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All right, good morning.
So today, we are going to talk

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about what is both a basic
device in itself,

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the amplifier,
and it also serves as a real

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key example of both nonlinear
analysis and small signal

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analysis.
So, today, dependent sources

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and amplifiers.

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So, let me first spend a few
seconds just pointing out to you

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some of the key points from our
previous lectures.

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I also want to point out that
each chapter in the course notes

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has a summary at the end of it.
And if you take a quick scan of

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the summary at the end of each
chapter, it highlights the major

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takeaway points from each
chapter.

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It stresses what's important,
and if you have to remember a

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few things, what are those
things to remember?

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So, to quickly review,
we talked about a few primitive

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elements: resistors,
voltage sources,

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and so on.
And by now, you should have the

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facility to play around with
these device elements.

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And then we talked about the
Node method, and this is kind of

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the workhorse of 6.002.
When in doubt,

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use the Node method.
OK, and this will work both for

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linear circuits and nonlinear
circuits.

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OK, so if you see a problem,
or if you see a situation in

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real life that requires
analysis, then as a first step,

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you should try to think of
whether you could apply some of

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the key intuitive shortcut
methods, superposition.

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One of my favorites,
the Thevenin method,

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the Norton method,
or the method that involves

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composition, that is very
quickly analyzing circuits that

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have resistors in series and
parallel.

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OK, so if you can apply one of
these quick, intuitive,

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shortcut methods,
go do so.

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If you can't,
then usually you can resort to

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the Node method irrespective of
whether the circuit is linear or

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nonlinear.
So the last week was focused on

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the nonlinear method or
nonlinear circuits,

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and we spent the first lecture
talking about a straightforward

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application of the Node method,
which gave us a bunch of

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nonlinear equations that we had
to solve.

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In the last lecture,
we talked about the small

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signal trick.
What we said is if you look at

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the whole space of nonlinear
circuits, then within that

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space, if we focus on small
variations, small perturbations

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about an operating point,
then even the behavior of

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nonlinear circuits in that small
regime would be linear.

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So small signal method.

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And as an example,
I showed you how I could take a

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highly nonlinear device like the
garage door opener LED,

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and using that,
build a pretty nice transmitter

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that would transmit music.
And as long as we kept the

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signal small,
and operated the device in a

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region where its transfer curve
was relatively smooth,

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and I biased,
or set the operating point

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appropriately,
I would get a linear,

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small signal response.
OK.

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So today, we're going to do a
couple things.

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We're going to look at
dependent sources.

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And the reading for this is
section 2.6 of your course

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notes.
And, the dependent source will

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be a new element in your tool
chest.

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We will also do amplifiers,
and amplifiers are in section

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7.1 and section 7.2 of your
course notes.

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So, before I begin with
dependent sources,

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I'm just a huge believer in
motivating things with real

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world examples.
OK, so let me start by

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motivating: why we need an
amplifier?

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Why do we need to do things
like this?

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Or why do we even bother?
And, spend a few minutes really

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getting you to appreciate that
amplification is fundamental.

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OK, it's as foundational to
life as high fat potato chips

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and stuff like that.
So, let's do some basic

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examples here.
So first, let me talk about,

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why do we need to amplify
signals.

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Why amplify?
Why do we care about building

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an amplifier?
So, an amplifier,

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think of a little box,
and apply some sort of small

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input.
And I get a larger output.

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In this example,
this may be a voltage with a

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swing of 10 mV,
and in this case,

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the output might be another
voltage with a swing of,

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say, 100 mV.
And commonly,

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the amplifier,
in addition to an input and an

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output, input port and output
port, may also contain the power

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port, OK, so that I can apply a
power supply to the amplifier

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because commonly as an amplifier
signal, I'm looking for a power

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gain as well,
an increase in the power

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provided by the output.
So, that's an abstract

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definition of an amplifier,
and let's take a look at an

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example of why we may need this.
So let's say I have a small,

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useful signal,
and let's say the signal has 1

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mV peak to peak.
And, I'm looking to transmit

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the signal over a wire to some
other point.

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But let's say that in this
environment, I get a bunch of

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noise that is in a noisy
environment.

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And in this environment,
let's assume that some noise

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may get superimposed.
And if I have a 1 mV signal,

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and 10 mV of noise,
then what I end up with at the

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output is something that looks
like this.

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And it's really hard to
distinguish my 1 mV signal from

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that large amount of noise.
On the other hand,

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if I do the following,
if I took the signal and passed

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the signal to an amplifier,
and I amplified the signal to

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be a much larger version of the
same signal, let's say in this

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particular situation 100 mV peak
to peak signal.

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OK, so I magnified the signal
by a factor of 100.

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OK, let's say it's a linear
amplifier, I linearly amplified

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signal to be 100 mV,
then in that case,

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if I had a noise on top of
this, it's going to be less

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discernible.
The signal will look like this.

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OK, my 10 mV noise would add on
to it.

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But, this is still pretty
decent.

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I can still recognize the
input.

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And so, this is one application
of amplification.

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If I need to send something
from point A to point B as an

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analog signal,
then an amplified signal is

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less prone to noise attacks than
a small signal.

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Not surprisingly,
a large number of devices that

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are used in everyday life have
amplifiers built into them.

