 we have now reached a point in this course at which we can do a realistic analysis of a synchronous machine under certain circumstances. Before we of course go on to do that in this particular lecture, let us first have an overview of what we are planning to do and what we have done so far in this course. So, today's lecture is actually being pursued after we have achieved the per unit model of a synchronous machine using standard parameters. Note that we have already completed in this course the general analysis of dynamical systems in particular Eigen analysis and numerical integration. We embarked upon the somewhat tedious job of modeling a synchronous machine. First of all, we obtained the basic synchronous machine model in the DQ frame and then we went on to obtain a per unit model using standard parameters which can be obtained from measurements. We will of course after a couple of lectures or three lectures, we will move on to understanding the modeling of excitation, prime mover systems, transmission line, loads and other components. Of course these will not take as much time as the modeling of a synchronous machine. There after we will of course go ahead and do the stability analysis of a interconnected power system and also understand the basis of power system stability analysis tools. And of course after we have done all the analysis, we will also think of methods to improve system stability. We will today do a short circuit analysis of a synchronous machine using the models that we have developed. So, if you look at what we did in the previous few lectures, we normally get a set of data called the standard parameter set of data. In the D axis, you have got x t, x leakage, t d dash, t d double dash, t d 0 dash and t d 0 double dash. And of course, alternatively you may be given instead of two time constants, you may be given this what are known as x d dash and x d double dash, but they are all interrelated. So, you could be either given the first set of data or the second set of data or both. On the q axis of course, you have got x q, t q dash, t q double dash, t q 0 dash and t q d 0 double dash. In lieu of two of these time constants, you could be given the sub what are known as x q dash and x q double dash. We also of course, will be given the stator resistance which usually is very small. In some cases, we may even be able to neglect it. Recall that in the previous lecture, I actually derived the model two equations in per unit form. So, we will be actually using these per unit equations to do the short circuit analysis of a generator. So, before we do that, let us look at the standard values of or rather typical values of standard parameters. For example, synchronous reactants. In fact, we are going to give some names to all these parameters x d in per unit. Remember that we can if we are in working in per unit, we can interchangeably use l d and x d. That is l d in per unit is the same as x d in per unit and so on for all the reactants. So, synchronous reactants in per unit for hydraulic units is lower. You look at this x d is 0.6 to 0.1.5 whereas, so thermal units it can be quite high. It is almost 2.3 per unit. We shall of course, see the implications of having such a high synchronous reactants later when we discuss excitation systems. x q of course, in hydraulic units are of course, salient pole units whereas, thermal units are round rotor have a round rotor. So, you will find that the effect of saliency is not there in thermal units or saliency is not manifested in thermal units. There may be a little bit of saliency because the field winding is housed in slots which are basically in the quadrature axis. So, if you look at even a round rotor machine, the field winding is housed in slots on the q axis. So, this does not look very round, but anyway I hope you get the idea. So, your field winding is effectively bound if you look at it in a 3D scale. So, your field winding is in fact, bound in this fashion. So, field winding is bound like this. So, in this axis q axis even though it is notionally a round rotor machine, you will find x q slightly less than x d even in a round rotor machine. So, you know you will rarely find a machine which is x d equal to x q, you will not find such a machine. Of course, in a hydro generator, there is actually a proper salient pole and field windings are concentrated and bound on the pole. So, you will find a very distinct kind of x q will be very distinctly less than x d. The quantities x d dash and x d double dash x q dash and x q double dash are called transient and sub transient reactances of the machine. Now, you will find an interesting feature here that one of the parameters is missing for the hydraulic units. So, x q dash in fact is not given for the hydraulic unit. The reason is that synchronous machines which are driven by hydraulic turbines normally stays salient pole machines can be represented very well by just one damper winding on the q axis because of which we need not. So, you know your transfer functions you know psi d s upon psi q s upon i q s is only a first order. So, you do not really have to define the time constants t q dash and t q 0 double dash or alternatively you do not have to define x q dash and t q 0 dash. So, one set of parameters reduces because you can represent the generator driven by a hydraulic turbine by just one