ENGN UMR0141 | UMR0141 | AFAL-TR-64-339 I Technical Report No. 158 Locked Instability and Forced Oscillations in Automatic Phase Control Systems E. M. Aupperle COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering v.%~~ ~ The University of Michigan Air Force Avionics Laboratory Research and Technology Division Air Force Systems Command Wright-Patterson Air Force Base, Ohio Project 4043 Task 404302 Air Force Project Engineer Lt. Larry J. Baumgardner December 1964

TECLtNICAL REPORT AFAI,-TRI-64-339 6098-2-1T Technical Report No. 158 LOCKED INSTABII,ITY AND FORCEDI) OSCI lLATI ONS IN AUTOMATIC PIlASE CONTROlI SYSTEMS by 1'. M. Aupperle Approvedby:/" is P. Ristenbatt for COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan Submitted in partial fulfillment of the requirements for the degree of Instrumentation Engineer in The University of Michigan D)ecember 1964 THE UNIVERSITY OF MICH1GAN ENGINEERING LIBRARY

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FOREWARD This report was prepared by the Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, Michigan on Air Force Contract No. AF 33(615)-1058, under Task No. 404302 of Project 04043. The work was administered under the direction of the Air Force Systems Command, Wright-Patterson Air Force Base, Ohio. Lt. L. Baumgardner served as the Project Engineer for the Air Force Avionics Laboratory. lis assistance and guidance are gratefully acknowledged,

Acknowledgments The author wishes to thank Professor Lawrence L. Rauch for his comments, technical guidance, and encouragement. Further, the author wishes to thank Mr. Ralph M. Olson for his patient help with the experimental portions of this work. Special gratitude is owed to Mrs. Martha Lee Barton, Miss Dianne Bloom, and Mr. Gary Greenlee for preparing this manuscript for publication. Finally, the author is appreciative of the support the Air Force Avionics Laboratory has given this work.

ABSTRACT This study presents an analysis and expands the understanding of the characteristics of the periodic response from an Automatic Phase Control (APC) system that may result from: a) A constant-frequency input signal within the system capture range when the system gain is excessive (locked instability oscillations), or b) A pair of constant-frequency input signals, both of which occur simultaneously within the passband of the APC system (forced oscillations). The absence of any additive random signal, such as noise, is assumed. The above cases are treated using the same technique. First, the nonlinear differential equation is determined which describes the APC system under excitation. Second, a periodic system response function is assumed and is expressed as a Fourier series. Third, this periodic series expression is inserted into the system differential equation and the nonlinear term(s) is expanded. Finally, the dependence of the arbitrary coefficients of the assumed response function on the system parameters is found by equating the equal-frequency terms that result. Although only the constant and first harmonic terms of the assumed Fourier series are retained for the complete set of calculations, the relationships thus determined agree closely with experimental observations. A principal result for the first of the two excitations is that minima of both system gain and phase shift are required for oscillation. The oscillation frequency, if oscillations occur, will be that (constant) frequency at which the transfer function of the system's low-pass filter has a phase shift of -n/2 radians. The magnitude of the periodic system response is an even function of the difference between the input signal frequency and the system oscillator's open-loop frequency. Its maximum magnitude occurs when the difference in these frequencies is zero, and decreases rapidly for increasing absolute values of the difference frequency. It is also shown that the static system phase error always exceeds in absolute value the error that would exist if no oscillations occurred. Finally, a technique is discussed for determining the APC system gain and the magnitude of the transfer 111

function of the system's lowpass filter at the frequency of oscillation from three easily performed measurements. Forced oscillations may occur in any APC system, and the analysis is considerably more complex than with the single excitation. The frequency of oscillation is related to the difference in the frequencies of the two input signals, e.g., it may be equal to an integer multiple or submultiple of this difference. The magnitude of the response is a function of the system lowpass filter characteristic, the system gain, difference in the frequencies of the two input signals and their amplitude ratio, and the difference in the frequencies of the input signal having greater amplitude and the open-loop system oscillator signal. This dependence is portrayed using several computer-evaluated examples since a direct calculation is not feasible. The system phase error is affected somewhat by the second input signal, but is primarily dependent on the initial frequency difference as in the single excitation case. The applications of this study are a) to provide the designer with a means for analyzing the susceptibility of his APC system to a locked periodic response, and b) to provide guidelines for the control of the locked periodic response. An example of the former is presented in detail in this study, and two examples of the latter are reviewed briefly. iv

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS vi LIST OF SYMBOLS ix SECTION 1: INTRODUCTION 1 1.1 Statement of the Problem 1 1.2 Review of the Literature 3 1.3 Method and Topics of Investigation 4 SECTION 2: MATHEMATICAL FORMULATION OF THE SUSCEPTIBILITY PROBLEM 6 2,1 Derivation of the APC System Defining Equation 6 2,2 Alternate APC System Defining Equations 9 2.3 Development of the Interference Susceptibility Equations 11 SECTION 3: LOCKED INSTABILITY OSCILLATIONS OF THE APC SYSTEM 16 3,1 Theoretical Determination of the Instability Characteristics 16 3.1.1 Instability Considerations 16 3.1.2 Characteristics of the Locked Oscillations 20 3.1.3 Stability Analysis for the Locked Oscillations 28 3.2 Experimental Comparisons with the Theoretically Determined Oscillation Characteristics 37 3.3 Comments and Conclusions 44 SECTION 4: SECONDARY SIGNAL INTERFERENCE SUSCEPTIBILITY 46 4,1 Introduction to the Interference Problem 46 4.2 Derivation of the Interference Susceptibility Coefficient Equations 47 4.3 Susceptibility Characteristics of the Ideal Integrator 51 4,4 Susceptibility Characteristics of Three Other Lowpass Filters 59 4.5 Stability Analysis of the Forced Oscillations 80 4.6 Alternate Analytical Approach 81 4.7 Case of System Insensitivity to Target Signal 85 Amplitude SECTION 5: SUPPLEMENTARY EXPERIMENTAL PROGRAM 90 5.1 Summary of the Experimental Program 90 5.2 Experimental APC System 92 5.3 Experimental and Comparative Theoretical Results 96 SECTION 6: CONCLUSIONS AND APPLICATIONS 108 APPENDIX A. Newton-Raphson Method for Simultaneous Nonlinear Equations 113 APPENDIX B. Implementation of the Newton-Raphson Method 115 APPENDIX C, Derivation of Some Fourier Coefficients 119 REFERENCES 123

LIST OF ILLUSTRATIONS Figure Page 1.1 The automatic phase control system. 1 3.1 The graphical solution of Eqs. 3.30 and 3.31. 24 3.2 An expansion of Fig. 3.1 near the origin. 25 3.3 B and AO versus H**Ar for H*G = 1.75. 26 3.4 Experimental curves of B and AOr versus Awr for three input signal levels. 40 3.5 Theoretical curves of S and AOr versus Awr corresponding to the curves in Fig. 3.4. 41 3.6 Direct comparison of the theoretical curve and experimental points shown for the largest value of input signal levels in Figures 3.4 and 3.5. 42 3.7 Periodic system response waveforms. 43 4.1 The harmonic relation between B and G /TWf2 53 4.2 The B values for which G/TrWf2 is minimum. 54 4.3 The limiting values of B for infinite G /Tuf2 55 4.4 The first subharmonic relationship between B and G /TWf2. 56 4.5 The a characteristics for an APC system with an ideal-integrator lowpass filter. 57 4.6 The system response characteristics for an APC system with an ideal-integrator lowpass filter. 58 4.7 Characteristics of the single pole filter. 61 4.8 System response for the single pole filter with G equal to 0,956 x 10 radians/second. 62 4.9 System response for the single pole filter with G equal to 1.20 x 105 radians/second. 62 4.10 System response for the single pole filter with G equal to 1.35 x 105 radians/second. 63 4,11 Characteristics of the single pole-single zero filter. 64 4.12 System response for the single pole-single zero filter with C equal to 0.956 x 105 radians/second. 65 4.13 System response for the single pole-single zero filter with G equal to 1.20 x 105 radians/second. 65 4.14 System response for the single pole-single zero filter with C equal to 1.35 x 105 radians/second. 66 vi

LIST OF ILLUSTRATIONS (Cont.) Figure Page 4.15 Characteristics of the three pole filter. 67 4.16 System response for the three pole filter with G equal to 0.956 x 105 radians/second. 68 4.17 System response for the three pole filter with G equal to 1.20 x 105 radians/second. 68 4.18 System response for the three pole filter with G equal to 1.35 x 105 radians/second, 69 4,19 AO characteristics for the three pole filter with G equal toS1.35 x 105 radians/second. 70 4,20 AOr characteristics for the three pole filter with G equal to 1.35 x 105 radians/second. 71 4.21 System variables dependence on Aur for Aus/2r equal to 3 kc/sec. 73 4.22 System variables dependence on Aur for Au /2,r equal to 7 kc/sec. 74 4.23 System variables dependence on Awr for Au /2r equal to 10 kc/sec. 75 4.24 System variables dependence on Awr for Aus/27 equal to 12 kc/sec. 76 4.25 System variables dependence on Awr for Awr/27 equal to 12.9 kc/sec. 77 4.26 System variables dependence on Aur for Aus/27 equal to 15 kc/sec. 78 4.27 System response as a function of Aur and Aus. 79 4.28 The harmonic relation between 8 and G'/TAus2 89 5.1 Experimental test configuration. 91 5.2 APC system circuit diagram. 93 5.3 The experimental APC system. 94 5.4 System oscillator frequency versus voltage relation. 95 5,5 System magnitude response for the experimental circuit with G equal to 1.355 x 105 radians/second. 97 5.6 System magnitude response for the experimental circuit with C equal to 1.685 x 105 radians/second. 98 5.7 System magnitude response for the experimental circuit with C equal to 1.89 x 105 radians/second. 99 5.8 Periodic forced system response waveforms. 101 5.9 Periodic forced system response waveforms. 102 5.10 Periodic forced system response waveforms. 103 5.11 Characteristics of the two-pole filter. 104 5.12 System magnitude response for the two-pole filter with G equal to 1.335 x 105 radians/second. 105 vii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 5.13 System magnitude response for the two-pole filter with G equal to 1.685 x 105 radians/second. 106 5.14 System magnitude response for the two-pole filter with G equal to 1.89 x 105 radians/second. 107 B.1 Flow chart of the Newton-Raphson method program. 116 viii

LIST OF SYMBOLS First Used Symbol ADescription on Page A A constant matrix 30 EL System oscillator signal peak magnitude 6 Er Reference signal peak magnitude 6 Es Secondary signal peak magnitude 6 Et Constant bias voltage 6 Fx A nxn matrix with columns defined by aF/ax; (i, =1,2,..., n) 29 F A column vector of functions dependent on x and possibly t 29 A real constant defined by H* = ITr(wf)I/wf 23 il(s) The APC system transfer function 4 11(w) The APC system transfer function 7 Rr(w) The normalized system transfer function, i.e., 1i(w) = 1I(w)/li(O) 8 G The APC system zero frequency gain (input signal amplitude sensitive case) 8 G' The APC system zero frequency gain (input signal amplitude insensitive case) 10 G The APC system gain for an ideal integrator lowpass filter 20 M(x) A matrix with elements f ij(x), (i, j=l,2,..., n) 113 T The fundamental period of ~(t) 30 a A complex constant 4 al,., a4 Four real constants 31 b A complex constant 4 c An arbitrary real constant 30 e (p=0,1,2,...) The unknown amplitude coefficients of the assumed Fourier series representing the system response 11 f. (x) The ij-th element of M(x) defined by fij(x) = af(x)/axj 113 ix

LIST OF SYMBOLS (Cont.) First Used Symbol Description on Page f A column vector of functions dependent on x 113 i An integer variable (i=l,2,..., n) used as a subscript 29 k As a real variable defined by 2/n/l+n 82 k As an integer variable (k=l,2,3,..,) used as a subscript 30 km The gain constant of the system multiplier (phase detector) 7 k' A real constant associated with the multiplier (phase detector) 7 k0 Gain constant of the system oscillator 7 m An integer variable: (m=1,2,3,...) 49 n As a constant: a given positive integer 29 n As an integer variable: (nul,2,3,...) 49 p An integer variable (p=12,3,j...) used as a subscript 11 s A complex variable 4 t An independent variable; time in seconds 6 t' An independent variable: time in seconds 13 to A real constant: dimensionally seconds 13 tl and t2 Two specific values of time 30 x A column vector of dependent variables which are functions of time 29 y A column vector of dependent variables which are functions of time LOr A phase constant defined by AO = 0' - o 14 r r r 0 AO The critical value of AO for which B becomes zero rc r for any fixed value of gain 38 A ro An initial estimate of A0r required by the NewtonRaphson method. 115 AOs A specific phase angle defined by AO = er - As/Wfo 48 AO An initial estimate of AO required by the Newtonso Raphson method 115

LIST OF SYMBOLS (Cont.) First Used Symbol Description on Page Ar Essentially the difference in the radian frequencies of the reference and open-loop system oscillator signals. Specifically, Ar - r o - km G/k 14 Au' Essentially, the reference and open-loop system r oscillator signals. Specifically, Aw' = - w - k' c'/k 15 r r o m m A rc The critical value of A r for which 8 becomes zero for any fixed value of gain 38 Aws The difference in the radian frequencies of the secondary and reference signals, i.e., Aus = us - r 12 I(wf) A real variable representing phase shift as a function of wf 50 a1 and a2 A pair of real variables 86 A real dimensionless variable defined by 8 = k el/f 21 8* A specific value of 8 52 Bmax The maximum value of B for any fixed system gain 38 00 An initial estimate of B required by the NewtonRaphson method 115 y A dimensionless real constant constrained by Osysl 17 E An arbitrary real positive constant 30 c(t) System response function 2 e(t) The rms value of c(t) 37 rms The ratio Es/Er 6 ni The i-th arbitrary constant (i=l1,2,...,n) 35 60 An integration constant 12 o (p-1,2,3,...) The unknown phase coefficients IP of the assumed Fourier series representing the system response 11 Or Reference signal phase angle 6 t' A real constant: dimensionally radians 13 9s Secondary signal phase angle 12 xi

LIST OF SYMBOLS (Cont.) First Used Symbol Description on Page eo(t) System phase error 6 0r(t) Reference signal phase function 6 es (t) Secondary signal phase function 6 X An unknown complex variable 18 Xi The i-th associated multiplier, defined by xi = e 1 (i=1,2,..., n) 30 A real variable 30 Pi The i-th characteristic exponent (i=1,2,..., n) 30 T An arbitrary positive time constant 17 T1' T2 A pair of arbitrary positive time constants 19 The singular point values of ~(t) found from the APC system defining equations 16 0p The principal value of % 16'np A nonprincipal value of ~ 17 ~(t) A phase function defined by ~(t) = Or(t) - eo(t) 8 ~(t) A phase function defined by ~(t) = ~(t) - p 17 ~(t) A column vector representing the real solution of a system of equations 29 A column vector representing the real solution of a system of equations 30,(t) A phase function defined by p(t) = es(t) - er(t) 10 W Independent variable: radian frequency in radians per second 7 uf The fundamental radian frequency of the periodic response 11 System oscillator open-loop radian frequency 6 WI The zero-bias, radian frequency of the system oscillator 7 0r Reference signal radian frequency 6 us Secondary signal radian frequency 12 xii

1. INTRODUCTION 1.1 Statement of the Problem The response of the Automatic Phase Control (APC) system to certain classes of input signals with and without additive wideband noise has been studied extensively. Nevertheless, the many nuances in the analysis of this relatively simple nonlinear feedback system reveal why some questions remain to be answered in greater detail. This study presents an analysis and expands our understanding of the characteristics of the periodic response that may result from: a) A constant-frequency input signal within the system capture range when the system gain is excessive, or b) A pair of constant-frequency input signals, both of which occur within the passband of the APC system, i.e., the secondary-signal interference problem. The absence of any random signal, such as noise, is assumed for the purposes of this study. Before expanding this statement of the problem, it is helpful to review briefly the operation of the APC system. The elements of the system are shown in the block diagram of Fig. 1.1. In the usual analysis, a single input signal (also referred to as the reference 2 Er sin [cot+ Or(t)] Phase Lowpass e(t) + - Detector sin [ + (t)] (Multiplier) Filter Voltagecontrolled Oscillator Eo co) ot+ o(t) Et Fig. 1.1 The automatic phase control system Manuscript released by author December, 1964 for publication as a RTD Technical Report.

