Hence,
(7–26)
As seen from Figure 7–92, the lead compensator is basically a high-pass filter. (The
high frequencies are passed, but low frequencies are attenuated.)
Lead Compensation Techniques Based on the Frequency-Response Approach.
The primary function of the lead compensator is to reshape the frequency-response
curve to provide sufficient phase-lead angle to offset the excessive phase lag associated
with the components of the fixed system.
Consider the system shown in Figure 7–93. Assume that the performance specifica-
tions are given in terms of phase margin, gain margin, static velocity error constants,
and so on. The procedure for designing a lead compensator by the frequency-response
approach may be stated as follows:
1.Assume the following lead compensator:
Define
Then
The open-loop transfer function of the compensated system is
where
Determine gain Kto satisfy the requirement on the given static error constant.
2.Using the gain Kthus determined, draw a Bode diagram of G 1 (jv),the gain-
adjusted but uncompensated system. Evaluate the phase margin.
3.Determine the necessary phase-lead angle to be added to the system. Add an
additional 5° to 12° to the phase-lead angle required, because the addition of the
G 1 (s)=KG(s)
Gc(s)G(s)=K
Ts+ 1
aTs+ 1
G(s)=
Ts+ 1
aTs+ 1
KG(s)=
Ts+ 1
aTs+ 1
G 1 (s)
Gc(s)=K
Ts+ 1
aTs+ 1
Kc a=K
Gc(s)=Kc a
Ts+ 1
aTs+ 1
=Kc
s+
1
T
s+
1
aT
(0 6 a 6 1)
vm=
1
1 aT
Section 7–11 / Lead Compensation 495+– Gc(s) G(s)Figure 7–93
Control system.