Fast Calculation and Analysis of the Equivalent Impedance of a Wireless Power Transfer System Using an Array of Magnetically Coupled Resonators

Transcription

1 Progress In Electromgnetics Reserch B, Vol. 80, , 018 Fst Clcultion nd Anlysis of the Equivlent Impednce of Wireless Power Trnsfer System Using n Arry of Mgneticlly Coupled Resontors José Alberto 1, *, Ugo Reggini 1, Leonrdo Sndrolini 1, nd Helen Albuquerque Abstrct In this pper the equivlent impednce of resontor rrys for wireless power trnsfer systems is obtined in closed-form from continued frction expression. Using the theory of difference equtions, the continued frction is described s the generl term of complex sequence defined by recurrence, nd its convergence is nlyzed. It is shown tht the equivlent impednce cn be esily found in closed-form in terms of the system prmeters. In this wy, the obtined closed-form expressions my help electricl engineers to quickly predict the behviour of system with the chnges of its prmeters. Some numericl exmples of the theoreticl results re given nd discussed. Finlly, the nlyticl formule obtined in this work re vlidted with mesurements nd good greement is observed. 1. INTRODUCTION Recently, mny studies hve been focused on wireless power trnsfer WPT systems to trnsfer power without electricl contct in severl pplictions [1 3]. In situtions of high mislignment or distnce between the emitter nd receiver coils, the use of intermedite resontors [4 10] cn increse the efficiency of these systems nd mke it possible to trnsmit power over longer distnces. These resontors cn be rrnged in rrys on plne with prllel xes, or in domino configurtion. In this wy we cn build device cpble of chrging smll electronic device, or electric vehicle, in severl positions long the rry (plnr rry) or be ble to chrge or power device t longer distnce (domino rry). Usully one of the resontors of the rry is connected to the power source nd trnsmits power through mgnetic coupling to lod connected t the other end of the rry or to one or more receivers over the rry. Mgnetoinductive wve theory (MIW) hs been used to study these rrys [11, 1], which cn lso be nlysed using circuit modelling pproch: fter removing the resontor of the rry connected to the source, the rest of the rry cn be described with n equivlent impednce. This equivlent impednce llows one to nlyse the power delivered by the source to the loded rry (the rry fcing receiver or terminted in lod, i.e., using the lst resontor of the line s receiver), to exmine impednce mtching of the source to the loded resontor rry nd cn be useful for the design of the rry nd its power source. This equivlent impednce is described s continued frction in [11, 13]. Continued frctions ply n importnt role in vriety of brnches of pplied sciences, s in [14, 15]. Although in [11] it is simply cknowledged tht n nlyticl study of this frction seems not to be possible, in this pper, by performing mthemticl study of the continued frction s the generl term of complex Received 17 Jnury 018, Accepted 7 Mrch 018, Scheduled 10 April 018 * Corresponding uthor: José Alberto (jose.mmede@unibo.it). 1 Deprtment of Electricl, Electronic, nd Informtion Engineering, University of Bologn, Vile del Risorgimento, I Bologn, Itly. Deprtment of Mthemtics of University of Coimbr, Coimbr, Portugl.

