 the last lecture on this modeling of electrical machine scores. We looked at the four coil primitive machine where you had the stator, a and beta axis and then you had one coil of the stator on the a axis, another coil of the stator on the beta axis and then you had coils referred from the rotor, one on the a axis, one on the beta axis and we saw how the electrical system equations for this have been developed. We wrote down the electrical system equations and then the torque equation and from this description we saw how various representations for various DC machines can then be arrived at. We have seen the expressions for a separately excited machine and then we saw the description for a series machine, series DC machine and then a compound machine and a machine with inter pole windings, all these different varieties can then be developed from a description similar to this. And then we started looking at how these machine equations can be used for any application and we saw that these machine equations are basically what is called as large signal models in the sense that one can apply a fairly large DC voltage, a fairly large torque on the shaft and we expect that the equations that we have derived enable us to determine how the machine is going to behave. But a large class of applications fall under the range of control system applications where it is not really of use to have large signal representations, we would rather have small signal models which enable us to get a linearized system description which is useful to design a closed loop control system. And we started looking at what the small signal models are, small signal models are obtained from large signal models by certain approximations, one can use the method of small perturbations which are essentially disturbances that you apply to the input signals to the machine and in response to these small disturbances, the output variables of the system also show some disturbances and we can derive a relationship between the disturbances that you apply at the input and the disturbance response that the system so gives and one can approximate the relationship between these disturbances to be a linear system. Since this approach is applied under various application areas, we will see how this can be done for the case of a separately excited DC machine. So if we look at the large signal model for a separately excited DC machine, we had simplified the equations that we developed for this four coil system, we eliminated two of them and we obtained the equations for a separately excited DC machine which can be written as follows. You have in the separately excited DC machine a field excitation system and then the armature VF is applied voltage to the field and VA is the voltage to the armature, this can then be written as RF plus P times LF 0 MSR into ?R and RA plus P times LA, field current and the armature current, this equation we had already derived in the earlier lectures along with that the generated electromagnetic torque PE is then given by MSR into IF into IE. So these equations we already know from this of course one can obtain the steady state expressions the steady state model which you would have studied in earlier lectures on electrical machines in the first course on electrical machines the steady state model is easily obtained by having P equal to 0 because it is under steady state D by DT goes to 0 because all are DC variables and so in the steady state case you have VF equals RF into IF this term goes to 0 because LDI by DT is not there so you have RF into IF and then the armature voltage is nothing but MSR into IF into ? plus RA into IE so this is nothing but your KE into ?R plus RA into IE these are the well known DC machine equations the torque is given by some K multiplied by IF into IE so these are well known. So in order to derive the small signal expressions now let us consider that the we have not actually written the mechanical equation of the system that is not there we will add that now the mechanical equation of the system would then represent JD ?R by DT that is this represents the acceleration of the machine is the result of the generated electromagnetic torque as mentioned here minus the load that is applied this is of two forms one may be the viscous friction drag that is there which is B times D ?R by DT that is B times ?R and then a load torque that may be applied which could have any form so the differential equations of the system are these two equations and this is the next equation the expression for TE is as given here so in this now we shall suppose that the armature voltage field voltage and the load torque these are the external inputs external actions that are impinged or enforced on the machine you apply an armature voltage you apply a field voltage and then some load is also applied on the machine. So in this so you have the load torque here you have the field voltage here and you have the armature voltage here now these VF, VA and TL we will assume that the machine is operating under some steady state so at that value of steady state or at that steady state the machine has an applied field voltage VF and applied armature voltage VA and we shall suppose that the load torque is TL so you have VF, VA and TL now over and above this there is a small disturbance that is now enforced or added so we have a small disturbance ?VA and then a small disturbance ?VF and the disturbance ?TL it is important to understand that we are looking at small disturbances so the disturbances are not large and these are really small very, very small so if this is the case in response to these small variations that you apply on the machine the machine is going to respond the response of the machine is observed as flow of IF some current in the field some current in the armature and the speed these are the responses that the machine gives to the applications of input at these three locations so in response to this then there was an armature current flowing steady state and in response to the disturbance you now have a change in the armature current similarly there was a field current flowing and in response to the input disturbances there is a change in the field current there was a machine speed ?R and in response to the disturbances there is a change in ?R it is