An improvement in the calculation of the magnetic field for an arbitrary geometry coil with rectangular cross section

Abstract

Hong Lei, Lian-Ze Wang and Zi-Niu Wu presented new simple and convenient solutions of the magnetic field for an arbitrary geometry coil with rectangular cross section. They treated two types of basic forms: the trapezoidal prism segment and curved prism segment. The curved prism segment has been divided into a series of small trapezoidal prism segments with the same cross section and its magnetic field is a vector sum of the individual fields created by each small trapezoidal prism conductor. For one trapezoidal prism conductor the magnetic field is obtained by 1-D integrals using Romberg numerical integration. In this paper, we give a completely analytical solution of these 1-D integrals that considerably saves the computational time especially in the computation of the magnetic field nearby the conductor surface, at the conductor surface and inside the conductor. From obtained analytical expressions the treatment of singularities can be easily done. Also, we tested four types of numerical integration (Gaussian, Romberg, Simpson and Lobatto) to find the most convenient singularity treatment of 1-D integrals. These obtained results are compared with those obtained analytically so that one can understand the advantage of the proposed approach. The paper points out on the accuracy and the computational cost. Copyright © 2005 John Wiley & Sons, Ltd.

Full text

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS
Int. J. Numer. Model. 2005; 18:493 –504
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jnm.594
An improvement in the calculation of the magnetic field for an
arbitrary geometry coil with rectangular cross section
Slobodan I. Babic
n,y
and Cevdet Akyel
z,}
E
´cole Polytechnique de Montre
´al, De
´partement de Ge
´nie E
`lectrique & de Ge
´nie Informatique, C.P. 6079,
Succ. Centre Ville Montre
´al, Que., Canada H3C 3A7
SUMMARY
Hong Lei, Lian-Ze Wang and Zi-Niu Wu presented new simple and convenient solutions of the magnetic
field for an arbitrary geometry coil with rectangular cross section. They treated two types of basic forms: the
trapezoidal prism segment and curved prism segment. The curved prism segment has been divided into a
series of small trapezoidal prism segments with the same cross section and its magnetic field is a vector sum of
the individual fields created by each small trapezoidal prism conductor. For one trapezoidal prism conductor
the magnetic field is obtained by 1-D integrals using Romberg numerical integration. In this paper, we give a
completely analytical solution of these 1-D integrals that considerably saves the computational time
especially in the computation of the magnetic field nearby the conductor surface, at the conductor surface
and inside the conductor. From obtained analytical expressions the treatment of singularities can be easily
done. Also, we tested four types of numerical integration (Gaussian, Romberg, Simpson and Lobatto) to find
the most convenient singularity treatment of 1-D integrals. These obtained results are compared with those
obtained analytically so that one can understand the advantage of the proposed approach. The paper points
out on the accuracy and the computational cost. Copyright #2005 John Wiley & Sons, Ltd.
KEY WORDS: computational electromagnetics; arbitrary geometry coils; rectangular cross section;
trapezoidal prism conductors; singularity treatment; Biot–Savart law; Gaussian’s,
Romberg’s, Simpson’s and Lobatto’s numerical integrations
1. INTRODUCTION
The 3-D magnetic field calculation of conductors with rectangular cross section has been the
topic of several papers. There are many applications where current-carrying conductors of
different cross sections are employed. Magneto-hydrodynamic, fusion reactor containment
vessels, levitation systems, magnetic resonance (MR) devices and shims are some applications in
which optimized arrangements of conducting segments are required. In these systems, currents
Received 1 April 2004
Revised 1 November 2004
Accepted 1 August 2005Copyright #2005 John Wiley & Sons, Ltd.
y
E-mails: radaslo@sympatico.ca, slobodan.babic@polymtl.ca
n
Correspondence to: Slobodan I. Babic, E
´cole Polytechnique de Montreal, De
´partement de Ge
´nie E
´lectrique & de Ge
´nie
Informatique C.P. 6079, Succ. Centre Ville Montre
´al, Que., Canada H3C 3A7.
z
E-mail: akyel@grmes.polymtl.ca
}
Member IEEE.

