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Fortescue transformation: positive, negative, and zero sequence quantities for unbalance analysis.
In Detego
The Fortescue transformation (1918) is one of the most powerful tools in power system analysis. It decomposes any set of unbalanced three-phase quantities into three balanced sets: positive sequence, negative sequence, and zero sequence. This decomposition dramatically simplifies the analysis of unbalanced faults, which are the most common type of fault on power systems.
Detego computes sequence components from the fundamental-frequency phasors of three-phase voltage or current channels. The computation supports multiple phase naming conventions and provides both individual sequence phasors and diagnostic ratios for fault classification.
The diagrams below illustrate the three sequence component sets. In a balanced system, only positive sequence exists. Faults introduce negative and/or zero sequence components whose magnitudes indicate the severity and type of unbalance.
Positive Sequence (V₁)
Negative Sequence (V₂)
Zero Sequence (V₀)
Fortescue sequence components. Positive sequence: balanced ABC rotation. Negative sequence: reversed ACB rotation (indicates unbalance). Zero sequence: all phasors in-phase (indicates ground path).
Fortescue operator a
Where
The operator a is the mathematical key to the transformation. Multiplying by a rotates a phasor by 120 degrees counterclockwise.
Positive sequence component
Where
The positive sequence represents the balanced ABC rotation component. It is the normal operating component -- in a perfectly balanced three-phase system, the positive sequence equals the phase magnitude and the other sequences are zero. During faults, the positive sequence magnitude drops (voltage) or remains dominant (current), and its value represents the portion of the system that is still operating normally.
Negative sequence component
Where
The negative sequence represents the reversed ACB rotation component. It is zero in a balanced system and appears only when there is unbalance between phases. Negative sequence is particularly useful because it is present in all unbalanced fault types (single line-to-ground, line-to-line, double line-to-ground) but absent during balanced three-phase faults and normal load conditions. This makes it a reliable indicator of asymmetric faults.
Zero sequence component
Where
The zero sequence represents the in-phase component -- all three phases moving together. In current, this corresponds to current flowing through the ground (earth) return path. Zero sequence is present only when there is a path for ground current, which means it indicates ground fault involvement. A line-to-line fault (no ground) produces negative sequence but no zero sequence, while a single line-to-ground fault produces both.
Detego provides three sequence component signals — Positive, Negative, and Zero — each with an output mode toggle in the Signal Builder that switches between Time Domain and DFT computation. Both modes produce the same result for a pure fundamental-frequency balanced system, but they differ significantly when harmonics, DC offset, or transients are present.
The time-domain method approximates the Fortescue phase rotation by shifting channels in time. A 120-degree rotation is equivalent to a time delay of (one-third of a cycle):
Time-domain positive sequence
Where
Negative sequence uses the opposite shifts: B advanced by T/3, C advanced by 2T/3. Zero sequence is simply (A + B + C) / 3 with no shifts.
This method preserves all frequency content including harmonics, DC offset, and transients. The resulting waveform matches the output of traditional DFR analysis tools (e.g., WaveWin, SIGRA). It is the default method in Detego and is recommended for general waveform inspection and fault analysis.
The phasor method uses a sliding one-cycle DFT to extract the fundamental-frequency phasor of each channel, applies the Fortescue matrix rotation in the complex plane, and reconstructs a clean sinusoidal waveform:
DFT-based sequence extraction
Where
The result is a pure sinusoid at the fundamental frequency. All harmonics, DC offset, and transient content are filtered out.
This method is useful for phasor-based analysis and relay modeling where only the fundamental-frequency component is relevant. It provides a clean, noise-free representation of the sequence quantity but does not capture transient behavior during fault inception or clearance.
Choosing between modes
Detego automatically detects and supports multiple phase naming conventions used across different regions and equipment manufacturers:
| Convention | Phase 1 | Phase 2 | Phase 3 |
|---|---|---|---|
| IEC / ANSI | A | B | C |
| European (IEC 60909) | L1 | L2 | L3 |
| Numeric | 1 | 2 | 3 |
| German / legacy | R | S | T |
Unbalance ratio
Where
The negative-to-positive sequence ratio quantifies the severity of the unbalance. In protection engineering, the following thresholds are commonly used:
The sequence component trend view steps through the recording one cycle at a time, computing sequence components at each step. This produces a time series of positive, negative, and zero sequence magnitudes, plus the negative/positive ratio, showing how the unbalance evolves through the fault event. The trend is particularly useful for identifying fault inception, classification changes (e.g., a line-to-line fault evolving into a line-to-line-to-ground fault), and fault clearance.
Fault classification