ABCD Parameters of Two Parallel Transmission Line

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

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 

ABCD Parameters: Two Parallel Transmission Line

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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