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UNIVERSITY COLLEGE LONDON DEPARTMENT OF PHYSICS AND ASTRONOMY
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I have read and understood the UCL code of assessment including those
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By signing below I declare that the accompanying piece of assessed work,
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Title of work submitted : Thermal Noise in a Resistor Student’s signature Date 16/12/11
PHAS 2440 - 1 - Sherman Ip
Thermal Noise in a Resistor
Sherman Ip*
Department of Physics and Astronomy, University College London
Date submitted 16th December 2011
A resistor produce random voltage, even if no current is passing through it, called thermal noise as a result of random movements of electrons in the resistor. The voltage of thermal noise is too small to be measurable using a typical voltmeter so a low-noise amplifier was used to amplify to voltage of the thermal noise in order for it to be measurable and investigable. A circuit was built to measure the amplified thermal noise of a resistor to obtain evidence that the magnitude of the thermal noise is dependent on the temperature and resistance of the resistor. Boltzmann’s constant was also obtained by investigating the gain of the amplifier as a function of frequency.
PHAS 2440 - 2 - Sherman Ip
I. INTRODUCTION
All conductors will produce thermal
noise, as a result of random motion of
electrons in the conductor, even if no current
is passing through it. 1
Thermal noise can be
measured using a voltmeter however the
thermal noise is very small and in many
cases negligible.
The thermal noise associated with a
resistance is given in Eq. (1).123
(1) (mean square thermal noise
, Boltzmann’s constant k,
absolute temperature T,
resistance R, frequency range
)
The mean thermal square noise must be
considered because instantaneous thermal
noise fluctuates rapidly. The mean square
thermal noise is defined in Eq. (2).2
∫
(2) (mean square noise ,
sampling time period T,
instantaneous noise
amplitude , time t)
Because thermal noise is very small, it
must be amplified to be measurable by using
a low noise amplifier; however this may
introduce additional noise as a result of all
electrical components having resistance
hence producing additional thermal noise.1
The combined mean square noise due to
several sources of noise is found by the sum
of the mean square of all the noises. This is
defined in Eq. (3).2
∑ (3) (mean square total noise
, mean square individual
noise )
Certain frequencies of the noise will be
amplified more than others as a result of the
gain of the low noise amplifier being
dependent of frequency. As a result Eq. (1)
is modified to consider amplification from
the low noise amplifier which is given in Eq.
(4).1
∫ ( )
(4) (mean square thermal
noise , Boltzmann’s
constant k, absolute
temperature T, resistance
R, gain G, frequency f)
The gain is defined in Eq. (5).123
(5) (gain G)
Eq. (4) requires an integral from 0 Hz to
∞ Hz which is impractical; usually the
integral limit is taken to the bandwidth
frequency which is where the gain tends to
zero for high frequencies.1
By considering additional noise produced
by the low noise amplifier and combining
Eq. (3) and Eq. (4) yields Eq. (6).
∫ ( )
(6) (mean
square
amplifier
noise )
The objectives of this investigation are to
obtain evidence of thermal noise by
considering Eq. (6) and to obtain an
experimental value of Boltzmann’s constant
k.
II. METHOD
Two experiments were conducted to
provide evidence that the mean square
thermal noise is proportional to resistance
and temperature separately.
Fig. (1) shows the schematic used in the
experiments to find how the mean square
thermal noise varies with resistance and
temperature of the resistor.
PHAS 2440 - 3 - Sherman Ip
Fig. (1). Schematic of the equipment setup used to
investigate RMS thermal noise.
The components’ details used in Fig. (1)
are shown in Table (1).
Label Component Detail 1 Thin metal film
resistors
(0 to 25.5) k Ω with
0.1% tolerance
1 Resistive probe 10 k Ω with 1%
tolerance
2 Amplifier Low noise amplifier
3 Oscilloscope Iso-tech ISR622
4 RMS Voltmeter ITT Instruments
MX579 Metrix
Table (1). Details of the component in Fig. 1.
In the first experiment (resistor
experiment), thin metal film resistors were
used with values ranging from 0 Ω to 25.5
kΩ, each has tolerance of 0.1%. At room
temperature, the root mean square (RMS)
total noise associated with each resistor was
measured using the RMS voltmeter.
Due to the random nature of thermal
noise, the readings from the RMS voltmeter
fluctuated mildly. A sample of 3 readings of
the RMS total noise from the RMS
voltmeter was taken. The mean and standard
deviation of the sample was worked out
which corresponds to the value and error of
the RMS total noise respectively.
