Find i L(t) for t>0 in the above circuit Solution Natural Response: When finding

Question

Find i L(t) for t>0 in the above circuit

Explanation / Answer

Natural Response: When finding the natural response of the circuit, all sources(voltage or current) should be killed. In this case, the 10 A source is open-circuited -- thus, only theseries connection of R, L, and C remains. We first find the state of the circuit by finding first the neperfrequency and the resonant frequency0. For a series connection, = R/2L = 2/[2(0.25)] = 4 np/s (nepers/s) 0 = 1/(LC)1/2 =1/[(0.25)(0.2)]1/2 = 4.472135955 rad/s Since 0 > , the circuit isunderdamped... This signals us to find the damping frequencyd: d = [02-2]1/2 = [(4.472135955)2-42]1/2d =2. Then the natural response is given by: iLn(t) = e-t(A cosdt + B sin dt ) iLn(t) = e-4t(A cos 2t + B sin2t ) where A and B are constants. Forced Response: To find the forced response, we look at the state of the circuit att = ... At t = , the capacitor is open-circuited and the inductor isshort circuited.. Therefore, iLf = iL() = 10A. Complete Response: Therefore, the complete response is: iL(t) = iLn(t) + iLf =e-4t(A cos 2t + B sin 2t ) + 10 A. Initial Values: In order to find the constants A and B, we first findiL(0) and diL(0)/dt. Since there are no sources present at t < 0 or t =0-, iL(0) = 0 andvC(0) = 0. (Capacitor voltage). Now to obtain diL(0)/dt, we investigate the circuit at t= 0+. Thus, the current source is present at thistime. Now doing KVL at the right loop, we have: -vC(0+) + 2iL(0+) +0.25 di(0)/dt = 0. Since vC(0+) =2iL(0+) =0. Thus, we end up with di(0)/dt = 0. Finding constants A and B: Since we already have the initial values, we can now find theconstants A and B. At t = 0, iL(t) = e-4t(A cos 2t + B sin 2t ) + 10A. iL(0) = A + 10 A = 0 Therefore, A = -10. diL/dt = -4e-4t(A cos 2t + B sin 2t )+ e-4t(-2A sin 2t + 2B cos 2t )  (product rule of differentiation) diL(0)/dt = -4A + 2B = 0 B = 2A = 2(-10) B = -20. Therefore, the complete response is: iL(t) = 10 - 10e-4t(cos 2t + 2sin 2t )

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