Pre-Phase A Report - Lisa - Nasa

4.2 Noises and error sources 95
is not significant. A charge limit is then derived by requiring that these spring constant
terms do not upset the nominal spring constant from the electrostatic control system by
more than 10 %.
Summary of charge limits. The above charge limits for the LISA proof mass are
summarised in Table 4.4 .. The most stringent limit comes from Lorentz force noise (note
a factor of 10 is included to allow for some electromagnetic shielding from the partial
titanium enclosure around the sensor). The next most critical effect is modification of the
spring constant.
Table 4.4 Summary of proof-mass charge limits and charge build up times
Effect Charge Limit Charging Time
[electrons/protons] [days]
Lorentz force acceleration noise 2×10 6 0.7
Electrostatic acceleration noise
Stochastic charge arrival 4×10 10 1.4×10 4
Displacement noise 5×10 8 187
Control voltage noise 2×10 11 7×10 4
Spring Constant 10 7 3.7
Charge measurement using force modulation. The force modulation technique
depends on applying oscillating potentials (dither voltages) to the electrode structure
around the proof masses which then exert forces on the charged proof mass via the third
term in equation 4.15. This induces an oscillatory motion in the proof mass which can
be detected capacitively. The amplitude and phase of the response give the size of the
charge and its sign. It turns out there is sufficient sensitivity in the displacement sensor
for the dither to be applied in the transverse direction. If two opposing electrodes are
used then the dither force is
Fd = − Q
∂C1 ∂C2
V1 + V2 , (4.21)
C ∂k ∂k
where C is again the total capacitance and ∂Ci is the capacitance gradient associated with
∂k
an individual surface. Assuming the proof mass is reasonably well centred within the
two opposing electrodes, such that ∂C1 ∂C2 ∂Co
= − = and we apply equal and opposite
∂k ∂k ∂k
voltages to the two sides (i.e. V1 = −V2 = Vd) then the dither force is
Fd = − 2QVd
C
∂Co
∂k
. (4.22)
The charge measurement sensitivity is here defined as the charge, Qs, at which the induced
acceleration just equals the system acceleration noise, ∆an :
Qs = m∆anC
−1 ∂Co
. (4.23)
2Vd ∂k
Corrected version 2.08 3-3-1999 9:33