Research of a Floquet s reference frame through first order recurrence equation

Autorização concedida ao Repositório da Universidade de Brasília (RIUnB) pelo autor, em 21 de

julho de 2014, com as seguintes condições: disponível sob Licença Creative Commons 3.0, que

permite copiar, distribuir, transmitir o trabalho e fazer uso comercial, desde que o autor e

licenciante seja citado. Não é permitida a adaptação desta.

Authorization granted to the Repository of the University of Brasília (RIUnB) by the author, at

July, 21, 2014, with the following conditions: available under Creative Commons License 3.0, that

allows you to copy, distribute, transmit the work and to make commercial use, provided the

author and the licensor is cited. It is not allowed to adaptation.

REFERÊNCIA

M. P OLOU J ADOF F , I . CaM ARGO ( *)

Research of a Floquet s referen ce fram e th rough

fir st ord er recu rren ce equation

ABST RACT

Th e a u t h o r s h a ve p r e v i o u s l y d e v e l o p e d a m e t h o d t o s o l v e p e r i o d i c c o e f f i c i e n t o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t h r o u g h t h e u s e o f s e c o n d o r d e r r e c u r r e n t r e l a t i o n s w h i c h y i e l d t h e v a l u e s o f t h e F l o q u e t ' s s m u l t i p l i e r s . In t h e p r e s e n t p a c e r , t h e y e x a m i n e t h e p o s s i b i l i t y o f u s i n g a f i r s t o r d e r r e c u r r e n t r e l a t i o n ; t h e y s t u d y on e e x a m p l e w h e r e t h i s Is p o s s i b l e .

I NTRODUCTI ON

I n pr e v i o u s pap e r s we hav e t r eat ed t he c a s e of t he fol l owi ng or d i n a r y di f f er ent i al equat i on wi t h per i odi c c oef f i c i ent s :

(1) RI + d( LI ) / dt = 0

wher e L i s a per i odi c mat r i x .

By us e of t he Fl oquet ' s t heor em, we hav e s hown t hat

wher e t he t he I p^ ' s sr | d ’ s ar e t o be d e ­ t er mi ned.

Subs t i t ut i ng ( 2) i nt o ( 1) y i el ds t he f ol l o­ wi ng s ec ond or der r ec ur r enc e r el at i on :

( 3)

wher e A and C ar e c ons t ant mx m mat r i c es , B (1 i s a mx m mat r i x whi c h i s f unc t i on of a and n.

I he k ey of t he met hod i s t o s how t hat ( 3) y i el ds a c ondi t i on whi c h ens ur es t he c onv er genc e of ex pans i on ( 2) ; f r om t hi s c ondi t i on, m v al ues of a c an be det er mi ned. Numer i c al appl i c at i ons hav e been gi ven.

(*■> Labor at oi r e

d'hietrotechmque des universités

Pa n s Vi

et

Xi

- Unité associé au CNRS n' 845

91405 - ORSAY

Now, i t i s a c ommon us e t o r epl ac e s ec ond or der di f f er ent i al equat i ons by f i r s t or der ones, at t he ex pens e of a l ar ger number of u n ­ k nowns . Ther ef or e, i t i s per t i nent t o r ai s e t he ques t i on : woul d i t be eas i er t o r epl ac e ( 3) by a f i r st or der r ec ur r enc e r el at i on s uc h as :

( 4a)

Ther ef or e, i t i s pos s i bl e t o r educ e t he s e­ c ond or der r ec ur r enc e t o a f i r st or der one, at l eas t i f A i s non s i ngul ar . I t i s t o be not ed t hat Mn i s al way s a f unc t i on of

a

Tor f i ni t e v al ues of n, and may be i ndependent of a when " n" goes t o pl us or mi nus i nf i ni t y.

L ' ENER&I A EL ETTRI CA - N. 1 0 - 1988 469

(2)

wher e M n woul d be a 2mx 2m mat r i x . I f t hi s i s pos s i bl e, i s i t pos s i bl e t o mak e us e of ( 4a) t o det er mi ne t he a^' s ?

