# Three-Phase Power Equations

## Electrical 3-phase equations

Most AC power today is produced and distributed as three-phase power where three sinusoidal voltages are generated out of phase with each other. With single-phase AC power there is only one single sinusoidal voltage.

### Real Power

Line to line voltage:

W_{applied}= 3^{1/2}U_{ll}I cos Φ

= 3^{1/2}UI PF (1)_{ll}

where

W_{applied}= real power (W, watts)

U= line to line voltage (V, volts)_{ll}

I = current (A, amps)

PF = cos Φ = power factor (0.7 - 0.95)

Line to neutral voltage:

* W _{applied} = 3 U_{ln} I cos Φ (2)*

*where *

*U _{ln} = line to neutral voltage (V, volts) *

For pure resistive load: *PF = cos Φ = 1*

**resistive loads**converts current into other forms of energy, such as heat**inductive loads**use magnetic fields like motors, solenoids, and relays

### Power Factor

Typical power factors:

Device | Power Factor |
---|---|

Lamp, fluorecent uncompensated | 0.5 |

Lamp, fluorecent compensated | 0.93 |

Lamp, incandescent | 1 |

Motor, induction 100% load | 0.85 |

Motor, induction 50% load | 0.73 |

Motor, induction 0% load | 0.17 |

Motor, synchronous | 0.9 |

Oven, resistive heating element | 1 |

Oven, induction compensated | 0.85 |

Pure resistive load | 1 |

#### Example - Pure Resistive Load

For pure resistive load and *power factor = 1* the real power in a *400/230 voltage (line to line / line to neutral) 20 amps* circuit can be calculated as

* W*_{applied}* = 3*^{1/2}* (400 V) (20 A) 1*

* = **13856** W*

* = **13.9** kW*

### Total Power

W = 3^{1/2}U I (2)

### Brake Horsepower

W_{BHP}= 3^{1/2}U I PF μ / 746 (3)

where

W_{BHP}= brake horse power (hp)

μ = device efficiency