*ELEKTROTEHNIKA IIPredavanje - 5Mjerenje otporaWheatstoneovim i Thomsonovim mostom,Kompenzatori i mjerenja kompenzatorimaDogovorena vrijednost (DV), MP, MO, POVremenski promjenljive struje i, Imax, Imin, Im, TFourieova analiza

*Wheatstoneov mostI u elektrotehnici imamo mostove. Naravno to nisu u pravom smislu mostovi ali elektrine mree u obliku kvadrata sa dvije dijagonale koje prelaze jedna preko druge nazivamo mostom.

Slika 5.1. Elektrina mrea koju nazivamo mostomSlika 5.2. Kao to Domovinski most premouje Savu tako se i dijagonale premouju

*Roen; 6 veljae 1802 Gloucester, Engleska Umro; 19 October 1875 Pariz, Francuska Dravljanstvo; Velika Britanija Polje; fizika Poznat po; Wheatstoneovom mostu, ifriranju, spektroskopiji i telegrafiji Ipak u Britanskoj wikipediji pie da je taj most prvi osmislio Samuel Hunter Christie, iz Kraljevske vojne akademije, Woolwich. Ipak danas je poznat samo kao Wheatstoneov most. Sir Charles Wheatstone

*Wheatstoneov mostU praksi obino horizintalnu dijagonalu (prikazanu plavo) crtamo povuenu prema dolje, pa se tada gubi dojam premoivanja. Uz to je uobiajeno, ali nije obavezno, da se gornja lijeva grana oznai brojem 1, gornja desna grana brojem 2, donja lijeva grana brojem 3, donja desna grana brojem 4, vertikalna dijagonala (prikazana crveno) brojem 5 i horizontalna dijagonala (prikazana plavo) brojem 0 ili 6.Obino se u vertikalnoj dijagonali smjeta nul-indikator a u horizontalnoj izvor elektrine energije, ali moe i obrnuto.

*Wheatstoneov mostOvakav most se naziva Wheatstoneov most.Napiimo za njega jednadbe konturnih struja;Slika 5.3. Shema Wheatstoneovog mosta s ucrtanim konturnim strujama

*Wheatstoneov mostDa bi izmjerili nepoznati otpor dovoljno je da poznajemo na pr. otpor R2 i odnos otpora R3 i R4 kod ega nam nije vano da znamo koliko iznose ti otpori.Zbog toga se vrlo esto umjesto otpora R3 i R4 koristi otporna ica s kliznim kontaktom.Poznavanjem duljine ice i oitavanjem udaljenosti kliznog kontakta od jednog kraja te ice moemo ustanoviti odnos ova dva otpora.Neka je l=a+b gdje je l ukupna duljina ice, udaljenost kliznika od lijevog kraja je a, a sa b oznaujemo udaljenost od desnog kraja ice.

*Wheatstoneov mostI evo ovdje na slici 5.4. prikazan je jedan takav Wheatstoneov most s kliznom icom.Kako je; l=a+b uvijek se moe pisati da je; a=l-b ili b=l-a pa dobivamo;Slika 5.4. Wheatstoneov most s otpornom icom i kliznim kontaktom

*Wheatstoneov mostJasno je da poloaj kliznika moemo oitati samo s nekom pogrekom a ne apsolutno tono.Odmah si moemo postaviti pitanje a kako pogreka oitavanja poloaja kliznika djeluje na tonost mjerenja nepoznatog otpora ?Da odgovorimo na to pitanje potrebno je da izraunamo sigurne granice pogreke mjerenog otpora s obzirom na tonost oitavanja poloaja kliznika.Kod ovog rauna biti e nam povoljnije koristiti oblik jednadbe koja ima jednostavniji nazivnik, tj.

*Wheatstoneov mostNa slajdu 3.38. imamo jednadbu po kojoj moemo izraunati sigurne granice pogreaka posredno mjerene veliine kad imamo vie parametara kojima poznajemo sigurne granice pogreke.Pa mi u jednadbi na prethodnom slajdu vidimo da imamo tri parametra; R2, l i b.U ovoj analizi ograniiti emo se na to da ustanovimo utjecaj samo jednog od ova tri parametra tj. utjecaj parametra b.Neka je na primjer sigurna granica pogreke oitavanja poloaja kliznika, tj. mjerenje duljine b; Gb=0,5mm, odnosno openito Gb.

*Wheatstoneov mostKako elimo uvijek mjeriti otpor s nekom pogrekom koja je po apsolutnom iznosu manja od maksimalno dozvoljene, to moramo nekako konstrukcijom osigurati da kliznik ne moe doi preblizu krajeva otporne ice.Ovo se moe relativno lako postii dodavanjem nekih graninika koji e mehaniki sprijeiti pomak kliznika prema kraju ice.No otporna ica je relativno skupa, zahtijeva i prostor da se postavi.To je razlog da se dijelovi ice na koje kliznik ne smije doi zamijene nekim otporom.

*Wheatstoneov mostTime dobivamo konstrukciju prikazanu shemom na slici 5.6.Slika 5.6. Wheatstoneov most s otpornom icom i njoj dodanim dodatnim otpornicima.

*Wheatstoneov mostKako rjeavamo ovakav most ?Pa jednostavno ili vrijednost dodatnih otpora pretvaramo u odgovarajuu duljinu ice ili vrijednosti duljine l i b pretvaramo u odgovarajue vrijednosti otpora. Ostalo je jednostavno.

A sada pogledajmo shemu tvornike izvedbe Wheatstoneovog mosta Iskra, Kranj !

*Wheatstoneov most Slika 5.7. Tvornika izvedba Wheatstoneovog mosta proizvod Iskra, Kranj

*Wheatstoneov mostIako prema jednadbama, pa i prema prethodnoj shemi (sa slike 5.7.) ispada da moemo s Wheatstoneovim mostom mjeriti i vrlo male otpore (shema sa slike 5.7. 50m) ipak to u stvari nije mogue.Odnosno mogue je ali je nedovoljno tono.Zato ?Pa zato jer na rezultat mjerenja kod malih otpora jako utjee iznos otpora kontakata. A taj je promjenljiv i iznosi najee negdje oko 0,5 do 3m. Moe naravno biti i manji ali i znatno vei.

*Wheatstoneov most za male otporeKako bi se smanjio utjecaj promjenljivih otpora kontakata mali otpori se izrauju s etiri prikljune stezaljke (slajdovi 1.70. do 1.73.). Takav otpor treba spojiti po ovdje prikazanoj shemi Sl. 5.8.Slika 5.8. Shema Wh. mosta za mjerenje malih otpora

*Wheatstoneov most za male otporeIako su ovim spojem otpori kontakata dodani otporima koji mogu biti znatno vei, pa nee utjecati na rezultat mjerenja ipak nastaje potekoa jer zbog toga mora biti veliki odnos izmeu otpora R3 i R4 ili otpora R2 i R4 a time je teko postii dovoljnu osjetljivost mosta.Sl. 5.9. Pregled poloaja otpora kontakata kod Wh. mosta za mjerenje malih otpora

*Thomsonov mostOva nemogunost postizanja dovoljne osjetljivosti ograniava upotrebljivost spoja s prethodna dva slajda.Stoga je razvijen Thomsonov most koji omoguuje kvalitetno mjerenje malih otpora uz veliku osjetljivost.Thomsonov most ima dio kroz koji moe prolaziti struja znatnih iznosa kako bi i na malim otporima stvorila potrebne i dovoljno velike padove napona, koji e omoguiti dovoljno tono mjerenje malih otpora.Pa pogledajmo shemu Thomsonovog mosta !

*Thomsonov most Slika 5.10. Shema Thomsonovog mosta. Debelo i crveno iscrtan je dio strujnog kruga mosta kroz koji protjee struja znaajnog iznosa. Ostatak mosta slui za naponske usporedbe dakle i potrebne struje su slabog intenziteta. Uoava se trokut otpora izmeu vorova B, D i E (otpori R3, R4 i RS gdje je otpor RS, otpor komadia spojne ice i iznosi svega nekoliko m). Kod Thomsonovog mosta su otpori R1 i R3 jednaki i promjenljivi a otpori R2 i R4 su takoer jednaki ali su konstantnog iznosa.Trokut otpora R3, R4 i RS se postupkom transfiguracije pretvara u odgovarajuu zvijezdu otpora.

*Thomsonov mostMoe se pisati da je;

U ravnotei je struja kroz nul-indikator jednaka nuli pa nas zbog toga niti ne zanima iznos otpora Rb.Nacrtajmo si novonastalu shemu nakon pretvorbe ovog trokuta otpora u odgovarajuu zvijezdu otpora.

Lako je uoiti sa slike 5.11. da smo na ovaj nain dobili najobiniji Wheatstoneov most.

*Thomsonov mostShema mosta nakon provedene transfiguracije je prikazana ovdje na slici 5.11. !Vrijednosti njegovih pojedinih otpora su;RW1=R1RW2=R2RW3=RX+RaRW4=Rc+RN Slika 5.11. Izgled sheme Thomsonovog mosta nakon provedene transfiguracije trokuta otpora izmeu vorova B, D i E. Thomsonov most se ovom transfiguracijom pretvara u Wheatstoneov most, a taj smo ve rijeili.

*KompenzatoriKompenzatori su mjerne naprave koje prvenstveno slue za mjerenje nepoznate elektromotorne sile energetski slabih izvora (itaj izvora s velikim unutarnjim otporom ili s vrlo malom elektromotornom silom).Svi kompenzatori proizali su iz dva osnovna spoja;

1. Potenciometarski postupak prema Poggendorffu (1841.g.)2. Ampermetarski postupak prema Lindeck-Rotheu (1899.g.)

*Johann Christian Poggendorff Roen; * 29. prosinac 1796 u Hamburgu Umro; 24. sijenja 1877 u Berlinu njemaki fiziar galvanometar oitavanje zrcalom naponsku kompenzaciju elektrostatski motor Pogendorfova optika varka itd.

*KompenzatoriPrincip potenciometarskog postupka vidi se sa slike 5.12. !Preklopka se postavi u poloaj 1 a kliznik (C) postavi se na vrijednost otpora (RP1) koja odgovara naponu UN.Kliznikom otpora Rr podesi se vrijednost struje tako da nul-ind. pokae otklon kazaljke jednak nuli.Slika 5.12. Kompenzator na principu potenciometarskog postupka

*KompenzatoriSada se preklopka prebaci na poloaj 2.Kliznik potenciometra RP pomie se dok se ponovno ne postigne ravnotea tj. dok ne postignemo da nul-indikator pokazuje nulu.Tada se oita vrijednost otpora RP2.Iz ovih dviju vrijednosti otpora potenciometra i poznatog iznosa napona UN moe se izraunati da je;

Napon izvora UX je izmjeren kod neoptereenog izvora (struja nul-indikatora IN=0).

*KompenzatoriNo ovaj nain ima nedostatak. Kod prebacivanja preklopke s poloaja 1 na poloaj 2, moe se promijeniti struja I (tj. Ip).A u raunu na prethodnom slajdu smo uzeli zdravo za gotovo da se kod toga ova struja ne mijenja, tj. da je sigurno konstantnog iznosa.Kako bi se uklonio ovaj nedostatak cijeli spoj se radi malo drugaije prema shemi prikazanoj na slici 5.13. na slijedeem slajdu.Po ovoj shemi struja se takoer moe promijeniti, ali moemo lako ustanoviti da li se je struja promijenila ili je ostala ista.

*KompenzatoriMjerenje se obavlja tako da se prvo preklopka prebaci na poloaj 1.Nakon toga se pomiui kliza podesi vrijednost otpora R1 tako da je pad napona U1=IR1 jednak poznatom naponu UN.Preklopka se prebaci u poloaj 2 i podesi otpor R2 na vrijednost da je U2=IR2, jednak naponu UX.Slika 5.13. Kompenzator s potenciometarskim postupkom kod kojeg se jednostavno provjerava konstantnost podeene struje I (Ip).

