Analysis of Mid-range Wireless Power Transfer Circuits Based on Reactance Function and Image Impedance

Transcription

1 Electrical and Electrnic Engineering 07 7(3): DOI: 0.593/j.eee Analysis f Mid-range Wireless Pwer Transfer Circuits Based n Reactance Functin and Image Impedance Yasuyuki Okumura * Masahir Yamamt Katsuyuki Fujii Naki Inagaki Faculty f Science and Engineering Nanzan University Aichi Japan Abstract This paper prpses a nvel methd t analyze the mid-range wireless pwer transfer circuits cmprised f square cils and capacitrs and maximize verall efficiency. The methd is based n the equivalent circuit mdel derived frm the reactance functin and image impedance. The mid-range circuits are prmising as an efficient wireless pwer transfer methd using near field electrmagnetic waves. Hwever n simple design and the analysis methds are available fr wireless pwer transfer. This paper assumes bth transmitter and receiver antennas are identical and electrically small. It analyzes the resnance frequencies f the even and dd mdes which are used t mdel the equivalent circuit describing the actual wireless pwer transfer circuits. This paper als analyzes the transfer efficiency using the equatin f the equivalent circuit simulatin sftware based n the mment methd and an experiment. It clarifies that the analytical values derived by the prpsed methd and the simulatin results mstly agree with the measured values. As a result the mid-range wireless pwer transfer circuits designed using the prpsed analysis methd achieve high efficiency. Keywrds Mid-range pwer transfer Reactance functin Image impedance. Intrductin In recent years near field cnnectin antennas are being widely used in applicatins such as radi frequency identifier (RFID) tags [] bdy-centric cmmunicatin [] and wireless pwer transfer [3]. Wireless pwer transfer realizes flexible pwer charging withut wires r plugs and its applicatin t electric vehicles is deemed essential since such vehicles need frequent charging. If high efficiency wireless pwer transfer circuits can be develped with large air gaps electric vehicles culd be charged in parking areas. Cnventinal wireless pwer transfer can be realized using three technlgies: electrmagnetic inductin micrwave transfer and the laser transfer. Only the electrmagnetic inductin is widely and cmmercially used but in limited envirnments where the transfer distance is less than cm. Hwever mid-range pwer transfer ver 0cm r s has been prpsed with the use f 0 t 40 khz signals. In additin mid-range pwer transfer f arund m has been prpsed thrugh the use f electrmagnetic resnance. As its theretical efficiency is arund 45 t 50% with a m air gap and ver 90% at under m it is a prmising technlgy [3-5]. Unfrtunately the current * Crrespnding authr: ykumura@nanzan-u.ac.jp (Yasuyuki Okumura) Published nline at Cpyright 07 Scientific & Academic Publishing. All Rights Reserved analysis apprach which uses mde cupling thery and a Q factr that describes the energy cnservatin f the circuit makes it difficult t design mre efficient circuits [6-0]. A simpler analysis thery f mid-range pwer transfer is needed t design circuits with less difficulty. This paper derives an equivalent circuit mdel frm the reactance functin and image impedance. Fr mid-range pwer transfer the electrmagnetic cupling thery is useful fr understanding the peratin principle which was develped frm circuit thery the micrwave thery and the filter thery. Mid-range transfer generally uses the frequency band f MHz t GHz and the antennas are electrically small. Althugh electrmagnetic inductin is based n the dynamic magnetic field shared by the transmitter and the receiver fr mid-range pwer transfer it is better t analyze electrical and magnetic fields. This paper assumes that bth transmitter and receiver antennas are identical and electrically small. T derive the equivalent circuit mdel fr the transmitter and the receiver antennas the reactance functin is analyzed using even and dd mdes excitatin f the antennas. Ignring the lss f the antennas the equivalent circuit is described by inductrs (L) and capacitrs (C) and the analysis f its image impedance identifies the frequency band that ffers high transmissin efficiency. This paper clarifies that the analytical values derived by the prpsed methd and the simulatin results basically agree with the measured values. As a result the mid-range wireless pwer transfer circuits designed herein achieve high efficiency.

2 78 Yasuyuki Okumura et al.: Analysis f Mid-range Wireless Pwer Transfer Circuits Based n Reactance Functin and Image Impedance. Analysis Mdel f Mid-Range Pwer Transfer Circuit Based n the Impedance Matrix The mid-range pwer transfer circuits cnsist f tw pairs f an excitatin cil and a resnance cil ne pair fr the transmitter and the ther fr the receiver as shwn in Figure. This paper assumes that bth pairs are identical i.e. the excitatin and resnance cils have the same shape and size. The RF generatr pwers the excitatin cil in the transmitter and the lad receives the pwer thrugh the excitatin cil in the receiver. This cnfiguratin was prpsed in [3 4] where the RF frequency is f the rder f MHz; the resnance cil maximizes the transmissin distance and efficiency. In the cnfiguratin the resnance cil is helically shaped and its resnance frequency equals that f the RF generatr. cnjugate image impedance is equal t the image impedance. Given z r +jx etc. the thery gives the image impedance fr transmitter side Z t and that fr the receiver side Z r as fllws; Z r θ + j θ jx () where ( θ θ ) t r x Z r j jx (3) r r + x θ + r r x rr rr θ x r x r r The maximum efficiency is achieved when impedance Z f the RF generatr is equal t Z t while impedance Z f the lad is equal t Z r []. (4) (5) Figure. One input and ne utput circuit Figure. Mid-range pwer transfer circuit cnfiguratin [3].. Efficiency f the Pwer Transfer Circuit [] The general equivalent circuit fr the input and utput circuit (the pwer transfer circuit) is shwn in Figure where the cils and the transmissin space between them characterize a 4-prt circuit with impedance matrix Z. This paper assumes the 4-prt circuit is linear reciprcal and passive which characterizes the reflectrs the scatterers and the absrbers f the radi signals. Given that the RF generatr and lad are cnnected t prt and prt respectively the efficiency f the pwer transmissin between the generatr and the lad is described as S S where S and S are elements f the scattering matrix [-3]. The design bjective fr the pwer transfer circuit is determining impedance Z f the RF generatr and impedance Z f the lad s as t maximize S S. Each impedance is described as a cmbinatin f resistance and reactance i.e. Z R +jx and Z R +jx. Given the abve ntatin efficiency is described as fllws [9]; 4 z RR S S () ( Z + z)( Z + z ) zz where z ij (i r j r ) is the element f Z. Efficiency S is maximized by impedance matching based n the thery f cnjugate image impedance []. Prvided the lad cntains nly the resistance cmpnent the.. Analysis f the Impedance Matrix T analyze the efficiency S all elements f Z must be assessed in the mid-range pwer transfer circuit as shwn by eq.(). The circuit mdel shwn in Figure 3(a) is a -prt circuit and thus linear reciprcal and passive. In this mdel the inductance f the excitatin cil and the resnance cil are dented as L and L respectively. Regarding the resnance cil capacitr C is inserted fr impedance matching. Mutual inductances M M 3 M 4 and M 34 and resistances R and R representing the lss are als cnsidered fr the circuit analysis as shwn in the figure. T derive Z f the circuit the mdel shuld be mdified t a 4-prt circuit as shwn in Figure 3(b). If prts 3 and 4 are shunted see Figure 3(b) it becmes identical t the circuit f Figure 3(a). First f all impedance matrix Z 4prt f the 4-prt circuit can be described using the angular frequency ωπf as fllws; z z z3 z4 z z z3 z z z z z z z z z Z 4 prt z3 z3 z33 z34 z3 z4 z33 z34 z4 z4 z43 z44 z4 z3 z34 z33 ' R + jωl jωm jωm3 jωm 4 ' jωm R + jωl jωm 4 jωm 3 ' jωm. 3 jωm4 R + jωl + jωm 34 jωc ' jωm4 jωm3 jωm34 R + jωl + jωc (6)

