 Good morning and welcome to the third lecture. In the last two lectures we actually saw the introduction to the course outline, need for FA analysis and also we understood the differences between analytical and numerical techniques. Today as I was mentioning to you yesterday, we will actually do revisiting of important concepts in electromagnetic and only those concepts will be covered which are more relevant to FM analysis. So the first you know and foremost thing that we always you know remember when we deal with electromagnetic is Maxwell's equations. So basically electromagnetic is study of phenomena of charges in either rest or motion and then you have depending upon whether the charges are at rest or motion, you have either electrostatics in which charges are static, current is 0 and electrostatic has also many applications like in high voltage engineering or energy storage. When it comes to magnetostatics, you have current is not equal to 0, charges are in motion, but they are moving with uniform velocity and but di by dt is 0 and that gives also many applications like in electromagnets, lunges, permanent magnets, although in permanent magnets you may not have current, permanent magnet itself acts as a source of magnetic field or even actuators is another example. And next is the time varying fields in which case charges are accelerated and rate of change of current is not 0. So all these above phenomena what I just described are described by Maxwell's equations. Now, the first law there is divergence D is equal to O v is Gauss's law for electricity, then the second one is divergence B equal to 0, Gauss's law for magnetics, del cross E is equal to minus dava B by dava T is the Faraday's law and finally del cross H is equal to J plus dava D by dava T is Ampere's Maxwell's law. Now, if you see the evolution of these laws, basically the divergence B equal to 0, the magnet kind of properties in terms of load stones were discovered many centuries ago, but only it was in 15th, 16th and 17th century more understanding came and finally around 1800 Faraday was the one who again postulated this Gauss's law of magnetism in the form divergence B equal to 0. Then, you know this Faraday's law basically as the name suggests it was discovered by experimentation by Faraday around 1831 and the Ampere's law del cross H is equal to J was again discovered by Ampere in somewhere around 1826 whereas this dava D by dava T term was introduced by Maxwell around 1861 that is why it is called as Ampere Maxwell's law and Maxwell's main contribution was this dava D by dava T term which explains current to the capacitors or wave propagation in freeze free and you know any medium and because of basically these last two discoveries one is Faraday's law and Maxwell's contribution of dava D by dava T you see tremendous progress that technology has made in many areas of electrical and electronics engineering. Apart from these four Maxwell's equations you have also continuity equation which is divergence J is equal to minus dava O V by dava T. Now this equation is not sort of independent of these Maxwell's equations because you know you can clearly see and we will derive later on the correlation between this continuity equation and this dava D by dava T term in fact Maxwell introduced this dava D by dava T term so that continuity equation is not violated and is always valid under all conditions. Apart from these Maxwell's equations you need what is known as Lorentz force equation for calculating forces on static and moving charges as well as you know static and moving charges where u is the velocity, v is the flux density and q is the charge and e is the electric field intensity. Now in order to calculate electromagnetic fields you also need to know material properties which are defined by these relations J equal to sigma i B equal to mu h and D is equal to epsilon e. Now let us quickly see what are all these vectors and scalars are and their units. Understanding units and matching units on both sides of a given equation is important to understand electromagnetic fields in a greater way. For example you know if you take we know the unit of h is ampere per meter unit of del is 1 over meter so the total this unit becomes ampere per meter square J we know is ampere per meter squares and then D by dt basically D is Coulomb per meter square Coulomb upon time is again ampere and that is why you get again ampere per meter square for this large term. So I suggest you know all those who want to learn electromagnetic understand phenomena whenever they see any equation it is good to match units on both sides so the understanding becomes better. So as you know listed here I think these are all well known you know quantities so those quantities which are marked by bar they are the vectors they have both magnitude and direction and there are some scalars also. So you have E D which are basically from the electricity or electrical field point of view B and H you have magnetic describing the magnetic fields. I will actually you know tell you the subtle differences between D and E, B and H little later as we progress in understanding more concepts. So then of course you have O V free volume charge density always remember many you know many people generally make mistake in understanding this O V. O V is a free volume charge density and not bound charge density that is the first thing more about it later. Also these material properties for isotropic medium they are just scalar numbers whereas when you have unisotropic medium that means properties of the media are changing with direction with x, y, z then what you have generally is a tensor you have to write the corresponding parameter whether it is dielectric constant epsilon or permeability mu you have to write in terms of a tensor 3 by 3 matrix as shown here. Now these 3 by 3 matrix has both diagonal as and off diagonal elements but these off diagonal elements will become 0 if material axis and coordinate axis they are aligned. If they are not aligned then of course you will have all the line terms so I hope you understand. Suppose you know you have a coordinate axis in one way and material is material axis are not aligned around the coordinate axis