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Schraeger_Uebergang.tex
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Schraeger_Uebergang.tex
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\subsubsection{Senkrechte Polarisation ohne $ \mu_r $}
\input{Figures/SchraegerUebergang_senk_Polarisation.tex}
$\vec{E}$-Feld senkrecht, $ \vec{H}$-Feld parallel. \qquad $ \mu_{r1} = \mu_{r2} =1$
%\[ \boxed{\texttt{mit } Z_{F0} = 120\pi \approx 377\si{\ohm}} \]
\begin{flalign*}
Z_{F0} &= 120\pi \,\si{\ohm} &
Z_{F(n)} & = Z_{F0}\cdot\frac{1}{\sqrt{\varepsilon_{r(n)}}}&
\frac{Z_{F1}}{Z_{F2}} & = \frac{\sqrt{\varepsilon_{r2}}}{\sqrt{\varepsilon_{r1}}}&
\end{flalign*}
%\[ n: \texttt{Brechungsindex} \quad ; \quad \theta_h = \theta_r\]
\textbf{Brechungsgesetz}: \qquad mit $ \theta_h = \theta_r\ $
\begin{flalign*}
\Aboxed{\frac{\sin\theta_t}{\sin\theta_h} & = \sqrt{\frac{\varepsilon_{r1}}{\varepsilon_{r2}}}} = \frac{\lambda_2}{\lambda_1}= \frac{\beta_1}{\beta_2}= \frac{n_1}{n_2} &
\sin\theta_t & = \sqrt{\frac{\varepsilon_{r1}}{\varepsilon_{r2}}}\cdot \sin\theta_h
\end{flalign*}
%\begin{align*}
% \dfrac{\sin \vartheta_{2}}{\sin \vartheta_{1}} = \dfrac{k_{h}}{k_{g}} & = \sqrt{\dfrac{\mu_{r 1} \varepsilon_{r 1}}{\mu_{r 2} \varepsilon_{r 2}}} = \dfrac{n_{1}}{n_{2}} = \dfrac{v_{p, 2}}{v_{p, 1}} = \dfrac{\lambda_{2}}{\lambda_{1}} \\
% % \alpha_{Bp} & = \tan^{-1} \left( \sqrt{ \dfrac{\varepsilon_{r1}}{\varepsilon_{r2}}} \right) \\
% % \alpha_{Bs} & = \tan^{-1} \left( \sqrt{ \dfrac{\mu_{r1}}{\mu_{r2}}} \right)
%\end{align*}
%\begin{itemize}
% \item magnetischer/elektrischer Reflexionsfaktor $[1]$
% \item magnetischer Transmissionsfaktor $[1]$
% \item elektrischer Transmissionsfaktor $[1]$
%\end{itemize}
\textbf{Fresnelsche Formeln}: \qquad $ \theta_h = \vartheta_{1} $ \quad $ \theta_t = \vartheta_{2} $
%\begin{equation*}
% \begin{aligned}
\begin{spreadlines}{2ex}
\begin{flalign*}
r_s & = r_{es} = r_{ms} = \\
& = \frac{Z_{F2} \cdot \cos \vartheta_1-Z_{F1} \cdot \cos \vartheta_2}{Z_{F2} \cdot \cos \vartheta_1+Z_{F1} \cdot \cos \vartheta_2}
% & = \frac{\cos\vartheta_1-\sqrt{^{\varepsilon_{r2}}/_{\varepsilon_{r1}}-\sin^2\vartheta_1}}{\cos\vartheta_1+\sqrt{^{\varepsilon_{r2}}/_{\varepsilon_{r1}}-\sin^2\vartheta_1}} \\
= \frac{\sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_1 - \sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_2}{\sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_2 + \sqrt{\varepsilon_{r1}}\cos\vartheta_1} \\
