---
title: "Transposition and inductance of transposed line"
subject: "Power System Analysis I"
chapter: 4
key: "semester_4/power_system_analysis_i:transposition-inductance-derivation"
source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/transposition-inductance-derivation
site: BCT Engineering
---

# Transposition and inductance of transposed line

Power System Analysis I · Chapter 4: Line Parameter Calculations

- Asked in: 2074 Bhadra, 6 marks; 2070 Magh, 8 marks (asked 2 times)
- Video: [Power Transmission|| 11 || Inductance of a Three-Phase Transmission line](https://www.youtube.com/watch?v=eOuMszZCfeY) — Dr. Ammar Arshad
- 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/transposition-inductance-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4

## Question

Why is transposition required in a three phase line with unsymmetrical spacing? Derive the expression for the inductance of such a transposed line.

## Short answer

Transposition equalizes the average inductance of each phase when spacings are unequal, preventing unbalanced voltage drops and interference. The inductance per phase becomes L = 2×10⁻⁷ ln(Deq/r') H/m, where Deq = ∛(D12 D23 D31) is the geometric mean distance and r' = 0.7788r. This yields a balanced, real inductance identical to an equilateral spacing Deq.

## Answer

#### Why transposition is needed
When the phase spacings are unequal ($D_{12} \ne D_{23} \ne D_{31}$), each phase has a different flux linkage. Its inductance therefore differs from the other phases and even contains an imaginary part. This causes unequal voltage drops, an unbalanced receiving-end voltage, and interference with nearby communication lines. **Transposition** means each phase takes each position for one-third of the length. This makes the average inductance of all phases equal, so the line becomes balanced and can be analysed per phase.

```text
          Sec I      Sec II     Sec III
 pos 1     a           c           b
 pos 2     b           a           c
 pos 3     c           b           a
        D12 between pos1-2, D23 pos2-3, D31 pos3-1
```

#### Derivation
Flux linkage of conductor a, of radius r, in a group of conductors (per metre):
$$
\lambda_a = 2\times10^{-7}\left[I_a\ln\frac{1}{r'} + I_b\ln\frac{1}{D_{ab}} + I_c\ln\frac{1}{D_{ac}}\right], \quad r' = 0.7788r
$$

**Section I** (a at 1, b at 2, c at 3):
$$
\lambda_{a1} = 2\times10^{-7}\left[I_a\ln\frac{1}{r'} + I_b\ln\frac{1}{D_{12}} + I_c\ln\frac{1}{D_{31}}\right]
$$

**Section II** (a at 2, b at 3, c at 1):
$$
\lambda_{a2} = 2\times10^{-7}\left[I_a\ln\frac{1}{r'} + I_b\ln\frac{1}{D_{23}} + I_c\ln\frac{1}{D_{12}}\right]
$$

**Section III** (a at 3, b at 1, c at 2):
$$
\lambda_{a3} = 2\times10^{-7}\left[I_a\ln\frac{1}{r'} + I_b\ln\frac{1}{D_{31}} + I_c\ln\frac{1}{D_{23}}\right]
$$

**Average** over the cycle:
$$
\begin{aligned}
\lambda_a &= \frac{\lambda_{a1} + \lambda_{a2} + \lambda_{a3}}{3}\\
&= \frac{2\times10^{-7}}{3}\left[3I_a\ln\frac{1}{r'} + (I_b + I_c)\ln\frac{1}{D_{12}D_{23}D_{31}}\right]
\end{aligned}
$$

For balanced currents, $I_b + I_c = -I_a$:
$$
\begin{aligned}
\lambda_a &= 2\times10^{-7}\left[I_a\ln\frac{1}{r'} - I_a\ln\frac{1}{\sqrt[3]{D_{12}D_{23}D_{31}}}\right]\\
&= 2\times10^{-7}I_a\ln\frac{\sqrt[3]{D_{12}D_{23}D_{31}}}{r'}
\end{aligned}
$$

**Inductance per phase:**
$$
L_a = \frac{\lambda_a}{I_a} = 2\times10^{-7}\ln\frac{D_{eq}}{r'}\ \text{H/m}, \qquad D_{eq} = \sqrt[3]{D_{12}D_{23}D_{31}}
$$

By symmetry, $L_b = L_c = L_a$. The transposed line behaves like an equilaterally spaced line with spacing $D_{eq}$ (the geometric mean distance), and the inductance is real and equal in all phases.

In mH/km: $L = 0.2\ln\frac{D_{eq}}{r'}$ mH/km per phase. For stranded or bundled conductors, $r'$ is replaced by the GMR $D_s$.

Example: horizontal spacing 4 m, 4 m, 8 m gives $D_{eq} = \sqrt[3]{4\times4\times8} = 5.04$ m.

## Check yourself

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

1. Why is transposition required on a line with D12 != D23 != D31?
   A. Each phase would have different flux linkage and inductance, causing unbalance
   B. To increase line voltage
   C. To lower conductor resistance
   D. To avoid the need for insulators
   Answer: A. Unequal spacing gives unequal (even complex) inductances, so transposition equalises them and makes per-phase analysis valid. (Key point: Why transposition)

2. How is the flux linkage of phase a in the transposed line obtained?
   A. Take the largest of the three section values
   B. Average the three section values
   C. Sum the three section values
   D. Use only section I
   Answer: B. Phase a takes each position for a third of the length, so the average of lambda_a1, lambda_a2, lambda_a3 is used. (Key point: Averaging step)

3. Which condition lets (Ib + Ic) ln(1/(D12 D23 D31)) be simplified?
   A. Ib + Ic = Ia
   B. Ib + Ic = -Ia for balanced currents
   C. Ib = Ic = 0
   D. Ia = 0
   Answer: B. For balanced three-phase currents Ia + Ib + Ic = 0, so Ib + Ic = -Ia. (Key point: Balanced current use)

4. What is the inductance per phase of the transposed line?
   A. 2e-7 ln(Deq/r') H/m, Deq = cube root of D12 D23 D31
   B. 2e-7 ln(r'/Deq) H/m
   C. 2e-7 ln(D12/r') H/m
   D. 2e-7 ln(Deq/r) with Deq the arithmetic mean
   Answer: A. The result looks like an equilateral line with spacing equal to the geometric mean distance. (Key point: Final result)

## Related in Power System Analysis I

- [Inductance of unsymmetrical transposed line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aunsymmetrical-inductance-10mm) — Power System Analysis I, chapter 4
- [Inductance of 3-conductor bundled line, 6.1 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainductance-three-subconductor-bundle-6-1m) — Power System Analysis I, chapter 4
- [Inductance of transposed double circuit, 0.9 cm radius](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Adouble-circuit-inductance-0-9cm) — 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
- [Inductance of 4-conductor bundle line, 6 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aquad-bundle-200km-12mm-6m) — Power System Analysis I, chapter 4, asked 2×
- [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

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