Resistors in Parallel

Understand parallel circuits and how to calculate total resistance

Overview

In a parallel circuit, the current divides and flows through multiple resistors simultaneously. The key characteristic of parallel circuits is that all resistors experience the same voltage.

Characteristic:

The total current divides among the branches, but the voltage across each resistor remains constant. This is the opposite of series circuits where voltage divides but current remains constant.

Total Resistance Calculation

Method 1: Using Ohm's Law

If the total current and voltage are known, we can calculate the total resistance directly using Ohm's Law:

\(\displaystyle R_{\text{total}} = \frac{U}{I}\)

Method 2: Using Conductivity Addition

When voltage and current values are unknown, we can add the individual conductivities (reciprocals of resistances). This is the most useful method for parallel circuits with known resistor values.

Key Principle:

In parallel circuits, conductivities add together. Since conductivity is the reciprocal of resistance, the reciprocals of the resistances add up.

\(\displaystyle G_{\text{total}} = G_1 + G_2 + G_3 + \ldots\)

Which can also be written as:

\(\displaystyle \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots\)

Key observation

The total resistance in a parallel circuit is always smaller than the smallest individual resistance. This makes sense because adding parallel paths reduces overall resistance.

Worked Example: Three Resistors

Calculate Total Resistance of Three Parallel Resistors

Given:

\(R_1 = 220\,\Omega\)
\(R_2 = 330\,\Omega\)
\(R_3 = 470\,\Omega\)

Step 1: Add the reciprocals

\(\displaystyle \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} = \frac{1}{220} + \frac{1}{330} + \frac{1}{470}\)

Step 2: Calculate the sum

\(\displaystyle \frac{1}{R_{\text{total}}} = 0.00455 + 0.00303 + 0.00213 = 0.00971\)

Step 3: Invert to get total resistance

\(\displaystyle R_{\text{total}} = \frac{1}{0.00971} = 103\,\Omega\)

Result: The total resistance is \(103\,\Omega\), which is indeed smaller than the smallest resistor (220Ω).

Special Case: Two Resistors in Parallel

For the common case of only two resistors in parallel, we can simplify the formula by combining fractions:

\(\displaystyle \frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_2 + R_1}{R_1 \cdot R_2}\)

By taking the reciprocal of both sides, we get a simpler formula:

Two Resistors in Parallel Formula

\(\displaystyle R_{\text{total}} = \frac{R_1 \cdot R_2}{R_1 + R_2}\)

This formula is often called the "product over sum" formula.

Example: Two Resistors

Calculate Total Resistance of Two Parallel Resistors

Given:

\(R_1 = 220\,\Omega\)
\(R_2 = 330\,\Omega\)

Using the two-resistor formula:

\(\displaystyle R_{\text{total}} = \frac{R_1 \cdot R_2}{R_1 + R_2} = \frac{220 \cdot 330}{220 + 330} = \frac{72,600}{550} = 132\,\Omega\)

Result: The total resistance is \(132\,\Omega\).

Formula Summary

General Formula (Any n Resistors)

\(\displaystyle \frac{1}{R_{\text{total}}} = \sum \frac{1}{R_i}\)

Two Resistors Only

\(\displaystyle R_{\text{total}} = \frac{R_1 \cdot R_2}{R_1 + R_2}\)

Using Ohm's Law

\(\displaystyle R_{\text{total}} = \frac{U}{I}\)
(if U and I are known)

Key Properties of Parallel Circuits

Same voltage across all resistors
Total current divides among resistors
Total resistance is smaller than any individual resistor
Conductivities (not resistances) add together
Adding more resistors in parallel reduces total resistance
Removing a resistor increases total resistance
If one branch fails (opens), others continue to work

Quick Calculation

Use the parallel resistor calculator to quickly determine total resistance for any combination of resistors:

Resistors in Parallel Calculator →

Resistors in Parallel
Resistors in Series
Resistor Color Coding

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