What Is Equivalent Resistance?

Equivalent resistance describes how an entire resistor network behaves when viewed from a chosen pair of terminals. Instead of tracking every resistor inside the network, you replace it with one resistor that has the same effect on the rest of the circuit.

The terminal pair matters: the same network can have different equivalent resistances when measured between different points. For a network made only of ideal, linear resistors, the equivalent resistance is the ratio of the voltage across those terminals to the current entering the network:

$$R_{\mathrm{eq}}=\frac{V}{I}.$$

What the definition means

Suppose you connect a voltage source across a resistor network. If the network draws the same current as a single resistor would draw at that voltage, that resistor is an equivalent resistance. The replacement preserves the network’s behavior at the selected terminals; it does not preserve the currents and voltages inside the network.

For example, if a two-terminal network draws $2\,\mathrm{A}$ when $12\,\mathrm{V}$ is applied, its equivalent resistance is

$$R_{\mathrm{eq}}=\frac{12\,\mathrm{V}}{2\,\mathrm{A}}=6\,\Omega.$$

A $6\,\Omega$ resistor connected in place of the network would draw that same $2\,\mathrm{A}$ under the same conditions.

Finding it for resistor networks

You can calculate equivalent resistance by repeatedly replacing simple groups of resistors with their equivalents.

For resistors in series, the same current passes through each resistor, and their voltage drops add. Therefore,

$$R_{\mathrm{series}}=R_1+R_2+\cdots+R_n.$$

For resistors in parallel, each resistor has the same voltage across it, and the branch currents add. Therefore,

$$\frac{1}{R_{\mathrm{parallel}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}.$$

For two resistors in parallel, this becomes $R_{\mathrm{parallel}}=R_1R_2/(R_1+R_2)$. More complicated circuits may require Kirchhoff’s current and voltage laws, symmetry arguments, or a circuit-analysis method such as nodal analysis.

Worked example

Consider a $6\,\Omega$ resistor in parallel with a $3\,\Omega$ resistor, with that parallel pair in series with a $4\,\Omega$ resistor. Find the resistance seen across the whole combination.

First reduce the parallel pair:

$$\begin{aligned} R_{\parallel}&=\left(\frac{1}{6\,\Omega}+\frac{1}{3\,\Omega}\right)^{-1}\\ &=\left(\frac{1}{6}+\frac{2}{6}\right)^{-1}\!\Omega\\ &=2\,\Omega. \end{aligned}$$

Then add the series resistor:

$$R_{\mathrm{eq}}=2\,\Omega+4\,\Omega=6\,\Omega.$$

As a check, connect a $12\,\mathrm{V}$ source across the combination. The total current should be $12\,\mathrm{V}/6\,\Omega=2\,\mathrm{A}$. The $4\,\Omega$ resistor drops $8\,\mathrm{V}$, leaving $4\,\mathrm{V}$ across the parallel pair. Its branches draw $4/6=2/3\,\mathrm{A}$ and $4/3\,\mathrm{A}$, respectively; together they draw $2\,\mathrm{A}$, as expected.

Equivalent resistance when sources are present

The simple ratio $V/I$ gives a single equivalent resistance across operating conditions when the terminal voltage–current relation is linear and passes through the origin. For a nonlinear network, $V/I$ is only the ratio at a particular operating point. If the network contains an independent source, its terminal voltage may not be proportional to its current. In that case, a linear network can often be represented by a Thevenin equivalent: an ideal voltage source in series with a resistance.

The Thevenin resistance of a linear network is the resistance seen looking into the terminals with independent voltage sources set to zero (short-circuited) and independent current sources set to zero (open-circuited). Dependent sources must remain active; if they are present, a test voltage or current is used to find the resistance. This is a related but more specific use of equivalent resistance than simply reducing a collection of resistors.

Common mistakes

• Ignoring the terminals: Equivalent resistance is always defined relative to a specified pair of points.
• Treating every network as a resistor-only network: A circuit with an internal source may need a Thevenin model, rather than a single resistor.
• Adding parallel resistances directly: In parallel, add reciprocals; direct addition applies to series resistors.
• Assuming the internal circuit stays unchanged: The equivalent resistor matches behavior at the terminals, not the distribution of current or voltage within the original network.
• Forgetting real components can vary: Temperature, frequency, and operating conditions can change a component’s resistance. The ideal-resistor formulas assume those effects are negligible.