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 $$\begin{gathered} E_{\rho }=–\frac{j}{{k_c}^2 }(\beta \frac{\partial E_z}{\partial \rho }+\frac{\mathrm{\omega \mu }}{\rho }\frac{\partial H_z}{\partial \phi }),E_{\phi }=-\frac{j}{{k_c}^2}(\frac{\beta }{\rho }\frac{\partial E_z}{\partial \rho }+\mathrm{\mu \omega }\frac{\partial H_z}{\partial \phi }, \\ {H}_{\rho }=\frac{j}{{k_c}^2 }(\frac{\mathrm{\omega \epsilon }}{\rho }\frac{\partial E_z}{\partial \phi }+\beta \frac{\partial H_z}{\partial \rho }),H_{\rho }=-\frac{j}{{k_c}^2 }(\mathrm{\omega \epsilon }\frac{\partial E_z}{\partial \phi }+\frac{\beta }{\rho }\frac{\partial H_z}{\partial \rho }) \end{gathered}$$

Let’s consider the TE waves, where $E_Z=0, H_Z\ne 0$. Considering the $H_z(\rho ,\phi ,z)=H_z(\rho ,\phi )e^{–\mathrm{j\beta z}}$ 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        $\left\{\begin{array}{l}{H}_z(\rho ,\phi )=(\mathrm{Asinn\phi }+\mathrm{Bcosn\phi })J_n(k_c,\rho )\\ E_{\phi }(z,\rho ,\phi )=\frac{\mathrm{j\omega \mu }}{k_c}(\mathrm{Asinn\phi }+\mathrm{Bcosn\phi })J_n‘(k_c,\rho )e^{–\mathrm{j\beta z}}\end{array}\right.$, here $J_n(k_c,\rho )$ and $J_n‘\; (k_c,\rho )$  are Bessel functions. The characteristics of the waveguide will be determined by the Bessel function roots p_nm. So the parameters of the TE mode transmission line are:

Cut-off wave number $$\begin{gathered} k_{\mathrm{nm}}=\omega \sqrt{\mathrm{\mu \epsilon }} \\ {k}_{\mathrm{cnm}}=\frac{p_{\mathrm{nm}}‘}{a} \end{gathered}$$ .

Propagation constant ${\beta }_{\mathrm{nm}}=\sqrt{k^2–{\frac{p_{\mathrm{nm}}‘}{a}}^2}$.
Cut-off wavelength  $$\begin{gathered} {\lambda }_{\mathrm{cnm}}=\frac{2\pi }{p_{\mathrm{nm}}‘} a \\ {\lambda }_{\mathrm{gnm}}=\frac{2\pi }{\beta } \end{gathered}$$ .
Phase velocity  $v_{\mathrm{pnm}}=\frac{\omega }{\beta }$.

Dielectric attenuation constant ${\alpha }_{dnm}=\frac{k^2\tan \delta }{2\beta }$.
$$\begin{gathered} E_{\mathrm{znm}}=0 \\ {H}_{\mathrm{znm}}=\; (\mathrm{Asinn\phi }+\mathrm{Bcosn\phi }\left)J_n\right(k_c,\rho )e^{–\mathrm{j\beta z}} \\ {E}_{\mathrm{\rho nm}}=-\frac{\mathrm{j\omega \mu }}{{k_c}^2\rho }(\mathrm{Acosn\phi }–\mathrm{Bsinn\phi }) J_n(k_c,\rho )e^{–\mathrm{j\beta z}} \\ {E}_{\mathrm{\phi nm}}=\frac{\mathrm{j\omega \mu }}{k_c}(\mathrm{Acosn\phi }+\mathrm{Bcosn\phi }) J_n‘\; (k_c,\rho )e^{–\mathrm{j\beta z}} \\ {H}_{\mathrm{\rho nm}}=\frac{–\mathrm{j\beta }}{k_c}(\mathrm{Acosn\phi }+\mathrm{Bcosn\phi }) J_n‘\; (k_c,\rho )e^{–\mathrm{j\beta z}} \\ {H}_{\mathrm{\phi nm}}=\; –\frac{\mathrm{j\beta }}{k_{c^2\rho }}(\mathrm{Acosn\phi }–\mathrm{Bsinn\phi }) J_n(k_c,\rho )e^{–\mathrm{j\beta z}} \end{gathered}$$

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 $$\begin{gathered} k_{\mathrm{nm}}=\omega \sqrt{\mathrm{\mu \epsilon }} \\ {k}_{\mathrm{cnm}}=\frac{p_{\mathrm{nm}}}{a} \end{gathered}$$ .
Propagation constant ${\beta }_{\mathrm{nm}}=\sqrt{k^2–\frac{{P_{\mathrm{nm}}}^2}{a^2}}$.
Cut-off wavelength $$\begin{gathered} {\lambda }_{\mathrm{cnm}}=\frac{2\mathrm{\pi a}}{p_{\mathrm{nm}}} \\ {\lambda }_{\mathrm{gnm}}=\frac{2\pi }{\beta } \end{gathered}$$.
Phase velocity $v_{\mathrm{pnm}}=\frac{\omega }{\beta }$.

Dielectric attenuation constant ${\alpha }_{dnm}=\frac{k^2\tan \delta }{2\beta }$.
$$\begin{gathered} E_{\mathrm{znm}}=\; (\mathrm{Asinn\phi }+\mathrm{Bcosn\phi }) J_n(k_c,\rho )e^{–\mathrm{j\beta z}} \\ {H}_{\mathrm{znm}}=\; 0E_{\mathrm{\rho nm}}=-\frac{\mathrm{j\beta }}{k_c}(\mathrm{Acosn\phi }+\mathrm{Bsinn\phi }) J_n‘\; (k_c,\rho )e^{–\mathrm{j\beta z}} \\ {E}_{\mathrm{\phi nm}}=\frac{\mathrm{j\beta n}}{{k_c}^2\rho }(\mathrm{Acosn\phi }–\mathrm{Bcosn\phi }) J_n(k_c,\rho )e^{–\mathrm{j\beta z}} \\ {H}_{\mathrm{\rho nm}}=\frac{–\mathrm{j\epsilon \omega n}}{{k_c}^2\rho }(\mathrm{Acosn\phi }–\mathrm{Bcosn\phi }) J_n(k_c,\rho )e^{–\mathrm{j\beta z}} \\ {H}_{\mathrm{\phi nm}}=\; –\frac{\mathrm{j\omega \epsilon }}{{k_c}^2\rho }(\mathrm{Acosn\phi }+\mathrm{Bsinn\phi }) J_n‘\; (k_c,\rho )e^{–\mathrm{j\beta z}} \end{gathered}$$

Impedance $Z_{TM nm}=\frac{\beta \eta }{k}$.

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.

Coaxial transmission line