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OK, good morning all.
So before we begin,

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I just thought I'd show you a
little news item that I happened

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to read that was very relevant
to what we covered recently in

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So you recall when we did the

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digital section a few days ago
last Thursday,

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we talked about a switch.
We talked about the MOSFET

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switch, which when turned on and
off, by input signals could help

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build gates which would then be
combined in tens of millions of

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quantities and go into chips
like the Pentium 4 and AMD

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Athlon 64, and so on it so
forth.

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So I just saw this news item
that I came across,

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and this says they are
rethinking the basic

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construction of the products.
It talks about the

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semiconductor manufacturers like
AMD, Intel, and others that

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build digital chips.
They are rethinking the basic

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construction of the products
down to the architecture of the

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transistor.
That's a MOS transistor,

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and the on/off switch inside
the chip.

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OK, now this might imply that
there is a single switch inside

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the chip, but no,
there's tens of millions of

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transistors, or tens of millions
of switches inside a chip.

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And pretty much any advancement
that can be made to the basic

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transistor can have a 10 million
to 20 million times effect

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because there are that many of
them on a single chip.

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So I thought that was very
appropriate.

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OK.
Let's dive into a quick review.

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So this week,
we had begun nonlinear

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analysis, and I just thought I'd
blast through a few animations

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that I've created,
trying to give you more insight

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into the behavior of some of the
things that we have done.

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Now first of all,
as I did the last time,

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let me try to put it in
perspective most of what you've

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learned thus far,
and what we will be learning

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today.
So the past week,

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we have been focusing on
nonlinear analysis.

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And as I pointed out,
here is how this fits into the

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big picture.
So, we had our 6.002 world,

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at what we said is that we are
engineers.

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We are going to devise our own
playground in which to play with

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our own rules.
And that's our playground.

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That's what we're going to
learn about in 002,

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and for that matter,
the rest of EECS at MIT.

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It's all within this playground
here.

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And this is the playground with
lumped circuit abstraction,

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and good old KVL,
KCl, node method,

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your basic composition rules
apply within this playground

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that directly come from
Maxwell's equations because you

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have made the lumped matter
discipline assumptions.

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OK, so then we said a large
part of the playground is

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linear, and some other much more
intuitive techniques apply

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within the linear portion of
that playground,

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techniques like the
superposition,

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Thevenin and Norton.
In most exercises,

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and quizzes,
and experiments,

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and so on that you do in real
life, you can pretty much apply

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these simple techniques.
Very rarely do you have to go

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into the node method for
circuits that are more

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complicated than single source
and a couple of elements.

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And then, there's the nonlinear
part.

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Remember, the reason I showed
this is that this is the same

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playground.
OK, linear and nonlinear are

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part of the same playground.
OK, even nonlinear elements are

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lumped circuit elements,
and they follow KVL,

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KCl, the node equation,
and so on.

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And then, last week we spent
some time talking about the

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digital abstraction.
So we focused on a smaller

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region of the playground.
And the assumptions we made in

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there were even tighter.
We said that it is part of the

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playground we shall only deal
with binary values.

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We'll digitize or lump values
into highs and lows,

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and that's where our circuits
are going to be.

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And these circuits,
when looked at as a whole,

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were nonlinear.
So, this is a simple NAND gate

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circuit.
And this is the input/output

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characteristic.
So, for example,

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if I hold B at zero,
and I apply a zero to one

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transition at A,
then this is the output that I

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will see at C.
So notice, this is decidedly

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nonlinear.
Then I said that,

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look, suppose we had to fix the
input values at a given set.

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OK, so let's say,
for example,

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I fix A at one,
and B at one.

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OK, and then look at the
circuit in this situation.

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What do I find?
What I find is that the entire

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digital set of circuits that we
were looking at move over into

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the linear space for a given set
of switch settings,

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OK?
So, when I set A 1 and B 1,

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A equal to one and B equal to
one, my NAND gate becomes like

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this.
OK, it's a simple resistive

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network with a voltage source,
VS.

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So, for a fixed set of inputs,
for a given set of inputs,

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if I don't change my inputs,
then my circuit looks like a

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linear circuit,
and my good old linear analysis

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techniques apply.
So that was last week.

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And this week,
we are looking at the nonlinear

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space.
And we looked at a couple of

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techniques in the nonlinear
space, analytical techniques and

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graphical techniques.
And then, I showed you an

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example.
OK, I showed you an example

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circuit that was something that
I would like to build involving

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the light emitting expo dweeb,
my little garage door opener

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thingamajig, and I wanted to
transmit music over that light

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beam.
I also showed you that it was

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highly distorted because it was
in the nonlinear space.

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So, today what I'm going to do
is introduce a new part of the

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playground.
There's a new part of the

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playground, and I'll show you a
technique whereby by focusing on

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this part of the playground and
disciplining ourselves in the

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kind of inputs we apply to
circuits, I'm going to show you

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that certain kinds of nonlinear
circuits also move over,

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when used in a particular way,
also move into the linear

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analysis domain.
OK, so let me leave that for

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now and go back into quickly
reviewing the motivating example

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of music that I had taken last
time.

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OK, so here was a little
example.

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So I have a music source,
VI, and I apply that.

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This device that I call the,
lightheartedly,

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the Light Emitting Expo Dweeb
has a current,

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VD, across it,
or a voltage,

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VD, across it,
and a current ID through it.

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And the light intensity,
I said, was proportional to the

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current.
And because of that,

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I was able to get the light to
impinge on a receiving device,

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which produced a current that
was proportional to the

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intensity of light falling on
it.

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And that signal would then be
amplified somehow.

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We haven't talked about all of
this stuff yet.

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This will happen next week.
But let's say we somehow

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amplify the signal and then
played out through a set of

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speakers.
All right, so if I had some

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sort of a music signal here,
then I could then transmit the

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music signal over to the side on
top of this light beam.

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But the problem,
as I said the last time,

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was that our device,
the Light Emitting Expo Dweeb

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had an exponential
characteristic,

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so that I had some trouble in
getting undistorted music.

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So, the characteristic of the
VI characteristics of my device

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looked like so.
The ID versus VD curve looked

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as follows.
OK, it was decidedly nonlinear.

