We discussed the basic operating principles of two kinds of excitation systems. One was a static excitation system and the other one was a brushless excitation system. In order to really go ahead and use these exciters or rather the study the effect of these exciters, we need to really model them mathematically and that will be the focus of today's lecture. So, today's lecture is focused upon excitation system modeling. The basic static excitation system which was studied in the previous class, we will just recap what we did. The static excitation system consists of a controlled rectifier. It is a thyristor based rectifier. The AC input of it is derived directly from the output of the or rather the stator voltages of the main generator and the rectified output is fed to the field winding. So, this kind of system is amenable to self excitation as I mentioned sometime back, but in a realistic situation you need to initially develop some voltage across the stator windings in order to start up the system. So, this is done usually with the station battery and is known as field flashing. Now, if you are going to more try to model this particular static excitation system for our stability programs or stability studies, we need to see how we can actually represent the various components in this system. Now, one of the important things which you should remember here is that a static excitation system draws a bit of power. In fact, it slightly loads the main generator. If you look at the block diagram here, you will find that it is loading the main generator, but remember that the amount of excitation power which is required is very small compared to the overall rating of the synchronous machine. So, we need not even consider the loading of a excitation system on the main generator. So, for example, a typical set of 2 10 megawatts under no load conditions, you have got excitation currents of approximately 1000 amperes and voltages of around 100 volts apply to the field winding. So, this is the under no load conditions or open circuited conditions and of course, under loaded conditions this may be 3 times as much. So, approximately 3000 volts and 300 volts and roughly 3000 amperes. This is the roughly under full load conditions. So, the excitation power required is not very very high. So, you look at this it is less than a megawatt. So, if you look at the no load excitation power requirement it is quite small. So, we really need not represent the load the loading of the synchronous generator on the excitation system on the generator itself. So, what we need to do of course, is however try to model the control rectifier itself. As you may have guessed the control rectifier is a very fast acting system and in fact, if I give a order to the control rectifier to change the voltage or the DC voltage it is practically implemented instantaneously. So, if you look at what basically it involves is changing the firing angle delay of the thyristors of the bridge and there is a switching if you use a 6 pulse thyristor bridge 3 phase bridge as is commonly known. Then every sixth of a cycle you can effectively change the firing angle delay. So, if you just have to have wait for a sixth of a cycle to change the firing angle delay and if such is the case then for most of the studies of our interest we need not model converter operation in detail. We can just treat the converter as some kind of instantaneous amplifier of the control signal. So, your control signal is something which you use to control the output which is fed to the field winding. So, the converter itself is almost instantaneously acting. So, we do not really have to model it in detail. Now, this of course, presumes that the kind of studies we are doing are essentially power system studies which do not require us to do such a detailed modeling. Of course, if you are involved in excitation system design itself then of course, you need to model the converter much in detail. But we will just at least see what are the limitations or what are the you know modeling intricacies which we need to consider whenever you make a mathematical model of a excitation system. Now, a static excitation system luckily the converter itself can be modeled as something as a kind of a static block. Static in the sense that as I mentioned some time back if I give a control signal this is a control signal let us call it v c then the d c output of the system immediately changes. So, this is one model of a converter. So, this converter model can be very very straight forward. So, you have you give a control signal and it immediately results in the change in d c voltage which is applied to the field. Now, of course, the d c voltage is applied to the field is related to the voltage which is applied at the ac terminals of the converter. So, if I desire to have a certain output voltage here I will appropriately change the control signal. Of course, there should be a certain mapping between the control signal and the d c voltage which is which appears here. In fact, this control signal essentially is the order to the converter to change its firing delay. So, if you look at if you look at the converter itself this is fed to the field winding this is actually derived from the terminal voltage of the synchronous machine. So, the main generator its output voltage stator voltage itself determines v ac. So, if I give a certain control signal say I say I want say 115 volts to appear here I can give an appropriate control signal and we will get this voltage here almost instantaneously. So, one thing is that you should map this v c to the d c voltage which appears here. So, this is one thing the other important thing which we should consider if you look at the slide the third point in the slide the