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PROFESSOR: So today I want to
start a new topic, circuits.

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No, that's good.

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That's good.

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Circuits are good.

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AUDIENCE: Yay.

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PROFESSOR: Thank
you, thank you.

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Much better.

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So just to provide some
perspective, I want to remind

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you where we are, how we got
here, and where we're going.

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So at the beginning of the
course, we promised that there

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were several intellectual
themes that

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we would talk about.

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Probably the most important
one there is

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designing complex systems.

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That's what we're
really about.

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We would like you to
be able to make

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very complicated systems.

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How do you think about parts?

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How do you think about
connecting them?

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How do you think about things
when you want to make

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something that's very
complicated?

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Part of that is modeling.

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We just finished a module
on modeling.

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So in order to make a complex
system, we'd like to be able

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to predict how it will behave
before we completely build it.

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Sometimes it's impossible
to build the

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entire system before.

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Sometimes it's impossible
to build prototypes.

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Sometimes you're stuck with
going with the design at

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launch, and figuring
out how it works.

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In those cases in specific, it's
very important to be able

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to model it, to have some
confidence that the thing's

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going to work.

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We're going to talk about
augmenting physical systems

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with computation.

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That's the module we're
about to begin.

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And we'll conclude by talking
about how to build systems

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that are robust to change.

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So we started with the idea
of, how do you make

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complicated systems?

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And we introduced this notion
of primitives, combination,

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abstraction, and pattern in
terms of software engineering.

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We did that because that's the
simplest possible way of

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getting started.

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It provided a very good
illustration of PCAP at the

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low level by thinking about
Python, by thinking about the

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primitive structures that Python
gives you, how those

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can be combined, how you can
abstract, how you can

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recognize patterns.

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But then we also built a higher
level abstraction,

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which was the state
machine idea.

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There the idea was you didn't
have to have state machines in

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order to build the brain for a
robot, but it actually turns

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out to be easy if you
do because there's

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a modularity there.

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You can figure out if each part
works independent of the

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other parts.

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And then you can be pretty sure
when you put it together

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the whole thing's
going to work.

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So that was kind of our
introduction to

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this notion of PCAP.

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Then we went on to think about
signals and systems.

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And that was kind of our
introduction to modeling.

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How do you make a model
of something

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that predicts behavior?

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So we transitioned from thinking
about, how do you

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structure a design to how do
you think about behavior.

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Today we're going to start
to think about circuits.

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Circuits are really going to a
more primitive physical layer.

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How do you think about actually
making a device?

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So the device that we'll think
about is a thing to augment

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the capabilities --

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the sensor capabilities
of the robot.

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We'll think about making a
light tracking system.

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And the idea is going to be
that you'll build a head.

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The head has a neck, so you'll
have to control the neck.

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The head has eyes.

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It will mount on top of the
robot, and that'll let you

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drive the robot around
looking for light.

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And you'll do that by
designing a circuit.

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So that's kind of the game plan
for the next three weeks.

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So today, what I want to do
is introduce the notion of

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circuits, introduce
the theory for how

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we think about circuits.

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And then in the two labs this
week, the idea will be to

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become familiar with the
practice of circuits.

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How do you actually
build something?

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How do you actually make
something work?

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So today's the theory.

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So the theory, the idea in
circuits is to think about a

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physical system as the
interconnection of parts and

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rules that connect them.

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In fact, the rules fall
into two categories.

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We're going to think about the
currents that go through parts

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and the voltage that develops
across parts.

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And we'll see that there's a
way of thinking about the

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behavior of the entire circuit
that integrates all of those

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three pieces.

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How does the part work?

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How do the currents that go
through the part work?

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And how do the voltages
that are produced

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across the part --

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how do they work?

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So I'll just start with two
very simple examples.

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The first is the most trivial
example you can think about.

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You can think about a flashlight
as a circuit.

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You close the switch,
current flows.

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Very simple idea.

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We will make a model of that
that looks like this.

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We'll think about the battery
being a source.

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In this case, a voltage
source.

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We'll think about the light
bulb being a resistor.

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We'll have two parts.

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We'll have to know the
current-voltage relationships

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for both of the parts.

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And we'll have to know the
ramifications for those

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currents and voltages when
you put them together.

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Very, very simple.

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The other simple example that
I want to illustrate is this

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idea of a leaky tank.

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Here the idea that I want to get
across is that the circuit

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idea is quite general.

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When we talk about circuits,
we almost always talk about

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electronic circuits.

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But the theory is by no means
limited to electronics.

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So for example, if we think
about a leaky tank, we think

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about a pipe spewing water
into a reservoir.

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Maybe that's the Cambridge
Reservoir.

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Maybe that's the water coming
out of the Woburn Reservoir.

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Maybe that's the demand put on
by the Cambridge people trying

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to take showers in
the morning --

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I don't know.

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So we think about flow
into a tank, a

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reservoir, and flow out.

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And we can make a model for that
in terms of a circuit.

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In the circuit there
are through

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variables and across variables.

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In an electronic circuit, the
through variable is current.

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Here the through variable
is the flow rate.

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So it's the flow of water in
and the flow of water out,

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represented here
by these things

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that look like currents.

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And the across variable for an
electronic device is voltage.

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The across variable for this
kind of a fluidic device, the

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across variable is pressure.

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So we think as this thing gets
ahead of that thing, as

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there's more stuff coming in
than going out, the height

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goes up and the pressure
builds.

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Same idea.

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So the point is that we'll
develop the theory for

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circuits in terms
of electronics.

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But you should keep in the back
of your mind that it's a

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lot more general idea than
just electronics.

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In fact, there are two
completely distinct reasons

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why we even bother
with circuits.

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One is that they're very
important to physical systems.

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If you're designing a power
network, of course you have to

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think about the way the power
network, the power grid works

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as a circuit.

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That's obvious.

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In electronics, of course, if
you're going to build a cell

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phone, you have to know how
the parts interconnect

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electronically.

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That's obvious.

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But probably the biggest use for
circuits these days is not

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those applications, although
those are very important.

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Circuits are also used
as models of things.

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So many models for complex
behaviors are in

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fact circuit models.

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So in terms of electronics, the
idea is that we want to

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get on top of electronics.

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We want to understand how
circuits work, so we can

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understand things like that.

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If you look at how complex
processors have got over my

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professional life, we start with
my professional life down

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at about 1,000 transistors
per processor.

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And today, we're up at
about a billion.

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That's enormous.

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Even in the stone ages when we
were designing things that had

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a thousand parts, we still had
trouble thinking about those

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thousands parts all at once.

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We still need PCAP.

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We still needed ways of
combining the activities of

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many things into a conceptual
unit that was bigger.

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Here, it's just impossible
if you don't have that.

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So that's one of the reasons
we study circuits.

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And the other reason is here.

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So here I'm showing a
model for the way

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a nerve cell works.

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This model is taken
from 6.021.

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The idea, this comes from the
study of the Hodgkin-Huxley

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00:09:15,280 --> 00:09:19,810
model for neural conduction,
arguably the most successful

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00:09:19,810 --> 00:09:23,020
mathematical theory
in biophysics.

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Which explains the completely
non-trivial relationship

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between how the parts from
biology works and the behavior

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00:09:33,260 --> 00:09:37,460
in terms of propagated action
potentials works.

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So the idea is that we
understand how this biological

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system works because
we think about it

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in terms of a circuit.

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That's the only successful way
we have to think about that.

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So what I want to do then is
spend today figuring out

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circuits At the very most
primitive level.

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The level that I'm going to
talk about in terms of

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00:10:01,330 --> 00:10:04,530
circuits is roughly analogous
to the level that we talked

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00:10:04,530 --> 00:10:09,020
about with Python when we were
thinking about how Python

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00:10:09,020 --> 00:10:14,590
provides utilities for
primitives, combinations,

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00:10:14,590 --> 00:10:17,260
abstractions, and patterns.

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00:10:17,260 --> 00:10:19,220
So I'm going to start at the
very lowest level and think

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00:10:19,220 --> 00:10:23,290
about, what are the basic
primitives, the smallest units

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00:10:23,290 --> 00:10:25,740
we'll ever think about
in terms of circuits?

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00:10:25,740 --> 00:10:29,450
And what are the rules by
which we combine them?

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So I'll start with the very
simplest ideas, the very

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00:10:33,090 --> 00:10:35,020
simplest elements.

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We will oversimplify things
and think about the very

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00:10:37,720 --> 00:10:41,010
simplest kind of electronic
elements as resistors that

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00:10:41,010 --> 00:10:42,640
obey Ohm's Law, V equals iR.

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Voltage sources, things that
maintain a constant voltage

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regardless of what you do.

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00:10:50,740 --> 00:10:53,290
And current sources, things
that maintain a constant

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00:10:53,290 --> 00:10:56,810
current regardless
of what you do.

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These things are, as I said,
analogous to the primitive

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00:10:59,640 --> 00:11:00,890
things that we looked
at in Python.

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00:11:00,890 --> 00:11:02,750
They're also analogous to the
primitive things that we

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00:11:02,750 --> 00:11:05,470
looked at in system functions.

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00:11:05,470 --> 00:11:07,900
Can somebody think of, when
we were doing difference

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00:11:07,900 --> 00:11:10,880
equations, what were the
primitives that we

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00:11:10,880 --> 00:11:12,680
started with --

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00:11:12,680 --> 00:11:15,770
when we started to study
difference equations?

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00:11:15,770 --> 00:11:17,260
What's the most primitive
elements

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00:11:17,260 --> 00:11:18,510
that we thought about?

238
00:11:20,937 --> 00:11:22,320
AUDIENCE: Delay.

239
00:11:22,320 --> 00:11:23,340
PROFESSOR: Delay, yeah.

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00:11:23,340 --> 00:11:24,740
So we thought about
things like--

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00:11:27,390 --> 00:11:28,640
so we had delay.

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00:11:31,810 --> 00:11:32,600
Anything else?

243
00:11:32,600 --> 00:11:33,330
AUDIENCE: Gain.

244
00:11:33,330 --> 00:11:34,300
PROFESSOR: Gain.

245
00:11:34,300 --> 00:11:35,550
Anything else?

246
00:11:38,463 --> 00:11:40,960
Add.

247
00:11:40,960 --> 00:11:44,130
So we had exactly three
primitives.

248
00:11:44,130 --> 00:11:47,850
And we got pretty far with
those three primitives.

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00:11:47,850 --> 00:11:51,000
We learned the rules for
interconnection.

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00:11:51,000 --> 00:11:52,340
We didn't really make a
big deal out of it.

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00:11:52,340 --> 00:11:54,320
We didn't formalize it, but the
rules for interconnection

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00:11:54,320 --> 00:11:58,430
were something like
every node has to

253
00:11:58,430 --> 00:12:01,810
have exactly one generator.

254
00:12:01,810 --> 00:12:03,830
You can't connect the
output of this to

255
00:12:03,830 --> 00:12:06,890
this, that's illegal.

256
00:12:06,890 --> 00:12:09,220
Every node has to
have one source.

257
00:12:09,220 --> 00:12:12,010
And every node can source
lots of inputs.

258
00:12:12,010 --> 00:12:14,050
That was kind of the rules
of the interconnect.

259
00:12:14,050 --> 00:12:15,505
The interconnects here will be a
little bit more complicated.

260
00:12:18,340 --> 00:12:20,960
So those are the elements
that we'll think about.

261
00:12:20,960 --> 00:12:23,370
And the first step's going to be
to think about, how do they

262
00:12:23,370 --> 00:12:25,800
interconnect?

263
00:12:25,800 --> 00:12:29,160
The simplest possible
interconnections are trivial.

264
00:12:29,160 --> 00:12:31,580
In the case of the battery, you
hook up the voltage source

265
00:12:31,580 --> 00:12:33,280
to the resistor.

266
00:12:33,280 --> 00:12:35,100
The voltage source makes
the voltage across

267
00:12:35,100 --> 00:12:37,220
this resistor 1 Volt.

268
00:12:37,220 --> 00:12:39,910
If we say the resistor is 1
Ohm, then there's 1 Amp

269
00:12:39,910 --> 00:12:40,745
current period.

270
00:12:40,745 --> 00:12:41,510
Done.

271
00:12:41,510 --> 00:12:43,750
Easy.

272
00:12:43,750 --> 00:12:47,160
Similarly, if we were to hook
up the resistor to a current

273
00:12:47,160 --> 00:12:49,120
source, we would get something
equally easy.

274
00:12:49,120 --> 00:12:51,060
Except now the current source
would guarantee that the

275
00:12:51,060 --> 00:12:56,450
current through the resistor
is an amp.

276
00:12:56,450 --> 00:12:59,360
Therefore, the voltage across
the resistor, by Ohm's law,

277
00:12:59,360 --> 00:13:00,700
would be a Volt.

278
00:13:00,700 --> 00:13:02,830
So we would end up with the same
solution for a completely

279
00:13:02,830 --> 00:13:04,940
different reason.

