Oscillations in Systems with Non-Linear Reactance 



By R. V. L. HARTLEY 



A theoretical study is presented of the properties of a condenser, one 

 plate of which is free to vibrate, when it is included in a circuit containing a 

 generator, the frequency of which is higher than the resonant frequency of 

 the plate and unrelated thereto. It is shown that the plate may be main- 

 tained in oscillation at a frequency at or near its mechanical resonance, at 

 the expense of the energy supplied by the generator, provided certain 

 conditions are satisfied. The most favorable condition is one in which the 

 plate is resonant at the frequency of its vibration and the electric circuit is 

 resonant at that of the generator, and at the difference between the generator 

 and plate frequencies, and is anti-resonant at their sum. Under these 

 conditions the generator voltage must exceed a threshold value determined 

 by the impedances and frequencies. This threshold voltage increases as the 

 conditions become less favorable. Expressions are given for the values of 

 the oscillations as functions of the voltage when the threshold is exceeded. 

 When the sum frequency is absent, the energies dissipated at the plate and 

 difference frequencies are in the ratio of the two frequencies. 



The oscillations described represent a special case of a class of similar 

 oscillations, all of which depend on the presence of a non-linear reactance. 

 Another special case is a molecular model capable of reproducing the main 

 features of the Raman effect. 



Introduction 



A TYPE of free oscillation has been found to occur in non-linear 

 coupled systems, which differs from the ordinary type in that 

 the supporting energy is drawn from an alternating sustained source, 

 rather than from a constant source, as in the ordinary vacuum tube 

 oscillator. The particular example of such oscillations to be described 

 here occurs in an electric circuit containing a condenser, one plate of 

 which is elastically supported so as to constitute a mechanically 

 resonant system. 



The possibility of such oscillations in a circuit of this kind was 

 discovered ^ in the course of a theoretical study of the possible use of 

 a moving plate condenser as a modulator in a carrier system. Such 

 use was suggested by the fact that, in a condenser, the mechanical 

 force on the plate is proportional to the square of the charge. In 

 this study, it was assumed that a generator of alternating electromotive 

 force of a relatively high carrier frequency was connected with the 

 condenser terminals, and an alternating mechanical force of a relatively 

 low frequency (corresponding to a Fourier component of a speech 

 wave) was applied to the movable plate of the condenser. The plate 

 was not assumed to be resonant. The non-linear relations between 



' Hartley; Phys. Rev., Vol. ii, p. 289, February, 1929. 



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