PhO 98 Theorecal Qeson.Elecrcy Problem (8 pons) Deren knd o oscllaon e s consder he elecrc crc n he gre, or whch mh, mh, nf, nf and kω. The swch K beng closed he crc s copled wh a sorce o alernang crren. The crren rnshed by he sorce has consan nensy whle he reqency o he crren may be vared. a. Fnd he rao o reqency m or whch he acve power n crc has he maxmm vale P m and he reqency derence o he reqences and power n he crc s hal o he maxmm powerp m. or whch he acve The swch K s now open. n he momen mmedaely aer he swch s open he nenses o he crrens n he cols and, and, (he crrens low as n he gre); a he same momen, he poenal derence on he capacor wh capacy s V : b. alclae he reqency o elecromagnec oscllaon n crc; c. Deermne he nensy o he elecrc crren n he B condcor; d. alclae he amplde o he oscllaon o he nensy o elecrc crren n he col. Neglec he mal ndcon o he cols, and he elecrc ressance o he condcors. Neglec he as ranson phenomena occrrng when he swch s closed or opened. Problem - Solon a. s s very well known n he sdy o crcs sng he ormalsm o complex nmbers, a complex ndcve reacance X, ( ) s aached o he ndcance - par o a crc sppled wh an alernave crren havng he plsaon. Smlar, a complex capacve reacance X s aached o he capacy. parallel crc wll be characerzed by hs complex admance. The admance o he crc represened n he gre s Elecrcy Problem - Solon Page rom
PhO 98 Theorecal Qeson Elecrcy Problem - Solon Page rom (.) The crc behave as has a parallel eqvalen capacy (.) and a parallel eqvalen ndcance (.) The complex admance o he crc may be wren as (.) and he complex mpedance o he crc wll be (.) The mpedance o he crc, he nverse o he admance o he crc s he modls o he complex mpedance (.6) The consan crren sorce spplyng he crc rnsh a crren havng a momenary vale sn, (.7) where s he eecve nensy (consan), o he crren and s he crren plsaon (ha can vary). The poenal derence a he acks o he crc has he momenary vale ϕ U sn (.8) where U s he eecve vale o he enson and ϕ s he phase derence beween enson and crren. The eecve vales o he crren and enson obey he relaon
PhO 98 Theorecal Qeson Elecrcy Problem - Solon Page rom U (.9) The acve power n he crc s U P (.) Becase as n he enonce, consan consan (.) he maxmal acve power s realzed or he maxmm vale o he mpedance ha s he mnmal vale o he admance. The admance (.) has as ncon o he plsaon - an he smalles vale mn (.) or he plsaon m (.) n hs case. (.) So, he mnmal acve power n he crc has he vale P m (.6) and occrs n he saon o alernave crren rnshed by he sorce a he reqency m m m π π (.7) To ensre ha he acve power s hal o he maxmm power s necessary ha P P m (.8) Tha s
PhO 98 Theorecal Qeson Elecrcy Problem - Solon Page rom ± (.9) The plsaon o he crren ensrng an acve power a hal o he maxmm power ms sasy one o he eqaons ± (.) The wo second degree eqaon may rnsh he or solons ± ± (.) Becase he plsaon s every me posve, and becase > (.) he only wo vald solons are ± ± (.) exs wo reqences ± ± π allowng o oban n he crc an acve power represenng hal o he maxmm power. π π (.) The derence o hese reqences s π (.) he bandwh o he crc he reqency nerval arond he resonance reqency havng a he ends a sgnal represenng rom he resonance sgnal. he ends o he bandwh he acve power redces a he hal o hs vale a he resonance. The asked rao s
PhO 98 Theorecal Qeson m m Becase nf mh resls ha m rad s and ( ) ( ) (.6) * 9 m (.7) The (.6) relaon s he answer a he qeson a. b. The ac ha mmedaely aer he sorce s deached s a crren n he cols, allow as o adm ha crrens dependens on me wll conne o low hrogh he cols. Fgre. The capacors wll be charged wh charges varable n me. The varaon o he charges o he capacors wll resls n crrens lowng hrogh he condcors lnkng he capacors n he crc. The momenary enson on he acks o he cols and capacors dencal or all elemens n crc s also dependen on me. e s adm ha he elecrcal poenal o he pons and D s () and he Elecrcy Problem - Solon Page rom
