0 (b) p <0 Fig. 4.12. Dependence of the Amplitude I on Detuning ((K I Fig. 4.13. Trajectories for Circuit Nonlinear Resistance, z -= /2,, P = 3, P = I ILLINOIS ENGINEERING EXPERIMENT STATION Fig. 4.14. Trajectories for z = 1.05, p = -'/2, it = -1 Fig. 4.15. Oscillations Corresponding to Drift Curve Shown in Fig. 4.14 Figure 4.14 shows the trajectories for z = 1.05. Because this is out- side the "region of synchronization," no singular points exist. There exists, however, one (and only one) drift curve shown dotted in Fig. 4.13. Figure 4.15 shows the oscillation corresponding to that curve. This figure shows a good agreement with some experimental results published by W. L. Barrow.'" *Parenthesized superscripts refer to correspondingly numbered entries in the Bibliography. V. SYNCHRONIZATION In this chapter the oscillatory circuit of Fig. 5.1 is discussed. This circuit is the equivalent of an ordinary plate-tuned oscillator with an external sinusoidal synchronizing voltage vs. When v, is zero, the circuit will oscillate with a frequency, 1 0 V'LC The frequency of oscillation w for v, 0- is not necessarily wo. As the frequency of the external voltage w, is varied and becomes approxi- mately equal to 00, w will suddenly change from wo to so. This phenomena is called locking, synchronization, or entrainment of frequency. It occurs not only if , a-- but also if ws/ýwo=p/q, where p and q are "small integers." This phenomenon is called subharmonic synchronization and has been discussed in detail in Bul. 400. If it is assumed that v, = V, cos w,t, then the voltage vi between points A and B is approximately sinusoidal, particularly if the losses in the resonant circuits are small vi - V cos (wt + €) It was shown in Bul. 400 that the circuit of Fig. 5.1 can then be reduced to that of Fig. 5.2 in which the non-linear element is replaced by an "equivalent network." The differential equations for V and q6 for this circuit are dV V [G + G (5.1) dt 2C d - 1 C (5.2) dt 2 - where Aw = wo - Ws These equations determine the transient behavior of the oscillator. Dividing Eq. 5.2 by Eq. 5.1 eliminates the time do_ _ 2XwC - c0oC, (5.3) dV v (G + Ge) This equation permits representation in a plane with V and 4 as coordinates, the V-4 plane. Steady-state oscillations (d V/dt = de/dt = 0) correspond to singular points. Besides singular points, the V-0 plane ILLINOIS ENGINEERING EXPERIMENT STATION C Fig. 5.7. Equivalent Circuit of a Tuned-Plate Oscillator with External Synchronizing Voltage QGe Ce L 6 C Fig. 5.2. Equivalent Linearized Circuit of Fig. 5.1 may also contain drift curves. They occur particularly if an oscillator is close to synchronization."" As an example, the special case w = 3w, is presented here. The fre- quency of the oscillator is three times that of the external voltage (frequency-multiplication). It is assumed that the current i of Fig. 5.1 can be expressed by a third degree polynomial of the voltage v i = av + 3v2 + 7v3 The equivalent conductance and capacitance for w=3o, and a third degree polynomial were calculated in Bul. 400. They are: 3 ly Vý3 G = a + 7(V2 + 2V,2) + - cos Ce = V sin 0 (5.4) 4co V Using these two expressions, Eq. 5.3 may be written as d zV2 +(V8/V)sin (5.5) dV 3V (V - Vo2) + V3 cos where 8wC Aco V,2 4 (G + a) _ 2V 3-y Introducing the dimensionless parameters X= V/Vo and X,/Vo, Eq. 5.5 can be simplified to d Xz Xz + 3 sinX dX X [3X (X2 - 1) + X,3 cos (5.6) The solutions of this differential equation can be represented as trajectories in the X-4 plane. The singular points in this plane are stable if they fall inside the region of stability in Fig. 5.3. Figure 5.4 Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS Fig. 5.3. Regions of Stability and Variation of X as a Function of p for p = 3, q = 1 oS 10 v Amplitude Fig. 5.4. Trajectories with Stable Singular Point r ILLINOIS ENGINEERING EXPERIMENT STATION shows the family of trajectories in the X-4 plane corresponding to a stable singular point (point A). This point is a focus. Figure 5.5 shows the trajectories in the X-0 plane for an unstable singular point (point B). This figure also shows a drift curve corresponding to "near synchronization." Amphliude Fig. 5.5. Trajectories with Unstable Singular Point VI. SIMULTANEOUS OSCILLATIONS One of the most controversial and also one of the more interesting phases of nonlinear mechanics is that of simultaneous oscillations in circuits with two degrees of freedom. In recent years a number of papers dealing with this problem have been written and it has been proven both theoretically and experimentally that simultaneous