METROLOGY 101: PISTON PROVER DESIGN - Cal Lab Magazine

More documents

Recommendations

Info

Metrology: Standardize and Automate! Mark Kuster Quantity Dimension ID Ex. Uncertainty Ex. DOF Time GUID[BIPM second] 0 s ∞ Velocity, Light in Vacuum GUID[CODATA c] 0 m/s ∞ Allan Variance GUID[NIST-F1(timestamp)] 10 -14 s/s 10 5 Unique Random Error NULL 10 -7 s/s 50 . . . . . . . . Table 5. Example Independent Uncertainties for a Length Measurement. Equation 1’s matrix, which we now denote C pf , gives the voltage, current, and phase sensitivities to the fundamental uncertainty vector; each sensitivity vector lies in one matrix row. To specify an uncertainty in the vector space, we multiply its sensitivities by the fundamental uncertainties componentby-component. So in vector form, the primary lab voltage measurement has the uncertainty vector does not appear in the list, except in the CODATA light velocity constant; all length measurements will trace back 0.9799 ∙ 3.274 mV 3.208 mV to time or frequency. u V = 0.1997 V/rad ∙ 379.1 µrad ( 2.209 mΩ ∙ 6.251 µA ) ( = 75.71 µV ) . 13.81 nV As long as we reference all uncertainty calculations to (5) 3.274 mVµ 0 µrad 0 µA where C zf = C zm C mf . Of course, a smart package would U f = 0 mVµ 379.1 µrad 0 µA (4) ( ) . use the phase information already present rather than 0 mVµ 0 µrad 6.251 µA recalculating it but we find it comforting to know the such an orthogonal vector space, we have no correlations to worry about. Every measurement has a certain sensitivity to one or more components in the vector space, the sensitivities determined by the derivatives of its measurement equation with respect to each component. We may represent a particular measurement’s uncertainty Thus we have the first uncertainty vector in the primary lab’s results. The numeric values mean nothing, however without the fundamental vector space definition. We may compute all uncertainty vectors at once as columns in a matrix by the simple propagation vector in that space with the component-by-component product of its sensitivities and uncertainties. To calculate U y = U x C yx T , (6) measurement uncertainty we compute the uncertainty vector’s magnitude (Euclidean length), u = ‖u‖ = √ ____ u T u , which defines a new vector space, though not orthogonal just the root-mean-square of its components. To combine unless C yx has a diagonal form. So the primary voltagecurrent-phase uncertainty space becomes multiple uncertainty vectors further down the traceability chain, we simply multiply each by the measurement’s sensitivity to that quantity’s error and add the resulting 3.208µmV −3.512 µA 0.6539µmrad vectors—no correlation complications, just multiply and U p = U f C pf T = 75.71 µV 6.189 µA 0.3714µmrad ( ) . add. 8 13.81 nV −6.250 µA −0.1013 µrad The combined uncertainty then just becomes the (7) length of the final vector (see Figure 1). To illustrate, let us compute the GUM H.2.4 example At the next traceability level we have the resistancereactance space by vector analysis. We note here that all the following calculations, though they take different forms, equate exactly to the GUM procedure. First, we define our U m = U f C mf T = ( 38.97 −263.9 −43.39 113.3 ) mΩ, orthogonal vector space by the primary lab’s independent 9 −40.63 −69.87 (8) uncertainty components, u f , and denote the corresponding where C error sources f 1 , f 2 , f 3 . The individual basis vectors take the mf = C mp C pf and C mp contains the sensitivity coefficients from the measurement equations relating form resistance and reactance to voltage, current, and phase. Finally, the customer computes the desired impedance 3.274 magnitude-phase uncertainty space u f 1 = ( 0 ) mV, u = 0 (3) f 2 ( 379.1 ) µrad, u = 0 f 3 ( 0 ) µA, 0 0 6.251 −208.6 mΩµ −653.9 µrad U (9) and comprise our fundamental vector space z = U f C z f T = 76.19 mΩµ 371.4 µrad ( ) , −80.82 mΩµ 101.3 nrad calculation will agree with prior data in all aspects. 8 Digital signal processor (DSP) chips love to multiply and add; with vector methods, they would make short work of even massive uncertainty problems. 9 This example has no SI traceability to correlate voltage, phase, and current errors. Cal Lab: The International Journal of Metrology 30 Apr • May • Jun 2013