*KompenzatoriNaravno kod ovog prebacivanja poloaja preklopke i pomicanja klizaa, moe se (iako nema vidljivog razloga) malo promijeniti vrijednost struje I (Ip).Da provjerimo, dovoljno je prebaciti preklopku ponovno u poloaj 1.Ako se struja I (Ip) nije promijenila nul-indikator e i nadalje pokazivati nulu.Ako nul-indikator ne pokazuje nulu cijeli postupak podeavanja poloaja jednog i drugog kliznika treba ponoviti.

*KompenzatoriPrincip ampermetarskog postupka vidi se sa slike 5.14. !Prikljui se izvor napona nepoznatog iznosa UX.Kliznikom potenciometra Rr reguliramo iznos struje.Poveavanjem (ili smanjivanjem) iznosa struje poveava se pad napona na otpornikoj normali. Kad e taj pad napona biti jednak nepoznatom naponu UX nul-indikator e pokazati nulu.Slika 5.14. Kompenzator na principu ampermetarskog postupka.

*KompenzatoriIz oitane vrijednosti struje I i iznosa otpora otpornike normale RN izrauna se iznos nepoznatog napona UX (i ovdje u neoptereenom stanju). UX=IRN

U tehnikoj mjernoj praksi koristi se za mjerenja tehniki kompenzator. To je naprava koja se sastoji od strujnog izvora (izvora konstantne struje), jednog kliznog otpornika malog iznosa otpora i serijskog spoja cca 10 preciznih otpora. Naravno tu je obavezno i nul-indikator.

*KompenzatoriSlika 5.15. Principijelna shema tehnikog kompenzatora

*KompenzatoriKad ukljuimo sklopku, strujni izvor (elektroniki sklop) poinje proizvoditi struju konstantnog i tono poznatog iznosa na primjer tono 1,000mA.Ova struja strujnog izvora prolazi kroz cijeli otpor potenciometra RP i kroz svih 10 preciznih otpora R.Preklopkom se bira odreeni broj otpora tj. time se podeava napon u skokovima od po IR.Nakon grubog podeavanja preklopkom vri se fino podeavanje kliznikom potenciometra (Ukupni otpor potenciometra jednak je iznosu jednog otpora R).Kad je prikljueni napon UX jednak naponu kompenzatora nul-indikator pokae 0.

*KompenzatoriMjerenje otpora tehnikim kompenzatorom.

Slika 5.16. Shema koja omoguuje mjerenje nepoznatog otpora tehnikim kompenzatorom. Prvo se mjeri pad napona na otporu RX (moe na RN) a nakon toga na poznatom otporu RN. Struja se kod toga ne smije promijeniti. Iz odnosa ova dva pada napona kod nepromijenjene struje moe se izraunati otpor RX.

*Tehniki kompenzatorSlika 5.17. Tehniki kompenzator

*KompenzatoriOstalo nam je jo da obradimo kako se s kompenzatorom mjeri nepoznata struja (tonije reeno kako se kontrolira ispravnost nekog ampermetra).

U tu svrhu obino emo koristiti tehniki kompenzator i vrlo toan mjerni otpornik otpornu normalu. Naravno i ampermetar kojeg kontroliramo.

*KompenzatoriSvu spomenutu opremu spojiti emo prema shemi prikazanoj ovdje na slici 5.18. !Regulacionim otpornikom Rr namjestimo na ampermetru neku eljenu vrijednost struje.Tehnikim kompenzatorom izmjerimo pad napona na otpornoj normali RN.Sada se po Ohmovom zakonu lagano izrauna toan iznos struje koja prolazi kroz ampermetar i utvrdi njegova pogreka.Slika 5.18. Mjerenje nepoznate struje tehnikim kompenzatorom (tj. kontroliranje tonosti ampermetra).

*Kompenzatori

Kod toga vidimo da nam se postotna pogreka mjerenja struje i postotna pogreka po kojoj odreujemo klasu ampermetra razlikuju.Za raun postotne pogreke po kojoj se odreuje klasa instrumenta treba u nazivniku uzeti dogovorenu vrijednost (krae DV).Kod raznih instrumenata je ta dogovorena vrijednost razliita vidi slijedee slajdove.

*Nul-metodeNa kraju ovog dijela predavanja napomenuo bih da neki autori mjerenja mostovima i kompenzatorima nazivaju mjerenja nulmetodama. Meni se lino ovaj naziv ne svia pa ga izbjegavam.No ipak vano je napomenuti da se ta mjerenja uvijek obavljaju tako da se struja kroz nul-indikator dovede na nulti iznos. Nul-indikator je naravno vrlo osjetljivi ampermetar sa nulom na sredini skale. On ne treba biti toan kod pokazivanja iznosa struje ali mora biti vrlo toan na podruju u blizini nule pa jedino nas taj dio pokazivanja zanima.Mjerenja mostovima i kompenzatorima su vrlo tona.

*Dogovorena vrijednostNajee se kao dogovorena vrijednost uzima vrijednost mjernog podruja, mjernog opsega, mjernog dometa a rijetko se uzima prava vrijednost mjerene veliine.Vrijednost mjernog podruja je vrijednost koju oitamo na poloaju preklopke instrumenta.To je najjednostavnije prikazati slikom.Slika 5.19. Preklopka prikazuje brojku 10mA (izmjenino), dakle na ovom instrumentu odabrano je izmjenino mjerno podruje od 10mA.

*Dogovorena vrijednostNa slici 5.20. prikazana je skala instrumenta s mjernim podrujem od na primjer 50mA.To je ono to bi proitali na poloaju preklopke.Ali mjerni opseg ovog instrumenta bi bio od prve do druge toke oznaene na skali.Taj dio se naziva mjerni opseg instrumenta i on samo u tom opsegu mjeri mjerenu veliinu s deklariranom tonou.Slika 5.20. Pokazni opseg ovog instrumenta je na primjer od 0 do 50mA. Ipak mjerni opseg je samo od 5 do 25mA (Oznaeno tokama na skali).Donja granica mjernog opsegaGornja granica mjernog opsegaGornja granica pokaznog opsega instrumenta koju oitavamo na poloaju preklopke i doivljavamo kao mjerno podruje

*Dogovorena vrijednostPonovimo !Gornja granica mjernog opsegaDonja granica mjernog opsegaGornja granica pokaznog opsegaMjerni domet je vrijednost koja odgovara gornjoj granici mjernog opsegaPokazni opseg je cijelo podruje skale na kojem se moe promatrati otklon kazaljkeSlika 5.21. Razlike izmeu pokaznog opsega, mjernog opsega i mjernog dometa instrumenta

*Dogovorena vrijednostZa instrumente s nulom na jednom kraju skale a gornjom granicom mjernog opsega na drugom kraju skale dogovorena vrijednost je vrijednost gornje granice mjernog opsega.Nula na jednom kraju skaleVrijednost gornje granice mjernog opsega na drugom kraju skaleDogovorena vrijednostSlika 5.22. Mjerni opseg i dogovorena vrijednost kako je to kod ogromne veine analognih mjernih instrumenata.

*Dogovorena vrijednostZa instrumente s nulom izmeu oba kraja skale dogovorena vrijednost jednaka je zbroju vrijednosti obiju granica, neovisno o predznaku.

Donja granica na pr. -70 dsk.Gornja granica na pr. +70 dsk.Nula izmeu oba kraja skaleSlika 5.23. U ovom primjeru bi dogovorena vrijednost bila vrijednost koja odgovara zbroju donje i gornje granice tj. vrijednost |-70|+|+70|=140 (dsk). naravno u odgovarajuoj vrijednosti mjernih jedinica 140810-8A=11,2A Osjetljivost instrumenta

*Dogovorena vrijednostA kako je kod ommetarskih skala ?

Slika 5.24. Pokazni opseg je od 0 oma. Mjerni opseg nije posebno oznaen ali s obzirom na izgled skale jasno je da mu je gornja granica cca 100 do 500 oma. Znai da je mjerni domet cca 100 do 500 oma. Ali koju vrijednost odabrati kao dogovorenu vrijednost ? Ovdje se kao dogovorena vrijednost uzima duljina skale na pr. 100mm, ili slino po donjoj recimo linearnoj skali 120 dsk..0120Ovdje bi oznaka klase tonosti trebala imati ovaj oblik; za -metarska podruja

*Dogovorena vrijednostKod frekvencmetara s jezicima kao dogovorenu vrijednost uzimamo vrijednost za svaki jeziac onu koja pie na kraju skale.47,5Dogovorena vrijednost za ovaj jeziac e biti 55 HzSlika 5.25. Kod frekvencmetara s jezicima kao dogovorena vrijednost uzima se vrijednost na kraju skale.

*Dogovorena vrijednostDa li je to stvarno tako ?Pa kad je klasa tonosti instrumenta oznaena jednostavno s brojkom bez ikakvih dodataka onda je to upravo tako.Slika 5.26. Ovaj instrument je klase 0,5 tj. smije grijeiti 0,5% od 55Hz bilo gdje na skali.

*Dogovorena vrijednostAli kod ovog frekvencmetra je oznaka klase u krunici.to to znai ?Pa samo to da je ovdje dogovorena vrijednost naznaena vrijednost svakog jezika posebno.Naznaena frekvencija ovog jezika je 7 Hz.Slika 5.27. Kako je oznaka klase tonosti instrumenta u krunici, a to onda znai da je dozvoljena pogreka u granicama 1% od stvarne vrijednosti, tj. naznaene vrijednosti jezika.To je dakle i dogovorena vrijednost

*Vremenski promjenljive strujeKad elektrina struja nije konstantnog iznosa ve se tijekom vremena mijenja onda kod nje razlikujemo razliite njezine vrijednosti. Pa vrijeme je da se upoznamo sa njima.Trenutna vrijednost elektrine struje je stvarna vrijednost elektrine struje u nekom odabranom, odnosno odreenom, trenutku vremena. Kako ovakav trenutak vremena esto nazivamo moment, udomaen je i naziv momentalna vrijednost elektrine struje.Pravilno je da se trenutne vrijednosti struje piu malim slovom.

*Vremenski promjenljive strujePromotrimo kako se sve moe mijenjati trenutna vrijednost struje.1. vrijednost struje se tijekom vremena ne mijenja, struja ima konstantni iznos i(t)=I.

Slika 5.28. Kad se struja tijekom vremena ne mijenja, tj. kad ima stalni (konstantni) iznos trenutne vrijednosti onda takvu struju nazivamo stalna istosmjerna struja, a zbog lijenosti obino samo kaemo istosmjerna struja.

*Vremenski promjenljive struje2. struja ima stalni smjer ali vrlo promjenljivi iznos takvu bi struju mogao imati neki instrument za biljeenje barometarskog pritiska ili temperature zraka.Slika 5.29. Vremenski promjenljiva neperiodika istosmjerna struja struja koju ne moemo matematiki opisati ali niti obuhvatiti bilo kakvom analizom osim da konstatiramo najveu i najmanju vrijednost u promatranom vremenu.

*Vremenski promjenljive strujeOvakav oblik i promjenu struje nemogue je teoretski opisati (matematiki aproksimirati), moemo kroz neki dui period vremena konstatirati njezin maksimalni i minimalni iznos, a eventualno i neki srednji iznos. I to je sve.

Na sreu osim kod biljeenja temperature, vlanosti i tlaka zraka obino se ne susreemo s takvim oblicima struje.

U pravilu se moe uoiti neka pravilnost periodinost. To znai da nam se nakon nekog vremena graf promjene struje ponavlja na potpuno jednak nain.

*Vremenski promjenljive strujeIpak ponekad se dogaa da ovakve struje mijenjaju svoj smjer. Tada govorimo da su to izmjenine (naizmjenine) struje.Ako nemaju nikakvu pravilnost vrijedi isto kao i ono to je ve reeno na slajdu 5.60, odnosno 5.61.Slika 5.30. Neperiodika izmjenina struja (nepotrebno je govoriti vremenski promjenljiva, jer kad kaemo izmjenina onda znai da mijenja smjer dakle da se i vremenski mijenja).

