ELK2 Formelsammlung V1
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ELK2_FS/formelsammlung.pdf
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@ -16,10 +16,9 @@
|
||||
\documentclass[color,german]{latex4ei/latex4ei_sheet}
|
||||
\usepackage{gensymb}
|
||||
\usepackage{cancel}
|
||||
|
||||
\usepackage{textcomp}
|
||||
|
||||
\usepackage{pbox} % Spaltenpaket
|
||||
|
||||
% set document information
|
||||
\title{Elektronik 2}
|
||||
\author{Mario Fleischmann}
|
||||
@ -58,11 +57,11 @@
|
||||
% ============================================================================================
|
||||
|
||||
% Thema 6: Gegentakt-, Leistungs-Verstärker ==================================================
|
||||
%TODO :\input{./thema6_gegentakt_verstaerker.tex}
|
||||
\input{./thema6_gegentakt_verstaerker.tex}
|
||||
% ============================================================================================
|
||||
|
||||
% Thema 7: Leistungselektronische Schaltungen, Lastschalter ==================================
|
||||
%\input{./thema7_leistungselektronik_schalter.tex}
|
||||
\input{./thema7_leistungselektronik_schalter.tex}
|
||||
% ============================================================================================
|
||||
|
||||
% Thema 8: Leistungselektronische Schaltungen, Schaltverhalten, Brücken und Treiber ==========
|
||||
@ -70,7 +69,7 @@
|
||||
% ============================================================================================
|
||||
|
||||
% Thema 9: Spannungsversorgung, DC-DC Wandler ================================================
|
||||
%\input{./thema9_spannungsversorgung.tex}
|
||||
\input{./thema9_spannungsversorgung.tex}
|
||||
% ============================================================================================
|
||||
|
||||
% DOCUMENT_END ===============================================================================
|
||||
|
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ELK2_FS/img/img_02_15_tp1_nichtinv.png
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ELK2_FS/img/img_02_16_tp1_inv.png
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ELK2_FS/img/img_02_17_hp1_nichtinv.png
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ELK2_FS/img/img_02_18_hp1_inv.png
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ELK2_FS/img/img_02_19_sallenkey_tp2.png
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ELK2_FS/img/img_02_20_sallenkey_hp2.png
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ELK2_FS/img/img_02_21_mfb_tp2.png
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ELK2_FS/img/img_02_22_mfb_hp2.png
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ELK2_FS/img/img_02_23_bandpass.png
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ELK2_FS/img/img_02_24_bandsperre.png
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ELK2_FS/img/img_02_25_bp2_kaskadiert.png
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ELK2_FS/img/img_02_26_bp2_inv.png
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ELK2_FS/img/img_02_27_bp2_mfb_inv.png
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ELK2_FS/img/img_02_28_bs2.png
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ELK2_FS/img/img_02_29_bs2_kerb.png
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ELK2_FS/img/img_03_01_logarithmierer.png
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ELK2_FS/img/img_03_02_exponentierer.png
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ELK2_FS/img/img_03_03_gleichrichter.png
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ELK2_FS/img/img_03_04_schmitt_trigger.png
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ELK2_FS/img/img_03_05_transfer_opv.png
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ELK2_FS/img/img_03_06_transfer_hysterese.png
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ELK2_FS/img/img_04_01_gegentakt_spannungsfolger.png
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ELK2_FS/img/img_04_02_gegentakt_verbessert.png
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ELK2_FS/img/img_07_01_schaltzeiten.png
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ELK2_FS/img/img_07_02_bjt_kennlinie.png
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ELK2_FS/img/img_07_03_bjt_schaltzeiten.png
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ELK2_FS/img/img_07_04_mos_dyn_modell.png
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ELK2_FS/img/img_07_05_gate_ladungsdiagram.png
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ELK2_FS/img/img_07_05_gate_testszenario.png
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ELK2_FS/img/img_07_06_kapazitiv.png
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ELK2_FS/img/img_07_07_kap_U_I.png
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ELK2_FS/img/img_07_08_ind_U_I.png
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ELK2_FS/img/img_07_09_ind_P_E.png
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ELK2_FS/img/img_07_10_snubber.png
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ELK2_FS/img/img_07_11_freilaufdiode.png
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ELK2_FS/img/img_07_12_inverter.png
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ELK2_FS/img/img_07_12_inverter_fazit.png
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ELK2_FS/img/img_07_13_cmos_inverter.png
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ELK2_FS/img/img_07_13_cmos_inverter_fazit.png
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ELK2_FS/img/img_07_14_gegentakt_spgfolger.png
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ELK2_FS/img/img_07_14_gegentakt_spgfolger_fazit.png
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ELK2_FS/img/img_07_15_totem_pole.png
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ELK2_FS/img/img_07_15_totem_pole_fazit.png
