Scaling in Electrostatic Forces|| MODULE 3 || KTU MEMS || LECTURE 19

Hi all, today we are discussing about scaling in electrostatic forces. So for that we have considered two parallel plates. In these two parallel electroconductive plates, a potential voltage V is applied. An electric field is generated there. We know the equation of the corresponding energy generated by that. U is equal to minus half Cv square. How can we write C? Epsilon zero epsilon r into A by D. Have you ever studied Epsilon A by D? Epsilon zero epsilon r into A is equal to W into L. Width into length is area. Divided by 2D. 2D is the distance between the plates. Now we are looking at the scaling here. L is the linear scale of the electrode. We are assuming L as the linear scale of the electrode. Epsilon zero and epsilon r are constant values. So, epsilon zero epsilon r is proportional to L raised to zero. WL is equivalent to L raised to 1. We can approximate the scaling voltage using the Paschen's effect. This figure illustrates the Paschen's effect. Voltage V drops with an increase in gap for D less than 5 micrometer. Voltage V increases with an increase in gap for D less than 10 micrometer. So, V is proportional to L. We are substituting these values into the potential energy equation. Potential energy U is equal to epsilon zero epsilon r into L raised to zero. W is equal to L raised to 1. V is equal to L raised to 1. L raised to 1 is the whole square. Whole divided by 2 is the constant. D is equal to L raised to 1. U is proportional to L raised to zero. L raised to zero is equal to 1. L raised to 1 is equal to L raised to 4. L raised to 1 is equal to L raised to 1. A ten times reduction of linear size of electrode. A thousand times reduction in potential energy. This is what we understood in the case of energy. A ten times reduction of linear size of electrode. A thousand times reduction in potential energy. Now, let us look at electrostatic phases in misaligned electrodes. If the direction of alignment is different, the width of the electrode can change. The length of the electrode plate can also change. The distance between the electrodes can also change. So, we can define electrostatic phases in three ways. First is Fd. Fd is the distance between the two plates. We can write it as minus dou u by dou d. What is u? It is epsilon zero epsilon r into WL V square divided by 2d. It is differentiated with respect to d. What is the derivative of 1 by d? It is minus 1 by d square. Instead of d, we can write it as minus 1 by d square. How will the equation change? It is minus 1 by 2 into epsilon zero epsilon r into WL V square divided by d square. Now, we can find out the values of these two. What is the derivative of epsilon zero? It is L raised to 0. What is the derivative of epsilon r? It is L raised to W. What is the derivative of L? It is L. What is the derivative of V? It is L. So, instead of V square, we will write L into the whole square. The whole divided by? Instead of d, we will write L. So, instead of d square, we will write L. In the numerator we have L raised to 4 and in the denominator we have L squared When we cut both of them, we get L squared Now let's look at the width For the width, we can write Fw as minus dou u divided by dou w We have taken w as the corresponding value We have taken u as dou by dou w of 1 by 2 epsilon 0 epsilon r into w L v squared divided by d When we take the derivative, we get w dou by dou x of x is 1 Similarly, when we go to the term w, all the other terms are there Then we get the equation like this When we substitute all the values, we get L squared When we take Fl, we get dou u divided by dou L which is equal to minus 1 by 2 into Here we can see the rest of the terms after L So we get minus 1 by 2 into epsilon 0 epsilon r into w v squared divided by d This is proportional to L squared We can conclude from this Electrostatic force Fd Fw Fl is proportional to L squared So a 10 times reduction in electrode linear dimension will cause 10 squared or 100 times reduction in the magnitude of electrostatic forces This is the scaling in the case of electrostatic