also important to understand that it is not that ?VA results in ?IA or ?VF results in ?IF or ?TL results in ?R these responses would be the result of these three disturbances being suitably modified by the machine and therefore ?IA would be the result of all these three being applied similarly ?IF would be the simultaneous result of all these three being applied one cannot isolate it as saying ?IA results only because of ?VA so how do we now go ahead now these equations that we have derived these equations are applicable under all situations these are large signal representations of the machine or any arbitrary signal representations of the machine and therefore these are valid whether they are large signals or whether the signal is small the only thing is this equation is inherently non-linear and therefore we would like to have a linearized representation linearized representations are highly useful when you want to design control systems and in these cases the disturbances applied are generally small one can look at the behavior of the system as a series of small signal disturbances that are added in closed loop so deriving equations that are valid for small signal disturbances becomes useful and we in order to derive these equations we look at applying these to the large signal equations and see if any simplifications can be done because these disturbances are small so let us look at that if you expand these equations and say that VF is now equal to this VF plus the ?VF so we have VF plus ?VF this should then be equal to RF plus P times LF multiplied by IF this IF is now the steady state plus the disturbance response so this is IF plus ?IF and this can be expanded to say that it is RF plus P times LF into IF plus RF plus P times LF into ?IF so I have basically expanded this into two terms similarly one can write for the armature VA plus ?VA please note that we are only expanding whatever equations we have already derived we are substituting in for the general field voltage and armature voltage the specific expressions as steady state plus disturbances that is all that we are doing so this is MSR into ?R plus ?R multiplied by IF plus ?IF plus RA plus P times LA into capital IA plus ?I we are using uppercase letters to determine the steady state expressions and the ? and lower case to represent the disturbance now along with this these are the electrical system equations along with this you need to look at the mechanical equation and there you have J times D by DT of ?R plus ?R is equal to TE is what you have and TE is MSR into IF plus IA and therefore this is MSR into IF plus ?IF multiplied by IA plus ?I this is the develop torque minus B times ?R plus ?R minus TL plus ?TF so these are the system equations this is how the model is going to look when you consider that the applied VF and VA here and the responses are then composed of a steady state part and a disturbance response part now in this then if you look at the steady state part of it you already know that VF equal to RF into IF and in this case P is equal to 0 if you are looking at an AC machine then P would not be equal to 0 some other term would be there so if you are looking at this particular case then you can see that VF is equal to RF into IF PLF is already 0 and therefore this term is equal to VF and therefore one can see ?VF must be equal to RF plus PLF into ?IF now it so happens in this machine therefore ?IF is purely dependent upon ?VF the other variables do not have an effect on the field this is for this particular case now in the next expression we can expand this term further as this term could be written as MSR into ?R into IF plus MSR into ?R into ?IF plus MSR into ?R into IF plus MSR into ?R into ?IF this expansion is for the first term and then you have the expansion for the second term which is RA plus P times LA into IA plus RA plus P times LA into ?I so the orange now gives the expanded version of the right hand side and we now look at the expression for steady state the steady state expression says that armature voltage VA uppercase is equal to MSR into IF into ?R this is nothing but ?R and then RA into IA one can of course add plus P times LA into IA because this in any way is 0 so if you now eliminate this part from this equation you see that VA is equal to MSR into ?R into IF so the first term goes away and then it is RA plus P times LA into IA and therefore this term also goes away the remaining terms if you look at it ?VA therefore has to be equal to MSR into ?R into ?IF MSR ?OmgR into IF plus MSR ?OmgR ?IF and then this term now out of these terms one can see that the first term here and the second term involve ?R into ?IF which is a product of a steady state quantity into the response due to the disturbance into disturbance response so that is the nature of this first term as well as the second term whereas if you look at the next term this is of the nature a disturbance response multiplied by another disturbance response and we have said that we are basically looking at small disturbances these are small disturbances and therefore you expect that the response to those disturbances is also small and therefore you have a small number multiplied by a small number here whereas here you may have a small number multiplied by perhaps a large number and therefore we neglect these kind of terms we neglect these in comparison to these kind of terms and here again there is only one small disturbance so this is fine so we can simplify this expression by neglecting terms that involve disturbances multiplied by other disturbance and therefore this term ?va simplifies to on the RHS this term this term and the last term similarly here in the next equation the left hand side can be written as JD ?R by dt plus JD ?R by dt that is the left hand side and on the right hand side this can be written as MSR into ?IF multiplied by ?IA plus MSR multiplied by ?IF into ?IA plus MSR multiplied by ?IF into ?IA plus MSR multiplied by ?IF into ?IA so these four terms are an expansion of this first term and then you have for the second term B ?R – B ?R and then you have – TL – ?TL now if you look at the first term here under steady state this Te would be MSR into IF into IA and therefore the steady state part here can be removed so the left hand side JD ?R by dt is then equal