Volume 49, Number 1, 2008
107
Mathematical Models for Evaluating
the Measurement Uncertainty of
Analogue AC Ammeters and Voltmeters
Dorin ŢICUDEAN, Ioan G. TÂRNOVAN and Gheorghe TODORAN
Abstract: In this article, are presented shortly the analogue AC ammeter and voltmeter, the calibration
method of this measuring apparatus and their admissible errors expression.
It is presented also the mathematical model for evaluating and expressing the measurement
uncertainty and the importance of this parameter for the quality of metrological measurements.
Keywords: analogue ammeter and voltmeter, admissible error, measurement uncertainty
1. Introduction
An analogue instrument is defined as a
measuring instrument whose output or
display (readout) is a continuous function of
the measurand or the input.
2. Calibration of AC analogue ammeter
and voltmeter
The ammeter is used to measure the
electric current flowing through an electric
circuit, and the voltmeter is used to measure
the electric voltage across two terminals of an
electric circuit.
A calibration means a set of operations
performed in order to determine the
relationship between the values displayed by
a measuring instrument, a measuring system,
or the values represented by a material or a
reference material, and the corresponding
values realized using standards.
The calibration of an AC analogue
ammeter and voltmeter consists in:
- verifying
of
technical
conditions
regarding the construction and the
operation of the instrument;
- verifying of metrological conditions.
The verification of the technical
conditions regarding the construction consists
in:
- verifying the overall status of the
instruments
(inscriptions,
general
cleanness, etc.);
- verifying the state of graduated scale,
pointer, dial, terminals, and protective
glass of the case.
The verification of the technical
conditions regarding the operation refers to:
- the panel used for calibration
- the friction associated with the moving
coil
- the settling time of the pointer
- the mechanical resonance
- the residual indication
The calibration of the analogue
ammeter and voltmeter is performed under
reference and repeatable conditions.
The intrinsic error represents the error
of a measuring instrument under reference
condition. The errors of indication
determined under reference conditions must
not exceed the permissible errors for any
indication within the measuring range under
the following conditions:
© 2008 – Mediamira Science Publisher. All rights reserved.
ACTA ELECTROTEHNICA
108
-
the thermal balance between the
instrument and the environment is set at
the reference temperature
the mechanical zero positioning of the
pointer is done
the instrument has been connected into a
circuit through which 80%...100% of the
current is flowing, for a period of time
between half an hour to one hour,
depending on the accuracy class of the
instrument.
The standard used for the calibration of
the instrument must meet the following
conditions:
- it is stable
- it operates within the current and
frequency range limits that are necessary
for the calibration
- for each value of the measurand used to
calibration the analogue ammeter and
voltmeter, the standard must have a
permissible error that is at least five time
lower than that of the instrument to be
calibrated.
An
instrument
with
permissible errors that are up to three
times lower than those of the instrument
to be calibrated when the values of the
current indicated by the standard will be
corrected (using the correction of the
standard).
The calibration is done for all the marks
of the graduated scale, typically using the
lowest measuring range and only for 3...5
scale marks, uniformly distributed, for the
other measuring ranges.
The intrinsic error is computed using
the expression:
I −I
εR(I) = i a ×100 [%] [for ammeter] (1.1)
Ic
εR(V) =
Vi − Va
×100 [%] [for voltmeter]
Vc
(1.2)
where:
Ii (Vi) – value of the current (voltage)
indicated by the ammeter (voltmeter)
Ia (Va) – value indicated by the standard
corresponding to the value of the
current Ii (voltage Vi) when the current
(voltage) is increasing (decreasing)
Ic (Vc) – fiducial value, which can be, for
example, the upper limit of the
measuring range or another clearly
stated value.
