 So, in the previous lecture we just had a quick introduction to the course and what are we expecting it expected to learn in this course. Now, let us see some interesting applications of finite element analysis. First let us see a case study on a transformer here in what we are doing is we are analyzing the magnetic circuit of a transformer and how the flux distribution and b-reptors they basically vary as a function of time. Here this is the one leg of core second leg of the core and third leg of course is not visible but it is there. This is the horizontal part of the core which is called as yoke. Now, if you see here in the middle joint that means the middle leg and the yoke joint the flux or the b vector basically it is is rotating as a function of time with time it is rotating. See here if you see in the main middle portion the flux is always in vertical direction because either directed upwards and after half cycle it will get you know directed vertically downwards. So, it basically you know the the direction is either vertically up or vertically down. But here in this joint portion the b vectors they rotate with time and that leads to additional losses what are called as rotational hysteresis losses. So, such kind of detailed analysis is possible only with finite element analysis. Next we will see a case study involving a rotating machine. Here you can see as in case of the transformer in the previous case we can study variation of field as a function of time as you know shown in this plot. Furthermore we can also calculate performance parameters for example we can calculate flux linkages of phases ABC with you know angle in degrees mechanical degrees. So, we can calculate such parameters quite easily and then we can further more calculate back AMF and other you know performance figures related to this rotating machine. Details of this we will see subsequently in the course. Before showing the further case studies I would like to you know mention that actual problems in equipment and machines are really complex. It involves many engineering fields. It doesn't involve just magnetic field or electric field, but there are thermal field, there are structural fields, acoustic fields, connected networks or circuits and so on. Now if we want to design, analyze and optimize a given electromagnetic device then all these engineering aspects need to be accounted for in our analysis and then the interactions between them need to be you know accounted accurately. For example let us see the interactions between magnetic field and thermal field. Alternating magnetic field leads to eddy and hysteresis losses in co-op that leads to temperature rise in the core and that is called as thermal field. This temperature rise may change permeability or conductivity if the temperature rise is significant and that in turn will change the magnetic field. So this is how the interactions go both ways. Now similarly we can analyze interactions between any other you know couple fields. In this course we are basically going to study magnetic fields interacting with the connected circuit. So basically for example in rush current in a transformer as will be explained in a further slide or you know finding the skin effect, proximity effect those kind of you know problems can be effectively solved by considering you know the given electromagnetic device being excited by external circuit. So as I said earlier we will be restricting in this course coupled circuit field analysis so this part. But for more details for more details on other engineering aspects you can refer this book which is dealing with transomers coupled fields in transomers and any other you know there are many other books and published literature which talk of coupled fields in rotating machines and other equipment. So in this slide we are going to analyze skin and proximity effects in conductors. In this first figure we have an isolated conductor carrying current at some frequency. As frequency increases the skin effect increases and current tries to become be more and more on the surface reducing the effective area and increasing effective AC resistance and hence the losses. This can be analyzed by coupled circuit field FEM analysis wherein this conductor is fed by an external voltage source or current source. How do we do that we will see later in the course. Same conductor when it is brought close to a metallic conducting plate then this skin effect gets skewed as shown here. This is because the alternating magnetic field produced by this current here by this conductor it induces eddy currents in this plate and because of this proximity effect and corresponding interaction the skin effect gets skewed and that further reduces the effective area of this conductor and it increases its effective AC resistance and losses. If we now see case of two parallel conductors first case is currents are carried in the same direction then again skin effect in both the conductors gets skewed as shown and the currents are concentrating in regions which are not facing each other. It can be proved that the impedance offered by these regions are lesser as compared to the other regions of this conductor. This is because of the fact that these regions they can be shown to link lower flux as compared to these regions and that's why they have lower inductance and lower impedance and that's why they carry higher currents. Case two currents are in opposite direction. Now here in this case the currents try to concentrate in regions which are facing each other and this is because of the fact that now here the flux is mostly concentrating in the region between the conductors as against the first case where flux was mostly around the conductors and flux was not there in the region between the two conductors. This basically reverses the phenomena. Now the inductance offered by these regions is much lesser as compared to other regions and that's why the current concentrates more in these regions and hence again the effective AC resistance goes up and the losses go up. So such analysis can be easily done by coupled circuit field analysis and in fact we will see in one of later lectures how this can be done. So finally we see how do we compute interest currents in a transformer or even an induction motor. Now here case study involving a transformer is shown there in a transformer is switched on to a voltage source and depending upon the instant of switching and the magnetic circuit condition phases ABC will draw currents and NUSC currents will be different because instant of switching will be favorable to one phase but may not be for others. For example here for phase A the instant of switching is unfavorable as compared to the other phases that's why the interest current is quite high as compared to B and C phases. Now these such kind of analysis can again be done by using transient voltage-fed coupled circuit field analysis involving magnetic nonlinearity. So there are three complications here first of all it is a transient analysis because with respect