 We are now in the 27th lecture of this course and we are focusing on the modeling of the automatic voltage regulator. Just to you know view things in perspective we have model two kinds of power apparatus in this course. One is the synchronous machine itself and also the excitation system which feeds the field voltage to the synchronous machine. Both of these are power apparatus. The dynamical system is also contributed by the excitation system controllers and the primary function of the excitation system controllers which are in fact feedback control systems which are made by us is to regulate the voltage at a terminal of a synchronous machine. We saw that if you look at the block diagram of a excitation system controller we also saw that we can implement other functions like improving the stability of you know the electromechanical system. This is something we have not shown yet but the in general an excitation system can also provide for this function and also it can change the field voltage if certain limits are hit. For example if the field current limit is hit or later on we will see that if the load angle becomes larger or delta becomes larger of the machine then also you can actually change the field voltage and try to rectify the situation. So in today's class what we will do is consider certain further transfer function blocks. We were discussing transfer function blocks which essentially make up the automatic voltage regulation system in a synchronous generator excitation system. The transfer function blocks we have considered so far are the simple first order transfer function and also wash out circuit. Both these blocks are very essential and important in the discussion of any practical control system. Remember that we are talking of transfer function block diagram because you will find that most of the representation of control systems will be in this form. So if you will open the manual of a synchronous machine in real life you will find that you know the nature of the AVR etcetera is expressed not in terms of state space equations or differential equations but in terms of block diagrams. So we have to interpret the state space relationships or the mathematical functions in terms of what is given to you in a block diagram. So today we will continue that so today's lecture is continuation of our discussion of the automatic voltage regulation. We are really looking at a few transfer function blocks. So the first transfer function block which we will discuss pertains to the regulator. In fact what we did was in the previous class was in fact a first order transfer function block whose block diagram is represented by the figure given at the bottom and the transfer function represent or rather the state space representation of this is given by this. So if somebody gives you 1 upon 1 plus s t you should write it down immediately like this. There is no unique state space representation of a transfer function. Also if you are given a transfer function like this it also means it is a linear system. It is a linear time invariant system. You can actually add some complexities to the transfer function block diagram which make it linear for example non-linear. For example we could have limiters etcetera which we will discuss shortly. The second transfer function which we discussed last time was in fact a washout circuit. This is in fact a system which is represented by the block diagram which is given below. The block diagram is kind of expanded but remember it really the block diagram is a manifestation of the state space equations which are given here on this sheet. You will find that there is a minor difference between the block diagram given previously and the block diagram given here in terms of its complexity but the functions can be in some sense opposite of each other whereas the first order transfer function block here has got a characteristics of a low pass filter. The washout block here has got the characteristics of a high pass filter in the sense that it has got a unity gain for high frequency and a zero gain for low frequencies. So it is essentially used in situations where you want to pass through transients but you want to block any offset or steady state input. Now the state space representation of this block diagram is as given on the sheet here. So you can have a look at that. We derived it in the previous lecture. Again there is no unique representation state space representation of this but this is the most common representation you will find. The step response of this transfer function is like this. In steady state if you give a step here in steady state you will get zero here eventually whereas the transient gain is one. So as soon as the step occurs this also responds. So in this sense this particular transfer function is different from the previous one. Of course these are transfer functions you will encounter but as I mentioned sometime back the regulator transfer function is usually slightly different. It does contain these blocks sometimes but the main regulator function of course is to drive the error between the set value and the actual value to zero. So what are the regulator transfer functions we shall see shortly but before we do that we will just look at one more transfer function which is very important and that is 1 plus s t 1 upon 1 plus s t 2. Now this is another transfer function which you can which you will encounter very often in practice. In fact if t 1 is greater than t 2 the numerator time constant is greater than the denominator time constant then it is known as a phase lead compensator. You will find that if you give a input u which is a sinusoid then in steady state y which is also sinusoid will lead the input u that is if t 1 is greater than t 2. So the block diagrammatic representation is of course given below here. So this particular compensator it could be a lead or a lag compensator depending on the relative values of t 1 and