The permissible errors of the measuring
instrument are expressed in terms of relative
errors (in percentages of a conventionally true
value):
Ε
εT(I) = + ×100 [%] [for ammeter] (2.1)
Ic
εT(V) = +
Ε
×100 [%] [for voltmeter] (2.2)
Vc
where:
εT – permissible relative error
Ε – module of the absolute permissible
Ic
error
(Vc) – fiducial
specifications)
value
(set
by
3. Data processing and the uncertainty
budget
For the calibration of an analogue AC
ammeter and voltmeter measurements are
made for each calibration point under
repeatability conditions using:
- same measurement method
- same operator
- same measuring instrument , operated
under the same conditions
- same location
- repeated measurements over a short time
interval
The mathematical model used to
determine the error is represented by the
equation:
Ammeter (Ix):
Ix = IE + δIE + δIET + δIED + δIR + δIZ + δIC
+ δIH + δIXT
(3.1)
Voltmeter (Vx):
Vx = VE + δVE + δVET + δVED + δVR + δVZ +
δVC + δVH + δVXT
(3.2)
Volume 49, Number 1, 2008
where:
►Ix (Vx) – the reading of the analogue
ammeter (voltmeter) to be calibrated
►IE (VE) – the mean value of the n values of
the current (voltage) indicated by the
standard ammeter (voltmeter)
n
IE =
∑I
i =1
i
[for ammeter] (4.1)
n
n
∑V
i =1
i
[for voltmeter] (4.2)
n
The uncertainty associated to IE[VE] is
u(IE) [u(VE)], and his expressions are:
VE =
n
u(IE) =
∑(I
i =1
− IE )
i
n
u(VE) =
[for ammeter] (5.1)
n × I E n × I max
+
100 = n × ( I E + Imax )
u(δIED) =
= 100
2 3
2 3
200 3
Dmax1
[for voltmeter] (5.2)
The uncertainty associated to δIE (δVE)
is u(δIE) [u(δVE)], and is given in the
calibration certificate.
►δIET (δVET) – the correction applied to IE
(VE), due to the deviation of the
ambient temperature from the reference
temperature (as the case may be)
δIET = (T – To)
[for ammeter] (6.1)
δVET = (T – To) [for voltmeter]
(6.2)
The uncertainty associated to δIET
[δVET] is u(δIET) [u(δVET)], and his
expressions are:
α1
2 3
α1
2 3
n × VE n × Vmax
+
100 = n × (VE + Vmax )
u(δVED)=
= 100
2 3
2 3
200 3
DmaxV
[for voltmeter] (8.2)
►δIE (δVE) – the correction applied to IE
(VE), as given in the calibration
certificate
u(δVET) =
The uncertainty associated to δIED
(δVED) is u(δIED) [u(δVED)], and is estimated
based on the history of the previous
calibrations of the standard. In case that there
are insufficient data or no data regarding the
history of the standard, the uncertainty is
estimated based on the specification from the
technical manual of the standard.
2
E
n ( n − 1)
u(δIET) =
►δIED (δVED) – the correction applied to IE
(VE), due to the drift of the standard,
which is determined based on the
history of the standard
[for ammeter] (8.1)
∑ (V − V )
i =1
where CET - temperature coefficient.
