Chapter 2

Coupled-Oscillator Arrays—Basic Analytical Description and Operating Principles

In this chaper we will show how to use the theory developed in Chapter 1 to mathematically describe a linear array of oscillators coupled to nearest neighbors. It was Karl Stephan who first showed that such arrays can be useful in providing excitation signals for a linear array of radiating elements in that if locking signals are injected into the end oscillators of the array, variation of the relative phase of the locking signals can be used to control the distribution of the phase of the signals across the array [1]. Later, Liao and York pointed out that by merely tuning the end oscillators of the array the phase distribution can be controlled without any external injection signals [28]. We will show that, while the equations and associated boundary conditions at the array ends can describe the nonlinear behavior of the array through numerical solution, if the inter-oscillator phase differences remain small, the equations may be linearized. The linearized version may be solved analytically for the dynamic behavior of the phase, and from this one may obtain the dynamic behavior of the beam radiated by the elements of this linear phased array antenna.

An important consideration in the analysis is the manner in which the oscillators are coupled. The coupling can be represented as a “coupling network” connected to the array of oscillators, and this network can be described in terms of its port ...

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