Effects of Multiple Influence Quantities on Rogowski-Coil-Type Current Transformers

This article presents a study on low-power passive current transformers. In particular, the performance of three Rogowski coils has been assessed when multiple influence quantities were acting on them: conductor position, frequency, and ambient temperature. First of all, their single effects have been assessed, then all their possible combinations have been tested. From the results, it can be concluded that the ratio error is mostly affected by the combination of uncentered positions and temperatures applied to Rogowski coils. Contrastingly, the phase error is substantially not influenced by any quantity. Therefore, the proposed set of tests could become a benchmark in Rogowski coil testing.


I. INTRODUCTION
T HE evolution toward smart grids has brought among the networks a variety of new intelligent electronic devices, measurement instruments, and power grid accessories. Their introduction should have no impact on both the operation of the network and the distribution system operators' (DSOs') efforts because of their management and maintenance [1], [2].
As for the medium voltage distribution network, it experienced a huge penetration of the so-called low-power instrument transformers (LPITs) [3] to substitute the traditional inductive ones. Such a new kind of transformer allows measuring the rated voltages and currents, providing low-power outputs (typically lower than 1 VA) which are, most of the time, already suitable for typical acquisition systems. Among the benefits obtained from their introduction, the LPITs have reduced dimensions, high robustness, and large bandwidths compared to the inductive ones. This make them suitable for a variety of new applications that arose, in recent years, among the networks [4]- [6].
To guarantee the reliability of such applications and of the equipment they involve, such equipment has to be subjected to several compliance tests defined by the Standards. With this aim, this article presents a complete series of tests performed Manuscript  on LPITs, in particular, on passive Rogowski coil-type, low-power current transformers (LPCTs), which could become benchmark-type tests in the future Standards. As a matter of fact, Standards provide a variety of tests for each kind of instrument. For example, IEC Standard 61869-10 [7] defines accuracy tests for the LPCT versus position, versus frequency, and versus temperature. Even literature contains several works on this critical topic, for example, [8] assesses the mutual inductance of the Rogowski versus primary conductor position. In [9], their performances are evaluated when the geometrical parameters are varying, while [10] studies the thermal expansion of the Rogowski as a principle cause of performance decrease. Finally, the single effect of the primary conductor position and of the electromagnetic fields on Rogowski measurements are analyzed in [11]- [13], respectively. Hence, in light of the aforementioned and by considering the growing importance of LPITs in smart grid operations, the authors made a further step toward achieving better knowledge of their behavior under typical influence quantities. This has been done by simultaneously assessing the effects of multiple influence quantities affecting the LPCTs. As a matter of fact, to the authors' knowledge, no accuracy performance study has been done so far either in the literature or in the Standards to understand the effects a quantity could have on Rogowski coils' performance, when combined with others. Hence, in light of this and by considering the key role of the LPCT accuracy analysis [14], [15], this article presents a full set of tests combining three different influence quantities: primary conductor position, frequency, and ambient temperature. Tests have been performed according to [7] when possible, otherwise they have been designed by starting from it. The input signal of the tests is always a sinusoidal waveform at a rated frequency (except for the frequency tests). Such way of proceeding fulfills the Standard's requirements. As a matter of fact, the Standards characterize the accuracy performance of an instrument transformer (IT) in terms of the ratio and phase errors at a rated frequency and under sinusoidal conditions. Therefore, up to now, the best way to assess if and how a combination of influence quantities affects the performance of an IT is to measure its ratio and phase errors, and this can be done only in the aforementioned conditions. Finally, the use of more complicated, even if more actual, signals would not have allowed appreciating the effects of the tested influence quantities. From the results, it is even more confirmed of the authors' proposal of using the presented tests as a benchmark for the future Standards.  This article is structured as follows: Section II briefly recalls the operating principle of the Rogowski coils. In Section III, the automatic measurement setup adopted for the experimental tests is fully described. All performed tests are detailed in Section IV, whereas Section V presents the experimental results obtained. Finally, conclusion and comments about the work are presented in Section VI.

