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ExaminetheStabilityMarginofCascadeCircui

By Wesley Arnold,2014-11-14 13:38
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ExaminetheStabilityMarginofCascadeCircui

Exa mine the Sta bil ity Margin of Ca sca de Circuits 1 2( ) ( ) YAN G Xi a n g2s he n g 杨相, F EN G Xi a o2ga n g 冯晓

    1 . Col lege of S cie n ce a n d Tec h n ol ogy , Ni n gbo U n i ve r s i t y , Ni n gbo 315211 , P . R . Ch i n a

    2 . Powe r M a n a ge m e n t P r od u ct - A n a l og Devi ces I n c . , S a n J ose , Ca l i f o r n i a 95134 , USA

    Abstract This p ap e r p r op os es a n e ngi ne e ri n g app r oac h t o e xa mi ne t he s t abili t y ma r gi n of cas cade ci r c ui ts . The p r op os e d meas u r e2 me nt i njects a n e xt e r nal p e r t u r bation c u r r e nt i nt o t he cas cade i nt e rf ace a nd meas u r es t he load2si de r esp ons e c u r r e nt . The s ys t e m s t a2 bili t y ma r gi n ca n be f i gu r e d out by comp a ri n g t he ma g ni t udes of t he p e r t u r bation c u r r e nt a nd t he r esp ons e c u r r e nt . The p r op os e d mea2 s u r e me nt is s t rictl y de ri ve d i n t he or y , i mp le me nt e d wi t h de t ails , a nd de mons t r at e d by e xp e ri me nt . Key words cas cade ci r c ui ts , s t abili t y ma r gi n , e ngi ne e ri n g meas u r e me nt .

    c u r r e n t ^i , t he s ys t e m s t a bili t y ma r gi n ca n be f i g u r e d L 1 Introduct ion out . Cas ca de is one of t he bas ic w a ys t o con nec t el ec t ric 2 Sta bil ity Margin of Ca sca de Circuit ci rc ui t s t o ge t he r . T yp ical cas ca de d ci rc ui t s i n a p p lica2 t i on i ncl ude p ow e r s upp l y a n d l oa d , i np ut f il t e r a n d The i de a of us i n g i mp e da nce r a t i o Z/ Zt o c hec k oi

    t he s t a bili t y of a cas ca de d ci rc ui t w as p r op os e d i n s wi t c h p ow e r con ve rs i on , m ul t i p l e2s t a ge a mp lif i e r , 2 1976 . As s how n i n Fi g . 1 , gi ve n t ha t t he s ou rcem ul t ip l e2s t a ge p ow e r con ve rs i on , dis t ri but e d p ow e r

    a n d loa d a r e s t a ble ci rc ui t i n t hei r s t a n d2alone op e r a2 s ys t e m , e t c .

    A f u n da me n t al conce r n of cas ca de d ci rc ui t s is t he t ion , t he ove r all s t a bili t y of t he cas ca de d ci rc ui t s is ove r all s ys t e m s t a bili t y . I n ma n y ap p lica t ions , t he de t e r mi ne d b y w he t he r Z/ Zs a t isf ies t he N y quis t oi cas ca de d ci rc ui t s a r e desi g ne d i n di vi duall y . Fo r e xa m2 c ri t e rion o r not .

    p le , i n a des kt op comp ut e r , sil ve r box p ow e r s upp l y

    a n d mot he r2boa r d a r e f r om diff e r e n t s upp lie rs . Us ual2 l

    y , t he mot he r2boa r d s up p lie r ass u mes t ha t t he p ow e r s

    upp l y is a n i de al one , w hile t he p ow e r s upp l y desi g n2 e

    r as s u mes a r esis t i ve l oa d . The r ef o r e , af t e r con nec t 2 i n g t w o w ell2desi g n ci rc ui t s , t he ove r all s ys t e m p e r2 Fig. 1 Cas cade ci r c ui ts f o r ma nce r e mai ns u nce r t ai n due t o t he s t r on g coup li n g

    The bloc k dia g r a m i n Fi g . 2 gi ves a si mp le e xp lai na2 be t w e e n ci rc ui t s . S ys t e m p e rf o r ma nce de g r a da t i on ,

    t ion of t his i mp e da nce s t a bili t y c ri t e rion . Wi t hout e ve n s ys t e m i ns t a bili t y w e r e r ep o r t e d i n a va ri e t y of 1 , 2 l os s i n g ge ne r ali t y , s ou rce a n d l oa d ci rc ui t s a r e mod 2 app lica t i ons .

    ele d as t w o2p o r t ne t w o r ks wi t h f ou r I/ O p a r a me n t e rs I n t his p ap e r , a p r ac t ical e n gi ne e ri n g app r oac h is

    ( )t r a nsf e r f u nc t ions A, A, Z, Zf o r e ac h , as de 2 p r op os e d t o e xa mi ne t he N y quis t loop gai n Z/ Zof v i i o oi

    s c ri be d b y cas ca de d ci rc ui t s . B y i njec t i n g s mall2si g nal e xt e r nal

    p e r t u r ba t ion c u r r e n t ^i i n t o t he cas ca de d i n t e rf ace a n d p AZ VVvo oi ( ) = . 1 - 1 me as u ri n g t he ma g ni t ude of t he l oa d s i de r es p ons e i i Z- A ioii

