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

Inductors in AC circuits exhibit unique behaviors where the voltage leads the current by 90 degrees, as described by the equation for voltage across an inductor: $I^{\max }\omega L\cos (\omega t+\frac{\pi }{2})$. The inductive reactance, akin to resistance, is given by $X=\omega L$. Understanding these relationships is crucial for analyzing AC circuits effectively.

Inductors in AC Circuits

Hey guys. In this video, we're going to talk about inductors and the role that they play in AC circuits. Alright? Let's get to it. Now remember that the current in an AC circuit at any time is going to be given by the equation that by now we've seen a bunch of times. Okay? Okay, this is going to tell us that the current is simply oscillating with an angular frequency of omega between some value positive i_max and some value negative i_max. Now the question is, how does the voltage across the inductor look? Well, remember that the voltage across an inductor, which we saw during our discussion of Faraday's law, is the inductance times delta I over delta t, which is the rate at which the current is changing. Okay. Now I can't show you how but using calculus you guys can arrive at the answer that the rate at which the current is changing looks like this.

So if I multiply this by the inductance, then I get the voltage across an inductor in an AC circuit at any time. This is going to be i_max times omega times L times cosine of omega t + π/2. Okay, so once again, remember the voltage across a resistor. The voltage across a resistor looks like i_max times R times cosine of omega t. So the angle that it operates at omega t is different than the angle that the voltage across the inductor operates at. This is some other angle theta prime which is omega t + π/2. Okay? So the current and the voltage across the resistor are in phase, right, their plots line up, but the current and the voltage across an inductor are not going to line up. They're going to be out of phase. If we plot the current across an inductor and the voltage across an inductor, you can see that the voltage across an inductor actually leads the current by 90 degrees. Okay? What's happening here is that the voltage is deciding to go up at a time when the current is 0. Okay? Then at a future time, the current starts to go up. But at this point, the voltage has already peaked. Then at a future time, the current peaks. Right? It's trying to match what the voltage is doing. But at this point, the voltage is already decreasing. So then at a future time, the current decreases but the voltage has already bottomed out. So you see the current is trying and trying and trying to match the voltage, but it's lagging behind, or we can say that the voltage leads the current. Either one is fine. Okay? Now, the maximum voltage across an inductor is going to look like i_max times omega L. This looks a lot like Ohm's Law where we said that the voltage across the resistor was I times the resistance. There appears to be a resistance-like quantity of omega L. That resistance-like quantity for capacitors we called the capacitive reactance. Now we are calling it the inductive reactance for inductors. The units are still ohms. Right? The same unit as for resistance. Alright. Let's do an example. An AC power source delivers a maximum voltage of 120 volts at 60 hertz. If an unknown inductor is connected to the source and the maximum current in the circuit is found to be 5 amps, what is the inductance of the inductor? Okay? What is the maximum current in an inductor circuit? This is just going to be the maximum voltage across the inductor divided by what they both have to share the maximum voltage. That's what Kirchhoff's loop rule says. So this is just going to be V_max, the maximum voltage by the source, divided by XL. And what is that inductive capacitance? Well, that is just going to be omega L. Okay so plugging this into our equation, we can say that i_max is simply V_max/omega L. And our unknown is the inductance. So what I want to do is multiply the inductance up and divide i_max over, and then I have inductance as V_max / omega i_max. Before I can continue though, we need to know what omega is. We're told the linear frequency is 60 hertz. Okay. Remember, this is linear frequency because the units are hertz. So the angular frequency, which is 2πf, is going to be 2π times 60 hertz, which is going to be about 377 inverse seconds. Now we can solve for the inductance. And the inductance is just going to be V_max / omega i_max. V_max is 120, right? Always look at and make sure that's a maximum voltage not an RMS voltage, divided by 377, which was our angular frequency. The maximum current in the circuit was 5 amps. So that's 5 and this whole thing is 0.064 henrys, the unit of inductance. Alright guys, that wraps up our discussion on inductors and AC circuits. Thanks for watching.

Inductors and Graphs

Guys, let's do a quick example. The voltage across and the current through an inductor connected to an AC source are shown in the following graph. Given the information in the graph, answer the following questions. Part A, what is the peak voltage of the AC source? B, what is the frequency of the AC source? And, C, what is the inductive reactance of the circuit?

