 Welcome to the 10th lecture of this course. We actually stopped you know at the wave equation in the previous lecture. So, we will start with this classical eddy current theory. So, this will be the first you know slide of 10th lecture. So, classical eddy current theory we basically solve diffusion equation a metallic plate excited by a surface you know field intensity this is time varying. So, here for example, here this is a you know what is known as semi-infinite slab conducting plate. So, why it is called a semi-infinite? Because it is infinite. So, this is in this is z direction and you know this is x direction and coming out of the paper is y direction. So, semi slab is infinite is infinite in extent in x and y direction and it is you know half infinite in z direction and then we are actually analyzing the diffusion of currents and eddy currents and diffusion of fields into this metal plate. So, now, here when I am writing this now we are this diffusion equation this diffusion equation we are actually writing it as you know d by dt is replaced by j omega. So, we are working in phasor with phasors now in frequency domain. So, now this you know if we consider e as only x directed because analytical solutions are generally simplified. So, you know as I have told in one of the first lectures also when you want to derive some simple expressions and understand the theory you have to do some approximation. So, e is assumed to be only in x direction and it is function of z. So, only as e into z when it you know field diffuses it changes with z. So, then if that being so then del square e will just get reduced to del d 2 e x by dz square minus and then minus j omega mu sigma e x is equal to 0. The solution that will satisfy this is this e x is equal to e x p this e x p is the amplitude into e raised to minus gamma z where gamma is propagation constant gamma is propagation constant which is given by alpha plus j beta alpha is the attenuation constant and beta is the phase constant. So, here alpha I know if you substitute back into this expression then you know you will get alpha as 1 plus j square root of pi x mu sigma. So, the you know theory of these eddy currents is you know available in many books. So, one of such books is you know mentioned here those who are interested to understand little deeper into how this derivation comes how this formula comes they can refer any of these such books. So, gamma is given by 1 plus j square root of pi f mu sigma. So, then since gamma is alpha plus j beta alpha and beta are given by square root of pi f mu sigma. So, going further e x then becomes e x p e raised to minus alpha z where it was earlier e raised to gamma z. But now gamma is alpha plus j beta and now when we go into time domain since beta is representing the phase constant then the expression becomes e x p e raised to minus alpha z this represent the attenuation is it not exponential decrease cos of omega t minus beta z. So, beta into z will give you the phase with beta being the phase constant. And then you know if you substitute in place of alpha you know the expression that we have got from the previous slide then you get e x p e raised to minus z square root of pi f mu sigma cos of omega t minus z square root of pi f mu sigma. So, now if you see at z is equal to delta now here you know delta is a skin depth here. So, that is at z equal to delta is equal to 1 over square root of pi f mu sigma which is called as skin depth. Then the magnitude of e x will be e raised to minus 1 e x p because this you know if z is 1 over square root of pi f mu sigma then this amplitude will become just e x p e raised to minus 1. And then at surface at the surface the magnitude will be just e x p because when z is equal to 0 here this will become e raised to 0 and then that will reduce to just e x p and here this is normalized with respect to e x p that is why here I am just marking it as 1. So, now if you see by this expression e raised to minus 1 as we go inside you know this conducting plate at one depth of penetration which is called as skin depth the magnitude become e raised to minus 1 of that at the surface. So, e raised to minus 1 is 0.368. So, it reduces to that. So, it is exponential dk and is a classical you know eddy current theory and the corresponding diffusion solution. Now, what is the importance of all this in this in the context of our course. So, for example, here you know if you calculate the skin depth for various materials that we encounter in electrical machines and equipment the for copper you know at 50 hertz if you substitute the parameter pi f at 50 hertz I am calculating mu is mu 0 into mu r right mu r for copper is 1 copper is diamagnetic material and mu r is very very close to 1 but less than 1 but for practical purposes we can take it as 1. Similarly, for aluminum it is 1 right. So, 1 and the conductivity of copper at you know room temperature we are taking it as 4.7 into 10 raised to 7 Siemens per meter and then you get 10.3 mm as the skin depth of copper right. Similarly, at 60 hertz it will be 9.4 because it is inversely positioned to square root of frequency. So, as frequency goes up the skin depth will drop right. For aluminum conductivity is 61 percent of that of copper right. So, the skin depth for aluminum is 13.2 mm at 50 hertz and then for mild steel which is a structural element used very commonly in machines and equipment. Relative permeability is 100 which is corresponding to saturation case while they are considering saturation because you know the skin depth is very small you know and if the flux that is impinging is getting concentrated in very small skin depth. So, you know the mild steel material is considered as sort of saturated and the corresponding saturation permeability is taken. Otherwise we know generally mild steel it is a type of steel with predominant iron the permeability will be