 So, welcome to 36 lecture. In this lecture we will see current fed couple circuit field analysis. In many of our problems in electrical machines and equipment, we have only current fed system. So, typically you know what happens most of our devices are voltage fed, but we simplify the analysis by making it as a current fed FE analysis, because current is generally known to us in the process of analysis we know already some current is flowing and then we analyze for the thing. So, generally we should remember that all our devices are typically voltage fed, but for analysis purpose we take it you know current fed cases for understanding in depth behavior of you know field distribution and the corresponding performance parameter. More about it when we see actual examples. So, now for a massive conductor when I say massive conductor that means we are actually considering eddy current in that conductor right. So, that means eddy current effects are considered and that is why this sigma is being considered and then a by dava t is being considered, because this is representing the induced voltage, induced electric field intensity and the corresponding induced voltage and eddy current. So, now we start with del cross h is equal to j our you know the Maxwell's equation with displacement current density being neglected, because we are in the low frequency you know region. So, del cross h is equal to j, j is replaced by sigma e and e is minus del b minus dava a by dava t. Now, here remember this del b is not on account of charge accumulation, but is on account of the voltage that is impressed across the terminals of the coil right. So, now here this you know going further this h is b upon mu and b is replaced by del cross h, this becomes del cross h becomes this is equal to sigma e and we invoke here Coulomb gauge, which we have seen in basics of electromagnetic is the Coulomb gauge divergence a equal to 0, divergence a in general is equal to minus mu epsilon dava v by dava t, which is called as Lorentz gauge, but since frequency is small you get divergence a equal to 0 and why divergence a equal to 0 has to be invoked because del cross of del cross of a is basically del divergence a minus del square a that is the vector identity and that is why del divergence a when it is made equal to 0 you are left with only minus 1 over mu del square a is equal to sigma e and now in place of sigma e we replace this e expression here and then we get this. Now, here one thing you should note I think probably I mentioned earlier also now this is representing the source current density and this is representing induced current density. So, going further now the functional for the above equation in the frequency domain is now here again we have for this term you get the well known expression we have seen it in the previous lecture also and this will give you the global coefficient first the global first the element coefficient matrix and then by combining all element coefficient matrices you will get the global coefficient matrix. Then you have this for the second term this sigma daba a by daba t daba by daba t will be replaced by j omega so sigma j omega a so sigma omega and then there that a become a square in the functional we have seen earlier when the functional form what we did we see take the other term except this on the right hand side the sign of that basically get reflected here in the functional term and here you have a 1 a is there so that becomes a square. This you have seen when we studied functional and then this half comes right because there is a square term here that is a half comes here so this a becomes a square and j omega sigma as it is right and then this del V is now del V is basically minus V by L so minus of del V is equal to V by L because see gradient of voltage is from low to high and is from high to low is it not that is the reason that del V will be minus of V by L because V by L is basically from like when you say V by L it is from positive to negative del V will be equal to minus V by L right. Effectively it is like E is nothing but V by L because E is from positive to negative so E is V by L so minus del so V by L has to be minus del V then that is substituted here so del V becomes minus V by L that is why you have minus sign and again here there is no a but then in the functional thing this whole sigma V by L will get multiplied by a right in the functional expression with minus sign as I just explained. I have already explained you this comes for this now the second term j omega sigma as it is here this a square becomes this after the FEM discrete direction right because a is replaced by summation i goes from 1 to 3 a ni aie and then further simplification you will get this with two summation term because there is a square term is it not and then this half is this half and then this for this third term sigma V by L is anyway constant a is replaced by summation ni aie i goes from 1 to 3 summation and then summed over all the elements so going further now we have this capital B E at the element level you will get sigma delta by 3 how this term is coming because basically you have sigma V by L here now L if you are doing 2D approximation L is equal to 1 right so it basically is basically it is just sigma V right and then V is variable unknown because here it is current rate system the voltage is unknown so here V is variable so now here only this sigma term is there L is 1 and this we know that this integral n reduces to delta by 3 and this aie in the process of minimization and this aie will go is it not so it is only summation n here summation integral n is it not and integral n dx dy is simply delta e by 3 and this aie will go in the process of minimization and here it is only sigma V because L is equal to 1 per meter depth we are doing so there is one typo here this a it should be ni here so this V is that is why sigma sigma by 3 into delta e and L is 1 right and that V V by L so L is 1 and then this V goes in this matrix when we form the global coefficient matrix so now this is the global coefficient global level matrix equation so that V e at the element level here now here this V becomes this V now you have to remember that all the elements will be associated with the corresponding one voltage the corresponding voltage will be one only one V voltage right so that is why here this V comes this is our normal global coefficient matrix into the corresponding a vector which is column vector of all the unknown magnetic vector potential minus