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Good morning,
all.

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Good morning.
I hope you guys did not spend

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all of last night celebrating
the Red Sox victory,

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but there is one more tonight.
OK.

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Let's see.
I trust the quiz went OK.

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What I will do today is take
off from where we left off on

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Tuesday.
And continue our discussion of

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the large signal and small
signal analysis of our

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amplifier.
Today the focus will be on

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"Small Signal Analysis".

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So let me start by reviewing
some of the material.

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And, as you know,
our MOSFET amplifier looks like

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this.

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One of the things you will
notice in circuits,

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as I have been mentioning all
along in this course,

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is that certain kinds of
patterns keep repeating time and

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time again.
And this is one such pattern.

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A three terminal device like
the MOSFET with an input and the

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drain to source port connected
to RL and VS in series in the

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following manner,
this is a very common pattern.

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There are several other common
patterns.

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The voltage divider is a common
pattern.

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We keep running into that again
and again and again.

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The Thevenin form,
a voltage source in series with

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the resistor is another very
common form.

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The Norton equivalent form,
which is a current source in

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parallel with a resistor is also
very common.

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And it behooves all of us to be
very familiar with the analyses

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of these things.
Voltage dividers in particular

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are just so common that you need
to be able to look at it and

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boom, be able to write down the
expression for voltage dividers.

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I would also encourage you to
go and look at current dividers.

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When you have two resistors in
parallel and you have some

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current flowing into the
resistors to find out the

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current in one branch versus the
other very quickly.

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The expression is very
analogous to the voltage divider

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expression.
And some of these very common

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patterns are highlighted in the
summary pages in the course

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notes, so it is good to keep
track of those and be extremely

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familiar with those patterns to
the point where if you see it

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you should be able to jump up
and shout out the answer just by

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looking at it without having to
do any math.

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So here was an amplifier.
And then we noticed that when

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the MOSFET was in saturation it
behaved like a current source.

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And this circuit would give us
amplification while the MOSFET

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was in saturation.
So we agreed to adhere to the

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saturation discipline which
simply said that I was going to

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use my circuit in a way that the
MOSFET would always remain in

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saturation in building things
like amplifiers and so on.

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And by doing that throughout
the analysis I could make the

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assumption that the MOSFET was
in saturation.

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I didn't have to go through --
Analysis became easier.

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I didn't have to figure out
now, what region is the MOSFET

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in?
Well, because of my discipline

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it is always going to be in
saturation.

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But in turn what we had to do
was conduct a large signal

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analysis.

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Again, in follow on courses you
will be given circuits like

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this.
In fact, this very circuit with

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a very high likelihood.
And you will be looking at more

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complicated models of the
MOSFET.

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Or you will be given the MOSFET
like this and,

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let's say in that course the
designers do not adhere to the

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saturation discipline,
in which case you have to first

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figure out is my MOSFET in its
triode region or in the

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saturation region?
And depending on the region it

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is in you have to apply
different equations.

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So it is one step more
complicated than in 002.

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In 002 we simplified our lives
by following a discipline.

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And let me tell you that
following a discipline is quite

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OK.
When it simplifies our lives

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and we can do good things with
it, it is quite OK to do that.

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We are not wimps or anything
like that.

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It is quite OK to have a
discipline and agree that we are

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going to play in this region of
the playground and build

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circuits in that manner.
By doing so,

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we could assume the MOSFET was
in saturation all the time.

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And analysis simply used a
current source model.

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By the same token,
what becomes important is to

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figure out what are the
boundaries of valid operation of

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the MOSFET in saturation?
To do that we conducted a large

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signal analysis.
And it had two components to

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it.
One of course was to figure out

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the output versus input
response.

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And what this usually does is
that it does a nonlinear

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analysis of this circuit.
If it is a linear circuit it is

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a linear analysis.
And figures out what the values

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of the various voltages and
currents are in the circuit as a

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function of the applied inputs
and chosen parameters.

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And the second step we said was
to figure out valid operating

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ranges --

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-- for input and corresponding
ranges for the other dependent

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parameters such as VO.
You could also find out the

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corresponding operating range
for the current IDS and so on.

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So by doing this you could
first analyze the circuit,

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find out the "bias" parameters,
find out the values of VI and

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VO and so on.
And then you could say all

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right, provided,
as long as VI stays within

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these bounds my assumption that
this is in saturation will hold

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and everything will be fine.
The reading for this is Chapter

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And today we will take the next

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step and revisit small signal
analysis.

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In the demo that I showed you
at the end of last lecture,

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I showed you an input
triangular wave.

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And the input triangular wave
gave rise to an output.

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And we noticed that we did have
amplification,

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I had a small input and a much
bigger output.

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I did have amplification when
the MOSFET was in saturation but

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it was highly nonlinear.
The input was a triangular wave

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and the output was some funny,
it kind of looked like a

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sinusoid whose extremities had
been whacked down and kind of

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flattened.
And its upward going peak had

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been shrunk.
So it was a kind of weird

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nonlinear behavior.
I will show that to you again

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later on.
And so it amplified but it was

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nonlinear.
And remember our goal of two

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weeks ago?
We set out to build a linear

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amplifier.
So today we will walk down that

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path and talk about building a
linear amplifier.

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So to very quickly revisit the
input versus output

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characteristic,
VI versus VO,

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this is VT and this is VS,
this is what things looked

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like.
Also to quickly review the

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valid ranges,
until some point here the

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amplifier was in saturation,
the MOSFET was in saturation

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and somewhere here I had VO
being equal to VI minus a

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threshold drop.
At that point the MOSFET went

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into its triode region and I no
longer was following the

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saturation discipline.
So therefore this is my valid

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region of operation.
We also know that the output

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was given by VS minus K (VI-VT)
all squared RL over 2.

