Journal of Electrostatics 72 (2014) 387e395Contents lists available at ScienceDirect Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat Calculation of electric and magnetic induced fields in humans subjected to electric power lines M. Talaat Electrical Power & Machines Department, Faculty of Engineering, Zagazig University, Egypt a r t i c l e i n f o a b s t r a c t Article history: Received 11 March 2013 Received in revised form 2 June 2014 Accepted 27 June 2014 Available online 11 July 2014 In this work, analysis of the human body exposed to high voltage electric and magnetic fields is presented. The distribution of the electric field is obtained by using Laplace's equation. This relates the surface charge induced on the body to the potential in a reciprocal Laplace problem, which is then calculated by charge simulation method coupled with genetic algorithms to determine the appropriate arrangement of simulating charges inside the human body. The magnetic field intensity along the vertical center line of the human is calculated. Exposure to external electric and magnetic fields at power frequency induces electric field, magnetic field and currents inside the human body. The presented model for simulating electric and magnetic fields are a three dimensional field problem and introduced different types of charges to simulate the different elementary geometrical shapes of human body. The particular strength of the charge simulation method in this application is its ability to allow a detailed representation of the shape and posture of the human body. The results have been assessed through comparison induced current, electric field, magnetic field and there distribution over the body surface, as estimated in other experimental and computational work. © 2014 Elsevier B.V. All rights reserved. Keywords: Magnetic field calculation Electric field simulation Induced fields and current Charge simulation method Genetic algorithms Human subject to electric power lines 1. Introduction The numerical analysis of electromagnetic field plays an important role in the understanding of electrical phenomena such as; flow in electrolytic solutions [1], exposures to high voltage power lines [2], treeing in solids [3], electrification and streamers in liquids [4], streamers in gases [5], and the design of high voltage insulation [6]. Numerical methods, such as finite element method (FEM) [1,6], charge simulation method (CSM) [2,3,7], charge density [8], Monte-Carlo method (MCM) [9], finite difference method (FDM) [10,11], and integral equation methods have been used to simulate the non-uniform electric fields. CSM is one of the most successful numerical methods used for solving electromagnetic field problems [2,3,5,7]. The interaction of electric and magnetic fields with humans has initiated public concern, due to the overlap between the power transmission lines and the settlement areas which lie very near or under the power transmission lines [2,7,12e15]. There has been a growing interest in determining the safe exposure level of humans to power frequency electric and magnetic fields [2,7,12,15]. Therefore, the simulation of electric and magnetic fields, in the space E-mail addresses:
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[email protected]. http://dx.doi.org/10.1016/j.elstat.2014.06.008 0304-3886/© 2014 Elsevier B.V. All rights reserved. between power lines and ground, is a prerequisite to assess the effect of power lines on human. The calculation of the induced electric and magnetic fields in human lead to substantial difficulty, due to the complex geometry of the human body. For this reason, several approximate solutions have been derived using CSM [2,7,12,15], MCM [9], FDM [10,11,16], moment method techniques [17], FEM [13], and boundary element method (BEM) [14]. This paper presented a three-dimension electromagnetic field simulation. CSM and the method of image are used for the electric and magnetic fields simulation in the human body. The electric and magnetic fields distribution are obtained from Laplace's equation by treating the human body as a good conducting medium. The surface charges on human body are simulated by a number of charges arranged inside the human body, such as ring charges, finite line charges [2,7]. In this model the simulated electric and magnetic fields is introduced different types of charges such as elliptical charges and segment ring charges [18], taking into consideration the different elementary geometrical shapes of human body. The optimum number, values, locations, and dimensions of these charges are achieved by using genetic algorithms (GAs) as a search optimization technique [2,4,7,19]. Series of vertical and inclined line charges [4,7,20] especially in the arms and unsymmetrical ring charges [21] especially in the legs. For