3 Electrical and Electrnic Engineering 07 7(3): Using elements z ij f Z 4prt the impedance matrix Z f the tw-prt circuit shwn in Figure 3(a) is given as fllws; z z Z z z z + z z z z z z ( z + z ) z + z z z z z33 z z33 z z33 z34 z z z + z z + z z z z + z z z z z34 z z where its derivatin prcess is described in the Appendix. By determining the RLC values and substituting the elements z ij f eq.(7) int eq.() efficiency S can be calculated. (a) Tw-prt circuit mdel (b) Fur-prt circuit mdel Figure 3. Analysis mdel f mid-range pwer transfer circuit 3. Analysis Based n the Equivalent Circuit Figure 4. Equivalent circuit mdel This sectin describes the ther analysis based n the equivalent circuit. The equivalent circuit f the mid-range pwer transfer circuit can be elucidated using the even and the dd mde resnance frequencies where the lss caused by the cpper f the circuit is negligible []. The equivalent (7) circuit used in this analysis is shwn in Figure 4 where identificatin means determining the values f L and C in Figure 4 given the values f L L C M M 3 M 4 and M 34 in Figure 3(a). 3.. Determinatin f the Values f Inductance and Capacitance [4-5] Using the resnant and the anti-resnant frequencies the L and C values f Figure 4 can be described as fllws; C L L 0 f pe e Llfe f se f 0 se e Le f pe ( ) e 4π Le0 fse fpe C L 4π L e f 0 s Le f p ( ) Le0 fs fp Le L L L e e e (8) (9) (0) () () (3) C C C (4) where f se and f pe are the resnant frequency and the anti-resnant frequency f the even mde and f s and f s are thse f the dd mde respectively. The value f L lfe is the inductance f the even mde in the lw frequency range. The analysis methd f these frequencies is prpsed in Sectin 3. and 3.3. The value f L lfe depends n the shape and the size f the cil and is derived by numerical analysis as described in Sectin 4. Once the equivalent circuit shwn in Figure 4 is identified all elements f the scattering matrix can be derived as fllws; impedance matrix Z can be derived frm the equivalent circuit as a ladder circuit which is cnverted int transfer matrix F and t scattering matrix S. In S element S means the transfer efficiency. 3.. Analysis f Resnant and Anti-Resnant Impedances The resnant and the anti-resnant frequencies f the even and the dd mdes can be derived by analyzing the impedance f the mid-range pwer transfer circuit mdel which will be described in Sectin 3.3. T analyze the impedance the mdel is mdified int a 4-prt circuit as described in Sectin. By inserting R R 0 int eq. (6) impedance matrix Z 4prt f the 4-prt circuit can be written as fllws;

4 80 Yasuyuki Okumura et al.: Analysis f Mid-range Wireless Pwer Transfer Circuits Based n Reactance Functin and Image Impedance Z4 prt z z z3 z4 z z z3 z4 z3 z3 z33 z34 z4 z4 z43 z44 jωl jωm jωm3 jωm4 (5) ω ω ω 4 ω j M j L j M j M3 jωm3 jωm4 jωl + jωm34 jωc jωm4 jωm3 jωm34 jωl + jωc V z z z3 z4 I V z z z3 z4 I. V3 z3 z3 z33 z34 I3 V4 z4 z4 z43 z44 I4 (6) In the abve equatin the current f prt i is dented by I i (i34) and the vltage f this prt is dented by V i. Prvided V 3 V 4 0 eq. (6) describes the mid-range pwer transfer circuit shwn in Figure 3(a). Even mde impedance Z e is calculated by deriving input current I e int prt prvided the same vltage E is given t prts and which is the cnditin f even mde peratin. On the ther hand regarding dd mde impedance Z input current I must be derived when vltages E and E are given t prt and prt respectively. Based n the abve definitin the even and the dd mde impedances are written as fllws; E Ze (7) Ie Z (8) I In the even mde the cnditins I I and I 3 I 4 are applied t eq. (6). By eq. (5) z z z 33 z 44 and z mn z nm (m n) which lead t the fllwing; Z e E z + z z3 + z4 I e 0 z3 + z4 z33 + z34 Ie3 ( z + z )( z + z ) ( z + z ) z E + z ω ( M3 + M4 ) jω L + M ωl + ωm ωc 34. (9) (0) In the dd mde by applying the cnditins f I -I and I 3 -I 4 t eq. (6) vltage E current I i (i r 3) and impedance Z are written as fllws; E z z z3 z4 I 0 () z3 z4 z33 z34 I3 Z ( z z )( z z ) ( z z ) z z ω ( M3 M4 ) jω L M ωl ωm ωc 34. () Based n the abve calculatin image impedance Z i can be derived as fllws; Zi ZZ e ω L M L + M ω ω ( M3 M4 ) L ωm 34 ωc ω ( M3 + M4 ). L + ωm 34 ωc ω (3) 3.3. Analysis f Resnant and Anti-Resnant Frequencies The resnant frequency is defined as the frequency that yields zer reactance and the anti-resnant frequency makes the reactance infinite. Therefre frm setting the cnditin Z e 0 and Z e in equatin (0) f se and f pe are derived as fllws; f se ( M ) 3 + M 4 π (4) C L + M34 L + M f pe π C L ( + M ) 34. (5) On the ther hand seting Z 0 and Z in equatin () f s and f p are derived as fllws; f s ( M ) 3 M 4 π (6) C L M34 L M f p π C L ( M ) 34. (7) Using these results the values f L and C in the equivalent circuit shwn in Figure 4 can be derived frm equatins (8) t (4). Prvided f se f p impedance matching is available fr any frequency f satisfying the cnditin; f pe < f < f s [5]. 