Design of a Miniature Kibble Balance for Kilogram-Scale Mass Calibration -- KBmini

TL;DR

This work presents KBmini, a tabletop Kibble balance designed to realize kilogramscale masses with E2-class accuracy in air. The design features a compact BIPM-type magnet delivering ~0.6 T in a well-defined air gap and a high-turn coil to achieve Bl ≈ 1400 Tm, paired with capacitive weighing, interferometric metrology, and a TMTP two-phase scheme to realize precise mass comparisons. A key innovation is multi-harmonic excitation: injecting odd harmonics into the drive current reshapes the coil velocity toward a near-square profile, with a second-harmonic correction addressing waveform tilt, yielding Δv/v < 5% over more than 60% of the motion cycle. These advances, along with a compact, integrated subsystem design, aim to make quantum-based mass realization accessible to a broader set of laboratories and industries, while highlighting necessary mitigations for thermal-magnetic and current-related biases and outlining future uncertainty evaluations.

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 $Δv/v< 5\%$ over 60% of the motion cycle.

Figures (8)

• Figure 1: The relative measurement uncertainty target of KBmini. The black curves show the standard relative measurement uncertainty of mass calibrations for E1, E2, and F1 classes. The dashed blue line is the velocity phase measurement uncertainty, $5\times10^{-8}$, not depending on the mass measurement range. The magenta curve is the weighing phase uncertainty. The current source has three ranges with matching sampling resistors, covering measurement ranges of respectively [1 g,12 g], [12 g, 120 g], and [120 g, 1000 g]. The uncertainty steps are range boundaries, and in each range, the weighing resolution is fixed, and hence the weighing uncertainty increases as the mass value scales down.
• Figure 2: The CAD model of KBmini.
• Figure 3: Measurement scheme of KBmini. The weighing measurement consists of two steps: mass-off and mass-on. In the mass-off step, with both the counterweight and test mass unloaded, a downward electromagnetic force $F_M = BlI_-$ balances the upward unbalanced force from counterweight $F_C$, establishing equilibrium: $-F_C + F_M = 0$. During the mass-on step, the test mass is loaded and the coil current is reversed, resulting in a new force balance: $-F_C + F_G -F_M = 0$, where $F_G = mg$. Subtracting the mass-off equation from the mass-on equation yields the expression $Bl = \frac{mg}{I_+ - I_-}$. In the subsequent velocity phase, the counterweight is loaded while the test mass is unloaded; the counterweight is chosen to exert a force $-F_C$ such that the total static force remains zero.
• Figure 4: Magnet-coil system design of KBmini. (a) presents the magnet's structure, with arrows indicating the magnetization direction. The red circled area indicates a ring-shaped compensation structure on both ends of the magnet. (b) depicts the system in an open, magnetically levitated state. (c) compares the measured magnetic flux density profile with results from a finite element analysis (FEA). (d) shows how the profile shape varies with different coil heights. (e) displays the magnetic profile within the velocity measurement range for a selected coil height of $h_c=30$ mm.
• Figure 5: Current effect of the KBmini coil obtained by FEA calculations. (a) displays the inductive force as a function of vertical position $z$, expressed as $\Delta F = \frac{I^2}{2} \frac{\partial L}{\partial z}$. The coil configuration corresponds to 24.5 ampere-turns, equivalent to the operating condition for a 1 kg standard mass load with $I = 3.5$ mA and $N = 7000$ turns. Although the FEA was performed in $\pm 5$ mm, as the coil moves in the sub-millimeter range, the linear fit (the dashed line) was conducted in the range of $|z|\leq1$ mm. (b) shows the residual inductive force after subtracting a linear component fitted within the $\pm 1$ mm range. (c) presents the relative magnetic field variation along $z$ for both positive and negative coil currents $\pm I$.