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\begin{document}

\section*{Trofazni ispravljači}

\[ v_1 = V_m \; \sin \left( \omega_0 t - \theta \right) \]

\[ v_2 = V_m \; \sin \left( \omega_0 t - \dfrac{2 \pi}{3} - \theta \right) \]

\[ v_3 = V_m \; \sin \left( \omega_0 t - \dfrac{4 \pi}{3} - \theta \right) \]

\[ v_k = V_m \; \sin \left( \omega_0 t - (k-1) \dfrac{4 \pi}{3} - \theta \right) \]

$k \in \left\{ 1,2,3 \right\}$

\[ v_k = V_m \; \sin \left( \omega_0 t - (k-1) \dfrac{4 \pi}{3} \right) \]

\[ v_k = V_m \; \cos \left( \omega_0 t - (k-1) \dfrac{4 \pi}{3} \right) \]

\clearpage

\subsection*{Trofazni diodni most sa prostim kapacitivnim filtrom}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{trofazni-C.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{ki1.ps}  
	\includegraphics[scale=1]{kvoltages1p.ps}  
	\includegraphics[scale=1]{kvoltages1m.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, struja prve faze i linijski naponi koji uzrokuju njeno proticanje.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{kvoltages.ps}  
	\includegraphics[scale=1]{kiout.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, linijski naponi i izlayna struja diodnog mosta.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{ki1v1.ps}  
	\includegraphics[scale=1]{ki2v2.ps}  
	\includegraphics[scale=1]{ki3v3.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, ulazni naponi i ulazne struje.}
\end{figure}

\begin{table}[!h]
	
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, linijski naponi koji dostižu maksimum, provodne diode i preneta naelektrisanja.}
	\centering
	
	\begin{tabular}{c|cc|c|c|c}
	maksimum & \multicolumn{2}{|c|}{provode} & $ \int_{t_a}^{t_b}{i_1(t) \; dt} $ & $ \int_{t_a}^{t_b}{i_2(t) \; dt} $ & $ \int_{t_a}^{t_b}{i_3(t) \; dt} $ \\
	\hline \hline
	$v_{12}$ & D1 & D4 & $Q$ & $-Q$ & $0$ \\
	\hline
	$v_{13}$ & D1 & D6 & $Q$ & $0$ & $-Q$ \\
	\hline
	$v_{21}$ & D3 & D2 & $-Q$ & $Q$ & $0$ \\
	\hline
	$v_{23}$ & D3 & D6 & $0$ & $Q$ & $-Q$ \\
	\hline
	$v_{31}$ & D5 & D2 & $-Q$ & $0$ & $Q$ \\
	\hline
	$v_{32}$ & D5 & D4 & $0$ & $-Q$ & $Q$ \\
	\hline
	\end{tabular}
	
	\label{tabela:buckccm}

\end{table}

\[ V_{OUTmax} = V_L = \sqrt{3} \; V_m = \sqrt{6} \; V_{RMS} \]

\[ C \left( 2 \Delta V_{OUT} \right) = \dfrac{T_0}{6} \; I_{OUT} \]

\[ \Delta V_{OUT} = \dfrac{I_{OUT}}{12 f_0 C} \]

\[ V_{OUT} = V_{OUTmax} - \Delta V_{OUT} = \sqrt{3} \; V_m - \dfrac{I_{OUT}}{12 f_0 C} \]

\[ i_{OUT} = \dfrac{I_{OUT}}{6 f_0} \sum_{k=-\infty}^{\infty}{\delta \left( t - k \; \dfrac{T_0}{6} \right)} \]

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{trofazni-kapacitivni-exp-phase1.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, napon i struja jedne faze.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{trofazni-kapacitivni-exp-output.ps}  
	\caption{Trofazni diodni most sa prostim kapacitivnim filtrom, izlazni napon i izlazna struja diodnog mosta.}
\end{figure}

