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## Performance Characteristics of Sensors and Actuators

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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

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.)

- 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 (cont.)

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

Transfer function (cont.)

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

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

- 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
- Impedance is important for interfacing
- Will only talk about electrical impedance

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 (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 (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 (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: 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: -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 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

- 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.)

- 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: 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 ±0.5C.

Example (cont)

- a. In terms of the input as ±0.5C
- 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 - introduced by the system

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
- Errors change along the transfer function
- Maximum error from ideal
- Average error
- Limiting curves follow ideal transfer function

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)

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.

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 (not necessarily constant).
- Same for actuators

Sensitivity (cont.)

- Example for a linear transfer function:
- Note the units
- a is the slope
- For a transfer function f(T) = aT + b:

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.)

- 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 (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

- 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

- Individual sensitivities
- Overall sensitivity
- But, x2=y1 (output of transducer 1 is the input to transducer 2) and x3=y2

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

- 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

- Output is in series
- Input in parallel (all sensors at same temperature)
- Outputs add up
- Noise multiplied by product of sensitivities

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

Hysteresis - Example

- If temperature is measured, at a rated temperature of 50C, 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

- 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 non-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 (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 (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

- 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: 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)

- 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.)

- 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

- 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.)

- 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:
- 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 (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

- 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 100C to measure temperature with accuracy of, say, ±0.5C.
- 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:
- 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:
- 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.)

- 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 (0.5C 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 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

- 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.

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