From 7fff36f153fb5e0bbd5920baac9a0018a40aa04d Mon Sep 17 00:00:00 2001 From: SevenOfNinePE Date: Thu, 23 Jan 2025 17:45:42 +0100 Subject: [PATCH 01/10] Ex07 Task2: Finish task and drawings (without solution) --- .../fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex | 12 +++++++----- exercise/tex/exercise07.tex | 15 +++++++++++++++ 2 files changed, 22 insertions(+), 5 deletions(-) diff --git a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex index e679b6a..71bf58d 100644 --- a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex +++ b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex @@ -81,13 +81,15 @@ (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) + (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 voltage arrows u2ab + (ju2ax) ++(0.2,0) to [open,v^=$$,voltage = straight] ++(0,-2.5) + (ju2ax) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2ab}(t)$} + % Add voltage arrows u2bc + (ju2b) ++(0.2,0) to [open,v^=$$,voltage = straight] ++(0,-2.5) + (ju2b) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2bc}(t)$}; \end{circuitikz} \end{center} diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index 5dbce2c..1947238 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -24,6 +24,21 @@ % \input{fig/ex07/Fig_ThreePhaseInverter_6StepMode} \input{fig/ex07/Fig_ThreePhaseInverter_6StepMode} +\begin{table}[ht] + \centering % Center the table + \begin{tabular}{ll} + \toprule + Input voltages: & $U_{\mathrm{1p},i}=\SI{255}{\volt}$ \quad $U_{\mathrm{1m},i}=\SI{255}{\volt}$ \\ + Internal voltages: & $u_{\mathrm{1ae}(t)} = \sqrt{2} \cdot \SI{220}{\volt} \cdot \sin(\omega_1t)$ \\ + Circular frequency: & $\omega_1 = \SI{2 \pi \cdot 30}{\frac{1}{\second}}$ \\ + Inductivity per phase: & $L= \SI{10}{\milli \henry}$ \\ + Phase angle between $u_{\mathrm{1ae}(t)}$ and $i_{\mathrm{1ae}^\mathrm{(1)}(t)}$ & $\alpha=\SI{30}{\degree}$ \\ + \bottomrule + \end{tabular} + \caption{Parameters of three-phase inverter in six-step mode.} + \label{table:ex07_Task2_ParametersOfTheCircuit} +\end{table} + \subtask{Create a table with all possible switching states for basic frequency clocking. Use the following notation: \\ From 060525582832a83dae6bc98da2298405e623011b Mon Sep 17 00:00:00 2001 From: Wallscheid Date: Thu, 23 Jan 2025 21:18:48 +0100 Subject: [PATCH 02/10] add to lec06 --- course_template | 2 +- lecture/main.ist | 2 +- lecture/main.tex | 4 +- lecture/tex/Lecture06.tex | 135 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 139 insertions(+), 4 deletions(-) diff --git a/course_template b/course_template index 235a1d7..14518cd 160000 --- a/course_template +++ b/course_template @@ -1 +1 @@ -Subproject commit 235a1d702edc252f9919bf8ec1201b10903f8233 +Subproject commit 14518cd3451f639b3b4923dc2ae2f5bc665fe88c diff --git a/lecture/main.ist b/lecture/main.ist index d3f3a74..47d31cf 100644 --- a/lecture/main.ist +++ b/lecture/main.ist @@ -1,5 +1,5 @@ % makeindex style file created by the glossaries package -% for document 'main' on 2025-1-22 +% for document 'main' on 2025-1-23 actual '?' encap '|' level '!' diff --git a/lecture/main.tex b/lecture/main.tex index 8e20ef4..b1191ec 100644 --- a/lecture/main.tex +++ b/lecture/main.tex @@ -5,7 +5,7 @@ \author{Oliver Wallscheid} \date{} -\includeonly{tex/temp} % build only selected sections +\includeonly{tex/Lecture06} % build only selected sections \begin{document} @@ -35,7 +35,7 @@ \include{tex/Lecture05} % Thyristor-based AC/DC converters \include{tex/Lecture06} % Transistor-based AC/DC converters -\include{tex/temp} % Temporary section +%\include{tex/temp} % Temporary section \section{Appendix} % Appendix \include{tex/dict} % English-German dictionary (subsection) diff --git a/lecture/tex/Lecture06.tex b/lecture/tex/Lecture06.tex index ebbcbaf..e360800 100644 --- a/lecture/tex/Lecture06.tex +++ b/lecture/tex/Lecture06.tex @@ -1055,6 +1055,141 @@ \subsection{Single-phase AC/DC bridge converter} \end{table} \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Outlook: multi-level converters %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Outlook: multi-level converters} + \begin{figure} + \centering + \begin{subfigure}{0.32\textwidth} + \centering + \begin{circuitikz} + \draw + (0,0) to [short, o-*] ++(1,0) coordinate (A) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{2}$}, voltage = straight] ++(0,-2) coordinate (B) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{2}$}, voltage = straight] ++(0,-2) coordinate (C) + to [short, -o] ++(-1,0) coordinate (D) + (B) to [short] ++(-0.5, 0) coordinate (E) + (E) node[rground, rotate=-90, name = gnd1]{}; + \draw + ($(B) + (1.5,0)$) node[cute spdt up arrow, xscale=-1] (Sw1) {} + (A) -| (Sw1-out 1.n) + (C) -| (Sw1-out 2.s) + (Sw1-in) to [short, -o] ++(1,0) coordinate (F); + \draw + (F |- C) node[rground, anchor = south, name = gnd2]{} + (F) to [open, v={$u_2(t)\hspace{0.3cm}$}, voltage = straight] (gnd2); + \end{circuitikz}\\[1em] + \begin{tikzpicture} + \begin{axis}[ + width=0.99\textwidth, + height=0.375\textheight, + xlabel={$t$}, + ylabel={$u_2$}, + xmin=0, xmax=1.2, + ymin=-1.15, ymax=1.15, + ytick={-1,0,1}, + yticklabels={$-\frac{U_\mathrm{dc}}{2}$,$0$,$\frac{U_\mathrm{dc}}{2}$}, + xticklabels={}, + grid=both, + axis lines=middle, + xlabel style={anchor=west}, + ylabel style={anchor=south}, + ] + \addplot[signalblue, thick] coordinates {(0,-1) (0.1,-1) (0.1,1) (0.6,1) (0.6,-1) (1.1,-1) (1.1,1) (1.2,1)}; + \end{axis} + \end{tikzpicture} + \caption{2-level half bridge} + \end{subfigure} + \begin{subfigure}{0.32\textwidth} + \centering + \begin{circuitikz} + \draw + (0,0) to [short, o-*] ++(1,0) coordinate (A) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{2}$}, voltage = straight] ++(0,-2) coordinate (B) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{2}$}, voltage = straight] ++(0,-2) coordinate (C) + to [short, -o] ++(-1,0) coordinate (D) + (B) to [short] ++(-0.5, 0) coordinate (E) + (E) node[rground, rotate=-90, name = gnd1]{}; + \draw + ($(B) + (1.0,0)$) node[rotary switch ->=3 in 90 wiper 90, xscale=-1, anchor=out 2](Sw1){} + (A) -| (Sw1-out 1.n) + (C) -| (Sw1-out 3.s) + (Sw1-in) to [short, -o] ++(1,0) coordinate (F) + (Sw1-out 2.w) to [short] (B); + \draw + (F |- C) node[rground, anchor = south, name = gnd2]{} + (F) to [open, v={$u_2(t)\hspace{0.3cm}$}, voltage = straight] (gnd2); + \end{circuitikz}\\[1em] + \begin{tikzpicture} + \begin{axis}[ + width=0.99\textwidth, + height=0.375\textheight, + xlabel={$t$}, + ylabel={$u_2$}, + xmin=0, xmax=1.2, + ymin=-1.15, ymax=1.15, + ytick={-1,0,1}, + yticklabels={$-\frac{U_\mathrm{dc}}{2}$,$0$,$\frac{U_\mathrm{dc}}{2}$}, + xticklabels={}, + grid=both, + axis lines=middle, + xlabel style={anchor=west}, + ylabel style={anchor=south}, + ] + \addplot[signalblue, thick] coordinates {(0,-1) (0.1,-1) (0.1,0) (0.35,0) (0.35,1) (0.6,1) (0.6,0) (0.85,0) (0.85,-1) (1.1,-1) (1.1,0) (1.1,0) (1.2,0)}; + \end{axis} + \end{tikzpicture} + \caption{3-level half bridge} + \end{subfigure} + \begin{subfigure}{0.32\textwidth} + \centering + \begin{circuitikz} + \draw + (0,0) to [short, o-*] ++(1,0) coordinate (A) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{3}$}, voltage = straight] ++(0,-1.0) coordinate (B) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{6}$}, voltage = straight] ++(0,-1.0) coordinate (C) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{6}$}, voltage = straight] ++(0,-1.0) coordinate (D) + to [capacitor, -*, v={$\frac{U_\mathrm{dc}}{3}$}, voltage = straight] ++(0,-1.0) coordinate (E) + to [short, -o] ++(-1,0) coordinate (F) + (C) to [short] ++(-0.5, 0) coordinate (G) + (G) node[rground, rotate=-90, name = gnd1]{}; + \draw + ($(C) + (1.75,0)$) node[rotary switch ->=4 in 90 wiper 90, xscale=-1, anchor=in](Sw1){} + (A) -| (Sw1-out 1.n) + (E) -| (Sw1-out 4.s) + (Sw1-in) to [short, -o] ++(1,0) coordinate (H) + (Sw1-out 2.n) |- (B) + (Sw1-out 3.s) |- (D); + \draw + (H |- E) node[rground, anchor = south, name = gnd2]{} + (H) to [open, v={$u_2(t)\hspace{0.3cm}$}, voltage = straight] (gnd2); + \end{circuitikz}\\[1em] + \begin{tikzpicture} + \begin{axis}[ + width=0.99\textwidth, + height=0.375\textheight, + xlabel={$t$}, + ylabel={$u_2$}, + xmin=0, xmax=1.2, + ymin=-1.15, ymax=1.15, + ytick={-1,0,1}, + yticklabels={$-\frac{U_\mathrm{dc}}{2}$,$0$,$\frac{U_\mathrm{dc}}{2}$}, + xticklabels={}, + grid=both, + axis lines=middle, + xlabel style={anchor=west}, + ylabel style={anchor=south}, + ] + \addplot[signalblue, thick] coordinates {(0,-1) (1/4,-1) (1/4,-1/3) (1/4+1/8,-1/3) (1/4+1/8,1/3) (2/4,1/3) (2/4,1) (3/4,1) (3/4,1/3) (3/4+1/8,1/3) (3/4+1/8,-1/3) (1,-1/3) (1,-1) (1.2,-1)}; + \end{axis} + \end{tikzpicture} + \caption{4-level half bridge} + \end{subfigure} + \end{figure} +\end{frame} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Rectifier operation for single-phase grids %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From 5e3a0c1b1614ef273373fb7e91b375a8e09f120e Mon Sep 17 00:00:00 2001 From: SevenOfNinePE Date: Fri, 24 Jan 2025 18:43:07 +0100 Subject: [PATCH 03/10] Ex07 Task2: Add calculations until subtask 4. --- .../ex07/Fig_ThreePhaseInverter_6StepMode.tex | 21 +-- exercise/tex/exercise07.tex | 124 +++++++++++++++++- 2 files changed, 133 insertions(+), 12 deletions(-) diff --git a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex index 71bf58d..ceea245 100644 --- a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex +++ b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex @@ -6,7 +6,7 @@ \begin{circuitikz} % 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 [short,o-o] ++(0.4,0) (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) @@ -51,17 +51,17 @@ % 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) + (ju2ae) to [sV=$u_\mathrm{2ae}$] ++(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) + (ju2be) to [sV=$u_\mathrm{2be}$] ++(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) + (ju2ce) to [sV=$u_\mathrm{2ce}$] ++(1.5,0) coordinate (ju2cn) % Add connection of u2in (ju2an) to [short,-*] (ju2bn) to [short,-] (ju2cn); @@ -80,16 +80,21 @@ (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 + % Add voltage arrow 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 + % Add voltage arrow u2ab (ju2ax) ++(0.2,0) to [open,v^=$$,voltage = straight] ++(0,-2.5) (ju2ax) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2ab}(t)$} - % Add voltage arrows u2bc + % Add voltage arrow u2bc (ju2b) ++(0.2,0) to [open,v^=$$,voltage = straight] ++(0,-2.5) - (ju2b) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2bc}(t)$}; + (ju2b) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2bc}(t)$} + % Add voltage arrow ua0 + (Gnd) ++(2,0) to [open,v^=$$,voltage = straight] ++(-1.6,0) + (Gnd) ++ (1.2,0.7) node[anchor=north,color=black]{$u_\mathrm{2a,0}(t)$}; + + \end{circuitikz} \end{center} diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index 1947238..9843a43 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -28,11 +28,11 @@ \centering % Center the table \begin{tabular}{ll} \toprule - Input voltages: & $U_{\mathrm{1p},i}=\SI{255}{\volt}$ \quad $U_{\mathrm{1m},i}=\SI{255}{\volt}$ \\ - Internal voltages: & $u_{\mathrm{1ae}(t)} = \sqrt{2} \cdot \SI{220}{\volt} \cdot \sin(\omega_1t)$ \\ + Input voltages: & $U_\mathrm{1p}=\SI{255}{\volt}$ \quad $U_\mathrm{1m}=\SI{255}{\volt}$ \\ + Internal voltages: & $u_{\mathrm{2ae}}(t) = \sqrt{2} \cdot \SI{220}{\volt} \cdot \sin(\omega_1t)$ \\ Circular frequency: & $\omega_1 = \SI{2 \pi \cdot 30}{\frac{1}{\second}}$ \\ Inductivity per phase: & $L= \SI{10}{\milli \henry}$ \\ - Phase angle between $u_{\mathrm{1ae}(t)}$ and $i_{\mathrm{1ae}^\mathrm{(1)}(t)}$ & $\alpha=\SI{30}{\degree}$ \\ + Phase angle between $u_{\mathrm{2ae}}(t)$ and $i_{\mathrm{2ae}^\mathrm{(1)}}(t)$ & $\alpha=\SI{30}{\degree}$ \\ \bottomrule \end{tabular} \caption{Parameters of three-phase inverter in six-step mode.