From 9798bcff2cb1c38ac79832d9e6391b70e16897bc Mon Sep 17 00:00:00 2001 From: Xaseiresh Date: Wed, 29 May 2024 13:09:07 +0200 Subject: [PATCH] feat(tex): :sparkles: write about the resistance choices --- TeX/Kapitel/Auslegung.tex | 323 ++++++++++++++++-- TeX/Kapitel/Grundlagen.tex | 7 +- .../entwicklung/cst_estatic/d_3t_b.png | Bin 91303 -> 94262 bytes .../entwicklung/cst_estatic/d_3t_t.png | Bin 111683 -> 114392 bytes .../entwicklung/cst_estatic/d_all.png | Bin 419902 -> 256465 bytes .../entwicklung/cst_estatic/d_flip.png | Bin 121027 -> 105345 bytes .../entwicklung/cst_estatic/d_probe_3t_t.png | Bin 0 -> 113034 bytes .../entwicklung/cst_estatic/d_probe_fc.png | Bin 0 -> 65365 bytes .../cst_estatic/d_streamlines_1206.png | Bin 0 -> 449412 bytes .../cst_estatic/d_streamlines_fc.png | Bin 0 -> 355559 bytes .../entwicklung/cst_estatic_shld/d_3t_t.png | Bin 0 -> 103695 bytes .../entwicklung/cst_estatic_shld/d_fc.png | Bin 0 -> 236112 bytes .../cst_estatic_shld/shielding.png | Bin 0 -> 27190 bytes .../shielding_capacitors.drawio.png | Bin 0 -> 138308 bytes .../cst_estatic_shld/shielding_potential.png | Bin 0 -> 297966 bytes 15 files changed, 300 insertions(+), 30 deletions(-) create mode 100644 TeX/grafiken/entwicklung/cst_estatic/d_probe_3t_t.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic/d_probe_fc.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic/d_streamlines_1206.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic/d_streamlines_fc.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic_shld/d_3t_t.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic_shld/d_fc.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic_shld/shielding.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic_shld/shielding_capacitors.drawio.png create mode 100644 TeX/grafiken/entwicklung/cst_estatic_shld/shielding_potential.png diff --git a/TeX/Kapitel/Auslegung.tex b/TeX/Kapitel/Auslegung.tex index 8681508..0d0ae09 100644 --- a/TeX/Kapitel/Auslegung.tex +++ b/TeX/Kapitel/Auslegung.tex @@ -6,6 +6,7 @@ Hiernach werden verschiedene Bauteile zur Auswahl gezogen, wobei die limitierend Eine Auswahl der Bauteile wird mit Hinsicht auf die Zielparameter des Designs durchgeführt. \section{Zielparameter} +\label{chap:tia_design_goals} Wie in Abschnitt \ref{chap:tia_in_ims} dargestellt, ist die Aufgabe eines TIVs im IMS, die Stromflüsse der Ionenpackete auf eine messbare Spannung zu verstärken. Hierbei soll der TIV die Form eines solchen @@ -42,14 +43,39 @@ In diesem Kapitel wird auf das Verhalten der passiven Bauteile eingegangen, und Effekte den Schaltkreis beeinflussen. Dies bezieht sich überwiegend auf den Rückkoppelwiderstand und die parasitären Kapazitäten der Schaltung. +\subsubsection{Thermisches Rauschen} + +Wie bereits in Kapitel \ref{chap:basics_parasitics} beschrieben, besitzen resistive Bauteile +ein thermisches Rauschen. In diesem Abschnitt wird der Einfluss des Rauschens untersucht. + +In einem TIV-Schaltkreis gibt es ein Bauteil mit hohem Widerstand: Der Rückkoppelwiderstand. +Somit wird vermutet, dass