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Merge branch 'master' of github.com:MaKeAppDev/FS-EM
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hofbi committed Jun 1, 2016
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38 changes: 20 additions & 18 deletions CMakeLists.txt
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@@ -1,32 +1,32 @@
cmake_minimum_required(VERSION 2.8)
cmake_minimum_required(VERSION 2.8)

project(FS NONE)
include(UseLATEX)

add_custom_target(
writegitid ALL
COMMAND ${CMAKE_CURRENT_SOURCE_DIR}/write-gitid.sh
WORKING_DIRECTORY ${CMAKE_CURRENT_BINARY_DIR}
writegitid ALL
COMMAND ${CMAKE_CURRENT_SOURCE_DIR}/write-gitid.sh
WORKING_DIRECTORY ${CMAKE_CURRENT_BINARY_DIR}
)

set(CHAPTERS
chapters/grundlagen.tex
chapters/permanentmagnete.tex
chapters/gleichstrommaschine.tex
chapters/wechselfeld.tex
chapters/synchronmaschine.tex
chapters/asynchronmaschine.tex
chapters/universalmaschine.tex
chapters/grundlagen.tex
chapters/permanentmagnete.tex
chapters/gleichstrommaschine.tex
chapters/wechselfeld.tex
chapters/synchronmaschine.tex
chapters/asynchronmaschine.tex
chapters/universalmaschine.tex
)

set(DRAWINGS
drawings/grundlagen/maschinen-querschnitt.tex
drawings/grundlagen/maschinen-querschnitt2.tex
drawings/permanentmagnete/gsm_querschnitt.tex
drawings/permanentmagnete/gsm_querschnitt.tex
drawings/gleichstrommaschine/flussverlauf_phi.tex
drawings/gleichstrommaschine/gsnsm_esb.tex
drawings/gleichstrommaschine/gsrsm_esb.tex
drawings/gleichstrommaschine/esb_pgsm.tex
drawings/gleichstrommaschine/esb_pgsm.tex
drawings/synchronmaschine/sm_esb.tex
drawings/synchronmaschine/zeiger_syn.tex
drawings/synchronmaschine/zeiger_syn_dq.tex
Expand All @@ -38,17 +38,19 @@ set(DRAWINGS
drawings/asynchronmaschine/zeiger_asyn_stromortskurve_i.tex
drawings/asynchronmaschine/zeiger_asyn_stromortskurve_p.tex
drawings/asynchronmaschine/zeiger_asyn_stromortskurve_s.tex
drawings/universalmaschine/esb_uvm.tex
drawings/universalmaschine/zeiger_uvm_stromortskurve.tex
drawings/universalmaschine/zeiger_uvm.tex
drawings/universalmaschine/esb_uvm.tex
drawings/universalmaschine/zeiger_uvm_stromortskurve.tex
drawings/universalmaschine/zeiger_uvm.tex
)

set(IMAGES
img/Logo.pdf
)

add_latex_document(formelsammlung.tex FORCE_PDF
INPUTS ${CHAPTERS} ${DRAWINGS}
add_latex_document(
formelsammlung.tex
FORCE_PDF
INPUTS ${CHAPTERS} ${DRAWINGS}
IMAGES ${IMAGES}
DEPENDS writegitid
)
25 changes: 20 additions & 5 deletions README.md
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@@ -1,11 +1,13 @@
# Elektrische Kleinmaschinen [![wercker status](https://app.wercker.com/status/d4cb3c1da011db50fd8bfceb7e5932e3/s "wercker status")](https://app.wercker.com/project/bykey/d4cb3c1da011db50fd8bfceb7e5932e3)
## Formelsammlung für Latex4ei
**Required custom Package on GitHub:** [scientific.sty](https://github.com/latex4ei/latex4ei-packages)

# Elektrische Kleinmaschinen
[![wercker status](https://app.wercker.com/status/d4cb3c1da011db50fd8bfceb7e5932e3/s "wercker status")](https://app.wercker.com/project/bykey/d4cb3c1da011db50fd8bfceb7e5932e3)

## Formelsammlung für Latex4ei
**Required custom Package on GitHub:** [LaTeX4Ei](https://github.com/latex4ei/latex4ei-packages)

### Links

Fresh Development Version: [TUM-Projekte](https://makeappdev.github.io/TUM-Projekte/)
Public Version: [latex4ei.de](http://latex4ei.de)
Public Version: [latex4ei.de](http://latex4ei.de)

