Current Divider

If two elements are in parallel, the voltage across them must be the same, but the current divides according to the resistances. A simple circuit with two (or more) resistors in parallel with a source is called a current divider.

Figure B: Parallel Resistors.

Mathmatic Formula

If a voltage V appears across the resistors in Figure B with only ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ for the moment then the current flowing in the circuit, before the division, i is according to Ohms Law.

${\displaystyle i={\frac {V}{R_{eq}}}}$

Using the equivalent resistance for a parallel combination of resistors is

${\displaystyle i={\frac {V(R_{1}+R_{2})}{R_{1}R_{2}}}}$ (1)

The current through ${\displaystyle R_{1}}$ according to Ohms Law is

${\displaystyle i_{1}={\frac {V}{R_{1}}}}$ (2)

Dividing equation (2) by (1)

${\displaystyle i_{1}={\frac {iR_{2}}{R_{2}+R_{1}}}}$

Similarly

${\displaystyle i_{2}={\frac {iR_{1}}{R_{2}+R_{1}}}}$

In general with n Resistors the current ${\displaystyle i_{x}}$ is

${\displaystyle i_{x}={\frac {iR_{1}R_{2}\cdots R_{n}}{(R_{2}\cdots R_{n}+\cdots +R_{1}\cdots R_{n-1})R_{x}}}}$

Or possibly more simply

${\displaystyle {\frac {i_{x}}{i}}={\frac {R_{eq}}{R_{x}}}}$

Where

${\displaystyle R_{eq}={\frac {R_{1}\cdots R_{n}}{R_{2}\cdots R_{n}+\cdots +R_{1}\cdots R_{n-1}}}}$