 present module in this course, we have been studying about various techniques to understand how a dynamical system behaves. Now, our treatment of these various methods is not very rigorous, but I trust whatever we are doing will give you some kind of introduction and reassurance in case you have already done it before. In particular in the past two lectures, we have been studying about numerical integration. Numerical integration is a very general tool which can be applied to the study of dynamical systems, unlike Eigen value and Eigen vector analysis which is applicable only for linear systems. Now, we have been studying the basic features of some of the numerical integration method. It is very important with the proliferation of a lot of simulation on numerical integration software. It is very important to know the characteristics of various numerical integration tools which are available and to do this, we have to kind of benchmark our behavior of dynamical systems and then compare it with what we get when we numerically integrate. Remember, numerical integration always involves some error because it is an approximation of a continuous time system. In the previous lecture, we started on analyzing how numerical integration methods behave when you are confronted with a stiff system. A stiff system is a system in which the various patterns in the response, the various patterns which you see in the response have widely varying time constants or widely varying rates of change. In a linear system, of course, one can correspond these fast and slow transients to large and small Eigen values. Very often in engineering, we do encounter such systems. In fact, if you do encounter such systems, you often are able to do modeling simplifications, a point which we discussed in about three lectures ago. So, today what we will do is consider the same system for which we did Eigen value analysis and see how it behaves or what answer we get when you numerically integrate and try to get the time response. Now, remember of course, at this point that we are you know doing the numerical integration of a linear system. The linear system response of course, is fully known in terms of simple functions like exponents and sinusoids. We can use our powerful Eigen analysis tools to obtain Eigen values and Eigen vectors and write down the system response. There is no need to do integration, but as I mentioned some time back that we use these to benchmark how our numerical integrations numerical methods work. Now, the system which we considered in that particular you know linear when we did our linear analysis, we considered a particular system which was very typical in the sense that it brought out the stiffness in the system. That system was basically an RLC circuit which was excited by our step in the input voltage. So, the system we were considering was this a relatively large inductor here and again a capacitor here. The system is a linear system and we can write down the differential equations in this form where A is and B is. So, this is our system and we of course, in a previous lecture we took out the time response of the system. The Eigen values of the A matrix when you do linear analysis are given by this and this. So, in fact there is a complex conjugate pair which will correspond to a damped sinusoidal response or damped oscillation and of course, there is an real Eigen value a negative Eigen value which will correspond to a pattern which will be seen in the response which is e raise to minus 0.1 t. It is very clear that the system is stable because the real part is negative of all the three Eigen values. Another issue which is important is that the rates of change associated with the pattern or mode corresponding to this Eigen value is very slow as compared to this. Look at this frequency it is extremely high. So, the kind of moment you will get in the response is going to be having a large rate of change for this pattern. So, your response is consisting of two patterns the fast mode and the slow mode. This is in fact typical of a stiff system you have got you know both fast and slow modes. Now, if one tries to numerically integrate this particular system then one may use Euler method. For example, if one wants to apply Euler method to this differential equation with a time step of h. In that case your iteration which you will I would not say an iteration if you know the value of x k you can get the next value of x k plus 1 the next sample of x by using this relationship. So, a x k plus b u k. Now, this implies we have seen in a previous lecture that I indicated that this system Euler method when it tries to numerically integrate x dot is equal to a x plus b u the discretized system which you get may be stable if is stable in case where lambda is the ith eigen value. The important point is this should be true for all eigen values. So, Euler method will be stable or rather the response which you get by using this relationship the samples which you get by using this relationship will be stable if for all eigen values this is satisfied lambda i s was the eigen value of a. So, this is the basic property of Euler method. Now, remember that the original system is stable, but Euler method under certain circumstances may not able to mimic the stability of the original system. It may show an originally stable system to be an unstable one because this relationship may not be satisfied and in Euler method for example, if I apply this to one of the eigen values that is lambda 1 is equal to this. Suppose, I want to check this relationship for this eigen value you will find that the relationship you will