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And because of that,
I was getting a lot of

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distortions in my signal,
and I showed you a little trick

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to plot, given an input waveform
at a transfer function such as

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here to plot the output
function.

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OK, let me show you another
little animation that I have

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created here for you that should
give you even more intuition in

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terms of how it happens.
So, this is a characteristic I

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showed you up here.
It's on both sides,

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but I guess it points to only
one unless I shuttle back and

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forth really fast.
So on average,

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I'll be in both places.
But anyway, so here's my ID

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versus VD characteristic.
And as I said,

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there's an exponential ID
versus VD curve.

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And I want to see what the
output looks like,

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for example,
a sinusoidal input.

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So I said, let's place the
input along a little graph,

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rotate it so,
and take a sinusoid,

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and apply a sinusoid to the
input, VI, which would also

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appear across the Light Emitting
Expo Dweeb.

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And then, what I wanted to see
was how the output looked.

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OK, so let me tell you that the
output is going to look like

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this.
OK, the output is going to look

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like so.
And, a little artifice to

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discover curves like this is to
think about a point here

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corresponding to the point on
the transfer curve here,

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because this is VD,
looking at the Y intercept.

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That's a value of ID,
and that's a value of ID here.

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And, time moves along here,
and time moves along here.

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So, I did this little
animation.

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You'd better be impressed.
It took me six hours to do it.

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So, here it goes.
So, let's say I start by

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focusing on this little point
that corresponds to this point

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on the transfer function,
which then, in turn,

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points to a time,
zero, this point on my ID

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curve.
OK, I hope this works.

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So, as my point moves down
[LAUGHTER], this was fun to do,

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I promise you.
So notice that as this point

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has the following excursion,
this had the following

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excursions here.
OK, all right.

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So let me pause that little
animation there.

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At the end of the lecture,
I'll put that up again if you

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like, and you all can come and
play with it.

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So, you can actually do this in
PowerPoint.

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It took me quite a bit of time
to figure out how to do it,

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though, but it's fun.
OK, so let me show you a little

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demo, and show you a sinusoid,
and show you what the output

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looks like if I apply a sinusoid
for VI.

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So, I'll show you ID as a
function of VI when VI is a

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sinusoid.
There you go.

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So, I applied my sinusoid VI,
and this is the current that I

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get.
And notice, this is the

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transfer function that I talked
about, the ID versus VD curve of

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my Light Emitting Expo Dweeb.
And I get this highly nonlinear

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transformation of the input as I
get to the output.

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OK, so that is a problem.
And then, I also played some

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music for you.
Let's do that,

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too.
I played some music for you.

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I applied the music as an input
to the circuit,

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and that's the output.
OK, that's the output that I'm

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observing at the amplifier.
It's highly distorted.

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OK, we can stop that.
There you go.

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OK, so that was my problem.
OK, so we had covered,

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we had gone this far last
Tuesday.

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I set the problem up for you,
motivated what we had to do,

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and showed you that I was able
to transmit music over my garage

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door opener, but I did not think
I could listen to that music for

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very long.
So, I challenged all of us to

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think about how a trick that I
could use to be able to transmit

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music and have a linear
response.

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So, did you people get time to
think about it?

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So how many people here think
they know the answer?

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It's OK, don't be modest.
Go ahead.

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Could you speak louder?
Yeah, you find another

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something, kind of element,
that's got the opposite graph

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so that when you add them
together.

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Oh, this guy wants to cheat.
No.

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He wants a new element.
So, no, no new elements.

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Pardon?
Build an MP3 encoder.

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Ah-ha, so that will happen much
later.

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Yes?
Digitize the signal before you

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send it to the LED?
Digitize the signal before you

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send it to the LED.
But in some sense,

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each of these solutions is a
huge sledgehammer approach to

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look at solving it.
There's a much simpler

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technique I can apply here.
Yeah?

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Add a voltage offset.
Ah, ah-ha, that might work.

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What else?
So let's say,

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here's my signal,
right?

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If I add a voltage offset,
that will just bump the signal

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up here.
Then the curve is still

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nonlinear.
But you're getting there.

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Well, I'll tell you what.
Let's pause here.

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00:15:50,000 --> 00:15:55,000
Let me quit while I'm ahead.
OK, so the answer here,

233
00:15:55,000 --> 00:15:58,000
folks, is Zen.
OK, what I want you to do is,

234
00:15:58,000 --> 00:16:03,000
so, in Zen, what you have to do
is you have to sit down in a

235
00:16:03,000 --> 00:16:08,000
courtyard, and look at a rock,
like a small rock on the

236
00:16:08,000 --> 00:16:12,000
ground.
And you got a focus on it till

237
00:16:12,000 --> 00:16:15,000
the rest of Earth kind of
vanishes.

238
00:16:15,000 --> 00:16:18,000
Just focus on the rock.
OK, now make like you're in a

239
00:16:18,000 --> 00:16:22,000
courtyard, and you're looking at
this little area here.

240
00:16:22,000 --> 00:16:25,000
Just look at this.
OK, and I'll give you ten

241
00:16:25,000 --> 00:16:26,000
seconds.
Sit down quietly,

242
00:16:26,000 --> 00:16:29,000
and no sounds.
Just stare at the spot here.

243
00:16:29,000 --> 00:16:33,000
OK, make believe this is your
little rock, and just stand

244
00:16:33,000 --> 00:16:38,000
there and think about it.
OK, I'll give you five seconds

245
00:16:38,000 --> 00:16:41,000
to do that.
Just stare at it.

246
00:16:41,000 --> 00:16:45,000
And very soon,
the answer should pop into your

247
00:16:45,000 --> 00:16:47,000
heads.
OK, what do you see?

248
00:16:47,000 --> 00:16:52,000
This guy, if I focus on this
really small region of the

249
00:16:52,000 --> 00:16:57,000
graph, this small little piece
looks more or less linear.

250
00:16:57,000 --> 00:17:02,000
OK, hmm, so that should give me
some insight.

251
00:17:02,000 --> 00:17:05,000
This whole thing,
the macrograph is nonlinear.

252
00:17:05,000 --> 00:17:09,000
But I focus on a little rinky
dinky piece of that graph like

253
00:17:09,000 --> 00:17:11,000
so, that appears more or less
linear.

