 I welcome you to the next class on modeling and analysis of electrical machine. In the last lecture we had looked at deriving expressions for inductance for a machine consisting of a salient for a machine consisting of a rotor and a stator which are both cylindrical in nature. We more specific the geometry looks like this you had a cylindrical stator inside which you have a cylindrical rotor you had a coil on the stator and we found out an expression for the self inductance of the stator coil. We had a coil on the rotor and we found out the self inductance of the rotor coil and then we also found out an expression for the mutual inductance between the two coils between the stator and the rotor coil. We found that the self inductance was of the stator coil is a constant in the sense it does not vary with the angle of the rotor in this case we are saying that the angle of the rotor is the angle made by the perpendicular joining these two lines. So this is the axis of the coil on the rotor similarly you take the axis of the coil on the stator and then this angle we call as the rotor angle ?r and therefore we found that the mutual inductance was a function of ?r whereas the self inductances are not a function of ?r. Welcome you to the next lecture on this course on modeling and analysis of electrical machine. In the last lecture we had looked at how one can derive expressions for the inductance of coils placed in an electrical machine where the stator is of the cylindrical variety and the rotor is also of the cylindrical variety. Here now we see how the geometry looks like we have a cylindrical stator this is your cylindrical stator and then you have a cylindrical rotor a coil on the cylindrical stator with one side lying here and another side lying here we derived the expression for the self inductance of this coil and we found that the self inductance of the stator coil is basically a constant. Similarly we also derived an expression for the self inductance of the rotor coil and we found that that was also a constant. We also derived an expression for the mutual inductance mutual inductance between the stator and the rotor coil and we found that the mutual inductance is a function of the rotor angle ?r the rotor angle being defined as the angle between the axis of this coil on the stator and the axis of this coil on the rotor the axis of the coil being defined as that angle along which the maximum mmf occurs which would then mean that the axis of the stator coil is going to lie at a line perpendicular to the line connecting the two coil sides which is this this is the axis of the stator coil and the axis of the rotor coil would lie along a line which is perpendicular to a line connecting the two coil sides and therefore that is the axis of the rotor coil and then the angle between these two this angle is then called as the rotor angle. Now we will look at the next variety of machines that are there which are salient stator and cylindrical rotor machine I am sorry we will look at we will look at cylindrical stator and salient rotor machine how does such a machine look like that is a cross section of such a machine look like that can be seen in the picture here you have a cylindrical stator you have a cylindrical stator you can notice that the inner surface is cylindrical and you have a salient pole rotor you can notice that the pole surface is here and here and the air gap between the pole surface and the cylindrical inner surface of the stator is small but as you go around in this region the air gap is pretty large similarly here also the air gap is very large so this distinguishes a salient pole rotor from that of a cylindrical rotor in the earlier case where the cylindrical rotor was considered where the geometry looks like this you can see that the air gap is uniform all around assuming that what we have drawn are exact circles then the air gap will really be uniform all around the machine whereas here the air gap is not uniform here the air gap is small whereas in this region and this region the air gap is large therefore salient pole machines are essentially non uniform air gap machine so in such a machine how do we derive expressions for inductances again we are interested in looking at one coil on the stator and we are having one coil on the rotor so here you see that there is a slot that has been described similarly another slot 180 degrees away you have a coil occupying these two slots one side of the coil lies here in this slot another side of this coil lies in this slot so this completes one coil on the stator similarly there is a coil on the rotor but now divided into two parts there is one winding on this side of the rotor another winding on this side of the rotor both connected in series and energized such that the flux flow aids each other and therefore this is equivalent to a single coil that has been wound around the rotor in this direction in this particular case what we are having is the coil on the stator is a hundred turn coil hundred turn coil on the stator and on the rotor it has a 30 turn coil and two of that that is what we have and this area all this is made of steel this is also made of steel whereas these regions are air region so in this case we would like to see what would be the expressions for the inductance of the coil on the stator inductance of the coil on the rotor and the mutual inductance between the two that is the exercise we would like to undertake now how to do that from our understanding of the inductances we know that inductance can be described as flux linkage in a particular coil by current that cost that flux linkage if you are looking at