From 2a853440db69195e0d5677d02c7c3289ffd66f2a Mon Sep 17 00:00:00 2001 From: fleischmannma75068 Date: Thu, 16 Jul 2020 17:58:53 +0200 Subject: [PATCH] ELK2 Formelsammlung V1 --- ELK2_FS/formelsammlung.pdf | Bin 0 -> 6836459 bytes ELK2_FS/formelsammlung.tex | 9 +- ELK2_FS/img/img_02_15_tp1_nichtinv.png | Bin 0 -> 146518 bytes ELK2_FS/img/img_02_16_tp1_inv.png | Bin 0 -> 243818 bytes ELK2_FS/img/img_02_17_hp1_nichtinv.png | Bin 0 -> 268175 bytes ELK2_FS/img/img_02_18_hp1_inv.png | Bin 0 -> 168581 bytes ELK2_FS/img/img_02_19_sallenkey_tp2.png | Bin 0 -> 224632 bytes ELK2_FS/img/img_02_20_sallenkey_hp2.png | Bin 0 -> 246996 bytes ELK2_FS/img/img_02_21_mfb_tp2.png | Bin 0 -> 376315 bytes ELK2_FS/img/img_02_22_mfb_hp2.png | Bin 0 -> 306336 bytes ELK2_FS/img/img_02_23_bandpass.png | Bin 0 -> 199289 bytes ELK2_FS/img/img_02_24_bandsperre.png | Bin 0 -> 167475 bytes ELK2_FS/img/img_02_25_bp2_kaskadiert.png | Bin 0 -> 347825 bytes ELK2_FS/img/img_02_26_bp2_inv.png | Bin 0 -> 227919 bytes 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a/ELK2_FS/formelsammlung.tex b/ELK2_FS/formelsammlung.tex index 38b414e..67337e1 100644 --- a/ELK2_FS/formelsammlung.tex +++ b/ELK2_FS/formelsammlung.tex @@ -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} 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\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. $-H_m (< 0$ + } + \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} diff --git a/ELK2_FS/thema5_nichtlinearer_OPV.tex b/ELK2_FS/thema5_nichtlinearer_OPV.tex index 69565c2..4a80f82 100644 --- a/ELK2_FS/thema5_nichtlinearer_OPV.tex +++ b/ELK2_FS/thema5_nichtlinearer_OPV.tex @@ -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} diff --git a/ELK2_FS/thema6_gegentakt_verstaerker.tex b/ELK2_FS/thema6_gegentakt_verstaerker.tex new file mode 100644 index 0000000..06b979a --- /dev/null +++ b/ELK2_FS/thema6_gegentakt_verstaerker.tex @@ -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} diff --git a/ELK2_FS/thema7_leistungselektronik_schalter.tex b/ELK2_FS/thema7_leistungselektronik_schalter.tex new file mode 100644 index 0000000..233c6cb --- /dev/null +++ b/ELK2_FS/thema7_leistungselektronik_schalter.tex @@ -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 ${U_{out}}^{(-)} = 0V$ \newline + ${I_{out}}^+ = UB/ R_{Dp}$ und ${I_{out}}^{(-)} = UB/ R_{Dn}$ (passiv Gate entladend) \newline + ${R_{out}}^+ = R_{Dp}$ und ${R_{out}}^{(-)} = R_{Dn}$ getrennt einstellbar → $\pm$ Schaltflanken getrennt} +\end{bluebox} +\begin{bluebox} + \item{Nachteile:} + \item{• „Miller-Effekt“ → hohes $\Delta Q_G$ → ggf. hohe Schaltzeiten (niedrige Dynamik) bei nicht sehr niederohmiger Ansteuerung} + 4\item{• Das Durchlaufen des „Verbotenen Bereichs“ (PMOS und NMOS aktiv → Querstrom) lässt sich nicht vermeiden. Der Querstrom ist aber durch $R_{Dp} + R_{Dn}$ begrenzt und bei kurzen Umschaltzeiten ($tr, tf \downarrow \downarrow$) ist auch $E_{on,off}$ akzeptabel klein!} +\end{bluebox} +} + +\subsubsection{Gegentakt-Spannungsfolger, Source-, Emitter-Folger} +\pbox{3cm}{\includegraphics[width = 3cm - 0.5cm]{img_07_14_gegentakt_spgfolger}} +\parbox{\textwidth - 3cm} { +\begin{bluebox} + \item{Vorteile, typ. Eigenschaften} + \item{• Kein Miller-Effekt → rel. kleines $\Delta Q_G$ → hohe Dynamik} + \item{• Kein „Verbotener Bereich“, d. h. kein Querstrom → High Imp. Bereich} + \item{• Typ. Strom-Treiber → hoher ${I_{out}}^{+(-)}$ → typ. niedriges ${R_{out}}^{+(-)}$} +\end{bluebox} +\begin{bluebox} + \item{Nachteile:} + \item{• Keine expliziten Low, High - Zustände, $|U_{in}| > |U_{out}|$, „Sp.