damper winding on the q axis. We will of course, later on see how you can you know have a lower order machine model how you can actually derive a lower order machine model from the synchronous machine model which we already have. So, hydraulic turbine in fact are an are an example of a lower order model of a synchronous machine. So, hydro turbine driven synchronous generators are examples of lower order synchronous machine they require a lower order synchronous machine model. So, you can look at so, t d 0 dash and t q 0 dash t d 0 double dash and t q 0 double dash are in fact called the open circuit time constants. Why they are called open circuit time constants is something we will discuss in this class itself. So, we will just hold a discussion for some time in the background we will have it a bit later. One thing you notice is that the field time constant what is known as the open circuit field time constant t d 0 dash can be very large in some cases can be as high as 10 seconds that is of course, because the resistance of the field winding is field winding is extremely small. So, you find that the time constant associated with the field winding in fact, this is the transient open circuit time constant is can be quite large and we shall see why that is so a bit later why it is called so. The stator leakage reactance of course, has a range 0.1 to 0.2 both in hydro turbine machines as well as thermal thermal units stator resistance of course, can be is very very small of course, t q 0 dash again as I mentioned sometime back in hydro hydraulic units generators connected to hydro turbines. One parameter is missing because synchronous machines in a hydro turbine a model by a lower order model just one winding damper winding on the q axis. Now, what we will do is do a simple example in this in this particular lecture which uses model 2 I will write down model 2 again in this class and uses the data which have shown here. What we will do is we will consider a round rotor machine. So, we will use two windings on the q axis a field winding and a damper winding on the d axis in addition to the d q 0 windings. So, we will take a synchronous machine which is running say at the base speed or the rated speed base speed is usually the rated speed. So, let us say omega is equal to omega base speed let us assume of course, the speed is maintained constant in this particular study. We will of course, do a study of electro mechanical transient a bit later in this course, but right now you assume that your speed is constant. So, you have got a machine which is rotating at a constant speed the machine initially the field winding is say short circuited that is no voltage is applied and what we do it at time t is equal to 0 we apply a voltage in the across the field winding. So, what I have done is we have a situation like this. So, you apply a voltage V f on the field winding. So, if you apply a voltage V f on the field winding of course, remember the current according to a convention comes out of this dot if you apply a voltage V f on the field winding you will of course, some voltage will start getting induced on the stator winding. So, what we will do is we will assume that the stator winding is open circuited we have in fact, done this analysis before we did the steady state analysis of this system before we have seen this if we apply a voltage to the field winding where the stator is open circuited you will find that after the voltage which is induced on the stator winding for example, V d f V d in fact, we found that V d is equal to 0 and V q turns out to be omega which is omega b in this case into m d f by R f. So, if you recall what we have done in the previous class instead of talking in terms of V f we shall now be talking in terms of the voltage induced on the stator winding if V f is applied. So, remember we will call this of course, E f d so under open circuit conditions. So, this is open circuit conditions now the equations of a synchronous machine under open circuited conditions can be given by putting i is equal to 0, but we will not really do that because after we finish our analysis of an open circuited machine we shall also short circuit the machine. So, what we will do in fact is represent the machine under open circuit as if it is connected to a very large resistance we do not actually set put the condition i d is equal to 0 and i q is equal to 0 and i 0 is equal to 0, but we put in fact, a large resistance across the machine. So, let us take consider a machine is connected in star. So, the machine stator winding a b c are connected in star to a load resistance R L. Now, R L is very large for under open circuit conditions and equal to 0 under short circuit conditions. So, we will formulate the equations once for all under these situations. So, what I will do is not we have already written down model 2 what I will do now is write down these equations in a kind of a state space form in fact it is the state space form. So, psi d psi q psi f psi h psi f psi h psi g and psi k is equal to a a matrix a into psi d psi q psi f psi h psi g and psi k plus let us call this a 1 plus a 2 i d i q psi d psi d psi q psi d psi q psi d psi plus b into v d v q and what are the other inputs e f d or e f d v d and v q I mean you can of course, shuffle around here a 1 and a 2 of course, an implicit assumption is that