Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 167
Magnetic Flux Density Reconstruction Method
M. Hadinec
Department of Theoretical and Experimental Electrical Engineering
Brno University of Technology, Czech Republic
Abstract— This article deals with the measurement of magnetic flux density around copper
specimens, which are created using printed circuit board technology. The current that flows in the
specimen creates magnetic field. Such magnetic fields can be mapped using magnetic resonance
tomography or by Hall probes. The method for reconstruction of magnetic flux density inside
the measured volume will be presented and analyzed. The reconstruction method is based on
Legendre polynomials approximation, using Least Mean Squares solver programmed in Matlab.
This reconstruction method could be usable for acceleration of the mapping process, which is
lengthy and can lead to many errors while measuring many points and many planes around
the specimen. Using proper approximation coefficients, it is possible to compute individual
approximated maps of magnetic flux density, so there is no need of fine mapping and meshing
inside desired volume. These coefficients can be obtained from values measured on the edge
of desired volume, using Laplaces’s equation solution.The measured and approximated maps of
magnetic flux density are compared and differences analyzed. The theoretical model of magnetic
flux density around measured specimen is made in COMSOL.
1. INTRODUCTION
The magnetic flux density was measured in several planes around specimens, using Hall’s probes.
The proper mapping of magnetic flux density around these specific specimens is important for
electric impedance tomography (EIT) research and other further numerical purposes [1]. The
mapping process is usually made manually and can obtain many errors. It would be helpful to
speed up this process by approximation and reconstruction of several planes in the desired volume
or by making simulation of magnetic flux density distribution in COMSOL. The aim of this work
is to show magnetic flux density reconstruction method based on Legendre’s polynomials, which
could be suitable for computing maps of magnetic flux density inside measured volume [2]. The
better speed of mapping process is obvious, but the approximation can put additional errors to
expected shape of magnetic field.
2. MATHEMATICAL MODEL
The mathematical model for reconstruction is based on solution of Laplace’s equation. We assume,
that the field inside analyzed volume is stationary (d/dt = 0) and there are no sources of magnetic
field. The Maxwell equations are reduced as
∇×B = 0
∇·B = 0
(1)
(2)
Each equation can be combined by application of Nabla operator and the result will be
∇(∇ × B) = ∇(∇ · B) − ∇2 B = ∆B = 0
(3)
where ∆ is the Laplace’s operator and this formula is Laplace’s equation. In our model, we assume,
that the desired magnetic flux density vector is oriented perpendicular to the surface of the copper
specimen.
Now it is suitable to use well known Dirichlet’s problem of finding a function, which solves a
specified partial differential equation in the desired volume. The solution takes prescribed values
measured on the boundary of the volume [3]. The Dirichlet’s problem is based on finding a unique
continuous function u, which is twice continuously differentiable in the desired volume and continuous on the boundary (if there is given function f that has values everywhere on the boundary of
a volume), such that u is harmonic in the interior and u = f on the boundary. The requirement is
called Dirichlet boundary condition. In NMR applications, the general solution could be obtained
as
∞ m=n
X
X
B(r, θ, ϕ) =
rn · Pn,m (cos θ) · [Cn,m · cos(m · ϕ) + Dn,m · sin(m · ϕ)]
(4)
n=0 m=0
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
168
where r is distance from origin of coordinates, Pn,m (cos θ) are Legendre’s polynomials of first order
a nth degree (for m = 0), resp. associated Legendre’s polynomials (for m = 1, 2, . . . n), m is
polynomial degree, θ is half of peak angle, ϕ is azimuth angle, Cn,m and Dn,m are approximation
coefficients [4]. The solution is converging inside sphere with radius, which is equal to number one.
The formulas for computing of Legendre polynomials for given order and degree were described
in [5] and are beyond the scope of this article. To obtain approximation coefficients, the Matlab
LSQR solver has to solve system of linear equations, which can be described according this formula




K

Bm1 Bm
hc1,1,1 hc1,1,2 · · · hc1,Q,Q hd1,1,1 · · · hd1,Q,Q S
C1
 Bm2 
 hc2,1,1 hc2,1,2 · · · hc2,Q,Q hd2,1,1 · · · hd2,Q,Q 
 .. 
 . 

 .
 . 
..
..
 . 
 .



.
.

 . 
 .
2 x1 
C

Q


 .
(5)
=
×

..
 ... 

 ..
 D1 
...
...
...
.




 . 
 . 

 .
..
 .. 
..
 .. 

