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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.

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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.

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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.