Estimation of uncertainty in measurement in precision ... - NOPR

Journal of Scientific & Industrial Research Vol. 64, September 2005, pp. 666-673

Estimation of uncertainty in measurement in precision calibration of DC high current source up to 100 A Shiv Kumar Jaiswal*, V N Ojha and Ajeet Singh JVS & DC Standards, National Physical Laboratory, Dr K S Krishnan Road, New Delhi 110 012 Received 03 March 2005; accepted 04 July 2005 In the present paper, calibration of precision DC high current source (> 2 A and up to 100 A), widely used in industries and research, has been discussed as case studies of transconductance amplifier and precision power supply calibration. Since transconductance amplifier requires boosting device for its operation, therefore effect of boosting device was added as an additional component of uncertainty. The various sources of uncertainty in measurement and their estimation based on Type A and Type B method as per ISO ‘GUM’ document are discussed in detail. The results are reported at coverage factor k=2 for approx 95% confidence level. The standards used for calibration are traceable to the ‘National Standards’ and their uncertainties were evaluated prior to use to avoid variation in results due to drift. Keywords: Calibration, Uncertainty in measurement, Current shunt, Digital voltmeter IPC Code: G01N37/00

Introduction The direct quantum standard of current based on single electron tunneling (SET) is at present in research stage1. Normally, current is realized from voltage drop across the reference resistors using Ohm’s law. This method is more precise than direct measurement by digital multimeter because it offers the choice of best artifact (i.e. standard current shunt) for current calibration. The standards of voltage and resistance used for measurement are traceable to ‘National Standards’ and in turn traceable to Josephson Series Array Voltage Standard of NPLIndia2,3 and Quantum Hall Resistance Standard of NPL-India4. High precision multifunction calibrator, a source of DC voltage, DC current and DC resistance, is used as a working standard at NPL and is widely used for calibration of industrial measuring instruments5. As a DC current source, it is normally used in the range of 1 µA to 2.2 A, however for large value of current used in various industrial sectors, this range can be further extended upto 100 A using transconductance __________ *Author for correspondence Tel: +91-11-25742610–12/ Extn 2233 & 2273 Fax: +91-11-25726938, 25726952 E-mail: [email protected]

amplifier or using precision power supply. As case studies, the precision high current sources have been calibrated at 20 A and 100 A using the traceable standard current shunt and high precision calibrated 8½ digit digital voltmeter. The various sources of the uncertainty in measurement and their estimation based on Type A and Type B evaluation of standard uncertainty as per ISO ‘GUM’ document is also discussed6-8. Type A is evaluated by the statistical analysis of a series of observations, and Type B is evaluated by other than statistical analysis of a series of observations. Methodology of Calibration The transconductance amplifier is a precision high current source whose output current is proportional to an input voltage. For the operation of the amplifier, it is energized by some DC voltage source such as direct volt calibrator, multifunction calibrator etc., therefore the effect of this boosting device is also taken in account for uncertainty evaluation. However, if one is using high precision power supply, the uncertainty due to effect of boosting device is not taken separately as it is in-built and included in its calibration. The output current of the transconductance amplifier/power supply is passed through the standard current shunt and the ‘voltage drop’ across the shunt is measured by using a high

JAISWAL et al: CALIBRATION OF PRECISION DC HIGH CURRENT SOURCE UP TO 100 A

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where, d= a constant; pi = a positive or negative number; and xi = an input estimate. Comparing Eq (1) with Eq (2) gives p1 = 1 and p2 = -1. The voltage V has three components of uncertainty as u1(V), u2(V) and u3(V). The resistance R has two components of uncertainty as u4(R) and u5(R). Where, u1(V) = standard uncertainty in voltage drop across the current shunt; u2(V) = standard uncertainty of digital voltmeter; u3(V) = standard uncertainty of precision calibrator; u4(R) = standard uncertainty of the current shunt; and u5(R) = standard uncertainty due to temperature effect of the current shunt. Uncertainty Equation There are two methods for solving Eq (1): (a) Absolute Method; and (b) Relative Method. Absolute Method

