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

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 having 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 would be called a branch voltage. That's a voltage that's associated with a particular element.

Now, we have the idea of something called a node voltage. This is still a voltage; it's not anything strange. 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, and these two resistors are connected to this current source, and that's a single distributed node. So, we'll call that node two.

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 a node that's connected to a lot of branches, a lot of elements. 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 are other kinds of ways to reference to indicate a reference node; that's a common way. You can draw one that looks like an upside-down T; that's another way to draw ground.

So, this symbol on a schematic indicates the reference node, and we've picked a reference node to be node three down here. A node voltage is measured between a node and the reference node. In our case, we have this voltage here; it is the voltage on node one, the node voltage on node one. We'll call it V1, and this voltage here is going to be called V2. In particular, these voltages are measured with respect to node three, so there's the minus and plus.

We're going to use these node voltages in the node voltage method. So, first, what I want to do, I want to label my components here. We're going to call this Vs and make it 15 volts. This resistor is going to be R1, and we'll give it a value of 4 kΩ. We'll call this R2, and we'll give it a value of 2 kΩ.

This is the same circuit that we analyzed when we did the application of the fundamental laws in another video. Oh, and the last guy here, the current source is I, and we'll make that one 3 milliamps. We've analyzed this circuit before; we used Kirchhoff'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, Kirchhoff'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. This method, it's not a theory; it's a method, so it's basically a sequence of steps that you go through to analyze a circuit, and I'll write the list right here.

The first step is to pick a reference node, and we already did that. The second step is to name the node voltages, and 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 third step is to solve the easy nodes, and I'll show you what that means in a second. The fourth step is to write KCL, Kirchhoff's current law equations, and 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, and now we're going to solve. 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.

Let's solve for V1. V1, I can just by inspection, I can say V1 is 15 volts. So, that's step three. The other node's not easy; the other node has lots of components, and something interesting is going on over here. So this was step three.

Let's label the steps: here's step one, here's step two, and here's step three. Now, we're ready to go to step four, and let me move up a little bit. Step four is to write the Kirchhoff's current law equations directly from the circuit.

We're going to do this in a special way. The current law, we're going to perform at this node here, at node two. We're going to write the current law for this, and 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. And we'll call this one; this one is already called I.

Now, let's write Kirchhoff'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 Kirchhoff's current law.

Let's do that right here, and we write I1 minus I2 minus I equals to zero. So, right now, we're working on step four. And now we're going to do this; 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 the 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 voltage flow—sorry, 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 over R2, and the last current is minus I; we'll just keep, we'll write that in terms of I like that, and that equals zero.

So, this means we have now completed step four. That is KCL written using the terminology of node voltages, and we can check off that we've done step four.