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Welcome to an Introduction to
Electrical & Computer Engineering.

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My name is Lee Brinton,

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

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

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voltage method.

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

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voltage.

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

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

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

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

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supernodes present in the circuit.

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First of all,
the similarities between elevation.

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

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

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In other words,
we establish our reference.

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Typically sea level is our
reference at zero feet.

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

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

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

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

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

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

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

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

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

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then Mount Olympus would be 5,400 feet,

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and sea level would be -4,100 feet.

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Elevations are all relative
to a reference, and

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similarly, that's true with voltages.

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

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voltage, and a no voltage.

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

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

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

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and a 12V drop across those resistors.

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

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

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

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

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

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

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

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Continuing on along here,

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

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

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

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

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just a drop across the single
element within the branch.

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

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

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

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

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

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

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we would have five different variables.

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

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

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The node voltage method involves,
rather than branch currents,

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it involves our defining node voltages.

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

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extraordinary nodes.

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

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

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

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another node.

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So we have one, two, three, four.

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I said that was, I made a mistake there.

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This is all one node, five nodes.

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

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

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we have three or
more branches coming to these.

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

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

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more branches come together,
and extraordinary or

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

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

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

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choose one of them as our reference, and

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define variables at the other

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two nodes, and

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then with those voltages equation

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

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

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In terms of V1 and this,
our voltage source,

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

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is V0 volts above our reference.

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Now, we can write an expression for

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

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

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

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resistors in terms of our node voltages,
or

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

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

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

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

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

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

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

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

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

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

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

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

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and the sum of those three
things must equal 0.

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Similarly, we write another KCl at node 2.

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

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the current leaving node
2 going this way is

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

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

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

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

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

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these equations other than
the different bias sign.

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

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

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

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

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leaving the node going in this direction,

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

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the sum of those three currents equal 0.

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So here we have two
equations with two unknowns,

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

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those two node voltages.