In this article we will discuss about:- 1. Insulation Resistance of a Single Core Cable 2. Capacitance of a Single Core Cable 3. Dielectric Stress.

**Insulation Resistance of a Single Core Cable:**

The cable conductor is provided with an insulation of suitable thickness so as to avoid leakage of current. The path for leakage current is radial, as shown in Fig. 11.11, through the insulating material. The opposition offered by the insulation to the leakage current is called the insulation resistance of the cable.

Consider a single core cable of conductor radius r_{1}, internal sheath radius r_{2}, length l and insulation material resistivity ρ.

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Let an elementary cylindrical section of the insulation of radius r and thickness dr be considered. Now the length through which the leaking current will flow is dr and area of x-section is 2πrl.

The insulation resistance offered to the leakage current by elementary cylindrical section of the insulation under consideration is ρdr/2πrl.

Insulation resistance of the cable,

i.e. the insulation resistance of the cable varies inversely as the length of the cable (R_{INS} α 1/l).

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If ρ is in Ω-m, r_{2}, r_{1}, and l are in metres then insulation resistance of the single core cable is in ohms. Average value of ρ for impregnated paper is about 5 × 10^{12} – 8 × 10^{12} Ω-m at 15°C and decreases exponentially with temperature so that

Ρ_{t} = ρ_{0}e^{-αt} … (11.2) α is about 0.04 or 0.05.

**Capacitance of a Single Core Cable****: **

The single core cable can be considered to be two coaxial cylinders of inner diameter d and outer diameter D. In actual cable, d represents the diameter of core and D represents the inner diameter of lead sheath which is at earth potential. Let the relative permittivity of dielectric in between the core and sheath be ∈_{r}.

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Let the charge per metre length of the cable on the outer surface of the core be + Q coulombs and on the inner surface of the lead sheath be -Q coulombs. For all practical purposes, the charge of + Q coulombs/m on the surface of the core can be assumed to be located along its axis. The metal sheath is earthed.

Surface area of the coaxial cylinder of radius x metres and length one metre is 2 x metres^{2}.

Therefore, electric field intensity at a point x metres from the centre of the inner cylinder,

Potential difference between the capacitor plates, (between core and sheath),

**Dielectric Stress in a Single Core Cable****: **

Under operating conditions, the insulation of a single core cable is subjected to electrostatic stress, called the dielectric stress. Potential gradient at any point is defined as the rate of increase of potential at that point and is the same as the dielectric stress at that point.

**Since single core cable is a form of cylindrical condenser, therefore, electric intensity at a distance x from the centre O of the cable is given by the expression: **

Since potential gradient = Electric intensity

Hence,

Substituting the value of Q from above expression in Eq. (11.4) we have

Since potential gradient g varies inversely as x (as obvious from the above expression), therefore, potential gradient will be maximum when x is minimum i.e., x = d/2 and potential gradient will be minimum when x is maximum i.e. x = D/2

**Maximum and minimum values of potential gradient are given by: **

Hence the ratio between the maximum potential gradient and the minimum potential gradient i.e.,

It is to be noted here that the Eq. (11.6) for the dielectric stress at the conductor surface is obtained on the assumption of a smooth cylindrical conductor, and thus only provides a nominal value for this stress. With the ordinary stranded conductors in common use, the dielectric stresses near the conductor are increased by about 20 per cent due to the greater curvature of the surface of the individual wires. This, however, can be avoided in practice by covering the conductor with a thin smooth metal sheath.

In high voltage single core cables the value of g is taken to the highest possible limit in order to keep down the overall diameter, but since the dielectric losses increase rapidly with potential gradient, the maximum permissible value of g is only about one-fifth of the breakdown value viz., 4-5 kV/mm.