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So, get a little cell phone,
and virtually every single cell

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phone contains an amplifier.
By the way, this is an all

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digital cell phone.
It's a Kyocera,

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I forget the number now.
It's completely digital.

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OK, although they say it's
completely digital,

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it turns out that a significant
fraction of the circuitry is

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analog, in particular,
so digital is sort of a

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marketing term to say that
there's something special about

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this.
But remember,

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there's a bunch of analog
stuff.

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So, here's my little antenna
from the cell phone.

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OK, and typically the first
thing that happens to a signal

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as it comes out of the antenna
in your cell phone is,

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look at cell phone circuits,
or cell phone systems would be

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something that looks like this,
OK, this, and may have a label

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LNA.
If someone were to take a guess

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at what LNA might stand for?
What's that?

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Linear amplifier.
That's pretty good.

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So that's LNA.
Close enough.

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A is correct.
It's amplifier.

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What does L and N stand for?
Low noise.

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OK, so this stands for low
noise amplifier.

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So, I get a really rinky dinky
small signal here,

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and then the low noise
amplifier amplifies a signal.

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And in real cell phones,
and for that matter,

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in your 802.11b,
or 802.11a, or 802.11g wireless

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cards, same thing.
Antenna, low noise amplifier,

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and then you may have a bunch
of processing.

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And commonly,
you have a bunch of analog

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processing.
And then, you convert the

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analog to a digital signal.
OK, I recall last week I asked

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somebody in class here,
how would we transmit the

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signal from point A to point B
without it being impacted way

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too much by noise,
and he said,

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oh, go digital.
Good point.

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OK, so if I go digital,
I can transfer the signal

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without noise being a real
factor.

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But the analog to digital
converters need the signal

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strengths to be a given value
before it can chop it up into

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digital levels.
OK, so an amplifier is very

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fundamental.
OK, and so in this case,

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what may be a signal of a few
tens of microvolts to be

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amplified to some large enough
value that it can be further

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processed.
So, that's application of

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amplification in the analog
domain.

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Let me talk about amplification
in the digital domain.

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So, that's in the analog
domain.

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This amplification is in the
domain that I have both analog

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and digital.
OK, and now let me talk about

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amplification in the digital
domain, OK?

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I'm going to argue that
amplification is absolutely

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foundational to the digital
domain.

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OK, the digital abstraction
would not occur if I did not

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have basic amplification.
OK, and the next minute and 37

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seconds I will prove that to
you, OK?

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So, let's do so.
So, let's suppose I have a very

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simple digital system,
and the system simply contains

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a pair of inverters.
So, if I send a one here,

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it's a zero here and a one
here, which is a very simple,

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trivial, digital system.
And here's the input.

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Here's the output.
And we said that for digital

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systems of this sort to work,
they have to follow a static

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discipline.
OK, our signals and our

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circuits must follow a
discipline for them all to work

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together.
And, the discipline we

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described comprised of signals
adhering to certain voltage

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thresholds so that all the
components in the system could

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agree on what comprised a zero,
and what comprised a one,

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OK?
So the way we did that was we

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said that you would have a
threshold called VIH,

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V input high,
and another threshold called

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VIL, V input low.
OK, and we said that this

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circuit must recognize signals
that are higher than VIH,

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3 V for example as a one,
and simultaneously,

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any signal that has a voltage
level less than VIL,

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say, two volts,
should be recognized as a zero.

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That was the input constraint.
On the output,

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it had a similar set of
constraints, where we had

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tougher constraints on devices,
where we said that the output

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had to satisfy a output low
constraint, output high

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constraint.
What this said is that for this

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circuit to be called a good
digital circuit that satisfies

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the static discipline,
signals that were ones here

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should be recognized as such.
And if I am producing a one as

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an output, then the signal level
should be higher than VOH.

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Similarly, if the signal's a
zero, then it should be less

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than VOL.
So as an example,

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this may be 2 V,
this may be 3 V,

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and this may be 4 V,
and this may be 1 V.

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OK, so input,
I should recognize 2 V and less

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as a zero, but at the output I
have to produce a very,

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very low value,
1 V.

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So, I have some noise margin.
So as an example,

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say if I made a plot of the
input/output,

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so I get my VIL here and VIH
here.

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This is time.
This would comprise a valid

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digital signal:
zero, one, zero,

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one, and so on.
OK, now, I had a tougher set of

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constraints at the output.
I would have VOL,

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VOH.
So, at the output,

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OK, I'm required to stretch the
ones and zeros to be further

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apart from each other so that I
get noise margin,

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and the corresponding signal
for our little circuit there

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would look like so.
Right, if this is a valid

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input, then this would be the
corresponding,

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valid output.
OK, and need I say more?

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OK, you can see that,
intuitively,

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look, there's amplification
happening here,

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and the reason is that VOL is
chosen to be less than VIL,

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and VOH is higher than VIH.
So therefore,

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the signal has to be stretched.
The signal has to be amplified.

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OK, and what's the minimum
amplification needed for the

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system to work?
The minimum amplification is if

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I had a signal that looked like
this.

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OK, that barely skimmed the
VIL, VIH level.

