Distortionless Transmission Line

A distortionless line does not distort the signal phase, but does introduce a signal loss since they are not super conductors. This is also known as the Heaviside condition. Phase distortion does not occur if the phase velocity Vp is constant at all frequencies.


By definition, a phase shift of 2π radians occurs over one wavelength λ.

Since

then

This tells us that in order for phase velocity Vp to be constant, the phase shift coefficient b, must vary directly with frequency w .

Recall

The problem now is to find b. This can be done as follows:

 

The 2nd and 3rd roots can be expanded by means of the Binomial Expansion. Recall:

In this instance n = 1/2. Since the contribution of successive terms diminishes rapidly, g is expanded to only 3 terms:

Since g = a + jb, equate the imaginary terms to find b.

Note that if then:

From this we observe that b is directly proportional to w. This means that the requirement for distortionless transmission is:

If we equate the real terms, we obtain: