Section 6–7 / Lag Compensation 321In examining these response curves notice that the compensated system designed by Method 1
exhibits a little bit larger overshoot in the step response than the compensated system designed
by Method 2. However, the former has better response characteristics for the ramp input than the
latter. So it is difficult to say which one is better. The decision on which one to choose should be
made by the response requirements (such as smaller overshoots for step type inputs or smaller
steady-state errors in following ramp or changing inputs) expected in the designed system. If both
smaller overshoots in step inputs and smaller steady-state errors in following changing inputs are
required, then we might use a lag–lead compensator. (See Section 6–8 for the lag–lead compen-
sation techniques.)6–7 Lag Compensation
Electronic Lag Compensator Using Operational Amplifiers. The configuration of
the electronic lag compensator using operational amplifiers is the same as that for the
lead compensator shown in Figure 6–36. If we choose in the circuit shown
in Figure 6–36, it becomes a lag compensator. Referring to Figure 6–36, the transfer
function of the lag compensator is given by
where
Note that we use binstead of ain the above expressions. [In the lead compensator we
usedato indicate the ratio which was less than 1, or 0<a<1.] In this
book we always assume that 0<a<1andb>1.
Lag Compensation Techniques Based on the Root-Locus Approach. Consider
the problem of finding a suitable compensation network for the case where the system
exhibits satisfactory transient-response characteristics but unsatisfactory steady-state
characteristics. Compensation in this case essentially consists of increasing the open-
loop gain without appreciably changing the transient-response characteristics. This means
that the root locus in the neighborhood of the dominant closed-loop poles should not
be changed appreciably, but the open-loop gain should be increased as much as needed.
This can be accomplished if a lag compensator is put in cascade with the given
feedforward transfer function.
To avoid an appreciable change in the root loci, the angle contribution of the lag
network should be limited to a small amount, say less than 5°. To assure this, we place
the pole and zero of the lag network relatively close together and near the origin of the
splane. Then the closed-loop poles of the compensated system will be shifted only slight-
ly from their original locations. Hence, the transient-response characteristics will be
changed only slightly.
R 2 C 2 AR 1 C 1 B,
T=R 1 C 1 , bT=R 2 C 2 , b=
R 2 C 2
R 1 C 1
7 1, Kˆc=
R 4 C 1
R 3 C 2
Eo(s)
Ei(s)
=Kˆc b
Ts+ 1
bTs+ 1
=Kˆc