HTumdamenltal Slip-n-mg SaeM ne - SLIDING MODE CONTROL OF PO"WER CONVERTERS

CHAPTER 5 CHAPTER 5

5.2 HTumdamenltal Slip-n-mg SaeM ne

147

d e s c rib e d la t e r , i t w ill be seen t h a t th e p e rfo rm a n c e e q u a tio n s o f a ll ty p e s o f r o t a t in g e le c t r ic a l m a c h in e s m a y be p u t in th e above fo rm .

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th e fu n d a m e n ta l m a c h in e as s ta te d e a r lie r .

F ig u re 5.2 show s th e s c h e m a tic o f th e fu n d a m e n ta l s lip - r in g m a c h in e . The s t a t o r c a r r ie s tw o w in d in g s A a n d B, o r th o g o n a l to e a c h o th e r . The w in d in g s A a n d B a re s h o w n as lu m p e d w in d in g s o r ie r ite d in th e d ir e c t io n o f t h e i r r e s p e c tiv e MMFs. The r o t o r c a r r ie s tw o lu m p e d w in d in g s a a n d £?, o r t h o g o n a l to e a c h o th e r . A g a in a a n d

@ a re sh o w n o r ie n te d a lo n g t h e i r re s p e c tiv e MMFs. The v o lta g e e q u a tio n f o r th e fu n d a m e n ta l m a c h in e m a y he w r it t e n as

VA Ra +pLa 0 pMAa pMAa V

VB 0 Rb +pLs pMBa pMBa i-B

v a PMaA P^aB Ra +pLa ⁰ i a

VP, ^, P^pA pMfjS ⁰ R p+pL p H.

(5 .1 6 ) A p p ly in g th e c o n d itio n t h a t th e m u t u a l c o u p lin g b e tw e e n th e s t a t o r

a n d r o t o r w in d in g s a re t r ig o n o m e t r ic fu n c tio n s o f th e r o t o r a n g le i9, Eq. (5.16) re d u c e s to

vA R^a + p L A ⁰ M P COS T? Jfcfp sin i? _V

0 Rb+pLb —M p sin i? M p cos _t_-_b

V a M p cos1? —M p sin Ra "F pRa ⁰ i'a

VP. M psin M p cos 0 R p + p L p H .

E q u a tio n (5 .1 7 ) is th e d e fin in in g e q u a tio n f o r th e fu n d a m e n ta l s lip - r in g m a c h in e . I t is s e e n t h a t th e Z m a t r ix o f e ve n th is s im p lifie d m a c h in e is a f u n c t io n o f th e r o t o r a n g le . F u r th e r m o r e th e " p " o p e r a to r o p e ra te s o n th e p r o d u c t o f th e v a r io u s c u r r e n t s a n d th e tr ig o n o m e tr ic fu n c tio n s . T h e e lim in a t io n o f th e r o t o r a n g le f r o m th e im p e d a n c e m a t r ix Z is t h e c e n t r a l fe a tu r e o f th e u n ifie d

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Fig. 5 .2 The f u n d a m e n ta l s lip rin g m a c h in e . The s ta to r c a rrie s w in d in g s A a n d B, w h ic h a re o rth o g o n a l to each other.

The ro to r c a rrie s two o rth o g o n al w in d in g s a an d The ro to r is ro ta tin g w ith a n g u la r v e lo c ity cj.

m a c h in e th e o r y . These d e ta ils a re e x p la in e d s u b s e q u e n tly in S e c tio n 5.4.

5 .3 B a sic C o m m u ta to r fia c M n e

In th e b a s ic s lip - r in g m a c h in e d e s c rib e d in S e c tio n 5.2, th e r o t o r MMF is a fu n c tio n o f th e r o t o r p o s itio n w h e n th e r o t o r c u r r e n ts a re dc. T h is p h y s ic a l fa c t is se e n m a th e m a tic a lly , in th e im p e d a n ce m a t r ix Z b e in g a fu n c tio n o f th e r o t o r a n g le i?. In c o n t r a s t to th is , in th e b a sic c o m m u ta to r m a c h in e , th e r o t o r MMF is in d e p e n d e n t of th e r o t o r angle tf. In th is s e c tio n th e a c tio n o f th e c o m m u ta to r , lea d ing to th e above r e s u lt is f i r s t e x p la in e d . T h e n th e b a s ic c o m m u ta to r m a c h in e is d e s c rib e d .