get is this which boils down to this even if I choose a time step of point double 0 1 you are not going to have this relationship satisfied. For example, you expect that this frequency is the imaginary part of course, corresponds to the frequency of the oscillation in radian per second. We expect that if I choose a 1 millisecond time step it should be able to you know mimic the response, but unfortunately that is not true because this relationship is not satisfied 1000 and 5 radians per second corresponds to around 1000 and 5 divided by 2 pi as a frequency the time period of this is of course. So, we expect that if you choose a 1 millisecond time step we should be able to mimic the response, but that is not true if you discretize using Euler method this relationship is not satisfied and you will find that the system which you have numerically integrated or rather the discrete time system which you get by discretizing the original continuous time system by Euler method will not be stable. So, you are getting a qualitatively wrong answer if I try to use Euler method with this time step of course, you may say let us reduce the time step further one can go on reducing this to for example, 10 raise to minus 5 6 or 7, but remember that time required to do this you know numerical integration will keep on increasing if I reduce the time step. So, if I want to simulate one second of the response if I choose h is equal to 1 millisecond in that I will I will require 1000 steps and if I choose 10 raise to minus 4 seconds you will require 10000 steps just for a 1 second simulation. So, the problem here of course, is that if I am interested in the slow response if I am primarily interested in how the system behaves the slow response of the system in that case if I use Euler method I will still be constrained to use a very small time step in order to prevent the faster mode from being unstable I mean the numerical integration should not display instability I mean that would be a qualitatively wrong answer and that is something which I do not want to have. So, Euler method has a problem in such a system to make your time step very very small. So, that is one issue which you should remember now of course, somebody may ask what is this interested in the slow transient not interested in the fast transient I mean what are the situations where you would be interested in the slow transient and not interested in the fast transient and so on. For example, let me give you a simple example you want to study what happens when you start a DC motor or you have got a DC motor running and the load torque on it the load torque on the DC motor suddenly changes and you are interested in how the speed varies. Now, the point is when you are trying to see how the speed of the DC motor varies what kind of transient are you interested in see if you look at the DC motor it has got some resistance it has got a small inductance it has got a small it has got inter winding capacitances and so on. So, if you model everything including the mechanical system the electrical system you will find that it becomes a stiff system because the electrical time constants or the electrical transients are much faster than the mechanical transient. So, if you model all the transients, but your interest is on the in seeing how the slow speed transient behaves you know the slow you know pattern in the response then you come up with a similar situation. So, whenever I say that you have got a stiff system and you are interested in the slow response you can remember this kind of example. So, if you have got a DC motor and you are interested in the speed transient and your model all the electrical all the electrical components of the system which are relatively faster then this particular situation does arise. So, let me reiterate we are thinking of a system which has got both fast and slow transients and we want to replicate the slow transients. So, that would be a particular situation which we may face of course, you may be interested in the fast and the slow transients that is another situation that is another thing you may encounter. If you are interested in the slow transients or the slow transient behavior of a system, but you are not interested in accurately representing the fast part of the response you know fast part of the response is there, but you are not very interested in the accurately you know getting the fast response. In that case Euler method is not a good idea because you have to really decide your time step based on the fast response and if you choose anything which is larger than what is mandated by that particular condition you will find that your system simulation blows up. So, that is one of the problem which you will face if you try to use Euler method. So, Euler method is taught, but rarely used. So, that is the basic interesting thing you will understand by experience. What we will do now I will just show you this particular aspect by doing a numerical simulation on Sylab. So, what I will do is I will just show you a clip of a program will be just I will just show you how you can run it and then I will display the result. So, if you look direct your attention to this particular program which I have used to simulate the system. So, this is a Sylab program we will of course I will tell you the main steps in the program you have given the A matrix of the system spec A will tell you the Eigen values if you are interested in them let us not print them out. This is a time step say you want to simulate the system for 2 just for 20 milliseconds using a time step of 0.001 just for 2 milliseconds and using Euler method. So, let