# Derive time-dependent current for circuit

1. Sep 7, 2011

### 0specificity

1. The problem statement, all variables and given/known data
Derive an expression for the current I(t) as a function of time for the following circuit:

____ \____
|.............. |
&.............. -
& ..............-
| ...............|
___________

This circuit may be a bit unclear - the && is an inductor L, the - - is a capacitor C, and the top part is a switch that is open for t<0 and closed at t=0. And the ... are filler.

2. Relevant equations

I have no idea I am struggling incredibly.

3. The attempt at a solution

2. Sep 10, 2011

What do you mean by filter???

3. Sep 10, 2011

### legendary_

1. Use Kirchoff's Voltage Law (sum of voltages around the loop of the circuit):

2. However, we need to determine the voltage across the inductor and capacitor.
Just do research, the voltage of the inductor is:

V_L = Ldi/dt

V_C = (1/C) integral (idt)

3. Using KVL: Ldi/dt + (1/C)integral(idt) = 0

4. We need to turn this one to differential equations. So we differentiate term by term

Ld^2i/dt^2 + (1/C)i = 0

5. Use methods of differential equation to solve for i with respect to t.

Lm^2 + (1/C) = 0

m^2 + 1/(L*C) = 0; m = +- j/(L*C)

I(t) = Acos(t/(LC)) + Bsin(t/(LC))

6. The constants A and B can be derived using initial conditions.

See here for more details:

http://en.wikipedia.org/wiki/LC_circuit