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Fig. 5 .3 The co n s tru c tio n o f a c o m m u ta to r ro to r. The brushes le a d c u rre n t in to the a r m a tu r e th ro u g h the c o m m u ta to r segm ents. The c o m m u ta to r segm ents a n d the connection to the ro to r con d ucto r is such th a t, the ro to r M M F is i n lin e w ith the brushes a n d in d e p e n d e n t o f the ro to r po sition.

F ig u re 5.3 shows th e c o n s t r u c t io n o f a c o m m u ta to r r o t o r . The r o t o r w in d in g s , t h e ir c o n n e c tio n to th e c o m m u ta to r s e g m e n ts a n d th e b ru s h e s le a d in g c u r r e n t in to th e r o t o r c o ils th r o u g h th e c o m m u ta to r s e g m e n ts a re a ll sh o w n . T he d ir e c t io n o f c u r r e n ts in th e in d iv id u a l r o t o r c o ils a re a ls o sh o w n . I t m a y be seen t h a t th e MMFs p ro d u c e d by th e in d iv id u a l w in d in g s o f th e r o t o r add up to give a n e t t r o t o r MMF a lo n g th e d ir e c t io n o f th e b ru s h e s . The d ir e c t io n of r o t o r MMF in a c o m m u ta to r fe d r o t o r is th e r e fo r e , in d e p e n d e n t o f th e r o t o r a n gle and s ta y s a lw a ys a lo n g th e d ir e c t io n o f b ru s h e s .

F ig u re 5.4 show s th e s c h e m a tic r e p r e s e n ta tio n o f th e c o m m u ta to r r o t o r . T he r o t o r w in d in g is s h o w n lu m p e d a n d fix e d in d ir e c tio n along th e b ru s h e s , in d ic a tin g th e f a c t t h a t th e d ir e c tio n o f

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151

Fig. 5.4 The schem atic re p re s e n ta tio n o f the c o m m u ta to r ro to r.

The ro to r w in d in g is show n lu m p e d a n d i n lin e w ith the brushes to in d ic a te the d ire c tio n o f the ro to r MMF.

F ig . 5 .5 The basic c o m m u tato r m a c h in e . The s ta to r a n d the ro to r each c a rry two orthogonal w in d in g s . The ro to r is supp lied through com m utato rs a n d hence the ro to r f ie ld is in dependent o f the ro to r p o sitio n .

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r o t o r MMF o f a c o m m u ta to r r o t o r is fix e d in sp a ce a lo n g th e d ir e c t io n o f th e b ru s h e s .

As h a d b e e n e x p la in e d in S e c tio n 5.2, th e g e n e r a liz e d r o t o r has to c a r r y a m in im u m o f tw o w in d in g s in o r d e r to be a b le to a r b it r a r ily s e le c t th e d ir e c t io n o f r o t o r MMF in th e a c tiv e p la n e o f th e m a c h in e . The b a s ic c o m m u ta to r m a c h in e , th e r e fo r e , is c o n c e iv e d w ith tw o o r th o g o n a l s ta t o r w in d in g s ( ju s t lik e th e s t a t o r w in d in g s o f th e fu n d a m e n ta l s lip - r in g m a c h in e ), a n d two o r th o g o n a l r o t o r w in d in g s s u p p lie d th r o u g h a s e t o f b ru s h e s a n d c o m m u ta to r s e g m e n ts . F ig u re 5.5 shows th e s t r u c t u r e o f s u c h a b a s ic c o m m u ta to r m a c h in e . A a n d B c o n s titu te th e o r th o g o n a l s t a t o r w in d in g s . The w in d in g s d a n d g o n th e r o t o r a re fe d th r o u g h a s e t o f b ru s h e s a n d c o m m u ta to r s e g m e n ts , a n d a re o r t h o g o n a l to e a c h o th e r . The v a rio u s w in d in g s a re s h o w n lu m p e d and o r ie n te d a lo n g th e d ir e c t io n o f t h e ir re s p e c tiv e MMFs.

