 We have been looking at deriving expressions for inductances in electrical machine and in the last lecture we had tried to derive an expression for the mutual inductance between the rotor coil and the stator coil for the case where the stator is a cylindrical structure but the rotor is a salient pole structure. And in order to derive the inductance the procedure we had adopted was first we excite the rotor coil when you excite the rotor coil there is a certain MMF and because of that flux is generated inside the machine and we had seen over the last couple of classes what happens to that flux waveform or the flux generated inside the machine as the rotor begins to rotate. We see that as the rotor rotates the flux density waveform travels along with the rotor there is a certain flux density distribution inside the machine that means as you travel around the inner circumference of the stator you encounter flux densities at various points and that then is called as a flux density distribution around the air gap or around the circumference of the stator inner circumference of the stator. And when the rotor rotates this flux density distribution also rotates along with it because the source of that excitation is from the stator. So we have excited the rotor coil we had then found out the flux density distribution we had found out the flux density distribution around the inner circumference of the stator and then this waveform was a quasi square wave in the sense it has a flat top over a certain angular distance and then it is 0 elsewhere and then a negative flat top and so on. In order to deal with this waveform we resolved this waveform resolved B of a into Fourier series thereby you have an analytical expression in terms of a and then now that you know the distribution of flux density around the circumference of the stator you then formulate an expression for flux linkage in order to do this we considered first an elemental flux linkage which means we looked at an elemental area around the inner circumference of the stator we found out how much flux was passing through that we know the expression for flux density and flux density multiplied by the area is then the flux passing through that elemental area multiplied by the number of turns is then the flux linkage and then we integrated this flux linkage expression integrate this flux linkage expression over the span of the coil over area spanned by the coil on stator coil span and then you derive the expression for flux linkage and we saw that this flux linkage has various terms relating to the fundamental and the third harmonic and the fifth harmonic and so on. And then we observed that these terms pertaining to the harmonics died down fairly rapidly and for most analysis it is sufficient to consider only the fundamental term and that fundamental term we found in the last lecture could be written as ?s is equal to this expression. So let us write down this once again it is – 8b hat ns l x r ? cos ? x cos ? r so what we have is ?s is equal to – 8b hat ns l x r ? x cos ? x cos ? r this is the flux linkage. Now we know that b hat is the peak flux density how to find out the peak flux density if we look at the rotor geometry which we saw in the last class I am scrolling back this is what this is the geometry that we have been considering in this geometry what is happening is that the flux lines the flux lines would travel along the rotor and then go into the stator and then come around and complete the flux loop in this manner. We have also noticed that in the air gap areas there is bound to be very little flux majority of the flux is going to go through the pole phases we had earlier seen a flux density distribution that is there in the system we saw how the flux lines go so these flux lines are going to go mainly here and we have also seen that the mmf or the value of h that is drop that is there along this flux path is predominantly concentrated in the air gap and we had derived an expression for that therefore what we can say is in if you apply amperes law around that loop of flux then what we have is h at the air gap multiplied by the air gap length this is along one air gap of one pole phase and then you have similarly another pole phase air gap so two times this must be equal to the number of turns in the rotor multiplied by I in the rotor and therefore from this what we have is hg equals nr x ir by two times lg and this is in the air gap and therefore we know that this is equal to b hat divided by or b divided by mu not and therefore the expression you have is b hat equals mu not nr x ir divided by two times lg so that is the expression you have so this can now be substituted into the expression for the flux linkage so you have ?s equals – 8b hat ns instead of b hat we are going to substitute the new expression therefore that is mu not nr x ir divided by two times lg multiplied by ns x l x r multiplied by cos ? x cos ?r and therefore one can now write this as ?s divided by ir is – 8 mu not ns nr x l x r x cos ? divided by two times lg x cos ?r which is then – the maximum value of the mutual inductance between the stator and rotor multiplied by cos ?r so this is then the mutual inductance between the stator and the rotor we are considering only the fundamental component of that so we find that it changes as cosine of the angle which finally leads to this value of mutual inductance equals msr hat may be a – sign and then cos ?r so we have derived then an expression for the variation of mutual inductance as the rotor changes its angle. Now in this system we have one coil on the stator one coil on the rotor which means electrically when we try to write down expressions for describing the system behavior we would have to find out the self inductance of this coil we have to find out the self inductance of this coil and then there is a mutual inductance between these two so these are the three inductances that we would encounter and what we have done now is we have derived an expression for the mutual inductance between the two coils we know how to derive an expression for the self inductance that is straight forward we have done that earlier for the cylindrical rotor case for the cylindrical stator case also it is very simple you have the if you take the machine and then you have a rotor which is there now if the