Learn Reactive Power in AC Circuits - Reactive Power Inductive Load and Power Factor Calculation

hello welcome to this lesson an AC circuit analysis the title of this lesson is called reactive power with an inductive load so we're going to talk about the concept of reactive power here and we're going to be talking about the concept when that applies to having an inductor inside of the load in the last section we had just a resistive network just resistors and we talked in great detail about what's going on with the power when we just have resistors so if you haven't watched that section go back and watch it and you to reacquaint yourself what's happening with the resistive network is we have a non-zero average we have an average amount of power delivered to that resistor and the it's it's oscillating the instantaneous power is oscillating up and down but it's always positive and so there's always an average value when you have a resistive Network here we'll talk about what happens we have inductive loads the next section will talk about what happens we have capacitive loads really excited to teach this because up until now we talked about resistors everyone can wrap their brain around resistors right because resistors are kind of the easiest thing to understand in circuits they obey Ohm's law V is equal to IR and you have some kind of familiarity with with what's going to happen when they sign your words go up and down and things are in phase you know the current the voltage you're in phase so it's kind of even though there was a lot of math in the last section it was very easy to follow I think if you watch it enough times and study it enough times what's really happening with the resistor but when you have inductors and when you have capacitors suddenly the current and the voltage across those things are no longer in phase and things get a little bit murky so we're going to take it slow we're not gonna do any problems we're gonna do theory for the next few sections and when you get to your problems you'll understand these terms you'll understand what's really happening so let's revisit this is what I call the granddaddy power equation I'm going to leave it on the board for very for a great many sections it's instantaneous power so if you start the stopwatch and time marches on this is the time dependence here this is a constant term these this is the time dependence you stick in the numbers with your phase angles for the current in the voltage which is going to be governed by your circuit that you have and this is going to be some kind of sinusoid over time and it's telling you that the instantaneous power is always changing but we want to zero in on the case when we have just an inductive load what's going to happen when there's just an inductor inside of the box last section we talked about what happens when there's a capacitor in the box now we're going to talk about if the load is purely inductive what happens and I can tell you right now actually it's something really cool that happens and I find this stuff really fascinating with circuit analysis so this is some some of the reasons I'm really excited about teaching it alright the first thing you need to remember there's very very few things I tell you in circuit theory to just remember all right one of them is Ohm's law V is equal to IR right you have to know that some of the real basic power equations just memorize them what I'm going to tell you here is something you should just remember when you have a purely inductive load all right the current okay through that inductor lags the voltage by 90 degrees exactly the current lags the voltage in an inductor or across an inductor by exactly 90 degrees current lags the voltage by 90 degrees I'm going to show you how to prove that to yourself but just remember it because if you can just remember it that these few little things I tell you to remember it's going to make your life so much easier so let's go down here and write that down and I'll explain what I'm talking about the current lags the voltage by I'm going to put the word exactly 90 degrees and I'll give you a hint when we get to capacitors it's also going to be 90 degrees but the current is not going to lag the voltage and a capacitor the current will lead the voltage by 90 degrees so the 90 degrees part is going to be easy to remember the leading versus the lagging is what always confused me when I learned this stuff the first first time finally you're just going to have to commit it to memory the current lags the voltage in an inductive across an inductive load when it's purely inductive now why do we care about that because in this power equation we see we have all these phase differences between the current and the voltage so when we say something like the current lags the voltage by exactly 90 degrees what we're really saying in words is the following we're saying that the phase angle of the current is equal to the phase angle of the voltage whatever it is minus 90 degrees make sure you understand this little equation here current lags the voltage that means whatever the voltage phase is you subtract 90 degrees and that is going to be what the current is so again make sure you understand whatever the voltage phase is shift at 90 degrees look to the left 90 degrees so you subtract 90 degrees that's what the current phase is going to be so the current lags the voltage by exactly 90 degrees so another way to rewrite this this is kind of the layman's terms way to put it but if you rearrange things a little bit what it's basically saying is theta V minus theta I is 90 degrees so again move the theta I over here move the 90 degrees over there boom you get this why do we why do we write it like this theta the amount of state I okay the reason we write it like that I can even put parentheses around here if you want the reason we do that is because this equation has theta V minus theta I everywhere so you see when the load is purely inductive you basically stick 90 degrees positive 90 degrees in here in here and in here and you should know that the cosine of 90 is zero and the sine of 90 is 1 so that's going to drastically simplify this equation and make it something that we can study when the load is inductive right remember back to the last section when the load was resistive theta V minus theta I was zero because the phase difference between current voltage there is no phase difference between them they're there in lockstep with one another when we say these guys are shifted by 90 degrees we mean the sinusoids are shifted in 90 degrees along their horizontal axis like that so we'll actually plug this number in and study it for a second but I want to talk for just a minute about this statement one more time current lags voltage by 90 degrees in an inductor because I guarantee you will forget it at