Electrical damping of a torsion balance

Volume 152, number 8

PHYSICS LETI’ERS A

4 February 1991

Electrical damping of a torsion balance Y.T. Chen’ and B.C. Tan InstituteforAdvanced Studies, University of Malaya, 59100 Kuala Lumpur, Malaysia Received 9 July 1990; revised manuscript received 27 November 1990; accepted for publication 29 November 1990 Communicated by J.P. Vigier

It is proved with a typical electrical circuit used in most torsion balance experiments that electrical feedback will not introduce an extra fluctuation force for the Brownian motion of a torsion balance; a reduced thermal noise level may be achieved by partial cooling ofcertain components of the electrical circuit without cooling the whole set of the balance at the equivalent temperature.

I. Introduction Thermal noise, as was discussed in a previous pa-

per [1], may present serious limitations for experiments usinga torsion balance, or any low dispersion linear oscillators. The question therefore is, whether it is possible to reduce this noise as much as we require in practice. It seems that there are two ways to do so; one is to reduce the environmental temperature T, another is to increase the mechanical quality factor Q of the oscillator. Usually, increasing the value of Q or reducingthe damping factor of the oscillator is a more effective and cheaper method. However, there are a few problems associated with the use of the above measures to reduce the Brownian motion of a torsion balance. First, cooling the whole system is not only expensive but also in the process external vibration is easily introduced; so the problem is, is there another method to reduce thermal noise without cooling the whole device? Second, we know that for an oscillator of very high Q some kind of man-made damping mechanism shall be accompanied to quickly stop the motion; then the problem is, what sort of extra fluctuation forces will be brought into the torsion balance system due to the introduction of the damping mechanism. Among different kinds of damping mechanisms, electrical damping is one of the most commonly used On leave from Corpus Christi College, University of Cambridge, Cambridge, UK. 0375-9601/91/$ 03.50 © 1991

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methods in laboratory gravitational experiments. In this paper, it will be shown that electrical damping will not bring extra fluctuation forces into the torsion detection system and that it is possible to reduce further the Brownian motion of a torsion balance by the method of partial cooling of some components of the electrical circuit in the damping System. Although there have been very few theoretical discussions about electrical damping in torsion balance experiments, there are some discussions of electrical damping in a galvanometer [2] and the bar antenna in a gravitational wave detector [31.Only electrical damping in a torsion balance experiment will be discussed here and we hope the theoretical results obtamed below will draw some attention from experimenters.

2. Electrical damping If an electrical signal is introduced into a torsion balance, two types of thermal noise are involved in the system, namely, fluctuation of mechanical origin and that ofelectrical origin. The purpose of the present discussion is to understand how these two sources of fluctuating force act together to contribute to the total thermal noise of a torsion balance. It is difficult to consider the general case with electrical feedback because electrical circuits may all be different. However, fig. 1 shows a typical method of

Elsevier Science Publishers B.V. (North-Holland)

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~

PHYSICS LETTERS A

[I]

E, and Q2 on the plane E2 (Qe) can be described as Qe=(V+’o)CiHft~”o)C’2]

‘~

T ~ T Fig. I. A typical circuit for the electrical damping of a torsion balance.

adding an electrical voltage V (in the case of electrical damping, the voltage Vis the velocity feedback signal) to the system with one of the moving arms of the torsion balance sandwiched between two symmetrical capacitive plates E, and E2. This circuit is used in most torsion pendulum experiments. The electrostatic force j acting on the system is (1)

~

fe=~V

where V1 and V2 are the voltages between the two electrodes and the beam which is grounded to the earth, C, and C2 are the capacities of E, and E2 relative to the beam respectively. Because of the symmetry of the electrodes, for a small displacement x of the oscillator, one can have C~=C0+ax, C 2=C0—ax,

=2VC0+V0ax. (5) It is necessary to consider the difference between the absolute values of the charges, because only this factor matters for the motion of the beam. In the situation considered, the beam of the torsion pendulum is in the zero-potential state, so that any potentials on E, and E2 with the same value (that is, Qe in eq. (5) is zero) will pull the beam with equal electrostatic forces. The non-zero value of Qe will cause a current in the resistor and the voltage equation can be described by RQ+ v—E(t) (6) —

where E( t) is the thermal voltage fluctuation inside the resistor R. Inserting eq. (5) into eq. (6) results 2RC0V+aV0Rx+V—E(t).

2mRC05~+(2HRC0 + m )I

2V~R)±+Ax

+(2ARC0+H+a

(3)

=aVoE(t)+2ñRC0+n(t).

~ V0,

(7)

Combining eqs. (4) and (7), the motion of the torsion balance is given by

(2)

where C0 and a are constant. From fig. 1, it is obvious that V1 = V+ I Vo~ V2 = V

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(8)

This indicates that, generally speaking, the torsion balance with electrical feedback will not obey a second-order differential equation. However, in the case of a torsion balance, ~ is usually very small, so that

where V0 is the fixed bias voltage across the plates E, and E2. Then eq. (1) becomes

eq. (8) can2V~R)j~+Ax=aV be simplified to

fe = I ( V1 V2) ( V1 + V2 ) a = VV0 a. Thus, the equation of motion of the torsion balance becomes

The above treatment to reduce the higher order differential equation to a second-order equation was used in the work of Ornstein et al. [2] where a galvanometer coil was studied. In the above study, for a galvanometer, the equation of motion is reduced to a second-order equation by neglecting the inductance of the coil. Analogous to that, for a torsion balance, the equation of motion can be reduced to a second-order equation by neglecting the capacity. In practice, the capacity of the feedback plates is very small. As a practical example, in the experiment of Chen et al. [4], the capacity of the feedback plates

—

mx+Hx+~~.x=aV0V+n(t).

