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Let's get started.

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Can you hear me back there?
Loud and clear.

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OK.
Let's get started.

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Before I begin,
just a couple of announcements.

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Brad Buren is one of our
students here and he needs a

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note-taker.
It's a paid position.

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So if you are interested you
can stop by after class and see

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him.
He's sitting right here out

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there, OK?
Second, just a reminder that

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6.002 does have prerequisites.
And the prerequisites are 8.02

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and 18.03.
So with that let me start off

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with the usual.
Do a quick review of what we've

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done so far.
So we started out life looking

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at the laws of physics and
Maxwell's equations and so on.

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And those were way too hard so
we said let's make life easy for

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ourselves.
So we chose to play in this

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playground in which we said we
shall adhere to the lumped

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matter discipline.
OK?

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The LMD.
So we are in that playground.

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So this entire course,
and for that matter large parts

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of EECS are within that
playground, within which the

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lumped matter discipline
applies.

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So as soon as we jumped into
the playground,

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the LMD playground,
we could take Maxwell's

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equations and abstract them out
into two very,

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very simple rules.
And the very simple rules were

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KVL and KCL.
KVL simply said that I can sum

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the voltages in any loop in a
circuit and the result then

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would be zero.
Similarly, I can sum the

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currents that enter or exit any
node and the sum will also be

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zero.
So what you can now do is,

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if you feel like,
you can go around and brag.

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Oh, yeah, we use Maxwell's
equations in everyday life and,

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yeah, it's good stuff.
And the key is that this is

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really an encapsulation of
Maxwell's equations within this

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playground that we are in.
So I talked about the first

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method of circuit analysis in
the last lecture.

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And that method simply took
the, wrote KVL for all the

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loops, wrote KCL for all the
nodes and wrote element vi

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relationships.
And together gave you a big

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bunch of equations.
And you sat down and grunged

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through the equations and you
solved for branch voltages and

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currents.
So we reviewed a second method

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of circuit analysis.
And I'll simply call it circuit

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composition.
The basic idea behind this

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method was to learn some simple
rules of how resistors add and

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conductances add and so on and
so forth and look at a circuit

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and simplify the circuit by
making series simplifications

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when the resistors are in series
and so on and so forth,

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and compose it and play around
with it till we end up with the

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current, the voltages that we
are looking for.

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This is the intuitive method.
And so a section in Chapter 2,

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I believe, of the course notes
discusses several examples using

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this method and attempts to make
a little bit formal the

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intuitive approach that is
applied in this method.

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So we then looked at the node
method.

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And the node method was simply
a particular way of applying KVL

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and KCL.
Node method,

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remember?
We took a ground node.

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Then we labeled the nodes of
the remaining voltages with

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respect to that ground.
Then we wrote KCL for each of

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the nodes.
And when we wrote KCL for each

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of the nodes,
remember, KVL was implicit in

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this expression that we used for
each of the currents that were

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exiting each node.
So if Ej was a node voltage,

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then Ej minus Ei multiplied by
the conductance Gi was the

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current that was going through
one of those,

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I should call it Gij.
This is a conductance that

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connects nodes i and j.
That gave us the KVL that fed

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into the same system.
So these are three methods.

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The node method,
by the way, is sort of the

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workhorse of the 6.002 industry.
And for that matter for all of

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the circuits industry.
When in doubt,

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apply the mode method,
you'll be OK.

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That applies to linear
circuits, nonlinear circuits,

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what have you.
What I'm going to do today is

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go through two more methods.
So notice that the first few

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lectures of this course,
the first three lectures simply

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comprise transitioning you from
the world of physics to the

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world of EECS.
And then two lectures on giving

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you a bag of tricks.
So we start you off with the

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sort of tools,
your mallets and chisels and so

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on and so forth.
And these five methods are your

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tools.
We'll look at two methods

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today.
One method is called the method

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of superposition and the second
method is called the Thevenin

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method.
And these methods apply only to

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linear circuits.
So we look at the subset of

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circuits that are linear,
and these two methods apply to

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only those circuits.
These are methods that combined

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with intuition really enables
you to solve very interesting

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circuits very,
very quickly.

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So let me do an example using a
usual node method.

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And then jump into introducing
the superposition methods and

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Thevenin methods using that same
example.

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So let me draw you an example
circuit here.

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So, again, I'm using this
example, I will use this example

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to introduce the method of
superposition and the Thevenin

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method.
So what I'm going to do is

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start off the usual way and
analyze the circuit using a

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method that you know now,
the node method.

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And what I'll do is write down
the node equations for this by

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applying the node method.
So if you recall the node

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method.
I choose a ground node.

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I'm going to choose this node.
It's got both the voltage

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source connected to it,
and it's also got many other

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edges impinging on it.
So I'm going to choose that as

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my ground node and I'm going to
label the other nodes with their

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voltages.
So this is an unknown.

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I'll label it as e.
I guess we just have one

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unknown e.
And I know the voltage of this

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node, and that is simply V.
Since it's V,

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there's a voltage source
between the ground node and that

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node.
So what I can do next is that I

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can write down the node equation
for this node and then go from

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there.
So let me go ahead and do that.

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So let me sum up the currents
going outside,

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going outwards.
So I have e minus v divide by

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R1, I have e minus zero divide
by R2, and I have minus i equals

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zero.
This is a node equation.

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The first thing I want you to
observe is that this equation is

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linear in V and i.
What I mean by linear is that

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you don't see terms like Vi or
V-squared and things like that.

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It's some constant times V plus
some constant times i equals

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some other constant.
So that's quite nice.

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So I'm going to rearrange the
terms in the following manner.

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I'll move the known sources to
the right-hand side and collect

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the coefficients of e on this
side, so I get one by R1 plus

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one by R2 over here.

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So stare at this for a moment
and notice again here I have e,

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my unknown node voltage,
there is some constant

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multiplier, and that equals some
function of V summed up with

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some function of i.
And, again, notice that this is

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a linear combination of V and i.
No multiplication terms and so

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on and so forth.
This is a pretty standard form

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in which we will represent
equations quite often.

