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Title: Minimization of the coil movement of the LNE watt balance
during weighing mode and estimation of parasitic forces and torques involved
Author(s): Thomas, Matthieu; Espel, Patrick; Briand, Yves; Genevès, Gérard; Bielsa, Franck;
Pinot, Patrick; Juncar, Patrick; Piquemal, François
Journal: Metrologia
Year: 2014, Volume: 51
DOI: 10.1088/0026-1394/51/2/S54
Funding programme: EMRP A169: Call 2011 SI Broader Scope
Project title: SIB03: kNOW Realisation of the awaited definition of the kilogram - resolving the
discrepancies
Copyright note: This is an author-created, un-copyedited version of an article accepted for
publication in Metrologia. The Publisher is not responsible for any errors or omissions in this
version of the manuscript or any version derived from it. The definitive publisher-authenticated
version is available online at http://dx.doi.org/10.1088/0026-1394/51/2/S54
EURAMET
Secretariat
Bundesallee 100
38116 Braunschweig, Germany
Phone: +49 531 592-1960
Fax:
+49 531 592-1969
[email protected]
www.euramet.org
Minimization of the coil movement of the LNE watt balance
during weighing mode and estimation of parasitic forces and
torques involved
Matthieu Thomas1, Patrick Espel1, Yves Briand1, Gérard Genevès1, Franck Bielsa1,
Patrick Pinot2, Patrick Juncar2, François Piquemal1.
1
2
Laboratoire National de Métrologie et d’Essais (LCM), 78197 Trappes, France
Conservatoire National des Arts et Métiers (LCM), 93210 La Plaine St-Denis, France
E-mail : [email protected]
Abstract. Alignments of watt balance experiments are necessary to achieve a relative
uncertainty at a level of few parts in 108. This article briefly describes the LNE watt balance
and concentrates on adjustments made to minimize the coil movements during weighing
mode. The parasitic forces and torques involved in these movements are estimated by a
mathematical model. Some of the calculated parasitic forces are compared to an evaluation
done by studying the yaw movement of the beam.
1. Introduction
The watt balance principle proposed by Kibble in 1976 [1] lies in the comparison of
virtual electromagnetic and mechanical powers in a two-phase experiment by means of a
moving-coil balance in order to establish a link between the Planck constant and the mass unit
in view of its redefinition.
The perfect alignment of the system results in an equality linking mechanical and
electrical quantities:
Fz ⋅ v z = U ⋅ I
(1)
whereby I is the current flowing through the coil winding, generating a vertical Laplace force
Fz which balances the weight of a standard mass (weighing mode). U is the voltage at the coil
terminals produced when the coil moves at vertical velocity vz (velocity mode). I is measured
by the voltage drop U ' it produces at the terminals of a known resistance R. Measuring
voltages and resistance with the Josephson effect and quantum Hall effect as standards
enables the Planck constant h and the standard mass m to be linked:
g ⋅ vz ⋅ R
h
= A⋅
m
U ⋅U '
(2)
whereby A is a calibration constant and g is the acceleration due to gravity. Kilogram can be
redefined by this way and a number of laboratories are engaged in this way or a similar one
[2] [3] [4] [5] [6] [7].
The fundamental equation of watt balances (1) is valid when the system is perfectly
aligned: no horizontal forces nor torques are exerted on the coil as well as the coil moves
precisely vertically, without rotating.
In practice, there can be no guarantee that all these requirements are met to the desired
measurement uncertainty (below 5 parts in 108 [8]). Taking into account the alignment
imperfections of the system, the equation (1) becomes:
r r r v
U ⋅ I = F ⋅ v + M ⋅ω
r
,
(3)
r
whereby v and ω are the linear and angular coil velocities which – when brought together –
induce the observed voltage U . The current I – which flows into the coil winding – produces
r
r
force F and torque M on the coil. The previous equation can be rewritten:
 F v
F y v y M x ω x M y ω y M z ω z 

Fz ⋅ v z = U ⋅ I ⋅ 1 −  x x +
+
+
+
Fz v z
Fz v z
Fz v z 
  Fz v z Fz v z
(4)
whereby Fx, Fy, vx and vy are the unwanted forces and the horizontal velocities of the mass
centre of the coil along the x- and y-axes, Mx, My, Mz, ωx, ωy and ωz are the unwanted torques
relative to the mass centre of coil and angular velocities about the three axes. These parasitic
quantities must be reduced as much as possible by careful tuning the balance. All watt
balances have their own strategies to minimize these misalignments [9], [10], [11].
During the two steps of the weighing mode, the coil is immersed in a magnetic circuit
and driven by opposite currents I − or I + . The principal objectives of this article are 1) to
study the effects of reversing the current on the displacements of the coil, 2) to minimize these
displacements between I − and I + and 3) to estimate the parasitic forces and torques
involved, which may need to access to rest position of coil with no current. In the latter, two
different approaches are proposed and compared: the first one is based on a mathematical
model of the coil suspension which gives access to parasitic forces Fx, Fy and torques Mx, My
[12] and the other one is based on an experiment which allows to estimate the unwanted
forces Fy by considering the yaw movement of the beam.
