Technical Report ECOM- 01870(E)- 8- T April 1967 THE PHASE SYNCHRONIZATION OF A PARAMETRIC SUBHARMONIC OSCILLATOR C. E. L. Technical Report No. 7695-178 Contract No. DA 28- 043-AMC-01870(E) DA-Project No. 1 PO 21101 A042. 01. 02 Prepared by James L. Cockrell 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 OF THIS DOCUMENT IS UNLIMITED THE UIbiV'Lih io i X... i & ENGINEERING LIBRARY

/2 C/l / /.).,.

ABSTRACT The general problem of phase determination for a parametric subharmonic oscillator is formulated. The analyses of several circuits are reduced to the solution of the canonical Mathieu equation. Solution is effected by the Floquet theorem and the parameters discussed in terms of the range of values encountered in experimental circuits. The transient portion of the solution produces arbitrary constants with two possible signs, yielding the two phase states of a growing oscillation. Conditions of oscillation are discussed, and the influence of signal and noise presented. Pump unbalance, as a primary signal masking source is a natural outgrowth of the analysis of the usual balanced pumping circuits. This condition is discussed in terms of coherent and incoherent quenching, that is, where the pumping is initiated at the same pump phase angle each quench cycles (incoherent quenching). The error probability where this pump unbalance is controlled by a variety of means is shown to be adequately low over a dynamic range significantly greater than generally reported in the literature. Experimental results are given demonstrating the various aspects of the analysis. Error probabilities of bistate phase detection are measured for noise and pump unbalance masking conditions and are shown to be represented adequately by the theory. iii

ACKNOWLEDGEMENTS The author wishes to express his gratitude to the many persons who have provided guidance, assistance, and support during the course of this study. In particular it was Dr. D. K. Adams who suggested the initial formulation of the problem and guided the early effort toward its solution. To Dr. Toshio Fujisawa, Osaka University, Japan, go special thanks for his suggestions relating to the noise analysis while he was in this country as a visiting professor. Also acknowledged is the guidance of Dr. Peter Khan and Professor Alan Macnee, the doctoral committee chairman, during the evolution of the analysis, experimental work, and preparation of the manuscript. The research conducted here was supported, in part, by the U. S. Army Electronics Command, Under Contracts DA-36-039 AMC00059(E), DA-28-043 AMC-00029(E), and DA 28-043 AMC-01870(E). Thanks are due to Mr. Herb Brett of USAECOM, Fort Monmouth, New Jersey, for his encouragement and assistance.

TABLE OF CONTENTS Page ABSTRACT iii ACKNOWLEDGEMENTS iv LIST OF SYMBOLS viii LIST OF ILLUSTRATIONS xvii LIST OF APPENDICES xix I. Introduction 1 1. 1 General Problem Area. Parametric Subharmonic Oscillator (SHO) 1 1.2 Prior Art 2 1.3 Statement of the Problem 5 1. 4 Topics of Investigation 5 II. Mathematical Formulation 8 2. 1 Type of Problem - Conceptual Formulation 8 2. 2 Circuit Models, Variables 8 2. 3 Reduction to Canonical Mathieu Form 14 2. 4 Circuit Relation to Matrix Parameters, a, g 18 2. 5 Method of Solution 21 III. Transient Solution 23 3. 1 Mathieu Form 23 3. 1. 1 The a-g Plane 23 3. 1. 2 /u and a Parameters 24 IV. The Steady-State Solution 35 4. 1 Derivation of General Steady-State Form 35 4. 1. 1 Signal Characterization 35 4. 1. 2 Phasor Description 38 4. 1. 3 Conversion Matrix 39 4. 1. 4 Gain 42 4. 1. 5 Passive Circuit Voltages 43 4. 1. 6 Signal Phase Relations 45 4. 1. 7 General Side Frequency Signals 47 4. 2 Reduction to Instantaneous Values 48 4. 3 Solution Under Special Cases 50 4. 4 Introduction of Pump Unbalance 51

TABLE OF CONTENTS (cont. ) Page V. General Solution 55 5. 1 Form of Solution for Constants 55 5. 2 Denominator - The Determinant of the Matrix 57 5. 3 Initial Conditions 57 5. 4 General Expression for A 60 5.4 1 (v -v ) 60 + ss+ 5. 4. 2 (v -v ) 62 + SS+ 5. 4. 3 General Form - Quasi-Degenerate Solution 65 5. 5 Signal A 69 5. 5. 1 Degenerate Form 69 5. 5. 2 Signal Power 71 5.6 Noise A 72 5. 6. 1 Excitation by Random Variables 72 5. 6. 2 Noise Power 74 G. eneral Expression for B 77 5. 7. 1 Signal B 79 5. 7. 2 Noise B 79 VI. Bistate Signal Detectibn by Subharmonic Oscillations 81 6. 1 Linear Superposition 82 6. 2 Detection Level - Superregeneration 82 6. 3 Error Probability 85 6. 3. 1 Signal and Noise 86 6. 3. 2 Signal and Pump Unbalance 88 6. 3. 2. 1 Coherent Quench Mode (CQM) 92 6. 3. 2. 2 Errors in CQM 93 6. 3. 2. 3 Incoherent Quench Mode (IQM) 94 6. 3. 2. 4 Error Probability in IQM 95 VII. Experimental Work 100 7. 1 Objectives Related to Study 100 7. 2 Circuit Descriptions 100 7. 2. 1 General 100 7. 2. 2 SHO Circuit 102 7. 2. 3 Quench Switch 104 7. 2. 4 Quench-Pump Synchronization 107 7. 2. 5 Signal Generation 107 7. 2. 6 Buffer Amplifier 109 7. 2. 7 Phase Detector 110 7. 3 Laboratory Methods 114 7. 3. 1 Setting the Value of g 114 7. 3. 2 Measurement of I.M 115 vi

TABLE OF CONTENTS (Cont.) Page 7. 4 Results 117 7. 4. 1 Basic Properties of Amplifier 117 7. 4. 2 Properties of the Associated SHO 126 7. 4. 2. 1 General 126 7. 4. 2. 2 Bistate Phase Positions 126 7. 4. 2. 3 Effect of Pump Level on Output 130 7. 4. 2. 4 Effect of Tuning on,g 130 7. 4. 2. 5 Effect of Signal Levefon Output 130 7. 4. 2. 6 Effect of Signal Phase on Output 132 7. 4. 3 Quantitative Relationships, Signal-to-Noise 135 7. 4. 3. 1 Signal Power Measurement 139 7. 4. 3. 2 Noise Power Measurement 140 7. 4. 3. 3 Signal-to-Active Band Noise 141 7. 4. 3. 4 Error Probability Determination-Noise 141 7. 4. 3. 5 Theory and Experimental CorrelationNoise 144 7. 4. 4 Signal to Pump-Unbalance 146 7. 4. 4. 1 Experimental A Contours 146 7. 4. 4. 2- Experimental Error Probabilities 148 7. 4. 5 Higher Order Subharmonic Oscillations 152 7. 4. 6 Magnetic SHO 152 VIII. Summary of Results 156 8. 1 Novel Features of the Research 156 8. 2 Incomplete Aspects 161 8. 3 Suggested Application 162 REFERENCES 183 DISTRIBUTION LIST 192 vii

LIST OF SYMBOLS Defined by, or Symbol Meaning First used in equation yV' d2 y where y is the dependent variable of the 2. 1 dz2 canonical Hill equation a constant coefficient of dependent variable of 2. 1 Hill equation 0(z) general periodic function of z 2. 1 g coefficient of parametric excitation for canonical 2. 1 Hill equation z. general independent variable for canonical Mathieu- 2. 1 Hi.. equat- o'e (t) signal voltage, function of time 2. 3 R series resistance of simple series circuit 2. 3 L series inductance of simple series circuit 2. 3 S(t) elastan.(c of series circuit, a fu:;ction of t:me, t 2. 3 i instantaneous current in series circuit 2. 3 q instantaneous charge associated with elastance 2. 3 F inverse inductance associated with series inductance 2. 3 L f a general function of the variable y 2. 6 K a function of t M a function of t X flux linkage associated with the inductance 2. 7 viii

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation G shunt conductance of a parallel circuit 17 coefficient of parametric excitation 2. 10 v instantaneous elastance voltage 2. 10 c SO constant part of pumped elastance 2. 10 2 pump angular frequency 2. 10 0 phase angle of pumped quantity at t = 0 2. 10 V a derived elastance voltage 2. 11 F0 constant part of pumped inverse!nduct!ance 2. 14 I derived current variable in series circuit with 2. 14 pumped inductance IL derived inductive current variable for parallel 2. 15 circuit with pumped inverse inductance V derived voltage variable for parallel circuit 2. 16 V lossless derived voltage variable 2. 17 e &Q0 resonant circuit quality factor 2. 18 W 0 natural resonant angular frequency of passive 2. 18 circuit. a' trial constant term in Mathieu equation 2. 20 g trial coefficient of parametric excitation in 2. 21 Mathieu equation 6 fractional excursion of w within passive resonance following curve 2. 23 ix

LIST OF SYMBOLS (cont. ) Defined by, or Symbol Meaning First used in equation A w angular frequency variation from resonance following to the half-power point 2. 23 t' translated time variable 2. 26 A arbitrary constant in solution equation 3. 2 (associated with growing term) B arbitrary constant in solution equation 3. 2 (associated with decaying term) characteristic expornent in solution to Mathieu 3. 2 equat ion,((z) unspecified periodic function of z 3. 2 a phase angle in Mathieu equation solution 3. 3 s3, 5.. sine term coefficient in series representation of 3. 3 0(z, ) c3, 5.. cosine term coefficient in series representation of 3. 3 * (z, ) 0 1 translated pump phase angle 3 5!~ L general dimensionless loss parameter 3. 10 1" pump phase angle purturbed by circuit losses 3. 13 is instantanious signal current 4. 1 wI lower side angular frequency 4. 1 Awt angular frequency excursion between w1 and 4. 1 we or wu w u upper side angular frequency 4. 1

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation 0 ~p phase anle of lower side frequency 4. 1 0 phase angle of upper side frequency 4. 1 Ie phasor representing lower side frequency signal 4. 2 current Iu phasor representing upper side frequency signal 4.2 current C circuit capacitance (time function) 4. 3 CO constant part of circuit capacitance 4. 3 V0 lower side frequency circuit voltage phasor 4. 5 V upper side frequency voltage phasor 4. 5 ~:* denotes complex conjugate 4. 4 fractional displacement of side frequencies from 4. 6 half-pump frequency L Hi complex pumping parameter 4.7c V( N lower side frequency voltage phasor in absence 4. 9 of pumping VN upper side frequency voltage phasor in absence 4. 9 pumping 9gt complex gain factor 4. 11 gm complex gain at frequency w1 4. 12a fractional excursion of signal frequency 4. 14 from half-pump frequency, W1 in the pumped, or active state xi

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation r' 1 detuned ~ 4. 14 I' complex pumping parameter in the detuned state 4. 14 phase factor associated with lower side frequency 4. 17 ~gou phase factor associated with upper side frequency 4. 17 phase factor for high gain situation 4. 20 0 2 consolidated phase angle (for convenience of 4. 20 manipulation) maximum value of, 4. 21 Omax 0 minimum value of g 4. 21 B0min 0 v (t) instantaneous -ir.:uit voltage in steady- 4. 23 state condition v (t) instantaneous lower side frequency circuit 4. 23 voltage v (t) instantaneous upper side frequency circuit voltage 4. 23 I degenerate signal current phasor 4. 25 Ot signal phase angle 4. 25 VV2(., circuit voltage at pump frequency 4. 29 C1 capacitance of one side of a balanced circuit 4. 29 C2 capacitance of the other side of an balanced circuit 4. 29 E1 drive voltage phasor of one side ok a balanced 4. 29 circuit xii

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation E 2 drive voltage phasor of the other side of a 4. 29 balanced circuit E common drive voltage phasor of a perfectly 4. 30 p balanced pump transformer e1 instantaneous pump voltage associated with E1 4. 30 e2 instantaneous pump voltage associated with E2 4. 30 c relative capacitance unbalance 4. 31,g growing term dimensionless parameter 5. 1 /ld decaying term dimensionless parameter 5. 1 v(0)+ instantaneous circuit voltage just after pumping 5. 1 begins v(0) instantaneous circuit voltage just time derivative 5. 1 just after pumping begins a,b, d, e coefficients of A and B in the pair of simultaneous 5. 5 equations in A and B c capacitance just after pumping begins 5. 9 C(0) and c capacitance just before pumping begins 5. 9 v circuit voltage just after pumping begins 5. 9 v circuit voltage just before pumping begins 5. 9 a, i3, y, y phase angles grouped for convenience of 5. 16 manipulation a' departure of Mathieu phase angle from 4 5. 29 radians al" a' perturbed by a detuning quantity 5. 30 xiii

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation all, a convenience angles. related to lower and upper 5. 32 side frequencies H " I1' purturbed by detuning 5. 34 Adeg value of A for the degenerate case 5. 37 Adeg value of A for the degenerate tuned case 5.38 6 =0 P signal power delivered to the tuned circuit 5. 41 P' signal power delivered to the detuned circuit 5 41 ( ) the mean value of 5.43 i1 sine component sum of upper and lower side 5o 44 frequency signal circuits i2 cosine component sum of upper and lower side 5. 44 frequency signal currents iN a root mean square signal current 5. 45 A, A due to excitations of differential bandwidth 5. 45 d' at + dN thermal noise power delivered to circuit within the bandwidth d' K Bottzmann constant, Joules oK 5. 47 T absolute temperature in degrees Kelvin 5. 47 df incremental frequency, cycles per second N total thermal noise associated with a passive 5. 51 resonant circuit N' thermal noise associated with an active resonant 5. 53 circuit xiv

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation AN root-mean-square value of A due to thermal 5. 56 noise aN normalized gaussian random variable 5. 57 Bdeg value of B for degenerate case 5. 62 Bdeg value of B for degenerate, tuned case 5. 63 6=0 B~, RMS value of B in differential band d[, 5. 64 BN RMS value of B due to thermal noise 5. 65 A A due to noise. Equal to A if thermal 6. 1 noise N noise is only noise source v 0 output circuit voltage peak 6.2 vi input circuit signal voltage peak 6. 2 vom maximum peak output voltage of oscillation 6.3 vim maximum input peak voltage 6. 3 gv voltage gain 6. 3 f maximum data frequency 6. 4 M figure of merit 6. 4 y01 a function of 1 and o"' in pump unbalance 6. 6 2 2 cons iderat ion A A due to pump unbalance 6. 8 pu e1(l) pump voltage for 1st quench cycle 6. 15 xv

LIST OF SYMBOLS (cont.) Defined by, or Symbol Meaning First used in equation e2(2...) pump voltage for 2nd, quench cycle 6.15 01(2) initial pump phase angle of 2nd quench cycle 6. 15 0 1(n) initial pump phase angle of nth quench cycle 6. 16 p0 minimum value of P p for zero error 6. 17 pu V envelope of peak voltage 7. 1 envelope V1' V2 successive observations of circuit voltage peak 7. 2 envelope v maximum steady-state voltage due to signal 7. 17 max phase shift v minimum steady-state voltage due to signal 7. 17 SS rmin phase shift C nth harmonic of time varying capacitance C III. 1 Vk kth harmonic of the capacitor voltage III. 1 Wok unidentified angular frequency III. 1 ci capacitive current III. 5 xvi

LIST OF ILLUSTRATIONS Figure Title Page 2.1 Series circuits. 11 2. 2 Parallel circuits. 11 2. 3 Pump-passive resonance relations. 19 3. 1 Stability of solutions. 26 3. 2 Iso ti - Iso a curves for first unstable Mathieu region. 30 3. 3 Oscillation criteria for a SHO circuit with losses. 31 4. 1 Side-Frequency Locations. 37 4. 2 Band center function vs pumping parameter. 44 4. 3 Balanced Pump Circuit. 53 6.1 Error probability density. 87 6. 2 Error probability vs signal-to-noise separation. 89 0 1 6. 3 A Contours in 2 96 6.4 Error probability for signal and pump unbalance in incoherent quenching. 99 7. 1 Block diagram of experimental layout for phase synchronization study. 101 7. 2 Fixed Bias-initial design. Differential Bias, for variable pump unbalance. 103 7. 3 Sub-Harmonic Oscillator Assembly. 105 7. 4 Quench Switch. 106 7. 5 Signal Attenuation and Delay. 108 7.6 Phase detector block diagram. 111 7. 7 Phase sensitive detector schematic. 112 xvii

LIST OF ILLUSTRATIONS (cont.) Figure Title Page 7. 8 Detector filter. 113 7.9 Evaluation of G. 116 7. 10 Maximum Steady State Response. 119 7. 11 Minimum Steady State Response. 122 7. 12 Maximum Steady State Response - High Gain. 123 7. 13 Minimum Steady State Response - High Gain. 124 7. 14 Sub-Harmonic Oscillation in Growing Region. 125 7. 15 Low level amplification of noise. 127 7. 16 Moderate level amplification of Noise. 128 7. 17 Sub-Harmonic Oscillation. 129 7. 18 Gain Variation with Pump Level. 131 7. 19 Output variation with signal level. 133 7. 20 Output variation with signal phase. 134 7. 21 Coherent Quenching. 136-7 7. 22 Saturated State-Coherent Quenching. 138 7. 23 Output phases for various signal input levels. 143 7. 24 Error probability vs signal-to-noise separation (Db.) 145 7. 25 Experimental A contours in coherent quenching. 147 7. 26 (a, b, c) Experimental error probability for incoherent quenching with pump unbalance. 149-51 7. 27 Higher Order Sub-Harmonic Oscillations. 153-4 8. 1 Digital commutating scheme for analog system. 165 xviii

LIST OF APPENDICES Page APPENDIX I: Removal of First Derivative in a General Second- Order Differential Equation 166 APPENDIX II: Detailed Plot of A- G Plane in the First Unstable Zone 168 APPENDIX III: The Steady-State Conversion Matrix 173 APPENDIX IV: Gain Function in Oscillation Region 176 APPENDIX V: Relation between a' and 6 179 APPENDIX VI: Phase-Shift Detuning Factor 181 xix

I. INTRODUCTION 1. 1 General Problem Area. Parametric Subharmonic Oscillator (SHO) The parametric subharmonic oscillator has been something of a curiosity to theoretical and experimental investigators for a period of at least one hundred years. As mentioned by E. Goto (Ref. 21) in 1959, and others before him, most of us have experienced the basic phenomena in attempting to cause our childhood swing to operate without any actual contact with the ground. In the child's heuristic manner he finds that appropriate bending (shortening) of the suspension at certain points in the cycle is effective in increasing the amplitude of oscillation. Quite appropriately named pumping, this effort patiently applied will cause the swing to proceed from no observable motion to dramatically violent oscillations. The phenomnon, though not new, is finding application in a world particularly sensitive to an inherent property of that phenomenon. The resulting oscillation possesses two intrinsically identifiable, stable states. Von Neumann (Ref. 45) and Goto (Ref. 22), at about the same time recognized that the subharmonic oscillation of order one-half observed in a parametrically excited resonant circuit always had one of two possible phase positions with respect to the pumping phase. The logical extension of these states to the "o" and "1" of the binary world quickly followed. It was further recognized that the device, being a resonant system, could be scaled to extremely high frequencies, with resultingly high possible binary data rates. Thus the principle held promise for exploitation in digital devices.

1. 2 Prior Art The volume of literature that bears closely on the general subjects of parametric amplifiers and the parametric subharmonic oscillator is large and stems from many sectors of the international literature. The emphases are identifiable into the subject categories which follow and present a variety of depth and elegance. Observations of subharmonic oscillations were reported in 1859 by F. E. Melde (Ref. 112) and in 1883 by Lord Rayleigh (Ref. 110). They had observed the sustained vibrations of a stretched wire at frequency f when the tension of the wire was varied by attachment to a tuning fork vibrating at frequency 2f. The oscillations were always phased so that as the wire moved away from the static position the tension was decreasing. Conversely as the wire moved toward the static position the tension was increasing and kinetic energy was being added to the system. A similar observation in a mechanical system was made by Stephenson in 1908 (Ref. 113) when he discussed the theory of the stability of an inverted pendulum by periodic movements of its point of support. The electrical analog of these observations have been subsequently reported by a great number of authors. Typical of the earlier effort is the work of R. V. L. Hartley (Ref. 28) in 1936 when he used the saturating effect of iron-cored inductors to produce parametric excitation. The work of Goto (Ref. 21) in Japan and a group of his colleagues has brought the understanding of the inductive SHO to a high level of practical importance by using ferrites. Computing schemes employing thousands of these devices have been built in Japan. In this country the

work has centered on devices constructed around the semiconductor diode as a voltage controllable capacitance. The primary reason for this emphasis is the high component quality factor (Q) obtainable into the gigahertz frequency region and possible extension of use to higher speed computers. Another general category of investigation profusely reported and bearing on the SHO is that of parametric amplification. The SHO is a degenerate form of the parametric amplifier and techniques used to analyze these devices are highly developed. In particular, (Ref. 15) methods of handling the steady-state analysis for perturbations about the degenerate condition have come from the study of double side-band parametric amplifiers. The analysis of the differential equations which result from parametric excitation have a long history. These discussions are included in a classic work by McLachlan (Ref. 103) published in 1946 and containing 226 references to the international literature. The refinement of methods of solution of initial value problems where the equations are of the MathieuHill type is evidenced by the work of L. A. Pipes (Ref. 56). The detailed study of oscillators invariably leads to large signal analyses where nonlinearities are the limiting influence. Much of the literature on the SHO concerns this region of operation since it is in the large-signal state that the operation is observed and logic operations effected. These studies deal with methods of handling the nonlinearities of the. reactive elements. Goto (Ref. 21) has used an interesting graphical presentation to display stable states at large signal levels. Wang (Ref. 46) developed a model of a thin magnetic film parametron (SHO) and demon

strated a computer solution to a high saturation steady-state level. The extreme nonlinearities of magnetic components were used in a unique analysis by Lavi (Ref. 26) in a dissertation at Carnegie Tech and later presented in a national publication (Ref. 25). The authors whose work most directly leads to the ideas and form of analysis used in this study are Bura (Ref. 17) and Phillips (Ref. 104). In particular Bura concerns himself with the expression for the operation of the parametric amplifier close to the degenerate condition. The results of his work differ considerably from some of the analytical and experimentally confirmed results of the study under discussion (c. f. paragraph 4. 16 and 7. 4). Phillip's contribution was primarily in the transient analysis and the evolution of the mathematical forms describing the phenomena as deriveable from actual circuit components. An additional segment of the prior art is concerned with the extension of the understanding of the SHO to applications in signal detection schemes and in computer technology. Radar front-ends have been built using the degenerate parametric amplifier (Ref. 23) in the amplifying mode. Wang and Wade (Ref. 10) have analyzed a similar device in the superregenerative mode. In the computer literature moderate frequency devices (10 to 50 MHz) are discussed and have assumed practical importance in computers in Japan (Ref. 14). In this country the effort has been to construct microwave computer elements using the semiconductor diode as the active element and to develop methods of handling information at the elevated data rates made possible by gigahertz pumping.