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ELK2_FS/img/img_07_16_gate_treiber.png
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ELK2_FS/img/img_07_17_optokoppler.png
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ELK2_FS/img/img_09_01_abwaertswandler.png
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ELK2_FS/img/img_09_02_synchronwandler.png
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@ -144,19 +144,19 @@
|
||||
|
||||
% Gegenkopplung und Mitkopplung
|
||||
% ----------------------------------------------------------------------
|
||||
\subsection{Kompensation der Ausgangs-Offset-Spannung}
|
||||
\subsection{Gegenkopplung und Mitkopplung}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_13_mitkopplung}}
|
||||
\parbox{\textwidth - \imagewidth}{
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
% Rückkopplungsfaktor %
|
||||
\begin{bluebox}
|
||||
\begin{basicbox}
|
||||
$\underline{k}
|
||||
= \frac{-\underline{u}_{id}\vert_{u_1 = 0}}{\underline{u}_2} \newline
|
||||
= \frac{\underline{u}(-)-\underline{u}(+)}{\underline{u}_2}\vert_{u_1 = 0} \newline
|
||||
= \underline{k}^{(-)} - \underline{k}^{(+)}$
|
||||
\end{bluebox}
|
||||
\end{basicbox}
|
||||
|
||||
\begin{emphbox}
|
||||
$\underline{k} = \frac{\underline{Z}_1}{\underline{Z}_1 + \underline{Z}_2} - \frac{\underline{Z}_3}{\underline{Z}_3 + \underline{Z}_4}$
|
||||
$\underline{k} = \frac{\underline{Z}_1}{\underline{Z}_1 + \underline{Z}_2} - \frac{\underline{Z}_3}{\underline{Z}_3 + \underline{Z}_4} > 0!$
|
||||
\end{emphbox}
|
||||
}
|
||||
|
||||
@ -222,7 +222,7 @@
|
||||
\subsubsection{Summierer (Invertierend)}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_07_summierer}}
|
||||
\parbox{\textwidth - \imagewidth}{
|
||||
$\underline{a}_{V,i} = \frac{R_2}{R_1}$ \newline
|
||||
$\underline{a}_{V,i} = -\frac{R_2}{R_1}$ \newline
|
||||
$\underline{z}_{in,i} = R_1$ \newline
|
||||
$\underline{z}_a = (R_2+\frac{R_1}{n})||\frac{\underline{z}_{a,OPV}}{1 + \underline{k} \cdot \underline{V}_{ud}}$
|
||||
}
|
||||
@ -270,10 +270,313 @@
|
||||
|
||||
\end{sectionbox}
|
||||
|
||||
\section{Filter}
|
||||
|
||||
\begin{sectionbox}
|
||||
|
||||
% Filter
|
||||
% Filter Grundlagen
|
||||
% ----------------------------------------------------------------------
|
||||
\subsection{Grundlagen}
|
||||
\subsubsection{TP1 / HP1}
|
||||
% Tiefpass 1. Ordnung %
|
||||
\begin{bluebox}
|
||||
$\underline{H}_{TP1}(\omega) = \frac{K_P}{1 + j\omega \cdot \tau_1}$ \quad\
|
||||
$\underline{H}_{TP1}(f) = \frac{K_P}{1 + j\cdot f / f_C}$ \quad\
|
||||
$f_C = f_{3dB} = \frac{1}{2\pi \cdot \tau_1}$ \newline
|
||||
|
||||
$\underline{H}_{HP1}(\omega) = \frac{K_P \cdot j\omega \cdot \tau_1}{1 + j\omega \cdot \tau_1}$ \quad\
|
||||
$\underline{H}_{HP1}(f) = \frac{K_P \cdot \frac{j\cdot f}{f_C}}{1+ \frac{j\cdot f}{f_C}}$ \quad\
|
||||
$f_C = f_{3dB} = \frac{1}{2\pi \cdot \tau_1}$ \newline
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{TP2 / HP2}
|
||||
% Tiefpass 2. Ordnung %
|
||||
\begin{bluebox}
|
||||
$\underline{H}_{TP2}(\omega) = \frac{K_P}{1 + j\omega \cdot \tau_1 + (j\omega \cdot \tau_2)^2}$ \quad\
|
||||
$\underline{H}_{TP2}(f) = \frac{K_P}{1 + j \cdot \frac{f}{f_C}/Q + (j \cdot \frac{f}{f_C})^2}$ \newline
|
||||
|
||||
$\underline{H}_{HP2}(\omega) = \frac{K_P \cdot (j\omega \cdot \tau_2)^2}{1 + j\omega \cdot \tau_1 + (j\omega \cdot \tau_2)^2}$ \quad\
|
||||
$\underline{H}_{HP2}(f) = \frac{K_P\cdot (\frac{j \cdot f}{f_C})^2}{1 + j \cdot \frac{f}{f_C}/Q + (j \cdot \frac{f}{f_C})^2}$
|
||||
\end{bluebox}
|
||||
|
||||
$f_C = \frac{1}{2\pi \cdot \tau_2}$ \quad \quad \quad\
|
||||
$Q = \frac{\tau_2}{\tau_1}$ % Güte %
|
||||
|
||||
% Grenzfrequenz %
|
||||
\begin{emphbox}
|
||||
\item{$\frac{f_{3dB(TP2)}}{f_C} = \sqrt{\sqrt{1+(\frac{1}{2\cdot Q^2}-1)^2} - (\frac{1}{2\cdot Q^2}-1)}$}
|
||||
\item{$\frac{f_{3dB(HP2)}}{f_C} = 1 / \sqrt{\sqrt{1+(\frac{1}{2\cdot Q^2}-1)^2} - (\frac{1}{2\cdot Q^2}-1)}$}
|
||||
\end{emphbox}
|
||||
Bei $Q = 1/\sqrt{2}$ : $f_C = f_{3dB}$
|
||||
|
||||
Resonanzüberhöhung:
|
||||
\begin{bluebox}
|
||||
\begin{center}
|
||||
\item{$|\underline{H}_{TP2,HP2}(F_C)| = K_P \cdot Q$}
|
||||
\item{$\varphi(\underline{H}_{TP2}(f_C)) = -90\degree$ \quad\
|
||||
$\varphi(\underline{H}_{HP2}(f_C)) = +90\degree$}
|
||||
\end{center}
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{TP - HP Transformation}
|
||||
% TP - HP Transformation %
|
||||
\begin{emphbox}
|
||||
$\underline{H}_{HP}(j\cdot f) = \underline{H}_{TP}(\frac{j\cdot f}{f_C} \to \frac{f_C}{j\cdot f})$
|
||||
\end{emphbox}
|
||||
|
||||
\subsubsection{Dimensionierungshinweise}
|
||||
% Dimensionierungshinweise
|
||||
\begin{bluebox}
|
||||
\item{GBW-Reserve ($V_{ud}$-Abstand):
|
||||
$|\underline{V}_{ud}(f_C)|/|H(f_C)| > 20dB ... \emph{40dB}$}
|
||||
\end{bluebox}
|
||||
für $f_C >> f_1$ : \emph{$|\underline{V}_{ud}(f_C)| = GBW / f_C$}
|
||||
\begin{bluebox}
|
||||
\item{Grenze des Normalbetriebs gemäß Vorgabe; für HP relevant:
|
||||
\quad $f_g \approx GBW \cdot |\underline{k}(f_g)|$}
|
||||
\item{Stabilität im Normalbetrieb: $Q > 0$}
|
||||
\item{Scharfer Übergang: $Q > 0,5 ... < 3$}
|
||||
\end{bluebox}
|
||||
|
||||
\subsection{Prinzip der Bandpass-, Bandsperren-Realisierung}
|
||||
% Bandpass / Bandsperre
|
||||
\parbox{0.5\textwidth}{
|
||||
Bandpass ($f_{3dB,HP} \leq f_{3dB,TP}$):
|
||||
\begin{center}
|
||||
\includegraphics[width = 0.5\columnwidth]{img_02_23_bandpass}
|
||||
\end{center}
|
||||
\begin{emphbox}
|
||||
\item{$\underline{H}_{BP}(f) = \underline{H}_{HP}(f) \cdot \underline{H}_{TP}(f)$}
|
||||
\end{emphbox}
|
||||
Für $f_{3dB,HP} << f_{3dB,TP}$ gilt: \newline $f_{3dB,u} \approx f_{3dB,HP}$ \newline $f_{3dB,o} \approx f_{3dB,TP}$
|
||||
}
|
||||
\parbox{0.5\textwidth}{
|
||||
Bandsperre ($f_{3dB,TP} < f_{3dB,HP}$):
|
||||
\begin{center}
|
||||
\includegraphics[width = 0.5\columnwidth]{img_02_24_bandsperre}
|
||||
\end{center}
|
||||
\begin{emphbox}
|
||||
