Checked against primary sources 2026-08-24

One sentence per configuration and you never confuse them again

In series, current is common. In parallel, voltage is common. Everything else follows from those two facts, including the check that catches your own arithmetic.

The two sentences

In a series circuit, the same current flows through every element, and the voltages across them add up to the source voltage.

In a parallel circuit, the same voltage sits across every element, and the currents through them add up to the total current.

That is the whole topic. Every formula you have seen for combining resistances is a consequence of one of those two sentences.

Which quantity is common, and which one splits

Say the configuration out loud as a pair: what is shared, and what divides. Series shares current and divides voltage. Parallel shares voltage and divides current.

That pairing answers a whole family of items without any arithmetic. Ask what happens to the other elements when one element changes.

• Series, add an element. Total resistance goes up, the one shared current goes down, and every element in the string sees less current. Nothing in a series circuit is independent of anything else.
• Parallel, add a branch. Total resistance goes down, total current goes up, and every existing branch carries exactly what it carried before, because the voltage across it has not moved.
• Series, open one element. Everything stops. The current had one path and there is now no path.
• Parallel, open one branch. That branch stops and the others do not notice.

How the split happens is worth one line each, because items are written on it. Voltage divides in proportion to resistance, so the largest element in a series string takes the largest share. Current divides inversely with resistance, so the smallest branch in a parallel set takes the largest share. People remember one of those and apply it to both.

In the parallel example below, the 4 ohm branch is the smallest resistance and carries 6 amperes, the most of the three. In the series version, the 6 ohm element is the largest resistance and takes 12 volts, the most of the three. Same circuit elements, opposite winners.

Resistance, the part people memorize unnecessarily

In series, resistances add. That follows from the first sentence: same current everywhere, voltages add, so resistances add.

In parallel, the reciprocals add. Also a consequence: same voltage everywhere, currents add, so conductances add, and resistance is the reciprocal of conductance.

Worked both ways, on the same circuit

Put 24 volts across three parallel branches of 12, 6 and 4 ohms. The branch currents are 2, 4 and 6 amperes, so the total is 12 amperes and the total resistance is 24 over 12, which is 2 ohms.

Now get there by reciprocals: a twelfth plus a sixth plus a quarter is one, two and three twelfths, which is six twelfths, which is a half. Invert it and you have 2 ohms. Same answer, arrived at without touching the current.

The series version of the same three resistors, wired 4 plus 6 plus 2 across the same 24 volts, is 12 ohms and 2 amperes, and the drops are 8, 12 and 4 volts. They sum to 24, which is the check.

Two shortcuts, and the exact conditions on each

Both of these answer parallel items with no fractions at all, and both are wrong outside their conditions, so learn the condition with the shortcut.

Equal branches: divide by how many there are

Two equal branches of 10 ohms give 5 ohms. Three equal branches of 9 ohms give 3 ohms. The rule is the branch value divided by the number of branches, and it holds only while the branches are equal to each other.

Two equal is the case that shows up most, and the answer is simply half. If a stem gives you two identical motors, two identical heaters or two identical anything on the same two wires, you already have the answer before you have finished reading the sentence.

Try it on unequal branches and it has nothing to work with, because there is no single branch value to divide. Average them instead, which is the repair people reach for, and the 12, 6 and 4 ohm set above gives 22 over 3, about 7.33 ohms. That is larger than the smallest branch, so the sanity check in the next section catches it before you can write it down.

Exactly two branches: product over sum

For two branches and only two, multiply them together and divide by their sum. Six and three ohms: eighteen over nine, which is 2 ohms. Confirm it by reciprocals, a sixth plus a third is a half, invert, 2 ohms.

It agrees with the halving rule wherever both apply. Two 10 ohm branches: a hundred over twenty, which is 5 ohms, the same answer.

The word exactly is the condition. Applied to the three-branch set above it gives 288 over 22, about 13.1 ohms, which is bigger than every branch in the circuit and therefore obviously wrong. If you have three or more branches, either work two at a time and then combine the result with the third, or go to reciprocals.

The sanity check that catches errors

Total parallel resistance is always smaller than the smallest branch. Always. No exception.

