Abstract—This paper describes a vibrating sample

magnetometer (VSM) intended to measure the magnetic moment of large permanent magnets. The reciprocity theorem was used to preview the pick-up coils output voltage. The finite element method was utilized to determine the magnetic field produced by a unitary current in the pick-up coils. The sensitivity of the magnetometer was determined by the measurements carried out with a well characterized sample.

Index Terms—Vibrating sample magnetometer, permanent magnets, magnetic measurements.

I. INTRODUCTION HIS paper describes a vibrating sample magnetometer (VSM) intended to measure the magnetic moment of large

permanent magnets. In industrial applications it is necessary to monitor the

quality of the large permanent magnets after being magnetized. Some of the produced specimens are subject to a complete measurement process, which includes the measurement of the reverse magnetization curve.

This process is possible when the material is magnetized in a single direction, even when the material coercive force is very large[1]. This is the case of the Nd-Fe-B material. However, for the great majority of the linearly magnetized magnets it will be sufficient to measure the total magnetic moment.

One accurate instrument to measure the magnetic moment of very small ferromagnetic samples is the Foner vibrating sample magnetometer[2,3]. This instrument, usually but not always[4], includes a large electromagnet to produce the necessary magnetomotive force to magnetize the sample. In our case the inclusion of this electromagnet is not necessary, because our purpose is to measure the remanent magnetic moment, being the specimen already magnetized.

To preview the electromotive force due to the movement of the magnetized sample we could use the finite element method to determine the sample field and the flux connected to the pick-up coils. However we preferred to use the reciprocity principle, because in this way it will be obvious that the condition to obtain linearity between the total sample

Manuscript received March 4, 2007. A. Lopes Ribeiro is with the Instituto de Telecomunicações, Instituto

Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (corresponding author’s phone: +351-218418376; fax: +351-8417672; e-mail: arturlr@ ist.utl.pt).

magnetic moment and the pick-up coils output voltage is to confine the sample movement in the pick-up coils central zone. In this zone the magnetic field produced by a hypothetical current in the coils must exhibit a component with a constant gradient along the movement direction.

II. EXPERIMENTAL SETUP In a vibrating magnetometer a magnetized sample is forced

to move in the vicinity of pick-up coils. The moving sample produces a variable magnetic flux in the coils and an electromotive force is originated in the circuit. This electromotive force appears as an output voltage at its terminals.

The experimental set-up is represented in figure 1.

Fig. 1. The experimental set-up. The magnetized sample

is placed in the center of the pick-up coils. In our setup a longitudinal movement is produced by a

mechanical actuator and the pick-up coils are of the Helmholz type. The magnetized sample is placed in the coils central point to obtain a linear relation between the total magnetic moment and the output voltage amplitude.

The Helmholz pick-up coils are represented in figure 2. They consist of two N-turn (N=100) circular windings of radius R=9 cm and separated by 2R. The coiling senses are opposite to each other. If a current were fed to these coils the magnetic field along the z-axis would be oriented in the z-direction and, in the central point, we would get a zero-field point. However, at this point, we will get the maximum gradient of the Bz component. This is the point where the sample magnet will be placed in order to vibrate along the z-direction, where the field gradient is known.

Vibrating Sample Magnetometer for Large Permanent Magnets

A. Lopes Ribeiro, Member, IEEE

T

1-4244-0987-X/07/$25.00 ©2007 IEEE.

ρ

z

null field

2R

2R

Fig. 2. The pick-up coils.

III. APPLICATION OF THE RECIPROCITY PRINCIPLE To determine the pick-up output voltage we apply the

reciprocity principle. In figure 3 a magnetized sample of volume V and magnetization M is represented in the presence of a contour σ.

Fig. 3. Magnetized sample close to the contour σ.