to MSR into IF into IA – B ?R – TL and therefore the relationship between the disturbance responses would be JD ?R by dt on the left hand side must be equal to the sum of this term which is a steady state multiplied by a disturbance response this is also steady state multiplied by a disturbance response whereas this one is disturbance multiplied by disturbance so small number multiplied by another small number therefore this can be considered to be very small in comparison to the others and therefore that can also be removed. Therefore now what we have is a model for the small signal machine or the small signal model for the machine small signal model for separately excited DC machine would then be ?VF equals RF plus PLF into ?IF and then ?VA equals MSR into ?R into ?IF plus MSR into ?R into IF and then plus RA plus PLA into ?IA and then you have JD by dt of ?R that is equal to MSR into IF into ?IA plus MSR into IA ?IF and then you have –B ?Or – ?T though this is the model this is a model for the small signal response of a separately excited DC machine normally it is useful to recast these expressions in terms of the so-called state variable format the state variable form which is of the form x dot equals ax plus bu where x is a vector of the state variables and then u is a vector of the inputs. So if you recast this the state variables can then be identified in the system as ?IF, ?IA and ?Or inputs in the system are ?VF, ?VA and ?TL and we need to recast this expression in this particular form the first equation is easily done you have p times ?IF that is nothing but x dot which is dx by dt so d by dt of ?IF equals 1 over LF into ?VF – RF into ?IF. So you take the term RF ?IF on the left hand side then you are left with p LF ?IF divide throughout by LF and this is what you get from the next equation we can get an expression for p ?IA as ?VA bring everything to the left hand side. So you have – MSR ?R ?IF – MSR ?IF ?Or – RA ?IA so all the terms are brought down then you have p LAIA alone and so divide this by LA and then you have p ?Or so which is already of that form we just have to divide the equation by j. And so you have MSR into IF into ?IA plus MSR into IA ?IF – B times ?Or – ?TA so this is an intermediate step we need to rearrange these equations in the form of AX with BU that can be done in the following manner the first equation is already arranged there is no further thing to change so 1 over or what we can do is write it in vectorial form directly. So you have ?IF ?IA and ?Or and you have the derivative of that so p of this is equal to you have to multiply it by the same vector so the first term would be – RF by LF and then it does not depend on IA does not depend on ?R so the first row the other two entries would be 0. In the next row you have an expression for p ?IA the first term is the term corresponding to ?IF so we identify the term from ?IF here so it is – MSR ?R divided by LA so this is – MSR ?R divided by LA the second term then corresponds to IA so that comes here so – RA by LA and then the third term would correspond to ?Or that comes here so – MSR into ?IF divided by LA so this is – MSR into ?IF divided by LA and then we go to the third equation the first term is that of ?IF MSR into IA by J the second term corresponds to ?IA that is here so MSR into IF divided by J that comes here MSR into IF divided by J and the third term corresponds to ?R so that is simply B by J so this is your A matrix and multiplies then ?IF, ?IA and ?Or and then you have a B matrix multiplying the input vector so the B matrix can be identified by looking at the those expressions relating to VF, VA and ?TL so the first term is here so you have 1 over LF does not depend on the other inputs as far as the armature is concerned you have 1 over LA does not depend on the other inputs and the last term would correspond to – 1 over J this multiplied by the input vector which is ?VF, ?VA and ?TA so we have now succeeded in recasting these small signal equations in the form of the state variable representation which is in the form X.AX with BU so this vector is X so you have X. this matrix is your A matrix and state vector again here B and U. Now having written it in this form it would be easy for somebody to study the roots of the system one can take then look at the determinant of the system and see where the roots are lying whether the system is stable or the system is unstable all that one can estimate but note that this model is going to depend upon values that you plug in for ?R, ?IA and ?IF that means as the steady state is going to change with different amounts of loading then the response to disturbances which is represented by this system of equations the response to disturbances can also change because it depends upon where the steady state is but nevertheless this is a useful tool in order to study the responses and where do we go from here normally one does not stop at this point one then looks at the transfer functions the system where does one use it now supposing you want to design a closed loop system for a DC machine or which is in other words if you want to look at how to design a DC drive the way the DC drive is normally arranged is that you have the DC machine that is your let us say DC machine the DC machine is going to have a mechanical shaft where you connect your mechanical load and therefore there is a certain speed at which the shaft is going to rotate so let us say that you are looking at an application where you want your machine to run at a fixed speed in spite of disturbances that may occur on the load or on the source in the sense the DC machine has to be supplied with DC voltage in order to run that the level of DC that you are applying may change with respect to time it may be at say some may be at 100 volts when you start the machine as you load the machine and you draw output from the source the source voltage may drop with respect to time but in spite of all that may be you would like to keep your speed of rotation fixed another situation is you have a machine rotating at a certain speed all of a sudden you throw some load on the machine normally the speed is expected to drop speed of the machine will drop and you then want to have a system which will bring the speed back to the normal value so how that is done