Node voltage method (steps 1 to 4) | Circuit analysis | Electrical engineering | Khan Academy

- [Voiceover] We're going to talk about

a really powerful way to analyze circuits

called the Node Voltage Method.

Before we start talking
about what this method is,

we're going to talk about a
new term called a node voltage.

So far, we already have the idea of

an element has a voltage
across it, and we refer to that

as an element voltage, or
if it's part of a circuit

and it's a branch of a circuit,

it'd be called a branch voltage.

That's a voltage that's associated

with a particular element.

So now we have the idea of
something called a node voltage.

This is still a voltage,
it's not anything strange,

but if we go over to our circuit here,

and we label the nodes.

Let's start labeling the nodes,
we'll call this node here

where this junction between
this resistor and this source,

we'll call this node one.

This is the junction between
these two components here.

There's another node that's
these two resistors connected

to this current source,
and that's a single

distributed node, so
we'll call that node two.

And down here, these three
components are connected together

in a junction, and that's node three.

To define a node voltage,
the first idea we need

is to define a reference node,
the idea of a reference node.

A good choice for the
reference node is usually one

that's connected to the
terminals of the power sources,

or it's the node that's
connected to a lot of branches,

a lot of elements, and node three here

is a good choice for a reference node.

The way we mark that is with
a symbol that looks like this,

a ground symbol, that's
called ground in this circuit.

There's other kinds of ways
to indicate a reference node.

That's a common way.

You can draw one that
looks like the ground,

connected to the ground.

Sometimes you'll see it
with just an upside-down T,

like that, that's another
way to draw a ground.

This symbol on a schematic
indicates the reference node.

We've picked a reference node
to be node three, down here.

So, a node voltage is
measured between a node

and the reference node.

In our case we have this voltage here,

is the node voltage on
node one, we'll call it V1.

This voltage here is
going to be called V2.

And in particular, these
voltages are measured

with respect to node three,
so there's the minus and plus

and minus and plus.

We're going to use these node voltages

in the Node Voltage Method.

First, what I want to do,

I want to label my complements here.

We're going to call this
Vs, and make it 15 volts.

This resistor's going to
be R1, and we'll give it

a value of 4kohms.

We'll call this R2, and we'll
give it a value of 2kohms.

This is the same circuit
that we analyzed when we did

application of the fundamental
laws in another video.

Oh, and the last guy here,
Is, current source Is,

and we'll make that one 3 milliamps.

We've analyzed this circuit before.

We used Kirchoff's Laws,
KVL and KCL, to figure out

what the voltages and
currents were in this circuit.

We're going to do the same analysis,

but this time we're
going to use what we call

the Node Voltage Method.

It's basically the same application

of the fundamental laws, we
use Ohm's Law, Kirchoff's Laws,

but it's in a really
clever, organized way,

that is really efficient.

Whoever thought this up was pretty bright

and I'm really glad
that they wrote it down

and shared it with us.

What I want to do first is just write down

what are the steps of this method?

It's not a theory, it's
a method, so it basically

a sequence of steps that you go through

to analyze the circuit.

I'll write the list right here.

First step is pick a reference node.

We already did that.

The second step is to
name the node voltages.

We already did that,
we named our nodes V1,

that node there is V1 and
that node there is V2,

with respect to the reference node,

which is down there at node three.

Whenever you talk about node voltages,

there's always an assumption
that one of the nodes

is a reference node.

The third step is to solve the easy nodes.

I'll show you what that means in a second.

The fourth step is to write KCL,

Kirchoff's Current Law equations.

The fifth step is to solve the equations.

That's the Node Voltage Method,

and we're going to go
through the rest of this,

we've done the first two steps.

What does it mean to solve the easy nodes?

The easy nodes are the
ones that are connected

directly to a source that
goes to the reference node.

That's an example of an easy node.

So V1 is an easy node.

So let's solve for V1.

By inspection, I can say V1 is 15 volts.

That's Step Three.

The other node's not easy,

the other node has lots of components

and something interesting's
going on over here.

So this was step three.

Let's label the steps.

Here's the Step One.

Here's Step Two.

And here's Step Three.

Now we're ready to go to Step Four,

let me move up a little bit.

Step Four is write the
Kirchoff's Current Law equations

directly from the circuit.

We're going to do this in a special way,

We're going to perform at
this node here, at node two.

We're going to write the
current law for this.

That means we got to
identify the currents.

There's a current, we'll
call that a current,

and that's a current.

Let me give some names to these
currents just to be clear.

We'll call this one I1
because it goes through R1.

We'll call this one here, I2
because it goes through R2.

This one is already Is.

Now let's write Kirchoff's
Current Law just in terms of I,

and we'll say all the
currents flowing into the node

add up to zero, so these
two have arrows going out,

so they're going to get negative signs

when we write Kirchoff's Current Law.

Let's do that right here.

And we write I1

minus I2

minus Is

equals to zero.

So right now we're working on Step Four.

This is the essence of
the Node Voltage Method.

This is where we do something new

that we haven't done before.

We're going to write these currents

in terms of the node voltages.

So we can write I1, I1 is
current flowing this way

through this current.

I1 equals V1

minus V2

over R1.

That's the current flowing in resistor R1,

in terms of node voltages.

The current flowing down through I2,

now we have to subtract I2,
so we just apply Ohm's Law

directly, which means
that the current in I2

is equal to V2 divided by R2.

The last current is Is, minus Is.

We'll write that in
terms of Is, like that,

and that equals zero.

This means we have now
completed Step Four.

That is KCL written using the
terminology of node voltages.

We could check off that
we've done Step Four.