Maximum efficiency formulation for inductive power transfer with multiple receivers

Transcription

1 LETTER IEICE Electroics Express, Vol1, No, 1 10 Maximum efficiecy formulatio for iductive power trasfer with multiple receivers Quag-Thag Duog a) ad Mioru Okada Graduate School of Iformatio Sciece, Nara Istitute of Sciece ad Techology, Takayama-cho, Ikoma, Nara 0 019, Japa a) thag@isaistjp Abstract: This paper presets closed-form expressios of optimal loads that achieve the maximum efficiecy for iductive power trasfer (IPT) system with multiple receivers We model the system as a passive N-port etwork ad joitly optimize both reactive ad resistive compoets of the loads The derived expressio allows oe to predict RF-to-RF efficiecy of IPT system with ay umber of receivers, uder ay couplig coditio ad through ay type of media Keywords: iductive power trasfer, multiple receiver, maximum efficiecy, optimal load Classificatio: Circuits ad modules for electroic istrumetatio Refereces IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 [1] T Imura ad Y Hori: Maximizig air gap ad efficiecy of magetic resoat couplig for wireless power trasfer usig equivalet circuit ad Neuma formula, IEEE Tras Id Electro 58 (011) (DOI: /TIE ) [] M Zargham ad P G Gulak: Maximum achievable efficiecy i ear-field coupled power-trasfer systems, IEEE Tras Biomed Circuits Syst (01) 8 (DOI: /TBCAS01119) [] T Ohira: Exteded k-q product formulas for capacitive- ad iductivecouplig wireless power trasfer schemes, IEICE Electro Express 11 (01) 0101 (DOI: 10158/elex110101) [] T Ohira: Maximum available efficiecy formulatio based o a black-box model of liear two-port power trasfer systems, IEICE Electro Express 11 (01) 0108 (DOI: 10158/elex110108) [5] T Zhag, et al: Optimal load aalysis for a two-receiver wireless power trasfer system, 01 IEEE Wireless Power Trasfer Cof (01) 8 (DOI: /WPT018900) [] T Imura: Simultaeous wireless power trasfer via magetic resoat couplig to multiple receivig coils, IEEJ Trasactio o Idustry Applicatios 1 (01) 5 (i Japaese) (DOI: 10151/ieejias15) [] M Fu, et al: Efficiecy ad optimal loads aalysis for multiple-receiver wireless power trasfer systems, IEEE Tras Microw Theory Tech (015) 801 (DOI: /TMTT01598) [8] S Kim ad B Lee: Aalysis of efficiecies for multiple-iput multiple-output 1

2 IEICE Electroics Express, Vol1, No, 1 10 wireless power trasfer systems, J Electromag Eg Sci 1 (01) 1 (DOI: /JKIEES0111) [9] T Imura: Cross-couplig cacelig method for wireless power trasfer via magetic resoace couplig, IEEJ Tras Idustry Appl 1 (01) 5 (i Japaese) (DOI: 10151/ieejias15) [10] C Zhog, et al: Reactace compesatio method to elimiate cross couplig for two-receiver wireless power trasfer system, IEICE Electro Express 1 (015) (DOI: 10158/elex ) [11] D Ah ad S Hog: Effect of couplig betwee multiple trasmitters or multiple receivers o wireless power trasfer, IEEE Tras Id Electro 0 (01) 0 (DOI: /TIE011990) [1] S Boyd ad L Vadeberghe: Covex Optimizatio (Cambridge Uiversity Press, 00) d ed 10 1 Itroductio IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 Efficiecy maximizatio based o load optimizatio for sigle-receiver iductive power trasfer (IPT) has bee studied extesively Efficiecy maximizatio for purely iductive couplig was ivestigated i [1] [, ] the exteded the solutio for geeral IPT system, i which the couplig ca be purely reactive (through air) or coductive (through tissue or biological media) Maximum efficiecy is the mathematically formulated based o passive two-port etwork model i [] These studies yield isights ito the system behaviors ad, at the same time, provided powerful tools for aalyzig more complicated IPT systems Efficiecy maximizatio for multiple-receiver IPT is oe of the most attractive topic Solutio for system with two receivers is studied i [5] ad the exteded i [,, 8] for system with arbitrary umber of receivers These works showed that the efficiecy icreases mootoically with the umber of receivers, meaig that simultaeous charig is more efficiet tha chargig i a time divisio maer As [,, 8] optimized oly resistaces of the loads, these works, however, do ot aswer a importat questio that whether simultaeous resoace at the receivig coils gives the maximum efficiecy Cross-couplig amog receivers is a importat issue which has bee ivestigated experimetally i literatures [9, 10, 11] However, to the best of our kowledge, closed-form expressio of optimal loads for a geeral sceario has ot bee derived so far Furthermore, the formulatios reported i literatures do ot cosider the case of couplig through coductive media Resolvig these issues requires the challegig joit optimizatio of both resistive ad reactive compoets of the loads This paper aims to derive closed-form expressios of optimal loads ad maximum efficiecy for a geeral multiple-receiver IPT system We model the system by passive N-port etwork ad joitly optimize load resistaces ad reactaces by applyig the first-order ecessary coditio [1] Verified by umerical aalysis, our expressio ca be used to predict RF-to-RF efficiecy for IPT system with ay umber of receivers, uder ay couplig coditio ad via ay type of media