limits of the converter. Normally whatever we desire to have at the output of the converter can be got it can be got provided v ac has got an adequate magnitude. In fact, suppose I want to get you know say 150 volts here or 110 volts here your v ac should have adequately high magnitude. Remember that you are feeding this voltage to the converter and this v d c here is dependent on v ac as well as the control signal or the firing delay angle effectively. So, if v ac of course, is too small if v ac is very small suppose I desire 100 volts here and I the v ac is only you know what appears across the converter ac side is only 70 volts you may not be able to achieve this 100 volts it depends on the conversion factor between the ac and dc voltages and the firing angle. So, you for you know you if for example, you had perhaps with I can perhaps put a lower value here to illustrate this if I put 10 volts v ac here whatever be the value of the firing angle you are not going to get 100 volts. So, in fact, those who are familiar with some basics of power electronics will recall that if you have got a 6 pulse thyristor bridge or thyristor bridge consisting of 6 thyristors then and this is v ac is the line to line r m s voltage then v d c is roughly 1.35 v ac v ac is the line to line r m s value at the terminals of this into cos of alpha. So, the maximum value v d c you can get out of this is 1.35 into v ac. So, there is a limitation now normally of course, one would design excitation systems such that the normal voltages which we require which we desire at the field winding are achievable with the kind of v ac we will have remember v ac is derived from the terminal of the generator which is stepped down. So, the step down ratio of the transformer is adjusted in such a way. So, that for all conceivable situations which are acceptable of course, we can get the value of v d c by appropriately choosing alpha, but there are certain situations like for example, if you have got a fault on the synchronous machine or there is a short circuit at the terminals near the terminals of a synchronous machine. So, the terminal voltage of the generator itself will dip the terminal voltage of the generator dips you will find that v ac is small and in some cases you may not be able to achieve the v d c which is desired. So, one of the ways you can model a converter is. So, this is a converter model this is a control signal this is the voltage applied to the field winding you need to appropriately put limits on the converter the limits really tell you that the voltages at the output of the converter cannot be changed beyond certain values for example, you could this could be roughly. So, these are the maximum values you can have for e f d. So, this is one e f d is nothing, but v d c I have just normalized e f d is a normalized value of v d c the basic idea is that if you want to get a certain v d c when v ac is very small say due to a fault you may not be able to achieve it because of these limits. Another interesting point which you should remember is of course, something I mentioned previously a converter can have negative voltages. So, you that is why you see this minus sign here. So, a thyristor base converter can have negative voltages v d c, but the important point is that current is always in this direction you cannot have current flowing in the opposite direction and in case you wish to allow current to flow in the opposite direction in the field winding you need to have separate shunting elements which are switched on as necessary. So, either these could be a switched kind of shunting element which will allow current to flow in this direction from the field winding this is the field winding of the generator. So, or you can have a non-linear varistor which allows current only in this direction in case the voltage across the field winding becomes very large. So, this kind of arrangement can allow reverse current flow in the field winding, but this converter itself does not allow any current to flow in this direction, but v d c of course, can be negative. So, that is an important and interesting capability of the thyristor bridge itself. So, this if you look at what I have just told you to summarize a control signal is there you have got the converter the converter is can be actually just a static model what I mean is that the field voltage is algebraically related to the control signal we do not have any differential equations or delay elements or anything of that kind the simplest model is a simple static or simple algebraic relationship between the control signal and the field voltage. The important things of course, are the limits of the static exciter the limits are dependent on the generator terminal voltage. One interesting point which of course, I did not mention in my discussion so far is that the voltage also gets limited in some sense because of the field current. Now, what do I mean by that if you look at the typical converter and which is fed form of source the v d c although I mentioned some time back it is roughly 1.35 v a c cos alpha for a 6 pulse bridge thyristor bridge where alpha is the controlled quantity. So, we can get the v d c we require if v a c is large enough and for a certain control delay angle, but one small thing one small issue which I did not consider in this is that v d c is also dependent on the i d c of the converter the current which is flowing out. So, it slightly drops so v d c slightly falls if i d c becomes larger this is because of what is known as commutation overlap effect due to source inductances. So, you may have actually some kind of regulation in some sense the v d c is not just a function of v a c, but it is also a function of i d c. Now, so this is normally does not matter because you could always adjust alpha so that we get v whatever v d c we require. However, at the limits for example, if v a c is very low in that case you will find that the limits are also determined by i d c. Let me just retrait what I said normally by changing alpha you can get whatever v d c you require irrespective of v a c and i d c. However, the minimum value of alpha is actually theoretically speaking 0, but typically 5 degrees also in such a case v d c also gets limited because of v a c and i d c. So, if I hit the minimum value of alpha you cannot control v d c any longer you cannot increase it beyond that point. So, one important point which you should keep in mind is of course, that v d c is limited not only by v a c, but by i d c also. So, v a c becomes very small it is likely that you will hit the limit and you will not be able to achieve what v d c you want. So, field current also comes into the picture because v d c is dependent on the field current as well though strictly speaking it is or rather practically speaking it is a weak relationship. Normally if a static excitation is fed the a c voltage is obtained simply by stepping down the voltages of the main generator the stator voltages of the main generator the so called commutating reactants or the source impedance is quite small it is just a leakage impedance of the transformer. So, in such a case we can almost neglect the limits the dependency of the limits on the field current. If you are not in the limits within the limits we can assume that the field voltage is simply as I said algebraically related to the control voltage. So, if I if I say I want to want a certain voltage and I give an appropriate control signal you will instantaneously obtain that particular field voltage if you know the mapping between the control signal and the field voltage. So, static excitation model is very simple and what we really need to spend more time on is the other kind of excitation system we will not really go into the gory details of this model, but let us at least try to correlate the various modeling the mathematical blocks which need to be present in the brushless excitation system. How is the brushless excitation different first thing is that the controlled rectifier is here thyristor based rectifier is here the control signals are given here the power is derived from a permanent magnet generator which rotates on the same shaft as the turbine and the generator. So, actually power is coming from the turbine in some sense for the excitation system, but one this is the point at which we control the voltage here after on the right hand side we have an AC generator this is called an excitation generator this is not the main generator this is the excitation generator a diode bridge and the field winding. Now, what you need to model really here are the converter model and the limits in fact the control rectifier which you see here is has similar characteristics as the static excitation system converter. So, there is nothing really difficult or you know different from the converter the static excitation system model here, but there are additional elements which have dynamical characteristics which need to be represented in a excitation system model. Now, if you look at what are those we need to model the exciter alternator the exciter alternator is not the main alternator remember then we also need to model the characteristics of a diode rectifier and we when we do this modeling we need to remember that a exciter a exciter runs on the same shaft of the generator the generator runs almost for most operating conditions for most of the studies we are going to do the generator will be near about the nominal speed. So, we can almost take assume that the permanent magnet generator which effectively generates voltages for the elements of this excitation system is actually running more or less at the nominal speed. So, we need not you know bring in this additional complication of making everything speed dependent here. Moreover, you will notice that since the control rectifier is here whatever voltage we desire can be achieved by giving an appropriate control signal subject to the limits the limits again are dependent on VAC here the VAC is derived from a permanent magnet generator. So, you can model this much this in much the same way as we have done for a static excitation system the voltage here is speed dependent, but as I mentioned some time back will almost be at nominal speed. So, the limits of this converter the limits of this converter is dependent on VAC here which we can for most for most purposes assume that that VAC is a constant not dependent on speed because speed variations considered for most studies will be doing in this course are not going to be much away from the nominal. Another interesting point here is that the actual field current may not be directly measurable. So, unless you have made some specific provisions you may not be having actual measurements of this field current. You may know what the excitation system generator this generator here what its field current is, but we will not be able to find out well in many cases we are not having the exact measurement or actual measurement of the field winding current of the main generator. So, this could be a possibility which you should consider that the measurement is not available, but this is accessible this current here this is the excitation generator current this is accessible. Another interesting point which you should note is the diode rectifier itself is slightly different from a thyristor based rectifier in a diode rectifier we do have similar effects as in a control rectifier thyristor based rectifier, but a diode rectifier itself gives us no control it is an