Home exercise 6.2.4: A low-loss microwave substrate

$1/0.809=1.24\mathrm{m}$.

Exactly, $\gamma =\mathrm{j}\omega \sqrt{{\mu }_0{\epsilon }_0{\epsilon }_r(1-\mathrm{j}\tan \delta )}=0.09319+\mathrm{j}93.195{\mathrm{m}}^{-1}$: $\alpha =0.09319Np/m$, the same as the estimate to four digits.

The relative error of the low-loss formula is of order ${\tan }^2\delta /8=5\times 10^{-7}$, invisible at four digits. Per wavelength the loss is $0.809\times 0.0674=0.055\mathrm{dB}$, close to the rule of thumb $27.3\tan \delta =0.055\mathrm{dB}$, Eq. (6.2.23): truly low-loss. On a real microstrip line only part of the field is in the substrate, and the conductor loss of the strip is usually larger than this dielectric loss at $3\mathrm{GHz}$. The dielectric loss grows linearly with $f$ at constant $\tan \delta$, which is why PTFE laminates replace FR-4 ($\tan \delta \approx 0.02$) at microwave frequencies.