Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913067938349056 |
|---|---|
| author | Bravo-Rodríguez, Juan Carlos del-Pino-López, Juan Carlos Casado-Machado, Francisco |
| author_facet | Bravo-Rodríguez, Juan Carlos del-Pino-López, Juan Carlos Casado-Machado, Francisco |
| contents | Unbalanced three-phase systems still lack a compact phasor-domain representation of power that makes phase asymmetry explicit while remaining consistent with established apparent-power definitions. This paper addresses that point through a complex-vector power formulation for sinusoidal steady-state operation. The proposed representation supplements the classical dot-product expression of complex power with the cross product of voltage and current phasors, thereby retaining the usual active and reactive terms while making explicit a cross-phase unbalance vector that captures antisymmetric interphase relations. In this way, apparent power is separated into intraphase and cross-phase contributions, and its norm is preserved under the power-invariant Fortescue transformation. The formulation is extended to three-phase four-wire systems by introducing equivalent coordinates that preserve the effective apparent-power norm for the chosen voltage reference. Only standard complex numbers and matrices are required. Numerical examples show operating conditions in which a non-negligible part of the apparent-power structure is associated with cross-phase unbalance and cannot be inferred from active and reactive power alone. The proposed formulation thus provides a compact phasor-based descriptor of unbalance that complements established apparent-power theories by making explicit a component that is not accessible from scalar apparent-power representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25473 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems Bravo-Rodríguez, Juan Carlos del-Pino-López, Juan Carlos Casado-Machado, Francisco Systems and Control J.2 Unbalanced three-phase systems still lack a compact phasor-domain representation of power that makes phase asymmetry explicit while remaining consistent with established apparent-power definitions. This paper addresses that point through a complex-vector power formulation for sinusoidal steady-state operation. The proposed representation supplements the classical dot-product expression of complex power with the cross product of voltage and current phasors, thereby retaining the usual active and reactive terms while making explicit a cross-phase unbalance vector that captures antisymmetric interphase relations. In this way, apparent power is separated into intraphase and cross-phase contributions, and its norm is preserved under the power-invariant Fortescue transformation. The formulation is extended to three-phase four-wire systems by introducing equivalent coordinates that preserve the effective apparent-power norm for the chosen voltage reference. Only standard complex numbers and matrices are required. Numerical examples show operating conditions in which a non-negligible part of the apparent-power structure is associated with cross-phase unbalance and cannot be inferred from active and reactive power alone. The proposed formulation thus provides a compact phasor-based descriptor of unbalance that complements established apparent-power theories by making explicit a component that is not accessible from scalar apparent-power representations. |
| title | Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems |
| topic | Systems and Control J.2 |
| url | https://arxiv.org/abs/2604.25473 |