When the voltage is supplied to a transistor vary very fast, and the

quasi-static operation assumption is no, not valid.

We have to allow for non-quasi-static operation.

This requires a different form of analysis as you will see.

We will make a transition from quasi static modeling to non-quasi static

modeling by using the multi segment idea I have presented in the previous video.

Remember that in order to help extend the validity of a quasi static model, we

split it into m segments. And the higher the speed were trying to

accommo to model, the larger the number of segments that we must split the device

into. We can take this idea to the limit and

allow m to become infinite, in which case the length of every subdevice here

becomes infinitesimal. And then we are led to non-quasi-static

model. Notice that in the general case, we can

expect that the current will be a function of position in the channel, and

of course the time, since the voltages are changing.

And the same we can say for the inversion layer charge per unit area.

It will depend on position and on time. Key to quasi, non-quasi-static modeling

is an equation called continuity equation, which expresses charge

conservation at the point in the channel. So let me go through a derivation of that

equation. Here we have a part of the channel, this

is the width w of the channel, this is a very small length delta x.

And we're going to consider the current that goes in, the current that goes out,

and the charge accumulation inside this section of the channel.

Now, let's call the current coming out i. The current coming in in general can be

different, let's say by amount delta i, because of the transit effects to the

channel. That means that a larger number of

carriers per unit time go in than come out, and therefore charge will accumulate

here. Let us say that in time delta t, the

amount of charge accumulation here is delta qi.

What is delta qi prime? It will be delta qi divided by the area

of the element as we look at it from above, which is W times delta x.

And what is delta qi? It is the charge that enters in time

delta t minus the charge that leaves in time delta t.

So it will be the current entering times delta t minus the current leaving times

delta t. And in this difference, you can see that

i delta t cancels out, and we end up with this equation.

Now if I rearrange the terms in this equation, I can write it in this form.

And now I can allow delta x to go to zero, in which case this becomes the

partial derivative of i with respect to x.

And I can allow delta t to go to zero, in which case this becomes the partial

derivative of qi prime with respect to t. So we end up with this equation, and this

is the continutiy equation. Again, this equation is a statement of

charge conservation. Now how do we apply this equation in the

non-quasi-static transient modeling of the device?

I will give a strong inversion example. We will write the equations we know from

DC analysis, and we will also write the corresponding equations in

non-quasi-static operation. First of all, the charge per unit area

according to the simplified source reference strong inversion model, is

given by this. You can look at the book.

But when we discuss strong inversion and the simplified model, you will see this

equation there. VFB and V0 plus gamma times this square

root is the local threshold voltage at the point in the channel.

Then the corresponding equation for non-quasi-static operation can be

obtained from this one, by replacing qi prime by q, small q capital I, meaning

the total charge. That is varying with respect to position

and time. Replace VGB by VGB of t, replace VCB by

VCB of x comma t, because now the channel body voltage will vary both with position

and time. And you do the same thing over here.