Kirchhoff's current law | Circuit analysis | Electrical engineering | Khan Academy

Up to now, we've talked about, uh, resistors, capacitors, and other components, and we've connected them up and learned about OHS law for resistors. We also learned some things about series resistors, like we show here the idea of Kirchhoff's laws. These are basically common sense laws that we can derive from looking at simple circuits.

In this video, we're going to work out Kirchhoff's current law. Let's take a look at these series resistors here. There's a connection point right there, and that's called a node, a junction. One of the things we know is that when we put current through this—let's say we put a current through here—and we know that current is flowing charge. So, we know that the charge does not collect anywhere. That means it comes out of this resistor and flows into the node, and then goes across, and it comes out on this side.

All the current that comes in comes out. That's something we know. That's the conservation of charge, and we know that the charge does not pile up anywhere. We'll call this current I1, and we'll call this current I2. We know we can just write right away that I1 equals I2. That seems pretty clear from our argument about charge.

Now, let me add something else here. We'll add another resistor to our node like that, and now there's going to be some current going this way. Let's call that I3. Now, this doesn’t work anymore; I1 and I2 are not necessarily the same. But what we do know is any current that goes in has to come out of this node. So, we can say that I1 equals I2 plus I3. That seems pretty reasonable.

In general, what we have here is if we take all the current flowing in, it equals all the current flowing out. That's Kirchhoff's current law. That's one way to say it, and in mathematical notation, we would say: I in (the summation of currents going in) equals the summation sign (the summation of I out). That's one expression of Kirchhoff's current law.

So now, I want to generalize this a little bit. Let's say we have a node, and we have some wires going into it. Here are some wires connected up to a node, and there's current going into each one. I'm going to define the current arrows. This looks a little odd, but it's okay to do, all going in. What Kirchhoff's current law says is that the sum of the currents going into that node has to be equal to zero.

Let's work out how that works. Let's say this is 1 amp, and this is 1 amp, and this is 1 amp. The question is, what is this one? What's that current there? If I use my Kirchhoff's current law expressed this way, it says that 1 + 1 + 1 + I (whatever this I here) has to equal zero. What that says is that I equals -3. So that says -3 amps flowing in is the same exact thing as +3 amps flowing out.

One amp, one amp, one amp comes in, and 3 amps flow out. Another way we could do it, equally valid—this is just three ways to say exactly the same thing. I have a bunch of wires going to a junction like this, and this time I define the currents going out. Let's say I define them all going out, and the same thing works. The sum of the currents this time going out—I'll go back over here, and I'll write in all the currents going in—has to equal zero as well.

You can do the same exercise. If I make all these 1 amp and ask what is this one here? What is I, here, the outgoing current? It's 1 + 1 + 1 + 1. Those are the four that I know, and those are the ones going out. So, what's the last one going out? It has to equal zero. The last one has to be -4 to equal zero, so this is a current of -4 amps.

So that's the idea of Kirchhoff's current law. It's basically we reason through it from first principles because everything that comes in has to leave by some route. And when we talked about it that way, we ended up with this expression for Kirchhoff's current law. We can come up with a slightly smaller mathematical expression if we say, let's define all the currents to be pointing in. Some of them may turn out to be negative, but then that's another way to write Kirchhoff's current law.

In the same way, if we define all the currents going out, you actually have your choice of any of these three anytime you want to use these. If we define them all going out, this is Kirchhoff's current law, and we'll use this all the time when we do circuit analysis.