JUNIOR PHYSICS LAB I
JUNIOR PHYSICS LAB I PHYS 331
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PHYS 331 Junior Physics Laboratory I Notes on Digital Circuits Digital circuits are collections of devices that perform logical operations on two logical states represented by voltage levels Standard operations such as AND OR INVERT EQUIVALENT etc are performed by devices known as gates Groups of compatible gates can be combined to make yesno decisions based on the states of the inputs For example a simple warning light circuit might check several switch settings and produce a single yesno output More complicated circuits can be used to manipulate information in the form of decimal digits alphanumeric characters or groups of yesno inputs These notes are intended to familiarize you with the elementary principles of this eld A Analysis of asynchronous logic Suppose we have a statement which can be true or false perhaps representing the presence or absence of a particle a light signal on or off a voltage present or absent or any other binary possibility For now we will ignore the physical meaning of the statement and ask how one would decide the logical truth or falsehood of combinations of such statements a subject called combinatoric logic If we denote the quottruth valuequot of a statementA by 0 or 1 the standard combinations are shown in the form of quottruth tablesquot in Fig 1 These basic combinations or Fig 1 Standard logic symbols and truth tables beam A B C D E F gt 0 0 0 0 1 0 0 0 1 0 0 0 Al Bl cl 0 1 0 0 1 0 0 1 1 0 0 0 A D B 1 0 0 0 1 0 F 1 0 1 0 0 0 1 1 0 1 1 1 C E 1 1 1 1 0 0 Fig 2 Logic to generate a stoppingparticle signal and corresponding truth table similar ones have been implemented in electronic circuitry where truth values can be represented by different voltage levels The standard circuit symbols are also shown in Fig 1 By combining the basic operations we can construct other logical functions For example suppose we wish to determine whether or not a particle has stopped in a target using the con guration of counters shown in Fig 2 A particle passing through a counter makes the corresponding output true and we assume the particle has stopped in the target if A and B are both true but C is false Formally we want to know when the compound statement A 0 B 0 5 is true An electronic implementation of the compound statement is also shown in Fig 2 together with a truth table Examination of the truth table shows that A 393 05 is true in exactly one situation which corresponds to the physically desired result Sometimes it is not obvious how to write down the required expression and implement it You might discover an implementation using only standard operations by trial and error but it is possible to be more systematic For example suppose we wish to make an exclusiveOR function using AND OR and NOT gates To do this we can try to combine some statements that are true for exactly one combination of A and B Consider the following A 03 is true only whenA l andB l A 0 is true only whenA l andB 0 l A OBis true only whenA 0 andB l AOEistrue onlywhenA 0 andB0 The middle two lines are the AB values for which the XOR is true so if we combine them with an OR we get the desired result A BZBAE 2 This procedure combining AND statements which are true for the desired true outputs is actually quite general although it may generate very cumbersome expressions It is sometimes desirable to transform a logical expression to some other form perhaps to simplify implementation or to take advantage of the devices available in a particular logic family Fortunately the logical operators define an algebra usually referred to as Boolean algebra The familiar commutative associative and distributive properties hold ABBA AOBBOA 3 AoBoCAoBoC ABCABC 4 AOBCAOBAOC 5 so one can use the normal rules of algebraic manipulation on logical expressions Two theorems called DeMorgan s laws are particularly useful ABAo ABZ 6 As an example suppose we wish to implement the exclusiveor expression using NAND gates which compute A OB We first attack the A 01 term by adding zero using the distributive law and then applying DeMorgan39s law Ao Ao AoZ A0Z AoAoB 7 Doing the same with ZOB we arrive at AeBBoMAoM 8 We then doublenegate and apply DeMorgan again to get AND H A B C A OR ED D A A XOR XOR B B Fig 3 NANDNOR implementation of some logic functions A BBomAom 9 which is the desired expression in terms of NAND functions This circuit is shown in Fig 3 along with several other examples Synthesis of desired logic functions can obviously become quite complex Fortunately there are far more sophisticated techniques available Some of these can be found in the book by Horowitz and Hill and others in electrical engineering texts B Time dependence In certain systems timing may become critical Gates require a finite amount of time to change their output in response to a change in the input signals gate delay In a complicated circuit it may happen that the inputs to a particular gate have been processed through different numbers of preceding stages and may not arrive at the same time This will cause the last gate to produce an electrically correct but logically wrong output at least transiently In situations where this causes problems it can be cured either by accurate matching of the signal delays or by clocking The matching approach is used where the logic must handle events in quotreal timequot as required in a particlecounting experiment The method is to add delay as needed to insure that all possible signals require the same amount of time to propagate through each stage in the system Timed logic circuits are sometimes called quotcombinationalquot or quotasynchronousquot logic since they produce an output as quickly as possible after a change in input Such circuits are obviously very difficult to adjust if they are at all complex The alternative to asynchronous logic is quotsynchronousquot or quotclockedquot logic In this scheme an additional input the clock is provided at each logical stage The circuits are designed to accept input on say a low to high transition of the clock signal and to change output state on the following high to low transition This scheme always leads to valid inputs at each successive stage as long as the clock period is longer than the longest propagation delay in the system The timing problem is then reduced to distributing the clock signal synchronously to all stages at the cost of a slower response to the inputs C Flip ops and counters Flip ops are circuits that can be put into one of two stable states with a pulse applied to a specified input The circuit then remains in that state until another pulse is received This constitutes a form of memory in that the circuit quotremembersquot that a pulse was applied sometime in the past An obvious application is in computer memory chips where millions of ip ops are put on a single integrated circuit The truth table for a simple example called a setreset SR ip op is shown in Fig 4 together with a realization using NOR gates The operation of the SR can be understood by reading through the truth table If both S and R inputs are zero the Q and Q outputs retain whatever state they are in If either S orR goes to 1 Q and Q change to the values indicated and remain in that state when S or R returns to zero If both S and R go to l the output is indeterminate so that combination of inputs is not allowed The limitations of the SR ip op led to the development of several other types of which only the JK will be examined here The JK ipflop can be toggled between its two states by applying pulses to a single clock input rather than to two inputs as with the SR More subtle differences are that the output is welldefined for all combinations of inputs and that the output changes only at a defined time in the clock cycle These advantages are achieved by linking two SR flip ops in a masterslave relationship as shown in Fig 5 Gates 3 and 4 control the transfer of data from master to slave while gates 7 and 8 control when the master will accept external Fig 4 SR ip op and a realization in NOR gates Fig 5 JK ip op constructed from NAND gates At level 1 on the clock pulse gates 3 and 4 close isolating slave from master At level 2 7 and 8 open connecting master to input and master may change state At level 3 gates 7 and 8 close