 So, we have been looking at salient pole alternators and from the basic large signal differential equations that are shown here we formulated equivalent circuits one for the direct axis and one for the quadrature axis and then for evaluating this term the second term here which is then represented as an equivalent circuit by the circuit within the green box. We separated or we specifically looked at the circuit within the green box both in the D and Q axis and then we wrote down an expression for the voltage which is S times Psi D of S in this form and then we tried to evaluate the expressions A and B, A turns out to be a fairly big expression but then as we go around trying to simplify and recombine various terms you end up with several expressions T1, T2 and T3 with the denominator expressed as a second order expression in S and then the numerator also we tried to expand and then combined the terms in various ways and then the numerator boiled down to a form which was looking like this several expressions T5, T4 and T6 were identified and the resultant expression A then looks as is written here similarly we worked on the Q axis equivalent circuit I mean expression for B first and that was then written in this form as a first order term here divided by the second order denominator multiplied by S times LMD by RF this we then applied a similar method of analysis to the Q axis circuit as shown here now Q axis contains only one source of excitation unlike that of the D axis where there is also a field excitation along the Q axis there is no field and therefore the equivalent circuit simply looks like this and with the Q axis we wrote down various terms and S times psi Q of S then could be simplified and written in this form where 1 plus ST7 by ST8 multiplied by S times LQS so T7 and T8 were identified so all these then are various all the expressions identified are then various time constants of the alternator which would manifest in the response produced by the alternator to any variations in the input voltage or the field voltage may for that matter so here we have summarized all the expressions that we have derived T1 is nothing but LLF plus LMD by RF this is basically the field all the terms of the field this is the inductance of the field and then the resistance L by R ratio and then T2 is then the one pertaining to the D axis damper T3 is 1 over RKD multiplied by LLKD plus LMD LLF by LMD plus LLF we have also seen that these terms may how they may be derived or identified directly from the equivalent circuit we found that all these terms can be written down by looking at the equivalent circuit considering the stator terminal as open so that is what we have seen just to briefly recapitulate if you look at the D axis equivalent circuit in this form that is the D axis circuit this is LLS you have LMD RKD LLKD LLF and RF with the voltage VF applied here as a function of S Laplace transform domain and so if you look at T1 and T2 are fairly straight forward so if you keep this open and you neglect the existence of this RKD and LKD neglect the damper then the circuit has this resistance and these two inductances then come in series as long as this is open right no current flows here so these two are in series so then the resulting T1 is then LLF plus LMD by RF and then the damper time constant itself if you consider only this branch turns out to be this and T3 is then if you consider this to remain open and if you say that you want to find out the time constant of the resulting circuit neglecting RF then you need to get at the equivalent inductance seen across the terminals of this damper resistance and that is exactly what this term is LMD in parallel with RF if this is a short and that occurring in series with LLKD so this is the expression so these three terms are then derived considering the stator terminal to be open this one that is T1 is derived neglecting damper whereas T3 is derived including damper similarly if you look at the numbers T4 so if you look at the numbers T4 T5 and T6 you can get T4 by considering that the stator terminal is shorted and then neglect the damper again so this is not there so if this is shorted then LMD occurs in parallel with LLS so the net inductance there is LLS plus LMD divided by LLS plus LMD that is what you have here and then that is occurring in series with LLF so plus LLF and divided by RF so this term T4 will arise if you neglect the damper circuit but short this side and similarly if you look at T6, T6 is then obtained as again considering this to be a short and you look at the inductance across RKD neglecting RF then what you have is if this is also the short circuit then you have LLS in parallel with LMD in parallel with LLF and that is this term all of them occurring in series that entire set occurring in series with LLKD so this and then divided by RKD that is what you have here so these terms are then derived by considering stator to be shorted and similarly T5 also can be written that way now T7 and T8 they belong to the Q axis and here again these two can be derived by considering the stator terminals to be shorted we already have the Q axis equivalent circuit here so if you consider this to be a short then one can derive these this expression LLS and LMQ then come across each other as can be seen here these if you short this this inductance is in parallel with this therefore the net inductance is product of these two divided by some of these two and that inductance comes in series with LLQ and therefore you have these two terms here. So here you see that both the terms are obtained that is this term is obtained by considering that the stator is shorted and this is nothing but the damper itself so these are all the various expressions that we have direct now to take the next step forward if you look at the expression that we have derived for s times ?d of s so s times ?d of s was written as a of s multiplied by id of s plus b of s multiplied by vf of s and a of s was of this form so let us look at the expression a of s a of s is s into an inductance multiplied by all these terms and therefore we can modify that a little and then write a of s as it