Development of a high precision electrostatic force balance for measuring quantity of dispensed fluid as a new calibration standard for the becquerel

The 2019 redefinition of the kilogram not only changes the way mass is defined but also broadens the horizon for a direct realization of other standards. The true becquerel project at the national institute of standards and technology is creating a new paradigm for realization and dissemination of radionuclide activity. Standard reference materials for radioactivity are supplied as aqueous solutions of specific radionuclides which are characterized by massic activity in the units becquerel per gram of solution, Bq/g. The new method requires measuring the mass of a few milligrams of dispensed radionuclide liquid solution. An electrostatic force balance is used, due to its suitability for a milligram mass range. The goal is to measure the mass of dispensed fluid of 1 mg–5 mg with a relative uncertainty of less than 0.05%. A description of the balance operation is presented. Results of preliminary measurements with a reference mass indicate relative standard deviations less than 0.5% for tens of tests and differ 0.54% or less from an independent measurement of the reference mass.


Introduction
Standard reference materials (SRMs) are usually supplied as aqueous solutions of specific radionuclides with a known massic activity.Massic activity is the activity of a given 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.radionuclide per mass of the aqueous sample, typically expressed in units of becquerel per gram [1].A complicated procedure involving multiple steps of sample preparation and characterization are required to certify SRMs, and in some cases, current methods do not provide low enough uncertainty for end applications [2,3].The true becquerel project at National Institute of Standards and Technology (NIST) aims for a new paradigm to realize and disseminate radionuclide activity using decay energy spectroscopy.This method, illustrated in figure 1, uses a superconducting transition edge sensor which is a cryogenic calorimeter measuring heat energy in the pico-Joule range [3,4].This allows simultaneous counting and energy measurement of individual decay events.The radionuclide solution is dispensed on a mm-scale gold foil sample with a few tens of µm in thickness to minimize its thermal capacity and maximize the energy resolution of the transition edge sensor measurement.After drying, the radionuclide residue is left on the surface and is encapsulated by folding the foil.For the full characterization of the solution in Bq/g the amount of solution dispensed to the sample, must be quantified.This requires a technique to measure milligramscale quantities of dispensed liquid with high accuracy.An electrostatic force balance (EFB) with an integrated liquid dispenser is proposed to determine the mass of a liquid droplet deposited to the gold foil sample.The balance measures the change in mass of a reservoir from which the liquid is dispensed to minimize evaporative effects.Electrical and optical metrology provide SI-traceability of the mass to fundamental constants (i.e.Planck constant, speed of light in vacuum and hyperfine splitting frequency of 133 Cs), as described in more detail in previous work [5].SI-traceability integrated in the measurement device itself avoids safety concerns related to exchanging of reference mass artefacts between radionuclide laboratories and mass calibration facilities where even the remote possibility of the presence radioactive materials is unacceptable.It also reduces the calibration chain, greatly simplifying the logistics of the calibration.
Globally, several EFB projects have significantly advanced the field of precision small force measurement.For example, NIST's EFB which operates in vacuum and measures mass of 1 mg with a relative combined standard uncertainty of 7 × 10 −6 [5] or the nanonewton force facility at the Physikalisch-Technische Bundesanstalt (PTB).Other balances at the national physical laboratory, the Industrial Technology Research Institute (ITRI), the National Institute of Metrology (NIM) of China and at NIST operate in air, achieving standard uncertainties around 10 −4 [6][7][8].These balances are traceable to the Planck constant and are used as primary standards or calibration tools for milligram masses, or atomic force microscope cantilevers, and use a four-bar linkage as the balance guide mechanism.In contrast, institutes such as the Korean Research Institute of Standards and Science (KRISS), the Federal Institute of Metrology (METAS) or the PTB have done measurements in the nano Newton force range using commercial ultramicrobalance [9][10][11].These instruments use reference masses for calibration.Also, a hybrid balance using electrostatic and magnetic forces is employed by for example the Technische Universität Ilmenau [12].Various capacitor designs are used, such as concentric cylinders, parallel plates or moving dielectrics.These balances are all designed to measure forces that can be repeatably applied to the instrument.
The challenge of the electrostatic balance in this paper is to measure mass of dispensed fluid.The larger total payload of around 50 g from the dispenser drives the guide mechanism's mechanical requirements and exceed the payload of conventional microbalances in the target mass and uncertainty range.For this purpose, a modular four-bar linkage design in combination with a sphere-flat capacitor design is used.The design of the four-bar linkage and the capacitor are described in more detail in subsequent sections of this paper.In contrast to previous work, where the differential force applied to balances can be repeated (i.e. by repeatedly loading a test mass), each dispensing process is unique [13].As a result, the exact same measurement cannot be reproduced to check the repeatability of the balances.Therefore, only the standard deviation of a single measurement obtained from the balance stability, rather than the mean of multiple mass measurements, can be used to characterize statistical uncertainty of the liquid mass measurement.Since no standard method is currently available for producing a liquid mass with sufficient accuracy, the balance accuracy and repeatability is therefore assessed using a test mass calibrated on a commercial balance.

