 So let us continue with the machine equations that we derived in the last lecture, we had derived expressions for a poor coil primitive machine, poor coil primitive one coil on the alpha axis of the stator, another coil on the beta axis of the stator and then we had fictitious coils, you had fictitious coil from the rotor being represented on the alpha axis of the stator and then the beta coil on the stator. For this we wrote down the system description in the operational impedance form, so let us write that again what we have is V alpha V beta V alpha again but that of the rotor V beta again but that of the rotor, so this forms the voltage vector this is equal to the first term is the self impedance term, the mutual impedance between the alpha and beta coils of the stator is 0, the mutual impedance between the two coils on the alpha axis and then 0 again. So here again 0 you have RS plus LS bar P 0 and then MSR bar P, so these two rows are expressions for coils that are real and fixed in space, these two equations are going to be for fictitious coils which were actually rotating but now you have considered them in the stator attached reference frame therefore they are pseudo stationary fictitious coils and therefore you will have speed EMF terms, this one is however EMF term that depends on DI by DT that is the transformer EMF and here you have the speed EMF term and this is P and then here again you have speed EMF term, you will have speed EMF terms in this equation also but then here the speed EMF terms were found to be negative, a transformer EMF here, another speed EMF here, this forms the machine description, we had written down the expression for the generated electromagnetic torque, TE was found to be MSR bar multiplied by I beta into IRalpha minus I alpha into IRb, so that is then the expression for the generated electromagnetic torque, this is as we said a four coil machine. Now let us take a look at DC machine, in the DC machine we have let us take the case of a two pole DC machine, what we have is a field winding on the field poles that is the pole structure and then you have the armature which is going to set here, so you have field windings here and therefore one may draw a set of axis, you can call this as your alpha axis and that as the beta axis and one can see that the field windings are now going to be producing an MMF along the beta axis just like what you have here, the field windings are fixed to the stator in the DC machine and therefore they are fixed in space, they are not rotating and these coils are fixed in space, they are not rotating, if you look at the armature in the DC machine, the armature consists of several conductors, the armature in the DC machine armature winding is a closed winding, it is closed upon itself and because of the placement of brushes and the arrangement of the brush plus commutator, brush plus commutator arrangement enables the armature to behave like two parts, the input current flowing in from here is going to split and then on one half of the DC machine, if you say that the flow of current is let us say outwards then in the other half it would be inwards, as the rotor is going to rotate, the way in which the armature distribution is going to happen remains the same, that means all conductors having reached this half as the rotor rotates will always have flow of current that are outwards, all conductors as they reach this half will have flow of current that are inwards, if it is outwards here, if it is inwards here then it is outward here, so this distribution in spite of the rotation of the rotor remains the same, you find that the armature is now going to be rotating but nevertheless because of this arrangement of having brushes here to the external world that is fixed with respect to the stator to the external world the armature looks like a fixed coil, always the armature is going to have in these parts flow of current is outside, in these parts the flow of current is inside and therefore in spite of the rotation of the armature from the outside world it looks like a fixed arrangement, which in other words means that the armature looks like a pseudo stationary coil, this is what we have been looking at in the earlier case also, the armature looks as if it is fixed but actually it is now moving and it is fixed along what axis the armature MMF because there is some flow of current in the armature it has to be generating an MMF and along what axis does it have the MMF that is going to depend upon where you are going to place this brush, if you place the brush here then the armature MMF acts along this axis, if you alter the axis of the brush then appropriately it acts along that axis, normally these are then located on the magnetic neutral axis which means that it is an axis that is going to be at 90 degrees to the field axis, which therefore means we again revert back to the same situation as we have seen here, you have the stator which is having the field winding, so this now becomes the field winding of the DC machine. The armature of the DC machine which is actually there on the rotating member of the DC machine because of the way the brush commutator arrangement is placed it looks like a pseudo stationary fixed MMF and this therefore would represent the armature of the DC machine, so we can see that this description can be used to describe a DC machine, in the normal DC machine there are only this fixed winding which is the field and the pseudo stationary winding which is armature and therefore if you want to reduce this description to model a DC machine all that we need to do we do not have V a, this is non-existent in the normal DC machine, so we strike off this equation it is not necessary and because this equation is not there I a is also not there we do not have this coil at all since I a is not there this also goes away we do not have this and looking at this axis we find that the field winding is there we do not have a pseudo stationary winding on this axis and therefore this equation is not necessary and because this coil is not there I a is also not there so we strike off this column also and looking at this expression we find that I a is non-existent therefore this term is not there and therefore the remaining set of equations that is this one and this one along with this term is representing the DC machine, so let us have that part alone and we will now put it down in the standard way in which the DC machine expressions are written V ? is nothing but the field of the DC machine we will call that as Vf and then Vr a vr a this is nothing but the armature of the DC machine, so we will call that as V a this is then given by you have for the field the self-term Rs plus Lsp and since this is the field