THEORETICAL ELEmRICAL POWER OUTPUT PER U N I T VCLUME OF PVF2 AND MECHANICAL-TO-ELECTRICAL CONVERSION EFFICIENCY AS F U N C T I O N S OF FRWUENCY V. Hugo Schmidt Phys ic s Department, Montana S t a t e U n i v e r s i t y , Bozeman, MT 59717, U.SR a b s t r a c t The e lectr ical energy ou tpu t per u n i t volume of po ly(v iny1idene f l u o r i d e ) (PVF2) i s c a l c u l a t e d us ing t h e d31 p i e z o e l e c t r i c c o e f f i c i e n t a p p l i c a b l e t o t h e bimorph bending mode of o p e r a t i o n S t r a i n s approach ing t h e y i e l d s t r a i n a r e considered. The c a p a c i t i v e p o r t i o n of t he source impedance i s assumed t o be c a n c e l l e d by a s u i t a b l e series induc to r . L i m i t a t i o n s caused by e l e c t r i c a l breakdown a r e cons ide red , because t h i s L C resonance effect can cause v o l t a g e s a c r o s s the sample g r e a t l y i n e x c e s s of t h e emf g e n e r a t e d d i r e c t l y by t h e bending, A t s e r i e s resonance , t h e power o u t p u t i s l i m i t e d by t h e r e s i s t i v e p a r t s of t h e PVF2 i n t e r n a l impedance and t h e i n d u c t o r impedance. The o u t p u t power per u n i t volume is c a l c u l a t e d from pub l i shed v a l u e s fo r t h e f r equency dependence of t h e l o s s y part of t h e p e r m i t t i v i t y f o r WF2. d e n s i t y is s u r p r i s i n g l y l a r g e a t f r e q u e n c i e s i n t h e Wz range just below t h e f r e q u e n c i e s a t which losses become l a r g e , and i s c a l c u l a t e d t o be a b o u t 100 W/cm3 a t 1 kHz. Also, t h e mechanica l - to- e l e c t r i c a l conve r s ion e f f i c i e n c y i s c a l c u l a t e d . Th i s e f f i c i e n c y c a n exceed t h e e? .ec t romechanica l coup l ing c o e f f i c i e n t of a b u t 1% cons ide rab ly . I ts c a l c u l a t e d v a l u e n e a r 70% is l i m i t e d o n l y by e l e c t r i c a l and mechan ica l losses. T h i s power FERTIN EN T MATERIAL PR O E R TIES A p i e z o e l e c t r i c polymer h a s seven i m p o r t a n t des ign parameters f o r purposes o f o b t a i n i n g t h e greatest p o s s i b l e e l e c t r i c a l o u t p u t and e f f i c i e n c y from a mechanica l d r i v i n g source. The mechanica l Y' p a rame te r s a r e Young's modulus Y, y i e l d s t r a i n 6 and mechanica l q u a l i t y f a c t o r Qm, w h i l e t h e e l e c t r i c a l ones a r e d i e l e c t r i c p e r m i t t i v i t y K. e l e c t r i c breakdown f i e l d E& and e l e c t r i c a l q u a l i t y f a c t o r 9,. The s e v e n t h i s the p i e z o e l e c t r i c c o e f f i c i e n t d31 r e l a t i n g e l e c t r i c f i e l d a c r o s s t h e s h e e t ( 3 d i r e c t i o n ) t o s t r a i n a l o n g the stretch d i r e c t i o n (1) w i t h i n t h e shee t . Young's modulus f o r PVFZ i s t h e r a t i o of t e n s i l e stress t o t e n s i l e s t r a i n which i s quoted' a s Y=1.5x109 N / m 2 or 2 . 