The switch in the circuit shown below has been opened for a long time before it is closed at t = 0. Find the current i(t) through the inductor for t ? 0. Assume R1 = 1 k?, R2 = 2 k?, R3 = 2 k?, R4 = 2 k?, Is = 5 mA, Vs = 6.4 V, L = 40 mH.
i(0) = I0 = Is × R1/(R1 + R2 + R3) = 1 mA
From Is,
Rb= R2 +(R3 || R4) = 3 k?
Iinf1 = [Is×R1/(R1 + Rb)]×R3/(R3 + R4) = 0.625 mA
From Vs,
Ra = R3 || (R1 + R2) = 1.2 k?
Iinf2 = [Vs × Ra/(Ra + R4)]/R3 = 1.2 mA
Iinf = Iinf1 + Iinf2 = 1.825 mA
Req = R3 +[ R4 || (R2 + R1)] = 3.2 k?
? = L/Req = 12.5 ?s, 1/? = 80000 (1/s)
i(t) = [Iinf + (I0 - Iinf)e-80000t] u(t) = [1.825 + (1 - 1.825)e-80000t] u(t) = [1.825 - 0.825 e-80000t] u(t) mA
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