dieser Widerstand eine dominierende Quelle des thermischen Rauschens ist. +Laut Gleichung \ref{eqn:thermal_voltage_noise} wächst die Amplitude des Spannungsrauschens mit der Wurzel des +Widerstandswertes, wodurch eine erste Vermutung ist, dass ein kleinerer Widerstand besser wäre. +Für einen TIV ist der Eingang jedoch ein strombasierter Eingang. Somit muss das Stromrauschen betrachtet werden. +Dies lässt sich berechnen wie folgt:\todo{Cite or explain this} + +\begin{eqnarray} + I_\mathrm{n,rms} & = & \frac{V_\mathrm{n,rms}}{R} \\ + I_\mathrm{n,rms} & = & \frac{\sqrt{4k_BTR\Delta f}}{R} \\ + I_\mathrm{n,rms} & = & \sqrt{\frac{4k_BT\Delta f}{R}}\label{eqn:thermal_current_noise} +\end{eqnarray} + +Laut Gleichung \ref{eqn:thermal_current_noise} ist somit ein {\em größerer} Widerstand von Vorteil, +um den Einfluss des thermalen Rauschens zu minimieren. Für das Design soll somit eine Maximierung des +gesamten Rückkoppelwiderstandes angestrebt werden. + \subsubsection{Parasitäre Rückkopplungskapazität} Der Rückkoppelwiderstand ist ein zentrales Bauteil des TIVs, welcher die Verstärkung des gesamten Schaltkreises festlegt. -Bisher wurde von einem idealen Widerstand ausgegangen, jedoch besitzen alle Bauteile eine parasitäre Kapazität, +Alle Bauteile eine parasitäre Kapazität, wie in Kapitel \ref{chap:basics_parasitics} festgelegt wurde. Abbildung \ref{fig:example_r_cp} in diesem Kapitel zeigt, dass diese Kapazität an hochohmigen Widerständen schon bei geringeren Frequenzen einen Einfluss auf die Bandbreite haben kann. +Im Falle des Rückkoppelwiderstandes sorgt die Verringerung der Impedanz für eine Verringerung +der Verstärkung des OpAmp, und somit für eine reduzierte Bandbreite des gesamten Verstärkers. Diese Einschränkung +darf nicht unter die in Kapitel \ref{chap:tia_design_goals} festgelegte Zielbandbreite fallen. Nun soll genauer auf den Ursprung der Kapazität, den zu erwartenden Wert, sowie mögliche Mitigationen eingegangen werden. Um dies zu erreichen, wird eine Simulation in dem Programm ``CST Studio Suite 2021'' eingerichtet. Dieses Programm erlaubt die Simulation @@ -82,10 +108,10 @@ Hierbei wird die Metallisierung nur auf einer Seite der Keramik, neben dem Wider Dies soll Streueffekte und Kapazitäten verringern. Das für diese Widerstandsart erstellte Modell ist in Abbildung \ref{fig:cst_model_1206_flipchip} dargestellt. -Mithilfe dieser Modelle können nun die kapazitiven Kopplungen bestimmt werden. +Mithilfe dieser Modelle werden nun die kapazitiven Kopplungen bestimmt. Hierfür wird der ``Electrostatic Solver'' genutzt, welcher die elektrischen Felder im statischen Zustand, -sowie die kapazitive Kopplung von Potentialen, berechnet. Die Widerstände werden hierbei auf einer Grundfläche aus FR4 platziert. -Dies entspricht dem Platinenmaterial einer reellen Platine, welches durch sein Dielektrikum auch Einfluss auf die Kapazitäten haben. +sowie die kapazitive Kopplung von Potentialflächen, berechnet. Die Widerstände werden hierbei auf einer Grundfläche aus FR4 platziert. +Dies entspricht dem Platinenmaterial