From b50520675d96fbe16e73747e5f0b409fe6b4cb5f Mon Sep 17 00:00:00 2001 From: Wallscheid Date: Mon, 23 Dec 2024 21:27:05 +0100 Subject: [PATCH] add to lec05 --- .vscode/ltex.dictionary.en-US.txt | 1 + course_template | 2 +- lecture/tex/Lecture05.tex | 445 +++++++++++++++++++++++++----- lecture/tex/dict.tex | 8 + 4 files changed, 387 insertions(+), 69 deletions(-) diff --git a/.vscode/ltex.dictionary.en-US.txt b/.vscode/ltex.dictionary.en-US.txt index 6bb1a49..7df0032 100644 --- a/.vscode/ltex.dictionary.en-US.txt +++ b/.vscode/ltex.dictionary.en-US.txt @@ -1,2 +1,3 @@ galvanically thyristor +line-commutated diff --git a/course_template b/course_template index 0a7de75..8e85156 160000 --- a/course_template +++ b/course_template @@ -1 +1 @@ -Subproject commit 0a7de75ce9cde8f3f20e5ddc65d2c3db6404afca +Subproject commit 8e851563602c2013d7833285e3780c7476e55e10 diff --git a/lecture/tex/Lecture05.tex b/lecture/tex/Lecture05.tex index 3924c25..67551c5 100644 --- a/lecture/tex/Lecture05.tex +++ b/lecture/tex/Lecture05.tex @@ -124,14 +124,18 @@ \section{Thyristor-based converters} grid=both, ] \nextgroupplot[ylabel = {$u_2(\omega t)$}] - \addplot[domain=0:pi, samples=50, signalblue, thick]{sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=10, signalblue, thick]{0}; - \addplot[domain=2*pi:3*pi, samples=50, signalblue, thick]{sin(deg(x))}; - \addplot[domain=3*pi:4*pi, samples=10, signalblue, thick]{0}; - \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed]{1/pi}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A]{sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=10, signalblue, thick, name path = B]{0}; + \addplot[domain=2*pi:3*pi, samples=50, signalblue, thick, name path = C]{sin(deg(x))}; + \addplot[domain=3*pi:4*pi, samples=10, signalblue, thick, name path = D]{0}; + \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed, name path = avg]{1/pi}; \node at (axis cs:3*pi/2,1/pi) [anchor=south] {$\overline{u}_2$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A and avg, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B and avg, soft clip={domain=pi:2*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=C and avg, soft clip={domain=2*pi:3*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=D and avg, soft clip={domain=3*pi:4*pi}]; - \nextgroupplot[ylabel = {$u_\mathrm{D}(t)$}, ymin=-1.15, ymax=1.15, height=0.4\textheight, xlabel = {$\omega t$}] + \nextgroupplot[ylabel = {$u_\mathrm{D}(\omega t)$}, ymin=-1.15, ymax=1.15, height=0.4\textheight, xlabel = {$\omega t$}] \addplot[domain=0:4*pi, samples=50, signalblue, dashed]{sin(deg(x))}; \addplot[domain=0:pi, samples=10, signalblue, thick]{0}; \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{sin(deg(x))}; @@ -168,15 +172,18 @@ \section{Thyristor-based converters} grid=both, ] \nextgroupplot[ylabel = {$u_2(\omega t)$}] - \addplot[domain=0:2*pi, samples=150, signalblue, thick]{max(sin(deg(x))*max(sign(x-\a),0),0)}; - \addplot[domain=2*pi:4*pi, samples=150, signalblue, thick]{max(sin(deg(x))*max(sign(x-\a-2*pi),0),0)}; - \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed]{1/(2*pi)*(1+cos(deg(\a)))}; + \addplot[domain=0:2*pi, samples=150, signalblue, thick, name path = A]{max(sin(deg(x))*max(sign(x-\a),0),0)}; + \addplot[domain=2*pi:4*pi, samples=150, signalblue, thick, name path = B]{max(sin(deg(x))*max(sign(x-\a-2*pi),0),0)}; + \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed, name path = avg]{1/(2*pi)*(1+cos(deg(\a)))}; \node at (axis cs:3*pi/2,{1/(2*pi)*(1+cos(deg(\a)))}) [anchor=south] {$\overline{u}_2$}; \draw[->] (axis cs:0,0.5) -- node[above]{$\alpha$} (axis cs:\a,0.5) ; \draw[->] (axis cs:2*pi,0.5) -- node[above]{$\alpha$} (axis cs:2*pi+\a,0.5) ; \draw[dashed, thick] (axis cs:2*pi,0) -- (axis cs:2*pi,1); + \addplot[shadecolor, opacity=0.3] fill between[of=A and avg, soft clip={domain=0:2*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B and avg, soft clip={domain=2*pi:4*pi}]; - \nextgroupplot[ylabel = {$u_\mathrm{T}(t)$}, ymin=-1.15, ymax=1.15, height=0.4\textheight, xlabel = {$\omega t$}] + + \nextgroupplot[ylabel = {$u_\mathrm{T}(\omega t)$}, ymin=-1.15, ymax=1.15, height=0.4\textheight, xlabel = {$\omega t$}] \addplot[domain=0:4*pi, samples=50, signalblue, dashed]{sin(deg(x))}; \addplot[domain=0:2*pi, samples=150, signalblue, thick]{max(sin(deg(x))*max(-sign(x-\a),0),0) + sin(deg(x))*max(sign(x-pi),0)}; \addplot[domain=2*pi:4*pi, samples=150, signalblue, thick]{max(sin(deg(x))*max(-sign(x-\a-2*pi),0),0) + sin(deg(x))*max(sign(x-3*pi),0)}; @@ -187,6 +194,11 @@ \section{Thyristor-based converters} \end{columns} \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Half-cycle rectification / M1C circuit %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{M1C circuit} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% M1C rectifier %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -265,11 +277,11 @@ \section{Thyristor-based converters} clip=false ] \nextgroupplot[ylabel = {$u_2(\omega t)$}] - \addplot[domain=0:pi, samples=50, signalblue, thick]{(x < \a) * 0 + (x > \a) * sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{(x - pi < \a) * 0 + (x - pi > \a) * -sin(deg(x))}; - \addplot[domain=2*pi:3*pi, samples=50, signalblue, thick]{(x - 2*pi < \a) * 0 + (x - 2*pi > \a) * sin(deg(x))}; - \addplot[domain=3*pi:4*pi, samples=50, signalblue, thick]{(x - 3*pi < \a) * 0 + (x - 3*pi > \a) * -sin(deg(x))}; - \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed]{1/(pi)*(1+cos(deg(\a)))}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A]{(x < \a) * 0 + (x > \a) * sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=50, signalblue, thick, name path = B]{(x - pi < \a) * 0 + (x - pi > \a) * -sin(deg(x))}; + \addplot[domain=2*pi:3*pi, samples=50, signalblue, thick, name path = C]{(x - 2*pi < \a) * 0 + (x - 2*pi > \a) * sin(deg(x))}; + \addplot[domain=3*pi:4*pi, samples=50, signalblue, thick, name path = D]{(x - 3*pi < \a) * 0 + (x - 3*pi > \a) * -sin(deg(x))}; + \addplot[domain=0:4*pi, samples=10, signalblue, thick,dashed, name path = avg]{1/(pi)*(1+cos(deg(\a)))}; \node at (axis cs:3*pi/2+0.4,{1/(pi)*(1+cos(deg(\a)))}) [anchor=north] {$\overline{u}_2$}; \draw[->] (axis cs:0,1) -- node[above]{$\alpha$} (axis cs:\a,1) ; \draw[->] (axis cs:2*pi,1) -- node[above]{$\alpha$} (axis cs:2*pi+\a,1) ; @@ -278,6 +290,10 @@ \section{Thyristor-based converters} \addplot[domain=0:4*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; \node at (axis cs:pi*2.5,-0.75) [anchor=south, signalbrown] {$u_{s,2}$}; \node at (axis cs:pi*3.5,-0.75) [anchor=south, signalgreen] {$u_{s,1}$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A and avg, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B and avg, soft clip={domain=pi:2*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=C and avg, soft clip={domain=2*pi:3*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=D and avg, soft clip={domain=3*pi:4*pi}]; \nextgroupplot[ylabel = {$u_\mathrm{T}(\omega t)$}, xlabel = {$\omega t$}, ymin=-2.15, ytick={-2, -1,0,1}, yticklabels={$-2\hat{u}_\mathrm{s}$, $-\hat{u}_\mathrm{s}$,$0$,$\hat{u}_\mathrm{s}$}, height=0.5\textheight] @@ -298,6 +314,11 @@ \section{Thyristor-based converters} \end{figure} \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% M2C circuit %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{M2C circuit} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% M2C converter: resistive load %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -392,9 +413,9 @@ \section{Thyristor-based converters} clip=false ] \nextgroupplot[ylabel = {$u_\mathrm{s}(\omega t)$}, title=CCM, height=0.475\textheight] % voltage CCM - \addplot[domain=0:pi, samples=50, signalblue, thick]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{2/(pi)*(cos(deg(\a)))}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A1]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=50, signalblue, thick, name path = B1]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg1]{2/(pi)*(cos(deg(\a)))}; \node at (axis cs:3*pi/2+0.4,{2/(pi)*(cos(deg(\a)))}) [anchor=north] {$\overline{u}_2$}; \draw[->] (axis cs:0,1) -- node[above]{$\alpha$} (axis cs:\a,1) ; \draw[->] (axis cs:pi,1) -- node[above]{$\alpha$} (axis cs:pi+\a,1) ; @@ -403,11 +424,13 @@ \section{Thyristor-based converters} \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; \node at (axis cs:pi*3/4,-0.75) [signalbrown, fill=white,inner sep=1pt] {$u_{s,2}$}; \node at (axis cs:pi*7/4,-0.75) [signalgreen, fill=white,inner sep=1pt] {$u_{s,1}$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A1 and avg1, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B1 and avg1, soft clip={domain=pi:2*pi}]; \nextgroupplot[title=BCM, height=0.475\textheight] % voltage BCM - \addplot[domain=0:pi, samples=50, signalblue, thick]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{2/(pi)*(cos(deg(\a)))}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A2]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=50, signalblue, thick, name path = B2]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg2]{2/(pi)*(cos(deg(\a)))}; \node at (axis cs:3*pi/2+0.4,{2/(pi)*(cos(deg(\a)))}) [anchor=north] {$\overline{u}_2$}; \draw[->] (axis cs:0,1) -- node[above]{$\alpha$} (axis cs:\a,1) ; \draw[->] (axis cs:pi,1) -- node[above]{$\alpha$} (axis cs:pi+\a,1) ; @@ -416,10 +439,12 @@ \section{Thyristor-based converters} \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; \node at (axis cs:pi*3/4,-0.75) [signalbrown, fill=white,inner sep=1pt] {$u_{s,2}$}; \node at (axis cs:pi*7/4,-0.75) [signalgreen, fill=white,inner sep=1pt] {$u_{s,1}$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A2 and avg2, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B2 and avg2, soft clip={domain=pi:2*pi}]; \nextgroupplot[title=DCM, height=0.475\textheight] % voltage DCM - \addplot[domain=0:2*pi, samples=200, signalblue, thick]{(x < \adcm +\b - pi) * sin(deg(x+pi)) + (x > \adcm + \b -pi) * (x < \adcm) * \ucdcm + (x > \adcm)* (x < \adcm + \b) * sin(deg(x)) + (x > \adcm + \b) * (x < \adcm + pi) * \ucdcm + (x > \adcm + pi) * sin(deg(x-pi))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{\ucdcm}; + \addplot[domain=0:2*pi, samples=200, signalblue, thick, name path = A3]{(x < \adcm +\b - pi) * sin(deg(x+pi)) + (x > \adcm + \b -pi) * (x < \adcm) * \ucdcm + (x > \adcm)* (x < \adcm + \b) * sin(deg(x)) + (x > \adcm + \b) * (x < \adcm + pi) * \ucdcm + (x > \adcm + pi) * sin(deg(x-pi))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg3]{\ucdcm}; \node at (axis cs:3*pi/2+0.4,\ucdcm) [anchor=north] {$\overline{u}_2$}; \addplot[domain=0:2*pi, samples=50, signalgreen, dashed]{sin(deg(x))}; \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; @@ -428,33 +453,47 @@ \section{Thyristor-based converters} \draw[->] (axis cs:0,1) -- node[above]{$\alpha$} (axis cs:\adcm,1) ; \draw[->] (axis cs:pi,1) -- node[above]{$\alpha$} (axis cs:pi+\adcm,1) ; \draw[dashed, thick] (axis cs:pi,0) -- (axis cs:pi,1); + \addplot[shadecolor, opacity=0.3] fill between[of=A3 and avg3, soft clip={domain=0:2*pi}]; \nextgroupplot[ylabel = {$i_\mathrm{L}(\omega t)$}, xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={$0$, } ] %current