somewhere around 1000 in unsaturated here. But for analysis we take mu R as 100 and the corresponding conductivity of mild steel is taken as 7 into 10 raised to 6 one order less than aluminum and copper and then the corresponding skin depth for mild steel is 2.69 much smaller than aluminum and copper and then you know also thing to notice typically you know the mild steel enclosures that we build the thickness is typically you know 5 mm, 10 mm you will never find till you know mild steel thickness 1 mm, 0.5 mm that will be too thin to be mechanically you know stable. So, when you use mild steel material almost all the loss will occur in the mild steel because you know the thickness or the plate is you know considerably higher suppose 10 mm. So, it is almost 4 times this we know that this you know exponential decay this is like typically in 5 exponential you know that is e raised to minus 5 if it is there it will basically you know or in 5 sorry in 5 skin depths this value of field will go almost to 0 like you know in circuit we have 5 time constant the we get almost final steady state value. Similarly, in case of in here in about 5 skin depth the value you can take it as you know almost equal to you know 0. So, we actually you know can say that if you use a mild steel material almost entire loss will get you know absorbed in the mild steel and in order to reduce that loss then you may have to use some kind of shielding. For example, if you have a mild steel plate in the vicinity of you know some source which is producing field and that field is impinging on this mild steel and causing eddy current losses in that what you could do is you could shield this mild steel by using aluminum or copper you know plate thin plate of at least you know something like if it is copper it should be at least 5 mm if it is aluminum it should be around 10 mm why it should be like that I will tell in the next slide. So, this aluminum or copper will basically shield is mild steel material. So, this is called as electromagnetic shielding right. So, more about it little later and you know the what will happen is this aluminum and copper the eddy currents induced in this aluminum or copper they will basically repel the incident field and a mild steel plate will be shielded from the incident field. So, that is you know the practical application of such materials. One may also use non-magnetic stainless steel in some application. Non-magnetic stainless steel means you know mu R is 1 relative permeability is 1 and conductivity is even lower than the mild steel because it is a high resistance material. So, it is 1.13 into 10 raise to 6. Now, here the skin depth for stainless steel you can see if you put it in this formula you will get 67 mm at 50 hertz. This is something important that means you know if you shield some electromagnetic device say transformer or any other device and you know you enclose it in stainless steel tank and that stainless steel tank is not going to have a thickness of 70 mm typically again the thickness will be some 10 mm, 5 mm, 15 mm. So, the field is going to come out of that is it not because you know in 1, 2 or 3 skin depth the field is not reducing to 0. So, in fact, field will come out of that is stainless steel enclosure. So, if suppose you know you are required to design an electromagnetically shielded product where in you know that your customer specifies that you know no field should be coming out of that device you cannot use stainless steel you know material as enclosure because the field will come out for the practical thicknesses that are used. So, this is again having a lot of practical you know significance. So, what I am trying to say is this theory of eddy currents is very important while you know designing machines and equipment. Now, we will see before going to the next slide we will see this slide. So, this is you know slide which shows you know this eddy loss as a function of material thickness. Now, you know it is again these three materials are shown which are commonly used aluminum which is representing non-magnetic but highly conducting material, mild steel it is conducting material but magnetic whereas aluminum is non-magnetic is it not. So, it is highly conducting non-magnetic conducting mild steel is aluminum is highly conducting and non-magnetic mild steel is conducting and magnetic and stainless steel is high resistant high resistance material and non-magnetic. So, you know all three materials are representing three different types of characteristics. And now you can also see here in this slide the you know the variation of loss with respect to plate thickness is considerably that pattern is considerably different in all these three materials. Now, I am not going to go into details of this you know but this has considerable influence in deciding you know which type of material to be used particularly for shielding purposes. And what should be the thickness of those materials and all that for those who are interested further in understanding this theory and application of this theory of eddy current they can refer the book that is you know mentioned here below in this slide. So, after seeing that let us go to the next slide. So, we will talk now about pointing vector another important vector in electromagnetic and it has good influence on the field computation that we generally do as part of this course or in general and it is useful for even high frequency electromagnetic when we are considering wave propagation. So, now P, pointing vector P is given by cross product of E and H. Now, in the case that we analyzed that conductor and field diffusing inside the conductor we assume their E as E only in x direction is it not. And if we actually use one of the Maxwell's equation that curl equation we will easily derive if E is only in x direction and is function of z is it not that is more important E x and