can flow in any direction. There are presently many types of numerical methods, which can give
solutions to these types of problems but they require substantial amounts of effort to set up input
data files. Both differential and integral methods (FDM, FEM, BEM) require considerable
amounts of time to define meshes and elements, particularly in 3-D problems [1–3]. Also, FEM
and BEM methods are routinely used for magnetostatic problems, but these methods have
accuracy problems near sharp surface discontinuities unless a high density of elements is used [4].
To avoid these problems and obtain fast magnetic routines for massive conductor of complicated
geometries we can use alternative 3-D calculations. In these cases, one needs to calculate a triple
integral (Biot–Savart integral) so that results can be obtained in analytical form or numerical form
or a combination of both. In most of calculations results are expressed over complete and
incomplete elliptical integrals of the first, second and third kind as the Heumman Lambda function.
Since it is impossible to evaluate the volume integral analytically for general geometric forms of
integration regions, the regions may be subdivided into elements of simpler forms for which it can
be performed. The accuracy of the field computation depends on the exact representation of these
regions as well as the current distribution. It is shown in Reference [5] that all coils irrespective of
their geometric complexity with rectangular cross sections can be divided into two types of basic
forms: the trapezoidal prism segment and curved prism segment. This statement permits one to
consider arbitrary geometry coils with rectangular cross section and the uniform current density
across this section for which the magnetic field is calculated by Biot–Savart’s law. It leads to 1-D
integrals that have to be solved using one of numerical integration. In Reference [5] Romberg
numerical integration has been used to solve these integrals. As an important engineering
application resorting to mathematical methods [11–31] we consider the problem of large scale
magnetic field computation that leads to completely analytical solutions of these integrals that can
have a significant influence on the computational cost and the accuracy. The proposed approach
has its real meaning in the case of very complicated configurations where the coil can be divided
into trapezoidal prism segments and curved prism segments. Also, these expressions permit one to
easily solve singular cases. In this paper four types of numerical integration are used (Gaussian,
Romberg, Simpson, Lobatto) for solving 1-D integrals. In the singularity treatment it is
recommended to use adapted numerical integrations such Simpson’s or Lobatto’s.
2. BASIC EXPRESSIONS
Let us consider an arbitrary coil [5] with rectangular cross section (Figure 1) that can be divided
into trapezoidal prism segments and curved prism segments (Figures 2 and 3).
The magnetic field created by a trapezoidal prism conductor with rectangular cross section
and the constant volume current J0(Figure 2) can be calculated at point Pðx;y;zÞin the local
co-ordinate system [5],
BxðPÞ¼m0J0
4pZd
dZðx0þdÞtan aþb
ðx0þdÞtan bbZc
c
ðzz0Þ
R3dz0dy0dx0
ByðPÞ¼0
BzðPÞ¼m0J0
4pZd
dZðx0þdÞtan aþb
ðx0þdÞtan bbZc
c
ðx0xÞ
R3dy0dx0dz0ð1Þ
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
S. I. BABIC AND C. AKYEL
494

Figure 2. Trapezoidal prism conductor with rectangular cross section: (a) simplified
schematic; and (b) vertical view.
Figure 1. Arbitrary geometry coil with rectangular cross section.
Figure 3. Segmentation of curved coil with rectangular cross section.
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
CALCULATION OF THE MAGNETIC FIELD 495

