The Influence of Electromagnetic Scattering from a Permeable Sphere on the Induced Voltage across a Rotating Eccentric Coil

doi:10.4236/jemaa.2011.31001

Paper Menu >>

Journal Menu >>

J. Electromagnetic Analysis & Applications, 2011, 3, 1-6

doi:10.4236/jemaa.2011.31001 Published Online January 2011 (http://www.SciRP.org/journal/jemaa)

The Influence of Electromagnetic Scattering from

a Permeable Sphere on the Induced Voltage across

a Rotating Eccentric Coil

Constantinos A. Valagiannopoulos

Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, Espoo, Finland.

Email: [email protected]

Received November 16th, 2010; revised December 20th, 2010; accepted December 27th, 2010.

ABSTRACT

Electromagnetic scattering and electromagnetic induction are research topics not directly associated to each other. In

this work, these two different concepts are combined in a model constituted by a rotating circular coil with a dielectric

spherical core at a fixed eccentric position. The scope of the analysis is to examine the effect of a permeable object on

the production of the alternating voltage. Methods and formulas from both scattering and induction have been utilized

for the derivation of the developed potential difference around the moving loop. Several graphs of the voltage output

with respect to the geometrical and material characteristics of the configuration, are presented and discussed.

Keywords: Electromagnetic Induction, Electromagnetic Scattering, Penetrable Dielectric Sphere, Rotating Loop

1. Introduction

The electromagnetic induction is defined as the develop-

ment of voltage across a closed conductor with time-

varying magnetic flux through it. The physics that govern

the inductive experiments have been mathematically

examined in a number of elementary treatises. The rela-

tionship between the various induction laws is summa-

rized in [1], where Cohn advocates that the combined use

of both motional and transformer induction assures the

validity of the produced results. A complete report re-

viewing the major developments and identifying impor-

tant trends in the broad field of geophysical electromag-

netic induction is given in [2]. Moreover, a rudimentary

study on induction inside a rotating coil surrounded by a

rigid conductor of finite or infinite extent has been pro-

vided in [3]. The analysis is based on integral solutions of

the field equations and leads to the conclusion that the

induced magnetic fields depend on the relative symmetry

of the rotator.

The electromagnetic scattering is defined as the modi-

fication of the incident field in the presence of an obsta-

cle through the fulfillment of the boundary conditions,

and numerous researches are performed on this topic. In

[4], the point-source scattering by an electrically large

conducting sphere has been discussed, where Bessel

functions of complex order are utilized. In addition, the

quasi-magnetostatic solution for a permeable prolate

spheroid under arbitrary excitation by a time-harmonic

primary field has been obtained by using the separation

of variables method with vector spheroidal wave func-

tions [5]. Finally in [6], a simplified solution is obtained

to the problem of a radiating loop in the presence of a

metallic core.

In this work, we combine the two aforementioned is-

sues (induction and scattering) by considering a structure

comprised of a rotating thin circular loop and an eccen-

trically positioned penetrable spherical core, illuminated

by a plane wave. The rotation happens around the eccen-

tric axis passing through the centre of the spherical cavity.

The magnetic vector potential at the position of the thin

closed wire is evaluated with use of spherical eigenfunc-

tion expansions and through the enforcement of the

boundary conditions. The variance of magnetic flux is

computed from the line integral of the electric field

around the metallic coil. The DC offset and the RMS

value of the produced voltage are represented in several

graphs with respect to the size of the sphere, the material

of the core and the excitation parameters as well. By in-

spection of the variations, one can reach various useful

and applicable conclusions. In particular, one can choose

the texture, the position and the radius of the scatterer

The Influence of Electromagnetic Scattering from a Permeable Sphere

on the Induced Voltage across a Rotating Eccentric Coil

that should be posed eccentrically inside the ring in order

to obtain maximum output voltage. The engineer could

utilize the dielectric core, when constructing an electro-

magnetic induction device, as an “optimizer” for regulat-

ing the produced potential.

