uc

gypt

Magnetic eld calculationElectric eld simulation

he hf thth

latio

frequency induces electric eld, magnetic eld and currents inside the human body. The presented

ctromaof elec], explectri

insulation [6]. Numerical methods, such as nite element method

initiated public concern, due to the overlap between the powertransmission lines and the settlement areas which lie very near orunder the power transmission lines [2,7,12e15]. There has been agrowing interest in determining the safe exposure level of humansto power frequency electric and magnetic elds [2,7,12,15]. There-fore, the simulation of electric and magnetic elds, in the space

method (BEM) [14].ctromagnetic eldsed for the electricy. The electric andlace's equation byedium. The surfaceber of charges ar-harges, nite line

charges [2,7]. In this model the simulated electric and magneticelds is introduced different types of charges such as ellipticalcharges and segment ring charges [18], taking into consideration thedifferent elementary geometrical shapes of human body.

The optimum number, values, locations, and dimensions ofthese charges are achieved by using genetic algorithms (GAs) as asearch optimization technique [2,4,7,19]. Series of vertical and in-clined line charges [4,7,20] especially in the arms and unsymmet-rical ring charges [21] especially in the legs. For these inclined and

Contents lists availab

le

.e l

Journal of Electrostatics 72 (2014) 387e395E-mail addresses: m_mtalaat@eng.zu.edu.eg, m_mtalaat@yahoo.com.(FEM) [1,6], charge simulationmethod (CSM) [2,3,7], charge density[8], Monte-Carlo method (MCM) [9], nite difference method(FDM) [10,11], and integral equation methods have been used tosimulate the non-uniform electric elds. CSM is one of the mostsuccessful numerical methods used for solving electromagneticeld problems [2,3,5,7].

The interaction of electric and magnetic elds with humans has

This paper presented a three-dimension elesimulation. CSM and the method of image are uand magnetic elds simulation in the human bodmagneticelds distribution are obtained fromLaptreating the human body as a good conductingmcharges on human body are simulated by a numranged inside the human body, such as ring cliquids [4], streamers in gases [5], and the design of high voltage moment method techniques [17], FEM [13], and boundary elementInduced elds and currentCharge simulation methodGenetic algorithmsHuman subject to electric power lines

1. Introduction

The numerical analysis of eleimportant role in the understandingas; ow in electrolytic solutions [1power lines [2], treeing in solids [3], ehttp://dx.doi.org/10.1016/j.elstat.2014.06.0080304-3886/ 2014 Elsevier B.V. All rights reserved.model for simulating electric and magnetic elds are a three dimensional eld problem and introduceddifferent types of charges to simulate the different elementary geometrical shapes of human body. Theparticular strength of the charge simulation method in this application is its ability to allow a detailedrepresentation of the shape and posture of the human body. The results have been assessed throughcomparison induced current, electric eld, magnetic eld and there distribution over the body surface, asestimated in other experimental and computational work.

2014 Elsevier B.V. All rights reserved.

gnetic eld plays antrical phenomena suchosures to high voltagecation and streamers in

between power lines and ground, is a prerequisite to assess theeffect of power lines on human.

The calculation of the induced electric and magnetic elds inhuman lead to substantial difculty, due to the complex geometryof the human body. For this reason, several approximate solutionshave been derived using CSM [2,7,12,15], MCM [9], FDM [10,11,16],Keywords:Available online 11 July 2014 arrangement of simulating charges inside the human body. The magnetic eld intensity along the ver-tical center line of the human is calculated. Exposure to external electric and magnetic elds at powerCalculation of electric and magnetic indsubjected to electric power lines

M. TalaatElectrical Power & Machines Department, Faculty of Engineering, Zagazig University, E

a r t i c l e i n f o

Article history:Received 11 March 2013Received in revised form2 June 2014Accepted 27 June 2014

a b s t r a c t

In this work, analysis of tsented. The distribution osurface charge induced oncalculated by charge simu

Journal of E

journal homepage: wwwed elds in humans

uman body exposed to high voltage electric and magnetic elds is pre-e electric eld is obtained by using Laplace's equation. This relates thee body to the potential in a reciprocal Laplace problem, which is thenn method coupled with genetic algorithms to determine the appropriate

le at ScienceDirect

ctrostatics

sevier .com/locate/elstat

unsymmetrical charges, a coordinate transformation is performed.Then, the electric and magnetic elds are calculated in the originalcoordinate system. The CSM in this application has the ability toallow a detailed representation of the shape and posture of thehuman body for grounded and an ungrounded case.

