System Response

This page demonstrates how systems respond: a motor on its own or in speed control (first order) or a mass spring system or motor position control (second order). The response is to a step, where it moves from zero to a final steady value, or to an impulse, where it starts at 0, is given a kick, and returns to 0. NB the mass-spring page shows an impulse response.

The different models are depicted and the user can click on model parameters to change them, and the user can select step or impulse. The associated differential equation is shown, and the response is given both algebraically and graphcally.

The responses are functions of exponentials, whose constants are associated with the roots of the 'auxilliary equation' which itself comes from the system differential equation.

For first order systems, there is one root of the auxilliary equation, r = -a, and the transient response has exp(-at).

For second order systems, it has two roots. If real, -a and -b, the transient response includes exp(-at) and exp(-bt). If complex, -a ± ib, the transient response is an exponentially damped sinusoid with exp(-at), cos(bt) and sin(bt).

Click on the relevant blocks in the diagram to change their parameters and see the effect.

If you select 'Show How Blocks Formed', the block diagram is developed in stages : click Next/Done button when it appears.