CCM - \addplot[domain=0:\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs1}; - \addplot[domain=\a:pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; - \addplot[domain=pi:pi+\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs1}; - \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg1}; + \addplot[domain=0:\a, samples=50, signalred, thick, name path = Ai1]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs1}; + \addplot[domain=\a:pi, samples=50, signalred, thick, name path = Bi1]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; + \addplot[domain=pi:pi+\a, samples=50, signalred, thick, name path = Ci1]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs1}; + \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick, name path = Di1]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi1]{\iLavg1}; \draw[<->] (axis cs:\a,0.75) -- node[above, fill = white]{$\beta=\pi$} (axis cs:\a+pi,0.75) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai1 and avgi1, soft clip={domain=0:\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi1 and avgi1, soft clip={domain=\a:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci1 and avgi1, soft clip={domain=pi:pi+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di1 and avgi1, soft clip={domain=pi+\a:2*pi}]; \nextgroupplot[xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={ } ] %current BCM - \addplot[domain=0:\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs2}; - \addplot[domain=\a:pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; - \addplot[domain=pi:pi+\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs2}; - \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg2}; + \addplot[domain=0:\a, samples=50, signalred, thick, name path = Ai2]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs2}; + \addplot[domain=\a:pi, samples=50, signalred, thick, name path = Bi2]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; + \addplot[domain=pi:pi+\a, samples=50, signalred, thick, name path = Ci2]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs2}; + \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick, name path = Di2]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi2]{\iLavg2}; \draw[<->] (axis cs:\a,0.6) -- node[above, fill = white]{$\beta=\pi$} (axis cs:\a+pi,0.6) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai2 and avgi2, soft clip={domain=0:\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi2 and avgi2, soft clip={domain=\a:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci2 and avgi2, soft clip={domain=pi:pi+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di2 and avgi2, soft clip={domain=pi+\a:2*pi}]; \nextgroupplot[xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={} ] %current DCM - \addplot[domain=\adcm:\adcm+\b, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm) - \u1*(cos(deg(x))-cos(deg(\adcm))))}; - \addplot[domain=0:\adcm+\b-pi, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm+pi) - \u1*(cos(deg(x+pi))-cos(deg(\adcm))))}; - \addplot[domain=\adcm+pi:2*pi, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm-pi) - \u1*(cos(deg(x-pi))-cos(deg(\adcm))))}; - \addplot[domain=\adcm+\b:\adcm+pi, samples=50, signalred, thick]{0}; - \addplot[domain=\adcm+\b-pi:\adcm, samples=50, signalred, thick]{0}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg3}; + \addplot[domain=\adcm:\adcm+\b, samples=50, signalred, thick, name path = Ai3]{1/\Lw*(-\ucdcm*(x-\adcm) - \u1*(cos(deg(x))-cos(deg(\adcm))))}; + \addplot[domain=0:\adcm+\b-pi, samples=50, signalred, thick, name path = Bi3]{1/\Lw*(-\ucdcm*(x-\adcm+pi) - \u1*(cos(deg(x+pi))-cos(deg(\adcm))))}; + \addplot[domain=\adcm+pi:2*pi, samples=50, signalred, thick, name path = Ci3]{1/\Lw*(-\ucdcm*(x-\adcm-pi) - \u1*(cos(deg(x-pi))-cos(deg(\adcm))))}; + \addplot[domain=\adcm+\b:\adcm+pi, samples=50, signalred, thick, name path = Di3]{0}; + \addplot[domain=\adcm+\b-pi:\adcm, samples=50, signalred, thick, name path = Ei3]{0}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi3]{\iLavg3}; \draw[<->] (axis cs:\adcm,0.5) -- node[above, fill = white]{$\beta<\pi$} (axis cs:\adcm+\b,0.5) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai3 and avgi3, soft clip={domain=\adcm:\adcm+\b}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi3 and avgi3, soft clip={domain=0:\adcm+\b-pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci3 and avgi3, soft clip={domain=\adcm+pi:2*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di3 and avgi3, soft clip={domain=\adcm+\b:\adcm+pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ei3 and avgi3, soft clip={domain=\adcm+\b-pi:\adcm}]; \end{groupplot} \end{tikzpicture} @@ -544,9 +583,9 @@ \section{Thyristor-based converters} clip=false ] \nextgroupplot[ylabel = {$u_\mathrm{s}(\omega t)$}, title=CCM, height=0.475\textheight] % voltage CCM - \addplot[domain=0:pi, samples=50, signalblue, thick]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{2/(pi)*(cos(deg(\a)))}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A1]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=50, signalblue, thick, name path = B1]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg1]{2/(pi)*(cos(deg(\a)))}; \node at (axis cs:pi,\uc1) [anchor=north, fill = white, inner sep = 2pt] {$\overline{u}_2$}; \draw[->] (axis cs:0,0) -- node[above]{$\alpha$} (axis cs:\a,0); \draw[->] (axis cs:pi,0) -- node[above]{$\alpha$} (axis cs:pi+\a,0); @@ -554,11 +593,13 @@ \section{Thyristor-based converters} \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; \node at (axis