*Vremenski promjenljive strujeNa slajdu 5.62. moemo jedino uoiti neke karakteristine toke. To su tako zvane nul toke. Toke u kojima je vrijednost struje jednaka nuli i uz to (to je vano) tu struja mijenja smjer. Ili iz pozitivnog prelazi u negativni ili obratno.No ako su i ove toke stihijski rasporeene na vremenskoj osi nemamo od njih nikakve koristi.Ipak vrlo je est sluaj da se nakon nekog perioda vremena T graf promjene struje ponavlja na potpuno jednak nain. Tada govorimo o periodiki promjenljivim strujama (= ili ~).

*Vremenski promjenljive strujeKad se struja vremenski mijenja tako da se nakon perioda T njezin graf na potpuno jednak nain ponavlja govorimo o periodikim strujama.Slika 5. 31. Periodika izmjenina struja (moe se naglasiti i to da nije sinusoidalna ali nije obavezno)

*Vremenski promjenljive strujeKod tih struja moemo analizom ustanoviti njihovu najveu, tj. maksimalnu vrijednost, te njihovu najmanju, to jest minimalnu vrijednost. Moemo ustanoviti i najkrae vrijeme nakon kojeg se graf struje ponavlja na potpuno jednak nain. To vrijeme nazivamo vrijeme trajanja periode struje (napona).

i(t)=i trenutna vrijednost strujeImax. najvea vrijednost struje u periodiImin. najmanja vrijednost struje u periodiT vrijeme trajanja periode

*Vremenski promjenljive strujeApsolutna vrijednost maksimalne i minimalne vrijednosti struje ne moraju biti jednake. Onu od tih apsolutnih vrijednosti koja je vea nazivamo tjemena vrijednost izmjenine struje i oznaujemo ju s indeksom m.

Im je tjemena vrijednost periodike strujeIm=max.(|Imax.|;|Imin.|)

Periodiki promjenljive struje mogu biti istosmjerne ili izmjenine.

*Sinusoidalne strujeOd svih valnih oblika najjednostavnija je za matematiko opisivanje i analiziranje izmjenina struja koja se vremenski mijenja po zakonu sinusa.Neki kau sinusne struje a neki ih nazivaju sinusoidalne struje.to se mene tie koristite naziv koji vam se vie svia. Ja lino smatram da je ispravnije govoriti o strujama koje su sinusoidalne jer je sinus samo njihova matematika aproksimacija a ne i prava zakonitost njihove promjene.

*Sinusoidalne strujeOvdje je prikazan graf sinusoidalne struje.

i(t)=Imsin(t)

Uoavamo da je ovdje Im=Imax.=|Imin.|. Da su obje poluperiode jednake tj. da je ISA=0.

Slika 5.32. Graf jedne sinusoidalne struje struje ija se jaina mijenja tijekom vremena po sinusnom zakonu.

*Sinusoidalne strujePozabavimo se prouavanjem svojstava sinusoidalnih struja iz dva razloga.

1. razlog je taj to je sinusoidalni valni oblik struje najvie koriteni valni oblik izmjenine struje vjerojatno u preko 90% sluajeva a moda i u 99%.

2. razlog je taj da se Fourierovom analizom bilo koja periodiki promjenljiva struja moe rastaviti na red sinusoida i nakon toga u linearnim mreama moemo doi do konanog rezultata metodom superpozicije raunajui za svaki harmonik posebno.

*Joseph Fourier

Roenje21. mart 1768 Auxerre, Yonne, FrancuskaSmrt16. maj 1830. (62. godina) Pariz, FrancuskaPrebivalite FrancuskaNarodnost FrencuzPoljeMatematika, fizika i historijaInstitucijacole Normale - cole PolytechniqueDiplomacole NormaleAkademski mentorJoseph LagrangeIstaknuti studentiGustav DirichletGiovanni PlanaClaude-Louis NavierPoznat poFourierova transformacija,Efekt staklenika

*Fourierova analizaNa prethodnom slajdu spomenuli smo da je 2. razlog za prouavanje sinusoidalnih struja;2. razlog je taj da se Fourierovom analizom bilo koja periodiki promjenljiva struja moe rastaviti na red sinusoida i nakon toga u linearnim mreama moemo doi do konanog rezultata metodom superpozicije raunajui za svaki harmonik posebno.Pa pogledajmo neke primjere !Pogledajmo kako to izgleda kod napona (ili struje) koji se mijenja kao izmjena pravokutnika (pravokutni impuls).

*Pravokutni impulsPreko Fourierovog integrala mogu se izraunati potrebni koeficijenti tonije reeno amplitude i poetne faze pojedinih harmonika.Za neke jednostavne sluajeve ove koeficijente nalazimo u matematikim i drugim prirunicima.U konkretnom primjeru koristio sam Matematiki prirunik za inenjere i studente autora I.N.Brontejn i K.A.Semendjajev, str. 646;a=100 (amplituda pravokutnog signala) ; x = vrijeme

*Trokutni naponA sada pogledajmo kako ista ta aproksimacija djeluje kod trokutnog napona !No ovdje krenimo obrnutim redosljedom pa za poetak evo matematikog rezultata Fourierove analize iz prije spomenutog prirunika sa prethodnog slajda.Naravno jednadba iz prirunika prilagoena je naim podacima. A=200/ u originalu je A=1, a x je u radijanima (u grafu u )

*Trokutni naponPa i bez 11. harmonika ova aproksimacija je gotovo savrena, ako ne gledamo na sam vrh trokuta, koji je malo zaobljen.

