MAE 170 Lecture 2 AD conversion, sampling rates Error - PDF

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					          MAE 170
          Lecture 2

A/D conversion and sampling rates
         Error Analysis
         Report Writing

          October 5, 2009
              What is due this week?

•    Worksheet (from Week 1)

•   Pre lab (summary of what you will be doing in lab. –
             not procedures!)

    - Please type, < 1 page

•   Prepare for Lab Quiz
              Objectives for this week

• Lab: Understanding A/D converter
   – Sampling resolution
   – Sampling rate

• Error Analysis

• Report Writing
A/D conversion and sampling

                Chapters 4 & 5
 Introduction to Engineering Experimentation
              Wheeler and Ganji
    What does an A/D converter do?
• Most instruments give an analog output
  – e.g. voltage varies continuously with subject of
• Computer deals with discrete (digital) data
  – Inherent error between analog signal and digital
     • How quickly can we capture a changing signal (how
     • How close is the digital value to the analog (how good)?
• Binary representation - # of bits corresponds to
  number of powers of 2 represented
 A/D conversion basics – Dynamic Range

• Dynamic range
  – The range between the lowest possible reading and the full-
    scale reading of a digital signal
  – In the case of a linear converter, directly related to the # bits
    in the conversion
          Bits   1      2      6      8      10     12
          2N =   2      4      64     256    1024   4096

  – Tells us how many pieces we can chop our signal into
  – Sometimes nonlinearity has to be introduced to
    accommodate range of transducer
  A/D Conversion Basics: Resolution
• Related to dynamic
  range, typically                          12


  – Lowest bit determines

                                ADC count

    resolution                              4


• Resolution = 1 /    2N,
                                                 0        2   4         6      8     10   12

  where N = # bits
          Bits   1      2   6                         8           10          12
          2N =   2      4   64                       256      1024           4096
         Res.    50    25   1.6                      0.39     0.1            0.024
    A/D Conversion Basics: Resolution
• For the DAC as the resolution gets bigger, the measurements
  become more coarse and the DAC can not resolve small
  voltage increments.

• On the other hand as the DAC resolution gets smaller, the
  measurements become more accurate because smaller
  voltages can be measured.

• So resolution in the context of the DAC is different than say
  for a microscope, where higher resolution implies you can
  resolve smaller things.)
   A/D Conversion Basics: Bandwidth
• Sample speed affects the measurement
  – Nyquist sampling criterion: sample at 2fmax or faster to
    prevent “aliasing” the signal due to undersampling
                  10 Hz Signal

• There are ten cycles in one second
            Sampling rate ~ 11 Hz

• The signal appears to be a sine wave
• One cycle appears in the time that 10 cycles occurred
  for the sampled data
• The frequency 1 Hz is the difference between the
  sampling frequency and sampled rate
           Sampling rate ~ 18 Hz

• Difference is 8 Hz
• The incorrect frequencies in the output data are
  called aliases
          Sampling rate theorem
• Any sampling rate greater than twice fm, the
  lowest frequency will be the same as the actual
• The restriction on the sampling rate is known as
  the sampling rate theorem
• This theorem states that the sampling rate must
  be greater than twice the highest frequency
  component of the signal in order to construct
  the original signal
                  Week 2 lab experiment:
                  "How fast, how good?"
• The resolution of our data
  acquisition card (DAC) is
  determined by the resolution of
  the A/D converter
   – Determine the resolution of the
     DAC using the digital multimeter
   – Hint: 6, 8, 10, 12, 16 bits are typical   Realize that any noise and
     resolutions                               an offset in the DMM may
                                               also affect your
   – Use the DAC to generate a voltage
       • Calculate the corresponding
   – Use existing "Generate"
            Week 2 lab experiment:
            "How fast, how good?"
• Send analog output of computer back to the
  – No influence of DMM on measurement
  – Use existing ""
  – Use "continuous run" mode for LabView data
• Sample a known signal at a known rate
  – "Sampling"
              Lab 2 objectives

• To determine the resolution of the DAC
• To investigate the importance of sampling rate
• To demonstrate your LabView VI is working
  Reporting experimental
   and associated error