(4)

To understand how the electrical signal V introduces the fluctuation and affects the mechanical movement, it is required to know the relation between the voltage V and the charges on the plates E, and E2 which cause the electrostatic force. In the above electrical system, for a large resistance R, the difference of the absolute values of the charges Q, on the plate 378

mI+(H+a

0E(t)+n(t)

.

.

(9)

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was only 0.5 pF, of the same order as a one meter coaxial cable. The mean square of the fluctuation of electrical origin and mechanical origin, denoted by and respectively, may be calculated from

<4>

=JIw(f)12G(/)df,

(10)

a

where G(f) is the power spectrum of the thermal fluctuation force and W(f) is the complex response function. W(f) can be defined for any linear instrument. For the motion of a torsion balance described by m

d2x -~—~

+ /3

dx -~-

+ )x= 0,

(11)

the response function is given by

~(J)=

I (12) A — mw2 + 1/3w where the damping factor /3 is a general factor, in the case of a system with feedback, i.e., in the case expressed by eq. (9), /3= H+ a2 V~R. The fluctuations of electrical origin and mechanical origin can be calculated separately as indicated below, In eq. (9), ifE(t)=0, then it is only necessary to consider the mechanical fluctuation. The power spectrum Gm of a purely mechanical thermal force, according to the generalized Nyquist theorem, is given by

U)

Gm (I) = 4HKb Tm,

(13)

where Tm is the temperature of the air. It follows that =

J

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GC(J)=4KbRTT, (15) where Tr is the temperature of the resistance. It follows that ai”o 8,tRK Tr dw 0

Kba2V~RT. (16) (H+a2V~R)2~ If Tm = Tr, then from eqs. (14) and (16), the total fluctuation is given by —

_+=KbT/A., (17) the same as for a purely mechanical system. Eq. (17) concludes that the electrical feedback system will not introduce any extra fluctuation to the system. Now we understand that, in experiments to measure an extremely weak gravitational force, on the one hand, in order to increase the factor of mechanical quality and to raise the signal to noise ratio, we should reduce the mechanical damping as much as we can; on the other hand, in order to make the measurement easy and convenient, electrical damping can be applied to the system without fear of introducing extra fluctuation forces. However, as indicated by eq.

(17), the total value of thermal fluctuation will not change. In section 3, a partial cooling method is proposed to reduce this value. 3. Partial cooling of the torsion balance The above discussion shows that the fluctuation of a system with electrical feedback is dependent only on the mechanical damping and the resistance of the circuit. This makes the method of partial cooling of the torsion balance possible.

2 ,~2~ifl

8~tHKbTmdw

Returning to eqs. (14) and (16), assuming that TrTm,

0

KbHTm (H+a2V~R)A~

(14)

In eq, (9), if n (t) = 0, a purely electrical fluctuation is being considered. The power spectrum Ge(f) of the thermal fluctuation of the voltage E( t), according to the Nyquist theorem, is given by

the total Brownian motion will become H (H+ a2 VO2R )A Kb Tm

a2V~R + (H+a2 V~R)A Kb Tr.

(18)

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Thus, the effective temperature can be defined as a2V~R Tr. (19) Te = H+a2 V~Tm + H+a2V~R It is easy to see that, if ______

Tr ~ Tm,

then Te ~ Tm.

This is very significant, because, with this method, the thermal noise level of K,,Te can be reached without cooling the whole system to the temperature Te. This is called the partial cooling method. To understand the meaning of partial cooling, i.e., how a cooler resistor is able to damp the Brownian motion of the oscillator, it may be of assistance to relate it to an actual physical mechanism. From everyday experience, it is known that an easy way to stop a swinging simple pendulum is to use the hands as a load to damp it. When the bob of the pendulum touches the hands, it transfers its energy to the hands and gradually the energy of the pendulum is dissipated and the motion stops. Unfortunately, generally speaking, this mechanism would not work for Brownian motion. As 15 known, the process of damping of the oscillator is the result of a thermal coupling between the oscillator and its environment which is the air around it or the load (e.g., the resistor in the feedback circuit). The oscillator transfers energy to the environment and reciprocally the environment transfers it to the

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oscillator. Under normal circumstances, because they are located in the same thermal bath at the same temperature, although the oscillator transfers energy to the load, the load will give the oscillator a backward reaction originating from the same thermal fluctuation. Therefore, the Brownian motion will not be reduced. However, in the case that the load is not located at the same thermal bath as the oscillator, the situation will be different. If the load is cooler than the oscillator, although the oscillator will transfer its thermal energy to the load, the load will transfer a lesser amount of energy backwards. The implication is that the Brownian motion of the oscillator will be damped.

Acknowledgement We would like to express our gratitude to Professor Air Alan Cook of the Cavendish Laboratory, University of Cambridge, for his help and valuable discussions. References [1] Y.T. Chen and A.H. Cook, Class. Quantum Gray. 7 (1990) 1225. [2] L.S. Ornstein, H.C. Burger, J. Taylor and W. Clarkson, Proc. R. Soc. A 115 (1927) 391. [3] H. Hirokawa, S. Hiramatsu andY. Ogawa, Phys. Lett. A 63 (1977) 199. [4] Y.T. Chen, A.H. Cook and A.J.F. Metherell, Proc. R. Soc. A 394 (1984) 47.