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1364 lines
25 KiB
TeX
1364 lines
25 KiB
TeX
\documentclass[a4paper, 10pt]{article}
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\usepackage[a4paper,left=1cm,right=1cm,top=1.5cm,bottom=1.5cm]{geometry}
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\usepackage{amsmath}
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\usepackage{textgreek}
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\usepackage{textcomp}
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\begin{document}
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\title{Formelsammlung Grundlagen der Elektrotechnik}
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\author{Marc V\"olkers}
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\date{November 2019}
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\maketitle
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\newpage
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\tableofcontents
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\newpage
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\section{EUE04}
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\subsection{Widerstand}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Temperaturabh\"angigkeit Widerstand (0 K = -273,15 \textcelsius )
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&
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{$\!\begin{aligned}
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\alpha &= \text{Temp. Koeffizient} \\
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\vartheta_0 &= \text{Betugstemp.}\\
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\vartheta &= Temperatur\\
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R_0 &= \text{Widerstand bei } \vartheta_0
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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R(\vartheta) &= R_0(1+\alpha\Delta\vartheta)\\
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\Delta\vartheta &= \vartheta - \vartheta_0
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\end{aligned}$}
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\\
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\hline
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Elektrische Verlustleistung
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&
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{$\!\begin{aligned}
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R_{TH} &= \text{thermischer Widerstand} \\
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\vartheta_U &= \text{Umgebungstemp.}\\
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\vartheta &= \text{Oberfl\"achentemp.}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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P_V = \frac{\vartheta - \vartheta_U}{R_{TH}}
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\end{aligned}$}
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\\
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\hline
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Gleichstromwiderstand arbeitspunktabh\"angig
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&
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&
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{$\!\begin{aligned}
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R_{AP} = \frac{U_{AP}}{I_{AP}}
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\end{aligned}$}
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\\
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\hline
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differentieller Widerstand (Wechselgr\"oßenwiderstand)
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&
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&
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{$\!\begin{aligned}
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r_{DAP} = \frac{\Delta U_{AP}}{\Delta I_{AP}}
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\subsection{Kondensator}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Kapazit\"at
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&
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{$\!\begin{aligned}
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\varepsilon &= \text{Permitivit\"at} \\
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A &= \text{Fl\"ache}\\
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d &= \text{Fl\"achenabstand}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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C = \frac{\varepsilon\cdot A}{d}
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\end{aligned}$}
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\\
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\hline
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Kapazit\"at
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&
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{$\!\begin{aligned}
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Q &= \text{Ladung} \\
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U &= \text{Spannung}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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C = \frac{Q}{U}
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\end{aligned}$}
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\\
