Stability criteria for an electrical or mechanical system with distributed parameters

Abstract

The equation C0 + C1z + (C2 + C3z) tanh z = 0 occurs in the theory of the stability of electrical or mechanical systems with distributed parameters or with an element producing a finite time delay. For the system to be stable all the roots of the equation must be negative or have negative real parts. Assuming C0 to be positive it is shown that C1 and C3 must also be positive and that C2/C1 must be greater than a certain critical value which is a function of C0/C3. These criteria are applied to determine the stability of a simple servomechanism in which the correction signal is delayed by a constant period before being returned to the control point.