Free Physics NEET Video Lectures for Alternating Current

00:00:01 [Music] hello in this and the next few lecture I will be discussing what is known as alternating current sometime back we have talked about direct current circuits and I would like to point out that in our day to day use it is the alternating current which is more prevalent than direct current and we will be discussing the various properties that go with it but before you do that let me remind you of what you have done earlier on Faraday's law in our discussion of Faraday's law of electromagnetic induction we have seen

00:01:32 that if we have a situation where the magnetic flux changes with time then an EMF is produced the EMF relationship is given by the mathematical statement of Faraday's law which save EMF is equal to minus M D Phi by DT we're 5 the flux to each turn and n is the number of if you recall our definition of flux was integral of B dot DS over a surface they formula is written usually with a minus sign and that is because this is a reminder of what is known as Linda's law the lend-lease law talks about the

00:02:48 direction of current induced in this so according to Lenz law now the direction of such a noose current you is always such that the magnetic field that such a current produces ah poses they change that produced in other words it tends to negate whatever change was being shot by the agency producing discussed let's push this idea a little further supposing we have a rotating coil in a uniform magnetic field so I will just give you a schematic diagram so I have a coil which is rotating in a constant

00:04:26 magnetic field so this is the rotating coil and let's suppose this is the direction of the magnetic field which is uniform and the coil of course moves around this axle and suppose the direction of B makes an angle theta with the perpendicular to the plane of the question so that we represent it like this so supposing this is the direction of the magnetic field and this is a section view so this angle is Theta so this is the way it's rotating so therefore what happens is this that the flux produced is remember this would be

00:05:26 dot a that is B dot DS so this B times a times cosine of theta which is of course a function of T and that is equal to BA cos Omega T because the coil is rotating with a uniform angular speed Omega so therefore the EMF generated by it according to Faraday's law is n J Phi by DT which is equal to NB a Omega sine where n is the number of turns in the coil now this I can write as e 0 sine Omega T so since you notice that this EMF is changing sinusoidally so therefore this will produce a potential difference

00:06:32 which also change the Sun to some sinusoidally so let me say that the potential the voltage V is given by an expression like VM sine Omega T now suppose I am to plot this voltage against time so this is the representative diagram in the sense that when I say T equal to zero it doesn't mean that is the instant when the voltage was switched on but any particular time you can take it as time T equal to zero and proceed for one cycle so supposing at my time T equal to 0 this is execute time my voltage

00:07:23 happens to be zero and I go through one cycle so this magnitude the maximum magnitude of it so this is my voltage V as a function of time and this is VM that's the maximum voltage so in this picture what I have done is to say V equal to 0 at time T equals now the voltage V returns to the same value so this is kind T equal to 0 after a time capital T passing through a sum cycle and this team after which the voltage at any point in the circuit returns to the value that it had kind capital T default

00:08:17 is known as the period so T by 4 after the initial time the voltage becomes maximum and this is if I to this point where it becomes maximum but in the negative direction is three ty4 and this is full cycle T now if you want to compare this gap with that of what happens in a DC circuit the DC means there is no time variation so this is the DC voltage so this definition is V of T plus T is equal to V of T now this Omega is related to the linear frequency remember the linear frequency R is equal to one over T and

00:09:30 Omega then is 2 pi times F which is equal to 2 pi of T now let me then write down the simplest possible AC circuit so the symbol for an AC remember in DC circuit I had a battery type of symbol but here this is given like this and this is V equal to VM sine Omega T and all that I have is a registrants image so I am looking at what happens when a alternating voltage is applied in a circuit which consists of a resistance only so let me assume Ohm's law so my current i-10 is given by this quantity VM divided by our

00:10:48 time sine omega-t which we will write as IM maximum ensign well I am is the maximum value of the current abusers now I had already shown you the where the voltage varies with time you notice that the variation of the voltage just for comparison sake was given by VM sine Omega T so what do you realize is that the time variation is identical there the currents maximum is VM divided by R so therefore the magnitude of the maximum current would depend upon what the resistors are so if I'm in the same

