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All right.
Good morning,

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all.

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You have two handouts,
lecture notes and an article on

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mixed signal chips.
A mixed signal stands for

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circuits that have both analog
and digital components to them.

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The reason I am giving you the
handout is that Lab 4 and also

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your last homework involve
designing and building a mixed

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signal circuit.
It's a real fun exercise.

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And I just wanted to tell you
that from past experience people

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who have taken 6.002 often view
the last lab as the single most

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fun thing they did in all of

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So, as you go into Lab 4,
you should be telling yourself

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I should be having fun,
I should be having,

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I should be having fun.
You have to positively psych

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yourself.
Otherwise, it's going to go by.

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And then you're going to say
boy, that was fun,

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I wish I had savored the moment
as I was doing it.

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All right.
Let's see.

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What do we do today?
Today's lecture is actually

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going to be a fair amount of
fun.

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We are going to blast through a
bunch of fun things.

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And some things that you will
be quite unprepared for.

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Until now, in the last two
lectures with op amps we talked

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about negative feedback.
That is applying some portion

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of the output voltage to the
negative input so that I could

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control this high strung device,
my op amp.

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Today, what we are going to do
is try to get a handle on what

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happens if we use positive
feedback.

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It's the usual curious child.
You tell them to do this,

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and of course they're going to
try to do this as well.

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And we are going to try to do
that and see what happens and

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look to see if we can build some
useful circuits.

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Today --

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As motivation,
let me do a quick review of a

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circuit that should now become
affixed in your brains in a

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standard pattern.
This is a circuit that gives

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you negative feedback.

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R1 and R2.

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And I apply a vIN.
By now you should be able to

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look at this pattern.
And this is your inverting

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amplifier pattern.
So, you should be able to write

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down by inspection this is
simply vIN or the minus vIN

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times R2 divided by R1.
This is an amplifier whose gain

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is controlled by the ratio of R2
and R1.

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This is a negative feedback
circuit because it is always fun

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to do the intuition thing and
say that look,

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if this voltage tends to go
more positive than I care then

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this negative input goes more
positive than I care.

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If that goes more positive then
the negative input v minus

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becomes more positive in the
plus input which yanks the

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output down.
So, there is a nice

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counteracting force that keeps
the output stable.

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Let's look at this circuit.
Being curious engineers,

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let's look at the opposite here
where I give myself some

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positive feedback in this op
amp.

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And it is going to be
interesting to analyze this

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because what we find out on the
face of it is not quite actually

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how it behaves.
We are going to spend most of

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the lecture today on
understanding the dynamics of

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circuits that look like this and
to see if we can build some fun

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and interesting circuits and
systems based on this kind of

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positive feedback.
It is positive feedback because

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I am feeding back a portion of
the output to the positive

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input.
And you should be able to stare

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at this and already begin to
intuit what should happen to

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this.
Let's think about it.

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This is zero.
Remember, with positive

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feedback, the famous v plus is
equal to v minus method doesn't

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apply anymore.
Let's apply very simple

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analyses.
If this is zero,

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let's say for example that this
output tends to go a little bit

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more positive.
This output,

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due to some noise or
perturbation,

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tends to go up a little bit.
If that goes up a little bit

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then because of feedback this
node tends to go up a little

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bit.
If this node tends to go up a

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little bit this exacerbates the
positive input here and this one

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goes cachunk,
whacks into the positive rail.

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Let's take the other point of
view and look at it intuitively.

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What if this one tries to droop
a little bit?

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If it droops a little bit then
the input at the plus terminal

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droops a little bit.
If that tends to go down a

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little bit, that makes the
output droop further and it goes

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and hits into the negative rail.
I can see that this circuit

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wants to hammer into the
positive rail or hammer into the

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negative rail because of the
positive feedback.

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It is like if you give
incredibly positive feedback all

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the time, and by positive
feedback I mean feedback

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encouraging the child to do
whatever the child is doing.

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It could be if he does bad
stuff you give a lot of positive

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feedback or good stuff you give
a lot of positive feedback then

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you are guaranteed to have a
very good child or a very bad

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child.
You are not going to have

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anybody in the middle.
Same way here.

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By giving positive feedback
you're going to drive this into

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the positive rail or drive this
into the negative rail.

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Now, I am going to analyze this
in two steps.

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First I am going to analyze
this using a method you've seen

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before which is replace the op
amp with its equivalent circuit

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and analyze it statically.
And by analyzing it statically

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we are going to show that the
simple static analysis will

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yield the following expression.
I put this in quotes,

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well, for a reason you will see
shortly.

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When I apply a plain and simple
static analysis here is what I

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find.
Let's go ahead with the

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analysis and see what is
basically different about these

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two.
And, first of all,

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I will confirm for you that our
naive analysis we have seen so

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far will give rise to that
expression.

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So, let's go ahead and analyze
that circuit.

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And to analyze that circuit
what I will do is replace the op

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amp with its equivalent circuit.
If you remember the op amp is

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characterized by the following
circuit, A times v+ minus v-,

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vOUT.
This is the equivalent circuit

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of my op amp.
And let me just impose that

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external circuit on this op amp.
I have grounded my v- terminal.

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My v+ terminal goes through a
resistor and a supply,

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the v into ground,
it's the resistance R1.

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This terminal goes to the
output through a resistor R2.

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So, this is the equivalent
circuit.

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And I can apply the same
good-old techniques I have

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learned about all through this
course to this circuit and see

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what vOUT looks like.
Very simply,

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vOUT is this expression here A
times v+ minus v-.

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And because of my ground
connection v- is zero.

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Then let me go ahead and
replace v+ with the voltage that

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relates vOUT and vIN.
What is v+?

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v+ is simply the current
through this part of the

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circuit, the current flowing
here times the resistance R1.

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That gives me the drop across
R1.

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And to that I add vIN and that
will give me V+.

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And then of course I multiply
this by the gain here.

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So, let me write down that
expression.

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The current through this is
simply vOUT minus vIN.

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That is the voltage drop
between these two points.

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I divide that by the resistance
R1 plus R2.

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That gives me the current
flowing through here.

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That times R1 is the drop
across resistor R1.

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And to that I add vIN and that
gives me the voltage v+.

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So, this is v+.
That is simply vIN plus the

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drop across the resistance R1.
Let me shuffle things around

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and put all the vOUT terms on
this side here.

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I get a 1+ for that vOUT and
let me move AR1 divided by R1

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plus R2 to the left-hand side.
And I pick up a minus sign.

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So, I get AR1 divide by R1 plus
R2.