some point in the future and I want to tell you if you forget on a test if it's leading or lagging how do you how do you you know figure it out for yourself what it comes from is if you remember back when we studied inductors the voltage across an inductor is L the inductance times di DT so this is the rate of change the derivative of the current flowing through that inductor so you see these are very interestingly related the voltage is not related to the current it doesn't matter the magnitude of the current the current can be a million or a micro amp or a nano amp okay but if it's changing really really really fast and you can have a very very large voltage across that inductor so even if you have micro amps but it's rapidly changing super super fast right then this di DT can be really really big even if the current is small so you see an inductor the voltage is related not to the current it's related to how fast the current is changing and that is why these guys are shifted out of phase by 90 degrees because if you go and draw it here let's go and actually draw the current okay so this guy is going to be time and this guy is going to be the current through our inductor so let's just say for the sake of argument that we're going to start out with a cosign let's just say we have a cosign because these are all cosine functions and we're going to plot the current so here is a well let's say it's a sign let's go like this start like this like this and so on so let's just say for example that this is the current flowing through the inductor at time zero it starts up like this and it goes down and it goes up and so on alright what is the voltage going to look like across that inductor okay and this would be the voltage what's it going to look like well we need to draw some lines to help us draw some things here at the maximum here I'm going to draw a little line here where it crosses zero I'm going to draw a little line here at the negative maximum we'll say it's right there and then here at zero we'll say it's right here and we can draw one more if you want at that maximum right here so what we're basically saying is that the voltage here that we're plotting is some constant times di DT now right here as its crossing through zero this guy has the maximum slope it's going to have the derivative is a maximum right here so the voltage is going to start at a maximum value and then what's going to happen is it's going to go the this guy's going up up up up up eventually it gets to the plateau here at the top the derivative is zero because the function is not changing anymore here so what happens it's it's it's going to go like this this guy's going to start to bend down and it's going to go through here and I'll just continue drawing it for you here goes down below to a negative maximum here up through here and then there's a maximum here and then it goes down like this so this is what it's going to look like make sure you understand that see see everywhere there is a peak the derivative should be zero here is a peak the derivative should be zero here is a peak the derivative should be zero here the derivative is a maximum positive number because it's sloping this way and as we go down this way the derivative also is a maximum but it's slanted the other direction in the negative slope direction so we should have a negative maximum here here the derivative a slope and back positive maximum here so you see what's happening is everywhere in between the thing is changing and it's just kind of crossing through these points so this is nothing new this is basically a sign and here's the derivative we know the derivative of sine is cosine that's what I've drawn here all right but what I want you to notice is that what we said here current lags voltage by exactly 90 degrees make sure you understand that it's a little easier to understand it if I actually kind of do a dotted line in the negative time direction like this right current lags voltage what that means is that any point in time right let's pick right here this is the voltage right here we're saying that it right at this point here is what the voltage is looking like the current is going to lag that guy by 90 degrees so if I take this guy and go 90 degrees to the left this is exactly what my current is doing at that point in time and that's what's basically happening if you pick any point along this voltage curve let's say this point right here this maximum right we say the current lags the voltage so we're basically the voltage hits the peak first and then a little bit later the current hits the peak the voltage here hits a trough here and a little bit later the current hits the trough so you see the current is lagging the voltage right the current is lagging the voltage by exactly 90 degrees because as the voltage gets to wherever it's going 90 degrees later okay then you have the current doing the same thing so it's a little bit of a delay there you can look at it a couple different ways if you take this guy shift it to the left 90 degrees that's what's happening at your on your curve at that point so if you forget that the current is lagging the voltage by 90 degrees you can derive it by drawing pictures but that takes time so I just recommend remember it inductors current lags voltage current lags voltage current lags voltage current lags voltage we get two capacitors it'll be on the one the confucian output it'll be a little bit different so this was an aside the point of this was to show you that theta V minus theta naught theta I is exactly 90 degrees and you should all remember that the cosine of 90 degrees is going to be equal to zero and the sine of 90 degrees is going to equal one so let's take this information we're going to go over to the right-hand side of the board and write down what the instantaneous power looks like now that we know that it's an inductive load what we have this first term is going to actually disappear because it'll be cosine of 90 degrees which is zero so this disappears right here's a cosine of 90 degrees so this disappears too so the only term that we have is this term but notice we have sine of 90 degrees sine of 90 degrees is 1 so the only thing here in this instantaneous power is the negative these coefficients here and then this guy at the end so what you have is negative VM I M over 2 times the sine of 2 times Omega T so this is switch colors this is really important this is the instantaneous power when they have an inductive load all right so I'm going to kind of circle this now I am going to kind of caution you here I'm giving you a lot of Theory here you're you're not going to use these instantaneous equations much when we're doing real circuit analysis we're going to use phaser techniques and other things but I'm trying to really show you where all this stuff comes from