280
00:13:04,940 --> 00:13:06,340
Here the voltage
is constrained.

281
00:13:06,340 --> 00:13:07,590
Here the current's
constrained.

282
00:13:10,390 --> 00:13:16,090
Just to make sure everybody's
with me, figure out, what's

283
00:13:16,090 --> 00:13:19,450
the current i that goes
through this resistor?

284
00:13:19,450 --> 00:13:22,040
Slightly more complicated
system.

285
00:13:22,040 --> 00:13:24,120
Take 20 seconds, talk to your
neighbor, figure out a number

286
00:13:24,120 --> 00:13:25,820
between (1) and (5).

287
00:15:23,520 --> 00:15:24,690
OK, so what's the answer?

288
00:15:24,690 --> 00:15:27,630
Everybody raise your hand with
a number (1) through (5).

289
00:15:27,630 --> 00:15:28,980
Come on, everybody vote.

290
00:15:28,980 --> 00:15:29,920
Come on.

291
00:15:29,920 --> 00:15:32,020
You can blame it on your
neighbor, that's the rules.

292
00:15:32,020 --> 00:15:33,790
You talk to your neighbor,
then you can blame dumb

293
00:15:33,790 --> 00:15:36,150
answers on your neighbor.

294
00:15:36,150 --> 00:15:41,100
OK, about 80% correct I'd say.

295
00:15:41,100 --> 00:15:42,410
So how do you think
about this?

296
00:15:42,410 --> 00:15:43,450
What's going to be
the current?

297
00:15:43,450 --> 00:15:44,420
How would you calculate
the current?

298
00:15:44,420 --> 00:15:45,670
What do I do first?

299
00:15:48,462 --> 00:15:48,930
Shout.

300
00:15:48,930 --> 00:15:52,772
If you shout, and especially
if my head's turned away I

301
00:15:52,772 --> 00:15:55,260
don't know who you are.

302
00:15:55,260 --> 00:15:56,200
AUDIENCE: Kirchhoff's law.

303
00:15:56,200 --> 00:15:57,240
PROFESSOR: Kirchoff's
law, wonderful.

304
00:15:57,240 --> 00:15:57,717
Which one?

305
00:15:57,717 --> 00:16:00,102
There's two of them.

306
00:16:00,102 --> 00:16:02,964
AUDIENCE: [UNINTELLIGIBLE].

307
00:16:02,964 --> 00:16:04,130
PROFESSOR: [UNINTELLIGIBLE].

308
00:16:04,130 --> 00:16:05,663
What loop do you want to use?

309
00:16:05,663 --> 00:16:06,430
AUDIENCE: Left.

310
00:16:06,430 --> 00:16:07,220
PROFESSOR: Left side.

311
00:16:07,220 --> 00:16:10,960
So if we use the left side loop,
we would conclude that

312
00:16:10,960 --> 00:16:12,310
there's a volt across
the resistor.

313
00:16:12,310 --> 00:16:15,288
So the current would be?

314
00:16:15,288 --> 00:16:16,764
AUDIENCE: 1 Amp.

315
00:16:16,764 --> 00:16:17,748
PROFESSOR: An amp.

316
00:16:17,748 --> 00:16:21,684
Where's the current come from?

317
00:16:21,684 --> 00:16:23,160
AUDIENCE: The voltage.

318
00:16:23,160 --> 00:16:28,080
PROFESSOR: The voltage source
just like before.

319
00:16:28,080 --> 00:16:29,880
So not quite.

320
00:16:29,880 --> 00:16:32,210
So the voltage source
establishes this

321
00:16:32,210 --> 00:16:33,710
voltage would be 1.

322
00:16:33,710 --> 00:16:36,850
That makes this current be 1.

323
00:16:36,850 --> 00:16:39,220
That would be consistent with
the current coming out of

324
00:16:39,220 --> 00:16:42,250
here, except we have to also
think about that 1 Amp source.

325
00:16:44,810 --> 00:16:46,780
So the question is, what's
does the 1 Amp source do?

326
00:16:46,780 --> 00:16:48,370
Nothing?

327
00:16:48,370 --> 00:16:52,240
It's just there sort of
for decoration or for

328
00:16:52,240 --> 00:16:55,170
[UNINTELLIGIBLE] so that we
can make an interesting

329
00:16:55,170 --> 00:16:56,580
question to ask in lecture?

330
00:16:56,580 --> 00:16:57,830
Maybe.

331
00:16:59,870 --> 00:17:01,040
So where's the current?

332
00:17:01,040 --> 00:17:02,689
Where's the 1 Amp that
goes through the

333
00:17:02,689 --> 00:17:05,184
resistor come from?

334
00:17:05,184 --> 00:17:07,180
AUDIENCE: [UNINTELLIGIBLE]
on the right.

335
00:17:07,180 --> 00:17:09,176
PROFESSOR: It comes
from the right.

336
00:17:09,176 --> 00:17:11,080
It comes from the current
over here.

337
00:17:11,080 --> 00:17:13,130
So the idea is that if this
current, ignore the voltage

338
00:17:13,130 --> 00:17:14,150
over here for the moment.

339
00:17:14,150 --> 00:17:16,560
If this current flowed through
the resistor, then you'd have

340
00:17:16,560 --> 00:17:19,130
1 Amp going through there, and
you'd have 1 Volt generated by

341
00:17:19,130 --> 00:17:22,069
that current which just happens
to be exactly the

342
00:17:22,069 --> 00:17:25,900
right voltage to match
the voltage

343
00:17:25,900 --> 00:17:28,930
from the voltage source.

344
00:17:28,930 --> 00:17:31,640
So if you think about this, the
voltage guarantees that

345
00:17:31,640 --> 00:17:35,720
this is 1 Volt, but so
does the current.

346
00:17:35,720 --> 00:17:38,980
In order to simultaneously
satisfy everything, all you

347
00:17:38,980 --> 00:17:41,310
need to do is have all of this
current go around and come

348
00:17:41,310 --> 00:17:44,240
down through that resistor.

349
00:17:44,240 --> 00:17:47,070
That will generate the volt, so
there's no propensity for

350
00:17:47,070 --> 00:17:50,020
more current to flow out of the
source because the source

351
00:17:50,020 --> 00:17:54,580
is 1 Volt and it's facing a
circuit that's already 1 Volt.

352
00:17:54,580 --> 00:17:56,310
So the idea was to try to give
you something that's

353
00:17:56,310 --> 00:18:00,260
relatively simple that you can
think through on your own, but

354
00:18:00,260 --> 00:18:01,530
not trivial.

355
00:18:01,530 --> 00:18:03,060
So the answer was 1 Amp.

356
00:18:03,060 --> 00:18:05,870
But the 1 Amp was not for
the trivial reason.

357
00:18:05,870 --> 00:18:10,280
The 1 Amp is because the current
from the right flows

358
00:18:10,280 --> 00:18:14,470
through the resistor and
makes the voltage be 1.

359
00:18:14,470 --> 00:18:15,900
So the right answer is 1.

360
00:18:15,900 --> 00:18:17,290
But for the reason that
you might not

361
00:18:17,290 --> 00:18:19,490
have originally thought.

362
00:18:19,490 --> 00:18:22,140
But more importantly, I wanted
to use that as a motivation

363
00:18:22,140 --> 00:18:25,013
for thinking about, how do we
think about bigger circuits?

364
00:18:27,670 --> 00:18:30,240
So when the simple circuit,
like two parts, it's no

365
00:18:30,240 --> 00:18:32,260
problem figuring out what the
answer's going to be.

366
00:18:32,260 --> 00:18:35,430
But when the circuit has even
three parts, it may require

367
00:18:35,430 --> 00:18:36,290
more thinking.

368
00:18:36,290 --> 00:18:38,480
And you may want to have a
more structured way of

369
00:18:38,480 --> 00:18:39,630
thinking about the solution.

370
00:18:39,630 --> 00:18:40,487
Yes?

371
00:18:40,487 --> 00:18:43,409
AUDIENCE: What would have
occurred if the current

372
00:18:43,409 --> 00:18:45,844
provider on the right
side was 2 Amps?

373
00:18:45,844 --> 00:18:47,792
PROFESSOR: Great question.

374
00:18:47,792 --> 00:18:51,660
Had this been 2 Amps, you can't
violate this voltage.

375
00:18:51,660 --> 00:18:54,570
So that would have
been 1 Volt.

376
00:18:54,570 --> 00:18:57,790
So that would have been 1 Amp
through the resistor.

377
00:18:57,790 --> 00:18:59,560
So then you're left with the
problem with this guy's

378
00:18:59,560 --> 00:19:03,330
pushing 2 and that guy's
only eating 1.

379
00:19:03,330 --> 00:19:06,660
But the rules for the voltage
source say eat or source

380
00:19:06,660 --> 00:19:10,140
however much current is
necessary in order to make the

381
00:19:10,140 --> 00:19:11,900
voltage equal to 1.

382
00:19:11,900 --> 00:19:16,070
So the excess amp goes through
the voltage source.

383
00:19:16,070 --> 00:19:20,400
So the voltage source is, in
fact, being supplied power

384
00:19:20,400 --> 00:19:23,350
rather than supplying
power itself.

385
00:19:23,350 --> 00:19:27,400
Had this been 2 Amps, some of
the power from this source

386
00:19:27,400 --> 00:19:30,060
would have gone into
the resistor.

387
00:19:30,060 --> 00:19:31,730
And some of the power from
this source would have

388
00:19:31,730 --> 00:19:34,030
actually gone into the
voltage source.

389
00:19:34,030 --> 00:19:38,150
So if the voltage source were,
for example, a model for a

390
00:19:38,150 --> 00:19:41,250
rechargeable battery,
that rechargeable

391
00:19:41,250 --> 00:19:43,720
battery would be charging.

392
00:19:43,720 --> 00:19:44,905
Does that make sense?

393
00:19:44,905 --> 00:19:47,240
So if there had been a mismatch
in the conditions,

394
00:19:47,240 --> 00:19:52,210
you still have to satisfy all
the relationships from all of

395
00:19:52,210 --> 00:19:54,470
the sources.

396
00:19:54,470 --> 00:19:56,770
AUDIENCE: What if the
voltage was larger?

397
00:19:56,770 --> 00:19:58,609
PROFESSOR: The same thing
would have happened.

398
00:19:58,609 --> 00:20:01,265
Except now the flowing
current would be in

399
00:20:01,265 --> 00:20:01,990
the opposite direction.

400
00:20:01,990 --> 00:20:04,620
Let's say that if the voltage
here had been 2 Volts, then

401
00:20:04,620 --> 00:20:07,530
the voltage would have required
that there is 2 Amps

402
00:20:07,530 --> 00:20:09,730
flowing here.

403
00:20:09,730 --> 00:20:13,590
1 Amp would come from here,
but another Amp

404
00:20:13,590 --> 00:20:15,130
would come from here.

405
00:20:15,130 --> 00:20:18,020
This voltage source will supply
whatever current is

406
00:20:18,020 --> 00:20:21,521
necessary to make its
voltage law real.

407
00:20:26,420 --> 00:20:26,495
Ok.

408
00:20:26,495 --> 00:20:30,570
In fact, what we'll do now is
turn toward a discussion of

409
00:20:30,570 --> 00:20:33,960
more complicated systems that
will let you go back and in

410
00:20:33,960 --> 00:20:37,430
retrospect, analyze all those
cases that we just did.

411
00:20:37,430 --> 00:20:39,240
And you'll be able to see
trivially how that has to be

412
00:20:39,240 --> 00:20:40,460
the right answer.

413
00:20:40,460 --> 00:20:46,460
So what I want to do now is
generate a formal structure

414
00:20:46,460 --> 00:20:47,960
for how you would
solve circuits.

415
00:20:47,960 --> 00:20:49,352
Yes?

416
00:20:49,352 --> 00:20:53,240
AUDIENCE: So did we know of
anything that could generate a

417
00:20:53,240 --> 00:20:56,800
current without generating a
voltage, like in real life?

418
00:20:56,800 --> 00:20:58,310
PROFESSOR: Can anything generate
a current without

419
00:20:58,310 --> 00:20:59,080
generating a voltage?

420
00:20:59,080 --> 00:21:01,130
That's a tricky question.

421
00:21:01,130 --> 00:21:05,280
If you think about something as
generating a current, then

422
00:21:05,280 --> 00:21:09,700
the voltage is not necessarily
determined by that part.

423
00:21:09,700 --> 00:21:12,030
So that's kind of illustrated
here.

424
00:21:12,030 --> 00:21:17,000
If this guy is generating a
current, this guy is not

425
00:21:17,000 --> 00:21:21,755
actually the element that is
controlling its own voltage.

426
00:21:24,560 --> 00:21:27,470
In general, if you want to speak
simultaneously about the

427
00:21:27,470 --> 00:21:32,000
current and voltage across the
device, you have to know what

428
00:21:32,000 --> 00:21:34,600
it was connected to.