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And just to label it,
this is often labeled G as the

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conductance matrix.
Of course this is e,

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our unknown node voltages,
and this is a linear sum of

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sources.
So this is a very standard way

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that we will represent
equations.

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We did that last week as well,
or rather on Tuesday where I

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took a conductance matrix,
multiplied that by a column

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vector of unknown node voltages
and equated that to some linear

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combination of my source
voltages.

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The reason the circuit is
linear is that I have only

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linear elements in the circuit.
I don't have any nonlinear

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elements.
And because of that I can

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rewrite this in the following
manner.

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I'm just going to express e as
a function of V and i and bring

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it over to this side.
So it's some function of i.

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So I get R1 R2 divide by R1
plus R2.

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And I bring R1 R2 to this side.
That's what I get.

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So stare at this for a few
seconds, very common form.

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My unknown node voltage is
equal to this stuff on the

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right-hand side.
The stuff on the right-hand

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side has a term multiplying the
source voltage V and some other

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term multiplying the current I.
And if I were to put this in

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sort of symbol-like form my
unknown node voltage is some

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constant times V1 plus some
constant times,

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is of the form constant times
the source current,

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constant times the source
voltage and so on.

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The units of As and Vs are
different because in this case A

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has no units because V is a
voltage.

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And so is e.
In this case V has units of

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resistance.
So that V times i gives me a

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voltage.
So stare at this equation for a

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few seconds and this should help
us build up some insight that

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will allow us to write down the
answer almost by inspection.

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I'm going to show you a method
now, in a few minutes,

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which will allow you to write
down the answer e just by

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starring at the circuit without
having to go through node

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equations and so on.
The more and more methods I

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teach you, the more you will be
able to do a lot of this

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completely by yourselves.
In this particular example it's

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a relatively simple circuit but
these methods would be

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particularly useful when you
have more complicated

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situations.
But before I go on let me spend

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a few minutes pontificating on
linearity.

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So that's a linear circuit.
And this equation gives me the

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unknown node voltage e as a
linear sum of source voltages

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and source currents.
Linearity implies two

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properties, the property of
homogeneity and also gives vice

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to the property of
superposition.

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Let's do homogeneity first.
What this says is if I have a

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circuit, some circuit and I feed
it some sort of inputs,

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A, then let's say my output is
S.

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If you're feeling hungry think
of these as apples and the

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circuit converts them into
applesauce.

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So what homogeneity says is
that what I can do is if I take

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each of my apples and instead of
feeding it an entire apple what

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if I give it three-quarters of
an apple?

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Say I multiple all my inputs by
some constant alpha,

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three-quarters.
What that says is that at the

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output instead of getting one
full bottle of applesauce I'm

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going to get three-quarters of a
bottle of apple sauce.

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So if I proportionately reduce
all the inputs and if this is a

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linear circuit then so shall my
output be reduced in the same

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proportion.
So that's homogeneity.

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Next, let's look at
superposition.

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The property of superposition
says the following.

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The same kind of circuit.
If I feed it apples then I get

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applesauce.
I take the same circuit,

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and this time around if I feed
the circuit a different set of

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inputs, say blueberries.
And let's say my output,

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oops, let me do it this way.
So as my output I get blueberry

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sauce, if such exists.
So apples applesauce,

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blueberries give me blueberry
sauce.

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Then what I'm going to get if I
mix up the two,

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so let's say I take my circuit,
the same circuit with a set of

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inputs and in this example one
output.

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Let's say I mix up my inputs
and some of my inputs in the

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following way,
here I feed an A1 plus B1 and

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here A2 plus B2 and so on then
at the output I am going to get

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a mush of apple sauce and
blueberry sauce.

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All this says is that if I
apply just apples I get

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applesauce.
If I apply just blueberries I

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get blueberry sauce.
Then if I were to figure out

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how this blender would have
worked had I fed in the

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combinations of apples and
blueberries, then for the

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purposes of understanding that
blender all I could have done

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was taken by two outputs and
just mixed them up together

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myself and that's exactly what
I'd get.

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So if I sum up the inputs my
outputs would also be the sum of

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the outputs with the inputs
applied by themselves.

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So let me take this here and
munge around with hit for a few

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seconds and get something
interesting out of it.

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So notice two inputs,
two inputs, outputs.

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In your notes I've given you
another template for the next

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00:18:14,000 --> 00:18:17,000
set of scribbles I'm going to
make here.

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00:18:17,000 --> 00:18:21,000
So use the next set of
templates on page three.

232
00:18:21,000 --> 00:18:25,000
What I'm going to do here is
something very simple,

233
00:18:25,000 --> 00:18:30,000
set one output to zero and feed
a voltage V1.

234
00:18:30,000 --> 00:18:35,000
So that's feed a voltage V1 and
set the other output to zero.

235
00:18:35,000 --> 00:18:38,000
And let's say I get Y1 as an
output.

236
00:18:38,000 --> 00:18:44,000
And in this case I set the
first voltage to zero and feed a

237
00:18:44,000 --> 00:18:48,000
different voltage V2 on the
second input.

238
00:18:48,000 --> 00:18:53,000
And let's say my output is Y2.
This is just a particular

239
00:18:53,000 --> 00:19:00,000
application of the superposition
principle I just outlined.

240
00:19:00,000 --> 00:19:03,000
Apply V1 set one output to
zero.

241
00:19:03,000 --> 00:19:09,000
Apply V2 set the original
output to zero.

242
00:19:09,000 --> 00:19:17,000
Then what I'm going to find is
that the answer will simply look

243
00:19:17,000 --> 00:19:24,000
like this, just replace for As
and Bs what I just did and we

244
00:19:24,000 --> 00:19:30,000
get V1 and zero here and we get
zero and V2 here.

245
00:19:30,000 --> 00:19:40,000
And as my output I'm going to
get exactly the sum Y1 plus Y2.

246
00:19:40,000 --> 00:19:43,000
This is simply a particular
application of superposition

247
00:19:43,000 --> 00:19:46,000
where what I'm saying is the
following.

248
00:19:46,000 --> 00:19:50,000
If you look at this circuit
here effectively what have I

249
00:19:50,000 --> 00:19:53,000
done?
Effectively what I've done is

250
00:19:53,000 --> 00:19:57,000
apply the voltage V1 on one
input and a voltage V2 on the

251
00:19:57,000 --> 00:19:58,000
other input.
V1 here.

