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Although the so-called magnetic geometrical factor, $Bl$, of a Kibble balance does not appear in the Kibble equations, it offers the precision link between electrical and mechanical quantities and furthers a quasi-quantum traceability path for mass metrology. This feature makes the magnet system, supplying the $Bl$ in Kibble equations, play a core role in Kibble balances. Following the open-hardware idea, we report here on the design, manufacture, assembly, optimization, and finally performance of a compact magnet system for the Tsinghua tabletop Kibble balance. Notably, the magnet system showcased in this study facilitates a straightforward upper levitation of splitting through a streamlined mechanism guide, substantially enhancing the ease of open and close operations. Experimental tests show the realized magnet systems can yield a high $Bl$ value (e.g., 400 Tm for a bifilar coil and 800 Tm for a single coil with a wire gauge of 0.2 mm) meanwhile a low volume/weight (40 kg) thanks to the uniformity improvement of magnetic profiles. Furthermore, important parameters related to systematic effects, such as the current effect, are checked, aiming for a final mass-realization accuracy at the $10^{-8}$ level., Comment: 12 figures, submitted to IEEE Trans. I & M

A high-precision current source, capable of supporting weighing measurements with a relative uncertainty at the $10^{-9}$ level, is essential for Kibble balance experiments. However, most current sources utilized in Kibble balances to date are homemade and not commercially available. In this paper, we introduce a digital-feedback, two-stage current source designed for the Tsinghua tabletop Kibble balance, relying solely on commercially available sources and voltmeters. A high-resolution, small-range current source is employed to digitally compensate for current output fluctuations from a large-range current source. Experimental tests show the proposal can offer an easy realization of a current source with nA/A stability to support Kibble balance measurements., Comment: 10 figures, submitted to IEEE Trans. Instrum. Meas

The Kibble balance requires a measurement of the local gravitational acceleration, $g$, with a typical relative measurement uncertainty of $10^{-9}$. In this paper, the determination of $g$ for the Tsinghua tabletop Kibble balance is presented. A polynomial fitting method is proposed for blind transfers of the absolute gravitational acceleration using relative gravimeters, showing agreement with the value obtained by the tide correction within a few parts in $10^{9}$. Horizontal and vertical gravity gradients are extracted by mapping the gravity distribution at different heights. The self-attraction effect of major components in the experiment, as well as some time-varying systematic effects, are modeled. The final determination of the gravitational acceleration at the mass position, with an uncertainty of 5.4 $\mu$Gal ($k=2$), is achieved for the Tsinghua tabletop Kibble balance experiment., Comment: 11 figures, submitted to IEEE Trans. Instrum. Meas

Magnetic-array-type current sensors have garnered increasing popularity owing to their notable advantages, including broadband functionality, a large dynamic range, cost-effectiveness, and compact dimensions. However, the susceptibility of the measurement error of one or more magnetic measurement units (MMUs) within the current sensor to drift significantly from the nominal value due to environmental factors poses a potential threat to the measurement accuracy of the current sensor. In light of the need to ensure sustained measurement accuracy over the long term, this paper proposes an innovative self-healing approach rooted in cyber-physics correlation. This approach aims to identify MMUs exhibiting abnormal measurement errors, allowing for the exclusive utilization of the remaining unaffected MMUs in the current measurement process. To achieve this, principal component analysis (PCA) is employed to discern the primary component, arising from fluctuations of the measured current, from the residual component, attributed to the drift in measurement error. This analysis is conducted by scrutinizing the measured data obtained from the MMUs. Subsequently, the squared prediction error (SPE) statistic (also called $Q$ statistic) is deployed to individually identify any MMU displaying abnormal behavior. The experimental results demonstrate the successful online identification of abnormal MMUs without the need for a standard magnetic field sensor. By eliminating the contributions from the identified abnormal MMUs, the accuracy of the current measurement is effectively preserved., Comment: 11 pages, 10 figures

The Kibble balance is a precision instrument for realizing the mass unit, the kilogram, in the new international system of units (SI). In recent years, an important trend for Kibble balance experiments is to go tabletop, in which the instrument's size is notably reduced while retaining a measurement accuracy of $10^{-8}$. In this paper, we report a new design of a tabletop Kibble balance to be built at Tsinghua University. The Tsinghua Kibble balance aims to deliver a compact instrument for robust mass calibrations from 10 g to 1 kg with a targeted measurement accuracy of 50 $\mu$g or less. Some major features of the Tsinghua Kibble balance system, including the design of a new magnet, one-mode measurement scheme, the spring-compensated magnet moving mechanism, and magnetic shielding considerations, are discussed., Comment: 8 pages, 9 figures

