From 1097175fee7ff087289b85b0bf891ede712a048d Mon Sep 17 00:00:00 2001
From: Roger Wolf <roger.wolf@kit.edu>
Date: Fri, 7 Mar 2025 16:18:22 +0100
Subject: [PATCH] refurbishing

---
 Operationsverstaerker/Datenblatt.md           |   8 +-
 Operationsverstaerker/README.md               |  54 ++-
 Operationsverstaerker/doc/Hinweise-OPV.md     | 439 ++----------------
 .../doc/Hinweise-Verstaerker.md               |   2 +-
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 24 files changed, 86 insertions(+), 417 deletions(-)
 create mode 100644 Operationsverstaerker/figures/OPV_Symbol.odg
 create mode 100644 Operationsverstaerker/figures/OPV_Symbol.png

diff --git a/Operationsverstaerker/Datenblatt.md b/Operationsverstaerker/Datenblatt.md
index f2ed197..c56f35c 100644
--- a/Operationsverstaerker/Datenblatt.md
+++ b/Operationsverstaerker/Datenblatt.md
@@ -1,10 +1,10 @@
-# Technische Daten und Inventar für den Versuch Operationsverstärker:
+# Technische Daten und Inventar für den Versuch **Operationsverstärker**
 
 Für die verschiedenen Aufgaben des Versuchs **Operationsverstärker** stehen Ihnen die folgenden Geräte zur Verfügung:
 
 - Ein Experimentiersteckbrett mit einem fest verbauten Transistor (vom Typ 2N2219A, npn) und 3 fest verbauten Operationsverstärkern (vom Typ LM741).
 - Verschiedene Verbindungskabel, Dioden, Widerstände, und Kondensatoren (**verwenden Sie nötigenfalls benachbarte Werte**, falls Ihnen die in der Dokumentation verwendeten Werte nicht zur Verfügung stehen sollten). 
-- Frequenzgenerator ($0.2\ \mathrm{Hz} \ldots\ 2\ \mathrm{MHz}$; Sinus, Rechteck oder Dreieck; $0 \ldots \pm10\ \mathrm{V}$).
-- Oszilloskop (Tektronix , 2 Kanal).
-- Multimeter.
+- Ein Frequenzgenerator ($0.2\ \mathrm{Hz} \ldots\ 2\ \mathrm{MHz}$; Sinus, Rechteck oder Dreieck; $0 \ldots \pm10\ \mathrm{V}$).
+- Ein Oszilloskop (Tektronix , 2 Kanal).
+- Ein Multimeter.
 
diff --git a/Operationsverstaerker/README.md b/Operationsverstaerker/README.md
index 7e56272..b5c1771 100644
--- a/Operationsverstaerker/README.md
+++ b/Operationsverstaerker/README.md
@@ -1,22 +1,26 @@
+
+
 <img src="../figures/Logo_KIT.svg" width="200" style="float:right;" />
 
 # Fakultät für Physik
 
 ## Physikalisches Praktikum P2 für Studierende der Physik
 
-Versuch P2-59, 60, 61 (Stand: Mai 2024)
+Versuch P2-191, 192, 193 (Stand: **Februar 2025**)
 
-[Raum F1-15](https://labs.physik.kit.edu/img/Praktikum/Lageplan_P2.png)
+[Raum F1-15](https://labs.physik.kit.edu/img/Klassische-Praktika/Lageplan_P1P2.png)
 
 
 
-# Operationsverstärker
+# Operationsverstärker (OPV)
 
 ## Motivation
 
  
 
-Ein [Operationsverstärker](https://de.wikipedia.org/wiki/Operationsverst%C3%A4rker) (OPV) ist ein aus Röhrenverstärkern oder Transistoren bestehender, heutzutage [integrierter Schaltkreis](https://de.wikipedia.org/wiki/Integrierter_Schaltkreis) zur Signalverstärkung, der eine sehr hohe Verstärkung aufweist und durch Gegenkopplung in seinem Verhalten kontrolliert werden kann. Das erste Patent wurde 1941 durch [Karl D. Schwartzel Jr.](https://en.wikipedia.org/wiki/Karl_D._Swartzel_Jr.) angemeldet. Der Begriff *operational amplifier*, von dem sich auch die deutsche Bezeichnung Operationsverstärker ableitet, wurde 1947 durch [John Ragazzini](https://de.wikipedia.org/wiki/John_Ralph_Ragazzini) geprägt. Sie geht auf den anfänglich überwiegenden Einsatz zur Durchführung einfacher mathematischer Operationen in den ersten analogen Rechenmaschinen und Computern zurück. Im Jahr 1968 entwickelte die Firma Fairchild Semiconductor den OPV vom Typ $\mu A741$, der sich seit dieser Zeit bis zum heutigen Tage (allerdings nur in geringer Stückzahl) in Produktion befindet und auch hier im Praktikum verwendet wird. Ein Schaltplan des $\mu A741$ ist in **Abbildung 1** gezeigt:
+Ein [Operationsverstärker](https://de.wikipedia.org/wiki/Operationsverst%C3%A4rker) (OPV) ist ein aus Transistoren bestehender [integrierter Schaltkreis](https://de.wikipedia.org/wiki/Integrierter_Schaltkreis) zur Signalverstärkung, der eine sehr hohe Verstärkung aufweist und durch Gegenkopplung in seinem Verhalten kontrolliert werden kann. Das erste Patent wurde 1941 durch [Karl D. Schwartzel Jr.](https://en.wikipedia.org/wiki/Karl_D._Swartzel_Jr.) angemeldet. Der Begriff *operational amplifier*, von dem sich auch die deutsche Bezeichnung **Operationsverstärker** ableitet, wurde 1947 durch [John Ragazzini](https://de.wikipedia.org/wiki/John_Ralph_Ragazzini) geprägt. Sie geht auf den anfänglich überwiegenden Einsatz zur Durchführung einfacher mathematischer Operationen in den ersten analogen Rechenmaschinen und Computern zurück. Im Jahr 1968 entwickelte die Firma Fairchild Semiconductor den OPV vom Typ $\mu A741$, der sich seit dieser Zeit bis zum heutigen Tage (allerdings nur noch in geringer Stückzahl) in Produktion befindet und auch hier im Praktikum verwendet wird. Ein Schaltplan des $\mu A741$ ist in **Abbildung 1** gezeigt:
+
+---
 
 <img src="./figures/OpAmpTransistorLevel_Colored.png" width="1000" style="zoom:100%;" />
 
@@ -24,29 +28,43 @@ Ein [Operationsverstärker](https://de.wikipedia.org/wiki/Operationsverst%C3%A4r
 
 ---
 
-OPVs sind aus der heutigen Elektrotechnik und Signalverarbeitung nicht mehr wegzudenken. Häufig werden sie neben der Verstärkung auch als [Impedanzwandler](https://de.wikipedia.org/wiki/Impedanzwandler) benötigt. In der physikalischen Messtechnik spielt diese Eigenschaft überall dort eine Rolle, wo ein Messgerät die Messgröße nicht beeinflussen darf. Nahezu jedes physikalische Experiment, das mit kleinen Signalen konfrontiert ist benutzt daher OPVs. Dort ist die Anforderung, dass der OPV eine sehr hohe Eingangsimpedanz $X_{e}$ (i.a. im Bereich von $10^{13}\ \Omega$) aufweist, so dass er das Signal nicht beeinflusst, eine hohe Verstärkung besitzt und eine niedrige Ausgangsimpedanz $X_{\mathrm{a}}$ hat, so dass er am andere Ende der Messanordnung die Messung mit dem Spannungsmessgerät nicht beeinflusst.
+OPVs sind aus der heutigen Elektrotechnik und Signalverarbeitung nicht mehr wegzudenken. Sie werden als Schalter, zur Verstärkung oder als [Impedanzwandler](https://de.wikipedia.org/wiki/Impedanzwandler) verwendet. In der physikalischen Messtechnik spielen sie überall dort eine Rolle, wo Signale klein sind und/oder ein Messgerät die Messgröße nicht beeinflussen darf. Nahezu jedes physikalische Experiment, das mit kleinen Signalen konfrontiert ist benutzt daher heutzutage OPVs. Die Anforderungen an den OPV sind: 
+
+- Gegebenenfalls hohe Verstärkung. 
+- Hohe Eingangsimpedanz $X_{e}$ (im Bereich von $10^{13}\ \Omega$!). Dadurch lässt sich z.B. die Rückwirkung eines angeschlossenen Voltmeters auf das Signal minimieren. 
+- Niedrige Ausgangsimpedanz $X_{\mathrm{a}}$. Dadurch lässt sich z.B. die Rückwirkung des OPV auf ein angeschlossenes Voltmeter minimieren.
+
+Im Praktikum begegnen Ihnen diese Anforderungen zum Beispiel in den folgenden Versuchen: 
+
+- [Photoeffekt](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Photoeffekt): Hier besteht die Herausforderung darin eine Spannung zu messen, die von nur wenigen Ladungsträgern auf einem Kondensator erzeugt wird. Es geht also um die Spannungsverstärkung. Ohne einen sehr hohen Eingangswiderstand würden die Ladungsträger aber auch sofort abfließen.
+-  [Franck-Hertz-Versuch](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Franck_Hertz_Versuch).  Hier besteht die Herausforderung darin einen Strom im Bereich weniger $\mathrm{nA}$ zu messen. Hier geht es also v.a. um die Stromverstärkung.
 