       Please read Chapters 2, 6 & 7
Introduction to Engineering Experimentation
             Wheeler and Ganji
        Noise in typical measurements
• Noise
   – Anything that obscures the intended signal
       • Frequency spectrum of noise?
       • Source of noise?
• Common types of "white noise"
   – Johnson noise (resistors)
       • Noise generated by thermal effects
           – VJ(rms) = (4kBTR f)1/2 = 0.4 µV for a 1kQ resistor, 10kHz bandwidth
   – Shot noise
       • Quantum nature of electrons gives a statistical fluctuation in the current
           – ISHOT(rms) = (2qIDC f)1/2 = 0.4 µA for 50 A current
   – Johnson noise and shot noise are based on physics and work basically
     the same whether your components are cheap or expensive
• Expect that your signal will fluctuate
      Other type of noise: 1/f sources
• Many other types of noise follow a 1/f decay
• Define the decibel (dB)
     • dB    10log10(X/X0)
  – There is equal power drop in noise per decade of frequency
     • Gain = 10log10(P1/P2)
     • P1 ~ 1/f1 and P2 ~ 1/f2
         – If f1 = 2f2 (frequency halving)
               • 10log10(1/2) = -3.01 dB
     • Hence, 1/f noise falls off at -3.01 dB/decade
  – P ~ V2
     • Gain = 20log10(Vout/Vin)
         – We will see later this is the fundamental property of an operational
           amplifier (op-amp)
    Measurement error & definitions

• Error = measured value - true value
  – Does not imply mistake in measurement
     • But mistakes in measurement cause error
• Experimenter usually doesn‟t know error of
  – What experimenter can estimate is the uncertainty
     • The uncertainty is an estimate (some level of confidence)
       of the limits of error in the measurement.
                Accuracy & precision
• Accuracy - closeness of agreement between measured an true
  value - used to identify uncertainty
• Precision - how often the instrument gives the same value
• The error of a measurement system (thermometer, voltmeter,
  etc) is usually 1/2 of the last precise digit. E.g. 1.1 is read, can
  say the error of the instrument is ± .5 V
   – A measurement may be precise, but not accurate

 Not precise                                            Precise
 Not accurate                                           Accurate

                                  Not accurate
         Confidence & uncertainty
• Example:
  – Voltage measurement
    • Could have a 95% confidence level that the
      uncertainty is ± 1V
       – Error is < 5% that the voltage varies > ± 1V

• Narrow uncertainty levels
  – Use of high quality, calibrated equipment
  – Make many measurements
     Random and systematic errors
• Systematic error = average of readings -
  true value
• Random error = reading - average value
  of readings
                          Range of
    True value
                        random error

                    X         XX X X   X

                 Systematic              Average of
                   error               measured values
                Systematic errors
• Consistent, repeatable errors
  – Affects the accuracy of your result
     • A major source is an uncalibrated instrument
        – XM = XT + C    XM = CXT        XM = f(XT)XT
           • Where M and T refer to measured and true
     • Human error
        – Misreading scale
           • Reading ˚F instead of ˚C
           • Measuring mV instead of µV
        – Not properly zeroing or using a wrong offset
        – Not taking note of atmospheric fluctuations
    Systematic errors (cont.)
• When measurement device alters what is
  to be measured
  –e.g. thermocouple alters temperature of bath

• When measuring device is affected by
  other things that what is to be measured.
  –e.g. metal ruler used inside and outside,
   humidity variations
        Systematic errors (cont.)
• Not obvious to experimenter
  – Compare measured values to theoretical
  – Compare measured values to values measured in
    another lab
• Minimize by
  – Taking careful measurements - eliminate human
  – Take time to calibrate the instruments
        Example of systematic error
• Calibration test, 10 measurements using digital
  voltmeter of a 6.11V battery
  – Readings: 5.98, 6.05, 6.10, 6.06, 5.99, 5.96, 6.02,
    6.09, 6.03, 5.99
     • Determine average = 6.03V
     • Systematic error = average - reported = -0.08V
• Thus, the battery is really 6.03 V
                  Random errors
• All experiments will have random error.
  – Affects the precision of a result, not its accuracy.
    Caused by lack of repeatability in the output
• Random errors are the major focus of your
  error analysis
  – Decrease uncertainty by repeated measurements
• Minimize by:
  – Eliminate uncontrolled variables
     • Shield, ground equipment from electrical noise and
       temperature variations
         Errors of a measuring system
• Precision of a measuring system
   – Given as tolerance or %
       • e.g. measuring device has a tolerance of 1.5%
         for a range of -100 to +100 V
           – Systematic uncertainty = B = ± 1.5 V
       • e.g. most resistors have a tolerance of 5%
           – 40    resistor is written as 40 ± 2