uncontrolled rectifier. Another interesting point which you should note is a diode rectifier voltage also cannot be negative. The diode rectifier voltage is a function of the AC voltage which is applied you know to its AC terminals as well as the actual DC current which is flowing. So, both these things determine what the DC voltage of diode bridges is. In fact, if you look at a thyristor bridge it is dependent on alpha VAC and IDC, but for a diode bridge a diode rectifier if it is a diode rectifier this alpha is practically 0 and you cannot change it. So, this is one thing as a result VDC cannot be negative in a diode rectifier you cannot have VDC negative that is one important and interesting point you should note. Now, unlike in a static excitation system which is fed from a transformer the source impedances of which are not very large the diode rectifier here is fed from a AC generator and AC alternator the excitation system alternator whose impedance can be significant. So, what I mentioned here is this effect of the loading on the DC voltage may be significant. So, this loading effect may be significant incidentally this K is not a constant it is dependent on IDC itself. So, in a especially when you have got large amounts of IDC this K will change. So, that is another complication which we have to consider again to summarize for a static excitation system fed from transformers this effect is very small, but for the brushless excitation system the source impedance that is the source impedance of the AC generator feeding the diode bridge may be large and then you have to consider this effect. Now, one of the important issues which you should keep in mind is that what is the range of a static exciter. So, whenever I am designing an excitation system I would need to know how much field voltage I need to apply to the synchronous generator in volts based on that I would have to decide the voltage rating of various elements. Now, one of the important things you should note which you will see in brushless excitation systems as well as in static excitation systems is that they are given a voltage range or a voltage capability the final voltage which you can actually get at the output of the exciter that is what is fed into the field winding of the main generator can be quite large. In fact, it may be 5 or 6 times the voltage which is required to get rated voltage at the output of the stator of the main generator under open circuited conditions as 1 per unit. So, to get let me just this was quite a mouthful. So, I will just reiterate what I said. So, if you have got the field winding and it say it takes 100 volts here to get the rated voltage here. So, if I want to get the rated voltage here say 15 kilo volts line to line RMS suppose I need to apply 100 volts. Now, an excitation system normally under open circuit conditions. So, under open circuited conditions suppose to get 15 kilo volts I need to put 100 volts. I have already mentioned to you if I load the machine I may have to put especially in steam turbine driven generators you may have to put almost 3 times 2 to 3 times the field voltage in order to get 15 k v under full load rated load conditions. So, there is a big range of voltages which you should you know budget for whenever you are designing the excitation system. In fact, another complication or another interesting point is that most of the excitation systems as I said may be rated for 6 to 7 times what is required to get 15 k v under open circuited conditions. Now, why is that why do I need to rate a static excitation system at 6 to 7 times it is capable under for short while for a short while of giving 600 volts. If 100 volts is what is required to get 15 k v under normal conditions. Now, why is that let me just amplify what I said what I mean to say here is of course that these limits here are such such that if it requires 100 volts to get the rated voltage under open circuited conditions at the terminal of the generator the limits here may be 6 or 7 times plus or minus 6 6 or 7 times of what is required So, what you should need to do is of course that you should appropriately rate your transformer. So, that it gives you an AC voltage which is sufficient to give you this 600 volts. So, this is what is normally done this is how it is done. Now, I am not told you why it is required to have such a large range of course as I mentioned sometime back you need to at least double the voltage of the field the field voltage if you start loading a generator and you expected rated voltages to appear. So, you should almost have double or triple the no load field voltage, but why 6 to 7 times the reason is such if you look at the field winding it is a slow acting winding what I mean by a slow acting winding if you just look at a synchronous generator under open circuit conditions the LFF by RF the time constant LFF by RF is extremely large for steam turbine driven it can be as or near about 10 seconds it could be as large as this. Now, if for example, if I just as a academic kind of example, if I give EFD if I give a step change of the field voltage from 0 to the voltage required to get rated voltage here suppose I just give from 0 to 100 volts I will get 15 KV in steady state using the example which I have given you previously I will get 15 KV in steady state, but the time required will be roughly the settling time for this voltage to appear here will be almost 40 seconds because of the large time constant of the field. So, under open circuit conditions you will require almost 40 seconds to achieve 15 KV even though I have given a step change here. So, this is this basically is not a very this becomes a very slow kind of system. So, what I will just illustrate what you need to do suppose I want 15 