2. Mathematical Concept

The configuration of the examined problem is shown in

Figure 1(a) where the spherical coordinate system (r, θ,

φ) and the equivalent Cartesian one (x, y, z) are also de-

fined. The origin O coincides with the center of a spheri-

cal volume (region 1) of radius a, filled with dielectric

material of relative permittivity ε1 and relative magnetic

permeability μ1. The scatterer is posed into vacuum (re-

gion 0) with intrinsic parameters (ε0, μ0). A thin circular

metallic loop of radius b > a, is shown on the x – y plane,

with its center K at (x, y, z) = (0, d, 0) with b > d, located

eccentrically to the spherical core. The closed wire is

rotated with respect to x axis with circular frequency

in the presence of an x-polarized plane wave E0,inc, with

magnitude Q (in V/m), advancing towards the negative z

semi-axis. Mind that the harmonic time dependence of

the incident field, is of the form exp(–iω0t); it has circular

frequency ω0 ≠ ω. In Figure 1(b), we present a side view

of the device as appeared from the positive x semi-axis,

when the frame is rotated by angle ωt, at an arbitrary

time t.

The polar radius R(φ) of the eccentric circular loop at t

= 0 (Figure 1(a)) is determined by applying the law of

cosines to the shaded triangle, yielding to:



222

sin cosRφdφbd φ . (1)

It is necessary to extract the parametric equation set of

the rotated coil denoted by









,, ,,xXφtyYφt



The azimuthal angle





,zZφt



0, 2φπ will play

the role of the parameterization variable even when the

object does not belong exclusively to x-y plane. At this

point, it is proper to make clear that there is no practical

importance in considering the filamentary loop as a torus

possessing non-negligible thickness. In the vast majority

of works concerning voltage production by electromag-

netic induction, the coil boundaries are constructed from

wires with infinitesimal transversal dimensions. Fur-

thermore, the assumed oscillation frequency is kept quite

low and thus the electrical size of the wire would be ex-

tremely small even it is not filamentary. In such a case,

the solution to the compound scattering problem from

both the dielectric sphere and the metallic torus would

lead to almost identical results to those assuming thin-

wire approximations.

As the closed wire is rotated with respect to x axis, the

corresponding coordinate X(φ, t) will be fixed, inde-

pendent from the angle ωt and equal to P(φ)cosφ. The

rest two equations are derived by projecting the other

edge of length P(φ)sinφ, positioned at angle ωt, upon the

axes x and z (see Figure 1(b)). Accordingly, one obtains

the following expressions:





,cXφtRφosφ (2a)









,sincoYφtRφφωts

(2b)









,sinsiZφtRφφωtn (2c)

The parametric representation of the curve in spherical

coordinates













,, Θ,, Φ,rPφtθφtφφt  for

each





0,φ 2π

, is given by:





Ρ,φtRφ



(3a)





Θ,arccos(sin sin)φtφωt (3b)





Φ,arctan(tan cos)φtφωt (3c)

According to Faraday’s law of induction [7], the in-

duced voltage across a closed metallic wire (W), is de-

fined as the line integral of the local electric field around

the loop. In case of a monochromatic electric field with

circular frequency ω0, the related formula is given below:



Re iωt

Ue d

















EW

(4)

It should be stressed that E0 does not denote the real,

time-dependent electric field into vacuum, but the corre-

sponding complex phasor. In case the field quantities are

expressed in terms of the spherical coordinate system (r,

θ, φ), the Cartesian components are given by [8]:



cos sincos cossin

sinsinsincoscos, ,

cossin 0,,

Erθφ

φθφ θφ

φθφ θφErθφ

θθ Erθφ























(5)

Once these functions are determined, the line integral

of (4) is particularized to give [9]:





000

Re,,, ,,,

iωt

xφyφzφ

Utee φtX φteφtY φteφtZ φtdφ











 (6)

