BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 201 
If the input is a unit step function, R(s) is 
1 
R(s) = = 
(s) = = 
For these assumptions, the continuous output C(s) is 
+ 0 
if 1 1 
06) = es ta) ee 
1 
n= 

where it is assumed that the sampling interval T is unity. 
The ripple component is given by the summation, and it will be 
approximated by considering only the first terms in which n = 1 and 

n = —1. The inverse of the ripple component so described is 
() = 24 : 4 : 
‘ (s + a)(s + juo) * (8 + a)(s — Jeo) 
which reduces to 
Dt) Ea 2 (cos wot 1 = sin Gui = C ~) 
The smooth component of the output is 
1 
— e—1 
c(t) £ s(s + a) 
which reduces to 
il — pat 
c(t) = = (1 — e*) 
The ripple component can be compared to the smooth output at each 
sampling interval and can be obtained as a percentage by dividing the 
ripple component p(t) by the smooth component c,(¢). 
A useful result can be obtained by consideration of the steady-state 
condition which expresses the ripple as a percentage of the steady-state 
smooth output. The former is obtained by taking only the sinusoidal 
terms in p(t), which can be combined to give 
Des(t) = = COs (wot + ¢) 
TET 
The steady-state smooth output obtained from c,(¢) is 
1 
Cab) — a 
The per cent ripple is obtained by dividing the steady-state ripple by 
the steady-state smooth output; hence, 
F 2a 
Tl le = ——_—. X 100 
ee a/a? + wo? 