 ## Lehrziele
 
 Wir listen im Folgenden die wichtigsten **Lehrziele** auf, die wir Ihnen mit dem Versuch **Operationsverstärker** vermitteln möchten: 
 
-- Sie lernen den physikalischen Grundbaustein des OPV, den [Transistor](https://de.wikipedia.org/wiki/Transistor), in seiner einfachsten und dem Röhrenverstärker ähnlichsten Form, als bipolarem Transistor, kennen. 
-- Sie lernen mit der Emitterschaltung die wichtigste [Transistorgrundschaltung](https://de.wikipedia.org/wiki/Transistorgrundschaltungen) für die Verwendung eines bipolaren Transistors als Strom- und Spannungsverstärker kennen. Das Prinzip der Gegenkopplung ([Negativen Rückkopplung](https://de.wikipedia.org/wiki/Negative_R%C3%BCckkopplung)) ist allgemein und auf den OPV übertragbar.
-- Sie lernen den OPV als elektrisches Bauelement mit seinen wichtigsten Grundschaltungen als invertierendem und nicht-invertierendem Verstärker kennen. 
-- Sie stecken ein Reihe komplexerer Schaltungen auf der Grundlage des invertierenden Verstärkers. 
-- Bei diesem Versuch steht nicht "halbleiterphysikalische Innenleben" der zu untersuchenden Bauelemente nicht im Vordergrund. Dieses werden Sie in späteren Vorlesungen genauer studieren können. Uns geht es um ein grundlegendes Verständnis der Vorgänge und den (sicheren) Umgang bei der Beschaltung dieser Bauelemente.
+- Sie lernen den **OPV als wichtiges aktives Bauelement** elektrischer Schaltkreise mit seinen wichtigsten Eigenschaften kennen. 
+- Sie verinnerlichen die Grundschaltungen des OPV als **invertierendem Verstärker, nicht-invertierendem Verstärker und Impedanzwandler**. 
+- Sie machen sich, im Rahmen weiterer algebraischer Schaltungen mit der **äußeren Beschaltung** des OPV und den **Goldenen Regeln** zur Beschaltung eines idealen OPV vertraut.
+- Sie erkennen in der Vorbereitung auf den Versuch den Nutzen von OPVs für physikalische Messungen und erfahren, wo für konkrete Messungen im Praktikum OPVs im Einsatz sind. 
 
 ## Versuchsaufbau
 
-Ein typischer Aufbau für den Versuch Operationsvertärker ist in **Abbildung 1** gezeigt:
+Ein typischer Aufbau für den Versuch Operationsverstärker ist in **Abbildung 2** gezeigt:
+
+---
 
 <img src="./figures/Operationsverstaerker.png" width="1000" style="zoom:100%;" />
 
-**Abbildung 1**: (Ein typischer Aufbau für den Versuch Operationsverstärker)
+**Abbildung 2**: (Ein typischer Aufbau für den Versuch Operationsverstärker)
 
 ---
 
-Alle Schaltungen werden mit Hilfe verschiedener Widerstände und Kondensatoren auf dem abgebildeten Schaltbrett aufgesteckt und mit dem Oszilloskop oder dem Multimeter untersucht. Zur Erzeugung eines Eingangssignals dient ein Frequenzgenerator. 
+Alle Schaltungen werden mit Hilfe verschiedener Widerstände, Potentiometer und Kondensatoren auf dem abgebildeten Schaltbrett aufgesteckt und mit dem Oszilloskop oder dem Multimeter untersucht. Zur Erzeugung eines Eingangssignals dient ein Frequenzgenerator. 
+
+## Was macht diesen Versuch aus?
+
+Bei diesem Versuch steht das "physikalische Innenleben" der zu untersuchenden Bauelemente nicht im Vordergrund. Dieses werden Sie in späteren Vorlesungen genauer studieren können. Uns geht es um ein grundlegendes Verständnis der Vorgänge und den (sicheren) Umgang bei der Beschaltung eines OPV. Die Grundschaltungen sollte jeder Physiker mit etwas Praxis aufbauen können. Einfache algebraische und komplexere Schaltungen geben Ihnen die Möglichkeit, sich mit der äußeren Beschaltung von OPVs und den dabei bestehenden Möglichkeiten weiter vertraut zu machen. In seinen Anfangsjahren war der OPV wirklich für die analoge Durchführung algebraischer und numerischer Operationen vorgesehen. In diesem Versuch dienen uns diese Operationen in erster Linie dazu Ihnen die Scheu bei der Beschaltung von OPVs zu nehmen. 
 
 ## Wichtige Hinweise
 
@@ -54,8 +72,8 @@ Alle Schaltungen werden mit Hilfe verschiedener Widerstände und Kondensatoren a
 
 # Navigation
 
-- [Hinweise-Verstaerker.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/doc/Hinweise-Verstaerker.md): Grundlagen zum Verständnis von elektronischen Verstärkern im Allgemeinen und Transistoren.
-- [Hinweise-OPV.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/doc/Hinweise-OPV.md): Grundlagen zum Verständnis der OPV im Allgemeinen und der verwendeten Schaltungen im Speziellen.
-- [Hinweise-Versuchsdurchfuehrung.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/doc/Hinweise-Versuchsdurchfuehrung.md): Wichtige Hinweise und **Tipps zur Versuchsdurchführung (NOCH IN BEARBEITUNG)**.
-- [Datenblatt.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/Datenblatt.md): **Wichtige technische Details** zu den Versuchsaufbauten.
-
+- [Operationsverstärker.iypnb](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstärker/Operationsverstärker.ipynb): Aufgabenstellung und Vorlage fürs Protokoll.
+- [Operationsverstärker_Hinweise.ipynb](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstärker/Operationsverstärker_Hinweise.ipynb): Kommentare zu den Aufgaben.
+- [Datenblatt.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstärker/Datenblatt.md): Technische Details zu den Versuchsaufbauten.
+- [doc](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Operationsverstärker/doc): Dokumente zur Vorbereitung auf den Versuch.
+- [figures](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Operationsverstärker/figures): Bilder, die für die Dokumentation des Versuchs verwendet wurden.
diff --git a/Operationsverstaerker/doc/Hinweise-OPV.md b/Operationsverstaerker/doc/Hinweise-OPV.md
index e3d4e46..e226597 100644
--- a/Operationsverstaerker/doc/Hinweise-OPV.md
+++ b/Operationsverstaerker/doc/Hinweise-OPV.md
@@ -1,444 +1,95 @@
-# Hinweise für den Versuch Operationsverstärker
+# Hinweise für den Versuch **Operationsverstärker**
 