it was function of z is it not if that is substituted in one of the curl equation you will easily get H only in y direction. So, H will be H y A y and the P will become then E cross H will be you know simply A x cross A y. So, that you will get in P in z direction and that is what we saw there when E was in x direction. E was in x direction the although I did not discuss that, but H was in y direction and that gave power flow in z direction. So, basically this z is the diffusion of power into the conductor. So, going further here we also need to understand what is known as intrinsic impedance which is you know intrinsic impedance of a medium which is given by in this case in general it is E upon H, but here for the considered case of diffusion problem it will be simply E x by H y because E is in x direction H is in y direction. And then if you use the expression that we have got earlier and if you refer the standard books on eddy current theory you can easily see this getting derived E x by H y is square root of j omega mu upon sigma and then you get then expressions for E x and H y as this this we had already written in the one of the previous slides and H y by using this you know intrinsic impedance formula H y you can easily determine as being this. Now, H also is you know there is a exponential variation e raise to minus z by delta as E, but there is phase difference between E x and H y. When it is a free space in case of you know wave propagation problem for example, a lossless case E x and H y there will be they will be in phase although they are orthogonal in space, but they will be in time phase. But here then since this is a lossy case sigma is there there is going to be a phase difference between E and H. So, now P z is equal to E x into H y. So, it is now you can see here when you multiply both this the power will be proportional to E raise to minus 2 z by delta. So, what is this power? This is basically representing eddy current loss in the metallic plate. So, what it is telling is that the although E and H they are varying with you know single exponential e raise to minus z by delta the power is varying with double exponential negative. So, you know you will have at z is equal to delta you will have power e raise to minus 2 into that of the surface value and e raise to minus 2 is 0.135 that means almost you know 86 percent of the power will be sort of lost in one skin depth. So, 86 percent of the power is lost in one skin depth. So, what it tells us is there is a more rapid variation of power as compared to E and H fields and that is important if you want to capture that variation then we have to use very fine mesh when we actually from next lecture onwards we are going to see that. So, we are going to see finite element theory. So, we will again discuss that point when we solve diffusion problem. So, we need to have sufficient number of elements in one skin depth at least two or three if you are using linear element. If you want to use only one or two finite elements then you will have to do quadratic approximation that is second order approximation in finite when you use finite element formulation. What we saw in the previous slide was basic theory of eddy current and basically it was you know we did if we did find solution to diffusion equation and based on that we calculated the expression found out the expression for skin depth and all that. Now, we will go further and find out how do we compute eddy current losses in conductors. Now, this is very common in electrical machines and equipment you have either you know conductors of copper or aluminum subjected to alternating field or you may have field because of current and leakage field in the winding hitting the structural parts conducting parts and causing eddy current. So, this is common in static as well as rotating machine. So, what we are going to see is we will calculate we will find out the basic formula for induced losses in winding conductors, co-laminations, joints and structural parts. There are two common cases of excitation. So, first you know is like this case wherein the conductor in question is very thick and it is excited on both sides by tangential B0 flux density. So, here this is electrically thick conductor whereas since it is you know T is thickness is much greater than skin depth. The second case is case of thin conductor when thickness is less than skin depth. So, you know for thickness greater than skin depth which is electrically thick conductor the loss per unit length in x and y direction. Now, remember this is z direction thickness is in z direction. So, the what you know have surface these two surfaces vertical surfaces they will be the x y plane. So, we are calculating loss per unit length in x and y direction. So, this T is in z direction. So, this formula is given by the formula is this eddy loss per meter square because it is per unit length in x and y direction. Now, the derivation of this is given in this reference below given below those who are interested they can you know find out how this formula is derived. Similarly, this formula below for electrically thin conductor also the corresponding derivation is given in this you know reference. Now, here H0 is the magnetic field intensity which is B0 upon mu 0 and this is the peak value. Now, this is the case of inductance limited eddy current because the thickness is high and the eddy currents will be induced because when thickness is small you will have resistance being high and though that high resistance will limit eddy current. But when thickness is large resistance is small, but inductance will be more. So, that is why the first case when thickness is more it is it is the case of inductance limited eddy current. Now, second case electrically thin T is less than skin depth you have again loss per meter square given by this formula. Of course, here sigma and delta have