Error sources in the force mode of the 'PB2' Planck-balance


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 53 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 53 - 57 

 
 

ERROR SOURCES IN THE FORCE MODE OF THE “PB2” PLANCK-

BALANCE 
 

N. Rogge1, C. Rothleitner2, S. Lin2, S. Vasilyan1, T. Fröhlich1, F. Härtig2, D. Knopf2 

 
1 Technische Universität Ilmenau, 98693 Ilmenau, Germany, norbert.rogge@tu-ilmenau.de  

2 Physikalisch-Technische Bundesanstalt (PTB), 38116 Braunschweig, Germany, christian.rothleitner@ptb.de  

 

Abstract: 

The PB2 Planck-Balance is a table-top Kibble 

balance, that is designed for the calibration of 

class E2 weights in a range of 1 mg up to 100 g. This 

work presents typical systematic errors which have 

to be considered during the calibration and will 

show results for measurements with small masses. 

Keywords: Kibble balance; measurement 

uncertainty; mass calibration; instrument 

adjustment 

1. INTRODUCTION 

Kibble balance experiments were originally 

designed for the determination of Planck constant 

and played an important role in the redefinition of 

the SI unit kilogram, which entered into force in 

May 2019. Today, they represent one of the two 

methods to achieve the lowest uncertainties for 

mass calibrations of 1 kg. Nonetheless, the working 

principle of a Kibble balance can also be used in 

smaller, less expensive and less complex table-top 

set-ups, when the user does not need to achieve 

higher accuracy. In the Planck-Balance (PB) project, 

two systems were developed in order to calibrate 

class E1 and E2 weights, where the system for the 

calibration of E2 is named “PB2” and is depicted in 

Figure 1. 

 
Figure 1: The PB2 set-up: balance and interferometer 

(right) and signal processing system (left). 

The calibration of weights is performed in two 

modes: velocity mode and force mode. In the 

velocity mode, the coil under test is moved 

sinusoidally relative to the magnetic field of the 

actuator with a frequency of typically 4 Hz and an 

amplitude of 20 µm. From the ratio of coil velocity 

𝑣 and induced voltage 𝑈ind the geometric factor 𝐵𝑙, 
which is the product of the magnetic flux density 𝐵 
and the wire length 𝑙, can be determined. This factor 
can also be derived from the ratio of the 

gravitational force 𝐹g  from a weight and the 

compensation current 𝐼  that is necessary to 
counterbalance the weight with an equal force 

generated by the actuator (electromagnetic force), 

which corresponds to the calibration with the use of 

a calibrated mass (force mode). This procedure of 

force equilibration is similar to the common 

calibration process of an electromagnetic force 

compensation (EMFC) balance. In a Kibble balance, 

both modes are combined in order to determine the 

mass of an unknown weight. However, this work 

that focuses on errors sources in the force mode, the 

results of the force mode measurements are 

presented as determinations of 𝐵𝑙 in order to be less 
dependent on the results of the velocity mode and to 

differentiate between the errors that occur in the 

different modes. 

Similar to the existing EMFC balances, the 

alignment along the gravitational acceleration g and 

air buoyancy has to be considered in the 

measurement during the force mode. Due to this 

different calibration method compared to a 

calibration of the compensating actuator with mass 

standards, additional effects like the alignment 

along velocity mode trajectory, coil current effect 

and coil displacement need to be taken into account. 

2. MEASUREMENT SET-UP 

The PB2 system [1] consists of a commercially 

available EMFC balance, an interferometer, a 

resistor and two digital multimeters (DMMs) of 

type Keysight 3458A-002. The DMMs are regularly 

calibrated against a programmable Josephson 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 54 

voltage standard that is available in PB2 laboratory, 

while the shunt resistor of type Vishay VHA-518 

was calibrated at Physikalisch-Technische 

Bundesanstalt (PTB). All these individual 

components form the PB2 system and its 

measurement infrastructure are based on a modular 

principle, so that upon necessity the procedures of 

replacements or the integration of additional devices 

are maximally simplified. For the purpose of 

characterisation of the system components, 

additional multimeters and current sources are 

available in the set-up that is shown in Figure 1. 