429
00:21:34,600 --> 00:21:36,020
Each part--

430
00:21:36,020 --> 00:21:37,790
we'll get to this in a moment
in case some of you are

431
00:21:37,790 --> 00:21:39,940
worried about launching ahead.

432
00:21:39,940 --> 00:21:42,300
We will cover this.

433
00:21:42,300 --> 00:21:44,300
This is very good motivation for
figuring out what's going

434
00:21:44,300 --> 00:21:46,690
to happen in the next
three slides.

435
00:21:46,690 --> 00:21:54,320
So each part gets to tell you
one relationship between

436
00:21:54,320 --> 00:21:55,790
voltage and current.

437
00:21:55,790 --> 00:22:00,450
Generally speaking, that's not
enough to solve for voltage

438
00:22:00,450 --> 00:22:01,030
and current.

439
00:22:01,030 --> 00:22:03,200
Voltage and current is
like two unknowns.

440
00:22:06,300 --> 00:22:11,000
Each element relationship
is one equation.

441
00:22:11,000 --> 00:22:14,660
So the current source gets to
say current equals x, current

442
00:22:14,660 --> 00:22:16,000
equals 1 Amp.

443
00:22:16,000 --> 00:22:20,400
It doesn't get to tell you
what the voltage is.

444
00:22:20,400 --> 00:22:22,850
So being a little more physical
to try to address

445
00:22:22,850 --> 00:22:27,480
your question more physically,
there are processes that can

446
00:22:27,480 --> 00:22:30,970
be extremely well modeled
as current generators.

447
00:22:30,970 --> 00:22:35,120
In fact, many electronic
semiconductor parts, like

448
00:22:35,120 --> 00:22:37,890
transistors, work more like
a current source than like

449
00:22:37,890 --> 00:22:39,790
anything else.

450
00:22:39,790 --> 00:22:42,590
So there are devices that behave
as though they were

451
00:22:42,590 --> 00:22:46,570
current sources, but they don't
simultaneously get to

452
00:22:46,570 --> 00:22:49,390
tell you what is their current
and what is their voltage.

453
00:22:49,390 --> 00:22:51,245
They only get to tell you
what is their current.

454
00:22:55,960 --> 00:22:58,630
So let's think about now, if
you had a more complicated

455
00:22:58,630 --> 00:23:02,930
system, how could you
systematically go about

456
00:23:02,930 --> 00:23:05,580
finding the solution?

457
00:23:05,580 --> 00:23:07,430
As was mentioned earlier,
there's something called

458
00:23:07,430 --> 00:23:08,110
Kirchhoff's law.

459
00:23:08,110 --> 00:23:10,060
And in fact, there's
two of them.

460
00:23:10,060 --> 00:23:12,500
Kirchhoff's voltage law and
Kirchhoff's current law.

461
00:23:12,500 --> 00:23:18,280
Kirchhoff's voltage law, in its
most elementary form, says

462
00:23:18,280 --> 00:23:21,790
that if you trace the path
around any closed path in a

463
00:23:21,790 --> 00:23:24,450
circuit, regardless of
what the path is--

464
00:23:24,450 --> 00:23:26,040
every closed path--

465
00:23:26,040 --> 00:23:30,650
the sum of the voltages going
around that closed path is 0.

466
00:23:30,650 --> 00:23:35,830
So for example in this
circuit, the red path

467
00:23:35,830 --> 00:23:39,130
illustrates one closed path
through the circuit.

468
00:23:39,130 --> 00:23:41,680
It goes up through the voltage
source, down through this

469
00:23:41,680 --> 00:23:44,900
resistor, and then down
through that resistor.

470
00:23:44,900 --> 00:23:49,160
Kirchhoff's voltage law says the
sum of the voltages around

471
00:23:49,160 --> 00:23:51,990
that loop is 0.

472
00:23:51,990 --> 00:23:56,150
That's written mathematically
here, minus v1 for here, plus

473
00:23:56,150 --> 00:23:59,850
v2 for here, plus v4
for here is 0.

474
00:23:59,850 --> 00:24:01,660
OK, where do the signs
come from?

475
00:24:01,660 --> 00:24:05,570
The signs came from the
reference directions that we

476
00:24:05,570 --> 00:24:09,730
assigned arbitrarily
to the elements.

477
00:24:09,730 --> 00:24:13,280
Before I ever do a circuits
question, I always assign a

478
00:24:13,280 --> 00:24:15,520
reference direction.

479
00:24:15,520 --> 00:24:17,125
Every voltage has a
positive terminal

480
00:24:17,125 --> 00:24:19,120
and a negative terminal.

481
00:24:19,120 --> 00:24:22,210
And I must be consistent in
order to apply these rules.

482
00:24:22,210 --> 00:24:27,800
These rules only work if I
declare a reference direction

483
00:24:27,800 --> 00:24:29,320
and stick with it.

484
00:24:29,320 --> 00:24:31,350
If midway through a problem
I flip it, I'll

485
00:24:31,350 --> 00:24:33,810
get the wrong answer.

486
00:24:33,810 --> 00:24:36,220
So the minus sign has to do with
the fact that as I trace

487
00:24:36,220 --> 00:24:41,730
this path, I enter the minus
part of this guy, but the plus

488
00:24:41,730 --> 00:24:43,200
part of that guy and that guy.

489
00:24:43,200 --> 00:24:47,250
So the sign of v1 is negated
relative to the others.

490
00:24:50,220 --> 00:24:53,510
A different way to think about
that is here, we can think

491
00:24:53,510 --> 00:24:56,330
that v1 is the sum
of v2 and v4.

492
00:24:56,330 --> 00:24:58,700
That's sometimes more intuitive
because if you

493
00:24:58,700 --> 00:25:03,280
started here, going through
this path you would end up

494
00:25:03,280 --> 00:25:08,540
with a voltage that is v1 higher
than where you started.

495
00:25:08,540 --> 00:25:11,580
Whereas starting here, you would
end up with a voltage

496
00:25:11,580 --> 00:25:15,030
here that's v4 higher than
where you started.

497
00:25:15,030 --> 00:25:17,400
And then by the time you got to
here, it would be v2 plus

498
00:25:17,400 --> 00:25:21,970
v4 higher than where
you started.

499
00:25:21,970 --> 00:25:25,030
You start one place and on one
route, you end up v1 higher.

500
00:25:25,030 --> 00:25:28,030
And in the other route, you
get v4 plus v2 higher.

501
00:25:28,030 --> 00:25:31,120
So it must be the case that v1
is the same as v2 plus v4.

502
00:25:33,620 --> 00:25:35,710
Those are absolutely
equivalent ways of

503
00:25:35,710 --> 00:25:36,990
thinking about it.

504
00:25:36,990 --> 00:25:38,650
So those laws are equivalent.

505
00:25:38,650 --> 00:25:40,600
If you think about it
a path, you think

506
00:25:40,600 --> 00:25:43,305
about some of the paths--

507
00:25:43,305 --> 00:25:43,780
no.

508
00:25:43,780 --> 00:25:47,700
The path coinciding with the
negative direction of some of

509
00:25:47,700 --> 00:25:49,750
the elements and the positive
direction of others.

510
00:25:53,030 --> 00:25:58,720
OK, how many other
paths are there?

511
00:25:58,720 --> 00:26:00,090
Take 20 seconds, talk
to your neighbor.

512
00:26:00,090 --> 00:26:04,630
Figure out all of the
possible paths for

513
00:26:04,630 --> 00:26:06,120
which KVL has to apply.

514
00:28:35,850 --> 00:28:39,450
OK, so everybody raise your
hand and show a number of

515
00:28:39,450 --> 00:28:44,130
fingers equal to the number
of KVL equations less two.

516
00:28:49,060 --> 00:28:51,000
Oh, very good.

517
00:28:51,000 --> 00:28:52,170
Virtually 100% correct.

518
00:28:52,170 --> 00:28:53,420
Why do you all say (5)?

519
00:28:55,760 --> 00:28:57,760
Which is to say 7.

520
00:28:57,760 --> 00:28:59,010
Why do you all say 7?

521
00:29:01,970 --> 00:29:03,650
So there's 3 obvious ones.

522
00:29:03,650 --> 00:29:05,660
I was expecting a
couple of 3's.

523
00:29:05,660 --> 00:29:07,230
This was supposed to be--

524
00:29:07,230 --> 00:29:09,930
OK, yeah, I do plot
against you.

525
00:29:09,930 --> 00:29:12,410
I was expecting some 3's.

526
00:29:12,410 --> 00:29:14,770
So there's 3 obvious paths
that are analogous to the

527
00:29:14,770 --> 00:29:15,620
first one we looked at.

528
00:29:15,620 --> 00:29:18,980
If I call the first path A, then
there's B and C which are

529
00:29:18,980 --> 00:29:22,030
the excursions around here.

530
00:29:22,030 --> 00:29:23,850
And you can write the equations
just the same.

531
00:29:23,850 --> 00:29:26,030
They each involve
three voltages.

532
00:29:26,030 --> 00:29:28,620
And they each go through, some
starting at the negative side

533
00:29:28,620 --> 00:29:32,720
and some starting at
the positive side.

534
00:29:32,720 --> 00:29:35,230
So those are in some sense,
the obvious ones.

535
00:29:35,230 --> 00:29:39,040
But there are others too.

536
00:29:39,040 --> 00:29:42,370
So one way to think about it,
what I'd like you to do is

537
00:29:42,370 --> 00:29:44,290
enumerate all the paths
through the circuit.

538
00:29:44,290 --> 00:29:46,640
I should have said all the paths
through the circuit that

539
00:29:46,640 --> 00:29:53,300
go through each element
one or fewer times.

540
00:29:53,300 --> 00:29:58,660
I don't want you to go through
the same element twice.

541
00:29:58,660 --> 00:30:02,280
So here's another path that
would go through elements at

542
00:30:02,280 --> 00:30:04,290
most one time.

543
00:30:04,290 --> 00:30:07,230
So up through here, over through
here, which didn't go

544
00:30:07,230 --> 00:30:08,050
through any elements.

545
00:30:08,050 --> 00:30:10,480
Down through that element,
across that, down through

546
00:30:10,480 --> 00:30:11,850
here, et cetera.

547
00:30:11,850 --> 00:30:14,090
And you get an equation
for that.

548
00:30:14,090 --> 00:30:14,950
Here's another.

549
00:30:14,950 --> 00:30:16,140
Here's another.

550
00:30:16,140 --> 00:30:18,830
Here's another.

551
00:30:18,830 --> 00:30:21,220
And if you try to think about a
general rule, a general rule

552
00:30:21,220 --> 00:30:24,410
is something like, how many of
those panels can you make and

553
00:30:24,410 --> 00:30:29,790
piece together where the loop
goes through the perimeter?

554
00:30:29,790 --> 00:30:31,970
You're not allowed to go through
an inner place because

555
00:30:31,970 --> 00:30:33,430
if you went through an inner
node, you'd have to

556
00:30:33,430 --> 00:30:34,770
go through it twice.

557
00:30:34,770 --> 00:30:37,550
If you wanted the path to go
through an inner element,

558
00:30:37,550 --> 00:30:40,340
you'd have to go through
that element twice.

559
00:30:40,340 --> 00:30:41,810
So in fact, the answer is 7.

560
00:30:41,810 --> 00:30:49,200
There are 7 different paths
according to Kirchhoff's

561
00:30:49,200 --> 00:30:51,970
voltage law, all of which the
sum of the voltages around

562
00:30:51,970 --> 00:30:53,505
those paths has to be 0.

563
00:30:56,080 --> 00:30:59,260
The problem is, of course, that
those equations are not

564
00:30:59,260 --> 00:31:00,625
all linearly independent.

565
00:31:04,030 --> 00:31:07,280
So if you just had a general
purpose equation solver-- and

566
00:31:07,280 --> 00:31:09,330
by the way, we'll write
one of those in week

567
00:31:09,330 --> 00:31:11,970
8 for solving circuits.

568
00:31:11,970 --> 00:31:15,400
If you just passed those 7
equations into a general

569
00:31:15,400 --> 00:31:17,410
purpose equation solver, it
would tell you there's

570
00:31:17,410 --> 00:31:20,010
something awry with your
equations because they're not

571
00:31:20,010 --> 00:31:22,800
linearly independent.

572
00:31:22,800 --> 00:31:27,140
So you can, however, think about
linearly independent in

573
00:31:27,140 --> 00:31:29,050
particularly simple cases.

574
00:31:29,050 --> 00:31:32,140
This network is a particular
kind of network that we call a

575
00:31:32,140 --> 00:31:34,070
planar network.

576
00:31:34,070 --> 00:31:37,680
A planar network is one that I
can draw on a sheet of paper

577
00:31:37,680 --> 00:31:41,570
without crossing wires.

578
00:31:41,570 --> 00:31:44,040
So I can draw this network
without crossing wires.

579
00:31:44,040 --> 00:31:45,410
I'll call it planar.

