3.1.1. Measure the output voltage U
2
of the resistive divider consisting of ten resistors of the same value for all division ratios d by means of: a) a DVM (digital voltmeter), b) a PMMC voltmeter (on 12 V measurement range).
Plot both the U
2
/ U
1
vs d curves in a common graph and explain the differences. The input voltage of the divider ( U
1
) should be 10 V.
3.1.2. Calculate the divider output resistance R
D from the measured values for the given division ratio d . Suppose the DVM input resistance be infinite.
3.1.3. Calculate the (coverage factor k r
= 2) of the evaluated divider output resistance supposing that the input resistance tolerancy of the PMMC voltmeter is less than 0.2 %.
U
1
= 10 V
0
R
1
R 2
R
3
R
4
R
-
5
+
R
6
R
7
R
8 R 9 R 10
U
2
DV
V
Fig. 3.1 Circuit connection
V - PMMC voltmeter, A.C.: ..., voltage range: ...
DV - digital voltmeter, model: ... accuracy ...
U1 - DC power supply, model: ...
RESISTIVE DIVIDER
If the output voltage of a resistive voltage divider is measured by a voltmeter, the input resistance of which is comparable to the resistance of the divider, the output voltage U
2 is significantly lower than the corresponding output voltage of the unloaded divider U
0
(Fig.
3.2a).
37

The real divider powered from DC voltage source U
1 and loaded by voltmeter with input resistance R
V
according to Fig. 3.2b can be substituted (using Thevenin theorem) by serial connection of DC voltage source U
0 and resistor R
D
according to the Fig. 3.2c.
Voltage U
0
is equal to the output voltage of unloaded divider and the resistance R
D
is given and parallel combination of divider resistors R
1
and R
2
. Output divider voltage U
2
that is loaded by voltmeter input resistance R
V
is given as
U
2
= U
0
R
V
R
V
+ R
D
(3.1) where R
D
is output divider resistance.
R
D
= R
2
II R
1
R
1
R
1
U
1
U
1
U
0
R
V
V U
2
R
2
U
0
R
2
R
V
V U
2
a) b) c)
Obr. 3.2 Resistive voltage divider: a) unloaded, b) loaded by R
V the voltage divider loaded R
V
, c) substitutive schematics of
(according to Thevenin theorem)
Voltmeter connection causes a methodical error of the following value
∆ U met
= U
2
− U
0
= U
0
R
V
R
V
+ R
D
− U
0
= U
0


R
V
R
V
+ R
D
− 1


= U
0
R
−
V
R
D
+ R
D
(3.2)
If the voltmeter input resistance is much higher than the output resistance of the measured voltage source (in our case output resistance R
D
of the voltage divider), the methodical error is neglectible. Therefore, using digital voltmeter (with input resistance R
DV
= 10 9 measure voltage U
2DV
that reaches voltage value for unloaded divider, U
0
≅ U
2DV
.
Ω) we
U
U
0
More methods can be used to evaluate output resistance of the divider. One of them is based on schematics on Fig. 3.2c and relation (3.1), where R
D
can be evaluated. It gives
U
2
R
D
=
R
V
(
U
2 DV
U
2
− U
2
)
= R
V


U
U
2 DV
2
− 1


(3.3)
0
0 I
2
Fig. 3.3 Loading characteristics of the divider
I
Another possibility is to start with loading characteristics of the
I resistor R
D
. Output characteristics is given as absolute value of slope of line:
R
D
=
U
2 DV
0 −
− U
2
I
2
=
U
2 DV
U
2
− U
2
R
V
= R
V


U
U
2 DV
2
− 1


(3.4)
38

Output resistance measurement uncertainty evaluation
If the fluctuations of measured values for each voltmeter are significantly lower than would correspond to its accuracy declared by its vendor (for PMMC voltmeter it is accuracy class, for digital multimeter it is the sum of errors of range and of value), the A-type uncertainty is negligible and it is sufficient to measure voltages U
2DV
and U
2
(that are needed for resistance
R
D
calculations) only once. In opposite case, each measurement must be repeated several times, average values of U
2 DV
and U
2
must be used instead and corresponding A-type uncertainties must be calculated.
One can expect that in our case the A-type uncertainties of U
2DV
and U
2
measurements are negligible the only sources of B-type uncertainty are inaccuracies of the used voltmeters and tolerance of the value of the input resistance of PMMC voltmeter.
The value of resistance R
D
can be calculated using (3.3), in general, we can say that
R
D
= f ( U
0
, U
2
, R
V
); uncertainty of its value is u
R D
=


∂ R
D
∂ U
2DV u
U 2 CV


2
+


∂ R
D
∂ U
2 u
U 2


2
+


∂ R
D
∂ R
V u
R V


2
( Ω ) (3.5) where u
R D
is resulting standard uncertainty of the R
D
( Ω ), u
U 2DV standard uncertainty of measurement of U
2CV
(V), u
U 2 u
R V standard uncertainty of measurement of U
2
(V), standard uncertainty of value of R
V
( Ω ).
After evaluating the partial derivations
∂ R
D
∂ U
2DV
=
R
V
U
2
;
∂ R
D
∂ U
2
= − R
V
U
U
2 D V
2
2
;
∂ R
D
∂ R
V
=
U
2 D V
U
2
− U
2 (3.6) we obtain the resulting standard uncertainty of the value of R
D u
R D
=


R
V
U
2 u
U 2DV


2
+


R
V
U
U
2DV
2
2 u
U 2


2
+


U
2DV
U
2
− U
2 u
R V


2
( Ω ) (3.7)
Each standard uncertainty is evaluated by the following way: u
U 2 D V
=
1
⋅ U
2DV
100
+
2
3
⋅ M
DV (V) (3.8) u
U 2
AC
100
M
3
(V) (3.9) u
R V
=
δ
R V
100
⋅ R
V
3
( Ω ) (3.10) where δ
1
is error in percent of digital voltmeter reading.
39

δ
2
error in percent of digital voltmeter full-scale range,
M
CV selected range of the digital voltmeter (V),
M measuring range of the PMMC voltmeter (V),
R V resistor R
V
tolerance (%).
Enhanced uncertainty U
R D
of resistor R
D
evaluation is obtained by multiplication of standard uncertainty by the coverage factor k r
(usually k r
= 2), it means U
R D
= k r u
RD
.
40