we will call it now as Rf plus Lf p and the second term is 0 there is no mutual inductance term the next term in the armature is the speed mf term msr into ?r so we will call that as mf a between the field and armature multiplied by ?r and then you have Ra plus L a p multiplied by If and Ia as the vector, so this in essence is the electrical system equation for the DC machine the mechanical torque the generated electromechanical torque is nothing but msr which is mf a multiplied by If multiplied by Ia, so this term becomes If and this term is Ia and therefore there we have, so this expression then models a separately excited DC machine. If we want to reduce it to steady state equations in your earlier electrical machine scores you would have seen the steady state equations for the machine where the flow of current is DC in those situations you do not really look at how the machine is going to accelerate or decelerate we simply say that given applied an armature Va the machine rotates at the speed, so we are looking at the machine having reach steady state, steady state in the case of the DC machine means that there is no change in the DC value and therefore p of If which is nothing but d by dt of If is 0, the armature is also going to have DC and therefore p of Ia is 0, so if we use this then what you have a steady state Vf is equal to Rf multiplied by If put that capital to denote DC and then the armature voltage is nothing but Mfa into If into ?r plus Ra into I and torque is nothing but Mfa into If into I, so these are the equations that you would have seen even earlier as equations for the separately excited DC machine. This term would then be called as the speed Emf term and this term is the torque constant and this would be the speed constant or rather the Emf constant, so we can see how one can derive the model for a DC machine starting from the same set of equations that we had. Now this is an expression for the separately excited DC machine one can of course derive the expression for a series DC machine if it is a series machine both the field and armature are going to be connected in series and therefore you have applied voltage V is nothing but Vf plus Va and then you have the armature current the net current flowing is equal to Ia which is the same as If, so all that we need to do is then add these two equations you then have this is Rf plus Lfp multiplied by I, If and Ia are the same I that is there that is going to flow this plus Ra plus La into p multiplied by I plus Mfa into ?r multiplied by I, so this can then be written as Rf plus Ra plus Lf plus La p multiplied by I plus Mf into ?r into I and then the expression for torque will then be Mfa into I square because both If and Ia are the same flow of current, so this then becomes the expression for the series machine one can look at other machines as well if you look at a compound DC machine in this case we have let us we do not need this figure in this case this machine has two field windings one is the main field winding and another is a weak field winding this weak field is normally connected in series and main field is normally in shunt, so it is an arrangement like you have the a axis and the ? axis you have a main field winding let us call this as f main and then you have another field winding this field is also located on the stator therefore it is not a pseudo stationary coil it is a really stationary coil let us call that as f series and then you have the armature which is pseudo stationary and present on the a axis this is your armature, so how does one derive a model for this kind of a system we do not really have this kind of system here in this case you see that there is one fixed coil one pseudo stationary coil on each axis whereas here we have two fixed coils and one pseudo stationary coil pseudo stationary coil is on a different axis, so how to write down the equation for this we can write down the equation for this machine by understanding the different by understanding the way in which these the original set of equations arose as we described yesterday we found how we wrote each one of these different terms we did that in the beginning of this lecture also, so we will use the same idea now there are these three coils, so we have VFS, VF main V field series and then we armature these are three voltages that you would be applying to these three coils, now the first term for VF main would be the self impedance and therefore this is RF main plus LF main V and then the second term would then be the mutual term between these two, so you have mutual we will call that as mutual between main and series field times P and then you need to look at how this is going to be affected by this current and this is a stationary coil therefore there is no speed EMF and this coil is at an axis 90 degrees to this there is no DI by DT MMF and therefore EMF and therefore this term is simply 0 if you look at the second row equation for VFS, VFS would have similarly M main and series field times P and then you have RFS plus LFS P and 0 again because this coil is at 90 degrees to this, no speed EMF because this is fixed and DI by DT is not there because it is at 90 degrees and then you write the equation for VA, now VA this is a fictitious coil and therefore it would have speed EMF due to those that are here and there are these two coils on this axis therefore there will be speed EMF terms due to both of them, so MMA into ?R and then MSA into ?R and then you have RR plus RA plus LA P and then you have IFM, IFS and so this is then the electrical system equation for this machine and then if you are going to look at the machine being operated like this normally what happens is you have the DC machine armature and then you have the series field this may be connected here and then this is the main field and here you have the series field this is your armature if this is the case then you can impose restrictions that arise from the way this is interconnected on these equations this means that IFS is equal to IA and the applied voltage V is equal to VFS plus VA and therefore you can get two equations one for this main field one for this entire set therefore we can say that V is the total applied voltage let us call this applied voltage as V then you have V and VA let us call VFM and V this is nothing but VFM is one the same as this I am going to write this in terms of this matrix this vector VFM, VFS and VA so VFM is the same as this VFM and this voltage V is nothing but VFS plus VA so it is 0, 1, 1 so this equation has now been written in this form similarly if you have that vector IFM, IFS and IA this can now be written in terms of a vector comprising of IFM and I where this I is the total current that is flowing into the