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 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
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
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.