Question

Cousider a causal LTI system implemented as the RLC cireuit shown below. In this cireuit, t) is the inpiat (t) across the ca d the system output where R 1Ω. L = 1H, and C-1F (a) Fnd tbe differential equation relating (t) and 311) (b) Determine the freqrecy respoe of this ten hy cotsidering the cutput of the system to inputs of the form r(,)

Answer 1

(a) By applying KVL we get L di(t)/dt + Ri(t) + y(t) = x(t) i = c dy(t)/dt implies i/c y(t) = 1/c integral i(t) dt (L d^2 y(t)/dt + R(dy(t)/dt + y(t) = x(t) L(d^2 y(t)/dt + R(dy(t)/dt + y(t) = x(t) (b) By applying Laplace transform on both sides we get (s^2 L_c + R Cs + 1) y(s) = x(s) y(s)/x(s) = 1/s^2 L_c + R cs + 1 put s = jw y(e^integral wt) = 1/(1 - w^2 L_c) + jw RC times (e^integral wt) \r\n (c) y (e^jw)/x (e^jw) = 1/(1 - w^2 Lc) + jw R_c since L = 1 c = 1, R = 1 w = 1 Y(e^jw) = 1/jwR_c times (e^jw) Vertical-bar Y(e^jw Vertical-bar = 1/jw R_c times (e^jw) Vertical-bar y e^jw Vertical-bar = 1/wRc Vertical-bar _w = 1 = 1 angle (e^jw) = -90 degrees Vertical-bar _w = 1 y(t) = sin (t - 90 degrees) = cos t y(t) = cos t rightarrow steady state output.