Sensor - Week 02 - Sensor Characteristic

7/23/2019 Sensor - Week 02 - Sensor Characteristic

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Set awan P. SaktiDe t. of Ph sicsBrawi a a Universit

SENSORS PERFORMANCECHARACTERISTICS


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Input and Output

Input: stimulus or measurand (temperature pressure, lightintensity, etc.)

Ouput: electrical signal (voltage, currentfrequency, phase, etc.)

Variations: output can sometimes be displacement

(thermometers, magnetostrictive and piezoelectricsensors). Some sensors combine sensing and actuation


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Transfer function

Relation between input and output Other names:

Input output characteristic function

transfer characteristic function Response characteristic


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Type of Transfer function

Linear or nonlinear Single valued or not

One dimensional or multi dimensional Single input, single output

Multiple inputs, single output In most cases: Difficult to describe mathematically (given graphically)

Often must be defined from calibration data

Often only defined on a portion of the range of the device


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Transfer function approximation

T1 to T2 - approximately linear Most useful range

Typically a small portion of the range

Often taken as linear

S =f(x)


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Data taken from Transfer function

Other data from transfer function saturation

sensitivity

full scale range (input and output)

hysteresis

deadband

etc.


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Transfer function (cont.)

Other types of transfer functions Response with respect to a given quantity

Impulse response, Step response, Linier response, etc

Performance characteristics (reliability curves, etc.)

Viewed as the relation between any two characteristics


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Impedance and impedance matching

Black box theory of two port devices Input impedance: ratio of the rated voltage and the resulting

current through the input port of the device with the output

port open (no load)

Output impedance: ratio of the rated output voltage and

short circuit current of the port (i.e. current when the output is

shorted)


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Sensor Impedance

Only output impedance of the sensor is relevant As it is mainly in electrical circuit, hence electrical impedance

is importance especially for further processing of signals


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Importance of Impedance

Impedance affects sensor (system) performance Example: 500 W sensor (output impedance) connected to

a processor Processor input impedance is infinite (Fig. b)

Processor input impedance is 500 W (Fig. c)



Strain Gauge Impedance Example

Case: impedance is 500 W at zero strain, 750 W atmeasured straina. sensor output: 2.5V (at zero strain), 3V at measured strain

b. sensor output: 1.666V to 1.875V

Result: Case b:

the sensor is loaded

Sensitivity is reduced (smaller output change for the same strain input)

Case a: Loading effect not exists (infinite impedance of the processor)

Better sensitivity than case b



Impedance matching

To avoid loading effect Often need a matching circuit : from high to low or from

low to high impedancesmo

Sometimes can be done directly (C-MOSdevices/processors have very high input impedances)

Voltage output: impedance is highneed high impedanceat processor

Current output: impedance is low - need low impedance atprocessor



Why Impedance Matching ?

Achieve maximum signal transmission with less (without)reflection

In addition to the previous: Conjugate matching (Zin=Zout*) - maximum power transfer

Zin=R+jX, Zout*=R-jX.No reflection matching (Zin=Zout) - no reflection from load Important at high frequencies (transmission lines)



Range and Span

Range: lowest and highest values of the stimulusSpan: the arithmetic difference between the highest and

lowest values of the stimulus that can be sensed withinacceptable errors

Input full scale (IFS) = spanOutput full scale (OFS): difference between the upper and

lower ranges of the output of the sensor corresponding tothe span of the sensor

Dynamic range: ratio between the upper and lower limitsand is usually expressed in db



Range and Span (Cont)

Example: a sensors is designed for: -30 C to +80 C tooutput 2.5V to 1.2V

Range: -30C and +80 C

Span: 80- (-30)=110 C Input full scale = 110 C

Output full scale = 2.5V-1.2V=1.3V

Dynamic range=20log(140/30)=13.38db



Range and Span (cont.)

Range, span, full scale and dynamic range may beapplied to actuators in the same way

Span and full scale may also be given in db when the

scale is large. In actuators, there are other properties that come into

play:

Maximum force, torque, displacement

Acceleration

Time response, delays, etc.



Accuracy, errors, repeatability

Errors: deviation from ideal

Sources:

materials used

construction tolerances ageing

operational errors

calibration errors

matching (impedance) or loading errors

noise

many others



Accuracy, errors (cont.)

