DISTRIBUTED ELEMENT MODEL IN TRANSMISSION LINES (PART III)

This is the last part of the Distributed Element Model in Transmission Lines, where we are analyzing the transmission lines as a function of the z coordinate, in the instant of time, t. As we mentioned in the previous post, the last thing to study are:
 Propagation constant and characteristic impedance of the line
 Solve and exercise calculating this parameters on a real line
So let’s have a look at these!
Propagation constant and characteristic impedance of the line
The propagation factor can be obtained from the general equations of the transmission line: is an eigenvalue. As a function of the distributed parameters, can be written as:
where α is the attenuation constant (Np/m) and β is the phase constant of the line(rad/m).
For a lossless line, we have that:
The characteristic impedance is the relation between the reflected or incident voltage and current waves:
As we saw in a previous post, we can write the characteristic impedance as a function of the primary line constants:
Where, for the case of a lossless line, we would have:
Exercise Explained:
A coaxial transmission line uses a frequency of 100 MHz and have the following distributed parameters:
R = 0.098 Ω/m,
L = 0.32 x 10^{ – 6} H/m,
G = 1.5 x 10^{ – 6} S/m,
C = 34.5 x 10^{ – 12} F/m
Find Z_{0} at this frequency and the propagation factor.
Solution:
We just need to apply the previous equations and we’ll obtain a quasi real value for Z_{0}, as it usually happens in the low losses lines.
Therefore, we will obtain:
Z_{0} = 96.3 – j 0.02 Ohms;
α= 573.81 x 10^{6} Np/m;
β = 2.05 Rads/m
Then,
γ= 573.81 x 10^{6} + j2.05 m^{1}
What’s next?
In our next posts, we’ll move onto the the parameters when the line is in a transient state which analyses what happens when there is a sudden change in the conditions of the line. In order to fully study this new block, we will provide the following 4 posts:
 Reflection coefficient at the load and at the source
 Standing wave ratio (SWR)
 Analysis of behavior of the voltage waves in the line: open circuit and short circuit cases
 Two solved exercises explained
After this block is finished, we’ll be ready to study and learn how to use the Smith Chart, a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. It can be used to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability.
We hope this sounds interesting! 😀