where the current density is J0¼J0j;J0¼constant and
R¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx0xÞ2þðy0yÞ2þðz0zÞ2
q
In the case of the curved prism segment, the prism segment is divided into a series of small
trapezoidal prism segments with the same cross sections, as shown in Figure 3. Then the
magnetic field created by the curved prism segment is the vector sum of the individual fields
created by each small trapezoidal prism conductor [5]. The magnetic field at point Pðx;y;zÞcan
be expressed as
BðPÞ¼X
n
i¼1
BWiðPÞð2Þ
In this paper, we give the complete analytical expressions of magnetic field (1) expressed over
elementary functions. These expressions are suitable for the fast and precise calculation of the
magnetic field either outside or inside the conductor as in its surfaces so that the singularities are
comfortable to be treated. Even though the components of magnetic field are obtained in the
local co-ordinate system it is necessary to use transformations given in Reference [5] to obtain
the magnetic field in the original co-ordinate system.
3. MAGNETIC FIELD CALCULATION
Double integrating (1) the components of the magnetic field at the point Pðx;y;zÞcan be
expressed over 1-D integrals [5],
BxðPÞ¼m0J0
4p½Vx1Vx2
ByðPÞ¼0
BzðPÞ¼m0J0
4p½Vz1Vz2þVz3Vz4þVz5Vz6ð3Þ
where
Vx1ðPÞ¼Zd
d
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
0þB2
2þC2
2
qB2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
0þB2
1þC2
2
qþB1
dx0Vx2ðPÞ¼Zd
d
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
0þB2
2þC2
1
qB2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
0þB2
1þC2
1
qþB1
dx0
Vz1ðPÞ¼Zb
b
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
1
qþC1
dy0Vz2ðPÞ¼Zb
b
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
4þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
4þB2
0þC2
1
qþC1
dy0
Vz3ðPÞ¼Zb
b2dtanðbÞ
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
1
qþC1
dy0
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
S. I. BABIC AND C. AKYEL
496

Vz4ðPÞ¼Zb
b2dtanðbÞ
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
2þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
2þB2
0þC2
1
qþC1
dy0
Vz5ðPÞ¼Zbþ2dtanðaÞ
b
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
3þB2
0þC2
1
qþC1
dy0
Vz6ðPÞ¼Zbþ2dtanðaÞ
b
ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
1þB2
0þC2
2
qC2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2
1þB2
0þC2
1
qþC1
dy0
A0¼x0x;A1¼ðy0bÞcotðaÞdx;
A2¼ðy0þbÞcotðbÞþdþx;A3¼dx;A4¼dþx
B0¼y0y;B1¼ðx0þdÞtanðaÞþby;
B2¼ðx0þdÞtanðbÞþbþy;C1¼cz;C2¼cþz
It is possible to solve all the integrals analytically in (1) [6,7] (Appendix) so that the components
of the magnetic field at the point Pðx;y;zÞbecome
BxðPÞ¼m0J0
4pX
n¼4
n¼1
ð1Þn1Sxn
BxðPÞ¼0
BzðPÞ¼m0J0
4pX
n¼4
n¼1
ð1Þn1Szn ð4Þ
where
Sxn ¼tsinh1tsinðyÞþRcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þQ2
pþRcosðyÞsinh1tþRsinðyÞcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2cos2ðyÞþQ2
p
(
þQtan1Q2sinðyÞtR cosðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þ2Rt sinðyÞcosðyÞþðR2þQ2Þcos2ðyÞ
p)
t¼L2
t¼L1
where
n¼1y¼a;R¼R1;Q¼Q2
n¼2y¼a;R¼R1;Q¼Q1
n¼3y¼b;R¼R2;Q¼Q2
n¼4y¼b;R¼R2;Q¼Q1
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
CALCULATION OF THE MAGNETIC FIELD 497

