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# Series resistors

Resistors connected head-to-tail are in series. The equivalent overall resistance is the sum of the individual resistance values. Written by Willy McAllister.
Components are in series if they are joined end to end like this:
We'll work with resistors to reveal the properties of the series connection.

## Resistors in series

Resistors are in series when they are connected head-to-tail and there are no other wires branching off from the nodes between components.
In the following image, start text, R, 1, end text, start text, R, 2, end text, and start text, R, 3, end text are in series:
Resistors in series share the same current.
The resistors in the following image are not in series. There are extra branches leading away from the nodes between resistors. If these branches carry current (orange arrows), then start text, R, 1, end text, start text, R, 2, end text, and start text, R, 3, end text do not share the same current.

### Properties of resistors in series

Here is a circuit with resistors in series:
Voltage source start text, V, end text, start subscript, start text, S, end text, end subscript is connected to the series resistor chain. Voltage v, start subscript, start text, S, end text, end subscript is some constant value, but we don't yet know the current i or how v, start subscript, start text, S, end text, end subscript splits up between the three resistors.
Two things we do know are:
• The three resistor voltages have to add up to v, start subscript, start text, S, end text, end subscript.
• Current i flows through all three resistors.
With this little bit of knowledge, and Ohm's Law, we can write these expressions:
v, start subscript, start text, S, end text, end subscript, equals, v, start subscript, start text, R, 1, end text, end subscript, plus, v, start subscript, R, 2, end subscript, plus, v, start subscript, R, 3, end subscript
v, start subscript, start text, R, end text, 1, end subscript, equals, i, dot, start text, R, end text, start subscript, 1, end subscript, v, start subscript, start text, R, end text, 2, end subscript, equals, i, dot, start text, R, end text, start subscript, 2, end subscript, v, start subscript, start text, R, end text, 3, end subscript, equals, i, dot, start text, R, end text, start subscript, 3, end subscript
This is enough to get going. Combining equations,
v, start subscript, start text, S, end text, end subscript, equals, i, dot, start text, R, 1, end text, plus, i, dot, start text, R, 2, end text, plus, i, dot, start text, R, 3, end text
We can factor out the current and gather the resistors in one place:
v, start subscript, start text, S, end text, end subscript, equals, i, left parenthesis, start text, R, 1, end text, plus, start text, R, 2, end text, plus, start text, R, 3, end text, right parenthesis
Since we know v, start subscript, start text, S, end text, end subscript, we solve for the unknown i,
i, equals, start fraction, v, start subscript, start text, S, end text, end subscript, divided by, left parenthesis, start text, R, 1, end text, plus, start text, R, 2, end text, plus, start text, R, 3, end text, right parenthesis, end fraction
This looks just like Ohm's Law for a single resistor, except the series resistors appear as a sum.
We conclude:
For resistors in series, the overall resistance is the sum of the individual resistors.

### Equivalent series resistor

We can imagine a new bigger resistor equivalent to the sum of the series resistors. It is equivalent in the sense that, for a given start text, V, end text, start subscript, start text, S, end text, end subscript, the same current start color #11accd, i, end color #11accd flows.
start text, R, end text, start subscript, start text, s, e, r, i, e, s, end text, end subscript, equals, start text, R, 1, end text, plus, start text, R, 2, end text, plus, start text, R, 3, end text
You may hear this slang expression for what is going on: From the "viewpoint" of the voltage source, the three series resistors "look like" one larger equivalent resistor. That means the current, start color #11accd, i, end color #11accd, provided by the voltage source is the same in both cases.
If you have several resistors in series, the general form of the equivalent series resistance is,
start text, R, end text, start subscript, start text, s, e, r, i, e, s, end text, end subscript, equals, start text, R, 1, end text, plus, start text, R, 2, end text, plus, point, point, point, plus, start text, R, end text, start subscript, start text, N, end text, end subscript

### Voltage distributes between resistors in series

We worked out the current i through the series connection, so what's left is to figure out the voltage across the individual resistors.
Do this by applying Ohm's Law to the individual resistors,
v, start subscript, start text, R, end text, 1, end subscript, equals, i, dot, start text, R, end text, start subscript, 1, end subscript, v, start subscript, start text, R, end text, 2, end subscript, equals, i, dot, start text, R, end text, start subscript, 2, end subscript, v, start subscript, start text, R, end text, 3, end subscript, equals, i, dot, start text, R, end text, start subscript, 3, end subscript
This becomes more interesting if you do an example, with real numbers. I encourage you to try this on your own before revealing the answer.
start text, P, R, O, B, L, E, M, space, 1, end text
a. What is the current start color #11accd, i, end color #11accd and what are the voltages across the three resistors?
b. Show the individual resistor voltages add up to start text, V, end text, start subscript, start text, S, end text, end subscript.

### Reflection

Based on the resistor voltages you just computed:
Problem 2
The largest resistor has the largest or smallest voltage?

problem 3
The smallest resistor has the largest or smallest voltage?

problem 4
The resistor with the highest current is ...?

