Uncertainties of measurement

Transcription

1 Uncertainties of measrement Laboratory tas A temperatre sensor is connected as a voltage divider according to the schematic diagram on Fig.. The temperatre sensor is a thermistor type B5764K [] with nominal resistance for 5 C R 00 Ω ± 0 %. The thermistor resistance dependence on temperatre is non linear and is given by characteristic No. 03 in Appendi. The connection is powered by a 9V reglated switching power spply type RQT666K [3]. The measred otpt voltage U (variable with temperatre) is measred with a voltmeter type MT-3 [5] on range 400 mv DC. Used resistor (nominal vale: R 70 Ω) is a precision metal film resistor with accracy ± 0.%. Tas: a) Calclate ncertainty of measrement type A for voltage U. b) Calclate type B ncertainty of temperatre measrement T by considering the following ncertainty sorces (for these tass consider the temperatre in the lab to be eactly 5 C): Fig. - Schematic diagram of temperatre sensor - Uncertainty cased by power spply voltage U0 variations B - Uncertainty of R calclation cased by ncertainty of power spply voltage U0, ncertainty of measrement of voltage U and R tolerance B Write reslts as Temperatre ± ncertainty and state the sed probability distribtion fnction and sed coverage factor. c) In conclsions write what other sorces of ncertainty cold be added to the calclations for a more precise reslt

2 Soltion Important note: Remember, yo have to really repeat the whole eperiment, it means to trn off and on the power spply. Use the switch on the fitre for this. It is not sfficient to jst read 9 following vales from the voltmeter. a) Uncertainty type A Uncertainty type A is obtained by repeating the eperiment and by statistical evalation. Tab. 3 - measred and calclated vales i U (mv) Sample mean n i i n () Estimated standard deviation of the mean ncertainty type A s A s n n i ( i ) ( ) n n () As the eperiment was repeated only 8 times, the interval for ncertainty type A has to be etended > corrected ncertainty type A needs to be calclated. In other words if the eperiment is repeated only 3 times the reslt is mch less sre than it is if the eperiment is repeated 00 times. The vales of the correction coefficient are given by the coverage factor from the Stdent s t- distribtion table. As at the end we will write the reslt in form: average ± ncertainty, we tae only ½ of the vale in Stdent s t-distribtion table. Those vales are shown in Tab. 4 directly. If the reslt wold be written as: average; ncertainty, then we wold tae the whole interval. Tab. Coverage factor for normal distribtion of ncertainty type A for p 95,45 % (ronded) i (nmber of measrements) (coverage factor),0,,,,,3,3,4,7,3 7,0 Corrected ncertainty type A is AK (3) A

3 b) Uncertainty type B - Uncertainty U0 in power spply voltage From manfactrer specifications on Fig. 3 it can see that the otpt voltage shold be 9,5 V ± 0,5 V when the power spply is nloaded. This assmption can be made as the circit crrent is very small; it was measred to be approimately 0 µa. As the manfactrer gives s an interval and not ncertainty, we have to calclate it. The interval 9,5 V ± 0,5 V is where the voltage is with certainty almost 00 %. If interval of the sed vale is nown then standard ncertainty is fond from U 0 (4) where is the semi-range (or half-width) between the pper and lower limits and is a divisor dependent on the shape of the probability distribtion fnction of or variable (rectanglar, normal, U-Shape etc.). Its vales are shown in Tab. 5 Tab. 5 - divisors for varios probability distribtion fnctions probability distribtion Rectanglar (for normal (Gassian) U- Shape trianglar fnction probability for σ %) (95,45 %) Coverage 3 factor 6 If we consider a normal (Gassian) probability distribtion of power spply otpt voltage, the ncertainty cased by power spply voltage variations will be U 0 (5) - Calclation of ncertainty R cased by ncertainty ncertainty U0 in power spply voltage, ncertainty of measrement of voltage U and R tolerance B To calclate ncertainty of measrement of R we have to first analyze the circit. The connection is a voltage divider composed of resistor R and temperatre sensor R. Otpt voltage of this divider is U f ( U 0, R, R) (6) (7) From circit analysis it is nown R U U U 0 R R R + R U 0 U (8)