The u n ifie d m a c h in e th e o r y c o n s is ts o f a p p ly in g a s e rie s o f m a th e m a tic a l tr a n s fo r m a tio n s , to re p la c e a n y r o ta tin g m a c h in e b y its e q u iv a le n t c o m m u ta to r ty p e o f m a c h in e , th e m o tiv a tio n b e in g th e im p e d a n c e m a t r ix o f th e e q u iv a le n t c o m m u ta to r m a c h in e is in d e p e n d e n t o f th e r o t o r a n g le , a n d h e n c e m u c h s im p le r to a n a ly z e . The v a rio u s tr a n s fo r m a tio n s t h a t f a c ilit a t e th is p ro c e s s a re d is c u s s e d in th e n e x t s e c tio n .

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153

5 .4 P a s s iv e T r a n s fo r m a tio n s

T h is s e c tio n d e s c rib e s th e p a s s iv e tr a n s fo r m a tio n s t h a t a re u s e d to u n if y th e d if fe r e n t k in d s o f r o t a t in g e le c t r ic a l m a c h in e s . T hese tr a n s fo r m a tio n s a re in tr o d u c e d w ith th e o b je c tiv e o f s im p lify in g th e a n a ly s is o f d if fe r e n t k in d s o f r o t a t in g m a c h in e s . In o r d e r to a p p re c ia te th e d iffic u ltie s in v o lv e d in s o lv in g th e g e n e r a l p e r fo r m a n c e e q u a tio n s , a th r e e p h a s e m a c h in e is f i r s t p re s e n te d .

F ig u re 5.6 show s s c h e m a tic o f a th r e e p h a s e s lip - r in g m a c h in e . T he s t a t o r c a r r ie s th r e e p h a s e w in d in g s R, Y a n d B. The r o t o r c a r r ie s th re e p h a s e w in d in g s r , y a n d 6. E q u a tio n (5 .1 8 ) is th e v o lta g e e q u a tio n f o r th is th r e e p h a se s lip - r in g m a c h in e , w r it t e n d o w n f r o m f i r s t p r in c ip le s .

V = Z I (5 .1 8 )

y

F ig . 5 .6 A p r a c tic a l three p h a s e s lip r in g m a c h in e . The s ta to r an d ro to r c a rr y b a la n c e d three p h a s e w in d in g s .

154

[ VR V]f V B Vt V y V b

Y

[ iy iy "h>

Rr +p Lr P M r y P Mr r P Mr t P ^ R y p M R h

P M y r

Ry+pLy pMyg pMyr pMyy pMy,

P ^ R R

pMsy

Rb+PLB

pMffr PMsy pMjsb

PMtR

pMj-y pMrB

Rr ^~pLr P-^ry

pMTb

pMyR pMyy

P M y R pMyr Ry+pLy pMry

P^bR pMby pMbB pMbr PMby

Rb

+ p h

The v o lta g e e q u a tio n c o n s is ts o f s ix c o m p o n e n t e q u a tio n s , one f o r e a c h o f th e th r e e p h a s e s ta t o r a n d r o t o r w in d in g s . The p re s e n c e o f th e d iffe r e n tia l o p e r a to r " p " m a k e s th is in to a s y s te m o f s im u lta n e o u s d if fe r e n t ia l e q u a tio n s . The d iffe r e n t ia l o p e r a to r o p e ra te s o n th e p r o d u c ts o f v a r io u s m u tu a l in d u c ta n c e s a n d c u r r e n ts , b o th o f w h ic h a re fu n c tio n s o f tim e . T h is is so b e c a u s e th e m u tu a l in d u c ta n c e s b e tw e e n th e s t a t o r a n d r o t o r w in d in g s a re d e p e n d e n t o n th e r o t o r a n g le a n d h e n c e o n tim e . The m a in o b je c tiv e o f th e p a s s iv e tr a n s fo r m a tio n s d e s c rib e d n e x t, is to s im p lify a n d e lim in a te th e d e p e n d e n c e o f th e a b o v e m a t r ix u p o n th e r o t o r a n gle -i?. F o r th e p u rp o s e o f s im p lic ity th is is do ne in tw o s te p s , a lth o u g h i t c a n be a c c o m p lis h e d in a s in g le s te p .