then you will have this off diagonal terms. Now what I am going to do is in typical book on basics of electromagnetics you will have you know various chapters you know on each chapter I am going to have one or two slides explanation because this is just a revisiting important concepts it is not you know I am not going to go into depth. So what only thing what I am going to do is typically you know the concepts which are sometimes difficult to understand that actually I am going to stress more. So first is you know when you come to vector algebra you have what is what are called as you know vector multiplication in terms of dot and cross product. Now dot product we all know it is simply a dot b is a b cos theta a b. Now you need to remember that this theta a b the angle is basically the smaller angle between always it is the smaller angle between the two vectors it is a smaller angle so that is one thing. Second is you know you can call this dot product as a 2D sort of 2D product 2 dimensional because you know you can always make a plane pass through any two vectors for example a and b vectors you can always make a plane pass through it and basically dot product is telling you the interaction of these two vectors and we know it is dot product when you take a dot product it gives the magnitude of one vector along the direction of the other. So it is basically the interaction between the two vectors and since those two vectors are in one plane this is called as sort of 2D product whereas later on we will see cross product is a three dimensional product because it requires all the three you know dimensions. Now little bit more understanding on this dot product suppose you know we take again this high voltage lead and ground configuration which we have seen in the previous lecture you have these you know equipotential lines as shown by these blue contours and then you have the electric field intensity lines given by these black contours. Now the integral E dot dl basically along any of these contours use the voltage difference between these two electrodes one is the V voltage V and other is ground. Now actually as if you take the line integral around along this black contours that means along the electric field intensity contours only if you take the line integral then the line integral reduces to simple scalar product because E and dl they are in you know they are along the same direction. So then it becomes you know very simple product to calculate if you take you know both the vectors along the same direction. So you can also observe that although this length is more the E integral E dot dl is same why because although dl is more what is going to be less along this is E. So E integral E dot dl is going to be same along this as well as along this this is the shortest sum to is this clear. Then but if you take a arbitrary path now you know if I take some arbitrary path between these two you know electrodes as shown here. Now you know since I do not have an easy expression of E as a function of space I do I need to have a numerical integration procedure. So I have from suppose you do FEM analysis and get electric field intensity values in the whole domain. You know electric field intensity at more or less every point in the domain. Why I am saying more or less because since you are using a numerical procedure as we will see later. Generally field values are calculated only at you know few points and then you do interpolation to calculate fields at other points. So what we have to do is in this case for arbitrary count we have to divide this into small small segments. So effectively we are doing sort of linearization over each segment as shown here suppose this one of those segments you know which is between nodes 1 and 2 which is called as ith segment. You know what is the electric field intensity which is given by E x i A x hat plus E y i A y hat and then you calculate the dot product for that segment. Now here what is the again there is some assumption the E is taken to be same all along this segment and at the center of that segment. So that again is a approximation. If somebody says oh this approximation is you know may lead to error then what you have to do is you have to reduce this segment length further and make it even smaller and again but you will have to assume E at the center of the segment or some other approximation or take E at both ends and take average. So these are all the ways of numerical approximation. How best you can actually further and further make it fine or discretized that will improve the decide the accuracy right. More about this you know the whole this course is on numerical methods and we will discuss this aspect in more detail when we see FEM theory. So by using you know calculating this dot product we will simply get a one scalar number which corresponds to the voltage drop across that ith segment. So you sum so this integral now the total integral this total integral along this arbitrary path will just then reduce to sum of voltage drop over all these segments. So these in a very simple way we have understood what is you know a numerical method. This is also a numerical technique right to you know find line integral by you know subdividing the one dimensional you know not one dimensional and arbitrary path because this is this is still not one dimensional because it has both dimensions there but what we are doing is we are this arbitrary shape we are dividing into segments and then we are actually doing this. Now an interesting thing in rotating machines to understand again the dot product is this is one coil is shown here right. So this coil is you know supplied by circuit. So this circuit can be you know maybe some source which is supplying the current. So current is flowing through this coil and what is shown here is the current the current density J current is going like this and coming back. So the correspondingly it is the J is shown as ampere per meter square. Now actually and the corresponding now we need to understand what is the concept of area as a vector. So now this loop has this area area is a vector and its magnitude is the area of that loop and the direction is given by the unit normal to the plane containing the