ILLIN I UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN PRODUCTION NOTE University of Illinois at Urbana-Champaign Library Large-scale Digitization Project, 2007. UNIVERSITY OF ILLINOIS ENGINEERING EXPERIMENT STATION Bulletin Series No. 400 ALMOST SINUSOIDAL OSCILLATIONS IN NONLINEAR SYSTEMS Part II : Synchronization JOHANNES S. SCHAFFNER Formerly Research Assistant Professor of Electrical Engineering Pulblished bi the UniecrsitY of Illinois, Urbana UNIVERS'TY 300--4-52--49621 O PRLINOIS PR YESS I' CONTENTS I. INTRODUCTION 5 II. OUTLINE OF THE METHOD 7 1. Definition of Terms 7 2. Linearization of the Oscillator 8 3. Equilibrium and Stability 9 [II. ORDINARY SYNCHRONIZATION 12 4. Internal 1:1 Synchronization 12 5. External 1:1 Synchronization 16 IV. SUBHARMONIC SYNCHRONIZATION 18 6. Oscillations in the Absence of Synchronism 18 7. Synchronized Oscillations 19 I. INTRODUCTION 5 8.- Particular Examples of Subharmonic Synchronization 24 9. Bandwidth of Synchronization 27 V. EXPERIMENTAL VERIFICATION 28 APPENDIX: SELECTED BIBLIOGRAPHY 32 FIGURES 1.1 Tuned-Plate Oscillator with External Sinusoidal Synchronizing Voltage 5 1.2 Entrainment of Frequency 6 2.1 General Oscillatory Circuits 7 2.2 Equivalent Linearized Oscillatory Circuits 10 3.1 Equivalent Linearized Circuit for Internal 1:1 Synchronization 13 3.2 Regions of Stability and Variation of X as a Function of p4 for p = 1, q = 1 14 3.3 Region of Stability and Variation of X as a Function of 6tyAw for p = 1, q = 1 15 4.1 Range of pp for Stable Synchronous Oscillations (X02 = 1) 23 4.2 Regions of Stability and Variation of X as a Function of pp for p = 1, q = 3 25 4.3 Region of Stability and Variation of X as a Function of pP for p = 3, q 1 26 5.1 Circuit Used for Experimental Verification of the Theoretical Results 28 5.2 Region of Stability for p = 1, q = 3 (XA2 = 1) 29 5.3 Region of Stability for p = 3, q = 1 (Xo2 = 1) 30 5.4 Bandwidth of Svnrhrnnization. (Avi')}. as a Funrtiom of 5 4 Bandwidth...... , as a .. .... .. . Q, Q/, for External 1:1 Synchronization I. INTRODUCTION In the absence of the external voltage vs the circuit of Fig. 1.1 is a tuned-plate oscillator. It consists essentially of a resonant circuit and an electron tube. This electron tube has a double function. It feeds power into the resonant circuit in order to compensate for the losses in the passive elements and because of its nonlinearity, it limits the amplitude of oscilla- tion. The frequency of oscillation in the first approximation is Wo = 1/vLC. 1. INTR In th( Fig. 1.o1 of a resc has a dou in order because o tion. Tl- is oo = 1 If the varied 01 Lation d( 1. 2. If becomes ; integers, ^o to ( 3Ce + K 2C 30 Necessary and sufficient conditions for this are that V +-- w >0 3CG 3C G V W +0 av -a v ýd (2.13) (2.14) Systems satisfying Eqs. 2.11 and Ineqs. 