Calibration of a nominal 10 kOhm standard resistor

The resistance of a four terminal standard resistor is determined by direct substitution using a long scale digital multimeter (7½digit DMM) on its resistance range, and a calibrated four terminal standard resistor of the same nominal value as the item to be calibrated as reference standard. The resistors are immersed in a well stirred oil bath operating at a temperature of 23 °C monitored by a centrally placed mercury-in-glass thermometer.

The resistors are allowed to stabilise before the measurement. The four terminal connectors of each resistor are connected in turn to the terminals of the DMM. It is determined that the measuring current on the 10 kOhm range of the DMM of 100 µA is sufficiently low not to cause any appreciable self-heating of the resistors. The measuring procedure used also ensures that the effects of external leakage resistances on the result of measurement can be considered to be insignificant.

Model Equation:

RX = ( RS + δRD + δRTS ) × rC × r - δRTX

List of Quantities:

Quantity Unit Definition
RX Ω resistance of the unknown resistor
RS Ω resistance of the reference
δRD Ω change of the resistance of the reference since its last calibration due to drift
δRTS Ω temperature related resistance deviation of the reference
rC   correction factor for parasitic voltages and instrument resolution
r   =RiX/RiS ratio of the indicated resistance for the unknown resistor and the reference resistor
δRTX Ω temperature-related resistance deviation of the unknown resistor

RS: Type B normal distribution
Value: 10000.053 Ω
Expanded Uncertainty: 5·10-3 Ω
Coverage Factor: 2
REFERENCE STANDARD: The calibration certificate for the reference standard gives a resistance value of 10000,053 Ohm ±5 mΩ (coverage factor k = 2) at the specified reference temperature of 23 °C.

δRD: Type B rectangular distribution
Value: +20·10-3 Ω
Halfwidth of Limits: 10·10-3 Ω
DRIFT OF THE STANDARD: The drift of the resistance of the reference resistor since its last calibration is estimated from its calibration history to be +20 mΩ with deviations within ±10 mΩ.

δRTS: Type B rectangular distribution
Value: 0 Ω
Halfwidth of Limits: 2.75·10-3 Ω
TEMPERATURE CORRECTION: The temperature of the oil bath is monitored with a calibrated thermometer to be 23.00 °C. Deviations from this value have been estimated to be within ±0.055 K, including temperature gradients in the oil bath. Thus the known value 5.0·10-6 K-1 of the temperature coefficient (TC) of the reference resitor gives limits ±2,75 mΩ for the deviation from its resistance value according to calibration, due to a possible deviation from the operating temperature.

rC: Type B triangular distribution
Value: 1.0
Halfwidth of Limits: 1.0·10-6
RESISTANCE MEASUREMENTS: Since the same DMM is used to observe both and the uncertainty contributions are correlated but the effect is to reduce the uncertainty and it is only necessary to consider the relative difference in the resistance readings due to systematic effects such as parasitic voltages and instrument resolution, which are estimated to have limits of ±0,5E-6 for each reading. The distribution resulting for the ratio rC is triangular with expectation 1,000 000 0 and limits ±1,0E-6.

r: Type A
Method of observation: Direct
Number of observation: 5
No. Observation
1 1.0000104
2 1.0000107
3 1.0000106
4 1.0000103
5 1.0000105

Arithmetic Mean: 1.0000105000
Standard Deviation: 160·10-9
Standard Uncertainty: 70.7·10-9
Degrees of Freedom: 4

δRTX: Type B rectangular distribution
Value: 0 Ω
Halfwidth of Limits: 5.5·10-3 Ω
TEMPERATURE CORRECTION: The temperature of the oil bath is monitored with a calibrated thermometer to be 23.00 °C. Deviations from this value have been estimated to be within ±0.055 K, including temperature gradients in the oil bath. From the manufacturor 10.0·10-6 K-1, thus the resistance variations of the unknown resitor due to a temperature variation is estimated to be within ±5,5 mΩ

Uncertainty Budgets:

RX: resistance of the unknown resistor
Quantity Value Standard
Uncertainty
Distribution Sensitivity
Coefficient
Uncertainty
Contribution
Index
RS 10000.05300 Ω 2.50·10-3 Ω normal 1.0 2.5·10-3 Ω 9.0 %
δRD 0.02000 Ω 5.77·10-3 Ω rectangular 1.0 5.8·10-3 Ω 48.1 %
δRTS 0.0 Ω 1.59·10-3 Ω rectangular 1.0 1.6·10-3 Ω 3.6 %
rC 1.000000000 408·10-9 triangular 10000 4.1·10-3 Ω 24.0 %
r 1.0000105000 70.7·10-9 normal 10000 710·10-6 Ω 0.7 %
δRTX 0.0 Ω 3.18·10-3 Ω rectangular -1.0 -3.2·10-3 Ω 14.5 %
RX 10000.17800 Ω 8.33·10-3 Ω

Results:

Quantity Value Expanded
Uncertainty
Coverage
factor
Coverage
RX 10000.178 Ω 0.017 Ω 2.00 95% (t-table 95.45%)

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