Skip to content

Series and Parallel Resistor Connections

In a circuit, a single, isolated resistor is rarely used. Multiple resistors can be combined to achieve an equivalent value. The two configurations are series and parallel.

Resistors in Series

In a series configuration, the resistors are connected one after the other.

The same current flows through all the resistors.

  • Example: \(R_1\) followed by \(R_2\) followed by \(R_3\) on the same circuit branch.

The equivalent resistance is calculated simply:

\[R_{\mathsf{eq}} = R_1 + R_2 + R_3 + \cdots\]

Therefore, connecting resistors in series increases the total resistance.

Resistors in Parallel

In a parallel circuit, the resistors are connected between the same two nodes.

They are "side by side" in the diagram. The same voltage is applied to each resistor, but the current is divided among the branches.

For two resistors in parallel:

\[\frac{1}{R_{\mathsf{eq}}} = \frac{1}{R_1} + \frac{1}{R_2}\]

For multiple resistors:

\[\frac{1}{R_{\mathsf{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots\]

In the special case of two identical resistors \(R\) in parallel:

\[R_{\mathsf{eq}} = \frac{R}{2}\]

In general, the equivalent resistance in parallel is always:

  • smaller than the smallest resistor in the group.

Intuitive Interpretation

We can reuse the hydraulic analogy:

  • Series: it’s like placing several narrow sections in a row in a pipe → the flow of water becomes more difficult → greater total resistance.
  • Parallel: it’s like placing several pipes in parallel between two reservoirs → the water has multiple paths → flow becomes easier → lower total resistance.

Key points:

Resistors in series

  • Same current through all resistors.
  • The equivalent resistance is the sum of the resistances:
  • \(R_{\mathsf{eq}} = R_1 + R_2 + \cdots\)
  • The total resistance is greater than each of the individual resistances.

Resistors in parallel

  • Same voltage across each resistor.
  • The equivalent resistance satisfies: \(\frac{1}{R_{\mathsf{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots\)
  • The equivalent resistance is smaller than the smallest of the resistors.
  • Special case: two identical resistors \(R\) in parallel → \(R_{\mathsf{eq}} = \frac{R}{2}\)

Equivalent Resistance Calculator

Name Resistance \((\mathrm{\Omega})\)
0 resistors
- Ω