this machine is operating under balance situations. So, 0 sequence voltages currents fluxes are all 0. So, the 0 sequence does not come in this equation. So, it is even connected to a balance load. Now, remember here that v d v q since I have connected a resistance is nothing but R L 0 0 R L a star connected resistance is there here it is easy to show that if you write down the voltage and the current relationships you can easily show that in fact v d and v q are related to i d and i q like this. How do you do it? Well, you take v a v b and v c and that is equal to an apply a transformation to the d q frame. So, you will in fact, get this relationship here. So, this is the this is a relationship you should keep in mind. We have these relationships and of course, we also have an additional relationship v d v q themselves can be written in terms of R L and i d and i q. So, actually this portion can be subsumed in a 2. So, these two variables can be subsumed in a 2 of course, if v d and v q if there was of course, an independent voltage source at the terminals of a synchronous machine then of course, you would have to define v d and v q, but as I mentioned sometime back v d and v q are in fact, having a related to i d and i q. So, what we will do now is write down these matrices which I have defined sometime back a 1 now a 2 is minus r a minus r l 0 0 minus r a minus r l this is multiplied by omega b. So, what I have done is actually of course, subsumed this v d v q into i d i q because they are related. So, that is why this r l is appearing here these in fact, equations are in per unit. So, that is an important note which you have to make now this b matrix which relates all the fluxes to e f d now the only input here is e f d remember now in this particular study we are I will call this b 2. So, this is relating flux psi to e f d remember e f d is the voltage is the steady state voltage which appears the line to line voltage which appears across the stator terminals if connected in star if v f is applied at the field winding and this is of course, in steady state under open circuit conditions and of course, e f d is in per unit. So, these are my equations. So, what I will do is do this step change in field voltage and show you the way the system evolves. So, what I will show you now is a program which has written in psi lab we have of course, recall that we have used psi lab in the previous examples when we are doing the analysis of dynamical systems. You can also use matlab or any other software in fact, you can write your own programs as well. So, I will of course, in the interest of saving time show you a simple program written in psi lab. So, you need to pay attention to what I have written here. So, this is the program. So, this is an Eigen analysis of an open circuited generator. In fact, it is the analysis of an open circuit generator. The radian frequency electrical frequency is 2 pi into 50 the speed of the machine is also the same. The parameters of the machine I have entered them I just may showed them to you sometime back. Only difference is of course, I have changed the time constant to 5 seconds because as I should show you the system settles down faster with a lower time constant. So, we do not have to simulate for a very long time. So, that is the only change, but I am using otherwise the parameters which I just mentioned to you in the previous slide just showed to you in the previous slide. Remember that I have been given the reactance is x t dash x t double dash x q dash x q double dash and open circuit so called open circuit time constants. You can get the time constant t d dash and t d double dash by using the relationship which I had given you sometime back. Remember that t d dash and t d double dash are related to t d 0 dash t d 0 double dash and x d dash and x q dash and x d. So, the formula which allow you to do that of course, are these in fact this is continued on the previous line. So, t d so we get these values of t d double dash and t d dash by solving the quadratic. Now, similarly you can find out the values of t q dash and t q double dash from x q x q dash t q 0 dash and t q d 0 double dash using these formulae. We of course, done this relationship in the previous class as well as the lecture previous to that. So, you can just refer to the formulae. So, I have written down of course, the state equations note that the states are these 0 sequence is totally decoupled. So, it is not considered here in case it is not visible I can increase the font we will do that we will just increase the font font size after selecting everything. So, I will just increase the font size to 24. So, it becomes a bit easier to view. So, we go back the show this again. So, I have programmed the matrices A 1. So, A 1 is this A 1 A 2 B 2 and A 3. So, this is of course, written in the Sylab syntax. Now, if I want to of course, get these equations in pure states based formulae, remember that the way the equations are written are these. So, psi dot is equal to A 1 into psi plus A 2 into I is nothing but the vector of I d and I q plus B 2 into E f d. So, if I want to actually get this in pure states based form I will write it as A into psi plus B 2 into E f d by substituting I in terms of psi. So, A will be equal to A 1 plus A 2 into A 3. So, we can get the Eigen values of A the matrix A remember it