} @@ -50,6 +50,50 @@ 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} + Each half bridge has got the 2 states '+1' and '-1', which results in $2^3 = 8$ combinations according table \autoref{stable:ex07_Task2_Switchingstates}. + The correct chronological order is displayed in table \autoref{stable:ex07_Task2_UsedSwitchingStates}. + \bigskip + \FloatBarrier + + \begin{minipage}{0.3\textwidth} + \begin{tabular}{|c|c|c|} % Each column is separated by a line + \hline + \bfseries $s_\mathrm{a}(t)$ & \bfseries $s_\mathrm{b}(t)$ & \bfseries $s_\mathrm{c}(t)$ \\ \hline + -1 & -1 & -1 \\ \hline + -1 & -1 & +1 \\ \hline + -1 & +1 & -1 \\ \hline + -1 & +1 & +1 \\ \hline + +1 & -1 & -1 \\ \hline + +1 & -1 & +1 \\ \hline + +1 & +1 & -1 \\ \hline + +1 & +1 & +1 \\ \hline + \end{tabular} + \noindent + \captionof{table}{Possible switching states.} + \label{stable:ex07_Task2_Switchingstates} + \end{minipage} + \hfill + \begin{minipage}{0.55\textwidth} + \begin{tabular}{|c|c|c|c|c|c|} % Each column is separated by a line + \hline + \bfseries $s_\mathrm{a}(t)$ & \bfseries $s_\mathrm{b}(t)$ & \bfseries $s_\mathrm{c}(t)$ + & \bfseries $+U_\mathrm{2a,0}$ & \bfseries $+U_\mathrm{2b,0}$ & \bfseries $+U_\mathrm{2c,0}$ \\ \hline + +1 & -1 & +1 & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ & $U_\mathrm{1p}$ \\ \hline + +1 & -1 & -1 & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ & $-U_\mathrm{1m}$ \\ \hline + +1 & +1 & -1 & $U_\mathrm{1p}$ & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ \\ \hline + -1 & +1 & -1 & $-U_\mathrm{1m}$ & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ \\ \hline + -1 & +1 & +1 & $-U_\mathrm{1m}$ & $U_\mathrm{1p}$ & $U_\mathrm{1p}$ \\ \hline + -1 & -1 & +1 & $-U_\mathrm{1m}$ & $-U_\mathrm{1m}$ & $U_\mathrm{1p}$ \\ \hline + \end{tabular} + \captionof{table}{Used switching states and voltages.} + \label{stable:ex07_Task2_UsedSwitchingStates} + \end{minipage} + \bigskip + \FloatBarrier + The switching state (-1,-1,-1) and (+1,+1,+1) are not used. In this case the chained voltages are zero. + This additional degree of freedom is applied at a higher switching frequency in order to reduce the amplitude of the + output voltage on average. In case of block switching, these switching states are not used, since + the switching only occurs twice per period. This results in the maximum possible voltage (square wave) at the output. \end{solutionblock} \subtask{The internal voltages $u_\mathrm{ea}(t)$, $u_\mathrm{eb}(t)$ and $u_\mathrm{ec}(t)$ are a symmetrical voltage system, @@ -57,11 +101,83 @@ 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} + In the case of a symmetrical three-phase consumer where the current sum at the consumer star point is zero, + the following results: + \begin{equation} + u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2c}(t)} = \SI{0}{\volt} \quad + i_{\mathrm{2a}(t)} + i_{\mathrm{2b}(t)} + i_{\mathrm{2c}(t)} = \SI{0}{\ampere}. + \label{eq:u2_i2_symgen} + \end{equation} + This leads to + \begin{equation} + u_{\mathrm{2a}}(t) = L \frac{\mathrm{d}i_{\mathrm{2a}}(t)}{\mathrm{d}t}+u_{\mathrm{2ae}}(t) + \quad u_{\mathrm{2a}}(t) = L \frac{\mathrm{d}i_{\mathrm{2a}}(t)}{\mathrm{d}t}+u_{\mathrm{2ae}}(t) + \quad u_{\mathrm{2a}}(t) = L \frac{\mathrm{d}i_{\mathrm{2a}}(t)}{\mathrm{d}t}+u_{\mathrm{2ae}}(t). + \label{eq:u2_i2_symL} + \end{equation} + Using \eqref{eq:u2_i2_symgen} leads to + \begin{equation} + u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2c}(t)} + = L \frac{\mathrm{d}}{\mathrm{d}t} \left( i_{\mathrm{2a}}(t)+i_{\mathrm{2b}}(t)+i_{\mathrm{2c}}(t) \right) + + \left( u_{\mathrm{2ae}}(t) + u_{\mathrm{2be}}(t) + u_{\mathrm{2ce}}(t)\right)=\SI{0}{\volt}. + \label{eq:u2_i2_symres} + \end{equation} + This derivation is valid under following conditions: + \begin{itemize} + \item The induction $L$ is constant. + \item The internal voltages $u_{\mathrm{2ae}}(t)$, $u_{\mathrm{2be}}(t)$ and $u_{\mathrm{2ce}}(t)$ + are purely sinusoidal (no harmonics), symmetrical (sum equals zero) and independent of the currents + $i_{\mathrm{2a}}(t)$, $i_{\mathrm{2b}}(t)$ and $i_{\mathrm{2c}}(t)$. + \end{itemize} \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)$ +\subtask{Calculate and sketch the voltages $u_\mathrm{2ab}(t)$, $u_\mathrm{2bc}(t)$, $u_\mathrm{2a}(t)$ and $u_\mathrm{2a,0}(t)$ depending on these switching states.} \begin{solutionblock} + The voltage $u_{\mathrm{2ab}(t)}$ is calculated by + \begin{equation} + u_{\mathrm{2ab}(t)} = u_{\mathrm{2a,0}(t)} - u_{\mathrm{2b,0}(t)}. + \label{eq:u2ab_gen} + \end{equation} + In similar way the voltage $u_{\mathrm{2bc}(t)}$ is calculated by + \begin{equation} + u_{\mathrm{2bc}(t)} = u_{\mathrm{2b,0}(t)} - u_{\mathrm{2c,0}(t)}. + \label{eq:u2bc_gen} + \end{equation} + The voltage $u_{\mathrm{2a}(t)}$ is obtained by + \begin{equation} + u_{\mathrm{2a}(t)} = u_{\mathrm{2ab}(t)} + u_{\mathrm{2b,0}(t)}. + \label{eq:u2a_1} + \end{equation} + Additional voltage $u_{\mathrm{2a}(t)}$ is obtained by + \begin{equation} + u_{\mathrm{2a}(t)} = u_{\mathrm{2ab}(t)} + u_{\mathrm{2bc}(t)} + u_{\mathrm{2c}(t)}. + \label{eq:u2a_2} + \end{equation} + The addition of \eqref{eq:u2a_1} and \eqref{eq:u2a_2} results in + \begin{equation} + 2u_{\mathrm{2a}(t)} = 2u_{\mathrm{2ab}(t)} + u_{\mathrm{2bc}(t)} + + \left( u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2v}(t)}\right) + - u_{\mathrm{2a}(t)}. + \label{eq:u2a_gen} + \end{equation} + Solving \eqref{eq:u2a_gen} with respect to $u_{\mathrm{2a}(t)}$ leads to + \begin{equation} + u_{\mathrm{2a}(t)} = \frac{2}{3} u_{\mathrm{2ab}(t)} + \frac{1}{3} u_{\mathrm{2bc}(t)} + \end{equation} + The voltage $u_{\mathrm{0,n}(t)}$ is obtained by + \begin{equation} + u_{\mathrm{0,n}(t)} = u_{\mathrm{2a,0}(t)} - u_{\mathrm{2a}(t)} + = u_{\mathrm{2a,0}(t)} - \frac{2}{3} u_{\mathrm{2ab}(t)} - \frac{1}{3} u_{\mathrm{2bc}(t)} + \end{equation} + Using \eqref{eq:u2ab_gen} and \eqref{eq:u2bc_gen} leads to + \begin{equation} + \begin{split} + u_{\mathrm{0,n}(t)} &= u_{\mathrm{2a,0}(t)} - \frac{2}{3} \left( u_{\mathrm{2a,0}(t)} - u_{\mathrm{2b,0}(t)} \right) + - \frac{1}{3} \left( u_{\mathrm{2b,0}(t)} - u_{\mathrm{2c,0}(t)} \right) \\ + u_{\mathrm{0,n}(t)} &= \frac{1}{3} \left( u_{\mathrm{2a,0}(t)} + u_{\mathrm{2b,0}(t)} + u_{\mathrm{2c,0}(t)} \right). + \end{split} + \end{equation} \end{solutionblock} \subtask{Decompose the voltage $u_\mathrm{a}(t)$ into a Fourier series and sketch the spectral lines related to the From 4ce7bd5cb0bc9d4749f6d55ed9d7c96c7c31e907 Mon Sep 17 00:00:00 2001 From: Wallscheid Date: Fri, 24 Jan 2025 20:17:56 +0100 Subject: [PATCH 04/10] add to lec06 --- lecture/main.ist | 2 +- lecture/main.tex | 2 +- lecture/tex/Lecture06.tex | 126 +++++++++++++++++++++++++++++++++++++- 3 files changed, 127 insertions(+), 3 deletions(-) diff --git a/lecture/main.ist b/lecture/main.ist index 47d31cf..dcc3373 100644 --- a/lecture/main.ist +++ b/lecture/main.ist @@ -1,5 +1,5 @@ % makeindex style file created by the glossaries package -% for document 'main' on 2025-1-23 +% for document 'main' on 2025-1-24 actual '?' encap '|' level '!' diff --git a/lecture/main.tex b/lecture/main.tex index b1191ec..7bb4bfa 100644 --- a/lecture/main.tex +++ b/lecture/main.tex @@ -5,7 +5,7 @@ \author{Oliver Wallscheid} \date{} -\includeonly{tex/Lecture06} % build only selected sections +%\includeonly{tex/temp.tex} % build only selected sections \begin{document} diff --git a/lecture/tex/Lecture06.tex b/lecture/tex/Lecture06.tex index e360800..7a30b28 100644 --- a/lecture/tex/Lecture06.tex +++ b/lecture/tex/Lecture06.tex @@ -92,6 +92,7 @@ \subsection{Single-phase AC/DC bridge converter} +1 & \text{upper position,}\\ -1 & \text{lower position.} \end{cases} + \label{eq:switching_function_VSI} \end{equation}}% \onslide<3->{Output voltage considering a voltage source at the input is: \begin{equation} @@ -179,7 +180,7 @@ \subsection{Single-phase AC/DC bridge converter} \draw let \p1 = (npn3.B) in node[anchor=east] at (\x1,\y1) {$T_3$}; \draw let \p1 = (npn4.B) in node[anchor=east] at (\x1,\y1) {$T_4$}; \end{circuitikz} - \caption{Full-bridge single phase AC/DC converter (identical to the one used in the DC/DC section in \figref{fig:DCDC-4Q-switch})} + \caption{Full-bridge single-phase AC/DC converter (identical to the one used in the DC/DC section in \figref{fig:DCDC-4Q-switch})} \label{fig:ACDC-4Q-switch} \end{figure} \end{column} @@ -1375,4 +1376,127 @@ \subsection{Rectifier operation for single-phase grids} \caption{Steady-state operation of the single-phase four-quadrant rectifier: (top) individual signals and (bottom) power oscillations at twice the grid frequency} \label{fig:recitifier_single_phase_transistor_power_oscillations} \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase AC/DC bridge converter %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Three-phase AC/DC bridge converter} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Idealized switch representation %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Idealized switch representation of a three-phase AC/DC bridge converter} + \begin{figure} + \begin{circuitikz} + \draw (0,0) node[cute spdt up arrow, xscale=-1] (Sw1) {}; + \draw (1.5,-1.5) node[cute spdt down arrow, xscale=-1] (Sw2) {}; + \draw (3,-3) node[cute spdt down arrow, xscale=-1] (Sw3) {}; + \draw (Sw2.in) to [short, -o] ++(3,0) coordinate (out2); + \draw (Sw1.in) to [short] ++(3.75,0) to [short, -o] (Sw1.in -| out2) coordinate (out1); + \draw (Sw3.in) to [short, -o, i=$i_{2\mathrm{c}}(t)$] (Sw3.in -| out2) coordinate (out3); + \draw (Sw3.in |- Sw2.in) to [short, -o, i=$i_{2\mathrm{b}}(t)$] (Sw2.in -| out2); + \draw (Sw3.in |- Sw1.in) to [short, -o, i=$i_{2\mathrm{a}}(t)$] (Sw1.in -| out2); + \draw (out1) to [open, v^=$\hspace{0.75cm}u_{2\mathrm{ab}}(t)$, voltage = straight] (out2); + \draw (out2) to [open, v^=$\hspace{0.75cm}u_{2\mathrm{bc}}(t)$, voltage = straight] (out3); + \draw ($(out3) + (1.5,0)$) to [open, v_=$\hspace{0.75cm}u_{2\mathrm{ca}}(t)$, voltage = straight] ($(out1) + (1.5,0)$); + \draw (Sw1-out 1.n) to [short, -*] ++(0,0.5) coordinate (int1); + \draw (Sw3-out 2.s) to [short] ++(0,-0.5) coordinate (int3); + \draw (int3) to [short] (int3 -| Sw1-out 2.s) coordinate (int4) to [short, *-] (Sw1-out 2.s); + \draw node[crossingshape, name=x1, rotate=-90] at (out1 -| Sw2-out 1.n) {}; + \draw (x1.west) to [short] (int1 -| x1.west) coordinate (int2) to [short] (int1); + \draw (x1.east) to [short] (Sw2-out 1.n); + \draw (int1) to [short] ++(-1,0) to [short, i_<=$i_1(t)$, -o] ++(-1,0) coordinate (in1); + \draw (int4) to [short, -o] ++(-2,0) coordinate (in2); + \draw (in1) to [open, v=$u_1(t)\hspace{0.5cm}$, voltage = straight] (in2); + \draw node[anchor = east, xshift=-0.3cm] at (Sw1) {$s_1(t)$}; + \draw node[anchor = east, xshift=-0.3cm] at (Sw2) {$s_2(t)$}; + \draw node[anchor = east, xshift=-0.3cm] at (Sw3) {$s_3(t)$}; + \draw (Sw2-out 2.s) to [short, -*] (int3 -| Sw2-out 2.s); + \draw node[crossingshape, name=x2, rotate=-90] at (out1 -| Sw3-out 1.n) {}; + \draw node[crossingshape, name=x3, rotate=-90] at (out2 -| Sw3-out 1.n) {}; + \draw (x3.west) to [short] (x2.east); + \draw (x3.east) to [short] (Sw3-out 1.n); + \draw (x2.west) to [short] (int2 -| x2.west) to [short,-*] (int2); + \end{circuitikz} + \caption{Idealized switch representation of a three-phase two-level AC/DC bridge converter} + \label{fig:idealized_switch_three_phase_bridge_converter} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Circuit realization %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Circuit realization} + \begin{figure} + \begin{circuitikz}[] + \draw (0,4) coordinate (A) to [capacitor, *-*, v = $\frac{u_1(t)}{2}$, voltage = straight] ++(0,-3) coordinate (Z) to [capacitor, *-*, v = $\frac{u_1(t)}{2}$, voltage = straight] ++(0,-3) coordinate (B) + (Z) node[rground, rotate=-90, name = gnd1]{} + (A) to [short, *-o] ++(-2,0) coordinate (Y) + (B) to [short, *-o] ++(-2,0) coordinate (X) + (Y) to [open, o-o, v = $u_1(t)\hspace{0.5cm}$, voltage = straight] (X) + (A) to [short, i=$i_{1}(t)$] ++(2,0) coordinate (E) + to [Tnpn, n=npn1, invert, bodydiode] ++(0,-2) coordinate (C) + to [short, *-] ++(1,0) to [crossing] ++(2,0) to [crossing] ++(2,0) + to [short, i=$i_{2\mathrm{a}}(t)$] ++(1,0) coordinate (U) to [short] ++(1,0) coordinate (G) + (C) to [short] ++(0,-2) + to [Tnpn, n=npn2, invert, bodydiode] ++(0,-2) coordinate (D) + (E) to [short, *-] ++(2,0) coordinate (I) + to [Tnpn, n=npn3, invert, bodydiode] ++(0,-2) + to [short] ++(0,-1) coordinate (F) + to [short] ++(0,-1) + to [Tnpn, n=npn4, invert, bodydiode] ++(0,-2) + to [short, -*] ++(-2,0) + to [short, -o] (B) + (I) to [short, *-] ++(2,0) + to [Tnpn, n=npn5, invert, bodydiode] ++(0,-2) coordinate (J) + to [short] ++(0,-2) coordinate (K) + to [Tnpn, n=npn6, invert, bodydiode] ++(0,-2) + to [short, -*] ++(-2,0) + (F) to [short, *-] ++(1,0) to [crossing] ++(2,0) to [short, i=$i_{2\mathrm{b}}(t)$] ++(1,0) coordinate (V) to [short] ++(1,0) coordinate (H) + (K) to [short, *-] ++(1,0) to [short, i=$i_{2\mathrm{c}}(t)$] ++(1,0) to [short] ++(0.5,0) coordinate(W) to [short] ++(0.5,0) coordinate (L) + (G) to [open, o-o, v^=$\hspace{0.75cm}u_{2\mathrm{ab}}(t)$, voltage = straight] (H) + (H) to [open, o-o, v^=$\hspace{0.75cm}u_{2\mathrm{bc}}(t)$, voltage = straight] (L) + ($(L) + (1.5,0)$) to [open, v_=$\hspace{0.75cm} u_{2\mathrm{bc}}(t)$, voltage = straight] ($(G) + (1.5,0)$); + \draw let \p1 = (npn1.B) in node[anchor=east] at (\x1,\y1) {$T_1$}; + \draw let \p1 = (npn2.B) in node[anchor=east] at (\x1,\y1) {$T_2$}; + \draw let \p1 = (npn3.B) in node[anchor=east] at (\x1,\y1) {$T_3$}; + \draw let \p1 = (npn4.B) in node[anchor=east] at (\x1,\y1) {$T_4$}; + \draw let \p1 = (npn5.B) in node[anchor=east] at (\x1,\y1) {$T_5$}; + \draw let \p1 = (npn6.B) in node[anchor=east] at (\x1,\y1) {$T_6$}; + \draw + (W |- D) node[rground, anchor = south, name = gnd2]{}; + \draw[->] ($(W) + (0.5,-0.2)$) to ($(W |- gnd2) + (0.5,0.2)$); + \draw[->] ($(V) + (0.5,-0.2)$) to ($(V |- gnd2) + (0.5,0.2)$); + \draw[->] ($(U) + (0,-0.2)$) to ($(U |- gnd2) + (0,0.2)$); + \draw ($(W)!0.5!(gnd2) + (0.5,0)$) node[anchor = west] {$u_{2i0}(t)$}; + \end{circuitikz} + \caption{Three-phase two-level AC/DC converter} + \label{fig:VSI_three_phase_two_level_bridge_converter} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Switching states and load-independent output voltages %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Switching states and load-independent output voltages} + Reutilizing the switching function definition \eqref{eq:switching_function_VSI}, the \hl{line-to-line voltages} can be expressed as + \begin{equation} + \begin{split} + u_{2\mathrm{ab}}(t) &= \frac{1}{2}\left(s_{\mathrm{a}}(t)-s_{\mathrm{b}}(t)\right)u_1(t),\\ + u_{2\mathrm{bc}}(t) &= \frac{1}{2}\left(s_{\mathrm{b}}(t)-s_{\mathrm{c}}(t)\right)u_1(t),\\ + u_{2\mathrm{ca}}(t) &= \frac{1}{2}\left(s_{\mathrm{c}}(t)-s_{\mathrm{a}}(t)\right)u_1(t). + \end{split} + \end{equation} + The \hl{line-to-neutral voltages} are given by + \begin{equation} + \begin{split} + u_{2\mathrm{a}0}(t) &= \frac{1}{2}s_{\mathrm{a}}(t)u_1(t),\\ + u_{2\mathrm{b}0}(t) &= \frac{1}{2}s_{\mathrm{b}}(t)u_1(t),\\ + u_{2\mathrm{c}0}(t) &= \frac{1}{2}s_{\mathrm{c}}(t)u_1(t). + \end{split} + \end{equation} \end{frame} \ No newline at end of file From 22e365826578b7d7b20edb4f9d67f6f7d770002b Mon Sep 17 00:00:00 2001 From: "[SilasElter]" <[SilasElter]> Date: Fri, 24 Jan 2025 22:30:31 +0100 Subject: [PATCH 05/10] Add Figures ex07 task2 --- .../Fig_ trigonometric_approach_triangle.tex | 25 +++++++++++++ .../fig/ex07/Fig_Voltage_U_um_excerpt.tex | 4 +- .../ex07/Fig_graphic_solutions_cos_terms.tex | 37 +++++++++++++------ ...Fig_standardization_to_fudamental_freq.tex | 27 ++++++++++++++ exercise/tex/exercise07.tex | 2 + 5 files changed, 82 insertions(+), 13 deletions(-) create mode 100644 exercise/fig/ex07/Fig_ trigonometric_approach_triangle.tex create mode 100644 exercise/fig/ex07/Fig_standardization_to_fudamental_freq.tex diff --git a/exercise/fig/ex07/Fig_ trigonometric_approach_triangle.tex b/exercise/fig/ex07/Fig_ trigonometric_approach_triangle.tex new file mode 100644 index 0000000..2b548df --- /dev/null +++ b/exercise/fig/ex07/Fig_ trigonometric_approach_triangle.tex @@ -0,0 +1,25 @@ +\begin{solutionfigure} +\centering +\begin{tikzpicture}[scale=1.2] + % Basislinie (horizontale Achse) + \draw[black, thick] (0,0) -- (6,0) node[midway, below] {\scriptsize \SI{311.13}{\volt}}; + + + % Oberer Vektor (mit Winkel) + \draw[black, thick] (0,0) -- (6.5,0.4) node[midway, above] {\scriptsize \SI{324.68}{\volt}}; + \def\drawArc#1#2#3{ + \draw[thick] (#1:0) -- (#1:#3) arc (#1:#2:#3) -- cycle; + } + % Beispiel: Kreissegment von 0 bis 60 Grad mit Radius 0.6 + \drawArc{0}{3.5}{3}{color=blue!70!black}; + \draw[blue,thick] (6.5,0.4) -- (6,0); + + \node at (6.2,0.11) [left] {\tiny \SI{120}{\degree}}; + \node at (6.7,0.11) [left] {\tiny \SI{60}{\degree}}; + \node at (3.5,0.11) {\tiny \SI{3.913}{\degree}}; + + +\end{tikzpicture} +\caption{Illustration for determining the angle using the sine theorem.} +\label{fig:Illustration for determining the angle using the sine theorem} +\end{solutionfigure} diff --git a/exercise/fig/ex07/Fig_Voltage_U_um_excerpt.tex b/exercise/fig/ex07/Fig_Voltage_U_um_excerpt.tex index e06280b..3091db1 100644 --- a/exercise/fig/ex07/Fig_Voltage_U_um_excerpt.tex +++ b/exercise/fig/ex07/Fig_Voltage_U_um_excerpt.tex @@ -12,7 +12,7 @@ axis line style={->}, % Pfeilspitzen an den Achsen xlabel={$\omega t$}, ylabel={$U_\mathrm{d}$}, - xmin=0, xmax=2*pi, + xmin=0, xmax=13/6*pi, ymin=-1, ymax=1, xtick={0, pi/3, 2*pi/3, pi, 4*pi/3, 5*pi/3, 2*pi}, xticklabels={0, $\frac{1\pi}{3}$, $\frac{2\pi}{3}$,$\pi$, $\frac{4\pi}{3}$, $\frac{5\pi}{3}$, $2\pi$}, @@ -27,7 +27,7 @@ mark=none, color=black, ] coordinates { - (0,1/3) (1/3*pi, 1/3) (1/3*pi, 2/3)( 2/3*pi, 2/3) (2/3*pi, 1/3) (pi, 1/3) (pi, -1/3) (4/3*pi, -1/3) (4/3*pi, -1/3) (4/3*pi, -2/3) (5/3*pi, -2/3) (5/3*pi, -1/3) (2*pi, -1/3) (2*pi, 1/3) + (0,1/3) (1/3*pi, 1/3) (1/3*pi, 2/3)( 2/3*pi, 2/3) (2/3*pi, 1/3) (pi, 1/3) (pi, -1/3) (4/3*pi, -1/3) (4/3*pi, -1/3) (4/3*pi, -2/3) (5/3*pi, -2/3) (5/3*pi, -1/3) (2*pi, -1/3) (2*pi, 1/3) (13/6*pi, 1/3) }; \end{axis} \end{tikzpicture} diff --git a/exercise/fig/ex07/Fig_graphic_solutions_cos_terms.tex b/exercise/fig/ex07/Fig_graphic_solutions_cos_terms.tex index e583461..083c8b7 100644 --- a/exercise/fig/ex07/Fig_graphic_solutions_cos_terms.tex +++ b/exercise/fig/ex07/Fig_graphic_solutions_cos_terms.tex @@ -2,7 +2,7 @@ % SwitchOnBehaviorAndSwitchOffBehaviorOfUI %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{figure}[htb] +\begin{solutionfigure}[htb] \centering \begin{minipage}[t]{0.45\textwidth} \centering @@ -55,20 +55,34 @@ \begin{tikzpicture} \draw[->] (-2,0) -- (2,0) node[right] {}; \draw[->] (0,-2) -- (0,2) node[above] {}; - \node at (-1.5,1.5) {$\cos\left(k \frac{\pi}{3}\right)$}; - \node at (-2,0) [left] {$2,6,\ldots$}; - \node at (2,0) [right] {$0,4,\ldots$}; - \node at (0,2) [above] {$1,5,9,13,\ldots$}; - \node at (0,-2) [below] {$3,7,11,15,\ldots$}; - + \node at (-2.5,1.5) {$\cos\left(k \frac{\pi}{3}\right)$}; + \node at (-1,0.2) [left] {$3,9,15,21\ldots$}; + \node at (1,0.2) [right] {$0,6\ldots$}; + \node at (2,1) [above] {$1,7,13,19\ldots$}; + \node at (-0.8,1.6) [below] {$2,8\ldots$}; + \node at (-0.8,-1) [below] {$4,10\ldots$}; + \node at (2,-1) [below] {$5,11,17,23\ldots$}; + \node at (-0.75,-0.3) [left] {-1}; + \node at (1,-0.3) [left] {0.5}; % Kreuz bei jedem markierten Punkt - \foreach \x/\y in {-1.5/0, 1.5/0, 0/1.5, 0/-1.5} { + \foreach \x/\y in {-1/0, 1/0, 0.5/1, -0.5/-1, 0.5/-1, -0.5/1} { \draw[thick] (\x,\y) +(-0.1,0.1) -- +(0.1,-0.1) % Diagonale des Kreuzes +(-0.1,-0.1) -- +(0.1,0.1); % Andere Diagonale des Kreuzes } + \draw[thick, color=blue!70!black] + (-0.5,1) -- (0.5,-1) -- cycle; + \draw[thick, color=blue!70!black] + (0.5,1) -- (-0.5,-1) -- cycle; + \node[shift={(0.1,0)}, anchor=west] at ({0.5*cos(60)}, {0.5*sin(60)}) [below] {\tiny $60^\circ$}; + \def\drawArc#1#2#3{ + \draw[thick] (#1:0) -- (#1:#3) arc (#1:#2:#3) -- cycle; + } + % Beispiel: Kreissegment von 0 bis 60 Grad mit Radius 0.6 + \drawArc{0}{63}{0.6}{thick, color=blue!70!black}; + \end{tikzpicture} - \hspace{1cm} % Abstand zwischen den beiden Diagrammen + \hspace{2.3cm} % Abstand zwischen den beiden Diagrammen \begin{tikzpicture} % Koordinatensystem zeichnen \draw[->] (-2,0) -- (2,0) node[right] {}; @@ -76,7 +90,8 @@ \node at (-1.5,1.5) {$\cos\left(k \frac{\pi}{3}\right)+1$}; % Beschriftungen an der x-Achse - \foreach \x in {0, 1, 1.5} { + \node at (-0.00009,-0.2) [left] {0}; + \foreach \x in { 1, 1.5} { \node at (\x, 0) [below] {\x}; } @@ -97,4 +112,4 @@ \end{tikzpicture} \caption{Graphical solution of the cos terms.} \label{fig:Graphical solution of the cos terms} -\end{figure} +\end{solutionfigure} diff --git a/exercise/fig/ex07/Fig_standardization_to_fudamental_freq.tex b/exercise/fig/ex07/Fig_standardization_to_fudamental_freq.tex new file mode 100644 index 0000000..4060230 --- /dev/null +++ b/exercise/fig/ex07/Fig_standardization_to_fudamental_freq.tex @@ -0,0 +1,27 @@ +\begin{solutionfigure}[htb] +\centering +\begin{tikzpicture} + % Achsen zeichnen + \draw[->] (0,0) -- (14,0) node[right] {$k$}; % x-Achse + \draw[->] (0,0) -- (0,5) node[above] {$\frac{\hat{u}_\mathrm{UM,k}}{\hat{u}_\mathrm{UM,1}}$}; % y-Achse + + % Ticks und Beschriftungen auf der x-Achse + \foreach \x in {1, 5, 7, 11, 13} { + \draw (\x,0.05) -- (\x,-0.05) node[below] {\x}; + } + \foreach \x in {2, 3, 4, 6, 8, 9, 10, 12} { + \draw (\x,0.02) -- (\x,-0.02); % kleinere Ticks + } + + % Ticks und Beschriftung auf der y-Achse + \draw (-0.05,4.5) -- (0.05,4.5) node[left] {1}; + + % Balken zeichnen + \foreach \x/\y in {1/4.5, 5/0.9, 7/0.6, 11/0.5, 13/0.4} { + \draw[thick] (\x,0) -- (\x,\y); % Balken + \draw[thick] (\x-0.1,\y) -- (\x+0.1,\y); % Querstrich oben + } +\end{tikzpicture} +\caption{Normalization to the amplitude of the fundamental oscillation.