---
title: "Distortionless line and telephone lines"
subject: "Electromagnetism"
chapter: 5
key: "semester_3/electromagnetism:distortionless-line-telephone"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/distortionless-line-telephone
site: BCT Engineering
---

# Distortionless line and telephone lines

Electromagnetism · Chapter 5: Transmission lines

- Asked in: 2067 Shrawan, 2+1 marks
- Video: [Distortionless Line and Its Conditions- Derivation - Transmission line](https://www.youtube.com/watch?v=3jjC30hw6T0) — Padmasri Naban
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/distortionless-line-telephone
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-5

## Question

What is a distortionless transmission line? Why are telephone lines required to be distortionless?

## Short answer

A distortionless line has attenuation constant α and phase velocity independent of frequency, achieved when R/L = G/C. Telephone lines require this to preserve speech fidelity, ensuring all voice frequency components are equally attenuated and travel at the same speed, preventing signal distortion.

## Answer

#### Distortionless transmission line

A **distortionless line** is one whose attenuation is the same for all frequencies and whose phase velocity is also the same for all frequencies, so a signal of any shape reaches the end with only a reduction in amplitude, not a change in shape.

Condition (Heaviside): **R/L = G/C**, i.e. RC = LG.

Then γ = √[(R + jωL)(G + jωC)] = √(RG) + jω√(LC), so

```text
α = √(RG)           (independent of frequency)
β = ω√(LC)          (linear in ω)
v = ω/β = 1/√(LC)   (independent of frequency)
Z₀ = √(L/C) = √(R/G)   (real, constant)
```

#### Why telephone lines must be distortionless

Speech contains many frequencies (about 300 Hz to 3.4 kHz). If α or v changed with frequency, the components would be attenuated unequally and would arrive at different times, so speech would sound muffled and unclear.

A telephone cable normally has high R, very small G and small L, so R/L ≫ G/C and the line is distorting. To make it distortionless, either L is increased (loading coils added at regular intervals, as Heaviside suggested) or G is increased (not preferred because it increases attenuation). Increasing L is the practical method, and gives a nearly constant α, v and Z₀ over the voice band.

## Check yourself

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

1. What is the condition for a distortionless line?
   A. R = G
   B. L = C
   C. R L = G C
   D. R/L = G/C (RC = LG)
   Answer: D. With R/L = G/C, gamma = sqrt(RG) + jw sqrt(LC). (Key point: Condition)

2. What do alpha and v become?
   A. alpha = w sqrt(LC) and v = 1/sqrt(RG)
   B. alpha = 0 and v = c
   C. alpha = sqrt(RG) and v = 1/sqrt(LC), both independent of frequency
   D. alpha proportional to frequency
   Answer: C. alpha is constant and beta is linear in w, so v is constant. (Key point: Result)

3. Why must telephone lines be distortionless, and how is it achieved in practice?
   A. To block DC; G is increased greatly
   B. Speech has many frequencies that must be attenuated and delayed equally; L is increased with loading coils
   C. To carry a single frequency; R is increased
   D. To reduce L; coils are removed
   Answer: B. Telephone cables have R/L much greater than G/C. Raising L is practical, while raising G increases attenuation. (Key point: Telephone lines)

## Related in Electromagnetism

- [Waveguide vs transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-advantages-over-transmission-line) — Electromagnetism, chapter 6, asked 3×
- [Derive input impedance of transmission line](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aderive-input-impedance-line) — Electromagnetism, chapter 5
- [Line at 120 MHz, γ v Z0](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-120mhz-primary-constants) — Electromagnetism, chapter 5
- [Lossless line, L Z0 β Γ](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Alossless-line-velocity-capacitance-30pf) — Electromagnetism, chapter 5
- [Two-wire 40 Ω line, input and output voltage](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atwo-wire-40-ohm-line-voltages) — Electromagnetism, chapter 5
- [Line with 50+j50 load, first maximum, power](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-50-j50-load-first-maximum) — Electromagnetism, chapter 5

## Related in other subjects

- [Hilbert transform properties; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-distortionless) — Communication Systems, chapter 2, asked 3×
- [Short note: Distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-distortionless-transmission) — Communication Systems, chapter 1, asked 2×
- [Impulse response, transfer function and distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adistortionless-transfer-function) — Communication Systems, chapter 1
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