self inductance of the stator then we are looking at flux linkage of the stator divided by current flowing in the stator if you are looking at self in which case we would call this as LS if we are looking at the self inductance of the rotor then we need to find out flux linkage of the rotor divided by current in the rotor on the other hand if we want to find out the value of mutual inductance what we could do is to find out MSR is flux linkage in the stator divided by current flowing in the rotor so you excite the rotor and find out due to that how much flux linkage is there on the stator then divide the two you get the mutual inductance we have also understood that if you want to find this out that is flux linkage in the stator divided by current in the stator now if you are going to excite the stator that is send some current through the coil here we first need to find out how much flux is going to be generated and we know that in order to find out how much flux is generated you need to know how the magnetic circuit looks like and we see that we intuitively understand from various flux patterns that we have seen in the earlier lectures that if you are going to energize the coil here let us say that in this case the flow of current out of this coil side is out of the plane of this figure and here the flow of current is such that it is into the plane of this figure then we know that in this loop flux will be flowing we can decide the direction in which flux is flowing by your rule of the right hand where you curl your fingers in the direction of the current flow then this finger points in the direction of flux so here current is coming out from the top side and going in to the bottom so there will be flux lines flowing from left to right in this figure which means now in the drawing that we have seen we are seeing here you expect to have flux lines that are flowing from left to right now in this direction you see that the air gap encountered for the flux lines is minimum you would expect therefore a large number of flux lines to go through this region to go through this region and here the air gap is very high here also the air gap is very high so you would expect very little flux lines on this region and therefore what we anticipate is that the flux density is here and of course in this iron region are likely to be pretty high whereas the flux density in this air gap region are likely to be very low at the same time let us also look at another situation where the rotor is not along this horizontal direction but imagine a situation where the rotor has rotated by 90 degrees and therefore this pole phase occurs here and this pole phase occurs here still because the coil is excited in the same way you expect the flux lines to have passed in this direction but now you have a fairly large air gap in this side and this side because the rotor has rotated the pole phase is here and here and therefore you would have a large air gap because of the large air gap now the flux that is generated is likely to be very low and therefore when you want to find out inductance as the ratio of flux linkage to the excitation that you give you would understand that how much of flux is generated for a given flow of current in the stator would depend on where the rotor is if the rotor has been put in this direction then you would expect a large amount of flux flow flux density levels could be very high on the other hand if the rotor has been rotated such that the pole phase occurs here and here then there is considerable air gap in the in the flux flow path and therefore the flux level itself would go down now let us look at an FEM simulation study so this figure now shows how flux lines have been generated for the rotor aligned horizontally remember this horizontal direction is then the direction of magnetic axis for this coil as well so what we find is a rotor is now aligned along the magnetic axis of this loop and you find that the flux density levels are quite high for example in this region this dark red corresponds to if you look at here somewhere in the region of 1.5 tesla here also it is in the region of 1.5 tesla in these regions of course the flux density is low because the flux lines tend to come here and then travel into the rotor so you would not expect much of flux at these regions so here therefore the flux density is very high and therefore the flux linkages are also going to be very high with this coil but as you now rotate the rotor let us consider the other situation as is seen on the right side here now the rotor has been rotated such that the pole phase occurs here and here and as you can see in this direction the air gap is very large and therefore if you had excited the stator you would have had very little flux lines here this plot has been generated by exciting the rotor and therefore this rotor flux lines are shown here and therefore in the case of exciting the stator the flux linkage in the stator would depend on the rotor position on the other hand if you are going to look at self inductance of the rotor where this plot itself becomes evident becomes of use now how to determine the self inductance of the rotor you excite the rotor coil find out the flux linkage with the rotor and now one can see that if it is the rotor that is going to be excited if you compare these two plot this plot also has been generated by exciting the rotor the results would have been similar if you excite the stator coil for this particular case now if you excite the rotor now you see that this region of high flux density here and here that is there the central limb has very high flux density and now the rotor