Library / Electrical / ELE-37
Electrical
Reactive power in a single-phase circuit
Power that swings back and forth between the source and the reactive parts of a load without being used up.
- Ref
- ELE-37
- Rev
- v1
- Field
- Electrical
- Result
- Q [var]
For a single-phase AC load, the reactive power is Q = VrmsIrms sin φ, where φ is the phase angle of the voltage relative to the current (positive for an inductive load such as a motor, negative for a capacitor). Reactive power is the energy that shuttles back and forth to the inductors and capacitors each cycle, and does no net work; it is measured in vars. Real power is V I cos φ and apparent power is V I.
Power that swings back and forth between the source and the reactive parts of a load without being used up. It is positive for an inductive load and negative for a capacitive one, and it is measured in var.
Calculator
- Real power, V I cos φ995.9 W
- Apparent power, V I1,150 VA
- Power factor, cos φ0.866
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 230 V, 5 A load at 30° draws 995.9 W of real power and 575 var of reactive power: a power factor of 0.866.
- Capacitors connected across an inductive load supply reactive power locally, reducing the current in the supply cable, which is the reason for power-factor correction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (var) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Irms RMS current | 0.1 to 100 A | 11.5 | 11,500 | 11,490 | +10% | +10 % |
| Vrms RMS voltage | 1 to 1,000 V | 2.5 | 2,500 | 2,497 | +10% | +10 % |
| φ Phase of voltage minus current (positive for an inductive load) | −90 to 90 ° | −1,150 | 1,150 | 2,300 | +3.6 ° | +10.7 % |
Sweep of Phase of voltage minus current (positive for an inductive load)
| Phase of voltage minus current (positive for an inductive load) (°) | Reactive power (var) |
|---|---|
| −90 | −1,150 |
| −67.5 | −1,062 |
| −45 | −813.2 |
| −22.5 | −440.1 |
| 0 | 0 |
| 22.5 | 440.1 |
| 45 | 813.2 |
| 67.5 | 1,062 |
| 90 | 1,150 |
Sweep one variable
Variables and how each one contributes
Reactive power
What the equation gives you. Everything else feeds it.
RMS voltage
The RMS voltage, in volts. Raises the reactive power in direct proportion. Raising it by 10 V (to 240 V) moves the result by +4.35 %, from 575 to 600 var.
RMS current
The RMS current, in amps. Raises the reactive power in direct proportion. Raising it by 0.5 A (to 5.5 A) moves the result by +10 %, from 575 to 632.5 var.
Phase of voltage minus current (positive for an inductive load)
The phase angle of the voltage ahead of the current, in degrees: positive for an inductive load, negative for a capacitive load. Raises the reactive power: a larger phase angle means more of the current is out of phase with the voltage. Raising it by 5° (to 35°) moves the result by +14.7 %, from 575 to 659.6 var.
Worked example with the values above
- Apparent power: V I = 230 × 5 = 1,150 VA.
- Sine of the phase angle: sin 30° = 0.5.
- Q = 1,150 × 0.5 = 575 var.
Assumptions and limits
- Sinusoidal voltage and current at one frequency, so a single phase angle describes the load.
- A single-phase circuit with the RMS values in the same units as the answer.
Common mistakes
- Getting the sign of φ wrong, so an inductive load appears capacitive.
- Using the formula with distorted (non-sinusoidal) waveforms, which need harmonic analysis.
Statistics and uncertainty
The phase angle is the hard quantity to measure: a 1° error at 30° changes Q by about 3 % and the real power by about 1 %. Use a power analyser rather than an RMS meter and a phase estimate.
Where it is used
- Sizing power-factor correction capacitors.
- Understanding why the current drawn by a motor exceeds what the real power alone needs.
Maths you need
- RMS values: the effective value of an AC quantity
- Phase angle: the lead or lag between voltage and current