LCv'' + RCv' + v = Vs

where: Vs is a stepped voltage at t=0

and: v is the voltage accross the capacitor

and: v'' and v' are the first and second derivitives respectivly

A solution is v = Ae^(αt)cos(βt) + Be^(αt)sin(βt) + Vs

where: α = -R/2L rad/sec

and: β = √(1/LC - R²/4L²) rad/sec

The wave form is a phase shifted sine wave with exponential ring down because the determinant is complex.

What are A and B?

Using initial conditions for both v(t=0) and i(t=0) = Cv'(t=0) I have come up with:

A = Vi - Vs and

B = Ii/Cβ - (Vi - Vs)α/β

where Ii is the current through the circuit at t=0 (initial current) and Vi is the voltage accross the capacitor at t=0 (initial voltage).

I've noticed that the oscillating frequency, β, induces an impedence accross C of 1/Cβ so that the initial current gives an initial voltage of Vi = Ii/βC = Ii.Xc

As of yet, I have not found a concise sollution on the web.

Does anybody have the correct sollution to an underdamped series LCR circuit?