 So let us begin by reviewing what we did in the last class we started out by looking at the model for a simple wire loop and we found that if this is excited by DC we can write a simple equation V equals R into I, however if it is excited by AC we need to add another term P times P, what we saw in the last class was that a current in a loop like this produces a magnetic field and this field exists all over space. If you have a wire loop here then the field that is generated exists all over space and of course decreases as you go further and further away from this wire loop. If you have only air around the loop then the field that is generated is quite small we saw the levels yesterday in the simulation that we showed you found that the field levels or the flux density levels were very small when it was just air around the loop. But the moment you introduce iron if you introduce iron in the vicinity of the loop then we find that iron causes a much higher field to be generated for the same current that is important for the same current you get much higher levels of flux density. And this much higher level of flux density exists primarily or maybe I should say exists within iron region in the region outside iron the field is still pretty small it is not that field does not exist outside the iron material field does exist but the levels are very small outside the iron area and this is the reason why in all electromagnetic equipment electromagnetic apparatus iron is used iron in the sense not the it may be an alloy of iron in some form iron is used okay and we saw that the field is described by various terms if you have a magnetic field you then speak about flux this is then measured in Weber given the symbol WB then you talk about flux linkage which is the number of turns linking this flux. So that is still Weber or to be more explicit sometimes it is written as Weber turns is not an SI unit so exactly if you want to say it is SI unit is Weber but one can say Weber turns to be more explicit to distinguish flux linkage from flux one may call it Weber turn and then you have flux density which is the number of flux lines in a given going through a given area if you have flux lines going through area you are looking at how many flux lines go through this area determines the flux density at that particular area therefore that is measured as Weber per meter square it is also called as by the name given here this is the symbol for this is just T and then you also have magnetic field intensity which is measured in ampere per meter or more explicitly ampere turns per meter. So this is given the symbol B and the field intensity is given the symbol H so these are all some of the terms by which you assess the field you describe the field and we know from our elementary physics that B and H for a given material are interrelated in some way so B you normally say BH loop right so B and H are interrelated by generally a non-linear shape so with all these known then you write down this equation V equal to RI plus PSI where PSI is the flux linkage and then we saw yesterday that in order to estimate how to determine in order to determine this value for PSI we went about saying that you looked at the equation N into I is equal to circular integral of H.dl which in the case of a simple iron core if you consider an iron core with a wire loop going around it then it is generating magnetic field inside you have magnetic flux lines that are going around and if you now define a loop that goes around this field around the line of magnetic flux then H.dl simplifies to H into length of this part and therefore this is the same as H into L provided the flux density is uniform everywhere and this then gave us an expression for H is then N into I by L and we then determine the flux that is generated as flux density multiplied by area of this core if you take a section and B is the same as µ into H therefore µ into H into AC and H is nothing but Ni by L so you have µ into N into I divided by L multiplied by AC this can be written as N into I divided by L by µ into area of the core and now you see that we have arrived at an expression of the form which is very similar to electrical circuits if you had an electric circuit excited by an EMFE having applied across a resistance then the flow of loop current I will be given by EMF divided by the resistance this looks very similar you have something that is flowing around the loop which is ? and that is given by the ratio of N into I divided by some number. So this expression indicates that the electromagnetic system that you have can simply be analyzed or looked at as an equivalent magnetic circuit similar to what you have as an electric circuit instead of looking at all the field equations and looking at integrals like this one can then simplify the system in this way so now you have in this the excitation source is known as magneto motive force magneto motive force which is MMF and this magneto motive force is applied across an entity which is called as reluctance and when magneto motive force is applied across the reluctance the ratio gives flux that is flowing this is the same as an electrical circuit where instead of an MMF you have an EMF electromotive force and electromotive force applied across a resistance then allows flow of current here a magneto motive force applied across a reluctance causes flux to flow so in this way one can simplify the analysis of this kind of a circuit by looking at the magnetic circuit and then if you substituted this expression into the equation V equal to RI plus PSI what we landed up with was V equals RI plus D by DT I this term P is an operator which symbolizes the operation of differentiation with respect to time it is an operator that symbol we did not use last class so now I am introducing this will be frequently using this symbol P so V equals RI plus P of PSI is nothing but N times Phi so N squared I divided by the reluctance R so I will call this as Ni divided by the reluctance R the reluctance I am giving the symbol R right so one can further write this as RI plus L into DI by DT where L then is N squared divided by R so that then is the inductance that is offered by this system