A method for evaluating the force factor in oscillating Kibble balances with application to the Planck-Balance

The Planck-Balance is a table-top version of a Kibble balance, developed in a collaboration between the PTB and TU Ilmenau. In contrast to most Kibble balances, where the motion range is of up to several centimeters in the velocity mode, in the Planck-Balance it is below 100 μm. For a reliable determination of the force factor, the voice coil is set into harmonic oscillation, resulting in an induced AC voltage. The amplitude of the first harmonic is then taken to determine the force factor. However, due to the non-linear magnetic field across the motion range, the first harmonic is subject to a bias, and higher harmonics are excited. Those higher harmonics contain the information about the relative magnitude of the bias in the first harmonic, and can be used for a correction. In this paper we present an analytical model and show its application with real measurements. The estimated relative bias amounts to −6.3⋅10−6 , with a residual relative uncertainty of 6.3⋅10−7 , for an oscillation amplitude of 20 μm. This method is especially of importance for Kibble balances with short motion range and working with harmonic oscillation. After such a correction is applied, a table-top version of a Kibble balance can be used as a conventional balance calibrated by means of E2 mass standards in the whole E2 class mass range.


Introduction
Since the redefinition of the international system of units SI in 2018 [1], the unit kilogram can be realized at any mass value. This is a paradigm shift, as for more than hundred years the whole mass scale was derived from a single mass value, 1 kg, realized through the international prototype kilogram. This * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. fact paves the way for innovations in mass metrology. In the future, there will be no need any longer to calibrate a mass balance by means of mass standards. A balance, which follows the principle of Kibble [2], is 'self-calibrating', so to speak. The traceability of the balance happens through the quantities time, length, and electrical current. Current is measured as a voltage drop at a precision resistor. The voltage, in turn, is traceable to a programmable Josephson voltage standard, and the resistance to a quantum Hall resistor. Kibble balances can be used for the dissemination of mass, not necessarily for calibrating mass standards [3]. A balance is usually calibrated by means of a set of mass standards of class E2, while E2 mass standards are calibrated with E1 mass standards. In the guideline OIML R111-1 [4] a table of the maximum permissible error (MPE) of a mass standard is given. This MPE says how much the true value of a mass standard may deviate from its nominal value of a certain class. Thus, if the Kibble balance has the same uncertainty as a mass of accuracy class E2, then a weighing done with such a Kibble balance is of same accuracy as a conventional balance, that has been calibrated against E2 mass standards.
A Kibble balance works generally in two modes. In a first mode, called velocity mode, the so-called force factor of a voice coil actuator is determined. This is necessary, as in the second mode, the force mode, the mass is determined by equilibrating the mechanical force of the mass under test, with the electro-magnetic force, produced by the voice coil actuator, which consists of a coil and a permanent magnet (hence, commonly called 'magnet-coil system'). The force factor (sometimes also called geometric factor) is a property of the voice coil actuator, and indicates how much current is necessary to produce a certain electro-magnetic force with which, in turn, the mechanical force of a weight piece will be compensated. Thus, the Kibble balance is a force comparator. In order to obtain the value of the mass, the force needs to be divided by the local gravitational acceleration, at the position of the center of mass of the mass piece 3 .
Generally, in Kibble balances the force factor is determined applying Faraday's law of induction. Here, the coil of the voice coil actuator is moved with a (usually) linear trajectory through the magnetic field of a permanent magnet (see, e.g. [5][6][7][8][9][10]). The induced DC voltage is measured synchronously to the position of the coil (from which its velocity is derived). The trajectory is divided into small regions, and for each region a value for the force factor is determined by dividing the mean induced voltage by the mean velocity needed to pass this region. From this data, the profile of the force factor along the whole range is determined. Since the force factor is generally not constant over the whole travel range, the induced voltage will not be constant. Of special importance is the value of the force factor at the position where the weighing in the force mode takes place, because in the force mode a feedback loop controls the current that is necessary to maintain the balance lever in equilibrium at this position. A polynomial fit is employed to give this estimated value from all data points determined over the whole trajectory.
For an accurate determination of the force factor via a linear trajectory, i.e. with a constant velocity, the trajectory must be of sufficient length. This is necessary, as the induced voltage is proportional to the coil velocity, and since an adequate integration time is necessary. In order to reduce the uncertainty in the induced voltage, the velocity must be as high as possible (typically some millimeters per second). A high velocity, in turn, decreases the integration time. Therefore, the trajectories are usually of several centimeters. In the Planck-Balance 4 (see [11]) we employ a standard load cell, as is used in commercial analytical balances, in force and velocity mode. Such load cells, however, have a limited total motion range of no more than 80 µm (maximum oscillation amplitude 40 µm) at the position of the load carrier. A determination by means of a linear trajectory is thus not practicable. We therefore decided to employ a sinusoidal trajectory, which was first proposed in [12], and was also employed by the TÜBİTAK National Metrology Institute (UME) [13] (recently, this group has switched to a linear trajectory [14]). This has the advantage that long integration times become possible. Moreover, the induced voltage can be increased by increasing the oscillation frequency. A disadvantage, however, is a step-wise determination of the force factor along the whole trajectory is not accurate. Furthermore, a calibration of an AC signal is not as trivial as a DC signal.
The non-linear magnetic field, over which the coil sweeps in our velocity mode, induces distortions in the induced voltage. As a consequence, the amplitude estimation of the induced voltage becomes biased, and leads to an erroneous estimate of the force factor, which must be used in the force mode. In this article, we aim at describing how we determine the force factor, and how we correct for a non-linear magnetic field.

