Power Factor Calculator

PF = P / S, where S = V × I, and Q = √(S² − P²)

P

The real power, in watts or kilowatts — the power that actually does work and shows up on the meter as energy consumed. It is the horizontal side of the power triangle, and it is the quantity a nameplate usually quotes, which is exactly why the other two have to be measured separately. In a purely resistive load such as a heater, all of the apparent power is real power and the power factor is 1

S

The apparent power, in volt-amperes, computed here as the supply voltage multiplied by the current the load draws. It is what the cables, the switchgear and the supply have to be sized for, regardless of how much of it does work. Volt-amperes rather than watts because in an AC circuit the product of RMS voltage and RMS current is not the rate at which energy is consumed unless the two are in phase

Q

The reactive power, in var, which is the part of the apparent power that is not real power. It is not lost energy: it is energy that flows into the magnetic field of a motor or the electric field of a capacitor and flows back again every half cycle. It does no work, but the current carrying it still heats the cables, which is why it is billed for indirectly through the power factor rather than through the kilowatt-hour meter

PF

The power factor itself, real power divided by apparent power, a number between 0 and 1 for an ordinary load. It cannot exceed 1, because real power can never exceed the apparent power it came from. Above about 0.95 the load is described as good; below about 0.85 utilities start to take an interest. Capacitive loads give a leading power factor and inductive ones a lagging one, and the distinction matters for correction even though this page reports only the magnitude

φ

The phase angle, in degrees, by which the current lags or leads the voltage. It is the same information as the power factor expressed as an angle rather than a ratio: the power factor is the cosine of this angle, so a power factor of 0.87 is an angle of about 29.6 degrees. Angle is the more useful of the two if you go on to vector sums, because adding currents in a circuit with a phase shift means adding vectors rather than numbers

Reach for it when someone says the current their equipment draws is higher than the rating plate suggested, because that is almost always a power factor question rather than a fault. A motor rated 8 kW at 230 V would draw 34.8 A if it were perfect; at a power factor of 0.87 it draws 40 A, and the extra 5.2 A is doing no useful work. Use it before sizing a cable or a breaker for a motor, a compressor, a welder or anything with a wound coil, since those are sized on current and the current is larger than the wattage implies. Do not use it for a resistive heater: that is a power factor of 1 and the power calculator is enough.

1. The defaults: 8 kW from a 230 V supply drawing 40 A

   1. Real power 8000 W, supply 230 V, current 40 A
   2. Apparent power is 230 × 40 = 9200 VA
   3. Power factor is 8000 / 9200 = 0.87
   4. Reactive power is √(9200² − 8000²) = 4543.1 var, and the phase angle is the angle whose cosine is 0.87, which is 29.59°

   Look at the current before you look at the power factor. This load is consuming 8 kW but drawing 40 A, and if it were purely resistive it would draw 8 000 / 230 = 34.8 A. The difference, 5.2 A, is the practical cost of a power factor of 0.87: every component in the path from the meter to the motor, including the cable, has to be rated for 40 A rather than 34.8 A. That is why the utility cares and why cables get sized on apparent power. You can check the four rows against each other: 8000² + 4543.1² is 9200² to within rounding.

2. A resistive load: the power factor is exactly 1

   1. Real power 2000 W, supply 200 V, current 10 A
   2. Apparent power is 200 × 10 = 2000 VA, which equals the real power exactly
   3. Power factor is 2000 / 2000 = 1
   4. Reactive power is √(2000² − 2000²) = 0 var and the phase angle is 0°

   This is a kettle, an immersion heater or an incandescent lamp: current and voltage in step, every amp doing work, nothing circulating. It is the ceiling of the scale and worth seeing once, because it shows that 1 is not a good score out of some larger number — it is the point where apparent power and real power are the same quantity, and the power triangle collapses to a line. Nothing in a real installation except a pure resistance reaches it, and a motor that claims 1.0 on its nameplate is quoting an ideal rather than a measurement.

3. A lightly loaded motor: 8 kW while drawing 50 A

   1. Real power 8000 W, supply 230 V, current 50 A
   2. Apparent power is 230 × 50 = 11500 VA
   3. Power factor is 8000 / 11500 = 0.696
   4. Reactive power is √(11500² − 8000²) = 8261.4 var, and the angle is 45.92°

   This is what a motor looks like when it is running well below its rated load, and it is the most common cause of a surprisingly low power factor in a real building. The motor still needs its magnetising current to hold the magnetic field up, and that current does not shrink when the mechanical load does — so as the real power falls, the power factor falls with it. The fix is not electrical at all in most cases: run the motor closer to its rating, or fit power factor correction capacitors sized to the actual load rather than to the nameplate.