2. Method of analysis

Description of the electric and magnetic elds emanating fromvarious transmissions line congurations have been adequately

part is given by Ref. [24]. The model given by Ref. [2] was used to

Muscle 0.86 434,930

M. Talaat / Journal of Electrost388Bone 0.04 12,320Skin 0.11 1136Heart 0.5 352,850Gland 0.11 56,558Blood 0.6 5259Lung 0.04 145,100Liver 0.13 85,673simulate the human body using CSM, see Appendix A.

Table 1Tissue conductivity and permittivity values [12].

Tissue Conductivity sU1m1 Relative dielectric constant rpresented in many papers and texts [22].

2.1. Charge simulation methods

Analytical solution of Laplace's equation used for calculatingelectric eld, can only be obtained for relatively simple chargedistributions and conductor congurations. However, most of thehigh voltage systems are complex so numerical techniques are usedto solve this problems. One of the most efcient end accurate nu-merical techniques for eld computations is the CSM. It consists ofreplacing the actual continuous surface charge distribution of theconductors by a discrete set of ctitious charge distribution placedinside the volumes occupied by the conductors. The exact positionsand values of the simulating charges are found so that the boundaryconditions of the particular congurations are satised to a certaindegree of accuracy.

In this paper the actual electric eld is simulated by a number ofdiscrete charges located in, (transmission lines, human body, andearth) [2,4,6,7,12,18e21]. Values of simulation charges are deter-mined by satisfying the boundary conditions at a number of con-tour points selected at the conductor surfaces. Once the values ofsimulation charges are determined, then the potential and electriceld of any point in the region outside the conductor can becalculated using the superposition principle.

The various conductivity and relative equivalent dielectric con-stant of humanTissue [13] is given inTable 1. From this table the largeconductivity and the large relative equivalent dielectric constant ofthe human body cause the external power frequency electric eldnear the human body to be perpendicular to the surface [23]. This iswhy the human body is treated as a conducting body.

In this model, surface charges on the high voltage line conduc-tors are simulated by innite line of charges located at each line axis[2,4,6e8].

The human body is modeled taking a representation ofboundary surface as a combination of certain elementary geomet-rical shapes: spheres, cylinders, boxes etc. these are juxtaposed orsuperposed as required.

Fig. 1 represents the schematic diagram of the engineeringdrawing of the human body with basic dimensions in centimetersas a three-dimension model; the average dimension of any humanLens 0.11 105,5502.2. Genetic algorithms

Genetic algorithms, (GAs), are a form of evaluation that occurson a computer. GAs are a searchmethod that can be used for solvingproblems andmodeling evolutionary systems. The basic idea of GAsis very simple. First, a population of individual is created in acomputer, and then the population is evolved with use of theprinciples of variation, selection, crossover and mutation untilsome termination criteria are reached [25].

In the present paper, GAs are used in the optimization of a varietyof variables. These variables include the optimumnumber of charges,n, used tond the requirednumberof simulated charges for eachpartof the human body and the transmission lines. Also the optimumlocation of charges, l , used to obtain the axial location of differenttypes of charges along the human height. Finally the optimum radiusof simulated ring charges, f, used for indicating the optimumradius ofring and segment ring of charges, see Appendix C.

2.3. Electric eld simulation

The vector of unknown charges Q is computed from the matrixequation:

Qj Pij1 Vi (1)

where, Pij is the potential coefcient calculated at the ith boundarypoint due to the jth simulation charge Qj and V is the appliedvoltage of transmission line.

For a given charge distribution, the calculated potential f, at anarbitrary point is a summation of the potentials resulting from theindividual charges,

fi X6k1

fk i 1;2;M (2)

where, k indicates thehumanpart (k1 forhead,k2 forneck,k3forwaist, k4 for arm, k5 for upper part of leg, k6 for lower partof leg). The estimated values of fk are presented in Appendix B.

The simulated charges are distributed uniformly according toshape dimensions except the axial location of the elliptical and ringcharges along the z-axis, the optimum location of these charges aredetermined by GAs, also the optimum radius of any ring charge, seeAppendix C.

