 in this next lecture on the modeling and analysis of machine scores, so let us go ahead and look at the elementary ideas on how we can model electrical machines. We know that all electrical machines have an electrical force through which you are going to give electrical supply and then there is a mechanical force at which point you are going to connect the load. This is what we had already seen in the last lecture and we start off from there. The aim of our exercise as I mentioned last time is to derive relationships between the electrical side, the inputs you give to the electrical side and the response of the system because of that. If you look at the electrical side, you know that an electrical machine has machine windings, if it is a DC machine it has an armature, if it is an induction machine it has three phase AC windings and alternator also has three phase AC and this is in its elementary form it is nothing but a wire loop having an input electrical port where you can then connect an AC supply or a DC supply. In the simplest form let us say we have a wire loop and then you have a DC supply and you are going to connect this to this wire loop. So how do you now write equations for this? It is very simple being a DC circuit it is very elementary what you would say is the applied voltage V is equal to the resistance of this loop multiplied by the flow of current let us call it IA and this as VA. So you have VA here and this wire loop has a resistance R there is a current that is flowing that is IA and these three are then related by this equation. Now this is fairly straight forward there is nothing else to do we have already arrived at the model for this wire loop as far as it is DC. Now supposing you are now going to vary this DC voltage in some manner and you want to find out how this current is going to change. You may say that well I have an equation VA equal to R into IA and if I change VA IA is going to change in proportion that is not really true because if you are going to change this applied voltage no doubt the current will change if you change low enough yes IA you will see is going to change in proportion to VA but if you are going to change this VA rapidly then you need to consider some more effects and to be more relevant to the case of electrical machines suppose you have an AC supply where this voltage is now going to change fairly rapidly in the case of AC supplies that we have in India you can the AC supply is going to change at the rate of 50 hertz. So if that is going to change then the relationship between VA and IA is not just this why it is not just that that is because any flow of this kind of IA in a wire loop is primarily going to generate in addition to the other effects it is going to generate a magnetic field. Now it is this magnetic field that it is going to generate that is what is going to cause the difference between a DC excited wire loop and an AC excited wire loop. Now let us look at an experiment which you might have done earlier to see what this magnetic field is now this is an experimental arrangement of an experiment that you may have done in your high school days an experiment intended to demonstrate the existence of a magnetic field. Now field is an idea which has been proposed in order to explain the observation of action that happens at a distance you do something that happens here some other action happens somewhere else even though apparently there is no physical connection. Now you are going to see an experiment where due to the existence of a magnetic field iron gets aligned along certain ways these then demonstrate that there is an effect of the flow of certain amperes in a loop on the iron which has no physical contact with the wire loop. So in this experimental arrangement what we have is a wire loop having several turns and in the horizontal plane of the wire loop I mean horizontal level we have then added some small particles of iron and upon excitation of the wire loop with DC flow of current and if you now gently tap this these small particles of iron they then tend to align themselves along certain lines which you can see you can see that there is several lines here and then similarly several lines here and in the center there are broader low lines here right you can see that they have aligned themselves and these lines are then known as the magnetic flux lines which then represent the existence of a magnetic field. Now it is easy to look at these arrangements in a simple loop like this but in an electrical machine this would be far more difficult because the arrangement of the loop is not so simple and therefore if you are going to look at electrical machines and you want to find out how the field lines like this arrange themselves one has to do it by some sort of simulation which we shall see a few examples of that now so that every time we do not have to look at these kind of experiments we can then do completely by simulation alone. So here we have some simulation results if you want to simulate how simulate and analyze how magnetic field distributions are there in a particular region of space you generally use simulation software which are which go under the generic name called finite element analysis based simulation software. And here we have the results from a simulation of the system that we have just now seen which is a coil this area designates the coil effectively this shows that if you have a plane and a coil that goes in and out of that the distribution of the lines is what you have seen on this plane okay. So this area which I have marked and read shows the section that you get if you slice this coil as you have seen here by this plane horizontally if you slice it then you see this rectangular area as the one having several turns and some flow of loop current through that in this case