• Reading error
   – Take the error to be 1/2 of the finest scale you
     can read
       • For a digital thermocouple
           – We can read this number to be 12.80 ± 0.05
       • For a 0.3 m ruler the finest division is 1mm
           – Estimate you can read the ruler to 1/2 of the
             minimum scale, or 0.5 mm
Statistical analysis - general definitions
• Population
  – Comprises entire collection of objects,
    observations under consideration
    • Examples: batch of light bulbs produced in a
      certain period
• Sample (this is what you have)
  – Representative subset of a population
    • Example: 10 light bulbs selected out of
      population of 1000 produced
   Measurement of central tendency

• Most common is the mean of the sample

                x1   x2   x3  xn   n
                         n         i 1   n

• Median
  – Exact value at center of data set
• Mode
  – Most frequently occurring value
   Measures of dispersion - spread or
          variability of data
• Deviation                    • Sample standard
        di        xi       x
• Average deviation               S
                                       n    xi   x
              n     di                i 1    n 1
              i 1      n
• For a Gaussian
  distribution, 68% of the
  data falls within
              x        d
• 60 temperature measurements
   Number of   Temperature
   Readings       (ūC)       Mean = {1x(1089+1092)+2x
       1          1089       (1094+1115)+3x1112+4x(1095+1
       1          1092
       2          1094       110)+…}/60 = 1103˚C
       4          1095
       8          1098       Median = 1104˚C
       9          1100
      12          1104       Mode = 1104˚C
       6          1105
       5          1107       Standard deviation S = 6˚C
       5          1108
       4          1110
       3          1112
       2          1115
          Normal (Gaussian) distribution:
              the 68 - 99.7% rule
68% of the observations fall within 1 of the mean
     between      and
95% of the observations fall within 2 of the mean
     between       and
99.7% of the observations fall within 3 of the mean
     between        and
                                    2           P(x)
               1           x
  P(x)              exp         2
               2            2

µ = population mean
  = standard deviation of the population mean          x
  Correlation of experimental data

• Correlation coefficient, rxy, used to
  determine if there is a functional
  relationship between two measured
  variables, x and y                                          n
                                                              i 1
                      (xi        x)(y i           y)      x
                i 1
    rxy   n                          n
                (xi         x)             (y i    y )2       n
          i 1                        i 1                            yi
                                                              i 1
               Least-squares fit
• Systematic approach to finding a linear
  – n pairs of data (xi,yi)
    • xi assumed to be error free
  – Seek to fit Y = ax + b
                                                                     x       x
  – Each xi has error ei      y           ei                 x

                                               x     x
  – ei = Yi-yi                        x            value of Yi
                                          x        value of yi

                        Least-squares fit
• Can solve for Y = ax + b
• Resulting equation is the least-squares
  best fit
• Measure of adequacy of fit
  – Coefficient of determination, r2 (should be
    close to 1)
                (axi         b     y i )2
 r2   1   i 1
                      (y i       y )2
                i 1
              Propagation of errors
• Following technique used to determine how
  error propagates through an experiment.
  Combines uncertainty of each step
  – M = result of calculation
  – X, Y, Z
     • Numbers used for calculation
  – SM
     • standard deviation of result
  – SX, SY, SZ
     • Standard deviation of numbers used in calculation
               Propagation of errors
• Addition and subtraction
                                             2     2    2
                                   SM       SX    SY   SZ

  – M = X + Y- Z                                  2         2        2
                                             SX        SY       SZ
• Multiplication and division      SM   M
                                             X         Y        Z
  – M = XY/(Z), M = XYZ
• Logarithm                   SX
  – M = log X    SM   0.434

