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.