THEORETICAL CHARACTERIZATION RIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING

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1 THEORETICAL CHARACTERIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING THEORETICAL CHARACTERIZATION RIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING Pof. Eg. Radu OLARU PhD 1, Eg. Robt GHERCA PhD Studt 1 1 Gh. Asachi Tchical Uivsity of Iasi, Faculty of Elctical Egiig. REZUMAT. I acastă luca pztăm u studiu totic al gatoilo lctomagtici ptu cupaa gii di vibaţii ii ambital. Gatoul st modlat ca u oscilato d odiul doi subamotizat. Optimizaa gatoului ptu obţia valoilo maxim al tasmisibilităţii ii şi putii lctic s-a făcut f pi cosidaa ui pulsaţii d zoaţă difită d pulsaţia atuală ca st folosită cut î î litatuă. Răspusul la itaa taptă a sistmului st util î aaliza gatoalo supus la vaiaţii ii buşt al amplitudiii vibaţiilo iilo alatoa. Cuvit chi: cupaa gii, gatoi lctomagtici, fom d zoaţă, oscilato subamotizat. ABSTRACT. I this pap w pst a thotical study of th lctomagtic gatos fo havstig gy fom ambit vibatios. Th gato is modld as a oscillato of scod od ud-dampd. dampd. Gato optimizatio fo achivig th maximum valus of tasmissibility ad lctical pow has pfomd by cosidig a soac fqucy difft to atual fqucy that is usually usd i litatu. Stp spos of th systm is usful to aalysis of th gatos subjctd to suddly vaiatios of th adom vibatio amplitud. Kywods: gy havstig, lctomagtic gatos, soac phomo, ud-dampd oscillato. 1. INTRODUCTION Egy havstig gatos fom vibatio, as altativ gy souc, hav bcom icasigly widspad bcaus vibatios a commo. Vibatiodiv gatos basd o lctostatic (capacitiv), pizolctic o lctomagtic (iductiv) tchologis hav b dmostatd [1]. Egy havstig gatos ca b oscillatoy o o oscillatoy. Th oscillatoy gatos hav a lastic suspdd itial mass that dampd oscillats du to th xtal applid vibatios. Ths ca b classifid as soat o o soat. Th soat oscillatoy gatos hav a soac fqucy wh th amplitud of th mass displacmt is lagst. I litatu th soat gatos hav b classifid as Vlocity Dampd Rsoat Gato (VDRG) ad Coulomb Dampd Rsoat Gato (CDRG) []. Th fist o dals wh th lctomchaical foc is popotioal to vlocity, whas th last is applicabl wh lctomchaical foc is a costat foc. Th maximum pow gatio at th soac fqucy fo both gatos is sam. Th two catgois of gatos (VDRG ad CDRG) ca wok ad as o soat gatos wh th dampd dvics oscillats with a difft fqucy tha th xcitatio fqucy of th vibatio. Th o oscillatoy gatos a calld as Coulomb Foc Paamtic Gato (CFPG). Such gato CFPG is o-lia; it dos ot pmit adjustmt o a soac fqucy, havig ot a suspsio with sot, th gato covtig mchaical gy ito lcticity wh it is at maximum acclatio. Class of gatos VDRG ca b implmtd usig lctomagtic phoma (th movmt of a pmat magt i a coil o vic vsa) whil th oth two (CDRG ad CFPG) usig lctostatic phoma (th movmt of th plats of a capacito i a dictio paalll to thm, thus maitaiig a costat distac btw th fixd ad mobil plats, dvlopig a costat foc to movmt). Whil th class VDRG icluds both mii ad mico gy gatig dvics, th classs CFPG ad CDRG icluds spcially th micopow gatos fabicatd usig MEMS tchiqu. I litatu th VDRG gatos a modld by a ducd scod od modl wh th output pow is maximum at soac, wh th xcitatio vibatio fqucy is matchd to th atual fqucy [],[],[4],[5]. This is tu oly wh th dampig Bultiul AGIR. /01 iui-august 1 Bultiul AGIR. /01 iui-august 75