## Written by
- Hofbauer, Markus
Expand All @@ -22,3 +24,16 @@ add upstream:
pull from upstream:

git pull upstream master

--------------------------------------

## Optional: Build with CMake
Download [UseLaTeX.cmake](https://cmake.org/Wiki/CMakeUserUseLATEX) and move to `/usr/share/cmake-X.X/Modules/.`

##### Steps to build:
```shell
mkdir build
cd build
cmake ..
make
```
156 changes: 90 additions & 66 deletions chapters/asynchronmaschine.tex
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Original file line number Diff line number Diff line change
@@ -1,9 +1,7 @@
\section{Asynchronmaschine}
\sectionbox{
\begin{sectionbox}
\subsection{Größen}
\tablebox{
\begin{tabular*}{\columnwidth}{p{4,5cm}cc}
\ctrule
\begin{tablebox}{p{4,5cm}cc}
Übersetzungsverhältnis & $\ddot{u}$ & $\unitof{\si{1}}$\\
Schlupf & $s$ & $\unitof{\si{1}}$\\
Kippschlupf & $s_K$ & $\unitof{\si{1}}$\\
Expand All @@ -16,11 +14,10 @@ \subsection{Größen}
Rotor-Statorwärmeverluste & $P_\text{Cu}$ & $\unitof{\si{\watt}}$\\
Magnetisierungsstrom & $\underline{I}_{1\mu}$ & $\unitof{\si{\ampere}}$\\
Rotor-Vorwiderstand & $R_{2V}$ & $\unitof{\si{\ohm}}$\\
\cbrule
\end{tabular*}
}
}
\sectionbox{
\end{tablebox}
\end{sectionbox}

\begin{sectionbox}
\subsection{ESB}
\begin{center}
\input{drawings/asynchronmaschine/asm_esb}
Expand All @@ -29,12 +26,12 @@ \subsection{ESB}
\subsubsection{Übersetzungsverhältnis}
Bei Schleifring-ASM gilt: $\quad M_{21} = M_{12} = M$

\symbolbox{
\begin{symbolbox}
\begin{center}
$\ddot{u} = \dfrac{L_{1h}}{M} = \sqrt{\frac{m_1}{m_2}}\cdot\frac{w_1\xi_1}{w_2\xi_2}\cdot\frac{1}{\xi_\text{Schr}} = \sqrt{\frac{m_1}{m_2}}\cdot\frac{w_{1,\text{eff}}}{w_{2,\text{eff}}}\cdot\frac{1}{\xi_\text{Schr}}$
\end{center}
}
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}X>{\centering\arraybackslash}X}
\end{symbolbox}
\begin{tabularx}{\columnwidth}{CC}
$R_{2,\text{ges}}' = \ddot{u}^2\cdot R_{2,\text{ges}}$ & $R_{2,\text{ges}}' = R_2' + R_{2V}'$\\
$\underline U_2 = \frac{1}{\ddot{u}}\cdot\underline U_{1i}$ & $L_{2\sigma}' = \ddot{u}^2\cdot (L_{2\sigma} + L_{2\text{Schr}})$\\
$\underline{I}_2' = \frac{1}{\ddot{u}}\cdot\underline{I}_2$ &
Expand All @@ -46,18 +43,21 @@ \subsection{Systemgleichungen}
0 &= R_{2,\text{ges}}\cdot\vec{i}_2 + \frac{\partial\vec\Psi_2}{\partial t}, & \vec\Psi_2 &= L_2\cdot\vec i_2 + M\cdot\vec i_1\cdot e^{-jp\vartheta_m}\\
J\frac{\diff\omega}{\diff t} &= M_i - M_R - M_L
\end{align*}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Wichtige Größen}
\subsubsection{Schlupf}
\emphbox{$s = \frac{n_\text{syn} - n}{n_\text{syn}} = \frac{\omega_\text{syn} - \omega_m}{\omega_\text{syn}} = \frac{\omega_1 - p\cdot\omega_m}{\omega_1} = \frac{\omega_2}{\omega_1}$}
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}X>{\centering\arraybackslash}X>{\centering\arraybackslash}X}
\begin{emphbox}
$s = \frac{n_\text{syn} - n}{n_\text{syn}} = \frac{\omega_\text{syn} - \omega_m}{\omega_\text{syn}} = \frac{\omega_1 - p\cdot\omega_m}{\omega_1} = \frac{\omega_2}{\omega_1}$
\end{emphbox}
\begin{tabularx}{\columnwidth}{CCC}
Gegenstrombremse & Motor & Generator\\
$s > 1$ & $1 > s > 0$ & $s < 0$
\end{tabularx}

