I have read and understood the UCL code of assessment ...zcapg66/work/Thermal Noise.pdf · thermal noise is very small and in many cases negligible. The thermal noise associated with

UNIVERSITY COLLEGE LONDON DEPARTMENT OF PHYSICS AND ASTRONOMY

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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