these inclined and (GAs). GAs are used in the optimization of a variety of variables. In this paper the actual electric field is simulated by a number of discrete charges located in. Table 1 Tissue conductivity and permittivity values [12]. [2] was used to simulate the human body using CSM. are a form of evaluation that occurs on a computer. f. used to obtain the axial location of different types of charges along the human height. human body. used to find the required number of simulated charges for each part of the human body and the transmission lines.11 0. most of the high voltage systems are complex so numerical techniques are used to solve this problems. n. crossover and mutation until some termination criteria are reached [25]. U¼ Mn X Qi ¼ 0:0 (4) i¼1 and for transmission lines the objective function is still the accumulated squared error but with only the simulated line charges.3.558 5259 145. From this table the large conductivity and the large relative equivalent dielectric constant of the human body cause the external power frequency electric field near the human body to be perpendicular to the surface [23]. One of the most efficient end accurate numerical techniques for field computations is the CSM.1. These variables include the optimum number of charges.550 2. GAs are a search method that can be used for solving problems and modeling evolutionary systems. The CSM in this application has the ability to allow a detailed representation of the shape and posture of the human body for grounded and an ungrounded case. k ¼ 2 for neck. y. The estimated values of fk are presented in Appendix B.86 0. surface charges on the high voltage line conductors are simulated by infinite line of charges located at each line axis [2.4. The used objective function used by GAs is simply the accumulated squared error. selection.11 0. used for indicating the optimum radius of ring and segment ring of charges.04 0. The various conductivity and relative equivalent dielectric constant of human Tissue [13] is given in Table 1. can only be obtained for relatively simple charge distributions and conductor configurations. 2.2. It consists of replacing the actual continuous surface charge distribution of the conductors by a discrete set of fictitious charge distribution placed inside the volumes occupied by the conductors.6. see Appendix C. (transmission lines. The ungrounded human body new objective function is simply the summation of factitious simulated charges inside the human body must be equal zero. Fig.4. k ¼ 3 for waist.930 12. Once the values of simulation charges are determined.100 85. Tissue Conductivity sðU1 m1 Þ Relative dielectric constant εr Muscle Bone Skin Heart Gland Blood Lung Liver Lens 0. zÞ2 (3) i¼1 where. In this model. l . Finally the optimum radius of simulated ring charges. In the present paper. The model given by Ref.320 1136 352. and then the population is evolved with use of the principles of variation. boxes etc. then the potential and electric field of any point in the region outside the conductor can be calculated using the superposition principle. Electric field simulation The vector of unknown charges Q is computed from the matrix equation: 1 ½Vi Qj ¼ Pij (1) where.18e21]. the average dimension of any human part is given by Ref. a population of individual is created in a computer.12. Genetic algorithms Genetic algorithms.13 0. zÞ2 (5) . Pij is the potential coefficient calculated at the ith boundary point due to the jth simulation charge Qj and V is the applied voltage of transmission line. see Appendix A. First. Talaat / Journal of Electrostatics 72 (2014) 387e395 unsymmetrical charges. a coordinate transformation is performed. these are juxtaposed or superposed as required. k ¼ 4 for arm. see Appendix C. which has the form [21]: U¼ M X ½V fi ðx. 1 represents the schematic diagram of the engineering drawing of the human body with basic dimensions in centimeters as a three-dimension model. the electric and magnetic fields are calculated in the original coordinate system.850 56.04 0. also the optimum radius of any ring charge.6 0. the calculated potential f. For a given charge distribution. Also the optimum location of charges. …M (2) k¼1 where.388 M. fi ¼ 6 X fk i ¼ 1. at an arbitrary point is a summation of the potentials resulting from the individual charges. cylinders. y. k ¼ 6 for lower part of leg).11 434. and earth) [2. Then. However. U¼ n X i¼1 ½V fi ðx. 2. This is why the human body is treated as a conducting body.673 105. [24].5 0. Charge simulation methods Analytical solution of Laplace's equation used for calculating electric field. The basic idea of GAs is very simple. 2.7.6e8]. The simulated charges are distributed uniformly according to shape dimensions except the axial location of the elliptical and ring charges along the z-axis. k ¼ 5 for upper part of leg. The human body is modeled taking a representation of boundary surface as a combination of certain elementary geometrical shapes: spheres. 