Different Circuit Models and Their Applications

This 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.

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.

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

(1.1)

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

 (1.2)

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],

  (1.3)

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.

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

  (2.1)

and thus

  (2.2)

Since β = 2π/λ, Eq. (2.2) becomes

  (2.3)

leading to

  (2.4)

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

  (2.5)

and thus

  (2.6)

leading to

  (2.7)

The spacing between adjacent minima and maxima is λ/4. The first minimum can be obtained from the first maximum as

  (2.8)

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],

  (3.1)

represented by point A in Figure 3.

Recall the phase-shifted load reflection coefficient [4]

  (3.2)

At point B, the total phase of (d), that is, (θ – 2βd), 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

  (3.3)

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 (θ – 2βd), 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

  (3.4)

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 λ/4 apart.

Back to Part 1. 
Back to Part 2.

References

1. Adamczyk, B., “Analysis of Transmission Lines in Sinusoidal Steady State, Part 1: Different Circuit Models and Their Applications,” In Compliance Magazine, October 2024.
2. Adamczyk, B., “Analysis of Transmission Lines in Sinusoidal Steady State, Part 2: Different Circuit Models and Their Applications,” In Compliance Magazine, November 2024.
3. Adamczyk, B., “Smith Chart and Standing Wave Ratio,” In Compliance Magazine, September 2024.
4. Adamczyk, B., “Smith Chart and Input Impedance to Transmission Line – Part 3: Input Impedance to the Line,” In Compliance Magazine, June 2023.