A semi-analytical method for the design of coil-systems for homogeneous magnetostatic field generation

Progress In Electromagnetics Research B, Vol. 37, 171–189, 2012
A SEMI-ANALYTICAL METHOD FOR THE DESIGN
OF COIL-SYSTEMS FOR HOMOGENEOUS MAGNETO-
STATIC FIELD GENERATION
M. A. Azpu´rua*
Instituto de Ingenier´ıa, Laboratorio de Electromagnetismo Aplicado,
Centro de Ingenier´ıa Ele´ctrica y Sistemas, Caracas, Venezuela
Abstract—This paper proposes a simple semi-analytical method for
designing coil-systems for homogeneous magnetostatic field generation.
The homogeneity of the magnetic field and the average magnitude
of the magnetic flux density inside of the volume of interest are the
objective functions chosen for the selection of the coil-system geometry
(size and location), number of coils and the number of turns of
each winding. The spatial distribution of the magnetostatic field is
estimated superposing the magnetic induction numerically computed
from the analytical expression of the magnetic field generated by each
coil, obtained using the Biot-Savart’s law and the current filament
method. The homogeneous magnetic field is synthesized using an
iterative algorithm based on TABU search with geometric constraints,
which varies the design parameters of the windings to meet the
requirements. The number of turns of each coil and gauge of wire
used for the windings is adjusted automatically in order to achieve
the target average magnitude of the magnetic induction under the
constraints imposed by power consumption. This method was used
to design a coil arrangement that can generate up to 10mT within a
volume 0.5m×0.5m×1m with 99% of spatial homogeneity, with square
loops of length less than or equal to 1.5m, and with a power dissipated
by Joule effect less than or equal to 1W per coil. The synthesized
magnetic field distribution was validated using Finite Element Method
simulation, showing a good correspondence between the objective
values and the simulated fields. This method is an alternative to design
magnetic field exposure systems over large volumes such as those used
in bioelectromagnetics applications.
Received 26 October 2011, Accepted 9 December 2011, Scheduled 14 December 2011
* Corresponding author: Marco A. Azpurua (marcoazpurua@gmail.com).

172 Azpu´rua
1. INTRODUCTION
Several applications require the generation of a very homogeneous
and controlled magnetic field, such as magnetic fluid hyperthermia
in cancer therapy [1], improvement of flight efficiency and even
maneuverability at hypersonic speed [2], testing the magnetic
properties and instruments of spacecraft [3], magnetic resonance
imaging [4], calibration of magnetic fields [5], plasma physics
experiments, particle accelerators, electromagnetic compatibility
testing [6] and bioelectromagnetics research among others.
Many experiments performed to study the effect of the magnetic
treatment on biological systems require devices that can generate
electromagnetic fields, to apply a controlled and repeatable exposure
dose to the samples involved in these experiments, but additionally
allowing the experimenter to observe, manipulate and easily locate
the sample. Therefore, the working volume should be conveniently
accessible and the electrical design and construction should be as
simple as possible. Thus, air-core coil arrangements has been
traditionally opted to guarantee a certain spatial distribution of
electromagnetic fields, when fed by an electric current. [7].
In this regard, the aforementioned applications require coil-
systems designed specifically to generate a spatial distribution of
magnetic field meeting the requirements of homogeneity, H, and
average magnitude of the magnetic flux density, B¯, within a certain
volume, V , containing the sample treated. In order to design such a
coil-systems, there are two main approaches, the local and the integral.
The local approach was developed for axially symmetric magnetic
coil-systems such as Helmholtz and Maxwell coils. It is based on the
assumption that in a space free from field sources, the magnetic field
satisfies Laplace equation, and in an ideal axially symmetric system
the field can be expressed as a series expansion about some point on
the axis, taken as the origin in terms of zonal spherical harmonics [8]:
Bz(R, θ) =
∞∑
n=0
CnRnPn (cos θ). (1)
A similar expansion with different coefficients is valid for the
radial component of the field Br(R, θ). Here R and θ are the
spherical coordinates of the point considered and Pn(x) is a Legendre
polynomial. The series converges within a sphere of radius R < R0,
where R0 is the distance from the origin to the nearest current loop.
The Cn coefficients depend on the system geometry and the current.
At θ = 0 (along the longitudinal axis of the coil-system) only the
field on the system axis is considered, and the series in (1) takes the

