Performance Characteristics of Sensors and Actuators

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Performance Characteristics of Sensors and
Actuators

• (Chapter 2)

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Input and Output

• Sensors
• Input stimulus or measurand (temperature
• pressure, light intensity, etc.)
• Ouput electrical signal (voltage, current
• frequency, phase, etc.)
• Variations output can sometimes be displacement
  (thermometers, magnetostrictive and piezoelectric
  sensors). Some sensors combine sensing and
  actuation

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Input and Output

• Actuators
• Input electrical signal (voltage, current
• frequency, phase, etc.)
• Output mechanical(force, pressure,
  displacement) or display function (dial
  indication, light, display, etc.)

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Transfer function

• Relation between input and output
• Other names
• Input output characteristic function
• transfer characteristic function
• response

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Transfer function (cont.)

• 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 (cont.)

• T1 to T2 - approximately linear
• Most useful range
• Typically a small portion of the range
• Often taken as linear

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Transfer function (cont.)

• 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
• Performance characteristics (reliability curves,
  etc.)
• Viewed as the relation between any two
  characteristics

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Impedance and impedance matching

• 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)
• These are definitions for two-port devices

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Impedance (cont.)

• Sensors only output impedance is relevant
• Actuators only input impedance is relevant
• Can also define mechanical impedance
• Not needed - impedance is important for
  interfacing
• Will only talk about electrical impedance

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Impedance (cont.)

• Why is it important? It affects performance
• Example 500 W sensor (output impedance)
  connected to a processor
• b. Processor input impedance is infinite
• c. Processor input impedance is 500 W

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Impedance (cont.)

• Example. Strain gauge impedance is 500 W at zero
  strain, 750 W at measured strain
• b sensor output 2.5V (at zero strain), 3V at
  measured strain
• c. sensor output 1.666V to 1.875V
• Result
• Loading in case c.
• Reduced sensitivity(smaller output change for the
  same strain input)
• b. is better than c (in this case). Infinite
  impedance is best.

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Impedance (cont.)

• Current sensors impedance is low - need low
  impedance at processor
• Same considerations for actuators
• Impedance matching
• Sometimes can be done directly (C-mos devices
  have very high input impedances)
• Often need a matching circuit
• From high to low or from low to high impedances

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Impedance (cont.)

• Impedance can (and often is) complex ZRjX
• In addition to the previous
• Conjugate matching (ZinZout) - maximum power
  transfer
• Critical for actuators!
• Usually not important for sensors
• ZinRjX, ZoutR-jX.
• No reflection matching (ZinZout) - no reflection
  from load
• Important at high frequencies (transmission
  lines)
• Equally important for sensors and actuators
  (antennas)

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Range and Span

• Range lowest and highest values of the stimulus
• Span the arithmetic difference between the
  highest and lowest values of the stimulus that
  can be sensed within acceptable errors
• Input full scale (IFS) span
• Output full scale (OFS) difference between the
  upper and lower ranges of the output of the
  sensor corresponding to the span of the sensor
• Dynamic range ratio between the upper and lower
  limits and is usually expressed in db

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Range and Span (Cont)

• Example a sensors is designed for -30 ?C to 80
  ?C to output 2.5V to 1.2V
• Range -30?C and 80 ?C
• Span 80- (-30)110 ?C
• Input full scale 110 ?C
• Output full scale 2.5V-1.2V1.3V
• Dynamic range20log(140/30)13.38db

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Range and Span (cont.)

• Range, span, full scale and dynamic range may be
  applied 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.

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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

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Accuracy, errors (cont.)

• Errors defined as follows
• a. As a difference e V V0 (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.

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Example errors

• Example A thermistor is used to measure
  temperature between 30 and 80 ?C and produce an
  output voltage between 2.8V and 1.5V. Because of
  errors, the accuracy in sensing is 0.5?C.

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Example (cont)

• a. In terms of the input as 0.5?C
• b. Percentage of input e 0.5/(8030)100
  0.454
• c. In terms of output. From the transfer
  function e 0.059V.

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More on errors

• Static errors not time dependent
• Dynamic 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

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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

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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

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Error limits - nonlinear TF

• Calibration curve may be used when available
• Lower errors
• Maximum error from calibration curve
• Average error
• Limiting curves follow the actual transfer
  function (calibration)

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Repeatability

• Also called reproducibility failure of the
  sensor or actuator 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.

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Sensitivity

• Sensitivity of a sensor is defined as the change
  in output for a given change in input, usually a
  unit change in input. Sensitivity represents the
  slope of the transfer function.
• Same for actuators

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Sensitivity

• Sensitivity of a sensor is defined as the change
  in output for a given change in input, usually a
  unit change in input. Sensitivity represents the
  slope of the transfer function.
• Same for actuators

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Sensitivity (cont.)

• Example for a linear transfer function
• Note the units
• a is the slope
• For the transfer function in (2)

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Sensitivity (cont.)

• Usually associated with sensors
• Applies equally well to actuators
• 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.)

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Sensitivity analysis (cont.)