-## Operationsverstärker
+## Einführung Operationsverstärker
 
-Ein [Operationsverstärker](https://de.wikipedia.org/wiki/Operationsverst%C3%A4rker) (OPV) ist ein aus mehreren Transistoren bestehendes [Netzwerk](https://de.wikipedia.org/wiki/Netzwerk_(Elektrotechnik)), das heutzutage i.a. in Form von [integrierten Schaltkreisen](https://de.wikipedia.org/wiki/Integrierter_Schaltkreis) realisiert wird. Das Schaltsymbol eines OPV ist in **Abbildung 10** gezeigt: 
+Ein [Operationsverstärker](https://de.wikipedia.org/wiki/Operationsverst%C3%A4rker) (OPV) ist ein aus mehreren Transistoren bestehendes [Netzwerk](https://de.wikipedia.org/wiki/Netzwerk_(Elektrotechnik)). Die Innenbeschaltung des OPV vom Typ $\mu\mathrm{A}741$, wie wir ihn für diesen Versuch verwenden, [hier](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/raw/main/Operationsverstaerker/figures/OpAmpTransistorLevel_Colored.png) gezeigt (Quelle [Wikipedia](https://commons.wikimedia.org/wiki/File:OpAmpTransistorLevel_Colored.svg)). Eine grobe Diskussion dieser Beschaltung findet sich in der Datei  [Hinweise-OPV-Innenbeschaltung.md](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/doc/Hinweise-OPV-Innenbeschaltung.md). In der Praxis werden OPVs i.a. als Blackbox mit wohl-definiertem Ein- und Ausgangsverhalten verwendet. Das Schaltsymbol ist in **Abbildung 1 (a)** gezeigt:
 
-<img src="../figures/OPVSymbolik.png" width="350" style="zoom:100%;"/>
+---
 
-**Abbildung 10**: (Schaltsymbol eines OPV und Definition der Ströme und Spannungen. Die Beschaltung im Inneren des OPV wird i.a. nicht eingezeichnet)
+ <img src="/home/rwolf/Data/Vorlesungen/2025/P2/students/Operationsverstaerker/figures/OPV_Symbol.png" width="750" style="zoom:100%;"/>
 
----
+**Abbildung 1**: (Schaltsymbol eines OPV mit Definition der relevanten Ströme und Spannungen (a) und Anschlussschema des OPV $\mathrm{\mu A741}$, wie er in diesem Versuch verwendet wird. $V_{S\pm}$ bezeichnet die Versorgungsspannung. NC steht für *not connected*. Die Beschaltung im Inneren, sowie die Versorgungsspannung(en) werden in Schaltbildern i.a. nicht gezeigt)
 
-Der OPV besitzt **mindestens fünf Klemmen**: 
+---
 
-- Einen (Minus) invertierenden und einen (Plus) nicht-invertierenden Signaleingang; 
-- einen Signalausgang und 
-- mindestens zwei (in Schaltbildern nicht gezeigte) Anschlüsse zur externen Spannungsversorgung. 
-- Je nach Komplexität des OPV können noch weitere Klemmen vorhanden sein. 
+**Abbildung 1 (b)** zeigt das Anschlussschema des $\mathrm{\mu A741}$. Ein OPV besitzt mindestens fünf Klemmen: 
 
-Im allgemeinen ist der Plus-Eingang als hochohmiger Spannungseingang ausgeführt; der Minus-Eingang ist je nach OPV-Typ ebenfalls ein hochohmiger Spannungs- oder ein niederohmiger Stromeingang. Entsprechend ist der Ausgang des OPV entweder als hochohmiger Strom- oder niederohmiger Spannungsausgang ausgelegt. Der OPV kann somit in guter Näherung als ideale [Strom-](https://de.wikipedia.org/wiki/Stromquelle_(Schaltungstheorie)#Ideale_Stromquelle) oder [Spannungsquelle](https://de.wikipedia.org/wiki/Spannungsquelle#Ideale_und_reale_Spannungsquellen) (relativ zum Massenpotential) angesehen und verwendet werden. 
+- Einen (Minus) invertierenden und einen (Plus) nicht-invertierenden Signaleingang. 
+- Einen Signalausgang. 
+- Mindestens zwei (in Schaltbildern nicht gezeigte) Anschlüsse zur externen Spannungsversorgung. 
+- Der $\mathrm{\mu A741}$ weist zudem zwei weitere Klemmen zur Feinabstimmung der Eingangsspannungen auf. 
 
-Für diesen Versuch verwenden wir OPVs vom Typ $\mu\mathrm{A}741$. Dabei handelt es sich um einen **Spannungsverstärker** (engl. *voltage feedback operational amplifier*, VFA) mit zwei hochohmigen Spannungseingängen und niederohmigem Spannungsausgang. 
+Im allgemeinen ist der Plus-Eingang als hochohmiger Spannungseingang ausgeführt; der Minus-Eingang ist je nach OPV-Typ ebenfalls ein hochohmiger Spannungs- oder ein niederohmiger Stromeingang. Der OPV kann i.a. in guter Näherung als [ideale Strom- oder Spannungsquelle](https://gitlab.kit.edu/kit/etp-lehre/p1-praktikum/students/-/blob/main/Elektrische_Messverfahren/doc/Hinweise-Spannungsquellen.md) (relativ zum Massenpotential) angesehen und verwendet werden. Die häufigste Verwendung ist die als **Spannungsverstärker** (engl. *voltage feedback operational amplifier*, VFA) mit zwei hochohmigen Spannungseingängen. 
 
-### Definition relevanter Größen
+## Definition relevanter Größen
 
-Der Stromfluss in den Plus-Eingang wird im Folgenden mit $I_{P}$ bezeichnet, der Stromfluss in den Minus-Eingang mit $I_{N}$, die anliegenden Spannungen werden entsprechend mit $U_{P}$ und $U_{N}$ bezeichnet; $U_{d}=U_{P}-U_{N}$ ist die Differenzspannung zwischen Minus- und Plus-Eingang; und $I_{a}$ und $U_{a}$ bezeichnen, je nach OPV-Typ den Strom oder die Spannung am Ausgang. Die Spannungsverstärkung erfolgt als 
+Der Stromfluss in den Plus-Eingang wird im Folgenden mit $I^{+}$ bezeichnet, der Stromfluss in den Minus-Eingang mit $I^{-}$, die anliegenden Spannungen bezeichnen wir mit $U^{+}$ und $U^{-}$; $U_{d}=U^{+}-U^{-}$ ist die Differenzspannung zwischen Minus- und Plus-Eingang; $I_{a}$ und $U_{a}$ bezeichnen den Strom und die Spannung am Ausgang. Die Spannungsverstärkung erfolgt als 
 $$
 \begin{equation*}
-U_{a} = v_{0}\left(U_{P}-U_{N}\right) + v_{\mathrm{Gl}}\,\frac{U_{P}+U_{N}}{2},
+U_{a} = v_{0}\left(U^{+}-U^{-}\right) + v_{\mathrm{Gl}}\,\frac{U^{+}+U^{-}}{2},
 \end{equation*}
 $$
-wobei man $v_{0}$  als **Leerlaufverstärkung** und $v_{\mathrm{Gl}}$ als **Gleichtaktverstärkung** bezeichnet. Beides sind OPV-spezifische Größen. In Datenblättern zu OPVs wird statt der Gleichtaktverstärkung auch die Gleichtaktunterdrückung (*common mode rejection ratio*, CMRR)
+wobei man $v_{0}$  als **Leerlaufverstärkung** und $v_{\mathrm{Gl}}$ als **Gleichtaktverstärkung** bezeichnet. Beides sind OPV-spezifische Größen. Im allgemeinen ist eine hohe Leerlaufverstärkung und eine niedrige Gleichtaktverstärkung erwünscht. In Datenblättern zu OPVs wird statt der Gleichtaktverstärkung daher auch die **Gleichtaktunterdrückung** (*common mode rejection ratio*, CMRR)
 $$
 \begin{equation*}
 G = 20\,\ln\left(\frac{v_{0}}{v_{\mathrm{Gl}}}\right)
 \end{equation*}
 $$
-angegeben.
+in [Dezibel $\mathrm{dB}$](https://de.wikipedia.org/wiki/Bel_(Einheit)), als Qualitätsmerkmal, angegeben.
 
 Bei der Verwendung von OPVs unterscheidet man zwei Hauptbetriebsarten: 
 
-- Den invertierenden Betrieb mit $U_{P}=0$ und $U_{a}=-v_{0}\,U_{N}$; sowie
-- den nicht-invertierenden Betrieb mit $U_{N}=0$ und $U_{a}=v_{0}\,U_{P}$,
+- Den **invertierenden Betrieb** mit $U^{+}=0$ (GND) und $U_{a}=-v_{0}\,U^{-}$; sowie
+- den **nicht-invertierenden Betrieb** mit $U^{-}=0$ (GND) und $U_{a}=v_{0}\,U^{+}$,
 
-das Eingangssignal $U_{e}$ liegt also entweder als $U_{P}$ am Plus- oder als $U_{N}$ am Minus-Eingang an während der jeweils andere Eingang auf Masse liegt.
+das Eingangssignal $U_{e}$ liegt also entweder als $U^{+}$ am Plus- oder als $U^{-}$ am Minus-Eingang an, während der jeweils andere Eingang auf Masse (GND) liegt.
 
