RF Electronic Devices and Systems

# Circular transmission lines

A circular waveguide is schematically depicted in Figure 5. To consider its parameters we will employ the polar coordinates with radial components ρ and angle φ.

So the corresponding electric and magnetic field components for the general case are

Let’s consider the TE waves, where . Considering the ${H}_{z}\left(\rho ,\phi ,z\right)={H}_{z}\left(\rho ,\phi \right){e}^{–\mathrm{j\beta z}}$ and applying the Laplace equation, we can get the magnetic field structure and the rest of the parameters of the waveguide, in a similar way we did before.

So        $\left\{\begin{array}{l}{H}_{z}\left(\rho ,\phi \right)=\left(\mathrm{Asinn\phi }+\mathrm{Bcosn\phi }\right){J}_{n}\left({k}_{c},\rho \right)\\ {E}_{\phi }\left(z,\rho ,\phi \right)=\frac{\mathrm{j\omega \mu }}{{k}_{c}}\left(\mathrm{Asinn\phi }+\mathrm{Bcosn\phi }\right){J}_{n}‘\left({k}_{c},\rho \right){e}^{–\mathrm{j\beta z}}\end{array}\right\$, here ${J}_{n}\left({k}_{c},\rho \right)$ and ${J}_{n}‘ \left({k}_{c},\rho \right)$  are Bessel functions. The characteristics of the waveguide will be determined by the Bessel function roots pnm. So the parameters of the TE mode transmission line are:

Cut-off wave number ${k}_{\mathrm{nm}}=\omega \sqrt{\mathrm{\mu \epsilon }}\phantom{\rule{0ex}{0ex}}{k}_{\mathrm{cnm}}=\frac{{p}_{\mathrm{nm}}‘}{a}$ .

Propagation constant ${\beta }_{\mathrm{nm}}=\sqrt{{k}^{2}–{\frac{{p}_{\mathrm{nm}}‘}{a}}^{2}}$.
Cut-off wavelength  .
Phase velocity  ${v}_{\mathrm{pnm}}=\frac{\omega }{\beta }$.

Dielectric attenuation constant ${\alpha }_{dnm}=\frac{{k}^{2}\mathrm{tan}\delta }{2\beta }$.

Impedance  ${Z}_{\mathrm{TEnm}}=\frac{\mathrm{k\eta }}{\beta }$ .

For the TM mode of the transmission line, applying the considerations and calculation above we can get the following results:

Cut-off wave number ${k}_{\mathrm{nm}}=\omega \sqrt{\mathrm{\mu \epsilon }}\phantom{\rule{0ex}{0ex}}{k}_{\mathrm{cnm}}=\frac{{p}_{\mathrm{nm}}}{a}$ .
Propagation constant ${\beta }_{\mathrm{nm}}=\sqrt{{k}^{2}–\frac{{{P}_{\mathrm{nm}}}^{2}}{{a}^{2}}}$.
Cut-off wavelength ${\lambda }_{\mathrm{cnm}}=\frac{2\mathrm{\pi a}}{{p}_{\mathrm{nm}}}\phantom{\rule{0ex}{0ex}}{\lambda }_{\mathrm{gnm}}=\frac{2\pi }{\beta }$.
Phase velocity ${v}_{\mathrm{pnm}}=\frac{\omega }{\beta }$.

Dielectric attenuation constant ${\alpha }_{dnm}=\frac{{k}^{2}\mathrm{tan}\delta }{2\beta }$.

Impedance .

The cut-off frequency for the waveguide can be obtained using the cut-off wavelength. The roots for Bessel functions can be found in mathematic catalogues.

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