Nobili et al. / Proposed noncryogenic, nondrag-free test of ...

7. The capacitance read out system

The displacements of the test masses are detected with a pair of capacitors in a circuit which is essentially an LC bridge, formed by two resonant coupled oscillators, as shown in Fig. 14 . The plates of the capacitors are sections of cylinders concentric to the test bodies (Fig. 15 ) and are supported by the frame shown in Fig. 2 .

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Fig. 14. Outline of the read-out circuit. The two variable capacitors  and  and the two halves of the inductor L form an LC bridge whose output is proportional to the difference between the two capacitances.

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Fig. 15. Each capacitor of the read-out system (see also Fig. 16 ) is formed by two surfaces, one for each of the two grounded masses, and one plate, to which a sinusoidal voltage is applied. Any differential displacement of the test masses with respect to the plates causes a loss of balance of the system and therefore an output signal.

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Fig. 16. The surfaces of the capacitors before and after: a) a common mode displacement and b) a differential mode displacement.

A voltage signal  of angular frequency  is applied to the bridge in order to shift the signal of interest to a frequency band with a  noise as small as possible. We consider . The circuit has the greatest sensitivity when  and when also the output circuit resonates at the same frequency . Since  we must have . Since the system has a bad heat dissipation, the amplitude of the signal should not exceed, say, 1 Volt in order to reduce power dissipation. We set  with . The two capacitors of the bridge  and  are shown in Fig. 15 . The signal is applied to the plates and the test masses are electrically grounded. Let us call a and b the initial distances from the plate to the inner and outer mass respectively, with . If a and b are small a simplified analysis can be carried out assuming zero curvature parallel plates, and the initial values of the capacity are then  (where  is the dielectric constant of vacuum). In the cylindrical geometry the algebra is somewhat more complicated but with no relevant changes in the results. Any displacement of the test masses will change  into  for  and  for . Such a displacement is the combination of a common mode and a differential mode displacement  (Fig. 16 ). Because of  the values of the capacity change into

and therefore . Similarly for  we have:

hence, . For the general displacement the total change of capacitance will be given by:

which at resonance determines the output signal through the relation:

where  is the electrical quality factor of the output circuit. For instance, if  and  we get  and , given the requirement to be sensitive to an EP displacement with (Section 2.1 ) to a  confidence level, i.e. . As for the thermal noise, laboratory tests have yielded:

and therefore, since , the differential displacement  due to thermal noise is:

This means that the measurement accuracy required by the experiment can be achieved with an integration time of about . Thus, it is ruled out for the GG experiment be limited by the performances of the capacitance read out. Care should be taken in keeping parasitic capacitances small. However, since they depend on the geometry of the system, the resulting perturbation will be DC. It is therefore enough to make sure that their effect does not exceed the sensitivity by several orders of magnitude. It must be stressed that the required accuracy of  refers to relative displacements - at the spin frequency - of the centres of mass of the test bodies, not to their surface irregularities. The latter will only give DC effects.

It is apparent from Eq. 48 that for the read out to be sensitive to the displacement caused by a possible EP violation of  the corresponding (differential) signal must be larger than the signal due to the largest possible displacement in common mode. Namely:

where  and the maximum common mode displacement is due to air drag and amounts to . Hence, the system must be balanced to , which means  . This level of balancing can be achieved actively by means of inch-worm piezoelectric actuators (Fig. 17 ) acting on the mechanical support of each capacitance plate to make their distance from the test masses as equal as possible. Inch-worm actuators are made of a combination of piezoceramics (no magnets) and can achieve relatively large displacements by a succession of very fine steps. Two inch-worms are needed for each plate, as shown in Fig. 2 . The driving signal for this active balancing is a constant voltage obtained by a proper analysis of the  signal. There is no danger to cancel an EP violation signal by actually making a and b different because the largest common mode effect - which is due to air drag - is variable in time. In any case, a phase check is able to tell whether the signal is due to air drag or EP violation. Inch-worms with 1 stepsize are commercially available.

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Fig. 17. Scheme of the inch-worm. Lateral piezoelectric actuators alternately fasten and release the extremities of the inch-worm to the sides of its container while the inner part is made to expand and contract by means of the other piezoelectrics. In this way the inch-worm can move on a relatively long path in successive very small steps.

We now consider the electrostatic force which affects each test mass. Let's take the inner test mass in the concentric initial configuration of Fig. 15 . As the mass moves by an amount x it is subject to an average force  given by:

This means that the electric forces simulate a spring with a negative constant 

which could in principle be used to increase the displacement produced by the signal and to reduce the natural frequency of the test masses, which would in turn reduce the integration time because of the smaller thermal noise effect (Section 6.2 ). However, since the coupled suspension of the test masses with gimbals appears to give a rather low natural frequency this possibility needs not to be exploited.

Although a more detailed analysis is needed we conclude that a capacitance read out system can reach the required precision of  (corresponding to a  confidence level in the displacement due to an EP violation with ) in a very short time and is therefore by far adequate to the task. It is also worth stressing that ground tests of the capacitance read out system are possible, not only for the sensitivity of the circuit, but also for the balancing and corresponding reduction of the common mode displacements. We are working on a laboratory experiment with concentric, cylindrical test masses in high speed supercritical rotation and a capacitance read out like the one envisaged for GG. It would be a ground test of the main components of the space experiment as well as, possibly, a valuable EP experiment in its own right.

Copyright 1998 Elsevier Science B.V., Amsterdam. All Rights Reserved.

Research Papers Available Online

       (Anna Nobili-