Working principle and balance setup
An EFB compensates a change in gravitational force with an electrostatic force using a capacitor.The gravitational force from a test mass can be calculated using where the electrostatic force F el is calculated by the capacitance gradient dC/dx and the change in applied voltage ∆V.In addition, the force of the guide mechanism F M causes force noise due to its stiffness and small deviations to the target position ∆x.This equation does not consider surface potential of the electrodes [5]. Figure 2 shows the principle of the balance system.
The balance operates with a sphere-flat capacitor configuration where the flat electrode is coupled to a modular flexure four-bar linkage, realizing approximately linear motion for the free end in the x-direction for small displacements.To regulate the balance position using feedback control, an interferometer laser beam reflecting from the upper surface of the flat electrode measures displacement.As liquid is dispensed, the weight of the dispenser's reservoir decreases, which is counterbalanced by the electrostatic force of the capacitor.Calculating the electrostatic force requires determination of the capacitance gradient during the capacitance measurement mode.In this mode, the capacitor is connected to a capacitance bridge while a voice coil actuator on the opposite side of the balance controls position.This permits feedback position control to change the separation between the sphere and flat while concurrently measuring capacitance of the sphere-flat electrode for determination of the capacitance gradient.In addition to the primary components, the balance mechanism features two counterweights, attached on its left side.The upper counterweight can be adjusted in the y-direction, moving the mechanism's effective center of mass through the central pivot point and tuning the force offset compensated by the capacitor.The lower counterweight, adjustable in the x-direction, acts as an inverted pendulum, allowing the adjustment of the balance stiffness.As illustrated in figure 3, both counterweight are mounted on piezo stages, enabling automated control of the their position.This setup enhances the balance sensitivity for mass offset and stiffness adjustments, as well as increasing operation speed and reducing user intervention.
The capacitor assembly is integrated into the center of the balance, facilitating easy access and replacement of the flexures, which, in turn, enables fundamental studies of a large number of different flexure geometries.The assembly also includes the fiber launch optics for a commerical fiber Fabry-Pérot interferometer with a nominal resolution of 1 nm.An integrated tip tilt stage in the fiber launch alows alignment of the interferometer beam to the local graviatational vector using a reference surface and bubble level.This is essential to minimize a cosine error in force measurment as discussed in previous work [14].A conservative estimated missalignment of 0.5 degree would lead to an relative error of 4.0 × 10 −5 which is within the targeted uncertainty budget.The piezo dispenser is attached to the bottom of the mechanism, and to minimize corner loading error, as further discussed in a subsequent section of this work, the dispenser's reservoir is aligned with the center of the spherical electrode along the balance x-axis.Electrical connection to the dispenser and capacitor electrode is made using miniature coaxial cable along the rigid parts of the balance, and gold wires with a diameter of 50 µm to bridge the gaps necessary for the flexures.The wires are connected near the flexures in order to minimize relative displacement of the wire connection points during balance motion.No increase in mechanism stiffness has been observed from addition of multiple bridging wires.An automated mass handler is centrally integrated, and also aligned with the sphere electrode.Considering each dispensing process as unique, the mass handler can be utilized to test the repeatability of the balance mass measurements and also allows cross-checking accuracy with reference masses.Although the general overview gives an impression of the balance mechanism, the complexity of the four-bar linkage and the capacitor requires a deeper description.The following sections will focus on these subcomponents, their design and measurements.

Four bar linkage
The four-bar linkage is a central feature of the balance mechanism and plays an important role as the mechanical guide mechanism in the balance.Its design and implementation are described below.

