Question: AC circuits and resonance

This question relates to Lab 6 but also to the lecture material (it was in the lab that I figured out that my understanding of the material was totally wrong).

It makes sense that in an A/C circuit the capacitor charges when there is a voltage and then discharges through the inductor when the voltage goes away. Also, it seems that the capacitor may or may not reach it's max charge depending of the frequency of the A/C generator (is that correct?). But why does the charging/discharging create a pattern with deteriorating voltage? It seems like the charge in the capacitor should just go up and down to the same extream values over and over. And how is this related to the resonance of the circuit at all? For some reason the frequency of these oscillations is the same as the resonance frequency, but it seems like this shouldn't be the case at all, I would have thought they would be very different values.

Let's see if I can help...

Your first comment is regarding the decay in the oscillations in the LC circuit. It's been awhile since I was a lab demonstrator, so I don't recall the setup exactly for this experiment, however, it sounds like there is a resistance in the circuit (whether there is a resistor there or not... it could be the fact that you have a "real" inductor which consists of a length of wire and will have some resistance). In that case, you have a damped harmonic oscillator system. The inductor and capacitor act as an oscillator (where the capacitor acts like a mass, and the inductor provides a restoring force like a spring), but the resistance provides a means for energy to leave the system, so the oscillations will decay. An analogous mechanical system would be a mass on a spring in molasses... for an entertaining visual. As for what you think it "should" do, you are correct, in that if you had a perfect capacitor and a perfect inductor with no resistances, the system would keep on oscillating forever.

As for resonance... this seems to confuse a lot of people every year. Here's a question and answer from a couple of years ago just to define what we are talking about:

Q: I am a little confused about frequency. What exactly is the resonant frequency and how and why does it effect the current compared to the frequency?

A: In an AC circuit, depending on the components in the circuit, the impedance (a sort of complex resistance) may depend on the frequency. Thus, if an oscillating voltage is applied, the amplitude of the current (peak amount of current that flows) may also depend on the frequency of the applied voltage. This is how the current depends on frequency.

In the case of an LRC circuit, there is a phenomenon called "resonance", whereby the amplitude response of the current has a maximum at the so-called "resonant frequency". Resonance is a common in oscillatory systems, where the amplitude response of the system, be it a mechanical system, or an electrical system, has a maximum at some frequency. At frequencies other than the resonance frequency the amplitude response of the system (the amplitude of the current in this case) will be smaller.

Now, in an electrical system, the resonance will be established by the capacitor and the inductor (the resistance provides the damping), and the values of these components will give the resonance frequency, which is the natural frequency at which the oscillating system operates at, and where it will have the greatest response if you drive it with a frequency. If you input an oscillating voltage to an LC circuit, the amplitude of the current should be very small until you get near the resonance frequency. However... I'm not sure if this is what you did?

If you input a square wave at a much much lower frequency than the resonance, then it will be like turning on and off the voltage, and you would see oscillations at the resonance frequency, which would decay until the next cycle when they would start again. Perhaps this is what you did in the lab.

I hope this is somewhat helpful. Without knowing what you did in lab, I can't be sure where the confusion lies. Please do comment (click the # comment link below... or email if that doesn't work) to follow up!