2 ~ 1 0 ~ p s i . w i t h our own measuremen t s2 Young's modulus i s a measure of s t i f f n e s s , which i s low f o r p l a s t i c s , a l l o w i n g d e s i g n s w h i c h g i v e t h e necessa ry f l e x i b i l i t y f o r coup l ing e f f i c i e n t l y t o wind and water energy s o u r c e s w i t h o u t requiring e x t r e m e l y t h i n s e c t i o n s . T h i s a g r e e s wel l The mechanica l q u a l i t y f a c t o r Q, is t h e r a t i o Y ' / Y " of t h e r e a l t o t h e i m a g i n a r y p a r t of Young ' s modulus, and i s t h e i n v e r s e of t h e mechanica l d i s s i p a t i o n f a c t o r D,. A h igh Q, i s i m p o r t a n t f o r good e f f i c i e n c y and even more f o r low l o s s e s which m i n i m i z e i n t e r n a l hea t ing . has been r e p o r t e d f o r a 25% TrFE ( t r i f l u o r c e t h y l e n e ) copolymer w i t h v i n y l i d e n e f l u o r i d e a t room t e m p e r a t u r e and 3.5 H z , ~ which s e e m s t o be t h e h i g h e s t f r e q u e n c y be low t h e MHz range f o r which Q, results a re r e p o r t e d . A v a l u e o f 200 f o r Q, The y i e l d s t r a i n i s t h e f r a c t i o n a l change i n l e n g t h A L / L beyond which t h e polymer w i l l no t return t o i t s o r i g i n a l l e n g t h whe t h e a p p l i e d stress i s removed. T h i s a t 3% s t r a i n (Sy=0.03) f o r WF2. The co r re spond ing y i I d s t ress a c c o r d i n g t o Hooke's l a w is S =Y6 =4.5~10' N J m 2 . T h i s h i g h y i e l d s t r a i n i s f o r t u n a t e a s i t a l l o w s f l e x i b l e d e s i g n s and a l a r g e o u t p u t vo l t age . Y Y Each of t h e t h r e e above mechan ica l p a r a m e t e r s h a s i t s e l ec t r i ca l analog. The r e l a t i v e d i e l e c t r i c p e r m i t t i v i t y K r e l a t e s t h e e l e c t r i c d i sp l acemen t D ( C / m 2 , or coulombs s t o r e d p e r s q u a r e meter of e l e c t r o d e cove r ing t h e d i e l e c t r i c ) t o t h e e l e c t r i c f i e l d E i n v o l t s per meter ( V / m ) o r newtons per coulomb ( N / C ) . For a n o r d i n a r y n o n p i e z o e l e c t r i c d i e l e c t r i c t h i s r e l a t i o n i s D . K O K E , where KO=8.85x10-12 C2/Nm2 i s t h e d i e l e c t r i c p e r m i t t i v i t y of vacuum. Measurements4p5 on WF2 y i e l d K=12 ( n e g l e c t i n g K ' s s m a l l l o s s y component). T h i s l a r g e va lue , 4 times t h a t t y p i c a l of p l a s t i c s , p roduces r e l a t i v e l y l a r g e c u r r e n t o u t p u t f o r a g iven p i e z o e l e c t r i c a l l y induced e l e c t r i c f i e l d . The e l e c t r i c a l q u a l i t y f a c t o r Qe i s t h e r a t i o K'/K"of t h e r e a l p a r t K' t o t h e i m a g i n a r y p a r t K" of t he d i e l e c t r i c p e r m i t t i v i t y K=K' - jK" . A h igh Qe r e d u c e s t n e sou rce impedance of t h e p i e z o e l e c t r i c g e n e r a t o r sys t em when t h e sys tem i n c l u d e s a n i n d u c t o r a s d e s c r i b e d below. It a l s o r e d u c e s h e a t i n g caused by d i e l e c t r i c losses. The va lue of Qe depends on f r equency , t empera tu re . and compos i t ion (amount of TrFE i n t h e copolymer) a s w e l l a s on p rocess ing , but a v a l u e of 50 i s t y p i c a l f o r f r e q u e n c i e s below 2 kHz.6 The e l e c t r i c breakdown f i e l d E b i n PVF2 i s near' 3x107 V i m . breakdown f i e l d i n a i r , so t h e m e t a l e l e c t r o d e s c o a t i n g t h e PVFZ s h e e t s should s t o p s h o r t of t h e edge. l e a v i n g a n uncoated f r i n g e a round t h e edge. The breakdown f i e l d i n WF2 i s much h i g h e r t han can T h i s i s much h i g h e r t h a n the be r eached by s i m p l y s t r e s s i n g a WF2 s h e e t