4. Physical Implementatin f the Mid-Range Pwer Transfer Circuit This paper describes the design and the analysis results f a pwer transfer circuit that transmits and receives 3.56 MHz radi signals in the Industry-Science-Medical (ISM)

5 Electrical and Electrnic Engineering 07 7(3): band which is a standard and cmmercially used band. The ISM band includes frequencies such as 30 t 35 khz 3.56 MHz 433 MHz 900 MHz and.45 GHz. One example f 3.56 MHz signal use is RFID which is a tuch-less integrated circuit (IC) that transfers infrmatin and electric energy. Since the wavelength f 3.56 MHz frequency radi is. meters and its radiansphere is 3.5 meters the antenna shuld be smaller than a ball with radius equal t the radiansphere. In the MHz frequency range with lnger wavelength the antenna size becmes larger than the abve which causes difficulties in the practical use. Althugh this paper fcuses n the 3.56MHz cnsidering the cnditin the analysis methds prpsed in the paper can be applied t this frequency band. In additin this paper assumes the characteristic impedance is 50 hm a cmmn cmmercial setting. The physical structure f the antennas is shwn in Figure 5 where each cil is a single square turn f cpper sheet. The transmitter and the receiver cnsist f tw cils where the inner cil transmits r receives the pwer and the uter ne is the resnant cil. In the transmitter and the receiver bth the inner and the uter cils lie n the same hypthetical plane and the center f each cil lie n the same hypthetical line. Each cil is made up f cpper sheet millimeter thick and 5 millimeter wide and a ceramic capacitr. The cpper sheet easy t prcess and is suitable fr antenna implementatin. The ceramic capacitrs ffer a flat reactance characteristic in the high frequency range and are small. Figure 5. Physical implementatin f pwer transfer circuit 4.. Analysis f the Inductance and Capacitance f the Antenna The square uter cil has length f a meters and that f the inner cil is b meters where bth cils are parallel t the x-y plane. The distance between the transmitting cil and the receiving cil is taken t be s millimeters. In the transmitter the excitatin cil is laded with the vltage surce and a resistr (R50Ω) and the resnance cil is laded with a capacitr (C Farad). The lad f the receiving cil is a resistr (R50Ω) and the lad f the resnance cil is a capacitr equal t that in the transmitter. Figure 6. Analysis mdel f mutual inductance f tw lines The self-inductance and mutual inductance can be derived frm Neumann s frmula. Since any pair f sides are parallel r rthgnal with each ther in the square shaped cil the inductance is determined nly by the parallel pair accrding Neunann s frmula. Therefre as a first step the mutual inductance M between the tw parallel lines shwn in Figure 6 is calculated as fllws; µ M x + d + xlg( x + x + d 4π ) a + d alg( a+ a + d ) (8) x + d x lg x + x + d a + d + alg a + a + d where dy -y a x -x a x 3 -x and x x x 3 y and y are crdinates f the end pints f the tw parallel lines. Regarding the square cil nly the parallel lines cntribute t the inductance since the inductance is accrding t Neumann s frmula expressed as the inner prduct f the vectrs alng the tw lines cmprising the cil. Therefre mutual-inductances M M 3 M 4 M 34 and self-inductances L and L can be derived frm equatin (8). Prvided the uter cil is cnfigured as a cpper line with radius r meters self-inductance L is ( + ) µ r L ar a+ a + r + alg. (9) π + + a a r Mutual inductance M f the tw uter cils (ne side is a meters in length) is written as fllws; µ a + s + a a + s M s a + s + a + s + alg π s( a+ a + s ) (30) where s is the gap between the tw cils. Althugh the cpper line f radius r is assumed in the abve equatins each cil is a single square turn f cpper sheet in the experiment. Therefre t match the abve analysis t the experimental cnfiguratin width w f the cpper sheet is transfrmed t equivalent radius r accrding t rw/4. T achieve resnance in the cils capacitance C is given by;

