I grid. I pvm + V pvm - I pv1 + V pv1 - V grid. Fig. 1. Conventional micro-inverter based PV system with central ESS. ACModulem MIV. ac grid.
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1 A Dual Acive Bridge DCDC Coverer wih Opimal DCLik Volage Scalig ad Flyback Mode for Ehaced LowPower Operaio i Hybrid PV/Sorage Sysems Shahab Poshkouhi ad Olivier Trescases Uiversiy of Toroo, 0 Kig s College Road, Toroo, ON, M5S 3G4, Caada shahab.poshkouhi@uoroo.ca Absrac Today s PV microiverers (MVs) provide a modular soluio for geeraio, however he eergy sorage archiecures remai ceralized, requirig a addiioal bidirecioal acdc coverer, wih complex cell balacig circuis. Disribuig sorage capaciy wihi he smar PV paels allows power flucuaios o be locally buffered, while miimizig he eed for addiioal power elecroics ad balacig circuis. The dualacivebridge (DAB) opology, which is adoped i his paper, provides bidirecioal power flow; however i geerally suffers from poor efficiecy a low power. is show ha wih a mior modificaio, he DAB ca be operaed as a worasisor flyback coverer for improved efficiecy. addiio, he dclik volage ca be dyamically adjused for he bes performace i DAB mode. The proposed corol scheme is demosraed o a 00 W prooype, wih up o 8% icrease i efficiecy a low power. dex Terms Phoovolaic (PV) microiverers, efficiecy, isolaed dcdc sage.. NTRODUCTON The coiuous declie of phoovolaic (PV) module prices, compouded wih aracive feedi ariffs i a variey of jurisdicios, is leadig o he rapid deployme of PV isallaios hroughou he world []. The iermie aure of PV ad oher reewable eergy resources, ad hus he eed for eergy sorage ad/or load sheddig, is a major challege i smallscale PV based grids. This is despie he fac ha powerqualiy ad oher grid sric requiremes such as frequecy variaios are reduced compared o coveioal grids. Lowpower dcdc microcoverers (MC) [2], [3] ad acdc microiverers (MV) [4], [5] provide highgraulariy Maximum Power Poi Trackig (MPPT) [6], [7] a he module or subsrig level. This leads o icreased robusess o clouds, dir, ad agig effecs, as well as irradiace ad emperaure gradies [7]. A coveioal MV based ac power sysem is show i Fig.. The Eergy Sorage Sysem (ESS), which is defiiely required for isladed operaio o he scale of oe or more houses for example, is usually based o a highpower ceralized bidirecioal acdc coverer, which is ierfaced o a baery bak or a flywheel [8], [9]. Exisig MV archiecures saisfy he eed for low capialcos ad expadable ac geeraio, while here is compellig argume o exed his echology o iclude smallscale disribued sorage. A ovel opology wih disribued sorage is proposed pvm V pvm pv V pv Ceral ESS ACModulem MV ACModule MV grid V grid ac grid ac Loads Fig.. Coveioal microiverer based PV sysem wih ceral ESS. i [0] for grid sabilizaio, while savig fuel, ad improvig he geeraor lifeime. MV egraed sorage helps o buffer he freque irradiace flucuaios, while also providig local backup power ad reacive power suppor [], [2]. A lowpower siglesage mulipor coverer for PV ad baery is proposed i [3], while a 3kW iercoecio of a baery pack ad a PV module hrough a isolaed dcdc coverer is discussed i [4]. The geeral archiecure of a wosage MV wih a iegraed ESS is show i Fig. 2. While wosage MVs have a slighly lower efficiecy ha siglesage MVs, he highvolage dc lik capaciace, C bus, ca be used for ac power decouplig i siglephase sysems [5], [6]. erfacig he lowvolage dc sorage, eiher baeries or ulracapaciors, direcly o he P is preferable for high efficiecy [4]. Lihiumio ulracapaciors [7], which offer 24 higher specific eergy ha coveioal Elecric Double Layer Capaciors ad ca wihsad more ha charge/discharge cycles, are a aracive fuure cadidae for shorerm MV iegraed sorage. The focus of his work is o he froed dcdc sage. The objecive of his paper is o ) discuss a bidirecioal isolaed dcdc sage for he PVbus coecio, which is al
