APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM

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1 Journl of ELECTRICAL ENGINEERING, VOL. 6, NO. 5, 0, APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM Alenk M. MILOVANOVIĆ Miroslv M. BJEKIĆ In this pper clcultions of the cpcitnce per unit length of one or multilyer dielectric lines re presented. Specil ttention is given to the clcultions of the cpcitnce per unit length of lines with rectngulr cross sections, whose electrodes my be in different or the sme lyers of two lyer dielectric line. For the purpose of performing the bove, severl numericl methods re used nd simple pproximte expressions re proposed. K e y w o r d s: lines with rectngulr cross section, multilyer medium, numericl methods, cpcitnce INTRODUCTION The problem of clcultion of the cpcitnce per unit length of lines with multilyer medium hs been evidenced in both theory nd prctice. When designing lines nd cbles with multilyer medium of different geometry, one should often be well cquinted with most ccurte vlues of the cpcitnce per unit length. Clcultion of the cpcitnce per unit length of lines with multilyer medium cn be performed using vrious nlyticl nd numericl methods such us the Chrge Simultion Method (CSM), Finite Element Method (FEM), Equivlent Electrode Method (EEM) [ 7], etc. All these methods, offering different degrees of precision, give results with stisfying ccurcy, but lso require extensive mthemticl work. This is serious drwbck nd difficulty tht engineers encounter in prctice. They re commonly very restricted in terms of time nd conditions for comprehensive numericl clcultions. The im of this pper is to provide review of the pplictions of different methods for clcultion of the cpcitnce of multilyer lines with rectngulr cross section nd to propose simple procedure for pproximte, but sufficiently exct, clcultion of cpcitnce per unit length. This review will be lso of gret help to PhD students who cn use this nlysis in reserch s well s in prctice. LINES WITH ONE LAYER MEDIUM The cpcitnce per unit length of lines with one lyer perfect dielectric medium is proportionl to the permittivity nd cn be expressed in generl s C = g ε, () where g is the coefficient of proportionlity which depends on the shpe, dimension nd mutul position of the electrodes. Fig.. b The squre coxil line In the cse of the squre coxil line, Fig., g = 8K(k) / K(k ), () where K(k) is the complete elliptic integrl of the first kind with modulus k, ( p p ) pp k = = p + p + pp, (3) nd complementry modulus k, k = k [7 9]. p is the connection with geometry of electrodes, K(p) K(p ) = /b + /b, p = p. (4) In order to clculte the pproximte vlues of the rtio K(k)/K(k ), Fig., the following simple formul cn be pplied [0] K(k) K(k ) ( π ln + k + 4 4k + k 4 4k ). (5) Besides expression () for the clcultion of g, the expression given in [] cn lso be used b g /b, 0.5 b 0.5, ( 6.33 ), b 0.5. (6) ln b University of Krgujevc, Technicl Fculty of Cck, Svetog Sve 65, Cck 3000, Serbi, Deprtment of Electricl Engineering, lenk@tfc.kg.c.rs; Deprtment of Power Engineering DOI: 0.478/v , ISSN c 0 FEI STU

2 50 A.M. Milovnović M.M. Bjekić: APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM K(k) / K(k ) In the cse of the coxil line with circulr inner conductor nd squre outer conductor, Fig. 3, the coefficient of proportionlity is [] g π ln ( ).079. (7) r Fig.. The rtio K(k)/K(k ).0 k The vlues of the coefficient of the proportionlity, g, in the cse of two wire line with rectngulr cross section, Fig. 4, for different rtios of d/ re presented in Tble nd Fig. 5. r Fig. 3. Coxil line with circulr inner conductor nd squre outer conductor Fig. 5. The cpcitnce per unit length of the line from Fig. 4, for different rtio d/ Electrode d Electrode Fig. 4. Two wire line with squre cross section Tble. The comprison of the results for coefficient of proportionlity, in the cse of squre coxil line, Fig., for different rtios /b /b g /b /b g 8 K(k) K(k ) /b g (MEE) g (FEM) Although pproximte, expressions () nd (6) do give stisfctory results, nd this cn be seen when they re compred with results obtined by other, numericl methods which give results with high precision. For tht purpose results were obtined by using Equivlent Electrode Method (Appendix A) nd Finite Element Method (Softwre pckge Femlb), Tble. The presented results re obtined using the Chrge Simultion Method [], Finite Element Method (Softwre pckge Femlb), Modified Equivlent Source Method (MESM) [5] nd pproximte nlyticl expression (8) which cn be used with stisfying ccurcy when d [], g ln π π ] [ ], d. (8) πd + ln + [ π(d+) After the cpcitnce is determined, dmittnce per unit length of the line hving one lyer perfect dielectric medium is Y = jωc = jωεg. (9) In the imperfect, liner medium with permittivity ε nd conductivity σ not only conductive currents, but lso displcement currents cn flow [6], so the first Mxwell s eqution cn be expressed s rot H = (σ + jωε) E = σ E = jωε E. (0) where: σ is complex conductivity, σ = σ + jωε; ε is complex permittivity, ε = ε ( ) j ωc ω nd ωc = σ/ε is ngulr frequency when densities of conductive current nd displcement current re equl. E nd H re the complex vectors of electricl nd mgnetic field strength.