has rotated to the next angular position and the rotor still has the same flux density levels one would therefore expect that the flux linkage in the rotor in this coil due to current flowing in the rotor is likely to be more or less same as what was observed here why is that happening because whether the rotor is in this position or in this position the flux lines that are going this way and this way they always encounter the same air gap whether it is here or in this position they always encounter the same air and therefore as the rotor is going to change we do not expect the flux levels to change significantly whereas if you had excited the stator in this position it encounters minimum air gap in this position if you had excited the stator it would encounter a very large air gap therefore inductances will change therefore when we talk of self inductance of the rotor in this particular case we do not expect the self inductance of the rotor to change with the rotor angle whereas we expect the self inductance of the stator to change with rotor angle how about mutual inductance in order to determine mutual inductance you excite the rotor just as has been done in these two plots and find out the flux linkage with the stator now how to find out flux linkage with the stator flux linkage with a particular wire loop like this which is placed on the stator is defined as if you have a loop of wire that is going around like this one side here and another side here in order to find out the flux linkage you need to find out how much flux is flowing through an area which lies in the plane of this loop you have the loop going around like this therefore this is the plane of the loop and you need to find out how much flux is flowing normal to the plane of the loop now if you look at the geometry here the plane of the loop would lie here and you have a large number of flux lines that are crossing the plane of the loop and therefore you would expect significant flux linkage arising due to current flowing in excitation in the rotor whereas now if you look at the figure on the right here also there is excitation in the rotor but now you see that all the flux lines are flowing like this the plane of the loop is here and you see that there is no flux line that is crossing the plane of the loop here you see that all the flux lines cross the plane of the loop normal to it here there is no flux line that is crossing the plane of the loop that means that the flux linkage of the stator coil in this position of the rotor will be 0 no flux linkage is there so on the one hand you have here a situation where flux linkage would be maximum on the other hand here you have flux linkage will be 0 so again mutual inductance this quantity we expected to be a function of the rotor position so let us now try to find out how to derive an expression for the mutual inductance we are going to excite the rotor and find out the flux linkage in the stator so in order to find out the flux linkage now what do we need to do we need to determine what is the flux density which is existing because of this excitation so if we now look at the next plot this plot shows how the flux density is there along the circumference of the rotor of the stator this x axis denotes 180 degrees of the circumference of the stator and the y axis denotes modulus of b that is there this plot has been drawn for the rotor position as shown on the left side here and this plot is drawn for the x axis starting from this position and travelling all the way along the circumference of the stator going up to this point so if you travel around the inner circumference of the stator along this arc and find out what is the flux density level you have at various points along this inner circumference then you end up with a graph that looks like this this is the distance travelled along the inner circumference this is the value of the flux density at various positions along the inner circumference you find now that at the place where you start that is here in this region the flux density levels are very low and then as you approach this the flux density level suddenly shoots up there is a very steep increase in the flux density level and then as you are in the pole phase region the flux density is almost flat and as you leave the pole phase region the flux density again rapidly drops down to reach zero levels at this end so this is the kind of flux density pattern that you have how does H field look like we know that H is equal to B by µ0 it is µ0 because we are looking at air gap right so B divided by µ0 if you find out what is the flux density in the air gap you can find out what is H in the air gap by taking this ratio now how to find out how much value of B will be there initially what we have is a geometry like this we know how many turns are there in this coil we know how many turns are there in this coil we know how much flow of current is there I am sorry in this coil we know how much turns are there in this coil we know how much flow of current is there in this coil so how to find out what will be the flux density levels in order to do that we need to be able to apply your Ampere's law Ampere's rule and in order to apply Ampere's rule you take an Amperian loop which we can conveniently take along the flux lines that go all the way here and then come back this way. So if you are going around this Ampere's loop then what you would write is remember that this Ampere's loop or Ampere's rule says that the circular integral h.dl is equal to the total current enclosed this expression we have used long ago in one of the lectures. If you look at this path that we are