PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Direct Current and Transient Analysis 197

The neper frequency is





1

2

6

RC

Hz

The complex frequencies are s 1 and s 2 , given by

sw 12 , 
^202

then 0 > (^) √

α^2 − w 02 and is clearly the overdamped case.
Then the solution vC(t) is of the form vC(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
LC
rad/s
The neper frequency is



1
2
2
RC
Hz
Then s 1 and s 2 are negative and repeated frequencies, which are given by
sw 12 ,
^202

Since 0 = (^) √

α^2 − w 02 , the case is critical damped.
Then the solution vC(t) is of the form vC(t) = A 1 e−αt + A 2 t e−αt
For R = 72 Ω, the resonant frequency is
w 0
1


2
LC
rad s/
The neper frequency is



1
2
1
RC 4
Hz
The complex frequencies s 1 and s 2 are given by s1,2 = −α ± j (^) √

w 02 − α^2 (complex conju-
gate), and is clearly the underdamped case. Then the solution vC (t) is of the form
vt e A wt A wt
()
t[ cos( 12 dd) sin( )]
where wd = (^) √

w 02 − α^2.
Let us use MATLAB to obtain and plot the solutions just presented, and compare them
with the analytical results.