A Phase Plane Approach to On-off Controlled Servomechanisms to Sinusoidal Input

A phase plane method for analizing an on-off controlled servomechanism to any sinusoidal input is proposed. In this method, the behaviour of a representative point is broken down into two separate movements, one of them is its rotation around the circumference of an elipse, whose radii are a and aω, where a, ω are the half amplitude and the angular frequency of the input respectively and the other is the travelling of this ellipse along a curve decided by the transfer function of the controlled element and the manipulated variable. The necessary conditions for generating the subharmonic oscillation in the output are stated as followed ; 1) The trajectory of the center of an ellipse is a closed curve. 2) The time for one cycle of the center of the ellipse is equal to an integral multiple of the period of the input. 3) The trajectory of the center of an ellipse converges into this closed curve as time goes on. 4) The representative point is always on one side of the switching line during a half period of the output. Practical examples of the first three conditions are calculated, where the on-off controlled servomechanism of the first or second order having symmetrical switching line is treated and some numerical results are shown through graphical figures. In these figures, the conditions that the output follows the input are indicated. These conditions are very valuable for designing on-off controlled servomechanisms so as to follow to given sinusoidal input.