Fundamental equations
There are two fundamental equations describing the Kibble balance experiment. In the force mode the mechanical force of a mass piece is counterbalanced by a voice-coil system as which is our first equation. Here, m denotes the mass, g the local gravity, B the magnetic flux density, L the coil length, and I the electrical current. Figure 1 shows a simplified schematic drawing of the Planck-Balance (PB1) at PTB. In this investigation we employ the lever ratio of the load cell. This means that the force due to the weight-placed onto the weighing pan-is not directly compensated by the force generated with the external voice coil actuator (2) but with the internal voice coil actuator (1) and the lever ratio of approximately 1 : 4.
To determine the mass m, the force factor, i.e. the product B · L, needs to be known (in the following, the quantity of the force factor B · L is denoted as 'BL'). This force factor is measured in the velocity mode by moving the coil (of the internal voice coil actuator (1)) sinusoidally through the magnetic field along the trajectory s(t) = S 0 · sin (2π f t), with the oscillation amplitude S 0 and the oscillation frequency f = const. The motion s(t) is directed along the vertical, and is measured with a laser interferometer at the external voice coil (2). This is leading to an induced voltage in the internal voice coil of our second equation. The time dependent velocity  (1) is powered with a DC current so that the balance is maintained in equilibrium when a mass piece is placed onto the weighing pan. The equilibrium position can be chosen for different set points of the position sensitive diode (PSD). Since at different set points the BL is different, also the required DC current will be different.
of the internal voice coil is the analytical time derivative of s(t), with the velocity amplitude v 0 = 2π f · S 0 . For a constant force factor, the induced voltage will also be a sinusoid of with the voltage amplitude U 0 . The force factor BL is then determined as follows. First, a linear least-squares fit is applied to U(t) and s(t), provided the frequency f is a known external quantity. This delivers the amplitudes U 0 and v 0 . 5 Then, these amplitudes are taken to calculate the force factor as It follows, considering equations (1) and (5), that (neglecting the uncertainties in the other parameters) i.e. a relative error in mass determination is of the same amount as the relative error in the BL estimation.
The position of the coil s can be observed at several set points along the vertical (force mode) or along the trajectory 5 The phases can also be obtained, but they are of no relevance for this investigation. s(t) (velocity mode). In either case, the position can be measured by means of the laser interferometer (output in meter) at the external voice coil actuator, or with a position sensitive diode (PSD-output in volt, denoted as u) close to the internal voice coil actuator (1) (see figure 1). Both outputs can be converted into one another (s = ku), in our case, with a conversion factor of k = 25.19 · 10 −6 m V −1 . That is, the PSD can be calibrated by means of the laser interferometer. At different positions s the PSD indicates a different voltage level, u(s). The PID control's set variable is the voltage output of the PSD. For clarity, in this investigation we decided to put the position in meter (s), as observed with the interferometer. It should be noted that the lever ratio drops out in the equations, when the measurement is performed as described above (see [15]).