was s times an inductance which is nothing but LLS plus LMD multiplied by a second order expression in s divided by a second order expression in s. Now if you multiply this inductance this inductance is effectively the direct axis self inductance and which we can write as s times xd of s where x is the direct axis reactance divided by the synchronous speed ? essentially we are saying x is nothing but ?s into L and therefore we are writing L as x divided by s and so that is what here is here this multiplied by the second order functions in s and therefore in this we can now identify this term this entire term as sorry this is not xd of s this entire term is then written as an expression xd of s it is called the direct axis operational impedance or the direct axis operational you know inductance. So xd of s is well we can call it as the direct axis reactance operational reactance which is then ? multiplied by LMD plus LLS this term is your xd the direct axis reactance multiplied by your second order term in s divided by the second order another second order term in s. Now let us look at the terms T5, T4 and T6 which we have consolidated earlier so if you look at these three terms T4, T5 and T6 we find that T5 and T6 are divided by I mean our expressions which are divided by RKD whereas T4 has RF in the denominator. Now normally in alternators this number RKD is considerably larger than this number RF and therefore both the terms T5 and T6 are really negligible compared to T4 and therefore we can say that T4 plus T5 is approximately equal to T6 plus T4 since T5 and T6 are negligible compared to T4 this term the s term in the numerator can approximately be written as T4 plus T6 itself which therefore means if you look at the numerator it is 1 plus s into T6 plus T4 plus s2 into T6 T4 and therefore the numerator can then be written in a simplified form as 1 plus st4 into 1 plus st6 this equality sign it is really approximately equal to this. Similarly if you look at the denominator you have s square T1, T3 and T1 plus T2 so let us look at the terms T1, T2 and T3 so here if you look at T1, T2 and T3 you find that T2 and T3 contain RKD in the denominator whereas T1 contains RF in the denominator and therefore T because RKD is much larger than RF both T2 and T3 are quite small compared to T1 and therefore in the term here T1 plus T2 is approximately the same as T1 plus T3 and therefore we can write this as s into T1 plus T3 and if we write that then the denominator expression can be simplified as 1 plus st1 into 1 plus st3. So XD of s can then be written as XDS this is nothing but the direct axis this is the direct axis reactance and then you have T4 which we write as T-D, T6 which we write as T double dash D, T1 is written as T-DO and T3 is written as T double dash DO the significance of dash and double dash in these expressions denotes double dash denotes an interval called as the sub transient period and single dash denotes an interval called as the transient period. We are simply redesignating T4 as T-D, T6 as T double dash D, T1 as T-DO and T3 as T double dash DO the names that are specifically given to these terms are T-D is called as the direct axis short circuit transient time constant and then T double dash D is then called as the direct axis short circuit sub transient time constant the terms must be self-explanatory by now direct axis obviously because it refers to the direct axis equivalent circuit short circuit because we have already seen that T4 and T6 can be obtained by considering that the stator terminals are shorted that is how we derived we said that one can look at the equivalent circuit so you can see here that T4 and T6 have been arrived at from the equivalent circuit by considering the stator to be shorted so those are short circuit transient because this term TD dash which is really T4 neglects the existence of dampers so if you go back here and see T4 we have said that dampers may be neglected in formulating this expression and therefore these are transient we will see why that is so in a moment and TD double dash which is T6 is again direct axis short circuit sub transient this means damper is also included now let us explain these terms a little more now if you look at the alternator and the alternator is rotating at synchronous speed and at synchronous speed the relative movement between the rotating magnetic field and the rotor itself is not there the rotating magnetic field rotates at synchronous speed the rotor also rotates at synchronous speed therefore if you look at the dampers which are there in the synchronous machine they may be specifically damper bars or they may be it may be the induced EMF in the rotor solid ion itself whichever be the case on this there is no induced EMF so if there is no induced EMF there are no eddy currents that are flowing and therefore it may be said that as far as the equivalent circuit is concerned the dampers may be looked at as an open circuit which means the damper does not really exist as far as the equivalent circuit is concerned that branch is simply neglected but however when there is a disturbance in the alternator may be a disturbance on the stator side or may be a disturbance on the rotor side there may be slight initially there would be an interval where there is a slight displacement or relative motion between the magnetic field of the magnetic field in the air gap and the rotor and this slight relative velocity causes induced EMF in the rotor itself or in the in the associated damper bars and this induced EMF then is going to result in certain flow and because of that these now impact the electrical behavior or the electromechanical behavior of the system. And therefore when there are disturbances damper currents flow now if you look at a large disturbance for example an alternator operating at some speed at the synchronous speed of course and if it is let us say subjected to an electrical fault which is a fairly large disturbance then because of the disturbance that is there there will be some relative motion between the rotor and that of the magnetic field and therefore there will be an induced EMF in the damper bars and some