Wave Guide
Parallel Plate Waveguide¶
TE Waves¶
\[
\begin{gather}
\vec E = E_0 \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_y}
\\\\
\vec H = - \frac{E_0}{\eta}\sin{\theta} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_x}
+\frac{E_0}{\eta}\cos{\theta} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_z}
\end{gather}
\]
- cutoff wavelength
\[
\begin{gather}
a = m \, \frac{\lambda_x}{2}=\frac{m\,\displaystyle \frac{\lambda}{\cos\theta}}{2}
\\\\
\implies \lambda =\frac{2a}{m}\cos\theta
\\\\
\implies
\lambda_c = \big[\lambda\big]_{\cos\theta=1} =\frac{2a}{m}
\\\\
\implies
\cos\theta = \frac{\lambda }{\lambda_c}
\end{gather}
\]
- guide wavelength (in \(z\) direction)
\[
\begin{gather}
\lambda_g = \frac{2\pi}{k_z}=\frac{2\pi}{k\sin\theta}=\frac{\lambda}{\sin\theta}
\\\\
\implies
\sin\theta=\frac{\lambda}{\lambda_g}
\end{gather}
\]
by the \(\lambda_c\) and \(\lambda_g\) we can also write the equations as
\[
\begin{gather}
\vec E = E_0 \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_y}
\\\\
\vec H = - \frac{E_0}{\eta}\frac{\lambda}{\lambda_g} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_x}
+\frac{E_0}{\eta}\frac{\lambda}{\lambda_c} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_z}
\end{gather}
\]
TM Waves¶
\[
\begin{gather}
\vec H = H_0 \cos{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_y}
\\
\end{gather}
\]
\[
\begin{align}
\vec E &= \eta H_0\sin{\theta} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_x}
+ \eta H_0\cos{\theta} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_z}
\\\\
&= \eta H_0\frac{\lambda}{\lambda_g} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_x}
+ \eta H_0\frac{\lambda }{\lambda_c} \sin{(k_x x)}\sin{(\omega t - k_z z)} \,\hat{a_z}
\end{align}
\]
Rectangular Waveguide¶
- \(\gamma\)
\[
\begin{gather}
k^{2} = k_x^{2} + k_y^{2} + k_z^{2} = k_x^{2} + k_y^{2} + (-j\gamma)^{2}
\\\\
\implies
\gamma^{2} = k_t^{2} - k^{2}
\end{gather}
\]
TM mode¶
tip
notice that we assuming perfect conductors.
thus,
\[
\begin{gather}
\vec E_n = 0
\\\\
\implies E_{zs}=0, \qquad\text{at boundary}
\end{gather}
\]
\[
\begin{gather}
E_{zs} = E_0\sin{\left(\frac{m\pi x}{a}\right)}\sin{\left(\frac{n\pi y}{b}\right)}\,e^{-\gamma z}
\end{gather}
\]
TE mode¶
\[
\begin{gather}
H_{zs} = H_0 \cos{\left(\frac{m\pi x}{a}\right)}\cos{\left(\frac{n\pi y}{b}\right)}\,e^{-\gamma z}
\end{gather}
\]
Resonators¶
- TM
\[
\begin{align}
E_{zs} &= \frac{E_0}{2}\sin{\left(\frac{m\pi x}{a}\right)}\sin{\left(\frac{n\pi y}{b}\right)}\left(e^{-jk_z z}+e^{jk_z z}\right)
\\\\
&= E_0\sin{\left(\frac{m\pi x}{a}\right)}\sin{\left(\frac{n\pi y}{b}\right)}\cos{\left(\frac{p \pi z}{d}\right)}
\end{align}
\]
- TE
\[
\begin{align}
H_{zs} &= H_0\cos{\left(\frac{m\pi x}{a}\right)}\cos{\left(\frac{n\pi y}{b}\right)}\sin{\left(\frac{p\pi z}{d}\right)}
\end{align}
\]
Q Factors¶
\[
\begin{gather}
Q = 2\pi\frac{W}{P_L\,T}= \frac{\omega W}{P_L}
\end{gather}
\]
recall
\[
\begin{gather}
Q = \frac{X}{R} = \frac{\omega L}{R} = \frac{\omega I^{2}L}{I^{2}R} = \frac{\omega W}{P}
\end{gather}
\]
Dominant Mode¶
- the mode with lowest \(f_c\) (i.e. largest \(\lambda_c\)).
Cylindrical¶
- \(\beta_c\)
\[
\begin{gather}
k^{2} = \beta_c^{2} + \beta^{2}
\end{gather}
\]
TM mode¶
\[
\begin{gather}
E_{zs} = J_n\big[\beta_c \rho\big] \,(A_n \cos{n\phi} + B_n\sin{n\phi})e^{\mp j\beta z}
\end{gather}
\]
TE mode¶
\[
\begin{gather}
H_{zs} = J_n\big[\beta_c \rho\big](A_n\cos {n\phi} + B_n \sin{n\phi})e^{\mp j \beta z}
\end{gather}
\]
Dielectric Slab Waveguide¶
- cutoff wavelength
\[
\begin{gather}
\lambda_c = \frac{2d\sqrt{\varepsilon_{r1} - \varepsilon_{r2}}}{m}\,\qquad m=0, 1, 2,\dots
\end{gather}
\]
thinking
\[
\begin{gather}
k\cdot 2d \cdot \cos{\theta_i} - 2\angle{\Gamma} = 2m\pi
\\\\
\implies
\frac{2\pi\sqrt{\varepsilon_{r1}}}{\lambda_0}\cdot d \cdot \cos{\theta_i} - \angle{\Gamma} = m\pi
\\\\
\implies
\angle{\Gamma} =
k\cdot d \cdot \cos{\theta_i} - m\pi
\end{gather}
\]
when cutoff \(\angle{\Gamma} = 0\)
\[
\begin{gather}
k_c \,d\cos{\theta_i} = m\pi
\\\\
\implies
\frac{2\pi\sqrt{\varepsilon_{r1}}}{\lambda_c}\,d\cos{\theta_i} = m\pi
\\\\
\frac{2\pi\sqrt{\varepsilon_{r1}}}{\lambda_c}\,d\sqrt{1-\frac{\varepsilon_2}{\varepsilon_1}} = m\pi
\\\\
\implies
\lambda_c = \frac{2d\sqrt{\varepsilon_{r1} - \varepsilon_{r2}}}{m}
\end{gather}
\]