: Pll(n) = (3. 33) -jW4C 1- yC 2 1 2:22(n) (3. 34) 5 e - 1 'Y2 1-y i12(n) -= 21(n) = 0 where the pumping "hardness" coefficient is defined by 2 C1 C-1 7'Y = 2 CO and 59

P11(4) = Pf(5) = 1 =j cW4 C0 22(4)= 22(5) = O in expressions (3. 33) and (3. 34), the quantity y2 appears 2) times for n even and 2 times for n odd. It is also shown in Appendix A that in the infinite frequency case, the expansion can be written in the simple form: 1 131(cc) = (3. 35) -j CC;4 C r21- ) =jCO(2 1) (00 (3. 36) 22 = /312() 21() = o (3. 37) Now Eq. 3. 32 can be inverted, and the resulting matrix examined to see which values of /311(n) and 2322(n) yield the proper impedance elements. From Eqs. 3. 15 and 3. 16 all the impedance matrix i S(m, n) elements nmust be of the form Z = where S is indem n (1) n 9 (m, n) pendent of frequency. The Zll(n) term of H -(n) can be computed by inspection. 60

[ +j ill(n)w4C0C1C1 Ji22(n) w5CoC1C+ 111(dn);322(n) Le45) C12 C 12] (3. 38) where 0(n) denotes the value of the determinate of the 9(n) matrix. But fronm the impedance matrix (Eq. 3. 15) S0(n) z () J= (3. 39) Combining Eqs. 3. 38 and 3. 39 S0(n) 3 i[CQ ji 411() Co C C - j322(n)5 c C0 CC1 ~'o(n) + 11(n) 322(n) c45C12 C 12 (3. 40) The same rProcedure can be done for Z22(n) and Z33(n). The results are from Z22(n) S(n) =l1 23 [C 2(n) 5 C C C1 C (3. 41) from Z33 JcWl W ) 11 ( 42) S(n) [CO2 + jl l(n) 4 CC1C - C1C1] 13.2)

Now the elastances in Eqs. 3. 40, 3. 41 and 3. 42 are all equal. Combining (3. 40), (3(3.41), (3. 42) 11(n) 4 = 322(n) 5 (3. 43) 322(1) (3. 44) J W cs0 + -2 Equations 3. 43 and 3. 44 are identical to Fqs. 3. 35 and 3. 36. This means that the infinite short circuit matrix is the only matrix which can yield the correct form of the open circuit matrix. Now that the infinite matrix case is known to be the correct one, all the terms may be found. The first few are written out in Eq. 3. 45. Equation 3. 45 is the inverse of Eq. 3. 29 only for the infinite frequency case. S O S1 S 1 -j +j -j W1 02 w3 S S S SO0 -2 -j +j -j (3.45) W1 W2 W3 S1 S S -j.... +j -J 01 002 3 62

where S = ~1 S0 = I(1~2ya+6Y4 +20Y~70y8+252y'0+...) 1 (1 + 3y2a 6 + 2 + 70 8 + 126y +...) Co C(1- - 4C)2: 2 (3. 46) 2C0 v4 C2 2 4 6 C 2 (12 A (1 +5 y+ 6 210y8 +35 + 12 +... ) o 4C12 2 C _ (13 +2 1 ) yC 2 226:2 (1 4 + 15.g4 + 5066 + 2 107 8 +. C0 2 C1 C-1 C 2 The following procedure provides further insight into the relationship between the open circuit and the short circuit equations: The capacitance considered in the short circuit matrix (Eq. 3. 29) has the form 63

C - CC + CeJ etJt (3. 47) The corresponding elastance is C C S _- 1 _ 1 ( + 1- e (3. 48) This i-ay be expanded and the coefficients of like exponentials collected. After some labor, the results can be expressed as an infinite Fourier series. S = S tljwLot S j2wt S jwt S -j2cwt S=S0 S1 2 S1 2 + S e (3. 49) where SC = 1 ( 1 + 62, 27 + 67 +, 8+ +70 (3. 50) C C0 - (1+ 3y 2 + 10jy4 + 35y" t-...) 2 C 2 S = (1 C4y2 + 154 + 56y...) 2 3 rhe elastance coefficients in Eqs. 3. 50 are identical with the elastance coefficients in the infinite short circuit matrix (Eqs. 3. 45 and 3. 46). A comparison of Eq. 3. 48 with Eq. 3. 49 shows that an infinite Fourier series in elastance is required to correctly represent a sinusoidal capacitance variation. It is now easily seen why only the 64

infinite elastance (open circuit) matrix is equivalent to a capacitance (short circuit) matrix. 3. 5 Summary In this chapter, the harmonic balance equations were developed. The results can be expressed either as an infinite set of impedance equations (3. 15), or as an infinite set of admittance equations (3. 22). These two sets of equations are the key to the entire paper, and the majority of the material in subsequent chapters is devoted to solving them. In the next two chapters, the elastance coefficients which are the individual elements of the matrix are computed. These elastance coefficients are then substituted in the matrix which is then inverted to obtain the input impedance. The open and short circuit approximations were discussed at some length. They clearly illustrate the assumptions made by other workers, and thus provide a background for this more exact study. 65

CHAPTER IV STABILITY 4. 1 Introduction The harmolnic balance equations in the preceding chapter fiorm the basis for the analysis of pumped diode input impedance. In deriving these equations, it is assumed that the pump voltage and charge are periodic and can be expressed in a Fourier series with the fundamental at frequency ap. The purpose of this chapter is to examine the validity of this assumption. Particular emphasis is placed upon those effects which result from nonlinear causes, and which affect the stability of the circuit. Although the linear voltage-current relations of the signal are the primary concern in this paper, it is important to study nonlinear instability effects for two reasons: (1) They affect the pump voltage-current relations. This in turn affects both the magnitude and phase of the Fourier coefficients of elastance, and thus the signal. (2) Under certain conditions, they cause the pump to become unstable. Thus, some pump voltage-frequency combinations are inaccessible. The next section is devoted to a discussion of the "Ferroresonant effect. " It is a particularly serious form of instability, since 66

it occurs frequently in a high Q tuned circuit driven near resonance. Almost all punmped varactor circuits use a tuning technique of this type to obtain a large pump voltage across the varactor with minimum pump power expenditure. In Section 4. 3, instabilities of a high Q tuned circuit are examined with the aid of Mathier's equation. 4. 2 The Ferroresonant Effect In this section, the "Ferroresonant" effect (Refs. 20, 21, 45, 46, 47) is examined using an approximate method which yields quantitive agreement with theory. The diode capacity is treated as a linear capacitor whose magnitude is dependent upon pump voltage magnitude. This allows the use of familiar impedance concepts to explain an inherently nonlinear phenomenon, In Appendix B, it is shown that the pump voltage across a reverse-biased junction diode may be expressed as: M VM = 1 + QM M- 1 I (B. 6) where QM is the normalized charge due to the pump, and the small signal capacity at the bias point is normalized to one farad. The exponent, M, not necessarily an integer, usually takes values ranging from 5 down to 2, depending upon the impurity distribution in the diode. M = 2 for an ideal abrupt (alloy) junction, and M= 3 for an ideal linear (diffused) junction device. 67

I 1-1 R C %-_ IN Fig. 4. 1. Parallel RLC networi,- excited by a current source 68

Consider now a circuit of the type shown in Fig. 4. 1. The diode capacity is tuned by the inductor, and the circuit is sinusoidally excited at a frequency near w0, the small signal resonant frequency of the circuit. In this case, w0 = (L) 2 because the small signal capacity has been normalized to one. If losses are small, the voltage is approximately sinusoidal. VM = a cos wt (4. 1) 0 < lal < 1 From Eq. B. 6 the charge can now be written as 1 M = (1 + a cos wt) 1 (4.2) Equation 4. 3 has the same form (except for a constant) as Eq. B. 9 where M N = (4.3) Thus, it may be expanded using the techniques developed in Appendix B, and the charge flowing at the fundamental frequency can be written in the form of Eq. B. 13. Q(w) = (a +.095 a3 +. 034 a5 +.025 a7 +...) cos t (4. 4) By Eq. 4. 1 this can be written as 69

Q() = (1 + ~095 a +.034 a4 +.025 a6 +. ) VM (4. 5) which can be put in the common form Q(a) VM (4. 6) where C(a) = (1 +.095 a2 +.034 a4 +.025 a6 +.o) Kirchhoff's current law applied to the current at frequency w which flows in the circuit of Fig. 4o 1 yields the following equation, d Q( )(t) VM dt VM(t) + + i (t) = 0 (4o 7) dt L R IN(t) ( where IIN = I cos(wt +r ). All terms in this expression are sinusoidal at frequency ow, Combine Eqso 4. 1, 4. 6, and 4. 7 and perform the indicated operations. The resulting relation may be expressed in exponential form: +jwt jaw C(a) ja a I ejT -2wL + 2R + +, jaw C(a) ja a Ie e- ji l t [ +'..+.a I e +j - 0 + 2w0 L 2R 0 +j cot ct The coefficients of ej and e must both equal zero. Setting the coefficient of e +jt to zero results in the familiar expression: 70

v j w L+ (4.8) + [1- c2 LC(a)] where P - j Q = P and V = ae cWO L For convenience, the analysis is conducted with Wz L2 iz =Vf = Ll (4.9) + [1-W2 LC(a)]2 Q2 From Eq. 4. 5, the capacitance C(a) may be represented by the approximate form C(a) = (1 + a IVi2) 2 C(V) (4.10) where a is small, and fv(2 1 Substituting this value in Eq. 4. 9 and differentiating with respect to frequency, there follows 2V2 2w LZ IN + 4V2 c LC(V) [1- w 2 LC(V)] - -V diVi IN Q2 dw 1 (4. 11).. + [1- w LC(V)]2 - 2w2 L 6C[1-w2 LC()] Q2 71

where the change in capacity is 5C = alV12 From inspection of the denominator of this expression, it is apparent that under certain circumstances, the slope of the response curve of V2 as a function of w will become infinite. A jump in output will occur as the frequency of the exciting current passes through the appropriate value. The circumstances under which this effect occurs are found by setting the denominator equal to zero and solving for the frequency. After some algebra, it is found that the required expression is I ~1+25C~q-1+ 6C 3+I] 2 1 + 2 C ~ 6C!1 - Q2 w - w0].................. (40 12) w 0= ~ 1 +36C 01+ 3C2) It follows that the jump frequencies are always less than the resonant frequency, and that these frequencies are real only if 1 >[ +()] [ (~C)] (4.13) Qa It is also apparent that the jump frequencies are a function of the magnitude of the applied signal voltage. Condition (4. 13) may be rewritten as 72

I C > (Q + 1)2 + 2 (4. 14) Q2 3 Output jumps occur at two frequencies when this inequality is satisfied, and at one frequency when equality occurs. Figure 4. 2 is a graphical plot of the relation: C = (Q + 1)2+ 2 (4. 15) Qz- 3 This curve divides the area into two regions, with no output jumps in the region below the curve and jumps at two frequencies in the region above it. It indicates that the ferroresonant effect may be avoided in a varactor diode circuit by reducing either the circuit Q or the applied signal level. Experimental measurements qualitatively confirmed the theoretical results. Figure 4. 3 shows three voltage-frequency curves for a diode-loaded tuned circuit. When the level is small, 6C is below the threshold value for jumps, and a conventional response occurs, as shown in Fig. 4. 3(a). As the level is increased by 10 db, the passband of the response becomes asymmetrical, as shown in Fig, 4. 3(b). Increase in the level by a further 10 db gives an output jump at one frequency, demonstrated in Fig. 4. 3(c). The voltage amplitude at which the jump in output occurs may also be found by setting the denominator of expression 4. 11 equal 73

1.0 1 0A 1Q Asymptote ' ac 0.01 0.001 I 0.01 10 100 1000 Fig. 4.2. Fractional increase in average capacity necessary to produce the ferroresonant effect 74

(a) (b) (c) Fig. 4. 3. Frequency response curves for parallel resonant circuit containing a varactor diode 751.

to zero,, Using the substitution c = O + 6 where S6 <K coo to simplify the algebra, the expression reduces to W ~ 3 -4 + 4 O O Q a V = (4. 16) 3 1 + This expression is plotted for various values of Q in Fig. 4. 4. Each value of Q defines a parabolic area, on the voltage-frequency plane, within which the circuit is unstable. The circuit is stable in the region between the parabola corresponding to a given Q and the coordinate axes. The experimental result of Fig. 4. 5 illustrates this effect clearly. The photograph in this figure shows the frequency response of a diode tuned circuit. The vertical axis is response voltage, and the horizontal axis is frequency, with the small signal resonant frequency co0 occurring at the right-hand edge of the photograph. The input frequency was swept continuously as the amplitude was slowly reduced. The light areas indicate conditions under which the circuit is stable, while the dark areas indicate circuit instability since the oscilloscope trace did not remain in the unstable regions long enough to record an image. 76

0.20 Locus of max. Fre, Q=o points \Asymptote 0.16 0. 12 \Q=10 ac C 0.08 Q=cc Asymptote - 0o 04 -0, 15 -0. 10 a -0.05 0 Fig. 4.4. Theoretical results showing region of ferroresonant instability 77

Fig. 4. 5. Experimental results showing region of ferroresonant instability 78

in Fig. 4.6. Series resonant circuit containing a varactor diode 79

4 3 Other Instabilities Consider a series LC circuit excited by a constant sinusoidal voltage source Ein cos wt (Fig. 4. 6). The normalized differential equation of the circuit is d2 Q E. cos wt = L. + + R sQ (4. 17) in dtNdtZ N where QN is the charge stored in the capacitor, i. e., the integral of the current, and VN is the voltage across the capacitor. For an abrupt (alloy) junction varactor diode, the relationship between AC voltage and charge was found in Section 2. 2 and can be written as QN(t)2 VN N + 4 (2. 12) Substituting this relationship, the differential equation for Q becomes d2NQN2 L [Ein cos wt - QN- (4. 8) dt2 This equation can be checked for stability by perturbing qt slightly, i. eo, letting QN change to QN + et, where et << QN. Substituting this into Eq. 4. 16 the following differential equation for e is obtained.. +RR t N dtz 5 dt t EL + ~ 80

If the solution for Et is stable, the circuit is stable. However, if Et is unstable (i. e., et becomes large without limit) the circuit is unstable, since it indicates that a very small perturbation in the circuit causes the charge and the current circulating in the circuit to increase without limit. If E. is small (i. e., E. << 1) then QN can be approximated by a sinusoidal function QN Cos Wt where q0 << 1 (4. 20) The differential equation for et then becomes d2E dEt0cos t _ a R -t dt + R t + (+ c2 0 (4. 21) a s dt L 2 This differential equation is a form of Mathieu's equation. It can be put in canonical form by making the following substitution for the independent variable, and thereby removing the first derivative term (Refs. 48, 49) Et(t) = f(t) e and then making the following substitutions: 81

wt 2 4 S W2L w2 -q0 W2L Then d2f d - + ( 4- 24 cos 2v) f = 0 (4. 22) dv2 This equation has been extensively studied and the results tabulated For a study of stability it is sufficient to note from the tabulated information that for <K< 1 (i. e., qO << 1) the solutions are unstable only for very small regions near: v/ = 0, 1, 2, 3,... n... 2w Since w =-, the circuit can therefore be expected to be unstable when 2 1 2 a = -o, 2w0, w0o 3 w0' 2 wO' 5 W0... (4. 23) The instability at w0 has already been discussed; it is the ferroresonant effect. Physically, the instabilities at the other frequencies are related to harmonic and subharmonic oscillations. Many of these oscillations have been observed experimentally. 82

The insights resulting from the simple analysis presented here are sufficient to understand both the numerical results of the theory and the phenomenon observed during the experiments. 83

CHAPTER V DEPLETION LAYER CAPACITY 5. 1 IntroductionIn Chapter III, a linear relationship (Fq. 3. 15) which relates signal currents to signal voltage was developed. The parameters in this expression are the Fourier coefficients of varactor elastance which for signals of sminall amplitude are completely independent of signal. Two iterative techniques for computing the coefficients of elastance are developed in Sectio-ns 5. 1 through 5.4 of this chapter. These procedures are applicable to situations where the diode is reversed biased in such a way that conduction and diffusion effects can be neglected, so that the diode can be characterized by its depletion layer capacity. Once these elastance coefficients are known, the input imnaedance at the signal frequency is computed using conventional matrix techniques which are outlined in Section 5. 5. The computation of the time varying elastance is the subject of the next three sections. Since the Fourier coefficients of this elastance are independent of both signal amplitude and frequency, they can be computed with any signal which is convenient. In this paper, the elastance is computed using a signal of zero amplitude; that is, all signal voltage sources are replaced with short circuits, and all signal current sources are replaced by open circuits. Of course, the pump 84

source remains in the circuit. The resultant simplified circuit is termed the pump circuit, and its use permits considerable simplification, for now only the effect of one source need be considered when computing the elastance. In determining the behavior of the pump circuit, only the "steady state" solution is considered. The circuit is presumed to be stable and to generate neither free nor subharmonic oscillations. This, of course, is at odds with the results of Chapter IV, where it was shown that instabilities are to be expected in this type of nonlinear circuit. The "steady state" solutions then, must be carefully examined to determine whether they are solutions of the physical problem. Under the "steady state" conditions, all currents and voltages in the circuit as well as the diode elastance can be expressed as a Fourier series with the fundamental at the pump frequency ow. However in much of the work which follows it is more convenient to deal with the charge stored in the diode junction. It is also periodic in co and can be expressed as QA(0) OC oc Q(t 2 + QA(n) cos nopt + n QB(n) sin nw t (5.1) or the equivalent exponential form QA(n) n= QB(n) jnw t (t) = [ i~ 2 -e p (5. 2) 85

.i1 + L o I Fs 1eul p Capacity Fig. 5. 1. Thevenin equivalent pump circuit

In much of this work it is expedient to express voltages and charges in terms of frequency, rather than time by using the conventional Laplace Transform. Using this notation, Qp can be written as ) L{Q (t)} = n Q (n) Because all currents, voltages, charges, etc. can be expressed in n=-c (5.-3) Because all currents, voltages, charges, etc. can be expressed in Fourier series form, the Laplace transform and the inverse transform exist, and in fact,can always be determined with relative ease. 5. 2 Pump Circuit Configuration Figure 5. 1 illustrates the pump circuit configuration used in this chapter. By referring back to the basic representation of a pumped varactor circuit shown in Fig. 1. 1, it can be seen that the linear portion of the circuit as seen from the diode terminals has been replaced by an equivalent Thevenin circuit. The nonlinear varactor diode remains unaltered, and is not included in the equivalent circuit. Because of the nonlinear element, the Thevenin equivalent impedance concept must be treated with some care. Usually, the entire circuit is linear, and voltages and currents are present at only one frequency, that of the voltage source. The introduction of the nonlinear diode complicates matters somewhat, for now voltage and currents may appear at frequencies other than that of the source. For each frequency component, the Thevenin equivalent circuit 87

must have the same (linear) voltage-current relation as the original circuit. Figure 5. 2 shows examples of two types of pumped diode circuits which are considered in this chapter. Figure 5. 2a shows a simple circuit composed of lumped elements. The capacity of the varactor diode is series resonant with the inductor L2 at a frequency near that of the pump source. Pump power is coupled into the circuit through the mutual inductance M1. It can usually be assumed that the pump generator is sinusoidal at frequency wp, and that losses in the circuit are small. Under these conditions, the Thevenin voltage generator is sinusoidal, and the equivalent impedance can be closely approximated by the network in Fig. 5.4. Here Rs represents the diode loss, while G accounts for both inductor loss and generator resistance; CS is the parasitic capacity of the inductor. In the ideal case when all losses and the parasitic capacitance can be neglected, the Thevenin impedance reduces to ZL(W) = jwoL (5. 4) Figure 5. 2b shows a simple distributed parametric circuit. A length of transmission line is shorted at one end and tuned to resonance near the pump frequency by a varactor diode placed across the opposite end. A loop near the shorted end couples pump power into the circuit. If the coupling is weak, and ohmic losses small, then the Thevenin impedance can be put in the form: 88