The diffusion of nonlinear loads and power electronic devices in power systems deteriorates the signal environment and increases the difficulty of measuring harmonic phasors. Considering accurate harmonic phasor information is necessary to deal with harmonic-related issues, this paper focuses on realizing accurate dynamic harmonic phasor estimation when the signal is contaminated by certain interharmonic tones, i.e. the out-of-band interference (OBI). Specifically, this work introduces the singular value decomposition into the Taylor-Fourier transform algorithm, deriving a general decomposition form for harmonic phasor filters, and yielding a set of adjustable parameters. Then these parameters are configured to minimize the negative influence of OBI on the dynamic harmonic phasor filters. Based on the recommended parameter values, the optimized harmonic estimator is obtained, and then tested under signals distorted by OBI, harmonic, noise, and in worse cases, frequency deviation, and some dynamic conditions. Test results show the proposal achieves good harmonic phasor estimation even under multiple interferences and dynamic scenarios, and has a much weaker dependence on OBI tones compared to conventional approaches., Comment: 10 pages, 14 figures, manuscript accepted for publication on IEEE Transactions on Power Delivery

Analytically the forces and torques on a coil in a field of magnetic flux density can be calculated one of two ways. The line integral can be conducted along the coil's wire, summing up the differential force contribution. For each differential line segment, the force is obtained as a cross product with the magnetic flux density. Alternatively, the coil's energy in the field is the current times the flux threading the coil. Hence, the energy is obtained by executing a surface integral over the coil's open surface. Here, a dot product of the differential surface element with the magnetic flux density is executed under the integral sign. The forces and torques can then be obtained from the negative derivative of the energy with respect to the appropriate coordinate. For yoke-based Kibble balances, the latter method is much simpler since most of the flux is contained in the inner yoke of the magnet and can be written as a simple equation. Here, we use this method to provide simple equations and their results for finding the torques and forces that act on a coil in a yoke-based magnet system. We further introduce a simple method that allows the calculation of the position and orientation difference between the coil and the magnet from three measurements., Comment: 16 pages, 15 figures, accepted for publication by Metrologia

The Kibble balance has become one of the major instruments for realizing the mass unit, the kilogram, in the revised international system of units (SI). Researchers at about a dozen national metrology institutes are actively working with Kibble balances that are capable of weighing masses with nominal values from 10g to 1kg. In the future, the design of smaller Kibble balances will play a more significant role. Smaller Kibble balances require smaller magnet systems, and here we investigate the scaling of systematic uncertainties with the size of the magnet system. We describe the size dependence of three magnetic effects: the coil-inductance effect, the yoke nonlinear effect, and the thermal effect. The analysis shows that the relative systematic effects become increasingly larger with smaller sizes. For small magnets the thermal effects become dominant and, hence, a good thermal design is imperative., Comment: 9 pages, 8 figures, submitted to IEEE Trans. Instrum. Meas

The magnet system is an essential component of the Kibble balance, a device that is used to realize the unit of mass. It is the source of the magnetic flux, and its importance is captured in the geometric factor $Bl$. Ironically, the $Bl$ factor cancels out and does not appear in the final Kibble equation. Nevertheless, care must be taken to design and build the magnet system because the cancellation is perfect only if the $Bl$ is the same in both modes: the weighing and velocity mode. This review provides the knowledge necessary to build a magnetic circuit for the Kibble balance. In addition, this article discusses the design considerations, parameter optimizations, practical adjustments to the finished product, and an assessment of systematic uncertainties associated with the magnet system., Comment: 69 pages, 20 figures

The increasing use of renewable generation, power electronic devices, and nonlinear loads in power systems brings more severe interharmonic tones to the measurand, which can increase estimation errors of P-class PMUs, cause misoperation of protection relays, and even threaten the stability of the power systems. Therefore, the performance of the P-class synchrophasor estimator under interharmonic interference should be evaluated and new estimation schemes that can improve the operational robustness are needed for various protection and control applications. In this paper, a synchrophasor estimator design for P-class PMUs that introduces singular value decomposition to the least square algorithm based on the Taylor series is proposed. By constructing an optimization with proposed adjustable parameters, finite impulse response filters with high transition band attenuation are designed. Numerical and experimental tests verify the proposed dynamic synchrophasor estimator performance and show an effective improvement in rejecting interharmonic tones, especially when a short time window and light computational burden are required., Comment: 10 pages, 10 figures, submitted to IEEE Access

As the complexity increases in modern power systems, power quality analysis considering interharmonics has become a challenging and important task. This paper proposes a novel decomposition and estimation method for instantaneous power quality indices (PQIs) monitoring in single-phase and three-phase systems with interharmonics and transient disturbances. To separate the interharmonic components, a set of new scaling filter and wavelet filter with narrow transition bands are designed for the undecimated wavelet packet transform (UWPT). Further, a two-stage decomposition method for multi-tone voltage and current signals is proposed. The Hilbert transform (HT) is applied to calculate the instantaneous amplitude and phase of each frequency component, which accordingly allows the monitoring of different PQI parameters. Numerical tests are conducted to check the performance of the proposed method. The test results show that compared to other conventional approaches, instantaneous PQIs estimated by the proposed method present significant advances for tracking transitory changes in power systems, and could be considered as a helpful tool for high-accuracy PQ detections., Comment: 13 pages, 8 figures, accepted for publication in IEEE Trans. Instrum. Meas

Employing very simple electro-mechanical principles known from classical physics, the Kibble balance establishes a very precise and absolute link between quantum electrical standards and macroscopic mass or force measurements. The success of the Kibble balance, in both determining fundamental constants ($h$, $N_A$, $e$) and realizing a quasi-quantum mass in the 2019 newly revised International System of Units, relies on the perfection of Maxwell's equations and the symmetry they describe between Lorentz's force and Faraday's induction, a principle and a symmetry stunningly demonstrated in the weighing and velocity modes of Kibble balances to within $1\times10^{-8}$, with nothing but imperfect wires and magnets. However, recent advances in the understanding of the current effect in Kibble balances reveal a troubling paradox. A diamagnetic effect, a force that does not cancel between mass-on and mass-off measurement, is challenging balance maker's assumptions of symmetry at levels that are almost two orders of magnitude larger than the reported uncertainties. The diamagnetic effect, if it exists, shows up in weighing mode without a readily apparent reciprocal effect in the velocity mode, begging questions about systematic errors at the very foundation of the new measurement system. The hypothetical force is caused by the coil current changing the magnetic field, producing an unaccounted force that is systematically modulated with the weighing current. Here we show that this diamagnetic force exists, but the additional force does not change the equivalence between weighing and velocity measurements. We reveal the unexpected way that symmetry is preserved and show that for typical materials and geometries the total relative effect on the measurement is $\approx 1\times10^{-9}$., Comment: Accepted for publication in Scientific Reports

A milestone revision of the International System of Units (SI) was made at the 26th General Conference on Weights and Measures that four of the seven SI base units, i.e. kilogram, ampere, kelvin, and mole, are redefined by fundamental physical constants of nature. The SI base unit founding the electrical measurement activities, i.e. ampere, is defined by fixing the numerical value of the elementary charge to $e=1.602\,176\,634\times10^{-19}$C. For electrical measurement, several major adjustments, mostly positive, are involved in this SI revision. In this paper, the main impacts of the new SI for electrical measurement activities are surveyed under the new framework., Comment: 10 pages, 3 figures, submitted to IEEE Trans. Instrum. Meas

Permanent magnets together with yokes to concentrate the magnetic flux into a cylindrical air-gap are widely employed in Kibble balances. These experiments require a uniform magnetic flux density along a vertical path, typically a substantial fraction of the length of the air-gap. Fringe fields that are present at both ends of the air-gap limit the region where the flux density does not change more than a certain relative fraction (here: $5\times 10^{-4}$) of the flux density in the center of the magnet system. By simply adding an iron ring with a rectangular cross-section to the inner yoke at each end of the air gap, the effects of the fringe fields can be counteracted, and, hence, the length of the region, where the flux density remains within a given tolerance band is increased. Compared to the alternative, employing a taller magnet, the proposed method yields a magnet system with an extended region of a uniform field without significantly increasing the mass of the magnet system. Potential applications include compact and table-top Kibble balances. We investigate possible adverse effects on the performance of the magnet system caused by the additional rings: magnetic field strength, coil-current effect, and a dependence of the radial field on the radial position in the field. No substantial disadvantage was found. Instead, the method presented here outperformed previously suggested methods to improve the radial dependence of the radial field, e.g., shorter outer yoke. In summary, adding rings to the inner yoke improves the uniformity of the field without a detrimental effect to function, cost, and form factor of the magnet system., Comment: 10 figures, accepted by IEEE Trans. Instrum. Meas

