Alternating Current - A Level Physics

hello today we are continuing in the a-level physics revision series looking at the subject of alternating current before we do so however it may be worthwhile just reminding ourselves what happens in relation to direct current you get direct current from an ordinary battery this is a 9-volt battery and it delivers a constant supply of nine volts very simply what that means is if we drew a graph of voltage against time then you would expect that as you measure the voltage as time goes by it would constantly stay at nine volts and if you were to put that battery in a circuit here's the battery the 9-volt battery and we put that in a circuit with a resistance and let's say the resistance is three ohms then we remember Ohm's law V equals I R if the battery voltage is nine volts then the current is I the resistance is three and that means the current must be three amps so if you put the 9-volt battery in a circuit with a three unresisted you would get a constant current of three amps so if we plot now current against time this is now current against time you would expect there to be a steady flow of three amps so to review the battery has a constant nine volts obviously in due course it would fade and die but initially it's going to have a constant nine volts and it will deliver a constant current which will be determined of course by the size of any resistance in the circuit that's direct current now how do we produce alternating current well you will find more details on the video I've already produced called electromagnetism part two in the a-level physics revision but I'm going to go over the ground again here what we're going to look at is a very simple electrical generator this is basically what drives the whole of our economy this is how we produce electricity we take a magnet this is the north and this is the South Pole of the magnet and between the magnet we put loop of wire this is my loop of wire and what we do is we put some kind of fins on the wire and then underneath here we have some kind of heating capacity so that the steam rises and causes the loop to flip over so this end of the loop goes downwards and this part of the loop goes upwards and the whole thing just spins around on an axis sort of along here it's a spinning wire caused by the heat which might be coal-fired nuclear fired wind powered some method that produces some energy that will cause this wire to twist within the presence of a magnetic field now that magnetic field of course will have field lines or flux lines and as that wire twists the wire cuts the flux line and we know from Faraday's law that where you have flux being cut there will be an EMF which is equal to minus D Phi by DT the rate of change of flux and as the wire moves and cuts the flux lines an induced current will flow in this wire which direction will it flow well we have to use the good old left hand rule and this you may remember we've seen this before the thumb is the motion the first finger is the field and the second finger is the current so if we say the field is in this direction it's going from north to south the motion in the wire is downward so we'll put the motion downwards the current is therefore flowing in this direction in the wire if we come here we've got the field still in this direction go from north to south this time the motion is upwards so the current will be coming down the wire like that and that's fine your current is just going round the wire like this and that's generating a wire that that's generating a current question will it be a constant current well let's look at this piece of apparatus as if it will end on looking at it flat on so that here is the North Pole of the magnet here is the South Pole of the magnet and if we're looking edge-on there's this wire looking at it edge on and this wire is represented by that there and the cable goes into the paper and we know that this is moving downwards and this one is moving upwards and we know I'll use a pencil for this these are the field lines going across from north to south so that's the flux now let's first of all think of this wire here as it is moving around in a circle because it's the whole thing is looping for the time being when it's in this position it is moving across the flux lines consequently it is cutting the flux it's the maximum point at which it cuts the flux and therefore it generates the maximum EMF so if we plot voltage against time then let's say that when it's here it's at this point here it will have a maximum voltage so let's give it a maximum voltage like that now let's consider what happens when that wire moves round to this point here it's now got as far as here and it is traveling for the time being in that direction in other words it's travelling along the flux line it's not cutting the flux line so there's no rate of change of flux so there'll be no EMF so when it gets to this point here on our chart there will be no EMF now what happens when the wire gets to this point here well once again it's going to be doing a maximum rate of change of flux but look something interesting is happening when the while was over here the current was flowing downwards but when the Y gets over here the cont will be flowing upwards there's a change in the direction so in other words when we get to this point here there will be a maximum voltage but in the other direction and then finally when the wire gets round here it will be traveling for the time being in this direction again in parallel with the lines of flux it's not cutting