Design of the four-bar linkage
Using a four-bar linkage, as shown in figure 4, as the balance guide mechanism reduces the corner loading error typically seen in beam balance configurations, while simultaneously enabling a compact, tabletop design [14].The symmetric design of the two linkages ensures an equivalent distribution of inertia on both sides when other components are added, reducing the system's sensitivity to external influences like seismic vibrations [15].
To compensate for an offset in force equilibrium and adjust the stiffness, counterweight masses are used as described in the previous section.Counterweights offer the potential for a higher degree of stability over time when compared to buckling springs, thereby minimizing potential drift in balance stiffness and force offset [16].The modularity of the design offers numerous advantages, such as the ability to use functional materials like copper beryllium for critical parts like the flexures, minimizing hysteresis effects.However, the modular design approach creates some challenges, particularly with respect to the tolerances in the positioning of the flexures caused by mounting.Position tolerances of the flexures can result in nonlinear motion, corner loading error, buckling, and friction in the clamping area.To overcome these challenges, an innovative design concept has been applied.The flexures are fabricated in a planar geometry from copper beryllium sheets.All four flexures required for each side of the 4-bar linkage are photo masked and etched from a single copper beryllium sheet, maintaining their connection to the main sheet throughout the installation process, see figure 5. Once installed, the flexures are detached with a knife from the main sheet at predefined cutting areas.This process ensures the precise positioning of the flexure sheets, defined by the etching mask during the etching process and the dowel pins used to attach the flexures during the installation process.Furthermore, by partially etching the flexure sheets in the third dimension, the compliant sections are defined by a reduced sheet thickness of 50 µm, while the thickness in the clamped section is 350 µm.This increses the bending stiffness in the clamped section significantly compared to the complaint section due to the 3rd order relationship between bending stiffness and thickness.Therefore, the clamped section can be assumed as rigid compared to the compliant section.This technique minimizes any unpredictable effects due to imperfections in the clamping area.To verify the four-bar linkage functionality, autocollimator measurements are presented in the next section to measure the rotation of the coupler and estimate the corner loading error.Additionally, stiffness and hysteresis measurements were conducted to characterize the four-bar linkage performance.

Corner loading error
An important part of qualifying the precision in the fourbar linkage motion involves measuring the pitch and yaw of the guide mechanism during translation.For this purpose, the four-bar linkage position is controlled stepwise to five positions within a range of 600 µm over a total measurement time of 4.5 h.The angular change of the fourbar linkage coupler is recorded using an open hardware autocollimator [17].Although the resolution of this autocollimator is lower than commercially available systems, it is sufficient for the current work.Figure 6 shows the results of this measurement.Subsequently, the relative corner loading error is calculated.This calculation estimates the impact of the pitch and yaw due to misalignments and manufacturing tolerances on the mass measurement.The relative corner loading error ∆F corner can be calculated using the following equation [14] In this equation, ∆L y and ∆L z signify the distance between the center of the applied electrostatic force and the center of the gravitational force of the dispenser's liquid reservoir.L y and L z represent the distance between the applied force of the capacitor and a virtual pivot point, calculated using the change in angles of rotation measured at the four-bar linkage's free end over the displacements shown in figure 6.These distances can be calculated by Over the total travel range of ∆x of 600 µm, a pitch of ∆ϕ z = 2 • 10 −6 rad and a yaw of ∆ϕ y = 8 • 10 −6 is measured.Using a conservative estimation of the distances ∆L y and ∆L z of 5 mm (the inner diameter of the fluid reservoir in the current design), the relative error due to corner loading is less than 0.007%, according to equation (2).Although this error does not limit the goal of 0.1% for k = 2, it may still be beneficial to implement an adjustment in a future four-bar linkage design to minimize corner loading error for applications that require even higher accuracy.