 ### Idealer und realer OPV
 
-Bei einem idealen OPV ist sowohl $v_{\mathrm{Gl}}$, also auch der Eingangswiderstand 0, während sowohl $v_{0}$ also auch der Ausgangswiderstand unendlich groß sind. Zudem ist die Verstärkung idealerweise von der Frequenz des Signals unabhängig. In der folgenden Tabelle sind einige charakteristische Eigenschaften von idealen und realen OPVs gegenübergestellt: 
+Bei einem idealen OPV ist sowohl $v_{\mathrm{Gl}}$, also auch der Ausgangswiderstand ($X_{a}$) 0, während sowohl $v_{0}$ also auch der Eingangswiderstand $X_{e}$ unendlich groß sind. Zudem hängt die Verstärkung idealerweise nicht von der Frequenz des Signals ab. In der folgenden Tabelle sind einige charakteristische Eigenschaften von idealen und realen OPVs gegenübergestellt: 
 
-| Eigenschaft                 | idealer OPV | realer OPV                               |
-| :-------------------------- | ----------- | ---------------------------------------- |
-| $v_{0}$                     | $\infty$    | $10^{5}\ldots10^{8}$                     |
-| $v_{\mathrm{Gl}}$           | $0$         | $0.1\ldots 3$                            |
-| $R_{d}$                     | $\infty$    | $10^{7}\ \Omega\ldots 10^{12}\ \Omega$   |
-| $R_{a}$                     | $0$         | $10\ \Omega\ldots 10^{3}\ \Omega$        |
-| $I_{P}^{(0)},\ I_{N}^{(0)}$ | $0$         | $0.1\,\mathrm{nA}\ldots 25\ \mathrm{nA}$ |
+| Eigenschaft           | idealer OPV | realer OPV                               |
+| :-------------------- | ----------- | ---------------------------------------- |
+| $v_{0}$               | $\infty$    | $10^{5}\ldots10^{8}$                     |
+| $v_{\mathrm{Gl}}$     | $0$         | $0.1\ldots 3$                            |
+| $R_{d}$               | $\infty$    | $10^{7}\ \Omega\ldots 10^{12}\ \Omega$   |
+| $R_{a}$               | $0$         | $10\ \Omega\ldots 10^{3}\ \Omega$        |
+| $I^{-(0)},\ I^{+(0)}$ | $0$         | $0.1\,\mathrm{nA}\ldots 25\ \mathrm{nA}$ |
 
-Dabei bezeichnen potentielle $I_{N}^{(0)}$ und $I_{P}^{(0)}$ Offsetströme auf den Eingängen aufgrund baulicher Asymmetrien. 
+Dabei bezeichnen die Größen $I^{-(0)}$ und $I^{+(0)}$ potentielle *offset*-Ströme auf den Eingängen aufgrund baulicher Asymmetrien und $R_{d}$ den Widerstand zwischen dem positiven und dem negativen Eingang. 
 
 Zudem ist die Verstärkung beim realen OPV oberhalb einer charakteristischen Grenzfrequenz $\nu_{\mathrm{G}}$ frequenzabhängig. Darunter garantiert der Hersteller i.a. ideale Unterdrückung der Frequenzabhängigkeit.  
 
-### Goldene Regeln
+## Goldene Regeln
 
-Zur (ungefähren) Dimensionierung, d.h. zur Beschaltung mit konkreten äußeren Widerständen, von OPV-Schaltkreisen betrachtet man den OPV als ideal. In diesem Fall gelten i.a. die folgenden **Goldenen Regeln**:
+Zur (ungefähren) Dimensionierung, d.h. zur Beschaltung mit konkreten äußeren Widerständen, von OPV-Schaltkreisen betrachtet man den OPV als ideal. In diesem Fall gelten die folgenden **Goldenen Regeln**:
 