T he f i r s t s te p is to a p p ly th e p h ase tr a n s fo r m a tio n Cj to b o th th e r o t o r a n d s ta to r w in d in g s . The p h a s e tr a n s fo r m a tio n C1 c o rre s p o n d s m a th e m a tic a lly to a c h a n g e o f v a ria b le s . T he a c tu a l m a c h in e v o lta g e s a n d c u r r e n ts a re re p la c e d b y s u ita b ly s c a le d , m o re c o n v e n ie n t, f ic t itio u s v o lta g e s a n d c u r r e n ts . The r e s u lt o f th is t r a n s fo r m a tio n is to re p la c e th e a c tu a l th r e e p h ase m a c h in e by its

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155

e q u iv a le n t fu n d a m e n ta l s lip - r in g m a c h in e d e s c rib e d in S e c tio n 5 .2.

The p h a se tr a n s fo r m a tio n Cl c o n s id e ra b ly s im p lifie s th e s y s te m e q u a tio n s . H ow e ve r, i t does n o t e lim in a te th e depe n d e nce o f th e im p e d a n c e m a tr ix u p o n th e r o t o r a n g le The e lim in a tio n o f th e r o t o r a n g le i? fr o m th e im p e d a n c e m a t r ix is th e essence o f th e g e n e ra liz e d m a c h in e th e o r y , a n d is a c c o m p lis h e d b y th e a p p lic a tio n to th e r o t o r w in d in g s o f a n o th e r p a s s iv e tr a n s fo r m a tio n . T h is s e c o n d tr a n s fo r m a tio n C2, c a lle d th e c o m m u ta to r tr a n s fo r m a tio n , is a r o t o r a n gle d e p e n d e n t tr a n s fo r m a tio n . I t a c c o m p lis h e s th e e lim in a tio n o f th e r o t o r a n g le fr o m th e im p e d a n c e m a t r ix b y re p la c in g th e m a c h in e v a ria b le s in to y e t a n o th e r s e t o f n ew v a ria b le s . In p h y s ic a l te rm s , th e o r ig in a l m a c h in e is re p la c e d b y its e q u iv a le n t c o m m u ta to r m a c h in e d e s c rib e d in S e c tio n 5.3.

A lth o u g h th e v o lta g e s as w e ll th e c u r r e n ts o f th e m a c h in e a re tr a n s fo r m e d in to new v a ria b le s , th e s e in d iv id u a l tr a n s fo r m a tio n s a re n o t in d e p e n d e n t o f e a c h o th e r , sin ce th e p o w e r in th e tr a n s fo r m e d d e s c r ip tio n has to be th e sam e as th e p o w e r in th e o r ig in a l d e s c rip tio n . I t is seen b e lo w th a t, w hen th e tr a n s fo r m a tio n is u n it a r y , th e p o w e r in v a ria n c e c o n d itio n lea d s to id e n tic a l tr a n s f o r m a tio n f o r b o th v o lta g e s a n d c u r r e n ts . The p rim e d a n d u n p rim e d s y s te m o f e q u a tio n s b e lo w r e p r e s e n t th e tr a n s fo r m e d a n d o r ig in a l s y s te m o f e q u a tio n s re s p e c tiv e ly .

V = Z I (5 .1 9 )

156

V - Z 'F (b.Su)

L e t

V = Cv V (5 . 2 1 )

/ = Q / ( 5 . 2 2 )

The c o n v e n ie n t fo r m o f r e p re s e n tin g p o w e r in m a t r ix f o r m is g ive n b y

P = V'>TI (5.23)

w h e re th e s u p e rs c rip ts * a n d T r e p r e s e n t th e tra n s p o s e a n d c o m p le x c o n ju g a te re s p e c tiv e ly .

P' = V ‘ Tr (5.24)

E q u a tin g P and P ', we g e t

V*TI - V’ T c ^ T C i l (5.25)

T h is lea d s to

Cjj*TCi = In ( I d e n t i t y m a t r i x o f o r d e r n ) (5.26)

CV* T = C r 1 (5.27)

When th e tr a n s fo r m a tio n s a re re a l, if Cv a n d Q a re b o th id e n tic a lly e q u a l to C, th e n

C T = C~l (5.28)

The phase tr a n s fo r m a tio n Cl a n d th e c o m m u ta to r tr a n s fo r m a tio n C2

b o th being p o w e r in v a r ia n t tr a n s fo r m a tio n s , possess th e above p r o p e r ty .