Measurement of the BL profile in the force mode
In a first step, we determine the relative force factor change along the total trajectory range. This is done by choosing different voltage set-points u i of the PSD for the PID control in the force mode. This means that the PID control controls the balance's equilibrium at different positions of the magnetic field of voice coil actuator (1). At each set-point we perform a mass determination with a mass piece of, say, 100 g. The absolute value of this mass piece is not important, but it is important that for each set-point the same mass piece is taken. The corresponding BL at each set-point position was then corrected to the same temperature, say 23 • C, as the temperature coefficient of the magnet was measured and thus known beforehand. This measurement campaign provides us with the profile of the force factor along the trajectory s = ku. In the following, we will also operate with the trajectory s, which can be equivalently expressed as PSD voltage u. Up to now, we do not know the accurate value of the force factor at any point, however, we know its profile, the function of which can be approximated as i.e. as a polynomial of second degree, with the polynomial coefficients b and c. The relative deviation of the BL from its value at s = 0 can be expressed as which is depicted in figure 2. The measured data are plotted with blue circles with their respective relative standard deviation as error bars. The red continuous line shows the quadratic least squares approximation to these data. Again, at this stage we are not interested in the exact absolute value of the BL, but in its relative profile. Its absolute value will be determined in the velocity mode. The parameters, as obtained from the fit, as well as the respective uncertainties, are shown in table 1.
, where BL(s = 0) is the value of the force factor at position s = 0. Column 3 and 4 show uncertainties of fit parameter.

Measurement of the BL profile in the velocity mode
As a consistency check, we also measured the force factor profile in the velocity mode (see figure 2, green stars). The oscillation was performed about different set-points. The variation of the different set-points was established by putting different masses onto the load carrier (as the flexure hinges function as mechanical springs). To be able to measure at different positions, we had to reduce the oscillation amplitude to about 11.5 µm, since-due to our flexure hinges-our total motion range is only approximately 80 µm. The uncertainty here is higher, as the oscillation amplitude is reduced, which, in turn, results in a lower induced voltage amplitude. Same applies to the velocity amplitude measurement. In table 1 the fit parameters are listed with their uncertainties. It can be seen that both measurements, FM and VM, agree to within 13%.

