Three-Phase Power Calculator
Real power of a balanced three-phase load from line quantities: P = √3 · VL · IL · cos θ.
Inputs
Drag the handle (or focus it and press ↑/↓) to reorder. Scientific notation such as 1.2e3 is accepted. Pick any unit; conversion is automatic. Use line-to-line voltage and line current for a balanced load. Pick any unit; conversion is automatic.
Results
Enter values and press Calculate. Your answer will appear here.
Real and reactive power vs. phase angle
For your VL and IL the apparent power S is fixed. P = S·cos θ falls and Q = S·sin θ rises as the angle grows. The dot is your operating point; the amber line is S.
What is being calculated
Common examples
| Case | S | P |
|---|
Illustrative inputs, not manufacturer data.
Effect of power factor at 400 V and 50 A
| cos θ | Real power P | Reactive power Q |
|---|
The apparent power stays at 34.64 kVA; only the split between real and reactive power changes.
Common nominal line-to-line voltages
| VL | Phase voltage (VL/√3) | Typical use |
|---|---|---|
| 208 V | 120 V | North American commercial |
| 400 V | 230 V | Europe and many other regions |
| 415 V | 240 V | UK, Australia, India (older nominal) |
| 480 V | 277 V | North American industrial |
| 11 kV | 6.35 kV | Medium-voltage distribution |
Unit conversions
| Quantity | Conversion |
|---|---|
| Power | 1 kW = 1,000 W · 1 MW = 1,000 kW · 1 HP (mechanical) = 745.70 W |
| Voltage | 1 kV = 1,000 V · 1 V = 1,000 mV |
| Current | 1 kA = 1,000 A · 1 A = 1,000 mA |
| Angle | 180° = π rad · 1 rad = 57.2958° |
| Energy | 1 kWh = 1,000 Wh = 3.6 MJ |
Formula
- P: real (active) power (W)
- VL: line-to-line voltage (V)
- IL: line current (A)
- cos θ: power factor, where θ is the angle between voltage and current
Example: VL = 400 V, IL = 50 A, cos θ = 0.85. S = √3 × 400 × 50 = 34,641 VA, so P = 34,641 × 0.85 ≈ 29,445 W ≈ 29.45 kW.
The formula assumes a balanced load, and it is the same for star and delta connections when written in line quantities. In a star system Vphase = VL/√3 and Iphase = IL; in a delta system Vphase = VL and Iphase = IL/√3. Either way P = 3·Vphase·Iphase·cos θ. Advanced mode also gives apparent power S = √3·VL·IL, reactive power Q = S·sin θ, output power with an efficiency, and energy for a number of hours. Use measured values for VL, IL and the power factor; for unbalanced or distorted loads a power meter is more accurate.
References
- Wikipedia, Three-phase electric power: line and phase quantities, power formula
- Wikipedia, Power factor: real, reactive and apparent power
- NIST SP 811, Guide for the Use of the SI: unit conversion factors
How to Use This Tool
- Pick a preset, or enter the line-to-line voltage VL and its unit.
- Enter the line current IL and the power factor cos θ.
- In Advanced, enter the phase voltage or the angle θ instead, and add efficiency or operating hours.
- Click Calculate.
- Read P in the highlighted result, then check S, Q and the power chart.
What This Tool Offers
- Basic and Advanced modes on one calculation engine
- Voltage, current, angle and power (W, kW, MW, HP) unit conversion
- Apparent, reactive and real power with phase angle
- Phase quantities, motor output, losses and energy (Advanced)
- P-and-Q-versus-angle chart and labelled three-phase diagram
- Reorderable inputs, presets and a mobile-friendly layout
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P | Real power (active power) | W (or kW) |
| V_L | Line-to-line RMS voltage | V |
| I_L | Line RMS current | A |
| cos θ | Power factor (pf) | dimensionless |
| θ | Phase angle between phase voltage and phase current | ° |
| √3 | Square root of 3 (1.732) | dimensionless |
What it means
For a balanced three‑phase system, the total real power is given by P = √3 * V_L * I_L * cosθ, where V_L is the line‑to‑line voltage, I_L is the line current, and cosθ is the power factor. This formula applies to both star (wye) and delta connections when line quantities are used. The √3 factor comes from the relationship between line and phase voltages (V_L = √3 * V_ph for star) and currents (I_L = √3 * I_ph for delta). Three‑phase power is the standard for industrial and large‑scale power distribution because it provides constant power and higher efficiency. This formula is essential for calculating the power in motors, generators, and transformers. It is also used for sizing cables, switchgear, and protective devices. Understanding three‑phase power is crucial for electrical engineers working in power systems. Example: A three‑phase motor with V_L=400V, I_L=10A, and power factor 0.8 draws P = √3 * 400 * 10 * 0.8 = 1.732*400*10*0.8 = 5542 W (about 5.54 kW).
Worked example
Three‑Phase Power – Practical Example
Real‑World| Parameter | Value |
|---|---|
| VL | 400 V |
| IL | 50 A |
| cos θ | 0.85 |
| Formula | P = √3 · VL · IL · cos θ |
Common mistakes
- V_L and I_L: Line‑to‑line voltage and line current – not phase quantities.
- Power factor cosθ: The angle between phase voltage and phase current – same as between line quantities.
- Balanced load: This formula assumes a balanced three‑phase system.
- Phase voltage: If you have phase voltage, use P = 3·V_ph·I_ph·cosθ.
- Units: V in volts, I in amperes → P in watts.
Applications
Three‑phase power using line values is P = √3 · V_L · I_L · cosθ, applicable to balanced star and delta systems. This formula is fundamental for industrial power systems, where three‑phase motors and loads are common. Engineers use it to size generators, transformers, and cables, to calculate the total power consumption of factories, and to assess power factor. By using line values, they avoid the need to know whether the load is star or delta. This formula also underpins the design of three‑phase distribution networks and the analysis of power factor correction. Understanding three‑phase power is essential for anyone working with large electrical loads.
- Power calculation in industrial and commercial installations
- Sizing of three‑phase transformers and generators
- Power factor correction in three‑phase systems
- Motor and load power estimation
- Educational understanding of three‑phase systems
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