The used objective function used by GAs is simply the accu-mulated squared error, which has the form [21]:

U XMi1

V fi x; y; z 2 (3)

where, M is the number of contour points. For an ungroundedhuman body each of the transmission lines and the human bodyhave an objective function. The ungrounded human body newobjective function is simply the summation of factitious simulatedcharges inside the human body must be equal zero.

U XMni1

Qi 0:0 (4)

and for transmission lines the objective function is still the accu-mulated squared error but with only the simulated line charges.

U Xn

V fi x; y; z 2 (5)

atics 72 (2014) 387e395i1

s o r n

of e

trostThe problem is now reduced to the determination of the opti-

Fig. 1. Representation of simulation subject in termsM. Talaat / Journal of Elecmumvalues of parameter subject to the satisfaction of the objectivefunction given by Equation (3) for grounded human body andEquations (4) and (5) for ungrounded human body, using GAs.

Also, the electric eld components Eri and Ezi at the contourpoint i are the vector sum of the eld contributions from all thesimulated charges, where Er represent the equivalent vector of Exand Ey.

Eri X6k1

Erk (6)

Ezi X6k1

Ezk (7)

The value of electric eld components Eri and Ezi for any part ofthe human body is calculated, see Appendix D.

The total electric eld at the ith contour point is expressed as,

Ei E2ri E2zi

q(8)

2.4. Induced electric eld, magnetic eld, and current calculation

Less attention has been paid to the magnetic eld, probablybecause of the disparity in magnitude between the electric andmagnetic elds. Even though it is known that the dielectric prop-erties of the human body are not totally isotropic at low frequenciesthey are assumed to be homogeneous for the analysis presentedhere. Extension to anisotropic media is straight forward.

The charge density rs at a boundary point on the human bodysurface at height z is expressed as,r E (9)lementary geometrical shapes, all dimensions in cm.

atics 72 (2014) 387e395 389where, En is the normal component of the electric eld calculated atthe boundary point, o is the permittivity of free space which equal8.854 1012 F/m, and r is the relative permittivity of the humanbody given in Table 1.

The polarization current density induced by an external eld ina homogeneous body is given by

Jk urs uorEn (10)

where, k indicates the human part. Also, in lieu of the fact that therelative permittivity of tissues is large at low frequencies, the hu-man body represented as a good conductor with o.

The value of the normal component of the electric eld at anyarbitrary point on the human surface is presented by:

En ETL Ein (11)

where, ETL is the normal component of the transmission lineselectric eld, which obtained from the presented simulation pro-gram, and Ein is the normal electric eld due to the induced po-larization current in the human body given by:

Ein Vfk uA (12)

where, A is the magnetic vector potential. The rst term of Eq. (12),E Vfk , can be obtained from the presented simulation pro-gram, and the value of A in the second term is given by;

A m4p

ZJkr r.dv (13)

At the boundary point, the unknown induced current density Jk,is expressed as,

Jk uor

1 uorLkETL Vfk (16)

The induced current IS just outside the boundary of a part of thebody, where, Skth, is obtained by integrating Jk over the surface areaof this part

ISk Z

JkdSk (17)

The area Sk of each part of human can be mathematicallycalculated from dimension given in Fig. 1.

3. Results and discussion

1 20 40 60 80 100 120 140 160 180 200-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Location of contour points along the human body surface

% P

oten

tial e

rror

h/200

M. Talaat / Journal of Electrostatics 72 (2014) 387e395390Fig. 2. The variation of the per cent potential errors along the human body.

1 20 40 60 80 100 120 140 160 180 200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Dev

iatio

n an

gle

in d

egre

e

h/200where, m is the permeability of the medium (for human bodym mo 4p 107 H/m), is the position of the observation pointand r is the position of the integration point, human body.-Substitution of Equations (11)e(13) into Equation (10) yields to

Jk uor ETL Vfk um

4p

ZJkr r.dv

0B@

1CA

264

375 (14)

The volume of any part of the human body can be obtained fromthe integration over area of the human part and its length, so thelast term can be simplied to LkJk.where,

Lk um

4p

1r r.dsdl (15)

3.2. Charge simulation method

Location of contour points along the human body surface

Fig. 3. The variation of the deviation angle along the human body.

Table 2Induced current in grounded and ungrounded human body.