Bias estimation and correction
After determination of the force factor profile along the trajectory, we can estimate an impact of the non-linear force factor onto the amplitude of the induced voltage. To this end, we start with the equation for the induced voltage U(t), which is where B(s(t)) denotes the magnetic flux density, as function of coil position s(t) = ku(t), and v(t) is the coil velocity, as function of time. For an estimation of the bias we assume a sinusoidal displacement of the coil-where we neglect possible non-linearities of the frequency generator-as with the oscillation frequency f. The coil velocity is then derived as Assuming a parabolic profile, as in equation (7), (9) becomes, after some trigonometric manipulation 6 , with y := 2π f t. Thus, the quadratic term c of the force factor introduces a relative bias of to the voltage amplitude U 0 (see equation (5)) of the fundamental frequency. As this amplitude is taken to calculate the force factor, the force factor will be subject to a bias, which now can be corrected. In other words, assume we oscillate in the velocity mode about the mid-point s = 0. Then, the measured BL (equation (5)) deviates from an instantaneous value. If the weighing in the force mode is now performed at set-point s = 0, and if this distortion is ignored, we would get a bias in the mass determination. If we take the BL(s), as shown in figure 2, the relative bias amounts to −6.30 · 10 −6 for an oscillation about the position s = 0, and for an oscillation amplitude S 0 = 20 µm (we usually leave the amplitude below 20 µm to avoid non-linearities in the mechanics). To put this into context, without applying any correction, the error contribution 6 A rigorous derivation of this equation will be published elsewhere.
due to the non-linear BL would be admittable to calibrate E1 mass standards of values 100 mg and smaller (case (A) in figure 3), i.e. MPE(E1)/6 (see [4]). After a correction is applied (case (B)), this range is extended to 5 g. Accordingly, an upper limit of 10 g without correction (case (C)) to use a Kibble balance as a conventional balance calibrated by means of E2 mass standards, is completely removed after correction is applied.
As the curvature of the BL profile changes with position, the bias is also a function of the point, about which the oscillation occurs. In this case, equation (12) must be modified. Considering only the amplitude of the first harmonic, we obtain for the bias as function of the oscillation mid-point. There are two terms in equation (14) that contribute to a bias relative to BL at s = 0. The first is just a change due to the changing BL along s in general. If we take the derivative with respect to position s, we note that the relative change is on the order of (b + 2cs = 2.89 · 10 −6 − 2 · 6.30 · 10 −8 µm · s) per micrometer, where we have taken the parameters as given in table 1. If the repeatability of oscillation is on the order of 25 nm, the relative error remains below 1 · 10 −7 . Figure 4 shows a temporal drift of the oscillation midpoint during a long-term measurement. It can be seen that the drift correlates well with the room temperature change. The total drift amounts to 1 µm, in this case. In an environment that is more temperature stable, the drift might be significantly smaller. However, there is the possibility of  applying a correction, as the DC component of the oscillation is known. Another method would be to control the DC component to return to the same mid-point by applying a DC current to the exciting voice coil, either by employing the position data, or the temperature data by derivation of a temperature coefficient.
The second term of equation (14) is a bias due to the nonlinear term of BL, similar to equation (12). This contribution, however, is negligible in our case, as this error is on the order of about 1.3 · 10 −8 µm. Figure 5 shows this bias compared to the shape of BL for different mid-point positions. Thus, a precise positioning during the velocity mode is not critical, when considering the position dependent bias, induced by the quadratic term c.
Equation (13) shows that the relative bias depends on the oscillation amplitude squared. In the case of the long-term  figure 6. The relative change in β would, in this case, i.e. with no good temperature stabilization, be on the order of 0.3%, and is thus negligible.
Finally, it is worth to add one more comment. The bias, which we derived above, does not depend on the sampling strategy employed. In [16] it was shown that if coherent sampling is applied, i.e. when only full cycles of the oscillation are analyzed, then the bias due to higher harmonics is zero. The origin of the higher harmonics, though, is the non-linear magnetic field along the trajectory. In case of non-coherent sampling we would introduce just an additional error. The estimate of the different harmonics would be subject to a further bias, and thus the bias due to the non-linear BL would be badly evaluated. An application of coherent sampling-or inclusion of all harmonics, if they are known-is of utmost importance to reach highest accuracy.

Uncertainty of the correction
After the evaluation of the magnitude of the bias, produced by the non-linearity of the magnetic field, a correction for equation (5) is applied as In relative units, this correction amounts to 6.30 · 10 −6 at s = 0, and an oscillation amplitude of S 0 = 20 µm, and the mispositioning errors (s ̸ = 0) can be ignored, as shown above. The uncertainty of this correction is given by the dominant standard error of the fit parameter c in the force mode and amounts to 10% (see table 1), or 6.3 · 10 −7 in relative units 7 . Comparing to the data given in figure 3, the uncertainty of this correction is negligible for calibration of mass standards of 5 g and smaller, of accuracy class E1, or 20 g and smaller, of accuracy class E2.

