Chapter 8 Objectives

Transcription

1 haper 8 Engr8 ircui Analyi Dr uri Nelon haper 8 Objecive Be able o eermine he naural an he ep repone of parallel circui; Be able o eermine he naural an he ep repone of erie circui. Engr8 haper 8, Nilon e

2 an ircui evie Tranien, naural, or homogeneou repone: Fae over ime; ei change. Force, eayae, paricular repone: Follo he inpu; Inepenen of ime pae. The oal repone ill be of he form: k k e The ircui circui conain boh an inucor an a capacior; Thee circui have a ie range of applicaion incluing ocillaor, frequency filer, fligh imulaion, moeling auomobile upenion, an more; The repone of circui ih D ource an iche ill coni of a naural repone an a force repone: v v f v n The complee repone mu aify boh he iniial coniion an he final coniion of he force repone. Engr8 haper 8, Nilon e

3 SourceFree Parallel ircui We ill fir uy he naural repone of econorer circui by looking a a ourcefree parallel circui: i i v i i i i v v ò v v v v Seconorer Differenial equaion Thi econorer ifferenial equaion can be olve by auming he form of a oluion: hich mean SourceFree Parallel ircui v Ae v v v A e Ae Ae Ae Thi i knon a he characeriic equaion. Engr8 haper 8, Nilon e 3

4 SourceFree Parallel ircui Uing he quaraic formula, e ge Define reonan frequency: Define amping facor: æ ç è æ ç è ö ø ö ø a Then:, a ± a SeconOrer Differenial Equaion Soluion We ill no ivie he circui repone ino hree cae accoring o he ign of he erm uner he raical., ± æ ö ç è ø, a ± a α > ω overampe:!# & ' * &. α ω criically ampe: α < ω unerampe: v Ae a v e B co B A e in Engr8 haper 8, Nilon e 4

5 Type of ircui epone Overampe Example Fin v in he circui a he righ. Ignore he curren arro. Given iniial coniion: v c, i A a α > ω herefore hi i an overampe cae, 6, a ± a Engr8 haper 8, Nilon e 5

6 Overampe ae coninue The oluion i of he form: v Ae Ue iniial coniion o fin A an A From v c a : v Ae Ae A A From K aken a : A e 6 i i v v 4 A 6A 4 i 6 A e 6A e Overampe ae coninue Solving he o equaion e ge A 84 an A 84 The oluion i: v 84e 84e 84 e e 6 6 V v v 84 e e 6 Engr8 haper 8, Nilon e 6

7 Example: Overampe ircui Fin v for >. v 8e 5, e, V for > riically Dampe ae α ω Fin v in he circui a he righ. Given iniial coniion: v c, i A /4f i v i 8.573Ω 7H i a.45 riically ampe hen α ω.45 The complee oluion i of he form: v Ae A e Engr8 haper 8, Nilon e 7

8 riically Dampe ae coninue Ue iniial coniion o fin A an A From v c a : v A e Ae A Therefore A an he oluion i reuce o v Ae Fin A from K a : i i v v i 4 4 A A.45 e A e.45 riically Dampe ae coninue Solving he equaion: A 4 The oluion i: v 4e.45 V v v 4e.45 Engr8 haper 8, Nilon e 8

9 riically Dampe Example Fin uch ha he circui i criically ampe for > an uch ha vv. Aner: 3.63 kω,.4ω Unerampe ae α < ω, a ± a For he unerampe cae, he erm inie he bracke ill be negaive an ill be a complex number. Define a Then, a ± j v A e v e a j a A e j A e a j A e j Engr8 haper 8, Nilon e 9

10 Engr8 haper 8, Nilon e j j e A A e e v a Uing Euler Ieniy q q q co j in e j in co in co in co in co B B e v A A j A A e v ja A ja A e v a a a in co B B e v a Unerampe ae coninue ooking a he magniue: A penulum i an example of an unerampe econorer mechanical yem. iplacemen Mechanical Analogue

11 Unerampe ae Example Fin v in he circui a he righ. Given iniial coniion: v c, i A /4f i v i.5ω 7H i a 6 α < ω herefore, hi i an unerampe cae a v i of he form: v e B co B in Unerampe ae coninue Ue iniial coniion o fin B an B From v c a : v e B co B in B Therefore B an he oluion i reuce o v e B in Fin B from K a : i i v v i 4 4 B B e co B e in Engr8 haper 8, Nilon e

12 Unerampe ae coninue Solving: B 97 The oluion i: v 97e in V v v 97e in V Unerampe Example Fin i for >. i e..7 co in 4.75 A Engr8 haper 8, Nilon e

13 Engr8 haper 8, Nilon e 3 Summary of Tranien epone Source Free Serie ircui ò i i i i i i v v v v v i v

14 omparing Serie an Parallel ircui Parallel v v v v A e Ae, ± æ ö ç è ø Serie i i i i A e Ae, ± æ ö ç è ø, a ± a, a ± a a a Serie ircui Soluion a a a a a If: α > ω overampe: α ω criically ampe: α < ω unerampe: i A e Ae A i e a A B co B in a i e Engr8 haper 8, Nilon e 4

15 Texbook Problem 8.49 Nilon h The circui conain no iniial energy. Fin v o for. v o 6 6e 4 co3.33e 4 in3 V Summary: Solving ircui. Ienify he erie or parallel circui;. Fin α an ω ; 3. Deermine heher he circui i overampe, criically ampe, or unerampe; 4. Aume a oluion naural repone force repone: A e Ae A e A e e V a B co B in V V f f f Overampe riically ampe Unerampe 5. Fin A, B, an V f uing iniial an final coniion. Engr8 haper 8, Nilon e 5

16 haper 8 Summary Shoe ho o eermine he naural an he ep repone of parallel circui; Shoe ho o eermine he naural an he ep repone of erie circui. Engr8 haper 8, Nilon e 6