LC Circuits

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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 and 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
− Aω2 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 …

ƒ =	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

Summary

• bullet

Problems

practice

1. Write something.
   • Answer it.
2. Write something.
   • Answer it.
3. Write something.
   • Answer it.
4. Write something completely different.
   • Answer it.

numerical

1. problems

Resources

• theremin
  • Robert Moog
    • Big Briar, Inc
    • Robert Moog Dies at 71; Created a Synthesizer That Revolutionized Music, New York Times, 23 August 2005
  • The Octopus Project - I Saw The Bright Shinies at YouTube
    • DJMattz0r
    • hotrob4000 (occasionally harsh audio)
    • JoshuaCarrShows
  • Theremin Home Page
  • theremin.org, Theremin Performer Eri
    • Star Trek Theme, QuickTime, WindowsMedia, RealAudio
  • ThereminWorld

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