Resistor series equivalent:
formula and worked example

Why series resistance simply adds

When resistors are connected end to end, there is only one path for current. The same current flows through every resistor, and each one drops its own share of the supply voltage. The more resistors you add, the harder it is to push current through the string, so the total resistance is the plain sum of the parts.

That makes series connections a handy way to reach a value you do not have in stock, to share a voltage between components, or to build a simple divider. It also means the series resistance is never smaller than the largest single resistor.

A useful picture is a line of people passing buckets of water along a chain. The flow is the same past every person, and each one adds a little effort. Adding another person makes the whole task harder by exactly their share.

The formula and what to enter

• R1, R2, R3: each resistor's value in ohms. Convert kilohms to ohms before adding, so 1.2 kΩ becomes 1,200 Ω.
• Result: the equivalent resistance in ohms, which is what the rest of the circuit sees.
• Extra resistors: the same pattern extends to any number by adding further terms.

If you are working backwards, the same relation tells you what to add. Needing 1,000 Ω and holding a 680 Ω resistor, you need a further 320 Ω in series, which could be a 330 Ω part, with the total landing at 1,010 Ω.

Worked example: 220, 330 and 470 ohms

Three resistors of 220, 330 and 470 ohms are connected in series.

1. 1R1: 220 Ω.
2. 2R2: 330 Ω.
3. 3R3: 470 Ω.
4. 4Add: 220 + 330 + 470 = 1,020 Ω.

Place this string across 12 V and the current is 12 ÷ 1,020 = 11.76 mA. The 470 Ω resistor drops about 5.5 V, the 330 Ω about 3.9 V and the 220 Ω about 2.6 V, which add back to 12 V.

Power follows the same pattern. At 11.76 mA the 470 Ω resistor dissipates about 65 mW, the 330 Ω about 46 mW and the 220 Ω about 30 mW, for a total near 141 mW, which matches 12 V times 11.76 mA. All three sit well below the rating of a typical quarter-watt part.

What the equivalent value tells you

The equivalent resistance sets the current from a given voltage by Ohm's law, and the share of each resistor sets how voltage divides. A resistor holding half the total resistance takes half the voltage, which is how voltage dividers work.

• Current is the same everywhere in the string.
• Voltage across each resistor is proportional to its value.
• Power in each resistor is the current squared times its value, so the largest resistor gets hot first.

Because the largest resistor takes the largest share of the voltage, a small error in a big resistor matters more in a divider than the same relative error in a small one. When designing a divider, spend your accuracy budget on the parts that carry the most voltage.

Which resistor moves the total

A 10% increase in R1, from 220 to 242 Ω, adds 22 Ω to the total and raises it from 1,020 to 1,042 Ω, about 2.2%. A 10% change in the largest resistor, 470 Ω, would have shifted the total by 47 Ω. In a series string the biggest resistor has the most influence, so that is where tolerance matters most.

Compared with a parallel connection the behaviour is very different. In parallel, a small resistor dominates the total; in series, the large one does. Remembering which resistor controls the result helps you spot an implausible answer quickly.

Mistakes with series resistance

• Using the parallel formula, where reciprocals are added, for a series string.
• Mixing kilohms and ohms in the same sum.
• Ignoring tolerance: five-percent parts can put the 1,020 Ω total anywhere within roughly 970 to 1,070 Ω.
• Forgetting wire and contact resistance in low-value or precision circuits.
• Overlooking power ratings, since each resistor must handle its own share of heat.

When to choose a series string

Series strings are a simple way to build an odd value from standard ones. Here, 1,020 Ω is close to a value on a standard series, and you might use a single 1 kΩ resistor instead if the tolerance allows. If you need a value above the largest one you stock, add resistors in series. For a value below it, you need a parallel arrangement instead.

• Convert every value to ohms.
• Add the nominal values.
• Compute the worst case from tolerances.
• Check each resistor's power.
• Measure the finished string if accuracy matters.