JSSSJournal of Sensors and Sensor SystemsJSSSJ. Sens. Sens. Syst.2194-878XCopernicus PublicationsGöttingen, Germany10.5194/jsss-7-339-2018Test equipment and its effect on the calibration of instrument transformersCalibration of instrument transformersMohnsEnricoenrico.mohns@ptb.dehttps://orcid.org/0000-0001-5098-9396RätherPeterPhysikalisch-Technische Bundesanstalt (PTB), Bundesallee
100, 38116 Braunschweig, GermanyEnrico Mohns (enrico.mohns@ptb.de)7May2018713393475December20175April201810April2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://jsss.copernicus.org/articles/7/339/2018/jsss-7-339-2018.htmlThe full text article is available as a PDF file from https://jsss.copernicus.org/articles/7/339/2018/jsss-7-339-2018.pdf
State-approved test centres in Germany supplying accuracy
tests for instrument transformers must, in the future, provide measurement
uncertainty budgets for their quality management systems. In this work,
the cause of ratio error and phase displacement of instrument transformers
are therefore discussed. The traceability to the national standards of PTB,
the attainable uncertainty, and the permitted error limits of test equipment
for testing instrument transformers are presented. Finally, an example of an
uncertainty budget for a current transformer of the class 0,2 S is
given.
Introduction
Instrument transformers that are used for the billing of electric energy are
connected to the energy meters if the currents or the voltages to be
measured are too high for the meters. This is generally the case when the
energy transmitted has to be measured in the medium-voltage or the
high-voltage power grids, since meters which can be directly connected are
designed for the nominal voltages of the low-voltage grid. However, in
applications in which the maximum expected currents are higher than 100 A,
transformer-connected meters are used with current transformers
connected before them. The task of the instrument transformers is thus to
reproduce the high primary voltage (or current) to be measured as a low
secondary voltage (or current) which is easy for the meters to detect.
Ideally, the reproduced secondary quantities should be exactly proportional
and in phase with the primary quantities. In practice, however, deviations
always occur. These deviations, which are caused by the instrument
transformers, lead to deviations of the electric energy measured by the
transformer-connected meter. In the different standardized classes, the
accuracy achieved under all conditions of use is reflected as the measure
for the metrological quality of an instrument transformer.
In the past, state-approved test centres therefore proved compliance with
the respective accuracy classes of the transformers by means of verification
before the transformers could be installed in the grids. Before this can
happen, the instrument transformer has to obtain type approval. The test
equipment used at the test centres are calibrated at PTB at regular
intervals and verified with regard to their compliance with minimum
requirements on their accuracy. This procedure is called a test
(“Prüfung”) as the result of this test is a test certificate which
contains both the results of the calibration and a statement that the
test equipment is verified according to the requirements, which are laid
down in the PTB Testing Instructions (PTB, 1979). In the state-approved test
centres, this procedure, based on compliance with limit values and on the
thus proven traceability of the test equipment to the national standards,
ensured – even without exhaustive mathematical evidence of the accuracy
achieved by the transformer measuring system set up with the test equipment
– that the deviations of the instrument transformers to be verified are in
sufficient agreement with the International System of Units (Sommer et al.,
2001).
With the new Measures and Verification Act (MEG, 2015), however, the
metrological prerequisites are also changing. In the test centres certifying
the conformity of an instrument transformer, not only must the test
equipment be traced to national standards via a test. Within the scope of
the quality management system, the test centre must also yield mathematical
evidence of the uncertainty achieved during the process of an instrument
transformer conformity assessment. This piece of evidence is given in the
form of an uncertainty budget pursuant to the rules (GUM, 2008). Based on
the measured deviations, their associated uncertainties and the rules for a
verification (JCGM, 2012), the test centre decide whether or not an
instrument transformer is within its accuracy class.
In this work, the ratio error and phase displacement of instrument
transformers are presented and discussed. Moreover, the standard measuring
systems used at PTB for the calibration of the test equipment of test
centres and of the measurement uncertainties achieved is illustrated.
Finally, an uncertainty budget for the assessment of an instrument
transformer in a state-approved test centre or a notified body is calculated
as an example (Mohns and Raether, 2017).