2 WORLD WORLD ENERGY ENERGY SYSTEM SYSTEM CONFERENCE CONFERENCE WESC - WESC 01 systm is vy low ( 0). If th dampig is icasd, th maximum pow ad fqucy at which it is obtaid may diff sigificatly fom th cas mtiod. I this pap w achiv a thotically study basd o aalytical modlig of dvics VDRG cosidd as uddampd (0 <1) systms. Th ffct of th dampig i obtaiig th lagst valus fo th mass itial amplitud, output pows ad voltag is atd, which is of gat impotac i th aalysis ad dsig of such gatos fo havstig gy fom vibatio. cm a lctical dampig cofficit ad mchaical dampig cofficit, spctivly.. THEORY OF VIBRATION-POWERED ELECTROMAGNETIC GENERATORS Elctomagtic iductio gato. I th followig w will mak fc to a gato that has a lvitatd magt (Fig. 1). Fig. 1. Picipl schm of lctomagtic gato with lvitatd magt. Fam xcitatio of th gato sults th vibatio of th magt lativ to statioay coil causig a iducd cut. O ca fomulat a gal modl fo th covsio of th kitic gy of a oscillatig magtic mass to lcticity basd o th lia systm thoy. This modl basd o th schmatic i Figu is th ducd scod od systm modl that is dscibd by quatio (1). d z dz d y m + c + k z m (1) wh, m is magt mass; z - mass lativ displacmt; y - th displacmt of th fam; k - th magtolastic costat of th magtic suspsio; c c + cm - th total dampig cofficit, wh c ad Fig.. Elctomagtic iductio gato modl. Th lctic dampig c xpsss th mchaism of covsio of th kitic gy ito lcticity, whil th mchaical dampig c m is du to ai dampig. Diftial quatio (1) ca b also wit: d z dz d y + z +, () c c wh is th total dampig atio; c mk c0 0 mk - citically dampig cofficit; k π f - atual agula fqucy (i ad/s), m wh f is th atual fqucy (i Hz). If c< c 0 o <1, th systm fom Figu ca oscilat du of a xtally aplid displacmt that has a ctai vaiatio i tim, y(t), big such a oscilattoy systm, without to b a idispsabl soat systm, fo which, as w will s, th must b aoth coditio. Bcaus th gy is xtactd fom th systm by lativ movmt fom mass ad systm w d th solutio of th quatio (1). Pfomig a Laplac tasfomatio o th q. (1) w gt to th tasf fuctio of th systm, G(s), Z( s ) ms G(s) () Y( s ) ms + cs + k Fo th mathmatical modl of th systm xpssd by th quatio () th tasf fuctio of th systm is: Bultiul AGIR. /01 iui-august 76 Bultiul AGIR. /01 iui-august

3 THEORETICAL CHARACTERIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING G( s ) s s s (4) + + Rspos of th systm to siusoidal vibatio. If it assumd that th iput is a siusoidal vibatio lik yysi(t), wh Y is th vibatio amplitud ad is th vibatio agula fqucy, substitutig this ito th govig quatio () sult: d z dz + + z Y si( t ) (5) It ca fid th fqucy spos of th systm dscibd by tasf fuctio G(s), by substitutig s by j i th quatio (4): G( j ) (6) + j Th modulus ad th phas agl of th fuctio G(j) pst th amplitud-spos ad th phasspos, spctivly, of th systm: Z( ) G() G( j ) Y ( ) ( ) + (7) φ()actg (8) Th, th paticula solutio (stady-stat solutio) is foud as: z p (t)z()si(t-φ) ( ) ( ) + Y si(t-φ) (9) atual agula fqucy,, that is tu oly wh th dampig is vy