## 28/03/2021

In this theory, the grading of cable or methods of uniform dielectric stress in the cable is given.

• The process of achieving uniform dielectric stress in the cable is called as grading of cables.
• The dielectric stress is maximum at the center of the core and its value goes on decreasing as we move from center to sheath of cable.
• The dielectric stress in the cable is undesirable due to following reasons.

1. The size of cable increases due to more thickness of insulation.
2. There are possibilities of breakdown of insulation.

## Methods of Grading of Cable

• There are two methods for achieving uniform dielectric stress in the cable. It is known as grading of cable.

• The uniform dielectric stress in the cable is achieved by using layers of different dielectric, it is called as capacitance grading.
• The uniform dielectric stress in this method is achieved by using different layers of dielectric such that the permittivity εr of any layer is inversely proportional to the radius of distance from the center.

( εr ) α ( 1 / x )

( εr ) ( x ) = Constant ……. ( 1 )

Where x = Distance from center

g = Q / 2πε0εrx

( g ) α Q / 2πε0 is also constant…. ( 2 )  ( ⸫ ( εr ) ( x ) = Constant )

• We can say that the value of dielectric stress at any point is constant and it is independent of distant from the center.
• The dielectric material having highest permittivity is used near the core and its value decreasing form core to the outer surface of cable.
• Let us consider that a cable is made of 3 layers of dielectric having outer diameter d1, d2 and D and relative permittivity εr1, εr2 and εr3 respectively.
• If the permittivity of dielectric materials are selected such that

εr1 > εr2 > εr3

( εr1d1 ) = ( εr2d2 ) = ( εr3D )

• The size of the graded cable is smaller than the non – graded cable for same safe potential.

Potential difference across inner layer

V1 = { Q Loge ( d1 / d ) / 2πε0εr1 }

V1 = gmax d Loge ( d1 / d ) / 2         { ⸫ gmax d  / 2 = Q / 2πε0εr1 }

Potential difference across centre layer

V2 = gmax d Loge ( d2 / d1 ) / 2

Potential difference across outer layer

V3 = gmax d Loge ( D / d2 ) / 2

Therefore, the potential difference between core and sheath is

V = V1 + V2 + V3

= gmax d Loge ( d1 / d ) / 2 + gmax d Loge ( d2 / d1 ) / 2  + gmax d Loge

( D / d2 ) / 2

If the cable had homogenous permittivity, the potential difference between core and sheath is given by V’

= gmax d Loge { ( d1 / d ) × ( d2 / d1 ) × ( D / d2 ) } / 2

V’ = gmax d Loge { ( D / d ) } / 2

It should be noted that the potential of the graded cable ( V ) is more than the non – graded cable ( V’ ).

OR

We can say that the size of the graded cable is less than the non – graded cable for a given safe working voltage.

• A homogenous dielectric material is used in this method of cable grading.
• The homogenous dielectric is divided into various layers by placing metallic inter-sheath between core and lead sheath.
• The inter-sheaths are held at constant potential whose value lies between core potential and earth potential.
• Let us consider that the core diameter d, lead sheath diameter D and two inter-sheath of diameter d1 and d2 are inserted into homogenous dielectric at constant voltage.
• Core diameter = d
• Lead sheath diameter = D
•  Voltage between core and inter-sheath = V1
• Voltage between inter-sheath 1 and inter-sheath 2 = V2
• Voltage between inter-sheath 2 and lead sheath = V3
• As there is definite potential difference between inner and outer layers of each inter-sheath, we can say that each inter-sheath can be treated as single core cable.

Maximum stress between core and inter-sheath1

g1max = V1 / ( d / 2 ) Log e ( d1 / d )

Maximum stress between inter-sheath1 and inter-sheath2

g2max = V2 / ( d1 / 2 ) Log e ( d2 / d1 )

Maximum stress between inter-sheath2 and lead sheath

g3max = V3 / ( d2 / 2 ) Log e ( D / d2 )

As the dielectric is homogeneous, the maximum stress in each layer is the same

g1max = g2max = g3max = gmax

V1 / ( d / 2 ) Log e ( d1 / d ) = V2 / ( d1 / 2 ) Log e ( d2 / d1 ) = V3 / ( d2 / 2 ) Log e ( D / d2 )

• As the cable behaves like three capacitors in series, all the potentials are in phase.
• The voltage between conductor and earthed lead sheath is

V = V1 + V2 + V3