\subsubsection{Drehzahl}
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}X>{\centering\arraybackslash}X}
\begin{tabularx}{\columnwidth}{CC}
synchrone Drehzahl & Nenndrehzahl\\
$n_\text{syn} = \frac{f}{p}$ & $n_N = n_s (1-s_N)$
\end{tabularx}
Expand All @@ -74,7 +74,9 @@ \subsubsection{Leistung}
\end{align*}

\subsubsection{Phase}
\emphbox{ASM immer induktiv $\Rightarrow \varphi > 0$}
\begin{emphbox}
ASM immer induktiv $\Rightarrow \varphi > 0$
\end{emphbox}
\begin{align*}
\varphi &= \varphi_{1Z}-\varphi_{1N}\\
\varphi &= \begin{cases}
Expand All @@ -83,22 +85,25 @@ \subsubsection{Phase}
\arctan(\frac{b}{a})-\pi & \text{für } a < 0, b<0
\end{cases}
\end{align*}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsubsection{Weitere Parameter}
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}X>{\centering\arraybackslash}X}
\begin{tabularx}{\columnwidth}{CC}
$L_{1\sigma} = \sigma_1\cdot L_{1h}$ & $L_1 = L_{1h} + L_{1\sigma}$\\
$L_{2\sigma}' = \sigma_2\cdot L_{1h}$ & $L_2' = L_{1h}\cdot (1+\sigma_2)$\\
\multicolumn{2}{c}{$L_\sigma = \sigma\cdot L_1 = L_{1\sigma} + \frac{\xi_\text{Schr}}{1 + \sigma_2}L_{2\sigma}'$}\\
$\rho_1 = \frac{R_1}{\omega_1 L_1}$ & $\rho_2 = \frac{R_{2,\text{ges}}}{\omega_1 L_2} = \frac{R_{2,\text{ges}}'}{\omega_1 L_2'}$\\
\multicolumn{2}{c}{$\Delta \rho_1 = \sqrt{1+\left( \frac{\rho_1}{\sigma}\right)^2}\cdot\sqrt{1+{\rho_1}^2}$}\\
\multicolumn{2}{c}{$\sigma = 1 - \frac{1}{(1 + \sigma_1)\cdot(1 + \sigma_2)} = 1 - \frac{M^2}{L_1 L_2}$}
\end{tabularx}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Statorstrom}
\emphbox{\[\underline{I}_1 = \frac{\underline{U}_1}{\omega_1 L_1} \cdot \frac{\rho_2 +js}{\rho_1\cdot\rho_2 -\sigma\cdot s+j(\rho_2 +s\cdot \rho_1)}\]
}
\begin{emphbox}
\[\underline{I}_1 = \frac{\underline{U}_1}{\omega_1 L_1} \cdot \frac{\rho_2 +js}{\rho_1\cdot\rho_2 -\sigma\cdot s+j(\rho_2 +s\cdot \rho_1)}\]
\end{emphbox}
Anlaufstrom:\\
$I_{1A} = |\underline{I}_1|(s=1) = \frac{U_1}{\omega_1 L_\sigma}\sqrt{\frac{1+{\rho_2}^2}{\left(1-\frac{\rho_1\cdot\rho_2}{\sigma}\right)^2 +\left(\frac{\rho_1+\rho_2}{\sigma}\right)^2}}$\\
Ideeller Kurzschlussstrom:\\
Expand All @@ -107,10 +112,11 @@ \subsection{Statorstrom}

