THE UNITARY THEORY OF THE ELECTRIC POWERS. Gheorghe MIHAI
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1 Aals of the versty of Craova, Eletral Egeerg seres, No. 3, 6 HE NARY HEORY O HE EECRC OWERS Gheorghe MHA versty of Craova, aulty of Eletrotehs, 7 eebal Blvd. E-al: gha@elth.uv.ro Abstrat Geeral physs approah s appled to aalyss of power opoets eletral systes uder susodal ad o-susodal odtos. hysal essee of atve, reatve ad dstorto are deterate. t s show that the all three powers are the dfferet aspets of the sae physal pheoeo: ehaal ato per te of Coulob fores or ertal fores. Keywords: eletr power, geeral physs.. NROCON 97, C. Budeau has trodued for the frst te the reatve ad dstortg powers, ot based by physal pheoeo, oly based by dtos. May studes have aalyzed the physal sgfaes, but the proble has ot resolved. hus, there are ay dtos lterature [], [], [3] ad dfferet approah. ths wor, we are startg wth followg physal fudaetal oepts for resolvg the eletral powers uder susodal ad o-susodal odtos: A) We a assoate a ass to ay eergy for, based by Este forula: E A) ore s the varato of the pulse uder te; A3) he stataeous power s the ultpled uder fore ad veloty, at a oet.. HE ACVE EECRCA OWER, et us osder a eletral urret ( that pass throw a ateral odutor. he urret desty J ( ad the ea veloty of eletral harges v ( at a oet te are relato: J ( ρv( () he fore exersed o the eletral desty of the eletral feld E( s: ( ρe( () A stataeous power s detered as: p( ( v( (3) By substtutg ( fro () (3) wll obta: sg () wll obta: p( ρe( v( (4) p ( J ( E( (5) hs relato (5) represets the power desty a pot stuated Ω doa, overg of the eletral urret. he stataeous total power the Ω doa s Ω tot ( p( dv (6) he stataeous total power has obtaed by substtutg (5) ad the eleetal volue dv dsdl (6): tot ( ( J ( ds )( E ( dl ) ( S he ea power averaged over te terval s: ed t t, t ( + ( ( (7) ( dt (8) t // Where ( ) ad ( are the orthogoal // t opoets of the ( relato wth u ( [5]. der susodal odtos: s ( s( ϕ) // (9) sg (8) ad (9) the ea power averaged over a perod s sg o-susodal odtos: + ( + ed osϕ () s s ( + α ) ( + α ϕ ) he ea power over a perod s ()
2 Aals of the versty of Craova, Eletral Egeerg seres, No. 3, 6 ed + osϕ () Aalysg () ad () the oluso draw s that the ea atve power s the ea wor doe by the ouloba eletral fores over a perod. 3. HE REACVE EECRCA OWER, he proess the rut wth R ad eleets oeted seres wll aalyzed. he stataeous agetally eergy of the feld s ( w ( (3) he aget eergy s a futo of the veloty of the eletral harges, fro the urret desty (. he lterature [4] show that the oytg vetor that depeds to eletral harges veloty does ot radated eergy. Aordg (3), the aget eergy stored the dutor s o-radate but s oly varable te. Aordg to A) odto, the feld ass assoated to (3) s ovg wth the speed of lght. ( ap ( (4) he assoated puls to the feld ass s p ( ( (5) ap ap( Aordg to A), the rate of the puls s equal to ertal fore: dp dap( (6) dt dt ( he stataeous power orrespodg to the deplaset of the feld ass s ( ( (7) he ertal fore ay be both postve ad egatve. We jo the postve sg of the ertal fore to the dutor loadg wth aget eergy ad the us sg the the dutor uload. he eessary power for load ad uload the dutor d delberated by the eletr geerator. We osder the fetve value of the ertal