Symmetrical Components Calculator
The unbalanced set 100∠0°, 80∠−120°, 90∠120° has a positive-sequence component of 90 ∠ 0°, with small negative and zero sequences.
90
- Negative sequence V₂
- 5.7735 ∠ -30°
- Zero sequence V₀
- 5.7735 ∠ 30°
- Unbalance factor |V₂|/|V₁|
- 6.415002991 %
- Check V₀ + V₁ + V₂
- 100 ∠ 0° (= V_a)
Step-by-step solution
- a = 1∠120°, a² = 1∠240°
- V₀ = (V_a + V_b + V_c) ÷ 3 = 5.7735 ∠ 30°
- V₁ = (V_a + aV_b + a²V_c) ÷ 3 = 90 ∠ 0°
- V₂ = (V_a + a²V_b + aV_c) ÷ 3 = 5.7735 ∠ -30°
Symmetrical Components Formula
V₁ = (V_a + aV_b + a²V_c)/3
V₀ = (V_a + V_b + V_c)/3, V₂ = (V_a + a²V_b + aV_c)/3, a = 1∠120°
Worked Example
Unbalanced phases V_a = 100∠0°, V_b = 80∠−120°, V_c = 90∠120°:
- a = 1∠120°, a² = 1∠240°
- V₀ = (V_a + V_b + V_c) ÷ 3 = 5.7735 ∠ 30°
- V₁ = (V_a + aV_b + a²V_c) ÷ 3 = 90 ∠ 0°
- V₂ = (V_a + a²V_b + aV_c) ÷ 3 = 5.7735 ∠ -30°
Zero sequence V₀: 5.7735 ∠ 30° · Unbalance factor |V₂|/|V₁|: 6.415002991 % · Check V₀ + V₁ + V₂: 100 ∠ 0° (= V_a)
GATE tip
Remember 1 + a + a² = 0 and a³ = 1. For a balanced set V₂ = V₀ = 0; a line-to-line fault has no zero sequence; a ground fault on a delta winding cannot pass zero-sequence current.
More Power Systems Calculators
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Per-Unit Calculator
Z_new = Z_old (kV_old/kV_new)² (MVA_new/MVA_old)
Z_base = kV²/MVA; I_base = MVA/(√3 kV); fault MVA = MVA_base/Z_pu
Example: A 0.1 pu reactance on 50 MVA, 11 kV, moved to a 100 MVA, 11 kV base: Impedance on the new base = 0.2 pu.
Frequently asked questions
What do the sequences mean physically?
Positive sequence rotates the normal way and produces useful torque; negative sequence rotates backwards and heats motors; zero sequence is in phase on all three lines and returns through neutral or earth.
How do faults use them?
A single line-to-ground fault connects the three sequence networks in series: I_f = 3V/(Z₁ + Z₂ + Z₀).
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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Scientific Converters. (2026, October 15). Symmetrical Components Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/symmetrical-components-calculator/
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