} +\label{fig:Normalization to the amplitude of the fundamental oscillation} +\end{solutionfigure} diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index 2d95368..62c982d 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -27,4 +27,6 @@ \begin{solutionblock} \input{fig/ex07/Fig_Voltage_U_um_excerpt} \input{fig/ex07/Fig_graphic_solutions_cos_terms} +\input{fig/ex07/Fig_standardization_to_fudamental_freq.tex} +\input{fig/ex07/Fig_ trigonometric_approach_triangle.tex} \end{solutionblock} \ No newline at end of file From 50179474cd2110907985eb8d61be19a57a4e99d1 Mon Sep 17 00:00:00 2001 From: Wallscheid Date: Sat, 25 Jan 2025 20:00:07 +0100 Subject: [PATCH 06/10] add to lec06 --- lecture/main.ist | 2 +- lecture/tex/Lecture06.tex | 169 +++++++++++++++++++++++++++++++++++++- 2 files changed, 169 insertions(+), 2 deletions(-) diff --git a/lecture/main.ist b/lecture/main.ist index dcc3373..10ab6d6 100644 --- a/lecture/main.ist +++ b/lecture/main.ist @@ -1,5 +1,5 @@ % makeindex style file created by the glossaries package -% for document 'main' on 2025-1-24 +% for document 'main' on 2025-1-25 actual '?' encap '|' level '!' diff --git a/lecture/tex/Lecture06.tex b/lecture/tex/Lecture06.tex index 7a30b28..cb451e1 100644 --- a/lecture/tex/Lecture06.tex +++ b/lecture/tex/Lecture06.tex @@ -1499,4 +1499,171 @@ \subsection{Three-phase AC/DC bridge converter} u_{2\mathrm{c}0}(t) &= \frac{1}{2}s_{\mathrm{c}}(t)u_1(t). \end{split} \end{equation} -\end{frame} \ No newline at end of file +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase converter with symmetric load in star connection %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase converter with symmetric load in star connection} + \begin{figure} + \begin{circuitikz}[] + \draw (0,4) coordinate (A) to [capacitor, *-*, v = $\frac{u_1(t)}{2}$, voltage = straight] ++(0,-3) coordinate (Z) to [capacitor, *-*, v = $\frac{u_1(t)}{2}$, voltage = straight] ++(0,-3) coordinate (B) + (Z) node[rground, rotate=-90, name = gnd1]{} + (A) to [short, *-o] ++(-2,0) coordinate (Y) + (B) to [short, *-o] ++(-2,0) coordinate (X) + (Y) to [open, o-o, v = $u_1(t)\hspace{0.5cm}$, voltage = straight] (X) + (A) to [short, i=$i_{1}(t)$] ++(2,0) coordinate (E) + to [Tnpn, n=npn1, invert, bodydiode] ++(0,-2) coordinate (C) + to [short, *-] ++(1,0) to [crossing] ++(2,0) to [crossing] ++(2,0) + to [short, i=$i_{2\mathrm{a}}(t)$] ++(1,0) coordinate (U) to [short] ++(1,0) coordinate (G) + (C) to [short] ++(0,-2) + to [Tnpn, n=npn2, invert, bodydiode] ++(0,-2) coordinate (D) + (E) to [short, *-] ++(2,0) coordinate (I) + to [Tnpn, n=npn3, invert, bodydiode] ++(0,-2) + to [short] ++(0,-1) coordinate (F) + to [short] ++(0,-1) + to [Tnpn, n=npn4, invert, bodydiode] ++(0,-2) + to [short, -*] ++(-2,0) + to [short, -o] (B) + (I) to [short, *-] ++(2,0) + to [Tnpn, n=npn5, invert, bodydiode] ++(0,-2) coordinate (J) + to [short] ++(0,-2) coordinate (K) + to [Tnpn, n=npn6, invert, bodydiode] ++(0,-2) + to [short, -*] ++(-2,0) + (F) to [short, *-] ++(1,0) to [crossing] ++(2,0) to [short, i=$i_{2\mathrm{b}}(t)$] ++(1,0) coordinate (V) to [short] ++(1,0) coordinate (H) + (K) to [short, *-] ++(1,0) to [short, i=$i_{2\mathrm{c}}(t)$] ++(1,0) to [short] ++(0.5,0) coordinate(W) to [short] ++(0.5,0) coordinate (L); + \draw (H) to [inductor, v^=$u_{2\mathrm{b}}(t)$, voltage=straight] ++(1.5,0) to [short,-*] ++(1,0) coordinate (N) + (L) to [inductor, v^=$u_{2\mathrm{c}}(t)$, voltage=straight] ++(1.5,0) to [short] (\tikztostart) -| (N) + (G) to [inductor, v^=$u_{2\mathrm{a}}(t)$, voltage=straight] ++(1.5,0) to [short] (\tikztostart) -| (N); + \draw let \p1 = (npn1.B) in node[anchor=east] at (\x1,\y1) {$T_1$}; + \draw let \p1 = (npn2.B) in node[anchor=east] at (\x1,\y1) {$T_2$}; + \draw let \p1 = (npn3.B) in node[anchor=east] at (\x1,\y1) {$T_3$}; + \draw let \p1 = (npn4.B) in node[anchor=east] at (\x1,\y1) {$T_4$}; + \draw let \p1 = (npn5.B) in node[anchor=east] at (\x1,\y1) {$T_5$}; + \draw let \p1 = (npn6.B) in node[anchor=east] at (\x1,\y1) {$T_6$}; + \draw (W |- D) node[rground, anchor = south, name = gnd2]{}; + \draw[->] ($(W) + (0.5,-0.2)$) to ($(W |- gnd2) + (0.5,0.2)$); + \draw[->] ($(V) + (0.5,-0.2)$) to ($(V |- gnd2) + (0.5,0.2)$); + \draw[->] ($(U) + (0,-0.2)$) to ($(U |- gnd2) + (0,0.2)$); + \draw ($(W)!0.5!(gnd2) + (0.5,0)$) node[anchor = west] {$u_{2i0}(t)$}; + \draw[->] ($(N) + (0,-1.25)$) to ($(N |- gnd2) + (0,0.2)$); + \draw ($(W)!0.5!(gnd2) + (1.8,0)$) node[anchor = west] {$u_{\mathrm{n}0}(t)$}; + \draw (N |- D) node[rground, anchor = south, name = gnd3]{}; + \end{circuitikz} + \caption{Three-phase two-level AC/DC converter with symmetric load in star connection} + \label{fig:VSI_three_phase_two_level_bridge_converter_load_star} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase converter with symmetric load in star connection (cont.) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase converter with symmetric load in star connection (cont.)} + Assuming a star-connected load, the three-phase currents sum up to zero: + \begin{equation} + i_{2\mathrm{a}}(t) + i_{2\mathrm{b}}(t) + i_{2\mathrm{c}}(t) = 0. + \label{eq:three_phase_current_sum_star} + \end{equation} + If the star point is not connected to ground, $u_{\mathrm{n}0}(t)\neq 0$ may occur leading to a load voltage of + \begin{equation} + u_{2\mathrm{a}}(t) = u_{2\mathrm{a}0}(t) - u_{\mathrm{n}0}(t), \quad u_{2\mathrm{b}}(t) = u_{2\mathrm{b}0}(t) - u_{\mathrm{n}0}(t), \quad u_{2\mathrm{c}}(t) = u_{2\mathrm{c}0}(t) - u_{\mathrm{n}0}(t). + \end{equation} + To calculate $u_{\mathrm{n}0}(t)$ one can utilize the load equation (assuming an inductive load): + \begin{equation} + u_{2i}(t) = L \frac{\mathrm{d}}{\mathrm{d}t} i_{2i}(t) + u_{\mathrm{n}0}(t) + \end{equation} + summing up to + \begin{equation} + 3u_{\mathrm{n}0}(t) + L \frac{\mathrm{d}}{\mathrm{d}t} \left(i_{2\mathrm{a}}(t) + i_{2\mathrm{b}}(t) + i_{2\mathrm{c}}(t)\right) = u_{2\mathrm{a}0}(t) + u_{2\mathrm{b}0}(t) +u_{2\mathrm{c}0}(t) + \end{equation} + and finally delivering the star-to-ground voltage as + \begin{equation} + u_{\mathrm{n}0}(t) = \frac{1}{3} \left(u_{2\mathrm{a}0}(t) + u_{2\mathrm{b}0}(t) +u_{2\mathrm{c}0}(t)\right). + \end{equation} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase converter with symmetric load in star connection (cont.) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}[b] + \frametitle{Three-phase converter with symmetric load in star connection (cont.)} + \vspace{1em} + \begin{table} + \renewcommand{\arraystretch}{1.25} + \centering + \begin{tabular}{c c c c c c c c c c c c c c} + \toprule + $\mathrm{No.}$ & $s_\mathrm{a}$ & $s_\mathrm{b}$ & $s_\mathrm{c}$ & $\frac{u_{2a0}}{u_1}$ & $\frac{u_{2b0}}{u_1}$ & $\frac{u_{2c0}}{u_1}$ & $\frac{u_{2a}}{u_1}$ & $\frac{u_{2b}}{u_1}$ & $\frac{u_{2c}}{u_1}$ & $\frac{u_{\mathrm{ab}}}{u_1}$ & $\frac{u_{\mathrm{bc}}}{u_1}$ & $\frac{u_{\mathrm{ca}}}{u_1}$ & $\frac{u_{\mathrm{n}0}}{u_1}$ \\ + \midrule + $0$ & $-1$ & $-1$ & $-1$ & $-\frac{1}{2}$ & $-\frac{1}{2}$ & $-\frac{1}{2}$ & $0$ & $0$ & $0$ & $0$& $0$& $0$&$-\frac{1}{2}$\\ + + $1$ & $+1$ & $-1$ & $-1$ & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $-\frac{1}{2}$ & $+\frac{2}{3}$ & $-\frac{1}{3}$ & $-\frac{1}{3}$ & $+1$& $0$& $-1$& $-\frac{1}{6}$\\ + + $2$ & $+1$ & $+1$ & $-1$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $+\frac{1}{3}$ & $+\frac{1}{3}$ & $-\frac{2}{3}$ & $0$& $+1$& $-1$& $+\frac{1}{6}$\\ + + $3$ & $-1$ & $+1$ & $-1$ & $-\frac{1}{2}$ & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $-\frac{1}{3}$ & $+\frac{2}{3}$ & $-\frac{1}{3}$ & $-1$& $+1$& $0$& $-\frac{1}{6}$\\ + + $4$ & $-1$ & $+1$ & $+1$ & $-\frac{1}{2}$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ & $-\frac{2}{3}$ & $+\frac{1}{3}$ & $+\frac{1}{3}$ & $-1$& $0$& $+1$& $+\frac{1}{6}$\\ + + $5$ & $-1$ & $-1$ & $+1$ & $-\frac{1}{2}$ & $-\frac{1}{2}$ & $+\frac{1}{2}$ & $-\frac{1}{3}$ & $-\frac{1}{3}$ & $+\frac{2}{3}$ & $0$& $-1$& $1$& $-\frac{1}{6}$\\ + + $6$ & $+1$ & $-1$ & $+1$ & $+\frac{1}{2}$ & $-\frac{1}{2}$ & $+\frac{1}{2}$ & $+\frac{1}{3}$ & $-\frac{2}{3}$ & $+\frac{1}{3}$ & $1$& $-1$& $0$& $+\frac{1}{6}$\\ + + $7$ & $+1$ & $+1$ & $+1$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ & $+\frac{1}{2}$ & $0$ & $0$ & $0$ & $0$& $0$& $0$& $+\frac{1}{2}$\\ + + \bottomrule + + \end{tabular} + \caption{Switching states and resulting voltages of the three-phase two-level AC/DC converter with symmetric load in star connection (with $2^3=8$ possible switching states)} + \label{tab:VSI_three_phase_two_level_bridge_converter_load_star} + \end{table} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase fundamental frequency modulation %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase fundamental frequency modulation} + \vspace{-0.1cm} + \begin{figure} + \begin{tikzpicture} + \def\a{0.33*pi} + \def\it{0.1*pi} + \begin{groupplot}[group style={group size=1 by 5, xticklabels at = edge bottom, vertical sep=0.25cm}, height=0.32\textheight, width=0.875\textwidth, xmin=0, xmax=3*pi, grid,clip = false, ymin = -0.8, ymax =0.8, xtick = {0, pi/2, pi, 3/2*pi, 2*pi, 5/2*pi, 3*pi}, xticklabels = {$0$,$\nicefrac{1}{2}\pi$, $\pi$, $\nicefrac{3}{2}\pi$, $2\pi$, $\nicefrac{5}{2}\pi$, $3\pi$}, ytick = {-1/2, 0, 1/2}, yticklabels = {$-\nicefrac{1}{2}$, , $\nicefrac{1}{2}$}, ylabel style={rotate=-90}] + + % ua0 + \nextgroupplot[ylabel = {$u_{\mathrm{a}0}(t)/U_1$}] + \addplot[signalblue, thick] coordinates {(0,-1/2) (\a,-1/2) (\a,1/2) (\a+pi,1/2) (\a+pi,-1/2) (2*pi+\a,-1/2) (2*pi+\a,1/2) (3*pi,1/2)}; + \addplot[signalblue, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-\a))}; + \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{a}0}=\frac{2}{\pi}$}; + \draw[->] (axis cs:0,0) -- node[above, fill=white, inner sep=1pt]{$\alpha$} (axis cs:\a,0); + + % ub0 + \nextgroupplot[ylabel = {$u_{\mathrm{b}0}(t)/U_1$}] + \addplot[signalgreen, thick] coordinates {(0,-1/2) (pi,-1/2) (pi,1/2) (2*pi,1/2) (2*pi,-1/2) (3*pi,-1/2)}; + \addplot[signalgreen, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(-x))}; + \draw[thin] (pi/3*5+\a+0.1,0.0) -- (pi/3*5+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{b}0}=\frac{2}{\pi}$}; + + % uc0 + \nextgroupplot[ylabel = {$u_{\mathrm{c}0}(t)/U_1$}] + \addplot[signalbrown, thick] coordinates {(0,1/2) (\a+pi/3,1/2) (\a+pi/3,-1/2) (\a+4*pi/3,-1/2) (\a+4*pi/3,1/2) (\a+7*pi/3,1/2) (\a+7*pi/3,-1/2) (3*pi,-1/2)}; + \addplot[signalbrown, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-4/3*pi-\a))}; + \draw[thin] (pi/3+\a+0.1,0.0) -- (pi/3+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{c}0}=\frac{2}{\pi}$}; + + % un0 + \nextgroupplot[ylabel = {$u_{\mathrm{n}0}(t)/U_1$}, height=0.27\textheight, ymin = -1/5, ymax =1/5, ytick = {-1/6, 0, 1/6}, yticklabels = {$-\nicefrac{1}{6}$, , $\nicefrac{1}{6}$}] + \addplot[signallavender, thick] coordinates {(0,-1/6) (pi/3,-1/6) (pi/3,1/6) (2*pi/3,1/6) (2*pi/3,-1/6) (3*pi/3,-1/6) (3*pi/3,1/6) (4*pi/3,1/6) (4*pi/3,-1/6) (5*pi/3,-1/6) (5*pi/3,1/6) (6*pi/3,1/6) (6*pi/3,-1/6) (7*pi/3,-1/6) (7*pi/3,1/6) (8*pi/3,1/6) (8*pi/3,-1/6) (9*pi/3,-1/6) (9*pi/3,1/6)}; + + % ua0 + \nextgroupplot[ylabel = {$u_{\mathrm{a}}(t)/U_1$}, , ytick = {-2/3, 0, 2/3}, yticklabels = {$-\nicefrac{2}{3}$, , $\nicefrac{2}{3}$}, xlabel={$\omega t$}] + \addplot[signalblue, thick] coordinates {(0,-1/3) (pi/3,-1/3) (pi/3,1/3) (2*pi/3,1/3) (2*pi/3,2/3) (3*pi/3,2/3) (3*pi/3,1/3) (4*pi/3,1/3) (4*pi/3,-1/3) (5*pi/3,-1/3) (5*pi/3,-2/3) (6*pi/3,-2/3) (6*pi/3,-1/3) (7*pi/3,-1/3) (7*pi/3,1/3) (8*pi/3,1/3) (8*pi/3,2/3) (9*pi/3,2/3)}; + \addplot[signalblue, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-\a))}; + \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{a}}=\hat{u}^{(1)}_{\mathrm{a}0}=\frac{2}{\pi}$}; + \draw[->] (axis cs:0,0) -- node[above, fill=white, inner sep=1pt]{$\alpha$} (axis cs:\a,0); + + \end{groupplot} + \end{tikzpicture} + \end{figure} +\end{frame} From 28d39a8b4b38439b409386b7eff0c65854e5a017 Mon Sep 17 00:00:00 2001 From: Wallscheid Date: Sun, 26 Jan 2025 17:45:36 +0100 Subject: [PATCH 07/10] add to lec06 --- .../lec06/PWM_three-phase_mod05_example.csv | 142 +++++++++ .../lec06/PWM_three-phase_mod1_example.csv | 137 +++++++++ .../lec06/PWM_three-phase_overmod_example.csv | 98 +++++++ lecture/fig/lec06/PWM_three_phase_ACDC.py | 123 ++++++++ lecture/main.ist | 2 +- lecture/main.tex | 2 +- lecture/tex/Lecture06.tex | 276 +++++++++++++++--- lecture/tex/dict.tex | 5 + lecture/tex/nomen.tex | 14 + 9 files changed, 764 insertions(+), 35 deletions(-) create mode 100644 lecture/fig/lec06/PWM_three-phase_mod05_example.csv create mode 100644 lecture/fig/lec06/PWM_three-phase_mod1_example.csv create mode 100644 lecture/fig/lec06/PWM_three-phase_overmod_example.csv create mode 100644 lecture/fig/lec06/PWM_three_phase_ACDC.py diff --git a/lecture/fig/lec06/PWM_three-phase_mod05_example.csv b/lecture/fig/lec06/PWM_three-phase_mod05_example.csv new file mode 100644 index 0000000..9fdb2da --- /dev/null +++ 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b/lecture/fig/lec06/PWM_three_phase_ACDC.py @@ -0,0 +1,123 @@ +import numpy as np +import os +from scipy.signal import find_peaks + +################################################### +# Some parameters # +################################################### + +# define the interval x=0...2*pi with xn points +xn = 10000 +x = np.linspace(0, 2*np.pi, xn) + +# pulse number (number of carrier periods per fundamental period) +N = 10 + +################################################### +# Helper functions / function definitions # +################################################### + +# generate a sawtooth carrier sequence c(w*t) between +1, -1 +def c_saw(wt, N): + return 1 - 2 * np.abs(np.modf((wt *N)/(2*np.pi))[0]) + +# generate a triangular carrier sequence c(w*t) between +1, -1 +def c_tri(wt, N): + return 4*(np.abs(np.modf((wt *N)/(2*np.pi))[0] - 0.5)) -1 + +# generate a fundamental and normalized sinusoidal reference d(t) +def d_sin(wt, phi = 0): + return np.sin(wt - phi) + +# generate a fundamental and normalized sinusoidal reference d(t) with overmod. amplitude +def d_sin_overmod(wt, phi = 0): + return 1.18*np.sin(wt - phi) + +# calculate complementary PWM-based switching signals +def s_comp(d, c): + return np.where(d>c, 1, -1) + +#calculate the integrated / summed error between the reference and the switching signal +def e(d, s, xn): + return np.cumsum(d - s)/xn*2*np.pi + + +################################################### +# Complementary switching PWM example # +################################################### + +c_example = c_tri(x, N) +d_a_example = d_sin(x) +d_b_example = d_sin(x, np.pi/3*2) +d_c_example = d_sin(x, np.pi/3*4) +s_a_example = s_comp(d_a_example, c_example) +s_b_example = s_comp(d_b_example, c_example) +s_c_example = s_comp(d_c_example, c_example) + +# e_comp_example = e(d_comp_example, (s_comp_example[0] - s_comp_example[1])/2, xn) + +# Compute the derivative of the signal +s_diff = np.diff(s_a_example) + +# Identify step indices (where derivative is non-zero) +step_indices = np.where(s_diff != 0)[0] + +# Find the two nearest samples for each step +nearest_samples = [0] +for step in step_indices: + nearest_samples.append(step) + nearest_samples.append(step+1) + +s_diff = np.diff(s_b_example) +step_indices = np.where(s_diff != 0)[0] +for step in step_indices: + nearest_samples.append(step) + nearest_samples.append(step+1) + +s_diff = np.diff(s_c_example) +step_indices = np.where(s_diff != 0)[0] +for step in step_indices: + nearest_samples.append(step) + nearest_samples.append(step+1) + +# identify peaks of the carrier +upper_peaks, _ = find_peaks(c_example) +lower_peaks, _ = find_peaks(-c_example) + +#combine the three sets of indices and add the first and last sample +idx_sum = np.unique(np.concatenate((nearest_samples, upper_peaks, lower_peaks, [0, len(x)-1]))) + + +# save the reduced data to a csv file +current_directory = os.path.dirname(os.path.abspath(__file__)) +save_path = os.path.join(current_directory, 'PWM_three-phase_mod1_example.csv') +np.savetxt(save_path, np.column_stack((x[idx_sum], s_a_example[idx_sum], s_b_example[idx_sum], s_c_example[idx_sum], d_a_example[idx_sum], d_b_example[idx_sum], d_c_example[idx_sum], c_example[idx_sum])), delimiter=',', header='wt, s1, s2, s3, d1, d2, d3, c', comments='') + + +################################################### + +# generate an exmaple carrier signal and plot it for N=10 +import matplotlib.pyplot as plt +plt.plot(x, c_example) +plt.plot(x, d_a_example) +plt.plot(x, d_b_example) +plt.plot(x, d_c_example) +plt.xlabel(r'$\omega t$') +plt.ylabel(r'$c(\omega t)$') +plt.title('Triangular carrier signal') +plt.grid() +plt.show() + +# add subplot for switching signal +plt.plot(x, s_a_example) +plt.plot(x, s_c_example) +plt.plot(x, s_b_example) +plt.xlabel(r'$\omega t$') +plt.ylabel(r'$s(\omega t)$') +plt.title('Switching signal') +plt.grid() +plt.show() + + + + diff --git a/lecture/main.ist b/lecture/main.ist index 10ab6d6..4289e8e 100644 --- a/lecture/main.ist +++ b/lecture/main.ist @@ -1,5 +1,5 @@ % makeindex style file created by the glossaries package -% for document 'main' on 2025-1-25 +% for document 'main' on 2025-1-26 actual '?' encap '|' level '!' diff --git a/lecture/main.tex b/lecture/main.tex index 7bb4bfa..978e00a 100644 --- a/lecture/main.tex +++ b/lecture/main.tex @@ -5,7 +5,7 @@ \author{Oliver Wallscheid} \date{} -%\includeonly{tex/temp.tex} % build only selected sections +\includeonly{tex/Lecture06.tex} % build only selected sections \begin{document} diff --git a/lecture/tex/Lecture06.tex b/lecture/tex/Lecture06.tex index cb451e1..1a3e51e 100644 --- a/lecture/tex/Lecture06.tex +++ b/lecture/tex/Lecture06.tex @@ -212,7 +212,7 @@ \subsection{Single-phase AC/DC bridge converter} \pgfmathsetmacro{\signalheight}{\cheight/(1.3)} % Start drawing the triangular signal - \draw[signalblue, thick] + \draw[signalred, thick] ($(blockBottomLeft) + (0.025, 0.1*\cheight)$) % Starting point with a margin \foreach \x in {1,...,\signalsteps} { -- ($ @@ -257,7 +257,7 @@ \subsection{Single-phase AC/DC bridge converter} \pgfmathsetmacro{\signalheight}{\cheight/(1.3)} % Start drawing the triangular signal - \draw[signalblue, thick] + \draw[signalred, thick] ($(blockBottomLeft) + (0.025, 0.1*\cheight)$) % Starting point with a margin \foreach \x in {1,...,\signalsteps} { -- ($ @@ -307,8 +307,8 @@ \subsection{Single-phase AC/DC bridge converter} % Top plot: duty cycle reference and carrier signal \nextgroupplot[ylabel = {$s^*(t), c(t)$}, legend pos=north east, legend columns=2] - \addplot[signalred, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_comp_example.csv}; - \addplot[signalblue, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_comp_example.csv}; + \addplot[signalblue, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_comp_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_comp_example.csv}; \legend{$s^*(t)$, $c(t)$} % top middle plot: individual switching signals @@ -344,9 +344,9 @@ \subsection{Single-phase AC/DC bridge converter} % Top plot: duty cycle reference and carrier signal \nextgroupplot[ylabel = {$s^*(t), c(t)$}, legend pos=north east, legend columns=2, ymin = -1.1, ymax =1.1] - \addplot[signalred, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_int_example.csv}; - \addplot[signalblue, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_int_example.csv}; - \addplot[signalred, thick, dashed] table[x=wt, y expr=-\thisrow{d}, col sep=comma] {PWM_single_phase_int_example.csv}; + \addplot[signalblue, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_int_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_int_example.csv}; + \addplot[signalblue, thick, dashed] table[x=wt, y expr=-\thisrow{d}, col sep=comma] {PWM_single_phase_int_example.csv}; \legend{$s^*(t)$, $c(t)$} % top middle plot: individual switching signals @@ -400,8 +400,8 @@ \subsection{Single-phase AC/DC bridge converter} xlabel style={anchor=west} ] \nextgroupplot[ylabel = {$s^*(t), c(t)$}, height=0.36\textheight] - \addplot[signalblue, thick] coordinates {(-0.1,-0.6) (0,-1) (0.5,1) (1,-1)(1.1,-0.6)}; - \addplot[domain = -0.1:1.1, samples = 10, signalred, thick] {\d}; + \addplot[signalred, thick] coordinates {(-0.1,-0.6) (0,-1) (0.5,1) (1,-1)(1.1,-0.6)}; + \addplot[domain = -0.1:1.1, samples = 10, signalblue, thick] {\d}; \node[anchor=west] at (axis cs:1.1,\d) {$s^*$}; \draw[<->] (axis cs:\t1,-1) -- node[above, fill=white, inner sep=1pt,yshift=2pt]{\footnotesize$\frac{T_\mathrm{s}(1-s^*)}{2}$} (axis cs:1-\t1,-1); \draw[<->] (axis cs:0,1) -- node[above, fill=white, inner sep=1pt,yshift=2pt]{\footnotesize$\frac{T_\mathrm{s}(1+s^*)}{4}$} (axis cs:\t1,1); @@ -463,9 +463,9 @@ \subsection{Single-phase AC/DC bridge converter} xlabel style={anchor=west} ] \nextgroupplot[ylabel = {$s^*(t), c(t)$}, height=0.36\textheight] - \addplot[signalblue, thick] coordinates {(-0.1,-0.6) (0,-1) (0.5,1) (1,-1)(1.1,-0.6)}; - \addplot[domain = -0.1:1.1, samples = 10, signalred, thick] {\d}; - \addplot[domain = -0.1:1.1, samples = 10, signalred, thick, dashed] {-\d}; + \addplot[signalred, thick] coordinates {(-0.1,-0.6) (0,-1) (0.5,1) (1,-1)(1.1,-0.6)}; + \addplot[domain = -0.1:1.1, samples = 10, signalblue, thick] {\d}; + \addplot[domain = -0.1:1.1, samples = 10, signalblue, thick, dashed] {-\d}; \node[anchor=west] at (axis cs:1.1,\d) {$s^*$}; \node[anchor=west, overlay] at (axis cs:1.1,-\d) {$-s^*$}; \draw[<->] (axis cs:\t1,-1) -- node[above, fill=white, inner sep=1pt,yshift=2pt]{\footnotesize$\frac{T_\mathrm{s}(1-|s^*|)}{2}$} (axis cs:1-\t1,-1); @@ -620,9 +620,9 @@ \subsection{Single-phase AC/DC bridge converter} % Top plot: duty cycle reference and carrier signal \nextgroupplot[ylabel = {$s^*(t), c(t)$}, ymin = -1.25, ymax =1.25, legend pos=north east, legend columns=2 ] - \addplot[signalred, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_overmod_example.csv}; - \addplot[signalblue, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_overmod_example.csv}; - \addplot[signalred, thick, dashed] table[x=wt, y expr=-\thisrow{d}, col sep=comma] {PWM_single_phase_overmod_example.csv}; + \addplot[signalblue, thick] table[x=wt, y=d, col sep=comma] {PWM_single_phase_overmod_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_single_phase_overmod_example.csv}; + \addplot[signalblue, thick, dashed] table[x=wt, y expr=-\thisrow{d}, col sep=comma] {PWM_single_phase_overmod_example.csv}; \legend{$s^*(t)$, $c(t)$} % top middle plot: individual switching signals @@ -652,12 +652,12 @@ \subsection{Single-phase AC/DC bridge converter} \frametitle{Overmodulation (cont.)} \begin{columns} \begin{column}{0.45\textwidth} - Considering a normalized input reference - $$ s^*(t) = A\sin(\omega t)$$ - one can distinguish two PWM operation areas: + Considering a normalized input reference + $$ s^*(t) = m\sin(\omega t)$$ + with the \hl{modulation ratio} $m$ one can distinguish two PWM operation areas: \begin{itemize} - \item $A \leq 1$: linear modulation, - \item $A > 1$: overmodulation. + \item $m \leq 1$: linear modulation, + \item $m > 1$: overmodulation. \end{itemize} \vspace{-0.25cm} \onslide<2->{% @@ -669,7 +669,7 @@ \subsection{Single-phase AC/DC bridge converter} \begin{figure} \begin{tikzpicture} \begin{axis}[ - xlabel={$A$}, + xlabel={$m$}, ylabel={$\hat{u}_2^{(1)}/U_1$}, ymin=0, ymax=4/pi*1.2, xmin=0, xmax=3, @@ -738,11 +738,11 @@ \subsection{Single-phase AC/DC bridge converter} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \frametitle{Fundamental frequency modulation (cont.)