Direct Current and Transient Analysis 201
For R = 75 Ω, the resonant frequency isw 0
1
2
LCrad/sThe neper frequency is
R
2 L75
184 1667. HzThe frequencies s 1 and s 2 aresw 12 , ^202then0 ^2 w 02and 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 isw 0
1
2
LCrad/sThe neper frequency is
R
2 L36
182Hzand s 1 and s 2 are repeated, given bysw 12 , ^202 then0 ^2 w 02and 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 isw 0
1
2
LCrad/sThe neper frequency is
R
2 L3
181
60 1667. Hzand 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.