252
00:19:58,000 --> 00:20:02,000
V2 here.
And the output is Y1 plus Y2.

253
00:20:02,000 --> 00:20:04,000
What I'm saying is look
backwards now.

254
00:20:04,000 --> 00:20:09,000
What I'm saying is that the
whole components of the output

255
00:20:09,000 --> 00:20:13,000
Y1 plus Y2 could individually be
derived in the following manner.

256
00:20:13,000 --> 00:20:18,000
I could get the component Y1 by
simply applying one of the

257
00:20:18,000 --> 00:20:21,000
voltages and setting the other
to zero.

258
00:20:21,000 --> 00:20:25,000
I can get the other component
Y2 by setting yet another input

259
00:20:25,000 --> 00:20:30,000
to zero and applying the voltage
V2 to get Y2.

260
00:20:30,000 --> 00:20:32,000
And sum then up and that's my
answer.

261
00:20:32,000 --> 00:20:36,000
This will become a lot clearer
with an example.

262
00:20:36,000 --> 00:20:40,000
Again, remember if I have a
bunch of inputs applied to a

263
00:20:40,000 --> 00:20:44,000
circuit, V1, V2 and so on,
and I get some output then what

264
00:20:44,000 --> 00:20:48,000
this is saying is that I can
alternatively find out the

265
00:20:48,000 --> 00:20:53,000
answer by applying just one
voltage, setting all the others

266
00:20:53,000 --> 00:20:57,000
to zero, measuring the output,
apply a second voltage,

267
00:20:57,000 --> 00:21:01,000
set all inputs to zero,
measure the output and sum of

268
00:21:01,000 --> 00:21:07,000
applesauce and blueberry sauce
and there you get the answer.

269
00:21:07,000 --> 00:21:13,000
Let's do an example.
And before we go into that I

270
00:21:13,000 --> 00:21:20,000
talked about setting voltage
sources and current sources to

271
00:21:20,000 --> 00:21:22,000
zero.
First of all,

272
00:21:22,000 --> 00:21:28,000
what does it mean to set a
voltage source to zero?

273
00:21:28,000 --> 00:21:38,000
This is the same as this.
Setting a voltage source to

274
00:21:38,000 --> 00:21:49,000
zero is simply replacing the
voltage source with a short,

275
00:21:49,000 --> 00:22:00,000
and setting a current source to
zero simply implies an open

276
00:22:00,000 --> 00:22:05,000
circuit.
So when I say zero that source,

277
00:22:05,000 --> 00:22:09,000
if it's a voltage source short
it, if it's a current source

278
00:22:09,000 --> 00:22:10,000
open it.

279
00:22:16,000 --> 00:22:19,000
I can take any two nodes in the
world and measure the potential

280
00:22:19,000 --> 00:22:22,000
difference across them.
So there may be some potential

281
00:22:22,000 --> 00:22:26,000
difference across these set by
the circuit that I haven't shown

282
00:22:26,000 --> 00:22:29,000
you on this side.
There might be some other

283
00:22:29,000 --> 00:22:33,000
circuit that is controlling the
voltage of these two nodes.

284
00:22:33,000 --> 00:22:38,000
The same with the short.
What's V going to be?

285
00:22:38,000 --> 00:22:42,000
But there is a V.
It's zero.

286
00:22:42,000 --> 00:22:48,000
So that's method four,
method of superposition.

287
00:22:48,000 --> 00:22:56,000
And this method says that the
output of a circuit --

288
00:23:03,000 --> 00:23:06,000
Again, remember I'm focusing on
linear circuits.

289
00:23:06,000 --> 00:23:10,000
Remember, I have this
playground where LMD applies.

290
00:23:10,000 --> 00:23:14,000
And within that playground I'm
playing in the south goal area.

291
00:23:14,000 --> 00:23:17,000
In the south goal area,
in that subset of the

292
00:23:17,000 --> 00:23:21,000
playground circuits are linear.
So in that part of the

293
00:23:21,000 --> 00:23:25,000
playground superposition applies
because there circuits are

294
00:23:25,000 --> 00:23:32,000
linear.
So the output of a circuit is

295
00:23:32,000 --> 00:23:42,000
determined by summing up the
responses to each source acting

296
00:23:42,000 --> 00:23:44,000
alone.

297
00:23:54,000 --> 00:23:58,000
Now, in this statement here
this source stands for

298
00:23:58,000 --> 00:24:01,000
independent source.
I haven't talked about

299
00:24:01,000 --> 00:24:04,000
independent versus dependent
sources.

300
00:24:04,000 --> 00:24:08,000
We'll talk about dependent
sources a few weeks from today.

301
00:24:08,000 --> 00:24:12,000
And just so you don't get
confused, for dependent sources

302
00:24:12,000 --> 00:24:17,000
you will be looking at Section
3.3.3 of your course notes to

303
00:24:17,000 --> 00:24:21,000
see how superposition works with
dependent sources.

304
00:24:21,000 --> 00:24:25,000
But remember we haven't covered
dependent sources yet.

305
00:24:25,000 --> 00:24:30,000
We will be covering them about
two weeks from now.

306
00:24:30,000 --> 00:24:36,000
So let's go back to our example
and apply the method of

307
00:24:36,000 --> 00:24:43,000
superposition to an example.
So the method says sum up the

308
00:24:43,000 --> 00:24:49,000
outputs of each of the
sub-circuits where I'm applying

309
00:24:49,000 --> 00:24:55,000
one source acting alone.
So let me just do this here.

310
00:24:55,000 --> 00:25:03,000
Let me start with the circuit.
And let me start with shutting

311
00:25:03,000 --> 00:25:06,000
I off.
So I have voltage V --

312
00:25:13,000 --> 00:25:16,000
I have R2.
And I'm shutting I off.

313
00:25:16,000 --> 00:25:21,000
So I have replaced this with an
open circuit.