R Diode Pump L2 Bias Voltage Fig. 5.2 a. Lumped circuit Bias ide Transmission Line Voltage [ _ Z (a ) E Pump Fig. 5. 2 b. Distributed Circuit Fig. 5. 2. Practic.- pumped varactor circuits

Z(w) = j XL(Co) + RL(c) + Rs (5.5) Except for frequencies very near the resonances of the line, the reactance takes the familiar tangent form: XL(O) - Z0 tan ( )c (5. 6) and for the common case of an air dielectric coaxial line the distributed losses can be computed using conventional approximate techniques (Ref. 50) 37. 6 x 10 (a+) (IwN + asinr)) R (W) (5. 7) VdJ cos (E;) where ZO is the characteristic impedance of the line W L is the radian frequency at which the line is one-quarter wavelength long a' and b' are the inner and outer conductor diameters in meters a is the conductivity of the material in mhos/meter. A comparison of Eqs. 5. 4 and 5. 6 indicates that although both impedances can be adjusted to resonate with the diode at the pump frequency, the impedances at harmonics of the pump can be considerably different. The inductor presents a higher impedance to each succeeding harmonic; 90

the line, because of the nature of the tangent function, need not follow such a regular pattern. Indeed, the line can be adjusted so that either a zero or pole of impedance falls at any harmonic. If the line length and the characteristic impedance are varied together in a manner such that the tuning of the pump fundamental remains constant, the tuning (and, therefore, the amplitudes) of the harmonics change. These simple circuits then provide a convenient means for studying the effects of higher harmonics upon a varactor circuit. Now referring back to the circuit of Fig. 5. 1 it is clear that Ein VI(t) + VL(t) (5.8) This equation is the basic pump circuit equation. The three voltages are discussed in the following three paragraphs. a. The Pump Voltage. The pump source is assumed to be sinusoidal at frequency ap. Thus, the open circuit Thevenin voltage, Ein(t), which would be present if the diode were removed, i. e., open circuited, is also sinusoidal at awp. The varactor DC voltage bias is taken into account by the relations developed in Chapter II and need not be included. In general, the time origin is arbitrary, and Ein(t) takes the form: E.i(t) = EA cos w t + E sin t (5. 9) 91

b. The Diode Voltage. In the abrupt junction reversed biased case, the voltage- charge relations for the diode were discussed in Section 2. 2. Equation 2. 12 is of particular value here: Q p(t)2 Vp(t) = Qp(t) + 4 (2.12) The subscripts have been changed from N to p. This minor change in notation was made to emphasize the fact that all the voltages and charges (currents) considered in this section are due to the pump only. Here, Vp and Qp are normalized according to the scheme in Section 2. 2 and must satisfy the inequalities of Eq. 2. 15. V > Qp(t) > -2 p_ - In the remainder of this chapter, this abrupt junction equation will be used. The choice of the abrupt junction was arbitrary; the linear diode (Eq. 2. 16), or measured data could have been used. c. The Network Voltage. Using the complex frequency notation introduced earlier, the voltage across the linear circuit can be easily expressed in terms of the charge. VLIN(s) =s Z(s) Qp(s) (5. 10) Here, the linear circuit components have been transformed in the 92

usual nm-anner into complex impedances. The impedances of the network at the pump and its harmonics are important enough to justify a special notation. The impedance of the linear network at the nth harmonic of the pump is denoted as Z(nwp) = R(n) + j X(n) (5. 11) where n 1 2, The voltage across the linear network in the time domain can now be found by combining Eqs. 5. 1, 5. 10 and 5. 11 oC VLIN(t) - 1 n F) [QB(n) R(n) )- QA(nn)n)] cos nw Pt Xn=1 w [QA(n) R(n) + QB(n) X(n)] sin nwPt (5. 12) The advantages of using the complex frequency notation are easily seen by comparing Eq. 5. 10 with Eq. 5. 12. A convenient form for the differential equation describing pump circuit behavior can be obtained by combining Eqs. 2. 12, 5. 8 and 5. 10. Q p2 (t) 0 = E (s) - s Z(s) Qp() - L Qp(t) + 4 (5. 13) 93

This can be solved for Qp(t) and rewritten. 2 Qp(t) -1 E. (s) L 4 + Qp(t) (5. 14) where the L -{ } notation denotes the inverse Laplace transform. This equation is closely related to Lalesco's Nonlinear Integral Equation (Ref. 51). However, the sequence method ordinarily used to solve this type of equation fails to converge in some instances. One circuit of particular interest for which the usual method does not converge is shown in Fig. 5. 2b. In the next section, a scheme which is useful in this case will be presented. The quantity that will be treated as the forcing function in Eq. 5. 14 has not yet been determined. It can be selected in a variety of ways, of which letting Ein be the forcing function, and allowing Qp(t) be the unknown is perhaps the most natural. In this chapter, however, a different choice is made. The forcing function is QA(1), the fundamental cosine component of charge at the pump frequency. QB(1) is set equal to zero; this is simply equivalent to choosing a time origin. Ein and all the remaining components of Qp(t) are now the unknowns. Physically, this is equivalent to replacing the voltage source in the circuit of Fig. 5. 1 by a current source in parallel with an ideal filter (Fig. 5.3). The filter short circuits all frequencies except wp; at this frequency it is an open circuit. The current source is 94

I. =j Q 17F1 I Z(w) tr VIN-I V Dio de Ideal filter is an open circuit at the pump frequency; a short circuit for all other frequencies. Fig. 5. 3. Pump circuit configuration used in Chapter V

lin(t) - -w QA(1) sin co t (5. 15) It is clear that under stable conditions nothing has been altered by making this change in driving functions, for by the "substitution theorem" (Ref. 52) a voltage source which is delivering a current of I amperes to a load mnay be replaced by a current source of magnitude I with no chkange in the external circuit. However this choice of a driving function along with the constraint of only a "steady state" solution does force the solution to the differential equation to be stable and periodic in wp. Unfortunately the physical circuit is not so constrained and as discussed in Chapter IV, can become unstable under certain conditions. Thus, an important step in characterizing the pump circuit behavior involves examination of the steady state solutions to determine whether, in fact, it is a stable solution of the physical problem. This can be carried out in a straightforward manner aiE. I by noting the behavior of in at a constant frequency. Whenever this term is positive; that is, whenever the magnitude of voltage in alE. 1 creases with current, the circuit is stable. However, if inl is negative or zero, that is, if the voltage magnitude decreases with current magnitude, the circuit is unstable, and the "steady state" solution is meaningless. This can be proven quite simply by recalling the results of Chapter IV, where the stability of a network driven by a current source of constant magnitude is determined by examining the expression for 96

XiVi -a V and searching for the conditions under which the expression a co becomes infinite; that is,where the denominator becomes zero. The dual is also true; the stability of a network such as Fig. 5. 1 which is driven by a voltage source can be determined by examining the expression for a I and searching for the conditions under which the denomiaw nator becomes zero (Ref. 45). Following the same technique as used in Chapter IV, the current at the fundamental frequency in the circuit of Fig. 5. 1 can be put in the following form: Ai. A(IE. i in I I in B (Ei n. i, )) (5. 16) B E in' I Iin I'co where A and B are functions of the indicated variables. Now this circuit is unstable when the denominator of in in equals zero, the voltage E. being held constant. That is, awo in when 0 B + I in allB- aAI (5. 17) in in Replacing the voltage source by a current source as is done in Fig. 5.3 does not change the voltage-current relationship of aIE. I in the network. If now the function a i with frequency being held conin stant is examined, this function has zeros at 97

o (B+Iin a [B -a iIn (5.18) This is, of course, the same condition under which the circuit by the voltage source becomes unstable. For the circuits cona I E. I sidered here, the in function has either one zero, or two zeros in which bound a negative region. These points correspond to the instability regions discussed in the previous chapter. 5. 3 An Iterative Technique Applicable to Single Tuned Circuits The preceding sections presented the basic pump circuit equation and explained the selection of the driving function and the dependent variable. In this section, an iterative technique is introduced for solving the equation. The basic scheme used to solve the equation is to (1) Substitute an estimate for Qp(t) into the right side of the equation and perform the indicated operations. The result is a "better" estimate for Qp(t). (2) This new estimate for Q(t) can be substituted back into the right side of the equation. (3) The process is repeated until the resulting estimate is as close to the correct answer as desired. Formally, the process can be set down as follows: The solution to Eq. 5. 14 is the limit of the infinite sequence of functions: Qo(t), Ql(t), Q2(t),... Qm(t)... 98

and Q (t) = Limit Qm(t) (5. 19) It is clear that in order to be a useful process, Qml(t) must converge uniformly to Q (t). A sufficient condition for convergence is that the error become smaller after each iteration. Thus 6 l+(t) < Kern(t)[ (5.20) for all m and all t. ec (t) is the error after m iterations, and E (t) is the error after m+1 iterations. Em(t) = Q (t) Qp(t) (5. 21) El(t) = Q1+l(t) - Qp(t) (5. 22) After nm iterations, the estimate for Qm (t) is computed from Eq. 5. 14, 5. 21, and 5. 22 using the procedure outlined above. 1 Qm+l(t) Qp(t) + E1(t) L - Q p(t) +EL) (t))2 (S ) L (Q()4 + Q(t) + e r(t) (5.23) Multiplying this out and substituting in the value of Qp as deterxmined by Eq. 5. 13, Eq. 5. 23 becomes 99

MEr+i(t)+S s iS L 2 mt) + 4 E(t + ) (5.24) Fromr this equation, an estimate can be mnade for the conditions under which the sequence will converge when I c+ l(t) I < l C m(t) i (5. 25) In order to make this estimate. sonze bounds must first be placed on Qp(t) and c (t). A lower limit for Qp(t) can be obtained fronm the fundamental equations derived for the abrupt junction diode. Qp(t) > -2 (5.26) One crude approximation of Q (t) is that it is sinusoidal. Certainly, this estimate should not be in error by more than a factor of 2. Then IQ (t) < 2 (5.27) In Eq. 5. 1, Qp(t) is written in the form of a Fourier series. The fundamental term has been selected as the driving function, and thus is a known quantity. It can be reasonably expected that the fundamental will be the largest term in the solution, and that the sum of all the harmonics will not be larger than the fundamental. Then, by Eq. 5. 27, and because the initial (and worst) estimate of Qp(t) is just this 100

fundamental term, an upper limit can be placed on the error IC m(t) 1 < 1 (5. 28) for all m. Now substitute Eq. 5. 25 into Eq. 5. 24. By Eq. 5. 28, the c (t)2/4 term is small compared with the other terms and may be neglected. After taking the transform 21c (s)l Ic (s) < m (5.29) m+1 Is Z(s) 1 Thus, iem+ (s) < IEm(s)l whenever Is Z(s) > 2. The equation is set up such that only the pump and its harmonics are present, and the iteration scheme is devised such that error term exists only at the harmonits. So QA(O)m xC xC Qm(t) = 2 QA (n) cosn t + Z E QB(n)m sinwpt n=- 1n= 1 (5. 30) EA(0) CC c m(t) = 2 + n EA(n) cos no t + C EB(n) sin no t n=l n= 1 (5. 31) Then, the sequence will converge if Inwp Z(nw p) > 2 (5. 32) for n= 2, 3, 4,... 101

With the aid of Eq. 5. 32, the convergence of the iteration process can now be examined for specific circuit configurations. a. Ideal Inductor. If the coil is ideal and all losses in the circuit can be neglected, then the Thevenin equivalent impedance reduces to an inductance and ZL(w) = j L (5. 33) The impedance at the nth harmonic of pump frequency is ZL(nwp) jnp L (5. 34) p The coil is usually chosen to resonate with the small signal diode capacity (1 farad) at a frequency near the pump. Then c 2 L 1 (5. 35) and nwp ZL(nc)l = n2 (5.36) for n = 2, 3, 4,... The criterion of Eq. 5. 32 is satisfied in this case, and the iteration procedure will converge for this circuit configuration. b. Inductor with Series Resistance. In practice, the diode is not ideal and its losses must be accounted for; in this event, the Thevenin equivalent circuit m1ay be approximated by an ideal inductor 102

in series with a resistance. The impedance at the nth harmonic is ZLR(nwp) = jnwp L+Rs (5.37) As in the previous example, the inductor is usually chosen to be resonant with the small signal diode capacity near the pump frequency. Then 2 2 Z n,Z(nw) 2 [n2] + [nw R. > [n 1 (5.38) for n= 2, 3, 4,... The criterion for convergence (Eq. 5. 32) is satisfied in this case for all values of resistance. c. Inductor with Parasitic Capacitance. The parasitic capacitance associated with all real inductors can be very important. The impedance of an ideal inductor increases linearly with frequency. It therefore presents a large inmpedance to the higher pump harmonics, and thus tends to prevent currents from flowing at these frequencies. The circuit approximates the so-called open circuit conditions, in which all currents except those at a few selected frequencies are prevented from flowing. In contrast, the impedance of a real inductor with parasitic 103

G Id bL ~ S Fig. 5.4. Equivalent circuit for an inductor with parasitic capacity 104

capacity goes toward zero above the self resonant frequency. This tends to short circuit the higher harmonics of pump. The circuit then approximates the so-called short circuit conditions, in which no voltages exist except at a few selected frequencies. If losses are ignored, the impedance of the circuit of Fig. 5.4 at the nth harmonic is jnw L Z(nwo ) = (5. 39) Coil P 1- n2 o 2LC p s In this case, there are two resonant frequencies. The first is the frequency at which the coil is series resonant with the small signal capacity of the diode. This is near the pump fundamental. 2 L — Pi i L (5.40) 1- 2 LC P s The second is co, the self-resonant frequency of the coil. It can be defined as 2 1 2c L. (5. 41) c LC Combining Eqs. 5. 39, 5. 40 and 5. 41 with the approximate criterion for convergence, (Eq. 5. 32), a condition for woc is obtained. Wc > V co (5.42) c 10p 105

This is a reasonable value which most coils satisfy. Other circuit configurations are easily checked for convergence in the same manner. In general, the larger the impedance, the faster the convergence, Tt should be clear that the sequence will not converge for all circuit configurations. For example, the criterion for convergence (Eq. 5. 32) could not be satisfied if a zero of impedance happened to occur at some harmonic of the pump. 5. 4 An Iterative Technique Applicable to Distributed Circuits An iteration technique for solving the pump circuit differential equation was developed in the previous section. It was shown that the sequence does converge to the correct result for single tuned circuits; it was also made clear in the discussion that the sequence does not converge for all circuit configurations. In particular, the sequence does not converge for the distributed circuit of Fig. 5. 2b. In this section an iterative technique is presented which was developed to solve this circuit. But, it is shown that the sequence will converge for any circuit. This new technique is more complex than the previously considered procedure, and generally, the computer running time is longer. The basic scheme used to solve the equation is to (1) Substitute an estimate for Qp(t) into the right side of Eq. 5. 13 and perform the indicated operations. The result is 6(t), the equation error (Ref. 53). 106

(2) Using the equation error 5(t) and the estimate for Qp(t), compute an approximate value for the "solution error" e(t) which is the difference between Qp(t) and the estimate. (3) A new estimate for Qp(t) can easily be computed knowing both the old estimate and the "solution error" associated with it. (4) This three-step process is repeated until the resulting estimate is as close to the correct answer as desired. Formally, the process can be set down as follows: The solution to Eq. 5. 13 is the limit of the infinite sequence of functions: Q0(t), Ql(t), Q2(t),..., Qm(t). and Q (t) Limit Qm(t) Step (1) of the iteration procedure is implemented by computing the equation error. After m iterations, Q (t) is the estimate for Qp(t), and 6m(t) the equation error is 6m(t) = Em(t) - L {s Z(s) Q,(s)} - (t) + 4 (5. 43) Of course, by Eq. 5. 13 the equation error is zero when the solution is found. Thus 107

mr(t) O when Q(t) = Qp(t) (5. 44) As before, Q (t) can be written as the sum of an error term plus the correct charge, where the error (hopefully) is small Qm(t) = Qp(t) + (5.(t)45) Em(t) l < 1 In Step (2) of the iteration procedure, an estimate for the solution error, em(t) is computed. Substitute (5. 45) into (5. 43) and use (5. 13) to obtain 6 A(t) in terms of e (t). m Qp(t) em(t) & (t) e (t) 2 m - L {s Z(s) e (s) This is an algebraic equation for em(t). For convenience, the solution will be written as e m(t) = -m(t) 1 + L- s Z(s)1 + 2 (5. 46) This is an expression for the "solution error." It is a function of the "equation error" and circuit parameters (both of which are known), and the solution (Q p(t) is unknown). Although Qp(t) is unknown, an estimate can be made for it: Qp(t) Qm(t) (5 47) 103

