\documentclass[pdftex,12pt,a4paper]{article}
\usepackage[pdftex]{graphicx}
\newcommand{\HRule}{\rule{\linewidth}{0.5 mm}}
\newcommand{\unit}[1]{\overline{\text{#1}}}
\usepackage[english]{babel}
\usepackage[europeanresistors]{circuitikz}
\usepackage{booktabs}
\usepackage{tikz}
%\usetikzlibrary{pgfplots.statistics}
\usetikzlibrary{chains}
\usetikzlibrary{shapes,arrows}
\usepackage{tensor}
\usepackage{float}
\usepackage{makecell}
\restylefloat{table}
\restylefloat{figure}
\usepackage[utf8]{inputenc}
\usepackage{fullpage}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{gensymb}
% \usepackage{slashbox}
\usepackage{amsmath}
\usepackage{float}
\usepackage{geometry}
\usepackage{layout}
\usepackage{caption}
\usepackage{rotating}
\usepackage{subcaption}
\usepackage{units} 
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{nicefrac}
\usepackage{textcomp}
\usepackage[version=3]{mhchem}
\usepackage{placeins}
\usepackage{listings}
\usepackage[absolute]{textpos}
\usepackage{graphicx}
\usepackage{epstopdf}

\usepackage{layout}
\usepackage{indentfirst}
\textheight = 710pt
\textwidth = 425pt
\oddsidemargin = 14pt
\topmargin = 7pt

\DeclareUnicodeCharacter{03A9}{\Omega}
\DeclareUnicodeCharacter{27E8}{\langle}
\DeclareUnicodeCharacter{27E9}{\rangle}
\DeclareUnicodeCharacter{394}{\Delta}
\DeclareUnicodeCharacter{B1}{\pm}
\DeclareUnicodeCharacter{222B}{\int}
\DeclareUnicodeCharacter{3B2}{\beta}

\newcommand*{\plogo}{\fbox{$\mathcal{PL}$}} % Generic publisher logo

%----------------------------------------------------------------------------------------
%	TITLE PAGE
%----------------------------------------------------------------------------------------

\newcommand*{\titleGP}{\begingroup % Create the command for including the title page in the document
\centering % Center all text
\vspace*{\baselineskip} % White space at the top of the page

\rule{\textwidth}{1.6pt}\vspace*{-\baselineskip}\vspace*{2pt} % Thick horizontal line
\rule{\textwidth}{0.4pt}\\[\baselineskip] % Thin horizontal line

{\LARGE \textbf{Protokoll} \\ \vspace{0.2 cm}
 Praktikum Quantenphysik}\\[0.2\baselineskip] % Title

\rule{\textwidth}{0.4pt}\vspace*{-\baselineskip}\vspace{3.2pt} % Thin horizontal line
\rule{\textwidth}{1.6pt}\\[\baselineskip] % Thick horizontal line

\scshape % Small caps
{\Large Noise Fundamentals}  \\ % Tagline(s) or further description
[\baselineskip] % Tagline(s) or further description
\large 9. Jan 2015\par % Location and year

\vspace*{2\baselineskip} % Whitespace between location/year and editors

{\small Gruppe von:} \\[\baselineskip]
{\Large Danny Milosavljevic \small (Verfasser) \par}
{\Large Daryoush Nosraty Alamdary \par}
 % Editor list
\vspace*{0.5 cm}
{\itshape{\large  Universität Wien \\ Fakultät für Physik}\par} % Editor affiliation

\vfill % Whitespace between editor names and publisher logo

{\scshape 2015} \\[0.3\baselineskip] % Year published
{\large Wien, Österreich}\par % Publisher

\endgroup}

\begin{document}
\titleGP
\newpage
\tableofcontents
\newpage

\section{Introduction}
We determine the Boltzmann constant by examining Johnson noise.
\subsection{Theory}