to isolate the master At level 4 gates 3 and 4 open allowing the slave to change state inputs The action of the gates during the clock cycle is explained in the gure The J and K inputs function like the S and R inputs to force a particular state of the ip op Finally the output is connected back to the input so that the circuit will toggle in response to clock pulses when J and K are held at logic 1 A binary counter can be made by connecting the output of one flip op to the clock input of another A chain of n ipflops will have 2quot different output states Each arriving clock pulse will either increment or decrement the output state by one unit depending on the device used A counter can be used to count the number of clock pulses arriving in a certain interval to divide an input frequency by n or to provide one output pulse for 2quot input pulses These last two applications are why counters are sometimes referred to as dividers Standard ICs often provide additional inputs to set the counter to a specific state or to control counting clear I Fig 6 Timing diagram for a binary counter Since counters and ip ops have inherent time dependence truth tables do not provide sufficient information about when changes of logic state occur They can be supplemented by a timing diagram as given in Fig 6 for a typical binary counter Using the diagram one can decipher how the outputs QAQD respond to pulse inputs on the quotclearquot and quotclockquot lines Note for example that QA always changes state on the falling 1 gt0 edge of the clock pulse D Logic families There are a number of commonlyavailable electronic logic families as summarized in Table I NIM logic is a special case included for completeness It will be considered more fully in the section on particle counting As you can see the types differ in their elementary function and in whether they respond to current or voltage signals Fanout refers to the ability of an output to drive more than one subsequent input but this is not usually a problem From a design perspective the speed of operation is often a deciding factor along with cost and the ability to construct very complex singlechip circuits Very Large Scale Integration You should also be aware that there are many readymade functions available in each of the commercial logic families These typically include multiinput gates ip ops adder circuits Table 1 Summary of Gate Characteristics Gate Type Charac TTL E L CMOS NIMgneg1 NIM os Basic func NAND ORNOR NOR Connection Current Voltage Voltage Current Voltage Logic 0 00 gt 08V 09V 0 gt 2V 1 gt 2mA 15 gt 2V 1 25 gt 50V 18V 7 gt 10V 4 gt12mA 3 gt 12V Fanout 10 25 1020 1 12 Gate delay 15ns 3ns 70ns 10ns 500ns Advantages Standard Fast Low power Modules Modules Cheap Very cheap Flexible Flexible Noise VLSI Z50 2 immunity Disadv Can t Noise Slow Costly Costly drive immunity Static Bulky Bulky cable sensitive Fanout and even complete CPUs There are also compatible interface circuits such as display drivers transmission line drivers analog to digital converters and so on For reasons of simplicity and reliability you should make use of specialized units whenever possible It is highly unlikely that you will ever need to build a logic function out of discrete transistors resistors etc so for the present we will treat the circuits as black boxes If you do a lot of digital design you should gain at least a rough idea of how the functions are carried out by the internal circuitry since that will help you use the building blocks more effectively E Connecting gates In real logic systems one wants to connect several gates together so we need to examine what happens when we do this with the TTL circuits used in our lab exercises When an output at logic high 5V is connected to the input of a similar gate a small current will ow due to the leakage currents of the input transistor in the driven gate Since leakage currents are fairly small a TTL gate can in fact drive several inputs into the quothighquot logic range The situation is a little different for a low signal since significant current must ow out of the driven input to hold it within the quotlowquot range The output stage of the driving gate does this by turning on its output transistor thereby providing a relatively lowimpedance path to ground As a practical matter though the transistor in the driving gate can only pass so much current and this limits the number of inputs that one circuit can drive to about 10 Similar considerations determine the number of inputs that each output can drive in the other logic families In general the drive capability of a gate is called the quotfanoutquot Within one logic family TTL ECL CMOS NIM an output will always drive inputs up to the fanout so electrical compatibility is assured simply by counting inputs Connections can sometimes be made between families if the electrical characteristics are well understood but it is usually necessary to use special devices to quottranslatequot logic levels It is sometimes tempting to connect several outputs to one input in the hope of getting some sort of summing action but this is unlikely to work Consider the situation where two outputs are connected with one in a low state and the other high The low output is trying to pull the connection low but the high output and the input of the next gate will supply substantial current perhaps exceeding the current sinking capability of the low output Even if no damage occurs the result is likely to be abnormal logic levels and unpredictable results F Conversion circuits For digital logic to be useful in many applications there must be a transformation to or from continuouslyvariable quotanalogquot signals This function is the province of digital to analog DA or analog to digital AD converters The actual design of such circuits demands complex Fig 7 Digital to analog converter using an R2R ladder network The switches are controlled by the digital input from most to least signi cant bit tradeoffs among cost speed and a multiplicity of possible error sources Here we will mention some of the basic principles so that you can rationally choose a commercial module when you need it The goal of a DA is to convert a multibit binary number to a proportional voltage or current General purpose schemes typically involve switching voltage or current sources on or off under digital control and adding the results One example is shown in Fig 7 in which a cleverlychosen resistor network provides a voltage proportional to the input number A unity gain ampli er buffers the resistor network from whatever load we wish to connect The accuracy and speed of conversion will depend on resistor tolerances the quality of the digitallycontrolled switches and the properties of the ampli er Analog to digital converters come in more varieties than DAs but most designs depend on a comparator The comparator circuit accepts a reference voltage and the unknown voltage and produces a logical one output when the unknown exceeds the reference The simplest AD implementations provide a reference voltage corresponding to each digital output and use one comparator for each level Logic circuitry determines the highestlevel comparator that is at logic one and produces a corresponding binary code These parallel or quot ashquot converters are fast but become too cumbersome when high accuracy is needed They do nd use where speed is paramount as in reading out tracking chambers in high energy physics experiments Successiveapproximation converters use only one comparator in combination with a DA converter The DA is given a digital code and the output is compared to the unknown voltage The largest digital code that does not exceed the unknown is taken as the digital output value The conversion obviously takes time but by using a binary search sequence only 71 trials are needed for 71bit output enabling a typical module to produce an eight to twelve bit conversion in a few microseconds Higher accuracy at lower speed is obtained from integrating converters typi ed by single and dualslope methods In the singleslope method a current source charges a capacitor The time required for the capacitor voltage to equal the unknown voltage is measured with a digital clock whose output is therefore proportional to the unknown voltage Singleslope integration is obviously sensitive to imperfections in