From (5. 46) and (5. 47) an estimate can be made for the "solution error"' r( Q (t-1 m(t) m() e (t) [1 + L {s Z(s)} + n (5. 48) m 2 In Step (3) of the iteration procedure, a new estimate for the solution, Qp(t), is computed. From Eq. 5.45 Q (t) Qm(t) - e(t) (5.49) Therefore, (hopefully) a better estimate than Q (t) can be made for Q p(t): Q (t) Qn m(t) - e(t) Qm(t) (5. 50) From (5. 49) and (5. 50) Qm+l(t) = Qp(t) + (t) - e' (t) (5. 51) Equations 5. 43 through 5. 50 complete one iteration (the mth) in the sequence. If the procedure is to be useful, the equation error after m+1 iterations should be smaller than the equation error after only m iterations. That is 16+1(t)l < 6m(t)l (5.52) 109

should hold for all t. Substitute (5. 51) into (5. 43), and use (5. 45) to obtain the equation error after m+1 iterations in terms of the previous equation error. After some manipulation m(t)Q m+l(t) - (t) [+ L1 {s Z(s)} + Q( -1 21 2 4 2(t) + L {s Z(s)} + 2t (5 53) 6m l(t) = 0[6 (t) 2 Therefore, providing the operator 1 + L- {s Z(s)} + 2 and its inverse exist, the equation error is reduced quadratically each iteration. The reason is that the expression for Qm+l(t) is stationary with respect to small variations in Qm(t). Once the sequence has converged to a value close to the correct solution, the variational property assures rapid convergence. Of course, when the error is large, the convergence is not as fast. In practice, the sequence converges to six-place accuracy in from 3 to 6 trials. Essentially, the iteration procedure described in this chapter is an extension of the so-called Newton-Raphson method. The nonlinear differential equation is first transformed into a set of simultaneous nonlinear algebraic equations. Each harmonic of the pump is 110

represented by two of these equations: one equation represents the cosine; the other represents the sine term. All the equations are coupled through the Q2/4 nonlinearity. The resulting set of nonlinear algebraic equations is then solved with the aid of a linearization procedure. Each approximation to the solution is found by using the preceding approximation as a starting point. A linearized set of equations is generated from the nonlinear set by expanding the nonlinear functions in a Taylor series about the approximate solution and using only the first two terms. The linear set of equations is then solved, and a (hopefully) better approximation to the solution is obtained. The process is repeated until the solution is as accurate as desired (or until it diverges ). The usefulness of the method depends upon how closely the linearized set of equations represents the nonlinear set in a neighborhood centered about the approximate solution but extending outward to include the actual solution. If the approximate solution is close to the actual solution, and if the function is smooth enough in a neighborhood which includes both, then e2 (t) and higher powers can really be ignored (as was done throughout this section). The sequence method will then converge fast enough to be a useful process. For the situation considered in this chapter, there are two reasons why the sequence will probably converge: (1) The varactor voltage-charge relationship (Q+ Q~ ) is a "mild" nonlinearity. In the region of interest (-2 < Q < +2), the 111

first two terms of the Taylor expansion adequately represent the function in a large neighborhood. The function is "smooth" enough so that the sequence will converge even with a bad initial approximation to the correct solution. (2) The initial approximation (the first estimate) is always close to the correct solution. This is a basic study, and, rather than just a single solution for one pump level, it is desirable to determine circuit behavior for a variety of pumping levels. A family of solutions can thus be constructed starting with the smallest pump magnitude and building up toward hard pumping. For each pump level, the initial approximation is the solution previously found for a slightly smaller pump amplitude. The choice of a current source for the driving function forces all the solutions to be stable, and the diode voltage increases continuously with pump level. This insures that the solution found for one pump level will be a reasonable initial approximation for a slightly higher pump level. 5. 5 Small Signal Equations The procedures developed in the preceding two sections determine the behavior of the pump circuit in the absence of signal. They implement the first step in the determination of small signal input impedance. The second step is straightforward, and is outlined only briefly here. Once the punmp circuit behavior has been determined, the 112

result can be expressed in terms of the charge circulating through the diode. Q (O) O c Qp = A + Z QA(n) cos n t + B(n) sinnpt (5.1) Q) 2 p Bp' By Eq. 2. 14, the small signal elastance (for an abrupt junction device) can also be written as a function of charge: Qp(t) S(t) = 1 + 2 (2. 14) Equations 2. 14 and 4. 6 can easily be combined and the resulting small signal elastance can be expressed as an exponential Fourier series: Xc jnwt S[Qp(t)] S e (3.5) np n This elastance is a function only of pump. In Chapter III, an expression (3. 15) is derived which relates signal voltages across the junction to signal currents through the junction. The elements in this matrix are the elastance coefficients of Eq. 3. 5. Now the circuit of Fig. 1. 1 may be redrawn to emphasize the signal circuit. This is shown in Fig. 5. 5. The diode is represented as a time varying elastance of the form (3. 5), where the coefficients 113

S(t) z I klSS Fig. 5. 5. Signal circuit.configuration used in Chapter V. 114

can now be computed. The relationship between the signal voltage appearing across the diode and the signal current is given by Eq. 3. 15. The linear portion of the circuit, including the impedance of the pump source can be replaced by a Thevenin equivalent impedance. This equivalent impedance, [Zs], must adequately model the circuit at the entire set of signal frequencies: W1' c2' W3 '.' The pump source mnay be removed since it is a linear current source and plays no direct role in the signal circuit. Its only purpose is to "pumpp" the diode, i. e., cause the diode elastance to vary periodically in time. The voltages around the circuit (Fig. 5. 5) can be written in terms of the currents: Vs = [IS] x is] + [Zs] x is] (5.54) V ] and iI are column matrices [S] is a square matrix of the form (3. 15) [Zs] is a diagonal matrix representing the impedance of the linear network. The nth element of the matrix, denoted Zn, is the impedance at the nth signal frequency, con. In the most common case, the signal source has only one frequency component (at co). The voltage matrix then reduces to 115

vT 0 Vs = (5. 55) However, currents -may still flow at all of the signal frequencies. The [ Z] and the IS] matrices can be combined as indicated in Eq. 5. 54 and then simplified with the aid of Eq. 5. 55. The result is Eq. 5. 56. The input impedance of a small amplitude signal can be computed from this matrix by ordinary techniques. The method used in this paper was to reduce the matrix to its lower triangular form, by using the Gauss-Jordon technique. The input impedance is the (1, 1) element of the new matrix. Any method, however, will involve a large amount of labor if the matrix is large. 116

S S S S 0 2 V1 1 j -j... 1 G 1 | 31 "2 3 4 01 02 003 04 1 S2 0 S 3 o I I-i- -j-I 3J I3 1 2 03 3 0 S S S0 S3 -2 13 S X o j -j +j 4 Jl1 "2 03 4 4 S S S -2 3 i o K0 00-j 15 S1 2 31 S3 - 3 S1 (5. 56) 5. 6 Summary In this chapter, a nonlinear differential equation which describes the behavior of the pump circuit was set up, and two iterative techniques for solving it were developed. These procedures involve the iteration of a sequence of solutions, and where properly used converge rapidly to the correct solution. The first technique (Section 5. 3) is useful for determining 117

the behavior of single tuned circuits in which the impedance at the harmonics is high compared with that of the diode. The second technique (Section 5. 4) converges for any circuit configuration. The solution to the pump circuit is expressed in terms of the charge Qp(t), but it is clear that the result can also be expressed in terms of voltage or current. Once it is known, the Fourier coefficient of the elastance can be easily computed. It was shown that these elastance coefficients can be substituted into the elastance matrix which describes the mixing process, and that the small signal input impedance can be obtained by manipulating the resulting impedance matrix. The numerical results which are presented in Chapter 7 were obtained by algorithms of the procedures described in this chapter programmed for a digital computer. The flow diagrams are presented in Appendices C and D. 118

CHAPTER VI DIFFUSION E FFE CTS 6. 1 Introduction The circuit effects of depletion layer capacity have been treated in the previous chapter. The results of that chapter are valid if the voltage across the varactor is negative (reverse) at all times, so that the diode junction can be accurately characterized with frequency independent depletion layer capacity. However if positive (forward) voltages are applied to the diode, a second type of charge storage contributes to capacity. It is called diffusion capacity, and the purpose of this chapter is to determine its circuit effects. In Chapter II, the physical origin of diffusion capacity was thoroughly discussed, and this capacity was shown to be significant only for forward voltages, that is, when there is some direct current flow. For reverse voltages, it is negligible compared with depletion layer capacity. Charge storage and diffusion effects are of interest primarily because in many cases, the pumping is hard enough to cause rectified direct current to flow through the diode. In most practical circuits, the varactor is operated with a reverse DC voltage bias on which the AC pump voltage is superimposed. This pump voltage is often so large that the diode is driven into forward conduction during each positive peak. In this situation, depletion layer capacity dominates the varactor characteristics most of the time, but for the 119

R S, M Pump _1 Generator V -T- s C v L1 [I + Signal Source L2 Bia Voltage Fig. 6. 1. Varactor circuit used to investigate diffusion effects 120

positive part of every cycle, diffusion effects become important. The organization of this chapter is similar to that of Chapter V. The behavior of the circuit in the absence of signal is treated first. Section 6.2 discusses a circuit representation which is useful when pump power is applied to the circuit. The nonlinear differential equation which results from this model is solved in Section 6. 3. The end result of this process is the determination of varactor voltage in the absence of signal. From this pump voltage, the input admittance of the diode can be computed for a signal of small amplitude. It is dependent upon the signal frequency, and is both timevarying at the frequency of the pump, and linear. In Section 6. 4, an admittance matrix is generated which relates signal currents to signal voltages. The input admittance (or impedance) at the input signal frequency is then found by manipulating the matrix. 6.2 Pump Circuit Representation The circuit of interest is shown in Fig. 6. 1. It has exactly the same schematic configuration as the circuit of Fig. 5. 2a. In Chapter V, however, only depletion layer (or barrier) capacity was of concern, and the voltage across the diode was always taken to be negative so that diffusion effects were negligible. In this chapter, there is no such restriction; both diffusion capacity and depletion layer capacity are of interest. From the viewpoint of the circuit external to the diode, 121

diffusion effects and barrier layer effects appear in parallel. Physically, there is only one potential across the junction, the effects of which are experienced by all charge carriers. However, there are two components of charge (and therefore current). The first component consists of the mobile charge carriers which are drawn out of the depletion region by the electric field; their departure leaves some fixed charge centers unneutralized. This form of charge storage constitutes depletion layer capacity. The second component of charge results from the mobile charge carriers which have enough thermal energy to overcome the field in the depletion region. These carriers travel across the junction and are stored on the opposite side as minority carriers. These carriers constitute the charge stored by the diffusion capacity. It is natural in this situation to treat voltage as the independent variable, and to reduce the pump circuit to its Norton equivalent (Fig. 6. 2). Here, the linear portion of the circuit as viewed from the pump source, is replaced by a current source in parallel with an admittance. This equivalent circuit must have the same currentvoltage relations at its terminals for all components of the pump. As in the previous chapter, it is assumed that the solution is periodic and stable, thus componerts exist only at the pump frequency and its harmonics. The diode is unchanged, although schematically, it can be split into two parts, a diffusion capacity and barrier capacity. The signal source is omitted entirely because this circuit is intended to 122

ILIN +d P__ "' Barrier )N Y(w) Diffusion Ca V 'W Capacity Capacity Diode Fig. 6. 2. Norton equivalent pump circuit!N 'LIN 'd, - [tilte I f i Diffusion / ' Barrier V E Capacity T. Capacity i+ N Diode Ideal filter is a short circuit at the pump frequency; an open circuit for all other frequencies Fig. 6. 3. Pump circuit configuration used in Chapter VI. 123

represent only the pump circuit. The currents in the circuit of Fig. 6. 2 must satisfy the elementary relationship N ILIN' Id Ip (6. 1) These four components of current can now be determined: a. The Current Source. The pump source is assumed to be sinusoidal at frequency w. Thus, the short circuit Norton curp rent, IN(t), is also sinusoidal at w. This is, of course, the current which would flow in the actual circuit if the diode was shorted. The varactor DC bias is also supplied by this source. In general, the time origin is arbitrary, and IN(t) takes the form IN(t) DC+ IA cos pt +I sin w t (6.2) N DC A p B p b. The Network Current. The current through the linear network can be determined in the usual manner ILIN(S) = Y(s) V p(s) (6.3) It is assumed that currents and voltages exist only at the pump frequency, wp, and its harmonics. The admittance of the linear network at the nth harmonic of the pump fundamental is conveniently denoted as Y(n) - Y(nw ) = G(n) + j B(n) (6. 4) 124

c. The Barrier Capacity Current. As in previous chapters it will be assumed that the varactor can be modeled with an ideal abrupt junction diode. In this case, a convenient charge-voltage relationship, in which voltage is the independent variable, is given by Eq. 2. 4. qt(t) = -2 C a ( - ~ ) + Q (2. 4) where Q is the constant of integration. In the absence of signal, qt(t) - QBias + Qp(t) Qc can be chosen so that Qp(t) = 0, when Vp(t) = 0. This selection reduces Eq. 2. 4 to the following form: Qp(t) = 2 Ca0 - 1) (6. 5) The current through the barrier layer is the derivative (with respect to time) of the charge. This can be written in terms of complex frequency Ip(s) s (s) (6. 6) 125

d. The Diffusion Current. As is the case for barrier layer capacity, the groundwork for the diffusion capacity model is set down in Section 2. 3. A relationship between the junction voltage and the excess (over the thermal equilibrium density) density of charge carriers at the junction was set down there. q - 1 (2. 26) P(ot) = p (eKT 1 (2.26) Because the voltage under stable operating conditions is a periodic function of time, this expression can easily be transformed from the time domain into the frequency domain: K L- {Vp(S)} P(O,s) L e L {V 1 (6. 7) The solution to the diffusion equation yields a linear relationship between the excess carrier density and current (Eq. 2. 30). In the frequency domain, this relationship can be written as = qA (1 +sT) P(, s) (6.8) d (s) T/qE \(6.8) K-( 1 + 5J+ 0 + ST) Equations 6. 7, 6. 8 and 2. 33 can be combined to form a useful current —voltage relationship. 126

I (S) (6. 9) (d + 1~)(1~ST) A nonlinear differential equation which describes the behavior of the pump circuit of Fig. 6. 2 can be obtained by substituting into Eq. 6. 1 the magnitudes of the various currents which are expressed in Eqs. 6. 2 6.3, 6.6 and 6.9. The result is 0 -IN(S) + Vp(s) Y(s)+ 2sCa 0 L - V (s)} q L-1V (s)} ~-p}(-+sT) L eKT p 1 (6.10) (11+ +(s 7) 6. 3 Iterative Techniques The technique for solving the nonlinear differential equation (Eq. 6. 10) employed is basically an extension of the Newton-Raphson approach used in the last chapter. But before describing the procedure for solving the equation, it is first necessary to make a few remarks as to the nature of the solution and the choice of the forcing function. Here, as in Chapter V, it is assumed that the circuit is stable and there are no oscillations of any kind. The solution to the differential equation can then be taken to be periodic, with the same period as the forcing function, and the pump voltage across the 127

varactor can be written as VA(O) x Vp(t) = 2 + [VA(n) cos nwpt +VB(n) sin nw t] (6. 11) The fundamental component of this voltage [VA(1) cos wpt + VB(1) sin wCpt ] is selected as the forcing function. VA(1) can be arbitrarily set equal to zero; this is simply equivalent to selecting a time origin. The physical significance of this rather unorthodox choice can be seen by comparing Figs. 6. 2 and 6. 3. The current source driving function IN(t), in Fig, 6.2 is replaced by a series combinaVA(0) tion of a voltage source, EN 2 + VB(1) sin wc pt, and an ideal filter. This filter is a short circuit at DC and at w p; it acts as an open circuit at all other frequencies. The current flowing through this voltage source-filter combination, plus the voltage components at harmonics of the pump frequency, will be treated as the unknowns. The term EN is selected to be the forcing function for three important reasons: (1) The fundamental component of pump voltage has more physical significance than the Norton equivalent current source. Only when Y(w) is much less than the impedance of the diode (the current pumping case) is the magnitude of the current source related to the degree of hardness with which the diode is being pumped. The pump voltage, on the other hand, is related to the diode capacity by Eq. 2. 1 128

C(t) a (2. 1) (1 Although there is no simple relationship between VB(1) and the magnitude of the capacitance coefficient at the pump frequency, there is at least a one-to-one correspondence. The hardness of pumping increases smoothly with increasing pump voltage. Even this does not always hold with the Norton current source. (2) It is assumed here that the diode is biased with a DC voltage source. It would be unreal to bias the diode with a DC current source. (3) The magnitudes of all the voltage components at harmonics of the pump change smoothly as the fundamental voltage component increases. It will be seen later in this section that the initial approximation used to initiate the convergence procedure must differ from the correct solution by less than a predetermined error, e max The value of m is "built into" the iterative procedure max and if the initial error is greater than emax' the sequence will not converge. It is therefore necessary to control the relationship between the initial approximation and the correct solution. To implement this, a family of solutions is constructed starting with the smallest pump magnitude and building up toward hard pumping. For each pump level, the initial approximation is the solution previously 129

determined for a slightly smaller pump amplitude. The voltage source forcing function insures that the solutions change smoothly, and thus guarantee that the solution previously found will be a reasonable initial approximation. Now that the forcing function has been established, an iteration procedure for solving Eq. 6. 10 can be described. The solution to Eq. 6. 10 is the limit of the infinite sequence of functions Vt(t), v;(t), vb(t),... Vm(t).. and Vp(t) = Limit V'(t) (6. 12) m- m The procedure for determining this sequence is identical with the Newton-Raphson technique described in Section 5. 4, except for one detail which is involved with the nature of the exponential term. It was explained in Chapter V that the usefulness of the Newton-Raphson method depends upon how accurately the nonlinear equations are represented by their linear approximations. Normally, the first two terms of Taylor's series are used as the approximation. This works well when the nonlinearity is not strong. This is the case in Chapter V, and also for the portion of Eq. 6. 10 which deals with depletion layer capacity. The function which describes diffusion capacity is more difficult to handle. The basic form of the function is 130

q v KT f(v) = e (6. 13) and at room temperature, q 40. (6. 14) KT It can be easily seen that the first two terms of Taylor's series fail to approximate this function accurately. In fact, if e max' the magnitude of the difference between the initial approximation and the correct solution, exceeds.020 volts, a Newton-Raphson sequence of the type used in Chapter V does not converge rapidly enough to be useful. To increase the range of convergence, two sequences of constants, Fm and Gm, are used to improve the accuracy of the linear approximation. In the mth iteration, the exponential function is approximated by f(V' +E) F f (Vm) + G f(V') E (6.15) m m m m m rather than the conventional two term Taylor's series approximation f(Vm +d) vf(Vm + - (6. 16) m The values of F and G are determined such that they minimize 131

Solid line is the function. e40V +40 Dotted line is the 1st two terms of Taylor's expansion around V=0. +30 Alternate solid and dotted line is improved linear approximation around V=0 according to m=4. +20 +10 / / / -10 / Fig. 6.4 Exponential function and linear approximations 132

the maximum error between V as computed using the exact relationship (Eq. 6. 13) and the approximate relationship (Eq. 6. 15) over the range V' - V V< V + e (6. 17) m max - - m max Figure 6. 4 illustrates this graphically. As the iterative sequence converges, i. e., as m increases, the error e max decreases, and the linear approximation can improve accordingly. A useful sequence for F and G is shown in Table 6. 1. m m It is assumed that the initial e is less than 200 mv. This can max always be achieved by constructing a family of solutions as explained earlier. If the initial e is greater than 200 mv, the sequence may max not converge, depending upon the relative magnitudes of the diffusion capacity and the depletion layer capacity terms. Iteration Max. Possible Max. Possible Numbe r, Error e Error Is m m m Reduced To 1 200 mv 208.0 7450 163 mv 2 163 mv 73.5 2040 127 mv 3 127 mv 20.7 629 94 mv 4 94 mv 10.4 272 62 mv 5 62 mv 3.2 95 30 mv 6 30 mv 1.38 50 8.2 mv 7 Use standard 1.0 40 and up Newton- Raphson method Table 6. 1. Iteration Sequence 133

I TIls] I[Ids] [is] [YS] I [Yd] [ c] CI [IVS ] Depletion Diffusion Layer Capacity Capacity _ Fig. 6. 5. Signal circuit configuration used in Chapter VI. 134

6. 4 Small Signal Equations In the preceding two sections, a method was set down for determining the voltage, Vp(t), across a varactor diode under conditions of hard pumping, but in the absence of signal. In this section, the signal is included, and the effect upon the signal of pumping the diode is determined. Figure 6. 5 shows the circuit redrawn in a manner which emphasizes the signal circuit. Admittance is used, because the depletion layer capacity appears in parallel with the diffusion capacity. The linear portion of the circuit, including the impedance of the pump source can be replaced by an equivalent (linear) admittance. This admittance must adequately model the linear circuit at the entire set of signal frequencies, nwp ~i co n O, 1,... (3. 7) or from Table 3. 1 w1' l2, W3,... (6. 18) The pump source is not shown, since it is a linear voltage source at wo with zero impedance, and cannot interact with any component of signal. The signal source is represented by a current source. The currents in the circuit of Fig. 6. 5 must satisfy the elementary relationship. 135