3 Journl of ELECTRICAL ENGINEERING 6, NO. 5, 0 5 Tble. The comprison of the results for cpcitnce per unit length in the cse of the line presented in Fig. 4 d/ C /ε C /ε C /ε C d/ /ε C /ε C /ε [5] (CSM) (FEM) [5] (CSM) (Eq. 8) If the rel vlue of permittivity in (9) is replced with the complex vlue, the dmitnce per unit length of the line hving one lyer imperfect dielectric medium is Y = jωεg = σg = G + jωc, () The suggested expression () is exct when the surfce of seprtion S is of uniform potentil. However, it is useful s very good pproximtion in the cse when the potentil on the surfce of seprtion is not uniform. The effective permittivity cn be clculted s where G = σc /ε = ω c C is conductnce. ε e = C g 3 (4) 3 LINES WITH TWO LAYER MEDIUM In the cse of the line with two lyer perfect dielectric medium, Figs. 6 nd 6b (the electrodes re in different lyers of multilyer dielectric line), the cpcitnce per unit length cn be determined by using pproximte expression (), Appendix B, C = (ε + ε )g 3 + ε ε ( ε + ε ε g 3 ) ε g, () where g nd g 3 re coefficients of the proportionlity of the lines which re formed by the existing electrode nd the electrode s shield coinciding with seprtion surfce S, nd g 3 is coefficient of proportionlity of the line when ε = ε, C ε = ε g 3, C ε = ε g nd C ε=ε = ε g 3. (3) nd depends on the electric chrcteristics of the existing lyers, conductor s shpe nd their mutul position. Exceptions re lines hving electrodes symmetric in reltion to the surfce of seprtion of dielectric lyers, where g = g 3 = g 3 nd = ( + ). (5) ε e ε ε As lyers inside the line re imperfect, the dmittnce per unit length, Y, is pproximtely Y = (σ + σ )g 3 + σ σ ( σ + σ σ g 3 ) σ g, (6) where: Y = G e + jωc e, G e = σ e g 3, C e = ε e g 3, σ = σ + jωε nd σ = σ + jωε. σ e nd ε e re effective permittivity nd effective conductivity, respectively. If the lines with two lyer imperfect dielectric medium re treted s lines with one lyer imperfect dielectric medium, complex effective conductivity is σ e = Y g 3 = G e + jωc e g 3 = σ e + jωε e, (7) Electrode Electrode Electrode ccordingly where σ + σ σ e = α + jβ, (8) () S Electrcode S (b) ε e = α(ε + ε ) + β (σ + σ ) (α + β ), β = β ω, (9) Fig. 6. Coxil line with: () circulr inner conductor, nd (b) squre outer conductor σ e = α(σ + σ ) + ωβ(ε + ε ) (α + β ), (0)

4 5 A.M. Milovnović M.M. Bjekić: APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM Electrode Electrode When the pproximte cpcitnce is clculted using (4), it is possible to determine the effective permittivity s ε e = C ε (ε + ε ) g = ε + (ε ε ) ( ) g /g. (5) S As the lyers of the lines re imperfect the dmittnce per unit length is pproximtely Fig. 7. The line with two lyer perfect dielectric medium, where electrodes re in the sme lyer Y = (σ + σ )g + σ σ σ + σ σ g, (6) e / / = 0. / = 0. / = 0.5 / = / = / = 0 Y = G e + jωc e, G e = σ e g, C e = ε e g, (7) σ = σ + jωε, σ = σ + jωε. In this cse dielectric complex effective conductivity is: σ e = Y σ g = (σ + σ ) σ + (σ σ ) ( ) g /g. (8) 4 THE LINE WITH THREE LAYER MEDIUM g / g Fig. 8. Effective permittivity of the line where electrodes re in the sme lyer Electrode Electrode S S 4 S S 3 Fig. 9. The line with perfect three lyer dielectric medium α = γ + γ + ω ω + ω [ ε ε γ ω + ω + ε ε γ ω + ω ], () β = ω ω [ ε γ ε ω + ε γ ] ω ε ω + ω, () ω ω = σ ε, ω = σ ε, γ = g 3 g nd γ = g 3 g 3. (3) In the cse when two lyer perfect dielectric medium exists, but the electrodes re in the sme lyer, Fig. 7, the cpcitnce per unit length cn be determined by using pproximte expression (4), C = (ε + ε )g + ε ε ε + ε ε g, (4) where g nd g re coefficients of proportionlity C ε = ε g nd C ε =ε = ε g. In the cse of the line with three lyer perfect dielectric medium, such s two wire line hving rectngulr cross section (Fig. 9), the pproximte expression for clcultion of the cpcitnce per unit length nd effective permittivity re C = (ε + ε )g 4 + ε ε [ ε + ε ε g 3 nd ( ε g + g 34 )] (9) ε e = C /g 4, (30) where g 4 nd g 3 re determined for the line composed by conductors defined by surfces S S 4, respectively S S 3 nd g, g 34 for the coxil line defined by surfces S S, respectively S 3 S 4. 5 EXAMPLES The ppliction of the proposed pproximte expressions (), (4), nd (9) will be illustrted through severl exmples nd the results obtined will be compred with results obtined by using different numericl techniques. Exmple. Results for effective permittivity of squre coxil line with two lyer perfect dielectric medium, Fig. 0, nd three lyer perfect dielectric medium Fig. 0b, obtined using proposed pproximte expressions () nd (9) re shown in Figs. nd. Some of the results obtined re compred with results obtined using FEM (Softwre pckge Femlb) in Tble 3. Coefficients of proportionlity g, g 3, g 3 in the cse of the line from Fig. 0 nd g, g 3, g 34, g 4 in the cse of the line from Fig. 0b re determined using expression (). The greement of the results is very good.

5 Journl of ELECTRICAL ENGINEERING 6, NO. 5, b c () (b) Fig. 0. Squre coxil line with multilyer dielectric medium Exmple. The dependency of effective permittivity of rectngulr coxil line with two lyer perfect dielectric medium ginst the rtio ε /ε for the different rtios /b, /b = /b = /b nd / = / = is shown in Fig. 3. The coefficients of proportionlity g, g 3 nd g 3 re determined using EEM. Also, the coefficients of proportionlity cn be determined using the nlyticl expressions given in []. For exmple, for the rectngulr coxil line presented in Fig. 4 expression (3) is suggested g = π ln +w/d b/d+t/d, d > 3t, w >.5b. (3) b b b e /.5 Fig.. Effective permittivity of squre coxil line with two lyer perfect dielectric medium (Fig. 0) for different rtios b/c nd c = = 0 b = 4b = b / Fig. 3. Effective permittivity of the rectngulr coxil line with two lyer perfect dielectric medium 0 Fig.. Effective permittivity of squre coxil line with three lyer dielectric medium (Fig. 0b) for different rtios 3 / nd / =, 4 / = 4 Tble 3. The comprison of the results for effective permittivity of the coxil line, Fig. 0b, when 4 / 3 = 3 / = / = ε /ε ε e /ε (FEM) ε e /ε (Eq.9 nd 30) Fig. 4. Rectngulr coxil line Exmple 3. For coxil line hving circulr inner conductor nd squre outer conductor (Fig. 5) coefficients of proportionlity g 3 nd g 3 re determined using expression (7) nd g = π ln ( r /r ). The results for effective permittivity of the coxil line obtined using expressions () nd (4) re compred with results obtined using FEM [Softwre pckge FEMM] in Tble 4.