Grafikon1

000

000

1.11111111111.41463878211.1808124857

2.22222222222.82884665152.3594861159

3.33333333334.24219282693.5339451033

4.44444444445.65424678924.7022380141

5.55555555567.06457841315.8625955319

6.66666666678.47275809817.0134830635

7.77777777789.87835689888.1536466853

8.888888888911.28094665649.2821511129

1012.680100128310.3984085939

11.111111111114.075391118811.502197872

12.222222222215.466394608912.5936726421

13.333333333316.852686885413.6733592066

14.444444444418.233845670214.7421433388

15.555555555619.609450248915.801246661

16.666666666720.979081599216.8521931341

17.777777777822.342322517917.8967665379

18.888888888923.698757748818.9369600714

2025.047974108519.9749194333

21.111111111126.389560612721.0128809326

22.222222222227.72310860122.0531063286

23.333333333329.048211861923.0978162097

24.444444444430.364466756124.1491237731

25.555555555631.671472339625.2089708805

26.666666666732.968830485726.279068219

27.777777777834.256146006727.360841307

28.888888888935.533026773928.4553839443

3036.799083836829.5634205196

31.111111111138.053931542330.6852783684

32.222222222239.297187651631.8208711107

33.333333333340.528473456932.9696936152

34.444444444441.747413896634.1308289251

35.555555555642.953637669635.3029671637

36.666666666744.146777348436.4844361082

37.777777777845.326469491237.6732428022

38.888888888946.492354752438.8671252628

4047.644077991840.0636130533

41.111111111148.781288383641.2600952255

42.222222222249.903639522442.4538939125

43.333333333351.010789529143.6423416664

44.444444444452.102401155344.8228604937

45.555555555653.178141885245.9930404571

46.666666666754.237684038147.1507156752

47.777777777855.280704866948.2940355723

48.888888888956.306886657649.4215293084

5057.315916825150.532161449

51.111111111158.307488009151.6253771165

52.222222222259.281298167452.701135097

53.333333333360.23705066853.7599276475

54.444444444461.174454379554.8027860596

55.555555555662.093223759655.8312713753

56.666666666762.993078942156.8474500092

57.777777777863.873745822757.8538544085

58.888888888964.734956141458.8534292538

6065.576447565559.849464082

61.111111111166.397963768360.8455135623

62.222222222267.199254508261.8453069962

63.333333333367.980075704262.852648909

64.444444444468.740189510663.8713128623

65.555555555669.479364389364.9049308356

66.666666666770.197375180665.9568806799

67.777777777870.894003171467.0301742589

68.888888888971.569036162168.1273489328

7072.22226853169.2503650241

71.111111111172.853501297470.4005118243

72.222222222273.462542181571.5783245596

73.333333333374.049205663572.7835145311

74.444444444474.613313040374.014914392

75.555555555675.15469247975.2704402184

76.666666666775.673179070476.5470716824

77.777777777876.168614878477.8408512546

78.888888888976.640848988479.1469029515

8077.089737553480.4594707197

81.111111111177.515143837681.7719761111

82.222222222277.91693825883.0770944729

83.333333333378.294998424284.3668484571

84.444444444478.649209175585.6327172545

85.555555555678.979462615886.8657595934

86.666666666779.285658146688.0567482189

87.777777777879.567702497989.1963132863

88.888888888979.825509756390.2750918824

9080.059001391191.2838807245

91.111111111180.268106278592.2137889849

92.222222222280.452760723293.0563881605

93.333333333380.612908477593.803855938

94.444444444480.748500759194.4491111122

95.555555555680.859496265194.9859367792

96.666666666780.945861185295.4090892593

97.777777777881.007569211995.7143904895

98.888888888981.044601548295.898801964

10081.056946913995.96047868

10081.056946913995.96047868

98.888888888981.044601548295.898801964

97.777777777881.007569211995.7143904895

96.666666666780.945861185295.4090892593

95.555555555680.859496265194.9859367792

94.444444444480.748500759194.4491111122

93.333333333380.612908477593.803855938

92.222222222280.452760723293.0563881605

91.111111111180.268106278592.2137889849

9080.059001391191.2838807245

88.888888888979.825509756390.2750918824

87.777777777879.567702497989.1963132863

86.666666666779.285658146688.0567482189

85.555555555678.979462615886.8657595934

84.444444444478.649209175585.6327172545

83.333333333378.294998424284.3668484571

82.222222222277.91693825883.0770944729

81.111111111177.515143837681.7719761111

8077.089737553480.4594707197

78.888888888976.640848988479.1469029515

77.777777777876.168614878477.8408512546

76.666666666775.673179070476.5470716824

75.555555555675.15469247975.2704402184

74.444444444474.613313040374.014914392

73.333333333374.049205663572.7835145311

72.222222222273.462542181571.5783245596

71.111111111172.853501297470.4005118243

7072.22226853169.2503650241

68.888888888971.569036162168.1273489328

67.777777777870.894003171467.0301742589

66.666666666770.197375180665.9568806799

65.555555555669.479364389364.9049308356

64.444444444468.740189510663.8713128623

63.333333333367.980075704262.852648909

62.222222222267.199254508261.8453069962

61.111111111166.397963768360.8455135623

6065.576447565559.849464082

58.888888888964.734956141458.8534292538

57.777777777863.873745822757.8538544085

56.666666666762.993078942156.8474500092

55.555555555662.093223759655.8312713753

54.444444444461.174454379554.8027860596

53.333333333360.23705066853.7599276475

52.222222222259.281298167452.701135097

51.111111111158.307488009151.6253771165

5057.315916825150.532161449

48.888888888956.306886657649.4215293084

47.777777777855.280704866948.2940355723

46.666666666754.237684038147.1507156752

45.555555555653.178141885245.9930404571

44.444444444452.102401155344.8228604937

43.333333333351.010789529143.6423416664

42.222222222249.903639522442.4538939125

41.111111111148.781288383641.2600952255

4047.644077991840.0636130533

38.888888888946.492354752438.8671252628

37.777777777845.326469491237.6732428022

36.666666666744.146777348436.4844361082

35.555555555642.953637669635.3029671637

34.444444444441.747413896634.1308289251

33.333333333340.528473456932.9696936152

32.222222222239.297187651631.8208711107

31.111111111138.053931542330.6852783684

3036.799083836829.5634205196

28.888888888935.533026773928.4553839443

27.777777777834.256146006727.360841307

26.666666666732.968830485726.279068219

25.555555555631.671472339625.2089708805

24.444444444430.364466756124.1491237731

23.333333333329.048211861923.0978162097

22.222222222227.72310860122.0531063286

21.111111111126.389560612721.0128809326

2025.047974108519.9749194333

18.888888888923.698757748818.9369600714

17.777777777822.342322517917.8967665379

16.666666666720.979081599216.8521931341

15.555555555619.609450248915.801246661

14.444444444418.233845670214.7421433388

13.333333333316.852686885413.6733592066

12.222222222215.466394608912.5936726421

11.111111111114.075391118811.502197872

1012.680100128310.3984085939

8.888888888911.28094665649.2821511129

7.77777777789.87835689888.1536466853

6.66666666678.47275809817.0134830635

5.55555555567.06457841315.8625955319

4.44444444445.65424678924.7022380141

3.33333333334.24219282693.5339451033

2.22222222222.82884665152.3594861159

1.11111111111.41463878211.1808124857

000

000

-1.1111111111-1.4146387821-1.1808124857

-2.2222222222-2.8288466515-2.3594861159

-3.3333333333-4.2421928269-3.5339451033

-4.4444444444-5.6542467892-4.7022380141

-5.5555555556-7.0645784131-5.8625955319

-6.6666666667-8.4727580981-7.0134830635

-7.7777777778-9.8783568988-8.1536466853

-8.8888888889-11.2809466564-9.2821511129

-10-12.6801001283-10.3984085939

-11.1111111111-14.0753911188-11.502197872

-12.2222222222-15.4663946089-12.5936726421

-13.3333333333-16.8526868854-13.6733592066

-14.4444444444-18.2338456702-14.7421433388

-15.5555555556-19.6094502489-15.801246661

-16.6666666667-20.9790815992-16.8521931341

-17.7777777778-22.3423225179-17.8967665379

-18.8888888889-23.6987577488-18.9369600714

-20-25.0479741085-19.9749194333

-21.1111111111-26.3895606127-21.0128809326

-22.2222222222-27.723108601-22.0531063286

-23.3333333333-29.0482118619-23.0978162097

-24.4444444444-30.3644667561-24.1491237731

-25.5555555556-31.6714723396-25.2089708805

-26.6666666667-32.9688304857-26.279068219

-27.7777777778-34.2561460067-27.360841307

-28.8888888889-35.5330267739-28.4553839443

-30-36.7990838368-29.5634205196

-31.1111111111-38.0539315423-30.6852783684

-32.2222222222-39.2971876516-31.8208711107

-33.3333333333-40.5284734569-32.9696936152

-34.4444444444-41.7474138966-34.1308289251

-35.5555555556-42.9536376696-35.3029671637

-36.6666666667-44.1467773484-36.4844361082

-37.7777777778-45.3264694912-37.6732428022

-38.8888888889-46.4923547524-38.8671252628

-40-47.6440779918-40.0636130533

-41.1111111111-48.7812883836-41.2600952255

-42.2222222222-49.9036395224-42.4538939125

-43.3333333333-51.0107895291-43.6423416664

-44.4444444444-52.1024011553-44.8228604937

-45.5555555556-53.1781418852-45.9930404571

-46.6666666667-54.2376840381-47.1507156752

-47.7777777778-55.2807048669-48.2940355723

-48.8888888889-56.3068866576-49.4215293084

-50-57.3159168251-50.532161449

-51.1111111111-58.3074880091-51.6253771165

-52.2222222222-59.2812981674-52.701135097

-53.3333333333-60.237050668-53.7599276475

-54.4444444444-61.1744543795-54.8027860596

-55.5555555556-62.0932237596-55.8312713753

-56.6666666667-62.9930789421-56.8474500092

-57.7777777778-63.8737458227-57.8538544085

-58.8888888889-64.7349561414-58.8534292538

-60-65.5764475655-59.849464082

-61.1111111111-66.3979637683-60.8455135623

-62.2222222222-67.1992545082-61.8453069962

-63.3333333333-67.9800757042-62.852648909

-64.4444444444-68.7401895106-63.8713128623

-65.5555555556-69.4793643893-64.9049308356

-66.6666666667-70.1973751806-65.9568806799

-67.7777777778-70.8940031714-67.0301742589

-68.8888888889-71.5690361621-68.1273489328

-70-72.222268531-69.2503650241

-71.1111111111-72.8535012974-70.4005118243

-72.2222222222-73.4625421815-71.5783245596

-73.3333333333-74.0492056635-72.7835145311

-74.4444444444-74.6133130403-74.014914392

-75.5555555556-75.154692479-75.2704402184

-76.6666666667-75.6731790704-76.5470716824

-77.7777777778-76.1686148784-77.8408512546

-78.8888888889-76.6408489884-79.1469029515

-80-77.0897375534-80.4594707197

-81.1111111111-77.5151438376-81.7719761111

-82.2222222222-77.916938258-83.0770944729

-83.3333333333-78.2949984242-84.3668484571

-84.4444444444-78.6492091755-85.6327172545

-85.5555555556-78.9794626158-86.8657595934

-86.6666666667-79.2856581466-88.0567482189

-87.7777777778-79.5677024979-89.1963132863

-88.8888888889-79.8255097563-90.2750918824

-90-80.0590013911-91.2838807245

-91.1111111111-80.2681062785-92.2137889849

-92.2222222222-80.4527607232-93.0563881605

-93.3333333333-80.6129084775-93.803855938

-94.4444444444-80.7485007591-94.4491111122

-95.5555555556-80.8594962651-94.9859367792

-96.6666666667-80.9458611852-95.4090892593

-97.7777777778-81.0075692119-95.7143904895

-98.8888888889-81.0446015482-95.898801964

-100-81.0569469139-95.96047868

-100-81.0569469139-95.96047868

-98.8888888889-81.0446015482-95.898801964

-97.7777777778-81.0075692119-95.7143904895

-96.6666666667-80.9458611852-95.4090892593

-95.5555555556-80.8594962651-94.9859367792

-94.4444444444-80.7485007591-94.4491111122

-93.3333333333-80.6129084775-93.803855938

-92.2222222222-80.4527607232-93.0563881605

-91.1111111111-80.2681062785-92.2137889849

-90-80.0590013911-91.2838807245

-88.8888888889-79.8255097563-90.2750918824

-87.7777777778-79.5677024979-89.1963132863

-86.6666666667-79.2856581466-88.0567482189

-85.5555555556-78.9794626158-86.8657595934

-84.4444444444-78.6492091755-85.6327172545

-83.3333333333-78.2949984242-84.3668484571

-82.2222222222-77.916938258-83.0770944729

-81.1111111111-77.5151438376-81.7719761111

-80-77.0897375534-80.4594707197

-78.8888888889-76.6408489884-79.1469029515

-77.7777777778-76.1686148784-77.8408512546

-76.6666666667-75.6731790704-76.5470716824

-75.5555555556-75.154692479-75.2704402184

-74.4444444444-74.6133130403-74.014914392

-73.3333333333-74.0492056635-72.7835145311

-72.2222222222-73.4625421815-71.5783245596

-71.1111111111-72.8535012974-70.4005118243

-70-72.222268531-69.2503650241

-68.8888888889-71.5690361621-68.1273489328

-67.7777777778-70.8940031714-67.0301742589

-66.6666666667-70.1973751806-65.9568806799

-65.5555555556-69.4793643893-64.9049308356

-64.4444444444-68.7401895106-63.8713128623

-63.3333333333-67.9800757042-62.852648909

-62.2222222222-67.1992545082-61.8453069962

-61.1111111111-66.3979637683-60.8455135623

-60-65.5764475655-59.849464082

-58.8888888889-64.7349561414-58.8534292538

-57.7777777778-63.8737458227-57.8538544085

-56.6666666667-62.9930789421-56.8474500092

-55.5555555556-62.0932237596-55.8312713753

-54.4444444444-61.1744543795-54.8027860596

-53.3333333333-60.237050668-53.7599276475

-52.2222222222-59.2812981674-52.701135097

-51.1111111111-58.3074880091-51.6253771165

-50-57.3159168251-50.532161449

-48.8888888889-56.3068866576-49.4215293084

-47.7777777778-55.2807048669-48.2940355723

-46.6666666667-54.2376840381-47.1507156752

-45.5555555556-53.1781418852-45.9930404571

-44.4444444444-52.1024011553-44.8228604937

-43.3333333333-51.0107895291-43.6423416664

-42.2222222222-49.9036395224-42.4538939125

-41.1111111111-48.7812883836-41.2600952255

-40-47.6440779918-40.0636130533

-38.8888888889-46.4923547524-38.8671252628

-37.7777777778-45.3264694912-37.6732428022

-36.6666666667-44.1467773484-36.4844361082

-35.5555555556-42.9536376696-35.3029671637

-34.4444444444-41.7474138966-34.1308289251

-33.3333333333-40.5284734569-32.9696936152

-32.2222222222-39.2971876516-31.8208711107

-31.1111111111-38.0539315423-30.6852783684

-30-36.7990838368-29.5634205196

-28.8888888889-35.5330267739-28.4553839443

-27.7777777778-34.2561460067-27.360841307

-26.6666666667-32.9688304857-26.279068219

-25.5555555556-31.6714723396-25.2089708805

-24.4444444444-30.3644667561-24.1491237731

-23.3333333333-29.0482118619-23.0978162097

-22.2222222222-27.723108601-22.0531063286

-21.1111111111-26.3895606127-21.0128809326

-20-25.0479741085-19.9749194333

-18.8888888889-23.6987577488-18.9369600714

-17.7777777778-22.3423225179-17.8967665379

-16.6666666667-20.9790815992-16.8521931341

-15.5555555556-19.6094502489-15.801246661

-14.4444444444-18.2338456702-14.7421433388

-13.3333333333-16.8526868854-13.6733592066

-12.2222222222-15.4663946089-12.5936726421

-11.1111111111-14.0753911188-11.502197872

-10-12.6801001283-10.3984085939

-8.8888888889-11.2809466564-9.2821511129

-7.7777777778-9.8783568988-8.1536466853

-6.6666666667-8.4727580981-7.0134830635

-5.5555555556-7.0645784131-5.8625955319

-4.4444444444-5.6542467892-4.7022380141

-3.3333333333-4.2421928269-3.5339451033

-2.2222222222-2.8288466515-2.3594861159

-1.1111111111-1.4146387821-1.1808124857

0-0-0

0-0-0

u(t)

1H

1 - 9

t

u(t)

Trokutni impuls i 1. + 3. + 5. + 7. + 9. harmonik

List1

Trokutni impulsiAAmpl.

63.661977236881.0569469139

tu(t)X'1H3H5H7H9H11H13H15H1 - 31 - 51 - 71 - 91 - 111 - 131 - 15

000000000000000000

000000000000000000

11.11111111110.01745329251.4146387821-0.47135475850.2825831365-0.20159912040.156544446-0.1278214430.1078925779-0.09324036270.94328402361.22586716011.02426803971.18081248571.05299104271.16088362061.067643258

22.22222222220.0349065852.8288466515-0.94141756650.5630156448-0.40019286220.3092342483-0.25094600620.2102545963-0.18012654871.88742908512.45044472982.05025186762.35948611592.10854010962.31879470592.1386681572

33.33333333330.05235987764.2421928269-1.40890001430.839163264-0.59282065020.454309677-0.36484939960.3018389913-0.25473740812.83329281263.67245607663.07963542633.53394510333.16909570383.47093469513.216197287

44.44444444440.06981317015.6542467892-1.8725207651.108924344-0.77661084780.5881984937-0.46534617070.3779511587-0.31198833413.78172602414.89065036824.11403952044.70223801414.23689184344.61484300214.302854668

55.55555555560.08726646267.0645784131-2.33100906661.3702458403-0.94882356640.7076039114-0.54874350220.4346895987-0.34797777084.73356934666.10381518685.15499162055.86259553195.31385202975.74854162845.4005638576

66.66666666670.10471975518.4727580981-2.78310823431.6211389383-1.1068915110.8095857724-0.61197690630.4691459062-0.36025309745.68964986387.31078880216.20389729117.01348306356.40150615726.87065206346.510398966

77.77777777780.12217304769.8783568988-3.22757909581.8596941901-1.24845825260.8916329448-0.65272283150.4795538553-0.34797777086.6507778038.51047199317.26201374058.15364668537.50092385387.98047770917.6324999383

88.88888888890.139626340211.2809466564-3.66320338732.0840960462-1.37141335730.951725155-0.66948404310.4653799359-0.31198833417.61774326919.70183931538.3304259589.28215111298.61266706999.07804700588.7660586717

9100.157079632712.6801001283-4.0887870932.292636673-1.47392384760.9883827332-0.66164463960.4273507014-0.25473740818.591313035310.88394970839.410025860710.39840859399.736763954310.16411465579.9093772476

