Resistors in parallel — the combined resistance from the branch currents

The column on the left holds the three inputs: Resistor 1, Voltage and Resistor 2. The top and bottom cards in the middle column are the branch currents, each worked out as voltage ÷ resistance. Total current, on the right, adds the two branch currents. Combined resistance, in the middle, divides the voltage on its left by the total current on its right.

The two resistors are sliders from 1 Ω to 40 Ω. You can edit the values directly; the four cards on the right-hand side update together.

In parallel, every branch gets the same voltage

In a parallel connection the current path splits, and the resistors sit side by side, one on each branch. Both branches are connected directly across the supply, so both get the same 12 V.

The current in each branch follows from Ohm's law, current = voltage ÷ resistance.

• The 4 Ω branch — 12 ÷ 4 = 3 A
• The 6 Ω branch — 12 ÷ 6 = 2 A

The current leaving the supply divides between the two branches and joins up again afterwards; none is gained or lost on the way. The total current is therefore the sum of the branch currents, 3 + 2 = 5 A.

From the total current to the combined resistance

The combined resistance is the single resistor that could replace the two parallel branches. A resistor that draws the same 5 A at the same 12 V looks exactly the same to the supply, and that resistor is 12 ÷ 5 = 2.4 Ω. Board 1's Combined resistance card does this division.

2.4 Ω is less than the smaller of the two resistors, 4 Ω. The branch currents show where the 2.4 Ω comes from. With the 4 Ω branch alone, 3 A flows. Adding the 6 Ω branch adds its 2 A, bringing the total to 5 A. More current at the same voltage means a smaller voltage ÷ current, so the combined resistance goes down.

Every branch carries some current above zero, so the total current is always larger than the current through the smallest resistor alone. That is why the combined resistance of a parallel circuit is always smaller than its smallest resistor.

Compared with series

The same 4 Ω and 6 Ω can instead be connected one after the other, in series. Board 2 works out the series resistance and current.

The inputs on the left are the same three as on Board 1. Series resistance, bottom right, adds the two resistors. Series current, top right, divides the voltage by that resistance.

In series there is only one path, and the current passes through both resistors in turn. The resistances add up along that path, so the series resistance is 4 + 6 = 10 Ω. At 12 V, the current is 12 ÷ 10 = 1.2 A.

In parallel, the same 12 V drove 5 A, more than four times the 1.2 A in series. Adding a resistor in series raises the combined resistance; adding one in parallel lowers it. In both cases the combined resistance is the voltage divided by the total current.

The reciprocal formula, 1/R = 1/R₁ + 1/R₂

The combined resistance of a parallel circuit is also written as 1/R = 1/R₁ + 1/R₂, where R is the combined resistance and R₁ and R₂ are the two resistors. The reciprocal formula expresses the same addition of branch currents.

With a voltage V, the branch currents are V/R₁ and V/R₂. The total current is their sum, and in terms of the combined resistance it is V/R. So V/R = V/R₁ + V/R₂, and dividing both sides by V gives 1/R = 1/R₁ + 1/R₂.

The reciprocal of a resistance is called its conductance, a measure of how easily current flows. Its unit is the siemens (S), which is the same as 1/Ω. Board 3 follows the formula step by step.

The two resistors are on the left, and the middle column takes the reciprocal of each. Total conductance, top right, adds the two conductances, and Combined resistance, bottom right, is the reciprocal of that total.

• Conductance of 4 Ω — 1 ÷ 4 = 0.25 S
• Conductance of 6 Ω — 1 ÷ 6 ≈ 0.167 S (0.166667 on the board)

Together they come to about 0.417 S (0.416667 on the board), and the reciprocal of that is 2.4 Ω, the same as on Board 1. Adding conductances gives the same result as adding the branch currents and dividing the voltage by their total.

Board 3 has no voltage card. The voltage was divided out when the formula was derived, so the combined resistance does not depend on it. On Board 1, 24 V gives branch currents of 6 A and 4 A and a total of 10 A, and the combined resistance is still 2.4 Ω.

For exactly two resistors the formula can be rearranged to R = R₁ × R₂ ÷ (R₁ + R₂): for 4 Ω and 6 Ω, 24 ÷ 10 = 2.4 Ω. With three or more resistors, the reciprocal form simply gains a term for each one: 1/R = 1/R₁ + 1/R₂ + 1/R₃, and so on.

Checking other numbers

On Board 1, with the voltage and the two resistors changed:

With two equal resistors, 4 Ω and 4 Ω, each branch carries 3 A. A single 4 Ω resistor would pass 3 A, so the two together pass twice as much, 6 A, and the combined resistance is half of 4 Ω: 2 Ω.

As Resistor 2 increases to 12 Ω, 20 Ω and 40 Ω, the combined resistance becomes 3 Ω, about 3.33 Ω and about 3.64 Ω, closer and closer to 4 Ω. The current through Resistor 2 gets smaller but never reaches zero, so the combined resistance never goes above 4 Ω.

At 24 V or 6 V the currents change in proportion to the voltage, and the combined resistance stays at 2.4 Ω.

With Resistor 2 at 4 Ω on Board 2, the series resistance is 8 Ω and the current 1.5 A. Against the parallel 2 Ω and 6 A, that is four times the resistance and a quarter of the current. With Resistor 2 at 4 Ω on Board 3, the total conductance is 0.5 S and the combined resistance 2 Ω, matching Board 1.

What this calculation covers, and what it does not

The boards treat the resistors and the supply as ideal. Each resistance stays the same whatever the current or the temperature, the supply voltage stays the same whatever current is drawn, and the wires have no resistance.

Real batteries and power supplies have some internal resistance, so their output voltage drops a little as the current rises. Adding branches in parallel increases the total current, and the voltage across each branch can end up slightly below the rated voltage. Real resistors also have a tolerance, so their actual values can differ slightly from the marked ones.

Each branch added in parallel increases the current drawn from the supply. Supplies and wires have a limit on the current they can carry, and the boards do not include that limit.

Read next — Average speed for a round trip — total distance over total time Calculates the average speed of a round trip with different speeds out and back, from the total distance and the total time.

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