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A rectangular waveguide with dimensions \$a=10~\mathrm{cm}, b=6~\mathrm{cm}\$ is filled with dielectric material with an unknown relative coefficient \$\epsilon_r\$. The guide is terminated by an unknown load. It is excited by \$TE_{3,0}\$ modes at a frequency of \$2.5~\mathrm{GHz}\$. How can I determine \$\epsilon_r\$?

I know the cutoff frequency is given by:

$$f=c\frac{\sqrt{(n/a)^2+(m/b)^2}}{2\sqrt{\epsilon_r}}$$

I am also told that the first minima are at \$s_a=5~\mathrm{mm}, s_b=30~\mathrm{mm}\$ away from the load. I know that the distance between the minima is \$\lambda/2\$, hence \$\lambda=50~\mathrm{mm}\$. Does that mean that \$c/(f\sqrt{\epsilon_r})=50\$? Because if it is, then I can easily find \$\epsilon_r\$ to be approx. 1.55. Is that correct?

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  • \$\begingroup\$ is the source impedance unknown as well? \$\endgroup\$ May 26, 2017 at 21:52
  • \$\begingroup\$ Yes, Tony, I don't know the source impedance. \$\endgroup\$
    – peripatein
    May 27, 2017 at 5:57
  • \$\begingroup\$ Nope, \$v=c/\sqrt{\epsilon_\mathrm{r}}\$ holds true for TEM modes only. \$\endgroup\$
    – carloc
    Dec 10, 2017 at 16:11

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of course standing wave minima was given thus $$\lambda /2 $$so you are almost correct ... calc error...

$$\epsilon_r = (\dfrac{c}{\lambda f })^2~~ \text{ so }~~\epsilon_r=5.76$$

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