2
n ( n − 1)
i
109
=
=
CET × (T − T0 )
2 3
[for ammeter] (7.1)
CET × (T − T0 )
2 3
[for voltmeter] (7.2)
where:
Imax (Vmax) – the upper limit of the
measuring range - current (voltage)
Dmax - drift coefficient based on the value
of the Imax (Vmax) and IE (VE)
n - number of the value measuring range
►δIR (δVR) – the correction applied to Ix
(Vx), due to the zero indication of the
analogue ammeter (voltmeter) to be
calibrated
The uncertainty associated to δIR (δVR)
is u(δIR) [u(δVR)], and his expressions are:
u(δIR) = α I 2 3
[for ammeter] (9.1)
u(δVR) = αV 2 3
[for voltmeter] (9.2)
where αI, αV - can be estimated: 0,1 division
►δIZ (δVZ) – the correction applied to IX
(VX), due to zero indication of the
analogue ammeter (voltmeter) to be
calibrated
The uncertainty associated to δIZ
(δVZ) is u(δIZ) [u(δVZ)], and his expressions
are:
u(δIZ) = Z I 2 3
[for ammeter] (10.1)
110
u(δVZ) = ZV 2 3
ACTA ELECTROTEHNICA
[for voltmeter] (10.2)
where ZI (ZV) – can be estimated: 0,1
division
►δIC (δVC) – the correction applied to IX
(VX), due to the reading error of the
operator
The uncertainty associated to δIC (δVC)
is u(δIC) [u(δVC)], is given by:
u(δIC) = CI 2 3
[for ammeter] (11.1)
u(δVC) = CV 2 3
[for voltmeter] (11.2)
where CI (CV) – can be estimated: 0,1
division
►δIH (δVH) – the correction applied to IX
(VX), due to the hysterezis of the
instrument to be calibrated
H(I1) = I X 1 − I Egrow ; H(I2) = I X 2 − I Edim
[for ammeter] (12.1)
H(V1) = VX 1 − VEgrow ; H(V2) = VX 2 − VEdim
[for voltmeter] (12.2)
and
1
H(I) = [ H(I1) + H(I2) ]
2
[for ammeter] (13.1)
1
H(V) = [ H(V1) + H(V2) ]
2
[for voltmeter] (13.2)
The uncertainty associated to δIH (δVH)
is u(δIH) [u(δVH)], is given by:
u(δIH) = H I 2 3
[for ammeter] (14.1)
u(δVH) = HV 2 3 [for voltmeter] (14.2)
►δIXT (δVXT) – the correction applied to IX
(VX), due to the deviation of the
ambient temperature from the reference
temperature (as the case may be)
δIXT = (T – To)
[for ammeter] (15.1)
δVXT = (T – To)
[for voltmeter] (15.2)
The uncertainty associated to δIXT
(δVXT) is u(δIXT) [u(δVXT)], and his
expressions are:
u(δIXT) =
u(δVXT) =
where CXT
α2
2 3
α2
2 3
=
=
C XT × (T − T0 )
2 3
[for ammeter] (16.1)
C XT × (T − T0 )
2 3
[for voltmeter] (16.2)
- temperature coefficient
After all these components of the
uncertainty (both type A and type B) were
estimated, the standard uncertainty uc(X) is
computed, using the expression:
uc(X) =
n
∑u
i =1
2
i
(17)
(Y )
The expanded uncertainty is given by:
U = k × uc(E)
(18)
where k represents the coverage factor.
4. Calibration of an analogue AC
ammeter
The calibration of an analogue AC
ammeter of nominal value 5 A is carried out
by direct comparison to a reference standard
of the same nominal value, using a digital
multimeter
whose
performance
characteristics
have
previously
been
determined.
Measuring appliance (gauge): Analogue AC
Ammeter, PsLL model
Series/Number:
6068321
Characteristics (parameters): 0–5 A; AC;
METRA manufacturer
0,2
Accuracy class:
Standards: Digital multimeter Keithley Nr.
0595894, CE 000356-DKD-K-39701/
06.2007
Method of calibration: direct comparison
Measured values:
Initial
terminals
Ambient
temp.
[°C]
27.5
27.5
Air
humidity
[%]
63.3
62.9
Atm.