is a linear system if you in fact, the speed dynamics are neglected then it is the absolutely a linear system. So, if we in fact, we can also compute the steady state if I want to know the steady state value of fluxes after the step has been given and we wait for long enough time you can set this psi dot is equal to 0 and get psi in terms of E f d. So, in fact, if E f d is 1 let us assume that the voltage which is applied at the field is such that E f d eventually is 1 per unit. So, E f d if it is assumed to be 1 that is what this says steady state value of variables with E f d is equal to 1 is simply equal to psi will be equal to minus of A inverse B 2. So, that is what is shown here on the screen here that the steady state values of course, the steady state values are something which we would like to know, but the main aim of the study is to obtain the transient behavior of the machine. Now, to obtain the transient behavior of a machine you do not actually have to simulate the system. So, what we will do here of course, is not simulate the system using a numerical integration technique, but since this is a linear equation that is psi dot is equal to A psi plus B 2 into E f d you know that psi of t is nothing, but e raise to A t into psi of 0 at the time 0 will assume all the fluxes are 0 initially. So, this vector psi will be 0 plus the convolution integral 0 to t e raise to A t minus tau into B 2 into E f d d tau. Of course, remember that E f d and B 2 both are constants. So, we can analytically obtain if you actually evaluate this will come out to be we will rewrite it on another page psi of t is nothing, but e raise to A t into psi of 0 plus or rather it will be minus of A inverse a 6 by 6 identity matrix minus e raise to A of t into B 2. So, this is the time response remember of course, that A is nothing, but e raise to A t is nothing, but we have done this before p e raise to p inverse where p is the Eigen vector matrix is the right Eigen vector matrix and this is nothing, but lambda 1 a diagonal matrix containing the Eigen values along its diagonal elements. So, this is what this is. So, e raise to is nothing, but n p suppose the n Eigen values of n by n matrix then this will be this of course, we assume here I am assuming that the matrix A is diagonalizable. In fact, it is as we shall see in a few moments from now. So, the if you look at the program it actually evaluates this it evaluates e raise to A t using this command it is gone on to two lines just because we have increase the font size. So, actually this statement is continued here so, you just continue this statement right up to this point. So, we evaluate x is in fact psi here psi is the states psi. So, e raise to A t into the initial condition of the states minus a inverse i the identity matrix into e raise to A t into B 2 and e f d of course, is 1. So, I have not written it down. So, e f d is an input which is equal to 1. So, after I do this analysis I can get the values of i d which is nothing, but the matrix A 3 into x the first term of A 3 into x. Similarly, I can get v d v q once I get v d v q i d i q and the states all the states you can also evaluate the torque of course, if it is an open circuited machine we expect torque to be 0. Remember of course, since this is a open circuited machine we have taken R L is equal to 1000. So, what we will do is run this program will quickly run this program. So, what I will do is execute this program. So, it takes a little bit of while because we are simulating this program for 30 seconds. Remember that I am not doing any numerical integration of equations I am directly evaluating the time response obtained analytically from the using the Eigen values and Eigen vectors. So, it has done this evaluation of the time response, but before I go ahead let us look at the Eigen values of this matrix. These are the Eigen values it is a stable system since all the Eigen values are negative. We can proceed further, but I think we can take this up in the next class of course, we have not gone to the point at which we will take our do our short circuit analysis of a synchronous machine. In fact, that is the next step we will in fact, first x we are first exciting the synchronous machine by a step change in the field voltage. There after once it reaches steady state we will give a short circuit to the machine. We have in fact, not completed our analysis of a short circuited generator. We will in fact, look at this program again rerun this program again and have a look at the Eigen values have a look at the time responses and see well whether it correlates well with what we think should be the response. The important point is of course, once the transient settle down we should come to a steady state which has been predicted already two lectures back that is the steady state analysis of a synchronous machine. So, after the transient I resound we should come to the corresponding steady state values. Now, one important thing which I intended to cover in this lecture we will do that in the next one is to understand why the time constants which I mentioned here t d 0 dash, t d 0 double dash and similarly, t d dash, t d double dash are known as open circuit time constants. And that is something of course, we have not discussed in this class we did not have time for doing it. We will do it in the next lecture.