Error: a. As a difference: e = VV0 (V0 is the actual value, V is that

measured value (the stimulus in the case of sensors or output in

actuators).

b. As a percentage of full scale (span for example) e =

t/(tmax-tmin)*100 where tmax and tmin are the maximum and

minimum values the device is designed to operate at.

c. In terms of the output signal expected.



Example: errors

)30(808.25.18.2

--

-

OV

Example: A thermistor is used to measure temperaturebetween30 and +80 C and produce an output voltage

between 2.8V and 1.5V. Because of errors, the accuracy in

sensing is 0.5C.



Example (cont)

a. In terms of the input as 0.5Cb. Percentage of input: e = 0.5/(80+30)*100 = 0.454%

c. In terms of output. From the transfer function: e= 0.059V.



More on errors

Static errors: not time dependentDynamic errors: time dependent Random errors: Different errors in a parameter

or at different operating times Systemic errors: errors are constant at all times

and conditions



Error limits - linear TF

Linear transfer functions Error equal along the transfer function

Error increases or decreases along TF

Error limits - two lines that delimit the output



Error limits - nonlinear TF

Nonlinear transfer functions

Error change along the

transfer function

Maximum error from ideal Average error

Limiting curves follow ideal

transfer function



Error limits - nonlinear TF

Calibration curve may beused when available Lower errors

Maximum error from

calibration curve Average error Limiting curves follow the

actual transfer function(calibration)



Repeatability

Also called reproducibility: failure of the sensor oractuator to represent the same value (i.e. stimulus or input)

under identical conditions when measured at different

times.

usually associated with calibration

viewed as an error.

given as the maximum difference between two readings taken

at different times under identical input conditions. error given as percentage of input full scale.



Sensitivity

Sensitivity of a sensor is defined as the change in outputfor a given change in input, usually a unit change in input.

Sensitivity represents the slope of the transfer function.

Same for actuators



Sensitivity

Sensitivity of a sensor is defined as the change in outputfor a given change in input, usually a unit change in input.

Sensitivity represents the slope of the transfer function.



Sensitivity (cont.)

Can be highly nonlinear along the transfer function Measured in units of output quantity per units of input

quantity (W/C, N/V, V/C, etc.)

For a linear transfer function, sensitivity is the slope ortransfer function.

Example:ddR

aT +b =1 dRdT

=a WC


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Sensitivity analysis (cont.)

A difficult taskthere is noisea combined function of sensitivities of various

components, including that of the transductionsections.

device may be rather complex with multipletransduction steps, each one with its own sensitivity,sources of noise and other parameters

some properties may be known but many may notbe known or may only be approximate.


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Sensitivity analysis (cont.)

An important task provides information on the output range of signals one can

expect,

provides information on the noise and errors to expect.

may provide clues as to how the effects of noise and errors

may be minimized

Provides clues on the proper choice of sensors, their connections

and other steps that may be taken to improve performance(amplifiers, feedback, etc.).


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Example - additive errors

Fiber optic pressure sensor Pressure changes the length of the fiber

This changes the phase of the output

Three transduction steps


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Example-1 - no errors present

Individual sensitivities

Overall sensitivity

But, x2=y1 (output of transducer 1 is the input to

transducer 2) and x3=y2

s1=dy1

dx , s2=dy

2

dx , s3=dy

3

dx

S =s1s2s3=

dy1dx

dy2dx

dy3dx

S =s1s2s3=dy3

dx


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Example -1 - errors present

First output is y1=y01 + y1. y01 = Output without error 2nd output

3rd output

Last 3 terms - additive errors

2=s2y

0

+y1

+y2

=y

0

+s2y1

+y2

3=s3y0+s2y1+y2 +y3=y

0+s2s3y1+s3y2+ y3


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Example -2 - differential sensors

Output proportional to difference between the outputs ofthe sensors

Output is zero when T1=T2

Common mode signals cancel (noise) Errors cancel (mostly)

E l 2 ( )


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Example -2 - (cont.)

s1=dy1dx1

, s2=dy2dx2

y=y1-y2=s1x1-s2yx

s =d y1-y2d x1-x2

E l 3 i i


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Example -3 - sensors in series

Output is in series Input in parallel (all sensors at

same temperature)

Outputs add up

Noise multiplied by product ofsensitivities

y=y1+y2+ y3+...+yn=(s1+s2+s3+...+sn)x =ns

S =ns

H i


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Hysteresis

Hysteresis (literally lag)- thedeviation of the sensorsoutput at any given pointwhen approached from twodifferent directions

Caused by electrical ormechanical systems Magnetization

Thermal properties

Loose linkages

H i E l


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Hysteresis - Example

If temperature is measured, at a rated temperature of50C, the output might be 4.95V when temperatureincreases but 5.05V when temperature decreases.

