Question

In the following circuit, the switch that had been OPEN for a sufficient time was shorted at t=0 In this case, the voltage v(t) at both ends of the capacitor and the current i(t) at the injector are obtained.

1) What is the Capacitor voltage v (0-) value and inductor current i (0-) value at t=0- just before t=0-

2) If sufficient time has passed since switching occurred, what is the value of v(00) and i(00) when t=oo?

3) In the case of t t0 in the RLC circuit above, obtain a characteristic equation for v(t) and obtain the two years. Use the results to determine which of these RLC circuits is under-, critical- or over-damping.

4) If t t0, obtain a v(t) expression.

switch to open 42 THI eu w 32n2 TSEV @ 12V tot

Answer 1

In the case of at t= to in the RLC citant above obtained a characteristic ' eaucation Hove vet) and obtain the two yeats. Use the results to determine RLC Clutcuit undet, Critical Which of these ovet damped Son: fore do By Jukning off the 12x Youtage source we obtain 42 IH je .5F 22 Apply kv2 at node: + 1 0 at Apply KVL in me 88 0 - uit I di + v=0 det from one eauation and ② 1 dav + 2x+ 2 dv df 1 dv 2 df tv=o 2 dei day det 2 + 5 dv dt +6y=0

Ves a Schlady state xesponse tu - 1.0206 a 21 S2 + 55+ 6zo 92+24 wnist wie Wn = ato 5 2 Cu xao = 5, 206 rá tu= 1 critically damped Tu > lovere damped undet damped. V = 4.6206 rovere damped if I = to obtain a 4 Vcd expression soln. from the burit ③ 92 +58 +6=0 3= -2,-3 Natural response Vn(d) = A e 2t + B e 2d Vn (A) Vsst VA) = 4 Yout from part- VOH) = 4+ A024 + B e 30 t = 0 out Vco)= 12 1 12 z 4 + A+B A+B=8

42 АН t je + 12V 22. •5 F E det IVCA = -2 À ë 20 - 3B e 30 c d = je du je Apply kel at node- ů (ot) = je (ot) + Xcot) o Jelot) + je 60+) = -6A dvcot) - 6 = - 12 V/sec - 12 V/sec - ③ det 5 VCHI = 4 + AC-24 + Be 3d 2 allad 12 2 0 dut 3 [from eam - 12 = -2A-9A A=12 B = -2 Yout Vef) = 4 + 12 e 2t - 4 é :30 et to