t o i t s e las t ic l i m i t . It becomes a d e s i g n l i m i t a t i o n i f t h e c a p a c i t i v e s o u r c e r e a c t a n c e of t h e p i e z o e l e c t r i c g e n e r a t o r i s r e s o n a t e d away by a se r ies i n d u c t o r (as d i s c u s s e d be low) , i n w h i c h case t h e v o l t a g e a c r o s s t h e WF2 s h e e t can be much g r e a t e r t h a n t h e p i e z o e l e c t r i c a l l y induced emf ( e l e c t r o m o t i v e force) . F i n a l l i , t h e p i e z o e l e c t r i c s t r a i n c o e f f i c i e n t d j l , quoted e s s e n t i a l l y v e r i f i e d by our measurements.2 d e s c r i b e s t h e p o l a r i z a t i o n P or elec r i c 3 p e r p e n d i c u l a r t o t h e s h e e t for each N/m2 of tensile or compress ive stress along t h e s t r e t c h d i r e c t i o n w i t h i n t h e sheet. Some ceramics and s ingle c r y s t a l s have c o n s i d e r a b l y higher p i e z o e l e c t r i c c o e f f i c i e n t s , b u t t h e i r stiff ness makes them i m p r a c t i c a l f o r wind, h y d r o e l e c t r i c . o r wave g e n e r a t o r a p p l i c a t i o n s . L a r g e r v a l u e s of d31 are be ing a t t a i n e d w i t h improved polymer materials, s p e c i f i c a l l y w i t h cop0 ymers of v i n y l i d e n e f l u o r i d e a n d t r i f l u o r o e t h y l e n e . a s 25 pC/N ( 2 5 ~ 1 0 - l ~ C/N), which was d i sp lacemen t D a s be ing 2 5 ~ 1 0 - l ~ C / m 8 i n d i r e c t i o n i PIE2 OELECTR I C GENERATOR CHAR ACTER I S T I C S To c a l c u l a t e t h e power from a g i v e n g e n e r a t o r des ign f o r a g i v e n b l ade d e f l e c t i o n a m p l i t u d e and o s c i l l a t i o n f r equency , we beg in w i t h t h e p i e z o e l e c t r i c e q u a t i o n s For n o n p i e z o e l e c t r i c materials f o r which d31=O. t h e s e are j u s t uncoupled mechanica l and e l e c t r i c a l equa t ions . t he rma l e x a n s i o n terms which a re no t ve ry important! They must be cons ide red i n a n e x a c t a n a l y s i s because our o p e r a t i n g f r e q u e n c i e s a r e h igh enough t h a t a d i a b a t i c r a t h e r t h a n i s o t h e r m a l c o n d i t i o n s ob ta in . The s u b s c r i p t s 1 and 3 refer t o t h e d i r e c t i o n s d e s c r i b e d above. From h e r e on, t h e s e s u b s c r i p t s w i l l be omi t t ed . We e l i m i n a t e stress S f r o m Eqs. (1) and ( 2 ) i n f a v o r of t h e more e a s i l y measured s t r a i n 6 , and s o l v e them s i m u l t a n e o u s l y t o o b t a i n We have o m i t t e d p y r o e l e c t r i c and D = K O K ( 1 - Y d 2 / KOK )E+ Yd6fKOKE+ Yds. ( 3 ) Here Yd2/KoK i s t h e d i m e n s i o n l e s s e l e c t r o m e c h a n i c a l coup l ing c o n s t a n t k2, PVFZ and can be n e g l e c t e d in this context . which i s s m a l l (=0.0088) f o r I n a p i e z o e l e c t r i c g e n e r a t o r o p e r a t i n g i n a bending mode, each volume e l emen t obeys t h e above e q u a t i o n s , and t h e e n t i r e g e n e r a t o r i s e q u i v a l e n t t o a s l a b of p i e z o e l e c t r i c material a s r e p r e s e n t e d i n Fig. 1, w i t h e l e c t r o d e d s u r f a c e s of a r e a wb, one grounded and t h e o t h e r connec ted t o t e r m i n a l T. If t h e g e n e r a t o r i s d r i v e n w i t h c o n s t a n t mechanica l a m p l i t u d e and f requency , t h e s l a b can be r e p l a c e d by t h e e q u i v a l e n t g e n e r a t o r c o n s i s t i n g o f an i d e a l emf E i n series w i t h a c a p a c i t o r C and r e s i s t o r R e as de r ived below and shown i n F ig . 1. 9' Q lRL F i g u r e I. S l a b o f e l e c t r o d e d p i e z o e l e c t r i c po lymer re p r e s e n t i n g p i ez o e l e c t r i c g e n e r a t o r o p e r a t i n g i n d31 mode. and i t s e q u i v a l e n t c i r c u i t coupled a t t e r m i n a l T t o a r e s o n a n t i n d u c t o r and a l o a d RL,. Symbols a re e x p l a i n e d i n t e x t . We assume a s i n u s o i d a l s t r a i n S=6,ejwt. The emf e t h e n is t h e open-c i rcu i t v o l t a g e found from Eq. ( 3 ) by s e t t i n g D=O: €=hE=-( Yhd/KgK)G, (4) where h i s t h e t h i c k n e s s of t h e shee t . The c u r r e n t is g i v e n by I=-wbdD/dt. so t h e s h o r t - c i r c u i t c u r r e n t I, found by s e t t i n g E=O i n Eq. ( 3 ) i s Is=- jwbYdS=e/Zs. ( 5 ) We see t h a t w h i l e f o r a g i v e n maximum s t r a i n 6, t h e emf i s independent of t h e a n g u l a r f r equency o, t h e c u r r e n t i s p r o p o r t i o n a l t o U, d e m o n s t r a t i n g t h e advan tage of d e s i g n i n g d e v i c e s t h a t o s c i l l a t e a t h i g h f requency . F i n a l l y , f rom Eqs. ( 4 ) a n d ( 5 ) . t h e s o u r c e impedance i s Z,=€/Is=- jh/wKoKwb=- j / w C . ( 6 ) s o t h e s o u r c e impedance i s s imply t h e c a p a c i t i v e r e a c t a n c e of t h e s l a b ' s capac i t ance C, where t h e s m a l l l o s s y p a r t of t h e d i e l e c t r i c p e r m i t t i v i t y h a s been neglec ted . T h i s i s j u s t i f i e d if t h e g e n e r a t o r i s connected d i r e c t l y t o a r e s i s t i v e load . The l o s s y component must be cons ide red i f t h e c a p a c i t i v e component of t h e s o u r c e impedance i s r e s o n a t e d away by a series i n d u c t o r t o i n c r e a s e t h e o u t p u t as d i s c u s s e d below. 539 If t h e g e n e r a t o r ( o r group o f g e n e r a t o r s f o r c e d mechan ica l ly t o o s c i l l a t e i n phase) i s connected t o a n i n d u c t o r of t h e c o r r e c t v a l u e t o g i v e series re sonance a t t h e o p e r a t i n g f r equency , t h e new source impedance w i l l just be t h e combined r e s i s t a n c e Rs=ReTRi of t h e g e n e r a t o r r e s i s t a n c e Re and i n d u c t o r r e s i s t a n c e Ri shown i n Fig. 1. t h e v a l u e 50 chosen above f o r Q = l / w C R e f o r FVF2 and choos ing a va lue of 33 for %i=oL/Ri of t h e series i n d u c t o r a t 1 kHz, t h e combined source Q v a l u e is Qs=20. of t h e c a p a c i t i v e sou rce r e a c t a n c e Xc=-j /wC, where t h e c a p a c i t a n c e C=KoK'wb/h i n acco rd w i t h Eq. ( 6 ) . Accordingly , From Thus Rs i s 1/20 of t h e magni tude Rs=h/ wbQsKoK'w (7) i s t h e s o u r c e r e s i s t a n c e of t h e g e n e r a t o r when e l e c t r i c a l resonance i s employed. We emphas ize h e r e t h a t L i s a n a c t u a l i n d u c t o r , and no t a n e q u i v a l e n t i n d u c t o r employed t o a n a l y z e p i e z o e l e c t r i c resonance. The e l e c t r i c a l r e sonance d e s c r i b e above h a s approx ima te a n g u l a r f r equency w=(LC)-lP2. Good d e s i g n s s h o u l d i n c o r p o r a t e mechanica l r e sonance a t t h e same f r equency t o maximize a m p l i t u d e of o s c i l l a t i o n , but we assume h e r e t h a t this f r equency i s f a r f rom t h e p i e z o e l e c t r i c resonance f r equency of t h e m a t e r i a l . With t h e dec reased s o u r c e impedance g i v e n i n 4. ( 7 ) , much l a r g e r c u r r e n t s a r e p o s s i b l e f o r t h e same emf and consequent ly t h e o u t p u t power w i l l i n c r e a s e acco rd ing ly . We must check t o see whether t h i s l a r g e c u r r e n t I can cause t h e breakdown f i e l d Eb t o be e x c e e d e d The f i e l d i s t h e v o l t a g e which i s a p p r o x i m a t e l y X c I f o r l a r g e Q,. d i v i d e d by t h e t h i c k n e s s h. The c u r r e n t I i s t h e emf E f rom 4. ( 4 ) d i v i d e d by R + R =Rs(l+r), r e s i s t a n c e and r is t h e r a t i o of l o a d t o sou rce r e s i s t a n c e . S ince t h e magni tude of t h e r a t i o X c / R s = Q s , we h a v e , u s i n g Eq. ( 4 ) : where RL i s t h e l o a d s . L E = Q , E / h( l+r )=QsYd6/KoK'( l+r ) . (8 ) If 6 i s se t a t t h e y i e l d s t r a i n 0 .03 , r a t 0 ( s h o r t c i r c u i t e d o u t p u t ) . and o t h e r pa rame te r s g i v e n above a r e s u o s t i t u t e d i n t o Eq. ( 8 ) . w e o b t a i n Emax=21.2x107 V/m. breakdown f i e l d 7 of ~ ~ = 3 x 1 0 ~ V/m, so breakdown must be cons ide red a s a p o s s i b l e l i m i t a t i o n o n o u t p u t power, a s d e s c r i b e d below. If e l e c t r i c a l resonance i s no t employed, t nen maximum f i e l d o c c u r s f o r o p e n - c i r c u i t e d load , i n which c a s e Eq. ( 4 ) a p p l i e s . The f i e l d magnitude oo r re spond ing to t n e emf a t y i e l d s t r a i n found from Eq. ( 4 ) is d e f i n e d a s Th i s i s l a r g e r t han t h e E. ,=E / n= Ye6 / KOK'=1.06x107 i'/ !& ( Y ) J Y :J so weakdown w i l l no t l i m i t the power o u t p n t if e l e c t r i c a l resonance i s n o t employed. 'THEORETICAL ELECTRICAL POWER GUTWT ?or t h e e l e c t r i c a l l y r e sonan t case, t h e power '4 t o t h e l o a d per u n i t volume of WF2 i s l i m i t e d e i t h e r by mechanica l y i e l d or e l e c t r i c a l breakdown, depending on polymer p a r a m e t e r s and t h e i n d u c t o r and l o a d r e s i s t a n c e s . For t h e y i e l d - l i m i t e d case, W =I RL/2wbh= [ % h / R s ( l + r ) 12Rsr/2wbh (10) Y Y u s i n g Eq. ( 7 ) f o r R,. l o a d r a t i o r , W f irst i n c r e a s e s l i n e a r l y , peaks a t r=l, and then dec reases . Note t h a t f o r i n c r e a s i n g Y For t h e b r e a k d o w r r l i m i t e d case. (11) = ( E b/QsR Rsr/ 2 wbh= r K O K wE 2/ 2Qs. Note t h a t W b i n c r e a s e s l i n e a r l y w i t h r f o r all r. I f t h e r e is a c r o s s o v e r from b r e a k d o w w l i m i t e d t o y i e l d - l i m i t e d power, i t must occur a t t h e r v a l u e rx a t which W b = W y . From Eqs. (10) and (ll), t h i s o c c u r s a t rx=Q E / E b - l . S Y (12 ) There are t h r e e p o s s i b i l i t i e s f o r t h e load- dependence of t h e power ou tpu t per uni t volume. F i r s t , if Eq. (12) i n d i c a t e s n e g a t i v e rx, t h e y i e l d - l i m i t e d case of 4. (10) is v a l i d f o r a l l r, and t h e maximum power a t r=l i s Wmy'QsK~K'wE 2 / 8 . Y (13 ) Second, i f O ( r (1, t h e maximum power i