Some Mathematics

Time Response

Significant Figures

Statistics

http://antwrp.gsfc.nasa.gov/apod/ap040117.html

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First Order Response

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

Step change

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Graphs of First order Responsesb(t)=bi+(bf-bi)(1-e-t/tau)

Two First Order Setpoint Changes: tau = .2 sec.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

Setpoint change from 2 to 5

Setpoint change from 0 to 3

5-3*exp(-t/.2)

3*(1-EXP(-A2/0.2))

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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(ωt)

• 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 100Ω, 5% resistor• Calculating the Voltage across the resistor:

V=IR = (100Ω +-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

2.10))102.

001.()05(.()102(.100 22 rmsV

52.2.10 rmsV

So, this describes the accuracy of the measurement

Volts

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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

N

XX i

1

2

n

di

xxd ii

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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.

2)(5.exp(**2

1

x

y

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Normal Distribution and Data

Normal Distribution with Mean =21.6 and StdDev=3.04 & temperature values in example

00.05

0.10.15

0.20.25

0.30.35

0.40.45

10 15 20 25 30

Temperature (degrees C)

y=

valu

e

27.1 0C

27.1 – 21.6 =5.5 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)