KV at the output the rated voltage at the output. So, what I do is I increase the field voltage from it is 0 value to it is rated value you will find that the field voltage increases from just show it with another rather the field voltage is a step then the terminal voltage will increase like this I will take almost 40 seconds for it to settle down the scale for this is of course 15 KV. Now, instead of doing this this is a very slow acting subsystem. So, what I do instead is what is known as field forcing what I do is I in order to make this rise very fast I give a step change not of 100 though I know that 100 yields 15 KV I give a step change in the field voltage this is the field voltage this is the stator terminal voltage stator terminal voltage. The field voltage I do not give a step of 100 what is required I give it give a much larger value say you know 5 or 6 times its value and then I reduce it to this. So, in such a case the synchronous generator terminal voltage will tend to rise like this and you can get a much faster response. So, by forcing the field by putting much more voltage then what is actually required we can overcome the effect of having a relatively slow acting field binding.  So, in order to give that large push the range of the excitation system is often the ceiling voltage as it is called is quite large. So, you may have give it as 5 6 to 7 times what is required to get the rated voltage under steady state conditions. So, this is one of the interesting points. Now, if I have really discussed the various models of rather the issues which need to be taken into account while modeling one of them is the limits then we talked about the diode rectifier the static excitation system. The static excitation system or the converter model and the diode rectifier model are static in the sense that we assume that these are instantaneous acting devices and there is no real dynamics associated with them the input and the output follow well defined relationships algebraic relationships. A controlled rectifier you can get the output which you want simply by giving the appropriate control signal subject to the ceiling voltages. Ceiling voltages are dependent on the AC voltages which are applied to the converter. So, as far as a controlled rectifier or thyristor based rectifier component in any excitation system is concerned it is just a static element with limits you get what you get what you want except subject to the limits. A diode rectifier the output voltage of a diode rectifier is simply algebraically related to the AC voltage which is applied and the current the DC side current because the effect of commutation overlap. Now, one of the components unfortunately or well which you need to pay much more attention on is the excitation system generator itself that is in an otherwise static excitation system model that is one element which requires you to model you know model the dynamics of the exciter. Now, so if you look at the kind of models which are recommended by the IEEE I refer you to this IEEE reference in which these system models are actually derived and I also refer you to the three books in which we have got a fairly good discussion of excitation systems a very good discussion of excitation systems existing in all these books may be the first two books have a larger treatment. Now, coming back to our excitation system models if you look at a brush legs exciter model as I mentioned sometime the control rectifier is a simple model the output is a function of the control signal. The diode rectifier the output field voltage is dependent on the current the output current as well as the exciter voltages this exciter alternator voltage. There is a limit here the converter limits are really determined by the AC inputs the maximum AC input you have I of the converter the diode rectifier limit on the other hand the field voltage the final output cannot go negative that is the basic limit of the diode rectifier. In fact, there is no upper limit in the sense that a diode rectifier is just dependent on the AC voltage and the field current there is no upper limit separately defined. There is no upper limit is simply a relationship between the AC the DC voltage is simply related to the AC voltage and the field current by an algebraic relationship, but it cannot go negative. So, actually that there is no top limit here that no need to specify any top limit, but that the bottom here we cannot go below 0 volts. Now, if you look at the excitation system models which are given in the reference which I have mentioned you will come across something like this. Now, before we really I would not really derive this complete model this is the standard brushless excitation system model what I will tell you is how you can come to it. Now, a converter model as I mentioned is straight forward the exciter alternator model is what is a bit tricky. And as I mentioned sometime back the diode rectifier itself you know is algebraically related to V AC as well as the AC voltage which appears here as well as the current which flows. So, the diode rectifier model has to be slightly if you have to pay some attention to it. The exciter alternator now somebody may argue that well this is an exciter alternator. Now, we need to model an alternator as we have done before in the sense when we have done synchronous machine modeling that is also an alternator. We have already come across you know a model which is fairly detailed I mean in the largest amount of detail you have considered is what is known as a 2.2 model in which you have considered a field winding and three damper windings in addition to the stator windings. Well, this excitation system alternator we do not take that approach we do not need to model it in so