1011.11111111110.174532925214.0753911188-4.50316371742.4837289504-1.55446152811.0007030483-0.62949268490.3674155252-0.18012654879.572227401412.055956351810.501494823711.50219787210.87270518711.240120712211.0599941635

1112.22222222220.191986217715.4663946089-4.90519748322.6559185507-1.61182576770.9883827332-0.57420962310.2886466768-0.093240362710.561197125813.217115676511.605289908812.593672642112.01946301912.308109695812.2148693332

1213.33333333330.209439510216.8526868854-5.29378644352.8078950072-1.64516139750.951725155-0.4978268650.1950818372-011.558900441914.366795449112.721634051613.673359206613.175532341513.370614178713.3706141787

1314.44444444440.226892802818.2338456702-5.66786550322.9385016873-1.65397146020.8916329448-0.40315114370.09151712790.093240362712.565980166915.504481854213.85051039414.742143338814.338992195214.43050932314.5237496857

1415.55555555560.244346095319.6094502489-6.02640933763.0467445951-1.63812461790.8095857724-0.2936613783-0.01673873760.180126548713.583040911416.629785506514.991660888615.80124666115.507585282715.490846545115.6709730938

1516.66666666670.261799387820.9790815992-6.36843520283.131799937-1.59785711070.7076039114-0.1733808397-0.12413657750.254737408114.610646396417.742446333416.144589222716.852193134116.678812294416.554675716916.809413125

1617.77777777780.279252680322.3423225179-6.69300562983.1930203903-1.53376923430.5881984937-0.0467293123-0.22517119260.311988334115.649316888218.842337278417.308568044117.896766537917.850037225617.62486603317.9368543671

1718.88888888890.296705972823.6987577488-6.99923099363.2299400304-1.44681639130.4543096770.0816393132-0.31466356150.347977770816.699526755319.929466785618.482650394418.936960071419.018599384618.703935823119.0519135939

18200.314159265425.0479741085-7.28627195173.2422778766-1.33829484830.30923424830.2070080505-0.3880263170.360253097417.761702156821.003980033419.665685185119.974919433320.181927483819.793901166920.1541542643

1921.11111111110.331612557926.3895606127-7.55334174493.2299400304-1.20982241160.1565444460.3247701459-0.44149889370.347977770818.836218867822.066158898120.856336486621.012880932621.337651078520.896152184721.2441299555

2022.22222222220.349065850427.723108601-7.79970835343.1930203903-1.063314309300.4305983567-0.47234029440.311988334119.923400247623.116420637922.053106328622.053106328622.483704685222.011364390822.323352725

2123.33333333330.366519142929.0482118619-8.02469650343.131799937-0.9009546398-0.1565444460.5206039582-0.47896959280.254737408121.023515358524.155315295423.254360655723.097816209723.618420167923.13945057523.3941879832

2224.44444444440.383972435430.3644667561-8.22768951823.0467445951-0.7251638117-0.30923424830.5914796377-0.46104697190.180126548722.136777237925.183521833124.458358021424.149123773124.740603410824.279556438924.4596829876

2325.55555555560.40142572831.6714723396-8.40813100782.9385016873-0.5385624615-0.4543096770.6406210235-0.4194911430.093240362723.263341331726.20184301925.663280557525.208970880525.84959190425.430100760925.5233411236

2426.66666666670.418879020532.9688304857-8.56552639482.8078950072-0.3439323854-0.58819849370.6662223841-0.3564322525024.403304090927.211199098126.867266712726.27906821926.945290603126.588858350626.5888583506

2527.77777777780.43633231334.2561460067-8.69944426942.6559185507-0.1441750697-0.70760391140.6673429815-0.2751026908-0.093240362725.556701737428.212620288128.068445218527.36084130728.028184288527.753081597727.659841235

2628.88888888890.453785605535.5330267739-8.80951757182.48372895040.0577315643-0.80958577240.6439416385-0.1796714009-0.180126548726.72350920229.207238152429.264969716728.455383944329.099325582828.919654181928.7395276332

27300.47123889836.7990838368-8.8954445992.2926366730.2587775536-0.89163294480.5968782523-0.0750301783-0.254737408127.903639237830.196275910830.455053464429.563420519630.160298771930.085268593629.8305311855

2831.11111111110.488692190638.0539315423-8.95698983082.08409604620.4559657657-0.9517251550.52788219690.0334570816-0.311988334129.096941711531.181037757731.637003523430.685278368431.213160565331.246617646830.9346293127

2932.22222222220.506145483139.2971876516-8.99398457611.85969419010.6463565784-0.98838273320.43948877590.1402293358-0.347977770830.303203075532.162897265632.80925384431.820871110732.260359886632.400589222432.0526114517

3033.33333333330.523598775640.5284734569-9.00632743491.62113893830.8271117032-1.00070304830.33494606160.2398134524-0.360253097431.522146022133.143284960333.970396663532.969693615233.304639676833.544453129333.1842000319

3134.44444444440.541052068141.7474138966-8.99398457611.37024584030.9955364976-0.98838273320.21809554230.3271047625-0.347977770832.753429320534.123675160735.119211658434.130828925134.348924467434.676029229934.3280514591

3235.55555555560.558505360642.9536376696-8.95698983081.1089243441.1491201359-0.9517251550.09323096410.3976287249-0.311988334133.996647838735.105572182836.254692318735.302967163735.396198127835.793826852735.4818385185

3336.66666666670.575958653244.1467773484-8.8954445990.8391632641.2855730396-0.8916329448-0.03505944490.4477702904-0.254737408135.251332749436.090496013437.37606905336.484436108236.449376663436.897146953736.6424095456

3437.77777777780.593411945745.3264694912-8.80951757180.56301564481.4028610104-0.8095857724-0.16206157230.4749592087-0.180126548736.516951919437.079967564138.482828574637.673242802237.511181229937.986140438637.8060138899

3538.88888888890.610865238246.4923547524-8.69944426940.28258313651.4992355547-0.7076039114-0.28310864470.4778017796-0.093240362737.79291048338.075493619539.574729174238.867125262838.584016618139.061818397838.9685780351

36400.628318530747.6440779918-8.565526394801.5732599501-0.5881984937-0.39375271070.4561522932-039.07855159739.07855159740.651811547140.063613053339.669860342740.126012635940.1260126359

3741.11111111110.645771823248.7812883836-8.4081310078-0.28258313651.6238306632-0.454309677-0.4899280840.41112049930.093240362740.373157375840.090574239341.714404902541.260095225540.770167141541.181287640841.2745280035

3842.22222222220.663225115849.9036395224-8.2276895182-0.56301564481.6501938013-0.3092342483-0.56810073980.34501472190.180126548741.675950004241.112934359542.763128160842.453893912541.885793172842.230807894742.4109344434

3943.33333333330.680678408351.0107895291-8.0246965034-0.8391632641.6519563507-0.156544446-0.62539817410.26122353460.254737408142.986093025742.146929761743.798886112443.642341666443.016943492343.278167026943.532904435

4044.44444444440.698131700852.1024011553-7.7997083534-1.1089243441.6290920358-0-0.65971495670.16404206270.311988334144.302692801943.193768457844.822860493744.822860493744.16314553744.327187599744.6391759339

4145.55555555560.715584993353.1781418852-7.5533417449-1.37024584031.5819417110.156544446-0.66979009540.0584518160.347977770845.624800140344.254554300145.83649601145.993040457145.323250361645.381702177645.7296799484

4246.66666666670.733038285854.2376840381-7.2862719517-1.62113893831.51120827880.3092342483-0.6552533731-0.05013466330.360253097446.951412086445.330273148146.841481426947.150715675246.495462302146.445327638846.8055807362

4347.77777777780.750491578455.2807048669-6.9992309936-1.85969419011.4179462120.454309677-0.6166389507-0.15615124620.347977770848.281473873446.421779683347.839725895348.294035572347.677396621547.521245375347.8692231461

4448.88888888890.767944870956.3068866576-6.6930056298-2.08409604621.30354583310.5881984937-0.5553657397-0.25416353650.311988334149.613881027847.529784981648.833330814749.421529308448.866163568748.612000032248.9239883663

45500.785398163457.3159168251-6.3684352028-2.2926366731.16971258830.7076039114-0.473685263-0.33914743680.254737408150.947481622348.654844949349.824557537650.53216144950.05847618649.719328749149.9740661572

4651.11111111110.802851455958.3074880091-6.0264093376-2.48372895041.01844162290.8095857724-0.3745989214-0.40674668350.180126548752.281078671549.797349721150.81579134451.625377116551.25077819550.844031511551.0241580602

4752.22222222220.820304748459.2812981674-5.6678655032-2.65591855070.85198803870.8916329448-0.2617477053-0.45349614790.093240362753.613432664250.957514113451.809502152152.70113509752.439387391751.985891243852.0791316065

4853.33333333330.83775804160.237050668-5.2937864435-2.80789500720.67283327520.951725155-0.139278404-0.4769994584054.943264224552.135369217352.808202492553.759927647553.620649243553.14364978553.143649785

4954.44444444440.855211333561.1744543795-4.9051974832-2.93850168730.48364811730.9883827332-0.0116912296-0.4760518386-0.093240362756.269256896353.330755209153.814403326454.802786059654.7910948354.315042991454.2218026288

5055.55555555560.87266462662.0932237596-4.5031637174-3.04674459510.287252881.00070304830.1163255464-0.4507018632-0.180126548757.590060042154.54331544754.83056832755.831271375355.947596921755.496895058555.3167685098

5156.66666666670.890117918562.9930789421-4.088787093-3.1317999370.08657536380.98838273320.2400678666-0.4022489687-0.254737408158.904291849255.772491912255.85906727656.847450009257.087517875956.685268907256.4305314991

5257.77777777780.90757121163.8737458227-3.6632033873-3.1930203903-0.11539279160.9517251550.3549887411-0.3331768441-0.311988334160.210542435457.017522045156.902129253557.853854408558.208843149557.875666305457.5636779713

5358.88888888890.925024503664.7349561414-3.2275790958-3.2299400304-0.31564070630.89163294480.4568653295-0.2470261177-0.347977770861.507377045758.277437015357.96179630958.853429253859.310294583359.063268465658.7152906948

54600.942477796165.5764475655-2.7831082343-3.2422778766-0.51118314510.80958577240.5419541121-0.1482128645-0.360253097462.793339331259.551061454659.039878309659.84946408260.391418194160.243205329559.8829522321

5561.11111111110.959931088666.3979637683-2.3310090666-3.2299400304-0.69910502050.70760391140.6071284478-0.0418022391-0.347977770864.066954701760.837014671460.137909650860.845513562361.4526420161.410839770961.0628620001

5662.22222222220.977384381167.1992545082-1.872520765-3.1930203903-0.87660485040.58819849370.64999346430.0667511636-0.311988334165.326733743262.133713352961.257108502561.845306996262.495300460562.562051624162.25006329

5763.33333333330.994837673667.9800757042-1.4089000143-3.131799937-1.0410365210.4543096770.66897405940.1718829104-0.254737408166.571175689963.43937575362.39833923262.85264890963.521622968363.693505878863.4387684706

5864.44444444441.012290966268.7401895106-0.9414175665-3.0467445951-1.18994873490.30923424830.66337277920.2682039615-0.180126548767.798771944164.75202734963.562078614163.871312862364.534685641564.80288960364.6227630543

5965.55555555561.029744258769.4793643893-0.4713547585-2.9385016873-1.32112155390.1565444460.63339544620.3507769122-0.093240362769.008009630866.069507943564.748386389664.904930835665.538326281865.889103194165.7958628314

6066.66666666671.047197551270.1973751806-0-2.8078950072-1.432599493500.58014359650.4153690839-070.197375180667.389480173465.956880679965.956880679966.537024276466.952393360466.9523933604

6167.77777777781.064650843770.89400317140.4713547585-2.6559185507-1.5227206743-0.1565444460.50557400310.45866949020.093240362771.3653579368.709439379267.186718704967.030174258967.53574826267.994417752268.0876581149