t 2 is a transfer function which is encountered quite often in practice. Now if you look at the how we got this particular block diagram you will just see that the transfer function y of s by u of s is basically 1 upon 1 plus s t 1 upon 1 upon 1 plus s t 2. You can represent this as k 1 plus k 2 upon 1 plus s t 2 and you can easily verify that k 1 is nothing but t 1 by t 2 and k 2 is nothing but t 1 minus t 2 by t 2. This is the first order block which we have seen already. So that is why we get this transfer function which or rather the block diagrammatic representation in terms of the integrator as shown in this figure. Now a thing about the steady state response of this to a step change if you give a step change to this transfer function 1 plus s t 1 upon 1 plus s t 2 what you will get at the output is depending suppose if t 1 is greater than t 2. What is the steady state gain of this transfer function? Well it is very apparent that this transfer function has got a steady state gain of 1 you just put s is equal to 0 here you will get what is known as the steady state gain for a step input. So if you have got a step unit step in that case your steady state value is equal to the output input. So the steady state value will be 1. So this is the steady state value this time and this is y. Now the transient value if t 1 is greater than t 2 you will see that for high frequencies which also define the transfer transient gain you will find that if you put s is equal to j omega and make j omega tend to infinity you will find that the gain of this is t 1 by t 2 which is greater than 1. So what you can expect is if you give a give a step input if I give a step input the output will be like this. So this is 1 this is t 1 by t 2 on the other hand in case t 1 is less than t 2 your output will be like this this is t 1 by t 2. So in this case it is called a lag compensator why it is a lag compensator and why it is why it is a lag compensator when t 1 is less than t 2 is something which you can easily try to find out by looking at the frequency response of this this I leave it to you. Now the state space representation of this is quite easy to find out in fact if you look at the block diagram which is given on your screen it is quite easy to derive this. So what comes before the integrator is your state the derivative of the state. So you can write this as d x by d t is equal to minus 1 by t 2 x minus 1 by t 2 into u. So this is actually the state space the differential equation the output is t 1 minus t 2 by t 2 into x this is minus of it and this is t 1 by t 2 into u. So this is what you get as a state space representation. So just remember that for every transfer function representation you can get a state space representation a state space representation is in in some sense more rich in the sense that it tells you you know the underlying differential equations it also tells you because looking at the Eigen values you can even tell the time response what the time response is going to be of course using the transfer function representation itself in the Laplace domain you can also get the time response depending on the Laplace transform the input. But working with the state space equations and differential equations is more convenient later on when we will be doing numerical integration as well as linearized analysis Eigen analysis of the system it is better to write everything in the state space form. But typically what you will be given in most of your manuals and you know worksheets of your synchronous generator excitation system you will find that it is usually the block diagrams will be in this form using transfer function blocks. Now the reason why I have discussed these three important transfer functions in fact we have not yet gone come to the main thing that is the regulated transfer function usually most of our controllers including regulators, stabilizers, limiters will be made out of transfer functions of this kind. The first one is a simple first order transfer function it is basically a low pass filter kind of characteristic wash out circuits which allow transients through but do not allow the steady state to go through because it has got low gain for low frequencies. This lead lag block on the other hand is something which you can by choosing the appropriate values of t 1 and t 2 get the frequency response of your choice of course you have only 2 degrees of freedom that is t 1 and t 2 here. Some of the obvious blocks which you will which I am not discussed explicitly are the gain block you know you just have a gain a summer a multiplier a multiplier is a non-linear block. So, it cannot be really you know you cannot form an integrated transfer function in case you have got multiplier blocks anywhere here you do not have multiplier blocks in the three transfer functions which have the block diagrams which have shown you here they have got only summers gains and integrators. But you could have under certain circumstances limiters and multipliers coming into the block diagram which make your system non-linear. So, the linear part of the system is usually shown with the block diagrams the transfer function block diagrams and I hope you have got now an idea about what kind of differential equations they represent and in some special cases you know how the they behave as well the three special transfer functions which I have described to you. The regulator itself will be made out of some of these blocks, but the typical structure of a plane regulator it could be containing some of the blocks which I have mentioned sometime back, but a plane regulator a plane p i regulator a proportional integrator regulator has got this kind of transfer function or this kind of representation. So, this is a p i regulator block diagram in which this is the simplest possible regulator in which in fact I would not call this the simplest regulator you can have just a proportional regulator as well the p i