The capacitor that was right once

A capacitor across a supply carries reactive power of −ωC|V|². A load carries some reactive power Q. Setting the sum to zero gives C = Q/(ω|V|²), and that is the whole of power factor correction: one division, no search, and an answer that is exact rather than approximately right.

Checking it takes one solve. For the twenty-ohm, fifty-millihenry load of the current that does no work, which draws 1,636 W and 1,285 var on a 230 V supply at 50 Hz, the capacitance is 77.311 µF. Adding it and solving again gives a power factor of 1.00000000, a reactive power of 0.0 var, and a real power unchanged to the last digit the arithmetic carries. Nothing about that is an approximation.

What “exact” is exact about #

The claim the calculation supports is narrow, and every word in it is doing work: this capacitance cancels this load’s reactive power at this frequency.

Change any of the three and the cancellation fails, and it fails in a way that has no restoring tendency. A capacitor does not know what it is cancelling. Its reactive power depends on the voltage across it and the frequency, both of which are held nearly constant by the supply, so it delivers the same 1,285 var whatever the load is doing. When the motor is unloaded and drawing a third of the reactive power, the capacitor is still delivering all of it, and the excess flows back into the supply.

That is the whole of overcorrection, and it is visible in the figure as the region where the two curves cross. The corrected installation is not gently less good outside its design point; it is on the other side of unity, with reactive power of the opposite sign, and it gets worse in exactly the same way an uncorrected one does — as the inverse square of the power factor in every conductor.

Why the shape changes rather than scaling #

The reason overcorrection is a practical problem rather than a curiosity is in how an inductive load actually varies.

A motor’s magnetising current is set by the voltage and the machine’s inductance, and it is very nearly the same whether the motor is doing work or spinning free. The current that does work is the one that varies with the load. So as the mechanical load falls, the real power falls a long way and the reactive power hardly falls at all — the triangle grows thinner rather than shorter, and the power factor of an unloaded motor is famously bad for exactly this reason.

Modelled here as a rising resistance in series with a fixed inductance, that behaviour comes out directly: at twenty ohms the load draws 1,636 W and 1,285 var, at eighty ohms it draws 637 W and about a fifth less reactive power. The capacitor is unchanged. The mismatch between what it delivers and what is wanted is therefore largest exactly when the installation is lightest — overnight, at weekends, whenever the plant is idling — and that is when the supply sees a leading power factor and a rising voltage.

The resonance nobody asked for #

There is a second consequence, and it is the one that does damage rather than merely wasting current.

A correction capacitor is not alone across the supply. The supply itself has inductance — the transformer’s leakage inductance and the cable’s, which is small but not zero — and a capacitance across an inductance is a resonant circuit. It has a frequency, and nothing in the sizing calculation looked at it.

For plausible numbers — a 77 µF capacitor and a few hundred microhenries of supply inductance — the resonance lands in the region of a few hundred hertz, which is to say somewhere between the fifth and the eleventh harmonic of a 50 Hz supply. Those harmonics are not hypothetical. Every rectifier on the installation produces them, and the figure in the next section shows what a rectifier’s current actually looks like.

At resonance, a circuit’s currents can greatly exceed the driving current — that is what the quality factor measures, and resonance and its bandwidth makes the point that the half-power width is exactly f₀/Q. A modest harmonic current at a frequency the installation happens to resonate at becomes a large circulating current between the capacitor and the transformer, and the capacitor is the part that fails.

This is the field’s clearest instance of the site’s rule. The correction calculation is exact and it is exact about a quantity — the reactive power at 50 Hz — that is not the quantity that decides whether the installation survives.