2 10 Alberto et l. sequence defined by recurrence, we obtin closed-form expression of the equivlent impednce for ny condition of the system. The generl term of the recursive sequence is determined through the resolution of liner homogeneous difference eqution with constnt coefficients nd its vlue cn then be clculted just knowing the initil conditions nd possible perturbtions on the system. Through the closed-form expressions, better insight of the behviour of the system with respect to the vrition of its geometricl nd electricl prmeters cn be chieved. Indeed, the results of this pper re useful for electricl engineers in the design of WPT system using n rry of resontors, especilly regrding impednce mtching nd the design of the power source. Therefore, s the nlysis is mde simpler nd fster through the closed-form expressions even for severl conditions of the system (different circuit prmeters, number of resontors, number receivers), designers cn sve time nd increse the clcultion precision compred with using electromgnetic simultion softwre. In this pper, description of the nlysed circuit is provided in Section. The closed-form expressions for the equivlent impednce of resontor rry with receiver over the rry t the end or t ny other position of the rry re presented in Section 3, together with numericl study of their vlues nd convergence. The mthemticl pproch used to develop these expressions, for simplicity of the presenttion, is detiled in the Appendix. Finlly, in the lst section of this pper experiments re crried out to vlidte the theoreticl results presented in this pper, by mesuring both the input impednce nd the input power nd their vrition with the equivlent impednce of the rry.. DESCRIPTION OF THE CIRCUIT The system tht we consider in this work, s shown in Fig. 1(), consists of n rry of n + 1 identicl resontors (cells) rrnged in plne long line to form plnr structure; two djcent resontors re spced by the sme constnt distnce nd re mgneticlly coupled with mutul inductnce M, wheres the coupling between nondjcent resontors is neglected, s done lso in [5, 11, 13]. Ech cell cn be described s n R-L-C series circuit, s in [16] where R represents the intrinsic resistnce of the cell, L its self-inductnce nd C the dditionl cpcitnce needed to tune the resonnt frequency of the cell. The impednce of ech cell is then given by Ẑ = R + jωl +1/(jωC), being ω =πf the ngulr frequency, tht becomes Ẑ = R t the resonnt ngulr frequency ω 0 =πf 0 =1/ LC. The cell of the resontor rry connected to source of sinusoidl voltge ˆV s nd internl resistnce R s is lbeled n +1, wheres the cell t the other end of the line is connected to termintion impednce ẐT nd lbeled 1. () (b) Figure 1. Equivlent circuit of system composed of n + 1 cells with () the impednce Ẑd inserted in the ith cell, representing the receiver over it nd (b) the sme system with n impednce Ẑeq representing the resontor rry (removing the resontor connected to the source) nd the receiver. When there is receiver bove the line, the receiver bsorbs prt of the power rriving from the source t the ith cell under it, with 1 i n, nd this fct represents perturbtion in the system. This perturbtion cused by the mgnetic coupling between the receiver nd the ith cell cn be represented by Ẑd, s seen in Fig. 1(), which is the impednce of the receiver seen from the cell which is under it [11, 13]. It is ssumed tht the receiver hs the sme electricl prmeters s the cells of the rry nd, consequently, hs the sme resonnt frequency.

3 Progress In Electromgnetics Reserch B, Vol. 80, The WPT system cn be further simplified by introducing n equivlent impednce Ẑeq, tht represents the impednce of the rry nd the receivers fter removing the resontor connected to the power source, s depicted in Fig. 1(b). As seen in [13], the equivlent impednce Ẑeq hs the following expression when the receiver coil is plced over ny cell of the resontor rry: (ωm) Ẑ eq = (ωm) Ẑ + (ωm)...+ Ẑ d + Ẑ + (ωm) (ωm) Ẑ (ωm) Ẑ + (ωm) Ẑ + ẐT. (1) Note tht we cn replce ẐT by Ẑ T in Eq. (1), in the prticulr cse where the first cell of the resontor rry is connected to lod nd therefore this cell is the receiver (Ẑ T = ẐT ) or receiver is plced over the first cell Ẑ T = ẐT + Ẑd. Moreover, in cse we re operting t the resonnt ngulr frequency ω 0, Eq. (1) cn be written with Ẑ = R, ẐT = R T, Ẑd = R d. The mthemticl nlysis developed to determine the vlue of the continued frction tht represents the impednce of n rry of resontors is illustrted in the Appendices. It is demonstrted tht the frction cn be rewritten s term of recursive sequence whose generl term cn be determined using the theory of liner homogeneous difference equtions [17]. In this wy, we get n expression for the vlue of the frction tht depends only on its initil conditions, number of terms nd order of the term which is ffected by the perturbtion. This expression is shown in the next section. 3. CLOSED-FORM EXPRESSIONS OF THE CONTINUED FRACTION The theoreticl results obtined in the Appendices re now pplied to WPT system composed of n rry of resontors, nd closed-form expressions for Eq. (1), considered for the two cses with ω ω 0 nd ω = ω 0, re found. Exmples of clcultions re crried out with the softwre MATLAB, nd some numericl results re presented nd discussed. In order to illustrte possible prcticl situtions, the vlues for R, L, C nd M used re the ones obtined through mesurements performed with the experimentl setup described in Section 4 (L =1.6 µh, C =93.1nF, R =0.11 Ω, M = 1.55 µh nd f 0 = 147 khz). To obtin the expression of the equivlent impednce Ẑeq, we write the generic vlues of the frctions in Eqs. (A7) nd (A11) in terms of the chrcteristics of the WPT system shown in Section, =(ωm), b = Ẑ, x 0 = Ẑ T = ẐT + Ẑd (which is reduced to ẐT when the receiver is not over the first cell), b = Ẑd + Ẑ. Equivlent impednce with resontor number 1 cting s receiver or with receiver over the first cell of the resontor line Ẑ eq = f n ((ωm) gẑ T )+gn(fẑ T (ωm) ) f n (f +Ẑ T ) gn (g +Ẑ T ) () where f = Ẑ Ẑ +4(ωM) nd g = Ẑ Ẑ + +4(ωM). Equivlent impednce with receiver over the resontor line t ny position (e.g., over the ith cell) ( Ẑ eq = (ωm) e 1 f n g i + e f i g n f i g i (e 3 f n + e 4 g n ) ) f n g i (e 5 f i + e 6 g i )+f i g n (e 7 f i + e 8 g i (3) )

4 104 Alberto et l. Figure. R eq versus the position of the receiver over n rry of 50 resontors for different vlues of R d, t the resonnt frequency f 0 = 147 khz nd for R T =1.5 Ω. The position of the receiver is 49 when over the cell next to the one connected to the source nd 1 when over the cell t the other end of the rry line. The position of the receiver long the longitudinl xis of the rry is lso represented in terms of the distnce d between the centres of two djcent resontors of the rry: 0 is when the receiver is over the first resontor. where the constnts e 1, e, e 3, e 4, e 5, e 6, e 7,nde 8 re described in Appendix A.5. In cse we re operting t the resonnt frequency (i.e., ω = ω 0 ) Ẑeq becomes R eq,ndeqs.() nd (3) cn be written with Ẑ = R, Ẑ T = R T, Ẑ T = R T, Ẑ d = R d nd ω = ω 0. In [18] n exmple is shown by pplying Eq. () repetedly in order to determine the equivlent impednce of resontor rry loded with two receivers. In [19] insted, Eq. () is used to nlyse the conducted emissions generted by resontor rry. In Fig. we cn see tht the equivlent impednce is ffected more significntly s the receiver gets closer to the cell next to the one connected to the source nd for higher vlues of R d Convergence of the Equivlent Impednce For the constnts nd b given in Appendix A.1, for n infinite number of resontors Eq. () converges to the following vlue: lim Ẑ eq = 1 ) ( Ẑ Ẑ n + +4(ωM). (4) We cn see tht Eq. (4) does not depend on the initil conditions, i.e., the impednce Ẑ T. The limit depends only on the electricl prmeters of the cells, the mutul inductnce M nd the ngulr frequency ω. For exmple, for frequency f = 165 khz different thn the resonnt frequency, we cn obtin nd plot the equivlent impednce Ẑeq (Fig. 3). For the resonnt frequency f 0 = 147 khz, the equivlent impednce is shown in Fig. 4. It cn be noticed tht s we increse the length of the resontor line, even for different vlues of the impednce Ẑ T (or resistnce R T ), the equivlent impednce Figure 3. Mgnitude nd rgument of the equivlent impednce Ẑeq versus the number of resontors of the rry t f = 165kHz nd for different vlues of Ẑ T.

