 We have studied so far modeling of various kinds of machines we looked at DC machines induction machines and then the synchronous machines and having studied one by one we have now reached the end of the course. So let us take a little while to look back at what we have studied so far and where do we go from here. So what we have been studying all along is modeling and analysis of electrical machines. We started out by looking at the element inductance. We have seen as we went through the course everywhere the inductance plays a very important part in determining the behavior of the machine. What we have seen is that the change of inductance with the rotor angle or change of inductance with respect to displacement is the one that is going to be responsible for generating torque or force as in the case if it is linear system. So we started out by looking at linear systems which were simple to analyze simpler system equations and then using that as an opportunity to formulate system equations we saw there that the force is really given by the change of inductance with respect to displacement. So if you look back or remember the first few lectures where we determined an expression was for force, force was written as ½ I2 x dl by dx. So inductance only if it can vary with respect to displacement you have generation of a force. So starting with that then we moved on to the next more sophisticated system more involved system that is the rotational system and in the rotational system we saw how this movement of the rotor and the variation of inductance is going to generate a force more importantly we saw that if you have a singly excited rotational system you will not be able to achieve movement for a long time the rotor might move and then stop after some time. And therefore in order to have continuous movement you require doubly excited systems at least one coil on the stator and one coil on the rotor if it has to move fully but then even with one coil on the stator and one coil on the rotor you really cannot achieve continuous movement because what we saw was for the generation of continuous movement you need to have at least a rotating field which can then make the rotor also move along with it and to generate a rotating field you need at least two coils on the stator and then two coils on the rotor as well. So we quickly moved on to the doubly excited systems or rather systems where there is a double excitation on the stator and a double excitation on the rotor so it is really four coil system that we spoke about and then we wrote down the system equations for that. So from the linear system linear motion system we went to the rotational systems and in the rotational system we then moved on to a four coil system that is two on stator and two rotor coils for this we then wrote down the electrical equations and the manner in which we proceeded from the voltage equations to arrive at an expression for the force generated or in case of rotational systems the generated electromagnetic torque that method we had established in the linear system itself you find out an expression for input electrical power and then determine the ways in which this input is then expended so some part of it goes off as I2R losses and then some part of it goes to change the magnetic field energy and then the remaining part has to be expressed as mechanical output and then knowing all these different entities we arrived at an expression for the mechanical output and that is the procedure that we have consistently followed all through the course. In between we also saw how a non-linear magnetic system could be analyzed so if one wants to analyze non-linear that is systems with materials having non-linear BH curves then you need to make use of different forms of expressions the equations which we formulate for the linear motion system and the linear magnetic rotational system are not really applicable here because the inductance is going to vary with respect to the excitation itself and therefore for analyzing this we then looked at ideas of field energy and then co-energy of the system and then from this how to derive the expression for force that is then looked at as the rate of change of co-energy with respect to displacement and so on force or for that matter the generated thought but then by enlarge in the machines that we encounter one may note that by enlarge the machines operated operate in the non-saturated region of the BH curve in some cases they may operate in the saturated region of the BH curve but for most analysis we have neglected the existence of iron going into deep saturation. So the subsequent analysis we had done assuming the assumptions that we made were that slotting was neglected the effects of slotting in the stator is neglected or in the rotor for that matter if you are going to consider inverted machines and then iron losses are neglected and then saturation was neglected with these assumptions then we tried to formulate machine equations we started with the four coil system for which we wrote down the voltage equations and from that we derived an expression for the generated electromagnetic torque and then we found that there are certain disadvantages to the representation of the system for the purpose of doing numerical computations the inductance being varying with respect to rotor angle renders the impedance description of the system to change with respect to rotor angle and therefore every time step that you use for computation the impedance is going to change the impedance representation V equal to Zi is then going to change because that is a function of rotor angle and therefore we then did something to get rid of the variation with respect to rotor angle in the system description what we did was we transformed transformation to