C From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.

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1 Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds on h magnic propris of h cor and h gomry of h coil. L ' N 2 U ' N 2 µa C call : For capaciors, i() is proporional o h ra of chang of v(). C Inducors sor nrgy in a magnic fild. C From Faraday's Law, h inducd volag is, v() ' d 6NN()> ' d di() 6Li()> ' L C Th ffc of lcromagnic inducion in h coil islf is calld slfinducion. (Slf) Inducanc Conncion o Magnic Circuis C An inducor : Nurns around a cor of lngh,, cross scional ara, A, and prmabiliy, µ. i() call, F m ' UN and F m ' Ni ˆ NN() ' N F m () U ' N 2 i() U ' Li() v() N() C This says : in an inducor flux linkag = NN() is proporional o i(). C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 1

2 Currn hrough an Inducor C Currn hrough an inducor is govrnd by, Ldi() ' v() o i() ' 1 v()d ' 1 v()d % 1 v()d L m L m Lm &4 &4 o ' i( o ) % 1 v()d Lm o Enrgy Sord in an Inducor C Enrgy sord in an (idal) inducor, )W() ' P()) W ' m v()i()d ' m L di() d i()d ' L m i()di() ' 1 2 Li 2 C This says h currn hrough an inducor canno chang insananously if h volag is boundd! C If currn flow hrough h inducor is consan hn h volag, v(), across h inducor is 0. Exampl ** Nonlinar Inducors ** v() (vols) i() (ma) 100 nn() (Wb) (sc) Slop = (sc) (sc) i() L v() L = 10 mh C If µ is consan (.g., an air cor) hn h inducanc, L, is consan and indpndn of i(). L ' N 2 N F m þ N F m A L ' N 2 µa C If h cor of h inducor has a nonlinar BH characrisic (.g., an iron cor) hn, ' µ dpnds on i()! C Th inducor is said o b nonlinar bcaus h inducanc, L, now dpnds on i()! C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 2

3 Inducors Conncd in Sris C Wha is h quivaln inducanc of Ninducors conncd in sris? i i 1 Inducors Conncd in Paralll C Wha is h quivaln of Ninducors conncd in paralll? i 1 () ' i 2 () '... ' i N (), KCL v 1 L 1 di 1 () ' di 2 () '... ' di 2 () v() ' v 1 () % v 2 () %... % v N (), ' L 1 di() % L 2 di() ' L 1 % L 2 %... % L N ' di() KVL %... % L N di() di() ' L q di() v v 2 : v N : i 2 L 2 i N L N i() ' i 1 () % i 2 () %... % i N (), di() ' di 1 () % di 2 () %... % di N () (KCL) Lik rsisors conncd in sris, h quivaln inducanc of inducors conncd is sris is, L q ' L 1 % L 2 %... % L N C Th volag across ach inducor is v() hrfor, v() ' L 1 di 1 () di 1 () ' v() L 1, ' L 2 di 2 () di 2 () ' v() L 2,... '... ' L N di N () di N () ' v() L N di() ' v() L 1 % v() L 2 %... % v() L N ' 1 L q v() 1 L q ' 1 L 1 % 1 L 2 %... % 1 L N C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 3

4 L Enrgy Sorag Explanaion Enrgy Sorag Cycl : Wha happns if E is suddnly swichd from 0 o E =, a im = 0? Sinc h currn canno chang insananously, h iniial currn is zro, i.., i( = 0) = i(0 ) = 0. W know : v () = i() and E = v () () þ v (0) = 0 and (0) =. As currn flow incrass þ v () incrass and () dcrass. W also know : () = L di()/ þ di/ dcrass. If () dcrass þ v () incrass þ i() incrass. Evnually di()/ = 0 þ () = 0 and v () =. Th L circui is said o b opraing in sady sa. Sinc () = 0 a sady sa h inducor bhavs lik a shor circui a sady sa! L Enrgy Sorag Analysis, i(0 )=0 C Assum L has an iniial currn, i(0 ) = 0, and a volag, E =, is applid across L a = 0. Givn, () ' v 0 &, v 0 ', ' L, $0 From, KVL, = v () (), ˆ v () = (), v () ' & From Ohms Law, v () = i(), &, $0 i() ' E & 0 (1 & ), $0 L Enrgy Sorag Transin, i(0 ) = 0 C Plos of h charg ransins for (), v () and i() follow. Noic ha () v () =, (KVL). v () / () i() / / ( / ) / C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 4

5 L Enrgy Dcay Explanaion L Enrgy Dcay Transin, = 0 Enrgy Dcay Cycl in L Circuis : An inducor is opraing wih an iniial currn of i(0 ) 0. Wha happns o (), v () and i() if h volag E is suddnly changd o E = 0 a im = 0? i() i(0 ) i(0 ) / W know : () v () = 0 and v () = i(). v () i() will no chang insananously hrough L þ i(0) = i(0 ) þ v (0) = i(0 ) = () þ () = i(0 ). i(0 ) i(0 ) / W also know : () = L di()/ þ di(0)/ = i(0 )/L þ i() dcrass from i(0 ). As i() dcrass þ v () dcrass from i(0 ) and () = v (). C Evnually, di()/ = 0 þ () = 0, v () = 0, i() =0. () i(0 ) i(0 ) / C Plos of h dcay ransins for i(), v () and () follow. Noic ha () v () = 0, (KVL). L Enrgy Dcay Analysis, = 0 C Assum L has a nonzro iniial currn, i(0 ) and E is consan = = 0 for > 0. Givn, () ' v 0 &, v 0 '&i(0 & ), ' L, $0 L Transins : Gnral Analysis C Compu h soluion for i(), v () and () whn E is swichd o a consan,. From, KVL, v () () = 0, ˆ v () ' i(0 & ) From Ohms Law, v () = i(), i() ' i(0 & ) &, $0 &, $0 i() % () ', (KVL) (1) C For an inducor, () ' L di() (2) C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 5

6 L Transins : Gnral Analysis L Transins : Gnral Analysis C Thrfor h L circui is modld by h firs ordr diffrnial quaion, i() % L di() ' C Assum an iniial currn of i(0 ) is flowing and E =, is suddnly applid a = 0. i(0 ) i(0 ) i() () i(0 ) i() () v () i(0 ) v () i(0 ) i(0 ) $i(0 ) # i(0 ) L Transins : Gnral Analysis Exampls Th compl soluion is givn by : () ' & i(0 & ) &, $0 v ' & () ' % i(0 & ) & &, $0 i() ' v ' % i(0& ) & &, $0 Th L im consan, ' L. C.P. Diduch, Elcrical and Compur Enginring, Univrsiy of Nw Brunswick, March 2001, diduch@unb.ca 6