\clearpage



\subsection*{Strujom opterećen trofazni diodni most}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{trofazni-strujni.ps}  
	\caption{Strujom opterećen trofazni diodni most.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{svoltages.ps}  
	\includegraphics[scale=1]{svout.ps}  
	\caption{Strujom opterećen trofazni diodni most, potencijali izlaznih terminala diodnog mosta i izlazni napon.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{si1v1.ps}  
	\includegraphics[scale=1]{si2v2.ps}  
	\includegraphics[scale=1]{si3v3.ps}  
	\caption{Strujom opterećen trofazni diodni most, ulazne struje i njima odgovarajući ulazni naponi.}
\end{figure}



\[ v_A = \max \left(v_1, v_2, v_3 \right) \]

\[ v_B = \min \left(v_1, v_2, v_3 \right) \]

\[ v_{OUT} = v_A - v_B \]

\[ V_{OUT} = \overline{v_{OUT}} = \left< v_{OUT} \right> = \dfrac{1}{T_0 /6} \int_{t_0}^{t_0+T_0/6}{v_{OUT}(t) \; dt} \]

\[ V_{OUT} = \dfrac{3}{\pi} \; V_m \int_{\pi/6}^{\pi/2}{ \left( \sin\left( \omega_0 t \right) - \sin\left( \omega_0 t - \dfrac{2 \pi}{3} \right) \right) \; d \, \omega_0 t} \]

\[ V_{OUT} = \dfrac{3 \sqrt{3}}{\pi} \; V_m \approx 1.65 \; V_m \approx 2.34 \; V_{RMS} \]

\[ V_{OUT} = \dfrac{3}{\pi} \; V_m \int_{-\pi/6}^{\pi/6}{\cos\left( \omega_0 t \right) \; d \, \omega_0 t}  = \dfrac{3 \sqrt{3}}{\pi} \; V_m \]

\[ P_{OUT} = V_{OUT} \; I_{OUT} = \dfrac{3 \sqrt{3}}{\pi} \; V_m \; I_{OUT} \]

\[ I_{RMS} = \sqrt{ \dfrac{2}{3} } \; I_{OUT} \]

\[ S = 3 \; V_{RMS} \; I_{RMS} = \sqrt{3} \; V_m \; I_{OUT} \]

\[ P = P_{IN} = P_{OUT} \]

\[ PF=\dfrac{P}{S} = \dfrac{3}{\pi} \approx 0.955 \]

\[ P_{IN} = \dfrac{3}{2} \; V_m \; I_{1m} \; \cos{\varphi_1} \]

\[ I_{1m} = \dfrac{2 \sqrt{3}}{\pi} \; I_{OUT} \]

\[ \varphi_1 = 0 \]

\[ I_{1RMS} = \dfrac{\sqrt{6}}{\pi} \; I_{OUT} \]

\[ THD = \sqrt{\dfrac{I_{RMS}^2}{I_{1RMS}^2} - 1} \]

\[ THD = \sqrt{\dfrac{\pi^2}{9} - 1} \approx 31.08\% \]

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{trofazni-strujni-exp-phase1.ps}  
	\caption{Strujom opterećen trofazni diodni most, napon i struja jedne faze.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{trofazni-strujni-exp-output.ps}  
	\caption{Strujom opterećen trofazni diodni most, izlazni napon i izlazna struja diodnog mosta.}
\end{figure}

\clearpage

\subsubsection*{Komutacija u diodnom mostu}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{komutacija.ps}  
	\caption{Komutacija u diodnom mostu, prenos provođenja sa D1 na D3.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{notches.ps}  
	\caption{Čvorovi (notches) u naponu prve faze ispravljača.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{komutacija-reduced.ps}  
	\caption{Komutacija u diodnom mostu, ekvivalentno kolo tokom prenosa provođenja sa D1 na D3.}
\end{figure}

\[ v_{12} = -\sqrt{3} \; V_m \; \sin \omega_0 t \]

\[ i_1(0) = I_{OUT} \]

\[ i_2(0) = 0 \]

\[ i_1 + i_2 = I_{OUT} \]

\[ \dfrac{d i_1}{dt} + \dfrac{d i_2}{dt} = 0 \]

\[ v_1 - v_2 - L \dfrac{d i_1}{dt} + L \dfrac{d i_2}{dt} = 0 \]

\[ v_{12} - 2 L \dfrac{d i_1}{dt} = 0 \]

\[ \dfrac{d i_1}{dt} = - \dfrac{\sqrt{3} \; V_m}{2 L} \sin \omega_0 t \]

\[ i_1 = I_{OUT} - \dfrac{\sqrt{3} \; V_m}{2 \omega_0 L} \left( 1 - \cos \omega_0 t \right) \]

\[ i_2 = \dfrac{\sqrt{3} \; V_m}{2 \omega_0 L} \left( 1 - \cos \omega_0 t \right) \]

\[ v_A = v_1 - L \dfrac{d i_1}{dt} = v_2 - L \dfrac{d i_2}{dt} = \dfrac{v_1+v_2}{2} = - \dfrac{1}{2} v_3 \]

\[ i_2( \Delta t) = I_{OUT} \]

\[ I_{OUT} = \dfrac{\sqrt{3} \; V_m}{2 \omega_0 L} \left( 1 - \cos ( \omega_0 \Delta t ) \right) \]

\[ \cos ( \omega_0 \Delta t ) = 1 - \dfrac{2 \omega_0 L \; I_{OUT}}{\sqrt{3} \; V_m} \]

\[ L = \dfrac{\sqrt{3} \; V_m}{2 \omega_0 \; I_{OUT}} \left( 1 - \cos ( \omega_0 \Delta t ) \right) \]

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{notch-ph1.ps}  
	\caption{Prenos provođenja sa D1 na D3, faza 1.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.5]{notch-ph2.ps}  
	\caption{Prenos provođenja sa D1 na D3, faza 2.}
\end{figure}

\[ \Delta t = 350 \; \mu \text{s} \]

\[ V_m = 140 \text{ V} \]

\[ L = 388 \; \mu \text{H} \]

\clearpage

\subsection*{Potpuno kontrolisan trofazni tiristorski most}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{trofazni-full.ps}  
	\caption{Potpuno kontrolisan trofazni tiristorski most.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{tvoltages.ps}  
	\includegraphics[scale=1]{tvout.ps}  
	\caption{Potpuno kontrolisan trofazni tiristorski most, potencijali izlaznih terminala mosta i izlazni napon, $\alpha = 30^\circ $.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{ti1v1.ps}  
	\includegraphics[scale=1]{ti2v2.ps}  
	\includegraphics[scale=1]{ti3v3.ps}  
	\caption{Potpuno kontrolisan trofazni tiristorski most, ulazne struje i njima odgovarajući ulazni naponi, $\alpha = 30^\circ $.}
\end{figure}


\[ v_{OUT} = v_A - v_B \]

\[ V_{OUT} = \overline{v_{OUT}} = \left< v_{OUT} \right> = \dfrac{6}{T_0} \int_{t_0}^{t_0+T_0/6}{v_{OUT}(t) \; dt} \]

\[ V_{OUT} = \dfrac{3}{\pi} \; V_m \int_{\pi/6+\alpha}^{\pi/2+\alpha}{ \left( \sin\left( \omega_0 t \right) - \sin\left( \omega_0 t - \dfrac{2 \pi}{3} \right) \right) \; d \, \omega_0 t} \]

\[ V_{OUT} = \dfrac{3 \sqrt{3}}{\pi} \; V_m \; \cos \alpha \approx 1.65 \; V_m \; \cos \alpha \approx 2.34 \; V_{RMS} \; \cos \alpha \]

\[ V_{OUT} = \dfrac{3}{\pi} \; V_m \int_{-\pi/6+\alpha}^{\pi/6+\alpha}{\cos\left( \omega_0 t \right) \; d \, \omega_0 t}  = \dfrac{3 \sqrt{3}}{\pi} \; V_m \; \cos \alpha \]




\[ V_{OUT} = \dfrac{3}{\pi} \int_{-\frac{\pi}{6}+\alpha}^{\frac{\pi}{6}+\alpha}{\sqrt{3} \; V_m \cos \left( \omega_0 t \right) \; d \omega_0 t} = \dfrac{3\sqrt{3}}{\pi} \; V_m \cos \alpha\]

\[ P_{OUT} = V_{OUT} \; I_{OUT} = \dfrac{3\sqrt{3}}{\pi} \; V_m \; I_{OUT} \cos \alpha\]

\[ S = \sqrt{3} \; V_m \; I_{OUT} \]

\[ PF = \dfrac{3}{\pi} \cos \alpha \]

\[ THD = \sqrt{\dfrac{\pi^2}{9} - 1} \approx 31.08\% \]