integrating which is drawn in blue line here this path is through iron within the rotor that is in this region the path passes through iron it crosses an air gap here and then goes again through iron crosses an air gap here and completes the loop. So if we were to split this if we were to write expand this integral what we would write is h in the air gap on the left hand side let me write it as hgl that is the air gap on the left hand side multiplied by the length of the air gap on the left hand side plus h in the rotor multiplied by the length of the path in the rotor plus h in the air gap on the right hand side multiplied by the length of the air gap on the right hand side these two lengths are identical that is how normally machine is made plus you will have h on the stator multiplied by the length on the stator these four terms refer to the region here which is the length which is the air gap on the left hand side and then path in the rotor air gap on the right hand side and then path through the iron. So that is the expression you get here and this must be equal to the total flow of current enclosed and how do you get the total current enclosed so you look at this path again and in this path if you travel around this path and you look at the current enclosed to the left side of you you find that there is current flowing here and then there is current flowing here there are same number of turns in both and each turn is carrying the same current so let us say that the total number of turns that are there in this coil plus this coil put together is NR and each turn is carrying the same current IR then the total current enclosed is given by NR multiplied by IR. So now in order to find out h we need to solve this equation and how do we solve this equation we find that there is some h here there is some h here and so on the sum total is what is equal to NR into IR and therefore it would be interesting to see along this loop that we have defined how is the variation of h the next plot shows this this is the modulus of h as we travel around the loop that we have described now one can see from this plot that initially this is a path that has been defined from here to here. So as you travel around the path you see that there is a big value for h somewhere here and then another big value for h here whereas in all the in between regions h is very small these two locations correspond to the air gap that you have here and here all the other areas are iron regions this is iron this is iron this is iron in all those regions you find that h is very very small therefore in comparison with this part and this part these two terms are negligible and therefore they can simply be struck of neglected in comparison with these two the value of h in both these regions since the machine is a symmetric machine length of the air gap is also the same one can expect that this value and this value is identical indeed that is what this graph also shows that these two levels are the same and therefore one can say reduce this expression to two times h into g multiplied by lg is equal to nr multiplied by ir please remember that this nr is the total number of turns in both the coils that we have placed on the rotor it is equivalent to having a single coil and therefore from this expression one can evaluate hg as nr multiplied by ir divided by two times lg and having got this expression we can now write an expression for the flux density in the air gap as nr into ir divided by two times lg and we know here that B is equal to µ0 into h so this multiplied by µ0 is what is going to give you your flux density now why does this happen that these two parts are negligible compared to these two areas now these two terms they correspond to value of h inside iron and inside iron the equation would look like h is equal to B by µ0 into µr there is a finite value of flux density however µr for iron is very high and therefore h is very low therefore wherever you have iron wherever you have a good iron material µr is likely to be high and h in the iron regions is likely to be much smaller compared to values of h in the air gap regions and therefore if you now look at the geometry here whether you take the loop that we have initially drawn that is along this line or whether you take a loop which is here at a different position in the air gap or whether you take a loop that is here all the iron regions are anyway going to consume very little value of h and irrespective of which loop you consider the total current enclosed is always the same and therefore in all these cases the final equation would still only be hg into lg in the left side of the air gap plus hg into lg on the right side air gap the together has to add up to n into i and therefore h all along this pole phase is expected to be the same and indeed that is what this plot shows h along the pole phase area is by enlarge constant and therefore B would also be the same fixed dc level at a fixed level and therefore B or h is constant but beyond that you see that this drops off pretty fast because you have large air gaps in that region and therefore h value is very small at the circumference B is also very small so that is how it looks now therefore if you plot the value of the flux density this line this graph is nothing but the modulus of flux density x axis refers to the angle around the circumference as you travel in the inner circumference of the machine so this plot says that if you start from here and then go around and come back here you would have traveled 360 degrees the rotor is not moving the rotor is at this position it is carrying some excitation we are traveling around