much detail. One of the reasons is of course that the excitation system alternator is of much smaller rating its characteristics are such we do not need to we can represent the gross effects of the exciter rather than modeling it as in as much detail as the synchronous machine the synchronous main synchronous generator itself. What we will do is we will get a kind of rough exciter model. So, what are the issues which you need to worry about? So, how is an excitation system is like a normal alternator. So, the field winding is here the converter output is fed to the field winding of this alternator exciter, exciter alternator it causes a current in the field winding and that changes the output voltage. Now, if you look at the various relationships you have one of the things you will notice is the current flow through this is obtained from a differential equation. So, first thing you will notice is that the current here the current here. So, this is in fact if you look at this this is the voltage and this is the current this is what you get of course this is not a linear exciter this is not a linear device you may have to consider saturation. So, better way of putting it is d psi by d t the field flux link with this winding is equal to v the voltage applied here minus i into r and the flux psi which is the flux linked with this field winding here is a function of it is a it could be a non if the machine is saturated it becomes non-linear the in general I can write this is i the depends on i as well as the current the load current of this exciter alternator. So, I will call this the load current of this exciter alternator. So, if you look at the relationship it is like this now of course this is the flux link with this winding we can consider the open circuit voltage which appears or rather the voltage which would have appeared under open circuit conditions is roughly proportional to. So, v o c here is roughly proportional to the flux psi. So, if you look at the various relationships which are there you have got the open circuit voltage the open circuit voltage is proportional to phi or the psi this d psi by d t is given by this differential equation psi itself is a function of this current as well as the load current which flows here the load current in fact is fed to the diode bridge which feeds the field winding. So, in fact, i l is proportional to the actual field current the field current of the main synchronous generator. So, what you have essentially is you can write this is i f here. So, this is the relationship you have got now if you try to model this then you need to have an integrator in the model you need to have an integrator in the model you will have to provision for this non-linear relationship because you may get have saturation effects. This is the open circuit voltage in fact due to armature reaction of this synchronous generator what actually appears here will be slightly different under loaded conditions. So, actual voltage which appears here is not v o c. In fact, this feeds a diode bridge you will have computation overlap phenomena which in some senses like armature reaction in fact it is due to source impedances. So, what you get here is not v o c is not proportional to v o c, but is proportional to well it is dependent on the generator reactance is the load currents the load currents i l is approximately you can say is proportional the you can say the r m s value of i l is proportional to i f for a diode bridge in steady state. Now, so a kind of a rough model of this kind can be used. So, if I give the output of the converter the what comes here is actually the output of the converter control rectifier this is the controlled signal this is the ac voltage coming out of the permanent, permanent magnet generator. So, this of course, is a completely controlled element, but here onwards these relationships actually determine what appears here eventually. In fact, so what appears eventually here which is fed to the field winding is a function of v o c as well as i l which is proportional to i f. So, this is what really you have you have got one integration which has to be performed of this differential equation. So, if you look at the IEEE model of this kind of excitation system will not derive it, but we should be able to identify the components corresponding to this. So, if you look at this figure here this integrator here is actually this is the one integrator which is required for obtaining the open circuit voltage of this exciter alternator. The open circuit voltage of this exciter alternator does not directly manifest as the field voltage there is an excitation the diode rectifier itself has got due to commutation overlap phenomena has got some regulation in the sense that what open circuit voltage is there across the excitation alternator does not directly appear is not directly related to E f d. You have to put a kind of a correction function here you have to multiplying f e x to e v e in order to get E f d. So, this is this what you see here is a component which takes into account the diode rectifier regulation. Remember that this correction factor which is multiplied with v e in order to take into account this rectifier commutation overlap is dependent on the field current I f d here as well as v e though you know the output voltage of the excitation armature. Now, the non-linear effects are taken into account using this function here and the effect of armature reaction is also taken into account here. So, although I am not derived this I am just directing you towards the various components as their model in this standard excitation system model. Now, the typical values of these the time constant T e here which is dependent on of course, the parameter of the excitation