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b/Images/Datavis/plots.yml index 951071f..297f61d 100644 --- a/Images/Datavis/plots.yml +++ b/Images/Datavis/plots.yml @@ -7,6 +7,18 @@ defaults: Cin: $C_{in}$ plots: + - load: Parasitics/Examples_R_Cp_RSweep.txt + loadtype: ltspice + step_parameter: $R$ + step_unit: $\Omega$ + + ofile: Parasitics/Examples_R_Cp_RSweep.png + + type: lt_sweep + y_key: 1/I(V1) dB + + title: Impedanzverläufe verschiedener Widerstände bei gleichbleibendem $C_p$ + ylabel: Impedanz (dB$\Omega$) - load: Parasitics/SingleStage_Cfp_Sweep.txt loadtype: ltspice step_parameter: $C_{fp}$ diff --git a/TeX/Eigenstaendigkeitserklaerung.tex b/TeX/Eigenstaendigkeitserklaerung.tex index 8e70c25..fe3ff89 100644 --- a/TeX/Eigenstaendigkeitserklaerung.tex +++ b/TeX/Eigenstaendigkeitserklaerung.tex @@ -20,6 +20,7 @@ Studienrichtung: & M.Sc. Energietechnik \\ Ich erkläre hiermit, dass ich die vorliegende Arbeit selbstständig angefertigt und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. \\ [2cm] +\todo{Can we add a no-LLM clause?} Hannover, den \today diff --git a/TeX/Kapitel/Grundlagen.tex b/TeX/Kapitel/Grundlagen.tex index 6e02c1f..e395c7f 100644 --- a/TeX/Kapitel/Grundlagen.tex +++ b/TeX/Kapitel/Grundlagen.tex @@ -47,7 +47,7 @@ Ein typischer Aufbau eines IMS ist in Abbildung \ref{fig:IMS_Schematic} dargeste \begin{figure}[h] \centering - \includegraphics[width=0.9\textwidth]{grundlagen/IMS_Schematic.drawio.png} + \includegraphics[scale=0.15]{grundlagen/IMS_Schematic.drawio.png} \caption{\label{fig:IMS_Schematic}Schematischer Aufbau einer IMS-Röhre nach \cite[Seite 3, Abb. 1.2.b]{Eiceman2013Oct}} \end{figure} @@ -61,7 +61,7 @@ Das Messergebnis eines IMS-Laufes wird as Spektrum bezeichnet, und wird meist al \subsubsection{Aufgabe eines TIV im IMS} -Wie in Kapitel \ref{chap:function_description_ims} beschrieben, beruht ein IMS auf der Messung der diskreten Ionenpakete, deren zeitlicher Versatz und Größe. Um die kleinen Ströme der Ionen, welche sich im Bereich von $\SI{100}{\pico\ampere}$ bis $\SI{10}{\nano\ampere}$ befinden, messen zu können, ist ein Verstärker notwendig. Dieser Verstärker wird als sog. Transimpedanzverstärker bezeichnet, da er als Eingangsgröße einen Strom hat, und eine Spannung als Ausgang gibt. Die Verstärkung wird somit in Ohm angegeben. Der TIV stellt hiermit ein zentrales Bauteil eines IMS dar, dessen Parameter maßgeblich die Qualität der Messungen beeinflusst. +Wie in Kapitel \ref{chap:function_description_ims} beschrieben, beruht ein IMS auf der Messung der diskreten Ionenpakete, deren zeitlicher Versatz und Größe. Um die kleinen Ströme der Ione im Bereich von $\SI{100}{\pico\ampere}$ bis $\SI{10}{\nano\ampere}$ messen zu können, ist ein Verstärker notwendig. Dieser Verstärker wird als sog. Transimpedanzverstärker bezeichnet, da er als Eingangsgröße einen Strom hat, und eine Spannung als Ausgang gibt. Die Verstärkung wird somit in Ohm angegeben. Der TIV stellt hiermit ein zentrales Bauteil eines IMS dar, dessen Parameter maßgeblich die Qualität