---
title: "Representing a 3-phase system by single phase"
subject: "Power System Analysis I"
chapter: 3
key: "semester_4/power_system_analysis_i:single-phase-representation-conditions"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/single-phase-representation-conditions
site: BCT Engineering
---

# Representing a 3-phase system by single phase

Power System Analysis I · Chapter 3: Computational Technique

- Asked in: 2068 Bhadra, 3 marks
- Video: [SIngle line Equivalent Circuit for a 3 phase balanced loads](https://www.youtube.com/watch?v=NB6ytxsXok4) — G Zolutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/single-phase-representation-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-3

## Question

Mention the conditions and advantages of representing a 3-phase system by a single phase system.

## Short answer

A balanced three-phase system can be represented by one phase if sources and loads are balanced, lines are transposed, and operation is steady state; delta loads are converted to equivalent wye. Advantages: only one circuit is solved, one-line diagrams and per-unit values apply, and three-phase power is three times the per-phase power.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which condition is required for the per-phase representation?
   A. The loads are unbalanced
   B. Sources and loads are balanced, with equal impedance in each phase
   C. Only a single load is connected
   D. The neutral carries current
   Answer: B. Balanced conditions make the other phases identical, shifted by 120°. (Key point: Balanced requirement)

2. Why can the neutral impedance be omitted?
   A. Neutral current is zero in a balanced system
   B. Neutral is always open
   C. Neutral voltage is large
   D. Because delta has no neutral current
   Answer: A. Balanced currents sum to zero. (Key point: Neutral current)

3. Which is an advantage?
   A. It can solve unbalanced faults directly
   B. It gives different pu in each phase
   C. It avoids the need for line-to-neutral voltage
   D. One circuit replaces three coupled circuits, and three-phase power is 3 x per-phase power
   Answer: D. Unbalanced faults need symmetrical components. (Key point: Advantage)

## Related in Power System Analysis I

- [Benefits of three-phase over single-phase delivery](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Athree-phase-benefits-over-single) — Power System Analysis I, chapter 1, asked 2×
- [P and Q of a 100 V single-phase source](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-source-power-100v) — Power System Analysis I, chapter 3
- [Inductance of composite-conductor single-phase line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-composite-conductors) — Power System Analysis I, chapter 4
- [Inductance of a single-phase two-wire line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asingle-phase-inductance-derivation) — Power System Analysis I, chapter 4
- [Transposition; single-phase capacitance with earth](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Atransposition-single-phase-earth-capacitance) — Power System Analysis I, chapter 4
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