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Formula & Calculator

Three-Phase Power (Line Values)

Calculates real power delivered in a balanced three-phase system using line voltage and current.

Written by the Vorzep Team

Power SystemsThree-Phase

Three-Phase Power Calculator P = √3 · VL · IL · cos θ

P = √3 · VL · IL · cos θ
P = real power (W)  ·  VL = line voltage (V)  ·  IL = line current (A)  ·  cos θ = power factor
⟹ Solve ⟷ P, VL, IL, cos θ
W
V
A
—
Please fix the errors above.
Solve for:
Presets:
Real Power (P) —
—
P: — VL: — IL: — cos θ: — S: — Q: —
✓ Copied!
Power (kW) —
Low (< 10 kW) Medium (10–100 kW) High (> 100 kW)
P = √3 · VL · IL · cos θ  ·  S = √3 · VL · IL  ·  Q = √3 · VL · IL · sin θ
P = √3·V_L·I_L·cos θ
Three-Phase Power (Line Values)

Variables

SymbolQuantityUnit
PReal power (active power)W (or kW)
V_LLine-to-line RMS voltageV
I_LLine RMS currentA
cos θPower factor (pf)dimensionless
θPhase angle between phase voltage and phase current°
√3Square 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
Scenario: A 400 V three‑phase motor draws 50 A with a power factor of 0.85. Calculate the real power.
ParameterValue
VL400 V
IL50 A
cos θ0.85
FormulaP = √3 · VL · IL · cos θ
1Substitute:P = 1.732 × 400 × 50 × 0.85
2Compute: 1.732 × 400 = 692.8; × 50 = 34640; × 0.85 = 29444 W
3Result:P ≈ 29.4 kW
Final Design P ≈ 29.4 kW ✓ Real power
Why: Three‑phase power is √3 times the product of line voltage, line current, and power factor – standard for industrial loads.

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