# Parallel Resistors:

## Introduction

If the total current into a set of parallel resistors is known, there is an easy way to determine the current through any individual resistor in the parallel combination. The appropriate formula is called the current divider formula, since the total current is divided among the individual resistors. For this reason, a pair of parallel resistors is commonly referred to as a current divider. Current dividers can be useful in electrical circuits. This section derives the current divider formula and provides examples of its use.

A basic current divider is a pair of resistors in parallel, as shown in Fig. 1. The resistors have resistance R1 and R2, and their individual currents are i1 and i2, respectively. The total current into the parallel combination is i, and the voltage across both resistors is v.1

• The current in each resistor is, by Ohm's Law:  ${i_1} = \frac{v}{{{R_1}}}$ and ${i_2} = \frac{v}{{{R_2}}}$ (1)
• Applying KCL at node “a” results in:  $i = {i_1} + {i_2}$ (2)
• Substituting equations (1) into equation (2) and solving for the voltage V results in:  $v = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}} \cdot i$ (3)
• Now, if we substitute equation (3) back into equations (1), we get:  ${i_1} = \frac{1}{{{R_1}}} \cdot \frac{i}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}}}$ (4)
• Which can be rearranged to give:  ${i_1} = \frac{{{R_2}}}{{{R_1} + {R_2}}} \cdot i$ (5)
• If we repeat the process for the current i2, we get:  ${i_2} = \frac{{{R_1}}}{{{R_1} + {R_2}}} \cdot i$ (6)

Equations (5) and (6) are the current divider relationships for two parallel resistances; so called because the current into the parallel resistance combination is divided between the two resistors. The ratio of one resistor's current to the overall current is the same as the ratio of the resistance of the other resistor to the total resistance.

## Important Generalization

The above results can be generalized for a series combination of N resistances:

• By Ohm's Law:  $v = {R_{eq}}i$

• The above results can be generalized for a series combination of N resistances:   $v = {R_{eq}}i$

• Substituting our previous result for the equivalent resistance for a parallel combination of N resistors results in:   $v = \frac{1}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + ...\frac{1}{{{R_N}}}}} \cdot i$

• Since the voltage difference across all resistors is the same, the current through the kth resistor is, by Ohm's law:   ${i_k} = \frac{v}{{{R_k}}}$

• Where Rk is the resistance of the kth resistor. Combining the above equations gives:  ${i_k} = \frac{{\frac{1}{{{R_k}}}}}{{\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + ...\frac{1}{{{R_N}}}}} \cdot i$ (7)

## Current Dividers in Terms of Conductance

It is often more convenient to provide the generalized result of equation (7) in terms of the conductance of the individual resistors. Recall that the conductance is the reciprocal of the resistance: $G = \frac{1}{R}$

Thus, the general equation in the note box above can be re-expressed as follows:

• The current through any resistor in a parallel combination of N resistances is proportional to the total current into the combination of resistors. The constant of proportionality is the same as the ratio of the conductance of the individual resistor value to the total conductance of the parallel combination. For example, the current through the kth resistance in a parallel combination of resistors is given by: ${i_k} = \frac{{{G_k}}}{{{G_1} + {G_2} + ...{G_N}}}i$
• Where i is the total current through the parallel combination of resistors.

#### Example:

For the circuit below, determine the current i.

#### Solution:

We know the total current into a parallel combination of resistances. Thus, we can use our current divider formula to obtain: $i = 12mA[\frac{{3k\Omega }}{{3k\Omega + 6k\Omega }}] \cdot 15V = \frac{3}{9} \cdot 12mA = 4mA$

## Important Points:

• If the total current through a set of parallel resistors is known, the current through any individual resistor can be determined by the concept of current division. The term current division comes from the fact that the current through a parallel combination of resistors is divided among the individual resistors.

• The ratio between the current through a particular resistor and the total current is the same as the ratio between the conductance of that resistor and the total conductance of the combination. If ik is the voltage across the kth resistor, iTOT is the total current through the parallel combination, Gk is the conductance of the kth resistor, and GTOT is the total conductance of the parallel combination, the mathematical statement of this concept is: $\frac{{{i_k}}}{{{i_{TOT}}}} = \frac{{{G_k}}}{{{G_{TOT}}}}$

1. Determine the value of I in the circuit below.
2. Determine the value of R in the circuit below which makes I = 2mA.

1. $I = 5mA\frac{{3k\Omega }}{{3k\Omega + 5k\Omega }} = \frac{{15}}{8}mA$
2. By the current divider relationship:    $I = 3mA\frac{R}{{R + 1k\Omega}}$
• Therefore, if I=2mA:    $\frac{R}{{R + 1k\Omega }} = \frac{2}{3}$