Position Grounded human body

Induced current(mA)

Induced electriceld (mV/m)

Induced magneeld (mT)

Top ofhead

18 0.3 1.6

Middleof neck

38 3 2.6

Middleof waist

130 1 3

Middleof legs

152 10 1.9The boundary conditions are checked over 200 points along thehuman height, h, in terms of the potential error, (the differencebetween the actual conductor voltage, V, and the potential calcu-lated due to the factious simulated charges, f), and deviation angle(the deviation in the eld angle from the normal position on theconductor surface). The accuracy of the simulation is satised forthe potential error, (not more than 0.1 %), and the eld deviationangle, (not more than 1 degree) [2,4,7,12,18e21] over the humanbody, as shown, in Figs. 2 and 3.

Ungrounded human body

tic Induced current(mA)

Induced electriceld (mV/m)

Induced magneticeld (mT)

18 0.29 1.6

35 2.2 2

66 0.6 2.1

20 0.9 1.563.1. Field calculation

In order to demonstrate the proposed approach, the height of thetwin bundle conductors of a three phase 380 kV line over the groundplane is 15mwhich is about ten times the height of the human bodyto make sure that the body have a negligible effect on the surfacecharge on the HV line conductors. Diameters and spacing betweensub-conductors are 27.7 mm and 400 mm respectively.The induced magnetic eld is evaluated based on Faraday's lawapplied to a cylindrical cross section, as a human body withcircumferential currents estimated by Ref. [26]:

B 2Jkusak

(18)

where, ak is the radius of the human body part expressed as cy-lindrical cross section. Then the magnetic eld strength H isexpressed as, H m1B.

3.3. Induced eld and current calculation

4. Conclusions

With the advent of high voltage power lines, it becomesincreasingly important to describe accurately the power line elec-

20 40 60 80 100 120 140 1600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Induced current along the human body (A)

)m(teef

evobathgiehna

muH

GroundedUngrounded

Fig. 4. Induced current distributions for ungrounded and grounded human body.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.5

1

1.5

2

2.5

3

Human height above feet (m)

Indu

ced

mag

netic

fiel

d (

T)

GroundedUngrounded

Fig. 7. Actual measured magnetic eld variation over human body for 220 kV powerlines [30].

M. Talaat / Journal of Electrostatics 72 (2014) 387e395 391Satisfaction of the boundary points at the chosen contour pointsresults in a set of equations whose solution determines the chargessimulating the body. Once the simulation charges are determined, theelectric eld, the induced charge, and current at the surface of the hu-manbodyaredeterminedasgiven inTable2 forgroundedhumanbody.

Table 2 gives the induced current, electric eld, and magneticeld distribution at the surface of a person standing in a 60 Hzperturbed eld for grounded and ungrounded bodies, these valuesare in accordance with that given by [12,27e30].

Computed induced current distributions for ungrounded andgrounded human of 1.8 m height standing in a vertical homoge-neous electric eld are illustrated in Fig. 4. Fig. 5 gives a comparisonof the proposed model with two different models [28,29], theagreement can be considered very acceptable.

Fig. 6 explains the calculated values of the induced magneticeld distribution for ungrounded and grounded human. Figs. 7 and8 give a comparison of the proposed model with magnetic eldvalues, which are obtained, from actual eld measurements, underdifferent actual 220 and 500 kV power transmission lines [30], theagreement can be observed between the measured values and theobtained from simulation program.Fig. 5. Computed (Gandhi & Chen) [28] and measured (Deno) [29] current distributionfor an ungrounded and grounded human of 1.77 m in height standing in a verticalhomogeneous electric eld of 10 kV/m at 50 Hz.Fig. 6. Induced magnetic eld distributions for ungrounded and grounded humanbody.tromagnetic eld interaction with life forms.

Fig. 8. Actual measured magnetic eld variation over human body for 500 kV powerlines [30].

This paper develops a simulation method for current, normalelectric eld and magnetic eld distribution induced on humanssituated in the vicinity of the power lines.

The technique is based on the charge simulation method andLaplace's equation to compute the external electric eld, magneticeld, induced charges and currents in a grounded and ungroundedhuman body standing beneath a 380 kv three phase high voltageoverhead transmission line, using the perturbed electric eld dis-tribution determined by CSM coupled with GAs.

The CSM calculation procedure described here offers a conve-nient and simple method for estimating induced elds and cur-rents. The particular strength of CSM in this application is its abilityto allow a detailed representation of the shape and posture of thehuman body.

