Variational Techniques Applied to Capture in Phase-Controlled Oscillators

The synchronization behavior of phase-controlled oscillators has been for many years the subject of extensive analytic studies. Although yielding significant and experimentally consistent results, these studies have been based primarily on laborious graphical procedures and linearizing approximations. In this paper we discuss and apply a variational formulation with which to analyze the synchronization phenomena of phasecontrolled systems more directly and somewhat more generally. The phase-controlled oscillator is represented by the block diagram of Fig. 1(a). The basic function of such a system is to establish and maintain, within certain limits, an approximately zero difference in frequency between the input and local carriers; i.e., the local oscillator is constrained so as to follow variations in the frequency of the input carrier. As indicated, this control is effected by applying the two carriers to a phase comparator and linear filter in cascade, and utilizing the resultant output to appropriately control the frequency of the local oscillator. The comparator output in most cases is a simple function of the phase difference - In general, A) 227