because you will probably have a few questions on it and more than anything else the books a lot of them are really hard to follow and what's happening with this math here all we're doing is taking this instantaneous equation we know what the phase difference is for an inductive load most of these terms drop away and here is what it really looks like for an instantaneous power in an inductor when you have purely inductive load now the rest of my lecture is going to be some comments about this very simplified equation because now we have just inductive load everything drops away so I'm not going to write too many things down but I'm going to make sure you understand everything alright the first thing is the most important thing is the average over one period actually I'm going to write this one down so I'm going to put the average power over one period is zero make sure that sinks into you okay remember we talked about how do you calculate the average power over period well you integrate the instantaneous power over DT over one period and then you divide by T we talked about that just in the last section so if you were going to figure out what the average power was here you would take this guy and you would integrate it but notice that it's just a sign right it's got two Omega T but you're integrating over one whole period so you've got a doubling of the frequency so you have this in the similar way for the resistor the frequency is doubled right but you're still integrating over a complete period so you're going to you're going to catch that sinusoid exactly where it starts if you start anywhere in integrate over one entire period so since it's a sinusoid it's just got a coefficient here some half the time it's spending on the positive side and half the time it's spending on the negative side you integrate that you get zero you get zero that means the average power over a period is zero that's totally different than a resistor remember go back to the last section we talked about the average power it was nonzero because when we integrated all this stuff this first term gave us a gave us an average power that we you know then used to to say here's what the average power is delivered to a resistor but in an inductor the average power is zero right if you look forever and ever and ever the average power is zero time for an inductive load so that also another another way to say this is that the there's no transfer of energy from electrical to non electrical form in an inductor so in a resistor we say that the power is delivered to the resistor and it rewrites it out as heat that's lost to the environment an inductor the energy that you deliver to the inductor during half the time get stored in the magnetic field in the inductor and then the other half the time the energy is kind of pulled out but it doesn't ever get dissipated to the environment it's not lost to heat it goes into the magnetic field then it comes out and kind of is transferred back to the circuit so it's a sloshing effect back and forth delivered and then taken from delivered and then taken from that'll be a little bit more clear in the other in here and second the instantaneous power is twice the frequency of the original frequencies that we're talking about so we that's kind of similar to the way it looks for the resistor and I already said this power is stored in the magnetic field and then later extracted so I want to talk about that power is stored and I'm using these terms loosely you know it's really energy is stored in the magnetic field in the magnetic field and later extracted all right now what do I mean by that notice that this instantaneous power is basically a sine function it's got a doubling of the frequency that means something has we have a negative sign there but that means sometimes this instantaneous power is going to be positive and sometimes the instantaneous power is actually going to be negative that did not happen for a resistor if you remember that thing was oscillating above the horizontal axis it was never negative but for an inductor everything else drops out this is what we're left with ladies and gentlemen half the time instantaneous power is positive half the time it's negative what does that physically mean what that physically means is when the instantaneous power is positive when it's greater than zero that means energy is stored in the magnetic field and what magnetic field are we talking about don't forget in an inductor is a coil of wire the reason it's coiled like that is to concentrate the magnetic flux inside of that coil so the energy is actually stored in this invisible thing we call the magnetic field and it's it's a real thing it's energy stored there that can do work later on down the line when does it come out is when during part of the cycle P that we calculate here is less than zero so when it's negative the energy is extracted from the magnetic field or from the inductor however you want to say that's really really really important and this is kind of where I start to get a little bit excited about teaching this kind of stuff it's it's really a beauty and symmetry what's going on here you have this device it's a coil of wire if you stick a current through it it concentrates a magnetic field inside the coil and magnetic fields we can't touch them or taste them or even see them but we know that they can store energy and then you say well how can something that's invisible store energy right how does that even work well what's happening is that all of these circuits we're talking about are sinusoidal that means they're driven by sources that oscillate up and down and that means that for the inductive components some of the time the power that you calculate across the terminals of that inductor is going to be a positive value and we know from our from our sign convention of power when you have a positive power that means energy is being delivered to that load for resistive it just get dissipated into space as heat but when we're talking about inductor it doesn't get dissipated it gets stored inside the magnetic flux inside of the coil of wire that we have right but then later on down the line the source swings the other direction and things oscillate back the other way and then current starts to come out of that inductor and then some of that energy is extracted from the magnetic field because then the magnetic field collapses down to zero again and as it happens that magnetic field which is storing energy is contributing to the current flowing out of that inductor as it collapses down and that's a very real thing I mean if you put a lot of energy in a magnetic field and then you unplug it you hook it up to something I mean you can see the current bleed out of that as the magnetic field collapses and so that's real energy transfer so it's kind of like a