Estimates of the electrode proximity and wall effects in theexperimental geometry are derived. The calculated induced eldand current in grounded and ungrounded humans conforms tothose reported earlier.

Appendix A. Human body modeling

The human body given by Ref. [2] can be divided into ve parts;head, neck, waist, arms, and legs. The head can be represented by ahemi-elliptical sphere at the top and a cylindrical shape for the

remaining of the head. The head can be simulated as ellipticalcharges [31] at the top part, then as ring charges at the remainingpart as shown in Fig. A.1.

Theneck canbe representedbya cylindrical shapewith simulatedring charges.While thewaist canbe representedbycuboid endswitha semi-cylinder at the edges in the direction of y-axis as shown inFig. A.2. This shape simulated by segment ring charges [17] at itssemi-cylinder edges and two nite line charges in parallel to y-axis.

The legs are divided into two parts; top part represented ascylindrical shape, and bottom part represented as truncated cone.The top part is simulated with xed ring charges diameters, and thebottom part is simulated with graduated ring charges diameters,see Fig. A.3.

The arm can be represented by inclined cylindrical shape withsimulated inclined vertical nite line charges, see Fig. A.3.

The ground surface in Fig. A.4 was represented by an inniteplane, while the transmission line was represented by innite lineof charge [2,7,12].

The potential calculated at the contour points chosen on thestressed transmission line is equal to the applied voltage V, whilethe potential calculated at the contour points chosen on the humanbody is equal to zero for a grounded body and for ungrounded bodythe summation of the ctitious charges in the human body must beequal zero [12].

Fig. A.1. Representation of the human head and different charges distribution.

wai

M. Talaat / Journal of Electrostatics 72 (2014) 387e395392Fig. A.2. Representation of the humanFig. A.3. Representation of the human arm and leg with dst with different charges distribution.ifferent charges distribution and transformation axes.

trostM. Talaat / Journal of ElecAppendix B. Potential calculation

The calculated potential, f, at an arbitrary point ith, wherei 1;2;M and,M is the number of contour points, on the powertransmission lines and the human surface body is given by;

f1 Xnj1

PijQj|{z}infinite line

f2 Xnn1

jn1PijQj|{z}

ring charges

f3 Xnn1n2

jnn11PijQj|{z}

inclined vertical finite lines

given by,Fig. A.4. Boundary conditions on potential with idealized point electrode.aj f1 ai

bj f1 biThe axial location of the jth ring charge along the z-axis in cy-

lindrical surface and truncated cone located at arms and legs isdetermined by [2,4,7] as,

zj zj1 n1 1 r l2 r

where, r is the radius which depends on the human simulation partaccording to dimension shown in Fig. 1.

The radius of any ring charge in cylindrical surface and trun-cated cone located at arms and legs is determined by

rj f2 rThe crossover operator used in GAs is the one-point crossover

with a crossover probability of 90%. As for the mutation operator(mutation probability: 10%), the Gaussian mutation operator isapplied for genes. The Gaussian mutation operator adds a randomnumber in a randomly selected gene. The objective function issimply the accumulated squared error. The convergence criterion issatised when the tness of the best solution found so far is lessf4 Xnn1n2n3

jnn1n21PijQj|{z}

elliptical charges

f5 Xnn1n2n3n4

jnn1n2n31PijQj|{z}

segment ring charges

f6 Xnn1n2n3n4n5

jnn1n2n3n41PijQj|{z}

horizontal finite line charges

Also n is the number of the innite line charge and its image fortwin bundle, n1 is the number of simulated ring charges located atthe head, neck, and legs, n2 is the number of simulated inclinedvertical nite line charges located at arms, n3 is the number ofsimulated elliptical charges located at head, n4 is the number ofsimulated segment ring charges located at waist, n5 is the numberof simulated horizontal nite line charges located at waist.

Appendix C. Genetic algorithms calculation

The axial location of the jth elliptical charges along the z-axislocated at the hemi-elliptical sphere part of the head dened by ithlocations is given by,

zj zj1 n3 1 x2

a2i y

2

b2i

! l1

x2

a2i y

2

b2i

!

where, a, b are the major and minor semi-axes of the ellipticalshape and themajor andminor semi-axes of the elliptical charges is

atics 72 (2014) 387e395 393than 1% away from the mean tness of the population in a speciciteration of the algorithm.

