performance characteristics of sensors and actuators n.
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Performance Characteristics of Sensors and Actuators. (Chapter 2). Input and Output. Sensors Input: stimulus or measurand (temperature pressure, light intensity, etc.) Ouput: electrical signal (voltage, current frequency, phase, etc.)

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input and output
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

input and output1
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.)

transfer function
Transfer function
  • Relation between input and output
  • Other names:
    • Input output characteristic function
    • transfer characteristic function
    • response
transfer function cont
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, numerically)
    • Often must be defined from calibration data
    • Often only defined on a portion of the range of the device
transfer function cont1
Transfer function (cont.)
  • T1 to T2 - approximately linear
    • Most useful range
    • Typically a small portion of the range
    • Often taken as linear
transfer function cont2
Transfer function (cont.)
  • Other data from transfer function
    • saturation
    • sensitivity
    • full scale range (input and output)
    • hysteresis
    • deadband
    • etc.
transfer function cont3
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
impedance and impedance matching
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
impedance cont
Impedance (cont.)
  • Sensors: only output impedance is relevant
  • Actuators: only input impedance is relevant
  • Can also define mechanical impedance
    • Impedance is important for interfacing
    • Will only talk about electrical impedance
impedance cont1
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
impedance cont2
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.
impedance cont3
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
impedance cont4
Impedance (cont.)
  • Impedance can be (and often is) complex: Z=R+jX
  • In addition to the previous:
    • Conjugate matching (Zin=Zout*) - maximum power transfer
      • Critical for actuators!
      • Usually not important for sensors
      • Zin=R+jX, Zout*=R-jX.
    • No reflection matching (Zin=Zout) - no reflection from load
      • Important at high frequencies (transmission lines)
      • Equally important for sensors and actuators (antennas)
range and span
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
range and span cont
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-1.2V=1.3V
  • Dynamic range=20log(140/30)=13.38db
range and span cont1
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.
accuracy errors repeatability
Accuracy, errors, repeatability
  • Errors: deviation from “ideal”
  • Sources of errors:
    • materials used
    • construction tolerances
    • ageing
    • operational errors
    • calibration errors
    • matching (impedance) or loading errors
    • noise
    • many others
accuracy errors cont
Accuracy, errors (cont.)
  • Errors: defined as follows:
  • a. As a difference: e = V – V0 (V0 is the actual value, V is the 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
Example: errors
  • Example: A thermistor is used to measure temperature between –30 and +80 C and produces an output voltage between 2.8V and 1.5V. Because of errors, the accuracy in sensing is ±0.5C.
example cont
Example (cont)
  • a. In terms of the input as ±0.5C
  • b. 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
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 - introduced by the system
error limits linear tf
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
Error limits - nonlinear TF
  • Nonlinear transfer functions
    • Errors change along the transfer function
    • Maximum error from ideal
    • Average error
    • Limiting curves follow ideal transfer function
error limits nonlinear tf1
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)
  • 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.
  • 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 (not necessarily constant).
  • Same for actuators
sensitivity cont
Sensitivity (cont.)
  • Example for a linear transfer function:
    • Note the units
    • a is the slope
  • For a transfer function f(T) = aT + b:
sensitivity cont1
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.)
sensitivity analysis cont
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
sensitivity analysis cont1
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.).
example additive errors
Example - additive errors
  • Fiber optic pressure sensor
    • Pressure changes the length of the fiber
    • This changes the phase of the output
    • Three transduction steps (current-light, pressure-displacement, light-voltage)
example 1 no errors present
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
example 1 errors present
Example -1 - errors present
  • First output is y1=y01 + y1. y01 = Output without error
  • 2nd output
  • 3rd output

Last 3 terms - additive errors

example 2 differential sensors
Example -2 - differential sensors
  • Output proportional to difference between the outputs of the sensors
  • Output is zero when T1=T2
  • Common mode signals (noise) cancel
  • Errors cancel (mostly)
example 3 sensors in series
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
  • Hysteresis (literally lag)- the deviation of the sensor’s output at any given point when approached from two different directions
  • Caused by electrical or mechanical systems
    • Magnetization
    • Thermal properties
    • Loose linkages
hysteresis example
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.
  • 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:
nonlinearity cont
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 non-linearities are small and the span is small (thermocouples, thermistors, etc.)
    • Gives an overall figure for nonlinearity
nonlinearity cont2
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
nonlinearity cont3
Nonlinearity (cont.)
  • c. use a linear best fit(least squares) through the points of the curve (2)
    • Take 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 from the following:
nonlinearity cont4
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
  • 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.)
frequency response
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 span (or 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)
frequency response cont
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 cont1
Frequency response (cont.)
  • Bandwidth: the distance in Hz between the half power points
    • 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 tolerance)
  • Resonant frequency: the frequency (or frequencies) at which the curve peaks or dips
half power points
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
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
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”).
response time cont
Response time (cont.)
  • Example: in a temperature sensor:
    • the time needed for the sensor’s 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
response time cont1
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
  • 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.
calibration cont
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 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
calibration cont1
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.
calibration cont2
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
  • 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.
resolution cont
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).
resolution cont1
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 that 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).
other parameters
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
other parameters1
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.
other parameters2
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.