Electrical Systems Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell

Transcription

1 Elctrcl Systms Ptr Avtbl Mchncl Engnrng Dprtmnt Unvrsty of Msschustts owll.45 Dynmc Systms Elctrcl Dr. Ptr Avtbl Modl Anlyss & Controls bortory

2 Elctrcl Systms Pssv Elctrcl Elmnts sstor, Inductor, Cpctor Elmntl ltons for Voltgs sstor V V Voltg sstnc ( ohm volt / mpr) Currnt sstors dsspt ht no nrgy storg.45 Dynmc Systms Elctrcl Dr. Ptr Avtbl Modl Anlyss & Controls bortory

3 Elmntl ltons for Voltgs cont. Inductor Cpctor V Voltg V d dt Inductnc Inductors gnrt consdrbl rsstnc c V c Not Cq/v c q-chrg Not lso dq/dt nd v c q/c t of of currnt ( Hnry rt of chng of on mp / sc) ( wll nduc n mf of on volt) V C c Voltg Cpctnc mp ( Frd volt c dt sc ) Currnt chng.45 Dynmc Systms Elctrcl 3 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

4 Elmntl ltons for Currnts sstor Inductor V V dt Cpctor C c dv dt c.45 Dynmc Systms Elctrcl 4 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

5 Elctrcl Systms Ohm s w stts tht th currnt n crcut s proportonl to th totl lctromotv (mf) forc ctng n th crcut nd nvrsly proportonl to th totl rsstnc n th crcut. Srs V currnt (mps); mf (volt); rsstnc (Ohm) A 3 B ffctv.45 Dynmc Systms Elctrcl 5 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

6 Elctrcl sstnc Prlll 3 3 ; ; 3 3 Snc Thn ff ffctv ff Dynmc Systms Elctrcl 6 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

7 Krchoff s w Two w s xst: Currnt w (nod lw) Voltg w (loop lw) Currnt w (nod lw) Th lgbrc sum of ll currnts ntrng nd lvng nod s zro ) ( Dynmc Systms Elctrcl 7 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

8 Krchoff s w Consdr th crcut V V V d dt V V V V Th nstnt th swtch s closd, th currnt (o)zro bcus th nductor cnnot chng from zro to fnt vlu nstntnously.45 Dynmc Systms Elctrcl 8 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

9 Krchoff s w Tk th plc Trnsform Snc (0)0 O [ si(s) (0) ] I(s) ( S )I(s) E I(s) S(S ) E S E S E S S ( ) Not tht E nd V r oftn usd for voltg.45 Dynmc Systms Elctrcl 9 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

10 Krchoff s w Th nvrs lplc gvs E E 0.63 (t) E ( ) t Prov tht th slop of th functon ntrscts th fnl vlu E t τ.45 Dynmc Systms Elctrcl 0 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

11 Voltg w (oop w) Th lgbrc sum of th voltgs round th loop n n lctrcl crcut s zro. V V V C C V V V VC V 0.45 Dynmc Systms Elctrcl Dr. Ptr Avtbl Modl Anlyss & Controls bortory

12 Voltg w (oop w) cont. Contnung wth ths loop, th dff. q. s d V ; V ; VC Cdt dt C Thn d t dt V dt c o Tkng drvtv to lmnt th ntgrl d d dv dt dt C dt O Thus d dt ω.45 Dynmc Systms Elctrcl d dt ; C ω dv dt n C n ς C Dr. Ptr Avtbl Modl Anlyss & Controls bortory

13 Voltg w (oop w) cont. Tkng th plc Trnsform (wth IC0) S ( S S )I(S) V (S) C Th Trnsfr Functon s rrngng cllng th rlton I(S) V(S) S ( CS CS ) I(S) CS V (S) V c cdt c S ( )S ( ) C.45 Dynmc Systms Elctrcl 3 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

14 Voltg w (oop w) cont. cllng th rlton V c cdt c Th Trnsfr Functon rltng th nput voltg to th voltg drop cross th cpctor s V V c (S) (S) CS I(S) V (S) C S ( )S (C).45 Dynmc Systms Elctrcl 4 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

