Pre-Phase A Report - Lisa - Nasa

70 Chapter 3 Experiment Description
Figure 3.9 Scheme of the capacitive sensing.
The negative electrostatic stiffness induced by the detection voltage is reduced along the
Z axis with respect to the two other ones by the electrode configuration and by the
distance between the mass and the cage increased to 15 mm. The capacitance variations,
measured by the sensor along this axial direction and due to the proof-mass motion, are
no longer induced by the variations of the gaps between the proof mass and the sensing
electrodes but by the variations of the proof-mass areas in view to these electrodes.
The electrostatic forces are generated by applying the same control voltage V on the opposite
electrodes (see Figure 3.10). This control voltage is generated from the output of
the corrector. Opposite DC voltages ±Vp are added in order to linearise the electrostatic
forces that become proportional to Vp when the configuration is symmetric. Both electrodes
attract the proof mass with forces F 1andF 2 proportional to the gradients of the
capacitances ∇Ci and to the square of the potential differences between the proof mass
and the electrodes. The resultant force is expressed by
F = F1 + F2 = 1
∇C2 V
2
2
2 + ∇C1 V 2
1 . (3.9)
Because of the geometrical symmetry we have
∇C2 = −∇C1 = ∇C, (3.10)
and the resultant force F , linearised by the use of biasing voltages ±Vp, is made proportional
to the control voltage V:
F =(2∇CVp) V. (3.11)
At low frequencies, inside the control bandwidth, the proof mass is kept motionless in the
accelerometer cage, and V is representative of the acceleration Γcage of this cage:
(2∇CVp) V ≈ mΓcage , (3.12)
with m being the mass of the levitated proof mass. In the above two expressions, the terms
in parentheses represent the instrument scale factor. The value of Vp is selected according
3-3-1999 9:33 Corrected version 2.08