15/04/2022

ABCD Parameters of Two Parallel Transmission Line

 

ABCD Parameters of Two Parallel Transmission Line or Networks

Let us consider that the two-transmission line or network is in parallel. In this theory, we find out equivalent A, B, C and D parameters of parallel networks or transmission line

Let

VS = Sending end voltage to neutral

VR = Receiving end voltage to neutral


abcd-parameters-of-two-parallel-networks.png

IS = Sending end current

IR = Receiving end current

IS1 = Sending end current of 1st transmission network

IS2 = Sending end current of 2nd transmission network

IR1 = Receiving end current of 1st transmission network

IR2 = Receiving end current of 2nd transmission network

A1, B1, C1 and D1 – Transmission line parameters of 1st transmission network

A2, B2, C2 and D2 – Transmission line parameters of 2nd transmission network

The sending end current

IS = IS1 + IS2 …… ( 1 )

The receiving end currents

IR = IR1 + IR2 …… ( 2 ) 

Generalized Constant of the transmission line is given by sending end voltage and sending end current equation

VS = AVR + BIR……. ( 3 )

IS = CVR + DIR……....( 4 )

First network

VS = A1VR + B1IR1…..…. ( 5 )

IS1 = C1VR + D1IR1……....( 6 )

Second network

VS = A2VR + B2IR2…..…. ( 7 )

IS2 = C2VR + D2IR2……....( 8 )

Compare equation ( 5 ) and ( 7 )

A1VR + B1IR1 = A2VR + B2IR2

A1VR – A2VR = B2IR2   B1IR1

( A1 – A2 ) VR = B2IR2   B1IR1….. ( 9 )

As IR = IR1 + IR2

So IR2 = IR – IR1

substitute value of IR2 into equation ( 9)

( A1 – A2 ) VR = B2 ( IR – IR1 ) –   B1IR1

( A1 – A2 ) VR = B2 IR – B2 IR1   B1IR1

( A1 – A2 ) VR = B2 IR – IR1 ( B1 + B2 )

IR1 ( B1 + B2 ) = B2 IR – ( A1 – A2 ) VR

Therefore

IR1 = B2 IR – ( A1 – A2 ) VR / ( B1 + B2 ) …. ( 10 )

Substitute value of IR1 into equation ( 5 )

VS = A1VR + B1IR1

     = A1VR + B1{ B2 IR – ( A1 – A2 ) VR / ( B1 + B2 ) }

     = A1VR + B1 B2 IR – B1( A1 – A2 ) VR / ( B1 + B2 ) }

     = A1VR + ( B1 / B1 + B2 ){ B2 IR – ( A1 – A2 ) VR  }

     = ( A1 – { B1 ( A1 – A2 )  / B1 + B2 }VR + ( B1B2 / B1 + B2 ) IR

 VS = {A1B2 + A2B1 / B1 + B2 }VR + ( B1B2 / B1 + B2 ) IR …( 11 )

Compare equation ( 11 ) with equation ( 3 )

VS = AVR + BIR

A = {A1B2 + A2B1 / B1 + B2 }

B = ( B1B2 / B1 + B2 )

Combine equation ( 6 ) and ( 8 )

IS = IS1 + IS2

IS = C1VR + D1IR1 + C2VR + D2IR2

IS = ( C1 + C2 ) VR + D1IR1 + D2IR2

Substitute IR2 = IR – IR1

IS = ( C1 + C2 ) VR + D1IR1 + D2 ( IR – IR1 )

IS = ( C1 + C2 ) VR + D1IR1 + D2 IR – D2 IR1

IS = ( C1 + C2 ) VR + ( D1 – D2 )IR1 + D2 IR

Substitute value of IR1 from equation ( 10 )

IS = ( C1 + C2 ) VR +

      ( D1 – D2 )( B2 IR – ( A1 – A2 ) VR / ( B1 + B2 ) +D2 IR

Combine VR and IR

IS ={ ( C1 + C2 ) – ( D1 – D2 )( A1 – A2 ) / ( B1 + B2 ) } VR  +

      B2( D1 – D2 ) / ( B1 + B2 ) IR + D2 IR

Simplify

IS ={ ( C1 + C2 ) – ( D1 – D2 )( A1 – A2 ) / ( B1 + B2 ) } VR  +

      {  B2D1 – B2D2 + B1D2 + B2D2 ) / ( B1 + B2 )} IR

IS ={ ( C1 + C2 ) – ( D1 – D2 )( A1 – A2 ) / ( B1 + B2 ) } VR  +

      { B1D2 + B2D1 / ( B1 + B2 )} IR

Compare this equation with equation ( 4 )

IS = CVR + DIR

C = { ( C1 + C2 ) – ( D1 – D2 )( A1 – A2 ) / ( B1 + B2 ) }

D = { B1D2 + B2D1 / ( B1 + B2 )}

 

You may also like to read these articles :

ABCD Parameter of the Transmission Line

ABCD Parameter of the Short Transmission Line

Size of Earthing Conductor in Rotating Machines

Importance of Load Factor & Diversity Factor

ABCD Parameters of Medium Transmission Line : Nominal π method

ABCD Parameters of Medium Transmission Line : Nominal T Method

ABCD Parameters of Medium Transmission Line : End Condenser  Method

 

 

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