## Definitions of Grading of cables

There is unequal stress distribution in cable is undesirable for the following reasons:

1. Insulation of greater thickness is required which increases the cable size.
2. Unequal stress distribution may leads to the breakdown of insulation.

To overcome this drawbacks it is necessary to have a uniform stress distribution in cables. Uniform stress distribution can be achieved by distributing the stress in such a way that its value is increased in the outer layers of dielectric. This is known as the grading of cables.

The process of achieving uniform electrostatic stress in the dielectric of cables is known as grading of cables. The electrostatic stress in a single core cable has a maximum value(gmax) at the conductor surface and goes on decreasing as we move towards the sheath. The maximum voltage that can be safely applied to a cable depends upon gmax i.e. electrostatic stress at the conductor surface. For the safe working of a cable having homogenous dielectric, the strength of dielectric must be more than gmax. If a dielectric of high strength is used for a cable, it is useful only near the conductor where stress is maximum. As we move away from the conductor, the electrostatic stress decreases, so the dielectric will be unnecessarily over strong.

The process of achieving uniformity in the dielectric stress by using layers of different dielectrics is known as capacitance grading.

In a capacitance grading , the homogenous dielectric is replaced by a composite dielectric. The composite dielectric consists of various layers of different dielectrics in such a manner that relative permittivity εr of any layer is inversely proportional to  its distance from the center. i.e

Under such conditions, the value of potential gradient at any point in the dielectric is constant and is independent of its distance form the center. It means that the dielectric stress in the cable is same everywhere and the grading is ideal one. It is impossible to use infinite number of dielectrics for a single cable so two or three dielectrics are used in the decreasing order of permittivity, the dielectric of highest permittivity being used near the core.

The capacitance grading can be explained by above figure. There are three dielectrics of outer diameters d1,d2 and D and of relative permittivity’s ε1,ε2 and ε3 respectively. The permittivity’s are such that ε1>ε2>ε3 and if the maximum stress to which each dielectric is subjected is constant then

For the potential difference across the inner layer, we have,

similarly,

and the pd between core and earthed sheath,

If the cable has homogenous dielectric ,then, for the same value of d, D and gmax ,the permissible potential difference between core and earthed sheath would have been,

Since V>V’ for given dimensions of cable, a graded cable can be worked at a greater potential than non-graded cable. For the same safe potential ,the size of graded cable will be less than that of non-graded cable.

In this method of cable grading ,a homogenous dielectric is used but it is divided into various layers by placing metallic intersheaths between the core and lead sheath. The intersheaths are held at suitable potentials which are in between the core potential and earth potential. This arrangement improves the voltage distribution in the dielectric of the cable and consequently more uniform potential gradient is obtained.

consider a cable of core diameter d and outer lead sheath of diameter D. Suppose the two intersheaths of diameters d1 and d2 are inserted into the homogenous dielectric and maintained at some fixed potentials.Let V1 ,V2 and V3 respectively be the voltages between core and intersheath 1 ,between intersheaths 1 and 2 and between intersheath 2 and outer lead sheath.  As there is a definite potential differences between the inner and outer layers of each intersheath ,therefore, each sheath can be treated like a homogenous single core cable.

Maximum stress between core and intersheath 1 is

similarly,

since the dielectric is homogenous, the maximum stress in each layer is the same i.e.

As the cable behaves like three capacitors in series, therefore, all the potential are in phase i.e. voltage between conductor and earthed lead sheath is

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