Lecture #13. Mutual Inductance

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Baldwin Poole
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1 ecure #3 ANNOUNCEMENTS Graded HW can be picked up in 78 Cry OUTNE Muual inducance Firs-rder circuis Naural respnse f an circui eang Chaper 6.4, Chaper 7. ecure 3, Slide Muual nducance Muual inducance ccurs when w circuis are arranged s ha he change in curren in ne causes a lage drp be induced in he her. Example: Cnsider inducr in he circui belw self-induced lage is ( /d) muually induced lage is M( /d) bu wha is he plariy f his lage? M g i i ecure 3, Slide

ecure 3, Slide Muual nducance Muual inducance ccurs when w circuis are arranged s ha he change in curren in ne

2 The D Cnenin f a curren eners he ded erminal f a cil, he reference plariy f he lage induced in he her cil is psiie a is ded erminal. f a curren leaes he ded erminal f a cil, he reference plariy f he lage induced in he her cil is negaie a is ded erminal. g i i M d M M d ecure 3, Slide 3 nduced Vlage Drp The al induced lage drp acrss an inducr is equal he sum f he self-induced lage and he muually induced lage Example (cn d): Apply KV lps g d i i M M M d d d ecure 3, Slide 4

f a curren leaes he ded erminal f a cil, he reference plariy f he lage induced in he her cil is negaie a g i i M d M

3 elainship beween M and, The alue f muual inducance is a funcin f he self-inducances: M = where k is he cefficien f cupling k (This equain is alid nly if is a cnsan funcin f i ) 0 k ecure 3, Slide 5 Tal Energy Sred in nducrs The al energy sred in inducrs and is w = i i Mi i if each cil curren eners he ded erminal (r if each cil curren leaes he ded erminal) The al energy sred in inducrs and is w = i i Mi i if ne cil curren eners he ded erminal and he her cil curren leaes he ded erminal ecure 3, Slide 6

inducrs and is w = i i Mi i if each cil curren eners he ded erminal (r if each cil curren leaes he ded erminal) The al energy

4 Firs-Order Circuis A circui which cnains nly surces, resisrs and an inducr is called an circui. A circui which cnains nly surces, resisrs and a capacir is called an C circui. and C circuis are called firs-rder circuis because heir lages and currens are described by firs-rder fferenial equains. s i s i C ecure 3, Slide 7 Naural espnse & Sep espnse The naural respnse f an r C circui is is behair (i.e. curren and lage) when sred energy in he inducr r capacir is suddenly released he resisie newrk (cnaining n independen surces). The sep respnse f an r C circui is is behair when a lage r curren he inducr r capacir is suddenly changed. ecure 3, Slide 8

and C circuis are called firs-rder circuis because heir lages and currens are described by firs-rder fferenial equains.

5 Naural espnse f an Circui Cnsider he fllwing circui, fr which he swich is clsed fr < 0, and hen pened a = 0: = 0 i Nain: 0 is used dene he ime jus prir swiching 0 is used dene he ime immeaely afer swiching The curren flwing in he inducr a = 0 is ecure 3, Slide 9 Sling fr he Curren ( 0) Fr > 0, he circui reduces i Applying KV he circui: Sluin: i( ) = i(0) e ecure 3, Slide 0

immeaely afer swiching The curren flwing in he inducr a = 0 is ecure 3, Slide 9 Sling fr he

6 Sling fr he Vlage ( > 0) i e ( ) = Ne ha he lage changes abruply: (0 ) = 0 fr > 0, ( ) = i = (0 ) = e ecure 3, Slide Sling fr Pwer and Energy Deliered ( > 0) i e ( ) = p = i w = 0 = e p( x) dx = = ( e ) ecure 3, Slide 0 e x dx

7 Time Cnsan τ n he example, we fund ha i( ) = e and ( ) = e Define he ime cnsan τ = A = τ, he curren has reduced /e (~0.37) f is iniial alue. A = 5τ, he curren has reduced less han % f is iniial alue. ecure 3, Slide 3 Transien s. Seady-Sae espnse The mmenary behair f a circui (in respnse a change in lage r curren) is referred as is ransien respnse. The behair f a circui a lng ime (many ime cnsans) afer he change in lage r curren is called he seady-sae respnse. ecure 3, Slide 4

Seady-Sae espnse The mmenary behair f a circui (in respnse a change in lage r curren) is referred as is ransien respnse.

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural