Electrical and current self-induction

Transcription

1 Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of self-inducion should be carried hose laws, which describe he reacion of such elemens of radio-echnical chains as capaciy, inducance and resisance wih he galvanic connecion o hem of he sources of curren or volage. These laws are he basis of he heory of elecrical chains. The resuls of his heory can be posponed also by he elecrodynamics of maerial media, since. such media can be represened in he form equivalen diagrams wih he use of such elemens. he moion of charges in any chain, which force hem o change heir posiion, is conneced wih he energy consumpion from he power sources. The processes of ineracion of he power sources wih such srucures are regulaed by he laws of self-inducion. Again le us refine very concep of self-inducion. By self-inducion we will undersand he reacion of maerial srucures wih he consan parameers o he connecion o hem of he sources of volage or curren. o he self-inducion le us carry also ha case, when is parameers can change wih he presence of he conneced power source or he energy accumulaed in he sysem. This self-inducion we will call parameric [].

2 Subsequenly we will use hese conceps: as curren generaor and he volage generaor. By ideal volage generaor we will undersand such source, which ensures on any load he lumped volage, inernal resisance in his generaor equal o zero. By ideal curren generaor we will undersand such source, which ensures in any load he assigned curren, inernal resisance in his generaor equally o infiniy. The ideal curren generaors and volage in naure here does no exis, since boh he curren generaors and he volage generaors have heir inernal resisance, which limis heir possibiliies. If we o one or he oher nework elemen connec he curren generaor or volage, hen opposiion o a change in is iniial sae is he response reacion of his elemen and his opposiion is always equal o he applied acion, which corresponds o hird Newon's law. if a our disposal is locaed he capaciy, and his capaciy is charged o a poenial difference, hen he charge Q, accumulaed in he capaciy, is deermined by he relaionship: Q, =. (.) The charge Q,, depending on he capaciance values of capacior and from a volage drop across i, we will call sill he flow of elecrical selfinducion. When he discussion deals wih a change in he charge, deermined by relaionship (.), hen his value can change wih he mehod of changing he poenial difference wih a consan capaciy, eiher wih a change in capaciy iself wih a consan poenial difference, or and ha and oher parameer simulaneously. If capaciance value or volage drop across i depend on ime, hen he curren srengh is deermined by he relaionship: dq d., I = = +

3 This expression deermines he law of elecrical self-inducion. Thus, curren in he circui, which conains capacior, can be obained by wo mehods, changing volage across capacior wih is consan capaciy eiher changing capaciy iself wih consan volage across capacior, or o produce change in boh parameers simulaneously. For he case, when he capaciy С is consan, we obain known expression for he curren, which flows hrough he capaciy: I =. (.) When capaciy wih he consan sress on i changes, we have: I =. (.3) This case o relae o he parameric elecrical self-inducion, since he presence of curren is conneced wih a change in his parameer as capaciy. Le us examine he consequences, which escape from relaionship (.). If we o he capaciy connec he direc-curren generaor I, hen sress on i will change according o he law: I =. (.4) Thus, he capaciy, conneced o he source of direc curren, presens for i he effecive resisance R =, (.5) which linearly depends on ime. The i should be noed ha obained resul is compleely obvious; however, such properies of capaciy, which cusomary o assume by reacive elemen hey were for he firs ime noed in he work []. 3

4 This is undersandable from a physical poin of view, since in order o charge capaciy, source mus expend energy. The power, oupu by curren source, is deermined in his case by he relaionship: P I =. (.6) The energy, accumulaed by capaciy in he ime, we will obain, afer inegraing relaionship (.6) wih respec o he ime: W I =. Subsiuing here he value of curren from relaionship (.4), we obain he dependence of he value of he accumulaed in he capaciy energy from he insananeous value of sress on i: W =. sing for he case examined a concep of he flow of he elecrical inducion Ф = = Q (.7) and using relaionship (.), obain: I dф Q = = d, (.8) i.e., if we o a consan capaciy connec he source of direc curren, hen he curren srengh will be equal o he derivaive of he flow of capaciive inducion on he ime. 4

5 Now we will suppor a he capaciy consan sress, and change capaciy iself, hen I =. (.9) I is eviden ha he value R = (.) plays he role of he effecive resisance []. This resul is also physically inelligible. This resul is also physically inelligible, since. wih an increase in he capaciance increases he energy accumulaed in i, and hus, capaciy exracs in he volage source energy, presening for i resisive load. The power, expended in his case by source, is deermined by he relaionship: = P (.) from relaionship (.) is eviden ha depending on he sign of derivaive he expendable power can have differen signs. When he derived posiive, expendable power goes for he accomplishmen of exernal work. If derived negaive, hen exernal source accomplishes work, charging capaciy. Again, inroducing concep he flow of he elecrical inducion obain Ф = = Q, Ф I =. (.) Relaionships (.8) and (.) indicae ha regardless of he fac, how changes he flow of elecrical self-inducion (charge), is ime derivaive is always equal o curren. 5