Part A: What is the peak voltage of the AC source? This is the voltage across the inductor, but we know that the voltage across the inductor is going to exactly match the voltage of the AC source in terms of the maximum voltage because they are connected together. And that's just what Kirchhoff's loop rule says. So, whatever the maximum voltage of the AC source is, that has to be the maximum voltage of the inductor. So, 10 volts is clearly the maximum voltage of this inductor. Therefore, the maximum voltage of the source, V_max, is just 10 volts. Part A, done.

Part B: We want to figure out what the angular frequency is. What this graph tells us about is time. From the time, we can find the period. And from the period, we can find the frequency. That's always how you want to approach these problems with time, as we've talked about before. Any instance of time tells you about the period, and the period can tell you about the frequency. From here to here, which is the time that we're given, is a full cycle. So we know that this time is a full period. The period is 0.1 seconds, and the frequency, which is just one over the period, is 1 0.1 seconds , which is 10 Hertz. Very straightforward and simple.

Part C: What is the inductive reactance in the circuit? Now, in order to do this, we need to figure out how to relate information on the graph to the inductive reactance. The only piece of information we haven't used yet is the maximum current. We know that the maximum current is 2.5 amps. And we know that this is going to be equal to the maximum voltage across the inductor divided by the inductive reactance. The maximum voltage across the inductor is equal to the maximum voltage output by the AC source divided by the inductive reactance. So if we want to solve for the inductive reactance, all we have to do is multiply this up and divide that down. The inductive reactance is going to be 10 volts divided by 2.5 amps, which is going to be 4 ohms. A very straightforward application of the formulas that we've used so far.

Alright, guys. That wraps up this example. Thanks for watching.

Will a frequency f = 60 Hz or ω = 75 s⁻¹ produce a larger max current in an inductor connected to an AC source?

• A. f = 60 Hz
• B. ω = 75 s^-1
• C. Both produce the same amount of maximum current
• D. More information is required.

Here’s what students ask on this topic:

What is the role of inductors in AC circuits?

In AC circuits, inductors play a crucial role by opposing changes in current. The voltage across an inductor leads the current by 90 degrees, which means the voltage reaches its peak before the current does. This phase difference is described by the equation for voltage across an inductor: $V_L=I_{\max }\bullet \omega \bullet L\bullet \cos (\omega t+\frac{\pi }{2})$. This behavior is essential for understanding how inductors affect the overall impedance and phase relationships in AC circuits.

How do you calculate the inductive reactance in an AC circuit?

The inductive reactance (X_L) in an AC circuit is calculated using the formula: $X_L=\omega \bullet L$, where $\omega$ is the angular frequency and $L$ is the inductance. The angular frequency is given by $\omega =\; 2\pi \bullet f$, where $f$ is the frequency in hertz. Inductive reactance is measured in ohms (Ω) and represents the opposition that an inductor offers to the AC current.

Why does the voltage lead the current in an inductor in an AC circuit?

The voltage leads the current in an inductor in an AC circuit because of the inductor's property of opposing changes in current. According to Faraday's Law, the voltage across an inductor is proportional to the rate of change of current through it. Mathematically, this is expressed as $V_L=L\bullet \frac{\mathrm{dI}}{\mathrm{dt}}$. As a result, the voltage reaches its maximum value before the current does, creating a phase difference of 90 degrees. This phase shift is crucial for understanding the behavior of AC circuits containing inductors.

How do you determine the inductance of an inductor in an AC circuit?

To determine the inductance (L) of an inductor in an AC circuit, you can use the relationship between the maximum voltage (V_max), maximum current (I_max), and angular frequency (ω). The formula is: $L=\frac{V_{\max }}{\omega }$. First, calculate the angular frequency using $\omega =\; 2\pi \bullet f$, where $f$ is the frequency in hertz. Then, plug in the values for V_max, I_max, and ω to find the inductance.

What is the difference between inductive reactance and resistance?

Inductive reactance (X_L) and resistance (R) both oppose the flow of current, but they do so in different ways. Resistance is a property of resistors and is constant, regardless of the frequency of the current. It is measured in ohms (Ω) and follows Ohm's Law: $V=I\bullet R$. Inductive reactance, on the other hand, depends on the frequency of the AC current and the inductance of the inductor. It is given by $X_L=\omega \bullet L$, where $\omega$ is the angular frequency. Unlike resistance, inductive reactance causes a phase shift between voltage and current.