cs:pi*1/4,0.75) [signalbrown, fill=white,inner sep=1pt] {$u_{s,2}$}; \node at (axis cs:pi*5/4,0.75) [signalgreen, fill=white,inner sep=1pt] {$u_{s,1}$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A1 and avg1, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B1 and avg1, soft clip={domain=pi:2*pi}]; \nextgroupplot[title=BCM, height=0.475\textheight] % voltage BCM - \addplot[domain=0:pi, samples=50, signalblue, thick]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; - \addplot[domain=pi:2*pi, samples=50, signalblue, thick]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{2/(pi)*(cos(deg(\a)))}; + \addplot[domain=0:pi, samples=50, signalblue, thick, name path = A2]{(x < \a) * -sin(deg(x)) + (x > \a) * sin(deg(x))}; + \addplot[domain=pi:2*pi, samples=50, signalblue, thick, name path = B2]{(x - pi < \a) * sin(deg(x)) + (x - pi > \a) * -sin(deg(x))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg2]{2/(pi)*(cos(deg(\a)))}; \node at (axis cs:pi,\uc1) [anchor=north, fill = white, inner sep = 2pt] {$\overline{u}_2$}; \draw[->] (axis cs:0,0) -- node[above]{$\alpha$} (axis cs:\a,0); \draw[->] (axis cs:pi,0) -- node[above]{$\alpha$} (axis cs:pi+\a,0); @@ -566,10 +607,12 @@ \section{Thyristor-based converters} \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; \node at (axis cs:pi*1/4,0.75) [signalbrown, fill=white,inner sep=1pt] {$u_{s,2}$}; \node at (axis cs:pi*5/4,0.75) [signalgreen, fill=white,inner sep=1pt] {$u_{s,1}$}; + \addplot[shadecolor, opacity=0.3] fill between[of=A2 and avg2, soft clip={domain=0:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B2 and avg2, soft clip={domain=pi:2*pi}]; \nextgroupplot[title=DCM, height=0.475\textheight] % voltage DCM - \addplot[domain=0:2*pi, samples=200, signalblue, thick]{(x < \adcm +\b - pi) * sin(deg(x+pi)) + (x > \adcm + \b -pi) * (x < \adcm) * \ucdcm + (x > \adcm)* (x < \adcm + \b) * sin(deg(x)) + (x > \adcm + \b) * (x < \adcm + pi) * \ucdcm + (x > \adcm + pi) * sin(deg(x-pi))}; - \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed]{\ucdcm}; + \addplot[domain=0:2*pi, samples=200, signalblue, thick, name path = A3]{(x < \adcm +\b - pi) * sin(deg(x+pi)) + (x > \adcm + \b -pi) * (x < \adcm) * \ucdcm + (x > \adcm)* (x < \adcm + \b) * sin(deg(x)) + (x > \adcm + \b) * (x < \adcm + pi) * \ucdcm + (x > \adcm + pi) * sin(deg(x-pi))}; + \addplot[domain=0:2*pi, samples=10, signalblue, thick,dashed, name path = avg3]{\ucdcm}; \node at (axis cs:pi,\ucdcm) [anchor=north, fill = white, inner sep = 2pt] {$\overline{u}_2$}; \addplot[domain=0:2*pi, samples=50, signalgreen, dashed]{sin(deg(x))}; \addplot[domain=0:2*pi, samples=50, signalbrown, dashed]{-sin(deg(x))}; @@ -577,33 +620,48 @@ \section{Thyristor-based converters} \node at (axis cs:pi*5/4,0.75) [signalgreen, fill=white,inner sep=1pt] {$u_{s,1}$}; \draw[->] (axis cs:0,0) -- node[above]{$\alpha$} (axis cs:\a,0); \draw[->] (axis cs:pi,0) -- node[above]{$\alpha$} (axis cs:pi+\a,0); + \addplot[shadecolor, opacity=0.3] fill between[of=A3 and avg3, soft clip={domain=0:2*pi}]; + \nextgroupplot[ylabel = {$i_\mathrm{L}(\omega t)$}, xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={$0$, } ] %current CCM - \addplot[domain=0:\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs1}; - \addplot[domain=\a:pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; - \addplot[domain=pi:pi+\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs1}; - \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg1}; + \addplot[domain=0:\a, samples=50, signalred, thick, name path = Ai1]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs1}; + \addplot[domain=\a:pi, samples=50, signalred, thick, name path = Bi1]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; + \addplot[domain=pi:pi+\a, samples=50, signalred, thick, name path = Ci1]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs1}; + \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick, name path = Di1]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs1 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi1]{\iLavg1}; \draw[<->] (axis cs:\a,0.75) -- node[above, fill = white]{$\beta=\pi$} (axis cs:\a+pi,0.75) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai1 and avgi1, soft clip={domain=0:\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi1 and avgi1, soft clip={domain=\a:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci1 and avgi1, soft clip={domain=pi:pi+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di1 and avgi1, soft clip={domain=pi+\a:2*pi}]; \nextgroupplot[xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={} ] %current BCM - \addplot[domain=0:\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs2}; - \addplot[domain=\a:pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; - \addplot[domain=pi:pi+\a, samples=50, signalred, thick]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs2}; - \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg2}; + \addplot[domain=0:\a, samples=50, signalred, thick, name path = Ai2]{1/\Lw*(-\uc1*x + \u1*(cos(deg(x))-1))+\iLs2}; + \addplot[domain=\a:pi, samples=50, signalred, thick, name path = Bi2]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x))+cos(deg(\a))) - \uc1*(x-\a))}; + \addplot[domain=pi:pi+\a, samples=50, signalred, thick, name path = Ci2]{1/\Lw*(-\uc1*(x-pi) + \u1*(cos(deg(x - pi))-1))+\iLs2}; + \addplot[domain=pi+\a:2*pi, samples=50, signalred, thick, name path = Di2]{1/\Lw*(-\uc1*\a + \u1*(cos(deg(\a))-1))+\iLs2 + 1/\Lw*(\u1*(-cos(deg(x-pi))+cos(deg(\a))) - \uc1*(x-\a-pi))}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi2]{\iLavg2}; \draw[<->] (axis