temporary battery you can't you know it's not practical to like power car with inductors or anything like that but as a temporary storage device you can store energy in an inductor when we're saying the power is positive part of the cycle here and then we can extract it whenever this is negative remember we we're talking about our circuit analysis we've done all along we've done power calculations and you when you for the resistors you always find the powers positive but for the sources even back in DC analysis you usually find you always find that the power that the source is delivering is always negative and we always said hey conservation of energy the power delivered by the source is equal to the power that's dissipated or absorbed by the resistive Network right so they have to add up they have to be equal and opposite so we have we've had the sign convention all along positive means the thing is absorbing something negative we've usually talked about negative power being sources that are that's delivering power to the circuit it's the same convention here when the inductor is put on the part of the cycle where the inductor has negative power it's just delivering energy much like a source much like a temporary source of energy in fact it is because it's had energy stored in its magnetic field so the way to think about it is inductors and capacitors we'll talk about in here in the next section there when you get into these AC analysis cases basically you have the sloshing around of energy sloshing or you can think of it as like a two buckets I'm pouring water in this bucket and pouring water in that bucket pouring water in this bucket pouring water in that bucket on the whole I'm not really I don't really have any average power deliver it anywhere on the whole because half the time I'm pouring water in this bucket and half the time I'm pouring water in this bucket so the water is just kind of staying here in my two hands it's not going anywhere that's why the average power is always zero because half the time the inductor is absorbing energy and the other half the time it's delivering energy back to the circuit that gave it the energy to begin with so it's like this pain pot can use them as many analogies as you want it's like ping-pong I hit the ball this way I hit the ball that way hit the ball this way the ball is still staying here on this table so average power is not really being irradiated to the environment or anything it's staying in the system but half the time I'm delivering and half the time I'm absorbing that's about as many analogies as I can get give you there but it's that's what's happening energy sloshing back and forth but the average power is zero over time because it's kind of always staying on the court here it's never going anywhere else so the last thing I'm going to leave you with is a little graph here we said this is the instantaneous power this is what it is let's draw a picture of that and then we'll close the section off talk about capacitors so this is time so this is the instantaneous power what would this look like well this is a constant so forget about it this is a sine we know what sines look like but we have a negative sign out there so it's going to look like an upside down sign it's going to instead of starting up like this it's going to start like this like this okay this is the instantaneous power it starts in the negative direction only because of this negative sign that's here at time zero that's what you have and so it starts here because it's negative and you can see that initially it's negative so the the inductor is delivering energy and then here it's absorbing and here's delivering and here is absorbing and here is delivery and so on and so forth I want to draw a picture of this for you because later on we talk about capacitors this will be flipped upside down and it'll look the same except it'll be it'll be upside down so I'm going to be drawing a lot of parallels between inductors and capacitors as we go forward here so the big takeaways and notice that the average power is zero this is the instantaneous power right but if you draw a line right through the middle of course you're just looking at the middle of the sine wave there so the average power is zero half the time we're delivering and half the time we're absorbing power in this abductor inductor so click-click ten second recap here's the instantaneous power the granddaddy equation then we figure out that for inductors current lags voltage which means theta the minus theta is positive ninety degrees we put 90 degrees in this term this term disappears we put ninety degrees in this term this term disappears we put 90 degrees in this term and we're left with this times this this is one this whole things one times this which is our instantaneous power equation for an inductive load because it's just a sign that is half the time positive half the time negative the average power is zero power stored in the magnetic field and later extracted when this part of the cycle is positive we're storing imaging in the magnetic field when the part of the cycle is negative we're extracting energy and it's kind of sourcing from that inductor to the rest of the circuit energy from the magnetic field and here's a plot of what's going on so try to internalize that as much as you can and relate that back to what we learned Section four resistors four resistors the power like this was a sine wave all right but it did not have an average of zero it was all shifted up and it was all positive up here that's because resistors really do dissipate energy and you're not doing any sloshing it can't store anything so you can't slosh back and forth you just deliver energy to that resistor and it gets radiated to the environment it's lost from the system that's why the instantaneous power is always positive for resistors but for inductors it can be positive and negative depending on what part of the cycle you're at so let me close here watch this a few times to get it follow me to the next section and we will make a very similar argument to show you how capacitors are very similar but yet obviously slightly different it's the cousin to the inductor so you'll see how that works in the next section

Channel: Math and Science

Views: 106,617

Keywords: reactive power, power factor, ac, imaginary power, ac circuit, ac circuit analysis, inductor, inductive load, power, energy, reactive power inductive load, power equation, power circuits, reactive, ac power, real power, reactive power explained, reactive power and real power, reactive power explanation, reactive power equation, reactive power definition, impedance, complex power, average power, electricity, circuit analysis, ac steady state, power factor formula

Id: QBtZQxBPE9M

Length: 25min 8sec (1508 seconds)

Published: Mon Nov 07 2016