Fourier and Laplace
421
Then from Table 4.2
v t
e
t u t
t
( )
sin(
) ( )
ϭ
120
7
7
⋅
⋅
Ϫ
and therefore,
i t
v t
R
e
t ut
i t
C
dv t
dt
d
dt
e
R
t
C
t
( )
( )
sin(
) ( )
( )
( )
ϭ
ϭ
ϭ
ϭ
6
7
7
1
40
120
7
Ϫ
Ϫ
⋅
⋅
s sin(
) ( )
( )
cos(
)
(
)
7
3
7
7
7
7
3
7
t u t
i t
e
t
t e
e
C
t
t
t










ϭ
ϭ
Ϫ
Ϫ
Ϫ
Ϫ sin
7 7
7
7
3
3
6
7
cos
( )
(
) sin(
)
( )
( )
( )
( )
t
t
i t
i t
i t
i t
e
L
R
C
L
t
Ϫ
Ϫ
Ϫ




−
u t
ϭ
ϭ Ϫ
7cos( 7 ) sin( 7 )
sin(
) ( )
( )
7
3
7
t u t
e
u t
t
Ϫ
Ϫ
− 



t
t
MATLAB Solution
% Script file: RLC _ parallel
t =-2:0.1:5;
t0 =0;
u = stepfun(t,t0);
vt =120/sqrt(7).*exp(-t).*sin(sqrt(7).*t).*u ;
figure(1)
subplot(2,2,1);
plot(t,vt);title(‘ v(t) vs.t (analytical)’);
ylabel(‘Amplitude (volts)’)
ir = 6/sqrt(7)*exp(-t).*sin(sqrt(7).*t).*u;
subplot(2,2,2);
plot(t,ir);title(‘ ir(t) vs.t (analytical)’);
ylabel(‘ Amplitude (amps)’)
ic=3/sqrt(7).*exp(-t).*(sqrt(7).*cos(sqrt(7).*t)- sin(sqrt(7).*t)).*u;
subplot(2,2,3);
plot(t,ic);title(‘ ic(t) vs.t (analytical)’);
ylabel(‘Amplitude (amps)’);xlabel(‘time(sec)’);
subplot(2,2,4);
il = 3*ones(1,71)-ir-ic ;
plot(t,il);title(‘il(t) vs.t (analytical)’);
ylabel(‘Amplitude (amps)’) ;xlabel(‘time (sec)’);
figure(2)
subplot(2,2,1)
num = [0 0 120];
den = [1 2 8];
vti = impulse(num,den,t);vti=[zeros(1,20) vti’];
plot(t,vti);title(‘ v(t) vs. t (sym. Laplace)’);
CRC_47760_CH004.indd 421
CRC_47760_CH004.indd 421
7/28/2008 12:26:17 PM
7/28/2008 12:26:17 PM