# Short Transmission Line (Phasor Diagram & Performance)

## What is Short Transmission Line?

A short transmission line is defined as a transmission line with an effective length less than 80 km (50 miles), or with a voltage less than 69 kV. Unlike medium transmission lines and long transmission lines, the line charging current is negligible, and hence the shunt capacitance can be ignored.

For short length, the shunt capacitance of this type of line is neglected and other parameters like electrical resistance and inductor of these short lines are lumped, hence the equivalent circuit is represented as given below. Let’s draw the vector diagram for this equivalent circuit, taking receiving end current Ir as reference. The sending end and receiving end voltages make angle with that reference receiving end current, of φs and φr, respectively.

As the shunt capacitance of the line is neglected, hence the sending end current and the receiving end current is same, i.e.

We can see from the short transmission line phasor diagram above that Vs is approximately equal to:

That means,

As it is assumed that:

As there is no capacitance, during no-load condition the current through the line is considered as zero, hence at no load condition, receiving end voltage is the same as sending end voltage.

As per dentition of voltage regulation of power transmission line,

Here, Vr and Vx are the per unit resistance and reactance of the short transmission line respectively.

Any electrical network generally has two input terminals and two output terminals. If we consider any complex electrical network in a black box, it will have two input terminals and output terminals. This network is called a two-port network. A two-port model of a network simplifies the network solving technique. Mathematically, a two-port network can be solved by 2 by 2 matrix.

A transmission as it is also an electrical network, and hence the transmission line can be represented as a two-port network.

Hence two-port network of the transmission line can be represented as 2 by 2 matrix. Here the concept of ABCD parameters comes into play. Voltage and currents of the network can be represented as:

Where, A, B, C and D are the different constants of the transmission network.

If we put Ir = 0 at equation (1), we get,

Hence A is the voltage impressed at the sending end per volt at the receiving end when receiving end is open. It is dimensionless. If we put Vr = 0 at equation (1), we get

Hence B indicates the impedance of the transmission line when the receiving terminals are short-circuited. This parameter is referred to as the transfer impedance.

C is the current in amperes into the sending end per volt on open circuited receiving end. It has the dimension of admittance.

D is the current in amperes into the sending end per amp on the short-circuited receiving end. It is dimensionless.

Now from the equivalent circuit, it is found that,

Comparing these equations with equation 1 and 2 we get, A = 1, B = Z, C = 0 and D = 1. As we know that the constant A, B, C, and D are mathematically related to a passive network as:

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

⇒ 1.1 − Z.0 = 1

So the values calculated are correct for a short transmission line. From the above equation (1),

When Ir = 0 that means receiving end terminals is open circuited and then from equation 1, we get receiving end voltage at no load.

and as per definition of voltage regulation of power transmission line,

## Performance of Short Transmission Line

The performance (i.e. the efficiency) of a short transmission line as simple as efficiency equation of any other electrical equipment, that means

Where R is the per phase electrical resistance of the transmission line. 