pressure
[mbar]
966.4
966.6
Volume 49, Number 1, 2008
111
4.1. Measuring values of current of the analogue ammeter to be calibrated [A]:
Vn
[A]
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
Run of values:
1
↑
↓
0,0735 0,0666
2,0298 2,0143
2,5425 2,5244
3,0717 3,0588
3,5485 3,5936
4,0558 4,0380
4,5288 4,5841
5,0412 5,0341
Run of values:
2
↑
↓
0,0676 0,0699
2,0102 2,0427
2,5537 2,5917
3,0346 3,0353
3,5755 3,5793
4,0217 4,0278
4,5468 4,5786
5,0041 5,0257
Run of values:
3
↑
↓
0,0674 0,0698
2,0542 2,0050
2,5988 2,5301
3,0256 3,0548
3,5637 3,5908
4,0138 4,0229
4,5574 4,5592
5,0968 5,0971
Run of values:
4
↑
↓
0,0675 0,0701
2,0540 2,0052
2,5985 2,5304
3,0255 3,0586
3,5638 3,5912
4,0142 4,0231
4,5573 4,5591
5,0967 5,0972
Run of values:
5
↑
↓
0,0679 0,0684
2,0543 2,0049
2,5986 2,5302
3,0254 3,0585
3,5642 3,5909
4,0139 4,0231
4,5575 4,5591
5,0966 5,0971
Run of values:
Scale
6
marks
↑
↓
0,0668 0,0674
0
2,0489 2,0146 40
2,5941 2,5483 50
3,0472 3,0564 60
3,5754 3,5952 70
4,0209 4,0239 80
4,5578 4,5589 90
5,0969 5,0984 100
where: Vn - current rating, [A].
4.2. Calculated values for evaluating of the uncertainty type A:
Mean value (average)
Vn
[A]
IUi
0,0685
2,0419
2,5810
3,0383
3,5652
4,0234
4,5509
5,0721
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
[A]
Standard deviation
(SIu)2
0,0000
0,0003
0,0007
0,0003
0,0001
0,0003
0,0001
0,0016
ICi
0,0687
2,0145
2,5425
3,0543
3,5902
4,0265
4,5665
5,0749
Standard
uncert.
uA (IX)
0,0021
0,0165
0,0256
0,0146
0,0081
0,0123
0,0116
0,0376
(SIc)2
0,0000
0,0002
0,0006
0,0001
0,0000
0,0000
0,0001
0,0012
Scale
marks
0
40
50
60
70
80
90
100
4.3. Valuation and observed values for evaluating of the uncertainty type B:
Vn
(A)
CET
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
T
0
[ C]
27,5
27,5
27,5
27,5
27,5
27,5
27,5
27,5
T0
[0C]
20,0
20,0
20,0
20,0
20,0
20,0
20,0
20,0
Valuation and observed quantity
Dmax
a
Z
div
α
0
[ C] [mA] [mA] [A]
[A]
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
7,5
0,01 0,001 0,0686 0,05
C
[A]
0,005
0,005
0,005
0,005
0,005
0,005
0,005
0,005
Hi
[A]
- 0,0002
0,0275
0,0385
- 0,0160
- 0,0250
- 0,0031
- 0,0156
- 0,0029
Scale
marks
0
40
50
60
70
80
90
100
4.4. Calculated values for evaluating of the uncertainty type B:
Vn
[A]
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
IE
0,00000
2,00001
2,50001
3,00001
3,50002
4,00002
4,50002
5,00002
δ IE
0,00000
0,00001
0,00001
0,00001
0,00002
0,00002
0,00002
0,00002
δ IET
0,00000
0,00000
0,00000
0,00000
0,00000
0,00000
0,00000
0,00000
Calculated quantity [A]
δ IED
δ IR
δ IZ
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
0,01
0,001 0,0686
δ IC
0,005
0,005
0,005
0,005
0,005
0,005
0,005
0,005
δ IDR
- 0,0001
0,0137
0,0193
- 0,0080
- 0,0125
- 0,0015
- 0,0078
- 0,0014
Scale
δ IXT marks
0,0000
0
0,0000
40
0,0000
50
0,0000
60
0,0000
70
0,0000
80
0,0000
90
0,0000 100
ACTA ELECTROTEHNICA
112
4.5. Calculated values of the uncertainty type B:
Vn
[A]
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
u(IE)
0,0150
0,0300
0,2500
0,3000
0,3500
0,4000
0,4500
0,5000
u(δIE)
0,0000
0,0050
0,0050
0,0050
0,0100
0,0100
0,0100
0,0100
u(δIET)
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
0,0000
Uncertainty type B
u(δIED) u(δIR) u(δIZ)
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
2,88675 0,2887 1,9796
u(δIC)
1,4434
1,4434
1,4434
1,4434
1,4434
1,4434
1,4434
1,4434
u(δIDR)
0,0000
0,0040
0,0056
- 0,0023
- 0,0036
- 0,0004
- 0,0022
- 0,0004
Scale
u(δIXT) marks
0,0000
0
0,0000
40
0,0000