2.14 will produce stable steady-state oscillations V 3G, e +K 2C 3V a 3C 2C 'V III. ORDINARY SYNCHRONIZATION 4. Internal I:1 Synchronization In Fig. 2.la, the voltage Vs is inserted in the resonant circuit in series with the inductance L. If the voltage v, = v cos ast synchronizes the oscillation, then the voltage v across the nonlinear element is v = V cos(6 t+t) (3.1) In this Chapter the subscript s of os is dropped, since for 1:1 synchronization w = w.s It is assumed that the voltage v and the current i passinf through the nonlinear network are related by i = av + ftv2 + yv3 (3.2) where a < 0, y > 0. This is a good approximation for most oscillators. It can then be shown that the equivalent impedanco of Fig. 2.2a consists of a conductance Ge only (Fig. 3.1) Ge = a + y2 (3.3) If no synchronizing voltage is present (vs 0), the conditiol for steady-state oscillation is G + Ge * 0 or 3 G + a +- yV2 = 0 (3.4) The value of V satisfying this equation is V0 4 G+a (3.5) Vo2 3 7 The discussion in this section is limited to oscillators foi which Vo2 > 0. In Fig. 3.1 the current passing through the inductance L is 1 iL =- [V sin(wt+O) - Vsin wt] (3.6) The same current would flow through the network consisting of L. G1 and C1 due to the voltage v = V sin (wt + 0) alone. The value for G1 and C1 are Theri state or where h The cone After ri Fig. 3.1. Equivalent Linearized Circuit for Internal 1:1 Synchronization Theri state or where h The cone After ri Bul. 400. NONLINEAR SYSTEMS 13 wCV G 8 sin 0 C, = C cos (3.7 V Therefore in the presence of a synchronizing voltage steady- state oscillations occur if C + G1 + Ge = 0 Aw Cl = 0 (3.8) w 2C where Aw = 1//LC - c. In terms of V and 4, these equations are 3 2 V G + a + -yV2 + wC-- sin 0 = 0 4 V (3.9) Aco Vs - - cos = 0 w 2V The conditions for stability are D3 C 3 V V-(G, +G ) + - yY2 " 2C 8 sin 3V a q 2 V 3 y(2V2 - V 2) > 0 (3.10) ac, a ac a -1 (G- + G) - - (Ge + G1 3 2 V = --yCV sin 0 + wC - > 0 2 V3 After rearranging the terms, the conditions for stability are 2V2 > V02 (3.11) 2wC V sin 0 <-- - 3y V3 13 (3.7) e steady- (3.8) ions are (3.9) (3.10) 2 -> 0 ty are (3.11) C ILLINOIS ENGINEERING EXPERIMENT STATION This last condition is satisfied automatically for sin 4 < 0. Foi sin 4 > 0, it can be simplified to (3.12) The equations for stability and equilibrium can be simplifiec somewhat through the introduction of dimensionless parameters V V, G+a X=- Xs =-V' A -31) X - - X = ---A V s V 2,-C Fig. 3.2. Regions of Stability and Variation of po for p = I, q = I of X as a Function Making these substitutions they become, for equilibrium, S1X A(X2-1) + - - sin 0 = 0 2 X - cos (P = 0 wa 2r and for stability (3.14) s'n2 < 1 Vo sin2 (<  -1) 2 V2 (3.13) Bul. 400. NONLINEAR SYSTEMS X2 > 2 and either sin p < 0 or if sin ( > 0 1 1 sin2p < - ) 2 X2 The phase angle p may be eliminated from these equations so that those for equilibrium reduce to 2 ý -4 2 (3.16) a[A2d 2(2e f2 + X (-)y to and those for stability to . (2 - 2 - 1) 0 (-T) + (x2 - l)(3X2 - 1) > 0 Fig. 3.3. Region of Stability NVAU for -A) A J and Variation of X as a Function of p = I, q = I (3.17) ILLINOIS ENGINEERING EXPERIMENT STATION The regions of stability may be represented in a plane with X Aw as the ordinate and either W or A as the abscissa. (Figs. 3.2 and 3.3). These figures also indicate the variation of X .with T- and 4, respectively. As stated before, these figures have been discussed extensively in the literature. 