5 Progress In Electromgnetics Reserch B, Vol. 80, converges lwys to Eq. (4). Furthermore, introducing n impednce Ẑ T equlto(4)intoeq.(),the equivlent impednce Ẑeq is constnt nd equl to Ẑ T regrdless of the number of resontors. Tking Eq. (4) into ccount, t the resonnt frequency the input impednce of the rry, i.e., the impednce seen from the source terminls, is R in = R + 1 ( ) R + R +4(ω 0 M) = 1 ( ) R + R +4(ω 0 M). (5) Eqution (5) coincides with the termintion resistnce tht ccording to the MIW theory provides mtching of the structure [11]. Thus, Eq. (5) cn be considered s the chrcteristic impednce of the line. Figure 4. R eq versus the number of resontors of the rry t the resonnt frequency f 0 = 147 khz nd for different vlues of R T. This lso mens tht when perturbtion Ẑd is present in the (j + 1)th resontor, nd the line is terminted with n impednce Ẑ T equl to Eq. (4), we cn determine Ẑeq with Eq. () by replcing Ẑ T with 1 ( Ẑ + Ẑ +4(ωM) )+Ẑd nd replcing n with n j. 4. EXPERIMENTAL RESULTS The theoreticl results obtined in this work were verified with the rry used in [19], composed of six resontors ech formed wrpping 6 turns of strnded wire (section of 3.31 mm ) round prllelepiped wooden core of 15 cm 15 cm squre bse. The resontor rry is shown in Fig. 5(). As done in [19], the resontor ws supplied by n H-bridge inverter, which uses n FSB44104A Firchild Semiconductor nd is powered by n AIM-TTI Instruments QPX100SP 100W DC Power Supply. An Arduino Due microprocessor controls the H-bridge nd sets the working frequency equl to the resonnt frequency f 0 of the resontors. The first resontor of the rry is terminted to resistive lod. An Agilent 4396B 100 khz 1.8 GHz Vector Network Anlyser (VNA) ws used to mesure the self-inductnce, intrinsic AC resistnce, dded cpcitnce of the resontors, which were verged between mximum nd minimum vlues s described in [19] (see Tble 1). Note tht s the resontors of the rry re rrnged in plne, the mutul inductnce between ech pir of djcent resontors, determined with the VNA s in [19], is considered negtive in Tble 1, s done in [11, 13]. () (b) Figure 5. Experimentl setup built in lbortory: () rry of resontors used for experimentl vlidtions, (b) rry of resontors connected to Vector Network Anlyser (VNA) for mesurements.

6 106 Alberto et l. Tble 1. Averge mesured circuit prmeter vlues of the resontor rry. Prmeter L (µh) C (nf) R (Ω) M (µh) f 0 (khz) Mesured vlues 1.6 ± ± ± ± ± Input Impednce of the Arry with Receiver over the First Cell (Or with the First Resontor Acting s Receiver) nd with Receiver over the Resontor Line t Any Position The mgnitude of the input impednce of the whole rry ws mesured using the sme VNA s shown in Fig. 5(b). Then, the vlue obtined is compred with the theoreticl result given by: Ẑ in = Ẑ + Ẑeq (6) which cn be quickly nd esily clculted using Eq. () or (3). The six resontors were used to form five different rry configurtions, with, 3, 4, 5 nd 6 resontors. For ech configurtion, from Eq. () with n =1,...,5, nd Eq. (6), we clculte Ẑin t the resonnt frequency f 0 = 147 khz for different vlues of R T (0.4 Ω, 1.5 Ω nd 10 Ω). These results were then compred with the input impednce obtined with the mesurements on the sme configurtions of the system nd re shown in Fig. 6. It cn be noticed tht for R T =1.5Ω,Z in tends to the vlue (5) with much smller oscilltions thn for the other two vlues of R T since R T =1.5Ω is the resistnce closest to 1 ( R + R +4(ω 0 M) )=1.38Ω which mkes the equivlent impednce constnt, confirming the theoreticl result indicted in Appendix A.4 nd Section 3.1. At frequencies different from the resonnt one, Ẑin is complex quntity. As n exmple, the results re reported in Tble for Ẑ T =1.5Ω. Figure 6. Ẑ in (Ω) t the resonnt frequency f 0 = 147 khz obtined with mesurements with VNA nd with Eq. (6) using Eq. () for different vlues of R T (0.4 Ω, 1.5 Ω nd 10 Ω) nd different number of resontors. Tble. Ẑ in (Ω) for different frequency vlues from 135 khz to 161 khz obtined with mesurements withvnandwitheq.(6)usingeq.()forẑ T =1.5Ω. f (khz) Mesurements with VNA Eqution (6) In cse there is receiver over the ith cell of the resontor rry, we cn clculte Ẑin in resonnce conditions from Eqs. (3) nd (6). In the experimentl setup, we connected 5 Ω resistor to the ith cell to represent the dditionl impednce R d. The results re shown in Fig. 7. The smll difference between