B into V which I explained just now and this is j omega da which is same as what we had seen earlier in diffusion equation solution j omega da so B basically is on account of this sigma this term is it not because this is the corresponding diffusion term in the original partial differential equation so that actually basically will lead to this j omega da this we have seen it earlier in when we have we solved diffusion equation we in fact had seen this j omega da so this is our global matrix equation and D at the element level is this before lecture 27 slide 6 we had exactly the same thing so this is this is D e and this is D global by appending all element level D matrices we will get this global D matrix as we have done earlier so current in the massive conductor is given by this now it is like this so there is a suppose there is a coil here which is fed by some oil now this coil conductor can be a massive conductor so this is the massive conductor through which this current is fed right now this current when it enters it will have not only this you know contribution from this source current but there will be there are going to be eddy current in this conductor massive conductor because we are considering the conductivity and there will be eddy current in this so the current inside this massive conductor will be superimposition of this source current and the corresponding eddy current induced right but the total current will remain same because KCL has to be satisfied with the total current here i current that you will get at any cross section here will also be i which is integral j ds at any cross section if you integrate whether it is this this or at the end the current will be still the terminal current okay but that terminal current now is a is due to the total current here which is combination of these two term one account of the source and one account of the induced effects right so now this is so integral j.ds is and j we are replacing by this right so this is the corresponding source current sigma V by L now what is L L is this length and what is V is the terminal voltage V is it not so V by L so V by L is voltage per minute length and remember this V is unknown the terminal voltage is unknown what is known here is the current that is fed in the previous case in the previous lecture what was the case some known voltage was applied to one of the winding and other winding was started and current was unknown in both the winding here it is a massive conductor in which eddy currents are induced is fed by a current source and terminal voltage is unknown right and that terminal voltage is represented by this capital V okay so now going further you have this we are just replacing we are just separating out the term and then we are replacing a by summation i goes from 1 2 3 ni ai e our usual FEM discretization procedure right and then here then we will get del V you substitute as V by L minus del V is V by L this becomes sigma V by L integral dx V by is simply the area of the triangle because this sigma V by L is constant right then you have sigma j omega sigma j omega delta by 3 again this ai this ai is now unknown that comes here as this column vector and then this delta by 3 sigma delta by 3 because this integral ni sorry integral ni dx dy is nothing but delta by 3 and then you have this in matrix notation in frequency domain you have you can write this equation where in now this g we have seen in the previous lecture sigma delta by 3 we are denoting by the same notation and again here since ai is column vector here at the element level this is taken as transpose this becomes rho rho into column so this gives you 1 by 1 it is 1 by 3 into 3 by 1 will be 1 by 1 which matches with the right hand side matrix dimension and now this T is at the element level is sigma sigma delta by L and this V goes in this variable matrix right and then we then we have this global level equation so this T becomes global here when you turn it to global what will change is only this was at the element level so this should be this is E here delta E when you take it to global level this will become sigma summation delta because it will all the element areas will get added that is the meaning of taking it to the global level so sigma summation delta E by L right and if L is equal to 1 then of course this becomes 1 so this is how you are this equation so this T in this equation will be this V is voltage which is unknown at the terminals and j omega G now this G becomes global by combining all these element level G matrices and A E becomes A global and I is still same I because I is only one value this O square this remains 1 by 1 so now we can write both the equations one for the circuit and other for field so the field equation is this and the circuit equation is this right so same two equations are written here field equation and is a circuit equation but you have to remember that in this coupled system of equation there are coupling parameter so in this circuit equation what is the coupling parameter A is it not so this is in the circuit equation this is representing the field whereas in this field domain equation what is representing the circuit D right these are the coupling parameter then only so these are not independent field and independent circuit equation because we are solving coupled circuit field there have to be coupling quantities between two equations so these are the coupling quantities here it is this in this case it is this A so A and V previously when in the previous lecture what were the coupling quantities A and I right and then you know we had two winding there so I basically what I1 and I2 so I1 I2 and A where the coupling quantities in voltage field is that we have seen earlier right so here V and A are the coupling quantities and that is what is the unknown right so now the global combined equations are this same just are written with one modification we are combining these two to one like we did in the diffusion case this become now we are calling as C global coefficient matrix complex right is basically combination of this matrix and this matrix so C subscript small c is capital C plus j omega c you can refer lecture 27 slide so this is the global system of equations now here clearly this the matrix dimensions are marked these n by n is n by 1 so this you know this is also n by 1 this is 1 by 1 these matrix together will be n in n plus 1 into 1 n plus 1 into 1 so this is also n plus 1 n plus n plus 1 into 1 right similarly this will be n plus 