to the electrical circuit in attempting to list this inductance we have implicitly made certain assumptions which we must be aware the inductance in determining the inductance we have implicitly assume that flux path is known and well defined for if it were not well defined you could not determine the area through which flux flows and you cannot determine the length of the flux path right so the first assumption is that the flux path is known and is well defined you cannot attempt to have this kind of an expression where there is a loop and then flux flows all over space right so flux path must be known the second assumption is that we have we have removed L by µ AC that is a reluctance out of the differentiation operation and therefore we are implicitly assuming that the magnetic circuit is linear which means that the relationship between if you plot I and then you plot the flux linkage as a result of flow of current then this relationship is a straight line this is another assumption that we are making in other words R is fixed it is just a number it does not change with respect to time or with respect to I with these if these two assumptions are made then we can define something called an inductance for the circuit and one can write the equation in this form RI plus LDI by DT it is good to address another issue here some of you may already be looking at that this expression V equals RI plus PSI this D by DT of PSI that is there is an expression of Faraday's law and may be some of you are already having the doubt well this law says that the EMF or the EMF induced is equal to minus d PSI by DT how come this equation says RI plus d PSI by DT so let us understand this now because we will be using this expression again and again so let us understand this in little more detail okay we have this figure here right so if you are now having a flow of current into this how do you determine which is the direction of the flow of flux in the core you will have to use your right hand rule which then says that you curve your fingers in the direction in which this current flows around this ion core which would be like this because current flows behind the core comes forward then goes behind so it curves around the core like this then this finger shows you the direction of flux in the core so the flux in the core is as shown that is this way now this law says that if this flux were to increase which means that this will increase if I increases you know that flux is simply dependent upon I N and R being fixed numbers Phi is directly proportional to I so if I increases flux will increase and this law says that if flux is going to increase it will cause an induced EMF so the issue is how is the induced EMF oriented right and we now have lenses law which says that the direction of induced EMF will be so as if you allow the induced EMF to circulate its own flow of I then that flow will oppose the increase in flux this is the statement of lenses law lenses law does not state that the EMF opposes the rate of change of EMF opposes the increase in flux EMF does not increase but if that EMF were allowed to circulate its own flow of current then that will oppose the increase in flux now if we want to therefore understand how this EMF which is orientation of EMF in this loop all we need to see is how should the direction of flow of current be in this loop in order to oppose flux lines flowing this way if I increases this way flux lines will increase in this direction and therefore you want to oppose this flux lines which means that current should cause flux lines flowing up which means that you need to have flux lines upwards so the fingers have to curl around this way which means that has to circulate a current flowing outwards like this and an EMF will be able to circulate a current flowing outwards only if it is positive here and negative here only then this EMF will circulate a current outwards so now we can see how to write this expression let us say you have some resistance and this is the direction of current and you have a voltage source here which is V plus minus and therefore if you now write the loop equation you have travelling around the loop this is your induced EMF E and the drop across the resistance since current is flowing this way is plus minus here so if you now write the loop equation you have minus V plus I into R plus E equal to 0 or in other words V equals I R plus d ? by dt the direction of d ? by dt we have decided by the arguments here and therefore what you need to put down here is only the magnitude of d ? by dt and therefore what you have is the same equation as what we have written so this equation is right you do not have a minus d ? by dt in this expression it assumes then that we know what is the direction of induced EMF which is determined in some way. So now let us get back to what we were discussing we have therefore seen that if the flux path is known and the magnetic circuit is linear one can define the number called reluctance the entity called reluctance right how does the situation now change supposing I have the same core with an air gap that is introduced air gaps are always used in electrical machines if there was no air gap you would not have anything which will be able to rotate everything is fixed so there has to be an air gap allowing something to move. So we need to look at what will happen if there is an air gap will the idea of having a magnetic circuit still hold and if so how to determine the magnetic circuit. So let us now look at the situation you have a coil and even here you find that when you excite this when there is a current that is flowing into this definitely you are going to have magnetic field that is generated the magnetic lines of flux now have to go through the score pass here come here cross the air gap and then complete this loop okay. So if this is the case let us apply the same rule ni equals integral of h.dl the loop that we need to consider is this loop around which flux flows and that now says n x i is integral of h.dl around this loop and if h is assumed to be the same all around this loop because area of cross section is same flux density is same then it amounts to same h in the