s g o v e r n e d by Eq. (117 f o r r < r x a n d Eq. (10) f o r r)rx, and t h e maximum power still o c c u r s a t r=l and i s s t i l l g i v e n by Eq. (13). Third , i f r >1, t h e maximum power i s s t i l l governed by Eq. 711) or (10) as d e s c r i b e d above. bu t t h e maximum power o c c u r s a t r=rX and i s g i v e n by (14) For t h e pa rame te r s g i v e n above, we a r e i n t h e t h i r d case , w i t h rx=6.06. For a f r equency of 1 kHz, somewhat below t h e f r equency a t which K" s t a r t s i n c r e a s i n g r a p i d l y for PVF2, Wmx=91W/cm3. a s i g n i f i c a n t power l e v e l , and so p o s s i b l e o v e r h e a t i n g of t h e material must be cons ide red i n the des ign process , even a f t e r t h e o p e r a t i n g power l e v e l i s reduced t o provide a f a c t o r of s a f e t y a g a i n s t y i e l d and breakdown The power cu rve a s a f u n c t i o n of l o a d r e s i s t a n c e i s shown i n F ig . 2 . This is I f e l e c t r i c a l r e sonance i s not eniployed. t k e power i s l i m i t e d by y i e l d and Eq. (13) z p p l i e s f o r maximum power, w i t h Q s r e p l a c e d by 2. For maxinuin power, t h e l o a d r e s i s t a n c e i s n u m e r l c a l i y equal t o t h e c a p a c i t i v e s o u r c e impedance. The 2 o c c u r s 'because t h e l o a d i s 90' o u t of phase w i t h t h e s o u r c e impedanc Thus t h e v o l t a g e a c r o s s each e l emen t i s e/ZlT2 i n s t e a d of € 1 2 . and t h e s q u a r e of v o l t a g e which i s p r o p o r t i o n a l t o power i s t w i c e a d g r e a t . The maximum p w e r f o r t h e above p a r a m e t e r s i s 19 W/crn3. provide e l e c t r i c a l resonance i n c r e a s e s t h e power per u n i t volume of WF2 a lmos t f i v e f o l d . Thus add ing a s e r i e s i n d u c t o r t o 54c %too, .. I I I 190 I 1 I F i g u r e 2. T h e o r e t i c a l maximum e l e c t r i c a l power o u t p u t d e n s i t y and e f f i c i e n c y f o r a PVF2-based g e n e r a t o r o s c i l l a t e d mechan ica l ly a t 1 kHz. EFFICIENCY The e f f i c i e n c y is t h e power t o t h e l o a d , d i v i d e d by t h e sum of t h e l o a d power and t h e e lectr ical and mechanica l power l o s s e s , which must add up t o t h e mechanica l i n p u t power. I n each case we cons ide r power per u n i t volume. The t h r e e power and l o s s components have t h e same r a t i o s independen t o f a m p l i t u d e of o s c i l l a t i o n , o r of whether y i e l d o r breakdown l i m i t s power output . Accord ingly , we can d r o p t h e s u b s c r i p t y f rom Eq. (10) and u s e Eq. (8) t o o b t a i n t h e o u t p u t power W O i n terms o f t h e s t r a i n 6 : W~=Qsrwy2d262/2KOKi ( l + r ) 2 . The e l e c t r i c a l l o s s power Wel i s g iven by Wel=WoRs/ R L = W o / r (15 ) (16 ) The mechanica l l o s s power Wml is t h e product of t h e a n g u l a r f r equency w and t h e mean energy l o s s per r a d i a n , which i n t u r n i s t h e energy s t o r e d d iv ideC by t h e mechanica l q u a l i t y f a c t o r Q,. energy s t o r e d per u n i t volume, i n a fo rmula ana logous t o t h a t f o r t h e energy s t o r e d i n a s p r i n g , i s Y b 2 / 2 . The Thus, Wd becomes W*=wYb2/ 2Qm. (17 ) T h e e f f i c i e n c y q t h e n can be w r i t t e n a s q= [ l+Wel/ Wo+Wd i W O ] -'= [ l+r-' +( rtl ) 2/ rF1-I ! 