7.7.10 Problems and solutions

The problems of this section use Hammerstad's analysis formulas ($u=w/h$)

Wheeler's synthesis, Eq. (7.7.7), the line constants $\beta =k_0\sqrt{{\epsilon }_{\mathrm{eff}}}$, $C^{\prime }=\sqrt{{\epsilon }_{\mathrm{eff}}}/(c{Z}_0)$, $L^{\prime }=Z_0\sqrt{{\epsilon }_{\mathrm{eff}}}/c$, the losses ${\alpha }_c\approx R_s/(Z_{0}w)$ with $R_s=\sqrt{\pi f{\mu }_0/{\sigma }_c}$ and ${\alpha }_d=q{k}_0\sqrt{{\epsilon }_{\mathrm{eff}}}\tan \delta /2$ with $q=(1-1/{\epsilon }_{\mathrm{eff}})/(1-1/{\epsilon }_r)$, Getsinger's dispersion, Eq. (7.7.12), and the limits $c/(4h\sqrt{{\epsilon }_r-1})$ (TE${}_1$ surface wave) and $c/[\sqrt{{\epsilon }_r}(2w+0.8h)]$ (transverse resonance). Constants: $c=3\times 10^{8}m/s$, ${\mu }_0=4\pi \times 10^{-7}H/m$, ${\epsilon }_0=10^{-9}/(36\pi )F/m$, ${\eta }_0=120\pi \Omega$, copper ${\sigma }_c=5.8\times 10^{7}S/m$, and $1\mathrm{Np}=8.686\mathrm{dB}$.