the only input that you can affect to the system to the DC machine is the armature or the field in many applications where you are not looking at running the DC machine at higher speeds you do not really change the field voltage so let us assume that you are keeping the field voltage fixed and therefore the only thing that you can affect in order to bring the speed back to its original level is to change the applied VA armature voltage is the only one that you can change so how will you then change you will then have to take the speed variable first of all in order to change the armature applied VA to the armature you need to know that the speed has followed otherwise why would you want to change so you need to have some kind of feedback mechanism you need to sense what the speed is doing and you have sense the speed and in order to know that it has gone below your set value above your set value you need to do some kind of comparison with a reference speed normally one calls it as omega r star and therefore if you now subtract your speed from the reference speed you get an error signal and once you get an error signal you know that something is not right an error will come if omega r star is different from omega r which is the situation when there is likely to be some disturbance so if this is different then based on this error you have to take some action and if you are going to take some action then that action will be done you will decide that action based on some algorithm so that algorithm is then put in what is called as a controller and this controller is then going to decide how much armature voltage to apply for that machine in order to overcome the effects of the disturbance and therefore the output from this controller has to in some way or the other affect the armature voltage that is going to be applied to the machine so normally this cannot be done just by this output from the controller you need one more block in between which is normally called as that is the converter so the DC machine then gets its armature voltage through this block you would probably supply this with DC voltage and this is then going to determine in what manner the DC voltage DC supply that is available is going to be conveyed to the DC machine based on what the controller say so this is then a typical close loop system. Now in order to know how your algorithm here should be designed you need to basically have an idea of how the speed is going to change if you change the armature voltage here for example if you change increase the armature voltage by a small amount what is going to happen to the speed only if you know that you will be able to design the algorithm that is going to sit inside that block so how does one do that you now look at these equations that we have derived you need to know how the speed is going to change when you change the delta VA when you change something at the armature so basically you are then looking at the response delta omega as a function of delta VA. So this is normally done in the Laplace domain as you would have seen in the earlier subjects on control systems so you have delta omega s as a function of the Laplace variable s divided by delta VA of s this then gives the transfer function between speed and VA. Now this DC machine system has three inputs delta VA and delta tf so when you are going to change delta VA obviously the response will be different if there is at the same time some change in vf and at the same time some change in delta tl also. So how does one define how the machine output will change for that you impose a restriction here while defining the relationship between delta omega and delta VA that this shall be evaluated with other inputs other inputs equal to 0 so do not give any inputs on the other for the others that means delta vf and delta tl you make it 0 and then consider what is going to be the response between omega r and delta VA. So how does one do that you now have to evaluate what delta omega r is so now it is easy to take the Laplace equivalent of this so you have s times d by dt if you then go over to the Laplace domain d by dt becomes s times so s into delta if and then delta ia delta omega r you must remember that the Laplace transform of the derivative of a function f of t is given by s times the Laplace transform of the function f of s – f at 0 and why are we not including that f of 0 we also suppose while arriving at this expression that initial conditions are 0 note again that when we say initial conditions we mean that the initial value of the disturbance responses are 0 initial value of delta if is 0 it is not that field current itself is 0 there is some field current that would flow under steady state that continues to flow delta if which is the change in the field current that is what is 0 and therefore when you do the Laplace transform of d that is d by dt of the vector it is not necessary to have delta if at 0 so this is nothing but your matrix a multiplied by the state vector and you have delta omega r here and then multiplied by the input vector this is now a function of s you have delta vf delta va and delta tf so one then has to solve this equation for delta omega r and note that in this case delta vf is 0 delta tl is 0 so you now have to solve for delta va delta va will then depend upon all these expressions and you will have delta omega r also in that so one has to solve for that expression we will leave it to you as an exercise to solve this and obtain the relationship between delta omega and delta va you will find that this can of course be written as delta omega r by delta va now as a function of s can be obtained as a numerator as a function of s by a denominator as a function of s and if you now find out the roots of the denominator one can now see where the roots are and whether the system is going to be over damped or under damped oscillatory or not all those can information can be obtained from this that means if you change the armature input voltage by a small amount whether the speed will be an over damped response whether it will oscillate and settle down all that kind of information can be obtained from this expression I will leave you to derive these expressions for now and look at the form of the expression in the next lecture we will stop here for today.