1. 3 Statement of the Problem In the various studies mentioned in the prior art the emphasis has been on the dynamic process of growth of the subharmonic oscillation. The authors have mentioned that a form of gain is possible in that a very small signal can positively establish the phase of growing oscillation. Only minimal attention has been given to optimizing that "gain" or finding the limitations of the gain. In none of the studies has significant attention been given to the mechanisms of phase-state determination. In such a binary signal detection scheme the resulting oscillation bears the information with a finite probability of error. Therefore, it is the purpose of this study: 1) to explore the mechanisms of phase determination and to identify the disturbing factors. 2) to express quantitatively, where possible, the limitations of sensitivity and error-probability in the determination of oscillation phase by a signal phase. 3) to demonstrate the validity of the analysis by suitable experimental procedures. 1. 4 Topics of Investigation The objectives lead naturally to limiting the study to very low levels of oscillation. In this region, where the phase determination actually occurs, the circuit elements are considered linear, and the pumped element

is a linear time-varying capacitance. Step quenching has been used throughout since in these days of high speed switching circuits it is a mode easily achieved and thus of practical importance. Along these lines progress has been made on these specific topics: 1) Formulation of the problem as an initial value problem where regions of consideration are (a) passive circuit, where only signal and noise are imposed on the circuit, (b) active circuit, where periodic pumping has been initiated abruptly and the values of (a) at the interface become the initial values of (b). 2) Consideration of circuit response to signal (degenerate condition) and noise (quasi degenerate condition). 3) Consideration of practical circuitry and the results of pump unbalance in nearly balanced pumping. 4) Consideration of coherent vs incoherent quenching. It is interesting to note that at the inception of the study the importance of pump unbalance in imposing limitations on the response of the SHO to signal phase information was not recognized. Nor has it been discussed in the literature. Appreciation of these limitations grew out of attempts to correlate laboratory results with the analysis evolved from topic 2 above. 1Quenching, a term conceived in the superregenerative amplifier field refers to the initiation and termination of the conditions permitting oscillation. These conditions can be inserted and removed smoothly, or abruptly, as in "step" quenching.

Finally as the pump unbalance problem was identified the concepts of coherent and incoherent quenching were found to divide these limitations into two families. The chapters which follow describe in detail the evolution of the analysis and the supporting experimental work. The mathematical formulation in Chapter II is not new but is glossed over without careful consideration by most authors. The material on the transient solution in Chapter III and in Appendices I and II has significant additions to material found in the literature. These additions permit a better understanding of operation away from the ideal "tuned, lossless situation. " Chapter IV deals with the steadystate solution which is necessary to the complete solution even in the oscillating region of operation since this is the "particular integral" of that solution. Chapters V and VI take the earlier results and develop analytical forms for error-probabilities under the various operating conditions. Chapter VII contains the experimental results which have proceded handin-hand with the development of the analytical results. The final chapter summarizes the results of the study and attempts to project the consequences of the findings into the present uses of the subharmonic oscillator and into some possible applications.

II MATHEMATICAL FORMULATION Possible circuit models of parametric subharmonic oscillators are discussed. The variables are chosen so that, after suitable changes, the canonical Mathieu equation results. In this process approximations are made and discussed. The general method of solution is presented. 2. 1 Type of Problem - Conceptual Formulation This discussion assumes that the oscillating system is controlled by external (information) signals where the level of oscillation is very low, and in the limit is in the vicinity of circuit noise. At this low level reactive circuit elements which are otherwise characterized by high level nonlinearities may be considered as linear elements. In particular when such elements are forced to vary periodically about a quiescent value by an external means a linear time - varying element is achieved. Such an element has the potential for transforming energy from the source of periodic variation (the pump) to the circuit to which the element is connected. (Ref. 15, 70, and others) The mathematical treatment of a resonant circuit having a linear time-varying reactive element as has been suggested by several authors (Ref. 21, 26), is closely related to the equations of Mathieu and Hill. Where the pumping is a general periodic function of the independent variable z the Hill equation results. (Ref. 103) y + [a - g(z)] y = 0 (2. 1) where a and g are constants. A special case of the Hill equation is the Mathieu form, which is

characterized by a simple harmonic pump frequency, thus: y"+ (a- 2g cos 2z)y = 0 (2.2) It is the purpose of this chapter to show that the Mathieu equation (2. 2) can be used to describe a resonant circuit characterized by moderately high passive Q and sinusoidal pumping of a reactive element. 2. 2 Circuit Models, Variables Consider the series resonant circuit with a linear time-varying elastance in Fig. 2. 1. The differential equation around the loop is e (t) = Ri + L + S(t)q (2, 3) Using the elastance charge q as the independent variable i = dq = q dt di d2q q(2. dt 2 q (2. 4) dt2 L = Eq. 2. 3 becomes re (t) = q + Rr + S(t)rq (2. 5) Eq. (2. 5) is an example of a canonical form which we shall restate in general terms f = y+ Ky+My (2.6)

Where K, M, and f in general are functions of independent variable t. As a second circuit example, consider the series resonant circuit where the inductance is pumped rather than the elastance or capacitance Fig. 2. lb. dX In this instance the inductance voltage is dt where the time variations are implicit in the constitution of the flux linkage, X. The circuit equation is written in terms of X as the dependent variable, while the inverse inductance, F, is identified as the pumped quantity. es(t) = + Rr(t)x + [Rr + sr(t)]x (2. 7) Equation 2. 7 is also in the form of 2. 6 although complicated somewhat by the time dependence of the K coefficient and the additional term in the M coefficient. The circuits of Fig. 2. 1 have their exact duals in parallel configurations and the equations for them can be written in the canonical form from Eqs. 2. 5 and 2. 7 at once. Parallel circuit, pumped inductance Si (t) = X+ GSX+ r(t)SX (2. 8) Parallel circuit, pumped capacitance is (t) = q + GS(t)q + [GS + FS(t)Jq (2. 9) It will be noted that in the examples presented up to this point the reactive elements are represented as elastance and inverse inductance. This

R r R S es(t) eS(t) eS( r(t) S(t S[1-isin(2w t+1)] (t) = = [1 1- sin(2w t + 01) (a) (b) Pumped Elastance Pumped Inverse-Inductance Fig. 2. 1 Series circuits. is(t)() GG S1 r(t) is(t) G rS(t) (a) (b) Pumped Inverse Inductance Pumped Elastance Fig. 2. 2 Parallel circuits.

is done for two reasons. First, the form of the equations is simpler because the terms all appear having only numerator values; and second, time variations are constrained to the numerator where the simple harmonic variations are represented in concise closed form rather than in a series of terms. The question arises, of course, about whether it is actually capacitance or elastance that is being pumped sinusoidally. The answer is related specifically to the particular element being pumped, and the manner of pumping. For small values of rq, the index of parametric excitation, these differences become small in any event. The primary effect is the reversal of sign as the explicit time variation is transferred from the denominator to the numerator. We have seen how the simple resonant circuit models having a single time-varying reactive element produce similar differential equations (2. 6). When the elastance is pumped the charge is the natural variable. For the pumped inductance the flux linkage is the natural variable. Neither of these quantities is particularly amenable to direct measurement in the laboratory nor do most workers in the field have facility in thinking specifically of these quantities. This problem has led several writers (e. g., Ref. 21) to find derived current and voltage variables which very nearly represent the actual current and voltage of the circuits and which can be represented by the form of Eq. 2. 6. For example, if in the series circuit with pumped capacitance we note that:

v V C C q = S (2. 10) SO [1 -S 1sin (2w1t + 1) then we can define a new elastance voltage V V c (2. 11) c [1 - t sin (2wcolt + 0 1)] from which V V V c c c q =SO - q = S (2.12) 0 0 0 and now Eq. 2. 5 becomes S res(t) = c + Rrc + S(t) rvc (2. 13) The variable Vc is very nearly the actual elastance voltage vc, differing from the actual only by the index of parametric excitation, ql, which appears as a time variation at the pump radian frequency 2wl, and its harmonics. Similarly, for the series circuit with pumped inductance the derived variable is the current I, and Eq. 2.7 becomes rF e (t) = I+ Rr (t) i + [Rr+ sT(t)] I (2. 14) and for the parallel circuits with pumped inductance and pumped capacitance Eq. 2. 8 and Eq. 2. 9 respectively become:

r S i = IL +GS i+ t) S IL (2.15) 0 s L * L So i = Vi+GS(t) + [GS+S(t)r] V (2.16) where V = ~ v derived from q = Cv = C V It has been shown that each of the simple resonant circuits having a pumped reactive element is described by the same form of Eq. 2. 6 and is given explicitly in Eqs. 2. 5, 2. 7,. 2, and 2. 9. In addition, a group of derived variables, closely approximating the actual variables are described by the same form. It follows that a reduction of the canonical form Eq. 2. 6 to a Mathieu equation is tantamount to reducing all the explicit representations to the same form. Therefore from this point on only a single form will be considered, and the particular one chosen is that most closely describing the circuit investigated in the laboratory. This circuit is the parallel circuit having a pumped capacitance. The equation representing the derived variable, V, the circuit voltage is the point of departure, Eq. 2. 16. It is worth noting that this equation and its dual, Eq. 2. 14, are somewhat more involved than Eqs. 2. 13 and 2. 15 because of the appearance of time variations of the coefficient of the first derivative. 2. 3 Reduction to Canonical Mathieu Form The Mathieu Eq. 2. 2 contains no first derivative and thus a change of variable in (2. 16) will be considered. This variable has the same dimensional form as V but is multiplied by an exponential factor which exactly compensates for the normal dissipation. This is a mathematical artifice and is described

in detail in Appendix I. Under this transformation Eq. 2. 16 becomes: S S(t)dt = V + [Gs(t)] GS(t) +s(t)r V (2.17) where 2 S S(t) dt V = VE e In order to proceed we consider the case of simple harmonic parametric excitation of the form S = S0[1 - 1 sin(2wlt + 01)] (2.18) r1 is the coefficient of parametric excitation. Circuit quality factor is variously stated as: R 1 S0C0 W0 QO - Gw0L G GS0 where; - SoF. When the passive circuit resonant radian frequency wg is factored out of the bracket and the substitutions of Eq. 2. 18 are made; Eq. 2. 17 becomes 15

G sdt S0e i S= V + W [1 - 71 sin(21t + 01)] 2 o s eO{ Q0 (2. 19) 1.-Ucos (2w olt + 01) + [1 - 71 sin(2co1t + 0 1)] V The terms in the bracket contribute primarily the two kinds of terms in the canonical Mathieu equation, namely a constant term, and a double frequency pumping term. Let us call the constant term a', and note: a' = 1- + (2. 20) 4Q02 8Q02 The double frequency coefficient we will call 2g' (to conform with McLachlan's canonical form of the Mathieu equation, c.f. ref. 103). This term then is: _ _ 1 1 2g' = -/[(1- 1 )sin(2w1t + 01)+ cos (2w1t+ 0) 2~0 Q0 1 2Q (2. 21) For moderate Q0 factors of 10 or larger this double frequency term is 2g' i 7 sin(2wlt + 01 + t an 1 Q) (2.22) (71 accurate to the order of 1 part in 8Q04 and 2g' 17)

In addition to the constant, and double-frequency terms, there is a 2 very small cos 4w1 term which is of the order of 77. If this very small 4Q2 (1 part in 1000) 4wco term is neglected the remaining equation is a dimensional form of the Mathieu equation. f = Ve + co2[a' - 2g' sin(2w1t + 01 + tan- 1 )] Ve (2. 23) where f S dt f S3 i 0 s We now consider the relation between the passive circuit resonance radian frequency wo and the half-pump radian frequency col in terms of a normalized dimensionless variable 6. (cc1 - cc0) _ 1 - max Acmaxo the half power point of the resonance curve, is known from passive circuit theory in terms of the center frequency and circuit Q factor to 0 be 1 Thus w0 = 6 (2.24) 2Q0 2 6 and 2 -* (1 - Q) (* read - is very nearly equal to —) And now if we let

a - a'(1- Q ) and g - g'(1 then (2. 25) a' 0 aw and 2g' 2 2g0 = 2gW and choose the origin of time t' such that cos 2w t' = sin (2 +t0 + t ) Eq. 2. 23 becomes f(t) = Ve+ w 2 [a - 2g cos 2w1t'] Ve (2. 26) Now let z - wlt' Ve(t') = V ( —) y(Z) = y" + [a - 2g cos2z] y (2. 27) 2 W1 2. 4 Circuit Relation to Matrix Parameters, a, g. As we shall see later, the transient solution to this equation (2. 27) is governed completely by the values of a and g. Therefore it is worthwhile emphasizing how these two quantities are related to the circuit parameters. In a circuit of moderate Q (e. g., Q0' 10) a is in the vicinity of unity 18~~

w 0 WI I] m I t 0 1 Fig. 2.3 Pump-passive resonance relations.

and is essentially the ratio of the squares of the passive circuit resonant frequency to the half pump frequency. A departure from this value occurs as we saw in Eq. 2. 20. Orders of magnitude are shown in the dimensionless form below a = (1 - )(1- )+ r (2. 28) 0 8Q02 detuning losses pumping Noting that within the normal pass band -1 < 6 < +1, the detuning has by far the greatest effect on a, but even that is a small perturbation, about unity. However, as we shall see, the effect is quite significant. The factor 2g is also affected by the detuning parameter 6, but the influence on the solution usually is small enough to be neglected. (Ref. 46) The fact that the ratio of a is not a function of pump frequency (i. e., detuning) but only of 71, the index of parametric excitation, is shown by Eqs. 2. 25, 2. 21, and 2. 22. The significance of this will be evident as the transient solution is discussed in the next chapter. Finally, the phase angle of the pump, is perturbed by a small angle (tan 1Q ) due to the losses. It has been shown above that the lossless Mathieu equation can be used to describe resonant circuits having a reactive element which is being pumped at approximately twice the resonant frequency of the passive elements. For circuit quality factors of 10 or above the terms of the final equation become quite simple. The dominating parameters are circuit detuning and 20

pumping level. A series of variable changes was made to convert the differential equation of the chosen circuit to the canonical Mathieu equation form. This allows the body of literature on that equation to be used in the transient solution. 2. 5 Method of Solution The solution of the final Eq. 2. 27 will be made in the classic manner of solving separately the complementary equation, finding a particular integral to accommodate the inhomogeneous member of the original equation, and adapting the constants to constraints appropriate to the problem. The complementary equation will be solved according to a classic reference on the Mathieu equation (Ref. 103) noting the particular range of circuit parameters involved, and evolving a graphic solution for the natural response of parameters of the system. The particular integral can be found using mathematical procedures appropriate to this class of problems, such as the so-called "variation of parameters" method. Such procedures will indeed produce a solution, as shown by Bura (Ref. 17) in a recent paper about the general degenerate form of parametric amplifier. However, these methods offer less insight into what actually happens in the circuit than the steady-state solution approach familiar to the engineer. This latter approach will be used following the general methods suggested by D. K. Adams in Ref. 15. The initial conditions which prevail when the circuit is switched from a passive circuit to an active, unstable, oscillating circuit provide the basis for imparting information to the active circuit. When thermal noise combines 21

with signal in the information handling process the problem becomes a statistical one where only the average performance over a large number of samples has a direct meaning. Included in this sampling process are other disturbing factors such as signal leakage and pump frequency unbalance. These can and frequently do have a predominant effect in determining the lowlevel signal limitations of the device as a sampled data digital signal amplifier. 22

III TRANSIENT SOLUTION 3.1 Mathieu Form The response of the active resonant systems presented in Chapter II has been shown to be constrained by the classic Mathieu equation. These systems are resonant circuits containing a single time-varying reactive element. The actual circuits contain dissipating members but by suitable changes of variable are represented by the lossless Mathieu equation, repeated for convenience below: y"+ [a - 2g cos2z]y = 0 (3.1) From the general theory of linear differential equations we learn that the solution is of the type y = A~eZ 0(z) + Be-TLZ 4(z) (3. 2) where A and B denote arbitrary constants, / is the characteristic exponent which depends on a and g, and 0 and V/ are periodic functions of z (Floquet's Theorem). In the words of Professor E. T. Whittaker in 1914 (Ref. 106) "While the general character of the solution from the function theory point of view is thus known its actual analytical determination presents great difficulties. " Professor Whittaker went on to introduce a new variable, o, which we shall also use here. This variable is the phase angle of the periodic portions of the solution Eq. 3. 2 and has a very definite physical significance. 3. 1.1. The a-g Plane. The "analytical determination" mentioned 23

above is not nearly so difficult in the particular case of interest as in the more general case. The stability of the general transient solution is best understood by referring to the a-g plane. Figure 3. 1 (Ref. 2-4) shows the various regions of stable solutions, the lines dividing these regions being periodic solutions. Of interest in the problem at hand are the unstable regions, and of particular concern is the plane in the vicinity of a = 1. For our problem we find that the area bounded by a = (1 ~.15) and g between 0 and.2 is sufficient to display the full range of solutions of the transient problem. 3. 1. 2 bi and a Parameters. Examination of the selected region of Fig. 3. 1 shows that the values of,u and a can be given explicitly in terms of the equation parameters (3. 1), a and g. The details of the development of Fig. 3. 2 are of considerable interest and are touched upon in an abbreviated form by Phillips (Ref. 104) in a recent paper. In Fig. 3. 2 the direct presentation of the variable, a, on the a, g, plane is a significant addition to this method of understanding the transient solution. The development of these curves is given in Appendix II. Noting that the characteristic exponent, p., is uniquely determined by the values of a and g, only the nature of 0(z) and \4(z) remains in order to understand the transient solution. Here the restricted region of a and g assist us considerably. Following Whittaker's development he supposed that 0(z) can be represented by 0(z, a) as a Fourier series introducing a as a phase angle, thus: 24

(z, o) = sin(z-ao) + s3 sin(3z-) + s5 sin(5z-c) ++..... cos(3z-) (3. 3) + c5 cos(5z- r) +.... The s's and c's are functions of g and X. The substance of Whittaker's development was to show that the missing terms in Eq. 3. 3 introduce nonperiodic terms in (z, (X) and that the coefficients sn and cn are convergent series in g and X. The coefficient of the sin (z-C) is arbitrarily chosen to be unity. The series expressions for the s's and c's are given in Appendix II and tabulated in table 3. 1 to show the maximum relative magnitudes. Table 3 1 Max. value for ~~~~Term ~g =.2 Term -Largest Term % of unity s3 -g/ 8 2. 5 C3 3 2 64 g sin 2a.19 S5 1 2 92 g.021 C5 2 — g sin 2a.0024 1 3 s7 - 9216 g.00009 35 4 c7 442368 g sin 2. 000013 Table 3. 1 shows that the s3 term is by far the largest, other than the reference term, sin(z-C). Thus for the region of the transient solution under discussion we find that it is nearly a pure sinusoid having at most a 3rd harmonic of 2. 5%. Consistent with the objective to find a comprehensive picture of the 25

Stable Unstable 16 Stable a _ U/ Unstable 9 - Stable Unstable _Unstable 5 10 15 Fig. 3. 1 Stability of solutions. 26

mechanisms involved in the phase synchronization in the simplest possible form only the sin(z- a) term will be used in the material which follows. The second independent solution is obtained by writing, -a in Eq. 3. 3 so that the complete solution to Eq. 3. 1 in the region of interest for this problem is: y = A c ez sin(z-a) + B E- 1 sin(z+a) (3. 4) 3. 2 Reversal of Variable Changes The changes made in Chapter II to arrive at the Mathieu equation are now reversed to restore the original variables. The individual changes are reversed one at a time in order to show the effect of the particular changes on the form of the final solution. w t' =z y(z) =Ve(t') Ve AE /wt' M1sin(w t' - c) + B - /1t' sin (wt' + c) (3. 5) from Eq. 2. 25 cot' = wlt+ —- and 0 +tan 2 V 2 1 - 1 s +. V A'E 1W1ltsin (w t + -a) + B 1t sin (w t + 2 + a) (3. 6) e 2 2 A' and B'; are merely new arbitrary constants. Ve is the lossless variable and is related to the true variable with losses by 27

Vf GS(t) V Ve (3. 7) The fact that S(t) is time-varying makes the form of V more difficult to relate to the problem. Using the substitution that GS0 = /Q -wot -Wv0 /7 0 U cos (2wt + 0 2Q0 1 4Q0 1 1 V - Ve e (3. 8) 0 Since is approximately unity the second exponential term is c1 simply a time perturbation of V and has values within the range, (1 ~ ). 4Q0 For values of 1 <. 2 and Q9 > 10 this perturbation is of the order of 25% or less. Using Eq. 3. 8 in 3. 6 (neglecting the second exponential in Eq. 3. 8) V(Iw0L)wlt. - (M A+L)wlt.0 V = A'e sin(colt + - or) + B'c sin(colt + - + () (3. 9) where - (01 + tan 1 1 2) and 0 (1- 2 ) ~21 0 /ML — E 2Q --- (3. 10) 028 2Q0 28

Equation 3. 9 is a solution to the homogeneous differential Eq. 2. 16 and but for the small variations introduced in paragraph 2. 2 is the actual elastance voltage. A new quantity has been introduced in /1L' which expresses explicitly the circuit losses, and the detuning. 3. 3 Criterion for Oscillation In Eq. 3. 9 the criterion for oscillation is that V grow without limit as t increases. Since pl is always positive in the region of interest this criterion is satisfied only when > /IL (3.11) In order to visualize this criterion one can identify the iso- / curves of Fig. 3. 2 as iso-Q0 curves. For example if Q0 = 10, from Eq. 3. 10 we note that /L. 05. Thus the pg =.05 curve of Fig. 3. 2 divides the a-g plane into an unstable region above g =. 05, and a stable region below. This is portrayed again in Fig. 3. 3 where the curves are laid out on the "a" axis (as ordinates versus g) and the iso- / curves are labeled with Q0 values. Each of the Q0 values divides the plane into stable and unstable regions for that particular Q0. Having defined stability regions it is desired now to visualize movement within the regions for a given problem. As discussed in paragraph 2. 4 and by Phillips (Ref. 2) the ratio of a- is determined by ar and is not a function of frequency. Thus, as the circuit is detuned with respect to the half-pump frequency (i. e., w0 with respect to wl) the ratio of a- is described by a straight line through the origin. The stability of the circuit is understood in terms of the intersections 29

0. 2 10~~~~~~~~0 C>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C 08~~~~~~0 0 ISO - first unstable Mathieu regionO 0 ~ ~ ~ ~ ~ ~ ~ ~ b~~~~~~ 14 12 10 8 6 0 0 0 02 04 6 8 1 1 -*(I,

Q0 Unstable for Q03 /1Circuit, = 31

of this straight line with the particular Q0 curve describing the passive circuit. If no intersection occurs, the circuit is stable. If the constant-T) line is tangent to the Q0 curve, oscillation at just one frequency is possible. For two intersections oscillation over a band of frequencies is possible. Using the Q02 of Fig. 3. 3 as a case in point and noting the increasing rj lines in the sequence 71' 772' 2 73 the geometry of the oscillation criterion becomes evident. In particular, we note asymmetry in the frequency excursion over which oscillation is possible for rj ='73 as shown by the unequal distance from a = 1 along the "a" axis. 3. 4 Simplifications under Various Operating Modes Equation 3. 9 is the transient solution to the original differential Eq. 2. 16. It is instructive, particularly as to understanding the substance of a given expression, if some of the members or terms representing small departures from the central representation are stripped away. Let us note the significance of constraining the passive resonant frequency (gv0) to the half-pump frequency (w1). Returning to Fig. 3. 2 we note that for a= 1, -45O (-~) r adians) and i -. -- then ML= 2&Q (3. 12) and Eq. 3. 9 becomes -Q 2 ) Xt " - (i + )w1Q)l t V 1 A'e 0 1 V = A'E 2Q)tsin(wlt + 2- )+ B"e cos(It + f) V= A'e (3.13) 32

where -1 1 0= 1 + tan and B" = -B' Q0 Note that 0 i' is essentially the pump phase angle and that for Q0 = 10 the phase shift in Eq. 3. 13 due to the losses is less than 3 degrees. Thus we see that the natural solution to the problem attacked is an exact subharmonic oscillation of order 2, whose phase angle is determined by the pump phase, and where the pump level and losses are explicitly represented in terms of a simple oscillation criterion. At this point it is worth examining Eq. 3. 13 qualitatively to note that the phase relationship of the solution is appropriate to the phase of the pump signal and the energy relationships which are occuring. In order to deliver energy continuously from the pump source to the tank circuit work must be done on the system during a portion of the cycle, but not extracted by the pump at other times. Using the simple example of Adams (Ref. 15) we note that work will be done on the system if at the time a voltage occurs across the capacitor the capacitance is decreased. Energy will not be extracted from the capacitor if at the time no voltage occurs across the capacitor the capacitance is increased. That such conditions prevail in the analytical forms above is shown simply by noting: capacitance variation from Eq. 2. 18 C = C0[1 + 7 sin 2(lt + -2-)] (3. 14) 33