An additional voice coil actuator (measurement 

actuator) that is attached to the load carrier of the 

EMFC balance forming a mechanically rigid 

connection generates the compensation force in the 

force mode and is characterised in the velocity mode. 

The internal actuator (drive actuator) is utilised to 

excite the coil of the measurement actuator in the 

velocity mode and is connected to a current source 

of type HP 3245 in force mode in order to produce 

a constant force offset that is necessary for the 

reduction of the coil current effect that is described 

in section 3. Furthermore, a mirror is attached to the 

load carrier, similarly forming a mechanically rigid 

connection, in order to determine the displacement 

that is used to calculate the velocity of the mirror 

and the measurement coil. For the sake of an 

accurate velocity and voltage measurement, a stable 

time base is required. The acquisition of voltage and 

displacement is triggered by a signal generator 

which is supplied with traceable time base by a 

GPS-disciplined oven-stabilised quartz oscillator. 

 
Figure 2: Schematic of the PB2 set-up. 

In contrast to a normal usage of an EMFC 

balance, the compensation force is generated by an 

actuator placed on the same side of the lever as the 

mass under test as shown in Figure 2. In the PB2 

set-up, this mass can be lifted and lowered remotely 

by an automated weight lifter to minimise 

disturbances caused by an operator of the device. 

Furthermore, a vacuum compatible piezo-based 

linear motor of type N-310 by Physik Instrumente is 

utilized in the weight lifter system that allows the 

PB2 system to operate in vacuum. Due to its 

working principle of a walking motion of 

alternatingly deflected piezo actuators, it does only 

generate mechanical noise during the active lifting 

and lowering process and not during the 

measurement when a compensation current is 

applied. Additionally, it does not generate magnetic 

fields that could disturb the measurement actuator. 

The weighing pan is designed as a fork structure 

as shown in Figure 3, while a fitting fork structure 

is used to lift the weight under test. The weight lifter 

is placed on a plate that can be adjusted in six 

degrees of freedom to allow a small gap between the 

forks without the risk of contact between them. This 

weight lifter design enables the investigation of a 

wide range of sizes and shapes of the mass standards 

including wire weights. 

 

Figure 3: CAD model of the weighing cell mechanics and 

the weighing pan with fork structure. 

The compensation current in the force mode is 

provided by a PID controller that is implemented on 

a digital signal processing system (DSP). The DSP 

is augmented by a power amplifier that drives the 

coil current of the measurement actuator in the force 

mode and a transimpedance amplifier circuit that 

converts the photo currents of the balance’s internal 

position sensor into a voltage signal that is utilized 

to indicate the successful compensation of the 

external force. Since the DSP that is currently used, 

does not allow the use of an external clock signal, 

the closed loop frequency of the implemented 

control algorithm is measured by a frequency 

counter that is connected to the reference clock. 

This provides a traceable time base that is crucial 

for the velocity mode.  

Since the measurements with the PB2 system are 

mostly performed in air, the air buoyancy of the 

weight needs to be corrected as described in 

OIML R 111-1. For this purpose, the density of the 

air has to be determined, which requires the 

measurement of air temperature, humidity and 

pressure. For this purpose, a calibrated climate 

module of type Sartorius YCM20MC is included in 

the PB2 system. 

 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 55 

3. COIL CURRENT EFFECT 

The necessary compensation current in the force 

mode is determined as the difference of the 

compensation currents of the loaded and the 

unloaded state of the balance. Since the coil current 

affects the magnetic field of the actuator, this 

current difference depends on the absolute values of 

both compensation currents [2]. In order to decrease 

the error that is caused by the coil current effect, the 

lever of the EMFC is balanced such, that the 

compensation currents in both loaded and unloaded 

states yield the same absolute value, but different 

polarity. This is ensured in the PB2 set-up by 

feeding a constant current 𝐼𝑖  through the internal 
actuator of the EMFC balance. Therefore, the 

balancing of the lever can be done quickly and 

automatically for any arbitrary value of the test mass. 