 - Die Differenzspannung zwischen den Eingängen des OPV ist Null: $U_{d}=0$;
-- Durch die Eingänge des OPV fließt kein Strom: $R_{d}=R_{P}=R_{N}=\infty$;
-- Bis zum maximal zulässigen Ausgangsstrom $𝐼_{a}^{\mathrm{max}}$ ist der OPV beliebig belastbar: $R_{a}=0$. Im Gegenzug hängt $U_{a}$ von der Last, also von $I_{a}$ ab. 
-
-### Darlington-Schaltung
-
-Die [Darlington-Schaltung](https://de.wikipedia.org/wiki/Darlington-Schaltung), als eines der einfachsten Netzwerke zur Signalverstärkung mit zwei Transistoren T1 und T2, ist in **Abbildung 11** dargestellt: 
-
-<img src="../figures/DarlingtonSchaltung.png" width="300" style="zoom:100%;"/>
-
-**Abbildung 11**: (Darlington-Schaltung aus zwei npn-Transistoren)
-
----
-
-Sie ist Bestandteil nahezu jedes OPV. Die Beschaltung erfolgt als Spezialfall eines [Emitterfolgers](https://de.wikipedia.org/wiki/Transistorgrundschaltungen#Emitterfolger), wobei der Emitter von T1 die Basis von T2 ansteuert. Für den **Stromverstärkungsfaktor** gilt:
-$$
-\begin{equation*}
-\begin{split}
-&I_{\mathrm{C,2}} = \beta_{2}\,I_{\mathrm{B,2}};\qquad
-I_{\mathrm{C,1}} = \beta_{1}\,I_{\mathrm{B,1}};\\
-&\\
-&\beta' = \frac{I_{\mathrm{C,1}}+I_{\mathrm{C,2}}}{I_{\mathrm{B,1}}} = \frac{I_{\mathrm{C,1}}}{I_{\mathrm{B,1}}}\,\left(1+\beta_{2}\right)\approx\beta_{1}\beta_{2},
-\end{split}
-\end{equation*}
-$$
-die Verstärkungsfaktoren von T1 und T2 multiplizieren sich also in erster Näherung. In der Praxis werden Kleinsignalverstärkungen von bis zu 50'000 erreicht. Die Darlington-Schaltung kann als einzelner sog. **Darlington-Transistor** betrachtet und verbaut werden. Je nach äußerer Beschaltung kann dieser zur Spannungs- oder Stromverstärkung genutzt werden.  
-
-### Innenbeschaltung eines OPV
-
-Die Innenbeschaltung eines OPV ist i.a. sehr komplex und wird in der Schaltungstheorie der Elektrotechnik behandelt. Wir diskutieren hier nur die wichtigsten Konzepte anhand der beispielhaften und stark vereinfachten Schaltskizze aus **Abbildung 12**:
-
-<img src="../figures/OPVInnenbeschaltung.png" width="1000" style="zoom:100%;"/>
-
-**Abbildung 12**: (Beispielhafte und vereinfachte Schaltskizze zur Diskussion der Innenbeschaltung eines OPV)
-
----
-
-Die Innenbeschaltung jedes OPVs kann grundsätzlich immer in drei Abschnitte unterteilt werden:
-
-- **Eingangsstufe**, 
-- **Verstärkerstufe**, 
-- **Ausgangsstufe**. 
-
-Wir werden jede Stufe im Folgenden etwas genauer diskutieren.
-
-#### Eingangsstufe
-
-Die Eingangsstufe besteht aus einem [Differenzverstärker](https://de.wikipedia.org/wiki/Differenzverstärker), an dem die beiden Eingangssignale mit den Spannungen $U_{N}$ und $U_{P}$ anliegen. In **Abbildung 12** liefern diese die Basisspannungen für zwei baugleiche pnp-Transistoren, die durch eine [Konstantstromquelle](https://de.wikipedia.org/wiki/Konstantstromquelle) versorgt werden. In unserem Beispiel soll zur Erklärung des Prinzips $U_{N}\gt U_{P}$ gelten. Für den pnp-Transistor liegt der Emitter (in diesem Fall als Quelle positiver Ladungen!) oben im Bild (siehe **Abbildung 2** [hier](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Operationsverstaerker/doc/Hinweise-Verstaerker.md)). Je größer die Basisspannung desto geringer $U_{\mathrm{BE}}$, desto geringer also auch die Verstärkung und damit $I_{\mathrm{C}}$. Für das hier ausgeführte Beispiel gilt also $I_{2}\gt I_{1}$. An die Kollektoren der Transistoren schließt sich die einfache Realisation eines [Stromspiegels](https://de.wikipedia.org/wiki/Stromspiegel) an. Dieser besteht aus zwei baugleichen npn-Transistoren mit kurzgeschlossener Basis. Ein weiterer Kurzschluss zwischen Kollektor und Basis an einem der Transistoren sorgt dafür, dass sich eine wohldefinierte, für beide Transistoren gleiche Basisspannung $U_{\mathrm{B}}\gtrsim -U_{D}$ einstellt. Ein Kurzschluss mit zwei baugleichen Widerständen sorgt dafür, dass die Emitter beider Transistoren ebenfalls auf gleichem Potential gehalten werden, so dass durch beide Transistoren der gleiche Strom $I_{3}=I_{4}$ fließt. Aufgrund dieser Konstellation fließt der überschüssige Strom aus $I_{2}$ als Differenzstrom
-$$
-\begin{equation*}
-I_{\mathrm{diff}}=I_{2}-I_{1}
-\end{equation*}
-$$
-(im Bild nach rechts) in die Verstärkerstufe ab. 
-
-Nach diesem Prinzip übersetzt der Differenzverstärker kleine Spannungsdifferenzen in einen dazu proportionalen (ggf. verstärkten) Strom $I_{\mathrm{diff}}$. Mit einem solchen Aufbau erreicht man eine sehr gute Gleichtaktunterdrückung. Ein Potentiometer im unteren Teil der Schaltung erlaubt es bauliche Unterschiede der Widerstände auszugleichen. 
-
-#### Verstärkerstufe
-
-Die **Verstärkerstufe** besteht in **Abbildung 12** aus einer Darlington-Schaltung, die $I_{\mathrm{diff}}$ aus der Eingangsstufe als $I_{\mathrm{B}}$ aufnimmt und in eine hohe Ausgangsspannung umsetzt. An dieser Stelle hängt die Spannungsverstärkung noch stark von der angeschlossenen Last ab. Wäre dies bereits der Ausgang des OPV, so würde die Verstärkung beim Anschluss einer bereits geringen Last stark abfallen.
-
-#### Ausgangsstufe
-
-Die **Ausgangsstufe** besteht aus einer Kollektorschaltung, die im Gegensatz zu den vorherigen Stufen keine Spannungsverstärkung aufweist. Als [Impedanzwandler](https://de.wikipedia.org/wiki/Impedanzwandler) erfüllt sie den Zweck das Signal mit einem niedrigen Innenwiderstand auch für hohe Ströme stabil, als ideale Stromquelle, an den Verbraucher weiterzugeben. Nach innen weist sie einen hohen Widerstand auf, wodurch die Verstärkerstufe wiederum nicht belastet wird. 
-
-Würde es sich um eine einfache Kollektorschaltung mit einem einzelnen Transistor handeln wäre die über $R_{E}$ permanent abfallende Leistung unwirtschaftlich hoch. Bei der hier dargestellten Schaltung handelt es sich um einen komplementären Emitterfolger ([Gegentaktendstufe](https://de.wikipedia.org/wiki/Gegentaktendstufe)), bei dem ein npn- und ein pnp-Transistor wechselseitig jeweils an eine Versorgungsspannung unterschiedlichen Vorzeichens angeschlossen sind. Bei einem positiven Signal ist der npn-Transistor offen und gibt das Signal weiter, während der pnp-Tansistor sperrt. Bei einem negativen Signal sind die Verhältnisse umgekehrt. Der Vorteil dieser Schaltung gegenüber der einfachen Kollektorschaltung besteht darin, dass im Arbeitspunkt $U_{\mathrm{B}}=0$ gewählt werden kann, so dass ohne Signal kein Ruhestrom fließt. Durch diese Wahl erreicht die Ausgangsstufe des OPV einen Wirkungsgrad von 78.5%, gegenüber 6.5% bei einer einfache Kollektorschaltung. Der Nachteil dieser Schaltung besteht darin, dass für jeden einzelnen Transistor erst ab einem Signal mit $|U_{\mathrm{B}}|\gtrsim |U_{D}|$ am Ausgang ein Strom fließt. Die daraus resultierende Verzerrung für kleine Eingangssignale bezeichnet man als **Übernahmeverzerrung**. Um diese weitestgehend auszuschließen weicht man von $U_{\mathrm{B}}=0$ als Arbeitspunkt ab und setzt beide Transistoren auf ein eigenes Potential mit der Differenz $\Delta U_{\mathrm{B}}=2\,U_{D}$. Dies wird durch die zwei Dioden im Schaltkreis vor der Ausgangsstufe erreicht, die als Spannungsteiler fungieren. Die Dioden haben zu den Transistoren äquivalente Kennlinien. Sind sie zudem wärmeleitend mit ihnen verbunden lassen sich selbst thermische Potentialdifferenzen zwischen Trasistoren und Dioden nahezu vollständig ausschließen.
-
-#### Reale Innenbeschaltung im OPV $\mu\mathrm{A}741$
-
-Im [hier](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/blob/main/Operationsverstaerker/figures/OpAmpTransistorLevel_Colored.png) dargestellten noch verhältnismäßig "einfachen" Schaltplan des $\mu\mathrm{A}741$ sind die einzelnen Stufen durch farbige Umrandungen gekennzeichnet: 
-
-- Die **Eingangstufe**, in der man den Differenzverstärker gut erkennen kann ist blau umrandet; 
-- die **Verstärkerstufe** mit einer Darlington-Emitterschaltung ist magenta-farben umrandet;
-- die **Ausgangsstufe** mit komplementärem Emitterfolger ist cyan-farben umrandet. Die Potentialdifferenz der komplementär beschalteten Transistoren wurde durch einen weiteren Transistor realisiert und ist grün umrandet; 
-- Schließlich sind durch rote Umrandungen einige **Stromspiegel** hervorgehoben.
-- Der Kondensator in der Mitte der Schaltung dient zur [Frequenzkompensation](https://de.wikipedia.org/wiki/Frequenzkompensation) und bestimmt $\nu_{\mathrm{G}}$.
+- Durch die Eingänge des OPV fließt kein Strom: $R_{d}=R^{+}=R^{-}=\infty$;
+- Bis zum maximal zulässigen Ausgangsstrom $𝐼_{a}^{\mathrm{max}}$ ist der OPV beliebig belastbar, d.h. $U_{a}$ hängt bis $𝐼_{a}^{\mathrm{max}}$ **nicht** von der Last (d.h. $I_{a}$) ab. 
 
 ### Einsatz und Dimensionierung
 
-Ohne äußere Beschaltung des OPV wäre aufgrund der hohen Verstärkung $U_{a}$ abhängig von $U_{e}$ entweder maximal oder Null. Aufgrund dieser Eigenschaft werden OPVs auch als **Schalter** oder [**Komparatoren**](https://de.wikipedia.org/wiki/Komparator_(Analogtechnik)) eingesetzt. Eine solche Anwendung, die Sie in **Aufgabe 4** dieses Versuchs sehen werden ist der sog. [Schmitt-Trigger](https://de.wikipedia.org/wiki/Schmitt-Trigger). 
+Ohne äußere Beschaltung des OPV wäre aufgrund der hohen Verstärkung $U_{a}$ je nach Eingangsspannung $U_{e}$ entweder maximal oder Null. Aufgrund dieser Eigenschaft werden OPVs auch als **Schalter** oder [**Komparatoren**](https://de.wikipedia.org/wiki/Komparator_(Analogtechnik)) eingesetzt.
 