fore both for loadg ad for uloadg of the dutor: ( dt (8) he reatve power s the ultpled betwee the speed of lght ad the fetve ertal fore: (9) he reatve power for a dutor Startg wth the susodal urret: ( s () he, the feld ass stored to aget feld of the dutor s: ap ( s () addto, the orrespodg pulse s p( s () he ertal fore at a oet geerated by the feld ass s: dp ω ( s (3) dt et osder the terval (, /4) whe s produed the eergy loadg of the dutor, ad alulate the fetve value of the ertal fore: / 4 ω ω s dt (4) he reatve power eessary to load the dutor s arare ω (5) Slarly, the reatve power eessary to uload the dutor s desarare / ω / 4 ω s dt (6) he total reatve power s the su betwee (5) ad (6): ω (7) Cosderg the eletr paraeters of the rut, the above relato a wrte: sϕ (8) 3.. he reatve power for a apator Startg wth the sae eletral urret (), we osder a apator rut. he teso for the apator s
3 Aals of the versty of Craova, Eletral Egeerg seres, No. 3, 6 u C ( dt C ωc os (9) he feld ass stored to eletr feld of the apator s: ap CuC ( ( os (3) ω C he ertal fore at a oet geerated by the feld ass s: ( s (3) ωc et osder the terval (, /4) whe s produed the eergy loadg of the apator, ad alulate the fetve value of the ertal fore: / 4 s dt (3) ωc ωc he reatve power eessary to load the apator s arare (33) ω C Slarly, the reatve power eessary to uload the dutor s desarare / / 4 s dt ωc ωc (34) he total reatve power s the su betwee (33) ad (34): (35) ω C 3.3. he reatve power C rut Coparg (3) ad (3) we observe that the ertal fore produed to dutor has opposte sg opare that the apator. he su of (3) ad (3) s..( ω s (36) ωc rez he reatve power s: / 4 ω ωc We a wrte the above relato as: ω s dt ωc (37) sϕ sϕ (38) 4. HE SORNG OWER et osder a eletral rut operated uder osusodal odtos: ( + s( t + α ) ω (39) + s( t + α ϕ ) ω (4) Aordg wth C. Budeau theory, the dstortg power s [( ) + ( ) ] ( ( ) ) osϕ ϕ Startg wth the followg relatos: (4) os( ϕ ϕ ) osϕ osϕ + sϕ sϕ (4) osϕ osϕ R R We a rewrte (4): sϕ sϕ ω ω [ ( ) ] We alulate the feld ass usg (4): ap where: ad: ( os α os (43) (44) ω (45) [( ) + α ϕ ] [( + ) + α ϕ ] α α ; α α + α ϕ + ϕ ϕ ; ϕ ϕ + ϕ (46) Startg to the dervatve futo of te for (46), we obta the ertal fore: 3
4 Aals of the versty of Craova, Eletral Egeerg seres, No. 3, 6 ( + ( ) s[ ( ) + α ϕ ] + ( ) [( ) ] (47) + ω s + + α+ ϕ + sg (8) for eah haro, we alulated the fetve value of the ertal fore: / Ø Ø ø ø Œ Œ Œ Œ / Œ Œ ( t )dt ( t )dt + Œ Œ ( ) - Œ Œ Œ Œº ß Œ Œ Ø ø Œ Œ Œ Œ / Œ Œ ( t )dt ( t )dt Œ Œ ( -) Œ Œ Œº Œº ß ß ag uder osderato the orthogoally of the trgooetr futos, we obta: ( ω ) ( ) ( ) + + ω.. (48) he above relato s the ultplato betwee the square power produed by the ertal fore ad the square of speed of lght. We alulate the followg expresso, startg wth the reatve ad dstortg powers gve by C. Budeau: s ( ω ) ϕ (49) [ ( ) ] ω (5) + ( ω ) + ( + )( ω) (5) We a observe that (48) ad (5) are detally. oluso, the square of the power gve by the fetve value of the ertal fore s equal to the su of the reatve ad dstortg powers, aordg C. Budeau ad S. ruze [], [3]. We deopose (48) usg physal rtera. 