OUTL I NED! THE METHOD

I ndeed, i f A i s non s i ngul ar , i t i s pos s i bl e t o wr i t e :

( 5a)

Si nc e i t i s al way s pos s i bl e t o wr i t e :

( 5b)

Po l o u j a d of f M. , Ca ma r g o I .

When " n" goes t o ♦<*>, s ome of t he ei genv al ues of M n ar e l ar ger t han uni t y , and o t her s ones ar e s mal l er . For t he ex pans i on ( 4) t o be c onv er gent , i t i s nec es s ar y t hat t he c ons t i t uent v ec t or s l ay i n t he s ubs pac e as s oc i at ed wi t h ei genv al ues l ar ­ ger t han one f or n = -<», and i n t he s ubs pac e as s oc i at ed wi t h ei genv al ues s mal l er t han one f or n =

<*>.

The r ol e o f t he i nt er medi at e ( i . e. f i ni ­ t e) v al ues of " n" i s t o ens ur e a t r ans i t i on bet ween t hos e t wo ex t r eme c as es . And s i nc e Mn i s a f unc t i on of a when n i s f i ni t e, t he t r ans i ­ t i on bet ween t he t wo ex t r eme c as es c annot be ens ur ed unl es s

a

as s umes s ome par t i c ul ar v al ues ; t hi s r emar k t hus pr ov i des t he equat i on whi c h d et er mi nes t he a^ ' s.

Thi s v er y s i mpl e pr oc edur e wi l l be now e x ­ pl ai ned on a pr ac t i c al exampl e.

PRESENTATI ON OF THE EXAMPLE

Let us c ons i der ( f i gur e 1) t wo c oi l s whos e r es i s t anc es and i nduc t anc es ar e c ons t ant , but mut ual i nduc t anc e i s a per i odi c f unc t i on of t i me.

The equat i ons ar e :

(6a)

w and 0 bei ng c ons t ant s . At t i me " t = U" when s wi t c h 5 i s c l os ed :

( 6c)

t i me c ons t ant s and mi ni mu m di s per s i on of t he t wo wi ndi nas :

( 7c)

I ndeed wi t h t hos e not at i ons , we get ( f or non z er o v al ues o f bot h " n" and " h" ) :

( 8a)

wher e

(8b)

s o t hat ( 4a) and ( 4b) bec ome :

( 9)

whi c h i s equi v al ent t o

(1 0a)

wi t h

!

I (10b)

I

and

(1 1a)

(11b)

t he ei genv al ues o f M(<*>) ar e X, - X, 1/ X, - 1/ X, whi c h c or r es pond t o t he f ol l owi ng ei genv ec t or s :

Fi g. 1 : Undamped uni f or m ai r gap mac hi ne

(11c)

Not e t hat (6a) i s i nhomogeneous , whi l e ( 1) i s homogeneous . Rel at i on ( 2) y i el ds :

( 7a)

wher e h = 0 c or r es ponds t o t he par t i c ul ar s ol u­ t i on o f t he i nhomogeneous equat i on

( aQ =

0). Fr om now on, we s hal l dr op t he i ndex " h" , i n or der t o al l ev i at e t he ex pr es s i ons . I n addi t i on, we s hal l us e t he f ol l owi ng v ar i abl es :

( 7b)

whi c h al l ow (6a) t o be r ewr i t t en i n t er ms of t he

r es pec t i v el y .

