Capacitors in Series and Parallel

Finds the total capacitance of several capacitors. In parallel they simply add; in series you add the reciprocals — the opposite of resistors.

Explanation

This is the total capacitance of several capacitors wired together. The rules are the reverse of the ones for resistors.

\text{Parallel}: \quad C = C_1 + C_2 + \cdots

\text{Series}: \quad \dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots

Resistors add in series and combine reciprocally in parallel. Capacitors do the opposite.

Example

Take capacitors of 10 μF, 20 μF and 30 μF.

In series:

\dfrac{1}{C} = \dfrac{1}{10} + \dfrac{1}{20} + \dfrac{1}{30} = \dfrac{11}{60} \qquad C = \dfrac{60}{11} = 5.45\ \mu\text{F}

Smaller than the smallest capacitor in the chain.

In parallel the total is simply $10 + 20 + 30 = 60$ μF.

Why the rules invert

Go back to the parallel plate formula, $C = \varepsilon \dfrac{S}{d}$ , and it becomes obvious.

Wiring in parallel is like widening the plates. $S$ grows, so the capacitance grows
Wiring in series is like pushing the plates apart. $d$ grows, so the capacitance falls

Losing capacitance in series sounds like a bad bargain, but it buys something: the voltage rating goes up, because the applied voltage divides itself between the capacitors.

In practice

Parallel — when you need more capacitance, as in a power supply smoothing bank
Series — when you need to withstand a higher voltage