254
00:17:11,000 --> 00:17:14,000
If it's small enough,
that appears linear.

255
00:17:14,000 --> 00:17:18,000
So, I'm staring at this,
and that appears linear.

256
00:17:18,000 --> 00:17:21,000
The question is,
how do I exploit this little

257
00:17:21,000 --> 00:17:24,000
small, little,
linear region to get a linear

258
00:17:24,000 --> 00:17:27,000
response from my device.
OK, so here's the trick that

259
00:17:27,000 --> 00:17:31,000
I'm going to use.
The little trick that I'm going

260
00:17:31,000 --> 00:17:35,000
to use is the following.
Notice that,

261
00:17:35,000 --> 00:17:42,000
let me call this voltage at the
center of this region capital

262
00:17:42,000 --> 00:17:43,000
VD.
What I can do,

263
00:17:43,000 --> 00:17:50,000
if I take my input signal,
and I just pointed out earlier,

264
00:17:50,000 --> 00:17:52,000
I bump it up.
I boost it.

265
00:17:52,000 --> 00:17:57,000
OK, so I apply a DC offset to
my input signal,

266
00:17:57,000 --> 00:18:02,000
like so.
So I apply some input signal,

267
00:18:02,000 --> 00:18:07,000
VI, which is also equal to the
VD if I look at a variable

268
00:18:07,000 --> 00:18:12,000
across the nonlinear element.
If I apply a DC offset,

269
00:18:12,000 --> 00:18:16,000
VI, and I superimpose the music
on top of that,

270
00:18:16,000 --> 00:18:20,000
let me call my music,
just to distinguish between the

271
00:18:20,000 --> 00:18:23,000
two, capital VI,
and the small vi.

272
00:18:23,000 --> 00:18:27,000
OK, that's my music.
So here's my capital VD,

273
00:18:27,000 --> 00:18:32,000
my DC offset.
And I want to superimpose my

274
00:18:32,000 --> 00:18:35,000
music on top of that.
OK, so I've gotten halfway

275
00:18:35,000 --> 00:18:38,000
there.
By superimposing my music here

276
00:18:38,000 --> 00:18:43,000
instead of having excursions out
here, I now have excursions out

277
00:18:43,000 --> 00:18:46,000
here.
OK, and so I'm using some

278
00:18:46,000 --> 00:18:50,000
portion of the graph here.
But that's still way beyond the

279
00:18:50,000 --> 00:18:55,000
small little element there.
So a second think that I do in

280
00:18:55,000 --> 00:19:00,000
addition to boosting up the
signal is shrink it.

281
00:19:00,000 --> 00:19:02,000
Think of boost and shrink,
BS.

282
00:19:02,000 --> 00:19:08,000
So what I want to do is boost
up the signal using a DC offset,

283
00:19:08,000 --> 00:19:13,000
and shrink the sucker.
OK, so I'm going to go with a

284
00:19:13,000 --> 00:19:19,000
small signal and bump it up.
OK, so now what happens is that

285
00:19:19,000 --> 00:19:24,000
small signal in its excursions,
only uses that little portion

286
00:19:24,000 --> 00:19:27,000
of the graph.
OK, again, remember:

287
00:19:27,000 --> 00:19:30,000
bump and shrink,
bump and shrink,

288
00:19:30,000 --> 00:19:37,000
two things, boost and shrink.
So what do you think of that

289
00:19:37,000 --> 00:19:38,000
trick?
So, by doing that,

290
00:19:38,000 --> 00:19:43,000
what happens is that signal
that has excursions here will

291
00:19:43,000 --> 00:19:46,000
produce a corresponding response
in this region,

292
00:19:46,000 --> 00:19:49,000
OK?
And I argue that since this is

293
00:19:49,000 --> 00:19:53,000
more or less like a straight
line, I invoke Zen here,

294
00:19:53,000 --> 00:19:57,000
and argue that this little
signal now gets transformed,

295
00:19:57,000 --> 00:20:02,000
and I get a linear response.
OK: boost and shrink.

296
00:20:02,000 --> 00:20:06,000
So in terms of my circuit,
let me draw it out for you.

297
00:20:06,000 --> 00:20:12,000
My Light Emitting Expo Dweeb,
and this whole signal was what

298
00:20:12,000 --> 00:20:16,000
I used to call V capital I,
and that's made up of two

299
00:20:16,000 --> 00:20:19,000
components now,
a bump offset,

300
00:20:19,000 --> 00:20:23,000
and a shrunk voltage VI.
It shrunk, so therefore I've

301
00:20:23,000 --> 00:20:27,000
used the small v and small i,
like, really,

302
00:20:27,000 --> 00:20:31,000
really small.
In the same manner,

303
00:20:31,000 --> 00:20:37,000
I get a VD ID across the LED,
and the corresponding values

304
00:20:37,000 --> 00:20:41,000
here will also have a DC offset
and a small response.

305
00:20:41,000 --> 00:20:44,000
Let me call that ID plus I
small d.

306
00:20:44,000 --> 00:20:49,000
I'll do all this mathematically
in a second as well,

307
00:20:49,000 --> 00:20:54,000
but first let me do it
completely intuitively so you

308
00:20:54,000 --> 00:20:57,000
get some insight into what's
going on.

309
00:20:57,000 --> 00:21:03,000
And, VD is simply capital VD
plus small vd.

310
00:21:03,000 --> 00:21:06,000
OK, and this is the same as VI,
I, and VI.

311
00:21:06,000 --> 00:21:09,000
OK, so what have I done?
I've done two things.

312
00:21:09,000 --> 00:21:11,000
I have said,
as an engineer,

313
00:21:11,000 --> 00:21:16,000
OK, I care about getting music
across my garage door opener.

314
00:21:16,000 --> 00:21:19,000
And I'll do what it takes to do
that.

315
00:21:19,000 --> 00:21:22,000
OK, so as an engineer,
I'll do two things.

316
00:21:22,000 --> 00:21:25,000
I'm going to bump my signal up
and shrink it.

317
00:21:25,000 --> 00:21:29,000
And the bumping and shrinking,
and I do it like this.