\item{$\underline{H}_{BS}(f) = \underline{H}_{TP}(f) + \underline{H}_{HP}(f)$}
|
||||
\end{emphbox}
|
||||
Für $f_{3dB,TP} << f_{3dB,HP}$ gilt: \newline $f_{3dB,u} \approx f_{3dB,TP}$ \newline $f_{3dB,o} \approx f_{3dB,HP}$
|
||||
}
|
||||
\begin{emphbox}
|
||||
\item{$f_{3dB,o} = f_m \cdot (\sqrt{1+(\frac{1}{2\cdot Q}^2)}+\frac{1}{2\cdot Q})$}
|
||||
\item{$f_{3dB,u} = f_m \cdot (\sqrt{1+(\frac{1}{2\cdot Q}^2)}-\frac{1}{2\cdot Q})$}
|
||||
|
||||
\end{emphbox}
|
||||
\end{sectionbox}
|
||||
|
||||
\setlength{\imagewidth}{4cm}
|
||||
|
||||
\begin{sectionbox}
|
||||
% Filter Grundschaltungen
|
||||
% ----------------------------------------------------------------------
|
||||
\subsection{Filter-Grundschaltungen mit OPV}
|
||||
%TODO
|
||||
\subsubsection{TP1, nichtinv.}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_15_tp1_nichtinv}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\begin{basicbox}
|
||||
$K_P = \frac{R_2+R_3}{R_2} (=\frac{1}{k})$ \quad \quad\
|
||||
$f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
|
||||
\end{basicbox}
|
||||
$f_g = GBW \cdot k$
|
||||
}
|
||||
\subsubsection{TP1, inv.}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_16_tp1_inv}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\begin{basicbox}
|
||||
$K_P = -\frac{R_2}{R_1}$ \quad \quad\
|
||||
$f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_2 \cdot C_2}$
|
||||
\end{basicbox}
|
||||
für typ. $f_g >> f_C \cdot (R_1 + R_2) / R_1$ gilt: \newline
|
||||
$f_g \approx GBW \cdot k(=1)$
|
||||
}
|
||||
\subsubsection{HP1, nichtinv.}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_17_hp1_nichtinv}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\begin{basicbox}
|
||||
$K_P = \frac{R_2+R_3}{R_2} (=\frac{1}{k})$ \quad \quad\
|
||||
$f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
|
||||
\end{basicbox}
|
||||
$f_g = GBW \cdot k$
|
||||
}
|
||||
\subsubsection{HP1, inv.}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_02_18_hp1_inv}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\begin{basicbox}
|
||||
$K_P = -\frac{R_2}{R_1}$ \quad \quad\
|
||||
$f_C (=f_{3dB}) = \frac{1}{2\pi \cdot R_1 \cdot C_1}$
|
||||
\end{basicbox}
|
||||
Allg. gilt: $\underline{k} = \frac{1+j\cdot f/f_C}{1+\frac{j\cdot f}{f_C}\cdot \frac{R_1+R_2}{R_1}}$ \newline
|
||||
für typ. $f_g >> f_C \to |\underline{k}(f_g)| \approx \frac{R_1}{R_1+R_2}$ \newline
|
||||
$f_g \approx GBW \cdot |\underline{k}(f_g)|$
|
||||
}
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
% Höhere Filter
|
||||
% ----------------------------------------------------------------------
|
||||
\subsection{Sallen-Key (nichtinv.)}
|
||||
\pbox{0.5\textwidth}{TP2:\newline \includegraphics[width = \imagewidth]{img_02_19_sallenkey_tp2}}
|
||||
\pbox{0.5\textwidth}{HP2:\newline \includegraphics[width = \imagewidth]{img_02_20_sallenkey_hp2}}
|
||||
\begin{multicols*}{2}
|
||||
\begin{basicbox}
|
||||
$K_P = 1 + \frac{R_4}{R_3} $ (a)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$f_C = \frac{1}{2\pi \cdot \sqrt{R_1 \cdot C_1 \cdot R_2 \cdot C_2}} $ (b)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$Q = \frac{1/(2\pi \cdot f_C)}{R_1 \cdot (C_2 - (K_P - 1) \cdot C_1) + R_2 \cdot C_2}$ (c)
|
||||
\end{basicbox} \newpage
|
||||
\begin{basicbox}
|
||||
$R_1 = \frac{1}{(2\pi \cdot f_C)^2 \cdot C_1 \cdot R_2 \cdot C_2} $ (d)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$R_2 = \frac{\frac{1}{2Q}\pm \sqrt{\frac{1}{(2Q)^2}+(K_P-1)-\frac{C_2}{C_1}}}{2\pi \cdot f_C \cdot C_2}$ (e)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$\frac{C_2}{C_1} \leq \frac{1}{(2Q)^2} + (K_P -1)$ (f)
|
||||
\end{basicbox}
|
||||
\end{multicols*}
|
||||
\begin{bluebox}
|
||||
\item{1. $K_P$ Vorgabe \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
|
||||
4. $C_1, C_2$ wählen (nF Bereich)}
|
||||
\item{2. $R_3$ wählen, (a) $\to R_4 = R_3 \cdot (K_P - 1)$ \quad \quad\
|
||||
5. (e) $\to R_2$, eingesetzt in (d) $\to R_1$}
|
||||
\item{3. $f_C, Q$ Vorgabe, (f) auswerten}
|
||||
\end{bluebox}
|
||||
Grenze des Normalbetriebs (TP, HP): $f_g \approx GBW \cdot k (= \frac{R_3}{R_3 + R_4})$ \newline
|
||||
$V_{ud}$ - Abstand (typ $>$ (20dB ... 40dB)): $\frac{GBW / f_C}{K_P \cdot Q} > 10 ... 100$?
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
\subsection{Multifeedback (MFB) (inv.)}
|
||||
\pbox{0.5\textwidth}{TP2:\newline \includegraphics[width = \imagewidth]{img_02_21_mfb_tp2}}
|
||||
\pbox{0.5\textwidth}{HP2:\newline \includegraphics[width = \imagewidth]{img_02_22_mfb_hp2}}
|
||||
\begin{multicols*}{2}
|
||||
\begin{basicbox}
|
||||
$K_P = -\frac{R_2}{R_1} $ (a)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$f_C = \frac{1}{2\pi \cdot \sqrt{R_3 \cdot C_1 \cdot R_2 \cdot C_2}} $ (b)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$Q = \frac{1/(2\pi \cdot f_C)}{C_1 \cdot (R_2+R_3+R_2\cdot R_3 / R_1)}$ (c)
|
||||
\end{basicbox} \newpage
|
||||
\begin{basicbox}
|
||||
$R_1 = \frac{1}{(2\pi \cdot f_C)^2 \cdot C_1 \cdot R_2 \cdot C_2} $ (d)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$R_2 = \frac{\frac{1}{2Q}\pm \sqrt{\frac{1}{(2Q)^2}-(1-K_P)\cdot \frac{C_1}{C_2}}}{2\pi \cdot f_C \cdot C_1}$ (e)
|
||||
\end{basicbox}
|
||||
\begin{basicbox}
|
||||
$\frac{C_1}{C_2} \leq \frac{1}{(2Q)^2 \cdot (1-K_P)}$ (f)
|
||||
\end{basicbox}
|
||||
\end{multicols*}
|
||||
\begin{bluebox}
|
||||
\item{1. $f_C, Q, K_P$ Vorgabe \quad \quad \quad \quad \quad\ \
|
||||
4. (e) $\to R_2$, eingesetzt in (d) $\to R_3$}
|
||||
\item{2. (f) auswerten \quad \quad \quad \quad \quad \quad \quad \quad \quad \
|
||||
5. (a) $\to R_1 = R_2/(-K_P)$}
|
||||
\item{3. $C_1, C_2$ wählen (nF Bereich)}
|
||||
\end{bluebox}
|
||||
Grenze des Normalbetriebs (TP): $f_g \approx GBW \cdot k(=1)$ \newline
|
||||
Grenze des Normalbetriebs (HP): $f_g \approx GBW \cdot k (= \frac{C_2}{C_1 + C_2})$ \newline
|
||||
$V_{ud}$ - Abstand (typ $>$ (20dB ... 40dB)): $\frac{GBW / f_C}{K_P \cdot Q} > 10 ... 100$?