2. Short description of the LNE watt balance
One of the main particularities of the LNE watt balance is that, during the velocity
mode, the force comparator and its suspension are moved as a single element by a translation
stage activated by a step motor so that the balance beam is not used as the element of the
movement generator. The translation stage is shown in the point 1 of figure 1. Its moving part
is linked to the support structure of the watt balance by means of 6 sets of flexure hinges
constraining its movement along only one direction. Adjusting the parallelism of the hinges
axes allows to both adjust non linearities and unwanted rotations around the horizontal axis.
Non linearities less than to 0.5 µm and unwanted rotations of about 5 µrad along the useful 40
mm of the total 72 mm stroke of the guiding stage have been demonstrated. Details of the
system can be found in [13].
1
2
Figure 1. Picture of the LNE watt balance.
3
5
4
8
Coil
Coil
winding
6
7
Figure 2. Schematic drawing of the force comparator (3), its suspension (4, 5) including the coil (6), the
magnetic circuit (7) and the mass exchanger (8).
Figure 3. A schematic of the force comparator with its suspensions. The force comparator, at its left end,
supports the 2-section tare suspension. At its right end, it supports the 2-section coil suspension and the 2-section
standard mass suspension. These two suspensions are linked together by a double monolithic electro-machined
gimbal. Note that: 1) the x-axis is horizontal and parallel to the longitudinal axis of the beam, 2) the y-axis,
which is directed into the figure, is horizontal and perpendicular to the x-axis and 3) the z-axis, which is defined
to be parallel to the local gravitational field, points upward.
The full suspension fixed to the lower plate of the guiding stage and the magnetic
circuit are represented in the lower section (point 2 of figure 1). The different elements are
represented in detail in figure 2.
The force comparator (point 3 – figure 2) is based on a home made aluminium alloy
beam [14], with two symmetrical arms l = 90 mm long, weighing about 200 g. The three
pivots are made of clamped 20 µm thick stainless steel flexure strips. The angular position of
the beam is servo controlled during weighing, but as central flexure strip has a small torsion
stiffness, the beam is likely to yaw when forces perpendicular to its longitudinal axis are
exerted at its ends.
A substitution method is adopted so as to cancel the effect of the relative difference in
terms of length of the two arms. In our experiment, the standard mass is 500 g and a tare mass
of 250 g is added to the tare suspension. The first equilibrium is realized with the standard
mass on the balance pan (mass down, disequilibrium of + 250 g and current I − ≈ −5 mA ). The
second is carried out without the mass on the pan (mass up, disequilibrium of – 250 g and
current I + ≈ +5 mA ). During the weighing mode, the current flowing through the coil is
permanently adjusted by a real time controller including a proportional-derivate-derivate
(PDD) algorithm driving a programmable current source. As a result, the beam is maintained
at its equilibrium position. The current is measured by the voltage drop (1 V) it produces at
the terminals of a 200 ohm resistance standard calibrated by comparison to the quantum Hall
resistance standard.
End flexure strips of the force comparator support on one side a simple suspension for
the tare mass (point 5 – figure 2) and a double one on the other (point 4 – figure 2) made of
two suspensions on the same vertical axis either for a standard mass of nominal value 500 g or
for the coil. The coil suspension used has two pivot points which allow to distinguish parasitic
forces and torques exerted on the coil by measuring the angular deviation of its segments
relative to the deviations with no current : the first segment of the suspension is only sensitive
to horizontal forces, whereas the second one is sensitive to horizontal forces and torques. In
practice, these angles will be deduced from the tilts and translations of the coil, and from the
geometry of the suspension [12]. The suspension is articulated in the middle with a clamped
flexure strips gimbal and linked at the top to the mass suspension by a double monolithic
electro machined gimbal (figure 3). This double gimbal reduces the effect of static and
dynamic coupling between these suspensions and concentrates the gravitational force and the
Laplace force on the same point of the beam.
The coil (point 6 – figure 2) is placed at the bottom of the suspension and inserted in
the magnetic circuit. When driven by the current, it provides the force that compensates the
weight difference between the two sides of the beam. The coil is made up of 8 superimposed
layers of a total of 684 winding of mean diameter 268 mm and the copper wire has a diameter
of 250 µm [15].
The linear and angular velocities of the coil during the velocity mode are measured by
an optical system composed of three optical position sensors and three interferometers. The
optical position sensors are Gaussian beam profilers which determine the position of the coil
in the horizontal plane (then vx and vy velocities) as well as its rotation about the vertical axis
(then ωz velocity). The three interferometers will determine the vertical position of the coil
(then vz velocity) as well as its rotation about the horizontal axis (then ωx and ωy velocities).