coil right. So that is why the way this area vector is shown is ds an hat where an hat is the unit normal to the plane of the coil. So now actually if you see here if you calculate b integral b dot ds because b dot ds is 0 here because you know it is integral b dot ds all along the all along the loop is 0 because b is parallel to the the coil plane. So direction of b is normal to the direction of coil and direction of coil is an hat an hat is normal to b. So that is why in this position of coil there is no flux linkage there is this clear. But you know even if flux linkage is 0 what about d phi by dt? d phi by dt is maximum is it not? So although phi phi is 0 the d phi by dt is maximum. So in fact the voltage induced in some motor or generator application involving such coil we get maximum induced voltage as the position of coil where is the time and we will see some you know applications of this little later because if yeah so this is ds the question that is asked by one of the participants is this ds is the you know small unit area of this loop. So the area of this you know the whole loop is this all this area enclosed by this coil right. So the total flux linked by this coil will be integral b dot ds right. Now coming to the cross product again we will make use of this same geometry. So j cross b now the simplest example to take again is you know Lorentz force or force density which is given by j cross b and we know the cross product is given by jb sin jb sin theta an. Now cross product leads to a vector and that is why it is a you can call it as you know a 3D product. As shown here you have j and b and theta again theta is a smaller angle between the two vectors and j cross b is basically you know you have to always use right hand rule. So j from the first vector you turn the fingers of right hand towards the second vector and the direction of the thumb will give you the direction of the cross product. So j cross b you will get force in downward direction right and jb sin theta is the area of parallelogram as shown here. So the two force components here so now if you actually find out the force directions on the two coil sides you have j and b. So in this case if you apply right hand rule the force will be on the in the downward direction whereas here the b direction is same now the current or j direction is reversed so the force becomes upward right. So now these two force components they form a couplet around this axis of the coil and then that couplet will actually produce a torque and this is the one of the main principles in rotating machines right. So that torque is given by r cross f where r is the distance of the coil side from the coil axis. You know note that these two coil sides will not actually contribute to the torque because you know they are along the so the force is there only on these two coils right. So that can be verified because this direction is along the b. So now going further this equation t is equal to r cross f can be with some mathematical manipulation which is given in all textbooks on electromagnetic. You will find that t is given by BIS sin alpha where S is do not get confused with this S here I have marked n and S these are the north and south poles whereas this S is the loop area. In this course you know there will be always because we are short of you know symbols and units you will find you know same letter is being used for more than one but I will try to you know differentiate when such thing is there. Then you know you can get the magnitude of the torque as BIS sin alpha and this again is quite widely used in machines. Next you know topic for discussion is coordinate systems. So there are three there are many coordinate system but most commonly used are Cartesian, cylindrical and spherical. Now spherical system is of almost no use for our discourse right because you know you do not have something which is you know very small you know entity and from there something is coming out like in for example in antennas spherical system is widely used because antenna can be hertz dipole can be considered as very small and the field radiates and there the spherical system. Those kind of applications spherical system is used but here we generally use only Cartesian and cylindrical system. So in Cartesian system it is basically intersection of threes in any any of this system basically a point gets defined by intersection of three surfaces in case of Cartesian it is basically x is equal to constant surface y equal to constant surface and z is equal to constant surface. So in Cartesian system is a very general system whenever no symmetries exist geometrical symmetries then you do not have choice but to use Cartesian because if you unnecessarily use cylindrical system that will become more complicated. Cylindrical systems are useful for you know when there is a symmetry exist for example here you have a coil which is circular and you know around this axis and if you enclose this into a cylindrical sort of cylinder with you know top and bottom surfaces then it becomes a perfectly cylindrical geometry and effectively you know you can get do this 2D analysis because this is a 2D analysis because there are two dimensions involved z and rho and you know the the the field distribution is independent of phi in this case the phi is into the paper is it not. So so it is perpendicular to the paper paper this paper in this case. So it is independent of phi that is why effectively when you do this 2D analysis the fields that you will get are really three dimensional fields because all along the phi direction which is you know normal to this plane of the paper the direction of the the magnitude of field is not changing right. So that is the advantage of using cylindrical system. But you know although you have you know Cartesian system is is you know most of the times you have no choice but to use Cartesian system. But you know even when you use Cartesian system you can do some approximations like what is shown here for example this is a squiggle case induction motor now with a you know rotor bars with skew. Now of course this what is shown here is a multi-slice model since it is a really a 3D model with skew of with some angle finite angle. So