Problem 7.3.4: Copper loss and power rating of WR-137

$a=0.03485\mathrm{m}$, $b=0.0158\mathrm{m}$, $f=7\mathrm{GHz}$, $\lambda =4.29\mathrm{cm}$, $k=146.6rad/m$, $L=25\mathrm{m}$, $E_{\max }=1.5MV/m$.

$f_{co,10}=\frac{3\times 10^8}{2\times 0.03485}=4.304\mathrm{GHz}$, $(f_c/f{)}^2=0.378$, $s=0.789$:

$R_s=\sqrt{\frac{\pi (7\times 10^9)(4\pi \times 10^{-7})}{5.8\times 10^7}}=21.8\mathrm{m\Omega }$ and $2b/a=0.907$:

$25\times 0.0542=1.35\mathrm{dB}$, or ${\mathrm{e}}^{-2\times 0.00624\times 25}=0.732$: $73\%$ of the input power reaches the antenna.

$10^{-1.355/10}=0.732$ agrees with ${\mathrm{e}}^{-2{\alpha }_{c}L}$. Halving the allowed field divides the power rating by four: $2.59/4=0.648\mathrm{MW}$.