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teen wire segments of a coil, as in (3).
∮
⃗F = I
d ⃗ l × ⃗ B = I( ⃗ L × ⃗ B) (3)
The stator field was obtained from a 3D magnetostatic
simulation from the software package COMSOL TM , as
shown in figure 4. The simulation results were rastered and
imported in Matlab. The mean value of the flux density in
the range from minimum to maximum deflection yielded
value of 0.9T in the z-direction. A driving current of 100mA
would generate therefore a lateral force of 7.2mN.
As depicted in the simplified actuation principle in figure
3, every coil actuation element requires one current source
and two voltage sources. For analogue sources, a floating
potential in the coil has to be taken in to account, which depends
on the driving current. From the sheet resistance of
the TC0302 metallization (< 2m#/sq) and the conductor
geometry of the coils and connections, the total track resistance
is calculated to be 0.71#. Assuming a maximum
current of 100mA, a maximum voltage shift of 71mV is obtained.
As the electrode voltages are above 25V, this shift is
assumed to be negligible. The power consumption is estimated
at 7.1mW.
C. Restoring forces
The restoring forces are generated using single-beam
flexures in the x-y plane because this method enables a feasible
way to interconnect the coils and guarantees a defined
zero-deflection position. Furthermore, the spring forces,
partially compensate for the quadratic dependence of the
pull forces of the electrostatic actuator with respect to the
deflection.
11-13
May 2011, Aix-en-Provence, France
The mechanical properties of the co-fired LTCC were obtained
from the manufacturers’ data sheets. For the HL2000
ceramic, the flexural strength is specified above 200MPa.
The Young’s modulus, derived from DuPont 941, is assumed
to be 120GPa. Tests with laser-structured doublebeams
(length 12mm, width 120!m / 80!m, height 120!m)
have shown that the flexures break at a deflection between
1.5mm and 2mm, which leads according to the strain/stress
curve of a beam with rectangular cross section to a force between
11mN and 15mN and stresses between 227MPa and
312MPa. Hence, the data determined by the manufacturer
using four-point measurements apply to the aspect ratio of
flexures, and a static maximum actuator stroke of 10!m can
be calculated to result in a stress of 15.5MPa for a vertical
point load. Figure 5 illustrates a flexure of one of the test
structures (80!m sample), which is the smallest LTCC
structure that could be manufactured with the laser.
As described in [5], the strain-stress curve of LTCC
shows nonlinear and strain-speed dependent characteristics.
Thus, in contrast to metals, only small deflections allow the
definition of a spring constant. Hence, the design was configured
to provide the longest possible beams while retaining
the 120º point symmetry of the device. This caused implicitly
a modification of the angle between the two beams
from 90º to 80º, as the via interconnections are fixed. The
spring constant for a single ended beam with the dimensions
l " b " h can be approximated using unified beam theory
[20]:
, k y = E hb3
4 l 3 (4)
Where k z is vertical spring constant, k y the lateral spring
constant and E the elastic modulus. For the inner beams
Fig. 5: Laser cut LTCC beam flexure used for stress test.
Fig. 6: Prototype magnet layers made of acrylic.
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