II. ROGOWSKI COILS
The Rogowski coil is an IT which works under the same principle as the typical inductive ones. The main difference consists of the material on which the conductors are wound: air for the Rogowski coil and iron for the traditional one. This aspect results in a linear behavior, in contrast to the inductive type which saturates due to the presence of the iron core. By considering Fig. 1, wherein the Rogowski coil is depicted, the primary conductor (whose current has to be measured) is placed inside the coil (of cross section S and radius R). Then, the output is a voltage proportional to the derivative of the primary current, and follows the equation: where i (t) and u s (t) are the primary current and the secondary output voltage, respectively. While M is the mutual inductance between the conductors. From (1), it can be observed that the output is 90 • -shifted from the input; this can also be seen by considering Rogowski's equivalent circuit shown in Fig. 2. It is composed of a series resistor and an inductor (R S and L S ), followed by the parallel straight capacitance and the high-impedance burden (C S and R B , respectively). As for their design, Rogowski coils are mainly divided into two categories: split-core and window-type. The former type can be opened to be placed around a conductor, while the latter type needs to be inserted over the conductor, which should be disconnected from its original place. Both types of  transformers can be rigid or flexible (high accuracy is obtained with the rigid ones) [7].
The use of Rogowski coils is typical among utilities and DSOs for various applications [16]- [19]; furthermore, the literature has been and is very vivid about their study. In particular, their modeling [20], [21] is a current and broad topic along with the design of new possible and innovative solutions [22], [23]. Furthermore, in light of their massive deployment among the networks, the evaluation and assessment of their accuracy are of paramount aspect [24], [26], which are dealt with in this work.

III. AUTOMATIC MEASUREMENT SYSTEM
In this section, the detailed description of the adopted measurement system is provided. Its simple schematic representation is depicted in Fig. 3. In the picture, the following elements can be distinguished: 1) Fluke Calibrator 6105 A: It is used as a current and a voltage reference source (Ī C andV C : current and voltage phasors) for all performed tests. Its main characteristics, including the accuracy ones, are listed in Table I. 2) Thermostatic Chamber: It allows varying its internal temperature in the range 5 • C-70 • C. In addition, a Chauvin Arnoux 863 thermocouple-based temperature sensor has been used to verify the desired temperature in each performed test. It features: −50 • C to +1300 • C measurement range, 0.1 • C resolution, and ±0.3% accuracy of the reading.

3) A Set of Three Rogowski Coil-Type Current Transformers:
From here on out, they are referred to as X, Y, and Z for the sake of privacy, and they are made by three different manufacturers. X is a window-type Rogowski, while Y and Z are of the split-core-type. The characteristics of Rogowski under test (RUT) are summarized in Table II. In addition, the RUTs come

4) A NI-9238 Data AcQuisition Board (DAQ) and Its USB
Chassis NI-9171: The DAQ main features are summarized in Table III. It has been used to collect the RUTs' output and the voltage phasor of the calibrator, used as the phase reference. Such a measurement setup has been adopted to perform the tests described in the following sections.

IV. EXPERIMENTAL TESTS
In this section, tests to assess the effects of several influence quantities on the RUTs have been described. With the same structure, the results are presented in Section V.

A. Resistive Burden Characterization
Before performing the main tests, the resistive burden connected to each RUT has been characterized to estimate its value. To this purpose, 200 measurements have been performed with the HP Digital Multimeter 3458a on three 22-k resistors.