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{t-Vout.ps}  
	\caption{Potpuno kontrolisan trofazni tiristorski most, zavisnost $V_{OUT}$ od $\alpha$.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{t-PFDPF.ps}  
	\caption{Potpuno kontrolisan trofazni tiristorski most, zavisnost $PF$ i $DPF$ od $\alpha$.}
\end{figure}





\clearpage

\subsection*{Polukontrolisan trofazni tiristorski most}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{trofazni-half.ps}  
	\caption{Polukontrolisan trofazni tiristorski most.}
\end{figure}




\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h30voltages.ps}  
	\includegraphics[scale=1]{h30vout.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, potencijali izlaznih terminala mosta i izlazni napon, $\alpha = 30^\circ $.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h30i1v1.ps}  
	\includegraphics[scale=1]{h30i2v2.ps}  
	\includegraphics[scale=1]{h30i3v3.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, ulazne struje i njima odgovarajući ulazni naponi, $\alpha = 30^\circ $.}
\end{figure}




\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h60voltages.ps}  
	\includegraphics[scale=1]{h60vout.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, potencijali izlaznih terminala mosta i izlazni napon, $\alpha = 60^\circ $.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h60i1v1.ps}  
	\includegraphics[scale=1]{h60i2v2.ps}  
	\includegraphics[scale=1]{h60i3v3.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, ulazne struje i njima odgovarajući ulazni naponi, $\alpha = 60^\circ $.}
\end{figure}




\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h90voltages.ps}  
	\includegraphics[scale=1]{h90vout.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, potencijali izlaznih terminala mosta i izlazni napon, $\alpha = 90^\circ $.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h90i1v1.ps}  
	\includegraphics[scale=1]{h90i2v2.ps}  
	\includegraphics[scale=1]{h90i3v3.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, ulazne struje i njima odgovarajući ulazni naponi, $\alpha = 90^\circ $.}
\end{figure}




\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h150voltages.ps}  
	\includegraphics[scale=1]{h150vout.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, potencijali izlaznih terminala mosta i izlazni napon, $\alpha = 150^\circ $.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h150i1v1.ps}  
	\includegraphics[scale=1]{h150i2v2.ps}  
	\includegraphics[scale=1]{h150i3v3.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, ulazne struje i njima odgovarajući ulazni naponi, $\alpha = 150^\circ $.}
\end{figure}




\clearpage

\[ V_A = \dfrac{3}{2 \pi} \int_{-\frac{\pi}{3}+\alpha}^{\frac{\pi}{3}+\alpha}{V_m \cos (\omega_0 t) \; d(\omega_0 t)} = \dfrac{3 \sqrt{3}}{2 \pi} \; V_m \; \cos \alpha \]

\[ V_B = \dfrac{3}{2 \pi} \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}{\left(- V_m \cos (\omega_0 t) \right) \; d(\omega_0 t)} = - \dfrac{3 \sqrt{3}}{2 \pi} \; V_m \]

\[ V_{OUT} = V_A - V_B = \dfrac{3 \sqrt{3}}{2 \pi} \; V_m \left( 1 + \cos \alpha \right) \]

\[ i_1 = i_{Q1} - i_{D1} \]

\[ i_{Q1} = \left\{ \begin{array}{ll}
I_{OUT}, & \dfrac{\pi}{6}+\alpha < \omega_0 t < \dfrac{5 \pi}{6} + \alpha  \\
0, & \text{otherwise}
\end{array} \right. \]