|
||||
|
||||
%\end{sectionbox}
|
||||
|
||||
%\begin{sectionbox}
|
||||
|
||||
% Bandpass / Bandsperre
|
||||
% ----------------------------------------------------------------------
|
||||
\subsection{Bandpass 2. Ordnung}
|
||||
Allgemeine Normalform (BP2):
|
||||
\begin{emphbox}
|
||||
\item{$\underline{H}_{BP2}(f) = \frac{H_m \cdot j \cdot \frac{f}{f_m}/Q}{1+j\cdot \frac{f}{f_m}/Q + (j \cdot\frac{f}{f_m})^2}$}
|
||||
\end{emphbox}
|
||||
\subsubsection{HP1, TP1 kaskdiert (nichtinv.)}
|
||||
\pbox{6cm}{\includegraphics[width = 6cm - 1cm]{img_02_25_bp2_kaskadiert}}
|
||||
\parbox{\textwidth - 6cm + 1cm}{
|
||||
$f_m = \sqrt{f_{3dB,u} \cdot f_{3dB,o}}$ \newline
|
||||
$Q = \frac{f_m}{f_{3dB,TP1} - f_{3dB,HP1}} = \frac{f_m}{B}$ \newline
|
||||
$H_m \approx K_{P,HP1} \cdot K_{P,TP1}$
|
||||
}
|
||||
\subsubsection{BP2, Invertierende Standardstruktur}
|
||||
\pbox{4cm}{\includegraphics[width = 4cm - 1cm]{img_02_26_bp2_inv}}
|
||||
\parbox{\textwidth - 4cm + 1cm}{
|
||||
$f_m = \frac{1}{2\pi \cdot \sqrt{(R_1||R_3)\cdot C_1 \cdot R_2 \cdot C_2}}$ \newline
|
||||
Für typ. $C_1 = C_2 = C$ : $Q = f_m \cdot \pi \cdot R_2 \cdot C$ \newline
|
||||
$H_m = -\frac{R_2}{2\cdot R_1}$
|
||||
}
|
||||
|
||||
\subsubsection{BP2, MFB (inv.)}
|
||||
\pbox{4cm}{\includegraphics[width = 4cm - 1cm]{img_02_27_bp2_mfb_inv}}
|
||||
\parbox{\textwidth - 4cm + 1cm}{
|
||||
$f_m = \frac{1}{2\pi \cdot \sqrt{(R_1||R_3)\cdot C_1 \cdot R_2 \cdot C_2}}$ \newline
|
||||
Für typ. $C_1 = C_2 = C$ : $Q = f_m \cdot \pi \cdot R_2 \cdot C$ \newline
|
||||
$H_m = -\frac{R_2}{2\cdot R_1}$
|
||||
}
|
||||
\begin{bluebox}
|
||||
\item{1. $f_m$ Vorgabe \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
|
||||
5. $-H_m (<<GBW \cdot f_m !) < 2 \cdot Q^2$ festlegen!}
|
||||
\item{2. $C_1 = C_2 = C$ wählen (nF-Bereich) \quad \
|
||||
6. $R_1 = R_2/(-2\cdot H_m)$}
|
||||
\item{3. $Q = \frac{f_m}{B}$ festlegen \quad \quad \quad \quad \quad \quad \quad \quad \
|
||||
7. $R_3 = 1 / (2\cdot 2\pi \cdot Q \cdot f_m \cdot C \cdot (1+\frac{H_m}{2\cdot Q^2}))$}
|
||||
\item{4. $R_2 = \frac{Q}{\pi \cdot f_m \cdot C}$}
|
||||
\end{bluebox}
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
|
||||
\subsection{Bandsperre 2. Ordnung}
|
||||
Allgemeine Normalform (BS2):
|
||||
\begin{emphbox}
|
||||
\item{$\underline{H}_{BS2}(f) = \frac{H_0\cdot (1+\frac{H_m}{H_0}\cdot j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2)}{1+j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2}$}
|
||||
\item{$\underline{H}_{BS2(ideal)}(f) = \frac{H_0\cdot (1+(j\cdot \frac{f}{f_m})^2)}{1+j\cdot \frac{f}{f_m}/Q+(j\cdot \frac{f}{f_m})^2} = H_0 \cdot (1-\underline{H}_{BP2,Kp=1}(j\cdot f))$}
|
||||
\end{emphbox}
|
||||
\subsubsection{HP1 + TP1 in Summe \emph{($K_{P,HP} = K_{P,TP}$! (=-1))}}
|
||||
\begin{center} \includegraphics[width = \imagewidth + 1cm]{img_02_28_bs2} \end{center}
|
||||
|
||||
Güte (Polqualität): $Q = \frac{f_m}{B}$ (wie bei BP2) \newline
|
||||
Durchlassverstärkung: $H_0 = K_{P,TP} (K_{P,TP}) \cdot K_{P,Addierer}$ \newline
|
||||
Resonanzverstärkung: $H_m = \frac{H_0 \cdot 2 \cdot f_{3dB,TP1}}{f_{3dB,TP1} + f_{3dB,HP1}}$ \newline
|
||||
Grenze des Normalbetriebs:\newline \newline
|
||||
$\underline{k}_1(f_{g1}) \approx \frac{R_{11}}{R_{11}+R_{12}} \to f_{g1} = GBW1 \cdot \underline{k}_1(f_{g1})$ \newline
|
||||
$\underline{k}_2(f_{g2}) \approx 1 \to f_{g2} = GBW2 \cdot \underline{k}_2(f_{g2})$ \newline
|
||||
$k_3 \approx \frac{R_{31} || R_{32}}{R_{31} || R_{32} + R_{33}} \to f_{g3} = GBW3 \cdot \underline{k}_3$
|
||||
|
||||
\subsubsection{BS2, Kerb- (Notch-) Filter}
|
||||
\pbox{5cm}{\includegraphics[width = 5cm - 1cm]{img_02_29_bs2_kerb}}
|
||||
\parbox{\textwidth - 5cm + 1cm}{
|
||||
$\underline{H}_{BS2(Notch)} = \underline{H}_{BS2(ideal)}$ \newline
|
||||
$Q = \frac{f_m}{B}$ \newline
|
||||
$H_0 = 1 + \frac{R_2}{R_1} \geq 1$ \newline
|
||||
$|\underline{H}(f_m)| = 0$ \newline
|
||||
Stabilitätsbedingungen: $1 \leq H_0 < 2$ ; $2 \geq 1/Q > 0$
|
||||
}
|
||||
\begin{bluebox}
|
||||
\item{1. $f_m$ festlegen \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \
|
||||
4. $Q = f_m / B(3dB)$ festlegen}
|
||||
\item{2. C wählen (nF-Bereich) \quad \quad \quad \quad \quad \quad \quad\
|
||||
5. $H_0 = 2 - \frac{1}{2\cdot Q}$}
|
||||
\item{3. $R = 1/(2\pi \cdot f_m \cdot C)$ \quad \quad \quad \quad \quad \quad \quad \
|
||||
6. $R_2 / R_1 = (H_0 - 1)$}
|
||||
\end{bluebox}
|
||||
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
\subsection{Zur OPV-Auswahl}
|
||||
Kleinsignalmäßig: GBW-Reserve
|
||||
\begin{emphbox}
|
||||
$\frac{GBW/f}{|\underline{H}(f)|} (spez. \frac{GBW/f_C}{|\underline{H}(f_C)|} bzw. \frac{GBW/f_m}{|\underline{H}(f_m)|}) > (typ. 10...100(=40dB)!)$
|
||||
\end{emphbox}
|
||||
Großsignalmäßig: Slew-Rate SR (Def. $\Delta U_{out} / \Delta t$)
|
||||
\begin{emphbox}
|
||||
$SR > \pi \cdot f_{3dB,max} \cdot U_{out,pp} !$
|
||||
\end{emphbox}
|
||||
\end{sectionbox}
|
||||
|
@ -1,7 +1,74 @@
|
||||
\setlength{\imagewidth}{4cm}
|
||||
% ============================================================================================
|
||||
\section{Nichtlineare OPV-Schaltungen}
|
||||
\subsection{test}
|
||||
\subsubsection{test2}
|
||||
|
||||
|
||||
% ============================================================================================
|
||||
\begin{sectionbox}
|
||||
\subsection{Logarithmierer}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_03_01_logarithmierer}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\subsubsection{Mathematisch def. Spannungsbereich ($U_1 \geq 0V$)}
|
||||
Für $U_1 >> IS \cdot R_1$ gilt:
|
||||
\begin{basicbox}
|
||||
$U_2 \approx -N \cdot U_T \cdot ln(\frac{U_1}{IS \cdot R_1})$ (= logarithm.)
|
||||
\end{basicbox}
|
||||
\subsubsection{Mathematisch nicht def. Spannungsbereich ($U_1 < 0V$)}
|
||||
\begin{basicbox}
|
||||
$U_2 = (V+)$
|
||||
\end{basicbox}
|
||||
}
|
||||
|
||||
\subsection{Exponentierer}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_03_02_exponentierer}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\subsubsection{Mathematisch def. Spannungsbereich ($U_1 > 0V$)}
|
||||
Für $U_1 > N \cdot U_T$ gilt:
|
||||
\begin{basicbox}
|
||||
$U_2 = -R_2 \cdot IS \cdot e^{\frac{U_1}{N\cdot U_T}}$ (= exponent.)