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Again assuming the MOSFET is in
saturation.

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It is very important to keep
stating this because this is

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true only when the MOSFET is in
saturation, when I am following

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the discipline.
Notice that this is a nonlinear

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relationship.
So VO depends on some funny

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square law dependence on VI.
The key here is how do we go

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about building our amplifier?
Take a look at this point here.

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At this point here let's say I
have a VI input.

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Corresponding output is VO.
Focus is this point.

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And left to itself this was a
nonlinear curve.

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Remember the trick that we used
in our nonlinear Expo Dweeb

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example?
We used the Zen Method.

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Remember the Zen Method?
We said look,

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this is nonlinear,
but if you can focus your mind

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on this little piece of the
curve here this looks more or

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less linear.
If I look at a small itty-bitty

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portion of the curve and I do
the Zen thing,

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and kind of zoom in on here.
This looked more or less

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linear.
This means that if I could work

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with very small signals and
apply the signal in a way that I

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also had a DC offset of some
sort.

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Then I would be in a region of
the curve, I would be

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delineating a small region of
the curve which would be more or

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less linear.
This was a small signal trick.

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And what we will do here is
simply revisit the small signal

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model.
Most of what I am going to do

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from here on will be more or
less a repeat of what you saw

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for the light emitting expo
dweeb.

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Just that here I have a three
terminal device,

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with a little bit more
complication.

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The equation is different.
I don't have to resort to a

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Taylor series expansion.
I will just do a complete

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expansion of this expression and
develop the small signal values

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for you.
Recall the small signal model.

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It had the following steps.
The first step will operate at

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some bias point,
VI, VO, and of course some

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corresponding point IDS.
This is Page 3.

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And then superimpose a small
signal VI on top of the big fat

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bias.
Remember the "boost"?

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So VI is the boost.
Boom.

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And above VI,
I have small signal VI that I

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apply.
And our claim is that response

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of the amplifier to VI is
approximately linear.

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The key trick with this is that
for my small signal model here,

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this is Page 3 here,
and Page 2.

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The key trick here is that with
the small signal model,

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I operate my amplifier at some
operating point,

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VO, VI.
I superimpose a small signal VI

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on top of small VI on top of big
VI.

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And then I claim that the
response to VI is approximately

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linear.
And let me just embellish that

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curve a little bit more.

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Notice that in this situation
this was my VI,

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which is my bias voltage,
this is VO, which is the output

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bias, and of course not shown on
this graph is the output

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operating current which is IDS.
One nice way of thinking about

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this is to redraw this and think
that your coordinate axes have

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kind of shifted in the following
manner.

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This is VI.
This is also on your Page 3.

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This is VT.
Remember this was the operating

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point, VO and VI.
And notice that we were

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operating in this small regime
of our transfer curve here.

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And in effect what we are
saying is that I am going to

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apply small variations about VI
and call those variations delta

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VI or small VI.
And the resulting variations

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are going to look like delta VO.
Also referred to as small V,

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small O.
So I will have small variations

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here.
And they give rise to

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corresponding small variations
there.

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One way to view this is as if
we are working with a new

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coordinate system.
Another way to view this is

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that so the capital VI and
capital VO correspond to my VI

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and VO as the total voltages in
my circuit, but at this bias

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point I can think of another
coordinate system here with

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small VI and VO out there.
And for small changes to VI,

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I can figure out the
corresponding small changes to

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VO.
Just that all the analysis I

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perform here is going to be
linear.

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And I will prove it to you in a
couple of different ways in the

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next few seconds.
When I am doing small signal

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analysis I am operating here in
this regime at some bias point.

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You have also seen this before.
How do I get a bias?

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This is my amplifier RL and VS.
This is Page 4.

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VO.
The way I get a bias is I apply

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some DC voltage VI and
superimpose on top of that my

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small signal small VI.
This is my DC bias that has

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boosted up the signal to an
interesting value.

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And because of that what I can
get is by varying VI as a small

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signal with a very small
amplitude, I am going to get a

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linear response here.
And I can draw that for you as

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well.

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This is my bias point here.
And if I vary my signal like so

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then my output should look like
this.

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This is point VI,
this is point VO,

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and this is my small signal VI
and this is my small signal VO

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and this is capital VO.
So this small thing here is VI.

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I would like to show you a
little demo.

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I will start with the same demo
I showed you the last time.

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I showed you the amplifier.
In the demo I am going to apply

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a triangular wave.
And initially I start with a

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large signal.
And you will see that the

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output looks really corny,
is going to look something like

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this.
That's large signal response.

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And then I will begin playing
with the input making it

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smaller, and you can see how it
looks yourselves.

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There you go.
So this is where I stopped the

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last time.
The last lecture I applied this

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input, time is going to the
right, and the purple curve in

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the background is the output.
It looks much more like a

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sinusoid with some flattening of
its tips.

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Nothing like an interesting
triangular wave.

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What I will do next is that let
me make sure I have enough of a

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boost here, enough of a DC
voltage so that I am operating

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at some point here.
I believe I already have that.

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Notice that I can shift up the
triangular wave input,

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or I can shift it down.
So let me bias it here.

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I have chosen a VI that's
about, I forget how many volts

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per division it is,
but I have chosen some VI here.

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And I biased it such that this
is the input.

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You get a nonlinear response.
It is amplified.

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It is much bigger.
What I will do next is make VI

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that I apply smaller and
smaller.

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I have already done the
boosting.