the capacitor and comparator Dualslope integration charges the capacitor for a fixed time with a current proportional to the unknown voltage The same capacitor is then discharged to zero by a known constant current The discharge time as measured by a digital clock again is the converted value Now the exact capacitor value is irrelevant and the comparator only needs to detect zero voltage which facilitates an automatic zeroing scheme Because of the accuracy up to 16 bits and relative noise immunity dualslope integration is used in all but the cheapest digital meters PHYS 331 Junior Physics Laboratory I Exercise on Analog Circuits In this exercise you will assemble and operate some simple transistor and op amp circuits The examples chosen are typical of those used elsewhere in our labs and in research Your write up should explain clearly what you did in each part following the general organization of this guide Include a schematic of each circuit as you built it using the schematics in the notes as a guide to standard practice Provide complete answers to any questions posed in the text A neatly hand written report with clear free hand figures is entirely acceptable Before starting the lab work you should review the topical notes on Analog Circuits available on the course web site Sections 201 209 215 401 410 and 1502 of The Art of Electronics by Horowitz and Hill will also be helpful A Circuit construction The most convenient way to set up test circuits is on a breadboard a large plastic block with sockets to mount transistors integrated circuits resistors capacitors etc Pre cut jumper wires are used to make connections by plugging in to the interconnected sockets An auxiliary circuit board plugs into the main board to provide connections for positive and negative supply voltages and for external signals The connections to the outboard can be understood by careful examination Pin identifications for semiconductor devices are posted near the work areas Connections to other components should be evident Note that the larger value capacitors are polarized as indicated usually by a negative sign near one lead Be sure to observe polarity in your circuit to avoid malfunction B Transistor circuits Two transistor types are provided 2N3904 NPN and 2N3906 PNP The manufacturer specifies 100 lt hFE lt300 maximum collector current of 200 mA and maximum collector emitter voltage 40 V The 12 V supply will be within ratings for the collector but you will destroy the transistor if you exceed IC 200 mA or apply supply voltage directly to the base even brie y 12V 004mi Vout Vi 100K 2N3904 2N3 904 a b Fig 1 Practical common emitter amplifier circuits with and without biasing 1 Common emitter amplifier Set up the N PN common emitter circuit of Fig 1a and drive it with sine wave inputs of various amplitudes Using the scope compare the input and output wave forms You should be able to qualitatively explain the severe distortion you will see and the additional clipping at large amplitudes in terms of the general characteristics of transistors Also try driving the input with a fairly large amplitude square wave to demonstrate that the circuit can be a logic inverter Now construct the biased common emitter circuit of Fig 1b Again drive the input with sine waves and qualitatively explain the output wave form including the DC component Be sure to determine the phase and amplitude of the output signal relative to the input so you can calculate the AC voltage gain and demonstrate that the amplifier inverts What happens when the input amplitude becomes large 2 Push pull follower Construct the push pull output stage shown in Fig 2a driving the input with a sine wave Sketch and compare the input and output wave forms and explain the distortion near the zero crossings Find the gain for large amplitude signals and determine the maximum amplitude before noticeable clipping Can you demonstrate a larger output amplitude from this circuit than from the biased common emitter Is the DC component essentially absent as claimed in the notes Next add the bias network shown in Fig 2b Biasing should remove or minimize the cross over distortion Does it Sketch the wave forms noting any remaining irregularities What is the maximum undistorted amplitude from this arrangement 12V 12v 10K 2N3904 2N3 904 2N3 906 12V a b Fig 2 Push pull amplifier circuits with and without biasing C Basic opamp circuits The type 741 op amp is supplied in an eight pin integrated circuit package It requires both positive and negative supplies in the range 10 to 36 V but no explicit ground connection There is provision for exactly zeroing the apparent input voltage but we will not use that feature The output can reach within a volt or so of either supply voltage and produce up to 20 mA of output current 1 Inverters and followers Following the schematic in the notes design and build an inverter with a gain of 10 The resistors should be chosen so that most of the current capacity of the op amp is available for driving the external load Verify that your circuit is an inverter with the specified gain at low frequencies and find the frequency at which the gain is reduced by a factor of 1 J2 This is called the 3 dB point because it is the frequency at which the power output is half the maximum Keep the output below a volt or two to avoid large amplitude effects It is also interesting to explore some of the limitations of the inverter How large an output amplitude will it produce before clipping or severely distorting the signal Try driving the input with a small amplitude square wave about 1 2 V at the output comparing input and output signals What is the maximum rate of change of voltage in voltsus that the amplifier can produce This is called the slew rate and is important in some applications Next set up a unity gain non inverting follower and again check the frequency response and slew rate Do you notice any difference from the inverter These characteristics will differ 10K Fig 3 An integratorfilter circuit for various types of op amp in various configurations and must be considered when designing circuits for specific applications 2 Integrator filter Set up an averaging filter using the component values in Fig 3 For an input signal of a volt or two plot the output amplitude vs frequency on logarithmic scales Is the 3dB point where you would expect from the component values Don t forget that a 279 It is also interesting to look at the phase shift as a function of frequency You can do this easily by displaying the input signal on channel 1 and the output on channel 2 of the scope At low frequencies the output is inverted as expected but you will notice an additional shift at higher frequency which you should describe Depending on the intended use the phase shift or equivalent time delay may be an important filter characteristic 3 Oscillator In the Wien bridge oscillator circuit shown in Fig 4 the RC network acts a frequency dependent voltage divider feeding 13 of the output voltage to the non inverting input when f 12nRC Since the RC network has no phase shift at this frequency this is positive feedback and the circuit will oscillate if the gain 1 RZRl is greater than 3 Fig 4 Wien bridge oscillator circuit Note that there is no connection to the top of R1 and that no signal source is needed to start the oscillation Avoid connecting the resistor slider to either the 12V supply or ground as damage will result for some settings 4 Construct the circuit as shown with R1 a 1 K82 variable resistor and R2 1 KS2 and observe the effect on the output of changing the variable resistor setting You should be able to find resistor settings for which the output voltage is essentially zero a stable amplitude sine wave or a badly clipped waveform Determine the variable resistance value needed to produce a stable sine wave and calculate the expected gain Is it approximately 3 This circuit is not very useful because the amplitude of the oscillations is quite sensitive to the amplifier gain 1 RZRl which may drift with time or temperature It can be stabilized by replacing R1 with a small incandescent light bulb As the amplifier output increases more current is drawn through the lamp causing the resistance of the metal filament to increase This negative feedback decreases the amplifier gain and the output is reduced until a steady