the stationary reference frame. So the requirement was that if inductance is if the inductance description is not to change with respect to rotor angle it or with respect to angle it can only happen if all the four inductances with which are associated with the four windings that we are considering in the system that we have they should all be stationary with respect to each other only if they are stationary then the inductances will not change with respect to of course since they are all of them are still there is no question of change with respect to anything if at all they can change they can change only with respect to excitation and we had assumed that the BH curve is linear therefore that is also not there. So the essence then is to have a reference frame in which all the four windings are stationary with respect to each other and that can be done by transforming the entire description to the stationary reference frame and we found that indeed in this reference frame the inductance matrix is constant there is no dependency of the rotor angle but having reached this point we found that this representation where you have two coils may be on the stator and the rotor coils which were rotating have now been transformed into the stator reference frame this approach or this the resulting set of equations could be used to model a variety of machine for example you have the induction machine where the rotor coils are indeed rotating but as a normal induction machine you observe everything from the stator and therefore this could be used to represent the induction machine this could also be used to represent a DC machine because in the DC machine you have the rotating armature and by virtue of the brush commutator arrangement the rotating armature is seen as if it is fixed on the stator right and therefore you are effectively converting this rotating coil on the rotor to a pseudo stationary fixed coil on the stator which is exactly what we have done by means of equations when we looked at this so we found that this system of equations could actually be used to represent DC machines as well it could also be used to represent synchronous machines because if we look at the case of an inverted synchronous machine an inverted synchronous machine is one where the field structure is on the stator and the rotor is now having the three phase winding which is rotating in the rotor it would behave for all analytical aspects as a normal synchronous machine would and this machine also what one can do is transform the rotating three phase or two phase as the case may be to the pseudo stationary stator reference frame and therefore this description could be used to describe an inverted synchronous machine as well in that sense we then called this as a primitive generalized machine from which one can get any machine description that we want by suitably interconnecting all the four coils in an appropriate way we also saw that this machine could then represent a DC machine DC separately excited machine or a series connected machine with or without inter pole windings and so on all these machine varieties can be adequately represented by this pseudo stationary by this equations in the pseudo stationary reference frame but then we were not happy enough with that we said that in the stationary reference frame all your excitations are DC and it would help if you can have a reference frame where I mean in the stationary reference frame all the excitations are AC and it would help if we can have a reference frame in which the excitations are really DC and therefore we did the transformation to the synchronous reference frame indeed we move to the arbitrary reference frame as an intermediate stage and then said that the arbitrary reference frame can have any arbitrary speed and that arbitrary value of speed that could perhaps be set to 0 or could be perhaps that to synchronous speed or it could be set to rotor speed and therefore you can have a machine description where you can attach yourself to any particular member that you want in an induction machine for example you could attach yourself to the rotor in which case you have a rotor attached reference frame or in the synchronous reference frame or stationary reference frame and if we choose the synchronous reference frame we saw that all the variables became DC and then we found that the synchronous reference frame offers us more benefits as well for example in the case of induction machines we saw that the synchronous reference frame gives us a means to align or to select a particular reference frame whose axis are aligned in some along some location some entity like the voltage or flux phasor or so on which gives us some additional advantages and we saw that these advantages have resulted in formulation of very nice control structures for induction machines namely field oriented control which is also called as vector control sometimes and therefore the transformation to the synchronous reference frame it happened to be of much more use than simply getting rid of the angle dependency for the inductances and if you have the induction machine in that reference frame we saw that the machine description also simplifies considerably and then in that reference frame the induction machine could be made to behave like a DC machine and this indeed is one of the most frequently used control schemes for induction machines today you have vector control schemes and then as an offshoot of analyzing these kind of equations we have more advanced vector control schemes where you have sensorless vector control schemes where you estimate the speed based on the machine model equations that we have derived. So the induction machine analysis we studied and then while looking at the induction machines we also said that the induction