|
||||
\end{basicbox}
|
||||
\subsubsection{Mathematisch nicht def. Spannungsbereich ($U_1 \leq 0V$)}
|
||||
\begin{basicbox}
|
||||
$U_2 = 0V$
|
||||
\end{basicbox}
|
||||
}
|
||||
|
||||
\subsection{Aktiver Präszisionsgleichrichter}
|
||||
\pbox{6cm}{\includegraphics[width = 6cm - 1cm]{img_03_03_gleichrichter}}
|
||||
\parbox{\textwidth - 6cm + 1cm}{
|
||||
\underline{Neg. Eingangsspannung} \newline
|
||||
D1 gesperrt, D2 aktiv
|
||||
\begin{basicbox}
|
||||
$A_V = -\frac{R_2}{R_1}$
|
||||
\end{basicbox}
|
||||
\underline{Pos. Eingangsspannung} \newline
|
||||
D1 aktiv, D2 gespert
|
||||
\begin{basicbox}
|
||||
$U_{2,Gleichr} = 0V$
|
||||
\end{basicbox}
|
||||
}
|
||||
|
||||
\subsection{Hysterese}
|
||||
\parbox{0.5 \textwidth}{
|
||||
\underline{Transferkennlinie des einfaches OPV}
|
||||
\begin{center}
|
||||
\includegraphics[height = 3cm]{img_03_05_transfer_opv}
|
||||
\end{center}
|
||||
Störung: $\pm \epsilon \approx \pm U_{ID,lin}/2$\newline(Größenordn. $100\mu V$)
|
||||
}
|
||||
\parbox{0.5 \textwidth}{
|
||||
\underline{Hysterese-Transferkennlinie}
|
||||
\begin{center}
|
||||
\includegraphics[height = 3cm]{img_03_06_transfer_hysterese}
|
||||
\end{center}
|
||||
$\Delta U_{Hysterese} >> \epsilon$ (bspw. $> 2\epsilon$)
|
||||
}
|
||||
\subsubsection{Schmitt-Trigger}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_03_04_schmitt_trigger}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
\begin{emphbox}
|
||||
\item{$U_1^+ = U_1(U_{Ref}, U_{ID}(=0\uparrow), U_2(=V-)) = \frac{(U_{Ref} + 0\uparrow) \cdot (R_1+R_2)}{R_2}-\frac{(V-)\cdot R_1}{R_2}$}
|
||||
\item{$U_1^- = U_1(U_{Ref}, U_{ID}(=0\downarrow), U_2(=V+)) = \frac{(U_{Ref} + 0\downarrow) \cdot (R_1+R_2)}{R_2}-\frac{(V+)\cdot R_1}{R_2}$}
|
||||
\item{$\Delta U_{Hysterese} = U_1^+ - U_1^- = \frac{[(V+)-V(-)]\cdot R_1}{R_2}$}
|
||||
\end{emphbox}
|
||||
Betriebsfrequenzgrenze: $f_g \lessapprox GBW/100$
|
||||
}
|
||||
\end{sectionbox}
|
||||
|
68
ELK2_FS/thema6_gegentakt_verstaerker.tex
Normal file
@ -0,0 +1,68 @@
|
||||
\setlength{\imagewidth}{4cm}
|
||||
% ============================================================================================
|
||||
\section{Leistungsverstärker}
|
||||
% ============================================================================================
|
||||
|
||||
\begin{sectionbox}
|
||||
\subsection{Gegentakt-Spannungsfolger}
|
||||
\begin{center} \includegraphics[width = 0.5\textwidth]{img_04_01_gegentakt_spannungsfolger} \end{center}
|
||||
$U_{in} > 0,7V$ : $U_{out} = U_{in} - 0,7V$ (npn aktiv) \newline
|
||||
$U_{in} < -0,7V$ : $U_{out} = U_{in} + 0,7V$ (pnp aktiv) \newline
|
||||
$-0,7V < U_{in} < 0,7V$ : $U_{out} = 0V$ (npn, pnp sperrt) (Übernahmeverzerrung!) \newline
|
||||
$D = \frac{T_{on}}{T} = 50\%$
|
||||
\subsubsection{Großsignalverstärkung, -Spannungen und -Ströme}
|
||||
$U_{out}(t) = \hat{U}_{out,max} \cdot sin(2\pi \cdot f \cdot t)$ \newline
|
||||
$\hat{U}_{out,max} \approx |UB\pm|$ \newline
|
||||
Großsignal-Spannungsverstärkung: $\frac{U_{out,eff}}{U_{in,eff}} = \frac{\hat{U}_{out}}{\hat{U}_{in}} \approx 1$ \newline
|
||||
Ausgangs- Eingangsstrom ($I_{in} = I_B$) \newline
|
||||
${I_{out}}^+ \approx \frac{{U_{out}}^+}{R_L}$ ; ${I_{in}}^+ (=I_{B,npn})\approx \frac{U_{out}}{R_L \cdot B_{npn}}$ \newline
|
||||
${I_{out}}^- \approx \frac{{U_{out}}^-}{R_L}$ ; ${I_{in}}^- (=I_{B,pnp})\approx \frac{U_{out}}{R_L \cdot B_{pnp}}$ \newline
|
||||
\subsubsection{Großsignal-Ein- und -Ausgangswiderstand}
|
||||
Für den Linearbetrieb gilt: \newline
|
||||
${R_{in}}^+ \approx R_L \cdot B_{npn}$ ; ${R_{in}}^- \approx R_L \cdot B_{pnp}$ \newline
|
||||
${R_{out}}^+ \approx R_G / B_{npn}$ ; ${R_{out}}^- \approx R_G / B_{pnp}$
|
||||
\subsubsection{Leistungsbilanz, Wirkungsgrad}
|
||||
\begin{basicbox}
|
||||
\item{$P_{out} + P_V = P_B + P_{in}$}
|
||||
\item{$\eta = \frac{P_{out}}{P_B + P_{in}}$}
|
||||
\end{basicbox}
|
||||
$P_{out}, P_{RL}$ : Nutzleistung ; $P_V$ : Verlustleistung
|
||||
$P_B$ : Abgegebene Leistung der Betriebsspannungsquellen ; $P_{in}$ : Abgegebene Leistung der Ansteuerquelle (oft vernachlässigbar)
|
||||
\subsubsection{Leistungsberechnungen bei sinusförmiger Ansteuerung}
|
||||
Leistungsabgabe einer einzelnen Betriebsspannungsquelle
|
||||
\begin{basicbox}
|
||||
$P_{UB+} (=P_{UB-}) = \frac{UB \cdot \hat{U}_{out}}{R_L\cdot \pi}$
|
||||
\end{basicbox}
|
||||
Mittlere abgegebene Wirkleistung der Doppelspannungsquelle:
|
||||
\begin{basicbox}
|
||||
\item{$P_B = P_{UB+} + P_{UB-} = 2\cdot \frac{UB \cdot \hat{U}_{out}}{R_L\cdot \pi}$}
|
||||
\item{$P_{B,max}(\hat{U}_{out} = UB) = 2\cdot \frac{UB^2}{R_L\cdot \pi}$}
|
||||
\end{basicbox}
|
||||
Mittlere Nutzleistung am Ausgang:
|
||||
\begin{basicbox}
|
||||
\item{$P_{out} = P_{RL} = \frac{\hat{U}_{out}^2}{2 \cdot R_L}$}
|
||||
\item{$P_{out,max}(\hat{U}_{out} = UB) = \frac{UB^2}{2 \cdot R_L}$}
|
||||
\end{basicbox}
|
||||
Ansteuerleistung des Gegentakt-Emitterfolgers:
|
||||
\begin{basicbox}
|
||||
\item{$P_{in} = {P_{in}}^+ {P_{in}}^-$ ; falls $B_{npn} = B_{pnp} \to P_{in} = \frac{P_{out}}{BF}$}
|
||||
\item{${P_{in}}^+ = \frac{{P_{out}}^+}{B_{npn}} = \frac{P_{out}}{2\cdot B_{npn}}$}
|
||||
\item{${P_{in}}^- = \frac{{P_{out}}^-}{B_{pnp}} = \frac{P_{out}}{2\cdot B_{pnp}}$}
|
||||
\end{basicbox}
|
||||
Transistorverlustleistung:
|
||||
\begin{basicbox}
|
||||
\item{$P_{V,npn} = P_{V,pnp} = 50\% \cdot (P_B - P_{out}) = \frac{\hat{U}_{out}}{R_L} \cdot(\frac{UB}{\pi} - \frac{\hat{U}_{out}}{4})$}
|
||||
\item{$P_{V,npn,max} = P_{V,pnp,max} (\hat{U}_{out} = \frac{2\cdot UB}{\pi}) \approx \frac{UB^2}{R_L\cdot \pi^2}$}
|
||||
\end{basicbox}
|
||||
|
||||
\end{sectionbox}
|
||||
\begin{sectionbox}
|
||||
|
||||
Wirkungsgrad:
|
||||
\begin{basicbox}
|
||||
\begin{center}
|
||||
{$\eta \approx \frac{\pi}{4} \cdot \frac{\hat{U}_{out}}{UB}$} \quad \quad\
|
||||
{$\eta_{max} \approx \frac{\pi}{4} \approx 0,78 (=78\%)$}
