What is voltage unbalance in 3-phase?

A condition where the three phase voltages are not equal in magnitude or are not exactly 120° apart. Even small imbalances (1–2 %) significantly heat induction motors because of the negative-sequence currents. NEMA defines a simple metric (max deviation from average); IEEE uses the more accurate symmetrical components ratio.

What are V₀, V₁, V₂?

The three "symmetrical components" of an unbalanced phasor set. V₀ (zero-sequence) — three equal voltages all in phase, sums to non-zero only when there's ground-current flow. V₁ (positive-sequence) — three balanced voltages 120° apart in normal phase rotation, this is the "useful" component. V₂ (negative-sequence) — three balanced voltages 120° apart in REVERSE rotation, causes counter-torque and heating in motors.

How do I calculate the voltage unbalance factor?

NEMA method: VUF = (max deviation of any single phase from the 3-phase average) / (3-phase average) × 100 %. Easy with three readings. IEEE method: VUF = |V₂| / |V₁| × 100 %. More rigorous; requires phase angles too. The two metrics agree at low imbalance and diverge slightly above ~3 %.

Why do unbalanced voltages damage motors?

Negative-sequence voltage produces a backward-rotating field in the motor air gap. This creates a counter-torque (small loss) and high-frequency rotor currents (large heating). At 3 % voltage unbalance, the negative-sequence currents can be 6–10× the rated rotor current — the rotor heats up but the torque output barely changes. Result: insulation degrades 2–3× faster, bearings fail prematurely.

What causes 3-phase voltage unbalance?

Single-phase load distribution — most common cause; when 1-φ loads are not balanced equally across three phases. Single-phasing — a blown fuse or open phase on the supply leaves the motor running on 2 phases (massive imbalance, motor burns out in minutes). Asymmetric impedance — different cable lengths or cross-sections per phase. Utility issues — unbalanced primary distribution feeders.

What is the maximum acceptable unbalance?

NEMA recommends ≤1 % for sustained operation of motors; tolerable up to 3 % with derate; never above 5 %. IEEE 1159 categorises voltage unbalance as a "long-duration variation" with limits of 0.5–2 %. ANSI C84.1 sets utility-supplied unbalance at the service point to 3 % maximum. Modern PFCs and converters add their own limits — check equipment specs.

How does NEMA MG 1 derate motors for unbalance?

NEMA MG 1 §14.36 publishes a derating curve: at 1 % VUF derate to 0.98 of nameplate; at 2 %, 0.95; at 3 %, 0.88; at 4 %, 0.82; at 5 %, 0.75 — and operation above 5 % is not recommended. Always pair the curve with a thermal protection device (NEMA Class C trip on overload).

What is the "a" operator in symmetrical components?

A complex unit-magnitude phasor at 120° angle: a = e^(j·2π/3) = −0.5 + j·0.866. Multiplying any phasor by a rotates it by 120° counter-clockwise. The Fortescue transformation matrix uses 1, a, and a² to extract each sequence from the original three phasors. a³ = 1, a² = e^(j·240°) = a*.

Calculator · Electrical · Fortescue 1918 · IEEE Std 1159 · NEMA MG 1

Unbalanced 3-phase calculator

Decomposes any unbalanced three-phase voltage set into the three Fortescue symmetrical components — zero-sequence, positive-sequence, and negative-sequence — and computes the voltage unbalance factor by both NEMA and IEEE definitions. Used to diagnose motor heating, ground faults, and power-quality problems. Reviewed by a licensed PE.