signal) is multiplied by the voltage-controlled, system-oscillator signal to produce an error signal proportional to the sine of the phase difference between the two phase-detector input signals. This error signal may then be modified by a linear lowpass filter, and, along with a constant tuning voltage, used to control the system oscillator. When the APC system is operating ideally, the system response function, c(t), (i.e., the output from the lowpass filter) is related linearly to the instantaneous frequency of the reference signal. Since the system oscillator is designed to have its instantaneous frequency related linearly to c(t), the instantaneous phase difference between the system oscillator and the reference signals must be constant. Thus, the APC system response, E(t), is said "to track," "to be synchronized with," or "to be locked to" the reference signal frequency. The characteristics of the lowpass filter greatly influence the performance of the APC system. Among other things, the filter limits the system bandwidth and restricts the type of input signals which the system can track satisfactorily. For further details concerning typical APC system operation, the reader is referred to Refs. 1-12. It is easy to understand why the APC system may be unstable for large system gains. Since the system oscillator already effects one ideal integration in the feedback loop, it is only necessary for the remainder of the loop to provide, at a finite frequency, an additional phase-shift of -X/2 radians to produce potential instability. Indeed, the lowpass filter usually introduces a substantial fraction, if not all, of the required, additional, negative phase shift. Stray circuit and/or component capacitance can also be counted on to provide negative phase shift. If the APC system does have a total phase shift of -v radians at a finite frequency, then the same system having sufficiently large gain will be unstable. It is easy to show (see Section 2) that, for some APC system designs, the system gain is proportional to the amplitude of the input signal. Hlence, in these cases, the system is large-signal-unstable when sufficient negative phase shift is present. This study demonstrates that the instability described above results in a periodic system response which will be referred to as "locked-instability oscillations." The relations are determined among this periodic system response, the system phase error, the system gain, the lowpass filter, and the difference in frequencies of the reference and the open-loop system oscillator signals. A periodic response function is also obtained, in general, when a pair of constant-frequency input signals is present in the passband of an APC system. This oscillation results from the two input signals, in contradistinction to the locked-instability

oscillation of the previous case and as such represents the response to a periodic forcing function. Therefore, no specific amount of phase shift and/or gain is necessary for the periodic system response to occur. It also follows that the response is dependent on the particular lowpass filter transfer function used; as a result, the influence of specific transfer functions must be studied individually. Several questions concerning the behavior of the APC system to the double input signal case quickly arise. How does the performance of the APC system compare with that of the conventional FM detector (the limiter-discriminator)? Can a periodic response exist at a frequency not equal to the difference in the frequencies of the two input signals? What maximum amplitude (if it exists) can the periodic response have for various ratios of the input signal amplitudes as the difference in input frequencies is varied? How is the periodic response affected by variation in the system gain? This study analyzes the forced periodic response of the APC system to a pair of constant-frequency input signals and answers the questions raised above. This is accomplished by determining the relationship of the periodic response to the various independent parameters. 1.2 Review of the Literature In 1949, J. Granlund (Ref. 13) analyzed the interference resulting from the simultaneous reception of two frequency-modulated signals by the limiter-discriminator FM detector. He employed quasi-stationary analysis of this co-channel interference problem to show that the interference signal frequency was equal to the difference in the frequencies of the two input signals. The magnitude of the interference signal was also shown to be proportional to the difference of input frequencies for any fixed ratio of input signal magnitudes. Granlund proved, both analytically and experimentally, that with proper limiter-discriminator design, the stronger signal could substantially suppress the weaker signal (30-db suppression with a 0.5-db difference in input signal strength). Although his paper does not directly apply to the APC system, it is a related device and affords an interesting comparison (see Section 4). Most of the early applications of the APC system focused on the horizontal-sweep and color subcarrier synchronization circuits in television receivers. Consequently, numerous papers appeared in the early nineteen-fifties treating the acquisition and tracking ability of the APC system to pulsed television synchronization signals (Refs. 1, 2, 14 and 15). Later treatments were concerned with:

a. Precision frequency control of high-power oscillators, b. spectral purity exceeding the reference; e.g., selecting one of many reference frequency harmonics, c. tracking frequency changing references. Papers, which appeared after the mid nineteen-fifties, described the APC system response to signal-plus-noise environments, threshold effects, additional work on acquisition and tracking, and transient behavior (Refs. 3-12 and 16). The first mention of potential APC system instability for a constant frequency reference with excessive loop gain was made by A. J. Viterbi in 1959 (Ref. 4). He found that, for a lowpass filter transfer function of the form H(s) = 1 + a + -, (1.1) s 2 where a and b are appropriate constants, the system would become unstable for sufficiently high values of b. However, he did not pursue the matter to investigate the nature of the system response when it is thus unstable. In 1960, T. J. Rey published a paper (Ref. 6) in which he demonstrated that a periodic response did occur for a constant frequency input signal within the system capture range. He derived a pair of equations characterizing this oscillation and indicated that the response is nearly sinusoidal for a system gain just in excess of the critical level. In this study the same analytical techniques as Rey used were employed, but a somewhat different pair of equations were found. The significance of these equations has also been expanded. The secondary signal problem was briefly considered by C. S. Weaver (Ref. 7) in 1961. lie asserts, without proof, that an interfering signal affects the APC system preceded by a limiter and a standard FM discriminator similarly. That is, the stronger signal (even if only incrementally) will completely capture the APC system. He shows that when the interference signal is weaker than the signal to be tracked, the loop tracking range is reduced. His results are obtained using standard linear operational techniques on a linearized APC system model. The results here do not overlap his, and are principally concerned with the characteristics of the output response, as a function of the various system parameters. 1.3 Method and Topics of Investigation The following technique is used to analyze both of the problems discussed in this study. First, the differential equation describing the loop operation is written for the

system shown in Fig. 1.1. Second, a periodic response function (in the form of a Fourier series) is assumed and inserted into the system differential equation. Finally, the dependence of the arbitrary coefficients of the response function (the form of which has been assumed) on the system parameters is found by equating terms of equal frequency. (T. J. Rey (Ref. 6) used this technique for a portion of his analysis.) The first and second steps of the above-described mathematical formulation are discussed in Section 2. Section 3 contains a treatment of the locked-instability problem as a special case of the interference problem, ie., the case where the magnitude of the second input signal is reduced to zero. The necessary and sufficient conditions for the occurrence of a periodic response are discussed. The magnitude of the response is shown to depend on the system gain, the difference in frequencies of the input and oscillator signals when the system loop is opened, and the frequency at which the lowpass filter has a phaseshift of -X/2 radians. It is also shown that the static system phase error always exceeds in absolute value the error that would exist if no periodic response occurred. The theoretically derived results are then compared with experimental results for a particular lowpass filter. The two-input signal problem is examined in Section 4. Since the character of the periodic response is dependent on the transfer function of the lowpass filter, it is necessary to examine specific examples. First the ideal-integrator case is analyzed by a graphical technique. The results of this aid in understanding the influence of other possible lowpass filters. The influences of three other lowpass filters are studied using a digital-computer solution of the coefficient equations derived in part in Section 2 and concluded in the earlier portion of Section 4. This solution is evaluated by means of the Newton-Raphson method for simultaneous non-linear equations. Sections 5 and 6, respectively, consider experimental details and data (including a comparison between an experimental and analytical study of a fourth filter), and some potential applications and conclusions based on this study. Appendix A briefly describes the mathematical basis of the Newton-Raphson method used for the computations required in Section 4, while Appendix B provides the implementation details of the particular routine employed in this study. In Appendix C, the two sets of Fourier coefficients required in Section 4 are derived.

2, MATHEMATICAL FORMULATION OF THE SUSCEPTIBILITY PROBLEM 2.1 Derivation of the APC System Defining Equation The well-known APC system model shown in Fig. 1.1 is characterized by an ordinary, nonlinear, differential equation of arbitrary order, and hence its complete analysis is impossible with present mathematical methods. The characteristics of the lowpass filter can be shown to determine the order of the system differential equation. Thus, system behavior is primarily a function of the lowpass filter. Let us now consider the form of the system equation when the input signal to the APC system consists of a phase-modulated reference signal of E 2 watts and a r phase-modulated secondary signal of Es watts. The combination may be, in general, represented by -/ Er sin[wtt+O (t)] +VTE sin[wot+ (t)], (2.1) where w is the radian center frequency of the APC system oscillator under open-loop conditions (when the oscillator input voltage is simply Et). er(t) and 0s(t), respectively, are phase functions of the reference and secondary signals. For example, if the reference signal has a constant radian frequency of wr with an initial phase angle of Er, then er(t)' (Wr - W) t+er (2.2) Es is set equal to nEr and the input signal is rewritten as rE (sin[wt+er(t)] + n sin[w t+e0(t)]) (2,3) The APC system oscillator signal may be expressed by /Y-Eo cos[wot+o (t)], (2.4)

where 0 (t) is the system phase error. This locally generated signal is multiplied by the input signal in the phase detector to yield a voltage expressed by kmEE fsin[2w0t + or(t) + eo(t)] + n sin[2wot + es(t) + 0o(t)] + sin[er(t) - eo(t)] + n sin[es(t) - 0(t)]), (2.5) where k is the gain (or loss) constant of the multiplier and is dimensionally (voltage). m Typically, the phase detector output is filtered (prior to the system lowpass filter) to the extent that the sum terms in expression 2.5 may be neglected. It is thus convenient, although not necessary, to view the phase detector as having an ideal lowpass filter output with a gain km and a cutoff frequency higher than the highest difference frequency but lower than the least sum frequency. In this event, the APC system lowpass filter input may be expressed by k E E sin[r(t) - Oo(t)] + n sin[0 t) - e(t) + k'EE, (2.6) m r o r 0 s m r o where the last term is used to account for any constant offset generated in the multiplier. k' is also dimensionally (voltage) 1 m Let the system lowpass filter have a transfer function H(w). Then the system response, e(t), is, in practice, given by c(t) = k'E EI(O) kEEH) in[(t) - 0o(t)] + n sin[ s(t) - 0o(t)] (2.7) The system response is added to a constant bias voltage, Et; the resulting sum voltage controls the instantaneous frequency of the voltage-controlled system oscillator (see Fig. 1.1). Since it is possible to design an oscillator whose frequency is linearly proportional to the applied voltage, such a relationship will be assumed. Thus, the instantaneous radian frequency of the system oscillator may be expressed as (from expression 2.4) + O(t) = ko[Et + E(t)] +w, (2.8) ~o o~~~~~

where ko is the gain constant of the system oscillator expressed in radians/volt and w' is the zero-bias, radian frequency. Consequently, wo, the radian frequency of the system oscillator under open-loop conditions, is equal to Wo koEt + WI' (2.9) and so eo(t) ko(2.10) Substitution of Eq. 2.7 into Eq. 2.10 produces Co(t) - kok'ErEoO(0) + k k mErEol() s in[r (t) - o(t)] + n sin[0s(t) - 0o(t)] (2.11) If a differential phase function, ~(t), is defined by ~(t) = r(t) - o(t), (2.12) then Eq. 2.11 can be expressed alternately as 0r(t) - k k'E E 11(0) = ~(t) + k k E Eo11() fsin[o(t)] r omro o In r + n sin[o(t) + es(t) - Or(r)] ), (2,13) and,further defining, G = kokmErEo 1(0), (2.14) T(w) = H(w)/H(O), (2.15) Len Equation 2.13 becomes Or(t) - G = (t) + G () sin[(t)] + n sin[(t) + (t) - r(t) (2.16)

Equation 2.16 is the fundamental differential equation for the operation of the APC system when all noise effects except the interfering secondary signal are neglected. In concluding this section, it is useful to point out that the order of Eq. 2.16 exceeds by one the degree of the denominator of tI(w). Thus, the second-degree transfer function given by Eq. 1.1 and analyzed by Viterbi (Ref. 4) results in a third-order differential equation. Even this complexity precludes analysis by the conventional phase-plane technique. The APC system properties with various lowpass filter combinations and with n equal to zero have been extensively studied (see, for example, Refs. 3, 4, 5, 9 and 17). 2.2 Alternate APC System Defining Equations In certain applications of the APC system, a balanced phase detector is used which is nearly insensitive to input-signal amplitude variation (Refs. 1 and 2). In this case, Eq. 2.16 is invalid because the loop gain constant expressly contains the referencesignal amplitude, and the secondary-signal amplitude is seen to modulate the loop gain periodically. To account for these effects, another treatment of the input signal can be employed; this is shown by the following development. The input signal given by expression 2.3 is repeated here for convenience /I Er (sin[wot + 0 (t) + n sin[w t + es(t)] (2.17) By applying trigonometric identities, it is not difficult to show (see, for example, Ref. 18) that Eq. 2.17 is equivalent to /2 ER/l+n2+2n cos[0 (t) - Or(t)] sin ~t +t r(t) + tan' nsin[0 (t) - Or(t)] ) l+ncos Les(t) - er(t) (2.18) This expression for the input signal is illuminating in two ways. First, the dependence of the input signal amplitude variation on n and on the instantaneous phase difference of the two signals is seen more specifically than in expression 2.17. Second, the instantaneous frequency of expression 2.18 is obtained by differentiating the argument of the sine function to yield n * cos[es(t) - er(t)] wo + or(t) + n [s(t) - Gr(t)] cs (2.19) 1+n2+2n cos[es(t) - Or(t)]

Expression 2,19 indicates that the instantaneous frequency is influenced, as is the amplitude, by n and the differences in phase and frequency of the input signals. These observations suggest introducing another phase function f(t) defined by $(t) = Es(t) - Or(t) (2.20) which permits rewriting expressions 2.18 and 2.19, respectively, as Er l+n22 cos(t) sin wot + er(t) + tan1 n conis(t) (2.21) and W+ (t + + nt(t) n + cos*(t) (2.22)!an2 + 2n cos*(t) If the APC system is made insensitive to the input-signal amplitude, expression 2.21 may be replaced by /sin [Lot + o (t) + tan 1 n csin (t) (2.23) 0 +r l n cos t) where iT is retained for subsequent convenience. Observe that this modified input-signal expression is equivalent to a single, phase-modulated reference signal expression. The implications of this equivalence are discussed in Section 4. The two input-signal expressions, 2.21 and 2.23, can now be used in conjunction with the APC system oscillator signal expression (2.4) to obtain alternate system equations. Since the procedure of these derivations is identical to that of the derivation given in Section 2.1, only the results are presented here. The defining equation for an input-signal-amplitude insensitive system is sint) F-PI) -G~(1 n sin(t(t) Or(t) -G = *(t) + G' IT(w) sin,(t) + tan (2.24) where G' = kokmEoH(O). (2.25)

and k and k' are dimensionless. m m The defining equation for a system with gain proportional to the input-signal amplitude is k' Or(t) - C= (t) + G T()/l + 2 + 2n cos(t) m sin L(t) + tan'-1,n sin(t) (2.26) which is equivalent to (repeating Eq. 2,16) Or(t) - G = ~(t) + G Rt() sin~(t) + n sin[4(t) + P(t). (2.27) m In concluding this section it is worthwhile to review briefly the assumptions implicit in Eqs. 2.24, 2.26 and 2.27. 1. The phase detector has a constant gain over the entire frequency range of the system response. 2, The system response consists only of the difference-frequencysignal terms resulting from the multiplication action of the phase detector. This may be thought of as a consequence of the phase detector and/or the system lowpass filter characteristics. 3. The voltage-controlled system oscillator has a linear frequency variation as a function of applied voltage. These assumptions are neither particularly restrictive nor difficult to achieve in practice. 2.3 Develo pment of the Interference Susceptibility Equations Under certain conditions, a periodic response occurs in the APC system while the system is locked in average frequency to the reference signal. The characteristics of the steady-state solution can then be determined from the system equation in the following way, Let co s(t) = e0 + >e cos (Pwft + Op) (2.28) where wf is the fundamental radian frequency of the periodic response and the ep and are constants to be determined from the system equation. Dimensionally, the e are expressed 11

in volts and the 0p in radians. From Eq. 2.10, it follows that 0o(t) =ke + k e cos (puft+ p (2.29) 0o00 pt 0) = ep f), and, upon integration, Go 8 (t) = 0 + k e t + ko P sin (pft + ), (2.30) o o p.1PWf p where 0O is the integration constant. From this point on, it is mathematically convenient to restrict the two input signals to sinusoids. Let 42 Er sin[k t + 0 (t)] =vi/ E sin[ rt + 0 ], (2.31)./ Es sin[uot + s(t)] = v Es sin[wst + 0s] (2.32) where wr and us are the respective radian frequencies, and 0r and 0s are the respective phase angles of the two input signals. This requires that 0r(t) = (r - Wo) t + or (2.33) (t) = (us - Wo) t + 0s (2.34) Thus, in this case, the instantaneous phase difference between the two input signals becomes (see Eq. 2.20) p(t) = Aust + es - er' (2.35) where Aus = us - ur. (2.36) A linear transformation in the independent variable t simplifies some of the subsequent

equations. Let t' = t + t, (2.37) 0 where t is such that i(t') = AWS(t' - to) + es - r = A st' (2.38) or t = r (2.39) o Aws Equation 2.33 then becomes Or(t') = (r ) t' - u0) (-u (Os Or) + 0 (2.40) or O r(t') (Wr - o) t' + 0', (2.41) where ~r 8~-(wr r r w (e u s -r)' (2.42) In the following equations, the system time reference will be t', though the prime will be dropped from t' for convenience. From Eqs. 2.12, 2.30, and 2.41, ~(t) is found to bel (t) = ( -r o) t' - -k e t -P sin(pft + ) (2.43) r o r o 0 0 oo puf p from which ~(t) is obtained by differentiation 1Note that Eq. 2.30 may be substituted here without a time translation, since its coefficients are yet to be determined. 13

4(t) r - -0 k0e - k0 ep cos(pft +p). (2.44) When the APC system is locked in average frequency to the reference signal, it follows, from Eq. 2.44, that Lr - W eo-. (2.45) 0 Under this last condition, Eqs. 2.38, 2.43, and 2.44, and the derivative of Eq. 2,41 may be substituted into Eq. 2.27 to yield Ar = -k p e cos(pw ft + e ) + G (w) (sin[AOr - ko pL sin (pwft + )] p0 = f p + n sin[Ae r - k 0 pi sin (pwft + Op)+ LWst]3 (2.46) where k' A = rW - o- G (2.47) m A0r = e - eo (2.48) The same substitutions into the equivalent equation (2.26) lead to =-o e cos(pwft + p) + G T(w).l+n22n cos(Ast) L sin 0o -k sin(pwt+t ) + tan l+ncos(AwSt) (2.49) r o pmf and into the input-amplitude-insensitive system equation (2.24) lead to 14

Aw =ko e cos (pw t + ) +' TT(w) sin[Or - ko r 0 pe f lr PWf l n sin(ws t) sin (pwft + ep) + tan1 1n sin(Aw t) 1 (2.50) where awu = ur'- O - G'. (2.51) Equations 2.46, 2.49, and 2,50 are the general, interference-susceptibility equations for the APC system. It will be recalled that the arbitrary constants, ep and ep, are to be determined by equating equal-frequency terms of these equations. Note that e0 has already been found (Eq. 2.45) and that 60 is contained in Ahe. Further consideration of these equations will be presented in Sections 3 and 4. The additional assumptions implicit in Eqs. 2.46, 2.49, and 2.50, relative to the more general system equations 2.24, 2.25, and 2.27, are: I. The possible existence of a steady-state system response. 2. The ability to express the steady-state solution in terms of a Fourier series. 3. Restriction of the two input signals to sinusoidal (constant frequency) waveforms. 4., The ability of the APC system to lock to the reference signal in average frequency simultaneously with the occurrence of a periodic response. The third assumption above is imposed to facilitate the mathematical formulation. Granlund (Ref. 13) used the same technique in his treatment of the interference problem for the conventional FM discriminator. The first and fourth assumptions will be discussed in Sections 3 and 4, The second assumption follows from the first and the fact that the system response is continuous. 15

3. LOCKED INSTABILITY OSCILLATIONS OF THE APC SYSTEM 3.1 Theoretical Determination of the Instability Characteristics The existence of a periodic response, when only a single, constant-frequency input signal is present, is dependent on both the APC system lowpass filter characteristic and loop gain. This dependence is characterized in this section by analyzing first the conditions necessary for system instability (failure to maintain a constant voltage proportional to the constant frequency of the input signal). Then it is shown that, for small oscillations and with a particular filter, the same conditions are compatible with a stable periodic response. In the process, the characteristics of the APC system periodic response are found, and the general technique for determining the stability of a periodic response is discussed. 3.1.1 Instability Considerations. Under the condition that the input signal to the APC system is a constant frequency sinusoid, the system defining equations, (2.26) and (2.27), both reduce to Ar = ~(t) + G Tr(w) sino(t), (3.1) where Awr is obtained from Eqs. 2.41 and 2.47. For the analysis in this section the gain, G, may be thought of as either dependent or independent of the reference-signal amplitude since only the value of gain and not its origin is significant. In this sense, Eq. 2.24 is equivalent to Eqs, 2.26 and 2.27 and also has the reduced form given by Eq. 3.1. When the lowpass filter transfer function has a finite value at zero frequency, Eq. 3.1 has singular points at (all) the solutions of i-1 r sins (3.2) and with all of the derivatives of 9 identically equal to zero. Let the principal value, ~p, of Eq. 3.2 be defined by the requirement that mlf~~ ~~~pl < s~/2. ~(3.3) [p[ _/

