# Some properties of parametric systems with great depth of modulation-I. Theory

## Abstract

A discussion is given of the mechanism by which electromagnetic energy may be increased in power and frequency by means of a resonant system whose natural frequency is slowly changed over a wide range. The radiation pump of classical thermodynamics is discussed, as a known example of such a system, but is shown to be unrealizable for practical reasons. Analogous effects are, however, shown to exist in a lumped, time-varying LC resonant circuit. A new expression for the energy stored in such a circuit is derived, on the basis of which it is possible to distinguish two different kinds of parametric effect. The first kind of effect occurs when the variation of the impedance ( L C) 1 2 is correlated with the second harmonic of the oscillation of the circuit. This is the familiar type of parametric effect. The second type arises whenever the natural frequency ω = (LC) -1 2 varies appreciably, and is the main subject of the paper. It is shown that if ω varies periodically at some frequency p, the single resonance of a time-invariant circuit splits up into a large number of resonances separated by p cycles. These resonances extend, roughly speaking, from the minimum to the maximum value reached by ω. They are coupled resonances, in that the excitation of one resonance causes a signal to appear at all the resonant frequencies. The network properties of the circuit are developed in terms of a set of admittances which describe this coupling; it is shown that an increase of power and frequency should be achievable using a pump frequency p which is much less than either the input or the output signal frequency. Practical limitations on the performance of the circuit are discussed, and interference from the parametric effect of the first kind is shown to be relatively unimportant. © 1962.