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
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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.
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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.
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