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\hline
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Verlustfaktor
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&
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{$\!\begin{aligned}
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tan\delta &= \text{Verlustfaktor}\\
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P &= \text{Wirkleistung} \\
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Q &= \text{Blindleistung}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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tan\delta = \frac{P}{Q}
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\end{aligned}$}
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\\
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\hline
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\end{tabular}\\
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\subsection{Spule}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Induktivit\"at
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&
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{$\!\begin{aligned}
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\Psi(I) &= \text{magnetischer Fluss?} \\
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I &= \text{Stomst\"arke}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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L(I) = \frac{\Psi(I)}{I}
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\end{aligned}$}
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\\
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\hline
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Induktivit\"at
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&
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{$\!\begin{aligned}
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\mu_0 &= \text{mag. Feldkonstante} \\
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&= 4\pi*10^{-7} Vs/Am \\
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\mu_R &= \text{Permeabilit\"atszahl} \\
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A &= \text{durchsetzte Fl\"ache}\\
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n &= \text{Anzahl Windungen}\\
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l &= \text{mag. Wegl\"ange}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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L = \mu_0\cdot \mu_R\cdot \frac{A\cdot n^2}{l}
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\end{aligned}$}
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\\
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\hline
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Verlustfaktor
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&
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{$\!\begin{aligned}
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tan\delta &= \text{Verlustfaktor}\\
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G &= \text{Spuleng\"ute} \\
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P &= \text{Wirkleistung}\\
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Q &= \text{Blindleistung}\\
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R &= \text{Widerstand}\\
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\omega &= \text{Winkelgeschwindigkeit}\\
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&= 2*\pi*f\\
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f &= \text{Frequenz}\\
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L &= \text{Induktivit\"at}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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tan\delta = \frac{1}{G} = \frac{P}{Q} = \frac{R}{\omega\cdot L}
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\newpage
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\subsection{Diode}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Strom-Spannungs-Abh\"anigkeit
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&
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{$\!\begin{aligned}
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I_D &= \text{Diostenstrom}\\
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U_D &= \text{Diodenspannung}\\
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I_S &= \text{Sperrstrom}\\
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U_T &= \text{Thermospannung}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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I_D = I_S(e^{\frac{U_D}{U_T}}-1)
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\end{aligned}$}
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\\
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\hline
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Thermospannung
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&
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{$\!\begin{aligned}
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k &= \text{Bolzmannkonstnant}\\
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&= 1,38066*10^{-23} Ws/K\\
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e_0 &= \text{Elementarladung}\\
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&= 1,602189*10^{-19} As\\
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T &= \text{absolute Temperatur in K}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_T = \frac{k * T}{e_0}