Voltage variation from Eq. 3. 13 (growing exponential part) IJ~gWlt B V = A'E g sin(t+ 2) (3. 15) 0" When the capacitor voltage is zero [ (wlt + 2-) = 77a) the capacitance 0" is at CO but increasing. When the capacitor voltage is maximum (wlt + 2) = (2 + 1) 2 J the capacitance is at CO but decreasing. These relationships are shown vividly by the photographs of experimental transients in Chapter VII, Figs. 7. 10 and 7. 11. For the problem at hand, that of understanding the manner and limits of controlling the system, the growing exponential term is the most important, since, after a significant number of time constants of this term, the decaying term will be negligible. In conventional superregenerative devices the growing exponential term is used as a measure of the magnitude of A. Of particular interest in this study is the phase ambiguity of the growing output as represented by the two possible signs of A. The subharmonic oscillator phase is by its very nature bistable. 34

CHAPTER IV THE STEADY-STATE SOLUTION The steady-state solution is that part of the general solution of a differential equation which results from the non-homogeneous member in the original equation. In the particular circuit under consideration this member represents the driving or signal function. The signal is characterized in terms most suitable for the degenerate parametric device being studied, so that both signal and noise can be described by the same analytical form. For our problem of understanding the mechanism of the transfer of information from the signal to the growing oscillation, the initial conditions are the vital link between the passive circuit and active circuit. Prior to initiation of pumping the circuit will be assumed to be under the steady-state constraint of the signal (information and noise). After initiation of pumping this signal experiences amplification via the active circuit parameters. In both cases we will be using the instantaneous values of the steady-state solution in the time domain. The generation of these forms produces a gain function which has a value both below and above a threshold for oscillation and as given by Adams (Ref. 15) contains explicitly the information about that threshold. 4. 1 Derivation of General Steady-State Form 4. 1. 1 Signal Characterization. The signal current, is, of Fig. 2. 2b, is now characterized in a form which is convenient for the 35

quasi- degenerate parametric amplifier. The juxtaposition of frequencies under consideration is shown in Fig. 4. 1. Each signal current at a frequency in the vicinity of w1 but below c1 (named wo, a lower side frequence), separated from w1 by Awc together with the time-varying capacitance generate a response at (2w1 - wd). This becomes part of Bu' the uprer side frequency. This frequency at wu', together with the timevarying capacitance generate a response at (2wl - ), which is back exactly at the original lower side frequency, cw. In addition to this amplitude characteristic there is also a phase characteristic peculiar to the signal but not the noise. Accordingly we will represent the signal is by this double side-frequency, each component carrying an explicit phase angle. i cos t Cos (wt+ O) + II i cos (out + 0U) (4. 1) where o W - oI- W c - 0 C + AC u 1 4 and side frequency phasors are: (4. 2) jo u u 36

/!~ ~~~~- I6 /0S1 - /( u- 1)= (O Po Q) / \ Fig. 4. 1 Side-Frequency Local ions. 37

4. 1. 2 Phasor Description. We now describe the currents at the lower and upper side frequencies associated with a time varying capacitance. For a general monochromatic sine pump of the form: C = C0 [1+ 71 sin(2w1t + 01)] (4.3) the expression for the currents is shown in Appendix III to be 1I1 WC c 71C E V (4.4) -j 1 Lcou C0-j Wc C0 V ui C o u (C subscripts identifying values related to the capacitance. ) When the reactive element of Eq. (4. 4) is placed in the circuit of Fig. 2, 2b with the driving function of Eq. (4. 1) the equilibrium is expressed by: G l +y jW c 2 X C V oG o 2 o (4.5) Iu* _o G~TV JC* w C e G + jWc C V L U 2 -jw U O U The source current phasors are represented explicitly in the lefthand member. Equation 4, 5 carries all the information about the steady38

state, and when the conversion matrix is inverted the side frequency voltages are given explicitly, Some effort will now be expended to expressing Eq. (4. 5) in terms which will be significantly more descriptive of the circuit and much more useful in the development which follows, 4. 1. 3 Conversion Matrix. In general, the resonant frequency of the circuit, w0, departs from the half-pump frequency, wc This variation can be specified, as in Fig, 4. 1, as a fractional part, 6, of the halfbandwidth. In addition, symmetrically disposed about the half-pump frequency the side-frequencies appear. Their displacement is described similarly by the fraction, 4, of the half-bandwidth. If these components are to have significant amplitudes, their frequency displacements must be restricted to the region -1 - (6 + t)' +1. From Fig. 4. 1 we write = c0+ AW- _ - - _ 1 + 2 — 0) CW =0 + aC + AcW- W (1 2 Where QO is the quality factor of the passive circuit and is defined: W0 C r gQ G W G (4. 7a) 0- G =3co —G 39

C02:_ (4.7b) In order to simplify the notation let us introduce a complex pumping parameter, 17, where: n7QO J01 1 =- 2 e (4. 7c) Now, if the conductance, G, is factored out of all the elements of the conversion matrix, the elements of resultant matrix are dimensionless. -G 1(~]K11(4.8) _ I 1 +j(6+ ) V u - u* We would like to develop a "gain" expression. In order to do this the side frequency voltages appearing in the absence of pumping are needed. Define V V N G[1 + j(6- )] (4.9) VuN G[1 + j(6+t)] where the "N" in each case signifies that this is the normalized circuit voltage that is observed in the absence of pumping. 40

aVIN l | 1 + lj(6-:) V2 (4. 10) V V LVuN 1-j(.) Vu* Inverting (4. 10) now gives the desired expression for the voltages produced at the side frequencies, the gain appearing explicitly as the reciprocal of the determinant of the conversion matrix. -H F V,F 1 a1 + j(&- ) 2V N gl [ = gI (4. 11) V V u 1- j(A+) uN where (4. 12) 1 +( 2_52- )- j2 When ~ is zerd, the degenerate condition is defined, and g, has a maximum value, gm, thus: g 1 (4. 12a) 1- - I +Sa'

Equations 4. 11 can be thought of as consisting of three separate factors; gain, passive circuit voltage, and phase response. Each will be examined in the paragraphs which follow, according to the description by magnitude and phase angle. 4.1.4 Gain. g5 is examined in detail in Appendix IV. It is given the gain significance in that it is a factor representing amplification when 1I, the pumping parameter is not zero. When II is zero (no pumping) gi is unity and the passive circuit voltages result. Equations 4. 11 and 4. 12 show g5 is the only factor which can produce gain, and that for oscillation to exist g4 must be able to go to infinity. From Eq. (4. 12) we see that two conditions must be satisfied for g to increase without limit, They are: = 0 (4. 13) 11 12 = 1 + 62 In the problem under study the steady-state term is not used when the band center gain is infinite, but at pumping levels above this point. When the gain is defined as in (4, 12) and the second condition of (4, 13) is exceeded ( >i(1 > 1) the gain as such has less meaning (i. e., relating to a growing oscillation) but is still an essential part to the general solution of the original differential equation. The magnitude of gm decreases with increasing pump level (71) and the sign is everywhere negative. (cf. Fig. 4.2). We note that for values of II//1+62 in the vicinity of 1 42

the effective band is narrowed as the gain increases and (4. 12) can be expressed simply g 1 j1 tan 1 (4.14) gg -g M jeyt,m E (4. 14) where - 2g m and 1 6 Further define il1'i - - then g — the primed notation (1', H') introduced to allow implicit expression of the detuning, 65 4. 1, 5 Passive Circuit Voltages. The right-hand column matrix of (4. 11) carries the information about the passive circuit, the elements being defined in (4. 9). When V2N and VuN are expressed as magnitudes and angles they become 43

20 15 - 10 Stable 5 Pumping Parameter E ~5 ~~ I1+6z 1 1.5 1 1.5 2 -5 Oscillating -10 I 15 I -20 Fig. 4. 2 Band center gain vs. pumping parameter. 44

j[O~ - tan (6- I)] V = VfN G /lj+(67)2 jtan (6-a) G (4. 15) j[O - tan l(6+t)] u u VuN = - G l+(/i7 ejtan ) G 1+(+)G We will be discussing an active band where the values of ~ << 1, and the expressions (4. 15) may be approximated: j[0; -tan- 6] 1I I - VV G ~J (4. 16) j[0 - tan-i 6] T ue u'uN The forms (4.16) are the usual ones for passive resonant circuits showing the explicit effect of the detuning. Note particularly that because of the assumption of small ~ the phase angle sign is the same for both side frequencies. 4. 1. 6 Signal Phase Relations. The off-diagonal terms of Eq. (4. 11) carry explicit phase information. If the passive circuit voltages are factored out the equations become: 45

Vi = g5 VQNg0Q (4.17) V * * Vu g VUN 0u where g o and glu are the factors associated with the lower and upper side frequency voltages respectively and which predominantly show the effect of the signal phase position. V* g = - (1+ j-1)) (] (4. 18) * [ (1 1- I(*) VN] Consider the case when the excitation currents IIQ I and }IUI Eq. (4. 14) are equal and make up a double side frequency signal. For this case Eq. (4. 18) becomes: JIB1- e - 0u + tan (6+~)] = lI-ille (4. 19) -j[l - 0 O + tan (5-5)] _ l- u g~u 1.......46

Within the active band at nominal gain levels (gm 10) the value of << 1. Under this condition we can write a single expression for the two equations of (4.18): _ _ _l j02 j02 =g g0= g0 = 1 -- E = 1- IIII eC (4.20) gofou Q) /1+62 where 02 = (1 -0 - 0u + tan 6) Equation 4. 20 is visualized as the sum of two phasors of nearly j0o2 equal magnitude. The maximum magnitude occurs when e = -1, i.e., j0 2 02 = f rradians. The minimum occurs when e = +1, i.e., 02 = 0 radians. The ratio of the maximum to the minimum is 90max +iIII,] max I 4 gm (4.21) 0mmn Thus we see that the go term has a profound influence on the steady state circuit voltage under the particular type of signal discussed above (i. e., III = iIUI). 4. 1.7 General Side Frequency Signals. When the side frequency signal currents are not equal in amplitude the simple visualization of Eq. 4. 20 is no longer possible. Under noise excitation the independence of the side frequency "signal" currents must be preserved in order that random amplitudes and phases can be ascribed explicitly. The development 47

which follows contains all the information of paragraph 4. 1. 6 but loses the conceptual picture of the discrete frequency excitation. Expanding Eq. (4. 11) using (4.14), and (4.16) gm VU = gm [ j(0 -1 6-tan -') j(0! - 0 - tan- l'( u (4.22) ugm j(01- 0 + tan -') j(O - tan 1 + tan' + 1iE'u 4, 2 Reduction to Instantaneous Values The instantaneous values of the upper and lower side frequencies are derived directly from Eq. (4. 22) and since both represent voltages within the passband the net circuit voltage is the sum of the lower and upper side frequency components. Vss(t) = VL(t)+ Vu(t) where 48

v2(t) = Re(Vf e (4. 23) V(t) = Re(V (423t) from which vs(t) = [gm iI cos(wot+O tan-1 -tan 1_,) G( 1+6 2)(1+') II'I cos(wut +0 - 0+tan-'); i {-II'1 cos(wt + 01- 0 - tan-1,) + cos(wut+ U tan1 6 + tanI')] (4.24) Equation 4. 24 describes the instantaneous steady state circuit voltage in terms of circuit values, signal constraints, and circuit adjustments. It applies to all pumping levels (II') including zero. Where the point-by-point time domain value of the solution to the original differential equation is needed in the vicinity of t= 0 this is the equation which must be handled. In looking at this Steady state term in view of the solution to the transient problem and recognizing that after a few cycles the only term of significance in the general solution will be the growing exponential we note that the significance of Eq. (4. 24) is identifying the value of 49

vss(t) at t = +0 so that the arbitrary constants of the differential equation solution can be evaluated. This will be done in paragraph 5. 4 of the next chapter. 4.3 Solution Under Special Cases When the side frequencies approach w1 exactly, the degenerate amplifier results. Under this condition the separate signals coincide in magnitude and phase. Thus in Eq. (4. 24) we set: = 0 Q 0u = 0s (4. 25) u S ils II = Ilu = 2 where 0s and Is are the phase angle and phasor associated with the degenerate signal. Equation 4. 24 becomes gm VsM IS cos(w t) = - III I cos(w1lt + 01- s] (4. 26) In order to visualize just what Eq. (4. 26) means, we may make another simplification by letting 01 (pump phase angle) be zero and the detuning, b, be zero. For moderate gains (II -> 5) the steady-state voltage 50

becomes approximately: - gn vms(t) -G I s iln t sin i s (4. 27) 7T 31T Equation (4.27) is a maximum whens is 2 or 2. Recalling that the signal currents are given by Eq. 4. 1 we see that the instantaneous signal current which will produce the maximum steady-state response is is -11 I sin w t (4. 28) 5 1 -- simply a signal current in phase, and of the same sign as the preferred steady-state response. 4A 4 Introduction of Pump Unbalance One of the direct results of the experimental phase of this study is the realization that voltages at the pump frequency do appear across the resonant circuit and these unbalance voltages very significantly affect the minimum signal response of the system. Although the pump frequency voltage does not experience amplification it can have a magnitude which completely dominates the observed voltages. Effectively the pump unbalance presents a coherent interference which the information carrying signal must exceed in order to achieve efficient phase determination. 51

This aspect of the problem arises because of imperfections in the balance of the circuit used to accomplish pumping. From Fig. 4. 3 which is a typical balanced circuit, the pump frequency voltage appearing across the circuit is simply C1E1 C2E2 V 2CO 1 (4.29) 1 C +C - 1 2 2 1 where E1 and E2 are the phasor pump voltages associated with e1 and e2. Resistance terms in an actual circuit do not change the values much since the reactive terms dominate. From Eq. (4. 29) an unbalance of the capacitances or pump voltages e1 and e2 in magnitude or phase, causes a voltage of pump frequency to appear across the circuit. Resistance unbalance in the actual circuit would introduce second order unbalance in amplitude and phase. Since e1 and e2 can be generated by bifilar secondary windings on a toroidal transformer, the degree of symmetry in the pump voltage source can be very high. The rest of the circuit is made mechanically and electrically symmetrical to insure a high degree of pump rejection. Matching the capacitances is much more difficult, and herein lies the principle source of unbalance. If we assume that the capacitances are connected to produce positive circuit capacitance variations for positive pump voltages, and 52

C1 7~ L ~ 2 C2 eI 1 e2 2c1 2c1 Fig. 4. 3 Balanced Pump Circuit.

el = e2 iEi sin (2w t + 0) (4. 30) the instantaneous value of pump unbalance voltage can be written va = c IEi sin (2 ~ t + 1) (4.31) Where we define the relative capacitance unbalance as C1- C c 1) (4. 32) 1Ci+C2 -4w 2 since w 2L(C1+C2) = 1, by the resonance condition (Ew. 4. 7), the relative capacitance imbalance is: 4 (C1- C2) c = 3 (C1 + C2) (433) 54

CHAPTER V GENERAL SOLUTION The general solution is derived. The dependence of the values of the constants on the initial conditions is shown. From these values the solution for the degenerate signal at exactly o1 radians per second is presented. Finally the solution under the noise-forcing function is derived. 5. 1 Form of Solution for Constants The general solution for the circuit voltage is the sum of the steady-state and transient, solutions. Thus, from Eqo (3. 9): v(t)= vss(t) + A e in co t + 2 - Id lt(2 + B e in cot + 2 r (5.1) whereg = (/ / zL) / d ( + L) The constants A and B are the only quantities yet undetermined in the solution. Two independent conditions are required in order to determine these coefficients. The ones available are the initial value and initial derivative of the circuit voltage. These quantities are determined by independent energy considerations at the time of the initiation of pumping. 55

Thus, when t = 0 (just after pumping begins) from Eq. 5. 1 + V(0) = v(0) +A sin( -a)+ B sinj}+ o) + Ss 2 (O)+ \(O) +Aw Co - a+ 1g sin(-n 2 o +B w Cos(+- + - Adsin(2+ )O (5.2) The steady-state terms become a part of the constant terms in a pair of simultaneous equations in A and B. Thus, when thp coefficients of A and B are labeled a, b, d, and e (5. 2) becomes (v(O)+- v s(0)) = aA+bB (5.3) ((O) v (0)O) = d A + e B ~0+ Sss The solutions for A and B follow from Cramer's rule directly. (0)+ - v(0)) b a (v(o) - vSs(0)) (0)+ Vss(O)+) e d (o)+ Ss(O)) A = -. 4 a b a b d e d e 56

a = sin 2 — ) b = sin(- +or d = 1 [cos( 2 or) + sin(- - ) e = los(- + a) d sin(' + (5.)5) 5, 2 Denominator - The Determinant of the Matrix The determinant, d be reduces to: ai = c [sin(-2) - 1j cos 2a + sin + tan Q (5.6) hen the detuning, 6, is zero the phase angle, cr, very closely -,T approaches. c. f. Fig. 3. 2 for a = 1, then (5. 6) reduces to b + i sin(1 + tan1 )] (5.7) Noting that /i << 1, the denominator for the tuned case is very nearly w 1. 5. 3 Initial Conditions Time is marked from the beginning of pumping. That is, for t < 0 the circuit elastance (Fig. 2. 2b) is SO, and capacitance S = C0. At t = 0 the elastance value switches to the functional form 57

S = So[1 -7 sin (2 w1t+ 01)] sin1t+ )] (5.8) Co [ 1 + rj sin (2 wit + 0)] Since all parameter values discussed in this paragraph are evaluated arbitrarily close to t = 0 we use the notation C ()+ = c+ and C(O) = c (5. 9) V(0)+ = v and V(O) = v Thus c = C = C = C(1 + sin O1) In general, the capacitance experiences a discontinuity, and a resulting voltage discontinuity. The charge associated with C cannot change abruptly, so therefore: c c v = cV and v = v - (5.10) - + + + - C or v+ "v (1 - 1/ sin O1) (5. 11) v is evaluated from the steady-state solution before t = O, that is, for no pumping. Since v7 << 1 we note that the discontinuity in the circuit voltage, v, is small and that whenever 01 = nin there is no 58

discontinuity. Equation (5. 11) is one of the values required to solve Eq. (5.4). Recalling that we are concerned with the parallel circuit of 2. 2b, the initial derivative is obtained from consideration of instantaneous currents at the instant pumping begins. (t = 0) The current in the inductance must be continuous for a constant inductance. If we require the signal current to be continuous at t = 0, then a current discontinuity in C is matched by a current discontinuity in G. Thus d(v c) d(v c ) G(v -v ) dt dt (5. 12) + dt dt Solving Eq. (5. 12) for v+ v (vG v + + - v ) (5.13) C+ - C + c C + + + which after some manipulation using Eqs. (5. 9) and (5. 11) becomes + ([1- Csin 1)1 + (- sin (1-7sinl)(2wlr cos Ol) + v i LJ (5. 14) and since 7 << 1 we note + v( - C 7sin 1 + 2w1 cos 1) + v (5. 15) This is the second value required to solve Eq. (5. 4). 59

5. 4 General Expression for A 5. 4 1 (v - vs+). The terms (v - vs+) and (v - vss+) of + SS+ + SS+ + SS+ Eq. (5. 4) are now evaluated, We proceed directly from Eq. (4. 24), the steady-state solution and include the pump unbalance term of Eq. (4. 3 1). In order to simplify the expression and keep all the information about the various phase shifts intact, Eq, (4. 24) is evaluated using the new angular parameters:. -1 -15, c - t - tan 6 tan j3 -- 1- 01 +tan ~' (5. 16) -I -1 y -= 0 - tan 6 +tan ~' _ U m - v(t) = I {cos(wt +a) - III'I cos(COut + 3) G (1+ 2)( 1+ 2) lul {cos(wt +?) - I I! cos(wt + 0) }] + c iE i sin(2w1t +01) (5. 17) hen t = 0 + 60

s+v.. 1Il4gm (coS a - I I cos 3) S G 2J(1+ )(1+('2) + III (cosT- III'1 cos0)] + c IE 1 sin01 (5 18) The derivative of the steady state is written from (5. 17) and evaluated at u Q t =0. Noting that and- 1. 1 1 -gwm 1 vs+ Sma) 1 [ III (sin c - ]II' sin j3) G ( 1+6 2)(1+(, 2 6) + u I(sin y- III sin 0)] + 2wl c IE I cos01 (5.19) Using Eq. 5. 11 for v+ and evaluating v from Eq. 5. 17 by noting that it applies before t = 0 if we note that gm= 1. II' = 0, and iEpi = O 1 - 1 sin 01 v+~ = [ly? cosa+ II cosy (5.20) +- - u G (1+62)(1+',2) Now using Eqs. 5. 20 and 5. 18 (v+ - v ) is writtenm Note that v is not multiplied by a "gain" factor and that the effect of the discontinuity at t = 0 (manifested by the 77 sin 01 term) is of second order since we have assumed small changes in C due to pumping.