it becomes a three dimensional analysis but what is generally done is you divide this model into number of slices and then for each of those slices you assume that there is no skew and then properly join these models through maths and you know at the interfaces these are the interfaces between the models and then get the total solution. So here you know this is the way you can actually reduce the complexity of 3D modeling by using say multiple 2D models and that kind of thing. Now remember you one may get confused whether this is a cylindrical you know symmetry no it is not because there are slots and this you know as you go along this phi direction the slots are not actually you know they are not 2 and 2 slots along the phi side. So this does not have cylindrical symmetry along the phi direction so that should be born in mind. Now this is another example how would we reduce the complexity this is a you know a transformer a large power transformer. Here actually we can you know what is done is although you know as you can see again here there is no cylindrical symmetry at all because around this axis of the coil the core is not rotating core is more or less a planar structure so there is no cylindrical symmetry but what is done is we have exploited three field symmetries the field is symmetrical about this half of the height of the transformer. So it is symmetrical it assumes symmetrical although you know the clearances at the top and bottom are in practice different but that again is the approximation is made. So if you do that approximation then there is a symmetry of field along this you know middle plane so that is first symmetry second symmetry is on this plane vertical plane here of the coils here and the third symmetry is on the other side of the coils you understand so this is the coil. So coil is model only one fourth along the phi direction so it is a quarter model along the phi direction because we are exploiting the field symmetry here as well as here at the two ninety degrees you know point separated by phi and then there is a one more symmetry we have exploited in the vertical direction so one fourth in the phi direction and one half in the vertical direction so that is why it becomes one-eighth of the model and that gives a big relief in terms of computational efforts when you go from full model to one-eighth model right. Now this is again you know rotating machine with you can see now here there are no when there are later on we will see there are what are called as Dirichlet and Neumann boundary conditions so whenever those you know kind of in fact previous example on transformer will be exploited Newman conditions at those three you know symmetry planes I will explain you later what is Dirichlet and Neumann condition but when those conditions are not possible to be exploited then you can use in particularly in rotating machines you can use what are known as periodic boundary conditions so you do not have to model the entire you know 360 degrees geometry so you can model only a sector and then impose what are known as periodic boundary conditions and in the post-processing stage you will get the entire field distribution how do we do that there will be a separate lecture in this course on how to basically use periodic boundary conditions and improve the analysis that we will see later. Now next topic is Dell operator so in the Cartesian system and in cylindrical system these are the expressions of Dell operator always remember see here why this one of our rho appears first of all this rho is distance right do not confuse this rho with rho v which is the volume charge density right so this one upon rho rho d phi will give you the distance that is why you know then it has to more because this d rho and d z they are distances so rho d phi also is distance along phi direction so this way you know you whenever you see any expression you again I am trying to tell you that you match the units then the understanding becomes better. Now this Dell operator when it operates on scalar you get vector when it operates on vector with a dot product you get a scalar when it operates on a vector with cross product it you get vector when you when it operates on tensor I will explain you what is what the tensor means already I have explained you little bit when we discussed permeability tensor again now this t is denoting tensor not the torque right just be sure about that so divergence of tensor gives you vector and divergence of gradient of phi gives you del square phi which is a scalar right remember that a a dot del where a is vector or a cross del is also an operator so this whole thing will operate for example a dot a a bar dot del phi so del phi then becomes vector dot product a a bar will give you a dot product so a bar dot del is an operator and not vector okay now coming to what is you know this tensor business so now here any vector f which has three components can be written in a in a column matrix form as given here f x f y f z when you write it like that you do not have to actually you know write a x a x hat a y hat and a z hat so that is implicit and assume so now when you actually say force force is divergence of tensor so effectively what we are doing it is the del operator is d d by dx d by dy and d by dz what is implicit here is here a x hat a y hat and a z hat but since we are written it in a matrix form those you know unit vectors are not mentioned similarly this tensor here it has got this t x x what is implicit here is a x a x hat here a y hat a z hat similarly a x hat a y hat a z hat like that so now when you actually take the product of these two matrices these three will get multiplied with these three so this this is a x hat here a x hat so it becomes just a this product is it clear right so now this is only the f x part of the total force and similarly f y and f z so this is how you get you know f force as divergence of tensor again this will be required in one of the lectures later when we calculate forces on conductors or magnetic magnetic system by using you know this tensor concept so this ends this lecture what we have discussed here is some of the aspects of electromagnetic in the next lecture we will see further thank you