Si nc e X > 1 , i t i mmedi at el y appear s t hat c onv er genc e of ( 7a) c annot be ens ur ed i f XR has a c omponent al ong uj or U2, For v er y l ar ge v a ­ l ues o f " n" ; c onv er s el y , c onv er genc e o f ( 7a)

c annot be ens ur ed i f Xn has a c omponent al ong U3 or u^ , f or n < 0 ( and | n| v er y l ar ge) . Thi s means t hat X mus t l ay i n t he pl ane ( uj , u^ ) f or n > 0, and Xn i n t he pl ane ( u- j , ^ ) f or n < 0. i n ot her wor ds :

470 _{L ' ENERGI A EL ETTRI CA - N. 1 0 - 1 988}

Re s e a r c h of a Fl o q ue t ' s r efe r e n c e fr a me t h r o u g h a fir s t o r d e r r e c u r r e n c e e q u a t i o n

( 1 2 )

wh e r e N" i s a v er y l a r ge v al ue of n" wi l l t her ef or e y i el d one f our l i ne equat i on t o d e t er ­ mi ne a, C, D, E, F, t hat i s t o s ay t o det er mi ne

f our unk nowns ( s i nc e one among C, D, E, F c an be c hos en ar bi t r ar i l y ) . Thi s equat i on i s non l i ­ near ; we k now t hat t her e ar e t wo t i me const ant s, t her ef or e we mus t f i nd t wo c onv eni ent s et s of ( a, C, D, E, F) .

We hav e c ompl et ed t he s ol ut i on i n t he c as e of a v er y hi gh d0/ dt , t hen f or v ar i ous v al ues of d0/ d t .

CTASECI FHI GHROI ATI ONSPEED

I f " w" i s l ar ge* enough, a wi l l be negl egi bl e when c ompar ed t o " j nw" , and

( 14)

ev en f or n = 1.

Ther ef or e, equat i on ( 13) c an be us ed ei t her as

( 15a)

or as

( 15b)

Si nc e t he di r ec t i ons o f Xœ and ar e k nown, and s i nc e M(0) i s a f unc t i on of a, t he f ol l owi ng equat i on def i nes C, D, E, F and ct :

( 16a)

Rec al l i ng t hat a Q and b Q ar e f unc t i ons of a and r ear r angi ng, we get

( 16b)

The det er mi nant of ( 16b) obv i ous l y c anc el s when ei t her one of t he f ol l owi ng c ondi t i ons i s f ul f i l l ed :

( 17a)

whi c h i n t ur n y i el d ei t her one of t he f ol l owi ng v al ues o f a :

( 17b)

whi c h ar e t he wel l k nown v al ues gi v en by Bouc he- r ot [1] .

CASEOF L OWROTAI I ONSPEEDS

For l ow v al ues of w, i t i s per mi s s i bl e t o use agai n ( 13) wi t h a " l ar ge enough" v al ue of N. Let us c al l :

( 18)

P+ and p - ar e f unc t i on of bot h a and N, and may be wr i t t en P+( a, N) and P “ ( a, N) . Thus

i s t he equat i on whi c h gi v es a and t he v al ues of C, D, E, F. Thi s equat i on wi l l be s ol v ed by s u c ­ c es s i v e i t er at i ons . To t hi s end, l et us c al l a" an appr ox i mat e v al ue of a, and a' a v al ue of a

whi c h i s a bet t er appr ox i mat i on t han a" . I n ( 19a) we may r epl ac e t he s quar e mat r i x by :

( 19c)

whi c h i s mor e gener al t han ( 16b) . Thus, i f we hav e an appr ox i mat i on of a, ( 19c) wi l l y i el d anot her one. I f we s t ar t wi t h one of t he v al ues ( 17) , c onv er genc e i s obt ai ned af t er 2 or 3 i t e­ r at i ons. We s hal l s ay t hat t he v al ue of N whi c h has been us ed was " l ar ge enough" i f t he pr oc es s c onv er ges t o t he s ame v al ue o f a f or N and f or N+ 1 .

I t i s t o be not ed t hat t he ei gendi r ec t i on ( E, D, C, F) pr ov i ded by ( 19c) mus t obv i ous l y be t he s ame as t he ei gendi r ec t i on pr ov i ded by ( 16b) . Ther ef or e, t he s ol ut i on of ( 19) f or a' when a" i s k nown may be gr eat l y s i mpl i f i ed.