6268.88888888891.082104136271.56903616210.9414175665-2.4837289504-1.5901415971-0.30923424830.41242677290.47845855780.180126548772.510453728570.026724778268.436583181168.127348932868.539775705769.018234263569.1983608122

63701.099557428872.2222685311.4089000143-2.292636673-1.6338571712-0.4543096770.30412465980.47372190170.254737408173.631168545371.338531872369.704674701169.250365024169.554489683970.028211585670.2829489937

6471.11111111111.117010721372.85350129741.872520765-2.0840960462-1.6532156982-0.58819849370.18464729350.44470232240.311988334174.726022062472.641926016270.98871031870.400511824370.585159117871.029861440271.3418497743

6572.22222222221.134464013873.46254218152.3310090666-1.8596941901-1.6479285869-0.70760391140.05838494560.39288735960.347977770875.79355124873.933857057972.28592847171.578324559671.636709505272.029596864772.3775746355

6673.33333333331.151917306374.04920566352.7831082343-1.6211389383-1.6180746561-0.8095857724-0.07002279420.32093304160.360253097476.832313897875.211174959573.593100303572.783514531172.713491736973.034424778573.3946778759

6774.44444444441.169370598874.61331304033.2275790958-1.3702458403-1.5640989589-0.8916329448-0.19585750210.23252773760.347977770877.84089213676.470646295874.906547336874.01491439273.819056889974.051584627574.3995623983

6875.55555555561.186823891475.1546924793.6632033873-1.108924344-1.4868061489-0.951725155-0.3144953020.13220309180.311988334178.817895866377.708971522376.222165373475.270440218474.955944916475.088148008275.4001363423

6976.66666666671.204277183975.67317907044.088787093-0.839163264-1.3873484838-0.9883827332-0.4215767730.02510173270.254737408179.761966163478.922802899477.535454415776.547071682476.125494909576.150596642276.4053340503

7077.77777777781.221730476476.16861487844.5031637174-0.5630156448-1.2672086482-1.0007030483-0.5131671385-0.0832863380.180126548780.671778595880.108762951178.841554302977.840851254677.327684116177.244397778177.4245243268

7178.88888888891.239183768976.64084898844.9051974832-0.2825831365-1.1281776503-0.9883827332-0.5859008527-0.18740516180.093240362781.546046471681.263463335180.135285684779.146902951578.561002098878.373596937178.4668372997

72801.256637061477.08973755345.2937864435-0-0.9723281223-0.951725155-0.637105269-0.2819176213082.38352399782.38352399781.411195874780.459470719779.822365450779.540447829479.5404478294

7381.11111111111.27409035477.51514383765.66786550320.2825831365-0.8019834215-0.8916329448-0.664898849-0.36197902-0.093240362783.183009340883.465592477382.663609055981.771976111181.107077262180.745098242180.6518578794

7482.22222222221.291543646577.9169382586.02640933760.5630156448-0.619682995-0.8095857724-0.6682602997-0.4234854211-0.180126548783.943347595684.506363240383.886680245383.077094472982.408834173281.985348752181.8052222034

7583.33333333331.30899693978.29499842426.36843520280.839163264-0.4281445224-0.7076039114-0.6470661027-0.4632840143-0.254737408184.66343362785.50259689185.074452368684.366848457183.719782354583.256498340183.001760932

7684.44444444441.326450231578.64920917556.69300562981.108924344-0.2302234012-0.5881984937-0.602095052-0.4793347291-0.311988334185.342214805386.451139149386.220915748285.632717254585.030622202484.551287473384.2392991392

7785.55555555561.34390352478.97946261586.99923099361.3702458403-0.0288701792-0.454309677-0.5349996375-0.4708148077-0.347977770885.978693609387.348939449687.320069270486.865759593486.330759955885.859945148285.5119673774

7886.66666666671.361356816679.28565814667.28627195171.62113893830.1729134306-0.3092342483-0.4482453226-0.4381609803-0.360253097486.571930098388.193069036688.365982467288.056748218987.608502896387.17034191686.8100888186

7987.77777777781.378810109179.56770249797.55334174491.85969419010.3721192994-0.156544446-0.3450199496-0.3830470778-0.347977770887.121044242888.980738432989.352857732389.196313286388.851293336788.46824625988.1202684882

8088.88888888891.396263401679.82550975637.79970835342.08409604620.5657777266-0-0.2291166-0.3082982317-0.311988334187.625218109789.709314155990.275091882490.275091882490.045975282489.737677050789.4256887166

81901.413716694180.05900139118.02469650342.2926366730.7510017110.156544446-0.1047942159-0.2177460582-0.254737408188.083697894590.376334567591.127336278591.283880724591.179086508690.961340450490.7066030423

8291.11111111111.431169986680.26810627858.22768951822.48372895040.92502998960.30923424830.0233788979-0.11603225-0.180126548788.495795796790.97952474791.904554736792.213788984992.237167882992.121135632991.9410090842

8392.22222222221.448623279280.45276072328.40813100782.65591855071.08526820170.4543096770.1506929394-0.0083706437-0.093240362788.86089173191.516810281792.602078483593.056388160593.207081099993.198710456293.1054700935

8493.33333333331.466076571780.61290847758.56552639482.80789500721.22932756470.58819849370.27246967340.0997200407-089.178434872491.986329879693.215657444393.80385593894.076325611494.176045652294.1760456522

8594.44444444441.483529864280.74850075918.69944426942.93850168731.35506048510.70760391140.38423433680.20269908880.093240362789.447945028592.386446715793.741507200894.449111112294.83334544995.036044537895.1292849005

8695.55555555561.500983156780.85949626518.80951757183.04674459511.46059257470.80958577240.48188006620.29528780780.180126548789.669013836992.715758432194.176351006894.985936779295.467816845495.763104653295.9432312019

8796.66666666671.518436449280.94586118528.8954445993.1317999371.54435059330.89163294480.56181880750.37274011210.254737408189.841305784292.973105721294.517456314595.409089259395.970908066896.343648178996.598385587

8897.77777777781.535889741881.00756921198.95698983083.19302039031.60508590150.9517251550.6211131610.43108580650.311988334189.964559042793.15757943394.762665334595.714390489596.335503650596.76658945797.0785777911

8998.88888888891.553343034381.04460154828.99398457613.22994003041.6418930760.98838273320.65758431820.46733409830.347977770890.038586124493.268526154794.910419230795.89880196496.556386282197.023720380497.3716981512

901001.570796326881.05694691399.00632743493.24227787661.65422340641.00070304830.66989212330.47962690480.360253097490.063274348793.305552225394.959775631795.9604786896.630370803397.109997708197.4702508055

901001.570796326881.05694691399.00632743493.24227787661.65422340641.00070304830.66989212330.47962690480.360253097490.063274348793.305552225394.959775631795.9604786896.630370803397.109997708197.4702508055

9198.88888888891.588249619381.04460154828.99398457613.22994003041.6418930760.98838273320.65758431820.46733409830.347977770890.038586124493.268526154794.910419230795.89880196496.556386282197.023720380497.3716981512

9297.77777777781.605702911881.00756921198.95698983083.19302039031.60508590150.9517251550.6211131610.43108580650.311988334189.964559042793.15757943394.762665334595.714390489596.335503650596.76658945797.0785777911

9396.66666666671.623156204480.94586118528.8954445993.1317999371.54435059330.89163294480.56181880750.37274011210.254737408189.841305784292.973105721294.517456314595.409089259395.970908066896.343648178996.598385587

9495.55555555561.640609496980.85949626518.80951757183.04674459511.46059257470.80958577240.48188006620.29528780780.180126548789.669013836992.715758432194.176351006894.985936779295.467816845495.763104653295.9432312019

9594.44444444441.658062789480.74850075918.69944426942.93850168731.35506048510.70760391140.38423433680.20269908880.093240362789.447945028592.386446715793.741507200894.449111112294.83334544995.036044537895.1292849005

9693.33333333331.675516081980.61290847758.56552639482.80789500721.22932756470.58819849370.27246967340.0997200407089.178434872491.986329879693.215657444393.80385593894.076325611494.176045652294.1760456522

9792.22222222221.692969374480.45276072328.40813100782.65591855071.08526820170.4543096770.1506929394-0.0083706437-0.093240362788.86089173191.516810281792.602078483593.056388160593.207081099993.198710456293.1054700935

9891.11111111111.71042266780.26810627858.22768951822.48372895040.92502998960.30923424830.0233788979-0.11603225-0.180126548788.495795796790.97952474791.904554736792.213788984992.237167882992.121135632991.9410090842

99901.727875959580.05900139118.02469650342.2926366730.7510017110.156544446-0.1047942159-0.2177460582-0.254737408188.083697894590.376334567591.127336278591.283880724591.179086508690.961340450490.7066030423

10088.88888888891.74532925279.82550975637.79970835342.08409604620.56577772660-0.2291166-0.3082982317-0.311988334187.625218109789.709314155990.275091882490.275091882490.045975282489.737677050789.4256887166

10187.77777777781.762782544579.56770249797.55334174491.85969419010.3721192994-0.156544446-0.3450199496-0.3830470778-0.347977770887.121044242888.980738432989.352857732389.196313286388.851293336788.46824625988.1202684882

10286.66666666671.78023583779.28565814667.28627195171.62113893830.1729134306-0.3092342483-0.4482453226-0.4381609803-0.360253097486.571930098388.193069036688.365982467288.056748218987.608502896387.17034191686.8100888186

10385.55555555561.797689129678.97946261586.99923099361.3702458403-0.0288701792-0.454309677-0.5349996375-0.4708148077-0.347977770885.978693609387.348939449687.320069270486.865759593486.330759955885.859945148285.5119673774

10484.44444444441.815142422178.64920917556.69300562981.108924344-0.2302234012-0.5881984937-0.602095052-0.4793347291-0.311988334185.342214805386.451139149386.220915748285.632717254585.030622202484.551287473384.2392991392

10583.33333333331.832595714678.29499842426.36843520280.839163264-0.4281445224-0.7076039114-0.6470661027-0.4632840143-0.254737408184.66343362785.50259689185.074452368684.366848457183.719782354583.256498340183.001760932

10682.22222222221.850049007177.9169382586.02640933760.5630156448-0.619682995-0.8095857724-0.6682602997-0.4234854211-0.180126548783.943347595684.506363240383.886680245383.077094472982.408834173281.985348752181.8052222034

10781.11111111111.867502299677.51514383765.66786550320.2825831365-0.8019834215-0.8916329448-0.664898849-0.36197902-0.093240362783.183009340883.465592477382.663609055981.771976111181.107077262180.745098242180.6518578794

108801.884955592277.08973755345.29378644350-0.9723281223-0.951725155-0.637105269-0.2819176213-082.38352399782.38352399781.411195874780.459470719779.822365450779.540447829479.5404478294

10978.88888888891.902408884776.64084898844.9051974832-0.2825831365-1.1281776503-0.9883827332-0.5859008527-0.18740516180.093240362781.546046471681.263463335180.135285684779.146902951578.561002098878.373596937178.4668372997

11077.77777777781.919862177276.16861487844.5031637174-0.5630156448-1.2672086482-1.0007030483-0.5131671385-0.0832863380.180126548780.671778595880.108762951178.841554302977.840851254677.327684116177.244397778177.4245243268

11176.66666666671.937315469775.67317907044.088787093-0.839163264-1.3873484838-0.9883827332-0.4215767730.02510173270.254737408179.761966163478.922802899477.535454415776.547071682476.125494909576.150596642276.4053340503

11275.55555555561.954768762275.1546924793.6632033873-1.108924344-1.4868061489-0.951725155-0.3144953020.13220309180.311988334178.817895866377.708971522376.222165373475.270440218474.955944916475.088148008275.4001363423

11374.44444444441.972222054874.61331304033.2275790958-1.3702458403-1.5640989589-0.8916329448-0.19585750210.23252773760.347977770877.84089213676.470646295874.906547336874.01491439273.819056889974.051584627574.3995623983

11473.33333333331.989675347374.04920566352.7831082343-1.6211389383-1.6180746561-0.8095857724-0.07002279420.32093304160.360253097476.832313897875.211174959573.593100303572.783514531172.713491736973.034424778573.3946778759

