Chapter 6, Problem 6.1Q

8. Obtain the inverse Laplace transform of each of the followingfunctions by first applying the partial-fraction-expansionmethod.(a) F1(s) =6(s+2)(s+4)(b) F2(s) =4(s+1)(s+2)2(c) F3(s) =3s3 +36s2 +131s+144s(s+4)(s2 +6s+9)(d) F4(s) =2s2 +4s−10(s+6)(s+2)2

12.12 In the circuit of Fig. P12.12(a), is(t) is given by the waveform shown in Fig. P12.12(b). Determine iL (t) for t≥ 0, given that R₁ = R₂ = 2 2 and L = 4 H. is() R₁ R2: (a) Circuit is(t) 8A- 8e-21 elle (b) is(t) Figure P12.12 Circuit and waveform for Problem 12.12. iL

12.12 In the circuit of Fig. P12.12(a), is(t) is given by thewaveform shown in Fig. P12.12(b). Determine iL(t) for t ≥ 0,given that R1 = R2 = 2 W and L = 4 H.

12.4 Determine the Laplace transform of each of the following functions by applying the properties given in Tables 12-1 and 12-2 on pages 642-643. (a) fi(t)=4tet u(t) (b) f2(t)=10cos (12t+60°) u(t) *(c) f3(t) = 12e−3(t−4) u(t −4) (d) f4(t) = 30(e³ +e³t) u(t) (e) fs(t)=16e2t cos 4t u(t) (f) f6(t)=20te 2 sin 4t u(t)

a) Calculate the values of v and i. + 803 1A Va 82 b) Determine the power dissipated in each resistor. 1A Va (a) + I 50 V 0.2 S (b) + D + 1 Α υ€ 20 Ω 50 V 250 ΩΣ ia (c) (d) Copyright ©2015 Pearson Education, All Rights Reserved

Exercise 3-12: Find the Thévenin equivalent of the circuit to the left of terminals (a, b) in Fig. E3.12, and then determine the current I. 502 502 0.6 Ω 20 V | + <302 Ω ΣΙΩ b 2025 Ω 15A Figure E3.12

2. Consider following feedback system. r(t) e(t) y(t) K G(s) 1 where G(S) = s²+as+b In above, K, a and b are constants. Select the values of K, a and b in a way so that (i) (ii) (iii) the closed loop system is stable, steady-state error of the closed-loop system for step input is 0.2, the closed-loop response has 20% overshoot and 2 seconds as settling time.

4. Answer the following questions. Take help from ChatGPT to answer these questions (if you need). But write the answers briefly using your own words with no more than two sentences, and make sure you check whether ChatGPT is giving you the appropriate answers in the context of class. a) What is the advantage of the PI controller over the proportional controller? b) What is the advantage of the PD controller over a proportional controller? c) In the presence of noise, what problem do we face implementing the derivate part of the PID (or PD) controller? To address this, what do we usually use? d) What are the forms of lead compensator and lag compensator? How do these two types of compensators differ?

3. Consider the following closed-loop system as shown in the figure. 16 Ge(s) s(s + 4) Suppose Ge(s) is a PID controller with Kp = 1, KD = 2 and K₁ = 3. a) Find the controller transfer function G₁(s). b) Find the open-loop transfer function. c) Find the closed-loop transfer function.