Worked example 7.2.2: Coax or waveguide for a 10 GHz feed

$R_s=26.1\mathrm{m}\Omega$ at $10\mathrm{GHz}$, $\eta =377/\sqrt{2.1}=260.1\Omega$, $a=0.46\mathrm{mm}$, $b=1.49\mathrm{mm}$, $\ln (b/a)=1.175$ (so $Z_0=\frac{\eta }{2\pi }\ln \frac{b}{a}=48.7\Omega$).

$k=2\pi f\sqrt{2.1}/c=303.5rad/m$ and ${\alpha }_d=303.5\times 2\times 10^{-4}/2=0.030Np/m=0.26dB/m$. The total is $1.32dB/m$.

A: $30\times 1.32=39.5\mathrm{dB}$, so only $1000\times 10^{-3.95}=0.11\mathrm{W}$ arrive (and the cable could not even dissipate the rest). B: $30\times 0.1084=3.25\mathrm{dB}$, so $1000\times 10^{-0.325}=473\mathrm{W}$ arrive.

$f_c^{T{E}_{11}}\approx \frac{3\times 10^8}{\pi (1.95\times 10^{-3})(1.449)}=33.8\mathrm{GHz}$: the coax is single-mode at $10\mathrm{GHz}$.

Air (${\epsilon }_r=1$, ${\alpha }_d=0$) and single-mode: $a+b<c/(\pi f)=9.55\mathrm{mm}$. With $b=3.6a$: $a=2.08\mathrm{mm}$, $b=7.47\mathrm{mm}$ and

$4.3\mathrm{dB}$ over $30\mathrm{m}$, still worse than WR-90.

Part (d) is self-consistent: $f_c^{T{E}_{11}}=c/[\pi (a+b)]=10.0\mathrm{GHz}$ exactly at the limit, so a real design would stay somewhat smaller and lose a little more.