To adapt this internal offset current 𝐼𝑖  to the 
changing 𝐵𝑙  of the internal drive actuator, an 
adjustment procedure is performed regularly. A 

small current Δ𝐼𝑖  is added to the previously used 
offset current 𝐼𝑖0 and for both currents through the 
drive actuator, the differences ∆𝐼𝑚 of the absolute 
values of the necessary compensation current in 

loaded (𝐼𝑚+)  and unloaded (𝐼𝑚−)  state are 
determined: 

∆𝐼𝑚 = |𝐼𝑚+| − |𝐼𝑚−| . (1) 

From these measurements the new internal 

current 𝐼𝑖  is calculated by linear interpolation to 
obtain a vanishing ∆𝐼𝑚: 

𝐼𝑖 = 𝐼𝑖0 +
∆𝐼𝑚(𝐼𝑖0)∙Δ𝐼𝑖

∆𝐼𝑚(𝐼𝑖0)−∆𝐼𝑚(𝐼𝑖0+Δ𝐼𝑖)
 . (2) 

The procedure is repeated before a new set of 

force mode measurements is taken to compensate 

for the temperature sensitivity and the general drift 

of the drive actuator. 

4. MAGNETIC PROFILE 

For the calibration of mass standards with a 

Kibble balance, the 𝐵𝑙 that is determined in velocity 
mode is used to convert the current difference in 

loaded and unloaded state into a gravitational force. 

To achieve a low combined uncertainty, it needs to 

be ensured that the position of the coil in loaded and 

unloaded state is the same as the position for which 

the 𝐵𝑙  was determined in the velocity mode. 
Assuming a given uncertainty of the coil positioning, 

the error that is caused by this effect can be reduced 

by adjusting the coil position to be in apex of the 

magnetic flux density of the magnet system. The 

current PB2 system does not provide the possibility 

to adjust the immersion depth of the coil and relies 

on the manufacturing tolerances of the voice coil 

actuator and the coil mounting. Therefore, the actual 

profile was acquired experimentally by determining 

the geometric factor 𝐵𝑙 at different coil positions. 

As shown in Figure 4, the apex was not exactly met 

and the 𝐵𝑙 changes with approximately 0.5 ppm/µm, 
which is sufficient for a comparison of the 𝐵𝑙 
determined in force and velocity mode respectively, 

since the repeatability of the coil positioning is in 

the order of nanometres. The shown profile was 

determined in force mode while changing the set 

point position of the controller. To reduce a change 

of the profile due to the coil current effect, the 

symmetry of compensation currents was adjusted 

for each position considering the different restoring 

force of the differently displaced balance mechanics. 

 

Figure 4: Relative change of 𝐵𝑙  depending on coil 
position. 

5. ALIGNMENT 

In contrast to the calibration of a standard EMFC 

balance, the alignment error cannot be compensated 

by having the same angle of misalignment during 

the calibration with mass standard and weighing 

process. Therefore, the force measurement system 

needs to be aligned parallel to the vector of free-fall 

acceleration 𝑔 . For alignment purposes, mirror 
cuboids that possess a reflective coating on all sides 

are attached to the load carrier of the balance 

(Figure 5). The angles between the surfaces of these 

optical alignment cuboids are measured and well 

known. In accordance to the calibration certificate 

they agree better than 21.8 µrad with a right angle. 

During the alignment process, a second 

interferometer is used to measure the orthogonal 

displacement of the side of the mirror cuboid, thus 

measuring the horizontal displacement, while the 

load carrier is driven up and down by an excitation 

of the lever. The interferometer beams need to be 

aligned to better than 0.4 mrad so that the 

interferometers work properly. From the ratio of 

horizontal to vertical displacement measured by the 

first interferometer, the trajectory of the guiding 

parallel lever system and its misalignment can be 

identified. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 56 

 

Figure 5: Schematic of the alignment axes with an 

alignment cuboid. 

A measurement of the horizontal displacement is 

shown in Figure 6, which sums up a set of six 

repeated measurements of both horizontal axes in x- 

and z-direction. The shown horizontal 

displacements correspond to an angular 

misalignment of 2.8 mrad and 2.2 mrad, 

respectively. The resulting cosine errors of 3.9 ppm 

and 2.4 ppm of the determination of the 

displacement amplitude are therefore well known 

and can be corrected during the data analysis. This 

procedure needs to be repeated every time the angle 

between the trajectory of the guided coil and the 

normal vector changes, which occurs when the 

balance or the mirrors are temporarily unscrewed 

during redesign and readjustment of the system. 