\subsubsection{Johnson Noise}

All circuits exhibit noise. Different sources of noise exist:
\begin{itemize}
\item Interference: Presence of an (unwanted) signal which is added to the desired signal.
\item Technical Noise: Noise that exists only because of faults in our measurement apparatus.
\item Fundamental Noise: Noise that is intrinsic to our measurement apparatus and cannot be removed.
\end{itemize}

We are interested in measuring fundamental noise, more specifically in Johnson Noise. Johnson Noise was first theoretically described by Nyquist.
Johnson Noise is white noise similar to blackbody radiation, just limited to one spatial dimension, along a transmission line.\par
The available spectral power density of Johnson Noise between two resistors separated by a transmission line to a first approximation is $ P_f $ as follows (similar to Rayleigh-Jeans - modes' boundary conditions force that an integer multiple of half the mode's wavelength fits on the transmission line)\cite{Nyquist1928}:
\begin{eqnarray*}
P_f = k_B T \\
P_J := ∫df k_B T = k_B T (f_2 - f_1) = k_B T Δf
\end{eqnarray*}
A better analysis takes into account that an ultraviolet catastrophe does not happen (but would have to in this description). Similar to Planck one gets:\par
\begin{eqnarray*}
β := \frac{1}{k_B T} \\
P_f = k_B T \frac{β h f}{\exp(β h f) - 1} \\
P_f = h f \frac{1}{\exp(β h f) - 1}
\end{eqnarray*}
The limiting case is contained if $ β h f << 1 $ (Taylor series of the exponential):
\begin{eqnarray*}
P_f \approx h f \frac{1}{1 + β h f - 1} \\
P_f \approx h f \frac{1}{β h f} \\
P_f \approx h f \frac{k_B T}{h f} \\
P_f \approx k_B T \\
P_J \approx k_B T Δf
\end{eqnarray*}
This power is available (like a voltage source in series) to the noisy resistor and the measurement apparatus. 
% The measurement apparatus tries to maximize power transfer and so its equivalent resistance will be equal.
\begin{eqnarray*}
P_J = U\indices{_0^2}/R = U\indices{_0^2}/R
\end{eqnarray*}
Let $U_J$ be the voltage across our noisy resistor, $R$ be the resistance ($ \pm 0.1 \% R $) of it, $k_B$ be the Boltzmann constant, $T$ be the absolute temperature ($ \pm 5 K $).\par
For an open load on the transmission line, the available voltage is twice $U_0$:
\begin{eqnarray*}
U_J = 2 U_0 \\
U\indices{_J^2} = 4 U\indices{_0^2} \\
⟨U\indices{_J^2}/R⟩ = 4 P_J = 4 k_B T Δf \\
⟨U\indices{_J^2}⟩ = 4 k_B R T Δf
\end{eqnarray*}
We thus arrived at Johnson Noise:\par
$ ⟨U\indices{_J^2}⟩  = 4 k_B R T \Delta f $ (Nyquist's formula)\cite{Baak2010}\par
$ ⟨U_J⟩  = 0 $\par

\subsubsection{Amplifier Noise}
Unfortunately, our measurement also contains amplifier noise which we eventually have to subtract.\par
Amplifier noise:\par
Assumed to be independent of R.\par
Assume: $ ⟨U_N⟩  = 0 $\par

Total noise (by assumption):\par
$ ⟨U^2⟩ := ⟨U\indices{_J^2}⟩ + ⟨U\indices{_N^2}⟩ $\par
$ ⟨U⟩ := 0 $\par

\subsubsection{Effective Bandwidth}
The filters used do not cut off at exactly the corner frequencies but rather weaken the signal steadily starting from there going outside. In order to be able to use the Nyquist formula one has to first find an equivalent "sharp" filter which would pass the same noise power as the real filter and use its corner frequencies to calculate $Δf$.
\input{Effective_bw.inc}
Uncertainities in $Δf$ are ($\pm 4\%$), except for the rightmost column where they are ($\pm 8\%$) and for the next-to-rightmost column where they are ($\pm 6\%$).\par
Source: \cite[Section 1.5]{Baak2010}.\par

