 we have been looking at modeling the synchronous machine and in the last lecture we had derived equations in the synchronous reference frame for the machine. Synchronous reference frame is also the rotor attached reference frame for the synchronous machine because the rotor of the synchronous machine rotates at synchronous speed and since in the rotor of the synchronous machine there is a source of excitation. We attach the synchronous reference frame axis accordingly the axis along which the main field of the alternator exists is then the d axis and the axis 90 degrees to that is the q axis. Similarly the machine analysis field had developed starting from the synchronous machine and the direct axis has always been identified with that of the main field and the q axis has been identified with an axis that is 90 degrees to it. It is only subsequently that the machine analysis method were applied to the induction machine also and it is only beyond the 1960s that these machine equations were used for doing machine control of the induction machine. So that is what we have been looking at we have been seeing the synchronous machine modeling in the synchronous reference frame and in the later part of the last lecture we looked at referring the rotor variables also to the stator number of turns in the synchronous reference frame and we were seeing how the equations simplify as a result of that. So let us look at those equations again we start with the synchronous machine equations you had vds, vqs, v0s and v field instead of writing an elaborate description we will combine all of them into an operational impedance matrix so that makes it a little more easy to write. So this is the d axis applied voltage which consists then of a stator resistance drop and then plus the direct axis synchronous inductance v where the direct axis synchronous inductance is then going to consist of a leakage inductance of the stator plus 3 by 2 times Ld which we then called as Lmd which is the magnetizing inductance three phase magnetizing inductance along the direct axis and this together is then the d axis synchronous inductance so that is what we had and then here it was – Lqs into ?s where Lqs is defined as the leakage inductance of the stator plus 3 by 2 times Lq this being defined as Lmq and then this together is then the quadrature axis synchronous inductance and then there is no contribution here from the 0 axis term and then there is a d axis pdmf that is induced which was root of 3 by 2 times msr multiplied by ?s so that forms the last entry here what you have is Lds times ?s and then rs plus Lqs into p and then you have this is q and 0 so 0 here and then there is no q axis induced emf sorry this is along the direct axis so this is p and then here you have the speed emf so this is root 3 by 2 into msr into ?s and then along the 0 axis rs plus Lls into p nothing there here you have root 3 by 2 into msr into p there is no pdmf term because this is not the fictitious coil and then on the 0 sequence term there is nothing here and then you have rf plus Lf into p this matrix multiplies ids iqs i0s and iq where Lf is the self inductance of the field on the rotor that then consists of leakage inductance and then a magnetizing inductance of the field. Now we started in the last lecture to refer the rotor variables to the stator turns and when we do that we said that it is referred to the stator turns under the assumption that if you now look at the d axis the d axis has the field coil this is on the rotor and then it has a fictitious coil derived from the stator that is the ds coil so this is ds coil and this is the field coil and since these two lie along the same axis we assume that the flux that is linking the field links the ds coil also and if you now multiply the mutual inductance by the ratio of number of turns what you get is the magnetizing inductance along the d axis which we then called as which is the same as LMD that is the magnetizing inductance along the d axis so with that what happens is you vf dash becomes the referred stator turns referred field voltage and here what you get is 3 by 2 times MSR into the turns ratio which then becomes LMD similarly here you get LMD into P and this becomes LMD into ?S as far as the field is concerned you get a turns ratio from replacing if by if dash and another equivalent term comes by replacing vf by vf dash so you get turns ratio squared in this term which then means RF is referred by the turns ratio squared to the stator turns so it becomes then RF dash and LF becomes referred by the turns ratio squared which is again LF dash but in LF dash what happens is the leakage inductance gets referred by turns ratio so LF dash is nothing but LLF dash which is LLF multiplied by the turns ratio squared into 3 by 2 of course LMF is the magnetizing inductance of the field and when you multiply that by the turns ratio squared what you get is the magnetizing inductance of the d axis coil and therefore this is nothing but LMD here so LF dash is LLF dash plus LMD where LMD is then given by this expression along with this we then