-Verstärkung“ $\approx$ 1} + \item{• Keine festen Source-, Emitter-Potentiale → floatende Steuersp.: $|V_G|, |V_B| = Fkt. (U_{out})$} +\end{bluebox} +} + +\subsubsection{Totem-Pole Struktur} +\pbox{3cm}{\includegraphics[width = 3cm - 0.5cm]{img_07_15_totem_pole}} +\parbox{\textwidth - 3cm} { +\begin{bluebox} + \item{Vorteile, typ. Eigenschaften} + \item{• Nur ein Transistor Typ erforderlich, z. B. NMOS, bzw. npn} + \item{• Kein „Verbotener B.“ → High Imp.} + \item{• High Imp.-B. durch nichtüberlappende in1-, in2- Ansteuerung einstellbar} +\end{bluebox} +\begin{bluebox} + \item{Nachteile:} + \item{• Miller-Effekt bei M2} + \item{• 2 Eingangssignale in1, in2 erforderlich} +\end{bluebox} +} + +\end{sectionbox} +\begin{sectionbox} + +\subsubsection{Gate-Treiber (Bootstrap Prinzip)} +\pbox{4cm}{\includegraphics[width = 4cm - 0.5cm]{img_07_16_gate_treiber}} +\parbox{\textwidth - 4cm}{ +\begin{basicbox} + \item{$V_{CC} = U_{GS,on}(M2)$} + \item{$V_{BS} = U_{GS,on}(M1)$} + \item{$V_B = V_{BS} + V_S (=U_{out})$} +\end{basicbox} +\begin{emphbox} + $C_{Boot} \geq \frac{Q_G (M1) + Q_{etc} (\approx 0)}{V_{CC} - V_{F,D} - {U_{out}}^- - U_{GS,on,min}} \approx \frac{Q_G (M1)}{V_{CC} - (U_{GS,on,min}}$ +\end{emphbox}} +Treiberleistung: +\begin{basicbox} + \item{$P_{out,High,LowSide} \approx V_{CC} \cdot Q_{G,High,LowSide} \cdot f$} + \item{$P_{V,dyn.} = P_{V,CMOS} \approx V_{CC} \cdot Q_{CMOS} \cdot f$} +\end{basicbox} + +\subsection{Optokoppler} +\begin{center} + \includegraphics[height = 2cm]{img_07_17_optokoppler} +\end{center} +\begin{emphbox} + $I_C (I_B, I_F) \approx BF \cdot I_B + \frac{CTR\%}{100} \cdot I_F = BF \cdot I_B + rel\_CTR \cdot I_F$ +\end{emphbox} +\end{sectionbox} +\vfill\null +\columnbreak diff --git a/ELK2_FS/thema9_spannungsversorgung.tex b/ELK2_FS/thema9_spannungsversorgung.tex new file mode 100644 index 0000000..321c6ae --- /dev/null +++ b/ELK2_FS/thema9_spannungsversorgung.tex @@ -0,0 +1,45 @@ +\setlength{\imagewidth}{5cm} +% ============================================================================================ +\section{Spannungsversorgung, DC-DC Wandler} +% ============================================================================================ +\begin{sectionbox} +\subsection{Abwärtswandler} +\pbox{\imagewidth}{\includegraphics[width = \imagewidth - 0.5cm]{img_09_01_abwaertswandler}} +\parbox{\textwidth - \imagewidth}{ + \begin{basicbox} + \item{$dI_L/dt = U_L/L$} + \end{basicbox} + \begin{emphbox} + \item{Eingeschalteter Zustand:\newline $\Delta {I_L}^+ \approx \frac{U_{in} - U_{out}}{L} \cdot T_{on} > 0$} + \item{Ausgeschalteter Zustand:\newline $\Delta {I_L}^- \approx \frac{- U_{out}}{L} \cdot T_{off} < 0$} + \end{emphbox} +} +\subsubsection{Voraussetzungen} +Stationär: +\begin{emphbox} + $\int_0^{T_{on}}\Delta I_L(t)\cdot dt = - \int_{T_{on}}^{T_{on}+T_{off}}\Delta I_L(t) \cdot dt$ +\end{emphbox} +Nicht lückend: +\begin{emphbox} + $I_{L,Mittel} = I_{out} = I_{RL} < \frac{\Delta I_{L,pp}}{2}$ +\end{emphbox} + +\subsubsection{Berechnung} + \begin{emphbox} + \item{$D \approx \frac{U_{out}}{U_{in}} \to U_{Out} \approx D \cdot U_{in} (\neq f(R_L)!)$} + \item{$I_{out} = \frac{U_{out}}{R_L} = I_{out,DC} = I_{L,Mittel}$} + \item{$\Delta I_{L,pp} = \frac{U_{in} - U_{out}}{L} \cdot \frac{D}{f_{CLK}}$} + \end{emphbox} + Bei realer Lastkapazität (inkl. ESR) gilt: $\Delta U_{Cout}(t) = \Delta U_{Cout}(t) + \Delta U_{ESR(Cout)}(t)$ +\subsubsection{Dimensionierung} + \begin{emphbox} + \item{$L \geq \frac{U_{in}-U_{out}}{\Delta I_{L,pp}}\cdot T_{on}$} + \item{$C_{out} > \frac{\Delta I_{L,pp}}{\Delta U_{out,pp}\cdot 8 f_{CLK}}$} + \end{emphbox} + +\subsection{Synchron-Wandler} + \begin{emphbox} Kein lückender Betrieb! \end{emphbox} + \begin{center} + \includegraphics[width = 6cm]{img_09_02_synchronwandler} + \end{center} +\end{sectionbox}

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