314
00:25:21,000 --> 00:25:25,000
So I is zero.
Let me call the node voltage eV

315
00:25:25,000 --> 00:25:32,000
to reflect that component of the
node voltage that arises due to

316
00:25:32,000 --> 00:25:37,000
V acting alone.
And you should look at this

317
00:25:37,000 --> 00:25:42,000
pattern here and very quickly be
able to write the answer for

318
00:25:42,000 --> 00:25:46,000
patterns like this voltage,
the two resistors.

319
00:25:46,000 --> 00:25:49,000
That's called a resistive
divider.

320
00:25:49,000 --> 00:25:52,000
It will appear again and again
and again.

321
00:25:52,000 --> 00:25:56,000
And eV is simply V times R2
divided by R1 plus R2.

322
00:25:56,000 --> 00:26:02,000
That's still my ground node.
So the voltage here is simply

323
00:26:02,000 --> 00:26:07,000
this voltage divided by the two
resistors to give you the

324
00:26:07,000 --> 00:26:12,000
current multiplied by R2 to give
you the voltage across this R.

325
00:26:12,000 --> 00:26:16,000
Remember this pattern.
You apply voltage divider

326
00:26:16,000 --> 00:26:21,000
patterns probably more times
than any other pattern that you

327
00:26:21,000 --> 00:26:24,000
might imagine.
So that's with the V acting

328
00:26:24,000 --> 00:26:27,000
alone.
Now, let me do I acting alone.

329
00:26:27,000 --> 00:26:31,000
So for I acting alone --

330
00:26:43,000 --> 00:26:49,000
And what I do this time around
is replace this with a short,

331
00:26:49,000 --> 00:26:53,000
replace the voltage source to
the short.

332
00:26:53,000 --> 00:27:00,000
And let me call this voltage eI
for the component of the voltage

333
00:27:00,000 --> 00:27:04,000
due to the current I.
And eI, in this case,

334
00:27:04,000 --> 00:27:08,000
is simply given by yet another
pattern here,

335
00:27:08,000 --> 00:27:12,000
the current across a pair or
resistors is simply the

336
00:27:12,000 --> 00:27:17,000
effective resistance multiplied
by the current so it's i and the

337
00:27:17,000 --> 00:27:21,000
effective resistance is R1,
R2 or R1 plus R2.

338
00:27:21,000 --> 00:27:24,000
That's eI.
That's a component that node

339
00:27:24,000 --> 00:27:31,000
due to the current I.
Now, so the method says that.

340
00:27:31,000 --> 00:27:38,000
Then take these components,
sum them up and there you have

341
00:27:38,000 --> 00:27:44,000
the answer.
So E is simply ev plus ei.

342
00:27:44,000 --> 00:27:49,000
The components of V and I
acting alone,

343
00:27:49,000 --> 00:27:56,000
just simply V times R2 divided
by R1 plus R2 plus R1,

344
00:27:56,000 --> 00:28:00,000
R2.
There we go.

345
00:28:00,000 --> 00:28:02,000
Fortunately,
the fates have been kind to us

346
00:28:02,000 --> 00:28:06,000
and the answer is the same as
the answer we obtained with the

347
00:28:06,000 --> 00:28:08,000
node method.
No surprise here.

348
00:28:08,000 --> 00:28:11,000
So this is actually an
incredibly simple method.

349
00:28:11,000 --> 00:28:14,000
So you can take a very complex
circuit.

350
00:28:14,000 --> 00:28:18,000
What have you really done here?
You can take a very complex

351
00:28:18,000 --> 00:28:22,000
circuit and you can solve a very
complex circuit by breaking it

352
00:28:22,000 --> 00:28:25,000
down into many simple individual
sub problems.

353
00:28:25,000 --> 00:28:30,000
You will do this in EECS time
and time and time again.

354
00:28:30,000 --> 00:28:33,000
Whether it's in software
systems or hardware systems or

355
00:28:33,000 --> 00:28:35,000
what have you,
you're often times building

356
00:28:35,000 --> 00:28:38,000
complicated systems.
Remember doom on this side?

357
00:28:38,000 --> 00:28:41,000
And the way and when you put
these things together,

358
00:28:41,000 --> 00:28:44,000
let's say a large software
system, is you don't write the

359
00:28:44,000 --> 00:28:47,000
whole piece of software starting
main and grunge down.

360
00:28:47,000 --> 00:28:50,000
You build a lot of little
components and tie the

361
00:28:50,000 --> 00:28:53,000
components together.
In the same manner here you

362
00:28:53,000 --> 00:28:57,000
take a big circuit and you find
its behavior for each source

363
00:28:57,000 --> 00:29:00,000
acting alone.
Lots of little inky dinky

364
00:29:00,000 --> 00:29:04,000
simple little circuits.
And you will see examples in

365
00:29:04,000 --> 00:29:09,000
your homework where you're given
a big circuit or because it set

366
00:29:09,000 --> 00:29:14,000
all the Is to zero and the other
Vs to zero the whole circuit

367
00:29:14,000 --> 00:29:18,000
almost vanishes and all that
you're left with is a little

368
00:29:18,000 --> 00:29:21,000
resistor or two.
So this is the very,

369
00:29:21,000 --> 00:29:24,000
very powerful method.
I'd like to do a little

370
00:29:24,000 --> 00:29:28,000
demonstration for you.
And what I'm going to show you

371
00:29:28,000 --> 00:29:36,000
is the demo is a vat of water.
Actually, I'll tell you what it

372
00:29:36,000 --> 00:29:42,000
is in a second.
But assume it is salt water for

373
00:29:42,000 --> 00:29:46,000
now.
I'll apply two voltages.

374
00:29:46,000 --> 00:29:52,000
In this case I'm going to apply
a sinusoid.

375
00:29:52,000 --> 00:29:59,000
That's not very good.
A sinusoid and a triangular

376
00:29:59,000 --> 00:30:03,000
wave.
And what I'm going to do is

377
00:30:03,000 --> 00:30:06,000
measure the response at this
site.

378
00:30:06,000 --> 00:30:08,000
Now, this is a vat of salt
water.

379
00:30:08,000 --> 00:30:13,000
And I'm going to tell you it
behaves like a linear system.