also have the expression for the electromagnetic torque torque was given as I ds x Iqs x Lds – Lqs plus root 2 by root 3 by 2 Msr x If x Iqs now that you replace it by the stator referred turns this term becomes then LMD in IF dash so these are the equations referring the machine to the stator turns and in the synchronous reference frame in this description we have assumed that damper windings damper bars have not been considered here we saw in the beginning of the lectures on alternators that if you look at the alternator poles if this is the alternator pole and then you have the coils of the field that are wound here what we said was in the pole phase you have bars that are introduced which run along the length of the pole and these bars are all shorted here shorted on this side with the result that if there is going to be any change in the speed from synchronous speed either more or less then there will be induced EMF in these bars and there can be currents that flow in these bars and the shorting ring connects it to the bars on the next pole phase so there is going to be flow of currents all around and this causes the loop that thereby forms here causes a source of excitation along the Q axis and definitely there is there are loops formed here which cause a source of excitation along the D axis and therefore because damper bars are there it will affect the machine model by providing sources of excitation both along D and Q and therefore that is represented as follows let us draw the stator axis so this is your AS axis DS and then the CS axis and then you have the rotor axis so let us say this is the axis of the rotor so you have the rotor angle and in the along 90 degrees to this axis you have the quadrature axis of the rotor so this is your D axis and Q axis what we had done earlier was you had the DS axis DS excited which is the stator winding converted to the synchronous reference frame which gives you a DS and QS coil this is QS coil apart from this in the model that we had written down we had a field excitation which was there along the D axis field now if one wants to account for the presence of dampers one has to consider at least one more coil on the D axis which is a representation of the field that is generated by currents flowing on the D axis due to this and one more coil at least on the Q axis which accounts for the MMF that generated along the Q axis. Now since these are always shorted by the shorting rings we normally consider these two coils to be shorted and the normal symbol that is used to represent these two are KD coil and KQ coil so if one is now going to include the effect of KD and KQ how do we modify these system of equations that we have derived so far. So if one has to do that remember we are going to write down the equation in the synchronous reference frame and we make use of the fact that we are writing these equations in a stator referred way that means we have already done this exercise where VF is replaced by VF dash IF is replaced by IF dash and so on similarly there will be flow of I KQ in this coil and I KD in this coil there are certain number of turns here and certain number of turns here and when you refer everything to stator turns then the equations really become easy to write to modify that let us extend these set of equations let us extend this set of equations what we are now going to have is in addition to all this you have VKD dash and VKQ dash so we augment the first equation set by adding these two rows this is extended now what happens as a result of this we know from the way in which we have analyzed or derived these synchronous machine equations and indeed the generalized machine equations valid even for the induction machine that the terms here are of different varieties one is the stator the self coil resistance terms and the self coil inductance terms and then the mutual DI by DT terms and then you have PDMF term now self coil resistance terms are bound to be there and they will always occur along the diagonal like this self coil inductance terms will always be there they will also occur along the diagonal in this manner mutual DI by DT terms we have seen that they will occur between one coil and another coil provided they lie along the same axis for example here if you take voltage of the D axis term you have a mutual inductance due to the field coil which was the mutual between this DS and F now you have one more coil KD this will also cause a mutual inductance and because we are now writing it in a stator referred manner you remember that mutual inductance multiplied by the ratio of number of terms gives us the magnetizing inductance and that magnetizing inductance is along the D axis and therefore if we want to write include the effect of KD coil on the stator voltage stator D axis equation it will arise as an LMD multiplied by P and then you have PDMF terms PDMF terms arise in one coil provided it is a fictitious one from the excitations on the axis 90° away now in this case KD is I mean we are looking at the equation for the DS term and this coil is definitely a fictitious