7 Progress In Electromgnetics Reserch B, Vol. 80, Figure 7. Ẑ in (Ω) t the resonnt frequency f 0 = 147 khz obtined with mesurement using the VNA nd with Eq. (6) using Eq. (3) versus different positions of the receiver. experiments nd clcultions comes probbly from the imperfections in the mnufcturing of the coils, s their self-inductnce nd resistnce cn be slightly different. Nevertheless, there is clerly good greement between the results, thus vlidting experimentlly the theoreticl developments presented in this work nd showing their prcticl pplicbility to impednce mtching. 4.. Determintion of the Input Power Using the Equivlent Impednce As referred in the introduction of this pper, the vlue of equivlent impednce could be used, for given voltge source, to determine the power delivered from the source to loded rry. Then, ssuming tht we re working t the resonnt frequency, the power delivered from sinusoidl voltge source with given RMS vlue V s, s represented in Fig. 1 (considering R s = 0), cn be given by: P in = V s (7) R eq + R where R eq is determined with Eq. () or with Eq. (3). When H-bridge resonnt inverter is used, the rry is fed with squre voltge wve. This mens we cn set V s = V s1 in q. (7), with V s1 determined s in [0]: V s1 = 4 π V sq (8) where V s1 is the RMS of the fundmentl component of the inverter output squre wve v in,ndv sq is the mesured mplitude vlue of the squre wve, whose duty cycle is ssumed to be 0.5. Then, through mesurements, the input power is obtined s the verge vlue in period of the product of the instntneous voltge nd current mesured t the terminls of the inverter: P in,exp = 1 T T 0 v in (t) i in (t)dt (9) where T =1/f 0 is the period of the wveforms, nd v in (t) ndi in (t) re the mesured instntneous vlues of the input voltge nd current. The product of the instntneous voltge nd current nd its verge vlue in period were clculted with the oscilloscope using the mthemticl functions of its internl softwre. We cn then compre the vlues clculted with the theoreticl formul (7) with the experimentl ones obtined with Eq. (9). This is done first considering n rry of 6 resontors terminted with different vlues of R T (0.4 Ω, 1.5 Ω, 5 Ω nd 10 Ω) nd then n rry of 6 resontors terminted with R T =1.5Ω with receiver in different positions. The results of these two comprisons re presented in Figs. 8 nd 9 which show, for given voltge source, the power P in delivered by the source to 6-resontor rry versus the termintion impednce nd the receiver position, respectively. Fig. 8 shows tht P in increses for n incresing vlue of R T ;Fig.9showsthtforfixedR T,the power delivered by the source hs lrge oscilltions depending on the position of the receiver. These exmples show some of the possible prcticl pplictions of the study of the equivlent impednce on

8 108 Alberto et l. Figure 8. Comprison of P in (W ) t the resonnt frequency f 0 = 147 khz obtined with mesurements using Eq. (9) to the input power clculted through the developed formule ((7) using V s =4.9V determined with Eqs. (8) nd ()) versus R T. Figure 9. Comprison of P in (W ) t the resonnt frequency f 0 = 147 khz obtined with mesurements using Eq. (9) to the input power clculted through the developed formule ((7) using V s =4.9V determined with Eqs. (8) nd (3)) versus the position of the receiver. determining nd predicting the input power (nd thus the input current) delivered by given voltge source to the resontor rry for different conditions of the system (vrible R T or vrible position of the receiver) which cn be useful for the design of the rry nd its power source. 5. CONCLUSIONS In this pper, through the ppliction of the theory of liner homogeneous difference equtions, n explicit closed-form expression of the equivlent impednce is developed, which depends on the electricl prmeters of the resontor rry, termintion impednce, number of resontors, position nd impednce of receivers. Moreover, from the mthemticl nlysis of the convergence of the recursive sequence tht defines the continued frction, it is found tht for n rbitrrily lrge number of resontors, the equivlent impednce is given only by the electricl prmeters of the resontor rry, nd tht it does not depend on the termintion impednce nd the number of receivers over the line. Furthermore, by terminting the resontor rry with the equivlent impednce of n infinite number of resontors, the equivlent impednce of the resontor rry is constnt for ny number of resontors. It is lso shown tht the recursive sequence used to model the system hs n oscillting behviour. Then, in order to illustrte the proposed closed-form formule, some exmples of equivlent impednce clcultion for different system configurtions re provided. Moreover, the theoreticl formule for the equivlent impednce nd for the efficiency re vlidted with experiments, nd the comprison shows very good greement.

