Three-Phase Power
P = √3 × VLL × IL × cos(φ)
Quick answer
This calculator solves P = √3 × VLL × IL × cos(φ) for Total Power. With Line-to-Line Voltage 12 V, Line Current 2 A, Power Factor 0.5, the result is 20.78 W. It solves in 4 directions, so you can enter the values you have and get the one you need.
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Formula
Description
Three-phase power systems are the standard for industrial and commercial power distribution because they deliver constant instantaneous power (unlike single-phase which pulsates) and use less conductor material for the same power delivery. The √3 factor arises from the 120-degree phase relationship between the three phases. Line-to-line voltage is √3 times the phase voltage. Three-phase motors are simpler and more efficient than single-phase equivalents. This formula gives the total real power delivered by all three phases combined.
Variables
- P — Total three-phase real power (W)
- VLL — Line-to-line RMS voltage (V)
- IL — Line current RMS (A)
- cos(φ) — Power factor
Practical Notes
Common three-phase voltages: 208V (US commercial), 380V (EU), 400V (EU newer), 480V (US industrial). Three-phase apparent power is S = √3 × VLL × IL (without power factor). For balanced loads, each phase carries one-third of the total power. Variable frequency drives (VFDs) convert single or three-phase input to three-phase output for motor speed control.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| P | Total Power | W | µW, mW, W, kW |
| VLL | Line-to-Line Voltage | V | pV, nV, µV, mV, V, kV |
| IL | Line Current | A | pA, nA, µA, mA, A, kA |
| cosPhi | Power Factor | — | — |
Step by step
- 1. What you know: Line-to-Line Voltage 12 V, Line Current 2 A, Power Factor 0.5
- 2. The relation: P = √3 × VLL × IL × cos(φ)
- 3. Substitute: VLL = 12 V, IL = 2 A, cosPhi = 0.5
- 4. Result: Total Power = 20.78 W
Related quantities from the same inputs
- P = V × I = 24 W
Worked examples
Minimum range
Given: Line-to-Line Voltage 0.33 V, Line Current 2 mA, Power Factor 0.025
Result: Total Power = 28.58 µW
Low range
Given: Line-to-Line Voltage 3.3 V, Line Current 20 mA, Power Factor 0.25
Result: Total Power = 28.58 mW
Typical
Given: Line-to-Line Voltage 12 V, Line Current 2 A, Power Factor 0.5
Result: Total Power = 20.78 W
High range
Given: Line-to-Line Voltage 0.23 kV, Line Current 100 A, Power Factor 0.8
Result: Total Power = 31.87 kW
Maximum range
Given: Line-to-Line Voltage 2.3 kV, Line Current 1 kA, Power Factor 8
Result: Total Power = 31870 kW
Solving for each variable
Solve for Total Power
Line-to-Line Voltage 0.33 V, Line Current 2 mA, Power Factor 0.025 → 28.58 µW
Solve for Line-to-Line Voltage
Total Power 10 mW, Line Current 2 mA, Power Factor 0.025 → 115.5 V
Solve for Line Current
Total Power 10 mW, Line-to-Line Voltage 0.33 V, Power Factor 0.025 → 0.6998 A
Solve for Power Factor
Total Power 10 mW, Line-to-Line Voltage 0.33 V, Line Current 2 mA → 8.748
How Total Power changes with Line-to-Line Voltage
| Line-to-Line Voltage (input) | Total Power (result) |
|---|---|
| 1.2 V | 2.078 W |
| 3 V | 5.196 W |
| 6 V | 10.39 W |
| 9 V | 15.59 W |
| 12 V | 20.78 W |
| 18 V | 31.18 W |
| 24 V | 41.57 W |
| 48 V | 83.14 W |
| 84 V | 145.5 W |
| 120 V | 0.2078 kW |
The same result in other units
| Unit | Value |
|---|---|
| mW | 20780 |
| W | 20.78 |
| kW | 0.02078 |
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<p><a href="https://vbatpower.com/formulas/three-phase-power/">Three-Phase Power</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Three-Phase Power?
Use P = √3 × VLL × IL × cos(φ). With Line-to-Line Voltage 12 V, Line Current 2 A, Power Factor 0.5, Total Power works out to 20.78 W.
How do you calculate v line v phase?
Rearrange for Total Power: with Line-to-Line Voltage 3.3 V, Line Current 20 mA, Power Factor 0.25, Total Power is 28.58 mW.
What is the formula for Three-Phase Power?
The formula is P = √3 × VLL × IL × cos(φ). Three-phase power systems are the standard for industrial and commercial power distribution because they deliver constant instantaneous power (unlike single-phase which pulsates) and use less conductor material for the same power delivery.
How do you find Line-to-Line Voltage when you know the other values?
Rearrange for Line-to-Line Voltage: with Total Power 10 mW, Line Current 2 mA, Power Factor 0.025, Line-to-Line Voltage is 115.5 V.
How do you find Line Current when you know the other values?
Rearrange for Line Current: with Total Power 10 mW, Line-to-Line Voltage 0.33 V, Power Factor 0.025, Line Current is 0.6998 A.
What units does Three-Phase Power use?
Total Power is shown in mW, W, kW. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
Related Concepts
All Power & Efficiency formulas →Need more features?
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