2. Values of simulation charges are determined by satisfying the boundary conditions at a number of contour points selected at the conductor surfaces. k indicates the human part (k ¼ 1 for head. the optimum location of these charges are determined by GAs. Method of analysis Description of the electric and magnetic fields emanating from various transmissions line configurations have been adequately presented in many papers and texts [22]. For an ungrounded human body each of the transmission lines and the human body have an objective function. The exact positions and values of the simulating charges are found so that the boundary conditions of the particular configurations are satisfied to a certain degree of accuracy. M is the number of contour points. Representation of simulation subject in terms of elementary geometrical shapes. magnetic field. Talaat / Journal of Electrostatics 72 (2014) 387e395 389 Fig. The polarization current density induced by an external field in a homogeneous body is given by Jk ¼ urs ¼ uεo εr En (10) where. probably because of the disparity in magnitude between the electric and magnetic fields. ðE ¼ Vfk Þ. A¼ m 4p Z J k dv . A is the magnetic vector potential. where Er represent the equivalent vector of Ex and Ey. (12).854 1012 F/m. ETL is the normal component of the transmission lines electric field. which obtained from the presented simulation program. Also. where. all dimensions in cm. 1. εo is the permittivity of free space which equal 8. Induced electric field. and εr is the relative permittivity of the human body given in Table 1. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ei ¼ Er2i þ Ez2i (8) rs ¼ εo εr En (9) where. Even though it is known that the dielectric properties of the human body are not totally isotropic at low frequencies they are assumed to be homogeneous for the analysis presented here. see Appendix D. Eri ¼ 6 X Erk (6) k¼1 Ezi ¼ 6 X Ezk (7) k¼1 The value of electric field components Eri and Ezi for any part of the human body is calculated. k indicates the human part. in lieu of the fact that the relative permittivity of tissues is large at low frequencies.M. The first term of Eq. Extension to anisotropic media is straight forward. the electric field components Eri and Ezi at the contour point i are the vector sum of the field contributions from all the simulated charges. Also. The problem is now reduced to the determination of the optimum values of parameter subject to the satisfaction of the objective function given by Equation (3) for grounded human body and Equations (4) and (5) for ungrounded human body. r r (12) (13) . The charge density rs at a boundary point on the human body surface at height z is expressed as.4. and Ein is the normal electric field due to the induced polarization current in the human body given by: 2. The total electric field at the ith contour point is expressed as. and current calculation Ein ¼ Vfk uA Less attention has been paid to the magnetic field. En is the normal component of the electric field calculated at the boundary point. using GAs. the human body represented as a good conductor with ε ¼ εo. can be obtained from the presented simulation program. and the value of A in the second term is given by. The value of the normal component of the electric field at any arbitrary point on the human surface is presented by: En ¼ ETL þ Ein (11) where. and the potential calculated due to the factious simulated charges. 1.1 %).6 18 0.2 -0.04 % Potential error 0. where.8 3.12.2 2 130 1 3 66 0. r r (14) The volume of any part of the human body can be obtained from the integration over area of the human part and its length.1 152 10 1. [26]: 0. the height of the twin bundle conductors of a three phase 380 kV line over the ground plane is 15 m which is about ten times the height of the human body to make sure that the body have a negligible effect on the surface charge on the HV line conductors.1.1 (16) 1 20 40 60 80 100 120 140 160 180 200 Location of contour points along the human body surface ×h/200 Fig.9 20 0. and deviation angle (the deviation in the field angle from the normal position on the conductor surface). Then the magnetic field strength H is expressed as. Field calculation -1 1 20 40 60 80 100 120 140 160 180 200 Location of contour points along the human body surface ×h/200 Fig.6 B¼ Deviation angle in degree 0.3 1. The accuracy of the simulation is satisfied for the potential error. 2 and 3.02 Jk ¼ 0 uεo εr ðE Vfk Þ 1 þ uεo εr Lk TL The induced current IS just outside the boundary of a part of the body. 3.where. Charge simulation method J 7 k dvC A5 . 