Three-Phase Power Calculator

Calculate three-phase active, reactive, apparent power, and line/phase quantities.

EU standard 400V
0-1
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How Power Is Shared in a Balanced Three-Phase System

Three-phase power comes from three sinusoidal phase voltages separated by 120°, so a balanced load receives a nearly steady total power rather than the strongly pulsating instantaneous power of one phase alone. When line-to-line voltage VL and line current IL are known, the total apparent power is S=√3VLIL. Active power is the portion that performs net work, P=S cosφ, while reactive power Q=S sinφ represents energy exchanged with electric or magnetic fields in inductive or capacitive loads.

The √3 factor comes from the geometry of three phase quantities, not from an arbitrary correction. Connection type changes the relationship between line and phase values. In a star connection, Vphase=VL/√3 and Iphase=IL. In a delta connection, Vphase=VL and Iphase=IL/√3. The total line-quantity formula P=√3VLILcosφ is valid for either balanced connection.

S=√3VLIL,   P=S cosφ,   Q=S sinφ
SymbolMeaningWhy it appears / units
VLLine-to-line RMS voltageV; measured between two line conductors.
ILRMS line currentA; current carried by each line conductor in a balanced system.
SApparent powerVA; combines active and reactive demand.
PActive powerW; average power converted to mechanical work, heat, light, or other net output.
QReactive powerVAR; field energy exchanged each AC cycle.
cosφPower factorDimensionless; equals P/S for sinusoidal balanced loads.

A power factor close to 1 means most apparent power is active power. A lower power factor requires more current for the same useful kilowatts, increasing conductor losses and equipment loading. For unbalanced systems or strongly distorted waveforms, phase-by-phase analysis and true power-factor measurements are needed instead of the balanced sinusoidal formulas.

Worked Examples

Example 1: Star: VL=400V, IL=10A, PF=0.85
S=√3×400×10=6928VA
Result: P=5.89kW, Q=3.65kVAR
Typical 3-phase motor operating point
Example 2: Delta: VL=230V, IL=15A, PF=0.95
S=√3×230×15=5981VA
Result: P=5.68kW
High PF load
Example 3: Motor load at 480 V
VL=480V, IL=25A, PF=0.80 → S=√3×480×25
Result: S≈20.78kVA, P≈16.63kW, Q≈12.47kVAR
The 0.80 power factor means only 80% of the apparent-power magnitude appears as active power.
Example 4: Delta phase quantities
VL=415V, IL=30A, delta connection
Result: Vphase=415V, Iphase≈17.32A
In delta, each phase sees the line voltage, while line current is √3 times phase current.

Common Mistakes

⚠️
Using phase voltage in the line-voltage formula

The √3VLIL expression expects line quantities. Mixing phase and line values introduces an extra or missing factor of √3.

⚠️
Treating power factor as efficiency

Power factor describes the relationship between active and apparent power. Motor or transformer efficiency compares useful output power with real input power; the two quantities are not interchangeable.

⚠️
Using degrees directly in a sine formula without finding the phase angle

If power factor is given, first use sinφ=√(1−PF2) or find φ=cos−1(PF). Do not substitute the power-factor number itself for sinφ.

Frequently Asked Questions

Star vs delta for same load?
Same 3-phase load: star has V_phase=V_line/√3 (lower), delta has V_phase=V_line (higher). Motors can be star-delta started: star reduces starting current to 1/3 of delta starting current.
Three-phase vs single-phase?
3-phase delivers power continuously (never zero). Single-phase power pulsates at 2× frequency. 3-phase generators and motors are simpler and more efficient. 3-phase uses less copper for same power transmission.
Why does three-phase power use √3?
The factor comes from the 120° phase separation and the vector relationship between line and phase quantities. In a balanced system, combining the three phase powers and expressing the result with line-to-line voltage and line current produces the √3 multiplier.
Is kVA the same as kW?
No. kVA is apparent power, while kW is active power. For a balanced sinusoidal load, kW=kVA×power factor. At unity power factor the numerical values match, but with inductive or capacitive loads the kVA value is larger.
Can the same formula be used for star and delta?
Yes, if VL and IL are line quantities and the load is balanced. Connection type matters when converting between line and phase voltage or current, but total balanced three-phase power still uses √3VLILcosφ.
What changes when the three phases are unbalanced?
The compact balanced formula no longer captures the complete system. Calculate the complex power of each phase using its own voltage, current, and phase angle, then add the three phase powers. Neutral current may also become important in a four-wire system.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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