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
**The dielectric stress in the cable is undesirable due to following reasons.**

- The size of cable increases due to more thickness of insulation.
- 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 / 2πε_{0}ε_{r}x

( g ) α Q / 2πε_{0 }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 d
_{1}, d_{2}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_{r1}d_{1} ) = ( ε_{r2}d_{2}
) = ( ε_{r3}D )

### 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

_{e} ( d1 / d ) / 2πε_{0}ε_{r1 }}

V1 = g_{max} d
Log_{e} ( d1 / d ) /
2 { ⸫ g_{max} d /
2 = Q / 2πε_{0}ε_{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.*

### 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 d
_{1}and d_{2}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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