coil because it is a representation of actually fixed coils on the stator we are moving away to the synchronous reference frame and therefore it is fictitious and therefore this will have PDMF terms and PDMF terms will arise due to coils that are in the axis 90° away and you already have one PDMF term that is – LQS into ?S due to the QS coil and now you need to provide another PDMF term due to the KQ coil and therefore this is LMQ into ?S now this PDMF term arising due to IQS is this is actually a source of excitation that is present in the stator itself it is not on the rotor and therefore the inductance term that arises in the PDMF term is the stator inductance of the Q axis whereas here this coil is present on the rotor and therefore the term that comes here is the magnetizing inductance or the mutual inductance between the stator and the rotor so that is what is going to come here and then we examine what is going to happen in the Q axis coil of the stator the Q axis coil of the stator will then have a PDMF term due to the D axis on the axis on the rotor which will then be LMD into ?S and then between the Q axis stator and the Q axis damper you would then have DI by DT term so that is LMQ into P0 axis is not affected by this operation so those two will be 0 and then you have VFD which is the equation for the field coil so the field coil is actually on the rotor so it is not a fictitious coil therefore it will not have PDMF term it will not have DI by DT terms due to excitation 90° away but it will have DI by DT terms connecting these two and of course its own self term which is there so this is your D axis column KD axis and this is your KQ axis column and therefore the field with the KD axis will then have an LMD into P with the Q axis it would be 0 and then we come to the D axis equation that is KD equation so this along with the direct axis would have an LMD into P they are on the same axis with respect to the Q axis of the stator there is not anything because this is not a fictitious coil it is directly on the D axis and then with respect to the 0 axis term there is nothing with respect to the field it has an LMD into P and then with respect to this one you have RKD this is the self term so there is a resistance of the damper and then a self inductance of the damper referred to the stator turns and then with the Q axis of the damper there is not any term there similarly here this would be 0 you have LMQ into P with the 0 axis term there is not anything and then this is with the field there is no term here with the D axis there is no term here you have RKQ dash plus LKQ dash into P so the matrix is this matrix is then multiplied by the vector of currents so that is assumed to be present here let me not write that again that is IDS, IQS, I0S, IF dash, IKD dash and IKQ dash so that then is the model of the synchronous machine having one coil to represent the damper along the D axis and one coil along the Q axis referred to stator turns and as usual one can write the expression for torque by using the relation I transpose GI you have to separate out the speed MF terms into the G matrix and then multiplied as I transpose GI one gets the torque so in you can see that if you want to augment the description by having more coils to represent the damper windings some more coils if you want now those can be added very easily by understanding how this representation is there is one more aspect that I think I have missed the speed MF term here is negative so that is then the synchronous machine equation one can look at developing this in terms of flux variables and let us look at how it can be written in terms of flux variables for a simpler description not having this elaborate form so to develop these equations or to represent it in flux variables we will again neglect the damper windings neglecting damper because it is just simpler to write otherwise one can still define the flux linkages as PdS which is the flux linkage of the stator winding along the D axis is Lds times Ids that is the self inductance direct axis synchronous inductance multiplied by Ids plus LMD x If dash the mutual inductance multiplied by the field current stator referred with respect to turns and then you have LMD x I kd I think following our notation we should be consistent so let us put all the rotor variables as superscript so this will be kd dash and kq dash so I kd dash if there are more representations along the D axis for the dampers then you would have this as kd dash 1 and then may be I kd dash 2 I kd dash 3 and so on so that would be the form and similarly psi qs could be written as the quadrature axis synchronous inductance multiplied by Iqs there is no field along the q axis in the rotor so after this there is there are only the dampers and therefore LMq x I kq dash if there are more representations for the damper you will have more such term so to develop the equations in the flux variables let us consider a simpler form we will neglect the damper bars and therefore we are going to consider only a subset of these equations which is up to this so now that