 So, welcome to lecture 37, now we will go to next level of complication, we will now consider transient analysis. So, transient magnetic problems for again in 2D, the governing equation will be this. So, del square 1 upon mu del square A is equal to minus j 0 plus sigma dA by dt. So, we are taking into account the conductivity and the corresponding diffusion term. So, the global system of equation will be C A as a function of time d D d d d daba by daba t of A t is equal to dt. So, this is representing the diffusion term and this sigma is directly reflected into this capital D. This as usual Laplacian gets reflected into this C, capital C global coefficient matrix which is function of geometry and material properties into column vector A, no vector of matrix, column matrix of magnetic vector potentials which are function of time t. Now, most popular time discretization technique is this. What you do is, you basically this A t plus delta t that means magnetic vector potential at t plus delta t minus A of t divided by delta t, you express as beta times daba A by daba t at t plus delta t plus 1 minus beta daba A by daba t at time t. So, this is the most popular discretization scheme and what is beta? Beta is just a number between 0 and 1. So, if beta is 0 then if you substitute it here, this will go to 0 and then this beta is 0. So, this remains just daba A by daba t at time t is equal to A t plus delta t minus A t upon delta t. So, what we have done is, we have expressed different partial derivative with respect to time as a function of forward difference because we are taking derivative of A with respect to time at t as a function of difference of A t plus delta t minus A t. So, this is a forward difference because this is a forward in time. So, basically A at t plus delta t minus A at t is taken divided by delta t is taken as daba A by daba t at t. The backward difference will be daba A by daba t at t plus delta t is equal to A t plus delta t minus A t upon delta t. So, here this what we have done, this partial derivative at t plus delta t is a function of this backward difference because now we are at t plus delta t. So, minus A t, this becomes backward difference. Here since we were at time instant t, this was forward difference. So, this is backward difference beta is equal to 1 and then third is Krang-Nikolson in which we take sort of average of partial derivatives of A with respect to time at t plus delta t and t and then average of this is this. So, for Krang-Nikolson method beta is 1 by 2. So, if you substitute beta is equal to 1 by 2 here, you will get this equation. So, again starting with this equation which we have seen on the last slide, if we multiply both sides of this equation by beta at time instant t plus delta t, then we will get this expression. So, beta times this, so this is this term beta times this is equal to B, beta times this term taken on the right hand side that is why it becomes minus plus beta times this at t plus delta t because we are doing this at t plus delta t. And then multiplying the same equation on both sides by 1 minus beta at t, you will get this 1 minus beta times this is equal to minus of 1 minus beta into this term taken on the right hand side plus 1 minus beta into B of t. Now, multiplying equation A by matrix D, so we multiply this A by matrix D. So, then we will get here D, here also we will get D matrix, here also we will get D matrix, why we have to do that because in these two expressions now you have D here, D times this. So, we are bringing it to that form. So, we are multiplying both the sides by D and then we substitute these two equations in that then we will get this total expression where this D times this is coming here and then this beta times this and this is being replaced by because here D times all three are multiplied by D all these three terms by D and D times this and D times this are replaced by these two expressions, is it clear? See here we multiplied it by D all the terms and now what we have already got D times beta dA by dA by t this and D times this we already have these expressions now. So, D times this will be equal to this plus this. So, that is what we have got now these four terms are on this side and D times this is on the right hand side. So, we have just substituted this and this in place of these two here. Now, what we will do is we will combine this term suitably in this whole equation. So, terms marked in blue are combined because this is in A t plus delta t then this is marked in green is B D plus delta t and this is B t. So, that we are combining together because both are in terms of B and then third is in black. So, this is A times t sorry A of t and this is also A of t right. So, these two terms we are combining together and then we are also dividing by beta here the whole equation we are dividing it by beta and then we are combining this blue, green and black term in this equation. So, for example, this just remains C A t plus delta t because this whole equation is divided by beta right. So, that then other terms also similarly are adjusted and arranged. Now, we are actually starting with this this equation is rewritten here and now you can see the three terms. Three schemes that emerge are for beta equal to 0 beta equal to 1 and beta is equal to half. So, if beta is equal to 0 then you have forward Ehlers method you substitute beta equal to 0 here this term will go this term also will go and what will remain is only D times this is equal to this beta is 0. So, this plus 1 times B of t. If beta is equal to 1 then that will give backward Ehlers method and then these two terms 1 and minus beta 1 minus beta they will become 0. So, these two terms will go. So, D times this will become simply this is beta is 1. So, C into A of t plus A at t plus delta t plus beta is 1. So, then it is just B at t plus delta t and then if beta is equal to half which leads to Kank Nikolson method. So, there is a if beta is equal to half then it leads to Kank Nikolson method. So, there is