Yoke-based permanent magnetic circuits are widely used in Kibble balance experiments. In these magnetic systems, the coil current, with positive and negative signs in two steps of the weighing measurement, can cause an additional magnetic flux in the circuit and hence a magnetic field change at the coil position. The magnetic field change due to the coil current and related systematic effects have been studied with the assumption that the yoke material does not contain any magnetic hysteresis. In this paper, we present an explanation of the magnetic hysteresis error in Kibble balance measurements. An evaluation technique based on measuring yoke minor hysteresis loops is proposed to estimate the effect. The dependence of the magnetic hysteresis effect and some possible optimizations for suppressing this effect are discussed., Comment: 10 pages, 9 figures, Manuscript of IEEE Trans. Instrum. Meas

Damping devices are necessary for suppressing the undesired coil motions in the watt/joule balance. In this paper, an active electromagnetic damping device, located outside the main magnet, is introduced in the joule balance project. The presented damping device can be used in both dynamic and static measurement modes. With the feedback from a detection system, five degrees of freedom of the coil, i.e. the horizontal displacement $x$, $y$ and the rotation angles $\theta_x$, $\theta_y$, $\theta_z$, can be controlled by the active damping device. Hence, two functions, i.e. suppressing the undesired coil motions and reducing the misalignment error, can be realized with this active damping device. The principle, construction and performance of the proposed active damping device are presented., Comment: Accepted by IEEE Trans. Instrum. Meas., 8 pages, 11 figures

Permanent magnets with yokes are widely used in the watt and joule balances to measure the Planck constant for the forthcoming redefinition of the unit of mass, the kilogram. Recently, a permanent magnet system has been in consideration for a further practice of NIM-2, the generalized joule balance. In this paper, an analytical model to design the permanent magnet system is presented. The presented model can be solved to obtain the preliminary parameters and then is used as guidance for FEA software to optimize the parameters of such magnetic system. As an instance for the application of the designing model, the design of the permanent magnet system for NIM-2 is described and the special design of opening shape makes the misalignment of the top and middle yokes has little influence on the vertical component of the magnetic field., Comment: Accepted for publication in IEEE Trans. Instrum. Meas., 5 pages, 9 figures

The watt balance is an experiment being pursued in national metrology institutes for precision determination of the Planck constant $h$. In watt balances, the $1/r$ magnetic field, expected to generate a geometrical factor $Bl$ independent to any coil horizontal displacement, can be created by a strict two dimensional, symmetric (horizontal $r$ and vertical $z$) construction of the magnet system. In this paper, we present an analytical understanding of magnetic field distribution when the $r$ symmetry of the magnet is broken and the establishment of the $Bl$ conservation is shown. By using either Gauss's law on magnetism with monopoles or conformal transformations, we extend the $Bl$ conservation to arbitrary two dimensional magnetic planes where the vertical magnetic field component equals zero. The generalized $Bl$ conservation allows a relaxed physical alignment criteria for watt balance magnet systems., Comment: 5pages, 6 figures, submitted to MST

A yoke-based permanent magnet, which has been employed in many watt balances at national metrology institutes, is supposed to generate strong and uniform magnetic field in an air gap in the radial direction. However, in reality the fringe effect due to the finite height of the air gap will introduce an undesired vertical magnetic component to the air gap, which should either be measured or modeled towards some optimizations of the watt balance. A recent publication, i.e., {\it Metrologia} 52(4) 445 [1], presented a full field mapping method, which in theory will supply useful information for profile characterization and misalignment analysis. This article is an additional material of [1], which develops a different analytical algorithm to represent the 3D magnetic field of a watt balance magnet based on only one measurement for the radial magnetic flux density along the vertical direction, $B_r(z)$. The new algorithm is based on the electromagnetic nature of the magnet, which has a much better accuracy., Comment: 19 pages, 10 figures, submitted to Metrologia

In the joule balance developed at National Institute of Metrology (NIM), the dynamic phase of a watt balance is replaced by the mutual inductance measurement in an attempt to provide an alternative method for the kg redefinition. But for this method a rather large current in the exciting coil, is needed to offer the necessary magnetic field in the force weighing phase, and the coil heating becomes an important uncertainty source. To reduce coil heating, a new coil system, in which a ferromagnetic material is used to increase the magnetic field was designed recently. But adopting the ferromagnetic material brings the difficulty from the nonlinear characteristic of material. This problem can be removed by measuring the magnetic flux linkage difference of the suspended coil at two vertical positions directly to replace the mutual inductance parameter. Some systematic effects of this magnet are discussed., Comment: 7 figures, 10 pages, accepted for publication in IEEE Transactions on Instrumentation and Measurement