The 2 ohms above is below the 4 ohm branch, so it passes. If your parallel answer comes out bigger than one of the branches you have made an arithmetic error, and you have caught it in two seconds without rechecking any of your work.

The equivalent check in series runs the other way: the total is always larger than the largest element, and 12 ohms clears the 6 ohm element comfortably.

Run both checks on every answer, including the ones you got from a shortcut. That is the habit that makes a shortcut safe to use under a clock, because a shortcut used outside its conditions produces a number the check refuses.

What this is worth on the papers

Two subject lines on the content outline have theory in the name. On the journeyman knowledge portion, Definitions, Theory, and Plans takes 3 items of the 56 scored. On the journeyman calculations portion, Calculations and Theory takes 2 of the 24. That is 5 of the 80 scored items a journeyman sits across both papers. A master candidate gets 7 and 2, out of 70 and 30.

Those two lines are not the measure of this topic, and reading them as one is how candidates decide theory is not worth studying. Series and parallel is machinery rather than subject matter. It sits underneath any item where loads share a circuit, any item about voltage at the far end of a long run, any item that describes two conductors doing the job of one, and every load calculation on the paper.

The practical consequence is about speed rather than about coverage. Nobody is going to ask you to state the reciprocal rule. They are going to hand you a circuit on a paper that allows 110 minutes for 26 items, a little over four minutes each once the unscored pretest items you cannot pick out are counted in, and the arithmetic is the part you should not still be thinking about.

Where this shows up beyond theory questions

Loads on a branch circuit are in parallel

Every receptacle and every luminaire on a circuit sits across the same two conductors, which is why unplugging one does not affect the others and why each one gets full voltage rather than a share of it.

It is also the arithmetic behind every load calculation you will ever do. The circuit current is the sum of the branch currents, because that is what parallel means. A load calculation is a very long parallel-circuit problem wearing a code book.

Control devices are in series

A contactor coil with a stop button, a float switch, a limit switch and an overload contact wired ahead of it is a series string, and every one of those devices has to be closed for the coil to pull in. Open any single one and the load drops out, which is the behavior the arrangement was chosen for.

That is also the troubleshooting shape. On a series string, one open kills everything and the fault could be any of the devices. On a parallel arrangement, one dead load points straight at that load or its connection, because everything else is still working.

Paralleled conductors

Run two conductors as one and the current divides by impedance rather than by intention, so a set whose paths are not identical loads one conductor harder than its partner while the overcurrent device ahead of them sees only the total and never notices.

That is why NEC 310.10(G) puts conditions on a paralleled set, and why those conditions are about the conductors being identical to each other rather than about any single dimension you could check by eye. The list is longer than most people remember it being, so read it in your own book rather than from memory, and read it beside our page on the paralleled set, which works through it condition by condition.

Overcurrent devices in parallel, which the code very nearly forbids

Fuses and circuit breakers are permitted in parallel only where they were factory assembled in parallel and listed as a unit, and individual devices or combinations of them may not otherwise be connected in parallel (NEC 240.8).

The physics is the reason. Two devices joined at both ends share current in inverse proportion to impedance, exactly as two conductors do, so the current through each one is not the half of the total that its rating was chosen against. Paralleling conductors is a design decision the code manages with conditions on the conductors. Paralleling the protection is a different thing, and the section closes it except where a manufacturer has built and listed the pair as one device.

Voltage drop on a long run

The conductor resistance sits in series with the load, so the load gets the source voltage minus whatever the run took. On a 240 volt circuit whose conductors drop 7 volts, the equipment at the far end sees 233.

What this page cites

• NEC 310.10(G) Conductors in parallel, and the conditions a paralleled set has to satisfy. Still at this address in the 2026 edition. source
• NEC 240.8 Fuses or circuit breakers in parallel. Permitted only where factory assembled in parallel and listed as a unit. Read from this repository's transcription of the 2026 text of Article 240 in data/nec2026-art240.js. source
• PSI candidate information bulletin, TDLR electrical examinations Updated 9 July 2026. Publishes the content outline for each portion. Definitions, Theory, and Plans carries 3 of the 56 scored items on the journeyman knowledge portion and 7 of 70 for a master, and Calculations and Theory carries 2 of the 24 scored on the journeyman calculations portion and 2 of 30 for a master. source