What a series connection does to current and voltage
Components are in series when they form a single unbroken path, so that the same charge must pass through every one of them in turn. That gives you the defining property: the current is identical in every element. There is nowhere else for it to go.
Voltage behaves the opposite way. The supply voltage is shared out among the elements, and each one's share is set by Ohm's law at the common current: Vn = I·Rn. A large resistance drops a large voltage; a small one drops a small voltage. Kirchhoff's voltage law states that the drops around any closed loop sum to the applied voltage, so those individual shares must add back to exactly what the supply provides.
Because the drops divide in proportion to resistance, a series chain is a voltage divider whether you intended it to be one or not. Adding a resistance to a chain always raises the total resistance and therefore lowers the current, which is why a series resistor is the standard way to limit current into an LED, a meter movement or a charging capacitor.
The other thing to notice is what a series connection does to reliability. One open element breaks the entire path — a single failed lamp in an old-style series Christmas string extinguishes the whole set. That is why practical wiring puts loads in parallel and reserves series connections for switches, fuses, current-sense shunts and deliberate current-limiting elements.
Why the resistances simply add
Take two resistors carrying the same current I. The first drops I·R₁ and the second drops I·R₂. The total voltage across the pair is therefore I·R₁ + I·R₂ = I·(R₁ + R₂). Compare that to V = I·Req and the equivalent resistance must be R₁ + R₂. The argument extends to any number of elements, which is the entire derivation.
Two consequences follow that are worth carrying around. First, order is irrelevant: swapping resistors along the chain changes nothing about the total, the current or any individual drop. Second, the total is always larger than the largest single resistance — the mirror image of the parallel rule, and an equally good sanity check.
The voltage-division form is the one you will use most: Vn = V·Rn/Req. Because it depends only on a ratio of resistances, the fraction each resistor takes is fixed by the values alone. Scaling every resistance in a chain by ten leaves every voltage exactly where it was and divides the current by ten — which is precisely how you reduce a divider's current draw without changing its output. The voltage divider calculator works that two-resistor case in detail, including what happens when you connect a load.
Power distributes along the chain as Pn = I²·Rn. Since I is common, the largest resistance always dissipates the most power in a series chain. That is the reverse of the parallel case, where the smallest resistance runs hottest, and mixing the two rules up is a reliable way to specify the wrong part.
Worked example: 100 Ω, 220 Ω and 330 Ω on a 12 V supply
Three resistors sit in a single loop across 12 V. Work through every quantity by hand.
- Add the resistances. 100 + 220 + 330 = 650 Ω. Check the sanity rule: 650 Ω is above 330 Ω, the largest single value. Good.
- Find the current. I = 12 ÷ 650 = 0.0184615 A, or 18.46 mA. This same current flows in all three resistors.
- Voltage across R1. 0.0184615 × 100 = 1.8462 V.
- Voltage across R2. 0.0184615 × 220 = 4.0615 V.
- Voltage across R3. 0.0184615 × 330 = 6.0923 V.
- Check Kirchhoff's voltage law. 1.8462 + 4.0615 + 6.0923 = 12.0000 V, exactly the supply. If your three drops do not sum to the supply, something is wrong upstream of this step.
- Power in each resistor. P = I²R, and I² = 0.00034083. So R1 takes 0.034083 W, R2 takes 0.074982 W, and R3 takes 0.112473 W.
- Total power, two ways. 0.034083 + 0.074982 + 0.112473 = 0.221538 W, and V·I = 12 × 0.0184615 = 0.221538 W. They agree.
Read the proportions: R3 is 330/650 = 50.8% of the total resistance and takes 50.8% of the supply voltage and 50.8% of the power. All three percentages are the same number, because with a common current every quantity that matters scales with R. Practically, all three parts are comfortable in 1/4 W packages here — the worst case, 0.112 W, is under half of a 1/4 W rating even before you apply the usual derating.
How to read the result
Check three things every time.
Does the total exceed the largest resistance? It must. If it does not, you have either mistyped a unit or accidentally computed a parallel combination.
Do the drops sum to the supply? The table on this page splits the supply into shares that add to 100% by construction, so any discrepancy in your own hand calculation points to a rounding error in the current — carry more digits in I than you need in the answer.
Is the hottest resistor within its rating? The highest-dissipation output tells you which value drives the wattage specification. If you buy all six parts the same, size them all from that number, doubled for margin. A common surprise is a current-limiting resistor in a low-voltage supply: 12 V across a 68 Ω dropper at 176 mA is 2.1 W, which needs a wirewound part, not a 1/4 W film resistor.
Beyond that, series resistance is the natural way to think about several practical problems. A long run of conductor is a series resistance in every branch circuit, and its drop is subtracted from the voltage the load actually sees — the National Electrical Code addresses this in informational notes recommending that branch-circuit voltage drop be held to about 3%, and the voltage drop calculator handles conductor material, length and size. Similarly, a battery's internal resistance sits in series with everything it powers, which is why terminal voltage sags under load.
