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In the previous video we determined a set of nine capacitance parameters that we

claimed to be a complete set. In this video I will show you how we can

derive a complete quasi-static small signal equivalence circuit involving

those nine parameters. This is the set of charging currents that

we decided to keep in our modeling effort in the previous video.

Now, you can take these equations and transform them into a different set of

equations that would be more suitable for deriving an equivalent circuit for our

small-signal model. The, the types of transformations you

have to do are related to Kirchhoff's voltage law and I can give you an example

here, let us take three of the terminals gate, drain and source.

We define a source voltage with respect to the arbitrary reference which we have

called ground and this source voltage is Vs.

Correspondingly we have a gate voltage Vg and a drain voltage Vd.

These are three voltages from terminals to ground.

But can also define terminal to terminal voltages, for example, this is Vgs, this

Vds and this Vdg. Notice that way we use the arrow here the

arrow points from the from point to the to point, so when I say the voltages,

from drain to gate, or between drain and gate, vdg the arrow is defined as

pointing from d to g. Now, we can write some relations between

these voltages by using Kirchhoff's voltage law.

So let us for example, go around the loop from d to the ground to s and back to d.

We have v sub d which can be written as vds plus vs.

This relation here,we can also write Vds is equal to Vdg plus Vs.

This comes from this loop here. And, Vdg is equal to minus Vgd of course.

So Vdg is the voltage from here to there Vgd is the voltage here to there.

So one is the opposite of the other. So with type of transformations, if you

play with these equations and you use the relations between capacitance parameters

which we derived in the previous video, you end up with another set of equations,

the algebra will be skipped but it can be found in the book.

These are the transformed equations and we have introduced for convenience three

new parameters, C sub m is Cdg minus Cgd, Cmb is Cdb minus Cbd, and Cmx is Cbg

minus Cgb. You may recall that I mentioend in the

previous video that in general, Cxy can be different, different from Cyx.

The difference between these corresponding capacitances Cdg minus Cbd

is called Cm. The same for the body related

capacitances and the body gate and gate body capacitances.

This, so this in a sense, these are the capacitance parameters that express the

non reciprocity of the behavior between two terminals.

And I will be more specific in a moment. So now, if you look at the total number

of parameters, capacitance parameters we have here, we still have nine.

We had said already that nine is the number of independent capacitance

parameters that we can have for the four terminal device we are considering.

So the total number of capacitance parameters here counting, Cm, Cmb, and

Cmx in these three equations is nine. Now you can take these equations and

derive a small signal equvialent circuit. If you do that and you're the

corresponding transport current in the small signal model, you get this

equivalent circuit. Now, this equivalent circuit may seem too

complicated at first, but actually it is easy to see the role of each of these

elements. Then you first repeat the Cm, Cmb and Cmx

are defined by these equations. All nine capacitance parameters are here.

So this is a complete quasi static model. I'm going to compare this model to the

one we had before. This is the simple capacitance model we

have, with only five capacitance. And this is the new model we have

derived. So what are the differences here?

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There is a new current source, cmdvgsdt, another one cmbdvbsdt, a third one

cmxdvgbdt, and finally, there is a new capacitance csd.

These are the four new capacitive elements.

And I call them capacitive, okay, this clearly is a two-terminal capacitor, this

here is a transcapacitance source, just like this is a transconductance source,

this is a transcapacitance source. What is the role of Cmdvgsdt?

If you don't have this, then you're back to this situation which shows you that

the behavior between gate and drain is represented by two plate capacitor Cgd.

You may recall that Cgd was defined as minus dqgdvd.

In other words if you vary the voltage Vd here, you get a variation the gate charge

qg here and that is being modeled by Cgd. But, in the previous video, we also

defined something called Cdg, which is, supposed to be modeling the variation of

the drain charge when we vary the gate voltage.

And we said that that coming from another partial derivative can in general

expected to be different. So, if you don't have this extra source

here, the behavior from gate to drain, or from drain to gate, is represented by the

same capacitance Cgd and would have been predicted to be the same.

While it turns out that the very high frequencies, the two are not the same, so

you need this extra source, to take care of this difference.

So, if you're varying the gate, the drain fold, then the variation in the gate

current is speeding that can be carried off by Cgd.

But if instead you're varying the gate voltage, or you're looking at the

variation of the drain current, only part of this behavior is modeled by Cgd and

the difference is being taken care of by this transcapacitance source.

So, at very high frequencies where the voltage is very fast, and Vvgsdt cannot

be assumed to be negligible, this source comes into play and help you model, helps

you model the non reciprocal, so to speak.

Rough, very roughly speaking behavior between gate and drain.

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Similarly, you can say the same thing for Cbd, Cmb, Dvbsdt is there to model the

non-reciprocal behavior between drain and body.