50
0,0000
60
0,0000
70
0,0000
80
0,0000
90
0,0000 100
4.6. The budget of the uncertainty for the current values Ix:
Vn
0,0
[A]
Quantity Estimate Uncertainty
X(i)
x( i )
u( xi )
Estimate for type B
0,00000
0,0300
IE
0,00000
0,00000
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
-0,0001
-0,0001
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0000
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,0150
0,00000
0,00000
2,8868
0,2887
1,9796
1,4434
0,0000
0,0000
normal
0,0021
6,61336
43,73648
IX
Vn
2,0
[A]
0,084
Quantity Estimate Uncertainty
u( xi )
X(i)
x( i )
Estimate for type B
2,00001
0,0600
IE
0,00001
0,00001
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
0,0040
0,0137
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0003
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,0300
0,00500
0,0000
2,8868
0,2887
1,9796
1,4434
0,0040
0,0000
normal
0,0165
6,63735
44,05474
IX
2,089
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,01500
0,00000
0,00000
2,88675
0,28868
1,97959
1,44338
-0,00004
0,00000
1
uC
K
U
0,00205
6,61336
2
13,23
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,03000
0,00500
0,00000
2,88675
0,28868
1,97959
1,44338
0,00396
0,00000
1
uC
K
U
0,01648
6,63737
2
13,27
Volume 49, Number 1, 2008
Vn
2,5
[A]
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
2,50001
0,5000
IE
0,00001
0,00001
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
0,0193
0,0193
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0007
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,2500
0,00500
0,0000
2,8868
0,2887
1,9796
1,4434
0,0056
0,0000
normal
0,0256
6,85895
47,04587
IX
Vn
3,0
[A]
2,605
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
3,00001
0,6000
IE
0,00001
0,00001
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
- 0,0080
- 0,0080
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0002
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,3000
0,00500
0,0000
2,8868
0,2887
1,9796
1,4434
- 0,0023
0,0000
normal
0,0146
6,90108
47,62515
IX
Vn
3,5
[A]
3,077
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
3,50002
0,7000
IE
0,00002
0,00002
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
- 0,0125
- 0,0125
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0001
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,3500
0,01000
0,0000
2,8868
0,2887
1,9796
1,4434
- 0,0036
0,0000
normal
0,0081
6,95479
48,36911
IX
3,572
113
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,25000
0,00500
0,00000
2,88675
0,28868
1,97959
1,44338
0,00556
0,00000
1
uC
K
U
0,02562
6,85900
2
13,72
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,30000
0,00500
0,00000
2,88675
0,28868
1,97959
1,44338
- 0,00231
0,00000
1
uC
K
U
0,01462
6,90110
2
13,80
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,35000
0,01000
0,00000
2,88675
0,28868
1,97959
1,44338
- 0,00361
0,00000
1
uC
K
U
0,00807
6,95479
2
13,91
ACTA ELECTROTEHNICA
114
Vn
4,0
[A]
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
4,00002
0,8000
IE
0,00002
0,00002
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
0,0015
0,0015
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0002
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,4000
0,01000
0,0000
2,8868
0,2887
1,9796
1,4434
- 0,0004
0,0000
normal
0,0123
7,00795
49,11147
IX
Vn
4,5
[A]
4,083
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
4,50002
0,9000
IE
0,00002
0,00002
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
