Capacitors in AC Circuits with Phasors | Doc Physics

hey if I hook up a capacitor to a battery let's say I've got this D battery here and this sides positive and this side is negative and here I have a parallel plate capacitor and my intention is to put a wire between here and then a wire between here and as soon as I connect that see as soon as I connect that the voltage on the capacitor will gradually increase like this voltage will go like this and approach some some maximum level probably the voltage of the battery if this is a function of time but if I'm going to also want to know what's happening to the current see the current immediately will be huge they'll immediately be a very large current as soon as I connect that there'll be a huge current and then the current will taper off and approach zero where these guys are acting very differently right the current in this circuit is going to be falling to zero and the voltage will be rising up and and that's because we can intuitively understand this because as soon as I make this connection here there is no force against the electrons initially going this direction there's no force initially against the current going that direction but as the capacitor becomes more and more charged than the voltage difference between the capacitor and the battery is approaching zero because well the voltage of the and the voltage of the capacitor is approaching the voltage of the battery so we want to think now about what happens if instead we connect a capacitor to an alternating source I'm going to connect a capacitor to an AC source and I want you to think about what this means see if I connect it to a DC source then the capacitor actually looks like a break in the circuit it will just prevent any current from flowing but if I connect it to a very fast AC source it's almost as if the capacitor isn't there because as soon as I begin to charge the capacitor the current from the source changes directions and then discharges it and can charge it and discharge it and charge and discharge it so it's just as if there were a wire right here so let me summarize that little observation I will say at oh and of course I'm going to use Omega to symbolize how fast I'm sloshing right here this is the frequency the angular frequency of my of my AC power source at low Omega capacitors look like what and at high Omega this is very important so I'm going to ask you to fill those in you have to think about what it means capacitors and I don't need to say look I could say they act like what in a circuit this is very important to think about in a DC circuit a capacitor looks like a break in the circuit and so at low Omega that's what capacitors look like it looks as if the circuit is broken but at high omegas sloshing back and forth shoo-shoo shoo-shoo the capacitor doesn't affect the sloshing at all assuming the capacitor is large enough right so that it doesn't reach its maximum voltage very quickly capacitors that are tiny reach the maximum voltage very quickly and could still impede the flow of charge through this part of the circuit of course there aren't actually charges going that direction or that direction it's just everywhere along here and everywhere along here so as long as you're outside the capacitor it looks as if there's just a wire right there so here we put wire and there we put break or here you could put short and here you could put break a break in a shorter very different things think about that for a moment now as we go into this we're going to have to do a little bit of thumb we're going to have to do a little bit of calculus and I think that's probably very good for us so let's do it you know that the voltage across a capacitor is the charge on the capacitor divided by its capacitance and this charge is actually the integral of the current going into the capacitor so as current goes into a capacitor it fills up the capacitor and that's what gives the capacitor charge on its positive plate so let's write that of again the voltage on the capacitor charge / capacitance so that's going to be well it's going to be the integral well it'll be one over C times the integral of current over time and with my AC generator now I'm going to say that I have current that does depend on time and I'm going to say that the current is well this is current as a function of time it's going to be IMAX times the sine of Omega times T that's my current I plug that in right there and see what happens it's going to be some really enjoyable calculus 1 over C times the integral of IMAX times the sine of Omega T over time IMAX as a constant and pulls out and we just have to do the integral of the sine of Omega T it gives us the negative cosine and then we have to do the chain rule for integrals oh man so we get negative IMAX over C and and then we have cosine of Omega T and we've also got to note what the integral of that sucker is and that would be oh man I think I'm going to get divided by Omega right here so that's really lovely and I could write it just slightly differently I'm going to write it as oh gosh well this is negative cosine here but I happen to know that negative cosine of theta is the sine of theta this is spelled this is really lovely sine of theta minus PI by 2 all right so that's a substitution I'm about to use and I'm just going to clean things up a little bit I'm going to say that this is 1 over Omega times C times the maximum current possible times the sine of well what do we have to put we have to put Omega t minus PI over 2 this equation is a very powerful equation we call this constant here phase-shift and thats related to the term fazer of course and what we're seeing is that the fazer for voltage is not facing the same direction as the fazer for the current so first of all we can identify some maximum values when the sine function is at a maximum it just disappears it becomes 1 so then we say that V cap is 1 over Omega C times IMAX on the cap so that would be the maximum voltage so I can say V Max is 1 over Omega C times IMAX I hope that you believed that particular observation and I want to make a substitution also because I like I mean it's one of my favorite electrical equations I like the idea of a resistor where V is I times R and for a resistor we also find that V Max is I max times the resistance of the resistor so what if capacitor is acted kind of like resistors and and they sort of do because they are sometimes able to act like a line in the circuit a short circuit sometimes they're able to act like a break in the circuit so that's sort of a resistance it's not exactly a resistance we call it a reactance so I want to say that this is V V is equal to I times something so I'm going to define 1 over Omega C to be the reactance of a capacitor and it's abbreviated with the letter X this is the reactance of the capacitor and you'll find that it actually has the same units of resistance this is radians per second and that is the capacitance in farad's and you get units of ohms Wow go figure try that out it's really beautiful so we have this analogy of Ohm's law for well we can write it here the analogy of Ohm's law for capacitors is that voltage on a