Standing Wave Diagram (Part 1)

In this post we are going to study the transmission lines characteristics that can be measured with the voltage amplitudes (as a function of the position) in the line.
On a daily basis, engineers do these measurements with a device named slotted line a lossless coaxial transmission line with a a longitudinal slot cut into the line. There is a voltmeter that can be moved along the line so we can measure the maximum and minimum voltages and their relation: the Voltage Standing Wave Ratio (VSWR). The slotted line is placed between the voltage source and the impedance that we want to measure:

Source

From the equations studied in our previous post, we can deduce the equation of the complete voltage wave. If the load is placed in z=0, all the positions along the slotted waveguide will be in the negative z-axis values.

If  V_0  is the amplitude of the input wave, the total phasor voltage is:

In other words, this is the summation of the incident and reflected wave. In order to determine the standing wave diagram, we will take the module of this phasor:

where the maximum values are given by 2\beta_0 z  + \phi = 2k\pi and the minimum values are given by 2\beta_0 z  + \phi = (2k+1)\pi (these values make the cosine in the previous equation being maximum and minimum respectively).

Once we know the maximum and minimum amplitudes, we can obtain the standing wave ratio as:

Where S=1  means that there is no reflected wave (matching impedances) and S=∞ means that there is a pure standing wave (maximum mismatching).

In addition, we know that because of the cosine periodicity, the distance between two consecutive nodes in the wave would be half of the wave length:

Before moving to see some key examples, let’s analyze the total voltage wave equation. In order to do that, let’s write it in its instantaneous real form:

This expression is very important because it tells us that the total voltage wave is the summation of the travelling wave (the first term in the summation) plus a standing wave (the second term in the summation). Therefore, the portion of the incident wave that it’s reflected and propagated back to the slotted line interferes with an equivalent portion of the incident wave to form a standing wave. The rest of the incident wave (that doesn’t interfere) is the part of the travelling wave that appears in this equation.

It’s possible to find the maximum amplitude in the line where the two terms of the equation are summed to give V_0(1+|\rho (0)|) . The minimum amplitude can be observed where the standing wave reaches zero, remaining only the travelling wave amplitude V_0(1-|\rho (0)|) , which is consistent with the maximum and minimum values analyzed earlier in the module of the wave’s phasor.

If you want to know how to apply these concepts in practical examples, don’t miss our next post where we will be showing you some Matlab exercises (solved!) and you will be able to plot the diagram 🙂

 

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