1. Control System Lab
Object : To design 2nd order electrical network and study its transient
response for step input and following cases- (a) underdamped system
(b) overdamped system (c) critically damped system.
Theory: Consider a simple series RLC network. V1(s) is input voltage
and V2(s) is output voltage across capacitor C. Then transfer function of
network is given as
G(s) = V2(s)/V1(s)
R L
V1(s)
C V2(s)
1 ܴ 1
‫ ܩ‬ሺ‫ݏ‬ሻ ൌ ൜ ൠ /ሾ‫ ݏ‬ଶ ൅ ൜ ൠ ‫ ݏ‬൅ ൜ ൠሿ
‫ܥܮ‬ ‫ܮ‬ ‫ܥܮ‬
ோ ଵ
Characteristic equation =‫ ݏ‬ଶ ൅ ቄ ቅ ‫ ݏ‬൅ ቄ ቅ
௅ ௅஼
ଵ
Here, Wn^2=[ ]
௅஼
భ
ோ ஼ మ
Z= ‫כ‬ ቂ ቃ
ଶ ௅

2. Program:
L=1;
R1=1;
C1=1;
R2=2;
C2=1;
R3=2;
C3=4;
t=0:0.1:25;
%
n1=inv(L*C1);
d1=[1 R1*[inv(L)] inv(L*C1)];
sys1=tf(n1,d1)
[y1,x1]=step(sys1,t)
%
n2=inv(L*C2);
d2=[1 R2*[inv(L)] inv(L*C2)];
sys2=tf(n2,d2)
[y2,x2]=step(sys2,t)
%

3. n3=inv(L*C3);
d3=[1 R3*[inv(L)] inv(L*C3)];
sys3=tf(n3,d3)
[y3,x3]=step(sys3,t)
%
clf
subplot(311);plot(y1,t);title('z=0.5')
subplot(312);plot(y2,t);title('z=1')
subplot(313);plot(y3,t);title('z=2')
Output:
Transfer function 1:
1
-------------
s^2 + 2 s + 1
Transfer function2:
1
-----------
s^2 + s + 1
Transfer function3: 0.25/[s^2 + 2 s + 0.25]