Page 342 - Mechanical Engineers' Handbook (Volume 2)

5 Approaches to Linear Systems Analysis 333

Table 6 Tabulated Values of the Response of a First-Order
System to a Unit Step Input
t y(t) ˙ y(t)
0 0 1
0.632 0.368 1
2 0.865 0.135 1
3 0.950 0.050 1
4 0.982 0.018 1
5 0.993 0.007 1
6 0.998 0.002 1

dy y y(t)
(t) ss
dt
Finally, note that after an interval of four time constants, the response is within 98% of the
steady-state value, that is,
y(4 ) 0.98168(y y )
0
ss
For this reason, T 4 is called the (2%) setting time.
s

Figure 16 Response of a first-order system to a unit step input, showing the relationship to the time
constant.