the flux so there's no voltage so when we get to this point in time here there will be no voltage now you might think that on that basis what's actually happening is it's pretty much a straight line and that the voltage is varying rather like that in fact what is happening is the change in the voltage is running according to the sine wave I'll I'll describe that mathematically in a moment but what it actually looks like is this is it's a sine wave and so consequently what is happening is that as the wire turns in the magnetic field it is constantly changing the voltage and the associated current it starts at a peak voltage it drops to zero it goes to the negative peak voltage it drops to zero it goes to the positive peak voltage and on and in the UK in the United Kingdom the domestic electricity supply does that 50 times per second it's 50 Hertz the frequency is 50 Hertz so 50 times a second the domestic supply that comes out of the plug in the wall will go from - will go from positive voltage to zero - - voltage to zero 50 times in a second so let's instead of using a battery let's put an alternating current source in a circuit here is my resistance and here is my that's R and here is my voltage supply which is an alternating voltage supply and what we now know is that that will produce an alternating voltage which goes as a sine wave constantly up and down as I said in the UK 50 times a second and it will reach a peak voltage which we'll call v-0 then it will fall to zero and then it will go to a negative peak voltage which will be minus v-0 and that means that the current in this circuit is constantly changing direction it starts off going in that direction then it goes in that direction then it goes in that direction then it goes in that direction and it changes 50 times a second and the mathematics that describe this says that the voltage at any time T this is of course time is equal to v-0 which is the maximum voltage times the sine of Omega times time or if you want to talk about the current i t is equal to the maximum current times the sine of omega t what's this omega t is just a time but what's this omega well omega is a fairly standard definition it two pi times the frequency and the frequency is the number of full waves per second and as I told you in UK that's 50 times per second so the frequency would be 50 in the UK we also know that frequency is 1 divided by the period the reason is the frequency is the number of full waves which pass a given point the waves are moving in that direction the number of full waves that pass a given point in one second is the frequency the period is the time it takes for one wave to go past that point so the frequency is simply the inverse of the period and if Omega is 2 pi F then Omega must also be 2 PI over T since F is 1 over T so now let's have a look at this formula let's do the voltage um and we'll take just this term sine Omega T and we'll ask ourselves what the value of that will be depending on the value of T so let's start off with T equals zero well if T equals zero I think it's pretty obvious that sine Omega T is going to be the sine of zero which is zero what happens if T is equal to capital T in other words the full period well in that set in those circumstances sine Omega T will be the sine of Omega which is 2 pi over T times T because the little T is equal to T the T's cancel and you get the sine of 2 pi the sine of 2 pi is 0 what happens when T is T over 2 half the period well the sine of Omega T will be the sine of Omega which is 2 pi over capital T times T which is T over 2 and that's the sign of T's cancelled twos cancel is the sine of pi and the sine of pi is also zero what happens when T is equal to T over for a quarter of the period well sine Omega T will be the sine of Omega which is 2 pi over capital T the period times the time which is T over for the t's cancel and now you've got the sine of PI over 2 and the sine of PI over 2 is 1 and what happens when T is 3 T over 4 in other words three quarters of the period well the sine of Omega T will be the sine of omega 2 pi over capital T times the time which is 3 T over 4 so it's the sine of 3 PI over 2 1 and 1/2 pi and that is the sine of 1 and 1/2 pi is minus 1 so now if we plot what we've just found let's say this is naught this is T this is T over 2 this is T over 4 and this is 3t over 4 so we're going quarter T 1/2 oh dear quarter T 1/2 T 3/4 T and T we can put our values in when T was 0 the sine Omega T is 0 when T was T over 4 sine Omega T was 1 let's say that that's the value of 1 when T was t over 2 sine Omega T was 0 when T was 3 T over 4 sine Omega T was minus 1 so that's going to be down there and when T is T the sine of Omega T is 0 so what we've got is the sine wave we were looking for and of course it continues to do that so in other words this formulation of sine Omega T is what gives us this shape of sine-wave now we're going to look at the power in an AC circuit and you'll remember that from DC power is equal to voltage times current and since we remember that Ohm's law says that V equals IR we can if we want substitute for either V or I in this formula and we can say that if V is IR then the power is V which is IR times I which is I squared R or if you like you can say that I is V over R and substitute in here so you've got V times V over R which is V squared divided by R so you can express the power in any way you like you can have it V times I or I squared R or V squared over R they are all the same thing so the power is equal to let's take this one first I