Conventional instrument transformers and definition of the deviation of
transformers
Instrument transformers are used to scale a high primary quantity Ap (primary voltage Up in the
case of voltage transformers, primary current Ip
in the case of current transformers) as a small secondary quantity
As (Us, Is) that is easily measurable for meters. The ratio
Ap/As is the
transformer ratio of the instrument transformer. Ideally, the ratio of the
primary to the secondary quantity is exactly proportional and in phase. The
actual ratio then corresponds to the rated transformer ratio
Kn=Apn/Asn (Ramm et al., 1998). The instrument transformers
used in Germany, but also in large parts of Europe, are mainly conventional
transformers that work inductively, i.e. according to the principle of a
classical transformer. The secondary rated values are usually normalized to
100 V for voltages and to 1 or 5 A for currents.
In practice, however, deviations always occur in the magnitude and the
phase. These are called ratio error ε and phase displacement
δ. Figure 1a shows the phasor diagram of the actual secondary
quantity, As, of the ideally secondary quantity
Ap/Kn and of the resulting phase
displacement δ between these quantities. The complex deviation
ΔAs is also shown. If the phasors are
normalized to Ap/Kn (Fig. 1b), it then
geometrically yields the ratio error ε as the difference
between the vector length and the value 1. It is as follows:
ε=AsApKn-1=AsApKn-1δ=argAs-argAp.
The complex measurement error E is also shown with its
real part α and its imaginary part β; it is not really
relevant for the definition of these transformer measurement errors. This
complex measurement error, however, allows measurement procedures according
to the difference method to be derived graphically. It becomes
immediately understandable that in the case of small phase displacements (as
are required for instrument transformers), the quantities ε and
α, as well as δ and β, practically agree.
(a) Phasor diagram of an instrument transformer with converted
primary quantity Ap/Kn, secondary quantity
As, and phase displacement δ. The complex
difference ΔAs is shown in green. (b) Phasor
diagram following from the figure on the left by normalization with
Ap/Kn, with the ratio error ε and
the complex measurement error E with the associated
real part α and imaginary part β (green).
The sources of the measurement errors of instrument transformers are easily
derived from the equivalent circuit of a transformer as shown in Fig. 2a
and from the impedances of the T-equivalent circuit compiled from this
and converted into those of the secondary side as shown in Fig. 2b.
The values are converted by means of the squared turns ratio
u2= (Np/Ns)2. The corresponding longitudinal and
transversal impedances are then given as the following:
Ym′=1RFe′+j⋅ω⋅Cw′-1ω2⋅Mh′,Zp′=Rp′+j⋅ω⋅Lσ,p′,Zs=Rs+j⋅ω⋅Lσ,s,
where it may be assumed that only the iron loss resistance RFe and the
main inductance Mh depend on the signal amplitude as a result of the
properties of the magnetic core material used. An equivalent winding
capacitance Cw is also taken into account; it is relevant mainly for
voltage transformers with a high primary number of windings.
(a) Equivalent circuit of an instrument transformer with discrete
components. (b) simplified T-equivalent circuit with primary and
secondary longitudinal impedance, transversal impedance, and burden
Zb.
For a voltage transformer, the complex measurement error E0 yielded as the no-load error for Zb→∞
is thus
E0≈εu,0+jδu,0=UsUp′-1=Um′Up′-1=11+Ym′Zp′-1,E0≈-Ym′Zp′=-Rp′RFe′-Lσ,p′Mh′+ω2Lσ,p′Cw′+jω-Lσ,p′RFe′-Rp′Cw′+Rp′ω2Mh′,
and in case of an additional load in the form of a burden, the additional
error ΔEb is obtained by
ΔEb=-Zp′+ZsZb,Δεb=Re{ΔEb},Δδb=Im{ΔEb}.
The total ratio error and phase displacement is
εu=εu,0+Δεb,δu=δu,0+Δδb.
From this, it is possible to derive the following principle-induced
properties for voltage transformers at power frequency and without turns
correction:
The open-circuit error is negative (-Rp′/RFe′).
The phase displacement is usually positive (+Rp′/ωMh′).
If, however, capacitive current dominates via Cw compared to inductive
current via Mh (high winding number on the primary side), then all
relations are inversed.
The introduction of a burden leads to an even more negative error, which is
practically independent of the signal amplitude of the voltage
transformer.
