We describe the fabrication and implementation of a dynamic Kibble balance experiment to enable the measurement of mass without the use of a scale, modifying the experimental design of a recently published approach. In this experiment, the voltage induced by a pellet, consisting of two opposed magnets passing through a coil, is used to determine mass via measurements of the induced voltage, pellet velocity, and the current required to levitate the pellet. In particular, we simplified the electronic timing by the use of two coils separated by a known distance to allow the measurement of velocity without the use of photogates. Thus, the total cost for the equipment fabrication is under $25 USD. For data collection, this equipment is then paired with either an oscilloscope, or a smart pulley system, equipment that is available in most first-year physics laboratories. Finally, we use expressions for the induced electromotive force (EMF) of a magnet falling through a coil to fit the distance dependence of the EMF generated during the velocity mode part of the experiment.

Abstract Tabletop version Kibble balances are a significant developing trend for mass realizations following the revised International System of Units. A key innovation through the miniaturization of the Kibble balance from a large-scale instrument into a tabletop device is making the quantum-based realization of mass accessible to a wider range of calibration laboratories and industries. This paper presents a tabletop Kibble balance design at Tsinghua University targeting E2-accuracy class mass calibrations from 1 g to 1 kg. For calibrating a mass of 1 kg, for instance, the required relative standard measurement uncertainty must be below 0.27 ppm to meet E2-accuracy class. Major components and features of the proposed system are discussed. A novel method of multi-harmonic excitation is proposed to improve the coil-motion linearity during velocity measurement. We show that injecting odd-order harmonics into the motion-driving current can significantly improve the uniformity of the coil’s moving velocity, while the second-order component can address the asymmetry between upward and downward movements. This achieves a flat velocity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>v</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>5</mml:mn> <mml:mi mathvariant="normal">%</mml:mi> </mml:mrow> </mml:math> over 60% of the motion cycle.

Abstract This article presents the development of a novel, active measuring head for coordinate measuring machines that is capable of measuring forces and torques in six degrees of freedom (6 DoF) during tactile sensing. It uses a so-called flotor as carrier of the stylus, which is kept in levitation by six electromagnetic voice coil actuators (VCA) in an hexapod-like arrangement. These VCAs function simultaneously as actuators and as force sensors, which are calibrated in situ according to the Kibble balance principle. Additionally, a position and orientation measurement system is integrated, consisting of three two-dimensional position sensors that can detect the position and orientation of the flotor in 6 DoF. The article describes the basic functionality as well as the most important technical functional elements and key design features. These features aim to optimise the metrological properties of probing force and position measurement in order to achieve the lowest possible measurement uncertainty. Furthermore, the structure of the peripheral electronics system for the controlling of the coordinate measuring head is described. So far, the conceptual and structural design has been implemented, two prototypes have been manufactured and put into operation. At the time of publication, initial metrological investigations are being carried out.

Abstract Kibble balances are complex electromechanical instruments that enable the determination of mass within the SI by linking it to the Planck constant h, the defining constant of the mass unit. The LNE has been developing its own Kibble balance since 2002, with the most recent improvements focusing on the implementation of a contactless linear motor for the dynamic phase and the fine adjustment of the beam’s orientation with respect to the horizontal plane for the static phase. In 2024, two mass calibration campaigns were carried out using the LNE Kibble balance: first with an iridium standard (DB1), and then with a platinum-iridium standard (W1). Both artefacts have a nominal mass of 500 g, and their masses were determined with relative standard uncertainties of 3.1·10⁻⁸ and 3.5·10⁻⁸ respectively (k = 1).

The 2019 redefinition of the International System of Units (SI) established the kilogram based on the Planck constant, eliminating reliance on a physical artefact and significantly enhancing mass metrology. The Kibble balance is one of the key experiments enabling this realization, linking mass measurements to fundamental electrical standards. This work presents a cost-effective Kibble balance prototype developed at Inmetro's Ultrasound Laboratory (Labus), designed using 3D printing and controlled via a Raspberry Pi microcontroller. Experimental results demonstrate the feasibility of using simplified Kibble balances for metrological applications, achieving mass measurements from 100 mg to 600 mg with a relative uncertainty of 7 %. Future developments will focus on sensor calibration and mechanical stability improvements to further enhance accuracy and precision.