damper currents will flow. However because the resistance of the damper circuit is pretty high these induced EMFs causing some currents to flow this induced behavior will decay down pretty fast and because RKD is high the damper currents die down fairly fast and after that still the effect of the external disturbance may be there and in that interval the rest of the circuit still undergoes or has these fields the effect of the disturbance and because of this effect the entire interval following the disturbance we can divided in a way into two distinct sub intervals first a region where there is an effect of the damper bars and second an interval where there is no effect of the damper bars and then lastly the region where you have only the end steady state response of the system the first interval where effect of dampers are there effect of dampers if they are felt that region is called as the sub transient period the second region where the effect of dampers is over then this region is called as the transient period and then when all these have decayed down the response that is left is the final steady state because of this kind of names that are given TD double dash which is the number that is derived by considering the damper to be in circuit this is then called as the short circuit sub transient entity and TD dash is then the short circuit transient one similarly if you look at the numerator I mean if you look at the denominator you have T1 being replaced by T dash DO and T dash DO is then called as the direct axis open circuit transient time constant and T double dash DO is called as the direct axis sub open circuit sub transient time constant the explanation for this is along similar lines we find that T double dash DO has been derived by considering the damper circuit damper the elements of the damper to be in circuit and T dash DO was obtainable from the equivalent circuit by neglecting the elements of the damper and we are obtaining that by considering the stator to be open circuited and therefore you have open circuit transient or the sub transient expression. So the numbers the expression T4, T6, T1 and T3 have been redesignated in these following form. So if you now look at XD of S in this format you find that XD of S is equal to XDS multiplied by all these things which is a function of the Laplace transform variable S. Now if all disturbances have died down and the system has reached steady state you may obtain the behavior by assuming that the Laplace transform variable S allowing S to go to 0. So in this if you allow S to go to 0 you should get the behavior or the equivalent entity in steady state and we find that under steady state then XD of S tends to XDS which is nothing but the direct axis reactance which is again what we expect that this expression if we are to give it the name operational direct axis reactance then when there are no disturbances it should be equal to the direct axis reactance which indeed it is so from the expression. But however if you look at the effect of the other entries that are there T dash DO, T double dash DO and all that we have divided the entire response duration of the alternator into three subdivisions. The first one where there will be an initial phase where there are fast variations in the system and another interval where the effect of the dampers has been neglected. So if there are going to be fast variations in the system initially then we may approximate the expression by allowing S to be very high and if S is very high we find that XD of S now tends to if S is going to be high then in all those expressions in the 1 plus ST term since S is high one can be neglected and therefore if you look at this expression XD of S now becomes XDS multiplied by T dash D multiplied by T double dash D divided by T dash DO multiplied by T double dash DO. So this is an approximation to XD of S under fast varying situations where damper effect is also there. Now this term is then given the name direct axis sub transient reactance it is called or given a special symbol XD double dash. So XD double dash is a simplification of XD of S under situations where the where S may be assumed to be large and where there are fast varying effects in the system now after sometime fast varying effects may not be there but still there is a variation and it is not yet the region of steady state in that situation we may assume that there are no dampers and in that case S may be large but no dampers and therefore XD of S becomes the expression this XDS multiplied by T dash D divided by T dash DO and this is then called as direct axis transient reactance this is given the symbol X dash D. Now these are representative or these give an indicative idea of how fast or of the magnitude of disturbances that may be there in a fast varying system in case of this alternative. Now in the same manner we may look at the expression for XQ of S now remember that XQ if we had this expression here where S times ?Q of S is written as S x LQ multiplied by these terms one may instead of having LQ of S LQS one may write this as XQS divided by ? let us put that down here so this may be written as S times XQS divided by ?S multiplied by 1 plus ST7 divided by 1 plus ST8 multiplied by IQS and in this case we can define similar to what we have done we have defined an operational impedance operational term XD of S similar to that we may define a new term XQ of S as XQ of S as XQS multiplied by 1 plus ST7 divided by 1 plus ST8. Now as before we can redesignate this as XQS multiplied by 1 plus STQ double dash divided by 1 plus T double dash QO and why is that so if you look at the Q axis equivalent circuit you have the leakage inductance and then the magnetizing inductance along Q axis and then the dampers so this is RKQ LLKQ LMQ and LLS now there are if you assume that the dampers are present which means we are talking about the sub transient period then if you look at the stator being open right then we can derive this expression T8 that is what we have seen earlier that is T8 is LMQ plus LLKQ by RKQ this can be done by keeping the circuit open why is it open now if this is open as it is then the effect across RK