For a current transformer, the complex measurement error E results in
E≈εi+jδi=IsIp′-1=Zm′Zm′+Zs+Zb-1=11+Ym′Zs+Zb-1,E≈-Ym′Zs+Zb=E0+ΔEb,
where
E0≈εi,0+jδi,0=-Ym′Zs=-RsRFe′-Lσ,sMh′+ω2Lσ,sCw′+jω-RsCw′-Lσ,sRFe′+Rsω2⋅Mh′,ΔEb=-Ym′Zb.
The terms with Cw are, however, practically negligible. The consequence
is that current transformers usually exhibit a wide band behaviour and still
transmit frequency fractions in the range of a few kHz with sufficient
accuracy. For current transformers, the no-load operation properties at
supply frequency are nearly identical with those of voltage transformers.
The application of a burden also leads to negative errors; in this case,
however, the error is no longer independent of the signal amplitude.
Especially at small measuring points (1 to 5 % In), the
permeability of the iron core is considerably smaller. This leads to a
reduced inductance Mh which, in turns, results in more significant
changes ΔEb than at full signal amplitude
(20 to 200 % In).
Testing equipment and traceability at PTB
Instrument transformers that are admissible for verification belong to the
accuracy classes 0.5–0.2–0.1 (both current and voltage transformers),
or 0.5 or 0,2 S (current transformers only). Every single instrument
transformer that is admissible for verification is tested by a
state-approved test centre for measuring instruments for electricity or by a
notified body before being mounted into the network in order to find out
whether it complies with all the requirements laid down in the type-approval
certificate (conformity assessment). The accuracy test is carried out to
answer the question as to whether the instrument transformer complies with
the maximum permissible error on verification (PTB, 1979).
The tests are carried out by comparison with instrument transformers of
“higher quality” – so-called standard transformers. Their ratio error
amounts to less than 0.02 %, which makes them 10 times as accurate as,
for instance, a class 0,2 S transformer. The comparing instrument in a
transformer measuring system is a transformer measuring bridge. The
secondary quantities of the instrument transformer to be tested and of the
standard transformer are connected to this bridge. From these quantities,
the bridge determines the ratio error and the phase displacement of the
instrument transformer compared to those of the standard transformer. The
instrument transformer is thereby subjected to a test burden which
corresponds to the measurement burden and, usually, to one-quarter of the
burden of the transformer. For the test equipment, the standard
transformer, the transformer measuring bridge, and the test burden, PTB
supplies high-precision calibration systems which will be briefly described
together with their main properties.
Test equipment for standard transformers
The transformer measuring bridge used at PTB for current and voltage
transformers works according to the difference method, which consists of
measuring the difference between the secondaries of the PTB's standard and
the transformer under test. Figure 3 illustrates this principle with the
example of the measurement of current transformers (Kahmann et al., 2017; Ramm and Moser,
1995; Xu, 2015); the same applies analogously to the measurement of
voltage transformers (Ramm and Moser, 1996; Badura, 2015).
Principle of a measuring bridge for current transformers according
to the difference method.
PTB's standard current transformer TN and current transformer TX
under test are connected in series on the primary side and supplied with the
desired primary current Ip by the current source.
Thereby, the same transformer ratio Kn is selected for TN as for
TX in order to attain secondary currents IX
and IN that are roughly identical. Analogously to
Fig. 1, the transformer measuring bridge determines the complex current
IN and the differential current ID=IX-IN by means of suitable, highly sensitive current sensors
and a two-channel sampling system. The sampled values are then resolved into
spectral values according to their magnitude and phase angle (i.e. complex
resolving) in the frequency range with a discrete Fourier transform. From
the values at the fundamental frequency, it is then possible to calculate
the complex transformer difference ED=ID/IN between TX and TN, and from this difference
the ratio error εD and the phase displacement δD can be calculated. Here, it is important to keep in my mind that
these errors still contain the low errors of PTB's standard transformer. The
errors of TX with the correction for TN are:
εX=εD+εN,δX=δD+δN.
The current transformer under test is thereby subjected to a burden
ZX in such a way that this burden corresponds to the load due to the
measurement leads and the transformer measuring bridge at the test centre.
The standards used by PTB for calibration are current comparators whose
measurement errors are in the range between less than 10-6 and
10-5 (Kusters and Moore, 1964; Mohns et al., 2017). The range of the primary
rated currents of these references is between 0.1 and 60 000 A. The
measurement uncertainty that is typically assigned when a standard current
transformer is calibrated for the verification of instrument transformers is
approximately ±0.003 % or 0.003 crad (centi-radian, which corresponds to 0.1′, angular
minute), with a confidence interval of approx. 95 % corresponding to k= 2.