(v - V ) = I [1i cOso a o- g (C a - ICos + Ss+ G (1+6 2)(1+~'2) + lIu1 {cos- gm(cosv- ILI'1 cos0)}] - c E IsinOH (5.21) In Eq. 5. 21 the effect of the initial conditions (v+) and the steadystate expression can be consolidated by noting that: (1- gm)= -IH'i2gm (5.22) (v+ Ss- vI +I(o) =I i (-'I cos a + os + cos ) G (1+56)(l+t') + I (-![I'i cosy + cos0)J - c IEpI sin 1 (5. 23) 5, 4. 2 (i+ - s+). v+,the voltage derivative just after pumping starts, is obtained from Eq. 5. 15. Note that G is very nearly equal to W1 C QO V; = {v - 2 1v (2Qv sin 1 + cos 0 (5. 24) v and v are evaluated from the steady-state solution just before t = O. Since 1 << 1 the last factor of Eq. 5.24 is very nearly 2Qo cos 81 and Eq. 5, 24 becomes: 62

-Wlo + = [III (sin a + 2 cos 0 cos a) G(1+62)(1+~'2) + II I (siny + 2r/ cos 01 cos )](5.25) Equations 5. 25 and 5. 19 are now combined to give: (V+-;SSt. (K - 1s+) [I {(sin a + 271 cos 01 cos a) gm (sina- IiI'l sini )4 + I iu {(sin y + 2r1 cos 01 cos y) - gm (siny- l II'I sin0)}] 2w cIE os (5.26) 1 p 1o1. If the terms representing the effect of the initial conditions are consolidated again with those of the steady-state by using Eq. 5. 22, Equation 5. 26 becomes: = + 1+g....7 [Il t lIn'i sina+sin3 + 2 cos Cos Gll+6+L'z (_ + Ii( I(II' I sin y + sin 0 + n 2 cos 0 cos -2w c I Ep cos 01 (5 27) 63

In the previous equations all the terms of Eq. 5. 4 have been developed in terms of the circuit parameters and independent side frequency forcing terms. Substituting in Eq. 5. 4 the equations 5. 21, 5. 5, 5. 27, and 5. 6 gives a general expression for A A = (v-v ( + a d sin - sin + -(v-+)sin + A - (V -v s+) lLsin(-2c)- / cos 2a + sin(1 + tan 2Q1)] (5.28) A change in notation here will help to reveal the significance of various operating conditions on Eq. 5. 28. 0 is the pump phase angle which includes a phase shift due to circuit loading (tan- 1 ) and another phase shift due to the form of the transient solution ( — ). c.f. Eq. 3.9. Since a is very nearly -- when the passive circuit is tuned to the half-pump frequency (w0 = W1), it is convenient to define: a' - + 4+ (5. 29) a' is approximately the departure of a from the tuned value. Recalling the 0" notation of Eq. 3. 13, Eq. 5. 28 becomes 64

A = (v+ -vs+) sin + o) + Ad Cos 2+ a') + v + v ) cos( 2+ a') -11 cos 2o' + /(- sin 2a' + sin 0'7) When the small phase shift due to the finite Q0 found in 08 is combined with the phase shift of the transient solution due to detuning in the form -,('" -a' + 1 tan 2 QO and the small perturbation of the value of A due to the "A" term of the denominator is neglected we have: A=(v-v+) sin ( —+) 0 ( + v2 ( coss +o) cos 2a' (5. 30) 5. 4. 3 General Form - Quasi-Degenerate Solution. Equations 5. 23, 5. 27, and 5. 30 are now brought together to provide an explicit form for A. It will be convenient to separate the pump-unbalance terms and express them at the end of the equation. 65

A = s +III; gnc- [{il;I (-II1 cosa+cosl)+ II (-1II']cosy+cos0)l U cos 2&' G 1+5)(1+'2) sin ( + or) + ~/. Co( 2 d o) 1+ ii (In1' sin a - sin j3 - i+'i cos 0 cos a) m +II i (II'ii sin7- sino cos 01 cos)> cos(E- + -c E i sin0 tsin + o' + ld Cs 2 T - 2c IE cos Cos + + + gdC~S(+~(' -2clE IcosO1 c U??) cos 2u' (5. 31) In order to establish dominant effects a judgement is imposed at this point in the development. Recalling Eqs. 3. 12 and 3. 13 where Ai, Ig, and [d are defined we note that id << 1. In addition the term 2r < 1. d d IigI Neglecting the contributions of these terms, which produce only very small phase shifts, the remaining terms combine using the sine of difference angles identity. Introducing a new angle notion the expression for A is: a ( I 2tan C) a (tan tan 6 +tan

A]IT'~ =' AI eI iII'WI sin (a a" - tan-) + sin (a + ") cos 2a' G(1+&2)(1+V'2) + 11l {II'lsin(a- r"- tan 6) + sin(a + r")} +, y1') u u u cos 2or' 2 (5. 32) where y -2-, (" is a pump unbalance factor to be considered later. Now when the sine functions are expanded by separating aQ and a from the rest of the angular expression, and the results grouped by u sines and cosines of af and au: A = II 2 g II isinaI {iH'Ilcos(a"+tan 6) + cos a" cos 2a' G 1+6 2)( 1+ 2) IIQI CcoS { H'iI' sin(a" + tanI 6) - sin a"} + 11 I sina III'lIcos(a""+tan i6+cosa") - 1I cosa {II' I sin (a" + tan-1 6) - sin r"}] - pump unbalance terms (5. 33) In Eq. 5. 33 the signal current magnitudes appear as independent sine and cosine components of the a phase angles associated with the upper and lower side frequencies. A common factor for the sums of the components has appeared also. Performing the cosine operation on (a" + tan1 6) 67

and defining: II" - II' II (5. 34) /1+62 (62) III'i cos(a" +tan 1) + cos r" = (1+' "! (cosa" I ) sino") I ) ost - (1+ IH" s nn I l I sin (" + tan 6) sin a" = (1+ i l)[( 1 i) (-sin a") + -6 )1 cos Ca ] (5. 35) (1- I') 1"1) ff??! s( "+tag r In Applying Eq. 5. 35 to Eq. 5. 33 and noting that ~ 1, )gsin a[( 1 sin osa" I l sin cos cos 2u('?/l+'_ G (lI ~o~a +f1 icos~)I ~ cos +I i coa )-2c I E I 19 i CosaCY~ + a y id + (5.36) 2 u COjJU \ 1+ co cos 2r' Equation 5. 36 is the complete expression for the most significant constant, the coefficient of the growing exponential term. After a few cycles of oscillation before any saturation occurs the only term of consequence will be the growing term. (The decaying term is characterized by the exponent, td' which is much greater than the exponent of the growing term, i.e., Izd >> ig.) In the phase synchronization problem we are interested in "A" 68

under two conditions. The one is for the signal excitation, which in this specific instance is a current at a frequency of exactly w1 radians per second. This is the completely degenerate case and is found from the foregoing expressions by letting 0' = 0, and noting that ]II = II = ] II/2 and 0f = 0u. This degenerate condition is equivalent to coincidence of equal side frequency source currents at identical phase positions. The other condition of interest is that of the value of A due to noise within the pass band where the source currents become Gaussian random variables whose mean values are zero and whose variances are identified with the circuit noise. In this study the noise energy will be assumed to be associated only with the resistance member of the load, that is, thermal noise, whose energy distribution within the band is uniform and whose phase angles are equally likely. The derivation of Eq. 5. 36 where the source currents are explicit in (v+ - vss+) and (v+ - vss+), preserves the independent and explicit representation of the source currents. We will see that, for the signal alone, a coherence of phase position of these source currents can be specified and the value of A is strongly controlled by the particular phase chosen. For the noise sources no such phase coherence nor amplitude equality are possible and the input information must be handled differently. 5. 5 Signal A 5. 5. 1 Degenerate Form. In the degenerate case, 69

a=a = a = - s; u s 0s 2 ) and Eq. 5. 36 becomes Il"i (1+ II"n)gm I sl li /0' deg cos 2a' G s 2 1 i I si (S t 2 (r1 + os 2a' II" 2' s ) The above is simplified significantly by noting that the general usage will be in the tuned or nearly-tuned case, i. e., for small 6. It is shown in Appendix VI that the term (cos" a"" Ini sin or + (5 cos a 1 cos2 2r' and +61"1' -, 1+11'I) cos6 cosv" - o sin al" The assumptions made here are that the Q > 10, that gm > 5, and 6 < 0. 5. Although the numbers chosen are somewhat arbitrary the approximation is of the order 4 which is about 1-1/2 percent for the 70

worst case. Hence r 1 -i -2c IE I 0 A __ 4 )m s sin 0 - - tan +. L. y C a deg G s2 2 cos 2j' 2 (5.37) and for the tuned condition this reduces to 2 gm II 01 /01 Ade= G sin 0s m- - 2ce1E y( 2 c") (5.38) deg G s2 c 6=0 It is convenient to relate the relative phase positions of s and 0 to the one which produces a maximum response. Accordingly let WC~ - 0s-~i~ 7 (5 39) $s 2 2 5 9 Using Eqo 5. 39 in Eq. 5. 38, and also that for high gains I II[ 1, 2 gm I Is (1 A deg = G cosp- 2clE I y( 1 " (5~ 40) 6=0 5. 5. 2 Signal Power. The signal power delivered to the tuned circuit (6 = 0) is here defined as Ps. 11 12 II P Ps 2G from which G G 71

When the passive circuit is detuned, the power, P' is reduced, according to 6 I1 12 P Pt s s = ______(5. 41) S 2G(1+62) (1+62) Using Eq. 5. 41 in Eq. 5. 37, A = 2(1 362 ) g sin (s 2 tan1 6-) 2clE 1 0 cos 2a' 2 When 6 = 0, the tuned case, 6=0 m 5.6 Noise A 5. 6. 1 Excitation by Random Variables. The excitation sources, 4. 1, and seen finally in Eq. 5. 36 will now be considered to come from a thermal noise generator. For this situation it and il are identified as independent Gaussian random variables having phase angles a, and ax each of which has uniform probability of occurrence over the range 0 - 2ur. Over the narrowband of = to (C + dC) both at the lower and upper side frequency position these sources will dissipate power characteristic 72

of a normal narrowband noise process. Such a process is described in detail by Middleton (Ref. 114) and in particular the independent components at each sideband are shown to have equal variances (mean-squared values). Thus: <(1l1 sina )2>* = <('IQ cosa)2> read symbol <> as "the mean" <(luI sinaQ)2> = <(IIU cosau)2> (5.43) The components of IIQ I and l Ui which appear together in Eq. 5. 36 form a linear combination of independent Gaussian random variables and are here identified similarly: IIl sina, + I I sina i= (5.44) IQ i cos at +!IU cosa = i2 Writing the variances of i1 and i2 with the aid of Eq. 5. 43, <i1> = <(i f I sin aQ)2> + <(IIUI sin au)2> and <i22> = <(II) i cOS at)2> +<(IUI cos Ca)2> <i12> = <i22> - <iN2> which defines iN.

We seek the statistical description of AU due to the excitation sources of bandwidth d' at ~i', and thus from Eq. 5. 36: 2A~,2> ~ < ] iN 2> os( i _" a ln"l sin a" cos 2a' G + <iN2> + III cos C (5 45) 5. 6. 2 Noise Power, We now identify <iN2> with the circuit thermal noise. First note that per Eqo 4. 1, iN was derived from peak rather than effective value currents. Also each of the components of i1 and i2 (i.e., Il9 sin a.'and II1 1 cos a., etc.) are identified with just half of the noise energy within the band d,'. Thus if we allow dN to be the total thermal noise source power occurring within the d,' bandwidth dN _-<(IIQI sina9Q)2> <(lI I sin a)2> (5. 26N- f I -- = u u (5. 46) ~2 2G 2G from which dN = 2>. From the thermodynamical argument of Nyquist (Ref. 114) dN = 4 KT Idfl (5. 47) Returning to Eqs. 4. 6 and 4. 14 we evaluate df. f 2 (1 2 4g6 (1+)) (5. 48) 72Q 4 g Q0 74

W, (1+ 6' idfI d (5.49) Exactly the same value of |dfj results from the upper side frequency, and thus KT w0 G(1+62 ) d' <iN2 = 2GdN = Q g (5. 50) 0 m It has been shown (Ref. 114 that the total thermal noise, N, associated with a passive resonant circuit is: KT w N = Q (5. 51) Using this normalization Eq. 5. 50 becomes <iN>2 = NG(1+6) di' (5.52) The active circuit bandwidth is significantly narrowed as evidenced by the relationship between the descriptive parameters, 5 for the passive bandwidth, and' for the detuned active bandwidth. Accordingly let us further normalize Eq. 5. 52 in terms of the noise, N', associated with the active band. Let Ndf N (1+62) N' = = (5. 53) df 2 gm from which <iNK1 = 2N'G di' (5. 54) 75

Substituting Eq. 5. 54 into Eq. 5. 45 and again using the simplifications following Eq. 5. 39, 2 <A > = [2(1.-)/ gm lG (1+,2) If each A, is treated as an independent Gaussian random variable which results from the currents i1 and i2 within the interval C' to (W' + d(') the mean squared A which results from all the noise source currents over the whole band is simply the sum of the mean-squared components. <AN = [ -62) 32 2GN (5.56) N,~,. [2( 3 4') m ]2 2N' The integration has been carried out from -co to +oo since noise within dC' at + C' is generated by source currents at +' and -' just as noise within - d' is generated by source currents at -' and +'. It will be convenient to associate the value of AN with the normalized Gaussian random variable, aN, having mean of zero and variance unity. Then (5. 56) becomes: AN =a<AN>2 7 2(1 3 2) ) aN (5.57) 76

and for the tuned case (6 = 0) /2N AN = 2gm G aN ((= 0) 5. 7 General Expression for B The coefficient of the decaying transient term of the general solution 5. 1 is now sought, primarily to demonstrate that this portion of the solution does not dominate observed effects during growing oscillations. From Eq. 5.4 (v+-vs+)(-) cos ") + sin(- + (v -'v + + +)sin( S'n B- ] w, cos 2a' + >i(sin 07 - sin 2.')] (5.58) Recalling Eqs. 5. 23 and 5. 27 - os a") + 11 sin gm cos 2' [I(-I'Icosa+cos) G1+I + Iui (-II'I cosy + cos0)] -clE sin 0 + cos 2a' I [II (II' sina-sing- iI g cos 01os + Ii 1 (HIlsinu sin0 cos 1 gmoicelCosv)]-2cIEp cosoiS (5. 59) 177

Equation 5. 59 can be simplified considerably by using 1a and au as in Eq. 5. 32 and by neglecting the terms arising from the factors /g and 2f -. Both of these quantities are very much less than unity. II'I gm In this vein Eq. 5. 59 becomes 1IflI gm B H= gIn j'1 [I'Icos(aox+' "- tan 16)- cos(a, - r")] G cos 2r' + [IClcos(au+"tan)- [lcI'lcos (a - a")] + 2' os or sin3 n " cos3 (5. 60) Performing the operations within the parentheses reveals the excitation components used in evaluating A. Gcos11111g~~cr' ~[ l I ", 1+ {i II " B= m ~ (1I 1cosai +11 cosau) [(Is ir~ ) cos ( " + (cIilI) sin a" 2'p (cos a sin + sin a" cos3 (5. ) cos 2x' 2 2/ ( Equation 5. 61 is very similar to the general expression for A. The major difference, and a highly significant one is the effect of detuning. When the circuit is tuned A is accentuated, whereas B is minimized. The converse is also true. 78

5. 7. 1 Signal B. Two special cases of B are of interest. The first is the degenerate or Signal B. VWhen C is zero Bdeg becomes: I HI'"I1(1+ I ITI1)g I1 (Ir(HnI-i) 6 IrI B ZD m S COS, (I risl cos'? + sincr' Bdeg = G cos 2c' s 2s- )(1+ 1 11) (1+ I 1!) sin] + sin ( h[- sincr" + coI'l 1 s 2-' + 11 H(i+"i) cos 2c 1E [0' + 2'pft (cos Cr sin' - sin o" cos - cos 2al 2 2 (5.62) When the circuit is tuned B nI(1~In 1) o1\ (-) ~ ~2c IE01 deg G gm os 2 (111 + 2c IEI sin 2 (6=0) (5. 63) For the instances when the signal-pump phase difference is adjusted to give maximum growing component, (A), the value of B is [II I-1 negligible. The appearance of the l+11l 1 term further makes the B term negligible under all but the adjustment of (/ - -2 to nir radians. Although both B and A will be very small under this particular condition they will be of comparable value. 5.7. 2 Noise B. Proceeding as in Section 5. 6. 1 and introducing the random excitation currents i1 and i2 defined in 5. 44 the mean-square value of B at a particular value of (' is: 79

<B,> = [11....]... i[ >4(+ 1I+III,) cosa+ + liI,s in Z"}2 G cos 2 al 1t' C I + <i2> sin a" + 1ln"l cos cr" (5. 64) Again identifying the excitation currents with the therma 1 noise in the passive and active circuits an expression for <BC,>2 identical with Eq. 5. 55 but for one term is produced. The exceptional term is 54 given below and is of the order of -2 compared with the corresponding value of unity for the case of the A value. The term of the B expression and the relative value of < B noise> 2/ <Anoi> 2 is: noise noise <B >2 rt 12 <AN>2 [( i"- ) C'}cos or" + sin a'j N + [-isin + i" + (5. 65) 64 For the assumptions under Fq. 5. 55 the 2 is about 3 percent for the worst case. Thus the value of B produced by the noise is negligible compared with the value of A produced by the noise. 80

CHAPTER VI BISTATE SIGNAL DETECTION BY SUBHARMONIC OSCILLATIONS In this chapter we assume that the parametric device under study is connected to a signal source for the purpose of detecting and amplifying bistate phase information. That is, the signal frequency is assumed locked exactly to the pump source at half the pump frequency, and also that the information is being carried by the two stable phase states of the oscillation. We have seen how the growing oscillation of the device can have one of the two phases represented by either sign of the coefficient, A, of the growing term. Further we note that the signal is most effective in producing a value of A when it bears the same phase relation to the pump as the natural response of the system. In the absence of noise the growing oscillation will always be in phase with the signal. However, when a noise signal in the form of a thermal agitation current is superimposed on the signal there develops a probability that, at the instant of initiation of pumping, (the time the sign of A is determined) the noise exceeds the signal and the growing oscillation can become 7r radians out of phase with the signal. When this does happen an error in detection occurs. The probability of error varies widely with the phase and magnitude of the signal relative to the noise occurring within the system. These effects are also related to system adjustments, and to the pecularities of the actual physical circuit construction.

6. 1 Linear Superposition We have seen in Chapter V (Eqs. 5. 5, 5. 6, 5. 7) how coefficients of a growing and a decaying oscillation arise in a pumped resonant circuit. The exponents of growth and decay have been expressed in Eq. 5. 1 as ijg and td and are the difference and sum of the natural circuit gi and loading parameter pL. In moderate to high gain situations (gm > 5),g is very much less than ad. Thus when an oscillation is observed for many cycles after initiation but before any saturation has occurred the effect of the decaying term is completely lost. The oscillation grows at the rate determined by 1Ag, and with the initial magnitude and phase determined by A. By Eq. 5. 36 A is linearly dependent on the excitation current whether this current is signal or noise. Accordingly the principle of linear superposition is applied and the value of A due to both signal and noise is simply: A = A +A (6. 1) deg. noise When Anoise as shown in Eq. 5. 45 is a Gaussian random varinoise able the noise can conceivably dominate the situation and determine the sign of A for any value of Adeg. 6. 2 Detection Level - Superregeneration We will find that the significance of the noise term is felt only for signal levels of the order of the noise. In moderate Q circuits like those constructed using present day varactor components these noise 82

levels can be at power levels as much as 10 of easily obtainable steadystate oscillation levels. This, in effect, means that the device can be considered an amplifier for signals of the order of the noise up to the level of the saturated oscillation. The general concept is known as superregeneration and is useful whenever very high gain per stage is the dominant design requirement. For the purpose of this study critical examination of waveforms, growing rates etc. have been limited to portions of the quench cycle well below saturation where the linear analytical treatment applies. Using the principle of superregeneration only one bit of information (in the digital sense) is obtainable during each quench cycle, i. e., during the time between initiation of the pump and the final extraction of the information by phase detection at a high level. Thus the information rate is directly proportional to the quench frequency. For a given gain this information rate is limited by the growth rate of the oscillation. Thus we see the familiar concept of gain-bandwidth arising. It is useful at this point to evaluate just what this quality factor is for the device when used in the detection of bistate phase information. The amplitude of output voltage of the device at any tinme is related to the input voltage according to the growing exponential: v = v.,E/g3u1lt vo = v. e g(6.2) 83

where t is the time interval since the quench cycle started. If there is a maximum output voltage v to which the oscillation is allowed to om grow, the voltage gain is then related to the maximum input voltage v. im v co t m = g = Ilm (6. 3) im gv is the same for all signal levels which are below vim and faithfully preserves both signal and noise levels during the growing process. Information can be recovered if data is sampled over each half cycle (Ref. 115) and thus if T is the sampling period and the quench cycle is symmetrical in time the maximum frequency of the data is f = 2, i. e., the bandwidth. t of Eq. 6. 3 becomes T/2 and the m 2T i m gain-bandwidth figure of merit is: i /pg wloT/2 M = 1 e (6. 4) For the tuned case Ag is related quite simply to circuit parameters and adjustments by Eqs. 5. 1, 2. 5, and 3. 10 IJ~ (- ) (6. 5) Mg 2Q0 ( 2gm (Recall that gm is negative in the oscillating region. ) When operating as a superregenerative amplifier Eqs. 6. 4 and 6. 5 show that as the steady-state gain (gm) increases in magnitude 84

(due to pump adjustment) /g is smaller and the circuit gain gv is lower. When the pump is turned off at the end of the firsthalf cycle of the quench the oscillation decays at the natural passive decay rate characterized by Qo and at the natural frequency w. By Eq. 3. 10 pL >> pg and the natural decay is rapid compared with the forced growth. It should be noted in passing that because of the disparity between pg and AL the quench cycle need not be a symmetrical square waveform in time, but could be scaled to the relative values of,g and AJ L' Thus the bandwidth could possibly be increased by a factor of nearly two. It will be shown that vim, for bistate phase signals of the order of the thermal noise produce good probabilities of correct phase synchronization, and that only a few decibels of signal power over the active circuit noise power are required to produce excellent phase synchronization. In addition to these properties the active bandwidth of the circuit is completely adjustable by electrical means, thereby providing a simple method for exploiting the full range of the gain-bandwidth figure of merit in either gain or bandwidth. 6. 3 Error Probability The result of the preceding is to allow the expression of the error probability during the detection of bistate phase information. 85

Using Eq. 6. 1 (linear superposition), Eq. 5. 42 (Ad e), and Eq. 5. 57 (A noise A 2(1 - i36 g[ A_ gm Ps sin s- 2 -tan 2) + 2' aN - 2clEI1+62 y 1 (6.6) 6. 3. 1 Signal and Noise. One case of particular interest will be that when the pump unbalance is zero and the detection is concerned only with signal and thermal noise. We will designate an error whenevery the sign of A differs from the sign of Adeg i. e., the sign of the signal. To facilitate this understanding assume that Ade g is positive, i. e., sin(s 24 -tan1 2 >0 There will now be an error if: 2 sin( 1 tan1 )aNK 2 sinks- 2 tan 2) + 2 aN < 0 or \| sin(s — 2 - tan - ) < -aN (6.7) Recall that aN is the normalized Gaussian random variable, and has a symmetric probability density about zero. (See Fig. 6.1) 86

2 -aN O(aN) = f 0(aN) daN = 1 +aN 0 -aN error probability P' 2 s (1 + 6 2) sin() - tan1 N' s Fig. 6. 1 Error probability density. 87

Thus when aN is positive there is no time when the left member of Eq. 6. 7 (being positive) is less than - (aN). Accordingly the upper limit for the error probability which follows from these assumptions is 0. 5. When aN is negative, - (aN) is positive and there is always a finite probability, as evidenced by the area under the probability density curve of the normalized variable aN, that the value of - (aN) (i. e., the noise) is greater than the signal. This area is tabulated in publications on probability functions and when related to the actual signal-to-normalized noise ratio (PS/N,) of Eq. 6. 7 results in the curves of Fig. 6. 2. The two cases shown demonstrate the effect the optimum adjustments, tuning and signal phase displacement, have on the error probability. From Eq. 6. 7 it can be noted that detuning can be compensated by readjusting the relative phase position between the signal and the pump phase. The effect of serious departure from optimum signal-pump phase alignment is shown in Curve 2 of Fig. 6. 2. sin(0 2 - tan1 ) = 6. 3. 2 Signal and Pump Unbalance. Another interesting case arises when the pump unbalance is the dominant effect masking the signal. Recall Eq. 6. 6 for the pump unbalance terms A = - 2c I E/ IJt72 y(' a) (6. 8) 88

Signal-to-Noise Separation (Db.) - 40 -30 - 20 - 10 0 +10 + 20 +30 50. 30. 20. o~~~~~ 1. QC Q, 2. o 0 o X0 0. 5 Optimum Signal-Pump Phase Curve 1 Curve 2 Fig. 6. 2 Error probability vs signal- - 0. 2 to-noise separation. 0. 1 89

The term of interest here is y(-, (a") which has been carried along in the development in this form in order to focus attention more 01, simply on other effects. y( —, i t' was introduced in Eq. 5. 32 of the general solution development and is: 2 [sin ( + + d cos' + cos 1os + cr (6.9) Following Eq. 5. 31 a judgement was made regarding retention of terms involving "d' and for the signal terms, these effects were deemed very small compared with more prominent effects. For pump unbalance consideration we will be attempting to minimize the effect and thus will need to retain second-order effects to improve that possibility. Doing so in Eq. 6. 9 and using the cosine of difference angles identify: y = [cos - d' + cos 01 cos- + v + i d sin 01 cos + ")j (6. 10) Separating out the effects of a" we find y dominated by cubic terms in 2. Grouping in this fashion Eq. 6. 10 becomes: 0 0 y = (cos O" + Ad sin or") cos3 2 + (sin a" - Ad cos a") sin 2 1 1 + Ad cos a" sin d sin a" cos 2 (6. 11) 90

In the tuned and nearly-tuned situation both /ud and sin c" can be expressed in terms of Q0, since by Eq. 5. 30 17 21~~~~~ ~ sin C" = sin (+ + 2 tan =2Q a is the phase angle in the transient solution developed in Chapter III. Reference to Fig. 3. 2 shows that in the tuned situation ais about - - radians. Further, for reasonable values of steadystate gain, gm, used in this study the departure of C from -, (a'), within the region of oscillation allowed by the pump level, is a few degrees at the most. Thus for the tuned and nearly-tuned situation sin c" is very small compared with unity and simply related to Q0. By Eqs. 5.1 and 3.10, and for moderate gains 1 1 3 1 1.~ 1 1.1 y = cos 2Q+ sin 2-+ Q sin -- (6.12) (the terms neglected being of order 1 2Q91 91

