EE201 Lecture 25 P 1 2nd Order Circuits with Constant Inputs Differential equations for RLC series circuits R vs t iL t L v t c C iL as circuit variable d2 iL t dt2 R d iL t L dt 1 iL t F 1 vc t F 2 LC vc as circuit variable d2 vc t dt2 R d vc t L dt 1 LC EE201 Lecture 25 P 2 Parallel RLC circuits iL R L vc t C iL as circuit variable d2 iL t dt2 1 d iL t RC dt 1 iL t F vc t F 4 LC 3 vc as circuit variable d2 vc t dt2 1 d vc t RC dt 1 LC EE201 Lecture 25 P 3 General form of differential equation d2 x t dt2 b d x t dt c x t F General solution form x t xn t xF xn t solution satisfying homogenous differential equation i e F 0 xF constant accounting for non zero forcing function Eqn 5 has the characteristic equation s2 bs c 0 b b2 4c s1 s2 2 Therefore 3 cases will be considered 5 EE201 Lecture 25 P 4 Case 1 s1 s2 are real and distinct Solution x t K1 e s1 t to K2 e s2 t to xF xF x Solve for K1 K2 using x 0 K1 K2 xF x 0 s1 K1 s2 K2 Case 2 s1 s2 are complex and distinct s1 j d s2 j d Solution x t e t to A cos dt B sin dt xF EE201 Lecture 25 P 5 xF x x 0 A xF evaluated at to 0 x 0 A dB Case 3 s1 s2 and are real Solution x t K1 K2t e s1 t to xF xF x where x 0 K1 xF x 0 s1 K1 K2 EE201 Lecture 25 P 6 Example Find vc t for t 0 assuming vc 0 1V and iL 0 3 2 A iL t 2 3 vc t 1H 0 5 F 10u t V Step 1 Set up initial conditions vc 0 vc 0 1V iL 0 iL 0 3 2 A Circuit at t 0 iL 0 3 2 A iR 0 2 3 ic t vL 0 v 0 c 1V 10V EE201 Lecture 25 P 7 From KVL vL 0 10 vc 0 9 V Solving for iR 0 iR 0 vc 0 R 3 2 A From KCL ic 0 iL 0 iR 0 0 A Step 2 Set up solution form for vc t with source zero parallel RCL EE201 Lecture 25 P 8 s2 1 RC s 1 LC s2 3s 2 0 s1 1 s2 2 Case 1 solution vc t K1 e s1t K2 e s2t vF vc t K1 e t K2 e 2t vF Step 3 Solve for unknown constants At t vc 10V 2 3 vF vc 10V From initial conditions K1 and K2 are found EE201 Lecture 25 vc 0 K1 K2 vF P 9 K1 K2 9 v c t 1 C ic 0 2 ic 0 0 v c 0 K1 2 K2 0 K1 18 K2 9 Step 4 Obtain final solution vc t K1 e s1t K2 e s2t vF vc t 18e t 9 e 2t 10 V t 0
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