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157

The p h a s e tr a n s f o r m a t io n Cx a n d th e c o m m u ta to r tr a n s f o r m a t io n Cs a re g iv e n in Eqs. (5 .4 .1 2 ) a n d (5 .4 .1 3 ).

Cx = V27T3

1 / V 2 1 / V 2 1 /V 2 1 - 1 / 2 - 1 / 2 0 V 3 / 2 - V3/ 2

h 0 O '

C2 = 10 sin i? cos 'i?

[Q c o s '& —s in iS

(5 .2 9 )

(5 .3 0 ) T h e tr a n s f o r m e d im p e d a n c e m a t r ix is g iv e n b y

Z = (5 .3 1 )

A p p e n d ix 1 g iv e s in g r e a t e r d e ta il h o w th e s e t r a n s fo r m a tio n s w h e n a p p lie d to th e v a r io u s m a c h in e s le a d to th e d e s ire d s im p lific a tio n s . F r o m th e g e n e r a l e q u a tio n s b r o u g h t o u t in A p p e n d ix 1, th e p e r fo r m a n c e e q u a tio n s a n d e q u iv a le n t c ir c u it s o f a fe w ty p e s o f r o t a t in g e le c t r ic a l m a c h in e s a re d e riv e d in th e fo llo w in g s e c tio n s .

5 .5 DC M a c h in e

In th is s e c tio n , th e p e r fo r m a n c e e q u a tio n s d e riv e d f o r th e g e n e r a l c o m m u t a to r m a c h in e in th e A p p e n d ix 1 a re u s e d as a s t a r t in g p o in t to a r r iv e a t th e e le c t r ic a l e q u iv a le n t c i r c u i t o f a s e p a r a te ly e x c ite d dc m o t o r d riv e .

F o r th e g e n e r a l tw o p h a s e c o m m u ta to r m a c h in e th e v o lta g e e q u a tio n s ( ta k e n f r o m E q. (A.7) o f A p p e n d ix 1) a re g iv e n b y

158

VA Ra + LaP ⁰ Ma p 0 _{u )}

V B 0 Rb + LqP ⁰ M gp ZB^.

Vd Ma p —MqV>t Rd +£dP ~L q U r u

. M AUr M gp Rq + L qp

The le a d in g d ia g o n a l te r m s a re th e v o lta g e d ro p s due to re s is ta n c e s a n d s e lf in d u c ta n c e s . The te rm s c o n ta in in g M p te r m s a re th e in d u c e d v o lta g e s due to th e m u tu a l c o u p lin g a n d d z /d t. These te rm s a re c a lle d th e tr a n s fo r m e r v o lta g e s . The te r m s c o n ta in in g uT a p p e a r o n ly o n r o t o r v o lta g e s a n d a re due to r a t e o f c h a n g e o f flu x lin k a g e due to r o t a t io n . These v o lta g e s a re te r m e d as r o t a r y v o lta g e s o r g e n e r a to r v o lta g e s .

T he v a r io u s in d u c ta n c e te r m s m a y be m e a s u re d in a p h y s ic a l m a c h in e b y e x c itin g one w in d in g a t a tim e , a n d m e a s u rin g th e v o lta g e s a p p e a rin g in th e o t h e r w in d in g s as a r e s u lt o f tr a n s f o r m e r a c tio n a n d r o t a t io n r e s p e c tiv e ly .

T he above r e s u lt f o r th e g e n e r a l m a c h in e m a y n o w be a p p lie d f o r th e p a r t ic u la r case o f th e s e p a r a te ly e x c ite d dc m o to r s h o w n in F ig. 5.7. The o p e r a tio n o f th e dc m o to r is as fo llo w s . The fie ld c u r r e n t iA in th e fie ld w in d in g s e ts u p a fie ld MMF in lin e w ith th e fie ld w in d in g . The fie ld MMF is s ta tio n a r y in sp a ce . The a rm a tu re c u r r e n t zq se ts u p a n a r m a tu r e MMF s t a tio n a r y in spa ce and in lin e w ith th e b ru s h e s . The fie ld MMF a n d th e a r m a t u r e MMF a re o r th o g a n a l to e a c h o th e r . The m e c h a n ic a l to r q u e p ro d u c e d is p r o p o r t io n a l to th e p r o d u c t o f th e fie ld a n d th e a r m a tu r e MMF’s.