In the second experiment (temperature
experiment), the resistors were replaced with
a 10 kΩ ±1% resistive probe. The resistive
probe was immersed in different substances:
liquid nitrogen, dry ice, cold water, room air
and boiling water. The temperature of the
substances was measured using a
temperature probe.
Because both the readings of RMS total
noise and temperature fluctuated much more,
a sample of 5 readings of the RMS total
noise, from the RMS voltmeter, and the
temperature, from the temperature probe,
were taken; again the mean and standard
deviation corresponds to the value and error.
From Eq. (6), a graph of the mean square
total noise against each associated resistance
and temperature should produce a linear
graph. The Pearson’s coefficients r were
obtained from each of these graphs and were
tested if they were sufficient evidence to
show that the mean square thermal noise is
proportional to the resistance and
temperature.
The integral ∫ ( )
(area under the
gain squared curve) was worked out by
using the setup of the equipment shown in
Fig. (2).
Fig. (2). Schematic of the equipment setup used to
obtain the gain curve.
The following components used in Fig.
(2) are shown in Table (2).
Component Details AC Generator Iso-tech Synthesized
function generator GFG2004
R1 2200 Ω ± 1%
R2 10 Ω ± 1%
Standard Digital
Voltmeter reading Vin
Black Star 3210MP
Multimeter
RMS Voltmeter
reading Vout
ITT Instruments MX579
Metrix
Table (2). Details of the component in Fig. 2.
PHAS 2440 - 4 - Sherman Ip
Resistors were used to make two
potential dividers to significantly reduce the
voltage suitable for the low noise amplifier.
As a result Eq. (5) was modified to consider
the two potential dividers as shown in Eq.
(7).
(7) (gain G, constant A)
( ) No units
Readings of Vout and Vin were taken from
the two voltmeters, shown in Fig. (2), to
obtain a value for gain for each associated
frequency up to the amplifier’s manufacturer
bandwidth of ~40 kHz. The errors
correspond to half the least significant figure
on the voltmeter’s display.
The graph of the gain squared against
frequency was numerically integrated using
the trapezium rule. The value of the integral
and the gradients of the graphs in Table (3)
were used to obtain two experimental values
of Boltzmann’s constant k.
Graphs Gradient b =
against resistance ∫ ( )
against temperature ∫ ( )
Table (3). Gradients of regressions from Eq. (6).
By considering Eq. (6), the intercept for
both graphs in Table (3) corresponds to the
mean square amplifier noise.
III. RESULTS
Fig. (3) and Fig. (4) show the results of
how the mean square total nose varies with
resistance and temperature of the resistor
respectively. Table (4) and Table (5) show
the regression statistics and the critical
values of r at the 0.5% significant level of
Fig. (3) and Fig. (4) respectively.46
Gradient b ( ) Intercept a ( ) r
rcricitcal6
Table (4). Regression statistics for the graph in Fig.
(3)
Gradient b ( ) Intercept a ( ) r
rcricitcal6
Table (5). Regression statistics for the graph in Fig.
(4)
-0.005
0.000
0.005
0.010
0.015
0.020
0 5 10 15 20 25 30
Me
an S
qu
are
d T
ota
l No
ise
(V
²)
Resistance (kΩ)
Fig. (3) Mean Square Total Noise against Resistance
0.000
0.002
0.004
0.006
0.008
50 100 150 200 250 300 350 400
Me
an S
qu
are
d T
ota
l No
ise
(V
²)
Temperature (K)
Fig. (4) Mean Square Total Noise against Temperature
PHAS 2440 - 5 - Sherman Ip
The values of r were bigger than the
critical values at the 0.5% significant level
therefore there is sufficent evidence to show
that there is a linear reationship between the
mean square total noise with resistance and
temperature at the 0.5% significant level. 46
Fig. (5) shows a typical thermal noise
shown on the oscillcope during the
experiement.
Fig. (5) Waveform of typical thermal noise from a
resistor. The camera’s exposure time is much longer
than the sampling time of the oscillscope hence a few
waveforms appear on the oscillscope in the image.
Fig. (6) shows the graph of the gain of the
amplifier squared as a function of frequency.
Using the trapezium rule, the area under
the gain squared curve was worked out to be
( ) Hz. Following from
this, two experimental values of
Boltzmann’s constant were calculated as
shown in Table (6).
Experiments
Resistor ( ) J.K-1
Temperature ( ) J.K-1
Combined ( ) J.K-1
Accepted5 J.K
-1
Table (6). Accepted value of k to 3 significant figures
and the experimental values of Boltzmann’s constant
k which were combined together using weighted
means.