a type over here. If beta is equal to half then you get this. So, then you get the average of those two terms. So, none of these terms goes to 0 and then you get average of the corresponding terms. So, now, let us see how do we implement this for example, in Poisson's equation and then coupled to circuit equation. In time domain. So, now our Poisson's equation is this del square A is equal to minus mu times j naught. So, j you replace by n i by s as we had done earlier in previous lectures and then this leads to matrix equation C times A minus B times i equal to 0 where B at the element level for the ith node is n by s delta by 3. So, that you can see in the previous lectures on you know current driven system we had similar you know expression and voltage driven. This is voltage driven because i is variable. So, i is unknown. So, this will be voltage driven case. So, now the corresponding circuit equation is this. This also we had seen it earlier same you know circuit coupled with finite element model this is the field region and this is the circuit region. So, this is a voltage driven circuit coupled to field model. So, now the circuit equation this also we had seen u is equal to n d psi by dt r into i plus l d i by dt and this n d psi by dt is the corresponding terminal voltage here at the device. Then if you actually you know apply a free formulation then u becomes g dash d a by dt plus r external i plus l external d i by dt this we had seen earlier. So, now the two equations become C A minus B i equal to 0 and u is equal to this. These two equations also we had seen earlier they are identical to what we had seen earlier for voltage fed circuit coupled to field model. In sequence it over. So, combining now these two equations then you can get this matrix equation. So, you have in the first equation there is no d a by a by d a by t and d a by i by d a by t. So, that is why you have 0 0 here and in the second equation you have g dash getting multiplied by a d a by a by d a by t is to be d a by a by d a by t and similarly l external is getting multiplied to d a by i by d a by t. So, this one and then plus now the you know multiplicants to a and i. So, they are C here minus B and in this equation it is second equation it is r external into i and then on the right hand side you have u in the second equation in the first equation it is 0. Now, actually we had got the same equation earlier for time harmonic case, but it is not in time harmonic. So, this equation now is in time domain, but same equation you know if we converted to time harmonic case wherein we replace d a by d a by t at j omega then we get this and this is this equation we had in fact seen earlier. So, here what we have done d a by d a by t is replaced by j omega. So, that is why j omega gets multiplied with g dash and now this since it is there is no d a by d a by t. So, now a and i become common and d a by d a by t terms get multiplied by j omega. So, then we get a simpler equation which we had seen earlier. So, since now we are dealing with time domain and transient simulation we will be working with this equation. Now, following the method of discretization what we did you know if you remember we multiplied one equation what we had. So, here also we multiplied there were you know 2 2 see here this equation we multiplied both sides by beta at t plus delta t and then multiplied by 1 minus beta at time t. So, same thing we are doing here this equation we are going to multiply both sides by beta at t plus delta t. So, you get this and then multiply both sides by 1 minus beta at t. So, you are then getting. So, that is why here it is t plus delta t. So, here it is t plus delta t, here it is t. So, all these superscripts are at t plus delta t here they are at t. So, both the equations are identical only the variables are at t plus delta t here they are at t here and here it is beta here it is 1 minus beta. So, now we add both these equations this whole equation and this equation we add together then we get this. So, what we are doing we just can you just see here these terms are common is it not these are common multiplicants. So, they are taken out and what is common to them is in the in this bracket. So, it is here beta times this here it is beta times this 1 minus beta times this and then you get this entire equation. Now what we will do is we will use that equation A which was there on the first slide and it was beta times dA by A by dA by t at t plus delta t plus 1 minus beta into dA by A by dA by t at t is equal to this difference divided by time delta t. So, this actually whatever we have it here in this equation we replace it by so you get this. So, now same equation is written here again this is the same equation what we have got here is written here and now the final step what we have to do is because this is a transient simulation. So, we have to bring t plus delta t terms on one side and all t terms on the other side because we are marching in time. So, we already have values at time t and we want to calculate values at values of variables at t plus delta t. So, we are just rearranging this equation by taking t plus delta t terms on the left hand side and all the variables and they are multiple currents at t time t on the right hand side. So, this is our final equation for implementation in finite element method. So, what we are going to do is we when we get when we discretize all these by using finite element procedure all these terms by our usual procedure they are known C is known is it not these are global coefficient matrix V is also known G is known R external L external are known everything is known and beta depending upon what we select which scheme we select beta will either it can be 0 it can be 1 or it can be half. So, depending upon that beta value will be so the nine accordingly it will become forward backward or Krang-Nicholson scheme. So, with this we complete this theory of transient FE formulation in the next lecture we will go to non-linear formulation. Thank you.