the usual notation sigma is the conductivity and delta is the skin depth that we have you know already studied delta is 1 over square root of pi mu sigma. So, P e in this case is given by angular frequency square then B0 square P cube sigma upon 24. But generally you know in electrical machines and transformer you would finally, would want eddy loss expression in terms of loss per unit volume because generally it is easier to calculate the volume of the winding conductor. So, that is why you further divide this by T in z direction because this earlier this formula is per unit length or per unit length in x and y. You further divide by unit further by the dimension in z direction and then you will get you know that one T will go and this is the famous formula for eddy loss per unit volume frequency angular frequency square B0 square T square upon 24 times resistivity. So, this is as I mentioned earlier here thickness is small. So, resistance will be high and this will be a case of resistance limited eddy current. How do we reduce the eddy current loss? That you can do by subdividing the conductor. So, you can divide subdivide the conductor and you know you can. So, you eventually you know you reduce the thickness to such value which is possible from the point of your mechanical consideration because lesser the thickness you make there to be other design considerations as well like you know mechanical strength and all that. So, as long as that is permitted you can reduce the thickness go on reducing the thickness and of course that conductor with that small thickness should be commercially available. So, that is the reason that at very high frequencies you generally do not have single strip conductor available in very thin dimensions. So, then there you use conductor in the form of foil a very thin foil of aluminum or copper foil which is of the you know the thickness is quite small of the order of say you know 0.5 mm or 1 mm like that. So, you know that you can use or at very high frequencies suppose there is some power electronics application you know and you know your frequency is something like more than 10 kilowatts or up to even 50 kilowatts in what is known as Litz wire is used. In that Litz wire you have thousands of small, small strands which are you know continuously sort of transport to reduce the circulating current loss. See because when you use number of parallel conductors you are reducing the eddy current loss, but those parallel conductors if they are not linking the same flux then there will be circulating current loss between these parallel conductors because flux and therefore inductance and impedance of these parallel conductors will not get equalized and there will be circulating currents and corresponding circulating current loss. So, that is the reason that whenever you use large number of you know conductors in parallel to reduce the eddy current loss you have to also transpose them continuously possible and reduce the circulating current loss also. So, we will go further now maybe before going further let me you know tell you these configurations which are practically you know possible suppose you know you take a case of transformer is a magnetic circuit window core window and you have two windings and we have seen earlier also there is a leakage field like this. It is a transformer window, this is the LV winding, this is the HV winding. Now if you actually see this winding if I sort of zoom this winding this LV winding. So, this may be made up of conductors or turns like this. The conductors are stacked in a disc form in one disc this will be another disc and so on. So, many discs are there. Now if you take now one particular turn or one particular turn of this disc which is of this form now and there is this leakage field. Now you can see this leakage field is incident tangentially on in both parts, both surfaces. So, this is the case what we are actually. So, when this thickness of this conductor is very small as compared to a smaller or even comparable with skin depth that approximation is good enough. Then you have this thin conductor case because when this conductor is very thick for example, in case of what are known as bar winding where in winding is made up of very thick conductor where you know it is like a bar they cannot be easily bent also. Then that becomes the case of electrically thick conductor and then the corresponding this formula has to be used which is you know in that formula for electrically thick conductor. Whereas when this thickness is much smaller or comparable or almost equal to the skin depth then you know you can use this formula for the thin conductor. So, the corresponding theory you can you know see in this book that is given below. So, we will go to the next. Now you know we have more or less covered many topics in basics of electromagnetic. I wanted you to little bit understand this pointing vector so that some of the very common doubts that we have can get cleared. For example, let us take a case of transmission line. Now transmission line you know we generally say the power flows through conductor is a source and this is say load and this is a transmission line. We generally say that you know power flows through this transmission line. So, generally power actually is not carried by the conductor power is carried by the surrounding field. So, I will you know explain you both the cases the power flows in two directions. So, one is along the line in z direction where the power is flowing along the z direction from source to load is it not. But that power is carried by this E and H field. So, there is a E field vertically down between the two conductors and the H field is shown by dot and cross because current is flowing in this direction. Now actually if you take you know E cross H for this H here is vertically down H is into the plane E will be in this direction. Up