Ez2 Xnn1

jn1fzijQj

(2006) 81e95.

trost|{z}ring charges

Ez3 Xnn1n2

jnn11fzijQj|{z}The number of generations used is 100, the population size is 10.The optimum values of parameter subject to the satisfaction of theobjective function given by Equation (3) for grounded human bodyand Equations (4) and (5) for ungrounded human body, using GAswere n 2 2, n 1 120, n 2 20, n 3 5, n 4 20, n5 2 20,l 1 1.0025, f 1 0.0157, l 2 0.056, and f2 0.00215.

Appendix D. Field calculation

The electric eld components Eri and Ezi , can be calculated as:

Er1 Xnj1

frijQj|{z}infinte line

Er2 Xnn1

jn1frijQj|{z}

ring charges

Er3 Xnn1n2

jnn11frijQj|{z}

inclined vertical finite lines

Er4 Xnn1n2n3

jnn1n21frijQj|{z}

elliptical charges

Er5 Xnn1n2n3n4

jnn1n2n31frijQj|{z}

segment ring charges

Er6 Xnn1n2n3n4n5

jnn1n2n3n41frijQj|{z}

horizontal finite line charges

Ez1 Xnj1

fzijQj|{z}infinte line

M. Talaat / Journal of Elec394inclined vertical finite lines[11] Michael E. Potter, Michal Okoniewski, Maria A. Stuchly, Low frequency nitedifference time domain (FDTD) for modeling of induced Fields in humansclose to line sources, J. Comput. Phys. 162 (2000) 82e103.

[12] M. Abdel-Salam, M. Abdallah, Transmission-line electric eld induction inhumans using charge simulation method, IEEE Trans. Biomed. Eng. 42 (11)(1995) 1105e1109.

[13] M. Bencsik, R. Bowtell, R. Bowley, Electric elds induced in the human body bytime-varying magnetic eld gradients MRI: numerical. calculations and cor-relation. analysis, J. Appl. Phys. Phys. Med. Biol. 52 (2007) 2337e2353.

[14] Dragan Poljak, Youssef F. Rashed, The boundary element modelling of thehuman body exposed to the ELF electromagnetic elds, Eng. Anal. Bound.Elem. 26 (2002) 871e875.

[15] J.D. Tranen, G.L. Wilson, Electrostatically induced voltages and currents onconducting objects under EHV transmission lines, IEEE Trans. PAS-90 (2)(1971) 768e776.

[16] H.C. Barnes, A.J. McElroy, J.H. Charkow, Rational analysis of electric elds inlive working, IEEE Trans. PAS-86 (1967) 482e492.

[17] R.J. Spiegel, High-voltage electric eld coupling to humans using momentmethod techniques, IEEE Trans. Biomed. Eng. 24 (5) (1977) 466e472.

[18] M.M. El Bahy, S.A. Ward, R. Morsi, M. Badawi, Onset voltage of a particle-initiated negative corona in a co-axial cylindrical conguration, J. Phys. D.Appl. Phys. 46 (2013) 1e10.

[19] A. El-Zein, M.M. El Bahy, M. Talaat, A simulation model for electrical tree insolid insulation using CSM coupled with GAs, in: IEEE Annual Report Con-ference CEIDP 2008, 2008, pp. 645e649.

[20] D. Utmischi, Charge substitution method for three-dimensional high voltageEz4 Xnn1n2n3

jnn1n21fzijQj|{z}

elliptical charges

Ez5 Xnn1n2n3n4

jnn1n2n31fzijQj|{z}

segment ring charges

Ez6 Xnn1n2n3n4n5

jnn1n2n3n41fzijQj|{z}

horizontal finite line charges

where, frij and fzij are the r and z eld coefcients of the charge Qjcalculated at the ith contour point. The value of frij represent theequivalent vector of fxij and fyij .

References

[1] M. Talaat, A. El-Zein, A numerical model of streamlines in coplanar electrodesinduced by non-uniform electric eld, J. Electrost. 71 (3) (2013) 312e318.

[2] M. Talaat, Calculation of electrostatically induced eld in humans subjected tohigh voltage transmission lines, Electr. Power Syst. Res. 108 (2014) 124e133.