The same sensors are used to measure the position of coil during weighing mode.
The magnetic circuit (point 7 – figure 2) is of the loud speaker type. The hard material
is a ring of 60 individual Sm2Co17 permanent magnets inserted between two XC48 steel
plates. The geometry of the yokes defines a 9 mm wide and 90 mm deep air gap where the
magnetic induction field about 0.94 T has a M-shape profile along the vertical direction. The
minimum of this M-shape defines the vertical position where the coil is placed during the
weighing mode to minimize the influence of its positioning error. The geometry of the
magnetic circuit, the choice of the different materials, the influence of magnetostriction, the
machining and the mounting of the circuit have been extensively described in [16]. During
mounting, the orientation of its radial magnetic field was made horizontal with an uncertainty
of 10 µrad [17].
A mass exchanger (point 8 – figure 2) has been developed to bring the mass to the
mass pan and to lift it when necessary during the weighing mode. The horizontality of the
mass exchanger is adjusted to allow the self centring of the mass on the pan. The vertical
velocity of the mass exchanger is reduced to a minimum value of 10 µm/s so as to minimize
the mass pan oscillations to the utmost.
3. Minimization of coil displacement during weighing mode
In a watt balance, it is unlikely that the coil remains at the same location during the
two measurement steps of the weighing mode.
Using the LNE 2-section suspension, the effects of forces and torques are separated.
Therefore, the less the coil moves and the less the beam yaw, the smaller the parasitic
horizontal forces and torques should be. The objective is to minimize as much as possible the
coil displacement and the beam yaw from mass down to mass up.
The horizontal components Fx and Fy depend on the tilt of the coil winding relative to
the magnetic circuit. Torques Mx and My related to the mass centre of coil depend on the
horizontal position of the coil in the magnetic circuit. These parasitic quantities have two
main effects: they can disturb the force comparator and affect the value of watt ratio error
term. Only the second effect has been taken into account in this article.
Therefore, the purpose of the study is to adjust the coil winding position relative to the
magnet along the x- and y-axes and relative to its magnetic plane – which coincides with the
horizontal at better than 10 µrad [17] [18] – in order to reduce the displacement of the coil
between the two currents of the weighing mode.
3.1. Experimental set-up
The three coil optical detectors, based on propagation properties of Gaussian beams,
are associated to the coil so as to measure its positions in the horizontal plane. Their principle
is based on the interception of a laser beam by a screen fixed to the object whose position
must be determined. The power transmitted, measured by a photodiode, is proportional to the
integral of the transmitted part of the Gaussian beam, i.e. proportional to a Gauss error
function. The dynamic range and the sensitivity of the detector may be tuned by adjusting the
diameter of the beam. In addition to these three detectors, three capacitive sensors (which are
used in place of the interferometers for this experiment) are placed at 120° from each other,
above the coil in order to measure the tilt of the coil compared with the plane defined by the
three sensors. These six detectors enable to follow the displacement of coil the with a
resolution far better than 1 µm, and rotation of coil a resolution far better than 1 µrad. These 6
detectors are collectively named “position sensors” and give access to the linear displacement
of coil ( ∆x, ∆y along x- and y-axis, figure 3) and to its angular displacement ( ∆β x , ∆β y about
x- and y-axis).
Two capacitive sensors are also used to monitor the yaw ( ∆φ about z-axis) movement
of the beam.
3.2. Study of the coil displacement
To start with, the position of the coil in the horizontal plane Oxy is only ensured by the
careful construction of the suspension.
3.2.1 Translating the whole suspension
The first step of the experiment is to translate the whole suspension in the horizontal
plane by moving the central flexure strip of the beam over a 800 µm distance by 200 µm
steps. In order to achieve this, a translation table based on a flexure strip pantograph was
designed and inserted between the translation stage of the watt balance and the beam. Its total
travel range is 1.5 mm.
For each 200 µm displacement of the central flexure strip of the beam, the positions of
the coil according to the position sensors, defined by the coordinates (x,y), are measured for
both the mass down (when current is I-) and the mass up (when current is I+). The five
displacements recorded, labelled from A to E, are presented in figure 4. First, the graph shows
clearly that for all the displacements studied, the coil is not located in the same place for the
two measurements using currents I+ and I–. Secondly, whatever the position of the coil in the
horizontal plane, this difference in position remains similar and is about 75 µm.
Figure 4. Linear displacement of the coil when translating the central flexure strip of the beam over a 800-µ m
distance.
At the same time, the angular positions of the coil defined by the coordinates (βx , βy),
are recorded for each step of the translation. Results are shown in figure 5. The change of the
position of the coil in the horizontal plane causes a change in the torques acting upon the coil
and therefore a tilt of the coil as one could have expected. For the five displacements shown,
the decrease of the angular displacement (from 500 µrad to 100 µrad) is clearly visible.
