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

Home , Magnetic hysteresis

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 compared 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 Sèvres, relationship of magnetic flux density change at the coil France. Email: [email protected]; [email protected]. position, ∆Ba, due to the coil current I and the yoke BH SmCo flux 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 Coil flux Coil 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 Inner yoke Outer yoke 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. twice) and inner/outer yokes. Fig. 1 presents the magnetic Note that (1) is not accurate for absolute field calculation, flux distribution created by the SmCo magnets and the but it is good enough for modeling the field change where coil current in the air gap region. The total magnetic field the total effect is small (verified in section II-D). will increase when the coil flux has the same direction as Since the normal component of magnetic flux density is the SmCo flux. Otherwise, when two flux directions are continuous across the yoke-air interface, the magnetic flux opposed, the magnetic field in the air gap is reduced by density at the coil position in two steps of the weighing the coil flux. measurement can be written as In a Kibble balance, the weighing position is usually set (B B ) + (B B ) at an extreme point (which has the flattest profile, i.e. B = B + yi+ − yiv yo+ − yov a+ av 2 ∂Bar/∂z = 0, where Bar is the radial magnetic flux density (Byi Byiv) + (Byo Byov) in the air gap, and z is the vertical axis) in order to minimize Ba = Bav + − − − − (2) − 2 Eq. (2), by combining with the weighing equations, ∆B mg m g B I = − c a+ + L mcg Ba I = − (3) C W1+ − − L where L denotes the coil wire length, m the mass of V ∆H the artefact, mc the counter mass (including the reading W2 of the weighing cell or a mass comparator), g the local A − gravitational acceleration, gives the effective magnetic field density seen by the coil in the weighing phase as B W1 mg − Baw = (I+ I )L W2+ − − (Byi+ Byiv)I+ (Byi Byiv)I (a) = Bav + − − − − − 2(I+ I ) − − ∆B (Byo+ Byov)I+ (Byo Byov)I + − − − − − . (4) 2(I+ I ) E − − Conventionally, as the currents in the weighing phase are set C symmetrically, i.e. , then (4) can be simplified I+ = I = I D as − − V ∆H

