Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 253

Now back to the main question. Let R = R 1 + R 2 . We have v i ð t Þ¼ Vu ð t Þ!
V i ð s Þ¼ 1 = s. Using L rules with zero IC's we get:

V C ¼ I =ð Cs Þ; V L ¼ LsI :
v i ð t Þ¼ v R þ v C þ v L ! 1 = s ¼ IR þ I =ð Cs Þþ LsI

ð 1 = L Þ
s 2 þð R = L Þ s þ 1 =ð LC Þ

! I ¼

If D \ 0, we get: I ¼ð 1 = x o L Þ½ x o = F ð s Þ! i ð t Þ¼ð 1 = x o L Þ e at sin ð x o t Þ u ð t Þ:
If D [ 0, we get I = (1/L)[a 1 /(s + b 1 )+a 2 /(s + b 2 )] where a 1 = 1/b, a 2 =-1/b,
hence

i ð t Þ¼ð 1 = bL Þ½ e b 1 t e b 2 t u ð t Þ:

Tutorial 19

Q1: Find the transfer function of the feedback system shown below. All signals
are voltages.

X ( s )
x ( t )

D ( s ) Y ( s )

y ( t )

G ( s )

Solution: First, we define important auxiliary points [i.e., e(t) and r(t)] as shown

below.

E ( s )

X ( s )
x ( t )

D ( s ) Y ( s )

y ( t )

e ( t )

R ( s )

G ( s )

r ( t )

Now we write the system equations as follows:

e ð t Þ¼ x ð t Þþ r ð t Þ ð 1 Þ

This equation is in the time-domain. It is easier to transform to the s-domain using
L , since the convolution in the t-domain would be transformed into a simple
multiplication in the s-domain. Taking the L of both sides of (1) we get:

E ð s Þ¼ X ð s Þþ R ð s Þ ð 2 Þ