0 -0. where.390 M. so the last term can be simplified to LkJk. in terms of the potential error. is the position of the observation point and r is the position of the integration point. ak is the radius of the human body part expressed as cylindrical cross section. H ¼ m1 B.7. “human body”.6 3. as shown.04 Z -0.02 -0. 2. the unknown induced current density Jk. Diameters and spacing between sub-conductors are 27. Table 2 Induced current in grounded and ungrounded human body.06 ISk ¼ -0. in Figs. is expressed as. 0. is obtained by integrating Jk over the surface area of this part -0. um 1 dsdl ∬ 4p r . The variation of the per cent potential errors along the human body.4 2Jk usak (18) 0. Position Grounded human body Induced current (mA) Top of head Middle of neck Middle of waist Middle of legs Ungrounded human body Induced electric field (mV/m) Induced magnetic field (mT) Induced current (mA) Induced electric field (mV/m) Induced magnetic field (mT) 18 0.2 where. r (15) The boundary conditions are checked over 200 points along the human height. The variation of the deviation angle along the human body.4 -0.6 35 2.08 -0. V.2.56 .Substitution of Equations (11)e(13) into Equation (10) yields to 2 0 6 Bum Jk ¼ uεo εr 4ETL Vfk @ 4p Z 13 3. Results and discussion -0.18e21] over the human body. Lk ¼ In order to demonstrate the proposed approach. (not more than 1 degree) [2. The induced magnetic field is evaluated based on Faraday's law applied to a cylindrical cross section.7 mm and 400 mm respectively. (not more than 0.29 1. as a human body with circumferential currents estimated by Ref. Talaat / Journal of Electrostatics 72 (2014) 387e395 At the boundary point. f).6 2.6 38 3 2.06 0. Jk dSk (17) The area Sk of each part of human can be mathematically calculated from dimension given in Fig. h.4. and the field deviation angle.9 1. (the difference between the actual conductor voltage. Skth. m is the permeability of the medium (for human body m ¼ mo ¼ 4p 107 H/m). it becomes increasingly important to describe accurately the power line electromagnetic field interaction with life forms.8 Human height above feet (m) Fig.4 1.2 Satisfaction of the boundary points at the chosen contour points results in a set of equations whose solution determines the charges simulating the body.8 1 1. 4. the agreement can be observed between the measured values and the obtained from simulation program. the induced charge. Figs. Induced field and current calculation 0. Fig. Induced current distributions for ungrounded and grounded human body. Induced magnetic field distributions for ungrounded and grounded human body. from actual field measurements.6 1. and magnetic field distribution at the surface of a person standing in a 60 Hz perturbed field for grounded and ungrounded bodies.2 1. Actual measured magnetic field variation over human body for 500 kV power lines [30]. 8. Conclusions With the advent of high voltage power lines. and current at the surface of the human body are determined as given in Table 2 for grounded human body.4 2.5 Induced magnetic field (µT) Human height above feet (m) 1.6 0. Talaat / Journal of Electrostatics 72 (2014) 387e395 1.6 1.M. electric field. the agreement can be considered very acceptable.4 0.5 Grounded Ungrounded 1 0. 4. Fig. 7.8 3 Grounded Ungrounded 1. Fig. 3. these values are in accordance with that given by [12. 5. 0.77 m in height standing in a vertical homogeneous electric field of 10 kV/m at 50 Hz.27e30]. Fig.8 0. under different actual 220 and 500 kV power transmission lines [30]. Computed (Gandhi & Chen) [28] and measured (Deno) [29] current distribution for an ungrounded and grounded human of 1. Once the simulation charges are determined. 6. Fig. which are obtained. Actual measured magnetic field variation over human body for 220 kV power lines [30]. 5 gives a comparison of the proposed model with two different models [28. the electric field.3. .2 0 20 391 2 1.4 Fig. 6 explains the calculated values of the induced magnetic field distribution for ungrounded and grounded human. Computed induced current distributions for ungrounded and grounded human of 1. 4.29]. 7 and 8 give a comparison of the proposed model with magnetic field values.2 1 0. Table 2 gives the induced current.8 m height standing in a vertical homogeneous electric field are illustrated in Fig.6 0.5 40 60 80 100 120 140 160 Induced current along the human body (µA) 0 0 0. Talaat / Journal of Electrostatics 72 (2014) 387e395 This paper develops a simulation method for current. A. Human body modeling The human body given by Ref. normal electric field and magnetic field distribution induced on humans situated in the vicinity of the power lines. and the bottom part is simulated with graduated ring charges diameters. The neck can be represented by a cylindrical shape with simulated ring charges. then as ring charges at the remaining part as shown in Fig. The