Baw Bav (Byi+ Byiv) + (Byi Byiv) − = − − − A Bav 4Bav (Byo+ Byov) + (Byo Byov) + − − − (5) 4Bav Note that in Kibble balances, only the magnetic field change between two measurement phases, as presented in (5), can introduce a measurement error. Eq. (5) links the magnetic (b) flux density change at the coil position to the magnetic Fig. 2. Magnetic status change of the yoke in the weighing phase. flux density change at the inner/outer yoke boundaries. (a) shows the yoke BH working point change due to the coil flux When the yoke field change is a linear function of the during mass-on and mass-off. In (b), the linear component of the minor loop is removed, yielding a flat normalized minor loop. Note that ∆By current, i.e. Byi+ Byiv = (Byi Byiv), Byo+ Byov = and ∆H are defined as the magnetic flux density and magnetic field − − − − − y (Byo Byov), the magnetic field change at the coil change compared to the yoke status in the velocity measurement, i.e. − − − ∆By = By Byv, ∆Hy = Hy Hyv. position is averaged out. However, the yoke magnetic flux − − density change as a function of the coil current (or H field change) is not linear (even without hysteresis), and the upper yoke is shifted to W following the blue minor loop, residual nonlinear term should be evaluated. 1+ and the lower yoke BH working point moves to W2+ along the magenta loop. With the negative current (mass-off), the C. and minor hysteresis loops ∆By(∆Hy) BH upper yoke is working at W1 and the lower yoke would be − In the Kibble balance magnet, a good design would keep at W2 . − the yoke permeability, µ, at or close to the maximum point Based on (5), the average magnetic flux density change of the µ(H) curve, where the B(H) curve has a large static for the inner yoke can be written as slope B/H. When the yoke magnetic field H is slightly (Byi+ Byiv) + (Byi Byiv) shifted by the coil flux in the weighing phase, a minor BH − − − loop, - - - , is Byi1+ + Byi2+ Byi1 + Byi2 (Hyv,Byv) (Hy+,By+) (Hy ,By ) (Hyv,Byv) = Byiv + − − Byiv formed. Fig. 2(a) presents an example− − of the minor BH 2 − 2 − hysteresis loop in the weighing measurement. The shape Byi1+ + Byi1 Byi2+ + Byi2 = − B + − B (6). of the minor loop can be different but the global slope of 2 − yiv 2 − yiv the loop is always positive (determined by the differential On the right side of (6), the first term denotes the non- permeability µ ). For either the inner or outer yoke, the d linearity of H increasing curve (dH/dt > 0) and the coil flux with mass-on will increase the horizontal magnetic second term presents the nonlinearity of H decreasing curve field in one half of the yoke boundary, while it will decrease (dH/dt < 0). (Byi1+ + Byi1 )/2 is the averaged magnetic the field of the other half of the yoke boundary by the same − flux density of W1+ and W1 , which equals the magnetic amount. For example, the magnetic field in the upper half − flux density at point A. (Byi2+ + Byi2 )/2 is the magnetic of the yoke boundary is shifted by , while the field of − ∆ flux density at point B. Then (6) can be rewritten as the lower half is shifted by ∆ . AsH shown in Fig. 2(a), − H with plus current (mass-on), the BH working point of the (Byi+ Byiv) + (Byi Byiv) = (AV + BV ), (7) − − − − where AV and BV denote the line lengths of AV and BV . ∆Hy (mass off) Since the two minor loops in Fig. 2(a) have the same shape, H and hence BV = AC. Then (7) is simplified to z (Byi+ Byiv) + (Byi Byiv) = (AV + AC). (8) − − − − Eq. (8) can be simplified by normalizing the hysteresis ∆Hy (mass on) curves as shown in Fig. 2(b), i.e. two end points of the Coil region original minor loop are rotated to be aligned with the Hysteresis effect ∆Hy axis. It can be mathematically proven (as shown in B Appendix) that z (AV + AC) = (VD + VE) = (∆ 2), (9) − − F H where denotes a function related only to the even-order Fig. 3. The magnetic field change due to the coil current and its related terms ofF yoke magnetic field change due to the coil current magnetic hysteresis effect. At the coil center z = 0 mm, the ∆Hy