7.4.11 Problems and solutions

The problems of this section use the results collected here. In a circular guide of radius $a$ the TM${}_{mn}$ modes have $E_z=C{J}_m(h\rho )\cos m\varphi {\mathrm{e}}^{-\mathrm{j}\tau z}$ with $h=x_{mn}/a$, the TE${}_{mn}$ modes $H_z=C{J}_m(h\rho )\cos m\varphi {\mathrm{e}}^{-\mathrm{j}\tau z}$ with $h=x_{mn}^{\prime }/a$, and

With $s=\sqrt{1-(f_c/f{)}^2}$: $\tau =ks$, ${\lambda }_g=\lambda /s$, $v_p=v/s$, $v_g=vs$, $Z_{TM}=\eta s$, $Z_{TE}=\eta /s$, and below cutoff $\tau =-\mathrm{j}\alpha$, $\alpha =\sqrt{h^2-k^2}$. The transverse fields are Eq. (7.4.17) and Eq. (7.4.20), the conductor losses Eq. (7.4.23), with $R_s=\sqrt{\pi f{\mu }_0/{\sigma }_c}$ and $1$ Np $=8.686$ dB. Constants: $c=3\times 10^8$ m/s, ${\eta }_0=120\pi =377\Omega$, ${\mu }_0=4\pi \times 10^{-7}$ H/m, ${\epsilon }_0=10^{-9}/(36\pi )$ F/m, copper ${\sigma }_c=5.8\times 10^7$ S/m.