Home exercise 7.2.4: Losses in a polystyrene-filled guide

$v=c/1.6=1.875\times 10^{8}m/s$, $f_{co,1}=v/2b=9.375\mathrm{GHz}$, $f_{co,2}=18.75\mathrm{GHz}$: TEM, TM${}_1$ and TE${}_1$ propagate. $f/f_{co,1}=1.6$ and $s=\sqrt{1-0.625^2}=0.781$.

$\eta =377/1.6=235.6\Omega$, $k=2\pi f/v=502.7rad/m$ and $R_s=\sqrt{\pi f{\mu }_0/{\sigma }_c}=32.0\mathrm{m}\Omega$ at $15\mathrm{GHz}$.

The totals are TEM $0.655+0.118=0.773dB/m$, TM${}_1$ $0.839+0.302=1.141dB/m$ and TE${}_1$ $0.839+0.118=0.957dB/m$. The TEM mode has the lowest loss. The ratio of the conductor losses is

The dielectric loss is independent of the plates and the conductor loss of the dielectric's $\tan \delta$, so the bars split as drawn. The TM${}_1$ conductor loss is exactly twice the TEM value divided by $s$, $2\times 0.118/0.781=0.302$.