  – M = ln X     SM
• To calculate the power consumption in a
  resistive electric circuit, P = IV
  – V = 100 ± 2 V
  – I = 10 ± 0.2 A
    • P = 1000 W
• What is the error in the calculated power?
  – SM = 1000[(2/100)2 + (0.2/10)2]1/2 = 28.3 W
    • Then P = 1000 ± 28.3 W
Reporting measurements: significant figures

• A piece of data should be reported with no more
  significant figures than are known
  – Your calculator is happy to carry along lots of points after the
    decimal place!
  – The last significant figure in any stated answer is typically of
    the same order of magnitude as the uncertainty
  – Several extra digits may be carried through calculations, but
    rounding should happen with the final answer
     • The final answer may have no more precision than the least
       precise component of the calculation!
                     Significant figures
Always use units when recording and reporting
Always record data to the proper number of
  significant figures
   leading zeros--never significant           e.g. 0.0000000001
   captive zeros--always significant          e.g. 1.0000000001
   trailing zeros--significant only if the number contains a
                              decimal point
             1000000000        (?)      1.0000000000 x 109 (11 sig. figs.)
                                          1.0000 x 109 (5 sig. figs.)
                                          1.0 x 109 (2 sig. figs.)
                Significant figures
When numbers are multiplied or divided,
the number of significant figures in the product
or quotient cannot exceed that of the least
precise number used in the calculation

             e.g. 1.0034 cm x 2.0 cm = 2.0068 cm2 = 2.0 cm2

                                (calculation)            (report)

In addition and subtraction, the sum or the
difference cannot be stated to more places after
the decimal than the term with the least number
of places after the decimal.
              e.g. 1.0 liter + 0.001 liter = 1.0 liter
           Significant figures
• Be aware of the limitations imposed on the
  number of significant in your result by the
  magnitude of your error.
• Only one of these reported values for a
  weight uses the correct format. Which
          20.15 gms        20.2 ± 2 gms
          20.2 ± 1.5 gms
     Reporting error in an experiment
• For single data points, estimate each source of error
  as well as you can, state the likely error sources if
• For data where replicate measurements are possible,
  typically an error estimate is given by 2S, where S
  is the standard deviation in the measurement
  – Why? This is the 95% confidence interval
• Add to this any separate estimates of error that may
  not be evenly distributed around the best estimate of
  the measured value
                                 REPORT M ± 2SM
         Expectations for reporting

• We expect that you will state error estimates for
  all of the data in your reports, including in the
  report and HOW you estimated the error

• Any reports submitted without a discussion
  of error will be NOT ACCEPTED!
   Laboratory Report Writing

    Please read Chapter 12 (sec. 12.2.7)
Introduction to Engineering Experimentation
             Wheeler and Ganji
                Key concepts in writing
• Concepts related to readers and writers
   – Purpose
   – Why are you writing this document?
      • Goals
          – to persuade, inform, document?
      • Academic purpose
          – Display of knowledge
   – Audience
      • Who is reading your document?
      • Consider multiple readers and readers' purposes and background
        knowledge, etc.
• Concepts related to texts
   – Features of content, organization, language and format are determined
     by your audience and your purpose
   – Content
  – The information contained in your document
• Main goal is to communicate to an audience
      Important points about your
           laboratory report
• Your audience is well known
• To make sure that you understand the
  material and ideas
• The report should be clear and coherent
• The report should be typed on a computer
• Details of the logical process
        Writing as part of a team
• If different people are writing different
  sections, one person should edit the final
• Team writing needs careful planning
• Groups should agree on the outline of the
  report before drafting starts
• All of the authors should read and approve
  the final version
      Structure of your lab report
• 4 page maximum of body of report
  – Including text, figures and tables
  – 1 inch margins around each page
  – Use 11 point Times or Times New Roman font or
    10 point Ariel or Georgia font
• Do NOT use a double-column page format
  (use single column)
• Appendix to include raw data
            choosing the main headings
• Main choice of headings
   –   Title page                                   separate page
   –   Abstract                                     separate page
   –   Introduction
   –   Theory
   –   Methods and procedures
   –   Results                                      4 pages maximum
   –   Discussion
   –   Conclusions
   –   Error analysis (can be in Discussion)
   –   Tables and figures
   –   Appendices
        • Raw data, lengthy procedures, graphs that are too long for the body of the
            Outline of the report
• Write each heading at the top of a sheet of
• Write the main points you can think of under
  each heading
• Find all your notes, figures and tables from the
• Remember it is very important to write every
  detail of your experiment
You must keep careful records
             Important points
• Decide which figures you need
• Make lines and curves clear, label and
  differentiate them clearly
• Label axes simply and clearly
• Mark scale calibrations clearly
• Number and identify the figures in the text
                  Title page
• The title answers the question
  – What is this report about?
• The title page should be
  – Concise
  – Informative
  – accurate
   Title page example
            Laboratory 1
Dynamic Behavior of Electrical Networks

