\documentclass[a4paper,12pt]{article}
\usepackage [czech]{babel}
\usepackage {labak}

% klystron
\def\kl#1{klys\-tr\'{o}\-n#1}

\begin{document}
\header{Meranie vlastnost\'{\i} smerov\'{y}ch odbo\v{c}n\'{\i}c}

\podnadpis{\'{U}lohy}
%% Ulohy: {{{

\begin{enumerate}
	\item Ur\v{c}te absol\'{u}tne hodnoty v\v{s}etk\'{y}ch koeficientov
		$S$ matice pre re\'{a}lnu smerov\'{u} odbo\v{c}nicu.
	\item Ur\v{c}te bo\v{c}n\'{y} \'{u}tlm a smerovos\q t
		odbo\v{c}nice.

\end{enumerate}
% }}}

\podpodnadpis{Potrebn\'{e} pom\^{o}cky a pr\'{\i}stroje}
% pomocky {{{

Reflexn\'{y} \kl{},
dr\v{z}iak \kl{u},
stabilizovan\'{y} zdroj nap\"{a}t\'{\i} pre \kl{},
feritov\'{y} izol\'{a}tor,
smerov\'{a} odbo\v{c}nica,
dr\v{z}iak di\'{o}dy s~detektorom,
kalibrovan\'{y} atenu\'{a}tor,
meracie vedenie,
impedan\v{c}n\'{y} jednokol\'{\i}kov\'{y} transform\'{a}tor,
prisp\^{o}soben\'{a} z\'{a}\q ta\v{z} 2 ks,
galvanometer 2 ks
% }}} pomocky


\podnadpis{Vypracovanie}

Smerov\'{a} odbo\v{c}nica predstavuje mikrovlnn\'{y} osmip\'{o}l,
vi\v{d} obr. \ref{odbocnica}. Vlastnosti tak\'{e}hoto osmip\'{o}lu
mo\v{z}no pop\'{\i}sa\q t pomocou matice rozptylu $S$
\begin{equation}
	||S|| =  \left| \matrix{
	S_{11} & S_{12} &  S_{13} & S_{14} \cr
	S_{21} & S_{22} &  S_{23} & S_{24} \cr
	S_{31} & S_{32} &  S_{33} & S_{34} \cr
	S_{41} & S_{42} &  S_{43} & S_{44} \cr
	} \right|
	\label{rov1}
\end{equation}
Po uv\'{a}\v{z}en\'{\i} podmienok, \v{z}e k~ramen\'{a}m s\'{u}
prisp\^{o}soben\'{e} z\'{a}\q ta\v{z}e a~\v{z}e matica je symetrick\'{a}
a unit\'{a}rna (izotropnos\q t a bezstratovos\q t dielektrika) mo\v{z}no
maticu previes\q t do tvaru
\begin{equation}
	||S|| =  \left| \matrix{
	0  &  p  &  0  &  jq \cr
	p  &  0  &  jq &  0  \cr
	0  &  jq &  0  &  p  \cr
	jq &  0  &  p  &  0  \cr
	} \right|
	\label{rov2}
\end{equation}
pri\v{c}om $p^2 + q^2 = 1$.

Pomocou tejto matice mo\v{z}no uk\'{a}za\q t, \v{z}e nevyhnutnou
a~posta\v{c}uj\'{u}cou podmienkou na to, aby \q lubovo\q ln\'{y}
bezstratov\'{y} osmip\'{o}l mohol by\q t pokladan\'{y} za smerov\'{u}
odbo\v{c}nicu je, aby bol prisp\^{o}soben\'{y} zo strany ka\v{z}d\'{e}ho
ramena.

\begin{figure}[!hbt]
	\begin{center}
		\includegraphics [angle=0, width=0.8\textwidth]{odbocnica.eps}
		\caption{}
		\label{odbocnica}
	\end{center}
\end{figure}

Vlastnosti smerovej odbo\v{c}nice sa v praxi mierne l\'{\i}\v{s}ia od
vlastnost\'{\i} ide\'{a}l\-nej smerovej odbo\v{c}nice. Dov\^{o}dom s\'{u}
najm\"{a} nenulov\'{e} odrazy v pr\'{\i}slu\v{s}n\'{y}ch ramen\'{a}ch,
\v{c}o znamen\'{a}, \v{z}e koeficienty $S_{ii}$ s\'{u} nenulov\'{e}
a~\v{z}e existuje ist\'{a} v\"{a}zba medzi ramenami 1~a~3 a~taktie\v{z}
aj medzi ramenami 2~a~4. Vlastnosti smerovej odbo\v{c}nice sme
sk\'{u}mali na meracej aparat\'{u}re v zapojen\'{\i} pod\q la sch\'{e}my
na obr\'{a}zku \ref{schema}.

