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% Rectifier Tables: Monophasic Controlled % 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

Monophasic Controlled Full Wave Rectifier, R vs RL load

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

\hline
What & Controlled FWR R load & Controlled FWR RL load \\
\hline

Circuit Diagram
&
$$ % AC to DC Full Wave rectifier, by Diego Trapero
\begin{circuitikz}[scale = 0.5, transform shape]
	\draw
	%Sinusoidal source and connections
	(0,2) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](0,4)
	(0,2) to[short](4,2)
	(4,2) node[circ] {}
	(0,4) to[short](2,4)
	(2,4) node[circ] {}
	%Rectifier branch left
	(2,0) to[thyristor, l^=$T3$](2,2)
	(2,2) to[short](2,4)
	(2,4) to[thyristor, l^=$T1$](2,6)
	%Rectifier branch right
	(4,0) to[thyristor, l^=$T4$](4,2)
	(4,2) to[short](4,4)
	(4,4) to[thyristor, l^=$T2$](4,6)
	%Load and connections
	(2,6) to[short](8,6) %upper node
	(2,0) to[short](8,0) %lower node
	(8,6) to[R](8,0) %load
	;
\end{circuitikz}
$$
&
$$ % AC to DC Full Wave rectifier, by Diego Trapero
\begin{circuitikz}[scale = 0.5, transform shape]
	\draw
	%Sinusoidal source and connections
	(0,2) to[sinusoidal voltage source, v^>=$v_i$, i^>=$i_i$](0,4)
	(0,2) to[short](4,2)
	(4,2) node[circ] {}
	(0,4) to[short](2,4)
	(2,4) node[circ] {}
	%Rectifier branch left
	(2,0) to[thyristor, l^=$T3$](2,2)
	(2,2) to[short](2,4)
	(2,4) to[thyristor, l^=$T1$](2,6)
	%Rectifier branch right
	(4,0) to[thyristor, l^=$T4$](4,2)
	(4,2) to[short](4,4)
	(4,4) to[thyristor, l^=$T2$](4,6)
	%Load and connections
	(2,6) to[short](8,6) %upper node
	(2,0) to[short](8,0) %lower node
	(8,6) to[L](8,3) %load
	(8,0) to[R](8,3)
	;
\end{circuitikz}
$$
\\
\hline

$v_o$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:0.53,samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=0.53:pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=pi:(pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):2*pi,samples=20, line width=2]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=2*pi:(2*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):3*pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=3*pi:(3*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0.53:(pi+0.53),samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):(2*pi+0.53),samples=20, line width=2]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):(3*pi+0.53),samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
\\

$\bar{v_o}(\alpha)$
&
$$ \bar{v_o} = \frac{V_i}{\pi} ( \cos{\alpha} + 1 ) $$
&
$$ \bar{v_o} = \frac{2 V_i}{\pi} \cos{\alpha} $$
\\
\hline

$v_R$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:0.53,samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=0.53:pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=pi:(pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):2*pi,samples=20, line width=2]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=2*pi:(2*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):3*pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=3*pi:(3*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:4*pi,samples=100, line width=2]{0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
	\end{axis}
\end{tikzpicture}
\\
\hline



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

$i_o(t)$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:0.53,samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=0.53:pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=pi:(pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):2*pi,samples=20, line width=2]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=2*pi:(2*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):3*pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=3*pi:(3*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:4*pi,samples=100, line width=2]{0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
	\end{axis}
\end{tikzpicture}
\\

$\bar{i_o}$
&
\begin{circuitikz}[scale = 0.6, transform shape]
	\draw
	(0,0) to[voltage source, l=$\bar{v_o}$](0,2)
	(0,2) to[diode](2,2)
	(2,2) to[R](2,0)
	(0,0) to[short](2,0)
	(0,0) node[ground](){}
	;
\end{circuitikz}
$$\bar{i_o} = \frac{\bar{v_o}}{R} = \frac{\frac{V_i}{\pi} ( \cos{\alpha} + 1 )}{R}$$
&
\begin{circuitikz}[scale = 0.6, transform shape]
	\draw
	(0,0) to[voltage source, l=$\bar{v_o}$](0,2)
	(0,2) to[diode](2,2)
	(2,2) to[R](2,0)
	(0,0) to[short](2,0)
	(0,0) node[ground](){}
	;
\end{circuitikz}
$$\bar{i_o} = \frac{\bar{v_o}}{R} = \frac{\frac{2 V_i}{\pi} \cos{\alpha}}{R}$$
\\
\hline

$i_i(t)$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:0.53,samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=0.53:pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=pi:(pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):2*pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=2*pi:(2*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):3*pi,samples=20, line width=2]{sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=3*pi:(3*pi+0.53),samples=20, line width=2]{0};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{sin(deg(x))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
		\addplot[no marks,red,-] expression[domain=0:0.53,samples=20, line width=2]{-0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
		\addplot[no marks,red,-] expression[domain=0.53:(pi+0.53),samples=20, line width=2]{0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
		\addplot[no marks,red,-] expression[domain=(pi+0.53):(2*pi+0.53),samples=20, line width=2]{-0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
		\addplot[no marks,red,-] expression[domain=(2*pi+0.53):(3*pi+0.53),samples=20, line width=2]{0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
		\addplot[no marks,red,-] expression[domain=(3*pi+0.53):4*pi,samples=20, line width=2]{-0.55+0.015*sin(deg(x))+0.01*sin(2*deg(x))+0.025*sin(3*deg(x))};
	\end{axis}
\end{tikzpicture}
\\
\hline

$v_{T1}(t)$
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
&
\begin{tikzpicture}[>=stealth, scale = 0.6, transform shape]
	\begin{axis}[
		xmin=0,xmax=13,
		ymin=-1,ymax=1,
		axis x line=middle,
		axis y line=middle,
		axis line style=->,
		%xlabel={$t$},
		%ylabel={$y$},
		ymajorgrids,
		xmajorgrids,
		xtick={0,3.14,6.28,9.42, 12.56},
		xticklabels={0, $\pi$, $2\pi$, $3\pi$, $4\pi$},
		ytick={},
		yticklabels={}
		]
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{sin(deg(x))};
		\addplot[no marks,gray,-] expression[domain=0:4*pi,samples=100]{-sin(deg(x))};
	\end{axis}
\end{tikzpicture}
\\
\hline

Power
&
\vspace{30mm}
&
$$P = V_{1\text{ RMS}} I_{1\text{ RMS}} \cos{\varphi_1}$$
$$P = V_{ip} I_o \frac{2\sqrt{2}}{\pi} \cos{\alpha}$$
\\

Apparent Power
&
\vspace{30mm}
&
$$S = V_{\text{RMS}} I_{\text{RMS}}$$
$$S = \frac{V_{ip}}{\sqrt{2}} I_o$$
\\

Power factor
&
\vspace{30mm}
&
$$\text{PF} = \frac{2\sqrt{2}}{\pi} \cos{\alpha}$$
\\
\hline

\end{longtabu}