11572.22222222222.007128639873.46254218152.3310090666-1.8596941901-1.6479285869-0.70760391140.05838494560.39288735960.347977770875.79355124873.933857057972.28592847171.578324559671.636709505272.029596864772.3775746355

11671.11111111112.024581932372.85350129741.872520765-2.0840960462-1.6532156982-0.58819849370.18464729350.44470232240.311988334174.726022062472.641926016270.98871031870.400511824370.585159117871.029861440271.3418497743

117702.042035224872.2222685311.4089000143-2.292636673-1.6338571712-0.4543096770.30412465980.47372190170.254737408173.631168545371.338531872369.704674701169.250365024169.554489683970.028211585670.2829489937

11868.88888888892.059488517471.56903616210.9414175665-2.4837289504-1.5901415971-0.30923424830.41242677290.47845855780.180126548772.510453728570.026724778268.436583181168.127348932868.539775705769.018234263569.1983608122

11967.77777777782.076941809970.89400317140.4713547585-2.6559185507-1.5227206743-0.1565444460.50557400310.45866949020.093240362771.3653579368.709439379267.186718704967.030174258967.53574826267.994417752268.0876581149

12066.66666666672.094395102470.19737518060-2.8078950072-1.4325994935-00.58014359650.4153690839070.197375180667.389480173465.956880679965.956880679966.537024276466.952393360466.9523933604

12165.55555555562.111848394969.4793643893-0.4713547585-2.9385016873-1.32112155390.1565444460.63339544620.3507769122-0.093240362769.008009630866.069507943564.748386389664.904930835665.538326281865.889103194165.7958628314

12264.44444444442.129301687468.7401895106-0.9414175665-3.0467445951-1.18994873490.30923424830.66337277920.2682039615-0.180126548767.798771944164.75202734963.562078614163.871312862364.534685641564.80288960364.6227630543

12363.33333333332.1467549867.9800757042-1.4089000143-3.131799937-1.0410365210.4543096770.66897405940.1718829104-0.254737408166.571175689963.43937575362.39833923262.85264890963.521622968363.693505878863.4387684706

12462.22222222222.164208272567.1992545082-1.872520765-3.1930203903-0.87660485040.58819849370.64999346430.0667511636-0.311988334165.326733743262.133713352961.257108502561.845306996262.495300460562.562051624162.25006329

12561.11111111112.18166156566.3979637683-2.3310090666-3.2299400304-0.69910502050.70760391140.6071284478-0.0418022391-0.347977770864.066954701760.837014671460.137909650860.845513562361.4526420161.410839770961.0628620001

126602.199114857565.5764475655-2.7831082343-3.2422778766-0.51118314510.80958577240.5419541121-0.1482128645-0.360253097462.793339331259.551061454659.039878309659.84946408260.391418194160.243205329559.8829522321

12758.88888888892.2165681564.7349561414-3.2275790958-3.2299400304-0.31564070630.89163294480.4568653295-0.2470261177-0.347977770861.507377045758.277437015357.96179630958.853429253859.310294583359.063268465658.7152906948

12857.77777777782.234021442663.8737458227-3.6632033873-3.1930203903-0.11539279160.9517251550.3549887411-0.3331768441-0.311988334160.210542435457.017522045156.902129253557.853854408558.208843149557.875666305457.5636779713

12956.66666666672.251474735162.9930789421-4.088787093-3.1317999370.08657536380.98838273320.2400678666-0.4022489687-0.254737408158.904291849255.772491912255.85906727656.847450009257.087517875956.685268907256.4305314991

13055.55555555562.268928027662.0932237596-4.5031637174-3.04674459510.287252881.00070304830.1163255464-0.4507018632-0.180126548757.590060042154.54331544754.83056832755.831271375355.947596921755.496895058555.3167685098

13154.44444444442.286381320161.1744543795-4.9051974832-2.93850168730.48364811730.9883827332-0.0116912296-0.4760518386-0.093240362756.269256896353.330755209153.814403326454.802786059654.7910948354.315042991454.2218026288

13253.33333333332.303834612660.237050668-5.2937864435-2.80789500720.67283327520.951725155-0.139278404-0.4769994584054.943264224552.135369217352.808202492553.759927647553.620649243553.14364978553.143649785

13352.22222222222.321287905259.2812981674-5.6678655032-2.65591855070.85198803870.8916329448-0.2617477053-0.45349614790.093240362753.613432664250.957514113451.809502152152.70113509752.439387391751.985891243852.0791316065

13451.11111111112.338741197758.3074880091-6.0264093376-2.48372895041.01844162290.8095857724-0.3745989214-0.40674668350.180126548752.281078671549.797349721150.81579134451.625377116551.25077819550.844031511551.0241580602

135502.356194490257.3159168251-6.3684352028-2.2926366731.16971258830.7076039114-0.473685263-0.33914743680.254737408150.947481622348.654844949349.824557537650.53216144950.05847618649.719328749149.9740661572

13648.88888888892.373647782756.3068866576-6.6930056298-2.08409604621.30354583310.5881984937-0.5553657397-0.25416353650.311988334149.613881027847.529784981648.833330814749.421529308448.866163568748.612000032248.9239883663

13747.77777777782.391101075255.2807048669-6.9992309936-1.85969419011.4179462120.454309677-0.6166389507-0.15615124620.347977770848.281473873446.421779683347.839725895348.294035572347.677396621547.521245375347.8692231461

13846.66666666672.408554367854.2376840381-7.2862719517-1.62113893831.51120827880.3092342483-0.6552533731-0.05013466330.360253097446.951412086445.330273148146.841481426947.150715675246.495462302146.445327638846.8055807362

13945.55555555562.426007660353.1781418852-7.5533417449-1.37024584031.5819417110.156544446-0.66979009540.0584518160.347977770845.624800140344.254554300145.83649601145.993040457145.323250361645.381702177645.7296799484

14044.44444444442.443460952852.1024011553-7.7997083534-1.1089243441.62909203580-0.65971495670.16404206270.311988334144.302692801943.193768457844.822860493744.822860493744.16314553744.327187599744.6391759339

14143.33333333332.460914245351.0107895291-8.0246965034-0.8391632641.6519563507-0.156544446-0.62539817410.26122353460.254737408142.986093025742.146929761743.798886112443.642341666443.016943492343.278167026943.532904435

14242.22222222222.478367537849.9036395224-8.2276895182-0.56301564481.6501938013-0.3092342483-0.56810073980.34501472190.180126548741.675950004241.112934359542.763128160842.453893912541.885793172842.230807894742.4109344434

14341.11111111112.495820830448.7812883836-8.4081310078-0.28258313651.6238306632-0.454309677-0.4899280840.41112049930.093240362740.373157375840.090574239341.714404902541.260095225540.770167141541.181287640841.2745280035

144402.513274122947.6440779918-8.5655263948-01.5732599501-0.5881984937-0.39375271070.4561522932039.07855159739.07855159740.651811547140.063613053339.669860342740.126012635940.1260126359

14538.88888888892.530727415446.4923547524-8.69944426940.28258313651.4992355547-0.7076039114-0.28310864470.4778017796-0.093240362737.79291048338.075493619539.574729174238.867125262838.584016618139.061818397838.9685780351

14637.77777777782.548180707945.3264694912-8.80951757180.56301564481.4028610104-0.8095857724-0.16206157230.4749592087-0.180126548736.516951919437.079967564138.482828574637.673242802237.511181229937.986140438637.8060138899

14736.66666666672.565634000444.1467773484-8.8954445990.8391632641.2855730396-0.8916329448-0.03505944490.4477702904-0.254737408135.251332749436.090496013437.37606905336.484436108236.449376663436.897146953736.6424095456

14835.55555555562.58308729342.9536376696-8.95698983081.1089243441.1491201359-0.9517251550.09323096410.3976287249-0.311988334133.996647838735.105572182836.254692318735.302967163735.396198127835.793826852735.4818385185

14934.44444444442.600540585541.7474138966-8.99398457611.37024584030.9955364976-0.98838273320.21809554230.3271047625-0.347977770832.753429320534.123675160735.119211658434.130828925134.348924467434.676029229934.3280514591

15033.33333333332.61799387840.5284734569-9.00632743491.62113893830.8271117032-1.00070304830.33494606160.2398134524-0.360253097431.522146022133.143284960333.970396663532.969693615233.304639676833.544453129333.1842000319

15132.22222222222.635447170539.2971876516-8.99398457611.85969419010.6463565784-0.98838273320.43948877590.1402293358-0.347977770830.303203075532.162897265632.80925384431.820871110732.260359886632.400589222432.0526114517

15231.11111111112.65290046338.0539315423-8.95698983082.08409604620.4559657657-0.9517251550.52788219690.0334570816-0.311988334129.096941711531.181037757731.637003523430.685278368431.213160565331.246617646830.9346293127

153302.670353755636.7990838368-8.8954445992.2926366730.2587775536-0.89163294480.5968782523-0.0750301783-0.254737408127.903639237830.196275910830.455053464429.563420519630.160298771930.085268593629.8305311855

15428.88888888892.687807048135.5330267739-8.80951757182.48372895040.0577315643-0.80958577240.6439416385-0.1796714009-0.180126548726.72350920229.207238152429.264969716728.455383944329.099325582828.919654181928.7395276332

15527.77777777782.705260340634.2561460067-8.69944426942.6559185507-0.1441750697-0.70760391140.6673429815-0.2751026908-0.093240362725.556701737428.212620288128.068445218527.36084130728.028184288527.753081597727.659841235

15626.66666666672.722713633132.9688304857-8.56552639482.8078950072-0.3439323854-0.58819849370.6662223841-0.3564322525-024.403304090927.211199098126.867266712726.27906821926.945290603126.588858350626.5888583506

15725.55555555562.740166925631.6714723396-8.40813100782.9385016873-0.5385624615-0.4543096770.6406210235-0.4194911430.093240362723.263341331726.20184301925.663280557525.208970880525.84959190425.430100760925.5233411236

15824.44444444442.757620218230.3644667561-8.22768951823.0467445951-0.7251638117-0.30923424830.5914796377-0.46104697190.180126548722.136777237925.183521833124.458358021424.149123773124.740603410824.279556438924.4596829876

15923.33333333332.775073510729.0482118619-8.02469650343.131799937-0.9009546398-0.1565444460.5206039582-0.47896959280.254737408121.023515358524.155315295423.254360655723.097816209723.618420167923.13945057523.3941879832

16022.22222222222.792526803227.723108601-7.79970835343.1930203903-1.0633143093-00.4305983567-0.47234029440.311988334119.923400247623.116420637922.053106328622.053106328622.483704685222.011364390822.323352725

16121.11111111112.809980095726.3895606127-7.55334174493.2299400304-1.20982241160.1565444460.3247701459-0.44149889370.347977770818.836218867822.066158898120.856336486621.012880932621.337651078520.896152184721.2441299555

162202.827433388225.0479741085-7.28627195173.2422778766-1.33829484830.30923424830.2070080505-0.3880263170.360253097417.761702156821.003980033419.665685185119.974919433320.181927483819.793901166920.1541542643

16318.88888888892.844886680823.6987577488-6.99923099363.2299400304-1.44681639130.4543096770.0816393132-0.31466356150.347977770816.699526755319.929466785618.482650394418.936960071419.018599384618.703935823119.0519135939

16417.77777777782.862339973322.3423225179-6.69300562983.1930203903-1.53376923430.5881984937-0.0467293123-0.22517119260.311988334115.649316888218.842337278417.308568044117.896766537917.850037225617.62486603317.9368543671

16516.66666666672.879793265820.9790815992-6.36843520283.131799937-1.59785711070.7076039114-0.1733808397-0.12413657750.254737408114.610646396417.742446333416.144589222716.852193134116.678812294416.554675716916.809413125

16615.55555555562.897246558319.6094502489-6.02640933763.0467445951-1.63812461790.8095857724-0.2936613783-0.01673873760.180126548713.583040911416.629785506514.991660888615.80124666115.507585282715.490846545115.6709730938

16714.44444444442.914699850818.2338456702-5.66786550322.9385016873-1.65397146020.8916329448-0.40315114370.09151712790.093240362712.565980166915.504481854213.85051039414.742143338814.338992195214.43050932314.5237496857