der Messungen beeinflusst. Folgende Aufgaben werden an den TIV eines IMS gestellt: \begin{itemize} @@ -71,5 +71,114 @@ Folgende Aufgaben werden an den TIV eines IMS gestellt: \item Genügend Bandbreite zur korrekten Abbildung der Spitzen der Ionenpackete \end{itemize} +\section{Grundlegende Parasitäreffekte} + +In diesem Kapitel wird auf die parasitären Effekte weiterer Bauteile eingegangen, die im folgenden relevant sind und bei der Auslegung der Schaltung beachtet werden müssen. + +\paragraph*{Leckströme:} Diese treten bei fast allen Schaltungsaufbauten auf. Sie entstehen durch die hohen aber endlichen Oberflächenwiderstände des PCB-Materials sowie durch durch Verunreinigungen. Diese erlauben es kleinen Leckströmen zwischen Zweigen der Schaltung zu fließen, und können bei Zweigen mit hoher Impedanz störend wirken.\todo{Find a citation for this} + +\begin{figure} + \centering + \includegraphics[scale=0.25]{grundlagen/Examples_Leakages.drawio.png} + \caption{\label{fig:example_leakages}Schematische Darstellung eines PCBs mit Anschlüssen zu Bauteilen (goldene Pads) und Leiterbahnen (dunkelgrün) mit verschiedenen Leckstrompfaden entlang der Oberfläche (schraffiert dargestellt). Leckströme fließen überwiegend zwischen freigelegten Kupferflächen, können zudem auch durch Oberflächenladungen in einem Isolator wie dem PCB-Lötstopplack entstehen.} +\end{figure} + +\paragraph*{Parasitäre Kapazitäten:} Diese entstehen ebenfalls durch den physikalischen Aufbau der Schaltung. Die Nähe von Leitungen oder Kontakten zueinander, oder zu einer Kupferebene wie z.B. der Erdungsebene, erstellt eine leichte kapazitive Kopplung hierzwischen. Dieser Effekt verursacht Kapazitäten von $\SI{10}{\femto\farad}$ bis hin zu einigen $\SI{}{\pico\farad}$. Abbildung \ref{fig:example_parasitic_c} zeigt einige dieser Kapazitäten auf. + +\begin{figure}[h] + \centering + \includegraphics[scale=0.2]{grundlagen/Examples_Capacitances.drawio.png} + \caption{\label{fig:example_parasitic_c}Schematische Darstellung einer Schnittfläche eines PCBs mit Leitungen und einem Bauteil, mit einigen parasitären Kapazitäten eingezeichnet. Diese Kapazitäten entstehen sowohl zwischen zwei Leitungen, Leitungen und Erdflächen, sowie den Anschlüssen eines Bauteiles.} +\end{figure} + +Wichtig ist dieser Effekt in Kombination mit hochohmigen Eingängen und Widerständen. So wird z.B. die Impedanz eines $\SI{100}{\mega\ohm}$ Widerstandes bereits ab wenigen zehn Kilohertz maßgeblich durch die eigene parasitäre Kapazität beeinflust. Diese liegt bei der Standardbaugröße ``1206'' im Bereich von ca. $\SI{30}{\femto\farad}$, und bildet einen RC-Pass-Filter mit einer Eckfrequenz von $\SI{53.05}{\kilo\hertz}$. Abbildung \ref{fig:example_r_cp} zeigt beispielhaft die Verläufe verschiedener Widerstandsimpedanzen über die Frequenz, und wie diese durch die parasitäre Kapazität einbrechen. + +\begin{figure} + \centering + \includegraphics[scale=0.8]{datavis/Parasitics/Examples_R_Cp_RSweep.png} + \caption{\label{fig:example_r_cp}Impedanzverläufe verschiedener