The Electromagnetic Field Propagation in a Spherical Core

doi:10.4236/jemaa.2012.412067

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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.

Email: [email protected]om, [email protected]

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:

JjD





 (1)

EjB





 (2)



 (3)





 (4)

Taking the curl of (1) and substituting (2), we obtain





 

 H (5)

where



andHKH kj



 

   (6)

Equation (5) can be written in spherical coordinates





,,r





22 2

1d11 1

sin

dsin sin

rrrr



 























(7)

The Electromagnetic Field Propagation in a Spherical Core

482

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



1d d

11 1

sin ,

sin sin

rkr

Rr r



 

  





 



 



 



(8)

where



is constant.

Therefore, the following equation is satisfied:

d2d 0





 





, (9)

 

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





Ykr, . From the Equation (11), we get 0

D

 

RrCJ kr (12)

where





kr is Bessel function.

The solution for (10) is simplified as follows:



  



214- !!e

llmlmPCos







  (13)

where , and

0ml 0,1, 2,3, 4,l











is a

spherical harmonic function.



cos



is a Legendre’s

associated polynomial defined by the expression



 

2d1d

2! !dd

mmll

lml

Px xx

lxx





 







 (14)

where cosx



.

From Equations (12) and (13) we got the magnetic field as













,, ,

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



Er H

 





 (16)

Assuming





,,Er





and





,,Er





represent the



and components of the electrical field E then



Er jr





 







 (17)



Er jr



 



 

(18)

From Equation (14) we find

 

nn l

HCJ krY











, (19)

where

  



2! 12

sns

ns kr

Jkr sns















 (20)

and

 

HCJ kr













(21)

From Equations (19)-(21) in (17) and (18) we have

  

rnn

Er CJkr



  





(22)

and

  

,, ,

nn l

Er CJkrY













(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







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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The Electromagnetic Field Propagation in a Spherical Core

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