THE PERIODIC SOLUTIONS OF THE DIFFERENTIAL EQUATION OF A RESISTANCE-CAPACITANCE OSCILLATOR

A third-order differential equation is considered of a form which arises in con-nexion with a resistance-capacity oscillator, the equation being normalized to the form, , ^g.. (D'+l)\D + —)+cD*\x+g(D) £ culi*-* & = g(D)2Bcosa>t, where e and (x. are small parameters which in the main part of the discussion are re-lated by e = (i1, and g(D) is a particular polynomial operator of the third degree. The procedure previously applied to a second-order equation with unsymmetrical non-linear damping, is used to obtain the periodic solutions having the period of the forcing term when a> is near to 1, i.e. for the case of fundamental resonance. It is shown that the resonance curves are of the same form as those obtained for the second-order equation and that the stability of the periodic solutions is determined by variational equations which are likewise identical in form to those previously obtained for the second-order equation. 1. Derivation of the differential equation THE equations of a triode oscillator may be written in the form (cf. (1)) v0 = en+<xZia, Zia = e a- V where va, vg are the anode and grid voltages, Z is the impedance of the linear circuit at the anode terminals when the grid terminals are open, a is the voltage transfer ratio from anode to grid through the linear circuit, and ea, eg are the effective e.m.f.s in the anode and grid meshes of the linear circuit. The currents and voltages are increments from the quiescent point, and it is assumed that the circuit is biased so that no grid current flows. If it is assumed further that ia is a function of mvn + v,,, with m the amplification factor of the valve, and if we set v = mvu+v,,, c — nie;l-t-e,, then we obtain [l~ni\)Zi,, < •>\ IQunrl. j o u r n. Mech. »nd Appllwl Mnlh.. \ ol. X. I'l. I < l*>57)