### Summary

Resistors in series share the same current.
start text, R, end text, start subscript, start text, s, e, r, i, e, s, end text, end subscript, equals, start text, R, 1, end text, plus, start text, R, 2, end text, plus, point, point, point, plus, start text, R, end text, start subscript, start text, N, end text, end subscript
Voltage distributes amongst series resistors, with the largest voltage across the largest resistor.

## Want to join the conversation?

• what grade is this information for
i know its open to all grades but i want to know what specific grade this is for i am 13 years old am i behind or what •   You aren't behind at all! Electrical Engineering is university level material. But if you're interested, don't let that discourage you! It's never too early to play around and start learning some circuitry. Some of the math might be a little daunting (we're talking calculus and higher), but a lot of circuitry can be designed and analyzed with just a bit of algebra and determination.

If you're really interested in engineering, talk to a physics teacher at your school. I'm sure they'd love to walk you through building a couple circuits. Some high schools have electronics classes or even robotics clubs that are a lot of fun.

Engineering is fun, challenging, and extremely rewarding, whether you want to study it professionally or if you just want to build gadgets to mess with. Good luck!
• There's an error while calculating I for R2. R1 is written twice. Thanks for your great article. • Why in series current is not the same? • I looked at your skill check questions and I found them leading to mistaken identification. Yes, there were resistors in series, and resistors in parallel, but a number of questions were intermixing series and parallel circuits together. If you would have asked for Rt, following the answers you were looking for, you would have arrived at incorrect answers. I would be HAPPY to help out with your circuit problems. OBEARCC • Is there a paragraph specifically about only series circits • I'm completely new at this. Could someone explain why we know that "The three resistor voltages have to add up to ​Vs"? I don't understand this. Couldn't you just take out a resistor, in which case the other resistors would have less voltage across them than at the battery?
(1 vote) • Think about the properties of the voltage source and the resistors.
The voltage source says, "No matter what, the voltage at my terminals is Vs. I will provide whatever current is required by my connections."

Resistors are not so rigid. They obey the much more flexible Ohm's Law, which says, "The ratio of my voltage to current is a constant ratio, determined by my value R." v/i = R.

You remove one of the three resistors and just have two resistors connected in series to the voltage source. The voltage source still says, "My voltage is Vs." The two resistors are connected across Vs. The two of them have to split up that voltage such that Ohm's Law is still obeyed for each of them.

I like analogies. Suppose you have three rubber bands of different stretchiness (resistance). Tie them together in series and stretch out the chain of rubber bands a fixed distance apart (Vs). Each rubber band will adjust to some different length to take up the stress (voltage on each rubber band).

Now remove one rubber band. Stretch out the remaining two to the same distance. They will stretch to a different longer length to span the same distance. Series resistors sort of do the same thing.
• can you wach a VIDEO!/!/!/!
(1 vote) • I would like to understand how to apply Ohm's law even with just single resistor but given a constant maximum mA current. I have a device that is giving off a maximum of 5V, 44 mA and it's sinking into a device which can a accept up to 40 mA. How do I size the resistor to arrive at a maximum pull of 20 mA or 10 or 40 exactly?

- H
(1 vote) • That depends on the voltage that you want to be dropped across the load when the current is 40mA and how the current through the load varies with applied voltage. Using a resistor to limit current through a load implies some voltage is dropped in that resistor, the larger the resistor the larger the voltage drop.

If I take the common example of an LED, which will have a certain maximum current rating and forward voltage drop at that current. For example, 20mA at 2V, in this case, to power the LED from a 5V supply you could use a resistor to drop the difference. The resistor's value would be (5-2)/0.02 = 150 ohms. If the supply voltage decreases however, the current through the LED will go down, the LED will get dimmer. To keep current constant you would need a device whose resistance varied depending upon output current, a constant current source, these usually take the form of an integrated circuit of some sort but can be made using discrete transistors.
• It seems like two of the problems show the wrong answer. In problem 2, doesn't the largest resistor (1200 ohms) have the largest voltage (6V)? The answer shows the opposite. In problem 4, don't all the resistors share the same current? The answer is showing the middle one??
(1 vote) • Hi! I understand the voltage goes from negative to positive. I'm confused on why in the diagrams the battery has one orientation of negative to positive while the resistors have the opposite orientation. Shouldn't they be in the same direction? Thanks!
(1 vote) • When someone hands you a circuit it has no labels.

Your first step is to scribble labels for voltage and current on the diagram. The direction of those labels (both current arrow and voltage polarity) is arbitrary (there is no "right" way to do it).

Usually you will annotate a voltage source with a voltage name that's in the same polarity as the source (so the + of the orange voltage label lines up with the black + on the voltage source). For the resistors, there is no "natural" starting polarity, so your choice of how you draw the voltage labels is up to you.

Your best bet is to think about the current arrow and the voltage label together. They should be chosen so the current arrow flows into the + voltage terminal of the battery (this is the "Sign Convention for Passive Components"). It makes Ohm's Law easier to apply.

Other than this tip you can orient the resistor voltage labels any way you want. When you work out the answers for V and I you will get positive or negative values. Your original labels tell you which way is positive.