The current that does no work

Every solve on this site returns twice. Once as node voltages and branch currents, and once as a pair of numbers that must agree: the power dissipated in the resistors, computed as |i|²R and never touching a node voltage, against the power delivered by the sources, computed as Re(v·conj(i)) and never touching a resistance. That second pair is the check described in what a network answers — the one that constrains the voltages against the currents, which current law alone leaves open.

It has been running on every figure in this collection since the first one, and its output has been thrown away every time. The number it computes is the real power.

What the imaginary half is #

A solve at s = jω returns complex node voltages and complex branch currents. The product S = V·conj(I) is therefore complex too, and both of its parts mean something.

The real part is the average power over a cycle: energy that goes into the circuit and does not come back, because a resistor converts it to heat. The imaginary part is energy that goes in during one part of the cycle and comes out during another, because a reactance stores rather than dissipates. Averaged over a whole cycle it is exactly zero, and that is why it can be dismissed. It should not be, because averaged over a cycle is not the same as not there: the current that carries it is in the wire the whole time.

For the load in the figure the numbers are 1,636 W of real power and 1,285 var of reactive. The apparent power — the product of the root-mean-square voltage and the root-mean-square current, which is what the two meters on a switchboard read — is 2,080 VA, and it is the hypotenuse of those two because S is a complex number and |S| is its magnitude.

The ratio of the first to the third is the power factor: 0.786. What it says, in the most concrete form available, is that the cable carries 9.044 A and 7.113 A of it does something.

Phasors have been carrying this since the second essay #

Nothing new is being introduced here. Three voltages that close draws a series circuit’s voltages as arrows in the complex plane and shows that they sum to the source not because their magnitudes add — they do not, and the sum of the magnitudes exceeds the source by a wide margin — but because their directions differ. The angle between the current and the voltage in that picture is the whole of the power factor.

The angle in the triangle is 38.15°, and cos 38.15° = 0.786. A reader who has the phasor picture already has this one; what is added is a convention about which product to take and a name for each part of the answer.

That convention is worth stating explicitly, because it is the first place on this site where the absolute size of a quantity matters. Everywhere else the answers are ratios — a Bode magnitude is the same number whichever scaling produced it — and a factor of two hiding in a definition would cancel out. Here it would not. Phasors on this page are root-mean-square, so a 230 V source means 230 V RMS and S = V·conj(I) with no factor of a half. The alternative convention, peak phasors and S = ½V·conj(I), is equally correct and puts a half in front of every product in the file, which is one more place for it to go missing.

Three routes, and the third touches no impedance #

The reactive power is computed three ways in the figure above and they are required to agree.

The first sums V·conj(I) over the sources. The second sums the same product over the elements — |i|²R for a resistor, jωL|i|² for an inductor, −j|v|²ωC for a capacitor — and never looks at a source. Those two agreeing is a real check, and it is a stronger one on the imaginary half than on the real: real power balance is very nearly guaranteed by current law alone, while reactive balance requires the energy stored in every reactance to be consistent with the voltages across it.

The third route is the interesting one. Reactive power is also $2\omega(W_L - W_C)$, where $W_L$ is the energy stored in the inductor and $W_C$ the energy stored in the capacitor. That expression contains no impedance at all — only stored energy and a frequency — and it is what makes “reactive” a physical statement rather than the name of an imaginary part. It agrees with the other two to a part in 10⁹, which is the tolerance the figure asserts.

A quantity computed three ways by routes that share no arithmetic is a quantity that is very unlikely to be wrong, and that is the whole reason for going to the trouble.

There is one detail in those three that is easy to pass over and is load-bearing. The sign of the reactive power is not a convention chosen for convenience; it falls out of the conjugate. Taking S = V·conj(I) puts a lagging current — one that peaks after the voltage, which is what an inductor produces — into the positive imaginary direction, and a leading one into the negative. The opposite convention exists in the machines literature, and the entire content of the words leading and lagging is which of the two is in use. It matters here because the correction capacitor computed later takes Q to zero by having the opposite sign, and a routine that silently took a magnitude would size it correctly and be unable to report the one condition worth reporting: an installation that has overshot and is now leading.

The element-by-element route is also where a subtle assembly error would show. A sign reversed in one stamp gives a smooth, plausible response and passes every visual inspection. It does not pass this: the sources and the elements would disagree about where the energy went, and the disagreement is returned as a number rather than hidden behind a magnitude.

Why a number with no work in it is expensive #

A reactive current does no work and it is not free, for a reason that has nothing to do with economics and everything to do with |i|²R.

The cable between the supply and the load has resistance. The current in that cable is the whole current, 9.044 A, not the 7.113 A that does something, and the loss in the cable is proportional to the square of it. At a power factor of 0.786 the cable dissipates 1/0.786² = 1.62 times as much as it would at unity for the same delivered work — sixty-two per cent more heat, for nothing.

The same argument sizes every piece of equipment between the generator and the load. A transformer rated at 2,080 VA is fully loaded by this circuit while delivering 1,636 W. Its core saturates on volts and its windings heat on amps, and neither of them knows or cares about the angle between them. That is why apparent power has its own unit: a volt-ampere is not a watt, and equipment is rated in the first.

The size of the effect is worth holding on to, because the power factor is a number between zero and one and small-looking departures from one are not small. A power factor of 0.9 — which sounds close enough to unity to ignore — costs 1/0.9² = 1.23, so twenty-three per cent more heat in every conductor carrying it. At 0.7 the figure is a factor of two. The loss goes as the inverse square of the power factor, so the penalty accelerates exactly where an installation is least able to absorb it, and a supply designed with a comfortable margin at 0.95 has none at all at 0.7.