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Boom, that's a boost.
So I have boosted up your VI

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already.
Next is I am going to shrink

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it, and hopefully you will see
that if all that I am saying is

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truthful here you will see a
triangular response.

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Let's go try it out.
Watch the yellow.

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I am going to shrink the yellow
and make it smaller and smaller.

269
00:19:42 --> 00:19:47
There you go.
It is great when nature works

270
00:19:47 --> 00:19:52
like you expect it to.
I have never seen a triangular

271
00:19:52 --> 00:19:57
wave looks so pretty in my life.
It is awesome.

272
00:19:57 --> 00:20:03
Look at this.
Here is a tiny triangular wave.

273
00:20:03 --> 00:20:08
And the output is also a
triangular wave but it is much

274
00:20:08 --> 00:20:10
more linear.
Yes.

275
00:20:10 --> 00:20:12
Question?
What's that?

276
00:20:12 --> 00:20:18
The question is that the output
here is only as big as the input

277
00:20:18 --> 00:20:22
used to be before.
That's a good question.

278
00:20:22 --> 00:20:27
What I have done here is I am
showing you a laboratory

279
00:20:27 --> 00:20:31
experiment.
And let's assume that this

280
00:20:31 --> 00:20:35
input is the input I am getting
from some sensor in the field.

281
00:20:35 --> 00:20:38
Assume that this is my input,
not what I had before.

282
00:20:38 --> 00:20:41
Assume that this is my input to
begin with and this is the

283
00:20:41 --> 00:20:44
amplified output.
What I can also do is I can

284
00:20:44 --> 00:20:47
also change the bias.
And we will see this at the end

285
00:20:47 --> 00:20:49
of the lecture,
in the last ten minutes of

286
00:20:49 --> 00:20:51
lecture.
How do you select a bias point?

287
00:20:51 --> 00:20:55
By changing your bias point you
can change the properties of an

288
00:20:55 --> 00:21:00
amplifier to give you a preview
of upcoming attractions.

289
00:21:00 --> 00:21:02
Let me ask you,
what do you think should happen

290
00:21:02 --> 00:21:06
if I change the bias point?
I have not shown you the math

291
00:21:06 --> 00:21:09
yet, so intuitively what do you
think should happen?

292
00:21:09 --> 00:21:13
If I increase the bias what do
you think is going to happen?

293
00:21:13 --> 00:21:14
Yes.
Good insight.

294
00:21:14 --> 00:21:17
Higher bias will be more
amplification.

295
00:21:17 --> 00:21:20
Let's see if our friend is
correct.

296
00:21:20 --> 00:21:33


297
00:21:33 --> 00:21:36
Let me set a higher bias.

298
00:21:36 --> 00:21:44


299
00:21:44 --> 00:21:45
Not necessarily,
I guess.

300
00:21:45 --> 00:21:47
You're actually right,
by the way.

301
00:21:47 --> 00:21:50
I am playing a trick on
everybody here.

302
00:21:50 --> 00:22:02


303
00:22:02 --> 00:22:05
As I change my input bias.
Notice that under certain

304
00:22:05 --> 00:22:10
conditions my output becomes
smaller and gets more distorted.

305
00:22:10 --> 00:22:14
Under other conditions what is
going to happen to my output is

306
00:22:14 --> 00:22:19
that it is becoming smaller and
is going to get distorted again.

307
00:22:19 --> 00:22:23
So there are a bunch of funny
effects happening that reflect

308
00:22:23 --> 00:22:26
on the bias point,
but for an appropriate choice

309
00:22:26 --> 00:22:31
of bias point as I increase the
bias the amplification should

310
00:22:31 --> 00:22:34
increase.
And I will show you that in a

311
00:22:34 --> 00:22:36
few minutes.
But it is a complicated

312
00:22:36 --> 00:22:38
relationship.
Yes.

313
00:22:38 --> 00:22:44


314
00:22:44 --> 00:22:47
This is finally getting fun.
Here is the question.

315
00:22:47 --> 00:22:50
Professor Agarwal,
we love your song and dance,

316
00:22:50 --> 00:22:53
but if you really want to get a
high signal at the output and

317
00:22:53 --> 00:22:58
you want to amplify your big
input signal how do you do it?

318
00:22:58 --> 00:23:02
So the question is let's say I
have an input that is this big

319
00:23:02 --> 00:23:06
here, if it is this big,
I have shown you how I can get

320
00:23:06 --> 00:23:11
things that are this big,
but what if my input was this

321
00:23:11 --> 00:23:13
big?
How do I get an output that is

322
00:23:13 --> 00:23:16
this big?
Well, I will use one of those

323
00:23:16 --> 00:23:21
learned by questioning methods
and have you tell me the answer.

324
00:23:21 --> 00:23:24
Someone tell me the answer.
How do I do that?

325
00:23:24 --> 00:23:28
Yes.
Use another amplifier.

326
00:23:28 --> 00:23:34
So the answer is I will use one
amplifier to go from here to

327
00:23:34 --> 00:23:37
here.
And the suggestion is use

328
00:23:37 --> 00:23:41
another amplifier to go from
here to here.

329
00:23:41 --> 00:23:45
And, in fact,
I believe that you may have a

330
00:23:45 --> 00:23:50
problem in your problem set
where you will do that.

331
00:23:50 --> 00:23:54
And so you have only yourselves
to blame.

332
00:23:54 --> 00:24:01
So how do you make this work?
What you have to do is this VI

333
00:24:01 --> 00:24:05
has to be much smaller than the
bias point VI on this one.

334
00:24:05 --> 00:24:09
I have to build a different
amplifier, choose a different

335
00:24:09 --> 00:24:14
set of parameters such that VI
prime, which is the VI for this

336
00:24:14 --> 00:24:18
guy, is much less than V capital
I prime for this guy.

