The capacitor in this circuit is initially charged to 8 V before the switch is closed at time t = 0. Suppose VB = 30 V, R = 220 惟, and L = 90 mH, C = 50 碌F.
(a) Determine the following initial condition for the capacitor's voltage drop vC:
dvCdt
(0+) = V/s
(b) Determine the roots of the characteristic equation for vC.
(If the roots are real, state the smaller-magnitude one first; if they are complex, state the one with positive imaginary part first.)
s1 = + j 路 s-1
s2 = + j 路 s-1
(c) Determine the voltage drop vC(t) as a function of time t (in seconds), for t > 0. Identify the numerical values of the parameters in whichever of the following forms is appropriate:
vC(t) = k1 路 e鈥揳t + k2 路 e鈥揵t + k3 (with |a| < |b|), or
vC(t) = e鈥揳t 路 [k1cos(bt) + k2sin(bt)] + k3.
a = s-1
b = s-1
k1 = V
k2 = V
k3 = V