armature circuit so IFM is this IFS is nothing but I and IA is also I so using these two equations one can now try to simplify this in order to represent this machine interconnection so how do we get that all that we need to do is in order to get VFM and V we need to multiply this vector by this one which is the same as multiplying this matrix on the left hand side by this and therefore you have VFM and V is equal to 1, 0, 0, 0, 1, 1 times that equation RFM plus LFM into P, MMS into P, 0 series field and then you have the speed EMF terms main field and armature, series field and armature and then the armature resistance and then you have this vector that vector in terms of the net input currents that are flowing that is one is the armature circuit current and field current that expression we have already derived so instead of this vector you substitute this term on the right hand side which means that you now have 1, 0, 0, 1, 0, 1 multiplied by IFM and IA or I so if you now do this operation this matrix multiplied by this multiplied by that you get the resultant equation for the system as interconnected in this manner and that expression would be so let us finish this operation you have VFM and V so let us do this matrix multiplied by this first which means this first term is RFM plus LFM into P remaining terms are 0 and then you have MMS into P the last term has come here you have MMS plus MMA into P and then you have RFS plus LFS into P plus MSA into ?R that is the second term and then you have 0 and RA plus LA into P this multiplied by 1, 0, 0, 1, 0, 1 multiplied by this vector IFM and I so simplifying this again what we have is RFM plus LFM into P the remaining terms are 0 and then you have here MMS into P then you have MMS plus MMA into P and then you have RFS plus RA plus LFS plus LA into P plus MSA into ?R the whole thing multiplied by IFM and I so this would then be the electrical equation for the machine interconnected in this manner. Now if there was some other form of interconnection this method of interconnection incidentally you would recollect is called as a long shunt one can interconnect in other ways you can have a short shunt connection and accordingly one can modify these equations. So this is a way in which one can derive expressions for the electrical system equation for other types of DC machine as well now the DC machine descriptions do not just stop here now apart from these varieties of DC machines there are other artifacts that are introduced in the DC machine there are inter pole windings that are sometimes introduced there are also compensating windings that are introduced the inter pole and this they are both designed so as to nullify the effects of armature MMF and therefore they are placed on the stator but and along the same axis as that of the armature MMF and designed to oppose the armature MMF that means therefore in machines that have this you have another equivalent winding here which is now going to oppose the armature they are connected in series with the armature therefore if this is going to be the armature the armature current flowing into this let us say you have a dot point here and then this coil has a dot point here since this MMF has to oppose the armature MMF but it has to be connected in series what one would do is take this here and this winding is now connected here this is brought out so this is the armature interconnection now a similar arrangement occurs in this case also in the case of these machines in the case of compound machines you have two varieties one is called as cumulative compounding another is the differential compounding this would mean that if you have the main field here and the armature field here we know that these two are going to be connected in series so the issue is whether the main field is going to I mean the this field winding is going to oppose the main field winding or aid the main field winding so in that case so you have the main field here and you have dot point here this is what we had so we need to now do an interconnection of this winding with this winding the way we had achieved it here is we have said implicitly in the equations that we have derived we have said that the net voltage V is the sum of this voltage plus this voltage that means we are saying that all these are aiding each other and therefore we have implicitly written it for this case now if you want a differential compounding machine this armature is going to flow here and the flow of the armature current has to now be connected here and taken out this way so that whatever armature current flows demagnetizes the main field what we had implicitly done was connected this way so the armature voltage was applied between these two terms so accordingly we have written it if it was the other way around then you would have had a situation where Ia is equal to minus of Ifs then with that if we derive the expressions you would get the equation for the differential compounded case we had instead said Ia is equal to Is and therefore you have the case of the other variety so in this manner then one can derive expressions for all the varieties of DC machines including those which have inter poles or this one both these are meant to oppose the armature MMF and therefore they are present along the same axis so both of these would be represented by a coil here if both are individually there then you would need these two coils here in order to represent the machine. So one can see that this description that we have derived though we started out from the case of an induction machine we can use this description to arrive at models for the DC machine as well these DC machine expressions we have not a priori made any assumptions on what is the nature of the voltage that you are applying here they may be steady DC they may be varying DC they may be DC that is switched on switched off in any manner that you may apply these equations still hold and the mechanical expressions we have not again made any assumption on what kind of loading is there or anything like that so these are fairly general equations applicable for any sort of study on the DC machine behavior these are known as large signal equations large signal model because one can use this to study the response let us say for example you have a DC machine and let us say this is the machine that you have and this is then connected to some load through the shaft and what we want to see is that if you are going to switch on a DC voltage directly to this all of a sudden you switch on a DC voltage how does this entire system behave how does the armature current flow how does the field current flow how the speed starts increasing from the time you switch on the supply depending on the load