# Lecture 12 Instrument Uncertainty

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1 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, u o, is set by international (ISO) convention to equal one-half of its resolution. It is the smallest uncertainty that an instrument can have. Thus, u o = ±resolution/2 (1) 1

2 Design-stage uncertainty, u d, further considers the instrument uncertainty, where u d = u 2 o + u 2 I, (2) with u I is the instrument uncertainty. The instrument uncertainty is the composite of all of the elemental instrument uncertainties, e i s, where u i = N i=1 The e i s typically result from errors in 1. hysteresis 2. linearity 3. sensitvity 4. zero-shift 5. repeatability 6. stability, and 7. thermal drift e 2 i. (3) The equations for each of these errors in their dimensionless form are on pp

3 Figure 2: The resolution of a ruler. For example, hysteresis error, is expressed as e H = e H,max = y up y down max, (4) where y up y down max is the maximum absolute difference between y up and y down 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). 3

4 EXAMPLE PROBLEM: A pressure transducer manufacturer states the following under accuracy data about their 0 in. H 2 O to 0.5 in. H 2 O 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. H 2 O? 4

5 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. 5

6 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 u t (x) = t ν,p S x / N, (5) where t ν,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 %, t ν,p = This is the same coverage factor that was assumed in the large scale approximation. 6

7 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. H 2 O to 5 in. H 2 O. The smallest marked division on the manometer s scale is 0.2 in. H 2 O. 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. H 2 O and 0.2 in. H 2 O, respectively. Assuming 95 % confidence, determine (a) the zero-order uncertainty, u 0, (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, u c. 7

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