rotor is excited and the rotor were going to rotate the flux generated by the rotor would be independent of where the rotor is it always sees a constant air gap along this region and we already know that there is not much of flux that is available in the air gap and therefore we can easily say that the flux that is thereby generated is given by the mmf which is nr x ir that is the mmf divided by the total reluctance and the reluctance is given by the reluctance of air gap one plus reluctance of air gap two air gap one being under pole phase say one and reluctance two being the air gap under pole phase two. So if you add these two up you get the total reluctance each reluctance is then described by length of the air gap divided by mmf into area of the air gap and since this is equal on both sides the total reluctance is then two times this value and therefore you can write the flux that is produced as nr x ir divided by two times length of the air gap multiplied by mmf into area of the air gap and the flux linkage is nothing but nr x ? because we are considering now the self inductance of the rotor it is nr x ? which means nr x ir x ? x ag by two times lg which means that the self inductance lr is nothing but ? let us call this ? r ? r by r which is nr x m0 ag divided by two times lg and therefore we know the self inductance of the rotor coil. So we have derived mutual inductance we have derived self inductance of the rotor now we need to derive an expression for the self inductance of the stator coil. So in order to derive an expression for the self inductance of the stator coil we need to excite the stator coil. So the procedure will be very similar to what we had done earlier now excite the stator coil and then find flux linkage in stator and then flux linkage per unit ampere is your self inductance. Now the question is how do we find out the flux linkage in the earlier approach what we did was in order to find out the flux linkage we found out the flux density distribution around the stator and then resolved it into Fourier series found out the flux passing through an elemental area and then did an integration. Now in this case and the whole method was based on our observation of how the flux density distribution in the machine is going to appear as you rotate the rotor we had started from there. Now let us see what will happen if you excite the stator and the rotor begins to move. So here you have some pictures of how the flux density flux lines are going to look like in the case of a salient pole rotor and a cylindrical stator you can see that the stator is cylindrical this is your cylindrical stator and here you have the salient pole rotor. So with the rotor located at this particular angle and if you excite the stator coils the stator coils are here if you excite the stator coils then the flux lines can be seen to be like this you can see that flux lines concentrate around the stator coil here and here it is also evident that there are plenty of flux lines here there is very little flux lines parsley distributed over here so one can infer that the flux density in these regions would be high flux density in these regions would be high flux density in all these areas would be low it would decrease as you travel from here this region would have a very low flux density and the flux density again increases as you travel to this area in this area you can see that there is very little flux so here again there is a dip in flux density and then flux density again rises and then falls down this is the situation for the rotor like this. Now let the rotor move so you come to the next position the rotor has rotated a little anticlockwise and you see that again under the pole phase there is a rather high flux density distribution but now it is occurring in this part of the pole phase below the slot it is a lower flux density and here the flux density is low and it picks up as you come here flux density here is high here it is high and as you go around it gradually falls this region would have a very low flux density let the rotor rotate a little more again you have a rather high flux density region here and then the flux density region falls and you have low flux density and high flux density regions here rotor rotates a little more now you have high flux density regions again near around the rotor area whereas these areas now have very little flux density this again high flux density here low flux density rotor rotates a little more you can see that the high flux density areas seem to be traveling along with the rotor now the flux density areas I mean areas that are in other zones these are all small rotor rotates a little more again you can see that this higher flux density area is traveling along with the rotor in these areas flux density is pretty small one more rotation so here it is so we have seen at different angles how the flux lines look like it appears as if the machine always looks at the entire air gap area there is this all this region is air all here is air air gap here is small whereas air gap here is large and one can see that the machine appears to look at it as if you see you are exciting the stator and you want to find out the flux that is thereby developed and always wherever this air gap is low the flux levels developed are high and wherever this air gap is high the flux levels developed are low and the region where air gap is high or air gap is low will depend upon where the rotor is if the rotor is at this angle as shown here this region of the stator circumference has low air gap and as the rotor moves this region of the stator circumference now becomes one with the high air gap so where the air gap is where the air gap is low flux levels seem to be high and it is very difficult to predict how this flux density distribution is going to change what will be the wave shape of this flux density distribution unlike in the case where the rotor was excited and the rotor was moving it is true that the flux density in the pole phase was always high and that region move along with the rotor but now the situation is slightly different even within the phase of the rotor the flux density distribution can change for example now you see that in the middle of the rotor pole phase the flux density level is low whereas on these two sides it is high but since flux