Since this is done regularly in the recent 

devolvement process of PB2, there are no 

investigations on the long-term stability of these 

angles yet. 

An additional horizontal displacement of about 

maximum 6 nm occurs due to the parallel lever 

system guides the load carrier on a circular 

trajectory. The horizontal velocity caused by this 

effect, which generates an additional induced 

voltage, occurs mainly at the maximum and 

minimum of the sinusoidal motion and is close to 

zero when the vertical velocity is at its maximum 

and minimum. Therefore, the voltage that is induced 

due to this horizontal motion contributes only a 

small error to the determination of the amplitude of 

the sinusoidal induced voltage caused by vertical 

motion and is rejected by the sine fitting of the data. 

 

 

 

Figure 6: Horizontal displacement of the load carrier.   

With an autocollimator and the reflective surface 

of a liquid pool, the normal vector of the top surface 

of the mirror cuboid can be adjusted relative to 𝑔. 
The water pool is only present during the adjustment 

procedure and regular checks of the mirror 

orientation. Since the orientation of the trajectory 

relative to the mirror surface is known from the 

horizontal displacement, the trajectory adjustment 

follows from the adjustment of the top surface. 

6. RESULTS 

The determination of 𝐵𝑙 in force and velocity 
mode was performed with several different E1 mass 

standards in a range from 1 mg to 50 g. The 

systematic deviation relative to the result with a 

calibrated 20 g standard as well as the statistical 

uncertainty over time period of 8 hours are shown 

in Figure 7. The statistical uncertainty is composed 

of the relative experimental standard deviations of 

the mean of the determinations of 𝐵𝑙 in velocity 
mode and force mode. While the contribution to the 

statistical uncertainty of the velocity mode 

experiment is independent from the weight under 

test, the relative standard deviation increases for 

smaller mass values. The maximum permissible 

error (MPE) of E1 and E2 standard weights 

according to OIML R-111 is also shown for 

comparison. 

Compared to the MPE of E2 standards, the 

results are quite satisfying for the higher nominal 

values. However, for the smallest standards, there 

are deviations that need to be identified in future 

investigations. 

 

 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 57 

 

Figure 7: Deviation between force and velocity mode. 

7. SUMMARY AND OUTLOOK 

The main error sources in the force mode of the 

PB2 system were identified and a good agreement 

of force and velocity mode was achieved for most 

of the measurement range. Future work will aim at 

identifying the deviations in small mass range and 

improving the statistical part of the uncertainty. This 

can be achieved by improving the controller 

hardware that is generating a significant fraction of 

the noise on the compensation current due to its 

resolution of 16 bit for ADC and DAC. A signal 

processing system based on a combination of 

microprocessors, a FPGA chip and 24 bit 

Delta-sigma ADC is currently under development. 

In the range of masses greater than 50 g, the 

system performance is limited by nonlinear 

deformation phenomena that are subject to ongoing 

research. Further investigations are also planned 

regarding the long-term stability of the mechanical 

system and its adjustment status. 

ACKNOWLEDGEMENT 

The research of this project is funded via the 

program “Validierung des technologischen und 

gesellschaftlichen Innovationspotenzials – VIP+”, a 

program of the German Federal Ministry of 

Education and Research (BMBF), and is managed 

by VDI/VDE Innovation + Technik GmbH. The 

authors thank also L. Günther, J. Schleichert und M. 

Spoors for fruitful discussions. Furthermore, the 

authors want to acknowledge the help of Working 

Group 1.11 of PTB for the calibration of mass 

standards and Working Group 2.61 for the 

calibration of standard resistors. 

8. REFERENCES 

[1] N. Rogge, S. Lin, C. Rothleitner, S. Vasilyan, 
“Excitation frequency dependent deviations during 

the “velocity mode” of Bl measurements in the 

Planck-Balance”, in Proc. of 23rd IMEKO TC4 

International Symposium, Xi’an, China, 2019. 

[2] S. Li, F. Bielsa, M. Stock, A. Kiss, H. Fang, 
“Coil-current effect in Kibble balances: analysis, 

measurement, and optimization”, Metrologia, 

vol. 55, pp. 75-83, 2018 

 
 

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