Widerstandswerte bei gleicher parasitärer Kapazität $C_p = \SI{30}{\femto\farad}$} +\end{figure} + +\paragraph*{Thermisches Rauschen:} Dieses Rauschen, genannt Johnson-Nyquist-Rauschen, betrifft resistive Komponenten. Es wird verursacht durch die thermische Bewegung von Ladungsträgern, und bildet ein weißes Rauschen aus. Das Rauschen lässt sich über die Formel $V_{\mathrm{n,rms}} = \sqrt{4k_BTR\Delta f}$ berechnen. Hierbei ist $V_{\mathrm{n,rms}}$ der RMS-Wert des Rauschens, $k_B$ die Boltzmann-Konstante, $T$ die Temperatur, $R$ der Widerstand des betrachteten Bauteils und $\Delta f$ die Bandbreite, über welche gemessen wird. Abbildung \ref{fig:example_r_noise} zeigt den schematischen Aufbau eines rauschenden Widerstandes. \todo{Insert citation} + +\begin{figure} + \centering + \includegraphics[scale=0.15]{grundlagen/Schematic_Resistor.drawio.png} + \caption{\label{fig:example_r_noise}Schematische Darstellung eines realen Widerstandes nach \cite{WikipediaResistors2024May}} +\end{figure} + \section{Grundlagen des Operationsverstärkers} -\section{Grundlegende Parasitäreffekte} \ No newline at end of file + +Im folgenden werden die Grundlagen eines Operationsverstärkers (auch genannt OpAmp) dargelegt. Hierbei wird nicht auf den exakten internen Aufbau eingegangen, sondern das relevante Verhalten sowie einige Parasitäreffekte beschrieben. + +Ein klassischer Operationsverstärker ist ein elektronisches Bauteil, welches vielseitige Anwendungen in einer Schaltung findet. Er kann als verstärkendes oder filterndes Bauteil aufgebaut werden, sowie differenzierend oder integrierend wirken. Die grundlegende Operation eines OpAmps ist bei jeder Verschaltung jedoch äquivalent:\\ +Er besitzt zwei Eingänge, positiv und negativ, und einen Ausgang (siehe Abbildung \ref{fig:example_opamp}). +Dieser Ausgang wird entsprechend der Formel $V_{\mathrm{out}} = A_\mathrm{ol} * \left(V_+ - V_-\right)$ getrieben, wobei $A_{\mathrm{ol}}$ der sog. Open-Loop-Gain, also die offene Verstärkung, ist. Für einen idealen OpAmp kann dieser Wert als quasi unendlich angenommen werden. Mithilfe eines Rückkoppelpfades wird das Ausgangssignal meist an den negativen Eingang zurück geführt. Der OpAmp wird somit den Ausgang so treiben, dass es keine Differenzspannung zwischen den Eingangssignalen gibt. Mit korrekter Auswahl der Rückkopplung können quasi-beliebige Transferfunktionen eingestellt werden. Abbildung \ref{fig:example_opamp_amplifier} zeigt einen simplen Verstärker-Schaltkreis, welcher das Eingangssignal um den Faktor 10 skaliert. + +\begin{figure} + \centering + \includegraphics[scale=0.25]{grundlagen/OpAmp.drawio.png} + \caption{\label{fig:example_opamp}Schematisches Symbol eines idealen OpAmps} +\end{figure} + +\begin{figure} + \centering + \includegraphics[scale=0.25]{grundlagen/OpAmp_10x.drawio.png} + \caption{\label{fig:example_opamp_amplifier}Beispielhafte Verstärkerschaltung mit einem OpAmp} +\end{figure} + +Ein realer OpAmp kann für viele Anwendungen als nahezu ideal angesehen werden. Da in dieser Arbeit jedoch mit hohen