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I pick up that.
And on the left-hand sign I end

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up with vIN, and my vIN here is
a function of the vIN that I

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have here.
I have an A multiplying both

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the vINs.
And then I get a one for this

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vIN here and there is a minus
sign, so I get a minus R1

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divided by R1+R2.
That is the expression that I

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have.
Let me go ahead and simplify

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that a little further and move
this whole thing down here.

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That gives me my expression as
a function of vIN.

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What I will do is,
let me continue here.

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vOUT=vIN A(1-R1/(R1+R2)).
By the way, you may be

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wondering why I am going through
so laboriously what is seemingly

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a very simple exercise.
The reason I want to do is it I

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want to very carefully show you
that the result produced by this

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exercise is exactly that.
No magic here.

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No cheating.
We are going to get exactly

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that.
And then stare at it and say

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huh, how did that happen?
And then we are going to try to

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figure out how it actually
behaves following that.

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I divide this by 1-AR1/(R1+R2).
And by now you should be

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familiar with the technique of
ignoring small numbers when I

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have a big number next to it.
So, AR1/(R1+R2) can be very

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much larger than one because A
is very large.

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So, I can ignore my one there.
And then what I am going to do

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is multiply the numerator and
denominator by R1+R2.

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Oh, this A and this A is going
to cancel out.

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This A and this A will then
cancel out.

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And then I multiply the
numerator and denominator by

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R1+R2, so this R1+R2 vanishes.
I get R1+R2 here.

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R1+R2 minus R1 is simply R2.
And then down here I get a R1

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and then I have a minus sign out
there.

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Notice that vOUT we have found
to be equal to vIN R2 divided by

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R1.
That is not wrong.

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That is correct.
Technically that is correct.

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But you will see in a few
seconds that in practice that

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that's rarely what you are going
to see happen.

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And we will try to understand
why that is so.

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What we have done so far,
if you stare at these two

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panels here, first of all,
we know that the inverting

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amplifier has the expression for
vOUT up there.

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And through this laborious
exercise we have also shown that

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even with positive feedback,
if I take a static view of the

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circuit --
If I take a snapshot of the

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circuit and simply analyze it as
a static circuit,

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I get the same expression vOUT.
But what we are going to do is

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when I explain to you that look,
a small perturbation in vOUT is

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going to drive the op amp to the
positive and negative rail,

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that is where the insight
begins to show.

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That if everything were magical
and I could somehow exactly keep

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things just so that will be
true.

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I will be able to build that
positive feedback circuit where

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the output is equal to R2/R1
vIN.

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But remember even the slightly
amount of perturbation is going

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to send the op amp scurrying off
to the positive rail or the

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negative rail.
How do we analyze that?

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How do we analyze the behavior
of a circuit that based on a

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small perturbation begins to
move one place or another?

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We want to analyze the dynamics
of the op amp.

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And to analyze the dynamics
what I need to do is give you a

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slightly more detailed view of
the operational amplifier.

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If the operational amplifier is
not moving instantaneously

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between the plus and minus rail,
I need to give you a more

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detailed model that encapsulates
the behavior of the op amp.

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And so let me do that.
If you want to study the

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dynamics of an op amp --
By dynamics I mean how an op

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amp moves as I perturb the input
or the output and so on.

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To capture the dynamics of the
op amp we build a slightly more

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involved circuit,
so v+ and v-.

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This is what we've seen before,
two terminals and dependent

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source that amplifies the
difference input here by a large

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amount.
Instead what we are going to do

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here is something slightly
different and interpose the

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following circuit in the middle
here.

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This is a model of the dynamics
of an op amp.

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We are going to impose a small
RC circuit in here.

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This is R.
This is C.

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And I am going to call the
voltage across the capacitor v*.

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Notice what I have done is
rather than say this is Av+

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minus v- I am breaking it apart
in two dependent sources,

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the first dependent source,
which is simply v+ minus v-,

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and there is a RC time constant
surrounding it and then here I

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simply add on my gain Av*.
Notice that if it turned out

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that the resistance here,
for example,

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was zero then v+ minus v- would
appear across v* and this would

230
00:17:14,000 --> 00:17:16,000
be A(v+ - v-),
what you have seen before.

231
00:17:16,000 --> 00:17:20,000
It is always good to take a
look at circuits and look at

232
00:17:20,000 --> 00:17:24,000
what happens when some component
goes to an extreme value.

233
00:17:24,000 --> 00:17:29,000
This would give you your basic
op amp circuit.

234
00:17:29,000 --> 00:17:33,000
What I would like to do next is
analyze the following circuit to

235
00:17:33,000 --> 00:17:38,000
understand how positive and
negative feedback work together.

236
00:17:38,000 --> 00:17:42,000
And by understanding that then
be able to explain how a

237
00:17:42,000 --> 00:17:47,000
positive feedback circuit works
or a negative feedback circuit

238
00:17:47,000 --> 00:17:49,000
works.
Here is what I will do.

239
00:17:49,000 --> 00:17:54,000
This part simply corresponds to
my positive feedback circuit,

240
00:17:54,000 --> 00:17:56,000
R2, R1.
So, that is my positive

241
00:17:56,000 --> 00:18:00,000
feedback circuit.
And I will do the same thing on

242
00:18:00,000 --> 00:18:02,000
this side.

243
00:18:09,000 --> 00:18:12,000
All I am doing is applying both
a positive feedback through R2

244
00:18:12,000 --> 00:18:15,000
and R1 and negative feedback
through R4 and R3 and

245
00:18:15,000 --> 00:18:19,000
representing the dynamics of the
op amp and then standing back

246
00:18:19,000 --> 00:18:22,000
and ee, all right,
let's see what happens to you.

247
00:18:22,000 --> 00:18:25,000
So, I am sticking positive
feedback, negative feedback,

248
00:18:25,000 --> 00:18:30,000
the dynamics of the op amp here
and let's see what happens.

249
00:18:30,000 --> 00:18:38,000
What I would like to do is
impose this circuit on top of

250
00:18:38,000 --> 00:18:46,000
this op amp model.
To save myself some effort,

251
00:18:46,000 --> 00:18:54,000
let me just go ahead and modify
this circuit directly.

252
00:18:54,000 --> 00:19:02,000
I get an R2 here,
an R1 here, and then up here I

253
00:19:02,000 --> 00:19:09,000
get an R4, R3 here.
The math is going to be just a

254
00:19:09,000 --> 00:19:14,000
little bit grubby but the result
is actually pretty spectacular.