Progress In Electromagnetics Research B, Vol. 37, 2012 173
form of a Maclaurin’s series,
Bz(z) =
∞∑
n=0
Cnzn, (2)
where the coefficients, Cn, are given by,
Cn = 1n!
dnBz (z)
dzn
∣∣∣∣
z=0
. (3)
In practice symmetric systems are used, because all the odd
coefficients in (1) are zero. Therefore, the local design approach is used
to calculate the relationship between the electrical current intensity, I,
the number of turns in the windings and their location on the axis, such
that the even coefficients of (2) are canceled (C2=0, C4=0, C6=0, . . . ),
reducing the variability of the magnetic field around the center of the
array and thus improving the homogeneity.
The local approach in the synthesis of magnetic field has been
widely used to propose several coil-systems of circular or square
windings such as those that have been reviewed in [9] and in [10],
because of the simplicity of the analytical function that describes the
magnetic flux density along their axis.
However, the fundamental flaw of the local approach is that the
synthesis condition is restricted to the central point. The fields start to
deviate from the required value in an uncontrolled way as the distance
from the reference point increases. In practice, the requirements should
be met for all the points within the working volume, but neither its
size nor its shape are included in the synthesis conditions. If a given
magnetic field distribution is required for a large volume, this approach
is not recommended.
On the other hand, the integral approach requires that when
designing a coil-system the magnetic field generated should minimally
deviate from the desired one over the whole working volume. For this
purpose, the strength of the field sources (current system) are chosen
and located in space so that the magnetic field they create is as close
as possible to the required one.
The determination of the unknown current density distribution
placed outside the working volume and generating the desired magnetic
field, is an inverse problem. This inverse problem can have more than
one solution. The determination of the particular solution, that will be
adopted for the design, constitutes a synthesis problem. This problem
has been solved through local or global, deterministic or stochastic,
optimization techniques [11].
In this regard, the integral method has at its foundation
the numerical minimization of an objective function determined by

174 Azpu´rua
the design goals, such as the mean-square deviation of the axial
components of the magnetic field within the working volume, proposed
by [8], or the magnetic field homogeneity, as has been used in [1, 12].
In addition, the integral approach is typically combined with
numerical integration and simulation techniques (i.e., the Finite
Elements Method) to calculate the approximated spatial distribution
of magnetic induction, since for some geometries the exact analytical
expression is complex to obtain or cannot be directly evaluated.
As expected, the main weaknesses of the integral approach to
the synthesis of the magnetic field are the large calculation time and
computational cost that are associated with the solution of a highly
non-linear inverse problem where the objective function can be non-
convex, stiff, non-differentiable, ill-conditioned, etc. [11].
As an alternative for the homogeneous magnetostatic field
synthesis, this paper proposes a simple semi-analytical method for
designing coil-systems. To this end, first the homogeneity of the
magnetic field will be defined (Section 2), later to explain in detail the
method (Section 3) and its implementation as an algorithm (Section 4),
which will allow formulating and resolving a specific design example
(Section 6) whose results will be discussed and validated (Sections 7
and 8) to conclude the matter.
2. MAGNETIC FIELD HOMOGENEITY
The homogeneity of the magnetic field, H, is a measure of the
variability of the magnetic field within a defined region of space.
Different approaches have been taken to quantify H. One of the most
common is the one that defines the non-homogeneity of the field in
terms of the variation in magnetic flux density at a given point in space
within the volume of interest, regarding the value of magnetic induction
in the central point of the coil system [12]. Hence, that definition for
the magnetic field homogeneity considers that it is a point-dependent
index, not providing global information about the homogeneity within
the volume of interest, and consequently the aforementioned definition
is not useful to describe the whole volume of interest.
Therefore, for the purposes of this paper, it was considered that
a better definition of H is given by (4) [1], because it provides a single
global index that summarizes the maximum change in the magnetic
field magnitude within the entire working volume, V , with respect to its
average value. That fact is important in biomagnetic experimentation
because allows controlling the doses and treatments applied to the
samples being investigated.