The cost of the fix, and the fact that it has one #

The reactive power can be cancelled, and this is the part of the subject where the arithmetic is unusually clean. A capacitor across the supply carries reactive power of the opposite sign, −ωC|V|², so the capacitance that takes Q to zero is Q/(ω|V|²) — one division, no iteration. For the load above that is 77.3 µF, and adding it takes the power factor to 1.000000 and leaves 0.0 var of a 1,636 VA apparent power.

The real power does not change by a part in 10⁹, which is the point: the load still does exactly what it did, and the cable now carries 7.113 A instead of 9.044 A.

That figure is the argument of the capacitor that was right once and is left here as a warning rather than developed. What matters at this point is only that the fix exists and is exact where it was computed.

The other thing that is not free #

There is a second trade in this area, and it is usually taught a chapter away from the first: the load that draws the most power from a source is not the load that uses it most efficiently, and the two cannot both be had.

The peak is at a ratio of one, and the efficiency there is exactly one half: the source burns as much as the load receives. Nine times the source resistance gives ninety per cent efficiency and thirty-six per cent of the available power. A tenth of it gives nine per cent efficiency and thirty-three per cent of the power — the same power, wasted eleven times over.

Which of those is wanted is a question about the application. A radio receiver’s front end wants the first, because what it is short of is signal and it has power to spare. A power station wants the second, because it is selling the thing that gets wasted. The figure’s only claim is that the choice is real and that the arithmetic is not a matter of opinion: the maximum was searched for, not asserted, and the efficiency at it came out as a half to eight decimal places.

Where the phasor picture gives out #

Everything above assumes one frequency. That assumption is invisible because it is built into the word phasor, and it is worth making visible, because for a large fraction of what is plugged into a wall socket it is false.

A phasor represents a sinusoid. If the current is not a sinusoid it has no single phasor, the angle between voltage and current is not defined, and cos φ is a calculation performed on a quantity that does not exist. What does still exist is the ratio of real power to apparent power — that only needs the two root-mean-square values and an average, none of which cares about shape — and the two numbers part company completely.

For a load conducting only near the peaks of the supply, which is what a rectifier with a reservoir capacitor does, the displacement factor is exactly 1.000000 and the true power factor is 0.780. A phasor calculation on that load reports a perfect result. The meter reports a cable carrying forty-three per cent more current than the real power requires. Both are right about what they measure, and only one of them is measuring the thing that decides the cable.

That is three phases and the wire that carries nothing and the field’s third essay between them; it is named here so that the triangle above is not read as a general description of power. It is a description of power at one frequency, in a circuit whose current is a sinusoid, and those conditions are stated rather than assumed.

What was actually added #

Very little machinery, which is the pleasant part. The solver already returned complex voltages and complex currents; the verifier already summed one of the two products. Real, reactive and apparent power are read out of quantities that were being computed anyway and discarded.

What is new is a convention — root-mean-square phasors, so that S = V·conj(I) — and the observation that a check written to catch assembly errors was, incidentally, computing the quantity a utility bills for.

The site’s rule applies to this field like any other, and here the edge is not a frequency but a shape. Every number on this page is exact for a sinusoidal current at 50 Hz and means nothing for a current that is not one. That is a strong condition and it is stated on every figure, because the alternative — a power factor quoted with no indication of what it assumes — is how an installation ends up with a correction capacitor that makes things worse.

How far from a sinusoid an ordinary load is #

The condition sounds mild until the loads are named, and three measurements in this collection say that the commonest load in a modern installation violates it completely.

The direct voltage that is a sawtooth marches the circuit behind almost every mains-powered appliance and finds it drawing 157 milliamps of average current as a 2.098 ampere pulse over 28.8° of each half cycle — a crest factor of 13.4 that gets worse as the reservoir capacitor is made larger. A displacement factor computed on that current’s fundamental is 1.000000, which is a perfect result about the wrong thing: the true power factor is 0.780, and the same hundred watts is drawn as 0.558 A instead of 0.435.

The neutral that carries more than a line shows what that shape does three phases up. Three balanced loads of exactly this kind put √3 times a line current down a conductor sized on the assumption that balanced loads put nothing there — because a harmonic of order three is shifted by 360° between phases, which is no shift at all, so the third harmonics add rather than cancelling.

And what a meter multiplies by says the instrument most likely to be pointed at it is 35.9 per cent low on a rectifier drawing its current in sixty degrees, being exactly right for a sinusoid and wrong for everything else by the ratio of two form factors.

So the condition stated on this page is not a technicality about pathological waveforms. It is violated by a laptop charger, and the three quantities that go wrong — the power factor, the neutral current and the meter reading — are the three an installation is designed and billed on.

The repair for a lagging power factor does not help and can hurt. The capacitor that was right once is where that is measured: the division that sizes a correction capacitor is a division of reactive power, which is a sinusoidal quantity, so for a load whose displacement factor is already 1.000000 it returns zero correction — for the load with the worst true power factor on the installation. And the capacitor fitted for the other loads is a low impedance at every harmonic the rectifiers are producing, which is the resonance the series case buys arriving in the shunt position.

Which is why the distinction between displacement and true power factor is worth an essay rather than a footnote. They are equal for a sinusoid, they are the same word in most of the literature, and the difference between them is a whole class of load for which every standard remedy is either useless or harmful.

The remedy that does work is not a correction at all but a change to the load: draw the current as a sinusoid in the first place, which is what an active power-factor correction stage does and which is a switching converter rather than a capacitor. That is outside what this field measures, and naming it is the honest end of the argument — the quantity separated on this page can be cancelled at the installation and can only be removed at the equipment.