Stiffness & hysteresis
Minimizing the stiffness of the four-bar linkage is important to keep the force noise due to the guide mechanism as low as possible to achieve low force uncertainties.Therefore, an inverted pendulum is implemented to adjust the mechanism stiffness.An analytical description of the balance stiffness can be formulated from the differential equation of motion using the Lagrange equations [14].Solving this equation for the factor of the state variable that represents the stiffness K, we can derive a solution for K as follows.
where k b represents the bending stiffness of a single flexure in the four-bar linkage.The term m m gL xm indicates the combined gravitational stiffness of the mechanism components, while m cm gL xcm signifies the gravitational stiffness of the counter mass m cm .The adjustable distance L xcm can be changed in nanometer increments using the piezo stages.During the dispensing process or while picking and placing the reference mass, it is not guaranteed that the balance will maintain its null position.This deviation can induce mechanical hysteresis, which in turn can influence the mass measurement [18].Therefore, the flexure sheets are made from copper beryllium, having a loss factor at least 25 times less than aluminum alloys to significantly minimize hysteresis effects [19].Figure 7 displays a stiffness measurement for two different balances, where the balance position was controlled over a range of 300 µm or 500 µm, with five substeps for a balance with copper-beryllium and aluminum flexures, respectively.At each position step, the measured voltage and capacitance gradient was used to calculate force.As shown in the left graph, the mechanism stiffness measured is less than 0.07 N m −1 .In a regular mass measurement, as further described in the next section, the standard deviation of the mean in position is typically less than 20 nm for a measurement time of 120 s.According to equation (1), this causes a force uncertainty of 1.4 nN and will be considered in the total uncertainty budget as flexure elastic force.Comparing the measurement results of the modular copper beryllium four-bar linkage on the left side, with a similar measurment of monolithic four-bar linkage made from aluminum alloy 7075, it is noticeable that hysteresis effects have been reduced significantly.Although the measured stiffness of the mechanisms shown in figure 7 are different, and slightly different displacement ranges are examined, the difference in mean measured force between the forward and backward motion still indicates a difference of up to 2 µN for a displacement of ±150 µm.Loading or unloading the balance with a 5 mg mass typically causes displacements of around 10 µm.Hysteresis effects due to smaller displacements will be considered in future work.Since the capacitance gradient is necessary to convert the applied voltage to force, the next chapter will provide a detailed examination of the capacitor and its role in this mechanism.

Capacitor
The capacitor is one of the main components in an EFB.Its capacitance gradient defines the electrostatic force.Its design must therefore reflect the desired range of forces while ensuring the force is applied in the appropriate location.

Capacitance theory
The electrostatic force in general can be described by the following equation [20] where V is the vector of voltage (and its related transform) potential in our system, C the capacitance matrix and x the relative position of the electrodes along the travel path.The capacitance is measured by a capacitance bridge, which has three terminals, LOW (1), HIGH (2) and GND (3), illustrated in figure 8. Therefore, the capacitance matrix of our system is a 3 × 3 matrix, where the row sum is equal to zero and can be written for a closed system as [21].
where C c12 , C c13 and C c23 describes the cross capacitance between the three terminals.The voltage vector is a 3 × 1 vector, describing the applied voltage on each electrode During the mass measurement two of the three electrodes are grounded and a high voltage is only applied to one electrode, leading to V 2 = V 3 = 0. Consequently, the electrostatic force can be formulated as The bridge only measures the capacitance between LOW (1) and HIGH (2), denoted by C c12 , leaving the capacitance gradient dC c13 /dx unknown, which also can be called as parasitic capacitance.To solve this issue, the wiring could be switched to allow measuring the capacitance C c13 .This measured value can then be used to correct the calculated force.However, this solution necessitates additional measurements, which amplifies the total uncertainty and leads into a more time-intensive process, since dC c13 /dx must be measured frequently to capture possible capacitance changes over time.A more straightforward solution is to design the capacitor in a way that reduces the capacitance gradient dC c13 /dx to be insignificantly small.This concept will be further discussed in the subsequent section.

Design of the sphere-flat capacitor
The design and functionality of the sphere-flat capacitor in the EFB system is presented in this section.Figure 8 shows the cross section of the capacitor.The flat electrode is a goldcoated optical window with a specified flatness of 60 nm attached electrically isolated via a connector to the four-bar linkage coupler.The connector also provides electrical isolation between the flat electrode and the mechanism.Since the coupler is connected to the four-bar linkage the flat electrode can move linearly in the x-direction.The top surface of the flat electrode acts as a mirror for the interferometer displacement measurement.
Below the flat electrode is the sphere electrode, also electrically isolated from the mechanism.During the mass measurement an electrical potential of up to 1000 V is applied to the sphere electrode, leading to a maximum force of around 200 µN.This potential causes an electrostatic field and therefore an electrostatic force between the sphere and flat electrode.To calculate this force, equation ( 8) requires two capacitance gradients dC c12 /dx and dC c13 /dx where C c12 can be measured by the capacitance bridge.Given that the threeterminal capacitance bridge rejects capacitance to ground, it doesn`t measure C c13 .By design, this capacitance gradient is reduced to a point where it can be approximated as zero.
To accomplish this, a cylindrical shield surrounds the sphere, shielding the sphere electrode and all moving parts of the balance which are connected to ground during the capacitance measurement.Consequently, the electrostatic force can be expressed as The sphere-flat capacitors provide certain benefits compared for example to a cylindrical capacitor.Invariance in capacitance to the sphere's rotation and lateral displacements in the y and z directions of the sphere and flat electrode results in reduced requirement for adjustments.Furthermore, this configuration also contributes to less capacitance drift from elongation of the mechanism due to thermal expansion compared to concentric or plate capacitors.The nonlinear variation of capacitance with electrode separation in a sphere-flat capacitor complicates evaluation of the fitting function describing the capacitance gradient of the capacitor.Therefore, the following section is focused on the capacitance measurement and fitting function.