9 Progress In Electromgnetics Reserch B, Vol. 80, The mthemticl pproch developed in this work gives consistent theoreticl bsis tht cn be used by electricl engineers not only s powerful tool for designing resontor rrys for WPT systems with given properties nd behviour, but lso more specificlly to design the power source tht feeds the resontor rry nd to study the mtching of the source with the rry. In fct, knowing the rry equivlent impednce nd its possible vritions, the current nd ctive power delivered by given voltge source cn be predicted ccurtely, thus sving time comprtively to other numericl or simultion softwre. ACKNOWLEDGMENT H. Albuquerque cknowledges finncil ssistnce by the Centre for Mthemtics of the University of Coimbr UID/MAT/0034/013, funded by the Portuguese Government through FCT/MEC nd co-funded by the Europen Regionl Development Fund through the Prtnership Agreement PT00. APPENDIX A. A.1. Vlue of the Frction without Perturbtion The continued frction in Eq. (1) for the prticulr cse where the receiver is over the first cell, cn be rewritten using generic letters for ny number n + 1 of resontors (with n 0): x n = (A1) b + + b + p 0 q 0 with, b, p 0,q 0 C where q 0 0, nd b both not equl to 0, nd in cse 0thenp 0 0. The previous frction in Eq. (A1) is the nth term of the following recursive sequence (with k 1): x k =, with x 0 = p 0. (A) b + x k 1 q 0 The term x 0 corresponds to the termintion impednce of the first cell, e.g., Ẑ T or ẐT in Eq. (1). In this wy, lbelling the n + 1 cells of the rry from 1 to n + 1, 1 is the first rry cell nd n +1is the cell connected to the source. Thus, noting by x k = p k, (A3) q k we verify by induction tht {p n } nd {q n } re sequences defined by the following recurrence reltions: being p 0 nd q 0 fixed nd p n = bp n 1 + p n q n = bq n 1 + q n for n, (A4) p 1 = q 0. (A5) q 1 = bq 0 + p 0 Eqution (A4) of the type p n bp n 1 p n = 0 is liner homogeneous second order difference eqution with constnt coefficients nd its solution is given by p n = m 1 λ n 1 + m λ n supposing tht λ 1 nd λ re distinct solutions of the eqution λ bλ =0: λ 1 = b b +4 ; λ = b + b +4 (A6) in which m 1 nd m re constnts tht should be determined using the initil conditions. Anlogous considertions cn be done for {q n }.

10 110 Alberto et l. Concluding, the generl term of the sequence {x n } = {pn} {q n} is given by ( b ) n ( b +4 b + ) n b x n = ( b ) n ( b b +4 b + ) n (A7) b 1 + b +4 where 1,, b 1 nd b re determined by the initil conditions x 0 nd x 1, nd re given in Eq. (A.3), ssuming, for simplicity tht p 0 = x 0 nd q 0 =1. A.. Vlue of the Frction with Perturbtion in the ith Term We determine now the vlue of expression (1), which is the equivlent impednce of multiple resontor system for receiver plced over the ith cell of the receiver line. To do this, we rewrite the frction s generic continued frction of the nth order with perturbtion b b in the step i n of the recursive sequence: x n = with, b, b,p 0,q 0 C. (A8) b b + b b + p 0 q 0 To solve this frction, we split the frction in two continued frctions (one with i nd the other with n i terms). After clculting the vlue of x k for the (i 1)th term, we determine the ith vlue x i using the perturbtion b nd, finlly, using x i s n initil vlue, we compute the vlue of the frction, with the lst n i vlues. So, we strt using Eq. (A7) to determine the term of (i 1)th order: 1 (b ) i 1 ( b +4 + b + ) i 1 b +4 x i 1 = b 1 (b ) i 1 ( b +4 + b b + ) i 1 (A9) b +4 where 1,, b 1 nd b re determined by the initil conditions x 0 nd x 1 s done before in Eq. A.1. After, we hve to clculte the vlue of x i = b = y 0 ; nd = y 1. (A10) + x i 1 b + y 0 Then, Eqs. (A10) re the initil conditions used to determine the vlue of the frction y n i = x n : c 1 (b ) n i ( b +4 + c b + ) n i b +4 y n i = d 1 (b ) n i ( b +4 + d b + ) n i. (A11) b +4 The constnts c 1, c, d 1 nd d re computed using the initil conditions y 0 nd where it is ssumed tht p 0 = y 0 nd q 0 = 1. These constnts re given in A.3 Expression (A11) represents the vlue of the frction in Eq. (1) for n + 1 resontors with the perturbtion in the ith term, (receiver fcing the ith resontor, i.e., i = 1 corresponds to the first resontor nd i = n to the resontor next to the one connected to the source).

