 The last lecture we had looked at the induction machine equation being rewritten in a more compact form using the space vector notation. We saw how a large system of equations having four independent variables neglecting the zero sequence components of course where the input voltages are Vds, Vqs and then the rotor voltages Vdr and Vqr with the responses ids, iqs, idr and iqr all of them can be combined into one single need space vector based equation. That is a useful form of representation in order to derive and understand control structures which one would learn if you were to study a course on motor drives. Having done that we started looking at the modeling of alternators and in the alternators we said that primarily the stator of the alternator looks very similar to that of the induction machine both are three phase wound stators distributed winding and the rotor is quite different from that of the induction machine. The rotor contains a source of excitation which is absent in the case of induction machine the rotor now has a field winding and then we said that the synchronous machine rotor predominantly for large machine is of two varieties one is the cylindrical rotor stator I mean cylindrical rotor of the synchronous machine and the other is the salient pole variety. In the last lecture we were looking at how the cylindrical rotor of the induction machine would look like and we saw a representation of the cylindrical rotor machine which we have here now just to briefly look back the rotor basically has a cylindrical surface which can be seen here you see that the outer surface of the rotor is by enlarge a smooth cylinder except for the slots that are there in which the rotor coils have to be house but for those slots it is a very smooth cylindrical surface and this arrangement is the shown I mean the arrangement shown is for a four pole cylindrical rotor and we have one pole that is here another pole phase here a third pole phase here and they all have to generate a magnetic field so around this pole you have rotor field coils that are wound through these two slots and then through the next two slots for generating the rotor field here you have a field winding in these two slots connected in series with the winding here and then the third pole phase and then the fourth field winding they may all be connected in series to ensure that the same current flows through them and then the direction in which field current flows is so adjusted that if you get a south pole field here due to flow of current in the orientation that is shown the flow of current in the adjacent one is such so arranged that it generates a north pole field and then here it is a south pole and then here it is a north pole field and these green lines represent show how the field lines are going to flow in the rotor structure so you have north pole field that is flux lines that are coming in to the south pole and then flowing along the rotor out through the north pole phase and similarly here as well so this gives rise to a four pole field structure here is the center shaft which will then be driven by a suitable mechanism this whole cylindrical structure is then inserted into the stator bore and the whole arrangement then forms your synchronous machine if this were not the case the alternative arrangement is then a salient pole rotor which would then look like this so here we see a salient pole rotor structure the center hollow represents the place where the shaft will be there the shaft is not shown here whereas in the earlier rotor case you had the shaft coming out and now here what we see is explicit poles if we call this as a north pole and then here you have a south pole this one would then be a north pole and this would be a south pole and you find that this pole surface is smooth here again the pole surface is smooth whereas between this pole and the next pole there is only a big air gap that is seen this region there is a big air gap similarly between this pole and this pole there is a big air gap here here there is an air gap and here there is an air gap so between adjacent to poles there is a large amount of air gap that is available in this kind of a arrangement field is then wound around each pole you see that coils have been wound around this and this winding is then connected in series with coils that are wound around this pole in series with coils that are wound here and then wound here. So this then provides the way to excite this field system so some flow of current is then given through these windings because of which this may generate a north pole south pole and so on and this structure is then rotated so that the field that is generated bodily rotates along with the rotor the field is again a DC field excitation just like in the case of a cylindrical rotor arrangement. Now apart from this we also see that there are some round holes that are provided in the pole shoe this region is called as the pole shoe and this that is this region is called as a pole shoe and this is then the pole phase so in this there are round holes that are provided these holes run through the pole along the length of the machine so along these dotted lines inside this hole runs through the pole similarly on each pole these holes have been provided it might actually be a slot for example this pole phase if we are to draw the pole phase here it may be a slot like this in this manner so by enlarge the surface is smooth but for these small openings