regulator here shown here obtains the error between the set point and a measured value multiplies it with or gives it a gain of k p this is by mistake it has been written as k i usually the representation k p k p is a proportional gain you also multiply it by an integral gain k i and then integrate it then sum the output of the integrator as well as the proportional gain and get your output y. So, the transfer function representation of a p i controller is the simplest possible p i controller is this. In fact you can have what are known as p i d derivative controllers as well by adding a derivative block this is u ref. So, this is a p i controller you can also have a p i d controller which in principle is something like this plus k d into s this is the derivative s denotes a derivative. So, this is again the summing junction which sums the reference value and the actual measured value of course, if you look at these trans these components of your block transfer function block there is one point which I must make at this point is that usually it is not possible using causal or a real physically realizable system to make a derivative you can chew on this on what I have just said you cannot physically realize the derivative function using causal systems. So, in such a case a derivative is actually an approximate is approximately realized by using a transfer function of this kind with t very small. So, if you are having transients which are much slower than this t then this behaves almost like a derivative. Similarly, this a plane gain without any dynamics is often not used usually if you have a plane gain just a gain here any noise or distortions in the measured value will get amplified by this gain k. So, usually instead of just a proportional gain you will have a proportional gain with a low pass filter kind of first order transfer function. So, this is what a p i d controller this is a p i d this is a practical p i d controller. Now, I am not really told you why this is a regulator transfer function the reason is simply that you take out a difference between the set point and the actual value and you try to amplify it and change the output. So, why is it a regulator because if you have a for example, in a excitation system you have got you are the voltage regulator the set point is given by you it compares it with the actual voltage suppose you have got a p i controller which has said. So, this error is amplified it is also amplified and integrated in case there is an integral controller and given the output of this is the control signal which is fed to the controlled excite control rectifier of a excitation power apparatus that in turns changes the field voltage of the synchronous machine the synchronous machine field voltage changes the terminal voltage of the machine. So, this is e f d and this is a control signal we will call it v c this is how the system works. So, if you have got an amplifier it tries to change this till this error goes down to 0. Now, of course there is a catch here should error here become 0 in steady state if you use a p i controller the answer is yes look at if you look at the block diagram which is given on a screen note that you have got an integrator at the bottom here this is a integrator. Now, if you have got an integrator what is the job of an integrator well it integrates. So, it integrates whatever appears at its input here which is just after the proportional integral gain. So, the thing is that if you want to reach steady state you should stop integrating. Now, what does that mean if I if you are in steady state it means that the all the variables reach a steady value. Now, if a integrator has got an input which is non zero there is no way you will be in steady state because the integrator keeps on integrating whatever is there the input. So, if you are using a p i regulator if you are using a p i regulator in that case the input to the integrator has to become 0 the input to the integrator is nothing but k i into the error that is u ref minus u measured. So, it follows that in case you are in steady state and you are using a p i regulator then the steady state error is 0. So, you can say that if your system is working well it is stable that is of course not something which I have proved the point is that if you have designed your system well your feedback control system well it is stable then in steady state if you are using a p i controller and a controller which has got an integral component as shown there in that case the steady state error is driven to 0 is it. So, just to do a quick example suppose I have got a p i controller say this has got a gain of 100 this is a proportional controller this is the integral controller suppose this gain is 500 and this is added here you have got an integrator here of the p i controller. So, this is a i channel a p channel this is your input this is u this is u ref this is u measured suppose the system which you are trying to this is just an arbitrary system suppose I have got a system of this kind this is a p i controller which is trying to get the output y equal to u ref. So, this y has to become we wanted to become equal to u ref. So, if you have got a system like this you will find that in steady state this error which appears here this is the error has been driven to 0 in steady state. So, in fact if u ref is a step input from 0 to 1 in steady state this error has to be 0. So, this will be 0 if this has to be 0 and this is a feedback system of this kind y also has to be 1 y has to be 1 if y has to be 1 can you tell me what is the steady state value out here well it is going to be 1 remember that the steady state gain of the transfer function 1 upon 1 plus s t is 1. So, if this is 1 what is the value here see this is 0 the error is 0 if you just multiply anything with 0 of course it is going to be 0 here. So, and this is 1. So, the output of the integrator is 1 the output of the integrator is 1 does