[ Is] = ls] + Ids + I is] (6. 19) a. The Driving Source. I is a current source, and is independent of the signal voltage, but the three other signal currents must be determined from the voltage-current relationships ( the admittances ) of the three elements. b. The Depletion Layer Current. The admittance of the depletion layer capacity is represented by a matrix of the form of Eq. 3.22. [i] = [C] x [Vs (6. 20) The individual elements of this matrix can be computed by first solving Eq. 6. 10 for Vp(t), the pump voltage across the diode as outlined in the previous section. In the abrupt junction case, the time varying capacity can then be computed by substituting Vp(t) into Eq. 2. 1. C C(t) = a, (2. 1) -1 VP(t)/ Since the pump voltage is periodic, the resulting capacity may be written as an exponential Fourier Series: cc jnw t C[Vp(t)] = V C e (6. 21) [1n36 136

The capacitance coefficients in Eq. 6. 21 when multiplied by the appropriate frequency form the individual elements in the matrix of Eq. 3.22. c. The Network Current. The signal current through the linear network can be written as the product of a voltage matrix and an admittance matrix: Ils] = [Ys] x [v (6. 22) The admittance of the linear network can be represented as a diagonal matrix of the form Eq. 6. 23. The elements of this matrix are just the admittances (or the complex conjugate of the admittances) at the appropriate signal frequency. Recall that in deriving Eq. 3. 22, it was necessary to take the complex conjugate of all the even rows in order to get a consistent set of signal voltages. It is desirable to have the admittance matrices compatible, and so the elements in all the even rows in Eq. 6. 23 have also been conjugated. 0 0 0 0 Y1 0 Q O 0 Y* 0 0 0 [Ys] = Y 0 0 (6.23) 0 O O Y* 137

d. The Diffusion Current. The admittance of the diffusion capacity is also a matrix; the individual elements of which can be computed by using the pump voltage, Vp(t). It was mentioned in Chapter II that parametric mixing occurs in diffusion capacity due to the nonlinear nature of the Boltzmann equation (Eq. 2. 26). Mixing the nonlinear diffusion capacity is fundamental to the input admittance, and is described here in some detail. The Boltzmann equation relates the excess density of mobile charge carriers at the junction to the instantaneous junction voltage. P(O,t)= n (eKTe - 1) (2.27) When both pump and signal are present, the total voltage across the junction consists of both pump and signal components: vt(t) = Vp(t) + vs(t) (6. 24) n +vc ( VA(n) j n- VB(n)) jcot +C vt(t) Z 2 e + n=-c m=-cc p+1 j(mw +w )t +c j(mw -wol)t e p + V( ) e p (6.25) By exactly the same reasoning used in Chapter III, the pump voltage may be assumed to be much greater than the signal voltage. That is, 138

Vp(t) >> v (t) (6. 26) Equations 6. 24 and 6. 25 may be substituted in Eq. 2. 27 and the result simplified because of Eq. 6. 26. The result is q +OC A( ) i VB(n) jno t P(O,t) = PneKT [C. 2 -. e J n ---- ocn +cc j(mw p+aW 1) t +Oc +KT n =_C (mC +W M) e C C (m -T1) *e 19 e P A Be + 2 n= - oc (6. 27) Now the individual terms in Eq. 6. 27 may be identified. The first term on the right hand side is just the carrier density due to pump in the absence of signal. This component of carrier density is P0t jnP neo t ne [n — (A(n)+jTT V(n) jp p (t) A On e n=- oc n=- cc (6.28) The coefficients of this series, the p 's, can be determined by first solving the pump circuit differential equation (Eq. 6. 10) for the pump voltage, and then expanding Eq. 6. 28. 139

The last term on the right hand side of Eq. 6. 27 is the signal component of carrier density. By using the definition of Eq. 6. 28, it may be written as,,+OC j(nwo+ l) + ++xc j(nmw -w l) t n=- cc n=- oc p 1 (+oc j(n+ o +co jw KT P V (t V(nco +co) mn=-c p 1 +OC jj(nw [( n) w1] t (nc -) e(6.30) n=-oc p and and j(no +cot + j(noomponents identified. n=-c ' p 1 n=-oc p1 KTc Pn (moo +co ) im, n=-oc p +(nOCw KT m [( n-m)wp iw] (6. 31) 140

The relationship between the charge carrier density and the current was derived in Section 2. 3. It is a linear relation which resembles an expression for admittance and it can be written as Pn(1 + jC) aw IT n Tci (6.32) The current due to diffusion at any signal frequency is found by combining Eq. 6. 31 and Eq. 6. 32. q Pn[1 + j(nwp~ co 1) )7] wp 1 KT (1-+ 1 [1+j(np~1)T]) m= - Cnm) (6. 33) This expression relates any one component of diffusion capacity signal current to all the components of signal voltage. It is convenient here to combine all the components of current into a single expression which has the same form as the matrix of Eq. 3. 22. Ids] = Yd] x Iv (6.34) Whe re 141

q Pn(l+ iw1T) p q n (l+w1T) 1 q Pn( + jo 1) p_ 1 KT 1+ i1+ 3(1+jw1T) 1+xf1+f3(lljrwT) T1+ Aj1i7jW1T) KT 1+1(1jw KT KT3(1-wr)T 1 + jWlT + 1 [Yd] ( q n (l 3 jWT)p1 q n(l +j3T) p2 |KT + 1+3(1+jw 3T) 1 + d+ ( -jwT) q 4n(T io4) P 2 KT 1+ 1+f(1-J4) + w 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The final form of the admittance seen by the current source is obtained by combining Eqs. 6. 19, 6. 20, 3. 22, 6. 23, and 6.34. The result is written out in Eq. 6.35. 1 and vS are column matrices, but to determine the input impedance, signal is applied only at one frequency, w1; and thus only one term of the I matrix is nonzero. However in computing the noise of a parametric amplifier, it is necessary to account for noise which is generated at frequencies other than wl, but is parametrically mixed with the pump and appears at the output. This can be done by assuming that the shot noise in the diode and the thermal noise in all the resistors are uncorrelated at all the signal frequencies. Noise due to each of these sources at any signal frequency can then be represented by an independent current source. In this case, I is a column matrix consisting of non-zero terms each of which represents the total noise generated at a particular signal frequency. Is] = [Ys] x [V + [Yd] x [Vs + [C] x [vs (6. 35) That is, Is] = [Yt] x [Vs whe re [Yt] [Ys] + [Yd] + [C] The first few terms of the complete admittance matrix [Yt] are written out in Eq. 6. 36. 143

q Pn(l+jCwOIT) q P(l+jw co1) Y1 K T 1+ 1+ +j17 KT 1+ +1 + 1+j11T q Pn(1- jw2'T) CP(-j2) KT 1+P12K P+- i7C1 - 2 KT + J 2 PO W2CO [ti= q Pn(l1+jw3r) q P n(1+jw37) KT 1~j3+ P1 +j 3C1 +qT 1+2 3 2 1 + 1++ jf3cr 73 4 (6. 36) The C's and p's are determined by solving Eq. 6. 10; 3 and Pn are constants of the diode n

The input admittance at co can be obtained by manipulating this matrix. A number of schemes can be used, however, the method used here is to reduce the matrix to its lower triangular form. The desired admittance is then just the (1, 1) element of this new matrix. 6. 5 Summary In this chapter, the results of the previous chapter were extended to the hard pumping case. Both depletion layer and conduction effects have been accounted for. A nonlinear differential equation which describes the pump circuit was set up, and an iterative technique for solving it was presented. This iterative technique requires the use of a digital computer. Once the behavior of the pump has been determined, the signal is added to the circuit. The signal circuit behavior is expressed in a matrix formulation, the elements of which are determined by the pump circuit. The small signal input impedance is then obtained by manipulating this matrix. The numerical results which are presented in Chapter VII were obtained by an algorithm of the procedures outlined in this chapter. The flow diagram of this program is similar to the one presented in Appendix D. 145

CHAPTER VII EXPERIMENTAL WORK AND CONCLUSIONS 7. 1 Introduction Throughout the course of this research, a great deal of experimental work has been performed. This work was of considerable aid in the development of the theoretical portions of the thesis, particularly in the determination of accurate mathematical models for the varactor diode and its associated circuitry. The experiments were performed concurrently with the development of the theory, and as the work progressed, the theory and circuit model were constantly altered to conform with the observed facts. The result of this process is an accurate characterization for pumped varactor circuits, which can be utilized by an engineer to determine circuit behavior. Three circuits were built and tested. The first two served mainly in the development of the depletion layer capacity theory of Chapter V. These two circuits are termed the "Lumped Circuit" and the "Distributed Circuit. " The third circuit is termed the "Diffusion Circuit" and aided in the development of the theory presented in Chapter VI. A description of these circuits and the measurement methods is found in the next section. In Section 7. 3, the results of the experiments are presented along with the corresponding theoretical results. In general, 146

Spectrum V.H. F. Analyzer Reference HP851A Signal HP608D V.H.F. Test Coupler L.P. Pump Circuit HGR874GA Filter Source I. lGR 1215B Twin- Tee D.C. < L e d 4 D. C. Bridge Supply GR 812A Communicatio H.F. Receiver 10 Db Signal Source R-390-URR R133Pad GR 13 30A Fig. 7. 1. 3Block Diagram of mleasurement apparatus 147

MC M Coax Cf Cf 2 Coax Tro To DC Hi To Pump and Supply Side To Spectrum Analyzer Referenee of Bridge Signal Fig. 7. 2. Circuit used for measurement on lumped circuit and diffusion circuit 148

To Spectrum Analyzer CI Coax,ITe2 -d -, To Hi SideCf3 d- -- Of Bridge CRZ tDoD CB11 C CB To DC Coax Supply To Pump and Reference Signal Fig. 7. 3. Circuit used for measurements on distributed circuit 149

N b c 0. 4 1.3 /-amp a 0. 3 Curve a - distributed network Curve b- diffusion network Curve c - 1st order approximation 0. 2 -0.1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 +1 Resonant Frequency Shift % of co Fig. 7.4. Pumping hardness as a function of resonant frequency shift 150

excellent agreement is obtained between the experimental results and the theoretical results. The conclusions of this study are discussed in Section 7. 4. 7. 2 Experimental Methods A block diagram of the measurement apparatus is shown in Fig. 7. _. The experimental setup is the same for each of the three circuits. The impedance of the varactor under test is measured at a High- Frequency signal (co I) on the parallel tee bridge, using the Communication's Receiver as a null-detector. The High-Frequency signal is of course supplied to the varactor circuit through the bridge. The frequency of this signal is in the 0. 5 MHz to 10 MHz range. DC bias is supplied to the varactor through a microamp meter which monitors the change in DC current with pump powe r. It is necessary to make all the input impedance measurements under "small signal" conditions; that is, the signal components must be restricted to the set of frequencies (3. 7). The presence of any components in the set (3. 11) indicates that the small signal conditions are not satisfied. The spectrum analyzer is used to observe the amplitudes of these signal components, and if any undesired higher order sidebands are present, the signal source is attenuated until small signal conditions are obtained. In practice, the undesired signal components are always kept at least 20 Db down from the largest 151

component in the signal set. The spectrum analyzer is also used in conjunction with the VHF reference signal to determine the pumping hardness. A definition of pumping hardness is A (3.31) or alternatively A iS1,^ Slj (7.1) where the S's and C's are the Fourier coefficients of the exponential series. An adjustable inductive coupler permits a low amplitude VHF reference signal (W1) to be added to the pump power (W1). As explained below, this reference signal is useful in determining the pumping hardness. The frequency of both the reference signal and the pump are in the 60 MHz to 1.20 MHz range. At these frequencies, there is no direct method to measure y. However, an indirect method proved to be quite satisfactory. It is known that the average capacity of the varactor increases with pump voltage. Hence, the resonant frequency of the tuned circuit decreases with pump amplitude. To measure y then, we simply insert the small amplitude VHF reference signal, and note the frequency of peak response. This will usually be slightly lower (0- 10%) 152

than the unpumped resonant frequency of the circuit. In some cases, however it may actually be higher because the signal response depends not only upon the average capacity of the diode but also, and to a lesser extent, upon parametric mixing with the pump. The value of y can be obtained by comparing the measured shift in resonant frequency with a plot of "y versus shift" which can be computed using the theory. This method of experimentally determining y is simple and reasonably accurate. The only difficulty lies in finding the peak response points, for the response curve tends to flatten out (the circuit Q appears to become lower) as pumping increases. Plots of "y versus shift" are shown in Fig. 7. 4. Curve "a" is for the distributed network which is described in (Fig. 7. 3; Table 7. 2) when it is pumped at resonance. Curve "b" is for the diffusion capacity circuit (Fig. 7. 2; Table 7. 1) when pumped at resonance. For a resonant frequency shift greater than about 7.5 %, curve "b" has little meaning, for beyond this point the diode is drawing a measurable amount of DC current. It is much easier to read this current, and correlate it with y than it is to measure the shift in resonant frequency. In fact, measurement of DC current is the accepted method of determining y. However, it has not only the disadvantage of yielding no qualitative information when pumping is not hard enough to draw rectified current, but also may be inaccurate due to reverse avalanche current (Ref. 54). 153

Curve "c" is a first order approximation. It is derived in the following paragraph using the approximate relations developed in Appendix B. For an abrupt junction diode (N= 2) and small current so that only the first terms of the expansions are important (B. 12) and (B. 13) can be combined with (B. 7) and (B. 9) to obtain CO = (1 +.5a2) CB (7. 2) C1 = (-. 5a) CB (7.3) The shift in resonant frequency is given by -6 Cw 2CO (7. 4) Combining (7. 4) and (7. 2) and using (7. 3) in the definition of y in Eq. 3.31, an approximate relation between y and 56 is obtained 5 -.38y2 (7. 5) c0 This is the approximate relationship plotted in Fig. 7. 4(c). Figure 7. 2 shows the test circuits which were used for the lumped circuit and for the diffusion capacity measurements. The DC bias is returned to ground through the bridge. Element values 154

for both circuits are given in Table 7. 1. Diffusion Capacity Lumped Circuit Circuit Diode Type CR 1 PC 1430 D PC 0622 C Nominal Diode Capacity 7. 2 pf @ -4 volts 14. 6 pf @ -4 volts Bias Voltage VBias - 18. 0 volts -2.0 volts Diode Capacity at VBias 3. 7 pf 18. 4 pf Inductor L1 0. 93 iih 0. 179 /Ah Inductor Stray Capacity Cs 0. 9 pf 0. 75 pf Inductors L2, L3 Negligible Negligible Mutual Inductances M1, M2 Negligible Negligible Cfl 11000 pf 11000 pf Cf2 1000 pf 1000 pf Small Signal Resonance co0 76. 0 MHz 84 MHz Small Signal Q 100 56 Table 7. 1. Lumped and Diffusion Capacity Circuit Parameters Figure 7.3 shows the test circuit built for measurements on a distributed circuit. Element values are given in Table 7.2. The body of the cavity is machined of brass. DC bias is returned to ground through the bridge, while the cavity itself is at potential VBias from ground. 155

Distributed Circuit Diode Type CR2 PC 0642 C Nominal Diode Capacity 13. 4 pf @ -4 volts Bias Voltage VBias -25.0 volts Diode Capacity at VBias 5 7 pf I. D. 0. 244 in. O.D. 0. 5625 in. Length d 26.5 in. CB1 10 pf Cf2, Cf3 1000 pf CB3, CB4 4000 pf Small Signal Resonance So 100 MHz Small Signal Q 140 Table 7. 2. Distributed Circuit Parameters 7.3 Experimental and Comparative Theoretical Results a. The Pump Circuit Solution. As described in the previous two chapters, the first step in determining the behavior of a pumped varactor circuit is to solve the pump circuit equation, that is, determine the circuit behavior with pump power applied, but with the signal amplitude at zero. The solution to this nonlinear equation takes the form of a nonsinusoidal periodic waveform, which if desired can be 156

+1 Time - Electrical Degrees,O -1 Bias Level -3 -4 -5 -6 Diffusion Circuit c = 84 MHz -7 Fig. 7. 5. Diode voltage waveforms 157

40 36 y=.45 32 28 24 20 16 12 Time - Electrical Degrees 4' 90 180 270 360 -42 -- -4 -8 -20 -24 -28 Diffusion Circuit -32 - Ow =84 MHz -36 -40 - Fig. 7.6. Total current waveforms 158

40 36 w = 74 MHz 32 P - =.45 28 D. C. Current =16 j/a 24 20 - - \ Current 16 12 0 Ole -- ".....<Time-Electrical Degrees 0! 90 180 270 360 -4 _ ~u I 7 - > \\ I - -8 -12 -24 -3 ~~~~~~~~~I-28 -32 / - s _ / Fig. 7. 7a. Barrier layer capacity current waveform 159

16 14 12 10o w | \ p = 74 MHz 8 y =.45 6 - 5 \ Current D. C. Current = 16 pLa 0 I4 - +1 Time - Electrical Degrees 90 180 270 360 O0 -2/ -4 L -6 '- 6 -8 -2 -3 Fig. 7. 7b. Diffusion current waveform 1-0 Volts / -5 160

expressed as a Fourier series. A few computed waveforms for the "Diffusion Circuit" are illustrated in Figs. 7.5 and 7.6. Figure 7. 5 shows the voltage waveform across the diode for three values of pumping. The most notable feature is the clipping which occurs as the diode is driven into the region near zero voltage. Here, the capacity-voltage curve has the greatest curvature. As the pump level increases, the clipping becomes more pronounced as the diode is driven into forward conduction during the peak of the cycle. The characteristics for all the diodes used in the experiments are summarized in Appendix E. The total current wave shape is shown in Fig. 7. 6. By comparing 7. 5 with 7. 6, it can be seen that the current waveshape is more nearly sinusoidal than the voltage wave shape. This can be understood by noting that the inductor presents a high impedance to the harmonic components of current, and tends to open circuit them. For this particular circuit, the current source approximation, in which only the fundamental component of current is allowed to flow is a quite accurate approximation. The currents shown in Fig. 7. 6 are the total currents flowing through the diode. The two coiiyonents of current are shown in Fig. 7. 7. The portion of current due to barrier capacity is shown in Fig. 7. 7a, while that current due to forward conduction and diffusion effects are shown in Fig. 7. 7b. The diffusion currents for y =. 15 and y = 0. 30 are negligible. 161

The distinction between these components is important because the diffusion current produces shot noise, while the current flowing through the barrier capacity is noise free. The double-spike waveform of the diffusion current occurs because the carrier lifetime is long co-pared with a period of the pump cycle. Hence, few of the carriers injected across the junction during the positive portion recombine, and when the voltage begins to decrease, they are swept back across the junction. In this regard, it is interesting to note that the current in the negative spike may actually be larger than the current in the positive spike. An avalanche effect may be initiated by the carriers coming back across the junction. The result may be a net reverse rather than a forward current (Ref. 54). This phenomenon was not observed during the course of these experiinents. b. The Elastance Coefficients. After the solution to the pump circuit has been comnputed, the next step in determining the behavior of a pumped varactor circuit is to calculate the elastance coefficients. Some theoretical results of these calculations is shown in Figs. 7.8 through 7. 12. The magnitudes of the harmonic elastance coefficients for a variety of conditions are plotted as a function of S1/SO, that is as a function of y. No direct experimental verification of this portion of the theory was obtained, for at these frequencies, it is impossible to measure the time varying capacity directly. Figure 7.8 illustrates the harmonic elastance coefficients 162

10-1 Abrupt Junction Diode -- 0= 100MHz w = 100 MHz = 106 MHz Qc 100 10-2 Cd 10-3 S3/S0 *-4 10 u" ~ / S4/S 10 50 0.1 0.2 0.3 0.4 Fundamental Elastance Coefficient Fig. 7. 8. Harmonic elastance coefficients for a lumped circuit 163

