Alternating Voltages and Currents - Online Tutor, Practice Problems & Exam Prep

Alternating current (AC) circuits involve currents that change direction, unlike direct current (DC) circuits, which flow in a single direction. AC is produced by alternating voltages, typically sinusoidal, represented as $v=Vmax\cos (\omega t)$. The current mirrors this sinusoidal pattern, with angular frequency $\omega =2\pi f$. Understanding these principles is crucial for analyzing AC circuits effectively.

Alternating Voltages and Currents

Hey, guys. In this video, we're going to start talking about alternating currents and circuits that contain alternating currents, which we would call AC circuits. Alright, guys. Let's get to it. Now before, all we had considered were direct currents, which are currents that only move in a single direction. Circuits containing direct currents, we call DC circuits. And a very simple example of a DC circuit was a battery connected to a resistor. That battery with a constant voltage would produce a constant current through that resistor that only pointed in a single direction. Okay? Now, when we consider alternating currents, which are currents that move in alternating directions, we need to consider different voltages. Obviously, a constant voltage like a battery across a resistor cannot produce a current that moves in anything other than a single direction. Now what we mean by alternating directions is we mean back and forth, left to right, or any two opposite directions up and down, etcetera. Okay? Alternating currents are not produced by constant voltages; they are produced by alternating voltages. And the only alternating voltage we're going to consider is this sinusoidal alternating voltage. Okay? Given by V_max∙cos⁡t. Now I want to talk a little bit about notation. Notice that this V is capitalized and this V is lowercase. A very common type of notation is that any value that changes with time is going to be given by the lowercase letter that typically represents that value. So V for voltage, I for current, P for power, they are typically given by the lowercase letter of that value. Now the maximum value or the amplitude of this oscillation is typically given by the capital of that value. I want to be extra specific and I'm giving often the time dependence explicitly, and I will often explicitly denote whether it is the maximum value. This is because notation varies wildly between professors and between textbooks, so I want everything to be super clear. Okay? Now something that's very, very important. One of the most fundamental things to remember about alternating current circuits, which I'll from now on call AC circuits, is that the alternating voltage always produces a particular type of alternating current. It's going to match the exact same sinusoidal pattern of that alternating voltage. So we'll say that the change of the current with respect to time is going to be some maximum current times cos⁡ωt. Okay? It matches that same exact sinusoidal pattern. This is a cosine. This is a cosine. Okay? Now, what is omega? Omega is just the angular frequency of these alternations. Okay? Remember that omega is related to a linear frequency by 2πf. Okay? So, if I were to say that some alternating source, by the way the symbol in a circuit diagram for an alternating source is this, if I have some alternating source that will produce a current in this direction and then a current in this direction and it flips directions twice a second, that tells me that the frequency is 4 hertz. Okay. The reason is that if it flips in this direction twice a second, then it'll flip in that direction and then back 4 times a second, etcetera. Either way, that'll tell me what the frequency is, and then I can find my angular frequency. Okay? That's what the angular frequency is. Now the current in an alternating circuit is always going to be of this form because alternating current circuits, AC circuits, are what we call driven circuits. Okay? The angular frequency of the source drives the current to look like this, and it will always look like this. And this is going to be a common theme as we go through these discussions on AC circuits. Okay? So here's a little plot of what the voltage and the current is going to look like in an AC circuit. Okay? It's a cosine, so it starts at some maximum and then decreases. Exactly the same for current as it does for voltage. And they're just going to oscillate between the positive of a maximum value and the negative of a maximum value. What the negative voltage means is it's just a reversed polarity, and what the negative current means is it's just a current that points in the opposite direction. Okay? Let's do a quick example. In North America the frequency of AC voltage coming out of a household outlet is 60 Hertz. If the maximum voltage delivered by an outlet is 120 volts, what is the voltage at 0.4 seconds? Okay. Now this frequency is given in hertz. Hertz are the units for linear frequency. Sometimes this can be a little bit ambiguous as to what the question means. Is it linear frequency or angular frequency? The units for linear frequency are hertz and the units for angular frequency are second inverse. Okay. So that's typically how you can tell them apart. So if the frequency is 60 2π×60 hertz, which is going to be 377 second-inverse. Okay, and all we have to do is apply our equation for voltage as a function of time to find what the voltage is at a particular time. This is our equation for the voltage at any time. Our maximum voltage we're told is 120 volts. And this is going to be omega which is 377 times our time, which is 0.04 seconds. And this whole thing is going to equal negative 97 volts. Okay? So the magnitude of the voltage is 97 volts, and the negative implies that the polarity is opposite of what it originally had. Alright, guys. That wraps up this introduction into alternating voltages and alternating currents. Thanks for watching.

An AC source produces an alternating current in a circuit with the function $i\left(t\right)=\left(1.5A\right)\cos\left\lbrack\left(250s^{-1}\right)t\right\rbrack$. What is the frequency of the source? What is the maximum current in the circuit?

