Series and Parallel Resistor Calculator

This interactive series and parallel resistor calculator allows you to obtain the equivalent total resistance ($R_{eq}$) for resistive circuits with any number of components. Below is the detailed physics and mathematics of these fundamental configurations.

What is Series and Parallel Resistance?

Resistors in an electrical circuit can be associated in two basic ways to modify the total or equivalent resistance seen by a power source:

• Series Connection: Resistors are coupled one after another, sharing a single common node per pair, so that the entire electrical current ($I$) flows sequentially through each of them without branching.
• Parallel Connection: Resistors are connected to the same two common nodes, all subjected to the same voltage or potential difference ($V$), while the total current is split among the different branches.

These associations allow you to adjust the resistance values available in the laboratory to those required by specific circuit designs.

Series and Parallel Resistor Formulas

The equations for calculating equivalent resistance are derived directly from Ohm's and Kirchhoff's Laws:

Resistors in Series

Since the current is constant throughout the entire path and the total voltage is the sum of the individual voltage drops, the equivalent resistance is the direct sum of each resistor:

$R_{eq} = R_1 + R_2 + R_3 + \dots + R_n$

Resistors in Parallel

Since the voltage is the same for all components in parallel and the total current is the sum of the individual currents, the equivalent resistance is calculated by the reciprocal of the sum of the reciprocals of the individual resistances:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$

Or equivalently:

$R_{eq} = \frac{1}{\sum_{i=1}^{n} \frac{1}{R_i}}$

For the simplified case of only two resistors in parallel, the following quick expression is commonly used:

$R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}$

Step-by-Step Calculation Example

Case 1: Series Connection

Suppose we connect three resistors in series with values of $R_1 = 100\,\Omega$, $R_2 = 220\,\Omega$, and $R_3 = 330\,\Omega$. We apply the direct formula: $R_{eq} = 100\,\Omega + 220\,\Omega + 330\,\Omega = 650\,\Omega$

Case 2: Parallel Connection

Suppose now we connect the same three resistors in parallel. We apply the general formula for three components: $\frac{1}{R_{eq}} = \frac{1}{100} + \frac{1}{220} + \frac{1}{330}$ $\frac{1}{R_{eq}} \approx 0.01 + 0.00454 + 0.00303 = 0.01757\,\Omega^{-1}$ $R_{eq} = \frac{1}{0.01757} \approx 56.9\,\Omega$

Frequently Asked Questions

Why is the equivalent parallel resistance always lower than the lowest of the individual resistors?

In a parallel configuration, adding more branches creates additional paths for the electric current to pass. This reduces the overall opposition to the flow of electrons, resulting in a total equivalent resistance that is lower than the value of the component with the lowest resistance in the set.

What happens if a resistor burns or opens in each configuration?

• In series: If the only available path is broken, the circuit opens completely, and current stops flowing through all components.
• In parallel: If one branch opens, the others continue to operate independently with the same supply voltage, although the total circuit current will decrease.

Can I combine series and parallel configurations in the same circuit?

Yes. This type of arrangement is known as a series-parallel or combination circuit. To calculate its equivalent resistance, you solve the individual parts step by step according to their connection (series or parallel) until the entire circuit is reduced to a single equivalent resistor.