580
00:31:45,410 --> 00:31:49,080
And it turns out that
Kirchhoff's voltage laws for

581
00:31:49,080 --> 00:31:54,200
the innermost loops are always
independent of each other.

582
00:31:54,200 --> 00:31:56,490
That's kind of obvious because
as you go to a --

583
00:31:56,490 --> 00:31:58,940
so each loop contains at least
one element that some other

584
00:31:58,940 --> 00:32:01,460
loop didn't have.

585
00:32:01,460 --> 00:32:04,620
So that's kind of the reasoning
for why it works.

586
00:32:04,620 --> 00:32:07,510
So if you think about this
particular loop, which we

587
00:32:07,510 --> 00:32:13,370
included in the 7, you can think
about that as being the

588
00:32:13,370 --> 00:32:17,020
sum of the loops this way, the
A loop and the B loop.

589
00:32:17,020 --> 00:32:20,360
Because if you write KVL for the
A loop and KVL for the B

590
00:32:20,360 --> 00:32:25,980
loop and add them, you end up
deriving KVL for the more

591
00:32:25,980 --> 00:32:28,220
complicated path.

592
00:32:28,220 --> 00:32:30,190
And if you think about what's
going on, it's not anything

593
00:32:30,190 --> 00:32:31,880
terribly magical.

594
00:32:31,880 --> 00:32:36,850
This path is the same as the A
path added to that path, where

595
00:32:36,850 --> 00:32:40,900
I went through this element down
when I did the A path and

596
00:32:40,900 --> 00:32:45,610
up when I did the B path.

597
00:32:45,610 --> 00:32:48,230
So those parts canceled out.

598
00:32:48,230 --> 00:32:50,420
That was the rule that I was
talking about how I don't

599
00:32:50,420 --> 00:32:54,480
really want to go through the
same element twice when I'm

600
00:32:54,480 --> 00:32:56,970
applying KVL.

601
00:32:56,970 --> 00:33:00,280
So the idea then is that there's
a systematic way, an

602
00:33:00,280 --> 00:33:05,170
easy way to figure out
all the KVL loops.

603
00:33:05,170 --> 00:33:06,765
You just think about
all the possible

604
00:33:06,765 --> 00:33:08,360
paths through the circuit.

605
00:33:08,360 --> 00:33:12,210
You do have to worry about
linearly independent.

606
00:33:12,210 --> 00:33:14,440
In the case of planar networks,
that's pretty

607
00:33:14,440 --> 00:33:15,220
straightforward.

608
00:33:15,220 --> 00:33:18,510
Planar networks, you can always
figure out the linearly

609
00:33:18,510 --> 00:33:21,770
independent KVL equations by
looking at the smallest

610
00:33:21,770 --> 00:33:24,140
possible loops.

611
00:33:24,140 --> 00:33:26,960
The loops with small area.

612
00:33:26,960 --> 00:33:28,250
OK, so that's half of it.

613
00:33:28,250 --> 00:33:28,960
That's KVL.

614
00:33:28,960 --> 00:33:32,650
The other Kirchhoff's law is
KCL, Kirchhoff's Current Law.

615
00:33:37,250 --> 00:33:41,760
There we are thinking of
the flow of current.

616
00:33:41,760 --> 00:33:44,770
So the flow of current is
analogous to the flow of

617
00:33:44,770 --> 00:33:47,560
incompressible fluid.

618
00:33:47,560 --> 00:33:49,960
Water, for example.

619
00:33:49,960 --> 00:33:52,310
If you trace the amount of water
that flows through a

620
00:33:52,310 --> 00:33:56,560
pipe that goes into a Y, then
the sum of the flows out has

621
00:33:56,560 --> 00:33:58,660
to equal the flow in.

622
00:33:58,660 --> 00:34:02,710
If that weren't true, the water
would be building up.

623
00:34:02,710 --> 00:34:05,860
So we think about pipes as
transporting the flow of water

624
00:34:05,860 --> 00:34:08,270
without allowing it to
build up anywhere.

625
00:34:08,270 --> 00:34:10,956
That's precisely how we
think about wires

626
00:34:10,956 --> 00:34:13,650
in electrical circuits.

627
00:34:13,650 --> 00:34:16,540
The wires allow the transport
of electrons but don't allow

628
00:34:16,540 --> 00:34:20,770
the buildup of electrons.

629
00:34:20,770 --> 00:34:22,820
OK, do electrons build up?

630
00:34:22,820 --> 00:34:24,000
Sure.

631
00:34:24,000 --> 00:34:28,219
But in our idealized world, we
say they don't build up in the

632
00:34:28,219 --> 00:34:31,130
wires, they build
up in a part.

633
00:34:31,130 --> 00:34:33,010
And we'll have a special
part that allows the

634
00:34:33,010 --> 00:34:34,570
electrons to build up.

635
00:34:34,570 --> 00:34:36,749
So we're not excluding the
possibility that they build

636
00:34:36,749 --> 00:34:41,840
up, we're just saying that in
this formalism, we don't allow

637
00:34:41,840 --> 00:34:44,540
the electrons to build
up in the wires.

638
00:34:44,540 --> 00:34:47,090
So for the purpose of the wires,
current in is equal to

639
00:34:47,090 --> 00:34:47,630
the current out.

640
00:34:47,630 --> 00:34:50,949
The net current in is 0.

641
00:34:50,949 --> 00:34:57,360
So we will think then, about
the circuit having nodes.

642
00:34:57,360 --> 00:35:01,300
The nodes are the places where
more than one element meets,

643
00:35:01,300 --> 00:35:03,980
two or more elements meet.

644
00:35:03,980 --> 00:35:07,790
And we will apply KCL
at each node.

645
00:35:07,790 --> 00:35:10,650
So for example, in this simple
circuit where I would have

646
00:35:10,650 --> 00:35:18,010
three parts connected in what
we would call parallel, they

647
00:35:18,010 --> 00:35:20,330
share a node at the
top and they share

648
00:35:20,330 --> 00:35:22,620
a node at the bottom.

649
00:35:22,620 --> 00:35:24,870
So even though it looks like
there's multiple interconnects

650
00:35:24,870 --> 00:35:27,290
up here, we say that's
one node.

651
00:35:30,430 --> 00:35:32,940
And we would say that the sum of
the currents into the node

652
00:35:32,940 --> 00:35:35,810
is equal to the sum of
the currents out.

653
00:35:35,810 --> 00:35:38,960
So if I lab all of the possible
currents that come

654
00:35:38,960 --> 00:35:42,160
out of that node, I would
have i1, i2, i3.

655
00:35:42,160 --> 00:35:43,685
i1 goes through the first
one, the second one

656
00:35:43,685 --> 00:35:45,100
and the third one.

657
00:35:45,100 --> 00:35:48,300
And so I would conclude from
Kirchhoff's current law that

658
00:35:48,300 --> 00:35:51,340
the sum of i1, i2,
and i3 is 0.

659
00:35:51,340 --> 00:35:51,720
OK.

660
00:35:51,720 --> 00:35:53,670
Easy, right?

661
00:35:53,670 --> 00:35:59,380
As I said, we're going to make
an abstraction where the

662
00:35:59,380 --> 00:36:03,620
electrons don't build
up in the wires.

663
00:36:03,620 --> 00:36:06,190
They don't even build
up in the parts.

664
00:36:06,190 --> 00:36:07,330
They do get stored
in the parts.

665
00:36:07,330 --> 00:36:10,230
That's a little confusing,
we'll come back to that.

666
00:36:10,230 --> 00:36:12,540
If they don't build up in the
parts, then the current that

667
00:36:12,540 --> 00:36:14,720
goes in this leg has to
come out that leg.

668
00:36:17,920 --> 00:36:23,930
If that's true, then i1 is i4,
i2 is i5, i3 is i6, and we end

669
00:36:23,930 --> 00:36:27,030
up with another equation down
here, which turns out to be

670
00:36:27,030 --> 00:36:28,900
precisely the same as
the one at the top.

671
00:36:33,380 --> 00:36:35,790
Everybody's happy with that?

672
00:36:35,790 --> 00:36:37,550
So we're thinking about this
just the way we would think

673
00:36:37,550 --> 00:36:39,650
about water flow.

674
00:36:39,650 --> 00:36:42,840
If there's water flow into a
part, it better be coming out.

675
00:36:42,840 --> 00:36:46,750
If there's water flow in a pipe,
the water that goes into

676
00:36:46,750 --> 00:36:49,830
the pipe better come out
of the pipe someplace.

677
00:36:49,830 --> 00:36:51,700
So here is an arbitrary network

678
00:36:51,700 --> 00:36:53,340
made out of four parts.

679
00:36:53,340 --> 00:36:57,070
How many linearly independent
KCL equations are there?

680
00:38:20,810 --> 00:38:24,740
So how many linearly independent
KCL equations are

681
00:38:24,740 --> 00:38:26,280
in that network?

682
00:38:26,280 --> 00:38:30,280
Everyone raise your hand, some
number of KCL equations.

683
00:38:34,125 --> 00:38:36,250
OK, I'm seeing a
bigger variety.

684
00:38:36,250 --> 00:38:38,170
I see (1)'s, (2)'s, and (3)'s.

685
00:38:38,170 --> 00:38:39,200
I don't see any (4)'s.

686
00:38:39,200 --> 00:38:40,515
That's probably good.

687
00:38:43,400 --> 00:38:46,600
So how do you think about the
number of linearly independent

688
00:38:46,600 --> 00:38:47,850
KCL equations?

689
00:38:52,945 --> 00:38:55,860
So the first thing to do
is to label things.

690
00:38:59,430 --> 00:39:01,660
So you have to have reference
directions before you can sort

691
00:39:01,660 --> 00:39:03,250
of think about things.

692
00:39:03,250 --> 00:39:04,540
So we have four elements.

693
00:39:04,540 --> 00:39:08,900
We would be expecting to see
four element currents.

694
00:39:08,900 --> 00:39:10,800
The same current that goes
into an element has

695
00:39:10,800 --> 00:39:12,460
to come out of it.

696
00:39:12,460 --> 00:39:14,080
So there's element current
1, 2, 3, and 4.

697
00:39:17,490 --> 00:39:21,230
There are three nodes,
so we might be

698
00:39:21,230 --> 00:39:24,680
expecting three KCL equations.

699
00:39:24,680 --> 00:39:28,190
Here's one node from which you
would conclude that the sum of

700
00:39:28,190 --> 00:39:31,520
i1 and i2 better be 0.

701
00:39:31,520 --> 00:39:33,520
Here's a node from which you
would conclude that the

702
00:39:33,520 --> 00:39:38,890
current in i2 better
be i3 plus i4.

703
00:39:38,890 --> 00:39:41,440
And here is a node from which
you would conclude that i1

704
00:39:41,440 --> 00:39:45,230
plus i3 plus i4 is 0.

705
00:39:45,230 --> 00:39:48,380
So I can write one KCL equation
for every node,

706
00:39:48,380 --> 00:39:50,840
that's not surprising.

707
00:39:50,840 --> 00:39:52,820
But if you look at those
equations, you'll see that

708
00:39:52,820 --> 00:39:54,390
they're not linearly
independent.

709
00:39:54,390 --> 00:39:57,710
In fact, if you solve
this one for i2--

710
00:39:57,710 --> 00:39:59,150
it's already solved for i2.

711
00:39:59,150 --> 00:40:03,910
Stick that answer up here, you
get i3 plus i4 added to i1 is

712
00:40:03,910 --> 00:40:06,590
0, which is just the same
as that equation.

713
00:40:06,590 --> 00:40:11,540
So of those three equations,
only two of them are linearly

714
00:40:11,540 --> 00:40:13,090
independent.

715
00:40:13,090 --> 00:40:14,525
The answer to that
problem was (2).

716
00:40:18,820 --> 00:40:20,920
And there's a pattern.

717
00:40:20,920 --> 00:40:23,690
So think about the pattern in
terms of figuring out the

718
00:40:23,690 --> 00:40:26,320
number of linearly independent
KCL equations that are in a

719
00:40:26,320 --> 00:40:27,735
slightly more complicated
network.

720
00:42:00,512 --> 00:42:05,880
So what's the answer
here, how many KCL

721
00:42:05,880 --> 00:42:07,270
equations are in this network?

722
00:42:10,040 --> 00:42:11,290
Wow.

723
00:42:13,160 --> 00:42:18,020
Well, I'm not getting any of the
answers I would have said.

724
00:42:18,020 --> 00:42:20,776
What does that mean?

725
00:42:20,776 --> 00:42:23,550
Ah, I'm forgetting to add 2.

726
00:42:23,550 --> 00:42:24,270
That's my problem.

727
00:42:24,270 --> 00:42:27,370
OK, now I'm getting some of
the answers that I would

728
00:42:27,370 --> 00:42:27,980
expect to get.

729
00:42:27,980 --> 00:42:30,080
OK, got it.

730
00:42:30,080 --> 00:42:31,730
I confused myself.

731
00:42:31,730 --> 00:42:33,910
OK, the vast majority say (1).

