---
title: "Economics of 1-φ vs 3-φ delivery"
subject: "Power System Analysis I"
chapter: 1
key: "semester_4/power_system_analysis_i:economics-single-vs-three-phase"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/economics-single-vs-three-phase
site: BCT Engineering
---

# Economics of 1-φ vs 3-φ delivery

Power System Analysis I · Chapter 1: General Background

- Asked in: 2071 Magh, 5 marks
- Video: [Why we Use 3ph? Why Not 4ph, 5ph??| Why Transmission done with 3ph| 1ph vs 3ph System](https://www.youtube.com/watch?v=3Zs6NqEY4mU) — Electrical With Abhishek Joshi
- 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/economics-single-vs-three-phase
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-1

## Question

Compare the economic aspects of 1-φ and 3-φ power delivery systems. Give suitable mathematical relationship to justify the argument.

## Short answer

For same power, distance, loss and voltage, three-phase 3-wire needs only 75% of the conductor of single-phase 2-wire, since Vol3/Vol1 = 3/4. Three-phase also gives more machine output per frame and cheaper transformers and motors.

## Answer

For the same power, distance, line loss and maximum voltage, a 3-phase system needs less conductor material and so costs less.

#### Single-phase 2-wire line
Power $P$, voltage $V$ (between conductors), p.f. $\cos\phi$, length $l$, area $a_1$:
$$
\begin{aligned}
I_1 &= \frac{P}{V\cos\phi},\qquad W = 2I_1^2\frac{\rho l}{a_1} = \frac{2P^2\rho l}{a_1V^2\cos^2\phi}\\
a_1 &= \frac{2P^2\rho l}{WV^2\cos^2\phi},\qquad \text{Vol}_1 = 2a_1l = \frac{4P^2\rho l^2}{WV^2\cos^2\phi}
\end{aligned}
$$

#### Three-phase 3-wire line (line voltage $V$)
$$
\begin{aligned}
I_3 &= \frac{P}{\sqrt3V\cos\phi},\qquad W = 3I_3^2\frac{\rho l}{a_3} = \frac{P^2\rho l}{a_3V^2\cos^2\phi}\\
a_3 &= \frac{P^2\rho l}{WV^2\cos^2\phi},\qquad \text{Vol}_3 = 3a_3l = \frac{3P^2\rho l^2}{WV^2\cos^2\phi}
\end{aligned}
$$

#### Comparison
$$
\frac{\text{Vol}_3}{\text{Vol}_1} = \frac{3}{4} = 0.75
$$
So 3-phase 3-wire needs only **75 %** of the conductor of 1-phase 2-wire (with the same voltage between conductors).

#### Other economic points
| Aspect | 1-phase | 3-phase |
|---|---|---|
| Conductor | 100 % | 75 % |
| Machine output per frame | 1 | ≈ 1.5 |
| Transformer cost per kVA | Higher | Lower |
| Motor | Needs starting aid | Self-starting, cheaper per kW |
| Power | Pulsating | Constant |
| Use | Small domestic loads | Generation, transmission, industry |

Hence 3-phase delivery is more economical for generation, transmission and distribution; 1-phase is only used for small loads tapped from the 3-phase system.

## Check yourself

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

1. What is Vol3/Vol1 for the same P, V, loss and p.f.?
   A. 4/3
   B. 1/2
   C. 3/4
   D. 1
   Answer: C. Vol1 = 4P^2 rho l^2/(W V^2 cos^2), Vol3 = 3(...), so ratio is 0.75. (Key point: Conductor volume ratio)

2. What is the line current of a 3-phase line of power P at line voltage V?
   A. P/(sqrt3 V cos phi)
   B. P/(V cos phi)
   C. sqrt3 P/(V cos phi)
   D. 3P/(V cos phi)
   Answer: A. I3 = P/(sqrt3 V cos phi). (Key point: Three-phase current)

3. What is the loss in a 3-phase line?
   A. 2 I^2 rho l/a
   B. I^2 rho l/(3a)
   C. sqrt3 I^2 rho l/a
   D. 3 I^2 rho l/a
   Answer: D. Three conductors each lose I^2 rho l/a. (Key point: Loss expression)

4. How does a 3-phase machine's output compare with a 1-phase of same frame?
   A. Same
   B. Half
   C. About 1.5 times
   D. 3 times
   Answer: C. A 3-phase machine of the same size gives about 1.5 times the output. (Key point: Machine output)

## 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×
- [Vs from P+jQ and reactive power flow direction](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Avs-from-pq-reactive-flow) — Power System Analysis I, chapter 6
- [Evolution of power systems and milestones](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aevolution-power-system) — Power System Analysis I, chapter 1
- [Infinite bus and HVDC vs HVAC comparison](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainfinite-bus-hvdc-vs-hvac) — Power System Analysis I, chapter 1
- [Conductor volume vs voltage and power factor](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aconductor-volume-voltage-pf) — Power System Analysis I, chapter 1
- [Power delivered from sending end data, nominal-π](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Anominal-pi-power-delivered) — Power System Analysis I, chapter 5

## Related in other subjects

- [Compare AM systems; one method of DSB-FC generation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-types-compare-dsbfc-method) — Communication Systems, chapter 2
- [Justify higher voltage/more circuits for more power](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Ajustify-higher-voltage-more-circuits) — Transmission and Distribution System Design, chapter 2
- [Justify: series motor power; shunt motor for traction](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Ajustify-series-motor-power-shunt-traction) — Utilization of Electrical Energy, chapter 4
- [Financial vs economic analysis; both BCR](https://www.bctengineer.com/question.md?key=semester_6%2Fengineering_economics%3Afinancial-vs-economic-bcr-300000) — Engineering Economics, chapter 3