expression can be written as Vds is RS plus LSL how have you written DS Lds x P x Ids plus – Lqs x ?s x Iqs and then plus LMd x P I f dash and then you have Vqs is LMd x ?s x Ids plus RS plus Lqs x Lqs P plus LMd ?s x I f dash so that is the equation for the stator and what we see here is you have Lds x Ids so let us rewrite that Vds we will take the stator resistance term out so that is LS x Ids and then you have Lds x Ids plus LMd x I f which is the same as your stator d axis flux linkage so that can be written as P x Ids and then you have – Lqs ?s x Iqs Lqs x Iqs is Iqs that is what we see here Iqs is Lqs x Iqs and therefore you have – ?s x Iqs and then we write the expression for Vqs that is again RS x Iqs and then you have LS Iqs Lqs which is nothing but therefore P x Iqs and then you have ?s x Ids so this is a simpler representation of the machine equation represented by flux variables this is often used in order to do computations with this the expression for the generated torque is Ids x Iqs x Lds – Lqs plus LMd x I f dash x Iqs so this is what you have this can be written as now let us split the terms Ids Iqs x Lds – Ids Iqs x Lqs plus LMd x I f dash x Iqs now these two terms together one can combine this and you see that that can be written as Iqs x Ids – Ids x Lqs Iqs now this term is nothing but your ?ds as we have defined and therefore this can be simplified as Iqs x ?ds – Ids this term is nothing but ?qs so that is ?qs so torque can again be written in terms of product of flux x I so these are different ways in which one can represent the synchronous machine equations which is useful for study though we have been representing the electrical equations and the expression for T in order to describe the dynamics of the system one must not forget that you need the mechanical equation also and the mechanical equation takes the form generally this is your generated torque remember again all the equations that have been written are being written with motor convention we may be analyzing synchronous machines normally when we study the first course on electrical machine synchronous machines are not studied under this convention they are studied using the generator convention whereas we have used this convention in order to maintain the continuity of what we had started as induction machines and indeed nowadays there are synchronous machines that are being used for control purposes at least as synchronous motor in which case this is a convention that one would be using to study these machine and therefore using this convention the generated Te is output from the machine is a mechanical output so this mechanical output – the load torque is going to result in an increase in speed load torque make then consist of a fixed component and in addition viscous friction drag components which are to the exponent of ?1 ?1 or ?2 and so on all those terms are going to come into here so we have been looking at lot of these machine equations how to develop these machine equations and so on however if you look back on whatever we have done we have not made any reference to the conventional way in which you would look at a machine let us say in a first course in induction machine the first course in induction machine normally starts or deals with the steady state analysis and starts by developing equivalent electrical circuit for example if you take the induction motor what you would do is to study the induction motor you would develop an equivalent electrical circuit which is a per phase electrical circuit and that electrical circuit is developed based on a an understanding of how the machine would work namely you have a rotating magnetic field and then you have therefore some current flowing in the rotor and that voltage induced in the rotor depends on the number of turns and therefore so on and so forth right whereas here in the development that we have done we have not made any reference to the equivalent circuit of the induction machine at all we have directly started by assuming that the stator of the induction machine has three phases distributed in a certain manner and then we found out an expression for distributed winding inductances we have developed it from a totally different angle so what is then the relationship between whatever we have derived now and what we have understood from the first machine course regarding the equivalent circuit and the performance of the induction machine so do these two match at all will just give the same equivalent circuit that what we have been using in the first course on electrical machine I am sure some of you would have had this doubt. So let us look at how to do steady state analysis before we make use of this equation for more sophisticated applications we need to understand that these equations indeed boil down to what we have been studying in the first course when we did steady state analysis so that is what we will now look at when we say steady state analysis what we automatically assume is we are looking at sinusoidal steady state analysis that means the voltages that are applied to the machine are