V; t=0 R Vo Transfer function : LS 71,2 2RC R2C52 +25+R 3 + 5 + 1 하는 RC radis ± LC 2R 1-4R²C 2RC V L

Vi(t) (a DC input voltage) is connected to the circuit at t=0, Vo(t) (across L and C) is the output. 

The answer to the first three questions are provided in the picture. 

a) Find the impedance of L and C using s=jω and find the second order transfer function after Vi(t) is connected where s is a complex number.

b) What is the canonical second order charecteristic polynomial?

c) In terms of R,L,C derive an expression for ωn and ζ and find the poles of the system.

d) what are the possible general solutions for Vo(t) considering this system could be overdamped, critically damped or underdamped. set up to find initial conditions. start by considering the circuit at t < 0, t = 0 and finding a second order D.E. and set up the poles of the system and v(t) at t tends to infinity. be as specific as possible with the answers.