us use assume that the initial conditions on the states are 0, but the system is excited by a step in the input. So, we are getting some kind of forced you are having a forcing function. Suppose I numerically integrate using Euler method I delete this. So, what you have here is the Euler method x is equal to 1 plus h into x plus h into u that b is kind of absorbed in this vector. So, I am not written it separately. So, I will comment the discretization by trapezoidal rule. So, what we have here is this program and of course, I will plot the values once I have simulated them. So, if I run this program I will save it and I run this program this is the response I get the green and the blue denote the inductor currents the small inductor and the large inductor the small inductor is blue the large inductor is green and red is the capacitor voltage. It may not be very clear in your screen, but the value here is 1200 1200 and this time of course, is 0 to 0.02 seconds. So, you see this response is kind of blowing up you get a blowing up response. So, this is one of the problems which you will find in Euler method that when you have got a fast transient you have to the time step you may choose really has really may not satisfy the stability condition and you will get spurious response. Before we go ahead and you know try out the other methods let us quickly look at what is the actual behavior of the system. The correct time response is this how did I get this correct time response? The correct time response if you recall is not obtained by numerical integration, but by simply evaluating the response at various time steps from the analytical functions that we derived using eigenvalues and eigenvectors. If you recall the response of I a 1 using eigenvalue analysis is 10 minus 10 e raise to into sin of 1005 t I 2 is 0. So, if you recall the response of I a 1 this is approximate not exact, but this was analytically derived using eigenvalues and eigenvectors. So, this is the response for this RLC circuit I 1 I 2 and V c. So, this is the time response but this is derived analytically. So, this what we are showing on the screen is the same response which is evaluated by simply plugging in t into the these functions which have just written down and so this is the correct time response. So, this is what I should be getting and at us if I try to simulate this just from 0 to 0.5 seconds this is what I should be getting this is V c and this is I 1 and I 2 green and blue. So, this is the correct time response and just to re plot what I got just now using Euler method with 0.001 and a simulation just 20 milliseconds you see the beginnings of the fact that this discrete time system obtained by using Euler method is just blowing up. So, we are nowhere going to be nowhere close to either this response or that response. So, Euler method is giving a horribly wrong solution. In fact, it is giving a qualitatively wrong answer also it is not only inaccurate it is giving a qualitatively incorrect you know conclusion or inference about stability. If I use backward Euler method remember the backward Euler discretization is done for x dot is equal to A x plus B u by the following u k of course, is constant u is constant. So, we will not have less trouble about this, but what you get eventually by just solving this I am skipping a few steps is 1 minus A h inverse x k plus 1 minus A h inverse B h u k. So, this is basically the iteration which I will have to use in case I want to use backward Euler method. So, this is backward Euler method. Now, if I use backward Euler method the important thing of course, here is that you have to take out the inverse of a matrix for linear. So, this involves extra computations it is not as straight forward as the explicit method that is the forward Euler method which was an explicit method. Now, normally taking out an inverse is not a big problem I mean it does not is not fraught with problems as far as computation is concerned provided the system is small. If a system is very large then computation of inverse can be quite intensive. For example, it is very common in a power system to have a size of the order of the system may be thousand you may be having thousands of states. So, in that case to try to compute this may be a bit may be fraught with a lot of hurdles because inversion is a computationally intensive you know operation and at every time step you have to do this you know this particular function that is x k plus 1 getting x k plus from from x k. Now, since 1 minus A h inverse has is appearing at every step you know you can compute it once in your program and then simply do a matrix multiplication when you are running this algorithm to implement backward Euler method. So, you can actually take out the inverse and keep it beforehand and only perform matrix multiplications. But again if you are working with very large systems it is not a good idea to compute the inverse explicitly. The reason being that if A is sparse the inverse of I minus A h I is incidentally an identity matrix something which I did not mention earlier. This particular matrix is the inverse may not be sparse. So, you may even have to store a very large number of values if you are going to in explicitly compute the inverse and keep it stored beforehand. What would be a pragmatic thing to do would be to compute the l u factors the lower and upper factors of l u factors of 1 I minus A h and just do backward and forward substitution during each iteration. So, the l u factors of course, you may have to do ordering of the states etcetera. So, that you get the l u factors as sparse if A is sparse. But