Figure 1: In star connection, VL = √3 × VP and IL = IP. In delta connection, VL = VP and IL = √3 × IP. Total power is the same: P = √3 × VL × IL × PF.
Table of Contents
What Is Three Phase Power?
Three phase power is an electrical system that uses three alternating current waveforms, each offset by 120°. It is the standard for power generation, transmission, and distribution worldwide because it delivers more power for the same amount of copper, produces a smoother torque in motors, and naturally balances the load across the supply.
The three phases are conventionally labelled A (or R), B (or S), and C (or T). In a balanced system all three carry equal current at equal voltage with 120° phase spacing. The Voltage Phasor Calculator can visualise the three phasors and verify that they sum to zero in a balanced system.
Star (Wye) vs Delta Connections
There are two ways to connect a three-phase load or source:
Star (Wye, Y): Each winding connects between one line and a common neutral point. The line voltage is √3 times the phase voltage, and line current equals phase current. This connection provides a neutral for single-phase loads and is standard for distribution transformers.
Delta (Δ): Each winding connects between two lines, forming a triangle. The line voltage equals the phase voltage, and line current is √3 times the phase current. Delta is common for motor connections and distribution without a neutral.
Star: VL = √3 × VP, IL = IP
Delta: VL = VP, IL = √3 × IP
Both: Ptotal = √3 × VL × IL × PF
Key Formulas
| Quantity | Formula | Unit |
|---|---|---|
| Total real power | P = √3 × VL × IL × cos(θ) | W |
| Total apparent power | S = √3 × VL × IL | VA |
| Total reactive power | Q = √3 × VL × IL × sin(θ) | VAR |
| Per-phase power | Pphase = VP × IP × cos(θ) | W |
| Power factor | PF = P / S = cos(θ) | — |
Worked Example — UK Industrial Motor (400V)
Step 1 — VP = 400 / √3 = 230.9 V
Step 2 — IP = IL = 25 A (star: line = phase current)
Step 3 — S = √3 × 400 × 25 = 17.32 kVA
Step 4 — P = 17.32 × 0.85 = 14.72 kW
Step 5 — Q = 17.32 × sin(31.8°) = 9.13 kVAR
This is a typical 15 kW (20 HP) induction motor. The 0.85 power factor means reactive power correction could save about 3.5 A per line. The Power Factor Calculator can size the correction capacitors.
Worked Example — US Commercial Building (480V)
Step 1 — VP = 480 / √3 = 277.1 V
Step 2 — S = √3 × 480 × 50 = 41.57 kVA
Step 3 — P = 41.57 × 0.80 = 33.26 kW
In the US, 480V three-phase is standard for commercial and light industrial use. The 277V phase voltage is commonly used for fluorescent lighting on the same system.
Worked Example — Delta-Connected Heating Load
Step 1 — VP = VL = 400 V (delta: line = phase voltage)
Step 2 — IP = 30 / √3 = 17.32 A
Step 3 — P = √3 × 400 × 30 × 1.0 = 20.78 kW
With PF = 1.0, all power is real — no reactive component. This is typical for resistive heating elements. Each heater element sees 400V and draws 17.3A.
Why Three Phase?
Three phase power offers several advantages over single phase. It delivers 73% more power using only 50% more conductors (3 wires vs 2). Three-phase motors are simpler, cheaper, self-starting, and produce smoother torque than single-phase equivalents. The instantaneous power in a balanced three-phase system is constant — unlike single-phase where power pulsates at twice the line frequency, causing vibration.
For transmission, three-phase allows higher power transfer per kilogram of copper, reducing infrastructure costs. This is why virtually all power generation and transmission worldwide is three-phase.
Balanced vs Unbalanced Loads
A balanced load draws equal current in all three phases at the same power factor. The neutral current is zero. Most three-phase motors and heaters are balanced. An unbalanced load — common in buildings with single-phase loads distributed across phases — creates neutral current and unequal phase voltages. Severe imbalance can overheat neutral conductors and damage motors.
The Role of the Neutral
In a star (wye) system, the neutral connects the star point of the load to the star point of the supply. It provides a return path for any imbalance between the three phases and establishes the reference voltage for single-phase loads. In a delta system there is no neutral — all loads are connected between pairs of lines.
The Phase Angle Calculator can help analyse the phase relationship between individual loads on different phases.
Frequently Asked Questions
What is the √3 factor?
Can I convert between star and delta?
What is the UK three-phase voltage?
How do I measure three-phase power?
What happens if one phase is lost?
Can I use this calculator for generators?
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