Capacitance of Single Core Cable

• A single core cable is shown in the Figure. 
• It is equivalent to two long co – axial cylindrical. The core of the conductor is inner cylinder whereas the lead sheath is outer cylinder. 
• The lead sheath is at earth potential.
• Let us consider that the core diameter is d meter and inner sheath diameter is D meter. 
• The charge per axial length of the cable is Q coulombs and e is permittivity of the insulation material between core and lead sheath.

Core diameter = d

Sheath diameter = D

Charge per axial length of cable = Q

Permittivity of insulating material between core and lead sheath = ε

Permittivity ε = ε₀ε_r

Where ε₀ = Absolute permittivity

               = 8.854 × 10 ^{– 12} Farad / meter

           ε_r = Relative permittivity

• Consider a cylinder of radius x meter and axial length 1 meter. The surface area of cylinder = ( 2πx )  × ( 1 )  = 2πx meter²
• Electric flux density at any point P on the cylindrical surface is

D_x = Q / 2πx Coulomb / meter²

Electric Intensity at point P

E_x = D_x / ε

    = Q / 2πx ε

    = Q / 2πx ε₀ε_r

• If unit positive charge moves from point P through distance dx in the direction of electrical field, the work done is E_x dx. 
• Therefore the work done for unit positive charge moves from conductor to sheath is given by

            

 V = ( Q / 2π ε₀ε_r ) Log_e ( D / d )

Capacitance of cable

C = Q / V

   = Q / { ( Q Log_e ( D / d ) / 2π ε₀ε_r ) }

   = ( 2π ε₀ε_r )  / Log_e ( D / d )

   = ( 2π × 8.854 × 10 ^{– 14} × ε_r ) / Log_e ( D / d )

   = (  ε_r × 10 ^{– 9} ) / 41.4 Log_e ( D / d ) Farad / meter

If the length of cable is L meter, the capacitance of cable is

                    C = (  ε_r L × 10 ^{– 9} ) / 41.4 Log_e ( D / d ) Farad

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