18) i n wh ich F i s a " F i g u r e o f s e r i t " f o r t h e g e n e r a t o r , g iven by F=QmQ k2 ( 1 9 ) and k2 i s t h e e l e c t r o m e c h a r u c a l coup l ing cons t an t 41d2/KOK which i s 0.0088 f o r PVF2. For our cnosen v a l u e s of 200 f o r Q, and 20 f o r Q t h e f i g u r e of merit i s 35.2, mucn l a r g e r t han 8 i t s e l f . l o a d r e s i s t a n c e r a t i o r =6.06 g i v i n g maximum power, t h e e f f i c i e n c y i s 0 .719 , o r 71.5%. As s e e n i n F ig . 2 , t h e e f f i c i e n c y i s q u i t e f l a t over a l a r g e r a n g e of r. It i s maximum a t rm=(F+1)1/2=6.02, which i s A t t h e c o i n c i d e n t a l l y v e r y nea r rd If no series i n d u c t o r i s used t o a c h i e v e resonance , t h e v a l u e of Qs=2 d i s c u s s e d above must be used i n Eq. (18). r=Qe=50, and e f f i c i e n c y 1=0.0633, o r only 6.33%. The e f f i c i e n c y peaks a t 36.09 f o r rm=2.13, the power d e n s i t y i s v e r y low, only 1.6 W/cm3. A t maximum power of 1 9 W/cm3, bu t h e r e If the g e n e r a t o r sys tem i s synchron ized t o t h e l i n e and i t s o u t p u t i s f e d i n t o t h e 60 Hz l i n e i n s t e a d of a l o a d resistor RL' t h e above power ou tpu t and e f f i c i e n c y e q u a t i o n s still ho ld s o l o n g a s a n i n d u c t o r i s used t o a c h i e v e e l e c t r i c a l resonance. One s imply r e p l a c e s R L w i t h VII, where V i s l i n e v o l t a g e a n d I i s c u r r e n t . A t 60 Hz a n i n d u c t o r Qi of on ly 2 0 c a n be expec ted , bu t Q r e m a i n s a t 50, s o t h e combined Qs becomes 14.2 and rx i n Eq. (12) becomes 4.05. Then t h e maxim power d e n s i t y f rom Eq. (14) becomes 5.1 W/cm and t h e e f f i c i e n c y from 4. (18) becomes 67%. A t t h i s r x t h e r a t i o E / V = ( R - R ) /RL=l+l/rx=1.25, so maximum power is ach ieved w i t h a g e n e r a t o r emf 25% h i g h e r t h a n l i n e vo l t age . tt" 9 L DISCUSSION A t f r e q u e n c i e s n e a r 1 kHz, e l e c t r i c a l power o u t p u t s approach ing 100 watts p r c u b i c c e n t i m e t e r of PVF2 a t e f f i c i e n c i e s nea r 70% can t h e o r e t i c a l l y be o b t a i n e d from mechan ica l ly -d r iven d e v i c e s i f t h e c a p a c i t i v e source impedance i s r e s o n a t e d away by a ser ies i n d u c t o r . The l o a d m u s t be prope r ly matched t o t h e gene ra to r . The power o u t p u t w i l l be reduced by t h e product of t h e s a f e t y f a c t o r s by which t h e d e v i c e i s o p e r a t e d below both t h e y i e l d s t r a i n and e lectr ical breakdown l i m i t s . Power and e f f i c i e n c y are t a b u l a t e d below f o r d i f f e r e n t f r e q u e n c i e s , w i t h and wi thou t employing e lectr ical resonance . F req . , Hz Resonant? Power, 'd/cm' E f f i c i e n c y , W 1000 Yes 91 72 1000 no 1 9 6 . 3 60 Yes 5 .1 67 Table L Maximum power d e n s i t y and cor responding e f f i c i e n c y f o r v a r i o u s o p e r a t i n g c o n d i t i o n s d e s c r i b e d more f u l l y i n t e x t . T h e on iy expe r imen ta l test of t hese power o u t p u t p r e d i c t i o n s is provided by d a t a from t h r e e PVFa-based wind ?Enera to r s which we developed, !WO r o t a t i n g d e s i g n s and one o s c i l l a t i n f O d e s i g n . l An o u t p u t of 0.012 W/cm3 was ach ieved a t 1 8 . 