Progress In Electromagnetics Research B, Vol. 37, 2012 175
H (Bmax, Bmin, B¯
) = 1− Bmax −BminB¯ , (4)
where Bmax and Bmin are the maximum and minimum values taken by
the magnitude of the magnetic flux density within the working volume,
respectively. Thus, H is measured as the maximum deviation of the
magnitude of magnetic flux density in relation to the average value of
the magnitude of ~B, within V .
3. A SEMI-ANALYTICAL METHOD FOR
HOMOGENEOUS MAGNETIC FIELD SYNTHESIS
The method is based on the assumption that it is possible to
estimate, with sufficient accuracy within the working volume, the
spatial distribution of the magnetic flux density generated by a coil-
system formed byN independent coils, ~B, by superposing the magnetic
flux density generated by each coil, ~Bi, as shown in (5).
~B =
N∑
i=1
~Bi. (5)
An approximated analytical expression for each ~Bi is obtained
using the Biot-Savart’s law and simplifying the path of the ith
current, Ii, considering that it follows a close trajectory formed by
line segments, as given by (6).
~Bi = µ04pi
∮ Iid~li × ~r
r3 , (6)
where, µ0 is the magnetic permeability of free space (considering that
the working volume is filled with air), d~l is a vector, whose magnitude
is the length of the differential element of the wire, and whose direction
coincides with the one of the current, and ~r is the displacement vector
in the direction pointing from the wire element towards the point at
which the field is being computed.
Considering that the symmetry of the windings favors homogene-
ity, it is recommendable, from an analytical and practical standpoint,
the construction of coil arrangements with symmetric geometry in
which each one of the coils maintains a regular and similar form within
the coil set. Thus, in most applications, coil systems have either a
rectangular or a circular cross section. In this regard, each ~Bi is an-
alytically calculated through (6) at any point space, (x, y, z), for the

176 Azpu´rua
Figure 1. Simplified geometry of
a rectangular current loop.
Figure 2. Simplified geometry of
a circular current loop.
particular cases of a rectangular and a circular current loop, as shown
in Figure 1 and Figure 2, respectively.
Additionally, the calculations are simplified considering that all
the coils are thin, and therefore, assuming that all the turns of each
winding contribute equally to the total magnetic flux density.
Consequently, the expression of the magnetic flux density for the
rectangular coil geometry, is given by (7),
~Bi = ni (Bix, Biy, Biz) , (7)
where ni is the number of turns of the ith coil and Bix, Biy and Biz are
the rectangular components of ~Bi in R3, which are given by (8), (13)
and (18) respectively,
Bix = µ0Ii4pi (B
∗
ix1 +B∗ix2 +B∗ix3 +B∗ix4) , (8)
where, B∗ix1, B∗ix2, B∗ix3 and B∗ix4 are given by (9), (10), (11) and (12)respectively.
B∗ix1 =
z − hi
(x− ai)2 + (z − hi)2
y + b√
(x− ai)2 + (y + bi)2 + (z − hi)2
, (9)
B∗ix2 =
z − hi
(x− ai)2 + (z − hi)2
−(y − b)√
(x− ai)2 + (y − bi)2 + (z − hi)2
, (10)
B∗ix3 =
z − hi
(x+ ai)2 + (z − hi)2
y − b√
(x+ ai)2 + (y − bi)2 + (z − hi)2
, (11)

