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Ex07 Task2: Add remaining subtask and update Fig_ThreePhaseInverter_6…
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@SevenOfNinePE
SevenOfNinePE committed Jan 22, 2025
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141 changes: 84 additions & 57 deletions exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex
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\begin{figure}[htb]
\begin{center}
\begin{circuitikz}
\def\vd{1.5cm} % vertical distance AC sources
\def\hd{1.5cm} % horizontal distance diode bridge
\def\h1d{5.0cm} % horizontal position first diode string
% Base point for voltage supplies
\coordinate (orig) at (0,0);
% Voltage sources and neutral connection
\draw
% draw the neutral connection
(0,0) to [short, -*] ++(0,-1.5) to [short] ++(0,-1.5)
% draw first phase ua
(0,0) to [sinusoidal voltage source, v^<=$u_{1\mathrm{a}}(t)$] ++(1.5, 0) to [short, i=$i_{1\mathrm{a}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (A)
% draw second phase ub
(0,-1*\vd) to [sinusoidal voltage source, v^<=$u_{1\mathrm{b}}(t)$] ++(1.5, 0) to [short, i=$i_{1\mathrm{b}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (B)
% draw third phase uc
(0,-2*\vd) to [sinusoidal voltage source, v^<=$u_{1\mathrm{c}}(t)$] ++(1.5,0) to [short, i=$i_{1\mathrm{c}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (C)
%thyristor bridge
% Add thyristor T1
(\h1d,0) to [thyristor, l=$T_1$, name=D1] ++(0,1.25) coordinate (D1top)
% Add thyristor T2
(\h1d,-4.25) coordinate (D2bot) to [thyristor, l=$T_2$, name=D2] ++(0,1.25) to [short] (\h1d, 0)
% Add connection to junction A
(\h1d, 0) to [short, *-] (A)
% Add thyristor T3
(\h1d+\hd,0) to [thyristor, l=$T_3$, name=D3] ++(0,1.25) coordinate (D3top)
% Add thyristor T4
(\h1d+\hd,-4.25) coordinate (D4bot) to [thyristor, l=$T_4$, name=D4] ++(0,1.25) to [short] (\h1d+\hd, 0)
% Add thyristor T5
(\h1d+2*\hd,0) to [thyristor, l=$T_5$, name=D5] ++(0,1.25) coordinate (D5top)
% Add thyristor T6
(\h1d+2*\hd,-4.25) coordinate (D6bot) to [thyristor, l=$T_6$, name=D6] ++(0,1.25) to [short] (\h1d+2*\hd, 0)
% Add connection to junction B
(B -| D3) to [crossing, *-, mirror] ++(-2*\hd,0) -- (B)
% Add connection to junction C
(C -| D5) to [short, *-] ++(-\hd/2,0) to [crossing, mirror] ++(-\hd,0) to [crossing, mirror] ++(-\hd,0) -- (C)
% Add wire T1-T3-T5
(D1top) to [short, -*] (D3top) to [short, -*] (D5top) to [short, -] ++(0.5,0) coordinate (jL1)
% Add inductor L and motor current
(jL1) to [L, l=$L$, name = L] ++(2,0) to [short,i=$\overline{i}_\mathrm{mot}$] ++(0.5,0) coordinate (jL2)
% Add DC-motor and motor voltage
(jL2) ++ (0,-3) node[elmech](motor){M}
(jL2) to (motor.north)
(motor.bottom) to (D6bot -| \tikztostart) to (D6bot)
% (jL2) to [R, l=$R$, name = R, v_>=$\overline{u}_\mathrm{mot}$, voltage = straight] (D6bot -| \tikztostart) to (D6bot)
% Add voltage U1p
\draw (0,0) coordinate (U1p) to [open, o-o, v = $U_1p\hspace{0.5cm}$, voltage = straight] ++(0,-2.5) coordinate (Gnd)
(Gnd) to [short,o-o] ++(1,0)
(Gnd) to [open, -o, v = $U_1m\hspace{0.5cm}$, voltage = straight] ++(0,-2.5) coordinate (U1m)
% Add current
(U1p) to [short, o-, i=$i_1(t)$] ++(2,0) coordinate (jT1c)
% Add T1
(jT1c) to [Tnpn, n=T1, invert, bodydiode] ++(0,-2) coordinate (jT1e)
% Add connection to u2a
(jT1e) to [short, *-] ++(1,0) to [crossing] ++(2,0) to [crossing] ++(2,0) to [short,-] ++(1,0) coordinate (ju2a)
% Add junction to T2
(jT1e) to [short] ++(0,-1) coordinate (jT2c)
% Add T2
(jT2c) to [Tnpn, n=T2, invert, bodydiode] ++(0,-2) coordinate (jT2e)
% Add connection to T3
(jT1c) to [short, *-] ++(2,0) coordinate (jT3c)
% Add T3
(jT3c) to [Tnpn, n=T3, invert, bodydiode] ++(0,-2) coordinate (jT3e)
% Add junction to ju2b
(jT3e) to [short] ++(0,-0.5) coordinate (jmu2b)
% Add connection to u1b
(jmu2b) to [short,*-] ++(1,0) to [crossing] ++(2,0) to [short,-] ++(1,0) coordinate (ju2b)
% Add junction to T4
(jmu2b) to [short] ++(0,-0.5) coordinate (jT4c)
% Add T4
(jT4c) to [Tnpn, n=T4, invert, bodydiode] ++(0,-2) coordinate (jT4e)
% Add connection to T5
(jT3c) to [short, *-] ++(2,0) coordinate (jT5c)
% Add T5
(jT5c) to [Tnpn, n=T5, invert, bodydiode] ++(0,-2) coordinate (jT5e)
% Add junction to T6
(jT5e) to [short] ++(0,-1) coordinate (jT6c)
% Add T6
(jT6c) to [Tnpn, n=T6, invert, bodydiode] ++(0,-2) coordinate (jT6e)
% Add connection to T4
(jT6e) to [short, -*] (jT4e)
% Add connection to T2
(jT4e) to [short, -*] (jT2e)
% Add connection to U1m
(jT2e) to [short, -] (U1m)
% Add connection to u1c
(jT6c) to [short,*-] ++(2,0) coordinate (ju2c)
% Add connection to u2a inductor
(ju2a) to [short,-] ++(0,2) coordinate (ju2ax)
% Add u2a inductor
(ju2ax) to [L, l=$L$, name = L] ++(2,0) coordinate (ju2ae)
% Add u2ae
(ju2ae) to [sV=$u_\mathrm{1ae}$] ++(1.5,0) coordinate (ju2an)
% Add u2b inductor
(ju2b) to [L, l=$L$, name = L] ++(2,0) coordinate (ju2be)
% Add u2be
(ju2be) to [sV=$u_\mathrm{1be}$] ++(1.5,0) coordinate (ju2bn)
% Add connection to u2c inductor
(ju2c) to [short,-] ++(0,-2) coordinate (ju2cx)
% Add u2a inductor
(ju2cx) to [L, l=$L$, name = L] ++(2,0) coordinate (ju2ce)
% Add u2ce
(ju2ce) to [sV=$u_\mathrm{1ce}$] ++(1.5,0) coordinate (ju2cn)
% Add connection of u2in
(ju2an) to [short,-*] (ju2bn) to [short,-] (ju2cn);