Metrology: Standardize and Automate! Mark Kuster Figure 1. Combining two uncertainty vectors (blue) in an orthogonal uncertainty vector space (red). The reader may verify all the above uncertainty vectors by computing their magnitudes (lengths-RSS the components in each column) and comparing to the uncertainties in the previous example. For DOF, we use the standard Welch- Satterthwaite (W-S) formula—we have no correlation issues as long as we retain and compute from the fundamental DOF. Therefore, we use the sensitivity coefficients from the quantity of interest to the fundamental quantities. For example, the impedance has fundamental sensitivity coefficients and so after the units cancel out, the W-S formula yields C Z f = ( −63.71 A −1 201.0 Ω/rad −12.93 kΩ A −1 ), (10) ___________________________________________________ 0.236 v Z = 4 ____________________ ( −63.71 ∙ 3.274 × 10 −3 ) 4 __________________ 3.9956 + ( 201.0 ∙ 3.791 ×10−4 ) 4 0.5565 + ( −12.93 ∙ 6.251 × = 6.0. (11) ____________________ 10−6 ) 4 −2.2996 If we ever want the correlation between two error sources, we may calculate the “co-linearity” or normalized dot product of the two associated uncertainty vectors as u i ∙ u j ρ i , j = ________ ‖u i ‖ ‖u j ‖ . (12) Note that the equation yields zero for any two fundamental uncertainty vectors. For the resistance-reactance correlation, the formula works out to 0.03897 ∙ −0.2639 + −0.04339 ∙ 0.1133 + −0.04063 ∙ −0.0699 ρ R , X = ___________________________________________ 0.07107 ∙ 0.29558 = −0.588. (13) To recover any covariance matrix from an uncertainty vector set U, for example to use the GUM matrix uncertainty framework [14], we calculate all the vector lengths and cross-lengths by the single matrix product Σ = U T U. (14) To implement this method, an MII certificate would store or point to the nominal values, vector space definition, fundamental uncertainties and DOF, along with the sensitivity coefficients between each measurement and the fundamental in full precision. It might optionally store them in sparse or compact array format. A more sophisticated tactic would optionally store the measurement equations instead of the sensitivity coefficients. Either way, software would then Apr • May • Jun 2013 31 Cal Lab: The International Journal of Metrology
Page 1 and 2: METROLOGY 101: PISTON PROVER DESIGN
Page 3 and 4: Volume 20, Number 2 www.callabmag.c
Page 5 and 6: EDITOR’S DESK PUBLISHER MICHAEL L
Page 7 and 8: Electric Voltage & Current Voltage
Page 9 and 10: CALENDAR SEMINARS: Electrical Jun 3
Page 11 and 12: CALENDAR Oct 17-18 Introduction to
Page 13 and 14: INDUSTRY AND RESEARCH NEWS New NIST
Page 15 and 16: Since elementary school, you’ve h
Page 17 and 18: NEW PRODUCTS AND SERVICES Fluke Cal
Page 20 and 21: NEW PRODUCTS AND SERVICES Low-Tempe
Page 22 and 23: NEW PRODUCTS AND SERVICES CAL-TOONS
Page 24 and 25: METROLOGY 101 Piston Prover Design
Page 26 and 27: METROLOGY 101 III. Clearance Seal P
Page 28 and 29: Metrology: Standardize and Automate
Page 38 and 39: A New Primary Standard for the Real
Page 46 and 47: AUTOMATION CORNER Could the Tried a
Page 48: Discover the “Blue Box” Differe

METROLOGY 101: PISTON PROVER DESIGN - Cal Lab Magazine

Create successful ePaper yourself

Delete template?

Save as template?