B. Accuracy Versus Position Tests
The first set of tests aimed to verify the effects of both the position of the internal and external conductors on the accuracy of the RUTs. To this purpose, according to [7], four different positions have been tested. As clarified by Fig. 4, they are referred to as A, B, C, and D. For the first three positions, [7] defines the position factor (PF) as where d max and d min are the maximum and minimum distances between the primary conductor and the Rogowski window. The PF ranges between 0 and 1. Position A is the rated one, where the internal conductor is centered with respect to the RUT, hence it has a PF of 0. Pictures of the four test configurations adopted. Each of them describes a different relative position between the LPCT and the internal and/or external conductor. As for positions B and C, they are not-zero PF, 0 < PF < 1, and 1, respectively. In particular, in B, the conductor is completely bent over the RUT, whereas in C, the conductor is perpendicular to the RUT but attached to it, hence not centered at all. The last position is D, where an external conductor is attached to the outer part of the RUT. Moreover, as for D, [7] states that the transmitting cables of the LPCT must be 90 • with respect to the external conductor. To better clarify this aspect, in Fig. 5, the correct positioning is depicted.
Afterward, for the four test configurations, a primary current I C = 100 A (at 50 Hz and 22 • C) has been injected with the calibrator through the primary conductor and measured with the 3 RUTs. Their outputs (Ū S ) have been acquired without using any integrator in-between to avoid any interference with the RUT performance evaluation. Then, 100 measurements of U S have been collected, and 100 values of ratio and phase errors (ε and ϕ) have been computed as follows: where |Ū S | and |Ī C | are the modules of Rogowski's output voltage and the primary current phasors, respectively. As for k, it is the nominal ratio of the RUTs, whileÛ S andV C are the phases of the related abovementioned phasors. Afterward, the mean value of the 100 measurements of ratio and phase errors (φ) have been computed (for all performed tests).
Then, the described tests have been repeated at 48 and 51 Hz. Such values have been adopted from [7] to tackle the harshest conditions, which refer to the use of the LPCTs for protective purposes. For the frequency tests, [7] states that the obtained ratio errors must be corrected as follows: where CF is the correction factor obtained as the ratio between the rated and actual frequencies, f r and f a , respectively

C. Accuracy Versus Temperature Tests
The second set of tests wanted to assess the effects of the working temperature on the accuracy of the RUTs. To this purpose, the temperatures defined for the tests are 5 • C, 22 • C, and 40 • C. The upper limit has been defined according to [7], while the lower one is in accordance with a typical outdoor average ambient temperature in Italy during cold seasons. Therefore, each temperature has been set on the thermostatic chamber and maintained for 8 h. This is to ensure proper thermal stability for both the chamber and the RUTs. Once such a condition has been obtained, 100 measurements ofŪ S have been acquired for the four test configurations and for the three frequencies (48, 50, and 51 Hz). Again, from the measurement results, ε and ϕ have been computed for each test configuration. In light of the aforementioned, an overall amount of 36 tests have been performed. For the sake of clarity, and for a better comprehension of Section V, they have been numerated and are listed in Table IV.

A. Resistive Burden Characterization Results
Table V collects the mean valuesR and the related combined uncertainty u c of the three resistors (R X , R Y , and R Z ). As for u c , it has been calculated, according to the Guide, as the expression of uncertainty in measurement [27], as where u a and u b are the uncertainties evaluated with type A and type B methods, respectively. In particular, u b has been computed by starting from the accuracy specifications of the multimeter 3458a used for the resistance measurements: 2 · 10 −6 error on the reading and 2 · 10 −7 error on the range. As for u a , as is well-known, it is computed by dividing the variance of the mean value measured by the number of measurements. From Table V, it is possible to highlight the low uncertainty associated with the resistors' values.

B. Results of the Accuracy Versus Position Tests
By considering that no calibration coefficients were provided by the manufacturers of the LPCTs, test 1 has been used as a reference test to determine the actual ratio of the three RUTs (K X , K Y , and K Z ). They are listed in Table VI along with their associated combined uncertainty [computed according to (7)]. In addition, all ratio errors presented in the following have been computed by taking the ratios in Table VI as the rated ones. Hence, for the sake of comparison, test 1 ratio error is always set at value zero.
Moving to the aim of the subsection, in Fig. 6, the results of the accuracy versus position tests are shown at 50 Hz and at room temperature, 22 • C (1, 2, 3, and 4). In the graph and in all the following ones, the standard deviation of the ratio error (obtained from the mean of 100 measurements) is not presented for the sake of brevity. As a matter of fact, it was always in the order of 10 −5 for all performed tests. As it can be seen, the window-type RUT (X) is almost not affected by the PF of the conductor, whereas Y and Z are sensitive to PF = 1 (position C) and to the presence of an external conductor (position D), respectively. The phase error of this four set of results has not been plotted for the sake of brevity, because it has not been affected by the PF. Moreover, it was always in the order of fraction of milliradians, for the three RUTs.
In light of the position-test results, it can be concluded that the conductor position is critical for the Rogowski performance. As confirmed in [8], the changes in the conductor position cause a variation of the mutual inductance M between conductors. Therefore, according to (1), it results in a different output voltage (by starting from the same input current), hence in the overall accuracy of the Rogowski coil. However, this issue is typically solved by using external accessories (usually insulating materials) aimed at keeping the conductor centered with respect to the Rogowski. However, as experienced by the authors in many in-field applications, this is not always possible, hence compensating solutions should be adopted as it has been demonstrated in [8].
By adding the contribution of another influence quantity, the frequency, the related results are depicted in Fig. 7 (dotted lines refer to 48 Hz, while the solid ones to 51 Hz). From it, a general comment is that the results confirm the overall trend (and absolute values) obtained from Fig. 6. However, aside from the case of X, which is not affected by the frequency, it is possible to appreciate its negative effect, which increases the one due to the positions tested. As for ϕ, neither frequency affects it, confirming what was already obtained from the 50-Hz cases. As a final comment on this first set of results, it can be stated that at 50 Hz (rated frequency), positions C and D are critical for the split-core type Rogowski. As a matter of fact, ε significantly overcomes the limits declared by the manufacturers (±1%). Instead, for frequencies different from the rated one, even position B becomes critical. In particular, the Y accuracy is noncompliant for positions B and C, whereas the Z one for positions B and D. It is worth emphasizing that in all the frequency test results, proper CF has been applied.

