What is the formula for 3-phase power?

S = √3 × V_LL × I in volt-amperes. P = S × cos φ. Q = S × sin φ. The √3 comes from the geometric relationship between line and phase quantities in a balanced 3-φ system. With V_LL in volts and I in amperes, divide by 1 000 to get kVA.

kW vs kVA vs kVAR — what is the difference?

kW (kilowatts) = real power, the work the load actually does. kVA (kilovolt-amperes) = apparent power, what the alternator and conductor must carry. kVAR (kilovolt-amperes reactive) = the part that circulates between the source and reactive elements without doing work. They are related by the right-triangle: kVA² = kW² + kVAR².

How do I calculate amps from kW in 3-phase?

I = (kW × 1 000) / (√3 × V_LL × cos φ). A 100 kW load at 400 V 3-phase, PF 0.85: I = 100 000 / (1.732 × 400 × 0.85) = 170 A. The PF in the denominator is critical — at PF 0.7 the same kW load draws 207 A; at PF 1.0 it draws 144 A.

Why is √3 in the formula?

In a balanced 3-phase system, the vector sum of the three phase voltages (each 120° apart) gives the line voltage as √3 times the phase voltage. Total power across all three phases adds the contribution of each, leading to S = 3 · V_phase · I_phase. Express V_phase in terms of V_LL (V_LL = √3 · V_phase for wye) and the result is S = √3 · V_LL · I_line.

Single-phase vs three-phase — which delivers more power?

Three-phase delivers √3 (≈ 1.73) times the power for the same line current and line voltage as single-phase. Equivalently, 3-φ uses smaller conductors for the same power. Industrial and commercial gear above ~5 kW is almost always 3-phase for this efficiency advantage.

How do I correct a low power factor?

Add capacitor banks (sized in kVAR) at the load or the bus. The capacitor supplies reactive power to the inductive load, so the alternator sees only real power. For a 100 kW load at PF 0.75 corrected to 0.95: required kVAR ≈ 100 × (tan(arccos 0.75) − tan(arccos 0.95)) = 55 kVAR. See the Power Factor Correction calculator for exact sizing.

What voltages are common in 3-phase?

Worldwide: 400 V line-to-line (230 V line-to-neutral) for European / Australian / Asian commercial. 480 V for North American commercial / industrial. 208 V for North American light commercial. 600 V for Canadian industrial. Utility distribution: 4 160 V, 12 470 V, 13 800 V, 25 kV, 34.5 kV.

Calculator · Electrical · IEEE Std 100 · Tesla 1888

Three-phase power calculator

Balanced 3-phase power calculator: from line-to-line voltage, line current, and power factor, returns real (kW), reactive (kVAR), and apparent (kVA) power, plus the phase angle. Solves for any one missing variable given the other three. With PDF report. Reviewed by a licensed PE.

Three-phase power — at a glance

Real power formula (3-phase): P = √3 · V_LL · I · cos φ in watts. Result in kW when V is in volts, I in amps, divided by 1000.
Apparent power formula (3-phase): S = √3 · V_LL · I in volt-amperes (VA). Divide by 1000 for kVA — what the alternator and conductor must carry.
Reactive power formula (3-phase): Q = √3 · V_LL · I · sin φ in volt-amperes reactive (VAR). The component circulating in inductors and capacitors without doing work.
Why √3 in 3-phase formulas?: In a balanced wye, V_LL = √3 · V_LN. Total power across three phases works out to P = √3 · V_LL · I · cos φ.
Power triangle relationship: S² = P² + Q² and PF = cos φ = P / S. The three quantities form a right triangle with the phase angle φ between V and I.
Amps from kW (3-phase): I = (kW × 1000) / (√3 · V_LL · cos φ). 100 kW at 400 V, PF 0.85 → I = 170 A.
Neutral current in balanced 3-phase: In a balanced 3-phase wye load, the three phase currents sum to zero, so I_N = 0 A. Imbalance or harmonics push neutral current up.
kVA from kW and PF: kVA = kW / PF. A 75 kW load at PF 0.87 needs an alternator rated 86.2 kVA minimum.

Use the calculator

Pick what you want to solve for, enter the known values and the power factor, and the calculator returns kW, kVAR, kVA, and phase angle. Switch to the Y/Δ tab for connection conversion, or the Motor tab for NEC 430 motor branch sizing.