the simulation has been done for a 50 turn loop having i equal to 10 ampere flowing through each turn. Now with this you can see how the field lines are this is the simulation result of how the magnetic field lines are going to look like and here you can see the magnetic field lines as you have seen in the experiment the field lines go circular near this and then they take bigger loops here and here still bigger loops if you allow it this will go and complete somewhere far away similarly on the other side this goes and completes far away so you have this outputs can be simulated now what we see here are magnetic flux lines which denote the existence of a magnetic field now in addition to this you also describe something called magnetic flux density which is an indication of how many flux lines are there in a given area if you now take an area which is going to be at this place for example if you take here the magnetic flux lines are going to be loops like this as you have seen here and if you now take an area which is equal to which is inclined at an angle 90 degrees to the flow of these lines then how many such lines are there in a given unit area that is indicated by the magnetic flux density what I want to say here is the magnetic flux density is nothing but flux lines per unit area so this is generally given the symbol B so B denotes the magnetic flux density if there are more flux lines in a given area that means flux density is higher if there is less flux lines and flux density is small so obviously in the regions here you find that the flux density is more because there are more flux lines and as you go outside far away from this loop you find that the flux lines are fewer so in all these areas flux density is small now an indication of this is also there from the simulation software and that you see in the next plot which shows how the magnetic flux density looks like now this indicates how the software has given an output regarding the flux density in different areas you see that if in areas where you have deep red which is here deep red areas deep red areas the flux density as you can see from here is about 6.179 x 10-4 flux density is measured in the units of Weber per meter square is also called as in the units of Tesla whereas flux itself flux lines flux is measured in Weber which is abbreviated as WB and flux density is abbreviated as E now you see that the flux density in regions here are higher than the flux in regions outside so if you go far away the flux density decreases you can see on the scale here flux density which is in the regions of red is about 6.179 x 10-4 and as you go down to the green regions it falls down to 1.5 and so on and finally in the regions which are really far away you have flux density which are very very small very little flux is there now these levels of flux are really small and not of much use in electrical machine you want higher flux and if you want higher flux you need to change the medium in which this is being the magnetic field is being established and for that what one does is to put iron now the next plot here shows that the region in between this wire loop is being filled with iron and now you can see that flux density in the iron is very high whereas in the regions around it is very low if you look at the levels the deep red areas here now have a flux density of about 1.2 Weber per meter square as is indicated by the legend here and whereas regions that are close to the edges here that goes to about 0.4 Weber per meter square and in the regions outside it is really small as compared to what is there inside now if you look back at the earlier figure nowhere in the area around the wire loop you have B which is in the range of 1.2 Weber per meter square the maximum is only 0.7 x 10-3 which means the flux levels the flux density that are obtained in a pure air when it is around this wire loop it is very small whereas when you put iron the flux density levels dramatically go up also you notice another thing that most of the flux is now confined to the iron and it appears as if there is not anything outside of the iron that is not really so if we now focus on the areas which have small flux density and zoom into the areas that have smaller flux density now you see the next plot you see that there is a flux density variation even in those areas which appear to be uniformly low in this region this appears uniformly low flux density is not really so there is a gradation there also you can see that around the wire loop areas this is the wire loop one section and this is the other section which is going in. So here there is a slightly higher flux density here there is a slightly higher flux density whereas if you go around flux density is still lower and you find some flux lines that are encircling this section of the wire loop not all the flux lines as you see here if you see it appears that all the flux lines go through iron only but whereas on deeper look you can see that there are some flux lines that are circling this area of wire alone. So if you now put iron then you have much higher level of flux and then most of that higher level of flux is confined to iron areas which are not iron spaces that are not iron have much lower density of flux. So with this understanding let us now get back to our equations so in effect from all these things what we have seen is that even a DC current flowing through a wire loop generates magnetic field higher the flow of amperes more the magnetic field that is going to be generated and the magnetic field is enhanced by the presence of iron right. So if you are now going to take let me draw this iron structure symbolically you have iron and then you had a wire loop this was what we had simulated in the simulation software you have certain number of turns here and you had certain