there may be load that is applied there may not be load that is applied may be after analysis you would find that the speed increases in this manner in some manner and settles down and may be you had studied the acceleration when there was no load applied after the speed having settled down may be you throw upon a sudden load on to the system then how does the speed behave may be there will be a speed dip and then it may come back and settle at some other value may be the same value as earlier may be different value how do these systems behave that can be analyzed by looking at all these equations this is a fairly large voltage that has been applied all of a sudden so these are called as large signal model in many applications however large signal models are not really necessary especially when you are looking at a system being controlled by a closed loop kind of operation you have let us say let us consider a separately excited DC machine simple case then you have the field winding that is there now any time you want to do a closed loop control of the DC machine for the purpose of regulating speed or may be the angular location of the shaft you would use some form of converter may be an SCR bridge which may then be given an AC supply so this is an AC to DC converter that is going to give supply to the DC machine in this case you would then let us say you want to look at an application where you regulate the speed of the DC machine then probably you would sense the speed and then based on comparison with certain reference you want to regulate the speed at some value you derive an error signal and then depending on what this error is you would then appropriately control this converter so that the speed remains regulate. Now in this kind of a system if you want to design this controller it is not really necessary to look at the large signal equations many times designing the controller based on small signal model so called small signal models works well enough and another advantage of this small signal model is that one can use this to get a linear linearized description of this machine. Now if you look at the equations for the DC machine you will find that they are not really linear equations why they are not linear because you have this term I ? which is the field excitation which is something that may change in a general DC machine if you are going to put a controller here as well it could change and then that is going to be multiplied by this ?r you see this equation this is the armature voltage equal to this pdmf term multiplied by this I ? so this becomes this makes the system a non-linear system on the other hand if you say that the field current is going to be maintained same irrespective of what you do to the armature then this becomes a fixed variable at this becomes a fixed number and therefore this is no longer a system that is non-linear but if you consider all these things put together then yes it becomes a non-linear system. So if you have a non-linear system in order to do control on this it is usual to consider a linearized model and what we do to get a linearized model is to apply or look at the method of small perturbations the idea is that you have a non-linear system that is working about which is operating and it is in steady state let us say for example in this DC machine you would have applied an armature voltage va that is a steady DC you would have applied a field voltage that is vf that is again a steady DC and the system is operating with a certain load torque that is a fixed load torque under this situation suppose there is a small disturbance that is applied to this va such that the armature voltage va becomes the steady state value which was there that is va this va plus another small disturbance that is delta va what kind of disturbance it is it could be anything it could be a simple small step that has been applied it could be a sinusoidal vary sinusoidally varying disturbance that is applied on the va anything but it is a small disturbance it is much smaller much much smaller compared to va in response to this kind of a disturbance the system will have to do something the amount of armature va that it draws it was having a flow of current va in response to the small disturbance there would be a small disturbance in va as well so there would be some disturbance current that is flowing delta va because this applied voltage this applied disturbance is small you expect that this is also small in comparison with a now this is only one way in which a disturbance could be applied you could have a disturbance in the field voltage or you could have a disturbance in the load torque on the load there may be some disturbance that is applied so the disturbance response that is going to arise delta ia may be a result of disturbance applied at any of these inputs so if you could say that va is disturbed by delta va vf was operating at vf it could experience a small disturbance delta vf the load torque tl was operating at some tl0 and then undergoes a delta tl variation in response to all this not only does the armature flow of current change the speed at which the rotor was rotating that would also undergo a change from some omega r and respond with some delta omega note that when we say that there is a response to the disturbance input that was applied we do not really mean that delta ia is a fixed number that is for example if the initial flow of current was 5 ampere and this delta va is let us say 0.1 into u of t which means it is a step of 0.1 level this does not mean that ia is going to go from I mean ia would be 0.1 into u of t or 0.1 by r into u of t that is not true ia will have some waveform in response to this disturbance that is applied it was flowing at 5 ampere now you have applied a small disturbance to the applied armature voltage in response to that this current may start doing something like this and finally settle down somewhere so we are looking at this entire variation as delta ia it is not just the value at which it is going to settle down so delta ia is really a function of time similarly delta omega r is a function of time it may be oscillating and settle down may be small oscillation may be slightly larger in any case because these disturbances we are considering to be very very small we expect that the responses are also very very small. So under this situation with small disturbances that are applied and responses also being small let us try in the next lecture to see how these equations may be simplified and derive an expression that is the so called small signal model of the machine which would then be useful for various applications such as closed loop control as we discussed a little earlier we will stop with this for today.