Define Onp as the only nonprincipal value of Eq. 3.2 that satisfies the conditions j< lmlT c i or 4p =. (3.4) _ I I Onp I < 7 or np =. All other nonprincipal values of Eq. 3.2 are then displaced from either 0p or ~np by plus and minus integer multiples of 2v. Thus, to demonstrate system instability, it is sufficient to show that both 4p and ~np are unstable singularities. It is convenient to define ~(t) = M(t) - p. P(3.5) Substitution of the equation into Eq. 3.1 yields A =r = ~(t) + G iT(w) [sin ~(t) cos p + cos ~(t) sin.p]. (3.6) Note that cos ~p is nonnegative due to the inequality (3.3). The equation analogous to Eq. 3.6 for ~np has a factor cos 0np which is nonpositive. At this point it is necessary to assign a particular form to the lowpass filter transfer function. Initially, let II(s) = + YTs (3,7) where y is A dimensionless constant constrained to values 0 < y < 1, (3.8) and T is an arbitrary, positive time constant. This transfer function represents a filter with a real zero to the left of a negative real pole. The limiting cases,y equal to 0 and 1, respectively, correspond to a simple RC integrator and no filter at all. Substitution of Eq. 3.7 into Eq. 3.6 results in the following second-order differential equation A=r = ~(t) + TO(t) + G(sin4(t) cos,p + cos T(t) sin,p + y~ cos ~(t)'(t) cos p - yr sin 4(t)>(t) sin,p). (3.9) 17

For 7(t)<<l, this equation is equivalent to ~(t) + (T + Cy cosop )(t) + T COS 0p (t) - Gy sin p~ (t)=(t) 0. (3.10) p T p p The question of the stability of the solution ~(t) = ~(t) = T(t) = 0, (3.11) for nonlinear differential equations such as Eq. 3.10, is discussed in Chapter 13 of Ref. 19. The following theorem is proven in that reference. Theorem: Let at least one characteristic root of the characteristic polynomial of the linear portion of the differential equation have its real part positive. Let the nonlinear portion have an order not greater than the linear portion and be real, continuous, and of "order zero" as its arguments approach zero. Then the identically zero solution is not stable. It follows that the conditions for instability of Eq. 3.10 depend on values of A which are solutions to 1 C 2 + ( + Gy cos )A + COs p = (3.12) or 1 + G y T cos 1 + G y cos 2 A = -.... -— cos. (3.13) 2T 2t COS p (313) Since cos p is nonnegative, both roots are clearly negative for all values of y and T, and the singular point ~p is not unstable. Indeed, it may be shown to be asymptotically stable for all mp <Ir/21. Notice also that the singular point np is always unstable when ~p <I1/21 since cos ~np is then negative, and one root will be positive. One concludes from the above discussion that no filters of the form given in Eq. 3,7 will exhibit instability oscillations for any value of loop gain. As an extension of the lowpass filter transfer function given by Eq. 3.7, consider the class of filters with an additional pole 18

1 + yTls (H(s) =I+T 1 12 (I+(3.14) where y, as before, is defined by Eq. 3.8, and T1 and T2 are arbitrary, positive time constants. In this case, the equation analogous to (3.10) is..1+T2,. 1 + GYT G cos T1 T 2) 4(t) T 1T() + T1T2 ) t fGy sin \ l(t) i(t) = O (3.15) 2 with the third-degree characteristic polynomial + l. +T2 X2 IG 1T 2 + (. (3.16) By applying Routh's criterion to this polynomial, it may be shown that instability at the cp singular point requires that _ +,T2 G > (3.17) cos $p [L-Y) T1T2 - Y T1 2] and that thp factor in brackets be positive. The singular point, $np' is again unstable for all values of G. Thus, APC systems with filters of the form given in Eq. 3.14 will exhibit instability oscillations, provided y is not too large and the loop gain is sufficient. The most significant point of the above discussion, and the examples, is that the question of APC system stability for a constant-frequency, sinusoidal input signal can be determined solely from the linear portion of the system equation, provided the appropriate linear portion exists. This fact permits the use of all the various linear techniques on the characteristic polynomial for the determination of system stability. Consequently, the Nyquist requirement discussed intuitively in Section 11. is appropriate; i.e., if the lowpass filter contributes an additional phase shift of -7/2 radians, the APC system is potentially unstable. The instability question is essentially the same if the lowpass filter transfer function is infinite at zero frequency. In this case, the input to the lowpass filter must 19

be identically zero when the system is stably locked to a constant-frequency reference. Notice that this requires km' (see, for example, Eq. 2.6) to be zero and zero phase difference between the reference and local signals. Thus, in Eq. 3.1, Aur is zero, and the singular points occur at values of 4 equal to 0, j~i, 2r.... Hlere, p = 0 and np (3.18) For the class of filters given by Il(s) =- + (3.19) Ts where y and T have the same definition as in Eq. 3.7, the equation analogous to (3.10) is...0 G (t) + ~ G (t) + - ~(t) 3 0, (3.20) where G0 = k k mEEr for amplitude-sensitive phase detectors or Go = k k E for amplitudeinsensitive phase detectors. Clearly, the system is stable for positive y. When y is zero, the characteristic polynomial has two purely imaginary roots. For completeness, the singular point, 4np' is unstable for all values of y. It follows that periodic oscillations may exist for the H(s) given in Eq. 3.19 when y is zero. Again, the possibility of instability oscillations can be determined readily from linear analysis. 3.1.2 Characteristics of the Locked Oscillations. In this section, periodic response, e(t), will be assumed for all APC systems whose equations have an appropriate linear portion and do not have any stable singular points. The frequency of this response will be assumed equal to the frequency at which the system has a total phase shift of 7 radians. The justification for the existence of such a stable periodic solution is postponed until Section 3.1.3. In this section, the characteristics of the assumed periodic response characteristics are determined. In Section 2.3, three equations were developed from the system defining equations under the condition that c(t) has the general periodic form given in Eq. 2.28. For the case of a single input signal, (recall that here G includes G') the Eqs. 2.46, 2.49 and 2,50 reduce to 20

Awr = -ko 0 e cos (pwft + 0 ) + G TT(w) sin [A0r e - o pf sin (pwft + p)] (3.21) where wf is the fundamental radian frequency of the oscillation. Implicit in Eq. 3.21 is the condition that the average frequency of the periodic response is equal to the reference frequency (see Eq. 2.45). This may be justified in the present case if the periodic solution is indeed stable. Clearly, if some average frequency difference existed, the consequent change in phase would preclude the continued encirclement of a singular point. Thus, the legitimacy of the results of this section rests solely on the stability of the periodic solution. The presence of the final term of Eq. 3.21 makes the solution of this equation using the general periodic form extremely difficult. When only the first term of the sum is retained, Eq. 3.21 becomes keI Aw = -koe1 cos (wft + 01) + G T(w) [A0r - sin (wft + 01)]. (3.22) Since the choice of the origin of the time scale is entirely arbitrary here, we selected t such that 01 is zero. Also, let ke. o. (3.23) Wf Equation 3.22 can be written as Air =-fB cos (wft) + G TT(w) sin [AOr - B sin (ft)].(3.24) The Bessel function expansion of the sine factor yields 21

sin [AOr - B sin (wft)] - sin (AOr) [Jo(B) + 2J2(a) cos (2uft) +...] (3.25) - cos (AOr) [2J 1() sin (wft) + 2J3(B) sin (3wft) +...]. Only the constant and fundamental terms of this expansion may be retained meaningfully when substituting back into Eq. 3,24, since higher-order harmonics were dropped previously. The following pair of equations results, for filters with a finite gain at zero frequency AWr = G Jo() sin (AOr)' (3.26)2'fa = 2G I f(uf)I J1(B) cos (AOr)' (3.27) when the constant and fundamental terms of Eq. 3.24 are equated. Recall that I(Wf) has, by assumption, an amount of phase shift consonant with equating the fundamental coefficients as indicated. For filters with an infinite zero frequency gain, Aur and AOr are equal to zero and the single equation required is WfB = 2 G III(f)l JI1(). (3.28) Since Eq. 3.28 is a special case of the finite-gain equations, consider now the information available from the pair 3.26, 3.27. When the APC system is locked to an unmodulated sinusoidal reference signal (recall that these equations were derived under this assumption), and is not oscillating (fB=O), only Eq. 3.26 has any significance. For this special case, A'r = G sin (AOr). (3.29) Notice that, here, A r is equal to *p in Section 3.11; i.e., Ahr is the static, system phase error. This equation shows that the phase error is dependent on the zero frequency loop gain, 2Rey (Ref. 6) indicates that sin (AGr) is identically zero; however, this is necessary only when Au is zero. 22

G, and reference frequency offset, Awr. Several authors (e.g., Refs. 1, 2, 4, 6) have shown that the loop will remain locked under the above conditions for A8r < r/2 radians. Hence, Awr must be less than G to insure maintenance of lock, and 2G is defined as the loop "holding range." It follows that the greater G is made, the greater the holding range becomes. This is true up to a point for lowpass filters which provide a total system phase shift of n radians, at a finite frequency. For these filters, the system will begin to oscillate as G is increased. The characteristics of the periodic system response can be determined from the dependence of B and GAr on BA, G, and IH(wf)I. Equations 3.26 and 3.27 cannot be solved directly for B and A r' but these equations can easily be solved for the independent variables in terms of B and AOr as shown below: a* G = (3,30) and Jo (5) tan (AOr) H* Awr = 20-(B) (3.31) where the factor H* is a constant completely determined by the specific lowpass filter used and is defined by IlT(wf) I }1H.~~~~~~* =~. ~(3,32) Wf The solution of Eqs. 3.30 and 3.31 is accomplished graphically. (See Figs. 3.1 and 3.2, for plots of these equations.) Observe that Fig. 3.2 is an enlarged plot of Fig. 3.1 near the origin. From these plots, for example, curves showing the dependence of a and AOr on H*Awr can be determined for constant values of HI*G. This is done by selecting a value of tI*G and reading the values of B and H*Awr for each value of AOr from the curves in Figs. 3.1 and 3.2. One example is given in Fig. 3.3 for H*G equal to 1.75. Only half of the curve is shown here since the curve is symmetric about the B axis. The dashed line indicates the values that AOr would assume for this gain if no locked instability oscillations existed. 23

5 41 2 ~3 ~4 ~5 ~5 cd 7*t i6 ~5 i~4 i~ 3 ~i2 ~ 0~ 1 2 3 4 5 9 H*G1.90 Fig.H~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 31 Fig. 3.1 The graphical solution of Eqs. 3.30 and 3.31.

4 3-_ C: 2%0 030 ~ i1 5 ~1. 25 i1 ~0. 75 ~0. 5 0. 25 o.25 5 75 1 1. 25 1. 50 1.75 2 H*Awr H*G Fig. 3.2. An expansion of Fig. 3.1 near the origin.

2.25 L 900 2. 00 800~ 1. 75 - 700 1. 50 \ 600 AO r (2 1. 25 50 o1. 00 40~. 75// 300 50 // 200 *25 L 1 E o00 00 0 o25.50.75 1o 00 1. 25 1o 50 1. 75 H*Awr Fig, 3.3. B and Aer versus tH*tr for H*G - 1.75. 26

It is worthwhile at this point to write the system response and system oscillator phase in terms of B and AOr. For the assumptions made in this section, Eq. 2.28 becomes Wt - - a C(t) = Wr o + f — Cos Wtft (3.33) 0 0 and Eq. 2.30 becomes 0 (t) -= r Oer + (-r - O) t + a sin wft (3.34) Note that here er is equal to 0'. Since B and A0r are functions of Auwr, C, and H*, the APC system's oscillation is completely characterized in terms of the independent variables. We conclude this treatment of the case of finite system gain at zero frequency with some observations on Figs. 3.1 - 3.3. The peak value of the locked instability oscillations occurs for Aur equal to zero (see Fig. 3.3). For increasing l4Arl, B decreases and eventually becomes zero. For still greater Au r[,the system remains stably locked until IAOrl reaches 900, just as if no oscillation had existed. Notice the abrupt transition in the Aer vs. III*Auri curve just as S becomes zero. For the larger values of IH1*Aurl, the AOr curve is simply the inverse sine relation (Eq. 3.29), whereas for the smaller values of the independent variable the effect of Jo(B) (see Eq. 3.26) is quite pronounced. By reference to Figs. 3.1 and 3.2, the question of when oscillations begin can be answered readily. Since the peak value of B occurs when AO r(Ar = O) is zero, clearly S must be zero for all values of H*G less than unity. Once H*G exceeds unity, B varies continuously with Au r It should be remarked that the holding range continues to increase as 1i*G increases above unity. The first-order theory developed above indicates that for any finite gain, G, there should be a finite region of H*Aur in which the locked instability oscillations cease before the loop loses lock. This can be seen from the following. From Eq. 3.31 the frequency at which B goes to zero is simply BJo(a) tan (hAer) H*Awr = B-O (25 1 ( r) tan (Ar). (3.35) From Eq. 3.30 the gain, H*G, at this point is 27

H =G Z a 1 (3.36) BH GIo- 2J1(8) cos (0 r) cos (A) 3.36) 840 r Hience, ~H'BUO = tan (AG ) <,1 11* (3.37) rt*A r -COS (Ar) e *G B The implication of these observations is that the holding range can be made arbitrarily large at the expense of loop oscillation, with the magnitude of oscillation decreasing to zero prior to the loop losing lock. Due to the truncations made in developing the firstorder equations these large-gain results are of questionable validity. Furthermore, this result has not been observed experimentally; actually, abrupt jumps are observed between locked and unlocked oscillations. Finally, a few brief comments are in order concerning the behavior of systems with filters having infinite gain at zero frequency. When H(w) has a unique frequency at which the APC system has a total phase shift of 7 radians, the value of a is found from the GOH* product and the curve for Aer 0 in the way discussed previously. Ilere, of course, there is just the single value of 6 which is independent of the reference signal frequency, Ideally, the holding range is infinite. For H(s) = 1/Ts, the system has a phase shift of w at all real frequencies and Eq. 3.28 becomes G G 2= (8)X 2. (3.38) = 2Ji(B) Wf The indication is that as the magnitude of oscillation increases, the frequency of oscillation decreases for a constant value of G 3.1.3 Stability Analysis for the Locked Oscillations. The validity of the analysis made in Section 3.1.2 is primarily dependent on the existence of a stable periodic response around the unstable singular point at p = ~p (or any of the other singular points displaced from 0p by an integer multiple of 2n). Although the mathematical theory necessary to investigate this validity is available, the required calculations are very lengthy for all but the simplest filters, In this section, the general stability analysis technique is 28

outlined briefly. An example calculation is also carried out for the lowpass filter transfer function given in Eq. 3.14. The question of orbital stability may be examined by the first variation technique (see Ref. 19 for additional details). The theory for this analysis, discussed in the literature, is concerned with a system of n first-order differential equations. The same formulation is presented here. If the column vector;(t) is a real solution of the system of equations x w F(t,x) (3.39) for 0 - t < a, where the column vector F is analytic in x for each t, then the first variation equation is y = F [t, ~(t)] y, (3.40) where Fx[t, 0(t)] is a matrix composed of the columns 3F/axi (i=1,2,..., n). A special case of Eq. 3.39 occurs when P does not depend on t. In this case, the first variation equation becomes y = FX[4(t)] y. (3.41) Notice that both Eqs. 3.40 and 3.41 are linear equations with, in general, time-dependent coefficients. The APC system equation for the case of a single input signal of constant frequency does not depend on time and hence takes the latter form. If ~(t) is a periodic solution of Eq. 3.39, it follows that Eq. 3.41 represents a system of linear differential equations with periodic coefficients. The stability characteristics of the solutions can be shown to depend on the real parts of the characteristic exponents for this system. Since ~(t) satisfies the variation equation, the characteristic exponent associated with it may be taken as zero. Thus, for an n-th order system, the remaining n-l characteristic exponents determine the orbital stability of ~(t). The solution x = ~(t) may be regarded as a closed curve, or orbit in x space with t as a parameter. If n-l characteristic exponents of Eq. 3.41 have negative real parts, then the closed orbit is asymptotically stable in the sense that any solution of Eq. 3.39 which comes near a point of the orbit tends to the orbit as t approaches infinity. This is called asymptotic orbital stability. The following theorem is proved in Ref. 19. 29

Theorem: Let n-1 characteristic exponents of Eq. 3.41 have negative real parts. Then, there exists an ~ > 0 such that if a solution;*(t) of Eq. 3.39 satisfies 1;*(t2) - 0*(tl)I<c for some tl and t2' there exists a constant c such that lim I;*(t) - *(t + c)l 0 o. (3.42) t-*om Thus, not only is there asymptotic orbital stability but each solution near the orbit possesses an asymptotic phase c. Although the technique just outlined provides the desired answer to the stability question, its application is hindered by the two requirements: 1. That a periodic solution already has been found. 2. That the remaining n-1 characteristic exponents can be found. For the present application, the first of these is at least approximately met. The second requirement can be fulfilled only with considerable effort. However, a method does exist that permits determining each characteristic exponent from the known elements of the matrix, F [;(t)]. This method is described in detail in Chapter XVII of Ref. 20 and applies to those cases where each element of FX [(t)] can be expanded in the power series form F[ (k) k Fx[(t)] = A + F [(t)l, (3.43) where A is a constant matrix independent of u, and p is sufficiently small. A perturbation technique is employed to determine each characteristic exponent in the form of Pi 0) + )" + (2) 2 + (3.44) 0i" 0 i i i A noteworthy feature of this method is that any one root can be approximated to an arbitrary degree without carrying along any approximations to the other roots. One additional fact is helpful in determining the characteristic exponents. There is an associated multiplier, Xi, for each characteristic exponent, Pi, defined by PiT,i' e, (3.45) where T is the fundamental period of *(t). It is proven in Ref. 19 that 30

n I. = exp trFx[O(s)] ds. (3.46) Since one of the Xi is known to have unity value, only n-2 characteristic exponents need be found by the perturbation method. The remaining Ai can be found from Eq. 3.46. The characteristic exponent associated with the unity value Xi equals zero. The requirement that each of the remaining pi have a negative real part is equivalent to requiring that each associated Ai be less than unity in absolute value. The remainder of this section, an example of the above stability analysis is carried out for the lowpass filter transfer function given in Eq. 3.14. The third-order differential equation that results for this choice of filter is T1T2$(t) + (T1 + T2) ~(t) + [1 + GyTl cos;(t)] ~(t) 1 1 2 r For simplicity, Awr will be assumed to have zero value. This avoids making the change of variables defined by Eq. 3.5, but does not limit the generality of the following results. Using the definitions T1 + T2 1 aI =. a2 = 1 1 2 1 2 T1T2 T1T2 a G 4 (3.48) 3 T1T2 and o(t) = (t), (3.49) then, Eq. 3.47 reduces to the following system of equations: 31

oIt) = 2 l(t) +l(t) = *2(t) 0 >2(t) = -a3 sin +o(t) - [a2 + a4 cos o0(t)],l(t) - al2(t) (3.50) This system has the form of Eq. 3.39. The first variation equation is thus y(t) 1o0 1 0 yl(t) Y2(t) 0 0 1 IY2(t) y3(t) -a3cos 0o(t) + a4sin yo(t)Ol(t) -a2 - a3cos 4o(t) -al y3(t) I (3.51) The next step in the stability analysis is to insert the known solution into this coefficient matrix. From Eqs. 2.12, 2.41, and 3.34, it may be shown that ~(t) = -B sin Wft (3.52) for any lowpass filter. Substitution of this solution into the coefficient matrix of Eq. 3.51 yields the periodic matrix: 0 1 0 0 0 1 -a3cos[ sin (ft) ]-a4sin[asin(wft) ] wfcos(wft) -a2 - a4 cos[Bsin(wft) ] -a1 (3.53) This matrix has a period of 7/Wf. If the periodic terms are now replaced by their Bessel function expansion and the Bessel functions are, in turn, expanded in terms of a, the matrix takes the required form, i.e., that of Eq. 3.43 with P u 2. The constant matrix A and the first F) [1(t)] matrix are 32