6 54 A.M. Milovnović M.M. Bjekić: APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM Tble 4. Effective permittivity of the coxil line presented in Fig. 5, for different rtios ε /ε nd /r, when /r = 4 ε e /ε ε /ε /r = 8/7 /r = 8/6 /r = 8/5 /r = 8/4 /r = 8/3 Eq. () FEM Eq. () FEM Eq. () FEM Eq. (9) FEM Eq. (9) FEM Tble 5. Effective permittivity of coxil line presented in Fig. 6, for different rtios ε /ε nd /, when /r = 4 Tble 6. Comprison of the results for C /ε, when d / = 8 nd d / = 8.5 ε e /ε ε /ε / = 8/6 / = 8/5 / = 8/4 / = 8/3 Eq. () FEM Eq. () FEM Eq. () FEM Eq. () FEM C ε /ε /ε C /ε Eq. (9) FEM r r g 4 re determined using expression () nd expression (8) respectively. Obtined results re presented in Tble 6 nd Fig. 8. Fig. 5. Coxil line hving circulr inner conductor nd squre outer conductor Exmple 4. For coxil line presented in Fig. 6, coefficients of proportionlity g, g 3 nd g 3 re determined using expressions (7) nd () respectively. The comprison of the results for effective permittivity of the coxil line, obtined using expressions () nd (4) nd using FEM (Softwre pckge FEMM) is presented in Tble 5. Electrode Electrode d d Fig. 7. The line with three lyer dielectric medium r Fig. 6. Coxil line hving circulr inner conductor nd squre outer conductor Exmple 5. In the cse of the line presented in Fig. 7 the coefficients of proportionlity g, g 34 nd g 3, Fig. 8. The rtio C /ε for different rtios / nd ε /ε, when d / = 8

7 Journl of ELECTRICAL ENGINEERING 6, NO. 5, 0 55 Tble 7. Comprison of the results for C /ε 0, when d/ = 0, Fig. 9 C /ε 0 ε /ε h/ = h/ = Eq. (4) CSM FEM Eq. (4) CSM FEM Exmple 6. The next exmple presents the line in which the electrodes re in the sme lyer of the two lyer dielectric medium, Fig. 9. The cpcitnce per unit length is determined using pproximte expression (4), CSM nd FEM (Softwre pckge Femlb). For clcultion g nd g CSM is lso used. The obtined results re compred in Tble 7. Electrode h d Electrode Fig. 9. Two wire line bove infinite dielectric surfce 6 CONCLUSION This pper presents n instructive review of different techniques for clcultions of the cpcitnce per unit length of lines with multilyer medium, especilly lines with rectngulr cross section. Severl numericl methods (CSM, EEM, MESM) nd two progrm pckges [7, 8] re used, nd some simple equtions re proposed which permit n pproximte, but sufficiently ccurte, evlution of the cpcitnces. Expressions for effective permittivity nd conductivity re lso suggested. The proposed expressions (), (4) nd (9) re postulted nd their vlidity is then tested ginst severl chrcteristic exmples. A very good greement between obtined results is demonstrted. It is worth mentioning, moreover, tht the expressions re lwys exct in cses of one lyer medium, in cses when the surfce of seprtion of the existing lyer is of uniform potentil s well s in cses when permittivity of one lyer hs high vlue nd this medium behves s conductive medium. The proposed expressions cn simplify the solving of problems s nd when they present, nd represent useful tool in everydy engineering prctice. APPENDIX A A. Equivlent Electrode Method Appliction The bsic ide of the method [3, 4] is tht n rbitrrily shped electrode cn be replced by finite system of the equivlent electrodes (EE) locted on the body surfce. The rdius of EE is equl to the equivlent rdius of electrode prt which it substitutes. Also, the potentil nd chrge of EE nd of the rel electrode prt re equl. So, it is possible using the boundry condition tht the electrode is equipotentil, nd forms system of liner equtions with chrge densities of EE s unknowns. After solving this system, unknown chrge densities re determined nd ny other quntity of interest cn be esily clculted in stndrd wy. In the cse when the