this slot is actually used to drive a bar through it so there is a bar that is driven through this slot and which comes out here and similarly a bar here which comes out of this bar that comes out of this and so on so in each hole through this holes that are provided you then have bars that are coming out these bars are known as damper bars what is done with these bars is if these bars once you provide all these bars here here and so on then there is a ring that is usually used in order to short all these bars together there is a ring that shorts all the bars here and there is another ring that shorts all the bars on the other side as well so these damper bars actually form a short circuited cage like structure you see that there are bars here which are shorted on either side so it forms a cage like structure much like the squirrel cage of an induction machine you have seen you know what a squirrel cage rotor looks like the squirrel cage rotor looks essentially like this it is a set of bars running through the rotor with a shorting ring on either side they are called end rings and that is the rotor which is used in the induction machine the induction machine does not have a field winding whereas here the main source of rotor field is the field winding these bars are provided as the name says to damp out the rotor movements now what happens is if you want to for example you know that the synchronous machine rotor runs at synchronous speed and if you want to supply a higher load on the electrical side you have to increase the input on the mechanical side as well which is what is responsible for higher electrical output now when you supply an increased mechanical input what will happen to the rotor is that the rotor because of a higher input now will accelerate a little and as it accelerates the rotor tends to move away from it is synchronous locked position as it begins to accelerate then the angle between the voltage vector of supplied on the state or side and that of the field it increases and therefore more mechanical input is able to absorb and deliver a higher electrical output ultimately the rotor still has to run at synchronous speed if it is connected to the grid so the rotor will momentarily accelerate and then decelerate again but settle down at a slightly different position during this acceleration and deceleration the rotor may actually oscillate and these oscillations are to be damped out and it is for that reason that this arrangement is given so whenever the speed goes away from synchronous speed this damper bars act very much like the squirrel cage of an induction machine in the squirrel cage of an induction machine EMF can be induced only when the speed is different from that of the synchronous speed and that is exactly what is happening here if the rotor accelerates the speed is different from that of the synchronous speed and now an induction machine action comes into play which now tries to bring the rotor which tries to decelerate the rotor and the rotor locks back into synchronous so this arrangement is also symbolically represented in the figures that I can show here so you see here in this picture that in these slots bars have been inserted in all the slots and then in the next picture so in the next picture here you see that rings have been put on either side which shot all these bars so this ring is the end ring and these bars are the ones that are meant to provide the damping action so this then describes how the salient pole rotor will be of course these bars are not restricted to the salient pole arrangement alone even if the rotor is cylindrical one would need to one may desire a damping action in which case this arrangement would need to be there even for the cylindrical rotor case now the question is how do we model this set of that is this alternator so let us start looking at the equations that one can use in order to describe this so in the alternator then the stator that is available is cylindrical and you have windings that are placed distributed on the stator and a phase winding here and then you will have the b phase winding here and c phase so b phase and c phase what we do as we have been doing before we draw spatial axis in order to describe the axis of each phase and so the a phase winding would then have an axis here so this is the a phase axis and then you have the b phase axis and then you have the c phase axis the rotor because it is associated with a field structure it will generate a north pole along some direction and a south pole along some other direction so we associate an axis of we associate an axis with the rotor to denote where the rotor angle is where the rotor is angularly located and that axis we will associate with the north pole of the field that is there in the rotor for analysis we are going to consider a two pole alternator so there is only one north pole and one south pole and that north pole is associated with an axis which we will then call as the rotor axis so the rotor axis is let us say the rotor axis can be drawn here so this is the axis of the rotor and this angle which the axis makes with respect to the axis of the a phase is then called as the rotor angle. Now we need to write down equations that describe the behavior of this machine so as before what we need to start out with is the voltage equation we need to write down expressions for the voltage that is applied to the machine and there are three phases so you have VAS, VBS and VCS so