    As s how n b y t he t hic k a r r ows , t he i m p e da nce r a t io Recei ve d Dec . 10 , 2002 ; Re vis e d May 10 , 2003 Z/ Zis ac t uall y t he loop gai n of a vol t a ge2c u r r e n t o1 i2 Pr oject s upp or t e d by t he Scie nce Fou ndation of Zhejia n g M n nici2

    cas de d ci rc ui t s s ys t e m is de t e r mi ne d b y t his l oop gai n . / Zi n t he cas e of s mall s i g nals . I n a ddi t i on , s i nce Z oi

    ( ) does not ha ve t he c ha nce t o ci rcl e - 1 . 0 p oi n t , Acco r di n g t o t he N y quis t c ri t e ri on , t he bas ic s ys t e m

    s t a bili t y r e qui r e me n t on Z/ Zis t ha t i t does not e n2 con di t ional s t a bili t yis als o a voi de d . oi

    ( ) ci rcle - 1 , 0 p oi n t on t he S2p la ne . Mo r e ove r , p r op2

    e r s t a bili t y ma r gi n is als o r e qui r e d f o r bot h s ys t e m r o2

    bus t nes s a n d d y na mic p e rf o r ma nce .

    The mos t s t r ai g htf o r w a r d ap p r oac h t o e xa mi ne s ys2

    t e m s t a bili t y ma r gi n is t o me as u r e t he i m p e da nce of

    s ou rce a n d l oa d ci rc ui t s Za n d Z, a n d t he n t o calc u2 o i

    la t e t he s t a bili t y ma r gi n of t he cas ca de loop gai n Z/ o

    Z. D ue t o i t s comp le xi t y i n me as u r e me n t a n d calc ula 2 i

    t i on , t his ap p r oac h is i ncon ve ni e n t i n a p p lica t i on . A

    r ela t i vel y s i mp l e r e n gi ne e ri n g app r oac h is p r op os e d .

     Fig. 4 S t abili t y ma r gi n f or cas cade loop gai n Z/ Z oi

    Z o 1 ( )R?- .3 e Z 2 i

    2 . 2 Theoretical verif ication of the proposed

    mea surement

    Fo r t he p u rp os e of b r e vi t y , t he f ollowi n g va ria ble is def i ne d Fig. 2 The s t abili t y of cas cade ci r c ui t ( ω) ( ω) | Zj/ Zj| o i Δ (ω) ( )D 4 . = ( ω)( ω)| 1 + | Z j / Z j 2 . 1 Proposed mea surement a pproach o i

    As ill us t r a t e d i n Fi g . 3 , a n e xt e r nal s mall2s i g nal s i2 ( ) 1 If t he cas ca de l oop gai nZ/ Z, is out of t he oi n us oi dal p e r t u r ba t i on c u r r e n t ^i is i nj ec t e d i n t o t he p (ω) s ha dow e d r e gi on i n Fi g . 4 , t he n t he r e is D < 1cas ca de i n t e rf ace , w he n t he s ys t e m is i n t he s t e a d y wi t hi n all f r e que nc y r a n ges , a n d vice ve rs a , i . e . , s t a t e of a n op e r a t i n g p oi n t , a n d t he s mall2s i g nal r e2 ( ω)Z j o 1 ω) (( )R5 ?- Ζ D < 1 . sp ons e c u r r e n t i n l oa d s i de ^i is me as u r e d . eL ( ω)Z j 2 i

    The a bove r ela t ions hi p ca n be ve rif ie d b y conf o r mal

    mapp i n g s how n below . As s how n i n Fi g . 5 , t he u n2

    ( ( ) ) s ha dow e d a r e a i n S2p a ne Fi g . 5 a ca n be mapp e d

    i n t o t he i ns i de a r e a of a u ni t ci rcl e i n t he W2p la ne ( ( ) ) Fi g . 5 bby t he f oll owi n g conf o r mal mapp i n g . Fig. 3 Pr op os e d s t abili t y ma r gi n meas u r e me nt f or cas cade ci r2 S ( ) w = . 6 c ui ts S + 1 ( ) If t he f ollowi n g i ne quali t y Eq . 2 is vali d wi t hi n The r ef o r e , f o r a n y f r e que nc y p oi n t on t he cas ca de

    ωt he e n t i r e f r e que nc y r a n ge , t he n t he N y quis t c u r ve l oop gai n Z/ Z, if i t s t a ys out s i de t he s ha dow e d r e 2 oi

    of Z/ Zs t a ys on t he ri g ht si de of R= - 1/ 2 as oi e gion i n Fi g . 4 , t he n t he f ollowi n g i ne qali t y is vali d . s how n i n Fi g . 4 , a n d vice ve rs a . I n ot he r w o r d , i n 2 ( ω) ( ω) | Zj/ Zj| o i S ω) (< 1 . = D | w | = = ( ) e quali t y 2is a necess a r y a n d s uff icie n t con di t ion f o r ( ω) ( ω)S + 1 | 1 + Z j / Z j | o i ( ) i ne quali t y 3. ( )7