15 Oprtonl Amplfr An OP-AMP s n lctronc mplfr wth sngl output nd vry hgh voltg gn (0 5 to 0 6 ) low currnt! o In Gnrl ( ) K Th OP-AMP mplfs th dffrnc n voltgs nd nd s oftn rfrrd to s dffrntl mplfr In ths confgurton, n OP-AMP s nhrntly unstbl to stblz t ngtv fdbck s oftn usd..45 Dynmc Systms Elctrcl 5 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

16 Oprtonl Amplfr An OP-AMP s th most wdly usd nlog lctronc dvc B B A V os A o z D o z o o V os ( ) mv A B Dfntons: A A, B Sgnl Ground Input Voltgs,B Bs Currnts Z D Dffrntl Input Impdnc A Opn oop Gn Chsss Ground Z o Output Impdnc V os Offst Voltg ± V s Voltg Supply o ( V )A A B os.45 Dynmc Systms Elctrcl 6 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

17 Oprtonl Amplfr Voltg Followr ( V )A o A B os Assum A nd V os 0 o o o Output follows nput Sourc supplyng works nto n nfnt mpdnc thus no currnt s drwn o dos not drw currnt or consum powr Somtms clld buffr mplfr.45 Dynmc Systms Elctrcl 7 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

18 Oprtonl Amplfr Consdr n nvrtng mplfr shown Snc only nglgbl currnt flows nto th mplfr, must qul ( ' ) o ' o ( ' ) Snc K >> thn ' 0. Thus th pproxmt modl s: o o Gn s.45 Dynmc Systms Elctrcl 8 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

19 Oprtonl Amplfr Consdr non-nvrtng mplfr shown Snc And f ' o K >> >> K Thn n pproxmt modl s o Gn 0 or s For ths crcut K( ( K ) 0 0 ).45 Dynmc Systms Elctrcl 9 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

20 ow-pss Fltr Trnsfr Functon X X () s o () s o τs Elctrcl τ C Frquncy rspons functon s vlutd t jτω X C Mchncl K B B τ K X o o s jω.45 Dynmc Systms Elctrcl 0 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

21 ow-pss Fltr Pssv styld fltrs hv vrous dvntgs: Vry low nos output No powr rqurd Vrstl Wd dynmc rng.45 Dynmc Systms Elctrcl Dr. Ptr Avtbl Modl Anlyss & Controls bortory

22 Hgh-Pss Fltr Trnsfr Functon X X () s o () s o τs τs Elctrcl C τ C Frquncy rspons functon s vlutd t jτω τs X Mchncl B K B τ K X o o s jω.45 Dynmc Systms Elctrcl Dr. Ptr Avtbl Modl Anlyss & Controls bortory

23 Bnd-Pss Fltr Cscd low pss nd hgh pss fltr togthr τs τ s τ s τ > τ.45 Dynmc Systms Elctrcl 3 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

24 Intgrton (Approxmton Usng ow Pss) C o o ( jω) jωτ Now f ωτ >> thn Or o ( jω) o () s jωτ dt τs o τ.45 Dynmc Systms Elctrcl 4 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

25 Dffrntton (Approxmton Usng Hgh Pss) C o o ( jω) jωτ jωτ Now f ωτ << thn Or o ( jω) jωτ o () s τs o BEWAE!! Ths crcuts r snstv to nos upon dffrnttng d τ dt.45 Dynmc Systms Elctrcl 5 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

26 Dffrntton (w/ow Pss Fltr) C o o () s Cs Cs Accurt for Cω << But mplfs hgh frquncy nos by.45 Dynmc Systms Elctrcl 6 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

27 Dffrntton (w/ow Pss Fltr & Nos Attn.) C C o o () s C s ( C s )( C s ).45 Dynmc Systms Elctrcl 7 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

28 ElctroMchncl Systms Armtur-controlld DC motors r populr. Fld Controlld DC motors r not s populr. Elmntl ltons Elctrcl nd mchncl prts r coupld. Th mchncl moton of th rotor rltv to th sttor crts n lctromotv forc voltg (EMF ffct). Th bck EMF voltg cross th DC motor s proportonl to ngulr spd of th rotor. K ω K θ& whl th torqu dvlopd by th motor s proportonl to th currnt b T K T.45 Dynmc Systms Elctrcl 8 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

29 ElctroMchncl Systms T K T whr b bck mf voltg θ ngulr dsplcmnt θ & ω ngulr vlocty T torqu ppld to rotor (xrtd by V K mf constnt of motor ( ) K PM oz f K torqu constnt of motor ( ) t Convrson: n - ozf V 0.74 A K PM Amp motor).45 Dynmc Systms Elctrcl 9 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