For uncorrelated input quantities, the combined standard uncertainty uc is given as follow: N N 2 2 uc (y) = Σ [(∂f/∂xi) u (xi)] = Σ [ci2 u2(xi)], i=1 i=1 2

Fig 1 — Block diagram for calibration of transconductance amplifier

precision digital voltmeter (Fig. 1). The value of current is calculated from the ratio of voltage drop across the current shunt to the resistance of the current shunt as per Ohm’s law (I=V/R). As case studies, the calibration of 20 A is discussed in detail using transconductance amplifier and for 100 A, the precision power supply is used.

where, uc(y) = combined standard uncertainty; u(xi)=components of uncertainty; ci=∂f/∂xi=sensitivity coefficients; and N = number of uncertainty components. The corresponding sensitivity coefficients [calculated from Eq (1)] are: c1 = c2 = ∂Ix /∂V = 1/R; c3 = 1 (because amplifier’s transconductance is 1 S); and c4 = c5 = ∂Ix /R = -V/ R2 The combined standard uncertainty is given as: uc2(Ix) = (c1)2 u12(V) + (c2)2 u22(V) + (c3)2 u32(V) + (c4)2 u42(R) + (c5)2 u52(R)

Mathematical Model Used for the Evaluation of Uncertainty The following mathematical model is used to estimate the uncertainty in measurement.

Relative Method

Ix = f (V, R) = V/R = V.R-1

N [uc(y)/y] = Σ [{ pi.u(xi)/xi}2] i=1

…(1)

where, Ix = current output of transconductance amplifier/precision power supply; V = voltage drop across standard current shunt; and R = resistance of standard current shunt. Eq (1) is in the form of y = d.Πxipi

…(2)

…(3)

…(4)

For uncorrelated input quantities, the relative combined standard uncertainty uc(y)/y is given as follows: 2

…(5)

where, uc(y)/y = relative combined standard uncertainty; u(xi)/xi = components of uncertainty in relative form; pi = a positive or negative number; and N = number of uncertainty components.

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Average value V = (ΣVi)/ n = 1.000380 V.

Table 1 — Experimental observation Input voltage to transconductance amplifier: 20.000000 V Nominal output current of transconductance amplifier: 20.00000 A No. of observation

Measured voltage drop across current shunt V

1

1.000390 V

2

1.000388 V

3

1.000387 V

4

1.000386 V

5

1.000384 V

6

1.000380 V

7

1.000376 V

8

1.000373 V

9

1.000369 V

10

1.000367 V

Experimental standard deviation s (V) = 8.30×10-6 V. Standard deviation of mean u (V ) = s (V ) = s(V)/√n = 2.62×10-6 V. So, Type A uncertainty u1 (V) = 2.62×10-6 V. Relative type A uncertainty u1 (V)/V = 2.62×10-6 V/ 1.000380 V = 2.62×10-6 and degree of freedom ν1 = n-1 = 10-1 = 9. The measured current is: Ix = V/R =1.000380/ 0.0500237 = 19.99810 A Type B Evaluation of Standard Uncertainty

V = 1.000380 V

There are four components of uncertainty for Type B evaluation as follows:

The sensitivity co-efficient for Eq. (1) is given by (a) Standard Precision Calibrator

Sensitivity co-efficient = pi (y/xi)

…(6)

The relative combined standard uncertainty is given as: [uc(Ix)/Ix]2 = (p1)2 [u1(V)/V]2 + (p2)2 [u2(V)/V]2 + (p3)2 [u3(V)/V]2 + (p4)2 [u4(R)/R]2 + (p5)2 [u5(R)/R]2