• A difficult task
• there is noise
• a combined function of sensitivities of various
  components, including that of the transduction
  sections.
• device may be rather complex with multiple
  transduction steps, each one with its own
  sensitivity, sources of noise and other
  parameters
• some properties may be known but many may not be
  known or may only be approximate. Applies equally
  well to actuators

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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, x2y1 (output of transducer 1 is the input
  to transducer 2) and x3y2

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Example -1 - errors present

• First output is y1y01 ?y1. y01 Output
  without error
• 2nd output
• 3rd output
• Last 3 terms - additive errors

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Example -2 - differential sensors

• Output proportional to difference between the
  outputs of the sensors
• Output is zero when T1T2
• Common mode signals cancel (noise)
• Errors cancel (mostly)

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Example -2 - (cont.)
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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 of sensitivities

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Hysteresis

• Hysteresis (literally lag)- the deviation of the
  sensors output at any given point when
  approached from two different directions
• Caused by electrical or mechanical systems
• Magnetization
• Thermal properties
• Loose linkages

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Hysteresis - Example

• If temperature is measured, at a rated
  temperature of 50?C, the output might be 4.95V
  when temperature increases but 5.05V when
  temperature decreases.
• This is an error of 0.5 (for an output full
  scale of 10V in this idealized example).
• Hysteresis is also present in actuators and, in
  the case of motion, more common than in sensors.

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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
  transfer function or from the calibration curve
• A few methods to do so

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Nonlinearity (cont.)

• a. by use of the range of the sensor/actuator
• Pass a straight line between the range points
  (line 1)
• Calculate the maximum deviation of the actual
  curve from this straight line
• Good when linearities are small and the span is
  small (thermocouples, thermistors, etc.)
• Gives an overall figure for nonlinearity

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Nonlinearity (cont.)

• b. by use of two points defining a portion of the
  span of the 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

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Nonlinearity (cont.)

• c. use a linear best fit(least squares) through
  the points of the curve
• Take n points on the actual curve, xi,yi,
  i1,2,n.
• Assume the best fit is a line yaxb (line 2)
• Calculate a and b from the following

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Nonlinearity (cont.)

• 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

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Saturation

• Saturation a property of sensors or actuators
  when they no longer respond to the input.
• Usually at or near the ends of their span and
  indicates that the output is no longer a function
  of the input or, more likely is a very nonlinear
  function of the input.
• Should be avoided - sensitivity is small or
  nonexistent
• In actuators, it can lead to failure of the
  actuator (increase in power loss, etc.)

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Frequency response

• Frequency response The ability of the device to
  respond to a 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 the Bode diagram of the device)

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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

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Frequency response (cont.)

• Bandwidth the distance in Hz between the half
  power points
• Half-power points eh0.707e, ph0.5p
• 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

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Half power points

• Also called 3db points
• Power is 3db down at these points
• 10log0.5?3db or
• 20log (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

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Frequency response (example.)

• Bandwidth 16.5kHz-70Hz16.43 kHz
• Flat frequency range 10kHz-120Hz9880 Hz
• Cutoff frequencies 70 Hz and 16.5 kHz
• Resonance 12 kHz

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Response time

• response time (or delay time), indicates the time
  needed for the output to reach steady state (or a
  given percentage of steady state) for a step
  change in input.
• Typically the response time will be given as the
  time needed to 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).

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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 to capacitances 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

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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

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Calibration

• Calibration the experimental determination of
  the transfer function of a sensor or actuator.
• Typically, needed when the transfer function is
  not known or,
• When the device must be operated at tolerances
  below those specified by the manufacturer.
• Example, use a thermistor with a 5 tolerance on
  a full scale from 0 to 100?C to measure
  temperature with accuracy of, say, 0.5?C.
• The only way this can be done is by first
  establishing the transfer function of the sensor.
  

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Calibration (cont.)

• Two methods
• a. known transfer function
• Determine the slope and crossing point (line
  function) from two known stimuli (say two
  temperatures) if the transfer function is linear
• Measure the output
• Calculate the slope and crossing point in VaTb
• 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

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Calibration (cont.)

• Two methods
• b. Unknown transfer function
• Measure the output Ri at as many input values Ti
  as 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.

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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
  design documents
• Errors due to calibration must be evaluated and
  specified

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Resolution

• Resolution the minimum increment in stimulus to
  which it can respond. 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.

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Resolution (cont.)

• Resolution is determined by the whole system, not
  only by the sensor
• The resolution of the sensor may be better than
  that of the system.
• The sensor itself must interact with a processor,
  the limiting factor on resolution may be the
  sensor or the processor.
• Resolution may be specified in the units of the
  stimulus (0.5?C for 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).

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Resolution (cont.)

• In digital systems, resolution may be specified
  in bits (1 bit or 6 bit resolution)
• In analog systems (those that do not digitize the
  output) the output is continuous and resolution
  may be said to be infinitesimal (for the sensor
  or actuator alone).
• Resolution of an actuator is the minimum
  increment in its output which it can provide.
• Example a stepper motor may have 180 steps per
  revolution. Its resolution is 2?.
• A graduated analog voltmeter may be said to have
  a resolution equal to one graduation (say 0.01V).
  ( higher resolution may be implied by the user
  who can easily interpolated between two
  graduations.

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Other parameters

• Reliability a statistical measure of quality of
  a device which indicates the ability of the
  device to perform its stated function, under
  normal operating conditions without failure 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

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Other parameters

• Deadband the lack of response or insensitivity
  of a device over 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
  this insensitivity is acceptable.
• Example, an actuator which is not responding to
  inputs around zero may be acceptable but one
  which freezes over a normal range may not be.

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Other parameters

• Excitation The electrical supply required for
  operation of a sensor or actuator.
• 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 or premature failure of the
  device.