squared well we said that I at any at any given time the current is constantly changing with alternating current and it is equal to I is equal to R I naught sine Omega T that's I so I squared will be that term squared and if you want the power you've then gonna multiply that by the resistance and that's the same as V squared which will be V naught sine Omega T all squared divided by R so the power is this term squared times R or this term squared divided by R this is simply the formula for the current at any given time given that it's constantly changing so if we now plot the power against time we see that we've got a squared term an I squared term so the we are going to get sine essentially the sine wave squared and that's going to look like this it's never going to be negative it's not going to go under here it's never going to do that because sine squared will always be positive multiply any number by itself and you get a positive term so the power is going to look like this where the maximum of course is going to be I naught squared R which is when sine Omega T is 1 then the maximum power will be I naught squared R or if you like venal squared over R when sine Omega T is 1 or minus 1 because minus 1 squared is still 1 you're going to get the maximum power in each case so they just remind you the voltage is probably going to look something like that but when you square it you're always going to get a positive term but we want to know what is the average power the power is constantly changing but what's the average power you're getting out of here well for that we need to know what the average of a sine squared term is so let's just take this is simply sine squared Omega T against T and that's going to produce something that looks like that where the maximum of course is 1 because the maximum of sine Omega T is 1 so the maximum of sine squared Omega T is 1 and the minimum is going to be 0 and what we want to know is what is the average value of sine squared Omega T over the entire cycle and the answer is that the average which we depict by these angle brackets the average of sine squared Omega T over the entire cycle is exactly 1/2 so what we can say is that the average value of sine Omega T in both these terms is 1/2 so the average value of power the average value of is going to be I naught squared R divided by two because all we've done is to take me I naught squared R and we've said the average value of that is 1/2 or we can simply say here that the average value of power is going to be V naught squared over R times the average value of that which is 1/2 so this is going to be equal to V naught squared over 2 R so that's the average power that you get out of an alternating current circuit the actual power is constantly varying but on average you get half of the sine term we now move on to the concept of the root mean square so once again let's look at our alternating current this is voltage and this is the sine wave which in the UK will go at 50 times per second from the maximum to 0 to the negative maximum and what we've just shown is that the average power that you get from this will be equal to I naught squared R over 2 which is the same as V naught squared over 2 R so if we say that power is V squared over R and we're looking at the average value of V squared over R where don't forget V is constantly changing but the average value of V squared over R is what we're looking for that will equal V naught squared over 2 R the RS cancel and that gives us that the average value of V squared is equal to V naught squared divided by 2 and that means if you take the square root of both sides the average value of V the voltage is equal to be the square root of V naught squared which of course is just V naught divided by the square root of 2 and that's a root-mean-square the average value of the voltage in an alternating current where it's where the voltage is constantly changing the average value of that voltage is equal to the maximum value divided by the square root of two and the same thing applies if you say let power equal I squared R then you've got the average power is I squared R and that equals I naught squared R over 2 I naught squared R over 2 now the RS cancel and that gets the average of I squared is I naught squared over 2 and so if you take the square root of both sides you get that the average value of I the average current is equal to the square root of I naught squared which is just I naught divided by the square root of 2 and there's another root mean square term what does this actually mean well let's take this formula here the average voltage is the maximum voltage divided by the square root of 2 what that means is that in the UK the plug in the wall that you get your electricity from is said to be 230 volts what that means is that's the average voltage that you're getting the average voltage is equal to the maximum voltage divided by the square root of 2 which means that the maximum voltage is equal to 230 times the square root of 2 and that is approximately 325 volts so what we're actually saying is that in the United Kingdom when you plug in at the wall you're getting an average of 230 volts but in fact the actual peak is 325 volts here and minus 325 volts here so it's an even bigger voltage than you think it is at its peak and before we move on we better just clarify exactly what we mean by this average term here because if you look at the sine wave obviously the true average of a sine wave over an entire wavelength there is the wavelength the true average is zero because it is