machine representation that we had really derived was from a two phase frame and therefore we looked at how to transform the three phase machine to two phase machine and here we came across two modes of doing this one is the non power invariant arrangement or transformation and then we saw the power invariant transformation power invariant transform and it is this transform that we have used subsequently in the course. So this then provides the link between the three phase and the two phase machine and we saw how the machine description that we had derived going from the basic ideas of single coil winding a distributed winding and so on all the way up to the set of machine equations that we have we had made no reference really to the equivalent circuit which one studies in the first course on induction machine and therefore we tried to establish the relationship between the equivalent circuit and the equations that we have derived and we really found that these two really agree with each other the equivalent circuit can be derived under the assumption or by simplifying the machine equations that we have derived for the steady state case and from the steady state case we found that the resulting equations can be represented as an equivalent circuit which is nothing but the very familiar equivalent circuit of the induction machine. So this once again reinforces the fact that equivalent circuit of the induction machine is something that is valid only for steady state case sinusoidal steady state analysis can be done using the equivalent circuit one cannot analyze non steady state conditions by using the equivalent circuit. So we saw all this for the induction machine and then we moved on to the synchronous machine we derived expressions for the salient pole synchronous machine if the synchronous machine is not salient pole but a cylindrical rotor the same formulations that we use for the induction machine would still suffice but if it is a non cylindrical rotor structure if the salient pole structure then we went on to derive expressions based on the direct axis and quadrature axis equations we in effect used what was called as the two reaction theory and then we represented the machine equations in the natural reference frame from which we went on to the synchronous reference frame because the synchronous machine is made to rotate always at the synchronous speed. So it would be of more use to have a synchronous reference frame from which we do the study and from the synchronous reference frame equations again we established what happens under steady state conditions which led us to the analysis of the phasor diagrams and so on and then we looked at the transient analysis of induction machine of the synchronous machines as well where dampers played a very important role damper winding and then as a way of application of the machine equations that we had studied we looked at the sudden short circuit of the alternator we looked at numerical example simulation results for both the induction machine and the synchronous machine in order to understand the relevance of the equations that we had studied and the utility of the models that we had seen now all this we have done where do we go from here net now as far as the machine equations are concerned in the case of induction machines we found that we had used values or we had basically used these terms we had used the stator resistance and then the stator inductance the rotor inductance rotor resistance how do you determine these numbers now these numbers may be determined from the block rotor test and no load test for an induction machine right as far as the synchronous machine is concerned we had many more any more such terms apart from these we have rkd rkq lkd lkq and so on how does one determine all these things and indeed in the analysis of synchronous machines that we had seen what we had done was we had used one damper circuit on each axis now is it enough to have one damper circuit on each axis or do we more or do we need more because how many such circuits are necessary is not very evident there is no physically identifiable single circuit which says this is my d axis d axis for the damper or this is my q axis for q axis electrical circuit this is actually an effect which we are trying to reproduce by means of modeling at as a short circuited winding on the d axis and q axis so maybe in order to represent these effects more accurately we may need more such representations along each axis so in the research in the literature if one sees people have looked at two circuits for the damper along each axis it has been reported in the literature that this gives a more accurate behavior of the alternator when subjected to disturbances rather than the single equal single circuit representation along each axis single circuit for damper so if we are going to have two then you will have rkd1 rkq1 lkd1 and lkq1 similarly you will have rkd2 rkq2 lkd2 and lkq2 so how does one go about deriving the numbers for all these entries from a physical machine because after all our goal overall in this course has been how to derive a model for a physically existing machine so you have a machine and you want to represent the behavior of the machine or estimate the behavior of the machine by means of suitable equations we have derived but one has to plug in the right numbers unless you put in the right numbers this set of equations may not represent reproduce the behavior of the machine that you are looking at it may not even reproduce anything if you put arbitrarily numbers for all these terms it may not give you any meaningful answer at all and therefore one has to have a way of estimating what all these different machine parameters are and plug them into these equations and if you are going to have more expressions like this the system becomes more and more involved the equations are now many more equations are there and it will then become very difficult to solve them analytical even for this case for the case where you have two damper circuits along each axis one may look at some of the references listed for this course there are analytical expressions describing all the different expressions that we have all the different things that we are derived like sub transient reactances and then all those different entities right. So one can look at analytical expressions but they are very very involved so those who are interested further to see I would urge you to look at these references and try deriving those expressions all along then we had assumed that saturation is neglected but this may not be acceptable if especially one is going to study the response of systems to very large excitation or large disturbances the machine as designed one cannot really design a machine such that it never goes into the area where iron gets into saturation if one wants to design a machine such that it is unsaturated under all operating situations you would probably design a grossly over designed machine and therefore physical machines will enter into the saturated region at least the rated value or near above. And especially in the case of synchronous machines this does play an important role because machine is invariably in the reasonably saturated region not really into deep saturation but has crossed the knee point of the BH curve. So if this is the case how can this be considered in the equations that we have derived. So this is often done by empirical extensions to the equations that we have derived and again there are lot of literature available on how saturation can be effects of saturation can be reasonably incorporated into the models of machines that we have derived. Of course once you incorporate this into the machine model you cannot do any analytical solution of the equations one has to resort to numerical simulations in order to get at the behavior of the machine even without that the equations are fairly involved and analytical solutions as we have seen the case of synchronous machines can become very very involved. So one can look at what these extensions are in order to represent the case of saturation in machines saturation of course happens in synchronous machines as well as induction machines also. So the methods for representing saturation are the same in both cases if it is a salient pole alternator if it is a salient pole synchronous machine then there is a small simplification in the sense that saturation is assumed to occur only on D axis because this is where your air gap is small and therefore it is likely that saturation occurs here on the Q axis you have a large air gap and therefore it is unlikely that that axis will go into saturation so no saturation on Q axis. But on the other hand if you have a cylindrical rotor machine then you can have saturation both on the D and Q axis and therefore one has to devise ways of accounting for the extra excitation that is necessary to produce the flux and that can be done by various ways as one can see in the literature. These machine models that we have derived are useful again in various domains some of them we have seen control systems synchronous machine models are very very useful in large system studies stability studies and so on. Now to understand the relevance of these equations better and to get a feel for how these things work it would be useful to do certain assignments and here I have listed some assignment areas which could be of use I would strongly urge that one does all these assignments to get a good grip on what is the meaning of all the equations that we have derived to really appreciate that these equations will indeed give a good estimate of the machine behavior. So the first assignment is the start up of a DC machine you know that in order to start a DC machine I am sure you would have done experiments in the UG electrical machines lab where you start a DC machine and you normally connect a series resistance and start with the high resistance and slowly reduce the resistance in order to get the machine to start. Now why would you use the resistance use the resistance because if you simply throw the DC machine on to the supply that is if you connect your full rated voltage to the machine all at once it will then draw a heavy in rush current how much in rush current will it draw for how long in rush current will last how does the speed accelerate when does the in rush current come down is it in any way feasible to have a circuit to reduce the in rush current and at the same time allow acceleration as well all these things can be studied by making use of the equations that we have derived. So the first assignment that one can do is to start the DC machine on no load with full voltage and see how it is behaving after all if you do it all these are essentially simulation experiments so please do not go to the lab and do this on a physical machine so do this on simulation again on simulation I would urge you to write your own simulation code and not use any predefined simulation models that are existing in one software or the other writing your own code is what will ultimately help you to understand how this equation behaves and what are the various issues involved in simulation as well so you must develop your own code to do all these things and since you are developing your own code you are not going to really have an equipment that will blow up on the face and therefore one can start the machine with full supply voltage and then how does the machine behave how does the machine behave if you start on no load with reduced voltage how do you compare the two how does the acceleration duration for acceleration how long is it with reduced voltage how much is it with full voltage and what happens when you start the machine with load as against no load and with load on reduced voltage so all this will give a good idea of the dynamics of DC machines as against what we have studied and experienced in the lab the second experiment is the study of regenerative breaking of the DC machine regenerative breaking is an approach where the machine was already running and you want to reduce the speed so if you want to reduce the speed essentially what it means is that we stored energy in the rotating masses of the system it was at some level and you want to bring it down to a lower level so that stored energy which is the difference between these two energy levels has to be taken away from the system and one can choose to dissipated in breaks in mechanical breaks you put them on and friction dissipates the energy that is one way or you can try to recover the energy back into the electrical form right so how does one do it so this can be done if the armature voltage is now reduced while operating on load so one can as the speed reduces you also go on reducing the armature voltage and therefore the machine then sends current back into the source and therefore it speeds starts following so how does this happen suppose you want to break the machine fast then how will you operate this regenerative breaking suppose you want to break the machine slowly how will you operate the regenerative breaking so all this can be studied with the help of these equations one has to program it in such a program your code in simulation code in such a way that as the speed falls you also reduce the supplied DC voltage and then see how it works and the other thing is supposing you do not want to exceed two times the rated current of the machine why two times well that is an arbitrary number maybe you can try to do the simulation by taking more than this amount of armature current so if you allow more armature current to flow how does it work if you allow lesser armature current to flow how does it work so all this can be studied by doing this assignment again the goal of these assignments is not just to do some simulation but to interpret these results appropriately and to understand that physical behavior of machine can be studied with the help of these equations all these exercises are in some way related to what really can happen physically the next assignment is on induction machine so how to determine what will happen to the induction machine when you start now starting of induction machine I am sure you would have heard of start delta starters that are used why start delta starters that are used and how does the machine behave even when those are used one can study using in this assignment how much in rush current that the induction machine take and if you can model the grid also appropriately after all the induction machine gets three phase supply from your electrical utility supply so if you can model the grid also appropriately by using a source and a suitable impedance in series one can then look at what will be the voltage drop that will be experienced at the point where the machine is thrown on to the grid so the direct online start of an induction machine is then the subject of this assignment and what would happen if the machine were to be started while on load that is another aspect to be studied load is given we will shortly show you the machine data also which is suitable for that the next thing is suppose you have an induction machine and then there is a fault at the stator supply side of the induction machine and then the circuit breaker or something trips so that the fault can be removed at some place and then the circuit breakers which is on back the induction machine has not yet gone down to 0 speed and then you have a switching off and then a switching on again so if you switch on the supply when the machine is still not gone really down to 0 speed then you have a situation which is not really a direct online start because in a direct online start machine initially is at 0 speed so when the recloser happens how does the system behave that is the fourth assignment the fifth assignment is we have derived small signal models both for the induction machine and the mean we derived the small signal model for the DC machine we had made certain assumptions that some things are small and some things can be neglected and so on so how does the small signal model how does an estimate of the behavior using the small signal model really compared with the unapproximated large signal model equation itself the same disturbance that you apply to the small signal model can also be given to the large signal model of course in the response of the large signal model one has to subtract out the steady state response in order to compare these two if you compare these two one can get an idea of how accurate the small signal model is or how inaccurate the small signal model is how does it really relate to the large signal model so that can be done one can do it for the DC machine and for the induction machine this is an exercise that we have done short circuit of the alternator with dampers I have shown you a lot of results from simulation and analysis so the analysis involved a lot of approximations some terms were neglected some terms were done in some approximated in some way and we arrived at certain equation so how do these equations compare with respect to a simulation of the actual set of equations themselves without doing any approximation one can today use the actual model and simulated and get the entire performance and we have an analytical expression that gives us the guidelines of how different terms affect this behavior