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Y Z Bai, H Yin, L Liu, D Y Tan, Z B Zhou ([email protected])Center for Gravitational Experiments, School of Physics, Huazhong Univ. of Science & Technology, Wuhan, China
An electrostatic torsion pendulum aiming at improving the angular detection sensitivity without increasing torque noise floor is presented. Theoretical analysis shows that it
could be used to release requirement of angular measurement, and is useful for gravitational experiments with much higher precision requirement. In this poster, the
principle of the electrostatic pendulum system is introduced, and its sensitivity and noise analysis are presented.
i. Introduction
A typical torsion balance is very sensitive to probe force or torque which induced by weak signals, and its resolution is limited by the thermal noise of the pendulum and the readout noise, namely, the angular detection level.
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Improving the angular detection sensitivity of a torsion pendulum by an electrostatic spring
The torsion pendulum plays a paramount role in the field of precision measurement and gravitational experiments due to its high-precision sensitivity. Such as investigating the performance of a gravitational sensor and charge management for LISA.
ii. A Typical Torsion Balance and Its Potential Sensitivity
The torsion balance is set in a high vacuum chamber, where viscous damping can be ignored. The motion equation of the system in this case can be written as
m th(1 )I k i
where φ is the structure loss angle, τth is a random force with a white spectral density, and is presented as follows
2th B
m m
4
Re[ ( )]
( )
k TR
R Z
Z iI k i k
2 B m
th 22 2m m
4
( )
k T k
k I k
2 2 2n th eq
s 2m m
1( )H
k I ik
The power spectrum of the minimum detectable torque can be obtained as follows
22eq2 n B m
min 2 2
s s
4
( ) ( )
k Tk
H H
iii. Electrostatic Torsion Pendulum
The schematic of the electrostatic pendulum is shown in Figure 2, where one pair of electrodes are added in sided of the test mass, where de is the distance between the test mass each electrode, le is the vertical distance from the centre of the electrode to the fiber, Vf is the voltage applied on the electrodes to adjust the performance the system, and Se=axe aye is the area of each electrode.
The power spectrum of the thermal fluctuation of the electrostatic pendulum, θth,e2
meff th,e e,n
eff
m e
20 f 3 2
e 3
(1 )
( 12 )6
eff
yexe xe e
e
kI k i
k
k k k
V ak a a l
d
2 B m
th,e 22 2eff m
4
( )
k T k
k I k
Resolution of electrostatic torsion pendulum can be given by
s,e 2eff m
1( )H
k I ik
2eq2 2B m
min,e e,n2
s,e
4
( )
k Tk
H
Compare the electrostatic torsion pendulum with the typical balance as follows
at low frequency: ω<<ω0 at high frequency: ω>>ω0
22
s s,e
2 2min,e min
( ) ( )H H
22
s s,e
2 2min,e min
( ) ( )H H
iv. ExampleThe torsion pendulum presented by Washington University to study the charge effects for gravitational-wave is used as an example to be discussed, whose main parameters are listed in Table I, and a couple electrodes are added as add additional electrostatic spring, whose parameters are listed in Table II.
The motion equation of the electrostatic torsion pendulum is given by
m th,e e e,n(1 )I k i
2 10 f 2
1e 2 22
d2
e xe
e xe
l aye
l ae e
V a x xx
d x d x
2e,n 0 f f ,n e2 /xe ye ea a l V V d
Assuming the angle measurement noise is 510-8 (1+10-2/f)1/2 rad/Hz1/2. Resolutions of the torsion pendulum system with or without an electrostatic spring are shown in Figure 3. The electrostatic noise of electrostatic pendulum induced by fluctuation of voltage 10 μV/Hz1/2 is within 10-15 Nm/Hz1/2, which can be neglected. If the requirement of torque detection is 510-15 Nm/Hz1/2 above 0.1mHz, the requirement of the angle measurement device is shown in Figure 4.
Conclusions:The electrostatic spring can release the requirement of the angular detection of a torsion pendulum, which is very important for the much higher precise torsion pendulum experiments, such as to investigate the effects of a test mass for LISA and advanced LISA projects.
Ground testing for LISA (University of Trento)
Charge management (University of Washington)
Figure 1
Figure 2(a)
Figure 2(b) Top view
Figure 3
Figure 4
where τe is the electrostatic torque, and τe,n is the random torque induced by applied voltage fluctuations.
References:1.M Hueller, A Cavalleri et al, Torsion pendulum facility for ground testing of gravitational sensors for LISA, Class. Quantum Grav. 19 (2002)1757-1765.2.S.E.Pollack, M.D.Turner, S.Schlamminger, et al, Charge Management for gravitational-wave observatories using UV LEDs, Phys. Rev. D. 81, 021001(R)(2010).3.Q. L. Wang, H. C. Yeh, Z. B. Zhou et al, Improving the sensitivity of a torsion pendulum by using an optical spring method, Phy. Rev. A, 80, 043811(2009)