} - \onslide<1->{The fundamental frequency modulation leads to a pulse pattern synchronized with the fundamental output voltage $\hat{u}_2^{(1)}(t)$, i.e., there are exactly \hl{two switching actions per fundamental} period $$ f_\mathrm{s} = 2\frac{\omega}{2\pi} = \frac{\omega}{\pi}.$$}% + \onslide<1->{The fundamental frequency modulation leads to a pulse pattern synchronized with the fundamental output voltage $\hat{u}_2^{(1)}(t)$, i.e., the \hl{switching frequency matches the fundamental voltage frequency} $$ f_\mathrm{s} = \frac{\omega}{2\pi}.$$}% \onslide<2->{The fundamental output voltage amplitude can be derived from the corresponding \hl{Fourier coefficient}} \begin{equation} \begin{split} - \onslide<2->{\hat{u}_2^{(k)} &= \frac{1}{\pi} \int_{\alpha}^{\alpha + \pi} u_2(t)\sin(k (\omega t-\alpha)) \mathrm{d}\omega t}\onslide<3->{ = \frac{2}{\pi} \int_{}^{\pi/2} U_1 \sin(k \omega t) \mathrm{d}\omega t} \\ + \onslide<2->{\hat{u}_2^{(k)} &= \frac{1}{\pi} \int_{\alpha}^{\alpha + \pi} u_2(t)\sin(k (\omega t-\alpha)) \mathrm{d}\omega t}\onslide<3->{ = \frac{2}{\pi} \int_{0}^{\pi/2} U_1 \sin(k \omega t) \mathrm{d}\omega t} \\ & \onslide<4->{= \frac{2}{\pi} \left[-\frac{U_1}{k} \cos(k \omega t)\right]_{0}^{\pi/2}} \onslide<5->{= \frac{2}{\pi} \left[\frac{U_1}{k} \left(\cos(0) - \cos(k(\pi/2))\right)\right]} \\ &\onslide<6->{= \frac{4}{\pi} U_1 \frac{1}{k}, \quad k=1,3,5,7,\ldots} \end{split} @@ -1491,7 +1491,7 @@ \subsection{Three-phase AC/DC bridge converter} u_{2\mathrm{ca}}(t) &= \frac{1}{2}\left(s_{\mathrm{c}}(t)-s_{\mathrm{a}}(t)\right)u_1(t). \end{split} \end{equation} - The \hl{line-to-neutral voltages} are given by + The \hl{line-to-ground voltages} are given by \begin{equation} \begin{split} u_{2\mathrm{a}0}(t) &= \frac{1}{2}s_{\mathrm{a}}(t)u_1(t),\\ @@ -1634,36 +1634,246 @@ \subsection{Three-phase AC/DC bridge converter} \begin{groupplot}[group style={group size=1 by 5, xticklabels at = edge bottom, vertical sep=0.25cm}, height=0.32\textheight, width=0.875\textwidth, xmin=0, xmax=3*pi, grid,clip = false, ymin = -0.8, ymax =0.8, xtick = {0, pi/2, pi, 3/2*pi, 2*pi, 5/2*pi, 3*pi}, xticklabels = {$0$,$\nicefrac{1}{2}\pi$, $\pi$, $\nicefrac{3}{2}\pi$, $2\pi$, $\nicefrac{5}{2}\pi$, $3\pi$}, ytick = {-1/2, 0, 1/2}, yticklabels = {$-\nicefrac{1}{2}$, , $\nicefrac{1}{2}$}, ylabel style={rotate=-90}] % ua0 - \nextgroupplot[ylabel = {$u_{\mathrm{a}0}(t)/U_1$}] + \nextgroupplot[ylabel = {$u_{2\mathrm{a}0}(t)/U_1$}] \addplot[signalblue, thick] coordinates {(0,-1/2) (\a,-1/2) (\a,1/2) (\a+pi,1/2) (\a+pi,-1/2) (2*pi+\a,-1/2) (2*pi+\a,1/2) (3*pi,1/2)}; \addplot[signalblue, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-\a))}; - \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{a}0}=\frac{2}{\pi}$}; + \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{2\mathrm{a}0}=\frac{2}{\pi}$}; \draw[->] (axis cs:0,0) -- node[above, fill=white, inner sep=1pt]{$\alpha$} (axis cs:\a,0); % ub0 - \nextgroupplot[ylabel = {$u_{\mathrm{b}0}(t)/U_1$}] + \nextgroupplot[ylabel = {$u_{2\mathrm{b}0}(t)/U_1$}] \addplot[signalgreen, thick] coordinates {(0,-1/2) (pi,-1/2) (pi,1/2) (2*pi,1/2) (2*pi,-1/2) (3*pi,-1/2)}; \addplot[signalgreen, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(-x))}; - \draw[thin] (pi/3*5+\a+0.1,0.0) -- (pi/3*5+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{b}0}=\frac{2}{\pi}$}; + \draw[thin] (pi/3*5+\a+0.1,0.0) -- (pi/3*5+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{2\mathrm{b}0}=\frac{2}{\pi}$}; % uc0 - \nextgroupplot[ylabel = {$u_{\mathrm{c}0}(t)/U_1$}] + \nextgroupplot[ylabel = {$u_{2\mathrm{c}0}(t)/U_1$}] \addplot[signalbrown, thick] coordinates {(0,1/2) (\a+pi/3,1/2) (\a+pi/3,-1/2) (\a+4*pi/3,-1/2) (\a+4*pi/3,1/2) (\a+7*pi/3,1/2) (\a+7*pi/3,-1/2) (3*pi,-1/2)}; \addplot[signalbrown, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-4/3*pi-\a))}; - \draw[thin] (pi/3+\a+0.1,0.0) -- (pi/3+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{c}0}=\frac{2}{\pi}$}; + \draw[thin] (pi/3+\a+0.1,0.0) -- (pi/3+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{2\mathrm{c}0}=\frac{2}{\pi}$}; % un0 - \nextgroupplot[ylabel = {$u_{\mathrm{n}0}(t)/U_1$}, height=0.27\textheight, ymin = -1/5, ymax =1/5, ytick = {-1/6, 0, 1/6}, yticklabels = {$-\nicefrac{1}{6}$, , $\nicefrac{1}{6}$}] + \nextgroupplot[ylabel = {$u_{2\mathrm{n}0}(t)/U_1$}, height=0.27\textheight, ymin = -1/5, ymax =1/5, ytick = {-1/6, 0, 1/6}, yticklabels = {$-\nicefrac{1}{6}$, , $\nicefrac{1}{6}$}] \addplot[signallavender, thick] coordinates {(0,-1/6) (pi/3,-1/6) (pi/3,1/6) (2*pi/3,1/6) (2*pi/3,-1/6) (3*pi/3,-1/6) (3*pi/3,1/6) (4*pi/3,1/6) (4*pi/3,-1/6) (5*pi/3,-1/6) (5*pi/3,1/6) (6*pi/3,1/6) (6*pi/3,-1/6) (7*pi/3,-1/6) (7*pi/3,1/6) (8*pi/3,1/6) (8*pi/3,-1/6) (9*pi/3,-1/6) (9*pi/3,1/6)}; + \draw[<->] (axis cs:\a,0) -- node[centered, fill=white, inner sep=1pt]{$T_\mathrm{s}$} (axis cs:\a+2*pi,0); % ua0 - \nextgroupplot[ylabel = {$u_{\mathrm{a}}(t)/U_1$}, , ytick = {-2/3, 0, 2/3}, yticklabels = {$-\nicefrac{2}{3}$, , $\nicefrac{2}{3}$}, xlabel={$\omega t$}] + \nextgroupplot[ylabel = {$u_{2\mathrm{a}}(t)/U_1$}, , ytick = {-2/3, 0, 2/3}, yticklabels = {$-\nicefrac{2}{3}$, , $\nicefrac{2}{3}$}, xlabel={$\omega t$}] \addplot[signalblue, thick] coordinates {(0,-1/3) (pi/3,-1/3) (pi/3,1/3) (2*pi/3,1/3) (2*pi/3,2/3) (3*pi/3,2/3) (3*pi/3,1/3) (4*pi/3,1/3) (4*pi/3,-1/3) (5*pi/3,-1/3) (5*pi/3,-2/3) (6*pi/3,-2/3) (6*pi/3,-1/3) (7*pi/3,-1/3) (7*pi/3,1/3) (8*pi/3,1/3) (8*pi/3,2/3) (9*pi/3,2/3)}; \addplot[signalblue, thick, domain = 0:3*pi, samples = 100, dashed] {4/pi/2*sin(deg(x-\a))}; - \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.6,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{\mathrm{a}}=\hat{u}^{(1)}_{\mathrm{a}0}=\frac{2}{\pi}$}; + \draw[thin] (pi+\a+0.1,0.0) -- (pi+\a+0.4,0.25) node[right, anchor=west, fill = white, inner sep = 1pt] {$\hat{u}^{(1)}_{2\mathrm{a}}=\hat{u}^{(1)}_{2\mathrm{a}0}=\frac{2}{\pi}$}; \draw[->] (axis cs:0,0) -- node[above, fill=white, inner sep=1pt]{$\alpha$} (axis cs:\a,0); \end{groupplot} \end{tikzpicture} \end{figure} \end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase fundamental frequency modulation (cont.) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase fundamental frequency modulation (cont.)} + From the previous figure and voltage equations, we can summarize the following observations: + \begin{itemize} + \item Due to the fundamental frequency modulation, the switching frequency of the inverter is identical to the fundamental frequency: $f_{\mathrm{s}} = \nicefrac{\omega}{2\pi}$. + \item The star-to-ground voltage $u_{\mathrm{n0}}(t)$ shows a rectangular signal pattern with triple fundamental frequency. + \item Consequently, it does not influence the fundamental output voltage, that is, the fundamental components of the line-to-ground voltage $u_{2i0}(t)$ as well as the load voltage $u_{2i}(t)$ are identical: $\hat{u}^{(1)}_{2i0}=\hat{u}^{(1)}_{2i}$. + \end{itemize}\vspace{-0.25cm} + \begin{varblock}{Note on the star point} + The previous analysis assumed a non-connected star point, which comes with certain advantages, e.g., on the rejection of current harmonics. If, however, the star point would be connected, the three-phase converter can be interpreted and analyzed as three independent single-phase converters (each driven by a half bridge). + \end{varblock} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase pulse width modulation (PWM) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase pulse width modulation (PWM)} + \begin{figure} + \begin{circuitikz} + \def\cwidth{1.5} + \def\cheight{1} + \draw[->] (0,0) to node[above]{$s_1^*(t)$} ++(1.0,0) node[adder, anchor = west, name=add1]{}; + \draw node[ctrlblock, anchor = west, minimum width = \cwidth cm, minimum height = \cheight cm](carrier) at (-1.5,-4) {}; + \path (carrier.south west) coordinate (blockBottomLeft); + + % Triangular signal pattern within block + \begin{scope} + % Define the number of signal steps + \def\signalsteps{6} + + % Compute step width and height of the triangular pattern + \pgfmathsetmacro{\stepwidth}{\cwidth/\signalsteps} + \pgfmathsetmacro{\signalheight}{\cheight/(1.3)} + + % Start drawing the triangular signal + \draw[signalred, thick] + ($(blockBottomLeft) + (0.025, 0.1*\cheight)$) % Starting point with a margin + \foreach \x in {1,...,\signalsteps} { + -- ($ + (blockBottomLeft) + + (\x*\stepwidth - 0*\stepwidth, {0.1*\cheight + mod(\x, 2)*\signalheight}) + $) + }; + \end{scope} + \draw[->] (carrier.east) to [short, l=$c(t)$] (carrier.east -| add1.south) coordinate (c1) -- (add1.south) node[anchor = north west] {$-$}; + \draw[->] (add1.east) -- ++(4,0) node[ctrlblock, anchor = west, minimum width = \cwidth cm, minimum height = \cheight cm](comp1){}; + + \draw node[ctrlblock, below=0.2cm of comp1, minimum width = \cwidth cm, minimum height = \cheight cm](comp2){}; + \draw node[ctrlblock, below=0.2cm of comp2, minimum width = \cwidth cm, minimum height = \cheight cm](comp3){}; + \draw[<-] (comp2.west) -- ++(-2.5,0) node[adder, anchor = east, name=add2]{}; + \draw[<-] (comp3.west) -- ++(-1,0) node[adder, anchor = east, name=add3]{}; + \draw[<-] (add3.west) to node[above]{$s_3^*(t)$} ++(-1,0); + \draw[<-] (add2.west) to node[above]{$s_2^*(t)$} ++(-1,0); + \draw[->] (c1) to [short, *-] (c1 -| add2.south) coordinate (c11) to [short] (add2.south) node[anchor = north west] {$-$}; + \draw[->] (c11) to [short, *-] (c1 -| add3.south) to [short] (add3.south) node[anchor = north west] {$-$}; + + % Comperator block #1 + \begin{axis}[at={(comp1)}, scale only axis, width = 0.8*\cwidth cm, height = 0. 8*\cheight cm, anchor = center, xtick=\empty, ytick={0,1}, axis lines=middle, ymax=1.25, ymin = -1.25, font = \footnotesize, extra y ticks={-1}, extra y tick style = {yticklabel shift = -0.75cm}] + \addplot[thick, signalblue] coordinates {(-1,-1) (0,-1) (0,1) (1,1)}; + \end{axis} + % Comperator block #2 + \begin{axis}[at={(comp2)}, scale only axis, width = 0.8*\cwidth cm, height = 0.8*\cheight cm, anchor = center, xtick=\empty, ytick={0,1}, axis lines=middle, ymax=1.25, ymin = -1.25, font = \footnotesize, extra y ticks={-1}, extra y tick style = {yticklabel shift = -0.75cm}] + \addplot[thick, signalblue] coordinates {(-1,-1) (0,-1) (0,1) (1,1)}; + \end{axis} + % Comperator block #3 + \begin{axis}[at={(comp3)}, scale only axis, width = 0.8*\cwidth cm, height = 0.8*\cheight cm, anchor = center, xtick=\empty, ytick={0,1}, axis lines=middle, ymax=1.25, ymin = -1.25, font = \footnotesize, extra y ticks={-1}, extra y tick style = {yticklabel shift = -0.75cm}] + \addplot[thick, signalblue] coordinates {(-1,-1) (0,-1) (0,1) (1,1)}; + \end{axis} + \draw[->] (comp1.east) -- ++(0.5,0) node[anchor = west]{$s_1(t)$}; + \draw[->] (comp2.east) -- ++(0.5,0) node[anchor = west]{$s_2(t)$}; + \draw[->] (comp3.east) -- ++(0.5,0) node[anchor = west]{$s_3(t)$}; + \end{circuitikz} + \caption{Three-phase PWM (note: a distinction between interleaved and complementary PWM is not relevant here, as the three-phase converter operates on a half-bridge basis while the previously considered single-phase converter was based on a full bridge. While independent and phase-shifted carriers per phase could be also used in the three-phase converter, this is typically not utilized due to increasing current harmonics.)