to time we want to plot current second is it is you know nonlinear analysis because magnetic circuit is nonlinear it saturates and hence the current drawn can be quite high and thirdly this is coupled circuit field analysis because voltage some voltage source is feeding this transformer and this transformer is switched on to that voltage source so again it is coupled circuit field analysis. So it is transient nonlinear coupled circuit field analysis how do we you know do such kind of analysis we will see later in the course. So now having understood the importance of finite element method in general let us understand to some extent the difference between analytical and numerical techniques. The analytical techniques are the ones which typically give you exact or close form solutions like for example here you have a parallel plate capacitor I know excited by some voltage difference V and you know it is just electric field intensity is just V by D. So in this course you know when I am explaining many of the concepts I will take help of very very simple devices like capacitor, inductor, transformer and explain you concepts. So here you know this E is equal to V by D is basically a exact or close form solution but here this close form solution was possible because we neglected the fringing at the ends. If I actually you know ask you to consider this fringing also then the problem becomes suddenly very very difficult and you know analytical formulation becomes very very complex. So but the advantage of this close form solutions if available is that dependence of the field variable on the governing influencing parameters is exactly known. For example here we know that electric field intensity is inversely possible to the distance. So looking at the equation you can directly understand the dependence and that is the advantage whereas when you use a numerical technique like FEM you do not get that you know dependence readily. There what you have to do is you have to do then number of parametric studies you have to vary the parameters in FEM analysis and then generate curves like the one I showed in the last lecture that rotor bar breakage versus resistive resistive versus torque. Those kind of graphs have to be generated and then you will get the dependence but that will be after series of parametric studies whereas in case of exact solution through analytical formulation you readily have governing equation and influence is readily seen. So now let us increase little bit complexity of the geometry. Now we are going to cylindrical you know coaxial system of two cylindrical conductors inner cylindrical conductor is at voltage v and outer conductor is at ground potential. Now again by using you know starting from Laplace's equation you can easily derive this analytical formula. So now the analytical formula becomes little bit complex but still it is very much manageable but such analytical solutions are possible only for one dimensional and two dimensional problems and that too without geometrical or vertical complexities. Now the same actually geometry if I modify little bit and make this outer cylindrical conductor as straight one suddenly the problem becomes almost very very difficult or impossible for you know for doing hand calculations. Yes but you know researchers earlier they found ways out to find solution for these also. So what they what did they do they basically you know use what is known as method of images. So method of images is basically what you are doing is you are replacing this ground by having an image conductor with a negative potential here. So that you will get potential here which is of 0 and this is a voltage is minus u here the voltage is plus u. So now since you are having a potential on this line which is 0 you have effectively replaced you can effectively replace this ground by image conductor. So one problem in terms of you know geometrical complexity you have eliminated by using image conductor. Secondly what then you have to do is this cylindrical conductors also the original conductor and the image conductor you can replace by equivalent line charges one with rho L and one with minus rho L and then you know you but you can replace them as long as you ensure boundary conditions. What are the boundary conditions that on this circle here and this circle here you should get plus u and minus u. So that is why this plus rho L and minus rho L has to be the position has to be adjusted so that you get equipotential surfaces or contours in 2D plane here and then you can then eliminate these conductors also. So now then the your geometry reduces to simply two line charges. Now you have then you know for line charge you have the analytical solution very simple straight forward solution is available and then you can use those formulae for positive and negative line charge and superimpose and at any point in the whole domain you can calculate the potential or electric field intensity. So this is how you do it but this is not without lot of you know involve mathematics. If you see this reference and appendix of this lot of you know maths is involve to derive this formula for E p and what is E p? E p is this point here the point where maximum stress occurs. So electric field intensity at this point is then given by this formula where gamma is given by S by R S is this distance between center and this ground plane and R is this radius and here M is given by square root of S square minus R square where M is the position of this line charge with respect to positions of this two line charges with respect to this ground plane. Now if we go a step further if we make the electrode shapes even more complicated like you know one of the electrodes is having a sharp point or the electrode is something like this then of course the analytical methods become very very difficult. Even for such things you will find in the literature they have the researchers have done some approximations and they have come out with some you know approximate formula for such cases but then they become geometry dependent. Every time you encounter such complicated arrangement of electrodes you will have to derive a separate formula. The main advantage of numerical techniques is that formulations in this techniques are geometry independent. That means let the geometry be of any shape the FE procedure is the same. So that is the main advantage of finite element method or any such numerical technique. So what is the FEM finite element method procedure? So first you have to draw geometry like the one here. So the geometry is the previous case wherein we analyzed a high voltage lead near the ground electrode. So first you have to draw the geometry which is very straight forward in any of the CAD packages or whatever. Now the next is you have to make this problem bounded. As I mentioned during the first lecture finite element method is a method which requires a bounded structure. So you have to make this problem which is closed. Now where should this fictitious boundary be placed? It should