value is zero and the magnetic hysteresis error at this point is also zero. ∆ . When the minor loop is flat, we can rewrite (6) as Above or below the coil, both ∆Hy = ∆ and the hysteresis effect H reach maxima. The hysteresis| effect| seenH by the coil should be the (Byi+ Byiv) + (Byi Byiv) average in the coil region. − − − = ∆ ∆ , (10) − Bi|dH/dt>0 − Bi|dH/dt<0 where ∆ i dH/dt<0, ∆ i dH/dt>0, as shown in Fig. 2(b), To evaluate the hysteresis effect, the real magnetic field denote theB | magnetic fluxB | density change on normalized H distribution along the vertical direction is required. A study decreasing and H increasing curves at ∆Hy = 0, i.e. the of the field change based on the finite element analysis magnetic flux density at points D and E respectively. (FEA) is presented [23]. A similar analysis can be applied to the outer yoke. The As shown in Fig. 3, the magnetic flux density change magnetic flux density change in the yoke boundary due to caused by the coil current is a step function along the ver- the magnetic hysteresis is, tical direction: At two ends, the field change stays constant (opposite directions) and in the coil region, the magnetic (Byo+ Byov) + (Byo Byov) − − − field change is a linear function of the coil vertical position = ∆ ∆ . (11) z. This result can be easily modeled by Ampere’s law: Above − Bo|dH/dt>0 − Bo|dH/dt<0 or below the coil, the ampere-turns of the coil, i.e. NI (N Substituting (10) and (11) into (5) yields is the total number of the coil winding), is fixed. In the Baw Bav coil region, the ampere-turns are NIz/h , where h is the − c c Bav half height of the coil. The main mmf drop is horizontally ∆ i dH/dt>0 + ∆ i dH/dt<0 along the air gap (twice), then the magnetic flux density = B | B | produced by the coil current at the air gap center − 4Bav r = rc can be calculated as ∆ o dH/dt>0 + ∆ o dH/dt<0 B | B | (12)  − 4B µ0NI av  sign(z), z hc  2δ | | ≥ Since the BH working point of inner and outer yoke bound- ∆B = (14) aries is not far separated (magnetic flux density difference  µ0NIz < 0.1 T), (12) can be approximated as  , z < hc 2δhc | | Baw Bav where is the width of the air gap, the permeability of − δ µ0 Bav vacuum, sign(z) the sign of z. This calculation is checked ∆ + ∆ using an FEA example. In the simulation, an air gap with = B|dH/dt>0 B|dH/dt<0 , (13) − 2Bav similar parameters of the BIPM Kibble balance magnet is used: The radii of the inner and outer yoke boundaries are where ∆ is the yoke magnetic flux density change (normal- r = 118.5 mm and r = 131.5 mm. The height of the air gap ized curve)B with B (B + B )/2. In order to evaluate i o y yi yo is 80 mm. The air gap width is δ = 13 mm. The coil, 10 mm the hysteresis effect,≈ the minor BH loops and the yoke in width and 20 mm in height (h = 10 mm), is placed at the magnetic field change due to the current, i.e. ∆H, should c geometrical center of the air gap, i.e. mm and be known. rc = 125 z = 0 mm. The ampere-turns of the coil, NI, are 14 A, which can generate 4.9 N magnetic force (corresponding to the weight D. Yoke magnetic field change due to coil current of a 500 g mass). The magnetic profile change due to the coil flux in Fig. 4 compares the magnetic flux density distribution the weighing phase has been studied in [20]. The slope obtained by FEA and an analytical model (14). The result of the magnetic profile change can be measured directly produced by the analytical model agrees well with the FEA by the force-current ratio at different positions or by the calculation. Fig. 4 also presents the magnetic flux density voltage-velocity ratio with a moving current-carrying coil. distribution at the inner and outer yoke boundaries, ri and 3 10− region. If the magnetic flux density change in the yoke 1 · ∆ is described by polynomial forms of ∆ 2 as B H X i ∆ = κi∆ , (17) 0.5 B H i=2,4,... I (r ,FEA) I (r ,FEA) + c + i a following gain factor should be added to the maximum I (rc,FEA) I (ri,FEA) /T 0 − − effect (at z = hc) due to the average in the coil range, i.e. B I (r ,model) I (r ,FEA) ±

