31. Alternating Current

Alternating Voltages and Currents

1

concept

## Alternating Voltages and Currents

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Hey guys, In this video, we're going to start talking about alternating currents and circuits that contain alternating currents, which we would call a C circuits. Alright, guys, let's get to it now. Before all we had considered were direct currents which air currents that Onley move in a single direction, Circuits containing direct currents we call D C circuits and a very simple example of the D. C circuit was a battery connected to a resistor. That battery, with a constant voltage, would produce a constant current through that resistor that Onley pointed in a single direction. Now, when we consider alternating currents which air 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 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 voltage is they are produced by alternating voltage is and the Onley alternating voltage we're going to consider. Is this Sinus soy idol 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 in this V is lower case. A very common type of notation is that any value that changes with time is going to be given by the lower case letter that typically represents that value. So v for voltage I for current P for power, they're typically given by the lower case letter of that value. Now the maximum value or the amplitude of this oscillation is typically given by the capital of that value. I wanna 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 all from now on called a C circuits, is that the alternating voltage always produces a particular type off alternating current. It's going to match the exact same Sinus, or it'll pattern off that alternating voltage, so we'll say that the current the change of the current with respect to time is going to be some maximum current times. Cosine of omega t it matches that same exact Sinus oil pattern. This is a cosine. This is a co sign. Now, what is Omega? Omega is just the angular frequency of these Alternations. Okay, remember that Omega is related to a linear frequency by two pi times f. Okay, so if I were to say that some alternating source, by the way, the sim, the symbol in a circuit diagram for an 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 four hertz. Okay. The reason is is that if it does, it flips in this direction twice a second. Then it'll flip in that direction and then back four times a second, etcetera. Either way, that will tell me what the frequency is. And then I confined 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 A C circuits or what we call driven circuits. Okay, The angular frequency of the source drives the current toe look like this, and it will always look like this. And this is gonna be a common theme as we go through these discussions on a C circuits. Okay, so here's a little plot off. What the voltage and the current is going to look like in an A C 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 gonna oscillate between the positive of the maximum value and the negative of the 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 a C voltage coming out of a household outlets is hertz. If the maximum voltage delivered by an outlet is 120 volts, what is the voltage at 0.4 seconds? Okay, Now, this frequency has given it hurts, hurts or 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 hurts. And the units for angular frequency, Our second inverse. Okay, so that's typically how you can tell them apart. So if the frequency is 60 hertz than the angular frequency, which is two pi times f is gonna be two pi times 60 hertz, which is gonna be 377 in verse seconds. And all we have to do is apply our equation for voltage as a function of time to find what the voltages 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 gonna be omega, which is 3 77 times our time, which is 0.4 seconds. And this whole thing is going to equal negative 97 volts. Okay, so the magnitude of the voltages 97 volts and the negative implies that it's in the Sorry. It has the opposite polarity off. What it originally had. Alright, guys, that wraps up this introduction in tow. Alternating voltage is an alternating currents. Thanks for watching.

2

Problem

ProblemAn AC source produces an alternating current in a circuit with the function ?(?) = (1.5 ?) cos[(250 s^{−1} )?]. 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

3

example

## AC Circuit Graphs

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Hey, guys, let's do an example about a C circuits current and voltage in in a see circuit or graft in the following figure. What are the functions that describe these values? Okay, so just remember that voltage as a function of time in a sea circuits, is going to be equal to some maximum voltage produced by the source times cosine of Omega T War Omega is the frequency of the source and I of t the current produced by the source is gonna 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 the's maximum values are, right. And then what The angular frequency off this oscillation is, Once we know those three values, right, the angular frequency for both of these functions is gonna 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 amplitude of these oscillations. So what's the amplitude of the voltage oscillation. 11 volts. This is the max. What's the amplitude of the current oscillation? It's 2.5 amps. This is actually negative. Imax. That's why this is 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 0.0.0.5 seconds, not half a second 0.5 seconds. Okay, well, this distance right here is half of a cycle off. 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 positive to the negative. Amplitude is half of a cycle, and that takes one half of a period. So that time 10.5 seconds is actually half of the period. So if you say one half of the period is 10. 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. 0.5 times two is just one. Now we wanna find angular frequency from the period. Okay, remember that the angular frequency is divided is defined this two pi f, which is the same as two pi over t. So this is two pi over 0.1 seconds, which is gonna be 62.8 inverse seconds. Okay, so now we know all three of our values. We know that the angular frequency 62.8 seconds in verse. We know that the maximum voltage is 11 volts, and we know that 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 will say that the Matt sorry. The current as a function of time is going to be 11 volts, which is the maximum. Sorry, this is the voltage is a function of time. The voltage is a function of time. Is the maximum voltage which is 11 volts times the co sign of thank you Lor frequencies 62 8 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 seconds times time. And these are our answers. All right, guys, Thanks for watching.

4

Problem

ProblemThe 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}