1 into n plus 1 because this is n into n so we have added 1 row and 1 column so this becomes n plus 1 into n plus 1 right and then when we solve so what is known here I is known and you know you can get the whole right hand side this is known this is all known right and because t is this is function of sigma this is just delta by 3 is it not g is simply sigma delta by 3 right so all these quantities are known this is only this is function of only geometry and material property right so this is also known this is our usual global coefficient matrix so all these are known so you take inverse and then you will get these unknown quantities magnetic vector potential values in whole field domain and voltage across this across this term okay so now let us see some examples so now this is skin effect of a current carrying conductor right so exactly the same geometry suppose this is a conductor and we want to find out the skin effect with current fade at some frequency right now here the question comes generally the voltage we have voltage sources feeding the conductor so if suppose you want to actually analyze these as a voltage fade if current is not known then you have to go back to the previous formulation and then excite this conductor by a voltage source and then voltage will be known and this current will be unknown right but you will get the same result but here generally what we do we simplify the analysis because we want to understand when some current is fed into a massive conductor with some frequency of excitation how the currents get redistributed due to skin effects that is why we generally feed for analysis purpose we feed into the conductor some current as frequency because current fade systems are easier to analyze as compared to voltage fade systems and that is why we have this current fade FEM formulation okay so now this these are usual of a skin effect which we have obtained by using the code developed in the formulation described you can clearly see the skin effect current is trying to be at the more at the surface and it is you know less current is less inside the inner part of the conductor so these you know can be simply understood by you know I think I previously also I might have explained that if you can analyze this conductor by as made up of small small annular conductors you will find that the inside elements of the this whole conductor are linking more flux as compared to the outside the inductance and the corresponding impedance offered by this inner part of the conductor is more as compared to outer part and hence current tries to be at the surface now actually same conductor which is fed by some current source near that conductor we if we bring in some you know conducting part which is a plate here a mild steel plate in which eddy currents are induced now you can see this skin effect gets skewed because of this induced currents in this plate you have the skin effect skewed so how do we understand this this is like you know in a very simplified explanation this you can consider as two winding one is LV winding one is HV winding because this is the source these are you know induced currents are there in this so the flux will be maximum in this part between the two winding and then you will know from the field plot that the flux that is linked by this part of the conductor is more as compared to the flux linked by this part that is why current tries to concentrate more on this that you can understand from the field plot and I think we have seen in basics also this proximity effect basically increases further the AC resistance because effective area comes down further because current is now trying to concentrate more and more on this part of the now you suppose if you have two conductors then and here the currents are in the same direction so if the currents are in the same direction and if these two conductors are side by side so here the geometry is something like this these two conductors are there side by side and you have this boundary and currents are in the same direction suppose a dot the corresponding your field will be like this and then you will observe that you can see this part of the two conductors facing each other they will basically they are linking most of the flux because all the flux is being linked by this inner part of the conductors which are facing each other and that is why the corresponding impedance offered is more and that is why current is less whereas this faces and the parts which are not facing each other you will have some flux contours which are cutting some flux contours which will be passing like this and like this those will not be basically linked by this parts and that is why the corresponding impedance will be will be lower and hence current carrying will be so if you really want to see the corresponding flux plot and understand what I was meaning by this you can actually see this virtual lap this experiment at this website so you can clearly see the flux plot there and then verify for yourself that in this case these two parts these parts they are linking more flux as compared to these parts and that is why the current distribution is like this. The opposite case when currents are in opposite direction now this is similar to this case this geometry is similar to this geometry so here also the currents were concentrating in the faces which are facing each other in the areas which are facing each other same thing is happening here currents are concentrating in the regions which are facing each other again this is the same geometry the geometry is same only thing it is dot and cross for this case right. So this way you can understand skin and proximity effect and this of course are very representative simple example but in you know many of our devices like transformers motors which are fed there by power electronic circuits with you know frequencies significantly in excess of 50 hertz with lot of harmonics this skin and proximity effect will increase substantially and then this formulation can be used to find out the increase in losses and temperature rise in windings subjected to you know high frequency harmonic current in such windings. So always remember that the complexity of the analysis further goes up because you will have to then model individual conductors in the winding right and then but the formulation remains the same the efforts in terms of geometry in terms of meshing will go up but the efforts in terms of formulation remain the same that is the advantage of every analysis.