Question

Problem 5. (20 points) The expression of voltage across the capacitor in s-domain is V1(s) = s(s +0.5) (s + 2) For t0, derive the time-domain expression for vi(t) 1 H 15? 15 v

Please solve this for me, if you're pretty sure about the solution

Answer 1

v_1(s) = 5(5 + 3)/s(s + 0.5) (s + 2) Step-1: - using partial fractions 5s + 15/s(s + 0.5) (s + 2) = A/s + B/s + 0.5 + C/s + 2 5s + 15 = A[s + 0.5] [s + 2] + B[s] [s + 2] + C[s] [s + 0.5] 5s + 15 = A[s^2 + 2.5s + 1] + B[s^2 + 25] + C[s^2 + 0.5s] Comparing coefficients 0 = A + B + C 5 = 2.5A + 2B + 0.5C 15 = A an solving we get A = 15, B = -16.67, C = 1.67 therefore V_1(s) = 15/s - 16.67/s + 0.5 + 1.67/s + 2 Step 2: - Apply inverse laplace Transform L^-1[V_1(s)] = V_1(t0 = L^-1{15/s - 16.67/s + 0.5 + 1.67/s + 2} V_1(t) = 15L^- (1/s) - 16.67L^-1 (1/s + 0.5) + 1.67L^-1 (1/s + 2) V_1(t) = 15 - 16.67 e^-0.5t + 1.67 e^-2t