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\end{aligned}$}
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\\
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\hline
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Diodenspannung bei linearisiertem Verlauf
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&
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{$\!\begin{aligned}
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U_F &= \text{Flussspannung}\\
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r_D &= \text{Widerstand}\\
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I_D &= \text{Diodenstrom}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_D = U_F+r_DI_D
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\end{aligned}$}
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\\
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\hline
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Verlustleistung
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&
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{$\!\begin{aligned}
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U_F &= \text{Flussspannung}\\
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r_D &= \text{Widerstand}\\
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I_D &= \text{Diodenstrom}\\
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T &= \text{Periodendauer}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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P_V &= \frac{1}{T}\int_{0}^{T}u_D(t)i_D(t)\\
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&= U_F\frac{1}{T}\int_{0}^{T}i_D(t)dt+r_D\frac{1}{T}\int_{0}^{T}i_D^2(t)dt\\
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&= U_F\bar{I}_D+r_DI^2_{DEFF}
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\subsection{Transistor}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Emitterstrom
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&
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{$\!\begin{aligned}
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I_C &= \text{Kollektorstrom}\\
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I_B &= \text{Basisstrom}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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I_E = I_C+I_B
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\end{aligned}$}
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\\
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\hline
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Kollektor-Emitter-Spannung
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&
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{$\!\begin{aligned}
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U_{CB} &= \text{Kollektor-Bsis-Spannung}\\
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U_{BE} &= \text{Basis-Emitter-Spannung}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_{CE} = U_{CB}+U_{BE}
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\end{aligned}$}
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\\
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\hline
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Kollektor Widerstand
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&
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{$\!\begin{aligned}
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U_B &= \text{Betriebsspannung}\\
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U_{CEA} &= \text{Kollektor-Emitter}\\
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&\text{Spannung Arbreitspunkt}\\
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I_{CA} &= \text{Kollektorstrom}\\
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&\text{Arbeitspunkt}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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R_{C} = \frac{U_B-U_{CEA}}{I_{CA}}
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\end{aligned}$}
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\\
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\hline
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4 Quadranten Kennlinienfeld Arberitspunkt in der Mitte
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&
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{$\!\begin{aligned}
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U_{CEA} &= \text{Kollektor-Emitter-}\\&\text{Spannung Arbeitspunkt}\\
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U_B &= \text{Betriebsspannung}\\
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I_{CA} &= \text{Kollektorstrom}\\
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&\text{Arbeitspunkt}\\
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R_C &=Kollektor Widerstand\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_{CEA} &= \frac{U_B}{2}\\
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I_{CA} &= \frac {U_B}{2R_C}
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\end{aligned}$}
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\\
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\hline
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Verstärkung
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&
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{$\!\begin{aligned}
