Time Constants of First Order Systems

The time constant of first order systems is often easy to find.  The time constants of some typical first order systems.

If the input force of the following system is a unit step, find v(t).  Also shown is a free body diagram.

The time constant can be defined as the time it takes for the step response to rise up to 63% or 0.63 of its final value. We refer to this as t = 1/a. If we take reciprocal of time constant, its unit is 1/seconds or frequency.

We call the parameter “a” the exponential frequency. Because the derivative of e-at is -a at t = 0. So the time constant is considered as a transient response specification for a first-order control system.

We can control the speed of response by setting the poles. Because the farther the pole from the imaginary axis, the faster the transient response is. So, we can set poles farther from the imaginary axis to speed up the whole process.

Rise Time of a First Order Control System

The rise time is defined as the time for the waveform to go from 0.1 to 0.9 or 10% to 90% of its final value. For the equation of rising time, we put 0.1 and 0.9 in the general first-order system equation respectively.

For t = 0.1

For t = 0.9

Taking the difference between 0.9 and 0.1

Here the equation of rising time. If we know the parameter of a, we can easily find the rise time of any given system by putting “a” in the equation.