Topic open: AC Circuits

The following topic is now open for questions: RC, RL, LC, LRC circuits (and any AC circuits)

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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!

Question: difference between RC, RL, LC

How do you tell the difference between RC, RL, LC circuits?

Each of these circuits, and what they qualitatively do in DC and AC circuits...

RC - a resistor and capacitor in series. Exhibits charging behaviour with characterisitic time constant with DC voltage source. Acts as a high pass filter (allows high frequency currents, but not low frequency currents) in AC circuits.
RL - a resistor and inductor in series. Acts as a short with a DC voltage source, but smooths out rapid variations in current. Acts as a low pass filter (allows low frequency currents, but not high frequency currents) in AC circuits.
LC (and RLC) - an inductor and capacitor (and resistor) in series. If initially charged, has oscillitory behaviour (damped if also has a resistor). Has resonant behaviour with AC driving voltage (damped if also has a resistor).

Question: LC in parallel

How to calculate the resistance of a capacitor and an inductor connected in parallel?
(-XcXL/(XL-Xc))^2 ?

This is a little bit trickier that considering elements in series because the inverse of the impedances have to be added. I'm pretty sure it is beyond the scope of this course to need to do this calculation, but I will put it here anyway. I'm also not sure how you would arrive at the answer without using the appropriate complex represenations of the impedance, so I will use them but try to be careful to explain.

So, for elements in parallel:

1/Z_tot=1/Z_C+ 1/Z_L+ 1/Z_R ...[1]

We will consider just an LC circuit (with ideal components), so R=0. Now, in complex notation Z_Cand Z_L are:

Z_C=1/iωC ...[2]
Z_L=iωL ...[3]

So substituting [2,3] into [1]:

1/Z_tot=iωC+1/iωL

or, using the fact that 1/i=-i:

1/Z_tot=iωC-i/ωL
1/Z_tot=i(ωC-1/ωL) ...[4]

so, we can now replace 1/ωC with X_C and ωL with X_L:

1/Z_tot=i(1/X_C-1/X_L) ...[5]

Now we can simplify the part in the brackets, giving us:

1/Z_tot=i(X_L-X_C)/X_CX_L ...[6]

or taking the inverse of the fraction on both sides (and once again using 1/i=-i):

Z_tot=-iX_CX_L/(X_L-X_C) ...[7]

Now, if we had also had a resistor in parallel in the circuit (R), we would end up at equation [7] with both real and imaginary parts. To find X_tot we would have to find the magnitude of the resultant phasor. Since Z_tot is purely imaginary, we need only to find the magnitude of this number, which we can express as:

X_tot=-X_CX_L/(X_L-X_C) ...[8]

Which is pretty much what I think you suggested. Note that if we hadn't used the imaginary numbers we would not have been able to get the right signs and would have come up with a different answer.

Looking back at eqn. [7], let's think about what this represents... In the imaginary plane Z_tot would be a phasor pointing along the y-axis. For values of X_L and X_C which make Z_tot positive, the impedance could be considered "inductive", and inversely if Z_tot were negative, the impedance could be considered "capacitive".

Let's consider two extreme cases...

If the frequency is high, X_L is very large, and X_C is very small (let's say negligible). In the numerator of Z_tot the frequency cancels out, so this is a constant. If we look at the denomenator, X_L >> X_C, so the denomenator is approximately X_L. Since X_L is large, Z_tot will be small and negative (capacitive). In fact if you substitute in equations [2,3] ignoring X_C in the denominator, you will recover Z_C. This is consistant with the inductor providing a break in the circuit, and the capacitor providing a short.

Similarly, if the frequency is very low, X_C becomes very large, and X_L very small. In this case, the result is small and positive (inductive), and indeed again, if you work it out you will recover Z_L as Z_tot. This is consistant with the capacitor providing a break in the circuit, and the inductor providing a short.

To summarize:

ω→∞ Z_tot→Z_C and current flows through the capacitor
ω→0 Z_tot→Z_L and current flows through the inductor

These limiting cases can be worked out simply as a thought experiment (consider which branch of the circuit has a small resistance and which has a large resistance) and are often more insightful than getting through the algebra.

Hope this helps.

Topic Open: RC, RL, LC, LRC circuits

The following topic is now open for questions: RC, RL, LC, LRC circuits (and any AC circuits)

To pose a question, please post a comment to this post by clicking on the comments link below.

Question: Impedance of LR circuit with "real" inductor

Just had a question about LR circuits. Just say there was a circuit consisting of a resistor and inductor. And just say the inductor had a resistance of its own. How do you find the total impedance of the circuit. This was similar to one of the CAPA questions. My friend told me to find the impedance of just the resistor and inductor and then add the resistance of the inductor to Z. I am confused on why you do this? Why couldn't you just add the resistance of the inductor and resistor first, then use the formula Z = sqrt(R^2 + XL ^2), R being the resistance of the inductor and resistor combined?

Actually, you are right! You can (and should) just add the resistance of the inductor to the resistance of the resistor then find the impedance including the inductive term.

Here's how it works. A real inductor will have some inherent resistance due to the fact that it is a (usually fairly large) coil of wire. To analyze a circuit, we would replace a "real" inductor with and "ideal" inductor and a resistor (to represent the pure resistance of the inductor). So, in an LR circuit you would go from having 1 resistor and 1 "real" inductor in series to having 2 resistors in series and one "ideal" inductor also in series. You would then find the net resistance of the circuit, and this would be "R" in the formula you quote above.

From this you would find the magnitude of the inductance from Z=sqrt(R²+X_L²)
(this formula comes from the fact that R and X_L are 90^o out of phase, so the magnitude is the hypotenuse of the two "phasors" see: Basics of AC circuits/RLC circuit example). Since R and X_L do not add linearly, I do not think you should get the right answer if you go about it in the opposite order (except maybe in some special circumstance).

EDIT: A similar question was also asked regarding an LRC circuit. In this case, the methodology is the same. The inductor with a resistance is replaced by a resistor and an inductor in series. The resistance of the whole cicuit is calculated, and then the magnitude of the impedance is determined. In the case of an LRC circuit, Z is given by:

Z=sqrt(R²+X_L²+X_C²)

Otherwise, the whole problem is the same.

Hope this helps!