Power in AC Circuits - Online Tutor, Practice Problems & Exam Prep

In AC circuits, only resistors emit average power, while capacitors and inductors store energy without net power output. The average power is calculated as half the maximum power, represented by the equation P_avg = 1/2⁢V_max⁢I_max. This leads to the relationship P = Irms⁢R for average power, emphasizing the significance of RMS values in AC circuits.

Power in AC Circuits

Hey guys. In this video, we're going to talk about power in AC circuits. What elements are emitting power? What elements are not emitting power? What the average power is, and things like that. Alright? Let's get to it.

In AC circuits, the only element to have an average power not equal to 0 is? What do you guys think it is? It's the resistor. This is because whatever energy enters a capacitor or an inductor equals the energy that leaves it. Capacitors and inductors are elements that store energy. A capacitor stores charge to store electric potential energy, and an inductor stores its current to store magnetic potential energy. Okay, but either way, they only store energy. The resistor is what's actually bleeding the energy from the circuit. Okay? So if you were to plot the other elements' power as a function average, all the peaks would cancel all the valleys, and you would have no power on average. Okay? The maximum power of a resistor is going to be that maximum voltage across the resistor times the maximum current. Okay?

Now, as a function of time, you can say that the power equals the current as a function of time squared times the resistance. This gives us the following graph of power and current versus time. You can see that current, like we expect, is going to have no average value because anything that's positive, any peak above the horizontal cancels with the negative peaks below the horizontal, but power stays above the horizontal. It just bounces above the horizontal. So it's always positive and therefore it has nothing to cancel it out. So, on average, it is absolutely not 0. Okay?

The average power emitted by an AC circuit is going to be one-half of the maximum power. This is because the power's peaks are completely symmetric. So the average is going to be one-half of the maximum. So it's going to be one-half \( V_{\text{max}} \times I_{\text{max}} \) and if you substitute the maximum values for their RMS values you find that this actually equals \( V_{\text{rms}} \times I_{\text{rms}} \). So the average power depends upon the RMS voltage current, which is an interesting result. Right? We're not talking about the RMS power here. We're talking about the true average of the power and it doesn't depend upon the average of the voltage or the average of the current. It can't because those are 0, but it does interestingly enough, depend upon the RMS values of the voltage and the RMS value of the current. Okay?

Let's do a quick example. An AC source operating at a maximum voltage of 120 volts is connected to a 10 ohm resistor. What is the average power emitted by this circuit? Is it equivalent to the RMS power, which would be \( I_{\text{rms}}^2 \times R \)? Okay? Don't forget that the average power we're just going to say is one-half times the maximum voltage times the maximum current. Okay, so first, what's the maximum current? Well, that's just the maximum voltage which we know, divided by R, which is 120 volts, that's the maximum voltage, right? Divided by 10 ohms, which is 12 amps. Okay? So we can say that the average power is one-half times 120 volts, right, which is the maximum voltage times 12 amps, which is the maximum current and that whole thing is going to equal 720 watts. Now if I take this maximum current I can then say that the RMS current is the maximum current over the square root of 2, which is \( \frac{12}{\sqrt{2}} \), which is going to be 8.49 amps. That I can take and I can find \( I_{\text{rms}}^2 \times R \) is \( 8.49^2 \times 10 \) which is indeed 720 watts. So, yes, that does match up. Okay? And this is something we touched upon earlier that the average power depends upon RMS values. So this is a form of power emitted by a resistor. We should absolutely be able to just plug in RMS values for it and get the average power out.

Alright guys, that wraps up our discussion on power in AC circuits. Thanks for watching.

Here’s what students ask on this topic:

What is the average power emitted by a resistor in an AC circuit?

The average power emitted by a resistor in an AC circuit is given by the equation:

$P_{\mathrm{avg}}=\frac{1}{2}{V}_{\max }{I}_{\max }$

This can also be expressed using the RMS values of voltage and current:

$P=V_{\mathrm{rms}}{I}_{\mathrm{rms}}$

In this context, RMS stands for Root Mean Square, which is a statistical measure of the magnitude of a varying quantity. The average power depends on these RMS values because they represent the effective values of voltage and current in AC circuits.

Why do capacitors and inductors not emit average power in AC circuits?

Capacitors and inductors do not emit average power in AC circuits because they only store energy rather than dissipate it. A capacitor stores energy in the form of electric potential energy by accumulating charge, while an inductor stores energy in the form of magnetic potential energy by maintaining current. The energy that enters these components is equal to the energy that leaves them, resulting in no net power output. Therefore, the average power for capacitors and inductors is zero.

How is the maximum power of a resistor in an AC circuit calculated?

The maximum power of a resistor in an AC circuit is calculated using the maximum voltage across the resistor and the maximum current through it. The formula is:

$P=V_{\max }{I}_{\max }$

Here, $V_{\max }$ is the maximum voltage and $I_{\max }$ is the maximum current. This product gives the peak power that the resistor can dissipate at any instant in time.

What is the relationship between average power and RMS values in AC circuits?

The average power in AC circuits is directly related to the RMS (Root Mean Square) values of voltage and current. The relationship is given by:

$P=V_{\mathrm{rms}}{I}_{\mathrm{rms}}$

This equation shows that the average power depends on the effective values of voltage and current, which are represented by their RMS values. RMS values are used because they provide a measure of the equivalent DC value that would produce the same power dissipation in a resistor.

How do you calculate RMS current from maximum current in an AC circuit?

To calculate the RMS current from the maximum current in an AC circuit, you use the following formula:

$I_{\mathrm{rms}}=\frac{I_{\max }}{\sqrt{2}}$

Here, $I_{\mathrm{rms}}$ is the RMS current and $I_{\max }$ is the maximum current. This formula accounts for the fact that the current in an AC circuit varies sinusoidally, and the RMS value provides a measure of the effective current that would produce the same power dissipation as a DC current.