Answer Happy

Q2 Laplace Transforms for Circuit Analysis A first-order Butterworth filter is shown in transfer function form in Figure Q2(a). x(t) Wc y(t) KS S + wc ed ks ed Figure Q2(a) A first-order Butterworth filter with cut-off frequency We rads? Answer all parts of this question, rks Use of mathematical software, such as MATLAB, is permitted. red Enter all numbers rounded to 3 significant figures unless otherwise advised. ges rks ered Q2(i) Impulse response and step response For the system shown in Figure Q2(a) arks ered a) If the cut-off frequency of the filter is to be fc = 0.5 kHz, what is the corresponding value of wc rads?? Round your answer to 3 significant figures. Sul Impulse Response The impulse response of the filter shown in Figure Q2(a) will be

Impulse Response The impulse response of the filter shown in Figure Q2(a) will be h(t) = We exp(-wet). This is graphed for we = 1 rads (f = 0.159 Hz) in Figure Q2(b) Impulse response S d 0.9 0.8 0.7 0.6 Ks ed 05 0.4 03 ks od 0.2 0.1 ks ed 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure Q2(b) Impulse response h(t) for a Butterworth filter with wc = 1 rads? The time constant of a first order system can be computed from the time-response for h(t) which will be like the graph illustrated in Figure Q2(b). The time constant is defined as the value of t = T for which exp(-we) = exp(-1). For the case illustrated, we = 1 rad/s and the time constant t = 1 s. This point is shown on Figure Q2(b). b) For the system of Figure Q2(a) with fc = 0.5 kHz, what is the value ofr in milliseconds? Round your answer to 3 significant figures,

b) For the system of Figure Q2(a) with fe = 0.5 kHz, what is the value of t in milliseconds? Round your answer to 3 significant figures, ks ed c) What is the value of the impulse response h(t) when t = 0.17? ks red Round your answer to 3 significant figures. -ks red rks ered Q2(ii) System Response by Convolution The response of a system is given by the time convolution of the system impulse response h(t) with the input r(t) such that y(t) =h(t) * 2(t) rks red Convolution in the time domain is multiplication in the Laplace transform (complex frequency domain): Y(s) = H(s)X(3) The next few questions will take you through the determination of the step response of the system of Figure Q2(a). d) The system input is z(t).= 2uo(t). Give the Laplace transform of the input X(s).

e) For the system of Figure Q1(a) with fc = 0.5 kHz, use convolution to write down an expression for y(s). carks wered marks wered f) Use the inverse Laplace transform to write down the system response y(t). marks wered anges marks swered marks wered Q2 (iii) Step Response The unit step response of the Butterworth filter (z(t) = u(t)) is illustrated in Figure Q2(a): h(t) * Uo(t) = (1 -exp(-wet)) uo(t). This gives the well-known first-order response illustrated for we = 1 rad/s in Figure Q2(c). Step response 0.9 08 007 0.6

Q2 (iii) Step Response The unit step response of the Butterworth filter («(t) = u(t))is illustrated in Figure Q2(a): h(t) * wy(t) = (1 - exp(-wet)) 20(t). This gives the well-known first-order response illustrated for We = 1 rad/s in Figure Q2c). Step response 0.9 0.8 0.7 0.6 0.5 (8) Pn. 04 토 0.4 0.3 02 Rise Time Trs 0.1 1 0 015 1.5 2.5 3 Figure Q2(c) - Step response of a first-order Butterworth filter with we = 1 rad/s. As for the previous question, the time constant in the figure is t = 1s, and the corresponding value of the step response is marked on the graph by the solid red line. g) Att = 1 ms, what is the value of the step response as a percentage of the final value? (Give your answer as an integer percentage. E.g. if 0.5 of the final value would be 50%.) Round your answer to the nearest integer % Submit part 1 mark

Rise Time Trs 0.1 0 0 0.5 1 2 2.5 3 1.5 1 Figure Q2(c) - Step response of a first-order Butterworth filter with we = l rad/s. As for the previous question, the time constant in the figure is t1s, and the corresponding value of the step response is marked on the graph by the solid red line. Att = tms, what is the value of the step response as a percentage of the final value? (Give your answer as an integer percentage, E.g. if 0.5 of the final value would be 50%.) Round your answer to the nearest integer % Submit part 1 mark Unanswered h In control theory, rise time T. is defined as the time taken for the step response of a system to move from 10% to 90% of the final value (indicated in Figure Q2(b) by the green dashed lines). Compute the rise time in milliseconds for the system with cut-off frequency fc = 0.5 kHz and input z(t) = 2uo(t). Round your answer to 3 sig-figs. Round your answer to 3 significant figures. Submit part 2 marks Unanswered Submit all parts 10 marks Created using Numbas, developed by Newcastle University