Transcript Performance Characteristics of Sensors and Actuators

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.)
Variations: output can sometimes be
displacement (thermometers,
magnetostrictive and piezoelectric sensors).
Some sensors combine sensing and
actuation
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
Relation between input and output
 Other names:

 Input
output characteristic function
 transfer characteristic function
 response
Transfer function (cont.)
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Linear or nonlinear
Single valued or not
One dimensional or multi dimensional
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Single input, single output
Multiple inputs, single output
In most cases:
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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
Transfer function (cont.)
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T1 to T2 - approximately linear
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Most useful range
Typically a small portion of the range
Often taken as linear
S = f(x)
Transfer function (cont.)
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Other data from transfer function

saturation
 sensitivity
 full scale range (input and output)
 hysteresis
 deadband
 etc.
Transfer function (cont.)
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Other types of transfer functions
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Response with respect to a given quantity
Performance characteristics (reliability curves, etc.)
Viewed as the relation between any two characteristics
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.)

Sensors: only output impedance is relevant
 Actuators: only input impedance is relevant
 Can also define mechanical impedance
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Not needed - impedance is important for
interfacing
Will only talk about electrical impedance
Impedance (cont.)
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Why is it important? It affects performance
Example: 500 W sensor (output impedance)
connected to a processor
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b. Processor input impedance is infinite
c. Processor input impedance is 500 W
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:
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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 (cont.)

Current sensors: impedance is low - need low
impedance at processor
 Same considerations for actuators
 Impedance matching:
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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 (cont.)

Impedance can (and often is) complex:
Z=R+jX
 In addition to the previous:

Conjugate matching (Zin=Zout*) - maximum power
transfer
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Critical for actuators!
Usually not important for sensors
Zin=R+jX, Zout*=R-jX.
No reflection matching (Zin=Zout) - no reflection
from load
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Important at high frequencies (transmission lines)
Equally important for sensors and actuators (antennas)
Range and Span
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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)
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 (cont.)
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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:
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Maximum force, torque, displacement
Acceleration
Time response, delays, etc.
Accuracy, errors, repeatability
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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.)
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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.

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.
Example (cont)
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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
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
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Linear transfer functions
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Error equal along the transfer function
Error increases or decreases along TF
Error limits - two lines that delimit the output
Error limits - nonlinear TF
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Nonlinear transfer
functions
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Error change along
the transfer function
 Maximum error from
ideal
 Average error
 Limiting curves
follow ideal transfer
function
Error limits - nonlinear TF
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Calibration curve
may be used when
available
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Lower errors
Maximum error from
calibration curve
Average error
Limiting curves
follow the actual
transfer function
(calibration)
Repeatability
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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.
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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 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.)
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Example for a linear transfer function:
 Note the units
 a is the slope

For the transfer function in (2):
d aT + b = 1
dR

dR = a
dT
W
C
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.)
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A difficult task
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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 (cont.)
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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
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Fiber optic pressure sensor
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Pressure changes the length of the fiber
This changes the phase of the output
Three transduction steps
Example-1 - no errors present
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Individual
sensitivities
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Overall sensitivity
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But, x2=y1 (output of
transducer 1 is the
input to transducer
2) and x3=y2
s1 =
dy1
,
dx1
s2 =
S = s 1s2 s3 =
dy2
,
dx2
dy1 dy2 dy3
dx1 dx2 dx3
S = s 1s2 s3 =
dy3
dx1
s3 =
dy3
dx3
Example -1 - errors present
First output is y1=y01
+ y1. y01 = Output
without error
 2nd output
y2 = s 2 y 0 + y1

1
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+ y2 = y20 + s 2y1 + y2
3rd output
y3 = s 3 y20 + s2 y1 + y2 + y3 = y30 + s2 s3 y1 + s3 y2 + y3
Last 3 terms - additive
errors
Example -2 - differential
sensors
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Output proportional to
difference between the
outputs of the sensors
Output is zero when
T1=T2
Common mode signals
cancel (noise)
Errors cancel (mostly)
Example -2 - (cont.)
s1 =
dy1
,
dx1
s2 =
dy2
dx2
y = y1 - y2 = s1 x1 - s 2yx
s=
d y1 - y2
d x1 - x2
Example -3 - sensors in series
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Output is in series
Input in parallel (all
sensors at same
temperature)
Outputs add up
Noise multiplied by
product of sensitivities
y = y1 + y2 + y3 + ... + yn = (s1 + s2 + s3 + ... + s n)x = nsx
S = ns
Hysteresis
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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
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Magnetization
Thermal properties
Loose linkages
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.
Nonlinearity
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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.)

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
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 (cont.)

c. use a linear best fit(least squares)
through the points of the curve
 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:
n
n• xiyi a=
i=1
n
n• xi2 i=1
n
• xi • yi
i=1
n
n
i=1
n
2
xi
i=1
•
•
b=
i=1
n
xi2
• yi
i=1
n
-
n• xi2 i=1
n
n
• xi • xiyi
i=1
n
• xi
i=1
i=1
2
Nonlinearity (cont.)
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d. use the tangent to the curve at some
point on the curve
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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

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
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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)
Frequency response (cont)

Important design parameters
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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.)

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 error)
 Resonant frequency: the frequency (or
frequencies) at which the curve peaks
or dips

Half power points
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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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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 (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.)

Example: in a temperature sensor:
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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:
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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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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
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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.)
Two methods:
 a. known transfer function:
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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 (cont.)

Two methods:
 b. Unknown transfer function:

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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 (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

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.)
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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 (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.
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 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 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.