∆ + c + o

I (rc,model) I (ro,FEA) Z 1 i P − − X κix i=2,4,... κi/(i + 1) 0.5 K = P dx = P , − κ κ 0 i=2,4,... i=2,4,... i i=2,4,... i (18) 1 where x is the normalized hysteresis effect ranged from 0 − 30 20 10 0 10 20 30 − − − to 1. Note that the gain factor K is equal to 1/3 when z/mm ∆ is described only by the quadratic term, i.e. i = 2. AnotherB conclusion obtained from the above analysis is that Fig. 4. The coil magnetic flux density distribution along the vertical the magnetic hysteresis does not rely on the coil height direction at different radii. (2hc), because the maximum effect value and K are both independent of hc. ro. It can be observed that for larger radius, the magnetic flux density is lower. The maximum magnetic flux density E. Measurement and evaluation technique (above or below the coil) is 0.7138 mT, 0.6766 mT and Using the above analysis, the measurement and evalua- 0.6431 mT respectively at ri = 118.5 mm, rc = 125 mm and tion technique for the magnetic hysteresis effect is proposed ro = 131.5 mm. These values confirm the 1/r distribution as follows. First, the yoke minor hysteresis loops, centered of the coil magnetic flux density in the air gap. The result to the yoke BH working point (air-yoke boundaries or an presented in Fig. 4 provides a good check on the approxima- averaged magnetic flux density close to Byv), need to be tion given in (1). The difference between the result obtained measured. The purpose of this step is to determine the by (1) and the 1/r relationship (real distribution) is only coefficient κi in (17). In Kibble balances, since the yoke 0.3%. H field change due to the current is tiny and cannot be Knowing the ∆B value in the air gap center rc based on measured directly, here a fitting method is suggested: 1) (14), we can then solve the magnetic flux density at both measure a group of minor hysteresis loops, H centered yoke-air boundaries following the relationship. Since 1/r to Hyv and ∆ changing as a variable; 2) normalize the the horizontal component of the magnetic flux density is H hysteresis loops measured; 3) fit VD = ∆ dH/dt>0 and continuous in both yoke-air boundaries, the magnetic field B| VE = ∆ dH/dt<0 as functions of ∆ ; 4) find a best-fit changes at the inner and outer yokes are solved respectively B| H order and calculate κi values for both H increasing and H as decreasing curves. In this way, both ∆ (∆ 2) and the gain factor K, i.e. (17) and (18), are solved.B H  NIrc  sign(z), z hc The next step is to follow (15) and (16) and calculate 2µ r δ | | ≥  r i the H field change at yoke boundaries during weighing ∆Hyi = (15) measurements. Note that in this step the permeability of  NIrcz  , z < hc the yoke needs to be measured. Knowing ∆H (z) in the 2µrrihcδ | | y coil region during mass-on and mass-off, the yoke magnetic flux density change ∆ in both directions can be calculated B  NIrc based on (17). Then using (12) or (13), the magnetic  sign(z), z hc 2µrroδ | | ≥ hysteresis effect can be evaluated. ∆Hyo = (16)  NIrcz III. Experimental measurement: An example  , z < hc 2µrrohcδ | | A. Experimental setup where µr is the relative yoke permeability. As shown in Fig. In this section, we give an estimation of the magnetic 3, at the coil center z = 0, ∆Hy is zero, and the magnetic hysteresis effect based on an experimental measurement of hysteresis effect at this point is also zero. It is shown in (9) yoke minor loops. The yoke material used is a Ni-Fe alloy that the magnetic hysteresis effect is related only to the even (50:50). The sample has a inner diameter of r1 = 55 mm order of yoke H field change. Therefore, at z = h , both and a outer diameter r = 70 mm. The thickness of the ± c 2 ∆Hy and the magnetic hysteresis effect reach a maxima. yoke ring is 15 mm. Note,| | we define the maximum magnetic field change at yoke The measurement circuit is presented in Fig. 5. A signal boundary as ∆ . As a result, the hysteresis effect seen by generator, which can output up to 100 mA current, is used the measurementH should be the average value in the coil to supply the required current. In order to reduce the 80 3 60 µd

