Parallel Circuits

Parallel is series with every rule turned over. Whatever is constant in one divides in the other, so learning them as a pair is far more reliable than learning two lists.

The mirror

Enlarge

Zoom in

Read that table across. Current and voltage swap roles, and so do the two resistance rules. That symmetry is the whole structure of the topic.

Why the total resistance falls

The result that surprises people: adding a resistor in parallel lowers the total resistance.

The reason is worth holding rather than memorizing. Each new branch is another path for current. More paths means more total current for the same voltage, and more current at the same voltage means less total resistance.

Think of doorways out of a room. Adding a second doorway does not make leaving harder.

So the total is always below the smallest branch, and that is the check that catches every arithmetic error in this topic.

Calculating the total

Two equal resistors: half of one.

Two 10-ohm resistors in parallel give 5 ohms. Three equal resistors give a third of one, and n equal resistors give one nth.

Two unequal resistors: product over sum.

R total = (R1 x R2) / (R1 + R2)

A 6-ohm and a 3-ohm resistor in parallel.

Product is 18, sum is 9, so 18 divided by 9 is 2 ohms.

Check: 2 is less than 3, the smaller branch. Correct.

More than two: use reciprocals.

1/R total = 1/R1 + 1/R2 + 1/R3, then flip the result.

Three 6-ohm resistors in parallel.

They are equal, so it is 6 divided by 3, which is 2 ohms. Reaching for reciprocals when the resistors are equal is doing unnecessary work.

Always run the check. A parallel total larger than the smallest branch means an error, every time.

How current divides

Each branch gets the full supply voltage, so each branch's current is that voltage divided by that branch's resistance.

A 12-volt supply across a 4-ohm and a 12-ohm resistor in parallel.

• 4-ohm branch: 12 divided by 4 is 3 amperes.
• 12-ohm branch: 12 divided by 12 is 1 ampere.
• Total current: 3 plus 1 is 4 amperes.

Check with the total resistance: product over sum is 48 over 16, which is 3 ohms, and 12 divided by 3 is 4 amperes. Agreed.

The lower resistance carries the larger current, and it is inversely proportional: the 4-ohm branch has one third the resistance and three times the current.

Current takes every available path, not just the easiest one. The phrase "current takes the path of least resistance" is misleading - it takes all of them, with more going through the easier one.

Why household wiring is parallel

Three reasons, all of them asked.

Every outlet gets the full supply voltage. In series each device would get only a share, and the share would change as devices were switched on.

Devices work independently. Switching off one lamp does not affect the others.

A failure in one branch leaves the rest working.

The cost is total current. Each device added in parallel lowers the total resistance and raises the total current drawn, which is why a circuit can be overloaded by plugging in too much, and why breakers exist.

Batteries in parallel

Voltage stays the same; capacity adds.

Two 1.5-volt cells in parallel give 1.5 volts, and they last about twice as long as one. That is the mirror of series cells, which add voltage and do not extend run time.

Cells in parallel must be the same voltage, or the higher one drives current backward into the lower one.

What you can skip

Across all 1,358 Electronics Information questions in our bank:

• Network theorems never appear by name. The parallel rules here and the series rules in the previous lesson carry every circuit question.
• Awkward reciprocal arithmetic. Nearly every parallel resistance question uses equal resistors (divide by the count) or two resistors (product over sum). The two three-branch questions use values that divide cleanly (2, 3 and 6 ohms; 4, 6 and 12 ohms), so there is no need to drill messy fractions.
• Big series-parallel networks. The handful of combination questions have one parallel pair and one series resistor. Reduce the pair first, then add.

Where people lose points

Adding parallel resistances. They combine reciprocally.

Getting a parallel total above the smallest branch. Always run the check.

Using product over sum with three resistors. That form is for two only.

Thinking voltage divides between parallel branches. It does not; it is the same across each.

Thinking adding a branch reduces the total current. It increases it.

Saying current takes only the path of least resistance.

Work one in under a minute

A 20-ohm and a 5-ohm resistor are in parallel across a 10-volt supply. Find the total resistance and the total current.

Product over sum: 100 over 25 is 4 ohms.

Check: 4 is less than 5. Correct.

Total current: 10 divided by 4 is 2.5 amperes.

Verify by branches: 10 over 20 is 0.5 A, and 10 over 5 is 2 A, and 0.5 plus 2 is 2.5. Agreed.

Where this leads

Between them, series and parallel cover every circuit question at this level, and the current-raising property of parallel is why circuit protection exists.