Pre-Phase A Report - Lisa - Nasa

4.2 Noises and error sources 93
electrode. The proof-mass potential, VT , due to any applied potentials, Vi, will be
VT = 1
C
n
ViCi , (4.13)
i=1
where the summation is over all electrodes and the total capacitance is C = n
i=1 Ci .If
a free charge, Q, resides on the proof mass then the arrangement will have a total energy,
E, which is a combination of stored energy in the electric fields, energy associated with
the ‘batteries’ providing the applied potentials, and the interaction energy between the
free charge and the applied potentials. This is given by
E = 1
2
i
Ci (Vi − VT ) 2 + 1 Q
2
2
C + QVT +
QBiVi . (4.14)
Forces will act on the proof mass primarily through the capacitance gradients. In any
direction, k,
Fk = − ∂E 1 ∂Ci 2
= Vi +
∂k 2 ∂k Q2
2C2 n
∂C Q ∂Ci
− Vi , (4.15)
∂k C ∂k
where we have assumed for simplicity that the electrostatic control system would normally
ensure that
i CiVi = 0, i.e. that
VT =0and ∂VT
∂k
= 1
C
n
i=1
i
i=1
∂Ci
Vi . (4.16)
∂k
The first term represents the electrostatic suspension and control forces, the second term
is from spurious forces due to the interaction of any free charge with the surrounding
electrode structures, and the third contains the interplay between the free charges and
the applied potentials on the surrounding electrodes.
All terms involve the gradient of the capacitance which must be evaluated for the specific
geometry of the proof mass and its surrounding electrode structure. For a simple
displacement type sensor geometry, as used on the transverse degrees of freedom of the
proof mass, the total capacitance gradient, ∂C,
comes from differencing the gradients from
∂k
the opposing surfaces with the result that ∂C A
=4ɛ ∂k g3 δk, whereAis the electrode area,
g is the gap between the electrode and the proof mass, ɛ is the permitivity of free space
and 2δk is the difference in the gaps at the two opposing sides; i.e. the asymmetry in
the arrangement. For an overlap type sensor geometry, as proposed for the sensing in
the sensitive direction, the total capacitance gradient is again determined by asymmetries
between the two ends. However in this case it is differences in the transverse dimensions
of the electrodes and the transverse gaps which are important. For the LISA configuration
each micron asymmetry in each of the critical dimensions gives rise to a capacitance
gradient of typically 6 − 9×10−13 F/m.
The total proof-mass capacitance with respect to its surroundings is ≈ 70 pF. The electrostatic
force acting on the proof mass due to free charge on it is then (using the second
term in equation 4.15) ≈ 2.3×10−30 n2 e (N µm−1 ), where ne is the number of free charges
on the proof mass. The corresponding acceleration is ≈ 1.8×10−30 n2 e (ms−2 µm−1 ).
Corrected version 2.08 3-3-1999 9:33