\[ i_{D1} = \left\{ \begin{array}{ll}
I_{OUT}, & \dfrac{7 \pi}{6} < \omega_0 t < \dfrac{11 \pi}{6}  \\
0, & \text{otherwise}
\end{array} \right. \]

za $0 < \alpha < {\pi}/{3}$

\[ i_1 = \left\{ \begin{array}{ll}
I_{OUT}, & \dfrac{\pi}{6}+\alpha < \omega_0 t < \dfrac{5 \pi}{6} + \alpha  \\
-I_{OUT}, & \dfrac{7 \pi}{6} < \omega_0 t < \dfrac{11 \pi}{6}  \\
0, & \text{otherwise}
\end{array} \right. \]

\[ I_{RMS} = \sqrt{\dfrac{2}{3}} \, I_{OUT} \]


za ${\pi}/{3} < \alpha < {\pi}$

\[ i_1 = \left\{ \begin{array}{ll}
I_{OUT}, & \dfrac{\pi}{6}+\alpha < \omega_0 t < \dfrac{7 \pi}{6}  \\
-I_{OUT}, & \dfrac{5 \pi}{6} + \alpha < \omega_0 t < \dfrac{11 \pi}{6}  \\
0, & \text{otherwise}
\end{array} \right. \]

\[ I_{RMS} = \sqrt{1 - \dfrac{\alpha}{\pi}} \, I_{OUT} \]

u oba slučaja

\[ I_{1S} = \dfrac{\sqrt{3}}{\pi} \left( 1 + \cos \alpha \right) I_{OUT} \]

\[ I_{1C} = - \dfrac{\sqrt{3}}{\pi} \, \sin (\alpha) \, I_{OUT} \]

\[ I_{1m} = \dfrac{\sqrt{6}}{\pi} \sqrt{1 + \cos \alpha} \; I_{OUT} = \dfrac{\sqrt{12}}{\pi} \cos \dfrac{\alpha}{2} \; I_{OUT} \]

\[ I_{1RMS} = \dfrac{\sqrt{3}}{\pi} \sqrt{1 + \cos \alpha} \, I_{OUT} = \dfrac{\sqrt{6}}{\pi} \cos \dfrac{\alpha}{2} \, I_{OUT} \]

\[ DPF = \cos \varphi_1 = \dfrac{I_{1S}}{I_{1m}} = \sqrt{\dfrac{1 + \cos \alpha}{2}} = \cos \dfrac{\alpha}{2}  \]


za $0 < \alpha < {\pi}/{3}$:

\[ PF = \dfrac{3}{2 \pi} ( 1 + \cos \alpha) = \dfrac{3}{\pi} \cos^2 \dfrac{\alpha}{2} \]

\[ THD = \sqrt{\dfrac{2 \pi^2}{9\left(1+\cos\alpha\right)}-1} = \sqrt{\left( \dfrac{\pi}{3 \cos \dfrac{\alpha}{2}} \right)^2 - 1}\]



za ${\pi}/{3} < \alpha < \pi$:

\[ PF = \sqrt{\dfrac{6}{\pi \left(\pi-\alpha \right)}} \; \cos^2 \dfrac{\alpha}{2} \]

\[ THD = \sqrt{\dfrac{\pi \left(\pi - \alpha \right)}{6 \cos^2 \dfrac{\alpha}{2}} - 1} \]


\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h-Vout.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, zavisnost $V_{OUT}$ od $\alpha$.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h-S.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, zavisnost prividne snage od $\alpha$.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h-PFDPF.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, zavisnost $PF$ i $DPF$ od $\alpha$.}
\end{figure}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=1]{h-THD.ps}  
	\caption{Polukontrolisan trofazni tiristorski most, zavisnost $THD$ od $\alpha$.}
\end{figure}














\clearpage

\subsection*{Nekontrolisan naponom opterećen trofazni ispravljač}

\begin{figure}[!h]
	\centering
	\includegraphics[scale=0.75]{trofazni-LV.ps}  
	\caption{Nekontrolisan naponom opterećen trofazni ispravljač.}
\end{figure}


\end{document}