This is an error of 0.5% (for an output full scale of 10V inthis idealized example).

Hysteresis is also present in actuators and, in the case ofmotion, more common than in sensors.

N li i


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Nonlinearity

A property of the sensor (nonlinear transfer function) or: Introduced by errors

Nonlinearity errors influence accuracy.

Nonlinearity is defined as the maximum deviation from the

ideal linear transfer function. The latter is not usually known or useful

Nonlinearity must be deduced from the actual transferfunction or from the calibration curve

A few methods to do so:

N li it ( t )


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Nonlinearity (cont.)

a. by use of the range of the sensorPass a straight line between the range points (line

1)Calculate the maximum deviation of the actual

curve from this straight lineGood when linearities are small and the span is

small (thermocouples, thermistors, etc.)Gives an overall figure for nonlinearity

N ( )


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b. by use of two points defining a portion of the span ofthe sensor/actuator.

Pass a straight line between the two points

Extend the straight line to cover the whole span

Calculate the maximum deviation of the actual curve from this

straight line

Good when a device is used in a small part of its span (i.e. a

thermometer used to measure human body temperatures Improves linearity figure in the range of interest

N l ( )


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c. use a linear best fit(least squares) throughthe points of the curveTake n points on the actual curve, xi,yi, i=1,2,n.Assume the best fit is a line y=ax+b (line 2)Calculate a and b

N l ( )


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d. use the tangent to the curve at some point on the curve Take a point in the middle of the range of interest

Draw the tangent and extend to the range of the curve (line 3)

Calculate the nonlinearity as previously

Only useful if nonlinearity is small and the span used very small

S t ti


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Saturation

Saturation a property of sensors or actuators when they nolonger respond to the input.

Usually at or near the ends of their span and indicates thatthe output is no longer a function of the input or, more likelyis a very nonlinear function of the input.

Should be avoided - sensitivity is small or nonexistent

S t ti C diti


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Saturation Condition

saturated

saturated

F


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Frequency response

Frequency response: The ability of the device to respond toa harmonic (sinusoidal) input

A plot of magnitude (power, displacement, etc.) as a function

of frequency

Indicates the range of the stimulus in which the device is

usable (sensors and actuators)

Provides important design parameters

Sometimes the phase is also given (the pair of plots is theBode diagram of the device)

Freq enc response (cont)


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Frequency response (cont)

Important design parameters

Bandwidth (B-A, in Hz) Flat frequency range (D-C in Hz)

Cutoff frequencies (points A and B in Hz)

Resonant frequencies

Frequency response (cont )


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Frequency response (cont.)

Bandwidth: the distance in Hz between the half powerpoints Half-power points: eh=0.707e, ph=0.5p

Flat response range: maximum distance in Hz over which

the response is flat (based on some allowable error) Resonant frequency: the frequency (or frequencies) at

which the curve peaks or dips

Half power points


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Half power points

Also called 3db points Power is 3db down at these points:

10*log0.5=-3db or

20*log (sqrt(2)/2)=-3db

These points are arbitrary but are now standard.

It is usually assumed that the device is useless beyond the

half power points

Frequency response (example )


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Frequency response (example.)

Bandwidth: 16.5kHz-70Hz=16.43 kHz

Flat frequency range: 10kHz-120Hz=9880 Hz

Cutoff frequencies: 70 Hz and 16.5 kHz

Resonance: 12 kHz

Response time


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Response time

response time (or delay time), indicates the time needed forthe output to reach steady state (or a given percentage ofsteady state) for a step change in input.

Typically the response time will be given as the time neededto reach 90% of steady state output upon exposure to a unit

step change in input. The response time of the device is due to the inertia of the

device (both mechanical and electrical).

R ti ( t )


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Response time (cont.)