In order to discover the action of Eqs. 6. 8 and 6. 12 some observations must be made about the operating scheme of the device. Focusing attention on 01, the initial pump phase angle, two situations can arise. If 01 is to be a constant some means must be provided to initiate the quench cycle at exactly the same time in the pump cycle for each quench cycle. Recall that the signal which appears in the tank circuit during the time the pump voltage is disconnected from the SHO bears a fixed-phase relation to the pump. In a sense the pump is a master clock of the system which is always running. When these "'means" are used the quench mode is described as coherent (CQM). The alternative is to allo~w complete phase freedom between pump and quench. This quench mode is described as incoherent (IQM). Each operating mode has its peculiarities as they affect signal detection properties. They are discussed in what follows. 6. 3. 2. 1 Coherent Quench Mode (CQM). Assume 0! is constrained to be a constant for each quench cycle in Eq. 6. 12. The "means" used to accomplish this are incidental to the consideration and will be discussed in the next chapter on the experimental work. Under this condition A is determined in the absence of signal or noise, simply by system adjustments. If c, the fractional capacitance unbalance, is built into the system by combinations of varactor assymmetry and bias assymmetry the value of A is determined by the particular value of 01 chosen. As various values of 01 are chosen the variations of A are 92

demonstrated by what we will call here as "A Contours in -2 (Fig. 6. 3). The shape of these are determined by the factor y of Eq. 6. 12. The 3 01 01 primary effect is the large cos 2 lobe at -2 = n X radians and the 01 iT gentle slope in the vicinity of = (2n + 1) 2 radians. 6. 3. 2. 2 Errors in CQM. We can add signal information to the pump-imbalance using Eq. 6. 6. To use a quantity similar to the noise power the pump-unbalance power, Ppu, is (c E IE)2 Ppu 2 G (6. 13) and for signal and pump-unbalance 2P (1+62) (1- 3) 1 -1 1 A= 2 pu+ sin(s 1 - tan ) (6. 4) Presumably the phase relationship between signal and pump can be maintained constant as 01 of y is varied. Under these assumptions the sign of A is determined by the difference between y and the rest of the bracket quantity. The general effect is to offset the baseline of y by increments depending essentially on Ps/Ppu for magnitude, and sin (Os 2 ) for sign. Thus the sign of A is either all correct or all incorrect. The presence of noise confuses the situation if all effects are considered simultaneously. However, the adjustment of y to the 01 value of zero by selecting the appropriate - offers a means of separating these effects. Indeed, as we shall see in discussing experimental results, 93

the appearance of a spurious signal would have made limiting thresholds much more difficult to measure had it not been for this ability to inject a compensating condition by using the variation of 01 in the coherent quenching mode. 6. 3. 2. 3 Incoherent Quench Mode (IQM). When means for establishing each quench cycle the same value of 01 in y of Eq. 6. 8 are not provided in the system incoherent quenching prevails. This was the situation in the early experimental work beforeany understanding of the importance of the pump unbalance situation had been achieved. The treatment follows the analytical forms for coherent quenching when the angle 01 is allowed to vary from quench cycle to cycle. To visualize the variation of 01, recall from Eq. 4. 30 the pump voltage, el, for the first quench cycle. e1(1) = IE Isin(2w1t + 01) (6. 15) P where the parenthetical (1) denotes the first quench cycle. The quench 27T angular frequency is w and thus after a full quench period (i. e., q q seconds) later the pump is turned on again, e1(2)= I Eplsin 2 w(1t + c + 0 where the initial phase angle is 01(2) =2 ) 2ir+ 01 94

In general, the n-th quench cycle will have the initial phase angle. 2 01(n) = (n- 1) 2 ~1 + 01 (6. 16) q 6. 3. 2. 4 Error Probability in IQM. The effect of the pump unbalance in incoherent quenching is to mask the signal just as effectively as does the noise. Moreover, the pump unbalance effect can be considerably greater than the noise and as we shall see has characteristics which set it apart from the noise. Qualitativelv one notes 2 W1 from Eq. 6. 16 that when is not an integer all values of the first Cq term are possible. This is equivalent to sampling in time all the values of in Fig. 6. 3. Wherr no signal is present the sign of y is equally distributed between plus and minus. This 50-50 split is exactly the same as when thermal noise determines the phase in the absence of pump unbalance and signal. We will now develop an expression for the error probability of signal detection for this incoherent quenching mode. Return to Eq. 6. 14. Let us decide as was done in the case of noise that an error occurs when the sign of A differs from the sign of the signal term. In contrast to the noise situation the masking quantity, y, for pump unbalance has a maximum value. Thus there is a minimum value of Ps/Ppu for which there is zero error. Identify this minimum value as po and note that this must be just sufficient to make the left term of the bracket expression equal to unity, since Ymax 1. 95

1. 0 0. 8 Fig. 6. 3 A Contours in X.:6 y cos 1 sin3 -+ sin / 24 LO (y subscript identifies value of Q) Y) 10 0/ Y10 2 C 0 120 > 20 40 60 80 00 140 160 180 200 220 24 -780 300 320 340 360 2 0 Degrees -.4 -.6 -. 8 -1.0

Accordingly 4 gm | sin(s 0 2- tan~ \6)= 1 (6. 17) gm PO s 2 2 Now assuming that the system adjustments ituning, gain, H(is ~~ 2)are being held constant and that the signal terms are positive, an error will occur when: P /P S PU- < y (6. 18) Po As P /Ppu is reduced below p, decimal fractions are generated. When this is expressed in decibels the correspondence is given in Table 6. I below. P /P P /P ~s/ pu in db s pu 0 1. 000 -0. 1 0. 989 -0. 2 0. 977 -0. 5 0. 943 - 1 0. 896 - 2 0. 796 - 5 0.563 -10 0. 316 -20 0. 100 -30 0.032 -40 0.010 97

The error probability is obtained by applying the fraction of Table 6. I, and the criterion of Eq. 6. 16 to the particular A contour of Fig. 6. 3 applicable to the system construction and adjustment. Although the analytical form for y does not lend itself to this manipulation.the graphical approach is very direct. For a particular ordinate Ps /)puthe width of the A contour is measured in units and Po). The fraction is the error probability. compared with 27r radians (3600). The fraction is the error probability. Figure 6. 4 presents these calculations based on the two contours given in Fig. 6. 3 and a third one based on a spurious signal ys present in the Y1o contour. This latter represents the kind of situation which might be encountered in an operating situation. Two significant differences separate the effects of thermal noise from pump unbalance as seen in incoherent quenching. The first is the slope of the error probability curve for low-error rates. The thermal noise, characterized by a Gaussian distribution presents a finite probability of error at increasing ratios of signal-to-noise. As evidenced by Eq. 6. 15 the error probability for pump unbalance approaches a definite value of signal-to-pump unbalance as an asymptote. The other basic difference is the production of a jog in the curve of error probability vs. signal-to-pump unbalance due to the condition of zero slope at = 2 2 radians. Small increases in signal-to-pump unbalance cause large changes in error probability. 98

Signal-to-Pump Unbalance (Db.) 30 -4o- 20 -10 0 1 —-': —- " —.-Y. -<.. I'! ~50 30 20 10 5 Q 3 (Normalized to zero error) 2 I Fig. 6. 4 Error probability for signal and 0 pump unbalance in incoherent quenching. o _1..5.3.2 99

CHAPTER VII EXPERIMENTAL WORK 7. 1 Objectives Related to Study The objectives of the experimental phase of the study are summarized by the following four statements: To demonstrate: 1. Basic properties of a parametric amplifier operating in the quasi-degenerate, and degenerate modes. 2. Properties of the associated parametric sub-harmonic oscillator. 3. Quantitative relationships of phase synchronization by a signal in the presence of thermal noise, including the error probability. And to determine: 4. The effects of incoherent and coherent quenching as they relate to the phase synchronization process. 7. 2 Circuit Descriptions 7. 2. 1 General. The experimental layout for the study is shown in Fig. 7. 1. The sub-harmonic oscillator, SHO, in the center receives energy at angular frequency, 2wl, through the quench switch from the pump oscillator. The nominal pump frequency for the study was 106 Hz. The quench switch, controlled by the quench oscillator allows periodic application of pump energy to the SHO. The pump oscillations are counted down; by 2 to provide a signal frequency; and then by 128 more to provide 100

Scope R. F. Millivoltmeter Phase Ou tSpectrum Shift..... Analyser Z3 Pump c F 2 1 ench Sub- imBuffeer R-F sc c Harmonic Limi. 1 A Switch e Amplifier VTVM coQ SYNC Fig 71Bo2 Scope oISYNC Phase DET 101uench ~~Reference SYNC Q uench Delay in Osc. Tek 105 Counter Fil SYNC Out Fig. 7. 1 Block diagram of experimental layout for phase synchronization study. 101

a fixed time relationship between the start of the pump and the start of the quench. Attenuators and phase shift circuits in each line allow independent adjustment of the phase difference along the appropriate line. The output of the SHO is fed through a limiting circuit and buffer amplifier and on to the various points of observation such as oscilloscope, VTVM, etc. It is interesting to note that the input and output are connected to the same point in the SHO circuit. The common point functions as the input before pumping starts and as the output after pumping starts. Time references for I and wQ are taken from the pump and counting circuits. Of particular significance in the entire circuitry is the flow of information to the right, through the SHO, and, because the ultimate signal levels are so low compared with the operating levels of the circuitry, the paths back to the SHO. The backward paths provide undesired signal coupling and tend to mask the final measurements. Each block is discussed in detail below. 7. 2. 2 SHO Circuit. The two circuits tested in this study are given in Figs. 7. 2a and 7. 2b. The basic configuration is parallel resonant, being tuned approcimately to. 5 megacycles per second. The variable reactance elements are D1 and D2, which are Pacific Semiconductor Type 27E "Varicaps. " This varactor has a nominal value of 27//uf at -4 volts and a range of 7- 70/zif. The two circuits differ in the backbiasing technique. The original circuit uses Mallory RM1-R cells having a fixed output of about 1. 35 volts. The pump-unbalance study revealed 102

Fig. 7. 2 SHO Circuits. I I J B1 B2 iI Signal-Output I Shield L ~1Pump BNC (a) Fixed Bias-initial design Signal I BNC I B1 B2 1 1 I i A? ~~' Shield () Differential Biast, for variable pump unbalance.

small difference in D1 and D2 which are compensated by the differential bias circuit of Fig. 7. 2b. In this latter circuit it is possible by adjusting R2 to balance out the pump voltage appearing across the circuit. T1 is a small ferrite toroidal transformer designed to give a 1:1 "unbalancedto-balanced" transformation. A separate signal BNC is incorporated in the circuit of Fig. 7. 3 in order to establish within the shield box the final attenuated signal level. The resistance R1 is an arbitrarily applied load resistance. Its value is low enough to dominate the usual loads of high impedance measuring devices and to control the aggregate circuit Q. Measurements made on the assembled circuit are given below: Resonant frequency 482. 2 KHz Loaded Q 10.8 Equivalent circuit resistance 14. 8 Kilohms (Conductance = 67. G6mhos) Circuit inductance. 45 mhy Equivalent Z3 565 Kilohms The complete assembly is enclosed in an aluminum shield box as shown in the closeup, Fig. 7. 3. 7. 2. 3 Quench Switch. The quench switch is designed to operate with the Tektronix Type 105 square-wave generator to produce electronic interruption of the 1 megacycle pump energy into the SHO. Fig. 7. 4 gives the details of the circuit. The transistor chosen is a 300 mc unit which is capable of shutting off the pump well within a single cycle. The external positive voltage provides the back bias necessary to assure Q1 being "off" when the pump is to be "on. " The characteristic of the Type 105 square-wave generator provides essentially zero volts during part of the cycle, and an adjustable negative voltage during the alternate part of the cycle. 104

i:: -:;PS ra~ _,. —:,,.r Fig. 7. 3 Sub-Harmonic Oscillator Assembly 105

Pump in Switched Pump BNC 180 Output BNC IN 740 2N71 1 B Quench in BNC 1, 33k~ R 3F+ 4 3v dc I [ 10 Fig. 7.4 Quench Switch 106

7. 2. 4 Quench-Pump Synchronization. The need to provide fixed relationships between the start of the pump and the start of a quench cycle necessitated the use of a counting-down process to derive the quench trigger. The first scale-of-two provides the angular frequency w1. Harvey-Wells modules provided this and the remainder of the flip-flops required to derive a trigger pulse every 256 pump cycles. In order to allow this trigger to start the quench cycle at any arbitrary point of the pump cycle a continuous delay of 0 to 2 microseconds is used (HarveyWells Delay C Module) between the counters and the external trigger point of the 105 square wave generator. In some parts of the experimental work extremely low quench frequencies were used, and it was found helpful to provide external trigger pulses less often. For example, the count of 256 provides about 3750 trigger pulses per second. If the desired quench frequency is 100 Hz there will be 37+ trigger pulses each quench cycle. This large number is sufficient to cause considerable difficulty in effecting the solid synchronization needed. An additional count of 2, 4, 8, or 16 was used for the very low quench frequencies so that, at most, only a few trigger pulses occurred each quench cycle. 7. 2. 5 Signal Generation. The seed signal is at exactly w1 radians per second and is derived from the Harvey-Wells modules counting down the pump frequency of 2w1 radians per second. The phase relationship between pump and signal is a dominant parameter [( S 2- ) as per equation 5. 38] and is adjusted, in steps and continuously between steps, by the delay mechanisms shown in Fig. 7. 5. The various resistances 107

2w Point of 1 measurement (- 20MV. P-P) Continuous Delav l Cl Step 330 Atten. 5 5K Counter -V Delay 1 220 390 50 50 Fig. 7. 5 Signal Attenuation and Delay. 108

used are to accomodate loading levels and impedance requirements of the counting and delay components. The step delay box is Type 602 from Advance Electronics Co., and provides.25 micro-second per step in 10 steps. This amounts to about Tr/ 4 radians of the signal frequency per step. The continuous delay is Type 509 also from Advance Electronics and has a full scale delay of 1 micro-second. The signal level is monitored at the input to the 100 Db attenuator. At this point the square wave from the Harvey-Wells counter is reduced about 46 Db from the original 4 volt, peak-to-peak value. The value of signal at the output of the attenuator is reduced by the attenuator setting and considered further as a square wave entering the resonant circuit of the SHO via 565 Kohms located within the SHO box. Since the resonant impedance presented by the SHO is of the order of 15 Kohm (<< 565K) the signal is injected essentially as a current. The resonant circuit develops a voltage at only 1!. The details of the calcuations follow later in paragraph 7. 4. 3. 1. 7. 2. 6 Buffer Amplifier. The output of the SHO, also being the input, must be carefully isolated from all the measurement and observation circuits since these circuits use large signals having components at signal frequency as time references. The isolation device used was the Hewlett-Packard Model 450A amplifier operating in the 20 Db (x10) gain position. The output is limited to ~ 10 volts. In order to insure that the input never exceed 1 volt a pair of silicon diodes was used across the SHO for some of the measurements. 109

7. 2.7 Phase Detector. The final question during any one quench cycle is "which of the two possible phase states actually occurred?" During the initial stages of the investigation it was sufficient to answer this question by observing the SHO output on an oscilloscope which was externally triggered by a voltage at c1 radians per second. As the number of trials required became greater and as the ability to make the trials rapidly was realized through the use of electronic quenching a new scheme was required. Thus the particular variety of phase detector suitable to this problem was developed. It is shown in block diagram form in Fig. 7. 6 and schematicly in Figs. 7. 7 and 7. 8. The input oscillation is isolated by an emitter follower, amplified and limited to a standard voltage form similar to that used in the Harvey- Wells modules. At the detector stage the signal voltage is compared through a simple "AND" logic circuit with a reference phase taken from the counter which supplies signal information. The output is isolated and inverted by a final stage. When the signal and reference disagree in phase the output is a 15 volt square wave at frequency co1. When the signal and reference disagree in phase the output is zero volts. When filtered by two simple R-C stages the average output is -7. 5 volts for agreement, and zero volts for disagreement. In the case of rapid quench cycles at frequency fq cycles per second, a counter observing the output of the filter will count successes, fs. The errors are, of course, f - fs After a great many trials we see that q s the error probability will approach the average value of f - f. 110~~~~~~~~~n

Reference Input Signal Input 7. 6 Phase deter blc iarm Emiter Amplif ier LmtrDetector InetrFilter Follow er Swit ch Switch Fig. 7. 6 Phase detector block diagram.

[7I i -15 -15 S $_ A -;_ W4 -4 - 4 1 — 4 1 SigLnal 3i4. 7k Inut 22k coax —1 [ [ IMiniatdre R1 D BNC 4. -7k -2 JR 6 - 1I~~1 R -4 Refe 8 1500 R2 12.l R14 D 6 44. 5 3 - I f - GD_____ SilBNC LI_ _ + I 74 -1 in 5 - G I20 I I Ou i 15 1 Ref er-nce e i Inp"t R IiC C 4 5 R8 1500 R 14 [. L1 L 12 L 3 GND Shield'V~A,- ~~~~~D's IN 740 -15 in -4 in +10 in Q(- 2N2401 Fig. 7. 7 Phase sensitive detector schematic. Reference!~- 1 ",~112

From phase 51, 5k detector R' To 002 C _ 2 counter 100k Fi. 7. 8 Detector filter.

7. 3 Laboratory Methods 7. 3. 1 Setting the Value of gri In all of the experimental work where quantitative measurements are made it is necessary to know the value of grrl the steady state gain term. The method used exploits the appearance of the gain in the growing exponential. After the decaying exponential term has disappeared we note from equation 5. 1 that the envelope of the circuit voltage is: V Ae bLgw1 (7. 1) envelope If we make two observations of the envelope the ratio of their amplitudes at times t1and t2 is: V2 g1 (t2- t1) (7.2) V1 Now arbitrarily select V2/V1 to be a simple value, 2, and solve 7. 2 for the spacing in time, (t2-t1): V2 log V 1 (t2-tl) = (7.3) 2 gwllog e /g contains the gain term and is found by noting: g = -L

For the tuned case =) 1 MI -4 A1L 2Q0 (7. 4.) sI Q 1 I 2 r gm 1 lIin2 After appropriate substitutions (g 4Qo gm (recall that gm is negative in the oscillating region), and (7. 5) 4Q0 log 2 (t2 l)x2 gm w1 log e Using the parameters in 7. 2. 2 the value of the time interval (t2 - t1) to double the envelope is: (t2 -t1)2 = (988 106)gm (7. 6) (t2- tl)x2 101 gm micro-seconds. This is remarkably simple to perform in the laboratory. If a gain of 10 is desired the circuit is carefully tuned, and then the pump level is adjusted until 100 M/ seconds are required to double the envelope. Care must be taken to insure that the measurement is made well below saturation. Fig. 7.9 showing the increasing waveform on a time scale of 20 /J sec / cm demonstrates the SHO with a steady state gain term of 10. 7. 3. 2 Measurement of g. Evaluation of / follws from 7. 4 115

time base 20 gsec/major div. Fig. 7. 9 Evaluation of Gm

For the tuned case T 1 MIJ T 4 M/L 2Q0 (7. 4.) Q0o 1 2 gm - n2 After appropriate substitutions g = - Qogm (recall that gm is negative in the oscillating region), and (7. 5) 4Q0 log 2 (t2- t1)x2 = gm c-1 log e Using the parameters in 7. 2. 2 the value of the time interval (t2 - t1) to double the envelope is: (t -t)x2 = (988.10-6) gm (7. 6) (t2 t1)2 101 gml Imicro-seconds. This is remarkably simple to perform in the laboratory. If a gain of 10 is desired the circuit is carefully tuned, and then the pump level is adjusted until 100 Ai seconds are required to double the envelope. Care must be taken to insure that the measurement is made well below saturation. Fig. 7.9 showing the increasing waveform on a time scale of 20 MI sec / cm demonstrates the SHO with a steady state gain term of 10. 7. 3. 2 Measurement of fL. Evaluation of f follws from 7.4 115

1 = g+ L = IL( 2gm L= 1. 0463 (7. 7) UL = 2(10 8) A =.0463(1 + — l ) The gain values used in the experimental work ranged from 5 to 100. Over this range I varies only slightly from. 051 to. 0466 7. 4 Results The following paragraphs relate directly to the objectives of 7. 1. 7. 4. 1 Basic Properties of Amplifier. When the configuration of Fig. 7. 1 is pumped below the level of oscillation the seed signal is amplified. If the seed sig'nal at w1 radians/sec. is in the preferred phase position it will be amplified depending on the exact pump level. If the signal is 2 radians from the preferred position the signal is attenuated. The theoretical description for the tuned degenerate case is obtained from Equ. 4. 24 and is: vs(t) = gm lcos(1tO )- + ) cos (l 1t+ 01 0) 6 = 0 (7.8) ~' = 0 The preferred signal phase by inspection is where: ( 10 s) S + ff (7. 9) (oI o)117o

In this condition a maximum steady-state term results and is: g vss (t) = G I I (+ II) cos(wlt + s) Ssna s 1G. max. 2g G II I cos(t + 0s) (7. 10) (since H 1) Recall from Eq. 4. 1 that the degenerate signal is is(t) = Il I cos(wlt + Os) and that the pump voltage producing the capacitance variations from Eq. 4. 3 is: el IEpI sin2(co1wt+ 0 1) Using the constraint for maximum response, i. e., (7. 9) el = IE I sin 2(w t + s + ) (7. 11) resp. max. Fig. 7. 10 demonstrates these time relationships. Note that the signal and steady-state response are in-phase and so related to the pump that the response nodes (wt = + = + nir) occur at positive slope nodes of the pump and the response peaks (w1t + S = nfr) occur at negative slope nodes of the pump. Checking Eq. 7. 11 for (wlit+ s) = (2 + n7r): e = I E I sin 2(n+l1) = 0 (i.e. nodes) (7. 12) r response nodes

I 11111!11111i1 II...m.ui III Fig. 7. 10 Maximum Steady State Response I Pump. 5v/cm II Signal. 05 v/ cm III S.S. Response. 2 v/cm voltage gain = 10 (Gm = 5) time base. 5g sec/ cm

e1 = 2-wl1E pcos 2(-wlt + Os + ) el =21IEplcos 2(n + l)4 = 2wlEp{I (i. e. positive slope) (7. 13) and for (wcot + ) = nr e =E IEsin (2n + 1) = O(i. e. nodes) (7. 14) response peaks el = 22olEplcos (2n + 1)r = -2wll1Ep (i. e. negative slope) (7. 15) response peaks When the signal is adjusted signal radians with respect to the pump the output is a minimum. The theoretical condition by inspection of Eq. 7. 8 is (0 1- s)= 0 (i. e. 0 1 = 20 ). Accordingly the steady-state response becomes VsS(t) G ITl I(1 -II) cos(olt + s) min. and recalling Eq.. 4. 14 I1 I 2G cos (t+0 ) (7.16) With respect to the signal the pump has shifted so that eresp =IEpl sin 2(cot + 0 ) resp. p s min. 120

Referring to Fig. 7. 11 for this condition, response nodes (w 1t+ s 7T = + r7Tr) correspond to negative slope nodes of the pump, and response peaks correspond to positive slope nodes of the pump. The minimum condition here is actually a loss(g = 1/ 2). Small departures from the minimum phase position present signals having a component in the preferred phase and thus experience the gain of the system. The phase shift of the response with respect to the pump is apparent only when the signal in the preferred position actually goes to zero. The degenerate amplifier acts like a phase selective amplifier, — amplifying the signal component in the preferred phase and attenuating the signal in the quadrature phase. The ratio of the maximum to minimum responses due to signal phase change alone is of some interest. Using Eq. 7. 10 and 7. 16. v ss max. = 4g v m ss min. Since gm can be arbitrarily large, the signal phase has a profound effect on the steady state ouput of the device. * Figs. 7. 13 and!7. 14 demonstrate the same properties described above using a much higher adjusted circuit gain. The device is on the threashold of oscillation and the effect of the phase position of the signal is very pronounced. In stark constrast with statements of Bura in Ref. 17.

