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 independent of the 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 real 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. Line characteristics The secondary constants of a line meeting the Heaviside condition are consequently, in terms of the primary constants: Attenuation, nepers/metre Phase change constant, radians/metre Phase velocity, metres/second Characteristic impedance The characteristic impedance of a lossy transmission line is given by In general, it is not possible to impedance match this transmission line at all frequencies with any finite network of discrete elements because such networks are rational functions of jω, but in general the expression for characteristic impedance is irrational due to the square root term. However, for a line which meets the Heaviside condition, there is a common factor in the fraction which cancels out the frequency dependent terms leaving, which is a real number, and independent of frequency. The line can therefore be impedance-matched with just a resistor at either end. This expression for is the same as for a lossless line () with the same L and C, although the attenuation (due to R and G) is of course still present. Practical use A real line, especially one using modern synthetic insulators, will have a G that is very low and will usually not come anywhere close to meeting the Heaviside condition. The normal situation is that To make a line meet the Heaviside condition one of