Worked example 7.5.1: Which coaxial cable up the mast?

$\eta =377/1.5=251.3\Omega$, $v=2\times 10^8$ m/s. RG-58: $a=0.405$ mm, $b=1.475$ mm, $\ln (b/a)=1.293$. RG-213: $a=1.125$ mm, $b=3.625$ mm, $\ln (b/a)=1.170$.

At $2.4$ GHz: $R_s=12.8$ m$\Omega$, $k=75.4$ rad/m, ${\alpha }_d=0.0075$ Np/m $=0.066$ dB/m for both cables.

At $5.8$ GHz ${\alpha }_c$ grows by $\sqrt{5.8/2.4}=1.555$ and ${\alpha }_d$ by $5.8/2.4$ (to $0.158$ dB/m). RG-58: $0.836+0.158=0.994$ dB/m, $19.9$ dB, $1.0\%$ arrives. RG-213: $0.342+0.158=0.500$ dB/m, $10.0$ dB, $10\%$ arrives.

RG-58: $\frac{2\times 10^8}{\pi \times 1.88\times 10^{-3}}=33.9$ GHz. RG-213: $\frac{2\times 10^8}{\pi \times 4.75\times 10^{-3}}=13.4$ GHz.

Single-mode at $28$ GHz requires $a+b<\frac{v}{\pi f}=2.27$ mm, with $b/a={\mathrm{e}}^{50/40}=3.49$ for $50\Omega$. Hence $b=1.77$ mm, $a=0.51$ mm, an RG-58-size cable. At $28$ GHz $R_s=43.7$ m$\Omega$ and its loss is $1.53+0.76=2.3$ dB/m, $46$ dB over $20$ m.

$Z_0=40\ln (b/a)$: $51.7\Omega$ for RG-58 and $46.8\Omega$ for RG-213 (its stranded inner conductor makes it nominally $50\Omega$). The cable of part (d) has $f_c=2\times 10^8/(\pi \times 2.27\times 10^{-3})=28.0$ GHz, exactly at the limit.