ACTA IMEKO  September 2014, Volume 3, Number 3, 68 – 72  www.imeko.org    ACTA IMEKO | www.imeko.org  September 2014 | Volume 3 | Number 3 | 68  Comparison of milligram scale deadweights to electrostatic  forces  Sheng‐Jui Chen, Sheau‐Shi Pan, Yi‐Ching Lin   Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu, Taiwan 300, R.O.C.      Section: RESEARCH PAPER   Keywords: Deadweight force standard; electrostatic force actuation; capacitive position sensing; force balance  Citation: Sheng‐Jui Chen, Sheau‐Shi Pan, Yi‐Ching Lin, Comparison of milligram scale deadweight forces to electrostatic forces, Acta IMEKO, vol. 3, no. 3,  article 14, September 2014, identifier: IMEKO‐ACTA‐03 (2014)‐03‐14  Editor: Paolo Carbone, University of Perugia   Received May 13 th , 2013; In final form August 26 th , 2014; Published September 2014  Copyright: © 2014 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits  unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited  Funding: This work was supported by the Bureau of Standards, Metrology and Inspection (BSMI), Taiwan, R.O.C.  Corresponding author: Sheng‐Jui Chen, e‐mail: sj.chen@itri.org.tw    1. INTRODUCTION  Micro- and nano-force measurement is of great interest in recent years among several national measurement institutes (NMIs) [1-6]. The Center for Measurement Standards (CMS) of the Industrial Technology Research Institute (ITRI) has established a force measurement system based on electrostatic sensing and actuation techniques. The system is capable of measuring vertical forces up to 200 N based on a force balance method. The system mainly consists of a flexure stage, a three-electrode capacitor and a digital controller [7]. The schematic drawing of the system is shown in figure 1. The three-electrode capacitor is used simultaneously as a capacitive position sensor and an electrostatic force actuator. The position of the center electrode is measured by comparing the capacitances between upper capacitor C1 and lower capacitor C2 formed within the three electrodes (see figure 2). The differential capacitance was detected using an inductive- capacitive resonant bridge circuit. The position detection is performed at a radio frequency (RF), say, 100 kHz, a frequency depending on the capacitance values and the design of the sensing bridge circuit. For electrostatic force actuation, the top and bottom electrodes are applied with two high voltage, audio frequency sinusoidal signals to generate a compensation electrostatic force fe to balance the force under measurement fm. The balance condition fm = fe is maintained by the digital controller by keeping the flexure stage at its zero deflection position. Some parts of the force measurement system were upgraded for performance improvements. A new design of Figure 1. Schematic drawing of the force measurement system.   ABSTRACT  This paper presents a comparison of milligram scale deadweights to electrostatic forces via an electrostatic sensing & actuating force  measurement system.  The electrostatic sensing & actuating force measurement system is designed for measuring force below 200 N  with an uncertainty of few nanonewton.   The force measurement system consists of three main components: a monolithic flexure  stage, a three‐electrode capacitor for position sensing and actuating and a digital controller.  The principle of force measurement used  in this system  is a static force balance,  i.e. a force to be measured  is balanced by a precisely controlled, electrostatic force. Four  weights  of  1  mg  to  10  mg  were  tested  in  this  comparison.  The  results  of  the  comparison  showed  that  there  exist  extra  stray  electrostatic forces between the test weights and the force measurement system. This extra electrostatic force adds a bias force to the  measurement result, and was different for each weight. In principle, this stray electrostatic force can be eliminated by  installing a  metal  housing  to  isolate  the  test  weight  from  the  system.  In  the  first  section,  we  briefly  introduce  the  electrostatic  sensing  and  actuating force measurement system, and then we describe the experimental setup for the comparison and the results. Finally, we  give a discussion and outlook.  ACTA IMEKO | www.imeko.org  September 2014 | Volume 3 | Number 3 | 69  copper-beryllium flexure stage was installed in the system, which has a counter weight balance mechanism and a lower stiffness of 13.08 N/m. Figure 2 shows a picture of the new flexure stage, where the counter weight and the gold-plated Cu- Be flexure stage are visible. A new set of gold-plated, polished electrodes was assembled as a three-electrode capacitor and put into operation. The capacitance gradient for the new three- electrode capacitor was measured. 2. EXPERIMENTAL SETUP  In this experiment, the compensation electrostatic force is compared to the deadweight by weighing a weight with the electrostatic sensing and actuating force measurement system. 