Heaviside Condition - Derivation

Derivation

The transmission function of a transmission line is defined in terms of its input and output voltages when correctly terminated (that is, with no reflections) as

where represents distance from the transmitter in meters and

are the secondary line constants, α being the attenuation in nepers per metre and β being the phase change constant in radians per metre. For no distortion, α is required to be constant with angular frequency ω, while β must be proportional to ω. This requirement for proportionality to frequency is due to the relationship between the velocity, v, and phase constant, β being given by,

and the requirement that phase velocity, v, be constant at all frequencies.

The relationship between the primary and secondary line constants is given by

which has to be of the form in order to meet the distortionless condition. The only way this can be so is if and differ by no more than a constant factor. Since both have a real and imaginary part, the real and imaginary parts must independently be related by the same factor, so that;

and the Heaviside condition is proved.