…(7)

where, u1(V)/V = relative standard uncertainty in voltage drop across the current shunt; u2(V)/V = relative standard uncertainty of digital voltmeter; u3(V)/V = relative standard uncertainty of precision calibrator; u4(R)/R = relative standard uncertainty of the current shunt; and u5(R)/R = relative standard uncertainty due to temperature effect of the current shunt. From Eq. (1), p1 = p2 = p3 =1 and p4 = p5 = -1. Eq. (4) and Eq. (7) give standard uncertainty of Eq. (1). Two methods are same except their different ways of evaluation. In relative method, it is easy to handle the equation having large numbers of components in product/quotient form. Evaluation of Uncertainty Type A Evaluation of Standard Uncertainty

For evaluation of Type A uncertainty, 10 numbers of observations were taken (Table 1).

Prior to use, the precision calibrator was calibrated using DC voltage reference standard, voltage divider and high sensitive digital voltmeter/nanovoltmeter to avoid variation in results due to drift. The following mathematical model was used to evaluate the uncerainty5: VX = (VZ + δVDVM − δVD) r

…(8)

where, VX = voltage applied by the calibrator; r = ratio of the voltage divider; VZ = average voltage of in-house DC voltage reference standard; δVD = difference in voltage between the in-house DC voltage reference standard and divided voltage of the calibrator; and δVDVM = non-linearity and instability of sensitive DVM/Nanovoltmeter. The uncertainty of standard precision calibrator at 20 V was estimated as ± 4.0×10-5 V at k = 2 for approx 95 % confidence level. Since the probability distribution is normal, therefore the relative standard uncertainty u2(V)/V in precision calibrator is: U2(V)/V = 4.0 ×10–5 V/(2×20 V) =1.0 ×10–6 and degree of freedom ν2 = ∞.

at k = 1


(b) Standard Digital Voltmeter (DVM)

Prior to use, the precision 8½ digit digital voltmeter was calibrated by using precision calibrator to avoid variation in results due to drift. The following mathematical model was used to evaluate the uncertainty8: Vx = Vstd + ∆Vx

…(9)

where, Vx is the voltage indicated on digital voltmeter under calibration; Vstd is the voltage applied from the standard precision calibrator; and ∆Vx is resolution of digital voltmeter under calibration. The uncertainty of standard DVM at 1 V was estimated as ± 3.0 ×10-6 V at k=2 for approx 95 % confidence level. Since digital voltmeter was calibrated at the point of its use, so non-linearity component of uncertainty does not come into picture. Assuming probability distribution as normal, the relative standard uncertainty u3 (V)/V in voltage drop measured by DVM is: U3(V)/V = 3.0 ×10-6 V/ (2×1 V) = 1.5 ×10–6

at k = 1

and degree of freedom ν3 = ∞. (c) Standard Current Shunt (SCS)

The SCS was calibrated using automatic direct current comparator (DCC) bridge. The following mathematical model was used to evaluate the uncertainty9: RX = (RS + δ RD + cS δ tS) r.rl – cX δ tX

…(10)

where, RX = measured value of the unknown resistor; RS = value of the reference resistor; δRD = drift in the value of reference resistor since its last calibration; δ tS = temperature deviation of the reference resistor; r = the ratio of the unknown resistance to reference resistance (i.e. RX /RS); rl = non-linearity and instability of the high resistance bridge; δ tX = temperature deviation of the unknown resistor; and cS & cX = temperature co-efficient of reference resistor and unknown resistor respectively. The actual resistance value of the nominal 0.05 Ω SCS from its calibration is 0.0500237 Ω ± 0.0000001 Ω at k = 2 for approx. 95 % confidence level. Assuming normal distribution, the relative standard uncertainty u4(R)/R in SCS is:

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u4 (R)/R=1.0×10-7Ω/(2×0.05Ω)=1.0×10-6 at k = 1 and degree of freedom ν4 = ∞. (d) Temperature Effect of Standard Current Shunt