positive for half the cycle and precisely the same negative amount for the other half of the cycle so the true average is zero what we mean when we talk about the average voltage is that of course the voltage coming out of the wall is always giving you some power and that power is the equivalent of this voltage we derived it from the power term so it's the voltage that effectively gives you that power if if instead of having a variable voltage you had that as a direct current voltage then it would produce the same amount of power so we need to understand that it's not the average of the sine wave because the average of a sine wave over a complete cycle is zero what we mean by this is that that's the average term derived from the power to give you the equivalent voltage in DC terms that would give you the same power we now move on to the concept of transformers which I have also dealt with in my videos on electromagnetism in the a level physics revision series but I'll just cover the subject again here for completeness what we often want to do is to transform voltages for example if you take the voltage that comes out of the wall in the United Kingdom at 230 volts and you want to charge your mobile phone there is no point plugging 230 volts into the mobile phone because you will cause it a lot of damage what you will need to do is to put that through a transformer which will transform the voltage down to something like 12 volts or 9 volts or whatever the mobile phone needs and then you can plug that into your mobile phone and then you can charge it so you need a transformer that will step the voltage down from 230 volts to 12 volts there's also the situation that we transmit electricity across the country on pylons over mini miles on cables and they that current is sent at something like 10,000 volts I'll tell you why in a minute but you don't want 10,000 volts being fed through to your home so that needs to go through a transformer that will deliver 230 volts which is what you want for your domestic supply so again you need to step down from 10,000 volts to 230 volts how do you do that well you take a coil of wire and this has an alternating voltage across it and let the number of terms in that wire be called n P P because this is known as the primary and now take another wire which is not connected to any power supply at all and let the number of turns in that wire be called NS because this is called the secondary then what we know is that if we've got an alternating current going through this primary the changing current will produce a changing magnetic field and this wire or this coil is in the presence of a changing magnetic field so in other words there will be a change of flux in the coil in this secondary and that means that there will be an induced alternating current in the secondary and we know that the EMF at any rate in the secondary the induced EMF is equal to minus D Phi by DT where D Phi by DT is the rate of change of flux and what we have shown before or is that the voltage in the primary is equal to the number of turns in the primary times the chain the rate of change of flux and that the voltage in the secondary is equal to the number of turns in the secondary times the same rate of change of flux and so you can see that D Phi by DT is equal to VP divided by n P and D Phi by DT is V s divided by n s so consequently VP divided by MP which is d Phi by DT is equal to V s / NS which is equal to divided by DT or putting it another way VP divided by V s is MP / NS the voltage in the primary divided by the voltage in the secondary is equal to the number of turns in the primary divided by the number of turns in the secondary so if you want to step up in other words if you want a higher voltage here than in the primary then you need fewer turns in the primary and more turns in the secondary and if you want to step down so if you want VP to be VP is greater than V s so VP is stepping down to be s MP must be greater than n s so you need more turns in the primary then you have in the secondary and by the same proportion so for example if you're going from 230 volts in the primary to 12 volts in the secondary you'd need 230 coils in the primary and 12 coils in the secondary but it's important to notice that this equals I s / I P make sure you don't get those confused here VPS on the top MPs on the top this is on the bottom in s isn't on the bottom but when it comes to the current it's the other way up s is at the top and P is at the bottom why is that because the power in this circuit must equal the power at least it must not exceed the power in this circuit so if we're assuming the Transformers operating at 100% then the power in the primary will be the voltage of the primary times the current in the primary and that will equal the voltage in the secondary times the current in the secondary in other words you can't increase the power you can only step the voltages down and up but the total power which is voltage time current must remain the same and if that's the case then you can see that VP divided by vs is equal to is divided by IP p on the top s on the bottom s on the top P on the bottom which is what we've got here and that's the way in which power remains constant and that's assuming you get a hundred percent power conservation which you won't because inevitably you'll get some heat loss in the transformer how do we manufacture a transformer well it's usually based on an iron bog what we do is we take piece of iron with a hole in the middle so it's a square shape and we put our primary