We have seen how we can derive an all-region model for the MOS transistor.

In this video, we will see how that model can be simplified.

We will call the result the simplified all-region model.

So here is the situation so far. In our development of models for the MOS

transistor, we have just describe the complete all region model.

And now we will start with a simplified all region model which we will present in

two forms, a body reference one and a sourced referenced one.

As you remember, we had some terms, sub cs to the one half and cs to the three

halves. Those terms, in the complete origin

model, originated in the expression for QB, the depletion region charge, that

contained the square root of[UNKNOWN] in it.

Okay. In a cab model, you'd like to be as

computationally efficient as possible, because, when you analyze a circuit,

let's say a transient response of a circuit, you may have

Hundreds or thousands of transistors. And the, you may want to evaluate the

transient response at thousands of points.

So, as we can understand, computational efficiency for every point is of

paramount importance. So, it's good to try to eliminate these

terms if you can do it without losing too much accuracy.

So that's what we will be doing. So let's plot now, QB versus CS.

Instead of QB I'm going to divide by C ox.

So then the result, what I'm plugging basically here is gamma square root of

psi s. You can see that because we have a square

root. The variation is very gradual.

This thing almost looks like a straight line.

And the surface potential varies between the value psi o at the source and psi l,

the value at the drain. So we only need to be accurate between

these 2. Now, because this thing looks so much

like a straight line, what we're going to do is we're going to choose a point

there. And do an expansion around that point.

So this point I will assume is at the surface potential psi e.

And I'm going to take the square root and expand it in a series and thus keep the

first two terms. So the first term, the 0 of the term,

will be this one. And the second order term would be the

derivative at that point, which turns out to be this, times the difference between

psi s and the point where we do the expansion, psi s minus psi se.

So I would neglect the higher order terms.

This, of course, will give me a linear variation, so it will approximate.

This smoothly varying square root by a straight line.

Now, it is common to write the equation I just showed you in this form.

We introduce a quantity, alpha. And if you look at the development of the

equation in the previous slide, you find that alpha has to be of the value shown

here, on the upper right hand corner. So we will be using this form for QB over

C ox and we will rederive our model based on this equation.

So you go again through the development of the drift and diffusion components for

the drain source current, but using this expression for QB, and you find that you

get this result for IDS1, the component into drift.

And this result for IDS2, the component due to diffusion.

And as we expected, we have gotten rid of square root of psi l and square root of

psi o and similarly the 3 half powers of these two quantitites.

So I leave the algebra to you but it, again it's very instructive to go through

the development again and make sure you do get these equations.

Now, this point psi se where we expanded, we didn't say exactly where it was and

there are different possibilities for that.

And depending on where we expand, how we choose the value of psi e we end up with

different models. I will show you two choices.

The first one will be such that we develop the nice body referenced model.

So we're going to choose psi e as the average value between psi o and psi l.

So that's a reasonable choice. You take the surface potential value of

the source in the drain, you take the average value of the two, and you call

the result psi m, and you choose, you choose that as your expansion point.

So here is what we're doing. We are choosing this point as our

expansion point. So now when you do an expansion and you

keep the 0 and first order terms you end up with a straight line that is show by

the broken line over here, and you can see that it approximates the square root

rather well throughout. Now when you plug in this value for psi e

in the equations I showed you before, you end up with this.

Equation for the drift component of the current, and this equation for the

diffusion component of the current. They're very simple.

Of course there is some complexity related to the fact that psi sm appears

over here, but certainly these equations are simpler than before.

In the value for alpha sub m from the development I showed you, turns out to be

this one. Again you should verify every step of

this derivation for yourselves. This, now, is the basis of the PSP

computer-aided design model. These curves compare the full and the

simplified all-region models, and you can see that they're practically the same.

One is the line, the other is the points and there's practically no difference

between them. So now let's go to another expansion

point. That, and we get the source referenced

version of our model from that. So I'm going to choose now my expansion

point as the surface potential at the source.

In other words, this point here. Now when you expand and you keep the

first two terms of expansion, then you're approximating the square root by a

straight line, that is tangent, at that point.

Like this. So you can see that, you get a little

error, near psi sl we're talking about curve A for now.

Now when you plug in this value, into the general expansion term equation that I

showed you, you end up with IDS1 and IDS2, as shown here.

Where alpha 1, shown here and there, it turns out to be this, again I'm skipping

the algebra. Note the following, you see the

difference between the rain and storms surface potentials here, here and there.

And this is typical of short reference model.

We call this a short reference model because we get the expansion around psi

o. The surface potential at the, the channel

end near the source. Now, as you, as I already mentioned

before the expansion around here gives you a, a straight line which is tangent

to the square root at psi so. Now if we're going to approximate the

square root by a straight line, we might as well use the straight line that gives

us the least error and that would be, something like, curve B.

So, you pass another straight line, still it passes throughpsi so and it still

approximates the square root, pretty well, but it actually approximates it

better because this straight line has been designed in such a way, as to give

you minimal overall error. So, that corresponds to a smaller slope,

and it corresponds to a smaller value of alpha 1 than what does, I show you here.

So if you lower alpha slightly, you get better performance out of this model.

Finally there is a third choice, which is historically one of the first choices

that was made, or rather the assumptions were such that it's equivalent to having

chosen this straight line to approximate the square root, which is actually a

pretty poor choice. Unfortunately, as you will see, this is

the choice that leads to the very popular square low models that you'll find in

circuit books. From this you'll see that you cannot

expect much accuracy of those models and that is actually the case.

Now, I remind you. We're considering this structure, and all

of the models we have presented so far, are of this form.

They give you the drain source current as a drift and diffusion, and a diffusion

component, the sum of the two. And they depend on the surface potential

of the source and not the drain. And to find those from the externally

applied voltage, VSP and VDB, you have to solve two implicit equations.

And you have to do that numerically. The set of the three equations together

can represent any of the models I showed you.

The only thing that is different is the actual functions that give you IDS1 and

IDS2. Notice that in the models we presented so

far, we do not have the current as an explicit function of the terminal

voltages. Not yet.

We have them as a function of the internal surface potential, psi so and

psi so. So by now we have seen three different

all region models, a complete one and two simplified ones.

One body reference, the other source reference.

We can now start focusing on particular regions of the inversion and there we

will see that we can have current expressions that, that are explicit

functions of the externally applied voltage.

Before I conclude, I would like to mention that they are also charge based

models. You can start from any of the surface

potential base models that I have presented, and you can show that the

drift component can be written in this form, and the diffusion component can be

written in this form. And there are ways to evaluate Qi 0 and

Qi l, which are, are the inversion charge charges per unit area, near the source

and near the drain. We will not have time to go through these

models, but they are described in the text.

popular models like EKV or ACF are based on these expressions.