318
00:21:29,000 --> 00:21:33,000
I shrink my signal,
the music signal here,

319
00:21:33,000 --> 00:21:38,000
and add a DC offset.
OK, and I claim that the music

320
00:21:38,000 --> 00:21:41,000
I listened on the other side
now, provided I have enough

321
00:21:41,000 --> 00:21:45,000
amplification there,
is going to be undistorted.

322
00:21:45,000 --> 00:21:49,000
OK, so far I've showing this to
you completely intuitively using

323
00:21:49,000 --> 00:21:50,000
little sketches,
no math.

324
00:21:50,000 --> 00:21:53,000
I promise you,
I'll give you a bunch of math

325
00:21:53,000 --> 00:21:56,000
in a few seconds,
but just get the basic idea,

326
00:21:56,000 --> 00:22:00,000
and get the intuition behind
it.

327
00:22:00,000 --> 00:22:04,000
So let's go back to our demo
and take a look.

328
00:22:04,000 --> 00:22:08,000
So remember,
BS, right, bump and shrink.

329
00:22:08,000 --> 00:22:12,000
So what I'm going to do is
first of all,

330
00:22:12,000 --> 00:22:18,000
let me bump up the signal.
So, what I'll do is I want to

331
00:22:18,000 --> 00:22:23,000
add an offset to my input,
and let me bump it up.

332
00:22:23,000 --> 00:22:28,000
Let me shrink it first.
It'll make the point a little

333
00:22:28,000 --> 00:22:32,000
clearer.
So, the big input,

334
00:22:32,000 --> 00:22:35,000
green, is a big input.
Let me shrink it.

335
00:22:46,000 --> 00:22:49,000
OK, so I've made my input
small, and in the middle of that

336
00:22:49,000 --> 00:22:52,000
picture out there,
you see the region of the

337
00:22:52,000 --> 00:22:54,000
transfer curve that's being
articulated.

338
00:22:54,000 --> 00:22:58,000
OK, this region of the curve is
being articulated by the small

339
00:22:58,000 --> 00:23:00,000
signal.
It's a much smaller signal.

340
00:23:00,000 --> 00:23:03,000
And the output is still
distorted because I have to do

341
00:23:03,000 --> 00:23:08,000
two things: bump and shrink.
I've only shrunk.

342
00:23:08,000 --> 00:23:13,000
OK, let me bump it up now.
What's the yellow curve?

343
00:23:13,000 --> 00:23:20,000
It's going to get linear.
It's going to get proportional

344
00:23:20,000 --> 00:23:25,000
to the input.
Then I'm bumping it up now.

345
00:23:25,000 --> 00:23:30,000
I can make it smaller,
make it even smaller,

346
00:23:30,000 --> 00:23:34,000
there you go.
Isn't that fantastic?

347
00:23:34,000 --> 00:23:37,000
So, I'm making nature do my
bidding here,

348
00:23:37,000 --> 00:23:40,000
OK?
So, this is one of those,

349
00:23:40,000 --> 00:23:44,000
when I learned electronics and
so on many, many years ago,

350
00:23:44,000 --> 00:23:48,000
this was one of those really
big ah-ha moments for me,

351
00:23:48,000 --> 00:23:51,000
saying, wow,
that stuff is cool.

352
00:23:51,000 --> 00:23:55,000
It's something that I couldn't
think about myself,

353
00:23:55,000 --> 00:23:59,000
and it's not obvious,
and by being disciplined and

354
00:23:59,000 --> 00:24:03,000
creative in how I use circuits,
I can do really,

355
00:24:03,000 --> 00:24:07,000
really cool things.
OK, remember this as a big

356
00:24:07,000 --> 00:24:11,000
ah-ha moment for you.
So, here's my little signal

357
00:24:11,000 --> 00:24:15,000
that I've shrunk and bumped up,
and my output is a sinusoid,

358
00:24:15,000 --> 00:24:18,000
and not this funny,
distorted waveform.

359
00:24:18,000 --> 00:24:22,000
And notice that this is the
region of the curve that is

360
00:24:22,000 --> 00:24:26,000
being articulated.
So, I can make the signal even

361
00:24:26,000 --> 00:24:29,000
smaller if I like.
OK, and what I'd like to do

362
00:24:29,000 --> 00:24:33,000
next is play music for you,
and if you don't believe your

363
00:24:33,000 --> 00:24:38,000
eyes, you can at least believe
your ears.

364
00:24:38,000 --> 00:24:43,000
Let me go to the distorted
signal again,

365
00:24:43,000 --> 00:24:48,000
switch to music,
and raise it up.

366
00:24:48,000 --> 00:24:56,000
OK, now what we'll do is shrink
the music signal and then bump

367
00:24:56,000 --> 00:25:01,000
it up.
Can I turn the volume down a

368
00:25:01,000 --> 00:25:06,000
little bit?
That's good.

369
00:25:06,000 --> 00:25:13,000
OK, so if I shrunk the volume a
little bit, and let me bump it

370
00:25:13,000 --> 00:25:17,000
up, now.
[MUSIC PLAYS] Just remember

371
00:25:17,000 --> 00:25:23,000
this as a big ah-ha moment.
OK, the signal is really,

372
00:25:23,000 --> 00:25:26,000
really small.
I like that.

373
00:25:26,000 --> 00:25:33,000
I like the enthusiasm.
OK, so the signal's very small,

374
00:25:33,000 --> 00:25:38,000
and I get a more or less linear
response.

375
00:25:38,000 --> 00:25:42,000
OK.
All right, so that's intuition,

376
00:25:42,000 --> 00:25:46,000
and the approach that I've
taken is called,

377
00:25:46,000 --> 00:25:50,000
it's variously called small
signal analysis,

378
00:25:50,000 --> 00:25:54,000
incremental analysis,
small signal method,

379
00:25:54,000 --> 00:25:59,000
small signal discipline,
whatever you want.

380
00:26:08,000 --> 00:26:13,000
OK, this simply says that by
boosting and shrinking my

381
00:26:13,000 --> 00:26:20,000
signal, I get a response that's
more or less linear even when I

382
00:26:20,000 --> 00:26:25,000
have a nonlinear device.
And this technique is called

383
00:26:25,000 --> 00:26:31,000
the small signal approach.
So, just to focus on that a

384
00:26:31,000 --> 00:26:36,000
little bit longer,
switch to page five of your

385
00:26:36,000 --> 00:26:42,000
notes and let me draw something
out for you.

