A coil system for VIRGO providing a uniform magnetic field gradient

Physics Letters A 171 ( 1992 ) 162-166

PHYSICS LETTERS A

North-Holland

A coil system for VIRGO providing a uniform magnetic field gradient L.E. H o l l o w a y Department of Physics, University of Illinois, Urbana, IL 61801, USA

C. Bradaschia, E. Calloni, M. Cobal, R. del F a b b r o , A. di Virgilio, S. Galeotti, A. G i a z o t t o , H. K a u t z k y , V. Montelatici, M. Morganti, D. Passuello, S. Braecini, R. F l a m i n i o INFN, Sezione di Pisa and Dipartimento di Fisica, Universititdi Pisa, Pisa, Italy

L. di Fiore INFN, Sezione di Napoli, Naples, Italy

and W. Velloso University of S~o Paolo, S~o Paulo, Brazil

Received 29 September 1992; accepted for publication 2 October 1992 Communicatedby J.P. Vigier

The mirrors of VIRGO, a proposed gravitationalwave interferometer,must be aligned and locked onto an interference fringe. The special requirements of the force transducer lead us to the need for a coil system which has a very uniform magnetic field gradient. We describe the design, construction, and testing of a coil systemwhich provides a magnetic field gradient with deviations from uniformity less than a few parts in 104 over a 10 mm diameter volume.

The motivation for this work lies in the control of mirror positions in V I R G O [ 1 ], a proposed gravitational wave interferometer. The mirrors of VIRGO must be aligned and locked onto an interference fringe. As the expected gravitational wave signals are extremely small the method of applying the force must not introduce noise into the system. This leads to special requirements on the force transducer system. (1) The method must be non-intrusive. For example attaching an elaborate mechanism to a mirror would degrade its Q and serve to introduce unacceptable thermal noise and spurious mechanical resonance into the system. (2) The force must be independent of small variations in the distance between the mirror and force transducer. This is to prevent coupling between the 162

seismic motion of the ground and the mirror. One method which has been suggested [2] is to fasten small permanent magnets, m, to the mirror and use the forces on them due to an external magnetic field gradient, F = (m. V)B. A concern, however, especially for low frequency gravitational wave detection, is introduction of seismic noise into the interferometric system due to the fact that the coil system is mechanically coupled to ground. That is, seismic vibrations in the coil system can be transmitted to the mirror if the magnetic field gradient is not uniform. In order to motivate this more explicitly, let us consider the derivative of the force along the magnetic axis, F ' = m ( O B ' / d z ) . Then seismic noise of amplitude Az produces an extra force A F = m ( # B ' / Oz)Az. Of course it is possible to find a position where

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Volume 171, number 3,4

PHYSICS LETTERS A

the second derivative of the magnetic field is zero, but this region is small and any misalignment will introduce a non-negligible error. This additional force is applied directly to the mirror and competes with the Riemann force due to a gravitational wave. The Riemann force is given by Fx=Mht22L, where M is the mirror mass, h is the gravitational wave amplitude, £2 is the angular frequency of the wave and L is the interferometer arm length. The measurability condition gives AF< FR. The seismic spectral amplitude is given roughly by ~kZ,= 5 X 1 0 - 7 / P 2 m/qrH--z, where v is the frequency. The smallest spectral Riemann force expected to be observable by VIRGO at 10 Hz is fiR---2 × 10-12 N / x / ~ ; the limitation being due to thermal noise in the suspension. Hence it follows that (OB'/dz)/B' must be less than F R / ( F ) A z over the range of operation. Here, ( F ) = 2 × 10 -3 N is the average force needed to keep the interferometer in lock. This leads to a requirement that the non-uniformity in linearity of the coil system be less than a few parts per thousand over a 5 to 10 mm range. The problem is thus well posed: design a coil system which provides a uniform magnetic field gradient within a specified volume. Consider a coil of radius a, with N turns carrying current L The on-axis field and the first few derivatives are given by

