sensors

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Sensors and Transducers

Terminology and fundamentalconcepts

Measurement in Mechatronic systems

◆ Measurement is at the heart of a mechatronicsystem.❖ Allow system to determine its physical state❖ Take appropriate action

• What are some examples?

◆ Need to choose best sensor for task at hand.◆ Must reckon with performance, physical size,

input requirements, outputs, life, cost, etc.

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Functional Model for a MeasurementSystem

adapted from, Doeblin, E. O., Measurement Systems: Application and Design, McGraw-Hill, New York, 1975.

MeasurandPrimarySensingElement

VariableConversionElement(s)

VariableManipulationElement(s)

DataTransmissionElement(s)

DataPresentationElement(s)

Observer/Controller

Functional Model for a MeasurementSystem, cont.

◆ Strain gage scale

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General Transducer Characteristics

◆ See the manufacturer’s data sheets◆ Design characteristics

❖ Range : the __________ limits of measurand valuesthe transducer will respond to within specifiedperformance tolerances.

❖ Overrange : the __________ measurand valuebeyond which the transducer will sustain permanentchange in performance.

◆ Electrical characteristics❖ Excitation: what is required to __________ the

sensor.❖ Output: output signal characteristics

General Transducer Characteristics, cont.

◆ Mechanical design characteristics:❖ Size, weight, configuration, key dimensions, pin-outs,

etc.

◆ Static performance characteristics❖ How output changes in response to a constant input❖ Determined by (static) calibration

◆ Dynamic performance characteristics❖ How quickly the output changes in response to

changes in the input❖ Determined by step, ramp, and frequency response

tests

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Static performance characteristics

◆ Calibration❖ The process of applying _____________ of the

measurand to the sensor and measuring the output.• Inputs must be known to be “true” - traceable to primary

standards➤ Why?

• Measurand should be applied in an increasing mannerfollowed by a decreasing manner.

➤ Why?

Calibration Example

Pressure sensor output

0

2

4

6

8

10

0 20 40 60 80 100 120Pressure, psi

Out

put,

volts Increasing

Decreasing

Best fit line

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Static performance characteristics, cont.

◆ Error❖ The ___________ between the measured and

actual value of a quantity.• There is always error in every measurement• Need to choose sensors that give an acceptable level of

error

❖ Systematic error - remains the same for eachmeasurement

• ex. high Z vs. low Z on function generator

❖ Random error - varies from measurement tomeasurement


◆ Accuracy❖ Specification on the _______________ to be

expected from a transducer or measurementdevice

• Typically expressed as a percent of full scale output(% FS0)

• Other expressions are used, so be aware!

➤ % of reading

➤ % of span➤ absolute (in terms of measurand units)

★ can be determined from %FSO specification

• In general, accuracy specification will includecontributions from several error sources (e.g., linearity,repeatability, hysteresis, etc.)

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Accuracy Example (due to linearity only)


0.000

2.000

4.000

6.000

8.000

10.000

0 20 40 60 80 100 120Pressure, psi

Out

put,

volts

Press. Vout true Vout act. Error Accuracy Vout act. Error Accuracypsi volts volts Volts % FSO volts Volts % FSO

0 0.000 0.200 -0.200 2.0 0.304 -0.304 3.05 0.417 0.420 -0.004 0.0 0.189 0.228 2.3

10 0.833 0.999 -0.165 1.7 0.223 0.611 6.115 1.250 2.366 -1.116 11.2 0.451 0.799 8.020 1.667 2.387 -0.721 7.2 0.743 0.923 9.225 2.083 3.627 -1.544 15.4 0.656 1.427 14.330 2.500 3.942 -1.442 14.4 1.164 1.336 13.435 2.917 3.896 -0.979 9.8 1.490 1.427 14.340 3.333 5.055 -1.721 17.2 1.531 1.802 18.045 3.750 5.577 -1.827 18.3 2.227 1.523 15.250 4.167 5.931 -1.765 17.6 2.698 1.468 14.755 4.583 6.360 -1.777 17.8 2.848 1.736 17.460 5.000 7.189 -2.189 21.9 2.853 2.147 21.565 5.417 8.073 -2.657 26.6 3.632 1.785 17.870 5.833 7.851 -2.017 20.2 4.404 1.429 14.375 6.250 7.964 -1.714 17.1 4.183 2.067 20.780 6.667 8.500 -1.834 18.3 4.875 1.792 17.985 7.083 8.401 -1.318 13.2 5.424 1.659 16.690 7.500 8.668 -1.168 11.7 5.756 1.744 17.495 7.917 9.115 -1.198 12.0 6.663 1.253 12.5

100 8.333 9.447 -1.113 11.1 7.285 1.048 10.5105 8.750 9.620 -0.870 8.7 8.088 0.662 6.6110 9.167 9.293 -0.126 1.3 8.638 0.529 5.3115 9.583 9.732 -0.149 1.5 9.007 0.576 5.8120 10.000 9.876 0.124 1.2 9.805 0.195 2.0

What is the accuracy of this sensor?as %FSOas an absolute accuracy


◆ Repeatability (reproducibility, precision)

❖ The maximum difference between output readingswhen the same measurand value is appliedconsecutively under the same conditions and in thesame direction

• Typically expressed as %FSO, either as:

➤ (max value - min value)/full scale

➤ (max deviation from avg. - avg.)/full scale• As determined by two calibration cycles unless stated

otherwise

➤ Better statistical value obtainable with more cycles

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Repeatability Example

What is the repeatability of this sensor?