3-phase — at a glance

Amps from kW (3-phase formula): I = (kW × 1000) / (√3 · V_LL · cos φ). Example: 50 kW at 400 V, PF 0.85 → I ≈ 85 A.
3-phase motor amps (NEC 430.6): NEC 430.6(A) requires you to use the NEC Table 430.250 FLA, not the nameplate, for branch sizing. A 10 HP 460 V motor: table FLA = 14 A → wire/breaker sized on 14 A.
How many wires in 3-phase?: 3 wires in a delta system (3 hots, no neutral). 4 wires in a wye system (3 hots + neutral). Plus a separate ground (EGC) in either case per NEC.
3-phase vs 1-phase power: 3-phase delivers √3 ≈ 1.732 × the power for the same line voltage and current as single-phase, with smaller conductors. Standard above ~5 kW.
Convert 3-phase kW to 1-phase amps: Per phase: I_1ph = (kW × 1000 / 3) / (V_LN · cos φ). The total kW divides equally across three balanced phases.
How to test a 3-phase motor: Megger the windings to ground (>1 MΩ), check winding-to-winding resistance (should match within 5%), and verify rotation with a phase-rotation meter before energising.
Acceptable voltage unbalance: NEMA recommends ≤1 % for continuous motor operation; tolerable up to 3 % with derate per NEMA MG 1 §14.36; never above 5 %.
Line vs phase quantity (Y vs Δ): Y: V_LL = √3·V_LN, I_line = I_phase. Δ: V_LL = V_phase, I_line = √3·I_phase.

Use the calculator

Enter the magnitude (V) and angle (degrees) of each line-to-neutral voltage. The calculator returns V₀, V₁, V₂ in polar form and the voltage unbalance factor by both NEMA and IEEE definitions. Warnings appear at 2 % and 5 % thresholds per NEMA MG 1.

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

Unbalanced 3-phase — symmetrical components

Enter phase magnitudes (V) and angles (degrees) for the three line-to-neutral voltages.

V_a magnitude[Va]

V_a angle[°]

V_b magnitude[Vb]

V_b angle[°]

V_c magnitude[Vc]

V_c angle[°]

Symmetrical components per Fortescue 1918: any unbalanced 3-phase set decomposes into V₀ (zero-seq, ground-fault current), V₁ (positive-seq, useful power), V₂ (negative-seq, motor heating).

Result

— kW

Pick a mode and enter values.

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

The four symmetrical-component formulas

Eq. 01 — Fortescue decomposition SI · Fortescue 1918

\begin{bmatrix}V_{0}\\V_{1}\\V_{2}\end{bmatrix} = \frac{1}{3}\begin{bmatrix}1&1&1\\1&a&a^{2}\\1&a^{2}&a\end{bmatrix} \begin{bmatrix}V_{a}\\V_{b}\\V_{c}\end{bmatrix}

a: unit phasor at 120°: a = e^(j·120°), —
V₀: zero-sequence, V
V₁: positive-sequence, V
V₂: negative-sequence, V

The three symmetrical components are linear combinations of the three original phase voltages. Reverse: V_a = V₀ + V₁ + V₂; V_b = V₀ + a²·V₁ + a·V₂; V_c = V₀ + a·V₁ + a²·V₂. Total energy preserved.

Eq. 02 — Voltage unbalance factor (NEMA) — · NEMA MG 1 §14.36

\text{VUF}_{NEMA} = \frac{\max\,|V_{i} - V_{avg}|}{V_{avg}} \times 100\%

V_avg: (V_a + V_b + V_c) / 3, V

Eq. 03 — Voltage unbalance factor (IEEE) — · IEEE Std 1159

\text{VUF}_{IEEE} = \frac{|V_{2}|}{|V_{1}|} \times 100\%

V₁: positive-sequence magnitude, V
V₂: negative-sequence magnitude, V

NEMA is simple — needs only magnitudes. IEEE is rigorous — needs angles too but gives the actual ratio of harmful negative-sequence to useful positive-sequence. Use IEEE for motor derate calculations; NEMA for quick on-site checks.

Eq. 04 — NEMA MG 1 motor derate factor — · NEMA MG 1 §14.36

F_{derate} \approx 1 - 2 \cdot (\text{VUF}\,\%)^{2} / 100

F_derate: multiply nameplate HP by this, —

Empirical fit: a 2 % unbalance derates a 100 hp motor to ~92 hp; a 4 % unbalance derates to ~68 hp. Operate motors above 5 % unbalance only with prior approval from the manufacturer.

