Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 383
Measurement of Magnetic Flux Density by NMR Using
Unsymmetrical Spin Echo
T. Křı́ž1 , K. Bartušek1, 2 , and R. Kořı́nek1, 2
1
Department of Theoretical and Experimental Electrical Engineering
Brno University of Technology, Kolejnı́ 4, Brno 612 00, Czech Republic
2
Academy of Science of the Czech Republic, Institute of Scientific Instruments
Královopolská 147, Brno 612 64, Czech Republic
Abstract— Post processing of magnetic flux density by measured by nuclear magnetic resonance is described in this article. Unsymmetrical spin echo was used to a magnetic field measurement. Magnetic flux density was measured outside the conductive specimen. The cooper
specimen was made on surface of Printed Circuit Board (PCB). Direct current source was connected to the measured specimen.
1. INTRODUCTION
The measurement was assembled to obtain magnetic field values in neighborhood of specimen
connected to the current source. The measured values will be used to reconstruction of specimen
conductivity. The reconstruction of specimen conductivity from magnetic field values is inverse
problem. For this purpose the EIT regularization methods are used. Details can be found in [1].
All magnetic flux density components must be measured in case of conductivity reconstruction
task.
There is a lot of different ways for magnetic field measurement. Most of these methods are
based on Hall effect. The magnetic field components were measured in three different times but
it is very difficult to move the measuring head precisely to measuring points. It is better to use
nuclear magnetic resonance (NMR) to measure magnetic field because one component of magnetic
flux density B is measured in whole specimen at the same time.
There is one very important condition — to use low level of direct current. The level of source
current has to be established with view to possibility of changes in specimen material properties.
The NMR approach was chosen to measure the magnetic field. It is possible to measure small
values of magnetic field by NMR. Gradient echo method (GE) or unsymmetrical spin echo method
(SE) can be use for magnetic field measurement. Both methods have advantages and disadvantages.
The GE is very sensitive to basic magnetic field B0 changes but it is necessary to unwrap the image
phase. The phase unwrapping is very difficult and it is multivalent in the case of neighboring pixels
with phase change higher than 2π. Disadvantage of SE method is smaller sensitivity to changes
of basic magnetic field B0 . It isn’t necessary to unwrap of phase correction the phase of image
obtained by this method. The values of phase are between 2π.
2. MEASUREMENT ASSEMBLY
Several specimens have been prepared for experimental measurement. Two basic shapes of specimens were chosen. The first shape is circle and second shape is ring. Both types of specimens were
prepared with and without defects. The specimens diameter is d = 34 mm. These specimens were
created on printed circuit board (PCB) with copper thickness of 35 µm. Direct current source was
connected to specimens. The current value was adjusted to I = 27 mA.
To obtain magnetic field map it is necessary to place suitable material over specimen. Distilled
water of level 2 mm was used as measured medium. Magnetic field distribution has been measured
only in this layer. In order to measure the distribution of magnetic flux components the specimen
was placed into the working place of NMR tomography. Specimen arrangement for measure the By
component of the magnetic field by tomography is shown in Fig. 1(a). In order to measure the Bx
component the specimen was rotated in perpendicular position in relation to the previous one.
Frequency encoded image represents the result of NMR measurement. Using inverse Fourier
transformation it is obtained a complex image. The phase and module image is necessary to
calculate the magnetic field.
The main information about magnetic field distribution is encoded in phase image. The interval
of phase is (π; −π). It is necessary to phase correction of unwrap the phase image. Initial point for
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
384
(a)
(b)
Figure 1: Specimen arrangement, (a) y-component, (b) z-component.
Figure 2: Phase images.
Figure 3: Magnetic flux density component Bx , By and Bz — phase correction.
phase unwrapping is identified from module image. A differential measurement has been used for
background signals suppression and to obtain a magnetic field map around the specimen. Differential measuring was used to obtain magnetic flux density values. Four measurements were taken for
each component of the magnetic field. Two measurements for each time shift of the spin echo value
+Tp and −Tp were taken. One of the measurements was taken for a positive current direction and
the second fort the negative current direction. Information about magnetic flux density is encoded
in phase image if unsymmetrical spin echo is used. Changes of values of magnetic field relatively
to static magnetic field B0 are given by following Equation (1):
µ
¶
µ
¶
∆ϕ+T e − ∆ϕ−T e +
∆ϕ+T e − ∆ϕ−T e −
∆B = 0.5
− 0.5
,
(1)
γ · TE
γ · TE
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 385
Figure 4: Magnetic flux density component Bx , By and Bz — unwrap.
Figure 5: Magnetic flux density component Bx , By and Bz — calculated.
(a)
(b)
Figure 6: Comparison of results phase (a) correction and (b) unwrap dash line measured data.
where ∆B is change of B0 , TE is time echo, ∆ϕ is coded phase and γ is gyro-magnetic spin ratio.
3. NUMERICAL SIMULATION
In order to verify the measured values by NMR tomography numerical models were built for each
specimen. Geometrical models were discretized by linear triangles elements. The models consist
of approximately 5000 elements and of 2600 nodes. The electrodes were placed on y axis outer
elements. The source current was adjusted to 27 mA. The copper conductivity was adjusted to
σ = 59.59 MS/m. Direction of current corresponded to negative y ax. Electrical potential was solved
by finite element method. The current was calculated on each element. These current values were
used for calculation of magnetic field components. For magnetic flux density components evaluation
was used Biot-Savart’law [3]. Components of magnetic flux density were calculated by averaging
of values measured two millimeters above models. Average value was calculated from forty values
for each element. There were used triangle meshes. We suppose that the surface current density K
is constant on each element. The x- and y-component of magnetic flux density in examined point,
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
386
which is given by coordinates [xi , yi , zi ], can be calculated by means of superposition principle.
Bix =
NE
∆Sj
µ0 X
Rijz 3 Kjy ,
4π
Rij
j=1
Biy = −
NE
∆Sj
µ0 X
Rijz 3 Kjx ,
4π
Rij
i = 1, . . . , N E
(2)
j=1
where ∆S is element area, K is surface current density component, x, y, z are element center
coordinates and R is distance between centers of elements.
4. MEASUREMENT AND SIMULATION RESULTS
The x-, y- and z-components are shown in Fig. 4. Comparing measured and simulated magnetic
field distribution we can see that distribution of magnetic fields is corresponding. Diversity between
measured and simulated values is due to signal phase periodicity in measured layer.
5. CONCLUSIONS
Comparing measured and simulated results it is obvious that the measured data obtained by NMR
tomography aren’t suitable to use as input data for conductivity reconstruction. Our future work
will be focused on improvement of magnetic field measurement by NMR tomography.
ACKNOWLEDGMENT
The research described in the paper was financially supported by project of the BUT Grant Agency
FEKT-S-11-5/1012 and projekt CZ.1.07.2.3.00.20.0175, Elektro-výzkumnı́k.
REFERENCES
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tomography (MREIT): Conductivity and current density imaging,” Journal of Physics: Conference Series, Vol. 12, 140–155, 2005.
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