Parallel Resistance

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Parallel Resistance Calculation

TotalR = 1 / ∑(1 / Rn...Rn)

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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

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