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I_{C} &= \text{Kollektorstrom}\\
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I_B &= \text{Basisstrom}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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B &= \frac{I_C}{I_B}\\
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\end{aligned}$}
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\\
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\hline
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Arbeitspunkteinstellung \"uber Spannungsteiler
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&
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{$\!\begin{aligned}
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I_{Q} &= \text{Querstrom}\\
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I_B &= \text{Basisstrom}\\
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R_{1/2} &= \text{Spannungsteiler}\\ & \text{Widerstände}\\
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U_{BEA} &= \text{Basis-Emitter-}\\ & \text{Spannung Arbeitspunkt}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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I_Q &= 10\cdot I_B\\
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R_1 &= \frac{U_B - U_{BEA}}{I_Q+I_B}\\
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&= \frac{U_B-U_{BEA}}{11\cdot I_B}\\
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R_2 &= \frac{U_{BEA}}{I_Q} = \frac{U_{BEA}}{10\cdot I_B}
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\newpage
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\subsection{Feld Effekt Transistor (FET)}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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Drain-Source Abschn\"urgrenze Strom
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&
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{$\!\begin{aligned}
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I_{DSS} &= \text{Maximaler Drain}\\ & \text{Strom}\\
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U_{DSP} &= \text{Drain Source}\\ & \text{Abschn\"urspannung}\\
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U_{DSS} &= \text{Maximale Drain-Source}\\ & \text{Spannung}
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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I_{DSP} &= I_{DSS} \left( \frac{U_{DSP}}{U_{DSS}} \right) ^ 2
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\end{aligned}$}
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\\
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\hline
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Drain-Source Abschn\"urgrenze Spannung
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&
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{$\!\begin{aligned}
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U_P &= \text{Pinch-off-Spannung}\\
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U_{GS} &= \text{Gate-Source Spannung}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_{DSP} &= U_{GS} - U_P
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\end{aligned}$}
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\\
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\hline
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Abschn\"urgrenze Strom anhand von Pinch-off-Spannung
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&
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{$\!\begin{aligned}
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I_{DSP} &= \text{Drain-Source}\\ & \text{Abschn\"urstrom}\\
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U_P &= \text{Pinch-off-Spannung}\\
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U_{GS} &= \text{Gate-Source Spannung}\\
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I_{DSS} &= \text{Maximaler Drain-Source}\\ & \text{Strom}
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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I_{DSP} &= I_{DSS} \left( \frac{U_{GS}}{U_{p}}-1 \right) ^ 2
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\end{aligned}$}
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\\
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\hline
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Steilheit (A/V)
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&
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{$\!\begin{aligned}
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U_P &= \text{Pinch-off-Spannung}\\
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U_{GS} &= \text{Gate-Source Spannung}\\
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I_{DSS} &= \text{Maximaler Drain-Source}\\ & \text{Strom}
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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S &= \frac{dI_{DSP}}{dU_{GS}} = \frac{2I_{DSS}}{U_P} \left( \frac{U_GS}{U_P}-1 \right)
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\subsection{Operationsverst\"arker}
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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|
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Gegengekoppelt Ausgangsspannung
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&
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{$\!\begin{aligned}
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\nu_D &= \text{Differenzverst\"arkung}\\