How to diagnose unbalance, step by step

Measure all three phase-to-neutral voltages. Use a 3-phase recording meter (Fluke 1738, Dranetz HDPQ) to capture magnitude and phase angle of each line-to-neutral voltage simultaneously. A simple multimeter measures only magnitude — not enough; angles between phases matter.
Pick V_a as the angular reference. Set V_a angle = 0°. Express V_b and V_c as offsets from this reference. Ideally V_b = −120°, V_c = +120°. Real measurements deviate by a few degrees, which is what unbalance analysis captures.
Compute the three sequence components. V₀ (zero-sequence) = ⅓ × (V_a + V_b + V_c). V₁ (positive-sequence) = ⅓ × (V_a + a·V_b + a²·V_c). V₂ (negative-sequence) = ⅓ × (V_a + a²·V_b + a·V_c). Where a = e^(j·120°). The calculator does this automatically.
Read the voltage unbalance factor. NEMA: VUF = max deviation from average / average × 100. IEEE: VUF = |V₂| / |V₁| × 100. NEMA is easier with a simple voltmeter; IEEE is more rigorous and is what motor derate tables use.
Check NEMA MG 1 motor derate. NEMA MG 1 §14.36: motors must be derated by ~5 % at 3 % unbalance, ~25 % at 5 % unbalance. Negative-sequence currents heat the rotor without contributing to torque — directly causing premature insulation failure.
Find and fix the source. Sources of unbalance: single-phase loads connected to one phase only, broken transformer winding, single-phase loss (open phase), unequal cable impedance, regenerative loads. Once identified, redistribute single-phase loads across all three phases or repair the open path.

NEMA MG 1 unbalance derate table

VUF (%)	Motor derate factor	Recommendation
0	1.00	Perfect balance — operate at full nameplate
1	0.98	Acceptable for continuous operation
2	0.95	Within NEMA limit; periodic monitoring
3	0.88	Significant derating required; investigate cause
4	0.82	Excessive — motor lifetime reduced 50 %+
5	0.75	Maximum allowable — replace, redistribute, or stop
> 5	—	Operation prohibited per NEMA MG 1

Worked example: unbalanced commercial bus

Measurements at a commercial 230/400 V bus: V_a = 230 ∠0°, V_b = 220 ∠−118°, V_c = 235 ∠122°. Compute symmetrical components and recommend action.

Step	Calculation	Result
V_avg	(230 + 220 + 235) / 3	228.3 V
Max deviation	\|220 − 228.3\|	8.3 V
VUF (NEMA)	8.3 / 228.3 × 100	3.6 %
V₀ via Fortescue	complex sum / 3	~ 4 ∠65°
V₁ positive-sequence	—	~ 228 ∠ 1°
V₂ negative-sequence	—	~ 8 ∠ −80°
VUF (IEEE)	8 / 228 × 100	3.5 %
Motor derate (NEMA MG 1)	~ 0.85 at 3.6 % VUF	−15 % capacity
Action	Investigate single-phase load distribution; aim < 2 % VUF

Variants and special cases

Single-phasing (open-phase fault)

One supply phase opens (blown fuse, broken cable, contact failure). The motor tries to run on 2 phases; voltage unbalance jumps to 33 %+. Motor draws 2–3× rated current and burns out in minutes unless thermal overload trips. Most expensive single failure mode in 3-phase systems — modern motor protectors include phase-loss detection.

Sequence components for fault analysis

Symmetrical components originally derived for short-circuit analysis (Fortescue 1918). Modern protection relays compute sequence currents continuously: ground faults show up as I₀ > threshold, phase-to-phase faults show up as I₂ > threshold, while normal balanced load shows only I₁. Each sequence has its own protection function.

Negative-sequence-only protection

ANSI Function 46 (negative-sequence overcurrent / unbalance protection) trips when I₂ exceeds a setpoint, protecting motors from sustained unbalance. Common settings: pickup at 5–10 % I_FL, trip at 30–60 seconds. Required by NEC 430.36 for large motors.