3 IEICE Electroics Express, Vol1, No, 1 10 System model ad problem formulatio 1 System model We cosider a sigle frequecy IPT system with oe trasmitter ad ðn 1Þ receivers as illustrated i Fig 1 The coils are surrouded by arbitrary media ad there exists cross-couplig amog the receivig coils The system ca be modeled by liear reciprocal N-port etwork where port 1 is the iput ad ports to N are the outputs I practice, port 1 is usually coected to a siusoidal power source ad ðn 1Þ passive loads are coected to ports from to N The power trasfer efficiecy of the system, or RF-to-RF efficiecy, is the ratio of total output power from ports to N to iput power to port 1 The efficiecy depeds o the load impedaces ad the N-port etwork parameters I order to achieve the maximum efficiecy, oe eed to be able to freely choose the values of load impedaces I practice, load impedaces basically vary whe the loads are i operatios Therefore, matchig etworks or DC-DC coverters are usually iserted betwee the passive loads ad the output ports i order to covert load impedaces to the optimal values Here, sice we are iterested i the values of loads which is see from the output ports, we express the load impedace by voltage-to-curret ratios ad hide the power source, actual passive loads ad matchig etworks from Fig 1 Fig 1 System model of geeral multiple-receiver IPT Curret-voltage relatio of the system is expressed by Z-parameters as V 1 Z 11 Z 1 Z 1N I 1 V Z 1 Z Z N I ¼ 5 5 ; ð1þ 5 IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 V N Z N1 Z N Z NN where I 1, V 1 are complex-valued curret ad voltage phasors of the iput port ad I, V ( f; ;Ng) are those of the outputs I this model, we characterize the couplig betwee output port ( f; ;Ng) ad iput port 1 by the term Z 1 ¼ R 1 þ jx 1, where R 1 ad X 1 respectively demostrate the coductive I N

4 IEICE Electroics Express, Vol1, No, 1 10 (or direct) path ad the iductive path of the couplig The cross-couplig amog receivig coils is represeted by the term Z m ¼ R m þ jx m, where m f; ;Ng ad m Load impedace at the output port is determied by Z L ¼ V =I Problem formulatio Without loss of geerality, we take I 1 as the referece phasor ad formulate the efficiecy i terms of I From Eq (1), the real part of V 1 ca be expressed as V re 1 ¼ R 11I 1 þ XN ¼ ðr 1 I re X 1I im Þ; where I re ad I im are the real ad imagiary parts of I, respectively Real ad imagiary parts of V are ðþ V re ¼ R 1I 1 þ XN m¼ ðr m Im re X mim im Þ; ðþ V im ¼ X 1 I 1 þ XN m¼ R m Im im þ X mim re : ðþ Therefore, iput power to port 1 is give by " # P 1 ¼ 1 V 1 re I 1 ¼ 1 R 11I1 þ XN I 1 ðr 1 I re X 1I im Þ : ð5þ The output power from port is ¼ P ¼ 1 re ðv I re thus the total output power is determied by P out ¼ XN ¼ P ¼ 1 ¼ þ V im I im Þ; ðr 1 I 1 I re þ X 1I 1 I im Þ 1 R m ði re I m re þ I im I m im Þ: ¼ m¼ Fially, RF-to-RF efficiecy is formulated as ðþ ðþ ¼ P out P 1 ¼ ¼ ðr 1 I 1 I re þ X 1I 1 I im ÞþXN R m ði re I m re þ I im I m im Þ ¼ m¼ R 11 I 1 þ XN ¼ I 1 ðr 1 I re X 1I im : ð8þ I Eq (8), I 1 is a parameter The efficiecy η ca be viewed as a fuctio of I re ad I im re im The output currets I ad I take values from 1 to 1 give that all output power is o-egative, ie, P 0 8 f; ;Ng Maximum efficiecy ad optimal loads IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 We derive the maximum value of the multi-variable fuctio η by applyig the firstorder ecessary coditio [1] o I re 8 f; ;Ng, we have ad I im Whe the coditio is satisfied,

5 IEICE Electroics Express, Vol1, No, re im ¼ 0: From equatios (8) ad (9) it follows a system of simultaeous equatios Solvig this system of equatios gives us the curret values, from which we ca derive the maximum efficiecy Sice it is very complicated to solve these equatios directly, we do this i a subtle way re ¼ ð@p out=@i reþp 1 ð@p 1 =@I reþp out P ; im ¼ ð@p out=@i imþp 1 ð@p 1 =@I imþp out P : ð11þ 1 From this it follows that 8 f; ;Ng, Let R rr ¼ m¼ m¼ ð9þ R m Im re ¼ R ð1 þ ÞI 1 1 ; ð1þ R m I im m R R N 5 ; ¼ X 1 ð1 ÞI 1 : ð1þ IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 R rt ¼ R N R NN R 1 R N1 5 ; X rt ¼ X ; Ire r ¼ Equatios (1) ad (1) ad be expressed as Ir re Ir im I re I re N ¼ R 1 rr R rtð1 þ ÞI 1 =; ¼ R 1 rr X rtð1 ÞI 1 =: 5 ; Iim r ¼ I im I im N 5 : ð1þ Substitutig Eq (15) ad Eq (1) i to Eq (8), we get a equatio of η A þ B 1 A B þ A þ B ¼ 0 ð1þ which yields two solutios ¼ ð pffiffiffiffiffiffiffiffiffiffiffi p 1 þ A ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi p 1 BÞð 1 þ A þ ffiffiffiffiffiffiffiffiffiffiffi 1 BÞ pffiffiffiffiffiffiffiffiffiffiffi p ð 1 þ A ffiffiffiffiffiffiffiffiffiffiffi ; ð18þ 1 BÞ where A ¼ 1 X T rt R R 1 rr X rt; B ¼ 1 R T rt 11 R R 1 rr R rt: ð19þ 11 The coditio that 0 <1 is satisfied oly with the solutio received whe replacig the symbol i Eq (18) by This solutio is the maximum efficiecy ð15þ ð1þ 5

6 IEICE Electroics Express, Vol1, No, 1 10 pffiffiffiffiffiffiffiffiffiffiffi p 1 þ A ffiffiffiffiffiffiffiffiffiffiffi 1 B max ¼ pffiffiffiffiffiffiffiffiffiffiffi p 1 þ A þ ffiffiffiffiffiffiffiffiffiffiffi : ð0þ 1 B Whe the efficiecy reaches its maximum, port currets are give by I I N 5 ¼ R 1 rr ½R rtð1 þ max ÞþjX rt ð1 max ÞŠI 1 =: Ad port voltages V ¼ Z 1I 1 þ P N m¼ Z mim Thus, the optimal loads are Z LðoptÞ ¼ R LðoptÞ þ jx LðoptÞ ð1þ Z 1I 1 þ Z m Im ¼ V =I ¼ m¼ I : ðþ Case study 1 IPT through air ad without cross-couplig Couplig through air is characterized by R rt ¼ 0, R rr ¼ diagðr ; ;R NN Þ ad the coditio of o cross-couplig is characterized by X m ¼ 0 (8m ) I this case, A ¼ 1 P N X1 R 11 ¼ R, B ¼ 0 ad the maximum efficiecy becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 X1 1 R 11 ¼ R max ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðþ 1 þ 1 X1 þ 1 R 11 ¼ R The resistace ad reactace of the optimal load are vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u R LðoptÞ ¼ R 1 þ 1 X t N X1 ; ðþ R 11 R ¼ X LðoptÞ ¼ X : ð5þ Maximum efficiecy ad optimal load resistace give by equatios () ad () matches perfectly with the formulatios i [,, 8] Optimal load reactace determied by Eq (5) cofirms the optimality of the commo practice of resoatig out the receivig coils As will be show i sectios ad, this commo practice loses its optimality whe couplig is through coductive media ad/or whe the cross-couplig exists IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 IPTthroughairadwithcross-couplig I this case, also R rt ¼ 0, R rr ¼ diagðr ; ;R NN Þ, thus A ¼ 1 P N X1 R 11 ¼ R B ¼ 0 As a result, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 X1 1 R 11 ¼ R max ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 X1 þ 1 R 11 ¼ R ad ðþ

7 IEICE Electroics Express, Vol1, No, 1 10 Fig Two-receiver IPT system with cross-couplig R LðoptÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ R 1 þ 1 X t N X1 R 11 R ¼ ðþ This result implies that the cross-couplig does ot affect the maximum efficiecy ad optimal load resistace However, i this case, sice X m 0 (8m ), the optimal load reactace becomes IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 X LðoptÞ ¼ X XN m¼;m X m X m1 R X 1 R mm ; meaig that resoatig out the receivig coils is ot optimal uder the existece of cross-couplig Aother importat observatio ca be draw from Eq () is that the efficiecy also icreases with the umber of receivers I order to verify the effectiveess of equatios () (8), we perform electromagetic field aalysis for a IPT system with two receivers show i Fig I this aalysis, all coils are made from copper wire with diameter of 1 mm ad with coductivity of 5:8 10 S/m Operatig frequecy f ¼ 1:5 MHz, which is a idustrial scietific ad medical (ISM) bad I this system, coil 1 is the trasmittig coil while coil ad coil are the receivig Sizes ad positios of coils are illustrated i the figure By computer simulatio, Z-parameters for the system are observed as Z ¼ 0:9 þ j9: j1:0 j1:0 j1:0 0:9 þ j9:1 j:1 j1:0 j:1 0:9 þ j9:1 ð8þ 5; ð9þ idicatig that there exists cross-couplig betwee coils ad as Z ¼ Z ¼ j:1[ω] We the carry out a exhaustive search to fid the optimal loads as well as maximum efficiecy Because receivig coils have the same size ad distace to the trasmittig coil, we assume all loads as Z L ¼ ZL ¼ RL þ jx L We varied the loads R L, X L ad obtaied efficiecy η as show i Fig The simulatio result shows that optimal loads ad maximum efficiecy are 0 j100 [Ω] ad ¼ 0:9 These values are very tight to the theoretically optimal values 19:85 j100 [Ω] ad max ¼ 0:9 calculated by equatios () (8) with Z- parameters give i Eq (9)