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U_D &= \text{Differenzspannung}\\
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U_{P} &= \text{Spannung Eingang}\\
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U_{N} &= \text{Spannung invert. Eingang}\\
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U_E &= \text{Eingangsspannung}
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_A &= \nu_D*U_D =\nu_d(U_P-U_N) \\
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&= \nu_D(U_E-k_RU_A)\\
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&= \frac{\nu_D}{(1+\nu_D\cdot k_R)}\cdot U_E\\
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& \sim \frac{1}{k_r}*U_E
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\end{aligned}$}
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\\
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\hline
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\end{tabular}
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\newpage
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\section{ EUE05}
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\subsection{Grundlagen digitale Schaltungstechnik}
|
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\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
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\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
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\hline
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|
|
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|
Spannungspegel
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&
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{$\!\begin{aligned}
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U_{AHMIN} &=\text{Ausgang} \\ & \text{Spannung High Min}\\
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U_{EHMIN} &= \text{Eingang} \\ & \text{Spannung High Min}\\
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U_{AHMAX} &=\text{Ausgang} \\ & \text{Spannung Low Max}\\
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U_{EHMAX} &= \text{Eingang} \\ & \text{Spannung Low Max}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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U_{AHMIN} &> U_{EHMIN}\\
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U_{ALMAX} &< U_{ElMAX}
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\end{aligned}$}
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\\
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\hline
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|
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St\"orabst\"ande / St\"orschwellen
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&
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{$\!\begin{aligned}
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\Delta U_{LS} &=\text{St\"orabstand Low}\\
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\Delta U_{HS} &=\text{St\"orabstand High}\\
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\text{...} &\text{\ (siehe Spannungspegel)}
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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\Delta U_{LS} &= U_{ELMAX} - U_{ALMAX} > 0\\
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\Delta U_{HS} &= U_{AHMIN} - U_{EHMIN} > 0\\
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\end{aligned}$}
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\\
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\hline
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Eingangslastfaktor (FAN-IN)
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&
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{$\!\begin{aligned}
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I_E &=\text{Eingangsstrom Bauteil}\\
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I_{ES} &=\text{Eingangsstrom} \\ & \text{Standardeingang} \\ & \text{gleiche Schaltkeisfamilie}\\
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\end{aligned}$}
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&
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{$\!\begin{aligned}
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\eta_E &= \frac{I_E}{I_{ES}}
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\end{aligned}$}
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\\
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\hline
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Ausgangslastfaktor (FAN-OUT)
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&
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{$\!\begin{aligned}
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I_A &=\text{Ausgangsstroml}\\
|
|
I_{ES} &=\text{Eingangsstrom} \\ & \text{Standardeingang} \\ & \text{gleiche Schaltkeisfamilie}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\eta_A &= \frac{I_A}{I_{ES}}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Delay Eingangs-/Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
t_{DLH} &=\text{Zeitdifferenz Low High}\\
|
|
t_{DHL} &=\text{Zeitdifferenz High Low}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
t_D = \frac{1}{2}(t_{DLH} + t_{DHL})
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Speed-Power-Produkt
|
|
&
|
|
{$\!\begin{aligned}
|
|
P_{v} &=\text{Mittlere Verlustleistung}\\ & \text{Umschaltvorgang} \\