255
00:19:14,000 --> 00:19:18,000
So, all I have done is replace
the op amp with its internal

256
00:19:18,000 --> 00:19:21,000
circuit out here.
And now we are going to take a

257
00:19:21,000 --> 00:19:26,000
look at what happens to op amp
dynamics when there is a small

258
00:19:26,000 --> 00:19:29,000
perturbation.
Let's develop an equation of

259
00:19:29,000 --> 00:19:33,000
this circuit containing a
capacitor using techniques that

260
00:19:33,000 --> 00:19:38,000
we already know.
Just to give you some insight

261
00:19:38,000 --> 00:19:42,000
into what you're going to see,
notice that if I make a small

262
00:19:42,000 --> 00:19:46,000
perturbation in the voltage
across the capacitor,

263
00:19:46,000 --> 00:19:50,000
let's say I make a small
perturbation to the capacitor

264
00:19:50,000 --> 00:19:55,000
voltage let's say by applying
some initial condition kind of

265
00:19:55,000 --> 00:19:59,000
thing onto the capacitor.
Then let's say that the output

266
00:19:59,000 --> 00:20:03,000
changes to some value K.
So, the change on the capacitor

267
00:20:03,000 --> 00:20:07,000
must have been K divided by A.
And what you are going to see

268
00:20:07,000 --> 00:20:11,000
is what happens to the op amp
when the initial condition on

269
00:20:11,000 --> 00:20:14,000
the capacitor is such that this
output gets perturbed to the

270
00:20:14,000 --> 00:20:16,000
value K.
Let's write an equation for

271
00:20:16,000 --> 00:20:19,000
this little circuit and see what
happens.

272
00:20:19,000 --> 00:20:22,000
Recall our goal was to
understand what happens when I

273
00:20:22,000 --> 00:20:24,000
perturbed the output a little
bit.

274
00:20:24,000 --> 00:20:29,000
Here I perturbed the output
such that its value goes to K.

275
00:20:29,000 --> 00:20:33,000
And I can perturb the output by
changing what happens at the

276
00:20:33,000 --> 00:20:36,000
capacitor.
Let me write the equation for

277
00:20:36,000 --> 00:20:41,000
this circuit now and then to
understand what happens to this

278
00:20:41,000 --> 00:20:45,000
capacitor circuit if I let go
after giving it a small

279
00:20:45,000 --> 00:20:48,000
perturbation.
What I am going to do is let me

280
00:20:48,000 --> 00:20:53,000
start by writing the good old
equation for this little circuit

281
00:20:53,000 --> 00:20:56,000
here.
And that equation is simply the

282
00:20:56,000 --> 00:21:02,000
voltage here v+ minus v- equals
the voltage across the RC.

283
00:21:02,000 --> 00:21:08,000
So, v+ minus v- will be equal
to the voltage drop across the

284
00:21:08,000 --> 00:21:13,000
resistor plus that across the
capacitor.

285
00:21:13,000 --> 00:21:17,000
The voltage across the
capacitor is v*.

286
00:21:17,000 --> 00:21:24,000
The voltage across the resistor
is the current through the

287
00:21:24,000 --> 00:21:31,000
capacitor C dv*/dt times R.
So, v* plus RC dv/dt is equal

288
00:21:31,000 --> 00:21:37,000
to v+ minus v-.
RC dv*/dt plus v* is v+ minus

289
00:21:37,000 --> 00:21:40,000
v-.
You have done this millions of

290
00:21:40,000 --> 00:21:43,000
times before,
but yet again.

291
00:21:43,000 --> 00:21:49,000
This voltage here is equal to
the drop across these two,

292
00:21:49,000 --> 00:21:54,000
and the drop across these two
is v*, the drop across C,

293
00:21:54,000 --> 00:22:00,000
plus the current through the
capacitor C dv/dt times the

294
00:22:00,000 --> 00:22:05,000
resistance R.
Or you can apply the node

295
00:22:05,000 --> 00:22:10,000
method as well and get the same
expression.

296
00:22:10,000 --> 00:22:16,000
Now, we also know here that vO
divided by A is v*.

297
00:22:16,000 --> 00:22:24,000
I can go ahead and replace this
guy here, v* by vO divided by A.

298
00:22:24,000 --> 00:22:29,000
RC/A dvO/dt.
Recall, I want the dynamics of

299
00:22:29,000 --> 00:22:35,000
vO so let me just get an
expression in vO.

300
00:22:35,000 --> 00:22:43,000
So, I get vO divided by A plus
v+ minus v- equals.

301
00:22:43,000 --> 00:22:51,000
Now, I want an expression in
vO, an equation in vO,

302
00:22:51,000 --> 00:23:00,000
so I need to express v+ and v-
in terms of vO.

303
00:23:00,000 --> 00:23:09,000
What are these expressions?
The expression for v- is vO and

304
00:23:09,000 --> 00:23:16,000
this voltage divider,
so it's vOR3/(R3+R4).

305
00:23:16,000 --> 00:23:25,000
And just for simplicity,
let me call this some constant

306
00:23:25,000 --> 00:23:30,000
gamma minus.
This is some fraction

307
00:23:30,000 --> 00:23:37,000
R3/(R3+R4).
And let me call that fraction

308
00:23:37,000 --> 00:23:42,000
gamma minus.
Similarly, v+ is vO R1/(R1+R2).

309
00:23:42,000 --> 00:23:45,000
And let me call that gamma
plus.

310
00:23:45,000 --> 00:23:51,000
All I am doing is replacing v+
and v- in terms of vO.

311
00:23:51,000 --> 00:23:57,000
So, effectively,
what I have here is v+ is some

312
00:23:57,000 --> 00:24:03,000
fraction of vO.
That's the best intuitive way

313
00:24:03,000 --> 00:24:09,000
of thinking about it,
some fraction of vO.

314
00:24:09,000 --> 00:24:14,000
And v- is some fraction of vO
as well.

315
00:24:14,000 --> 00:24:21,000
And I just stick these.
I now have an expression in vO.

316
00:24:21,000 --> 00:24:29,000
Don't get psyched by gamma plus
and gamma minus.

317
00:24:29,000 --> 00:24:34,000
Simply read this as if it is an
F1 and F2 if you would like.

318
00:24:34,000 --> 00:24:39,000
So, vO times some fraction
minus vO times some other

319
00:24:39,000 --> 00:24:42,000
fraction.
I am feeding back some fraction

320
00:24:42,000 --> 00:24:48,000
of the output to the positive
and to the negative terminals.

321
00:24:48,000 --> 00:24:52,000
Then, just moving things around
a little bit,

322
00:24:52,000 --> 00:24:55,000
dividing throughout by A
divided by RC.

323
00:24:55,000 --> 00:25:00,000
So, I divided by A divided by
RC.

324
00:25:00,000 --> 00:25:08,000
Plus vO divided by RC.
And what I am going to do here

325
00:25:08,000 --> 00:25:15,000
in a second, vO gamma plus minus
gamma minus.