5. External 1:1 Synchronization In Fig. 2.1b, it can be assumed that the voltage v across the resonant circuit is approximately sinusoidal; that is v1 i ' V cos(wt+<) (3.18) If the oscillator is synchronized by the external voltage, the frequency of oscillation is w, and as in Section 4 the subscript of w, is dropped. The voltage across the nonlinear element is v = V cos(wt+p) + V8 cos Wt - (V + V, cos c) cos(wt+4) + V. sin P sin(wt+g) (3.19) It is again assumed that the current i and the voltage v are related by i = av + +V2 + yv3 (3.2) where a < 0, y > 0. Using Eqs. 3.19 and 3.2, the current i can be expressed as a function of time i = I1 cos(wt+c) + 12 sin(ot+0) + components at frequency 3w (3.20) The circuit of Fig. 2.1b can then be replaced by that of Fig. 2.2b where 1 12 Ge -, C, = - (2.7) V WV The values of Ge and Ce are V. 3 G, = (1 +-- cos 4) [a +-y (V2 + 2VV, cos q + V,2)] V 4 (3.21) Ce -" V sin q [a + -4(V2 + 2VVs cos p + V 2)] WV 4 If V, << V, these equations can be simplified considerably to Ge a +- yV2 4 (3.22) V 3 V C= -1- sin p (a +--V2) = -(-3 sin G2 V 4 wV Bul. 400. NONLINEAR SYSTEMS As before the conditions for steady-state oscillations are G + G =0 AW. C (3.23) e 0 S-2C where Aw = 1//LC - w. In terms of V and 4, Eqs. 3.23 are 3 G + a + - V2 = 0 Ao Vs (3.24) - 2OV sin Q Ge = 0 The conditions for stability are 'Ge 3Ce V"JF + W -W > 0 3G 3C, 3Ge 3Ce (3.25) 3v 3¢- 3- 3 V > 0 3G Since e = 0, these inequalities are satisfied if 3G, a3C -V > 0, 7 > 0 (3.26 or if y > 0 cos 0 > 0 (3.27) The permissible variation in P is, therefore, 7T 77 - < ^ < 2 The variation in frequency corresponding to this is Ao VW G V 1 o V C Vo where Q = cC/G is the quality factor of the resonant circuit. Thus the maximum variation in frequency is proportional to the ratio of amplitudes of the external synchronizing voltage to the internal voltage. Consequently if the external voltage increases, the bandwidth of synchronization increases likewise. In addition the maximum frequency variation is inversely proportional to the Q of the circuit. Therefore for small Q, the frequency can be entrained over a large band; but for high Q, the bandwidth of synchronization is relatively small.(7) IV. SUBHARMONIC SYNCHRONIZATION For external subharmonic synchronization the voltage v1 across the resonant circuit of Fig. 2.1 is again assumed to be approxi- mately sinusoidal v1 ^ V cos(Wt+k) (3.17) where w is approximately 1/ILC. For synchronization, w and the frequency ws of the external voltage v, are related by Eq. 1.1. The voltage across the nonlinear network is-then v = V cos(at+() + Vs cos wst (4.1) It is assumed that this voltage and the current i passing through the nonlinear network are related by a rapidly convergent power series S= av + /3v2 + yv3 + Sv4 +... (2.1) If this expression is combined with Eq. 4.1, then the current can be expressed as a function time I = I1 cos(wt+4) + 12 sin(wt+0) + components at frequencies other than w (2.6) The circuit of Fig. 2.lb can then be replaced by that of Fig. 2.2b where again Ge = 1 C = 12 (2.7) V e wV Methods for calculating Ge and Ce have already been discussed in University of Illinois Engineering Experiment Station Bulletin 395, Chapter II.