337
00:24:18 --> 00:24:22
It's a design question.
You need to design it in a way

338
00:24:22 --> 00:24:26
that the signals of interest
need to be much smaller than the

339
00:24:26 --> 00:24:32
bias voltage of this amplifier.
So you may have to use much

340
00:24:32 --> 00:24:34
higher supply voltages.
My amplifier,

341
00:24:34 --> 00:24:38
I believe, has a 4 volt supply
or 5 volt supply.

342
00:24:38 --> 00:24:42
You might have to use an
amplifier with a much bigger

343
00:24:42 --> 00:24:45
supply, different values of RL
and so on.

344
00:24:45 --> 00:24:49
And I know that the course
notes also have some exercises

345
00:24:49 --> 00:24:53
and problem sets that discuss
that in more detail.

346
00:24:53 --> 00:24:55
Yes.
This is even more fun.

347
00:24:55 --> 00:24:59
The question is,
good question.

348
00:24:59 --> 00:25:03
The question is why do you need
this guy here?

349
00:25:03 --> 00:25:05
Just use this guy,
right?

350
00:25:05 --> 00:25:09
Why do you need this guy?
Big guys rule,

351
00:25:09 --> 00:25:13
right?
Who needs the little guys?

352
00:25:13 --> 00:25:17
Well, let me use the Socratic
method again.

353
00:25:17 --> 00:25:20
Why don't you give me the
answer?

354
00:25:20 --> 00:25:25
You guys are smart.
Why do you need little guys?

355
00:25:25 --> 00:25:30
Why do you need the small guy
here?

356
00:25:30 --> 00:25:34
Anybody with the answer?
Yeah.

357
00:25:34 --> 00:25:39
The big guy may not be as
sensitive.

358
00:25:39 --> 00:25:43
I like that.
You know what?

359
00:25:43 --> 00:25:50
He is almost correct.
I will show you why in a

360
00:25:50 --> 00:25:54
second.
Anything else?

361
00:25:54 --> 00:25:59
Any other reason?
Yes.

362
00:25:59 --> 00:26:08


363
00:26:08 --> 00:26:10
Bingo.
That is another good answer.

364
00:26:10 --> 00:26:13
So let me address both the
answers.

365
00:26:13 --> 00:26:18
The answer given was that look,
this amplifier is amplifying

366
00:26:18 --> 00:26:22
the signal by a certain amount,
by a factor of 7.

367
00:26:22 --> 00:26:27
And I have designed this such
that this amplifies a signal by

368
00:26:27 --> 00:26:31
a factor of maybe 10.
So in all I am getting an

369
00:26:31 --> 00:26:34
amplification of 70.
This would be a great design

370
00:26:34 --> 00:26:37
question for lab next year.
I give you a bunch of

371
00:26:37 --> 00:26:40
components and ask you to design
an amplifier given the

372
00:26:40 --> 00:26:43
constraints with the highest
amount of amplification.

373
00:26:43 --> 00:26:46
It turns out that when you
design your amplifier,

374
00:26:46 --> 00:26:50
in order to meet the saturation
discipline and so on,

375
00:26:50 --> 00:26:53
you have to choose values of RL
and VS and stuff like that and

376
00:26:53 --> 00:26:57
be within power constraints so
the amplifier doesn't blow up

377
00:26:57 --> 00:27:00
and stuff.
And by the end of it all you

378
00:27:00 --> 00:27:02
are going to get a measly 7X
gain out of it.

379
00:27:02 --> 00:27:05
The same way here,
to be able to deal with a very

380
00:27:05 --> 00:27:08
small signal here and get some
amplification,

381
00:27:08 --> 00:27:10
another set of values and you
get 10X.

382
00:27:10 --> 00:27:12
So they multiply.
It is much harder to build one

383
00:27:12 --> 00:27:14
amplifier with a much larger
gain.

384
00:27:14 --> 00:27:17
You know what?
I just realized that we will be

385
00:27:17 --> 00:27:20
looking at this in the last five
or seven minutes of lecture.

386
00:27:20 --> 00:27:23
I am going to show you what the
amplification depends upon.

387
00:27:23 --> 00:27:25
It depends upon K.
It depends upon RL.

388
00:27:25 --> 00:27:30
It depends upon VI.
Now the question is I have had

389
00:27:30 --> 00:27:33
all this time to think about how
to stitch in sensitive into

390
00:27:33 --> 00:27:37
this, and I believe I can.
It turns out that when you have

391
00:27:37 --> 00:27:41
large voltages and so on and you
have practical devices,

392
00:27:41 --> 00:27:45
it turns out that the more
current you pump through devices

393
00:27:45 --> 00:27:48
they tend to produce noise of
various kinds.

394
00:27:48 --> 00:27:52
So very powerful amplifiers are
not very good at dealing with

395
00:27:52 --> 00:27:55
really tiny signals because they
have some inherent noise

396
00:27:55 --> 00:27:58
capabilities.
And so I guess that is

397
00:27:58 --> 00:28:03
sensitive.
It is sensitive to noise.

398
00:28:03 --> 00:28:07
Another question?
Yes.

399
00:28:07 --> 00:28:22


400
00:28:22 --> 00:28:24
Ask me the question again.
I didn't follow.

401
00:28:24 --> 00:28:35


402
00:28:35 --> 00:28:38
Let me just explain it.
It turns out that I will not be

403
00:28:38 --> 00:28:41
able to pass this through the
big amplifier to begin with

404
00:28:41 --> 00:28:45
because it is just going to give
me a gain of just a factor of 7.