core multiplied by the length of the flux path within the core plus h in the air gap multiplied by length of the flux path within the air gap. So this is the simplification of this equation which is an integral equation. Now h that is there in any given region whether it is ion or whether it is in the air can always be written as flux density in that region divided by µ in that region µ x lc plus b in the air gap divided by µ in the air gap multiplied by l and this can in turn be written as flux in the core divided by area of the core which is nothing but b is flux per unit area multiplied by µc x lc plus ?g divided by area of the air gap µ of the air gap multiplied by lg but you now see that flux in the core all the flux lines that are flowing in this core has to flow through this air gap there is no other way or we assume that there is no other way very little flux may cross somewhere here or somewhere there right but such flux would be negligible which does not go through this path. So flux that is flowing through the air gap is the same as flux that is flowing through the core it is all in the same loop and therefore one can call ?c equal to ?g let us take it out as ? then it is lc by µc x ac plus lg divided by µ of air gap multiplied by area of the air gap and we have already seen this number l by µa is what you already have here l by µa which we defined as the reluctance. So one can now write this as ? multiplied by reluctance of the core plus the reluctance of air gap and therefore you get the expression ? is nothing but the total mmf divided by reluctance of the core plus reluctance of the air gap in the earlier case you had ? equals ni by r and therefore what we can say is that the circuit here looks like an mmf n x i applied across a reluctance which is r there is a single value for r because there is only one core that is all there is but now in the path of the flux you have the reluctance of the core plus the reluctance of the air gap that is coming in series in this loop and therefore you now have the mmf that is generated here and then you have the reluctance of the core reluctance of the air gap both occurring in series rc and rg right. So in this manner one can reduce a given system into an equivalent magnetic circuit it is important to remember all these expressions because we will be using them again and again when you describe various electrical machines okay. Now one may have an iron core which is a little more sophisticated or more involved not a simple geometry like this but a more complicated geometry let us say that you have an iron core which looks like this this is now an iron core and you have a winding that is here there is a current flowing in this. So how does one now determine what the equivalent circuit is is very simple now if you excite this imagine how the flux lines will flow the flux lines in this case obviously you would have flux lines going down and now this flux that comes here has one path this way and another path this way so these flux lines will half of them will flow like this another half would now flow this way right. So the equivalent circuit will then be an MMF that is generated here since it is allowing flux to go downwards MMF is generated here flux is flowing through this path therefore there is a reluctance of the central limb which is equal to length of the central limb divided by µ of the material multiplied by the area of cross section of this limb and then there is flux coming from here and therefore there is a reluctance there is flux coming from here there is a reluctance here that is the same flux coming through this you have a reluctance here and then a reluctance here then a reluctance here and here the values of all the reluctances that we have drawn will depend upon the length area of cross section of this and the material of course. Now you may have different areas here and here these three areas need not be the same area of section need not be the same it may be different depending on how you design the right so appropriately one has to find out what this is and then the next issue that arises is what length do you take because this limb extends all the way from there to there so do you take that length or do you take only this length. So when we say length in L by µa the length refers to is normally taken to be the mean length mean length in the sense you will have some flux lines that are going close to this and there will be some flux lines that are going here so the mean length in this part of the core will be if you draw a line connecting these two parts and then you take the midpoint of this draw a line connecting these two take a midpoint here then this length is considered to be the mean length similarly here take this midpoint this will then be the mean length and here that would be the mean length right one can look at that as the mean length of the way through which flux goes so using this one can determine an equivalent electrical circuit for analyzing a magnetic circuit this is then your magnetic circuit and this is an electrical equivalent of this circuit once you have this you know the mmf ni and you know all the resistances here so one can do a very simple network analysis to solve this and find out how much flux is going through each one right. So in this manner one can now understand how much flux density is there in various parts and all these things so if you know all this then one can find out what is the inductance of this loop now as I said inductance is an important aspect in electrical machines and it is also necessary to understand something more about inductance. Now let me draw this again you have an iron core and you have a coil that is wound around this now if you have a flow of current here it is going to generate flux we have been seeing it again and again it is going to generate flux and all of the flux generated by this excitation through this current I all of it is going to somehow link these turns some of the flux lines will go through this core and as we saw yesterday the flux density around the core is not really 0 as you go further and further away the flux reduces flux density reduces but near the core the flux density is not