386
00:26:47,000 --> 00:26:52,000
OK, so what I have here,
this is my offset VD,

387
00:26:52,000 --> 00:26:59,000
and from the VD offset I have
my little signal V small d,

388
00:26:59,000 --> 00:27:06,000
and the total signal is called
V capital D.

389
00:27:06,000 --> 00:27:10,000
Offset, small signal,
and that's my total signal.

390
00:27:10,000 --> 00:27:14,000
OK, notice the offset is all
capital.

391
00:27:14,000 --> 00:27:19,000
The total signal is small v
capital D, and the music or the

392
00:27:19,000 --> 00:27:25,000
small signal is small v small d.
Similarly, the output is going

393
00:27:25,000 --> 00:27:30,000
to look like this,
and here I get an offset in the

394
00:27:30,000 --> 00:27:35,000
output ID.
I get a corresponding signal,

395
00:27:35,000 --> 00:27:40,000
I small d, and I get a total
signal, I capital D,

396
00:27:40,000 --> 00:27:43,000
OK?
The cool thing to notice is

397
00:27:43,000 --> 00:27:48,000
that the signal here,
the output signal here

398
00:27:48,000 --> 00:27:54,000
corresponding to the input
signal, the music signal,

399
00:27:54,000 --> 00:27:59,000
VD, is small I small D,
and that is more or less

400
00:27:59,000 --> 00:28:02,000
linear.
OK, and I can even plot the

401
00:28:02,000 --> 00:28:10,000
signal like so.
This is my input,

402
00:28:10,000 --> 00:28:15,000
v capital D.
That's T.

403
00:28:15,000 --> 00:28:26,000
This is VD, V small d.
That is my total input.

404
00:28:26,000 --> 00:28:36,000
And similarly,
I have an output.

405
00:28:36,000 --> 00:28:44,000
And this is my output ID.
And, that looks like this,

406
00:28:44,000 --> 00:28:49,000
I capital D,
small i small d,

407
00:28:49,000 --> 00:28:58,000
total signal I capital D.
OK, so that's the small signal

408
00:28:58,000 --> 00:29:02,000
method.
So, let me summarize that for

409
00:29:02,000 --> 00:29:03,000
you.

410
00:29:12,000 --> 00:29:16,000
There are three steps to the
method.

411
00:29:16,000 --> 00:29:21,000
So, first of all,
operate at some DC offset.

412
00:29:21,000 --> 00:29:27,000
This is also called DC bias,
and in that example it's VDID.

413
00:29:27,000 --> 00:29:33,000
OK, so I choose an operating
point that bumps up the

414
00:29:33,000 --> 00:29:39,000
operation in some region of
interest.

415
00:29:39,000 --> 00:29:45,000
The second step is to
superimpose small signal on top

416
00:29:45,000 --> 00:29:53,000
of VD, capital V capital D,
to superimpose a small signal,

417
00:29:53,000 --> 00:30:00,000
and the third step is observe
the response --

418
00:30:00,000 --> 00:30:06,000
-- and the response,
small i small d,

419
00:30:06,000 --> 00:30:14,000
that's the music part of the
response, ID,

420
00:30:14,000 --> 00:30:23,000
is approximately linear.
OK, three steps to the method

421
00:30:23,000 --> 00:30:32,000
here, and just remember this
notation.

422
00:30:32,000 --> 00:30:39,000
And, my notation in the small
signal model is as follows.

423
00:30:39,000 --> 00:30:46,000
My total signal ID is the sum
of two signals,

424
00:30:46,000 --> 00:30:50,000
I capital D plus small i small
d.

425
00:30:50,000 --> 00:30:54,000
This is called the total
signal.

426
00:30:54,000 --> 00:31:03,000
That's called the DC offset.
And this is the superimposed

427
00:31:03,000 --> 00:31:07,000
small signal.
OK, total signal,

428
00:31:07,000 --> 00:31:12,000
DC offset, plus the small
signal.

429
00:31:12,000 --> 00:31:17,000
And sometimes,
especially when doing math,

430
00:31:17,000 --> 00:31:24,000
and so on, we may oftentimes
represent ID as a delta,

431
00:31:24,000 --> 00:31:29,000
I capital D,
OK, to show that ID is

432
00:31:29,000 --> 00:31:37,000
incremental change in the value
of I capital D.

433
00:31:37,000 --> 00:31:41,000
And because of that,
this method is also often

434
00:31:41,000 --> 00:31:46,000
called the incremental method,
incremental analysis.

435
00:31:46,000 --> 00:31:51,000
OK, so far what I've done is
given you some intuition.

436
00:31:51,000 --> 00:31:55,000
I've developed a small,
simple method,

437
00:31:55,000 --> 00:31:59,000
given you some insight into why
we use this method,

438
00:31:59,000 --> 00:32:05,000
and also shown you some
demonstrations that show that

439
00:32:05,000 --> 00:32:09,000
when I bump and shrink,
and observe the response,

440
00:32:09,000 --> 00:32:15,000
I do get a more or less linear
response.

441
00:32:15,000 --> 00:32:21,000
So let me now do this
mathematically and show you that

442
00:32:21,000 --> 00:32:27,000
mathematically,
you can also derive your

443
00:32:27,000 --> 00:32:32,000
response to be a linear
response.

444
00:32:32,000 --> 00:32:38,000
This is page seven.
So, I know that ID is some

445
00:32:38,000 --> 00:32:47,000
function of the diode voltage.
F was my nonlinear function.

446
00:32:47,000 --> 00:32:54,000
OK, so my function F was a
nonlinear function.

447
00:32:54,000 --> 00:33:00,000
So therefore,
ID was nonlinearly related to

448
00:33:00,000 --> 00:33:03,000
VD.
So, let's do the math.

449
00:33:03,000 --> 00:33:07,000
So as a first step,
what we did was replace VD by a

450
00:33:07,000 --> 00:33:11,000
DC offset, the small signal
method, a DC offset,

451
00:33:11,000 --> 00:33:13,000
plus a small incremental
change.