The arithmetic is worth doing once explicitly, because the frequency it lands on is not a matter of opinion. A distribution transformer’s leakage inductance seen from its secondary is of the order of a hundred microhenries for a small installation. With 77 µF across it, ω₀ = 1/√(LC) gives about 11.4 krad/s, which is 1.8 kHz — the thirty-sixth harmonic, comfortably above anything a rectifier produces in quantity. Scale the installation up, though: ten times the capacitance for ten times the load, on a transformer with a tenth the leakage inductance, and the frequency stays put. Scale it the other way — a large bank on a weak supply, which is the case that actually gets built — and it falls as the square root of the product. A bank of 800 µF on 400 µH resonates at 281 Hz, which is the fifth and sixth harmonic of a 50 Hz supply and precisely where rectifier current lives.

The engineered answer to that is a detuning reactor: a small inductance deliberately put in series with each capacitor to move the resonance below the lowest harmonic present, usually to about 189 Hz for a 50 Hz supply so that the fifth harmonic at 250 Hz sees an inductive rather than a capacitive circuit. The reactor is typically around seven per cent of the capacitor’s reactance and it costs real money and real losses. That is the price of the fact this essay is about: the sizing calculation is exact, and the component it sizes cannot be installed on its own.

Why the exactness is the trap #

There is a general pattern here worth separating from the electrical detail, because it recurs across this collection and it is the reason this field’s second essay is about a fix rather than about a problem.

An approximate answer carries its own warning. A curve fitted to five points invites the question of what happens at the sixth; a series truncated at three terms invites the question of the fourth. The error is visible in the form of the answer, so the reader knows to ask.

An exact answer carries no such warning, and the exactness is doing something quite specific: it is telling the truth about the model. C = Q/(ω|V|²) is not an approximation to the capacitance that cancels the reactive power of a fixed load at a fixed frequency; it is that capacitance, and the solved network agrees to a part in 10⁹. Every digit of that agreement is a statement about the model and not one word of it is a statement about the installation.

The same shape appears elsewhere on this site and is worth collecting. Resonance and its bandwidth finds that the half-power width of a series resonance is exactly f₀/Q at every quality factor tested — and that the band is not centred on the resonance, which the usual picture draws as though it were. The exact relation and the false one sit side by side, and the exact one gives no clue that its neighbour is wrong. The quarter-wave transformer in the lines field reflects 5×10⁻¹⁷ of the incident wave at the frequency it was cut for, and something quite different a few per cent either side.

An exact model with an unstated domain is more dangerous than an approximate one with a stated error bar, and this field supplies the cleanest example of it in the collection.

The current that made the harmonics #

The measured displacement factor for that waveform is 1.000000 and the true power factor is 0.7803, and the two are related exactly: the true power factor is the displacement factor times the distortion factor, and the identity holds to the arithmetic on every waveform tested.

What that means for correction is unusually stark. A capacitor cancels reactive power, which is a displacement effect. This load has no displacement to cancel — its cos φ is already one — so a correction capacitor does nothing for its power factor at all, while still forming a resonance with the supply and still amplifying the harmonics that made the problem in the first place.

An installation that measures cos φ, finds it acceptable, and concludes that no correction is needed has reached the right conclusion by an argument that would have been just as confident if it had been wrong. An installation that measures cos φ, finds it poor because of a motor, fits a capacitor, and has rectifiers on the same board has fitted a resonant circuit to a harmonic source.

What a correction calculation cannot be asked #

It is worth being precise about what the sizing routine on this site refuses to do, because the refusal is the honest part.

Given a real power, a reactive power, a voltage and a frequency, it returns a capacitance. It does not take a load model, because it does not need one — and that is exactly why it cannot warn about anything. Everything that goes wrong above goes wrong because the four numbers it was given describe one operating point of a system that has many, and the routine has no way of knowing that.

The general shape is the one the divider and its load sets out. A component’s effect is a property of the component and of its surroundings, and a calculation that takes only the component’s own parameters gives an answer that is right about the component and silent about the installation.

It is worth naming what would have to be supplied for the routine to be able to warn about any of this, because the list is short and none of it is exotic.

The range of the load, not one point of it. Two solves rather than one — heaviest and lightest — would put both ends of the figure above in front of whoever is choosing the capacitance, and would turn “0.23 leading at light load” from a discovery into an input.