380
00:30:13,000 --> 00:30:17,000
If you view each little
particle, or each little

381
00:30:17,000 --> 00:30:22,000
cubic-centimeter or whatever of
water, it'll behave like little

382
00:30:22,000 --> 00:30:25,000
resistor.
So this vat of salt water

383
00:30:25,000 --> 00:30:30,000
behaves like big distributed
resistor in the following

384
00:30:30,000 --> 00:30:32,000
manner.

385
00:30:39,000 --> 00:30:41,000
And so on.
This of this big mesh of little

386
00:30:41,000 --> 00:30:44,000
resistors, but it's all
resistors.

387
00:30:44,000 --> 00:30:47,000
It's a linear circuit.
So I'm going to apply two

388
00:30:47,000 --> 00:30:51,000
voltages, a triangular and a
sinusoid, and we're going to

389
00:30:51,000 --> 00:30:54,000
observe the output.
And what do you expect to see

390
00:30:54,000 --> 00:30:57,000
there?
You will see the superposition

391
00:30:57,000 --> 00:31:01,000
of the two, which is you'll see
a sinusoid.

392
00:31:01,000 --> 00:31:07,000
And then you'll see the jagged
triangular thing articulating

393
00:31:07,000 --> 00:31:13,000
the sinusoid pattern.
What I'm going to do right now,

394
00:31:13,000 --> 00:31:18,000
don't put any water yet.
This is the vat of nothing

395
00:31:18,000 --> 00:31:21,000
right now.
It's all empty.

396
00:31:21,000 --> 00:31:25,000
Can we show the screen on this
side?

397
00:31:25,000 --> 00:31:29,000
The oscilloscope screen?

398
00:31:34,000 --> 00:31:35,000
OK.
Oh, there you go.

399
00:31:35,000 --> 00:31:38,000
So this is the screen of the
oscilloscope now.

400
00:31:38,000 --> 00:31:43,000
Notice that I have a sinusoid
and I have a triangular wave and

401
00:31:43,000 --> 00:31:46,000
the output is zero.
And the reason is there is

402
00:31:46,000 --> 00:31:49,000
nothing in this vat.
It's empty.

403
00:31:49,000 --> 00:31:53,000
So previously when I taught
this course I would get

404
00:31:53,000 --> 00:31:57,000
saltwater and pour saltwater.
Then we discovered a much

405
00:31:57,000 --> 00:32:01,000
better source of water that
conducted electricity like one

406
00:32:01,000 --> 00:32:05,000
real mean fluid.
Cambridge water.

407
00:32:05,000 --> 00:32:10,000
It just works very pleasantly.
It just conducts electricity

408
00:32:10,000 --> 00:32:14,000
like nothing at all.
And I've been thinking of using

409
00:32:14,000 --> 00:32:18,000
Charles River water next time
and see what happens,

410
00:32:18,000 --> 00:32:23,000
although there we'd probably
get some biological organisms

411
00:32:23,000 --> 00:32:26,000
doing strange things at you.
But go ahead.

412
00:32:26,000 --> 00:32:30,000
Our friendly demonstration
expert, Lorenzo,

413
00:32:30,000 --> 00:32:34,000
will pour some water into the
vat.

414
00:32:34,000 --> 00:32:39,000
And you should begin seeing the
output being a superposition of

415
00:32:39,000 --> 00:32:42,000
the two.
So as he pours,

416
00:32:42,000 --> 00:32:44,000
there you go,
do you see that?

417
00:32:44,000 --> 00:32:50,000
So you do see the sinusoidal
articulation and the jagged wave

418
00:32:50,000 --> 00:32:53,000
form.
And just to have some more fun,

419
00:32:53,000 --> 00:32:58,000
what I can do is increase one
of the voltages.

420
00:32:58,000 --> 00:33:01,000
And you'll see --

421
00:33:06,000 --> 00:33:10,000
Now you know what would have
happened if I had used Charles

422
00:33:10,000 --> 00:33:13,000
River water.
So my output keeps increasing

423
00:33:13,000 --> 00:33:17,000
as I increase the corresponding
wave form.

424
00:33:24,000 --> 00:33:26,000
I could do this,
this is fun.

425
00:33:26,000 --> 00:33:30,000
So let me pause there and go
onto the next topic.

426
00:33:30,000 --> 00:33:34,000
So that little demonstration
showed you that even something

427
00:33:34,000 --> 00:33:39,000
as simple as this physical
entity vat of water behaves like

428
00:33:39,000 --> 00:33:42,000
a linear system,
and we can model that linear

429
00:33:42,000 --> 00:33:46,000
system as a set of resistors.
Unbeknownst to you,

430
00:33:46,000 --> 00:33:50,000
right now, in the past ten
seconds I introduced a new

431
00:33:50,000 --> 00:33:53,000
concept.
It's called subliminal

432
00:33:53,000 --> 00:33:56,000
advertising.
So one of the things we do in

433
00:33:56,000 --> 00:34:02,000
EE a lot is model real systems.
So often times if I wanted to

434
00:34:02,000 --> 00:34:06,000
look at the behavior of salt,
behavior of a vat of water,

435
00:34:06,000 --> 00:34:11,000
I can model it as a set of
resistors for certain kinds of

436
00:34:11,000 --> 00:34:14,000
activities.
Just hold that thought for some

437
00:34:14,000 --> 00:34:17,000
time later in your careers.
All right.

438
00:34:17,000 --> 00:34:21,000
That's method four,
the superposition method.

439
00:34:21,000 --> 00:34:25,000
Remember, it is methods like
this that will make your life

440
00:34:25,000 --> 00:34:29,000
really, really,
really easy.

441
00:34:29,000 --> 00:34:34,000
If you find that you are having
to do a lot of grunging homework

442
00:34:34,000 --> 00:34:38,000
or something,
just step back and think

443
00:34:38,000 --> 00:34:41,000
superposition,
think Thevenin or think

444
00:34:41,000 --> 00:34:45,000
composition rule.
There must be a simpler way

445
00:34:45,000 --> 00:34:48,000
usually.
Let's do the next method.

446
00:34:48,000 --> 00:34:51,000
This is called the Thevenin
method.