0 1 0 A 0 0 1 (3.54) -a3 -(a2 + a4) -a 0 0 0 F(1) 0 0 0 x a3 a4wf a4 |T- [1 - COS(2wft)] - - sin(2w ft) l - Cs(2wft) 0. (3.55) The method given in Ref. 20 for determining the characteristic exponents may now be initiated. Before this is done, however, recall that one of the pi may be taken as zero since ~(t) is a solution of Eq. 3.51. Let p1 equal zero and hence X1 is unity. From Eq. 3.46, X2x3 - exp J -al ds = exp (.f a (3.56) Since wf is the radian frequency at which the APC system has a phase shift of -ir radians, wf is uniquely determined by the lowpass filter time constants. Indeed, it is easy to show that Eq. 3.14 (the transfer function of the lowpass filter) will provide the necessary additional phase shift (-Tr/2 radians) in the APC system when f1 (3.57) I(1 - y) T I1t2 -YT2 Substitution back into Eq. 3.56 yields (see also Eq. 3.48) 33

('1 + T2 ) ~) y'2' X2,3 - exp [- 1 22. (3.58) It will be recalled from Section 3.1.1 that the expression under the radical sign must be positive for system instability. It follows that the product 2AX3 is equal to some number less than unity. Since both 1X21 and IX31 must be less than unity, it is necessary to find only one or the other to answer the question of orbital stability. In determining any characteristic exponent using the method given in Ref. 20, the initial step is to find the roots of the characteristic polynomial of the constant coefficient matrix, A. This polynomial is simply the polynomial 3.16 with cos lp equal to unity. Let G be chosen so that the loci of two of the roots of this polynomial are imaginary, in the light of their dependence on G. This requires that T1 + T, G = _ 1 2 (3.59) [(1 - y) t1T2 - Y~12] and the three roots for this gain are Pi(0) (0) " j f (0) () p3() -a (3.60) The superscript zero is used here to denote the zero-order approximation to pi. The next step in the method is to select a particular pi and solve the appropriate system of linear, constant-coefficient equations to determine the associated YiP) For p this system is 34

(0) (0) (0) y1 a1y1 y2 =0 (0) a (0) ) =(3.61) 2 1 2 -3 y (0) + (0) (0) y3 3y1 + (a2 + 4)2 0 The superscript zero again indicates that this system is the zero-order approximation. The general solution of this system has the form (0) (a1 + jWf)t Y1 C11 c12 C13 enl (0) =jWfC1 -j a n2(a1 - jwf)t Y2 | |j 11 -jwfc12 -alc13 y(0) W |2 2 c a2 c132 ( 3 f 11 f 1 13 3 (3.62) where the ni are arbitrary constants. It is necessary, however, that the yjo) be periodic, and consequently, n1 and n2 are set equal to zero. Since c13 is also arbitrary, let n3 be unity in value, whereupon y(0) = 3 y1 = -ac13 (0) = -a1c13 (0) = a2c (3.63) Y3 21 13 The next step in the perturbation method is to solve the appropriate first-order system. The result for our example is given below: (1) - a (1) y(l) _p (1) c 1 ay1 2 3 13 (1) _ay(l) y(l). p(1) alc15 35

+ ay1 (a2 a4) y2 ) 1) 23 + [1 - cos(2wft] c (1 2 a4 y2 3 a1 c13 T+ f-[13 a4wf a4 (3.64) 2- - sin (2wft) c13 - [1 - cos(2wft)] alc13. Notice that the zero-order solutions, together with the nonzero elements of the FM1 matrix, are used as forcing functions on the yi system. This system may now be solved, and the values of pl) and the yil) determined as periodic functions of time as well as functions of the constants of the A matrix. This general procedure is repeated for higher-order approximations to the characteristic exponent and to the associated periodic Yi functions. For the purposes of this study, only p3 will be determined. This is accomplished through solution of Eq. 3.64, using the variation-of-parameters technique. It is not difficult to show that (1) a3 - a 1a4 P3 (3.65) 4 (f2+a12) Thus, p(1) is positive provided a3 -ala4 > 0, (3.66) which requires that (see Eq. 3.48) T1T2(1-y) - yT12 > 0. (3.67) This last condition is also necessary for the instability of an APC system subject to a constant-frequency input signal (see Eq. 3.17). From Eq. 3.44, the value of p3 is given by a3 - ala4 P3 a + B2 +... (3.68) 4(wf2+a12) Clearly, p3 becomes less negative as B increases, and?3 becomes larger. Consequently, X2 must decrease (from Eq. 3.58). It can also be seen that i2 is equal to unity (since o20) equals -jmf) when B is equal to zero. Both A2 and k3 are less than unity, for positive 36

which provides the criterion for orbital stability when this filter is employed. The purpose of this section has been to establish the conditions for orbital stability. As is apparent, the required calculations are lengthy, particularly for higherorder APC systems where it is necessary to calculate several of the roots. All of the details for handling the perturbation calculations may be found in Chapter XVII of Ref. 20. 3.2 Experimental Comparisons with the Theoretically Determined Oscillation Characteristics Experimentation with an APC system has confirmed the analysis of locked instability oscillation characteristics presented in Section 3.1.2. Experimental data pertinent to these instability oscillations are given in this section for an APC system having finite gain at zero frequency. Details concerning the experimental circuit and procedure are discussed in Section 5. According to the theory of Section 3.1.2, the APC system response, c(t), is given by Wr -O a fB E(t) = + k — cos Wft (3.69) o o for small values of S and for all lowpass filters providing sufficient phase shift. The response, ~(t), may easily be measured experimentally for various input conditions. Since the instability oscillation frequency, wf, and the system oscillator gain constant, ko, may also be found easily, B may be calculated directly from the experimental data. Assuming that the periodic term of Eq, 3,69 is measured with an rms-reading voltmeter, then ( is found by k = * (1.414) 2, C(t) (3.70) ff rms where c(t)rms is the measured rms value. When this equation is used, a plot of R versus the frequency offset, Awr, can be drawn from experimental data on any APC system. The reader will recall that B is a function of G and Jri(cf), as well as AWr. ITF(wf) is a constant for any given APC system. Thus, a family of S vs. Awr curves may be drawn for a specific system with the system gain, G, as a parameter. The angle A0r may also be found experimentally. A direct way to determine AOr is based on Eq, 2.48, which indicates that L0r is the difference in phase of the reference and system oscillator signals. An alternate method, based on Eq, 3,26, is as follows. Let 37

MArc be the smallest frequency offset at which B is zero. This value may be found accurately by experiment. From Eq. 3.26, AOrc is given by AWrc =G sin (Arc) (3.71) Dividing Eq. 3.26 by Eq. 3.71 yields Ar sin (AOr) Arc Jo(6) sin (A) (372) Finally, solving for AOr yields aO sin(aOrc) Wr A = sin (a)AW. (3.73)'rc This last equation permits AOr to be plotted, given the c(t)rms values. Again, a family of AOr vs. Aur curves exists, with parameter G, for each APC system. Another useful relationship may be found from Eq. 3.30. a assumes some maximum value when AOr is equal to zero. Let this value be defined as Bmax" Thus, max H*G = max (3.74) 1 max Since 8/2J1(8) equals unity when 8 equals zero, it follows from Eqs. 3.30 and 3.74 that A1 (8 max) cos AO = aa. (3.75) max This permits determining AOrc without measuring the system gain. Furthermore, if Eq. 3.75 is used to find AOrc then Eq. 3.71 can be used to determine G, since re Aw G sin (Arc) (3.76) Finally substitution of Eq. 3.76 and Eq. 3.75 into Eq. 3.74 yields fBmax sin AO rc Wf II )I =f2 i Aw ~ rc - ufr tan AO (3.77) 2aWf J (T T rc 1 max rcr These relations among the system parameters are useful in checking the results of various measurements and evaluating those quantities that are difficult to measure directly. 38

Three sets of values of c(t) versus Au were obtained, for three values of rms r reference-signal magnitude, on the APC system shown in Fig. 5.2 of Section 5,. Because this system is sensitive to input amplitude, the three sets of data correspond to different values of system gain, G. As shown in Fig. 5.4,the system oscillator gain, ko, has a value of 42.8 kc/volt. The instability frequency was measured to be 12.94 kc. The values of B as a function of Awr were found from Eq. 3.70 and plotted in Fig. 3.4. Also shown in Fig. 3.4 are the Aer curves which were found with aid of Eqs, 3.73 and 3.75. The three theoretical curves shown in Fig. 3.5 were constructed for comparison. 11*G was determined from Eq. 3.74 for each value of system gain. The curves shown in Fig. 3.2 were then used to find B and hAr in the manner that the curves in Fig. 3.3 were constructed. The close agreement between the two sets of curves shown in Figs. 3,4 and 3.5 is demonstrated in Fig. 3.6 which further substantiates the validity of the theoretical results, In Fig. 3.6, the solid curves are taken directly from Fig. 3.5 (the theoretical prediction for the highest value of input signal) and the points represent actual data values from which were constructed the experimental curves in Fig. 3.4. The three photographs shown in Fig. 3.7 portray the periodic portion of the system response. Notice that though the first two of these responses are essentially sinusoidal with time in appearance, the third definitely contains some energy in the higher-order harmonics. The high-frequency ripple evident in these photographs is introduced by the system oscillator and is at the reference-signal frequency of approximately 450 kc. A'check on the experimental values of B found from ko, Wf, and e(t)rms can be made by measuring J (B) [or J1(B)] directly. This can be done conveniently with an appropriate spectrum (or waveform) analyzer. As an example, the three values of Bmax found with the aid of Eq. 3.70 (see Fig. 3.4) are 0.69, 1.01, and 1.38. The corresponding values of amax found from J (Bmax) are 0.70, 0.96, and 1.33. Finally, the experimentally determined values of H*G, G, and ITT(wf) I, found from Eqs. 3.74, 3.76 and 3,77, respectively, are listed below for the three gain levels. Input Signal Level H*G G IT(Wf) I (volts rms) _______________ (kc) 0,.320 1.06 32.75 0,.427 0.350 1.14 34,0 0,434 0.400 1.,28 38.5 0,430 39

900 goo AO r 800 1. 40 70~ 1. 30 | \ Input Signal Level 0. 400 volts rms 1o 20 - - 600 1. 10 4000 volts rms 1. 00 O. 350 volts rms - 500 0.350 volts rmsvls 90 0 35. 320 volts rms,$.~~~~~~70 - ~~ 4.70 0. 320 volts rms.60 -300 50 o40 - - 200 30 - ~20 - / -100 10 00 0 3 6 9 12 15 18 21 24 27 30 33 36 39 AW (kilocycles/second) Fig. 3.4 Experimental curves of B and A8r versus for three input signal levels. 40

1. 30 [ \ H*G 1. 28 1O. 4 \ // / - 605 1* 1.5. 28 j1. 20 - \ \ 0 \ 30 1\ 10 -20 (0 90 9 corresponding to the080 40 0~. 4170 1.06 6, 60 -300 0. 50 - 0. 40 200 0O 30 0O 20 - 10l 0O 10 0 3 6 9 12 15 18 21 24 27 30 33 36 39 Aw r (kilocycles/second) 41

90~ 80O 1. 40- 700 1.30 1. 20- H*G 1.28 60~. Input Signal Level i. 1. 10;Ej \ 0. 400 volts rms "1. 00 - 50 0. 90 0 0.80 -- 40~ 0. 70 0.60 —- 30 0.50 0. 40 — -- 200 0. 30 0. 20 _ 100 0, 10 00 0 3 6 9 12 15 18 21 24 27 30 33 36 39 A wr (kil-ocycles/second) Fig. 3.6. Direct comparison of the theoretical curve and experimental points shown for the largest value of input signal level in Figures 3.4 and 3,5. 42

a) Input signal level 0.320 volt rms Zero frequency offset Vertical scale 0.20 volt/division *Horizontal scale 20 ps/division b) Input signal level 0,350 volt rms Zero frequency offset Vertical scale 0.20 volt/division Horizontal scale 20 us/division c) Input signal level 0.400 volt rms Zero frequency offset Vertical scale 0.20 volt/division Horizontal scale 20 us/division Fig. 3.7. Periodic system response waveforms. 43

Notice that IH(wf) J is nearly independent of the input signal level, as was expected. The values of G shown above also represent the maximum frequency offset at which the APC system will remain locked (see Section 3.1.2). Although a linear relationship between G and the input signal level was assumed in the analysis, this is not quite appropriate to the experimental circuit. The nonlinearity was introduced by the experimental phase detector, where diodes were employed as envelope detectors. These diodes influence -the phase detector gain. Since the diode characteristics are sensitive to the input signal level, then the phase detector gain and, hence, G, varies nonlinearly with the input signal level. 3.3 Comments and Conclusions The most significant result of this section is the relatively simple expression for system response for the locked instability oscillations given by Eq. 3.33. This equation relates ~(t) to the input signal parameters and the appropriate APC system constants. The simplicity and general applicability of Eq. 3.33 stems from the fact that knowledge of only the constant I1* is necessary to predict behavior, rather than detailed knowledge of the system lowpass filter characteristics. This fact permits the designer of APC systems to determine rapidly the possibility of instability oscillations as a function of system gain, G. It is worthwhile to compare here the requirement found in Section 3 1 2, i.e., that H*G must be greater than unity for oscillation to exist, with the gain requirements necessary for system instability with an input signal of constant frequency as found in Section 3.1.1. This can be done conveniently, for the example lowpass filter transfer function given by Eq. 3.14, by setting y equal to zero. In this case, H* is given by T1+T2 1 (T12 + 22)12 + T12T22W4 (3.78) WWf where wf is found from Eq. 3.57 as 1 (3.79) Hence, the product H*G is given by T1T2('T1 + r2) G 12 G ti*G =........ (3.80) 1 122 T1 + t2 T12+2T1T2+r24

Since fl*G must have a value greater than unity for oscillations to exist, it follows that r1 +'2 G > (3,81) T1t2 As expected, inequality 3.81 is identical to inequality 3.17 in Section 3.1.1 when y and ap are zero. It is also possible to show that the same result holds for nonzero y and ep; however, the calculations are considerably more involved. Note that, although the theoretical development of this chapter is premised on an input signal of constant frequency, the results appear to apply also for slowly modulated input signals. That is, it has been experimentally observed that the same instability oscillations occur when the input signal is (slowly) modulated. Indeed, if an audio modulation signal is present and the system gain is then increased until oscillations result, the audio signal may still be successfully demodulated. In this case, the system response consists of the audio signal and some oscillations. Assuming that the oscillation frequency is sufficiently high, it may be filtered out without appreciably affecting the audio information, Since the system holding range increases with gain, a trade-off between the oscillation amplitude and the system bandwidth may be considered. 45

4. SECONDARY SIGNAL INTERFERENCE SUSCEPTIBILITY 4.1 Introduction to the Interference Problem In contrast to the specific requirements for the occurrence of locked instability oscillations, the simple existence of a second periodic input signal leads to a periodic response of an APC system. Although neither special gain nor phase conditions are necessary for the existence of oscillations, both the system gain and phase shift substantially influence the character of the periodic response. In fact, the frequency dependence of the transfer function of the lowpass filter over a wideband must be accounted for in the analysis presented in this section. In the treatment of locked stability oscillations in Section 3, it was sufficient to know only the value, IH(wf)I. Indeed, as will be demonstrated, provision for the effect of the secondary signal in the interference susceptibility equations precludes a relatively simple analytical treatment of this problem. In this section, an extension of the theoretical techniques of Section 3.1.2, which yielded the dependence of the periodic response on system parameters and input signal characteristics, are applied to the secondary signal problem. The initial work makes use of the first of the three interference susceptibility equations derived in Section 2, i e,, Eq. 2.46. The reader will recall that this equation is premised on the assumption that the APC system is sensitive to the input signal amplitude. A set of three equations in three unknowns is found by equating the appropriate coefficients of Eq. 2.46. These "coefficient equations" are used to determine the dependence of the periodic response on the various significant system and signal parameters. This dependence can be examined in detail only for particular choices of lowpass transfer functions. The initial choice of lowpass transfer characteristic selected is that of the ideal integrator. The three coefficient equations for this characteristic reduce to a single equation which can be solved graphically. This example provides useful insight into the general character of the solutions for more practical system filters. Next, three other APC systems are studied for selected lowpass transfer characteristics. For these studies, the three coefficient equations were solved using a digital computer, 46

This section continues by considering briefly the stability question of these "forced" oscillations. The procedure is very similar to that presented in Section 3. Also included is a discussion of an alternate approach to the determination of the response characteristics in terms of the input signal and system parameters using the second of the interference-susceptibility equations. This section concludes with a short treatment of the case when the system is insensitive to input signal amplitude. 4.2 Derivation of the Interference SuscePtibility Coefficient Equations In Section 2.3, three equations were developed from the defining equations of the system under the restriction that E(t) has the general periodic form given by Eq, 2.28, The first of these, Eq. 2.46, will be considered further in this section, and is repeated below for the reader's convenience. co Air = -k Z e cos (pw ft + ) r o P P p pal + GfT() (sin[Ar -- P sin (Pft + )]P (4.1) p=l PWf X eP + n sin[Ae0 - kO p sin (pwft + 0p) + AWst] Implicit in Eq. 4.1 is the condition that the average frequency of the periodic response is equal to the reference frequency (see Eq. 2.45). As in Section 3, this assumption is justified whentthe periodic solution is stable, Clearly, if some average frequency difference existed, the consequent increasing (decreasing) phase value would cause a spiraling solution path as opposed to the required closed-loop path. The validity of the calculations made in this section thus rests solely on the stability of the periodic solution. Following the same approach as employed in the last chapter, we will simplify Eq. 4.1 by retaining only the first term of the sum expressions; i.e., we assume that the waveform of the periodic response is sinusoidal. The simplified equation may be written as AWr' -Baf cos (Wft + e1) + d(w) sin[Or - sin (wft + e1)] + n sin [aer - 8 sin (wft + 01) Ast], (42) where 8 is defined as before by Eq. 3.23. If we replace the independent variable t by

t a t' - (4,3) Wf and define 60 = AO -. el (4.4) s r w f1' after the prime notation is dropped, Eq. 4.2 becomes ar = -6 Wf cos (wft) + Gi(m) (sin[Aer - B sin (wft)] + n sin [AOs + Awst - 8 sin (wft)]). (4.5) The Bessel function expansion of the sine terms in Eq. 4.5 is sin [Aer - 8 sin (wft)] + n sin [Aes + AwSt - 8 sin (w ft)] = [sin (AGr) + n sin (AOs + Aw St)] cos [8 sin (wft)] - [cos (A r) + n cos (AOS + Awst)] sin [8 sin (wft)] = [sin (AO r) + n sin (AOs + Awst)] [Jo(B) + 2J2(8) cos (2wft) + [cos (ADr) + n cos (AO8 + Awt)] [2J1(B) sin (wft) + 2J3(B) sin (3wft) +...]. (4.6) Substitution of this expansion into Eq. 4.5 yields Awr + Bwf cos (wrft) = GF(w)([sin (AOr) + n sin (AOs + Awst)] [Jo(B) + 2J2(B) cos (2wft) +...] - [cos (A r) + n cos (AGo + Aw s] * [2J1(B) sin (ft) + 2J3() sin (ft) +.... (4.7) 48

