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 *V _{p}* 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 *V _{p}* 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: