Monday, 6 November 2017

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

VR = Receiving end voltage

Vs = Sending end voltage

Is = Sending end current

IR = Receiving end current

Thus we can write,

Vs = AVR + BIR …………………..(1)

Is = CVR + DIR ……………………(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 IR = 0.

From equation (1).

A = Vs / VR

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

From equation (2),

C = Is / VR

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 VR = 0.
From equation (1),

B = Vs / IR

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

From equation (2),

D = Is / IR

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 ParametersValueUnit
Voltage Ratio, AVs/VRUnit less
Short Circuit Resistance, BVs / IROhm
Open Circuit Conductance, CIs / VRMho
Current Ratio, DIs / VRUnit 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 ADBC = 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.

short-transmission-line-ABCD-Parameters


From the figure above,

Is = IR  ………………………(3)

and Vs = VR + IRZ  …………..(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 π Model.

Nominal T Model

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

Nominal-T-model-ABCD-Parameters


From the figure above,

Vs = V1 + IsZ/2    (These are in phasor from)

But V1 = VR + IRZ/2  (These are in phasor from)

Again,

Ic = Is - IR

   = V1where Y = Shunt admittance

   = Y (VR + IRZ/2 )

Therefore,

Is = IR + YVR + YIRZ/2

   = YVR + IR(1 + YZ / 2)  ……………………….(5)

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

Vs = VR + IRZ/2 + IsZ/2  
    
Putting the value of Is in the above expression, we get

 Vs = (1 + YZ / 2)VR + (Z + YZ2 / 4) IR ……………(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 π Model

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

Nominal-Pi-model-ABCD-Parameters


As clear from the figure above,

Is = IL + IC2

Is = IL + VsY/2     where Y = Shunt Admittance = jωC

Again,

IL = IR + IC1

    = IR + VRY/2    where Y = Shunt Admittance = jωC

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

Vs = VR + ILZ

Putting the value of IL, we get

Vs = VR + (IR + VRY/2) Z

Vs = VR + (IR + VRY/2) Z

     = (1 + YZ/2) VR + (YZ / 2) IR  ………………………………(7)

Now,

Is = IL + VsY/2
    
Putting the value of IL we get,

Is = (IR + VRY/2) + VsY/2    

Now, putting the value of Vs we get,

Is = (IR + VRY/2) + [(1 + YZ/2) VR + (YZ / 2) IR ]xY/2 

   = (1+ YZ/2)IR + Y (1+ YZ/4)VR  ………………………………(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 π 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.

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