The Influence of Electromagnetic Scattering from a Permeable Sphere

on the Induced Voltage across a Rotating Eccentric Coil

0,inc

b + d

–

b + d

(φ = π/2)

(φ = 0)

–

a O φ

R(φ)

(

, μ

)

(ε

, μ

)

0,inc

b + d

(θ = 0)

(

, μ

)

(ε

, μ

)

b – d

–

(θ = π/2)

(a) (b)

Figure 1. The physical configuration of the investigated device: (a) As viewed from the positive z semi-axis. (b) As viewed

from the positive x semi-axis. The red arrow denotes the moving direction of the incident plane wave, not the actual vector of

the electric field.

where subscript φ corresponds to the azimuthal partial

derivative of the related function. The small-e functions





000

eee are the electric field components evaluated

around the moving circular loop:





,Ρ,,Θ,,Φ,

eφtE φtφtφt (7a)

 



,Ρ,,Θ,,Φ,

eφtEφtφtφt



(7b)

 



,Ρ,,Θ,,Φ,

eφtE φtφtφt (7c)

Thus, the only prerequisite to use expression (6) in

computing the induced voltage, is the explicit form of the

total electric field into vacuum, expressed in spherical

coordinates.

The electric field into vacuum background is com-

prised of the incident and the scattering component E0 =

E0,inc + E0,scat, where E0,inc = xQexp(-ik0z) and k0 =

000





. The electric field into vacuum region is com-

puted with use of spherical eigenfunctions and the follow-

ing series expansion [10]:

 



cos 0

21 cos

ik rθ

einPθjkr

 











(8)

The symbol





Px corresponds to the Legendre

function of degree n, order m and argument x. The

spherical Bessel jn(x) and the spherical Hankel of the first

kind hn(x) are well-known [11]. The Riccati functions are

defined as









zxdzxdx 



, where zn(x) is the

spherical Bessel or Hankel function. Once the boundary

conditions at r = a are imposed, the respective scattering

components of the electric field are given by:

 





,,coscos n

rscat n

hkr

Erθφ QφSnP θkr





 (9a)

 



0, 0

cos cos

,, cos11sin

θscat n

hkr

Sn dPθTnPθ

ErθφQφhkr

nn dθkrn nθ







 









 (9b)

 



0, 0

cos cos

,, sin1sin 1

φscat n

hkr

Sn PθTndPθ

ErθφQφhkr

nn θkrnndθ

















 (9c)

where:

  



















 



110101

10101

nnnn n

nnn n

jkajkaεjkajka

Sninjkahkaεhkajka



  (10a)

 















 



10101

nnn n

jkajka μjkajka

Tn injkahka μhkajka



  (10b)

The Influence of Electromagnetic Scattering from a Permeable Sphere

on the Induced Voltage across a Rotating Eccentric Coil

The wavenumber in region 1 is defined by 10 11





As one can notice, the mathematical formulation

above is divided in two subsections; the first one con-

cerns the electromagnetic induction and the second one

examines the electromagnetic scattering. The induced

voltage around the loop (first subsection) is solely depen-

dent on the electromagnetic field and the shape/position

of the coil. Successful manipulation of the boundary

value scattering problem (second subsection) gives the

required electromagnetic field in explicit form. Accord-

ingly, there are two successive series of algebraic opera-

tions which help us understanding the combined induc-

tion/scattering consideration. In this sense, the two con-

cepts are not treated independently each other, because

the solution to the scattering problem is a prerequisite to

solving the induction one. That is why the dielectric core

cannot be present in our configuration, in case one ignores

the scattering procedure. In particular, the only mecha-

nism through which the permeable sphere participates in

the investigated context is the modification of the elec-

tromagnetic field externally to it.

It is also known [7] that the developed voltage around

a loop with area (S) is proportional to the time rate of

change of the magnetic flux through it, namely:



Re iωt





 





BS

(11)

where Β0 is the magnetic field into vacuum. The area

integral above will be used alternatively for the evalua-

tion of the induced potential difference in order to vali-

date our results.