10 µr

/ 40 1 r µ 20 0 0 5 10 15 20 1 H/Am−

/T 0 B

Fig. 5. Electrical circuit for measuring main and minor hysteresis loops in the yoke material. Bmax Hmax 1 average − inﬂuence of eddy current and skin eﬀect, the frequency (a) of the signal used in the measurement is set at 0.1 Hz. 80 60 40 20 0 20 40 60 80 − − − − 1 The primary winding, with a total number of turns N1, H/Am− is excited by the signal generator. The current through 1 4 the primary winding is measured by the voltage drop on 3 3 µd

10 µr

/ 2 a standard resistor, Rs = 25Ω. According to Ampere’s law, r

µ 1 the magnetic ﬁeld through the core (yoke) is calculated as 0 0.5 0 50 100 150 200 1 N H/Am− H = 1I (19) π(r1 + r2)

/T 0 The induced voltage, U, of the secondary winding is mea- B sured against a voltmeter, which can be written in Faraday’s Bmax law as 0.5 dB − Hmax U = N2s + u0 (20) average − dt (b) where s is the yoke sectional area and u0 an offset in the 1 − 300 200 100 0 100 200 300 measurement. The magnetic flux density given by (20) can − − − 1 be written as H/Am− 1 Z B = (U u0)dt (21) −N2s T − Fig. 6. Measurement result of the main BH hysteresis loops in the yoke sample. (a) and (b) respectively show the measurement result Note that in the measurement, u0 is an unknown quantity. with and without heat treatment. The main magnetization BH curve But in practice, we can choose a constant u0 value that is calculated by averaging the B-maximum and H-maximum points. makes the averaged B field in a period (T ) equal to zero In the subplot of each graph, the µrH curve and the µdH curve are when the excitation current has no dc component. presented.

B. Measurement results values. In this case, the main magnetization curve can be With the measurement of H and B fields, respectively easily obtained by connecting these maximum field points. presented in (19) and (21), the BH hysteresis loop of the However, after the heat treatment, sharp edges disappear sample can be determined. As a comparison, two cases of in the main BH loops before reaching saturation. This is measurement for the yoke material, with and without the probably caused by electromagnetic resistance effects, e.g. heat treatment ( 1150◦C in hydrogen for 4 hours), were skin effect ( For the low frequency range, the additional made. Note that both≈ measurements were carried out in the phase shift is proportional to √µˆ where µˆ is the average same yoke piece. permeability of the BH loop). In order to suppress the bias We first measured the main BH hysteresis loops. In this related to this effect, as shown in Fig. 6, a curve averaged measurement, the signal generator supplies a sine voltage by the B-maximum point and the H-maximum point is of 0.1 Hz without the dc component. The amplitude of the used to present the main magnetization. Accordingly, the excitation, i.e. H field, is slowly increased to the maximum relative permeability µr as a function of H, and the relative (respectively 320 A/m and 76 A/m before and after the differential permeability µd as a function of H can be material heat treatment). The main BH hysteresis loops calculated, as shown in the subplots of Fig. 6. Before the of the yoke sample with and without heat treatment are heat treatment, the maximum permeability is about 2900 shown in Fig. 6 (a) and 6 (b). It can be seen that the at H 60 A/m while the maximum µd is about 3700 at hysteresis shape of the sample has a significant dependence H ≈30 A/m, while after the heat treatment, µ has a ≈ r to the heat treatment: Without the heat treatment, the maximum value of 51000 at H = 8.5 A/m and µd reaches main BH loops have a sharp edge, where the B and H the maximum at H = 5 A/m. It is concluded that heat fields meet at the same BH point and reach both maximum treatment improves the permeability of the yoke sample by 0 5 10 15 20 25 30 0.1 100

50 0

0 0.1 − /T 50 B /mT − B ∆ 0.04 ∆

∆ (measurement) 20 B|dH/dt<0 ∆ (measurement) B|dH/dt>0 0.02 ∆ (cubic ﬁt) B|dH/dt<0 ∆ (cubic ﬁt) 10 B|dH/dt>0 0

0.02 0 − 0 5 10 50 100 150 200 0 0.2 0.4 0.6 0.8 1 2 1 2 4 1 ∆ /(Am− ) 10 H/Am− H ·

Fig. 8. Measurement and ﬁt results of ∆ (∆ 2) functions in both H Fig. 7. The measurement result of minor hysteresis loops. The upper B H subplots are original measurement results with a linear component increasing (dH/dt > 0) and H decreasing (dH/dt < 0) directions. The while the lower are normalized hysteresis loops as described in Fig. 2. upper and lower plots show the results of the yoke sample with and The left and right are two independent measurements with and without without the heat treatment, respectively. heat treatment.