einer reellen Platine, welches durch sein Dielektrikum auch Einfluss auf die Kapazitäten hat. Der Flipchip-Widerstand wird hierbei mit den Kontakten nach unten zeigend simuliert. Bei dem Standard-1206 Gehäuse werden zwei Anbringungsmöglichkeiten (Widerstandsbelag nach oben und nach unten) getestet. @@ -101,35 +127,76 @@ werden zwei Anbringungsmöglichkeiten (Widerstandsbelag nach oben und nach unten In der Simulation werden die metallisierten Enden der Widerstände auf unterschiedliche potentiale gelegt, um das E-, D- und Potentialfeld berechnen zu können. -Hierbei wird $\pm\SI{0.5}{\volt}$ gewählt, um ein Gesamtpotential von $\SI{1}{\volt}$ auf zu bauen. +Hierbei wird $\pm\SI{0.5}{\volt}$ gewählt, um ein Gesamtpotential von $\SI{1}{\volt}$ auf zu bauen, wobei +die Auswahl der Potentialwerte auf die von CST berechnete Kapazität keinen Einfluss nimmt, und lediglich zur +Visualisierung dient. + +\begin{table}[h] + \centering + \caption{\label{table:para_r_cf}Ergebnisse der Kapazitätsberechnung} + \begin{tabular}{ |l|r|r| } + \hline + Typ & Parallelkapazität & Erdkapazität \\ + \hline + 1206, Film obig & $\SI{46.81}{\femto\farad}$ & $\SI{89.95}{\femto\farad}$ \\ + 1206, Film unten & $\SI{46.93}{\femto\farad}$ & $\SI{90.17}{\femto\farad}$ \\ + Flipchip & $\SI{40.84}{\femto\farad}$ & $\SI{84.36}{\femto\farad}$ \\ + \hline + \end{tabular} +\end{table} + +Die Ergebnisste sind in Tabelle \ref{table:para_r_cf} dargestellt. Deutlich zu erkennen ist eine +Verringerung der parasitären Kapazität bei der Flipchip-Technologie. Die Anbringung des Standard-1206 +Widerstandes hat nur eine kleine Auswirkung auf die Kapazität, wobei die normale Anbringung (Film obig) +etwas besser scheint. Zusätzlich wurde die Kapazität in das Vakuum bzw. Erde berechnet. +Dies beeinflusst nicht direkt die Übertragungsfunktion des Widerstandes, trägt jedoch zu z.B. +der Eingangskapazität bei. Zudem scheint es keine großen Unterschiede bei der Anbringung des +1206-Widerstandes zu geben, wofür im Folgenden nur noch die Standard-Anbringung betrachtet wird. + +Mithilfe der ersten Kapazitätswerte und der in Kapitel \ref{chap:tia_design_goals} bestimmten Bandbreite +lässt sich nun ein oberer Grenzwert des Rückkoppelwiderstandes berechnen. +Dies ergibt aus der Gleichung der Grenzfrequenz eines RC-Filters, beschrieben in Gleichung \ref{eqn:max_rf}. +Die berechneten Grenzwerte der Widerstände sind in Tabelle \ref{table:para_r_max} dargestellt. + +\begin{eqnarray} + f_c & = & 2\pi\cdot \left(R_f \cdot C_f\right)^{-1} \\ + \SI{30}{\kilo\hertz} & \leq & f_c \\ + \SI{30}{\kilo\hertz} & \leq & 2\pi\cdot \left(R_f\cdot C_f\right)^{-1} \\ + R_f & \leq & 2\pi\cdot \left(\SI{30}{\kilo\hertz}\cdot C_f\right)^{-1}\label{eqn:max_rf} +\end{eqnarray} + +\begin{table}[h] + \centering + \caption{\label{table:para_r_max}Obere Grenzwerte der Widerstandsauswahl} + \begin{tabular}{ |l|r| } + \hline + Typ & Grenzwert \\ + \hline + 1206, Film obig & $\SI{113.3}{\mega\ohm}$ \\ + 1206, Film unten & $\SI{133.0}{\mega\ohm}$ \\ + Flipchip & $\SI{129.9}{\mega\ohm}$ \\ + \hline + \end{tabular} +\end{table} + +Für den gesamten TIV ist nach Kapitel \ref{chap:tia_design_goals} eine Gesamtverstärkung +von ca. $\SI{2}{\giga\ohm}$ gewünscht, und entsprechend des vorherigen Kapitels ist ein möglichst großer +Rückkoppelwiderstand vorteilhaft. Somit wird nun mithilfe der Simulationen nach der Quelle +dieser Kapazität gesucht, und Möglichkeiten zur Verringerung dieser (und somit Steigerung der Widerstandsgrenze) gesucht. Abbildungen \ref{fig:cst_r_potentials} und \ref{fig:cst_r_ds} zeigen die Ergebnisse der Feldsimulationen auf. Vor allem die Darstellung des D-Feldes gibt Hinweise auf die Positionen der parasitären Kapazitäten, da sich die auf einer leitenden Fläche befindende Ladung wie folgt berechnen lässt:\todo{Quote Maxwell?} -\begin{table}[h] - \centering - \begin{tabular}{ |c|c|c| } - \hline - \multicolumn{3}{|c|}{\caption{Ergebnisse der Kapazitätsberechnung}} - Typ & Kapazität \\ - \hline - 1206, Film obig & $\SI{57\decimalcomma 07}{\femto\farad}$ \\ - 1206, Film unten & $\SI{60\decimalcomma 41}{\femto\farad}$ \\ - Flipchip & $\SI{51.13}{\femto\farad}$ \\ - \hline - \end{tabular} -\end{table} - \begin{equation} - \iint \mathbf{D} \cdot dS = \iiint \rho_f dV + \iint \mathbf{D} \cdot dS = \iiint \rho_f dV\label{eqn:integral_d} \end{equation} Durch Bestimmung der Flussrichtungen des D-Feldes lassen sich somit die Quellen der Ladungen bestimmen. Dies ist zum Verständnis der Kapazität und der späteren Verminderung dieser -nützlich. +nützlich.\todo{Rewrite this more understandably} -\begin{figure}[h] +\begin{figure}[p] \centering \begin{subfigure}[b]{1\linewidth} \centering @@ -162,7 +229,7 @@ nützlich. -\begin{figure}[h] +\begin{figure}[p] \centering \begin{subfigure}[b]{1\linewidth} \centering @@ -176,29 +243,227 @@ nützlich. \begin{subfigure}[t]{.25\linewidth} \centering \includegraphics[width=1\textwidth]{entwicklung/cst_estatic/d_3t_t} - \subcaption{Schnittfläche des D-Feldes in der Mitte des nach oben zeigenden 1206 Widerstandes} \end{subfigure}\hfill% + \subcaption{Schnittfläche des D-Feldes in der Mitte des nach oben zeigenden 1206 Widerstandes} + \end{subfigure}\hfill% \begin{subfigure}[t]{.25\linewidth} \centering - \includegraphics[width=1\textwidth,clip,trim={0 0.8cm 0 0}]{entwicklung/cst_estatic/d_3t_b} + \includegraphics[width=1\textwidth,clip,trim={0 0.4cm 0 0.4cm}]{entwicklung/cst_estatic/d_3t_b} \subcaption{Schnittfläche des D-Feldes in der Mitte des herunterzeigenden 1206} \end{subfigure}\hfill% \begin{subfigure}[t]{.25\linewidth} \centering - \includegraphics[width=1\textwidth,clip,trim={0.2cm 0 0cm 0}]{entwicklung/cst_estatic/d_flip} - \subcaption{Schnittfläche des D-Feldes in der Mitte des Flipchip} + \includegraphics[width=1\textwidth,clip,trim={0cm 0.4cm 0cm 0.4cm}]{entwicklung/cst_estatic/d_flip} + \subcaption{\label{fig:cst_d_flipchip}Schnittfläche des D-Feldes in der Mitte des Flipchip} \end{subfigure}\hspace{0.1\linewidth} \caption{\label{fig:cst_r_ds} Die D-Feldstärken der elektrostatischen Simulation in verschiedenen Ansichten.