## 27/03/2021

### 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 = ε0εr

Where ε0 = Absolute permittivity

= 8.854 × 10 – 12 Farad / meter

εr = 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

Dx = Q / 2πx Coulomb / meter2

Electric Intensity at point P

Ex = Dx / ε

= Q / 2πx ε

= Q / 2πx ε0εr

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

V = ( Q / 2π ε0εr ) Loge ( D / d )

Capacitance of cable

C = Q / V

= Q / { ( Q Loge ( D / d ) / 2π ε0εr ) }

= ( 2π ε0εr )  / Loge ( D / d )

= ( 2π × 8.854 × 10 – 14 × εr ) / Loge ( D / d )

= (  εr × 10 – 9 ) / 41.4 Loge ( D / d ) Farad / meter

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

C = (  εr L × 10 – 9 ) / 41.4 Loge ( D / d ) Farad

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