so one can now compare the accuracies of these two using this and then the short circuit performance of the induction machine so if there is going to be a short on the induction machine stator terminals the induction machine also to some extent sends current into the short circuit so how long does it do how does the machine itself behave one can study that aspect of it and then you have an inverter fed induction machine in some of the numerical examples that I had shown when we discuss the induction machine we had seen some waveforms of that one you can now do the simulations yourself you can take an inverter or alternatively determine the waveforms generated by an inverter and then feed that to the machine model that you have and then determine what the response is going to be does it have a smooth electromagnetic torque smooth speed or is there are there ripples why there are ripples how much is how much is the frequency of the ripples all this can be seen and related to the inverter switching action in this assignment and then we come to breaking an induction machine one of the ways of breaking an induction machine is to reverse the phase sequence of the stator terminal if you reverse the phase sequence then the stator MMF or the stator field starts rotating in the opposite direction to the rotor and that really brings about a very severe breaking action in the induction machine. So but it will bring the induction machine to a stop very fast but this is not advisable on a normal machine because it will subject the shaft to a very severe stress and therefore this is not normally used but at least however one can study it in simulation to see what really will happen during this kind of a setup and then study of the grid connected alternator effect of varying field current and effect of varying the mechanical input. So how does the alternator behave when it is connected to the grid in simulation you cannot just take the alternator and give a supply voltage because you have to start the alternator and run it. So please take care while doing these assignments that you have to first somehow establish the conditions where the alternator is brought up to speed and synchronized in the proper way and then one can go about studying the effect of field current variation and the mechanical input variation. So one can see how the load angle will change and then whether if you change the field current whether it will become leading or lagging all these effects can really be seen. So we already we will get a feel for how these equations can represent the steady state behavior as well already we have seen how the alternator equations can represent the steady state by looking at phasor diagram. Now this is another dimension to understanding these equations we had looked at xd and xq representations for the alternator. Now in the lab if one wants to determine xd and xq one would do a slip test on the alternator. So here again one can do the slip test in simulation you would already know what are the xd, xq values for the alternator model that you have used. So compare the results of what you get by a slip test and what you have put in the alternator. Now you need to use machine data for all these exercises. So here are some machine data. So here we have the data for a DC machine which is a 240 volt DC machine with the field rating of 300 volts, 1050 rpm is the rated speed. The armature resistance and self inductance are given, field resistance and self inductance are given and the mutual inductance between the armature and field is also given along with that inertia and viscous friction coefficient is given. This torque is a fixed torque if you are going to model your mechanical equation as Te is the electromagnetic torque that is generated – T mechanical is the accelerating torque then this T mechanical can be split as Tf plus the viscous friction coefficient into omega. Now this is the resultant accelerating torque which is j d omega by dt. So this is what is going to be the mechanical equation. Remember in all the assignments that we have done the mechanical system also has to be modeled because we are looking at speed changes and speed changes can be modeled only through this mechanical system. And then we have a model some data for the induction machine this is a 320 kilowatt induction machine 6600 volt rated at 50 hertz the rated speed is 1487 rpm and it is a 4 pole and it is a number of pole pairs so it is a 4 pole machine stator resistance leakage inductance rotor resistance leakage inductance all referred to stator and then the 3 phase magnetizing inductance this is really 3 by 2 times your magnetizing inductance per phase inertia is also given and friction coefficient is given this friction coefficient refers to the value of B in this case one can assume Tf to be 0. So with this machine data one can now try to do the simulations and then here we have data for the synchronous machine this is indeed the synchronous machine that we had looked at for the numerical example on the full short circuit for the synchronous machine. So this is a star connected to MVA synchronous machine 400 volt and 50 hertz it is a 4 pole machine obviously because it is 1500 rpm and then the various data are given stator resistance leakage inductance and so on and then the damper values are also given here so one can make use of all these machine data in order to do all your assignments. So I wish you all the best with all your assignments I would strongly urge you to do all these assignments to get a good grip on the equations and the utility of all these equations and models that we had studied. I hope you enjoyed listening to this course and I hope you benefit from the material presented in this course we will stop here for today and as well as for this course.