number of turns here right. We are now looking at exciting this and what we have seen is that the flux levels in the iron here are dramatically higher than the flux levels had there been no iron and those dramatically higher flux levels are confined to iron whereas rest of the space is not having very high flux density levels so which means that if I now shape this iron material in different ways you would still have high flux densities in those regions of iron and which therefore means by providing an appropriate shape to this iron I can make this high flux density area occupy whichever space I want however I want to shape it for example if I am now going to have an iron core that looks like this and then put certain number of turns here I can still be sure that in all these areas you will have high flux density so shaping of high flux density regions is another important aspect in electrical machines. So we have now seen that current reduces magnetic field use of iron increases dramatically the magnetic field for the same excitation for same current that flows there is no use in saying that by increasing this amount of flow of current you are increasing the field for a given current with and without the iron there is a dramatic difference and because iron high field density exist in the iron and not anywhere else you can now shape iron in order to get high flux density regions where desired so these are all important aspects of the generation of magnetic field inside a machine so now that there is a magnetic field invariably iron is used in electrical machines because you want high magnetic field for low flow right you do not want to generate large excitation and have a large magnetic field you want to generate as much magnetic field as you can for small current. Now if there is a field then and if you are going to change the excitation the field is also going to change and if the field is going to change then there are some basic laws of electromagnetism which you will know you would have heard as Faraday's laws they now come into existence which says that whenever there is a change in the field associated with a loop there is an induced EMF to be more exact this law says that an EMF is induced in a wire loop which is equal to the rate of change of flux linkage in that loop associated with that loop and how do you define this term flux linkage is defined as the number of let us call that as ? the flux linkage as ? then ? is defined as the number of turns in the loop which is n multiplied by the flux lines that are going through that loop which is 5. So if this is going to undergo a change then an EMF is induced in the loop which is proportional or which is equal to the rate of change of flux linkage and therefore the simple equation that we had VA equals RIA can now be written as VA equals R times IA plus the rate of change of flux linkage. So a new term has to be added if one is going to consider that the excitation is going to change with respect to time. So how do we now find out how much is the value of ? for that we take recourse to again laws of electromagnetic there are laws of electromagnetic which you can refer to books on electromagnetic let us start with a simple equation that is necessary for us we will look only at what is required for us there is one law which says that the integral over a closed loop of h.dl both being vectors is equal to the surface integral of j.da where j is the current density in the region da in the region a over which we are going to integrate h is then called as magnetic field intensity. Now how does this apply to the situation we have here you have an iron loop now if you see this arrangement let me draw a three dimensional extension of this so that one can understand what is going on three dimensional picture and this wire loop now goes around like this okay. So if you now take a section of this along the horizontal and look at that what you would see is an iron core with a coil here this coil here and then this core this is the simulation that we have done in the FE analysis we had taken a coil here and a coil here allowing this current to flow and all that we have seen. Now in this you are going to have the magnetic field generated and we have seen that the magnetic flux lines go around this iron core in this manner we have seen this in simulation in the graphs that we have in the pictures now h.dl around a closed loop so this is now your closed loop around which flux flows so if you integrate h along this closed loop this equation says that that must be equal to if you look at the surface enclosed by this loop which is this in the surface enclosed by this loop you find out what is j that is how flow of amperes is distributed in that and then integrate that if you integrate it around this right. You would basically get number of turns times multiplied by whatever current is flowing if I call this as Ia this Ia is what is flowing around the loop and if you integrate over that surface it is nothing but j which is the ampere density in each of these turns right multiplied by the area of each of those turns is nothing but Ia itself in the rest of the areas there is no flow it is open so all these integration would give 0 there is flow of current only in the region where there is this wire loop so this is nothing but n x Ia and this h x integral that is h.dl integral around this closed loop okay h is the same inside this ion material assuming that the sectional area of the ion remains the same and therefore this left side boils down to h multiplied by the length of this loop which is the same as the length of the route through which the flux flows inside the ion core. So you have basically h.hxlc is equal to n x Ia which therefore means that h is nothing but n x Ia divided by n this term n x Ia is then called as the magneto