MBS4108 Tutorial 3 Laplace Transform 1 . Find the Laplace Transform of the following, by definition, a) At) = eat where a is a constant b) f(t) = cos(4t) c) f(t) = sin(4t) (Hint: cos wt = Re (e jot ) and sin wt = Im(e jot ) 2. Using the result of Q1(d) and the property of "First Shifting", determine the Laplace Transform of a) At) = 2e -2sin(4t) b) At) = e-4 cos(2t) c) At ) = Be -21 3 . Using the Laplace table to find the Laplace Transform of 5+ 13 +- 4 sin 3t -2t Inverse Laplace Transform 4. Find Z-1 3 2 } and Z-1, S+ 2 - } S 2 + 4 5. Find Z-1{ - s+ 6 } and 2-1{ s+ 5 $2 + 65 + 10 $2 + 45 + 8 6 . Find Z-1 /s + 11 S 2 +5 - 6 Block Diagram Manipulation Problems 7. Find the overall transfer functions of the systems described by the block diagrams. (a) s(s + 1) 10 2 . (d) (7) 1 S + 1 2 K 5-1 K S+ 1 s + 2 (c) 3 + 1 [e) (g)MBS4108 Tutorial 3 10. 11. Convert the block diagrams to the equivalent systems with unity feedback. (a) (b) Derive an equation relating the inputs 6,(s), Qai(s) and 842(s) to the output of system. Q4:(S) 8a0(S) The forward, feedback and closed-loop transfer function for the following system are: The transfer function from d(s) to Y(s) 1 MBS4108 Tutorial 3 13. For the following block diagrams, determine the following: (i) the transfer function of the forward path (ii) the transfer function of the feedback path (iii) the closed-loop transfer function from R(s) to Y(s) (a) R(s) E(s Y (s ) + 1.2 2.0 3.2 (b ) R(s) E(s) Y(s) + 2.5 3.2 2.0 4.0 (c) R(s ) E(s) + Y (s) K1 K2 K3 H1