PhO 98 Theorecal Qeson poenal o he pons and B s zero. hrogh he ndcance passes he varable crren havng he momenary vale, he relaon beween he crren and poenals s d (.8) The crren passng hrogh he second ndcance has he expresson, d (.9) on he posve plae o he capacor havng he capacy acks o he capacor he elecrcal enson s and s socked he charge q (.) Dervng hs relaon resls dq d (.) B dq (.) q, hen a he becase he elecrcal crren appears becase o he dmnshng o he elecrcal charge on capacor plae. onseqenly d (.) nalogos, or he oher capacor, d (.) onsderng all obaned resls d d respecvely d d d d (.) (.6) Denong he momenary nensy o he crren lowng rom pon B o he pon, hen he same momenary nensy has he crren hrogh he pons and D. For he pon he Krchho rle o he crrens gves Elecrcy Problem - Solon Page 6 rom
PhO 98 Theorecal Qeson (.7) For B pon he same rle prodces (.8) onsderng (.7) and (.8) resls (.9) and dervng d d d d (.) ha s d d d ( ) Usng he symbols dened above d (.) (.) Becase he enson obeys he relaon above, ms have a harmonc dependence on me sn ( ) (.) The plsaon o he enson s (.) Takng no accon he relaons (.) and (.6) resls ha and d d d sn d sn resls ha ( sn( )) ( ) ( sn( )) ( ) ( ) ( ) (.) (.6) Elecrcy Problem - Solon Page 7 rom
PhO 98 Theorecal Qeson ( ) ( ) M N (.7) n he expresson above,, M, N and are consans ha ms be deermned sng nally conons. s remarkable ha he crrens hrogh capacors are snsodal b he crrens hrogh he cols are he sm o snsodal and consan crrens. n he rs momen V,, Becase he vales o he ndcances and capaces are,h,h nf nf he eqvalen ndcance and capacy s H H respecvely nf From (.) resls (.8) (.9) (.). (.) rad s (.) 9 The vale o he plsaon allows calclang he vale o he reqesed reqency b. Ths reqency has he vale Hz (.) * π π * c. he momenary enson on crc s lke n (.), one may wre Elecrcy Problem - Solon Page 8 rom
PhO 98 Theorecal Qeson Elecrcy Problem - Solon Page 9 rom sn sn (.) From he crrens (.7) s possble o wre N M (.) On he oher sde s possble o express (.9) as N M (.6) n deny as D B α α (.7) s valable or any vale o he argmen α only D B (.8) onsderng (.8), rom (.6) resls N M (.9) For he las eqaon resls ha he crc oscllae wh he plsaon n he relaon (.) ddng relaons (.) and consderng (.) and (.9) resls ha (.6)
PhO 98 Theorecal Qeson The nmercal vale o he amplde o he elecrcal enson resls by smmng he las relaons rom (.) and (.6) sn( ) ( ) ( ( )) ( sn( )) ( ) ( ) (( ) ) The nmercal vale o he elecrcal enson on he acks o he crc s (,) 6V nd conseqenly rom (.) resls sn( ) sn( ) 6 and 6 (.6) (.6) (.6) ( ) (.6) 6 lso g( ) arcg From (.) M N ( ) ( ) ( ) he correspondng nmercal vales are (.6) (.66) Elecrcy Problem - Solon Page rom
PhO 98 Theorecal Qeson M,, N,, The relaons (.7) becomes 6 6 6 6 ( arcg( ) ) ( arcg( ) ), 6, 6 ~, ~, (.67) (.68) The crrens hrogh he cols are he sperposon o snsodal crrens havng deren ampldes and a drec crren passng only hrogh he cols. Ths drec crren has he consan vale, (.69) as n he gre.. * * Fgre. The alernave crrens hrogh he cols has he expressons ~ ~ 6 6 ( arcg( ) ) ( arcg( ) ) (.7) Elecrcy Problem - Solon Page rom
PhO 98 Theorecal Qeson The crrens hrogh he capacors has he orms ( 6 ( arcg( ) )) 6 ( ) arcg ( 6 ( arcg( ) )) 6 ( arcg( ) ) The crren has he expresson (.7) 8 6 ( arcg( ) ), The vale o he nensy o crren s he answer rom he qeson c. The nal vale o hs crren s (.7) 8 6,, 6 (.7) d. The amplde o he crren hrogh he ndcance ~ 6 max arcg( ) s 6 max, (.7) represenng he answer a he qeson d. * * Proessor Dela DVDESU, Naonal Deparmen o Evalaon and Examnaon Mnsry o Edcaon and esearch- Bchares, omana Proessor dran S.DFNE,PhD, Facly o Physcs Unversy o Bchares, omana Elecrcy Problem - Solon Page rom