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The aim of this paper is to show that the developed inductive current transformer may ensure the required wideband transformation accuracy and it may be applied, as the reference source, in the measuring system for the evaluation of the transformation accuracy of inductive current transformers for harmonics of distorted current. This device ensures 5 A and 1 A RMS secondary currents to provide the opportunity to use the differential measuring setup. Such solutions are characterized by the significantly reduced measurement uncertainty in relation to the comparative measurements made between two current/voltage channels. The problems required to ensure the high wideband transformation accuracy, including the self-generation phenomenon of the low order higher harmonics to the secondary current and a too-low frequency range of operation, were overcome in the design process. The values of its ratio error and the phase displacement of the developed reference wideband inductive current transformer did not exceed ±0.2%/° up to 1 kHz, ±0.4%/° from 1 kHz up to 1.5 kHz and ±0.5%/° from 1.5 kHz up to 3 kHz, as is required to perform the test procedure in accordance with the optional requirements for the inductive current transformers defined in the new edition of the standard IEC 61869-1. [ABSTRACT FROM AUTHOR]

This paper presents the evaluation of tested inductive CTs' accuracy for distorted current harmonics in accordance with the optional accuracy class WB1 introduced by the new edition of the standard IEC 61869-1 published in the year 2023. The tests were performed in compliance with the interpretation sheet IEC 61869-2:2012/ISH1:2022. Therefore, the resistive and the resistive–inductive loads of the secondary winding of tested inductive CTs were used, as this was required for the given test conditions. The results indicate that the units designed for the transformation of a sinusoidal current of a frequency of 50 Hz ensure the high wideband transformation accuracy of the distorted current harmonics, as demanded by the power quality monitoring and distorted electrical power and energy requirements. The key to this is proper design using modern magnetic material(s) for the magnetic core and its oversizing in relation to the requirements for a given accuracy class defined for the transformation of sinusoidal currents with a rated frequency. Both tested inductive CTs with a rated primary current RMS value equal to 300 A, class 0.2 and 0.5, ensured compliance with the requirements of the WB1 wideband accuracy class. [ABSTRACT FROM AUTHOR]

In this review article, ideas for the development of measuring methods and setups to test inductive instrument transformer accuracy for the transformation of distorted current or voltage are analyzed. This article proposes limiting values for ratio error and phase displacement at harmonics. Moreover, the manuscript discusses some remarks about the required test waveform of voltage and current, as well as the operating conditions of the inductive instrument transformer during the test. This includes designation of the number of harmonics, their RMS values and phase angles in relation to the fundamental component, as well as the required load of the VT's and CT's secondary winding. The aim of this work is also to show the common requirements and principles of developed methods and measuring setups to justify the proposed solution. [ABSTRACT FROM AUTHOR]

Abstract Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. Graphical Abstract

The Croatian height transformation model (HTMV) enables the transformation of absolute heights of discrete points with known horizontal positions from the Trieste 1875 reference system used throughout the former Republic of Yugoslavia and in Croatia known as HVRS1875 to the new height reference system of the Republic of Croatia (HVRS71) and vice versa. The HTMV is realised as a grid transformation model based on separate modelling of datum and distortion components comprised in height differences. The quality of the transformation model is assessed by comparing the difference in height values at benchmarks that were not used in the model derivation against height differences at the benchmark locations interpolated from the model. The results indicate a high external and internal accuracy of the transformation process, which points to the adequacy of the HTMV utilisation for engineering purposes. [ABSTRACT FROM AUTHOR]

This paper presents a newly developed method to determine the values of current error and phase displacement for the transformation of distorted current harmonics by the inductive current transformers. This approach eliminates the necessity for the utilization of an expensive, high-current supply system for the measuring setup. In this method, the secondary winding is fed by the distorted voltage with RMS values of harmonics calculated in order to reproduce the operation point of the inductive current transformer on the magnetization characteristic of its magnetic core, as in primary winding excitation conditions. This proposed approach is successfully verified with the typically used primary current excitation method, where the secondary currents of the reference and tested current transformers are compared in the differential measuring setup. It was confirmed that the inductive CT with current error and phase displacement for transformation of distorted current harmonics determined in the rated ampere-turns conditions may be effectively used in the measuring setup as the reference source of the primary current. [ABSTRACT FROM AUTHOR]

