Capacitors in Series

Capacitors in series

Wire capacitors in series and you add up the reciprocals to get the reciprocal of the equivalent capacitance: 1 / Ctotal = 1/C₁ + 1/C₂ + … + 1/Cn. With just two of them that collapses to C = (C₁ · C₂) / (C₁ + C₂). Notice that C_total is always smaller than the smallest capacitor in the chain, because putting them in series cuts the capacitance down. Every capacitor carries the same charge Q, yet the voltage splits inversely with capacitance, so the smaller C ends up with more of it. Energy adds straightforwardly: Etotal = E₁ + E₂ + …. Example: put two 220 µF / 16 V capacitors in series and you get C = 110 µF rated, on paper, for 32 V. Reality is harsher. Skip the balancing resistors across each capacitor and the leakage currents won't match, so one capacitor ends up well past its rated voltage and gives out.

Applications: high-voltage banks and capacitive dividers

People mostly reach for series capacitors when they need to raise the voltage rating — stacking 450 V electrolytics so the bank can take the 800 V DC link of three-phase inverters and variable-frequency drives. Put balancing resistors (usually somewhere between 100 kΩ and 1 MΩ) across every capacitor, no exceptions. You'll also see series used in capacitive voltage dividers that sample AC for metering, in tuned LC filters when the lower capacitance you want simply isn't sold as a single part, and in RF circuits, where small precise capacitances get assembled out of larger standard values.

FAQ

Why does series reduce capacitance? Capacitance grows with plate area and shrinks as the plates move apart. Stringing capacitors in series widens the gap between the outermost plates without giving you any more area to work with.

Can I mix different values in series? You can, but keep in mind the smaller capacitor soaks up proportionally more voltage. Hand a 10 µF and a 100 µF a combined 110 V and the 10 µF will see 100 V while the 100 µF only gets 10 V. It's an easy way to over-volt the small one.

Why are balancing resistors needed? No two real capacitors leak at exactly the same rate. Without parallel resistors holding the voltages even, one of them charges past its rating and blows up.

Does polarity matter? With electrolytics it does. Line them all up the same way, or switch to bipolar (non-polarized) types.