Again following the procedures employed in the last chapter, we retain only those terms in Eq. 4.7 which are constant or vary with the fundamental periodic response radian frequency, Wf. This truncation is justified for APC systems having sufficient attenuation of higher frequencies. The decision of which terms of Eq. 4.7 are to be retained is based on the relationship between Aws and wf. In Ref. 21, Stoker states that four relationships may exist between the forcingfunction frequency and the natural system frequency in nonlinear systems. In general, one may expect that mf; 7 WS' (4.8) where m and n are integers. When m and n are both unity, the system response oscillation is said to be harmonic, When m equals one and n is greater than one, the oscillation is termed subharmonic. When' n equals one and m is greater than one, the oscillation is called ultraharmonic. When neither m nor n are equal to unity, the system oscillation is said to be ultra-subharmoni c. In the experimental circuit discussed in Section 5, all but the last of the above forms of oscillation have been observed. However, the harmonic and subharmonic cases predominate. For this reason, the relations between wf and Aus in the remainder of this analysis is assumed to be given by AWs = nwf n = 1, 2, 3,.,.. (4.9) Substituting Eq. 4.9 into Eq. 4.7, and retaining only the constant and fundamental terms, we obtain Amr + Bf cos (Wft) u GT(w) CJo(B) sin (Aer) - 2J1(a) cos (Aer) sin (aft) +?I [Jn1(8) sin (wft + A5S) + Jn(a) sin (A s) - Jn+l(a) sin (wft - Aes)] n a 1, 2, 3, (4.10) Separately equating the constant and fundamental terms of Eq. 4.10 yields the following pair of equations 49

Awr = G[Jo(6) sin (A r) + n Jn(B) sin (Aes)], (4.11) SWf cos (wft) - GIT(wf) [-2J1(B) cos (AGe) sin (wft) + n J 1 (B) sin (wft + A 5) n Jn+l () sin (wft - AGs)] (4.12) It is now noted that for small 8, J n+1() is small compared with Jn l(B). Thus, Eq. 4.12 can then be approximated by Bwf cos (wft) = GH(wf) [-2J1(8) cos (AOr) + n Jnl(B) cos (AOe)] sin (wft) + [nJnl(B) sin (Aes)] cos (wft), (4.13) Now, if T(wf) is written in the form 7i(wf) = IT(wf~)I exp [jQl(wf)], (4.14) Eq. 4.13 can be written as Bwf cos (wft) = GF(wf) [-2J() cos (Aer) + n Jn l() cos (Aes)] sin [w ft + Q (wf)] + [r Jn- (8) sin (AOs)] cos [umt + Q (Wf) (4,15) Equating coefficients in Eq. 4.15 yields the following two equations 1 Jn- 1(B) sin (AOs) cot (W f) a Jn 1(B Cos' ( s " (B) cos (Ar)' (4.16) BWf U GIR(wf)| [4J12(8) cos2 (Aer) - 4J1(8) cos (AOr) n Jnl1(B) cos (AGs) + 12 j2 1(8)]1 2 (4.17) SO

Equations 4.11, 4.16 and 4.17 are the coefficient equations from which the dependence of AOr, Aes, and a on n, Aurs Aws, G and T(w) can be found. This dependence information in turn permits predicting the secondary signal interference susceptibility characteristics of the APC system, at least for small B. Unfortunately, these three coefficient equations are quite complex and cannot be solved conveniently even by graphical techniques. In Sections 4.3 and 4.4, solutions of these equations will be found for various lowpass filter characteristics. In concluding this section it is necessary to emphasize a fact that is implicit in the development above. The reader will recall that there are neither special gain nor phase-shift requirements for the existence of oscillations in the present case. Nevertheless, the applicability of this study is limited to values of APC system gain such that locked instability oscillations will not occur. This restriction prevents the simultaneous occurrence of two distinct modes of oscillation and, consequently, permits writing the form of the oscillatory system response in terms of a single Fourier series, as was assumed originally. 4.3 Susceptibility Characteristics of the Ideal Integrator In the previous section, three coefficient equations in three unknowns were derived for those APC systems whose gain is proportional to the input signal level. If, in addition, an ideal integrator3 is assumed for the system's lowpass transfer characteristic, then these three equations reduce to the following single equation, G co B ~~~~~~~~(4.18) 2 +Jn 1B)' where G is equal to kokmEoEr as before (see Eq. 3.20) and T is the integrator time constant. This single equation follows from the fact that AOs must be equal to either 0 or i radians 3In Section 3.1.1 it was shown that the assumption of an ideal integrator lowpass filter characteristic produced an unstable system and, consequently, the coefficient equations developed in Section 4.2 do not apply. With an arbitrary, small amount of damping, however, the integrator system is stable (see the comments following Eq. 3.20). The reader may choose to view this section as either an approximate solution to an almost-ideal integrator case, or an exact solution to the ideal integrator case with an approximate model. In either case, the purpose of this section is to illustrate the basic susceptibility characteristics for this particularly simple case. In the following section, rigor is achieved at the expense of considerable additional complexity. 51

as required by Eq. 4.13 whenever the lowpass transfer characteristic has a phase shift of r/2 radians, and the fact that dAr must be equal to zero, since Awr is zero (see Eq. 4.11). Thus, Eqs. 4.11 and 4.16 are identically equal to zero, and Eq. 4.17 reduces to Eq. 4.18. Notice also that Eq. 4.18 reduces to Eq. 3.38 when n becomes equal to zero. The solution of Eq, 4.18 for 6 can be obtained graphically for each value of n. Figure 4.1 is a plot of a versus G /TWf2 for various values of n and for n equal to one (the harmonic solution). The three values of n shown (0.1, 0.2, and 0.355) correspond, respectively, to secondary reference signal power levels of -20 db, -14 db, and -9 db. Three aspects of these curves warrant comment. The first, which aids in constructing these curves, concerns the intersection of all curves for the various values of n. It is easy to determine the point of intersection. Let 6* be the value of the ordinate at this point and equate 4. 19) 2J ('*) + nJO(B*) 2J l(*) o(.*) (4.19)* The solution of this equation for nonzero n requires that JO (*) equal zero. The first zero of Jo(6) occurs for S* equal to 2.4048. The corresponding abscissa value is given by:*/2J1(6*) and is equal to 2.317. The second aspect concerns the curves in Fig. 4.1 that result from subtracting the nJ (B) term in the denominator of Eq. 4.18. All of these curves have a minimum value of 0 G/T f2. The coordinates of this minimum point may be determined by equating the following expression to zero d[GC/Tw f2] 3J10() - nJo(B) - 6[2Jo(M) + nJl() ] ~ -.............0 (4.,20) [2J1 (B) - rJo()]2 and solving for n 2[2J 1 () - J o(B) rn jo (8) + J (8).(4.21) A graphical solution of Eq. 4.21 is presented in Fig. 4,2. As n increases, the value of B at which the minimum value of G/Twf2 is reached also increases. The corresponding value of G/TWf2 is again found from Eq. 4.18 once the value of 8 has been found. The locations

2. 50 2.25 2. 00 1.75 (Radians) 1. 50I I!. 25 1. 00 / II 0.75 I /I \ ~~ r/ =.2 r 77 =k.2i~ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 G/TCOWf2 Fig. 4.1. The harmonic relation between B and G /Tm~2

2.0 1. 6 (Radians) 1.2 0. 8 0. 4 I.. I I 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 4.2. The B values for which G /TWf2 is minimum. of these minimum values are useful for constructing the graph shown in Fig. 4.1. A further significance to these minima is that as G /Twf2 is decreased, S must exhibit a jump discontinuity for each fixed value of n. This point will be seen more clearly in the construction of Fig. 4.5 below. The third aspect concerns the limiting value of the curves in Fig. 4.1 for high values of G/TWf2. From Eq. 4.18 it is clear that this occurs for values of S such that 2J1(a) - nJo(B) O. (4.22) The solution of 5 as a function of n for this equation is given in Fig. 4.3. Again this information is useful in constructing the original graph shown in Fig. 4.1.

The next step in determining the interference susceptibility characteristics of an APC system with an ideal-integrator lowpass filter is to plot B versus G./Twf2 for various values of N and for n equal to two (the first subharmonic solution). These curves, shown in Fig. 4.4, lack distinctive features due to the relatively simple form of the denominator in Eq. 4.18. In principle, one next can construct the appropriate curves for n equal to three, etc.; however, the contribution from these higher order subharmonic solutions will be relatively negligible. This statement follows from two facts. First, the secondary signal will now be essentially outside the passband of the APC system, and second, the shape of the higher-order theoretical curves restricts any appreciable contribution. Using the curves given in Figs. 4.1 and 4.4, B can be plotted as a function of Aws (the difference in the frequencies of the reference and secondary signals) with n as a parameter. The dependence of A0st AOr, and 5 on Aws, Awr, n, G, and l(w) is then 1. 0 0. 8 (Radians) 0. 6 0. 4 0. 2 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Fig. 4.3. The limiting values of B for infinite G /Twf2. 55

2. 5 2.0 (Radians) 1. 5 / 7 i7=. 355 o~~~~~~~~~~~ ~~~~// 1.0'":~~~~~I! i/ 0.5 ii I" I /r / I o. 0. 5 1.0 1.5 2. 0 2. 5 3.0 GjT7Cof2 Fig. 4.4. The first subharmonic relationship between 8 and Gc/Twf2

2. 0 G / = l sec-2 1. 6 00 (Radians) 1. 2 0. 8 =. 355 0. 4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 Aw (radians/second) Fig, 4.5, The B characteristics for an APC system with an idealintegrator lowpass filter. established for this particular lowpass filter. From AOs5 AOrt and 8, the values of 01o 00o and el can be determined to complete the solution. The reader will recall that both Aur and AOr are zero for the ideal-integrator lowpass transfer characteristic, while AOs is equal to either zero or w radians. The value of A0s is dependent on Aus as will be seen shortly. Figure 4.5 shows 8 versus Aus for the three values of n. It was constructed as follows. The ratio G /T was assigned unity value, and Aws was allowed to range between 0 and 2.8 radians per second. With Aw5 equal to wf9 Fig. 4.1 was used to find B as a function of Aus over its range for each value of n. The limiting values of 6, for Aus equal to zero, were read from Fig. 4.3. Superimposed on this plot (the second group of peaks) are the values of B obtained from Fig. 4.4 (first subharmonic solution) with Aus equal to 2uf. The 57

periodic system response frequency is equal to Aus except within the range of the second peaks, where wf is half of Aws. In Section 3, it was sufficient to plot B, el being known immediately since B and e are proportional when the system oscillation frequency is constant (see Eq. 3.23). In the present case, wf is not constant. The curves of k el versus Aw, shown in Fig. 4.6, are obtained directly from Fig. 4.5 by multiplying each ordinate value by the appropriate value of wf. These two sets of curves differ in one important aspect —the periodic system response amplitude decreases to zero (nearly linearly for low frequencies) with decreasing As', whereas B approaches a limiting value. The variable AOs is equal to 0 radian for the negative sign in Eq. 4.18 and 7 radians for the positive sign. Translating this to either Fig. 4.5 or 4.6, it can be seen that AOs is equal to zero for values of Aws from zero to the occurrence of the first jump, i.e., 0.8 cps for n = 0.355. For all higher frequencies, AO is equal to 7. 2.0 1.6 2.<-2 Ws (radians/ = 1 second) 8 0. 8~~~~ ~58 \= 0. 355 40.1 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 AWo (radians/second) 58

In conclusion, the reader is reminded that the purpose of this section has been to introduce the basic characteristics of secondary-signal susceptibility interference. As will be seen theoretically in Section 4,4, and experimentally in Section 5, these results, although somewhat extreme, show qualitative similarity to the results for other transfer characteristics. One indeed finds essentially abrupt jumps in the system response amplitude, and rapid changes in Aes, for certain lowpass filters. A more complex graphical analysis of the three, original, coefficient equations can be carried out for any H(w) that has a negative r/2 radian phase shift at a finite frequency. This analysis is applicable to only a narrow range of frequencies about the critical frequency above. Here the original three equations reduce to T(wf)G (4.23) f 2J 1(a) Cos(A) J n.AO v n and -T(wf)AWr BJo(a) sin (Aer),.of (4.24) 2J1(a) cos (A r) njn 1(). (424) f 1 r n-l The procedure for solving these equations is very similar to that used previously in this section and in Section 3, 4.4 Susceptibility Characteristics of Three Other Lowpass Filters In this section, we return to the problem of the effect of more practical lowpass filter characteristics. For such filters, none of the three coefficient equations developed in Section 4.2 is identically equal to zero. The first step in the analysis procedure for any given filter is the specification of H(w) in terms of a magnitude and phase function of frequency. The sequence of subsequent steps is somewhat arbitrary. We have chosen to specify next the system gain, G. Finally, the three dependent variables (AOer Aes, and a) are solved for in terms of Awr and Awst with n serving as a parameter. The technique for solving the three coefficient equations used in this study is an extension of Newton's method for finding the zeros of a function in one variable, This extension, known as the Newton-Raphson method, is discussed in Appendix A. Both of these methods are iterative procedures and thus are easily implemented for solution on a digital computer. A flow chart for the Newton-Raphson method solution to Eqs. 4,11, 4,16, and 4.17, 59

along with the derivation of the required matrix coefficients for the method, are presented in Appendix B. In the remainder of this section, the computer solution for three specific lowpass filters are presented as examples. The first solution is for a stable APC system with a lowpass filter having a single real pole at -6.06 x 104 radians/sec. This value corresponds to the design value for the particular lowpass filter incorporated in the experimental circuit shown in Fig. 5.2. The square of the magnitude of the filter transfer function and the cotangent of the filter phase characteristic were computed, and are plotted over a portion of their range in Fig. 4.7. These two filter properties are required in the program based on the Newton-Raphson method (see Appendix B). Notice that the system is necessarily stable, since the cotangent never reaches zero at a finite frequency. The three sets of curves shown in Figs. 4.8-4.10 correspond, respectively to three distinct selections of system gain. Each of these sets shows kel1 (equal to AwsB) plotted versus LAs, for Awr equal to zero and for the values of n indicated. Only the harmonic solution has been calculated from theoretical considerations and plotted here. All of these curves are far less peaked, and lack the abrupt jumps, obtained with the ideal integrator characteristic. Nevertheless, there is a maximum value to the periodic system response, which increases both in amplitude and frequency with increasing loop gain, for each value of n. The same general characteristics are observed for nonzero values of Aur, as will be seen in a later example.,For the second example solution, a zero located at -19.15 x 104 radians/sec. is added to the pole of the first example in the filter characteristic. This type of filter frequently is found in practical APC systems, the zero being located to optimize the system's signal-to-noise characteristics (see Ref. 4 for the appropriate design equations), The squared magnitude of the transfer function, and the cotangent of its phase angle for this filter are shown for this in Fig. 4.11. This system is also stable; indeed, it is more so since the cotangent always has a smaller value than in the last example. The same system gains were chosen for this example to facilitate comparison. The Figs. 4.12-4.14 are the curves of koel versus Aws with Awr equal to zero. The curves are even less peaked and of lower amplitude in this example than in the previous one. Notice, however, that as before, the maximum system response amplitude, and its frequency of occurrence along the Aus axis, increase with increasing system gain. 60

1.0 Magnitude Squared 0. 8 0.6 0.4 -2 0.2 -3 0 4 8 12 16 20 ws /27T (kc/s) Fig. 4.7. Characteristics of the single pole filter,

10 O~ 8 Aw s/27 (kc/s) Fig. 4.8. System response for the single pole filter with G equal to 0.956 x 10 radians/second. ko e1 o a 2 08s/28 (kc/s) Fig. 4.9. System response for the single pole filter with G equal to 1.20 x 105 radians/second. 62

c 6 4k el.1 8 16 24 Aow s/2r (kc/s) Fig. 4.10. System response for the single pole filter with G equal to 1.35 x 105 radians/second. In the third example solution, the zero in the filter characteristic of the second example is replaced with a pair of complex conjugate poles. The filter transfer function is then given by 1.984 x 1015 (s+6.06x104)(s+1,207xlO~+j 1.348x105)(s+1,207x105- j 1.348x105) (4.25) This particular filter characteristic was originally chosen as an approximation to the composite lowpass filter characteristic of the experimental system shown in Fig. 5. Consequently, a relatively complete theoretical study of this case was performed and is presented here for its general interest. Subsequent experimental studies indicated, however, that the actual filter characteristic was not as accurately approximated in the example as was thought. Once again, the square of the filter transfer characteristic magnitude, and the cotangent of the filter phase angle were computed and are plotted over a portion of their range in Fig. 4.15. It is interesting to compare the three sets of filter curves for these three examples. The magnitude curves are nearly identical, particularly the first 63

Magnitude Squared 88.6 ~44 Cotangent of the Phas( nl -1.2~~~~~~~~~~~~~~~~~~~~~~~~~~~~-3 0 4 8 12 16 2 Aw /2ir (kc/s) S Fig. 4.11. Characteristics of the single pole-single zero filter.

?I 6 6 2 2 24 0 S Fig. 4,12. System response for the single pole-single zero filter with G equal to 0.956 x 105 radians/second. koel )6 4 O 8 n =.355 -2-I 0 13 S m16 2s / (kc/s) ig. 4.13 System response fo the single pole-single lter wi G equal to 1.20 x 105 radians/second. 65

ko el C o.355 a) 1~~~~~~~~~~~~~~~~~~~~~~~~~7.2 2) l]. 24 16 sL/2 (kc/s) _oi___- 1 0 System response for the single ole-sinle ero filter Fig. 4.14. yt 135 x 10rad nsc with G equal to 1 anm while the phase cu rves differ ntb' in c t hi psren uand third, "fr ke. nsyastem Y to -7/2 radians at afreuen gain must be limited in order for phase shift is equal mo that the system... ~ edbl uns table. This means thate is potentially hn ation analysis to aply e the same as the values the forced Oscill Ues of gain selected for this example" ayto 5 db,,yhe three oa ey correspond respectivel ehe thre-s In this caSe theyolations to exist. Figures sed in the first two orloc.ed instabilitY o,:- of s n than is required to zero and for the value and 0 db less g~" for Amr equa graphed. less gain ~ vesu 6 O-Oted an" rphJ are again plots of 1oel. solution has been comp 4.16-4c18h os-in, only the first har-oie i a litude, than in th VreviouslY chosen* Again, siderablY, greater n hat n prev i o usly'' - - and are consis case is some yhes ves are much more pae respo nse th ~~~The ge reral appearance Setor... _.ticlaryfo earlier exampleS. h en.., case in the last scxn' simlarto hatof heideal-integrathighest system gain.