data. The fit is then used to interpolate the ∆ value in the a factor of around 17. weighing measurement of the Kibble balance,B where ∆ in With the same experimental configuration, if a dc compo- this case should be calculated based on the magneticH field nent is added to the excitation current , it is then allowed change due to the coil current, i.e. ∆ = ∆B/(µ µ ). Note to measure the minor hysteresis loopI of the sample. 0 r BH that ∆B 0.6 mT is shown in Fig. 4H in the BIPM magnet The measurement without heat treatment was made with system. ≈ the field centered at 130 A/m. The field amplitude H H Table I presents the fitting result and parameters used for varies from 65 A/m to 200 A/m by changing the ac exci- evaluating the magnetic hysteresis error. It is observed from tation amplitude. The set point after heat treatment is at the calculation that when ∆ is small, the non-linearity of H = 6.3A/m and the H field changes in the range of 1 A/m (22) is mainly contributed byH the quadratic term (> 99.9% to 12 A/m. These configurations, where the B field is about in both cases). Also, the gain factor K approaches 1/3 with 0.4 T, are close to the real working point of a Kibble balance a difference below 1 10 5. Note that although the high magnetic circuit. − order terms contribute× weakly to ∆ when ∆ is small, The measurement result of the minor hysteresis loops they cannot be simply removed duringB the fit. BecauseH the is shown in the upper subplots of Fig. 7. As discussed in ∆ value is much larger in the fit, and these higher order section II, the linear component of the B field change does termsH can have a significant contribution. In Table I, ∆ not contribute to the hysteresis effect, therefore, we removed 1 and ∆ are defined as the interpolation values accordingB the linear component and normalized these minor hysteresis 2 to (22)B in the weighing measurement along the H decreasing loops as shown in the lower subplots. It can be seen from and increasing directions, i.e. ∆ = ∆ , ∆ = the measurement result that the non-linearity of the minor 1 dH/dt<0 2 ∆ . It is seen from theB calculationB| that theB main BH loops behavior differs in two H changing directions, dH/dt>0 contributionB| comes from ∆ , and ∆ has an opposite sign and the normalized loop has also a dependence on the heat 1 2 with a smaller amplitude. AsB shown inB (13), it is reasonable treatment. to consider that the inner and outer yokes share the same BH working point to simplify the calculation. Using K C. An evaluation of the hysteresis effect 1/3, the magnetic hysteresis effect presented in (13) can be≈ Knowing the minor hysteresis loops as shown in Fig. written as 7, we can obtain the ∆B value at the H field center Baw Bav ∆ 1 + ∆ 2 (∆H = 0), i.e. ∆ , as a function of the H field change − B B . (23) y B ≈ − 6B ∆ . Fig. 8 presentsB ∆ (∆ 2) functions in H increasing av av (dH/dtH > 0) and H decreasingB H (dH/dt < 0) directions. In Following (23), the total magnetic hysteresis effect is both directions, we use a cubic fit, i.e. formed by the residual of combining ∆ and ∆ , in which B1 B2 2 4 6 ∆ 2 cancels the major part of the ∆ 1 component. With ∆ = χ2∆ + χ4∆ + χ6∆ , (22) B B 9 B H H H Bav = 0.4 T, (23) yields a bias of ( 21.0 2.4) 10− 2 − ± 9 × to model the ∆ (∆ ) function. It can be seen in Fig. (with heat treatment) and ( 16.8 1.4) 10− (without 8 that the fit ofB (22)H can well represent the measurement heat treatment) in the Kibble− balance± measurement.× Note 2 TABLE I 10− Evaluation results of the magnetic hysteresis error. 2 · (a) Parameters Unit Before HT After HT Bav,Byv T 0.4 0.4 1 /T

Hyv A/m 130 6.3 B

µr rel. 2400 48700 ∆ µ rel. 1600 62700 d 0 ∆By T 0.0006 0.0006 ∆ A/m 1.989E-01 9.804E-03 ∆H2 (A/m)2 3.958E-02 9.612E-05 0 50 100 150 200 250 300 H 2 χ2,dH/dt<0 T/(A/m) 1.871E-06 2.578E-03 H 1 4 /Am− χ4,dH/dt<0 T/(A/m) 2.271E-11 -1.033E-05 2 6 10− χ6,dH/dt<0 T/(A/m) -1.246E-15 5.604E-07 · 2 (µ ) χ2,dH/dt>0 T/(A/m) -8.519E-07 -2.053E-03 (b) B r 4 1.5 χ4,dH/dt>0 T/(A/m) 2.917E-10 3.300E-05 /T (µd) 6 0 B χ6,dH/dt>0 T/(A/m) -1.684E-14 -5.996E-08 ∆ 1 T 7.40E-08 2.478E-07 B 1 ∆ 2 T -3.37E-08 -1.973E-07 dH/dt<