oscilla- tions are possible. (171 19) For simultaneous oscillations, the voltage between points A and B of Fig. 6.1 is, approximately, vi = V1 cos (wit + 0i) and the voltage between B and C is v2 = V2 cos (Wt + 42) (6.1) where 1 # C2. A further distinction between wi and W2 gives rise to the necessity for distinguishing between synchronous and asynchronous simultaneous oscillations. For synchronous oscillations the frequencies Wi and 02 are related by PWi = qW2, where p and q are small integers, and comes about as a result of an effect of mutual synchronization. When the sum (p+q) is large, the synchronization becomes so weak that it can no longer counteract the continuous disturbances caused by line fluctuations, noise, etc. For asynchronous oscillations no relation P01 = qgc2 is preserved over an appreciable length of time. Mathemati- cally it is assumed in this case that the ratio Wi/w2 is constant but irra- tional. Asynchronous simultaneous oscillations were discussed briefly in Bul. 395. The oscillations in a circuit with two degrees of freedom can be represented in a four-dimensional phase space with (vi, dvi/dt, v2, dv2/dt) as coordinates. For a successful treatment, however, it is imperative that the oscillations of any system be represented in a plane or at most in a three-dimensional space. The four dimensions of the present case can be reduced to three if it can be assumed that vi and v2 are sinusoidal. The oscillations can then be represented in a space having as coordinates Vi, V2 and, for synchronous oscillations, the angle = p - q42 which is defined as the phase angle between vi and v2. An oscillation which has amplitudes Vi, V2 and phase angle 4 corresponds then to a point in this space. ILLINOIS ENGINEERING EXPERIMENT STATION Fig. 6.1. Oscillator with Two Degrees of Freedom The voltage v of Fig. 6.1 is v = v1 + v2 = V1 cos (wit + 01) + V2 COS (w2t + 02) This voltage applied to the nonlinear element will produce a current i = In cos (wit + 01) + 112 sin (wit + 0i) + 121 cos (W2t + 02) + 122 sin (w2t + 02) + terms at frequencies other than w1 or W2. (6.2) As shown in Bul. 395, Chapter I, the circuit of Fig. 6.1 can then be replaced by the two circuits of Fig. 6.2 where Gie = TIll. CI 112 V,1 WVl (6.3) G2e = 121 Ce - = 122 V2 W2 V2 where the equivalent conductances and capacitances are functions of V1, V2, and 4. If p and q are small integers, then the expression 4 = pfl -q02 is determined independent of a shift of the time axis. However, if either p or q is irrational, 4 can still be adjusted arbitrarily close to any value Fig. 6.2. frequent Circuit of Fig. 6.1 Fig. 6.2. Equivalent Circuit of Fig. 6.1 Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS by a suitable shift of the time axis. Since this shift is a purely mathe- matical operation and cannot have any influence on the oscillator, it is clear that the equivalent impedances for asynchronous oscillations cannot be functions of 4. The transient and steady-state behavior of the system can be calculated from the circuits of Fig. 6.2. The conductances of Fig. 6.2 will damp the oscillations and change the amplitudes. The capacitances C1 and C2 will change the frequencies, wi and 02, and cause a shift of the phase angle, 0. The corresponding differential equations are for asynchronous oscil- lations dVi Vi dV = - (G( + G-e) dt 2C1 1 dV V (G2 + G2) (6.5) dt 2C2 and for synchronous oscillations dV1 V ± d - V- (G, + Gie) (6.6) dt 2C1 dV2- V2 (G + G2e) (6.7) dt 2C2 d A-P , C (6.8) dt 2 C1 where C6 = C1- C2 By dividing Eqs. 6.5 it is possible to eliminate the time between them, giving dVi _ V1C2 (G + G) (6.9) dV2 V2C1 (G2 + G2) This differential equation can be solved by the methods of isoclines in a plane with Vi and V2 as coordinates. A point in this plane corre- sponds to an oscillation with the amplitudes, Vi and V2. If the system is not disturbed, then the representative point will move along a tra- jectory. Figures 6.3 and- 6.4 show typical groups of trajectories. As shown in Bul. 395, stable simultaneous steady-state oscillations are not possible if the current and voltage across the nonlinear element are related by (6.10) i = av + fv2 + yva ILLINOIS ENGINEERING EXPERIMENT STATION Fig. 6.3. Trajectories for Asynchronous Oscillator and i = av + jav2 + Yv' Fig. 6.4. Trajectories for Asynchronous Oscillator and i = av + Gv' + Yv' + Bv + ev Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS The trajectories shown in Fig. 6.3 correspond to those obtained from such a polynomial. The point A shown in this figure is one that would ordinarily correspond to