C. Results of the Accuracy Versus Temperature Tests
In this section, the effects of working temperature variation on the accuracy of the RUTs are assessed. To this purpose, Fig. 8.
Ratio error results for tests 1, 13, and 28. Accuracy versus temperature, 50 Hz. let us start from the basic position A, where the LPCT is centered with respect to the internal conductor. Hence, Fig. 8 shows the results of tests 1, 13, and 28 (position A, at 50 Hz). From the picture, it can be concluded that X, the windowtype Rogowski, is almost not affected by the temperature when working at 50 Hz. Contrastingly, for Y and Z, the split-core-type ones, the temperature significantly reduces their accuracy. In particular, at 40 • C, the ratio error increased up to 1 order of magnitude. However, for all the RUTs, either at 5 • C or 40 • C, ε remains within the accuracy limits provided by the manufacturers and is listed in Table II.
As for the computed phase errors, they are listed in Table VII along with their associated combined uncertainty. From the table, it emerges that even the temperature does not affect ϕ for all the studied RUTs, and they are always contained within the accuracy limits.
In accordance with what was already done in the previous subsection B, the abovementioned results are now evaluated at frequencies different from the rated one. All results are depicted in Fig. 9, where the dotted lines represent the 51-Hz tests (5, 14, and 29), whereas the 48 Hz ones (9, 15, and 30) are represented by solid lines. The first comment that arises from the graph is the overall confirmation of the trend observed in  Fig. 6 for the tests at 50 Hz. Second, both 48-and 51-Hz tests provide almost the same results (in absolute value terms) for each tested temperature.
As for the evaluation of the combined effects of temperature and frequency, Figs. 8 and 9 must be compared. From the comparison, it can be stated that the significant contribution to accuracy worsening is provided by the temperature. As a matter of fact, the frequency contribution is negligible and cannot be distinguished from the temperature one. Moving to the phase error evaluation, in the position A studied in this subsection, it can be concluded that ϕ is not affected either by the temperature or by the frequency. Hence, the results are not reported for the sake of brevity.
As an overall comment on the effects of temperature, this quantity seems to have a critical effect on the Rogowski performance. This can be associated with two different phenomena affecting the RUT when the temperature varies: changes in its geometry and thermal expansion of the copper windings. Both are confirmed to have an effect on the Rogowski performance [9], [10], hence two possible solutions to mitigate such effects might be: 1) using an external cage for the Rogowski with thermal properties aligned with the working temperatures and 2) the development of compensating (hardware or software) techniques to consider the effects of temperature on the Rogowski output. As for this last point, [27] and [28] describe two interesting research works that suggest how to consider the effect of temperature when dealing with Rogowski's measurements.