3. Numerical Results

Prior to proceeding to the numerical results, we should

determine the intervals into which the input parameters

belong. The radius of the metallic loop can take values

within the range 0.5 m < b < 2.5 m (in most cases equal

to b = 1 m). The frequency of the excitation wave is

moderate (usually equal to ω0 = 200π rad/sec), being

taken between the limits: 2π rad/sec < ω0 < 400π rad/sec.

When it comes to the spherical scatterer, its relative per-

mittivity does not affect substantially the results and thus

is kept constant throughout the numerical simulations (ε1

= 5). The relative magnetic permeability of the sphere

possesses usual magnitudes, that is 1 < μ1 < 2.5 and nor-

mally is assigned the value μ1 = 2. To the query “why

does the permeability vary?”, one shall respond that the

characteristics of the sphere’s material are treated as de-

grees of freedom in designing the device and therefore

can vary. As far as the values of μ1 are concerned, they

are chosen close to those owned by common composite

materials [12]. Instead of the radius of the sphere, we use

the normalized parameter









0,1abd as the core

should be kept internal to the rotating ring. The eccen-

tricity ratio









0,1dbd is also utilized to quantify

the relative transposition of the scatterer. In the following

graphs, two quantities are mainly represented; the DC

offset and the RMS value of the induced voltage, defined

below:



UUt

dt

(12a)



rms dc

UUtU





dt

(12b)

The amplitude of the plane wave Q is a trivial parame-

ter and therefore is chosen high enough to give realistic

values for the output voltages.

In Figure 2(a), the RMS component of the produced

voltage is shown as function of the normalized sphere

0.1 0.2

0.3

0.4 0.5

0.6

0.7

0.8 0.9

normalized sphere radius a/(b – d)

b= 0.5

b= 1.0

b= 1.5

b= 2.0

RMS voltage U

rms

(V)

1.4

1.2

0.8

0.6

0.4

0.2

(a)

0.1 0.2

0.3

0.4 0.5 0.6

0.7

0.8 0.9

normalized sphere radius a/(b – d)

= 1.0

= 1.5

= 2.0

= 2.5

RMS voltage U

rms

(V)

0.31

0.3

0.29

0.28

0.27

0.26

0.25

0.24

0.23

(b)

Figure 2. The RMS induced voltage Urms as function of the

normalized radius of the spherical scatterer a/(b – d), (a) for

various sizes of the loop b (μ1 = 2), (b) for various magnetic

permeabilities of the core μ1 (b = 1 m). Plot parameters: d =

0.2 m, ε1 = 5, ω = 200π rad/sec, ω0 = 200π rad/sec, Q = 105

V/m.

The Influence of Electromagnetic Scattering from a Permeable Sphere

on the Induced Voltage across a Rotating Eccentric Coil



radius for constant eccentric position d corresponding to

several sizes of the rotating loop. The maximum mag-

netic flux through the circular wire (and implicitly the

induced voltage) is proportional to the size of this ring.

Note also that the normalized radius of the sphere plays

rather unimportant role when it is kept low. On the con-

trary, when the scatterer gets close to the frame, there is

an amplifying effect on the measured quantity, which

gets more significant for larger loops. This is a natural

result because for fixed b,d, the available manoeuvring

area for the sphere to move gets restricted when the of

the loop is small. In Figure 2(b), the RMS value of the

produced voltage is represented with respect to the same

variable



abd for various magnetic permeabilities

of the sphere. One could also notice the exponentially

increasing behavior of the curves which remarks the

beneficial influence of the scatterer’s radius on the in-

duced voltage. In addition, this upward sloping trend is

proportional to the magnetic permeability of the spheri-

cal core, while there is no variation when the obstacle is

magnetically inert. In other words, the size and the mag-

netic density of the dielectric core provide the design

engineer with two additional degrees of freedom in con-

structing an efficient voltage generator.