B B| Total effect 10−9 -16.8 -21.0 Uncertainty(k = 2) ×10−9 2.8 4.8 ∆ × 0.5 HT=heat treatment 1,000 1,500 2,000 2,500 3,000 µr or µd that here the uncertainty (k = 1) is mainly from the ∆ 2 H Fig. 9. The magnetic hysteresis dependence on the yoke permeability. determination: The maximum error for ∆By determination (a) presents the measurement result of normalized minor loops at three is 3.9% obtained from figure 4, and the µ uncertainty is different H positions: H = 65 A/m, H = 130 A/m and H = 227 A/m. assigned by the standard deviation of the measurement, (b) shows the relationship between the peak values of three loops, i.e. ∆ , as functions of the yoke permeability µr and the 4.2% and 1.0%, respectively for samples with and without B|dH/dt<0 differential permeability µd. heat treatment. Since the two effects are comparable, ap- parently, the magnetic hysteresis effect cannot be limited by the yoke heat treatment. A minus sign means that the sensitive to the heat treatment, or the yoke permeability, µ. yoke hysteresis will lower the magnetic field at the coil This is because when µ is increased, on the one hand, ∆ position in the weighing measurement. In a Kibble balance, becomes smaller, but on the other hand, the coefficientsH of the magnetic flux density decrease in the weighing phase the fit (mainly χ2) become larger. Also, from the measure- will be compensated by a feedback current, which increases ment result, it seems that the symmetry of the normalized the realized mass in the new SI. minor loops is better with a larger yoke permeability. It would be also interesting to analyze the hysteresis error IV. Discussion change when the yoke working point shifts along a fixed The magnetic hysteresis effect could have a dependence BH curve. First, the magnetic field change in the yoke on the chemical composition of the yoke material. In the is inversely proportional to µr and hence the hysteresis 2 following discussion, we do not focus on the yoke material effect is proportional to 1/µr. However, the sensitivity of itself, but the related parameters when the yoke magnetic the magnetic flux density change ∆ as a function of ∆ property is known. Some dependence and possible optimiza- can also be related to the yoke permeability.B In order toH tion of the magnetic hysteresis effect are summarized as investigate this dependence, we measured the magnetic follows. density change with a fixed ∆ at three different H Eqs. (15) and (16) show that the yoke magnetic field locations (65 A/m, 130 A/m andH 227 A/m) of the sample change in the weighing measurement is proportional to the without heat treatment (The sample without heat treat- coil ampere-turns, NI, and inversely proportional to the ment provides better resolution during the measurement). air gap width δ. Because the hysteresis error is a square The measurement result of three normalized minor loops is effect of the yoke H field change, a smaller NI or a larger shown in Fig. 9(a). It can be seen that the magnetic flux δ in the electromagnet system greatly helps to reduce the density change has an obvious dependence on the H field. effect. Since NI mg/(2πrcBa), if the test mass m is fixed, To clarify the relationship between the yoke magnetic flux ∝ 2 the hysteresis effect is then proportional to 1/(Barcδ) . For density change and its permeability, we calculated the peak example, the 1/(B r δ)2 value of the NIST-4 system [21] is value of three minor loops, i.e. ∆ , as functions of a c B|dH/dt<0 only 1/25 of the BIPM value, and hence the hysteresis of the yoke static permeability µr and the yoke differential 9 NIST-4 Kibble balance should be much weaker (< 1 10− ) permeability µ . The results are plotted in Fig. 9(b). A × d if a similar material is used. linear relation between ∆ dH/dt<0 and µd is obtained, The working point of the yoke, or the yoke permeability therefore, we conclude thatB| the hysteresis effect is approxi- µr also appears in the analysis of the hysteresis effect. An mately proportional to µd. Combining the two conclusions interesting conclusion obtained from the result in Table I above, the magnetic hysteresis effect is proportional to a 2 is that the magnetic hysteresis error, in fact, is not very permeability ratio µd/µr. It can be easily observed that 2 close to the working yoke permeability the µd/µr value is in this work. It is assumed that the yoke magnetic status stable, and a slight change of the yoke permeability during is repeatable in a full measurement cycle, e.g. velocity- the weighing measurement will not significantly affect the weighing (mass on/off), which in reality may shift with the magnetic hysteresis error. environmental change and the repeatability of the current Except for the two-mode, two-phase scheme, a Kibble ramping. Besides, in the weighing measurement, it probably balance can also be operated with a one-mode, two-phase needs several mass on/off cycles to stabilize the magnetic scheme [12], [13], or one-mode, one-phase scheme [14]. state and the first measurement may differ from the ones Under a one-mode measurement, the current is through after. The estimation assumes a weighing position at z = 0, the coil during both weighing and velocity measurement where the hysteresis effect, in fact, is minimum, because phases. Compared to the two-mode, two-phase scheme, the when the weighing position is chosen shifted from z = 0, coil current change and hence the yoke magnetic status the H field change will increase in one vertical end of the change between weighing and velocity measurements is coil, and decrease on the other. The major effect (quadratic much less (at least two magnitudes smaller), therefore, the term) in this case is no longer symmetrical, which will lead magnetic hysteresis effect in the above examples is negligible to a larger average factor K in the coil region. 9 (< 1 10− ) and should not be a limitation in the BIPM one-mode,× two-phase measurement. Appendix In Fig. 2(b), the H increasing curve (dH/dt > 0) and the V. Conclusion H decreasing curve (dH/dt < 0) in the original loop are The yoke magnetic hysteresis error is a part of the respectively written in forms of polynomials, i.e. current magnetization effect which arises from the BH nonlinear characteristic of the yoke material. Understanding X∞ ∆B = λ ∆Hi , (24) its mechanism helps to characterize the performance of a y|dH/dt>0 i y i=0 yoke-based magnetic circuit, and may also lead optimization in designing such systems. In this paper, we presented X∞ ∆B = γ ∆Hi , (25) both a theoretical analysis and a practical technique for y|dH/dt<0 i y evaluating the magnetic hysteresis error based on measuring i=0 yoke minor hysteresis loops. where λi and γi are polynomial coefficients of two curves. Theoretical analysis shows the magnetic hysteresis er- As (24) goes through point V, i.e. ∆B = 0 when y|dH/dt>0 ror is a nonlinear current effect. The yoke status change ∆H = 0, and hence λ0 = 0. Therefore, (24) can be rewritten is mathematically described by a normalized minor loop. as