-+2 Lt 0 Magnitude Squared Cotangent of the Phase Angle.8.6.4.22 0 4 8 12 16 20 Fig. 4.15. Characteris/2ti (kc/s) Fig. 4.15. Characteristics of the three pole filter.

M Rn=. 3 355 0 6 = _A2 4 koel | / / 71=. 1 koel 0 8 16 24 aw S/21r (kc/s) Fig. 4.16. System response for the three pole filter with G equal to 0.956 x 105 radians/second. 1 = 355 koel 8 2 — /s 668 1W 0 0 8 16 24 68

1 =.355 10 =.2 k0e1 8 C: 4 0 8 16 24 0 8 16 A /2 (kc/s) Fig. 4.18. System response for the three pole filter with G equal to 1.35 x 105 radians/second. For this filter the variables A0s and Aer have also been plotted versus Aws, for Awr equal to zero. Each of the two sets of curves shown in Figs. 4.19 and 4.20 is for the greatest value of system gain. Similar curves, of less amplitude, occur for lower values of system gain. Observe that AOs has the general appearance of a step function, as was found for the ideal integrator. Also, AOr is nearly equal to zero except in the immediate vicinity of 12.9 kc, the frequency at which the filter has a phase shift of -i/2 radians. Notice further that the required values (AO5 - w radians and Ar = 0 radian) of S r these two dependent variables at 12.9 kc has been found by the computer solution, which serves as a check on the computational method. 69

4. 8 4. 0 AOS 3. 2 Cd rCd 2. 4 1=355'a 1=. 1. 6.8 0 8 16 Aw /2iT (kc's) 24 32 S Fig. 4. 19. AOs characteristics for the three pole filter with C, Sulo13x ainscn5 equal to 1.35 x 10 radians/second.

.2.1 A 0 r 0 -.2 77.355 -.3 0 8 16 2432 Aw /277 (kc/s) 24 S Fig. 4.20. Aer characteristics for the three pole filter with G equal to 1.35 x 105 radians, second.

Figures 4.21-4.26 (each at a constant value of Aus) demonstrate the behavior of the three dependent variables as a function of Awr. All of these curves are for the intermediate value of system gain and for n equal to 0.2. In these curves B, rather than k el (equal to AusS), is shown. Since Aws is a constant for each of these curves, koe1 also can be found readily. Comparison of the AOs curves should be performed cautiously since the scale for A0O is not always identical between curves. One observes from these curves that AOr varies with Aur in much the same way that it does when instability oscillations occur, i.e., with an almost inverse sine relationship. Nevertheless, AOr deviates from this single input signal relationship whenever B is nonzero. Furthermore, AOr generally is not equal to zero for Aur equal to zero. This means that AOr is not symmetric about the origin and,consequently, B and AOs are not even functions. For relatively low values of n (such as n = 0.2), however, both B and AO are almost even and AOr is almost odd about Awr equal to zero and so little can be gained by also plotting these curves for negative values of Au. Finally, for this filter, Fig. 4.27 represents the system response plotted as a function of both positive ACr and Au for n equal to 0.2 and G equal to 1.2 x 105 radians/sec, r s the intermediate gain value. From the previous discussion, it follows that this surface is essentially symmetric about the koel - Aus plane. It also can be shown that this surface is symmetric about the k~el I Awr plane. For these reasons,it is sufficient to examine only the first quadrant. The maximum system response is seen to occur for Au equal to zero and for Aus/2n equal to approximately 12 kc (see also Fig. 4.17). As Aur increases, the peak decreases slightly, spreads out, and occurs at lower values of Aus. The above description represents the general behavior of the harmonic solution for all filters. The peaking exhibited in Fig. 4.27 always exists but is of lesser magnitude for the inherently stable filters. To a great extent the over-all system response characteristics can be found from the Aur-equal-to-zero graphs shown extensively throughout this section. In this section, the interrelations of the various system and input signal parameters to the system response have been presented with the aid of three examples, since any general direct interpretation of the coefficient equations appears impossible. One final interesting observation can be made from these three examples. For all filters and system 72

S S.4 -1.0 -.8 -.4.2 8 A r/27 (kc/s) 12 16 Fig. 4.21. System variables dependence on A. for 6s/27 equal to 3 kc/sec. r

.2.4- 1.0 be~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t.6 -.8.8 -to 02,a~~~~.8 -.6kc/S) Fig. 4,,22. system variables dependenceo Lw fr 5ws/1T equal to 7 kc/sec. m~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~1.0 -.4 Q,4~1. 2 -\,2,~~~~~~~~~~~~~~2,~~ —---- 4 8 h~~~ar/2~ (kc/S) F~ig. 4.22. System variables dependence on hwr for bws/2n equal to 7 kc/sec.

.7 _. A~O9 0~~~~~~~~~~~~~ 1.1-_1.0 AO ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~A sa tt~~~~~~~~~~~~~~~~~~~~,8QOL r~~~~~~~~~~~~~~~~~~~~~~~A ~~~~~~~~~~~~~~~~~~~~~~~~~~~. 6,2 0 ACVr/27T (kc/s) 16 Fig. 4.23. System variables dependence on Aw for Aw //2 equal to 10 kc/sec. r r~~~~~~~~~~~~

2. 5 - S~~~~~~~~~~~~~~~~~~ 1. 0 -2.9 1.0 r0 3.1.8 a 6.6.4.2 oL / I I I l!I I II C 0 ~~4 ~8 CO/27 (kc/s) 12 16 Z~w/22T (kc/s) r' Fig. 4.24. System variables dependence on Aur for Aw /2T equal to 12 kc/sec.

ASX (Radians) L - tor/2 or equal to 1.0.8 6..6 Qz 4.4 2.2 4 8 12 16 Aw /2ir (kc/s) Fig. 4.25. System variables dependence on Aw for Awr/27 equal to 12.9 kc/sec.

?[+.2 8 S~~~~~8~ ~ ~ ~~S.8~ s sX.62 / ~~~~~~~~~~~~7+.6 6 C Akfr.4 6 4 r I _ _ _ IACr/2r (kc/s) 126 0 r Fig. 4.26. System variables dependence on A for Aw /2,r equal to 15 kc/sec. r

ko e (104 Radians/Sec) 10 f and 14 of Aw and Aw (kc/s) 20~ 1I I IIIIIIIIX /C (kc/s

gains evaluated, the system response amplitude is equal to nhAs for small Aws, when lAr is equal to zero. Granlund (Ref. 13) found an identical relation for the interference response in a conventional FM discriminator circuit. 4.5 Stability Analysis of the Forced Oscillations. The validity of the analysis performed in Section 4.2 is primarily dependent on the existence of a stable, periodic, system response when the secondary interference signal is present. The stability theory for this forced case does not differ greatly from the locked oscillation case of Section 3.1.3. Once again, although the mathematical theory for answering the question of stability is available, the required calculations for each case are very lengthy. The general theory is presented briefly below. (For a more complete discussion see Section 3.1.3). The orbital stability of the forced oscillations may be established by the first variation technique. The theory for this analysis (see Ref. 19) is described in terms of a system of n first-order differential equations. Let the column vector ~(t) be a real solution of x a F(t,x) (4.26) for 0 < t < a, where the column vector F is analytic in x for each t. Then the first variation equation is Y = Fx[t,p(t)] y, (4.27) where Fx[t,4(t)] is a matrix composed of the columns aF/axi(i=1,2,..., n). In the event that ~(t) is periodic of least period T, and F is periodic of period T in t, then Eq. 4.26 has a periodic coefficient matrix of period T. This is precisely the form assumed by the APC system equation when a secondary interference signal is present. The following orbital stability theorem is proved in Ref. 19 for this case. Theorem: If the characteristic exponents associated with the equation of first variation (Eq. 4.27) all have negative real parts, then the periodic solution W(t) of Eq, 4.26 is asymptotically stable as t approaches infinity. The application of the stability theory just outlined is identical to that carried out in Section 3.1.3 except that for an n-th order system it is necessary to find all n 80

characteristic exponents instead of n-l. Since each must be found separately, this increases the work for a system of given order. 4.6 Alternate Ana!ztical Approach In this section, the second, interference-susceptibility equation, Eq. 2.49, is examined further. This equation is equivalent to Eq. 2.46 (also Eq. 4.1), but it does differ substantially in form. The difference in form turns out to be a mixed blessing. The coefficient equations that result from this latter susceptibility equation are even more complex than those found in Section 4.2. On the other hand, an estimate of the system response can be made directly from this form for low values of Aus and Awr. When all but the first term of the sums in Eq. 2.49 are neglected, following the technique employed in the earlier work, it may be written as A r = wf -o (t 0f + GT) Vl * + 2n cos (t 01) + Gt) sin L1 nl sin (AxS t) Aer a sin (waft 01) + tan 1 + n osin (Au t) (4 28) where a is defined by Eq. 3.23. The next step is to expand the two periodic functions of Awst into their respective Fourier series representations. It is then possible to obtain the desired coefficient equations. The Fourier series expansion of the seemingly innocuous function, l++2ncos t is difficult'. Since it is an even function, the expansion contains only cosine terms. The first three coefficients are derived in Appendix C. The results are + n2 + 2n cos (Ast) = 2 (1 + n) E(kr/2) + - [F(ki/2) + (k2 - 2) D(k,n/2)] cos (Awst) + 2T [15k2E(k,r/2) - 8(1 + k2) F(k,o/2) + 16(1 - k2 + k4) D(k,i/2)] cos (2AKst) +..}(4.29) where 81

k = 2 Vn- (4.30) Ek ) The complete elliptic integral of the second kind, (431) E(k,r/2) = The complete elliptic integral of the second kind, (4.31) F(ko/2) = The complete elliptic integral of the first kind, (4.32) D(k,7/2) = [F(k,n/2) - E(k,#/2)]/k2. (4.33) The higher order coefficients become progressively more complex expressions in terms of the first two kinds of elliptic integrals. The second function in Awst has a relatively simple Fourier series representation. It is convenient to differentiate the function first. dd r~-1 - n sin (Aust) n cos (Aust)+n2 d-r tan- o 1 =s. A... (4.34) dt l+n cos (Aust) s l+n2+2n cos (As t) The Fourier series expansion of this latter function also is given in Appendix C, with the result that for n2 less than one, n cos (A t)+n2 os (n ) (4.35) A I (-n) cos (n A t). (4.35) S l+n2+2n cos (Aust) nal Integration yields the desired series tan1 +sn Cos (Ast) = F. (n sin (n a wt). (4.36) tan l+rl (Au t) n s s n=l When the constant and fundamental terms of Eqs. 4.29 and 4.36 are substituted into Eq. 4.28, the harmonic solution becomes +Ar B Au cos (A st + ( ) G(w)2 E + 2 [F + (k2 - 2)D] cos (Ast sin[Ar (4.37) + (n - B cos 81) sin ({Ast) - 8 sin 01 cos (Ahst)] 82

where E = E(k,I/2), F = F(k,7/2), and D = D(k1,/2). (4.38) It is now possible to expand the sine function in terms of Bessel functions, and then equate coefficients as was done in Section 4.2. The result of this operation is an even more complex set of coefficient equations than the earlier ones, i.e., Eqs. 4.11, 4.16, and 4.17. Since these sets of equations should yield essentially the same result, this development will not be conducted further. Some interesting relations can be found from Eq. 4.37. For low values of Aus and Amr the dependent variable 01 is essentially zero as can be determined from the analysis in the earlier section of this chapter. The product GcT(u) is nearly equal to the real number G for low values of Aus and for the commonly used lowpass filters. Finally, it may be shown that for small n 2(l+n) E + [F + (k2 - 2)D] cos (Abst)) 1 + n + -ns cos (Awst). (4.39) Thus, for small AuS, Aw, and n, Eq. 4.37 takes the form Air + 3 Aw Cs ( Gl+ + cos (Au t)] sin[A +r + 1 cos r sin[be + (n-B) sin (Awst)].(4.40) The Bessel function expansion of the sine term is sin[Aer + (n-8) sin (Awst)] = sin (AOr) cos[(n-B) sin AwSt)] + cos (A2O sin[(n-B) sin (Aust)] = sin A(hr) [Jo(n-B) + 2J2(n-8) cos (2Aust) +...] + cos (bAr) [2J1(n-B) sin (Amst) +...]. (4.41) 83

Substitution of the first terms of this expansion into Eq. 4.40 yields Ar + B Aus cos (Awst) = G[l+n+ r1 COS (us t)] [sin AOJ (n-B) + cos AOr2Jl(n-B) sin (Au wst)]. (4.42) r 1 From this last equation it follows, by equating the constant and periodic terms separately, that AWr - G(l+n) J (n-6) sin A0r (4.43) and B Aus cos (Aust) = G(l+n) cos Aer2J (n-S) sin (Aust) + +Tn Jo(n-B) sin Aer cos (Ast). (4.44) The requirement for Eq. 4.44 to hold is that the first term on the right-hand side have a zero coefficient. This means that Ji(n-B) must be zero which, in turn, implies that a is equal to n. Hlence, from Eq. 4.43, Ar sin Aer (L+n) G' (4.45) and from Eqs. 4.44 and 4.45, n Awr n Awr a Aws r..... r (4.46) (l+rn)2 Note that since Eq. 4.44 is not applicable when Aws is equal to zero, Eq. 4.46 contains only the system response's dependence on Aur. Because 8 is essentially equal to n for low values of Aur and As, one can write more generally n Au koel = s -— n rAs+ (4.47) where the second term on the left-hand side has significance for only nonzero values of Au8 84

These relations can be confirmed from the curves given in Section 4.4. Indeed, both Eq. 4.45 neglecting the n term and Eq. 4.47 with Aur equal to zero previously were observed and remarked upon. The dependence of k el on Aur can be seen in Fig. 4.21, for example. 4.7 Case of Sxstem Insensitivit to Input Signal Amplitude The third interference susceptibility equation (Eq. 2,501 developed in Section 2, 3, pertains to APC systems designed to have a fixed system gain, i.e., systems which are insensitive to input signal amplitude. In this section, the coefficient equations appropriate to this case are developed. Once again, the complexity of these equations requires a computer solution.' Here, as before, only the first term of the sum expressions in Eq. 2.50 will be retained. Employing this simplification, the equation may be rewritten as Ar = -Baf cos (uft + 01) + G'IT(u) sin [ Ae r1 nsin ( rW t) - sin (ft + 01) + tan 1 nsin (Ast) (4.48) Substitution of the first term of the expansion given in Eq. 4.35 into Eq. 4.46 yields, for the harmonic solution, Arw + a Ads cos (A~st + 01) = G'IT(w) sin [AO Aur + 8 Au cos 1 r - 8 sin (hAst + 1) + n sin (Aust)].(4.49) The Bessel function expansion of the sine factor is given by sin[A6r - B sin(Arst + 61) + n sin (Aust)] = sin (A8r) cos [(n- 8 cos 1) sin ( t) - 8 sin 01 cos (Aust)] + cos (Ahr) sin [(n - 8 cos 61) sin (Aust) - B sin 61 cos (Aust)] = sin (AOr)cos[(n - B cos O1) sin (Aust)] cos[B sin 01 cos (Amst)] + sin[(n - 8 cos 01) sin (Aust)] sin[B sin 01 cos (Amst)] 85

+ cos (AOr) sin[(n - B cos 61) sin (Awst)] cos [B sin e1 cos (AsSt)] - COS [n - COS 0) sin (Awst)] sinl1 sin 01 os sin (Ar) [JO(al)+2J2(al)cos (2wSt) +...] [Jo(a22)-2J2(a2)cos (2ASt) +...] + [2J1(al) sin (Awst) +...] [2J(a2) cos (Awst) -...]| + cos (Ar)[2J(al) sin (At) +...] [J(a2) 2J2(a2) cos (2 t) - [J(al) + 2J2(a1) cos (2Aw t) +...][2Jl(a2) cos (Aw t) -...](4.50) where "a, n - O cos O1, (4.51) a2 B sin 1. (4.52) When only the principal constant and first harmonic terms of this expansion are retained, the sine factor becomes for small al and a2, sin[A8r - B sin (ALst + 01) + n sin (A st)] sin (AOr) Jo(al) Jo(a2) + 2 cos (AOr)[Jo(a2) J1(al) sin (ast) - Jo(a1) J1(a2) cos (Ast)]. (4.53) When this is used in Eq. 4.49, the result is Ar' + (n a) As cos ( wst) - a2 asA sin (A st) G'iT(u) sin (A0r) Jo(al) Jo(a2) + 2 cos (aer) [Jo(a2) Jl(al) sin (ASt) - Jo(al) Jl(a2) cos ( (4.54)

The three coefficient equations follow from this result and Eq. 4.14; they are Awl = G' sin (A0r) Jo(al) Jo(a2), (4.55) a2 Jo (a2)J (al) n'al J (al)l (a2) tanQ(as). a gJo'(a) (4.56) (n-1+ io (1J1aP (2) Ai[ (n-al)2 + a22] 2 4G'21T(Aws) 12 cos2 (~Ar) 0[J2(a1)J12(a2) + Jo2(a2)J12(al)]. (4.57) These three coefficient equations can be solved using the same technique that was employed in Section 4.4 for the original set of equations. Although this has not been done, some insight into the nature of the solution can be found by examining the ideal-integrator case. On the basis of earlier work, it is thought that this particular example will give an exaggerated indication of the general character of the system response amplitude as a function of Aw. When H(s) equals 1/Ts, then Eq. 4.54 reduces to (recall that Ar' must be zero and so Aer is equal to zero) 2G' (n-al) cos (Awst) -a2 sin (Ac t) - [-J (a2) (a1) (Acs)2T T cos (Aust) - Jo(al)Jl(a2) sin (A5 t)] (4.58) Equating coefficients yields 2G' (n-al) = ( 2 J (a2)Ja), (4.59) 2G a2 = - J(al)Jl(a2). (4.60) (Acs) 2T A solution to this pair of equations is given by a2 equal to zero and the single equation 87

