An Investigation of Magnetic Hysteresis Error in Kibble Balances Shisong Li, Senior Member, IEEE, Franck Bielsa, Michael Stock, Adrien Kiss and Hao Fang

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An Investigation of Magnetic Hysteresis Error in Kibble Balances Shisong Li, Senior Member, IEEE, Franck Bielsa, Michael Stock, Adrien Kiss and Hao Fang An Investigation of Magnetic Hysteresis Error in Kibble Balances Shisong Li, Senior Member, IEEE, Franck Bielsa, Michael Stock, Adrien Kiss and Hao Fang Abstract—Yoke-based permanent magnetic circuits currents (masses). Experimental measurements of Kibble are widely used in Kibble balance experiments. In these balances at the National Research Council (NRC, Canada) magnetic systems, the coil current, with positive and and the National Institute of Standards and Technology negative signs in two steps of the weighing measurement, can cause an additional magnetic flux in the circuit (NIST, USA) yielded a nonlinear current term with a few 9 and hence a magnetic field change at the coil position. parts in 10 in their systems [17], [18]. In [19], it was shown The magnetic field change due to the coil current and that the linear current effect is mainly contributed by the related systematic effects have been studied with the coil inductance change at different vertical positions, and assumption that the yoke material does not contain a significant linear magnetic profile change, proportional to any magnetic hysteresis. In this paper, we present an explanation of the magnetic hysteresis error in Kibble the coil current, has been experimentally observed. Further, balance measurements. An evaluation technique based different profile changes in weighing and velocity phases on measuring yoke minor hysteresis loops is proposed were experimentally measured in one-mode schemes [20]. to estimate the effect. The dependence of the magnetic The linear change of the magnetic profile can cause a bias hysteresis effect and some possible optimizations for when there is a coil vertical position change during two steps suppressing this effect are discussed. of weighing, i.e. mass-on and mass-off. This bias, which is Index Terms—Kibble balance, magnetic circuit, BH also closely related to parameters of the magnet system, in hysteresis loop, measurement error. general, should be carefully considered in the measurement. So far, all studies of the current effect are made based I. Introduction on an assumption that the yoke has a fixed magnetization The Kibble balance, formerly known as the watt balance curve and does not contain any hysteresis. In reality, the [1], is one of the major instruments for realizing the unit of yoke used to build the Kibble balance magnet, as reported mass, i.e. the kilogram, in terms of the Planck constant h in [21], has a considerable magnetic hysteresis. The exci- under the revised International System of Units (SI) [2]. tation (current) change during the weighing measurement A Kibble balance establishes a relationship between the can shift the yoke working point and hence could bring a mass of an artefact and the Planck constant by comparing magnetic field change at the coil position compared to the mechanical power to electrical power, which can be viewed field measured in the velocity phase. In this paper, theoret- as a bridge linking classical and quantum mechanics [3]. ical analysis and experimental investigations are presented Detailed principles and descriptions of a Kibble balance can to build an evaluation technique for potential errors caused be found in recent review papers, e.g. [4]. by the magnetic hysteresis. The analysis assumes that the In Kibble balance experiments, a sub-Tesla magnetic magnet yoke is left in a magnetized state at the end of field with a good vertical uniformity is required. After the weighing phase. We note this magnetization state is many years of optimization and practice, all the ongoing not unavoidable. It can practically be erased by applying a Kibble balances in the world have chosen to use yoke- decaying oscillatory waveform to the coil current at the end based permanent magnetic circuits [5]–[14]. One of the main of the weighing phase, as implemented in the NPL (National effects of such a magnet system is the coil-current effect, i.e. Physical Laboratory, UK) and NRC Kibble balances [5], the coil current interacts with the main magnetic circuit and [6]. The remainder of the paper is organized as follows: In can cause a change in the magnetic field at the measurement section II, a theoretical analysis of the magnetic hysteresis arXiv:1912.06694v2 [physics.ins-det] 17 Dec 2019 position. effect is presented. In section III, an experimental example Theoretical and experimental studies have been made on is taken for an estimation of the magnetic hysteresis error. the current effect: In [15], [16], the static nonlinear effect Some discussions on the dependence of the effect, as well as was studied, and