Two-resistor series chains on a 12 V supply
| R₁ | R₂ | Total | Current | Drop across R₁ | Drop across R₂ |
|---|---|---|---|---|---|
| 100 Ω | 100 Ω | 200 Ω | 60.000 mA | 6.000 V | 6.000 V |
| 100 Ω | 220 Ω | 320 Ω | 37.500 mA | 3.750 V | 8.250 V |
| 100 Ω | 470 Ω | 570 Ω | 21.053 mA | 2.105 V | 9.895 V |
| 220 Ω | 330 Ω | 550 Ω | 21.818 mA | 4.800 V | 7.200 V |
| 470 Ω | 470 Ω | 940 Ω | 12.766 mA | 6.000 V | 6.000 V |
| 1 kΩ | 2.2 kΩ | 3.2 kΩ | 3.750 mA | 3.750 V | 8.250 V |
| 1 kΩ | 4.7 kΩ | 5.7 kΩ | 2.105 mA | 2.105 V | 9.895 V |
| 2.2 kΩ | 3.3 kΩ | 5.5 kΩ | 2.182 mA | 4.800 V | 7.200 V |
| 10 kΩ | 10 kΩ | 20 kΩ | 0.600 mA | 6.000 V | 6.000 V |
Rows with the same value ratio produce identical drops at very different currents — 100 Ω with 220 Ω and 1 kΩ with 2.2 kΩ both give 3.750 V and 8.250 V, but the second draws one tenth of the current. That is how you make a divider stingier without moving its output.
Mistakes that produce a wrong series result
- Applying the reciprocal formula. Series resistances add directly. Reciprocals belong to the parallel resistance calculator, and using them here gives an answer far too small.
- Mixing units. Entering 4.7 for a 4.7 kΩ resistor with ohms selected leaves it contributing almost nothing to the total. Set the unit selector before typing.
- Sizing every resistor from the total power. Each part only dissipates I²R for its own value. Size from the individual figures, and from the largest of them if you buy one part number.
- Forgetting the source's internal resistance. A coin cell with tens of ohms of internal resistance is itself a significant series element, and it will not deliver the current this calculation predicts.
- Ignoring lead and contact resistance in low-value chains. Below about an ohm, connector and trace resistance is a real fraction of the total and your measured current will fall short.
- Assuming a series divider holds its output under load. As soon as you draw current from the junction, the lower resistor is effectively paralleled and the ratio changes.
- Treating a series string of LEDs as resistive. LEDs in series do share one current, which is the correct way to run them, but their combined forward voltage subtracts from the supply before any Ohm's-law calculation begins.
Where series resistance is used, and what to reach for next
Four jobs account for most deliberate series resistors.
Current limiting. Putting a resistor in series with a device fixes the maximum current at (Vsupply − Vdevice)/R. This is exactly how an LED is driven — see the LED resistor calculator for the forward-voltage subtraction that has to happen first.
Making a value you do not have. Two E24 values in series reach many intermediate resistances exactly, and unlike the parallel case the arithmetic is trivial to do in your head.
Current sensing. A small, precise shunt in series lets you infer current from the voltage across it. Keep the shunt small enough that its own drop does not disturb the circuit, and remember it dissipates I²R.
Sharing voltage stress. Series resistors split a high voltage among parts, so no single component sees more than its rated working voltage. Ordinary film resistors have a maximum working voltage independent of their power rating, and a chain is the standard way past it.
For real networks you will usually alternate between this page and the parallel calculator, collapsing the circuit inward one group at a time. Where a network is neither — a Wheatstone bridge, for instance — no amount of series-parallel reduction will resolve it, and you need the wye-delta resistance conversion calculator or a full nodal analysis. For the underlying single-element relationships, the Ohm's law calculator covers every rearrangement, and the electrical power calculator handles what happens once alternating current and power factor enter the picture.
Key terms
- Kirchhoff's voltage law
- The algebraic sum of voltages around any closed loop is zero — equivalently, the drops in a series chain sum to the applied voltage. It follows from energy conservation.
- Voltage drop
- The potential difference across one element, I·R. In wiring practice the term usually refers to the unwanted drop along the conductors themselves.
- Voltage division
- The rule that each element in a series chain takes a fraction of the supply equal to its share of the total resistance: Vₙ = V·Rₙ/R_eq.
- Internal resistance
- The resistance inherent to a source, in series with everything it drives. It is why a battery's terminal voltage falls as current rises.
- Shunt
- A low-value precision resistor placed in series specifically so that the voltage across it indicates the current through it.
- Maximum working voltage
- The highest voltage a resistor may have across it regardless of power dissipation, set by the physical construction. Series chains are used to stay within it.