} \end{figure} +Deutlich zu erkennen ist der Grund der geringeren Kapazität des Flipchip in Abbildung \ref{fig:cst_d_flipchip} +im Vergleich zu dem Standardwiderstand. Durch die geringere metallisierte Oberfläche ist die D-Feld-Intensität +innerhalb der Keramik des Widerstandes verringert, und befindet sich näher an der Unterseite. +Bei den Standardwiderständen liegt eine homogene Ausbreitung des D-Feldes in der Keramik vor. +Die D-Feld-Intensität innerhalb des PCB-Materials ist bei allen drei Widerständen gleich, und scheint ebenfalls +einen großen Einfluss auf die Kapazität zu besitzen. -\subsubsection{Thermisches Rauschen} -\subsubsection{Parasitäre Eingangskapazität} + +CST erlaubt die Berechnung des Feldflusses durch eine gegebene Fläche, welches dem +Flächenintegral der Gleichung \ref{eqn:integral_d} entspricht. +Somit können die Ladungsanteile berechnet werden, welche durch das D-Feld verursacht werden. +Abbildung \ref{fig:d_field_probe_all} zeigt die Flächen, welche zum integrieren verwendet wurden. +Die entsprechenden Ergebnisse der Integration sind in Tabelle \ref{table:d_field_integration} dargestellt. + +\begin{figure}[h] + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[width=0.9\textwidth]{entwicklung/cst_estatic/d_probe_fc.png} + \subcaption{\label{fig:d_field_probe_flipchip}Integrationsflächen des Flipchip-Widerstandes} + \end{subfigure}% + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[width=0.9\textwidth]{entwicklung/cst_estatic/d_probe_3t_t.png} + \subcaption{\label{fig:d_field_probe_1206}Integrationsflächen des 1206-Widerstandes} + \end{subfigure} + \caption[D-Feld Integrationsflächen]{\label{fig:d_field_probe_all}Die in CST genutzten Integrationsflächen (grün) zur + Berechnung des D-Feld-Durchflusses} +\end{figure} + +Angemerkt werden muss hierbei, dass die Simulation auch die Kapazität in das Vakuum simuliert. +Die somit berechneten Ladungen entsprechen nicht nur der Parallel-, sondern auch der Erdkapazität. +Dies erklärt die leichten Diskrepanzen der berechneten Kapazität und der berechneten Feldstärken. +Aus diesem Grund können die berechneten Feldstärken nur als Richtlinie für die Verteilung der Felder genutzt werden. + + +\begin{table}[h] + \centering + \caption{\label{table:d_field_integration}Ergebnisse der Feldintegration bei $\SI{1}{\volt}$ Potential} + \begin{tabular}{ |c|r|r| } + \hline + Typ & Feld im Keramik-Kern & Feld im PCB \\ + \hline + 1206 & $\SI{17.85}{\femto\coulomb}$ & $\SI{17.19}{\femto\coulomb}$ \\ + Flipchip & $\SI{15.99}{\femto\coulomb}$ & $\SI{17.89}{\femto\coulomb}$ \\ + \hline + \end{tabular} +\end{table} + + +Zu sehen ist, dass sich ein erheblicher Anteil des Feldes, circa 50\%, durch das Material des PCBs bewegt. Dies +trifft auf sowohl den Standard-Widerstand als auch den Flipchip zu. + +\subsubsection{Mitigation der Parallelkapazität} +Im Folgenden wird untersucht, ob durch eine bestimmte Platzierung von Elektroden im PCB-Material +die Parallelkapazität verringert werden kann.\todo{Find a citation for this.