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

V_S = Sending end voltage to neutral

V_R = Receiving end voltage to neutral

I_S = Sending end current

I_R = Receiving end current

I_S1 = Sending end current of 1^st transmission network

I_S2 = Sending end current of 2^nd transmission network

I_R1 = Receiving end current of 1^st transmission network

I_R2 = Receiving end current of 2^nd transmission network

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

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

The sending end current

I_S = I_S1 + I_S2 …… ( 1 )

The receiving end currents

I_R = I_R1 + I_R2 …… ( 2 ) 

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

V_S = AV_R + BI_R……. ( 3 )

I_S = CV_R + DI_R……....( 4 )

First network

V_S = A1V_R + B1I_R1…..…. ( 5 )

I_S1 = C1V_R + D1I_R1……....( 6 )

Second network

V_S = A2V_R + B2I_R2…..…. ( 7 )

I_S2 = C2V_R + D2I_R2……....( 8 )

Compare equation ( 5 ) and ( 7 )

A1V_R + B1I_R1 = A2V_R + B2I_R2

A1V_R – A2V_R = B2I_R2 –   B1I_R1

( A1 – A2 ) V_R = B2I_R2 –   B1I_R1….. ( 9 )

As I_R = I_R1 + I_R2

So I_R2 = I_R – I_R1

substitute value of I_R2 into equation ( 9)

( A1 – A2 ) V_R = B2 ( I_R – I_R1 ) –   B1I_R1

( A1 – A2 ) V_R = B2 I_R – B2 I_R1 –   B1I_R1

( A1 – A2 ) V_R = B2 I_R – I_R1 ( B1 + B2 )

I_R1 ( B1 + B2 ) = B2 I_R – ( A1 – A2 ) V_R

Therefore

I_R1 = B2 I_R – ( A1 – A2 ) V_R / ( B1 + B2 ) …. ( 10 )

Substitute value of I_R1 into equation ( 5 )

V_S = A1V_R + B1I_R1

     = A1V_R + B1{ B2 I_R – ( A1 – A2 ) V_R / ( B1 + B2 ) }

     = A1V_R + B1 B2 I_R – B1( A1 – A2 ) V_R / ( B1 + B2 ) }

     = A1V_R + ( B1 / B1 + B2 ){ B2 I_R – ( A1 – A2 ) V_R  }

     = ( A1 – { B1 ( A1 – A2 )  / B1 + B2 }V_R + ( B1B2 / B1 + B2 ) I_R

 V_S = {A1B2 + A2B1 / B1 + B2 }V_R + ( B1B2 / B1 + B2 ) I_R …( 11 )

Compare equation ( 11 ) with equation ( 3 )

V_S = AV_R + BI_R

A = {A1B2 + A2B1 / B1 + B2 }

B = ( B1B2 / B1 + B2 )

Combine equation ( 6 ) and ( 8 )

I_S = I_S1 + I_S2

I_S = C1V_R + D1I_R1 + C2V_R + D2I_R2

I_S = ( C1 + C2 ) V_R + D1I_R1 + D2I_R2

Substitute I_R2 = I_R – I_R1

I_S = ( C1 + C2 ) V_R + D1I_R1 + D2 ( I_R – I_R1 )

I_S = ( C1 + C2 ) V_R + D1I_R1 + D2 I_R – D2 I_R1

I_S = ( C1 + C2 ) V_R + ( D1 – D2 )I_R1 + D2 I_R

Substitute value of I_R1 from equation ( 10 )

I_S = ( C1 + C2 ) V_R +

      ( D1 – D2 )( B2 I_R – ( A1 – A2 ) V_R / ( B1 + B2 ) +D2 I_R

Combine V_R and I_R

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

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

Simplify

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

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

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

      { B1D2 + B2D1 / ( B1 + B2 )} I_R

Compare this equation with equation ( 4 )

I_S = CV_R + DI_R

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

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

 

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