simultaneous oscillations, but it is a saddle point and is therefore unstable. On the other hand, if the current- voltage relationship is a fifth-degree polynomial and if certain other conditions are satisfied, then stable simultaneous oscillations are possi- ble. Such a condition is represented by point B in Fig. 6.4. This figure also shows that as a rule asychronous simultaneous oscillations are not self-starting. In general the representation of the synchronous oscillations in the Vi-V2-4 space is not practicable. For certain conditions, however, Eq. 6.8 can be reduced to dt= f (V1, V2) sin 0 (6.11) where f(Vi,V2) >0 for VI >0, V2 >0. All trajectories will then approach the 4 = v plane and a trajectory starting in this plane will remain in it. Figure 6.5 shows a typical group of trajectories in the 0= ~ plane. The simultaneous oscillations corresponding to point A in this figure are stable and self-starting. Fig. 6.5. Trajectories for Synchronous Oscillations (eW = 3W2) VII. THE LIMITATION OF THE AMPLITUDE OF OSCILLATIONS BY LAMPS The amplitude of oscillations in an oscillator containing only linear elements will either decrease or increase indefinitely. It can reach a stationary value other than zero only if the oscillator contains a non- linearity of some sort, in most cases an electron tube. The electron tube has a double function: it delivers a-c power to the passive elements and limits the amplitude of oscillation. In the oscillator discussed here, these functions have been separated. The electron tube delivers the a-c power, but the amplitude is limited by a lamp inserted in the resonant circuit. The temperature of this lamp depends on the power dissipated in it and, hence, on the amplitude of oscillation. The resistance of the lamp varies as a function of fila- ment temperature. For small amplitudes of oscillation more power will be generated in the electron tube than is dissipated in the lamp, and as a consequence the amplitude will increase. Due to this increase, the temperature of the lamp, and also its resistance, will become greater. At a certain am- plitude the resistance of the lamp will be so great that the power gener- ated in the electron tube will be equal to that dissipated in the lamp, and steady-state conditions for amplitude are achieved.""4 Fig. 7.1. Oscillator An idealized oscillator using a lamp to limit the amplitude of oscilla- tion is shown in Fig. 7.1. Actual oscillation using lamps to limit their amplitudes are usually more complex."', 16) The variation in resistance of a conductor with the temperature is RT, = RTr (1 + aTi) (7.1) Bul.421. OSCILLATIONS IN NONLINEAR SYSTEMS where a is a constant depending on the material; RT, the resistance at the ambient temperature To; and Ti= T- To the difference between the temperature of the conductor and the ambient temperature. It would be difficult to treat the circuit of Fig. 7.1 in full generality; therefore a number of simplifying assumptions must be made. It is assumed that no grid current flows and that only the linear portion of the triode characteristic is used. The plate-current is then a function of the plate voltage only: ip = gm 9. + -- V = gm-v = - GovP (7.2) where (M 1) Go = gm L- is a constant with the dimensions of a conductance. Equation 7.2 shows that the triode of Fig. 7.1 can be replaced by the equivalent negative conductance -Go. (Fig. 7.2) The lamp has been replaced in this figure by a resistance RT, The circuit of Fig. 7.2 can be further simplified. Since the frequency of oscillation w differs by only a very small amount from 1/ VLC , the conductance -Go can be replaced by a resistance -Ro in series with the inductance L (Fig. 7.3). L Ro = -- Go The total resistance of this resonant circuit is R = -Ro +- RT = -Ro+RT (1 +aTi) = - R1 (1 -/3T) (7.3) where Ri = Ro - RT R, It is assumed that the quality factor Q of the circuit is high. The current i can then be approximated over a few cycles by i = I cos ct (7.4) 34 ILLINOIS ENGINEERING EXPERIMENT STATION 0 R(T7) Fig. 7.2. Oscillator with Triode Replaced by Negative Conductance The amplitude of current I will vary slowly and, for a short period of time, it can be assumed constant. The total energy stored in the circuit of Fig. 7.3 is LP/2. During one cycle, it changes twice from electromagnetic to electrostatic energy and back again, but the total amount remains approximately constant. A slow change is caused, however, since some of the energy is dissipated (or generated) in the resistance R. L dP R1 Sd = 2 (1 - ) (7.5) 2 dt 2 A similar differential equation can be found for the thermal energy