Equivalent T and π Network of Long Transmission Line

Equivalent T and π Network of Long Transmission Line

In case of analysis of medium transmission line, we have used nominal T and nominal n method. These methods are said to be nominal methods as they do not take into account the actual conditions i.e. instead of taking the parameters of the lines as uniformly distributed they are considered to be concentrated or lumped at one point in the transmission line network.

For transmission line of lengths more than 160 km, the parameters of the line are considered to be distributed and voltages and currents at sending end are obtained by rigorous method as,

We will now derive equivalent T and equivalent n networks for such long lines. Let us first consider equivalent T network.

1. Equivalent T Representation of Long Transmission Line

Consider the following network shown in Fig. 2.13.1 to determine the equivalent-T network.

From analysis of this network carried out earlier we have,

Comparing above equations with those obtained for along transmission line using rigorous analysis.

Consider the shunt branch of equivalent-T network as shown in Fig. 2.13.1.

Its admittance is given as,

This indicates that the shunt branch of equivalent T can be obtained by multiplying the shunt branch of nominal T (lumped shunt admittance) by the factor (sin ɤ l / ɤ l)

Now let us determine series impedance of the equivalent-T network. 

This indicates that the series branch of the equivalent-T network can be determined by multiplying the series branch of the nominal-T (lumped series impedance) network with factor tanh (ɤ l / 2) / (ɤ l / 2). The equivalent network is shown in Fig. 2.13.2.

It can be seen that for small values of y l , the ratio of tanh(y l / 2)/(y l / 2) and (sinh ɤ l) / ɤ l almost approach unity and the nominal circuits represent the medium length lines quite accurately.

By taking into consideration only the first order terms, we can get the A, B, C, D parameters of nominal-T or n representation. Also, it can be assumed that  Y²Z / 6 ≈ Y²Z / 4 for small Y².

2. Equivalent π Representation of Long Transmission Line

Consider the equation of series branch impedance,

This shows that the series branch impedance of equivalent n network can be obtained by multiplying series branch of nominal π (lumped series impedance) by the factor (sinh ɤ l) / ɤ l.

This indicates that the shunt branch of the equivalent л network can be determined by multiplying the shunt branch of the nominal-л (lumped shunt admittance) network by factor tanh (ɤl / 2) (ɤl /2)

The equivalent network is shown in the Fig. 2.13.4.

Review Question

1. Explain Т and л equivalent representation of long transmission line.