Question

6. In a simple RCL series circuit with R = 100 Ω, C = 0.0004 F(farad), and L-1 H (henry) and the impressed voltage v) 30. Find the charge Q) on the capacitor in the circuit at any time tif the initial current i(0) 2.A and the initial charge on the capacitor is 0(0)-0 C(coulomb) The Second Order ODE for RCL series circuit is given by where Qis the charge, and the current I =a- de dt

Answer 1

For an RCL circuit R = 100 Ohm c = 0.004 F, L = 1 H, v(t) = 30 also i(0) = 2 A, Q(0) = 0 implies dQ/dt (0) = 2A, Q(0) = 0 (Since I = dQ/dt) the second order ODE for RCL circuit is d^2Q/dt^2 + RlL dQ/dt + Q/LC = 0 d^2Q/dt^2 + 100/1 dQ/dt + 1/(1) (0.0004) Q = 0 d^2Q/dt + 100 dQ/dt + 1/(1)(0.0004) Q = 0 d^2Q/dt^2 + 100 dQ/dt + 14/10000 Q = 0 d^2 Q/dt^2 + 100 dQ/dt + 2500 Q = 0 implies D^2Q + 100DQ + 2500Q = 0, where d^2/dt^2 = D^2 implies (D^2 + 100D + 2500)Q = 0 implies P(D)Q = 0 where P(D) = D^2 + 1000 + 2500 its auxiliary equation is p(n) = 0 m^2 + 100m + 2500 = 0 (m + 50)^2 = 0 (m + 50)^2 = 0 m + 50 = 0, twice m = -50, twice thus e^-50t, te^-50t are solutions of (1)