system hs severl electrodes, or when multilyer medium exists, it is convenient to use Green s functions for some electrodes or for strtified medium. In the cse of squre coxil line Fig. A the interior electrode is replced by N cylindricl EE, with circulr cross section hving the rdius e, chrged by line chrge per unit length q n, n =,,..., N. The rdius of the EE is equl to the equivlent rdius of the electrode prt which it substitutes, in this cse thin flt strip conductor, e = x /4 = /4N. The position of these EE re x = ±x n, y = ±y n, x n = x / + (n ) x, y n = /, n =,,..., N/ nd x n = /, y n = / ( x / + (n ) x ), n = N/, N/ +,..., N. (A) (A) In similr wy, the shield is replced by M cylindricl conductors (EE), of circulr cross section hving the rdius e = x /4 = b/4m, chrged by q m, m =,,..., M. The xes of these EE re x = ±x m, y = ±y m, x m = x / + (m ) x, y m = b/, m =,,..., M/ (A3)

8 56 A.M. Milovnović M.M. Bjekić: APPROXIMATE CALCULATION OF CAPACITANCE OF LINES WITH MULTILAYER MEDIUM y Q M Q M After solving liner eqution (A5 A7) the unknown line chrges of the EE re determined nd the cpcitnce per unit length cn be clculted s q N q N C = 4 N n= q n. (A8) U Q Q q q q q Q Q x APPENDIX B B. Determintion of the proposed expression q N q N In the cse of the line with two lyer perfect dielectric medium, Figs. 6 nd 6b, when seprtion surfce S is uniform potentil, the cpcitnce per unit length cn be exctly determined s Fig. 0. The squre coxil line nd x m = b, y m = b/ ( x / + (m ) x ), m = M/, M/ +,..., M. (A4) So, the totl number of the EE ccordingly unknown chrges hving to be determined is N +M. Using the condition tht equivlent electrodes hve the sme potentil s the electrodes they represent, φ nd φ, φ φ = U, system of liner equtions re obtined U + φ = M m= N n= q n πε ln[ r r r 3 r4 e δ ] mn q m πε ln[ r r r 3 r 4 e δ mn], x = x n, y = y n, n =,,..., N, (A5) C = C + C 3 = ε g + ε g 3, (B) where g nd g 3 re coefficients of the proportionlity of the lines which re formed by the existing electrode nd the electrode s shield coinciding with surfce S. After certin elementry trnsformtion, nd considering tht the g 3 = g + g 3, (B) where g is coefficient of proportionlity of the line 3 when ε = ε, C = ε ε ( ε + ε ε g + ) ε g 3 + ( ε + ε g 3 g 3 (B3) the expression (B) cn be expressed in form of (B4) s is proposed in this pper C = (ε + ε )g 3 + ε ε ( ε + ε ε g 3 ) ε g. (B4) ), φ = M m= N n= q n πε ln[ r r r 3 r4 e δ ] mn q m πε ln[ r r r 3 r 4 e δ ] mn x = x m, y = y m, m =,,..., M. (A6) For clcultions of the cpcitnce per unit length, in the cse when two lyer perfect dielectric medium exists, but the electrodes re in the sme lyer of multilyer dielectric line, Fig. 7 expression (B5) is proposed C = (ε + ε )g + ε ε ε + ε ε g. (B5) N M q n + q m = 0, n= m= (A7) r 0 r 0 << H, h, d r 0 r = (x + x n ) + (y y n ), r = (x + x m ) + (y y m ), H d h r = (x x n ) + (y + y n ), r = (x x m ) + (y + y m ), r 3 = (x + x n ) + (y + y n ), r = (x + x m ) + (y + y m ), r 4 = (x x n ) + (y y n ), r = (x x m ) + (y y m ), nd δ mn is Kronecker symbol. Fig.. Two wire line bove infinite dielectricl surfce

9 Journl of ELECTRICAL ENGINEERING 6, NO. 5, 0 57 This expression is formed ccording to the expression (B4) nd firstly, we cn check its ccurcy in exmple of two wire line bove infinite dielectricl surfce presented in Fig. B, for which nlyticl expression exists C = ( ln d + ε ε hh ) ln. (B6) ε π r 0 ε + ε d + 4hH In this cse the coefficients of proportionlity g nd g re g π = ln(d/r 0 ) nd π g = ln ( hh / r 0 d + 4hH ). (B7) Expression (9) proposed for the exmple of the line with three lyer perfect dielectric medium, Fig. 9 is obtined similrly s expressions (B4) nd (B5). Acknowledgement The uthors of this pper re very grteful to