B(z) = p.oNI a3 2a (z2q-a2! 3/2' p.oNI

2a 2 (z2~_a2)S/2,

B"(z)=-

p.oNI 3aS ( a 2 - 4 z 2) 2a 3 (z2+a2)7/2 ,

0.0010

•~

ltoNI 15za6(3a2-4z 2) B"(z)-

2a 4

it is necessary to add additional correction coils. If these coils are placed anywhere on a cone whose apex is at the position of the permanent magnet and whose opening half-angle has a tangent of 2.0, then the second derivative of the force vanishes automatically. It remains to find currents and positions of additional coils to cancel the third, fourth, etc., derivatives. There are many possible geometries which one can use. We use the following simple geometry. Consider a primary coil with radius a placed a distance ½a from the permanent magnet. 'The first correction coil is placed at ~a from the magnet, the second at ~a, and so on. The corresponding radii of the correction coils are ½a, ~a, etc. The solution for the currents for the two coil case is: Ix = 4 I0 and 12 = - ~ I 0 , where I0 is the current in the single coil configuration. The solution for the currents for the three coil case is: I1 = ~I0, 12 = - 410, and 13 = ~-~I0.1 In this configuration the second, third and fourth derivatives vanish. Figure 1 shows the deviation from uniformity as a function of the distance, on axis, from the center of the system. The radius of the primary coil is taken to be 100 mm in this example. The results for one, two, and three coil geometries are shown. We now consider the effects of imperfect alignment of the coil system. There are two effects when the permanent magnet is not on the axis of the coil. The first effect is a torque on the magnet due to a radial component of the magnetic field and arises

3za 4

B'(z)=-

(Z2.I.a2)9/2

7 December 1992

0.0000

'

'~

I

. . . .

--

I

~"

. . . .

I

.

\

,

where z is the axial distance from the center of the coil. The requirement that the first derivative of the force (second derivative of the field) be zero leads to z = ½a. This ensures that small displacements of the coil cause no additional forces on the permanent magnet. In order to increase the size of the region where the force on a permanent magnet is uniform, i.e. insensitive to larger displacements in the coil position,

. . . .

o :,3

-0.0005

\

I

•~

r

®

~

i . . . . -0'001010

I

-5

,

.

,i

i ,

I

0

,

~.

,

,

I

~

,

,

6

10

Axial Distance from Origin Cram) Fig. 1. Deviation in ma~etic field gradient from the nominal value at the center of the permanent magnet. The deviation is defined as D = B ' ( z ) / B ' ( 0 ) - I. Tbe dot-dashed, dasbed and solid curves correspond to one, two and three coil systems respectively. The radius of the primary coil is 100 ram.

163

Volume 171, number 3,4

PHYSICS LETFERS A

7 December 1992

Table l Deviations from constant magnetic field gradient. The longitudinal, z, and transverse, p, dimensions are given in ram. The entries correspond to the deviation in parts per 10000. z (ram) single coil