Based on this description, the yoke magnetic flux density X∞ i ∆By dH/dt>0 = λi∆Hy. (26) change ∆By is modeled by the yoke H field change ∆ , | while ∆ is linked to the coil magnetic field in the air gapH i=1 H following a continuous boundary condition. In this way, the Eqs. (25) and (26) have crossing points at ∆Hy = ∆ , hysteresis error is quantitatively related to the coil current. then we have ± H Experimental measurement of a soft yoke sample has X∞ X∞ been carried out to check the proposed theory. The mea- λ ∆ i = γ ∆ i, (27) i H i H surement and proposed evaluation technique showed how i=1 i=0 the hysteresis effect is related to even orders of the yoke field change caused by the coil current. An evaluation of the X∞ X∞ λ ( ∆ )i = γ ( ∆ )i. (28) hysteresis effect based on this experimental determination i − H i − H i=1 i=0 yields an effect of about 2 parts in 108 under a configuration of the two-mode, two-phase scheme in the BIPM system. As is known in the analysis, the yoke magnetic flux density The one-mode scheme has the advantage of suppressing the change due to the yoke hysteresis is (AV + AC), which − magnetic hysteresis error. As observed the effect depends based on (24)-(28) can be written as closely on the coil ampere-turns, the width of the air gap, " # X∞ X∞ and the yoke property. As demonstrated and discussed in (AV + AC) = γ ∆ i + γ ( ∆ )i λ γ − i H i − H − 0 − 0 this paper, the non-linear current effect can be significant i=0 i=1 in Kibble balances (especially for magnet systems with a X = γ ∆ i. (29) small gap), which should be checked or optimized carefully: i H i=0,2,4,... 1) conventionally, the nonlinear magnetic effect can be determined experimentally by weighing different masses; 2) as It can be seen from (29) that the yoke magnetic flux presented in [6], with an appropriate mechanical design, the density change due to the hysteresis contains only even hysteresis error can be removed by ramping the weighing order terms of ∆ . current slowly to zero before each velocity measurement. The normalizedH hysteresis curves in Fig. 2(b) are ob- In the end of the paper, we would like to acknowledge tained by removing a linear component that through two some unaddressed consequences of the magnetic hysteresis end points of the original hysteresis curves. First, the line through point A and two end points of the original loop [5] I. A. Robinson, “Towards the redefinition of the kilogram: a curve is solved as measurement of the Planck constant using the NPL Mark II watt balance,” Metrologia, vol. 49, no. 1, pp. 113-156, 2011. X∞ X∞ [6] B. M. Wood, et al, “A summary of the Planck constant determi- γ ∆ i + γ ( ∆ )i nations using the NRC Kibble balance,” Metrologia, vol. 54, no. i H i − H i=0 i=0 3, pp. 399-409, 2017. ∆By = [7] D. Haddad, et al, “Invited Article: A precise instrument to de- 2 termine the Planck constant, and the future kilogram,” Rev. Sci. X∞ i X∞ i Instrum., vol. 87, no. 6, pp. 061301, 2016. γi∆ γi( ∆ ) [8] H. Baumann, et al, “Design of the new METAS watt balance H − − H experiment Mark II,” Metrologia, vol. 50, no. 3, pp. 235-242, 2013. + i=0 i=0 ∆H . (30) 2∆ y [9] M. Thomas, et al, “A determination of the Planck constant using H the LNE Kibble balance in air,” Metrologia, vol. 54, no. 4, pp. Then the normalized H increasing and H decreasing curves 468-480, 2017. can be then written as [10] C. M. Sutton C M and M. T. Clarkson, “A magnet system for the MSL watt balance,” Metrologia, vol. 51, no. 2, pp. S101-S109, 2014. ∆ = ∆B ∆B D|dH/dt>0 y|dH/dt>0 − y [11] D. Kim, et al, “Design of the KRISS watt balance,” Metrologia, vol. 51, no. 2, pp. S96-S100, 2014. X∞ i X∞ i γi∆ + γi( ∆ ) [12] H. Fang, et al, “The BIPM Kibble Balance for the realization of H − H the redefined kilogram,” 2018 Conference on Precision Electro- X∞ i=0 i=0 = λ ∆Hi magnetic Measurements (CPEM 2018), Paris, France, 2018. i y − 2 i=1 [13] I. A. Robinson, et al, “Developing the next generation of NPL Kibble balances,” 2018 Conference on Precision Electromagnetic X∞ X∞ Measurements (CPEM 2018), Paris, France, 2018. γ ∆ i γ ( ∆ )i i H − i − H [14] A. Picard, et al, “The BIPM watt balance: improvements and i=0 i=0 ∆H . (31) developments,” IEEE Trans. Instrum. Meas., vol. 60, no. 7, pp. − 2∆ y 2378-2386, 2011. H [15] S. Li, Z. Zhang and B. Han, “Nonlinear magnetic error evaluation of a two-mode watt balance experiment,” Metrologia, vol. 50, no. ∆ dH/dt<0 = ∆By dH/dt<0 ∆By 5, pp. 482-489, 2013. D| | − [16] S. Li, S. Schlamminger, J. R. Pratt, “A nonlinearity in permanent- ∞ ∞ X i X i magnet systems used in watt balances,” Metrologia, vol. 51, no. γi∆ + γi( ∆ ) ∞ H − H 5, pp. 394-401, 2014. X i i=0 i=0 [17] C. A. Sanchez, et al, “A determination of Planck’s constant using = γi∆Hy − 2 the NRC watt balance,” Metrologia, vol. 51, no. 4, pp. S5-S14, i=0 2014. X∞ i X∞ i [18] D. Haddad, et al, “Measurement of the Planck constant at the γi∆ γi( ∆ ) National Institute of Standards and Technology from 2015 to H − − H 2017,” Metrologia, vol. 54, no. 5, pp. 633-641, 2017. i=0 i=0 ∆H . (32) − 2∆ y [19] S. Li, et al, “A permanent magnet system for Kibble balances,” H Metrologia, vol. 54, no. 5, pp. 775-783, 2017. Based on (31) and (32), the magnetic field change in the [20] S. Li, et al, “Coil-current effect in Kibble balances: analysis, normalized hysteresis curves can be written as measurement, and optimization,” Metrologia, vol. 55, no. 1, pp. 75-83, 2018. [21] F. Seifert, et al, “Construction, measurement, shimming, and per- (VD + VE) − formance of the NIST-4 magnet system,” IEEE Trans. Instrum. Meas., vol. 63, no. 12, pp. 3027-3038, 2014. = ∆ dH/dt>0(∆Hy = 0) ∆ dH/dt<0(∆Hy = 0) − D| − D| [22] S. Li, et al, “Coil motion effects in watt balances: a theoretical = ∆ dH/dt>0 ∆ dH/dt<0 check,” Metrologia, vol. 53, no. 2, pp. 817-828, 2016. − B| − B| [23] S. Li, et al, “Field analysis of a moving current-carrying coil in X∞ X∞ = γ ∆ i + γ ( ∆ )i OMOP Kibble balances,” 2018 International Applied Computa- i H i − H tional Electromagnetics Society Symposium (ACES2018), Den- i=0 i=0 ver, USA, 2018. X = γ ∆ i. (33) i H i=0,2,4,... A comparison of (29) and (33) yields X (AV + AC) = (VD + VE) = γ ∆ i. (34) − − i H i=0,2,4,...

References [1] B. P. Kibble, “A measurement of the gyromagnetic ratio of the proton by the strong ﬁeld method,” Atomic Masses and Fundamental Constants 5, pp. 545-551, Springer, 1976. [2] Resolution 1, 26th General Conference on Weights and Mea- sures (CGPM), 2018. https://www.bipm.org/utils/common/pdf/ CGPM-2018/26th-CGPM-Resolutions.pdf [3] D. Haddad, et al, “Bridging classical and quantum mechanics,” Metrologia, vol. 53, no. 5, pp. A83-A85, 2016. [4] I. A. Robinson and S. Schlamminger, “The watt or Kibble balance: a technique for implementing the new SI deﬁnition of the unit of mass” Metrologia, vol. 53, no. 5, pp. A46-A74, 2016.