 So welcome to this lecture 31. In this lecture and part of next lecture, we will study permanent magnet sum theory and corresponding FE formulation. Now as you are aware permanent magnets, the use of permanent magnets is increasing day by day in rotating machine and that is why understanding permanent magnet theory and corresponding FE formulation is important. Now first we in this slide we are basically comparing magnetic characteristics of soft and hard magnetic material. If you actually recall our lecture on pyrromagnetic materials, paramagnetic and diamagnetic, there we know we discuss only those three materials, dia, para and pyrro and then permanent magnets we did not discuss and at that time I said that we will study permanent magnets as part of the course later because it requires more detailed understanding and corresponding FE formulation is also not set forward. So we are now at right point to discuss permanent magnets. Now first of all we all know that for soft magnetic materials the BH loop is narrow and for hard magnetic materials BH loop is wider which is obvious now from these two figures. Now there are many interesting things here to note. First of all you should know this our basic equation is B is equal to mu 0 H plus M R. Now this M R when we are actually talking of this M R this is the at this point. At this point let me see here since you are another thing to note is whenever we talk of magnetic characteristics they are really N versus H characteristic not really B versus H. This is basically the magnetization versus H because we have seen basic magnetic dipole theory. So magnetic dipole is what there are millions of bound current loops in magnetic material and those current loops can be represented by one vector which is perpendicular to that loop area is it not and that is the vector M small M and then when external field is applied then M gets aligned with the external field. This we have seen in basic right. So basically it is M versus H when we say magnetization process is basically M is getting aligned with H but generally whenever we talk of magnetic characteristic we talk in terms of B and H and then we say hysteresis loop we generally say B versus H but we should always remember it is really the M versus H and then B versus H characteristic then are obtained by using such expression. So is the first point. Now here just to match the units because mu 0 times M will have the unit of B like mu 0 times H as unit of B similarly mu 0 times M will have unit of B because M and H they have the unit of ampere per meter. So that is why here what we have plotted mu 0 this blue one is mu 0 M versus H whereas this orange one is B versus H. Now since you know here along whole this part for all these H values because these X axis is H it is in saturated state is it not. So M is that means all the domains are aligned along the applied field that is the meaning of this being constant there is no further magnetization. So when that being the case that is when it is saturated mu will be equal to mu 0 is it not. So then that is why B versus H characteristic will go as sort of straight line with this slope being equal to mu 0 clear. So this slope is mu 0 only when it comes out of saturation here that means when you suppose initially you have taken some permanent magnet material which was in raw state it was not fully magnetized and then you apply external field H and take that material into saturation that means you increase H value substantially. So then you are somewhere here and then in terms of B you are here and then you reduce the excitation to 0 then it will come to this point is it not. When H is external H is reduced to 0 this will be the residual B which is nothing but mu 0 times M R M residual magnetization residual. No M does not reduce that is the important signal it is in saturation state when H is reduced substantially in negative direction then only M will reduce here can you see here the M starts reducing only when H has become this much negative and that is the main difference between the your normal soft magnetic material and hard magnetic material the HC is very high and here in this case the M starts reducing much before in the first quadrant itself the M starts reducing here you can see the slope is continuously changing the you know it is downward slope but here the slope is more or less flat right. So here the another interesting thing to note here is B goes to 0 first then N goes to 0 that means when you go on applying the you know H when H becomes negative negative and negative more and more negative first B reduces to 0 because one of these two terms is 0 because H is negative. So when this term cancels with this B will reduce to 0 is it not we are reducing H and making it more and more negative after having magnetized the magnetic material common magnetic material this negative term will cancel at some point with this and then B will reduce to 0 that is this point. And then further you take H further negative then you are bringing you know those magnetic domains they start getting you know reversed and then you know you have this M reducing. So now here the permeability is now different than mu 0 it is substantially you know high. So this portion and this portion permeability is much different than mu 0. So you have the same characteristics have been plotted here right and now if you zoom this second quadrant here right and this part is zoom so the exact this in vertical direction it is stretched now this line orange line in second quadrant is this line so this BH characteristic in second quadrant right and this is M versus mu 0 M versus H characteristic right and then as I have already explained mu R is close to 1 here only when this comes out of saturation mu R will not be that means beyond this point mu R will not be 1 it will be high. Here up to this point more or less you can consider this as saturated right and mu R is close to 1. To say the same thing in this zone when it is more or less flat characteristic dM by dH is close to 0 is it not because there is not much change in M with change in H. So dM by dH is equal to 0 which means chi M is equal to 0 1 plus chi M is mu R so chi M is close to 0 so mu R is close to 1 that means it is mu is equal to mu 0 right in this zone or this in this entire this entire this zone right this blue line when it is horizontal here chi M is equal to 0 and that is a corresponding B versus H this is also mu 0. Clear? Now here another interesting thing now we generally know divergence B equal to always 0 right because magnetic monopoles do not exist right and all that maybe that we have seen. Now let us understand what happens to divergence H and divergence M. Now this is just a magnet which is kept in like air and because of that now there will be B vector like this. So this is like this will be which pole north pole is it not and this is south pole always remember the B vector basically outside the magnet they will go from N to S inside the magnet they will go from S to N okay. The corresponding H you can see it is something interesting happens H field actually is reverse inside the magnet and it is along the B H is along the B outside the magnet why it through that happen it has to satisfy this equation so divergence B is always equal to 0. Now divergence B what is B? B is nothing but mu 0 H plus mu 0 M R and here actually this is not really exactly equal to mu 0 but it is almost because that chi M is almost equal to 0 that is why mu will be almost equal to mu 0. Generally but you know if you see the text and you know practical properties permanent magnets have mu R of something like 1.05 it is close to 1. So that sort of confirms that mu can be taken as mu 0 okay. So then if we agree that we can write here both as mu 0 then you know this mu 0 gets cancelled because right side it is 0. So divergence H will be equal to negative of divergence of M R right. Now this is again why I am writing here M R because this is second quadrant operation. Why second quadrant? Because permanent magnet is magnetized and it is just sort of kept in air. So moment so this outside air is going to act as load on this permanent magnet is it not? So this permanent magnet is going to operate in second quadrant because this whether you know you are not physically seeing any load but this air is load right. Keep this you know permanent magnet shorted to a magnetic material then of course H will be 0 and it will not act as a load right and it will actually the operating point will shift at this point right more about it little later. So now in this slide let us analyze this magnetic circuit to understand those flux plot little bit more. Now this is a permanent magnet here right N S N North Pole South Pole and this is a core material with high permeability almost you can assume tending to infinity and there is a air gap here right. So now here as we have shown B is of course continuous red one. Divergence B is always equal to 0 right. M R is basically from south to north right. Now H if you see H is blue and remember this M R is a residual that means this we are again we are talking second quadrant that means the magnet is is still in saturated condition the magnetization has not happened by any means right and this is like a M R is a DC, DC bias we are actually talking in this region second quadrant right moment we say M R it is basically second quadrant with this saturation condition and the BH characteristic is linear right. Here if this is a high permeable say ferromagnetic material then here also M will be there right but then we can always M is if you are assuming linearity right then M will be proportional to H right and that is why then we will get B is equal to mu H right and hence if divergence B equal to 0 and since mu is constant and linear is it not. So here everywhere divergence H also will be equal to 0 only if this we consider as nonlinear then mu will change in general with space and then divergence H will not be equal to 0 and divergence H will then be related to the corresponding change in mu with space but here we are assuming that you know this ferromagnetic high permeable material is linear right that is why divergence H as well as divergence B is equal to 0 outside the permanent magnet and now here if you see this permanent magnet model there are two approaches one is you know in literature we will see there is a ampere end model wherein this you know whole permanent magnet can be represented by one single current loop. Now that single current loop is how that also gets formed suppose you take a cross section of a permanent magnet you know there are number of these bound current loops is it not for this current loop the current is like this for this bound current loop current is like this. So here there will be current cancellation is it not. So what will remain is only the outside current internal current will get cancelled and that is why you will get one equivalent sort of current loop this is only in one Z suppose some cross section suppose you take a cross section here so we are talking of this. So along the vertical height there will be so many single current loop and that will add to M is this clear. So this is this ampere end model right there is another model which helps us to explain this you know divergence M and divergence H not being equal to 0 at this boundaries and that is called as that is based on magnetic charges although magnetic isolated magnetic charges do not exist here you know they are of use to us and here actually what we are doing is this magnetic