\subsection{Experimental Setup}
\subsubsection{Connections}
\tikzstyle{box} = [rectangle, draw, fill=blue!20, text height = 20mm]
\begin{tikzpicture}[every join/.style={ultra thick}, node distance = 40mm]
\node [box,label=Preamp, text width = 20mm] (preamp) { Gain $ G_1 = 600 $ };
\node [box, below = 5mm of preamp] (resistor) { $R$ };
\node [box, right = 5mm of preamp, label=High pass filter, text width = 20mm] (highpass) { $ f_1 $ };
\node [box, right = 5mm of highpass, label=Low pass filter, text width = 20mm] (lowpass) { $ f_2 $ };
\node [box, right = 5mm of lowpass, label=Amp, text width = 20mm] (amp) { Gain $ G_2 $, linear in range (-10 V, 10 V) };
\node [box, right = 5mm of amp, label=Multiplier, text width = 20mm] (multiplier) { square and time-average over 1s };
\node [box, right = 5mm of multiplier, label=Multimeter, text width = 20mm] (dmm) { };
\node [box, above = 10mm of multiplier, label=Oscilloscope, text width = 20mm] (osci) { AC };
\draw[-] (resistor.north) to (preamp.south);
\draw[-] (preamp.east) to (highpass.west);
\draw[-] (highpass.east) to (lowpass.west);
\draw[-] (lowpass.east) to (amp.west);
\draw[-] (amp.east) to (multiplier.west);
\draw[-] (multiplier.east) to (dmm.west);
\draw[-] (amp.east) to (osci.west);
\end{tikzpicture}

A (changeable) resistor $R$ (with tolerance ($\pm 1\% R$)) is connected to a preamplifier (with gain $G_1 = 600$) which is connected to filters which is connected to the main amplifier (gain $G_2$) which is connected to a multiplier (which is used for squaring and time-averaging the result over $1 \unit{s}$) which is then connected to a digital multimeter.
After the resistor is set up, the noise in it is then measured using the digital multimeter.
We use AC coupling in order to minimize error (influence of DC offset).

\subsubsection{Influence of Gain on Measured Values}
Let $G_1$ be the preamplifier gain, $G_2$ be the amplifier gain (see table below), $U_{met}$ be the measured (amplified) noise in:\par
\begin{eqnarray*}
G_1 := 600 \\
G := \frac{G_1 G_2}{\sqrt{10 V}} \\
U_{met} := G^2 ⟨U^2⟩ \\
U_{met0} := G^2 ⟨U\indices{_N^2}⟩ \\
ΔU_{met} := U_{met} - U_{met0}
\end{eqnarray*}

\subsubsection{Determination of Boltzmann's Constant}
We are interested in $k_B$, so first rearrange to get access to $⟨U^2⟩$, then substitute in order to get $⟨U\indices{_J^2}⟩$, then substitute in order to get $k_B$.\par
\begin{eqnarray*}
\frac{U_{met}}{G^2} = ⟨U^2⟩ \\
\frac{U_{met}}{G^2} = ⟨U\indices{_J^2}⟩ + ⟨U\indices{_N^2}⟩ \\
\frac{U_{met}}{G^2} - ⟨U\indices{_N^2}⟩ = ⟨U\indices{_J^2}⟩ \\
\frac{1}{G^2}\left(U_{met} - U_{met0}\right) = ⟨U\indices{_J^2}⟩ \\
\frac{1}{4 R T G^2 \Delta f}\left(U_{met} - U_{met0}\right) = k_B
\end{eqnarray*}

\section{Results}
\subsection{Measurements and Analysis}

\subsubsection{Correcting for Amp Noise}
First we measured just the amp noise (equal in first approximation to $U_{met} = U_{met0}$ measured on a resistor with $R = 1 Ω$).\par
\input{Amp_Noise.inc}
Compare \cite[Section 1.3]{Baak2010}.\par