2.1. Deadweight  We used five wire weights with nominal mass values and shapes of 1 mg-triangle, 2 mg-square, 5 mg-pentagon and 10 mg-triangle to generate vertical downward forces. These weights meet the metrological requirement of the OIML class E1 and were calibrated against standard weights using a mass comparator balance. The calibration results are compiled in Table 1. The forces can be derived from the calibrated mass values and the local acceleration of gravitation g = 9.78914 m/s2 as fw = m (1-a/w) g, where a and w are densities of the air and the weight, respectively. These weights were loaded and unloaded by a DC motor actuated linear translation stage. 2.2. Electrostatic sensing & actuating force measurement system  As shown in Figure 3, the compensation electrostatic force fe generated by the force measurement system is determined by the following equation: 2 22 2 11 2 1 2 1 VSVSfe  (1) where S1, S2 are the capacitance gradients of the top and the bottom capacitors C1, C2 and V1, V2 are voltage potentials between the top and the bottom capacitors, respectively. Using the parallel-plate capacitor as the model for capacitor C1 and C2,                               432 00 1 1)( d x d x d x d x d A xd A xC  (2)                               432 00 2 1)( d x d x d x d x d A xd A xC  (3) where 0 is the vacuum permittivity, A is the effective area of the electrode and d is the gap distance between electrodes when the center electrode is vertically centered. The capacitance gradients S1 and S2 can be expressed as                          432 0 1 1 54321)( d x d x d x d x S dx dC xS (4)                             432 0 2 2 54321)( d x d x d x d x S dx dC xS (5) where S0 = 0A/d2 is the capacitance gradient at x=0. The electrostatic force can be written as )()( 2 1 )( 22 2 10 2 2 2 10 VVS d x VVSxf e  (6) The voltages V1 and V2 contain the RF detection signal Vdsindt, audio frequency high voltage actuation voltages Va1sinat, Va2sinat and the electrodes’ surface potentials vs1, vs2, namely 111 sinsin saadd vtVtVV   (7) 222 sinsin saadd vtVtVV   (8) The high voltage actuation signals are provided by a full range 10 V 16-bit resolution digital-to-analog converter within the digital controller and an ultra low-noise high- voltage amplifier. To make the electrostatic force linearly proportional to a control voltage vc, we set )(11 cba vVAV  (9) )(22 cba vVAV  (10) Figure 2. Picture of the new flexure stage.  Table 1. Mass calibration result. Nominal mass (mg) Conventional mass (mg) Uncertainty, 95% confidence (mg) 1 1.00096 0.0003 2 2.00116 0.0003 5 5.00124 0.00065 10 10.0021 0.00048 C1 C2 d  x d + x V1 V2 C1 C2 d  x d + x V1V1 V2V2 Figure 3. Three‐electrode capacitor for electrostatic force actuation.   ACTA IMEKO | www.imeko.org  September 2014 | Volume 3 | Number 3 | 70  where A1, A2 are amplification factors of the high-voltage amplifier. The term Vb is a constant and determined by the value of S0 and the upper limit of the force measurement range. Taking the gain difference between channels of the high-voltage amplifier, substituting equations (7)-(10) for V1 and V2 in equation (6), we obtain an equation for the electrostatic force fe     terms)(ac )( 220 222 00 2 00   dsbccbe bVvSVvbaASvVASf , (11) where a is the gain difference fraction, i.e. a=(A1A2)/(A1+A2), A0 is the mean gain factor, b is the offset fraction x/d, vs2 = (vs12-vs22)/2 and vc is the control voltage. The high frequency AC terms at audio and RF frequencies can be omitted because they cause only negligible ac displacement modulations on the flexure stage. Parameter a can be tuned to very close to zero by adjusting the gain of the DAC within a software program. After the tuning, parameter a was measured to be smaller than 5105 contributing to a negligible force uncertainty. Instead of using an optical interferometer, the position of the center electrode is measured by the difference between C1 and C2 with a differential capacitance bridge circuit [7]. Hence, any deviation of the center electrode from the vertical center position can be detected by the bridge circuit. With a commercially available optical interferometer, the offset adjustment could be quite difficult and ambiguous. The effect of parameter (a+b) can be tested by setting vc = 0, modulating Vb with a square wave profile and observing the displacement signal of the flexure stage. For Vb=2.0, we did not observe the displacement due to the modulated Vb. The remaining factors S0, vc and vs dominate the uncertainty of the electrostatic force fe. The capacitance gradient S0 was measured using a weight of 1 mg and a set of optical interferometer. The weight of 1 mg was cyclically loaded and unloaded to the system by a motorized linear stage to produce a deflection modulation. The deflection was measured by the optical interferometer and the corresponding capacitance variation was measured