Thi s pr oc es s has been us ed f or a = 0. 09,

Sg = 2. s, 6f = 1. 8 ; f or eac h v al ue of w, t wo v al ues and o f a ar e f ound : t hey ar e pl ot ­ t ed i n f i gur e 2.

Fi g. 2 : Var i at i on of a as a f unc t i on of f r equenc y

L ' ENERGI A EL ETTRI CA - N. 1 0 - 1988 ₄₇₁

( 19a) Wr i t i ng ( 4a) , ( 9) and ( 10a) as :

( 13)

( 19b)

Pol ouj adof f M. , Camar go I .

CONCL USI ON

So me l i n e a r f i r s t o r d e r p e r i o d i c c o e f f i c i e n t o r d i n a r y d i f f e r e n t i a l e q u a t i o n s ma y b e s o l v e d t h r o u g h a s e c o n d o r d e r r e c u r r e n t r e l a t i o n wh i c h we h a v e s t u d i e d i n our p r e v i o u s wor k . I n t he p r e s e n t paper , we h a v e s h o wn t hat i t i s p o s s i b l e t o r e p l a c e t hi s r e l a t i o n by a f i r s t o r d e r one, u n d e r t he c o n d i t i o n t hat a g i v e n ma t r i x be non s i n gul ar . We h a v e s h o wn h o w t o s o l v e s u c h a f i r s t o r d e r r e c u r r e n t r e l at i on. We h o p e t hat t hi s wi l l s i mp l i f y t he s t u d y o f mo r e c o mp l i c a t e d p r o b l e ms .

REF ERENCES

[ 1] P. Bo u c h e r o t : " Les p h é n o mè n e s é l e c t r o ma g n é ­ t i q u e s qui r és ul t ent , de l a mi s e e n c o u r t - c i r c u i t

b r u s q u e d ' u n a l t e r n a t e u r " . Bu l l e t i n de l o So c i é ­ t é I n t e r n a t i o n a l e d e s El e c t r i c i e n s , 3ème s ér i e, t ome 1, n° 10, pp. 55 5 - 5 9 5 , 1911.

[ 2 j S. L e f s c h e t z : " Di f f e r e n t i a l e q u a t i o n s : g e o me t r i c t heor y " . Dov e r , Ne w Yor k , Î 977.

[ 3] E. Pi l l et , M. Po l o u j a d o f f , J . P. Ch a s s a n d e : ¡ " Ti me c o n s t a n t s o f u n s y mme t r i c a l s h or t c i r c u i t s o f s y n c h r o n o u s ma c h i n e s " . I EEE Tr a n s a c t i o n on PAS, v ol . PAS- 99, n° 3, pp. 12 9 8 - 1 3 0 5 , 1980. [ 4] I . Ca ma r g o : " Mé t h o d e ma t r i c i e l l e p o u r l a r é s o l u t i o n d ' é q u a t i o n d i f f é r e n t i e l l e à c o e f f i ­ c i e n t s p é r i o d i q u e s " . Ra p p o r t d e DEA, Gr e n o b l e , 1985.

[ 5] M. Po l o u j a d o f f , E. Pi l l et , I . Ca ma r g o : " I mp r o v e d d e t e r mi n a t i o n o f Fl o q u e t ' s r e f e r e n c e f r ame f or t he p h a s e t o g r o u n d s h or t c i r c u i t of a n o n s a l i e n t p o l e d a mp e d a l t e r n a t o r " . I EEE Tr a n s a c t i o n o n CAS, v ol . CAS- 3 4 , n D

8

, 1987.

A u to r izz . Tribunale d i M ila n o 24 luglio 1948, N , 275 de! R egistro D irettore responsabile:P r o f . I n g . G i o r g i o C a t e n a c c i

Proprietário: A s s o c i a z i o n e E l e t t r o t e c n i c a e d E l e t t r o n i c a I t a l i a n a

S t a m p a : A r t i G r a f i c h e S t e f a n o P i n e l l i - V i a R . F a r n e t i , 8 2 0 1 2 9 M i l a n o