A Lead compensator has a Transfer function of the form
, where and
Pole-zero plot of Lead compensator:-
i.e, the pole is located to the left of the zero.
- A lead compensator speeds up the transient response and increases margin of stability of a system.
- It also helps to increase the system error constant through a limited range.
Realization of Lead compensator as an Electrical Network:-
The lead compensator can be realized by an electrical Network.
Assume impedance of source is zero and output load impedance to be infinite .
The transfer function is
by comparing this equation with the transfer function of lead compensator has a zero at and the pole is .
from the pole and .
therefore the transfer function has a zero at and a pole at .
the values of the three parameters , and C are determined from the two compensator parameters and .
using the EQN(II)
there is an additional degree of freedom in the choice of the values of the network components which is used to set the impedance level of the N/w.
the gain is
D.C gain at is which is less than 1.
attenuation is used to determine the steady state performance.
while using a lead N/w , it is important to increase the loop gain by an amount of .
A lead compensator is visualized as a combination of a N/w and an amplifier.
Note:-“lead” refers to the property that the compensator adds positive phase to the system over some appropriate frequency range.
Frequency-response of a lead compensator:-
, let .
the frequency response of lead compensator is
to find at which frequency the phase is maximum , differentiate w.r to and equate it to zero.
implies , which is invalid because .
, at this lead compensator has maximum phase given by
at , .
when there is a need for phase leads of more than , two cascaded lead networks are used where each N/w provides half of the required phase.
for phase leads more than , decreases sharply and if single N/w is used will be too low.
Choice of :-
In choosing parameters of compensator depends on and C . The value may be anything but for there is a constraint. It depends on inherent noise in Control systems.
from the lead N/w , it’s been observed that the high frequency noise is amplified by while low frequencies by unity.
more (or) less should not be less than 0.07.
Procedure for bode-plot of a lead compensator:-
Step 1:- Sketch the Bode-plot of the uncompensated system with the gain k. Set the value of k according to the steady-state error requirement.
Measure the gain cross over frequency and the phase margin of uncompensated system.
Step 2:- using the relation
Additional phase lead required = specified phase margin- Phase Margin of uncompensated system.
is a margin of safety required by the fact that the gain cross over frequency will increase due to compensation.
for example :- is a good assumption for -40 dB/decade.
(or) -60 dB/decade.
Step 3:- Set the maximum phase of the lead compensator Additional phase lead required and compute .
Step 4:- Find the frequency at which the uncompensated system has a gain of dB, which gives the new gain cross over frequency.
with as the gain cross over frequency the system has a phase margin of .
where as with as the gain cross over frequency the system has a phase margin of .
Step 5:- Now . find the value of and the transfer function of lead compensator .