Be Careful of Self and Mutual Inductance Formulae
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1 B Ca o S an ta Incc Foa Hn K t. o Ectca an Cot Ennn Unv. o conn - aon hn@ca.wc. Cha Chn-Pn Chn t. o Ectca an Cot Ennn Unv. o conn - aon chn@n.wc. BSRC Incc ha bn nt a an n toba o ntconnct o th nt naton hh- ta VSI n. a t accat an at ncc tacton o cca oc. Foa-ba ata ncc tacton tho aoach hav bn w aot o ncc tacton to t ct an cnc. Howv th a o an ncc oa n th tat an th c o ta coaon o tho oa to chc th accac an acabt. In th a w ca coa va w-nown ata an ta ncc oa n ta to chc th an o acabt an vat. Snc o oa a ona v o oth acaton wh th aat non a t nt o th VSI acaton th a not nca tab an accat o a th concn an o VSI acaton. a t can t b o wh an tho oa.. Intocton th coc nc o VSI tchnoo ow ov a ht ncc ct to b anoth na ntt an hnc bco an ot n acto o hh oanc VSI n [][7][8][9]. Ecv ncc w ca naccat a taton na nn nctv con no an /t o o th ow v nt. Fo th aon accat ncc tacton o at oc o th nt naton hh- VSI n. to th ct o obn th cnt tnn ath Pata Ent Evant Cct PEEC aoach ha bn oo b Rh [] to ata ncc nta o oo ncc a bac cct ana nt. Snc t ntocton PEEC ha bn w n th VSI to t ct an coatbt wth th na cct aton nn ch a SPICE. h th caacc tacton at-ovn tchn ata an ta ncc can b a ob n cot an ntaton oa. Fo th aon va oa-ba ncc tacton aoth hav bn oo. Howv na ach tho nt oa nc th a o an tn ncc oa n th tat. Unotnat ot o th tn ncc oa a ona v o oth acaton wh th aat non a t nt o th VSI acaton. Hnc th vat an accac o F.. Sn Concto tho oa n to b ca v o th VSI acaton. Unotnat th c o cohnv t o tho oa n th VSI. In th a w ca t va ot oa cac ncc oa ch a Rh [] Gov [] Ho [] an FatHn []. Each oa tt an coa n tnv VSI aat an. thoh o oa hav t o th oc naccat o ntab t o atv hot an on an o concto. a t can t b o wh an tho oa. h oanaton o th a a oow: h vao ncc an ta ncc oa a ntoc n cton an cton ctv. In cton tnv cotatona t an coaon a ov. h concon o th a nt n cton 5.. S Incc Etacton In th cton w b a o oa ncc oa o ata ncc tacton ncn Rh [] Gov [] Ho [] an FatHn []. on wth PEEC Rh [] ov ncc oa to actat PEEC vont. Gov [] Gotc an c tho an ov at oa o ncc cacaton. Ho [] ov act oa o cacatn ncc. In th FatHn [] aca an ncc oa wa o to-accat ncc tacton acaton.. Rh Foa Gvn a n concto wth wth thcn an nth a hown n F.. Rh v th oown ncc oa [].
2 µ n π wh [ n ] n [ n ] n n 5 5 n [ ]} n 7 n Snc th o o th abov oa bco a o v a va o an v a va o th con oa vn o nnt thn concto whch accat o th ca whn <.. h oown oa vn ba on th aton that o thcn µ n π [ ] n.. Gov Foa Gov at th ncc oa b Gotc an c tho a oow: nth n th oa a n cntt.. n o h at t o ob o ab o th vn va o o. It obv that o atv a t. h abov oa v natv va whn th nth o concto v hot n th taton th oown o von o Gov oa can b n th an.. n.5 ab. Cont o th Gotc an c o o o o FatHn Foa In th w FatHn aca [] th oown ncc oa. µ w t nh nh nh π w at t aw t w w t nh nh w t at a t w aw a t w w t nh nh w t at a t w aw a w t t w nh nh w t at aw a t w aw at a w t t w w t w t a w t a t w a a t at t a t t at at a a w aw w a w w aw aw a a aw at a aw aw at at a a aw a at a wh w aw w t at t a w t w t. Ho Foa In [] Ho ob a ncc oa wth th aton that concto cont o th t ant an ncc can b ob o ta ncc b vo -nta. ho th oa a to th oa o Ho ta ncc n cton.. h oa act whn th cnt ow thoh th concto a no.