it mean that the input has to have a certain value well no this is the value which the integrator has integrated up to the input to the integrator in steady state has to be 0 otherwise the integrator will integrate whatever input comes and change this value. So, this is the steady state values in case you have got step change given to the system of this kind which has got a p i regulator and a thing to be controlled has a transfer function upon 1 upon 1 plus s t. In our case you will have to replace this 1 upon 1 plus s t by the dynamical system corresponding to the excitation power apparatus and a synchronous generator the y is nothing but the terminal voltage. So, this plant is very simple in our system the plant which we are trying to control will consist of the excitation system apparatus as well as the synchronous generator. So, this is just a toy example in actual practice for our systems will have a complicated plant which has to be controlled. So, a regulator is you can say trying to control a plant. Now, so a regulator if you look at it consists typically of a proportional controller or a proportional integral controller or a proportional integral derivative controller. If you have got a proportional controller for example, something simple like this simply again and here is the output this is a proportional controller. So, this is u ref and u remember a regulator is defined as something which is trying to get a measured value equal to the set point value. So, this is all a regulator. Now, this is a proportional controller. Now, a proportional controller to have any non-zero output it has to have a non-zero input. So, it follows that if you are using a proportional controller in that case steady state error between the set point and the thing you want to follow the set point is not 0 not equal to 0 why because in case you want to have this to have any control signal which is going to affect your excitation apparatus then this has to be non-zero. Now, if this gain is very large then to get a certain control signal in order to obtain the voltage you desire or near the voltage you desire in that case the error need not be too large. So, if I use a larger gain in a proportional controller then the steady state error is going to be lower because to get the same value of the excitation required to get a certain value of u you require a smaller value of error. So, in a proportional controller steady state error is not equal to 0, but a high gain proportional controller will have lower corresponding steady state error. Now, as in any control system design it is not guaranteed that your system is going to be stable for any kind of gain. So, you actually have to do a control system design in order to ensure that your system is stable under various situations. In fact, we have I am sure you have done a course on control system design sometime in your previous in the previous years. Now, this particular system which I showed you which is shown here on the sheet is in fact it you can show that it is going to be stable if t is greater than 0 then you can show that this particular system is stable or unity feedback system is stable for any value of P n i. So, we have we can have a system of this kind which is stable you can just verify this that at least if I got just a proportional controller it is easy to show that the system is always stable. So, this is something you can just check out is it proportion is it stable with just a proportional controller is it stable with a proportional integral controller and what are the gains for which it is it is going to be stable in terms of this time constant t. So, this is a separate you know subject of control system design which is related to power system dynamics if you are going to do power system dynamics you should know a bit about a little bit or at least about control system design and stability. Remember that just using a proportional controller or proportional integral controller is sometimes not adequate that in that context you may actually have to use the transfer functions which I have discussed before. For example, instead of a plane proportional integral controller you may not get a stable performance. So, you may have to add a lead block or a lag block in order to improve the performance of the P or P i controller which you will use. So, this is one thing which you may see in a you know a control system that in addition to the proportional or proportional integral controller you also have these blocks which try to improve the response what do I mean by improved response well one of the things you should ensure with your regulator is that if for example, I give a step change this is one way to specify the performance of a regulator if I give a step change how much time does you require to settle down. So, you is actually determined by a fairly complex processes remember that for a AVR this will the output of this is the control signal to the excitation apparatus. The excitation apparatus itself may have very significant dynamics as we see in a brushless excitation system then that determines the field voltage. The field voltage again field voltage change results in a terminal voltage change of the synchronous generator is a fairly complicated way because you will have to actually solve all the differential equations either numerically or if we linearize it around an operating point you can even do a linearize kind of analysis. What I want to say is that eventually the you know automatic voltage regulator is going to be determined by fairly complicated set of dynamical processes. So, eventually a response is going to be something like this it could be something like this. So, for a step change if you look at this for a step change in the input you could have for example, so if this is your voltage reference you actually we could be like this. Now obviously you should design your system. So, that it settles fast if a system settles