Permanent magnets with high-permeability yokes have been widely used in watt balances for supplying a robust and strong magnetic field at the coil position. Subjected to the mechanical realization, only several $B_r(z)$ (radial magnetic field along the vertical direction) profiles can be measured by coils for field characterization. In this article, we present an algorithm that can construct the global magnetic field of the air gap based on $N(N\geq1)$ additional measurements of $B_r(z)$ profiles. The proposed algorithm is realized by polynomially estimating the $B_z(r)$ function with analysis of basic relations between two magnetic components in air gap, i.e., $B_r(r,z)$ and $B_z(r,z)$, following the Maxwell's equations. The algorithm, verified by FEM simulations, can characterize the three-dimensional contribution of the magnetic field for a watt balance magnet with acceptable accuracy, which would supply basic field parameters for alignment and misalignment corrections., Comment: 15 pages, 7 figures, submitted to Metrologia

The development of the joule balance method to measure the Planck constant, in support of the redefinition of the kilogram, has been going on at the National Institute of Metrology of China (NIM) since 2007. The first prototype has been built to check the feasibility of the principle. In 2011, the relative uncertainty of the Planck constant measurement at NIM is $7.7\times10^{-5}$. Self-heating of the coils, swing of the coil, are the main uncertainty contributions. Since 2012, some improvements have been made to reduce these uncertainties. The relative uncertainty of the joule balance is reduced to $7.2\times10^{-6}$ at present. The Planck constant measured with the joule balance is $h=6.6261041(470)\times10^{-34}$Js. The relative difference between the determined h and the CODATA2010 recommendation value is $5\times10^{-6}$. Further improvements are still being carried out on the NIM-1 apparatus. At the same time, the design and construction of a brand new and compact joule balance NIM-2 are also in progress and presented here., Comment: 15 pages, 14 figures, accepted for publication in IEEE Trans. Instrum. Meas

An inertial mass measurement project, which is expected to precisely measure the Planck constant, $h$, for possible comparisons with known gravitational mass measurement projects, e.g., the watt balance and the Avogadro project, is being carried out at the National Institute of Metrology, China. The principle, apparatus, and experimental investigations of the inertial mass measurement are presented. The prototype of the experiment and the Planck constant with relative uncertainty of several parts in $10^{4}$ have been achieved for principle testing., Comment: 9pages, 12 figures, accepted for publication in IEEE Transactions on Instrumentation and Measurement

The Planck constant $h$ is one of the most significant constants in quantum physics. Recently, the precision measurement of the numeral value of $h$ has been a hot issue due to its important role in establishment for both a new SI and a revised fundamental physical constant system. Up to date, two approaches, the watt balance and counting atoms, have been employed to determine the Planck constant at a level of several parts in $10^8$. In this paper, the principle and progress on precision measurement of the Planck constant using watt balance and counting atoms at national metrology institutes are reviewed. Further improvements for the Planck constant determination and possible developments of a revised physical constant system in future are discussed., Comment: 24 pages, 15 figures, accepted for publication in Chinese Physics B

Watt balances are used to measure the Planck constant and will be used in the future to realize mass at the kilogram level. They increasingly rely on permanent magnet systems to generate the magnetic flux. It has been known that the weighing current might effect the magnetization state of the permanent magnetic system used in these systems causing a systematic bias that can lead to an error in the result if not accounted for. In this article a simple model explaining the effect of the weighing current on the yoke of the magnet is developed. This model leads to a nonlinear dependence of the magnetic flux density in the gap that is proportional to the squared value of the coil current. The effect arises from changing the reluctance of the yoke by the additional field produced by the coil. Our analysis shows that the effect depends on the width of the air gap, the magnetic flux density in the air gap, and the $BH$ curve of the yoke material. Suggestions to reduce the nonlinear effect are discussed., Comment: 16 pages 9 figures. Accepted for publication in metrologia

The magnet system is one of the key elements of a watt balance. For the new watt balance currently under construction at the National Institute of Standards and Technology, a permanent magnet system was chosen. We describe the detailed construction of the magnet system, first measurements of the field profile, and shimming techniques that were used to achieve a flat field profile. The relative change of the radial magnetic flux density is less than $10^{-4}$ over a range of 5 cm. We further characterize the most important aspects of the magnet and give order of magnitude estimates for several systematic effects that originate from the magnet system., Comment: v2: changed layout; 13 pages, 15 figures. published in IEEE Trans. Instrum. Meas. 63, 3027-3028 (2014), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6827217&isnumber=6940066