Figure 5. Angular displacement of coil when translating the central flexure strip of the beam over a 800-µm
distance.
3.2.2 Tilting the coil
The second step of the experiment consists in tilting the coil from a stationary place in
the horizontal plane. For that, the position of the flexure central strip of the beam is
definitively determined by the xy-translation table and some weights are placed on the
periphery of the coil in order to modify the angle between the coil and the magnetic circuit –
notice that as a small effect, the initial position of the coil is also modified because of the
change of the centre of mass of the coil (thus torque). These modifications generate different
horizontal forces exerted on the coil. For all the forces, the positions of the coil are measured
for the mass down (when current is I–) and the mass up (when current is I+) together with its
tilts.
The variation of coil initial tilt of 1 mrad yields a variation of angular displacement
from 200 µrad to 400 µrad but yields to a linear displacement from 50 µm to 200 µm. As
expected, there is an inversed effect compared to the previous experimental case (3.2.1):
tilting the coil mainly affects horizontal forces. Thus the angular displacement is less affected
than the linear displacement.
3.2.3 Minimizing the displacement during weighing mode
The two previous experiments have shown that the translation of the coil in the air gap
of the magnetic circuit mainly modifies its angular displacement whereas its tilting modifies
mainly its linear displacement during the two steps of the weighing mode. By combining the
two effects, the linear and angular displacements of the coil have been considerably reduced.
By successive adjustments, coil displacements have been limited to linear and angular
variations as small as 1.5 µm ( ∆x = −0.9 µm and ∆y = 1µm ) respectively along the x- and yaxis) and 0.7 µrad ( ∆β x = 0.5 µrad and ∆β y = 0.5 µrad ) respectively about the x- and y-axis)
when currents is set to I– or I+, well below the values of 75 µm and 500 µrad measured before
adjustments. This displacement of the coil is associated with a yaw movement of the beam of
–1.1 µrad about its central flexure strip.
This displacement of coil and beam will be refered below as “adjusted coil”.
4. Estimation of parasitic forces and torques involved in weighing mode
The displacements of the coil end the beam during the two steps of the static phase
have been minimized but not nulled. It means that some parasitic forces and torques still exert
on the coil and have nearly the same values during these two steps. The objective is now to
estimate these unwanted quantities using the mathematical model of the coil suspension given
in appendix A. Then, the study is completed by some experiments to compare these results to
other experimental data.
4.1 Mathematical model of the coil suspension considering coil equilibrium position with no
current
4.1.1 Simplifying assumptions and conventions
To keep the mathematics to a manageable complexity, some assumptions have been
made. Normally, the model of the suspension would include:
i)
the end flexure strip of the force comparator. The movement of the extremity
of this element is calculated to be far less than 1 µm and then its contribution
is negligible relative to the amplitude of the coil movement.
ii)
two gimbals with stiffness. The gimbals have a stiffness around 0.01 Nm/rad
[11] which gives torques of 10 µN ⋅ m considering a rather large 1-mrad tilt of
the coil. Therefore all terms involving these stiffness can be neglected.
iii)
the standard mass suspension. A double monolithic electro machined gimbals
is used to link together the coil and the standard mass suspensions. It reduces
the effect of static and dynamic coupling between these two suspensions. So,
the movement of the standard mass suspension should not affect the coil
suspension which has been checked by experiment. That is why, this part has
been suppressed.
With the assumptions above, the suspension model to estimate the unwanted forces
and torques only includes the coil suspension as schematized on figure 6a. Sections 1 and 2
with masses m1, m2 and lengths l1, l2 are connected by gimbals. Their mass centres are
respectively noted G1 and G2 and their positions from the above gimbal are noted l1G and l2G.
Figure 6. a) The schematic of the coil suspension with its lengths, mass centres G and the optical centre λ. b)
Forces and torques reduced to masse centre, weights acting for each section of suspension together with
measured movements ( ∆y and ∆β x ) of section 2 along y-axis and measured beam yaw ( l ⋅ ∆φ ) (green
arrows).
The coil winding – wounded around a Delrin support – is attached at the end of the
section 2. The position sensors allow us to obtain linear movement of an arbitrary reference
point called λ (figure 6b) and angular movement of the coil.
Finally, Fz is the known vertical component of the Laplace force, Fy and M x are the
unwanted and unknown force and torque which are reduced to the mass centre G of the coil –
see figure 6b for a schematization in Ozy plane. Note that similar notations are used in x-z
plane.
Besides, the coil winding being assumed circular and immersed in a radial magnetic
flux, the torque Mz is zero. Therefore, the only parameters considered in the mathematical
model are Fx, Fy, Mx and My.
All these parameters are known and their numerical values are tabulated in table A1.