-Um den OPV als Verstärker zu betreiben verhindert man dieses Verhalten durch äußere Beschaltung, mit der man, analog zum Transistor, $v_{0}$ durch Gegenkopplung kontrolliert reduziert. Bei der Gegenkopplung wird ein Teil von $U_{a}$ mit invertiertem Vorzeichen so auf den Eingang des OPV zurückgeführt, dass die Schaltung insgesamt Veränderungen des Eingangssignals entgegenwirkt. Bei allen Verstärkerschaltungen wird daher immer der Ausgang auf den Minus-Eingang gekoppelt. Bei Gegenkopplung steigt $U_{a}$ nur so lange an, bis $U_{d}$ auf Null abfällt. **Analog zum Transistor hängt der Verstärkungsfaktor $v_{U}$ der resultierenden Schaltung nicht mehr von $v_{0}$, sondern nur noch von der äußeren Beschaltung ab.** 
+Um den OPV als Verstärker zu betreiben verhindert man dieses Verhalten durch äußere Beschaltung, mit der man, analog zum Transistor, $v_{0}$ **durch Gegenkopplung kontrolliert reduziert**. Bei der Gegenkopplung wird ein Teil von $U_{a}$ mit invertiertem Vorzeichen so auf den Eingang des OPV zurückgeführt, dass die Schaltung insgesamt Veränderungen des Eingangssignals entgegenwirkt. Bei allen Verstärkerschaltungen wird daher immer der Ausgang auf den Minus-Eingang gekoppelt. Bei Gegenkopplung steigt $U_{a}$ nur so lange an, bis $U_{d}$ auf Null abfällt. **Analog zum Transistor hängt der Verstärkungsfaktor $v_{U}$ der resultierenden Schaltung nicht mehr von $v_{0}$, sondern nur noch von der äußeren Beschaltung ab.** 
 
 Die ungefähre Dimensionierung des Schaltkreises erfolgt unter Anwendung der oben erwähnten [**Goldenen Regeln**](https://de.wikipedia.org/wiki/Operationsverst%C3%A4rker). Die exakte Justierung wird daraufhin experimentell vorgenommen. 
 
-### Nicht-invertierender Verstärker
-
-Beim nicht-invertierenden Verstärker (**Elektrometerverstärker**) handelt es sich um eine Verstärkerschaltung, bei der das Ausgangssignal die gleiche Polarität aufweist, wie das Eingangssignal. Das Eingangssignal liegt also auf dem Plus-Eingang des OPV, während der Minus-Eingang auf Masse liegt. Ein Teil des Ausgangssignals wird zur Rückkopplung auf den Minus-Eingang zurückgeführt. Die Grundschaltung ist in **Abbildung 12** gezeigt: 
-
-<img src="../figures/OPV_Grundschaltung1.png" width="400" style="zoom:100%;"/>
-
-**Abbildung 13**: (Grundschaltung des nicht-invertierenden Verstärkers)
-
----
-
-Die Verstärkung ergibt sich aus der ersten goldenen Regel ($U_{d}=0$), wonach am Minus-Eingang des OPV ebenfalls $U_{P}$ anliegt. Nach den [Kirchhoffschen Regeln](https://de.wikipedia.org/wiki/Kirchhoffsche_Regeln) ergibt sich für die Spannungsverstärkung: 
-$$
-\begin{equation*}
-\begin{split}
-&U_{e} = I\, R_{1};\qquad U_{a} = I\, (R_{1} + R_{2})\\
-&\\
-&v_{U} = \frac{U_{a}}{U_{e}} = \frac{R_{1}+R_{2}}{R_{1}} = 1+\frac{R_{2}}{R_{1}}
-\end{split}
-\end{equation*}
-$$
-Die Eingangsimpedanz $X_{e}$ der Schaltung lässt sich mit einer Erweiterung der Schaltung, wie in **Abbildung 14** gezeigt bestimmen: 
-
-<img src="../figures/OPV_Grundschaltung1_XE.png" width="1000" style="zoom:100%;"/>
-
-**Abbildung 14**: ((Rechts) Schaltung zur Bestimmung der Eingangsimpedanz $X_{e}$ des OPV mit Ersatzschaltbild (links))
-
----
-
-Vor den Plus-Eingang des OPV wird ein bekannter Messwiderstand $R_{M}$ geschaltet, über den die abfallende Spannung $U_{M}$ gemessen wird. Aus der Messung von $U_{P}$ und $U_{M}$, sowie aus der Kenntnis von $R_{M}$ lässt sich $X_{e}$ wie folgt bestimmen:
-$$
-\begin{equation*}
-\begin{split}
-&U_{e} = I\, (R_{M}+X_{e});\qquad U_{M}=I\, R_{M}\\
-&\\
-&U_{e} = U_{M}\left(1+\frac{X_{e}}{R_{M}}\right);\\
-&\\
-&X_{e} = R_{M}\left(\frac{U_{e}}{U_{M}}-1\right).
-\end{split}
-\end{equation*}
-$$
-Die Beschaltung innerhalb des blau gestrichelten Kastens wird im Ersatzschaltbild durch $X_{e}$ ersetzt. 
-
-Die Bestimmung der Ausgangsimpedanz $X_{a}$ ist weniger offensichtlich, da der OPV, als quasi ideale Stromquelle, das Ausgangssignal nachregelt wobei $U_{a}$ von der jeweiligen Last abhängt. Eine immer noch relativ einfache Methode, $X_{a}$ zu bestimmen ist in **Abbildung 15** gezeigt: 
-
-<img src="../figures/OPV_Grundschaltung1_XA.png" width="1000" style="zoom:100%;"/>
-
-**Abbildung 14**: ((Links) Schaltung zur Bestimmung der Ausgangsimpedanz $X_{a}$ des OPV mit Ersatzschaltbild (rechts))
-
----
-
-Es ist davon auszugehen, dass $X_{a}$ klein ist. Schließt man ein Potentiometer mit regelbarem Widerstand $R_{M}$ zum Ausgang des OPV parallel, ist davon auszugehen, dass $U_{a}$ vor allem über $X_{a}$ abfällt:
-$$
-\begin{equation*}
-U_{a} = \left(\frac{1}{X_{a}}+\frac{1}{R_{M}}\right)^{-1}\,I\approx X_{a}\,I;\qquad \text{mit: }R_{M}\gg X_{a}.
-\end{equation*}
-$$
-Regelt man $R_{M}$ kontinuierlich nach unten bis $R_{M}\lesssim X_{a}$ fällt $U_{a}$ zunehmend über $R_{M}$ ab. An dem Punkt, an dem $U_{a}$ auf die Hälfte des ursprünglichen Wertes abgefallen ist gilt:
-$$
-\begin{equation*}
-\begin{split}
-&\frac{U_{a}}{2} = \left(\frac{1}{X_{a}}+\frac{1}{R_{M}}\right)^{-1}\,I; \\
-&\\
-&\text{mit:}\\
-&\\
-& \vphantom{\left(\frac{1}{X_{a}}+\frac{1}{R_{M}}\right)^{-1}}
-I=\frac{U_{a}}{X_{a}} \\
-&\\
-&\frac{1}{2} = \frac{R_{M}}{R_{M}+X_{a}}; \\
-&\\
-&\vphantom{\left(\frac{1}{X_{A}}+\frac{1}{R_{M}}\right)^{-1}}
-X_{a}=R_{M}\\
-\end{split}
-\end{equation*}
-$$
-Der Strom $I$ bleibt trotz der variierenden Belastung am OPV-Ausgang gleich, weil der OPV sich näherungsweise, wie eine ideale Stromquelle verhält. 
-
-Die Beschaltung innerhalb des blau gestrichelten Kastens wird im Ersatzschaltbild durch $X_{a}$ ersetzt. 
-
-### Invertierender Verstärker
-
-Die Grundschaltung des invertierenden Verstärkers ist in **Abbildung 16** gezeigt: 
-
-<img src="../figures/OPV_Grundschaltung2.png" width="450" style="zoom:100%;"/>
-
-**Abbildung 16**: (Grundschaltung des invertierenden Verstärkers)
-
----
-
-Das Eingangssignal liegt in diesem Fall auf dem Minus-Eingang des OPV, während der Plus-Eingang auf Masse liegt. Das Ausgangssignal $U_{a}$ wird zur Rückkopplung teilweise ebenfalls auf den invertierenden Eingang zurückgeführt. 
-
-Zur Berechnung der Verstärkung verwenden wir wieder die erste goldene Regel ($U_{d}=0$), wonach am Minus-Eingang des OPV die gleiche Spannung anliegt, wie am Plus-Eingang. Da der Plus-Eingang jedoch auf Masse liegt muss dies auch für den Minus-Eingang gelten. Man spricht in diesem Fall von **virtueller Masse** am Minus-Eingang. Nach den [Kirchhoffschen Regeln](https://de.wikipedia.org/wiki/Kirchhoffsche_Regeln) muss $U_{e}$ vollständig über $R_{1}$ abfallen, das gleiche gilt für $U_{a}$ und $R_{2}$:
-$$
-\begin{equation}
-\begin{split}
-&U_{e} = I\,R_{1},\qquad U_{a} = -I\,R_{2};\\
-&\\
-&U_{a} = -\frac{R_{2}}{R_{1}}\,U_{e}.\\
-&\\
-&v_{U} = \frac{U_{a}}{U_{e}} = -\frac{R_{2}}{R_{1}}
-\end{split}
-\end{equation}
-$$
-Das Minuszeichen in Gleichung **(1)** folgt aus der entgegengesetzten Stromrichtung vom jeweils signalführenden, nicht geerdeten Pol auf Masse. Durch das Vorzeichen wird die Invertierung des Eingangssignals, als Phasenverschiebung um $\pi$ ($e^{i\pi}=-1$) explizit sichtbar. 
-
-Der invertierende Verstärker ist weiter verbreitet, als der nicht-invertierende Verstärker. Die Verstärkung der Schaltung ist intuitiver zu berechnen und die Schaltung lässt sich intuitiver erweitern. Im Rahmen von **Aufgabe 3** werden Sie einige einfache Erweiterungen der Grundschaltung untersuchen, die im folgenden kurz eingeführt werden.
+## Essentials
 