452
00:33:13,000 --> 00:33:17,000
OK, by doing the math,
let me simply use the delta VD

453
00:33:17,000 --> 00:33:21,000
notation to show you that I'm
dealing with small increments,

454
00:33:21,000 --> 00:33:25,000
and also because in the
mathematics community,

455
00:33:25,000 --> 00:33:28,000
when you learn about some of
these techniques,

456
00:33:28,000 --> 00:33:32,000
they will use the incremental
change notation,

457
00:33:32,000 --> 00:33:38,000
which is the delta VD notation.
In electrical engineering,

458
00:33:38,000 --> 00:33:41,000
we use a small v,
small d notation.

459
00:33:41,000 --> 00:33:46,000
So, this is a large DC offset,
and this is a small change

460
00:33:46,000 --> 00:33:50,000
about that offset.
So, you folks have taken math

461
00:33:50,000 --> 00:33:54,000
courses before,
and been looking at finding out

462
00:33:54,000 --> 00:33:59,000
the value of a function,
which is a small change for an

463
00:33:59,000 --> 00:34:04,000
input value, which is a small
change about a big input value

464
00:34:04,000 --> 00:34:09,000
or a big DC point is Taylor's
expansion.

465
00:34:09,000 --> 00:34:12,000
OK, so let's use Taylor's
series expansion,

466
00:34:12,000 --> 00:34:16,000
OK, and substitute VD plus
delta VD into this,

467
00:34:16,000 --> 00:34:21,000
and see what ID looks like.
Again, let me tell you where

468
00:34:21,000 --> 00:34:24,000
I'm going with this.
ID equals F of VD.

469
00:34:24,000 --> 00:34:28,000
This is a nonlinear function,
OK?

470
00:34:28,000 --> 00:34:33,000
I claim that by replacing VD,
the input, with the DC offset

471
00:34:33,000 --> 00:34:37,000
plus a small value,
the resulting response to the

472
00:34:37,000 --> 00:34:40,000
small value will be linear,
OK?

473
00:34:40,000 --> 00:34:46,000
So what I'm going to do next is
replace VD with this sum here,

474
00:34:46,000 --> 00:34:50,000
and then do the math,
and show you that the response

475
00:34:50,000 --> 00:34:53,000
corresponding,
or the change in ID

476
00:34:53,000 --> 00:34:58,000
corresponding to the change in
VD is going to be linear.

477
00:34:58,000 --> 00:35:04,000
All right, so let's expand this
function using Taylor's series

478
00:35:04,000 --> 00:35:10,000
near the DC offset point,
capital V capital D.

479
00:35:10,000 --> 00:35:13,000
OK, so ID is simply,
by Taylor's series,

480
00:35:13,000 --> 00:35:19,000
I want to find out a value of
the function close to V capital

481
00:35:19,000 --> 00:35:21,000
D.
OK, so I take the value of the

482
00:35:21,000 --> 00:35:26,000
function at that point,
and then I add a few terms in

483
00:35:26,000 --> 00:35:34,000
my Taylor's series expansion.
The first term is simply the

484
00:35:34,000 --> 00:35:43,000
good old Taylor's series stuff.
OK, the first term is the first

485
00:35:43,000 --> 00:35:49,000
derivative of the function times
the change.

486
00:35:49,000 --> 00:35:57,000
And then, the second one is
second derivative.

487
00:36:09,000 --> 00:36:11,000
OK, and then I get higher order
terms.

488
00:36:11,000 --> 00:36:16,000
So this is nothing new here.
This is good old Taylor series

489
00:36:16,000 --> 00:36:19,000
expansion, and again,
let me tell you where I'm

490
00:36:19,000 --> 00:36:22,000
going.
I want to look at the response

491
00:36:22,000 --> 00:36:27,000
for an input that looks like
this, and I want to show you at

492
00:36:27,000 --> 00:36:30,000
the end of the day that the
response in ID,

493
00:36:30,000 --> 00:36:34,000
the effect on ID of using an
input like this is as if that

494
00:36:34,000 --> 00:36:39,000
effect, the incremental change
is linearly related to the small

495
00:36:39,000 --> 00:36:44,000
input, delta VD.
So here's my Taylor's series

496
00:36:44,000 --> 00:36:47,000
expansion for delta V.
Now remember,

497
00:36:47,000 --> 00:36:52,000
I told you that delta VD is
much, much smaller than V

498
00:36:52,000 --> 00:36:54,000
capital D.
OK, it's a very,

499
00:36:54,000 --> 00:36:59,000
very small quantity.
But that quantity is really

500
00:36:59,000 --> 00:37:03,000
very small.
Then what I'm going to get is

501
00:37:03,000 --> 00:37:07,000
that my output is,
I can begin to ignore my second

502
00:37:07,000 --> 00:37:09,000
order terms.
OK, delta VD is very,

503
00:37:09,000 --> 00:37:13,000
very, very small.
Then, what I'm going to do is

504
00:37:13,000 --> 00:37:18,000
that ignore higher order terms.
So I'll go and ignore higher

505
00:37:18,000 --> 00:37:20,000
order terms.
They'll all go to zero.

506
00:37:20,000 --> 00:37:25,000
Remember, I can do this because
by design I've chosen delta VD

507
00:37:25,000 --> 00:37:29,000
to be very, very,
very small.

508
00:37:29,000 --> 00:37:32,000
OK, remember,
we are engineers.

509
00:37:32,000 --> 00:37:37,000
I've chosen it in a way that
this is very small.

510
00:37:37,000 --> 00:37:42,000
OK, so I'm telling you that's
the case, and under those

511
00:37:42,000 --> 00:37:48,000
conditions, I can ignore second
higher order terms,

512
00:37:48,000 --> 00:37:53,000
in which case I am left with
this expression here.

513
00:37:53,000 --> 00:38:00,000
So let me rewrite this.
Let me rewrite this down here.

514
00:38:15,000 --> 00:38:19,000
OK, I've just copied this
turnout, I've ignored all these

515
00:38:19,000 --> 00:38:24,000
terms here, and so I have a more
or less equal to sign that

516
00:38:24,000 --> 00:38:27,000
remains.
So what I'm going to do is when

517
00:38:27,000 --> 00:38:32,000
I apply a small input of this
form to a large DC offset,

518
00:38:32,000 --> 00:38:37,000
my output is also going to look
like some output offset with a

519
00:38:37,000 --> 00:38:43,000
change in the output offset.
And let me call the output

520
00:38:43,000 --> 00:38:49,000
offset I capital D,
and some small change in the

521
00:38:49,000 --> 00:38:54,000
output delta ID.
OK, we'll make sure we can

522
00:38:54,000 --> 00:38:59,000
convince ourselves that this is
indeed the case.