|
|
t_{DHL} &=\text{Zeitdifferenz High Low}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
W = P_V\cdot t_D
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
|
|
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Analog-Digital- / Digital-Analog Umsetzer}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
|
|
|
|
Eingangsgr\"oße (Meist Spannung)
|
|
&
|
|
{$\!\begin{aligned}
|
|
z &=\text{bin\"ar kodierte Zahl} \\ & \text{am Ausgang}\\
|
|
Q &= \text{Quatisierungseinheit} \\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_E = z\cdot Q
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Maximale Anzahl der Quantisierungsstufen
|
|
&
|
|
{$\!\begin{aligned}
|
|
N &=\text{Bitzahl}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
Z_{MAX} = 2^N
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Tastverh\"altnis DA Z\"ahlverfahren
|
|
&
|
|
{$\!\begin{aligned}
|
|
z &=\text{Eingangszahlenwert}\\
|
|
n &=\text{Aufl\"osung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
TAST = \frac{z}{2^n} = \frac{z}{z_{MAX}+1}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Mittelwert der Impulsfolge DA Z\"ahlverfahren
|
|
&
|
|
{$\!\begin{aligned}
|
|
TAST &=\text{Tastverh\"altnis}\\
|
|
U_{REF} &=\text{Eingangsspannung}\\
|
|
z &=\text{digitaler Zahlenwert}\\
|
|
Q &=\text{Quantisierungseinheit}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\bar{U}_A = TAST\cdot U_{REF} = zQ
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|
|
|
|
\newpage
|
|
\section{LEL01}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
|
|
|
|
Arithmetischer Mittelwert&
|
|
|
|
&
|
|
{$\!\begin{aligned}
|
|
\bar{u} = \frac{1}{T}\cdot \int_{0}^{T}i(t)dt
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Effektivwert
|
|
&
|
|
|
|
&
|
|
{$\!\begin{aligned}
|
|
\\
|
|
U_{eff} = \sqrt{\frac{1}{T}\cdot \int_{0}^{T}u^2(t)dt)}\\
|
|
I_{eff} = \sqrt{\frac{1}{T}\cdot \int_{0}^{T}i^2(t)dt)}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Mischgr\"oße aus Gleich und Wechselanteil
|
|
&
|
|
{$\!\begin{aligned}
|
|
\bar{x} = \text{Gleichgr\oßenanteil}\\
|
|
x_{\sim}(t) = \text{Wechselgr\"oßenanteil}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u(t) = \bar{u} + u_{\sim}(t)\\
|
|
i(t) = \bar{i} + i_{\sim}(t)\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Mischgr\"oße Wechselanteil
|
|
&
|
|
{$\!\begin{aligned}
|
|
x_{\sim}(t) = \text{Wechselgr\"oßenanteil}\\
|
|
\hat{x} = \text{Scheitelwert}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_{\sim}(t) = \sum_{v=1}^{\infty}\hat{u}_v\cdot sin(v\omega t + \varphi_v)\\
|
|
i_{\sim}(t) = \sum_{v=1}^{\infty}\hat{i}_v\cdot sin(v\omega t + \varphi_v)
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gesamteffektivwert Mischgr\"oße
|
|
&
|
|
{$\!\begin{aligned}
|
|
X_{d} &= \text{Gleichgr\"oßen-}\\ & \text{effektivwert}\\
|
|
X_{eff\sim} &= \text{Wechselgr\"oßen-}\\ & \text{effektivwert}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff} &= \sqrt{U_d^2 + U_{eff\sim}^2}\\
|
|
I_{eff} &= \sqrt{I_d^2 + I_{eff\sim}^2}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Welligkeit ( w=0 -\textgreater reine Gleichgr\"oße; w -\textgreater $\infty$ reine Wechselgr\"oße)
|
|
&
|
|
{$\!\begin{aligned}
|
|
X_{eff} &= \text{Gesamteffektivwert}\\
|
|
X_{d} &= \text{Gleichgr\"oßen-}\\ & \text{effektivwert}\\
|
|
X_{eff\sim} &= \text{Wechselgr\"oßen-}\\ & \text{effektivwert}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
w_u &= \frac{U_{eff\sim}}{U_d} = \sqrt{ \left( \frac{U_{eff}}{U_d} \right)^2-1}\\
|
|
w_i &= \frac{I_{eff\sim}}{I_d} = \sqrt{ \left( \frac{I_{eff}}{I_d} \right)^2-1}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Klirrfaktor (Qualit\"at der erzeugten Wechselspannung k=0 -\textgreater rein Sinusf\"ormig)
|
|
&
|
|
{$\!\begin{aligned}
|
|
X_{eff} &= \text{Gesamteffektivwert}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\\
|
|
k_u &= \frac {\sqrt{\sum_{v=2}^{\infty} U_{eff,v}^2}} {U_{eff}}\\
|
|
k_i &= \frac {\sqrt{\sum_{v=2}^{\infty} I_{eff,v}^2}} {I_{eff}}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Berechung einfacher Kurvenverl\"aufe (diskrete Werte)
|
|
&
|
|
{$\!\begin{aligned}
|
|
D &= \frac{t_1}{T} \text{Tastverh\"altnis}\\
|
|
U_D &=\text{diskrete Spannung}
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\\
|
|
\bar{u} = U_d\cdot D\\
|
|
U_{eff} = U_d\cdot \sqrt{D}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Kondensator Strom-Spannungs-Beziehung
|
|
&
|
|
{$\!\begin{aligned}
|
|
i_C &= \text{Kondensatorstorm}\\
|
|
u_C &= \text{Kondensatorspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
i_C &= C\cdot \frac{du_C}{dt}\\
|
|
u_C &= \frac{1}{C}\cdot \int_{0}^{t}i_C(\tau)d\tau+i_C (t=0)
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Spule Strom-Spannungs-Beziehung
|
|
&
|
|
{$\!\begin{aligned}
|
|
i_L &= \text{Spulenstrom}\\
|
|
u_L &= \text{Spulenspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_L &= L\cdot \frac{di_L}{d_t}\\
|
|
i_L &= \frac{1}{L}\cdot \int_{0}^{t}u_L(\tau)d\tau + u_L (t=0)
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Scheinleistung
|
|
&
|
|
{$\!\begin{aligned}
|
|
P &= \text{Wirkleistung}\\
|
|
Q &= \text{Blindleistung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
S &= \sqrt{P^2+Q^2}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Verschiebungsfaktor
|
|
&
|
|
{$\!\begin{aligned}
|
|
P &= \text{Wirkleistung}\\
|
|