The supply’s inductance. One number, available from the transformer’s nameplate as a percentage impedance, and enough to place the resonance to within a factor of two. Without it the resonant frequency is not merely unknown; it is not even a quantity the calculation has a symbol for.

The shape of the current. Not a phasor but a waveform, or at minimum the amplitude of the harmonics present. That is what decides whether the resonance matters, and it is the one input a 50 Hz measurement cannot produce at all.

None of those three is difficult to obtain and none of them is in the formula, and that gap is the essay. A calculation is not made careless by being simple; it is made careless by returning a number of the same apparent authority whether or not the conditions it assumed are present.

What holds and what does not #

Three statements, in decreasing order of how far they can be trusted.

The capacitance is exact. C = Q/(ω|V|²) takes the reactive power to zero and the solved network confirms it to a part in 10⁹, with the real power unchanged to the last digit. There is no approximation in it.

It is exact at one point. The same capacitor across a load of three or four times the resistance gives a power factor below 0.5 leading, and the figure sweeps sixty-one loads to show the shape of the failure rather than asserting that one exists. The measured worst case on the default axis is 0.23.

It cannot see the resonance or the harmonics. Both of those need information the sizing calculation was never given: the supply’s inductance, and the shape of the current rather than its phasor. Neither is available from the four numbers that produce the capacitance, and neither shows up in any check performed on the result.

The unusual thing about this corner of the subject is how clean the wrong answer looks. There is no residual to inspect, no iteration that fails to converge, no tolerance to widen. The number is exact, the meter agrees with it on the day of commissioning, and everything that goes wrong afterwards goes wrong somewhere the calculation never looked.

The three places it looks afterwards #

Naming them is the practical half of this essay, because each has a measurement elsewhere in this field and each is the thing that actually goes wrong.

Light load. The capacitor’s current does not fall with the load and the lagging current does, so a feeder correctly corrected at full load is over-corrected at light load — and the far end that rises computes what over-correction costs: past a computable angle of leading load the voltage at the far end goes above the source’s, 28.35 degrees for a line of fifty ohms and a hundred of reactance, and with no reactance in the line there is no such angle at all because the rise is a partial resonance and needs both halves.

Harmonics. The correction capacitor is a low impedance at every harmonic and the supply’s inductance is a high one, so the pair resonates. The reactance cancelled, and the resonance it buys computes where — the line frequency times the root of the fraction cancelled, so a quarter compensation resonates at exactly half the line frequency and a ninth at exactly a third — and the neutral that carries more than a line shows that the harmonics to be resonant at are present and additive rather than hypothetical.

And the load’s own shape. The division that sizes the capacitor is a division of reactive power by a voltage squared, and reactive power is a sinusoidal quantity. The current that does no work states the condition; a rectifier load has a displacement factor of 1.000000 and a true power factor of 0.780, so the calculation returns zero correction needed for a load whose power factor is the worst on the installation.

An exact answer to a question that stops being asked #

The failure this essay is about has an unusual shape and it is worth naming, because the collection has one other instance of it and the pair makes the shape recognisable.

The calculation is exact. Not accurate, not adequate — exact, one division, no iteration, no residual. What changes is not the answer but the question: the load moves, the frequency spectrum changes, the season turns, and the quantity the answer was the answer to is no longer the quantity anybody has.

An exact answer to a different question is the other instance, and it is the cleaner one because there the exactness is measurable on both sides of the boundary: a reconstruction returns an alias to 1.6 parts in a thousand above half the sample rate, which is the same accuracy it returns the input with below it, while being wrong about the input by twice the amplitude. Its error is not a degradation but exactness about something else.

The two together say what to look for. A calculation with no residual has nothing to warn with, so the only available check is on its premises rather than on its output — and a premise that was true when the calculation was made and is not checked again is the failure mode that no amount of numerical care prevents. For a correction capacitor the premises are a load, a frequency and a waveform, and all three are properties of somebody else’s equipment.