The Physics
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Opus in profectus

# LC Circuits

## Discussion

### lc circuit

Begin with Kirchhoff's circuit rule.

 V = L dI + q dt C

Take the derivative of each term.

 dV = L d2I + 1 dq dt dt2 C dt

The voltage of the battery is constant, so that derivative vanishes. The derivative of charge is current, so that gives us a second order differential equation.

 0 = L d2I + 1 I dt2 C

Rearrange it a bit…

 d2 I = − 1 I dt2 LC

and then pause to consider a solution.

We need a function whose second derivative is itself with a minus sign. We have two options: sine and cosine. Either one is fine since they're basically identical functions with a 90° phase shift between them. Without loss of generality, I'll choose sine with an arbitrary phase angle (φ) that could equal 90° if we let it. Or it could be equal to some other angle. The other parameters in a generic sine function are amplitude (I0) and angular frequency (ω).

The basic method I've started is called "guess and check". My guess is that the function looks like a generic sine function…

I = I0 sin(ωt + φ)

and the check is to pop it back into the differential equation and see what happens.

 d2 I0 sin(ωt + φ) = − 1 I0 sin(ωt + φ) dt2 LC − ω2I0 sin(ωt + φ) = − 1 I0 sin(ωt + φ) LC

Basically everything cancels but one parameter — angular frequency.

 ω = 1 √LC

An LC circuit is therefore an oscillating circuit. The frequency of such a circuit (as opposed to its angular frequency) is given by…

 f = ω = 1 2π 2π√LC

So what? How is this useful?

An audio crossover circuit consisting of three LC circuits, each tuned to a different natural frequency is shown to the right. The inductors (L) are on the top of the circuit and the capacitors (C) are on the bottom. On the left a "woofer" circuit tuned to a low audio frequency, on the right a "tweeter" circuit tuned to a high audio frequency, and in between a "midrange" circuit tuned to a frequency in the middle of the audio spectrum.

RC circuits are basically filters.

### rcl circuit

I need to write this part.