} + \label{fig:three-phase_pwm} + \end{figure} +\end{frame} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase PWM example (with ref. modulation index $m=0.5$) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase PWM example (with ref. modulation index $m=0.5$)} + \vspace{-0.1cm} + \begin{figure} + \begin{tikzpicture} + \pgfplotsset{table/search path={fig/lec06}} + \begin{groupplot}[group style={group size=1 by 5, xticklabels at = edge bottom, vertical sep=0.25cm}, height=0.31\textheight, width=0.875\textwidth, xmin=0, xmax=2*pi, grid,clip = false, ymin = -1.1, ymax =1.1, xtick = {0, pi/2, pi, 3/2*pi, 2*pi}, xticklabels = {0,$\nicefrac{1}{2}\pi$,$\pi$, $\nicefrac{3}{2}\pi$, $2\pi$}, ytick = {-1, 0, 1}, yticklabels = {$-1$, $0$, $1$}] + + % Top plot: duty cycle reference and carrier signal + \nextgroupplot[ylabel = {$s_{i}^*(t), c(t)$}] + \addplot[signalblue, thick] table[x=wt, y=d1, col sep=comma] {PWM_three-phase_mod05_example.csv}; + \addplot[signalgreen, thick] table[x=wt, y=d2, col sep=comma] {PWM_three-phase_mod05_example.csv}; + \addplot[signalbrown, thick] table[x=wt, y=d3, col sep=comma] {PWM_three-phase_mod05_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_three-phase_mod05_example.csv}; + \node[signalblue, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt] at (axis cs: pi/2, 1) {\small $s_1^*$}; + \node[signalgreen, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=-3mm] at (axis cs: pi/2+pi/3*2, 1) {\small $s_2^*$}; + \node[signalbrown, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=3mm] at (axis cs: pi/2+pi/3*4, 1) {\small $s_3^*$}; + \node[signalred, anchor=south, fill = white, inner sep = 1pt] at (axis cs: pi/10, -0.6) {\small $c$}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_1(t)$}] + \addplot[signalblue, thick] table[x=wt, y=s1, col sep=comma] {PWM_three-phase_mod05_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_2(t)$}] + \addplot[signalgreen, thick] table[x=wt, y=s2, col sep=comma] {PWM_three-phase_mod05_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_3(t)$}] + \addplot[signalbrown, thick] table[x=wt, y=s3, col sep=comma] {PWM_three-phase_mod05_example.csv}; + + % bottom plot: line-to-line voltage + \nextgroupplot[ylabel = {$\frac{u_{2\mathrm{ab}}(t)}{U_1}$}, xlabel={$\omega t$}] + \addplot[signalblue, thick] table[x=wt, y expr=(\thisrow{s1}-\thisrow{s2})/2, col sep=comma] {PWM_three-phase_mod05_example.csv}; + \end{groupplot} + \end{tikzpicture} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase PWM example (with ref. modulation index $m=1$) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase PWM example (with ref. modulation index $m=1$)} + \vspace{-0.1cm} + \begin{figure} + \begin{tikzpicture} + \pgfplotsset{table/search path={fig/lec06}} + \begin{groupplot}[group style={group size=1 by 5, xticklabels at = edge bottom, vertical sep=0.25cm}, height=0.31\textheight, width=0.875\textwidth, xmin=0, xmax=2*pi, grid,clip = false, ymin = -1.1, ymax =1.1, xtick = {0, pi/2, pi, 3/2*pi, 2*pi}, xticklabels = {0,$\nicefrac{1}{2}\pi$,$\pi$, $\nicefrac{3}{2}\pi$, $2\pi$}, ytick = {-1, 0, 1}, yticklabels = {$-1$, $0$, $1$}] + + % Top plot: duty cycle reference and carrier signal + \nextgroupplot[ylabel = {$s_{i}^*(t), c(t)$}] + \addplot[signalblue, thick] table[x=wt, y=d1, col sep=comma] {PWM_three-phase_mod1_example.csv}; + \addplot[signalgreen, thick] table[x=wt, y=d2, col sep=comma] {PWM_three-phase_mod1_example.csv}; + \addplot[signalbrown, thick] table[x=wt, y=d3, col sep=comma] {PWM_three-phase_mod1_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_three-phase_mod1_example.csv}; + \node[signalblue, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt] at (axis cs: pi/2, 1) {\small $s_1^*$}; + \node[signalgreen, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=-3mm] at (axis cs: pi/2+pi/3*2, 1) {\small $s_2^*$}; + \node[signalbrown, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=3mm] at (axis cs: pi/2+pi/3*4, 1) {\small $s_3^*$}; + \node[signalred, anchor=south, fill = white, inner sep = 1pt] at (axis cs: pi/10, -0.6) {\small $c$}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_1(t)$}] + \addplot[signalblue, thick] table[x=wt, y=s1, col sep=comma] {PWM_three-phase_mod1_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_2(t)$}] + \addplot[signalgreen, thick] table[x=wt, y=s2, col sep=comma] {PWM_three-phase_mod1_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_3(t)$}] + \addplot[signalbrown, thick] table[x=wt, y=s3, col sep=comma] {PWM_three-phase_mod1_example.csv}; + + % bottom plot: line-to-line voltage + \nextgroupplot[ylabel = {$\frac{u_{2\mathrm{ab}}(t)}{U_1}$}, xlabel={$\omega t$}] + \addplot[signalblue, thick] table[x=wt, y expr=(\thisrow{s1}-\thisrow{s2})/2, col sep=comma] {PWM_three-phase_mod1_example.csv}; + \end{groupplot} + \end{tikzpicture} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Three-phase PWM example (with ref. modulation index $m=1.18$) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Three-phase PWM example (with ref. modulation index $m=1.18$)} + \vspace{-0.1cm} + \begin{figure} + \begin{tikzpicture} + \pgfplotsset{table/search path={fig/lec06}} + \begin{groupplot}[group style={group size=1 by 5, xticklabels at = edge bottom, vertical sep=0.25cm}, height=0.31\textheight, width=0.875\textwidth, xmin=0, xmax=2*pi, grid,clip = false, ymin = -1.1, ymax =1.1, xtick = {0, pi/2, pi, 3/2*pi, 2*pi}, xticklabels = {0,$\nicefrac{1}{2}\pi$,$\pi$, $\nicefrac{3}{2}\pi$, $2\pi$}, ytick = {-1, 0, 1}, yticklabels = {$-1$, $0$, $1$}] + + % Top plot: duty cycle reference and carrier signal + \nextgroupplot[ylabel = {$s_{i}^*(t), c(t)$}, ymin = -1.2, ymax =1.2] + \addplot[signalblue, thick] table[x=wt, y=d1, col sep=comma] {PWM_three-phase_overmod_example.csv}; + \addplot[signalgreen, thick] table[x=wt, y=d2, col sep=comma] {PWM_three-phase_overmod_example.csv}; + \addplot[signalbrown, thick] table[x=wt, y=d3, col sep=comma] {PWM_three-phase_overmod_example.csv}; + \addplot[signalred, thick] table[x=wt, y=c, col sep=comma] {PWM_three-phase_overmod_example.csv}; + \node[signalblue, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt] at (axis cs: pi/2, 1.1) {\small $s_1^*$}; + \node[signalgreen, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=-3mm] at (axis cs: pi/2+pi/3*2, 1.1) {\small $s_2^*$}; + \node[signalbrown, anchor=north, fill = white, inner sep = 1pt, yshift=-1pt, xshift=3mm] at (axis cs: pi/2+pi/3*4, 1.1) {\small $s_3^*$}; + \node[signalred, anchor=south, fill = white, inner sep = 1pt] at (axis cs: pi/10, -0.6) {\small $c$}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_1(t)$}] + \addplot[signalblue, thick] table[x=wt, y=s1, col sep=comma] {PWM_three-phase_overmod_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_2(t)$}] + \addplot[signalgreen, thick] table[x=wt, y=s2, col sep=comma] {PWM_three-phase_overmod_example.csv}; + + % top middle plot: individual switching signals + \nextgroupplot[ylabel = {$s_3(t)$}] + \addplot[signalbrown, thick] table[x=wt, y=s3, col sep=comma] {PWM_three-phase_overmod_example.csv}; + + % bottom plot: line-to-line voltage + \nextgroupplot[ylabel = {$\frac{u_{2\mathrm{ab}}(t)}{U_1}$}, xlabel={$\omega t$}] + \addplot[signalblue, thick] table[x=wt, y expr=(\thisrow{s1}-\thisrow{s2})/2, col sep=comma] {PWM_three-phase_overmod_example.csv}; + \end{groupplot} + \end{tikzpicture} + \end{figure} +\end{frame} \ No newline at end of file diff --git a/lecture/tex/dict.tex b/lecture/tex/dict.tex index ef3adbf..4192742 100644 --- a/lecture/tex/dict.tex +++ b/lecture/tex/dict.tex @@ -331,6 +331,11 @@ \subsection{English-German dictionary} description={Wechselrichtersperrzeit} } +\newglossaryentry{mod_ratio}{ + name={modulation ratio}, + description={Aussteuergrad} +} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Build glossary %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/lecture/tex/nomen.tex b/lecture/tex/nomen.tex index 7d5b88d..61ba29b 100644 --- a/lecture/tex/nomen.tex +++ b/lecture/tex/nomen.tex @@ -11,6 +11,20 @@ \subsection{Nomenclature} sort = 010 } +\newglossaryentry{amplitude}{ + type=nomen, + name={$\hat{x}$}, + description={(fundamental) amplitude of a signal $x(t)$}, + sort = 011 +} + +\newglossaryentry{amplitude_harmonic}{ + type=nomen, + name={$\hat{x}^{(k)}$}, + description={$k$-th harmonic amplitude of a signal $x(t)$}, + sort = 012 +} + \newglossaryentry{vectorial_signal}{ type=nomen, name={$\bm{x}(t)$}, From b822ed43ee856b989bd8e041e8567e8442d62178 Mon Sep 17 00:00:00 2001 From: SevenOfNinePE Date: Mon, 27 Jan 2025 11:14:31 +0100 Subject: [PATCH 08/10] Ex07 Task2: Align task description and figure to lecture. --- .../ex07/Fig_ThreePhaseInverter_6StepMode.tex | 31 ++++++-- exercise/tex/exercise07.tex | 79 ++++++++++--------- 2 files changed, 66 insertions(+), 44 deletions(-) diff --git a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex index ceea245..5ef33ed 100644 --- a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex +++ b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex @@ -6,14 +6,14 @@ \begin{circuitikz} % 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] ++(0.4,0) + (Gnd) node[rground, rotate = 270 ](){} ++(0.4,0) (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) + (jT1e) to [short, *-] ++(1,0) to [crossing] ++(2,0) to [crossing] ++(2,0) to [short,-] ++(3,0) coordinate (ju2a) % Add junction to T2 (jT1e) to [short] ++(0,-1) coordinate (jT2c) % Add T2 @@ -25,7 +25,7 @@ % 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) + (jmu2b) to [short,*-] ++(1,0) to [crossing] ++(2,0) to [short,-] ++(3,0) coordinate (ju2b) % Add junction to T4 (jmu2b) to [short] ++(0,-0.5) coordinate (jT4c) % Add T4 @@ -44,8 +44,10 @@ (jT4e) to [short, -*] (jT2e) % Add connection to U1m (jT2e) to [short, -] (U1m) - % Add connection to u1c - (jT6c) to [short,*-] ++(2,0) coordinate (ju2c) + % Add 2. Ground symbol + (jmu2b) ++(4.7,-2.6) node[rground](){} coordinate (Gnd2) + % Add connection to u2c + (jT6c) to [short,*-] ++(4,0) coordinate (ju2c) % Add connection to u2a inductor (ju2a) to [short,-] ++(0,2) coordinate (ju2ax) % Add u2a inductor @@ -63,7 +65,11 @@ % Add u2ce (ju2ce) to [sV=$u_\mathrm{2ce}$] ++(1.5,0) coordinate (ju2cn) % Add connection of u2in - (ju2an) to [short,-*] (ju2bn) to [short,-] (ju2cn); + (ju2an) to [short,-*] (ju2bn) to [short,-] (ju2cn) + % Add connection point u2n + (ju2bn) to [short,-o] ++(0.6,0) coordinate (ju2n) + % Add 3. Ground symbol + (ju2n) ++(0,-2.6) node[rground](){} coordinate (Gnd3); % Add component name of transistors @@ -91,8 +97,17 @@ (ju2b) ++(0.2,0) to [open,v^=$$,voltage = straight] ++(0,-2.5) (ju2b) ++ (0.9,-1) node[anchor=north,color=black]{$u_\mathrm{2bc}(t)$} % Add voltage arrow ua0 - (Gnd) ++(2,0) to [open,v^=$$,voltage = straight] ++(-1.6,0) - (Gnd) ++ (1.2,0.7) node[anchor=north,color=black]{$u_\mathrm{2a,0}(t)$}; + (Gnd2) ++(-0.7,3.4) to [open,v^=$$,voltage = straight] ++(0,-3.6) + (Gnd2) ++ (-1.4,1.2) node[anchor=north,color=black,rotate = 90]{$u_\mathrm{2a0}(t)$} + % Add voltage arrow ub0 + (Gnd2) ++(0,2.9) to [open,v^=$$,voltage = straight] ++(0,-3.01) + (Gnd2) ++ (-0.7,1.2) node[anchor=north,color=black,rotate = 90]{$u_\mathrm{2b0}(t)$} + % Add voltage arrow uc0 + (Gnd2) ++(0.7,2.4) to [open,v^=$$,voltage = straight] ++(0,-2.42) + (Gnd2) ++ (0,1.2) node[anchor=north,color=black,rotate = 90]{$u_\mathrm{2b0}(t)$} + % Add voltage arrow un0 + (Gnd3) ++(0,2.8) to [open,v^=$$,voltage = straight] ++(0,-3.01) + (Gnd3) ++ (-0.7,1.2) node[anchor=north,color=black,rotate = 90]{$u_\mathrm{2n0}(t)$}; diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index 580a913..fb73610 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -18,7 +18,7 @@ An symmetrical switching rectifier in three-phase bridge topology shall supply a symmetrical three-phase consumer. The consumer is simulated by an inductance and -a sinusoidal counter voltage per phase. The inverter is operated with a basic frequency clock. +a sinusoidal counter voltage per phase. The inverter is operated with a basic switching frequency. The switching elements are considered as ideal. % \input{fig/ex07/Fig_ThreePhaseInverter_6StepMode} @@ -40,15 +40,16 @@ \end{table} -\subtask{Create a table with all possible switching states for basic frequency clocking. +\subtask{Create a table with all possible switching states for basic switching frequency. 