11 Progress In Electromgnetics Reserch B, Vol. 80, A.3. Constnts 1,, b 1, b, c 1, c, d 1, d 1 = x 0 bx 0 b +4 ; = x 0 + bx 0 b +4 ; b 1 = 1 x 0 + b b +4 ; b = 1 + x 0 + b b +4 ; c 1 = y 0 y 0b b +4 ; c = y 0 + y 0b b +4 ; d 1 = 1 y 0 + b b +4 ; d = 1 + y 0 + b b +4. A.4. Convergence of the Continued Frction Following the study of the frction, we cn predict its behviour for n infinite number of terms, in other words, its vlue for n. Supposing tht z bz = 0 hs distinct roots z 1 = b + b +4 nd z = b b +4, with z 1 > z,wehve ( ) z n < 1, so lim z =0. (A1) nd z 1 n x n = z1 n + 1z n b z1 n + b 1z n, thus lim x n = lim n n z ( z z 1 b + b 1 ( z z 1 ) n ) n = b. (A13) This proves very importnt fct: for fixed, b C, the vlue of the frction is the sme, /b = 1 ( 4 + b b) when the number of terms is infinite, not depending of the initil condition x 0. So if we tke x 0 equl to this vlue {x n } becomes constnt sequence. Moreover, for finite number of perturbtions, the behviour of the frction t infinity remins the sme. A.5. Determintion of the Constnts of the Frction ( ) ( ) e 1 = Ẑ d (ωm) fẑ T ; e = Ẑ d (ωm) gẑ T ) e 3 = g (Ẑ ( ) Ẑ d ẐT +(ωm) h + Ẑd +ẐT ) e 4 = f (Ẑ ( ) Ẑ d ẐT +(ωm) h + Ẑd +ẐT ( ) ( ) ) e 5 =(ωm) 4(ωM) + f (Ẑ + Ẑ d + h + Ẑd ẐT ( ( )) ( ( )) e 6 = Ẑd fẑẑt +(ωm) f +ẐT ; e 7 = Ẑd gẑẑt (ωm) g ẐT ( ) ( ) ) e 8 =(ωm) 4(ωM) + g (Ẑ + Ẑ d + h + Ẑd ẐT Note: for resonnce conditions Ẑ, ẐT nd Ẑd re replced by R, R T nd R d, respectively. REFERENCES 1. Olvitz, L., D. Vinko, nd T. Svedek, Wireless power trnsfer for mobile phone chrging device, Proc. 35th Int. Convention MIPRO 01, , My 01.. Nguyen, M. Q., Z. Hughes, P. Woods, Y. S. Seo, S. Ro, nd J. C. Chio, Field distribution models of spirl coil for mislignment nlysis in wireless power trnsfer systems, IEEE Trns. on Microwve Theory nd Techniques, Vol. 6, No. 4, , Apr Li, S. nd C. C. Mi, Wireless power trnsfer for electric vehicle pplictions, IEEE Journl of Emerging nd Selected Topics in Power Electronics, Vol. 3, No. 1, 4 17, Mr. 015.

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