Distortionless line and telephone lines

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

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.