Capacitors in Series Calculator

Series capacitors combine by reciprocals — the total is always less than the smallest — and the string’s voltage divides in inverse proportion to capacitance. Enter up to three capacitances and the total voltage, and this calculator returns the combined capacitance, the shared charge and each capacitor’s voltage.

Reciprocals, and the Tyranny of the Smallest

Wire capacitors in series and the effective plate separation adds — each dielectric stacks — so capacitance divides like parallel resistors: 1 ÷ C = 1 ÷ C₁ + 1 ÷ C₂ + …, and the combination is always smaller than the smallest member. The charge bookkeeping is the elegant part: the string's interior plates are isolated islands, so every capacitor ends up holding the same charge Q, and since V = Q ÷ C, the voltages divide inversely with capacitance — the smallest capacitor, least able to hold charge at a given voltage, is allotted the largest share of the string's voltage. That inversion is the page's practical payload: series strings are built to reach high voltages (two 200 V capacitors making a 400 V part), but the sharing is unequal by design and treacherous by tolerance, which the calculator's per-capacitor voltage readout exists to audit before the string is ever energised.

Worked Example: 10, 22 and 47 Microfarads at 100 V

The defaults: reciprocals 0.1000 + 0.04545 + 0.02128 = 0.16673 per µF, so C = 6.00 µF — less than the 10 µF smallest, as the rule demands. At 100 V the shared charge is Q = 6.00 × 100 = 599.8 µC, and the voltages fall out as Q ÷ C each: 60.0 V across the 10 µF, 27.3 V across the 22 µF, 12.8 V across the 47 µF — summing to 100.1 by rounding, the calculator's figures being the precise ones to quote. A builder reaching 100 V service from 63 V-rated parts sees the trap instantly: the string's rating is NOT set by the sum of ratings, because the 10 µF member is already at 95% of its own limit while the others idle. The readout's voltage column is the design's conscience, printed in volts.

Balancing Resistors: The String's Chaperones

Real high-voltage strings add a high-value resistor across each capacitor, and the reason is leakage: in practice each capacitor leaks its charge at its own rate, so over minutes the ideal inverse-capacitance sharing drifts toward a division set by the leakage resistances — unpredictable, and capable of parking nearly the whole bus voltage on the tightest capacitor until it fails (often noisily, and cascading the rest). The balancing resistors swamp the leakages with a defined divider current, pinning the shares near the calculator's ideal. Electrolytic strings in photoflash and DC-link service are never shipped without them. The calculator computes the ideal the resistors defend; this section names the defence, because a page that prints only the ideal of a high-voltage circuit owes the reader its practical chaperone. The resistors' values are chosen by a rule worth keeping with the page: their divider current should run at least ten times the worst leakage spread, so the defined division swamps the undefined one, while their dissipation stays a trivial tax on the supply.

The Voltage Multiplier's Staircase

The Cockcroft–Walton ladder that first split the atom runs on series capacitors arranged as a charge escalator: AC drives the string, diodes gate the transfers, and each stage's capacitors hold shares that sum to many times the input peak — the voltage division of this page, harvested rather than feared. Its descendants charge camera flashes, power photomultipliers and bias a thousand laboratory instruments, all from the bookkeeping that a series string's members hold one charge and split the voltage in inverse proportion. The calculator's per-member readout is the multiplier designer's stage audit, run at DC after the switching settles — the same numbers, with the diodes' choreography politely assumed complete.

Tolerance: The Share-Out With Real Parts

Catalogue capacitances are promises with margins — film parts at ±5 or ±10%, electrolytics at ±20% — and the shares the calculator divides shift by those same margins: the member that arrives 20% low takes proportionally more of the bus voltage than the nominal arithmetic promised, exactly when its rating may also be the batch's weakest. Sound string design therefore runs this page's arithmetic at the tolerance corners (smallest plausible C₁, largest plausible bus) rather than at the nameplate centre — the difference between a datasheet exercise and a design that survives its own production run. The calculator makes the corner cases two extra runs, which is all that worst-casing ever is: the same formula, fed the unfriendly numbers on purpose.