10 -Abrupt Junction Diode WO = 100 MHz w- - = 100 MHz P o = 220 MHz C Qc= 100 10-2 l10 C) EC Cl) S3/S0 a) / % 10 4-4 -4 10 _ / O~~~S4/So 1-5 0 0.1 0.2 0.3 0.4 S Fundamental Elastance Coefficient Fig. 7. 9. Harmonic elastance coefficients for a lumped circuit with distributed capacity 164

101 10 2!0-3 _- t I / Abrupt Junction Diode j~j S3/So m t / ~~Q = 100 0 a) w / co = 100 MHz wcoo = 100 MHz En P4 310-4 10 Fundamental Elastance Coefficient SDiode Fig. 7. 10. Harmonic elastance coefficients for a distributed circuit 165

991 apo.p lenp3' ue ioJ s~uaolitJJoo a9ou-stla D.UOtUzJH '11 ~L O~L.I OS/IS 4umalo.ooD aouelse2 t Ipua uepunj $ '00 000 T *0 ~s/s ZHI OZnZ = ' 001 = ZHN 001 = a o 4in-Ij padnn / O C Os/Cs ~sl m 0I _0 I

10 B 10-2 01 C) ~ c) -- o I D o / A - Abrupt junction diode, co = 100, = 100, distributed cirit 4-1-4 4Q = 100, lumped circuit W 106 10 C) C --- C - Abrupt junction diode, co = 100, Q = 100, lumped circuit ode, w = 220 D - Diode No. PC1430D, wco = 100, Q = 100, lumped circuiP wc = 220 10-5 o 1.2 0.2 0.34 Fundamental Elastance Coefficient S1/S0 Figo 7. 12. Comparison of harmonic elastance coefficients for different circuit conditions 167

for a lumped circuit of the type shown in Fig. 7. 2. The diode is assumed to be an ideal abrupt junction device having a characteristic of the form (Eq. 2. 12). The self-capacity of the coil is assumed to be negligibly small. Figure 7. 9 represents the same conditions, except that here the self-capacity of the coil has assumed a practical value, making the coil self-resonant at a frequency just above the second harmonic of the pump, The effect of the resulting pole of impedance can be clearly seen by comparing Fig. 7.8 with Fig. 7. 9. It virtually open circuits the second harmonic, prevents current from flowing and lowers the harmonic content of the elastance. Figure 7. 10 illustrates the harmonic elastance coefficients for a distributed circuit of the type shown in Fig. 7. 3. As in Figs. 7. 8 and 7. 9, the diode is assumed to be an abrupt junction device. However in contrast to the circuit conditions of Fig. 7. 9, this circuit has an infinity of series type resonances or zeros. The most important of these resonances lies near the third harmonic of the pump. This zero of impedance virtually short circuits the third harmonic, which results in a large component of current flowing at this frequency and raises the harmonic content of the elastanceo These figures indicate that the magnitudes of the harmonic elastance coefficients are strongly dependent upon the circuit impedance at the harmonic frequencies. From Appendix B, note that if the commonly used open circuit approximations were used, all the harmonic elastance coefficients would be zero. 168

Figure 7. 11 illustrates the harmonic elastance coefficients for the same conditions as those in Fig. 7.9 except that data on an actual diode rather than an idealized abrupt junction device was used for the computations. Each of the deep nulls in the curves is accompanied by a 1800 shift in phase. Figure 7. 12 is intended to summarize the results in Figs. 7. 8 through 7. 11.The second harmonic elastance coefficient is plotted for each of the conditions described above,, The curves illustrate that elastance is dependent upon both the details of the capacity-voltage characteristic, and the circuit impedance at the harmonics of the pump. c. The Determination of Input Impedance. The third step in the computation of input impedance is to insert the elastance coefficients into a small signal matrix, and manipulate this matrix to obtain the desired quantities. Some theoretical results along with the corresponding experimental values are illustrated in Figs.7.13through7.19. In contrast to the pump voltage, pump current, and elastance waveshape which are most difficult to measure experimentally, it is relatively easy to measure the input impedance by using the techniques described in the previous section. This in fact is one reason for emphasizing the input impedance. It is easy to measure, and, as will be shown, is also a sensitive function of the parameters which influence circuit behavior. These include the hardness of pumping, the diode capacity-voltage characteristics, the circuit impedances at the pump harmonics, the termination of the sidebands of both the pump 169

0 =.10 -20 - 40-0 0 -60 80170 0 y=. 3 Experimental y -=. 2 Experimental o y=. 1 Experimental -140 Theory Pump Frequency = 76 MHz - 160 0 1 2 3 4 5 Signal Frequency % of wo 170

+120 +100 ClO o +80 +60 +240 I. E -=0.3 -20 -40 I I I I I 0 1 2 3 4 5 Signal Frequency % of wo FigO 7. 13b. Imaginary part of the input admittance of the lumped circuit 171

fundamental and its harmonics, the electric fields in the diode junction the minority carrier density, the minority carrier lifetime, and the diode series resistance. For these reasons, the input impedance is an excellent media for exploring pumped varactor circuit behavior. Figure 7. 13 illustrates the input admittance of the "Lumped Circuit" for three values of y. The pumping was not hard enough to draw rectified current or to cause diffusion effects to become important. All the results,then,are due to nonlinear depletion layer capacity. The experimental data is indicated by circles, while numerical results are shown as solid-lines. The abscissa quantities in all these figures is signal frequency (w1) normalized to the small signal resonant frequency of the circuit and expressed in percent, that is: 100x(w 1/w0). The ordinate quantities are the input admittance of the varactor at the signal frequency. Both the real and imaginary parts of the admittance are expressed in A- mhos. The real part of the unpumped diode admittance is zero L- mhos, while the imaginary part is indicated by the straight dashed line. On each graph, data is shown for several values of y. The unpumped admittance of the circuit is 3.7 pf in parallel with 0 p- mhos. Pumping is done at the small signal resonant frequency of the circuit (cv0 = 76 MHz). The most striking feature exhibited by these curves is a series resonant type of effect —except that the signs of the real part as well as the imaginary part are opposite front those of a passive circuit. 172

This phenomenon can be understood (at least intuitively) by considering only the behavior of the pump lower sideband (wp - wa) By the Manley-Rowe relations, Section 2. 4, the power dissipated at this sideband reflects as a negative resistance into the signal frequency. The magnitude of power at the lower sideband is determined both by y and the impedance; the maximum flow of power, of course, occurs for large y and when the circuit is resonant at the frequency of the lower sideband. From Eq. 7. 5, the resonant frequency of the circuit is seen to decrease with y. Thus, as pump is increased, both the amplitude of the negative resistance curve, and the frequency at which it peaks, increase. The bandwidth of the curve is essentially determined by the Q of the unpumped circuit. For purposes of argument, the effect of other sidebands can be ignored, for they become further from resonance as pumping is increased, In the curve of Fig. 7. 13, the amplitude of the (y=. 3) curve has a minimum at approximately 5. 5 % (4. 2 MHz). Figure 7. 14 illustrates the results of three different methods for computing the input admittance. The Lumped Circuit with a value of y= 0. 2 is used as the example. The first method is the computer solution described in this paper. The results for this solution can be seen to yield results which correlate well with experiment. The second method is the first order open circuit theory which assumes a sinusoidal elastance variation and the existence of 173

Pump rExperiment o Frequency = 76 MHz First Order Theory - - y = 0o 2 First Order Theory Corrected for Shift in Resonance -- - - - - +10 Computer Solution 0 - 20 - \ -40 -50 % / -60 I~d - 80 1 / 0 2 3 4 5 Signal Frequency To of wp Fig. 7- 14a. Comparison of methods for computing the real part of 174

9LJ I.nolos padtunj aql jo awou-uu!upu 4ndu. aoq Jo pl.id AXIuia.tu.I aq Sul. ndtuoi a oj spoqtaw jo uosl.edu oD ~qYt ~L 'S co o, % ouanbalae IeuSlS IE Z `~~I II / — " OZI I I -/,'I I - I I -, 0 - 0t /~ II I = '

+20 Pump Frequency= 100 MHz +10 o 0 E =15 -10 -20 20 /a 7'y=. 2 Experimental 4- A/ y=. 15 Experimental -30 a o y=. 1 Experimental Pq/ - Theory - 40 -=O 2 -50 _ I I I I 0 1 2 3 4 5 Signal Frequency % of cop Fig. 7. 15a, Real part of the input admittance of the distributed circuit 176

+180 Unpumped Admittance +160, /// I ~~~/ - +1 0 / 1 h// ~ + 120 +20 - +60 /2 3 4 5 Signal Frequency of w Fig~ 7. 15b. Imaginary part of the input admittance of the distributed circuit 177

Experiment o First Order Theory First Order Theory Corrected for Shift in Resonance-____. Computer Solution -20 -2 o \ / 40 -40 S!.' \ ~~~~~I Pump Frequency = 100 MHz -60 o 1 y = 0.2 PI I ~ -80 - I - 100 \\ I / I 'L.I' l l 0 1 2 3 4 5 Signal Frequency % of wp Fig. 7. 16ao Comparison of methods for computing the real part of the input admittance of the distributed circuit 178

200 I ' _ I/ reIn~~~~~~ I AXI 160 120 Im 0 = 40 ICl 80 S I I -.S\ 1 E \ I -40 \ I 0 0I I' I - 80 \ I 0 1 2 3 4 5 Signal Frequency % of wp Figo 7. 16bo Comparison of methods for computing the imaginary part of the input admittance of the distributed circuit 179

only three signal currents (at wl, wp-w1, and wp+W ). The results using this method do not yield accurate results. The third method is the above first order open circuit theory which assumes a sinusoidal elastance variation, and the existance of only three signal currents, but is modified slightly to account for the variation of the average capacity and the resulting shift in resonant frequency with pump power. This is done by using the first order Eqs. 7. 2 and 7. 5 to compute the average capacity and the shift in resonant frequency. Except for a small error in the resonant freqeuncy shift, this approximate method can be seen to yield quite accurate results for this circuit. Figure 7. 15 shows the admittance of the "Distributed" circuit (Fig. 7. 3, Table 7. 2). The unpumped admittance of this circuit is 5.7 pf. Curves are shown for pumping at the small signal resonance ( c0 = 100 MHz). In this case, the resonances exhibited by the lumped circuit (Figs. 7. 13 and 7. 14) are masked by several extraneous effects. First of all, the shift in resonant frequency with pump is not as great for the distributed circuit as for the lumped circuit. This can be seen by comparing Fig. 7. 4(a and b). Also, due to the multiple resonances of this structure, sidebands other than (u p- c 1) and (wp + Cl) are important. The tuning of these sidebands change with pump in a manner different from the lower sideband, making this case difficult to understand intuitively. The difficulty in understanding this circuit can be seen 180

0 - -20 E0/ U -40 Distributed Circuit ~ -60 Pump Frequency= 100 MHz rHo y = 0,.2 A-4 Experiment o -80 k Computer Solution (21x 21) Computer Solution (3 x 3) Computer Solution (5 x 5) Computer Solution (7x7)............ 100 0 1 2 3 4 5 Signal Frequency % of wp Fig. 7. 17a. Effect of harmonics upon the real part of the input admittance 181

200 160. I - Cd 80 E / 0 CPd/ -40 -80 I I I I I 0 1 2 3 4 5 Signal Frequency % of wP Fig~ 7. 17b. Effect of harmonics upon the imaginary part of the input admittance 182

by comparing Fig. 7. 14 with Fig. 7. 16. In Fig. 7. 14 which illustrates the Lumped Circuit, the three current open circuit theory is seen to yield good results when modified by a shift in resonant frequency. However in Fig. 7. 16 in which the distributed circuit is exhibited, this simple modification yield results which are no better than the conventional theory. This can be at least qualitatively explained by referring to Fig. 7. 12 where it is seen that the coefficients of the higher harmonics of elastance are much larger in the Distributed circuit than in the Lumped Circuit. Thus, sidebands around the harmonics of pump have a greater magnitude and therefore play a larger role in the determination of input impedance. Figure 7. 17 illustrates this concept. The input impedance of the Distributed circuit for y= 0. 2 was computed using various size impedance matrices of the form (Eq. 5. 57) and is plotted here along with the experimental results. The (3 x 3) matrix includes the effects of only w 1 (wp- 1), and (Wp+W 1). It differs from the first order short circuit theory only by the inclusion of a shift in average capacity, and the addition of two second harmonic elastance terms. The (5 x 5) matrix includes, in addition to these effects, signal currents at (2wp -1) and (2p - W1)' as well as some elastance terms at the third and fourth harmonics. The (7 x 7) matrix includes signal currents at (3p - wl) and (3wp +w 1), and additional higher harmonic elastance terms. It can be seen that even a (7 x7) matrix differs to some extent from the "correct"' solution which in this work is assumed to be obtained from a (21x21) matrix. For this circuit, 183

+50 o Y=.1 0-50 y= o 3 -100 - C. -4-j, - 150 Pump Frequency = 84 Mc ~ | Theory c.n Experiment y=. 1 o r - 200 - Experiment y=. 2 v Experiment y=.3 o Experiment y=.35 + Experiment y=.40 * -250 Experiment y=.46 * -300 I I I I 1 2 3 4 5 Signal Frequency % of cp Fig. 7. 18a. Real part of the input admittance of the diffusion circuit 184

+50 y o35 ID =.1 ga DC @ y= 40 IDC = 1.3 ga DC @ = -46 IDC = 20 Aa =-o35 4 -40 + ~ -50-50 0 + -2100 4-l -200 -250 -300 I I I 1 2 3 4 5 Signal Frequency % of wp Figo 7. 18b. Real part of the input admittance of the diffusion circuit 185

+800 - +700 +600 - PS +500 CI~~~ I/ E~ +400 o -0o 1 - lot +200 D,, III. _3 100 0 1 2 3 4 5 Signal Frequency % of wp Figo 7, 18c. Imaginary part of the input admittance of the diffusion circuit 186

+700 - +600 o +500 Unpumped / Admittance +400 / +300 0 t +200 / p — / +100. 46 I! I I 0 1 2 3 4 5 Signal Frequency % of wp Figo 7o 18do Imaginary part of the input admittance of the diffusion circuit 187

the solutions obtained using matrix sizes greater than (13 x 13) are all virtually identical, indicating that signal sidebands around pump harmonics above the 6th are negligible. The admittance of the "Diffusion" circuit (Fig. 7. 2; Table 7. 1) is shown in Fig. 7. 18, for 6 values of y. The unpumped admittance of this circuit is 18. 4 pf. The curves shown are for pumping at the small signal resonance of the circuit (w0 = 84 MHz). For y <. 3, diffusion capacity is negligible, and as illustrated in Fig. 7. 18 (a and c), this circuit exhibits the same sort of resonances as the "Lumped" circuit. However, for y >.3, a measurable D. C. current flows through the diode, diffusion and charge storage effects become important, and the results [Fig. 7. 18(b and d)] are no longer subject to simple interpretations. It should be pointed out here that, whenever diffusion effects are important, the concept of y to denote pumping level must be approached with some care,, The parameter y as ordinarily used and as defined in Eq. 7. 1 refers to a time varying, but frequency independent capacitor. This is a valid concept for dealing with depletion layer capacitance. In one sense, diffusion capacity is not really a capacitor at all; it is merely an admittance with a positive imaginary part, onto which we have applied the name capacity. Mathematically, it must be considered as a time varying, frequency dependent admittance, nothing more. The parameter y as used in Fig. 7. 18 refers only to the depletion layer portion of the diode admittance; diffusion 188

-100 0 d) -150 - 200 o ~s. \It: Diffusion Circuit PY | Pump Frequency= 84 MHz 250 = 0o 4 DC current 1. 3 amps c~ j=0= p Signal Frequency = of w\ -ol -Oo,- \ -300 -- Experiment o -350 I I! I I 1 2 3 4 5 Signal Frequency % of wp Fig~ 7O 19ao Effect of the diode model upon the real part of the input admittance 189

+500 +400 _ Unpumped Admittance 0 +300 -e; +200 = - + 100 0 1 2 3 4 5 Signal Frequency % of cop Fig~ 7. 19b. Effect of the diode model upon the imaginary part of the input admittance 190

effects were ignored in computing it. Figure 7. 19 illustrates the effect of the varactor diffusion current model upon the input admittance. The admittance is plotted for three values of junction electric field. This field is seen to have a large effect upon the real part, but little effect upon the real part. Some understanding of this very complex situation may be obtained by recalling that /i=0 corresponds to a retarding electric field with an infinite magnitude. This is the ideal step recovery diode case (Appendix E). The high electric field prevents carriers from diffusing into the n-region; this raises the Q of the diffusion capacity, and results in low losses which are reflected in a small real part of the admittance. On the other hand, c = x corresponds to zero electric field in the n- region. This is the situation in an ideal p - n junction diode. The Q of the diffusion capacity is always less than one in this type of diode. The result is higher losses which are reflected in a larger real part of the admittance. The carrier density and lifetime have similar effects upon the circuit performance. It is interesting to note that the voltage waveform and the elastance coefficients are affected very little by changes in electric field, carrier density, and carrier lifetime. Changes in these quantities affect the input impedance mainly by altering the terminations of sidebands at harmonics of the pump. 191

In constructing the set of curves (Figs. 7. 13 - 7. 19), every effort was made to plot the results using a consistent set of iy's. This, however, proved to be impractical. For example, the maximum value of y obtainable in the lumped circuit was limited by the onset of diffusion effects. For harder pumping, D. C. current flows and diffusion effects become appreciable. Also, y is limited by pump power, The oscillator used as the pump source has a maximum output (into a matched load) of 200 mW. In addition, the circuit tuning changes as the pump power is varied. The maximum value of y obtainable in the distributed circuit was limited by these effects. In general, the agreement between the theory and experiment can be seen to be excellent. There are a few minor discrepancies, however, probably caused by some idealizing of the varactor diode model. 7. 4 Conclusions This paper has presented the theoretical development and the experimental results of an investigation into pumped varactor circuits. In view of the evidence presented in the previous section, it may be concluded that the theoretical development given in this work is adequate to analyze a pumped varactor circuit with some accuracy. The quantity of experimental verification is insufficient to be totally conclusive, but enough data was presented to indicate that the analysis 192

is valid for a wide variety of situations. The small amplitude pump restrictions of previous analyses have been totally eliminated by using numerical techniques to solve the nonlinear differential equations which describe the behavior of varactor diode circuits which are driven by large amplitude sources. The reactive mixing between the non-sinusoidal pump and the signal was investigated by using a matrix formulation. All significant mixing products were taken into account by using a large matrix. This matrix was then manipulated by using numerical techniques. The only assumptions made were certain general assumptions as to the nature of the diode, and an assumption that the signal amplitude (not the pump amplitude) is small. The work has been exploratory in nature, and it is difficult to summarize precisely, However, it is possible to make a few generalizations. The most practical of these deal with the character of circuit model and the nature of analysis which should be used when studying pumped varactor circuits, An engineer, who has some computer time at his disposal,can utilize the methods of this paper to analyze practical circuit configurations, He can do this with some confidence, because the theory yields good correlation with experiment, correlation which is better than any previous work. Specifically, it is possible to make the following practical suggestions on modeling a pumped varactor circuit: 193

