# Derivation of the Parallel Resistance Formula

### Derivation of the Parallel Resistance Formula

It’s easy to see where the series resistance formula comes from; add a resistor, add its resistance.  It’s more difficult to see where the parallel resistance formula comes from.  However, the formula can be derived using a simple combination of KCL and Ohm’s law.  Take the simple circuit shown in Figure 11.   KCL says that the total current in the circuit must be equal to the currents I1 and I2.

From KCL:           $I_{s} = I_{1} + I_{2}$

Using Ohm’s law gives equations for Is, I1, and I2.

From Ohm’s Law: $I_{s} = \frac{V}{R_{1} || R_{2}}$  $I_{1} = \frac{V}{R_{1}}$   $I_{2} = \frac{V}{R_{2}}$

Plug these three equations back into the first equation and solve for R1||R2 to derive the formula for two parallel resistors.

$\frac{V}{R_{1} || R_{2}} = \frac{V}{R_{1}} + \frac{V}{R_{2}}$

$\frac{1}{R_{1} || R_{2}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$ Which gives us…

$R_{1} || R_{2} = (\frac{1}{R_{1}} + \frac{1}{R_{2}})^-1 = \frac{R_{1}R_{2}}{R_{1} + R_{2}}$

There you have it, a technical explanation of the parallel resistance formula.

Written by Ryan Eatinger (reatinge@ksu.edu).  Thank you!