Question

3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential equation given below: a causal system is dx(t) dse)+540+6y(t) = x(t) +T Assuming that the initial conditions are zero and using the Laplace transform determine [5 Marks] 15 Marks the following: a- Transfer function H(s) of the system. b- Impulse response h(t) of the system. Y (s) X(s)

Answer 1

Given equation is d^2 y(t)/dt^2 b + 5 dy(t)/dt + 6 y(t) = x(t) + d x (t)/dt according to Laplace transform. df(t)/dt leftarrow sF(S) - f(w) d^2 f(t)/dt^2 leftarrow s^2 F(s) - s f(0) - f' (0) but according to qusect6ion, initial values are zero. s^2 y(s) + 5 s y(s) + 6 y (s) = x(s) + s X cs) (s^2 + 5s + 6) y(s) = y(s) [1 + s] therefore y(s)/x(s) = s + 1/s^2 + 5 s + 6 hence H(s) = s + 1/s^2 + 5 s + 6 = (s + 1)/(s b + 2) (s + 3) H(s) = (s + 1)/(s = 2) (s + 3) = A/s + 2 = B/s + 3 \r\n A = lim_s rightarrow - 2 (s + 2) H (s) = lim_s rightarrow - 2 (s + 1)/(s + 3) A = (- 2 + 1)/(- 2 + 3) = -1 and B = lim_s rightarrow - 3 (s +