Coverage: through class #7 material (updated June 14, 2017) Terminology/miscellaneous. for a linear instrument:

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1 Boston University ME 310: Summer 017 Course otes to-date Prof. C. Farny DISCLAIMER: This document is meant to serve as a reference and an overview, and not an exclusive study guide for the course. Coverage: through class #7 material (updated June 14, 017) Terminology/miscellaneous Precision Accuracy Resolution Full Scale Input, Output Significant figures Static sensitivity for a linear instrument: K = f(x) x K = output input where f(x) is the output (the functional response of the instrument) and x is the input parameter. Probability density function p(x) p(x) = 1 σ π exp [ ] (x x) σ The absolute probability percentage P is defined over a defined interval of deviation ±δx about the average value x: P ( x δx x x + δx) = x+δx x δx (1) () (3) p(x)dx (4) Convert to a simplified scale by defining β, the normal variate for all samples x, and z 1, the normal variate for a particular sample value x 1 : β = x x σ ; z 1 = x 1 x ; σ dx = σdβ (5) and substituting into Eqn??, P ( z 1 β z 1 ) = 1 π z1 z 1 exp( β /) dβ (6) or the more common version to reflect the symmetrical distribution assumption (note change in the integration bounds and constant term): P ( z 1 β z 1 ) = π z1 1 0 exp( β /) dβ (7)

2 Analog-Digital Conversion (ADC) The ADC process discretizes the analog signal at discrete time points (sampling frequency) and discrete voltage amplitudes. Amplitude discretization: The resulting amplitude discretization corresponds to n = V F SR V res (8) where n is the number of bits (an integer value, commonly even). The full scale range must be greater than the amplitude of the analog signal in order to properly resolve the signal. Temporal discretization: The ADC can only sample incoming data at a certain rate. This rate is generally adjustable, and should be set according to the yquist frequency. The yquist sampling criterium states that at least two data points are required to in order to resolve the highest frequency present. This results in: f s f max (9) (10) f yq = f s = 1 (11) t s Aliasing: Effect occurs when f yquist is lower than one or more frequencies in the analog signal. The energy from these higher analog frequencies appears as energy at a frequency lower than f yquist. ADC Uncertainty: Due to its discrete nature, there are two types of uncertainty related to ADC, both of which act like resolution-based bias sources: Temporal: The digitizer is sampling the signal at discrete time points, so the amplitude at any given time has an inherent uncertainty: U B,t = ± t s Magnitude resolution: The vertical resolution due to the number of bits and full scale range introduces a resolution error: (1) U B,V = ± V res (13)

3 Uncertainty analysis Precision Uncertainty: Inherent scatter in a measurable x about a mean x due to uncontrolled fluctuations. 1. Infinite Statistics: Large sample sets (sample size > 63) σ x U P,x = ±z 1 ; (14) z 1 = x 1 x ; (15) σ (xi x) σ x =. Finite Statistics: Small sample sets. Degrees of freedom ν = -1 (16) S x U P,x = ±t ν,95% ; (17) (xi x) S x = (18) ν ote: The subscript P stands for precision (not pressure!). If the measurement were for a pressure, it would be denoted U P,p and if it were for a voltage, it would be denoted U P,V. Bias Uncertainty: Systematic uncertainties from instrumentation. Affects every data point in a data set of measurable x. Common bias sources: Resolution: Arises from rounding error. U B,resolution,x = ±resolution/ (19) Hysteresis: Arises from discrepancy in output due to direction-dependent loading effects. Commonly found in elastic systems. U B,hysteresis,x = ±%FS (0) Linearity (also referred to as onlinearity): Arises from instrument response, when output deviates from a linear response over the full scale range. U B,linearity,x = ±%FS (1) Sensitivity (also referred to as a Calibration Error): Arises from drift in sensitivity over time. Total bias uncertainty: U B,sensitivity,x = ±%FS () U B,x,total = (U B,x,i ) (3) 3