Abstract Temperature changes affect the coercivity of permanent magnets, thereby impacting the Bl factor and potentially introducing systematic errors in Kibble balance measurements. While the thermal-magnetic effect is negligible in large magnet systems, it increases substantially as the magnet size decreases, posing an engineering challenge for tabletop Kibble balance systems. We discuss the mechanism of thermal-magnetic effects through finite element analysis, which has not been sufficiently emphasized in previous studies. A bifilar-coil power regulator is proposed to eliminate thermal-magnetic errors in Kibble balances. The approach aims to keep the power of the internal heating source—coil ohmic power—constant over time, allowing the Bl drift to be mitigated through ABA or ABBA measurement sequences. Experimental results obtained from a small magnet with an outer diameter of 40 mm validate the proposed approach. The findings demonstrate that in air, the thermal effect under the new scheme is reduced to 0.9% of that observed in the conventional two-mode, two-phase measurement scheme and to 13.3% of that in the traditional one-mode, two-phase scheme. In a vacuum with air pressure approximately 0.02 Pa, these reduction numbers are 0.04% and 20.6%, respectively. The proposed approach can eliminate the influence of thermal-magnetic effects on the measurement results, thus further breaking down the limitations on the minimum size of tabletop Kibble balances.

Experimental verification of coil motion misalignment insensitivity in a Kibble balance

Abstract The Planck-Balance (PB) is a compact Kibble balance, designed for the realization of the unit of mass at values much smaller than 1 kg across multiple orders of magnitude. Recent advancements enable the PB at PTB to measure mass with uncertainties equivalent to calibration accuracy classes E1 (up to 5 g), E2 (up to 50 g), and F1 (up to 100 g), while operating in air. Type B uncertainties limit the relative combined standard uncertainty for a mass of 100 g to the minimum achievable value of 4.5 ⋅ 10 − 7 . A measurement campaign was conducted using six mass standards (1 mg, 10 mg, 100 mg, 1 g, 10 g, and 100 g), yielding results consistent with the values specified in their E1 calibration certificates. The presented results demonstrate the feasibility of metrologically traceable small mass measurements without reliance on mass standards. Unique design features of the PB will be discussed, including a rigidly mounted coil that minimizes geometrical alignment requirements, and an oscillatory velocity mode that achieves a high signal-to-noise ratio, even with a movement range of less than 0.2 mm. This approach could offer an alternative to calibration and dissemination using mass comparators.

The magnet-moving measurement scheme in Kibble balances avoids displacing force-sensitive components, such as the weighing cell, and enables a broader magnetic profile measurement range during the velocity phase. However, this mechanism introduces the risk of asymmetry in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bl</i> measurement due to external magnetic flux, leading to a potential systematic error in the final measurement results. Using the Tsinghua tabletop Kibble balance magnet as a case study, this paper investigates the error mechanism through finite element analysis and experimental investigations. An evaluation method combining external weak-field measurements with attenuation factor analysis is proposed to assess external magnetic flux errors in magnet-moving measurement schemes. The findings demonstrate that selecting an optimal weighing position can reduce the far-end flux effect to the order of 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−9</sup>. In contrast, the near-end flux effect can be quantified by monitoring the magnetic field surrounding the magnet system. In the Tsinghua Kibble balance system, we show that with proper control of external flux sources, the relative error can be reduced below 1 × 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−8</sup> without requiring additional magnetic shielding.