|
||||
\end{center}
|
||||
\end{basicbox}
|
||||
\end{sectionbox}
|
352
ELK2_FS/thema7_leistungselektronik_schalter.tex
Normal file
@ -0,0 +1,352 @@
|
||||
\setlength{\imagewidth}{4cm}
|
||||
% ============================================================================================
|
||||
\section{Leistungselektronische Schaltungen, Lastschalter und Treiber}
|
||||
% ============================================================================================
|
||||
\begin{sectionbox}
|
||||
\setlength{\imagewidth}{\textwidth - 4cm}
|
||||
\begin{center}
|
||||
\includegraphics[width = \imagewidth - 0.5cm]{img_07_01_schaltzeiten}
|
||||
\end{center}
|
||||
|
||||
Typisch: $R_{on} << R_L$ (ideal: $R_{on} = 0 \Omega$) \newline
|
||||
Invertierend (bzgl. out-Spannung):
|
||||
\begin{basicbox}
|
||||
{$U_{out}({U_{in}}^- (=0V)) = {U_{out}}^+ (=UB)$} ;
|
||||
{$U_{out}({U_{in}}^+) = {U_{out}}^- (\approx 0V)$}
|
||||
\end{basicbox}
|
||||
Ausgangsspannung:
|
||||
\begin{basicbox}
|
||||
{$U_{out,on} = {U_{out}}^- = \frac{R_{on}}{R_{on} +R_L} \cdot UB \approx \frac{R_{on}}{R_L} \cdot UB$}\newline
|
||||
{$U_{out,off} = {U_{out}}^+ = UB$}
|
||||
\end{basicbox}
|
||||
Laststrom:
|
||||
\begin{basicbox}
|
||||
{$I_{RL,on} = \frac{UB}{R_{on}+R_L} \approx \frac{UB}{R_L}$} ;
|
||||
{$I_{RL,off} = 0A$}
|
||||
\end{basicbox}
|
||||
|
||||
\subsection{BJT-Lastschalter, -Sättigungsschalter}
|
||||
Der BJT-Sättigungsschalter entspricht der Emitterschaltung (open-collector-Treiber) \newline
|
||||
\setlength{\imagewidth}{0.5 \textwidth}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 0.5cm]{img_07_02_bjt_kennlinie}}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 0.5cm]{img_07_03_bjt_schaltzeiten}}
|
||||
\newline Genaue Berechnung des Kollektorstroms: $I_{C,on} = \frac{UB}{R_L + R_{on}}$
|
||||
\subsubsection{Typ. Vorgehen zur Dimensionierung}
|
||||
\begin{bluebox}
|
||||
\item{1. Übersteuerungsfaktor ü festlegen (typ. 2...6)}
|
||||
\item{2. Einschaltstrom: $I_{C,on} = I_{C,\text{ü}} = \frac{UB - U_{CE,on}}{R_L} \approx \frac{UB}{R_L}$}
|
||||
\item{3. Basisstrom (übersteuert): $I_{B,\text{ü}} = \text{ü} \cdot \frac{I_{C,\text{ü}}}{BF}$}
|
||||
\item{4. Basiswiderstand: $R_B = \frac{{U_{in}}^+ - U_{BE}(=_0,7V)}{I_{B,\text{ü}}}$}
|
||||
\item{5. $R_{on} = \frac{U_{CE,on}}{I_{C,on}} = \frac{U_{CE,sat}}{\text{ü} \cdot I_{C,\text{ü}} (=I_{C,normal})}$}
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{Schaltzeiten, Dynamisches Verhalten}
|
||||
Wichtige Beziehungen:
|
||||
Übersteuerungsfaktor ü (= Verhältnis vom tatsächlichen Basisstrom zum Normal-Ansteuerstrom)
|
||||
\begin{basicbox}
|
||||
ü = $\frac{I_{B,\text{ü}}}{I_{C,\text{ü}} / BF} \approx \frac{({U_{in}}^+ - U_{BE}(=0,7V))/R_B}{I_{C,\text{ü}} (=UB/R_L) / BF}$
|
||||
\end{basicbox}
|
||||
Ausräumfaktor a (= Verhältnis vom tatsächlichen Ausräum-Basistrom zum Normal-Ansteuerstrom)
|
||||
\begin{basicbox}
|
||||
$a = \frac{-I_{B,a}}{I_{C,\text{ü}} / BF} \approx -\frac{({U_{in}}^- - U_{BE}(=0,7V))/R_B}{I_{C,\text{ü}} (=UB/R_L) / BF}$
|
||||
\end{basicbox}
|
||||
|
||||
\end{sectionbox}
|
||||
\begin{sectionbox}
|
||||
|
||||
Für die Schaltzeiten gilt dann:
|
||||
\begin{bluebox}
|
||||
\item{$t_{d,on} = \tau_r (=TF \cdot BF) \cdot ln(\frac{\text{ü}}{\text{ü}-0,1})$}
|
||||
\item{$t_r = \tau_r \cdot ln(\frac{\text{ü}-0,1}{\text{ü}-0,9})$}
|
||||
\item{$t_{d,off} = t_s = \tau_s(= (TR + TF(1+\frac{1}{BR}))\cdot BR \approx TR \cdot BR) \cdot ln(\frac{a+\text{ü}}{a+1})$}
|
||||
\item{$t_f = \tau_f (=TF \cdot BF) \cdot ln(\frac{a+0,9}{a})$}
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{Erhöhung des Ausräumfaktors zur Verbesserung der Dynamik}
|
||||
Bedingung für symmetrische Schaltflanken:
|
||||
\begin{emphbox}
|
||||
a $\approx$ ü - 1
|
||||
\end{emphbox}
|
||||
Dann gilt:
|
||||
\begin{emphbox}
|
||||
${U_{in}}^+ + {U_{in}}^- \approx 2 \cdot U_{BE} + \frac{UB}{BF} \cdot \frac{R_B}{R_L}$
|
||||
\end{emphbox}
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
|
||||
\subsection{MOS Last- (Leistungs-)Schalter}
|
||||
\begin{bluebox}
|
||||
\item{A) Vorwärtsbetrieb ($U_{DS} \geq 0V$, Body-Diode gesperrt)}
|
||||
\item{\quad 1. Sperrbetrieb ($U_{GS} < U_{th}$) : NMOS-FET gesperrt}
|
||||
\item{\quad 2. Linear-, Ohm'scher, Trioden-Bereich ($U_{GS} > U_{th}, U_{DS} < U_{DSsat}$)}
|
||||
\item{\quad 3. Sättigungsbereich ($U_{GS} > U_{th}, U_{DS} > U_{DSsat} (=U_{GS}-U_{th})$)}
|
||||
\item{B) Rückwärtsbetrieb ($U_{DS} < 0V$) : Body Diode aktiv}
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{DC-Verhalten}
|
||||
\underline{A.1) Sperrbereich (OFF)} : $I_D = 0$ \newline
|
||||
\underline{A.2) Ohm'scher Bereich (ON)}:
|
||||
\begin{basicbox} $I_D \approx \beta_n \cdot U_{D's} \cdot (U_{GS} - U_{th})$ \end{basicbox}
|
||||
\underline{A.3) Sättigungsbereich (abgeschnürt)}:
|
||||
\begin{basicbox} $I_D = (\beta_n / 2) \cdot (U_{GS} - U_{th})^2$ \end{basicbox}
|
||||
\begin{emphbox} $U_{GS,sat} = \sqrt{2\cdot I_{D,sat} / \beta_n} + U_{th}$ \end{emphbox}
|
||||
\underline{B) Sperrbereich} : $-I_D = +I_{SD} = I_{Diode}$
|
||||
|
||||
\subsubsection{ON-Widerstand $R_{DS,on}(U_{GS})$ im Ohm'schen Bereich}
|
||||
\begin{basicbox}
|
||||
\item{$R_{DS,on} = R_{ch}(U_{GS}) + R_{DS,etc}$}
|
||||
\item{$R_{ch}(U_{GS}) = \frac{1}{\beta_n \cdot (U_{GS}-U_{th})}$}
|
||||
\end{basicbox}
|
||||
Der Nominal-Wert entspricht in guter Näherung nur den parasitären Widerständen:
|
||||
\begin{emphbox}
|
||||
$R_{DS,on(nom)} \approx R_{DS,etc} = konst.$
|
||||
\end{emphbox}
|
||||
|
||||
\subsubsection{Sicherer Arbeitsbereich SOA (Safe Operating Area)}
|
||||
\begin{bluebox}
|
||||
Im statischen Fall gelten folgende Betriebsgrenzen: \newline
|
||||
$I_{Dmax}, P_{Vmax}, U_{DSmax}, T_{Jmax}$ \newline
|
||||
Bei Pulsansteuerung darf der max. Drainstrom-Puls bis $I_{DM,Puls}$ betragen, je nach Pulsbreite.