326
00:25:15,000 --> 00:25:24,000
And I have multiplied by A
divided by RC throughout.

327
00:25:24,000 --> 00:25:34,000
Finally, collecting all the vO
terms I get vO times one divided

328
00:25:34,000 --> 00:25:42,000
by RC plus A divided by RC.
I got a plus sign here so I

329
00:25:42,000 --> 00:25:47,000
will just reverse these two guys
in there, gamma minus minus

330
00:25:47,000 --> 00:25:51,000
gamma plus equals zero.
All I have done here is simply

331
00:25:51,000 --> 00:25:56,000
grunged through some math to
express this equation in terms

332
00:25:56,000 --> 00:25:59,000
of vO.
And just to make it even

333
00:25:59,000 --> 00:26:04,000
simpler, I will just replace
this thing by one divided by T,

334
00:26:04,000 --> 00:26:09,000
much as we did for first order
equations.

335
00:26:09,000 --> 00:26:19,000
What I end up with is
dvO/dt+vO/T=0.

336
00:26:29,000 --> 00:26:33,000
Despite all the grubbiness,
I end up with something that is

337
00:26:33,000 --> 00:26:36,000
very, very familiar to all of
us.

338
00:26:36,000 --> 00:26:41,000
I went through a bunch of
gyrations to substitute for v+,

339
00:26:41,000 --> 00:26:45,000
v- and v*, but at the end of
the day I got the simple

340
00:26:45,000 --> 00:26:48,000
expression which was
dvO/dt+vO/T=0.

341
00:26:48,000 --> 00:26:52,000
Where capital T is the time
constant of the circuit,

342
00:26:52,000 --> 00:26:57,000
and the time constant of the
circuit relates to the

343
00:26:57,000 --> 00:27:01,000
expression in there
1/RC+A/RC(gamma minus - gamma

344
00:27:01,000 --> 00:27:05,000
plus).
The gamma minus and gamma plus

345
00:27:05,000 --> 00:27:09,000
are the respective portions of
the output fed back to the

346
00:27:09,000 --> 00:27:12,000
negative input and the positive
input.

347
00:27:12,000 --> 00:27:15,000
Now, as we all know,
based on very simple intuition

348
00:27:15,000 --> 00:27:20,000
that we can completely predict
the behavior of a first order of

349
00:27:20,000 --> 00:27:24,000
an RC circuit once we know what
the initial condition of the

350
00:27:24,000 --> 00:27:28,000
capacitor is and once you know
the time constant.

351
00:27:28,000 --> 00:27:32,000
That's it.
We know, we are masters at the

352
00:27:32,000 --> 00:27:36,000
fact that the capacitor is going
to behave like this.

353
00:27:36,000 --> 00:27:40,000
It is going to be exponential.
And I do know that the time

354
00:27:40,000 --> 00:27:42,000
constant capital T.
What's here?

355
00:27:42,000 --> 00:27:45,000
It is simply the initial
condition.

356
00:27:45,000 --> 00:27:49,000
There is no drive input.
I am not driving this with any

357
00:27:49,000 --> 00:27:52,000
input here.
There is no input drive

358
00:27:52,000 --> 00:27:55,000
anywhere here.
This is simply the natural

359
00:27:55,000 --> 00:27:58,000
dynamics of the system.
And, recall,

360
00:27:58,000 --> 00:28:03,000
I start off with bumping the
capacitor voltage such that the

361
00:28:03,000 --> 00:28:06,000
output starts off being K.
That is it.

362
00:28:06,000 --> 00:28:10,000
You should be able to write
down this expression and the

363
00:28:10,000 --> 00:28:14,000
form of the response simply
based on this.

364
00:28:14,000 --> 00:28:18,000
So, this is what I bumped up
the output to be by perturbing

365
00:28:18,000 --> 00:28:22,000
the capacitor voltage.
My output response based on

366
00:28:22,000 --> 00:28:25,000
this equation is going to look
like that.

367
00:28:25,000 --> 00:28:30,000
Let's try to understand what
that means.

368
00:28:30,000 --> 00:28:32,000
It is actually quite a lot of
fun.

369
00:28:32,000 --> 00:28:37,000
How do we plot that response?
You all learned that the way to

370
00:28:37,000 --> 00:28:40,000
plot the response is plot the
initial value,

371
00:28:40,000 --> 00:28:43,000
plot the final value,
and go cachoock,

372
00:28:43,000 --> 00:28:45,000
right?
It's pretty simple.

373
00:28:45,000 --> 00:28:48,000
I am going to start at K.
I know that.

374
00:28:48,000 --> 00:28:53,000
I am going to start at K and I
am going to go and find out what

375
00:28:53,000 --> 00:28:57,000
the steady state value is.
Here is where the interesting

376
00:28:57,000 --> 00:29:01,000
stuff comes in.
The final value on the

377
00:29:01,000 --> 00:29:06,000
capacitor depends a lot on
whether T is positive or

378
00:29:06,000 --> 00:29:09,000
negative.
In my RC circuits that I looked

379
00:29:09,000 --> 00:29:13,000
at what was T?
In the very simple RC circuit

380
00:29:13,000 --> 00:29:18,000
we looked at what was capital T?
What was the time constant?

381
00:29:18,000 --> 00:29:19,000
RC.
This was RC.

382
00:29:19,000 --> 00:29:24,000
This was a positive quantity.
When capital T is positive my

383
00:29:24,000 --> 00:29:27,000
output is going to look like
this.

384
00:29:27,000 --> 00:29:33,000
When T is positive.
And T is positive when this

385
00:29:33,000 --> 00:29:39,000
expression is positive.
And if A is so large that I can

386
00:29:39,000 --> 00:29:43,000
ignore the 1/RC term,
if A is very,

387
00:29:43,000 --> 00:29:50,000
very large and I can ignore the
left-hand term here then T is

388
00:29:50,000 --> 00:29:57,000
positive when gamma minus is
greater than gamma plus.

389
00:29:57,000 --> 00:30:00,000
So, when gamma minus is greater
than gamma plus,

390
00:30:00,000 --> 00:30:04,000
I have a stable circuit,
this is the good-old stuff we

391
00:30:04,000 --> 00:30:08,000
have seen before.
Now things begin to make sense.

392
00:30:08,000 --> 00:30:11,000
Intuitively,
what am I saying here?

393
00:30:11,000 --> 00:30:15,000
All the gammas and other pieces
of crapola aside,

394
00:30:15,000 --> 00:30:18,000
what am I really saying here in
English?

395
00:30:18,000 --> 00:30:22,000
What I am saying here is that
if the portion of the output fed

396
00:30:22,000 --> 00:30:27,000
to the negative input is greater
than that fed to the positive

397
00:30:27,000 --> 00:30:32,000
input then I have net negative
feedback.

