Difference between revisions of "Waveguides"

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2D and 3D models of metallic waveguides
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Introduction

$\rightarrow$ To run the model, open main.pro with Gmsh.

Description of the model

Classical waveguides

General waveguide

Let us consider a hollow cylindrical waveguide of arbitrary cross-sectional shape that has a principal axis in the $z$-direction. The elementary solution of this problem reads \begin{align} {\bf E}(x,y,z,t) &= {\bf E}(x,y) \: e^{i(\pm kz-\omega t)} \\ {\bf H}(x,y,z,t) &= {\bf H}(x,y) \: e^{i(\pm kz-\omega t)} \end{align} where the new unknowns are governed by \begin{align} \left[\nabla_t^2 + (\mu\varepsilon\omega^2 - k^2)\right] \left\{\begin{array}{x}{\bf E}\\{\bf H}\end{array}\right\} = 0 \end{align} where $\nabla_t$ is the transverse part of the Nabla operator. Parallel and transverse fields It is useful to separate the fields into components parallel to and transverse to the $z$-direction: \begin{align} {\bf E} &= {\bf E}_z + {\bf E}_t && \text{with } {\bf E}_z = {E}_z \hat{\bf z} \\ {\bf H} &={\bf H}_z + {\bf H}_t && \text{with } {\bf H}_z = {H}_z \hat{\bf z} \end{align} Some well-known cases: Transverse electromagnetic (TEM) waves: if ${E}_z=0$ and ${H}_z=0$ everywhere Transverse magnetic (TM) waves: if ${H}_z=0$ everywhere Transverse electric (TE) waves: if ${E}_z=0$ everywhere If both parallel fields are vanishing (TEM case), the transverse fields are the solution of an electrostatic problem in two dimensions. If at least one parallel field is non-vanishing, the transverse fields are \begin{align} {\bf E}_t &= \frac{i}{\mu\varepsilon\omega^2-k^2} \left[\pm\:k\:\nabla_t{E}_z - \mu\omega\:\hat{\bf z}\times\nabla_t{H}_z\right] \\ {\bf H}_t &= \frac{i}{\mu\varepsilon\omega^2-k^2} \left[\pm\:k\:\nabla_t{H}_z + \varepsilon\omega\:\hat{\bf z}\times\nabla_t{E}_z\right] \end{align} In TEM, TM and TE cases, the transverse fields are related by \begin{equation} {\bf H}_t = \pm\frac{1}{Z} \hat{\bf z}\times{\bf E}_t \end{equation} where the wave impedance $Z$ is given by \begin{equation} Z= \left\{\begin{array}{ll} \sqrt{\frac{\mu}{\varepsilon}} &\quad \text{(TEM case)} \\ \frac{k}{k_0} \sqrt{\frac{\mu}{\varepsilon}} &\quad \text{(TM case)} \\ \frac{k_0}{k} \sqrt{\frac{\mu}{\varepsilon}} &\quad \text{(TE case)} \end{array}\right. \end{equation} with $k_0=\omega\sqrt{\mu\varepsilon}$. Eigenvalue problem For a waveguide with perfectly conducting borders, the non-vanishing parallel field of TM and TE cases is governed by, respectively, \begin{align} \left[\nabla_t^2 + \gamma^2\right] {E}_z &= 0 \\ \left[\nabla_t^2 + \gamma^2\right] {H}_z &= 0 \end{align} with $\gamma^2 = \mu\varepsilon\omega^2 - k^2$, and is subject to the homogeneous boundary condition ${E}_z=0$ (TM case) or ${\bf n}\cdot\nabla{H}_z = 0$ (TE case). These equations define eigenvalue problems. There is a spectrum of eigenvalues $\gamma^2_\ell$ and corresponding solutions $\left.E_z\right|_\ell$ or $\left.H_z\right|_\ell$, $\ell=1,2,...$, which form an orthogonal set. For a given frequency $\omega$, the wave number $k$ is determined for each $\ell$: \begin{equation} k_\ell = \sqrt{\mu\varepsilon\omega^2-\gamma^2_\ell} = \sqrt{\mu\varepsilon} \sqrt{\omega^2-\omega^2_\ell} \end{equation} where $\omega_\ell$ is the cutoff frequency, defined by \begin{equation} \omega_\ell=\frac{\gamma_\ell}{\sqrt{\mu\varepsilon}} \end{equation} This frequency defines the nature of waves: If $\omega>\omega_\ell$, $k_\ell$ is real and the waves are travelling modes. If $\omega<\omega_\ell$, $k_\ell$ is imaginary and the waves are evanescent modes.