CALC.019 3-Phase · Power · Y/Δ · Unbalanced · NEC 430 Motor

Balanced 3-phase power

Solve for[?]

V_LL[V]

Line current I[A]

Power factor[cos φ]

For 3-phase: S = √3·V_LL·I, P = S·cos φ, Q = S·sin φ. Lagging PF (induction motors) is positive φ; leading PF (over-corrected systems) is negative.

Result

— kW

Pick a mode and enter values.

FORMULA · S = √3 · V_LL · I SOURCE · NEC 430 · IEEE STD 100 · FORTESCUE 1918

The four 3-phase formulas

Eq. 01 — Apparent power S SI · IEEE Std 100

S = \sqrt{3} \cdot V_{LL} \cdot I

S: apparent power, VA
V_LL: line-to-line voltage, V
I: line current, A

Eq. 02 — Real and reactive power SI · IEEE Std 100

P = S \cdot \cos\varphi, \qquad Q = S \cdot \sin\varphi

P: real power, W
Q: reactive power, VAR
φ: phase angle, rad

Eq. 03 — Power triangle SI · Steinmetz 1897

S^{2} = P^{2} + Q^{2}, \qquad PF = \cos\varphi = \frac{P}{S}

PF: power factor, —

Eq. 04 — Line current from kW SI · IEEE Std 100

I = \frac{P \cdot 1000}{\sqrt{3} \cdot V_{LL} \cdot \cos\varphi}

I: line current, A
P: real power, kW
V_LL: line voltage, V

How to compute 3-phase power, step by step

Identify what you know. You have any two of: line-to-line voltage (V_LL), line current (I), real power (P_kW), plus the power factor (PF). The calculator solves the third quantity.
Use line-to-line voltage, not line-to-neutral. Three-phase formulas use V_LL by convention. For a wye system, V_LL = √3 × V_LN. Most nameplates show line-to-line — confirm the value matches the bus you are measuring.
Pick a realistic power factor. Pure resistive load (heater): PF = 1.0. Modern motor at full load: 0.85–0.90. Lightly loaded motor: 0.50–0.70. Industrial mixed: 0.75–0.85. Datacenter / IT: 0.95+. Setting PF too high underestimates current and oversizes everything.
Read kW, kVAR, kVA from the result. P (kW) = useful work. Q (kVAR) = reactive component circulating in the inductors / capacitors. S (kVA) = total apparent power the alternator and conductor must carry. Their relationship: S² = P² + Q².
Cross-check the line current. For 3-phase: I = (kVA × 1000) / (√3 × V_LL). The same current flows in every line; the alternator winding sees the per-phase current depending on Y or Δ configuration.
Use the result for sizing. Conductor and breaker are sized on line current (with continuous-load 1.25 × factor). The transformer is sized on kVA. The fuel / cooling load on the prime mover is set by kW. Each downstream calculation pulls a different output from the same balanced 3-phase calc.

Common 3-phase voltages and applications

V_LL	V_LN	Region / use
208 V	120 V	NA light commercial (delta-wye from 480 V)
240 V	139 V	NA legacy industrial / open-delta
400 V	230 V	Europe / AU / NZ / Asia commercial standard
415 V	240 V	UK commercial (declining; aligning with 400 V)
480 V	277 V	NA commercial / industrial standard
600 V	347 V	Canadian industrial
2 400 V	1 386 V	Industrial MV motors
4 160 V	2 400 V	Industrial substation feeders
12 470 V	7 200 V	Utility primary distribution

Worked example: 75 kW air handler at 400 V

A factory air handler runs a 75 kW (≈ 100 hp) induction motor at 400 V 3-phase, PF 0.87 lagging. Compute line current and apparent power.

Step	Calculation	Result
S = P / PF	75 / 0.87	86.2 kVA
I = (kVA × 1000) / (√3 × V)	(86.2 × 1000) / (1.732 × 400)	124.4 A
φ = arccos(0.87)	—	29.5°
Q = S × sin φ	86.2 × 0.493	42.5 kVAR
Sanity: P² + Q²	75² + 42.5² = 5 625 + 1 806 = 7 431, √7 431	86.2 matches S
Wire (NEC 430.22, 1.25 × FLA)	1.25 × 124.4 = 155.5 A	2/0 AWG Cu (175 A)

Where 3-phase is used

Industrial motors

Three-phase induction motors above ~3 kW dominate factories, pumps, conveyors, compressors. Naturally produce a rotating magnetic field with no extra capacitor or starting circuit.