is passing out from this side and entering back into this side the flux density levels here if they are greater than 0 at some high value the flux density levels here will be less than 0 at some high value whereas now as the rotor rotates even under the pole phase you see that the flux density distribution are going to change so one can sort of say that the machine is looking at what is happening inside as if there are two different air gaps that are available one air gap that is related to what is happening in the small air gap area another air gap which is giving a response or flux generation which is related to what is happening in the large air gap area and therefore in order to analyse how the system is going to behave what is normally done is that you have the stator of the machine which is cylindrical and you have the rotor of the machine which is salient pole as you travel around the inner circumference of the stator the air gap length dramatically changes in as you approach the pole phases the air gap is small as you move away from the pole phases air gap is large and that is the kind of varying air gap that is seen by the stator coil and therefore in order to analyse this kind of a system what is done is that you assume that the machine or the coil on the stator assume that the coil on the stator sees a geometry with two air gaps that is it appears to the stator as if there is one rotor which is always moving one cylindrical rotor which is always moving such that the air gap is small why it is so we have seen in the pictures here that there is irrespective of where the rotor is there is always a region nearby the rotor which has high flux density so these regions always behave as if the air gap is small and this region moves as the rotor rotates and there are certain regions where the air gap flux density is always low and that region also moves as the rotor rotates and therefore we then say that the stator winding looks always at one air gap that is small and another cylindrical rotor which has an air gap which is large so let us call the small air gap as gd and let us call this large air gap as gq which means that we are saying that there are hypothetically two rotors in the machine both of which are cylindrical now this has an effective air gap which is gd and then there is another rotor inside the machine which is also cylindrical and has another air gap gq now the net response of the flux is the response produced due to both these rotors having an air gap gd and gq how much of flux is going to act along the air gap gd how much of mmf is going to act along the air gap gd and how much along gq is what we have to now determine for that what we do is how we ultimately of course one has to see if we do all this whether the estimate of the inductance of the stator is similar to what we are actually observing if it is similar to what we are observing then perhaps we can say our assumptions and procedure is alright. Now that we are looking at a stator that is cylindrical and having considered the rotor as effectively being composed of two cylindrical rotor structures in order to find out what the mmf is we will have to now consider what happens with the cylindrical rotor structure and therefore if we now look at the here is your stator and here is the rotor assume that they are nice circles now you have a coil here and coil here in a slot in the earlier case to find out the mutual inductance we started with the flux density distribution similarly here we can think of an mmf distribution to start with it is more convenient to start with an mmf distribution how to find out the mmf distribution if you have a cylindrical stator and a cylindrical rotor like this you will find flux lines that are going to cross like this and come around similarly here and one can now write the expression for amperes law around one of these loops if you go around any one of these loops then you will have hg x lg let us say we are looking at the d axis mmf so lgd this would then be lgd or if we look at the actual mmf itself let us look at it that way hg x lg and two times there is one hg x lg here another hg x lg here this is equal to nrns x is and therefore hg x lg is equal to ns x is by two we can therefore get hg as ns x is by two lg this would give you hg the h x l itself is then the mmf that is dropped across one air gap here you have a similar mmf that is dropped across the other air gap here flux lines let us say are travelling into the rotor whereas here flux lines are travelling away from the rotor so if you call one mmf as positive the other mmf is obviously negative because it is now causing flux to flow outside this is causing flux to flow inside so if you draw this at any angle along the circumference of the stator or the rotor any loop that you take will always give the same equation for example let us say you take another loop here it would still give the same experience hg x lg at this point we have said that the flux travels normal so hg x lg and then again hg x lg in absolute terms should be equal to that so what we have is if you now plot the actual mmf variation around the circumference that is there this is your angle a as you travel around the circumference and this is let us say mmf this mmf is constant at a value equal to ns x is by 2 as you move around a so what you would have is a equal to 0 to a equal to that if you call that as one sign well we have called let us take it this way that would be your mmf distribution each of magnitude ns is by 2 if that is ns is by 2 here it will be – ns is by 2 it depends on what is the convention you choose if you choose to call flux lines going from rotor to stator as positive then flux lines going from stator to rotor will be negative if you call flux lines going from rotor to stator as negative then flux lines going from stator to rotor will be plus so one of these convention if we draw it this way it is ns is by 2 and – ns is by 2 this is the flux density distribution so I mean this is the mmf distribution so what we see is that if the rotor structure is cylindrical one can write down the way mmf changes as you travel around the circumference of the air gap and what we have seen earlier in the expressions for the rotor mutual inductance rotor to stator mutual inductance we have been saying that it is sufficient to