Series resistors | Circuit analysis | Electrical engineering | Khan Academy

Now that we have our collection of components, our favorite batteries and resistors, we can start to assemble these into some circuits. Here's a circuit shown here; it has a battery and it has three resistors in a configuration that's called a series resistor configuration. Series resistors is a familiar pattern, and what you're looking for is resistors that are connected head to tail to head to tail.

So these three resistors are in series because their successive nodes are all connected one after the other. So, that's the pattern that tells you this is a series resistor connection. We're going to label these resistors here; we'll call this R1, R2, and R3, and we'll label this as V. The unknown in this is: what is the current that's flowing here? That's what we want to know. We know V; we want to know I.

Now, one thing we know about I is that it flows down into the resistor R1. All of the current comes out of the other end of resistor R1 because it has to; it can't pile up inside there. All that goes into here and all that comes out of R3, and I returns to the place it came from, which is the battery. So, that's a characteristic of series resistors, in this series configuration, is they are head to tail.

That means all the components, all the resistors, share the same current. That's the key thing. The thing that we don't know that's different between each resistor is the voltage here and the voltage here. Let's call that V1, this is V2 plus minus, and this is V3 plus minus. So, in general, if these resistors are different values because they have the same current going through them, Ohm's law tells us these voltages will all be different.

So the question I want to answer with series resistors is: could I replace all three of these with a single resistor that caused the same current to flow? That's the question we have on the table right now. So, we make some observations. We have Ohm's law, our friend Ohm's law, and we know that means V = I * R for any resistor. That sets the ratio of voltage to current.

There's another thing we know about this, which is that V3 plus V2 plus V1, those are the voltages across each resistor. Those three voltages have to add up to this voltage because of the way the wires are connected. So, the main voltage from the battery equals V1 plus V2 plus V3; we know that's for sure.

Now, what we're going to do is we're going to write Ohm's law for each of these individual resistors:

• V1 = I * R1
• V2 = I * R2
• V3 = I * R3

Now, you can see if I had four, five, or six resistors, I would have four, five, or six equations just like this for each resistor that was in series. So now, what I'm going to do is substitute these voltages into here, and then we'll make an observation.

So let's do that substitution. I can say V = V1 + V2 + V3, and because it's the same I on every resistor, I can write V = I * (R1 + R2 + R3). Now, what I want to do is take a moment here and compare this expression to this one here, the original Ohm's law.

Right, there's Ohm's law: V = I * R. I can come up with a resistor value, a single resistor that would give me the same Ohm's law, and that is going to be called R_series. Let's draw it over here. Here's our battery, and I'm going to say there's a resistor that I can draw here, R_series, that's equivalent to the three resistors here.

It's equivalent in the sense that it makes I flow here; that's what we mean by equivalent. So, in our case, to get the same current to flow there, I would say V = I * R_series, in which case what I've done is I've said that R_series is the sum of these three things: R1 + R2 + R3. This is how we think about series resistors; we can replace a set of series resistors with a single resistor that's equivalent to it if we add the resistors up.

Let's just do a really fast example to see how this works. I'm going to move the screen. Here's an example with three resistors; I've labeled them 100 ohms, 50 ohms, and 150 ohms. What I want to know is the current here, and we'll put it in a voltage. Let's say it's 1.5 volts, just a single small battery. So what is the equivalent resistance here?

One way to figure this out and to simplify the circuit is to replace all three of those resistors with a series resistor, R_s. As we said here, it's the sum: 100 + 50 + 150, and that adds up to 300 ohms. So that's the value of the equivalent series resistor right here.

If I want to calculate the current I, I = V / R; in this case, it's R_series, and that equals 1.5 divided by 300. If I do my calculations right, that comes out to 0.005 Amps, or an easier way to say it is five milliamps. So that's I; and now that I know I, I can go ahead, and I can calculate the voltage at each point across each resistor because I know I. I know R; I can calculate V.

So, V1, which is the voltage across that resistor, V1 = I * R1 as we said before. So it's 5 milliamps times 100 ohms, which equals 0.5 volts. Let's do it for the other one: V2 = I (the same I this time) times R2, 5 milliamps times 50 ohms, and that equals 0.25 volts.

Finally, we do V3, and this equals the same current again times 150 ohms, which is equal to 0.75 volts. So we've solved the voltage and the current on every resistor, so this circuit is completely solved.

Let's do one final check: let's add this up; 0.5 + 0.25 + 0.75 = 1.5 volts. That's very handy because that is the same as that. So indeed, the voltages across the resistors did add up to the full battery that was applied.

There's one more thing I want to point out; here's an example of some series resistors, and that's a familiar pattern. You'll say, "Oh, those are series resistors." Now be careful, because if there's a wire here going off and it's doing this, or there's a wire here connected to this node here, this looks like they're in series, but there might be current flowing in these branches here.

If there's current flowing out anywhere along a series branch, anywhere along what looks like a series branch, then this I may or may not be the same as this I, and it might not be the same as this. So you've got to be careful here. If you see branches going off, your series resistors are not in series unless these have zero current. If that's zero current and if that is zero current, then you can consider these in series.

So that's just something to be careful of when you are looking at a circuit, and you see things that look like they’re in series but have little branches coming off. So, a little warning there.

So that's our series resistors. If you have resistors in series, you add them up to get an equivalent resistance.