(revised 4/27/01)THERMAL NOISEAdvanced Laboratory, Physics 407,University of WisconsinMadison, Wisconsin 53706AbstractThe aim of this experiment is to observe the thermal noise in aresistor, to verify that the mean square noise voltage is proportionalto the absolute temperature, and to obtain an an experimental valuefor the Boltzmann constant.1TheoryJ.B. Johnson discovered that any resistor exhibits a small random alternat-ing e.m.f. (now called Johnson noise) and that the noise is dependent on thetemperature. Nyquist assumed that the noise was due to the thermal agi-tation of the electrons in the resistor and, using thermodynamics, developedthe expression1¯V2= 4kTZf2f1Real [Z(f)]df (1)where•¯V2is the mean square noise voltage.• k is Boltzmann’s constant• T is the absolute temperature• Z(f) is the impedance of the device (a resistor) at the frequency f• f1and f2are the frequency limits between which the noise is acceptedby the measuring device.If the device is a simple resistor, this becomes¯V2= 4kT R(f2− f1). (2)In this experiment, we will not measure¯V2directly but will use an amplifier.The amplifier output V1is related to the input V byV1= A(f)Vwhere A is the frequency dependent amplification. Hence¯V2= 4kT RZf2f1A2(f)df. (3)2ApparatusResistorThe resistor assembly is mounted in a copper tube (for good thermal conduc-tion) on the end of a thin stainless steel tube (for lows thermal conduction).Two 500 kΩ metal film resistors in series are used and so R=1 MΩ. Thenominal precision of the resistors is 1%.Amplifier (Stanford Research Systems Model SR560)This solid state amplifier has an input impedance of 100 MΩ shunted by 25pF and can be run either DC or AC coupled. There is a wide range of gainand band-width settings. The noise figure for the amplifier is < 4nV/√Hzat 1 kHz. This is to be compared to the Johnson noise value of 129 nV/√Hzat T = 300◦K for a 1 MΩ resistor.RMS Voltmeter (Schlumberger Model SI 7061)The Schlumberger has a true RMS AC voltage func tion. The voltmeterwill be used on the most sensitive scale (100 mV). This voltmeter also hasa General Purpose Interface Bus (GPIB) computer interface with a GBIPconnector on the back of the voltmeter. One of the laboratory computershas a GPIB interface card which can be used to read the voltmeter. Thereis a computer program called “C:\QB45\THERMAL.EXE” which is usedto read the meter and perform an average over N readings. The programfirst prompts for the desired number of readings from which to form anaverage. You would typically use N= 5 −10. The program then asks for thetemperature reading which is then typed in on the command line. Each timethe program is run to perform a sequence of readings at a given temperature,the result is added to a sequential file called “RESULT.DAT”. This file canbe edited and printed if desired.Function Generator (Stanford Research Systems Model Model DS345)The function generator is used to produce a sinusoidal output of variable fre-quency and amplitude for measuring the frequency response of the combinedpreamp and voltmeter system. It is used together with an in-line attenuatorof 104to produce amplifier inputs that are comparable to the thermal noisevoltage level.3Oven (Leybold 200 watt)The oven is operated from a variable autotransformer (General Radio “Variac”)and is used to heat the resistor up to 150◦Higher temperatures will damagethe thermocouple.ThermocoupleAn iron-constantan thermocouple junction is mounted next to the resistorand is used to measure the temperature of the resistor. A Keithley Model610C Electrometer is used to measure the thermocouple voltage. The Hand-book of Chemistry and Physics contains thermocouple tables.ProcedureThe resistor output connection is the two BNC connectors on the resistorassembly box, corresponding to the two ends of the resistor. These outputsare connected to the A and B inputs of the preamplifier which is run in theA–B differential mode to eliminate common mode noise. The suggested bandwidth settings are 100 to 1000 Hz, and the suggested gain is 100. Use thepreamp in the “low noise” setting.1. Connect the resistor assembly to the preamp as described above. Lookat the output of the preamp on the scope. The signal should be purewhite noise with no extraneous 60 Hz pickup. The actual RMS voltagemeasurements are made using the Schlumberger 7061. Calculate thethermal noise expected at the output of the preamplifier at room tem-perature. For a simple calculation assume that the amplifier frequencyresponse is simply (flow− fhigh) where A is the amplifier gain and flowand fhighare the the 6 db low and high roll-off frequenc ies respectively.Is the actual measured voltage reasonable?2. Take data above room temperature using the variac controlled oven.Do not exceed 150◦C. The experimental fluctuations in the measuredvoltage will give you guidance as to how to space the data in temper-ature. Your data will be used to determine the slope and intercept ofthe V2vs. T curve. The intercept at 0◦K will be a measure of thenoise contributions from other than thermal noise. The slope is usedto determine the Boltzmann constant.43. On the basis of the high temperature data decide if the result is im-proved by taking data well below room temperature. In principle wecan use liquid nitrogen to cool the apparatus, but this is a difficultmeasurement.4. You may later find it nece ssary to directly measure the frequency re-sponse of the combined preamp-voltmeter system. This is most con-veniently done by inserting the attenuator pad between the functiongenerator and the input to the preamp. The purpose of the attenuatoris to be able to work at the same approximate voltage level as the ac-tual noise signal since the voltmeter response may b e somewhat voltagedependent.5. There are several checks you should make. The resistor value shouldbe measured at the highest and lowest tempe ratures used during theexperiment to check that the res istor maintains the same value. thevalue should be 1 MΩ to 1%. Also, at some point, the amplifier noiseshould be measured directly by shorting the A and B inputs and readingthe output voltage. This noise voltage should be significantly smallerthan the thermal noise voltages you are measuring from the resistor.6. For analysis, first calculate the band-width integral analytically or nu-merically, using exact analytic expressions for 6 db/octave RC roll-offfilters. This procedure will