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) | |
| d | I = | ωI0 cos(ωt + φ) |
| dt | ||
| d2 | I = | −ω2I0 sin(ωt + φ) |
| dt2 |
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 above. 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.
LC circuits are basically filters. They have a natural fequency and when that natural frequency equals some driving frequency, resonance occurs and that frequency is amplified above all others.
rcl circuit
Begin with Kirchhoff's circuit rule.
| V = L | dI | + IR + | q |
| dt | C |
Take the derivative of each term.
| dV | = L | d2I | + R | dI | + | 1 | dq | |
| dt | dt2 | dt | 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 much like the one derived earlier, but now also with a first order term in the middle.
| 0 = L | d2I | + R | dI | + | 1 | I |
| dt2 | dt | C |
Let's guess and check. I guess that the solution to this is a combination of exponential and sinusoidal functions. Let's try a cosine rhis time.
| I = | I0e−αt | cos(ωt + φ) | |||||||||||||
| d | I = | −I0e−αt | [ | α | cos(ωt + φ) | + | ω | sin(ωt + φ) | ] | ||||||
| dt | |||||||||||||||
| d2 | I = | I0e−αt | [ | α2 | cos(ωt + φ) | + | αω | sin(ωt + φ) | + | αω | sin(ωt + φ) | − | ω2 | cos(ωt + φ) | ] |
| dt2 | |||||||||||||||
| d2 | I = | I0e−αt | [ | (α2 − ω2) | cos(ωt + φ) | + | 2αω | sin(ωt + φ) | ] | ||||||
| dt2 |
Substitute into our three-part differential equation (after factoring out the common terms) and check to see what happens.
| 0 = | I0e−αt | { | L | [ | (α2 − ω2) | cos(ωt + φ) | + | 2αω | sin(ωt + φ) | ] | ||||
| −R | [ | α | cos(ωt + φ) | + | ω | sin(ωt + φ) | ] | |||||||
|
[ | cos(ωt + φ) | ] | } | ||||||||||
Eliminate the exponential part, since it can never equal zero. Combine like terms in the sinusoidal part.
| 0 = | [ | L(α2 − ω2) −Rα + 1/C | ] | cos(ωt + φ) | ||
| + | [ | 2αωL − ω | ] | sin(ωt + φ) | ||
When is this expression true?
− [2αωL − ω] sin(ωt + φ) = [L(α2 − ω2) −Rα + 1/C] cos(ωt + φ)
Or this one?
| − | sin(ωt + φ) | = | L(α2 − ω2) −Rα + 1/C |
| cos(ωt + φ) | 2αωL − ω |
Duh… I forgot how to do this.
Try this instead