if you go E basically is vertically up and it that you know E may eventually you know go to the reference like this it may turn. But at this point E will be normal to the conductor because we have seen earlier E is always normal to the conductor surface. So, at this point at least E will be normal later on it may it may turn immediately, but that is okay. So, now E has reverse this direction but H also has reverse this direction it is now dot and this is vertical E is vertically up again E cross H if you apply right hand rule E will be along the this direction. Right. So, here who is carrying the energy and the corresponding power it is basically the E and E field which is all in this space here and all this space E field is there everywhere is it not as well as the H field is there. So, this E and H field together and the corresponding cross product is basically carrying the energy. So, the fields carry the energy. Now you may wonder you know then what is this current you know the let us what is this current doing and what is happening inside the conductor right. So, now let us sort of zoom this conductor this is this conductor zoom here. Now the here this I am taking this E which is this E in the horizontal direction which is basically due to the voltage drop is it not voltage drop across the conductor because there is a finite conductivity and then there is a corresponding voltage drop and then there will be corresponding electric field intensity right. So, that you know electric field intensity is now in this direction right and then you know you have this current is still in the same direction and the corresponding you will have you know cross and dot H fields within the conductor earlier we were considering H fields outside the conductor in the vicinity here H fields we are considering inside the conductor. So, it will be dot and cross and now again if you calculate you know E cross H you will get you know for this horizontal component and this dot H field P will be down and for this two components we will be up. So, that means these two you know vertical components of P they are representing conductor loss because of finite conductivity of the conductor right. So, basically you know what we understood here is there are two two components of E one we one two sets of component one set of component we considered vertical because of potential difference between the you know top conductor and the bottom conductor and the other you know E field we considered along the conductor basically is representing the voltage drop and then we clearly understood that the power and the corresponding energy is carried by the fields in the by the you know fields in the space surrounding the conductor. So, all these space here here that is carrying the power from the source to the load let us see another example of transformer. Now, this is the transformer with you know LV and HE windings as usual with currents flowing in dot and cross and the corresponding leakage field will be as shown here the leakage field will be like this right. Now, actually if you zoom this now this part here the gap between the two windings and the corresponding leakage field here the leakage field is vertically up for the dot and cross shown is it not. So, H field is vertically up here in the gap between the LV and HE winding E is into the paper because here LV is the primary winding yeah. So, E will be into the into the paper right and so will be the and remember this again you know E is not necessarily always is associated with the conductor by Maxwell's equations del cross E is equal to minus daba V by daba T there will be you know curling of E field in the whole space surrounding that dB by dt. So, that is why there will be you know E E field not only in the in the winding space but it will be also there in the gap between the two winding okay. So, now so this E is into the paper and H is vertically up. So, now you can see again E cross H the power flows from LV to HE what core is responsible is only for electromagnetic induction. So, core basically is efficiently inducing voltages in LV and HE winding because without core also you could have you know voltage induced in secondary winding is it not only thing it will not be efficient right. So, what actually is responsible for power flow is both E and H field E field we understand because of Faraday's law del cross is equal to minus daba V by daba T the corresponding H is basically once the current flow start flowing in the secondary then only you know power will be transferred to the secondary if not when secondary is loaded then only power will be transferred. Moment the secondary is loaded you have this corresponding leakage field here as shown here and then the whole space of the windings wherever this leakage field is there crossed with E E cross H will result into power flow from primary to secondary. Now we will see another you know example of pointing vector and again here it is capacitor right. Now this is you know two capacitor plates right and capacitor is charged by some source it is not sure it is a DC source maybe okay and then you know capacitor plate get charged to plus and minus plus and minus charges. Now carefully observed earlier also I told the charges are marked with specific you know logic here you can see you know the charges are more or less equi-spaced at the end they are concentrated because electric field intensity is high at the ends because of you know sharp corners turning is same right. On this side on the top plate also there will be charges so we generally show we have been showing capacitor plate as only you know thin and we used to show only charges on one side but here charges will be on both sides on the top surface as well as bottom surface. But top surface they will be you know sparsely placed they because you have more coulombic force in the inside of the capacitor is