2nd year laboratory script
E17 Johnson Noise
Department of Physics and Astronomy
2nd Year Laboratory
E17 Johnson Noise
Scientific aims and objectives
To determine an accurate value for the Boltzmann constant, k, by measuring the
electrical power generated from thermal fluctuations.
• To compare the accuracy of k measured by assessing the power generated at
different temperatures and with that measured using different resistances
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Learning Outcomes
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To appreciate differences in language and methods of original scientific
investigations
• To be able to use Excel to numerically evaluate an integral from raw data.
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Apparatus
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Safety instructions
Unit comprising of amplifier,
attenuator, resistor holder, and
R.M.S voltmeter
Signal generator
Oscilloscope
Frequency meter
Metal-film resistors
Liquid nitrogen
Dry ice
Ice
Hotplate
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Gloves and goggles must be worn
when handling liquid nitrogen.
Care must also be taken when
handling dry ice and hot water
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2nd year laboratory script
E17 Johnson Noise
Task 1 - Pre-session questions
The original scientific paper by Johnson (Phys. Rev. 32 (1928) 97-109) is provided. This paper
is the information source with which to answer the pre-lab questions. The questions will help
you pick out the key information from the paper and interpret it into a method that you will
apply using the modern equipment available in the 2nd year lab.
Task 2 – The experiment
Boltzmann’s constant can be determined by measuring the voltage produced across a
conductor with resistance R at temperature T. However because these voltages are very
small they must be amplified and it is the amplification process that adds complications to
the measurements. The thermal fluctuations result in voltages with many different
frequencies. The average of these different frequencies can be measured using an RMS
meter, but the integrated gain of the amplifier must be calibrated by measuring the gain of
known frequencies. A 100:1 attenuator allows a 10 mV sine wave (4 - 11 kHz) that is amplified
by 105 to be measured on an oscilloscope with 10 V peak-to-peak range.
Once the amplifier has been calibrated and its integral gain determined, the effect of
temperature and resistance on voltage can be measured. The measurements need to be
corrected for the internal noise of the instrument, V02 , by short-circuiting with a copper wire
of essentially zero resistance. A variety of metal-film resistors provide a method of changing
R, and ice, dry ice, liquid nitrogen and a hot-plate for use with a beaker of water provide a
method for changing T. When investigating the effect of T, the temperature stable resistance
rods, rather than the metal foil resistors should be used. Two different resistance rods are
available.
You must also become familiar with the modifier M that is used to correct the impedance of
the conductor where the conductor’s resistance is similar to the input resistance of the RMS
meter. In general when M < 1.001 a value for k can be determined. When M > 1.001 a value for k
will be much less precise, but the capacitance of the meter can be estimated.
Task 3 – Reporting
You should report your experimental method in your own words showing how you determine
the integral gain of the amplifier and your method for determining k. You should report your
experimental precision of any value of k you determine. You should also be able to provide
values of k, and a comparison and explanation of the differences in precision for k determined
using both resistance and temperature methods. You should compare your experimental
value with both the accepted value for k and Johnson’s original paper.
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E17 Johnson Noise
Appendix of supporting information
1.
2.
3.
4.
1.
Electrical noise due to thermal fluctuations
Determination of integral gain
Compound errors table
Pre-lab questions
Electrical noise due to thermal fluctuations
The average noise power from a conductor of resistance R is given by
Pf =
V f2
4R
= kT∆f
(A1)
Where Pf is the fraction of the total power having component frequencies between f
and f + ∆f , V f2 is the mean square of the fluctuating potential, k is Boltzmann’s
constant, and T is the absolute temperature of the conductor. Thus:
V f2 = 4kTR∆f
(A2)
When a resistor is connected across the input terminals of an amplifier and the
amplifier has with a frequency dependant voltage gain A(f), the noise voltage at the
amplifier output is given by
V 2 = V02 + 4kTR ∫ [A( f )] df
∞
2
(A3)
0
Additionally, one must also consider the noise contribution from the amplifier itself.
Consider the following diagram that illustrates the addition components within the
amplifier that contribute to the noise signal.