Capacitance measurement and fitting function
The process of determining the electrostatic force as expressed by equation ( 9) requires calculation of the capacitance gradient dC c12 /dx.To accomplish this, two steps are required.The empirical measurement of capacitance at different fixed x-axis positions, and subsequent mathematical modeling using a fitting function to determine the capacitance gradient.
In the first step, a capacitance bridge is connected as shown in figure 7. It is essential to note that during this phase, the balance position cannot be feedback controlled by the capacitor due to the active connection to the capacitance bridge.Therefore, a voice coil is attached to the system, which acts as an auxiliary actuator, shown in figure 3.This enables the mechanism to move and hold the flat electrode at various positions.
In the second step the capacitance vs. position data is fitted, using an appropriate mathematical model.Considering the unique characteristics of a sphere-flat configuration, Maxwell's equation describing a sphere-flat capacitor is used [21].To accommodate nonidealities, a polynomial term is added to the Maxwell's equation.Thus, the fitting function for the capacitance C can be written as )) Here, ε represents the permittivity of air, r the sphere electrode's radius, x g the gap between the sphere and flat electrode, and p i refers to the coefficients of the polynomial term.To fit these parameters, a least squares method is used.To capture the characteristics of the sphere-flat capacitor, an additional constraint is introduced to ensure that dC/dx 2  g remains positive at all times.For mass measurement, the flat electrode is moved to five positions along the x-axis while capacitance is measured at each position, as shown in figure 9. To verify the fitting function, the capacitance is also measured for a larger number of positions in a separate experiment, with a step size of 2 µm.The determination of the fitting function for the true capacitance gradient is more important than evaluating the Maxwell model for the capacitance, since the capacitance gradient used to calculate the electrostatic force is empirically determined.The derivative of the linear fit between these small sub-steps approximates the true capacitance gradient value, allowing an evaluation of the fitting function.
Despite the presence of noise in the relative difference of the capacitance gradient derived from the fitting function and the closely spaced 2 µm sub-step measurements, no discernable trend indicative of a systematic deviation is identified.Therefore, the fitting function can be used to describe the capacitor behavior.The final validation of the fitting function can be done by a mass measurement, described in the following section.