 Department of Mechanical and Aerospace
    University of California, San Diego
                 MAE 170

        Names of group members
             Day and time
            Group number
            Date submitted
• The abstract is an abbreviated, accurate
  representation of the content of the report
• It should be
   – Informative
   – Quantative
   – Short
      • Typically one paragraph
• Do not refer to anything not in the main body
• Write complete sentences that follow each other
• Use the third person (as with the rest of the report)
                                  An example

The purpose of this experiment is to calibrate a pressure transducer, an accelerometer, and determine
the spring constant k. The calibration of the pressure transducer measures a sensitivity of 8.5 mV/cm
H2O with a 15% error. The sensitivity of the accelerometer is 497.24 mV/g, with a 0.6% error. Two
methods namely static and dynamic were used. The static spring constant was determined to be 59.5
0.1 N/m, and the dynamic spring constant was 52.9 0.1 N/m.
• The main questions to be answered
  – Why did you do the work?
  – What is the purpose?
• Deal with these questions interestingly and as
  simple as possible
• Tell your readers briefly what you examined
• Indicate your experimental approach
• Cite the published work, lab hand outs, etc.
I. Introduction
    LRC circuits are present in a large number of modern devices, such as in the radio. In the laboratory,
in order to gather information from signals recorded in the presence of noise, using RC circuit filters to
eliminate noise becomes an important application. The filtered signal can be further processed by
amplification with LC cir-cuit resonance.2
    This experiment serves to not only examine the behavior but also the use of LRC circuits. The first
goal was to investigate the response of a first-order RC circuit to signal waveforms and to apply the time
constant of the voltage response towards finding the unknown value of a capacitor in the circuit. Different
frequency responses in low pass and high pass filters were recorded. The phenomenon of resonance
frequency of a LC circuit was also observed. The LC circuit was further used to demonstrate a method of
finding inductance.

                         Good way of citing someone else work

Good way of numbering equations
            Experimental procedure
• Motives
  – Apparatus/experimental set-up
  – Procedure
    • Step by step organization
• Organization
  – Paragraph unity
  – Informative headings
• Language issues
  – Past tense
  – Passive and impersonal subjects
Pr o c e d u r e
                                                                           Use past tense
Th e fir st s t e p in th is e xpe ri m en t is to c a lib ra t e a p r es s ur e t r a n sd u c er. T he        Set u p in f ig ur e 3 i s ne e d e d fo r th is pa r t o f t he e xp e r im en t.
se t u p in f ig ur e 1 is ne e d e d for t h e c al ib r at io n.

