LEAD, LAG & LAG-LEAD COMPENSATOR
THEORY:
Compensator are some kind of currective subsystems which are
introduced to force choosen plant to meet the given specifications. Their
performance of the
main purpose is to compansate for the deficiency in the
plant. There are three types of compansators.
1. Lead compensator
2. Lag compensator
3. Lag-Lead compensator
LEAD COMPENSATOR:
Lead compensator speeds up the transient response and increase the
margin of the stability of system. It also helps to increase the system error
constant through to a limited extent. A high pass filter is often reffered as a
phase lead controller. It is so called since positive phase is introduced to the
system over some frequency range. These ideas related to filtering and
phase shift are useful in design of system carried out in frequency domine.
The transfer function of such a compensator is given by
G(s) =(s +z) =(s+ 1/r) ......(1)
(st pe) (s + l/ar)
Where =z/p. <1, r>0
The s-plane representation of the lead compensator is as shown fig.1.
Which has a zero at s = -1/ rand a pole at s = -1/ar with a zero closer to the
origin than the pole.
j
Fig.1. The s-plane representation of the lead compensator
R,
Fig.2. Electric lead network
Assuming that the circuit is not loaded, KCL for the output node
yields,
Cd (e;- e)+ 1 (e;-e) =1e ......2)
dt R R2
Laplace Transforming this equation (with zero initial conditions)
gives
Cs[ E(s) -E(s)]+1 (E E 1 E,(s)
R, R2
The transfer function is
Es) = R =
s+ 1/RC) .....3)
E{s) R+ R1/sC
RË + 1/sC [Rz/RË+R] R,C
If we define
= RC
and = R/R, +R, < 1,
we recognize that the transfer function (3) has the same form as that of lead
compensator in equation (1)
The sinusoidal transfer function of the lead compensator is given by
G. (jo) =!+ jor; a <1, ....4)
+ jaon
Since we have a <1, the network output leads the sinusoidal input
under steady state and so the name lead compensator.
slope 20 dbydecade
db 20 log 1/a
Gj) | |10 log l/a
90
45°
LG,(ju)
-INar Wat
Fig.3. Bode plot of phase lead network
From equation (4), it is evident that the compensator provides a phase
lead betvween the output and input, given at any frequency, 0,given by
(o)= tan (or)- tan (aor) ...5)
LAG COMPENSATOR:
Lag compensator improves the steady state behaviour of a system,
while nearly preserving a transient response. A low pass filter is often
reffered as a phase lag. controller. It is so called since negative phase is
introduced to the system over some frequency range. The phase lag control
bring in more attenuation to a system, the stability margen will be improved
but suffer from lesser bandwidth.
The transfer function of such a compensator is given by
G(s) = (s+ z) = (s+ 1/r) ......6)
(st p) (s+ 1/ßr)
Where B =z/p. > 1, r> 0
The s-plane representation of the lag compensator is as shown figure.
Which has a zero at s = -1/r and a pole at s = -1/Br with the zero located to
the left of the pole on the negative real axis.
Fig.4. The s-plane representation of the lag compensator.
w
RË
Fig. Electric lag network.
Applying KCL for the loop yields equation
iR,+ 1J'idt + iRz=e; ......()
C
Laplace Transforming this equation (with zero initial conditions)
gives
(R, + R2t1)I(s) =E (s) ......(8)
Cs
The output ýoltage is given by
Eo(s) = (R2+1 ) I(s)
Cs
The transfer function of the lag network is therefore given by
E(s) = R, +1/sC
E(s) R, + R + i/sC
s+(1/RC) .....9)
R,tRll st
R2 (R +R2) RC
R2
From equations (6) and (9)
f = RC
and B= (R + R) /R2 >1,
The sinusoidal transfer function fo the lag compensator is given by
G. (jo) =1+jor ; ß>1, ......(10)
1+jß or
Since we have B >1,the network output lags the sinusoidal input
under steady state and so the name lag comnpensator.
dope -20 oldecade 20 log B
|G,jm)
LG,jao)
-90
log co
Fig.6. Bode plot of phase lag network
From equation (10), it is evident that the compensator provides a
phase lag between the output and input, given at any frequency, o given by
(o)= tan (or) - tan (Bor) .......(11)