I *EEUElEEE Erl.l..EE..... III Fig. 7. 11 Minimum Steady State Response I Pump. 5 v/ cm II Signal. 05 v/ cm III Response. 05 v/ cm voltage gain Time-base. 5 gisec/cm = 1/2 122

III iese/llti n = Fig. 7. 12 Maximum Steady State Response - High Gain I Pump.5v/cm II Signal. 05 v/ cm III Response. 2 v/ cm voltage gain 72 (Gm = 36) Time base. 5 usec/cm 123

I i UI........ Fig. 7. 13 Minimum Steady State Response - High Gain I Pump. 5v/ cm II Signal. 05v/ cm III Response. 2v/ cm Time base. 5/t sec/cm 124

wfiiuuuulYrrii II N KI Fig. 7. 14 Sub-Harmonic Oscillation in Growing Region I Pump. 5 v/ cm II Signal.01 v/cm III Response. 5 v/ cm, 1/ 3 saturation value 125

7. 4. 2 Properties of the Associated SHO 7. 4. 2. 1 General. As the pumping parameter III' increases toward unity the amplifier gain increases toward positive infinity. As the point 1 is passed g becomes negative and decreases in absolute value as III'I increases. As shown in Fig. 4. 2 the region beyond unity is the oscillating region, and the transient solution possesses a positive exponent. When a dominant signal is present in the circuit before the oscillation begins the oscillation will assume the phase of the signal. Fig. 7. 14 shows the signal and grown oscillation in phase and related to the pump exactly as was the steady-state response. The magnitude continues to grow until the conditions for growth are balanced by nonlinearities in the system. The mechanisms for this large-signal behavior are amply treated in other works (Refs. 21, 25 and 46). It is sufficient to observe here that the final phase position slightly lags the transient portion of the oscillation. This will be evident in some of the oscilloscope pictures which follow. 7. 4. 2. 2 Bistate Phase Positions. Equation 5. 36 gives the coefficient, A, of the growing exponential. The sign of A is determined by the instantaneous sum of signal effects whether these effects are desired or undesired (noise). The magnitude observed at any point in the growth interval depends on the initial magnitude. A noise input, having a distribution of magnitudes and signs, will cause a cloud of rising exponentials. Each has its nodes at the same points relative to the pump waveform. This is pictured at three successively higher pump levels in Figs. 7. 15 to 7. 17. 126

III Pump 1 v/ cm II Output 10 mv/ cm 127

II Fig. 7. 16 Moderate level amplification of noise. I Pump 1 v/ cm II Output. 2 v/ cm 128

IIFig. 7. Sub-harmonic oscillation. Fig. 7. 17 Sub-harmonic oscillation. I Pump 1 v/ cm II SHO 2 v/cm time base. 5b sec/cm 129

Fig. 7. 15 trace I shows the pump in the two states controlled by the quench switch. Trace II shows the noise amplified only slightly, i. e. an oscillation is not occurring because the pump level is not quite high enough. The noise, occurring with random magnitude and phases experiences gain in only the preferred phase. However both signs appear. Fig. 7. 16 has a slightly higher pump level, and consequently higher gain. Fig. 7. 17 results when the pump level is increased just to the oscillation state. The darker extreme traces of the SHO trace show the oscillation has saturated. The background cloud includes all the rising exponentials of the superimposed traces. 7. 4. 2. 3 Effect of Pump Level on Output. For the tuned case the magnitude of the positive exponent in the growing portion of A is given by equation 7. 5. For small variations in pump level within the oscillating region the more pronounced effect is the inverse relation between pump and gain, and thus the direct relation between /g and pump. This is demonstrated by the waveforms in Fig. 7.18. 7. 4. 2. 4 Effect of Tuning on pg. Fig. 3. 3 for a constant pump level, r7, shows the variation of p with detuning. Recalling that a = 1 is the tuned situation, the approximate symmetry of the plot shows decreasing M for detuning in either direction. For the experimental work the tuning was accomplished by varying the pump frequency to produce the fastest rise. This procedure is very sensitive to detuning and becomes very simple to perform. 7. 4. 2. 5 Effect of Signal Level on Output. When the signal level is significantly above the noise the major manifestation when viewing an 130

*I~~~~~~rnu..in...umin~~~~~~~~~~~~~~~~~~~~~~ininrninin~~~~~~~~~~~~~~~~~mmrninrn..J~~~~~~~~~...... Il Fig. 7. 18 Gain Variation with pump level. I Gain 10 II Gain 5 III Gain 20 131

entire quench cycle is the control of the location of the rising portion on the time axis. The value of A determinesthe initial value of the exponential, the rising shape being the same. If the oscillation is allowed to saturate a simple model (Ref. 11) shows the quench envelope to be proportional to the logarithm of the signal. Thus the term "logarithmic mode" of operation is understood. When the signal is varied over a 60 Db range in 10 Db steps a linear variation of the location of the rising portion results. Fig. 7.19 demonstrates this property. For a -60 Db signal, (below an arbitrary reference) noise influences the start of the exponential in a random manner as evidenced by the haze on the exponential. 7. 4. 2. 6 Effect of Signal Phase on Output We have seen in paragraph 7. 4. 1 how the steady state gain depends upon the phase relationship between pump and signal. Now we see the corresponding variation of the output of the sub -harmonic oscillator for signal phase variation. In Fig. 7. 20 if we describe "0" phase as the phase adjustment to give maximum response (as per equation 5. 40), variations to 2 produce output variations as shown. Note the appearance of the noise when the signal phase is in quadrature to the preferred phase. 7. 4. 2. 7 Coherent Quenching. A development of particular significance to this study has been the matter of coherent quenching. Coherent quenching is achieved when, on successive quench cycles, the pumping begins with exactly the same pump phase angle. For this case not only is the value of A determined, but also the instantaneous value of the growing oscillation is determined. Thus each cycle of the oscillation will track on 132

Db. -1 I -20 III - 40 V -50 VI -60 VII Fig. 7. 19 Output variation with signal level. 133

M fl,...~r...... z z z......~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... Fig. 7. 20 Outpu variation w SiSi:Ei ii:7 E E: Eith i:i Esi:i.E d iS E f:s P rE ig a Wi psi i i i i i i i.iiiiiiii.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i-iiii~iiii r: rj-i- _ ~.::::::: _::::::::::: / ii iii ~ ~/4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i,-ij T::: itH tH 0 iT;: i 0 0 0 i ff; t:! S:::::::::E t E i::::: fiX iL: 00::f:: i T:;::E t:: E:#; E i 0 75 i~~~i T ti::::::: --::: -: -: -.- j.. -: 1 i: --—: -.. —: -' — - S t:: i iCEfit i~i i i70:i: ii7-i'::::::- iW:i:i i:iii -i;: i i- iS;ik.-.....- ig;.:-: S: t.:E: S-1'4W::':-:-:. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i:- -_:- i-: -:: --:i: i: -: -:k 7 f i-:- --- -- - -:: 7 ii- —: --: ---- ---- --—:- ---: —--— i: iiii ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: -F::::::::i: ~~~~~~~~~~I-I:E 1111 - _. E::::: 0. — 5:::diE i:i i::-: if::f:0i:E!i00 -fE;SCEffigi:7: _ g:: ~i (:4:;!: - -: -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- ---- -- mmff~~...................................................................................................................................ii_:(:_i ii: i~iii~iii i: ii::_::::_-i~it::i:0:!::S ~ ~::::f d::::::: ii:::: i-:_ii:i:::i;::: -: thf fE!: E E::E 7::::::::: iE:: i~ ~iE:::::::::i~t::EREE::i! i::i~i~iEEREL iE:::::i::i:::::::::::E:::::::::::::::::::::::E::::::.:::::::::: -:::::::::::::::::::::-::::::.::::::: ~~~~~~~~P: r::::i:::E::::::-: -:-:::::::::::::::f::E iEV-i:::::::::::E::: i:EC: __ -..: —.-..::-::::E::::::i::::::::::::i:::::::::i::i:i:: i::::::::::E::::E: IV3 / - - 7:: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.-:j:-:::::-::1~ 1 Cp _:::i- _.... _... _E. v,,E-::.:.ft, 41.............. -:::_....; -:_::::-:- -ii -:_-::i —:i. i:-i11 _ _~~~~~~~~~~~~~~~ V~~~~~~~~~~~~~~~~~~~~~~~: —- i~-iii-i ~r/2 Fig. 7. 20 Output variation with signal phase. 134

superimposed traces. The following series of traces demonstrates the phenomenon. Fig. 7. 21 (a) shows three entire quench cycles. The pump is not completely "off" during any of the cycle but is well below the level required for oscillation. The brightened interval at the start of the second burst of pump energy is selected for display in (b). This waveform is made on repetitive quench cycles and we note the identical pump initiative phase each time. The SHO trace immediately starts to show an oscillation. If the brightened portion of (a) is moved to the right to the saturated portion the resulting delayed trace, (c), is perfectly superimposed. Fig. 7. 21 (d) shows both stable states superimposed by making one exposure with a signal in one state, and a second exposure with the signal delayed mr radians. When the oscilloscope is synchronized with the pump (not with the quench as in 7. 21) both pump states appear in the pump trace and the unusual superimposed trace of Fig. 7. 22 results. The growth of the oscillation is exactly related to the framing rate since the quench is ultimately tied to the pump, and thus the sync pulse. If this coherence is not established the saturated region has the same envelope as Fig. 7. 22, but instead of superimposed traces, the envelope is filled with the continuous range of possible values of the same phase. (much as in Fig. 7. 17) 7. 4. 3 Quantitative Relationships, Signal-to-noise. This section concerns the experimental demonstration of the relationships set forth in the theoretical derivation of paragraph 6. 3. 1. Here the error probability 135

Fig. 7. 21 Coherent Quenching (a) Entire quench cycle II (b) II Pump initiation 136

II I Oscillation II "I, 22~~~~~~~saturation *EMB EB E Two stable states n — Em. (o Fig. 7. 21 (cont.)

Fig. 7. 22 Saturated State-Coherent Quenching 138

for phase synchronization of the oscillation in the presence of thermal noise is related to the normalized signal-to-noise ratio, or as is more commonly used, the signal-to-noise separation in decibels. Looking at Fig. 7. 1 we see that the experimental problem is to measure the signalto-noise ratio and then to measure the error probability as evidenced by sufficient trials to give statistical confidence to the result. Each phase presents particular problems of absolute measurement (accuracy) and repeatability (precision). Illustrative of these problems, a sample data point will be carried along. 7. 4. 3. 1 Signal power measurement. The signal power delivered to the oscillator circuit is simply E 2G where Es is the effective signal voltage across the resonant circuit and G is the circuit conductance. Examining the signal path on Figs. 7. 1 and 7. 5,and the measurement problem we note: 1. Lowest level of absolute measurement is made by the oscilloscope at the input to attenuator 1. 2. 0-100 Db attenuator feeds SHO assembly. 3. A series resistor (with associated shunt capacitance) feeds the final SHO circuit described by measurements in paragraph 7. 2. 2. Examining each of these in detail: 1 a. The lowest level of actual observation was the attenuated and phase shifted square-wave output of the first HarveyWells count-down circuit. This measurement was made using the Tehtronix oscilloscope on a 50 mv/ cm dual trace preamplifier unit, and was the peak-to-peak value. The initial reading for the sample data point was 28 mv, peakto-peak. The effective value of the signal power is developed only from the fundamental component of the input square wave since the higher harmonics are rejected by the 139

tuned oscillator circuit. A simple Fourier description of the square wave reveals that the effective value of the fundamental is #J2/ 1 (i. e..45) times the peak- to- peak value of the square wave. Thus the lowest level observed was 12. 6 mv (effective). Estimated errors, - - less than. 6 Db. 2a. The signal next was attenuated by a 0-100 Db step attenuator. This is the only variable link. The particular unit used had a 50 ohm characteristic impedance and estimated accuracy of. 5 Db. The particular data point being followed here is typified by a 50 Db setting. 3a. The series resistor (nominal value 1 megohm) Z forms a fixed attenuator with the conductance of the loaaed oscillator circuit. The voltage ratio of this attenuator measured at the resonant frequency of the SHO circuit (approx. 482KC), was found to be 39. 2/1. This is almost exactly 32 decibels. The attenuation is sensitive to frequency since the element shunting the one megohm resistor is the lead-to-lead capacitance (about. 5 M 4). However, over the narrow range of frequencies of interest the variation of attenuation is less than. 2 Db. From these figures the signal voltage across the SHO circuit for the data point in question is 82 decibels below the measured value of 12. 6 my. This is coincidently almost exactly 1 microvolt. (i. e. 10 6V) The signal power then is. 68.10 watts. 7. 4. 3. 2 Noise Power Measurement. The noise power was not measured directly but rather taken from the theoretical value depending on the circuit parameters. It was stated in Eq. 5. 51 that the passive resonant circuit noise power is KT W0/Q0. Using the numbers of paragraph -15 7. 2. 2 the passive noise power is 1. 16' 10 watts. The power used in the expression 6. 7 is normalized by the steady-state gain factor 2gM (c. f. Eq. 5. 53) In the data point being examined here, the gain as determined by the procedure of paragraph 7. 3. 1 is 10. The active band power is 140

N -16 N' = = 580- 10 watts 7. 4. 3. 3 Signal-to-Active Band Noise. The signal to active band noise ratio is.68. 1 1 1.17 * 580 1016 or a positive separation of. 67 Decibels 7. 4. 3. 4 Error Probability Determination-Noise. Knowing the signal phase, and as described in 7. 2. 7,having a phase detector device to measure correct phase identification or errors, it is a matter of accurately counting the number of trials and the number of times the oscillation grew with the correct phase over a selected period of time. Since the noise as used in 6. 7 is normalized for the actual steady-state gain factor, gm, by equation 5. 53 the value of gm must be known. This is pre-set by the method described in 7. 3. 1. Now both the quench oscillator frequency (the trials) and the phase detector filtered output (the successes) are counted over a preselected interval of time. It was convenient to count over a 1second interval and repeat the counting procedure at least five times. Thus for a quench frequency of about 200 Hz at least 1000 trials were made. For 1 % error probability, this means only 10 errors in the total run. This number is marginal to obtain good confidence in the result but was sufficient to locate the trend. For higher error rates correspondingly better statistics result. For no signal entering the system, the theoretical error probability is. 5 or 50 %.

Because the passive thermal noise is over 100 Db below the normal saturation levels, and because the active circuit noise is further depressed by the gain, it became extremely difficult to insure that there was no signal leakage into the tank circuit. After all available measures were taken to isolate the tank from extraneous signal, it was amply evident that the phase was not being determined by thermal noise. The method used to check this was to shift the phase of the signal by m radians. The output phase, if determined by a control signal,will shift phase. If determined by noise, primarily both phases occur and signal phase shifting only alters the density of one phase position compared with the other. The final method used to make the very lowest level measurements was to provide an auxiliary signal balancing the leakage exactly equal in magnitude but opposite in phase in order to allow the residual noise to be the final phase determining quantity. A convenient method for doing this was to use coherent quenching, and to phase shift the pump slightly to make use of the arbitrarily small controlled signals available through the action 01 of thel- quantity of equation 5.42. Fig. 7. 2 3 (a, b, and c) illustrates the appearance of oscilloscope waveforms of the error situation for various signal levels. These traces are delayed sweep traces showing a short section of the growing oscillation and the presence of oscillation of both possible phases. The decibel notation is that of the variable attenuator setting. For - 100 Db both output phases occur, and the lines of the two phases appear nearly equal. Increasing the signal (-80 Db, etc. ) decreases the error phase trace 142

-100 Db (a) MBB E lEHEE/B -80 Db HUE,41l A*Rs~ E-60 Db -lll'/i/i-~gl/i — 55 Db (b) 1 —-— g~l~~,~'~a~s ll"~~~ ~~ -50 Db Exposure I/_,Jl _K ~ SJl', l45 Db Fig. 7. 23 Output phases for various signal input levels. 143 _tS | w i in" 1zMZMEXERW31 1~~~~~~~~

density, and increases the signal phase density. This proceeds through the signal levels to -45 Db. At this point it appears that there is no error phase. However, if the oscilloscope camera is allowed to integrate the effect over a period ten times the original period, Fig. 7. 23 c shows the appearance of the error phase. Even the -45 Db signal generates significant error. The sample data point under discussion in the previous paragraphs resulted from the following data QUENCH FREQUENCY 185.0 hertz ATTEN. SETTING ERROR COUNTS AVE. PERCENT ERROR -50 Db 32, 39, 29, 38 34.5 18.65 -100 Db (MIN. SIGNAL) 92. 75 50.1 95, 96, 92, 88 For each point it was found necessary to establish that the minimum signal (-100 Db attenuator setting) actually produced an error probability of nearly 50%0. If it did not, the balancing signal was readjusted to produce the 50% error probability, the actual signal inserted, the run made, and then an additional run was made with the minimum signal again. The 01 adjustment of phase of - was difficult and tended to drift during a run. The normal run consisted of a series of 5, 1 second counts, interspersed with reading times of about 2 seconds. Thus the total run time was about 15 to 20 seconds. 7. 4. 3. 5 Theory and Experimental Correlation-Noise. Fig. 7. 2 4 shows the sample data point and other data points with respect to the 144

Decibels -40 -30 -20 -10 0 +10 +20 o0 _ 40 _ -30 m20. Sample Data Point 5 0o _P Theoretical 4j ~ o g 10 Experiimental Points - DQ L g= 20 Experimental Points 2 (2 runs) 2,M 1.0 0. 5 Fig. 7. 24 Error probability vs signalto- noise separation (Db.) 0. 2 0. 1 145

theoretical curve. Considering the range of well over 100 Decibels through which the oscillation moves the deviations shown are remarkably small. Both the curve shape and level are significant in judging the harmony between theory and experiment. The shape is determined by the spectral distribution of the masking signal, while the level relates to absolute values and basic accuracy of measurement. As we shall see in the following section,when the masking signal is caused by a different phenomena the shape of the ensuing error probability curve is quite different. 7. 4. 4 Signal to Pump-Unbalance 7. 4. 4. 1 Experimental A Contours. As pointed out in Chapter VI 1 for coherent quenching (CQM), variation of - produces a phase determining constraint. This has been used in the previous paragraph to inject a controlled signal to balance a spurious signal arising in the apparatus. The phase determining constraint produces what was called an A contour. Theoretical A contours were plotted in Fig. 6. 3. In the course of the experimental work a number of these A contours were measured as given in Fig. 7. 25. Particularly worthy of note is the variety of experimental contours achieved with various system adjustments. Curve #1 is that achieved on the original SHO (paragraph 7.2.2with no special attempt to insure absence of spurious signals. Curve #2 is the same device having a balance signal so adjusted in magnitude and phase to cause zero A at the expected zero point of curve Y10 of Fig. 6. 3. Finally curve #3 is of a later SHO having a passive Q0 of 30. 146

Fig. 7. 25. ExperimentalA contours in coherent quenching CURVE 1 o SHO # A 1. 0 N. ~2 o - - SHO # 1 (Adjusted Zero) 3 --- SHO# 2 (Tuned slightly high) m% \.5 B ~~~~/d X C %, 2/ -4~ ~~~%2E 4 To of 1/ 2 Cycle'~ ~~~~~~~~~~30 60 d o —--— FlO 90 10 20~~~~~,. 40o 80 o o 242 34 0? 70 -.5-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J -1. 0~~~~~~~~~~~~~~~~~~~~~~~~~~

Each of the above curves demonstrates the sharp major lobe and 0 flat portions of the curve characteristic of the cos3 2. The departure of the zero slope portion from the axis is dependent on a combination of things. They are listed below and have varying degrees of importance depending on the particular effect predominating. 1. Spurious signal present 2. Minor detuning 3. Removed spurious signal by amplitudes and phase adjustment 4. Removed spurious signal by phase only adjustment 5. Q of the SHO 6. Pumping level It is worthy of comment here that it has been singularly difficult to isolate each of the above effects because of the serious interaction and the extremely low signal levels which are effective in the active circuit. 7. 4. 4. 2 Experimental Error Probabilities. A group of experimental error probability curves vs signal-to-pump unbalance were run during the course of the investigation. It was determined early in the study that the amount of signal unbalance present in the sample SHO produced a curve about 30 decibels above the expected signal-to-noise curve. The curves in Fig. 7. 26 have been normalized at the point of zero error. They each show the trends of the pump unbalance interference. That is, the asymptote at zero error, the relatively sharper break toward higher errors and the jog in the curve due to the displaced zero slope of 148

Signal-to-pump unbalance separation (Db.) -60 -50 -40 -30 -20 50,, 50 40 20 10 gm= 5 "'O" Phase signal 5 4 a) 3 Q.5.4.3 Fig. 7. 26 (a) Experimental error probability. for incoherent quenching with pump unbalanc e..1 149

Signal-to-pump unbalance separation (Db.) -60 -50 -40_. -30 -20 -10 _ 50 _ 40 - 30 5 3.,) 0 "I" Phase signal "7r Phase signal Ch11 5.5.4.3 Fig. 7. 26 (b) Experimental error probability 2 for incoherent quenching with pump unbalance. 150

Signal-to-pump Unbalance Separation (Db.) -60 -50 -40 -30 -20 -10 0 40 30 20 10.5.4 3 for incoher 20 "Ounb Phase signallance. 151.4.3 Fig. 7. 26 (c) Experimental error probability 2 for incoherent quenching with pump unbalance.

