ECE 421 - Fall 2021 Homework 1 Due Monday, Sep. 13 1. Obtain the Laplace transform of the following functions (A) ( f (t) = 0 t cos t for t < 0 for t ≥ 0 (B) ( 0 e−0.5t sin 2t for t < 0 for t ≥ 0 ( 0 sin(t + π4 ) for t < 0 for t ≥ 0 f (t) = (C) f (t) = 2. Based on the finial-value theorem, find the steady state of a function f (t), whose Laplace transform is given by F (s) = 3. Find the transfer function F (s) = 5 s(s + 8) U (s) for the following electric system. Y (s) where u(t) is the input voltage, y(t) is the output voltage. 1 Student Name: θ2 (s) θ1 (s) and F2 (s) = for the following mechanical 4. Find the transfer functions F1 (s) = F (s) F (s) system, respectively where T (t) is the input torque, θ1 (t) is the output angle for the first function, and θ2 (t) is the output angle for the second function. (Hint. Use free-body diagrams on J1 and J2 respectively to obtain two equations. For F1 (s), try to cancel out θ2 ; For F2 (s), try to cancel out θ1 .) 2