With the adoption of the revised International System of Units (SI), the Kibble balance has become a pivotal instrument for mass calibrations against the Planck constant, h. One of the major focuses in the Kibble balance community is prioritizing experiments that achieve both high accuracy and compactness. The Tsinghua tabletop Kibble balance experiment seeks to develop a compact, high-precision, user-friendly, cost-effective, and open-hardware apparatus for mass realization, specifically within the kilogram range. This paper reports on the progress of the Tsinghua tabletop Kibble balance project over the past two years. Various aspects of the Tsinghua tabletop system, including electrical, magnetic, mechanical, and optical components, are summarized. Key achievements, such as the construction and characterization of the magnet system, determination of absolute gravitational acceleration, investigation of a capacitor-sensor-based weighing unit, and development of a high-precision current source, are presented to provide a comprehensive understanding of the experiment’s status.

The Kibble balance realizes the kilogram by linking mechanical and electrical quantities via a magnet system. In an improved BIPM-type magnet design by Tsinghua University, an open/close surface was incorporated, facilitating operation. However, an unavoidable mechanical air gap at the splitting plane introduces asymmetry in the magnetic flux density profile, degrading field uniformity. This study proposes a two-step yoke compensation method to restore symmetry by adjusting the upper outer yoke’s inner radius and the splitting gap height. Finite element simulations show linear relationships between asymmetry and these parameters, enabling predictive compensation. Experimental results confirm that sequential tuning successfully eliminates asymmetry and recovers the designed uniform field range. The method provides an effective solution for enhancing magnetic field quality in openable Kibble balance magnets.

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 quasiquantum 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-8}$ </tex-math></inline-formula> level.

In a Kibble balance experiment, the mass value is determined by calculating the geometric factor of the magnetic circuit - coil set up. This factor is calculated using two different interleaved methods: the weighing phase and the dynamic phase. This paper presents improvements in the noise measurement of the geometric factor in both the dynamic and static phases from 2017 to 2024, ultimately reducing the noise associated with the determination of the value of a mass of the LNE Kibble balance to a value of 4·10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−8</sup> for an integration time of 1 day (a 6 fold improvement with respect to 2017).

Abstract A uniform magnetic flux density that is effective in the movement range of the coil is essential for accurate Kibble balance experiments. By utilizing Hopkinson’s law and Kirchhoff’s circuit laws, basic formulas have been derived, providing a method to calculate the magnetic flux density distribution in the airgap of a cylindrical magnet system. A parabolic outer contour of the inner yoke has been found to be a suitable solution to achieve uniformity. Experiments show, that this approach results in a relative change of magnetic flux density in the order of 3 ⋅ 10 − 4 per 8 mm movement range, using a magnet system with a mass of only 2.3 kg. Therefore, the system will be integrated into the upgraded version of PTB’s Planck-Balance—a compact variant of a Kibble balance—aiming for sub 1 ⋅ 10 − 6 accuracy level determination of the geometric factor. The solution described provides a comparatively easy means to design a cylindrical magnet system, using only one permanent magnet disc, without the use of complex simulation software.

We studied the anelastic aftereffect of a flexure being used in a Kibble balance, where the flexure is subjected to a large excursion in velocity mode after which a high-precision force comparison is performed. We investigated the effect of a constant and a sinusoidal excursion on the force comparison. We explored theoretically and experimentally a simple erasing procedure, i.e. bending the flexure in the opposite direction for a given amplitude and time. We found that the erasing procedure reduced the time-dependent force by about 30%. The investigation was performed with an analytical model and verified experimentally with our new Kibble balance at the National Institute of Standards and Technology employing flexures made from precipitation-hardened Copper Beryllium alloy C17200. Our experimental determination of the modulus defect of the flexure yields 1.2×10−4 . This result is about a factor of two higher than previously reported from experiments. We additionally found a static shift of the flexure’s internal equilibrium after a change in the stress and strain state. These static shifts, although measurable, are small and deemed uncritical for our Kibble balance application at present. During this investigation, we discovered magic flexures that promise to have very little anelastic relaxation. In these magic flexures, the mechanism causing anelastic relaxation is compensated for by properly shaping and loading a flexure with a non-constant cross-section in the region of bending.

