9. Measurement of resitances AE1B38EMB 9. MEASUREMENT OF RESISTANCES Task of the measurement 1. Measurement of low resistances using voltmeter-ammeter method (Ohm’s method). Connect the measuring circuit according to Fig.1. Use suitable measurement procedure to exclude influence of thermo-electrical voltages. Calculate measured resistance R X from the measured voltage and current and estimate the expanded uncertainty of measured resistance (coverage factor k = 2). 2. Measurement of low resistances using series comparison method. Connect the measuring circuit according to Fig. 2. Measure the voltage drop across the resistance standard R S and across the measured resistor R X . Use suitable measurement procedure to exclude influence of thermo-electrical voltages. Calculate measured resistance R X and estimate the expanded uncertainty of measured resistance (coverage factor k = 2). 3. Measurement of medium resistances by an R → U converter. Connect a resistance-to-voltage converter according to Fig.3. (U r = 10 V, R N1 = 10 kΩ). Derive transfer function of the converter and verify the converter function (use a resistance box as the measured resistor). Find the approximate highest value of a resistance measureable by this converter. Schematic diagras RP A DC POWER SUPPLY A RX DV RP DC POWER SUPPLY Fig. 1 Measurement of low resistance by V-A method R N1 Ur RX RS DV DV Fig. 2 Measurement of low resistance by series comparison method RX + U2 Fig. 3 R → U convereter 1 of 3 9. Measurement of resitances AE1B38EMB List of the equipment used DC power supply A RP RX RS DV C - type ...; - moving coil ammeter, accuracy class ..., range: …; - current limiting resistor ... - measured resistor; - resistance standard ... , tolerance ... %; - digital voltmeter, type ..., error ... + ..., measurement range ...; - commutator Theoretical background V-A method Resistance of the measured resistor is found from the Ohm’s law, i.e. as R X = U/I. The fourterminal connection of the measured resistor eliminates influence of the terminal resistances and resistances of connecting wires. Averaging results of measurement by both polarities of the power supply (i.e. R X = (R X1 + R X2 )/2) allows to eliminate influence of thermo-electrical voltages. Since the V – A method is indirect measurement of resistance, standard uncertainty of measurement of resistance R X for each polarity of current can be estimated using the relation 2 u RX1, 2 where 2 2 U 1, 2 1 RX RX u I 1, 2 uU 1, 2 2 u I 1, 2 uU 1, 2 I I U 1, 2 I 1, 2 2 (1) u U1,2 are standard uncertainties of measurement of voltages U 1,2 , u I1,2 are standard uncertainties of measurement of currents I 1,2 , subscripts 1, 2 correspond to the two polarities of current. Since both absolute values of currents and of voltages are approximately the same before and after power supply commutation, it is possible to suppose that u RX1 u RX2 , and it is therefore sufficient to find the uncertainty of the measured resistance for one power supply polarity only. The resulting standard uncertainty of measurement can be found as 2 u RX 2 u u RX1 RX2 2 2 2 u R2 X1 u RX1 4 2 (2) and the expanded uncertainty of measurement is found by multiplying this value by coverage factor k. Note: Taking into account the fact that the measured resistance is low and that the input resistance of the digital voltmeter is 10 M or higher, the methodical error caused by input resistance of the voltmeter can be disregarded in this measurement. 2 of 3 9. Measurement of resitances AE1B38EMB Series comparison method When measuring the low resistance using series comparison method measuring circuit is connected according to Fig. 2. Voltage U RS across resistance standard R S and voltage U RX across the measured resistor R X are measured using the same digital voltmeter. The measured resistor R X can then be found as RX U RX RS U RS (3) As in the previous case it is necessary to eliminate influence of the thermo-electrical voltages on the result of measurement. This can again be achieved by measuring voltages U RS and U RX for both polarities of power supply (measure first voltage U RS for both polarities, and afterwards voltage U RX for both polarities). The resulting value of the measured resistance is here also found from measurements for both polarities using averaging, i.e. R X = (R X1 + R X2 )/2. Standard uncertainty of measurement of the resistance R X can be found for each polarity of current as 2 u RX1,2 2 2 RX R RX X u RS uURX1,2 uURS1,2 U RS U RX RS 2 2 U RX1,2 RS U RX1,2 RS u u RS uURX1,2 U RS1,2 U U U2 R R S1,2 S1,2 RS1,2 2 (4) where u URS1,2 and u URX1,2 are standard uncertainties of measurement of voltages U RS1,2 and U RX1,2 , which can be found from the error of the used digital voltmeter; u RS RS / 3 R S RS 100 3 RS is standard uncertainty of resistance R S ; is tolerance of this standard resistor in %. As in the previous case it can be supposed that there is u RX1 u RX2 and it is therefore sufficient to estimate only one of these uncertainties. The resulting standard uncertainty of the averaged value can be found again from the relation (2) and the expanded uncertainty can be found by multiplication of this value by coverage factor k. Note: As in the case of the V-A method, influence of terminal resistances and resistances of connecting wires are eliminated using four-terminal (four-wire) connection of resistors (in this case of both the measured resistor and the resistance standard). Information about resistance to voltage converter – see lecture. 3 of 3