And, Cmx takes care of the non-reciprocal behavior between gate and body.

More details are found in the book. We'll come back to Csd in a moment.

So now, we can find expressions for the new capacitance parameters we have

defined by differentiating charge expressions as we have done in the past,

always assuming quasi-static operation. For example, we obtain for Cm this

relation Cox is the total gate capacitance, it's the capacitance for

unitary at times the area of the gate. And this a function of eeta, eeta is the

degree of non-saturation we have used it in the past.

So, if you derive expressions which is done the book, eh, you can get certain

plots that help you see what is happening.

This is, we're assuming now strong inversion.

In non saturation Cdg vie, varies like this and then it becomes a constant value

in saturation, but not 0 unlike what Cdg was doing.

And the other composite are shown here. The solid line is an accurate model based

on surface potential formulation and the broken line is the strong inversion

model. So you can see that the, the model does a

pretty decent job in strong inversion. Now, just like we have done in the past,

you can reason physically about the behavior of these capacitances.

As an example, I will take Csd. So, notice that Csd comes out to be

negative. Now, how can we physically, help explain

this fact? Let us recall what Csd is.

Csd is defined to be minus dqs,dvd. Which means that you look at the charts

going into the source when you vary the voltage at the drain, you take the ratio

of the two changes with a minor sign and that is Csd.

Alright let's try to do that. I'm going to increase the voltage at the

drain over here by a small amount. When you increase the voltage at the

drain you reduce the gate to drain voltage.

And at the same time you increase the drain to body voltage.

Both of these effects will show you that the inversion layer here becomes less

strongly inverted. In fact, if you keep increasing the drain

voltage eventually you get pinch-off here.

In the simplified picture we have discussed for strong inversion in the

past. But in general, when the drain voltage

goes up, the inversion layer charge near the drain decreases in magnitude.

And the total inversion layer charge decreases in magnitude as well.

Now the inversion layer charge is negative so when you say decreases in

magnitude, it means that it becomes less negative which is equivalent to saying it

becomes more positive. So, delta Qi, the change and the

inversion layer charge is positive. We have said in the past that delta Qi is

shared by a delta Qd and a delta Qs. Both of those are positive, and

contribute to a positive delta Qi. So delta Qs is positive, which means that

when you vary the drain voltage by delta vd, corresponding to this denominator,

there is a delta Qs which is positive. And because it was a minus sign in the

definition of the capacitance, the whole thing becomes negative that's why Csd is

negative. It is very instructive to try to, reason

in a similar manner for the other capacitances as well.

You can try this or you can consult the book about it.

Now, some other capacitances, plotted versus Vds are shown here, these are the

three capacitances that I mentioned representing non-recicprocal behavior

between two, two terminals, Cm, Cmb and Cmx.

For, for the strong inversal model in fact, Cmx is predicted to be 0 and very

often, we neglect it. Now, here are the capacitances versus

Vgs, so we start from weak inversion, moderate inversion, and strong inversion.

in weak inversion, all of these capacitances, except for Cgb are

negligible because they have to do something with the channel charge but the

channel charge but the channel charge is negligible in weak inversion and the gate

see's the body directly. So Cgb is definitely is not 0.

Then it goes down and eventually goes towards 0 for reasons we have discussed.

Deeply and strong inversion, non saturation.

The other capacitances behave in this way.

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These are the non-reciprocal capacitances calculated from the derivatives that we,

that we have used for the definition of these parameters.

You can see that Cdg and Cgb are very different, Cdb and Cbd are very

different, and Cgb and Cbg are similar. Now, one may wonder, is it possible that

Cdg is different from Cgd? Okay, the derivatives blindly, if blindly

used, give us different values. But would a measurement actually predict

this? these are measurements coming from a

reference given in the book, and indeed you find that Cgd and Cdg are not the

same. Now, if that still bothers you, it is

because you associate A symbol C, with two subscripts, has a two-plate capacitor

between the corresponding two terminals. Please do not do that.

Think of Cxy as it, as it results from definition of it, which was a partial

derivative. Now, Cgd and Cdg, each of them model

different effects. One models the effect of the drain

voltage change on the gate charge, and the other models the effect of the gate

voltage change on the drain charge. Just like, when you vary the gate

voltage, you have a variation in the drain current even at very low

frequencies. But when you vary the drain voltage in

such ratio for example, you don't see a variation in the gate current.

The same thing holds for the charges. So these two effects are different and

they're modeled by different quantities, that we call Cdg and Cgd.

Finally, if you include short channel effects, things become very complicated

analytically. But if you have an accurate model, you

can still predict them or you can measure them.

And they have the same general shape. You still notice that they are

non-reciprocities and so on. but of course the, it is very difficult

to analytically predict the, the values of these capacitances, in the presence of

short-channel effects, using simple equations.