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OK, so if signal were this high
peak to peak,

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VIH minus VIL,
and what's the absolute minimum

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signal at the output?
It would look something like

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this.
OK, barely skimming VOL and

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VOH, OK, so the corresponding
output level would be VOH minus

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VOL.
OK, so this is the absolute

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minimum amplification that my
digital circuit has to provide.

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00:17:19 --> 00:17:23
OK, and notice,
VOH is larger than VIH.

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VOL is smaller than VIL.
Therefore, this quantity needs

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to be greater than one.
OK, so I've shown you both a

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simple, graphical,
intuitive explanation,

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00:17:38 --> 00:17:45
and this is a slightly more
formal proof that even the

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digital circuit really requires
to have amplification built into

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it, if it is to satisfy valid
static disciplines.

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Yes?
Yes.

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00:18:00 --> 00:18:03
The question is,
is that the same as gain?

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00:18:03 --> 00:18:08
Good question.
Yes, the term amplification has

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00:18:08 --> 00:18:11
many, many variants.
You could say gain.

254
00:18:11 --> 00:18:16
You could say amplification.
You could say increase in

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00:18:16 --> 00:18:20
signal strength,
and so on and so forth.

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00:18:20 --> 00:18:24
And in fact,
when talking about low noise

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00:18:24 --> 00:18:29
amplifiers, people sometimes
talk about having the low noise,

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00:18:29 --> 00:18:35
high gain amplifier at the
input stage.

259
00:18:35 --> 00:18:39
OK, so let me pause there in
terms of motivation.

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So, I believe I've motivated
every which way:

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00:18:44 --> 00:18:47
pure analog,
analog/digital,

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00:18:47 --> 00:18:50
and digital.
OK, so I've covered every

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00:18:50 --> 00:18:55
single base here.
And so, we need amplification.

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00:18:55 --> 00:19:00
OK, so let's look at how to
build a fundamental,

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00:19:00 --> 00:19:04
primitive device called the
amplifier.

266
00:19:04 --> 00:19:08
Before we do that,
however, let me take a quick

267
00:19:08 --> 00:19:13
detour.
It will be convenient for me,

268
00:19:13 --> 00:19:19
as I show you how to build an
amplifier, to introduce a new

269
00:19:19 --> 00:19:24
device, a new element,
called the dependent source.

270
00:19:24 --> 00:19:30
OK, let me introduce a new
device for your arsenal of

271
00:19:30 --> 00:19:38
devices, along with resistors,
You learned about a MOSFET,

272
00:19:38 --> 00:19:45
a switch, voltage source,
current source,

273
00:19:45 --> 00:19:54
and now a dependent source.
So, a dependent source looks

274
00:19:54 --> 00:19:59
like this, OK,
has an output port,

275
00:19:59 --> 00:20:07
and has a control port.
So, a dependent source in its

276
00:20:07 --> 00:20:12
simplest form has two ports:
an input port and an output

277
00:20:12 --> 00:20:14
port.
Remember, a port is a

278
00:20:14 --> 00:20:19
convenient pairing of terminals,
and I apply signals to such

279
00:20:19 --> 00:20:23
terminal pairs.
But this is a abstract diagram

280
00:20:23 --> 00:20:27
for a dependent source,
and to get a little bit more

281
00:20:27 --> 00:20:34
specific, let me show you an
example of a dependent source.

282
00:20:34 --> 00:20:36
So, let's say,
here's my input,

283
00:20:36 --> 00:20:40
and I label the terminal
variables for the input.

284
00:20:40 --> 00:20:45
VC is the voltage applied to
the input, and IC is the current

285
00:20:45 --> 00:20:50
into this terminal here.
And, here is the symbol for the

286
00:20:50 --> 00:20:54
dependent source.
Much like a current source or a

287
00:20:54 --> 00:20:59
voltage source has a circle
around it, the corresponding

288
00:20:59 --> 00:21:04
symbol for a dependent source is
like so.

289
00:21:04 --> 00:21:06
So this example,
for instance,

290
00:21:06 --> 00:21:09
is a dependent,
current source.

291
00:21:09 --> 00:21:14
I can apply the corresponding
output variables,

292
00:21:14 --> 00:21:20
I0, OK, and I can say that the
current, I, is some function.

293
00:21:20 --> 00:21:24
In this example,
I've designed the example that

294
00:21:24 --> 00:21:30
the current through the current
source, I, is some function of

295
00:21:30 --> 00:21:36
the input voltage or the control
voltage, VC.

296
00:21:36 --> 00:21:40
OK, so notice that the current
through a current source,

297
00:21:40 --> 00:21:43
the current through this
current source,

298
00:21:43 --> 00:21:45
I, is some function of another
variable.

299
00:21:45 --> 00:21:49
OK, in this example,
it's the voltage across its

300
00:21:49 --> 00:21:51
control port.
Not surprisingly,

301
00:21:51 --> 00:21:57
this device is called a voltage
controlled current source --

302
00:21:57 --> 00:22:10


303
00:22:10 --> 00:22:14
-- or a VCCS.
So, in like manner I can also

304
00:22:14 --> 00:22:19
devise other forms of sources.
You can think of this is a

305
00:22:19 --> 00:22:24
device where a voltage controls
an output current.

306
00:22:24 --> 00:22:27
You can think of all other
combinations,

307
00:22:27 --> 00:22:33
current controlling current,
voltage controlling voltage,

308
00:22:33 --> 00:22:38
current controlling voltage,
and so on.

