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(revised 1/25/07)
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\bf THERMAL NOISE\\[30pt]
\rm Advanced Laboratory, Physics 407,\\[10pt]
University of Wisconsin\\[10pt]
Madison, Wisconsin 53706
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\begin{abstract}
The aim of this experiment is to observe the thermal noise in a 
resistor, to verify that the mean
square noise voltage is proportional to the absolute temperature, 
and to obtain an an experimental value for the Boltzmann constant. 

\end{abstract}
\newpage
\newcommand{\ub}{\underline}
\subsection*{\ub{\bf{Theory}}}

J.B. Johnson discovered that any resistor exhibits a small random 
alternating e.m.f. (now called Johnson noise) and that the noise is 
dependent on the temperature. Nyquist assumed that the noise was 
due to the thermal agitation of the electrons in the resistor and, 
using thermodynamics, developed the expression$^1$
\begin{equation}
\bar V^2 = 4kT \int^{f_2}_{f_1} {\rm Real}\ [Z (f)] df
\end{equation}
where 
\begin{itemize}
\item
$\bar V^2$ is the mean square noise voltage. 
\item
$k$ is 
Boltzmann's constant
\item
$T$ is the absolute temperature
\item
$Z(f)$ is the impedance of the device (a resistor) at the 
frequency $f$ \item{}$f_1$ and $f_2$ are the frequency limits 
between which the noise is accepted by the measuring device.
\end{itemize}

If the device is a simple resistor and the frequency spectrum is flat,
this becomes: 
\begin{equation}
\bar V^2 = 4kTR 
(f_2 - f_1).
\end{equation}
In this experiment, we will not measure $\bar V^2$ directly but 
will use an amplifier. The amplifier output $V_1$ is related to the 
input $V$ by: 
$$V_1 = A(f)V$$
where $A(f)$ is the frequency dependent amplification. Hence: 
\begin{equation}
\bar 
V^2 = 4kTR \int^{f_2}_{f_1} A^2 (f) df.
\end{equation}

\newpage
\subsection*{\ub{\bf{Apparatus}}}
\subsubsection*{\ub{Resistor}}

The resistor assembly is mounted in a copper tube (for good thermal
conduction) on the end of a thin stainless steel tube (for lows 
thermal conduction). Two 500 k$\Omega$ metal film resistors in 
series are used and so R=1 M$\Omega$. The nominal precision of the 
resistors is 1\%. 
\subsubsection*{\ub{Amplifier} (Stanford Research Systems Model SR560)} 

This solid state amplifier has an input impedance of 100 M$\Omega$ 
shunted by 25 pF and can be run either DC or AC coupled. There is a 
wide range of gain and bandwidth settings. The noise figure for 
the amplifier is $\approx 4\ {\rm n}V/\sqrt{Hz}$ at 1 kHz. This is to be 
compared to the Johnson noise value of 129 nV/$\sqrt{Hz}$ at T = 
300$^{\circ}$ K for a 1 M$\Omega$ resistor. 

\subsubsection*{\ub{RMS Voltmeter} (Keithley Model 2100)} 

The Keithley 2100 is connected by USB to the PC.\\

\noindent A Visual Basic program SCHLUM.EXE (icon on desktop) is used to 
read the meter and record the readings. The program makes five measurements
when you ``Start Read'', computes the mean and standard deviation, and writes 
the result to a file. The temperature that you have entered in the 
temperature box also is written to the file. You can set the file name
to a different name if desired. When the file logging is set to ``Append''',
subsequent readings are appended to the file. Below is the Windows 
screen that appears when SCHLUM.EXE is opened. 
\newpage
\includegraphics[width=4.5in]{thermalfig1.pdf}

\subsubsection*{\ub{Oven} (Leybold 200 watt)}

The oven is operated from a variable autotransformer (General Radio 
``Variac") and is used to heat the resistor up to 150$^{\circ}$ 
Higher temperatures will damage the thermocouple.

\subsubsection*{\ub{Thermocouple}}

An iron-constantan thermocouple junction is mounted next to the 
resistor and is used to measure the temperature of the resistor. A 
Keithley Model 197 digital multi-meter is used to measure the 
thermocouple voltage. The Handbook of Chemistry and Physics 
contains thermocouple tables. 