really 0 it is small but not 0 so you will have as we saw yesterday some flux lines that go here. So not all of the flux is confined to iron core a majority of the flux is confined to the iron core but a small part lies outside the iron core as well right but all of them link this loop and therefore if you want to find out the flux linkage ? you have to obtain it as n multiplied by all the flux that links this whether it goes through the core or whether it is in the air everything has to be considered in the expression that we wrote earlier v equal to r i plus p ? and then when we derived the expression for ? as ? equals b ac and all that we had implicitly made another assumption that apart from these two there is one more assumption that all flux is in the core otherwise this expression would only give you the flux in the core and these additional flux lines that are there the effect of that should also be incorporated in that equation v equal to r i plus p into ? we had not looked at that and therefore the implicit assumption is that these fluxes do not matter but in reality they are there whether you want to neglect it or not is a different issue but they are there now this is fine but invariably in electrical machines you do not have just one wire loop there are many more and therefore what happens if you now put another wire loop here let me draw that with another colour a second wire loop that goes around like this. Now there are some important differences you can now see that the flux lines that are drawn here outside in the air do not link this loop whereas flux lines that are in the iron do link this loop and therefore when you talk of flux linkage of this coil let us call this coil as a and this second coil as b so when you talk of flux linkages of a it is n into ? a but now this ? a can be thought of as n into flux that is linking both a and b which goes through the iron I will call that therefore as mutual flux and flux that is not linking a and b which links only a alone I will call that as leakage flux so there is a difference now overall it is the flux linkage of a but we can imagine or divide that into two parts one is mutual and one is leakage now if there were one more suppose you extend this into a third limb and put another coil here frequently machines are three phase machines so you may have one more coil here and then there will be some flux lines that not only go here but there would be some flux lines here also so there is now a difference there is flux that is generated here some part of it links this coil some part of it links this coil and this plus this together is what is flowing here apart from whatever is there inside so now you cannot just call it as mutual plus leakage right mutual refers only to flux that is mutual between coil a and coil b now you have coil c as well so how do you do that so we then distinguish this as not mutual flux but n into ? leakage refers to flux that links only this coil and nothing else and then you call it as ? magnetizing this flux is the flux that goes links everything else apart from a so this flux would then be all the flux that goes here it links out of this flux there is a few lines that link here a few lines go there but leakage links only this and therefore you know that the effect of flux is what is felt on the electrical circuit or defined from the electrical circuit side as an inductance so this leakage flux on the electrical circuit is represented as a leakage inductance L leakage normally called as LL and this part is then represented as LM which is the inductance for magnetize this is your this is the magnetizing inductance this is leakage inductance and these two together leakage inductance plus magnetizing inductance is then called as the self inductance self inductance which is normally given the symbol LS can then be written as LL plus LM now if you are referring to flux that links coil a and specifically coil b then such flux is called as mutual flux and therefore the effect of mutual flux on the electrical circuit is brought out as a mutual inductance so mutual inductance only refers to the interaction between two coils whereas this one LM is the effect of flux that a particular coil generates which then affects everything else other than the leakage flux part of it so that is mutual induct how do you now derive expressions for this we can use the same ideas that we had earlier now in order to understand this let us take a simpler situation and consider only two coils now if you look at the flux lines that are going here one can see that the flux lines going there are may be given by N into I is the MMF that is generated here divided by the reluctance of the core reluctance of this flux part and we know that the inductance is given by N square by RC and from this one can get the magnetizing inductance as NA square divided by RC I am using the expression that we derived earlier may be to explain this Ni by RC is the flux and then flux linkage is Ni is NA into I because flow of current is through a so NA into I flux linkage is NA square into I by RC and from this you get the inductance as NA square by RC now if you look at mutual inductance the flux that is generated here links the second coil also and therefore the flux linkage is NA into NB into I by RC and therefore the inductance which is L mutual is NA NB by RC so there is a relationship between mutual inductance and LM one if you know this one can find out this provided you know what NB is so in this lecture we have gone further into the idea of inductances and we have seen how the electromagnetic device can be analyzed or thought of on simpler terms how the reluctance can be defined and using that how different inductances can be arrived at though the electromagnetic phenomenon is the same for our utility we can look at the magnetic field into various groups the leakage and flux that links different things and therefore we can imagine the system to have leakage inductance and then the mutual inductance and magnetizing inductance in the next lectures we will see how these are to be used and the importance of inductance as I said will be known in the next lecture so we will close with this for today.