-#### Addierer   
+Was Sie ab jetzt wissen sollten:
 
-Das Schaltbild des Addierers ist in **Abbildung 17** gezeigt:
+- Operationsverstärker haben einen **invertierenden und einen nicht-invertierenden Eingang**. Im allgemeinen werden Sie zur **Spannungsverstärkung** eingesetzt. 
+- Ein OPV zeichnet sich durch eine große **Leerlaufverstärkung ($v_{0}$), geringe Gleichtaktverstärkung ($v_{\mathrm{Gl}}$), große Eingangs- ($X_{e}$) und kleine Ausgangsimpedanz ($X_{a}$)** aus. 
+- Für die äußere Beschaltung mit Widerstanden (Dimensionierung) gelten zur groben Abschätzung die **golgenen Regeln**. Diese sollten Sie benennen können. 
 
-<img src="../figures/OPV_Grundschaltung2_add.png" width="450" style="zoom:100%;"/>
+## Testfragen
 
-**Abbildung 17**: (Schaltbild des Addierers)
-
----
-
-Die Logik folgt derjenigen zur Berechnung von $v_{U}$ für die Grundschaltung. Im Eingangsschaltkreis addieren sich alle Teilströme zum Gesamtstrom
-$$
-\begin{equation*}
-I = \sum\limits_{i=1}^{n}I_{i};\qquad \text{mit: } I_{i}=\frac{U_{ei}}{R_{ei}}.
-\end{equation*}
-$$
- Für $U_{a}$ gilt analog zu Gleichung **(1)**: 
-$$
-\begin{equation*}
-\begin{split}
-&U_{a} = -I\,R_{2}= -\left(\sum\limits_{i=1}^{n}{\frac{U_{ei}}{R_{ei}}}\right)\,R_{2},
-\end{split}
-\end{equation*}
-$$
-d.h. $U_{a}$ ist die (negative) gewichtete Summe der $U_{ei}$, mit den Gewichten $R_{2}/R_{ei}$. Belegt man alle Widerstände mit den gleichen Werten ist
-$$
-\begin{equation*}
-U_{\mathrm{a}}=-\sum\limits_{i=1}^{n}U_{ei}.
-\end{equation*}
-$$
-
-#### Integrierer
-
-Das Schaltbild des Integrierers ist in **Abbildung 18** gezeigt:
-
-<img src="../figures/OPV_Grundschaltung2_int.png" width="450" style="zoom:100%;"/>
-
-**Abbildung 18**: (Schaltbild des Integrierers)
-
----
-
-Der Widerstand $R_{2}$ aus der Grundschaltung wird in diesem Fall durch den Kondensator $C$ ersetzt. Für das Ausgangssignal gilt: 
-$$
-\begin{equation}
-\begin{split}
-& U_{e} = I\,R_{1};\qquad U_{a} = \frac{Q}{C} = -\frac{1}{C}\int I\,\mathrm{d}t = -\frac{1}{C\,R_{1}}\int U_{e}\,\mathrm{d}t,
-\end{split}
-\end{equation}
-$$
-das Ausgangssignal entspricht also dem (negativen) Integral des Eingangssignals. 
-
-Der Widerstand $R_{S}$ in **Abbildung 18** wird in der Schaltung eingeführt, um zu verhindern, dass der Kondensator durch ein Gleichspannungssignal aufgeladen wird. Wählt man $R_{S}$ geeignet groß, kann er (den [Kirchhoffschen Regeln](https://de.wikipedia.org/wiki/Kirchhoffsche_Regeln) entsprechend) in den Betrachtungen aus Gleichung **(2)** vernachlässigt werden. 
-
-#### Differenzierer
-
-Das Schaltbild des Differenzierers ist in **Abbildung 19** gezeigt:
-
-<img src="../figures/OPV_Grundschaltung2_diff.png" width="450" style="zoom:100%;"/>
-
-**Abbildung 19**: (Schaltbild des Differenzierers)
-
----
-
-Im Vergleich zum Integrierer werden Widerstand und Kondensator vertauscht. Für das Ausgangssignal gilt: 
-$$
-\begin{equation*}
-\begin{split}
-U_{e} = \frac{Q}{C};\qquad U_{a} = R_{2}\,I = -R_{2}\,\frac{\mathrm{d}Q}{\mathrm{d}t} = -&\underbrace{C\,R_{2}\vphantom{\frac{\mathrm{d}U_{e}}{\mathrm{d}t}}}\frac{\mathrm{d}U_{e}}{\mathrm{d}t},\\
-&\equiv\tau
-\end{split}
-\end{equation*}
-$$
-das Ausgangssignal entspricht also der (negativen) Ableitung des Eingangssignals. Das Produkt $\tau=C\ R_{2}$ wird als Zeitkonstante bezeichnet.
-
-### Komplexere Schaltungen mit Operationsverstärkern
-
-In **Aufgabe 4** werden Sie drei komplexere Schaltungen mit OPVs aufbauen:
-
-- Einen idealen Einweggleichrichter;
-- einen Generator für Drei- und Rechtecksignale;
-- ein Beispiel für die analoge Lösung einer Differenzialgleichung 2. Ordnung.
-
-Alle drei Schaltungen werden wir im folgenden kurz erklären.
-
-#### Idealer Einweggleichrichter
-
-Ein Einweggleichrichter lässt aus einem bipolaren Signal immer nur einen unipolaren Anteil durch. Sie kennen eine solche Schaltung womöglich bereits aus dem P1 Versuch [Oszilloskop](https://gitlab.kit.edu/kit/etp-lehre/p1-praktikum/students/-/tree/main/Oszilloskop). Die einfachste Realisation mit Hilfe einer Diode mit Durchlass für die positive Halbwelle eines sinusförmigen Eingangssignals ist in **Abbildung 20** gezeigt: 
-
-<img src="../figures/Einweggleichrichter_Diode.png" width="600" style="zoom:100%;"/>
-
-**Abbildung 20**: (Schaltbild eines einfachen Einweggleichrichters mit Hilfe einer Diode mit Durchlass für ein positives Signal)
-
----
-
-Die Eingangsspannung $U_{e}$ fällt über die Diode $D$ und den Widerstand $R$ ab. Das gleichgerichtete Signal kann als über $R$ abfallende Spannung $U_{a}$ abgegriffen werden. Ein Nachteil dieser Schaltung besteht darin, dass nicht die komplette Halbwelle des Signals wiedergegeben wird, da an der Diode immer zusätzlich die Diffusionsspannung ($U_{D}\approx -0.7\ \mathrm{V}$ im Fall einer Siliziumdiode) abfällt. 
-
-Die Schaltung eines idealen Einweggleichrichters die sich mit Hilfe eines OPV realisieren lässt und die gesamte Halbwelle wiedergibt ist in **Abbildung 21** gezeigt: 
-
-<img src="../figures/Einweggleichrichter_OPV.png" width="600" style="zoom:100%;"/>
-
-**Abbildung 21**: (Schaltbild eines idealen Gleichwegrichters mit Hilfe eines OPV)
-
----
-
-Die Schaltung enthält zwei Gegenkopplungszweige, von denen je nach Vorzeichen von $U_{e}$ immer nur einer stromführend aktiv ist. 
-
-- Für $U_{e}>0$ sind $D_{-}$ und $R_{-}$ stromführend; 
-- für $U_{e}<0$ sind es $𝑅_{+}$ und $𝐷_{+}$. 
-
-Am Ausgang des OPV steigt $U_{a}$ solange an, bis $R_{+}$ bzw. $R_{-}$ gerade den Eingangsstrom führt, was eine Überhöhung von $U_{a}$ um $2\ U_{D}$ bewirkt. Da aber an den Dioden immer genau diese Überhöhung abfällt, lässt sich als $U_{a}^{(-)}$ ($U_{a}^{(+)}$) die komplette negative (positive) Halbwelle abgreifen.
-
-#### Generator für Drei- und Rechtecksignale
-