if you take for example VAS the expression can very easily be written as the stator resistance we will assume that the three phases are identical except that they are displaced in space of course the resistance of each phase is the same so you have RS multiplied by IAS plus P x PSIAS which is PSIAS is the flux linkage of the stator A phase and derivative of that gives you the induced voltage so P x PSIAS and now therefore we will have to write down an expression for PSIAS PSIAS is now going to comprise of flux that is produced due to current flowing in A phase plus due to current flowing in B and C and due to current flowing in the rotor as well so how do you get these expressions so we know that flux linkage is equal to inductance multiplied by the flow of current so self inductance multiplied by current flowing in the A phase let us call it LAS and then mutual inductance between A and B multiplied by the current flowing in B phase plus mutual inductance between A and C multiplied by the current flowing in C phase plus mutual inductance between A phase and the rotor we will call the rotor as the field winding that is the normal nomenclature that is used for synchronous machines so we will put that as MAF multiplied by IAS now we need to derive expressions for each one of these LM, MAF and so on so how to derive expressions so we need to consider a particular rotor geometry first and what we will do is we will consider the salient pole rotor why should we consider the salient pole if you look remember back when we started discussing about distributed field and determination of inductances and so on we wrote down expressions for inductance for the salient pole rotor and cylindrical stator machine arrangement and if you remember we have in order to describe the salient pole rotor we had inductances which were given symbols as LD and LQ LD represents the direct axis inductance and LQ represents quadrature axis inductance the direct axis inductance then is indicative of the smaller air gap that is present along the field pole and the quadrature axis inductance is indicative of an inductance that would arise due to the larger air gap that is present 90 degrees to the field pole these two are distinct and different because the rotor is salient pole because in a salient pole machine the air gap along the direct axis is smaller than the air gap which is at 90 degrees to it because you have a large air gap in between two poles if you take a cylindrical rotor structure the air gap wherever you look at whether it is at the pole that is at the pole phase or at 90 degrees to the pole the air gap is always the same and therefore you expect LD to be equal to LQ so in the resulting expressions of inductances that we have determined if you simply say LD is equal to LQ you get equations for the cylindrical rotor machine if you keep LD and LQ distinct you get equations for the salient rotor machine and therefore it is good to derive equations for the salient pole machine and the cylindrical rotor machine equations can be obtained very simply by making LD equal to LQ so we will start by looking at expressions for the salient rotor machine. In the case of the salient rotor machine again we have derived all the expressions earlier so we just have to remember and recollect those equations back so if you have an arrangement like this where there is a particular phase axis a salient pole rotor so let me symbolically draw the salient pole rotor this way salient pole rotor having it is rotor axis oriented here then we have already determined an expression for the self inductance of the stator A phase so let me write that down what it would look like so the self inductance LAS is then given from the expressions that we have derived earlier we had a LD plus LQ by 2 plus LD minus LQ by 2 into cos of 2 times ?R if you remember back this is the expression that we wrote where LD and LQ are inductances that take into account equivalent air gaps along the direct and quadrature axis now this expression talks about flux that is generated due to excitation of the A phase flux that is generated and crosses into the rotor before coming out so the inductance that is described here essentially is a magnetizing inductance so this is the magnetizing inductance component of the self inductance in addition there would be leakage flux generated by exciting the rotor and this leakage flux is something that does not cross over into the rotor and link the other coils as well it links only the A phase and there is no analytical expression for it what is normally done is you simply add a term leakage inductance to it LL is a leakage inductance of the A phase so this is the expression that we had already derived so going forward we need to get the expression for the inductance of the phase B phase B has an axis here so this expression can be derived very easily from the earlier expression because it is the same phase which is displaced at a different angle the expression for the A phase and the rotor here I mean inductance of the A phase when the rotor is located here is that where ?R is the angle between the axis of the phase and the rotor axis now you have the B phase axis here and the rotor makes an angle with the B phase axis which is this angle and it is behind the B phase in the earlier case when we determine the expression the rotor axis was ahead of the A phase but now it is behind the B phase and this angle is what you need to determine for the angle ?R