The yaw movement of the beam is measured ( ∆φ ), as well as the coil displacement
(∆x, ∆y, ∆β x , ∆β y ) with the same sensors described in section 3.1
4.1.3. Equations
This section is the implementation of [12]: it consists in evaluating forces and torques
acting on the coil winding by measuring the linear and angular displacements of section 2 of
suspension and the yaw movement of the beam. Defined by the coordinates (∆x, ∆y), (∆βx,
∆βy) and ( ∆φ ) they represent the displacements of the coil and of the beam from its position
of equilibrium at rest – when no current flows through it – to its position of equilibrium with a
current I flowing through it, as shown in figure 6b for Ozy plane.
From elementary mechanical considerations on coil equilibrium, it is possible to
deduce a relation between unknown horizontal forces and torques applied on coil
(Fx , Fy , M x , M y ) , known displacement (∆x, ∆y, ∆β x , ∆β y ) of the coil and know yaw of the
beam ( ∆φ ). The development of calculations is detailed in appendix A.
Therefore, all the unwanted forces and torques Fx , Fy , M x , M y can be calculated
from measured quantities.
All quantities (forces, torques, positions, angles) can be distinguished by a subscript “–
” when they refer to equilibrium position of the coil when the mass is down ( I − current), and
by subscript “+” when the mass is up ( I + current).
4.1.4. Additional experiment and parasitic forces and torques estimation from mathematical
model
When coil is at its “adjusted position”, its displacement from
I − -current to I + -
current was reduced to the limit of the uncertainties of determination of coil position by the
sensors (section 3.2.3). In order to evaluate parasitic forces and torques by means of the
mathematical model, the equilibrium position of the coil when no current flows through it
( I 0 = 0 mA ) must be known.
This was done by an additional experiment: the I 0 = 0 mA reference position was
measured by carrying a weighing with an almost null current: by manually removing the
250g-tare mass and lifting up the standard mass. Then displacement from I 0 -current to I − current or to I + -current was then obtained by carrying out a “standard” double weighing.
Results are shown an table 1.
Displacement from…
… I 0 to I −
… I 0 to I +
∆x / µm
+ 0.7
– 0.2
∆y / µm
+ 0.6
+ 1.7
∆β x / µrad
– 4.3
– 3.8
∆β y / µrad
– 8.9
– 8.4
∆φ / µrad
+ 36.6
+ 37.7
Table 1. Displacement of the coil and the beam from I 0 -current to
I − -current and from I 0 -current to
I + -current.
The total displacement from I − -current to I + -current measured here is the same that
was measured directly in section 3.2.3.
Then, horizontal forces and torques exerted on the coil when fed by + 5 mA and –
5 mA can be calculated from equations (A6-A9, A12 and A13) given in the appendix. Results
are shown on table 2.
I − ≈ −5 mA
I + ≈ +5 mA
F x / µN
– 100
– 80
F y / µN
– 100
– 150
M x / µN ⋅ m
– 20
+ 10
M y / µN ⋅ m
– 80
– 60
Table 2. Calculations of the parasitic forces and torques exerted on the coil suspension after adjustment at I −
equilibrium position and at I + equilibrium position. An estimation of associated uncertainties is given in section
4.3 below.
Sensitivity coefficients linking coil displacements and parasitic forces and torques can
be found at the end of appendix A2. As an example, the sensitivities of Fx and My to linear
and angular displacements from position I = 0 mA to position I = 5 mA are :
Fx+ = +96
µN
µN
⋅ ∆x + 18
⋅ ∆β y ,
µm
µrad
(6)
M y+ = +21
µN ⋅ m
µN ⋅ m
⋅ ∆x + 10
⋅ ∆β y .
µm
µrad
(7)
4.2 Estimation of Fy parasitic force considering beam yaw
After having adjusted the coil to reduce its displacements to the minimum during the
weighing
mode,
we
obtained
an
angular
displacement
about
x-axis
of
∆β x = β x+ − β x− = −1.1µrad . The parasitic forces F y+ and F y− are proportional to the current
flowing into the coil winding and to the tilt of the coil winding relative to the plane of the
magnetic circuit, θ x+ and θ x− , thus the θ x ’s are the quantities of interest. There is no obvious
relations between the measured angle β x and the actual angle θ x of coil winding magnetic
plane relative to magnetic circuit magnetic plane. However angular differences are equals:
θ x+ − θ x− = β x+ − β x− = −1.1µrad
(8)
To estimate the θ x+ and θ x− values and also parasitic forces F y+ and F y− , an other equation is
necessary.
For that, let us consider the yaw movement of the beam (figure 7). The quantities
involved in this movement are the forces that are perpendicular to the longitudinal beam axis
( F y ) and the vertical torque ( M z ). The magnet, coil and wiring have been arranged carefully
to avoid producing such torque. So, as a first approximation, it is neglected as it is for the
mathematical model.
Besides, the horizontal forces along beam axis ( Fx ), and the horizontal torques ( M x
and M y ) do not influence the yaw movement of beam.