technique is based on the charge simulation method and Laplace's equation to compute the external electric field. top part represented as cylindrical shape. and legs. A. [2] can be divided into five parts.3. While the waist can be represented by cuboid ends with a semi-cylinder at the edges in the direction of y-axis as shown in Fig.1. head. Representation of the human waist with different charges distribution. A. Fig. The particular strength of CSM in this application is its ability to allow a detailed representation of the shape and posture of the human body. and bottom part represented as truncated cone. Representation of the human arm and leg with different charges distribution and transformation axes. The calculated induced field and current in grounded and ungrounded humans conforms to those reported earlier.12]. Fig. Estimates of the electrode proximity and wall effects in the experimental geometry are derived. while the transmission line was represented by infinite line of charge [2. The top part is simulated with fixed ring charges diameters. Representation of the human head and different charges distribution.3. while the potential calculated at the contour points chosen on the human body is equal to zero for a grounded body and for ungrounded body the summation of the fictitious charges in the human body must be equal zero [12]. The ground surface in Fig.392 M. A. The CSM calculation procedure described here offers a convenient and simple method for estimating induced fields and currents. waist. A.7. neck.3. . The legs are divided into two parts. A. The head can be represented by a hemi-elliptical sphere at the top and a cylindrical shape for the remaining of the head. Fig. see Fig. see Fig. induced charges and currents in a grounded and ungrounded human body standing beneath a 380 kv three phase high voltage overhead transmission line. The potential calculated at the contour points chosen on the stressed transmission line is equal to the applied voltage V. using the perturbed electric field distribution determined by CSM coupled with GAs. Appendix A. magnetic field.2. The arm can be represented by inclined cylindrical shape with simulated inclined vertical finite line charges.4 was represented by an infinite plane. arms.1. The head can be simulated as elliptical charges [31] at the top part.2. A. A. This shape simulated by segment ring charges [17] at its semi-cylinder edges and two finite line charges in parallel to y-axis. The convergence criterion is satisfied when the fitness of the best solution found so far is less than 1% away from the mean fitness of the population in a specific iteration of the algorithm. the Gaussian mutation operator is applied for genes.4. zj ¼ zj1 þ ðn3 1Þ Fig. As for the mutation operator (mutation probability: 10%). on the power transmission lines and the human surface body is given by. The axial location of the jth ring charge along the z-axis in cylindrical surface and truncated cone located at arms and legs is determined by [2. aj ¼ f1 ai Appendix B. n3 is the number of simulated elliptical charges located at head. Talaat / Journal of Electrostatics 72 (2014) 387e395 f4 ¼ nþn1X þn2 þn3 393 Pij Qj j¼nþn1 þn2 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} elliptical f5 ¼ charges nþn1 þn 2 þn3 þn4 X Pij Qj j¼nþn1 þn2 þn3 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} segment f6 ¼ ring charges nþn1 þn2X þn3 þn4 þn5 Pij Qj j¼nþn1 þn2 þn3 þn4 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} horizontal finite line charges Also n is the number of the infinite line charge and its image for twin bundle.7] as.4. n5 is the number of simulated horizontal finite line charges located at waist.M. Genetic algorithms calculation The axial location of the jth elliptical charges along the z-axis located at the hemi-elliptical sphere part of the head defined by ith locations is given by. Boundary conditions on potential with idealized point electrode. n4 is the number of simulated segment ring charges located at waist. neck. The objective function is simply the accumulated squared error. Appendix C. where i ¼ 1. x2 y2 þ 2 a2i bi ! þ l1 x2 y2 þ 2 a2i bi ! where. r is the radius which depends on the human simulation part according to dimension shown in Fig. 1. . zj ¼ zj1 þ ðn1 1Þ r þ l2 r f1 ¼ n X Pij Qj where. at an arbitrary point ith. The radius of any ring charge in cylindrical surface and truncated cone located at arms and legs is determined by j¼1 |fflfflfflfflffl{zfflfflfflfflffl} infinite line f2 ¼ rj ¼ f2 r nþn X1 Pij Qj j¼nþ1 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ring charges nþn 1 þn2 X f3 ¼ Pij Qj j¼nþn1 þ1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} inclined vertical finite lines The crossover operator used in GAs is the one-point crossover with a crossover probability of 90%. b