} +Durch korrekte Platzierung von Elektroden mit festgelegtem Potential kann theoretisch das D-Feld auf diese umgeleitet +werden, wodurch das PCB-Material selbst eine kleinere Teilhabe an der parasitären Kapazität des Widerstandes haben sollte. + +Ein erster Versuch hierfür wird aus zwei symmetrischen Elektroden aufgebaut, welche unterhalb der Kontakte der +Widerstände aufgebaut werden, und auf das selbe Potential wie die entsprechenden Kontakte gelegt werden. +Abbildung \ref{fig:r_symmetric_shielding} zeigt den Aufbau der im folgenden verwendeten Abschirmungselektroden und +deren Potentiale. + +\begin{figure}[h] + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[clip,trim={4.8cm 0 7.2cm 0},width=0.9\linewidth]{entwicklung/cst_estatic_shld/shielding.png} + \caption{Konstruktion der Schirmungselektroden} + \end{subfigure}% + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[clip,trim={0 0 0.4cm 0},width=0.9\linewidth]{entwicklung/cst_estatic_shld/shielding_potential.png} + \caption{\label{fig:r_symmetric_shielding_potential}Potentialfeld der Schirmungselektroden} + \end{subfigure} + \caption{\label{fig:r_symmetric_shielding}Schnittbild durch das Simulatiosmodell mit eingebauten Abschirmungselektroden} +\end{figure} + +Da es bei diesem Aufbau vier Potentiale gibt, sind auch entsprechend mehr Kapazitäten zu beachten. +Abbidlung \ref{fig:r_shielding_capacitances} zeigt alle Kapazitäten, welche von einem Kontakt sichtbar sind. + +\begin{figure}[h] + \centering + \includegraphics[width=0.6\linewidth]{entwicklung/cst_estatic_shld/shielding_capacitors.drawio.png} + \caption{\label{fig:r_shielding_capacitances}Schematische Darstellung der Kapazitäten, welche einer der Widerstandskontakte sieht.} +\end{figure} + +Von Interesse sind die Parallelkapazität der Widerstandskontake, $C_\mathrm{r,p}$, +welches der im vorherigen Kapitel beschriebenen Kapazität entspricht, sowie den +Kapazitäten $C_\mathrm{sa,rb}$ und $C_\mathrm{sb,ra}$, welche zwischen dem Widerstand und den +Schirmungselektroden entstehen. Durch den hier verwendeten Aufbau sind diese Kapazitäten symmetrisch, +und werden im Folgenden als $C_\mathrm{r,sp}$ bezeichnet. +Die Kapazitäten $C_\mathrm{sa,ra}$ und $C_\mathrm{sb,rb}$ sind nicht +relevant für die Bandbreite, da die Schirmelektrode auf das Potential des anliegenden Widerstandes getrieben wird, können +jedoch z.~B. die Eingangskapazität erhöhen. Sie werden im Folgenden als $C_\mathrm{r,s}$ bezeichnet. +Ebenso ist die Kapazität zwischen den Schirmelektroden nicht relevant, da diese separat getrieben werden und nicht hochohmig sind. + +\begin{table}[h] + \centering + \caption{\label{table:shielding_capacitances}Parasitäre Kapazitäten mit Abschirmungselektroden} + \begin{tabular}{ |c|c|c|c|c| } + \hline + Typ & $C_\mathrm{r,p}$ & $C_\mathrm{r,sp}$ & $C_\mathrm{r,s}$ & $C_\mathrm{r,g}$ \\ + \hline + 1206 & $\SI{5.64}{\femto\farad}$ & $\SI{28.16}{\femto\farad}$ & $\SI{194.25}{\femto\farad}$ & $\SI{17.71}{\femto\farad}$ \\ + Flipchip & $\SI{3.51}{\femto\farad}$ & $\SI{23.39}{\femto\farad}$ & $\SI{183.53}{\femto\farad}$ & $\SI{15.99}{\femto\farad}$ \\ + \hline + \end{tabular} +\end{table} + + +\begin{table}[h] + \centering + \caption{\label{table:shielding_charges}Ergebnisse der Feldintegration mit Abschrimung bei $\SI{1}{\volt}$ Potential} + \begin{tabular}{ |c|r|r| } + \hline + Typ & Feld im Keramik-Kern & Feld im PCB \\ + \hline + 1206 & $\SI{13.25}{\femto\coulomb}$ & $\SI{10.37}{\femto\coulomb}$ \\ + Flipchip & $\SI{11.35}{\femto\coulomb}$ & $\SI{9.22}{\femto\coulomb}$ \\ + \hline + \end{tabular} +\end{table} + +Tabelle \ref{table:shielding_capacitances} zeigt, dass die nun berechneten gesamten +Parallelkapazitäten ($C_\mathrm{r,p} + C_\mathrm{r,sp}$) wesentlich +geringer sind als diejenigen ohne Abschirmung. Dies wird ebenfalls durch eine erneute Ladungsberechnung +mit der in \ref{fig:d_field_probe_all} aufgezeigten Integrationsflächen bestätigt, dessen Ergebnisse in +Tabelle \ref{table:shielding_charges} dargestellt sind. +Sowohl die vom Kern als auch die im PCB Verursachten Ladungen wurden verringert, was darauf schließen +lässt dass die Abschirmungselektroden einen größeren Einfluss haben als erwartet. +Abbildung \ref{fig:shielding_d_field} zeigt die Schnittbilder der D-Felder mit Abschirmungselektroden auf. + +\begin{figure}[h] + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[clip,trim={0 0 0 0},width=0.9\linewidth]{entwicklung/cst_estatic_shld/d_3t_t.png} + \caption{Schnittbild des 1206-Widerstandes} + \end{subfigure}% + \begin{subfigure}[t]{.5\linewidth} + \centering + \includegraphics[clip,trim={0.5cm 0 0.5cm 0},width=0.9\linewidth]{entwicklung/cst_estatic_shld/d_fc.png} + \caption{Schnittbild des Flipchip} + \end{subfigure} + \caption{\label{fig:shielding_d_field}Schnittbild des D-Feldes durch das Simulatiosmodell mit eingebauten Abschirmungselektroden} +\end{figure} + +Die Abschirmungselektroden sind somit in der Lage, die parasitäre Parallelkapazität des Widerstandes deutlich +zu verringern. Hierdurch jedoch entstehen größere Kapazitäten zu den jeweiligen Schirmungselektroden, welche somit +auf das gleiche Potential wie den entsprechenden Widerstandskontakt getrieben werden müssen, um negative Effekte auf die +Bandbreite zu vermeiden. +Mit der verringerten Parallelkapazität lassen sich somit größere Widerstände verwenden. Die erneut berechneten +Grenzwerte sind in Tabelle \ref{table:para_rshield_max} aufgelistet. + +\begin{table}[h] + \centering + \caption{\label{table:para_rshield_max}Obere Grenzwerte der Widerstandsauswahl mit Abschrimung} + \begin{tabular}{ |c|r| } + \hline + Typ & Grenzwert \\ + \hline + 1206 & $\SI{156.96}{\mega\ohm}$ \\ + Flipchip & $\SI{197.22}{\mega\ohm}$ \\ + \hline + \end{tabular} +\end{table} + +Da die berechneten Werte noch nicht der in Kapitel \ref{chap:tia_design_goals} festgelegten +Verstärkung entsprechen, werden zusätzlich noch andere Möglichkeiten zur Verringerung der +Parallelkapazität hinzu gezogen. +Eine dieser Möglichkeiten ist die Nutzung mehrerer Widerstände in Reihenschaltung. +Hierdurch wird der effektive Widerstand der Gesamtschaltung erhöht und die Parallelkapazität +verringert, entsprechend: + +\begin{eqnarray} + R_\mathrm{tot} & = & \sum_{i=1}^{n}{R_i} \\ + C_\mathrm{tot} & = & \left(\sum_{i=1}^{n}{\frac{1}{C_i}} \right)^{-1} +\end{eqnarray} + +Und mit einer Vereinfachung, dass alle Widerstände gleich gewählt sind, ergibt sich: + +\begin{eqnarray} + R_\mathrm{tot} & = & R\cdot n \\ + C_\mathrm{tot} & = & \frac{C}{n} \\ + f_\mathrm{c,tot} & = & 2\pi\cdot \left(R_\mathrm{tot}\cdot C_{tot}\right)^{-1} \\ + f_\mathrm{c,tot} & = & 2\pi\cdot \left(Rn \cdot \frac{C}{n}\right)^{-1} \\ + f_\mathrm{c,tot} & = & 2\pi\cdot \left(R\cdot C\right)^{-1}\label{eqn:r_series_frequency} +\end{eqnarray} + +Aus Gleichung \ref{eqn:r_series_frequency} lässt sich erschließen, dass die Grenzfrequenz +der Gesamtschaltung der Grenzfrequenz eines einzelnen Widerstandes entspricht. +Dies bedeutet, dass bei Auswahl eines geeigneten Einzelwiderstandes eine beliebig hohe +Gesamtimpedanz bei gleicher Bandbreite kreiert werden kann. +Zu beachten ist jedoch, dass die einzelnen Zweige dieser Widerstandsschaltung +hochimpedante und somit empfindliche Potentiale dar stellen. +Parasitäre Kapazitäten z.B. zu Erde, wie diejenigen in Tabelle \ref{table:shielding_capacitances} dargestellt, +können an diesen Potentialen ebenfalls die Bandbreite beeinflussen. +In der realen Schaltung wird somit nur eine begrenzte Anzahl an Widerständen in Reihe geschaltet. +Die genaue Menge ergibt sich aus der praktisch unterbringbaren Größe innerhalb der PCB-Schaltung. \subsection{Effekte des OpAmp} + \subsubsection{Verstärkungs-Bandbreiten-Produkt} \subsubsection{OpAmp-Rauschen} + \section{Untersuchung von Kompensationsmöglichkeiten} \section{Design der Schaltung} diff --git a/TeX/Kapitel/Grundlagen.tex b/TeX/Kapitel/Grundlagen.tex index d5ed136..81733fa 100644 --- a/TeX/Kapitel/Grundlagen.tex +++ b/TeX/Kapitel/Grundlagen.tex @@ -101,7 +101,12 @@ Wichtig ist dieser Effekt in Kombination mit hochohmigen Eingängen und Widerst \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} +\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 folgende Formel berechnen: +\begin{equation} + V_{\mathrm{n,rms}} = \sqrt{4k_BTR\Delta f}\label{eqn:thermal_voltage_noise} +\end{equation} + +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 diff --git a/TeX/grafiken/entwicklung/cst_estatic/d_3t_b.png b/TeX/grafiken/entwicklung/cst_estatic/d_3t_b.png index f2e7371bdcd123b9c6e3777f7bab1bf293e931a0..cc85ef0b7b05ec4d1ba591caae77f2dc93167f00 100644 GIT binary patch literal 94262 zcmXt9bv)dE{6FohneMTWGtKFqsngw?Zq7K}&6v*9r)%1YnQl%`Z2Hv39mDYR`#m1N ze?Nb{KkvsYp7o9cX{rzt&=UXv0Akg5iaG!Q1{wgsjKjx$`cfj>C;|Ym0#p^{^!zMN z;32iHYF=Bd*Ww4g0b2n(e-fX~Oy>Q0VLOv3wUW#zKJc3~lT5MsbCjGibu+6h=*4HA z2Y_0tZ1xAX#PG%sAXai^l5rO58+MLY29&Wc2~`B}|HC6(^^6S(uwz_*{{5;)^n4>A z7^H8$c)Z?P9a7u6ENx#Da9yx^egZdWc+lR^FRDJXa6bLiJ+pN>{m4b}H=FtKUmHc& znn?GQclQs;kwL*8_Z%i&T+_CX6FnwyADCpa_Kdr+L2U859Xl2D-^_%#+uV(UI0y8Ds>Ua3%k_E&R?qBFV&guWc?hlTJcq8(u z5qzI$Mp$+N=qXT-sMYhc!;pV`kIPfX9?f+Rr&Y7v_j+tJLyJvaqgz9l@2jFOEXNva z>oP=MhF|u2W(@zbt|{ED*;8eJZKDhyk3OZJxs`=<2Jo=l*oeFQ;O``&GNZPt4yj|P 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