• A. f=39.3 Hz
• B. f=250 Hz
• C. f=393 Hz
• D. f=1570 Hz

AC Circuit Graphs

Hey guys, let's do an example about AC circuits. Current and voltage in an AC circuit are graphed in the following figure. What are the functions that describe these values? Okay. So just remember that voltage as a function of time in AC circuits is going to be equal to some maximum voltage produced by the source times cosine of omega t where omega is the frequency of the source, and I of t, the current produced by the source, is going to be some i_max, which is the maximum current produced by the source, times cosine of omega t. So in order to find these functions, all we need to do is find what these maximum values are, right? And then what the angular frequency of this oscillation is. Once we know those three values, right, the angular frequency for both of these functions is going to be the same. Once we know those three values we can plug them into our functions and be done with it. Okay?

Now remember that the maximum voltage and the maximum current according to these equations right above me are just the amplitudes of these oscillations. So what's the amplitude of the voltage oscillation? 11 volts. This is V_max. What's the amplitude of the current oscillation? It's 2 and a half amps. This is actually negative i_max. That's why this says negative 2 and a half amps because you're at the negative amplitude. The only question remaining is what's the angular frequency? Well, we are told that from this point up here to this point down here takes 0.05 seconds. Not half a second, 0.05 seconds. Okay?

Well, this distance right here is half of a cycle. A full cycle would be starting from the amplitude, coming down to the negative amplitude, and going back up to the positive amplitude where you started. Going from to the negative amplitude is half of a cycle and that takes 1 half of a period. So that time 0.05 seconds, is actually half of the period. So if you say one half of the period is 0.05 seconds, then we can just multiply this 2 up to the other side, and we can say that the period is 0.1 seconds. Okay. 0.05 times 2 is just 0.1. Now we want to find angular frequency from period. Okay. Remember that the angular frequency is defined as 2πf which is the same as 2π/T. So this is 2π<\/mn>0.1<\/mn> seconds<\/mtext><\/mrow><\/mfrac><\/math> which is gonna be 62.8 inverse seconds. Okay?

So now we know all three of our values. We know that the angular frequency is 62.8 inverse seconds, we know that the maximum voltage is 11 volts, and we know that the maximum current is 2 and a half amps. So all we have to do is plug in those three values to the two equations above me. And we'll say that the voltage as a function of time is: V(t) = 11 volts × cos(62.8⁻¹ seconds × t). The current as a function of time is the maximum current, which is 2 and a half amps, times cosine of, once again, the angular frequency which is 62.8⁻¹ seconds × t). And these are our answers.

All right, guys. Thanks for watching.

The current in an AC circuit takes 0.02 s to change direction. What is the angular frequency of the AC source?

• A. 6.4×10⁻³ s⁻¹
• B. 0.04 s⁻¹
• C. 0.25 s⁻¹
• D. 157 s⁻¹

Here’s what students ask on this topic:

What is the difference between AC and DC circuits?

AC (Alternating Current) circuits involve currents that change direction periodically, typically following a sinusoidal pattern. This is represented by the equation $v=V_{\max }\cos \left(\omega t\right)$. DC (Direct Current) circuits, on the other hand, have currents that flow in a single direction, usually produced by a constant voltage source like a battery. In DC circuits, the current remains constant over time. Understanding these differences is crucial for analyzing and designing electrical systems effectively.

How is the angular frequency related to the linear frequency in AC circuits?

The angular frequency $\omega$ in AC circuits is related to the linear frequency $f$ by the equation $\omega =2\pi f$. The linear frequency $f$ is measured in Hertz (Hz), which indicates the number of cycles per second. The angular frequency $\omega$ is measured in radians per second (s^-1), and it describes how quickly the current oscillates in terms of radians.

What is the significance of the maximum voltage in an AC circuit?

The maximum voltage, often denoted as $V_{\max }$, in an AC circuit represents the peak value of the voltage waveform. It is the highest voltage that the AC source can deliver at any point in time. This value is crucial for determining the amplitude of the sinusoidal voltage and current waveforms. For example, in a household outlet in North America, the maximum voltage is typically 120 volts, which means the voltage oscillates between +120V and -120V.

How do you calculate the voltage at a specific time in an AC circuit?

To calculate the voltage at a specific time $t$ in an AC circuit, you can use the equation $v=V_{\max }\cos \left(\omega t\right)$. Here, $V_{\max }$ is the maximum voltage, and $\omega$ is the angular frequency. For example, if the maximum voltage is 120V and the angular frequency is 377 s^-1, the voltage at $t=0.04$ seconds would be $v=120\cos (3770.04)=-97$ volts.

What is the role of angular frequency in AC circuits?

The angular frequency $\omega$ in AC circuits determines how quickly the current and voltage oscillate. It is given by the equation $\omega =2\pi f$, where $f$ is the linear frequency. The angular frequency is measured in radians per second (s^-1) and is crucial for defining the time-dependent behavior of the sinusoidal voltage and current waveforms in the circuit. It ensures that the current and voltage follow the same sinusoidal pattern, maintaining the AC circuit's functionality.