1. K.-D. Sommer 09/02 san_diego.ppt Modelling of Measurements for the Evaluation of the Measurement Uncertainty Klaus - Dieter Sommer Manfred Kochsiek San Diego, California August 4 - 8, 2002

2. K.-D. Sommer 09/02 san_diego.ppt Uncertainty of Measurement (I) Problem:Experience has shown that Measuring processes can never be controlled perfectly... Operating conditions are not infinite-precisely known... A measurable quantity can never be described by a single value only ! Solution:Description of the (imperfect) knowledge by probability distributions and their moments, and respectively

3. K.-D. Sommer 09/02 san_diego.ppt Uncertainty of Measurement (II) Y : Measurand y :Expectation value of the measurand U : Expanded uncertainty of measurement Definition of the measurement uncertainty (GUM): Parameters associated with the result of a measurement, that characterizes the dispersion of values that could reasonably be attributed to the measurand. Interval expected to encompass a large fraction of values that could be attributed to Y

4. K.-D. Sommer 09/02 san_diego.ppt Modelling the Measurement - Basic Step of the GUM Procedure Taking up and gaining knowledge about the measuring process and the involved quantities Evaluating the quantities Type - A Type - B  Modelling the measurement  SRC IND... establishes the relationship between the input quantities and the measurand...is the basis for uncertainty propagation calculus...is the most difficult problem of uncertainty evaluation

5. K.-D. Sommer 09/02 san_diego.ppt Modelling the Measurement Basic Idea Measuring- process Analyzing the cause-and-effect relationship Model equation Taking up the knowledge about the measuring process Knowledge about the measuring process Expressing the cause-and- effect relationship in mathematical terms and converting it into the model equation...

6. K.-D. Sommer 09/02 san_diego.ppt Bases of the Modelling Concept Presented Non-reactive chaining in cause-and-effect direction Description of all disturbances and imperfections by deviations Determining the structure and the branches of the model Modelling Concept „Classical“ Measuring Chain K1K1 K2K2... Method of Measurement Direct Measurement Substitution Difference Method

7. K.-D. Sommer 09/02 san_diego.ppt Setting Up a Cause-And-Effect Relationship Principle Procedure Causal quantity Cause-and-effect direction ac a)Transference of the causal quantity to the scale b) Taking into account the scale error c)Reading the scale b Indicated quantity

8. K.-D. Sommer 09/02 san_diego.ppt Modelling Concept A Four-Steps Procedure (I) 1st Step Describing the measurement Identifying the causal quantity and the measurand (... not necessarily the same!) identifying the measurement method used Formulating the cause-and-effect relationship of the (fictitious) ideal measurement 2nd Step SRC

9. K.-D. Sommer 09/02 san_diego.ppt Considerating of all disturbances, imperfections and possible effects of incomplete knowledge Introducing them by means of deviations from the ideal measurement Converting the cause-and-effect relationship of the real measurement into the model equation Modelling Concept A Four-Steps Procedure (II) 3rd Step 4th Step SRC

10. K.-D. Sommer 09/02 san_diego.ppt Example: Determination of an Electrical Resistance (I) 1.Description of the measurement: Determination of the electrical resistance R X of a resistor, that is supplied by a constant current I. The voltage over the resistor is measured by a digital voltmeter. Measurand: Measurement method: Direct measurement Resistor Voltmeter

11. K.-D. Sommer 09/02 san_diego.ppt Example: Determination of an Electrical Resistance (II) 2. Cause - and - effect relationship of the ideal measurement: Assumptions: - I* is infinite-precisely known - V* X is equal to V IND Cause-and-effect relationship: SRCINDUTRANS Cause – and - effect direction

12. K.-D. Sommer 09/02 san_diego.ppt 3. Introducing of all disturbances and imperfections: - constant current - instrumental error - deviation of the current - „ effective“ input voltage - voltage over the resistance - deviation due to th resolution - thermal voltage - resistance - deviation of the temperatur Example: Determination of an Electrical Resistance (III) Resistor Voltmeter

13. K.-D. Sommer 09/02 san_diego.ppt Example: Determination of an Electrical Resistance (IV) Cause - and - effect relationship of the real measurement: graphically: mathematically: SRCINDUTRANS

14. K.-D. Sommer 09/02 san_diego.ppt Example: Determination of an Electrical Resistance (V) 4.Model equation: The model equation is obtained by converting the cause-and-effect relationship of the real measurement.... all these involved quantities must be evaluated by means of probability distributions [ Methods A and B ]

15. K.-D. Sommer 09/02 san_diego.ppt Example: Determination of an Electrical Resistance (VI) Evaluation of the Involved Quantities QuantityKnowledge Expectation Standard Kind of available value uncertainty Distribution nominal value: 10 ·10 -3 A max. deviation: ± 0,1 · 10 -6 A max. deviation: ± 2 °C deviation range: 0... 3 mA 16 observations mean 1,1065 V SD 2,2 · 10 -3 V...... 10 mA 0 1,5 mV 1,1065 V 0 0,058 ·10 -6 A 1,15 °C 0,87 mV 0,28 mV single value rectangular normal........................

16. K.-D. Sommer 09/02 san_diego.ppt Role of the Method of Measurement (I) Structure and chaining sequence are determined by the method used! Generic structure of direct measurements: Generic structure of direct comparisons of indicating measuring instruments: SRC TRANSINDU SRC TRANS X INDU X TRANS S INDU S Evaluation X-path: S-path:

17. K.-D. Sommer 09/02 san_diego.ppt Example: Direct Comparison of Thermometers (I) 1Description of the measurement: Determination of the instrumental error of an liquid-in-glass Thermometer at 20°C Measurand: Instrumental error Method used :Direct comparison 2Cause-and-effect relationship of the ideal measurement: standardinstrument to be calibrated SRC INDU X INDU S

18. K.-D. Sommer 09/02 san_diego.ppt 3Cause-and-effect relationship of the real measurement: SRC TRANSINDU X INDU S - (local) temperature of the bath liquid - deviation of the temperature of the instrument to calibrated from the temperature of the standard - instrumental error of the standard - deviations due to imperfect reading Measurand Example: Direct Comparison of Thermometers (II)

19. K.-D. Sommer 09/02 san_diego.ppt Modelling Concept Conclusion and Limits Benefits: straightforward and widely applicable concept based on the classical approach of the measuring chain modular treatment of complex problems possible Limits: Systems described by recursive algorithms No limits: pretty complex measuring processes and systems