# Two Parallel Transmission Line

Let us consider that the two-transmission line or network is in parallel. As two transmission lines are parallel, sending end voltage and receiving end voltage for both transmission line is equal however current through both transmission lines is different. 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

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

## ABCD Parameters: Two Parallel Transmission Line

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 )}