TRANSACTIONS IEEE ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 4, APRIL 1999
471
Continuum Modeling of the Dynamics of Externally Injection-Locked Coupled Oscillator Arrays
Ronald J. Pogorzelski, Fellow, IEEE, Paolo F. Maccarini, and Robert A. York, Member, IEEE
AbstractMutually injection-locked arrays of electronic oscillators provide a novel means of controlling the aperture phase of a phased-array antenna, thus achieving the advantages of spatial power combining while retaining the ability to steer the radiated beam. In a number of design concepts, one or more of the oscillators are injection locked to a signal from an external master oscillator. The behavior of such a system has been analyzed by numerical solution of a system of nonlinear differential equations which, due to its complexity, yields limited insight into the relationship between the injection signals and the aperture phase. In this paper, we develop a continuum model, which results in a single partial differential equation for the aperture phase as a function of time. Solution of the equation is effected by means of the Laplace transformation and yields detailed information concerning the dynamics of the array under the inuence of the external injection signals. Index Terms Beam steering, coupled oscillators, injection locked, phased array.
Fig. 1. A coupled oscillator array with the pth oscillator locked to an externally derived signal.
I. INTRODUCTION MUTUALLY coupled array of electronic oscillators can be made to oscillate in a mutually synchronized mode, in which the relative phases of the oscillators form a welldened distribution useful in terms of exciting an array of radiating elements to achieve spatial power combining, beam formation, and steering of the radiated beam. Two methods of achieving this have been suggested. Having demonstrated that the ensemble of oscillators would oscillate at the average of the free-running (tuning) frequencies of the oscillators, Liao and York [1] showed that antisymmetrical detuning the oscillators at the ends of a linear array (or on the perimeter of a two-dimensional array) results in a linear phase progression across the array, which could be exploited to steer the beam without the use of phase shifters. Alternatively, Stephan [2] proposed and demonstrated externally injection locking the end oscillators with signals phase shifted with respect to each other to achieve a similar linear phase progression using only a single phase shifter. Such systems have, in the past, been analyzed by numerical solution of a system of nonlinear differential equations describing the oscillator coupling [2][4]. Pogorzelski et al. [5] have analyzed the
Manuscript received June 23, 1998; revised December 4, 1998. This work was sponsored by the Ball...
105 - MTT (47) April 1999 Continuum Model Ext