30 Armtur-Controlld DC Motors ν b J k,c J d c d θ, ω θ& T (t) c dmpng of shft c d dmpng of dsk rmtur currnt f fld currnt (constnt) J d mss nrt of dsk J mss nrt of rotor k torsonl stff shft rmtur nductnc rmtur rsstnc ν rmtur voltg.45 Dynmc Systms Elctrcl 30 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

31 Armtur-Controlld DC Motors gd Shft ump th rotor, shft nd dsk togthr so tht J J r J d On nput, ν Two outputs, nd θ For th lctrcl crcut, th loop mthod gvs V drop 0 ν ν b ν 0 Thn, ν ( K ω K θ& ) So tht d dt d dt b K θ & ν b.45 Dynmc Systms Elctrcl 3 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

32 Armtur-Controlld DC Motors For th mchncl prt T Thn T B θ & J & θ d t d J && θ B θ& K Systm Equtons T J && θ Bdθ& K t T d Kθ & ν dt O J 0 T T J θ B d θ & (Not: Voltg s nput to systm but currnt s nput to mchncl prt) && θ θ& 0 Bd 0 0 K t θ T d d 0 K 0 ν dt dt.45 Dynmc Systms Elctrcl 3 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

33 33 Dr. Ptr Avtbl Modl Anlyss & Controls bortory.45 Dynmc Systms Elctrcl Armtur-Controlld DC Motors Exprssng th qutons usng (snc ) ω θ ω & t d T K B J ω ω& K dt d ν ω ν ω ω t d T K K B dt d 0 0 J &

34 Armtur-Controlld DC Motors plc Trnsform Th plc Trnsform gvs Js K B d K t s Ω(s) T I (s) V (s) (s) Aftr som mnpulton, th chrctrstc quton s gvn by ( B J ) s B K K 0 J s d d t Aftr puttng ths n stndrd form, ω n ζω n ( B K K ) d J ( B J ) d J t.45 Dynmc Systms Elctrcl 34 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

35 Armtur-Controlld DC Motors plc Trnsform Th trnsfr functons of ntrst r Ω(s) V (s) J s K t ( Bd J ) s Bd KK t θ(s) V (s) K J Bd J Bd K s s s J J t K t Not: A mor complx modl whch ncluds flxbl shft cn lso b dvlopd..45 Dynmc Systms Elctrcl 35 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

36 Exmpl - Crcut Attnuton A low frquncy sgnl (t 3 Hz) s to b msurd but s contmntd by hghr frquncy componnt such s 60 Hz nos for nstnc. A frst ordr low pss fltr (C crcut) sg n C V out scop Fltr V n V out Th fltr must b slctd to ttnut th 60 Hz nos wthout sgnfcntly ffctng th ctul sgnl ttnuton by no mor thn fctor of Dynmc Systms Elctrcl 36 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

37 Exmpl Crcut Attnuton Fltr Trnsfr Functon Vout (s) H(s) V (s) Cs Snc H(jω) n Th mgntud of th frquncy rspons functon s V V out n.45 Dynmc Systms Elctrcl ( ωτ) τ C tm constnt 0.06 s ( τ π( 3Hz) ) At 60 Hz, th mpltud s ( π) wth C 0.06s- possbl vlus [ ] C 0.µ F 30kΩ 37 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

38 Forc Voltg Anlogy Mchncl Elctrcl Forc f v Voltg Mss m Inductnc Vscous Frcton c sstnc Sprng Constnt k /C cprocl of Cpctnc Dsplcmnt x q Chrg Vlocty m & x cx& kx x& f (t) Currnt && q q& q C ν.45 Dynmc Systms Elctrcl 38 Dr. Ptr Avtbl Modl Anlyss & Controls bortory

39 Forc Currnt Anlogy Mchncl Elctrcl Forc f Currnt Mss m C Cpctnc Vscous Frcton c / cprocl of sstnc Sprng Constnt Dsplcmnt Vlocty m & x cx& kx k x x& f (t) / ψ ν cprocl of Inductnc Mgntc Flux Voltg C ψ && ψ & ψ.45 Dynmc Systms Elctrcl 39 Dr. Ptr Avtbl Modl Anlyss & Controls bortory