2 ready preseed i [0] ad 2) demosrae a ovel lowpower operaig mode ad dyamic dclik opimizaio scheme o maiai high efficiecy over a broad power rage. This is crucial i ay commercial MV archiecure; for example he Europea Efficiecy idex dedicaes 32% of he oal evaluaio weigh o operaio below 30% of he raed power [8]. Dcdc Sage Focusofhiswork egraed Sorage erface MV Archiecure STG wih egraed V STG Sorage Dcac Sage ac ac grid Fig. 2. Twosage MV archiecure wih iegraed sorage [0].. PROPOSED DAB ARCHTECTURE AND PRNCPLE OF OPERATON The proposed dcdc archiecure is show i Fig. 3. This coverer is a modified DualAciveBridge (DAB) ha ierfaces V pv wih he dc lik,. A. DAB Mode The DAB opology was seleced based o () galvaic isolaio, (2) sofswichig operaio ad (3) simple phaseshif power corol [9], [20]. addiio, he DAB opology is bidirecioal, herefore he sorage ca be used o rasfer eergy o/from oher elemes i he grid. The average power from V pv o, P, is P = V P V ϕ( ϕ ), () ω s L DAB π where is he rasformer s urs raio, L DAB is he DAB iducace, which is he sum of rasformer s leakage iducace, L leak, ad a opioal exeral iducace, L ex. ϕ is he phaseshif bewee he wo bridges, ad ω s = 2πf s, where f s is he swichig frequecy. The swichig waveforms of he DAB coverer are show i Fig. 4. The slopes of he DAB iducace curre, LDAB, i swichig saes ad are respecively calculaed as s = V P V, (2) L DAB s 2 = V P V. (3) L DAB wosage MV archiecures, is geerally regulaed o a fixed volage by he iverer sage. The referece volage, Vbus, is usually chose o opimize efficiecy a he omial operaig poi [7]. ca be show ha he DAB coverer achieves uro ZeroVolageSwichig (ZVS) ad maximum efficiecy whe = V P V, as he reacive circulaig curre is miimized [9]. PV PV i Ci i Ci M M3 M M3 Primary Side M2 M4 egraed Sorage erface M2 M4 egraed Sorage erface D VLDAB LDAB L DAB : M5 M6 Cbus Lm Vx STG V STG Lm Vx2 Secodary Side M7b D VLDAB LDAB L DAB : M5 M6 Cbus Lm Vx STG V STG Lm OFF ON Swichig Fig. 3. a) Proposed modified DAB dcdc archiecure for improved low power efficiecy. b) Swich cofiguraio i Flyback mode. order o miimize he losses i he DAB, he referece DC lik volage, Vbus, is dyamically adjused i he iverer sage such ha Vbus = V MP P, where V MP P is he PV MPP volage. is wellkow ha V MP P udergoes a relaively low flucuaio of abou 30% durig he course of a ypical day [2]. This is i coras o he PV curre a MPP, MP P, which is proporioal o irradiace ad hus has largescale flucuaios, especially o cloudy days. B. Flyback Mode A ypical PV geeraio sysem speds more ha wohirds of he ime operaig below 50% of is raed power [7]. The coveioal DAB coverer suffers from relaively poor efficiecy a low power due o high swichig ad drive losses [9], hece he eed for a dedicaed lowpower mode. By drivig M ad M 4 simulaeously o he primary side, he coverer ca be operaed as a worasisor flyback coverer (2Tflyback) [22]. The swich cofiguraio i Flyback mode is show i Fig. 3. The swiches M ad M 4 remai acive, M 8 is kep o ad all oher swiches are off. The secodaryside bridge is slighly modified wih he addiio of oe swich, M 7b, o achieve bidirecioal blockig capabiliy i Flyback mode. The chargig ad dischargig slopes of he mageizig Vx2 M7 M7b M7 M8 M8
3 c,4 c 2,3 c 6,7 c 5,8 V x V x2 LDAB Sae: c,4 c 2,3,57 c 8 Lm LDAB D Sae: s T o = D T s s 3 T s/2 s 3 s 5 s 6 s 4 D 2 T s Circulaig Charge s 7 T s s i pri (0) Ts j 2 p V V T s i pri (φ) i pri (0) i pri (φ) Fig. 4. Swichig waveforms i DAB mode, ad Flyback mode. iducace curre, Lm, are give by s 3 = s 2 s 2 V P V L DAB L m, (4) s 4 = L m. (5) The DAB iducace circulaes charge i every swichig period i his mode. The chargig slope of LDAB is he same as s 3 ad he dischargig slope is s 5 = V P V. (6) L DAB Fially, he oupu diode s curre, D, delivers charge o he bus wih he followig slopes i swichig saes ad s 6 = s 4 s 5, (7) s 7 = s 4. (8) The 2Tflyback opology exhibis several advaages over DAB mode for low power codiios, icludig less swichig ad gaedrive losses (wo swichig devices versus ie i he DAB mode). Ulike he more coveioal sigle rasisor flyback opology, he body diodes of M 2 ad M 3 clamp he drai volage o he swichig devices M ad M 4, which reduces Elecromageic erferece (EM) ad limis he blockig volage raig o he primary swiches o V P V. The Flyback mode is operaed wih fixed oime, T o, i Pulse Frequecy Modulaio (PFM) mode [23], where T o is give by T o = D T s, (9) where D is he duy cycle i Flyback mode, ad T s is he swichig period. The correspodig waveforms of he coverer are show i Fig. 4. There are wo ihere limiaios o he 2Tflyback opology: ) D mus be less ha 50% i order o avoid rasformer sauraio, ad 2) mus be less ha V pv, o esure ha he body diode of M 5 rasfers power o whe he primaryside swiches are off. As a resul, eeds o be reduced i Flyback mode. The presece of L DAB resuls i addiioal losses, sice i circulaes curre i a swichig period. The eergy capured i L DAB is rasferred back o he ipu capaciace, C i, i he 2Tflyback opology, as opposed o a coveioal flyback scheme, which does o provide a reur pah for he charge i he leakage iducace. addiio, L DAB resuls i he sof uro of he oupu diode. The Flyback mode exhibis uidirecioal power rasfer. The coverer ca operae wih reverse power flow by addig aoher swich o he primary side. This addiioal swich is o icluded i he experimeal prooype, as he efficiecy i DAB mode is sesiive o coducio losses a he lowvolage, highcurre primaryside. While possible, reverse power capabiliy is o sricly eeded i lowpower Flyback mode; he DAB ca be preveed from operaig i his codiio by adopig bursmode corol isead, albei a slighly lower efficiecy ha Flyback mode. C. Dual Mode Corol The cocepual corol diagram of he coverer is show i Fig. 5. c 8 deoe he gaig volages for swiches M 8. The DAB mode is adoped if P is higher ha a hreshold value, P hresh, or if P is egaive, i which case he sorage is charged direcly from he bus. DAB mode, ϕ is corolled o regulae he power flow o/from he dcac sage, while he sorage eleme s SaeofCharge (SOC) ad MPPT operaio ca be corolled by he dedicaed ierface coverer.
4 Flyback mode, T s is adjused by he coroller, G c2 (s), i order o regulae P o P. Assumig ha he mageizig iducace of he rasformer, L m, is much larger ha L DAB, he power flow is give by G c2(s) P = (V P V D ) 2 T s 2L m. (0) clk rese P PWM φ c 4 P * G c(s) T s Freewheelig couer c 58 PhaseShif Codiio P<0 0 P P hresh <P P hresh Flyback mode To Eable(e) (DABMode) 0(Flyback Mode) (DABMode) 0 c 37 c 8 c c 2 c 8 DAB mode Fig. 5. Simplified cocepual corol diagram.. EFFCENCY ANALYSS c 8 Mode Selecor This secio discusses he domia power losses i he DAB ad Flyback modes. A. Coducio Losses As show i Fig. 4, he curre io he leakage iducor ad primary widig of he rasformer approach a perfec rapezoid whe = V pv. The followig equaio for he RMS curre a he primary side of rasformer, pri, ca be obaied [24] pri = 3π (i pri(0) 2 γ i pri (ϕ) 2 (ϕ γ) (π ϕ)(i pri (0) 2 i pri (ϕ) 2 i pri (0)i pri (ϕ)), () where γ = M U X e c 8 i pri (0) i pri (0) i pri (ϕ), (2) ad i pri (0) ad i pri (ϕ) are he isaaeous curres of he rasformer a he primary side a imes = 0 ad = ϕ ω s, respecively. These ca be easily calculaed cosiderig he symmery of he rasformer curre [24] i pri (0) = (πv P V (π 2ϕ) ), 2L DAB ω s (3) i pri (ϕ) = ((π 2ϕ)V P V π ). 2L DAB ω s (4) Similar calculaios ca be doe for he rasformer s secodary side RMS curre, sec. Two primary ad wo secodary swiches are coducig a each isace i DAB mode. Thus, he coducio losses i his mode are approximaed by P DAB,cod = (2R o,pri R LDAB ) 2 pri (5) 2.5R o,sec 2 sec, where R o,pri ad R o,sec are he primary ad secodary side swiches oresisaces respecively, ad R LDAB is he lumped widig resisace of he rasformer ad iducor. The facor of 2.5 o he secodary side comes from he fac ha, here are wo backoback swiches, M 7, ad M 7b o oe leg i he secodary side o suppor he flyback operaio. Neglecig he clampig diodes coducio ierval, he RMS curre for he wo swiches, M,M 4, ad i he oupu diode, D, i he 2Tflyback coverer ca be obaied as [25] M,M 4 = P D ( D ), (6) D = D 2T s D2 2 L m 3, (7) where D 2 is approximaed by he followig D 2 = V P V D. (8) The oal coducio loss i Flyback mode is P F lbk,cod = (2R o,pri R LDAB ) 2 M (9) V F D R o,sec 2 D, where V F is he oupu diode s forward volage. B. Swichig Losses The swiches i DAB mode ca be ured o realizig Zero Volage Swichig (ZVS) [24]. However he uroff losses are o elimiaed. The oal swichig losses i his mode ca be approximaed by P DAB,sw = 2 f s off (V pv (i pri (0) i pri (ϕ) (20) (i sec (0) i sec (ϕ)), where off is he uroff ime of he MOSFETs. Swiches M ad M 2 exhibi hard swichig a uroff i Flyback mode. Thus, he correspodig swichig loss i Flyback mode ca be approximaed as P F lbk,sw = V 2 P V D off 2(L DAB L m ). (2) The swich drive losses are o cosidered here; However, here are ie swiches acively drive i DAB mode compared o oly wo swiches acively swichig i Flyback mode. Furhermore, f s is much lower i Flyback mode he i DAB mode, which furher reduces he swich drive losses. C. Core Losses Core losses are prese i he highfrequecy rasformer i boh DAB ad Flyback modes ad ca be approximaed usig he Seimez equaio [26] P core = kf α s B β peak, (22) where B peak is he peak flux desiy, ad k, α, ad β are he Seimez parameers, which deped o he core, ad are
5 usually foud by curve fiig. Core losses cosiue a low perceage of he oal losses i DAB mode [24] due o highfrequecy acac operaio. However, core losses are domia i Flyback mode. This is due o high peaks i core volage, ad lower frequecy operaio, which icreases B peak i he rasformer. This aalysis eglecs he ski effec i all coducors, which ca be sigifica, especially i DAB mode due o high frequecy operaio. D. Loss Compariso i Two Modes The calculaed loss breakdow for P = 0 W ad P = 40 W is show i Fig. 6. The coducio losses i all acive ad passive elemes are lumped ogeher. The swichig losses also iclude he drive losses. Flyback mode, he swichig losses are reduced by a leas 0, mosly by elimiaig he uroff losses o he high volage side, a a cos of margially icrease i coducio losses. The rasformer ad iducor core loss is slighly higher i Flyback mode, due o higher B peak. The core losses i Flyback mode icrease rapidly wih he power due o higher B peak ad f s. This is o he case for he DAB coverer, i which he core losses remai almos cosa over he power rage. V. EXPERMENTAL RESULTS A prooype of he sysem show i Fig. 3 was fabricaed o a cusom Pried Circui Board, wih DAB power raig of 00 W. The specificaios of he prooype are lised i Table. The coverers are digially corolled usig a o TABLE MV PROTOTYPE SPECFCATONS Parameer Value Uis Raed Power, P om 00 W Dcdc Sage Swichig Frequecy, f s DAB Mode 95 khz Flyback Mode 2050 khz Fixed OTime, T o 8 µs pu Capaciace, C i 300 µf Bus Capaciace, C bus 00 µf DAB ducace, L DAB 4.2 µh Mageizig ducace, L m 32 µh Bus Volage Rage, DAB mode V Flyback mode 70 V Trasformer Turs Raio, 9 Power Loss(W) Power Loss(W) DAB Flyback DAB Flyback P=0W P=40W 44 V V LDAB 0 V 44 V LDAB 5.5 A c 8 c 4 c 4 8 A LDAB 70 V V LDAB 5.5 A 0 Coducio Swichig Core Fig. 6. Simulaed power losses for P = 0 W, ad P = 40 W. Fig. 7. Seadysae waveforms of he coverer i DAB mode a V P V = 22 V ( LDAB :5 A/div), ad Flyback mode a V P V = 25 V ( LDAB :5 A/div).
6 board FPGA. A cusom plaar rasformer was desiged o reduce he weigh ad profile of he prooype. The seadysae waveforms i DAB ad Flyback modes a P = 70 W, ad P = 5 W, are show i Fig. 7 ad, respecively. is adjused o V P V i DAB mode o achieve opimal efficiecy, as i is illusraed by he fla porios i LDAB. The closedloop dyamic respose of Flyback mode for a sep LDAB i 0.75 A 0.35 A Fig. 8. Sep respose of Flyback mode wih he iegraed sorage ierface off: P : 9. W 9.5 W ( i :0.2 A/div, LDAB : 0 A/div). chage i P, while he dedicaed iegraed sorage coverer is off, is show i Fig. 8. f s is icreased i Flyback mode by he coroller o accommodae he higher ipu power. h (%) DAB Mode Flyback Mode P (W) Fig. 9. Measured efficiecy, η, of he coverer. The measured efficiecy of he coverer, η, i boh modes is show i Fig. 9. A peak efficiecy of 94% is achieved i DAB mode, while Flyback mode has a superior efficiecy up o P = 40 W. The power is limied i Flyback mode due o he maximum duy raio of 50%. However, he desig is such ha he wo efficiecy curves iercep a a poi close o he maximum rasferrable power i Flyback mode. The coverer operaes a he edge of Discoiuous Coducio Mode (DCM) a he iercep poi, such ha he operaio is swiched o DAB mode a higher power. V. CONCLUSONS A ovel DAB swichig scheme was iroduced for he dcdc sage of module iegraed power coverers for PV applicaios. The modified flyback swichig scheme exhibis 8% higher efficiecy ha DAB mode a 0 W, which comes a he cos of a addiioal swich. While Flyback mode exhibis more core losses ad slighly more coducio losses compared o DAB mode, he swichig losses are sigificaly reduced by elimiaig mos of he swichig acios, ad reducig he frequecy. ACKNOWLEDGEMENT This work was suppored by Solaro Semicoducor, he Oario Ceres of Excellece, he Naural Scieces ad Egieerig Research Coucil of Caada, he Caadia Foudaio for ovaio ad he Oario Research Fud. The auhors also hak Ray Orr, Be Bacque, Mihai Varla, ad Chris Gerolami for discussios relaed o aogrids ad microiverers. REFERENCES [] P. Deholm, R. Margolis, T. Mai, G. Brikma, E. Drury, M. Had, ad M. Mowers, Brigh fuure: Solar power as a major coribuor o he u.s. grid, EEE Power ad Eergy Magazie, vol., o. 2, pp , 203. [2] R. K. Heser, C. Thoro, S. Dhople, Z. Zhao, N. Sridhar, ad D. Freema, High efficiecy wide load rage buck/boos/bridge phoovolaic microcoverer, i EEE Applied Power Elecroics Coferece ad Exposiio, 20, pp [3] B. York, W. Yu, ad J.S. Lai, A iegraed boos resoa coverer for phoovolaic applicaios, EEE Trasacios o Power Elecroics, vol. 28, o. 3, pp , 203. [4] R. Erickso ad A. Rogers, A microiverer for buildigiegraed phoovolaics, i EEE Applied Power Elecroics Coferece ad Exposiio, 2009, pp [5] Ephase m90 microiveer, Ephase Daashee, 2009, available hp://ephaseeergy.com. [6] N. Femia, G. Lisi, G. Peroe, G. Spaguolo, ad M. Vielli, Disribued maximum power poi rackig of phoovolaic arrays: Novel approach ad sysem aalysis, EEE Trasacios o dusrial Elecroics, vol. 55, o. 7, pp , July [7] S. Poshkouhi, V. Palaiappa, M. Fard, ad O. Trescases, A geeral approach for quaifyig he beefi of disribued power elecroics for fie graied mpp i phoovolaic applicaios usig 3d modelig, EEE Trasacios o Power Elecroics, vol. 27, o., pp , 202. [8] L. Xu ad D. Che, Corol ad operaio of a dc microgrid wih variable geeraio ad eergy sorage, EEE Trasacios o Power Delivery, vol. 26, o. 4, pp , 20. [9] G. Suvire, M. Molia, ad P. Mercado, mprovig he iegraio of wid power geeraio io ac microgrids usig flywheel eergy sorage, EEE Trasacios o Smar Grid, vol. 3, o. 4, pp , 202. [0] S. Poshkouhi, M. Fard, H. Hussei, L. Dos Saos, O. Trescases, M. Varla, ad T. Lipa, A DualAciveBridge based BiDirecioal Microverer wih egraed ShorTerm Lio UlraCapacior Sorage ad Acive Power Smoohig for Modular PV Sysems, i press. [] M. Alam, K. Muaqi, ad D. Suao, Miigaio of roofop solar pv impacs ad eveig peak suppor by maagig available capaciy of disribued eergy sorage sysems, EEE Trasacios o Power Sysems, vol. PP, o. 99, pp., 203.