405
00:28:45 --> 00:28:49
However, if I have a signal
that is this big to begin with

406
00:28:49 --> 00:28:51
then I may just need this
amplifier.

407
00:28:51 --> 00:28:54
I don't need the smaller guy.
If my signal was this big to

408
00:28:54 --> 00:28:58
begin with, if I had a strong
sensor that produced a strong

409
00:28:58 --> 00:29:01
signal to begin with,
yeah, I can deal with just a

410
00:29:01 --> 00:29:04
single stage.
I don't need to two stages.

411
00:29:04 --> 00:29:09
It is all a matter of design.
And it is actually a fun design

412
00:29:09 --> 00:29:10
exercise.
Given a budget,

413
00:29:10 --> 00:29:13
dollars, right?
You go to your supply room and

414
00:29:13 --> 00:29:18
look at the parts that you have
and you go to build what you

415
00:29:18 --> 00:29:20
have to build with the parts
that you have.

416
00:29:20 --> 00:29:25
And so sometimes you need to
build two amplifiers to get the

417
00:29:25 --> 00:29:27
gain or build a signal
amplifier.

418
00:29:27 --> 00:29:30
It's all a design thing.
All right.

419
00:29:30 --> 00:29:34
Moving on to Page 7.
That brings us to the small

420
00:29:34 --> 00:29:36
signal model.

421
00:29:36 --> 00:29:51


422
00:29:51 --> 00:29:58
Page 5.
What I showed you up on the

423
00:29:58 --> 00:30:05
little demo was that provided
the signal input in this example

424
00:30:05 --> 00:30:11
VI was much smaller than capital
VI out there as I shrank my

425
00:30:11 --> 00:30:17
input, I was able to get a more
or less linear response at the

426
00:30:17 --> 00:30:21
output.
And so to repeat my notation at

427
00:30:21 --> 00:30:28
the input, the total input is a
sum of the operating point input

428
00:30:28 --> 00:30:35
plus a small signal input.
This is called the total

429
00:30:35 --> 00:30:41
variable.
This is called the DC bias.

430
00:30:41 --> 00:30:47
It is also called the operating
point voltage.

431
00:30:47 --> 00:30:53
And this is called my small
signal input.

432
00:30:53 --> 00:31:02
It is also variously called
incremental input.

433
00:31:02 --> 00:31:06
This is more a mathematical
term relating to incremental

434
00:31:06 --> 00:31:09
analysis or perturbation
analysis.

435
00:31:09 --> 00:31:14
So VI, call it small signal,
call it small perturbation,

436
00:31:14 --> 00:31:17
call it increment,
whatever you want.

437
00:31:17 --> 00:31:23
Similarly, at the output I have
my total variable at the output

438
00:31:23 --> 00:31:28
a sum of the output operating
voltage and the small signal

439
00:31:28 --> 00:31:32
voltage.
I do not like using Os in

440
00:31:32 --> 00:31:39
symbols because big O and small
O is simply a function of how

441
00:31:39 --> 00:31:43
big you write them.
It is not super clear.

442
00:31:43 --> 00:31:48
And in terms of a graph,
let me plot the input and

443
00:31:48 --> 00:31:53
output for you.
Let's say this is the total

444
00:31:53 --> 00:31:57
input and that is the total
output.

445
00:31:57 --> 00:32:03
I may have some bias VI.
And corresponding to that I may

446
00:32:03 --> 00:32:07
have some bias VO.
Hold that thought for a second

447
00:32:07 --> 00:32:12
while I give you a preview of
something that we will be

448
00:32:12 --> 00:32:15
covering in about three or four
weeks.

449
00:32:15 --> 00:32:19
Notice that as I couple
amplifiers together,

450
00:32:19 --> 00:32:23
the output operating point
voltage of this amplifier in

451
00:32:23 --> 00:32:28
this connection becomes the
input operating point voltage of

452
00:32:28 --> 00:32:32
this amplifier,
right?

453
00:32:32 --> 00:32:34
So when they connect this
output to this input,

454
00:32:34 --> 00:32:38
the output operating point
voltage becomes coupled to the

455
00:32:38 --> 00:32:42
input here so it becomes the
input operating point voltage

456
00:32:42 --> 00:32:44
here.
Now I have a nightmare on my

457
00:32:44 --> 00:32:46
hands.
As I adjust the bias of this

458
00:32:46 --> 00:32:48
guy, the bias of this guy
changes, too.

459
00:32:48 --> 00:32:51
The two are dependent.
It is a pain in the neck.

460
00:32:51 --> 00:32:55
And we being engineers find
ways to simplify our lives.

461
00:32:55 --> 00:32:58
And you will learn another
trick in about three or four

462
00:32:58 --> 00:33:02
weeks.
And that trick will let you

463
00:33:02 --> 00:33:07
decouple these two stages in a
way that you can design this

464
00:33:07 --> 00:33:11
stage in isolation,
go have a cup of coffee and

465
00:33:11 --> 00:33:16
then come back to this stage and
design this stage in isolation.

466
00:33:16 --> 00:33:22
For those of you who want to
run ahead and think about how to

467
00:33:22 --> 00:33:26
do it, think about it.
What trick can you use to get

468
00:33:26 --> 00:33:30
them in isolation?
Moving on.