by a calibrated precision capacitance bridge. To reduce the effect from seismic noise and drift noise from the optical interferometer or the flexure stage itself, both deflection x and capacitance variation C are measured from the difference between average values of mass loaded data and two adjacent mass unloaded data. The capacitance gradient S0 was obtained by calculating the ratio of C/x which is shown in Figure 3. Using (2) and (3), the capacitance gradient estimated by C/x can be expressed as )1( 1 )( 0 32 2 0 0 d x S d x d x d x d A x CxC S                                 (12) From (12), S1 deviates from S0 by a small portion of S0x/d. Using the nominal design value of d = 0.5 mm, the ratio x/d is 0.15 %. This ratio can be reduced by using a smaller x for measuring the capacitance gradient. The measured capacitance gradient S has a mean value of S = 2.87610-8 F/m and a standard deviation of S = 0.00810-8 F/m. Therefore, the standard uncertainty of the capacitance gradient is 12104)(   N Su S  F/m with N = 369 in this measurement. The uncertainty u(vc) of the control voltage vc is calculated using the DAC resolution of 0.3 mV as 088.0)32/(3.0)( cvu mV which contributes 1 nN. For the surface potential noise vs, the current actuation scheme prevents the surface potential effect from being coupled to and amplified by the control voltage vc as the case in the previous electrostatic actuation scheme [7] where vs was amplified as Svcvs. The surface potential is reported to range from 20 mV to 180 mV [8, 9]. Taking vs = 0.18 V for example and S = 2.876  10-8 F/m, the surface potential induced electrostatic force is about 0.9 nN. 2.3. Null deflection control  The force under measurement fm is balanced by fe by the null deflection control. Figure 5 shows the block diagram of the null deflection control. The transfer functions of the main components, namely the flexure stage, capacitive position sensor, loop filter and the electrostatic force actuator, are represented by G, H, D and A respectively. The term xn represents a deflection noise which may be contributed by the seismic vibration noise and the thermal noise of the flexure stage itself. The relation between fe and fm appears to be )( )(1 )( )( )(1 )( sF sT sT sX sT HDA sF mse     (13) where T(s) = GDHA is the open-loop transfer function of the control system, and Fe(s), Xs(s) and Fm(s) are the Laplace transforms of fe, xs and fm, respectively. Within the control bandwidth, i.e. for T(s) >> 1, the relation between fe and fm can be approximated as )( mne fkxf  (14) 0 50 100 150 200 250 300 350 400 -2.91 -2.9 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 x 10-8 Measurement index C ap ac it an ce g ra di en t (F /m ) 0 50 100 150 200 250 300 350 400 -2.91 -2.9 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 x 10-8 0 50 100 150 200 250 300 350 400 -2.91 -2.9 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 x 10-8 Measurement index C ap ac it an ce g ra di en t (F /m ) Figure  4.  Capacitance  gradient  calculated  from  C/x.  The  mean  capacitance gradient S0 = 2.87610 ‐8  F/m, standard deviation S = 0.00810 ‐ 8  F/m and standard deviation of the mean  12104  N S  F/m (N = 369 in  this measurement).   ACTA IMEKO | www.imeko.org  September 2014 | Volume 3 | Number 3 | 71  where k is the stiffness of the flexure stage. Equation (11) shows that the null deflection control automatically generates a compensation force fe to balance the force under measurement fm. To reduce the influence form the noise xn, fm is measured in a short period of time by comparing fe(t0) before fm is applied and fe(t1) after fm is applied: mnneee ftxtxktftff  )]()([)()( 0101 thus ntem kxff  . (15) The term xnt represents the temporal variation of xn during the measurement time frame. From one deflection measurement data set taken for 8-hr, using a window of 300 s to evaluate xnt, we obtained a standard deviation of 0.33 nm for xnt. With a measured value k of 13.0 N/m, the standard deviation of the xnt equivalent force noise is 4.3 nN. Table 2 lists the main sources of uncertainty of the measured fm. 2.4. Weighing process  Each weight was loaded for 100 seconds and unloaded for 100 seconds. The compensation electrostatic force was calculated from the control voltage vc. Figure 6 shows the control voltage vc acquired during one weighing cycle. The voltage difference vc was determined from one weight loaded segment and its two adjacent weight unloaded segments as 22 21 cAcA cBc vv vv  . (16) The weighing cycle was repeated for a long period of time in order to evaluate the stability and uncertainty of the system. 