The forces F y+ and F y− are directly proportional to the angles θ x+ and θ x− [19]:
Fy+ = + Ar ⋅ θ x+ and Fy− = − Ar ⋅ θ x−
(9)
where Ar = n ⋅ I ⋅ π ⋅ r0 ⋅ Br0 , (n = 684 turns, I = ± 5 mA, r0 = 133 mm and Br0 = 1 T) is a
constant equal to 1.43 N/rad in the LNE watt balance .
Figure 7. Horizontal forces Fy+ and Fy– produce beam yaw. Torques are not drawn.
The next section describes the experiments we have carried out to establish the
missing equation with the help of the beam yaw leading to an evaluation of Fy– and Fy+,
without the need to measure the equilibrium position of coil with no current.
4.2.2 Study of the beam yaw
Movements of the beam balance have been studied during weighing phase. Since the
beam is maintained at its equilibrium position in the two configurations (with and without the
mass on the pan, I– and I+), the only beam movement of interest here is the yaw. The yaw
∆φ is defined as the beam rotation about the vertical (z-axis). It is due to the torsion of the
central flexure strip under the effect of horizontal forces applied at the end of the beam
balance.
The first measurement with no particular adjustment (A label in figure 4) yields an
angular deviation ∆φ of about – 200 µrad. The conclusion is that some horizontal forces act
perpendicular to the end of the beam and these forces, respectively Fy– and Fy+, have different
values during the two steps of the weighing mode.
In order to modify the amplitudes of the forces, one weight of about 2 grams has been
placed at different locations on the periphery of the coil. For each position, the angular
deviation ∆φ of the beam is measured. Results are shown in figure 8. When the coil is
ballasted along the x-axis which represents the longitudinal axis of the beam (cases 3 and 7),
component Fy of the Laplace force remains substantially the same, as the torque acting on the
central flexure strip: the position of the beam balance is unchanged compare to the previous
position. In addition, when the coil is ballasted along the y-axis – which represents the
transverse axis of the beam (cases 1 and 5) – the amplitude variation of the quantity
(F
+
y
)
− Fy− is maximum and the beam is deviated from its original position with an angle of
600 µrad. When the weights are placed at 45° to the x- and y-axis (cases 2, 4, 6 and 8), the
angular deviation take values halfway between the two previous configurations. We clearly
see here that the yaw movement of the beam can be controlled by ballasting (tilting) the coil.
Figure 8. Modification of beam yaw ∆φ by ballasting coil.
As previously explained, the forces involved in the yaw movement of the beam, respectively
Fy+ and Fy-, are proportional to the current flowing into the coil and to the tilt of the coil
winding relative to the plane of magnetic circuit, respectively θ x+ and θ x− . Thus, the beam
(
)
yaw must be proportional to the difference Fy+ − Fy− of the forces and therefore the sum
(θ
+
x
)
+ θ x− of the angles. As there is no access to these angle, but only to the measured β x+ and
β x− (the angular position of coil with respect to the position sensors), the result is:
∆φ =
(
)
(
)
(
l
l
l
⋅ F y+ − F y− = ⋅ Ar θ x+ + θ x− = ⋅ Ar β x+ + β x− + 2 ⋅ β xoff
K
K
K
)
(10)
whereby β xoff is the “angular offset” defined by the difference between the unknown
angle θ x (angular position of the coil winding relative to the magnetic circuit) and the
measured angle β x (angular position of the coil relative to the position sensors). l is the
length of the beam from the central strip to the end strip ( 9 cm ) and K is the torsion constant
of the central strip (unknown value).
Figure 9. Fine tuning of the beam yaw ∆φ by fine variation of ballasting position. Black circles are experimental
data whereas the red line corresponds to the linear fit.
To determine unknown quantity of interest ( β xoff and K) from equation (10), experiments
have been carried out to study the evolution of the angular deviation ∆φ of the beam due to
yaw movement as a fonction of the measured quantity β x+ + β x− . For that, a 2 gram mass is
moved about the coil periphery to tilt the coil and change the forces Fy+ and Fy– involved in
the movement of the beam. Results are shown in figure 9. Analysis of this figure provides
these informations:
(a) Linear regression. The straight line of a simple linear regression fits very well the
data. This means that the relationship between ∆φ and β x+ + β x− is linear as expected.
The interception at the horizontal axis yields twice the value of the angular offset
( β xoff = 1015 mrad ). So that, the missing equation to estimate the angles θ x+ and θ x− is:
θ x+ + θ x− = β x+ + β x− + 2 × 1015 µrad .
