Resolution of the paradox of the diamagnetic effect on the Kibble coil

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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_A$$\end{document}NA, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 10^{-8}$$\end{document}1×10-8, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 1\times 10^{-9}$$\end{document}≈1×10-9.


A YOKE
In this section, we compare the magnetic field of a current-carrying coil in free space and in a Kibble balance magnet. Figure 1(a) shows the effect that the current in the coil has on the measured Bl in force and velocity mode. We present a summary of the finding that have been described in [1]. Plotted on the vertical axis is the difference of Bl measured in different scenarios from a velocity phase measurement (U/v) 0 = B 0 l where the coil is moving through the magnetic field without any current. The two profiles (F/I) off and (F/I) on are used in most Kibble balances for weighing -usually the two measurements in weighing phase, mass-on and mass-off, are carried out with equal and opposite currents, I off = −I on . The profiles (U/v) off and (U/v) on are profiles determined in the velocity phase, respectively with plus and minus currents.
The bifilar coil used in the BIPM balance allows current to be present during the velocity mode.
The data in figure 1(a) shows that the change in Bl is a linear in coil position z, as well as in coil current I in the weighing measurement. As discussed in the article, the field gradient ∂ z B produces a diamagnetic force bias that can not removed by mass-on and mass-off measurements.
The interesting conundrum is, can the magnet field gradient that is produced by the coil itself produce a force on itself.
To clarify the paradox we study two scenarios: 1) the coil in free space and 2) the coil inside an air gap surrounded by iron yokes as it would in a Kibble balance . Figure 1 [5] with an average field in the air gap of B 0 = 0.45 T. To obtain the freespace calculation, the yoke relative permeability is simply set to one. Using the same geometry and mesh, results in a trustworthy comparison of both scenarios. To one side of the coil the magnetic flux density is higher to the other it is lower. As was shown above, it is not the local gradient of the field that provides information on the force. The volume average over the coil must be considered. The volume average, shown as the blue and red line in the figure, is no longer independent of z as it was in the free space scenario.
The external force on the coil depends on the average field. The experimental observation, shown in figure 1(a) agrees well with the FEA calculation presented in 1(c).
The comparison of average fields produced by a current-carrying coil in free space and in a yoke shows that in the latter the broken symmetry will lead to an energy redistribution and hence a magnetic field change. A physical picture can be provided as follows: In the yoke B changes because the coil current magnetizes the yoke, and the magnetized yoke produces an additional magnetic field at the coil position.
The diamagnetic force arises from F χ = χV µ0 B∂B z , where the last factor is the derivative of the volume average field with respect to the coil position. As shown in figure 1(c), the slopes of the red and blue lines and, hence the gradient ∂B z is constant within reasonable ranges of z. Accordingly, the diamagnetic force is constant as a function of z, but changes direction when the current is reversed. Because of the latter fact, the diamagnetic force does not cancel in the Kibble balance experiment and can lead to a systematic bias.
In conclusion, the statement that 'a current-carrying coil can not produce a measurable force on itself' holds for free space system, but not necessarily for a coil inside a yoke, unless the yoke moves with the coil. We note that this finding does not contradict Newton's third law as there is an equal and opposite force on the yoke.

EFFECTS
The diamagnetic effects described in the main article are very small. In force mode, an additional bias of 10 µN on top of 10 N is produced. It would be impossible with finite element analysis to detect, let alone to calculate with any uncertainty, this additional bias. We first developed the analytical equations of the diamagnetic effect in force and velocity modes. However, we wanted to verify our analytic results with numerical calculations. In searching for ways to do this, we invented differential FEA (dFEA). With differential FEA, it was possible to calculate the size of the effect with a relative uncertainty of about 0.1%, corresponding to f 10 nN. We could further show that the diamagnetic effect produces the same bias in force and velocity mode within that uncertainty.
Conventional finite element analysis, FEA, is a powerful technique to solve a variety of engineering problems that would be difficult or impossible to solve analytically. The basic idea of FEA is to divide the domain under investigation into small elements, and then to approximate the solution of the problem by a linear combination of calculations in each element. In FEA simulations, small errors are inevitable. Either discretization or numerical errors cause these problems.
The computer simulates a discrete system, while the real system in nature is continuous. Numerical errors can occur, for example, by rounding when two large numbers are subtracted or by ill-conditioned matrices. In our experience, problems in magnetostatic can be calculated with relative errors of a few parts in 10 3 on a standard PC in reasonable time. The effects discussed in our work are of the order 1 × 10 −6 . Hence, they are three orders of magnitude smaller than what we consider reliable results obtainable by FEA.
With dFEA, small effects and their uncertainty can be calculated using commercial FEA packages without exponentially extending the computation time. The technique takes about five times longer than a single calculation because, as the reader shall see below, the same model has to be calculated about five times with different χ. The idea is simple. As mentioned above, the uncertainties of the calculations are 1000 times larger than the effect, so if one can increase the effect by a factor of 1000, the effect could be detected. In our case, where the effect scales with the susceptibility, one only needs to multiply the susceptibility of the part that is investigated by a factor of 1000 in question by a factor of 1000 to achieve that amplification. So, if χ nom is the susceptibility of the part in question, here the coil, χ exag ≈ 1000 × χ nom is used in the finite element calculation. It is safe to change the susceptibility because it occurs as a linear parameter, and the underlying physics does not change. In the end, all one has to do is scale the measured effect, say the force, by χ nom /χ exag to obtain the size of the previously immeasurable effect.
A successful implementation of dFEA relies on three good practices: 1. It is not sufficient to perform one calculation with χ exag and scale the result by χ nom /χ exag .
Usually there are small calculation biases that would be scaled and falsify the result. See, for example, the middle graph of figure 2 , where the force calculated with zero current is not exactly zero. The best practice is to calculate the desired effect for several χ values and then interpolate the size of the effect to the nominal χ. For example, we would like to calculate the diamagnetic effect for χ nom = 1 × 10 −5 . We use five value for χ exag , namely -0.01, -0.005, 0, 0.005, 0.01. We then plot the desired function, for example the force F , as a function of χ exag , subtracting F (χ exag = 0). This function is linear in χ exag and the value at χ nom can be obtained from a linear fit to the data.
2. It is of utmost importance to keep all other parameters constant when using dFEA. If additional parameters were changed for the calculation with different χ exag , they would cause a change in the calculation result that would then, falsely, be attributed to being driven by the susceptibility. This point applies, especially to the meshing. The domain should be meshed only once before the various χ exag are assigned to the components.
3. The third practice applies to calculations where current is involved. In this case, the best results are obtained by calculating the system twice, with and without current. The difference between the two results gives the desired effect. Again, other than the current, nothing can change between the calculations, most importantly, not the meshing. The curves obtained with dFEA are not only smooth, but they are also physical. They represent the coil inductance force [1], a linear force curve over z. A physical result can be obtained from the original noisy result. The standard deviation of the data calculated with dFEA is 100 times smaller than standard FEA data.
In summary, dFEA allows one to calculate a relative diamagnetic force of 1×10 −6 with meaningful uncertainties. The method relies on FEA calculations with several exaggerated susceptibilities, the largest one about 1000 times the size of the nominal susceptibility.