Editing Heaviside condition (section)

==Practical use==
[[File:Mu-metal_cable.svg|thumb|upright=2|An example of loaded cable]]
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

:<math>\frac{G}{C} \ll \frac{R}{L}</math> by several orders of magnitude.

To make a line meet the Heaviside condition one of the four primary constants needs to be adjusted and the question is which one.  ''G'' could be increased, but this is highly undesirable since increasing ''G'' will increase the loss.  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.  This leaves increasing ''L'' which is the usual solution adopted.

''L'' is increased by loading the cable with a metal with high [[permeability (electromagnetism)|magnetic permeability]].  It is also possible to load a cable of conventional construction by adding discrete [[loading coil]]s at regular intervals.  This is not identical to a distributed loading, the difference being that with loading coils there is distortionless transmission up to a definite [[cut-off frequency]] beyond which the attenuation increases rapidly.

Loading cables is no longer a common practice. Instead, regularly spaced digital [[Repeater|repeaters]] are now placed in long lines to maintain the desired shape and duration of pulses for long-distance transmission.
===Frequency-dependent line parameters===
[[File:Typical Good Transmission Line Parameter Ratioes.png|thumb|Typical transmission line parameter ratios|left|350px]]

When the line parameters are frequency dependent, there are additional considerations.<ref name="Miano"/>{{rp|132}}  Achieving the Heaviside condition is more difficult when some or all of the line parameters depend on frequency.  Typically, R (due to skin effect) and G (due to dielectric loss) are strong functions of frequency.  If magnetic material is added to increase L, then L also becomes frequency dependent.

The chart on the left plots the ratios <math> \tfrac {R_{\omega}} {\omega L_{\omega}} ({\color{blue}\text{blue} }) \text{ and } \tfrac {G_{\omega}} {\omega C_{\omega}}  ({\color{red}\text{red} })</math> for typical transmission lines made from non-magnetic materials.  The Heaviside condition is satisfied where the blue curve touches or crosses a red curve.  

The knee of the blue curve occurs at the frequency where <math> R_{\omega} = \omega L_{\omega} </math>.

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 knee of each curve occurs at the frequency where <math> G_{\omega} = \omega C_{\omega} </math>.  The reciprocal of this frequency is known as the [[Relaxation (physics)#Dielectric relaxation time|dielectric relaxation time]] of the dielectric.  Above this frequency, the value of G/(ωC) is the same as the [[loss tangent]] of the dielectric material.  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.  

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