There was insufficient evidence to show
that the mean experimental values of k, from
the resistor experiment and the temperature
experiment, do not correspond to each other
at up to the 57% significant level. †46
Obtaining two insignificantly different
experimental values of k using different
methods increases the reliability of the
combined value of k.
However there was sufficient evidence to
show that the mean experimental combined
value of k does not corresponds to the
accepted value of k at the 0.01% significant
level. †46
IV. CONCLUSION
From the two experiments, there was
sufficient evidence to show that the mean
square total noise is proportional to
temperature and resistance of the resistor.
An experimental value of k was obtained to
be ( ) J.K-1
which does
not correspond to the accepted value of k by
at least 7 standard errors away. It however
has a percentage error of 4% which is a good
value of precision.
Both experimental values of k are an
overestimate but correspond well with each
other which may suggest a systematic error
because all experimental values of k are
consistently an overestimate.
0
5
10
15
20
25
0 10 20 30 40
Gai
n²
(x1
0⁸
No
un
its)
Frequency (kHz)
Fig. (6) Gain² against frequency
PHAS 2440 - 6 - Sherman Ip
By considering Eq. (4), the mean square
thermal noise must be an overestimate or the
integral of the gain curve squared must be an
underestimate for k to be an overestimate.
The resistors may produce excess noise
which contributes to systematic error
making the measured RMS total noise an
overestimate.
However there is not enough evidence to
suggest additional excess noise has been
produced in the experiment. The intercepts
in Table (4) and Table (5), which
corresponds to the mean square amplifier
noise by considering Eq. (6), has very small
negative values with relatively large errors.
This means the noise in the amplifier is not
significant enough to cause a systematic
error, as a result of the intercept having a
high probability of taking positive small
values and zero; therefore there is not
enough evidence to suggest that there is
additional noise produced in the experiment.
The gain curve in Fig. (6) mostly has an
decreasing gradient for increasing frequency,
as a result this makes the trapezium rule an
underestimate. The numerical integration
can be improved by using more trapezium
strips or use a different method for
numerical integration, for example the
Simpson’s rule.
By consider the gradients in Table (4)
and Table (5) and assuming k to take the
value of the accepted k, the area underneath
the gain curve should be ( )
Hz which is significantly different from the
value obtained by using the trapezium rule
on Fig. (6) of ( ) Hz; this
suggest the integral of the gain squared had
varied.
A typical 20 kΩ resistor in room
temperature will produce thermal noise of
magnitude V with bandwidth
~40 kHz by using Eq. (1). However in Fig.
(2), the input voltage going into the low
noise amplifier has magnitude of . This may suggest the integral of the
gain squared has decreased due to the
increase of voltage, compared to the thermal
noise produced by the resistor, by using the
setup in Fig. (2).
The experiment should be repeated but
instead use an AC voltage of magnitude
~0.1 V, to reproduce voltage with
magnitudes comparable with thermal noise
to investigate the gain curve and use smaller
intervals of frequency to measure the gain
therefore increases the accuracy of the
numerical integration.
*E-mail : [email protected] †Significant level is such that it is the probability of a random normal distributed variable to take value to accept the
alternative hypothesis and rejects the null hypotheses below, i.e. significant level corresponds to the accuracy of the
value:
H0 : Experimental value = Accepted value or another experimental value
H1 : Experimental value ≠ Accepted value or another experimental value 1 Electronic Noise and Low Noise Design; pg. 72, 78
P.J. Fish, 1993, ISBN 0-333-57310-2 2 Operational Amplifiers; pg. 51, 52, 53
George Clayton & Steve Winder, 2003, 5th
Edition, ISBN 0-750-65914-9 3 Circuits, Amplifiers and Gates; pg. 62
D.V. Bugg, 1991, ISBN 0-750-30110-4 4 Statistical Treatment of Experimental Data; pg. 32, 296
J.R. Green & D. Margerison, 1978, Vol. 2, ISBN 0-444-41725-7 5 Physics for Scientist and Engineers with Modern Physics; pg. i
Raymond A. Serway & John W. Jewett, 2010, 8th
Edition, ISBN 1-439-04875-4
6 OCR MEI Structured Mathematics Examination Formulae and Tables (MF2, CST251, January 2007)
Unpublished but on world wide web, used for statistical tables
http://www.mei.org.uk/files/pdf/formula_book_mf2.pdf