732
00:42:33,910 --> 00:42:35,640
How do you get that?

733
00:42:35,640 --> 00:42:36,890
Which is 3.

734
00:42:42,574 --> 00:42:45,740
So again, you think of how in
this circuit there are four

735
00:42:45,740 --> 00:42:51,880
nodes, A, B, C, D. So we can
think about writing a KCL

736
00:42:51,880 --> 00:42:53,060
equation for each one.

737
00:42:53,060 --> 00:42:56,450
If we go to A, A has three
currents coming out of it --

738
00:42:56,450 --> 00:42:57,360
1, 2, 3.

739
00:42:57,360 --> 00:43:02,910
So the sum of those has
to be 0, et cetera.

740
00:43:02,910 --> 00:43:05,010
And if you think about those
equations, they're not

741
00:43:05,010 --> 00:43:06,450
linearly independent either.

742
00:43:09,010 --> 00:43:11,250
If you work through the math,
you see that there's exactly

743
00:43:11,250 --> 00:43:15,330
one of those equations that
you can eliminate.

744
00:43:15,330 --> 00:43:17,350
So you're left with
three linearly

745
00:43:17,350 --> 00:43:20,250
independent KCL equations.

746
00:43:20,250 --> 00:43:22,210
And so there's a pattern
emerging here.

747
00:43:22,210 --> 00:43:23,460
Somebody see the pattern?

748
00:43:25,540 --> 00:43:26,520
1 minus.

749
00:43:26,520 --> 00:43:27,770
Can somebody prove
the pattern?

750
00:43:30,560 --> 00:43:31,520
So there's a pattern here.

751
00:43:31,520 --> 00:43:35,810
The pattern is take the number
of nodes and the number of

752
00:43:35,810 --> 00:43:37,910
independent KCL equations
as one less.

753
00:43:41,480 --> 00:43:43,398
So the challenge is,
can you prove it?

754
00:43:46,390 --> 00:43:47,640
And by the theory
of lectures--

755
00:43:50,858 --> 00:43:51,356
AUDIENCE: Yes.

756
00:43:51,356 --> 00:43:53,350
PROFESSOR: Yes.

757
00:43:53,350 --> 00:43:55,510
And by a corollary of the theory
of lectures, the way

758
00:43:55,510 --> 00:43:57,010
you would prove it is?

759
00:43:57,010 --> 00:43:57,970
AUDIENCE: On the next slide.

760
00:43:57,970 --> 00:43:58,980
PROFESSOR: On the next slide.

761
00:43:58,980 --> 00:44:00,230
Exactly.

762
00:44:02,960 --> 00:44:04,210
So how do I prove it?

763
00:44:06,958 --> 00:44:08,930
Yeah?

764
00:44:08,930 --> 00:44:11,395
AUDIENCE: Whenever you take
minus 1, you just add all the

765
00:44:11,395 --> 00:44:12,650
[UNINTELLIGIBLE] together
[UNINTELLIGIBLE].

766
00:44:12,650 --> 00:44:12,910
PROFESSOR: Yeah.

767
00:44:12,910 --> 00:44:15,312
So there's something special
about the last one.

768
00:44:15,312 --> 00:44:17,304
Why should there be something
special about the last one.

769
00:44:17,304 --> 00:44:19,296
AUDIENCE: Because the
circuit's closed.

770
00:44:19,296 --> 00:44:22,504
PROFESSOR: Because the
circuit's closed.

771
00:44:22,504 --> 00:44:23,760
That's right.

772
00:44:23,760 --> 00:44:26,230
So the idea is to sort
of generalize the way

773
00:44:26,230 --> 00:44:27,480
we think about KCL.

774
00:44:29,690 --> 00:44:30,710
So we start with a circuit.

775
00:44:30,710 --> 00:44:33,840
We think about having
four nodes here.

776
00:44:33,840 --> 00:44:37,270
It's certainly the case that
KCL holds for each node.

777
00:44:37,270 --> 00:44:41,376
So here's KCL for that node.

778
00:44:41,376 --> 00:44:46,830
But now if you think about KCL
for this node, and then add

779
00:44:46,830 --> 00:44:51,180
them, that looks like
a KCL equation.

780
00:44:53,850 --> 00:44:57,200
But it applies to
a super node.

781
00:44:57,200 --> 00:45:02,520
Imagine the node defined by the
black box, and think about

782
00:45:02,520 --> 00:45:05,293
the net currents into or
out of the black node.

783
00:45:08,470 --> 00:45:13,940
This current i2, which leaves
the red node, enters the green

784
00:45:13,940 --> 00:45:17,890
node, but doesn't go through
the surface of the

785
00:45:17,890 --> 00:45:20,520
black node at all.

786
00:45:20,520 --> 00:45:23,600
That's exactly the current
that's subtracted out when we

787
00:45:23,600 --> 00:45:26,986
added the red equation to
the green equation.

788
00:45:26,986 --> 00:45:28,940
Does that make sense?

789
00:45:28,940 --> 00:45:34,090
So KCL says, oh, if all the
currents at a node have to sum

790
00:45:34,090 --> 00:45:40,380
to 0, and if elements have the
same current coming out and

791
00:45:40,380 --> 00:45:48,560
going in, then if you draw a
box around an element, what

792
00:45:48,560 --> 00:45:51,000
goes into the element
is the same as what

793
00:45:51,000 --> 00:45:52,240
comes out of the element.

794
00:45:52,240 --> 00:45:55,340
It doesn't change the net
current through the surface.

795
00:45:55,340 --> 00:45:58,890
So the generalization of the KCL
equation, KCL says the sum

796
00:45:58,890 --> 00:46:00,450
of the currents into
a node is 0.

797
00:46:00,450 --> 00:46:06,690
The generalization says take any
closed path in a circuit,

798
00:46:06,690 --> 00:46:11,630
the sum of the currents going
across that closed path is 0.

799
00:46:11,630 --> 00:46:16,600
So if we apply that rule again,
think about node 3.

800
00:46:16,600 --> 00:46:20,470
If we add the result of node 3
to the black node, which was

801
00:46:20,470 --> 00:46:25,170
the sum of 1 and 2, we get
the new green curve.

802
00:46:25,170 --> 00:46:27,340
We get the new green equation.

803
00:46:27,340 --> 00:46:29,720
And what that says is the sum
of the currents going across

804
00:46:29,720 --> 00:46:33,120
the green super node--

805
00:46:33,120 --> 00:46:36,530
OK, so what's going on? i1 is
coming out of it, i4 is coming

806
00:46:36,530 --> 00:46:38,330
out of it, i5 is coming
out of it.

807
00:46:38,330 --> 00:46:41,540
So the sum of i1, i4,
and i5 has to be 0.

808
00:46:46,570 --> 00:46:49,030
Well, KCL says the sum of the
currents coming out of a

809
00:46:49,030 --> 00:46:50,890
node must be 0.

810
00:46:50,890 --> 00:46:54,990
The super KCL says the sum of
the currents coming out of any

811
00:46:54,990 --> 00:47:00,180
closed region is also 0.

812
00:47:00,180 --> 00:47:03,840
But the interesting thing about
this closed region is

813
00:47:03,840 --> 00:47:06,425
that it encloses all but
one of the nodes.

814
00:47:09,230 --> 00:47:11,310
That's always true.

815
00:47:11,310 --> 00:47:14,150
Regardless of the system,
regardless of the circuit, you

816
00:47:14,150 --> 00:47:19,090
can always draw a line that will
isolate one node from all

817
00:47:19,090 --> 00:47:22,030
the others.

818
00:47:22,030 --> 00:47:25,825
So what that proves is that you
can always write KCL for

819
00:47:25,825 --> 00:47:28,920
this node in terms of
KCL for those nodes.

820
00:47:31,870 --> 00:47:31,965
Ok.

821
00:47:31,965 --> 00:47:34,770
So there's a generalization then
that says that you can

822
00:47:34,770 --> 00:47:40,120
always write KCL
for every node.

823
00:47:40,120 --> 00:47:42,130
They will always be linearly
dependent.

824
00:47:42,130 --> 00:47:43,580
So you can always
throw away one.

825
00:47:47,780 --> 00:47:49,690
So in some sense now,
we're done.

826
00:47:49,690 --> 00:47:51,840
We've just finished
circuit theory.

827
00:47:51,840 --> 00:47:55,060
We talked about how every
element has to have a law.

828
00:47:55,060 --> 00:47:56,470
A resistor is Ohm's Law.

829
00:47:56,470 --> 00:47:58,820
A voltage source says that the
voltage across the terminals

830
00:47:58,820 --> 00:48:00,295
is always a constant.

831
00:48:00,295 --> 00:48:02,730
A current source says that the
current through the current

832
00:48:02,730 --> 00:48:04,530
source is always a constant.

833
00:48:04,530 --> 00:48:07,380
So every element tells
you one law.

834
00:48:10,170 --> 00:48:12,760
We know how to think
about KVL.

835
00:48:12,760 --> 00:48:18,930
So we know the rule for how the
across variables behave.

836
00:48:18,930 --> 00:48:21,230
What's the aggregate behavior
of all the across variables?

837
00:48:21,230 --> 00:48:24,140
Well, KVL has to be satisfied
for every possible loop.

838
00:48:24,140 --> 00:48:26,430
The loops don't have
to be independent.

839
00:48:26,430 --> 00:48:28,620
You have to worry about whether
they're independent.

840
00:48:28,620 --> 00:48:31,020
The only simple rule we came up
with-- we'll come up with

841
00:48:31,020 --> 00:48:32,415
another one in a moment.

842
00:48:32,415 --> 00:48:35,340
The only simple rule that we
came up with was for planar

843
00:48:35,340 --> 00:48:38,410
circuits, where the innermost
loops were linearly

844
00:48:38,410 --> 00:48:39,720
independent of each other.

845
00:48:39,720 --> 00:48:44,280
And you have to write KCL for
all the nodes, except one.

846
00:48:44,280 --> 00:48:45,530
One of them never matters.

847
00:48:48,110 --> 00:48:50,210
So in some sense, we're done.

848
00:48:50,210 --> 00:48:52,450
What we would do to
solve the circuit,

849
00:48:52,450 --> 00:48:54,310
think about every element.

850
00:48:54,310 --> 00:48:59,240
For every element assign a
voltage, a reference voltage.

851
00:48:59,240 --> 00:49:02,570
For every element,
assign a current.

852
00:49:02,570 --> 00:49:04,460
Make sure they go in the
right direction.

853
00:49:04,460 --> 00:49:07,460
We always define currents
to go down

854
00:49:07,460 --> 00:49:09,070
the potential gradient.

855
00:49:09,070 --> 00:49:11,920
They always go in the directions
through the element

856
00:49:11,920 --> 00:49:14,840
from the positive
to the negative.

857
00:49:14,840 --> 00:49:18,610
So for every element, assign
a current and a voltage.

858
00:49:18,610 --> 00:49:22,420
We have 6 elements, that's
12 unknowns.

859
00:49:22,420 --> 00:49:25,370
Now we dig and we find
12 equations.

860
00:49:25,370 --> 00:49:28,980
In this particular circuit, we
found those 12 equations.

861
00:49:28,980 --> 00:49:31,505
There were three KCL equations,
one for each of the

862
00:49:31,505 --> 00:49:32,550
inner loops.

863
00:49:32,550 --> 00:49:35,420
There were three KCL equations,
one for each node

864
00:49:35,420 --> 00:49:38,080
except one.

865
00:49:38,080 --> 00:49:41,020
There were 5 Ohm's law
equations, one for each one of

866
00:49:41,020 --> 00:49:41,750
the resistors.

867
00:49:41,750 --> 00:49:45,390
There was one source equation
for the voltage source.

868
00:49:45,390 --> 00:49:49,120
12 equations, 12 unknowns,
we're done.

869
00:49:49,120 --> 00:49:52,900
The only problem is a
lot of equations.

870
00:49:52,900 --> 00:49:54,500
It's not a very complicated
circuit.

871
00:49:54,500 --> 00:49:56,430
We've only got 6 elements.

872
00:49:56,430 --> 00:49:59,580
I tried to motivate this in
terms of studying networks

873
00:49:59,580 --> 00:50:04,390
that had 10 to the 9 elements.

874
00:50:04,390 --> 00:50:06,750
This technique is not
particularly great

875
00:50:06,750 --> 00:50:08,920
at 10 to the 9.

876
00:50:08,920 --> 00:50:11,240
It would probably work.

877
00:50:11,240 --> 00:50:13,130
But we would probably
be interested in

878
00:50:13,130 --> 00:50:15,970
finding simpler ways.

879
00:50:15,970 --> 00:50:20,430
So there are simpler ways you
might imagine, and we'll

880
00:50:20,430 --> 00:50:24,680
discuss two of them
just very briefly.

881
00:50:24,680 --> 00:50:27,160
The dumb way that I just talked
about is what we call

882
00:50:27,160 --> 00:50:30,190
primitive variables,
element variables.