pure sinusoids and therefore the responses flow of IAS or IBS ICS they will also be pure sinusoids and you do not have any speed variation right. So this means that speed has settled down both voltages and currents are pure sinusoids current transients have settled down have gone to 0 no disturbances so everything is a nice sinusoid that is there. So let us look at the induction machine equation so let me write down the equations in the synchronous reference frame you have VDS VQS VDR and VQR remember that when we are using a balanced excitation 0 sequence is absent and therefore we need not really represent the 0 sequence most cases of machine analysis 0 sequence may indeed be neglected since it is balanced here you have RS plus LS x P ?S x LS and you have LM x P LM x ?S – ?R now that we have just now finished the synchronous machine modeling it would be appropriate to draw your attention to these two terms there is no LMD and LMQ here but just an LM this arises because the induction machine has a perfectly cylindrical arrangement in the rotor there is no saliency and therefore you do not have LMD and LMQ coming here and then you have – ?S LS RS less LS in the P similarly here there is no direct axis synchronous inductance of the quadrature axis synchronous inductance it is just the self inductance of the stator LM ?S so this would then be the machine equation with IDS IDR similarly so that is the machine equation. Now the equivalent circuit of the induction machine as we draw in the normal way refers to the per phase equivalent circuit per phase equivalent circuit in the stator reference frame that is the then the actual phase variables that are applied so we have to transform these equations into the normal ABC reference frame and how does one do that we have VDS VQS V0S and so on so if you want to transform those equations you have VDS VQS V0S is of course V0 so we need not really consider that so let us look at VDS and VQS and this has to now be transform to the aß reference frame and then the three phase frame so how to transform VDS and VQS to the aß frame you remember that the transformation VD VQ from aß was cos ? and then sin ? – sin ? and cos ? so in this case when you want to invert this you will get the transpose of that matrix so you have cos ? – sin ? and then sin ? and cos ? that would then be in the aß frame and then from the aß frame you want to transfer to the ABC frame and there again you have to transpose whatever system matrix that we use so that is root of 2 x 3 x 1 and 0 remember this when you go from aß to dq what we had used was I mean ABC to aß what we had used was root of 2 x 3 x 1-1-1-2 0 root 3 x 2 and – root 3 x 2 1 over root 2 and 1 over root 2 now since we are taking the first row first column to be the first row you have 1 0 and 1 by root 2 and 0 sequence is not considered that 1 over root 2 can be neglected and this will now give us your ABC if you put 3 rows here and so on. But then what we are interested in is a per phase equivalent circuit which means it is enough to consider the a phase alone and if we are going to consider the a phase alone we need to consider the first row here and the first row here is going to be obtained by multiplying this first row by the first column and the first row by what you get here and therefore that is this column this term into this – this term into this second row is not really important. So essentially then what you have here is vds cos ? – vqs sin ? the second term that is going to come here which is vds sin ? vqs cos ? is not important because that is going to multiply 0 and therefore what we can say is that vas the stator a phase voltage is nothing but root of 2 by 3 times vds cos ? – vqs sin ? this is vas now similarly one can do the same thing for ids and iqs as well you can substitute here ids and iqs same way you transform it reverse in the reverse sense and get back ias, ibs and ics again we are looking at a per phase equivalent circuit so it is sufficient to consider ias alone and therefore you would have another equivalent relation ias is equivalent equal to root 2 by 3 x ids cos ? – iqs sin what we started out with the equation description here is the induction machine in the synchronous reference frame that means a reference frame rotating at synchronous speed ? is the angle that the reference frame d axis makes with respect to the stator a axis and since the frame rotates at synchronous speed one can therefore say ? must be equal to synchronous speed multiplied by time assuming that at t equal to 0 the d axis coincides with that of the stator a phase axis so using this description then one can try to derive the steady state equivalent of the induction machine we have done this for stator a phase v and i similarly one has to write down the expression for vdr I mean v ar and iar so once you get expressions for these things we can try to process it and see whether we can derive an equivalent circuit like representation from this in order to do steady state analysis so that part of it we will look at it in the next lecture we will stop with this for today.