Capacitors combine the opposite way round from resistors. In parallel they add — 100 µF, 47 µF and 10 µF give 157 µF — and in series they follow a reciprocal sum, so two 10 µF capacitors give 5 µF. That is the mirror image of resistors, and it follows from what each quantity measures rather than from a convention worth memorising separately.
Resistance opposes current, so extra paths reduce it. Capacitance stores charge, so extra plates increase it: parallel capacitors are effectively one capacitor with a larger plate area.
What does each arrangement give you?
Different things, which is why both are used deliberately.
| Arrangement | Total capacitance | Voltage rating | Typical use |
|---|---|---|---|
| Parallel | Adds | The lowest of the set | More storage, lower ESR |
| Series | Reciprocal sum | Adds up | Higher working voltage |
Parallel is how you get bulk capacitance and low equivalent series resistance at once, which is why supply rails carry a large electrolytic with small ceramics beside it — the electrolytic holds the charge and the ceramics handle the fast transients the electrolytic is too slow for.
Series is how you get a voltage rating no single part offers, and it comes with a trap. The voltage only divides evenly if the capacitors are matched and their leakage currents are similar; in practice leakage differs, the string drifts, and one part ends up carrying more than its share. Balancing resistors across each capacitor are the standard fix.
What does the RC time constant tell you?
How fast the pair responds. Tau is resistance times capacitance, and a 10 kΩ resistor with a 10 µF capacitor gives τ = 100 ms — 63.2 per cent of the way to the final value in one time constant, and effectively settled after five.
| Time | Charged to |
|---|---|
| 1τ | 63.2% |
| 2τ | 86.5% |
| 3τ | 95.0% |
| 4τ | 98.2% |
| 5τ | 99.3% |
The shape is the thing to internalise: charging is fastest at the start and slows continuously, so the last few per cent take as long as the first sixty. A 470 µF capacitor through 1 kΩ has τ = 470 ms and is effectively full after 2.35 seconds.
It also never quite reaches the supply voltage, which is why five time constants is the working definition of "charged" rather than a number anyone derived from a requirement.
Why is the same pair also a filter?
Because time and frequency are two views of the same physics. In the time domain an RC pair is a delay or a debounce characterised by tau; in the frequency domain it is a first-order low-pass filter with a −3 dB cutoff at 1 ÷ (2πRC) and a roll-off of 20 dB per decade.
The 10 kΩ and 10 µF pair above has a cutoff of 1.59 Hz. That is the same statement as τ = 100 ms, said the other way round, and which description is useful depends only on whether you are thinking about a step or a signal.
Inrush is the practical consequence. A large reservoir capacitor looks like a short circuit at the instant power is applied, so the initial current is limited only by whatever resistance is in the path — which is why equipment with a big supply either has a series resistor that is shorted out once charged, or trips the breaker on the way up.
How do inductors fit in?
As the mirror of capacitors. A capacitor opposes changes in voltage and passes high frequencies; an inductor opposes changes in current and blocks them. Inductive reactance is 2πfL and rises with frequency rather than falling, so a 10 mH inductor presents 62.8 Ω at 1 kHz and ten times that at 10 kHz.
Pair one with a capacitor and you get resonance. A 10 mH inductor with a 1 µF capacitor resonates at 1,592 Hz — the frequency where the two reactances are equal and opposite, which is the basis of every tuned circuit.
The practical warning is stored energy. An inductor carrying one amp holds 5 mJ, and interrupting that current produces a voltage spike large enough to destroy whatever switched it. That is what a flyback diode across a relay coil is for.
Questions people ask
Why put a small ceramic next to a big electrolytic? Because they are good at different things. The electrolytic supplies bulk charge slowly; the ceramic has far lower series resistance and inductance and can respond to fast current demands the electrolytic cannot.
Does series double the voltage rating? Only with balancing. Two 400 V capacitors in series are nominally 800 V, and without balancing resistors the string can put 500 V across one of them and fail.
Is charging the same as discharging? The same exponential shape and the same time constant, running the other way. It reaches 36.8 per cent of the starting voltage after one tau.
Is a charged capacitor dangerous? A large one at a high voltage, yes, and it stays charged after the power is off. Equipment with big reservoir capacitors has bleed resistors for that reason, and they do not always work.
Parallel adds, series does not, and the resistor beside it decides how fast any of it happens. The capacitor parallel calculator and capacitor series calculator handle the combinations, the RC time constant calculator and capacitor charge time calculator cover the two views of the same pair, and the inductor calculator does reactance, resonance and stored energy.