883
00:50:30,190 --> 00:50:33,600
If you write all the element
variables, v1, v2, v3, v4, v5

884
00:50:33,600 --> 00:50:37,480
v6, all of the element currents,
i1, i2, i3, i4, i5,

885
00:50:37,480 --> 00:50:40,630
i6, write all the equations,
you can solve it.

886
00:50:40,630 --> 00:50:43,620
However, if you're judicious,
you can figure out a smaller

887
00:50:43,620 --> 00:50:46,600
number of unknowns and a
correspondingly smaller number

888
00:50:46,600 --> 00:50:47,340
of equations.

889
00:50:47,340 --> 00:50:49,640
One method is called
the node method.

890
00:50:52,180 --> 00:50:54,400
When we're thinking about the
individual elements, the thing

891
00:50:54,400 --> 00:50:56,465
that matters is the voltage
across the element.

892
00:50:59,230 --> 00:51:02,130
However, that's not
the easy way to

893
00:51:02,130 --> 00:51:06,140
write the circuit equations.

894
00:51:06,140 --> 00:51:10,110
A much easier way is not to tell
me the voltage across an

895
00:51:10,110 --> 00:51:13,360
element, but instead tell me
the voltage associated with

896
00:51:13,360 --> 00:51:14,610
each of the nodes.

897
00:51:18,710 --> 00:51:23,400
If I tell you the voltage
associated with every node,

898
00:51:23,400 --> 00:51:25,930
the important thing about that
way of defining the variables

899
00:51:25,930 --> 00:51:29,040
is that you're guaranteed that
from those variables, you can

900
00:51:29,040 --> 00:51:32,360
tell me the voltage
across every part.

901
00:51:32,360 --> 00:51:35,630
So for example, in this circuit,
this voltage source--

902
00:51:35,630 --> 00:51:42,140
so if I call this one ground,
we'll always have a magic node

903
00:51:42,140 --> 00:51:43,410
called ground.

904
00:51:43,410 --> 00:51:47,490
It is not special
in the least.

905
00:51:47,490 --> 00:51:50,780
It's just the reference
voltage.

906
00:51:50,780 --> 00:51:51,610
I'll come back to that.

907
00:51:51,610 --> 00:51:54,400
I'll say words in a minute about
what the reference is.

908
00:51:54,400 --> 00:51:58,100
We always get to declare
one node to be ground.

909
00:51:58,100 --> 00:52:01,270
We get one free node.

910
00:52:01,270 --> 00:52:04,830
It's a node whose voltage we
don't care about because it's

911
00:52:04,830 --> 00:52:06,380
the reference for
all voltages.

912
00:52:06,380 --> 00:52:09,260
It's a node whose current we
don't care about because we

913
00:52:09,260 --> 00:52:12,370
get to throw away one node when
we do current equations.

914
00:52:12,370 --> 00:52:15,400
So we have one special mode
called ground, about which we

915
00:52:15,400 --> 00:52:16,660
don't care too much.

916
00:52:16,660 --> 00:52:18,900
Except that it's the most
important node in the circuit.

917
00:52:18,900 --> 00:52:20,650
Except for that, we don't
care about it.

918
00:52:20,650 --> 00:52:23,430
So this guy's ground.

919
00:52:23,430 --> 00:52:26,080
We think about its
voltage being 0.

920
00:52:26,080 --> 00:52:30,360
Then this voltage supply
makes that node be v0.

921
00:52:30,360 --> 00:52:32,250
I don't know what that is,
so I'll call it e1.

922
00:52:32,250 --> 00:52:36,340
And I don't know what that
is, so I'll call it e2.

923
00:52:36,340 --> 00:52:39,600
So if I tell you the voltage on
all of those nodes, ground

924
00:52:39,600 --> 00:52:44,550
voltage is 0, the top voltage is
v0, the left voltage is e1,

925
00:52:44,550 --> 00:52:45,670
the right voltage is e2.

926
00:52:45,670 --> 00:52:51,700
From those four numbers, 0 and
3 nontrivial numbers, you can

927
00:52:51,700 --> 00:52:54,940
find all of the component
voltages.

928
00:52:54,940 --> 00:52:59,460
So for example, the voltage v6,
the voltage across R6 is

929
00:52:59,460 --> 00:53:00,710
e2 minus e1.

930
00:53:03,420 --> 00:53:06,740
The voltage v4, the voltage
across the R4

931
00:53:06,740 --> 00:53:10,470
resistor is e1 minus 0.

932
00:53:10,470 --> 00:53:15,690
So if I tell you all the node
voltages, you can tell me all

933
00:53:15,690 --> 00:53:17,250
of the element voltages.

934
00:53:17,250 --> 00:53:20,530
And in general, there's fewer
nodes than there are

935
00:53:20,530 --> 00:53:21,560
components.

936
00:53:21,560 --> 00:53:23,820
OK, that's great.

937
00:53:23,820 --> 00:53:26,990
So instead of naming the volts
across the elements, we'll

938
00:53:26,990 --> 00:53:30,115
name the voltages at the nodes
because there's fewer of them.

939
00:53:32,750 --> 00:53:36,620
Then, all we need to do in the
node method is write the

940
00:53:36,620 --> 00:53:42,780
minimum number of
KCL equations.

941
00:53:42,780 --> 00:53:46,060
We know we only have two
unknowns, e1 and e2.

942
00:53:46,060 --> 00:53:47,970
And it turns out-- and you
can prove this, but I

943
00:53:47,970 --> 00:53:50,990
won't prove it today.

944
00:53:50,990 --> 00:53:54,000
It turns out that you need
two KCL equations.

945
00:53:54,000 --> 00:53:57,670
Two unknowns, e1, e2,
two KCL equations.

946
00:53:57,670 --> 00:54:01,400
And it turns out those two KCL
equations are exactly the KCL

947
00:54:01,400 --> 00:54:03,400
equations associated
with the two nodes.

948
00:54:05,950 --> 00:54:10,990
So the current leaving
e1, so KCL at e1 --

949
00:54:10,990 --> 00:54:14,780
well, there's a current
that goes that way.

950
00:54:14,780 --> 00:54:19,080
Well, that's the voltage drop
in going from e1 to v0, e1

951
00:54:19,080 --> 00:54:21,380
minus v0, divided by R2.

952
00:54:21,380 --> 00:54:23,610
That's Ohm's law.

953
00:54:23,610 --> 00:54:28,600
So this term represents the
current going up that leg plus

954
00:54:28,600 --> 00:54:30,930
the current that goes through
this leg, which is e1

955
00:54:30,930 --> 00:54:33,780
minus e2 over R6.

956
00:54:33,780 --> 00:54:36,450
Plus the current going in
that leg, which is e1

957
00:54:36,450 --> 00:54:39,250
minus 0 over R4.

958
00:54:39,250 --> 00:54:40,980
The sum of those three
currents better be 0.

959
00:54:43,640 --> 00:54:46,460
Analogously, the sum of the
currents at this node must be

960
00:54:46,460 --> 00:54:50,070
0, and the equation looks
virtually the same.

961
00:54:50,070 --> 00:54:56,230
Because v0 is known, so it
didn't add an unknown.

962
00:54:56,230 --> 00:55:00,250
v0 was set by the voltage,
by the voltage source.

963
00:55:00,250 --> 00:55:04,000
So I have two equations,
two unknowns, solved.

964
00:55:04,000 --> 00:55:04,630
Done.

965
00:55:04,630 --> 00:55:07,040
So rather than solving 12
equations and 12 unknowns, I

966
00:55:07,040 --> 00:55:09,300
can do it with two equations
and two unknowns.

967
00:55:09,300 --> 00:55:11,640
That's called the node method.

968
00:55:11,640 --> 00:55:15,610
One of the most interesting
theories about circuits is

969
00:55:15,610 --> 00:55:19,110
that every simplification that
you can think about for

970
00:55:19,110 --> 00:55:22,250
voltage has an analogous
simplification that you can

971
00:55:22,250 --> 00:55:23,050
think about in current.

972
00:55:23,050 --> 00:55:24,030
That's called duality.

973
00:55:24,030 --> 00:55:26,430
We won't do that because it's
kind of complicated.

974
00:55:26,430 --> 00:55:27,790
But it's kind of
a cute result.

975
00:55:27,790 --> 00:55:29,500
If you can think of a
simplification that works in

976
00:55:29,500 --> 00:55:31,920
voltage, then there is
an analogous one, and

977
00:55:31,920 --> 00:55:33,200
you can prove it.

978
00:55:33,200 --> 00:55:36,860
In fact, you can formally derive
what it must have been.

979
00:55:40,090 --> 00:55:42,970
This is a rule for how you can
simplify things by thinking

980
00:55:42,970 --> 00:55:45,115
about voltages in aggregate.

981
00:55:47,640 --> 00:55:50,480
Rather than thinking about the
element voltages, think about

982
00:55:50,480 --> 00:55:51,890
the node voltages.

983
00:55:51,890 --> 00:55:56,510
The analogous current law is
rather than thinking about the

984
00:55:56,510 --> 00:55:59,770
currents through the elements,
the element currents, think

985
00:55:59,770 --> 00:56:01,950
about loop currents.

986
00:56:01,950 --> 00:56:04,110
OK, that's a little bizarre.

987
00:56:04,110 --> 00:56:07,430
So we name the loop, the current
that flows in this

988
00:56:07,430 --> 00:56:12,590
loop, IA, the current that flows
in this loop, IB, and

989
00:56:12,590 --> 00:56:14,780
the current hat flows
in this loop, IC.

990
00:56:14,780 --> 00:56:16,280
What on earth is he doing?

991
00:56:16,280 --> 00:56:20,250
Well, the element voltages are
some linear combination of

992
00:56:20,250 --> 00:56:24,070
those loop currents.

993
00:56:24,070 --> 00:56:26,460
And in fact, the coefficients in
the linear combination are

994
00:56:26,460 --> 00:56:27,660
one and minus one.

995
00:56:27,660 --> 00:56:32,320
So the element current I4, the
current that flows through the

996
00:56:32,320 --> 00:56:37,780
R4 resistor is the sum of
IA coming down minus IC,

997
00:56:37,780 --> 00:56:39,670
which is going up.

998
00:56:39,670 --> 00:56:45,500
So there's a way of thinking
about each element current as

999
00:56:45,500 --> 00:56:49,790
a sum or difference of
the loop currents.

1000
00:56:49,790 --> 00:56:52,180
Everybody get that?

1001
00:56:52,180 --> 00:56:54,600
So instead of thinking about
the individual element

1002
00:56:54,600 --> 00:56:56,190
currents, I think about
the loop currents.

1003
00:56:56,190 --> 00:57:00,900
And now, I need to write
three KVL equations.

1004
00:57:00,900 --> 00:57:06,730
So in the node method, I named
the nodes and had to write two

1005
00:57:06,730 --> 00:57:08,480
KCL equations.

1006
00:57:08,480 --> 00:57:11,205
Here, I named the loop currents
and I have to write

1007
00:57:11,205 --> 00:57:15,590
three KVL equations,
one for each loop.

1008
00:57:15,590 --> 00:57:16,760
It's completely analogous.

1009
00:57:16,760 --> 00:57:19,420
If you write out a sentence,
what did you do?

1010
00:57:19,420 --> 00:57:21,760
I assigned a voltage to every
node, and I wrote

1011
00:57:21,760 --> 00:57:22,910
KCL of all the nodes.

1012
00:57:22,910 --> 00:57:26,040
Then if you turn the word
"current" into the word

1013
00:57:26,040 --> 00:57:30,540
"voltage," the word "node"
into the word "loop," you

1014
00:57:30,540 --> 00:57:33,920
derive this new method.

1015
00:57:33,920 --> 00:57:37,340
So this says that if I write KVL
at the A loop, think about

1016
00:57:37,340 --> 00:57:40,580
spinning around this loop, as
I go up through the voltage

1017
00:57:40,580 --> 00:57:43,910
source, so I go in the negative
terminal here.

1018
00:57:43,910 --> 00:57:46,090
So that's minus v0.

1019
00:57:46,090 --> 00:57:48,940
As I go down through this
resistor, I have to use Ohm's

1020
00:57:48,940 --> 00:57:51,770
law, so that's R2 times
the down current.

1021
00:57:51,770 --> 00:57:57,610
Well, the down current is
IA down minus IB up.

1022
00:57:57,610 --> 00:57:59,380
So I went up through here,
down through here.

1023
00:57:59,380 --> 00:58:01,160
Now I go down through
this one.

1024
00:58:01,160 --> 00:58:04,580
When I go down through that one,
according to Ohm's law,

1025
00:58:04,580 --> 00:58:07,530
that's R4 times the current
through that element.

1026
00:58:07,530 --> 00:58:13,490
That current-- well, it's
IA down and it's IC up.

1027
00:58:13,490 --> 00:58:18,650
So this is the KVL equation
for that loop.

1028
00:58:18,650 --> 00:58:23,030
I write two more of them, and I
end up with three equations

1029
00:58:23,030 --> 00:58:25,590
and three unknowns.