Szn ¼tsinðyÞsinh1LþRsinðyÞcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þR2cos2ðyÞ
ptsinh1LsinðyÞþRcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þL2
p
(
Rcos2ðyÞsinh1tcosðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2þ2LR sinðyÞcosðyÞþR2cos2ðyÞ
p
RsinðyÞcosðyÞtan1tðLþRsinðyÞcosðyÞÞ
RcosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2cos2ðyÞþL2þ2RL sinðyÞcosðyÞþR2cos2ðyÞ
p
þLtan1tðLsinðyÞþRcosðyÞÞ
Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2cos2ðyÞþL2þ2RL sinðyÞcosðyÞþR2cos2ðyÞ
p)
t¼Q2
t¼Q1
where
n¼1y¼a;R¼R1;L¼L2
n¼2y¼a;R¼R1;L¼L1
n¼3y¼b;R¼R2;L¼L2
n¼4y¼b;R¼R2;L¼L1
L1¼dx;L2¼dx;Q1¼cz;Q2¼cz;
R1¼ðdþxÞtanðaÞþby;R2¼ðdþxÞtanðbÞþbþy
Preceding expressions can be easily used in the singularity treatment. It is necessary to find the
limit regarding a singular point and the corresponding variable. For example, if the singular
point Pðx;y;zÞ¼Pðd;b;cÞis taken into consideration it is necessary to find limits of all
components when x!d;y!band z!c:In the limits, the choice of variable is not important.
It is to be noted that the geometry of the conductor changes from a trapezoidal prism to a
rectangular prism when a¼b¼0:In analytical expressions all values of ða;bÞA½0;p=2Þare
included so that it is not necessary to find limits when a¼0orb¼0 as in Reference [5]. The
cases a¼p=2 or/and b¼p=2 can be mathematically treated but they have not real meaning
because conductor begins infinitely long.
4. RESULTS
4.1. Magnetic field calculation outside the conductor
Let us solve the magnetic field created by the trapezoidal prism for which b¼c¼d¼1m;
a¼p=3;b¼p=6 and J¼1:0105A=m2:The presented analytical approach will be compared
to the approach given in Reference [5] where the magnetic field has been obtained using some of
numerical integration (Gaussian’s, Romberg’s, Lobatto’s or Simpson’s) for 1-D integrals. The
point of calculation is outside the conductor ðx¼y¼z¼2mÞ:In Table I values of the
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
S. I. BABIC AND C. AKYEL
498

magnetic field obtained by previous approaches are shown. Also the corresponding
computational time of the calculation is given.
All results are in excellent agreement with considerably small computational time but the
analytical approach is considerably faster.
4.2. Magnetic field calculation on the conductor surface
Let us calculate the magnetic field at the singular point x¼y¼z¼1m:In the case of singular
points, it is necessary to solve 1-D integrals using Gaussian numerical integrations [7,8] or some
of adaptive numerical integrations such as Simpson’s or Lobatto’s [9,10].
The analytical approach gives the magnetic field of 53:581000397 mT:The computation was
obtained practically instantly.
Using Gaussian numerical integration of 1-D integrals (3) [7] it is not possible to obtain the
accurate results with few Gaussian points, Table II. It means that the satisfactory accuracy can
be reached by increasing the number of integration points.
From Table II it is evident that the best accuracy can be reached with a large number of
integration points that considerably increase the computational cost. We can also use Gaussian
numerical integration [8] when the choice of the tolerance (we use the tolerance Eps ¼
2:220446049250313 1016 given in MATLAB programming) can play the significant role on
the computational cost. Using Gaussian numerical integration of 1-D integrals (3) [8] we obtain
the magnetic field of 53:580990419 mT with the computational time of 2.36 seconds. For
singular cases it is recommended to use Gaussian numerical integration [8] because of its
extreme rapidity and high accuracy.
If one of the adaptive numerical integration is used it is recommended to calculate the
magnetic field at very close point to the singular point, for example x¼y¼z¼1:0000001;
because of singularities and function oscillations.
Table I. Comparison of computational efficiency.
Approach Magnetic field (mT) Computational time (s)
This work 15.5533805 }
1-D (Gauss [7]) 20 points 15.5533805 0.11
1-D (Gauss [8]) e¼1:00 10515.5533805 0.21
1-D (Romberg [7]) 15.5533805 0.12
Table II. Comparison of computational efficiency.
NGaussian points B(mT) Computational time (s) Error (%)
50 53.56450441 0.422 0.030787
100 53.57683549 1.273 0.007731
300 53.58053458 32.472 0.000869
600 53.58088377 332.253 0.000217
1000 53.58095842 1498.528 0.000078
1200 53.58097123 2617.848 0.000054
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
CALCULATION OF THE MAGNETIC FIELD 499