\subsubsection{Johnson Noise Dependence on Bandwidth}
Then we measured how the (total) noise depends on bandwidth $(f_2 - f_1)$.\par
\input{Noises_bw.inc}
Compare \cite[Section 1.5]{Baak2010}.\par

Then we calculated the Johnson Noise component (by interval arithmetic) and finally the Boltzmann constant $k_B$ from all this:

\input{1.5.inc}

We calculated this overall $k_B$ by using interval intersection.\par

\subsubsection{Johnson Noise Dependence on Resistance}
Then we measured how the Johnson noise depends on resistance.\par
Selected bandwidth was ($f_1 = 100 Hz, f_2 = 100 kHz$) and main amplifier gain was $G_2 = 1500$. Therefore, effective bandwidth $Δf = 110961 Hz$ and $U_{met0} = 0.265 V$ for $G_2 = 1000$ due to amp noise (see table above).\par
\begin{eqnarray*}
\frac{10 V U_{met}}{G\indices{_1^2} (1500 Hz)^2} = ⟨U\indices{_J^2}⟩ + ⟨U\indices{_N^2}⟩ \\
\frac{10 V U_{met0}}{G\indices{_1^2} (1000 Hz)^2} = ⟨U\indices{_N^2}⟩ \\
\frac{10 V}{G\indices{_1^2}}\left(\frac{U_{met}}{(1500 Hz)^2} - \frac{U_{met0}}{(1000 Hz)^2}\right) = ⟨U\indices{_J^2}⟩ \\
\frac{10 V}{G\indices{_1^2}}\left(\frac{U_{met}}{(1500 Hz)^2} - \frac{0.265 V}{(1000 Hz)^2}\right) = ⟨U\indices{_J^2}⟩ \\
\end{eqnarray*}
\input{1.4.inc}
%\begin{figure}[H]
We verified the linear dependence on R in our (small) sample (C is an arbitrary constant):\par
\input{1_4}
%\includegraphics[scale=1.2]{{1.4}.eps}
%\caption{$⟨U\indices{_J^2}⟩$ depending on $R$}
%\end{figure}

Compare \cite[Section 1.4]{Baak2010}.\par

\subsubsection{Two-temperature Johnson Noise Measurement}
We selected the following settings:\\
\begin{eqnarray*}
f_1 := 100 \unit{Hz} \\
f_2 := 3.3 \unit{kHz} \\
\Delta{f} = 3554 \unit{Hz}
\end{eqnarray*}

Then we measured Johnson noise once at room temperature ($T = 295 K$) 
and once at $T = 77 K$.\par
Compare \cite[Section 2.2]{Baak2010}.\par

First we measured the resistor at room temperature.\par\par
\begin{eqnarray*}
T := 295 \unit{K} \\
\end{eqnarray*}
\input{2.2a.inc}
Then we measured the resistor in thermal conduct with liquid nitrogen,
\begin{eqnarray*}
T := 77 \unit{K} \\
\end{eqnarray*}
\input{2.2b.inc}

We interpolate absolute zero temperature:\par
\input{2_2}\par

\subsection{Conclusion and Summary}
We conclude that the Boltzmann constant $k_B$ is:\par
\input{k.inc}\par
Also, with high enough resistances it can be seen that the origin of the Kelvin scale is well-chosen and that specifying the temperature in Kelvin is useful for blackbody radiation.
\begin{thebibliography}{9}
\bibitem{Baak2010} David Van Baak, Georg Herold, Jonathan Reichert, Johannes Majer, Stefan Putz, Christian Koller: Noise Fundamentals NF1-A
\bibitem{Nyquist1928} H. Nyquist: Thermal Agitation of Electric Charge in Conductors
\end{thebibliography}
\end{document}