398
00:30:32,000 --> 00:30:36,000
I have net negative feedback.
I am feeding the output back to

399
00:30:36,000 --> 00:30:39,000
both the positive and negative
inputs.

400
00:30:39,000 --> 00:30:44,000
And if my negative input has a
stronger effect then I am going

401
00:30:44,000 --> 00:30:49,000
to see the op amp output decay
down to a value that I expect

402
00:30:49,000 --> 00:30:54,000
which is going to be zero.
Notice that since I am not

403
00:30:54,000 --> 00:30:58,000
applying any input here,
I expect the stable point for

404
00:30:58,000 --> 00:31:03,000
this to be output going to zero.
I don't have any input there.

405
00:31:03,000 --> 00:31:06,000
Let's take a look at another
situation.

406
00:31:06,000 --> 00:31:08,000
What happens when the opposite
is true?

407
00:31:08,000 --> 00:31:12,000
What happens when gamma minus
is less than gamma plus?

408
00:31:12,000 --> 00:31:15,000
When I feedback more,
what happens when I do this,

409
00:31:15,000 --> 00:31:18,000
when gamma plus is greater than
gamma minus?

410
00:31:18,000 --> 00:31:21,000
The opposite is true.
This means that I am feeding

411
00:31:21,000 --> 00:31:25,000
back more to the positive input.
A bigger proportion goes to the

412
00:31:25,000 --> 00:31:30,000
positive than the negative.
What happens then?

413
00:31:30,000 --> 00:31:34,000
Then what happens is capital T
becomes negative.

414
00:31:34,000 --> 00:31:39,000
We cannot see this happening on
the RC circuit because capital T

415
00:31:39,000 --> 00:31:42,000
is equal to RC,
but here we have a more

416
00:31:42,000 --> 00:31:47,000
complicated circuit and capital
T can go negative.

417
00:31:47,000 --> 00:31:52,000
If capital T goes negative then
this whole thing in the exponent

418
00:31:52,000 --> 00:31:56,000
there goes positive.
If that goes positive what

419
00:31:56,000 --> 00:32:02,000
should the output look like?
It should take off into

420
00:32:02,000 --> 00:32:05,000
never-never land.
There we go.

421
00:32:05,000 --> 00:32:11,000
I start off at zero and a make
a small perturbation,

422
00:32:11,000 --> 00:32:17,000
and the output should go as t
divided by capital T.

423
00:32:17,000 --> 00:32:23,000
The dynamics of this it goes
berserk, so it is net positive

424
00:32:23,000 --> 00:32:27,000
feedback.
This is called a stable

425
00:32:27,000 --> 00:32:31,000
situation.
This is unstable.

426
00:32:31,000 --> 00:32:36,000
What happens when capital T
goes to infinity?

427
00:32:36,000 --> 00:32:41,000
When capital T goes to
infinity, spend five seconds

428
00:32:41,000 --> 00:32:45,000
thinking about what it means
physically.

429
00:32:45,000 --> 00:32:51,000
What does it mean for the time
constant of an RC circuit to go

430
00:32:51,000 --> 00:32:56,000
to infinity?
That means that your R and C

431
00:32:56,000 --> 00:33:00,000
are very, very,
very large.

432
00:33:00,000 --> 00:33:03,000
That means that circuit is
going to be very,

433
00:33:03,000 --> 00:33:05,000
very sluggish.
Think elephant.

434
00:33:05,000 --> 00:33:08,000
A big time constant.
I want to move a leg.

435
00:33:08,000 --> 00:33:11,000
It takes a while to do that.
Think big.

436
00:33:11,000 --> 00:33:15,000
Big time constant.
So, everything is going to

437
00:33:15,000 --> 00:33:18,000
happen really slowly.
It's like moving in molasses.

438
00:33:18,000 --> 00:33:22,000
Big time constant.
Everything is going to happen

439
00:33:22,000 --> 00:33:26,000
really, really slowly.
If gamma minus is greater than

440
00:33:26,000 --> 00:33:30,000
gamma plus with a huge time
constant it is going to look

441
00:33:30,000 --> 00:33:35,000
like this.
And the output is going to look

442
00:33:35,000 --> 00:33:38,000
like this.
I make T even larger.

443
00:33:38,000 --> 00:33:42,000
All right.
It is going to like this.

444
00:33:47,000 --> 00:33:51,000
I make these so large that T
tends to zero,

445
00:33:51,000 --> 00:33:57,000
T tends to infinity in which
case I get this situation.

446
00:33:57,000 --> 00:34:00,000
The output goes dah.
OK?

447
00:34:00,000 --> 00:34:02,000
Very slow.
Very lethargic.

448
00:34:02,000 --> 00:34:06,000
Big time constant.
T tends to infinity.

449
00:34:06,000 --> 00:34:10,000
And so if this is stable,
this is unstable,

450
00:34:10,000 --> 00:34:13,000
this is called corresponding
neutral.

451
00:34:13,000 --> 00:34:18,000
And there is a mechanical
analog to all of this.

452
00:34:18,000 --> 00:34:23,000
You can show that this
situation is akin to let's say I

453
00:34:23,000 --> 00:34:30,000
had a physical well of the sort
and I had a ball in there.

454
00:34:30,000 --> 00:34:33,000
I let the ball go.
Then the ball will come down

455
00:34:33,000 --> 00:34:36,000
here and settle down in a stable
state.

456
00:34:36,000 --> 00:34:40,000
Any small perturbation of the
ball will get it to come down

457
00:34:40,000 --> 00:34:44,000
and settle down here.
The unstable situation is this

458
00:34:44,000 --> 00:34:49,000
situation where I have a ball
sitting up here where any small

459
00:34:49,000 --> 00:34:54,000
perturbation will get it to zip
down to a positive rail or to a

460
00:34:54,000 --> 00:34:57,000
negative rail.
So, this is an unstable

461
00:34:57,000 --> 00:35:02,000
equilibrium situation.
And exactly the reason we got

462
00:35:02,000 --> 00:35:06,000
this analysis in the static
situation is that this can

463
00:35:06,000 --> 00:35:08,000
happen.
If I do this circuit here and

464
00:35:08,000 --> 00:35:13,000
don't perturb it then I could
get the output sitting at zero,

465
00:35:13,000 --> 00:35:17,000
but the slightest perturbation,
boom, it is going to fall down

466
00:35:17,000 --> 00:35:20,000
or go up.
What about the neutral

467
00:35:20,000 --> 00:35:23,000
equilibrium state?
That can be modeled like a

468
00:35:23,000 --> 00:35:27,000
table top and the ball is here.
It doesn't matter where you go.

