ABCD Parameters of Transmission Line

ABCD parameters are generalized parameters of transmission line. These parameters are quite helpful for calculation and analysis of transmission line. In any four terminal network, the input voltage and current can be expressed in terms of output voltage and current provided the network should be passive, linear and bilateral. Fortunately, a transmission line is a four terminal network and above all it is a passive, linear and bilateral network. Therefore, input voltage and current of transmission line can be expressed in terms of output voltage and current.

For a transmission line, it is better to call input voltage and current as Sending end voltage and current. Similarly, output voltage as receiving end voltage and current.

Let

V_R = Receiving end voltage

V_s = Sending end voltage

I_s = Sending end current

I_R = Receiving end current

Thus we can write,

V_s = AV_R + BI_R �������..(1)

I_s = CV_R + DI_R ��������(2)

Here A, B, C and D are constants and known as Generalized Circuit Constants of transmission line. The values of constants can easily be calculated from the above equations. For calculating the values of the ABCD parameters, we will consider two cases. Bold sign in the entire post means vector form of the quantity.

Case (1): When Receiving end is open.

As the receiving end is open, therefore load current through the line will be zero. Thus I_R = 0.

From equation (1).

A = V_s / V_R

Notice that the unit of A is unitless and hence called voltage ratio.

From equation (2),

C = Is / V_R

From the above expression of constant C, it is clear that, the unit of C is Mho. This constant is known as open circuit conductance.

Case (2): When receiving end is shorted.

As the receiving end is shorted, therefore load end voltage will be zero. Thus V_R = 0.

From equation (1),

B = V_s / I_R

Notice that the unit of B is same as that of impedance i.e. Ohm and called short circuit resistance.

From equation (2),

D = I_s / I_R

Thus D is a dimension less quantity and known as current ratio.

To summarize, ABCD Parameters for a four terminal passive, linear and bilateral network is given as

Generalized Circuit Parameters	Value	Unit
Voltage Ratio, A	Vs/VR	Unit less
Short Circuit Resistance, B	Vs / IR	Ohm
Open Circuit Conductance, C	Is / VR	Mho
Current Ratio, D	Is / VR	Unit less

Features of ABCD Parameters

Following are some of the important points about ABCD parameters, which shall always be kept in mind:

• The constants A, B, C and D are usually complex numbers and hence a phasor.

• For a given system,

        A = D

        and AD � BC = 1

Determination of ABCD Parameters for Transmission Line

We will determine the value of these parameters for Short as well as Medium Transmission lines.

Short Transmission Line � ABCD Parameters

As we know that, the effect of shunt capacitance in short transmission line is neglected, therefore the equivalent circuit for single phase for short transmission line can be shown as below.

From the figure above,

I_s = I_R  ���������(3)

and V_s = V_R + I_RZ  ����..(4)

Comparing equation (3) & (4) with equation (1) & (2), we get

A = 1, B = Z, C = 0 and D = 1

Thus we see that, A = D and AD � BC = 1

Medium Transmission Line � ABCD Parameters

We will find the ABCD parameters for two models of medium transmission line i.e. for Nominal T Model and Nominal p Model.

Nominal T Model

The equivalent circuit diagram for Nominal T model for single phase of transmission line is shown in figure below.

From the figure above,

V_s = V₁ + I_sZ/2    (These are in phasor from)

But V₁ = V_R + I_RZ/2  (These are in phasor from)

Again,

I_c = I_s - I_R

   = V₁Y  where Y = Shunt admittance

   = Y (V_R + I_RZ/2 )

Therefore,

I_s = I_R + YV_R + YI_RZ/2

   = YV_R + I_R(1 + YZ / 2)  ���������.(5)

Putting the value of V₁ in the expression of Vs, we get

V_s = V_R + I_RZ/2 + I_sZ/2  

    

Putting the value of I_s in the above expression, we get

 V_s = (1 + YZ / 2)V_R + (Z + YZ² / 4) I_R �����(6)

Comparing equation (6) with (1), we get

A = 1 + YZ / 2

B = Z + YZ² / 4 = Z(1 + YZ / 4)

Comparing equation (5) with (2), we get

C = Y

D = 1 + YZ / 2

Thus the ABCD Parameters for Nominal T model of medium transmission line are

A = D = 1 + YZ / 2

B = Z(1 + YZ / 4)

C = Y

Nominal p Model

The equivalent circuit diagram for Nominal p model for single phase of transmission line is shown in figure below.

As clear from the figure above,

I_s = I_L + I_C2

I_s = I_L + V_sY/2     where Y = Shunt Admittance = j?C

Again,

I_L = I_R + I_C1

    = I_R + V_RY/2    where Y = Shunt Admittance = j?C

Now, we will calculate the value of sending end voltage and current in terms of receiving end voltage and current.

V_s = V_R + I_LZ

Putting the value of I_L, we get

V_s = V_R + (I_R + V_RY/2) Z

V_s = V_R + (I_R + V_RY/2) Z

     = (1 + YZ/2) V_R + (YZ / 2) I_R  ������������(7)

Now,

I_s = I_L + V_sY/2

    

Putting the value of I_L we get,

I_s = (I_R + V_RY/2) + V_sY/2    

Now, putting the value of V_s we get,

I_s = (I_R + V_RY/2) + [(1 + YZ/2) V_R + (YZ / 2) I_R ]xY/2 

   = (1+ YZ/2)I_R + Y (1+ YZ/4)V_R  ������������(8)

Comparing equation (7) with (1),

A = 1 + YZ/2

B = YZ / 2

Comparing equation (8) with (2),

C = Y (1+ YZ/4)

D = 1 + YZ/2

Thus the ABCD Parameters for Nominal p model of medium transmission line are

A = D = 1 + YZ/2

B = YZ / 2

C = Y (1+ YZ/4)

The conformity of AD � BC = 1 can be checked easily and left for the reader. Hope you understand this topic clearly.