8 IEICE Electroics Express, Vol1, No, 1 10 Fig Efficiecy vs load resistace ad reactace IPT through coductive media ad without cross-couplig Couplig through coductive media is characterized by R rt 0 Thus A ¼ 1 P N X1 R 11 ¼ R, B ¼ 1 P N R 1 R 11 ¼ R ad sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 X1 1 1 R 1 R 11 ¼ R R 11 ¼ R max ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð0þ 1 þ 1 X1 þ 1 1 R 1 R 11 R 11 R LðoptÞ ¼ R ¼ R ð1 max ¼ ÞðR 1 þ X 1 ÞR ð1 þ max Þ R 1 þð1 maxþ ; ð1þ X1 IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 X LðoptÞ max R 1 X 1 R ¼ ð1 þ max Þ R 1 þð1 maxþ X X1 : ðþ Oe should ote that max i equatios (1) ad () is determied by Eq (0) This result idicates that eve whe the couplig is via coductive media, the efficiecy also icreases with the umber of receivers However, resoatig out the output ports loses its optimality as Eq () shows that the optimal reactace is differet from X Numerical aalysis is performed for a IPT system with two receivers show i Fig I this system, all coils are made from 1 mm 1 mm square solid rods of copper with coductivity of 5:8 10 S/m ad the operatig frequecy f ¼ 1:5 MHz Coil 1 is the trasmittig coil while coil ad coil are the receivig All coils are placed o mm-thick acrylic substrates with relative permittivity of ad coductivity of 1: 10 5 S/m The medium betwee trasmittig ad receivig coils is sea water with relative permittivity of 810 ad coductivity of 5S/m I this ivestigatio, thickess of the medium is varied as d ¼ 10, 15, 0, 5, 0 mm Z-parameters for the three-port etwork are obtaied via simulatio ad show i Table I The orm of tras-impedace betwee ports ad is observed to be very small, cofirmig that the cross-couplig betwee is egligible Because receivig coils have the same size, we assume loads impedaces as Z L ¼ ZL ¼ R L þ jx L Fig 5 shows the variatio of η with R L ad X L whe d ¼ 10 mm The simulatio result shows that optimal loads ad maximum efficiecy are Z L ¼ 8

9 IEICE Electroics Express, Vol1, No, 1 10 Fig Two-receiver IPT through coductive media without crosscouplig Table I Z-parameters measured by simulatio of system i Fig d [mm] R 11 [Ω] X 11 [Ω] R [Ω] X [Ω] R [Ω] X [Ω] R 1 [Ω] X 1 [Ω] R 1 [Ω] X 1 [Ω] Fig 5 Efficiecy vs load resistace ad reactace whe d ¼ 10 mm IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 0:80 j8:0[ω] ad ¼ 0:8, respectively These values are very close to the theoretically optimal value Z LðoptÞ ¼ 0: j:99[ω], max ¼ 0:88 calculated by equatios (0) () Fig demostrates the simulated efficiecy ad the efficiecy that is theoretically estimated by Eq (0) for various values of d The two values are show to be very tight to each other, cofirmig the optimality of the derived solutios 9

10 IEICE Electroics Express, Vol1, No, 1 10 Fig Efficiecy whe load are optimized usig equatios (1) ad () 5 Coclusio ad discussio This paper derived closed-form expressios of optimal loads that maximize RF-to- RF efficiecy for geeral multiple-receiver IPT system Verified by umerical aalysis, the derived expressios give some iterestig isights ito the system behavior as follows Cross-couplig, i fact, does ot affect the maximum efficiecy May previous ivestigatios have reported that cross-couplig decreases the system efficiecy However, i such ivestigatios the loads are ot optimal As mathematically show i sectio, whe the loads are optimized by the derived expressios, the impact of cross-couplig ca be perfectly mitigated Resoatig out the receivig coils is optimal oly for the case of couplig through air ad without cross-couplig amog receivers This commo practice loses its optimality whe the media is coductive ad/or whe the crosscouplig exists The maximum efficiecy icreases with the umber of receivers, regardless of couplig coditio ad/or electrical property of the surroudig media Experimetal studies are required to further cofirm the derived expressio as well as the aforemetioed observatios Ackowledgmets This work was supported by DAIHEN Corporatio IEICE 01 DOI: 10158/elex Received September 19, 01 Accepted October 1, 01 Publicized November, 01 Copyedited November 5, 01 10