SKEDSOFT

Unit-step response of first-order systems

Fig: 1 (a) Block diagram of a first-order system; (b) simplified block diagram.

Consider the first-order system shown in Figure 1(a). Physically, this system may represent an RC circuit, thermal system, or the like. A simplified block diagram is shown in Figure 1 (b). The input-output relationship is given by

Since the Laplace transform of the unit-step function is 1/s, substituting R(s) = 1/s into the above equation, we obtain

Note that the smaller the time constant T, the faster the system response. Another important characteristic of the exponential response curve is that the slope of the tangent line at t = 0 is 1/T, since

The output would reach the final value at t = T if it maintained its initial speed of response. From Equation above we see that the slope of the response curve e(t) decreases monotonically from 1/T at t = 0 to zero at t = ∞.