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I'm in an intro electromagnetics class. It makes sense to me that reflection coefficient at the load has an imaginary component, because I am thinking of it like "the reflected wave is some percentage of the amplitude of the incident wave, and it's shifted by some constant phase shift".

However I cannot understand why reflection coefficient along the T-line (i.e. reflection coefficient as a function of distance) has a changing phase angle. I'm thinking if you freeze-frame a T line at some moment in time and look at the voltage across the length of the T line, the reflected wave and the incident wave may be offset, but the offset will be the same for the entire length of the T line, because the two waves have the same frequency. I graphed this to hopefully help explain my line of thinking.

enter image description here

So why does the angle of the reflection coefficient depend on where you are on the T line? I would love a graphical explanation, I feel like I am getting caught in the math and not understanding what is really going on.

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So why does the angle of the reflection coefficient depend on where you are on the T line?

Because the reflected wave is still produced at the same location (call it x = 0), and then travels back to wherever you're measuring the reflection. As it travels, it accumulates phase.

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Because waves have a length to them and physically move through the medium. In an ideal feedline you would have 360 degrees of phase for every wavelength of distance (since it will go through a full cycle in that space). If it helps try to visualize water waves. If a water wave is coming towards you and you happen to be at the peak of the wave at the ocean and you look ahead you will see the trough of the wave half a wavelength in front of you. So to someone half a wavelength in front of you in the ocean they would be at the part of the same wave that is 180 degrees phase difference from yourself. It is no different from a feedline.

One thing that is very important to say here is that the way you are measuring the phase difference on the diagram you posted is completely incorrect. You are treating it as if it would be waves on an oscilloscope screen. The diagram has physical space as the x dimension with the waves moving in the opposite directions, so the way you went about it is no longer valid. Now if the waves were both moving in the same direction then the way you are indicating phase on there would be correct, but they are not. Think about it, because they are moving in opposite directions the two peaks you have labeled would be moving away from each other, as such if you measured it that way the values would constantly be changing from one moment to another cycling through 0 degrees to 360 degrees over and over again. The reflection coefficient, however, is a fixed value that wont change so long as the two waves remain consistent.

So basically for those two waves there are points on the x axis where they are in phase and points where they are out of phase. For example to find the points where the two waves are always 180 degrees out of phase find the points where each wave crosses the x-axis nearest to each other, then find the point halfway between that. At that point the two waves are always opposite of each other at the same part of their cycle. Similarly if you want to find the points where the two waves are always in phase then find the peaks and do the same.

It is important to understand that because these waves are moving in opposite directions it makes no sense to say the waves as a whole are in phase or out phase in this context. The phase relationship only matters at specific points, and is different at every point.

Here is an image of the actual waves in motion that might make this clearly.

enter image description here

The red line is the voltage that would be resulting from the combination of the two waves as they move. See those points where the red line is always 0, that is where the two waves are always 180 degrees out of phase. Since both waves are the same amplitude that means they always cancel each other out at that point.

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    \$\begingroup\$ Thanks, that helps some. So a reflection coefficient of 0.5 angle 30 at the load (k=0m) means "At this point, 50% of the incident wave is reflected and has a 30 degree phase shift." But what does it mean in English if you measure the reflection coefficient at k=1m? \$\endgroup\$ – K4KFH Sep 7 '20 at 16:28
  • \$\begingroup\$ I cant fit my answer to this in a single comment. Can you please post it as a new question and I will be happy to give a detailed answer. Short answer is.. The reflection coefficient at that point tells you the ratio of the forward wave at that point to the reflected. so if R is 30 degrees it means the reflected signal is 30 degrees ahead of the forward signal at that point. If |R| is 0.25 it means the reflected wave is 0.25 the amplitude of the forward wave. \$\endgroup\$ – Jeffrey Phillips Freeman Sep 7 '20 at 20:07
  • \$\begingroup\$ Thank you! I would like to hear a more detailed answer. I posted this question for you: electronics.stackexchange.com/questions/520552/… \$\endgroup\$ – K4KFH Sep 8 '20 at 21:13
  • \$\begingroup\$ @K4KFH Wonderful, transmission line questions are my absolute favorite. I am going to bed right now but I promise I will answer it in about 8 hours in glorious detail :) \$\endgroup\$ – Jeffrey Phillips Freeman Sep 8 '20 at 21:15
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    \$\begingroup\$ Thank you! The updated answer here helps a lot. \$\endgroup\$ – K4KFH Sep 9 '20 at 22:05
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We calculate the reflection coefficient as a function of place (=distance from an end of the line). In point X the ratio of the phasors of the reflected and incident and waves is just that coefficient. Its angle depends on X because the reflection comes from the load and the incident wave comes from the input of the line. The R.coefficient at X is calculated with those phase angles that the waves have at X, not with the values at the ends of the line.

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