In this video, we will see how the results of our general analysis can be simplified to result in equations that would be valid in the weak inversion region. I will start by showing you again the plot of surface potential versus gate body voltage, but the emphasis now is in, in weak inversion, where this goes almost like a straight line. I don't care about moderate and strong inversion for the purposes of this video. Weak inversion is defined as the point where the surface potential from the point where the surface potential is phi F to the point that it is 2 phi F. And the corresponding values for VGB can easily be calculated. The top of the weak inversion region would be marked with an M. This is the beginning of the moderate inversion region, hence the M. So, in the general inversion case, we have derived this equation. This equation, I remind you, gives you the inversely or charged versus the surface potential, in all regions of inversion. And now I would like to simplify this equation for the case of weak inversion. In weak inversion, the surface potential is between phi F and 2 phi F. So, psi S minus 2 phi F, for almost all of the weak inversion region, is a negative number, and this exponential becomes very, very small. Phi t times this exponential is a very small, quantity. I will call it xi, and because it is very small, I will be able to make some simplifications. Using xi for this quantity you can write the quantities inside the parentheses like this, root of xi S plus xi minus root of xi S. If you, take a power series expansion of this, and maintain only the first two terms, you end up with this. Okay? So, this is the value of this when xi is zero, this is the, first derivative of this, multiplied by xi. And this one, is this. So, now this and this cancel each other out and you end up with this result, that all of this, all of this quantity inside the parentheses here, can be simplified to this. So, doing this, and plugging in for xi the value here, we have this. So, this is now the simplified equation in weak inversion, and it is clear that reversely are charged depends almost exponentially on the surface potential. There is a slowly varying square root of cs also in the denominator. But the, by far, the drastic, variation of this QI is due to this exponential. Now in the weak inversion region, you have only very few electrons, so you can practically assume that the depletion region only contains acceptor atoms. So you can use the same approximation so as, as for the depletion region where no electrons were present. Back then we had shown that the surface potential for the depletion region case simplifies to this, and we had called this CSA, and this is the approximation we will use for surface potential. Csa, when you plot it, is this. You see it here, it's this one. This is the exact surface potential, and this is CSA. You can see that they deviate From each other in the moderate inversion region, but in the weak inversion region, they nearly coincide. So we're justified to use the simple expression for psi sa in the weak inversion region. So now I will go here and help them replace psi s, here and there, by psi sa, and we end up with this equation. To emphasize that PSA is a fraction of the externally applied gate by the voltage, I show it explicitly here. Now we can simplify things further because eventually we want to develop an equation that gives you the inversion linear charts as a function of VGB itself. So how can we simplify this equation further? Here is a plot again of the surface potential psi s, versus VGB. This is psi sa, and we're going, as I already said, to approximate psi s by psi s a. I'm going to concentrate on this point, the top of the weak inversion region. And at that point, the slope of either of these two, I will call 1 over N0. The reason is simply historical, okay? So I'm calling it 1 over N0 so that the equations that will come out have the standard form that you may have already seen in is, is certain literature. So now, if I assume that this is a straight line, and from the plot, it almost is, instead of talking over here about differences between csa and 2 phi f, which basically mean, let's say you're at this point, here at this point. This is CSA at this point. And you're taking the difference here, between 2PhiF and CSA. Instead of talking about these differences, you can relate them to differences between VGB and VMO. You can relate in it, them to this difference. So then we can write that CSA minus 2 phi F is equal to VGB minus VMO the value here times the slope and once we have this I can replace this in here and the equation becomes very simple. So this one I replace by this over here, and this equation also contains a square root of CSA, and I'm going to use as an approximation, the value of CSA, at this point n, which is actually two phi f, you can see here. So now I have an explicit equation for the [unknown] charge, as a function of a gate body voltage. Now, you may wonder, if I wanted to make an approximation for the weak inversion region, why did I choose to expand around this point rather than the middle point, and you will be right to wonder. You can also expand around that point, but it turns out that you do not gain much in accuracy. And the result I have obtained above, will turn out to be convenient later on. So again, we have an explicit equation for the inversion layer charge, as a function of the gate voltage, and this equation, will turn out To develop into the corresponding transistor equation for the quick inversion current versus gate voltage. Let us plot this equation, and also plot the strong inversion equation we derived in the previous video, and see how they compare to the exact equation. So, here I'm plotting not Qi, but the magnitude of Qi and in fact I'm taking the logarithm of it so that I can show several orders of magnitude over here, and I'm plotting this versus VGB. The exact equation for inversion, which we have shown, is this solid line. This is the weak inversion equation that I showed you before, which showed that magnitude of qi is proportional to the exponential of vgb minus vmo over a null. Because this is exponential, when you plot it on the log axis, it becomes a straight line. It is this here. As you can see, it matches the exact result pretty well in the weak inversion region. Now, the strong inversion equation, that was this one, which we developed in the video before this one, turns out to be this. As vgb approaches vt zero, this quantity goes to zero and the logarithm goes to minus infinity, this is why this becomes, like this. We don't care what it does here, because we said this is the strong inversion equation. The strong inversion region is over here. So over here, the two, this equation really, matches well, the exact result. So, we have the strong inversion equation that matches the exact result in the strong inversion. We have the weak inversion equation that matches the str-, the exact result in the weak inversion region. But, both the weak inversion equation and the strong inversion equation miserably fail, in moderate inversion. This is why, we cannot say that below strong inversion, we have weak inversion. Sometimes, people talk about sub threshold behavior. Meaning weak inversion, and above threshold behavior, meaning strong inversion. That will not do, especially in these days of very low voltage circuits. Moderating version is a key region, and we have to be careful how we handle it. We cannot pretend that it is not there. The only problem is, we cannot find a simple expression for it. So we will have to be using, the complete expression, that we have already derived. In this video we have seen how the results of our general analysis, can be simplified to lead to equations that are valid in that weak inversion region. In the next video we will conclude the, study of the two-terminal structure, by talking about small-signal capacitance.