\subsection*{\ub{\bf{Procedure}}}

The resistor output connection is the two BNC connectors on the 
resistor assembly box, corresponding to the two ends of the 
resistor. These outputs are connected to the A and B inputs of the 
preamplifier which is run in the A--B differential mode to 
eliminate common mode noise. The suggested bandwidth settings are 
100 to 1000 Hz, and the suggested gain is 1000. Use the preamp in 
the ``low noise'' setting. 
\begin{enumerate}
\item
Connect the resistor assembly to the preamp as described above. 
Look at the output of the preamp on the scope. The signal should be 
pure white noise with no extraneous 60 Hz pickup. The actual RMS 
voltage measurements are made using the Keithley 2100.
Calculate the thermal noise expected at the output of the 
preamplifier at room temperature. For a simple calculation assume 
that the amplifier frequency response is simply $A(f)=A$ for $f_ {low} \leq f \leq f_{ high}$, where $A$ is the amplifier gain and $f_{low}$ and 
$f_{high}$ are the the 6 db low and high roll-off frequencies 
respectively. Is the actual measured voltage reasonable? 

\item
Take data above room temperature using the variac controlled oven. 
Do not exceed 150$^{\circ}$ C. The experimental fluctuations in the 
measured voltage will give you guidance as to how to space the data 
in temperature.  Determine the slope and 
x-intercept of the V$^2$ vs. T curve. The x-intercept should be less than zero; its 
magnitude is called the ``noise temperature'' and is a measure of the noise contributions from other than 
thermal noise.  The slope is used to determine the Boltzmann 
constant.  With the Variac set at 120 V, it will take about 35 min 
to reach 150$^{\circ}$ C. One recommended way to take data is to
take data at about  20$^{\circ}$ intervals while the temperature is increasing. 
For each measurement switch the variac off (why?), but do not forget to 
turn it on again. Be sure not to move anything while making the
temperature measurements. 

\item
On the basis of the high temperature data, decide if the result is 
improved by taking data well below room temperature.
Liquid nitrogen can be used to cool the resistor assembly.
Currently we are not making cold measurements.
\item
There are several checks you should make. Measure the resistor value at the highest and lowest temperatures used during the 
experiment to check that the resistor maintains the same value, which should be 1 M$\Omega$ to 1\%.\\

Also, measure the amplifier noise directly by shorting the A and B 
inputs and reading the output voltage.  It should be 
significantly smaller than the thermal noise voltages you measured from the resistor.
Make appropriate unit conversions, and compare the amplifer noise you measured to the noise figure for the
amplifer given earlier.  Then, convert the amplifier noise voltage to a noise temperature, and compare to the
noise temperature you deduced from your thermal noise voltage measurements.  Are there other noise sources present in your set-up?

\item
For analysis, first estimate the bandwidth integral analytically 
or numerically, using exact analytic expressions for 6 db/octave RC 
roll-off filters. This procedure overestimates the bandwidth 
integral, since the system input capacitance 
reduces the bandwidth integral by lowering the  high frequency
roll-off. 

It is also possible to simulate the complete circuit with a SPICE circuit
simulation program. The frequency response obtained is shown below. 
\begin{center}
\includegraphics[width = 4.0in]{thermalfig2.pdf}
\end{center}

Measure the bandwidth directly using a Stanford Instruments Model
DS345 Function Generator. The output of the Function Generator should 
be connected through the small insertion box which contains a 1M$\Omega$
resistor to correctly simulate the output impedance of the thermal noise
circuit. Doing this will correctly account for the 
low-pass filter representing the front-end input impedance and capacitance.
After the bandwidth has been measured, use SigmaPlot to 
numerically calculate the bandwidth integral. 

Once you have obtained the bandwidth integral, k can be determined 
by using the measured value of the slope of the V$^2$ vs. T curve 
together with eqn.(3). Your final value should include a full 
error analysis.
\end{enumerate}
%\subsection*{\ub{References}}
\begin{thebibliography}{99}
\bibitem{ref1}
R.E. Simpson, ``Introductory Electronics for Scientists and 
Engineers", 2nd ed., (Allyn and Bacon, Inc., 1974), Sec.\ 8.3.
\bibitem{ref2}
P. Horowitz and W. Hill, ``The Art of Electronics, Second Edition'',
(Cambridge University Press, 1989), pp. 430f, 453.
\end{thebibliography}

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