(11)
(b) Slope. The slope of the linear regression is – 0.447. According to equation 10, the
slope is also equal to the ratio l ⋅ Ar K . Consequently, it is possible to deduce the
angular stiffness K. Calculation yields K = 0.29 Nm/rad . This torsion constant has
been also evaluated using an other method based on the pseudo-period of the angular
stiffness of the pendulum (composed of the beam, the tare mass suspension and the
coil supension). The motion is described by:
J
d 2α
= Kα (12)
dt 2
where J is the moment of inertia of the mass and Kα is the torque acting the pivot
point. The period of oscillation T0 of this system can therefore be defined as:
J
K
T0 = 2π
(13)
This period has been determined experimentally and its value is 2.65 s. The moment
of inertia has been calculated at J = 0.0936 kg m 2 . Consequently, equation (13) yields
an angular stifness of 0.5 Nm/rad which is of the same order of magnitude than the
previous value.
(c) Sensitivity coefficient. The relation
(F
+
y
)
− F y− ⋅ l = ∆φ ⋅ K gives to the sensitivity
(
)
coefficient linking the yaw movement of beam to the differential force Fy− − Fy+ a
value of − 3.2 µN/µrad .
Additionnally, the same experiments have been carried out between a current of 0 mA
and a current of 5 mA. Results was exactly comparable, with notably the sensitivity
coefficient between the yaw movement and the F y+ or F y− forces having a value of
− 3.2 µN/µrad .
4.2.3. Estimation of the forces by mean of the beam yaw
After adjustments of the coil at section 3.2.3 and this study of the beam yaw
movement, the two following relations linking the angles of the coil winding when currents
are I + and I − (respectively β x+ and β x− ) have been measured:
β x+ − β x− = θ x+ − θ x− = −1.1 µrad
β x+ + β x− = θ x+ + θ x− − 2 ⋅ β xoff = −2022 µrad
(14)
which leads to θ x+ ≈ θ x− about 4 µrad. These angles are the tilt of the coil relative to magnetic
plane of magnetic circuit.
From equation (9), the forces Fy+ and Fy– are roughly + 5 µN and − 7 µN . These
experimental results are respectively 100 µN and 150 µN away from the values obtained from
the mathematical model (see table 2) for this particular case.
4.3 Comparison between the two estimations of Fy
Comparison between Fy+ forces estimated by the mathematical model and by the mean
of the beam yaw is shown on figure 10 below. This experiment, which involves rather larger
values of Fy+, was done between a current of 0 mA and a current of 5 mA because
mathematical model needs the position of coil with no current. Ballasts were moved on the
periphery of the coil to obtain different movement of yaw of the beam. For each of these
points, an evaluation of forces by the mean of the beam yaw (see section 4.2.3) and by the
mathematical model (see section 4.1.4) was done.
A linear regression fits very well the data, with a slope of 0.8885. This difference to 1
will be investigated, but is not relevant for the evaluation of small forces as in the “adjusted
coil”.
This comparison between two different ways of estimate Fy+ forces shows that the
half-width of prediction band is roughly 300 µN. Therefore we can use a standard uncertainty
of 300 µN for parasitic forces Fy+ which leads to a standard uncertainty of 100 µNm on
torques Mx+. Standard uncertainties associated to all parasitic forces and torques evaluated by
the mean of the mathematical model should be of the same magnitude.
Figure 10. Comparison between Fy+ forces estimated by the mathematical model and by the mean of the
beam yaw. Uncertainty given for mathematical model (150 µN, k = 1) considers only a standard uncertainty of 1
µm associated with measurements done by optical and capacitive sensors. This uncertainty was estimated by
using a Monte-Carlo analysis.
5. Summary and future works
The fundamental equation describing the operation of LNE watt balance applies to a
coil perfectly aligned. In practice, this can not be realized within the desired uncertainty of the
measurement. Imperfections in the watt balance experiment lead to the presence of unwanted
forces and horizontal velocities along the x- and y-axis, and unwanted torques and angular
velocities about the three axis. Those parasitic terms act upon the linear and angular
movement of the coil between the two steps of the weighing mode.
The main objectives of the experiments described in this article were 1) to study and to
minimize the linear and angular movement of the coil and 2) to estimate parasitic forces and
torques involved:
–
After adjustments of position and tilt of the coil, it was possible to obtain linear
and angular displacements of the coil as small as 1.5 µm and 0.7 µrad between the
two steps of the weighing mode. The yaw movement of the beam is then limited to
–1.1 µrad.
–
The mathematical model of the coil suspension, developed a few years ago at LNE
[12], has been used to estimate all the parasitic forces and torques which exert on
the coil at its adjusted position by using the position of coil and beam when no
current flows through it.
–
An experiment has been carried out to evaluate some of these parasitic forces. It is
based on the study of the beam yaw movement. This movement is directly
dependant upon the parasitic force Fy.
–
The unwanted forces F y+ estimated by both methods are comparable.
Acknowledgments
The authors are pleased to acknowledge the helpful criticism and comments given during the
preparation of this paper by I. Robinson.