469
00:35:27,000 --> 00:35:32,000
There you are.
How many people saw the

470
00:35:32,000 --> 00:35:37,000
Buckaroo Bonzi thing?
Possibly well before your time.

471
00:35:37,000 --> 00:35:39,000
OK.
I have this table here.

472
00:35:39,000 --> 00:35:45,000
No matter what I do to it,
it just goes and settles down

473
00:35:45,000 --> 00:35:49,000
where it is, and that is neutral
equilibrium.

474
00:35:49,000 --> 00:35:55,000
But what this gives you is a
fun view of the dynamics of the

475
00:35:55,000 --> 00:36:02,000
operational amplifier as I make
small perturbations to it.

476
00:36:02,000 --> 00:36:06,000
And the even more interesting
thing here is you have the tools

477
00:36:06,000 --> 00:36:11,000
based on your first order RC
analysis to analyze the dynamics

478
00:36:11,000 --> 00:36:15,000
of a simple op amp circuit.
OK, so much for theory.

479
00:36:15,000 --> 00:36:18,000
Now let's get to some action
here.

480
00:36:18,000 --> 00:36:19,000
All right.
Fine.

481
00:36:19,000 --> 00:36:22,000
That is really pretty,
good and so on,

482
00:36:22,000 --> 00:36:26,000
but what can you do for me?
What good does this property do

483
00:36:26,000 --> 00:36:30,000
for me?
What can I build?

484
00:36:30,000 --> 00:36:34,000
What we will do is look at the
op amp circuit and focus on the

485
00:36:34,000 --> 00:36:37,000
situation where I have net
positive feedback.

486
00:36:37,000 --> 00:36:42,000
In particular just look at this
circuit with R1 and R2 and send

487
00:36:42,000 --> 00:36:46,000
both to infinity.
So, I have no negative feedback

488
00:36:46,000 --> 00:36:50,000
and I ground this terminal here
and take a look at what happens

489
00:36:50,000 --> 00:36:55,000
to a circuit with positive
feedback and see if I can build

490
00:36:55,000 --> 00:37:01,000
some interesting circuits.
What you are going to do is

491
00:37:01,000 --> 00:37:06,000
build on a circuit called the
basic comparator.

492
00:37:06,000 --> 00:37:11,000
What is that?
If I have an op amp that looks

493
00:37:11,000 --> 00:37:18,000
like this, and remember a VS
rail and minus VS supply there,

494
00:37:18,000 --> 00:37:23,000
this is v+, this is v-,
I can build a very basic

495
00:37:23,000 --> 00:37:29,000
comparator by doing the
following.

496
00:37:29,000 --> 00:37:34,000
All the circuits I am going to
show you are going to build on

497
00:37:34,000 --> 00:37:39,000
this basic little circuit.
What I am going to do is

498
00:37:39,000 --> 00:37:43,000
consider applying an input to
the v- terminal,

499
00:37:43,000 --> 00:37:48,000
applying some sort of an input
and taking a look at how the

500
00:37:48,000 --> 00:37:52,000
output behaves.
So, I apply some input vIN.

501
00:37:52,000 --> 00:37:57,000
And if I just do that,
if this is v+ minus v- here

502
00:37:57,000 --> 00:38:03,000
then I am going to get something
that goes like this.

503
00:38:03,000 --> 00:38:09,000
That is when this is positive
here then this guy is going to

504
00:38:09,000 --> 00:38:16,000
go to the VS rail and this guy
is going to go to the minus VS

505
00:38:16,000 --> 00:38:19,000
rail.
In terms of the,

506
00:38:19,000 --> 00:38:24,000
if I plot the same thing,
in terms of vIN,

507
00:38:24,000 --> 00:38:29,000
and this is vOUT,
if I plot the thing in terms of

508
00:38:29,000 --> 00:38:36,000
vIN then notice that as vIN
increases this guy should go to

509
00:38:36,000 --> 00:38:42,000
a negative rail.
So, in terms of vIN it looks

510
00:38:42,000 --> 00:38:45,000
like this.
What this says is that as the

511
00:38:45,000 --> 00:38:50,000
input becomes more and more
positive applied to v- then the

512
00:38:50,000 --> 00:38:55,000
output goes to minus VS,
and if the input becomes more

513
00:38:55,000 --> 00:39:00,000
and more negative then the
output goes to VS.

514
00:39:00,000 --> 00:39:05,000
This is what is called a very
basic comparator circuit.

515
00:39:05,000 --> 00:39:10,000
It compares the two inputs and
goes up if the input is in one

516
00:39:10,000 --> 00:39:16,000
direction and goes to the other
rail if the input is in the

517
00:39:16,000 --> 00:39:21,000
opposite direction.
So supposing I feed this- I can

518
00:39:21,000 --> 00:39:26,000
plot this is a function of time.
Let's say I plot vIN.

519
00:39:26,000 --> 00:39:32,000
Let's say I feed some vIN here.
Let me just call this.

520
00:39:32,000 --> 00:39:37,000
I feed some vIN to this circuit
here, then what do you expect

521
00:39:37,000 --> 00:39:41,000
the output to look like,
the output wave form?

522
00:39:41,000 --> 00:39:45,000
For all positive vINs the
output is negative.

523
00:39:45,000 --> 00:39:50,000
So, my output vO is going to be
negative as long as vIN is

524
00:39:50,000 --> 00:39:54,000
positive.
And when vIN becomes negative

525
00:39:54,000 --> 00:39:59,000
this one shoots up and behaves
like this.

526
00:39:59,000 --> 00:40:02,000
This is minus VS.
That is plus VS.

527
00:40:02,000 --> 00:40:07,000
This is my input vIN.
Then this guy is going to be my

528
00:40:07,000 --> 00:40:11,000
output.
As vIN is positive output slams

529
00:40:11,000 --> 00:40:15,000
to the negative rail.
When vIN becomes negative the

530
00:40:15,000 --> 00:40:19,000
output slams to the positive
rail.

531
00:40:19,000 --> 00:40:24,000
So, that is quite nice.
And so such a circuit is pretty

532
00:40:24,000 --> 00:40:28,000
useful to me.
Let's say, for example,

533
00:40:28,000 --> 00:40:32,000
I want to build a little
digital circuit that is fed ones

534
00:40:32,000 --> 00:40:34,000
and zeros.
I can use a comparator to turn

535
00:40:34,000 --> 00:40:37,000
my vIN voltage into a sequence
of ones and zeros.

536
00:40:37,000 --> 00:40:41,000
When vIN is positive I produce
a zero and when vIN is negative

537
00:40:41,000 --> 00:40:44,000
I produce a one.
I can get this one,

538
00:40:44,000 --> 00:40:48,000
zero, one, zero sequence coming
out corresponding to the values

539
00:40:48,000 --> 00:40:50,000
of vIN being greater or less
than zero.