it not that is why the charges positive and negative charges will be concentrated on the inner surfaces of these two plates right. So that is the only few charges are shown on the these two surfaces top surface of this and bottom surface of this plate and there will be e field lines like this also right. Now actually if you see when this you know capacitor is being charged right so you have you know if you see generally we say that you know power we may tend to say that the power and the energy is flowing into the capacitor like this in the vertical direction. So actually power flows again you apply e-classification as concept here you know for example take on this line outside line e is vertically down h is into the paper for this current right and the p will be like this here e is at this angle h is still into the paper e is like this. So energy is being fed into the capacitor through the fields in the whole space. Now here e is in the same direction what is reversing here is h because h will be dot here and that is a power wheel power direction will be reverse right. So I hope this makes the thing clear so we will go further. So this is the you know last slide of our basics of electromagnetic module. So what we are doing is we are summarizing Maxwell's equation. So you know Maxwell's equations here are written in differential as well as integral form right. So I do not have to read this you all know why now all these equations also we time to time we converted those differential form of equation into integral form as you know listed here right. Now one question may come into our mind that you know what about electromagnetic boundary conditions under time varying field because we initially saw boundary conditions for electrostatic field then we saw magnetostatic field. Now you know we will see what happens when time varying terms like this d psi by dt or this term dA by d by dA by t term they come into picture what happens to the boundary condition. Now although we did not derive earlier any of the boundary conditions for you know understanding the contribution of this time varying term I have just you know derived here for one case that is we will derive the boundary conditions for tangential component of electric field intensity. Now if you read any textbook standard textbook on electromagnetic they will take you know boundary like this with you know two material this is material two here and here material one with E2 and E1 as the electric field intensities. Now you take this counter at the interface and that counter you take that and that delta H is very small. So now here you have this counter and the corresponding area dS is this and you know you are the A n vector will be vertically downward that is into the paper in this case right and but since delta H tends to 0 dS also will tend to 0 because one of the you know of this rectangle rectangular sort of contour if you assume this as rectangle since delta H tends to 0 the area will tend to 0. So this you know contribution of this because as we see as we have seen earlier and I told you that boundary conditions are derived by using integral form of equation is it not differential form would not help because differential form of equations are telling you what is happening at a point in space but here we need to integrate something along the surface so you have to use integral form. So integral form when we use this integral close line integral E dot dl is this now here dS will tend to 0. So that is why this time derivative term does not contribute to boundary condition. So you get your usual you know each E1 t is equal to E t2 which can be easily derived now because integral E dot dl is equal to 0 the tangential component of E2 is opposite to this you know delta W as a vector because the integration you are going taking along this direction whereas E tangential is along this direction right. So that is why you get a minus sign here E minus E2 t delta W right. So then you that is why finally you get E1 t as E2 t right just for the sake of completeness I derived this so that you also understand that the time varying term does not contribute in deciding the boundary condition right. And then using the same concepts for example this dn d1n is equal to d2n can be derived by using you know pillow box right at the interface and then you know you do the surface integral and all that you can see any standard textbook on electromagnetic and remember this is in absence of surface charges right. So d1n is equal to d2n similarly you have b1n is equal to b2n and H1 t minus H2 t previously you know when we studied magnetof static I equated this to 0 because that time you know the surface current we were not considering because surface current will generally come at high frequencies now since we have studied the skin depth and the theory of eddy current we now know that as the frequency will increase right that the skin depth will become smaller and smaller and in the limit it will tend to be a you know surface current the eddy current will be on the surface only. So in such cases then it is very much relevant to consider those surface current by we know k and in that case H1 t will be H1 t minus H2 t will be equal to k right and k here is the only magnitude because these H1 t and H2 t are magnitude. So I think with this we will we have completed basics of electromagnetic so we almost you know took 8 hours of of an hour in about 4 hours we have roughly 4 hours we have completed basics of electromagnetic and as you might have noticed they are not just simple basics as described in any textbook but everywhere we understood the relevance of whatever theory that was covered in terms of finite element analysis in general the relevance was there in most of the cases. So I think we are all set now to get into finite element analysis of machines and equipment from the next lecture. Thank you.