coni]guration

0

1

2

3

4

5

6

7

8

9

-7 -6 -5 -4 -3 -2

-169 -123 -84 -53 -29 -13

-167 -121 -82 -51 -28 -11

-161 -115 -77 -46 -22 -6

-1

-3

-1

3

-137 -92 -54 -24 -1 14 23 25 21 11 -4 -26 -54 -86 - 124 -166 -213

-120 -75 -37 -7 14 29 38 40 35 24 7 -14 -42 -75 - 113 -156 -204

-98 -53 -17 12 33 48 56 57 52 40 23 0 -28 -61 - 100 -144 - 192

-72 -28 7 35 57 70 78 78 72 60 42 18 -11 -45 -85 -129 - 178

-42 0 35 63 83 96 102 102 95 82 63 38 8 -26 -67 -112 - 162

-8 33 67 94 113 126 131 129 121 107 87 61 30 -5 -47 -93 - 143

-11 -5 -1 0 l 1 0 0

-12 -4 0 3 3 3 l 0

-11 -1 4 7 7 5 2 0

-1

-2

-2 -2 -1 0 2 5 7 10

-4 -3 -2 0 2 6 10 13

0 -3 -12 -27 -48 -75 -106 - 143 -184 -230

-1 -11 -26 -47 -73 -105 - 142 -183 -229

6 2 -6 -22 -43 -69 -101 - 138 -180 -226

-151 -105 -67 -37 -13 2 11 14 10 0 -14 -36 -63 -95 - 132 -174 -221

-7 -6 -5 -4 -3 -2 -1 0

5 2 0 0 0 0 0 0

4 2 0 0 0 0 0 0

3 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0

-1 -1 -1 0 0 0 0 0

-5 -3 -1 0 0 0 0 0

-8 -4 -2 0 0 0 0 0

1

0

0

0

0

0

0

0

0

2 3 4 5 6 7 8 9

0 0 0 0 0 -1 -2 -4

0 0 0 0 0 -1 -2 -4

0 0 0 0 0 0 -1 -3

0 0 0 0 0 0 -1 -2

0 0 0 0 0 0 0 0

0 0 0 0 1 1 1 1

0 0 0 0 1 2 3 3

-l -1 0 0 2 4 5 6

0 1 2 3 4 5 6 7 8 9 three coil configuration

p (ram)

1

f r o m Ir=mXB. T h e r a d i a l c o m p o n e n t o f t h e m a g netic field can be related to the derivative along the axis b y u s i n g V - B = 0. A n e x p a n s i o n o f Bp a b o u t p = 0 g i v e s Bp~ -½pB'. H e n c e t h e t o r q u e o n t h e m a g n e t is g i v e n b y I TI = ½PFw h e r e p is t h e t r a n s v e r s e disp l a c e m e n t f r o m t h e coil axis a n d F is t h e l o n g i t u dinal force on the permanent magnet. T h e s e c o n d effect o f t r a n s v e r s e m i s a l i g n m e n t is a s m a l l r e d u c t i o n i n t h e z c o m p o n e n t o f t h e force. T h i s arises from a term in the magnetic field of the form 164

8pB'. A g a i n , o n e c a n r e l a t e t h i s t e r m t o d e r i v a t i v e s i n z b y u s i n g b o t h V . B = 0 a n d V × B = 0. T h e l a t t e r e q u a t i o n g i v e s t h e r e l a t i o n 0 p B z = O~Bp. E x p a n d i n g Bz a n d Bp a b o u t pffi 0 i n a p o w e r series i n p a n d m a k i n g u s e o f t h e r e l a t i o n s g i v e n b y t h e d i v e r g e n c e a n d curl, one obtains B'~(p)"B'~(O)-~p2B~3)+~p4B~5) + .... T h u s i f t h e t h i r d d e r i v a t i v e o f B~ w i t h r e s p e c t to z vanishes then the variation in axial force with r e s p e c t t o t r a n s w r s e m i a a l i ~ n m e n t is f o u r t h o r d e r i n p. T h i s is a n i m p o r t a n t p o i n t i n t h a t it e n s u r e s a u n i -

Volume 171, number 3,4

PHYSICS LETTERS A

form gradient for both longitudinal and transverse displacements. Table 1 shows the deviations from uniformity in the z-p plane for the one and three coil configurations. In the three coil case the deviations are less than one or two parts in 104 in a cylindrical volume 15 m m long and 15 m m in diameter. The effect o f a small rotation, t~, o f the coil about a vertical axis is the equivalent to a rotation o f the permanent magnet on axis by the same angle tx and a transverse misaLignment o f p = ½o~a. Summing these two effects gives a torque o f value ~=t~B=m+ ½olaB'~m=am(B=+½B'~a). These two terms are o f opposite sign and are related by the divergence of B. For a single coil with the permanent magnet placed at z = ½a the result is z = ~aaB'~m= ~aaF. Thus the torque induced by a rotation o f the coil is equivalent to that given by a transverse misalignment o f an amount p = ~txa. The second effect o f a coil rotation is to produce a transverse force on the permanent magnet. This force is given by Fp = ½aB'm= ½txF~. The third effect is a reduction o f the longitudinal force by cos a. Thus, ~F/F= 1 - cos ot _ ½ot2. We now describe a practical implementation o f these ideas. The field on the axis o f a coil with rectangular cross section is obtained by integrating the formula for a single turn to give B z l o "~

7 December 1992 16~x 33 turns

P

T

6 x 8 turns

P.'j;:7 n'

-E

E

0

0

E E

Z

o O o4

12.5

IN! 25.1 mm

goNI 2(Z2 --Zl) (a2 --t21) 51.2 mlIT--'~ I

L

in

2

2

--Zl

In

2

2

•

The field is calculated for a point at the origin. The coil cross sectional bounda~es are at z = z~ and z = z2 along the z axis and at radii p=a~ and p=a2. The position along the z axis where the second derivative vanishes is no longer given by the simple relation z = ½a but is a complicated function more easily determined numerically than by explicit evaluation o f a formula. A computer program determined the following optimized three coil configuration shown in fig. 2. The coil system consists o f three windings. The same current I flows in all three windings but is reversed in the middle one. The wire diameter is 1 ram. Table 2 lists the coil parameters. The total resistance o f the windings is R = 7.5 fL The self inductance is approximately 72 m H which gives a reactance o f

Fig. 2. Details of the three coil system. The wire diameter is 1 ram. All dimensions are in nun.