Power grids are a combined system where the electrical energy produced by the power plants is transmitted to consumers. This forms a specific interdependence where the recipients have a significant impact on the power quality. Therefore, the nonlinear loads connected by households and industrial customers cause current and voltage distortion in the power networks. This creates the need for accurate measurement of nonsinusoidal voltage and current composed not only from the fundamental component but also containing higher harmonics, interharmonics, and subharmonics. In order to ensure high transformation accuracy of distorted current and voltage, the inductive instrument transformers have to be tested in these conditions. Many papers describe their behavior during the transformation of sinusoidal current or voltage. Nowadays, the scientific field in this scope is focused on the evaluation of their exploitation properties for distorted signals. The common problem of inductive instrument transformers is the self-generation of low-order higher harmonics to the secondary current or voltage. In the case of the inductive VTs, an additional problem results from the resonance caused by the parasitic capacitance of the primary winding. The proposed solutions to compensate for the values of current or voltage errors and phase displacement of inductive instrument transformers are also analyzed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that the developed inductive current transformer may ensure the required wideband transformation accuracy and it may be applied, as the reference source, in the measuring system for the evaluation of the transformation accuracy of inductive current transformers for harmonics of distorted current. This device ensures 5 A and 1 A RMS secondary currents to provide the opportunity to use the differential measuring setup. Such solutions are characterized by the significantly reduced measurement uncertainty in relation to the comparative measurements made between two current/voltage channels. The problems required to ensure the high wideband transformation accuracy, including the self-generation phenomenon of the low order higher harmonics to the secondary current and a too-low frequency range of operation, were overcome in the design process. The values of its ratio error and the phase displacement of the developed reference wideband inductive current transformer did not exceed ±0.2%/° up to 1 kHz, ±0.4%/° from 1 kHz up to 1.5 kHz and ±0.5%/° from 1.5 kHz up to 3 kHz, as is required to perform the test procedure in accordance with the optional requirements for the inductive current transformers defined in the new edition of the standard IEC 61869-1.

This paper presents the evaluation of tested inductive CTs’ accuracy for distorted current harmonics in accordance with the optional accuracy class WB1 introduced by the new edition of the standard IEC 61869-1 published in the year 2023. The tests were performed in compliance with the interpretation sheet IEC 61869-2:2012/ISH1:2022. Therefore, the resistive and the resistive–inductive loads of the secondary winding of tested inductive CTs were used, as this was required for the given test conditions. The results indicate that the units designed for the transformation of a sinusoidal current of a frequency of 50 Hz ensure the high wideband transformation accuracy of the distorted current harmonics, as demanded by the power quality monitoring and distorted electrical power and energy requirements. The key to this is proper design using modern magnetic material(s) for the magnetic core and its oversizing in relation to the requirements for a given accuracy class defined for the transformation of sinusoidal currents with a rated frequency. Both tested inductive CTs with a rated primary current RMS value equal to 300 A, class 0.2 and 0.5, ensured compliance with the requirements of the WB1 wideband accuracy class.

Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. [ABSTRACT FROM AUTHOR]

In this review article, ideas for the development of measuring methods and setups to test inductive instrument transformer accuracy for the transformation of distorted current or voltage are analyzed. This article proposes limiting values for ratio error and phase displacement at harmonics. Moreover, the manuscript discusses some remarks about the required test waveform of voltage and current, as well as the operating conditions of the inductive instrument transformer during the test. This includes designation of the number of harmonics, their RMS values and phase angles in relation to the fundamental component, as well as the required load of the VT’s and CT’s secondary winding. The aim of this work is also to show the common requirements and principles of developed methods and measuring setups to justify the proposed solution.

This paper presents a newly developed method to determine the values of current error and phase displacement for the transformation of distorted current harmonics by the inductive current transformers. This approach eliminates the necessity for the utilization of an expensive, high-current supply system for the measuring setup. In this method, the secondary winding is fed by the distorted voltage with RMS values of harmonics calculated in order to reproduce the operation point of the inductive current transformer on the magnetization characteristic of its magnetic core, as in primary winding excitation conditions. This proposed approach is successfully verified with the typically used primary current excitation method, where the secondary currents of the reference and tested current transformers are compared in the differential measuring setup. It was confirmed that the inductive CT with current error and phase displacement for transformation of distorted current harmonics determined in the rated ampere-turns conditions may be effectively used in the measuring setup as the reference source of the primary current.

Power grids are a combined system where the electrical energy produced by the power plants is transmitted to consumers. This forms a specific interdependence where the recipients have a significant impact on the power quality. Therefore, the nonlinear loads connected by households and industrial customers cause current and voltage distortion in the power networks. This creates the need for accurate measurement of nonsinusoidal voltage and current composed not only from the fundamental component but also containing higher harmonics, interharmonics, and subharmonics. In order to ensure high transformation accuracy of distorted current and voltage, the inductive instrument transformers have to be tested in these conditions. Many papers describe their behavior during the transformation of sinusoidal current or voltage. Nowadays, the scientific field in this scope is focused on the evaluation of their exploitation properties for distorted signals. The common problem of inductive instrument transformers is the self-generation of low-order higher harmonics to the secondary current or voltage. In the case of the inductive VTs, an additional problem results from the resonance caused by the parasitic capacitance of the primary winding. The proposed solutions to compensate for the values of current or voltage errors and phase displacement of inductive instrument transformers are also analyzed.