small, as w will still show. To fid th soac fqucy at which th magitud of th lativ movmt of th itial mass is th maximum, xpss th q. (10) usig th atio btw th xcitatio fqucy ad th atual fqucy, / : T() ( ) + 4 Th fuctio (1) has a maximum fo (1) 1, (1) wh is obtaid th maximum valu of th tasmissibility: 1 T max (14) Thfo, th soac agula fqucy is, (15) ad th amplitud of th mass displacmt o th soac amplitud is giv as: Y Z (16) Fom qs. (1) ad (15) sults that th soac phomo oly occus wh <1/ (pactically, <0.5). This is th spos of th systm to siusoidal vibatios. Fo domais which studis how popagatio o damp vibatio, oft us th tm of tasmissibility, T(), to dsigat th atio of th amplitud of th vibatio tasmittd to th xcitatio vibatio amplitud: Z( ) T() (10) Y + ( ) ( ) Fo, th amplitud of th itial mass is: Y mk Z( ) Y (11) c I litatu is cosidd that th soac occus wh th xtal vibatio fqucy is matchd to th Fig.. Displacmt tasmissibility vsus fo difft valus. Bultiul AGIR. /01 iui-august Bultiul AGIR. /01 iui-august 77

4 WORLD WORLD ENERGY ENERGY SYSTEM SYSTEM CONFERENCE CONFERENCE WESC - WESC 01 Th displacmt tasmissibility is optimizd wh th xcitatio fqucy is qual to th soac fqucy, q. (15). This ca b wll obsvd fom Figu. Th maximum tasmissibility is obtaid wh 1. I oth wods, th fam vibatio is totally coupld to magt mass. I this way, th lativ motio of th movig magt lativ to coil is maximizd, q. (16). This is a impotat sult. As will b s, th gato givs th maximum pow at th mchaical soac fqucy. Rspos of th systm to stp iput. Usig a stp iput displacmt, y(t) y 0, fo t 0, wh y 0 is th stp amplitud (Fig. 4), th itia foc o th lvitatd magtic mass is thotically ifiit at th iitial momt, accodig to th scod tm of th q. (5), as wll as th lativ movmt z i ( t ) that ca b xpssd by:, t 0 z i ( t ) (14) 0, t > 0 Both th foc ad th lativ displacmt hav cotay sss to y 0. Du to this impuls z i ( t ), if <1, th systm oscillats with th amplitud gadually dcasig to zo. Th aalitical xpssios of th stp spos fo th th cass, uddampd systm (<1), citically dampd systm (1) ad ovdampd systm (>1) a [6]: t si ( ) t + actg z i t ; t zt (t ) [ ( 1+t ) ] zi ( t) ; 1 t 1 sh t + acth z ( t) ; i > 1 (15) wh z i ( t ), giv by q. (14), is th focd compot o th compot of stady stat of th systm. I pactical cass, w cosid that th foc ad displacmt impulss hav limitd valus at t0, z 0 ad F 0, spctivly, ad latioship btw ths is giv of q. (), z 0 F 0 /. Stp spos fo th pactical cass is plottd i Figu 5. Fig. 4. Focd compot of th stp spos of th systm. Fig. 5. Stp spos cuvs fo difft valus. Fo 0<<1, wh th systm is uddampd, also to oscilattig gatos, th fqucy of oscillatio is calld th dampd fqucy o igig fqucy, d (16) ad th dampd piod T d will b: Td π / d (17) Bcaus th dampig cofficit c of th systm is gally difficult to b thotically dtmid (th a som xpimtal mthods fo this), th valu of th dampig atio is ot accuatly kow. Cosid that th magtic mass of lctomagtic gato psts a dampd oscillatig spos wh 0 < < 1, o c< c c. F oscilatios of th magtic mass fo a stp iput displacmt y 0 a show i Figu 6. Th logaithmic dcmt δ fo uddampd oscilatios is dfid as th atual log of th atio of ay two succssiv amplituds, z1 π δ l, 0< <1 (18) z which shows th dampig dpdc oly of th dampig atio. d 4 Bultiul AGIR. /01 iui-august 78 Bultiul AGIR. /01 iui-august