2G' n-a1 - )2 J(a1). (4.61) Since a2 equals B sin 01, and B is assumed to be positive, this requires that el be equal to either zero or i radians. In those cases, a1 must equal n-B or n+B, respectively. Substitution of these values into Eq. 4.61 yields, respectively, cB. (4,62) (As )2T 2J1(Bqn) This equation is similar to Eq. 4.18 (for the harmonic solution Eq, 4.18 has n equal to one and Auf equal to Aw5) developed from the earlier analysis. Equation 4.62 is plotted in Fig. 4.28 for the lower two values of n. When this figure is compared with Fig. 4.1 (a plot of Eq. 4.18), the similarity is more prominent. A close inspection of Figs. 4.1 and 4.28 reveals jump discontinuities (see Fig. 4.6) which occur, respectively, at essentially the same abscissa values. (These discontinuities result from the existence of minimum abscissa values, for the right-hand branches of the curves.) The peak ordinate value (maximum B value), however, is slightly less in the case of system insensitivity to input signal amplitude. Based on the above brief investigation, it appears that the use of a balanced phase detector or other mechanisms for eliminating dependence of the APC system gain on the input signal level does not materially affect the system response to a secondary signal. Recall that the above theory depends on the value of n existing at the input of the APC system's multiplier. The effect of a limiter, or other nonuniform signal processor preceding the APC system, must be considered separately. 88

2. 5 2.0 (Radians) 1. 5 1.0 3 0 1.0 2. 0 GI/T Aws2 3.0 Fig. 4.28. The harmonic relation between S and G'/TAs2. 89

5. SUPPLEMENTARY EXPERIMENTAL PROGRAM 5.1 Summary of the Experimental Program In Section 4, a general theoretical analysis of the APC system's response to a secondary interference signal was developed and interpreted. The purpose of this section is to examine a particular APC system and compare experimental findings with the theoretical predictions. Since the experimental work is not limited to a system response consisting of a single sinusoidal waveform, the data can be extended to include values of n near unity and system gains approaching the locked oscillation level. The latter information provides additional insight to the effect of secondary-signal interference. An input-amplitude-sensitive APC system was constructed for the experimental program. The details of the system are described in Section 5.2. The experimental tests performed on the system are summarized with the aid of the block diagram shown in Fig. 5.1. This experimental configuration was used for checking both the locked instability oscillation analysis of Section 3 and the secondary signal interference analysis. In the former case, the signal generator supplying the secondary signal was turned off. The following procedure was used for the experimental study of locked instability oscillation. The combination of reference signal generator level and attenuator setting was adjusted to yield a locked periodic oscillation (as opposed to the oscillations which occur just outside the locking range). This input signal amplitude was maintained as the reference signal frequency was varied across the entire passband of the APC system. The amplitude (both dc and ac) and frequency of the APC response were recorded for each reference frequency test point. Data were obtained for three levels of system gain. The results of these measurements have been presented in Section 3.2. In measurements of the influence of a secondary signal, the following procedure was employed. A minimum attenuation of 20 db was maintained between the output of each signal generator and the APC system in order to insure adequate isolation of the two sources. At the beginning of each experimental run, the maximum attenuation available (a total of 80 db) was inserted at the output of the secondary signal generator. With the minimum attenuation of 20 db at the output of the reference signal generator, the output level and frequency of this generator were adjusted so the system verged on locked instability oscillations 90

Signal Generator AC Signal Generator (Reference) VTVM (Secondary) HP 606A HP 400D HP 606A Adjustable Adjustable Attenuator Attenuator Daven 650 Daven 650 APC System AC VTVM Oscilloscope RMS, Average, or DC Peak Reading VTVM Tektronix 561A Ballentine 321 HP 412A Counter HP 523B Fig. 5.1. Experimental test configuration.

with Awr equal to zero; i.e., with the reference signal frequency equal to the open-loop system-oscillator frequency. An identical calibration procedure was then followed for the secondary signal generator. The next step was to set up the particular values of G. Aw ro and n required for the current run. The system gain, G, was adjusted by inserting the appropriate value of attenuation in the output of the reference signal generator; e.g., 21 db would represent a gain 1 db less than the gain required for oscillation. Awr was set by tuning the reference signal generator frequency to the desired value with the aid of a counter. The value of n, the ratio between the secondary and reference signal levels, was set by inserting the required attenuation at the output of the secondary signal generator. After these preliminary steps, the secondary signal generator frequency was varied across the passband of the APC system. The difference in the frequencies of the two signal generators, i.e., AWs, was recorded for each test point. The system response frequency, ac rms voltage level, and dc voltage level were recorded for these values of Aws. This entire procedure was carried out for several combinations of G. AuWr and n. Some experimental results are given in Section 5.3. 5.2 Experimental APC System Before discussing the experimental data for the secondary interference signal case, it is convenient to consider briefly the circuit details of the APC system. This will further an understanding of the specific cause of system instability when an excessive input signal is present and of the particular response of the system to the secondary signal. Furthermore, the system's over-all lowpass filter transfer function is determined approximately for use in a theoretical comparison. A study of the circuit diagram for the APC system, shown in Fig. 5.2, reveals that the required multiplication function is performed by a balanced phase detector. The resultant error signal is amplified by a differential amplifier and then filtered by a singlesection RC filter. This lowpass filter is isolated with emitter-follower circuits. The filtered signal is then applied to the system oscillator, an astable multivibrator, in such a way as to control its frequency. The system oscillator output is approximately a square wave and is applied to one of the two inputs of the phase detector. The input signal is applied to the other detector input as indicated. In addition, an ac-coupled amplifier is provided for the system response signal. For experimental flexibility, a switch is included which can effectively remove the RC lowpass filter. (A photograph of this circuit is shown in Fig. 5.3.)

-4 - 15 - 15 - 15 15 30:30:30 CTP 740 740 pot K. 2 Input l 470SA ~; 1. 5K +100AuioOut t - 1f + -O + ~1 ~~~~10 10K.005 * —---- 15f Frequency - j ] l are 2N711B +10 Afu 3.K 47OK I o f -010 1K Fig. 5.2. APC system circuit diagram.

One of the important experimental parameters is ko, the gain constant of the system oscillator measured in cycles (or radians) per second per volt. This parameter can be measured easily by applying a dc voltage to the oscillator input and noting the resultant frequency. The data points shown in Fig. 5.4 represent a series of these measurements, and the curve drawn through these points is a straight line approximation. The excellent frequency linearity of this astable multivibrator with voltage can be predicted by theory over an appropriate range of voltage. From this curve, the slope, ko, is determined as 42.8 kc per volt. Fig. 5.3. The experimental APC system. Another important circuit property is the over-all lowpass filter transfer function. If the stray capacitance of wiring and components is ignored, then the transfer function consists of a pair of isolated real poles. One of these is due to the RC filter of the system; the other is a consequence of the envelope detection circuit of the phase detector. Mlore precisely, this latter pole is caused by the series diode resistance and effective secondary resistance of the transformer in the circuits coupling to the inputs of the differential amplifier. The presence of this second pole in the transfer function causes this system to be potentially unstable. Since the diode resistance depends on the input voltage level, so does the location of the pole. For this reason, it is difficult to assign directly by circuit analysis, or to determine experimentally, the appropriate time constant for this 94

580 560 540 520 r 500 /Slope 42. 8 kc/s/volt 480 460 440, * -7.0 -8. 0 -9. 0 - 10. 0 System Oscillator Bias Voltage (volts) Fig. 5.4. System oscillator frequency versus voltage relation. pole. An indirect determination follows. Assuming that iT(s) for this system has the form (s) =- 1 ( (5.1) (1.T1S) (l+T2s) and recalling from the experimental work in Section 3.2 that Iff(wf) is approximately equal to 0.43 when wf/2r is equal to 12.94 kc, then it follows that 95

0,.43 =1 (5.2) j(1+jT12ir x 12.94 x 10 +j 2 x 12 At this frequency, the transfer function must introduce a phase shift of -n/2 radians. This requires that 1T2 (27 x 12.94 x 103)2 = 1. (5.3) Solving Eqs. 5.2 and 5.3 for T1 and t2 yields T1 7.02 x 10'6 second T2 21.6 x 10 6 second. (5.4) Where T2 is the time constant of the RC filter of the system and T1 is the time constant of the pole produced by the phase detector. A direct experimental evaluation of T2 can be made easily, as a check on the above result. This yields a value of 20.8 x 10 second. Although it is unlikely that the transfer function given by Eq. 5.1, with the values of T1 and T2 indicated by Eq. 5.4, is an exact description of the actual filter characteristic, the model given is not unreasonable. It should be reiterated that it is difficult to measure experimentally the actual lowpass filter characteristics. This difficulty is caused by the phase detector (multipliei circuit which introduces phase shift and attenuation in generating the control signal. This filtering action cannot be measured conveniently since the control signal does not exist explicitly in front of the diode circuits, and is modified by the detector filter at subsequent points. For this reason (in addition to the difficulty of experimentally evaluating iT(s)) the phase-detector gain constant, km, and hence the system gain, G, are also difficult to obtain directly. Certainly, one of the useful applications of the locked instability oscillation analysis is the indirect evaluation of G and ItT(wf)l from simple measurements. 5.3 Experimental and Comparative Theoretical Results The experimental procedure outlined in Section 5.1 for the secondary-interferencesignal case was employed to obtain the data shown graphically in Figs. 5.5-5.7. Each of these three sets of curves was made for a constant value of system gain and for A,,)r equal to zero. The gains for these three sets are, respectively, 3 db, 1 db, and O db less than that required 96

12 k0e1 10 C.) I0~~~~~~~~~r =.355 4 2 04 12 16 20 24 28 32 4 aw /2z (kc/s) Fig. 5.5. System magnitude response for the experimental circuit with G equal to 1.355 x 105 radians/second.

12 ko e1 10-.3 = 55'0 4 X4 20 4 8 h2a> /2u (kc/s) agnitude response for the experimental circuit with G equal t

12 koe1 10.1 = 355 8 r=.2 6 77 Cd) 4 2 0 4 8 12 16 20 24 28 32 36 40 Aws/27r (kc/s) Fig. 5.7. System magnitude response for the experimental circuit with G equal to 1.89 x 105 radians/second.

for the initiation of locked oscillations. The ordinate and abscissa quantities, k el and Au /2r, are those used in Section 4 for the similar theoretical curves. The values of n used here are also the same as in Section 4. In general, these curves reveal the presence of the harmonic, the first subharmonic, and a series of smaller ultraharmonic system response peaks. The harmonic response peak occurs on the various graphs between 7 and 11 kc. The first subharmonic response peak occurs at approximately twice the frequency of the harmonic peak. The frequency of system response within the subharmonic peak was found to equal A s/4T, as expected. The absence of a second subharmonic response also agrees with the theoretical predictions. Although no attempt was made to analyze the ultraharmonic case theoretically, it is reasonable to expect that techniques similar to those used in this study could be applied. These peaks are relatively insignificant for the lower value of n and G. This changes dramatically, however, for the higher values of these parameters. Frequencies of system response within these 2Aw 3Au $ s peaks were,successively, 2- kc, 2-y- kc, etc. The photographs in Figs. 5.8 - 5.10 show the system response for conditions as stated individually on each picture. For lower values of n the response is essentially sinusoidal. Nevertheless, it is clear that nonsinusoidal periodic waveforms occur for appreciably high values of n. These effects are pronounced for high values of G. For this reason, the applicability of the theoretical results developed in Section 4 is limited. The high-frequency ripple evident in these pictures is due to imperfect filtering of the system oscillator at the point of observation. For the purposes of comparison, the analytical techniques developed in Section 4 were applied to a model of this experimental circuit. The system lowpass transfer function given by Eq. 5.1 (using the time constant values given in Eq. 5.4) was assumed for the model. The two filter quantities required by the Newton-Raphson method, i.e., the square of the magnitude of the transfer function and the cotangent of its phase angle, are plotted in Fig. 5.11. The harmonic solutions for the same values of gain used in Figs. 5.5-5.7 are shown respectively in Figs. 5.12-5.14. These latter curves evidence system behavior similar to that observed previously, but they have noticeably more amplitude than the experimental curves. An explanation of this may be that Eq. 5.1 does not adequately describe the actual 100

a) The harmonic peak for: G=-1 db (relative to the gain required for system instability) n0O.355 Frequency of oscillation (i.e., Aws/2-) = 7.92 kc/s Vertical scale 0.1 volt/ division Horizontal scale 50 psec/ division b) Halfway between the harmonic and first subharmonic peaks for the values of G and n used in (a) Frequency of oscillation (i.e., Aws/27) = 11.75 kc/s Vertical and horizontal scales are identical c) First subharmonic peak for the values of G and n used in (a) and (b) Frequency of oscillation (i.e., As /4r) = 7.85 kc/s Vertical and horizontal scales are identical Fig. 5.8, Periodic forced system response waveforms, 101

a) The harmonic peak for: G = -1 db, n = 0.2 Frequency of oscillation (i. e., As/2.R) = 9.52 kc/s Vertical scale 0.1 volt/ division Horizontal scale 50 usec/ division b) Halfway between the harmonic and first subharmonic peaks for the values of G and n used in (a) Frequency of oscillation (i.e., AW /2f) = 14.30 kc/s Vertical and horizontal scales are identical c) First subharmonic peak for the values of G and n used in (a) and (b) Frequency of oscillation (i.e., Aus/4w) = 9.52 kc/s Vertical and horizontal scales are identical Fig. 5.9. Periodic forced system response waveforms. 102

a) The harmonic peak for: G a -1 db, n = 0.1 Frequency of oscillation (i.e., As /2r) = 10.00 kc/s Vertical scale 0.1 volt/ division Horizontal scale 50 usec/ division b) The first subharmonic peak for the values of G and n used in (a) Frequency of oscillation (i.e., AWs/4rn) = 10.52 kc/s Vertical and horizontal scales are identical c) System response with Aw /2= equal to and for: G = -1 10.00 kc/s db, n 0.56 Vertical and horizontal scales are identical Fig. 5.10. Periodic forced system response waveforms. 103

system transfer function. From the work in Section 4.4 it can be seen that relatively slight changes in the phase angle of the transfer function can influence quite noticeably response characteristics. A second factor is that the actual response is nonsinusoidal for the higher values of n and G; this violates a basic theoretical assumption. +2 1.0 +1 Magnitude Squared. ~~~~~~~~~~~~8 L ani SquaredCotangent of the Phase Angle.6 -1.4 -2.2 -3 2 4 6 8 10 12 14 16 18 20 Aw /27r (kc/s) Fig. 5.11. Characteristics of the two-pole filter. 104

10 k0e1 8 _ 7=.355 4 8 12 16 20 24 28 32 /kt s/2~ (kc/s) Fig. 5.12. System magnitude response for the two-pole filter with G equal to 1.335 x 105 radians/second.

r0 =. 355 kel l 2 8 7. 1 C. C0 2 4 8 12 16 20 24 28 32 Aw s,/2i (kc/s) Fig. 5.13. System magnitude response for the two-pole filter with G equal to 1.685 x 105 radians/second.

12 =. 3 55 ke l o 1 6 2 o 4 8 12 16 20 24 28 32 Aw S/27r (kc/s) Fig. 5.14. System magnitude response for the two-pole filter with G equal to 1.89 x 105 radians/second.

6. CONCLUSIONS AND APPLICATIONS Results of this study facilitate an understanding of the causes and effects of the locked, periodic responses of APC systems beyond that previously achieved. Two distinct types of periodic response were considered —locked instability and forced oscillations. The system designer is provided with the tools necessary to determine the possibility of either form of periodic response by his circuit, and with design guidelines for controlling these oscillations where needed. Several conclusions and applications based on the findings of this study are presented below. The periodic response of an APC system to a single, constant-frequency input signal within the system's capture range has been referred to in this study as a locked instability oscillation. An APC system will exhibit locked instability oscillations only when the following two requirements are met: 1) The transfer function of the system's lowpass filter must introduce a phase shift of -Tr/2 radians at a finite frequency, and 2) The total system gain, including the attenuation through the lowpass filter, must exceed unity. The frequency of oscillation, if it exists, will be the finite frequency at which the first requirement is met. When locked instability oscillations exist, the magnitude of the periodic response is an even function of Auw, the difference between the input (reference) signal frequency and system oscillator's open-loop frequency. The response is maximum for Aur equal to zero and decreases quite abruptly to zero for increasing IAur]. The system phase error is an odd function of Aur and has zero value for zero Auwr Except when Awr is equal to zero, the static system phase error always exceeds in absolute value the error that would exist if no oscillations occurred. A typical example of both of these relationships was given in Fig. 3.3. It is generally true that as the system gain increases, the magnitude of the system response increases for any given value of Awrs and the range of Awr over which the magnitude is nonzero also increases. The effect on the system phase error is also accentuated as the gain increases. 108

An additional, interesting aspect of the locked instability oscillation analysis is the ability to determine the system gain and the magnitude of the lowpass filter's transfer function at the frequency of oscillation. Both of these quantities can be found from three easily performed measurements. Since direct measurement of the gain and complete filter characteristics is often difficult, this indirect method is of considerable value. The three measurements needed are those of 1) The frequency of oscillation, 2) The maximum magnitude of oscillation, and 3) The value of Awr at which the oscillations cease. The specific relationships between these measured quantities and the quantities of interest were developed in Section 3.2 (see Eqs. 3.74-3.77). The second type of periodic response considered in this study, i.e., forced oscillations, results when the APC system is subjected to two constant-frequency input signals. In the particular problem analyzed, it is assumed that the greater amplitude input signal is the reference signal and that the other signal differs only slightly in frequency from the reference. It is then shown that the system exhibits a periodic response, i.e., a forced oscillatory response. Under the above conditions, the average frequency of the system oscillator is equal to the reference frequency for low-level periodic responses. (The frequency of the response oscillations should not be confused with the system oscillator frequency.) In contrast to locked stability oscillations which exist only for certain ranges of system parameters, all APC systems exhibit forced oscillations. The peak magnitude of the system response with forced oscillations is related, however, to the stability of the system. Based on the examples of Section 4.4, it can be seen that as the phase shift of the lowpass filter approaches and exceeds -i/2 radians, the response magnitude increases. The general characteristics of the forced oscillations are considerably more complex than those of the locked instability oscillations. Three dependent variables are defined for this case —the magnitude of the response, the system phase error (between the reference signal and the system oscillator signal), and the phase relations between the periodic response and the input reference signals. The third variable is usually of little interest and will not be discussed here any further (see Section 4 for additional details). The four independent variables for any given APC system are: 109

