Journal of Electromagnetic Analysis and Applications, 2012, 4, 481-484
http://dx.doi.org/10.4236/jemaa.2012.412067 Published Online December 2012 (http://www.SciRP.org/journal/jemaa)
481
The Electromagnetic Field Propagation in a Spherical Core
Osama M. Abo-Seid1, Ghada M. Sami2
1Mathematic Department, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt; 2Mathematic Department, Faculty
of Science, Ain shams University, Cairo, Egypt.
Received September 15th, 2012; revised October 16th, 2012; accepted October 26th, 2012
ABSTRACT
A simple and explicit derivation for the electric and magnetic fields in the ferromagnetic cores has been studied. An
improved model for analyzing the distribution of electric and magnetic fields in a toroidal core is given. This leads to a
basis system for the theoretical analysis of iron loss in the magnetic cores, so, the theoretical results have been evaluated. A
method is used to perform numeric calculations of the magnetic field produced by the eddy current and displacement
current due to the electric field which shield the magnetic flux from the inner portion of the core cross section. The re-
sults obtained from this work could be used to evaluate the skin effect in the conductors of a winding.
Keywords: Electromagnetic Field; Wave Propagation; Magnetic Flux; Magnetic Cores
1. Introduction
Considerable interest has been given to the study of elec-
tric and magnetic fields distribution in a toroidal core.
Some of these studies calculate the core loss theoretically
[1]. In their calculation, the magnetic field strength pro-
duced by the winding current is assumed the constant
around the perimeter of the cross section. Other studies
have been applied to measure the magnetic core loss [2].
Also, Abo-Seida et al. [3] gave an explicit derivation for
the electromagnetic transient, Abo-Seida et al. [4] de-
rived the transient fields of a vertical electric dipole on
an M-layered dielectric medium. The transient field of
the magnetic dipole on a two-layered conducting earth
has also been studied [5,6]. Abo-Seida [7] has studied the
far field of a vertical magnetic dipole. Wait [8] derived
the electromagnetic fields of a traveling current line
source.
We noticed that the diameter of the core cross section
is very small compared with the innermost radius of the
core. In this paper, we calculated the distributions of the
electric and magnetic fields in the torodial core with
spherical cross section based on Maxwell’s equation as
in Figure 1(a) and take the boundary conditions into
consideration. The magnetomotive forces are affected by
the external current, the eddy and displacement current in
the area r < R as in Figure 1(b), but on the boundary r =
R, the magnetic field intensity is determined only by the
exciting current. The induced eddy current and dis-
placement current due to the electric field will shield the
flux from the inner portion of the core section resulting
in a flux skin effect analogous to the skin effect in the
conductors of winding at high frequency. With the in-
crease of frequency, this phenomenon becomes more ob-
vious.
2. Formulation of the Problem
The physical model is illustrated in Figures 1(a) and (b).
A frequency domain analysis was performed in this work.
It is assumed that all field vectors and all currents and
charge densities vary sinusoidally with time at a single
angular frequency. Maxwell’s equations are then written
as follows:
H
JjD


 (1)
EjB


 (2)
0B

 (3)
D


 (4)
Taking the curl of (1) and substituting (2), we obtain


2
H
j    H (5)
where

22
andHKH kj

     (6)
Equation (5) can be written in spherical coordinates


,,r

 as
2
2
22 2
1d11 1
sin
dsin sin
0
H
rH
rrrr
kH

 
2











(7)

The Electromagnetic Field Propagation in a Spherical Core
482
A
B
B'
A'
φ
R
2
ˆ r
o
θ
o
R
R
1
Cross section
through BB'
Cross section
through AA'
(a) (b)
Figure 1. The physical model of the problem.
With theassumption of

,, ,Hr RrY


  
and
using separation variable method we have

22
2
22
1d d
dd
11 1
sin ,
sin sin
R
rkr
Rr r
Y
Y

 
  


 

 

 