\begin{figure}[!hbt]
	\begin{center}
		\includegraphics [angle=0, width=0.8\textwidth]{schema.eps}
		\caption{}
		\label{schema}
	\end{center}
\end{figure}

Koeficienty matice $S_{ii}$ s\'{u} koeficienty odrazu v ramene $i$
(ke\v{d} s\'{u} ostatn\'{e} ramen\'{a} pripojen\'{e} na
prisp\^{o}soben\'{e} z\'{a}\q ta\v{z}e), tak\v{z}e absol\'{u}tnu hodnotu
koe\-fi\-cien\-tu $S_{ii}$ mo\v{z}no ur\v{c}i\q t meran\'{\i}m PSV v ramene
$i$ merac\'{\i}m veden\'{\i}m a~prepo\v{c}tom pod\q la vz\q tahu
\begin{equation}
	S_{ii} = |\varrho_i| = \frac{r_i -1 }{r_i + 1}
	\label{rov3}
\end{equation}
kde $r_i$ je PSV v ramene $i$.

Koeficienty matice $S_{12}$ a~$S_{14}$ mo\v{z}no ur\v{c}i\q t
meran\'{\i}m v\'{y}konov $P_2$ a~$P_4$ v ramen\'{a}ch 2~a~4 pri
zn\'{a}mom vstup v\'{y}kone $P_1$ a pri za\q ta\v{z}en\'{\i}
ostatn\'{y}ch ramien prisp\^{o}soben\'{y}mi z\'{a}\q ta\v{z}ami.
Pre koeficienty potom platia vz\q tahy
\begin{eqnarray}
	|S_{12}| & = & \sqrt{\frac{P_2}{P_1}}
	\label{rov4} \\
	|S_{14}| & = & \sqrt{\frac{P_4}{P_1}}
	\label{rov5}
\end{eqnarray}

Koeficient $S_{13}$ by pre ide\'{a}lnu smerov\'{u} odbo\v{c}nicu mal
by\q t nulov\'{y}, ale v~na\v{s}om pr\'{\i}pade ho ur\v{c}\'{\i}me
taktie\v{z} z~merania v\'{y}konov v ramen\'{a}ch 1~a~3, tak\v{z}e pre
koeficient $S_{13}$ potom plat\'{\i} vz\q tah
\begin{equation}
	|S_{13}| = \sqrt{\frac{P_3}{P_1}} \\
	\label{rov6}
\end{equation}

Pre koeficienty $S_{21}$ a~$S_{23}$ platia vz\q tahy
\begin{eqnarray}
	|S_{21}| & = & \sqrt{\frac{P_1'}{P_2'}}
	\label{rov7} \\
	|S_{23}| & = & \sqrt{\frac{P_3'}{P_2'}}
	\label{rov8}
\end{eqnarray}
kde $P_2'$ je v\'{y}kon vstupuj\'{u}ci do ramena 2~a~$P_1', P_3'$
v\'{y}kony vystupuj\'{u}ce ramenami 1~a~3. Ak $S_{ii} \neq 0$, vz\q tahy
\rov{4}, \rov{5}, \rov{6}, \rov{7} a~\rov{8} platia iba pribli\v{z}ne.

Smerovos\q t odbo\v{c}nice $D$ sa definuje ako
\begin{equation}
	D \doteq 20 \log \frac{|S_{14}|}{|S_{13}|} = 20 \log \sqrt{\frac{P_4}{P_3}}
	\label{rov9}
\end{equation}
a odbo\v{c}n\'{y} \'{u}tlm
\begin{equation}
	C =  -20 \log |S_{14}| = -20 \log \sqrt{\frac{P_4}{P_1}}
	\label{rov10}
\end{equation}
pre v\'{y}kon $P_1$ vstupuj\'{u}ci do ramena 1~a~pre
prisp\^{o}soben\'{e} ostatn\'{e} ramen\'{a}.

\noindent Nami nameran\'{e} hodnoty s\'{u} n\'{a}sledovn\'{e}:
%\begin{tabular}[t]{r@{=}l}
%$|S_{11}|$ & $0.0145919$ \\
%$|S_{22}|$ & $0.011027$
%\end{tabular}
%\newline
\begin{equation}
	||S|| =  \left| \matrix{
	S_{11}=0.015   & S_{12}=0.977 & S_{13}=0.110  & S_{14}=9.441  \cr
	S_{21}         & S_{22}=0.011 & S_{23}        & S_{24}=0.110  \cr
	S_{31}         & S_{32}       & S_{33}        & S_{34}        \cr
    S_{41}         & S_{42}       & S_{43}        & S_{44}=0.015  \cr
	} \right|
	\label{rov11}
\end{equation}
Odbo\v{c}n\'{y} \'{u}tlm je $C =  -20 \log |S_{14}| = 19.5$
\newline
Smerovos\q t je $D = 20 \log \frac{|S_{14}|}{|S_{13}|} = 38.67$


\end{document}

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