- 0,0078
- 0,0022
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0001
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,4500
0,01000
0,0000
2,8868
0,2887
1,9796
1,4434
- 0,0022
0,0000
normal
0,0116
7,05615
49,78932
IX
Vn
5,0
[A]
4,577
Quantity Estimate Uncertainty
u(xi)
X(i)
x( i )
Estimate for type B
5,00002
1,0000
IE
0,00002
0,00002
δIE
0,00000
0,0000
δIET
0,0100
0,0100
δIED
0,0010
0,0010
δIR
0,0686
0,0686
δIZ
0,0050
0,0050
δIC
- 0,0014
- 0,0014
δIDR
0,0000
0,0000
δIXT
Estimate for type A
0,0014
δIx
Probability
distribution
Standard
uncertainty u(xi)
normal
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
rectangular
0,5000
0,01000
0,0000
2,8868
0,2887
1,9796
1,4434
- 0,0004
0,0000
normal
0,0376
7,10798
50,52473
IX
5,085
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,40000
0,01000
0,00000
2,88675
0,28868
1,97959
1,44338
- 0,00045
0,00000
1
uC
K
U
0,01226
7,00796
2
13,80
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,45000
0,01000
0,00000
2,88675
0,28868
1,97959
1,44338
- 0,00225
0,00000
1
uC
K
U
0,01165
7,05615
2
14,11
Sensitivity
Uncertainty
coefficient Ci contribution ui (y)
1
1
1
1
1
1
1
1
1
0,50000
0,01000
0,00000
2,88675
0,28868
1,97959
1,44338
- 0,00042
0,00000
1
uC
K
U
0,03759
7,10808
2
14,22
115
Volume 49, Number 1, 2008
4.7. Summarization of the final values for the current Ix :
Current rating (Vn)
[A]
0,0
2,0
2,5
3,0
3,5
4,0
4,5
5,0
Uncertainty
(U)
[mA]
13,23
13,27
13,72
13,80
13,91
14,02
14,11
14,22
Conventionally true
values (VC)
[A]
0,084
2,089
2,605
3,077
3,572
4,083
4,577
5,085
5. Conclusions
REFERENCES
In
order
to
have
consistent
measurements, a metrological traceability
chain has to be established for the
measurement results, that is a chain of
alternating
measuring
systems
with
associated measurement procedures and
standards. A metrological traceability chain is
defined through a calibration hierarchy from
the measurement result to the stated
metrological reference, which means that
establishing a metrological traceability chain
involves a series of comparisons with
standards of higher and higher accuracy.
Within each of these comparisons, the
uncertainty associated with the corresponding
reference standards has to be significantly
lower than the uncertainty associated with the
analogue ammeter and voltmeter to be
calibrated. If follows that no traceability
chain can be established without estimating
the corresponding uncertainty for each
performed measurement. In order to have a
coherent metrological system, it is also
important to use procedures (ISO Guide –
GUM) to evaluate and calculate the
measurement uncertainty.
1. EN 60051-1, Direct acting indicating analogue
electrical measuring instruments and their
accessories. Part 1: Definitions and general
requirements common to all parts.
2. EN 60051-2, Direct acting indicating analogue
electrical measuring instruments and their
accessories. Part 2: Special requirements for
ammeters and voltmeters.
3. EA - 4/02, Expression of the uncertainty of
Measurement in Calibration.
4. SR ENV 13005/2003, Guide to the expression of
uncertainty in measurement.
5. VIM 13251/1996, International vocabulary of
basic and general terms in metrology.
Eng. Dorin ŢICUDEAN
Regional Direction of Legal Metrology
Metrological Laboratory of Turda
Prof. Dr.ing. Ioan G. TÂRNOVAN
Prof. Dr.ing. Gheorghe TODORAN
Department of Electrical Measurements
Faculty of Electrical Engineering
Technical University of Cluj-Napoca, Romania