Table 2 Parameters for the three coil configuration. Dimensions are in mm.

turns per layer number of layers total turns inner radius outer radius near distance far distance

Coil 1

Coil 2

Coil 3

33 16 528 92 108

8 6 48 47 53

1 1 l 25 25 12.5 12.5

34.7

21.1

67.7

29.1

165

Volume 17 l, number 3,4

PHYSICS LETTERS A

X = 4.5 fl at 10 Hz. With this coil the value of the force on a 1 T permanent magnet consisting of a 4 mm thick disk of diameter 14 mm is equal to 1.8× 10 -2 N/A. A study of the effect of a small longitudinal displacement of the secondary coil with respect to the main coil showed that with ordinary fabrication tolerances, the distortion of the field gradient would not be acceptable. A solution to the problem was to insen two trimming turns, one on either side of the second coil, carrying equal but opposite currents. By adjusting the current in the trimming turns one can, in effect, move the average position of the coil/trimmer combination to the required value. Three complete sets of three coils were constructed and tested. A test pickup coil was made in order to directly measure the magnetic field gradient. It consisted of two windings, of 800 turns each, connected in opposition. The wire diameter was 0.1 mm. Each winding had inner and outer diameters of 12 and 14 mm respectively and a width of 4 mm. The windings were mounted coaxially 6 mm apart, center to center. A small difference between the two coils serves to make a minor, constant, offset in the measurements. Measurements were made by exciting the three coil system with a 1 kHz current. Signals representing the excitation current and the test coil voltage were fed into a dual channel signal analyzer with a dynamic range of more than 80 dB. Variations in the ratio of the two signals were plotted as a function of test coil position with respect to the three coil system. The optimum trim coil currents were determined empirically; they varied substantially from set to set reflecting the imperfections in the fabrication process. Results for one, typical, coil system is plotted in fig. 3. The results are, evidently, in good agreement with the theoretical predictions. In conclusion, we

166

7 December 1992

0.0010

....

I ....

I ....

0.0005

I ....

x Data x

"0 •~'~ r.,.

0

0.0000

"

"

x

x

,

.

x

t

-0.0005

I~1

I -o.oolo

,

-10

~

,

,

I

-5

,

,

a

i ,

I

0

,

~,

I

5

....

10

Axial Distance From Origin (ram) Fig. 3. Comparison of measured deviations of the magnetic field gradient with theory. The solid curve is for the three coil system. The dot-dashed curve is for comparison with a single coil.

have demonstrated that it is possible to significantly extend the volume over which a magnetic field gradient can be uniform by using secondary coils which cancel higher derivatives. The performance of the prototype coil system meets the VIRGO requirements.

References [ 1 ] C. Bradaschia, R. del Fabbro, A. di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. PassueUo, A. BriUet, O. Cregut, P. Hello, C.N. Man, P.T. Manh, A. Marraud, D. Shoemaker, J.Y. Vinet, F. Barone, L. di Fiore, L. Milano, G. Russo, J.M. Aguirregabiria, H. Bel, J.P. Duruisseau, G. le Denmat, Ph. Tourrenc, M. Capozzi, M. Longo, M. Lops, I. Pinto, G. Rotoli, T. Damour, S. Bonazzola, J.A. Marck, Y. Gourghoulon, L.E. Holloway, F. Fnli~ni~ V. lafolla and G. Natale, Nucl. Instrum. Methods A 289 (1990) 518. [2] D. Shoemaker, W. Winkler, K. Maischberger, A. Rudiger, R. Schilling and L. Schnupp, Prngress with the Garching 30meter prototype for a gravitational wave detector, MPQ 100 (1985).