state is reached Demonstrate this effect by using a light bulb for R1 and the variable resistor for R2 Adjust R2 to get stable oscillations and then note the effect of R2 values somewhat larger or smaller than the stable setting It is also interesting to watch the oscillations build up after the power is turned on when R2 is at the stable setting You should be able to see the effects of the thermal response time of the lamp quite clearly Historical note The use of a light bulb to stabilize an oscillator was invented by William R Hewlett and patented by him in 1942 A variable frequency audio oscillator based on this circuit was the first product from the Hewlett Packard company 4 Power booster Construct a unity gain inverter using 10K 2 input and feedback resistors Find the maximum output voltage with an open circuit and again when the output is driving a 10092 resistor What is the output current capability of the op amp Is this consistent with the specification for the 741 Now construct the circuit shown in Fig 5 and again check the output 10K 12v 2N3 904 gt 2N3906 ijoad 12V 10K Fig 5 An op amp inverter with power output stage 5 voltage for RM 2 00 and for RIM 1009 Do you see a substantial increase in output current capacity as predicted This circuit can also be used to demonstrate the use of feedback to suppress distortion Sketch the output waveform of the circuit as built Then reconnect the feedback resistor to the normal inverter configuration from the op amp output to the inverting input Sketch the new output waveform and comment on the differences you see For a more dramatic demonstration connect a small loudspeaker to the output of the unity gain inverter What is the maximum output voltage before the signal distorts Now drive the speaker with the power booster circuit and again check the maximum undistorted amplitude You should also hear a large increase in acoustic output Note Do these tests very quickly to minimize overheating of the transistors Real amplifiers need heat sinks to keep the transistors from melting PHYS 331 Junior Physics Laboratory I Notes on Noise Reduction When setting out to make a measurement one often nds that the signal the quantity we want to see is masked by noise which is anything that interferes with seeing the signal Maximizing the signal and minimizing the effects of noise then become the goals of the experimenter To reach these goals we must understand the nature of the signal the possible sources of noise and the possible strategies for extracting the desired quantity from a noisy environment Figure 1 shows a very generalized experimental situation The system could be electrical mechanical or biological and includes all important environmental variables such as temperature pressure or magnetic eld The excitation is what evokes the response we wish to measure It might be an applied voltage a light beam or a mechanical vibration depending on the situation The system response to the excitation along with some noise is converted to a measurable form by a transducer which may add further noise The transducer could be something as simple as a mechanical pointer to register de ection but in modern practice it almost always converts the system response to an electrical signal for recording and analysis In some experiments it is essential to recover the full time variation of the response for example the timedependent uorescence due to excitation of a chemical reaction with a short laser pulse It is then necessary to record the transducer output as a function of time perhaps repetitively and process the output to extract the signal of interest while minimizing noise contributions Signal averaging works well when the process can be repeated and we will examine the method in some detail Alternatively the signal of interest may be essentially static Consider for example measuring the optical absorption of a beam of light traversing a liquid as a function of temperature or optical frequency Any time variation is under the experimenter s control and does not convey additional information so it is only necessary to extract the essentially steady signal from whatever noise is present For this situation drifts and instabilities in the transducer are often important along with noise introduced in the experimental system The usual strategies to deal with this combination of problems include chopping ltering and ultimately lockin detection all of which we will consider Excitation lt Exper1mental system Fig 1 Schematic of a generalized experiment Response noise noise A Sources and characteristics of noise Noise can be roughly categorized as follows quotInterferencequot is at a de nite frequency Examples would be 60 Hz fields from power lines high frequency fields from nearby radio or TV transmitters and scattered laser light in an optics set up Broad band or quotwhite noisequot is more or less uniform over a wide range of frequencies It can arise from multiple sources which blend together to give a uniform appearance or ultimately and inevitably from the thermal vibrations of the apparatus quotFlickerquot or quotlf noisequot is a broad band noise that increases with decreasing frequency It is quite commonly observed but the level depends on the system For example all electrical resistors of the same value exhibit the same level of white noise but the additional lf contribution depends on the material The overall situation is summarized in Fig 2 which shows a typical noise spectrum and some interference sources The appendix to these notes includes a brief discussion of power spectra The type of noise af icting the measurement as well as the measurement requirements will suggest an appropriate strategy First one should attempt to strengthen the signal if possible This might involve using a stronger optical source larger telescope bigger force or whatever else might be possible and appropriate After that one can try to reduce external interference by putting the apparatus in a metal box to block high frequency disturbances by proper grounding to minimize powerline pickup by cleaning lenses to reduce scattered light and so on Once the 10 temperature variations mechanical vibration 10 5 mterferences region of Spectral density arbitrary units 10 39 icker and burst noise contact arcin 2 harmonics TV and 10 radar computer amfm broadcast 1 power supphes 10 39 lf noise gt white norse level x region of thermal and shot noise 1 0 0 I I I I I I 10394 10392 100 102 104 106 108 Frequency Hz Fig 2 Schematic spectrum of noise and interference signal has been enhanced and shielded as much as possible it is appropriate to turn to signal recovery procedures B Signal averaging Signal averaging exploits the fact that if one makes a measurement many times the signal part will tend to accumulate but the noise will be irregular and tend to cancel itself More formally the standard deviation of the mean of N measurements is smaller by a factor of NI than the standard deviation of a single measurement This implies that if we compute the average of many samples of a noisy signal we will reduce the uctuations and leave the desired signal visible There are of course a number of complications and limitations in practice We model the averaging process by assuming a signal voltage V30 which is contaminated with a noise voltage Vnt that is comparable or larger than VS The voltage actually measured is then the sum of these two parts V0 VsU VnU 1 If we record Vt over some time interval we have a record which contains information about the time variation of VS The crucial point is to average many such records of Vt so that the noise which is equally likely to be positive or negative tends to cancel out while the signal builds up This obviously requires that we start each measurement at the same relative point in the signal a requirement we will consider when we implement the averaging To calculate the actual noise reduction we need to compute the standard deviation of the average of Vt after N records have been averaged For convenience we assume that the noise is characterized by V 0 and standard deviation in If the noise mean is not zero we can simply include 17 as part of the signal and subtract it later The signal is assumed to repeat exactly from trial to trial so is 0 With these assumptions 70 V50 and q on W 2 which is the expected outcome