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}$\\ +\bigskip 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. -} +Calculate and sketch the voltages $u_\mathrm{a0}(t)$, $u_\mathrm{b0}(t)$ and $u_\mathrm{c0}(t)$ +depending on these switching states.} \begin{solutionblock} Each half bridge has got the 2 states '+1' and '-1', which results in $2^3 = 8$ combinations according table \autoref{stable:ex07_Task2_Switchingstates}. The correct chronological order is displayed in table \autoref{stable:ex07_Task2_UsedSwitchingStates}. @@ -77,7 +78,7 @@ \begin{tabular}{|c|c|c|c|c|c|} % Each column is separated by a line \hline \bfseries $s_\mathrm{a}(t)$ & \bfseries $s_\mathrm{b}(t)$ & \bfseries $s_\mathrm{c}(t)$ - & \bfseries $+U_\mathrm{2a,0}$ & \bfseries $+U_\mathrm{2b,0}$ & \bfseries $+U_\mathrm{2c,0}$ \\ \hline + & \bfseries $u_\mathrm{2a0}$ & \bfseries $u_\mathrm{2b0}$ & \bfseries $u_\mathrm{2c0}$ \\ \hline +1 & -1 & +1 & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ & $U_\mathrm{1p}$ \\ \hline +1 & -1 & -1 & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ & $-U_\mathrm{1m}$ \\ \hline +1 & +1 & -1 & $U_\mathrm{1p}$ & $U_\mathrm{1p}$ & $-U_\mathrm{1m}$ \\ \hline @@ -97,15 +98,15 @@ \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$. +i.e. the following is always applicable: $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} In the case of a symmetrical three-phase consumer where the current sum at the consumer star point is zero, the following results: \begin{equation} - u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2c}(t)} = \SI{0}{\volt} \quad - i_{\mathrm{2a}(t)} + i_{\mathrm{2b}(t)} + i_{\mathrm{2c}(t)} = \SI{0}{\ampere}. + u_{\mathrm{2a}}(t) + u_{\mathrm{2b}}(t) + u_{\mathrm{2c}}(t) = \SI{0}{\volt} \quad + i_{\mathrm{2a}}(t) + i_{\mathrm{2b}}(t) + i_{\mathrm{2c}}(t) = \SI{0}{\ampere}. \label{eq:u2_i2_symgen} \end{equation} This leads to @@ -117,7 +118,7 @@ \end{equation} Using \eqref{eq:u2_i2_symgen} leads to \begin{equation} - u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2c}(t)} + u_{\mathrm{2a}}(t) + u_{\mathrm{2b}}(t) + u_{\mathrm{2c}}(t) = L \frac{\mathrm{d}}{\mathrm{d}t} \left( i_{\mathrm{2a}}(t)+i_{\mathrm{2b}}(t)+i_{\mathrm{2c}}(t) \right) + \left( u_{\mathrm{2ae}}(t) + u_{\mathrm{2be}}(t) + u_{\mathrm{2ce}}(t)\right)=\SI{0}{\volt}. \label{eq:u2_i2_symres} @@ -131,65 +132,71 @@ \end{itemize} \end{solutionblock} -\subtask{Calculate and sketch the voltages $u_\mathrm{2ab}(t)$, $u_\mathrm{2bc}(t)$, $u_\mathrm{2a}(t)$ and $u_\mathrm{2a,0}(t)$ -depending on these switching states.} +\subtask{Calculate and sketch the voltages $u_\mathrm{2ab}(t)$, $u_\mathrm{2bc}(t)$, $u_\mathrm{2a}(t)$ and $u_\mathrm{2a0}(t)$ +depending on the switching states.} \begin{solutionblock} - The voltage $u_{\mathrm{2ab}(t)}$ is calculated by + The voltage $u_{\mathrm{2ab}}(t)$ is calculated by \begin{equation} - u_{\mathrm{2ab}(t)} = u_{\mathrm{2a,0}(t)} - u_{\mathrm{2b,0}(t)}. + u_{\mathrm{2ab}}(t) = u_{\mathrm{2a0}}(t) - u_{\mathrm{2b0}}(t). \label{eq:u2ab_gen} \end{equation} - In similar way the voltage $u_{\mathrm{2bc}(t)}$ is calculated by + In similar way the voltage $u_{\mathrm{2bc}}(t)$ is calculated by \begin{equation} - u_{\mathrm{2bc}(t)} = u_{\mathrm{2b,0}(t)} - u_{\mathrm{2c,0}(t)}. + u_{\mathrm{2bc}}(t) = u_{\mathrm{2b0}}(t) - u_{\mathrm{2c0}}(t). \label{eq:u2bc_gen} \end{equation} - The voltage $u_{\mathrm{2a}(t)}$ is obtained by + The voltage $u_{\mathrm{2a}}(t)$ is obtained by \begin{equation} - u_{\mathrm{2a}(t)} = u_{\mathrm{2ab}(t)} + u_{\mathrm{2b,0}(t)}. + u_{\mathrm{2a}}(t) = u_{\mathrm{2ab}}(t) + u_{\mathrm{2b0}}(t). \label{eq:u2a_1} \end{equation} - Additional voltage $u_{\mathrm{2a}(t)}$ is obtained by + Additional voltage $u_{\mathrm{2a}}(t)$ is obtained by \begin{equation} - u_{\mathrm{2a}(t)} = u_{\mathrm{2ab}(t)} + u_{\mathrm{2bc}(t)} + u_{\mathrm{2c}(t)}. + u_{\mathrm{2a}}(t) = u_{\mathrm{2ab}}(t) + u_{\mathrm{2bc}}(t) + u_{\mathrm{2c}}(t). \label{eq:u2a_2} \end{equation} The addition of \eqref{eq:u2a_1} and \eqref{eq:u2a_2} results in \begin{equation} - 2u_{\mathrm{2a}(t)} = 2u_{\mathrm{2ab}(t)} + u_{\mathrm{2bc}(t)} - + \left( u_{\mathrm{2a}(t)} + u_{\mathrm{2b}(t)} + u_{\mathrm{2v}(t)}\right) - - u_{\mathrm{2a}(t)}. + 2u_{\mathrm{2a}}(t) = 2u_{\mathrm{2ab}}(t) + u_{\mathrm{2bc}}(t) + + \left( u_{\mathrm{2a}}(t) + u_{\mathrm{2b}}(t) + u_{\mathrm{2v}}(t)\right) + - u_{\mathrm{2a}}(t). \label{eq:u2a_gen} \end{equation} - Solving \eqref{eq:u2a_gen} with respect to $u_{\mathrm{2a}(t)}$ leads to + Solving \eqref{eq:u2a_gen} with respect to $u_{\mathrm{2a}}(t)$ leads to \begin{equation} - u_{\mathrm{2a}(t)} = \frac{2}{3} u_{\mathrm{2ab}(t)} + \frac{1}{3} u_{\mathrm{2bc}(t)} + u_{\mathrm{2a}}(t) = \frac{2}{3} u_{\mathrm{2ab}}(t) + \frac{1}{3} u_{\mathrm{2bc}}(t) \end{equation} - The voltage $u_{\mathrm{0,n}(t)}$ is obtained by + The voltage $u_{\mathrm{0,n}}(t)$ is obtained by \begin{equation} - u_{\mathrm{0,n}(t)} = u_{\mathrm{2a,0}(t)} - u_{\mathrm{2a}(t)} - = u_{\mathrm{2a,0}(t)} - \frac{2}{3} u_{\mathrm{2ab}(t)} - \frac{1}{3} u_{\mathrm{2bc}(t)} + u_{\mathrm{0,n}}(t) = u_{\mathrm{2a0}}(t) - u_{\mathrm{2a}}(t) + = u_{\mathrm{2a0}}(t) - \frac{2}{3} u_{\mathrm{2ab}}(t) - \frac{1}{3} u_{\mathrm{2bc}}(t) \end{equation} Using \eqref{eq:u2ab_gen} and \eqref{eq:u2bc_gen} leads to \begin{equation} \begin{split} - u_{\mathrm{0,n}(t)} &= u_{\mathrm{2a,0}(t)} - \frac{2}{3} \left( u_{\mathrm{2a,0}(t)} - u_{\mathrm{2b,0}(t)} \right) - - \frac{1}{3} \left( u_{\mathrm{2b,0}(t)} - u_{\mathrm{2c,0}(t)} \right) \\ - u_{\mathrm{0,n}(t)} &= \frac{1}{3} \left( u_{\mathrm{2a,0}(t)} + u_{\mathrm{2b,0}(t)} + u_{\mathrm{2c,0}(t)} \right). + u_{\mathrm{0,n}}(t) &= u_{\mathrm{2a0}}(t) - \frac{2}{3} \left( u_{\mathrm{2a0}}(t) - u_{\mathrm{2b0}}(t) \right) + - \frac{1}{3} \left( u_{\mathrm{2b0}}(t) - u_{\mathrm{2c0}}(t) \right) \\ + u_{\mathrm{0,n}}(t) &= \frac{1}{3} \left( u_{\mathrm{2a0}}(t) + u_{\mathrm{2b0}}(t) + u_{\mathrm{2c0}}(t) \right). \end{split} \end{equation} \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: +amplitude of the fu +ndamental 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} -\input{fig/ex07/Fig_standardization_to_fudamental_freq.tex} -\input{fig/ex07/Fig_ trigonometric_approach_triangle.tex} -\end{solutionblock} \ No newline at end of file + \input{fig/ex07/Fig_Voltage_U_um_excerpt} + \input{fig/ex07/Fig_graphic_solutions_cos_terms} + \input{fig/ex07/Fig_standardization_to_fudamental_freq.tex} + \input{fig/ex07/Fig_ trigonometric_approach_triangle.tex} +\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} +\end{solutionblock} From 79fdf5d711a6402929c95a449097dd42c3dc6d34 Mon Sep 17 00:00:00 2001 From: SevenOfNinePE Date: Mon, 27 Jan 2025 11:24:26 +0100 Subject: [PATCH 09/10] Ex07 Task2: Add final correction according alignment to lecture. --- exercise/tex/exercise07.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index fb73610..f455d9d 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -17,7 +17,7 @@ \task{Symmetrical 3-phase switching rectifier} An symmetrical switching rectifier in three-phase bridge topology shall supply a -symmetrical three-phase consumer. The consumer is simulated by an inductance and +symmetrical three-phase consumer in star connection. The consumer is simulated by an inductance and a sinusoidal counter voltage per phase. The inverter is operated with a basic switching frequency. The switching elements are considered as ideal. @@ -132,8 +132,8 @@ \end{itemize} \end{solutionblock} -\subtask{Calculate and sketch the voltages $u_\mathrm{2ab}(t)$, $u_\mathrm{2bc}(t)$, $u_\mathrm{2a}(t)$ and $u_\mathrm{2a0}(t)$ -depending on the switching states.} +\subtask{Calculate and sketch the voltages $u_\mathrm{2ab}(t)$, $u_\mathrm{2bc}(t)$, $u_\mathrm{2a}(t)$ and +the star-to-ground voltage $u_\mathrm{2n0}(t)$ depending on the switching states.} \begin{solutionblock} The voltage $u_{\mathrm{2ab}}(t)$ is calculated by \begin{equation} From 485c5d1819a68346439497bafa8f044c55d65bee Mon Sep 17 00:00:00 2001 From: SevenOfNinePE Date: Mon, 27 Jan 2025 12:09:52 +0100 Subject: [PATCH 10/10] Ex07 Task2: Additional update of task description to simplify solution --- .../ex07/Fig_ThreePhaseInverter_6StepMode.tex | 3 +++ exercise/tex/exercise07.tex | 21 ++++++++++++++++++- 2 files changed, 23 insertions(+), 1 deletion(-) diff --git a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex index 5ef33ed..b9540ac 100644 --- a/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex +++ b/exercise/fig/ex07/Fig_ThreePhaseInverter_6StepMode.tex @@ -86,6 +86,9 @@ (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 arrow u1 + (U1p) ++(0.3,0.5) to [open,v^=$$,voltage = straight] ++(0,-6)coordinate (Uges) + (Uges) ++ (0.3,3) node[anchor=north,color=black]{$U_\mathrm{1}$} % Add voltage arrow 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) diff --git a/exercise/tex/exercise07.tex b/exercise/tex/exercise07.tex index f455d9d..bab3737 100644 --- a/exercise/tex/exercise07.tex +++ b/exercise/tex/exercise07.tex @@ -28,7 +28,7 @@ \centering % Center the table \begin{tabular}{ll} \toprule - Input voltages: & $U_\mathrm{1p}=\SI{255}{\volt}$ \quad $U_\mathrm{1m}=\SI{255}{\volt}$ \\ + Input voltages: & $U_\mathrm{1}=\SI{510}{\volt}$ \quad $U_\mathrm{1p}=U_\mathrm{1m}=U_\mathrm{1}/2$ \\ Internal voltages: & $u_{\mathrm{2ae}}(t) = \sqrt{2} \cdot \SI{220}{\volt} \cdot \sin(\omega_1t)$ \\ Circular frequency: & $\omega_1 = \SI{2 \pi \cdot 30}{\frac{1}{\second}}$ \\ Inductivity per phase: & $L= \SI{10}{\milli \henry}$ \\ @@ -190,6 +190,25 @@ \label{sub:DecomposeVoltage} } \begin{solutionblock} + In the case of odd and alternating functions corresponding to $f(x)=-f(x+\pi)$ the Fourier coefficients are: + \begin{equation} + \begin{split} + a_\mathrm{k} &= 0 \\ + a_\mathrm{k} &= \frac{4}{\pi} \int_0^{\pi/2} f(x)\sin(x) \mathrm{d}x \quad k=\mathrm{odd} \\ + f(x) &= \sum_{k}^{} \left( b_k \sin(kx) \right). + \end{split} + \end{equation} + The coefficients $b_k$ are the amplitudes of the respective harmonic. The voltage $u_{\mathrm{2a}}(t)$ needs + only to be integrated up to $\pi/2$. Only the terms with odd order numbers are taken into account. + \begin{equation} + \begin{split} + a_\mathrm{k} &= \frac{4}{\pi} \int_0^{\pi/3} f(x)\sin(x) \mathrm{d}x \quad k=\mathrm{odd} \\ + f(x) &= \sum_{k}^{} \left( b_k \sin(kx) \right). + \end{split} + \end{equation} + + + \input{fig/ex07/Fig_Voltage_U_um_excerpt} \input{fig/ex07/Fig_graphic_solutions_cos_terms} \input{fig/ex07/Fig_standardization_to_fudamental_freq.tex}