Clearly, Vi = IZ i and V f = I ( R + Z i ) , and thus the input and output voltages (Vi and Vf
respectively) are related by
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Vi =
Vf
(1 + R
E17 Johnson Noise
(A4)
Zi )
Where the impedance Zi comprises of the parallel combination of the input capacity
Ci and the input resistance Ri. Briefly, the voltage across the amplifier Vi is equal to the
output voltage Vf modified inversely by a term that depends on the resistance outside
the amplifier and the impedance (based upon the internal resistor and capacitor) of
the amplifier. Suitable manipulation of this complex modifier leads to the modified
version of A3, namely
4kTR ∫ [ A( f )] df
∞
V =V +
2
2
0
2
0
(A5)
M
Where the modifier M is given by
2

R
2
M = 1 +  + (2πfCi R )
 Ri 
(A6)
At low resistor values, R (typically below 33 kΩ), the modifier, M, tends to unity and
therefore the modified equation A5 simplifies to that given by A3. Note that in the
current experimental setup, Ri = 10 MΩ. It is in this low resistance region that the
linear relation between voltage and the variable (resistance or temperature) allows a
determination of k. The higher resistance regime however allows one to estimate the
input capacity Ci. The frequency used to make an estimate of Ci should be that at the
peak gain of the amplifier.
5.
Integral Gain
The integral gain, defined by
I = ∫ [A( f )] df
∞
2
(A7)
0
and can be determined by evaluating the area under an amplitude-frequency curve.
Various methods exist for doing this, but the preferred technique available to you is
the trapezium method. The curve is approximated by a series of trapeziums of width
∆t. The total area is thus the sum of each trapezium over the whole frequency range.
You should use your experimental data points to determine the trapezium widths and
then use excel to perform the calculation of area for each trapezium.
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3.
E17 Johnson Noise
Compound errors
f
x
Proportional error
∆f 2∆x
=
f
x
∆f n∆x
=
f
x
xn
∆f = nx n −1∆x
x+ y
(∆f ) 2 = (∆x) 2 + (∆y ) 2
(∆f ) 2 ( ∆x) 2 + (∆y ) 2
=
f2
( x + y) 2
xy
(∆f ) 2 = y 2 (∆x) 2 + x 2 (∆y ) 2
(∆f ) 2  ∆x   ∆y 
=   +  
f2
 x   y 
x/ y
4.
Absolute error
∆f = 2 x∆x
2
(∆f ) 2 =
(∆x) 2 x 2
+ 4 (∆y ) 2
2
y
y
2
2
2
2
(∆f ) 2  ∆x   ∆y 
=   +  
f2
 x   y 
Pre Lab questions
1. In the abstract of Johnson’s 1927 Phys Rev. paper he defines an equation relating
the noise current in a conductor to temperature and resistance. This is his
equation:
is it:
a. Equivalent to V 2 = V02 + 4kTR ∫ [A( f )] df
∞
2
0
b. Approximately equivalent to V 2 = V02 + 4kTR ∫ [A( f )] df
∞
2
0
4kTR ∫ [ A( f )] df
∞
c. Equivalent to V = V +
2
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2
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E17 Johnson Noise
4kTR ∫ [ A( f )] df
∞
d. Approximately equivalent to V = V +
2
2
0
2
0
M
e. Not equivalent to any of these equations.
2.
Can you write down a more precisely equivalent equation to the ones
identified above? (If you can identify the approximation that has been made,
can you discuss why and how appropriate it might be – you might want the
return to this point following the experiment) (page 99 in the paper might
help)
3.
Figure 3 shows Johnson’s calibration plot for his amplifier – how the gain varies
with input frequency. Did Johnson make an appropriate number of
measurements to determine the integral gain of the amplifier? How much
more precise would the determination have been if Johnson doubled the
number of data points?
(a)
Approximately 10 times
(b)
Approximately 5 times
(c)
Approximately the same
On page 101 Johnson describes the resistances that he used to determine k.
4.
Why is he using these funny materials – why not just use metal film resistors?
(a)
They were not available in 1927
(b)
He wanted to see if the resistor material affected the thermal fluctuations
(c)
He was just a curious kind of guy.
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