447
00:34:51,000 --> 00:34:57,000
To derive this method let me
start by applying superposition

448
00:34:57,000 --> 00:35:03,000
to some circuit.
So let's say I have some

449
00:35:03,000 --> 00:35:09,000
arbitrary network N.
Assume it's a linear network

450
00:35:09,000 --> 00:35:15,000
and the network has a whole
bunch of goodies in it.

451
00:35:15,000 --> 00:35:22,000
It has a bunch of resistors,
it has a bunch of voltage

452
00:35:22,000 --> 00:35:30,000
sources, and it has a bunch of
current sources.

453
00:35:30,000 --> 00:35:33,000
Many current sources.
Many voltage sources.

454
00:35:33,000 --> 00:35:37,000
Many resistors.
Some jumbled voltage sources,

455
00:35:37,000 --> 00:35:43,000
current sources and resistors.
And I look at two nodes in this

456
00:35:43,000 --> 00:35:46,000
network.
Here are two nodes in the

457
00:35:46,000 --> 00:35:50,000
network, two points in the
network were elements connect.

458
00:35:50,000 --> 00:35:56,000
I'm looking at those two nodes
and all I want to do is the

459
00:35:56,000 --> 00:36:00,000
following.
I want to figure out if I take

460
00:36:00,000 --> 00:36:05,000
a rinky-dinky little current
source and apply it there,

461
00:36:05,000 --> 00:36:10,000
all I want to figure out is
what is V and what is I.

462
00:36:10,000 --> 00:36:15,000
There is this mongo box out
here, a black box of resistors,

463
00:36:15,000 --> 00:36:20,000
voltage source and current
sources, too many to count.

464
00:36:20,000 --> 00:36:24,000
I pick two nodes,
apply a current source,

465
00:36:24,000 --> 00:36:29,000
and all I care about is what is
the voltage that I will measure

466
00:36:29,000 --> 00:36:35,000
by applying it here.
Notice the current here will be

467
00:36:35,000 --> 00:36:40,000
I because the current here is I.
And I apply it here.

468
00:36:40,000 --> 00:36:43,000
I want to measure what the
voltage is.

469
00:36:43,000 --> 00:36:48,000
Now, with the insight you've
obtained from superposition,

470
00:36:48,000 --> 00:36:53,000
you should be able to jump up
and state the form of the

471
00:36:53,000 --> 00:36:57,000
answer.
So by superposition we know the

472
00:36:57,000 --> 00:37:01,000
following.
We know that the effect of the

473
00:37:01,000 --> 00:37:06,000
circuit will be the same as the
sum of components being added

474
00:37:06,000 --> 00:37:07,000
up.
Sum of component,

475
00:37:07,000 --> 00:37:11,000
sum of component,
a bunch of components added up.

476
00:37:11,000 --> 00:37:16,000
Each component will be the
response of one source acting

477
00:37:16,000 --> 00:37:18,000
alone.
So if I can figure out the

478
00:37:18,000 --> 00:37:23,000
effect of one source acting
alone and put that down here,

479
00:37:23,000 --> 00:37:28,000
and do the same thing for all
the sources, that's what I will

480
00:37:28,000 --> 00:37:32,000
get.
So for the source Vm it's a

481
00:37:32,000 --> 00:37:35,000
linear circuit.
So I know that my answer is

482
00:37:35,000 --> 00:37:40,000
going to be, in the final answer
is going to be a Vm term and

483
00:37:40,000 --> 00:37:44,000
it's going to be multiplied by
some alpha M term.

484
00:37:44,000 --> 00:37:47,000
I know that.
It's a linear circuit so I know

485
00:37:47,000 --> 00:37:52,000
that the answer shall have a
term Vm multiplied by some

486
00:37:52,000 --> 00:37:54,000
constant.
Simple, I know that.

487
00:37:54,000 --> 00:38:00,000
Similarly, the same is true
for, oh, this is the term Vm.

488
00:38:00,000 --> 00:38:05,000
And what I can do is I can
measure just this effect by

489
00:38:05,000 --> 00:38:09,000
setting all the other sources to
zero.

490
00:38:09,000 --> 00:38:15,000
So I can set all the other
current sources to zero and all

491
00:38:15,000 --> 00:38:19,000
voltage sources,
except for this one,

492
00:38:19,000 --> 00:38:23,000
and I can get that answer.
So, similarly,

493
00:38:23,000 --> 00:38:30,000
for every voltage source I am
going to get a term.

494
00:38:30,000 --> 00:38:34,000
So for every single voltage
source, M1, M2,

495
00:38:34,000 --> 00:38:39,000
M3 and so on I'm going to get
such a term and they're all

496
00:38:39,000 --> 00:38:44,000
going to sum up.
Similarly, I'm going to get a

497
00:38:44,000 --> 00:38:48,000
term for In.
And I know there will be an In

498
00:38:48,000 --> 00:38:53,000
term, and I know it's going to
be some constant beta

499
00:38:53,000 --> 00:38:57,000
multiplying In.
In this example of ours here,

500
00:38:57,000 --> 00:39:01,000
in this example,
remember alpha was this and

501
00:39:01,000 --> 00:39:08,000
beta was this constant here.
There's some constant beta,

502
00:39:08,000 --> 00:39:11,000
some constant alpha.
And because I have a whole

503
00:39:11,000 --> 00:39:16,000
bunch of current sources there's
going to be such a term for each

504
00:39:16,000 --> 00:39:19,000
one of them.
And each one of these terms,

505
00:39:19,000 --> 00:39:24,000
Vm, In will be the voltage I
would see here if I set all the

506
00:39:24,000 --> 00:39:28,000
other Vms to zero and I set all
the other current sources,

507
00:39:28,000 --> 00:39:33,000
except for that one to zero.
What am I missing?

508
00:39:33,000 --> 00:39:36,000
Is that it?
The response here,

509
00:39:36,000 --> 00:39:39,000
V here.
Am I missing anything here?

510
00:39:39,000 --> 00:39:43,000
Is that it?
Now, don't all yell at once.

511
00:39:43,000 --> 00:39:47,000
What am I missing?
Current source i,

512
00:39:47,000 --> 00:39:50,000
exactly.
So if I have a current source i

513
00:39:50,000 --> 00:39:55,000
then there's an effect of this
current as well.

