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Do anyone know how to derive the microstrip stub equations for all four microstrip below ?

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Impedance of a transmission line v. frequency

Stolen from: https://www.edn.com/design/test-and-measurement/4424529/So-you-think-you-understand-transmission-lines-

But anyways, that's the impedance of some transmission line that's open on the other end. You can see it decreases in impedance initially, which is the hallmark of a capacitor. Then it sharply decreases and starts going back up, which is a resonance and can be modeled by a inductor in series with the capacitor.

If you look at the impedance of a transmission line:

$$ Z_{in}(l) = Z_0 \frac {Z_L + jZ_0 \tan(\beta l)} {Z_0 + jZ_L \tan(\beta l)} $$

where \$ \beta = \frac {2 \pi} {\lambda}\$.

Try simplifying and graphing that equation when \$Z_L = 0\$ (shorted stub) and \$Z_L = \infty\$ (open stub).

You'll see the kind of behavior I described--increasing or decreasing impedance wrt frequency, and a resonance at \$l = \frac \lambda 4\$.

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  • \$\begingroup\$ ok, thanks for your reply. However, I am looking for derivation for each of four cases in my original question. You only explained the first case in your answer. \$\endgroup\$ – kevin Sep 30 '19 at 7:15
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    \$\begingroup\$ Are you having trouble with something particular in the derivation or do you not know where to begin? I think the general community rules dictate I'm supposed to guide you but not do all the work. \$\endgroup\$ – hatsunearu Sep 30 '19 at 19:21
  • \$\begingroup\$ David Pozar's book (Microwave Engineering) does not derive the analytical expression for the rest of the cases. He just used smith chart (which is not I want in this question) to do the explanation. \$\endgroup\$ – kevin Oct 1 '19 at 1:24
  • \$\begingroup\$ I assume you get how to derive why open and shorted stubs look like inductors and capacitors when l << lambda? \$\endgroup\$ – hatsunearu Oct 1 '19 at 23:54
  • \$\begingroup\$ Ah, okay I get it now. I am fairly certain the behavior of a stub isn't EXACTLY supposed to match the lumped element version; I believe the lumped element versions just capture the fact that it resonates in a particular way such that the impedance drops to 0 or rises to infinity at a certain frequency. \$\endgroup\$ – hatsunearu Oct 2 '19 at 0:08

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