Progress In Electromagnetics Research B, Vol. 37, 2012 177
B∗ix4 =
z − hi
(x+ ai)2 + (z − hi)2
−(y + b)√
(x+ ai)2 + (y + bi)2 + (z − hi)2
, (12)
and,
Biy (x, y, z) = µ0Ii4pi
(B∗iy1 +B∗iy2 +B∗iy3 +B∗iy4
) , (13)
where, B∗iy1, B∗iy2, B∗iy3 and B∗iy4 are given by (14), (15), (16) and (17)
respectively.
B∗iy1 =
z − hi
(y − bi)2 + (z − hi)2
x+ a√
(x+ ai)2 + (y − bi)2 + (z − hi)2
, (14)
B∗iy2 =
z − hi
(y − bi)2 + (z − hi)2
x− a√
(x− ai)2 + (y − bi)2 + (z − hi)2
, (15)
B∗iy3 =
z − hi
(y + bi)2 + (z − hi)2
x− a√
(x− ai)2 + (y + bi)2 + (z − hi)2
, (16)
B∗iy4 =
z − hi
(y + bi)2 + (z − hi)2
x+ a√
(x+ ai)2 + (y + bi)2 + (z − hi)2
, (17)
and finally,
Biz (x, y, z) = µ0Ii4pi (B
∗
iz1 +B∗iz2 +B∗iz3 +B∗iz4) , (18)
where, B∗iz1, B∗iz2, B∗iz3 and B∗iz4 are given by (19), (20), (21) and (22)respectively.
B∗iz1 =
(x− ai)(y − bi)√
(x− ai)2 + (y − bi)2 + (z − hi)2
(β1 + β2) , (19)
B∗iz2 =
(x− ai)(y + bi)√
(x− ai)2 + (y + bi)2 + (z − hi)2
(β1∗ + β2) , (20)
B∗iz3 =
(x+ ai)(y − bi)√
(x+ ai)2 + (y − bi)2 + (z − hi)2
(β1 + β2∗) , (21)
B∗iz4 =
(x+ ai)(y + bi)√
(x+ ai)2 + (y + bi)2 + (z − hi)2
(β1∗ + β2∗) , (22)
where β1, β2, β1∗ and β2∗ are given by,
β1 = 1(y − bi)2 + (z − hi)2
(23)

178 Azpu´rua
β2 = 1(x− ai)2 + (z − hi)2
(24)
β1∗ = 1(y + bi)2 + (z − hi)2
(25)
β2∗ = 1(x+ ai)2 + (z − hi)2
(26)
Furthermore, the spatial distribution of magnetic flux density
generated by a circular current coil is given by [9, 13],
~Bi = ni (Biρ, Biϕ, Biz) , (27)
where ni is the number of turns of the ith coil and Biρ, Biϕ and Biz
are the vector components of ~Bi in the cylindrical coordinate system,
which are given by (28), (29) and (30) respectively,
~Biρ = −µ0Iik (z − hi)4piρ√aiρ
(
K(k)− 2− k
2
2 (1− k2)E(k)
)
, (28)
~Biϕ = 0, (29)
~Biz = − µ0Iik4pi√aiρ
(
K(k) + k
2 (ai + ρ)− 2ρ
2ρ (1− k2) E(k)
)
. (30)
In (28), (29) and (30), K(k) and E(k) are the complete elliptic
integrals of the first and second kind respectively, and are defined by,
K(k) =
∫ pi
2
0
dα√
1− k2sin2α
, and, (31)
E(k) =
∫ pi
2
0
√
1− k2sin2α dα, (32)
where, k is,
k =
√
4aiρ
(ρ+ ai)2 + (z − hi)2
. (33)
Hence, by having analytical simplified expressions for ~Bi, ~B can be
numerically computed for any arrangement of rectangular or circular
coaxial coils, and thus make the estimation of H within the working
volume.
In this sense, the problem of synthesis requires iterative calculation
of ~B and H, to find a geometry such that would meet the design
requirements, optimizing the dimensions of the system. Consequently,
~B is numerically evaluated over a discrete set of points P. P is a set