The temperature affects the resistance value of the current shunt. For determination of temperature coefficient of the current shunt, the current shunt was calibrated using automatic DCC bridge at three temperatures9, 23, 25 and 27 °C. The temperature coefficient factors (α & β) were calculated from the following widely used working formula10: R23 = R25 [1+ α (t23 – t25) + β (t23 – t25)2]

…(11)

and R27 = R25 [1+ α (t27 – t25) + β (t27 – t25)2]

…(12)

where, R23 = resistance of the current shunt at 23°C; R25 = resistance of the current shunt at 25 °C; R27 = resistance of the current shunt at 27 °C; α = slope of the curve (ppm/°C) at 25°C; and β = rate of change of slope of the curve (ppm/°C2). The calculated values of α and β are –1.17 ppm/°C and –0.069 ppm/ °C2 respectively. From the values of α and β, the current shunt is characterized for the other temperature using the following formula: Rt = R25 [1+ α (t – t25) + β (t – t25)2]

…(13)

The resistance of current shunt with respect to temperature (Fig. 2) is decreasing, as the values of temperature co-efficient factors (α & β) are negative for this shunt. For working temperature of 25 ± 1°C, temperature co-efficient of the standard current shunt was taken as –1.51 ppm/°C ≈ –2 ppm/°C as the maximum value (Table 2). Since upper and lower limit of this uncertainty component is given, therefore, distribution was assumed as rectangular. For this distribution, relative standard uncertainty, u5(R)/R, due to temperature effect of standard current shunt is: u5(R)/R=(2 ppm/°C×1°C)/√3=2.0×10-6/√3=1.15×10–6 at k=1 and degree of freedom ν5 = ∞. Relative Combined Standard Uncertainty

The relative combined standard uncertainty is calculated as follows: [uc(Ix)/Ix]2=(p1)2[u1(V)/V]2+(p2)2[u2(V)/V]2+(p3)2[u3(V)/V]2 +(p4)2[u4(R)/R]2+(p5)2[u5(R)/R]2uc(Ix)/Ix = 3.527×10–6


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Fig 2 — Resistance versus temperature curve of 0.05 Ω current shunt Table 2 — Temperature characterization chart for 0.05 Ω current shunt Alpha: –1.17 ppm/°C Beta: –0.069 ppm/°C2 Temperature °C

Resistance ohm

Deviation from value at 25 °C ppm

22.0

0.05002419

+ 4.11

23.0

0.05002405

+ 2.61

24.0

0.05002399

+ 1.37

25.0

0.05002392

+ 0.00 (Ref. temperature)

26.0

0.05002384

-1.51

27.0

0.05002376

-3.17

28.0

0.05002367

-4.96

Effective Degree of Freedom (νeff)

Effective degree of freedom of relative standard uncertainty uc(y)/y associated with the output estimate ui(y) is given by Welch-Satterthwaite formula6: [uc(y)/y]4 νeff = ——————— N {∑ [ui(y)/y]4/νi } i=1

νeff = 29.54 ≈ 29

Truncating to the lower side to get the maximum value of the coverage factor (k) and thereby the practical expanded uncertainty. Expanded Uncertainty (U)

The relative expanded uncertainty is given by U(y)/y = k × [uc(y)/y]. where k is a coverage factor. From student’s t-distribution6, the value of coverage factor k for approx 95 % confidence level and for νeff=29 is 2.11. U(Ix)/Ix = 2.11×3.527×10–6 = 7.441×10–6 U(Ix) = 7.441×10–6 × Ix = 7.441×10–6 ×19.99810 A = 1.49×10–4 A The complete uncertainty budget is summarized in Table 3. Results The measured value of current of nominal 20 A current source alongwith expanded uncertainty is 19.99810±0.00015 A (Table 4). The reported expanded uncertainty of measurement is stated as the combined standard uncertainty multiplied by the coverage factor k=2.11 which for a normal distribution corresponds to coverage probability of approx 95 %. In the similar way, uncertainty estimation of the precision high


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Table 3 — Statement of the uncertainty budget in transconductance amplifier calibration at 20 A (Relative Method) Quantity Xi

