Useful Tip: You may often find the following partial fractions expansion technique useful: (s² +as+b− s² − as) /b s (s²+as+b) 1 s (s²+as+b) for any pair of values of a and b=0. = Problem 1 Consider a parallel RLC circuit, driven by an input current source i(t). We would like to analyze the current i(t) that flows through the inductor, i.e., in this problem we consider i(t) as the output of the plant. The governing equation of the parallel RLC circuit is given by: i(t) where i(t)= + = IR(t) | i(t) i(t) = ir(t) + iz(t) + ic(t), v(t) v(t) 3 R iz(t)= 1 s+a bs b(s²+as+b) H(s): ic(t) = = [v(7) dr, ic(t) = Assume zero initial condition, i.e., the initial voltage across the parallel elements is zero, or v(t) = 0. Also, assume the following values R = 1, L = 1, C = 1. dv(t) dt a) Starting from the plant equations above, show that the input-output transfer function is: IL(S) I(s) where IL(s) and I(s) denote, respectively, the Laplace transforms of i(t) and i(t). = C 1 s²+8+1