5 THEORETICAL CHARACTERIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING so th total pow dvlopd i a lctomagtic damp will b: m Y P () 1 + Fig. 6. Ilustativ with dampd oscillatios fo calculus of th logaithmic dcmt (18), dampig atio (19) ad atual fqucy (0). Eq. (18) ca b usd to xpimtally dtmi th atio dampig, by masuig th atio of two succsiv amplituds z 1 ad z : δ, δ >0 (19) 4π + δ Also, th atual fqucy ca b dtmid by masuig th dampd piod T d (Fig. 6), usig th latioship: 4π + δ (0) T d Pow aalysis. Th lctical avag pow of th vibatio-iducd gato ca b divatd as: m Y P (1) 1 + wh is th lctical dampig atio. This is th gal fom of th xtactd pow fom a soat lctomagtic gato. I this fom, o oth quivalt foms, th latioship is wll kow i th litatu. Pow losss a xpssd i a simila latioship i which appas th mchaical dampig atio m, m m Y P m, () 1 + wh c m m + is th total dampig atio. Wh pow output P is at lagst if <<1 ad q. () has th simplifid foms: P Y m 4 ma P 4 (4) (5) Rlatioship (5) uss th acclatio amplitud of th xtal vibatio Y. Similaly th lctical pow at th atual fqucy ca b xpssd as: ad A m Y P (6) 4 m A P (7) 4 Elctical pow is maximisd whv possibl wh m, that is th pow fom th lctical dampig is qual to th mchaical losss. I this cas latioship (7) givs: P m Y 16 m (8) Rlatioships (4)...(7) show that th pows i th oscillatig systm a thotically ifiit wh ad th total dampig is qual to zo. Howv, i som cass, wh th dampig is otabl ad th soat systm is aig th limit of xistc of zoac ( ), th maximum lctical pow ca b foud i latio to th soac phomo dscibd at th bgiig of th chapt. I od to fid this pow, th atio of th fqucs / is substitutd ito q. (1) ad aagd as: Bultiul AGIR. /01 iui-august 5 Bultiul AGIR. /01 iui-august 79

6 WORLD WORLD ENERGY ENERGY SYSTEM SYSTEM CONFERENCE CONFERENCE WESC - WESC 01 6 m Y P (9) ( ) + 4 Usig th soac atio 1, lctical pow i soac is giv by: m Y P, 0 <0,7 (0) 4 ( )( ) By usig q. (6), o ca wit: P P 1 (1) ( )( ) Eq. (1) shows that th lctical pow to soac fqucy is bigg tha th pow at atual fqucy, if >0. Oly wh 0 th two pows a qual. Fo xampl, with valus of 0., 0. ad 0.4, th pow atio P P is 1.1, 1.4 ad 1.75, spctivly. Ow quatio (0) fo lctical pow to soac fqucy compis th pow xpssio to atual fqucy, q. (1), th last big obtaid fom (0) fo th paticulaly cas wh ad hav gligibl valus lativ to th uit. Th lctical dampig atio ca b witt as [], B l () m ( R + Rc ) wh B is th avag flux dsity; l th lgth of th coil; R th load sistac; R c - th coil sistac. Th voltag i soac V max ca b ow xpssd as: BlY Vmax P ( R + Rc ) () ( )( ) Eq. () shows that th output voltag of th gato is maximum i soac, ad popotioal to, whil th cospodig lctical pow P is popotioal to th thid pow of.. CONCLUSIONS modlig of th lctomagtic gatos, systm spos to siusoidal vibatio ad to stp iput displacmt, ad pow aalysis. I th fqucy spos of th gato systm aalisd fo th wokig soat domai wh th dampig atio has th valus 0<<0.7, w stablishd th xpssios fo th soac fqucy ad th soac amplitud of th itial mass, qs. (15), (16), which dpd o. I th litatu ddicatd to gy havstig gatos th soac phomo is