6 8 Yasuyuki Okumura et al.: Analysis f Mid-range Wireless Pwer Transfer Circuits Based n Reactance Functin and Image Impedance C ω L where ω is the resnance angular frequency. 4.. Designing the Antenna Figure 7. Analysis results f resnance frequencies Table. Design parameters f pwer transfer circuit Width f resnance cil a[mm] 400 Width f excitatin cil b[mm] 68 Width f cpper sheet w[mm] 5 Gap f cils s[mm] 700 Frequency[MHz] 3.56 Lad impedance R[Ω] 50 Even mde inductance in lw freq. L lfe [mh] 0.36 Figure 8. Analysis results f Image impedance (3) parameters f the designed antennas are summarized in Table. Using eq. (3) the relatin between frequency and image impedance Z i is described in Figure 8 where the image impedance is real in the 3.49 t 3.75MHz range and becmes 50hm at 3.56MHz. 5. Mdel fr Simulatin and Experiment This sectin utlines the simulatin mdel and the experimental setup. 5.. Simulatin Mdel The electrmagnetic field analysis sftware FEKO [6] was used t analyze the pwer transfer efficiency by simulatin. FEKO ffers a numerical methd fr cmputatinal electrmagnetics in a wide range f electrmagnetic prblems. The simulatin mdel reprduces the transmitter and the receiver antennas shwn in Figure 5. Antenna material is cpper with cnductivity f 57.6x0 6 S/m and relative permeability f The simulatin is carried ut nt nly t determine the pwer transfer efficiency but als identify the equivalent circuit parameters described in Sectin 4. The inductance and the capacitance values f the equivalent circuit L e0 L e L e C e and C e are summarized in Table based n the simulatin. Table. Identificatin results f equivalent circuit Inductance L e0 [mh] 0.35 Inductance L e [mh] 7.60 Inductance L e [mh] 88.4 Capacitance C e [nf] 8.9 Capacitance C e [nf] Experimental Setup The experimental setup is shwn in Figure 9 where the square antennas are munted n wden frames. The pwer transfer efficiency between the transmitter and the receiver antennas was measured using an Agilent E507C netwrk analyzer. The design parameters f the antennas fr mid-range pwer transfer are variables such as; width w f the cpper sheet side length f the square resnance cil a side length f the square excitatin cil b and the gap between the transmitter and the receiver cils s. As initial cnditins this paper assumes that w a and s are 5mm 400mm and 700mm respectively and the peratin frequency f is 3.56MHz. By using equatins (9) and (30) self-inductance L and mutual inductance M can be calculated and used in eqs. (4) t (7) t calculate resnance frequencies f pe f se f p and f s respectively. The relatin between b and the fur resnance frequencies is shwn in Figure 7. Applying the cnditin f se f p described in Sectin 3 side length f the square excitatin cil b is 68mm. All Figure 9. Experimental setup

7 Electrical and Electrnic Engineering 07 7(3): Mdel fr Simulatin and Experiment This sectin describes the results f transfer efficiency S tgether with reflectin cefficient S derived by the analysis based n the impedance matrix in Sectin the analysis using the equivalent circuit in Sectin 3 the simulatin and the experiment. The results related t frequency are shwn in Figure 0. Regarding the accuracy f the abve values the significant figures f the analysis and the simulatin are fur digits. The trace nise f the netwrk analyzer is 0.003dBrms in the experiment. The results derived by the analysis simulatin and experiments shw almst the same characteristics; that this the tw analysis methds intrduced in this paper agree with the experiment and simulatin results. Maximum efficiency (S ) is dB at 3.56MHz by the analysis f the impedance matrix (Sectin ) dB at 3.56MHz by the analysis f the equivalent circuit (Sectin 3) -0.87dB at 