charges here we are you will take this as plus and this as minus right so plus charge magnetic charges here and minus charges here right and these whenever in magnetics if we have to represent any field by using magnetic charges they cannot appear in isolation so they will always appear in pair. So moment I say plus here I have to put minus here because there are north and south poles and right and isolated north and south poles will not appear so similarly isolated positive or isolated negative charges will not appear so they have to appear in pair. So here it is plus and here it is minus. So now here you can say here you know because there are now magnetic charges now you know this electric and magnetic fields become exactly sort of they become analogous now we know in case of you know parallel plate capacitor for example here you have say plus and here minus so electric field intensity goes from positive to negative is it not H is in blue so from positive to negative inside the magnet like positive to negative E field inside capacitor and outside H field is again from positive to negative fictitious magnetic charges. So perfect analogy is established so that is why sometimes this magnetic charge model is helpful in you know understanding field distribution inside and outside permanent magnet right. So going further so now we were again discussing this slide I was mentioning that you know this H field now this is B field it is continuously you know it is continuous all along H field is reversing its direction as we just now discussed and when you observe that here in the middle it is perfectly vertical at the ends it becomes little slanted and that nicely merges with outside H field right because there is no discontinuity here. So there is no sudden change in field H field at this interface here there is a sudden change here and here because there are magnetic charges equivalent magnetic charges so there has to be sudden change in the value of field like you know you have in capacitor for example lines originate or terminate on charges right. So here on this and this horizontal line you have sudden discontinuity sudden change of H field direction of H field but not along this there is a nice gradual change you know H field inside and outside and here as we have already seen outside H field nicely you know merges with the corresponding B field and inside it actually opposes the M field which is vertical here and always remember when we are showing it like this we are in second quadrant that means magnetic saturated condition and this we are talking of this U0 MR moment I say MR that means it is constant right and M is not proportional to H MR is not proportional to H because it is constant only when it comes out of saturation in there the proportionality will fold right. So again as I have just now explained we are talking of second quadrant and then we are talking of now B versus H characteristic so again just to refresh ourselves here B versus H characteristic is essentially linear okay except the end portion here where it starts dropping because there is a corresponding reduction in M because it comes out of saturation so only at the end of the second quadrant there is a reduction that actually you know change here we are not actually representing here because later we will see the characteristics because this point will never be really read because this point will mean perfect open circuit which is not possible in case of permanent magnetic more about it after some time okay. So here what need thing to remember is we are modeling it as just a straight line without this you know invert going curve at diagonal okay. So now P is equal to mu 0 H plus M plus mu 0 MR now see this mu 0 MR is that saturation because of that saturation this M is due to the inherent because there are still you know some because mu R is not exactly equal to 1 mu R is something like 1.05 or 1.06 for permanent magnet that not being equal to 1 is being represented by this M there is still some you know scope for moment you say 1.05 that means the material is not fully so there are still some magnetic domains which can get further aligned is it not and that has to be represented so that is basically represented by this as capital M whereas this capital MR is represented all those domains which are already aligned and in saturation condition clear okay. So now going further so now this is the same thing we can write it as mu 0 1 plus chi M H where chi M into H is capital M and then mu 0 MR and then as we have been discussing chi M is if we take almost approximately equal to 0 that means then mu R is approximately equal to 1 right then of course then this M this term will not appear so then it will be simply mu 0 H right because XM will be 0 then M capital M will be 0 okay. So now so MR is residual magnetization magnetization of permanent magnet VR is the residual flux density at this point so H is due to external magnetic load or free current these are again important to understand due to external magnetic load that means permanent magnet just kept in air moment it is kept in air the surrounding air is acting as load right and that is why that you know that is basically H due to that okay. Second is due to free current if you pass the current in this direction and take the magnet to saturation or if you demagnetize then that would represent this H is it not in either direction if you buy external source if you take the magnet either in the first quadrant or in the third quadrant then again you know it will be this then that H would be represented by that source current okay. So now H is equal to 0 almost equal to 0 under short circuit right so it will be then B is equal to mu 0 MR that is equal to VR but D is generally you know this short circuit also H