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Circuits, electronics and noise
Balanced reactive power from two wattmeters
Total reactive power of a balanced three-phase load from the two wattmeter readings, with W₁ = V I cos(30° − φ) and W₂ = V I cos(30° + φ), for lagging φ positive.
- Ref
- ELN-44
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Q [var]
For a balanced three-phase load the two wattmeter readings are W₁ = V I cos(30° − φ) and W₂ = V I cos(30° + φ), with V the line voltage, I the line current and φ the load angle (lagging positive). Subtracting them gives W₁ − W₂ = V I sin φ, and the total reactive power is Q = √3 (W₁ − W₂). Swapping the meters changes the sign, so the connection must be known. This applies to balanced loads only.
Total reactive power of a balanced three-phase load from the two wattmeter readings, with W₁ = V I cos(30° − φ) and W₂ = V I cos(30° + φ), for lagging φ positive. Swapping the meters reverses the sign.
Calculator
- Total real power, W₁ + W₂3,000 W
- Apparent power, √(P² + Q²)3,464 VA
- Load angle φ30 °
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.
- With readings of 2,000 W and 1,000 W the load takes 3,000 W and 1,732 var, at a load angle of 30°.
- Equal readings mean a unity power factor, with no reactive power, and equal and opposite readings mean a purely reactive load.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (var) | Local step | Change in result |
|---|---|---|---|---|---|---|
| W1 Wattmeter 1 reading | −10,000 to 10,000 W | −19,050 | 15,590 | 34,640 | +400 W | +40 % |
| W2 Wattmeter 2 reading | −10,000 to 10,000 W | −13,860 | 20,780 | 34,640 | +400 W | −40 % |
Sweep of Wattmeter 1 reading
| Wattmeter 1 reading (W) | Reactive power (var) |
|---|---|
| −10,000 | −19,050 |
| −7,500 | −14,720 |
| −5,000 | −10,390 |
| −2,500 | −6,062 |
| 0 | −1,732 |
| 2,500 | 2,598 |
| 5,000 | 6,928 |
| 7,500 | 11,260 |
| 10,000 | 15,590 |
Sweep one variable
Variables and how each one contributes
Reactive power
What the equation gives you. Everything else feeds it.
Wattmeter 1 reading
The reading of wattmeter 1, in W. Raises Q in direct proportion, with a factor of √3. Raising it by 10 % (to 2,200) moves the result by +20 %, from 1,732 to 2,078 var.
Wattmeter 2 reading
The reading of wattmeter 2, in W. Lowers Q in direct proportion, with a factor of √3. Raising it by 10 % (to 1,100) moves the result by −10 %, from 1,732 to 1,559 var.
Worked example with the values above
- Difference: W₁ − W₂ = 2,000 − 1,000 = 1,000 W.
- Multiply by √3 = 1.7321.
- Q = 1,732.1 var.
Assumptions and limits
- Sinusoidal steady state with RMS values, and a balanced three-phase load where stated.
- Accurately connected meters in the standard two-wattmeter arrangement.
Common mistakes
- Using peak voltage where RMS is needed.
- Ignoring the sign of one wattmeter reading, which can be negative for power factors below 0.5.
- Using line-to-line voltage where the phase voltage is needed, or the reverse, a factor of √3.
Statistics and uncertainty
Meter errors, supply voltage fluctuation and unbalance all move these results. Wattmeter readings carry the meter’s accuracy class, and for the balanced-load formulas any unbalance gives an error, so check the readings against the total power.
Where it is used
- Power-factor correction in plants and workshops.
- Measuring power in three-phase motors and checking delta loads and cables.
Maths you need
- RMS values: the equivalent steady value of an AC quantity
- Real, reactive and apparent power: P, Q and S
- Three-phase systems: line and phase quantities differ by √3