Understanding RLC Circuits: Calculating Current in a Series Circuit

[kings13]

A series RLC circuit has an electromotive force given by E=200e^(-100t) V, a resistor of 80 ohms, an inductor of 0.2 H, and capacitor of 5x10^-6 F. If the initial current and charge on the capacitor are zero, find the current at any time t>0.

How on Earth do i start this?!

 

[nucl34rgg]

First, write a differential equation using KVL. Call the voltage across the cap, v(t). Then, the voltage across the inductor is Ldi/dt. The voltage across the resistor is iR. The driving voltage is E. Now, since i is going into a cap, we know that i=Cv', where v' is dv/dt. Then, putting it all together and dividng by LC gives v''+(R/L)v'+v/(LC)=E/(LC), where E is the EMF driving the circuit. Now, before you can solve this equation, you need to figure out your initial values. What does i(0)=0 imply about v'(0)? Use i(t)=Cv'(t) to find this initial value. Now, what does q(0)=0 imply about v(0)? Note that C=q(t)/V(t), so V(t)=q(t)/C.

Now you have a second order differential equation and you have two initial values to solve it. You can use any method you'd like to solve this equation, but you may find it difficult because of the driving term, E. I would suggest a Laplace transform. Then, if you get a weird answer in the s domain, use partial fractions to break it into parts which you can easily invert.