S_1 &= \text{Spulenspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\lambda = \frac{P}{S} = \frac{I_1}{I}\cdot cos\varphi_1
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
W\"armestr\"omungsfeld Temperaturdifferenz
|
|
&
|
|
{$\!\begin{aligned}
|
|
R_{th} &= \text{W\"armewiderstand } ( \frac{K}{W} )\\
|
|
P_v &= \text{Verlustleistung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\Delta\vartheta &= R_{th}\cdot P_V\\
|
|
\vartheta_j - \vartheta_a &= (R_{thjc} + R_{thcs} + R_{thsa})\cdot P_V
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
transienter W\"armewiderstand
|
|
&
|
|
{$\!\begin{aligned}
|
|
P\Delta\vartheta(t) &= \text{Temperaturdifferenz}\\
|
|
P_v &= \text{Verlustleistung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
Z_{th} = \frac{\Delta\vartheta(t)}{P_V}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\newpage
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
|
|
|
|
Stromrichter M1U Diodenspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_S &= \text{Eingangsspanmnung}\\
|
|
u_d &= \text{Ausgangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_D = u_S-u_d = u_S - i_d\cdot R\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gleichrichter M1U Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff} &= \text{Effektivwert}\\ & \text{Eingansspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d = \frac{\sqrt{2}U_{eff}}{\pi}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gleichrichter M2U Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff1} &= \text{Effektivwert}\\ & \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d = \frac{2\sqrt{2}U_{eff1}}{\pi}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gleichrichter M3U Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff1} &= \text{Effektivwert}\\ & \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d = \frac{3\sqrt{3}\sqrt{2}U_{eff1}}{2\pi}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gleichrichter B2U Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff1} &= \text{Effektivwert}\\ & \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d = \frac{4\sqrt{2}U_{eff1}}{\pi}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Gleichrichter B6U Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{eff1} &= \text{Effektivwert}\\ & \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d = \frac{3\sqrt{3}\sqrt{2}U_{eff1}}{\pi}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Tiefsetzsteller Einschaltzeit
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{Steuer} &= \text{Steuerspannung}\\
|
|
\hat{U}_{SZ} &= \text{S\"agezahnspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
t_{ein} = \frac{U_{Steuer}}{\hat{U}_{SZ}}\cdot T_s\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Tiefsetzsteller Mittelwert Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{d} &= \text{Eingangsspannung}\\
|
|
t_{ein} &= \text{Einschaltzeit}\\
|
|
T_S &= \text{Schaltperiode}\\
|
|
D &= \text{Tastgrad}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{0} = \frac{t_{ein}}{T_S}\cdot U_d = D\cdot U_d\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
LC Tiefpassfilter Spannungsteiler
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_{0} &= \text{Mittelwert}\\ & \text{Ausgangsspannung}\\
|
|
u_{0F} &= \text{Filtereingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\frac{U_{0}}{u_{0F}(t)} = \frac{1}{1-\omega ^2 LC}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
LC Tiefpassfilter Eckfrequenz
|
|
&
|
|
{$\!\begin{aligned}
|
|
f_E &= \text{Eckfrequenz}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\omega_E = 2\pi\cdot f_E = \frac{1}{\sqrt{LC}}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
LC Tiefpassfilter Oberschwingungen unterdr\"ucken
|
|
&
|
|
{$\!\begin{aligned}
|
|
f_E &= \text{Eckfrequenz}\\
|
|
f_S &= \text{Schaltfrequenz}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
f_E= \frac{1}{2\pi\cdot \sqrt{LC}} \text{ mit } \frac{f_E}{f_S} = 0,01\\
|
|
L = \frac{1}{C(2\pi\cdot 0,01\cdot f_S)^2}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Hochsetzsteller Zusammenhang Eingansspannung und Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_0 &= \text{Ausgangsspannung}\\
|
|
U_d &= \text{Eingangsspannung}\\
|
|
D &= \text{Tastverh\"altnis}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\frac{U_0}{U_d} = \frac{T_S}{T_S-t_{ein}} = \frac{1}{1-D} \\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\section{LEL02}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
|
|
|
|
Wechselrichter mit Halbbr\"ucke Scheitelwert Ausgangswechselspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\hat{U}_{0,1} = \frac{2}{\pi}\cdot U_d \\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Wechselrichter mit Vollbr\"ucke Scheitelwert Ausgangswechselspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\hat{U}_{0,1} = \frac{4}{\pi}\cdot U_d \\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Sinusf\"ormige Pulsweitenmodulation Ausgangsspannung
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
u_{Steuer} &= \text{Steuerspannung}\\
|
|
\hat{U}_\Delta &= \text{Amplitude} \\ & \text{Dreiecksspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_0 = U_d\cdot \frac{u_{Steuer}}{\hat{U}_\Delta}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Sinusf\"ormige Pulsweitenmodulation Aussteuergrad
|
|
&
|
|
{$\!\begin{aligned}
|
|
\hat{U}_{Steuer} &= \text{Amplitude} \\ & \text{Steuerspannung}\\
|
|