6 Le us examine one addiional process, which earlier he laws of inducion did no include, however, i i falls under for our exended deerminaion of his concep. From relaionship (.7) i is eviden ha if he charge, lef consan (we will call his regime he regime of he frozen elecric flux), hen sress on he capaciy can be changed by is change. In his case he relaionship will be carried ou: = = cons, where С and - insananeous values, and and - iniial values of hese parameers. The sress on he capaciy and he energy, accumulaed in i, will be in his case deermined by he relaionships: W =, (.3) ( ) =. I is naural ha his process of self-inducion can be conneced only wih a change in capaciy iself, and herefore i falls under for he deerminaion of parameric self-inducion. Thus, are locaed hree relaionships (.8), (.) and (.3), which deermine he processes of elecrical self-inducion. We will call heir rules of he elecric flux. Relaionship (.8) deermines he elecrical selfinducion, during which here are no changes in he capaciy, and herefore his self-inducion can be named simply elecrical self-inducion. Relaionships (.3) and (.9-.) assume he presence of changes in he capaciy; herefore he processes, which correspond by hese relaionships, we will call elecrical parameric self-inducion. 6

7 . urren self-inducion Le us now move on o he examinaion of he processes, proceeding in he inducance. Le us inroduce he concep of he flow of he curren selfinducion ФL, I = LI. If inducance is shorened oued, and made from he maerial, which does no have effecive resisance, for example from he superconducor, hen Ф = L I = cons, L, I where L and I- iniial values of hese parameers, which are locaed a he momen of he shor circui of inducance wih he presence in i of curren. This regime we will call he regime of he frozen flow []. In his case he relaionship is fulfilled: I I L L =, (.) where I and L - he insananeous values of he corresponding parameers. In flow regime examined of curren inducion remains consan, however, in connecion wih he fac ha curren in he inducance i can change wih is change, his process falls under for he deerminaion of parameric self-inducion. The energy, accumulaed in he inducance, in his case will be deermined by he relaionship W L I cons L L = = L. Sress on he inducance is equal o he derivaive of he flow of curren inducion on he ime: 7

8 dф I L d. L, = I = L + I le us examine he case, when he inducance of is consan. L designaing ФI = L I, we obain (.) on he ime, we will obain: I = L. (.) dф = I. Afer inegraing expression d I =. (.3) L Thus, he capaciy, conneced o he source of direc curren, presens for i he effecive resisance which decreases inversely proporional o ime. R L =, (.4) The power, expended in his case by source, is deermined by he relaionship: P =. (.5) This power linearly depends on ime. Afer inegraing relaionship (.5) on he ime, we will obain he energy, accumulaed in he inducance of W L L L =. (.6) Afer subsiuing ino expression (.6) he value of sress from relaionship (.3), we obain: WL = L I. 8

9 This energy can be reurned from he inducance ino he exernal circui, if we open inducance from he power source and o connec effecive resisance o i. Now le us examine he case, when he curren I, which flows hrough he inducance, is consan, and inducance iself can change. In his case we obain he relaionship Thus, he value = I R L. (.7) L = (.8) plays he role of he effecive resisance []. As in he case he elecric flux, effecive resisance can be (depending on he sign of derivaive) boh posiive and negaive. This means ha he inducance can how derive energy from wihou, so also reurn i ino he exernal circuis. Inroducing he designaion ФL = LI and, aking ino accoun (.7), we obain: dф L d =. (.9) Of relaionship (.), (.6) and (.9) we will call he rules of curren self-inducion, or he flow rules of curren self-inducion. From relaionships (.6) and (.9) i is eviden ha, as in he case wih he elecric flux, he mehod of changing he flow does no influence evenual resul, and is ime derivaive is always equal o he applied poenial difference. Relaionship (.6) deermines he curren self-inducion, during which here are no changes in he inducance, and herefore i can be named simply curren self-inducion. Relaionships (.7,.8) assume he presence of 9

10 changes in he inducance; herefore we will call such processes curren parameric self-inducion.. Менде Ф. Ф. Новая электродинамика. Революция в современной физике. Харьков, НТМТ,, 7 с.