3. The Laplace transform is a powerful technique that provides a universal technique for the solution of differential equations. a. Find the inverse Laplace transform of a system with transfer function: 2 (s + 1)² (s + 2) b. A signal has an s-domain representation as given below. What is h(t)? G(s) = H(s) S+6 (s + 2)³ [10 marks] [10 marks]

i. Determine from first principles the Laplace Transform of t+2e=t -6s+85-56 ii. Show that f (t) = {4sin (2t) – 6e“} =F+4)(3-4)

A system defined by h(t) = 5e-5t u(t) is driven by a sinusoidal signal with frequency 5 rps and amplitude 4 V. How much is the amplitude of the output signal in volt? (hint: use L-transform of h(t))

2. An input function is called doublet pulse as shown in the figure. (a) Find the Laplace transform of this function (b) What is the response of a process having a first-order transfer function K/(ts + 1) to this input? Draw the dynamic response. (c) What is the response of an integrating process K/s to this input? Draw the dynamic response. xDp(t) h 0 -h tw 2tw V/(-) Ş

Answer All of the Questions Below 1. Find the inverse Laplace Transform of the following function below. i. F(s) = s+ 5 s? + 8s + 65 ix. F(s) = 49 7s + 21 ii. F(s) 5s? + 14s + 2 (s-2)(s+4)? %3D X. F(s) = 3s (s+3)² + 64 iii. F(s) 3s + 4 s? + 16 %3D 2s - 3 2s2 - 8s + 16 xi. F(s) = iv. F(s) = _ 5s² + 14s + 2 (s-2) (s+3)? xii. F(s) =, s? + 12 s (s+1)(s+3) v. F(s) =. s + 5 (s+3) + 9 %3D 10s? + 4 s (s+1)(s+2)² xii. F(s) = vi. F(s) 8s + 2 s? + 3s + 2 xiii. F(s) = 1 s (s? + 6s + 9) vii. F(s) = s? + 3s + 1 (s? + s) xiv. F(s) = 3s + 2 s? + 3s + 2 viii. F(s) 8(s + 2) (s+1)(s+3)(s+5)

- 7s 10s –13 3. Find the inverse Laplace transform of F(s)=- (s – 10) – 27 (s-13(6s +3)(3s – 7)'

Q2: The signal f(k) has Z transform as the following: 276 – z5 + 373 + 2z2 F(Z) Z7 + 226 + Z5 +Z4 + 0.5 Compute f(3). Ans: f(3) = 8