00:11:56 diagram plotting the different plotting both the current and the voltage in the same time let's do that so I have a time diagram x-axis is time and on y-axis I will plot both the voltage and the current obviously because their scales are different their units are different so I will have two different scales on that so let me first plot the voltage so this is one cycle but let me just write down our cyclops perfect so this is your VN supposing in the same bag so this is yeah supposing on the same diagram I

00:13:06 also plotted the current in other words the current will be plotted but with this scale being different so voltage is in volts and I will be plotting here the current will be in amperes and depending upon the value of our I will get a different value of I maximum but the important point to note when the voltage becomes maximum the current becomes maximum and y 0 so therefore my plot of the current would be something lately each part which is identical this doesn't look identical because of friggin doing so this is your eyes

00:14:05 so the point that I am trying to make is this time here or let Omega D doesn't matter point that I am trying to make is this current I becomes maximum or minimum at the same time as the voltage now one of the things which is frequently done is to plot what is known as a phasor leg now a phasor diagram in basically a polar curve with its x-axis being a reference line at time T equals e so let me just plot this and I will go on explaining what is the favored so this is some reference line I have

00:15:04 taken with respect to this time that what I call it the initial time I am plotting everything so I assume that at time T equal to zero I have a vector of length V M which is aligned with this so axis so let me put it like this imagine one end of that vector is at the point of origin and it has a length equal to VM and endpoint is a so this is so let me say that Oh a vector has a magnitude equal to BM now what I assume is this that keeping the end or fixed this vector rotates about an axis which

00:15:58 is perpendicular to the plane of the paper passing through with the angular velocity Omega so that at time T equal to T the angle that should have swept is Omega T and this vector then lines up like this so the magnitude still remains the end but it goes to the point B where this point this angle is omega times t now suppose I take my variation of voltage with time as I have said a little while back V of T is equal to VM sine Omega T this tells me that if you take the projection of this vector V OB at time T

00:16:56 equal to C then this gives you the instantaneous values of voltage now supposing instead I have taken VT equal to VM cos Omega T then they make projection along the x-axis would have given me that now let's look at what happens to the current now in a phasor diagram like this I plot both the current and the voltage in the same diagram but since current and voltage have different units of measurement I can choose my scale appropriately to make these lengths of vector the way I want so so what we do is this that

00:17:45 suppose I write I of T for a purely resistive circuit to be given by I M sine Omega T remember the current and the voltages are in case them as all time the current phaser is lined up along the direction in which the voltage phasor is lined up and suppose I decide on a scale in which the current magnitude is given by the length of the vector Rho C so OC magnitude is I M then the projection of this OC at time T equal to T is gives me the instantaneous value of the current now an important point to take home from

00:18:41 this is for a purely resistive circuit the current isn't raised with the voltage now let us look at what is the average value of subtle correct but before I do that let me define what is meant by the average of a quantity over a cycle so suppose I have a time dependent quantity f of T the average of F of T over a penis it is written like this or you could also write it like a bear of you whatever there is no standard way of doing it is one over T integral from 0 to t of F T so let us look at a time dependent

00:20:12 quantity such as current which is given by I of C equal to I M spying upon it so average of I of you to look at this definition would be I am 1 over T integral from 0 to T sine Omega T DT you recall that integral of Phi Omega T is minus cos Omega T by Omega so I choose I am 1 over Omega T and if you take the limit it is cosine of zero that one minus cosine of Omega constructed so ad to find out what is this recall by definition of time period I have Omega times capital T is equal to 2 pi so therefore what I have here is cosine

00:21:21 of 2 pi + but cosine of 2 pi has the same value as cosine of 0 so therefore this quantity equal and this would also be true for a function such as also valid for you could have sine 2 Omega T 3 Omega 2 etc or even cosine Omega T cosine 2 Omega t that's there is another relationship that we would require as a golem how much is average of fine scale so let us plug this into the definition average of sine square Omega T yes one over T integral sine square Omega T DT from zero to two am you recall your