514
00:39:55,000 --> 00:40:00,000
And so I write down i there,
too.

515
00:40:00,000 --> 00:40:03,000
It's going to be some constant
multiplying I.

516
00:40:03,000 --> 00:40:07,000
And that constant is going to
look like a resistor,

517
00:40:07,000 --> 00:40:11,000
right, because this circuit
contains current sources,

518
00:40:11,000 --> 00:40:15,000
voltage sources and resistors.
If I've shorted all my voltage

519
00:40:15,000 --> 00:40:19,000
sources and opened all my
current sources,

520
00:40:19,000 --> 00:40:22,000
what's left in here?
Just a whole caboodle full of

521
00:40:22,000 --> 00:40:25,000
Rs.
It's just going to look like

522
00:40:25,000 --> 00:40:30,000
some resistance R.
And that's what I get here.

523
00:40:30,000 --> 00:40:35,000
So this is what V is going to
look like and that's a form.

524
00:40:35,000 --> 00:40:40,000
So let's take a look at these
components.

525
00:40:45,000 --> 00:40:47,000
Let's focus on the easy part
first.

526
00:40:47,000 --> 00:40:51,000
What does this look like?
This component looks like an I,

527
00:40:51,000 --> 00:40:54,000
it looks like a current and has
some resistance.

528
00:40:54,000 --> 00:40:57,000
What is that resistance given
by?

529
00:40:57,000 --> 00:41:01,000
Supposing I gave you this
network and this currency source

530
00:41:01,000 --> 00:41:05,000
and I asked you tell me R.
How would you measure R?

531
00:41:05,000 --> 00:41:09,000
What you would do is open all
the current sources,

532
00:41:09,000 --> 00:41:14,000
short all the voltage sources,
put a ohmmeter in there and

533
00:41:14,000 --> 00:41:17,000
measure the resistance R.
That's R.

534
00:41:17,000 --> 00:41:21,000
OK, so we understand this term.
What about this term here?

535
00:41:21,000 --> 00:41:25,000
Can someone tell me the units
of this term here,

536
00:41:25,000 --> 00:41:27,000
this big thing here?
Voltage.

537
00:41:27,000 --> 00:41:32,000
This is a voltage.
This is a voltage.

538
00:41:32,000 --> 00:41:37,000
iR is a voltage.
So this does behave like a

539
00:41:37,000 --> 00:41:40,000
voltage.
And it behaves like some

540
00:41:40,000 --> 00:41:45,000
voltage V.
So notice that as far as this

541
00:41:45,000 --> 00:41:52,000
current I is concerned the rest
of the universe looks like a

542
00:41:52,000 --> 00:42:00,000
resistor and a voltage source
behaving in some manner.

543
00:42:00,000 --> 00:42:04,000
And let me just call it Vth for
now, and you'll know why in a

544
00:42:04,000 --> 00:42:05,000
second.

545
00:42:21,000 --> 00:42:26,000
The voltage has a form,
some voltage plus Ri.

546
00:42:26,000 --> 00:42:32,000
So, in other words,
as far as this I is concerned

547
00:42:32,000 --> 00:42:39,000
this whole network here N full
of all the nice stuff is

548
00:42:39,000 --> 00:42:43,000
indistinguishable to this I
here.

549
00:42:43,000 --> 00:42:50,000
So my I is sitting out there
injecting a current into two

550
00:42:50,000 --> 00:42:55,000
nodes.
If I am i, I'm looking at this,

551
00:42:55,000 --> 00:43:03,000
this network looks no different
than a voltage source in series

552
00:43:03,000 --> 00:43:11,000
with the resistor R.
Notice that the equation for

553
00:43:11,000 --> 00:43:18,000
this simple circuit is this,
so I is given by V minus Vth

554
00:43:18,000 --> 00:43:22,000
divided by R.
Just remember.

555
00:43:30,000 --> 00:43:32,000
It's a circuit.
In other words,

556
00:43:32,000 --> 00:43:37,000
Agarwal sitting here cannot
tell the difference if I'm

557
00:43:37,000 --> 00:43:42,000
measuring the voltage here
between a circuit that looks

558
00:43:42,000 --> 00:43:47,000
like a Vth in series to the
resistor or this huge mess of

559
00:43:47,000 --> 00:43:51,000
voltage sources and current
sources and so on.

560
00:43:51,000 --> 00:43:54,000
Now, we will talk about Vth and
R.

561
00:43:54,000 --> 00:43:59,000
R is called the resistance of
the network as seen from the

562
00:43:59,000 --> 00:44:04,000
port with all the sources shut
off.

563
00:44:04,000 --> 00:44:06,000
And similarly Vth,
what is Vth?

564
00:44:06,000 --> 00:44:09,000
Vth is the open circuit
voltage.

565
00:44:09,000 --> 00:44:12,000
In other words,
if I apply the voltage here

566
00:44:12,000 --> 00:44:17,000
this is the response of all the
current sources and all the

567
00:44:17,000 --> 00:44:23,000
voltage sources acting together.
So it's as if I took this out

568
00:44:23,000 --> 00:44:27,000
and simply measured my V here as
if I didn't exist,

569
00:44:27,000 --> 00:44:31,000
correct?
Because this is the component

570
00:44:31,000 --> 00:44:35,000
of i.
So if I opened i and measured

571
00:44:35,000 --> 00:44:40,000
V, I would get that big term on
the left-hand side.

572
00:44:40,000 --> 00:44:43,000
That's my Vth.
So that inspires the next

573
00:44:43,000 --> 00:44:48,000
method called the Thevenin
method.

574
00:44:58,000 --> 00:45:04,000
In this method what I'm going
to do is take some circuit,

575
00:45:04,000 --> 00:45:07,000
I'm on Page 9,
with a mess of stuff.

576
00:45:07,000 --> 00:45:13,000
It's a big mess of stuff.
And if I care to look at its

577
00:45:13,000 --> 00:45:20,000
impact on something else that I
add from the outside then as far

578
00:45:20,000 --> 00:45:25,000
as the outside world is
concerned this is

579
00:45:25,000 --> 00:45:32,000
indistinguishable from a circuit
that looks like this.

