Heaviside condition
The importance of the Heaviside condition is that it showed the possibility of dispersionless transmission of telegraph signals.
[1]: 131 In some cases, the performance of a transmission line can be improved by adding inductive loading to the cable.
A transmission line is dispersionless, if the velocity of signals is independent of frequency.
A transmission line is distortionless if it is dispersionless and the attenuation coefficient is independent of frequency.
This was a major problem on the first transatlantic telegraph cable and led to the theory of the causes of dispersion being investigated first by Lord Kelvin and then by Heaviside who discovered in 1876 how it could be countered.
To prevent intersymbol interference it was necessary to reduce the transmission speed of the transatlantic telegraph cable to the equivalent of 1⁄15 baud.
An analogous Heaviside condition for dispersionless propagation in left-handed transmission line metamaterials cannot be derived, since no combination of reactive and resistive elements would yield a constant group velocity.
The relationship between the primary and secondary line constants is given by If the Heaviside condition holds, then the square root function can be carried out explicitly as: where Hence Velocity is independent of frequency if the product
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 complex due to the square root term.
A real line 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 the four primary constants needs to be adjusted and the question is which one.
Decreasing R is sending the loss in the right direction, but this is still not usually a satisfactory solution.
R must be decreased by a large number and to do this the conductor cross-sections must be increased dramatically.
This not only makes the cable much bulkier, but also adds significantly to the amount of copper (or other metal) being used and hence the cost and weight.
Decreasing the capacitance is difficult because it requires using a different dielectric with a lower permittivity.
Gutta-percha insulation used in the early trans-Atlantic cables has a dielectric constant of about 3, hence C could be decreased by a maximum factor or no more than 3.
L is increased by loading the cable with a metal with high magnetic permeability.
Instead, regularly spaced digital repeaters are now placed in long lines to maintain the desired shape and duration of pulses for long-distance transmission.
[1]: 132 Achieving the Heaviside condition is more difficult when some or all of the line parameters depend on frequency.
There are three red curves indicating typical low, medium, and high-quality dielectrics.
Pulp insulation (used for telephone lines in the early 20th century), gutta-percha, and modern foamed plastics are examples of low, medium, and high-quality dielectrics.
The curve is depicted as flat on the figure, but loss tangent shows some frequency dependence.
The value of G/(ωC) at all frequencies is determined entirely by properties of the dielectric and is independent of the transmission line cross-section.