(15) 6. Oscillations in the Absence of Synchronism For some frequencies the external voltage will not synchronize the oscillator. No relation p) = qw8 will then be preserved over any appreciable length of time and it can be assumed that W/w, is irrational. The equivalent impedances for )/w, irrational have been calculated in Bulletin 395, Chapter II.15) They are Gc = a + - y(V2+2V,) + . S(4.2) Ce = 0 Bul. 400. NONLINEAR SYSTEMS The prime indicates that no synchronism is present. To faci- litate further calculations, it is assumed that neglecting all terms of Ge other than the first two will not lead to appreciable errors. This is true for most oscillators. Steady-state oscillations are possible if a < 0, y > 0 and G + G' = 0 (4.3) or 3 G + a + y(V2 + 2V,) = 0 (4.4) The value of V that satisfies this equation is v2 4 G+a Vo2 = - 7 - 2V,2 (4.5) Vo2 has a physical significance only if it is positive, otherwise it is just a parameter satisfying Eq. 4.5. If Vo2 > 0, then the oscillations of the system are called "free" oscillations; if V02 < 0, "forced" oscillations. It is desirable that Eq. 2.1 be changed to a form correspond- ing to Eq. 11.1 of Bulletin 395.(15) aO i = Io axx (4.6) 1 where the ax and x are dimensionless. This can be done by defining v a 3 y V I \V \ 2=V 28 3 7 So2 V 1- 2- o -Vo 13 (4.7) Similarly, dimensionless variables corresponding to V and vs are defined as V V V I XX ' -- X G0 = (4.8) for free oscillations Xo2 = 1; for forced oscillations X2 - 1. An expression that will be helpful in the next section is 3 2) G + Ge = G + a +-(V2 +2,2) (4.9) Combining this with Eq. 4.5 and making the substitutions of Eq. 4.8 3 G + G'e -4a 3G(X2- Xo2) (4.10) 7. Synchronized Oscillations Locking phenomena occur if the two frequencies c and ws are related by an equation (1.1) pw = qoW, ILLINOIS ENGINEERING EXPERIMENT STATION where p and q are small integers. The special case p = q is not considered in this chapter. Some of the combination frequencies (nw+±mn) are now identical with w; as a consequence I1 and 12 and therefore Ge and Ce may contain additional terms. In fact these terms are necessary for the mechanism of synchronization. A double-prime is used to in- dicate such additional terms. For synchronization the equivalent impedances are G = G' + C" Ce = Ce (4.11) It is assumed that the series Xaxx converges rapidly. It is 1 then sufficient to consider only the first term that contributes to G" and C", As shown in Bulletin 395, Eq. 10.19,(15) this term is 1oa xx = p+q-l (4.12) and the corresponding G" and C" are (Bulletin 395, Eqs. 11.4 and 11.5) akGo h G" -i (q) xp-2 Xs cos P( ahC G (.4.13) C" = 2-l(q) Xp-2 X q sin cp These can be simplified by introducing a coefficient aX X 77 = q^r(q) G" and C" are then e e G' = 7Go XP-2Xsq cos pq C (4.14) C" -r-Xp-2 Xq sin p0 Some values for 77 are presented in Table I. Equations 4.14 correspond to a first-order approximation. As stated above, none of the terms Ioakxx where X = (p+q-l) will contribute to G' and C'e. This does not hold if approximations higher than the first are considered. For example, for 5w = w, (p = 5, q = 1) the contribution of Io a3x towards G" and C" is for a second-order approximation 27 G 2 G" a 2 -G X3Xs cos 5 27 G 2 (4.15) Ce = 128 " X3X, sin 50 snC Bul. 400. NONLINEAR SYSTEMS TABLE I Values of ) p q 1 2 3 4 5 1 ... 1/2 a2 1/4 a3 1/8 a4 1/16 a5 2 a2 .. 1/2 a4 ... 3/16 a6 3 3/4 a3 3/4 a4 ... 15/32 a6 21/64 a7 4 1/2 a4 ... 5/8 a6 ... 7/16 a8 5 5/16 a5 15/32 a6 35/64 a7 35/64 a8 ... The contribution due to a first-order approximation would be zero since X = 3 is smaller than (p+q-1) = 5. In most cases the terms of Eq. 4.15 are small compared with those corresponding to Ioa5x5 and a first-order approximation, particularly if the circuit has a large Q (G/wC small). They can therefore be neglected. As a result of Eqs. 4.14 the total equivalent impedances are then Ge = Ge + G" = Go [al + a3(X2+2X2) + rXP-2Xsq cos pc G (4.16) Ce = C" = - -- XP-2 X sin p As shown in Chapter II, steady-state oscillations can occur if G+Ge =0 A C, (2.11) o 2C where Ao) 1 q --r= / O In terms of X and 4, these equations are X2 - X 2 + --_ " X2 cos pc = 0 (4.17) Aco G -- + 7 G - XP-2Xq sin pb = 0 (4.18) w 2wCo If the phase angle is eliminated, then these two equations combine into 3 2 =o £] 2 [r a(X2-Xo )] + [2-L 2 = [77XP-2X q] (4.19) Equations 4.18 and 4.19 permit a representation of X as a func- tion of either pc or Aw/w. For the purpose of this bulletin, p is the more adequate independent variable. ILLINOIS ENGINEERING EXPERIMENT STATION The conditions for stable steady-state oscillations are 3G aC X - + " - > 0 (4.20) aC eG Ce aG e +e e > 0 The first of these two inequalities is, in terms of X and c, X3G 3C X-e + a)w c BX 34 ax 4€ = Go a3X2 + ?(p-2)oX-2Xq Cos pc - 7pXp-2Xs Cos p4] = Go [-a X2 - 27,X-2X, cos p4] 23 -7 = Go[a3X2 +ta3(X2-Xo2)] > 0 (4.21) Since a3 > 0, this inequality is satisfied if X2 > X 2 (4.22) 2 o For forced oscillations (X02 = -1), Ineq. 4.22 is satisfied automatically. The second part of Ineq. 4.20 is WCe aGe + 3Ce 'Ge 3 3X ~X 3 G 2 3 O.- {[77Pp-2X cos pq] [--aX2 + (p-2)X^2Xq cos p4] + (TpXp-2Xs sin pf] [7(p-2)Xp-2X"q sin p4]} G 2 3 - -pXp-'1 q9[-a3 cos p - 2(p-2)XP-4 Xs] > 0 (4.23) co 2 This inequality is satisfied if 7 cos p0 < - 2 p- XP-4X q (4.24) 3 a For Ineq. 4.24 three different cases of subharmonic synchron- ization have to be distinguished: p = 1, p = 2 and p > 2. a. p = 1 If p - 1, then the right-hand side of Ineq. 4.24 is positive. The range of the phase angle 4 for which the oscillation is stable is therefore larger than 7T. For positive 7 it extends Bul. 400. NONLINEAR SYSTEMS approximately from 7/2 to 37/2 and for negative - from -7/2 to 77/2 (Fig. 4.1). Equation 4.17 shows that for Xo2 = -1 (forced oscillation) the possible variation of 4 for which X is real is smaller than 77. Inequality 4.24 is therefore automatically satisfied. For Xo2 = 1 (free oscillations), 4 may however vary over a much wider range than 7 as indicated in Fig. 4.1, and therefore Ineq. 4.24 must be considered. The critical regions are shaded in Fig. 4.1. In these regions 7 cos 4 is positive. Using Eq. 4.17, Ineq. 4.24 can be simplified to: cos2 - <1 X 2 2 X2 (4.25) Fig. 4.2 shows the region of possible steady-state oscillations defined by Ineqs. 4.22 and 4.25 for ] > 0. The regions for q < 0 can easily be obtained by a phase shift of 1800. 6. p = 2 For p = 2, Ineq. 4.24 reduces to 7 cos 20 < 0 (4.26) The total variation of (24) is therefore 77 for Xo2 = 1 and smaller for XJ2 = -1 (Fig. 4.1). c. p > 2 If p > 2, then the right-hand side of Ineq. 4.24 is negative. Therefore the variation of the phase angle for which the oscilla- tions are stable is smaller than 77 (Fig. 4.1). Using Eq. 4.17, Ineq. 4.24 transforms to cos2 p4 > 2 - 2 X2 Fig. 4.1. Range of p4 for Stable Synchronous Oscillations (Xo2 = I) (4.27) 0 L_? I 0 4.! -77 -.77 - 31r n7 - 2 ~ ILLINOIS ENGINEERING EXPERIMENT STATION Equation 4.17 shows that for this variation and for Xo2 = 1, X > 1 and therefore Ineq. 4.22 need not be considered. The region of possible steady-state oscillations is indicated ir Fig. 4.3 for p = 3. For Xo2 = -1, Ineq. 4.27 can be satisfied for p = 3 only (Fig. 4.3b). For p ' 4, Ineq. 4.27 cannot possibly be satisfied. Therefore, for a first-order approximation no forced subharmonic oscillations are possible for p =4. This, of course, would notbe true if more than the first two terms of Eq. 4.2 were considered. 8. Particular Examples of Subharmonic Synchronization The discussion in this section deals with three examples of subharmonic synchronization: o = 3w ,, 2w = Ws, 3w = W. a. 0 = 3w8 (p = 1, q = 3) I For this example 7 from Table I is -a3. Eq. 4.17 is then 1 X3 X2 X2 cos V 3X The regions for stability for Xo2 = 1 are indicated in Fig. 4.2a. For Xo2 = -1, all steady-state oscillations are stable as shown in Fig. 4.2b. Fig. 4.2 also shows the variation of X with pc for X. = constant. The variations of the phase angle are -77/2 < A4 < T/2 (X2 = 1) 0 < A0 < 77 (Xo2 = -1) b. 20 = w, (p = 2, q = 1) From Table I 7 is a2 and Eq. 4.17 is X2 = Xo2 _4 a2X cos 20 3 a3 As shown in Fig. 4.1, the range of 20 is ~T for Xo2 = 1. For Xo2 = -1, all steady-state oscillations are stable. c. 3w = w, (p = 3, q = 1) For this case 77 is-a3. Eq. 4.17 is then X2 = Xo2 - X Xs cos 34 For Xo2 = 1 the region of stability is shoun in the shaded area of Fig. 4.3a; for Xo2 = -1, in Fig. 4.3b. The variation of X Bul. 400. NONLINEAR SYSTEMS 25 0 4- H- 0 C) Cf 0 0 41 C' L- 0 C 0 L. U- 26 ILLINOIS ENGINEERING EXPERIMENT STATION 0 0 0, cn 'U C4 C 4- 0 C) ·- Bul. 400. NONLINEAR SYSTEMS with pp is also shown in Fig. 4.3. As can be seen from these figures, the total change in phase is 7/2 < A3 < 7 (Xo2 = 1) 0 < A3 < 7T/2 (Xo2 = -1) 9. The Bandwidth of Synchronization So far in this bulletin the limits of stable synchronization have been expressed in terms of phase. For practical purposes it is, however, more important to know the maximum permissible de- viation in frequency, or the bandwidth of synchronization (ao/w)o It is of course possible to find precise expressions, but in most cases an approximate value is satisfactory. Such an approximate value can be found if it is assumed that X = 1 and that the phase angle varies by an amount 7 either from -7/2 to 7/2 or from 7T/2 to 37/2. The bandwidth of synchronization is then (from Eq. 4.14) Aw G aG a ( q-) - x - ( ) X q (4.28) o 2- wC This equation shows that the bandwidth of synchronization be- comes rapidly smaller as k = p+q-1 increases. V. EXPERIMENTAL VERIFICATION In the discussion of the theory some questionable approxima- tions have been made. It was therefore necessary to investigate how well the theoretical results are confirmed by the experiment. The circuit used in the experiment is shown in Fig. 5.1. In this circuit, designed by Dr. Giuseppe Francini, a con- ventional transitron circuit served as the nonlinear element. The operating point of the tube was adjusted by the potentio- meters R1 and R2 and slight changes in the current-voltage characteristic were made by changing tubes. The synchronizing voltage