The novelty of the paper contributes to the increase of the reliability of the methods used for evaluation of the inductive CTs accuracy of transformation of the distorted current. The results of performed analyses shows that the values of current and phase errors of transformation of the low order higher harmonics by inductive CT depends significantly from the phase angle between transformed higher harmonics of the distorted primary current and the self-generated higher harmonics of the secondary current. Moreover, higher harmonics of the distorted primary current may increase the peak value of the magnetic flux density in the magnetic core. Then the impact of the self-generated low order harmonics resulting from the nonlinearity of the magnetic core on values of current and phase errors of transformation of the higher harmonics of the distorted primary current is increased. Proposed evaluation procedure enables determination of the maximum values of current and phase errors of transformation of the higher harmonics of the distorted primary current. It is essential to determine the accuracy class. In such approach the impact of the secondary current's distortion caused by the nonlinearity of the magnetic core and the influence of the harmonic distortion on the peak value of the magnetic flux density are considered. Therefore, transformation accuracy of tested CT is ensured for specified range and type of the load of the secondary winding, rms values of primary current and its distortion.

Highlights • The requirements for testing accuracy of the wideband voltage dividers are discussed. • The conditions having influence on ratio error and phase displacement are analyzed. • It is explained why the course of frequency characteristics depends from applied voltage. • It is shown that leakage current should be determined to provide proper level of insulation. • Proposed zero comparative method ensures accuracy of the divider for rated input voltage. Abstract The general concerns of the studies presented in this article are determination of the requirements and development of the alternative indirect method for testing transformation accuracy of distorted voltages and higher frequency signals by the wideband resistive-capacitive voltage dividers. The increase of the rms value of the input voltage causes increase of the internal temperature and the probability of malfunction, including initiation of the partial discharges, additionally increased with frequency of the higher harmonics. Changes of these conditions may cause considerable variations of the equivalent capacitance and resistance of the wideband voltage dividers. Therefore, their testing with decreased rms value of distorted input voltage, as permissible by the standard IEC 61869-11, may result in determination of significantly different frequency characteristics of ratio error and phase displacement, than it is for rated conditions. In order to ensure transformation accuracy for rated input voltage of the wideband voltage divider the zero comparative method may be used. Applied rms value of voltage to each part of the wideband voltage divider under construction must result from its rated input voltage. Developed procedure provides conditions needed to obtain rated equivalent capacitance/resistance of each part of the wideband voltage divider during its development. This ensure its required transformation accuracy for rated input voltage. However, to preserve the proper level of its insulation to ground the test to exclude additional increase of the leakage current with increase of the input voltage must be made. The principle of application of proposed idea is verified by determination of the frequency characteristics of ratio error and phase displacement of tested wideband voltage dividers for their rated input voltage. Therefore, they may be used as a reference voltage source for testing transformation accuracy of voltage transformers and other dividers. [ABSTRACT FROM AUTHOR]

The modern spatial data coordinate basis (SDCB) is built taking into account the variety of existing and used today geodetic networks, models of physical fields of the Earth, cartographic models, as well as coordinate systems (СS). One of the requirements for SDCB from the standpoint of system analysis is the requirement of integrity, which presupposes the unity of the determination of coordinates, that is, the consistency of the results of determining the coordinates of the same points in different CSs. The article is devoted to the monitoring of the accuracy characteristics of the available software for coordinate transformations in terms of single-stage and multi-stage transitions between ellipsoidal coordinates of different systems.

In this paper the concept is discussed for evaluation of accuracy of inductive current transformer (CT) for transformation of distorted current higher harmonics. In proposed method the composite error (transformer correction factor) is determined for various operating conditions of tested CT in respect of primary currents rms value, distortion and secondary winding load active power. In this practical approach distorted primary current generated for the tests contains multiple harmonics. The results of FFT analysis of measured composite error are used to simultaneously evaluate tested CT maximum current error and phase displacement of transformation for each of the distorted current harmonic. This enables reduction of both examination time and its cost. The usage of measured differentially composite error, for evaluation of accuracy of inductive CT for transformation of distorted current higher harmonics, ensures high accuracy of obtained results, at low costs of required measuring apparatus. Presented methodology, allows determination of the optimum range of tested CT's secondary winding load, at which it can be used for transformation of distorted currents with maximum accuracy. [ABSTRACT FROM AUTHOR]

A circuit design version of an optoelectronic transducer of displacements into time interval with insertion of the carrier signal into the radiation source is proposed. It is shown that amplitude errors associated with the instability of the light flux as well as the error of the single-channel raster modulator are compensated in the newly developed device. [ABSTRACT FROM AUTHOR]