This result is often phrased in terms of a voltage signal to noise ratio given by 1 VHW 3 Fig 3 Sampling of sinusoidal waves Note that there is no way to distinguish between the two sine waves on the basis of the regularlyspaced samples This relation is the basis for the common statement that averaging N records improves the original signalnoise ratio by a factor of l Note that this improvement is bought at the expense of measurement time Speci cally the signal to noise ratio improves only as the square root of time spent averaging a result we will see again The averaging process could be done using a continuous record of Vt but it is more common to sample the input with an analog to digital converter at regular intervals At so that the required computations can be done digitally Unfortunately the sampling process introduces the possibility of aliasing as illustrated in Fig 3 Evidently input frequencies from zero to l2At are all distinguishable but if higher frequencies are present they will be confused with some lower frequency The highest distinguishable frequency is called the Nyquist frequency lZAI Experimentally it is important to avoid aliasing by lowpass filtering the input before sampling and choosing At small enough that the Nyquist frequency is well above the pass limit Several factors limit the degree of SN enhancement attainable by averaging For example digital systems measure inputs to a specified accuracy perhaps 1214 bits and do arithmetic to a similar fixed precision Comparable effects occur in analog averaging schemes Ultimately this must limit the accuracy of the computed average There may also be time or voltagedependent drifts in the instrument which have the effect of adding spurious variation to the true signal It is also possible for the signal itself to change with time perhaps due to internal changes in the system under study or the slow variation of external conditions Whatever the cause one always finds that the gain in SN does not increase indefinitely with averaging time C Filtering and modulation In the case of a quasistatic signal where we do not need to measure the actual waveform it may be possible to use a frequencyselective filter to suppress noise Consider again the power spectrum of noise shown in Fig 2 If the desired signal occupies a fairly narrow range of frequencies somewhere in this region we could improve the situation by measuring the voltage only in that range and throwing out uctuations at all other frequencies For example in a laser scattering experiment we might use an optical filter at the laser frequency to reject room light and other disturbances Any noise contributions that happen to fall in the same frequency range as the signal will still interfere but we will have removed a great deal of other noise The filters used are commonly either lowpass or bandpass A lowpass filter rejects all inputs above a certain frequency and is therefore most useful when the signal is essentially DC A DC voltmeter is a good example of this approach in that it integrates the input for some finite time effectively canceling any input component that changes sign during the integration period Alternatively if the signal has a known periodicity it may be possible to design a filter to reject all inputs outside of a small band of frequencies near the signal as in the laser example above If the signal is inherently static it may be advantageous to impose a periodic variation Referring again to Fig 2 the spectral density of noise tends to decrease with increasing frequency so measurements made at higher frequencies are inherently less noisy The output of DC amplifiers also tends to drift with temperature and other environmental variables adding further uncertainty to the measurement These problems can be addressed by interrupting or modulating the excitation so that the signal portion of the transducer output has a definite frequency In the laser light scattering experiment the modulation might be done by interrupting the laser beam with a toothed wheel The electrical output of the detector can then be filtered to minimize noise and other disturbances added by the detector or amplifiers The improvement in signal to noise ratio due to filtering can be easily quantified for the simple situation shown in Fig 4 The input noise spectrum is assumed to be at or quotwhitequot from zero frequency up to the bandwidth BI The signal Sm is visible as a bump on the total input spectrum and has a meansquare intensity The total meansquare intensity of the noise is just the area under the spectral density curve or PNBI so the meansquare signal to noise ratio at the input is 2 SNRI S I 4 PNBI By design the filter transmits all the signal but attenuates strongly over most of the noise bandwidth If the effective bandwidth of the filter is BO the meansquare signal to noise ratio at the output will be signal components 411 power h by transmission 4 30 gt Fig 4 Power spectrum of signal and noise above and the percenttransmission characteristic of a bandpass lter below 2 SNRO 3 0 5 PNBO and the improvement in meansquare SNR is just the ratio of bandwidths SN B R0 1 6 SNRI BO Instruments are usually designed to display rms voltage rather than meansquare voltage so the improvement read on the output meter will be the squareroot of this value As always there are some practical limits For maximum enhancement it is obviously desirable to make the filter bandwidth as small as possible but it must be left broad enough to pass all signal frequencies Depending on the implementation very narrowband filters may also drift off the desired frequency with disastrous consequences for the signal In a favorable situation with broadband noise as assumed here it might be possible to gain a factor of ten or so in rms SNR by filtering D Lock in detection For bandwidth minimization to be effective we require that the signal occur within a restricted range of frequencies preferably well above lf noise and away from interference Oscillator Dr Transducer Experim ental system Signal channel Fig 5 Block diagram of a phasesensitive detection system connected to an experiment sources We then insert some sort of lter most commonly electronic before our measuring apparatus to pass the signal frequency range and block other frequencies The lockin detector is a particularly ingenious implementation of these principles because it allows us to create a very narrowband lter which will automatically follow small changes in modulating frequency The basic principle behind the lockin detector is shown in Fig 5 The experimental system is our apparatus and sample as before We want to measure some property of this system as a function of an external parameter like temperature or magnetic eld which we can slowly 39sweep over a desired range An external oscillator is used to modulate a system parameter in such a way that the quantity we wish to measure varies at frequency 00 This could be the same parameter that is being varied slowly such as the wavelength of the incident light in our optics example or an independent parameter such as the intensity of the light The object in either case is to impose a variation at a on the desired signal without modulating the noise that you want to eliminate The response of the system is converted to a varying voltage with an appropriate transducer and applied to the phase detector as VS which includes noise from the experiment and transducer The phase detector multiplies this voltage by a reference voltage V at frequency 00 producing a product voltage VrVS Now comes the crucial point The desired signal and the reference voltage are at the same frequency and are always in phase so a Fourier analysis of their product contains a constant term Other inputs like noise will be at different frequencies and phases Fourier analysis of their products will disclose timevarying components but almost no DC part The multiplier voltage is ltered by the lowpass circuit which passes the DC signal voltage but suppresses the timevarying noise voltages to yield the final output V0 For completeness we note that the slow sweep must be slow enough that all of its Fourier components can get through the low pass lter Otherwise the desired signal will be partially removed along with the noise There are several advantages to lockin detection First by modulating