Mass measurment
The evaluation of the balance accuracy necessitates conducting a mass measurement.For this purpose, a reference wire mass of 4.954 mg is used, with a relative standard deviation of 0.02%, as measured using 35 weighings with a commercial high precision ultra-microbalance.The balance uses an internal E1 reference mass for calibration, sufficient for preliminary comparisons with the dispensing EFB.The weightings of the reference wire mass were done next to the experimental setup under similar environmental conditions such that buoyancy correction is not necessary when comparing the two mass measurements at the current uncertainty.Further uncertainties and traceability of the reference mass will be considered in future work in the context of determining true mass.The electrostatic balance software allows an automated measurement cycle.This cycle starts with several capacitance gradient measurements as discussed in the previous section.Following this, the capacitors connection switches automatically from the capacitance bridge to a high voltage amplifier, operating typically around 600 V.A real-time control system maintains a null position by outputting a voltage to the amplifier.At typical operation, the gap between the two electrodes is about 500 µm.The wire mass is then automatically placed on the balance using piezo stages, as shown in figure 3, returning to a home position afterward.During each mass measurement, the voltage across the electrodes is monitored in three stages: unloaded, loaded, and re-unloaded, enabling a differential weighing method [22].Once a series of mass measurements are completed, the capacitance bridge is reconnected for additional capacitance measurements.The force calculation uses the mean capacitance gradient calculated from the capacitance measurements taken before and after each set of mass measurements.Figure 10 provides a comprehensive overview of one measurement cycle, displaying results for both mass measurement and capacitance gradient, alongside corresponding temperature and humidity data.The interferometer data are compensated for changes in refractive index based on the temperature, barometric pressure and humidity measurements.
In total, 65 mass measurements were conducted in this series, yielding a mean value of 4.964 mg, with a relative standard deviation of 0.365%.The standard deviation of the measurement is therefore seven times greater than the target relative standard uncertainty of 0.05%.Additionally, a deviation of 0.2% between the measured mean value and the reference mass is observed for the trial shown in figure 10.This discrepancy could be due to systematic errors, such as the parasitic capacitance gradient or mechanism hysteresis.The capacitance gradient was measured 56 times for the data series To determine the combined standard uncertainty of the balance, all type A and B uncertainties need to be characterized.For a preliminary examination only, main contributors were discussed in this paper and summarized in table 2.
Uncertainties in the displacement, voltage, or capacitance measurements are typically in the order of 10 −6 or less according to experience in previous work [5].To determine the statistical uncertainties due to hysteresis, requires further investigation.However, the main contributor of uncertainties is the standard deviation in weighing, in the current state of the project.Plans are in place to relocate the balance to a better controlled environment characterized by lower seismic and acoustic noise levels and improved environmental stability to reduce the standard uncertainty in mass measurements.Efforts will also be made to refine the control loop to minimize force noise and modify the design to incorporate eddy current damping and better draft shielding.In addition, a design optimization process of the replicable flexure sheets is ongoing, to improve the balance static and dynamic behavior.

Conclusion and outlook
This paper presents a tabletop EFB for mass metrology traceable to physical constants through integrated electrical and dimensional measurements.It is designed to measure mass of dispensed liquid for a novel method of producing SI-traceable SRMs for radionuclide activity.The working principle and experimental setup of the EFB are elucidated, demonstrating a new modular design that uses etched flexure sheets to minimize side effects of mounting and clamping.The functionality of this design to confine corner loading errors to an acceptable range has been verified.Moreover, the design of a sphere-flat capacitor was introduced, which, due to its inherent symmetry, significantly reduces the adjustment effort compared to cylindrical or flat-flat capacitor designs.This approach also aims to minimize thermomechanical drift effects in the capacitance gradient.A critical milestone was reached when conducting a mass measurement using a reference mass.Although the current standard deviation is larger than the targeted uncertainty, steps are being undertaken to improve the precision.These include plans to relocate the balance to an environment with lower noise levels and enhanced environmental stability as well as adding acoustic isolation, thereby reducing the standard uncertainty in mass measurements.In the future, additional type A and type B uncertainty characterizations will be performed.Once the performance of the balance has been verified using a reference mass, measurements of the dispensed liquid mass of the radionuclide solution will be conducted.Thus, the outlined methodology and advancements offer a step towards reliable characterization of radionuclide standards.

Figure 1 .
Figure 1.Sample preparation for decay energy spectroscopy.

Figure 6 .
Figure 6.Measured pitch and yaw of the for-bar linkage coupler using an autocollimator.

Figure 7 .
Figure 7. Stiffness and measurement.Left chart displays a stiffness and hysteresis measurement of the modular four-bar flexure mechanism with copper beryllium flexure sheets.The right chart displays, and stiffness and hysteresis measurement of a similar monolithic four-bar linkage made from aluminum alloy 7075.The difference in force between the up (blue) and down (red) sweep for the same position is displayed in the lower charts.

Figure 9 .
Figure 9. Capacitance measurement and verification of the fitting function.The top chart shows capacitance data used for the fitting function in black, the fitting function in blue, and as reference the measured capacitance of the sub steps in gray, not used for the fit.The middle chart shows the capacitance gradient of the fitting function in blue, and the capacitance gradient of the sub-steps obtained by finite differences of two successive measurements.The bottom chart shows the relative differences between the capacitance gradient of the fitting function and the finite differences.

Figure 10 .
Figure 10.Mass and capacitance measurement cycle.First chart from the top shows mass measurements in blue, second chart shows capacitance gradient for the setpoint used in the mass measurement in yellow, third and fourth chart show changes in temperature and humidity during the mass and capacitance measurement.The horizontal solid lines represent the mean values of the measured data with the dashed lines as standard deviation.

Table 2 .
Main uncertainty components in the EFB mass measurements for a 5 mg mass.