Direct Current and Transient Analysis
201
For R = 75 Ω, the resonant frequency is
w 0
1
2
ϭ
ϭ
LC
rad/s
The neper frequency is
ϭ
ϭ
ϭ
R
L
2
75
18
4 1667
.
Hz
The frequencies s 1 and s 2 are
s
w
1 2
2
0
2
, ϭ
Ϯ
Ϫ
Ϫ
then
0
2
0
2
Ͼ Ϫ w
and the solution i(t) is the overdamped case given by i(t) = A 1 e
−s 1 t
+ A 2 e
−s 2 t
.
For R = 36 Ω, the resonant frequency is
w 0
1
2
ϭ
ϭ
LC
rad/s
The neper frequency is
ϭ
ϭ
ϭ
R
L
2
36
18
2 Hz
and s 1 and s 2 are repeated, given by
s
w
1 2
2
0
2
, ϭ
Ϯ
ϭ
Ϫ
Ϫ
Ϫ
then
0
2
0
2
ϭ Ϫ w
and the solution i(t) is the critical-damped case given by i(t) = A 1 e
−αt + A 2 te
−αt
.
For R = 3 Ω, the resonant frequency is
w 0
1
2
ϭ
ϭ
LC
rad/s
The neper frequency is
ϭ
ϭ
ϭ ϭ
R
L
2
3
18
1
6
0 1667
.
Hz
and the complex frequencies are s 1 and s 2 given by s 1,2 = −α ± j √
_______
w 0
2
− α
2 , clearly the
underdamped case.
Then the solution i(t) is of the form
i t
e
A
w t
A
w t
t
d
d
( )
[ cos( )
sin( )]
ϭ
ϩ
Ϫ
1
2
where w d = √
_______
w o
2
− α
2 .
CRC_47760_CH002.indd 201
CRC_47760_CH002.indd 201
7/23/2008 1:39:00 PM
7/23/2008 1:39:00 PM