consider the fundamental component alone so let us do that here also we will take the fundamental component of this mmf the fundamental component of this mmf waveform is going to look like this and it is an easy matter to find out the amplitude of the fundamental component you have to resolve the square wave into its Fourier series and look at the amplitude of the fundamental component and that is a very standard exercise which I would leave it to you to do the expansion of the Fourier series and to find then that the amplitude of the fundamental let us call that as f hat the amplitude of the fundamental is given by 4 by pi into ns is by 2 this is the amplitude of the fundamental component of the mmf distribution around the circumference of the stator and therefore one can find right now mmf distribution as a function of a that means as you travel around the circumference what is the expression for the mm that is nothing but f hat into sin of a so that means that at a equal to 0 f f is 0 as a becomes pi by 2 it reaches the maximum value of f hat and then so on it is a sinusoidal distribution. Now this is the distribution that is there along the circumference inner circumference of the stator but this mmf now acts along 2 air gaps that we have now called the air gap length we have called as lgd and lgq gd and gq and how much of this mmf acts along those 2 air gaps we have seen that as the rotor is going to rotate the flux levels move along with the rotor and one part of this mmf then acts along the low air gap area another part acts along the high air gap area as the rotor moves so we then say that you resolve this so if you have the stator geometry stator and then let us say you have the rotor like this the mmf distribution that we have written is an mmf that varies from here is your a equal to 0 and then you travel along at a equal to 90 degrees you reach the maximum value of mmf a equal to 90 degrees is the maximum mmf and then as you go along you reach a equal to 180 degrees you get the minimum value of mmf and so on. Now the maximum value of mmf occurs at a equal to 90 degrees in order to find out how much of this mmf acts along the rotor which has a small air gap area and that part which has a large air gap length we resolve this mmf into 2 parts one which is always acting along the rotor axis that means one which always acts along this axis and another which always acts along this axis now if we do that then this mmf will always see a small air gap and this part of the mmf will always see a large air gap that is in essence why we are why we decided to have 2 air gaps that are acting in the machine that seem to be acting in the machine. So this axis let us call as the d axis this axis let us call as the q axis the word d generally stands for direct axis direct referring to the axis of the rotor pole of the pole structure and q standing for quadrature axis. So how does one resolve this into these 2 parts now the mmf itself the stator generated mmf we see that it has a maximum value on this axis maximum value in the sense it reach the absolute value of the mmf is maximum along this axis you can see that the mmf reaches a maximum at a equal to 90 degrees and again it reaches a maximum at a equal to 270 degrees so that means at this position so mmf is maximum either here or here and the location where the mmf is maximum we use that to define the axis of that mmf so this then this line is defined as the axis of the stator mmf and that place where the maximum amplitude instantaneous amplitude occurs greater than 0 this is the reference axis is chosen as a reference axis in order to define the rotor angle that is your ?r. Now we say therefore that the rotor is at an angle 0 if this rotor bar where horizontal and we say that the rotor angle is equal to 90 degrees if the rotor bar where arranged in this manner so that is then the rotor angle if this is the rotor angle then by geometry what one can see is that this angle is also the rotor angle because these 2 axis are oriented 90 degrees to each other so with this then what we want to do is to try and resolve this mmf stator mmf along these 2 axis so in order to do that if the maximum value occurring here is f at then fd which is the resolved value of the stator mmf acting along the d axis is given by f hat that is the peak fd hat f hat x cos of ?r f hat is the maximum value of the stator mmf acting along this axis you want to consider the component of that along the d axis so that is f hat x cos ?r is that value and the component acting along q axis is f hat x sin ?r this acting along q axis so we have now arrived at expression for denoting the peak value of the mmf acting along the d and q axis we now have to define the spatial distribution of this you see that the mmf generated by the stator is spatially distributed that means as you travel around space what is the variation of mmf that you would encounter that was given by this expression f hat x sin a now by resolving these 2 expressions describe the peak and therefore what would the spatial distribution be fd as a function of a will then be this peak value fd hat multiplied by cos of a – ?2 ?r this would then describe the variation of that fd mmf in a sinusoidal fashion about that fq of a will then be fq hat multiplied by cos of a – ?r so we can write this as fd of a equals fd hat multiplied by cos of a – ?2 – ?r which is fd hat multiplied by cos of a – ?r – ?2 which is fd hat multiplied by cos of a – ?r which is equal to fd hat multiplied by this is cos of 90 – ? so I can call this as – of a – ?r – of a – ?r and therefore this is fd hat x sin of a – ?r and then what you have is fq a is fq hat x cos of a – ?r so these are 2 expressions for the spatial variation of fd and fq now these mmf are now going to act along their respective air gaps fd acts along the smaller air gap saying lgd and fq acts along the larger air gap lgq the net flux at any given angle will be the flux cos by the d axis mmf and this q axis mmf we need to find out the flux linkage of the stator winding with the flux is produced by these 2 and then as usual do the process of integration in order to find the total flux linkage and then flux linkage divided by unit current is going to give us the total inductance so these steps we will see in the next lecture to come we will stop at this point.