309
00:22:38 --> 00:22:43
So, another example,
I give you another dependent

310
00:22:43 --> 00:22:50
source, and in this situation,
my output current is controlled

311
00:22:50 --> 00:22:52
by an input current,
VC.

312
00:22:52 --> 00:22:57
IC rather.
And I claim that I for this one

313
00:22:57 --> 00:23:02
is some function of a current,
IC.

314
00:23:02 --> 00:23:06
OK, it's another dependent
source where the output current

315
00:23:06 --> 00:23:09
for its output port is related
to the current,

316
00:23:09 --> 00:23:11
IC.
And, this is a current

317
00:23:11 --> 00:23:15
controlled current source.
OK, it's a current controlled

318
00:23:15 --> 00:23:18
current source.
And, if I had lots of time on

319
00:23:18 --> 00:23:22
my hands, and I was wanting to
kill time, I'd sit around

320
00:23:22 --> 00:23:25
drawing for you,
other types of dependent

321
00:23:25 --> 00:23:28
sources.
I would draw for you a current

322
00:23:28 --> 00:23:33
controlled voltage sourced,
and I could also draw for you a

323
00:23:33 --> 00:23:37
voltage controlled voltage
source.

324
00:23:37 --> 00:23:42
OK, so that's an abstract
diagram for such a source.

325
00:23:42 --> 00:23:48
And so, let's do a few examples
involving elements like this.

326
00:23:48 --> 00:23:53
To begin, just so you can build
up your intuition,

327
00:23:53 --> 00:23:57
let me start by doing a very
simple circuit,

328
00:23:57 --> 00:24:01
involving an independent
current source,

329
00:24:01 --> 00:24:09
OK, just so we can relate back
to what we've been doing so far.

330
00:24:09 --> 00:24:14
So, let's say I have some
resistor, and I have a standard

331
00:24:14 --> 00:24:17
current source with current I
nought.

332
00:24:17 --> 00:24:20
This is an independent current
source.

333
00:24:20 --> 00:24:24
Remember the circle?
And, some resistor,

334
00:24:24 --> 00:24:29
R, and let's say I care about
the voltage across the resistor.

335
00:24:29 --> 00:24:35
OK, so I have a current I
nought flowing through it.

336
00:24:35 --> 00:24:39
So, I can very quickly write
down VR as, simply,

337
00:24:39 --> 00:24:43
I0 R.
OK, it's the drop across the

338
00:24:43 --> 00:24:48
resistor when a current I nought
flows through it.

339
00:24:48 --> 00:24:53
OK, so this is what you've been
used to doing.

340
00:24:53 --> 00:24:57
Correspondingly,
I can do an example with a

341
00:24:57 --> 00:25:03
dependent current source.
And, as an example,

342
00:25:03 --> 00:25:07
I'll use a voltage controlled
current source.

343
00:25:07 --> 00:25:13
OK, a voltage controlled
current source is a dependent

344
00:25:13 --> 00:25:20
current source whose output
current depends on the voltage

345
00:25:20 --> 00:25:25
applied at the control port of
the current source.

346
00:25:25 --> 00:25:30
So let me build a little
circuit.

347
00:25:30 --> 00:25:34
OK, so here's my current.
And let's say it's VC IC for

348
00:25:34 --> 00:25:37
the control port,
and similarly,

349
00:25:37 --> 00:25:41
let's say my current I here is
some function of the control

350
00:25:41 --> 00:25:44
port voltage.
And let's say,

351
00:25:44 --> 00:25:47
to be specific,
there is some K over VC,

352
00:25:47 --> 00:25:50
some function.
OK, there are a variety of

353
00:25:50 --> 00:25:55
dependent sources that can be
built, and here's a hypothetical

354
00:25:55 --> 00:26:01
device where the output current
is mathematically related to the

355
00:26:01 --> 00:26:07
input in the following manner.
So, let me build a circuit of

356
00:26:07 --> 00:26:11
the following form.
So, let's add the resistor,

357
00:26:11 --> 00:26:14
R, and here's my circuit,
OK?

358
00:26:14 --> 00:26:18
And, as before,
let me look to figuring out

359
00:26:18 --> 00:26:21
what VR is.
So, notice that I have to

360
00:26:21 --> 00:26:27
supply some voltage at the input
so that the output can depend on

361
00:26:27 --> 00:26:34
the input because right now I
don't know what the input here.

362
00:26:34 --> 00:26:38
So what I'll do is let me apply
VR over here.

363
00:26:38 --> 00:26:41
OK, so let me make this
connection.

364
00:26:41 --> 00:26:46
OK, let me make the connection
from here to here.

365
00:26:46 --> 00:26:52
What I've done is I've applied
VR at the control port of the

366
00:26:52 --> 00:26:58
dependent current source.
OK, and I often draw a circuit

367
00:26:58 --> 00:27:03
like this.
This looks pretty messy.

368
00:27:03 --> 00:27:10
I will often draw the circuit
like so: R, VR.

369
00:27:10 --> 00:27:26


370
00:27:26 --> 00:27:30
OK, short form circuit drawing
would look like this.

371
00:27:30 --> 00:27:34
This is a complete drawing that
I show you the explicit

372
00:27:34 --> 00:27:38
connections of the control port,
but oftentimes,

373
00:27:38 --> 00:27:43
when the control port does not
have any other impact in the

374
00:27:43 --> 00:27:48
circuit, you can eliminate,
don't explicitly show the

375
00:27:48 --> 00:27:51
control port.
Rather, you can simply show the

376
00:27:51 --> 00:27:57
dependence of the output current
on whatever circuit variable you

377
00:27:57 --> 00:28:00
have in mind.
So, you can draw the diamond

378
00:28:00 --> 00:28:04
like this, and see its current
is some function of VR.

379
00:28:04 --> 00:28:10
VR in this is case is K divided
by VR, OK?

380
00:28:10 --> 00:28:14
OK, so let's go ahead and
analyze this little circuit

381
00:28:14 --> 00:28:18
here, and look at what this
might give us.

382
00:28:18 --> 00:28:21
Our goal, as before,
is to find out the value,

383
00:28:21 --> 00:28:23
VR.
So, in this case,

384
00:28:23 --> 00:28:28
let's apply the Node method to
this node, and sum the currents

385
00:28:28 --> 00:28:34
into that node to be zero.
OK, so sum the currents going

386
00:28:34 --> 00:28:39
into that node to be zero.
The current going down is

387
00:28:39 --> 00:28:44
simply VR divided by R.
OK, and that is equal to the

388
00:28:44 --> 00:28:47
current that is going out of the
node.

389
00:28:47 --> 00:28:50
And so that is equal to F of
VR.

390
00:28:50 --> 00:28:54
And I know that F of VR is
given by K divided by VR.

391
00:28:54 --> 00:29:00
OK, a simple application of the
Node method.

392
00:29:00 --> 00:29:04
So then, I collect VR's on the
left hand side,

393
00:29:04 --> 00:29:10
and I get VR squared is K times
R, OK, and VR is simply the

394
00:29:10 --> 00:29:13
square root of KR.
There you go:

395
00:29:13 --> 00:29:17
I'm done.
OK, I've gone ahead an applied

396
00:29:17 --> 00:29:22
the Node method to this,
and when have to figure out the

397
00:29:22 --> 00:29:27
current here,
I simply reflect the fact that

398
00:29:27 --> 00:29:32
it depends on VR like so,
and I just go ahead and solve

399
00:29:32 --> 00:29:37
the circuit.
Remember, the workhorse of the

400
00:29:37 --> 00:29:40
circuit industry,
the Node method,

401
00:29:40 --> 00:29:42
when in doubt,
apply it.

402
00:29:42 --> 00:29:46
It simply works.
And notice, this is a nonlinear

403
00:29:46 --> 00:29:48
circuit.
OK, the dependence is

404
00:29:48 --> 00:29:53
nonlinear, and I get the
response like so.

405
00:29:53 --> 00:30:00
So, to plug in some numbers,
supposing K was 10 to the minus

406
00:30:00 --> 00:30:05
3 amperes per volt,
and R was one kilo ohm,

407
00:30:05 --> 00:30:12
then I can plug the numbers in
and the kilo here cancels with

408
00:30:12 --> 00:30:18
the 10 to the minus 3,
and I get VR equals 1 V.

409
00:30:18 --> 00:30:24
OK, this simply says,
if I build a circuit like this,

410
00:30:24 --> 00:30:30
then this voltage here will be
1 V.

411
00:30:30 --> 00:30:34
So, again, as long as you
remember that the dependent

412
00:30:34 --> 00:30:38
source is simply another little
circuit element,

413
00:30:38 --> 00:30:43
OK, and you usually draw just
the output port for dependent

414
00:30:43 --> 00:30:48
sources, and reflect the way
that the control affects the

415
00:30:48 --> 00:30:52
current, that'll suffice,
and you get,

416
00:30:52 --> 00:30:57
through the application of the
Node method, the variable you're

417
00:30:57 --> 00:31:02
interested in.
Let's do another example,

418
00:31:02 --> 00:31:08
OK, of another fun current
source, a voltage controlled

419
00:31:08 --> 00:31:12
current source,
and look at it this way.

420
00:31:12 --> 00:31:17
So, let's say I have a
resistor, and I have a current

421
00:31:17 --> 00:31:22
source, a resistor,
RL, and this goes to some,

422
00:31:22 --> 00:31:26
I apply a VS here.
Remember this short form

423
00:31:26 --> 00:31:33
notation; that's simply applying
a supply VS between that node

424
00:31:33 --> 00:31:39
and the ground.
OK, and let us say the current

425
00:31:39 --> 00:31:45
IV through the device is some
function of the current at its

426
00:31:45 --> 00:31:49
control port.
OK, so I'm not going to show

427
00:31:49 --> 00:31:53
you that.
But remember that the device

428
00:31:53 --> 00:31:58
already looks like this,
that there is a control port

429
00:31:58 --> 00:32:02
here.
I'm not showing that to you.

430
00:32:02 --> 00:32:05
And let us say that I apply
some voltage,

431
00:32:05 --> 00:32:09
VI, to the input port.
The reason we often don't show

432
00:32:09 --> 00:32:13
the input port is for many
practical dependent sources,

433
00:32:13 --> 00:32:17
the input has no other effect
on the circuit.

434
00:32:17 --> 00:32:19
So, for example,
in this case,

435
00:32:19 --> 00:32:22
the input has infinite
resistance looking in.

436
00:32:22 --> 00:32:25
So therefore,
if I apply a VI here,

437
00:32:25 --> 00:32:28
it doesn't draw any current
from VI.

438
00:32:28 --> 00:32:32
I simply apply the voltage,
VI.

439
00:32:32 --> 00:32:38
It doesn't affect the circuit
in any other way except in terms

440
00:32:38 --> 00:32:42
of how it controls the current
ID.

441
00:32:42 --> 00:32:49
So let's say the current ID is
some function of VI because VI

442
00:32:49 --> 00:32:55
is applied at the control port.
OK, and as I pointed out

443
00:32:55 --> 00:32:59
before, I oftentimes,
just for clarity,

444
00:32:59 --> 00:33:06
just to show this dependent
source explicitly.

445
00:33:06 --> 00:33:10
OK, so let's work the example.
So as I said,

446
00:33:10 --> 00:33:16
I'm going to choose ID to be F
of VI, and let's pick some

447
00:33:16 --> 00:33:22
specific parameters here.
Let's say it's K by two VI

448
00:33:22 --> 00:33:28
minus one, both squared.
OK, and let's say this is true

449
00:33:28 --> 00:33:33
for VI less than equal to one
volt.

450
00:33:33 --> 00:33:38
And let us also say that ID
equals zero for VI less than one

451
00:33:38 --> 00:33:40
volt.
OK, it's a dependent source,

452
00:33:40 --> 00:33:45
and it can have various forms
of dependences on the input.

453
00:33:45 --> 00:33:49
And, I just picked an example
of some hypothetical,

454
00:33:49 --> 00:33:53
or as yet, hypothetical
dependent source,

455
00:33:53 --> 00:33:58
the current through which is
related to the input using a

456
00:33:58 --> 00:34:02
square law relation,
VI minus one all squared as

457
00:34:02 --> 00:34:08
long as VI is greater than one.
And if VI is less than one,

458
00:34:08 --> 00:34:11
then the current is simply
zero, it shuts off.

459
00:34:11 --> 00:34:16
So, I can go ahead and apply.
So, let's say I want to find

460
00:34:16 --> 00:34:19
out V0 versus VI.
So, I care about finding out

461
00:34:19 --> 00:34:22
V0.
V0 is the voltage of this node

462
00:34:22 --> 00:34:26
with respect to ground.
OK, so it's a slightly more

463
00:34:26 --> 00:34:32
complicated circuit than you saw
up here, than you saw up there.

464
00:34:32 --> 00:34:35
So, let's go ahead and do this
example.

465
00:34:35 --> 00:34:39
Start by applying the workhorse
of the circuits business,

466
00:34:39 --> 00:34:43
the Node method,
and let's start with doing this

467
00:34:43 --> 00:34:46
for VI.
Let's first do it for VI

468
00:34:46 --> 00:34:50
greater than one,
notice the behavior of this is

469
00:34:50 --> 00:34:53
different for different ranges
of VI.

470
00:34:53 --> 00:34:58
So let's first do it for VI
greater than or equal to one and

471
00:34:58 --> 00:35:02
apply the Node method.
Node method says sum the

472
00:35:02 --> 00:35:06
currents going into this node;
we know the voltage at this

473
00:35:06 --> 00:35:07
node.
It's VI.

474
00:35:07 --> 00:35:09
We know the voltage at this
node.

475
00:35:09 --> 00:35:11
It's VS.
OK, the only unknown is V

476
00:35:11 --> 00:35:14
nought.
And so, let's go ahead and

477
00:35:14 --> 00:35:16
write the node equations for
that node.

478
00:35:16 --> 00:35:20
So, the current going up,
let me simply equate the

479
00:35:20 --> 00:35:24
current going up to the current
that has been supplied by this

480
00:35:24 --> 00:35:27
particular node here.
And, that should equate that

481
00:35:27 --> 00:35:31
the two of them should sum to
zero, the current going up plus

482
00:35:31 --> 00:35:36
the current going down should
sum to zero.

483
00:35:36 --> 00:35:38
So, I get V0 minus VS divided
by R.

484
00:35:38 --> 00:35:43
That's the current going up.
Plus, the current going down

485
00:35:43 --> 00:35:46
must sum to zero,
plus ID must sum to zero.

486
00:35:46 --> 00:35:51
And ID is going to be K divided
by two VI minus one all squared.

487
00:35:51 --> 00:35:56
That must equal zero.
Straightforward application of

488
00:35:56 --> 00:36:00
Node method, current going up
plus the current going down at

489
00:36:00 --> 00:36:05
this node should equal zero
because the total current

490
00:36:05 --> 00:36:09
leaving the node must be zero,
OK?

491
00:36:09 --> 00:36:13
So I can go ahead and simplify
this, multiply it throughout by,

492
00:36:13 --> 00:36:17
I call this RL here.
So, multiply it throughout by

493
00:36:17 --> 00:36:22
RL, and move all of this to the
other side, so I get VS divided

494
00:36:22 --> 00:36:24
by RL, multiply it throughout by
RL.

495
00:36:24 --> 00:36:28
I get VS at this side.
I take this term to the other

496
00:36:28 --> 00:36:31
side.
This becomes a minus.

497
00:36:31 --> 00:36:34
RL multiplies here,
so I get KRL.

498
00:36:34 --> 00:36:39
That's the expression I get.
V nought is VS minus KRL

499
00:36:39 --> 00:36:43
divided by two times VI minus
one all squared.

500
00:36:43 --> 00:36:48
Let me put a box around this
because I will be referring to

501
00:36:48 --> 00:36:54
this more times in 6.002 for a
variety of reasons than probably

502
00:36:54 --> 00:36:59
any other equation on Earth.
OK, this is the first time you

503
00:36:59 --> 00:37:03
saw it.
You saw it here.

504
00:37:03 --> 00:37:06
OK, mark it down.
You'll smile every other time

505
00:37:06 --> 00:37:10
you look at it in quizzes,
and you will find out why this

506
00:37:10 --> 00:37:14
comes up very often in 6.002.
So, I'll just give you a few

507
00:37:14 --> 00:37:18
seconds to savor this big moment
in your 6.002 life.

508
00:37:18 --> 00:37:21
All right, OK,
so it's pretty simple actually.

509
00:37:21 --> 00:37:23
I mean, there's really not
much.

510
00:37:23 --> 00:37:28
A lot of this stuff is just a
plain old, simple application of

511
00:37:28 --> 00:37:31
the Node method,
and things just fall out.

512
00:37:31 --> 00:37:35
It's just so simple.
So, the V nought,

513
00:37:35 --> 00:37:41
I apply the Node method,
I get V nought for this

514
00:37:41 --> 00:37:47
nonlinear circuit.
I can also it for VI less than

515
00:37:47 --> 00:37:49
one.
For VI less than one,

516
00:37:49 --> 00:37:54
when VI is less than one,
what happens?

517
00:37:54 --> 00:37:58
ID is zero.
OK, since ID is zero,

518
00:37:58 --> 00:38:01
think of this as an open
circuit.

519
00:38:01 --> 00:38:06
OK, so there's no voltage drop
across RL.

520
00:38:06 --> 00:38:13
And, this voltage V nought is
equal to VS.

521
00:38:13 --> 00:38:15
So, I like to see things in
pictures.

522
00:38:15 --> 00:38:18
I'm not an equations kind of
person.

523
00:38:18 --> 00:38:21
I'm much more of a graphical
person.

524
00:38:21 --> 00:38:25
So, let me draw a little graph
to show how V nought,

525
00:38:25 --> 00:38:29
to see the form of V nought,
and then let's study that

526
00:38:29 --> 00:38:34
little system a little bit more
carefully.

527
00:38:34 --> 00:38:39
So, this is page seven,
and we plot V nought versus VI

528
00:38:39 --> 00:38:42
for you.
And let's take a look at how

529
00:38:42 --> 00:38:46
this really simple circuit
looks.

530
00:38:46 --> 00:38:50
This has got nothing.
It's got an RL resistor

531
00:38:50 --> 00:38:55
connected to a supply,
and a dependent current source,

532
00:38:55 --> 00:39:01
and I apply some voltage VI at
the input.

533
00:39:01 --> 00:39:04
It's a very,
very simple circuit.

534
00:39:04 --> 00:39:08
So, let's see.
So as long as VI is less than

535
00:39:08 --> 00:39:14
one, the output stays at VS.
OK, that makes intuitive sense,

536
00:39:14 --> 00:39:18
right?
As long as the current here is

537
00:39:18 --> 00:39:21
zero, this is like an open
circuit here.

538
00:39:21 --> 00:39:26
If this is an open circuit,
then effectively,

539
00:39:26 --> 00:39:31
V nought is simply the voltage
VS.

540
00:39:31 --> 00:39:35
V nought simply appears here.
If you want to grunge through

541
00:39:35 --> 00:39:39
KVL and KCL, go ahead.
VS minus RL times the current

542
00:39:39 --> 00:39:43
is V nought, and the current is
zero so it's,

543
00:39:43 --> 00:39:45
yes.
So, this is simply VS.

544
00:39:45 --> 00:39:50
When VI goes above one volt,
fun stuff begins to happen.

545
00:39:50 --> 00:39:54
OK, when V nought goes above
one volt, then this equation

546
00:39:54 --> 00:39:58
applies because VI is greater
than one.

547
00:39:58 --> 00:40:02
This equation applies.
And, when VI is a one,

548
00:40:02 --> 00:40:06
one minus one is zero.
This term cancels out,

549
00:40:06 --> 00:40:08
so this is VS.
OK, phew!

550
00:40:08 --> 00:40:12
So, I start off here.
As VI increases,

551
00:40:12 --> 00:40:15
what happens now?
As VI increases,

552
00:40:15 --> 00:40:19
this term here becomes
increasingly negative,

553
00:40:19 --> 00:40:23
OK, subtracting from VS.
OK, so I get some behavior like

554
00:40:23 --> 00:40:26
this.
V nought begins to drop.

555
00:40:26 --> 00:40:31
And it makes intuitive sense,
right?

556
00:40:31 --> 00:40:35
As ID begins to increase,
the voltage here will begin to

557
00:40:35 --> 00:40:40
drop because I'm drawing more
and more current through RL.

