Question

Consider an air-filled rectangular waveguide with a = 6 cm and b = 9 cm. The y component of the $T{E}_{mn}$ mode is
$E_y=12sin\left(\frac{2\pi x}{a}\right)cos\left(\frac{2\pi y}{b}\right)sin(10\pi \times {10}^{10}t-\beta z)$V/m. The intrinsic impedance is ________$\Omega$

Answer 1

Mode is $T{E}_{mn}=T{E}_{23}$

${\eta }_{TE}=\frac{{\eta }_0}{\sqrt{1-{\left(\frac{f_0}{f}\right)}^2}}$

${\eta }_0=120\pi$

$f=\frac{10\pi \times {10}^{10}}{2\pi }=5\times {10}^{10}Hz$

$f_c=\frac{C}{2}({\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2)=\frac{3\times {10}^8}{2\times {10}^{-2}}{({\left(\frac{2}{6}\right)}^2+{\left(\frac{3}{9}\right)}^2)}^2$

$=\frac{3\times {10}^{10}}{2}.\frac{\sqrt{2}}{3}=\frac{1}{\sqrt{2}}\times {10}^{10}Hz$

$\frac{f_0}{f}=\frac{1}{5\sqrt{2}}$

${\eta }_{TE}={\eta }_0.\frac{1}{\sqrt{1-{\left(\frac{1}{5\sqrt{2}}\right)}^2}}=\frac{{\eta }_0}{\sqrt{1-\frac{1}{50}}}=\frac{{\eta }_0\sqrt{50}}{7}$

$=\frac{5{\eta }_0\sqrt{2}}{7}=380.82\approx 381\Omega$