4 Total Uncertainty: Combine bias and precision sources for each measurable x using root-sum-square (RSS) method U total,x = (U P,x,i ) + (U B,x,i ) = U P,x,total + U B,x,total (4) Class 5 Error propagation: Combining uncertainty from multiple measurables x, y, z, where f(...) represents the generic analytical expression that relates x, y, z to the resultant parameter R: U total,r = U total,r = f(u x, U y, U z ) (5) ( f ) ( ) ( ) f f x U x + y U y + z U z (6) where each individual measurable uncertainty (U x, etc) is the total uncertainty for that parameter. Error propagation can be used in a single-instrument setting for the purpose of converting its input and output units. Measurement analysis Input/output impedance: Recall the voltage divider scenario that the function generator and oscilloscope describe: [ ] R L V L = V S (7) R S + R L where V is the generic instrument voltage, R is the generic instrument impedance, L represents the load (the downstream measurement instrument, or the oscilloscope), and S represents the source (the signal source instrument, or function generator). Linear system modeling Given 3 linear instruments 1,, and 3 in line with each other, an instrument-specific static sensitivity K i and static offset O i, and an input parameter x, the instrument I output is y, instrument II output is z and instrument III output is W. In generic functional response terms: y = f 1 (x); z = f (y); W = f 3 (z) = g(x); (8) Plugging in for the instrument-specific sensitivities: W = f 3 (f (f 1 (x))) = g(x); (9) y = K 1 x + O 1 ; z = K y + O ; W = K 3 z + O 3 ; (30) W = K sys x + O sys ; (31) where K sys = K 1 K K 3, O sys = O 3 + K 3 O + K K 3 O 1 (3) 4

5 Resistance measurement methods RTDs Strain gauge/wheatstone bridge Bridge constant Compensation techniques Linear System Calibration Linear Regression Analysis: The method of least squares error gives the following relationship for a linear system (with additional terms for an appropriately-nonlinear relationship): y fit = f(x) = a 0 + a 1 x (33) x a 0 = i ym,i x i xi y m,i ; a 1 = x i y m,i x i ym,i (34) B B B = x i ( xi ) (35) where x i is each control point (input) value, y m,i is each measured response (output) point value, y fit is the calibration value for a given input x, and is the number of unique control points involved in the calibration. ote that it is helpful at times to invert the functional relationship to instead solve for x in terms of a measurement of y: x = f 1 (y) = y a 0 a 1 (36) For an exercise involving multiple output measurements per control point, a weighted scheme is used: wi x i wi y m,i w i x i wi x i y m,i a 0 = (37) B w wi wi x i y m,i w i x i wi y m,i a 1 = (38) B w B w = w i wi x i ( wi x i ) (39) w i = 1 U i where U i is the precision uncertainty value per control point. Calibration uncertainty: Potentially the most confusing of the uncertainty calculations. The goal is to understand the relationship between an output y (likely a voltage) and a physical input parameter x for an uncalibrated instrument A. Remember that A is being used to measure x, so x is unknown, and y is simply a voltage (and thus not helpful on its own). The functional response of A may only be known via use of a calibrated 5 (40)

6 instrument C that is capable of knowing what the input x actually is. C tells us what x is, and a regression analysis can then reveal how to get the constants in Eq?? for instrument A. It gets confusing because both instruments can have bias uncertainties, and both may have their own precision uncertainties. We re dealing with multiple units (for x and for y), so a final, total uncertainty value can only be obtained via propagation of the multiple uncertainties to a final uncertainty in units of x or of y. Try to clearly differentiate which uncertainties belong with each instrument; keeping careful track of units helps a good deal. Here U meas refers to the uncertainties from the uncalibrated instrument A and U cal refers to those from the calibrated instrument C. Obtaining a total uncertainty from C is familiar and straightforward: U cal,x = ± ( U B,x,cal,i) + U p,x,cal,max (41) Remember that these will all be in units of x. Here the precision uncertainty would come from the scenario where you ve taken multiple measurements of x i at each control point value, for the purpose of accounting for scatter in the measurement. This would give rise to values of U p,x,cal ; since only one such value would be considered for the uncertainty, you should take the maximum such value over the that you ve calculated. For the uncalibrated instrument A, the various uncertainty parameters will likely be in units of y, or voltage. The new component of the analysis here involves the precision uncertainty of instrument A. The precision uncertainty is now a scatter of the data about a linear trend (the calibration curve), known as the precision uncertainty of the fit U p,fit that relies on a standard error of the fit S yx : U p,y,fit = ±t ν,95% Syx (4) (yfit,i y m,i ) S yx = (43) ν The number of statistical degrees of freedom in this case (specifically, calibration uncertainty analysis) ν is (I sometimes notate ν here as ν fit as a reminder of the calibration aspect), and is the number of control points on the x axis. The parameter y fit,i represents the known/calibrated output value at each control point. The uncertainty from instrument A may be then found per the usual method: U meas,y = ± ( U B,y,i) + U p,y,fit (44) Since the values of the calibration for instrument A depend on the inherent uncertainties of instrument C, the two uncertainties U meas and U cal must be combined, to obtain a total uncertainty. Error propagation must be used to properly combine the different units of x and y: ( f ) U total,y = ± x U cal,x + Umeas,y (45) 6

7 ( f 1 or: U total,x = ± y U meas,y ) + U cal,x (46) where f represents the calibration equation y = a 1 x + a 0 and f 1 is the inverse of that equation, or x = y a 0 a 1. 7