Derivation of the Parallel Resistance Formula

by Jeff Schuler

July 27, 2009

12:21 am

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Electrical Engineering Concepts

parallel resistance formula

parallel-resistance-kcl

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 I_s, 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!

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