O1 the Cos3 curve. 7. 4. 5. Higher Order Subharmonic Oscillations. In the course of the experimentation the question was asked, "could higher order subharmonics be demonstrated with the particular experimental apparatus used in this study?" Fig. 3. 1 showing the stability regions for Mathieu equation solutions provides fuel for the inquiry. We note the increasingly narrow approach of the unstable regions to the g axis as the higher orders are sought. However the refinement of construction and adjustability of a second sub-harmonic oscillator, along with an electronic counter for precision location of pumping and tuned frequencies made possible the location of higher order harmonics. The procedure was to tune the passive circuit exactly to 0. 5 Mhz which now represents the frequency at which sub-harmonics will occur. Pumping was then performed at 500 Khz for the second sub-harmonic oscillation, and at 250 Khz and 125 Khz for the third and fourth. By the balanced pump provided in the construction, the pump voltage does not appear directly across the resonant circuit. The higher order sub-harmonics are shown in Fig. 7. 27 along with the pump trace in each case. 7. 4. 6. Magnetic SHO. In order to demonstrate some of the same properties as shown for the parallel-resonant, pumped capacitance device a series-resonant, pumped inductor SHO was built. The magnetic couplings from pump to SHO and from signal source to SHO offer some advantages to the varactor device studied intensively. The particular torroidal ferrites used were essentially square-loop materials and exhibited 152

01 the Cos3 2 curve. 7. 4. 5. Higher Order Subharmonic Oscillations. In the course of the experimentation the question was asked, "could higher order subharmonics be demonstrated with the particular experimental apparatus used in this study?" Fig. 3. 1 showing the stability regions for Mathieu equation solutions provides fuel for the inquiry. We note the increasingly narrow approach of the unstable regions to the g axis as the higher orders are sought. However the refinement of construction and adjustability of a second sub-harmonic oscillator, along with an electronic counter for precision location of pumping and tuned frequencies made possible the location of higher order harmonics. The procedure was to tune the passive circuit exactly to. 0. 5 Mhz which now represents the frequency at which sub-harmonics will occur. Pumping was then performed at 500 Khz for the second sub-harmonic oscillation, and at 250 Khz and 125 Khz for the third and fourth. By the balanced pump provided in the construction, the pump voltage does not appear directly across the resonant circuit. The higher order sub-harmonics are shown in Fig. 7. 27 along with the pump trace in each case. 7. 4. 6. Magnetic SHO. In order to demonstrate some of the same properties as shown for the parallel-resonant, pumped capacitance device a series-resonant, pumped inductor SHO was built. The magnetic couplings from pump to SHO and from signal source to SHO offer some advantages to the varactor device studied intensively. The particular torroidal ferrites used were essentially square-loop materials and exhibited 152

f = 1300 Hz. 5v/ cm SHO = 650 Hz 1. Ov/ cm (a) 1st. Sub- Harmonic f = 635.13 Hz lv/ cm II... mIlEE p SHO = 635.13 Hz 1v/ cm (b) 2nd. Sub-Harmonic Fig. 7. 27 Higher Order Sub-Harmonic Oscillations 153

f = 416. 3 Hz 2vjcm p f = 624 2 Hz lv/cm (c) 3rd. Sub-Harmonic C...... fp = 314. 3 Hz 2v/cm f = 608. 6 lv/ cm |............. S H O (d) 4th. Sub-Harmonic Fig. 7. 27 Higher Order Sub-Harmonic Oscillations (Cont.) 154

marked hysteresis. In this latter respect, they were difficult to use. The Japanese in their Musasino-I (Ref. 37) computer have evolved ferrite devices as the basic pumped element and have demonstrated the remarkable reliability possible using such active elements. 155

CHAPTER VIII SUMMARY OF RESULTS 8. 1 Novel Features of the Research Although the parametric subharmonic oscillator has been the subject of quite a body of authors, a number of facets of the study presented here are novel and represent new analyses and new understanding. These will be summarized here and then discussed briefly. Since no study is an isolated affair, ncr complete in comprehension, these aspects will be mentioned. Finally an application of particular interest to the writer will be presented and discussed in conceptual form. In any study which treats a specific problem and draws on a body of prior literature for its starting place,the problem of notation and analytical forms to be used must be faced. In this study the usual amount of new forms are introduced and at times they may appear to be confusing. However, from the author's viewpoint there are reasons for the progression of forms as they relate to comprehension, sophistication, and elegance. Indeed, in reviewing the possible direction the presentation might have taken at various junctures it would have been possible to avoid some of the pain of the development. But comprehension usually is a painful process, and so some of the pain is evident in the rational used. 156

The specific features of the research which are novel are listed below: 1. Treatment of the statistical problem of phase synchronization of the parametric sub-harmonic oscillator in the presence of signal and thermal noise. 2. Involvement of pump unbalance in the solution for synchronization statistics in the presence of signal. 3. Presentation of the quantitative significance of coherent and incoherent quenching in realizing minimum signal power for phase synchronization. 4. Treatment of detuned transient solution of the active resonant circuit by detailed graphical plot relating differential equation parameters to form of solution. 5. Treatment of the steady-state solution of the quasi-degenerate amplifier- oscillator (where signal frequency, circuit passive resonant frequency, and half-pump frequency do not coincide). 6. Experimental demonstration of the theoretical work of points 1 through 5, with particular emphasis on: 6. 1 Arbitrarily low level signal recognition in the presence of noise. Quantitative data on penetration of passive circuit noise level. 6. 2 Practical aspects of building parametric oscillator circuits for noncoherent quenching. 157

6. 3 Superiority of coherent quenching to achieve maximum dynamic range between signal level and final oscillator output level. 8. 1. 1 The treatment of the statistical problem of the phase synchronation of the parametric subharmonic oscillator was the basic'objective of the study. To do this two developments were made. The first was the degenerate solution, for the signal at exactly half the pump frequency. Although the development had been made in the prior art by Bura(Ref.17) it contained a number of errors significantly affecting the result. These were corrected and verified experimentally. The second development was the quasi-degenerate solution, posed in such a form to be applicable to random noise signal and then unified with the first development. These were combined with the transient solution and initial conditions to form the general solution. The results are typified by the curve of Fig. 7. 24 which displays the theoretical and analytical work together. By way of using numbers, it has been shown here that at least 120 decibels of signal gain are possible in the presence of thermal noi se using the device under discussion for detection of binary phase signals. This is significantly higher than any statement found in the literature. 8. 1.2 The involvement of the pump unbalance in the solution for the synchronization statistics was not an original objective of the study. It became a necessary part of fulfilling the basic objective, however, as the original error statistics suggested a noise level about 30 decibels 158

higher than thermal noise. As the phenomenon became more understandable and as the equipment to demonstrate pump unbalance and coherent quenching was assembled, this whole shubject became an integral part of the study and its understanding. The concept of "A Contour" as presented in Fig. 6. 3 as a basis for understanding coherent quenching, and as a basis for error probability determination of incoherent quenching is new. So far as the writer knows there is no mention of this aspect of dynamic range limiting anywhere in the literature. The functional form developed and the demonstration of experimental A contours having these general characteristics as shown in Fig. 7. 25 are essentially in harmony. 8. 1. 3 In presenting the quantitative significance of coherent and incoherent quenching in realizing minimum signal power for phase synchronization, alternatives for design are offered. It was shown, for example, that the ultimate limit for dynamic range is the active-band noise. The masking effects of pump unbalance can be minimized either by extremely careful construction and matching procedures if incoherent quenching is used, or by the use of coherent quenching and appropriate selection of initial pump phase angle. Of course, another alternative is to do both, which was done in the experimental procedures discussed. 8.1. 4 At the inception of the study there were no detailed graphical presentations of the Mathieu solutions in the vicinity of the a-g plane occupied by this problem. Since then, Phillips (Ref. 104) presented the concept 159

of iso-u curves in the vicinity of the first unstable Mathieu region. Included in the present study are terms neglected by Phillips which distort his iso-u curves and give improved prediction for criteria for oscillation. Also in the present study the phase angle of the transient terms are shown on the graphical representation as iso-Ua curves. This makes quite simple the complete visualization of the Floquet solution to the differential equation with time varying parameters used here. By this graphical form, for example, it is possible to show that for a particular steady state gain used (i. e. a particular pump levelrj) the phase angle of the growing term is constrained to lie between particular values close to - r/4, and what the angles will be when oscillation ceases due to detuning. 8. 1.,5 The quasi-degenerate amplifier is one where signals are expected within a small band about the half-pump frequency. Bura (Ref. 17) treated this problem for discrete signals. Where many signals are present simultaneously (noise) the signal is conceived rather differently because of the expected interaction about the half-pump frequency. Using a signal form suggested by D. K. Adams the analysis produced a solution form which could be used equally for signal or noise by choosing appropriate constraints on amplitude and phase. By retaining departures from ideal centering of pump frequency, and passive resonant frequency the effect of these departures are expressed quantitatively right through the final analysis. Although these departures were not 160

used extensively, the forms are believed adequate and they provide a resource for further work in this particular vein. 8. 2 Incomplete Aspects Work such as presented in this study has many natural extensions. It appears to the writer that they will be worthy of investigation as the need for particular performance arises or as the state of-the-art in some aspect changes significantly to open up some new dimension. As the picture of pump unbalance became clearer it was also obvious that many aspects of quench initiation had not been studied. The abrupt initiation used here was limited to the relative abruptness of a tra;m..f.or turning on or off. M ore or less drastic initiation may have other consequences and these could be studied as a means of answering more exhaustively the question of what to do about unavoidable pump unbalance. The subject of noise on the pump excitation was discussed in formulating the problem. This has not been discussed in this study although the forms used are generally amenable to this consideration. Similarly noise in amplitude or phase of the quench input, particularly for coherent quenching, is suitable for further study. Application of the details of this study to the magnetic SHO would be of interest. Certainly there are properties particular to such circuits that would bear further investigation. 161

Finally, the study of the higher order sub-harmonic solutions would be fascinating just for the sake of knowing more about the function and performance in these relatively little known regions. 8. 3 Suggested Application Although not a direct part of the study, it is interesting to speculate just how the fruits of such a study might be used. The following is a description of such an application, actually conceived before the study, and for whose realization the subject study was an integral part. The problem area is instrumentation of low-level transducers such as may be found in a great many distributed processes where a large number of sample points must be measured against known limits or set-points. A high pressure steam boiler or large nuclear reactor would be cases in point. Such processes normally operate close to limits in several parameters and must be monitored closely for satisfactory compromise between efficiency,safety, and destructive damage. A number of schemes for performing these measurements are being used currently, each having problems in one or more areas such as: 1. Low level metallic commutating contacts 2. Multiple precision amplifiers 3. Commutated low level amplifiers having long settling times The scheme proposed uses the combination of two solid-state devices; the second-harmonic magnetic modulator (SHMM), and the parametric subharmonic oscillator (SHO). For one data sampling 162

point Fig. 8. 1 shows the proposed layout. The operating scheme is this. The SHMM, pumped at frequency f, generates frequency 2f, proportional in amplitude and phase to the magnitude and polarity of the difference between the d-c levels of two currents, the signal, S1, and the set point, S2. This transducer principle is applicable to lim iting d-c power levels of the order of 10 watts/hz and is singularly free of temperature dependence (Ref. 116). The SHO, pumped at frequency 4f, oscillates at 2f having a phase dependent on the phase of the received information from the SHMM, and whose amplitude, short of saturation, depends on the magnitude of the received information. Two basic questions can be asked of the signal: 1. What is the value of the variable? 2. Is it above a set-point? In answering 1, the SHO would be quenched at such a rate to place it in the linear operating mode, where average envelope carries amplitude information. In answering 2 the system need only decide which phase is present in the output. The primary reason for offering such a hybrid system for commutation problems isthat the equipment which would have to be duplicated for each point sampled is: 1. solid state 2. not temperature sensitive in basic mode of operation 3. inexpensive 4. simple and highly reliable 163

The rest of the equipment needed for such a system as shown in Fig. 8. 1 would not need to be duplicated at each point, and could be engineered for design reliability by normal practices. In this application the unique properties of two devices have been married, one for low level d-c, transduction and conversion to low level a-c, and the other for extremely large stage gain to levels easily transmitted to and understood by a computer. 164

Coherence link Master C4 Quench Oscillator Switch -.- Delay -r,2 Geea t Modulator.Phase 8 Excitation ShAt _Switch Signal I 2nd Harmonid n st S1 Magnetic SHO16 St " Modulator [ i l l [ Phase Computer LL _ = I Detector Digital Programmer duplicated at each Output sampling point. Set Point Generator Fig. 8. 1 Digital commutating scheme for analog system. 165

APPENDIX I REMOVAL OF FIRST DERIVATIVE IN A GENERAL SECOND-ORDER DIFFERENTIAL EQUATION 1 Given y +Ky +My = f (I. 1) where K, M, f, and y are functions of the independent variable t. It is desirable in some instances to re-express this relation in terms of a new variable, a, such that the resulting equation has no first derivative. Let y - ab (I. 2) y = ab + ab y = ab + 2ab + ab where a is the new variable with the above constraint and b is a function to be determined. Substitute (I. 2) in (I. 1) and collect terms in orders of a: ab + a(2b + Kb) + a(b'+Kb + Mb) = f (I. 3) now imposing the desired constraint that the first derivative vanish 2b + Kb = 0 (I. 4) As per Kaplan, (Ref. 60) and presented here for completeness. 166

and e-' K dt b = K KK b = ( )b K22 (I. 3) now becomes +2- +M) a = f J Kdt (I. 5) and when K is a constant(I. 5) simplifies further to K2 K a + ( —- M) a = fe (I.6) 167

APPENDIX II DETAILED PLOT OF A-G PLANE IN THE FIRST UNSTABLE ZONE This appendix concerns the solution of the canonical homogeneous Mathieu equation y" + (a- 2g cos 2z) y = 0 (II. 1) The solution according to Floque's theorem is of the form: y = A ElZ 0(z) + B e M Z(z) (0 and 4 periodic in z) (II. 2) a, 0 and g/ are very difficult to express in terms of "a" and "g" alone. Whittaker (Ref. 106) and McLachlan, using a slightly different form (Ref. 103 ) expressed "a" and "i"' in terms of power series in "g", the coefficients of terms being functions of a new dependent variable, cr. Also the coefficients of the series: 0(z, a) = sin(z- r) + s3 sin(3z- r) + s5 sin(5z-cr).... + C3 cos (3z-c) + c5 cos (5z-a)..... (II. 3) (i. e., the s's and c's) are shown to be power series in g, the coefficients of the g terms being functions of the new variable, a. The term cos (z- cr) is missing as a consequence of the periodicity requirement on 0. 168

In the notation of McLachlan the representations of /i, a, S's and c's are: a = l-g cos 2a + 4 (-1+ 2 cos4a) + 64cos2 1 4 1 11 (I. 4) 16 (-3 2 cos 4) (.4) or a = 1-g cos 2cr- /2 + g s3 1.33 3 4 1~sin r+ g (II. 5) g sin 2a 128 g sin 2a - 1024 g sin 4a -. (. 5) or t= -2 g(- sin 2a+ c3) 1 1 2 1 g (_ 14 5 cos 4) + s3 = - - g cos 2cr 512 3 2 3 g sin4+... C3 64 g sin 2or 512 g3 sin 4c + = 192 g 112 g3 cos 2r +... c5 = - 2304 g3 sin 2or+... (II. 6) 169

All the basic information is given in II. 4 to II. 6. On the a-g plane two sets of curves are needed, namely iso- a (constant value of a) and iso- i (constant value of A/). Iso- a The iso-a curves are the most directly obtained since for a chosen value of a the a-g is a simple numerical relation according to II. 4. A family of iso- a curves results when a is specified in discrete values over the total range -90~ - a < 0. The boundary of the region is defined by the relation _ 3 4 a = 1 + g- + g (II. 7) 8- -64 1536...... the upper signs corresponding to a= 0 the lower signs corresponding to a=-90~ Errors Within the region shown on Fig. 3. 2 the ordinates are values of the departure from the value of a = 1, i. e., (a-l). The maximum contribution of each term is given in Table II. 1. Table II. 1 Absolute Value Fraction of % Term Maximum Contribution Principal Value, g g =. 2 1 g 1 100. 2 3/8 g2 3/8 g 7.5 3 g3/64 g2/64.06 4 g4 65/(96)(16) g3 65/(96)(16).034 170

Retaining only 2 terms assures that the value of (a- 1) is within. 1 percent. Iso- M Equation (II. 5) is not suitable as it stands for determining an iso- j curve since particular values of ji cannot be designated in terms of a and g. The procedure chosen is to select the number of terms in (II. 4) and (II. 5) to assure a desired accuracy, and then eliminate the variable a between them. Table II. 2 shows the rate of convergence of Eq. II. 5. Table II. 2 Absolute Value Fraction of Max % for Term (Maximum) Principal Value g'. 2 1 g/2 1 100. 2 3/128 g3 3/64 g2. 19 3 3/'1024 g4 3/512 g3.0047 The problem is simplified by the very fast convergence of (II. 5). Thus we shall use: = - g sin 2c (correct to better than.2%) (II. 8) a-1 = - g cos 2a- 3 g + - cos2 2 (correct to better than. 1%) (II. 9) from which

(a-l)= + g 1- g (II. 10) g For each value of ji and g chosen there are two values of (a-1), but note they are not symmetrical with respect to (a = 1). Referring back to Fig. II. 1 shows the extent of the asymmetry for large values of g. For the region chosen (O < g <. 2) the departure from symmetry is 2 to 10 percent at the large g values. The data for Eqs. II. 9 and II. 10 were accumulated with the aid of the IBM 7090 and are presented in Fig. 3. 2. 172

APPENDIX III The Steady-State Conversion Matrix Consider a time varying capacitance n=+cc j2nw t C= Ce III. 1 n n= -cc The fundamental pumping frequency is 2wl. This notation is used to simplify consideration of the degenerate parametric amplifier. Due to charges flowing in and out of C there will be a voltage produced having the general form: n=+oc jcokt V = j Vke III. 2 n=-oC where Vk = V* k (condition for real voltage; * is the complex conjugate). The general charge description is given in terms of the product of III. 1 and III. 2 +cc +~c j(2nwl + wk)t q= L L C Vk -cC -cC III. 3 n k C = C* (condition for real capacitance) n -n In the restricted instance of a resonant circuit, where the frequency components are limited to a small band, the number of significant terms in III. 3 is small. Specifically if the discussion is centered about those frequencies in the vicinity of wl the only combinations of indices can can produce these are: 173

(lower side frequency) wu (upper side frequency) k= C =l Wk -W III. 4 n=1 w = co n= O w =w k -u k u If the terms of III. 3 represented by III. 4 are summed and the time derivative taken the capacitive currents results. jw t jW t jC t jW t c = j Co V + C. + jw C Vu*e + jC Ve + j u III. 5 and in matrix form where row 1 is at the lower side frequency, and row two at the upper side frequency: I1 jWC jw C V ~I*u ~_ C* -jw C * For a general monochromatic sine pump: C = C[1 +17 sin (2w1t+ 01)] III. 7 = 2 (sin 01- j Cos 0 1) and III. 6 becomes I je~~~o S L L o0 1CE Lv ( Q a o 2CO: *1 III. 8 -I c2 co;e -jC C V1C o u o 6C Since the introduction of the angle 0 1 is tantamount to an arbitray choice of the origin of the time axis, t, a new time axis, t', can be chosen such that 174

0 will now be zero. Ir: j oC co 2C C V of 0 2 0 III. 9 1 ~* c2 C -jwo C * u u2 o u o where the time variable has been chosen such that 2colt' = (2w1t + 0 1) 175

APPENDIX IV Gain Function in Oscillation Region The gain function, Eq. 4. 12, is repeated here and examined in detail for the specific region of interest, the region of oscillation. 1 gt I 1 12 IV. 1 1+(62 2 ) - j2~ where 6 = fractional detuning = fractional signal departure from half-pump frequency, w H='Qo jo 1 2 The conditions for g to be in the oscillating region are seen from IV. 1 to be: 0 =0 IV. 2 lI2 > 1 + 2 It will be convenient to think of a new variable H', the detuned pumping factor where II Simplifying IV. 1 produces (1 +62 - 2) - j2 IV.3 g (1 + 62 - 12 -2 ) j2 176

And now if we actually factor out the detuning parameter (1 + 62 ) from numerator and denominator and use the detuned pumping parameter nI "2 2j 1 - _ j g = 1 + 62 + 62 IV 4 1 - Il2 2 - j 1+6~ 1+6 When 4 = 0, g = gm gm 1-!'12 IV. 5 from which 42 2 + 62 1 62 t = gm 9 g42 2g 2 IV. 6 1- -- m 1+ 62 1 + 62 2gm4 For small values of X, where m is of the order of unity IV. 6 1 + 62 becomes much simpler and recognizable as a simple resonance form: g I.* 1 1 j tan-' t;=g l _~j gm + /1 + ((') IV. 7 where 2g m 1+62 1+62 Here ( is to the active circuit what 4 is to the passive circuit, and' to the active detuned circuit what is to the passive detuned circuit. 1 + 62 Since:g~ an -- ar aes voltage gain expressions, a relative powrer gain form is achieved try, the ratio of the squares of f5 and gm, thus: 177

g] =k= 1 1 IV. 8 -m,) 1+ +'2 1+ C 2 (1 +62 )2 Solving IV. 8 for ( in order to relate relative bandwidth to detuning yields = 1(k _ 1) (1 +62) IV. 9 For a chosen fractional response, k, the bandwidth increases with detuning, and from IV. 5 the band center gain, gm, decreases. 178

APPENDIX V Relation between a' and 6 a', introduced in equation 5. 29, is the departure of the Mathieu 7T 71 solution phase angle from4 radians. (- radians being very nearly the tuned phase angle. ) 6 is introduced in Chapter 2 in connection with detuning the passive circuit from half the pump frequency. (cf equation 2. 24) Combining 2. 20 and 2. 25 we note that a 1- 6 V. 1 Qo From Appendix II for small values of /i II. 5 is written: a- = -gcos 2a V. 2 - g sin 2r' From 2. 22 2g' - 2g - 17(1- ) V.3 Q = - 2' ) sin 2' V. 4 26 6 6 sin 2&'. 5 i i II - A1+2 179

oo b CM to

APPENDIX VI Phase-Shift Detuning Factor In the development of A for both the signal and the noise a factor arises that will be evaluated here. This factor appears first in component form in (5. 37) and is: 11II' - I t \2' 1 " Ifv I 2os CS" a 1+ I. sin Ca") d+I, cos o') VI. 1 The evaluation proceeds on the basis of some simplifying assumptions. 1 1 angle tan 2Q 2Q< 1. a" CT' + tan 2Q0 and use a" a'. 2. For 6. 5 v.v-e use co 2C' - from Appendix V sin 2a' = Performing the operations indicated in VI. 1 the factor becomes 1 4/2 C _' - 26/16 IJ"I 2 cos 2c; os'2 r _26' 1 cos Ca' sin Cr' + + l) VI. 2 181

using the double angle identities 22 cos2 C' 1 - VI. 3 2 cos a' sina' A 6(1- 6- ) and the relations for a medium to high gain system by'=~l+6'~' - 1 In"l? 1 2 VI. 4 iI"I ~ 1 62 +1 = (1 -.... VI. 2 now becomes 62 1 2 1 \4 a 1 VI 5 1+62 In addition the ratio of the two factors ofVI. 1 are used for an inverse tangent function. 1+ I"I cos T T 4 6 8 -+ 611 COS i 1 62 cos ("+ +1II"l sin cr" 62 6 2 4 VI4 6 V182 6 182

REFERENCES Superregenerative Devices 1. E. H. Armstrong, "Some Recent Developments in Regenerative Circuits, " Proc. IRE, Vol. 10, August 1922, p. 244. 2. B. B. Bossard, "Superregenerative Reactance Amplifier," Proc. IRE? Vol. 47, No. 7, July 1959, 1. 1269. 3. B. B. Bossard, E. Frost, and W. Fishbein, "X Band Superregenerative Parametric Amplifier, " Proc. IRE, (Cotrr'espj, Vl. 48, July 1960, pp. 1329-1330. 4. H. A. Glucksman, "Superregeneration, Proc. IRE, Vol. 37, May 1949. 5. A. G. Jordan and R. Elco, "Esaki Diodes as Superregenerative Detectors," Proc. IRE, Vol. 48, No. 11, November 1960, p. 1902. 6. J. K. Pulfer (Radio and E. E. Div., Nat. Research Council, Ottawa), "Microwave Superregenerative Amplifier, " The Microwave Journal, Vol. 4, No. 2, February 1961, p. 61-p. 66. 7. H. E. Stockman, "Tunnel Diode Superregenerative Parametric Motor," Proc. IRE, Vol. 49, October 1961, p. 1586 (Corresp). 8. D. N. Thomson, "Microwave Superregenerative Amplifiers, " Proc. NEC, Vol. 16, 1960, p. 753. 9. C. P. Wang, "Superregeneration in a Cavity-Type Parametric Amplifier, " Tech. Report No. 318-1, April 24, 1961, Electron Devices Lab., Stanford Electronics Laboratory. 10. C. P. Wang and G. Wade, "Noise Figure and Sensitivity of a Superregenerative Parametric Amplifier, " IRE Trans. on Circuit Theory, Vol. CT-9, December 1962, No. 4, p. 320. 11. J. R. Whitehead, "Superregenerative Receivers," (Book) Cambridge U. Press, 1950. 12. J. J. Younger, A. G. Little, H. Heffner, G. Wade, "Parametric Amplifiers as Superregenerative Detectors, " Proc. IRE, Vol. 47, July 1959, pp. 1271-72. 183

Subharmonic Oscillators 13. I. Ageyta, F. Borgini, D. R. Crosby, "A Computer Sub-system using Kilomegacycle Subharmonic Oscillators, " Proc. IRE, Vol. 49, January 1961, p. 128. 14. I. Abeyta, F. Borgini, D. R. Crosby, "A Computer Subsystem using Kilomegacycle Subharmonic Oscillators, " Proc. IRE, Vol. 49, No. 1, January 1961, p. 128. 15. D. K. Adams, "A Study of Double-Sideband Reactive Mixers," Technical Report No. 134, Cooley Electronics Laboratory, Dept. of Electrical Engineering, The University of Michigan, Dec. 1962. 16. A. T. Bolint (U. of Buffalo), "An Improved Magnetic Modulator for Satic Subharmonic Generators, " 1959 Proceedings Spec. Tech. Fonf. on Nonlinear Magnetics and Magnetic Amplifiers; p. 167. 17. P. Bura, "The Degenerate and Quasi-Degenerate Mode of Parametric Amplification, " IRE Transactions on Circuit Theory, September 1960, pp. 200- 210. 18. W. J. Cunningham; "Nonlinear Analysis, " McGraw-Hill Book Co., Inc., New York, N. Y., 1958, pp. 245-280. 19. R. W. Damon and J. R. Eshbach, "Theoretical Limitations to Ferromagnetic Parametric Amplifier Performance, " IRE Trans, Vol. 10778, January 1960, pp. 409. 20. R. A. Elco, J. Nee, "The Reactance Tube as a Parametric Frequency Divider and Amplifier, " (EE Dept., Carnegie Inst. of Tech. ), Proc. IRE, Vol. 49, March 1961, p. 624. 21. E. Goto, "The Parametron, A digital Computing Element which Utilizes Parametric Oscillations, " Proc. IRE, Vol. 47, August 1959, pp. 1304-1316. 22. E. Goto, "On the Application of Parametrically Excited Nonlinear Resonators," J. Institute of Electrical Communication Engineers (Japan), Vol. 38, October 1955, pp. 770-775. 23. J. C. Greene and W. D. White (Airborne Instruments Lab., Melville, N.. ), "Radar Sensitivity with Degenerate Parametric Amplifier Front End," Proc. IRE, Vol. 49, April 1961, pp. 804-807. 24. J. Hilibrand, C. W. Mueller, C. F. Stocker, R. D. Gold (RCA, Princeton), "Semiconductor Parametric Diodes in Microwave Computers," IRE Trans. on Electronic Computers, September 1959, pp. 287-297. 184