Ratio error of a class 0,2 S current transformer (Mohns, 2018).
Description of the symbols used.
QuantityDescriptionUnitεXResult: ratio error of the transformer (X)%εNRatio error of the standard (N)%εDIndicated difference of the bridge for the ratio error X-N%dWMUncertainty in the transformer measuring bridge%dMPInfluence of the test point%sBSensitivity of the transformer (X) due to a burden change% VA-1ΔBError of the burden: measured burden – rated burdenVA
Uncertainty budget of the ratio error for a class 0,2 S instrument
transformer at a signal amplitude In of 100 % and a rated burden Sn= 10 VA.
The relations stated for PTB's calibration facility for current transformers
can, in principle, be applied analogously to the calibration facility for
voltage transformers. It is also based on the difference method. Two voltage
transformers connected in parallel on the primary side are connected to the
primary voltage Up, and, on the secondary side, to the voltage
transformer measuring bridge. The secondary voltage UN of the transformer used as a reference TN, and the
difference voltage UD=UX-UN are detected by means
of suitable voltage sensors (complex detection). Similar to the current
transformer measuring system, the transformer measurement of TX is
calculated according to Eq. (8) from the complex measurement error ED=UD/UN.
A set of standard voltage transformers at PTB provide a
range of the primary rated voltages from 100 V up to 400/√3kV. The
measurement uncertainties assigned for calibrating standard voltage
transformers are also 0.003 % or 0.003 crad (k= 2).
Test equipment for standard burdens
During the accuracy test, standard burdens simulate the load that will later
be induced in the network by the instrument transformer due to the energy
meters and cabling connected to the secondary side. Characteristics of a
burden are the rated apparent power Sn and the rated power factor
cos βn. The deviation between the actual apparent power and
Sn may not exceed 3 %; for the phase angle β, the limit
amounts to ±3 crad on the basis of βn (PTB, 1979).
The standard values of the burden steps for the verification of current
transformers are based on the rated power values for transformers that are
admissible for verification and are usually in the range between 1 and
30 VA. The burden power factor cos β describes the ratio of active to
the apparent power. Below 5 VA, the load is purely ohmic; at 5 VA and above,
the load is ohmic-inductive, i.e. the voltage phasor precedes the current
phasor (by 36.9∘ at cos β= 0.8). The standard values of
the burden steps for the verification of voltage transformers cover a vast
range that extends to up to 300 VA.
PTB's measuring facility for standard burdens (Braun et al., 1993; Ni,
2015) is based on the measurement of the complex impedance via the ratio of
the voltage U to the current I.
Suitable sensors are connected in such a way that no loading of the burden
is caused by the sensors. They cover the range from 1 mA to 10 A (current
sensors) and from a few millivolts to 300 V (voltage sensors). An analogue power
amplifier supplies a test power of up to 700 VA. The achievable measurement
uncertainty with this calibrator is from 0.02 to 0.05 %. The
measurement uncertainties attributed to standard burdens are, however,
higher – from 0.1 to 0.5 % (k= 2). This is due to the properties
of the burden under test determined during the calibration, such as the
stability (influences of heating up) or the repeatability.
Test equipment for transformer measuring bridges
The transformer measuring bridge for the verification (conformity
assessment) of an instrument transformer compares either the secondary
currents IX and IN or the secondary voltages UX and UN of
the transformer under test and of the standard transformer according to the
magnitude and the phase, pursuant to the definitions Eq. (1). The differences
between X and N which should hereby be shown by the measuring bridge are in
the range of up to ±1.5 % and ±2.7 crad (class 0.5 S at
1 % In). Hereby, the admissible deviations according to PTB (1979)
are based on the maximum permissible error limits of a class 0,2 S current
transformer. They amount to one-tenth of the error limits required there,
i.e. ±0.02 % (±0.03 crad) in the case of measuring bridges
for voltage transformers, and at test points of 20 to 200 % in the
case of measuring bridges for current transformers. In addition, increased
limits of ±0.035 % (±0.045 crad) and ±0.075 %
(±0.09 crad), respectively, are permitted for the test points 5
and 1 %.