(8)
where

is constant.
Therefore, the following equation is satisfied:
2
22
d2d 0
d d
RR
kR
rr rr


 


, (9)
 
2
22
1dd1 d
sin, ,
sin ddsindYY


 
 




(10)
The solution of Equation (9) is expressed as follows:
  
nn nn
RrCJkrDYkr (11)
Since the area of interest includes and , 0r0r


n
Ykr, . From the Equation (11), we get 0
n
D
 
nn
RrCJ kr (12)
where


n
J
kr is Bessel function.
The solution for (10) is simplified as follows:

   
,
214- !!e
m
l
mi
l
Y
llmlmPCos
m


   (13)
where , and
0ml 0,1, 2,3, 4,l

,
m
l
Y




is a
spherical harmonic function.

cos
m
l
P

is a Legendre’s
associated polynomial defined by the expression

 
22
2d1d
11
2! !dd
mmll
m
lml
Px xx
lxx


 



 (14)
where cosx

.
From Equations (12) and (13) we got the magnetic field as






,, ,
m
nn l
HrCJ krY

  

, (15)
Based on the expression of

,,Hr


, the formula
used to calculate the electrical field is deduced as follows:
considering Equation (11) together with J = σE, D = εE,
the following expression is deduced

1
,,
Er H
j
 


 (16)
Assuming


,,Er

 and


,,Er


 represent the
ˆ

and components of the electrical field E then
ˆ
r

11
,,
r
H
Er jr 

 



 (17)

1
,,
H
Er jr

 

 
(18)
From Equation (14) we find
 
,
m
nn l
HCJ krY
r



 
, (19)
where
  

1
0
12
1
2! 12
sns
n
s
ns kr
Jkr sns







 (20)
and
 
,
m
l
nn
Y
HCJ kr






(21)
From Equations (19)-(21) in (17) and (18) we have
  
,
11
,,
m
l
rnn
Y
Er CJkr
jr

  


(22)
and
  
1
,, ,
m
nn l
Er CJkrY
j





  (23)

The Electromagnetic Field Propagation in a Spherical Core 483
(a) (b) (c)
Figure 2. (a) Amplitude of H at f = 5 kHz; (b) Amplitude of Eθ at f = 5 kHz; (c) Amplitude of Er at f = 5 kHz.
(a) (b) (c)
Figure 3. (a) Amplitude of H at f = 300 kHz; (b) Amplitude of Eθ at f = 300 kHz; (c) Amplitude of Er at f = 300 kHz.
The electrical and magnetic fields in toroidal core have
been calculated using Equations (15), (22) and (23) based
upon the typical values [9,10].
3. Numerical Results
The magnitude of the frequency rate of change of the
electromagnetic field due to propagation in a spherical
core is computed for different frequencies, and is shown
in Figures 2(a)-(c) and 3(a)-(c). In these figures, the am-
plitude of the tangential component


E

of electrical
field is much larger than the normal component (Er) at
high frequency. From the Figures 2(a) and 3(a), it is
found that the magnetic field strength in the core is very
small when r is small. With the increase of r, the mag-
netic intensity increases too. The magnetic field strength
reaches the maximal value when r = R as shown in Fig-
ure 3(a), 0

. From Figures 3(b) and (c), we can reach
the conclusion that the components of the electrical field
intensity increase with the increase of radius at same
angle

.
The E

component, it reaches the maximal value at r
= R and 2π

.
However for the normal component, it reaches the
maximal value somewhereπ2π

 . With the increase
of frequency, this phenomenon becomes clearer. Based
on the equations of electrical and magnetic fields, core
losses are calculated in this paper.
4. Conclusion
The distributions of the electrical and magnetic fields in
the toroidal core with spherical cross section have been
calculated based on Maxwell’s equations. Based on the
equations of electrical and magnetic fields core losses
were calculated. It is found that the losses obtained by
taking the boundary conditions into consideration are
larger than those without considering it.
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