-Das Schaltbild eines Drei- und Rechteckgenerators ist in **Abbildung 22** gezeigt:
-
-<img src="../figures/OPV_Generator.png" width="600" style="zoom:100%;"/>
-
-**Abbildung 22**: (Schaltbild eines Drei- und Rechteckgenerators mit Hilfe von OPVs)
-
----
-
-Es handelt sich dabei um eine selbsterregende Schaltung, an der kein explizites Eingangssignal anliegt; das Ausgangssignal wird allein aus den anliegenden äußeren Betriebsspannungen der OPVs abgeleitet. Dabei entstehen periodische
-Ausgangssignale obwohl an den OPVs nur Gleichspannungen anliegen. In der Schaltung fungiert der linke OPV als Schwellenwertschalter ([Schmitt-Trigger](https://de.wikipedia.org/wiki/Schmitt-Trigger)) mit der Referenzspannung 0, bei dem das Ausgangssignal $U_{a}^{\Box}$ über $R_{a}^{\Box}$ teilweise auf den Plus-Eingang $E$ zurückgeführt wird: Liegt an $E$ ein positives (negatives) Signal an gibt der OPV (im Sättigungsbetrieb) die positive (negative) Betriebsspannung aus. Der OPV rechts im Bild fungiert als Integrierer. 
-
-Zur weiteren Klärung der Vorgänge gehen wir von einer positiven Betriebsspannung am Ausgang des Schmitt-Triggers ($U_{a}^{\Box}>0$) aus, die über $R_{1}$ auf den Minus-Eingang des Integrierers geführt wird. Das Ausgangssignal des Integrierers ist negativ ($U_{a}^{\Delta}<0$) und wird über $R_{a}^{\Delta}$ ebenfalls auf $E$ zurückgeführt. 
-
-Zunächst wirkt auf $E$ vor allem der (positive) Signalanteil aus $U_{a}^{\Box}$. Dieser Zustand besteht solange, bis $C$ hinreichend aufgeladen ist, sodass das negative Signal aus $U_{a}^{\Delta}$ überwiegt. Von diesem Zeitpunkt an wird $U_{a}^{\Box}$ negativ, am Integrierer liegt ein negatives Eingangssignal an, $U_{a}^{\Delta}$ wird positiv. Der Kondensator $C$ wird positiv geladen und sobald der nun positive Signalanteil aus $U_{a}^{\Delta}$ den negativen Signalanteil aus $U_{a}^{\Box}$ an $E$ erneut überwiegt kehrt die Schaltung in ihren ursprünglichen Zustand zurück. 
-
-An den eingezeichneten Klemmen lassen sich $U_{a}^{\Delta}$ als periodisches Drei- und $U_{a}^{\Box}$ periodisches Rechtecksignal abgreifen. Für den Versuch gehen wir von der folgenden Belegung der Widerstände und des Kondensators aus: 
-$$
-\begin{equation*}
-R_{a}^{\Box}=10\ \mathrm{k\Omega}; \quad
-R_{a}^{\Delta}=5.6\ \mathrm{k\Omega}; \quad
-R_{1} =100\ \mathrm{k\Omega}; \quad
-C  =1\ \mathrm{nF}.
-\end{equation*}
-$$
-
-#### Analoge Lösung einer Differentialgleichung 2. Ordnung
-
-Eine homogene Differentialgleichung 2. Ordnung hat die Form: 
-$$
-\begin{equation}
-\ddot{U}(t) + 2\gamma\, \dot{U}(t) + \omega_{0}^{2}U(t) = 0.
-\end{equation}
-$$
-Die Lösung $U(t)$ einer solchen Differentialgleichung lässt sich mit Hilfe von zwei Integrierern und einem weiteren invertierenden Verstärker, wie in **Abbildung 23** gezeigt, auf analoge Weise ermitteln: 
-
-<img src="../figures/OPV_Diffgleichung.png" width="600" style="zoom:100%;"/>
-
-**Abbildung 23**: (Schaltbild zur Darstellung der Lösung einer homogenen Differenzialgleichung 2. Ordnung mit Hilfe von OPVs)
-
----
-
-In der Schaltung wird $U_{a}$ auf den Minus-Eingang des ersten Integrierers zurück gekoppelt. Nach Gleichung **(2)** ist das Ergebnis 
-$$
-\begin{equation*}
-\begin{split}
-& U_{a}^{(1)} = -\frac{1}{C_{1}\,R_{1}}\int U_{a}\,\mathrm{d}t.
-\end{split}
-\end{equation*}
-$$
- Der Ausgang des ersten liegt auf dem Minus-Eingang des zweiten Integrierers
-$$
-\begin{equation*}
-\begin{split}
-U_{a}^{(2)} &= -\frac{1}{C_{2}\,R_{2}}\int U_{a}^{(1)}\,\mathrm{d}t = \frac{1}{C_{2}\,R_{2}}\frac{1}{C_{1}\,R_{1}}\iint U_{a}\,\mathrm{d}t. \\
-\end{split}
-\end{equation*}
-$$
-Auf der rechten Seite des Netzwerks addieren sich die Spannungen $U_{a}$, $\hat{U}_{a}^{(1)}$ und $\hat{U}_{a}^{(2)}$, wobei der Symbolzusatz $\hat{\cdot}$ kennzeichnet, dass es sich um die Anteile der Spannung nach Abzug der über die Widerstände $R_{2,5,\mathrm{pot}}$, bzw. $R_{3,4}$ abgefallenen Spannungen $U_{a}^{(1)}$ und $U_{a}^{(2)}$ handelt. Das Symbol $R_{\mathrm{pot}}$​ steht für ein regelbares Potentiometer. Für den Versuch gehen wir von der folgenden Belegung der Widerstände und Kondensatoren aus: 
-$$
-\begin{equation*}
-R_{1}=R_{2}=10\ \mathrm{k\Omega}; \quad
-R_{3}=R_{4}=5.6\ \mathrm{k\Omega}; \quad
-R_{5} =1\ \mathrm{M\Omega}; \quad
-R_{\mathrm{pot}} =10\ \mathrm{k\Omega,\ (regelbar)}; \quad
-C_{1}=C_{2}=470\ \mathrm{nF}.
-\end{equation*}
-$$
-Aus der Belegung des Netzwerks lässt sich mit Hilfe der [Kirchhoffschen Regeln](https://de.wikipedia.org/wiki/Kirchhoffsche_Regeln) die Beziehung zwischen den Widerständen, Kapazitäten und Vorfaktoren $\gamma$ und $\omega_{0}$ aus Gleichung **(2)** bestimmen. Für die Durchführung des Versuchs ist dies nicht notwendig. 
-
-Dadurch, dass $R_{\mathrm{pot}}$ regelbar ist können Sie $\gamma$ für den Versuch variieren und den Schwing-, Kriech- und aperiodischen Grenzfall der gedämpften Schwingung experimentell einstellen.  
+1. Was macht eine Ideale Spannungsquelle aus?
+2. Was sagt die dritte Goldene Regel zur Dimenionsierung von OPVs über $X_{a}$ aus?
 
 # Navigation
 
 [Main](https://gitlab.kit.edu/kit/etp-lehre/p2-praktikum/students/-/tree/main/Operationsverstaerker)
+
diff --git a/Operationsverstaerker/doc/Hinweise-Verstaerker.md b/Operationsverstaerker/doc/Hinweise-Verstaerker.md
index 9bb37b6..50ff50d 100644
--- a/Operationsverstaerker/doc/Hinweise-Verstaerker.md
+++ b/Operationsverstaerker/doc/Hinweise-Verstaerker.md
@@ -1,4 +1,4 @@
-# Hinweise für den Versuch Operationsverstärker
+# Hinweise für den Versuch **Operationsverstärker**
 
 ## Verstärker
 
diff --git a/Operationsverstaerker/figures/Einweggleichrichter_Diode.odg b/Operationsverstaerker/figures/Einweggleichrichter_Diode.odg
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HcmV?d00001

-- 
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