system alternator like the field winding resistance itself k c which is a factor which takes into account the commutation overlap phenomena which makes the field voltage dependent on the field current k d is a factor which corrects the output voltage of the excitation alternator it is it is basically trying to represent armature reaction and k e which is typically 1. So, this is effective and s e is a saturation function here. So, this is basically the excitation system model 1 clarification I need to give here these typical values which are given here are obtained in this block diagram assuming that e f d and I f d are normalized. So, e f d and I f d and the exciter alternator field current are normalized what do I mean by normalized e f d is assumed to be 1 which is actually consistent with the notation we have been following. So, far e f d is 1 if the open circuit voltage of the main synchronous generator line to line voltage is the rated value or the base value. So, instead of talking of e f d in volts or field voltage in volts will be using e f d which satisfies this. So, the mapping is known. So, the this e f d is in per unit I f d is also 1 per unit if in steady state it results in the open circuit voltage which is equal to the rated voltage and also in addition the exciter current this exciter alternator current I also is taken to be 1 per unit in case this is satisfied. So, the gains which are shown here in this slide here k c k d k e are assuming that such a normalization has been done. So, we have normalized what do you mean by normalized divided the values by divided for example, the actual field voltage by a value such that e f d is 1 per unit when you get the rated voltage under open circuit conditions at rated speed. Similarly, I f d is defined and similarly I also which is the excitation system alternator current field current. So, what I mean to say is that if your field voltage is 100 volts 100 volts is required to get the rated value 15 k v at the terminals under open circuited conditions then whatever.  So, whatever field voltage you are going to get under other conditions is normalized by this.  So, this is this is what I mean similarly water value of I f d you get it is divided by the value which flows under these conditions. So, suppose it is 900 amperes in that case I will have to normalize this by 900. So, if you do all this then what you get in this block diagram these k c k d t the typical values will be these. So, it is important to know what this normalization is. So, that is what I have just shown you here the field voltage the field current and the excitation system field current is normalized. So, that it becomes 1 when we get 15 kilo volts under open circuited conditions at the stator terminal voltage I hope that is clear. Now, one of the interesting points which I am not dealt with because this is something which you need to you know when you are doing a actual power system study what you will be doing is you will be giving e f d to the synchronous generator equations this main synchronous generator you will be giving e f d in per unit. The per unit is what I have just defined one in case you get open circuited voltage equal to the rated value under rated speed conditions. Now, the synchronous generator equations once you solve that is you numerically integrate the differential equations you will get all the fluxes psi f psi h psi g psi k psi d psi q and from this you can also get i d i q and so on. Now, the excitation system model requires you to tell what i f is the field current that is. Now, the field current is then we will normalize and then use this model this IEEE model which you have got. Now, what is the value of i f the field current now if you use the synchronous generator model which I have defined sometime back. So, I will just show it to you this is the d axis model in per unit which we have learnt in the previous class which this is something which we can rewrite by replacing psi d by the relationship the third relationship here which is an algebraic relationship. So, if I substitute this relationship in this and this equation what I get is this. So, I am writing this d psi h by d t purely in terms of psi h i d psi h and psi f. Now, this yields this particular equation similarly the d psi f by d t equation can be obtained in or rewritten in terms of i d and the state psi f and psi h in this particular manner. Now, if I define two currents i upper case f and i upper case h as follows. So, it is in fact suppose I define it in this fashion it is easy to see that you get equations which look like this. Now, what is the significance of this particular set of equations the point is that if you recall the original equations for the fluxes and damper winding fluxes and the field winding fluxes were these. I had mentioned sometime back that psi upper case f and psi upper case h are related to the actual field fluxes and the field the damper winding h damper winding fluxes by some transformation. But that relationship was never actually defined, but if you look at these equations along with this it looks very very familiar to these original equations. So, what it follows we do not do a very rigorous proof of this, but if you assume t d c double dash is equal to t d double dash in that case it is clear that what we have called as psi capital F or psi upper case f is actually proportional to the field flux and psi capital H is proportional to the damper winding h damper winding flux. And the i f and i h obtained from the following equations are such that i f and i upper case f is proportional to the field current and i upper case h is proportional to the damper winding current. So, although in the earlier lectures I did not actually give you any physical