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Series and parallel resistors

The total resistance of resistors connected in series is the sum of their individual resistance values.

R e q = ∑ i = 1 n R i = R 1 + R 2 + ⋯ + R n. {\displaystyle R_{\mathrm {eq} }=\sum _{i=1}^{n}R_{i}=R_{1}+R_{2}+\cdots +R_{n}.}

The total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.

R e q = (∑ i = 1 n 1 R i) − 1 = (1 R 1 + 1 R 2 + 1 R 3 + ⋯ + 1 R n) − 1 {\displaystyle R_{\mathrm {eq} }=\left(\sum _{i=1}^{n}{\frac {1}{R_{i}}}\right)^{-1}=\left({1 \over R_{1}}+{1 \over R_{2}}+{1 \over R_{3}}+\dots +{1 \over R_{n}}\right)^{-1}}

For example, a 10 ohm resistor connected in parallel with a 5 ohm resistor and a 15 ohm resistor produces 1 / 1/10 + 1/5 + 1/15 ohms of resistance, or 30 / 11 = 2.727 ohms.

A resistor network that is a combination of parallel and series connections can be broken up into smaller parts that are either one or the other. Some complex networks of resistors cannot be resolved in this manner, requiring more sophisticated circuit analysis. Generally, the Y-Δ transform, or matrix methods can be used to solve such problems.