the air gap to see what is the flux density at various angular locations if you do that therefore this is the plot that you expect we have seen initially that the flux density 0 then as you come to the pole phase the flux density becomes high and then as you cross the pole phase it is low again and then the flux density reverses and then comes back so this is how the flux density pattern around the air gap is going to look like now what happens as the rotor rotates we have seen that for this position of the rotor which is aligned along the axis of coil on the stator or for 90 degree later the flux density pattern as is seen on the right hand side of this figure is the same right so what we expect is as the rotor is going to rotate the flux density waveform also moves along with it indeed that is what you see here these plots show how the flux density variation around the circumference of the machine this x axis in this case again shows angle around circumference y axis shows flux density levels in each plot each of these plots has been generated for a particular angle of the rotor the first plot on the top shows the flux density variation around the circumference of the machine for an angle of the rotor ? r equal to 0 degree that means the rotor is located at a position which looks like this now as the rotor rotates the angle of the rotor is going to change so for an angle ? r equal to 30 degrees this is how the flux density plot looks like what one can readily observe from this is that this waveform and this waveform are identical except that you notice this waveform has shifted as compared to this because the rotor has moved along with the rotor this region where maximum flux density occurs has also moved otherwise the wave shape is the same if you look at the next plot this plot is drawn for an angle 60 degrees of the rotor the rotor has rotated even more and this plot also by enlarge you see it looks the same except for the small kink here the general shape of the plot is the same now why is this kink occurring before we get to that let us look at the next plot also this is drawn for 90 degree of the rotor position that means it is drawn for this position of the rotor on the right hand side now the rotor is at 90 degree position initially it was at 0 degree position now it is at 90 degree position so for varying rotor angles you can find that this waveform is shifting but you get this kink why is this kink coming if you notice this kink always occurs at the same position whether the rotor is here or rotor is here the x axis position of this kink remains the same and that is about 180 degree and what is there at 180 degrees at 180 degrees there is this slot you have starting from 0 degree here travelling around the circumference of the stator you reach 180 degree here and because a slot is opening on the stator inner surface effectively the air gap becomes larger in this position and therefore because the air gap is larger flux density falls down you get lesser flux in that area and the slot is not going to change its location because it is on the stator and therefore this kink always occurs at the same position along the x axis remember the x axis is the angle around the circumference of the stator the inner circumference of the stator so this effect of having a kink arises because of slotting that is there on the stator we have only one slot on the stator in that 180 degree span and therefore there is only one kink if you had more slots on the stator imagine that you have this position where you have one slot on the stator suppose you had one more slot here and one more slot here what would happen in this plot you would then have one more kink here and another kink here for our analysis we will neglect the slotting effects assume that the stator inner circumference is really uniform there are no slotting problems and therefore these kinks do not exist for our analysis our analysis we are making the simplification of neglecting slotting so with all that then this is the flux density waveform we are really going to have a flux density waveform that has very negligible flux density here and very rapidly builds up remains constant rapidly falls down and so on so we try to simplify this one more step and we assume that the wave shape of this flux density is square this is the approximation that we are going to make for our study that the flux density is square like this remember this is the flux density of flux generated by exciting the rotor we want to find out flux linkage in stator that is what we have to find out and we have also found that the flux density shape that is there this flux density wave shape rotates along with the rotor that can be seen here if rotor is at 0 degrees you have the flux density wave shape like this as the rotor goes to 30 degrees the flux density has moved by 30 degrees you see that this is probably somewhere around 75 degrees and this is come somewhere here so this is moved by 30 degrees and as the rotor moves another 30 degrees the flux density waveform is moved again by 30 degrees and so on and therefore the flux density wave shape rotates along with the rotor when we say that flux density wave shape what we mean is flux density around the circumference we are not talking about exciting the rotor with some varying current so as it does that we want to find out how flux linkage on the stator is also going to change which will then lead us to the value of mutual inductance. So in this class then we will stop at this point and then using this information let us see how to estimate the value of mutual inductance in the next lecture so we will stop here.