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.