Combination of Resistors

Series Combination

When resistors are connected end-to-end:

$$R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots$$

• The current is the same through each resistor.
• The total voltage is the sum of individual voltage drops: $V = V_1 + V_2 + V_3$

Parallel Combination

When resistors are connected between the same two points:

$$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$$

• The voltage across each resistor is the same.
• The total current is the sum of individual currents: $I = I_1 + I_2 + I_3$

Example

Three resistors of $2\,\Omega$, $3\,\Omega$, and $6\,\Omega$ are connected in parallel. Find the equivalent resistance.

$$\frac{1}{R_{\text{eq}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3 + 2 + 1}{6} = \frac{6}{6} = 1$$ $$R_{\text{eq}} = 1\,\Omega$$