469
00:33:30 --> 00:33:35
What I would like to do next is
address this from a mathematical

470
00:33:35 --> 00:33:39
point of view.
And much as I did for the light

471
00:33:39 --> 00:33:44
emitting expo dweeb analyze this
mathematically and show you that

472
00:33:44 --> 00:33:48
if VI is much smaller than
capital VI, I indeed get a

473
00:33:48 --> 00:33:52
linear response.
This time around I won't use

474
00:33:52 --> 00:33:57
Taylor series because it turns
out that this expression can be

475
00:33:57 --> 00:34:02
expanded fully.
So you don't have to buy into

476
00:34:02 --> 00:34:07
Taylor series and so on.
I am going to list everything

477
00:34:07 --> 00:34:11
down for you.
We know, to begin with,

478
00:34:11 --> 00:34:15
that VO for the amplifier is
VS-RLK/2 (VI-VT)^2.

479
00:34:15 --> 00:34:20
What I am going to do for this,
much as I did for the LED,

480
00:34:20 --> 00:34:26
what I'm going to do is derive
for you the output as a function

481
00:34:26 --> 00:34:32
of the input when the input VI
is very small.

482
00:34:32 --> 00:34:36
In other words,
when I substitute for VI,

483
00:34:36 --> 00:34:39
V capital I squared plus small
VI.

484
00:34:39 --> 00:34:46
Much as I did for the expo
dweeb, I want to substitute for

485
00:34:46 --> 00:34:50
VI a big DC VI.
So VI is much smaller than VI.

486
00:34:50 --> 00:34:56
And show you for yourselves
that the output response,

487
00:34:56 --> 00:35:03
V small O is going to be
linearly connected to VI.

488
00:35:03 --> 00:35:06
Notice that,
let me write another equation

489
00:35:06 --> 00:35:09
here.
This is a total variable.

490
00:35:09 --> 00:35:14
This simply says that if the
input is VI then the output is

491
00:35:14 --> 00:35:18
going to be VO,
which means that the operating

492
00:35:18 --> 00:35:22
point input voltage should
satisfy this equation,

493
00:35:22 --> 00:35:24
correct?
In other words,

494
00:35:24 --> 00:35:30
the operating point output
voltage V capital O should equal

495
00:35:30 --> 00:35:35
VS-RLK/2 (VI-VT)^2.
This is at VI equals capital

496
00:35:35 --> 00:35:38
VI.
This is very simple but may

497
00:35:38 --> 00:35:42
seem confusing.
All this is saying is that

498
00:35:42 --> 00:35:49
look, this equation gives me the
relationship between VI and VO.

499
00:35:49 --> 00:35:53
Therefore, if I apply capital
VI as the input,

500
00:35:53 --> 00:35:58
I'm given that my corresponding
output is capital VO,

501
00:35:58 --> 00:36:04
so they must satisfy this
equation, right?

502
00:36:04 --> 00:36:10
Those are bias point values and
that must satisfy this equation.

503
00:36:10 --> 00:36:12
Simple.
I know that.

504
00:36:12 --> 00:36:18
So hold that thought.
Stash it away in the back of

505
00:36:18 --> 00:36:22
your minds.
Now let me go through a bunch

506
00:36:22 --> 00:36:30
of grubby math and substitute
for VI in this expression here.

507
00:36:30 --> 00:36:35
Let me go ahead and do that.
VS-RLK/2((VI+vi)-VT)^2.

508
00:36:35 --> 00:36:41
When I do something that is
other than math I will wake you

509
00:36:41 --> 00:36:45
up.
I will just keep doing a bunch

510
00:36:45 --> 00:36:49
of steps that are pure math.
No cheating.

511
00:36:49 --> 00:36:52
No nothing.
Watch my fingers.

512
00:36:52 --> 00:37:00
When I do anything that is not
obvious math I will wake you up.

513
00:37:00 --> 00:37:06
Next I am going to simply move
VT over and rewrite this as

514
00:37:06 --> 00:37:13
follows, RLK/2((VI-VT)+vi)^2.
Again, I haven't done anything

515
00:37:13 --> 00:37:18
interesting so far.
I have just substituted this.

516
00:37:18 --> 00:37:25
I am just juggling things
around just to pass away some

517
00:37:25 --> 00:37:29
time, I guess.
All right.

518
00:37:29 --> 00:37:42
Next what I am going to do is
simply expand this out and write

519
00:37:42 --> 00:37:52
it this way RLK/2,
expand that out and treat this

520
00:37:52 --> 00:37:59
as one unit VS -
RLK/2((VI-VT)^2+

521
00:37:59 --> 00:38:06
2(VI-VT)vi+vi^2).
Nothing fancy here.

522
00:38:06 --> 00:38:11
This is like the honest board.
Nothing fancy here.

523
00:38:11 --> 00:38:14
Standard stuff.
Only math.

524
00:38:14 --> 00:38:20
I will move to this blackboard
here where I do some fun EE

525
00:38:20 --> 00:38:23
stuff.
Yes.

526
00:38:23 --> 00:38:28


527
00:38:28 --> 00:38:32
Good.
At least one person isn't

528
00:38:32 --> 00:38:34
asleep here.
Thank you.

529
00:38:34 --> 00:38:38
So just math here.
Nothing fancy.

530
00:38:38 --> 00:38:42
Plain old simple math.
I have not done any trickery.

531
00:38:42 --> 00:38:45
I still have all my ten
fingers.

532
00:38:45 --> 00:38:49
Now what I am going to do,
now watch me.

533
00:38:49 --> 00:38:55
I am not using Taylor series
here because this expression

534
00:38:55 --> 00:39:00
lends itself to this analysis.
Notice VI squared here.