540
00:40:50,000 --> 00:40:54,000
Now, one problem with something
like this is that this circuit

541
00:40:54,000 --> 00:40:59,000
can be quite messy in the
following situation.

542
00:40:59,000 --> 00:41:04,000
Suppose I superimpose a small
amount of noise in vIN.

543
00:41:04,000 --> 00:41:08,000
In particular,
let's say that I have some

544
00:41:08,000 --> 00:41:13,000
amount of noise on vIN.
I get a bunch of noise sitting

545
00:41:13,000 --> 00:41:17,000
around here.
What happens is that at this

546
00:41:17,000 --> 00:41:22,000
point where the value goes
negative, I do bump up.

547
00:41:22,000 --> 00:41:30,000
But when for a second I have my
input going above zero again --

548
00:41:30,000 --> 00:41:34,000
-- this output comes down again
and out here it goes up again.

549
00:41:34,000 --> 00:41:39,000
I get this nasty behavior at
the point where the input is

550
00:41:39,000 --> 00:41:42,000
around zero.
When the input is around zero,

551
00:41:42,000 --> 00:41:46,000
the input is meandering around
zero because of noise,

552
00:41:46,000 --> 00:41:51,000
I get a huge amount of up and
down glitches on the output.

553
00:41:51,000 --> 00:41:55,000
That's not very nice.
And we will do a little circuit

554
00:41:55,000 --> 00:42:00,000
that attempts to fix that little
problem.

555
00:42:00,000 --> 00:42:04,000
What we are going to do is use
positive feedback.

556
00:42:04,000 --> 00:42:11,000
And I am going to build you a
circuit that shows that we can

557
00:42:11,000 --> 00:42:15,000
eliminate this for small noise
on the input.

558
00:42:15,000 --> 00:42:19,000
So, let's build the following
circuit.

559
00:42:19,000 --> 00:42:23,000
So I still feed vi to the
negative input,

560
00:42:23,000 --> 00:42:30,000
but this time around I give it
some positive feedback.

561
00:42:30,000 --> 00:42:33,000
So, I give it some positive
feedback.

562
00:42:33,000 --> 00:42:38,000
And what I am going to do is
feedback a portion of vO to the

563
00:42:38,000 --> 00:42:42,000
positive input.
This is positive feedback.

564
00:42:42,000 --> 00:42:47,000
And, in particular,
let's assume that VS equals 12

565
00:42:47,000 --> 00:42:50,000
volts.
And to the negative one I

566
00:42:50,000 --> 00:42:54,000
connect -VS.
This guy is going to go between

567
00:42:54,000 --> 00:42:57,000
12 and -12.
And correspondingly because

568
00:42:57,000 --> 00:43:05,000
these two are equal this one is
going to go between 6 and -6.

569
00:43:05,000 --> 00:43:08,000
This is going to be a 12 or
-12.

570
00:43:08,000 --> 00:43:12,000
Remember, the top rail and the
bottom rail.

571
00:43:12,000 --> 00:43:16,000
And this one is going to be a
+6 or -6.

572
00:43:16,000 --> 00:43:21,000
And let's understand how this
circuit works when I apply an

573
00:43:21,000 --> 00:43:25,000
input vIN.
Let's start by saying that

574
00:43:25,000 --> 00:43:30,000
assume my input is zero for a
moment.

575
00:43:30,000 --> 00:43:35,000
And let's say my output starts
off being 12 volts.

576
00:43:35,000 --> 00:43:40,000
The output is 12 volts then the
input here is going to be 6

577
00:43:40,000 --> 00:43:44,000
volts.
In this case v+ is going to be

578
00:43:44,000 --> 00:43:47,000
6 volts.
The output is 12,

579
00:43:47,000 --> 00:43:52,000
v+ is going to be 6 volts.
And my circuit is sitting out

580
00:43:52,000 --> 00:43:57,000
there doing nothing.
Now, this started off being

581
00:43:57,000 --> 00:44:02,000
zero.
Let's say vIN increases.

582
00:44:02,000 --> 00:44:05,000
As vIN begins to increase what
happens?

583
00:44:05,000 --> 00:44:08,000
Well, nothing until vIN reaches
6 volts.

584
00:44:08,000 --> 00:44:11,000
Since this is 6,
vIN has to go up to 6 volts,

585
00:44:11,000 --> 00:44:16,000
has to equal this voltage
before I can flip the circuit.

586
00:44:16,000 --> 00:44:19,000
What happens when vIN is
greater than 6 volts,

587
00:44:19,000 --> 00:44:24,000
if vIN goes above 6 then I have
more voltage on a negative

588
00:44:24,000 --> 00:44:30,000
terminal than the positive so
the op amp flips its state.

589
00:44:30,000 --> 00:44:36,000
And vO gets to -12 volts.
When vi goes above 6,

590
00:44:36,000 --> 00:44:42,000
vO gets to 12 volts.
And what does v+ go to?

591
00:44:42,000 --> 00:44:50,000
In this state v+ goes to half
of -12 which is -6 volts.

592
00:44:50,000 --> 00:45:00,000
Now, this guy is sitting at -6
and this guy is sitting at -12.

593
00:45:00,000 --> 00:45:03,000
If this one keeps rising
nothing happens,

594
00:45:03,000 --> 00:45:07,000
so output can stay at -12.
So I am pretty safe.

595
00:45:07,000 --> 00:45:10,000
Then let's say v begins to come
down.

596
00:45:10,000 --> 00:45:15,000
As v begins to come down,
does anything happen when v

597
00:45:15,000 --> 00:45:19,000
gets to 6 again?
If v is equal to 6 what

598
00:45:19,000 --> 00:45:22,000
happens?
Nothing because this is at -6

599
00:45:22,000 --> 00:45:25,000
now.
So, there is still a huge net

600
00:45:25,000 --> 00:45:30,000
negative voltage here from v+ to
v-.

601
00:45:30,000 --> 00:45:36,000
And so therefore I sit at -12.
Oh, well, I keep coming down

602
00:45:36,000 --> 00:45:42,000
until I reach -6.
When I reach -6 here these two

603
00:45:42,000 --> 00:45:46,000
become equal.
And what happens when this

604
00:45:46,000 --> 00:45:51,000
becomes less than -6?
v- becomes less than -6.