The calibrator for transformer measuring bridges (Ramm et al., 1998) works
according to the principle of feeding errors divided into a real and an
imaginary part (see Fig. 1) by means of high-precision, electronically
error-compensated current transformers or voltage dividers. The measurement
uncertainties achieved with this calibration system are another 10 times
lower than the tolerance values required for transformer measuring bridges
under test. The working range covers currents of up to 10 A for IX and
IN, and secondary voltages of more than 200 V for UX and UN.
Uncertainty budget for the calibration of an instrument
transformerSimplified methodology of measurement uncertainty calculation
The basis of the measurement uncertainty analysis consists of statistically
obtained findings on error propagation. Contrary to a worst case estimation,
in which the error propagation of the maximum error is calculated somewhat
conservatively, measurement error calculation is based on the Gaussian error
propagation. The basic approach consists of identifying the physical model
of a system and transforming it into a mathematically utilizable
calculation model in the form of an analytical equation. Very generally, the
model equation y obtained is a function of several variables x1,
x2, x3 …with
y=fx1,x2,x3,….
The uncertainty u(xi) of a variable xi obviously influences the
result y. This result exhibits a deviation by the difference ∂y(u(xi)). This deviation, or rather this indeterminacy ∂y(u(xi)), is
∂y(u(xi))=∂fx1,x2,x3,…∂xi⋅u(xi),
and derives from linearization around a working point by means of a Taylor
series which is interrupted after the first order. The sensitivity
coefficient ci corresponds to the differential quotient ∂f/∂xi. Its sole purpose is improved legibility. The total
uncertainty u(y) is thus the geometrical addition
This is applicable
in cases where no correlation between variables xi exist.
of all
individual uncertainties ∂y(u(xi)):
uB(y)=∑∂y(u(xi))2=(c1⋅u(x1))2+(c2⋅u(x2))2+(c3⋅u(x3))2+….
This measurement uncertainty is a so-called type B uncertainty. In addition,
the type A contribution to the total uncertainty will later be added
geometrically to the type B uncertainty. Type A uncertainty essentially
describes the statistical variability in the mean value yave of the
measurand y to be determined which is yielded over the course of the
measurement. The estimated quantity for type A measurement uncertainty is
obtained by means of the standard deviation s(y), the number N of measured
values, and Student's t factor t(DOF, P) as
uA(yave)=t(DOF,P)⋅s(y)N=t(DOF,P)N⋅1N-1⋅∑i=1Nyi-yave2.
Student's t factor is listed in the table for the different confidence
intervals P and the degrees of freedom DOF =N-1 (GUM, 2008).
For a sufficient number of measured values (N > 10), this factor
is, however, approx. 2 for a confidence interval of 95 %, and approx. 1
for the confidence interval of approx. 68 % which is relevant for
calculating the standard uncertainty.
Description of the symbols used.
QuantityDescriptionUnitδXResult: phase displacement of the′transformer (X)δNphase displacement of the standard (N)′δDIndicated difference of the bridge′for the phasedisplacement X-NdWMUncertainty in the transformer′measuring bridgedMPInfluence of the test point′sBSensitivity of the transformer (X)′ VA-1due to burden changeΔBError of the burden: measured burden –VArated burden
Phase displacement of a class 0,2 S current transformer (Mohns, 2018).
Uncertainty budget of the phase displacement for a class 0,2 S
instrument transformer at 100 % In and Sn= 10 VA.
QuantityValueStandard uncertainty uDistributionSensitivity coefficient cVariance (cu)2Index in %δN0.00′0.05′normal12.50 ×-30.7δD-1.00′0.05′normal12.50 ×-30.7dWM0′0.58′rectangular13.33 ×-197.3dMP0′0.02′rectangular12.99 ×-40.1sB-0.36′ VA-1–ΔB0 VA0.17 VArectangular-0.360′ VA-13.88 ×-31.1Sum3.42 ×-1δx-1.0′U= 1.2′ (k= 2)Example of an uncertainty budget for class 0,2 S current
transformerUncertainty budget for the ratio error
First, the model equation for the ratio error εX must be
developed. According to Eq. (8), εX=εD+εN. Influences due to the unprecise setting of
the test point (e.g. 99 % instead of 100 %) and of the tolerances of
the standard burden used have to be taken into account. The model equation
then expands into
εX=εD+εN+dWM+dMP+sB⋅ΔB.