In Figure 3, we show the measured response with re-

spect to the permeability of the core μ1, for several ec-

centricity ratios



dbd. It should be stressed that the

eccentricity ratio affects crucially the RMS value of the

induced voltage which means that the degree of asymme-

try reinforces the recorded quantity. In this way, the per-

meable sphere could play the role of a tuner with which

the magnitude of the output is chosen at will by changing

the eccentric position of the sphere. Also, all the curves

coincide at μ1 = 1 and then increase gradually for mag-

netically denser construction materials. It seems that

when μ1 = 1, the dielectric core becomes completely

transparent to the incident electromagnetic field regard-

less of its position.

In Figure 4(a), the RMS component of the produced

voltage is shown in a contour plot with respect to the

rotation frequency of the loop and the oscillation fre-

quency of the incident plane wave. For increasing ω0, the

recorded quantity gets reinforced with a pace negatively

related to ω. Once the rotation frequency gets larger,

there is either a stability in the measured output (modest

ω0) or a magnitude boost (substantial ω0). It should be

also remarked that when ω is very low, rapid variations

in Urms are observed for little change of ω0. This chaotic

behavior is attributed to the fact that, in case ω  0, the

magnetic flux is considerably affected even by the

slightest variance in the frequency of the alternating field.

In Figure 4(b), the DC offset Udc is represented for the

1.5

core permeability μ

b – a

) = 0.1

b – a

) = 0.6

b – a

) = 0.8

b – a

) = 0.9

RMS voltage U

rms

(V)

0.2345

0.234

0.2335

0.233

0.2325

Figure 3. The RMS induced voltage Urms as function of the

core magnetic permeability μ1, for several eccentricity ra-

tios d/(

– a). Plot parameters: a = 0.2 m, b =1 m, ε1 = 5, ω =

200π rad/sec, ω0 = 200π rad/sec, Q = 105 V/m.

rotation frequency ω (rad/sec)

200 400

600

800 1000 1200

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1200

1000

800

600

400

200

excitation frequency ω

(rad/sec)

(a)

rotation frequency ω (rad/sec)

200 400

600

800

1000 1200

0.4

0.2

-0.2

-0.4

-0.5

1200

1000

800

600

400

200

excitation frequency ω

(rad/sec)

(b)

Figure 4. Contour plots of (a) the RMS induced voltage Urms

and (b) the DC component of the produced voltage Udc,

with respect to the rotation frequency ω and the excitation

frequency ω0. Plot parameters: a = 1/3 m, d = 1/3 m, b = 1 m,

ε1 = 5, μ1 = 2, Q = 105 V/m.

The Influence of Electromagnetic Scattering from a Permeable Sphere

on the Induced Voltage across a Rotating Eccentric Coil

same set of parameters. Note that in all the previous ex-

amples, the presence of the scatterer makes the produced

oscillating voltage to have nonzero average value owning

similar waveforms to Urms. This is not the case; when ω

is chosen close to ω0, there is a substantial increase for

growing frequencies. After numerical trials, we con-

cluded that the shape of the equal Udc levels in Figure

4(b) resemble rotated hyperbolas with narrow extent

following an asymptotic law of:

fundamental phenomena in electromagnetics (induction

and scattering) have not been studied yet.

The variation of the measured output is represented as

function of the sphere’s characteristic parameters and

several conclusions are drawn describing its effect on the

magnetic flux through the coil. An interesting expansion

of the present study would be to assume a spherical scat-

terer with inhomogeneities and/or anisotropies, or to

modify slightly the shape of the core using oblate sphe-

roid coordinates and functions. Also, closed wires of

arbitrary curvature rotating around arbitrary axes could

be also investigated with use of the same techniques.