& = \frac{\cos \vartheta_1-\sqrt{\frac{\varepsilon_{r 2}}{\varepsilon_{r 1}}-\sin ^2 \vartheta_1}}{\cos \vartheta_1+\sqrt{\frac{\varepsilon_{r 2}}{\varepsilon_{r 1}}-\sin ^2 \vartheta_1}}
\\
t_{es} & =\frac{2 \cdot Z_{F2} \cdot \cos \vartheta_1}{Z_{F2} \cdot \cos \vartheta_1+Z_{F1} \cdot \cos \theta_2} = \frac{2\cdot\sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_1}{\sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_2 + \sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_h}
\\
& = \frac{2 \cos \vartheta_1}{\cos \vartheta_1+\sqrt{\frac{\varepsilon_{r 2}}{\varepsilon_{r 1}}-\sin ^2 \vartheta_1}}
\\
& = 1+r_s
\\
t_{ms} & = \frac{2 Z_{F1} \cdot \cos \vartheta_1}{Z_{F2} \cdot \cos \vartheta_1+Z_{F1} \cdot \cos \vartheta_2} \\
&= \frac{2 \sqrt{\frac{\varepsilon_{r 2}}{\varepsilon_{r 1}} \cos \vartheta_1}}{\cos \vartheta_1+\sqrt{\frac{\varepsilon_{r 2}}{\varepsilon_{r 1}}-\sin ^2 \vartheta_1}}
\\
% & = (1 - r_s) \cdot \dfrac{\cos \vartheta_1}{\cos \vartheta_2} \\
& = \frac{Z_{F1}}{Z_{F2}}\cdot t_{es} = \sqrt{\frac{\varepsilon_{r2}}{\varepsilon_{r1}}}\cdot t_{es}
\end{flalign*}
\end{spreadlines}
% \end{aligned}
%\end{equation*}
%\begin{align*}
% r_s & = r_{es} = r_{ms} = \\
% & = \frac{Z_{F2} \cdot \cos \theta_h-Z_{F1} \cdot \cos \theta_t}{Z_{F2} \cdot \cos \theta_h+Z_{F1} \cdot \cos \theta_t} \\
% & = \frac{\cos\theta_h-\sqrt{^{\varepsilon_{r2}}/_{\varepsilon_{r1}}-\sin^2\theta_h}}{\cos\theta_h+\sqrt{^{\varepsilon_{r2}}/_{\varepsilon_{r1}}-\sin^2\theta_h}} \\
% & = \frac{\sqrt{\varepsilon_{r1}}\cdot\cos\theta_h - \sqrt{\varepsilon_{r2}}\cdot\cos\theta_t}{\sqrt{\varepsilon_{r2}}\cdot\cos\theta_t + \sqrt{\varepsilon_{r1}}\cos\theta_h} \\
% t_{ms} & = Z_{F1} \cdot \frac{2 \cdot \cos \theta_h}{Z_{F2} \cdot \cos \theta_h+Z_{F1} \cdot \cos \theta_t} \\
% & = (1 - r_s) \cdot \dfrac{\cos \theta_h}{\cos \theta_t} \\
% & = \frac{Z_{F1}}{Z_{F2}}\cdot t_{es} \\
% t_{es} & = Z_{F2} \cdot \frac{2 \cdot \cos \theta_h}{Z_{F2} \cdot \cos \theta_h+Z_{F1} \cdot \cos \theta_t} \\
% & = 1+r_s
%\end{align*}
%\subsubsection*{Beziehungen Polarisation}
\textbf{Beziehungen Polarisation}
\begin{equation*}
\begin{aligned}
E_r & = r_s \cdot E_h \\
E_t & = t_{es} \cdot E_h \\
H_r & = r_s \cdot H_h \\
H_t & = t_{ms} \cdot H_h \\
E_t & = H_t\cdot Z_{F2} \\
E_h & = H_h\cdot Z_{F1}
\end{aligned}
\qquad
\begin{aligned}
E_r & = r_p\cdot E_h \\
E_t & = t_{ep}\cdot E_h \\
H_r & = r_p\cdot H_h \\
H_t & = t_{mp}\cdot H_h \\
E_t & = H_t\cdot Z_{F2} \\
E_h & = H_h\cdot Z_{F1}
\end{aligned}
\end{equation*}
\textbf{Richtungssinn Felder (Hand-Regel)}
\begin{equation*}