The example stated refers to the 100 % testing point of a class 0,2 S
current transformer. For the standards, the uncertainty contributions are
used as those that are determined from the measured values and associated
uncertainties (for standard transformers) that are documented in PTB's test
certificates, from the admissible limit values (transformer measuring
bridge, standard burden), and from the results obtained during the testing of
the instrument transformer (see Fig. 4). The thus obtained results of this
uncertainty budget can also be used at signal amplitudes of > 20 % In.
Only for the testing points 1 and 5 % is it
recommended that this uncertainty budget be recalculated using the modified
numerical values. Table 1 lists and explains the quantity symbols used in
Eq. (9).
The following conditions and numerical values are given:
The transformer (X) is loaded with the rated burden 10 VA.
The mean value of the display of the transformer measuring bridge εD= 0.048 % from N= 10 measured values. The standard deviation
s of the displayed value is ±0.001 %, from which the type A
standard uncertainty u=s/√N=±0.0003% is
calculated.
Measurement error of the standard εN= 0.000 %. The
expanded measurement uncertainty in the standard is stated as being
U=±0.003 % (k= 2). The standard uncertainty is
u=U/2=±0.0015 %.
Via the admissible tolerance of a=±0.02 %, the uncertainty
in the transformer measuring bridge is considered as having a rectangular
distribution. The standard uncertainty is u=a/√3=±0.0115%.
The influence of the measuring point is calculated via one measurement at
99.0 and one at 100.0 %. From the measured ratio error difference of
0.005 %, a maximum change of 10 %, i.e. a=±0.0005 % with
a rectangular distribution, is estimated for the instrument transformer,
taking a tolerance of the testing point indication of ±0.1 % into
account. The standard uncertainty is u=a/√3=±0.0003%.
The burden sensitivity of the transformer is determined via the measurement
at one-quarter of the burden, which is necessary anyway. Due to the
measurement with 10 and 2.5 VA, the value indicated by the measuring
facility changes from εD= 0.048 % (10 VA) to
εD=+0.163 % (2.5 VA). The sensitivity is thus
sB= (0.048–0.163 %)/(10–2.5 VA) =-0.015 % VA-1.
The admissible tolerance of the standard burden is a=±3 %
(with a rectangular distribution), corresponding to ±0.3 VA. The
standard uncertainty in the burden is u=a/√3=±0.173VA.
The uncertainty budget presented in Table 2 is derived from these
assumptions and numerical values. This table makes several results obvious.
Firstly, the result for the instrument transformer is indicated as
εX= 0.048 %. The expanded measurement uncertainty
amounts to U(εX)=±0.024 % for k= 2, i.e.
for a confidence interval of 95 %. Secondly, it appears that the greatest
influence on the measurement uncertainty is caused by the measuring bridge
(index 93.5 %). By using the actual measurement errors determined and the
measurement uncertainties instead of the permissible tolerances of ±0.02 %
stated in the calibration certificate of the measuring bridge, it
might be possible to reduce the uncertainty contribution u(dBridge) to
0.003 to 0.005 %.
It must, however, be checked for each
individual case whether the individual measuring bridge really allows these
improved measurement uncertainties at all. In any case, at smaller test points
(1 %) and with class 0.5 S transformers, for instance, these relations
change. It can be assumed that the greatest influence is caused from the
greater burden sensitivity of the transformer.
Uncertainty budget for the phase displacement
Similar to the procedure described in Sect. 4.2.1 for the uncertainty in
the ratio error, the model equation for the phase displacement δX must be developed. According to
Eq. (8), δX=δD+δN. Influences due to the unprecise setting of the
test point and of the tolerances of the standard burden used have to be
taken into account. The model equation then expands to
δX=δD+δN+dWM+dMP+sB⋅ΔB.
The example stated refers to the 100 % testing point of a class 0,2 S
current transformer. For the standards, the uncertainty contributions are
used as those that are determined from the measured values and associated
uncertainties (for standard transformers) that are documented in PTB's test
certificates, from the admissible limit values (transformer measuring
bridge, standard burden), and from the results obtained during the testing of
the instrument transformer (see Fig. 5). The thus obtained results of this
uncertainty budget can also be used at signal amplitudes of > 20 % In.
