Watts, volt-amperes and VARs are three different quantities
On a DC circuit, power is simply voltage times current, and there is one number. On an AC circuit there are three, because voltage and current are sinusoids that need not peak at the same instant. IEEE Std 1459-2010 is the document that fixes the definitions, and its distinctions are the ones every nameplate, breaker schedule and utility bill relies on.
Real power, P, measured in watts, is the average of the instantaneous product of voltage and current over a cycle. It is energy that leaves the circuit as heat, light, torque or sound. This is what a kilowatt-hour meter integrates and what you pay for on a residential tariff.
Apparent power, S, measured in volt-amperes, is the plain product of RMS voltage and RMS current, with no regard for phase. It is the number your conductors, terminations, breakers, transformers and generators must be sized to, because copper heats on current alone and does not care whether that current is in phase with anything.
Reactive power, Q, measured in volt-amperes reactive, is the component that flows into the load during part of the cycle and back out during the rest. Inductance — motors, transformers, ballasts, solenoids — stores energy in a magnetic field and returns it; capacitance does the same with an electric field but with the opposite sign. Reactive power does no net work, yet it occupies conductor capacity all the same.
The three are related by a right triangle: S² = P² + Q². Power factor is the cosine of the angle at the origin, PF = P/S. A power factor of 1 collapses the triangle to a line and the three quantities become numerically identical, which is exactly what happens on DC and on a purely resistive AC load.
How each formula is built, and when to use which
Start from apparent power, because it needs the fewest assumptions: S = V·I with both values RMS. Nothing about the load's nature enters into it. If you clamp a meter on a conductor and read the voltage at the panel, you have S immediately.
Real power then needs one more piece of information — how far out of phase the current is. That is the power factor: P = V·I·PF. On a sinusoidal system with no harmonic distortion, PF = cos φ, where φ is the phase angle. Nameplates on motors and HVAC equipment print this figure; resistive loads such as heaters, water heaters, incandescent lamps and resistance-element ovens have a power factor of 1.00 and need no correction.
Reactive power falls out of the triangle: Q = √(S² − P²). The square root always returns a positive magnitude, so this form does not tell you whether the load is inductive (lagging) or capacitive (leading). Almost every industrial and commercial load is lagging, because motors and transformers dominate.
When you know resistance rather than current, the substitutions P = I²·R and P = V²/R follow directly from Ohm's law — see the Ohm's law calculator for the full set of rearrangements. On an AC circuit the corresponding quantity is impedance magnitude |Z|, which gives you apparent power as S = V²/|Z|; you still need the power factor to get watts.
Worked example: a 240 V motor drawing 20 A at 0.8 power factor
An electrician clamps 20 A on the supply to a single-phase motor fed from a 240 V circuit. The nameplate gives a power factor of 0.80. Work out every quantity by hand.
- Apparent power. S = 240 × 20 = 4,800 VA = 4.8 kVA. This is the figure the conductors and the breaker see.
- Real power. P = 4,800 × 0.80 = 3,840 W = 3.84 kW. This is the mechanical output plus the motor's own losses, and the figure the meter bills.
- Reactive power. Q = √(4,800² − 3,840²) = √(23,040,000 − 14,745,600) = √8,294,400 = 2,880 VAR. Notice the 3-4-5 proportions: 2,880 : 3,840 : 4,800 reduces to 3 : 4 : 5.
- Phase angle. φ = arccos(0.80) = 36.87°. The current peaks 36.87 electrical degrees after the voltage.
- Continuous-load sizing. If the motor runs three hours or more at a time, the National Electrical Code requires the conductors and overcurrent device to carry 125% of the load: 20 × 1.25 = 25 A.
- Energy at four hours a day. 3.84 kW × 4 h = 15.36 kWh per day.
Now the instructive part. Suppose you correct the power factor to 0.95 without changing the mechanical work being done. Real power stays at 3,840 W, but apparent power falls to 3,840 ÷ 0.95 = 4,042 VA, and the current falls to 4,042 ÷ 240 = 16.84 A. The same work, 3.16 A less conductor current. That is the entire argument for power factor correction, and the power factor correction capacitor calculator sizes the capacitor bank that does it.
How to read the result: which number sizes what
Use each output for the job it is meant for, and the confusion disappears.
Size conductors, breakers, transformers and generators from apparent power or current, never from watts. A 5 kVA load at 0.7 power factor delivers 3.5 kW but still draws the full 5 kVA of current. This is why transformers and UPS units are rated in kVA and not in kW — the manufacturer does not know your power factor, and the limiting factor is winding heating, which follows current.
Bill and budget energy from real power. Multiply kilowatts by hours to get kilowatt-hours. Residential tariffs almost universally meter real energy only, which is why domestic customers are not directly penalised for poor power factor. Commercial and industrial tariffs frequently are, either through a kVA demand charge or an explicit power-factor adjustment — read your own rate schedule rather than assuming.
