Theoretical Electrical Power Output Per Unit Voume of PVF2and Mechanical-To-Electrical Conversion Efficiency as Functions of Frequency

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 toe 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