I think I'm suffering from brain-drain. I'm trying to solve a 2nd order differential equation for the natural response of an under-damped parallel RLC circuit with initial conditions of \$I_L\$ and \$V_C\$ but I'm falling (apparently) at the final hurdle.
All the solutions on the web that I have found seem to skirt over the solution to finding B1 and B2 in this type of formula. Most (if not all the solutions) seem to solve for output voltage but I also want to know how the inductor current decays.
The best single picture I have found shows this: –
And, I am happy that I can use this to solve for the voltage waveform if my initial condition was just the capacitor voltage BUT, I need to know what B1 and B2 are for the intial conditions of BOTH capacitor voltage and inductor current.
So, in short: –
- What is the formula for \$i_L(t)\$ for a parallel RLC
- How do I formulate the B1 and B2 values taking into account BOTH initial conditions
- Ditto for \$v(t)\$