    ( ω) ( ω) ( )| ^i j| < | ^i j| .2 L p Me a nw hil e , s i nce W2p la ne ca n be mapp e d bac k i n t o ω) ( S2p la ne , vali da t i on of D < 1 als o i n dica t es t ha t As s how n i n Fi g . 4 , b y ke ep i n g N y quis t c u r ve Z/ oZ/ Zs t a ys out si de t he s ha dow e d f o r bi dde n r e gion i n oi Zi n t he ri g ht si de of R= - 1/ 2 , t he cas ca de s ys t e m i e

    S2p la ne .

    p r op os e d s ys t e m s t a bili t y ma r gi n me as u r e me n t a n d a n

    e xp e ri me n t e xa mp l e .

    The me as u r e me n t ca n be si m p l y ca r rie d out wi t h a

    ne t w o r k a nal yze r o r i mp e da nce a nal yze r . As s how n i n

    Fi g . 7 , t he s mall2si g nal p e r t u r ba t ion is t he f r e que nc y2

    s w ep t s i g nal f r om t he ne t w o r k a nal yze r a n d i nj ec t e d

    i n t o t he cas ca de i n t e rf ace via a c u r r e n t t r a nsf o r me r .

    Tw o c u r r e n t p r obes a r e r e qui r e d . O ne me as u r es t he

    ( ω) p e r t u r ba t i on c u r r e n t ^i ja n d s e n ds i t s out p ut s i g2 P nal t o t her ef e r e nce p r obe i np ut of ne t w o r k a nal yz2 Fig. 5 Conf or mal mapp i ng e r . The ot he r me as u r es t he loa d si de r es p ons e c u r r e n t ( ω) ^i ja n d s e n ds i t s out p ut si g nal t o t he t es t p r obe L ( ) 2B as e d on t he k nowle d ge of basic ci rc ui t s , me a 2

    i np ut of t he ne t w o r k a nal yze r . The ne t w o r k a nal yze r (ω) s u r e me n t of D ca n be f i g u r e d out as

    ( ω) ( ω) ge ne r a t es | ^i j| / | ^i j| acco r di n g t o t he r ef2 e L P ( ω)^i j L r e nce p r obe i np ut a n dr ef e r e nce p r obe i np ut s i g2 (ω) ( ) D =. 8 ( ω)^i j P nals .

    A de t ail e d p r oof is gi ve n i n t he f oll owi n g .

    The cas ca de ci rc ui t s a r e r e2d r aw n i n Fi g . 6 , w he r e

    ^i a n d ^i a r e s mall2s i g nal s i n us oi dal c u r r e n t s co r r e 2 s L

    sp on di n g t o p e r t u r ba t ion c u r r e n t ^i i n t he s ou rce a n d P

    loa d si des r es p ec t i vel y , w hile ^v is t he s mall2si g nal si2 b

    n us oi dal vol t a ge on t he cas ca de i n t e rf ace . Acco r di n g t o t he def i ni t ion of i mp e da nce , Fig. 7 Tes t e d cas cade d s ys t e m ( ω)^v j b ( ω) ( )9 Zj= , o ( ω)^ij s 3 . 1 Set the perturbation current

     ( ω)^v j b Us uall y , a ne t w o r k a nal yze r is cap a ble of p r ovi di n g ( ω) ( )Zj= , 10 i ( ω)^ij L f r e que nc y2s w e ep i n g si g nals f r om 10 mV up t o 1 V ( ω) ( ω) Δ| Zj/Zj| o i Ω wi t h 50 outp ut i mp e da nce . (ω)( ) D . 11 ( ω) ( ω) Zj/ Zj|| 1 + = o i The a mp li t ude of t he p e r t u r ba t ion c u r r e n t is s e t b y t

    he r esis t o r R a n d cap aci t o r C i n Fi g . 7 . The r esis t o r R ( )S ubs t i t ut ion of Eq . 9 ( )( )n d Eq . 10 i n t o Eq . 11 a

    li mi t s t he a mp li t ude of t he p e r t u r ba t ion c u r r e n t a t all gi ves

    f r e que nc y p oi n t s . The f u nc t i on of ca p aci t o r C is t o bloc ( ω) ( ω) | ^i j|| ^i j| L L (ω) )( D = = .12 k a n y DC c u r r e n t , w hic h ca n s a t u r a t e t he t r a ns 2 f o r ( ω)( ω) ( ω) | ^ij + ^i j| | ^i j| s L P me r . The mi ni m u m ca p aci t a nce of C s houl d g ua r a n2 t e e ( ) ( ) B as e d on Eq . 5a n d Eq . 8, t he p r op os e d s t a bili2 t ha t low f r e que nc y p e r t u r ba t ion c u r r e n t is not bloc t y ma r gi n me as u r e me n t ap p r oac h has be e n ve rif i e d . ke d . The r ef o r e , ass u mi n g 1 : 1 r a t io of t he c u r r e n t