-2.000

0.000

2.000

4.000

6.000

8.000

10.000

12.000

0 20 40 60 80 100 120Pressure, psi

Out

put,

volts

Press. First run Second run 1 - 2 1 - 2psi volts volts volts % FSO

0 0.042 -0.181 0.223 2.235 0.542 0.039 0.503 5.03

10 1.293 1.408 -0.115 1.1515 1.817 2.511 -0.694 6.9420 2.660 2.381 0.279 2.7925 2.923 3.397 -0.474 4.7430 3.850 3.961 -0.111 1.1135 4.543 4.464 0.078 0.7840 5.046 5.141 -0.095 0.9545 5.468 5.803 -0.335 3.3550 6.327 6.364 -0.038 0.3855 6.880 6.694 0.186 1.8660 6.700 7.190 -0.490 4.9065 7.291 7.009 0.282 2.8270 7.532 7.891 -0.359 3.5975 8.081 8.211 -0.130 1.3080 8.582 8.544 0.038 0.3885 9.112 8.734 0.378 3.7890 9.095 8.452 0.643 6.4395 9.394 8.615 0.779 7.79

100 9.102 9.116 -0.015 0.15105 9.386 9.284 0.102 1.02110 9.916 9.931 -0.016 0.16115 9.772 9.564 0.208 2.08120 9.849 10.016 -0.167 1.67


◆ Linearity❖ The measure of closeness of a calibration curve to a

specified straight line• Most sensors attempt to give nominally linear outputs• Typically expressed as “within X%FSO”

• There are many ways to specify linearity:

➤ Theoretical slope

➤ Terminal linearity➤ End-point linearity

➤ Independent linearity

➤ Least-squares linearity (Be careful! There are two possible least-squares lines. The method of absolute differences is a good compromise.)

• The general idea, “fit a straight line through the data, reportthe maximum deviation as the linearity”

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◆ Linearity, cont.

❖ The method of absolute differences (MAD)• MAD gives the equation of a line approximately midway

between the two possible least squares lines

• Plot the equation, y = mx + b, where,

m =−∑

−∑=

=

y y

x x

i cgi

N

i cgi

N1

1

b y mxcg cg

= −

xNcg

i 1

N

= ∑=

1x

i

yNcg

i 1

N

= ∑=

1y

i


◆ Sensitivity❖ The slope of the calibration curve❖ Typical units are output qty./measurand unit, e.g.,

mV/°C• ex. pressure sensor sensitivity

❖ Change in sensitivity error is expressed as a % ofFSO (sensitivity drift)

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◆ Hysteresis❖ The maximum difference in output at any measurand

value when the value is approached first withincreasing measurand and then with decreasingmeasurand.

• ex. pressure sensor

• Internal damping of sensing element produces a lag inaction

➤ ex. piezoelectric actuators• Expressed as a % FSO

• Friction error (stick-slip) is distinct, but sometimes calledhysteresis

➤ Can be alleviated by dithering (tapping, vibrating)

Hysteresis, (cont.)


0.000

2.000

4.000

6.000

8.000

10.000

0 20 40 60 80 100 120Pressure, psi

Out

put,

volts

10


◆ Resolution❖ The smallest measurable input change

• ex. potentiometer

• Will often depend on what the sensor is being read by➤ ex. A/D converter

Other Comments About Errors

◆ Zero (null) shift or drift◆ Changes due to temperature

❖ Mfg. calibration at a given temperature, e.g. 25 °C

◆ How to combine error sources to determineoverall accuracy❖ If error sources are independent, use root-

sum-of-squares method

etot n1

2

2

2 2

Where ei are the independent errors expressed in %FS

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Summary

◆ Data sheets contain key information onmechanical, electrical, and performancecharacteristics of sensors

◆ Static performance characteristics:❖ Accuracy❖ Repeatability❖ Linearity❖ Hysteresis❖ Sensitivity❖ Resolution

Sensor and Transducers

Part 2

Dynamic Performance

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Dynamic Performance of Sensors

◆ Response to inputs that change as a function oftime (time response)❖ Ideal response vs. actual response❖ ¿What is the error in measurement?

◆ Characterization of dynamic performancethrough:❖ Step response❖ Sinusoidal response❖ Ramp response❖ Impulse response

System Models

◆ Model the input/output relationship of atransducer using differential equations.

◆ First order model❖ Time response can be described with a first order

differential equation

❖ ex. temperature response of all sensors

a Q a Q b Q1 0 0

′ + =o o i

′ =Qd Q

o

o

bK

o

static sensitivity= =

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System Models, cont.

◆ Second order model❖ Time response can be described with a second order

differential equation: a Q a Q a Q b Q2 o 1 o o i

′′+ ′ + =0 0

❖ Mechanical systems with both potential and kineticenergy storage elements (springs and masses)

❖ Electrical systems that store energy in both magneticand electric fields (inductive and capacitiveelements)

System Models, cont.

◆ Second order model, cont.❖ Output (solution) has two parts:

• Transient (complementary)

• Steady state (particular)

❖ Transient solution will take on one of three formsdepending on roots of characteristic polynomial:

• Overdamped : ζ>1, sum of two decaying exponentials

• Critically damped : ζ=1, exponentially decaying ramp

• Underdamped : ζ<1, exponentially decaying sinusoid➤ Many transducers have ζ=0.7 ±0.1

❖ Steady-state part will take on the same form as theinput.

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