significance of the psi capital F and psi capital H fluxes it can be seen that in fact with this assumption that t d c double dash is equal to t d c t d double dash. In fact, these states are actually proportional to the original flux and h damper winding states. We have not really done a rigorous proof, but by just looking at the nature of the equations we can infer that. E f d is equal to 1 per unit implies the open circuit line to line RMS voltage of a star connected generator is 1 per unit. I f is obtained in per unit on generator base from the relationship with psi f psi h and I d this is something which we have discussed just sometime back i f and i h obtained from psi f and psi h. Now, you will of course, i you obtain i f in per unit on current on the generator current base. Now, when I mean generator current base it is nothing but the m v a base of the generator divided by the voltage base of the generator. So, you will get i f in per unit. Now, the question is that whenever you want to use it for a static excitation model or for any other excitation system model if you want the field current in amperes you would need to know the mapping between or the proportionality constant between i uppercase f and the field current in amperes that is i lower case f. So, one of the ways you can get this mapping very easily is to you know if you know v f and i f in voltage volts and amperes under open circuit rated conditions that is if I am running the machine at rated speed and I have got the voltage equal to the rated voltage that is 1 per unit at the generator terminals and I know the v f and i f under that situation. In that case effectively I have got the proportional proportionality constant which I am looking for because you can using the per unit model compute i f in per unit i uppercase f in per unit you can compute e f d in per unit.  So, the proportionality between the field quantities in volts and amperes the actual field currents and voltages in volts and amperes and those obtained from the per unit synchronous machine model can be obtained. So, you know the relationship this proportionality between i uppercase f in per unit and the field current in amperes. So, this can be easily got if this additional information is given that is the voltage the voltage field voltage and the field current under rated open circuit conditions if it is given to you can get this mapping. So, it is not a very tough task to actually get the field current in amperes under other situations as well because i f can be obtained in per unit from the basic synchronous machine model and then you know the proportionality constant. So, you can actually get the ampere value of the field current as well. So, if I know that i f if you look at what I am writing here i f suppose comes out to be x per unit when we get rated voltage under open circuit conditions x per unit. And if I know that the actual field current in i of is 900 amperes then I have actually got a mapping between i f and i f this i capital f in per unit and i f in amperes. So, I know what you know effectively I know the i f in amperes and then I can normalize it as I mentioned sometime back and then use this model this i triply model with these either typical or actual values these are the typical values assuming that e f d and i f are in fact normalized. So, this is one thing you have to keep in mind we now move on to the control systems associated with the exciter. The exciter and generator are in fact what is known as power apparatus they are power apparatus the excitation system as I mentioned sometime back needs to be controlled you know there is usually a closed loop control system exciter the exciter is convenient to control also because it can contains basically a controlled rectifier which can be controlled by a low voltage signal. So, the block which does this is known as a regulator the primary function of an excitation system is to regulate the voltage at the terminals of the generator which otherwise would vary very substantially with loading or during transient conditions. So, that is the main reason why you need to have an excitation system regulator.  Now, we will basically try to I will try to tell you the block diagrams associated with this control system remember a control system is designed by us it is not a high power apparatus it is something which is designed by us to get appropriate control performance or transient performances. Now, although I have said that regulation is the main function exciter needs to be controlled. So, that it stays within limits. So, there are limiters and protective circuits and you may also wish to use the leverage afforded by easily controllable excitation system by modulating it in a certain way. So, as to improve the transient performance. So, this is known as a stabilizing function. So, you notice that there is a power system stabilizer at the bottom of this figure which is also used to improve the transient performance of a power system itself. So, to summarize we have discussed the models associated we have not derived it, but discussed the model of the excitation power apparatus. In the next lecture what we will do is consider the dynamic models or the modeling of the other control systems associated with the exciter which are essentially required to improve the dynamic performance of not only the excitation exciter itself or the regulator itself, but of the power system as a whole. Of course, the interface of the generator to a power system etcetera we will have to wait for some time. What we will just discuss is the basic block diagram diagrammatic representation of the typical excitation system controllers which are used in the next lecture.


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