In Table III values of the magnetic field obtained by previous approaches are shown. Also the
corresponding computational time of the calculation and the absolute error of calculation
regarding the exact value are given.
All results are in excellent agreement with considerably small computational time but the
analytical approach is considerably faster.
4.3. Magnetic field calculation inside the conductor
Finally, let us calculate the magnetic field inside the conductor at point x¼y¼z¼0:5m:In
Table IV values of the magnetic field obtained by previous approaches are shown. Also the
corresponding computational time of the calculation is given.
From obtained results one can conclude that presented analytical approach is preferable to all
used numerical integrations (Gaussian, Romberg, Simpson and Lobatto) regarding the accuracy
and the computational time. Outside the conductor all numerical integrations give appro-
ximately the same accuracy for the approximately same computational time. The comparative
calculation was made using MATLAB programming on a personal computer with a Pentium III
700 MHz processor.
5. CONCLUSIONS
In this paper, we present a very efficient analytical method for the calculation of the magnetic
field for trapezoidal prism conductors with rectangular cross section. Even though this approach
appears messy because of enormous expressions that describe the magnetic field it gives fast and
accurate results for all points either regular or singular. Also these analytical expressions are
easy for numerical applications because they are all elementary functions. They have been
obtained from 1-D integrals, which also can be solved by using Gaussian, Romberg, Simpson or
Table III. Comparison of computational efficiency.
Approach Magnetic field (mT) Computational time (s) Error (%)
This work 53.581000397 }}
Gauss [8] 53.580990419 2.36 0.000019
Simpson [9] 53.580971262 0.22 0.000054
Lobatto [9] 53.580953263 0.16 0.000088
Table IV. Comparison of computational efficiency.
Magnetic field Computational time
Approach (mT) (s)
This work 34.99691567 }
1-D (Gauss [7]) 100 points 34.99691567 1.21
1-D (Gauss [8]) 34.99691567 2.30
1-D (Romberg [7]) 34.99691567 0.12
1-D (Simpson [9]) 34.99718946 0.22
1-D (Lobatto [9]) 34.99690773 0.22
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
S. I. BABIC AND C. AKYEL
500

Lobatto numerical integration. These numerical integrations can give very fast and accurate
results but it takes considerable time calculation (Gaussian numerical integration) if the
magnetic field is supposed to be calculated close to singular points, at singular points and inside
the conductor. This is why the proposed analytical approach is preferred to the numerical
integration of 1-D integrals.
Appendix A
For the component of the magnetic field Bxthe order of integration is dz0dy0dx0so that we
obtain
Iz1¼Zc
c
ðzz0Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ðz0zÞ2þðy0yÞ2þðx0xÞ23
qdz0¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þðy0yÞ2þðx0xÞ2
q
t¼Q2
t¼Q1
Iy1¼Zðx0þdÞtanðaÞþb
ðx0þdÞtanðbÞb
Ix1dy0
¼sinh1t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx0xÞ2þQ2
2
qsinh1t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðx0xÞ2þQ2
1
q
2
6
43
7
5
t¼ðx0þdÞtanðaÞþby
t¼ðx0þdÞtanðbÞby
Ix1¼Zd
d
Iy1dx0¼tsinh1tsinðyÞþRcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þQ2
p
"
þRcosðyÞsinh1tþRsinðyÞcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þR2cos2ðyÞ
p
þQtan1Q2sinðyÞRt cosðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þ2Rt sinðyÞcosðyÞþðR2þQ2Þcos2ðyÞ
p#
t¼L2
t¼L1
For the component of the magnetic field Bzthe order of integration is dy0dx0dz0so that we
obtain
Iy2¼Zðx0þdÞtanðaÞþb
ðx0þdÞtanðbÞb
ðx0xÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½ðz0zÞ2þðy0yÞ2þðx0xÞ23
qdx0
¼t
½ðx0xÞ2þðz0zÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þðx0xÞ2þðz0zÞ2
q
t¼ðx0þdÞtanðaÞþby
t¼ðx0þdÞtanðbÞby
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
CALCULATION OF THE MAGNETIC FIELD 501

