String efficiency of Three Suspension insulators

 

String of Suspension Insulator Disc

• Figure shows three suspension insulators. Each suspension insulator has two metallic links therefore it has its own capacitance and it is called as mutual capacitance or self capacitance.

Mutual capacitance ( Self capacitance )( C )

• It is defined as the capacitance due to suspension insulator itself. 
• Each metallic links acts as conducting plate and porcelain acts as dielectric material.
• The voltage across each insulator becomes same when considering only mutual capacitance of insulator.

Voltage across string V = V1 + V2 + V3

                          As V1 = V2 = V3

                               V1 = V / 3

                              V2 = V / 3

                             V3 = V / 3

Shunt capacitance ( C1 )

• It is a capacitance between each insulator and transmission line tower.

Explain : The voltage across line insulator is maximum and top of the insulator is minimum.

Effect of Shunt capacitance

• There is a capacitance between each insulator and transmission line tower. 
• The charging current flows due to shunt capacitance therefore the voltage across each insulator is different. 
• The charging current is maximum through line insulator and decrease towards top of the tower therefore the voltage across line insulator is maximum and its value decreases from bottom of the insulator to top of the insulator.

String efficiency

String efficiency = Voltage across string / ( n × Voltage across line insulator )

            Where n = Number of string

                           = Number of discs

Importance of string efficiency

• The string efficiency indicates voltage across string. 
• The string efficiency 100% means that the voltage across all the discs is same. However the string efficiency of 100% achieve is almost impossible.

Mathematically explanation

• Figure shows equivalent circuit of three suspension disc insulator. Let us consider that the voltage across top of the insulator to bottom of the insulator is V1, V2 and V3.
• Let

       C = Capacitance across disc

          = Self capacitance

       C1 = Shunt capacitance

• Let us assume that C1 = mC

                                       m = C1 / C

                           Where m = Constant

Applying Kirchhoff’s current law at point R

I2 = I1 + i1

ɷCV2 = ɷCV1 + ɷC1V1

ɷCV2 = ɷCV1 + ɷmCV1

ɷCV2 = ɷC ( V1 + m V1 )

V2 = V1 ( 1 + m  )……..( 1 )

Applying Kirchhoff’s current law at point Q

I3 = I2 + i2

ɷCV3 = ɷCV2 + ɷC1( V1 + V2 )

ɷCV3 = ɷCV2 + ɷmC ( V1 + V2 )

ɷCV3 = ɷCV2 + ɷmCV1 + ɷmCV2

ɷCV3 = ɷmCV1 + ɷCV2 + ɷmCV2

ɷCV3 = ɷC [ mV1 + V2 + mV2 ]

V3 =  mV1 + ( 1 + m )V2  …..( 2 )

From equation ( 1 )

V3 =  mV1 + ( 1 + m ) ( 1 + m )V1 

V3 =  mV1 + ( 1 + 2m + m² )V1 

V3 =  ( 1 + 3m + m² )V1………. ( 3 ) 

Voltage across string ( all insulators )

V = V1 + V2 + V3

    = V1 + V1 ( 1 + m  ) + ( 1 + 3m + m² )V1

    = [ 1 + ( 1 + m  ) + ( 1 + 3m + m² ) ]V1

    = [ m² + 4m + 3 ]V1

    = [ m² + 3m + m + 3 ]V1

   = [ m ( m + 3 ) + 1 ( m + 3 ) ]V1

   = [ ( m + 1 ) ( m + 3 )  ]V1……….( 4 )

V1 = V / ( m + 1 ) ( m + 3 )  ……….( 5 )

From equation ( 1 ) , ( 3 ) and ( 5 )

V1 = V / ( m + 1 ) ( m + 3 )  = V3 / ( 1 + 3m + m² ) = V2 / ( 1 + m  )

Voltage across top insulator

V1 = V / ( m + 1 ) ( m + 3 ) 

Voltage across middle insulator

V2 = V / ( m + 3 ) 

Voltage across line or bottom insulator

V3 = V ( 1 + 3m + m² ) / ( m + 1 ) ( m + 3 ) 

String efficiency for three insulators

      = Voltage across string / ( 3 × Voltage across line insulator )

      = V / ( 3 × V3 )

      = V / { 3 × [V ( 1 + 3m + m² ) / ( m + 1 ) ( m + 3 ) }

      = ( m + 1 ) ( m + 3 )  / { 3  ( 1 + 3m + m² ) }

Let us assume that m = 1

String efficiency = ( 1 + 1 ) ( 1 + 3 )  / { 3  ( 1 + 3 ( 1 ) + ( 1 )²  ) }

                           = ( 2 ) ( 4 )  / { 3  ( 1 + 3 ( 1 ) + ( 1 )²  ) }

                           = ( 8 )  /  ( 15 )

                           = 53.34%

Let us assume that m = 0

String efficiency = ( 0 + 1 ) ( 0 + 3 )  / { 3  ( 1 + 3 ( 0 ) + ( 0 )²  ) }

                           = ( 1 ) ( 3 )  / { 3  ( 1 + 0 + 0  ) }

                           = 3 / 3

                          = 1

                          = 100%

• It means that if the shunt capacitance is equal to zero, the string efficiency is 100%

Important point

• The maximum voltage is across the bottom of the string or say line insulator.
• Voltage across string is phase voltage not line voltage
• Line voltage = ( √ 3 )  × Phase voltage
• Greater the value of m, more non – uniform distribution of voltage across string and less string efficiency
• As the number of insulators increases, the string efficiency decreases and vice versa.

The string efficiency becomes 100% 

( 1 ) DC transmission line or

( 2 ) When the shunt capacitance becomes zero or

( 3 ) Charging current from insulator to transmission line tower is equal to zero

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