What is Long Transmission Line?

What is Long Transmission Line?

Definition of Long Transmission Line

A long transmission line is defined as a transmission line longer than 250 km (150 miles), which needs a different modeling approach.

A long transmission line is defined as a transmission line with a length greater than 250 km (150 miles). Unlike short transmission lines and medium transmission lines, long transmission lines need detailed modeling of their distributed parameters along the entire length. This makes calculating the ABCD parameters of transmission line more complex but allows us to find the voltage and current at any point on the line.

In a long transmission line the line constants are uniformly distributed over the entire length of line. This is because the effective circuit length is much higher than what it was for the former models (long and medium line) and hence we can no longer make the following approximations:

Ignoring the shunt admittance of the network, like in a small transmission line model.Considering the circuit impedance and admittance to be lumped and concentrated at a point as was the case for the medium line model.

Instead, we must consider the circuit impedance and admittance as distributed over the entire length. This makes the calculations more rigorous. For accurate modeling of these parameters, we use the circuit diagram of the long transmission line.

Here a line of length l > 250km is supplied with a sending end voltage and current of VS and IS respectively, whereas the VR and IR are the values of voltage and current obtained from the receiving end. Lets us now consider an element of infinitely small length Δx at a distance x from the receiving end as shown in the figure where.

V = value of voltage just before entering the element Δx.

I = value of current just before entering the element Δx.

V+ΔV = voltage leaving the element Δx.

I+ΔI = current leaving the element Δx.

ΔV = voltage drop across element Δx.

zΔx = series impedance of element Δx

yΔx = shunt admittance of element Δx

Where, Z = z l and Y = y l are the values of total impedance and admittance of the long transmission line.

Therefore, the voltage drop across the infinitely small element Δx is given by

Now to determine the current ΔI, we apply KCL to node A.

Since the term ΔV yΔx is the product of 2 infinitely small values, we can ignore it for the sake of easier calculation.

Therefore, we can write

Now derivating both sides of eq (1) w.r.t x,

Now substituting from equation (2)

The solution of the above second order differential equation is given by.

Derivating equation (4) w.r.to x.

Now comparing equation (1) with equation (5)

Now to go further let us define the characteristic impedance Zc and propagation constant δ of a long transmission line as

Then the voltage and current equation can be expressed in terms of characteristic impedance and propagation constant at

Now at x=0, V= VR and I= Ir. Substituting these conditions to equation (7) and (8) respectively.

Solving equation (9) and (10),We get values of A1 and A2 as,

Now applying another extreme condition at x = l, we have V = VS and I = IS.Now to determine VS and IS we substitute x by l and put the values of A1 andA2 in equation (7) and (8) we get

By trigonometric and exponential operators we know

Therefore, equation (11) and (12) can be re-written as

Thus compared with the general circuit parameters equation, we get the ABCD parameters of a long transmission line as,