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% Rectifier Tables: Triphasic Uncontrolled % Diego Trapero

Rectifier Tables

  1. Monophasic Uncontrolled Rectifiers with R load
  2. Monophasic Uncontrolled Full Wave Rectifier loads
  3. Monophasic Controlled Full Wave Rectifier, R vs RL load
  4. Triphasic Uncontrolled Rectifiers with R load
  5. Triphasic Controlled Half Wave Rectifier, R vs RL load
  6. Triphasic Controlled Full Wave Rectifier, R vs RL load

\newpage

Triphasic Uncontrolled Rectifiers with R load

\tabulinesep=1mm \begin{longtabu} to\linewidth{|X[0.6,m,c]|X[2,m,c]|X[2,m,c]|}

\hline
What & $3 \Phi$ Half Wave Rectifier & $3 \Phi$ Full Wave Rectifier \\
\hline

Circuit Diagram
&
\begin{circuitikz}[scale = 0.6, transform shape]
	\draw
	%Sinusoidal source and connections
	(0,5) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](2,5)
	(0,4) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](2,4)
	(0,3) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](2,3)
	(2,5) to[diode](4,5)
	(2,4) to[diode](4,4)
	(2,3) to[diode](4,3)
	(0,5) to[short](0,0)
	(4,5) to[short](4,3)
	(4,4) to[short](5,4)
	(5,4) to[R](5,0)
	(0,0) to[short](5,0)
	(0,0) node[ground](){}
	;
\end{circuitikz}
&
\begin{circuitikz}[scale = 0.6, transform shape]
	\draw
	%Sinusoidal source and connections
	(0,2) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](1,2)
	(0,3) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](1,3)
	(0,4) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](1,4)
	(1,4) to[short, -*](2,4)
	(1,3) to[short, -*](3,3)
	(1,2) to[short, -*](4,2)
	(0,2) to[short](0,4)
	%Rectifier branch A
	(2,0) to[diode, l=$D4$](2,2)
	(2,2) to[short](2,4)
	(2,4) to[diode, l=$D1$](2,6)
	%Rectifier branch B
	(3,0) to[diode, l=$D3$](3,2)
	(3,2) to[short](3,4)
	(3,4) to[diode, l=$D2$](3,6)
	%Rectifier branch C
	(4,0) to[diode, l=$D6$](4,2)
	(4,2) to[short](4,4)
	(4,4) to[diode, l=$D3$](4,6)
	%Load and connections
	(2,6) to[short](6,6) %upper node
	(2,0) to[short](6,0) %lower node
	(6,6) to[R](6,0) %load
	;
\end{circuitikz}
\\
\hline

$v_o$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
\\

Peaks/period
&
3 peaks/period
&
3 peaks/period
\\

Period
&
$$ \frac{2\pi}{3} $$
&
$$ \frac{2\pi}{6} = \frac{\pi}{3} $$
\\

Integration limits
&
$$ \frac{\pi}{6}, \frac{5\pi}{6} $$
&
$$ \frac{\pi}{3}, \frac{2\pi}{3} $$
\\

Load Voltage
&
Phase Voltage
&
Line Voltage
\\

$\bar{v_o}$
&
$$\bar{v_o} = \frac{1}{\frac{2\pi}{3}} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} { V_{PN} \sin{\theta} d\theta}$$
$$\bar{v_o} = \frac{3 V_{PN}}{2\pi} [ -\cos{\theta} ]_{\frac{\pi}{6}}^{\frac{5\pi}{6}}$$
$$\bar{v_o} = \frac{3 \sqrt{3}}{2\pi}  V_{PN}$$
&
$$\bar{v_o} = \frac{1}{\frac{\pi}{3}} \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} { V_{LL} \sin{\theta} d\theta}$$
$$\bar{v_o} = \frac{3 V_{LL}}{\pi} [ -\cos{\theta} ]_{\frac{\pi}{3}}^{\frac{2\pi}{3}}$$
$$\bar{v_o} = \frac{3}{\pi}  V_{LL}$$
\\
\hline

Diode table
&
\vspace{30mm}
&
\vspace{30mm}
\\
\hline

$i_o$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
\\
\hline

$i_i$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=6.5,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28},
		xticklabels={0, $\pi$, $2\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{sin(deg(x-2.09440))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-1.04719))};
		\addplot[no marks,gray,-] expression[domain=0:2*pi,samples=100]{-sin(deg(x-2.09440))};
	\end{axis}
\end{tikzpicture}
\\

\end{longtabu}

\newpage