Chapter 8, Solution 20.
For t < 0, the equivalent circuit is as shown below.
2
i
30
+
+
C v
v(0) = –30 V and i(0) = 30/2 = 15 A
For t > 0, we have a series RLC circuit.
= R/(2L) = 2/(2x0.5) = 2
o = 1/ 2241x5.0/1LC
Since is less than o, we have an under-damped response.
248
22
od
i(t) = (Acos(2t) + Bsin(2t))e-2t
i(0) = 15 = A
di/dt = –2(15cos(2t) + Bsin(2t))e-2t + (–2x15sin(2t) + 2Bcos(2t))e-t
di(0)/dt = –30 + 2B = –(1/L)[Ri(0) + vC(0)] = –2[30 – 30] = 0
Thus, B = 15 and i(t) = (15cos(2t) + 15sin(2t))e-2t A