As a rough orientation for power factor: purely resistive loads sit at 1.00; a modern induction motor at full load typically runs in the mid-0.8s and drops substantially at light load, which is why an oversized motor is doubly wasteful. Small unfiltered switch-mode supplies can be well below 0.7 with heavy harmonic distortion. Always prefer the nameplate figure, or a measurement with a true-RMS power meter, over a rule of thumb.
Finally, watch the phase-angle output as a plausibility check. If you enter a power factor of 0.5 you are claiming a 60° phase shift, which is an extremely reactive load. Values that low usually mean the nameplate has been misread or the load was measured unloaded.
Reference: current drawn at common voltages and power factors
| Supply voltage | PF 1.00 | PF 0.95 | PF 0.85 | PF 0.70 |
|---|---|---|---|---|
| 120 V | 8.33 A | 8.77 A | 9.80 A | 11.90 A |
| 208 V | 4.81 A | 5.06 A | 5.66 A | 6.87 A |
| 230 V | 4.35 A | 4.58 A | 5.11 A | 6.21 A |
| 240 V | 4.17 A | 4.39 A | 4.90 A | 5.95 A |
| 277 V | 3.61 A | 3.80 A | 4.25 A | 5.16 A |
| 480 V | 2.08 A | 2.19 A | 2.45 A | 2.98 A |
Single-phase only. For a three-phase load divide by an additional √3 and use the line-to-line voltage — the three-phase power calculator does that for you.
Mistakes that make a power calculation wrong
- Treating VA and W as interchangeable. They are equal only at unity power factor. Sizing a generator from watts when the load is a motor bank is how you buy a generator that trips on overload.
- Using peak instead of RMS values. On a sinusoid, peak is √2 times RMS. Feeding peak volts into these formulas overstates power by 41% for one substitution and 100% if you do it to both.
- Assuming a nameplate power factor applies at partial load. Induction motors lose power factor badly as load falls; the nameplate figure is a full-load figure.
- Applying single-phase formulas to a three-phase circuit. Three-phase power adds a factor of √3 when you use line-to-line voltage. Use the three-phase power calculator instead.
- Ignoring harmonic distortion. With non-sinusoidal current, PF is no longer simply cos φ, and Q = √(S² − P²) lumps distortion in with reactive power. IEEE Std 1459-2010 exists precisely to separate those terms.
- Forgetting the continuous-load multiplier. The Code requires branch-circuit conductors and overcurrent devices serving a continuous load to be sized at 125% of that load.
- Reading impedance as resistance. On AC the ohms field here is |Z|, the magnitude. A load with 12 Ω of impedance at 0.85 power factor has about 10.2 Ω of resistance and 6.3 Ω of reactance.
Where this fits, and when to use a different tool
This calculator covers DC and balanced single-phase AC with a sinusoidal waveform. Three other situations need their own treatment.
Three-phase circuits introduce a √3 factor between line and phase quantities: P = √3 · VLL · IL · PF. Almost all commercial motor loads are three-phase, so reach for the three-phase power calculator and, for the motor itself, the motor full load amps calculator.
Reactive components analysed individually — a capacitor's or inductor's contribution at a given frequency — belong in the RLC impedance calculator, which gives you resistance and reactance separately rather than just the magnitude.
Operating cost takes the real power figure and multiplies it by hours and by your tariff. The appliance energy cost calculator handles the tariff structure, including tiered and time-of-use rates.
One historical note that still bites: apparent power is written in volt-amperes rather than watts deliberately, so that a specification cannot be misread. When a transformer is stamped 75 kVA, that is a thermal limit on the windings. Loading it with 75 kW of real power at 0.8 power factor means 93.75 kVA of current — 25% over the rating — and the transformer will run hot regardless of how efficiently the load turns watts into work. Size from the transformer kVA sizing calculator with apparent power, not watts.
Key terms
- RMS
- Root mean square — the equivalent DC value that would produce the same heating in a resistor. For a sinusoid, RMS equals peak divided by √2. All AC voltage and current ratings are RMS unless stated otherwise.
- Power factor
- The ratio of real power to apparent power, between 0 and 1. For undistorted sinusoids it equals cos φ. Described as lagging for inductive loads and leading for capacitive ones.
- Volt-ampere (VA)
- The unit of apparent power. Numerically identical to a watt but reserved for the product of RMS volts and RMS amps, so that a rating cannot be confused with real power.
- VAR
- Volt-ampere reactive, the unit of reactive power. Utilities meter VAR-hours separately on large services.
- Continuous load
- A load whose maximum current is expected to persist for three hours or more. The National Electrical Code requires conductors and overcurrent devices serving one to be sized at 125% of the load current.
- Distortion power
- The part of apparent power that arises from harmonic content rather than phase shift. Separated out explicitly in IEEE Std 1459-2010; lumped into Q by the simple triangle used here.