Electrical Revolution: Most Economical Conductor Diameter

The maximum stress in the cable occurs at the surface of the conductor.
The dielectric strength of the cable should be more than the maximum stress in order to safe working of the cable.

Maximum stress on the cable

g_max = 2V / { d Log_e ( D / d ) }…..( 1 )

The internal sheath diameter D and working voltage have to be kept constant for design consideration therefore only variable parameter d is under consideration.
The most economical diameter of conductor is achieved when stress on the conductor becomes minimum.
The value of g_max becomes minimum when { d Log_e ( D / d ) } becomes maximum.

d /dd { d Log_e ( D / d ) } = 0

d {( d / D ) ( – D / d² ) } + { Log_e ( D / d ) } × 1 = 0

{ Log_e ( D / d ) } – 1 = 0

{ Log_e ( D / d ) } = 1…. ( 2 )

( D / d ) = e¹

( D / d ) = 2.718

D = 2.178d

d = D / 2.718….. ( 3 )

Maximum stress under this condition, using equation ( 1 ) and ( 2 )

g_max = 2V / d

If we consider maximum stress, the most economical conductor diameter

d = 2V / g_max….. ( 4 )

This is only theory point of view

Practically Cable Diameter

If we calculate most economical conductor diameter considering maximum stress on the cable, it should be too small from the point of view of current density. Therefore the cable diameter is decided by considering current density.
The diameter of cable is too much high for high voltage cable from point of view of current density considering above formula ( 4 ). The diameter of cable can be increased by
Use of aluminum conductor instead of copper conductor because the diameter of aluminum conductor is larger than that of copper conductor for same current capacity
Use of hollow conductor

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Electrical Revolution