605
00:45:51,000 --> 00:45:58,000
If this one goes below this
voltage, this is -6 and this is

606
00:45:58,000 --> 00:46:02,000
-7.
There is a net positive voltage

607
00:46:02,000 --> 00:46:06,000
between v+ and v-,
so this output swings to the

608
00:46:06,000 --> 00:46:10,000
positive rail like so.
We will spend a lot more time

609
00:46:10,000 --> 00:46:15,000
on this in the next few minutes
to really hammer the point home.

610
00:46:15,000 --> 00:46:19,000
What is interesting about this
is that even though the moment

611
00:46:19,000 --> 00:46:24,000
vi became more than 6,
I swung to the positive rail,

612
00:46:24,000 --> 00:46:28,000
and then I had to go all the
way back down to -6 before I

613
00:46:28,000 --> 00:46:34,000
could change state.
I had to go way down before it

614
00:46:34,000 --> 00:46:39,000
could flip again.
How can we make use of that?

615
00:46:39,000 --> 00:46:46,000
Well, let me draw you a little
vi versus vO diagram and then

616
00:46:46,000 --> 00:46:50,000
talk about how that can be
useful to us.

617
00:46:50,000 --> 00:46:55,000
This is vi, this is vO,
this is zero.

618
00:46:55,000 --> 00:47:00,000
Let's say this is 12,
-12, -6, +6.

619
00:47:00,000 --> 00:47:04,000
Let's plot that on the screen
and see what it looks like.

620
00:47:04,000 --> 00:47:08,000
As I told you,
the output was at 12 volts to

621
00:47:08,000 --> 00:47:11,000
begin with and my input was at
zero.

622
00:47:11,000 --> 00:47:15,000
So, my input kept increasing.
When the input hit +6 what

623
00:47:15,000 --> 00:47:20,000
happened to my output?
My output swung down to -12.

624
00:47:20,000 --> 00:47:23,000
As the input kept increasing
nothing happened.

625
00:47:23,000 --> 00:47:26,000
This was step one,
this was step two,

626
00:47:26,000 --> 00:47:31,000
step three.
My input kept increasing and

627
00:47:31,000 --> 00:47:36,000
output stayed at -12 volts.
Then what I said was well,

628
00:47:36,000 --> 00:47:41,000
let's bring the input down.
So, my input began to go down,

629
00:47:41,000 --> 00:47:44,000
step four, became more and more
negative.

630
00:47:44,000 --> 00:47:47,000
Nothing happened until I
reached -6.

631
00:47:47,000 --> 00:47:51,000
When I reached -6 I swung
positive, step five.

632
00:47:51,000 --> 00:47:54,000
Again, one, two,
three, four,

633
00:47:54,000 --> 00:47:56,000
five.
I am going up here.

634
00:47:56,000 --> 00:48:01,000
It came up here.
And nothing happens until I

635
00:48:01,000 --> 00:48:06,000
reach -6, but at -6 boom,
I switch to the positive rail.

636
00:48:06,000 --> 00:48:10,000
And as I get more and more
negative I stay there.

637
00:48:10,000 --> 00:48:13,000
Then again, as I start
increasing again,

638
00:48:13,000 --> 00:48:16,000
nothing happens until I reach
+6.

639
00:48:16,000 --> 00:48:19,000
Think of that as your seventh
step.

640
00:48:19,000 --> 00:48:24,000
What is spectacular about this
is that I seem to have a circuit

641
00:48:24,000 --> 00:48:30,000
that now has some knowledge of
where it came.

642
00:48:30,000 --> 00:48:33,000
If it is coming from here it
switches at +6,

643
00:48:33,000 --> 00:48:37,000
but if it is coming from here
it switches at -6.

644
00:48:37,000 --> 00:48:42,000
So, there seems to be sort of a
lag in the behavior of the

645
00:48:42,000 --> 00:48:46,000
circuit or some memory property
in the circuit.

646
00:48:46,000 --> 00:48:50,000
This kind of behavior is called
hysteresis.

647
00:48:50,000 --> 00:48:54,000
The word comes from magnetic
circuits where,

648
00:48:54,000 --> 00:49:00,000
or rather elements that you're
trying to magnetize.

649
00:49:00,000 --> 00:49:04,000
Where if you take a magnet and
move it over a piece of metal it

650
00:49:04,000 --> 00:49:07,000
may leave some residual
magnetism in it.

651
00:49:07,000 --> 00:49:11,000
And, in the same way,
that is called hysteresis.

652
00:49:11,000 --> 00:49:14,000
Same way here.
As the voltage increases it

653
00:49:14,000 --> 00:49:19,000
seems to leave some residual in
the circuit so that it effects

654
00:49:19,000 --> 00:49:22,000
when it shifts.
The good news with this is that

655
00:49:22,000 --> 00:49:27,000
now, if I take the same kind of
noisy wave form that I had

656
00:49:27,000 --> 00:49:32,000
before and do this --
If this is vi then what is

657
00:49:32,000 --> 00:49:38,000
going to happen is for vO I am
going to be negative at this

658
00:49:38,000 --> 00:49:42,000
point.
Nothing happens here because I

659
00:49:42,000 --> 00:49:47,000
have to get to -6 or +6 before
something happens.

660
00:49:47,000 --> 00:49:52,000
Out here I get to -6 and I
switch state and go up to +12.

661
00:49:52,000 --> 00:49:58,000
And then this one comes up
above -6 very slightly out

662
00:49:58,000 --> 00:50:01,000
there.
Nothing happens because the

663
00:50:01,000 --> 00:50:05,000
next change will happen only
when the input goes to +6.

664
00:50:05,000 --> 00:50:09,000
So, if eventually the input
gets to +6 and then I am going

665
00:50:09,000 --> 00:50:13,000
to change state again.
It is actually a really cool

666
00:50:13,000 --> 00:50:17,000
property and something that is
completely non-obvious.

667
00:50:17,000 --> 00:50:20,000
In the last 30 seconds let me
show you a quick demo.

668
00:50:20,000 --> 00:50:24,000
And, based on this property of
hysteresis, I have actually

669
00:50:24,000 --> 00:50:29,000
built a little circuit.
Let me do that first.

670
00:50:29,000 --> 00:50:34,000
Notice here that I am showing
you the input on the X axis vi

671
00:50:34,000 --> 00:50:38,000
and vO on the Y axis.
Notice how the output switches

672
00:50:38,000 --> 00:50:43,000
at +6 volts and switches at a -6
volts to +12 or -12.

673
00:50:43,000 --> 00:50:48,000
That's the hysteresis property.
And we can actually use this

674
00:50:48,000 --> 00:50:53,000
property to build a clock
circuit, which is on page 9,

675
00:50:53,000 --> 00:51:00,000
build an oscillator that sits
there and oscillates by itself.

676
00:51:00,000 --> 00:51:03,000
And you will see details of
that in recitation tomorrow.