their collegues from the University of Technology Ilmenu, Germny, who provided ccess to the softwre pckge Femlb during the uthors sty t their University. References [] ABDEL-SALAM, M. : Combined Method Bsed on Finite Differences nd Chrge Simultion for Clculting Electric Field, IEEE Trnsctions on Industry Applictions 5 No. 6, (989), [] MALIK, N. H. : A review of the Chrge Simultion Method nd its Applictions, IEEE Trnsctions on Electricl Insultion 4 No. (989), 3 0. [3] [4] VELIČKOVIĆ, D. M. MILOVANOVIĆ, A. : Electrosttic Field of Cube Electrode, Serbin Journl of Electricl Engineering No. (004), VELICKOVIĆ, D. M : Equivlent Electrodes Method, Scientific Review, Belgrde No. - (996), [5] DOBRICIC, M. : Distribution of Potentil Surrounding Two Wire Lines with Specil Retrospect the Method of Complex Potentil, PhD Disserttion, Technicl fculty of Cck, University of Krgujevc, 008. [6] MUSA, S. M. SADIKU, M. N. : Anlysis of Rectngulr Coxil Lines, IEEE Region 5 Technicl Conference, April 0-, Fyetteville, Arcnss, 007. [7] VELIČKOVIĆ, D. M. MILOVANOVIĆ, A. : Approximte Clcultion of Cpcitnce, Proceedings of VIII Interntionl IGTE Symposium on Numericl Field Clcultion in Electricl Engineering, Grz, Austri, September 998, pp [8] LIN, W. XIANG, Y. : Electrosttic Force on the Wlls of Rectngulr Coxil Line, Journl of Electrosttics 43 (998), [9] RIBLET, H. J. : Expnsion for the Cpcitnce of Squre in Squre with Comprison, IEEE Trns. Microwve Theory Tech. 44 No. (Feb 996), [0] UITTEKER, E. T. VATSON, D. N. : Kurs sovremennogo nliz (A Course of Modern Anlysis), Fizmtgiz, Moskv, 963. (in Russin) [] KAISER, K. L. : Trnsmission Lines, Mtching, nd Crosstlk, Tylor nd Frncis Group, Boc Rton, FL, 006. [] MILOVANOVIĆ, A. M. KOPRIVICA, B. M. VESKOVIC, M. M. : The Cpcitnce of Two Wire Line with Rectngulr Cross Section, Interntionl Conference on Electricl Systems Design nd Technologies, Hmmmet Tunisi, Nov 8-0, 008, CD Proceedings. [3] COSTAMAGNA, E. FANNI, A. : Anlysis of Rectngulr Coxil Structures by Numericl Inversion of the Schwrz-Christoffel Trnsformtion, IEEE Trns. Mgn. 8 (Mr 99), [4] GREEN, H. E. : The Chrcteristic Impednce of Squre Coxil Line, IEE Trnsctions on Microwve Theory nd Techniques MTT- (Nov 963), [5] CHEN, T. S. : Determintion of the Cpcitnce, Inductnce, nd Chrcteristic Impednce of Rectngulr Lines, IRE Trnsctions on Microwve Theory nd Techniques, 8 No. 5 (Sep 960), [6] VELICKOVIĆ, D. M. UHLMANN, F. H. BRANDISKY, K. STANTCHEVA, R. D. : Fundmentls of Modern Electromgnetics for Engineering, Technishe Universität Ilmenu, Germny, 005. [7] Softwre pckge Femlb, Version , [8] Softwre pckge FEMM, Received 9 December 00 Alenk M. MILOVANOVIĆ ws born in Cck, Serbi on December 965. She received the MSc nd PhD degrees in Electricl Engineering from the Fculty of Electronic Engineering of Nis in 999 nd from the Technicl fculty of Cck in 007, respectively. Since 99 she hs been with the Deprtment of Electronic nd Electricl Engineering of Technicl fculty of Cck, where now work s Assistnt Professor. Her reserch interest includes Computtionl electromgnetics nd Applied electrosttics. She is the uthor/couthor of more thn 30 ppers in journls nd proceedings nd the couthor of three textbooks for students. Miroslv M. BJEKIĆ ws born on 8 August 966 in Cck, Serbi. He received MSc nd PhD degrees in Electricl Engineering from the School of Electricl Engineering Belgrde (995) nd Technicl Fculty of Cck (006), respectively. Since 99 he hs been working with the Deprtment of Power Engineering of Technicl Fculty of Cck, where he works s Assistnt Professor. His speciliztion: Computtionl electromgnetics, Micromchines nd Step motors. He is the uthor/co-uthor of more thn 50 ppers in journls nd proceedings, nd the co-uthor of three textbooks for students.