be placed far away from this electrode as shown here. Now you can see here this electrode what we are interested is the stress distribution in this part. We are not interested in field distribution here but at the same time you need to take these boundaries three boundaries far away from this electrode. So that the boundary conditions on these three do not influence unnecessarily field conditions in this zone. So more about this we will understand when we get into details of FE procedure and its applications. So once you have this bounded structure then you have to discretize into finite elements and then approximate the solution in each element. Now these finite elements as shown here these are the triangular elements small small triangular elements the whole region is filled with such triangular elements. So for example if this is my lead the triangular elements would be something like this and so on. Now by triangular element yes we can use any other shape also but triangular element is much easier to code and that is why we are using this. Later on we will discuss more about this. Then approximate the solution in each finite element for example u which is the voltage is a plus bx plus cy in two dimensions the one of the approximations could be this. Then you derive the element coefficient matrix which represent the energy of that element finite element which is a function of its geometry and material properties right. So then you assemble all such element coefficient matrices and form what is known as global coefficient matrix. Then after having form global coefficient matrix which represent the energy of the entire system then you minimize the energy in one of the FE approaches. There are some other approaches also which we will discuss later but then that will give you typically for this case equation which is Ax is equal to 0. So this does not have unique solution unless you impose boundary conditions. So boundary conditions is that you know this lead is at potential v and this is at whole thing is at ground potential boundary. When you do that then you get the form of equation which is Ax is equal to b which can be easily solved to determine the unknown which is x which is the potentials at various points in the domain. Now using this finite element method procedure we can calculate the stresses at various points in the domain. For example here there are three cases you can see in this slide there are three cases one is coarse mesh, fine mesh and very fine mesh. This is the zoom part in very close to the lead and as you increase the mesh density or reduce the finite element mesh size you get much better result as can be observed from this table. So as you increase the mesh density that means you increase the number of nodes and finite elements from 796 to almost 12000 the stress maximum stress at this point which occurs at this point which is the minimum distance between this high voltage lead and ground. The stress comes more or less very close to the analytical value which is calculated from the formula that we saw in the one of the previous slides. Here you can see the uniform stress is this is the 100 kV is the kilo volts is the difference 40 mm is the clearance 6 mm is the radius of the electrode. So you have 2.5 kV per mm as the uniform stress if this you know was a if this were a parallel plate geometry. But because of this radius of the electrode this field distribution here is non-uniform and basically this E max 6.97 divided by this 2.5 which is uniform stress gives the enhancement factor what is called as enhancement factor. As compared to uniform field stress you are getting 6.97. So this is the enhancement in the field due to non-uniform field condition. Now quickly summarizing the difference between analytical and numerical techniques in analytical techniques closed form solutions are available here they are not available solutions is available are exact you need to do lot of improvements like mesh improvement in mesh size and all that to get accurate results that is very much possible. Dependence of field on influence factor can be readily seen here we have to do number of parametric studies complexities cannot be easily handled complexities in geometry and material can be handled applicable for 1 D or 2 D problems whereas there is no such limitation in case of numerical techniques. They can easily solve 3 D problems also with available commercial course or if you develop your own you know course even in 3 D. So, and so we are going to study in this course only finite element method which is based on typically either variational approach or wetted residual approach. So variational approach is energy based energy minimization approach whereas wetted residual approach is error minimization approach. So we will see both these methods but we will see more of variational approach because it is easy to understand because it is more closely related to the physics. Then the next method is finite difference method it basically has a procedure in which you have the whole geometry converted into a uniform grade and that is in fact it is one of its main limitations that you need to have uniform grade there is no such requirement in case of finite element method. But it is easy to implement because derivatives in partial differential equations they are converted into differences. So method is conceptually very simple but it becomes difficult to handle material interfaces in this method. In a one of the simplest you know applications for you know Laplace the potential at this point o or 0 is expressed as function of the surrounding 4 nodes and in case of Laplace equation the u 0 become just one fourth of u 1 plus u 2 plus u 3 plus u 4 it just average of surrounding 4 potentials. So it is very intuitive and very simple technique but as I said for practical problems it has some limitations. Then there are some integral methods, integral equation methods like charge simulation method, boundary element method or method of moments. As I mentioned previously in the first lecture that these methods are useful for open boundary problems and the involved maths to derive the final set of equation is high and they lead to full matrix whereas finite element method as we will see later is a highly sparse matrix and you can exploit that sparsity to your computational advantage. But the advantage of these integral equation methods is the order of the problem reduces by one and so depending upon the problem type you either have to use integral equation method or finite element method for your application. So we will see more of these methods particularly finite element method in the next set of lectures. So what we are going to do is now from the next lecture onwards we will see some basics of low frequency electromagnetic as relevant to finite element method and analysis of electrical equipment and machines. So having done that for about 8 lectures of half an hour approximately then we will get into the finite element method theory and then applications. Thank you.