Models of driven and mutually coupled relativistic magnetrons with nonlinear frequency-shift and growth-saturation effects

Abstract

The driven van der Pol-Duffing equation has been used to model the behavior of a relativistic magnetron driven by an external locking signal. The authors continue the study of the driven van der Pol-Duffing equation and present initial results of the investigation of coupled van der Pol-Duffing equations as models of mutually coupled relativistic magnetrons. A method is presented for determining the amplitude and phase of a signal in the slowly-varying amplitude approximation in the case that both the signal, X(t), and its time derivative, X(t), are available. When second-order differential equations and coupled systems of such equations are used as models of driven and coupled nonlinear oscillators, both X(t) and X(t) are available. In this case, it is possible to determine the amplitude and phase without averaging over a fast time scale. Thus certain dynamical information is retained that is lost if it is necessary to average over a fast time scale. In the case of two oscillators linearly coupled with time delays in the mutual drive configuration, the slowly varying amplitude and phase approximation has been used in order to simplify the problem. In general, behavior of the oscillator amplitudes, as well as the phase difference between oscillators, must be considered. An essential step in studying this system is the determination of stationary amplitudes and phase difference. In the case of zero frequency mismatch between oscillators and optimum coupling delay phase, stationary amplitudes and coupled power are obtained analytically as functions of coupling quality factor and ratio of oscillator growth rate to natural frequency. Unless these parameters are sufficiently large, the potential increase in coherent power delivered due to coupling will not be realized.