3 F.. Got o Concto n n n.8 wh Not that th abov ncton nton n to a 8 t nc t nvo th v o th aton. h nt an cot can b ch a than oth oa.. ta Incc Etacton In th cton w b a th abov-nton oa ncc oa o ta ncc tacton. Snc FatHn ao Gov ta ncc oa w on nt Gov oa.. Rh Foa Rh con th ca whn two aa concto a oton n aa th ctona c o th concto an a hown n F.. h nth o th two concto a an ctv. It wa a that th cton o cnt n two concto a th a aon t nth. Rh ob th oown oa: { } n π µ wh F.. a Gna oton o two concto ant F.. b wo a-nth aa concto ant v v v. Gov Foa hown n F. a vn two concto wth nth an c an acn ctv Gov v th oown ta ncc oa []. nth n th oa a n cntt. ] nh nh nh nh. α α α wh α n nh Not that n th ca whn th two concto a ova thn natv. h abov oa can ao b ob o th o ca ca o a-nth aa concto b an th aw o aton o ta ncc. ho th tota ta ncc can b ob b th aton an btacton o th ta ncc o o aa a-nth concto. { } 5. wh th ta ncc o two a-nth an aa concto wth nth an c a hown n F. b. In th ca th ta ncc can b cacat b. n. Concto Concto
4 F.. a ncc o noa nth.- n nttn ont to nton that athoh th oa nt o th Rh th a athatca vant.. Ho Foa h nc btwn Gov/Rh an Ho oa th nb o ant a. h Gov/Rh oa a on on ant o ach concto wh Ho oa a t ant o ach concto. a t Ho oa o accat an ch o cocat than Gov/Rh. h Ho ta ncc oa btwn two aa concto wth wth an thcn an an nth an ctv a oow []. n n n. wh Not that th abov nton n to a t nc t nvo th v o th aton. h nt an cot can b ch a than oth oa. F.. b ncc o noa nth - F.. c ncc o noa nth -. Cotatona Rt In th cton w nt th vaaton t o th a th abov-nton ncc oa n th w an o VSI acaton. h cotatona t o ncc wth wth a to.5 µ an thcn a to. µ a hown n F.. F. a b an c how th noa nth o concto / n.- - an - ctv. F. a how that n th an whn th noa nth hot < an oa nt th a bhavo o h o. Fo a th ncc va o Gov wthot tab ncan a th nth o own an th va o Gov t oa ca too ch. Ftho Gov t oa can vn nat natv ncc va whn th noa nth hot than.5. h Rh t oa bco at. F. a ao how that Ho an FatHn oa a th ot contnt an ab whn th noa nth hot. thoh th va Rh con oawth o thcn how contnc o th acaton an t ovtat b % coan to Ho an FatHn t. It t otn that th noa nth than. Fo a whn th concto wth
5 ab. Coaon o ncc oa Foa Shot nth nth a nth / < ~ / > Ho O O X FatHn O O O Rh X O X O O Rh <. <. % a % a Gov X O O Gov X O O O <. % a con th concto nth ha to b a than con to a noa nth a than. F. b how that whn th noa nth a wthn - a oa nt on bhavo an a th va a aot ntca. F. c how that athoh Ho oa accat o hot noa nth t ha a ocaton whn th noa nth a than. Sa bhavo ao occ o th Rh oa. o n th atona bhavo o th act oa ov b Ho w ca a th nca t o ach t o Ho oa. conc that th ao oc o o o Ho oa co o th nca on nc th hh o t ch a t cat h nb wh th va o oth t a a. a t can t b o whn w nt Ho oa. Fo a w ho avo ct aton o btacton oaton btwn a an a nb. t to wtch ov to Gov o FatHn oa whn th noa nth a than. a t w conc that on FatHn oa va an nca tab o a an o VSI acaton an a oth oa hav th acab an. ab a o concon. ca nt a th o ta oa an nt th nta t n F. 5 an F.. Snc th ta ncc o two concto wth nt nth an oton can b ob o th ca ca whn two concto a aa an wth a-nth. on nt th t o th ca ca. h thcn o both concto. µ an wth.5 µ o o nt. Snc Rh oa vant to Gov an FatHn Gov oa a w w Gov/Rh to nt th va o Gov Rh an FatHn. h atonh o th c to th ta ncc btwn two concto a hown n F. 5. h atonh o th nth to th ta ncc btwn two concto a hown n F.. Coan F. 5 an F. w now that ta ncc a wa ncton n th c o two concto ath than th nth. Un ncc oa a th ta ncc oa a contnt an th va a atch. h a o wthn.%. conc that a th ta F. 5. ta Incc v. c btwn concto F.. ta Incc v. nth o concto ncc oa a ab n th VSI acaton. 5. Concon In th a w tnv tt va w ata an ta ncc oa. h nta t how that on FatHn oa nca on o a an o VSI acaton o ncc wh a ta ncc oa a hown cnt atch wth ach oth. a t w t a attnton wh an th oa. REFERENCES []. E. Rh Incc Cacaton n a Co Intat Cct Envonnt IB J. R. vo St. 97. [] F. Gov Incc Cacaton: on Foa an ab ov Nw Yo 9. [] Ct Ho an Ca ov Eact Incc Eaton o Rca Concto wth caton to o Cocat Got Jona o Rach o th Natona Ba o Sa C. Ennn an
6 Intntaton Vo. 9C No Jn 95 []. Kaon. J. an J. K. ht FSHENRY: to-ccat - Incc Etacton Poa IEEE an. cowav ho ch. vo. No St. 99. [5]. E. Rh Sv o Cot- Ectca na o Intat Cct Intconncton IB J. R. vo. -9 Nov. 979 []. C. Cana J.. Pc an. P. Vaana Fncnnt ncc an c cacaton o thnona tct n hh- ntconnct t IEEE an. Coonnt Hb anactn ch. vo a. 99 [7] K.. Sha an Z. an Rtn-t ncc: actca aoach to on-ch ncc tacton n Po. IEEE Cto Intat Cct Connc [8] Z. H. C an. P SPIE: Sa ata ncc tacton n Po. -th n toaton Connc. 7- Jn 997. [9]. van H. J an. a How to cnt cat on-ch ncc ct: Intocn a nw cct nt K ICC.
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