\begin{aligned}
& \text{\textbf{Linke Hand}}\\
& \text{Daumen:}\quad \vec{E}\\
& \text{Zeigef.:}\quad \vec{S}_{av}\\
& \text{Mittelf.:}\quad \vec{H}
\end{aligned}
\qquad
\begin{aligned}
& \text{\textbf{Rechte Hand}}\\
& \text{Daumen:}\quad \vec{E}\\
& \text{Zeigef.:}\quad \vec{H}\\
& \text{Mittelf.:}\quad \vec{S}_{av}
\end{aligned}
\end{equation*}
\newcolumn
\subsubsection{Parallele Polarisation ohne $ \mu_r $}
\begin{center}
\input{Figures/SchraegerUebergang_para_Polaristion.tex}
\end{center}
$\vec{E}$-Feld parallel, $ \vec{H}$-Feld senkrecht. \qquad $ \mu_{r1} = \mu_{r2} =1 $\\
Stücke: $\vec{H}_h$ und $ \vec{H}_r$ zeigen in die selbe Richtung!\\
Sattler: $\vec{H}_h$ und $ \vec{H}_r$ zeigen in \textbf{entgegengesetzter} Richtung!\\
%\[ \boxed{\texttt{mit } Z_{F0} = 120\pi \approx 377\si{\ohm}} \]
%\begin{align*}
% Z_{Fn} & = Z_{F0}\cdot\frac{1}{\sqrt{\varepsilon_{rn}}} \\
% \frac{Z_{F1}}{Z_{F2}} & = \frac{\sqrt{\varepsilon_{r2}}}{\sqrt{\varepsilon_{r1}}}
%\end{align*}
%\[ n: \texttt{Brechungsindex} \quad ; \quad \theta_h = \theta_r\]
%\begin{align*}
% \frac{\sin\theta_t}{\sin\theta_h} & = \frac{\lambda_2}{\lambda_1}= \frac{\beta_1}{\beta_2}= \frac{n_1}{n_2} \\
% \sin\theta_t & = \sqrt{\frac{\varepsilon_{r1}}{\varepsilon_{r2}}}\cdot\sin\theta_h
%\end{align*}
%\begin{itemize}
% \item magnetischer/elektrischer Reflexionsfaktor $[1]$
% \item magnetischer Transmissionsfaktor $[1]$
% \item elektrischer Transmissionsfaktor $[1]$
%\end{itemize}
\textbf{Fresnelsche Formeln (Stücke)}: \qquad $ \theta_h = \vartheta_{1} $ \quad $ \theta_t = \vartheta_{2} $
\begin{equation*}
\setlength{\jot}{10pt}
\begin{aligned}
r_{ep} & = r_{mp} = r_{p} \qquad \qquad =-r_{p,[\texttt{Sattler}]}
\\
& = \frac{Z_{F1} \cdot \cos \vartheta_1-Z_{F2} \cdot \cos \vartheta_2}{Z_{F1} \cdot \cos \vartheta_1+Z_{F2} \cdot \cos \vartheta_2}
\\
% & = - \left( \frac{\varepsilon_{r2}\cos\vartheta_1-\sqrt{\varepsilon_{r2}\varepsilon_{r1}-{\varepsilon_{r1}}^2\sin^2\vartheta_1}}{\varepsilon_{r2}\cos\vartheta_1+\sqrt{{\varepsilon_{r2}\varepsilon_{r1}-{\varepsilon_{r1}}^2\sin^2\vartheta_1}}} \right)
% \\
& =\frac{\cos \vartheta_1-\sqrt{\frac{\varepsilon_{r 1}}{\varepsilon_{r 2}}-\frac{\varepsilon_{r 1}{ }^2}{\varepsilon_{r 2}{ }^2} \sin ^2 \vartheta_1}}{\cos \vartheta_1+\sqrt{\frac{\varepsilon_{r 1}}{\varepsilon_{r 2}}-\frac{\varepsilon_{r 1}{ }^2}{\varepsilon_{r 2}{ }^2} \sin ^2 \vartheta_1}} \\
t_{ep} & = \frac{2 \cdot Z_{F2} \cdot \cos \vartheta_1}{Z_{F1} \cdot \cos \vartheta_1+Z_{F2} \cdot \cos \vartheta_2} = (1-r_p) \cdot \dfrac{\cos \vartheta_1}{\cos \vartheta_2} \\