A new Kibble balance is being built at the National Institute of Standards and Technology (NIST). For the first time in one of the highly accurate versions of this type of balance, a single passive flexure mechanism is used for both modes of operation: the weighing mode and the velocity mode. The mechanism is at the core of the new balance design as it represents a paradigm shift for NIST away from using knife edge-based balance mechanisms, which exhibit hysteresis in the measurement procedure of the weighing mode. Mechanical hysteresis may be a limiting factor in the performance of highly accurate Kibble balances approaching single digit nanonewton repeatability on a nominal 100g mass, as targeted in this work. Flexure-based mechanisms are known to have very good static hysteresis when used as a null detector. However, for larger and especially longer lasting deformations, flexures are known to exhibit anelastic drift. We seek to characterize, and ideally compensate for, this anelastic behavior after deflections during the velocity mode to enable a 10−8 accurate Kibble balance-measurement on a nominal 100g mass artifact with a single flexure-based balance mechanism. A measurement of the anelastic after-effect after static excitation hints that the apparatus produced a result for anelastic relaxation comparable to previously published work. Furthermore, a series of oscillatory displacements similar to those occurring in a velocity mode of a Kibble balance measurement are imposed upon the flexure mechanism and show a significant anelastic relaxation torque resulting in multiple micronewton of force relaxation. The amplitude of this force relaxation could be reduced by counterbending the flexures before performing a force measurement.

We investigate the bias due to the beam divergence of the collimator in a free-fall absolute gravimeter of type FG5X. First, we measure the beam parameters with a Shack-Hartmann sensor. Then, we use the parameters to simulate the relative gravitational acceleration error of an FG5X gravimeter, which employs an unbalanced Mach-Zehnder laser interferometer. This investigation we do with four different commercial collimators, providing different divergence angles. We compare the results to real gravity measurements using the same collimators. The larger the divergence angle, and the bigger the relative length error, the bigger is the bias in the gravity measurements. A good agreement between theory and experiment is found, resulting in a relative bias of −2.77(24)⋅10−9 ( −2.72(24) μGal) for our standard collimator of type Thorlabs TC25APC, which is usually used for free-fall acceleration determinations. The outcome is also important for the realization of the SI unit kilogram via Kibble balance experiments that, on one side, employ laser interferometers for velocity measurements, and, on the other side, require accurate values of the gravitational acceleration. For example, if this divergence error is not corrected in the Kibble balance, then the mass determination would be biased by 2.77(24) μg kg−1 (numbers are valid only for our gravimeter with our collimator and fiber).

A high-precision current source, capable of supporting weighing measurements with a relative uncertainty at the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-9}$ </tex-math></inline-formula> 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 article, we introduce a digital-feedback two-stage current source (DTCS) 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 that the proposal can offer an easy realization of a current source with nA/A stability to support Kibble balance measurements.

The Kibble balance requires a measurement of the local gravitational acceleration, g, with a typical relative measurement uncertainty of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-9}$ </tex-math></inline-formula>. In this article, the determination of g for the Tsinghua tabletop Kibble balance is presented. A polynomial fitting method is proposed for referenceless transfers of the absolute gravitational acceleration using relative gravimeters, showing agreement with the value obtained by the tide correction within a few parts in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{9}$ </tex-math></inline-formula>. Horizontal gravity gradient (HGG) and vertical gravity gradient (VGG) are extracted by mapping the gravity distribution at different heights. The self-attraction effect of major components in the experiment and some time-varying systematic effects are modeled. The final determination of the gravitational acceleration at the mass position, with an uncertainty of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5.4~\mu $ </tex-math></inline-formula>Gal (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k=2$ </tex-math></inline-formula>), is achieved for the Tsinghua tabletop Kibble balance experiment.

This work shows the results obtained by Inmetro’s mass lab implementation of a simple Kibble balance approach for demonstration of its mass measurement principle based on the Planck’s constant. It is based on some modifications of Mimimal Watt Balance work reducing the required electrical measurements by using two loudspeakers mechanically coupled so that one of them acts as a linear actuator in order to perform BL measurements in the moving phase. The mean BL value obtained in weighing phase was 2.66 N/A and in moving phase 2.67 N/A. The maximum error of mass values measured for standard weights in the range from 5 g to 50 g was lower than 5%.