L3 1 1 Node Voltage

Introduces Node Voltage method of analyzing circuits

Welcome to an Introduction to
Electrical & Computer Engineering.

My name is Lee Brinton,

I'm an electrical engineering instructor
at Salt Lake Community College.

In this video we'll be talking about
ways to analyze circuits using the node

voltage method.

We're gonna start by making a, looking at
the similarities between elevation and

voltage.

We'll then discuss the difference between
branch voltages and node voltages.

We'll introduce the concept of nodes and
critical or extraordinary nodes.

We'll then look at the actual process of
analyzing circuits using node voltages,

and we'll look at how that applies
when they are dependent sources and

supernodes present in the circuit.

First of all,
the similarities between elevation.

If we wanna talk about the elevation of,
say Mount Olympus,

we've got to define first of all
what we're measuring relative to.

In other words,
we establish our reference.

Typically sea level is our
reference at zero feet.

Mount Olympus then is about
9,500 feet above sea level,

and the Salt Lake Valley Floor is
around 4,100 feet above sea level.

On the other hand,
if we're standing on the Valley Floor, and

looking up to the mountains to the east,
we'll see that the elevation,

the amount that they rise, can be
calculated by taking the higher elevation,

9,500 feet less the lower elevation
of 4,100 feet, and we've got a 5,400

foot change in elevation going from the
Valley Floor to the top of Mount Olympus.

On the other hand, if instead of
calling sea level our reference,

we made the Valley Floor our reference and
said then the elevation here = 0 feet,

then Mount Olympus would be 5,400 feet,

and sea level would be -4,100 feet.

Elevations are all relative
to a reference, and

similarly, that's true with voltages.

To understand that, let's make sure we
understand the difference between a branch

voltage, and a no voltage.

A branch voltage is the voltage
across a branch within a circuit.

In this case here,
we've got a 3V drop across this resistor.

Over here, we've got a 10V drop
across that resistor, similarly a 2V,

and a 12V drop across those resistors.

On the other hand,
if we wanted to talk about the voltage,

at a point in the circuit, we would need
to specify what we were referring it to.

So let's create,
we'll call this node here 0V,

and then as we come along this
branch we go up from 0, up 15V,

so at this point, the voltage there is
15V above 0, or above our reference.

Now as we traverse this branch here,
we drop 3V,

getting to a voltage here of 15-
3 = 12V above our reference.

Continuing on along here,

we drop another 10V to 2V
above our reference, and

then continuing on down across these
2V to back to our zero reference.

Thus the distinction a node voltage where
the voltage add a node is a voltage at

the node relative to some reference,
whereas the voltage across the branch is

just a drop across the single
element within the branch.

Up until now, as we've been analyzing
circuits, we've identified branch

currents and voltages, and
worked with those as our variables.

For example, we have a branch current
here, call it i1, we have another branch

current here, call it i2,
another here, i3, i4,

and perhaps referencing like that, i5,
and of course we know that i5 in this

case equals i0, but in order to analyse
this circuit using branch currence,

we would have five different variables.

And when using those variables we could
then write KDL and KCL equations, and

solve for any branch voltage or branch
current in that circuit we wanted to do.

The node voltage method involves,
rather than branch currents,

it involves our defining node voltages.

In order to do that, we need to specify or
to make the distinction between nodes and

extraordinary nodes.

A node is a point where two or
more branches are joined.

Here we've got a node, there's another
node here, there's another node there, and

here's a node, here's a node, and then
all the way along the bottom here is yet

another node.

So we have one, two, three, four.

I said that was, I made a mistake there.

This is all one node, five nodes.

So one, two, three, four,
five regular nodes,

and now we have this node here
where we have more than two,

we have three or
more branches coming to these.

Those types of nodes are referred to as
extraordinary nodes, or critical nodes.

In other words,
a node is a place where two or

more branches come together,
and extraordinary or

critical node is a place where three or
more branches come together.

In this case we have one, two,
three extraordinary or critical nodes.

Our approach is going to be then
to identify the critical nodes,

choose one of them as our reference, and

define variables at the other

two nodes, and

then with those voltages equation

at each of those critical nodes,
in terms of V1 and V2.

For example, let's start by summing
the currents leaving this node right here.

In terms of V1 and this,
our voltage source,

we can first of all note that,
the voltage at that node

is V0 volts above our reference.

Now, we can write an expression for

the current leaving this node by
taking the voltage of this side

minus the voltage at this side of those
resistors and dividing by the resistence.

In other words, we're specifying
the branch voltage across those two

resistors in terms of our node voltages,
or

thus we would write (V1-V0)/(R2+R3)

will be the current leaving our
first node going to the left.

Now let's do similarly for
the other two branches, and

sum those three currents together and
acknowledge that the sum of the three

currents leaving that node must
equal zero, just cuz current law.

So the current now leaving that node
coming downs through our one would be,

V1, the voltage at the top,
minus the voltage at this side,

which in this case is just 0,
divided by R1.

And, finally, the current leaving
node 1 going to the ride would be,

the voltage of the left hand side
of that resistor would be V1,

the voltage on the right
hand side would be V2, so

(V1-V2)/R4 represent the sum of
the three currents leaving that node,

and the sum of those three
things must equal 0.

Similarly, we write another KCl at node 2.

So this is node 1, and
then at node 2 we have,

the current leaving node
2 going this way is

going to be the voltage
at the right-hand side,

V2 minus the voltage of
the left-hand side V1 divided by R4.

Note right now, that the current
leaving node 2 going to the left

is equal, but of opposite sign to
the current leaving node 1, and

going to the right, and you'll notice
those two terms are the same in each of

these equations other than
the different bias sign.

This term in the first equation,
(V1-V2)/R4,

and that term in the second
equation (V2-V1)/R4.

All right continuing on, we now add
the current leaving node 2 going down.

that current will be (V2/R5) + the current

leaving the node going in this direction,

well actually the current is going in so
we'll subtract -I0 = 0,

the sum of those three currents equal 0.

So here we have two
equations with two unknowns,

it becomes simply a matter of
algebra at this point to solve for

those two node voltages.