3. RESULTS  Figure 7 shows the result of one weighing run for the weight of 1 mg. The measurement was done during three days. For this run, the measured electrostatic force was fe = (9,782.6  6.7) nN, where the given uncertainty is one standard deviation. The forces produced from the weights are estimated as fw = mg (1-air/mass), where the air buoyancy was taken into consideration. The comparison results are compiled in Table 3. In general, the electrostatic force has a smaller value than the deadweight. For comparisons of weights 1 mg and 10 mg, the force differences defined as fe-fw are similar and close to 10 nN, and they both are in triangle shapes with similar dimensions. For comparisons of the weight 2 mg and 5 mg, the force differences are rather larger, and they are in shapes of square and pentagon, respectively. The weight of 5 mg has the largest force difference of about 200 nN (20 g), and it is the biggest weight in terms of wire length and shape area dimensions. A possible explanation for this force difference is that there might be some extra electrostatic or magnetic force between the weight and its surroundings. Due to the size of the weight of 5 mg, it has the shortest distances to and possibly experiences the strongest electrostatic/magnetic interactions with its surroundings. 1 D A H G  Flexure stage (m/N) Capacitive position sensor (V/m) Loop filter (V/V) Electrostatic force driver (N/V) fmxn     vc fe V(x) 1 D A H G  Flexure stage (m/N) Capacitive position sensor (V/m) Loop filter (V/V) Electrostatic force driver (N/V) fmxn     vc fe V(x) Figure 5. Block diagram of the null deflection control. Some noise sources are omitted for simplicity.   Figure  6.  Capacitive  displacement  and  control  voltage  vc  during  one  weighing cycle.  Table 2. Uncertainty budget for measured fm  Source of uncertainty Standard uncertainty (N) Capacitance gradient S0 ef 4104.1 16-bit DAC resolution 9101  Surface potential vs 9108.1  Displacement noise xnt 9103.4  Combined standard uncertainty: 2429 )104.1()108.4()( em ffu   N Figure 7. A data run for 1 mg weighing.  ACTA IMEKO | www.imeko.org  September 2014 | Volume 3 | Number 3 | 72  4. DISCUSSION AND OUTLOOK  A force measurement system based on the electrostatic sensing and actuation techniques has been built and upgraded. The system is enclosed by a vacuum chamber which resides on a passive low frequency vibration isolation platform. The voltage actuation scheme has been modified to allow the decoupling between the surface potential vs and the actuation voltage leading to a reduction in the drift and bias of the compensation electrostatic force. The system is stable over a long period of time. However, the cause of the extra electrostatic/magnetic force observed in the weighing test is still unclear and investigation to that is underway. A new design of the apparatus’s housing is being fabricated, it was designed to isolate most of the apparatus from its surroundings and expose only the force loading area. In addition, other parameters such as alignment factors, the capacitance gradient and its frequency dependence will also be re-verified and studied further to find out the cause for the force difference. ACKNOWLEDGEMENT  This work was supported by the Bureau of Standards, Metrology and Inspection (BSMI), Taiwan, R.O.C. REFERENCES  [1] Newell D B, Kramar J A, Pratt J R, Smith D T and Williams E R, “The NIST microforce realization and measurement project”, IEEE Trans. Instrum. Meas. 52 (2003) 508. [2] Kim M-S, Choi J-H, Park Y-K and Kim J-H, “Atomic force microscope cantilever calibration device for quantified force meterology at micro- or nano-scale regime: the nano force calibrator (NFC)”, Metrologia 43 (2006) 389-395. [3] Leach R, Chetwynd D, Blunt L, Haycocks J, Harris P, Jackson K, Oldfield S and Reilly S, “Recent advances in traceable nanoscale dimension and force metrology in the UK”, Meas. Sci. Technol. 17 (2006) 467-476. [4] Choi J-H, Kim M-S, Park Y-K and Choi M-S, “Quantum-based mechanical force realization in piconewton range”, Appl. Phys. Lett., 90 (2007) 073117. [5] Nesterov V, “Facility and methods for the measurement of micro and nano forces in the range below 10-5 N with a resolution of 10-12 N (development concept)”, Meas. Sci. Technol., 18 (2007) 360- 366. [6] M-S Kim, J.R. Pratt, U. Brand and C.W. Jones, “Report on the first international comparison of small force facilities: a pilot study at the micronewton level”, Metrologia, 49 (2012), 70 [7] S-J Chen and S-S Pan, “A force measurement system based on an electrostatic sensing and actuating technique for calibrating force in a micronewton range with a resolution of nanonewton scale”, Meas. Sci. Technol., 22 (2011), 045104 [8] J.R. Pratt and J.A. Kramar, “SI realization of small forces using an electrostatic force balance”, Proc. 18th IMEKO World Congress, (17-22 September 2006, Rio de Janeiro, Brazil) [9] S.E. Pollack, S. Schlamminger and J.H. Gundlach, “Temporal extent of surface potentials between closely spaced metals”, Phys. Rev. Lett. 101 (2008), 071101 Table 3. Comparison results, unit in nN.  1 mg 2 mg 5 mg 10 mg fw 9797.12.9 19586.72.9 48950.56.4 97897.34.7 fe 9782.66.7 19527.04.1 48751.48.2 97886.416.5 e-fw -14.5 -59.7 -199.1 -10.9