This work was jointly funded by the European Metrology Research Program participating
countries within the European Association of National Metrology Institutes (Euramet) and the
European Union.
Appendix A. Mathematical model of coil suspension
A1. Equations
Notations and simplifications were given in section 4.1.1. In all the below equations,
the approximation cos(angle ) ≈ 1 and sin (angle ) ≈ angle is made.
Equations (A1), (A2), (A3) and (A4) are derived from the calculation of the torques of
the forces about O1 and O2 for the first and the second section of the suspension (figure 6b)
considering their angular deviations ( ∆α x , ∆α y , ∆β x and ∆β y ). Thus, for the section between
the double gimbals O1 at the top of the suspension and the gimbals O2 which supports the
suspension of the coil :
m1 ⋅ g ⋅ ∆α y ⋅ l1G + (Fz + m2 ⋅ g ) ⋅ ∆α y ⋅ l1 − Fx ⋅ l1 = 0 , in the plane Oxz
(A1)
m1 ⋅ g ⋅ ∆α x ⋅ l1G + (Fz + m 2 ⋅ g ) ⋅ ∆α x ⋅ l1 + F y ⋅ l1 = 0 , in the plane Oyz
(A2)
and for section between the gimbals O2 which supports the suspension of the coil and the coil
itself :
m 2 ⋅ g ⋅ ∆β y ⋅l 2G + Fz ⋅ ∆β y ⋅ l 2G − Fx ⋅ l 2G + M y = 0 , in the plane Oxz
(A3)
m 2 ⋅ g ⋅ ∆β x ⋅l 2G + Fz ⋅ ∆β x ⋅ l 2G + F y ⋅ l 2G + M x = 0 , in the plane Oyz
(A4)
These equations can be rewritten as:

l 
Fx = + (m 2 ⋅ g + Fz ) + m1 ⋅ g ⋅ 1G  ⋅ ∆α y
l1 

(A6)

l 
Fy = − (m2 ⋅ g + Fz ) + m1 ⋅ g ⋅ 1G  ⋅ ∆α x
l1 

(A7)
M x = − F y ⋅ l 2G − (m2 ⋅ g + Fz ) ⋅ l 2G ⋅ ∆β x
(A8)
M y = + Fx ⋅ l 2G − (m2 ⋅ g + Fz ) ⋅ l 2G ⋅ ∆β y
(A9)
All quantities needed to evaluate parasitic forces and torques are know (table A1) in
previous equation. It clearly appears that deviation of the upper section ( ∆α x , ∆α y ) is only
sensitive to parasitic forces, whereas the deviation of the second section is sensitive to
parasitic forces and torques.
However, the angular deviations of the suspension is actually not measured. The
quantities which are known are the linear and angular displacement of the coil ( ∆x , ∆y , ∆β x
and ∆β y ) as well as the yaw movement of the beam ( ∆φ ).
Equation (A5) and (A6) are then purely derived from geometrical considerations:
∆α y ⋅l1 + ∆β y ⋅ l2 = − ∆x
(A10)
∆α x ⋅l1 + ∆β x ⋅ l 2 = ∆y − l ⋅ ∆φ ,
(A11)
The angular deviations of the 2 sections of the suspension are then known from the
displacement of the coil (linear and angular) and from beam yaw:
− ∆ x − ∆β y ⋅ l 2
∆α y =
∆α x =
l1
∆y − l ⋅ ∆φ − ∆β x ⋅ l 2
l1
(A12)
(A13)
A2. Sensitivity coefficients
From the above equations, sensitivity coefficients linking displacement of the coil to
parasitic forces and torques are calculated for 5 mA.
Fx+ = +96
F y+ = +96
M x+ = −21
µN
µN
⋅ ∆x + 18
⋅ ∆β y ,
µm
µrad
µN
µN
⋅ ( ∆y − l ⋅ ∆φ ) − 18
⋅ ∆β x ,
µm
µrad
µN ⋅ m
µN ⋅ m
⋅ ( ∆y − l ⋅ ∆φ ) + 10
⋅ ∆β x ,
µm
µrad
M y+ = +21
µN ⋅ m
µN ⋅ m
⋅ ∆x + 10
⋅ ∆β y .
µm
µrad
Values are smaller in absolute value by roughly 10 % for –5 mA.
(A14)
(A15)
(A16)
(A17)
Designation
Symbol Value
Length of section 1
l1
0.311 m
Length of section 2
l2
0.189 m
Distance from double gimbals to G1
l1G
0.145 m
Distance from central gimbals to G2
l2G
0.221 m
Acceleration of gravitation
g
– 9.81 m/s²
Mass of the section 1 of the suspension
m1
0.460 kg
Mass of the section 2 of the suspension
m2
2.547 kg
Length of beam arm (from central
l
9 cm
flexure strip to end flexure strip)
Table A1. The geometrical parameters of the coil suspension. All quantities are known from the
construction of the suspension.
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