4.. he deoposto uder physs rtera Based by superposto prple, the eletral urret to a rut wth o-susodal perod voltage, s equal to the su of the urrets geerated by eah haro of voltage, f oly tself operato the rut. Based to rel. (7), the reatve power geerated by the haro s ω he reatve power geerated by all haros s total ω (5) herore, we obta a detally expresso that of C. Budeau. he dstortg power proposes by C. Budeau result fro rel.(5). he deoposg of the reatve power after S. ruze, two orthogoal opoets: reatve power ad dstortg power fro C. Budeau, has based o the superposto prple usg the eletr rut theory. Reatve power after C. Budeau represets the terato betwee the urret haros of the sae order,. stortg power s the terato betwee the urret haros of dfferet order,. 5. HE AAREN OWER a eletral rut two fores wor the Coulob fore, that s dsplae wth very sall veloty ad the ertal fore of the feld ass that s dsplae wth speed of lght. Our sope s to detere a Coulob equvalet fore that s dsplag wth the sae veloty wth the ertal fore of the feld ass. Addg these two fores, we obta a resultg fore that s dsplag wth speed of lght. Hs fetve value deteres the apparet power. he expresso that gves the equvalet eletr fore ust have the sae for that the teso apples: ehv ( s (53) he ostat has detered fro the followg relato:. ehv s ( ϕ) os osϕ (54) We substtute (54) to (53) ad we obta the equvalet eletr fore: ehv osϕ ( s (55) sg rel.(3), (7), (8) ad rel.(55) we obta the resultat fore fro the eletr rut: rez osϕ sϕ ( s + s (56) 4
5 Aals of the versty of Craova, Eletral Egeerg seres, No. 3, 6 he two fores gve rel.(55) ad rel.(3) are orthogoal uder a perod. he fetve value for the resultat fore gve to rel. (56) s. rez he apparet power s (57) S. rez (58) der o-susodal odto, rel. (56) beoes: rez ( sϕ osϕ s s (59) he frst ter represet the equvalet Coulob fore ehv ad the seod represet the ertal fore of the feld ass : ehv osϕ ( s (6) sϕ ( s (6) he two fores are orthogoally, that s: (, ( ehv he fetve value to the resultat fore gve (59) s. rez ehv + (6) he fetve value of the Coulob fore equvalet s ehv s dt osϕ osϕ (63) We substtute rel.(49) ad rel. (63) to rel.(6), that beoes: where: rez + + S ad he all results ay sythesze a dagra, whh presets the logal relatoshps betwee the fudaetal eleets of the theory preseted ad the osequees edate. Coulob law he equvalet eletr fore Atve ower 6. CONCSONS Relatvty theory he resultat fore Reatve ower Apparet ower S S++ Superpozto prple he ertal fore of the feld ass stortg ower ()he physal essee of power opoets s detered as the a a of the artle. hs proble we bee solvg usg oly the fetve values for the Coulob fores ad ertal fores over a perod. ()he equvalet fetve Coulob fores produe a eha wor that s the eletr atve power. (3)he fetve ertal fores of the feld ass ad superposto prple produe a ehaal wor that s reatve ad dstortg powers. (4)he fetve resultat fore that s obtaed fro equvalet Coulob fores a ertal fores of the feld ass produe a ehaal wor that s apparet power. Rerees [] C.. Budeau, ussae réatves et ftves, Edtura..E., Buurest, 97. [] M. Slo, hysal essee of power opoets, Eerga Elettra, vol.8, 4, Rerhe. [3] S. Sverso, ower easureet tehque for osusodal odtos, otoral hess, epartaet of Eletr ower Egeerg Chalers, versty of ehology Göteborg Swede, 999. [4].adau, E. fhtz, héore des haps, édtos Mr Mosou, 97. [5]. Reza, ear Spares Egeerg, Edtura ddata s pedagoga, Buurest,
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