remember storing the inverse of even a sparse matrix you know is a problem because the inverse may not be sparse. So, these are some of the issues which you may face if you are developing a program for large systems. But of course, right now we are talking of a third order system you can just as well take out the inverse you can even keep on taking out the inverse at every time step though it is not necessary. So, this is basically how you will implement backward Euler method trapezoidal rule is again similar you will get x k plus 1 minus x k by h is equal to A x k plus 1 plus A x k upon 2 plus B u k plus 1 by 2 plus B u k by 2. So, this is how you will discretize it again you will need to take out an inverse of a matrix. So, that is one of the critical features of trapezoidal rule as well as backward Euler or any implicit method. Remember since this is a linear system our job in implicit methods requires inversion of a matrix if your system is non-linear you may require even to do iterations to get x k plus 1 from x k using some method like n r that is Newton Raphson or Gauss-Seiden method. Now, coming back to the qualitative results if I used backward Euler method with h is equal to 1 millisecond and a simulation interval of 0.02 this is the response I get one thing you can notice here is that backward Euler seems to have killed the Euler in the oscillation very quickly. So, the fast transient which I expect to be seen in this 20 millisecond window is in fact seems to be very well damped damped out better damped then what in fact it actually is. So, you if you look at the original response it takes at least a second or so to damp out more than a second to damp out whereas, here backward Euler with a 1 millisecond time set the oscillation dies down very soon. So, if you look at if you just recall how this the correct time response should be like this in about 0.5 seconds your oscillation is gradually dying down. So, this oscillatory part of a transient takes at least a second or two to completely die out whereas, backward Euler has simply killed at oscillation and you do not even see an oscillatory response. On the other hand if I choose a large value of h in fact this h is more compatible with the slow mode that is e raise to minus 0.1 t you know the time constant corresponding to the decay time constant corresponding to e raise to minus 0.1 t is 10 seconds. So, it makes sense to choose h is equal to 1 second only if you are interested in the slow response. The interesting part which you see here of course is although the initial part of the transient is not captured very well the system is able to capture the slow transient nonetheless. This would not have been possible with forward Euler method because you would find that the fast transient is getting destabilized that is what we saw in one of those previous simulations yeah this one. So, we could not have of course used the same strategy with forward Euler method even though our main response of interest was the slow transient. So, backward Euler method in fact is a good method to use if you are not too worried about how the fast transient evolves. Of course, we would be worried if the fast transient were actually unstable, but if it is known that it is stable if you from your engineering judgment you are sure that the fast transients are not unstable. In that case if you are interested in the slow transient it is a good idea to use a method like backward Euler method with a large time step which is compatible with the slow transient. Trapezoidal rule with 1 millisecond and a simulation interval of point 1 second seems to capture the fast transient quite well. This is unlike backward Euler which introduces some damping extraneous damping into the original system. However, if I use trapezoidal method with h is equal to 1 although the response is not destabilized you are getting a highly inaccurate response as far as V c is concerned. So, trapezoidal rule is not very very good you cannot use it with very large time steps or time steps corresponding to the slow response if your system is very stiff like the one which we are encountering here. So, what is the solution to these issues? The solution to this problem is use variable step sizes. For example, you can use 1 millisecond for 1 second and 1 second h is 1 second for 30 seconds. So, variable time step sizes can be used in order to obtain a response. So, you can easily program it in psi lamp I will just show you the program. So, if you look at if you look at the program for variable time steps you will have to when you program define 2 time steps h 1 which is 1 millisecond let us say we simulate only for 1 second. Then after some time you can shift over to a large time step and simulate for a longer time and of course, you will have to program it appropriately. So, that you can get the appropriate response now. So, of course, one important point in this whatever simulation we are trying to do is that a choice of h is somewhat ad hoc I mean the choice of h 1 and h 2 is ad hoc and also it is ad hoc that we switch over from the smaller time step to the larger time step at 1 second by 1 second it could have been 10 seconds and so on. The point is in this particular case I do not know the response. So, since I know the response I can actually tell you at what point to switch, but implementing variable time step methods for systems in general may be a bit tricky. I mean if you know nothing about the system then how do you decide that you should switch over at 1 second from the fast so on small value to the larger value or even more importantly how do you choose these values h 