\hat{U}_\Delta &= \text{Amplitude} \\ & \text{Dreiecksspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
m_a = \frac{\hat{U}_{Steuer}}{\hat{U}_\Delta}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Sinusf\"ormige Pulsweitenmodulation Verh\"altnis von Schaltfrequenz zu Grundschwingfrequenz (muss gr\"oßer als 10 sein)
|
|
&
|
|
{$\!\begin{aligned}
|
|
f_S &= \text{Schaltfrequenz}\\
|
|
f_1 &= \text{Grundschwingfrequenz}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
m_f = \frac{f_S}{f_1}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Dreiphasiger Wechselrichter Ausgangsspannung erster Schalter geschlossen
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_{a0}(t) = \frac{U_d}{2}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Dreiphasiger Wechselrichter Ausgangsspannung erster unterer Schalter ge\"offnet
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_{a0}(t) = -\frac{U_d}{2}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Dreiphasiger Wechselrichter Ausgangsspannung erster unterer Schalter ge\"offnet
|
|
&
|
|
{$\!\begin{aligned}
|
|
U_d &= \text{Eingangsspannung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_{a0}(t) = -\frac{U_d}{2}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
\end{tabular}
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Frequenzumrichter in der Antriebstechnik}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
Kraft
|
|
&
|
|
{$\!\begin{aligned}
|
|
m &= \text{Masse}\\
|
|
a &= \text{Beschleunigung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
F = m\cdot a = m\cdot \frac{dv}{dt} \Rightarrow \frac{dv}{dt} = \frac{F}{m}\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Geschwindigkeit / Weg
|
|
&
|
|
{$\!\begin{aligned}
|
|
\omega &= \text{Winkelgeschwindigkeit}\\
|
|
r &= \text{Radius}\\
|
|
\varphi &= \text{Winkel}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
v &= \omega\cdot r \Rightarrow dv = r\cdot d\omega \\
|
|
s &= r\cdot \varphi \Rightarrow ds = r\cdot d\varphi\\
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Kraft mit Drehmoment
|
|
&
|
|
{$\!\begin{aligned}
|
|
r &= \text{Radius}\\
|
|
m &= \text{Masse}\\
|
|
v &= \text{Geschwindigkeit}\\
|
|
t &= \text{Zeit}\\
|
|
F &= \text{Kraft}\\
|
|
M &= \text{Drehmoment}\\
|
|
J &= \text{Tr\"agheitsmoment}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
F\cdot r &= M \Rightarrow F = \frac{M}{r} \\
|
|
\frac{dv}{dt} &= \frac{F}{M} = \frac{M}{r\cdot m} \Rightarrow \frac{1}{r}\cdot \frac{M}{r\cdot m} = \frac{M}{r^2+m}\\
|
|
J &= r^2\cdot m
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Bewegungsgleichung f\"ur rotierende K\"orper
|
|
&
|
|
{$\!\begin{aligned}
|
|
\omega &= \text{Winkelgeschwindigkeit}\\
|
|
r &= \text{Radius}\\
|
|
m &= \text{Masse}\\
|
|
t &= \text{Zeit}\\
|
|
M &= \text{Drehmoment}\\
|
|
J &= \text{Tr\"agheitsmoment}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\frac{d\omega}{dt} = \frac{M}{r^2\cdot m} = \frac{M}{J}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Tr\"agheitsmoment eines Volumens
|
|
&
|
|
{$\!\begin{aligned}
|
|
r &= \text{Radius}\\
|
|
m &= \text{Masse}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
J = \int_{V}r^2 dm
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
\end{tabular}
|
|
|
|
\newpage
|
|
\subsection{Erl\"auterung Drehmomentbildung Synchronmaschine}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
Lorenzkraft
|
|
&
|
|
{$\!\begin{aligned}
|
|
B &= \text{Magnetisches Feld}\\
|
|
I &= \text{Stromst\"arke}\\
|
|
l &= \text{Leiterl\"ange Wicklung}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
F_L = B\cdot I \cdot l
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
Elektrisches Drehmoment
|
|
&
|
|
{$\!\begin{aligned}
|
|
B &= \text{Magnetisches Feld}\\
|
|
I &= \text{Stromst\"arke}\\
|
|
l &= \text{Leiterl\"ange Wicklung}\\
|
|
r &= \text{Rotorradius}\\
|
|
i_a &= \text{Wicklungsstrom}\\
|
|
F_L &= \text{Lorenzkraft}
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
M_{el} = F_L\cdot r = B\cdot I \cdot l \cdot r= B\cdot i_a \cdot l \cdot r
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
\end{tabular}
|
|
|
|
\subsection{Asynchronmaschine}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
Statordrehfeld synchrone Drehzahl
|
|
&
|
|
{$\!\begin{aligned}
|
|
f_1 &= \text{Frequenz}\\
|
|
n_p &= \text{Kurzschlussl\"aufer}\\ & \text{Polzahl}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
n_s = \frac{f_1}{n_p}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
\hline
|
|
Schlupf
|
|
&
|
|
{$\!\begin{aligned}
|
|
n_s &= \text{Synchrondrehzahl}\\
|
|
n_p &= \text{Mechanische Drehzahl}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
s = \frac{n_s - n_m}{n_s}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
\end{tabular}
|
|
|
|
\subsection{Schaltnetzteile}
|
|
\begin{tabular}{|p{6cm}|p{5cm}|p{6.5cm}|}
|
|
\textbf{Beschreibung}&\textbf{Variablen}&\textbf{Formel}\\
|
|
\hline
|
|
Transformator Zusammenh\"ange
|
|
&
|
|
{$\!\begin{aligned}
|
|
u_1 &= \text{Eingangsspannnung}\\
|
|
u_2 &= \text{Ausgangsspannung}\\
|
|
N_1 &= \text{Windungsanzahl Eingang}\\
|
|
N_2 &= \text{Windungsanzahl Ausgang}\\
|
|
i_2 &= \text{Stromst\"arke Ausgang}\\
|
|
i_1 &= \text{Stromst\"arke Eingang}\\
|
|
\end{aligned}$}
|
|
&
|
|
{$\!\begin{aligned}
|
|
\frac{u_1}{u_2} = \frac{N_1}{N_2} = \frac{i_2}{i_1}
|
|
\end{aligned}$}
|
|
\\
|
|
\hline
|
|
|
|
|
|
|
|
|
|
\end{tabular}
|
|
|
|
|
|
\end{document} |