mcT, of the lamp dT1 12 mc d = -KT1 - RT, (7.6) where m is the mass of the heat element, c is the specific heat of the element, and K is the constant of thermal conductivity. For steady- state oscillations it is necessary that both I and Ti remain constant; i.e., dI dT --- --0 dt dt The values of I and T1 corresponding to steady-state oscillations are 2K 1 1o2 = f- ; T1 = 1 (7.7) These oscillations may be stable or unstable depending on whether they will or will not resume their original amplitude after being subjected to a small disturbance. This can be determined by calculating and examin- ing the conditions for stable steady-state oscillations from Eqs. 7.5 and 7.6. Such an examination shows that these oscillations are stable. Fig. 7.3. Equivalent Circuit of Fig. 7.2 Bul. 421. OSCILLATIONS IN NONLINEAR SYSTEMS The behavior of the nonsteady-state oscillations can be determined also from Eqs. 7.5 and 7.6. Dividing one of these equations by the other eliminates time as a variable and makes possible the representation of the transient oscillations in a plane with P and T1 as the coordinates 12 dT L -- T, + 2K [R0 - R, (1 - 3 - Ti)] dTi L 2K d - R1T 12(1 - T) (7.8) where T= K/mc is the thermal time constant of the lamp. This differential equation cannot be solved in closed form. An ap- proximate solution, however, can be found by the method of isoclines. In Fig. 7.4, a typical solution is represented by a group of trajectories in a plane with the amplitude of oscillation as the abscissae and the temperature as the ordinates, the I2-T1 plane. At a given time the oscillator will have an amplitude, I, and the lamp a temperature, Ti. The point in the P-Ti plane defined by these two values will be called the "representative point" since it represents the state of the oscillator. As temperature and amplitude vary, the representative point will move along a line or "trajectory" in the P-Ti plane. For example, if the oscillator starts from rest (point 0 in Fig. 7.4), it will move toward the point B along the dotted line. T, 1 0 Amph/>ude Fig. 7.4. Trajectories Describing the Transient Behavior of the Oscillator ILLINOIS ENGINEERING EXPERIMENT STATION The point B corresponds to steady-state oscillations with amplitude, Io, and temperature, Ti0. The steady-state oscillations corresponding to point B will be dis- turbed constantly because of noise, variations in circuit parameters, and change of the ambient temperature; but it can be seen that the representative point, after having been displaced from B, will always return to its original position. Consequently the steady-state oscillations corresponding to point B are stable. By a similar consideration it can be seen that point A is unstable since the representative point will never return to this point after it has been given a small displacement. By assuming the current i to be i = I cos ct it was possible to describe the oscillator by the equations 7.5 and 7.6. If no assumption regarding the current i is made, however, it is neces- sary to use three differential equations of the first order. For example, the. circuit of Fig. 7.3 leads to di w= y dy 1 1. dt L C(1-T1)y- dTt = T + m1 (Ro - Rl (1 - fT1))i2 (7.9) where i is the current through the resonant circuit. Similar to the 12-Ti plane it is possible to set up a i-y-Ti space. The solutions of Eqs. 7.9 could then be represented by a group of trajectories in this space. VIII. CONCLUSIONS Several oscillatory systems have been discussed in this bulletin with special emphasis on nonsteady-state or "transient" oscillations. For autonomous systems with one degree of freedom and any degree of nonlinearity, a complete representation of the transient oscillations is possible in the phase plane. For more complicated systems, this method of representation is either not possible or impracticable since it leads to spaces with four or more dimensions. A representation in a plane (which is no longer the phase plane) is, however, still possible in some important cases if it can be assumed that the oscillations are almost sinusoidal. As has been shown, the transient oscillations can then often be described by two first order differential equations: dX dt = fA (X, Y) dY = f2 (X,Y) (8.1) where X and Y may be an amplitude, phase angle, temperature, etc. These equations do not give a complete description of the oscillations, however, since they neglect the higher harmonics. This does not lead to appreciable errors if the losses in the resonant circuits are small (high Q) since then the amplitudes of the higher harmonics are also small. Even for low Q circuits