and this angle is nothing but 120°- ?R this angle is 120 because in a 3 phase machine the phase axis are displaced by 120° here is already ?R and therefore this must be 120- ?R but however with respect to the B axis the rotor axis is behind and therefore this angle is – so – 120- ?R therefore in the same expression if we substitute instead of ?R – 120- ?R you should get the expression for the self-inductance Bs and that is therefore Ld- Lq by 2 – Lq by 2 x cos of 2 x – of 120- ?R is nothing but ?R – 120 then plus the leakage induct and this is therefore this can be simplified as 2 ?R – 240° so 2 ?R – 240° similarly if we want to determine the inductance Lcs now if we take a look at this figure again you have the C phase axis here and the rotor is behind that of the C phase axis by this much angle and that angle is nothing but 240- ?R and by the same argument the rotor is now behind the C phase axis and therefore what you are looking at is – 240- ?R so let us put that down you have Ld- Lq by 2 – Ld- Lq by 2 x cos of 2 x ?R – 240° plus leakage inductance and 2 ?R – 240 is nothing but 2 x ?R – 480 you can add another 360° to it and therefore or subtract 360 so this becomes – 120° so these are the expressions for the self-inductance of the 3 phases apart from this then there is a field winding and so the field winding will have a self-inductance now is that inductance going to vary with rotor position or not now the self-inductances of the stator vary with respect to the rotor angle because of the saliency of the rotor obviously if this rotor were aligned along the a phase axis the inductance of the stator a phase would be maximum if the rotor were aligned 90° to the a phase axis inductance of the stator a phase would be minimum but as far as the rotor is concerned irrespective of where the rotor is located it always sees the same air gap so the flux that is generated by a unit excitation given to the rotor will always be the same or the flux linkage will always be the same irrespective of where the rotor is and therefore the inductance of the rotor coil is independent of the rotor angle therefore we can say that the field inductance self-inductance is LF is simply equal to a leakage inductance of the field plus we will call this as a magnetizing inductance of the field so this is a simple expression for the self-inductance of the field so the self-inductances are over now we need to determine expressions for the mutual inductance again let us look back on what we have done we have already studied the case where you have two stator windings let me redraw this figure again you have a stator and then you have a winding here which is let us say phase a and then you have another phase which is phase b and phase a would have its axis here this is the a phase axis and this is then the b phase axis the rotor is situated at an angle that is then your rotor angle so under this condition assuming that it is a salient pole rotor again where we had looked at the case where these two axis are displaced by an angle ? we have determined the expression for mutual inductance for these two coils and that mutual inductance is then given by Ld x cos ? cos of ? – ? ? is nothing but ? r – Lq x sin ? r sin of ? – ? r this was the expression for the mutual inductance between these two phases a and well one and the other now in the actual machine what we are having is three such phases so you have a phase b phase and your c phase so a b and c and we need to determine therefore the mutual inductance between a and b a and c and then b and c of course the mutual inductance between b and a will be the same as that of a and b so that is trivial there is nothing to direct so now if we want to get this for mab what do we do with this expression the figure is as it is there is no change in the figure a is here and b is here that is the situation for which we have derived this expression all that we need to do is substitute ? equal to 120 degrees because in a three phase machine the two axis are displaced by 120 degrees so mab is nothing but Ld x cos ? r x cos of 120 – ? r – Lq x sin ? r sin of 120 – ? r so this can be written as Ld x this is of the form cos a cos b which is nothing but cos of a plus b cos a – b divided by 2 so cos of a plus b would then give you cos 120 and then cos a – b will be cos ? r – this so 2 ? r – 120 and then – Lq this is sin a sin b and therefore that is nothing but cos a – b – cos a plus b by 2 of course so there is a by 2 here and here also by 2 cos of a – b is ? r – this so 2 ? r – 120 degrees cos a – b – cos a plus b so – of cos this plus this is 120 degrees so that is what you have which is nothing but so mab cos of 120 degrees cos of 120 degrees is – ½ so you have – ½ here so that simply gives us – Ld by 2 – Lq by 2 so this – and this – goes away but you still have this – here so you have – Ld by 4 – Lq by 4 and then you have Ld by 2 – Lq by 2 into cos 2 ? – 120 so you have plus Ld – Lq by 2 into cos of 2 ? – 120 so this expression can then be combined it is – of Ld by 4 so that is the expression for the mutual inductance between the phases a and b what about mutual inductance between phases a and c in order to determine mutual inductance between a and c the rotor position is still the same so and it is with respect to a phase so that is also not a problem. The only thing that we have to now do is ? is between a phase and c phase that is the angle we need to consider and ? for considering a and c is 240 degrees so