down fast it also means that the modes which are observable in the voltage are more stable they are more having real parts which are more on the left side of the of the complex plane and as a result of which they decay very fast. You also would like your rise time to be fast you do not want it to rise like this. The best possible response could be something like this you wanted to rise and settle down immediately. So, that kind of response you could want in the case of a excitation system remember that the conditions of the synchronous generator whether it is open circuited on no load or it is at half load or at full load will really change the kind of response you will get the plant of the system which is the excitation exciter power apparatus as well as the synchronous generator and the power system to which it is connected will really determine the response. So, what you need to do when you are trying to design a system is to use not only proportional integral controllers, but you may require to use a lead or a lag block in series with the proportional controllers in order to get some degrees of freedom the degrees of freedom are in fact the time constants t a and t b of the lead or lag compensator you get these degrees of freedom in order to improve the response. So, this is what is very important which you should know. So, if you look at a typical regulator it is not just consisting of proportional controllers typically you will find that it is consisting of v ref v a lead or a lag block usually you know depending on the situation you could use most likely you will use a lag block if you want to achieve some functionality and a p i controller in often you will find it is just a high gain proportional controller integral component is absent in many in many kinds of a v r. So, this is what you will normally see you may sometimes I will just redraw this what you will normally see is this is a summing junction of course, then you have got a lead or a lag block you could have a cascade of these get of another for example, if you require it if your time response has to be further tuned then you have got a p i controller, but usually a high gain p controller suffices. So, it has the system will have a steady state error here, but if it is a high enough gain the steady state error will be negligible. So, this is usually what your a b r is going to look like this is the control system. Now, sometimes it may not be adequate to get to get a very good response just with these blocks. So, in some some a v r's what you will find is that the excitation system is connected to this of course, the synchronous generator. The field current is taken as a feedback the field current is taken as a feedback and you have got this additional block of this kind which is fed into the summing junction here. So, you may have another block like this which takes a feedback of the field current of the synchronous machine in a brushless excitation system we may not be able to get the field current of the main generator, but you may be able to get it of the excitation system alternator. So, in some cases you may take input from this point. Now, this this may be an extra block which may be present in certain excitation system certain a v r's to improve the you know the stability of this control system. So, this is often called an excitation system stabilizer remember it has got a transfer function which is s k f upon 1 plus s t f. So, this is essentially like a wash out block with a slightly different gain of course, the steady state gain of this would be k f. So, this sorry this the transient gain would be k f and the steady state gain is 0. Now, one of the things you should remember though this excitation system stabilizer is being fed into this summing junction it is output in steady state is going to be 0. So, the output in steady state is going to be 0. So, it will not interfere with the regulation function out here. So, in steady state this will be 0. So, we will try to be driven to be v ref by this controller. So, this does not contribute anything at the summing junction if something gets if something non zero is contributed at the summing junction then the regulation function will get compromised, but this is not the case because in steady state the output of this is 0. So, you may find will not actually go into the design of the AVR itself, but you may find blocks like this in addition to the regulator the basic regulator which is a proportional controller. So, this is what our controllers typically look like there is another block which I need to discuss at this point we have already kind of got a flavor of that block before that is the limiter. Now, we have without much spending too much time if you recall this was the kind of you know symbolic representation of a limiter which was given. So, if you have got input u it will get clipped to the value specified here. So, if I specify this value as plus 1 and minus 1 in that case this suppose you have got an output y dash y will not be equal to y dash if y dash exceeds these limits it will just get clipped at that limit this is often called what is known as a soft limiter. In fact, we have used the limiter to model the converter static converter which is used in the excitation power apparatus. Now, this is simply a clipper it simply clips the output. So, if you find you have got something like this this is simply clipping the output which appears here, but remember it does not affect it does not affect the output y dash. So, it allows y dash to get any value you want, but it clips the value of y dash in order to get y. So, this is a soft limiter you can have another class of limiter which is called a hard limiter in order to do that let us take the example of a simple integrator. Suppose, I have got an integrator and I try to limit it this is a soft limit. So, if I got an input u it will be integrated in order to get y dash y will be the clipped value of