Page 476 - Mechanical Engineers' Handbook (Volume 2)

4 z-Transforms 467

where
1 e Ts
G (s)
h0
s
F*(s) ƒ(kT)e kTs (52)

k 0
In Eq. (51), G (s) may be considered the transfer function between the output H(s) and the
h0
input F*(s) (see Fig. 18). Thus the transfer function of the zero-order hold device is
1 e Ts
G (s) (53)
h0
s
First-Order Hold
The transfer function of a first-order hold is given by
1 e Ts Ts 1
G (s) (54)
h1
s T
From Eq. (47) for n 1,
h(kT ) a ƒ(kT) (55)
1
where 0 T and k 0, 1, 2,....By using the condition h[(k 1)T] ƒ[(k 1)T]
the constant a can be determined as follows:
1
h[(k 1)T] aT ƒ(kT) ƒ[(k 1)T]
1
or
ƒ(kT)(k 1)T]
ƒ
a
1
T
Hence Eq. (55) becomes
ƒ(kT) ƒ[(k 1)T]
h(kT ) ƒ(kT) (56)
T
where 0 T. The extrapolation process of the first-order hold is based on Eq. (56) and
is a piecewise-linear signal as shown in Fig. 19.

Figure 18 Sampler and zero-order hold.