4 Uncertainty R will be given by ncertainties of all variables in eqation(8), i.e. by ncertainty of U0, U and R. The individal components are calclated as follows - Uncertainty U0 in power spply voltage Uncertainty cased by power spply voltage variations B was calclated earlier as U 0 - Uncertainty cased by resistance R tolerance R To calclate ncertainty of resistance R, R, available manfactrer data is sed, R 70 Ω ± 0, % > R R 0,% /00 ± 0, 7 Ω. If we assme a normal (Gassian) distribtion of resistance for σ then R R (9) - Uncertainty of measrement of voltage U U From voltmeter manfactrer specifications on Fig. 5 and Fig. 6 it is fond that on range 400 mv, accracy is ±(0,5 % of reading + 4 digits), resoltion 00 µv. Considering the worst case (highest inaccracy) on this range (maimal voltage), we get ± (0,5 % of reading + 4 digits) ± ( 0,5 % 400 mv µv) ±,4 mv Uncertainty for a rectanglar distribtion is U accracy (0) 3 - Uncertainty of R calclation R In a general case, to calclate the ncertainty of a variable given as a fnction f of inpt variables,,..., n it is necessary to calclate a sqare root of sm of sqares of partial derivatives from fnction f by all variables,,..., n mltiplied by ncertainty of nominal vales of variables,...,, N N nn This is called law of propagation of ncertainty f f f B N K N () nn n Note.: Uncertainties of measrement always add together. If one ncertainty is mch larger and others are small, the large one will be dominant and no improvement in the smaller ncertainties in the whole measring chain will improve mch the combined ncertainty. It will be the weaest lin in the whole chain. In other words, the most significant part of total ncertainty will be cased by the largest ncertainty. As not correlated variables are assmed, in or case R f ( U 0, U, R), the ncertainty is

5 R R R R U 0 + U + R U 0 U R U R R U 0 U ( R U) ( ) R U 0 U 0 U 0 U 0 U R R U R U + U U ( U 0 U) U 0 U R U R R R U 0 U () (3) (4) (5) (6) R (7) The ncertainty R shows the ncertainty of R measrement based on nown properties of the circit. To recalclate this to ncertainty of temperatre measrement, the dependence of thermistor resistance as a fnction of temperatre has to be nown. From the manfactrer data on Fig. it was determined, that in the range 0 C to 50 C the dependence can be approimated with an eponential fnction R[ Ω] ln 0,044 T[ C] R[ Ω ] 695 e T [ C] 695 (8) 0, Uncertainty cased by resistance R tolerance RTOL Eqation (8) gives s an approimated dependency of resistance on temperatre. The manfactre vales are given with a tolerance ± 0 %. This tolerance of the nominal vale has also to be considered for temperatre ncertainty calclations. R 0 Ω ± 0 % > R 0% /00 ± 0 Ω. If we assme a normal (Gassian) distribtion of resistance for σ then ncertainty is R R TOL R (9) - Uncertainty of temperatre measrement T The ncertainty of temperatre T measrement is given by ncertainty cased by the properties of the connection given by ncertainty R and by ncertainty cased by thermistor R tolerance RTOL T T T R + RTOL R R (0)

6 T 5000 R 07 R T 5000 R R R 07 R T 5000 RTOL RTOL R 07 R () () (3) T (4) Write reslt as: Temperatre ± ncertainty. Uncertainty has been evalated assming normal distribtion and coverage factor (probability 95,45 %)

7 Appendi Fig. - Thermistor resistance - temperatre dependence

8 Fig. 3 - Power spply specifications

9 Fig. 4 - Accracy specifications Fig. 5 - MT-3 mltimeter accracy for DC voltage References [] Bell S.: A Beginner's Gide to Uncertainty of Measrement, online (9..00) on [] NTC thermistors for temperatre measrement, online (6..0) on B5764 K64.pdf [3] Power spply specifications, online (6..0) on [5]MT-3 online (6..0) on [6] Taylor B. N., Kyatt Ch. E.: Gidelines for Evalating and Epressing the Uncertainty of NIST Measrement Reslts, online (6..0) on [7] Appendi V. Uncertainties and Error Propagation, online (6..0) on