523
00:38:59,000 --> 00:39:05,000
Notice that this guy here,
F of capital V capital D is a

524
00:39:05,000 --> 00:39:12,000
constant.
That's a constant with respect

525
00:39:12,000 --> 00:39:17,000
to the incremental change,
delta VD.

526
00:39:17,000 --> 00:39:23,000
Similarly, this part here is a
constant.

527
00:39:23,000 --> 00:39:31,000
Notice that this term here is
the first derivative of the

528
00:39:31,000 --> 00:39:41,000
function evaluated at the DC
bias point, capital V capital D.

529
00:39:41,000 --> 00:39:45,000
OK, so this term is also a
constant with respect to delta

530
00:39:45,000 --> 00:39:47,000
VD.
So notice, then,

531
00:39:47,000 --> 00:39:52,000
I have a constant term plus a
constant term multiplying a

532
00:39:52,000 --> 00:39:54,000
small change,
delta VD.

533
00:39:54,000 --> 00:39:57,000
So what I can do next is,
in this case,

534
00:39:57,000 --> 00:40:02,000
given that I have a constant
term on both sides,

535
00:40:02,000 --> 00:40:05,000
and on this side it's a time
varying term,

536
00:40:05,000 --> 00:40:11,000
what I can do is equate the two
constant terms.

537
00:40:11,000 --> 00:40:14,000
I can go ahead and equate these
two terms.

538
00:40:14,000 --> 00:40:18,000
Remember, I have a constant
plus a time varying term,

539
00:40:18,000 --> 00:40:22,000
OK, if I'm assuming here that
delta VD, my little music signal

540
00:40:22,000 --> 00:40:26,000
is a time varying term.
So, this constant will equal

541
00:40:26,000 --> 00:40:31,000
this, so ID must equal F of VD.
And I know that's the case

542
00:40:31,000 --> 00:40:36,000
because the function evaluated
at the DC offset gives me the DC

543
00:40:36,000 --> 00:40:39,000
current ID.
And similarly,

544
00:40:39,000 --> 00:40:46,000
ID is equal to that component.
Delta ID is equal to D,

545
00:40:46,000 --> 00:40:48,000
F of --

546
00:41:00,000 --> 00:41:04,000
OK, so my incremental change in
the output is the first

547
00:41:04,000 --> 00:41:09,000
derivative multiplied by the
small change in the current.

548
00:41:09,000 --> 00:41:13,000
OK, so I'm pretty much done.
So, therefore,

549
00:41:13,000 --> 00:41:17,000
notice that delta ID is
proportional to delta VD.

550
00:41:17,000 --> 00:41:21,000
OK, and that's what I had set
out to show.

551
00:41:21,000 --> 00:41:26,000
Remember, I had set out to show
that provided my input is a

552
00:41:26,000 --> 00:41:31,000
small excursion around a large
DC offset,

553
00:41:31,000 --> 00:41:35,000
then my output could also be a
large DC offset with a small

554
00:41:35,000 --> 00:41:39,000
excursion on top of it where the
two excursions,

555
00:41:39,000 --> 00:41:43,000
the input excursion and the
output excursion would be

556
00:41:43,000 --> 00:41:48,000
linearly related like so.
OK, and the method is very

557
00:41:48,000 --> 00:41:51,000
simple.
I simply expanded the function

558
00:41:51,000 --> 00:41:53,000
about that point,
that DC point,

559
00:41:53,000 --> 00:41:58,000
neglected higher order terms,
and notice that my incremental

560
00:41:58,000 --> 00:42:03,000
term was simply the derivative
plus the incremental change,

561
00:42:03,000 --> 00:42:09,000
a derivative times the
incremental change in the input.

562
00:42:09,000 --> 00:42:12,000
Move onto page nine,
and I'd like to give you a

563
00:42:12,000 --> 00:42:16,000
quick graphical interpretation
of this.

564
00:42:16,000 --> 00:42:19,000
So I gave an intuitive
explanation earlier.

565
00:42:19,000 --> 00:42:24,000
This is a mathematical
explanation that shows you that

566
00:42:24,000 --> 00:42:28,000
the input could be linearly
related to the output,

567
00:42:28,000 --> 00:42:33,000
provided, the outputs would be
linearly related to the input,

568
00:42:33,000 --> 00:42:38,000
provided the input has a DC
offset, and small excursions

569
00:42:38,000 --> 00:42:43,000
about that DC offset.
So, let me give you some

570
00:42:43,000 --> 00:42:50,000
intuition in what you've really
done here, using a little graph

571
00:42:50,000 --> 00:42:54,000
here.
So, I'm going to plot ID versus

572
00:42:54,000 --> 00:43:00,000
VD, and notice that I have some
point here, V capital D,

573
00:43:00,000 --> 00:43:03,000
I capital D.
That's my DC bias.

574
00:43:03,000 --> 00:43:08,000
So, I have some DC bias point
here.

575
00:43:08,000 --> 00:43:14,000
OK, what is this?
That is simply the slope of the

576
00:43:14,000 --> 00:43:20,000
curve at that point.
OK, it's the slope of this

577
00:43:20,000 --> 00:43:28,000
curve evaluated at this point.
So this guy here is simply the

578
00:43:28,000 --> 00:43:34,000
slope of this curve evaluated at
ID VD.

579
00:43:34,000 --> 00:43:41,000
OK, now, what I care about is
this point here,

580
00:43:41,000 --> 00:43:50,000
and this point here.
So let's say that this is delta

581
00:43:50,000 --> 00:43:57,000
VD, all right,
and that corresponds to this

582
00:43:57,000 --> 00:44:03,000
point here.
So what I've done is taken the

583
00:44:03,000 --> 00:44:07,000
slope and multiplied that by
delta VD.

584
00:44:07,000 --> 00:44:12,000
So I've taken the slope,
and multiplied it by delta VD,

585
00:44:12,000 --> 00:44:17,000
OK, and that gives me this
component here.