16813.33333333332.932153143416.8526868854-5.29378644352.8078950072-1.64516139750.951725155-0.4978268650.1950818372011.558900441914.366795449112.721634051613.673359206613.175532341513.370614178713.3706141787

16912.22222222222.949606435915.4663946089-4.90519748322.6559185507-1.61182576770.9883827332-0.57420962310.2886466768-0.093240362710.561197125813.217115676511.605289908812.593672642112.01946301912.308109695812.2148693332

17011.11111111112.967059728414.0753911188-4.50316371742.4837289504-1.55446152811.0007030483-0.62949268490.3674155252-0.18012654879.572227401412.055956351810.501494823711.50219787210.87270518711.240120712211.0599941635

171102.984513020912.6801001283-4.0887870932.292636673-1.47392384760.9883827332-0.66164463960.4273507014-0.25473740818.591313035310.88394970839.410025860710.39840859399.736763954310.16411465579.9093772476

1728.88888888893.001966313411.2809466564-3.66320338732.0840960462-1.37141335730.951725155-0.66948404310.4653799359-0.31198833417.61774326919.70183931538.3304259589.28215111298.61266706999.07804700588.7660586717

1737.77777777783.0194196069.8783568988-3.22757909581.8596941901-1.24845825260.8916329448-0.65272283150.4795538553-0.34797777086.6507778038.51047199317.26201374058.15364668537.50092385387.98047770917.6324999383

1746.66666666673.03687289858.4727580981-2.78310823431.6211389383-1.1068915110.8095857724-0.61197690630.4691459062-0.36025309745.68964986387.31078880216.20389729117.01348306356.40150615726.87065206346.510398966

1755.55555555563.0543261917.0645784131-2.33100906661.3702458403-0.94882356640.7076039114-0.54874350220.4346895987-0.34797777084.73356934666.10381518685.15499162055.86259553195.31385202975.74854162845.4005638576

1764.44444444443.07177948355.6542467892-1.8725207651.108924344-0.77661084780.5881984937-0.46534617070.3779511587-0.31198833413.78172602414.89065036824.11403952044.70223801414.23689184344.61484300214.302854668

1773.33333333333.0892327764.2421928269-1.40890001430.839163264-0.59282065020.454309677-0.36484939960.3018389913-0.25473740812.83329281263.67245607663.07963542633.53394510333.16909570383.47093469513.216197287

1782.22222222223.10668606852.8288466515-0.94141756650.5630156448-0.40019286220.3092342483-0.25094600620.2102545963-0.18012654871.88742908512.45044472982.05025186762.35948611592.10854010962.31879470592.1386681572

1791.11111111113.12413936111.4146387821-0.47135475850.2825831365-0.20159912040.156544446-0.1278214430.1078925779-0.09324036270.94328402361.22586716011.02426803971.18081248571.05299104271.16088362061.067643258

18003.14159265360-00-00-0-0-00000000

18003.14159265360-00-00-0-0-00000000

181-1.11111111113.1590459461-1.41463878210.4713547585-0.28258313650.2015991204-0.1565444460.127821443-0.10789257790.0932403627-0.9432840236-1.2258671601-1.0242680397-1.1808124857-1.0529910427-1.1608836206-1.067643258

182-2.22222222223.1764992386-2.82884665150.9414175665-0.56301564480.4001928622-0.30923424830.2509460062-0.21025459630.1801265487-1.8874290851-2.4504447298-2.0502518676-2.3594861159-2.1085401096-2.3187947059-2.1386681572

183-3.33333333333.1939525311-4.24219282691.4089000143-0.8391632640.5928206502-0.4543096770.3648493996-0.30183899130.2547374081-2.8332928126-3.6724560766-3.0796354263-3.5339451033-3.1690957038-3.4709346951-3.216197287

184-4.44444444443.2114058237-5.65424678921.872520765-1.1089243440.7766108478-0.58819849370.4653461707-0.37795115870.3119883341-3.7817260241-4.8906503682-4.1140395204-4.7022380141-4.2368918434-4.6148430021-4.302854668

185-5.55555555563.2288591162-7.06457841312.3310090666-1.37024584030.9488235664-0.70760391140.5487435022-0.43468959870.3479777708-4.7335693466-6.1038151868-5.1549916205-5.8625955319-5.3138520297-5.7485416284-5.4005638576

186-6.66666666673.2463124087-8.47275809812.7831082343-1.62113893831.106891511-0.80958577240.6119769063-0.46914590620.3602530974-5.6896498638-7.3107888021-6.2038972911-7.0134830635-6.4015061572-6.8706520634-6.510398966

187-7.77777777783.2637657012-9.87835689883.2275790958-1.85969419011.2484582526-0.89163294480.6527228315-0.47955385530.3479777708-6.650777803-8.5104719931-7.2620137405-8.1536466853-7.5009238538-7.9804777091-7.6324999383

188-8.88888888893.2812189937-11.28094665643.6632033873-2.08409604621.3714133573-0.9517251550.6694840431-0.46537993590.3119883341-7.6177432691-9.7018393153-8.330425958-9.2821511129-8.6126670699-9.0780470058-8.7660586717

189-103.2986722863-12.68010012834.088787093-2.2926366731.4739238476-0.98838273320.6616446396-0.42735070140.2547374081-8.5913130353-10.8839497083-9.4100258607-10.3984085939-9.7367639543-10.1641146557-9.9093772476

190-11.11111111113.3161255788-14.07539111884.5031637174-2.48372895041.5544615281-1.00070304830.6294926849-0.36741552520.1801265487-9.5722274014-12.0559563518-10.5014948237-11.502197872-10.872705187-11.2401207122-11.0599941635

191-12.22222222223.3335788713-15.46639460894.9051974832-2.65591855071.6118257677-0.98838273320.5742096231-0.28864667680.0932403627-10.5611971258-13.2171156765-11.6052899088-12.5936726421-12.019463019-12.3081096958-12.2148693332

192-13.33333333333.3510321638-16.85268688545.2937864435-2.80789500721.6451613975-0.9517251550.497826865-0.19508183720-11.5589004419-14.3667954491-12.7216340516-13.6733592066-13.1755323415-13.3706141787-13.3706141787

193-14.44444444443.3684854563-18.23384567025.6678655032-2.93850168731.6539714602-0.89163294480.4031511437-0.0915171279-0.0932403627-12.5659801669-15.5044818542-13.850510394-14.7421433388-14.3389921952-14.430509323-14.5237496857

194-15.55555555563.3859387489-19.60945024896.0264093376-3.04674459511.6381246179-0.80958577240.29366137830.0167387376-0.1801265487-13.5830409114-16.6297855065-14.9916608886-15.801246661-15.5075852827-15.4908465451-15.6709730938

195-16.66666666673.4033920414-20.97908159926.3684352028-3.1317999371.5978571107-0.70760391140.17338083970.1241365775-0.2547374081-14.6106463964-17.7424463334-16.1445892227-16.8521931341-16.6788122944-16.5546757169-16.809413125

196-17.77777777783.4208453339-22.34232251796.6930056298-3.19302039031.5337692343-0.58819849370.04672931230.2251711926-0.3119883341-15.6493168882-18.8423372784-17.3085680441-17.8967665379-17.8500372256-17.624866033-17.9368543671

197-18.88888888893.4382986264-23.69875774886.9992309936-3.22994003041.4468163913-0.454309677-0.08163931320.3146635615-0.3479777708-16.6995267553-19.9294667856-18.4826503944-18.9369600714-19.0185993846-18.7039358231-19.0519135939

198-203.4557519189-25.04797410857.2862719517-3.24227787661.3382948483-0.3092342483-0.20700805050.388026317-0.3602530974-17.7617021568-21.0039800334-19.6656851851-19.9749194333-20.1819274838-19.7939011669-20.1541542643

199-21.11111111113.4732052115-26.38956061277.5533417449-3.22994003041.2098224116-0.156544446-0.32477014590.4414988937-0.3479777708-18.8362188678-22.0661588981-20.8563364866-21.0128809326-21.3376510785-20.8961521847-21.2441299555

200-22.22222222223.490658504-27.7231086017.7997083534-3.19302039031.0633143093-0-0.43059835670.4723402944-0.3119883341-19.9234002476-23.1164206379-22.0531063286-22.0531063286-22.4837046852-22.0113643908-22.323352725

201-23.33333333333.5081117965-29.04821186198.0246965034-3.1317999370.90095463980.156544446-0.52060395820.4789695928-0.2547374081-21.0235153585-24.1553152954-23.2543606557-23.0978162097-23.6184201679-23.139450575-23.3941879832

202-24.44444444443.525565089-30.36446675618.2276895182-3.04674459510.72516381170.3092342483-0.59147963770.4610469719-0.1801265487-22.1367772379-25.1835218331-24.4583580214-24.1491237731-24.7406034108-24.2795564389-24.4596829876

203-25.55555555563.5430183815-31.67147233968.4081310078-2.93850168730.53856246150.454309677-0.64062102350.419491143-0.0932403627-23.2633413317-26.201843019-25.6632805575-25.2089708805-25.849591904-25.4301007609-25.5233411236

204-26.66666666673.5604716741-32.96883048578.5655263948-2.80789500720.34393238540.5881984937-0.66622238410.3564322525-0-24.4033040909-27.2111990981-26.8672667127-26.279068219-26.9452906031-26.5888583506-26.5888583506

205-27.77777777783.5779249666-34.25614600678.6994442694-2.65591855070.14417506970.7076039114-0.66734298150.27510269080.0932403627-25.5567017374-28.2126202881-28.0684452185-27.360841307-28.0281842885-27.7530815977-27.659841235

206-28.88888888893.5953782591-35.53302677398.8095175718-2.4837289504-0.05773156430.8095857724-0.64394163850.17967140090.1801265487-26.723509202-29.2072381524-29.2649697167-28.4553839443-29.0993255828-28.9196541819-28.7395276332

207-303.6128315516-36.79908383688.895444599-2.292636673-0.25877755360.8916329448-0.59687825230.07503017830.2547374081-27.9036392378-30.1962759108-30.4550534644-29.5634205196-30.1602987719-30.0852685936-29.8305311855

208-31.11111111113.6302848441-38.05393154238.9569898308-2.0840960462-0.45596576570.951725155-0.5278821969-0.03345708160.3119883341-29.0969417115-31.1810377577-31.6370035234-30.6852783684-31.2131605653-31.2466176468-30.9346293127

209-32.22222222223.6477381367-39.29718765168.9939845761-1.8596941901-0.64635657840.9883827332-0.4394887759-0.14022933580.3479777708-30.3032030755-32.1628972656-32.809253844-31.8208711107-32.2603598866-32.4005892224-32.0526114517

210-33.33333333333.6651914292-40.52847345699.0063274349-1.6211389383-0.82711170321.0007030483-0.3349460616-0.23981345240.3602530974-31.5221460221-33.1432849603-33.9703966635-32.9696936152-33.3046396768-33.5444531293-33.1842000319

211-34.44444444443.6826447217-41.74741389668.9939845761-1.3702458403-0.99553649760.9883827332-0.2180955423-0.32710476250.3479777708-32.7534293205-34.1236751607-35.1192116584-34.1308289251-34.3489244674-34.6760292299-34.3280514591

212-35.55555555563.7000980142-42.95363766968.9569898308-1.108924344-1.14912013590.951725155-0.0932309641-0.39762872490.3119883341-33.9966478387-35.1055721828-36.2546923187-35.3029671637-35.3961981278-35.7938268527-35.4818385185

213-36.66666666673.7175513067-44.14677734848.895444599-0.839163264-1.28557303960.89163294480.0350594449-0.44777029040.2547374081-35.2513327494-36.0904960134-37.376069053-36.4844361082-36.4493766634-36.8971469537-36.6424095456

214-37.77777777783.7350045993-45.32646949128.8095175718-0.5630156448-1.40286101040.80958577240.1620615723-0.47495920870.1801265487-36.5169519194-37.0799675641-38.4828285746-37.6732428022-37.5111812299-37.9861404386-37.8060138899

215-38.88888888893.7524578918-46.49235475248.6994442694-0.28258313