1) The "Uhlir" model, with some modifications, adequately represents the actual diode. Good experimental - theoretical correlation is obtained by using the equivalent circuit of Figs. 2. 3 or 2. 4, and including in the junction characteristics the effects of diffusion as well as barrier layer capacity. For reverse bias, the use of the barrier layer capacity yields results which can be verified almost exactly with experiment, while for forward voltages, the parallel combination of barrier layer capacity plus diffusion capacity yields results which are almost as good. It is also clear that the variation of series resistance with voltage, which sometimes occurs in high quality varactors, must also be accounted for. 2) Under reverse bias conditions, the details of the capacity-voltage characteristics can be very important. A diode which appears to have only slight deviations from the ideal abrupt junction may actually perform in quite a different fashion when used in a circuit. Likewise, two diodes may have similar characteristics, but very dissimilar performances. The practical implications of this are obvious. 3) Under conditions of forward voltage, the diode junction model consisting of diffusion capacity in parallel with barrier layer capacity appears to give quite adequate results. The diffusion capacity model involves the assumption of a retarding electric field which is uniform with distance. The magnitude of this field may be varied from zero to infinity. Although this is an obvious 194

oversimplification, this model yields quite accurate results. 4) The concept of treating the linear portion of the network as an equivalent Thevenin or Norton impedance is most useful. This equivalent impedance appears across the terminals of the diode, and terminates the pump harmonics as well as the fundamental and its sidebands. The nature of this impedance, even at the harmonics, has been shown here to influence the circuit behavior significantly. The so-called open circuit and short circuit conditions are the limiting cases of this impedance. 5) The shift in average capacity is very important. For applications where accuracy may not be of the greatest importance, the three frequency open circuit model, modified to account for changes in tuning with pump level, may be good enough. It is also a useful concept in treating the ferro-resonant effect, and is helpful in the intuitive understanding of some experimentally observed phenomenon. This circuit model is the end product of evolution through a series of models. The first model tried was the first order model which consists of two sidebands and a sinusoidal capacitance (or elastance). In succeeding trials, more exact circuit representations were used, and the discrepancy between theory and experiment narrowed, The present model is not the most accurate model which could be conceived, but it does yield accurate results and is physically meaningful. 195

In addition to the work on the circuit model, mention should be made of the following original contributions. These results were effected while developing the circuit model, but are not logically placed with the above results. 1) Obtained relationships which clarified the connection between the open circuit and the short circuit equations. These relationships showed that even though sidebands at higher frequencies are terminated in open circuits or short circuits, and therefore carry no power, their presence affects the sidebands which do carry power. 2) Set up and then solved the nonlinear pump circuit differential equation using an accurate circuit model. Computed the reactive mixing between the non-sinusoidal pump and a small sinusoidal signal taking into account all significant components. 3) Devised a practical method for measuring the pumping coefficient, y, This method is independent of whether the pumping is hard enough to cause DC current to flow. It is fast, reasonably accurate, and requires only two additional pieces of equipment: a signal source, and a detector. Using this method, y can actually be measured while the circuit is being operated, 4) Obtained good experimental correlation of the analytical portion of the work,, The experimental work was carried out using measured values of y, while the diode parameters used in the analytical work were obtained using simpl RF bridge and pulse techniques. The good agreement tends to confirm the accuracy of 196

the circuit model and the validity of the analytical procedure. 5) Obtained relationships which clarified the ferroresonant or jump effect. By treating it from an impedance viewpoint, the present treatment avoids many of the complications associated with other interpretations. This work has presented a procedure for analyzing pumped varactor diode circuits. Using the procedures and models presented here, an engineer can analyze any circuit he chooses to devise. Useful as this may be, the design engineer is still left with a most difficult task. He must still base his design on simplified theory (Ref. 55); and although he can utilize the procedures of this paper to check this design, his method of procedure, should this initial design fail to be satisfactory, is not at all clear. The most profitable extension of this work lies in the development of advanced design tools to make the analysis more useful. The results of this work can also be used as a basis for the consideration of more practical pumped varactor circuits. The most important of these is the circuit consisting of a diode mounted in a'waveguide. Although all the concepts presented in this paper hold true for this circuit, the computation of the impedances at the diode terminals presents some difficulty. In a waveguide designed to propagate the pump fundamental, higher order modes of the pump harmonics will not be cut off. The impedance (both the real and the imaginary parts) must be derived from the solution of field problems 197

involving the excitation and the propagation of waves in an oversize waveguide. Work can also be profitably undertaken in the improvement of the diode model. The use of a more accurate diode model, especially in the forward-conduction region,would give even closer correlation between theory and experiment than was obtained here. 198

APPENDIX A INVERSION OF MATRICES USED IN CHAPTER III In Chapter III, the values of il(n), j312(n), 0322(n) were simply quoted. The computation will be carried out in this appendix. i31l(n), 312(n), 2 1(n), i22(n) are defined as the upper four elements of the inverse of ac22(n). Whe re -jw4C0 Q -jw4C1 0 0 0 0 jaw 5C0 0 jw 5C1 O0 - jw6C 1 0 -jW6C0 C -Jw6C1 0 a22(n) - 0 jw7C 1 C! jw 7C 0 jWi 1 o C -wC-l C -j8C 1 C - o 0 0 jwgC1 0 jw9Co (A. 1) The symbol n is the order of the original matrix (3. 29); a22(n) is of order n- 3 (n is greater than 3). The values of the 13's can be found by dividing the co-factor corresponding to that j3 by the determinant of the matrix The determinant will be computed first, Tile computation 199

can be easily done if the arrangement of zero and non-zero elements are used to advantage. Because there is only one non-zero element below the diagonal in each row (for rows below the second), the matrix can easily be put in upper triangular form. This can be systematized by starting from the top and working down. When this is done, the result is (See next page) where C1 C 1 CO The value of the determinate is now just the product of all the diagonal terms. a22(n) = F(n) CO ()( ( 1-72/ (A. 3) where n+1n 1 F(n) = (-jw4)(+j5). [(-1)+ 'n], and, appears.( [2 times. To complete the evaluation of fll(n), the co-factor B11(n) 200

- jc4C 0 -j4C1 0 0 o jw OJ5C0 J 5C 1 0 0 -jw 6C 0(1-y2) 0 -j6C1 a22(n) = 0 0 0 jW7C0(1-y2) 0 ~0 0 0 0 -Iw8C0 (1)0 0 0 0 0 jw9Co 1 1o) (A. 2)

of the (1, 1) element is also needed. This is a matrix of order n- 4, which has the same form as 1a22(n)i. jW5C O0 jw5C 1 0 0. -jW6C0 0 -j6C1 0 Bll(n) = jW7C1 0 jw 7C0 0 jw7C1 0 -jo8C1 0 -jw 8C 0 0 jW9 C1 0 j 90L (A. 4) This can be evaluated by the same procedure used to evaluate (A. 1). The result is 2( 2 2 Bl(n) = F'(n) Co(n 4)(1(-z) 1- 1 ) (A. 5) whe re F'(n) = (ji5) (-ij6).... [(-1)n+ j w and n,1 -1 7 )- - In y appears (?.. 2 1) times. 202

Gin A l2 ty2 Fig, A. 1. Resistive ladder network 203

Now that both the co-factor of the (1, 1) element, B11(n) and the determinate of the matrix la22(n) I are known, the inverse element can be found. Bll(n) 1 oll(n) a22(n) (A. 6) Only the first and last terms in 1a22(n)I are left after the division is carried out. y2 appears (n- 2) times for n even and (2 2) times for n odd. In the limit as n- oc, the expansion of Eq. A. 6 can be put in a very simple form. This can be accomplished by giving the expansion a physical interpretation (Ref 44). Conasider the ladder network Fig. A. 1 which consists of 1 ohm and (-72) ohm resistors. Except for the multiplicative factor 1j, the input admittance of this network can be put in the same form as Eq. A. 6, where the network n _ (n 5 has (2 2) nodes for n even and (- - ) nodes for n odd. For n - c, the admittance can be put in a very simple form by using standard image parameter theory. G. 1 (A. 7) in (1 ]i) 204

Then: 1(1) (A.8) _ 4Co)\ 4 ' The computation for 1322(n) is carried out in exactly the same manner. 322(n) = 1 (A.9) (jw5)C 0 -,Y 1-~2 The continued fraction expansion is carried out until y2 appears n 2) times for n even and (-n 2) times for n odd. In the infinite case P,2(n) (A. 10) (j 5) Co ( + ) (A. To evaluate:12(n), the co-factor of the (2, 1) element is needed. This can be shown by induction to equal zero for all n. 205

O -jW4C1 0 0 0 0 o -jw6C0 0 -jw6C1 0 0. B2 1(n) = Jw7C1 0 jw7C07C1 0 0 -jwo8C1 0 -jC j8C1 *0 o 0 jw9C1 0 jW 9C 0 (A. 11) For all n greater than 6, the form of the lower right hand corner is X O X 0 X 0 0 0 X 0 X 0 X 0 (A. 12) 0 0 X 0 X 0 jw 2C O O0 X 0 X 0 ~ ~* O O0 0 0 jWnC O jwnCO the X's simply denote the presence of a non-zero element. Now, the nth row can be multiplied by n 1( and then added to the 206

(n-2)th row. This operation changes the value of the (n- 2, n- 2) element, and makes the (n- 2, n) element zero. The determinate can be expanded along the nth column, which now has only one element. B2, 1(n) = j WnCO B, 1(n- 1) (A. 13) The (') indicates that the B2 1(n- 1) matrix has been changed by the alteration of its (n- 2, n- 2) diagonal term. However, if B2 1(n- 1) is equal to zero simply by virtue of the arrangement of zero and non-zero elements, then B2 1(n- 1) must also equal zero, because this arrangement has not been disturbed. By inspection, B2 1(4) = B2 1(5) = B2 1(6) = 0, because of their arrangement of zero and non-zero elements. Then B 1(4) = B1(5) = B 1(6) =0. Thus, by induction, B2, 1(n) equals zero for all n. Now B2 1(n) = 2,(n) (A. 14),1,2 i22(n ) and 1a22(n) I has been shown to be non-zero for all n. Therefore, 1, 2(n) = 0 for all n. '1 2(n) can be shown to equal zero for all n by the same reasoning. 207

APPENDIX B VOLTAGE AND CURRENT PUMPING In Chapter V, an iterative technique is used to compute the elastance of a pumped varactor. This technique is applicable to any circuit containing one varactor (see Fig. 5. 1), but it is laborious to execute and the results are in numerical form. In this appendix, two special cases, for which a power series form of solution can be obtained, are presented. 1) The linear network can be approximated by a sinusoidal voltage source with negligible internal impedance. In this case, currents may flow at harmonics of the pump frequency, but the voltage must remain sinusoidal (Ref, 56). 2) The linear network can be approximated by a sinusoidal current source with very high internal impedance. In this case, voltages may exist at harmonics of the pump frequency, but the current is constrained to be sinusoidal. The depletion layer capacity of a varactor diode is given by (see Section 2. 2) dqt C dv CM - - (2. 1) ~t /0 v\ M (124 208

The voltage of a pumped varactor can be written as the sum of a DC bias voltage plus a time varying voltage. That is,let Vt = VBias + K1 VM(t) (B. 1) then 1C 0 a 1 CM = (B.2) (V(-Bias)M ___ Now define CB as the small signal capacity at the bias point, and normalize the time varying voltage. 1 A Ca 0 B 1 ( - V M Bias K1 = - + VBias Then Eq. B. 2 can be rewritten as d =~ C 1 (B. 3) dvt CM = CB( +VM) and 209

1 M SM= C ( 1 + VM) (B. 4) whe re VM > -1 In the case of sinusoidal voltage pumping V =a cos t M whe re lal < 1 The charge stored in the diode can be found by integrating Eq. B. 3. q = CB(l1 - 1+ VM] + Qc (B. 5) where Qc is a constant of integration. The total charge equals a constant, or bias, charge plus a charge due to the pump. qt QBias + K3 QM(t) QBias can be chosen such that QM- = when VM= 0, and the time varying charge can be normalized 210

QBias CB(1l- ) Qc ~K = 1-1 3 = CB(1- ) Equation B. 5 can be written in terms of the new charge variables, and solved for VM M VM = [1 + QM] M 1 (B. 6) Combine Eqs. B. 3 and B. 6 to obtain an expression for the capacity in terms of the charge. CM = CB[++QM M1 (B. 7) i M- 1 M = [1+ QM] (B. 8) whe re QM > -1 For the case of sinusoidal current pumping: QM = a cos t where lal < 1 211

Under sinusoidal pumping conditions, the expressions for the capacity of the voltage pumped diode (B. 3), for the elastance of the voltage pumped diode (B. 4) for the capacity of the current pumped diode (B. 7), and for the elastance of the current pumped diode (B. 8) have the same form:,N =(ltt N-9 K N (1+ a cost) (B. 9) whe re N = M for voltage pumped capacitance N = -M for voltage pumped elastance N = M- 1 for current pumped capacitance N = -(M- 1) for current pumped elastance It is convenient to put this relationship in the form of a Fourier series. To do this, the expression (B. ) can first be expanded in a power series. KN = (N+1) (2N+1)... ([2-1] N+ 1) af(cos t)( K!N (-1)r (B. 10) The terms of the form (cos t)~ can be expanded by using trigonometric identities. 212

[ [1 + (-1) ] () 1! cos ( - 2k) t (cos t)12 2+1 (! ()! fO (-k)! (k) Using this identity, the expression for KN becomes c [1 + (-1) ] (N+1)(2N+1)... ([f-1] N+1) af KN Z + N =0._ -_ 2+1 (2)! (2)! N (N+1)(2N+1)... ([-1] N+1) a cos (f - 2k)t =-1 k=O (-1) N' 2'- (2-k)! (k)! This expansion can be put in the form C (N+1)(2N+1)... (12- 1 N+ 1) a2 KN + N =O 4 (2)! (f)! N2' (B. 11) (N+1)(2N+ 1)... ([Q + 2k - 1] N+ 1) af + 2k -.................. cos f t =1 N=0 22+2k- i +2k (- 1)f (2+k)! (k)! Equation B. 11 can easily be put into exponential form +OC KN = K ejft 2=-cc where the average value is K C (N+1)(2N+1).. ([22-1] N+ 1) a KO: -: - '- 22 (B. 12) =O0 4 (2)! (2)! N 213

and the coefficient of the t- harmonic is given by K - (- >a f (N+1)(2N+1)... ([ + 2k- 1] N+ 1) a2k (B 13) k=O 2k N2k( +k)k)!(k)! The values of the first 24 of these coefficients have been computed for various values of N. The results are presented in tabular form in Tables B. 1 through B. 8. The values of f are in the left column; the values of a are in the top row. 214

FOURIER COEFFICIENTS N = 1.000 a.1.2.3.4.5.6 7.9 0 1.005038 1.020621 1.048285 1.091089 1.154700 1.25000(0 1.40C280 1.666666 2.294155 1 -.050378 -.103104 -.160949 -.227724 -.309401 -.4166' 7 -.571829 -.833333 -1.437952 2.002525.010416.024712,047529.082904.138889.233516.416667,901292 3 -.00()127 -.001052 -.003794 -.009920 -.022214 -.046 296 -.095360 -.20 333 - 56492 4.000006.000106.000583.002070.005952.015432.038942.104167.354086 5 -.000000 -.000011 -.000089 -.000432 -.001595 -.005144 -.01 590 3 -.052083 -.221937 6.000000.000001.000014.000090.000427.001715.006494.026042.139103 7 -.000000 -.000000 -.000002 -.000019 -.000115 -.000 57 -.002652 -.013021 -.087191 8.000 000.000000.000000.000004.000031.000191.001083.006510.054650 9.000000 -.000000 -.000000 -.000001 -.00000 8 —,000064 -.000442 -.003255..34254 10.000000.000000.000000,000000.000002,000021.00C181.001628,021470 11.000000.000000 -.000000 -,000000 -.000001 -.000007 -.000374 -.000814 -.013457 12.000000.000000.000000.000000.000000.000002.000030.000407.008435 13.000000.000000.000000 -.000000 -.000000 -.000001 -.000012 -.000203 -.005287 14.000000.000000.000000.000000.000000.000000.000005.000102.003314 15.000000.000000.000000 000000 -.000000 -.000000 -.000002 -.00051 -.002077 16.000000.000000,000000.000000.000000.000003.000001.000025.001302 17.000000.000000.000000.000000 -,000000 -,000000 -.000000 -.000013 -,000816 18.000000 000000.000000.000000.00000 00 000000.000000.000006.000511 19.000000.0000 00. 000000.000000 -.00 00 00 -000000 -.00 0000 3 -.00 00 03 21 20.000000.000000.000000.000000.000000.000000.000000.000002.000201?1.000(00 0 000000.000000.000000.000000.000000 -.000000 -,000001 -.000126 22.00000 000.000 000.000 000.000000..0000000.00 00000.00o00007 23.000000.000000.000000.000000,000000.000000.000000 -.000000 -,000049 24.000000.000000.000000.000000.000000.00000 0000 0.000000.000000 31 Table B. 1 Fourier coefficients for N= 1.0

a.1.2.3.4.5.6.7.8.9 0 1.00188 5 1.007669 1. )17751 1.-032954 1.054649 1.085357 1.12S987 1.199854 1.331821 I -.025 1 1 8 -.050963 -.078364 -.108395 -.142612 -.183556 -.235984 -.310739 -.443336 2.000944.003863.009033.016999.028749.046116.072832.117946.212854 3 -. 000039 -.000325 -.0()01156 -.002959 -. 006 42 9 -.012842 -.024881 -.049450 -. 112405 4.000002.000029. 0')155.000541.001509.003751.008912.021716.062066 5 -.00000 -.000003 -.000021 -.000102 -.000364 -.001126 -.003280 -.009797 -.035174 6.000000.000000.000003.000019.000089.000344.001229.004499.020277 7 -.000000 -.000000 -.0o000 -.000004 -.000022 -.000107 -.000467 -.002092 -.011832 8.00(!O0.0000 0. 000000.000001.000006.000033.00C179.000981.006967 9.000000 -.000000 -.000000' -.000000 -.000001 -.00001 1 -.000069 -.000464 -.004131 10.00000().000000.000000.000000.000000.000003.000027.000221.002463 11.000000 000000 -.000000 -.000000 -.0000001 -.000010 -.000105 -.001475 12.000000.000000.000000.000000.000000.000000.000004.000050.000887 13.000000.000000.000000 -.000000 -.000000 -.000000 -.000002 -.000024 -.000535 14.000000.000000.000000.000000.000000.000000.000001.000012.000324 15.000000.000000.000000.000000 -.000000 -.000000 -.000000 -.000006 -.000196 16.000000.000000..)00000.00000.000000.000000.000000 000003.000119 1 7. 0000 0).00000.000000.0000.000000 -.000000 -.000000 -.000001 -.000073 18.0000.000000.000000.000000.000000. 000000.000000.000001.000044 19.00(000.000000.000000.00o0000.000000.000000 -.000000 -.000000 -.000027 20.00000.000000.000000.000000.000000.000000.000000.000000.000017 21.0(0000.00000.000000.000000.000000.000000.000000 -.000000 -.000010 22.000000.00000 000000.000000 000000.000000.000000.000000 000000.000006 23.0 00 000.000 000.0 00 000.000000.00000 0.00 0000 -.00000.00000 0 -.000004 24.0(.0000 0..000000.000000.000000. 000000. 000000.000002 Table B. 2 Fourier coefficients for N= 2.0