the right parameter we distinguish the quantity we wish to measure from other fluctuations which are not sensitive to the varying parameter Second the desired response is now at a controllable relatively high frequency Other disturbances such as power line interference and mechanical vibrations tend to occur at specific lower frequencies which we can avoid Finally the combination of the phase sensitive detector and lowpass filter allow us to work with extremely small bandwidth Even if we could directly construct a filter with similar bandwidth it would be formidably difficult to keep it centered on the oscillator frequency because of drifts in component values To do a quantitative analysis of the multiplier filter combination we imagine Fourier analyzing V and V to obtain their frequency components The reference voltage is a sine wave so it has only the single component v sinat r 7 while V has significant amplitude over a broad range vsasinat S 8 The components at and near a represent the desired signal plus any noise that happens to be at the same frequency Components at other frequencies are due to the noise we want to remove The result of multiplying one frequency component of V by V is then vwvcosw a w m cosw aw 01gt M 9 The phase detector output is applied to a low pass filter which will reject the second term entirely because it is at the reference frequency or above The filter will pass the first term provided that a 00 is sufficiently close to zero The signal occurs at 00 so the output from the phase detector filter combination due to the desired signal is just a constant amplitude cosw 10 which we will maximize by adjusting the relative phase of the reference to zero when we set up the instrument Incidentally the sensitivity to signal phase is sometimes useful in itself and is the reason the device is called a phasesensitive detector Other frequency components for which a 0 falls within the band pass of the filter will also appear in the final output The usual lowpass filter is an RC circuit with time constant 139 and 3dB bandwidth l L39 so signals within roughly ilT of a will pass to the output Effectively then the multiplier and lowpass filter act as a bandpass filter of width 2 centered at the reference frequency Note that if the reference frequency shifts somewhat the effective center frequency shifts as well It is sometimes said therefore that this is a 39lockin amplifier or detector One final point to consider is the response of the system to a temporal change in signal amplitude The response time of any linear filter is inversely proportional to the bandwidth This means that as we narrow the bandwidth the filter output takes longer and longer to respond to changes in the input An explicit calculation of the response of an RC filter to a step change in input is fairly easy with the result shown in Fig 6 Note that the output of even a single RC stage takes about 21to come within 10 of the final value Changes in signal level due to the slow sweep of an external parameter must therefore occur over many time constants to avoid significant distortion and the improvement in SNR comes at the expense of measurement time In fact since the bandwidth in inversely proportional to 1 Eq 6 tells us that the improvement in rms SNR is proportional to E Since the measurement time increases directly with T the SNR increases only as the square root of the measurement time exactly as in signal averaging fraction of final amplitude tT 0 l 2 3 4 5 Fig 6 Response of one solid and twostage dashed RC filters to a sudden change in input E Appendices l Meansquare quantities It is sometimes desirable to have a measure of the intensity of a timevarying voltage For a sinewave signal the intensity is conventionally taken as the square of the amplitude since the power dissipated in a resistor R connected to a sine wave voltage of amplitude A is proportional to A2 We can generalize this idea by de ning the intensity of an arbitrary timevarying voltage Vt as the timeaverage of the square of the instantaneous voltage T V2 JV2tdt 11 0 where the avelge runs over either one full cycle or over a long time if the voltage is irregular The quantity V2 is called the mean square voltage with units of volts squared and is exactly half of A2 for a sine wave It is easy to show that the average power dissipated by the voltage V in a resistor R is just VR further reinforcing the idea that this is an intensity The square root of W has units of volts and is called the root mean square or rms voltage This quantity is useful because it sometimes behaves like a voltage in circuit calculations but is not limited to sine waves 2 Power spectra The rms voltage describes the strength of a signal but is it sometimes helpful to have information about the time variation as well The Fourier transform T2 Fa JVteimtdt 12 iTZ is useful for that purpose To avoid issues of convergence the integral here runs over some long but finite time Like the set of Fourier series coefficients the complex function F1 gives the amplitudes and phases of the sine waves at frequencies a which will add up to the original signal We are usually interested only in intensity information not phase so we can employ the magnitudesquared of the transform Pa Fa2 13 where PM is called the power spectral density of the signal with units of voltszHz For real valued Vt the only case of physical interest PM is a an even function of a so all the physical information is contained in the positive frequency domain Integrating Pa over all frequencies we recover the meansquare voltage V ZIPw da 14 0 further reinforcing the connection to intensity More physically the power part of the name comes again from the proportionality to voltage squared and it is a spectral density because it is the voltage squared per unit frequency interval Effectively Pa tells us how much power our signal has at a specified frequency This information is particularly useful when considering the action of circuits which selectively attenuate or amplify certain frequency bands 3 RC filters One and twostage RC circuits as shown in Fig 7 are frequently used as lowpass filters Their frequency response can be derived by standard circuit analysis techniques For a sinusoidal input voltage K the output voltage from one stage is V V ZC 15 ZC R where ZC lz39aJC In practice we usually need only the amplitude ratio which is V l 0 2 12 16 Vi 1an where L39 RC is the time constant of the circuit This ratio is plotted in Fig 8 The lowpass R R R V CIV V C C V Fig 7 One and twostage RC filter circuits A unity gain amplifier prevents the second stage from loading the first stage 2 2 0 00 1039 1 10 Fig 8 Frequency response of single solid line and twosection dashed line RC lters ltering action is evident with the 3 dB point at a frequency of 000 UT The 3 dB point is the frequency at which the power output is reduced by a factor of 2 or equivalently the voltage is reduced by a factor of J5 When two RC circuits are connected in series with a buffer amplifier the output voltage of the first becomes the input to the second The final output is therefore just a product of factors like Eq 16 or E 17 V 1an2 for two identical sections Note that the low frequency response is essentially the same as for the single RC circuit but the attenuation falls off more rapidly with increasing frequency above 000 The time response of the filter can be found by setting up and solving the differential equations for the circuit For a singlestage RC with an input voltage going from zero to K at t 0 the output voltage is V0V1 e 18 The corresponding result for the twostage RC is V0 Vil Lle7 T 19 T Both of these relations have been plotted in Fig 6 Comparing Figs 6 and 7 we see that the improvement in highfrequency rejection with the twostage lter has been obtained at the cost of much longer response time The application will determine which factor is more important This is an example of a wellwritten formal report from the PHYS 331 or 332 lab Particularly pertinent features are called out in the marginal notes This document uses computerdrawn gures and typed equations to facilitate posting on the web but you may substitute handdrawn gures and equations if more convenient Graphs must be done on graph paper or by computer not as rough sketches Measurements of the Sodium Doublet by Grade A