Calculate real, apparent, and reactive power for delta and wye three-phase systems
In a balanced three-phase system, total real power is
P = √3 × VLL × IL × cosφ
The √3 arises because the three phases are 120° apart. Adding two phase voltages that are 120° apart gives a resultant √3 times larger than either, so line-to-line voltage is √3 times line-to-neutral voltage. Writing the total of three phase powers in terms of line quantities leaves that factor behind.
| Star (wye) | Delta | |
|---|---|---|
| Line voltage | √3 × phase voltage | = phase voltage |
| Line current | = phase current | √3 × phase current |
| Neutral available | Yes | No |
The total power formula is identical for both, which is the point: expressed in line quantities, the connection does not matter. In a 400 V system the line-to-neutral voltage is 400/√3 ≈ 230 V, which is why the same supply serves both three-phase machinery and single-phase outlets.
This relationship is also what makes star-delta starting work: in star, each winding receives only 1/√3 of the line voltage, so starting current and torque both fall to about a third.
Instantaneous power in a balanced three-phase system is constant rather than pulsating, because the three phases peak in sequence. A single-phase supply delivers power that varies between zero and twice its average at twice the supply frequency, which produces vibration and torque pulsation in motors. Three-phase machines run smoothly, are self-starting, and need no starting capacitor.
Three-phase also carries a given power with roughly 42% less line current than single-phase at the same voltage, so conductors, switchgear and transformers are all smaller for the same duty.
P = √3 × VLL × IL × cosφ for a balanced load, using line-to-line voltage and line current. The same formula applies to both star and delta connections.
From the 120° phase separation. Adding two voltages 120° apart gives a resultant √3 times larger, so line-to-line voltage is √3 times line-to-neutral voltage.
In star, line voltage is √3 times phase voltage and line current equals phase current, with a neutral available. In delta, line voltage equals phase voltage and line current is √3 times phase current, with no neutral.
Because line-to-neutral voltage is the line-to-line value divided by √3: 400/√3 ≈ 230 V. The same supply therefore serves both three-phase machinery and single-phase loads.
Instantaneous power is constant rather than pulsating, so motors run smoothly and are self-starting. It also carries the same power with about 42% less line current, allowing smaller conductors and switchgear.
In star, each winding sees only 1/√3 of the line voltage, so starting current falls to about a third. Starting torque falls by the same factor, since torque follows the square of voltage.
Written and reviewed by
Lead Electronic Engineer (R&D) · Power Electronics Specialist · M.E. Power Electronics
Founder of Power4All. Works in power-electronics R&D and writes the converter guides, calculators and simulators on this site.
Last updated