dscibd oly fo. Th stp spos of th gato systm fo a stp iput displacmt of vibatio, hlp i th aalysis of th soat (0<<0.7) ad o-soat (0.7<<1) oscillatoy gatos wh th xtal vibatios applid to th havst systm psts suddly vaiatios i amplitud at vaiabl tim itvals. W did ot fid i litatu th aalysis of th stp spos of th lctomagtic gatos. W foud th xpssio fo th lagst lctical pow of th gatos usig th soac fqucy, which is bigg tha th pow to atual fqucy, if >0, giv i litatu as th maximum lctical pow xtactd fom th systm. Fo xampl, if 0.4, th pow calculatd with ow xpssio (q. (0)) is with 75% gat tha that calculatd with th latioship kow i litatu, fo th maximum pow (1)). Ow gal quatio (0) fo lctical pow to soac fqucy icluds th pow xpssio to atual fqucy, as shows q. (1). Th basic thoy of th lctomagtic gatos fo havstig gy fom vibatio has b dvlopd. This thoy fs to: aalytic 6 Bultiul AGIR. /01 iui-august 80 Bultiul AGIR. /01 iui-august

7 THEORETICAL CHARACTERIZATION OF AN ELECTROMAGNETIC GENERATOR FOR VIBRATION ENERGY HARVESTING BIBLIOGRAPHY [1] Chalasai, S., Coad, J. M., A suvy of gy Havstig soucs fo mbddd systm, IEEESouthastCo 008, pp [] Mitchso, P.D., G, T.C., Yatma, E.M., Holms, A.S., Achitctu fo Vibatio-Div Micopow Gatos, Joual of Micolctomchaical Systms, Vol. 1, No., 004, [] Matu, L., Moll., F., Rviw of Egy Havstig Tchiqus ad Applicatios fo Micolctoics, Cofc o VLSI Cicuits ad Systms II, Svill, SPAIN, 005, Pts 1 ad Book Sis: PROCEEDINGS OF THE SOCIETY OF PHOTO- OPTICAL INSTRUMENTATION ENGINEERS (SPIE) Vol. 587, pp [4] Bby, S.P. t al., A mico lctomagtic gato fo vibatio gy havstig, J. Micomch. Micog. 17 (007) [5] Vulls, R.J.M. t al., Micopow gy havstig, Solid- Stat Elctoics 5 (009) [6] Voicu M., Itoduc i Automatica, Ed. POLIROM, 00. About th authos Pof. Eg. Radu OLARU, PhD Tchical Uivsity Gh. Asachi fom Iasi, Faculty of Elctical Egiig, Dpatmt of Egy Utilizatio, Elctical Divs ad Idustial Automatio, Blvd Dimiti Mago, o.67, Iasi, Romaia. mail:olau@.tuiasi.o Gaduatd at th Tchical Uivsity of Iasi, Faculty of Elctical Egiig, study pogam Elctical Egiig with th Elctomchaical Espciality. H civd th PhD dg i Elctical Egiig fom th sam uivsity i 1994 with a thsis cocig magtic fluid tasducs. H is cutly pofsso at th Dpatmt of Egy Utilizatio, Elctical Divs ad Idustial Automatio, Faculty of Elctical Egiig fom this uivsity. His mai sach itsts a coctd with lctomagtic dvics ad systms fo masuig, cotol ad actuatio. Eg. Robt GHERCA, PhD Studt Tchical Uivsity Gh. Asachi fom Iasi, Faculty of Elctical Egiig, Dpatmt of Egy Utilizatio, Elctical Divs ad Idustial Automatio, Blvd Dimiti Mago, o.67, Iasi, Romaia. obtghca@yahoo.com Gaduatd at th Tchical Uivsity of Iasi, Faculty of Elctical Egiig, study pogam Elctical Egiig with th Espciality i Elctic Machis. H is cutly wokig towad th Ph.D dg at th Dpatmt of Egy Utilizatio, Elctical Divs ad Idustial Automatio, Faculty of Elctical Egiig fom this uivsity. His sach itst is i th aa of Elctomagtic gatos fo havstig gy fom ambit vibatios. Bultiul AGIR. /01 iui-august 7 Bultiul AGIR. /01 iui-august 81

8 WORLD WORLD ENERGY ENERGY SYSTEM SYSTEM CONFERENCE CONFERENCE WESC - WESC 01 8 Bultiul AGIR. /01 iui-august 8 Bultiul AGIR. /01 iui-august