FOURIER COEFFICIENTS N= 3.000 a.1.2.3.4 5. 7.8.9 0 1.001117 1.004533 1.010465 1.019321 1.0318 18 1.(149209 1.073874 1.111075 1. 176859 1 -.016732 -.033865 -.051851 -.0712o6 -.099292 -.11811) -.149130 -.191060 -.259339 2. 000559.002282.005312. )009932.n16645.02& 360.04C877.064375. 110414 3 -.000022 -. 000179 -.000635-,00114 -. 4 -.03474 -.006849 -.0 1 302 8 -.025173 -. 054354 4.000001.000015.000081.000281.000776.001905.004443.010524.028562 5 - 0 0(000 - 000001 -. 00011 -. 000051 -.003180 -.000551 -.001575 -.004571 -.015579 6.000000.000000.000001.000009.000043.000163.00C572.002035.00a706 7 -o000,000 -.0 00 000 - 00 000 0 -.000002 -.00010 -.000049 -.000212 -.000922 -.004948 8.000000.000000.000000.000000.000003.0001 5.00C079.000423.002848 9.00(000. -.000000 -.000000 '-.000000 -.000001 -.0000n05.-.000030 -.000196 -.001655 10.000000.000000.000000.000000 000000.00001.000011 000091.000969 11.000000.000000 -.000000 -.0000000 -.000000 -.000000 -.000004 -.000043 -.000571 12.000000.000000.000000.000000.000009.000000.000002.000020.000339 13.000000.000000.000000 -.000000 -.000000 -.000000 -.000001 -.000010 -.000201 14.000000.000000.000030.000000.000000.000000.000000.000005.000120 15.000000.000000,000000.000000 -.00000 -.000000 -.000000 -.000002 -.000072 16.000000.000000.000000.000000 00000000.000000.000001.000043 17 000000 00000 0 000000.000000.000000 -.000000 -.000000 -.300001 -.0 00026 1R.000000.000000.000000. 000000.'000000.000000.000000.000000 000.0016 19.000000.000000.000000.000000.000000.000000 -.00COO -.o000 -.c001oo 20.000000.000000.000000.000000.000000.000000.00 0000.000000.000006 21.000000.000000.000000 000.000000. 000000.000000 -000000 -.000 000 4 22.00000.000000.000000.000000n.000000.000000.000000.000000.000002 23.000000.000000.000000.000000. 000000.000000.000000;000000 -.000001 24.000000.000000.000000.000000.000000.000000.000000.000000.000001 Table B. 3 Fourier coefficients for N=3. 0

FOURIER F COEFFICIFNTS N = 4.000 a.1.2.3.4.5.6.7.8.9 0 1.000785 1.0031 84 1.007338 1.C13516 1.()22184 1.(034150 1.n5C924 1.075779 1.118372 1 -.012544 -.025360 -.038752 -.053104 -.06(953 -.( 7126 -.109111 -.138051 -.183244 2.00C393.001602.003721.06 037.011574.013218.028008.043533.072923 3 -.000015 -.000121 —.0004?9 -.001087 -.002329 -.C04563 -.C08603 -.016401 -.034563 4.000001.000010,000053,C00184.000507.001237.002860.006682.017691 5 -.000000 -.000001 -.000007 -.000033 -.000116 -.0003 5 1 -.000994 -.002845 -.009458 6.000000.000000.000001.000006.000027.000 102.00C355.001:246.005200 7.0000 0 -.000000 -000000 -.00000 1 -,000006 -.000 03 -.000130 -.000557 -.002916 8.000000.000000.000000.000000.000002.000009.00C048.000253.001659 9.000000 -.000000 -.000000 -.000000 -.000000 -.000003 -.000018 -.000116 -.000954 10.000000.000000.000000.00C000.000000.000001.000007.000054.000554 C 11.00(000.000000.000000 -.000000 -.000000 -.000000 -.000003 -.000025 -.000324 12.000000.009000.000000.000000.000000.000000.000001,000012.000190 13.0000000.000000 *000000.000000 -.000000 -.00000) -.000000 -.000006 -.000112 14.000000.000000.000000.000000.o00000.000000 000000.00003.000067 15.000000.000000.000000.000000.000000 -.000000 -.000000 -.000001 -.000040 16.000000.000000.000000,000000.000000 000000 0 000000.000001.000024 17. 000000.000000.00 0000.000000.000000 000000 -.000000 —.000 14 18.000000.00000 0000. 0000 000000.000000.00 0000 00000.000000 000009 19.000000.000000.000000. 000000.000000.000000 -.000000 -.000000 -.000005 20.000000.00C000. 000000.000 000.000000.000000.000000. 000003 21.000000.000000.000000. 000000.000000. 000000.0000 -.000000 -. 0 00002 22 o000000.000000.000000.000000.000000.000000.000000.000000.000001 23.000000.000000 00.00000 000000.000000.00000 0.0 00000-.000001 24.000000,000000 000000 00000 00000 000000 000.000000.000000 Table B. 4 Fourier coefficients for N=4. 0

FOURI FP Cntf-FFICIENTS N = -1.000 a.1.2.3.4.5.6.7.8.9 0 1.0000000 1.000000 1.000000 1.000000 1.000000 1.00 000 1.000000 1.00000 1.0 5000.10)00.15()0000.200000.250000.300000.350000.400000.45000 2.00o000.000000.000000.000000.000000.000000.000000 o.i000.000000 3.000000.000000.ooo000000.000000. 000000.000000 000000.oo30oo00.oooooo00000 4.000000.00000o.000000.000000.000000.000000.000000.o00000.0o3000 5. 000000 000000.000000.000000 o000000.000000.000000. )000.o000000 6.000000.000000.00000 000000.000000 00oooooo0000.000000.0)ooo00.00oo00 7.000000.000000.000000.000000.000000.000000.000000.000oo0.0 0000 8.000000.000000.000000.000000.000000.000000.000000.o00000.000000 h3 9.000000.000000.0(0000.000000.000000.000000.000000.000000.000000 Ju 10.000 000.00*0000. 000000.000000.000000 *000000.009000.000000 1.0000.00000 0.000000.000000.000000.000000.000000..000000.0000000 12.000000.000000.000000.000000.00000000 000000.000000000 13.000000.000000.000000.000 000.00000 0.000000.000000.000000 14.000000.000000.000000.000000.000000.000000.000000.000000.000000 15.000000.000000.000000 000000.000000.000000.000000.000000.000000 16.000000.00000 0000000.000000.000000.000000.000000.000000.000000 17.0000000.000000.0000 000000.000000.000000.000000.000000.000000 18.800000.000000.000 000.00000 0.000000.000000.00000. 00000000.000000 19.000000.000000.000000.000000.000000.000000.000000.003000.000000 201.00cY003.000000.000000.000000.000000.000000.0o00000.00000.000000 21. 000000.900000.000000.000000.000000.000000.ooo0oo.000000 22.000000.000000.000000.000000.000000.000000.00000000.000000 23. 000000.0000 00.00000 0. 00000 0.000000. 00000 0.00000 0. 00 000 0 24.000000.0000000.000000.000000.000000.000000.000000.000000.0000 00 Table B. 5 Fourier coefficients for N=-1. 0

FOURIER CCEFFICIENTS N = -2.000 a.j.2.3.4.5.6.7.8.9 0.999374.997476.994251.989596.983343.975224.9h4798.951263.932819 1.025024.050190.* 15655.101596.128237.155886.185002.216381.251768 2 -.000313 -.001266 -.002895 -.005272 -.008511 -.012 04 -.018472 -.026129 -.037230 3 4000008.000064.000222.000549.001135.002118.003729.006415.011309 4 -.0000000 -000004 -.000021 -.000071 -.000190 -.000439 -.000945 -.001983 -. 04344 5.000000.000000.000002.000010.000035.000102.000269.000689.001881 6 -.000000 -.000000 -.000000 -.000002 -.000007 -.000025 -.000082 -.000257 -.000875 7.000000.000000,000000.000000.000001.000007.000026.000100.000428 8.000000 -.000000 -.000000 -.000000 -.000000 -.000002 -.000009 -.000041 -.00321 7 IW 9.000000.000000.000000.000000,000000.000001.000003.000017.000113 10.C000000.00000 -.0000 00 -00000 -.000000 -.000000 -.000001 -.000C07 -.000060 11.000000.000000.000000.000000.000000.000000.000000.000003.000032 12.0 000000 0 0 0 00000 0.000000.000000 -.000000 -.000000 -.0000001 -.000018 13.000000.00 000 0. 000 000.000 000. 0000 00 0 00 0000.00000.000010 14.0000000.000000.000000.000000. 000.00000 0 -.000000 -0.000000 -.000005 17.000000.000000.000000.000000. 000000.000000.000000.000003 16.000000.000000.000000.000000.000000.000000.000000 -.000000 -.000002 17.000000.000000.0000000.000000 00000.000000.000000.000000 000001 18.000 00000 000000.000000.000000.000000.000000 -.000000.0-.00000 1OO19.000000.000000.000000.000000.000000.000000.000000 000000o.00000 20.000000.000000.000000.000000.000000.000000.000000. OO~o0 -.0O000 2.000000.000000.000000 00.000000. 00000 0.000000.000000. OOOOO 22.000000.000000.000000.000000.000000.000000.000000.000000 -o. OOOO0 23.000000.000000.000000.000000.000000.000000.000000.000000.000090 24.000000.000000.000000.00000000.000000.000000.000000.0000 00 Table B. 6 Fourier coefficients for N=-2. 0

FOURIER CO[E FFICIENTS N = -3. 000 a. i.2.3.4.5.6.7.8.9 0.99"443.997753.9948b9.990683.9B5C12.977561.967833.954883.936473 1.016690.033522.050649.068254.;)86570.105924.126826.150202.178213 2 -.000279 -.001128 -.q02588 -.004733 -.007690 -.911668 -.017036 -.024519 -.035945 3.000008.000063.900221.000548.001-142.002151.003839.006738.012283 4 -.000000 -.000004 -.000023 -.000076 -.000204 -.000477 -.001041 -.00C2231 -.005073 5.000000.000000.000003.000012.000040.000116.000311.000814.002313 6 -.000000 -.00000000 000000 -.000002 -.000008 -.000030 -.00C099 -.000316 -.001121 7.000000.000000.000000.000000.000002.000003.000033.000127.000566 8.000000 -.000000 -,000000 -. 000000 -.000000 -.000002 -.0 CO 1 1 -.000053 -.000295,, 9.000000.000000.000000.000000 ~.000000.00000 1.000004.000023.000157 -, 10.00000.00000.000000 -.000000 -.000000 -.000000 -.000000 -.000001 -.000010 -.000085 11.000000 000 00 000.0 0000 0.0000000.000000.000000.000004.000047 12.000000.000000.000000.000000 -.000000 -.000000 -.000000 -.000002 -.000026 13.000000.000000.000000.000000. 000000.000000 00.000001.000015 14.000000.000000.000000.000000.000000 -.00000 -. 000000 -.000 00 0 -.000008 15.000000.000000.003000.000000.000000.000000.000000.000000.0000 05 16.000 0000(0.000000.000000.0000000.0000000 000000 -.OOCOOO -.000003 17.000000.000000.000000.000000.000000.000000.000 000.00000 2 18.000000.000000.000000. 300000.000000.000000. 000000 -.000000 -.00001 19.0000 00. 0000000.000000.000000.000000.000000.000000.000000 000001 20,000000,000000.000000.000000.000000.000000.000000C -.000000 -.000000 21.000000.000000.000000.000000.000000.00000 OOCOOO 000000.000000 000000 22.00000000.000000.000000. 000000.000 000.0 00000.0 0 000 0 -.000000 23.000000.0000 00.000000.000000.000000.000000.000000.000000 000000 24.000000.000000.000000.000000.000000.000000.00000.000000,000 Table B. 7 Fourier coefficients for N=-3. 0

FOURIER COEFFICIENTS N = -4. 000 a.1.2.3.4 5 6 7.8 09 0.099530.998102. ')9 5 6 2.992107.987271.98087. 9 7 2454.961088.944555 1.012521.025167.038076.051412.065336.00300.096634.115278. 130379 2 -.000235 -.300953 -.002190 -.304015 - n006545 -.009977 -.014660 -.021299 -.031720 3.000007 i, 000056.0001~?.000488.~001021.001934.003475.006 165.311445 4 -. 000000 -~0004 -. 00021 -.000070 -.000138 -000 442 -.00C973 -.002109 -.04891 5 0000 0 0000 000 0000,02 000011.000033.000110.000297.000789,02283 6 -.0000 00 -.00000 -. 000000 -.)00002 -.000008 - 0000?9 -.00C096 -.300311 -.301130 7.000000.00O000.000000.000000.}00002. 00008.000032.000128.000580 8.000000 -.000000 -.000)00 -.000000 -.000000 -.0000092 -.000011 -.000054 -,000306 h, 9.000000.000000.00000(.000000..000000.000001.000004.000023.000165 10.00000.00000 -. 000) -.000000 -0 - 000000 -..000000 -.000001 -.000'010 -.000090 11.000000.000000.000000.000000.000000.00 0000.000001 00)004.000050 12.000000. 000000.000000 -.000000 -.000000 -.000000 -.000000 -.000002 -000028 13.000000.000000.000000.000000.000000.000000.000000.000001.000016 14.000000.000000.000000.000000.000000 -.000000 -.000000 -,000000 -.000009 15.0000'00.000000.000000.000000.000000.000000.000000.000000.000005 16.000000.000000.000000.o00000.000000.000000 -.000000 -.000000 -.000003 17.009000.000000.00,)000.000000.000000.000000.000000.000000 o000002 18.000000.oooooo o0o0o 00 000000.000000.000000.000000 -.0ooooo -000001 19.000000.000000.000000.000000.000000.000000.000000 00000 000001 20.000000.000000.00000. 000000.000000.000000.000000 -00 -0000 -00000 21.000000.000000.000000.000000.000000.000000.000000.000000.000000 2 2.0 00000.00 0000.000000 000000 00 0000 000000.000000 0 -.0 0000 0 23.000000.000000.000000.000000.000000.000000. 000000.000000 000000 24.000000.000000.000000.000000.000000.000000 000000 0000000000 Table B. 8 Fourier coefficients for N=-4. 0

APPENDIX C THE COMPUTER FLOW DIAGRAM TO SOLVE THE PUMP CIRCUIT EQUATION FOR SINGLE TUNING This appendix presents the computer flow block diagram for the solution of Eq. 5. 14. This equation describes the behavior of the circuit in the absence of signal, and is valid for an abrupt junction diode, although it can be easily modified for any other type of reversed bias diode. -1 1 ~s) ~I~~~~IQ p2(t) Qp(t)) L (5.14) Qp(t) L s Z(s) [Ein(S) - L 4 + Qp(t) (5.14) The method of solution which is diagrammed here implements the iterative procedure described in Section 5. 3. It is useful whenever the impedance of the linear portion of the circuit is such that mnwo Z(nwop) > 2 (5.32) p p when n= 2, 3,... The notation used in the block diagram is consistent with that introduced in Chapter V. The computer block diagram is shown in Fig. C. 1. It computes both the waveform of the diode charge, Qp(t), and the Fourier analysis of this waveform. The input data needed includes QA(1), L, Rs, G and Cs. 223

The theoretical results presented in Chapter VII were obtained by an algorithm of this procedure programmed in MAD for an IBM 7090 computer. From four to twelve iterations were necessary to obtain one millivolt accuracy. This number of iterations varied depending upon both the pump level and the accuracy of the initial estimate, Q0(t). On the average, each iteration took about one second of computer running time. 224

Compute Read Daa G'(n) +j Y'(n) =, Compute Perform Compute A m+1 =-V(O) V, ~Q(t) = +Q (n) G(n) VA(n) o+ Y'(n) V(n)= () one period. = 0. Qp ml I _____________________ffor n= 2,3,... Co mputerIs Compute Qm+ l(t) Qm+(t) for Print Results sufficiently close to many evenly spaced Compute rTRUE | Q M? points over one period V (t), V (n), VA(n) m m = VM using Equation 5.30 Qp (t) QA(n)m+1, QB(n)m+1 using Equation Q p (t) Qm+1(t) E.() 5.14( gin Fig. C. 1. The computer flow diagram to solve the pump circuit equation for single tuning.

APPENDIX D THE COMPUTER FLOW DIAGRAM TO SOLVE THE DISTRIBUTED CIRCUIT EQUATION This appendix presents the computer flow block diagram for the solution of Eq. 5. 13. This equation describes pump circuit behavior in the absence of signal, and is valid for an abrupt junction diode. However, both the equation, and the program may be adapted to fit any type of reversed biased varactor. Q p2 (t) 0 = Ein(S) - s Z(s) Qp(s) - L Q p(t) + P ( (5.13) The method of solution diagrammed here implements the iteration procedure of Section 5. 4. It is useful for all circuit configurations which contain one varactor. However, it is more complex than the method of Appendix C, and generally, the computer running time is longer. The computer block diagram is shown in Fig. D. 1. The notation used is consistent with that introduced in Chapter V. Both the waveform of the diode charge Qp(t), and the Fourier analysis of this waveform are computed. The input data needed includes QA(1), Wp, and Z(s) [or at least enough data to determine Z(s ]. In Step (2) of the mth iteration, it is necessary to set up and solve the following set of equations for c m(t). 226

6m(S) = m(S) [-1- s Z(s)] - L Q (D. 1) where because only the "steady state" solution is of interest the equation error can be written as 6A(0)m cC &6t)= 2 + [6A(n) cos np t + 6B(n) sin nw t] n= 2 (D. 2) and the desired correction term is EA (0) xc e m(t) 2 m + [(n) cos nw t+c (n) sin nw t] 2 n2 A m p B m p (D. 3) The left side of Eq. D. 1 can be expressed in the time domain as a Fourier series. In the time domain the first two terms of the right side of the equation can be written as L {em(S) [1 + s Z(s)]} = [ (n)R(n) R '(n) X'(n)] cosnnwpt n=2 0C + L [Cem(n) R'(n) - c' (n) X'(n)] sin nw t n=2 (D. 4) where the Thevenin linear impedance at pump harmonic is expressed as Z(nwOp) = R(n)+ j X(n) (5.12) 227

and (-1 - Z(nco)p) - R'(n) + j X'(n) (D. 5) for n= 1, 2, 3, In the time domain, the last term on the left side of the equation (Eq. D. 1) can be written as the product of two infinite Fourier series. The result, of course, is a third Fourier series, whose coefficients are combinations of QAm(n), QBm(n), Eom(n), and ECm(n). An infinite set of simultaneous linear equations can now be generated by equating coefficients of the cos no t terms and the sin ncowt terms in Eq. D. 1. The n= 1 term can be omitted from this set of equations, because both & (t) and e (t) have no components m m at frequency wcop This infinite set of equations can be reduced to a finite set by truncating each of the series and neglecting all harmonics beyond the f th. The remaining equations can be put into the form of a (21 - 1)x(2! - 1) matrix. 228

I I I. E /1 2 ~ ~ ~ 7 f~l 2S- 1 Ak~ 5A( p72,1:p2,2 ER ~ EX(3) (3 7/1, I, ' ' ' /1, l~, 1 / + [! I [ '~~~~~~~~~~~~~~ I~~~~~~~~ I l --------------- I2,1 E l(2) A (2) '7k+1,2 B B II* C I(3) 6A(3) B I i I~~~~~~~ ~ I ~~~~~~~~~~~~~~~~~I I I I! IA4 I I ~~~~~~~~~~~~~~~~~~~~~~II I I I ~ [~~~~~~~~~~~~~~~~~~~D6

