Representing a 3-phase system by single phase

A balanced three-phase system can be solved as a single-phase (per-phase) equivalent: one phase plus a neutral, using line-to-neutral voltages. The other two phases have the same results shifted by ±120°.

Conditions

1. The sources are balanced: equal magnitudes and 120° apart, with positive sequence.
2. The loads and network are balanced: equal impedance in each phase, and lines are fully transposed so that mutual effects are equal.
3. The system operates in steady state under balanced (normal or three-phase symmetrical fault) conditions. Unbalanced faults need symmetrical components.
4. Δ-connected elements are replaced by their equivalent Y ($Z_Y = Z_\Delta/3$). Transformer phase shifts are ignored or added back later.

Under these conditions the neutral current is zero. So the neutral impedance carries no current and is left out of the per-phase circuit.

Advantages

• Calculation is reduced to one circuit instead of three coupled circuits. This saves time and effort.
• One-line (single-line) diagrams and per-phase impedance/reactance diagrams can be used.
• Per-unit quantities are the same for the per-phase and three-phase circuits, which makes the work simpler.
• Three-phase power is simply $3\times$ per-phase power: $S_{3\phi} = 3V_{ph}I_{ph}^*$.
• It is used directly in load-flow, short-circuit (symmetrical fault) and stability studies.