Node voltage method (step 5) | Circuit analysis | Electrical engineering | Khan Academy

And now we're down to solving this circuit. What I want to do now is put in the component values and solve this specific circuit. Let me move the screen up again. We'll leave the list of steps up there so we can see them. Let's go to work on this equation now. We have a little bit of algebra and we can plug in values where we need to. We can plug in 15 volts for D1, and for R1 we can plug in 4,000 ohms.

We can put in for V2. V2 is still unknown, and that's divided by 4K in this expression here. We still have V2 unknown over 2K. And is, let's put is over on the other side. Is was 3 milliamps. Let's just keep working at this now. V2 * 1 over 4K + 1 over 2K equals 3 milliamps. Oh, get my minus signs right. Minus sign over here.

Let's bring the constant term over to this side, so this is 15 volts divided by 4K. And continuing over here we have minus V2. Let's combine those two resistor terms, so it's going to be 1 + 2 over 4K equals 3 milliamps. And 15 volts divided by 4K is 3.75 minus 3.75 milliamps.

Moving on, minus V2 equals minus 0.75 milliamps times 4K over 3. Let's get rid of the two minus signs; we don't need those anymore. And V2 equals D dump dump one volt. That's good! We solved it. We solved our for our two voltages. We have one one here and we have one over here, and we can check off the last step.

So that's our first application of the node voltage method. I want to show you one more thing that is a powerful part of this technique. Let me quickly sketch the schematic again. So this was our schematic and we assigned node voltages. We assigned node voltages here V1 and V2, and we made this our reference node.

One of the things we did not do as part of the node voltage method, we did not use KVL to write equations around these loops. One of the features of the node voltage method is that the KVL equations, because we're using node voltages, the KVL equations are automatically satisfied.

And I'll show you why. I want to put one more label on here, which is the element voltage. We'll call this VR1; that's the element voltage across here. The element voltage here is just V2, so in this case for R2, V2 is the element voltage and the node voltage at the same time. VR1 is an element voltage.

And now we're going to write KVL starting from this point and going around the loop in this direction. What we have is, let's get all our labels on here, the loop voltages. We start with a rise of plus Vs, then we take away VR1, and then we take away V2, and that equals zero. So that would be the KVL equation for this circuit.

Now I'm going to plug in using node voltages. I'm going to write VR1, so I get plus Vs minus VR1 is node voltage V1 minus node voltage V2. V1 minus V2 minus V2 equals zero.

And we'll just do one more substitution. I forgot Vs and V1 are the same voltage, so this is actually V1 minus V1 minus V2 minus V2 equals zero. And if we look at this equation that goes plus V2 minus V2, this equation is automatically true.

If we write Kirchhoff's current law in terms of node voltages, it always turns out to be the case. That's why we don't bother to do it; we know it's going to be true. So that's a nice feature of the node voltage method. It's a really efficient way to write equations; you only write KCL equations.

And this is such a good method, in fact, that circuit simulators like you make come across a circuit simulator called SPICE. Almost every circuit simulator uses this node voltage method to do its computations.