535
00:39:00 --> 00:39:06
I made the assumption that VI
is much smaller than capital VI,

536
00:39:06 --> 00:39:11
so what I can do is assuming
that VT is small enough that VI

537
00:39:11 --> 00:39:15
minus VT is still a big number
compared to small VI,

538
00:39:15 --> 00:39:20
what I can do is ignore this in
comparison to the capital VI

539
00:39:20 --> 00:39:23
terms.
So I have a capital VI term

540
00:39:23 --> 00:39:26
here.
I am going to ignore VI

541
00:39:26 --> 00:39:29
squared.
So, for example,

542
00:39:29 --> 00:39:35
if capital VI was 5 volts and
small VI was 100 millivolts 0.1,

543
00:39:35 --> 00:39:40
so 0.1 squared is 0.01.
So it is comparing 0.01 to 5.

544
00:39:40 --> 00:39:44
So I am off by a factor of 500.
So now watch me.

545
00:39:44 --> 00:39:48
Now I begin playing some fun
and games here.

546
00:39:48 --> 00:39:52
I eliminate this,
and because I eliminate that it

547
00:39:52 --> 00:40:02
now becomes approximately equal.
What I do in addition is let me

548
00:40:02 --> 00:40:10
write down the output.
The total variable is the sum

549
00:40:10 --> 00:40:18
of the DC bias and some
variation of the output.

550
00:40:18 --> 00:40:27
And let me simply expand that
term and write it down again.

551
00:40:27 --> 00:40:34
VS-RLK/2(VI-VT)^2-RLK/2.
I get a two here.

552
00:40:34 --> 00:40:38
And I get VI-VT.
I won't forget the VI this

553
00:40:38 --> 00:40:41
time.
Again, from here to there

554
00:40:41 --> 00:40:45
nothing fancy.
This is the one step where I

555
00:40:45 --> 00:40:49
have used a trick.
I have said small VI is much

556
00:40:49 --> 00:40:54
smaller than capital VI,
and so I have simply expanded

557
00:40:54 --> 00:40:59
this out and written it here.
So do you see the obvious next

558
00:40:59 --> 00:41:07
trick here?
From star look at this guy.

559
00:41:07 --> 00:41:15


560
00:41:15 --> 00:41:20
I can cancel this out from star
because I know that at the

561
00:41:20 --> 00:41:25
operating point these two
expressions are equal,

562
00:41:25 --> 00:41:31
and so therefore I can cancel
out the operating point voltage

563
00:41:31 --> 00:41:38
and this.
What I am left with is small VO

564
00:41:38 --> 00:41:45
is simply minus RLK(VI-VT) times
vi.

565
00:41:45 --> 00:41:52
Only one place where I did
something funny.

566
00:41:52 --> 00:42:00
Other than that it is purely
math.

567
00:42:00 --> 00:42:05
So this is what I get.
Notice that this whole thing is

568
00:42:05 --> 00:42:11
a constant, minus RLK(VI-VT).
This whole thing is a constant.

569
00:42:11 --> 00:42:15
And so VO is equal to some
constant times VI.

570
00:42:15 --> 00:42:21
Let me just define some terms
for you that you will use again

571
00:42:21 --> 00:42:25
and again.
For reasons that will be

572
00:42:25 --> 00:42:30
obvious next lecture,
I am going to call this term

573
00:42:30 --> 00:42:32
here GM.

574
00:42:32 --> 00:42:37


575
00:42:37 --> 00:42:42
I am going to call this term a
constant, K(VI - VT).

576
00:42:42 --> 00:42:47
It is a constant for a given
bias point voltage.

577
00:42:47 --> 00:42:53
So I am going to call that GM.
And then I am going to call

578
00:42:53 --> 00:42:58
this whole thing A.
And of course this is VI.

579
00:42:58 --> 00:43:03
There you go.
I have my linear amplifier.

580
00:43:03 --> 00:43:09
A is the gain times small VI.
And the gain has these terms in

581
00:43:09 --> 00:43:12
it.
I just call this GM.

582
00:43:12 --> 00:43:17
You will see why later.
But notice that the gain

583
00:43:17 --> 00:43:21
relates to RL.
The size of the load resistor

584
00:43:21 --> 00:43:25
RL, how big it is,
1K, 10K, whatever.

585
00:43:25 --> 00:43:31
K, this is a MOSFET parameter,
and VI minus VT.

586
00:43:31 --> 00:43:36
That is a constant for a given
bias point voltage and small VI.

587
00:43:36 --> 00:43:39
So VO equals small VI.

588
00:43:39 --> 00:43:47


589
00:43:47 --> 00:43:50
I won't give you a graphical
interpretation,

590
00:43:50 --> 00:43:55
but I encourage you to go and
look at Figure 8.9 in the course

591
00:43:55 --> 00:43:57
notes.
And it gives you a graphical

592
00:43:57 --> 00:44:01
interpretation of that
expression.

593
00:44:01 --> 00:44:06
Move to Page 7.
Another way of looking at this,

594
00:44:06 --> 00:44:10
another way of mathematically
analyzing it,

595
00:44:10 --> 00:44:17
here I went through a full
blown expansion and pretty much

596
00:44:17 --> 00:44:20
deriving the small signal
response.

597
00:44:20 --> 00:44:25
What I can also do is take a
shortcut here.

598
00:44:25 --> 00:44:29
So let me just give you the
shortcut.

599
00:44:29 --> 00:44:36
You might find this handy.
VO=VS-KRL/2(VI-VT)^2.

600
00:44:36 --> 00:44:41
And my shortcut is as follows.
My small signal response is

601
00:44:41 --> 00:44:46
simply this relationship.
I find the slope at the point

602
00:44:46 --> 00:44:50
capital VI and multiply by the
increment.