motive force or MMF for short numerator is called as magneto motive force and the h is then measured in ampere per meter in SI units which is also sometimes written as ampere turns per meter because of the number of turns n that appears here okay having got this how do we now go ahead and find out what SI is we know that SI is nothing but number of turns times the flux that is flowing through the ion core you have lot of flux lines flowing through the ion core which we have seen in the simulated result also and those flux lines number of such flux lines multiplied by the number of turns in this is what is going to give you SI this in turn can be written as number of turns times flux density in the core we assume for now that the flux density in this core is uniform that means all through this area of the ion core remember this picture is drawn by taking a cross section of this so all through this area flux density remains the same if that is the case then flux is given by B into area of the core you must remember that flux density B is given by flux lines per unit area it is measured in Weber per meter square which is also called as Tesla so NB into AC B on the other hand we know from the physics of materials that B and H are interrelated depending on the material now let us assume therefore that B is equal to µ times H where µ is the permeability of the material you definitely remember that if the material is air that is then called as µ0 or which is the same as the free space µ0 has a value equal to 4p x 10 to the power of – 7 now B is µ x H and therefore if you substitute that then you have ? as N x AC x µ x H and now we have already derived an expression for H here so we substitute this there and therefore that is N x AC x µ x NIA L so multiplied by N x IA divided by L which is nothing but µ N2 AC x IA divided by L so if you have a flow of current IA as you have in this expression then you get a flux linkage which is µ N2 AC x IA divided by L and therefore if you substitute this in the expression for VA then you have VA equals R x IA plus d by dt of µ N2 AC x IA divided by L obviously in this expression N2 AC and L do not change with respect to time and therefore they can be taken out of the differentiation now the question is what you do with µ may be dependent on IA in a non-linear material and therefore it may not be feasible to take it out of the differentiation IA in turn depends upon T and therefore µ you can say depends upon T but in situations where it does not depend upon IA we say that those materials are linear materials in that case you can write this as R x IA µ N2 AC by L multiplied by DIA by dt and it is this term that you now call as inductance right so you can now write this in a simplified form as R IA plus L into DIA by dt which is the expression that you would have been very familiar with from the first courses on circuit analysis so this in essence is the birth of the inductor the inductor is an interpretation of the phenomena involving magnetic field on or it describes the effect of the magnetic phenomena on the electrical circuits and this inductance is going to be very important in determining the behavior of electrical machines as we shall see in the several lectures that are going to come. We then need to get an expression for ? so that we can substitute in this equation ? as we have seen earlier can be described as N multiplied by ? and which is nothing but N multiplied by the flux density into the area of the core we must remember that flux density is measured in units of Weber per m2 it is nothing but the number of flux lines per unit area so if you multiply that by the area of the core then what you get is flux and that multiplied by the number of turns is then your flux linkage B can now be written as µ H where µ is then the permeability of the material which is given by µ0 x µr where µ0 is the permittivity of free space µr is the relative permeability of the material so all this you would have seen definitely in an earlier course so you end up with ? equals N x µH x AC and we have already derived an expression for H which is nothing but N x IA divided by L so you substitute that expression here so it is µ x N2 x IA divided by L multiplied by AC this is then the expression for flux linkage and one can then substitute this expression in your expression for voltage so you have V equals R x IA plus d by dt of µ N2 IA x AC divided by L if you look at the terms here N2 AC and L are not going to change with respect to time and therefore they can be taken out of the derivative operation and this expression can be simplified as R x IA plus N2 AC by L multiplied by d by dt of µ x IA µ is something that may change with respect to operation which we will see later on but by and large in elementary electrical circuits electromagnetic circuits also µ is considered to be a fixed number and therefore that can also be taken out of the differentiation process so you have µ N2 AC by L and it is in fact this expression that is now known as the inductance and therefore you write this as R x IA plus L x IA by dt this is the expression that you would have seen in first courses on electrical networks where applied voltage if it is AC can be written as Ri plus Ldi by dt so this in effect is the birth of inductance the inductance then represents is an interface between electrical circuits and the magnetic field it is a representation of phenomena that involve electromagnetic field on the electrical network and this inductance as we shall see in the lectures to come is an important aspect in determining the behavior of electrical machine it is therefore necessary for us to understand what this inductance is and how to write equations for circuits involving inductances in the next few lectures with that we will end now and continue in the next lecture.