D. Evaluation of Temperature and Position Combined Effect on the RUT Accuracy
Among the novelties of the article, the evaluation of multiple influence quantities' effects on the LPCT performance is one of the most interesting ones. To this purpose, Figs. 10 and 11 show the results of the position and temperature combined tests.
By starting from Fig. 10, it contains the comparison between the tests performed at 22 • C (solid lines) and the ones performed at 5 • C (dotted lines). From the graph analysis, it results that the RUTs are affected by temperature even in rated position A. This causes X, Y, and Z to exceed their accuracy limits. Such a trend is then confirmed for the other positions tested and for all RUTs. In addition, by considering that the solid curves represent the computed ε obtained from the single effect of the conductor position, from the graph,  it is possible to quantify the temperature contribution to the overall value of ε.
Similar comments can be drawn from the graph in Fig. 11, where the comparison between the tests performed at 22 • C (solid lines) and the ones at 40 • C (dotted lines) is presented. However, compared to Fig. 10, a slight difference can be highlighted: higher temperatures seem to less affect the RUT performance. This is true for all the RUTs except for X, the window-type one, which is affected by both high-and low-temperatures. For the sake of the completeness, the phase error results obtained by all the above-mentioned test combinations are listed in Table VIII. However, as obtained for the previous tests, the phase displacement is not affected by the combination of temperature and conductor position.

E. Evaluation of Temperature, Position, and Frequency Combined Effect on the RUT Accuracy
The last set of test results concerns the combination of three influence quantities applied to the RUTs in order to evaluate their performance. The results are presented in Figs. 12-14 for the LPCTs X, Y, and Z, respectively. They show the ratio   errors of all possible test configurations, which include the temperature, frequency, and position variations. In particular, each set of columns represents a position, whereas the colors refer to the temperatures: blue, green, and red, for 5 • C, 22 • C, and 40 • C, respectively. From the pictures, it is possible to appreciate the ε trends due to multiple influence of the varying quantities. As for Fig. 12, it shows the negative effect of the temperature superimposed onto position B. In fact, the combination of these two influence quantities turns into a ratio around seven times greater than the allowed limit. On the contrary, working at a frequency different from the rated one does not result in significant variation of the RUT performance accuracy.
Similar comments on the frequency can also be stated for Figs. 13 and 14. From the Y results in Fig. 13, it can be concluded that positions B and C are particularly critical, whereas the presence of an external conductor (position D) does not affect at all the performance of Y. Moreover, a low temperature seems to be more critical in all the performed test, compared to the high one. Interesting results can also be drawn from Fig. 14. As a matter of fact, Z is sensitive to the presence of external cables. However, this sensitiveness seems to be reduced by a working temperature different from the rated one (22 • C). One more time, the frequency does not influence the RUT operation, whereas the temperature combined with the position effects results in critical results.
From the abovementioned results, it can be concluded that, on the one hand, the simultaneous presence of the influence quantities, temperature and position, causes a severe degradation of the LPCT performance. This is true for all the RUTs studied in this work. In addition, such a degradation brings the ratio error out of its bounds, hence not guaranteeing anymore the manufacturers' given accuracy. On the other hand, the phase error ϕ seems not to be affected by any of the influence quantities tested in this work.
In addition to the previous comments, the interesting and satisfactory results presented support the authors' idea of using the proposed tests as a benchmark for the Rogowski coil testing. Then, the study could be completed by assessing the Rogowski behavior with waveforms affected by all kinds of power quality issues (harmonics, interharmonics, dips, and so on.)

VI. CONCLUSION
This work presents a study on low-power passive current transformers, in particular, the Rogowski type. By starting from their related Standards, new tests have been proposed to assess their accuracy performance under the simultaneous influence of multiple quantities: frequency, position, and ambient temperature. The obtained results confirm the authors' hypothesis: the passive transformers suffer from the multiple presence of such influence quantities. In particular, all the tested devices exceeded their accuracy thresholds when the temperature and the position were varied from the rated one. This holds for the ratio error, whereas the phase displacement is completely insensitive with respect to the influence quantities applied. Along with the results, suggestions and comments on the possible technical solutions to be implemented in order to compensate the obtained results are provided.
In conclusion, the work wants to be a first step toward the idea of testing the accuracy of the LPCTs, not just considering one influence quantity at a time, but multiple ones. In addition, it can be observed that the simultaneous presence of more than one influence quantity does not always turn into worsening the accuracy performance of the LPCT. Furthermore, the described tests, in light of the obtained results, might become a starting point for improving the existing Standards.