'All-in-one' solution
For what we have described so far, the relative shape of the BL(s) needs to be determined first, in order to derive the bias. This requires repeated mass determinations (or velocity measurements) for different positions s, which is relatively time consuming. However, there is, in principle, a more straight forward way for determining the bias. As can be seen in equation (12) the quadratic term c of the BL generates a third harmonic. The amplitude of this harmonic is of the same amount as the relative bias 8 in the fundamental note. Consequently, if we know the amplitude of the third harmonic, we can immediately correct for the bias. To this end we need only to include the second and third harmonic terms in the least squares adjustment. This does not require the determination of BL(s) with the force mode. The information is contained in the induced voltage signal itself.Above we said 'in principle', because in practice, unfortunately, this is currently not possible, as the noise in the voltage signal is higher than 7 The relative standard error of squared S 0 in equation (13) after least-squares fitting does not exceed 0.1% and is negligible in the given example. A rigorous evaluation of uncertainty due to the applied correction is preserved for a practical case of weighing or calibrating mass standards. 8 It should be noted that we consider here only a quadratic function with negative curvature. An extension to higher degrees, with eventually changing curvature, is possible, but outside the scope of this text.  figure 7 shows the amplitude spectral density of a real signal, taken with our Planck-Balance (however, with an amplitude of 22.98 µm). In comparison, the red line shows the spectral density of synthetic data, with same oscillation amplitude and same BL(s) shape. It can be seen that the expected signal is buried in the noise in the respective frequency range. As the frequency is exactly known, we can, nonetheless, extract the third harmonic from such a noisy signal. With such a signal-to-noise ratio, our preliminary attempts reveal an amplitude of the third harmonic with a relative error of about 60%, which is insufficient for this application.

Conclusion and outlook
Running a Kibble balance with oscillatory motion is a nontrivial task, especially when the motion range is extremely short-in our case less than 100 µm. A non-linear force factor can introduce a non-negligible bias. For the investigated voice coil system the relative bias amounts to −6.30 · 10 −6 , for an oscillation amplitude of 20 µm. In this paper, we have presented two ways how the bias can be quantified and corrected. Both ways are based on an analytic solution that has been derived. The first is by conducting force (or velocity) mode measurements at different coil positions, and plotting the magnetic field as a function of the coil position. Here, only the relative field change is determined, so that the field can be normalized to the position, which is normally used for weighings. The absolute mass value, that is used for this step, is not needed to be known. After that, the field distribution is approximated by a second order polynomial. The quadratic term gives rise to the magnitude of the bias. When, later, the BL is quantified in the velocity mode, it can be corrected with this quadratic term. The second way is based on the nature that the non-linear magnetic field distribution generates higher harmonics in the induced voltage in the velocity mode. Since the bias in the fundamental note is of the same amount as the third harmonic, the amplitude of the third harmonic can be used to correct for the bias, without need for conducting force mode measurements.
Currently, however, this way is not viable due to the high noise level in the induced voltage signal. Further improvements are required in order to reduce the noise level. The remaining uncertainty after correcting for the bias is, in this investigation, on the order of 10% of the magnitude of the bias itself, i.e. about 6.3 · 10 −7 . The reached remaining relative uncertainty after correction is smaller than the relative uncertainties of E2 mass standards of any nominal mass value. The Planck-Balance could thus be used in a same way as conventional balance calibrated by means of E2 mass standards, when considering the non-linearity error of the magnetic field.
In our investigation a standard ('off-the-shelf') voice coil actuator has been used. A custom made voice coil actuator might have an even smaller non-linearity, and thus might not require any correction at all, or at least the remaining uncertainties become negligible.
In further studies we aim at investigating whether the shape of BL(s) changes in time, i.e. whether the parameters of the fitted function will change, e.g. due to environmental changes, such as temperature.