Analysis of Transmission Lines in Sinusoidal Steady State

EMC concepts explained

Different Circuit Models and Their Applications: Part 3

his is the third and final article discussing four different circuit models of transmission lines in sinusoidal steady state. In [1], Model 1 and Model 2 were presented. Model 1 was used to present the solution of the transmission line equations. Model 2 introduced the standing waves. Model 3 discussed in [2] led to the evaluation of the values of the minima and maxima of standing waves. This article uses Model 4 to determine the locations of the minima and maxima of standing waves. This determination is first done analytically, followed by the graphical method using the Smith chart.

1. Transmission Line Model 4

To present Model 4, it is helpful to recall Model 3, shown in Figure 1.

Figure 1: Transmission line circuit – Model 3

In Model 3, we are moving away from the source, located at z = -L to the load located at z = 0. Model 4 is shown in Figure 2.

Figure 2: Transmission line circuit – Model 4

In Model 4, we are moving away from the load located at d = 0, towards the source located at d = L. Model 4 is obtained from Model 3 by simply relating the distance variables according to

Model 3 was led to the expression for the magnitude of the voltage at any location z away from the source as

With the change of variables given by Eq. (1.1), Model 4 produced an expression for the magnitude of the voltage at any location d away from the load as [2],

In the next section, we will use this equation to determine the locations of the voltage maxima and minima in terms of the distance d away from the load.

2. Location of the Voltage Maxima and Minima – Analytical Solution

Examining Eq. (1.3), we deduce that the maximum magnitude of the voltage occurs when the cosine function equals 1 or its argument satisfies the condition

and thus

Since

= 2π/

, Eq. (2.2) becomes

leading to

The minimum magnitude of the voltage occurs when the cosine function equals -1 or its argument satisfies the condition

and thus

leading to

The spacing between adjacent minima and maxima is

/4. The first minimum can be obtained from the first maximum as

3. Location of the Voltage Maxima and Minima – Graphical Solution Using Smith Chart

To illustrate this graphical solution, consider a load with the normalized load impedance [3],

represented by point A in Figure 3.

Figure 3: Smith Chart and the voltage maxima and minima

Recall the phase-shifted load reflection coefficient [4]

At point B, the total phase of

(d), that is, (

– 2

d), is zero, or -2nπ, (n being a positive integer).

As stated earlier, the maximum magnitude of the voltage occurs when the cosine function in Eq. (1.3) equals 1 or its argument satisfies the condition

which is exactly the condition satisfied at point B. Thus, point B is the location of the voltage maxima.

At point C, the total phase of

(d), that is (

– 2

d), equals -π or – (2n + 1)π (n being a positive integer).

As stated earlier, the minimum magnitude of the voltage occurs when the cosine function in Eq. (1.3) equals -1 or its argument satisfies the condition

which is exactly the condition satisfied at point C. Thus, point C is the location of the voltage minima.

The corresponding minima and maxima are /4 apart.

References

Adamczyk, B., “Analysis of Transmission Lines in Sinusoidal Steady State – Different Circuit Models and Their Applications – Part I,” In Compliance Magazine, October 2024.
Adamczyk, B., “Analysis of Transmission Lines in Sinusoidal Steady State – Different Circuit Models and Their Applications – Part II,” In Compliance Magazine, November 2024.
Adamczyk, B., “Smith Chart and Standing Wave Ratio,” In Compliance Magazine, September 2024.
Adamczyk, B., “Smith Chart and Input Impedance to Transmission Line – Part 3: Input Impedance to the Line,” In Compliance Magazine, June 2023.

Share this story:

Dr. Bogdan Adamczyk is professor and director of the EMC Center at Grand Valley State University (http://www.gvsu.edu/emccenter) where he performs EMC educational research and regularly teaches EM/EMC courses and EMC certificate courses for industry. He is an iNARTE-certified EMC Master Design Engineer. He is the author of two textbooks, “Foundations of Electromagnetic Compatibility with Practical Applications” (Wiley, 2017) and “Principles of Electromagnetic Compatibility: Laboratory Exercises and Lectures” (Wiley, 2024). He has been writing “EMC Concepts Explained” monthly since January 2017. He can be reached at adamczyb@gvsu.edu.

Back to Issue