Verstärkungen und kleinen Strömen gearbeitet wird, müssen die parasitären Effekte des OpAmps mit beachtet werden. Diese sind wie folgt: + +\begin{itemize} + \item Eingangs-Leckströme. Die Eingänge eines realen OpAmp können kleine Ströme führen. Je nach Verstärker befinden sich diese im Bereich von $\SI{1}{\micro\ampere}$ bis hin zu $\SI{1}{\femto\ampere}$. Diese Leckströme können die Spannungen an den Eingängen, und somit das Messergebis, beeinflussen\cite{analogINBIAS2008}. + \item Parasitäre Kapazitäten. Ein OpAmp hat, bedingt durch die physikalische Auslegung des Bauteils, verschiedene ungewollte Kapazitäten sowohl gegen Masse, als auch zwischen den Kanälen selbst. Diese können die Transferfunktion beeinflussen\cite{tiOpAmpCap2000}. + \item Endliche Geschwindigkeit. Ein realer OpAmp kann auf Signaländerungen nur in endlicher Zeit reagieren. Hierdurch ergibt sich eine Grenze der Bandbreite in Relation zur Verstärkung. Dies wird als Verstärkungs-Bandpreitenprodukt\todo{Spelling OK?} charakterisiert\cite{Cox2002}. Im folgenden wird dies als GBWP aus dem Englischen ``Gain-Bandwidth-Product'' bezeichnet. Dies kann ebenfalls die Transferfunktion beeinflussen, siehe Abbildung \ref{fig:example_opamp_gbwp}. + Das GBWP gibt an, bei welcher Frequenz der OpAmp eine Verstärkung von 1 aufweist. Die effektive Bandbreite eines OpAmp kann somit durch Dividieren des GBWP mit der Verstärkung berechnet werden. + \item Rauschen. Ein realer OpAmp hat verschiedene Rauschquellen, welche in das Messsignal übergehen können. Dies sind Eingangsbezogenes Strom- und Spannungsrauschen \cite{tiNoise2007}, und sind in Abbildung \ref{fig:example_opamp_noise} dargestellt. Auf die genauen Quellen dieses Rauschens soll hier nicht weiter eingegangen werden, da diese durch die internen Schaltungen des OpAmp entstehen.\\ + Das Spannungsrauschen ist hierbei im unteren Frequenzbereich proportional zu $1/\omega$ und flacht ab einer Eckfrequenz zu einem konstanten Wert ab, während das Stromrauschen konstant anfängt und im höheren Frequenzbereich proportional zu $\omega$ zu nimmt.\todo{Add OpAmp noise graph} +\end{itemize} + +\begin{figure}[h] + \centering + \missingfigure{GBWP data here!} + \caption{\label{fig:example_opamp_gbwp}Darstellung des Limits der Bandbreite in Abhängigkeit des GBWP} +\end{figure} + +\begin{figure}[h] + \centering + \includegraphics[scale=0.25]{grundlagen/OpAmp_Noise.drawio.png} + \caption{\label{fig:example_opamp_noise}Schematische Darstellung der Rauschquellen eines OpAmp nach \cite{tiNoise2007}} +\end{figure} + +\section{Aufbau eines Transimpedanzverstärkers} + +\todo[inline]{Should we add integrating TIA as well?} + +Im Folgenden wird auf den grundlegenden Aufbau und die Funktionalität eines TIVs eingegangen. + +Ein TIV ist einer variante einer OpAmp-Verschaltung, dessen Aufgabe es ist, einen Strom in eine Spannung um zu wandeln. Somit wird die Verstärkung der Schaltung in $\Omega$ angegeben. Die grundlegende Schaltung ist hierbei in \ref{fig:example_tia_circuit}\todo{Add TIA example circuit} aufgeführt. + +\begin{figure} + \centering + \missingfigure{Add TIA Circuit} + \caption{\label{fig:example_tia_circuit}Grundlegender Schaltkreis eines Transimpedanzverstärkers.