One volume element dv of the magnetized sample located in

r’, produces an elemental variation of the vector potential δA in a point r of the contour given by

0 (1/ )4 or dvµδπ

= − ×∇A Μ , (1)

with | ' |or = −r r . Integrating for the whole volume V we get

0 (1/ )4 o

Vr dv

µπ

= − ×∇∫Α M (2)

The flux σψ connected with the contour σ may be expressed as the circulation of the vector potential A in σ:

dσσ

ψ σ= ⋅∫ A (3)

Replacing the vector potential A in (3) we obtain

0( )[ (1/ ) ]4 o

Vr dv dσ

σ

µψ σπ

= − ×∇ ⋅∫ ∫M (4)

Taking now into account that (1/ ) (1/ )o or r∇ = −∇' , meaning that in the gradient operator ( ')∇ the derivatives are now taken in relation to the spatial coordinates ( ')r of the magnetized sample, rotating the factors of the triple product in (4) and changing the integration order, we get

0( ) '(1/ )4 o

Vr d dvσ

σ

µψ σπ

= ∇ × ⋅∫ ∫ M . (5)

We must remember now that the magnetic field due to a current i in a contour σ is given by

3

' (1/ )4 4 oi id r dσ

σ σσ σ

π π−= × = ∇ ×−

∫ ∫r rH '

r' r, (6)

and the flux σψ in (5) can now be written as

0V

dvσ σψ µ= ⋅∫h M (7)

being σh the magnetic field normalized to the unitary current in σ. From (7), it is possible to obtain the electromotive force developed in σ, and resulting from the movement of the sample. In our case the sample is magnetized along the z-direction and moves along the same direction. The pick-up coils output voltage may be expressed in the form

zO z zV V

d hv dv v M dv

dt t zσ σψ µ µ0 0

∂ ∂= = ⋅ =

∂ ∂∫ ∫h

M . (8)

This result shows that, if the sample moves with a known time-varying velocity zv , the output voltage Ov will exhibit the same type of time variation, and will present an amplitude proportional to the total magnetic moment, if the space derivative of zh in (8) is constant in the sample volume.

To be sure that this linear relationship is obtained, a special attention must be paid to the pick-up coils field.

A. The Pick-up Coils Field The magnetic field along the z-axis of the pick-up coils is

given by [5]

/ /( )

( ) ( )z

IH z RR R z R R z

23 2 3 22 2 2 2

1 1= − 2 + − + +

, (9)

being I the current intensity in the coils. The space derivative of ( )zH z at the central point is

z

z

H Iz R2

=0

∂ 2=∂ 2

. (10)

To obtain a linear relationship between the sample magnetic moment and the coils output voltage, the derivative in (10) must be approximately constant in the volume occupied by the sample.

To determine the constancy of the field derivative mentioned above we used the finite element method. In figure 4 we show the magnetic field lines for a normalized geometry. The dashed lines represent the locus where the z-component of the magnetic flux density ( )zB z was determined. In figure 5 we represent the three profiles of

( )zB z determined for ρ =0, ρ =0.1R and ρ =0.2R respectively. These results show that is the sample volume is contained inside a cylinder with a radius not exceeding 20% of the coils radius, and height not exceeding 40% of that radius, we expect that the output voltage will be proportional to the total

magnetic moment, and with sensitivity independent of the sample size.

ρ, m

z,

0 1 2

1

2

Fig. 4. Magnetic field lines for a normalized geometry with radius R =1 m. The Bz(z) profiles were determined

along the dashed lines.

B z(z

), m

T

z, m

0.6

0.5

0.4

0.3

0.2

0.1

0 0 1

ρ = 0 ρ = 0.1R ρ = 0.2R

Fig. 5. Profiles of the vertical component of B for ρ =0,

ρ =0.1R and ρ =0.2R. The derivatives of the three curves are almost equal at z =0.

IV. EXPERIMENTAL RESULTS In figure 6 we represent the experimental setup used to

measure the pick-up coils output voltage. A function generator Agilent 33220A was used to drive the power amplifier Hewlett Packard 467A with a sinusoidal voltage.

Fig. 6. Experimental measurement setup.

