# Parallel Resistance

## Quick Reference

### Parallel Resistance Calculation

`TotalR = 1 / ∑(1 / Rn...Rn)`

## More Info

Resistors (or resistances) can be arranged in parallel, as in the following diagram:

### Conductance; the Siemens (S)

When resistors are arranged in this configuration, their total resistance is calculated by adding up the *conductance*, measured in *siemens* (S), which is defined as the reciprocal of resistance:

`Siemens = 1 / Resistance in Ω`

The letter `G`

is often used to denote conductance/siemens, so the units calculate as follows:

`G = 1/R`

R = 1/G

### Calculation

Therefore, total resistance is calculated by:

`TotalR = 1 / ∑(1 / Rn...Rn)`

### Calculation Steps

To calculate the resistance of a parallel resistor network, we have to:

- Convert each individual resistance to conductance
- Add the conductances together
- Convert the sum back to resistance

#### Example

Let's consider the same resistor values we had in series, and calculate them in parallel:

- R1 - 100Ω
- R2 - 5Ω
- R3 - 1KΩ

First, we need to convert each value to siemens:

`G1 = 1/100Ω = 0.01 S`

G2 = 1/5Ω = 0.2 S

G3 = 1/1KΩ = 1/1000Ω = 0.001 S

Once we have their conductance, we add them to get the total conductance:

`0.01 + 0.2 + 0.001 = 0.211 S`

Converting from `0.211`

siemens to ohms:

`Resistance = 0.211 S = 1/0.211 = 4.74Ω`

Total resistance with the same resistors as we had in series is now `4.7Ω`

in parallel.

## Parallel Resistor Banks

Sometimes, resistors in parallel come in banks of the same resistor values. In this case, there's a shortcut to calculate the total resistance:

`Total Resistance = Resistance of Each Resistor / Number of Resistors`

Therefore, (10), 5KΩ resistors in parallel would be:

`5,000Ω / 10 = 500Ω`

## Common Voltage, Different Current

In a parallel resistance circuit, the voltage at each resistor is the same, but the current flowing through each resistor is dependent on the amount of resistance that resistor has.

### Power Calculation

Since we know the voltage and resistance, we can use the `P = V^2 / R`

form of the power calculation equation, and just as with series resistance, we add an `n`

to specify power and resistance at resistor number `n`

:

`Pn = V^2 / Rn`

## Further Reading

For a more in-depth discussion about resistors and resistance, check out Part 4 of the Electronics Tutorial.