00:22:36 multiple angle formula which tells me that sine square Omega T is written as one minus cosine of two Omega T / - and I've just now told you that any multiple of sine or cosine Omega T integrates out two averages out to zero so therefore they only think that I am left with is which factor half DT education et so therefore this gives me one over two and that is because average cosine of zero negative people we'll be using this arugula so what we have shown in the processes the average current is zero

00:23:43 incidentally this doesn't imply that the power dissipated is zero because power is given by I square so average power dissipated in the circuit this average of I square ur and which is equal to I am square R average of sine square Omega T which I just now boo the average of sine squared Omega T is half over a cycle so it is I M square R divided baking so notice that this formula have some similarity with the power dissipated in DC Circuit's but for this factor now we could remedy this situation and make b2

00:24:47 formula very similar provided we define a new quantity which is known as the root means we're correct usually denoted by I RNA I can similarly define root mean square voltage but let's stick with this map as the name suggests the root mean square is take the square take the mean of the square of the thing and then take a square root so the way one defines it is like this it is square root of average of I square but we have just now seen that I square of T is I am square by two so therefore

00:25:56 I are NS happens to be a n divided by square root of two and likewise we can define a VRS equal to VM divided by square root of two now if you are plotting for example the current I of T remember that my current has a sinusoidal variation so this was my maximum IMAX this will mean square which is I M by square root of two is about seventy percent of this value because 1 over square root of 2 is about 0.707 so therefore my root mean square value is somewhere here one by this piece I am so once you start using loop means you

00:27:02 are current instead of the maximum current in this circuit you realize immediately that I can write P average as equal to I RMS square that is because there is a 1 over square root there to there which would take care of the factor of 2 find that and the formula would then be looking very similar to what we had seen in case of a DC Circuit now I would like to point out that when we talk about the voltage supplied to our homes for us in India for instance the supplied voltage is AC and berries generally the

00:27:54 supposed to be 240 volts but it generally varies between 220 to 240 is between 220 to 240 supposed to remain around 240 and the frequency linear frequency nu is taken to be 50 since many of our textbooks are of American origin I would like to point out that in the u.s. s the household supply is about 120 volts and the frequency is 60 and that is the reason why when you go abroad to a different country you need adapters to match the if you your equipments are designed for a particular voltage or a

00:29:05 frequency then you need adaptation so we have discussed already what happens when an AC source is connected to a resistor but that's not really a very interesting the alternating voltage they become more interesting when you put into the circuit other elements in particular the inductance and the capacitance about which you have learnt in your previous lecture so let me first talk about and alternating source applied to or connected to and a purely inductive load what it means is that the circuit does

00:30:08 not have any resistance no resistance and the only thing that is there in the circuit other than your AC source which will take it to the VM sine Omega T is an end up now once again I'll use the ketchups lawyer and you remember from your discussion of Faraday's law and the properties of inductance that inductance provides what is known as a back here so what we get is this and this back EMF provided by the inductance is minus L di by so therefore if I use the ketchup slide the circuit I had V of T at any instant of time

00:31:18 minus L di by DT is equal to Z which tells me that di by DT is VT over l but VT is known to be VM sine Omega T so this is V M over L and sine Omega T if I integrate this now so my I as a function of time will be V M over L sine Omega T which is equal to minus DM over L Omega Oh sign now here I have taken the constant of integration to the zero because we have seen that the voltage has more constant component and it is symmetrically populated around B so therefore my current should also not have any

00:32:28 constant purpose and should all show oscillate symmetrically about it so what is this in case of a purely resistive circuit the trigonometric form of the time variation of both the current and the voltage was identical but now I have a different and sin I can write this cosine function as sine of Omega t minus Phi by 2 right X square of the - minus Z so what I am getting in this process is that the amplitude current amplitude is given by V M over L M so this is current amplitude another interesting thing you notice is

00:33:42 that since the voltage is varying as I know Vega but the current in this circuit is varying as sine Omega t minus Phi right so what I notice is this that the current Falls behind the voltage or in our language there is a fav lag or PI by 2 with respect to voltage the purely resistive circuit had no such leg but now we have realized that the current lags behind the voltage this quantity which comes here as in the expression for current is given a name this Omega times L now if you compare this with the

