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\begin{document}

\section*{Upravljanje konvertorima}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{buck.ps}  
	\caption{Buck konvertor.}
\end{figure}

\[ \overline{i_S} = \left< i_S \right> = d \, i_L \]

\[ \overline{v_D} = \left< v_D \right> = d \, v_{IN} \]

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{usrednjen-buck.ps}  
	\caption{Usrednjen model buck konvertora.}
\end{figure}

Sistem jednačina stanja

\[ \dfrac{d i_L}{dt} = -\dfrac{1}{L} \, v_{OUT} + \dfrac{1}{L} \, d \, v_{IN} \]

\[ \dfrac{d v_{OUT}}{dt} = \dfrac{1}{C} \, i_L - \dfrac{1}{C} \, i_{OUT} \]
jednačina izlaza (mada je u pitanju ulaz, ali je u sistemu jednačina stanja $i_{IN}$ izlazna promenljiva)

\[ i_{IN} = d \, i_L \]

Nelinearnosti $d \, v_{IN}$ i $d \, i_L$

Mali signali, vrednost signala u radnoj tački i odstupanje

\[ d = D + \widehat{d} \]

\[ v_{IN} = V_{IN} + \widehat{v}_{IN} \]

\[ v_{OUT} = V_{OUT} + \widehat{v}_{OUT} \]

\[ i_L = I_L + \widehat{i}_L \]

\[ i_C = I_C + \widehat{i}_C = \widehat{i}_C \]

\[ i_{OUT} = I_{OUT} + \widehat{i}_{OUT} \]

\[ i_{IN} = I_{IN} + \widehat{i}_{IN} \]

Ustaljeno stanje

\[ \dfrac{d i_L}{dt} = 0 \]

\[ \dfrac{d v_{OUT}}{dt} = 0 \]

\[ 0 = -V_{OUT} + D \, V_{IN} \]

\[ 0 = I_L - I_{OUT} \]

\[ V_{OUT} = D \, V_{IN} \]

\[ I_{OUT} = I_L \]

Linearizacija

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{linearizovan.ps}  
	\caption{Linearizovan model buck konvertora.}
\end{figure}

\[ d \, v_{IN} = (D+\widehat{d}\,) ( V_{IN} + \widehat{v}_{IN} ) = D \, V_{IN} + V_{IN} \, \widehat{d} + D \, \widehat{v}_{IN} + \widehat{d} \; \widehat{v}_{IN} \]

\[ d \, i_{L} = (D+\widehat{d}\,) ( I_{L} + \widehat{i}_{L} ) = D \, I_{L} + I_{L} \, \widehat{d} + D \, \widehat{i}_{L} + \widehat{d} \; \widehat{i}_{L} \]

\[ \widehat{d} \; \widehat{v}_{IN} \approx 0 \]

\[ \widehat{d} \; \widehat{i}_{L} \approx 0 \]

\[ d \, v_{IN} \approx D \, V_{IN} + V_{IN} \, \widehat{d} + D \, \widehat{v}_{IN} \]

\[ d \, i_{L} \approx D \, I_{L} + I_{L} \, \widehat{d} + D \, \widehat{i}_{L} \]
linearizovan sistem jednačina stanja
\[ \dfrac{d i_L}{dt} = -\dfrac{1}{L} \, v_{OUT} + \dfrac{1}{L} \, D \, V_{IN} + \dfrac{1}{L} \, V_{IN} \, \widehat{d} + \dfrac{1}{L} \, D \, \widehat{v}_{IN} \]

\[ \dfrac{d v_{OUT}}{dt} = \dfrac{1}{C} \, i_L - \dfrac{1}{C} \, i_{OUT} \]

\[ i_{IN} = D \, I_{L} + I_{L} \, \widehat{d} + D \, \widehat{i}_{L} \]
ili
\[ \dfrac{d i_L}{dt} = -\dfrac{1}{L} \, v_{OUT} + \dfrac{1}{L} \, D \, v_{IN} + \dfrac{1}{L} \, V_{IN} \, \widehat{d} \]

\[ \dfrac{d v_{OUT}}{dt} = \dfrac{1}{C} \, i_L - \dfrac{1}{C} \, i_{OUT} \]

\[ i_{IN} = D \, i_{L} + I_{L} \, \widehat{d} \]
odnosno
\[ \dfrac{d (I_L+\widehat{i}_L)}{dt} = -\dfrac{1}{L} \, V_{OUT} -\dfrac{1}{L} \, \widehat{v}_{OUT} + \dfrac{1}{L} \, D \, V_{IN} + \dfrac{1}{L} \, D \, \widehat{v}_{IN} + \dfrac{1}{L} \, V_{IN} \, \widehat{d} \]

\[ \dfrac{d (V_{OUT}+\widehat{v}_{OUT})}{dt} = \dfrac{1}{C} \, I_L + \dfrac{1}{C} \, \widehat{i}_L- \dfrac{1}{C} \, I_{OUT} - \dfrac{1}{C} \, \widehat{i}_{OUT} \]

\[ I_{IN} + \widehat{i}_{IN} = D \, I_{L} + D \, \widehat{i}_{L} + I_{L} \, \widehat{d} \]

Primenom teoreme superpozicije

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{linearizovan-ac.ps}  
	\caption{Linearizovan model za male signale buck konvertora.}
\end{figure}

\[ \dfrac{d \widehat{i}_L}{dt} = -\dfrac{1}{L} \, \widehat{v}_{OUT} + \dfrac{1}{L} \, D \, \widehat{v}_{IN} + \dfrac{1}{L} \, V_{IN} \, \widehat{d} \]

\[ \dfrac{d \widehat{v}_{OUT}}{dt} = \dfrac{1}{C} \, \widehat{i}_L - \dfrac{1}{C} \, \widehat{i}_{OUT} \]
jednačina izlaza
\[ \widehat{i}_{IN} = D \, \widehat{i}_{L} + I_{L} \, \widehat{d} \]
sada je sistem linearan

Prenosne funkcije:

\[ H_1(s)=\dfrac{\widehat{v}_{OUT}(s)}{\widehat{d}(s)} = \dfrac{V_{IN}}{1+s^2LC} \]

\[ H_2(s)=\dfrac{\widehat{v}_{OUT}(s)}{\widehat{v}_{IN}(s)} = \dfrac{D}{1+s^2LC} \]

\[ H_3(s)=\dfrac{\widehat{v}_{OUT}(s)}{\widehat{i}_{OUT}(s)} = - \dfrac{sL}{1+s^2LC} \]


\end{document}