cs:\a,0.6) -- node[above, fill = white]{$\beta=\pi$} (axis cs:\a+pi,0.6) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai2 and avgi2, soft clip={domain=0:\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi2 and avgi2, soft clip={domain=\a:pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci2 and avgi2, soft clip={domain=pi:pi+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di2 and avgi2, soft clip={domain=pi+\a:2*pi}]; \nextgroupplot[xlabel = {$\omega t$}, ymin=-0.01, ytick={0, 1/2, 1}, yticklabels={} ] %current DCM - \addplot[domain=\adcm:\adcm+\b, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm) - \u1*(cos(deg(x))-cos(deg(\adcm))))}; - \addplot[domain=0:\adcm+\b-pi, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm+pi) - \u1*(cos(deg(x+pi))-cos(deg(\adcm))))}; - \addplot[domain=\adcm+pi:2*pi, samples=50, signalred, thick]{1/\Lw*(-\ucdcm*(x-\adcm-pi) - \u1*(cos(deg(x-pi))-cos(deg(\adcm))))}; - \addplot[domain=\adcm+\b:\adcm+pi, samples=50, signalred, thick]{0}; - \addplot[domain=\adcm+\b-pi:\adcm, samples=50, signalred, thick]{0}; - \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed]{\iLavg3}; + \addplot[domain=\adcm:\adcm+\b, samples=50, signalred, thick, name path = Ai3]{1/\Lw*(-\ucdcm*(x-\adcm) - \u1*(cos(deg(x))-cos(deg(\adcm))))}; + \addplot[domain=0:\adcm+\b-pi, samples=50, signalred, thick, name path = Bi3]{1/\Lw*(-\ucdcm*(x-\adcm+pi) - \u1*(cos(deg(x+pi))-cos(deg(\adcm))))}; + \addplot[domain=\adcm+pi:2*pi, samples=50, signalred, thick, name path = Ci3]{1/\Lw*(-\ucdcm*(x-\adcm-pi) - \u1*(cos(deg(x-pi))-cos(deg(\adcm))))}; + \addplot[domain=\adcm+\b:\adcm+pi, samples=50, signalred, thick, name path = Di3]{0}; + \addplot[domain=\adcm+\b-pi:\adcm, samples=50, signalred, thick, name path = Ei3]{0}; + \addplot[domain=0:2*pi, samples=10, signalred, thick,dashed, name path = avgi3]{\iLavg3}; \draw[<->] (axis cs:\adcm,0.5) -- node[above, fill = white]{$\beta<\pi$} (axis cs:\adcm+\b,0.5) ; + \addplot[shadecolor, opacity=0.3] fill between[of=Ai3 and avgi3, soft clip={domain=\adcm:\adcm+\b}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Bi3 and avgi3, soft clip={domain=0:\adcm+\b-pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ci3 and avgi3, soft clip={domain=\adcm+pi:2*pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Di3 and avgi3, soft clip={domain=\adcm+\b:\adcm+pi}]; + \addplot[shadecolor, opacity=0.3] fill between[of=Ei3 and avgi3, soft clip={domain=\adcm+\b-pi:\adcm}]; \end{groupplot} \end{tikzpicture} @@ -658,6 +716,11 @@ \section{Thyristor-based converters} \end{figure} \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Commutation %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Commutation} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Commutation %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -944,7 +1007,7 @@ \section{Thyristor-based converters} \end{equation} The output voltage lost in the process corresponds to \begin{equation} - \Delta u = \frac{1}{\pi}\int_{\alpha}^{\alpha + \kappa} u_\mathrm{s}\sin(\omega t) \mathrm{d}(\omega t) = \frac{u_\mathrm{s}}{\pi}\left[-cos(\omega t)\right]_{\alpha}^{\alpha + \kappa} = \frac{u_\mathrm{s}}{\pi}\left[\cos(\alpha) - \cos(\alpha + \kappa)\right]. + \Delta u = \frac{1}{\pi}\int_{\alpha}^{\alpha + \kappa} u_\mathrm{s}\sin(\omega t) \mathrm{d}(\omega t) = \frac{u_\mathrm{s}}{\pi}\left[-\cos(\omega t)\right]_{\alpha}^{\alpha + \kappa} = \frac{u_\mathrm{s}}{\pi}\left[\cos(\alpha) - \cos(\alpha + \kappa)\right]. \end{equation} Inserting \eqref{eq:overlap_angle_M2C} for $\kappa$ yields \begin{equation} @@ -956,6 +1019,11 @@ \section{Thyristor-based converters} Hence, the average output voltage is deviating by $\Delta u$ due to the commutation process. \end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Complex power analysis %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Complex power analysis} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% M2C: complex power analysis %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -1066,10 +1134,10 @@ \section{Thyristor-based converters} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% M2C: complex power analysis (cont.) %% +%% M2C: reactive power diagram %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[c] - \frametitle{M2C: complex power analysis (cont.)} + \frametitle{M2C: reactive power diagram} Rewriting the fundamental apparent power in terms of the active and reactive power yields: \begin{equation} \left(S_1^{(1)}\right)^2 = P_1^2 + \left(Q_1^{(1)}\right)^2 = I_2^2 \hat{u}^2_\mathrm{s0} \quad \Leftrightarrow \left(\frac{Q_1^{(1)}}{I_2 \hat{u}_\mathrm{s0}}\right)^2 + \left(\frac{P_1^2}{I_2 \hat{u}_\mathrm{s0}}\vphantom{\frac{Q_1^{(1)}}{I_2 \hat{u}_\mathrm{s0}}}\right)^2 = 1. @@ -1082,10 +1150,10 @@ \section{Thyristor-based converters} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% M2C: complex power analysis (cont.) %% +%% M2C: reactive power diagram (cont.) %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[c] - \frametitle{M2C: complex power analysis (cont.)} + \frametitle{M2C: reactive power diagram (cont.)