00:35:04 direct current circuit you realize that this is playing the role of register but there is a difference this quantity depends on Omega and this is given a name it's called inductive reactance usually dinner denoted by Excel so Excel is equal to Omega tensor so basically what happen until something like this the Excel supposin are plotting this not such a sequence then Excel is linear this is your acceleration but as the frequency increases the current decreases this is the way your current will let the current at any instant of

00:36:08 time would change provided I am plotting it against the frequent so this is called inductive reactor so what actually happen to this then as the frequency increases the inductance value of the reactant increases and as a result the current decreases let me plot the current and voltage as a function of time so let me go the voltage waveform so this is completes one cycle by size we also draw under to more than that so this is my time tea and what does plotted here is the voltage in this blue so this is V of T this is T by 2 this is

00:37:10 T this is of course 3t right now in the same block first let me write down what I have water here is voltage of course in words in the in the same figure I will be plotting the current but since current and voltages are measured in different units I can choose different scales for this and I've seen that for an inductor the current lags the voltage in phase by PI by 2 it means it lags by water of a cycle so let me make a note here that for the inductive circuit current relax the voltage in phase by PI by two

00:38:22 which means by one quarter of a circle so in this figure when the voltage is zero my current would be negative and maximum magnitude so let me plot it here and this is quarter of a cycle so therefore that Sun the voltage has reached maximum my current would become zero so the nice to divide this by t by 4 you so I cannot put be something like this well we don't exactly look like a ten come but since it's a free and going this sort of an acceptance so what I've got here is this this is the red curve

00:39:38 is Ayaka and you can see that it is lagging in phase by team effort so that it is maximum negative at time T equal to zero and voltage review and when the voltage has reached its reached its maximum the current is zero and then as the voltage remains positive and is decreasing and adjust about reached view the current has reached its maximum and this maximum is mine I am so let me write down in red I have current in our users and this magnitude is as media so let's look at how does the favor by Gramlich

00:40:40 this is for an inductive circuit so let me for reference reproduce the current and the voltage curve remember this one by voltage so this is by time thing and this is time T by 4 T by 2 3 T by 4 and of course time T equal to 0 I repeat that this doesn't mean that I am switching on the EMF at time T equal to zero but it is a representative curve starting from any origin that you have taken from the time and with respect to which I am drawing this now the corresponding current curve was something like this remember that this

00:41:46 is the way the current base so this is current and this voltage so if you look at this diagram and find out what is the phasor diagram look like for this so as before I take the initial line to the T equal to zero which is the reference line the voltage which are the magnitude beer is a vector of length OB so once again OB magnitude is equal to VL and at time T it makes an angle Omega times T with the exact as I have said several times the projection along the y axis gives me the instantaneous value of the voltage

00:43:00 now sense the current lag behind the voltage by PI by 2 it means that current becomes maximum effing water of a cycle after the voltage now the thing is is that because the lag the time lag between the voltage and the current P is PI by T so when this is really first quadrant the corresponding current will be in the fourth quadrant and this angle will be 90 degrees so this is my current there this is what I have represented by or C this so OC is magnitude is I and of course it automatically means

00:43:59 that if the voltage goes to the second quadrant imagine this entire thing being rigidly being rotated by certain angle taking OB to the second quadrant by that time the current would then come to the first one so current would also become positive so this is the way it is when voltage is positive the current is negative in the first quarter cycle in the next quarter cycle both of them happen to be positive in the third I have a negative from here and a positive from there and finally to conclude it both negative voltage as

00:44:47 well as current are both negative what about the power and an inductor so let's look at instantaneous power so power is given by I times V this is instantaneous so instantaneous current multiplied by instantaneous voltage and this is equal to I n sine Omega t minus Phi by 2 and V is equal to VM sine Omega T so it is equal to I am VM we have seen that this is minus cos Omega T and there is a cosine Omega T it is nothing but minus a n BN by 2 sine 2 which tells me that the power over a cycle the situation is very similar to what