Progress In Electromagnetics Research B, Vol. 37, 2012 179
of uniform sample points distributed over the working volume, which
generates a three-dimensional matrix of points (xi, yj , zk) by varying a
regular distance ∆ the coordinates of the points.
As a result of the spatial sampling, an approximated discrete
function of the sampled magnetic flux density in space is obtained
as,
~B[i,j,k] (~r) = ~B (~r)
∑
~r[i,j,k]∈P
δ3
(~r − ~r[i,j,k]
), (34)
where δ3(~r), is the unit impulse function in three dimensions and,
P ={~r[i,j,k] : ~r[i,j,k] = (∆i,∆j,∆k) /i, j, k ∈ Z
} , P ⊂ V. (35)
Thus, H is approximated to,
H = 1−
max
(∣∣∣ ~B[i,j,k] (~r)
∣∣∣
)
−min
(∣∣∣ ~B[i,j,k] (~r)
∣∣∣
)
∣∣∣ ~B[i,j,k] (~r)
∣∣∣
(36)
Finally, the semi-analytical method for homogeneous magnetic
field synthesis consists in an iterative evaluation of (36) following the
procedure explained above, in order to find a suitable configuration
that meets the design requirements. The iterative evaluation of H for
different geometries, shall follow a logical procedure that allows the
geometry parameters of the system to finish the convergence process
to a solution that meets the optimization objectives. This procedure
can be implemented through a heuristic search algorithm such as the
TABU search.
4. HOMOGENEOUS MAGNETIC FIELD SYNTHESIS
ALGORITHM
A simplified form of the algorithm for homogeneous magnetic field
synthesis using rectangular coils is shown in Figure 3. The design
configuration, X, is a vector set including the number of coil-windings,
N , the number of turns of each winding, n = {n1,n2, . . . , nN},
and the size a = {a1, a2, . . . , aN}, b = {b1, b2, . . . , bN} (considering a
rectangular coil-system) and position, h = {h1, h2, . . . , hN}, of each
coil-winding. As expected, H is iteratively evaluated for different
design configurations to find a solution that meets the requirements
of homogeneity. All the different design configurations form the
neighborhood, X∗.
The search for a suitable design configuration was implemented
using a TABU Search Algorithm, TSA. The TSA is a heuristic
approach for solving optimization problems by using a guided, local

180 Azpu´rua
The process is repeated for
all elements of the
neighborhood
The resolution of the neighborhood
is greater at each iteration but its
extent is less.
First, generate movements in N, then
perform movements in h, a and b and
finally n. The variation of the
configuration parametersis performed
under constraints.
It implies that the synthesis process has
been completed and as a result provides
constructive parameters that satisfy the
requiremets taking into account all the
constraints
Current configuration = Final
Configuration
Generate P and
Calculate B and H inside V
It implies that the synthesis process can
not end yet, because the solutions do not
meet the objective sought
NO
Figure 3. Simplified flow chart of the algorithm for homogeneous
magnetic field synthesis (rectangular coil system).
search procedure to explore the entire solution space without becoming
easily trapped in local optima. One characteristic of TABU search is
that it finds good near-optimal solutions early in the optimization run.
It does not require initial guesses, does not use derivatives, and is also
independent of the complexity of the cost function considered [15].
In this particular case, the neighborhood is uniformly distributed
over the possible values of each parameter, considering the constrains
of maximum and minimum coil size, maximum number of turns of each
coil and maximum number of coil windings. When the neighborhood

