## Mathematical Modelling and Simulation of Sub-Harmonics in Relay Control System

P.M. Nawghare (Botswana)

### Keywords

M1 Mathematical Modelling, Simulation, Sub-harmonics, Relay Control Systems

### Abstract

A relay control system represents the simplest possible non-linear system. Its practical applications occur in many electrical, mechanical, hydraulic and pneumatic systems due to advantages such as simplicity, lesser number of components, smaller size requirement of motor, lesser weight, space, cost and minimum rise time during transient period . However, it has one major disadvantage in the occurrence of sub-harmonic oscillations i.e. oscillations whose frequency is a fraction of the input frequency . x Outv N L(s)r(t) r In C FIG.1: Generalised Relay Control System An attempt is made to find out whether a sub harmonic will physically exist i.e. to see whether it is stable or in other words, whether the system after a disturbance will resume that periodic regime. A mathematical model i.e. a variation equation which defines a small deviation from the state of equilibrium is developed. The solution of this equation will indicate the existence of the sub harmonic. Further, the mathematical model is simulated into a system whose stability indicates the existence of the sub- harmonic for which routine methods can be used. It is shown that this simulation can be obtained directly from the system without evaluating the mathematical model equation. Fig. 2 shows the most general input-output characteristics of the relay element with dead zone ‘b’, hysterisis ‘h’ and output magnitudes +M1 & -M2 indicating non-symmetry. Evaluating the occurrence of sub- harmonic is same as finding whether the sub harmonic is stable or, in other words, whether the system after disturbance will resume that periodic regime . This necessitates the mathematical modelling of the sub harmonic under the condition of small disturbance from its state of equilibrium. v(t)