                                                                                                                                          Fig. 3 Se tup fo r a c cel e ro m et e r frequ e ncy r esp ons e t est

                                                                                                                 Fir s t, t he d is p la ce m en t o f t he sp r in g w it h o n ly t h e ac ce ler o m e t e r is t a k e n a s a
                                                                                                                 ref e r en ce p oi nt . T he s t ati c m e t h o d o f d et e r min in g k s t ar t e d w it h add in g
                                                                                                                 we igh t to t h e s pr in g, an d me a su r e m en t o f t o ta l d is p la c e me n t a n d w e igh t
                                 Fig. 1 Se t u p for p ress u r e m easu r e m ents                              a d d e d is r e co r d e d. T h is is r e pe a t e d fo r ab o u t fo u r t ime s a n d th e s t a t ic p a r t o f
                                                                                                                 ex p e ri m e n t is c o m p le te.
Bot h c o lum n s of w a t e r w e re po s it io n a t 10 0 c m. V ol tag e r ea din g is ta ke n as
on e of t h e c o lum n o f w at e r is l o we r a t in cr em en t o f 5 c m. Ab o u t 2 0                       Ne xt is t h e d y n am ic m e t h od . As bef o re, w e igh t is ad d e d to t h e s pr in g, b u t t h e
mea s u re m e n t s a re m a de an d d a ta is r e cor d e d. A b o u t a n ot h e r fiv e                      a c ce ler o m e t e r fr eq u e nc y r e sp o n s e VI is u s ed fo r th is p a r t o f t h e e xp e r im en t.
mea s u re m e n t s a re ta ke n to se e if hy s t e re si s o cc u r s. Th is c onc lu de s t h e              Aft e r w e igh t is ad d e d, t he w e igh t i s p u ll d o w n fo r a b o u t 5 cm a n d is r ele a se d.
ca libr at io n o f t he p r es s ur e t r a n sd u c er.                                                        Th e VI is ac t iva t e d as t he w eig h t is b e in g r elea s e. T he V I g e n er a te s a si ne wa v e
                                                                                                                 gr ap h, an d t he p e ri o d fr om t h e g r a p h is u sed t o c a lc ul a t e k. T hi s is r epe a t e d
Sec on d p a r t of t h e e xp e r im en t is t o c al ib r a te an a c ce le r o m e t e r. T he s e t u p in
                                                                                                                 for t wo m o re ti mes an d t h e e xp e r im en t i s co m p le t e d.
fig u re 2 is n e ed ed fo r t he c a lib ra t io n.

                                                                                                                                         Please reference it if you are
                                                                                                                                          using material from other
                               Fig. 2 s e tup fo r calibr a tion of ac c eler o me t er                                                            sources
Th e a cc el e r o m e t e r s t a r t e d a t 9 0 o an d mea s ur e m en t s ar e ma d e as t h e a n g le is
in c r e m en t a lly d ec rea si ng. Sa m e as t h e p r e ss u re tr an sdu ce r, a n ot he r fiv e o r s o
mea s u re m e n t a r e ma d e to te s t fo r hys t er e sis.
La s t p a r t of t h e e xp e r im en t is t o fi ne t he s p ri n g c o n st a n t k u si n g s ta ti c an d
d y na mi c m et h o ds.
               Data and results
• You are answering the question
  – What did you find and observe?

• Emphasize results that answer the question you
  are examining
  – Put secondary results after the most important ones

• Don't suppress valid results that appear to
  contradict your hypothesis
  – Suppressing such results is unethical
    • Explain why they are anomalous
             More on Results
• Don't repeat in the text all the numbers
  that are presented in tables and figures
• Don't repeat the table title and figure
  caption in the text

                            Note: no error bars

                                                                      Caption is not clear

Use Fig. instead of graph

                            No Y-axis title

                                    This is a good discussion point
• You are answering the general question
  – What do your findings mean?
• The discussion is where you answer specific
  question(s) you stated in the introduction
• Discuss any possible errors in your method and
• Do not refer to every detail of your work again
• A useful way to open the discussion is to use the end
  of the introduction as a starting point
• Mention the applications of the experiment at the end

This belongs in the
experimental procedures

There is no point in
writing a long
discussion if you are
just repeating text from
previous sections
• Distinguish between results and

• Introduce your conclusions by using a
  strong verb such as 'show' or 'indicate„

