Question

Derive the ODE for the RLC circuit provided using Kirchoff's voltage and current laws. Indicate the order of the equation and what conditions are required for a unique and completely defined solution to existing. You should solve for the voltage across the capacitor as the output, y(t), due to the excitation of the input voltage, x(t), and the initial conditions, y(0) = c0 and y'(0)=c1.

Compute the homogeneous solution to the ordinary differential equation for unknown values of R, L and C. You can keep the solution in exponential form or simplify the expression for cases where the response is under-damped, critically-damped and over-damped.

Answer 1

Page - 3 Homogeneous Sol" is y(t) = (co +(c +R (6) x Je ure

2 L ²+ & D + į = 0 [Page 2 9 D + 16) - 4x013 (5) 2 for unique and defined solution, we know, + (%) - 4 x0* = 0 = DER so, the required condition (B) - - 0 + I c Homogeneous solution of above equation, deyes + + tc y(t) = 0 soln, will be y(t) = (A + Bx) e mar lie m--R/227 & yt - A+ boje . Ty Ct = (A + Bx) e hoc By initial conditions y(o)= A = Col and y'C+) = Ben B (A + B) e in y'(o) = B - RAA ma C = B- B co (B - C, taco

| Page 1 Given, Series RLC ekt, 7(0) = co ? initial yo) = c conditions alt; = V مقدوا cyt) fig: 1 solution: - Derivation of ODE, Applying KvL in the elet, we get, x(t) - IR - L d I - Yt)= 0 - ♡ Also, y tt) = I lidt - ② putting epr. ① - ② We get, x (t) - R c d yt - L c d yet) - g (t) = 0 Rearranging egn. we have dy (t + ß dyt) + 1 ve y(t) = 1 L x (t) | at order of the above Differential equation is TWO (2) Now, characteristics equation of the above differential eqr. will be
For the series RLC circuit we can derive the Ordinary Differential Equation by KVL and then can find the unique solution of the Differential equation(making it homogeneous) by standard solution described in the solution.