% Add component name of transistors
\draw let \p1 = (T1.B) in node[anchor=east] at (\x1,\y1) {$T_1$};
\draw let \p1 = (T2.B) in node[anchor=east] at (\x1,\y1) {$T_2$};
\draw let \p1 = (T3.B) in node[anchor=east] at (\x1,\y1) {$T_3$};
\draw let \p1 = (T4.B) in node[anchor=east] at (\x1,\y1) {$T_4$};
\draw let \p1 = (T5.B) in node[anchor=east] at (\x1,\y1) {$T_5$};
\draw let \p1 = (T6.B) in node[anchor=east] at (\x1,\y1) {$T_6$};
% Add current arrows i2a, i2b and i2c
\draw (jT1e) ++(1,0) node[currarrow](i2a){}
(i2a) node[anchor=north,color=black]{$i_\mathrm{2a}(t)$}
(jmu2b) ++(1,0) node[currarrow](i2b){}
(i2b) node[anchor=north,color=black]{$i_\mathrm{2b}(t)$}
(jT6c) ++(1,0) node[currarrow](i2c){}
(i2c) node[anchor=north,color=black]{$i_\mathrm{2c}(t)$}
% Add voltage arrows u2an, u2bn and u2cn
(ju2ax) ++(0,-0.8) to [open,v^=$u_\mathrm{2a}(t)$,voltage = straight] ++(3.8,0)
(ju2b) ++(0,-0.8) to [open,v^=$u_\mathrm{2b}(t)$,voltage = straight] ++(3.8,0)
(ju2cx) ++(0,-0.8) to [open,v^=$u_\mathrm{2c}(t)$,voltage = straight] ++(3.8,0)
% Add voltage arrows u2ab and u2bc
(ju2ax) ++(0.2,0) to [open,v^=$u_\mathrm{2ab}(t)$,voltage = straight] ++(0,-2.5)
(ju2b) ++(0.2,0) to [open,v^=$u_\mathrm{2bc}(t)$,voltage = straight] ++(0,-2.5);