the main static non-linearity was found to possible ways to suppress the hysteresis error, are summa- be due to the yoke magnetic status change in the weighing rized in section IV. measurement. The effect was evaluated to be small com- pared to a typical Kibble balance measurement uncertainty, 8 II. Theoretical analysis e.g. 2 10− . The nonlinear current effect can be pre- cisely determined× by running the experiment with different A. Overview of the analysis The main purpose of this paper is to establish the Manuscript of IEEE Trans. Instrum. Meas. All authors are with the Bureau International des Poids et Mesures (BIPM), 92312 S`evres, relationship of magnetic flux density change at the coil France. Email: leeshisong@sina.com; fang@bipm.org. position, ∆Ba, due to the coil current I and the yoke BH related systematic errors, e.g. the magnetic field difference due to the current asymmetry of mass-on and mass-off [20]. Note that in an up-down symmetrical magnetic circuit design, the field extreme point locates at the vertical center of the air gap, i.e. z = 0, because, in this plane, the SmCo magnetic flux contains a purely horizontal component, i.e. Baz = 0. Note that in reality, the magnetic center could be shifted by non-ideal construction of the magnet, e.g. the asymmetry of the magnetization, mechanical assembly, etc, but the hysteresis error, which is the focus of this paper has a weak dependence on the field flat point in the central region and can be similarly analyzed. Without losing generality, in the following analysis, we assume that the weighing position is z = 0, and in this case (as shown in Fig. 1. The magnetic flux distribution in the air gap region. The blue Fig. 1) the coil flux has a symmetrical up-down distribution and red dashed lines denote respectively the SmCo flux and the coil in both inner and outer yokes. Since only the horizontal flux. Different colors (yellow, blue) marked in the yoke-air boundaries component of the magnetic flux density can generate a denote opposite signs of magnetic flux density change. vertical force, in the following discussion, the magnetic flux density denotes simply the horizontal component. Also, hysteresis. The analysis begins with a conventional two- in the following analysis, we take a typical cycled Kibble mode, two-phase measurement scheme, and is divided into balance measurement sequence, VW1W2V (V denotes the three independent steps: velocity measurement, W1 and W2 the weighing measure- 1) Linking the magnetic flux density change at the coil ments with different current polarities), as an example. Other measurement sequences can be similar analyzed. position ∆Ba to the magnetic field change in the yoke In the velocity measurement, there is no current in the ∆By, i.e. function ∆Ba = 1(∆By). 2) Modeling the magnetic fluxA density change of the coil and the magnetic flux density at the coil position is Bav. The BH working points at inner/outer yoke boundaries yoke ∆By, via yoke BH minor hysteresis loops, as a function of the magnetic field change in the yoke are respectively, (Hyiv;Byiv) and (Hyov;Byov). In the first weighing step, i.e. I = I (mass-on), the flux produced ∆Hy. i.e. ∆By = 2(∆Hy). + 3) Expressing the magneticA field change in the yoke, by the current shifts the magnetic flux density of inner and outer yokes to B and B , and in the second step ∆Hy, as a function of the coil current in the weighing yi+ yo+ phase, i.e. ∆H = (I). I = I (mass-off), the magnetic flux density in the inner y A3 − Using the three steps above, the magnetic field change at and outer yokes is changed to Byi and Byo . It is known that in such a magnetic− circuit,− the horizontal the coil position can be modeled in terms of coil current magnetic flux density in the air gap follows approximately in the weighing measurement, and the magnetic hysteresis a relationship, where is the radius of the focused effect can be evaluated accordingly. 1=r r position in the air gap [22]. In a typical Kibble balance magnet, the air gap width is usually much smaller than B. Linking ∆Ba to ∆By its radius, and hence the coil field gradient in the air gap In Kibble balances, the general idea of a yoke-based can be approximately considered linear along r direction. magnetic circuit, e.g. the BIPM-type magnet [21], is to This approximation allows us to write the magnetic field at compress the magneto-motive force (mmf) in a narrow air the coil position as a function of the inner and outer yoke gap formed by inner and outer yokes so that the magnetic boundaries as field generated is strong and uniform. In the weighing ∆B + ∆B ∆B yi yo ; (1) measurement, the coil is placed in the air gap with a current, a ≈ 2 and the newly created magnetic flux of the coil remains where ∆Byi and ∆Byo denote the magnetic flux density within the path composed of the air gap (goes through change respectively at the inner and outer yoke boundaries.
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