the results calculated from Eq. 8.1 are confirmed surprisingly well by experiments. The general procedure is to eliminate from Eqs. 8.1 by division: dX f- (X, Y) dY f (X,Y) The solutions of this differential equation can then be represented as a group of trajectories in the X-Y plane, in which an oscillation with parameters X and Y corresponds to a point lying on one of the tra- jectories in this plane. If the system is not disturbed, then this point will move along the trajectories defined by Eq. 8.2. Steady-state oscil- lations correspond to singular points of this equation. These oscillations 38 ILLINOIS ENGINEERING EXPERIMENT STATION are stable if after being subjected to a small disturbance the parameters X and Y will return to this point. Besides the singular points, the X-Y plane may also contain closed trajectories or drift curves. Four different oscillatory phenomena were treated by this method: parametric excitation, synchronization, simultaneous oscillations and amplitude limitation by means of lamps. The method can be applied to a number of other problems, such as oscillations in conservative systems with several degrees of freedom, self-modulating oscillators, and the motor-boating effect occurring in audio oscillators. The results obtained in Chapter V and VI were confirmed experi- mentally. The degree of correspondence between theory and experiment was good. APPENDIX: BIBLIOGRAPHY 1. N. Minorsky, "Introduction to Non-linear Mechanics," J. W. Edwards, Ann Arbor, 1947. 2. J. Stoker, "Nonlinear Vibrations," Interscience Publishers, New York, 1950. 3. L. Mandelstam and N. Papalexi, "Ueber einige nichtstationare Schwingunga- vorgange," Journ. Tech. Phys., USSR 1, 1934, p. 415 (in German). 4. McLachlan, "Theory and Application of Mathieu Functions," Oxford Uni- versity Press, 1947. 5. B. van der Pol and M. J. 0. Strutt, "On the Stability of Solutions of Mathieu's Equations," Phil. Mag., Vol. 5, 1928, p. 18. 6. W. L. Barrow, "On the Oscillations of a Circuit Having a Periodically Vary- ing Capacitance," Proc. IRE, Vol. 22, 1934, p. 201. 7. L. Mandelstam and N. Papalexi, "Expose des recherches recentes sur les oscillateurs non-lineaires," Journ. Tech. Phys., USSR 2, 1935, p. 18 (in French). 8. L. Mandelstam and N. Papalexi, "On the Excitation of Electrical Oscillations by Variation ofParameters," Journ. Tech. Phys., USSR 4, 1934, p. 5 (in Russian). 9. W. Gulajied and W. Migulin, "On the Stability of Oscillatory Systems with Periodically Varying Parameters," Journ. Tech. Phys., USSR 4, 1934, p. 48 (in Russian). 10. N. Minorsky, "Parametric Excitation," Journ. Appl. Phys., Vol. 22, Jan. 1951, p. 49. 11. M. L. Cartwright, "Forced Oscillations in Nearly Sinusoidal Systems," Journ. IEE, Vol. 95, 1938, p. 88. 12. A. Melikjan, "Ueber das Anwachsen der Amplitude bei Resonanzerscheinun- gen," Journ. Tech. Phys., USSR 1, 1934, p. 428 (in German). 13. A. Shibanko and S. Shelkov, "Qualitative Investigation of the Behavior of Electron-Tube Oscillators with Two Coupled Oscillatory Circuits by the Method of van der Pol," Journ. Tech. Phys., USSR 4, 1934, p. 158 (in Russian). 14. K. F. Teodorchik, "Auto-Oscillating Systems with Inertial Nonlinearity," Journ. Tech. Phys., USSR 16, 1946, p. 845 (in Russian). 15. L. A. Mescham, "The Bridge Stabilized Oscillator," Proc. IRE, Vol. 26, 1938, p. 1279. 16. F. B. Llewellyn, "Constant Frequency Oscillators," Proc. IRE, Vol. 19, 1931, p. 2063. 17. R. E. Fontana, "Internal Resonance in Circuits Containing Nonlinear Resist- ance," Proc. IRE, Vol. 39, 1951, p. 945. 18. L. V. Skinner, "Criteria for Stability in Circuits Containing Nonlinear Re- sistance," Ph.D. Thesis, University of Illinois, June 1948. 19. W. H. Huggins, "Multifrequency Bunching in Reflex Klystrons," Proc. IRE, Vol. 36, 1948, p. 624. The Engineering Experiment Station was established by act of the University of Illinois Board of Trustees on December 8, 1903. Its pur- pose is to conduct engineering investigations that are important to the industrial interests of the state. The management of the Station is vested in an Executive Staff composed of the Director, the Associate Director, the heads of the departments in the College of Engineering, the professor in charge of Chemical Engineering, and the Director of Engineering Information and Publications. This staff is responsible for establishing the general policies governing the work of the Station. All members of the College of Engineering teaching staff are encouraged to engage in the scien- itf*~ ~~~ ~ ^ - . i L . i1 - , e ^..i* ' '. * * '* r f ~,, ~k$;'l 44 tJI I' IA 4, iJ?'''t'>;7