580
00:45:45,000 --> 00:45:49,000
So what I can do is if I want
to figure out what's happening

581
00:45:49,000 --> 00:45:54,000
here then, for the purpose of my
analysis, this simple network

582
00:45:54,000 --> 00:45:59,000
here with R and Vth becomes a
surrogate for this entire mess.

583
00:45:59,000 --> 00:46:03,000
So for the purpose of finding
out the behavior at this point,

584
00:46:03,000 --> 00:46:07,000
I can take this huge mess and
replace it with its Thevenin

585
00:46:07,000 --> 00:46:10,000
surrogate or Thevenin
equivalent.

586
00:46:10,000 --> 00:46:14,000
This is called the Thevenin
equivalent of this big network.

587
00:46:14,000 --> 00:46:18,000
Let me do an example that will
make the method completely

588
00:46:18,000 --> 00:46:21,000
clear.
Again, remember in EECS,

589
00:46:21,000 --> 00:46:25,000
most of our lives are about how
can we make things so simple as

590
00:46:25,000 --> 00:46:30,000
being able to be analyzed by
inspection?

591
00:46:30,000 --> 00:46:36,000
And so this is a method that
takes you further down that

592
00:46:36,000 --> 00:46:40,000
path.
So let me use the same circuit

593
00:46:40,000 --> 00:46:45,000
that I've been using before,
my voltage V,

594
00:46:45,000 --> 00:46:48,000
R1, R2.
This is an R.

595
00:46:48,000 --> 00:46:55,000
I'm 55 minutes fast so we have
another three or four minutes.

596
00:46:55,000 --> 00:47:01,000
So this is my circuit.
And let's say all I care about

597
00:47:01,000 --> 00:47:05,000
is finding out i1.
That's all I care about.

598
00:47:05,000 --> 00:47:10,000
And what I'm going to do is I'm
going to box this up and see if

599
00:47:10,000 --> 00:47:14,000
I can replace that with its
Thevenin equivalent.

600
00:47:14,000 --> 00:47:17,000
So I'm going to box that up.

601
00:47:35,000 --> 00:47:39,000
What I'm saying is that I'm
going to box it up and replace

602
00:47:39,000 --> 00:47:42,000
it with this Thevenin
equivalent.

603
00:47:42,000 --> 00:47:45,000
I don't know what Vth and R are
at this point.

604
00:47:45,000 --> 00:47:48,000
I'm just calling it Rth for
fun.

605
00:47:48,000 --> 00:47:52,000
I don't know what these two
values are, but if I knew what

606
00:47:52,000 --> 00:47:57,000
these two values were I can
determine I really trivially as

607
00:47:57,000 --> 00:48:00,000
follows.
I can get i1 as simply V minus

608
00:48:00,000 --> 00:48:06,000
Vth divided by R1 plus Rth.
So if I knew Vth and Rth,

609
00:48:06,000 --> 00:48:11,000
I can write down i1 by
inspection in that manner.

610
00:48:11,000 --> 00:48:16,000
So next, finally,
how do I get Vth and Rth?

611
00:48:16,000 --> 00:48:22,000
You get Rth by looking at this
network and shutting off all the

612
00:48:22,000 --> 00:48:29,000
voltage sources and measuring
the resistance there.

613
00:48:29,000 --> 00:48:36,000
So I short my voltage source,
that's R1.

614
00:48:36,000 --> 00:48:43,000
Oops, wrong way.
I need to look this way.

615
00:48:43,000 --> 00:48:51,000
So looking this way,
that's what I get.

616
00:48:51,000 --> 00:48:56,000
So what's Rth?
Rth is simply R2.

617
00:48:56,000 --> 00:49:05,000
So I have opened my current
source.

618
00:49:05,000 --> 00:49:09,000
Similarly, for Vth,
remember all I want to do is

619
00:49:09,000 --> 00:49:13,000
look at the two nodes,
step back, put a voltmeter

620
00:49:13,000 --> 00:49:18,000
there, measure the voltage,
that's my open circuit voltage.

621
00:49:18,000 --> 00:49:24,000
So the way I do it is I take
the circuit and simply measure

622
00:49:24,000 --> 00:49:26,000
the voltage there.
That's R2.

623
00:49:26,000 --> 00:49:32,000
That's my current capital I.
And I simply want to measure

624
00:49:32,000 --> 00:49:37,000
the open circuit voltage here,
which is what?

625
00:49:37,000 --> 00:49:44,000
Just simply if I stand back and
I kind of gingerly measure the

626
00:49:44,000 --> 00:49:50,000
voltage here without disturbing
anything, I simply get IR2.

627
00:49:50,000 --> 00:49:56,000
So Vth is IR2 and Rth is R2 and
here is the formula for the

628
00:49:56,000 --> 00:50:02,000
current in this branch when I
apply a voltage source and a

629
00:50:02,000 --> 00:50:08,000
resistor R1 to this little
circuit here.

630
00:50:08,000 --> 00:50:12,000
OK, let's pause and let me
summarize this in about ten

631
00:50:12,000 --> 00:50:14,000
seconds.
I had this circuit here.

632
00:50:14,000 --> 00:50:18,000
I wanted to find out i1.
So what I said I'd do is take

633
00:50:18,000 --> 00:50:22,000
this complicated mess,
well, it's not a complicated

634
00:50:22,000 --> 00:50:26,000
mess but assume it is,
and replace with it a

635
00:50:26,000 --> 00:50:31,000
resistance Rth got by turning
off all the sources.

636
00:50:31,000 --> 00:50:35,000
And the voltage in series,
Vth, which I get simply by

637
00:50:35,000 --> 00:50:38,000
pulling this thing out,
taking my input,

638
00:50:38,000 --> 00:50:42,000
this part out and simply
measuring the open circuit

639
00:50:42,000 --> 00:50:43,000
voltage out there,
Vth.

640
00:50:43,000 --> 00:50:48,000
And then I replaced the whole
network with this new network

641
00:50:48,000 --> 00:50:51,000
that they call the Thevenin
network, and voila,

642
00:50:51,000 --> 00:50:54,000
I get the answer in a second.