 ..
.
.
2
D
Q x1
Bmg g×1.
hdg,1,1 hcg,1,2 · · · hcg,Q,Q hdg,1,1 · · · hdg,Q,Q g×2Q2
2Q2 ×1
where Q is a degree of approximation, Bm are measured values inside the volume. The left matrix
is created according to (4) equation from Legendre’s polynomials Pn,m (cos θ), distance r from origin
of coordinates and goniometric functions sin(mϕ) and cos(mϕ). The approximation accuracy is
highly dependant on Q, but it the highest degree of approximation depends on the density of
measured points of the volume. There is an analogy with Nyquist-Shannon sampling theorem. The
solution is obtained using LSQR method, which is implemented in Matlab. The matrix should be
large and sparse.
3. SPECIMENS AND FIELD MAPPING PROCESS
The specimens are realized by printed circuit board technology. The shape of two measured specimens is given by needs of EIT tomography methods. First specimen has full circle shape and second
specimen has large central gap. The dimensions of gaped specimen can be seen on Fig. 1(a), photo
of non-gaped specimen with Hall probe can be seen on Fig. 1(b). The power cables are attached
trough the isolation plate, direction of magnetic field created by the current inside these cables
should not influence the measured field.
The mapping of magnetic flux density was realized by Hall probe and fluxmeter F.W.BELL 9550.
While measuring magnetic fields up to 100 µT, the resolution is 0.1 µT. Two copper wires, which
are used for feeding the specimens are oriented perpendicular to mapping plane. Unfortunately,
there are still problems with influence of feeding wires near the specimen, the field is slightly
deformed. The input current was 5 A, the field was mapped in 5 planes, equally distributed above
the specimen. The mapping area can be seen from Fig. 1(a).
4. RESULTS
Before computing approximation coefficients, it was suitable to create COMSOL model of measured
specimen to compare measured fields with simulations. The meshing of this model was quite difficult
task, because of the proportions between input wires and specimen. The specimen thickness is
about 35 µm. Therefore, the meshing net has to be adjusted manually, especially around the edge
(a)
(b)
Figure 1: Copper specimens and dimensions of mapped area.
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 19–23, 2012 169
of specimen. If this optimization is not made, the solution doesn’t converge and we can not obtain
proper results.
As we can see from Fig. 2(b), the shape of the field has maximum value of magnetic flux density
around 64 µT and minimum around −64 µT. These values are in a good agreement with measured
fields. The higher value could be caused by absence of isolation desk in the model or just the
thickness of copper specimen is not exactly 35 µm.
The results of measuring and approximation can be seen on Fig. 3. The degree of approximation
was set to Q = 10, because with this degree we obtained the best results. If we raise the approximation degree above Q = 15, the convergence of linear equation system solved by LSQR method
(a)
(b)
Figure 2: Copper specimens and dimensions of mapped area.
(a)
(b)
(c)
(d)
Figure 3: Comparison of measured and computed maps for two different specimens.
PIERS Proceedings, Moscow, Russia, August 19–23, 2012
170
(a)
(b)
Figure 4: Comparison of measured and computed maps without one plane for not-gaped specimen.
stagnated. On Fig. 3(a) we can see comparison of measured field map (left) and approximated
field map (right) in the plane, which is 6 mm above the specimen. The other measured planes
have similar shape. In this case, the coefficients used for computing the approximated map were
obtained from all measured points. On Fig. 3(c) we can see field above gaped specimen. Fig. 3(b)
and Fig. 3(d) show absolute value of difference between measured and approximated values of magnetic flux density at each measured point. We can see, that the field of gaped specimen is more
“complicated” in shape and also the maximum approximation error is nearly twice bigger than
error of not-gaped specimen. Therefore is obvious, that shape of magnetic field has big influence
on approximation error. On Fig. 4(a) we can see also comparison of measured map and approximated map above non-gaped specimen, but the approximation coefficients are not computed from
all five planes, but only from four planes. Therefore, in this case the values of this approximated
map are fully reconstructed from other four planes. As we can see, the shape of magnetic field is
still very similar to original measured field, but the maximum absolute difference is nearly 20 µT,
which is quite big failure and should be better [6]. The error can be caused by unsuitable position of measured volume (size of planes is much bigger than distance between planes), because the
approximation obtained on the spherical surface has much better results.
5. CONCLUSIONS
The description of magnetic flux density reconstruction method is presented in this paper. The
method has been applied to reconstruction of magnetic flux density fields above different copper
specimens. The precision of computed field is highly dependant on the shape of measured field.
Although the error of reconstruction seems to be quite big, the shape of reconstructed field is very
similar to original. The error should be lower when choosing different distribution of measured
points, which is also aim of our future work. The results from simulation in COMSOL are presented
and are in good agreement with measured field.
ACKNOWLEDGMENT
This work was supported within project from Education for Competitiveness Operative Programme
CZ.1.07.2.3.00.20.0175 and project of the BUT Grant Agency FEKT-S-11-5/1012.
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