y dash. Of course, if y dash does not exceed any of the limits specified here for example, plus 1 or minus 1 this is an example of a limit in that case y dash is equal to y. So, only when the limit is exceeded that these limiters coming to play a limiter makes the system non-linear. So, although our transfer function representations etcetera are actually of linear time invariant systems when you have limiters included in the transfer function blocks effectively your system becomes non-linear. So, let us now look at another kind of limiter it is represented in this fashion. The difference between these two limiters this limiter and this limiter is that in case the output y exceeds the limit or tends to exceed the limit the integrator stops integrating. So, just try to chew on this statement the integrator stops integrating in case the limits are exceeded here the integrator does not stop integrate keeps integrating the output simply gets clipped. So, y dash and y need not be equal and the output gets clipped here if y dash exceeds the limits specified here say plus 1 and minus 1 say this could be anything in that case the integrator simply stops limiting in some sense you can say that it starts integrating 0 instead of u. It starts integrating again when there is a chance that the limit the output y can come out of its limits. So, if you for example, if the output is at plus 1 which is the limit specified here then this integrator stops integrating till u starts becoming negative and there is a chance for this y to come out or start coming out of the limit. You can take say an example of a simple system like this. So, if I got an integrator and I am trying to integrate input u say u is cos omega t. So, let us assume that this integration has been going on for a while I will just show it this way suppose u is cos omega t then of course, the output y would be. So, if you have got input as omega into cos omega t and you try to integrate it you will get sin omega t. So, omega into cos omega t suppose is this will eventually get sin omega t is something like this. So, this is your sin omega t this is your output y. Now, suppose I have got a soft limit which is put at plus 0.5 and minus 0.5 in that case the output would be clipped. So, what you will get in fact at y is not this something which is clipped at 0.5 plus 0.5 and minus 0.5. So, this is the response of a soft limiter. So, it just clips the output. So, what you will get at the output is what is in black that is this I will just darken it a bit. So, that you can see it. So, what you see is the integrator does integrate as per its rule. So, the red curve is what you get simply by integrating omega cos omega t, but what you get at the output is a clipped version of this. The integration operation itself is not affected. Now, if you on the other hand try to integrate by a using in the presence of a hard limiter. So, you have this is not very much well drawn, but anyway. So, this is cos. So, if you look at how this particular system behaves you will see that it integrates up to 0.5 this is actually will not meet at 0.5, but anyway will integrate up to 0.5 after that it gets limited. Now, it will go on staying at the limit. So, what happens is that the integrator itself stops integrating it is not that the integration is going on at this point we do not continue with the integration. We stop the integration and the output remains at this value this is exactly as we got before, but whenever the integrand this is u the input becomes negative there is a chance of coming out of a limit because if you integrate a negative quantity then you start decreasing from where you are. So, what will happen is you will come out of the limit right away. So, this is what happens in case you have got a hard limit. In fact, if you have got a soft limit it is this and a hard limit it is this. So, whenever you have got an integrator which is hard limited you will find that it stops integrating as soon as the head limit is it. In fact, this is desirable under certain circumstances. In this earlier circumstances circumstance with a soft limit what happens is the integrator goes on integrating and you may really go on although your output is being clipped the integration here is continuing and you may find it is staying much longer time to come out of its limit even though the input u has got become negative in this case. So, the input u has become negative at this point, but it comes out of the limit only at this point whereas, here with the hard limiter at this instance itself the integration resumes because there is a tendency to reduce the value because u has become negative. So, this is what is known as a limiter. So, you will find it most control systems will be designed with limiter in order to prevent see what happens if you are you know your limit has been reached you do not want to go on making the controller trying to do something anyway it is not getting implemented. So, it is a good idea to put limiters wherever feasible and wherever it is reasonable to do so. So, you will find typically have a limiter here on the hard limiter on the integrator and a soft limiter here which will clip the output as well of course, if you are just an integral controller which is hard limited you do not have to put a soft limiter again, but here since you have a combination of p and i you can have a hard limiter as well as a soft limiter. So, on the whole you will prevent the values from going out of range. So, this is what typically you will find in your AVR controllers. Now, suppose we want to take up now a simulation of automatic voltage regulator we have actually discussed the way control systems are typical control systems associated with the regulator are we have of course, not discussed about limiters etcetera right now we will focus on the regulator itself regulating function itself remember that V ref of the synchronous of this AVR can be modified by limiters