UOωωω ω









(13)

In Figure 5, we show in contour plot the relative dif-

ference between the estimation of the RMS voltage via

normal expression (4) and its evaluation through the al-

ternative Formula (11). This percent error is represented

as function of the quantities (a/b, d/b), where a + d  b,

and one can observe that its magnitude is very low (be-

low 0.0002%). As the truncation limit (number of terms

summed) in the series (9) is kept fixed, the error gets

more significant for more sizeable scatterers. Addition-

ally, the more eccentric is the position of the sphere, the

more substantial is the recorded quantity. As far as most

of the numerical results represented in previous figures

are concerned, they have been verified through the area

integral (11) and remarkable coincidence with (4) is ex-

hibited.

REFERENCES

[1] G. I. Cohn, “Electromagnetic Induction,” Electrical En-

gineering, Vol. 68, No. 5, May 1949, pp. 441-447.

[2] A. D. Chave and J. R. Booker, “Electromagnetic Induc-

tion Studies,” Reviews of Geophysics, Vol. 25, No. 5,

June 1987, pp. 989-1003. doi:10.1029/RG025i005p00989

[3] A. Herzenberg and F. J. Lowes, “Electromagnetic Induc-

tion in Rotating Conductors,” Philosophical Transactions

of the Royal Society of London, Series A, Mathematical

and Physical Sciences, Vol. 249, No. 5, 1957, pp. 507-584.

doi:10.1098/rsta.1957.0006

[4] C. A. Valagiannopoulos, “An Overview of the Watson

Transformation Presented through a Simple Example,”

Progress in Electromagnetics Research, Vol. 75, 2007,

pp. 137-152. doi:10.2528/PIER07052502

4. Conclusions [5] C. O. Ao, H. Braunisch, K. O’Neill and J. A. Kong,

“Quasi-Magnetostatic Solution for a Conducting and Per-

Meable Spheroid with Arbitrary Excitation,” IEEE Trans-

actions on Geoscience and Remote Sensing, Vol. 40, No.

4, 2002, pp. 887-897. doi:10.1109/TGRS.2002.1006370

In this work, we examine the induction of electromagnetic

voltage across a rotating circular loop, in the presence of

an eccentric dielectric sphere under a low-frequency,

plane-wave excitation. Similar topics combining two [6] C. A. Valagiannopoulos, “Single-Series Solution to the

Radiation of Loop Antenna in the Presence of a Con-

ducting Sphere,” Progress in Electromagnetics Research,

Vol. 71, 2007, pp. 277-294. doi:10.2528/PIER07030803

a/b

0 0.1

0.2

0.3

0.4 0.5

1.8

1.6

1.4

1.2

0.8

0.6

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

d/b

×10

[7] M. Sadiku, “Elements of Electromagnetics,” Oxford Series

in Electrical and Computer Engineering, 2001, p. 372.

[8] C. A. Balanis, “Advanced Engineering Electromagnet-

ics,” John Wiley & Sons, New York, 1989, p. 924.

[9] R. C. Wrede and M. R. Spiegel, “Advanced Calculus,”

Schaumm’s Outline Series, 2002, pp. 229-232.

[10] V. A. Erma, “Exact Solution for the Scattering of Electro-

Magnetic Waves from Conductors of Arbitrary Shape. II.

General Case,” Physical Review, 1968, pp. 1544-1553.

doi:10.1103/PhysRev.176.1544

[11] M. Abramowitz and I. A. Stegun, “Handbook of Mathe-

matical Functions,” National Bureau of Standards, 1970,

pp. 437-438.

Figure 5. Contour plot of the percent error in computing

the RMS induced voltage U

rms through (4) and (11), with

respect to the size of the scatterer a/b and its eccentric posi-

tion d/b. Plot parameters: b = 1 m, ε1 = 5, μ1 = 2, Q = 105

V/m.

[12] C. A. Valagiannopoulos, “On Smoothening the Singular

Field Developed in the Vicinity of Metallic Edges,” Inter-

national Journal of Applied Electromagnetics and Me-

chanics, Vol. 31, No. 3, 2009, pp. 67-77.