7 [2] L. Liu, H. Li, Z. Wu, ad Y. Zhou, A cascaded phoovolaic sysem iegraig segmeed eergy sorages wih selfregulaig power allocaio corol ad wide rage reacive power compesaio, EEE Trasacios o Power Elecroics, vol. 26, o. 2, pp , 20. [3] Y.M. Che, A. Huag, ad X. Yu, A high sepup hreepor dcdc coverer for sadaloe pv/baery power sysems, EEE Trasacios o Power Elecroics, vol. 28, o., pp , 203. [4] Z. Wag ad H. Li, A iegraed hreepor bidirecioal dcdc coverer for pv applicaio o a dc disribuio sysem, EEE Trasacios o Power Elecroics, vol. 28, o. 0, pp , 203. [5] H. Hu, S. Harb, N. Kuku, Z. She, ad. Baarseh, A siglesage microiverer wihou usig elerolyic capaciors, EEE Trasacios o Power Elecroics, vol. 28, o. 6, pp , Jue 203. [6] S. Kjaer, J. Pederse, ad F. Blaabjerg, A review of siglephase gridcoeced iverers for phoovolaic modules, EEE Trasacios o dusry Applicaios, vol. 4, o. 5, pp , Sep [7] Lihiumo Capacior, JSR Micro, 202, available a hp:// [8] Europea or CEC Efficiecy, available a hp://files.pvsys.com/help/idex.hml. [9] F. Krismer ad J. Kolar, Efficiecyopimized highcurre dual acive bridge coverer for auomoive applicaios, EEE Trasacios o dusrial Elecroics, vol. 59, o. 7, pp , 202. [20] H. Qi ad J. Kimball, Geeralized average modelig of dual acive bridge dcdc coverer, EEE Trasacios o Power Elecroics, vol. 27, o. 4, pp , 202. [2] M. Park ad.k. Yu, A sudy o he opimal volage for mpp obaied by surface emperaure of solar cell, i 30h Aual Coferece of EEE dusrial Elecroics Sociey, 2004, vol. 3, 2004, pp Vol. 3. [22] D. D. C. Lu, H.C. u, ad V. Pjevalica, A siglesage ac/dc coverer wih high power facor, regulaed bus volage, ad oupu volage, EEE Trasacios o Power Elecroics, vol. 23, o., pp , [23] R. Erickso ad D. Maksimović, Fudameals of Power Elecroics, Secod Ed. Spriger, 200. [24] H. Qi ad J. Kimball, Geeralized average modelig of dual acive bridge dcdc coverer, EEE Trasacios o Power Elecroics, vol. 27, o. 4, pp , April 202. [25] D. MurhyBellur ad M. Kazimierczuk, Twoswich flybackforward pwm dcdc coverer wih reduced swich volage sress, i Proceedigs of 200 EEE eraioal Symposium o Circuis ad Sysems (SCAS), May 200, pp [26] J. Reier, A. Brockmeyer, ad R. De Docker, Calculaio of losses i ferro ad ferrimageic maerials based o he modified seimez equaio, EEE Trasacios o dusry Applicaios, vol. 37, o. 4, pp , Jul 200.
R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder
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