Ch 30: Alternating CurrentWorksheetSee all chapters
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Concept #1: RMS Current and Voltage


Hey guys, in this video we're going to talk about these things called RMS values. Since current and voltage are changing continuously in AC circuits it's helpful to know average quantities of the current and the voltage, but it turns out that it's not actually useful to know the average specifically but more useful to know a type of average called the RMS, alright? Let's get started. A very common question to answer is in alternating circuit, sorry, alternating current circuits, what is the average of the current and average of the voltage? I have above me the two graphs of the voltage versus time and the current versus time, and you guys can see that for every peak I have above the horizontal for voltage, which represents a positive voltage or a voltage with a particular polarity, I have a symmetric or identical peak below the horizontal, which represents a negative voltage or a voltage with the opposite polarity. The same thing happens when I look at the current versus time graph above me, for every peak that I have thatÕs above the horizontal axis, which represents a positive current or a current in one direction, I have a symmetric or identical peak below the horizontal, which represents a current that's negative or current in the opposite direction, okay? And you're going to alternate between these positive and negative peaks forever, so what do you guys think is the average value of the voltage and the current it's going to be 0, and it's always going to be 0 because these positive peaks are always going to alternate with the negative peaks and the effect is always going to cancel itself out, okay? So, a much better average quantity is called it RMS value, okay? RMS is an acronym and it stands for the root mean squared, so the RMS value is the root mean squared value. Now, I space these words, because there's a little bit missing here, so that we really understand what an RMS is. It's the root of the mean of the squared value, that's what RMS means, that's what root means squared means; it's the root of the mean of the squared. So, I want to know the RMS value of X for instance, X can be anything, it could be voltage, it could be current, it could be power, it could be whatever. To find the RMS value, I first square it, so that's the first step, right? Then I average it or I take the mean of it, then I take the square root or the root, so it is the root of the mean of the squared, right? The root of the mean of the squared. This is very important that you do it in this order because if, for instance, you were to just average X, that's not going to be the same value as the RMS, because, for instance, what's the average of current? 0. So, if you then square that average it's still 0, and if they would then take the square root of that average squared it's still 0, okay? So, it has to be the root of the mean of the squared, alright? Now, luckily there are very easy relationships between the RMS currents and the maximum current, and the RMS voltage and the maximum voltage. The RMS of either is just the maximum value divided by square root of 2, or you could rewrite it and say that the maximum value of either is just the square root of 2 times the RMS value, okay? Let's do a quick example to illustrate this. If the RMS voltage of an outlet in the US is 120 volts, what is the maximum voltage of an outlet? If you complete a simple circuit with this AC source by connecting a 12 ohms resistor, what is the RMS and the maximum current in this circuit? Okay? So, three questions here, what's the maximum voltage given the RMS voltage? What's the maximum current? And what's the RMS current? So, letÕs take this one at a time. We know that V RMS equals 120 volts and we want to figure out what the maximum is, all we need to do is use this equation right here. The maximum voltage is just going to be the square root of 2 times the RMS voltage, so that's square root of 2 times 120 volts, which is about 170 volts, okay? One question down. Now we want to know what are the RMS and the maximum currents, I'm going to find the maximum current first, okay? Now, imagine for a second what this circuit looks like. I have an alternating source and I have a resistor, whatever voltage is across this alternating source, is the voltage across the resistor by KirchoffÕs loop rule, right? So, when the voltage is V MAX across the AC source, what's the voltage across the resistor? V MAX. So, what is the maximum current? It's going to be the current when the voltage across the resistor is V MAX, okay? So, we can just say that the maximum current is going to be V MAX divided by R, this is just Ohm's law applied to the resistor when the voltage across the resistor is V max, okay? So, this is going to be 120 divided by 12, sorry, not 120, 120 is the RMS voltage, 170 divided by 12, which is about 14.2 amps. Now, that I know the maximum current I can use this equation to find the RMS current very easily. The RMS current is just going to be the maximum current divided by the square root of 2, that's going to be 14.2 divided by the square root of 2 and that is about 10, okay? That wraps up this discussion on the RMS values for voltages and currents. Alright guys, thanks for watching.

Practice: An AC source operates with a 0.05 s period. 0.025 s after the current is at a maximum, the current is measured to be 1.4 A. What is the RMS current of this AC circuit?