10. Response Characteristics
performance measures quantifying the operation of a transducer
�definitions of response characteristics (§10.1)
�calibration and errors (§10.2)
�time scaling and frequency scaling (§10.3)
10.1 Response Characteristics Defined
specifications describing transducers, measures of operation range
the extent of measurement ability
from the smallest (without the signal being obscured by to the largest (without the system being overloaded or for sensors : the smallest to largest input signals
for actuators : normally specified in terms of the output signal and allowable range of input variable
cf. rangeability : some range of signal values over which the device is useful
accuracy cf.
a measure of the error and uncertainty 1. random error (
causes different output signals for apparently identical input signals 2. systematic error (
correlated with some system parameter, such as 3. hysteresis error (
systematic error depending on the direction of approach repeatability
the extent to which identical input signals will produce identical output signal related to accuracy but distinctly different
linearity
a measure of the manner in which the error varies with range in the transducer expressed as the deviation from a
1. best straight line Fig. 10.2
produces the minimum absolute error
2. best straight line through the origin Fig. 10.3 linearity with no output for no input
3. a line matching exactly at the terminal points Fig. 10.4 easy to draw but eliminated by the proliferation of computers 4. a line based on a theoretical calculation Fig. 10.5
used to predict performance
rarely used in determining the linearity of a transducer
distortion
the size of response at frequencies other than in response to excitation at (one of the measures of dynamic
nearly the complement to linearity (dynamic domain) harmonic distortion :
an excitation at frequency causes a response at , being a positive integer second harmonic distortion :
result of the transduction mechanism total harmonic distortion
the net output at all harmonic frequencies in the range divided by the output at the
intermodulation distortion:
output at frequency given excitation simultaneously at two frequencies
sensitivity (
a transfer function between the output signal and the input signal of a transducer applies over a range of operation or at a single input value of measurement frequency response
amplitude of the output versus frequency for a set level of input ex.
efficiency
a ratio of the output power (or energy) to the input power (or not important for sensors, but for
resolution ≈
the smallest signal step that can reliably be resolved (sensors) or imposed (actuators) related to the sensitivity and the amount of noise
noise
the amount of the output signal uncorrelated with the noise floor :
a measure of the magnitude of the time-varying output signal level even in the absence of an input signal
noise spectrum : a measure of the noise floor as a function of signal-to-noise ratio (
noise in terms of its comparison to a
transient response
cf. static or steady-state sinusoidal response
system : , system : response time (
a measure of how quickly it responds to a step change in the input signal
presented in terms of time for the output signal to first be within a given fraction (ex. 95%) of the final value it will achieve
cf. settling time :
a measure of time for the output signal to maintain within a given fraction (ex.
of the final value it will achieve overshoot :
a measure of how much its response goes beyond the final value before settling back down to it
usually quoted as a ratio. ex.
response time and overshoot tend to work against drift
the tendency for the output to vary monotonically and very slowly compared to times associated with the sensing or actuation
caused by influence quantities such as changing cf.
offset
nonrandom value of the transducer output when the input is others
geometric parameters : economic parameter :
10.1.1 example of transducer selection based on specifications ( skip )
10.2 Calibration
process by which a sensor or actuator is provided with a factual statement about the the instrument will have
standard laboratories and organizations
KRISS Korea Research Institute of Standards and Science ISO International Standard Organization
NIST National Institute of Standards and Technology ASTM American Society for Testing and Materials ANSI American National Standard Institute
calibration methods
1. comparison method :
the transducer performance is compared to that of a similar transducer with performance
2. absolute direct method :
a special calibrating machine is used to calibrate the transducer 3. absolute reciprocal method :
the transducer is calibrated using other transducers but the calibration of other transducer is rendered irrelevant through the use of
comparison method
Ex. calibration of a new accelerometer
signal →[shaker]-(input, )→[calibrated accelerometer, + new accelerometer, ] ↓ output ↓ output
() = () = (1) transfer function of the new accelerometer
=
(2)
advantages : , no need for specialized equipment
disadvantages : , i.e. more prone to errors than absolute methods errors in the comparison method
a. errors in measurement
= (actual transfer function) + (3)
= (actual value) + (4)
= ( + ) ( + ) = + + + (5)
: actual transfer function
+ + : errors due to
b. errors due to changing excitation
only when it is not possible to test simultaneously both the calibrated and new transducers
= = ( +)
=
=
(6)
: error due to changing excitation depending on
= relative difference in the
= relative size of the c. errors due to slipping out of range
a noise floor at high and low frequencies (Fig. 10.6) will cause serious distortion in the resulting ratio in those frequency regimes (Fig. 10.7).
absolute direct method
a special piece of calibrating equipment produces a well-controlled input signal to a particular type of transducer
Ex.1 accelerometer calibrator
Ex.2 pistonphone (microphone calibrator)
uses a high precision motor to move a piston at a single frequency
→ volumetric change of a cavity producing a known sound pressure, advantages :
disadvantages : , performed at only one frequency or a few frequencies
still need the comparison method to complete the calibration at all frequencies absolute reciprocal method
uses at least one reciprocal transducer and
relies on reciprocity relationships in determining the calibration developed for accelerometers and other vibration sensors Ex. microphone calibration
step 1 : Fig. 10.8
a reversible transducer B is placed a distance from loudspeaker C
open circuit voltage = (7)
: sound pressure at B produced by C step 2 : Fig. 10.9
transducer B is replaced by the new microphone A
= (8)
⇒ =
(9)
step 3 : Fig. 10.10
loudspeaker C is replaced by the transducer B
′ = (10)
: sound pressure produced at A by B
= (11)
: input current to B
: speaker sensitivity of B ⇒ =
′
(12) analysis
(9)×(12) =
′
(13) │ └ quantities related to B
measured quantities reciprocity relation to determine
(9.20) →
= -
(14)
(7) →
(=
) = -
(15)
: volume velocity produced at B by C
= -
(16)
= -
=
← (11)
=
⇒
=
: wavelength (17)
advantages : need for specialized equipment disadvantages : , process should be found for each transducer
10.3 Frequency and Time Scaling
prototype testing
larger or smaller prototype(scale-up or scale-down model) is calibrated and its response characteristics are then used to the response of
the full-scale transducer simulation
more useful in than prototype testing analog simulation
purely electrical version of the transducers (when possible) can be quite useful in identifying problems in transducer design
numerical (digital) simulation used to predict system
need a means of going from model results to predictions for behavior of a ( skip below )
