The use of the perturbation method for determining the pumping frequencies of parametric amplifiers
This paper demonstrates that an old but very powerful and very general mathematical technique, the perturbation method, may be used to investigate such a modern application as parametric amplification. All possible pumping frequencies are predicted without recourse to any specialized procedures. Introduction. One of the most powerful mathematical techniques for solving nonlinear differential equations, or linear differential equations with time varying coefficients is the perturbation method as introduced by Poincar<§ [1] and modified by Lindstedt [2]. In spite of this well known fact it is not used as often as it might be because often some less general but less lengthy means of solution [3] is available. If only the first order approximation is required, the method of Kryloff and Bogoliuboff may be used. If higher order approximations are required the KB method as generalized by Bogoliuboff and Mitropolsky [5] is useful. An excellent discussion of all methods can be found in the recent book of Minorsky [6]. It is often more difficult for the uninitiated in a particular field to learn and apply a specialized technique then it is for him to utilize an old familiar procedure. In the hope of making the perturbation method an even more familiar procedure, we apply it to the problem of parametric amplification. The parametric amplifier makes use of a nonlinear reactive element to achieve amplification; the name arises, because during operation, a parameter of the circuit is made to vary with time. The recent interest in this type of amplifier is due to its noise-free operation and its ability to function at the higher radio frequencies. Thus, today the reactive element is often a nonlinear (per se) capacitor. In the earlier electrical experiments, at low frequencies, the varying parameter was either a capacitor or an inductor whose value was changed by mechanical means. The degenerate case. As a review of the perturbation method, consider the so called degenerate case represented by a single circuit as shown in Fig. 1, in which R and C are constants but E and L are periodic functions of time. By Kirchhoff's second law we have j (Li) + Ri + | f idt = E. (1.1) In Eq. (1) let L = L0( 1 + K sin 2wt), L0 = const., 0 < K < 1, R = Kr, E = KA sin wt + KB cos ut. *Received January 6, 1964; revised manuscript received April 17, 1964. CARL A. LUDEKE [Vol. XXIII, No. 1