Page 668 - Mechanical Engineers' Handbook (Volume 2)

6 PID Controllers 659

Another simplified version of the PID controller is obtained by setting K 0. The
i
transfer function of this so-called proportional-differential (PD) controller can also be derived
from Eq. (33):
G Ks K
d

pd
s 1 p
d
K (K /K )s 1
d
d
p
p
s 1
d
K s 1
pd
p
s 1 (37)
d
This transfer function is the same as the first-order lead compensator of Eq. (16) and is
normally used in systems having an inherent integration. Note that the lead break frequency
can be a factor of 10 lower than the lag break frequency. In this case the term in the
d
numerator of Eq. (37) is small:
K d
K
d

d
pd
K p for K p (38)
Frequency responses of representative PID, PI, and PD controllers are given in Fig. 36,
which also shows the effects of the various controller parameters.
6.2 Systems Having No Inherent Integration
Electronic integrators are normally used in the forward loops of systems having no inherent
integrations, as explained in Section 2.1. Section 3 explains the various ways in which lead

Figure 36 Frequency response comparisons of PID, PI, and PD controllers.