Ix2¼Zd
d
ðx0xÞIy2dx0¼sinðyÞsinh1tþRsinðyÞcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz0zÞ2þR2cos2ðyÞ
q
2
6
4
sinh1tsinðyÞþRcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz0zÞ2þt2
q3
7
5
t¼L2
t¼L1
Iz2¼Zc
c
Ix2dz0¼tsinðyÞsinh1LþRsinðyÞcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þR2cos2ðyÞ
ptsinh1LsinðyÞþRcosðyÞ
cosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2þL2
p
"
Rcos2ðyÞsinh1tcosðyÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2þ2LR sinðyÞcosðyÞþR2cosðyÞ
p
þRsinðyÞcosðyÞtan1tðLþRsinðyÞcosðyÞÞ
RcosðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2cos2ðyÞþL2þ2RL sinðyÞcosðyÞþR2cos2ðyÞ
p
þLtan1tðLsinðyÞþRcosðyÞÞ
Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t2cos2ðyÞþL2þ2RL sinðyÞcosðyÞþR2cos2ðyÞ
p#
t¼Q2
t¼Q1
where L1;L2;Q1;Q2;R1;R2;a;b;and ywere previously given.
ACKNOWLEDGEMENTS
We would like to thank the Natural Science and Engineering Research Council of Canada (NSERC) that
supported this work
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AUTHORS’ BIOGRAPHIES
Slobodan I. Babic was born in Tuzla, Bosnia and Herzegovina. He received his Dipl
Ing degree from the Faculty of Electrical Engineering, University of Sarajevo, MSc
degree from the Faculty of Electrical Engineering, University of Zagreb, Croatia, and
PhD from the Faculty of Electrical Engineering, University of Sarajevo, Bosnia and
Herzegovina, in 1975, 1992 and 1980, respectively. From 1975, he was with the
Electrical Engineering Faculty of the University of Sarajevo, where he held an
Associate Professor position until 1994. Since 1997 he has been a Lecturer at E
´cole
Polytechnique de Montre
´al, Que
´bec, Canada. His major interests are in the
mathematical modeling of stationary and quasi-stationary fields, electromagnetic
fields in machines, magnetic materials, transformers, computational electromag-
netics, and field theory. He has published over 60 papers in these fields. Dr Slobodan
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
CALCULATION OF THE MAGNETIC FIELD 503

Babic has been cited over 30 times in international journals and conference proceedings. He is a member of
International Compumag Society.
Cevdet Akyel (M’81) was born in Samsun, Turkey. He received his Sup Ing degree
from the Technical University of Istanbul in 1971 and MScA and DScA degrees from
E
´cole Polytechnique de Montre
´al in Canada in 1975 and 1980, respectively. He had
engineering positions in 1972 and 1976 at Northern Telecom of Canada as a System
Engineer in radio telecommunications.
Since 1986, he has been a Professor of Electrical Engineering at E
´cole
Polytechnique de Montre
´al where he teaches electromagnetic theory and automated
microwave instrumentation. In 1991 he joined the Group of Poly-Grames involved in
space electronics and microwave advanced technologies at the same university.
His main interest in research is the permittivity measurement with microwave
active cavity methods, the characterization of materials (conductive polymers,
superconductivity ceramics, ferromagnetic materials, etc.), and high power micro-
wave measurement systems and applications.
Copyright #2005 John Wiley & Sons, Ltd. Int. J. Numer. Model. 2005; 18:493–504
S. I. BABIC AND C. AKYEL
504