} \begin{figure} \begin{tikzpicture} \tikzmath{ @@ -1134,4 +1202,245 @@ \section{Thyristor-based converters} \caption{Fundamental reactive power demand at some constant output current} \label{fig:M2C_reactive_power_demand} \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% M2C: complex power analysis incl. harmonics %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}[c] + \frametitle{M2C: complex power analysis incl. harmonics} + Extending the previous analysis of the complex power fundamental components to the total complex power, one can determine the \hl{total apparent power} as + \begin{equation} + S_1 = I_1 U_1 = \frac{1}{2}\frac{N_2}{N_1} I_2 U_1 = \frac{\pi}{2\sqrt{2}}S_1^{(1)}\approx 1.11 \cdot S_1^{(1)}. + \end{equation} + Interestingly, the apparent power is independent of the firing angle $\alpha$. The \hl{total reactive power} is given by + \begin{equation} + Q_1 = \sqrt{S_1^2 - P_1^2} = S_1^{(1)}\sqrt{\frac{\pi^2}{8} - \cos^2(\alpha)}=\frac{\sqrt{2}}{\pi} \frac{N_2}{N_1} I_2U_1\sqrt{\frac{\pi^2}{8} - \cos^2(\alpha)}. + \end{equation} + Alternatively, one could also determine the harmonic reactive power + \begin{equation} + Q_{1}^{(\mathrm{h})} = \sqrt{\left(S_1\vphantom{S_1^{(1)}}\right)^2 - \left(S_1^{(1)}\right)^2} = S_1^{(1)}\sqrt{\frac{\pi^2-8}{8}}=\frac{\sqrt{2}}{\pi} \frac{N_2}{N_1} I_2U_1\sqrt{\frac{\pi^2-8}{8}}. + \end{equation} + first and then determine the total reactive power as + \begin{equation*} + Q_1 = \sqrt{\left(Q_1^{(1)}\right)^2 + \left(Q_{1}^{(\mathrm{h})}\right)^2 } = S_1^{(1)}\sqrt{\sin^2(\alpha)+\frac{\pi^2-8}{8}}=\frac{\sqrt{2}}{\pi} \frac{N_2}{N_1} I_2U_1\sqrt{\sin^2(\alpha)+\frac{\pi^2-8}{8}}. + \end{equation*} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Higher-pulse number converters %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Higher-pulse number converters} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% M3C converter %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{M3C converter} + The previous diode-based rectifiers with higher-pulse numbers can be directly transferred to their controlled counterparts using thyristors, such as the 3-pulse converter shown in \figref{fig:M3C_topology_filter}. + \begin{figure} + \begin{circuitikz} + \def\vd{1cm} % vertical distance inductors + \def\htraf{0.75cm} % horizontal distance transformer coils + \draw (0,0) to [short, o-] ++(0.5,0) coordinate (L1astart) to [short] ++(0.5,0) to [L] ++(2,0) coordinate (L1aend) + (0,-1*\vd) to [short, o-] ++(1,0) coordinate (L1bstart) to [L] ++(2,0) coordinate (L1bend) + (0,-2*\vd) to [short, o-] ++(1,0) coordinate (L1cstart) to [L] ++(2,0) coordinate (L1cend) -- ++(0,-0.5*\vd) to (\tikztostart -| L1astart) + to [crossing] ++(0, 1*\vd) to [crossing] ++(0, 1*\vd) to [short, -*] (L1astart) + (L1aend) -- ++(0,-0.5*\vd) to (\tikztostart -| L1bstart) to [short, -*] (L1bstart) + (L1bend) -- ++(0,-0.5*\vd) to (\tikztostart -| L1cstart) to [short, -*] (L1cstart); + \draw let \p1=(L1aend) in (\x1 + \htraf, \y1) coordinate (L2astart) to [L, v^<=$u_{1\mathrm{a}}(t)$, voltage = straight] ++(2,0) to [short, i=$i_{1\mathrm{a}}(t)$] ++(0.5,0) coordinate (L2aend); + \draw let \p1=(L1bend) in (\x1 + \htraf, \y1) coordinate (L2bstart) to [L, v^<=$u_{1\mathrm{b}}(t)$, voltage = straight] ++(2,0) to [short, i=$i_{1\mathrm{b}}(t)$] ++(0.5,0) coordinate (L2bend); + \draw let \p1=(L1cend) in (\x1 + \htraf, \y1) coordinate (L2cstart) to [L, v^<=$u_{1\mathrm{c}}(t)$, voltage = straight] ++(2,0) to [short, i=$i_{1\mathrm{c}}(t)$] ++(0.5,0) coordinate (L2cend); + \draw (L2astart) to [short, -*] (L2bstart) to [short, -*] (L2cstart) -- ++(0, -1*\vd) -- ++(5,0) coordinate (Rend); + \draw[double, double distance=3pt, thick] let \p1=(L1aend), \p2=(L2cstart) in (\x1/2+\x2/2, \y1) -- (\x1/2+\x2/2, \y2); + \draw (L2aend) to [thyristor] ++(1.25,0) coordinate (D1end); + \draw (L2bend) to [thyristor] ++(1.25,0) coordinate (D2end); + \draw (L2cend) to [thyristor] ++(1.25,0) coordinate (D3end) to [short, -*] (D2end) to [short, -*] (D1end); + \draw (D1end) to [short] ++(0.5,0) coordinate (u2) to [short, i=$i_2(t)$] ++(0.75,0) to [L, l=$L$] ++(2,0) coordinate (Ctop) to [short, i = $i_\mathrm{R}(t)$] ++(1.5,0) to [R, l=$R$] (Rend -| \tikztostart) to (Rend); + \draw (u2) to [open, v^>=$\hspace{0.5cm}u_2(t)$, voltage = straight] (Rend -| \tikztostart); + \draw (Ctop) to [C, l=$C$, i = $i_\mathrm{C}(t)$, v = $u_\mathrm{C}(t)$, voltage = straight, *-*] (Rend -| Ctop); + \end{circuitikz}% + \caption{M3C topology with an input three-phase transformer, a resistive load and output filter} + \label{fig:M3C_topology_filter} + \end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% M3C converter (cont.) %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{M3C converter (cont.)} + \begin{columns} + \begin{column}{0.5\textwidth} + The M3C converter's firing angle $\alpha$ starts at the crossing of two adjacent input voltages, that is, where the voltage over the next thyristor becomes positive. For CCM and neglecting commutation and other parasitic effects, the M3C's average output voltage is + \begin{equation} + \begin{split} + \overline{u}_2 &= \frac{3}{2\pi} \int_{\frac{1}{6}\pi+\alpha}^{\frac{5}{6}\pi+\alpha} \hat{u}_1 \sin(\omega t) \mathrm{d}\omega t\\ + &= \frac{3}{2\pi} \hat{u}_1 \left[-\cos(\omega t)\right]_{\frac{1}{6}\pi+\alpha}^{\frac{5}{6}\pi+\alpha}=\ldots\\ + &= \frac{3\sqrt{3}}{2\pi}\hat{u}_1\cos(\alpha). + \end{split} + \end{equation} + \end{column} + \begin{column}{0.5\textwidth} + \begin{figure} + \begin{tikzpicture} + \tikzmath{ + real \uavg, \omegat1, \u1, \a; + \u1 = 1; + \a = 0.25*pi; + \uavg = \u1*3*sqrt(3)/(2*pi)*cos(deg(\a)); + } + + \begin{axis}[height=0.5\textheight, width=0.95\textwidth, xmin=0, xmax=2*pi, grid,clip = false, ymin = -1.1, ymax =1.1, xtick = {0, pi/3, pi/6*4, pi, 8/6*pi, 10/6*pi, 2*pi}, xticklabels = {}, ytick = {-1, 0, 1}, yticklabels = {, 0, }, ylabel = {$u(t)$}, yticklabels = { $-\hat{u}_1$, $0$, $\hat{u}_1$}, xticklabels = {0, $\frac{1}{3}\pi$, $\frac{2}{3}\pi$, $\vphantom{\frac{1}{1}}\pi$, $\frac{4}{3}\pi$, $\frac{5}{3}\pi$,$\vphantom{\frac{1}{1}}2\pi$},xlabel = {$\omega t$}] + + \draw[thin, signallavender] (axis cs:pi/6+\a+0.2,\uavg) -- node[signallavender, fill=white, inner sep=2pt, anchor = north, xshift=1mm] {$\overline{u}_2$} (axis cs:pi/6+\a+0.4,\uavg/3); + \draw[thin, signallavender] (axis cs:pi/3,0) -- node[signallavender, fill=white, inner sep=1pt, anchor = north east, xshift=2mm] {$u_2(t)$} (axis cs:pi/5,-0.4); + \addplot[domain=0*pi:2*pi, samples=100, signalblue, thick, name path=A]{sin(deg(x))}; + \addplot[domain=0*pi:2*pi, samples=100, signalgreen, thick, name path=B]{sin(deg(x-pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signalbrown, thick, name path=C]{sin(deg(x+pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signallavender, thick, dashed, name path=avg]{\uavg}; + \addplot[domain=0:2*pi, samples=150, signallavender, thick]{(x < \a + pi/6) * sin(deg(x+pi/3*2)) + (x > \a + pi/6)*(x < \a + pi*5/6)*sin(deg(x)) + (x > \a + pi*5/6)*(x < \a + pi*9/6)*sin(deg(x-pi/3*2)) + (x > \a + pi*9/6)*sin(deg(x+pi/3*2))}; + \draw[->] (axis cs:pi/6,1.2) -- node[above]{$\alpha$} (axis cs:{pi/6+\a},1.2); + \draw[thin, dashed] (axis cs:pi/6,1.5) -- (axis cs:pi/6,0); + \draw[thin, dashed] (axis cs:pi/6+pi/3*2,1.5) -- (axis cs:pi/6+pi/3*2,0); + \draw[thin, dashed] (axis cs:pi/6+pi/3*4,1.5) -- (axis cs:pi/6+pi/3*4,0); + \draw[->] (axis cs:pi/6+2*pi/3,1.2) -- node[above]{$\alpha$} (axis cs:{pi/6+\a+2*pi/3},1.2); + \draw[->] (axis cs:pi/6+4*pi/3,1.2) -- node[above]{$\alpha$} (axis cs:{pi/6+\a+4*pi/3},1.2); + \addplot[shadecolor, opacity=0.3] fill between[of=A and avg, soft clip={domain=pi/6+\a:5*pi/6+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B and avg, soft clip={domain=5*pi/6+\a:3*pi/2+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=C and avg, soft clip={domain=3*pi/2+\a:2*pi+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=C and avg, soft clip={domain=0:pi/6+\a}]; + \end{axis} + \end{tikzpicture} + \caption{Examplary firing angle for the M3C converter} + \label{fig:firing_angle_M3C} + \end{figure} + \end{column} + \end{columns} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% B6C converter %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{B6C converter} + \begin{figure} + \begin{circuitikz} + \def\vd{1.5cm} % vertical distance AC sources + \def\hd{1.5cm} % horizontal distance diode bridge + \def\h1d{6.0cm} % horizontal position first diode string + % Voltage sources and neutral connection + \draw (0,0) to [sinusoidal voltage source, v^<=$u_{1\mathrm{a}}$] ++(1.5, 0) to [L] ++(1.75, 0) to [short, i=$i_{1\mathrm{a}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (A); + \draw (0,-1*\vd) to [sinusoidal voltage source, v^<=$u_{1\mathrm{b}}$] ++(1.5, 0) to [L] ++(1.75, 0) to [short, i=$i_{1\mathrm{b}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (B); + \draw (0,-2*\vd) to [sinusoidal voltage source, v^<=$u_{1\mathrm{c}}$] ++(1.5,0) to [L] ++(1.75, 0) to [short, i=$i_{1\mathrm{c}}(t)$]++(0.75,0) -- ++(0.25,0) coordinate (C); + \draw (0,0) to [short, -*] ++(0,-1.5) to [short] ++(0,-1.5); + + %thyristor bridge + \draw (\h1d,0) to [thyristor, name=D1] ++(0,1.25) coordinate (D1top); + \draw (\h1d,-4.25) coordinate (D2bot) to [thyristor, name=D2] ++(0,1.25) to [short] (\h1d, 0); + \draw (\h1d, 0) to [short, *-] (A); + \draw (\h1d+\hd,0) to [thyristor, name=D3] ++(0,1.25) coordinate (D3top); + \draw (\h1d+\hd,-4.25) coordinate (D4bot) to [thyristor, name=D4] ++(0,1.25) to [short] (\h1d+\hd, 0); + \draw (\h1d+2*\hd,0) to [thyristor, name=D5] ++(0,1.25) coordinate (D5top); + \draw (\h1d+2*\hd,-4.25) coordinate (D6bot) to [thyristor, name=D6] ++(0,1.25) to [short] (\h1d+2*\hd, 0); + \draw (B -| D3) to [crossing, *-, mirror] ++(-2*\hd,0) -- (B); + \draw (C -| D5) to [short, *-] ++(-\hd/2,0) to [crossing, mirror] ++(-\hd,0) to [crossing, mirror] ++(-\hd,0) -- (C); + \draw (D1top) to [short, -*] (D3top) to [short, -*] (D5top) to [short, i=$i_2(t)$] ++(2,0) to [R, l=$R$, name = R] (D6bot -| \tikztostart) to (D6bot); + \draw (D2bot) to [short, -*] (D4bot) to [short, -*] (D6bot); + \draw ($(D5top)!.3!(D5top -| R)$) to [open, v^>=$\hspace{0.5cm}u_2(t)$, voltage = straight] ($(D6bot)!.3!(D6bot -| R)$); + \draw (A) to [open, v^>=$\hspace{0.75cm}u_{1\mathrm{ab}}(t)$, voltage = straight] (B); + \draw (B) to [open, v^>=$\hspace{0.75cm}u_{1\mathrm{bc}}(t)$, voltage = straight] (C); + \draw (-0.5,-2*\vd) to [open, v^>=$u_{1\mathrm{ca}}(t)\hspace{0.75cm}$, voltage = straight] (-0.5,0); + \end{circuitikz} + \caption{B6C topology with line chokes and a resistive load} + \label{fig:B6C_topology} +\end{figure} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Output voltage of a thyristor bridge converter with $p$ pulses %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Output voltage of a thyristor bridge converter with $p$ pulses} + \begin{columns} + \begin{column}{0.5\textwidth} + The average output voltage (under idealized CCM operation) of a thyristor bridge converter with $p$ pulses is given by + \begin{align} + \overline{u}_2 &= \frac{p}{2\pi} \int_{\alpha-\frac{\pi}{p}}^{\alpha+\frac{\pi}{p}}\hat{u} \cos(\omega t) \mathrm{d}\omega t\notag\\ + &=\hat{u}\frac{p}{2\pi}\left[\sin\left(\alpha+\frac{\pi}{p}\right)- \sin\left(\alpha-\frac{\pi}{p}\right)\right]\notag\\ + &= \hat{u}\frac{p}{\pi}\sin\left(\frac{\pi}{p}\right)\cos(\alpha\vphantom{\frac{\pi}{p}}). + \end{align} + Here, the maximum achievable voltage + \begin{equation} + \max_{\alpha} \overline{u}_2 = \hat{u}\frac{p}{\pi}\sin\left(\nicefrac{\pi}{p}\right) + \end{equation} + increases with