1) G: The system gain, 2) n: The amplitude ratio between the secondary and reference signals, 3) Ahr: The difference in the radian frequencies of the reference and open-loop system oscillator signals, and 4) Aws: The difference in the radian frequencies of the secondary and reference signals. In general, the magnitude of the system response will be zero whenever Aus is zero, provided that the sum of the input signals does not cause system instability (for systems sensitive to input-signal amplitude). Further, it has been shown that the response magnitude is essentially equal to nAus for small A s. Grandlund found this same behavior for the FM discriminator preceded by a wideband limiter (see Ref. 13). For this reason, it is expected that an APC system designed to operate properly as an FM demodulator will exhibit the same effectiveness against co-channel interference as does the limiter-discriminator combination. As Aus is increased further, the magnitude of the system response tends to peak in one or more places. The number of peaks and the degree of peaking are a function of the other three independent variables as well as of the lowpass filter characteristic of the system (as discussed previously). Usually, there are two dominant peaks —the harmonic and first subharmonic response peaks. The radian frequency of the response oscillation within these two peaks is, respectively, Aus and Aws/2. Since the first subharmonic peak occurs for values' of Aus approximately twice those of the harmonic peak, the frequency of system response in these peaks is nearly the same. The harmonic response prevails for values of Aus immediately below and above the first subharmonic peak. The consequent changes of oscillation mode may be, but are not necessarily, quite abrupt. The system response waveform is nearly sinusoidal in both of these peaks for low values of n and G. The existence of any significant higher-order subharmonic peaks was neither predicted by the theory nor observed experimentally. A number of lesser peaks may exist with values of Aus below that for occurence of the harmonic peak. These ultraharmonic peaks occur in increasing numbers as n is increased, for any fixed value of G. The radian frequencies at which these peaks occur are 1/2, 1/3, 1/4, etc., times the values of Aws for which the harmonic peak exists. The frequencies of system response within these peaks are, respectively, 2Aus, 3Aus 4Au5, etc. Here the response waveform was quite complex for the experimental system evaluated (see Section 5). Certainly, the existence of these ultraharmonic peaks jeopardizes the co110

channel performance of an APC system. The tendency toward this peaking decreases as G is decreased. Furthermore, it is expected that the magnitude of these peaks can also be decreased through choice of a suitable, relatively stable lowpass filter (such as the single zero-single pole filter of Section 4.4), The dependence of the system response magnitude on Aur is, in turn, quite dependent on Aws5 An example of this relation, for the harmonic peak with n and G fixed, was given in Fig. 4.27. In general, the harmonic peak decreases in amplitude, and occurs at lower values of Auw as Ar increases. Furthermore, this peak broadens with increasing fAr. The first subharmonic peak tends to decrease in relative size and to disappear eventually for increasing Aur9 as do the ultraharmonic peaks. In general, the effect of increasing n and/or G is to accentuate the characteristics already discussed. Modification of the phase error of the system by the presence of the forced oscillations is similar to that in the case of locked instability oscillation. There are some differences, however, in the two cases. For example, the phase error is not generally equal to zero when Aur is zero, although it does not deviate greatly from zero for even moderate values of n and G. The phase error remains an odd function of Aur (essentially the inverse sine of Aur/G), and is almost independent of Au.s These results have been verified both theoretically and experimentally. As previously stated, this study serves, a) to provide the designer a means for analyzing his system with regard to its susceptibility to a locked periodic response, and b) to provi'de guidelines for the control of the locked periodic response. An example of the former is given by the work of Sections 3.2 and S. From the theoretical work developed in this study, along with the analysis technique described in Appendix A, it is possible to determine analytically the system response characteristics for at least the harmonic and subharmonic response peaks. Likewise, the degree of locked instability oscillation, if any, can be determined for any given value of system gain. Specific applications where the guidelines are useful are in the design of receivers to reduce co-channel interference and of frequency synthesizers utilizing APC systems. It was mentioned previously that an APC system might effect co-channel rejection of a second FM signal similar to that of the limiter-discriminator circuit. Another related application of an APC system was recently discussed by Bridges and Zalewski (see Ref. 22), who proposed a technique to reduce co-channel interference in AM double-sideband systems — which may employ two APC systems. Each APC system is required to lock to one of the two received carrier signals and reject the other. These carrier signals are assumed to be at 111

nearly the same frequency. Since the object of the two APC systems is to provide a copy of their respective carriers without any (or, at most, with a minimum of) additional spectral components, results of this study dictate that considerable care must be taken in the design of the two lowpass filters in the system. Specifically, a particularly stable filter is needed to minimize the forced response peaks —at the possible expense of system holding range. In the design of a particular form of frequency synthesizer (see Ref. 23), APC systems are used to extract a single harmonic component from a reference frequency comb of many, harmonically related signals. It is desirable that the APC system bandwidth approach in value the harmonic separation of adjacent comb components, in order to insure that the system will lock to the desired signal. Beyond this objective, other spectral energy in the output of the APC oscillator should be minimized. Care and compromise in the design of the system filter are required to achieve these two conflicting objectives. 112

APPENDIX A NEWTON-RAPHSON METHOD FOR SIMULTANEOUS NON-LINEAR EQUATIONS The Newton-Raphson method is an iterative technique for the solution of simultaneous non-linear equations. The following brief description is abstracted from Ref. 24. We seek the solution(s) of the system f(x) = 0, (A.1) where x = column vector [xl,x2,...,n], (A.2) and f = column vector [fl(x), f2(x)...,fn(x)], (A.3) consisting of n real equations in the n real unknowns, x. Now define the matrix M(x) [fij(x)] 1 < i, j < n, (A.4) with elements afi(x) fij (x) = ix (A.5) = ax. Thus, det M(x) is the Jacobian of the system (A.1) evaluated for the vector x. With these definitions in mind, and with the starting vector xo = column vector to[x10, x20,.**, Xn0], (A.6) 113

the Newton-Raphson method is given by the iterative equation xk+l I xk + 6xk' k = 0, 1, 2, 3,..., (A.7) where 6xk is the solution vector for the set of simultaneous linear equations given by M(xk)6xk = -f(xk) (A.8) It can be proven (see Ref. 24) that if the elements of M(x) are continuous in a neighborhood of a point x' such that f(x') ~ o (A.9) and if det M(x') is nonzero and x is "near" x', then lim xk x'. (A.10) k~ Since the system (A.1) may have several solutions, the requirement that x be near x' guarantees that the selected starting vector will converge to the desired solution. 114

APPENDIX B IMPLEMENTATION OF THE NEWTON-RAPIISON METHOD) The Newton-Raphson method described in Appendix A easily may be programmed for the solution of a particular set of equations on a digital computer. The only possible programming complication lies in executing the required solution of a set of simultaneous linear equations (i.e., the solution of Eq. A.8). Since many efficient routines are available for this specific problem, even this usually entails only selecting an appropriate library subroutine. A flow chart of the program used for the calculations in Sections 4.4 and 5.3 is given in Fig. B.1. This program was designed to solve for the three unknowns B, Ae,, and A0s, as functions of Awr and Aws for fixed values of n and G, all for a given lowpass filter. The starting vector Bo, A ro and AOso, is supplied for the first values of Ahr and Awm analyzed. After the first solution is obtained, Aur is incremented and the solution of the first point is used as the new starting vector. This continues until the maximum desired value of Awr is reached. Then Amr is reset to its initial value and Aws is incremented. Here, the starting vector is the solution to the point with the initial value of Aur and the previous value of Aws. For this value of Aus, Aur again is run through all of its values. The program continues until the final values of Aus and Aur are reached. For each point on the Awr - &Ws plane, the solution vector is printed and so is the product, BA s. This latter value is proportional to the magnitude of the system response. Based on the experience of running this program on an IBM 7090 computer, it was possible to calculate approximately 125 points per minute. (Each point represents a new value of Aur and, when incremented, Aus.) Typically, between three and four iterations were required for each point when it was specified that each of the variables, 6xk, xk,and 6xk at any given iteration was less than one percent of the respective values of the elements of xk. The M(x) elements required by the Newton-Raphson method are determined from the coefficient equations derived in Section 4.2; namely, Eqs. 4.11, 4.16, 4.17. These are rewritten here in the format of Appendix A. 115

Read and Print Evaluate* Print 7, G, Awr Inc, Read Start Ar Ms w = M.1ax s r k o H Tcot I (AO ) EPSI, ITMAX Ino0 AOro, AOSO *For the harmonic solution Wf = <Es Through Q for Through P for K =0'1 r AO =AO0 Test =OB I If Det M~x) AWr Inc, r ro K > ITMAX A0r AO= A0so Awr > Acr Max Whenever 63 < EPSI* P F = 6 Evaluate an d =A ~86A0 tEvluem Solve Equation A. 8 andA + 6A the Elements 5A8r < EPSI* of the M(x) for, Ar r A Ar and Matrix r' s AO and s If Det M(x) Print /0 =Wev is Zero AO8 = T ohene v e or Print r= s Test = OB No Solution A f*AO so s Test = lB Fig. B.1. Flow chart of the Newton-Raphson method program.

flx) - Ar - G[Jo0B) sin (Aer) + nljn() sin (Aes)], (B. 1) n Jn- (l) sin thes) f2(x) = cot ( Wf) - g9- F c(B.2) nJnIa.2 1 ( f (x) = 8$2f2- G2ITr(f) 12[4J12 () cos2 (AOr) -4J1(8) cos (Aer)' n.n(B) cos (Aes) + n2Jn2L1(8)] (8.3) where x - column vector [B, ero, Aes]. (B.4) The respective partial derivatives (the matrix required elements) can be expressed as: fll(x) = -G{J1(8) sin (AOr) + n sin (Aes) [B Jnt() Jn-l)] (B5) f12(x) =G Jo(B) cos (Aer), (B.6) f13(x) = n GJn(a) cos (Aes) (8.7) 2nsin (Ae ) cos (Ae r)[Jo(B)Jni(8)- Ji(B)Jni(B)J()J n( )] (8) [nJn 1(B) cos (Aes) - 2J1(B) cos (Ar) ]2 -2nJ1(8)Jn-l() sin (Ase) sin (AOr) f (X) =os - co(B.9) 2oj2~~~~~~~~o~~~ (B2 nB. 10) 23) -(BJ 1 (a)cos (aet c os (AO) =_ I,,.....1,.... f2(x)= W n-In(a) Cos (Ae B 2J 10) Cos (es -)]2 f31(X) = G2 IT(wf) 12{8 cos2(Aer) J1(B)[Jo(8) - B Jl(8)] -4n cos (AOr) cos (A G) n()[J(8) - - 4n cos (AO) Cos (AO5) J1(8) l n-l 0 a I (B.11) )12. n- [BJn i B) - Jn(a )i + 2rn2Jn (B) [ (JnO-lt) Jn(B)hl -2Bwf (B.11) f32(x) * 4G2ITT(if) 2J1 () sin (Aer)[nJn1(8) cos (A) -2J1(8) cos (e r)], (8.12) f33(x) = 4nG2I]T(uf) 2Jl(B)Jn1(B) cos (0s) cos (ABr). (8.13) 117

Equations B.5-B.13 constitute the nine elements required for the 3 x 3 M(x) matrix in Eq. A.8 for the original three coefficient equations. Equations B,1-B.3 constitute the elements of the f(x) vector also required by Eq. A.8. In the program, all twelve of thse elements must be evaluated at each iteration in order to solve for the correction elements in 6xk. 118

APPENDIX C DERIVATION OF SOME FOURIER COEFFICIENTS In Section 4.6, the Fourier series expansion of two periodic functions are required. The appropriate coefficients for these series are developed below. The first function to be considered is the radical expression,l + + 2n cos AWst, with period 2/Aw s. The constant n is positive and less than unity. This function is even with respect to t and has a nonzero average value. The appropriate cosine series coefficients are given by ak - 1 + n2 + 2n cos x cos kx dx, k = O, 1, 2..., (C.1) where x = Aust. (C.2) With the aid of the identity cos x = 1 - 2 sin2() (C3) the a coefficient becomes 0 a = 4 + 2 + n2 4n sin2() dx/2 (C.4) 0 7' or a = J 1 n)2 - 4n sin2 dy (C.5) with the change of variables y = x/2. (C,6) 119

Let k2. 4n.(C. 7) (l+rn)2 Then Eq. C.5 may be written as r./2 a- 4(l+I - k2 sin2 y dy - (n E(k,R/2), (C,8) a0 r=. 0 where E(k i/2) is the complete elliptic integral of the "second kind." Although E(k,r/2) is a function of k (and hence n) it will be helpful to write it simply as E in the following development. Thus, the leading coefficient in the cosine series is just 2(l+n) ao/2 = 2(n) E. (C, 9) For small n this may be expanded in the form 3 n2 a /2 = (l+n)[1-n-...] (C.10) (1+n)2 which shows that the average value of this function for small n is similar to unity, The calculation of the coefficient a1 is carried out in much the same way. With the changes of variables (C.2, C.6 and C.7) and the identity (C.3), al may be written as ir/2 al J 1 - k2 sin2 y/(l - 2 sin2 y) dy (C.ll) or i 2 al _- 2 (2n) a1 4-2 | Esi2y2 s in sin y dy (C. 12) The integral of Eq, C.12 may be evaluated with the aid of the identity sin2 y - sin2 y dy = sin2 y(l - k2sin2y)dy (C,13) j +o + E 1 l- J 1k2 sin2 y With the aid of Ref. 25 the two terms of the right-hand side may be expressed by 120

7r/2 sin2.dy - 1 [F(k,/2) - El (C. 14) o/-k2 sin2y k2 and (r/2 | / sin y dy 2 (l2k2) (F(k, /2) - E) 1 J siTYd-2 (1,k2) F(k, /2) (C.15) O~/iI/l~T~i; 3k2 3k2 where F(k,nr/2) is the complete elliptic integral of the "first kind", abbreviated to F below. Combining these expressions yields a = 4(1+n)2 [(2 - k2) E + 2(k2 - 1)F]. (C.16) 3k27 For small n it may be shown that al1 is similar to n. The higher order coefficients may be found in exactly the same way. For example a2 is easily placed in the form rf/2 a2 = 4(l+) /1 - k2 sin2 k (1 - 8 sin2 y + 8 sin 4y) dy, (C.17) 2o from which it is not difficult but somewhat tedious to show that a 4(1+ [15k2 E - 8(1 + k2)F + 16(1 - k2 + k4)D], (C18) where D is defined as D =F E (C. 19) k2 The second function for which the Fourier coefficients must be found is ncos(Awst) + n2 A l c. In this case it is easy to derive the general coefficient. Once s l+n2+2ncos(Awst) again this function is even with respect to t, but has no constant term. This may be seen from its integral (Eq. 4,34) which has no term increasing linearly with t. The desired coefficients are given by ak= -...I cosx cos kx dx, k = 1, 2,..., (C.20) o ln2+2ncos x 121

where again x = Awt. (C.21) With the aid of the following two definite integrals found in Ref. 25, J cos kxdx (-n)k cos kxdx =. (-.. for n2 < 1 (C.22) o l+n2+2ncos x l-n2 and cosxcoskxdx = l+n2 k- I0~~ =.... (-n) for n < 1, (C.23) o l+n2+2ncos x 2 Eq. C.20 takes the form 2TnAw7r k-n l + n 1(n) =k s [ n2 (- -- n - 2 (C.24) 1-n2 1-n2 which can be simplified algebraically to ak k -(-n) a (C. 25) 122

REFERENCES 1. G. W. Preston and J. C. Tellier, "The Lock-In Performance of an AFC Circuit," Proc. IRE Vol. 41 (1953), pp. 249-251. 2. W. J. Gruen, "Theory of AFC Synchronization," Proc. IRE, Vol. 41 (1953), pp. 1043-1048. 3. R. Jaffe and E. Rechtin, Design and Performance of Phase-Lock Loops Capable of NearOptimum Performance Over aWide Range of Input Signal and NoiseLevels Progress Report No. 20-243, Jet Propulsion Laboratory, Dec., 1954. 4. A. J. Viterbi, Acquisition and Tracking Behavior of Phase-Locked Loops, External Publication No. 673, Jet Propulsion Laboratory, Julyt 1959. 5S H. T. McAleer, "A New Look at the Phase-Locked Oscillator," Proc. IRE, Vol. 47 (1959), pp. 1137-1143. 6. T. J. Rey, "Automatic Phase Control: Theory and Design," Prock IRE, Vol. 48 (1960), pp. 1760-1771. 7. C. S. Weaver, "Thresholds and Tracking Ranges in Phase-Locked Loops," IRE Trans. SET, Sept. 1961, pp. 60-70. 8. A. K. Rue and P. A. Lux, "Transient Analysis of a Phase-Locked Loop Discriminator," IRE Trans. SET, Dec. 1961, pp. 105-111. 9. D. L. Schilling, The Response of an Automatic Phase Control System to FM Signals and Noise, Research eport o. PIB -40-62 Poytecnic nst o Broolyn, June 19 10. J. A. Develet, Jr., "A Threshold Criterion for Phase-Lock Demodulation," Proc. IEEE, Vol. 51 (1963), pp. 349-356. 11. A. J. Viterbi, "Phase-Locked Loop Dynamics in the Presence of Noise by Fokker-Planck Techniques," Proc. IEEE, Vol. 51 (1963), pp. 1737-1753. 12. tH. L. VanTrees, "Functional Techniques for the Analysis of the Nonlinear Behavior of Phase-Locked Loops," Proc. IEEE, Vol. 52 (1964), pp. 894-911. 13. J. Granlund, Interference in FrequencZ-Modulated Reception, Technical Report No. 42, Mass. Inst. of Tech. Jan. 14. T. S. George, "Analysis of Synchronization Systems for Dot-Interlace Color Television," Proc. IRE, Vol. 39 (1951), pp. 124-131. 15. D. Richman, "Color-Carrier Reference Phase Synchronization Accuracy in NTSC Color Television," Proc. IRE, Vol. 42 (1954), pp. 106-288. 16 L. H. Enloe, "Decreasing the Threshold in FM by Frequency Feedback," Proc. IRE, Vol. 50 (1962), pp. 18-30. 17. C. S. Weaver, "Increasing the Dynamic Tracking Range of a Phase-Locked Loop," Proc, IRE, Vol. 48 (1960), pp. 952-953. 18. Reference Data for Radio Engineers, (International Telephone and Telegraph Corp., 1956). 19. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, (New York: McGraw-IHill Book Company, Inc., 195I). 20, F. R. Moulton, Differential Equations (New York: Dover Publications, 1958). 21. J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, (New York: Interscience Publishers, inc.,19503 123

REFERENCES (Cont.) 22. J. E. Bridges and R. A. Zalewski, "Orthogonal Detection to Reduce Common Channel Interference," Proc. IEEE, Vol. 52 (1964), pp. 1022-1028. 23. T. W. Butler, Jr. and E. M. Aupperle, "Solid-State Discrete-Frequency Synthesizer," IRE Trans. Inst., Sept. 1962, pp. 67-71. 24. B. Carnahan, 1i. A. Luther, and J. O. Wilkes, pplied Numerical Methods, Vol. I, (New York: John Wiley and Sons, Inc., 1964). 25. I. M. Ryshiki and I. S. Gradstein, Summen-, Produkt- und Integral-Tafeln Tables, (Berlin: VEB Deutscher Verlag Der Wissenschaften, l157). 124

DISTRIBUTION LIST No. of Copies 20 DDC Cameron Station, Bldg. 5 ATTN: TISIA 5010 Duke Street Alexandria, Virginia 3 AFAL Wright-Patterson AFB, Ohio ATTN: AVWW (Deception) 2 Research and Technology Division Wright-Patterson AFB, Ohio ATTN: SEPI SEPIR 1 Dr. B. F. Barton, Director Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 40 Project File Cooley Electronics Laboratory The University of Michigan Ann Arbor, Michigan 125

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