PLEASE NOTE: This version of the course has been formed from an earlier version, which was actively run by the instructor and his teaching assistants. Some of what is mentioned in the video lectures and the accompanying material regarding logistics, book availability and method of grading may no longer be relevant to the present version. Neither the instructor nor the original teaching assistants are running this version of the course. There will be no certificate offered for this course. Learn how MOS transistors work, and how to model them. The understanding provided in this course is essential not only for device modelers, but also for designers of high-performance circuits.
Small-Signal Modeling – Model Comparison
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MOS Transistors
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From the lesson
Small-Signal Modeling II
This material concludes our discussion of small-signal modeling; it shows how we can model the transistor at frequencies at which the simple model we derived last week is not valid. Such modeling is important in high-frequency applications, e.g. transceiver design.
Meet the Instructors
Yannis TsividisCharles Batchelor Professor of Electrical Engineering
Fu Foundation School of Engineering and Applied Science
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.
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