Lecture 12
Instrument Uncertainty
• Resolution is the smallest physically indicated division
that an instrument displays or is marked.
Resolution = 1 mm
Figure 1: The resolution of a ruler.
• Readability is the closeness with which the scale of an instrument is read by an experimenter. It is a subjective value
and does not enter into assessing the uncertainty of an instrument.
• The zero-order uncertainty of an instrument, uo, is set
by international (ISO) convention to equal one-half of its
resolution. It is the smallest uncertainty that an instrument
can have. Thus,
uo = ±resolution/2
1
(1)
• Design-stage uncertainty, ud, further considers the instrument uncertainty, where
r
ud = u2o + u2I ,
(2)
with uI is the instrument uncertainty.
• The instrument uncertainty is the composite of all of the
elemental instrument uncertainties, ei’s, where
ui =
v
u
uX
N
u
t
i=1
e2i .
(3)
• The ei’s typically result from errors in
1. hysteresis
2. linearity
3. sensitvity
4. zero-shift
5. repeatability
6. stability, and
7. thermal drift
• The equations for each of these errors in their dimensionless
form are on pp. 245-246.
2
Figure 2: The resolution of a ruler.
• For example, hysteresis error, is expressed as
eH = eH,max = |yup − ydown|max,
(4)
where |yup − ydown|max is the maximum absolute difference
between yup and ydown on the calibration curve, as illustrated
in the middle of Figure 2.
• Each dimensionless error can be made dimensional by multiplying it by the F SO, the output at full scale (the maximum
output value).
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EXAMPLE PROBLEM: A pressure transducer manufacturer states the following under “accuracy data” about their
0 in. H2O to 0.5 in. H2O pressure sensor/transducer.
• Accuracy as RSS non-linearity, hysteresis, and non-repeatability:
±0.14 % FS at constant temperature
• Non-linearlity: ±0.1 % full scale range output (best straight
line method)
• Hysteresis: 0.1 % FS
• Non-repeatability: ±0.02 % FS
• Thermal Effects (30 ◦F to 150 ◦F): zero-shift < ± 1 %
FS/100 ◦F and span shift < ± 1 % FS/100 ◦F
(a) Is the quoted “accuracy” for constant-temperature conditions correct as stated?
(b) What is the total uncertainty if the environmental temperature varies by 10 ◦F during an experiment?
(c) What is the overall uncertainty in the pressure for 100
measurements under steady-state and constant-temperature conditions estimated with 95 % confidence assuming that the manometer used to read the pressure has a resolution of 0.2 in. H2O?
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EXAMPLE PROBLEM: The manufacturer of the ADXL335
±3 g accelerometer with a sensitivity of 300 mV/g provides the
following information under “Sensor Input”.
• Nonlinearlity: ±0.3 % of full scale
• Cross-axis sensitivity: ±1 % of full scale
• Package alignment error: ±1 degrees
• Interaxis alignment error: ±0.1 degrees
The accelerometer is connected rigidly to a beam and aligned
such that its y-axis is vertically downward. The beam is then
rotated to exactly 60 ◦ from vertical. Determine the % uncertainty in (a) the angle and (b) the acceleration. Finally,
determine the overall uncertainty in g if the accelerometer output voltage is read by either (c) a DVM having a resolution of
0.1 V or (d) a 12-bit A/D converter with a range of 0 V to 5 V.
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Temporal Precision Uncertainty
• Even when an experiment is conducted under fixed operating conditions, a measurand’s signal may vary in time to
an extent. This is the result of uncontrolled ‘extraneous’
variables that change in time and affect the measurand.
• This uncertainty is called the temporal precision uncertainty. It can be estimated as
√
ut(x) = tν,P Sx/ N ,
(5)
where tν,P is Student’s t variable, which depends upon the
number of degrees of freedom, ν, and the % confidence, P .
• For this case, ν = N - 1.
• Student’s t variable is simply a coverage factor that fixes the
confidence limits (like k did before). Its values for various
%P and ν are given on p.196 and on the back, inside cover
of the text.
• For N = 61 (ν = 60) and P = 95 %, tν,P = 2.000. This
is the same coverage factor that was assumed in the large
scale approximation.
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EXAMPLE PROBLEM: An inclined manometer has a
stated accuracy of 3 % of its full-scale reading. The range of
the manometer is from 0 in. H2O to 5 in. H2O. The smallest
marked division on the manometer’s scale is 0.2 in. H2O. An
experiment is conducted under controlled conditions in which a
pressure difference is measured 20 times. The mean and standard deviation of the pressure-difference measurements are 3 in.
H2O and 0.2 in. H2O, respectively. Assuming 95 % confidence,
determine (a) the zero-order uncertainty, u0, (b) the temporal precision uncertainty that arises from the variation in the
pressure-difference during the controlled-conditions experiment,
and (c) the combined standard uncertainty, uc.
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