• Identify speculation by using 'might' with
  the verb

Co n c l us i on
Th e ide a of th is ex pe ri m en t is to un de r st an d t h e func ti on of a p re s su re
tr an sdu ce r an d a n ac ce le r o m e t er. Hy s te re ses do occ ur in t he t wo de vi ces, bu t
t he e ff ec t i f very s ma ll an d c a n be n e g lec t. T he s p ri n g co n s t a n t is d e te rmi n e d
by t w o d iff e r en t m et h o ds, s t a t ic a nd d y n a m ic. T he t wo m et h o ds ha v e si m ila r
re s u lts, b u t th e dyn ami c m et h o d se e m s to hav e m or e e r r o rs sin ce m o re
eq ui p me n t a n d s te p s a re invo lv e d. Over all , t he ap p li ca t io n s o f t h e se t wo
de vi ces a re c le a re r a ft e r u n d e r s t an d in g h o w t hes e t wo d ev ice s fun ct io n s
t hr oug h t h is e xp e r im en t.
                                      Error analysis
 Narrative describing sources of error
VI . Error A na l y s is
     The      gains for t h e non -inv erting a n d in v e r ti ng a m pli -fier ci rc uits d iff e re d sligh t ly fr om
the exp ec te d g ai n s. Thi s m ay h av e b ee n d u e t o t h e fac t tha t t h e DM M r e a di n g of t he
r e sis t ers u se d to b ui ld t h e ci rc uits w e r e sm al ler t h an the v a lu e s p r o v ide d by the fa ct o r y.
Alth o ugh t h e d if fer en c es w e r e w ithin t h e 5% e r r or ran ge , t h ey sti ll aff ec te d t h e fin a l gain of
the a m pl ifie r . Als o, t h e d iff e r e n ce s in g a in m ay b e du e t o u na cc oun te d r e sis t an c es in t he
wi r e s , o p -am p, a n d ot he r cir c uit e lem en ts, in ad dit ion to t h e r e s ist o rs u se d.
     The      o p -a m p ha d a sm all DC of fs et t h at d r ifte d t o m or e sign ifi c an ce ov er t ime . Th is
sho we d that t h e o p -a m p w as not ide al in that the r e w as d iffer ent ial v olt a ge e r r o r a n d n oi se
e xis t en t . Bu t t h e DC o ffse t w o ul d h a v e ha d lit t le eff ec t on t h e v ol t a g es m ea sur e d si nc e t h e y
w er e me as ur e d for p eak -t o -pe ak v a lu e s .
     Ext re m e fluc t u a tio n o n th e o sci llo s cope o cc u rr e d d u rin g D C of fs et m e a sur e m en t,
ca using d if ficu lty in p in p ointi n g a ce rt a in v a lu e.
     The       clip ping ef fe ct d e m ons t ra te d w ith t he non -in -v er t ing am pl ifier w a s expe ct ed t o
o c cu r at 3 0V p -p b u t ins t e a d o ccu r re d at 2 7.6 6 0. 0 5V . This m ay p a rtly b e du e to t he
osc illo sc o p e r e so lut ion. The ex act p oint o f cli p p ing w as di ffic ult to d e t e r mi n e by e y e sin ce
the o s cil los co p e sig n al s h ad thic k lin e s . Als o , t he ou tp ut v o lta ge m ay ha v e b e e n limi te d b y
int e rnal los s es s u ch a s h e ating within t h e o p -a mp it s el f. 4

    Then include your quantitative analysis
• Acknowledge briefly any substantial help
• This section can be placed before the
   – Example

We would like to thank Mike Watson, Nick Busan, and Dan Zemler for assistance and useful discussions..

  P. Bandaru and J. McKittrick. “Lecture 3: Dynamic Behavior of Electrical Networks.”<http://mae17>.
  P. Bandaru and J. McKittrick. “Experiment 3. Filters and LRC Circuits Lab Handout.”
• Lengthy material related to your report
• If you cite published work in the appendix,
  add the reference to your list of references
               Revising the first draft
• Examine the text for logical necessity, order, accuracy and
• Check that the tables and figures are necessary
• Check the accuracy of citations
• Check spelling and grammar
• Make sure figures and tables are listed chronologically and that
  each table and figures is referred to in the text.
         This week – objectives
• Determine the resolution of the DAC
• Investigate the importance of sampling
• Demonstrate that your LabView is working

• Please don't forget to write your pre-lab
             Experiment 2
          sample quiz questions
• To capture 150 kHz, theoretically what is the
  lowest sampling frequency?
• 100 Hz signal sampled at 10, 90 and 110 Hz
  results in _____, ______, ______
• Representing (by connecting the dots) a 100 Hz
  sine wave when sampling at 100, 200 and 2000
                Experiment 2
             sample quiz questions
• For a -5 to +5 volt range, what is the resolution for a
  N-bit converter where N = 10, 12 and 16?
   – Remember as the resolution becomes larger, the
     measurements become more coarse
                                      b a

   – As the DAC resolution gets smaller, measurements become
     more accurate
                    Next week
• Kirchoff's laws
• Filters
  – High pass filter
  – Low pass filter
• RLC circuits