7 ik and 144 V peak-peak o u t p u t w i t h o u t employing e l e c t r i c a l resonance. From Table 1 wi th ou tpu t of 19 W/cm3 reduced by t h e f r equency r a t i o 18.7/1000, an o u t p u t of 0.36 W/cm3 is expec ted a t y i e l d s t r a i n ampl i tude . Our result i s c o n s i s t e n t w i t h peak s t r a i n 18% of y i e l d s t r a i n because from 4s. ( 9 ) and (13) o u t p u t i s p r o p o r t i o n a l t o ( s t r a i n ) 2 . T h i s 541 peak s t r a i n i s close t o t h e s t r a i n e s t i m a t e d from s t r o b o s c o p i c o b s e r v a t i o n of t h e r o t o r . Cons t ruc t ion and testing of generators operating a t higher frequency w i t h larger s t r a i n l e v e l and employing e lectr ical resonance i s planned, t o provide a better test of pl-edict ions of Table L Piezoelectric polymers w i t h b e t t e r pi-operties are being deve loped The most promising i s a copolymer of v i n y l i d e n e f l u o r i d e (C€$CF2 monomer) w i t h trif 1 uor oethy l e n e (CHEF2 monomer )? 525148% copolymer h a s d t w i c e as l a r g e as for Its d i e l e c t r i c p e r m i t t i v i t y K’ is 19 i n s t e a d o f 12, and its Young’s modulus Y is 1.04 i n s t e a d of 1.5 i n u n i t s of 109 N/m2 c o n s t a n t k2’ is 0.0148 i n s t e a d of 0.0088. v a l u e s p r e d i c t improved e f f i c i e n c y and power dens i ty . cons ide rab ly l a r g e r t h a n t h e E /Qs term and s u b s t i t u t i n g for maximum power is n e a r l y p r o p o r t i o n a l t o Y, d, and 6y, w i t h on ly weak dependence on K‘ and Qs. one shou ld i n c r e a s e Q Q,, Y, and e s p e c i a l l y d, and should dec rease K . as Nigh a s 49~lO-~’;/N, so i t s e lec t romechan ica l coup l ing These From 4. (14) . n o t i n g t h a t t h e Ey term i s from 4. (h, we see t h a t t h e t o o and E b and To i n c r e a s e e f f i c i e n c y , t ’ CONCLUSIONS Experiments should be made t o de t e rmine whether t h e above power d e n s i t i e s can a c t u a l l y be approached. Tests t o d e t e r m i n e l i fe t ime a g a i n s t f a t i g u e , e l e c t r o d e failure, depo l ing or o t h e r f a i l u r e modes should be run, p r e f e r a b l y over a wide t e m p e r a t u r e range. Temperature rise from h e a t i n g caused by e lectr ical and mechanical losses should be monitored. Although t h e electromechanical coup l ing c o e f f i c i e n t of WF2 i s low compared t o t h a t of many ceramic and c r y s t a l l i n e p i e z o e l e c t r i c s , d e s i g n s based on mechanical and e l e c t r i c a l resonance should p rov ide g e n e r a t o r s w i t h h igh power ou tpu t and e f f i c i e n c y . materials i n c e r t a i n g e n e r a t o r a p p l i c a t i o n s because of i t s much greater f l e x i b i l i t y . WF2 i s p r e f e r a b l e t o t h e s e o t h e r ACKNCWLEDGEMENTS T h i s work was p a r t i a l l y suppor t ed by Montana Department of Natural Resources and Conse rva t ion Grant No. RAE-82-1017 and by a MONTS-NSF g r a n t . 1. 2 . 3 . 4. 5 . 6 . 7. 8 . 9 . 10 . 11. REFERENCES P. E. B l o o m f i e l d , R. A. F e r r e n , P. F. R a d i c e , H. S t e f a n o u , and 0. S. S p r o u t , Nava l R e s e a r c h Review3 Vol. 35. No. 5 , p. 1 (May 1 9 7 8 ) . Our unpublished work. N. Koizumi , N. Haikawa, a n d H. 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