Progress In Electromagnetics Research B, Vol. 37, 2012 181
is completely evaluated, the best design configuration found (the
one with the highest H) is selected and the algorithm generates a
new neighborhood around it, but with an increased resolution. The
neighborhood size remains constant. This process is repeated until
a solution is found or until the distance between the elements of the
neighborhood cannot be reduced (in practice this limit is set by the
manufacturing capacity).
The parameters nact, aact, bact, hact and Nact correspond to the
best design configuration found. An initial design configuration shall
be set. In this particular case, the initial design configuration was
automatically selected as the one with all the windings of the same
size, with the same number of turns and with their positions uniformly
distributed along the longitudinal axis of the coil system.
5. DESIGNING THE COIL-SYSTEM UNDER POWER
CONSTRAINTS
When designing magnetic coils, one important consideration is how to
keep them cool while all the electric power is dissipated in the wire by
the Joule effect. In some instances, no cooling is needed and in others,
the coils systems shall be air cooled or water cooled. It really depends
on the coil holder structure and airflow available [16]. Additionally,
current and power constraints must be considered when designing
magnetic coil systems, since there are limitations of the power supply
that will be used the feed the coil system. The power dissipated in the
ith coil by means of Joule effect, Pi, is given by:
Pi = Ii2Ri = Ii2ρCu lA ≈ Ii
2ρCuni 4lturnpid2 (37)
where Ri is the DC resistance of the ith coil, which is calculated
approximately from the electrical resistivity of the wire (copper, Cu),
ρCu, the length of the wire, l and the cross-sectional area of the wire,
A. The length of the wire is approximated as the average length of a
single turn, lturn, multiplied by the number of turns, ni. A is calculated
assuming a circular cross-section of diameter d.
Consequently, in order to choose a specific wire type for the coils,
Pi is calculated for the typical wire sizes of the American Wire Gauge
types [17]. Nevertheless, an additional alternative to reduce the power
consumption of the coil system is to increase the number of turns of
each coil winding while the current intensity is reduced proportionally
in the same amount. In that sense, a brute-force search algorithm is
used to find an appropriate combination of the number of coil winding
turns and wire gauge, which allows to keep the power consumption
below the constraint.

182 Azpu´rua
6. EXAMPLE OF A DESIGN PROBLEM
It is required to design a coil arrangement that can generate up to
10mT within a working volume 0.5m × 0.5m × 1m with 99% of
spatial homogeneity, with a tolerance of up to 1% in the results of
field homogeneity, in order to create magnetic field exposure system
intended to be used to apply magnetic treatment to vegetable seeds
to improve the germination rate and crops yield. The theoretical
model that explains the influence of a stationary magnetic field on
water relations in seeds, can be consulted in [18] and in a companion
paper, it is presented some experimental evidence to support that
hypothesis [19].
Since the working volume is a parallelepiped, square loops were
selected. The maximum side length of the coils must be 1.5m, the
maximum longitudinal length is set at 1.5m, and the maximum power
consumed by Joule effect shall be 1W per coil considering that the
windings could be constructed with enameled copper wire using either
type AWG 12, 10 or 8. The maximum number of coils is set at 7.
Parameterized geometry of the coil system to be designed under the
restrictions stated above, is shown in Figure 4.
Figure 4. Geometry of the coil system for homogeneous magnetostatic
field generation. The grey rectangle represents any vertical cross-
section of the working volume V , parallel to the x or y axis.
It is important to note that, in order to solve the problem,
the proposed method and algorithms were implemented using
MATLABTM scripting.

Progress In Electromagnetics Research B, Vol. 37, 2012 183
7. DESIGN PROBLEM RESULTS
The algorithm converged to a solution satisfying the requirements of
magnetic field homogeneity and magnetic flux density inside of the
volume of interest, reaching H = 98.7%, B¯ = 10.00mT, Bmax =
10.05mT and Bmin = 9.92mT for a six-coils (connected in series)
system with the constructive parameters and characteristics shown in
Table 1. The solution to the problem was found for ∆ = 2 cm.
The magnitude of the magnetic flux density generated by the
the designed coil system is shown in Figure 5. The calculations
were performed using different input currents, proportional to the
current required to achieve 10mT (Imax = 14.49A). The results are
in accordance with design specifications supplied to the synthesis
algorithm.
Table 1. Constructive parameters and characteristics of the designed
coil system.
Parameter Coil 1 Coil 2 Coil 3 Coil 4 Coil 5 Coil 6
Ii (A) 14.49 14.49 14.49 14.49 14.49 14.49
Ri (Ω) 4.75 2.06 1.81 1.81 2.06 4.75
Pi (W) 996 433 379 379 433 996
ai (m) 1 1 1 1 1 1
hi (cm) 72.9 36.7 11.6 11.6 36.7 72.9
ni (turns) 368 160 140 140 160 368
1 0. 5 0 0.5 10
2
4
6
8
10
12


