Home exercise 7.5.1: A 75\ \Omega foam-dielectric cable

$b/a=\exp (75\sqrt{1.5}/60)={\mathrm{e}}^{1.531}=4.62$, so $2b=4.62$ mm.

(exact root: $28.4$ GHz).

Solid polyethylene (${\epsilon }_r=2.25$) with the same $Z_0$ needs $b/a={\mathrm{e}}^{75\times 1.5/60}=6.52$, a thinner inner conductor, $2a=0.71$ mm for the same $2b$. Then $1/a+1/b$ is $3254$ instead of $2433$ m${}^{-1}$, so the foam cable has $25\%$ less conductor loss. Its dielectric loss ${\alpha }_d=\pi f\sqrt{{\epsilon }_r}\tan \delta /c$ is also smaller, by $\sqrt{1.5/2.25}=0.82$, for the same $\tan \delta$.

$\sqrt{L^{\prime }/C^{\prime }}=\sqrt{306\times 10^{-9}/54.4\times 10^{-12}}=75.0\Omega$ and $1/\sqrt{L^{\prime }{C}^{\prime }}=2.45\times 10^8$ m/s $=0.816c$.