Example: in a temperature sensor: the time needed for the sensors body to reach the temperature it is

trying to measure (thermal time constant) or

The electrical time constants inherent in the device due tocapacitances and inductances

In most cases due to both Example: in an actuator: Due to mass of the actuator and whatever it is actuating

Due to electrical time constants

Due to momentum

Response time (cont )


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Response time (cont.)

Fast response time is usually desirable (not always) Slow response times tend to average readings

Large mechanical systems have slow response times

Smaller sensors and actuators will almost always respond

faster We shall meet sensors in which response time is slowed down

on purpose

C lib ti


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Calibration

Calibration: the experimental determination of the transferfunction of a sensor or actuator.

Typically, needed when the transfer function is not known or,

When the device must be operated at tolerances belowthose specified by the manufacturer.

Example, use a thermistor with a 5% tolerance on a full scalefrom 0 to 100C to measure temperature with accuracy of,say, 0.5C.

The only way this can be done is by first establishing thetransfer function of the sensor.

Calibration Method


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Calibration Method

a. known transfer function: Determine the slope and crossing point (line function) from two known stimuli (say twotemperatures) if the transfer function is linear

Measure the output

Calculate the slope and crossing point in V=aT+b

If the function is more complex, need more points: V = aT + bT2 + cT3 + d

4 measurements to calculate a,b,c,d Must choose points judiciously - if linear, use points close to the range. If not, use equally

spaced points or points around the locations of highest curvature

b. Unknown transfer function: Measure the output R

iat as many input values T

ias is practical

Use the entire span Calculate a best linear fit (least squares for example)

If the curve is not linear use a polynomial fit

May use piecewise linear segments if the number of points is large.

Calibration (cont )


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Calibration (cont.)

Calibration is sometimes an operational requirement(thermocouples, pressure sensors)

Calibration data is usually supplied by the manufacturer

Calibration procedures must be included with the designdocuments

Errors due to calibration must be evaluated and specified

Resolution


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Resolution

Resolution: the minimum increment in stimulus to which it canrespond. It is the magnitude of the input change which results

in the smallest discernible output.

Example: a digital voltmeter with resolution of 0.1V is used

to measure the output of a sensor. The change in input(temperature, pressure, etc.) that will provide a change of

0.1V on the voltmeter is the resolution of the

sensor/voltmeter system.

Resolution (cont )


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Resolution (cont.)

Resolution is determined by the whole system, not only by thesensor

The resolution of the sensor may be better than that of thesystem.

The sensor itself must interact with a processor, the limitingfactor on resolution may be the sensor or the processor.

Resolution may be specified in the units of the stimulus (0.5Cfor a temperature sensor, 1 mT for a magnetic field sensor,0.1mm for a proximity sensor, etc) or may be specified as a

percentage of span (0.1% for example).

Resolution (cont )


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Resolution (cont.)

In digital systems, resolution may be specified in bits (1 bit or6 bit resolution)

In analog systems (those that do not digitize the output) theoutput is continuous and resolution may be said to beinfinitesimal (for the sensor or actuator alone).

Resolution of an actuator is the minimum increment in itsoutput which it can provide.

Example: a stepper motor may have 180 steps perrevolution. Its resolution is 2.

A graduated analog voltmeter may be said to have aresolution equal to one graduation (say 0.01V). ( higherresolution may be implied by the user who can easilyinterpolated between two graduations.

Other parameters


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Other parameters

Reliability: a statistical measure of quality of a devicewhich indicates the ability of the device to perform itsstated function, under normal operating conditions withoutfailure for a stated period of time or number of cycles.

Given in hours, years or in MTBF

Usually provided by the manufacturer

Based on accelerated lifetime testing

Other parameters


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Other parameters

Deadband: the lack of response or insensitivity of a deviceover a specific range of the input. In this range which may be small, the output remains

constant.

A device should not operate in this range unless thisinsensitivity is acceptable.

Example, an actuator which is not responding to inputsaround zero may be acceptable but one which freezesover a normal range may not be.

Other parameters


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Other parameters

Excitation: The electrical supply required for operation of a sensor oractuator.

It may specify the range of voltages under which the device should

operate (say 2 to 12V), range of current, power dissipation, maximum

excitation as a function of temperature and sometimes frequency.

Part of the data sheet for the device

Together with other specifications it defines the normal operating

conditions of the sensor.

Failure to follow rated values may result in erroneous outputs orpremature failure of the device.

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Sensor - Week 02 - Sensor Characteristic