The output voltage was measured using a Scitec Instruments 450S DSP Lock-in amplifier. The all system was monitored using a Tektronix TDS 2014 digital oscilloscope. A

photograph of the experimental measuring setup is shown in figure 7.

Fig. 7. The experimental measuring setup.

A calibration procedure was performed using a Nd-Fe-B

magnet of cylindrical shape, and with the magnetization along the axis. The properties of the calibration sample are summarized in table I.

Some preliminary tests were carried out. The amplitude and

frequency f (50< f <100 Hz) of the vibrations was varied. A linear relationship between the voltage applied to the mechanical driver and the pick-up voltage was verified, for a constant frequency. The resulting curve for the frequency

83 Hzf = is represented in figure 8. However, the vibration amplitude depends on the sample mass. For this reason a second small coil was included, to certify that the measurements were made with the same amplitude of movement.

It was verified that the output voltage decreases with frequency according to the curve presented in figure 9. This kind of variation is related to the mechanical properties of the actuator. We chose to operate at a frequency of 83 Hz and to drive the actuator with a voltage of 1 VRMS. By choosing a prime number to the frequency we expect to avoid signal corruption originated in the power supply system.

The obtained sensibility of the magnetometer, being the output voltage per unity of magnetic current, was

-1 -225.3 µVA m .

TABLE I ND-FE-B MAGNETIZED SAMPLE PROPERTIES Manufacturer OMRON External diameter (mm) 10 Width (mm) 10 Grade N30H Remanence (T) 1.18 Coercivity (kA/m) 915 Magnetization (kA/m) 939 Energy density (kJ/m3) 240 Total magnetic moment (Am2) 93.9

0

0,5

1

1,5

2

2,5

0 0,2 0,4 0,6 0,8 1 1,2

Driver Voltage (volt)

Pick

-up

volta

ge (m

V)

Fig. 8. Pick-up voltage versus driver voltage, for an operating frequency

f=83 Hz.

2

2,1

2,2

2,3

2,4

2,5

2,6

2,7

2,8

2,9

3

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Frequency (Hz)

Pick

-up

Volta

ge (m

V)

Fig. 9. Pick-up voltage versus frequency, when the driver voltage was set to

Vd=1 VRMS.

V. CONCLUSION In this paper we presented the design and construction of a

vibrating sample magnetometer to measure the magnetic moment of permanent magnets. The pick-up coils were designed using the reciprocity principle. Using this procedure it was possible to determine the maximum volume of the samples, which corresponds to the region where the magnetic field originated by currents in the pick-up coils is parallel to the sample magnetization and presents a constant gradient in that direction. The vibration movement on that direction originates an electromotive force in the coils that is proportional to the total magnetic moment.

The obtained results were consistent with previous modeling work [6] to determine the repulsion force between cylindrical magnets.

The obtained sensitivity is low, but this is not a disadvantage as the magnets under measurement exhibit large magnetic moments due to their strong magnetization and large volumes.

REFERENCES [1] J. Dudding et al, “A pulsed field magnetometer for the quality control of

permanent magnets”, in Journal of Magnetism and Magnetic Materials, 242-245 (2002), pp. 1402-1404.

[2] S. Foner, “Vibrating Sample Magnetometer”, in Review of Scientific Instruments, Vol. 27 (1956) N.7, p.548.

[3] S. Foner, “The vibrating sample magnetometer-Experiences of a volunteer”, in Journal of Applied Physics, Vol. 79 (8) (1996), pp. 4740-4745.

[4] O. Cugat et al, “A compact vibrating-sample magnetometer with variable permanent magnet flux source”, in Review of Scientific Instruments, Vol.65, N.11,(1994), pp. 3570-3573.

[5] M. A. Plonus, Applied Electromagnetics. New York: Mc Graw-Hill. 1978, ch. 6.

[6] A. Lopes Ribeiro, “Axial and transverse forces in an axisymmetric suspension using permanent magnets”, in Elsevier-Physica B, Vol. 384 (2006), pp. 256-258.

Copyright Information

© 2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists,

or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.