1 and h 2 you know here is an ad hoc switch from point 0 1 point rather 1 millisecond to 1 second. The key to this of course is that often when we are simulating system we know something about the system. So, that is one way you can actually use your engineering judgment and come to a particular conclusion about what time steps to use or you can do a bit of trial and error, but most industry grade programs will actually have some way of finding out you know the truncation error estimating the truncation errors at every step and if the errors at every step are not too large then they may even permit adaptively to start increasing the time steps. So, in a you know if you look at a commercial software or software is like MATLAB and other software they do implement variable time step methods and you would have one way of checking out or estimating the truncation errors and adaptively changing the time step. Here of course I do it in one shot I just change from 1 millisecond to 1 second. Now if I use the variable time step method the question is can I use variable time step method with Euler method. So, the question is can I use Euler method with variable step sizes in a step system? The answer is no. The point is that the moment I switch over from a small time step to a large time step the remnants of the fast mode may not be completely zero. In fact they never zero because of numerical precision can never be infinite. So, you will find that there are remnants of the fast response which are there in your. So, when there the fast response would not have completely died down when is rigorously speaking it never completely dries down. So, if I use Euler method the moment I switch over to a larger time step the faster transient even though we have waited for the fast transient to died out after sometime you will find the whatever remnants of the fast transient are there they will again start becoming unstable and you will find that the whole system blows up. So, that is the major issue which you will face when you use methods like Euler method with variables time step. So, Euler method actually is not suitable for variable time steps for a system. So, Euler method is you keep it in the background. So, for example, I actually tried to program and do the simulation for Euler method you see this I do not know whether it is clear on your screen, but it is 2 into 10 raise to 306. So, you know your by trying to implement this variable step method for just a few seconds has resulted in a complete blow up of the solution. So, forward Euler cannot be used in conjunction with the variable time step method. On the other hand backward Euler method if I use variable time step method it gives a reasonably good response for the first few first one second or so it gives a step really it does not capture the fast transient quite well anyway you know it does not capture the fast transient anyway the slow transient it is captured pretty well trapezoidal rule with this kind of thing works well because when you make the time step very small for the first few seconds it captures the fast transient correctly and also the slow transient. So, that is the basic deal you can say whenever you are using a variable time step method. So, these are these are the things you have to keep in mind. So, to summarize this part of the lecture for non-stiff system where stiffness is not there there is less of a worry that the system will become unstable if you choose your time step appropriately. Appropriately in the sense if you know something about the system you can choose a time step roughly to correspond to the fastest time constants in your system. So, if it is a non-stiff system and you know that well this exponential rate of change or the sinusoid is roughly going to be in this range then you can actually choose a time step use any method in fact of course, prefer higher order methods in case you are using explicit method. So, of course, avoid using Euler method is not a very good method to you because it is inaccurate as well. So, what one one can try to do is if you are coming with a non-stiff if you are encountering a non-stiff system and you have some rough idea about the time constants associated with the system or the frequencies of oscillatory response if any then you can choose a time step corresponding to the fastest such transient and you can use a higher order explicit method. Why explicit method? Because explicit methods are easier to implement they do not require inversions and or in a non-linear system they do not require iterations within a time step and so on. So, however, if one is faced with a stiff system. So, you can direct your attention to the screen if only a slow transient is of interest in that case you must look at implicit methods. So, if slow transients are of interest you can try to use backward Euler method methods like backward Euler method with larger time steps. So, if you see the this particular slide what I have written if the slow transients are of interest and the fast transients are known to be stable either from engineering judgment of some prior study with somebody else is done. If you know something about the system and you know that the fast transients are indeed stable and the slow transients are what you are really interested in you can use backward Euler with larger time steps. Larger I mean compatible with the time constants associated with the slow transients. If you have got a stiff system and both fast and slow transients are of interest you can use higher order explicit methods or implicit methods with the small time step. So, if both fast and slow transients are of