Estimates xi

Relative Limits ±∆xi/xi

Probability distribution (Type A or B)

Relative standard uncertainty u(xi)/xi (±)

Relative uncertainty contribution ui(y)/y=pi u(x)/xi (±)

Degree of freedom

Standard DVM

1V

3.0×10-6

Normal Type B

1.5×10-6

3.0×10-6

∞

Standard calibrator

20 V

2.0×10-6

Normal Type B

1.0×10-6

1.0×10-6

∞

0.05 Ω

2.0×10-6

Normal Type B

1.0×10-6

1.0×10-6

∞

—

2.0×10-6

Rectangular Type B

1.15×10-6

1.15×10-6

∞

Repeatability

1.000380 V

—

Normal Type A

2.62×10-6

2.62×10-6

9

Ix

19.99810 A

—

—

—

3.527×10-6

29

Standard current shunt Temp. effect of current shunt

U(Ix)/Ix

—

U(Ix)

—

— —

— —

-6

coverage factor k=2.11

7.44×10


-4

1.49×10 A

νi

29 29

Table 4 — Statement of the uncertainty budget in transconductance amplifier calibration at 20 A (Absolute Method) Quantity Xi

Estimates xi

Limits ± ∆xi

Standard DVM

1V

Standard calibrator


Standard uncertainty u(xi) (±)

Sensitivity coefficient ci

Uncertainty contribution ui(y) = ci u(xi) (±)

Degree of freedom

3.0×10-6 V

Normal Type B

1.5×10-6 V

19.99 S

3.0×10-5 A

∞

20 V

4.0×10-5 V

Normal Type B

2.0×10-5 V

1S

2.0×10-5 A

∞

0.05 Ω

1.0×10-7 Ω

Normal Type B

5.0×10-8 Ω

-399.78 A/Ω

2.0×1-4 A

∞

—

1.0×10-7Ω

Rectangular Type B

5.77×10-8Ω

-399.78 A/Ω

2.31×10-5 A

∞

Repeatability

1.000380 V

—

Normal Type A

2.62×10-6 V

19.99 S

5.25×10-5 A

9

Ix

19.99810 A

—

—

—

—

7.06×10-4 A

29

Standard current shunt Temp. effect current shunt

Expanded uncertainty U(Ix)

of

—

—

—

current source at 100 A was done using high precision power supply. Since the precision current source has inherent boosting device, therefore, effect of boosting device in the uncertainty budget is not warranted. The measured value of current of nominal 100 A current source alongwith expanded uncertainty is 99.6435 ± 0.0176 A (Tables 5 & 6) as evaluated by the two different methods.


-4

1.49×10 A

νi

29

Conclusions In the present work, the calibration of DC high current source, widely used in the industrial and research areas, is discussed in detail. As case studies, 20 A and 100 A current sources using transconductance amplifier and precision power supply were calibrated. In case of transconductance amplifier, the effect of boosting device is included in


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Table 5 — Statement of the uncertainty budget in precision high current source calibration at 100 A (Relative Method) Quantity Xi

Estimates xi

Relative limits ±∆xi/xi


Standard DVM

10 mV

2.0×10-5

0.0001 Ω

Standard current shunt Temp. effect of current shunt Repeatability Ix

Relative standard uncertainty u(xi)/xi (±)

Relative uncertainty contribution ui(y)/y=pi u(xi)/xi (±)

Degree of freedom

Normal Type B

1.0×10-5

1.0×10-5

∞

3.0×10-6

Normal Type B

1.5×10-6

1.5×10-6

∞

—

6.0×10-6

Rectangular Type B

3.46×10-6

3.46×10-6

∞

9.96475 mV

—

Normal Type A

7.54×10-5

7.54×10-5

9

99.6435 A

—

—

—

7.62×10-5

9.37

-4

9.37

U(Ix)/Ix

—

—

—


1.77×10

U(Ix)