558
00:40:40 --> 00:40:43
I'm dropping more and more
across RL.

559
00:40:43 --> 00:40:48
So more and more drops across
RL, so V nought begins to drop

560
00:40:48 --> 00:40:51
too.
So, it looks something like

561
00:40:51 --> 00:40:53
this.
I'll show you a little demo,

562
00:40:53 --> 00:40:58
but my claim is that you have
just seen an amplifier.

563
00:40:58 --> 00:41:03
Whoa.
You just saw an amplifier.

564
00:41:03 --> 00:41:07
So, I snuck an amplifier by
you, OK?

565
00:41:07 --> 00:41:11
So, I just snuck an amplifier
past you.

566
00:41:11 --> 00:41:18
I'll show you why in a second.
So, let's take a look at this

567
00:41:18 --> 00:41:23
waveform here.
Let's not worry about what

568
00:41:23 --> 00:41:29
happens way down here.
We'll talk about that a little

569
00:41:29 --> 00:41:36
later.
But, look at this curve here.

570
00:41:36 --> 00:41:44
I claim there is amplification
in the following sense.

571
00:41:44 --> 00:41:51
Focus on some change in the
input voltage,

572
00:41:51 --> 00:41:57
delta VI, OK,
and for that change in input

573
00:41:57 --> 00:42:06
voltage, I get some change in
the output voltage.

574
00:42:06 --> 00:42:09
OK, for some change in the
input voltage,

575
00:42:09 --> 00:42:13
delta VI, I get some change in
the output voltage.

576
00:42:13 --> 00:42:17
And guess what?
In this, at least the way I

577
00:42:17 --> 00:42:21
have drawn it,
delta V nought divided by delta

578
00:42:21 --> 00:42:26
VI, if I can find regions of the
curve where this is greater than

579
00:42:26 --> 00:42:32
one, then I have amplification.
OK, so what's that saying?

580
00:42:32 --> 00:42:37
What that's saying is that if I
apply some voltage here,

581
00:42:37 --> 00:42:42
OK, and I change that voltage
by a small amount from,

582
00:42:42 --> 00:42:46
let's say, 2 V to 2.1.
OK, I am going to find the

583
00:42:46 --> 00:42:50
output voltage.
Let's say I go from 2 V to 2.1

584
00:42:50 --> 00:42:53
here.
OK, abstractly out there,

585
00:42:53 --> 00:42:57
I might have an output that
goes from three to,

586
00:42:57 --> 00:43:03
let's say, two V perhaps.
OK, so for a 0.1 change here,

587
00:43:03 --> 00:43:07
I'm going to get a bigger drop
here, so from 3 V to 2 V,

588
00:43:07 --> 00:43:10
giving me an amplification in
this little circuit.

589
00:43:10 --> 00:43:14
OK, so we'll see this again and
again, and you'll really

590
00:43:14 --> 00:43:17
understand it.
So, I have a small change in

591
00:43:17 --> 00:43:21
the input, and I have a
corresponding larger change in

592
00:43:21 --> 00:43:23
the output.
So, I've shown you an

593
00:43:23 --> 00:43:26
amplifier.
I haven't shown you a linear

594
00:43:26 --> 00:43:29
amplifier.
There's an extra charge for

595
00:43:29 --> 00:43:32
that.
OK, that'll happen later.

596
00:43:32 --> 00:43:35
OK, all I've shown you so far
is an amplifier,

597
00:43:35 --> 00:43:37
and this happens to be a crummy
amplifier.

598
00:43:37 --> 00:43:40
It's a nonlinear amplifier
because, notice,

599
00:43:40 --> 00:43:43
this is not linear.
It's a nice little curve,

600
00:43:43 --> 00:43:45
and so it's not linear.
But, I promised you an

601
00:43:45 --> 00:43:49
amplifier, and I'm cheap,
and that's all you get for now.

602
00:43:49 --> 00:43:52
OK, we'll see linear stuff
later, but for now,

603
00:43:52 --> 00:43:55
I have a little amplifier.
So, let's do some real numbers,

604
00:43:55 --> 00:44:00
and plot some numbers down,
and also look at a demo.

605
00:44:00 --> 00:44:04
So, let's do an example.
Let's say VS is 10 V,

606
00:44:04 --> 00:44:10
that the K is two milliamps per
V squared, and let's say RL is

607
00:44:10 --> 00:44:12
five kilo-ohms,
OK?

608
00:44:12 --> 00:44:18
So, let me substitute these
values into that equation,

609
00:44:18 --> 00:44:21
and I get V nought is,
VS is ten.

610
00:44:21 --> 00:44:25
So, it's ten minus,
KRL divided by two.

611
00:44:25 --> 00:44:31
So, K is two milliamps.
Two milliamps times five

612
00:44:31 --> 00:44:36
kilo-ohms is ten divided by two
gives me five,

613
00:44:36 --> 00:44:40
and VI minus one squared.
That's what I have.

614
00:44:40 --> 00:44:46
I just plug in a bunch of
numbers, and that's what I get.

615
00:44:46 --> 00:44:52
So, what I'll do is let me just
do a little table for you,

616
00:44:52 --> 00:44:57
and plot using real numbers,
simply plot those values for

617
00:44:57 --> 45:00
you.