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I have been reading Pozar's Microwave Engineering, and in Section 4.6, he goes through the analysis to determine the reactance of a discontinuity between two coupled waveguides. In that chapter, he describes how using two waveguides with a difference in width, it is possible to determine the equivalent parallel inductance. In the same section, he claims that a change in height would result in a equivalent parallel capacitance.

Diagram from Pozar, Fig. 4.22, showing the discontinuity geometries and the equivalent circuits that can be used to model them.

However, I am unable to follow the analysis that he does for change in width and apply it to change in height. The main issue seems to be that the TE10 mode is only dependent on X and not on Y. Therefore, it almost appears that the change in height has no effect on the reflection, which obviously makes no sense.

As a reminder, the TE10 mode has transverse components of the form

$$E_y^i = \sin(\frac{\pi x}{a}) e^{-j \beta_1^a z}$$

$$H_x^i = \frac{-1}{Z_1^a} \sin( \frac{\pi x}{a}) e^{-j \beta_1^a z}$$

and at the boundary plane, continuity equations ought to be applied.

I think the main issue I have is with this following sentence regarding reflected modes in the second guide:

Because there is no y variation introduced by this discontinuity, TE_mn modes for m \neq 0 are not excited, nor are any TM modes. A more general discontinuity, however, may excite such nodes.

In the change in height case, there is only y-variation. I do not know how to apply that y-variation to this condition to determine what transmitted modes into guide 2 ought there to be!

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This method actually does not apply in this kind of discontinuity. See Pozar Problem 4.1, which directly shows that using this type of modal analysis results in a nonsensical result of having zero reflection due to the height change. Instead, a different kind of analysis needs to be done.

Refer to Marcuvitz v10, p308, section 5-26 - change in height of rectangular guide for approximate equations. Still no explanation for how the analysis is done, but good enough for me.

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