1030
00:58:25,590 --> 00:58:29,410
Both the node method and the
loop method resulted in a lot

1031
00:58:29,410 --> 00:58:33,330
fewer equations than
the primitives did.

1032
00:58:33,330 --> 00:58:38,400
I had 12 primitive unknowns,
6 voltages and 6 currents.

1033
00:58:38,400 --> 00:58:42,670
In the node method, I get the
number of independent nodes as

1034
00:58:42,670 --> 00:58:44,910
the number of equations and
unknowns, which is less than

1035
00:58:44,910 --> 00:58:46,520
the number of primitive
variables.

1036
00:58:46,520 --> 00:58:49,280
In the loop method, I have the
number of independent loops.

1037
00:58:51,840 --> 00:58:55,100
Which is again, smaller.

1038
00:58:55,100 --> 00:58:57,220
So the idea then is that we have
a couple of ways to think

1039
00:58:57,220 --> 00:59:00,360
about solving circuits.

1040
00:59:00,360 --> 00:59:04,450
Fundamentally, all we have are
the element relationships and

1041
00:59:04,450 --> 00:59:05,846
the rules for combination.

1042
00:59:05,846 --> 00:59:09,180
Oh, this is starting to sound
like PCAP, primitives and

1043
00:59:09,180 --> 00:59:10,380
combinations.

1044
00:59:10,380 --> 00:59:14,860
So the primitives are, how does
the element constrain the

1045
00:59:14,860 --> 00:59:17,260
voltages and currents?

1046
00:59:17,260 --> 00:59:19,470
We know three of those,
Ohm's law, voltage

1047
00:59:19,470 --> 00:59:21,480
source, current source.

1048
00:59:21,480 --> 00:59:24,380
And what are the rules
for combination?

1049
00:59:24,380 --> 00:59:27,010
Well, the currents add to the
node, and the voltages add

1050
00:59:27,010 --> 00:59:30,070
around loops.

1051
00:59:30,070 --> 00:59:33,720
OK, just to make sure you've
absorb all that, figure out

1052
00:59:33,720 --> 00:59:35,535
the current I for
this circuit.

1053
01:02:31,840 --> 01:02:33,810
OK, what's a good
way to start?

1054
01:02:33,810 --> 01:02:38,830
What should I do to start
thinking about calculating I?

1055
01:02:38,830 --> 01:02:40,190
OK, bad way.

1056
01:02:40,190 --> 01:02:42,520
Assign voltages and currents
to everything.

1057
01:02:42,520 --> 01:02:46,150
4 elements, that's 4 voltages,
4 currents.

1058
01:02:46,150 --> 01:02:47,620
That's 8 unknowns.

1059
01:02:47,620 --> 01:02:48,990
Find 8 equations, solve.

1060
01:02:48,990 --> 01:02:50,200
That'll work.

1061
01:02:50,200 --> 01:02:50,980
Bad way.

1062
01:02:50,980 --> 01:02:52,230
What's a better way?

1063
01:02:56,146 --> 01:02:57,396
OK, [UNINTELLIGIBLE PHRASE].

1064
01:03:00,986 --> 01:03:01,954
AUDIENCE:
[UNINTELLIGIBLE PHRASE].

1065
01:03:01,954 --> 01:03:03,890
PROFESSOR: It was on
the previous sheet.

1066
01:03:03,890 --> 01:03:06,820
[UNINTELLIGIBLE PHRASE].

1067
01:03:06,820 --> 01:03:08,020
AUDIENCE: KVL [UNINTELLIGIBLE].

1068
01:03:08,020 --> 01:03:09,660
PROFESSOR: KVL for where?

1069
01:03:09,660 --> 01:03:11,616
AUDIENCE: Loops.

1070
01:03:11,616 --> 01:03:12,594
PROFESSOR: Which loop?

1071
01:03:12,594 --> 01:03:13,572
AUDIENCE: Left loop.

1072
01:03:13,572 --> 01:03:16,506
PROFESSOR: So do KVL
on the left loop?

1073
01:03:16,506 --> 01:03:17,973
AUDIENCE: Yes.

1074
01:03:17,973 --> 01:03:19,440
PROFESSOR: OK, that's good.

1075
01:03:19,440 --> 01:03:22,090
But you have to tell me how
to assign variables.

1076
01:03:22,090 --> 01:03:24,273
Do you want 8 primitive
variables?

1077
01:03:28,540 --> 01:03:30,920
8 primitive variables are v1,
i1, v2, i2, v3, i3, v4, i4.

1078
01:03:36,130 --> 01:03:38,540
So that's what I mean by
primitive variables.

1079
01:03:38,540 --> 01:03:42,700
Or element variables is
another word for it.

1080
01:03:42,700 --> 01:03:44,555
What's a better way than using
element variables?

1081
01:03:47,552 --> 01:03:48,000
Yeah.

1082
01:03:48,000 --> 01:03:49,908
AUDIENCE: Create 2
loop equations.

1083
01:03:49,908 --> 01:03:52,902
PROFESSOR: Create 2 loop
equations, that's fantastic.

1084
01:03:52,902 --> 01:03:54,950
AUDIENCE: I1 for the first loop,
I2 for the second loop.

1085
01:03:54,950 --> 01:03:58,750
PROFESSOR: So if you do I1 going
around here, then I1 is

1086
01:03:58,750 --> 01:04:05,915
actually I. And if you do I2
going around here, what's I2?

1087
01:04:10,673 --> 01:04:11,670
AUDIENCE: [INAUDIBLE].

1088
01:04:11,670 --> 01:04:16,490
PROFESSOR: So if I think about
I2 spinning around this loop,

1089
01:04:16,490 --> 01:04:20,920
so the sum of I1 and I2
goes through that box.

1090
01:04:20,920 --> 01:04:23,200
But the only current that goes
through this box is?

1091
01:04:26,556 --> 01:04:27,806
AUDIENCE: [INAUDIBLE].

1092
01:04:29,993 --> 01:04:31,957
PROFESSOR: So the suggestion
is that I think about--

1093
01:04:43,250 --> 01:04:49,790
so if I have I1 here, but I know
that's I. Then I can see

1094
01:04:49,790 --> 01:04:51,310
immediately that since
the only current that

1095
01:04:51,310 --> 01:04:52,380
goes through here--

1096
01:04:52,380 --> 01:04:54,100
so if I have I1 and I2.

1097
01:04:54,100 --> 01:04:56,010
That was a very clever idea.

1098
01:04:56,010 --> 01:04:58,090
If you have I1 and I2, the only
current that goes through

1099
01:04:58,090 --> 01:05:02,470
here is I. So I1
must've been I.

1100
01:05:02,470 --> 01:05:04,640
The only current that
goes over here

1101
01:05:04,640 --> 01:05:07,390
must've been this guy.

1102
01:05:07,390 --> 01:05:11,340
So this must be minus 10.

1103
01:05:11,340 --> 01:05:14,130
So I could redo that this way.

1104
01:05:14,130 --> 01:05:19,130
I could say I've got
10 going that way.

1105
01:05:19,130 --> 01:05:21,410
That make sense?

1106
01:05:21,410 --> 01:05:26,340
So now I only have one unknown
which is I. So that's a very

1107
01:05:26,340 --> 01:05:28,310
clever way of doing it.

1108
01:05:28,310 --> 01:05:31,720
So what I could do
is showed here.

1109
01:05:31,720 --> 01:05:35,180
I have I going around one
loop and I have 10

1110
01:05:35,180 --> 01:05:37,950
going around that loop.

1111
01:05:37,950 --> 01:05:39,950
That completely specifies
all the currents.

1112
01:05:39,950 --> 01:05:43,080
So now all I need to
do is write KVL for

1113
01:05:43,080 --> 01:05:45,980
these different cases.

1114
01:05:45,980 --> 01:05:50,810
So if I write KVL for the left
loop, then I get going up

1115
01:05:50,810 --> 01:05:55,160
through here, that's minus 15,
and going down through here,

1116
01:05:55,160 --> 01:05:58,360
going to the right through
this guy is 3I.

1117
01:05:58,360 --> 01:06:03,090
Going down through this guy
is 2 times I plus 10.

1118
01:06:03,090 --> 01:06:05,450
Both of these are going down,
so you have to add them.

1119
01:06:10,010 --> 01:06:11,960
So I get one equation
and one unknown.

1120
01:06:11,960 --> 01:06:15,160
And when I solve it,
I get minus one.

1121
01:06:15,160 --> 01:06:16,410
That make sense?

1122
01:06:18,470 --> 01:06:19,900
There's an analogous
way you could have

1123
01:06:19,900 --> 01:06:21,150
done it with one node.

1124
01:06:24,080 --> 01:06:28,130
You could have said that the
circuit has a single node and

1125
01:06:28,130 --> 01:06:31,860
figured out KCL for
that one node.

1126
01:06:31,860 --> 01:06:34,170
KCL would be the sum of
the currents here.

1127
01:06:34,170 --> 01:06:35,770
There's a current that
goes that way, that

1128
01:06:35,770 --> 01:06:36,880
way, and that way.

1129
01:06:36,880 --> 01:06:39,240
And again, you end up with
1 equation and 1 unknown.

1130
01:06:39,240 --> 01:06:40,069
Yes?

1131
01:06:40,069 --> 01:06:41,319
AUDIENCE:
[UNINTELLIGIBLE PHRASE].

1132
01:06:45,558 --> 01:06:46,808
PROFESSOR: Correct.

1133
01:06:49,051 --> 01:06:51,430
If I thought about this current
going this way, it

1134
01:06:51,430 --> 01:06:53,890
would be minus 10.

1135
01:06:53,890 --> 01:06:59,050
If I flipped the direction,
then it's plus 10.

1136
01:06:59,050 --> 01:07:03,820
So the loop current has the
property that it's the only

1137
01:07:03,820 --> 01:07:05,290
current through this element.

1138
01:07:05,290 --> 01:07:07,220
So that has to match.

1139
01:07:07,220 --> 01:07:11,234
It's one of two currents that
go through this element.

1140
01:07:11,234 --> 01:07:13,186
AUDIENCE: You said that
everything that

1141
01:07:13,186 --> 01:07:15,626
[UNINTELLIGIBLE PHRASE].

1142
01:07:15,626 --> 01:07:17,578
PROFESSOR: This loop is
[UNINTELLIGIBLE].

1143
01:07:17,578 --> 01:07:18,066
Yes.

1144
01:07:18,066 --> 01:07:20,018
AUDIENCE: So why is the
[UNINTELLIGIBLE PHRASE].

1145
01:07:23,434 --> 01:07:24,410
PROFESSOR: Correct.

1146
01:07:24,410 --> 01:07:26,880
I want to have this
picture now.

1147
01:07:26,880 --> 01:07:29,570
So if I'm doing it with loops,
I have two loops.

1148
01:07:32,390 --> 01:07:35,100
The current through this element
is just I. The current

1149
01:07:35,100 --> 01:07:37,720
through this element is just
I. The current through this

1150
01:07:37,720 --> 01:07:40,020
element is just 10.

1151
01:07:40,020 --> 01:07:42,040
The current through this
element-- well, the sum of

1152
01:07:42,040 --> 01:07:43,560
these two currents go through
that element.

1153
01:07:46,730 --> 01:07:49,512
Does that make sense?

1154
01:07:49,512 --> 01:07:50,762
AUDIENCE: [INAUDIBLE]

1155
01:07:55,428 --> 01:07:57,893
PROFESSOR: This loop current
is just a fraction of the

1156
01:07:57,893 --> 01:08:01,344
current in the whole system.

1157
01:08:01,344 --> 01:08:05,550
So this loop current goes
through this element and

1158
01:08:05,550 --> 01:08:07,150
contributes to this element.

1159
01:08:07,150 --> 01:08:08,400
But so does that one.

1160
01:08:12,074 --> 01:08:15,150
OK, if you're still confused,
you should try to get it

1161
01:08:15,150 --> 01:08:17,170
straightened out in one of the
software labs or the hardware

1162
01:08:17,170 --> 01:08:18,670
lab, or talk to me
after lecture.

1163
01:08:18,670 --> 01:08:20,760
But the idea is to
decompose in the

1164
01:08:20,760 --> 01:08:22,720
case of the node voltages.

1165
01:08:22,720 --> 01:08:25,380
Think about the element
voltages in terms of

1166
01:08:25,380 --> 01:08:27,000
differences in the
node voltages.

1167
01:08:30,580 --> 01:08:32,439
In the case of the loop
currents, think about the

1168
01:08:32,439 --> 01:08:36,100
element currents in terms of
a sum of loop currents.

1169
01:08:40,029 --> 01:08:43,600
OK, so the answer is minus 1
regardless of how you do it.

1170
01:08:46,345 --> 01:08:46,472
Ok.

1171
01:08:46,472 --> 01:08:49,439
The remaining thing I want to
do today is think about

1172
01:08:49,439 --> 01:08:52,140
abstraction.