capacitor is and sorry I have to put Max's in here or rms is in here because it's not an instantaneous equation at all it's going to be I max times the reactive capacitance now my next plan is to show you how this works we've got we've got no I'm gonna have to use a different page because this is really important stuff um I'll set up my axes and I've got Y and X and this is where my phasor is going to live before I go on though I want to remind you this is what the phasor of a resistor looks like starts here and if the voltage is max then the current through the resistor will be max at that same time so right now I'm saying what does the y-component gives the value the Y component of the phasor at any instant in time gives the actual value of that variable so right now I've got no y component for anything and now I've got maximum remember this is this is everything here is rotating at Omega so that's the frequency of my power source and as I reach this point right here I get maximum value for current and maximum value for voltage and that's a resistor and then I get none and then I get negative current and negative voltage right because you're pushing the other way it's going the other way and then I get none again and so if I draw this graph for a resistor I'm going to find that the voltage might do this and the current one we want to use for current let's use orange for current the current would do this for instance where these guys have the same time dependence and they are we can call these in phase and they differ only by this this Ohm's law relationship this is going to be let's see this is V and this level is the Max and the orange is I and this value here is I max and we can define I max by Ohm's law these are so I is V over our IMAX is going to be V Max divided by the resistance of my resistor it's time to go to capacitors though so resistors are really really Pleasant everybody likes resistors because they're super simple it's time to go to capacitors and in a capacitor let me get you the same idea of a graph in a capacitor graph if the voltage starts out well let's see I'm actually going to say that the voltage on the capacitor is going to start out I mean it really doesn't matter where start will just start you let you go voltage big voltage small I'm zooming in a little bit I changed my angular frequency so that you can study this a little bit better so this is my voltage and this is all as a function of time sorry I should have labeled that up there now the interesting thing about a capacitor is that if you have a large voltage across the capacitor then that's when the current actually stops so let's go back to this diagram right here as I'm putting a really large voltage across the capacitor that means the capacitor is fully charged because Q is CV and when Q reaches a maximum that means the capacitor is going to be impeding the flow of charge you won't have any charge flowing at that time so the current is zero when the voltage reaches a maximum and we can also argue that the current is zero when the voltage reaches a minimum so we're going to have these zeros right here I want to also say that if the if the voltage is decreasing then you can imagine that will be what are we doing here this is going to be a graph of current ultimately and I'm going to say that the current will be going the opposite direction as the voltage decreases so there we've got voltage at a peak and the current now is going below zero and then it comes up and goes above zero and comes down and goes below zero and comes up and goes above zero so you've got this very interesting pattern notice in this case we can look at in terms of current the current is a minimum when the voltage across the capacitor is zero and the current is zero when the voltage across the capacitor is very negative if the voltage across the capacitor is very negative then you can bet that that will induce a current to start going the other direction so it's sloshing Chuchu Chuchu but the key fact is that they are out of phase we've already seen that they're out of phase I want to remind you that they're out of phase by this equation right here it says the voltage on a capacitor is the current max on the capacitor with this stuff right here this reactance but it's out of phase of the same Omega it's minus PI over 2 or 90 degrees or it's a quarter revolution out of phase that's what it means to be a quarter revolution out of phase these guys are in phase and these guys are a quarter out of phase so I'll say PI by two out of phase hi Kira and if I take this capacitor phase or you see that initially if I start it out like this I didn't actually start it out like this but look at this instant right here this is the instant right here where the voltage that's my purple is a oh no sorry this is maximum current oh that is right there yeah good so I get maximum current initially before the capacitor is charged right and there's no voltage on the capacitor as I start so I start charging the capacitor there's a huge current and the voltage rises up as I'm going to here the voltage is rising up because I'm charging my capacitor boom now the capacitor is fully charged so the voltage is huge but the current has dropped to zero because the capacitor is fully charged and the power supply starts and no let's go the other direction so the current actually starts unloading going away from the capacitor and back towards the power supply and that's here right now and now we have no voltage C the Y component of this phasor right here is at zero but we have a very negative current and if we continue going counterclockwise right here for instance we've got no current but we've got a very very negative voltage and as I continue spinning this around you see how these two things interact which one is leading would you say that the current is leading the voltage or that the voltage is leading the current now take that answer that you got right there the voltage leading the current and the current leading the voltage and see what you think if you look at this which one of these does it look like is leading does it look like the voltage is leading here or the current is leading I mean I guess if it's erased that direction then it looks like the voltage is leading but it's not what we're doing is we're scrolling across this so we're saying at this instant right here we've got that and then oh man see that's going down and now that's going up which one looks like it's leading now it looks to me like the orange is leading because the orange leaps up and then the purple is like okay we can go up and then the orange shoots down the Purple's like okay we can go down and then the orange shoots up and purple says okay we can go up all right so that's why it's consistent to say that the voltage lags the current for a capacitor and that will be our final statement here voltage lags current by a phase shift of PI over 2 they are 90 degrees out of phase I'm gonna go get some lunch I think you should have some

Channel: Doc Schuster

Views: 194,210

Keywords: help, solve, tutor, problem, understand

Id: P9xSEhPENqM

Length: 16min 57sec (1017 seconds)

Published: Mon Feb 04 2013