interest you can use higher order implicit or explicit method with a small time. That small of course corresponds to the fast transients the time constants of the frequencies associated with the fast transients even here even higher order explicit methods may be a problem. So, actually if you are encountering a stiff system I think it is very safe to use implicit methods with small time steps if both fast and slow transients are of interest. It is a bit risky to use explicit method because they do not they do not have very good stability properties. You will have to use extremely small time steps otherwise you may end up destabilizing some response especially true this is especially true with Euler like methods. Of course if this is what you want a fast and slow transients are of interest in a stiff system the best solution or a better solution would be to use a variable time step with in conjunction with backward Euler or trapezoidal. That is initially keep the time steps small so that you capture your fast transient well and then increase your time step and you know you can capture your slow transient even with the larger time step. Now of course the important thing implicit in all what I am trying to say is that we are trying to you know complete our numerical integration as fast as we can. Somebody may ask well you have seen this RLC circuit you know simple RLC circuit what you know to integrate numerically integrate this for say 30 seconds even with a time step of say you know 100 microseconds or 50 microseconds should not be a problem on today's computers. But this is not true when you consider larger order systems you know when you have got very large order systems and if you are forced to use a very small time step like you know 50 microseconds or 100 microseconds and you want to simulate for say 100s of seconds this really may be a big bottleneck and you may take hours sometimes to simulate the system this actually happens. So, if you those who are doing power electronic systems or you know large scale power system simulations would have encountered this problem if they use then inappropriate method. As I mentioned sometime back with the proliferation of so many you know software tools for numerical integration of circuits power systems and other systems control systems and so on it is very important to know this basic know the basic properties of these integration methods. So, although our you know treatment here has been very brief I mean the aim of introducing you to analysis methods right in the beginning of the course is to give you a feel of these kind of methods which we will now apply when we do the modeling of power system components. So, when once you finish our modeling of power system components we will directly use these tools like numerical analysis of Eigen value analysis later on in case you have forgotten what we have covered you can come back to these lectures and just revise. Now, one small point which I did not mention as far as numerical integration is concerned is that if you are faced with a stiff system if you are having a stiff system rather than do all this jugglery of using either very small test step sizes or variable time steps and these worrisome kind of things. One thing you can do right away when you are considering your system when you are modeling your system is to get rid of the fast transient get rid in the sense make modeling simplification. So, that your system is of lower order and it kind of you know kind of does not have the fast transient at all and this is something we have discussed before. If you are encountering a stiff system you can neglect the fast transients what you get because of that is that you are going to get a lower order system the differential equations corresponding to the states associated with the fast transients you know are converted to algebraic equations. So, you know what you are really doing is that the states for example, the inductor current of the capacitor voltage in this particular circuit the differential equations corresponding to the states are converted into algebraic equations. For example, we have done this before but this is as a revision in this particular circuit through participation matrix the participation matrix or through by engineering judgment. You know that the states associated with the fast transients are this and this that is 10 millihenry and 100 micro farad and state associated with the slow transient is this is something we have done before. So, why not use a modeling simplification you know for example, here you have got you can use the modeling simplification that this capacitor is actually open circuit d V c by d t is equal to 0 and d I l 1 by d t is equal to 0 in that case you are going to get effectively a circuit of this kind of lower order is just a single dynamical element or a single state. This will be an acceptable approximation provided you are interested only in the slow transient. So, this is something we have done before the point is that if I have got this system to begin with if I want to numerically integrate it I will have to worry about you know what method I am going to use what is the time step I am going to use the possibility of using variable time steps to speed up your numerical integration and so on. But if you look at this system this is a non-step system actually this particular system I can use Euler forward Euler backward Euler trapezoidal or say Runge-Kutta say fourth order method which I have just mentioned sometime in the lecture previous to the previous one. You can use all of these which say a time step of one second without you will get