Grading of Underground Cable

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

Grading of Cables

• 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.

Capacitance Grading

• 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 ε_r of any layer is inversely proportional to the radius of distance from the center.

( ε_r ) α ( 1 / x )

( ε_r ) ( x ) = Constant ……. ( 1 )

Where x = Distance from center

Now potential gradient

 g = Q / 2πε₀ε_rx

( 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 ε_r1, ε_r2 and ε_r3 respectively.  
• If the permittivity of dielectric materials are selected such that 

ε_r1 > ε_r2 > ε_r3

( ε_r1d₁ ) = ( ε_r2d₂ ) = ( ε_r3D ) 

Advantages of Capacitance Grading

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

       Potential difference across inner layer 

 V1 = { Q Log_e ( d1 / d ) / 2πε₀ε_r1 }

V1 = g_max d Log_e ( d1 / d ) / 2         { ⸫ g_max d  / 2 = Q / 2πε₀ε_r1 }             

Potential difference across centre layer

V2 = g_max d Log_e ( d2 / d1 ) / 2              

Potential difference across outer layer

V3 = g_max d Log_e ( D / d2 ) / 2    

Therefore, the potential difference between core and sheath is

V = V1 + V2 + V3  

       = g_max d Log_e ( d1 / d ) / 2 + g_max d Log_e ( d2 / d1 ) / 2  + g_max d Log_e

                                                                                               ( D / d2 ) / 2

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

       = g_max d Log_e { ( d1 / d ) × ( d2 / d1 ) × ( D / d2 ) } / 2

 V’ = g_max d Log_e { ( 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.      

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Inter- sheath Grading

• 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

g_1max = V1 / ( d / 2 ) Log _e ( d1 / d )

Maximum stress between inter-sheath1 and inter-sheath2

g_2max = V2 / ( d1 / 2 ) Log _e ( d2 / d1 )

Maximum stress between inter-sheath2 and lead sheath

g_3max = V3 / ( d2 / 2 ) Log _e ( D / d2 )

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

g_1max = g_2max = g_3max = g_max

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

Disadvantages of Inter-sheath Grading

• It is not very easy to set sheath potentials.
• The inter-sheath may be damaged due to transportation and installation.
• There may be considerable inter – sheath losses due to charging current.

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