Commercial HVAC and lighting

Office buildings supply 480 V or 400 V to step-down transformers feeding 208 / 120 V receptacle / lighting branches. Centralised chillers and rooftop units run at the higher voltage.

Utility transmission and distribution

The entire grid above the residential service drop is 3-phase: 138 kV, 34.5 kV, 12.47 kV, 4.16 kV. Single-phase exists only for the final transformer to a small residence.

Data centres and EV chargers

Modern data-centre PDUs run at 400/415 V 3-phase to reduce conductor size; high-power EV DC fast chargers (50 – 350 kW) tap directly off the 480 V 3-phase service through their own transformer.

The Tesla three-phase patent

The invention relates to a system of electrical distribution wherein the alternating currents of three or more phases are employed to produce in the receiving motor a rotary magnetic field. The motor requires no commutator or sliding contacts of any kind, and is thereby distinguished from the direct-current machines hitherto in use.

Tesla, N. — System of Electrical Distribution by Alternating Currents → US Patent 381 968, 1888

Related calculators and references

Frequently asked questions

What is the formula for 3-phase power?: S = √3 × V_LL × I in volt-amperes. P = S × cos φ. Q = S × sin φ. The √3 comes from the geometric relationship between line and phase quantities in a balanced 3-φ system. With V_LL in volts and I in amperes, divide by 1 000 to get kVA.
kW vs kVA vs kVAR — what is the difference?: kW (kilowatts) = real power, the work the load actually does. kVA (kilovolt-amperes) = apparent power, what the alternator and conductor must carry. kVAR (kilovolt-amperes reactive) = the part that circulates between the source and reactive elements without doing work. They are related by the right-triangle: kVA² = kW² + kVAR².
How do I calculate amps from kW in 3-phase?: I = (kW × 1 000) / (√3 × V_LL × cos φ). A 100 kW load at 400 V 3-phase, PF 0.85: I = 100 000 / (1.732 × 400 × 0.85) = 170 A. The PF in the denominator is critical — at PF 0.7 the same kW load draws 207 A; at PF 1.0 it draws 144 A.
Why is √3 in the formula?: In a balanced 3-phase system, the vector sum of the three phase voltages (each 120° apart) gives the line voltage as √3 times the phase voltage. Total power across all three phases adds the contribution of each, leading to S = 3 · V_phase · I_phase. Express V_phase in terms of V_LL (V_LL = √3 · V_phase for wye) and the result is S = √3 · V_LL · I_line.
Single-phase vs three-phase — which delivers more power?: Three-phase delivers √3 (≈ 1.73) times the power for the same line current and line voltage as single-phase. Equivalently, 3-φ uses smaller conductors for the same power. Industrial and commercial gear above ~5 kW is almost always 3-phase for this efficiency advantage.
How do I correct a low power factor?: Add capacitor banks (sized in kVAR) at the load or the bus. The capacitor supplies reactive power to the inductive load, so the alternator sees only real power. For a 100 kW load at PF 0.75 corrected to 0.95: required kVAR ≈ 100 × (tan(arccos 0.75) − tan(arccos 0.95)) = 55 kVAR. See the Power Factor Correction calculator for exact sizing.
What voltages are common in 3-phase?: Worldwide: 400 V line-to-line (230 V line-to-neutral) for European / Australian / Asian commercial. 480 V for North American commercial / industrial. 208 V for North American light commercial. 600 V for Canadian industrial. Utility distribution: 4 160 V, 12 470 V, 13 800 V, 25 kV, 34.5 kV.

Sources and methodology

IEEE. IEEE Std 100 — The Authoritative Dictionary of IEEE Standards Terms, 7th Ed., 2000.
NFPA. National Electrical Code (NEC) NFPA 70, 2023 Ed. Article 220 (loads), Article 430 (motors).
Steinmetz, C.P. Theory and Calculation of Alternating Current Phenomena, McGraw-Hill, 1897.
Tesla, N. U.S. Patent 381 968 — System of Electrical Distribution by Alternating Currents, 1888.
NEMA. NEMA MG 1 — Motors and Generators, 2021. § 14.36 voltage unbalance derate.