Every term in this matrix has the subscript m implied. The e' terms are coefficients of sines and cosines which are multiplied by constants [the Rt(n) and X'(n) terms] and by other sine and cosine terms [Qm(t) ] The resulting terms are, of course, also sine and cosine terms, whose coefficients are the 6's. The individual elements of the 77 matrix are constants plus the coefficients of products which have one of the forms: (a) [sin r] x [sin s] (b) [sin r] x [cos s] or (c) [cos r] x [cos s] Using the formula for products of trigonometric functions and after much labor, the matrix elements can be found to be r and s are dummy indices 2 < r < l, 2 <3 s < f 71, 1 = 2 QA(0) + 1 (D. 7), r = QA(r) (D. 8) 7r, 1 = QA(r) (D. 9) Q (r) (D. 10) 7rr+Q-_1, 1 = QB(r) (D. 11) 230

7r, r 2 QA(O) + 4 QA(2r) + R'(r) (D. 12) 1= 2 QA(0) I4 QA(2r) + R'(r) (D. 13) 7r -1 = - Q (2r)+ X'(r) (D. 14) The following hold only for s f r 7sr 4 QA(S+r)+I QA(ls-rl) (D. 16) 1 1 s+- 1, r+ - 1 = QA(s-r) - QA(s+) 4) (D.217) 1 1 1 7sr+-1 = 4 QB(2s-r)~4 QB(s+r) (D.18) + - 1, r = QB(s+r) + 4 QB( Is- r) (D. 19) In (D. 18) use + when s > r, - when s < r In (D 19) use - when s > r, + when s r. The theoretical results presented in this paper were obtained by an algorithm of this procedure programmed in MAD for an IBM 7090 computer. The number of iterations needed to compute each pump voltage waveform varies depending upon the accuracy of the initial For the data presented in this paper fronm three to eleven iterations were necessary to obtain one millivolt accuracy. The computer 23 1

running time for each iteration depends upon the size of the 71 matrix and whether diffusion effects are included in the program. For the distributed network and a matrix large enough to include the first eight harmonics between three and eight iterations are necessary. The running time for each iteration is approximately three seconds. The running time increases considerably when the matrix size is increased to include the first twenty harmonics, and diffusion effects are included. Under these conditions, the number of iterations averages nine to eleven, and the running time for each iteration is approximately ten seconds. 232

/Read Data ComputeCompute Qm(t) = QA(1) cos ~Vpt Start Qp(1), Wp, R'(n) + jX'(n ) or another -jnw Z() - 1 approximation 2 Compute 0F Perform | rCompute Are The TR3 V(t = - Harmonic Analysis A( )= R (n)QA(n)m+X'(n)Q (n),- VI(n) 6 (n) and tep A m A spaced points over V (t)- V ( n) a | n) B(n) = R'(n)Q B(n)m- X'(n)QA(n)(n) V(n) | |B FA iny one period for n = 0, 2,3,...small 19 | Insert the l lCSolve the Compute tCh3 Step Elements into Equations for QA(n)m~1 Q (n) -T ) C Sep ~ The r7 Matrix r E'(n) and E I Sn) A "m+1 A m A m=nm+ Step 2 Inserts theH A c () (1) use Equations n= 0,n) 3,. (nm+1 B m( B B(n)m D.-7 -sD.1fc9 D. 7 -D. 19 QA(1)m+1= QA(1) QB(1)m+l 0 | 2op (t) Q (t) = Q (t) Results Step Vp(t)= - + Qm(t) E ( by using (t), Qp(t), Ein(p) P P(o m by using e p einod 4in,Q(n) I Q (n) for many evenly spaced Eq. 5. 13 m B m points over one period for n= O, 1, 2,.. Fig. D. 1. The computer flow diagram to solve the distributed circuit equation.

APPENDIX E MEASUREMENT OF VARACTOR DIODES This appendix presents the measured characteristics of the diodes used in the experiments. A summary of the measurements which were performed is shown in Table E. 1, the data itself is shown in Fig. E. 2 and Table E. 2. It can be seen from Table E. 1 that a complete set of measurements was not performed on every diode. In every case, however, enough information was obtained to produce a model which represents the diode under circuit conditions. The measurement techniques are briefly described below. Data Presented In: Barrier Diffusion Diode Used In Laye r Q- V Capacity Number Circuit Capacity Relation Prn T, | PC 1430 D "Lumped" Fig. E. l(a) Fig. E. 2(a) PC 0642 C "Distributed" Fig. E. l(b) Fig. E. 2 (b) PC 0622 C "Diffusion" Fig. E. l(c) Table E. 2 Ideal abrupt Fig. E. 2 (c) junction Table E. 1. Varactor Measurements 234

48 - 44 - 40 36 32 28 24 20 16 12 8 C 4 0 -4 -8 - 12 - 16 -20 -24 -28 Dias Voltage Fig. E. 1. Measured depletion layer capacity 235

-0. 5 -1.5 / b a1t z II -2.0 I I I I I I 1.1 -1.0 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 0 +.1 Normalized Junction Voltage Fig, E. 2. Measured stored charge

Measurements of barrier layer capacity were made on the GR 821-A Parallel-T bridge. All these were taken at 1 MHz, and are ordinary RF bridge measurements, except for two minor points: a) The circuit was arranged so as to minimize the effect of the DC. bias supply on the HF signal. ByM supplying the diode bias through the bridge terminals, the entire bias supply circuitry was kept at RF ground. b) Great care was taken to ensure that the signal across A dqt the diode was really "small": that the differential quantity C dvt was really the quantity being measured. The plots of qt versus vt were obtained directly from the dqt dvt curves by graphical integration and are normalized according to the procedure in Section 2. 2. The techniques used to measure diffusion capacity were somewhat more novel. Using the diffusion capacity model (and the notation) introduced in Section 2. 3, the diode can be completely characterized if the following data is known: a) Pn the normalized carrier density ') T the carrier flletime c) 03 the electric field term A relation between j and P can be obtained from the forward DC n current-voltage relation. From evaluation of this data and using Eq. 2.34, it is possible to obtain the value of 237

R p Positive Voltag T +EF Pulse VF or VR Negative Voltage -ER Fig. E. 3, Test circuit for pulse measurements 238

t= 0 Excess Hole Density 0<t< T p 0 -Pn x Distance into n-region Fig. E. 4. Minority carrier density in base of diode 239

P (E. 1) The quantities 7 and 13 can be obtained from pulse measurements. The carrier lifetime T, of course, has been measured by several workers. However they have either used the assumption that i?=0 (Refs. 57, 58) or the assumption that 3=-oc (Refs. 59, 60, 61). The method used involves formulating an expression for the reverse current pulse which appears when the diode is switched from forward to reverse bias with a square wave. The circuit is shown in Fig. E. 3. The generator voltage, EF, and the resistor, Rp, can be considered large enough so that during forward conduction, the forward voltage drop, VF, across the diode may be neglected and F'F EIn RF (E. 2) F R It is further assumed that the duration of forward current is long so that steady state conditions are established. Before switching, the holes set up a distribution in the base as shown in Fig. E. 4. When the diode is switched to reverse bias, the n region becomes a source of positive carriers and reverse current flows until these carriers have been removed. By Eq. 2. 27 between the time the generator is switched (t- 0) and the tinme when the excess hole concentration at the junction becomes zero (t=Ts), the voltage across the diode is small. 240

0 < VR< VF for O t<T _ (E.3) Then between t = 0 and t = T, the current can be considered to be constant and equal to IR R (E. 4) P After time Ts, however, the voltage magnitude across the junction increases very rapidly as more carriers are removed, and the current magnitude drops rapidly to the normal reverse current value. The result is an almost rectangular pulse, of reverse current. To determine Ts, consider the behavior of holes in the base region. In the one dimensional situation they are governed by aD P(x, t) a P(x, t) P(x, t) a P( t) 24 ax2 - + -/E...t) (2.24) ax = at T ax The current through the diode is given by I(t) = -q DA aP(x,t) - q A Eli P(x, t) (2. 26) Total x x=0 x=0 where, as in Section 2.3, the junction is taken to be at the origin of the coordinate system. The base region extends indefinitely in the positive x direction. 241

P(c, t) = 0 At t= 0, the current is switched instantaneously from IF to -IR. A boundary value problem may be set up and then P(O, t) may be solved using conventional Laplace transform methods. The result is found (use Transform No. 42 in Ref. 62) after some labor P(0, t) -I + ' I~~~F R~~~ I + - + q[ + i;] (E. 5) by the definition of Ts P(0, T ) - and T 5i s T (s IF i + l +1 erf e ( +erf F +IR 1 + ~A f (E.6) For the case of 0 electric field (3= cc), this relation reduces to that of an ideal PN junction. IF T s F = erf (E. 7) IF + IR T Likewise, for the infinite electric field case (-3= 0) the function reduces to that of the ideal step recovery diode which is 242

modeled by a parallel RC circuit 1-e ( (SE.8)s F 1- e\ / (E. 8) 1F + IR Figure E. 5 shows the function IF/(IF+IR) plotted as a function of TS/T for various values of P. Both T and / can be estimated by using this graph in conjunction with a similar plot of measured values of IF/(IF+IR) versus Ts. The values of T and /3 for diode number PC0622C which were measured using this technique are presented in Table E. 2 along with an estimate of P. It should be noted that the curves near /3= 0 become very close together and an accurate estimate for /3 is difficult, and so the very good comparison of values obtained using pulse methods with those values which correlate with measured values of pumped varactor input impedance may be somewhat fortuitous. However, even if the comparison should not be this good, the use of this diode model and this means of measuring the diode constants still apparently have some merits. Measured Parameters of Diode PC 0622 C P r /3 n t0- 12 0. 5 /sec. 0. 1 Table E. 2. 243

1. 0 0. 9 0. 8 0.7 F I +1 0. 6 0. 5 0. 4 0. 3 0. 2 0 " 1~~~~~~~~~5 I I ~~~~~I i 0. 001 0 01 0. 1 T 1. 0 1 S T Fig. E. 5. Normalized switching time for diode with charge storage

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REFERENCES (Cont. ) 27. D. B. Leeson, Large-Signal Analysis of Parametric Frequency Multipliers, Stanford Electronics Laboratories Technical Report No. 1710-1. Prepared under Office of Naval Research Contract Nonr-225(31), AD 287 001. 1 Stanford, California: Stanford Electronics Laboratories, May 1962. 28. A. van der Ziel, Electronics, Boston: Allyn and Bacon, 1966. 29. Y. -F. Chang, Transient Phenomena in Semiconductor Diodes, Cruft Laboratory Technical Report No. 294. [Prepared for the Office of Naval Research, Contract No. 1866(16) NR-372-012. i Cambridge, Massachusetts: Cruft Laboratory, Harvard University, January 1959. 30. D. Koehler, "A New Charge Control Equivalent Circuit for Diodes and Transistors and Its Relation to Other Large Signal Models, " Internat. Solid-State Circuits Conf. Digest of Tech. Papers, 1965, 38-39.31. D. Koehler, "The Charge-Control Concept in the Form of Equivalent Circuits, " Bell System Tech. J., 1967, 46, 523-567. 32. J. L. Moll, Physics of Semiconductors, New York: McGraw-Hill, 1964, pp. 110-140. 33. L. P. Hunter, Handbook of Semiconductor Electronics, New York: McGraw-Hill, 1956, pp. 3- 1 to 3-17. 34. R. Fekete, "Varactors in Voltage Tuning Applications, " Microwave J., 1964, 7, No. 7, 53-61. 35. D. E. Crook, "A Simplified Technique for Measuring High Quality Varactor Parameters," Solid State Design, 1965, 6, No. 8, 31-33. 36. R. YV. Garver & J. A. Rosado, "Microwave Diode Cartridge Impedance," IRE Trans. MTT, 1960, MTT-8, No. 1, 104-107. 37. A. P. Epperly, "Varactor Fabrication for Microwave Applications," Proc. NEC, 1962, 18, 406-413. 38. B. J. Levin, "The Modified Series Model for an Abrupt-Junction Varactor Frequency Doubler, " IRE Trans., Microwave Theo. & Tech., 1966, MTT-14, 184-188. 39. S. T. Eng & W. P. Waters, "A Gold Bonded Germanium Diode for Parametric Amplification, " Proc NEC, 1959, 15, 83-91. 247

REFERENCES (Cont.) 40. A. P. Boile, "Application of Complex Symbolism to Linear Variable Networks," IRE Trans., 1955, CT-2, No. 1, 32-35. 41. H. Heffner, "Capacitance Definitions for Parametric Operation, " IRE Trans. MTT, 1961, MTT-9, No. 1, 98-99. 42. G. L. Matthaei, "A Study of the Optimum Design of Wideband Parametric Amplifiers and Up-Converters, " IRE Trans. on Micro. Theor. & Tech., 1961, MTT-9, No. 1, 23-.38. 43. M. Uenohara, "Noise Consideration of the Variable Capacitance Parametric Amplifier, " Proc. IRE, 1960, 48, No. 12, 1973-1987. 44. C. A. Desoer, "Steady State Transmission through a Network Containing a Single Time-Varying Element, "IRE Trans. on Circuit Theory, 1959, CT-6, 244-252. 45. A. A. Kharkevich, Nonlinear and Parametric Phenomena in Radio Engineering, New York: John F. Rider, 1962. 46. J. J. Stoker, Nonlinear Vibrations, New York: Interscience, 1950. 47. C. Hayashi, Forced Oscillations in Nonlinear Systems, Osaka, Japan: Nippon Printing and Publishing, 1953. 48. W. Kaplan, Operational Methods for Linear Systems, Reading, Massachusetts: Addison-Wesley, 1966. 49. J. L. Cockrell, The Phase Synchronization of a Parametric Subharmonic Oscillator, Cooley Electronics Laboratory Technical Report No. 178, Prepared for USAEC, Fort Monmouth, N. J. Contract No. DA 28-043 -AMC-01870(E)1 Ann Arbor, Michigan, The University of Michigan, April 1967. 50 S. Ramo & J. R. Whinnery, Fields and Waves in Modern Radio, New York: John Wiley and Sons, 1953. 51. V. Volterra, Lecons sur ies Equationes Integrales, Paris: Gauthier-Villar 1913 52. R. E. Scott, Linear Circuits. Reading, Massachusetts: Addison-Wesley, 1966 248

REFERENCES (Cont.) 53. C. S. Bunus, An Iterative Procedure for the Analysis of Nonlinear Networks, Stanford Electronics Laboratories Technical Report No. 6657-1 [Prepared under Office of Naval Research Contract Nonr-225(83), NR 373 360. ] Stanford, California: Stanford Electronics Laboratories, August 1965. 54. K. Siegel, "Anomalous Reverse Current in Varactor Diodes, " Proc. IRE, 1960, 48, 1159-1160. 55. B. Albrecht, J. Kliphuis, & D. Heuf, "Universal Performance Curves Speed Parametric Amplifier Design, " Microwaves, 1966, 5, No. 7, 20-24. 56. A. L. Helgesson, '"Varactor Charge-Voltage Expansions for Large Pumping Conditions," Proc. IRE, 1962, 50, No. 8, 1846-1847. 57. S. Krakaur, J. L. Moll, & R. Shen, "P-N Junction Charge-Storage Diodes," Proc. IRE, 1962, 50, No. 1, 43-53. 58. C. Culwell, E. H. Devletoglou, & L. E. Niemann, Jr., Negative Resistance Study —Phase I., Rome Air Development Center Technical Report RADC-TR-66-231, 1966, June. New York: Griffiss Air Force Base, Rome Air Development Center. 59. J. G. Gardiner & D. P. Howson, "Influence of Minority-Carrier Storage on Performance of Semiconductor-Diode Modulator, " Proc. IEE, 1964, 111, 1393-1400. 60. R. H. Kingston, "Switching Time in Junction Diode, " Proc. IRE, 1954, 25, 829- 834. 61. B. Lax & S. F. Neustadter, "Transient Response of a P-N Junction, J. Appl. Phys., 1954, 25, 1148-1152. 62. R. V. Churchill, Operational Mathematics, New York: McGraw-Hill Book Co., 1958. 249

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DISTRIBUTION LIST (Cont.) No. of Copies 16 Commanding General U. S. Army Electronics Command Fort Monmouth, New Jersey 07703 ATTN: AMSEL-EW AMSEL- PP AMSEL-IO-T AMSEL-RD- MAT AMSEL-RD-MAF (Record Copy) AMSE L- RD- LNA AMSE L- RD- LNR AMSEL-XL-D AMSEL-NL-D AMSEL-VL-D AMSEL-KL-D AMSEL-HL-CT-D AMSEL- BL-D AMSEL-WL 3 copies Dr. T.W. Butler, Director Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 7 Cooley Electronics Laboratory The University?f Michigan Ann Arbor, Michigan 48105 254

Security Classification DOCUMENT CONTROL DATA - R&D (Security clasification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) _a. REPORT SECURITY C LASSIFICATION Cooley Electronics Laboratory Unclassified The University of Michigan Z2b GROUP Ann Arbor, Michigan 3. REPORT TITLE Impedance Characteristics of Pumped Varactors 4. DESCRIPTIVE NOTES (Type of report and Inclusive date.) Technical Report No. 7695-189 November 1967 S. AUTHOR(S) (Last name, first name, initial) Oliver, David E. 6. REPORT DATE 7a TOTAL NO. OF PAGES 7b. NO. OF REF3 November 1967 278 62 6a. CONTRACT OR GRANT NO. 9a. ORIOINATOR'S REPORT NUMBER(S) DA 28-043-AMC-01870(E) b. PROJECT NO. C. E. L. Technical Report No. 7695-189 iPO 21101 A042. 01.02, c. b. OTH E R R PORT NO(S) (Any other numbers that may be assigned this report) d. ECOM-01870-189 10. AVAILABILITY/LIMITA TION NOTICES This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG. U. S. Army Electronics Command, Fort Monmouth N. J. Attn: AMSE L-WL-S 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Electronics Command AMSE L -WL- S Fort Monmouth, N. J. 07703 13. ABSTRACT A considerable amount of work has been reported on the analysis of pumped varactor diode circuits. However, all these studies have been limited in generality by two simplifying assumptions: the limitation of the pump voltage to low amplitudes, and the restriction of signal power flow to only a few sidebands. The first of these assumptions excludes the possibility of harmonic generation by the nonlinear diode, while the second presupposes the existence of ideal filters in the linear circuitry. This study is concerned with an analysis of pumped varactor diode circuits without these two restrictions. The input impedances of several basic pumped varactor diode circuits were analyzed theoretically by determining a suitable model for each circuit and hen predicting its impedance characteristics. The results showed good correlaion with experimental measurements, indicating the adequacy of the circuit mode and the validity of the analysis methods. It was shown that several circuit parameters significantly affect circuit behavior. A practical technique was developed for experimentally measuring the pump voltage amplitude across the diode junction. The most significant outcome of this study is the development of a circuit model which can accurately characterize a varactor diode circuit. D D *~JAN 64 1 473 Security Classification

Security Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Varactor Diode Impedance characteristics Linear networks Pump circuits INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report. report from DDC." 2a. REPORT SECURITY CLASSIFICATION: Enter the. over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accord- by DDC is not authorized." ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di- users shall request through rective 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional., markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. ized. report directly from DDC Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified.,, If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial. tory notes. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE: Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication, 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though 7a. TOTAL NUMBER OF PAGES: The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information. be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or'grant under which formation in the paragraph, represented as (TS), (S), (C), or (U) the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers,'task number, etc. or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected- so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical conassigned any other repcrt numbers (either by the originator text. The assignment of links, rules, and weights is optional or by the sponsor), also enter this number(s). 10, AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those Security Classification

UNIVERSITY OF MICHIGAN 3III 9III I 1 1111 IIIl II1 3 9015 03483 2181