603
00:44:50 --> 00:44:56
Slope times the increment gives
me the incremental change in VO

604
00:44:56 --> 00:45:01
as follows.
d/dI (VS-KRL/2(VI-VT)^2)

605
00:45:01 --> 00:45:07
evaluated at vI=VI times vi.
This is math again.

606
00:45:07 --> 00:45:14
I want to find out the change
in VO for a small change in VI,

607
00:45:14 --> 00:45:21
and I do that by taking the
first derivative of this with

608
00:45:21 --> 00:45:29
respect to VI substituting V
capital I and multiplying by the

609
00:45:29 --> 00:45:35
small change delta VI or small
VI.

610
00:45:35 --> 00:45:41
So this is simply the slope of
the VO versus VI curve at VI.

611
00:45:41 --> 00:45:47
And so therefore taking the
derivative here of this.

612
00:45:47 --> 00:45:51
This is a constant so it
vanishes.

613
00:45:51 --> 00:45:57
But twice 2 to cancel out,
so I get KRL(VI-VT) times small

614
00:45:57 --> 00:46:06
vi evaluated at capital VI.
So I get twice KRL,

615
00:46:06 --> 00:46:18
VI evaluated at capital VI,
so it is VI minus VT times

616
00:46:18 --> 00:46:23
small VI.
Same thing.

617
00:46:23 --> 00:46:32
Oh, and I have a minus sign
here.

618
00:46:32 --> 00:46:37
I get the same expression that
I derived for you up there,

619
00:46:37 --> 00:46:42
and this is just taking the
slope and going with it.

620
00:46:42 --> 00:46:46
And this, as I mentioned
before, this is A.

621
00:46:46 --> 00:46:52
The last few minutes let me
kind of pull everything together

622
00:46:52 --> 00:46:57
and also hit upon something that
many of your questions are

623
00:46:57 --> 00:47:02
touched upon.
And that all relates to how to

624
00:47:02 --> 00:47:07
choose the bias point.
So here I have taken an

625
00:47:07 --> 00:47:11
analysis approach.
When teaching we often teach

626
00:47:11 --> 00:47:15
you are given something,
you analyze it,

627
00:47:15 --> 00:47:20
but as you begin to master it
you can begin to design things

628
00:47:20 --> 00:47:25
where you can ask a lot of
questions and so on.

629
00:47:25 --> 00:47:30
And here what we have is an
analysis given a value of RLK,

630
00:47:30 --> 00:47:34
VI and so on.
How to choose the bias point

631
00:47:34 --> 00:47:38
becomes more of a design issue.
If you are designing an

632
00:47:38 --> 00:47:42
amplifier, you asked me the
question, how do I choose two

633
00:47:42 --> 00:47:46
small amplifiers versus one big
amplifier, that sort of stuff?

634
00:47:46 --> 00:47:50
It boils down to how do you
choose the bias point?

635
00:47:50 --> 00:47:54
How do you choose VI?
How do you choose RL and so on?

636
00:47:54 --> 00:47:59
What I would like to do is
touch upon some of these things.

637
00:47:59 --> 00:48:02
First of all,
gain or the amplification.

638
00:48:02 --> 00:48:06
One of the most important
design perimeters for an

639
00:48:06 --> 00:48:11
amplifier is what is the gain?
Let's say you get a job at

640
00:48:11 --> 00:48:16
Maxim Integrated Technologies,
and they say we would like you

641
00:48:16 --> 00:48:20
to build a linear power
amplifier for cell phones.

642
00:48:20 --> 00:48:23
You can say I know how to do
that.

643
00:48:23 --> 00:48:28
And then they say the next
stage needs a 100 millivolt

644
00:48:28 --> 00:48:32
input.
While this thing coming from

645
00:48:32 --> 00:48:36
the antenna is only a few tens
or a few hundreds of a

646
00:48:36 --> 00:48:39
microvolt.
So you sit down and say oh,

647
00:48:39 --> 00:48:43
my gosh, I need an
amplification of so much,

648
00:48:43 --> 00:48:48
and you go design an amplifier.
So gain tends to be a key

649
00:48:48 --> 00:48:51
parameter.
And notice that gain is

650
00:48:51 --> 00:48:55
proportional to RL.
It relates to VI minus VT,

651
00:48:55 --> 00:49:00
so proportional to VI.
It is also related to RL.

652
00:49:00 --> 00:49:07
The second point is the gain
point determines where I bias

653
00:49:07 --> 00:49:12
something.
If I choose my bias too high I

654
00:49:12 --> 00:49:17
get distortion,
or if I choose my bias too low

655
00:49:17 --> 00:49:20
I get distortion.

656
00:49:20 --> 00:49:27


657
00:49:27 --> 00:49:30
So depending on how I choose my
bias point, as a signal goes up

658
00:49:30 --> 00:49:33
it may begin clipping or begin
distorting.

659
00:49:33 --> 00:49:38
And I will show you a demo the
next time on that particular

660
00:49:38 --> 00:49:41
example.
So bias point will determine

661
00:49:41 --> 00:49:47
how big of a signal you can send
without getting too much

662
00:49:47 --> 00:49:50
distortion.
And the other thing is that,

663
00:49:50 --> 00:49:56
relates to how big of an input,
what is a valid input range?

664
00:49:56 --> 00:50:02
So let's say you have a signal.
And you want that signal to

665
00:50:02 --> 00:50:08
have both positive and negative
excursions of the same value.

666
00:50:08 --> 00:50:12
Then, depending on where you
choose a bias point,

667
00:50:12 --> 00:50:16
your input range may become
smaller or larger.

668
00:50:16 --> 00:50:22
And we will go through these in
the context of and amplifier and

669
00:50:22 --> 50:25
look at some design issues in
the next lecture.