Rectangular waveguide

Let us consider a rectangular waveguide $(x,y)\in[0,a]\times[0,b]$ that has a principal axis in the $z-$direction. For TM modes, the solutions for $E_z$ are \begin{align} \left.E_z\right|_{mn} &= E_0 \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) e^{ikz-i\omega t} && \text{with } m,n=0,1,2,... \end{align} For TE modes, the solutions for $H_z$ are \begin{align} \left.H_z\right|_{mn} &= H_0 \cos\left(\frac{m\pi x}{a}\right) \cos\left(\frac{n\pi y}{b}\right) e^{ikz-i\omega t} && \text{with } m,n=0,1,2,... \end{align} In both cases, the eigenvalues and the cutoff frequencies are, respectively, \begin{align} \gamma_{mn}^2 &= \pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right) && \text{with } m,n=0,1,2,... \\ \omega_{mn}^2 &= \frac{\pi}{\sqrt{\mu\varepsilon}}\sqrt{\frac{m^2}{a^2}+\frac{n^2}{b^2}} && \text{with } m,n=0,1,2,... \end{align} The complete solution for the TM10 and TE10 modes are, respectively, \begin{align} \begin{cases} \displaystyle E_z = E_0 \sin\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \\ \displaystyle E_x = i\frac{ka}{\pi} E_0 \cos\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \\ \displaystyle H_y = i\frac{\varepsilon\omega a}{\pi} E_0 \cos\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \end{cases} \quad\quad\text{and}\quad\quad \begin{cases} \displaystyle H_z = H_0 \cos\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \\ \displaystyle H_x = -i\frac{ka}{\pi} H_0 \sin\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \\ \displaystyle E_y =i\frac{\mu\omega a}{\pi} H_0 \sin\left(\frac{\pi x}{a}\right) e^{ikz-i\omega t} \end{cases} \end{align}

Discontinuities and networks

Discontinuity in a parallel-plate waveguide

See [1], section 4.6.1. Solution for the TE mode \begin{align} H_z &= H_0 e^{-jk_0 x} + R H_0 e^{jk_0 x} && \text{at } x=x_1 \\ H_z &= T H_0 e^{-jk_0 x} && \text{at } x=x_2 \end{align} Boundary conditions \begin{align} \partial_x H_z &= jk_0 H_z - 2jk_0H_0 e^{-jk_0 x} && \text{at } x=x_1 \\ \partial_x H_z &= -jk_0 H_z && \text{at } x=x_2 \end{align} Reflection and transmission coefficients \begin{align} R &= \left.\frac{H_z - H_0 e^{-jk_0x}}{H_0 e^{ jk_0 x}}\right|_{x=x_1} \\ T &= \left.\frac{H_z}{H_0 e^{-jk_0 x}}\right|_{x=x_2} \end{align} with $|R|^2+|T|^2=1$.

Waveguide with discontinuities

See [1], section 8.5. Solution for the TE$_{mn}$ mode \begin{align} {\bf E}(x,y,z) &= E_0 {\bf e}_{mn}(x,y) e^{-jk_{z_{mn}} z} + R E_0 {\bf e}_{mn}(x,y) e^{jk_{z_{mn}} z} && \text{at } z=z_1 \\ {\bf E}(x,y,z) &= T E_0 {\bf e}_{mn}(x,y) e^{-jk_{z_{mn}} z} && \text{at } z=z_2 \end{align} Reflection and transmission coefficients \begin{align} R &= \left.\frac{{\bf E}(x,y,z) - E_0 {\bf e}_{mn}(x,y) e^{-jk_{z_{10}}z}}{E_0 {\bf e}_{mn}(x,y) e^{jk_{z_{10}}z}}\right|_{z=z_1} \\ T &= \left.\frac{{\bf E}(x,y,z)}{E_0 {\bf e}_{mn}(x,y) e^{-jk_{z_{10}}z}}\right|_{z=z_2} \end{align} with $|R|^2+|T|^2=1$. Reflection and transmission coefficients (improved - the dominant mode $mn$ - using the orthogonality properties of modes) \begin{align} R &= \frac{e^{-jk_{z_{mn}}z_1}}{E_0} \frac{\int_{S_1}{\bf E}(x,y,z)\cdot{\bf e}_{mn}(x,y) \: dS}{\int_{S_1}{\bf e}_{mn}(x,y)\cdot{\bf e}_{mn}(x,y) \: dS} - e^{-2jk_{z_{mn}}z_1}\\ T &= \frac{e^{jk_{z_{mn}}z_2}}{E_0} \frac{\int_{S_2}{\bf E}(x,y,z)\cdot{\bf e}_{mn}(x,y) \: dS}{\int_{S_2}{\bf e}_{mn}(x,y)\cdot{\bf e}_{mn}(x,y) \: dS} \end{align} with $|R|^2+|T|^2=1$.

Some results

Here will be some snapshots and validation cases.

References

1. ↑ ^{1.0 1.1} J. Jin, The Finite Element Method in Electromagnetics. Second edition. John Wiley & Sons, 2002