Student with A Partner Physics 331332 February 30 2010 Based on a report by Dan Sutton Jon Denlinger and William Leaf Physics 450 Purdue University March 4 1982 I Introduction The intense yellow 589 nm optical emission line of sodium is known to be a doublet with a splitting of about one nanometer A more accurate measurement of this splitting would be useful for comparison with calculations of the energy levels in this multi electron atom The present paper reports such a measurement using a Michelson interferometer The next section describes the calibration and use of the interferometer Section 111 contains our results and Section IV provides some concluding remarks 11 Experimental Methods The layout of the interferometer is shown in Fig 1 A source of light is directed toward a beam splitter a cube consisting of two equal glass prisms glued together that reflects and transmits two equal beams The beams then travel toward two plane mirrors M1 and M2 situated perpendicular to each other The light is reflected back towards the beam splitter where the beams are deflected to the observer The tilts of the mirrors can be adjusted with fine screws to exactly superpose the two beams at the observing position Mirror M1 remains fixed to the base while M2 is attached to a micrometer screw driven by a reducing gear so that L2 can be precisely adjusted With an extended monochromatic source and parallel mirrors the observer sees a set of concentric circular interference fringes The condition for destructive interference at wavelength A in a medium of index n is 2ndcos0m mh 1 where d is the path difference ILZ Lll 0quot is the angle between the interfering rays and the optic axis and m is an integer Varying d causes fringes to appear or disappear at the center of the pattern at intervals of one half wavelength For a practical instrument cos 0quot 1 but m may be large and is not known accurately To measure a wavelength L2 is set so that an interference minimum dark fringe is at the center The mirror separation is then changed until N fringes have appeared or disappeared at the center Denoting the two settings by 12 and applying Eq 1 we get 2nd2 d1 m2 m1 NA 2 which relates the wavelength to measurable quantities Equation 2 will be used to calibrate the micrometer drive and to determine the average wavelength of the doublet When two wavelengths are present each will create a ring system according to Eq 1 The wavelength difference can be measured by finding mirror separations such that the rings of one system are centered between those of the other wavelength Since the light and dark fringes are of approximately equal width the fringes nearly disappear when this condition is met If a fringe disappearance is observed at separation d1 application of Eq 1 yields States the work done and places it within a context The gure shows the eometry conveniently Explains what is happening without extraneous details Minimum theory needed to define the quantities measured Equations numbered for easy reference Explain the phenomenon qualitatively and then derive needed relations 2nd1 m12tm112t 3 the 12 indicating that the fringe patterns are one half order out of step If d is now changed to bring the fringes into register and then a half order out of register again at d2 we get 2nd2 m2 m2 32 4 Taking the difference and solving to first order for the splitting we obtain A A 7f2nd2 d1 5 A first order solution is sufficient since AM is of order 103 Two additional points need to be considered before using these relations to Anticipates questions interpret data First the index of refraction of the medium air appears in the a thoughtful reader equations We avoid a precision determination of the index by using a known vacuum would ask wavelength and Eq 2 to calibrate the interferometer drive screw Neglecting the small effects of dispersion the index then cancels from subsequent calculations Second in measuring the sodium wavelength d is set so that the ring systems are superimposed Fringe counting then leads to a determination of the average wavelength of the doublet since the translation needed for this determination is much less than that needed to resolve the ring systems The light sources used in this work are commercially available A helium Standard enough not neon laser provided the standard wavelength 6329914 nm vacuum The sodium to need discussion source was a low pressure DC discharge lamp operated at ambient temperature III Measurements Calibration of the micrometer drive was accomplished by determining the States what was done displacement Ad needed to collapse 50 fringes Displacements are expressed in terms without apparatus of degrees of rotation of the wheel which drives the reducing gear system on the specific clutter micrometer screw Averaging over 5 trials we find I9 453 i 2 degrees All error estimates are the standard deviation of the mean This leads to a calibration constant Er 1 01 v are given and of 350 i 02nm per degree which compares favorably with the nominal value of deiS OfESlimdte 347 nm per degree determined from the gear ratio and the pitch of the micrometer SPECifiEd screw The average wavelength Z of the doublet is found by the same procedure For Other W0rk CitEd but five trials we find g 420 i 2 degrees which yields Ad 2 147 x 104 nm and quot0t 5361 in Place of Z 588 i 4 nm This result agrees with the value X 5893 nm found by other error EStimates experimenters The doublet splitting is found from measurements of Ad between points of minimum fringe contrast The displacement needed to measure the splitting is relatively large so the micrometer was used without the gear reduction mechanism Averaging 6 trials we determine a nominal average Ad 2 0294 t 0016mm This is corrected to an actual Ad by assuming that the difference between nominal and actual calibration constants found above is entirely due to pitch errors in the micrometer screw With this assumption Ad 2 0297 t 0016mm Finally application of Eq 5 with the value of X determined in the present experiment gives A 058 i 003nm Note that the uncertainty in this result is almost entirely due to the difficulty Helps reader in locating the position of minimum contrast understand limits of experiment IV Conclusions The average wavelength and splitting of the yellow sodium doublet have been Summary of results measured relative to a standard neon wavelength We find X 588 i 4nm and Not profound and A 058 i 003nm The accuracy of the latter result is limited by the poor doesn t 118851 to be definition of the positions of minimum fringe visibility A substantially more accurate determination of A would therefore require different experimental methods adjusting Not a great gure but SCFCWS M MI xed adequate to define terms L1 gear and beamsplitter Elcrometer 1V6 I light source M 2 movable v observer Short but descriptive Fig 1 Layout of Michelson interferometer caption This is an example of a very poor report from the PHYS 331 or 332 lab Particularly egregious features are called out in the marginal notes This document uses computerdrawn gures and typed equations to facilitate posting on the web but you may substitute handdrawn gures and equations if more convenient Graphs must be done on graph paper or by computer not as rough sketches Measurements of the Sodium Doublet by J Q Klutz with I M Lazy Physics 331332 February 30 2010 Plagiarized from the Physics 331332 lab manual Rice University September 1988 and from the Physics 350450 lab manual Purdue University March 1982 with data by Dan Sutton Jon Denlinger and William Leaf I Introduction Interferometers using various frequencies of electromagnetic radiation are Doesn t tell the employed to measure a wide variety of physical phenomena including lengths reader whatyou did wavelengths and indices of refraction Here we will concentrate on measuring in this experiment wavelengths and especially very small wavelength differences which occur in the optical spectra of atoms The shift we wish to measure is caused by spin orbit coupling in the sodium atom It is a measure of the power of the interferometric method that this splitting is about 1 103 In the Michelson interferometer two beams of light are folded and combined Irrelevant by mirrors to give broad interference fringes The open geometry and high light transmission make this instrument particularly suited