& = \frac{2 \sqrt{\frac{\varepsilon_{r 1}}{\varepsilon_{r 2}}} \cos \vartheta_1}{\cos \vartheta_1+\sqrt{\frac{\varepsilon_{r 1}}{\varepsilon_{r 2}}-\frac{\varepsilon_{r 1}^2}{\varepsilon_{r 2}{ }^2} \sin ^2 \vartheta_1}} \\
& = \frac{Z_{F2}}{Z_{F1}}\cdot t_{mp} = \sqrt{\frac{\varepsilon_{r1}}{\varepsilon_{r2}}}\cdot t_{mp} \\
t_{mp} & = \frac{2 \cdot Z_{F1}\cdot \cos \vartheta_1}{Z_{F1} \cdot \cos \vartheta_1+Z_{F2} \cdot \cos \vartheta_2}
= 1+r_p
\end{aligned}
\end{equation*}
%\begin{align*}
% r_p & = r_{ep} = r_{mp} = \\
% & = \frac{Z_{F1} \cdot \cos \theta_t-Z_{F2} \cdot \cos \theta_h}{Z_{F2} \cdot \cos \theta_t+Z_{F1} \cdot \cos \theta_h} = \\
% & = \frac{\varepsilon_{r2}\cos\theta_h-\sqrt{\varepsilon_{r2}\varepsilon_{r1}-{\varepsilon_{r1}}^2\sin^2\theta_h}}{\varepsilon_{r2}\cos\theta_h+\sqrt{{\varepsilon_{r2}\varepsilon_{r1}-{\varepsilon_{r1}}^2\sin^2\theta_h}}} \\
% t_{ep} & = Z_{F2} \cdot \frac{2 \cdot \cos \theta_h}{Z_{F1} \cdot \cos \theta_h+Z_{F2} \cdot \cos \theta_t} \\
% & = (1-r_p) \cdot \dfrac{\cos \theta_h}{\cos \theta_t} \\
% & = \frac{Z_{F2}}{Z_{F1}}\cdot t_{mp}\\
% t_{mp} & = \frac{2 Z_{F1}\cdot \cos \theta_h}{Z_{F1} \cdot \cos \theta_h+Z_{F2} \cdot \cos \theta_t} \\
% & = 1+r_p
%\end{align*}
\textbf{Fresnelsche Formeln (Sattler)}:
\begin{equation*}
\setlength{\jot}{10pt}
\begin{aligned}
r_p & = r_{ep} = r_{mp} \qquad \qquad =-r_{p,[\texttt{Stücke}]} \\
& = \frac{Z_{F2} \cdot \cos \vartheta_2-Z_{F1} \cdot \cos \vartheta_1}{Z_{F2} \cdot \cos \vartheta_2+Z_{F1} \cdot \cos \vartheta_1} \\
& = \frac{\sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_2 - \sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_1}{\sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_1 + \sqrt{\varepsilon_{r1}}\cos\vartheta_2} \\
t_{ep} & = \frac{2 Z_{F2} \cdot \cos \vartheta_1}{Z_{F1} \cdot \cos \vartheta_1+Z_{F2} \cdot \cos \vartheta_2} \\
& = \frac{2\cdot\sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_1}{\sqrt{\varepsilon_{r2}}\cdot\cos\vartheta_1 + \sqrt{\varepsilon_{r1}}\cdot\cos\vartheta_2}\\
& = (1+r_p) \cdot \dfrac{\cos \vartheta_1}{\cos \vartheta_2}\\
t_{mp}
% &
% = \frac{2 Z_{F1}\cdot \cos \vartheta_h}{Z_{F1} \cdot \cos \vartheta_h+Z_{F2} \cdot \cos \vartheta_t} \\
& = 1-r_p = \frac{Z_{F1}}{Z_{F2}}\cdot t_{ep}
\end{aligned}
\end{equation*}
%\begin{align*}
% E_r & = r_p\cdot E_h \\
% E_t & = t_{ep}\cdot E_h \\
% H_r & = r_p\cdot H_h \\
% H_t & = t_{mp}\cdot H_h \\
% E_t & = H_t\cdot Z_{F2} \\
% E_h & = H_h\cdot Z_{F1}
%\end{align*}