the number of pulses $p$. + \end{column} + \begin{column}{0.5\textwidth} + \begin{figure} + \begin{tikzpicture} + \tikzmath{ + real \uavg, \omegat1, \u1, \a; + \u1 = 1; + \a = 0.4*pi; + \uavg = \u1*6/pi*sin(deg(pi/6))*cos(deg(\a)); + } + + \begin{axis}[height=0.5\textheight, width=0.95\textwidth, xmin=0, xmax=2*pi, grid,clip = false, ymin = -1.1, ymax =1.1, xtick = {0, pi/3, pi/6*4, pi, 8/6*pi, 10/6*pi, 2*pi}, xticklabels = {}, ytick = {-1, 0, 1}, yticklabels = {, 0, }, ylabel = {$u(t)$}, yticklabels = { $-\hat{u}$, $0$, $\hat{u}$}, xticklabels = {0, $\frac{2\pi}{p}$, $\frac{4\pi}{p}$, $\frac{6\pi}{p}$, $\frac{8\pi}{p}$, $\frac{10\pi}{p}$,$\frac{12\pi}{p}$},xlabel = {$\omega t$}] + + \draw[thin, signallavender] (axis cs:pi/3,\uavg) -- node[signallavender, fill=white, inner sep=2pt, anchor = north] {$\overline{u}_2$} (axis cs:pi/3,-\uavg/6); + \draw[thin, signallavender] (axis cs:pi*0.965,0.66) -- node[signallavender, fill=white, inner sep=1pt, anchor = south west, yshift=2mm] {$u_2(t)$} (axis cs:pi*1.2,1.4); + \addplot[domain=0*pi:2*pi, samples=100, signalblue, thick, name path=A]{cos(deg(x))}; + \addplot[domain=0*pi:2*pi, samples=100, signalgreen, thick, name path=B]{cos(deg(x-pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signalbrown, thick, name path=C]{cos(deg(x+pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signalblue, thick, dashed, name path=D]{-cos(deg(x))}; + \addplot[domain=0*pi:2*pi, samples=100, signalgreen, thick, dashed, name path=E]{-cos(deg(x-pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signalbrown, thick,dashed, name path=F]{-cos(deg(x+pi/3*2))}; + \addplot[domain=0*pi:2*pi, samples=100, signallavender, thick, dashed, name path=avg]{\uavg}; + \addplot[domain=0:2*pi, samples=200, signallavender, thick]{(x < -pi/6+\a) * -cos(deg(x-pi/3*2)) + (x > \a -pi/6)*(x < \a + pi/6)*cos(deg(x)) + (x > \a+ pi/6)*(x < \a + 3*pi/6)*-cos(deg(x+pi/3*2)) + (x > \a + 3*pi/6)*(x < \a + 5*pi/6)*cos(deg(x-pi/3*2))+ (x > \a + 5*pi/6)*(x < \a + 7*pi/6)*-cos(deg(x))+ (x > \a + 7*pi/6)*(x < \a + 9*pi/6)*cos(deg(x+pi/3*2))+ (x > \a + 9*pi/6)*(x < \a + 11*pi/6)*-cos(deg(x-pi/3*2))+ (x > \a + 11*pi/6)*(x < \a + 13*pi/6)*cos(deg(x))}; + \draw[->] (axis cs:pi/6,1.2) -- node[above]{$\alpha$} (axis cs:{pi/6+\a},1.2); + \draw[thin, dashed] (axis cs:pi/6,1.5) -- (axis cs:pi/6,0); + \draw[thin, dashed] (axis cs:pi/6+\a,1.5) -- (axis cs:pi/6+\a,{-cos(deg(pi/6+\a+pi/3*2))}); + \draw[thin, dashed] (axis cs:pi/6*3+\a,1.5) -- (axis cs:pi/6*3+\a,{-cos(deg(pi/6+\a+pi/3*2))}); + \draw[<->] (axis cs:pi/6+\a,1.2) -- node[above]{$\nicefrac{2\pi}{p}$} (axis cs:pi/6*3+\a,1.2); + \addplot[shadecolor, opacity=0.3] fill between[of=E and avg, soft clip={domain=0:-pi/6+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=A and avg, soft clip={domain=-pi/6+\a:pi/6+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=F and avg, soft clip={domain=pi/6+\a:pi/6*3+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=B and avg, soft clip={domain=pi/6*3+\a:pi/6*5+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=D and avg, soft clip={domain=pi/6*5+\a:pi/6*7+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=C and avg, soft clip={domain=pi/6*7+\a:pi/6*9+\a}]; + \addplot[shadecolor, opacity=0.3] fill between[of=E and avg, soft clip={domain=pi/6*9+\a:pi/6*11+\a}]; + \end{axis} + \end{tikzpicture} + \caption{Generalized firing angle representation for a thyristor bridge converter with $p$ pulses and $\hat{u}$ being the line-to-line voltage amplitude} + \label{fig:firing_angle_thyristor_bridge_gen} + \end{figure} + \end{column} + \end{columns} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Section summary %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{Section summary} + This section provided an introduction to thyristor-based converters. The key takeaways are: + \begin{itemize} + \item In contrast to diode-based rectifiers: + \begin{itemize} + \item Can be controlled by varying the firing angle $\alpha$ (within its feasible range). + \item Can transfer power in both directions (rectifier and inverter operation). + \end{itemize} + \item Likewise diode-based rectifiers: + \begin{itemize} + \item Introduce harmonics in the output voltage and input current (i.e., require filters). + \item Typically do not operate at unity power factor (require reactive power). + \item Are line-commutated, as the external grid voltage is required to achieve the commutation. + \end{itemize} + \end{itemize} + Previous analyses based on diodes or thyristor-based converters were dealt with in varying detail level, but as they can be transferred analogously they are not explicitly shown due to time constraints. In addition, there are further interesting thyristor-based applications such as + \begin{itemize} + \item four quadrant thyristor converters (e.g., \href{https://en.wikipedia.org/wiki/Cycloconverter}{cycloconverters}) covering both voltage and current polarities, + \item specialized stacked topologies for \href{https://en.wikipedia.org/wiki/High-voltage_direct_current}{high-voltage DC transmission}. + \end{itemize} \end{frame} \ No newline at end of file diff --git a/lecture/tex/dict.tex b/lecture/tex/dict.tex index b662599..eb74df8 100644 --- a/lecture/tex/dict.tex +++ b/lecture/tex/dict.tex @@ -306,7 +306,15 @@ \section{English-German dictionary} description={Kommutierung} } +\newglossaryentry{choke}{ + name={choke}, + description={Drossel / Spule} +} +\newglossaryentry{line_choke}{ + name={line choke}, + description={Netzdrossel} +} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Build glossary %%