and stabilizing functions whenever there is a need to do so. So, for example, if the synchronous machine field current is exceeded you may wish to sacrifice the regulation function, but reduce the field voltage. So, as to reduce the field current. So, if some equipment limit is being hit then the limiter may wish to reduce the field voltage instead of carrying on with the regulation function. So, in some sense the V ref objective the objective of being making V as close to V ref as possible is compromised and you would rather maintain limits. We saw that if you want to improve the dynamics of one or more modes of the system you may actually put in stabilizing functions one of the function we saw as an excitation system stabilizer, but the main characteristic of a stabilizer is that it is output in steady state is 0. Now, we will move on to trying to simulate an automatic voltage regulation system along with AVR. So, if you really look at what is involved you have got the mathematical functions or the state space representation which describe how an AVR works. So, AVR of course, will require you to have a set point which is given it takes the feedback of voltage the terminal voltage of a generator. The excitation system gives the control signal to the excitation power apparatus that gives the field voltage to the synchronous machine. The synchronous machine output is of course, what you measure using a potential transformer which is fed to the AVR. Remember the AVR is not a power apparatus it is a control system. The AVR itself may take feedback signals like the field current. So, this is a typical structure of the system we will what we will do in this the next class is try to analyze a system of this kind. So, what we will the simplest thing we can do is suppose we take a static excitation system. So, if you have got a static exciter as shown in the screen on the screen then the converter model is usually a static one and the only thing which you need to represent is the field voltage limits the limits of the converter itself I am sorry. The field the limits of the converter are determined by the terminal voltage of the generator itself because the power apparatus of the excitation system is fed from the terminal voltage of the synchronous machine itself. So, if you look at this it is fed from the terminal voltage of the synchronous machine. So, the limits are effectively decided by the terminal voltage of the synchronous machine. So, the converter itself will assume that it is a static model it is a simple model in the sense that it implements whatever the control system tells it to do subject to the limit. So, if I want. So, we will kind of have let us say we will just modulate this way this is the terminal voltage of the synchronous machine these are the limits imposed on the converter output by the terminal voltage of the synchronous machine this is the field voltage. Let us assume that this field voltage is in per unit we have defined the per unit system before E F D is one if it results in an open circuit line to line voltage for a star connected synchronous generator at rated speed. So, it develops the rated voltage at the rated speed then we call that voltage as one per unit. So, we will not represented in volts, but in per unit. Now, the control signal which gives this E F D again can be expressed in terms of voltage it is a signal which is given to this converter, but if I say that one per unit of let us define on normalizer control signal in such a way that one per unit here a control signal of one per unit results in one per unit E F D then the gain of the converter becomes one. So, this is also represented in per unit the AVR this is the excitation apparatus the AVR let us assume is a simple proportional controller of this kind this is the error signal this is V ref this is V. So, this is what is your AVR and power excitation system block diagram K a is typically you know it could be say around 200 to 300 or even 400 per unit by per unit the gains are in per unit. So, what I will do is I will also do this sum. So, all these gains etcetera are expressed all in per unit. So, I have already described what this per unit system means as far as the field voltage the control signal the terminal voltage is expressed in per unit on the generator terminal voltage base that is the rated K V of the synchronous machine. So, if I represented this way then K is typically of this value it could be say 200 or 300 or 400 it is usually kept quite high. So, that to get a value say of E F D 1 here you the amount of steady state error required here is very small remember that now this T a is usually very small this is of the order of one cycle that is all say 20 milliseconds. So, this is the block diagram of excitation system with just the regulator included we could have included many more possibilities. In fact, in this course we will not do that will include a simple excitation system model using a static exciter and this will use to study the voltage regulation of a synchronous machine. What I will do is now incorporate these the equations which are you know embodied in this transfer function or these limiter and summer blocks write them as state space equations interface them with the synchronous generator equations connect the synchronous machine to another voltage source and try to regulate the terminal voltage of the synchronous machine. We will do one extra thing we will not connect the synchronous machine directly to a voltage source we will connect it via a model of a transmission line a very simple model of a transmission line. Modeling of a transmission line is something we will do in brief later in this course we will just take a simple model when we are studying this. So, with this we will come to the end of this lecture in the next lecture stay on for the incorporation of an AVR into the synchronous machine equations and the simulation of the voltage regulation action.