are the major and minor semi-axes of the elliptical shape and the major and minor semi-axes of the elliptical charges is given by. n2 is the number of simulated inclined vertical finite line charges located at arms. a. …M and. A. 2. n1 is the number of simulated ring charges located at the head. and legs. The Gaussian mutation operator adds a random number in a randomly selected gene. M is the number of contour points. Potential calculation bj ¼ f1 bi The calculated potential. f. Appl. Electrost. Talaat / Journal of Electrostatics 72 (2014) 387e395 The number of generations used is 100. line frij Qj j¼nþ1 References ring [1] M. Part 1 85 (12) (2002) 38e44. Talaat. IA-23 (1987) 474e480. Bound. can be calculated as: Er1 ¼ n X nþn X1 where. n 2¼ 20. Appl. [15] J. Transmission-line electric field induction in humans using charge simulation method. Res. 162 (2000) 82e103. Efficient calculation of current densities in the human body induced by arbitrarily shaped. Onset voltage of a particleinitiated negative corona in a co-axial cylindrical configuration. [2] M. Electr. El Bahy. IEEE Trans. n 3¼ 5. J.J. n 1¼ 120. 40 (2007) 3094e3101. A numerical model of electrical tree growth in solid insulation. Bencsik.0157. Badawi.00215.394 M.J. R. Herbert De Gersem. S. El Bahy. frij and fzij are the r and z field coefficients of the charge Qj calculated at the ith contour point. [10] Andreas Barchanski. Anal. Tranen. in: IEEE Annual Report Conference on Electrical Insulation and Dielectric Phenomena CEIDP. McElroy. A. [12] M. Comput. [7] M. IEEE Trans. in: IEEE Annual Report Conference CEIDP 2008. Bowley. Rashed. Phys. 46 (2013) 1e10. Biomed. Stuchly. 2008. Michal Okoniewski. Thomas Weiland. The boundary element modelling of the human body exposed to the ELF electromagnetic fields. A numerical model of streamlines in coplanar electrodes induced by non-uniform electric field. 108 (2014) 124e133. Abouelsaad. in: 3rd International Symposium on High Voltage Engineering (ISH). Electr.A. El-Zein. Wilson. 214 (2006) 81e95. M. f 1¼ 0.L. Charge substitution method for three-dimensional high voltage fields. M. and f2 ¼ 0. Pickles.H. [16] H. Phys. Appl. El-Zein. [13] M. Ind. Washizu. Abdallah. IEEE Trans. [19] A. Phys. [5] M. Abdel-Salam. The optimum values of parameter subject to the satisfaction of the objective function given by Equation (3) for grounded human body and Equations (4) and (5) for ungrounded human body. Phys. Power Syst. 52 (2007) 2337e2353. Talaat. Biomed. A. Masuda. The value of frij represent the equivalent vector of fxij and fyij . [17] R. M. 71 (3) (2013) 312e318. Youssef F. Electron. Potter. 2010. pp. Elem. M. 16 (6) (2009) 1724e1734. G. D. Bowtell. 26 (2002) 871e875. IEEE Trans. Talaat. J. M. M. High-voltage electric field coupling to humans using moment method techniques. pp. J. |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} charges nþn 1 þn2 X Er3 ¼ frij Qj j¼nþn1 þ1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} inclined vertical finite nþn1X þn2 þn3 lines frij Qj j¼nþn1 þn2 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} elliptical charges nþn1 þn 2 þn3 þn4 X frij Qj j¼nþn1 þn2 þn3 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} segment ring Er6 ¼ charges nþn1 þn2X þn3 þn4 þn5 frij Qj j¼nþn1 þn2 þn3 þn4 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} horizontal Ez1 ¼ n X finite line charges fzij Qj j¼1 |fflfflfflfflffl{zfflfflfflfflffl} infinte Ez2 ¼ line nþn X1 fzij Qj j¼nþ1 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ring Ez3 ¼ fzij Qj |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} infinte Er5 ¼ nþn1 þn2X þn3 þn4 þn5 horizontal finite line charges frij Qj j¼1 Er4 ¼ Ez6 ¼ j¼nþn1 þn2 þn3 þn4 þ1 |fflfflfflfflffl{zfflfflfflfflffl} Er2 ¼ segment ring charges charges nþn 1 þn2 X fzij Qj j¼nþn1 þ1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} inclined vertical finite lines .M. IEEE Trans. [14] Dragan Poljak. J. N. Pt. Talaat. Markus Clemens. 1979. [6] M. Eng. using GAs were n ¼ 2 2. Ez4 ¼ nþn1X þn2 þn3 fzij Qj j¼nþn1 þn2 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} elliptical Ez5 ¼ charges nþn1 þn 2 þn3 þn4 X fzij Qj j¼nþn1 þn2 þn3 þ1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Appendix D. n5 ¼ 2 20. R. D Appl. M. M. 645e649.H. A. [20] D. the population size is 10. Jpn. Med. Charge simulation modeling for calculation of electrically induced human body currents. El-Zein. Paper Number 11. Dielectr. Separation of small particles suspended in liquid by nonuniform traveling field. Spiegel. Phys. Charkow.056. Calculation of electrostatically induced field in humans subjected to high voltage transmission lines. Phys. Commun. [18] M. El-Bahy.01. R. n 4¼ 20. J. El Bahy. Monte-Carlo calculation of eletrically induced human-body currents. Badawi. 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