Capacitors in Series & Parallel

Intuition

Which way round the rules go

Anyone who has learned the resistor rules arrives here expecting the same arithmetic, and gets it — but attached to the opposite word. Capacitors in parallel add up. Capacitors in series combine the way parallel resistors do, and the total comes out smaller than any single one of them.

The reason is not a quirk of notation. Wire two capacitors side by side, and you have effectively built one capacitor with the plate area of both. More area means more charge held per volt, so the capacitances add — directly, no reciprocals involved.

Wire them one after the other instead, and the picture changes. The same charge has to pass through the whole chain, so each capacitor takes on the same amount of it, and each develops its own voltage in response. The voltages add up along the string, so the total voltage for that shared charge is larger than any one capacitor would have shown alone. More volts for the same charge is less capacitance, by definition. Stacking capacitors in series is much like moving the plates further apart.

Once the direction is clear the rest follows. A pile of small capacitors in parallel makes a bigger one. A chain of capacitors in series makes a smaller one, and it does so for the useful reason that the applied voltage is shared out among them rather than landing on any single part.

Practitioner

The two rules, on one pair of parts

For two capacitors side by side, the capacitances simply add:

For two in a chain, the combination is the product over the sum, the shape resistors use for the parallel case:

Working a mixed network uses the same discipline as a resistor network, and only the two rules change. Find a group that is unambiguously in series or unambiguously in parallel, replace it with its single equivalent, redraw, and repeat until one capacitance is left. The redrawing is the part people skip and the part that prevents mistakes.

For a general chain, the reciprocal of the total is the sum of the reciprocals — the same statement in the form that extends to any number of capacitors. The two-capacitance product-over-sum above is that rule collapsed for exactly two parts, and it does not generalise to three by inspection. Take them a pair at a time instead, which is what Layer 3 does.

Some useful shortcuts fall out. Equal capacitances in parallel multiply: ten identical parts give ten times one of them. Equal capacitances in series divide: three identical parts give a third of one. And a very small capacitance in series with a very large one gives almost exactly the small one, because the large one contributes hardly any voltage to the total — which is the reason a small coupling capacitor in series with a large reservoir behaves as though the reservoir were not there.

Engineer

What series division actually does to the voltage

The rule that matters practically is not the capacitance total, it is the voltage split, and it catches people because it runs the opposite way to intuition.

Charge cannot enter the junction between two series capacitors from anywhere else — it has nowhere to come from. Whatever charge flows into the top of the string flows through the whole of it, so every capacitor in a series chain carries the same charge. Each then shows a voltage equal to that charge divided by its own capacitance. The smaller capacitance therefore takes the larger share of the voltage.

Extending to three capacitors is a matter of applying the pair rule twice, and it is worth doing once explicitly to see how far the series result falls:

The duality with resistors is exact and worth stating properly, because it makes both sets of rules one set. A resistor's opposition to current rises with resistance; a capacitor's opposition to a changing current falls with capacitance. Series and parallel arithmetic follows the opposition, not the label, so capacitance combines as conductance does and everything lines up. That symmetry runs through the rest of this department, and it appears again in inductors in series and parallel, where the rules go back to matching resistors.

There are limits to the model. The rules above are DC and low-frequency statements about ideal capacitances. At AC the split between series capacitors is set by the capacitances as described; at DC, held for long enough, it is set instead by the parts' leakage resistances, which have nothing to do with the marked values — a case series circuits works through, and the reason series stacks across a high-voltage bus carry balancing resistors. The rules also ignore ESR and lead inductance, which is precisely what Layer 4 cannot afford to do.

Professional

Why real banks are built from several parts

Paralleling capacitors in a real design is rarely about reaching a capacitance figure. It is about the properties that come along with it.

A bank of several capacitors in parallel has the sum of their capacitances, but also a lower total series resistance and a lower total series inductance, since those are in parallel too. Four parts side by side give four times the capacitance and roughly a quarter of the ESR, which is why supply decoupling and switching-converter output stages use several medium parts rather than one large one.

The parts do not share the work equally, though, and that is the trap.

Mixing values in a parallel bank is a frequency decision rather than a capacitance one. A large electrolytic holds the bulk charge and stops being capacitive well below a megahertz; a small ceramic beside it has a hundredth of the capacitance and stays useful into the hundreds of megahertz. Fitted together they cover a range neither manages alone, which is why decoupling schemes stack a bulk part, a mid-value ceramic and a small one at the pin.

That pairing has a failure mode of its own. Between the self-resonant frequency of the large part, where it has gone inductive, and that of the small part, the two form a parallel resonance with a high impedance right in the band the designer was trying to decouple. The usual mitigations are to keep the values within about a decade of each other rather than spreading them widely, and to rely on the loss in real parts to damp the peak. Widely-spaced value pairs sound thorough and can be worse than a single well-chosen part.

Series connection in a design is almost always about voltage rather than capacitance, and it brings its own bookkeeping. The tolerance of each part feeds into the split, so parts that are nominally equal are not, and the balancing resistors have to be sized against the worst-case leakage spread as well as the tolerance. Series strings also halve the capacitance while doubling the voltage capability, so the stored energy per unit volume stays roughly where it was — the stacking buys voltage rating, not density.