Only for the testing points 1 and 5 % is it
recommended that this uncertainty budget be recalculated using the modified
numerical values. Table 3 lists and explains the quantity symbols used in Eq. (10).
The following conditions and numerical values are given:
The transformer (X) is loaded with the rated burden 10 VA.
The mean value of the display of the transformer measuring bridge δD=-1′ from N= 10 measured values. The standard deviation s of
the displayed value is ±0.158′, from which the type A standard
uncertainty u=s/√N=±0.05′ is calculated.
Measurement error of the standard δN= 0.00′. The expanded
uncertainty in the standard is stated as U=±0.1′ (k= 2). The
standard uncertainty is u=U/2=±0.05′.
Via the admissible tolerance of a=±1′, the uncertainty in the
transformer measuring bridge is considered as having a rectangular
distribution. The standard uncertainty is u=a/√3=±0.5773′.
The influence of the measuring point is calculated via one measurement at
99.0 % and one at 100.0 %. From the measured difference of 0.3′, a
maximum change of 10 %, i.e. a=±0.03′ with a rectangular
distribution, is estimated for the instrument transformer, taking a
tolerance of the testing point indication of ±0.1 % into account.
The standard uncertainty is u=a/√3=±0.0173′.
The burden sensitivity of the transformer is determined via the measurement
at one-quarter of the burden, which is necessary anyway. Due to the
measurement with 10 VA and 2.5 VA, the value indicated by the measuring
facility changes from δD=-1′ (10 VA) to δD=+1.7′ (2.5 VA). The sensitivity is thus
sB= (-1′–1.7′)/(10–2.5VA) =-0.36′ VA-1. The admissible
tolerance of the standard burden is a=±3 % (with a rectangular
distribution), corresponding to ±0.3 VA. The standard uncertainty in
the burden is u=a/√3=±0.173VA.
The uncertainty budget presented in Table 4 is derived from these
assumptions and numerical values. This table makes several results obvious.
Firstly, the result for the instrument transformer is indicated as δX=-1′. The expanded measurement uncertainty amounts to
U(εX)=±1.2′ for k= 2, i.e. for a confidence
interval of 95 %. Secondly, it appears that the greatest influence on the
measurement uncertainty is caused by the measuring bridge (index 97.3 %).
By using the actual measurement errors determined and the measurement
uncertainties stated in the calibration certificate of the measuring bridge
instead of the permissible tolerances of ±1′, it might be possible
to reduce the uncertainty contribution u(δBridge) from
0.2 to 0.4′. It must, however, be checked for each individual
case whether the individual measuring bridge really allows these improved
measurement uncertainties at all. In any case, at smaller test points (1 %)
and with class 0,5 S transformers, for instance, these relations change. It
can be assumed that the greatest influence is caused from the greater burden
sensitivity of the transformer.
Summary and findings of the presented uncertainty budgets
The approach introduced here was intended to develop an uncertainty budget
which is as simple as possible to elaborate by using permissible tolerances
of the test equipment according to the test certificates or according to
PTB (1979). The measurement errors (standard transformer) are only
corrected where this is necessary or feasible by simple means. In addition,
using the permissible tolerances of the bridge and of the standard burden
according to PTB (1979) instead of a tighter range, based on the actual
measurement errors, provides a margin in the conservatively calculated
measurement uncertainty which has thus been overestimated. What is also
interesting is the measurement capability index of the accuracy class of the
transformer and the measurement uncertainty which is ≈ 8 (amount
0.2 %/0.024 % and phase 10′/1.2′, respectively). This means that
the measuring system is approx. 8 times more accurate than the accuracy
class of the device under test. There is a consensus (Sommer et al., 2001; JCGM, 2012) that the minimum accepted measurement capability index is 3.
Vice versa, for the verification which determines whether the instrument
transformer is within its accuracy class or not, it means that measured
ratio error must be within ±7/8 of the class limits to have a
margin of 1/8 due to the system's measurement uncertainty. This leads to a
widened acceptance interval which greatly lowers the probability of a false
rejection of a conformal instrument transformer.
The data relevant to this work are the data for the two figures (Figs. 4 and 5; Mohns, 2018).
The authors declare that they have no conflict of interest.
This article is part of the special issue “Evaluating measurement data and uncertainty”.
It is not associated with a conference.
Edited by: Klaus-Dieter Sommer
Reviewed by: two anonymous referees
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