Related Tools

More from the Physics Calculators hub: the Current Divider Calculator — the resistor network whose arithmetic this one mirrors with a twist — and the Capacitor Energy Calculator, for what the combined string stores.

Frequently Asked Questions

Why does series reduce capacitance at all?

Because from the terminals' viewpoint the string is one capacitor whose dielectric is the series stack: the middle plates are floating islands whose induced charges cancel, so the charge that flows onto the string for a given voltage is limited by the tightest member — equivalently, the effective plate spacing is the sum of the spacings, and capacitance falls as spacing grows. The same picture explains the parallel case's addition (areas add, spacing common). The calculator's reciprocal rule is the algebra of that picture; a reader who holds the picture can re-derive the rule when memory fails, which is the only durable way to own it.

What if the capacitors have different initial charges?

Then the simple shares shift by those charges (the interior islands' net charge becomes an initial condition, and the voltage distribution carries an offset the calculator's zero-start assumption does not model) — which is why discharged-before-assembly is a workshop rule for series strings, and why a string snatched from a running circuit can balance at surprising voltages when re-energised. In practice the balancing resistors bleed the anomaly away in seconds to minutes. The calculator models the clean-start case it states, and this FAQ marks the offset's existence for readers meeting it in the wild.

Can I series capacitors of different voltage ratings?

Yes, with the shares audited — which is the calculator's core service: enter the values and the bus voltage, read each member's allotted volts, and require each to sit comfortably under its own rating (a 20–30% margin is workshop custom, since tolerances of ±20% on electrolytics shuffle the shares by comparable amounts). Mismatched ratings are legitimate engineering when the arithmetic is done; they are roulette when it is not. Two identical capacitors remain the lazy-safe choice, their shares equal by symmetry whatever the tolerances do in chorus.

What about series for AC — the nonpolarised case?

The same reciprocal law governs AC steady state (impedances of pure capacitances combine in series exactly as the DC-charged bookkeeping suggests), with the sharing set by the reactances — still inverse in the capacitances. The classic application is the two-capacitor trick for making a bipolar electrolytic from two polarised ones (back-to-back series, the pair behaving nonpolarised at half the single value) — a vintage-audio expedient whose voltage sharing the calculator audits as usual. Film and ceramic strings for snubbers and dividers run the same sums at radio frequencies, where the skin-depth page's effects start taxing the leads.

How does the energy divide?

Unequally, and instructively: each capacitor stores Q² ÷ (2C) with the same Q, so the smallest capacitor stores the most energy — the 10 µF of the worked example holds 18.0 mJ of the string's 30.0 mJ total. Total energy is conserved in the bookkeeping (the sum equals ½ C_total V², as it must), but its concentration in the smallest member is another reason that member is the string's fuse. The calculator's readouts (Q and the V shares) let the reader compute each store; the capacitor-energy page of this hub prices a single device's store directly.

Why do camera flashes and defibrillators use banks, not strings?

Because their problem is energy at a fixed voltage, and energy scales with capacitance — which series connection reduces — so they parallel their capacitors into banks (capacitances add, voltage rating unchanged) and accept the charging supply's voltage as the design's ceiling. The string and the bank are opposite tools for opposite scarcities: voltage scarce, series; capacitance scarce, parallel. The calculator covers the string; the bank's arithmetic (a plain sum) is stated here so the reader chooses deliberately — the choice itself being the first line of any capacitor-network design, and misreading which scarcity you have is the classic beginner's inversion.

Is there a two-capacitor shortcut formula?

There is: for exactly two capacitors the reciprocal law rearranges to product over sum, C = C₁ × C₂ ÷ (C₁ + C₂) — so 10 µF and 10 µF in series give 5 µF, and 10 µF with 22 µF give 6.9 µF, mental arithmetic a technician can do at the bench while the three-capacitor case waits for this page. The shape is the same as parallel resistors, the mnemonic generations of students carry. Beware the dual confusion the shortcut invites: resistors in series add, capacitors in series reciprocate; resistors in parallel reciprocate, capacitors in parallel add — the crosswise symmetry being exactly why a calculator, rather than a memory under exam pressure, should do the combining.