3.56MHz by the simulatin and -.54dB at 3.56MHz measured in the experiment. The measured value f S is smaller than the analysis and simulatin equivalents due t the cpper lss. The mid-range wireless pwer transfer circuits designed using the prpsed analysis methd achieved high efficiency. T increase the efficiency lw-lss material such as Litz Wire culd be used fr the antenna cil. with the measured values. As a result the designed mid-range wireless pwer transfer circuits achieved high efficiency. ACKNOWLEDGEMENTS This wrk was supprted by the Nanzan University Pache Research Subsidy I-A- fr the 07 Academic year. Appendix Prvided prts 3 and 4 are shunted see Figure 3(b) the circuit becmes identical t that f Figure 3(a). In this case the vltage and the current f each prt are given as fllws; V z z z3 z4 I 4 3 V z z z z I. (A) 0 z3 z4 z33 z34 I3 0 z4 z3 z34 z33 I4 In the tw-prt circuit the vltage and the current f each prt are given as fllws; V {[ ] [ ] [ ] I Z Z Z [ Z] } (A) V I where z z z3 z4 [ Z] [ Z] [ Z] z z z4 z3 (A3) z33 z34 [ Z ]. z34 z33 Eq.(7) is derived by inserting (A3) int (A). 7. Cnclusins Figure 0. S-parameter characteristics This paper prpsed a nvel methd fr analyzing the mid-range wireless pwer transfer circuits cmprised f a square cil and capacitr and maximizing verall efficiency. The methd is based n the equivalent circuit mdel derived frm the reactance functin and image impedance. This paper assumes that transmitter and receiver antennas are identical and electrically small. It analyzed the resnance frequencies f the even and dd mdes and used them t mdel the equivalent circuit f the actual wireless pwer transfer circuits. This paper als analyzed the transfer efficiency using the equatin f the equivalent circuit simulatin sftware based n the mment methd and an experiment. It clarified that the analytical values derived by the prpsed methd and the simulatin results mstly agree REFERENCES [] K. Uesaka and M. Takahashi Antennas fr Cntact-Less IC Card/RFID Tag Systems IEICE Trans. Cmmunicatins vl. J89-B n. 9 pp [] P. S. Hall and Y. Ha Antennas and Prpagatin fr Bdy-centric Wireless Cmmunicatins Artech Huse 006. [3] A. Kurs A. Karalis R. Mffatt J. D. Jannpuls P. Fisher and M. Sljacic Wireless pwer transfer via strngly cupled magnetic resnances Science vl.37 n.5834 pp [4] A. Karalis J. D. Jannpuls and M. Sljacic Efficient wireless nn-radiative mid-range energy transfer Annals f Phisics 33 pp [5] T. Ishizaki T. Kmri T. Ishida and I. Awai Cmparative study f cil resnatrs fr wireless pwer transfer system in terms f transfer lss IEICE Electrnics Express vl.7 n. pp

8 84 Yasuyuki Okumura et al.: Analysis f Mid-range Wireless Pwer Transfer Circuits Based n Reactance Functin and Image Impedance [6] T. Ohira Maximum available efficiency frmulatin based n a black-bx mdel f linear tw-prt systems IEICE Electrnics Express vl. n.3 pp [7] Q. T. Dung M. Okada Maximum efficiency frmulatin fr inductive pwer transfer with multiple receivers IEICE Electrnics Express vl.3 n. pp [8] B. Minnaert N. Stevens Single variable expressins fr the efficiency f a reciprcal pwer transfer system Internatinal Jurnal f Circuit Thery and Applicatins nline 06. [9] Q.T. Dung M. Okada kq-prduct frmula fr multiple-transmitter inductive pwer transfer system IEICE Electrnics Express vl.4 n.3 pp [0] M. Tamura Y. Watanabe I. Takan Waveguide-mde wireless pwer transfer in shielded space with aperture plane IEICE Electrnics Express vl.4 n.8 pp [] S. Rberts Cnjugate-image impedances Prc. I. R. E. and Waves and Electrnics vl.34 pp [] D. Pzer Micrwave Engineering Furth Editin Wiley 03. [3] G. Vendelin A.M. Pavi and U. L. Rhde Micrwave Circuit Design Using Linear and Nnlinear Techniques Wiley 995. [4] N. Inagaki S. Hri Fundamentals f wireless cnnectins with near field cupled antennas IEICE Trans. Cmmunicatins vl. J94-B n. 3 pp [5] N. Inagaki S. Hri Characterizatin f wireless cnnectin systems f resnant methd based n even and dd mde reactance functins and the image impedance IEICE Trans. Cmmunicatins vl. J94-B n. 9 pp [6] Altair Engineering FEKO hmepage