for wavelength measurements The Michelson is also frequently used with a laser source to measure distances up to a few meters with high precision 11 Theoretical Considerations The energy levels of atoms are largely determined by the Coulomb interaction This is not a theory between the electrons and the nucleus but other interactions may also be present paper Provide the These additional interactions manifest themselves as shifts or splittings of the minimum background observed spectral lines relative to the lines which would occur if the electrons moved needed to place the in a pure central potential The quotfine structurequot splitting arises from electron electron work in context Coulomb interaction and from the spin orbit interaction of the moving electrons with the nuclear electric field Smaller splittings called hyperfine structure arise from non electrostatic interactions with the nucleus For atomic species with several isotopes the differing nuclear mass and volume can cause a shift of spectral lines among the isotopes Alternatively if the nucleus has a spin there can be magnetic interactions which perturb and split some of the electronic levels The ground state of the sodium atom has one electron in an n3 orbital outside the closed n1 and n22 shells To a first approximation this electron moves in a spherically symmetric potential and its motion can be characterized by n s as for a hydrogen atom However the potential does not go as 1r down to the origin because of the inner electrons so states of a given n and different 6 are not degenerate The ground state denoted 2S has n23 60 s12 while the first excited state 213 has n23 61 s12 The bright yellow emission characteristic of sodium results from transitions 2P gtZS When examined with high resolution this emission line is seen to be two lines in close proximity indicating that one or both of these levels are further split One way to account for this splitting is the spin orbit interaction This coupling arises because in the reference frame of the electron the nucleus is a moving charge which generates a magnetic field The magnetic field interacts with the magnetic moment of the electron removing the s12 degeneracy The magnitude of the effective magnetic field is proportional to the orbital velocity and hence to the orbital angular momentum The coupling Hamiltonian can then be But you never carry out these calculations so this is pointless and there is no theoretical value to compare written as 52 39 s where E is related to the strength and radial gradient of the central potential The ground state of sodium has 0 so its energy is unaffected but the 213 state is split into two states with s32 12 The two observed lines then arise from transitions 21333 9812 and 213 12 gtZSU2 By comparing the calculated value of this energy difference with the measured splitting we can decide if this is an adequate explanation III Procedure This lab is an experiment with the Michelson interferometer We calibrated the interferometer with a helium neon laser measured the wavelength of the sodium doublet and measured the separation of the sodium doublet This was done with a modular Michelson interferometer There are two mirrors and a beam splitter mounted on magnetic bases One mirror is movable The beam splitter is the cube type two 45 degree prisms bonded together Michelson spectrometers can produce several types of fringes depending on the illumination Interference fringes can be either real or virtual and either localized or non localized As with ordinary images real fringes can be observed on a screen Virtual fringes formed by parallel or diverging rays can be observed directly by eye or telescope or they can be projected onto a screen with a lens Localized fringes appear to originate at some particular surface perhaps within the apparatus The imaging device eye telescope must be focused on that surface in order to see the fringes Non localized fringes fill an extended volume as in Young39s two slit experiment The diffuse sources we will use produce different types of fringes depending on the mirror adjustments By observing with the unaided eye 15 20 cm away from the spectrometer one tends to set the mirrors to produce virtual fringes localized at the last optical element in the interferometer This location can be verified by placing the telescope about a meter away and focusing it on the various optical elements By placing the eye very close to the spectrometer or using a telescope focused at infinity one can find virtual fringes localized at infinity This is the desired operating mode and the one assumed below Finally both spectrometers can produce real non localized fringes but the patterns are hard to see because of scattered light The interferometer was aligned by the lab instructor We were careful not to bump the mirrors One mirror can be moved with a large brass wheel The wheel is connected to a micrometer screw by means of a worm gear The micrometer screw in turn translates the mirror The gear ratio is 40 1 The brass wheel is marked in 2 degree graduations When the wheel is turned the mirror will move By measuring the number of degrees the wheel is turned the distance the mirror moves can be calculated since the micrometer screw moves 05 mm per turn In the first part we checked the motion calibration with the He Ne laser On the wall we saw red fringes from the interferometer We turned the brass wheel counter clockwise to remove any screw lash watching carefully how the fringes moved We wrote down the starting position of the brass wheel and then turned it counter clockwise very slowly We counted 50 fringes that collapsed and then stopped recording the reading of the brass wheel at the end This procedure was repeated In the second part we measured the average wavelength of the sodium doublet We replaced the laser with a sodium lamp and placed a piece of frosted glass in front of the beam splitter The instructor realigned the interferometer and we saw This is relatively useful but it belongs in the introduction A gure would be more helpful Mostly irrelevant You are not writing a treatise on interferometers But what does it look like and how does it work Too much detail and too little information here F ringes of what Again lots of useless detail but no explanation How many times Averaged Errors fringes when we looked into the beam splitter We did the measurements as in part one The final experiment was to measure the separation of the two lines in the sodium doublet We carefully removed the brass wheel from the interferometer and turned the micrometer screw until the fringes vanished This is called quotwash outquot We recorded the micrometer reading and then turned the micrometer until we found the next wash out This was repeated five times We did not have to back the mirror out to avoid going through zero path difference IV Data From the two readings of the brass wheel and the gear ratio we calculated the distance the mirror moved The equation is K22dn For the laser we found 2 6348 nm The expected wavelength is 6328161 nm The percent error between this and our average experimental value is 03 For the sodium we found amp1 5820 nm The expected wavelength is 589295 nm The percent error between this and our average experimental value is 124 The equation to use for the doublet separation is A A22d We found A 57878 nm but we expected 5967 The difference is 3 The total error was computed by the method of percent errors and found to be 00332 nm so we are OK V Conclusions The errors in this lab are due to the accuracy of the scales used in measuring d and to human error due to eyestrain in watching the fringes There is also a problem with slipping of the brass gear on the work gear We took special care in this lab to reduce the errors but we did not entirely succeed Nevertheless our measurements are pretty good and we learned a lot about atoms To add to the insult many parts of the text are a direct paraphrase of the lab instructions This is plagiarism Explain what is happening not what to turn on an apparatus the reader has never seen Source Derivation Symbol de nitions Is this a calibration Estimate your own uncertainty not deviation from others Source Derivation Significant figures What kind of error Colloquial phrasing Not true Fix the apparatus No obvious relation to the results reported