4. A corrected installation: 8 kW while drawing 36 A

   1. Real power 8000 W, supply 230 V, current 36 A
   2. Apparent power is 230 × 36 = 8280 VA
   3. Power factor is 8000 / 8280 = 0.966
   4. Reactive power falls to 2135 var and the angle to 14.94°

   Same work, 8 kW, but the current has dropped from 40 A to 36 A. Nothing about the motor changed; the reactive power it was drawing has been supplied locally by capacitors instead of being carried all the way from the supply. This is the entire point of power factor correction, and the number to notice is the current, not the power factor: the apparent power fell by 10%, so the cable, the transformer and the losses in both are all 10% smaller for free. Utilities charge industrial customers for exactly this gap, which is why correction pays for itself.

This is a single-phase calculation. A balanced three-phase load needs the square root of three in the apparent power and is not covered here. The current and voltage you enter must be RMS values measured at the same point and at the same time, since the ratio of two numbers read at different moments is meaningless. The page reports the magnitude of the power factor and not its sign, so a capacitive leading factor and an inductive lagging one look identical, though correcting them requires opposite actions.

Why is the current higher than the rating plate says?

Because the rating plate quotes real power, in watts, and the current is set by apparent power, in volt-amperes. An 8 kW motor on a 230 V supply would draw 34.8 A if the power factor were 1; at 0.87 it draws 40 A. The extra current is not doing work, it is circulating between the supply and the motor's magnetic field, but the cable has to carry it and the breaker has to tolerate it. So a load's current is always at least its wattage divided by its voltage, and for anything with a coil it is meaningfully more.

Does a low power factor waste electricity?

Not in the way it sounds. The reactive power is not consumed: energy flows into the motor's magnetic field and back out again every half cycle, so the kilowatt-hour meter, which counts real power, does not charge for it. What it does cost is current, and current costs money twice: once through the larger cable, transformer and switchgear the installation needs, and in many countries once more through a power factor penalty on the industrial tariff. So it is worth correcting, but the saving is in capacity and in the penalty, not in the energy itself.

What is a good power factor?

Above 0.95 is generally considered good, and around 0.9 is the lowest most utilities will accept without comment under a typical industrial tariff. Below about 0.85 the penalties start, and the exact threshold and the charge depend on the contract rather than on physics, which is why this page does not show a coloured badge: a domestic meter is not assessed on power factor at all, so a number that would be a problem for a factory is meaningless in a house. If you are being charged, read the tariff rather than a rule of thumb.

What is the power triangle?

It is the right triangle whose sides are real power, reactive power and apparent power, with apparent power as the hypotenuse. Because the two sides that do work and circulate work are at right angles, they add as vectors rather than as numbers: 8000 W of real power and 4543 var of reactive power give 9200 VA of apparent power, not 12543. The angle at the bottom is the phase angle, and the cosine of that angle is the power factor. Every relationship on this page is a statement about that triangle.

How do I correct a low power factor?

By supplying the reactive power locally, which in practice means capacitor banks connected near the load. Adding capacitance draws a leading current that cancels the lagging current the motor's inductance draws, so the supply only has to deliver the real part. The example on this page is the shape of it: the same 8 kW drawing 40 A comes down to 36 A once the reactive power is generated on site rather than imported. Correction has to be sized to the actual measured load, not to the nameplate, and over-correcting past 1.0 is itself a problem.

Does this work for three-phase supplies?

Not directly, and this is the page's main limitation. In a balanced three-phase circuit the apparent power is the square root of three times the line voltage times the line current, so using the single-phase product here understates it by about 73%. The power factor itself — real power divided by apparent power — is still the same ratio and comes out right if both are three-phase quantities. The reason the page does not offer a phase selector is that it would silently change what every other number means.

Can the power factor ever be greater than 1?

No, and the page refuses to report it. The power factor formula is real power divided by apparent power, and apparent power is the product of RMS voltage and RMS current, which is the largest real power those two values can produce. If the real power you enter exceeds the voltage times the current, the three numbers are mutually inconsistent — usually because the voltage was measured somewhere other than the load, or the current is an inrush or peak reading rather than an RMS one. The page reports the inconsistency instead of returning 1.2.

• IEEE Standard 1459-2010, Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions — the standard definitions of real, reactive and apparent power used here — IEEE
• Power factor — the definition as the cosine of the phase angle, the distinction between lagging and leading, and the reasons utilities penalise a low value — Wikipedia
• OpenStax College Physics 2e, §20.5 Alternating current versus direct current — RMS voltage and current, and why their product is not the energy consumption rate — OpenStax