Some Mathematics

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Some Mathematics

• Time Response
• Significant Figures
• Statistics

http//antwrp.gsfc.nasa.gov/apod/ap040117.html
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First Order Response
Step change

• b(t)bi(bf-bi)(1-e-t/tau)
• bi initial value of the exponential change
• bf final value of the exponential change
• tau the time constant of the first order process

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Graphs of First order Responsesb(t)bi(bf-bi)(1-
e-t/tau)
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2nd order response

• 2 time constant circuit
• Like an RLC electrical circuit
• Can be
• Pure 2 time constant exponential
• Damped sinusoid Ve-t/tauSin(?t)
• Higher order circuits are similar, but have
  additional time constants.

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Measurement Error

• DMM - one least significant digit
• So for a measurement of .102 amps, it is accurate
  to - 1 mA.
• Assume you have a 100O, 5 resistor
• Calculating the Voltage across the resistor
• VIR (100O -5)(.102A-.001A)
• How to calculate the error?
• Use Square Root of Sum of Squares technique

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Square Root of Sum of Squares
Volts
So, this describes the accuracy of the
measurement
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Significant Figures

• Some common mathematical rules used in
  electronics (for both homework and lab
  assignments) are
• Use 3 significant figures for solutions (for a
  result that is correct to 3 significant figures,
  you must use more than 3 significant figures in
  all of the intermediate calculations). e.g. if
  your answer is 12E-6, write 12E-6 or 1.2E-5, if
  your answer is 1.47678E-3, write 1.48E-3
• Use standard rounding e.g. if your answer is
  1.034567, write 1.03
• Give the numerical solution to problems (unless
  the problem asks for a formula or equation). e.g.
  don't write ?79, write 8.89, don't write 127/9,
  write 14.1 (calculators are required when doing
  homework, labs, and tests)

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Statistics

• Mean
• Average
• DC
• Standard Deviation
• rms

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An example using Excel
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STDEV using Excel
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Normal Probability (Bell) Curvehttp//www.cquest
.utoronto.ca/geog/ggr270y/notes/not09d.html

• This is probably the most important of all the
  probability distributions. Among other things,
  people's weights, heights and shoe sizes are
  normally distributed, as are annual rainfall and
  temperatures of a region, IQ scores, test scores,
  and most natural phenomena in general. Many more
  variables can be approximated very well by normal
  distributions.

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Normal Distribution and Data
27.1 21.6 5.5 0C
27.1 0C
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Summary

• 1st and 2nd order time response
• Measurement Error
• Square Root of Sum of Squares
• Significant Figures in Calculations
• Statistics
• Mean
• Standard Deviation
• Normal Probability Curve (Bell Curve)