|
||||
\end{bluebox}
|
||||
|
||||
\subsubsection{Einfach MOS-Lastschalter (Source-Schaltung), statisches Verhalten}
|
||||
\begin{emphbox}
|
||||
\item{$I_{D,on} \approx \frac{UB}{R_L}$}
|
||||
\item{$U_{DS,on} = R_{DS,on} \cdot I_{D,on} \approx R_{DS,on} \cdot \frac{UB}{R_L}$}
|
||||
\end{emphbox}
|
||||
|
||||
\end{sectionbox}
|
||||
\begin{sectionbox}
|
||||
|
||||
\subsubsection{Dynamisches Verhalten}
|
||||
\setlength{\imagewidth}{4cm}
|
||||
\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 1cm]{img_07_04_mos_dyn_modell}}
|
||||
\parbox{\textwidth - \imagewidth + 1cm}{
|
||||
$L_{D,S}$ und $R_{GG'}$ zu vernachlässigen (vereinfachtes Modell) \newline
|
||||
Eingangskapazität:
|
||||
\begin{basicbox} $C_{iss} = C_{GS} + G_{GD}$ \end{basicbox}
|
||||
Rückwirkungs- (Miller-) Kapazität:
|
||||
\begin{basicbox} $C_{rss} = G_{GD}$ \end{basicbox}
|
||||
Ausgangskapazität:
|
||||
\begin{basicbox} $C_{iss} = C_{DS} + G_{GD}$ \end{basicbox}
|
||||
}
|
||||
|
||||
Gate-Ladungs-Diagramm (Für genormtes Test-Szenario): \newline
|
||||
\pbox{7cm}{\includegraphics[width = 6cm]{img_07_05_gate_testszenario}}
|
||||
\parbox{\textwidth - 6cm}{
|
||||
}
|
||||
|
||||
\subsubsection{Berechnung der Schaltzeiten ($\Delta t = \Delta Q_G / I_G$)}
|
||||
\begin{emphbox}
|
||||
$U_{GS,on} = {U_{in}^+}$ \quad \quad
|
||||
$U_{GS,sat} = \sqrt{2\cdot I_{D,on} / \beta_n} + U_{th}$ \quad \quad
|
||||
$U_{GS,Miller} = \frac{U_{GS,sat} + U_{th}}{2}$
|
||||
\end{emphbox}
|
||||
|
||||
Einschaltzeit : $t_{on} = t_{d,on} + t_r$
|
||||
\begin{basicbox}
|
||||
\item{$t_{d,on} = \frac{+\Delta Q_G(t_{d,on})}{+I_G(t_{d,on})}$ ; für ohm'sche Belastung: $+I_G(t_{d,on}) = \frac{{U_{in}}^+ - U_{th}/2}{R_G}$}
|
||||
\item{$t_r = \frac{+\Delta Q_G(t_r)}{+I_G(t_r)}$ ; für ohm'sche Belastung: $+I_G(t_r) = \frac{{U_{in}}^+ - U_{GS,Miller}}{R_G}$}
|
||||
\end{basicbox}
|
||||
|
||||
Ausschaltzeit : $t_{off} = t_{d,off} + t_f$
|
||||
\begin{basicbox}
|
||||
\item{$t_{d,off} = \frac{-\Delta Q_G(t_{d,off})}{-I_G(t_{d,off})}$ ; für ohm'sche Belastung: $-I_G(t_{d,on}) = \frac{\frac{U_{GS,on} + U_{GS,sat}}{2}- {U_{in}}^-}{R_G}$}
|
||||
\item{$t_f = \frac{\Delta Q_G(t_f)}{-I_G(t_f}$ ; für ohm'sche Belastung: $-I_G(t_f) = \frac{U_{GS,Miller}-{U_{in}}^-}{R_G}$}
|
||||
\end{basicbox}
|
||||
|
||||
\subsubsection{Leitsungsbetrachtungen}
|
||||
Nutzleistung:
|
||||
\begin{basicbox} $P_{Nutz}(=P_{RL}) \approx D \cdot \frac{UB^2}{R_L}$ \end{basicbox}
|
||||
|
||||
Ansteuer- (Treiber-) Leistung (Wirkleistung $U_{in}$):
|
||||
\begin{basicbox} $P_{in} = \frac{1}{T} \cdot \int_T U_{in}(t)\cdot I_G(t)\cdot dt = \frac{E(U_{in})}{T}$ \end{basicbox}
|
||||
|
||||
Gesamte Wirkleistung des Schalters:
|
||||
\begin{basicbox}
|
||||
\item{$P_V = P_{V,stat.} + P_{V,dyn.}$}
|
||||
\item{$P_{V,stat.} = \frac{1}{T} \cdot \int_{T_{on}} U_{DS}(t) \cdot I_D(t) \cdot dt = \frac{E_{V,stat.}}{T} \approx D \cdot {I_{D,on}}^2 \cdot R_{on}$}
|
||||
\item{$P_{V,dyn.} = \frac{1}{T} \cdot \int_{t_{on}, t_{off}} U_{DS}(t) \cdot I_D(t) \cdot dt = \frac{E_{on} + E_{off}}{T} = E_{tS} \cdot f$}
|
||||
\end{basicbox}
|
||||
|
||||
Max. Wirkleistung aufg. max. zulässig innerer Temperatur:
|
||||
\begin{basicbox}
|
||||
\item{$P_V = \frac{T_J - T_A}{R_{thJA}}$}
|
||||
\item{$P_{Vmax} = \frac{T_{Jmax} - T_A}{R_{thJA}}$}
|
||||
\end{basicbox}
|
||||
\begin{emphbox}
|
||||
\item{$P_{Vmax} = Min.(P_{Vmax}(T_{Jmax}), P_{Vmax}(SOA))$!}
|
||||
\end{emphbox}
|
||||
|
||||
\end{sectionbox}
|
||||
\begin{sectionbox}
|
||||
|
||||
\subsubsection{Schalten von ohmsch kapazitiven Lasten}
|
||||
\pbox{4cm}{\includegraphics[width = 4cm - 0.5cm]{img_07_06_kapazitiv}}
|
||||
\pbox{6cm}{\includegraphics[width = 6cm - 0.5cm]{img_07_07_kap_U_I}}
|
||||
\begin{basicbox}
|
||||
\item{Einschalten: $I_{D,Peak} \leq \frac{U_{DS,off} (=UB)}{R_{on}}$}
|
||||
\item{Ausschalten: $\tau = R_L \cdot C_L$}
|
||||
\end{basicbox}
|
||||
\begin{emphbox}
|
||||
\item{$E_{tS} \approx E_{on} = E_{CL} = 0,5 \cdot C_L \cdot {U_{CL}}^2$}
|
||||
\item{$P_V = P_{V,dyn.} + P_{V,stat.} \approx \frac{1}{2} \cdot C_L \cdot UB^2 \cdot f + \frac{UB^2}{{R_L}^2} \cdot R_{DS,on} (={I_{D,on}}^2)\cdot \frac{T_{on}}{T}$}
