Direct Current and Transient Analysis
197
The neper frequency is
ϭ
ϭ
1
2
6
RC
Hz
The complex frequencies are s 1 and s 2 , given by
s
w
1 2
2
0
2
, ϭ
Ϯ
Ϫ
Ϫ
then 0 > √
_______
α
2
− w 0
2 and is clearly the overdamped case.
Then the solution v C (t) is of the form v C (t) = A 1 e
−s 1 t
+ A 2 e
−s 2 t
For R = 9 Ω, the resonant frequency is
w 0
1
2
ϭ
ϭ
LC
rad/s
The neper frequency is
ϭ
ϭ
1
2
2
RC
Hz
Then s 1 and s 2 are negative and repeated frequencies, which are given by
s
w
1 2
2
0
2
, ϭ
Ϯ
ϭ
Ϫ
Ϫ
Ϫ
Since 0 = √
_______
α
2
− w 0
2 , the case is critical damped.
Then the solution v C (t) is of the form v C (t) = A 1 e
−αt
+ A 2 t e
−αt
For R = 72 Ω, the resonant frequency is
w 0
1
2
ϭ
ϭ
LC
rad s
/
The neper frequency is
ϭ
ϭ
1
2
1
4
RC
Hz
The complex frequencies s 1 and s 2 are given by s 1,2 = −α ± j √
_______
w 0
2
− α
2 (complex conjugate), and is clearly the underdamped case. Then the solution v C (t) is of the form
v t
e A
w t
A
w t
t
d
d
( )
[ cos( )
sin( )]
ϭ
ϩ
Ϫ
1
2
where w d = √
_______
w 0
2
− α
2 .
Let us use MATLAB to obtain and plot the solutions just presented, and compare them
with the analytical results.
CRC_47760_CH002.indd 197
CRC_47760_CH002.indd 197
7/23/2008 1:38:59 PM
7/23/2008 1:38:59 PM