a reasonably accurate solution because this is not a non this is not a step system at all. So, often what we do is neglect the d i by d t corresponding to this and the d v by d t that is the current through this and get a non-step system. Again just in case you are worried about why am I emphasizing this point remember that if I am going to use implicit explicit methods it is important that the system should not be stiff otherwise you have to you will be constrained to use a very very very small time step and it will take a very long time to complete the simulation. Moreover some of a some explicit methods like Euler method are not even very accurate. So, the important thing is if given a choice a programmer will use explicit methods because it involves less of programming and less computations per time step. But implicit are more stable they do not give they do not show an unstable system to be a rather a stable system to be an unstable one and of course, I have mentioned that unfortunately implicit methods require more computations per time step. So, it does make sense sometimes to use explicit methods, but you should basically make modeling simplifications. So, that the fast or non-stiff or the stiff components of the system or the fast components of the system are effectively removed. So, that is the basic you know thing which modeling you know simplification which one should use if possible wherever possible. So, that you it permits you to use methods some sometimes it permits you to use explicit methods. But if you have no idea about the system you know you cannot make modeling simplification. So, you know if you start off with the system which you have no knowledge if you give for example, a synchronous machine to a mechanical engineer or a civil engineer you may not know all the you know you may not have that engineering judgment of the various transients involved or what transients to expect. So, in that case you may find it very difficult to make modeling simplifications. So, in that case then there is always an issue about which method to use and so on. But an interesting feature about modeling and you know what we will be doing next is that often we would be kind of making assumptions about the system from a general knowledge a general engineering sense about the system. For example, when you are modeling a synchronous machine and the main aim of the modeling is to study for example, loss of synchronism or electromechanical transients associated with the system. We will not be modeling the currents for example, through the inter winding compare capacitances of the stator windings. So, because we have a kind of engineering feel that this stray components like the inter winding distributed capacitances and so on may not the transients associated with them are fast very very very fast. So, unless you are really doing the analysis of ultra fast transients in a synchronous machine you may not need to model them at all. So, to some extent there is this engineering judgment. Now, before we end this particular lecture we just have a few 10 to 15 more minutes to go. Let us quickly summarize the first part of our you know our lectures you know the first part of this course was in fact, analysis of dynamical systems it was a more general analysis. We considered linear time invariant systems and we could really characterize the response in terms of modes. We could even understand the stability of such systems simply by looking at the properties of the A matrix in particular looking at modes could be characterized by the Eigen values and the Eigen vectors associated with the A matrix of the system. Non-linear and linear time variant system they should read as variant non-linear and linear time variant systems are difficult to analyze. Unfortunately, the only tool which is left with us when we are trying to analyze non-linear systems in general is numerical integration. There are of course, some specialized techniques which approximate the behavior of non-linear systems, but most of the times we will be using in fact, numerical integration to analyze non-linear system and an exception to that of course, is when you are having a non-linear system and we are analyzing its behavior for small disturbances around an equilibrium point. We can create or rather derive a linearized model from the non-linear system for the analysis of small disturbances from the equilibrium. Of course, once we linearize the system, we can use the tools of Eigen value and Eigen vector analysis. One of the key systems which we have not really considered right now is the linear time variant systems. I am please note that there is a small error here in the slide. It should read as linear time variant systems. An example of a linear time variant system as we shall see in the coming lecture is the synchronous machine itself. The flux as seen by the stator windings of a synchronous machine that is the rotor winding flux, the fluxes by three phase windings of a synchronous machine are in fact, time variant. The differential equations which come out when analyzing a synchronous machine are in fact, linear time variant and we shall be using a very powerful method or we shall be using a kind of a transformation of variables in order to derive a time invariant system from the time variant system. So, this is something of course, we are yet to come to this is just a kind of a curtain razor to what is to come. So, the modeling of a synchronous machine we shall start off soon and we will be entering in some sense into the domain of power systems slightly away from the kind of general analysis which we have done so far.