We have so far, derived several different models and now comes the time to compare

them, specifically in terms of their frequency range of validity.

Let us review the values models we have derived.

The first was the simple model that was, claimed to be valid up to medium

frequencies. It includes a constant trans-conductance

GM, and a few capacitance's, five of them.

Then we derive the complete quasi-static model which can be though of as resulting

from this with addition of certain elements.

Of particular interest, is this element here, a new control source, it is

controlled by VGS but, the derivative of VGS appears here and it appears in

parallel with the GM VGS source. We have already mentioned that in the

frequency domain, this converts to J omega CM VGS, so therefore, the combined

current of these two sources is GM minus J Omega CM times VGS.

And the minus appears because this direction is opposite from this

direction. Finally, we had the non quasi-static

model. First order non quasi-static model, which

again, can be thought of as resulting from these after adding certain elements,

like series resistors and inductor over here, all coming out of the math.

And instead of a constant trans conductance GM, we have a trans

admittance that has one plus J Omega tau one, where tau one depends on the

operating point in the denominator. And we have seen, that as omega goes up,

the tau centimeters goes down, which is a manifestation of the fact that at very

high frequencies, the device gives up. So, we have these three models and we

would like now to compare them. To do a complete job would take a very

long time, so what I will do is I will concentrate on the trans admittance of

the three elements, of the three models. And compare it versus frequency for one

specific operating point. And the operating point I will assume

that is that we are somewhere in the middle between VDS equal zero and VDS

equal to VDS prime. In other words, we're in the middle of

the non saturation region. So, the first model, which is the simple

model, has a constant trans conductance times VGS, and this is the control

current source that tells you how much the drain current will vary when you vary

a drift, the gate source voltage, VGS. The complete quasi-static model has, in

addition to the GM VGS, a control source, a J Omega CMVGS source going in the

opposite direction, and the resulting trans admittance is GM minus J Omega CM.

If you multiply this trans admittance by VGS, you get the current which is the sum

of the two currents taking the different direction of the current sources into

account. The third model is the non-quasi static

model which had the transmittance in lieu of the trans conductance, and it is GM

over one plus J Omega tau one VGS. And the fourth model we will consider is

a high order model. We have not derived this, but it can be

derived, and it is done in the literature.

You can find references in the book. Now, if you plot these parameters versus

frequency, both magnitude and phase you find these results.

Here we plot the magnitude of the transmittance, normalized to the trans

conductors. At very low frequencies, the trans

admittance reduces the trans conductors. For example, this trans admittance at

very low frequencies, where the imaginary part is negligibly small reduces to GM

and here, when Omega is very small, the whole thing also reduces to GM.

This is the phase of the trans admittance.

And let us see now, how the various models fair.

First of all, the first model that has a constant trans admittance sequence to GM,

predicts that it stays fixed no matter how high a frequency you're operating the

device at. And of course, this is wrong, because you

know that at very high frequencies, the device will give up.

So, this model does not do a very good job at very high frequencies, as

expected. Now let me for a moment, go to the

non-quasi-static model. The non-quasi-static model C, predicts

that the trans admittance magnitude will fall as you physically expect.

And in fact, it compares rather well to a high-order model that is more accurate

than C. They both predict that the magnitude of

the trans admittance will go down. In other words, the device becomes a poor

trans admittance element at very high frequencies.

Now let's go to the phase. The first model has a constant trans

admittance, therefore it's phase is zero, and it predicts, zero phase for

arbitrarily high frequencies. This model here, the non quasi static

model, predicts that the phase will have a negative value, which will become more

and more negative as we go to high frequencies, which is correct.

Because this represents the phase lag. Between the cause, the cause being, VGS

and the effect, the effect being the drained current, ID.

And we know that there is a phase lag for reasons we have already discussed.

So, it correctly predicts this, and it agrees rather well with the phased

predicted by the high order model. Now, when it comes to the phase, this

model here, the complete quasi static model, also predicts that there will be a

phase lag and the reason it predicts that is that, it's imaginary part is negative

and becomes larger in magnitude with frequency.

So both the non quasi static model and the complete quasi static model predict

the phase rather well. The big problem is the complete quasi

static model. When it comes to the magnitude, notice

that the way this thing is written, if you take the magnitude of it, which would

be the square root of the sum of the squares of the real part, and the

imaginary part, becomes larger and larger as the frequency is increased.

So it goes up as you see here. [BLANK_AUDIO].

Of course this is totally wrong, because it is telling you that the device is

becoming better and better as the frequency becomes higher and higher.

Whereas we know that the opposite is true.

So clearly, the complete quasi static model does a very poor job when it comes

to predicting the magnitude of a trans admittance.

In fact, even the simple model does a better job than that, at least it tells

you that the trans admittance doesn't go up.

So now, it's clear that at least as far as the trans admittance is concerned, if

you don't exceed the certain frequency, let's say up to Omega zero, the phase is

predicted well by both of these models, but unfortunately the magnitude starts

deviating. So, it turns out from such

considerations, for other wide parameters, you can very roughly get a

rule of thumb as to how good these models are, up to what frequency they are valid.

The simple model is about, good to about a tenth of omega zero.

I remind you omega zero was this quantity over here, and it is essentially the cut

off frequency, the intrinsic cut off frequency of a device.

So, for this model, you have to stay below one tenth of Omega zero, because

otherwise, the phase is not predicted accurately by it, it gives you zero

phase. So it is the phase error that makes you

limit the model to .1 Omega zero. When it comes to the complete

quasi-static models, go to about one third of Omega Zero and although it does

predict phase accurately even at higher frequencies, the magnitude starts

becoming higher and higher. So we have to limit this to about a third

of a Omega zero, to avoid getting this wrong prediction for the magnitude.

And finally, this one the non quasi static model turns out to be good up to

Omega zero. You can see that up to Omega zero, both

the magnitude and the phase are predicted relatively accurately.

All of this is just a very rough rule of thumb.

So, at low frequencies, I would use this model, at high frequencies, I would use

this model. And if I have to, in between I may use

this model, but I would rather go to this one directly if I can.

So in this brief video, we discussed how the models compare, and compare to each

other, and we found that there is a factor of three in terms of the upper

frequency limit of validity as you go form the simple model to the complete

quasi-static and to the non-quasi-static. In the next video, we will conclude the

small signal modelling discussion by giving some considerations for very high

frequency, or radio frequency models.

Small-Signal Modeling – Model Comparison

MOS Transistors

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