\subsubsection{Magnetisierungsstrom}
\[\underline I_\mu = \frac{\rho_2 + j\cdot s\cdot(\sigma - \sigma_1\cdot(1-\sigma))}{\rho_1\cdot\rho_2 - \sigma\cdot s + j\cdot(\rho_2 + s\cdot\rho_1)}\cdot\frac{\underline U_1}{\omega_1 L_1}\]
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Zeigerdiagramm}
\cookbox{Zeigerdiagramm}{
\begin{cookbox}{Zeigerdiagramm}
\item $\underline{U}_1$ auf reelle Achse legen und $\underline{I}_1$ einzeichnen
\item $R_1\underline{I}_1$ (gleiche Phasenlage wie $\underline{I}_1$)\\
$j\omega_1 L_{1\sigma}\underline{I}_1$ (eilt $\underline{I}_1$ um $\ang{90}$ voraus)
Expand All @@ -120,43 +126,50 @@ \subsection{Zeigerdiagramm}
\item $R_{2,\text{ges}}'\underline{I}_2'$ (parallel zu $\underline{I}_2'$)
\item $j\omega_1 L_{2\sigma}'\underline{I}_2'$ (eilt $\underline{I}_2'$ um $\ang{90}$ voraus)
\item $R_{2,\text{ges}}'\cdot\frac{1-s}{s}\cdot\underline{I}_2' = -\underline{U}_{1i} - R_{2,\text{ges}}'\,\underline{I}_2' - j\omega_1 L_{2\sigma}'\underline{I}_2'$
}
\end{cookbox}
\begin{center}
\input{drawings/asynchronmaschine/zeiger_asyn}
\end{center}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Stromortskurve}
\symbolbox{bei $R_1 = 0\qquad\qquad\qquad\tan(\mu) = s_K$}
\cookbox{Stromortskurve $R_1 = 0\wedge R_\text{Fe} = 0$}{
\begin{symbolbox}
bei $R_1 = 0\qquad\qquad\qquad\tan(\mu) = s_K$
\end{symbolbox}
\begin{cookbox}{Stromortskurve $R_1 = 0\wedge R_\text{Fe} = 0$}
\item $\underline{U}_1$ auf reelle Achse legen $\Rightarrow\varphi_{1U} = 0$
\item $R_1 = 0\Rightarrow \underline I_{10}$ und $\underline I_{1Ki}$ haben keinen Realteil
\item Kreismittelpunkt auf Im-Achse zwischen $\underline{I}_{1Ki}$ und $\underline{I}_{10}$
\item $\mu$ zwischen $P_0$ und $P_A$
}
\end{cookbox}
\input{drawings/asynchronmaschine/zeiger_asyn_stromortskurve_i}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsubsection{Schlupfgerade}
\cookbox{Schlupfgerade $R_1 = 0\wedge R_\text{Fe}\neq 0$}{
\begin{cookbox}{Schlupfgerade $R_1 = 0\wedge R_\text{Fe}\neq 0$}
\item (Bei $R_\text{Fe} = 0$) Mittelpunkt $M$ auf -Im Achse
\item Schlupfgerade an beliebiger Stelle einzeichnen
\item gesuchtes $s$ aus Längenverhältnis zu bekanntem Schlupf bestimmen
}
\end{cookbox}
\input{drawings/asynchronmaschine/zeiger_asyn_stromortskurve_s}
}
\sectionbox{

\subsubsection{Maßstab}
\symbolbox{
\begin{symbolbox}
\begin{tabular}{p{2.8cm}cc}
Strommaßstab & $m_I$ & $\unitof{\si{\ampere\per\centi\meter}}$\\
Leistungsmaßstab & $m_P = m_1\cdot U_1\cdot m_I$ & $\unitof{\si{\watt\per\centi\meter}}$\\
Drehmomentmaßstab & $m_M = \frac{m_P}{2\pi\cdot n_\text{syn}}$ & $\unitof{\si{\newtonmeter\per\centi\meter}}$
\end{tabular}
}
\end{symbolbox}
\end{sectionbox}

\begin{sectionbox}
\subsubsection{Ablesbare Werte}
\emphbox{$R_1\neq 0\wedge R_\text{Fe}\neq 0$}
\begin{emphbox}
$R_1\neq 0\wedge R_\text{Fe}\neq 0$
\end{emphbox}
\begin{tabular}{p{4cm}l}
Aufgenommene elektrische Leistung & $P_1 = \overline{PD}\cdot m_P$\\
Eisenverluste Stator & $P_\text{Fe} = \overline{CD}\cdot m_P$\\
Expand All @@ -165,47 +178,58 @@ \subsubsection{Ablesbare Werte}
Abgegebene mechanische Leistung & $P_m = \overline{PA}\cdot m_P$\\
Inneres Drehmoment & $M_i = \overline{PB}\cdot m_M$
\end{tabular}
\symbolbox{Definition Punkt D: Orthogonale Projektion von $P$ auf Im-Achse\\
\begin{tabularx}{\columnwidth}{lX}
$R_1 = 0$ & $B = C$ und $M$ auf Höhe von $P_0$\\
$R_\text{Fe} = 0$ & $C = D$ und $P_0$ auf -Im Achse
\end{tabularx}}
\begin{symbolbox}
Definition Punkt D: Orthogonale Projektion von $P$ auf Im-Achse\\
\begin{tabularx}{\columnwidth}{lX}
$R_1 = 0$ & $B = C$ und $M$ auf Höhe von $P_0$\\
$R_\text{Fe} = 0$ & $C = D$ und $P_0$ auf -Im Achse
\end{tabularx}
\end{symbolbox}

