Question

poirmts) l'igure output y ws an RG circuit with inpu Function Generator Model Capacitor Model i,(t) Figure 2: RC cireuit (a) (1 point) Sketch the circuit in the phasor domain by replacing the capacitor with its impedance represen- tation (b) (3 points) Using circuit analysis techniques, show that the frequency response function is F(u) UT 1 Specify the DC gain, K, and time constant, τ, in terms of the parameters Re, C, and R. (c) (2 points) Determine the magnitude, H and phase angle, LH(u) of the frequency response function in part (b) in terms of the parameters Re, C, and R. 2. (5 points) In part 1 you obtained the frequency response function by applying circuit analysis techniques to the phasor domain representation of the circuit In applications other than electronic circuits, for example control systems engineering, it is necessary to obtain the frequency response function directly from the ordinary differential equation that describes the input-output behavior of the system. Consider a system with input f(t), output y(t), and the ODE representation Q(D)y(t) dy P(D)/(t) dm itm = dn <d"-y df Suppose that the sinusoidal input generates the sinusoidal steady-state response y(t) = Y,n cos(wt +%) = Repeat} where the phasor representation of f(t) and y(t) are Y=Y,ne tivel

PLEASE SOLVE QUESTION NUMBER 2 ONLY, MAKE SURE ALL PARTS IN QUESTION 2 ARE SOLVED. THANK YOU

Answer 1

Source Transformation, Taking Laplace Transformation, F (S) - V_s/R_s + F (S)/R + L_C = 0 L_C = R_C Vertical-bar Vertical-bar 1/S C = R_C/S C R_C + 1 F (S) [1/R_S + 1/R + L_C] = V_S/R_S F (S) [R + L_C + R_S]/R + L_C = V_S \r\n F (S) = R + L_C/R + L_C + R_S V_S F (S) = R + R_C/1 + S C R_C/R + R_C/1 + S C R_C + R_S V_S F (S) = R (1 + S C R_C) + R_C/(R + R_C) (1 + S C R_C) + R_C V_S F (S) = (R + R_C) + S C R_C/(R + R_C + R_S) + (R + R_S) S C R_C V_S Y (S) = F (S) L_C/R + L_C = R_C/(1 + S C R_C)/R + R_C/(1 + S C R_C) F (S) Y(S) + R_C/R_C + R (1 + S C R