Question

5. (Five points) Consider an RLC circuit for which an inductor of L-1 H and capacitor C= 0.1 F are present. For the given forcing function f (t), use the Laplace transforms to determine the charge Q (t) and current I (t) in the circuit at time t if initially(0)-0 and I (0) 0. Determine the charge and current in the case when R-8 and f (t) 10e. Show all your work solving this equation

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Answer 1

Given that: - L Q^derivative + R Q^derivative + 1/c Q = f(t) L = 1 H, c = 0.1 f Q^derivative(t) = I(t) Q(0) = Q^derivative(0) = 0 implies Q^second derivative + R Q^derivative + 10Q = I(t) using Laplace transform S^2 Q^bar - S Q(0) - Q^derivative(0) + R (S Q^bar - Q(0)) + 10Q^bar = L {f(t)} Q^bar (S^2 + Rs) + 10) = L {f(t)} Q^bar = 1/(S^2 + Rs + 10) integral_0^infinity e^-st f(t) dt therefore now R = 8 f(t) = 10e^-t Q^bar = 10/(S^2 + 8s + 10) integral_0^infinity e^-st e^-t dt = 10/(S^2 + 8s + 10)(s + 1) = -10s/3(S^2 + 8s + 10) 10/3(S^2 + 8s + 10) + 10/3(s + 1) Q(t) = -10/3 e^-4t cos h . (squareroot 6 t) -5 squareroot 2/squareroot 3 e^- 4t sin h (squareroot 6 t) + 10/3 e^-t so correct = Q^derivative (t) = I(t) = 40/3 e^-4t cos h (squareroot 6