Direct Current and Transient Analysis 201
For R = 75 Ω, the resonant frequency is
w 0
1
2
LC
rad/s
The neper frequency is
R
2 L
75
18
4 1667. Hz
The frequencies s 1 and s 2 are
sw 12 , ^202
then
0 ^2 w 02
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
LC
rad/s
The neper frequency is
R
2 L
36
18
2Hz
and s 1 and s 2 are repeated, given by
sw 12 , ^202
then
0 ^2 w 02
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
LC
rad/s
The neper frequency is
R
2 L
3
18
1
6
0 1667. Hz
and the complex frequencies are s 1 and s 2 given by s1,2 = −α ± j (^) √
w 02 − α^2 , clearly the
underdamped case.
Then the solution i(t) is of the form
it e A wt A wt
()t [ cos( 12 dd) sin( )]
where wd = (^) √
w (^) o^2 − α^2.