the inductance across RKQ is nothing but LMQ plus LLKQ and therefore T8 is the same as T double dash QO which is this term if you assume that the stator is shorted shorted stator that would then have LLS in parallel with LMQ that occurring in series with LLKQ and that is really the term that you had here this we have already seen so that is the term that comes here as we have seen and this also assumes the existence of RKQ and LLKQ and therefore both these terms refer to the sub transient period and therefore T the expression which is equal to you know the expression T7 is designated as T double dash Q under open circuit situation under the situation of short circuit. So XQ of S in the expression of XQ of S then we find that only sub transient period is there only sub transient effects are there unlike the D axis there is no single dash term and as before whenever there is an interval where there is an interval where the disturbance variations are fast then there is an S missing here so whenever there is a disturbance that is occurring which is varying very fast then we may assume S to be large in which case XQ of S tends to the term XQS multiplied by TQ double dash divided by TQO double dash and this term is then called as the quadrature axis sub transient reactor and whenever there is steady state then S tends to 0 and we find that XQ of S tends to XQS which is the quadrature axis reactance note that in this case there is no quadrature axis transient reactance because there is no field winding in this. So we find that from the expressions of XDS and XQS both which are functions of the Laplace variable S we are able to derive specific terms which are relevant for the sub transient interval or the transient interval on the D axis and for the sub transient interval in the Q axis and we have assigned specific names for all the reactances that we encounter. So with all this then we need to go ahead and see where or how the usage of all these terms is going to be seen for that let us go to look at a big disturbance that is going to happen to the alternator which is a sudden short circuit of the alternator. Now out of the disturbances that can be visualized to the alternator this is in fact a very major and fairly severe disturbance. One can also see as we go along this is also one of the important evaluations that is recommended for the alternator you have to do this experiment to determine some of the important operational features of the alternator and as a part of this evaluation then you get the terms one it may be feasible or one can determine the terms XD double dash and XD dash from this experiment on the alternator. So how is this done if you then have I mean this is described in this manner that you have the alternator being driven by a prime mover it is on open circuit initially you allow it to reach full voltage and then all of a sudden apply a dead short on the alternator. Now in response to the short there will be an effect on the mechanical system as well but as far as the overall response is concerned because the mechanical systems are generally slower much slower than electrical systems we may look at the entire behavior again in the two parts one where the speed of the rotor is still the same as what it was and then another interval where the speed may change as a result of the effect of the short on the rest of the system. So we are now attempting to see what will happen to the electrical system if you are going to apply a dead short like this that is our focus of interest and therefore we may assume that speed is constant speed is assumed to be held fixed and you short circuit the terminals of the alternator under that conditions we are interested in electrical system transients that means how much flow of current will be there how high a flow of current how it will decay down when how long it will take to decay down what will be the steady state magnitude. So these are the things that are of interest and as we develop the expressions for this we will find that all these things depend very much on the terms that we have defined earlier like the direct axis impedances during the sub transient period and so on xd double dash xd dash all those things are going to play a part in this. So how we are going to do that we will start with the initial equation of the alternator that we have been dealing with all along in the earlier that we have seen that the differential equations of the machine as a fourth order system can then be represented as far as the state or side is concerned as vd of s is raid plus p times id of s minus omega r psi q and vq is given by ra iq plus p psi q plus omega r psi d note that both these expressions involve psi d and psi q and therefore they are interlink and one cannot really separate them off. Now if you assume that speed is held fixed then this term becomes a fixed number and therefore the resulting expression one can consider it to be one can look at it in the Laplace domain and then say vd is then vds id becomes the Laplace transform of id that is id s and then p times psi d now becomes s psi d and then omega r psi q becomes omega s omega r is nothing but the synchronous speed alternator is being assumed to be running at the synchronous speed and this psi q of s. Now we see that because you have s times id of s and s times psi q of s and we have already derived expressions for s times id and psi q in terms of the other exciting variables as far as the direct axis is concerned the variables where id and vf as far as q axis is concerned only iq was there. So we can simplify this s times psi q in terms of the other exciting variables and here also psi d and psi q can be replaced in terms of the other exciting variable. Therefore we can reduce this to an expression that depends on id and vf and similarly vq also to an expression that depends on id and vf also iq as well once we get that we can now try to do any experiment that we want like for example attempting to apply a dead shot at the terminals of the alternator and solve the system of equations to see what kind of response you will get this analysis we will do in the next lecture we will stop here for now.