|
||||
\end{emphbox}
|
||||
|
||||
\subsubsection{Schalten von ohmsch induktiven Lasten}
|
||||
\pbox{5cm}{\includegraphics[width = 5cm - 0.5cm]{img_07_08_ind_U_I}}
|
||||
\pbox{5cm}{\includegraphics[width = 5cm - 0.5cm]{img_07_09_ind_P_E}}
|
||||
\begin{basicbox}
|
||||
\item{Einschalten: $\tau = L_L / R_L$}
|
||||
\item{Ausschalten: $U_{DS} \approx -L_L \cdot \frac{dI_D}{dt}$ \newline pos. Begrenzung: Vorwärts- Durchbruchspg.; neg. Begrenzung: Body-Diode}
|
||||
\end{basicbox}
|
||||
\begin{emphbox}
|
||||
\item{$E_{tS} = E_{LL} = 0,5\cdot L_L \cdot {I_{LL}}^2$}
|
||||
\item{$P_V = P_{V,dyn.} + P_{V,stat.} \approx \frac{1}{2} \cdot L_L \cdot \frac{UB^2}{{R_L}^2} \cdot f + \frac{UB^2}{{R_L}^2} \cdot R_{DS,on} \cdot \frac{T_{on}}{T}$}
|
||||
\item{$P_{V,dyn.} > P_{V,stat.}$}
|
||||
\end{emphbox}
|
||||
|
||||
\underline{Kompensationsbeschaltung (Snubber)} \newline
|
||||
\pbox{3cm}{\includegraphics[width = 3cm - 0.5cm]{img_07_10_snubber}}
|
||||
\parbox{\textwidth - 3cm}{
|
||||
Dimensionierung:
|
||||
\begin{basicbox}
|
||||
\item{$C(=C_{sn}) = \frac{L_S \cdot {I_{D,on}}^2}{(U_{C,max - UB})^2}$}
|
||||
\item{$R(=R_{sn}) \approx \frac{T_{on}}{2,2 \cdot C_{sn}}$}
|
||||
\end{basicbox}
|
||||
}
|
||||
|
||||
\underline{Freilauf-Diode} \newline
|
||||
\pbox{2cm}{\includegraphics[width = 2cm - 0.5cm]{img_07_11_freilaufdiode}}
|
||||
\parbox{\textwidth - 2cm}{
|
||||
Zeikonstante: $\tau = \frac{L_L}{R_L}$ \newline
|
||||
Stationärer (eingeschwungener) Zustand: $t > 5 \cdot \tau$ \newline
|
||||
$I_{RL,max} = I_{L,max} = \frac{UB}{R_L}$ \newline
|
||||
$I_{L,mittel} = D \cdot I_{RL,max}$ \newline
|
||||
\begin{basicbox}
|
||||
$\Delta I_{L,pp} = \frac{U_{L,on} \cdot T_{on}}{L_L} = \frac{UB \cdot (1-D) \cdot D \cdot T}{L_L}$
|
||||
\end{basicbox}
|
||||
}
|
||||
\parbox{0.5 \textwidth}{
|
||||
$E_{on}, E_{off}$ bei ohmsch induktiver Last
|
||||
\begin{emphbox}
|
||||
\item{$E_{on} = I_{D,on} \cdot U_{DS,off} \cdot \frac{t_{fall}(U_{DS})}{2}$}
|
||||
\item{$E_{off} = I_{D,off} \cdot U_{DS,off} \cdot \frac{t_{rise}(U_{DS})}{2}$}
|
||||
\end{emphbox}}
|
||||
\parbox{0.5 \textwidth}{
|
||||
$E_{on}, E_{off}$ bei ohmscher Last
|
||||
\begin{emphbox}
|
||||
\item{$E_{on} = I_{D,on} \cdot U_{DS,off} \cdot \frac{t_{r}}{6}$}
|
||||
\item{$E_{off} = I_{D,on} \cdot U_{DS,off} \cdot \frac{t_{f}}{6}$}
|
||||
\end{emphbox}
|
||||
}
|
||||
$\eta = \frac{P_{RL}}{P_{RL}+P_{V,MOS}+P_{V,DF}+P_{in}}$
|
||||
\end{sectionbox}
|
||||
|
||||
\begin{sectionbox}
|
||||
\subsection{Treiber-Grundstrukturen}
|
||||
\setlength{\imagewidth}{\textwidth-3cm}
|
||||
|
||||
\subsubsection{Einfach Inverter, Level-Shifter}
|
||||
|
||||
\pbox{3cm}{\includegraphics[width = 3cm - 0.5cm]{img_07_12_inverter}}
|
||||
\parbox{\textwidth - 3cm}{
|
||||
${R_{out}}^- \approx R_{on,NMOS}$ \newline
|
||||
${R_{out}}^+ = R_D$ (!)
|
||||
\begin{bluebox}
|
||||
\item{Vorteile, typ. Eigenschaften:}
|
||||
\item{• Einfache Schaltungsstruktur}
|
||||
\item{• Typ. nur kleine $U_{in}$-Ansteuerung notwendig ($<< U_{out}$-Aussteuerungsbereich → fungiert auch als Levelshifter)}
|
||||
\item{• Typ. $R_{on,Transistor} << R_D(R_C)$}
|
||||
\end{bluebox}
|
||||
\begin{bluebox}
|
||||
\item{Nachteile:}
|
||||
\item{• „Miller-Effekt“ → hohes $\Delta Q_G$ → ggf. hohe Schaltzeiten (niedrige Dynamik) (BJT: übersteuerter Basisstrom)}
|
||||
\item{• Sehr unsymmetrische Ausgangswiderstände ${R_{out}}^{+(-)}$}
|
||||
\item{• Statische und dyn. Verlustleistung}
|
||||
\end{bluebox}
|
||||
}
|
||||
|
||||
\subsubsection{CMOS Inverter Struktur}
|
||||
\pbox{3cm}{\includegraphics[width = 3cm-0.5cm]{img_07_13_cmos_inverter}}
|
||||
\parbox{\textwidth - 3cm} {
|
||||
${R_{out}}^- \approx R_{Dn}$ \newline
|
||||
${R_{out}}^+ \approx R_{Dp}$
|
||||
\begin{bluebox}
|
||||
\item{Vorteile, typ. Eigenschaften:}
|
||||
\item{• Feste Source-Bezugspotentiale und damit nichtfloatende Steuerspannungen $U_{GS,n} (= U_{in})$, $U_{SG,p} (= UB - U_{in}) \neq Fkt.(U_{out})$}
|
||||
\item{• Definierte Schaltzustände: Low, High , Invertierender Schalter}
|
||||
\item{• ${U_{out}}^{+(-)} = UB$ und ${U_{out |