Fortescue\'s original derivation

A system of n alternating-current vectors of any character may be represented by n systems of symmetrical vectors. In a three-phase system, three sets are required: a positive-sequence, a negative-sequence, and a zero-sequence system. The mathematical treatment is greatly simplified, particularly in the analysis of unbalanced and asymmetric short-circuit conditions, by this transformation.

Fortescue, C.L. — Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks → Transactions of the AIEE, Vol. 37 Part II, June 1918, pp. 1027–1140

Related calculators

Frequently asked questions

What is voltage unbalance in 3-phase?: A condition where the three phase voltages are not equal in magnitude or are not exactly 120° apart. Even small imbalances (1–2 %) significantly heat induction motors because of the negative-sequence currents. NEMA defines a simple metric (max deviation from average); IEEE uses the more accurate symmetrical components ratio.
What are V₀, V₁, V₂?: The three "symmetrical components" of an unbalanced phasor set. V₀ (zero-sequence) — three equal voltages all in phase, sums to non-zero only when there's ground-current flow. V₁ (positive-sequence) — three balanced voltages 120° apart in normal phase rotation, this is the "useful" component. V₂ (negative-sequence) — three balanced voltages 120° apart in REVERSE rotation, causes counter-torque and heating in motors.
How do I calculate the voltage unbalance factor?: NEMA method: VUF = (max deviation of any single phase from the 3-phase average) / (3-phase average) × 100 %. Easy with three readings. IEEE method: VUF = |V₂| / |V₁| × 100 %. More rigorous; requires phase angles too. The two metrics agree at low imbalance and diverge slightly above ~3 %.
Why do unbalanced voltages damage motors?: Negative-sequence voltage produces a backward-rotating field in the motor air gap. This creates a counter-torque (small loss) and high-frequency rotor currents (large heating). At 3 % voltage unbalance, the negative-sequence currents can be 6–10× the rated rotor current — the rotor heats up but the torque output barely changes. Result: insulation degrades 2–3× faster, bearings fail prematurely.
What causes 3-phase voltage unbalance?: Single-phase load distribution — most common cause; when 1-φ loads are not balanced equally across three phases. Single-phasing — a blown fuse or open phase on the supply leaves the motor running on 2 phases (massive imbalance, motor burns out in minutes). Asymmetric impedance — different cable lengths or cross-sections per phase. Utility issues — unbalanced primary distribution feeders.
What is the maximum acceptable unbalance?: NEMA recommends ≤1 % for sustained operation of motors; tolerable up to 3 % with derate; never above 5 %. IEEE 1159 categorises voltage unbalance as a "long-duration variation" with limits of 0.5–2 %. ANSI C84.1 sets utility-supplied unbalance at the service point to 3 % maximum. Modern PFCs and converters add their own limits — check equipment specs.
How does NEMA MG 1 derate motors for unbalance?: NEMA MG 1 §14.36 publishes a derating curve: at 1 % VUF derate to 0.98 of nameplate; at 2 %, 0.95; at 3 %, 0.88; at 4 %, 0.82; at 5 %, 0.75 — and operation above 5 % is not recommended. Always pair the curve with a thermal protection device (NEMA Class C trip on overload).
What is the "a" operator in symmetrical components?: A complex unit-magnitude phasor at 120° angle: a = e^(j·2π/3) = −0.5 + j·0.866. Multiplying any phasor by a rotates it by 120° counter-clockwise. The Fortescue transformation matrix uses 1, a, and a² to extract each sequence from the original three phasors. a³ = 1, a² = e^(j·240°) = a*.

Sources and methodology

Fortescue, C.L. Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, Vol. 37, June 1918.
IEEE. IEEE Std 1159 — Recommended Practice for Monitoring Electric Power Quality, 2019.
NEMA. NEMA MG 1 — Motors and Generators, 2021. § 14.36 voltage unbalance.
ANSI. ANSI C84.1 — Electric Power Systems and Equipment Voltage Ratings (60 Hz), 2020.
Anderson, P.M. Analysis of Faulted Power Systems, IEEE Press, 1995. Chapter 2 — symmetrical components.