B: 5.004
z: 0
Position on the longitudinal axis, z (m)
Ma
gni
tud
e of
the
Ma
gne
tic F
lux
De
nsit
y, B
(mT
)
B: 10.01
z: 0
B: 2.502
z: 0
B: 1.001
z: 0
B: 9.946
z: 0.5
Imax 0.5 Imax 0.25 Imax 0.1 Imax
1 0. 5 0 0.5 10
2
4
6
8
10
12


B: 9.94
z: 0.5
Position on the longitudinal axis, z (m)M
agn
itud
e of
the
Ma
gne
tic F
lux
De
nsit
y, B
(mT
)
B: 5.002
z: 0
B: 2.501
z: 0
B: 1
z: 0
B: 10
z: 0
Imax 0.5 Imax 0.25 Imax 0.1 Imax
(a) (b)
- -- -
Figure 5. Magnitude of the magnetic flux density of the designed coil
system. (a) (x, y) = (0, 0). (b) (x, y) = (±0.25, ±0.25)m.

184 Azpu´rua
Position on the x axis (m)P
osit
ion
on
the
y a
xis
(m)
Position on the z axis (m)




Pos
itio
n o
n th
e
Tra
nsv
ers
e a
xis
(m)
Position on the x axis (m)Po
siti
on
on
the
y a
xis
(m)
Position on the z axis (m)




Pos
itio
n o
n th
e
Tra
nsv
ers
e a
xis
(m)
(a) (b)
(c) (d)
Figure 6. Magnitude of the magnetic flux density of the designed coil
system in the external faces of V . Note: The colorbar scale indicates
B in units of mT. (a) x = 0 or y = 0. (b) z = 0. (c) c = ±0.25m or
y = ±0.25m. (d) z = ±0.5m.
Figure 6 shows the magnetic flux density on the different faces
of the volume, V . The results also show that the volume of interest
is contained completely within the region of 99% homogeneity of the
magnetic field.
8. VALIDATION OF RESULTS
The validation of the results of the synthesis of magnetic field
was performed using COMSOL MultiphysicsTM 4.2 which is a
solver/simulation software package based on finite element method,
FEM. The simulation model was implemented specifically with the
AC/DC module. Figure 7 shows the 3D model of the coil system
designed.
Within the domain defined for V , Bmax, Bmin and B¯ were
calculated as derived values, obtaining 10.14mT, 9.86mT, 10.00mT
respectively, and in consequence the simulated value of H is 97.23%.
The results are satisfactory, considering the differences inherent in
the models used in the design process and simulation. Figures 8

Progress In Electromagnetics Research B, Vol. 37, 2012 185
and 9 shows good correspondence between the results obtained in
calculating the magnitude of the magnetic flux density using both
the proposed method and the FEM analysis within the volume of
interest. The relative difference between the average magnetic flux
density calculated with the proposed model within V and the one
obtained by simulation using COMSOL Multiphysics was 0.026%. The
average relative difference (measured in each grid point) between the
magnetic flux density within V calculated with the proposed model
and the one obtained by simulation using COMSOL Multiphysics was
0.1822%. The maximum relative difference between the simulated and
synthesized magnetic flux density, within the volume of interest, was
3.30%.
Figure 7. COMSOL multiphysics 3D model of the coil system
designed.
1 0. 5 0 0.5 13
4
5
6
7
8
9
10
11
Posición sobre el eje longitudinal z [m]Ma
gni
tud
e of
the
Ma
gne
tic
Flu
x D
ens
ity,
B (
mT)


FEM Simulated B
Synthesis Results
1 0. 5 0 0.5 1
4
5
6
7
8
9
10
11
12
Position on the longitudinal axis, z (m)Ma
gni
tud
e of
the
Ma
gne
tic
Flu
x D
ens
ity,
B (
mT)


FEM Simulated B
Synthesis Results
(a) (b)
----
Figure 8. Comparison of the magnitude of the magnetic flux
density synthesized and simulated. (a) (x, y) = (0, 0). (b) (x, y) =
(±0.25, ±0.25)m.