Friday, March 23, 2012

Two current divider rules

Wikipedia has the following current divider rule

$$I_X = \frac{R_T}{R_X + R_T} \times I_T$$

where as certain websites have the following rule:

$$I_X = \frac{R_{total}}{R_X} \times I_T$$

where

$$\frac{1}{R_{total}} = \frac{1}{R_X} + \frac{1}{R_T}$$

for two resistors, this simplifies to

$$R_{total} = \frac{R_X \times R_T}{R_X + R_T}$$

These two rules are equal:

\begin{align}
I_X &= I_T \times \frac{R_{total}}{R_X} \\
&= I_T \times R_{total} \times \frac{1}{R_X} \\
&= I_T \times \frac{R_X \times R_T}{R_X + R_T} \times \frac{1}{R_X} \\
&= I_T \times \frac{R_T}{R_X + R_T}
\end{align}

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