A Lead compensator has a Transfer function of the form

,    where      and

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

after simplification

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.

,   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

implies .

.

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.

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  .     (No Ratings Yet) Loading... ## Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.

Posted on Categories Control Systems