25. A. Lavi, L. A. Finzi, "An Analysis of the Magnetic Second Subharmonic Oscillator," Proc. IRE, Vol. 49, April, 1961, p. 779-787. 26. A. Lavi, "Large Signal Analysis of Parametric Subharmonic Oscillators," Magnetic Amplifiers, Techn. Report No. 28, Carnegie Inst. of Tech., Dept. of Electrical Engineering, Pittsburgh, Pa., September 19 59. 27. D. L. Hedderly, "An Exact Analysis of a Parametric Subharmonic Oscillator, " IEEE Trans. on Electron Devices, Vol. ED- 10, May 1963, No. 3, p. 134. 28. R. V. L. Hartley, "Oscillations in Systems with Nonlinear Reactance," BSTJ, Vol. 15, pp. 424-440, July 1936. 30. C. Hayashi, Y. Nishikawa, and M. Able, "Subharmonic Oscillations of Order One-Half,'IRE Trans. on C. T., Vol. CT-7, pp. 102-111, June 19 60. 31. J. Hilibrand and W. R. Beam, "Seminconductor Diodes in Parametric Subharmnonic Oscillators, " RCA Review, Vol. 20, pp. 229-253, June 1959. 32. D. S. Leeson, "Subharmonic Generation with Nonlinear Reactance - A Circuit Analysis, " Proc. IRE, Vol. 49, January 1961, p. 351. 33. H. T. McAleer, "A New Look at the Phase-Locked Oscillator, " Proc. IRE, Vol. 47, June 1959, p. 1137. 34. N. Minorsky, "Parametric Excitation, " Journal of Applied Physics, Vol. 22, No. 1, January 1951, pp. 49-54. 35. E. Mount and G. Begg, "Parametric Devices and Masers: An Annotated Bibliography, " IRE Trans. of MT and T, 8-9, p. 222, 1960-61. 36. S. Muroga, "Elementary Principles of Parametron and Its Applications to Digital Computers, " Datamation, Vol. 1, October 1958, pp. 31-34. 37. Muroga and Takashima, "The Parametron Digital Computer, MUSASINO1, " IRE Transactions on Electronic Computers, Vol. EC-8, September 1959. 38. S. Onyshkevych, W. F. Kosonocky, A. W. Lo, "Parametric PhaseLocked Oscillator Characteristics and Applications to Digital Systems, " IRE Trans. on Electronic Computers, Vol. EC-8, September 1959, pp. 277-286. 185

39. S. Oshima, H. Enomoto, S. Watanabe, Y. Koseki, "Analysis of Parametrically Excited Networks, " Committee on Electric Computers, I. E. C. J., September 1955. 40. N. S. Psywes, "Relaxation Oscillations at Multiple Frequencies of Even Subharmonics, " 1960 Proceedings of Special Tech. Conf. on Nonlinear Magnetics and Magnetic Amplifiers, Phil., Pa., October 26-28, 1960, p. 181. 41. N. Rouche (Lovanium U., Leopoldville, Rep. of Congo, Africa), "Oscillations of Parametric Subharmonic Oscillator, IRE Trans. on C. T., Vol. CT-9, March 1962. 42. F. Sterzer (RCA), "Microwave Parametric Subharmonic Oscillators for Digital Computing," Proc. IRE, Vol. 47, August 1959, pp. 1301-1412. 43. F. Sterzer and W. R. Beam, "Parametric Subharmonic Oscillators, " IRE-AIEE - U.of Pennsylvania Solid-State Circuits Conference, Phil., Pa., February 13, 1959. 44. H. Takahashi, "The Parametron," J. Inst. Elec. Commun. Engr. (Japan), Vol. 39, June 1956, pp. 586-590. 45. J. Von Neumann, "Nonlinear Capacitance or Inductance Switching, Amplifying, and Memory Organs, " U. S. Pat. No. 2, 815, 488 assig. to IBM, December 3, 1957. 46. C. P. Wang, "The Transient Build-Up and Phase Synchronization of Thin Magnetic Film Parametrons, " Proceedings of the 1963 Intermag. Conference, April 17, 18, 19, 1963, IEEE Corn. on Nonlinear Magnetics, PTG on I Electronics and Electron Computers, pp. 2-6-1 - 2-6-8. 47. R. L. Wigington, "A New Concept in Computing, " Proc. IRE, Vol. 47, April 1959, pp. 516-523. Time Varying Systems 49. W. R. Bennett, "New Results in the Calculation of Modulation Products, " BSTJ Vol. 12, pp. 228-243, April 1933. 50. S. Darlington, "An Introduction to Time-Variable Networks, " Proc. of Midwest Symp. on Circuit Analysis, U. of Ill., 1955, pp. 5-1 to 5-25. 186

51. B. Henning, "Flow Graph Analysis for Variable Parametric Networks," IRE Trans. of C. T., CT-9, No. 3, September 1962, p. 283 (Corresp.). 52. R. E. Kalman, "On the Stability of Time-Varying Linear Systems," IRE Trans. on Circuit Theory, Vol. CT-9, December 1962, No. 4, p. 420 (Corresp. ). 53. B. K. Kinariwala, "Analysis of Time-Varying Networks, " IRE Convention Record 9, Part 4, 1961, pp. 268-276. 54. J. R. Macdonald, D. E. Edmonson, "Exact Solution of a Time-Varying Capacitance Problem, Proc. IRE, Fegruary 1961, p. 453. 55. H. E. Meadows, "Solution of Systems of Linear Ordinary Differential Equations with Periodic Coefficients, " BSTJ, July 1962, Vol. 41, No. 4, p. 1275-94. 56. L. A. Pipes, "Matrix Solution of Equations of the Mathiew-Hill Type, " Journal of Applied Physics, Vol. 24, pp. 902-910 (1953). 57. L. A. Pipes, "Matrix Analysis of Linear Time-Varying Circuits, " IRE Transactions on Circuit Theory, Vol. CT-2, December 1953, pp. 91105. 58. C. Sato, "Stability Conditions for Resonant Circuits with TimeVariable Parameters, " IRE Transactions on Circuit Theory, Vol. CT-9, December 1962, No. 4, p. 340. 59. R. G. Smart, "Steady-State Analysis of Power-Frequency Relations in Time-Varying Systems, " Proc. IRE, June 1961, p. 1051. 60. W. Kaplan, "Operational Methods for Linear Systems," (Book), Addison-Wesley Publishing Co., Inc., 1962, pp. 481-482. Parametric Devices 61. 1960 Abstracts IRE, Oscillators - parametric, -using point contact diodes, 1525, June, p. 1197; Amplifiers - superregenerative parametric (88), February, p. 274 -. 62. D. K. Adams, "An Analysis of Four-Frequency Nonlinear Reactance Circuits," IRE Trans. on Microwave Theory and Techniques, Vol. MTT-8, No. 3, May 1960, pp. 274-283. 187

63. E. Ahlstrom, W. G. Matthei, W. W. Gartner, "Use of Surface-Barrier Photodiodes as Fast Response Photocapacitors, " Rice Scientific Inst., Vol. 30, July 1959, pp. 592-3. 64. J. T. DeJager and B. J. Robinson, "Sensitivity of the Degenerate Parametric Amplifier,' Proc. IRE, Vol. 49, July 1961. 65. E. P. Felch, V. E. Legg, F. G. Merrill, "Magnetic Modulators," Electronics, Vol. 25, February 1952, pp. 113-117. 66. L. J. Giacoletto, "Fundamental Limiting Noise of Depletion Layer Capacitance, " IRE Int. Convention Record, Vol. 98t3, 3-9, 1961. 67. J. C. Greene, W. D. White and R. Adler, "Radar Sensitivity with Degenerate Parametric Amplifier Front End, " Proc. IRE (Corresp.), Vol. 49, pp. 804-807, April 1961. 68. W. T. Hatley, Jr., "Simulating Parametric Amplifiers," IRE Trans. on Electron Devices, Vol. ED-8, No. 2, March 1961, p. 116-123. 69. H. Heffner, "Capacitive Definitions for Parametric Operation, " IRE Trans. Microwave Theory and Techniques, Vol. MTT-9, January 1961, pp. 98-9.' 70. H. Heffner and G. Wade, "Gain, Bandwidth, and Noise Characteristics of the Variable-Parametric Amplifier, " Journal of Applied Physics, Vol. 29, No. 9, September 1958, pp. 1321-1331. 71. C. R. Hurtig, "Dielectric Modulators Using the Space-Charge Capacitance of Junction Diodes, " IRE-AIEE-UP Transistor and Solid-State Circuits Conf., Phil., 1958. 72. F. V. Hyde and D. G. Tucker, "Parametric Amplifiers; Static and Dynamic Inductance and Capacitance and Their Significance in the Nonlinear and Time-Varying Approaches, " British Institution of Radio Engineers, Journal, Vol. 25, April 1963, pp. 353-55. 73. E. T. Jaynes "Nonlinear Dielectric Materials," Proc. IRE, Vol. 43, 1955, p. 1733. 74. J. M. Manley, "Some General Properties of Magnetic Amplifiers, " Proc. IRE, Vol. 39, March 1951, pp. 242-251. 75. J. M. Manley and E. Peterson, "Negative Resistance Effects in Saturable Reactor Circuits," Trans. Am. Inst. Elec. Eng., Vol. 56, December 1946, pp. 870-881. 188

76. W. P. Mason and R. F. Wick, "Ferroelectrics and the Dielectric Amplifier, Proc. IRE, Vol. 42, November 1954, pp. 1606-1620. 77. G. Sherane, F. Jona, R. Pepensky, "Some Aspects of Ferroelectricity, " Proc. IRE, Vol. 43, 1955, pp. 1738. 78. D. J. Soltz, "Low Level Parametric Amplifier, J. of Franklin Inst., Vol. 275, No. 2, February 1963, p. 79-87. 79. G. C. Spacek (G-M, Santa Barbara, Calif. ), "Double Pumped X Band Parametric Amp. with Extremely Large Gain-Bandwidth Products, " Proc. IRE, Vol. 50, June 1962, pp. 1534-35. 80. H. Suhl, "Proposal for a Ferromagnetic Amplifier in the Microwave Range," Phys. Rev., Vol. 106, pp. 384-385, April 15, 1957. 81. A. M. Vincent, "Dielectric Amplifier Fundamentals," Electronics, Vol. 24, December 1951, p. 84. 82. M. T. Weiss, "A Solid-State Microwave Amplifier and Oscillator Using Ferrites," Phys. Rev., Vol. 107, p. 317, July 1957. 83. L. A. Zadeh (U. of Calif., Berkeley), "Time Varying Networks I, " Proc. IRE, Vol. 49, October 1961, pp. 1488-1503. 84. "A Study of Microwave Superregenerative Amplifiers for Lightweight Radars, " Atronics Report 761 - 196-6, 13 June 1960, ASTIA No. AD 318- 865. 85. D. N. Thompson (General Atronics Corp. ), "Microwave Superregenerative Amplifiers, " Proceedings of the NEC, 1960, Vol. 16. 86. D. N. Thomson (General Atronics Corp., W. Conshohocken, Pa. ), "Maximum Sampling Rate for Superregenerative Amplifiers, " 1962 IRE Convention Record, Part 2 (Automatic Control and Circuit Theory), p. 77-87. 87. "A Study of Microwave Superregenerative Amplifiers for Lightweight Radars, " Atronics Report 761-196-6, 13 June 1960, ASTIA No. AD 318- 865. 88. P. Bura and D. M. Tombs, "Resonant Circuit with Periodically Varying Parameters, " Wireless Engineer, pp. 120-125, April 1952, May 1952. 189

89. S. V. Yadavalli (Stanford Research Inst. ), "Detection of a Sinusoid Immersed in Noise-Finite Integration Time and Linewidth," IEEE Proc., Vol. 51, May 1963, p. 865. 91. B. A. Kaufman, W. Pfieffer "On the Characteristics of Cylindrical Thin-Film Parametrons PDrnped to Saturation," Proceedings of IEEE, Vol. 51, May 1963, pp. 855. 92. E. R. Peressini, Et. al., "Experimental Study of Parallel Pumped Ferromagnetic Amplifier," J. Applied Phys., Vol. 33: 329-5, November 1962. 93. E. D. Banta, "Transient Analysis of a Parametric Oscillator," 1962 IRE International Convention Record, Part 2 (Automatic Control and Circuit Theory), 1962, p. 70-76. 94. K. W. Beer, "An Experimental Investigation into the Operation of the Parametric Phase- Locked Oscillator, " Radio Electronic Engineering, Vol. 25, No. 5, May 1963, p. 432-40. 95. Y. H. Ku, T. T. Yank, "Analysis of Parametrically Excited Systems," Franklin Inst. Journal, V. 274, No. 6, December 1962, p. 452-74. 96. A. R. Saha (Jadavpur University, Dept. of Telecom. Eng., Calcutta), "Transistor Parametric Subharmonic Generator, " Journal of Electronics and Control, July 1963, p. 21-39. 98. C. A. Desoer, "Linear Time-Varying G-C Networks: Stable and Unstable, " IEEE Transactions on Circuit Theory, Vol. CT- 10, June 1963, No. 2, p. 180. 99. A. P. Bekasov and 0. G. Smolianoff, "Transient Characteristics of Resonance-Type Parametric Amplifiers and Converters, " Radiotechnica i Electronika, Vol. 8, February 1963, p. 241-247 (Russian). 100. R. Spence and A. R. Boothroyd, "Discriminating Properties of the Synchronized Oscillator," Inst. EE Proc., 110:481-92, March 1963. 101. A. J. Viterbi, "On Coded Phase Coherent Communications, " JPL Technical Report No. 32-25, JPL Laboratory, California Institute of Technology, August 15, 1960. 102. P. N. Zanedvorov, "Transient Processes of Oscillator Phase Stabilization in an Oscillator Locked by a Weak Signal, " Radiotekhnika (USSR), Vol. 18, No. 4, p. 31-9, April 1963. 190

103. N. W. McLachlan, "Theory and Application of Mathieu Functions," (Book) Oxford, 1947. 104. R. F. Phillips, "Parametric Oscillation in a Damped Resonant System, " IEEE Transactions on Circuit Theory, December 1963, p. 512515. 105. E. D Banta, "Transient Analysis of a Parametric Oscillator," General Atronics Corporation report, Sponsored by Rome Air Development Center under Contract AF 30(602)- 2283. 106. E. T. Whittaker, "Proceedings of the Edinburgh Mathematical Society," July 1914. 107. L. A. Blackwell and K. L. Kotzebue, "Semiconductor-Diode Parametric Amplifiers, " (Book) Prentice-Hall 1961. 108. P. Penfield, Jr. and R. P. Rafuse, "Varactor Applications" (Book) The MIT Press, 1962. 109. F. E. Melde, Pogg., Annalen 108, 508 (1859). 110. Lord Rayleigh,'"On Maintained Vibrations," Phil. Mag. 15, p. 229235 Fifth Series, (April 1883). 111. L. Mandelstam, N. Papalexi. A. Andronov, A. Witt, and S. Chaikin, "Expose des recherches recentes sur les Oscillations Noniineaires, J. Techn. Phys. (USSR), 2,81-134 (1934). 112. W. W. Mumford, "Some Notes of the History of Parametric Transducers," Proc. IRE, Vol. 48, May 1960, p. 848-854. 113. A. Stephenson, Proc. Manchester Phil. Soc. 52, No. 8, (1908). Noise 114. Middleton, "Introduction to Statistical Communication Theory, " McGrawHill, 1960, pp. 348, 472, 476. 115. Ragazzine & Franklin, "Sampled-Data Control Systems, " McGraw-Hill Book Company, New York, 1958. 116. Williams, F. C., and Noble, SW, "The Fundamental Limitations of the Second-Harmonic Type of Magnetic Modulator as Applied to the Amplification of Small D. C. Signals," Proc. EE, Part II, Vol. 97, 1950, p. 445. 191

DISTRIBUTION LIST No. of Copies 20 Defense Documentation Center Attn: DDC- IRS Cameron Station (Bldg. 5) Alexandria, Virginia 22314 1 Office of Assistant Secretary of Defense (Research and Engineering) Attn: Technical Library, Rm. 3E1065 Washington, D. C. 20301 1 Bureau of Ships Technical Library Attn: Code 312 Main Navy Building, Rm. 1528 Washington, D. C. 20325 1 Chief, Bureau of Ships Attn: Code 454 Department of the Navy Washington, D. C. 20360 2 Director U. S. Naval Research Laboratory Attn: Code 2027 Washington, D. C. 20390 1 Commanding Officer and Director U. S. Navy Electronics Laboratory Attn: Library San Diego, California 92101 1 Commander U. S. Naval Ordnance Laboratory Attn: Technical Library White Oak, Silver Spring, Maryland 20910 1 Rome Air Development Center (EMTLD) Attn: Documents Library Griffiss Air Force Base New York 13440 192

DISTRIBUTION LIST (Cont.) No. of Copies 1 Systems Engineering Group (SEPIR) Wright- Patterson Air Force Base Ohio 45433 2 Electronic Systems Division (ESTI) L. G. Hanscom Field Bedford, Massachusetts 01731 2 Chief of Research and Development Department of the Army Washington, D. C. 20315 2 Commanding General U. S. Army Materiel Command Attn: R&D Directorate Washington, D. C. 20315 3 Commanding Officer U. S. Army Combat Developments Command Communications - Electronics Agency Fort Monmouth, New Jersey 07703 1 Commanding Officer U. S. Army Sec Agcy Combat Dev Actv Arlington Hall Station Arlington, Virginia 22212 1 Harry Diamond Laboratories Att: Library Connecticut Avenue and Van Ness Street Washington, D. C. 20438 2 Commanding General, U. S. Army Security Agency, Arlington Hall Station Arlington, Virginia 22207 Attn: IALOG and IARD 1 Chief, Mountain View Office Electronic Warfare Lab., USAECOM P. O. Box 205 Mountain View, California 94042 193

DISTRIBUTION LIST (Cont.) No. of Copies 1 Chief, Intelligence Materiel Dev Office Electronic Warfare Lab., USAECOM Fort Holabird, Maryland 21219 1 Chief Missile Electronic Warfare Tech Area EW Lab, USA Electronics Command White Sands Missile Range, N. M. 88002 1 Chief, Willow Run Office CSTA Lab. USAECOM P. O. Box 618 Ann Arbor, Michigan 48107 1 USAECOM Liaison Officer MIT, Bldg. 26, Rm. 131 77 Massachusetts Avenue Cambridge, Massachusetts 02139 USAECOM Liaison Officer Aeronautical Systems Division Attn: ASD L- 9 Wright-Patterson AF Base, Ohio 45433 1 USAECOM Liaison Officer Rome Air Development Center Attn: EMPL Griffiss Air Force Base, New York 13440 1 USAECOM Liaison Office U. S. Army Electronic Proving Ground Fort Huachuca, Arizona 85613 1 Stanford Electronics Laboratories Stanford University Stanford, California Attn: Dr. D. G. Grace 194

DISTRIBUTION LIST (Cont.) No. of Copies 16 Commanding General U. S. Army Electronics Command Ft. Monmouth, New Jersey 07703 Attn: AMSEL- EW AMSEL-WL- D AMSEL-WL-S (3 copies) AMSEL-WL- N AMSEL-WL- C AMSEL-WL- E AMSEL-HL- CT- DD AMSEL-RD- MAT AMSEL-RD- MAF AMSEL-RD- MAF- 2 AMSEL-RD- GFR AMSEL-XL-D AMSEL-RD- LNA AMSEL-KL- TM 1 Dr. T. W. Butler, Director Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 10 Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 48105 195

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA - R&D (Security classificatlon of title, body of abstract and Indexing annotation must be entered when the overall report ia clauified) ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION Cooley Electronics Laboratory Unclassified The University of Michigan 2b GROUP Ann Arbor, Michigan 48105. REPORT TITLE The Phase Synchronization of a Parametric Subharmonic Oscillator * DESCRIPTIVE NOT1S (Type of report and inclusive datee) April 1967 - 7695-8-T - TR 178. AUTHOR(S) (Lost name, first name, initial) Cockrell, James L. * REPORT DATE 7a TOTAL NO. OF PAGE 7b. NO. OF REFS April 1967 209 116 a. CONTRACT OR GRANT NO. 9e. ORIGINATOR'S REPORT NUMBER(S) DA 28- 043-AMC-01870(E) b. PROJECT NO. 1 PO 21101 A042. 01.02 C. 9b. OTHER REPORT NO($) (Any othernumbere that may be asaigned tis report, d. ECOM- 01870-8- T 0. AVA IL ABILITY/LIMITATION NOTICES Distribution of this report is unlimited. 1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Electronics Command Electronic Warfare Laboratory, AMSELIWL-S, Ft. Monmouth. N. J. 07703 3. ABSTRACT The general problem of phase determination for a parametric subharmonic oscillator is formulated. The analyses of several circuits are reduced to the solution of the canonical Mathieu equation. Solution is effected by the Floquet theorem and the parameters discussed in terms of the range of values encountered in experimental circuits. The transient portion of the solution procluces arbitrary constants with two possible signs, yielding the two phase states of a growing oscillation. Conditions of oscillation are discussed, and the influence of signal and noise presented. ID (AON 4 1473 UNCLASSIFIED Security Classification

UNCLASSIFIED Security Classification LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT INSTRUCTIONS ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements the contractor, subcontractor, grantee, Department of De- such as: nse activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this ie report. report from DDC." m. REPORT SECURITY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this 1 security classification of the report. Indicate whether report by DDC is not authorized" Restricted Data" is included. Marking is to be in accordice with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC a. GROUP: Automatic downgrading is specified in DoD Di- users shall request through ctive 5200. 10 and Armed Forces Industrial Manual. Enter e group number. Also, when applicable, show that optional arkings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ed. report directly from DDC Other qualified users REPORT TITLE: Enter the complete report title in all shall request through'pital letters. Titles in all cases should be unclassified.., a meaningful title cannot be selected without classificarn, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qual~mediately following the title. ified DDC users shall request through DESCRIPTIVE NOTES: If appropriate, enter the type of __ _ port, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical ive the inclusive dates when a specific reporting period is SerVices, Department of Commerce, for sale to the public, indi-:vered. cate this fact and enter the price, if known. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanain the report. Enter last name, first name, middle initial. tory notes. military, show rank and branch of service. The name of e principal au.thor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (payREPORT DATE: Enter the date of the report as day, ing for) the research and development. Include address. )nth, year; or month, year. If more than one date appears X 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 o. TOTAL NUMBER OF PAGES: The total page count it may also appear elsewhere in the body of the technical remould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall Lmber of pages containing information. be attached. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports ferences cited in the report. be unclassified. Each paragraph of the abstract shall end with.. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the ine applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U) e report was written. There is no limitation on the length of the abstract. How-,, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. ilitary department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms bproject number, system numbers, task number, etc. projct numer, system numerstask number, etc. or short phrases that characterize a report and may be used as i. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be al report number by which the document will be identified selected so that no security classification is required. Identid controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military unique to this report. project code name, geographic location, may be used as key ). OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical consigned any other repcrt numbers (either by the originator text. The assignment of links, rules, and weights is optional. by the sponsor), also enter this number(s). i AVAILABILITY/LIMITATION NOTICES: Enter any limations on further dissemination of the report, other than those UNC LASSIFIED Security Classification

3 9015 02827 4689