coil around this part here and we put our secondary coil around this part here and that's the number of turns in the primary that's the number of turns in the secondary and this would be obviously a step-down transformer because we've got three coils here we've got two coils here and consequently you'd have a smaller voltage out but a higher current then you put in I said earlier that we transmit electricity across the country on pylons with electrical wires now this might go on fourth you could go on for 100 miles of pylons and electric cables and I said that we to transmit at 10 kilovolts or more so here's our electrical generator which is what we started this video with the loop of wire which spins in a magnetic field and generates an EMF and an Associated current and here when we get to the end of our hundred mile journey we need to go to a transformer which we've just spoken about which produces 230 volts for the UK domestic market now why do we bother why don't we just send this at 230 volts across the pylons across the country and then we don't need to bother about the transformer at this end and the reason is that you get a very high power loss in the cable if you send it at voltages like 230 volts and let's see why we know that the power loss in this cable is going to be the voltage times the current which can also be written as I squared R the current times the resistance of this cable the cable will have although it's an electrical conducting cable probably made out of copper it will nevertheless have a resistance so there will be a power loss so let's just imagine for example that we transmit this current at 230 volts and let's say the current is 4 amps will actually call it 250 volts to make the maths a little easier so 250 volts and there's 4 amps the power will be 1,000 watts alternatively we can transform this to 10 kilowatts if we do that then the voltage goes up to 10,000 volts but of course the current must come down proportionately because the power remains the same so if we have multiplied the voltage by 40 we have to divide the current by 40 so now that current is simply point 1 of an amp it's still the same power it's still a thousand watts being transmitted across the powerlines but it's a much higher voltage and a much lower current now why does that help well let's imagine that the resistance of this hundred mile cable is 10 ohms I don't know what it would be but let's for the sake of argument call it 10 ohms the power loss in that cable is going to be I squared R where R is 10 ohms now let's have a look at the two options first we'll look at this one where we're sending it at 250 volts and 4 amps the power loss is going to be 4 M squared which is 16 times the resistance which is 10 is a hundred and 60 watts so of our thousand watts of power we lose 160 watts just transmitting it from one end of the country to the other on the other hand if we take this option the power loss is going to be the current squared which is naught point naught 1 times the resistance which is 10 and that brings us back to north point 1 watts so if you use 231 250 watt volts you'll get a power loss of 160 watts a huge amount 16% is lost if you go up to 10,000 volts your power loss is not 0.1 watts very very much smaller you don't waste as much electricity so that's the reason why we send the the electricity supply at the highest possible voltage so that we reduce the current and thus reduce the power loss now we come to the subject of rectification because you see there's a problem with alternating current and that is that the voltage and the current alternates as we've already seen in the form of a sine wave and that means that the domestic supply that comes out of the wall in the United Kingdom ranges from plus 325 volts at its peak to minus 325 volts when it's at the other end and it oscillates between the two at 50 times a second that's a huge change and although some domestic appliances don't mind some of them certainly do even if you transform this down so that it goes from say plus 12 volts to minus 12 volts if you put that through a computer you would blow all the components Computers can't handle constantly changing voltages and currents in this way what you need is a direct current and a direct voltage like we had with DC where the voltage and the current remain constant over time so the question is how do you go from here to here well one option is what's called half wave rectification we take our alternating current here is our resistance our appliance whatever it is we want to put it through but we also put in the circuit a diode and of course a diode permits current to flow in that direction only so consequently where we have an alternating current coming from the supply that looks like this oops a control sine wave that's what it supposed to look like what actually happens is when the current flows in this direction it's okay the diode will allow it to flow so that's when it's flowing in that direction but when it flows in this direction the diode won't let it flow so instead of the current going in the opposite direction no current flows at all and then when the current flows in this direction again the diode says okay you can go but when it comes to this direction the diode stops it so what you've got is what's called half wave rectification you've got a variation from zero to maximum back down to zero and then half the time no current at all and then the process repeats well that's better than nothing at least you haven't got any current flowing in the opposite direction but you've