% Add wire T2-T3-T6
(D2bot) to [short, -*] (D4bot) to [short, -*] (D6bot)
% Add voltage arrow u2(t) between Dtop and Dbot
(jL1) to [open, v^>=$\hspace{0.5cm}u_2(t)$, voltage = straight] (D6bot-|jL1)
% Add voltage arrow u2+n(t) between Dtop and neutral
(D1top) ++(-0.2,0) to [open, v_>=$u_\mathrm{2,p}(t)$, voltage = straight] ++(-5.5,0)
% Add voltage arrow u2-n(t) between Dbot and neutral
(D2bot) ++(-0.2,0) to [open, v_>=$u_\mathrm{2,m}(t)$, voltage = straight] ++(-5.5,0)
% Add voltage arrow between AC source a and b
(A) to [open, v^>=$\hspace{0.75cm}u_{1\mathrm{ab}}(t)$, voltage = straight] (B)
% Add voltage arrow between AC source b and c
(B) to [open, v^>=$\hspace{0.75cm}u_{1\mathrm{bc}}(t)$, voltage = straight] (C)
% Add voltage arrow between AC source a and c
(-0.5,-2*\vd) to [open, v^>=$u_{1\mathrm{ca}}(t)\hspace{0.75cm}$, voltage = straight] (-0.5,0);
\end{circuitikz}
\end{center}
\caption{Three-phase inverter in six-step mode.}
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44 changes: 41 additions & 3 deletions exercise/tex/exercise07.tex
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a sinusoidal counter voltage per phase. The inverter is operated with a basic frequency clock.
The switching elements are considered as ideal.

% \input{fig/ex07/Fig_ThreePhaseInverter_6StepMode}
\input{fig/ex07/Fig_ThreePhaseInverter_6StepMode}

\subtask{Aufgabe 4}

\subtask{Create a table with all possible switching states for basic frequency clocking.
Use the following notation: \\
$(s_\mathrm{a}(t),s_\mathrm{b}(t),s_\mathrm{c}(t))=\begin{cases}
s_i(t)= +1 & \text{upper position,}\\
s_i(t)= -1 & \text{lower position.}
\end{cases}$\\
Sketch the switching states in the correct chronological order for minimum one periode.
Calculate and sketch the voltages $u_\mathrm{a,0}(t)$, $u_\mathrm{b,0}(t)$ and $u_\mathrm{c,0}(t)$ depending on these switching states.
}
\begin{solutionblock}
\end{solutionblock}

\subtask{The internal voltages $u_\mathrm{ea}(t)$, $u_\mathrm{eb}(t)$ and $u_\mathrm{ec}(t)$ are a symmetrical voltage system,
i.e. the following always applies: $u_\mathrm{ea}(t)+u_\mathrm{eb}(t)+u_\mathrm{ec}(t)=0V$.
Show that this equation is also applicable for the voltages $u_\mathrm{a}(t)$, $u_\mathrm{b}(t)$ and $u_\mathrm{c}(t)$ under the same conditions.
}
\begin{solutionblock}
\end{solutionblock}

\subtask{Calculate and sketch the voltages $u_\mathrm{ab}(t)$, $u_\mathrm{bc}(t)$, $u_\mathrm{a}(t)$ and $u_\mathrm{a,0}(t)$
depending on these switching states.}
\begin{solutionblock}
\end{solutionblock}

\subtask{Decompose the voltage $u_\mathrm{a}(t)$ into a Fourier series and sketch the spectral lines related to the
amplitude of the fundamental signal up to order n=13. Hint: The following applies to the Fourier coefficients of an odd and alternating function:
\begin{align*}
b_k = \frac{4}{\pi} \int_{0}^{\frac{\pi}{2}} f(x)\sin(kx) \mathrm{d}x \quad k =\mathrm{odd} \quad \quad
\end{align*}
\label{sub:DecomposeVoltage}
}
\begin{solutionblock}
\input{fig/ex07/Fig_Voltage_U_um_excerpt}
\input{fig/ex07/Fig_graphic_solutions_cos_terms}
\end{solutionblock}


\subtask{Based on \autoref{sub:DecomposeVoltage}, calculate the fundamental amplitude $\hat{i}_\mathrm{a}^1$ using a vector diagram and complex alternating current calculations.
From this, determine the total active power converted in the load.}
\begin{solutionblock}
\input{fig/ex07/Fig_Voltage_U_um_excerpt}
\input{fig/ex07/Fig_graphic_solutions_cos_terms}
\end{solutionblock}

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