was supplied by a Hewlett-Packard oscillator. The ex- perimental results were obtained using resonant circuits tuned to a frequency of 6 kc and having a Q of about 35. Synchronization was observed with the aid of Lissajou figures on the screen of an oscilloscope. The voltages V and V, were measured directly across the oscillatory circuit and R . It was to be expected that the critical part of the theory was that dealing with stability rather than equilibrium. The conditions of stability for cases (3a = wS) and (a = 3ws) were checked in this experiment. The results of the experiment are shown in Figs. 5.2 and 5.3. It can be seen from these results that the correspondence between theory and experiment is reason- ably good. Equation 3.28 was also verified by experiment. Good corres- pondence was obtained for ratios of V,/V as high as 0.15. The experimental results for V,/V = 0.1 are shown in Fig. 5.4. Gen- erally, the maximum variation of the phase angle was somewhat larger than ?; therefore, the measured values of (Aw/w)o are somewhat too large. Bul. 400. NONLINEAR SYSTEMS ---------___- Fig. 5.1. Circuit Used for Experimental Verification of the Theoretical Results Fig. 5.2. Region of Stability for p=I, q = 3 (Xo2 = I) - - ILLINOIS ENGINEERING EXPERIMENT STATION Fig. 5.3. Region of Stability for p = 3, q = I (X2 = I) Bul. 400. NONLINEAR SYSTEMS 06" OE 03 a4 US 0h6 07 1/Q Fig. 5.4. Bandwidth of Synchronization, (A&/,w)o, as a Function of Q, G/cC, for External I:1 Synchronization 7 0 z o Theoret/ca/ Curve 0 Epementa/ Points nz'zzz/ _ _° z _ _ _ _ n- o / ^ ^^ "- APPENDIX: SELECTED BIBLIOGRAPHY 1. N. Minorsky, "Introduction to Nonlinear Mechanics," J. W. Edwprds, Ann Arbor, 1947. 2. J. J. Stoker, "Nonlinear Vibrations," Interscience Publishers Inc., New York, 1950. 3. E. V. Appleton, "The Automatic Synchronization of Triode Oscillators," Proc. of Cambridge Phil. Soc. 21, 1923, p. 231. 4. B. van der Pol, "Forced Oscillations in a Circuit with Nonlinear Resistance," Phil. Mag. 3, 1927. 5. J. Kobsarew, "Zur Theorie der Nichtlinearen Hesonanz," Journ. Techni- cal Physics URSS 2, 1935, p. 27. 6. H. Samulon, "Ueber die Synchronisierung von Hohrengeneratoren," Helve- tica Physica Acta, Vol. 14, 1941, p. 280. 7. R. Adler, "A Study of Locking Phenomena in Oscillators," Proc. IRE 34, 1946, p. 351. 8. D. L. Herr, "Oscillations in Certain Nonlinear Driven Systems," Proc. IRE 27, 1939, p. 396. 9. D. G. Tucker, "Forced Oscillations in Oscillatory Circuits and the Syn- chronization of Oscillators," Journ. AIEE 92, 1945. 10. M. L. Cartwright, "Forced Oscillations in Nearly Sinusoidal Systems," Journ. AIEE 95, 1948, p. 88. 11. R. Rjasin, "Einstellungs- und Schwebungsprozesse bei der Mitnahme," Journ. Technical Physics URSS 2, 1935, p. 194. 12. L. Madelstam and N. Papalexi, "Ueber Resonanzercheinungen bei Frequenz- teilung," Zeitschrift f. Physik 73, 1931, p. 223. 13. W. Migulin, "Ueber Autoparametrische Erregung der Schwingungen," Journ. Technical Physics URSS 3, 1936, p. 841. 14. R. L. Fortescue, "Quasi-Stable Frequency Dividing Circuits," Journal AIEE 84, 1939, p. 693. 15. J. S. Schaffner, "Almost Sinusoidal Oscillations in Nonlinear Systems, Part I: Introduction - Simultaneous Oscillations," Univ. of Ill. Engr. Exp. Sta. Bul. 395, 1951.