Chapter 3: Measurement Errors

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Chapter 3 Measurement Errors

• Agenda
• Systematic errors
• Random errors
• Total measurement systems errors

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Systematic Errors

• System errors due to the process of measurement
• used for measurement ? Improve instrument
  design
• Modifying inputs the
  conditions away from the ?zero and
  sensitivity drift ?
• Careful instrument design
• Method of opposing inputs the effect of a
  modifying input
• High-gain feedback reduce number of modifying
  inputs
• Signal filtering reduce periodic noise

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Other Sources of Systematic Errors

• in instrument components
  
• leads proper choice of connecting leads
  ?minimize resistance
• Thermal e.m.f.s thermoelectric effect (e.g.
  connection of thermocouple copper wire to
  Nickel-iron relay 40?V/oC)

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Example of Systematic Errors
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Random Errors

• Causes random, unpredictable variations ?
  calculate mean/median of the measurements,
  variance of data
• All error bounds placed on measurements can only
  be quantified in terms
• The distribution of measurement data about the
  mean value can be displayed graphically by
• Area under the frequency distribution curve
  gives the that the will lie between
  any

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Random Errors

• Mean of n measurements x1,x2,..xn is
• Median is value when the measurements
  are in
• Variance and standard deviation

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Random Errors

• The two commonly used graphic forms are and a
  .
• The normal distribution is the most popular form.
  
• The arithmetic of the distribution are equal
  and located at the .
• The normal probability distribution is about
  its mean.
• A normal distribution with a mean of 0 and a
  standard deviation of 1 is called the .
• Z value

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• Between
• 1? 68.26
• 2? 95.44
• 3? 99.74

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Distribution of Measurement Errors

• Assume that the errors are only random, the
  frequency of occurrence of each error level F(E)
  often follows normal distribution
• If the measurement errors E (x-?) then
• We have the error function

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Distribution of Manufacturing Tolerances

• The normal distribution can be extent to analyze
  tolerances in manufactured components rather
  than errors in process measurements
• Error of mean (sample deviation)

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Total Measurement Systems Errors

• Product Quotient P yz?(ab)yz Qy/z
  ?(ab)y/z where yy ?ay zz ?az
• Sum Difference S(y ?z)(1?f) where fe/(y
  ?z) and

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Examples

• Example 1 The following measurement in mA were
  taken of the current in a circuit
• 21.5, 22.1, 21.3, 21.7, 22.0, 22.2,21.8,
  21.4,21.9, 22.1
• Calculate the mean, median and standard
  deviations
• Solution
• Mean value ?(data values)/10 218/10 21.8mA
• We have 21.3lt21.4lt21.5lt21.7lt21.8lt21.9lt22.0lt22.1lt2
  2.1lt22.2.
• Thus, median (x5 x6)/2(21.821.9)/2 21.85mA

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Examples

• Example 2 An IC contain 105 transistors. The
  transistors have a mean current gain of 20 and a
  standard deviation of 2. Calculate (a) the number
  of transistors with a current gain between 19.8
  and 20.2, and (b) the number of transistors with
  a current gain greater than 17
• Solution
• (a) Pxlt20.2 Pxlt19.8 Pzlt0.1-Pzlt-0.1
• 0.5398 0.4602 0.0796? 0.0796105
  7960 trans.
• (b) Pxgt17 1-Pxlt17 1- Pzlt-1.5 Pzlt1.5
  0.9332
• ? 93.32, i.e. 93320 transistors have a gain gt17

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Examples

• Example 3 A circuit includes 2 resistors of
  nominal values 220 ohm and 330 ohm connecting in
  parallel. If each resistor has a tolerance of
  2. Calculate the total resistance of the
  circuit and its error
• Solution R1 2202, R2 330 2
• R R1//R2 (R1R2)/(R1 R2) Ra/Rb
• We have Ra 220330 (22)72600 4