what we need to do is substitute 240 here so what you have is Ld into cos ?r cos of 240 – ?r – Lq sin ?r sin of 240 – ?r so this is again the same kind of relations so you have Ld by 2 multiplied by cos of 240 degrees cos a plus b plus cos a – b and therefore cos of 2 ?r – 240 and – Lq by 2 into cos of 240 degrees – cos 2 ? – 240 so cos of 240 is again – half this is again – half so you have Ld plus Lq by 4 now you have – Ld plus Lq by 4 this is cos of a – b there is some problem the cos of a – b – cos a plus b so it should be the other way around cos of a – b is cos of 2 ?r – 240 degrees – cos a plus b is cos of 240 so this is – half then so you have – Ld plus Lq by 4 and then you have plus Ld – Lq by 2 into cos of 2 ?r – 240 degrees so this is the expression for mutual inductance between the a and c phases the third expression that we need is the mutual inductance between the b and c phases so how does one determine that so if you now look at this figure what we need is the inductance between b and c so the angle between b and c is 120 degrees just like it was between a and b so ? is 120 degrees but however the rotor is now lagging behind or lying behind the b axis by a certain angle and therefore the angle that we will have to consider is – of 120 – ?r therefore what we will have to write is mbc is Ld into cos of instead of where is that here instead of ?r here what you will need is – of 120 – ?r or ?r – 120 therefore cos of ?r – 120 degrees into cos of ? – ?r ? is 120 so 120 – ?r is now ?r – 120 so 120 – ?r – 120 and then – Lq sin ?r – 120 into sin of this expression and that is nothing but 240 – ?r so that is your expression for the mutual inductance between b and c phases so this should then simplify as Ld by 2 again you have cos a cos b so that is nothing but cos of a plus b so ?r so this let me rewrite this as 240 – ?r so you add these two so that would then be ?r and – ?r cancel out 240 – 120 is 120 degrees plus cos a – b so this – that that is cos of ?r – ?r is 2 times ?r – 120 – 240 is – 360 degrees and then – Lq by 2 into cos of a – b this – that is 2 times ?r – 360 – cos a plus b so – cos 120 degrees that is the expression you get which then simplifies to 2 ?r – 360 is nothing but 2 ?r itself so that is 2 ?r and this is as usual – ½ so you have – of Ld – Lq by 4 – Ld – Lq by 2 into cos of 2 ?r so that is then the mutual expression for the mutual inductance between the b and c phases so we now have self inductance of the a phase we have mutual inductance of a and b we have mutual inductance of a and c when we write the flux linkage for the b phase then you will get the mutual inductance for b and c as well and now we need to have an expression for the mutual inductance between the field and the a phase now as far as the field is concerned you are going to have a salient pole rotor that is rotating and we have said that the field produced by the salient pole rotor is primarily a dc field and the field bodily rotates with the rotor as the rotor moves if that is the case then the field generated by the field structure is not going to change with respect to time looked at from the rotor it always generates a constant field provided the excitation is also fixed and therefore the total flux generated by the field is the same field structure is the same but however if the field structure were aligned on this axis all of its flux will link phase a and as the structure moves rotates then less of its flux is going to link phase a therefore the mutual inductance will be maximum when the field structure is aligned along the axis and that value of maximum mutual inductance let us call it as MAF well let us call it as MAF itself right but however that value of mutual inductance is not going to remain as the rotor rotates the flux that links the stator a phase will be whatever is generated by the rotor field one part of it that lies along the a phase axis is what is going to link that field and therefore the total flux multiplied by cos of this angle is what is going to be responsible for linking phase a and therefore the mutual inductance will then be MAF multiplied by cos of ?R between the rotor field and the a phase of the stator by the same token if you then have the field somewhere else then the mutual inductance between the field and another phase of the stator will depend upon the angle between the two and therefore if the rotor is aligned here at ?R mutual inductance between rotor and the b phase will then be the same MAF because had the field been aligned along the b phase axis the flux linkage of the b phase would have been the same as what was therefore a phase and therefore the maximum value of mutual inductance for all the three phases with the rotor will remain the same the instantaneous mutual inductance between the rotor and the b phase axis will then be well since that mutual inductance is a common value between that of the stator and the rotor structure we will simply call it as MSR that is a better notation I think so this is between mutual inductance between a and field between b and field will then be MSR into cos of this angle which is 120- ?R will be MBF and then with the c phase it would be MSR cos 240- ?R will then be MBF so with this then we have derived all the expressions for inductances between the stator phases and between the rotor and the stator as well so we now have to put all of them together and formulate the voltage equations of the machine which we will do it in the next lecture.