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Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). Y(s) X(s) E(s) (a) Find the transfer functions H(s) = X (8) and He(s) = X(3) (b) Now set G(s) = 10. This is termed “proportional feedback,” where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) = sin 10t. = (ii) How do your answers change (provide a qualitative discussion) for x(t) = sint and x(t) = sin 100t? (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t→ ∞. Hint: You can simply use the final value theorem in (iii). 2 (c) Redo (b)(iv) for G(s) = 10+ ("proportional plus integral" feedback). S

Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). Y(s) X(s) E(s) (a) Find the transfer functions H(s) = X (8) and He(s) = X(3) (b) Now set G(s) = 10. This is termed “proportional feedback,” where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) = sin 10t. = (ii) How do your answers change (provide a qualitative discussion) for x(t) = sint and x(t) = sin 100t? (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t→ ∞. Hint: You can simply use the final value theorem in (iii). 2 (c) Redo (b)(iv) for G(s) = 10+ ("proportional plus integral" feedback). S

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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i only want part b and c please. i want a step by step solution and explanation 

Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). (a) Find the transfer functions H(s) = Y(s) X(s) and He(s) = = E(s) X(s)* (b) Now set G(s) = 10. This is termed “proportional feedback," where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) sin 10t. (ii) How do your answers change (provide a qualitative discussion) for x(t) sin 100t? = sint and x(t) (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t → ∞. Hint: You can simply use the final value theorem in (iii). (c) Redo (b)(iv) for G(s) = 10+ 2/3 ("proportional plus integral" feedback).
Transcribed Image Text:Consider a feedback-based causal LTI system with input x(t) shown in Figure 6.41, where G(s) is the transfer function of a "loop filter" and 1/s is an integrator. The idea is to use the error e(t) to drive these, so as to make y(t) track x(t). (a) Find the transfer functions H(s) = Y(s) X(s) and He(s) = = E(s) X(s)* (b) Now set G(s) = 10. This is termed “proportional feedback," where the feedback is propor- tional to the error. (i) Find y(t) and e(t) for x(t) sin 10t. (ii) How do your answers change (provide a qualitative discussion) for x(t) sin 100t? = sint and x(t) (iii) For x(t) = u(t) (unit step), find and sketch y(t) and e(t), and specify their asymptotic values as t∞. (iv) For x(t) = tu(t) (ramp starting at time zero), find the asymptotic value of the error e(t) as t → ∞. Hint: You can simply use the final value theorem in (iii). (c) Redo (b)(iv) for G(s) = 10+ 2/3 ("proportional plus integral" feedback).
x(t) — e(t) G(s) y(t) 1/s
Transcribed Image Text:x(t) — e(t) G(s) y(t) 1/s
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