Symmetrical Components Calculator (Positive, Negative and Zero Sequence)
Enter three phase currents or voltages as magnitude and angle to resolve them into positive, negative, and zero sequence components using Fortescue's transformation. Also returns the residual 3I0 a ground element sees, the negative sequence unbalance ratio, a phasor diagram, and a round-trip check.
For ACB rotation the calculator swaps the roles of phases B and C in the transformation, so a balanced ACB set resolves to pure positive sequence, the same convention most relays use when their rotation setting is ACB.
Both panels share one scale. Sequence components show the phase A reference; components too small to draw appear as a dot at the origin.
| Phase | Entered | Rebuilt from sequences |
|---|---|---|
| A | 100.00 A ∠ 0.00° | 100.00 A ∠ 0.00° |
| B | 95.00 A ∠ -125.00° | 95.00 A ∠ -125.00° |
| C | 105.00 A ∠ 118.00° | 105.00 A ∠ 118.00° |
Relays and meters differ in reference phase, rotation setting, and whether they report I0 or 3I0. Confirm the relay's convention before comparing its metering to these numbers. This is an educational and verification tool, not a protective relay settings study.
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How symmetrical components work, and what each sequence means
Fortescue's theorem says any three unbalanced phasors can be written as the sum of three balanced sets. The positive sequence set has equal magnitudes, 120 degrees apart, rotating in the normal order. The negative sequence set is the same but rotates in the opposite order. The zero sequence set is three equal phasors all in phase. Solve for the three sets once and the unbalanced problem turns into three balanced ones.
The tool that makes this work is the a operator, a unit phasor at 120 degrees: a = 1∠120° and a² = 1∠240°. Multiplying by a rotates a phasor 120 degrees forward. For ABC rotation:
- X0 = (Xa + Xb + Xc) ÷ 3
- X1 = (Xa + a·Xb + a²·Xc) ÷ 3
- X2 = (Xa + a²·Xb + a·Xc) ÷ 3
What each sequence means physically
- Positive sequence is the normal rotating field. In a healthy, balanced system it is the only component present, and it carries the useful power.
- Negative sequence rotates backward. In a motor or generator it induces double-frequency currents in the rotor and heats it without producing useful torque. Negative sequence overcurrent (46) and negative sequence voltage (47) elements are built on it, and it also exposes phase loss and reversed rotation.
- Zero sequence is three equal currents in phase, so it can only flow when there is a path to ground or a neutral. Ground overcurrent (51N) and directional ground (67N) elements respond to it, usually as 3I0. A delta winding traps zero sequence current inside the delta, so it does not pass through to the other side of a delta-wye transformer.
How fault types map to the sequence networks
- Single line-to-ground: positive, negative, and zero sequence networks connect in series, so I1 = I2 = I0 and the fault current is 3I0. Load the line-to-ground preset to see three equal 333.33 A components from a 1000 A fault.
- Line-to-line: positive and negative networks connect in parallel with I2 = -I1. There is no zero sequence because no current returns through ground.
- Three-phase: the fault is balanced, so only the positive sequence network is involved.
Worked example: the default currents
Ia = 100 A ∠ 0°, Ib = 95 A ∠ -125°, Ic = 105 A ∠ 118°, ABC rotation. In rectangular form Ia = 100.00 + j0.00, Ib = -54.49 - j77.82, and Ic = -49.29 + j92.71.
- Zero sequence: Ia + Ib + Ic = -3.78 + j14.89. Divide by 3: I0 = -1.26 + j4.96 = 5.12 A ∠ 104.26°. The ground element sees 3I0 = 15.36 A.
- Positive sequence: a·Ib = 95 ∠ -5° and a²·Ic = 105 ∠ -2°, so all three terms nearly line up near 0°. I1 = (100 ∠ 0° + 95 ∠ -5° + 105 ∠ -2°) ÷ 3 = 99.86 - j3.98 = 99.94 A ∠ -2.28°.
- Negative sequence: a²·Ib = 95 ∠ 115° and a·Ic = 105 ∠ -122°, which mostly cancel against Ia. I2 = 1.40 - j0.98 = 1.71 A ∠ -34.98°.
- Ratios: I2 ÷ I1 = 1.71 ÷ 99.94 = 1.71% negative sequence unbalance, and I0 ÷ I1 = 5.12%.
The same three magnitudes give 5.00% unbalance with the NEMA MG-1 method used in the current unbalance calculator. Most of that deviation is zero sequence, which a magnitude-only method cannot separate from negative sequence.
Symmetrical components: frequently asked questions
What are symmetrical components?
Symmetrical components, introduced by C. L. Fortescue in 1918, split any set of three unbalanced phasors into three balanced sets: positive sequence (same rotation as the system), negative sequence (opposite rotation), and zero sequence (three equal phasors in phase with each other). Adding the three sets back together reproduces the original phasors exactly. Protection engineers use them because each sequence has its own network and its own relay elements.
What does negative sequence current indicate?
Negative sequence current means the three phase currents are no longer balanced. Common sources are unbalanced load, an open phase or blown fuse, a high-resistance connection, and unbalanced faults. It produces a field rotating backward relative to the rotor, which induces double-frequency currents that heat motor and generator rotors. Negative sequence overcurrent (46) and negative sequence voltage (47) elements use it to protect rotating machines and to detect phase loss or reversed rotation.
Why is zero sequence current shown as 3I0 at the relay?
A ground or residual element measures Ia + Ib + Ic, either from a residual CT connection or a neutral CT. That sum equals 3I0, not I0. For a 1000 A single line-to-ground fault on phase A with no load current, I0 is 333.33 A and the ground element sees 3I0 = 1000 A. Some relays meter I0 while their 51N and 67N pickups are set in 3I0 terms, so confirm which quantity the manual uses before comparing.
How does phase rotation affect the result?
The equations assume a rotation. With ABC rotation, I1 = (Ia + a·Ib + a²·Ic) ÷ 3. If the system actually rotates ACB and the ABC equations are applied, positive and negative sequence swap: a perfectly balanced system reads as 100% negative sequence. Select the rotation that matches the system, or load the Reversed rotation preset to see the effect. Zero sequence does not depend on rotation.
How is the negative sequence ratio different from the NEMA unbalance definition?
NEMA MG-1 unbalance uses only magnitudes: the maximum deviation from the average divided by the average. The IEC and IEEE unbalance factor is the ratio of negative to positive sequence, which needs the angles as well. For the default currents on this page (100 A, 95 A, 105 A) the NEMA method gives 5.00% while I2 ÷ I1 is only 1.71%, because magnitudes alone mix zero and negative sequence content together. Use the current unbalance calculator when a specification calls for the NEMA figure.
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