**Concept:** 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, only considered with 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 the 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 the 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, etc, 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 times the cosine. 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're 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 the 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 all from now on call it 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 currents, the change of the current with respect to time is going to be some maximum current times cosine of Omega t, 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 2 PI times f, okay? So, if I were to say that some alternating source, by the way the symbol in a circuit diagram for alternating sources 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, is that, if it does, if it flips in this direction twice a second then it will flip in that direction and then back four times a second, etc. Either way, that will 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 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 household outlets 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 Hertz, then the angular frequency, which is 2 PI times f, is going to be 2 PI times 60 Hertz, which is going to be 377 inverse seconds, okay? And all we have to do is apply our equation for a 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 it's in the-, sorry, it has the opposite polarity of what it originally had. Alright guys, that wraps up this introduction into alternating voltages and alternating currents. Thanks for watching.

**Problem:** An AC source produces an alternating current in a circuit with the function 𝑖(𝑡) = (1.5 𝐴) cos[(250 𝑠 −1 )𝑡]. What is the frequency of the source? What is the maximum current in the circuit?

**Example:** 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 iMAX, 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 2.5 Amps, this is actually negative iMAX, that's why this says negative 2.5 Amps, because you're at the negative amplitude. The only question remaining is what's the angular frequency? Well, we're told that from this point up here to this point down here, takes half-,sorry, 0.05 seconds, not half a second, point 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 the positives of the negative amplitude is half of a cycle, and that takes 1/2 of a period, so that time 0.05 seconds, is actually half of the period. So, if you say 1/2 of the period is 0.05 seconds, then we can just multiply this to up to the other side, and we can say that the period is point-, sorry, yes 0.1 seconds, okay? 0.05 times 2 is just 0.1. Now, we want to find angular frequency from the period, okay? Remember, that the angular frequency is defined as 2 PI f, which is the same as 2 PI over T so this is 2 PI over 0.1 seconds, which is going to be 62.8 inverse seconds, okay? So, now we know all three of our values, we know that the angular frequency of 62.8 seconds inverse, we know that the maximum voltage is 11 volts and we know the maximum current is 2.5 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 mat-, sorry, the current as a function of time is going to be 11 volts, which is the maximum, sorry, this is the voltage as a function of time, the voltage as a function of time is the maximum voltage which is 11 volts times the cosine of the angular frequency 62.8 inverse seconds times time and the current as a function of time is the maximum current, which is 2.5 Amps, right? The amplitude of oscillations times cosine of, once again, the angular frequency, which is 62.8 inverse seconds times time and these are our answers. Alright guys, thanks for watching.

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