still got a massive variation in voltage here and then half the time actually no voltage at all so that's not ideal well we can solve that problem to some extent by using what's called a bridge rectifier once again we take our alternating current source and this time we come through what's called a bridge rectifier which looks a bit like this we draw it as a kind of diamond shape and we take the output here and what we do is we put in diodes like this and then we connect them like so now how does that help well let's just think when the current is flowing in this direction remember it's going to also not alternate when the current is flowing this direction it comes along here along here down here which way does it go it can go that way because the diode is against it it can go this way because the diode is okay in that direction so the current flows down here flows down here where does it flow here it can't go in this direction because the diode is against it it must therefore flow in that direction what happens when the current comes this way well it comes all the way along here up here until it gets to this point which way does it flow can't go that way because the diode is against it it can go that way what happens when it gets here can't go that way because the diode is against it it must go this way so no matter which way the current comes out of the alternating current supply it always ends up going in this direction by the time it gets beyond the bridge rectifier and so what you get from that is if this is the voltage or indeed the current then you get this in other words the current is always in the same direction but of course it's still varying from naught to the maximum and although that's helpful we still haven't got any negative current and we haven't got this half the time no current at all it's still not going to be very good for our computers so it's helpful but we're not there yet what's the solution what's the solution well what we do is we take the output of our bridge rectifier I won't bother drawing in the diodes it's exactly the same diagram as before but here is the output and what you will need to do is to put a capacitor a large capacitor in here and now this can be your computer which is earth your appliance what happens we know that the current is only going to flow in this direction but it's going to be and constantly changing current and voltage but if you put a capacitor here what happens as the current flows this capacitor will charge up and it is a big capacitor it will hold a lot of charge indeed it will charge to the maximum voltage v-0 now when the current starts to fall off so let's just draw this a little bit this is the voltage or the current against time this is what the voltage coming along this line is doing going from zero to maximum which is not much good for our computer but it's also charging this capacitor so what happens is that when the capacitor is charged this can't Falls but the sitter takes over as a kind of battery and maintains the voltage across the computer so whilst the current is going down the capacitor it would discharge a little but not much and then the current comes back up again to recharge the capacitor then the current falls down but the capacitor holds its voltage and then the same thing applies again so what you've got now is not a perfect direct voltage or direct current but almost there's a slight wobble in the voltage but nowhere near as much as there would be if you just had it like this and that will enable the computers to work much more satisfactory what kind of capacitor do you need to do that well a large one in our video on capacitors we showed what happens to a capacitor as it discharges if it has a maximum voltage of V zero then over time this is a plot against time the capacitor will discharge and it discharges according to the formula V V T that should be voltage in at any given time is equal to the maximum voltage V zero times e to the minus T over RC so the voltage after time T is equal to the maximum voltage times this term here so let's have a look and see what this term here does what we want to do is to keep the voltage as high as possible for as long as possible well if C is large then our C will be large why arrow that means that's large which means that 1 over RC will be small and that means that e to the T over RC will be small which means 1 over e to the T over RC which is essentially what this is will be large so if you keep see large then this term will be large and therefore you will get a much smaller fall-off in the voltage so this as it were decline in voltage would be for a small capacitor if you have a large voltage sorry a large capacitor then the fall-off will be very much less in other words you keep the voltage higher for longer and that's what you need to do here so what you need to do is to put a large capacitor which has which will charge up to a maximum voltage of V zero but it will slowly discharge and that means that as this alternating current varies the capacitor acts as it were as a DC supplier it effectively becomes battery supplying a DC supply whilst the current would go down and back up again it can smooth out that voltage and thus provide a pretty much constant voltage for the computer

Channel: DrPhysicsA

Views: 227,442

Rating: 4.907526 out of 5

Keywords: Alternating Current, A Level Physics, sine wave, power, root mean square, transformers, rectification, transmission, AC generator, electricity

Id: C_mxpy0bdoA

Length: 47min 46sec (2866 seconds)

Published: Mon Apr 08 2013