[3] A. El-Zein, M. Talaat, M.M. El Bahy, A numerical model of electrical tree growthin solid insulation, IEEE Trans. Dielectr. Electr. Ins. 16 (6) (2009) 1724e1734.

[4] S. Masuda, M. Washizu, M. Iwadare, Separation of small particles suspended inliquid by nonuniform traveling eld, IEEE Trans. Ind. Appl. IA-23 (1987)474e480.

[5] M.M. El-Bahy, M. Abouelsaad, N. Abdel-Gawad, M. Badawi, Onset voltage ofnegative corona on stranded conductors, J. Phys. D Appl. Phys. 40 (2007)3094e3101.

[6] M. Talaat, Electric eld simulation along silicone rubber insulators surface, in:Proceedings of the 17th International Symposium on High Voltage Engi-neering, (ISH 2011) Germany. Paper A-22, 2011, pp. 1e6.

[7] M. Talaat, Charge simulation modeling for calculation of electrically inducedhuman body currents, in: IEEE Annual Report Conference on Electrical Insu-lation and Dielectric Phenomena CEIDP, 2010, pp. 644e647.

[8] Osamu Fujiwara, Takanori Ikawa, Numerical calculation of human-bodycapacitance by surface charge method, Electron. Commun. Jpn. Part 1 85(12) (2002) 38e44.

[9] J.H. Pickles, Monte-Carlo calculation of eletrically induced human-body cur-rents, IEE Sci. Proc. 134 (9) (1987) 705e711. Pt. A.

[10] Andreas Barchanski, Markus Clemens, Herbert De Gersem, Thomas Weiland,Efcient calculation of current densities in the human body induced byarbitrarily shaped, low-frequency magnetic eld sources, J. Comput. Phys. 214

atics 72 (2014) 387e395elds, in: 3rd International Symposium on High Voltage Engineering (ISH),Paper Number 11.01, Milan, Italy, 1979.

[21] N.H. Malik, A review of the charge simulation method and its applications,IEEE Trans. Electr. Insul. 24 (1989) 3e20.

[22] Electric Power Research Institute, Transmission Line Reference Book 345 kVand Above, Fred Weidner & Son, New York, 1975, pp. 248e254.

[23] T. Matsumoto, Measurement for Human Exposure to AC Electric Fields, ISH,Braunschweig, Germany, 1987. Paper number 33.21.

[24] Aydn Tozeren, Human Body Dynamics: Classical Mechanics and HumanMovement, Springer-Verlag, New York, Inc., 2000.

[25] S. Forrest, Genetic algorithms: principles of natural selection applied tocomputation, Science 261 (5123) (1993) 872e878.

[26] Ronald J. Spiegel, High-voltage electric eld coupling to humans usingmoment method technique, IEEE Trans. Biomed. Eng. BME-24 (5) (1977)466e472.

[27] N.M. Maalej, C.A. Belhadj, External and internal electromagnetic exposures ofworkers near high voltage power lines, Prog. Electromagn. Res. C 19 (2011)191e205.

[28] O.P. Gandhi, J.Y. Chen, Numerical dosimetry at power line frequencies usinganatomically-based models, Bioelectromagnetics (Suppl. 1) (1992) 43e60.

[29] Don W. Deno, Currents induced in the human body by high voltage trans-mission line electric eld-measurement and calculation of distribution anddose, IEEE Trans. Power App. Syst. 96 (5) (1997) 1517e1527.

[30] Abdel-Salam Haz Hamza, Evaluation and measurement of magnetic eldexposure over human body near EHV transmission lines, Electr. Power Syst.Res. 74 (2005) 105e118.

[31] Miguel A. Furman, Compact complex expressions for the electric eld of 2-Delliptical charge distributions, Am. J. Phys. 62 (12) (1994) 1134e1140.

M. Talaat / Journal of Electrostatics 72 (2014) 387e395 395

Calculation of electric and magnetic induced fields in humans subjected to electric power lines1 Introduction2 Method of analysis2.1 Charge simulation methods2.2 Genetic algorithms2.3 Electric field simulation2.4 Induced electric field, magnetic field, and current calculation

3 Results and discussion3.1 Field calculation3.2 Charge simulation method3.3 Induced field and current calculation

4 ConclusionsAppendix A Human body modelingAppendix B Potential calculationAppendix C Genetic algorithms calculationAppendix D Field calculationReferences