00:46:14 happens in a mass spring system supposing I have a mouse on a frictionless table connected by a spring no fiction now as you set the system into motion the mass gets gains kinetic energy at the expense of the potential energy of the spring and later on returns that amount of energy as the potential energy of the spring and it loses its kinetic and excess conservative system because the only other thing here which could take away or dissipate power that is the friction which we have assumed that does

00:47:14 not exist you and I identical thing happens in case of your inductance in the circuit so inductor in one cycle part of a cycle would absorb power from the circuit actually retain it and return it to the surface in the next person the effect of what is said is that for an inductive circuit the power over a cycle is zero so for a an inductive circuit the hpower equal to zero oversight which is another way of saying that they appear li inductive circuit conserves energy and compare this a purely resistive

00:48:21 circuit for which the average power was shown to be equal to I square R and actually I RMS square so let me elaborate on this already let us plot the current voltage relationship in this case so this is x axis this time and in the same plot I'll be plotting both the voltage and the current and this blue line on the blue curve is my voltage bit which is obviously goes so pinky in volts I will also not now the current in the same curve in amperes so n IG in and yes and since I know that the current

00:49:34 foreign inductive circuit lags behind the voltage by quarter of a cycle the curve that I get from the current agency notice my scales are different because in one case I am plotting current in other cases what is a pitch so this is is now let's consider an instant of time for the modalities now since the current at that instance has a maximum so that this is my time till I say you and let me also give some labels to be this is d-max and in the red curve this will be slightly different scale a second this

00:50:40 should be eye so let's look at how does our current and voltage but in this cycle some tea by for two they like you it is yes notice my current is greater than you know that is the heat level that is di by DT with this put in place the by voltage V often associative no it can also be seen of course in the voltage service so it tells me that power which is I times V is greater than zero in this side so since the power is greater than here it implies that the energy is absorbed from Bishops you

00:51:55 so the various signatures are the common voltage plus current now let's go to the next one cycle from T by whoo-hoo treatable the current which has become maximum starts decreasing but it still remains positive so I is greater than zero but di by Vicki is lesson which as you can see from the curve and from the signature of B I by DT V of C becomes nigut so which means power P IV is less than zero which means the energy that was absorbed in the previous quarter circuit a return to this in the next such so this online

00:53:10 simulator so in this section as we have seen my voltage is negative got this in the next one cycle from 3 P by 4 to see my current is less than so is the identity which is VT evangelist but since both have negative my power is greater than their quick implies again that the energy is absorbed so decide from 55 power 2 T so here you know if this is negative and shows the current now if I went from 3 to 5 people the situation would be simply a replica off what happens from 0 C T by 4 and there you can see then the current is negative

00:54:36 the voltage is positive 1 second it means that whatever energy was absorbed in the previous water cycle is returned so this is positive let me close this discussion with a couple of examples suppose as a 48 million Riel inductor connected to some of this number five children cells as arithmetic become PD connected to 240 volts 50 Hertz I need to find out what is RMS current I do not have to tell you that whenever we are giving the value of voltage or current there are in the sense otherwise we would specifically point out that

00:55:55 these are pigments so in this case my first job is to find out what is my reactor my reactance is Omega times L Omega is 2 pi nu and nu is 60 or so to find the 50 the inductance is 48 million reach over 448 into 10 to the power minus 3 and that is 4.8 I which is a calculate works out to fifteen point zero eight of Oh so our - value of current is 250 divided by fifteen for zero it is let's take it as fifteen thirty will be sixteen average this value of connectivity other things is much higher than the typical

00:56:54 household which is generally limited to about 8 to 10 amperes but that's primarily because in this artificial stuff it that I have conferred I have not taken any serums normally resistances are there in the circuit which is limit the value so what we have done today is to define an alternating source voltage and seen that when this is connected to a purely resistive circuit the current and the voltage is are in phase but when you connect it to an inductive circuit then you find that the current lags behind the voltage the

00:57:43 second point is that in a purely resistive circuit there is a power dissipation which is given by the same formula as for DC excepting that I have to change my definition of current with I remember I square R where R a purely inductive circuit does not dissipate power whatever power it absorbs in a part of the cycle is removed returned back to the circuit in another [Music]