586
00:44:17,000 --> 00:44:22,000
OK, and so, this is the point
that I'm going to get.

587
00:44:22,000 --> 00:44:27,000
So in other words,
what I've done is approximated

588
00:44:27,000 --> 00:44:34,000
point A using the Taylor trick
by the point B.

589
00:44:34,000 --> 00:44:38,000
OK, so this is a point,
A, which is what I really want,

590
00:44:38,000 --> 00:44:44,000
and I've approximated that by
taking the slope of the function

591
00:44:44,000 --> 00:44:48,000
at V capital D,
and multiplying that by the

592
00:44:48,000 --> 00:44:53,000
change in the input to get the
corresponding Y offset,

593
00:44:53,000 --> 00:44:55,000
and that's the point that I
get.

594
00:44:55,000 --> 00:45:00,000
And notice that if I make this
delta VD small enough,

595
00:45:00,000 --> 00:45:05,000
then the error between these
two points becomes smaller and

596
00:45:05,000 --> 00:45:10,000
smaller.
So back to our example,

597
00:45:10,000 --> 00:45:16,000
so ID was a e to the BVD.
This was the relation for our

598
00:45:16,000 --> 00:45:21,000
Expo Dweeb, and let me just plug
in the values.

599
00:45:21,000 --> 00:45:26,000
So, ID plus small id.
Notice, I'm just shuttling back

600
00:45:26,000 --> 00:45:34,000
and forth between the notation
delta VD, and small v small d.

601
00:45:43,000 --> 00:45:51,000
OK, and so that is given by a e
to the BVD, oops,

602
00:45:51,000 --> 00:45:59,000
plus, I'm just writing that
equation up there.

603
00:45:59,000 --> 00:46:08,000
Let me call this equation X.
And so, I get the second term

604
00:46:08,000 --> 00:46:13,000
is the derivative,
ab times e to the BVD times

605
00:46:13,000 --> 00:46:18,000
delta VD, small VD,
and equating this term that the

606
00:46:18,000 --> 00:46:22,000
DC offset.
Notice that this is the DC

607
00:46:22,000 --> 00:46:27,000
offset in the output,
and the small signal,

608
00:46:27,000 --> 00:46:33,000
ID is, further notice that in
this particular example,

609
00:46:33,000 --> 00:46:37,000
what's that?
a e to the BVD.

610
00:46:37,000 --> 00:46:45,000
That's simply ID again.
It just happens to be that way

611
00:46:45,000 --> 00:46:50,000
in this example.
So, I get ID times BVD.

612
00:46:50,000 --> 00:46:56,000
So, for my input,
small id, my incremental change

613
00:46:56,000 --> 00:47:03,000
in the output is some ID times B
times VD.

614
00:47:03,000 --> 00:47:07,000
And notice that this is a
constant.

615
00:47:07,000 --> 00:47:15,000
And because that is a constant,
my small signal behavior ID is

616
00:47:15,000 --> 00:47:20,000
going to be linearly related to
the signal, VD,

617
00:47:20,000 --> 00:47:27,000
the input signal VD.
OK, in the last three minutes,

618
00:47:27,000 --> 00:47:34,000
I'd like to give you one
additional insight.

619
00:47:34,000 --> 00:47:38,000
So what we've shown so far is
if I have an offset and a small

620
00:47:38,000 --> 00:47:42,000
change above it,
then my output ID will be

621
00:47:42,000 --> 00:47:47,000
linearly related to my input.
Now let's stare at this thing

622
00:47:47,000 --> 00:47:49,000
again.
Let me rewrite it.

623
00:47:49,000 --> 00:47:51,000
It's some constant IDB times
VD.

624
00:47:51,000 --> 00:47:55,000
So, where have we seen such an
expression before?

625
00:47:55,000 --> 00:48:00,000
OK, where ID was some constant
times VD.

626
00:48:00,000 --> 00:48:03,000
OK, remember,
I equals V divided by R:

627
00:48:03,000 --> 00:48:06,000
Ohm's law.
What I want to show you now is

628
00:48:06,000 --> 00:48:10,000
how we constantly keep
simplifying our lives.

629
00:48:10,000 --> 00:48:15,000
The moment we hit some
complication and things get too

630
00:48:15,000 --> 00:48:18,000
painful to analyze,
as engineers,

631
00:48:18,000 --> 00:48:23,000
we come up with some clever
tricks to make an analysis and

632
00:48:23,000 --> 00:48:32,000
use of circuits simple again.
And so, notice that this is

633
00:48:32,000 --> 00:48:37,000
similar to some,
one by RD VD,

634
00:48:37,000 --> 00:48:43,000
where RD is simply one over
IDB.

635
00:48:43,000 --> 00:48:48,000
I'm just defining this to be
RD.

636
00:48:48,000 --> 00:48:59,000
And what that means is that I
can take a nonlinear circuit

637
00:48:59,000 --> 00:49:07,000
that looks like this.
OK, and what I can do is

638
00:49:07,000 --> 00:49:13,000
replace this by its incremental
equivalent, and build what is

639
00:49:13,000 --> 00:49:18,000
called a small signal circuit.
And I'll just introduce it

640
00:49:18,000 --> 00:49:22,000
here.
And we will revisit the circuit

641
00:49:22,000 --> 00:49:27,000
in much more gory detail a
couple of weeks from now.

642
00:49:27,000 --> 00:49:32,000
So, what I can do is build a
small signal circuit where I

643
00:49:32,000 --> 00:49:37,000
have all the small signal
variables, and replace a

644
00:49:37,000 --> 00:49:43,000
nonlinear device by a simple
little resistor whose value is

645
00:49:43,000 --> 00:49:47,000
given by IDB.
OK, so therefore,

646
00:49:47,000 --> 00:49:51,000
what I can do is take my
nonlinear circuit,

647
00:49:51,000 --> 00:49:55,000
and for small,
incremental changes,

648
00:49:55,000 --> 00:50:01,000
replace that circuit with this
equivalent small signal circuit,

649
00:50:01,000 --> 00:50:04,000
and go back to doing simple
stuff again.

650
00:50:04,000 --> 00:50:07,000
Thank you.