\input{drawings/asynchronmaschine/zeiger_asyn_stromortskurve_p}
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Drehmoment}
\emphbox{$M_K\sim \left(\frac{U_1}{f_1}\right)^2\qquad M_N\sim\Phi_\delta\frac{U_1}{f_1}$}
\begin{emphbox}
$M_K\sim \left(\frac{U_1}{f_1}\right)^2\qquad M_N\sim\Phi_\delta\frac{U_1}{f_1}$
\end{emphbox}
\begin{align*}
M_i = M_R + M_L + J\frac{\partial\omega}{\partial t}
\end{align*}
\subsubsection{Drehmomentgleichung}
\emphbox{\[M_i = 3p(1-\sigma)\frac{{U_1}^2}{{\omega_1}^2 L_\sigma}\frac{s \cdot s_K}{\Delta\rho_1{s_K}^2 + 2\frac{\rho_1}{\sigma}(1-\sigma)s_K s+\Delta\rho_1 s^2}\]
}
\begin{emphbox}
\[M_i = 3p(1-\sigma)\frac{{U_1}^2}{{\omega_1}^2 L_\sigma}\frac{s \cdot s_K}{\Delta\rho_1{s_K}^2 + 2\frac{\rho_1}{\sigma}(1-\sigma)s_K s+\Delta\rho_1 s^2}\]
\end{emphbox}
Kippmoment:\\
$M_K = M_i(s_K) = \frac{3}{2} p\cdot (1-\sigma)\frac{{U_1}^2}{{\omega_1}^2 L_\sigma}\left( \frac{1}{\Delta\rho_1+\frac{\rho_1}{\sigma}(1-\sigma)}\right)$\\
$(R_1 = 0): M_K = \frac{m_1U_1\frac{I_{1Ki} - I_{10}}{2}}{2\pi\cdot n_s}$\\
Kippschlupf: $s_K = \frac{\rho_2}{\sigma}\sqrt{\frac{1+{\rho_1}^2}{1+\left(\frac{\rho_1}{\sigma}\right)^2}}$

\symbolbox{
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}X>{\centering\arraybackslash}X}
\begin{symbolbox}
\begin{tabularx}{\columnwidth}{CC}
$s_K > 0\quad$ Motor & $s_K < 0\quad$ Generator
\end{tabularx}
}
}
\sectionbox{
\end{symbolbox}
\end{sectionbox}

\begin{sectionbox}
\subsubsection{Klossche Gleichung (Annahme $R_1 = 0$)}
\emphbox{\[\frac{M_i}{M_K} = \frac{2\cdot s_K\cdot s}{{s_K}^2 + s^2}\]}
\begin{emphbox}
\[\frac{M_i}{M_K} = \frac{2\cdot s_K\cdot s}{{s_K}^2 + s^2}\]
\end{emphbox}
\[s_{1,2} = s_K \frac{M_K}{M_i} \pm \sqrt{\left(s_K \frac{M_K}{M_i}\right)^2 - {s_K}^2}\]
Nur echte Lösung wenn gilt: \quad $s < s_K$
}
\sectionbox{
\end{sectionbox}

\begin{sectionbox}
\subsection{Symmetrische Komponenten}
\symbolbox{
\begin{symbolbox}
\begin{tabularx}{\columnwidth}{>{\centering\arraybackslash}Xc>{\centering\arraybackslash}X}
$s_m + s_g = 2$ & $s_m = s = \frac{n_s - n}{n_s}$ & $s_g = \frac{n_s + n}{n_s}$
\end{tabularx}
}
\end{symbolbox}

\subsubsection{Spannungen Mit- und Gegensystem}
\begin{tabular}{lc}
Mitsystem & $\underline{U}_m = \frac{1}{3}\cdot (\underline{U}_u + \underline{a}\cdot\underline{U}_v + \underline{a}^2 \cdot \underline{U}_w)$\\
Expand All @@ -216,7 +240,7 @@ \subsubsection{Spannungen Mit- und Gegensystem}

\subsubsection{Drehmoment mit Kompensation (Kippschlupf ändert sich)}
\[M_\text{ges} = M_m - M_g\]
\emphbox{
\begin{emphbox}
\[M = 3p\cdot (1-\sigma)\cdot\frac{{U_1}^2}{{\omega}^2 L_1} \cdot\frac{\rho_2 \cdot s}{(\rho_1\cdot\rho_2 -\sigma\cdot s)^2 +(\rho_2 +s\cdot\rho_1)^2}\]
}
}
\end{emphbox}
\end{sectionbox}
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