1173
01:08:52,140 --> 01:08:55,180
We've talked about the
primitives, which are things

1174
01:08:55,180 --> 01:08:59,750
like resistors, voltage sources,
and current sources.

1175
01:08:59,750 --> 01:09:03,100
Means of combinations,
that's KVL and KCL.

1176
01:09:03,100 --> 01:09:05,090
Now we want to think
about abstraction.

1177
01:09:05,090 --> 01:09:07,700
And the first abstraction that
we'll talk about is, how do

1178
01:09:07,700 --> 01:09:11,510
you think about one element
that represents

1179
01:09:11,510 --> 01:09:13,680
more than one element?

1180
01:09:13,680 --> 01:09:15,740
This is the same thing that we
did when we thought about

1181
01:09:15,740 --> 01:09:18,770
linear systems, when we did
signals and systems.

1182
01:09:18,770 --> 01:09:23,270
We started with R's and K's and
pluses, and we made single

1183
01:09:23,270 --> 01:09:29,020
boxes that had lots of R's and
pluses and gains in them.

1184
01:09:29,020 --> 01:09:32,590
What was the name of the thing
that was inside the box?

1185
01:09:32,590 --> 01:09:36,270
If we combined lots of R's,
gains, and pluses into a

1186
01:09:36,270 --> 01:09:39,860
single box, what would we call
the thing that's in the box?

1187
01:09:39,860 --> 01:09:41,019
AUDIENCE: [INAUDIBLE].

1188
01:09:41,019 --> 01:09:41,789
PROFESSOR: Shout again.

1189
01:09:41,789 --> 01:09:42,260
AUDIENCE: System function.

1190
01:09:42,260 --> 01:09:44,930
PROFESSOR: System function.

1191
01:09:44,930 --> 01:09:45,019
Right?

1192
01:09:45,019 --> 01:09:47,609
So we started with boxes
that only had

1193
01:09:47,609 --> 01:09:49,580
things like R's in them.

1194
01:09:49,580 --> 01:09:53,920
But eventually, we got boxes
that looked like much more

1195
01:09:53,920 --> 01:10:00,000
complicated things like that.

1196
01:10:00,000 --> 01:10:02,170
We thought about a system
function which was a

1197
01:10:02,170 --> 01:10:06,580
generalized box, that could have
lots of R's, or lots of

1198
01:10:06,580 --> 01:10:08,890
gains, or lots of
pluses in it.

1199
01:10:08,890 --> 01:10:14,810
And that was a way of
abstracting complicated

1200
01:10:14,810 --> 01:10:17,140
systems so they looked
like simple systems.

1201
01:10:17,140 --> 01:10:18,900
What we want to do here is the
same thing for circuits.

1202
01:10:18,900 --> 01:10:21,530
We want to have a single
element, a single circuit

1203
01:10:21,530 --> 01:10:25,620
element, that represents
many circuit elements.

1204
01:10:25,620 --> 01:10:29,240
And the simplest case of that
is for series in parallel

1205
01:10:29,240 --> 01:10:31,050
combinations of resistors.

1206
01:10:31,050 --> 01:10:34,890
It's very simple to think about
how if you had two Ohm's

1207
01:10:34,890 --> 01:10:39,220
law devices connected in series,
you could replace

1208
01:10:39,220 --> 01:10:42,490
those two with a single
resistor.

1209
01:10:42,490 --> 01:10:45,520
And the voltage-current
relationships measured at the

1210
01:10:45,520 --> 01:10:48,460
outside of the box would
be the same.

1211
01:10:51,140 --> 01:10:53,930
That's how we think about an
abstraction in circuits.

1212
01:10:53,930 --> 01:10:59,730
When is it that you can draw
a box around a piece of a

1213
01:10:59,730 --> 01:11:03,590
circuit and think about
that as one element?

1214
01:11:03,590 --> 01:11:06,350
The very simplest cases, the
series combination of two

1215
01:11:06,350 --> 01:11:09,940
resistors, same sort of thing
happens for the parallel

1216
01:11:09,940 --> 01:11:11,830
combination.

1217
01:11:11,830 --> 01:11:15,870
And that simple abstraction
makes some things very easy.

1218
01:11:15,870 --> 01:11:19,370
What would be the equivalent
resistance for a complicated

1219
01:11:19,370 --> 01:11:21,350
system like that?

1220
01:11:21,350 --> 01:11:24,070
Well, that's easy.

1221
01:11:24,070 --> 01:11:29,440
All you need to do is think
about successively reducing

1222
01:11:29,440 --> 01:11:32,170
the pieces.

1223
01:11:32,170 --> 01:11:35,050
Here I'm thinking about that
having four resistors.

1224
01:11:35,050 --> 01:11:39,020
I can just successively apply
series and parallel in order

1225
01:11:39,020 --> 01:11:41,760
to reduce that, make it
less complicated.

1226
01:11:41,760 --> 01:11:47,660
So I can think about combining
these two in series to get,

1227
01:11:47,660 --> 01:11:50,025
instead of two 1 Ohm resistors,
one 2 Ohm resistor.

1228
01:11:52,580 --> 01:11:56,640
Then I can think about these
two 2 Ohm resistors being

1229
01:11:56,640 --> 01:11:59,170
equivalently one parallel
1 Ohm resistor.

1230
01:12:02,810 --> 01:12:06,170
And so this whole thing looks
as though it's just 2 Ohms

1231
01:12:06,170 --> 01:12:09,560
from the outside world.

1232
01:12:09,560 --> 01:12:11,220
That's what we mean
by an abstraction.

1233
01:12:11,220 --> 01:12:15,430
What we're trying to do and what
we will do over the next

1234
01:12:15,430 --> 01:12:18,380
two weeks, is we'll think
about ways of combining

1235
01:12:18,380 --> 01:12:22,630
circuits so that we can reduce
the complexity this way.

1236
01:12:25,250 --> 01:12:28,070
Another convenient way of
thinking about reducing the

1237
01:12:28,070 --> 01:12:31,120
work that you need to do is to
think about common patterns

1238
01:12:31,120 --> 01:12:33,130
that result.

1239
01:12:33,130 --> 01:12:37,740
PCAP, Primitives, Combinations,
Abstractions.

1240
01:12:37,740 --> 01:12:40,120
So the series of parallel
idea was an abstraction.

1241
01:12:40,120 --> 01:12:42,330
A pattern, here's a
common pattern.

1242
01:12:42,330 --> 01:12:48,930
If you've got two resistors in
series, if the same current

1243
01:12:48,930 --> 01:12:52,250
flows through two resistors,
then there's a way of very

1244
01:12:52,250 --> 01:12:56,170
simply calculating the voltage
that falls across each.

1245
01:12:56,170 --> 01:13:01,090
So you can think about the sum
resistor, R1 plus R2 since

1246
01:13:01,090 --> 01:13:03,290
they're in series.

1247
01:13:03,290 --> 01:13:05,190
So that allows you then
to compute the

1248
01:13:05,190 --> 01:13:08,390
current from the voltage.

1249
01:13:08,390 --> 01:13:11,240
Then the voltage that falls
across this guy is by Ohm's

1250
01:13:11,240 --> 01:13:16,000
law, just the current
times its resistor,

1251
01:13:16,000 --> 01:13:18,150
which is like that.

1252
01:13:18,150 --> 01:13:19,970
And similarly with this one.

1253
01:13:19,970 --> 01:13:24,560
So you can see that some
fraction of this voltage v

1254
01:13:24,560 --> 01:13:26,930
occurs across the v1 terminal.

1255
01:13:26,930 --> 01:13:30,100
And some different fraction
appears across the v2

1256
01:13:30,100 --> 01:13:34,770
terminal, such that the sum of
the fractions is, of course,

1257
01:13:34,770 --> 01:13:38,280
v. That's what has to
happen for the two.

1258
01:13:38,280 --> 01:13:40,260
And there's a proportional
drop.

1259
01:13:40,260 --> 01:13:43,350
The bigger R1, the bigger is the
proportion of the voltage

1260
01:13:43,350 --> 01:13:45,860
that falls across R1.

1261
01:13:45,860 --> 01:13:49,620
So it's a simple way of thinking
about how voltage

1262
01:13:49,620 --> 01:13:51,120
drops across two resistors.

1263
01:13:51,120 --> 01:13:53,610
There's a completely analogous
way of thinking about how

1264
01:13:53,610 --> 01:13:58,690
current splits between
two resistors.

1265
01:13:58,690 --> 01:14:02,440
Here the result looks virtually
the same, except it

1266
01:14:02,440 --> 01:14:06,710
has kind of the unintuitive
property that most of the

1267
01:14:06,710 --> 01:14:11,710
current goes through the
resistor that is the smallest.

1268
01:14:11,710 --> 01:14:19,910
So you get a bigger current in
i1 in proportion to the R2.

1269
01:14:19,910 --> 01:14:23,040
So it works very much like the
voltage case, except that it

1270
01:14:23,040 --> 01:14:26,060
has this inversion in it, that
the current likes to go

1271
01:14:26,060 --> 01:14:29,870
through the smaller resistor.

1272
01:14:29,870 --> 01:14:33,670
OK, so last problem.

1273
01:14:33,670 --> 01:14:38,880
Using those kinds of ideas,
think about how you could

1274
01:14:38,880 --> 01:14:42,855
compute the voltage v0 and
determine what's the answer.

1275
01:16:30,480 --> 01:16:33,910
So what's the easy way to
think about this answer?

1276
01:16:33,910 --> 01:16:35,160
What do I do first?

1277
01:16:38,912 --> 01:16:40,409
AUDIENCE: Superposition.

1278
01:16:40,409 --> 01:16:41,407
PROFESSOR: So superposition?

1279
01:16:41,407 --> 01:16:43,902
That's one thing.

1280
01:16:43,902 --> 01:16:46,397
AUDIENCE: Simplify
[UNINTELLIGIBLE].

1281
01:16:46,397 --> 01:16:47,395
PROFESSOR: Simplify.

1282
01:16:47,395 --> 01:16:51,886
So what's a good
simplification?

1283
01:16:51,886 --> 01:16:52,884
Collapse?

1284
01:16:52,884 --> 01:16:56,127
AUDIENCE: You can put the two
[UNINTELLIGIBLE] in series and

1285
01:16:56,127 --> 01:16:57,375
treat them as one.

1286
01:16:57,375 --> 01:17:01,780
PROFESSOR: So you can treat this
as a series combination,

1287
01:17:01,780 --> 01:17:06,025
and you can replace the series
of 1 and 3 with a?

1288
01:17:06,025 --> 01:17:06,995
AUDIENCE:
[UNINTELLIGIBLE PHRASE].

1289
01:17:06,995 --> 01:17:07,480
AUDIENCE: 4..

1290
01:17:07,480 --> 01:17:08,935
PROFESSOR: 4.

1291
01:17:08,935 --> 01:17:12,330
So this can be replaced by
four [UNINTELLIGIBLE]

1292
01:17:12,330 --> 01:17:12,815
resistor.

1293
01:17:12,815 --> 01:17:13,300
Now what?

1294
01:17:13,300 --> 01:17:15,725
AUDIENCE: You can do the same
on the parallel one.

1295
01:17:15,725 --> 01:17:18,343
PROFESSOR: So you can replace
the parallel of the 6

1296
01:17:18,343 --> 01:17:19,506
and a 12 with a?

1297
01:17:19,506 --> 01:17:20,342
AUDIENCE: 4.

1298
01:17:20,342 --> 01:17:22,653
PROFESSOR: Amazing
-- with a 4.

1299
01:17:22,653 --> 01:17:25,599
So there's a 4 there and
there's a 4 there.

1300
01:17:25,599 --> 01:17:26,090
And the answer is?

1301
01:17:26,090 --> 01:17:27,072
AUDIENCE: It's half.

1302
01:17:27,072 --> 01:17:28,545
PROFESSOR: Half of whatever
it was by voltage

1303
01:17:28,545 --> 01:17:30,020
[UNINTELLIGIBLE] relationship.

1304
01:17:30,020 --> 01:17:32,760
So you think about this
becoming that.

1305
01:17:32,760 --> 01:17:35,120
You think about the parallel
becoming that.

1306
01:17:35,120 --> 01:17:38,200
You get a simple divide
by 2 voltage divider.

1307
01:17:38,200 --> 01:17:46,660
So the answer is 7 and 1/2,
which was the middle answer.

1308
01:17:46,660 --> 01:17:51,780
And so what we did today was
basically a whirlwind tour of

1309
01:17:51,780 --> 01:17:54,790
the theory of circuits.

1310
01:17:54,790 --> 01:17:58,060
And the goal for the rest of the
week is to go to the lab

1311
01:17:58,060 --> 01:18:00,580
and do the same sort of thing
with practical where you build

1312
01:18:00,580 --> 01:18:03,440
a circuit, and try to use some
of these ideas to understand

1313
01:18:03,440 --> 01:18:04,690
what it does.