—

—

—


1.76×10-2 A

νi

9.37

Table 6 — Statement of the uncertainty budget in precision high current source calibration at 100 A (Absolute method) Quantity Xi

Estimates xi

Limits ± ∆xi


Standard uncertainty u(xi) (±)

Sensitivity coefficient ci

Uncertainty contribution ui(y)=ci u(xi) (±)

Degree of Freedom

Standard DVM

10 mV

1.99×10-7 V

Normal Type B

9.96×10-8 V

9999.6 S

9.96×10-4 A

∞

Standard current shunt

0.0001 Ω

3.0×10-10 Ω

Normal Type B

1.50×10-10 Ω

-996395 A/Ω

1.49×10-4 A

∞

Temp. effect of current shunt

—

6.0×10-10Ω

Rectangular Type B

3.46×10-10Ω

-996395 A/Ω

3.45×10-4 A

∞

9.96475 mV

—

Normal Type A

7.52×10-7 V

9999.6 S

7.51×10-3 A

9

99.6435 A

—

—

—

—

7.59×10-3 A

9.37

—

—

—

1.76×10-2 A

9.37

Repeatability Ix Expanded uncertainty U(Ix)

the uncertainty budget. However, in the calibration of 100 A current source using high precision power supply, the effect of boosting device, which is inbuilt, does not come into picture. Uncertainty calculations can be done by using Relative Method (Table 3 for 20A & Table 5 for 100A) and Absolute Method (Table 4 for 20A & Table 6 for 100A) resulting into same results. However, it gives the user an opportunity to use either depending on the ease of use and his understanding of the sensitivity co-efficients. Further, the effect of the temperature on


νi

the resistance value of the standard current shunt and its evaluation was highlighted. For this purpose, the temperature co-efficient of standard current shunt was also determined (Table 2) using recently established high precision automatic DCC bridge and its effect on uncertainty in measurement was incorporated. Acknowledgements Authors thank Dr Vikram Kumar, Director and Dr P C Kothari, Head, Electrical and Electronic Standards of National Physical Laboratory, New


Delhi for their constant encouragement and support. Authors also thank Dr Sudhir Kumar Sharma for useful discussions.

5

Ojha V N, Singh A, Jaiswal S K & Sharma S K, Automatic and manual calibration of high precision multifunction calibrator, MAPAN, J Metrol Soc India, 18 (2003) 43-48.

6

Guide to the Expression of Uncertainty in Measurements, I edn (International Organisation for Standardization, Switzerland), 1995.

7

Ojha V N, Evaluation and expression of uncertainty in measurement, MAPAN, J Metrol Soc India, 13 (1998) 7184.

8

Guidelines for estimation and statement of overall uncertainty in measurement results, NABL Document No. 141 (Department of Science and Technology, Govt. of India, New Delhi) 2000.

9

Singh A & Ojha V N, Calibration of national standard of resistance using newly established automatic direct current comparator resistance bridge, MAPAN, J Metrol Soc India, 18 (2003) 49-55.

10

Resistance Standards Instruction Manual, 742A series (Fluke Corporation, Everett, USA) 1989, 1-9.

References 1

Keller M W, Standards of current and capacitance based on single-electron tunneling devices, Proc Fermi School CXL VI: “Recent Advances in Metrology and Fundamental Constants” (IOS Press, Amsterdam) 2000.

2

Ojha V N & Gupta A K, Josephson voltage standard at 1 volt level at NPL India, in CPEM-Digest (USA), edited by T L Nelson (Washington DC, USA) 1998, 558-559.

3

Sharma S K, Ojha V N, Singh A, Jaiswal S K & Sharma R, Traceability and calibration-A case study, MAPAN, J Metrol Soc India, 18 (2003) 57-61.

4

Ojha V N & Singh A, Traceability of DC resistance standard at NPLI, 5th International Conference on Advances in Metrology, Admet-2005 (NPL, New Delhi) February 23-25, 2005, 78.

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