1. Performance Characteristics of Sensors and Actuators
(Chapter 2) 1

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 2

3. 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.) 3

4. Transfer function Relation between input and output Other names:
Input output characteristic function
transfer characteristic function
response 4

5. 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 5

6. Transfer function (cont.)
T1 to T2 - approximately linear
Most useful range
Typically a small portion of the range
Often taken as linear 6

7. Transfer function (cont.)
Other data from transfer function
saturation
sensitivity
full scale range (input and output)
hysteresis
deadband
etc. 7

8. 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 8

9. 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 9

10. 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 10

11. 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 11

12. 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. 12

13. 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 13

14. Impedance (cont.) Impedance can (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) 14

15. 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 15

16. 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 16

17. 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. 17

18. 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 18

19. 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. 19

20. 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. 20

21. 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. 21

22. 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 22

23. 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 23

24. 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 24

25. 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) 25

26. 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. 26

27. 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 27

28. 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 28

29. Sensitivity (cont.) Example for a linear transfer function:
Note the units
a is the slope
For the transfer function in (2): 29

30. 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.) 30

31. 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 31

32. 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.). 32

33. Example - additive errors
Fiber optic pressure sensor
Pressure changes the length of the fiber
This changes the phase of the output
Three transduction steps 33

34. 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 34

35. Example -1 - errors present
First output is y1=y01 + y1. y01 = Output without error
2nd output
3rd output
Last 3 terms - additive errors 35

36. Example -2 - differential sensors
Output proportional to difference between the outputs of the sensors
Output is zero when T1=T2
Common mode signals cancel (noise)
Errors cancel (mostly) 36

37. Example -2 - (cont.) 37

38. 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 38

39. Hysteresis
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 39

40. 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. 40

41. 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: 41

42. 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 42

43. 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 43

44. 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: 44

45. 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 45

46. 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.) 46

47. 47

48. 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) 48

49. 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 49

50. 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 50

51. Half power points These points are arbitrary but are now standard.
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 51

52. 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 52

53. 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”). 53

54. 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 54

55. 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 55

56. 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. 56

57. 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 57

58. 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. 58

59. 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 59

60. 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. 60

61. 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). 61

62. 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. 62

63. 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 63

64. 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. 64

65. 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. 65