generally will not be exactly equal to 0 because there will be you know mu R for a magnetic material ferromagnetic material outside even if you you know take this as a shorted one is it not if you take this as a complete short circuit without this air gas mu R although we take infinite actually it is not infinite it is high but not infinite that is the reason that you will not have a perfect short circuit right. So then you will have something like maybe here like this the load line will be something like this okay close to short circuit if you actually short the permanent magnet by high permeable ferromagnetic material and I have already explained you for m H is the de-magnetization due to loading of free current right and H is equal to for H is equal to Hc when you actually reverse if you are having you know free current you are exciting by some free current and if you take it to this point then the the magnetic material will get permanent magnetic material will get de-magnetized and as we have seen earlier the permanent magnet is characterized by high very wide loop that is why Hc is very high because this is the Hc point it is curved yeah so we have assumed as I said we have assumed it as a linear this end portion we are assuming it as linear for simplicity okay so H for H is equal to Hc B equal to 0 and I already explained you B becomes 0 first and then m becomes 0 okay and then you know if we at H is equal to Hc then B is equal to 0 then if you substitute and B is equal to Br here then Br is equal to this is it not and that is why then the original expression for B can be modified like this so going further again just to re-emphasize this point for a ferromagnetic material in linear region B is equal to mu 0 H plus m this is in general but when you write in place of this if you are writing this then we are assuming linearity is it not so moment you say mu 0 mu r H that means we are assuming linearity and m is proportional to H and then H is equal to mu mu is 1 over permeability so 1 upon mu is mu so H then is mu times B and further for a permanent magnet then you have this B is equal to mu H plus mu 0 m r and then rearranging H will be mu that is 1 over permeability that is the relativity into B minus mu 0 m r then we can write del cross H is equal to j we know del cross is equal to j so then we can write in place of H we substitute this expression right is equal to j and then if we rearrange del cross mu into B is equal to j plus this term right and now we substitute B is equal to del cross A so now this del cross mu relativity into B is del cross mu del cross A is equal to j this j plus jm we are calling this as jm that is j due to permanent magnet then going further jm is this and then mu is as we have seen earlier now before going into the FEM formulation let us understand this load line you know carefully and through simple equation again we take this permanent magnet with a high permeable core material and with a air gap so the load is basically high permeable material with a small gap so effectively this gap is the only load right because the permeability of this can be assumed as infinity. So now apply Ampere's law for this loop so HPM Pm stands for permanent magnet HPM LPM plus HCR LCR plus H gap L gap all the ampere turns or MMFs are added to 0 mu R of core is infinity so HCR is close to 0 right so then you get this expression equation multiply both sides by mu 0 and take this from on this side then you get this equation right now flux is neglecting this fringing here neglecting fringing at the gap flux will be BPM, FEM, DCR, ACR, B gap, A gap flux density into the area corresponding cross sectional area right so then BPM is equal to B gap is equal to mu 0 H gap so here see mu 0 H gap is being replaced by BPM here so BPM L gap will be equal to this that is our BPM then becomes this okay so this is the load line equation right so it is like B versus H and the slope is minus mu 0 LPM by L gap and since it is a minus sign here we are in second quadrant H is negative and B is positive clear so now here let us understand the two extremes open circuit and short circuit now same expression for BPM is rewritten here so then BPM L gap is equal to minus HPM LPM into mu 0 now if L gap is 0 if L gap is 0 that means it is short circuit complete short circuit then you know if L gap is 0 HPM will be 0 because this is 0 so HPM will be 0 so short circuit condition in terms of reluctance right and then load line is parallel to B axis but which I said which is generally not possible because you know there is the mu the basic our assumption that mu core is infinite is not valid right that is the reason but here if that assumption is valid then of course you will get this parallel to B axis right so there is no ideal open circuit condition for permanent magnet yeah and one final point high source current or high thermal energy is required to demagnetize that means if you have to take this material into demagnetized state that means you are again bringing this out of the magnetization out of the saturation either you have to supply much higher current greater than corresponding H C to demagnetize or you have to basically you know increase the temperature maybe heat or whatever increase the thermal energy and then we have seen in basics of electromagnetic that for ferromagnetic materials if you increase the temperature too much something like 700 800 degree centigrade then the randomization becomes significant due to thermal agitation and that will basically make this all this domain disoriented in different direction and that way you can demagnetize right so that that is the last point that I wanted to mention. So we will stop at this in this lecture 31 we will in the next lecture we will see how to implement FEM for permanent magnet.