} +\end{figure} + +Die Funktionsweise ist wie folgt: +\begin{itemize} + \item Der OpAmp steuert den Ausgang, um die Differenz der Eingangsspannungen zu minimieren. Da der positive Eingang fest auf $\SI{0}{\volt}$ gelegt ist, wird der negative Eingang ebenfalls auf $\SI{0}{\volt}$ gesteuert. + \item Ein Eingangsstrom fließt in den Eingang des TIV. Durch den Strom kombiniert mit einer (parasitären) Eingangskapazität bildet sich eine Spannung aus. + \item Durch die aufbauende differenzielle Spannung am Eingang steuert der OpAmp eine neue Ausgangsspannung an. + \item Die Ausgangsspannung lässt über den Rückkoppelwiderstand $R_f$ einen Strom fließen. Dieser Strom gleicht den Eingangsstrom so aus, dass die Spannung am negativen Eingang zurück auf $\SI{0}{\volt}$ getrieben wird. Die Ausgangsspannung wird somit auf $R_\mathrm{f} \cdot I_\mathrm{in}$ getrieben. +\end{itemize} + +Die Vor- und Nachteile dieser Schaltungsart sind wie folgt: + +\begin{itemize} + \item[+] Leicht einstellbare Verstärkung. Der Rückkoppelwiderstand legt direkt die Verstärkung fest. + \item[+] Sehr hohe Verstärkungen sind durch Auswahl eines hohen Widerstandes möglich. + \item[+] Konstante Eingangsspannung. Der TIV-Eingang wird konstant auf $\SI{0}{\volt}$ getrieben. Hierdurch werden Effekte von z.B. parasitären Kapazitäten am Eingang verringert. Zudem können Abschirmungen an $\SI{0}{\volt}$, d.h. Erde, angeschlossen werden. + \item[-] Parasitäre Effekte begrenzen oft die Bandbreite. + \item[-] Ein OpAmp mit sehr hohem GBWP ist notwendig, um stabil zu bleiben. + \item[-] Durch die hohe Verstärkung ist die Schaltung sehr Rauschanfällig. +\end{itemize} \ No newline at end of file diff --git a/TeX/Literaturverzeichnis.bib b/TeX/Literaturverzeichnis.bib index 3630702..e598d0b 100644 --- a/TeX/Literaturverzeichnis.bib +++ b/TeX/Literaturverzeichnis.bib @@ -11,6 +11,15 @@ doi = {10.1016/j.ohx.2018.e00030} } +@book{Cox2002, + author = {Cox, James F.}, + title = {{Fundamentals of Linear Electronics: Integrated and Discrete}}, + year = {2002}, + isbn = {978-0-76683018-9}, + publisher = {Delmar Thomson Learning}, + url = {https://books.google.de/books?vid=ISBN0766830187&redir_esc=y} +} + @book{Eiceman2013Oct, author = {G. A. Eiceman, Z.Karpas, Herbert H. Hill Jr.}, editor = {}, @@ -18,4 +27,42 @@ title = {{Ion Mobility Spectrometry}}, year = {2013}, isbn = {978-1-4398-5998-8} +} + +@techreport{tiNoise2007, + author = {Texas Instruments}, + institution = {Texas Instruments}, + journal = {Digital Signal Processing Solutions}, + title = {{Noise Analysis in Operational Amplifier Circuits}}, + year = {2007}, + url = {{https://www.ti.com/lit/an/slva043b/slva043b.pdf}}, + eid = {SLVA043B} +} + +@techreport{tiOpAmpCap2000, + author = {Texas Instruments}, + institution = {Texas Instruments}, + title = {{Effect of Parasitic Capacitance in Op Amp Circuits}}, + year = {2007}, + month = sep, + url = {{https://www.ti.com/lit/an/sloa013a/sloa013a.pdf?ts=1715091777328}}, + eid = {SLOA013A} +} + +@techreport{analogINBIAS2008, + author = {Analog Devices}, + institution = {Analog Devices}, + title = {Op Amp Input Bias Current}, + year = {2008}, + url = {{https://www.analog.com/media/en/training-seminars/tutorials/MT-038.pdf}}, + eid = {MT-038} +} + +@misc{WikipediaResistors2024May, + author = {{Contributors to Wikimedia projects}}, + title = {{Johnson{\textendash}Nyquist noise - Wikipedia}}, + year = {2024}, + month = may, + note = {[Online; accessed 13. May 2024]}, + url = {https://en.wikipedia.org/w/index.php?title=Johnson-Nyquist_noise&oldid=1222085733} } \ No newline at end of file diff --git a/TeX/config/hyphenation.tex b/TeX/config/hyphenation.tex index 410dd0b..a7f612d 100644 --- a/TeX/config/hyphenation.tex +++ b/TeX/config/hyphenation.tex @@ -4,8 +4,6 @@ \hyphenation{IGBT-Durch-lass-span-nung} \hyphenation{Ei-gen-er-wär-mung} \hyphenation{STMCubeIDE} - - - +\hyphenation{Span-nungs-rau-schen} diff --git a/TeX/grafiken/grundlagen/Examples_Capacitances.drawio.png b/TeX/grafiken/grundlagen/Examples_Capacitances.drawio.png new file mode 100644 index 0000000..2e441d5 Binary files /dev/null and b/TeX/grafiken/grundlagen/Examples_Capacitances.drawio.png differ diff --git a/TeX/grafiken/grundlagen/Examples_Leakages.drawio.png b/TeX/grafiken/grundlagen/Examples_Leakages.drawio.png new file mode 100644 index 0000000..076b4c8 Binary files /dev/null and b/TeX/grafiken/grundlagen/Examples_Leakages.drawio.png differ diff --git a/TeX/grafiken/grundlagen/IMS_Schematic.drawio.png b/TeX/grafiken/grundlagen/IMS_Schematic.drawio.png index 3310eba..1c4e16a 100644 Binary files a/TeX/grafiken/grundlagen/IMS_Schematic.drawio.png and b/TeX/grafiken/grundlagen/IMS_Schematic.drawio.png differ diff --git a/TeX/grafiken/grundlagen/OpAmp.drawio.png b/TeX/grafiken/grundlagen/OpAmp.drawio.png new file mode 100644 index 0000000..7dd2b09 Binary files /dev/null and b/TeX/grafiken/grundlagen/OpAmp.drawio.png differ diff --git a/TeX/grafiken/grundlagen/OpAmp_10x.drawio.png b/TeX/grafiken/grundlagen/OpAmp_10x.drawio.png new file mode 100644 index 0000000..8b78515 Binary files /dev/null and b/TeX/grafiken/grundlagen/OpAmp_10x.drawio.png differ diff --git a/TeX/grafiken/grundlagen/OpAmp_Noise.drawio.png b/TeX/grafiken/grundlagen/OpAmp_Noise.drawio.png new file mode 100644 index 0000000..d72ef89 Binary files /dev/null and b/TeX/grafiken/grundlagen/OpAmp_Noise.drawio.png differ diff --git a/TeX/grafiken/grundlagen/Schematic_Resistor.drawio.png b/TeX/grafiken/grundlagen/Schematic_Resistor.drawio.png new file mode 100644 index 0000000..bcdfc32 Binary files /dev/null and b/TeX/grafiken/grundlagen/Schematic_Resistor.drawio.png differ diff --git a/TeX/grafiken/grundlagen/Schematic_Resistor_Noise.drawio.png b/TeX/grafiken/grundlagen/Schematic_Resistor_Noise.drawio.png new file mode 100644 index 0000000..cca6c84 Binary files /dev/null and b/TeX/grafiken/grundlagen/Schematic_Resistor_Noise.drawio.png differ