Lag compensator:-

A Lag compensator has a Transfer function of the form

, where and

Pole-Zero Plot of Lag compensator:-

i.e, the pole is located to the right of the zero.

Realization of Lag compensator as Electrical Network:-

The lag 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

after comparing the above equation with the transfer function of lag compensator has a zero at and has a pole at .

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 greater than 1.

Let the zero-frequency gain as unity, then the Transfer function is .

Frequency-response of Lag compensator:-

Note:-“lag” refers to the property that the compensator adds positive phase to the system over some appropriate frequency range.

, let .

the frequency response of lag compensator is

at .

has a slope +20 dB/decade with corner frequency .

slope is -20 dB/decade with corner frequency .

to find at which frequency the phase is minimum , differentiate w.r to and equate it to zero.

implies , which is invalid because .

.

, at this lead compensator has minimum phase given by

implies .

.

at , .

Choice of :-

Any phase lag at the gain cross over frequency of the compensated system is undesirable.

To prevent the effects of lag compensator , the corner frequency of the lag compensator must be located substantially lower than the of compensated system.

In the high frequency range , the lag compensator has an attenuation of dB, which is used to obtain required phase margin.

The addition of a lag compensator results in an improvement in the ratio of control signal to noise in the loop.

high frequency noise signals are attenuated by a factor , while low-frequency control signals under go unit amplification (0 dB gain).

atypical value of .

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:- find at which phase angle of uncompensated system is

+ given Phase Margin+ .

is a good assumption for phase-lag contribution.

Step 3:- find gain of the uncompensated system at and equate it to 20 log () and then find .

Step 4:- choose the upper corner frequency of the compensator to one octave to one decade below and find value.

Step 5:- Calculate phase lag of compensator at , if it is less than go to next step.

Step 6:- Draw the Bode plot of compensated system to meet the desired specifications.