Coupled-Oscillator Based Active-Array Antennas

Chapter 5

Causality and Coupling Delay

In the analysis presented in the preceding chapters, it was tacitly assumed that the coupling was implemented using nondispersive transmission lines characterized by a phase shift of Φ generally taken to be an integral multiple of 2π (plus π in the case of series resonant oscillators). However, the theory made no provision for the transit time through the coupling line. As a result, the solutions were noncausal. That is, each oscillator in the array responded immediately upon changing the tuning of an oscillator or the phase of an injection signal no matter what the distance between the excitation and the response. This is characteristic of the diffusion equation that arises from the continuum model. Heat conduction analyzed in this manner is similarly non-causal. Following Pogorzelski [47], we propose to remedy this situation by explicitly introducing time delay in the coupling. This time delay is determined by the physical length of the line and its propagation velocity.

5.1   Coupling Delay

A nondispersive transmission line introduces a pure time delay in that the signal applied at one end of the line is duplicated at the other end after the delay time. At that point the signal is reflected if the termination is not matched to the line impedance. For our analysis we will assume a matched termination. Now, if the analysis is done via Laplace transformation of the applied signal, the transform of the delayed signal is merely the original ...