C o m p a rin g F ig. 5.7 w ith th e g e n e ra l m a c h in e g iv e n in Fig. 5.5, we fin d t h a t th e fie ld w in d in g c o rre s p o n d s to th e s t a t o r w in d in g A. The a r m a tu r e w in d in g c o rre s p o n d s to th e q u a d r a tu r e a x is w in d in g q. No

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159

o th e r w in d in g s a re p re s e n t. The s im p lifie d v o lta g e e q u a tio n s a re

vA Ra + ⁰ 'i-A

V g .

MAar Rq^-rLqp (5.33)

The n e x t p e rfo rm a n c e e q u a tio n is t h a t o f to r q u e . The to r q u e is o b ta in e d b y a p p ly in g th e c o n s e rv a tio n o f e n e rg y . The v o lta g e e q u a tio n m a y be s e p a ra te d in to re s is tiv e d ro p s , in d u c e d tr a n s f o r m e r v o lta g e s a n d r o t a r y v o lta g e s .

V = R I+ L p I+ G v TI (5.34)

Ra ⁰ La ⁰ ro Ma

R =

0 Rq \L =

0 Lq C5 II o

The in p u t p o w e r is g iv e n b y

I TV = I rR I+ IrL p I+ItGut I (5.35)

T he RHS o f E q.(5 .3 5 ) g ive s th e p o w e r flo w th r o u g h th e m a c h in e . The f i r s t t e r m gives th e c o p p e r losse s. T he s e c o n d t e r m is th e r a te o f c h a n g e o f s to re d m a g n e tic e n e rg y . The la s t t e r m re p re s e n ts th e p o w e r t h a t is c o n v e rte d in t o m e c h a n ic a l o u tp u t, w h ic h is th e p r o d u c t o f to r q u e a n d a n g u la r v e lo c ity o f th e s h a ft. The a n g u la r v e lo c ity ur b e in g a s c a la r, th e to rq u e is g iv e n b y

T = f?GI = MAi Ai q (5.36)

The to rq u e g e n e ra te d b y th e m a c h in e d riv e s th e m e c h a n ic a l s y s te m c o u p le d to th e m o to r . The d y n a m ic s o f th e m e c h a n ic a l s y s te m is g iv e n b y

MAiAi q - JdaT/d t + B u r + T i (5.37)

w h e re , J = m o m e n t o f in e r tia ; B = f r ic t io n ;

160

Fig. 5 .7 A p r a c tic a l s e p a ra te ly excited dc m ach ine. The s ta to r c a rrie s the fie d w in d in g A. The ro to r is supppied through com m utato r. The s ta to r an d the ro to r M M F ’s are orthogonal to each other.

Fig. 5 .8 The e le c tric a l e q u iva e n t c irc u it o f the s e p a ra te ly excited dc m achine. The m e c h a n ic a l subsystem o f the m ach ine is represented by its analog e le c tric a l eq u ivalen t.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

161

Tl = lo a d to r q u e .

The c o m p le te s e t o f e le c t r ic a l a n d m e c h a n ic a l e q u a tio n s m a y be p u t c o n v e n ie n tly in t o th e e le c t r ic a l e q u iv a le n t c i r c u i t sh o w n in Fig. 5.8.

5 .6 SymchroiaoTms M a c h in e

In th is s e c tio n , th e p e r fo r m a n c e e q u a tio n s d e riv e d f o r th e g e n e ra l s lip - r in g m a c h in e (F ig . A . l ) in th e A p p e n d ix 1 a re u s e d as a s ta r tin g p o in t to a r r iv e a t th e p e r fo r m a n c e e q u a tio n s a n d e le c t r ic a l e q u iv a le n t c i r c u i t f o r th e s y n c h r o n o u s m a c h in e .

The g e n e ra l c o m m u t a t o r e q u iv a le n t o f th e th r e e p h a se s a lie n t p o le m a c h in e c o n s id e r e d in th e A p p e n d ix 1 is g o v e rn e d b y th e (ta k e n f r o m Eq. (A.7) o f A p p e n d ix 1) fo llo w in g e q u a tio n .

R

+L

&

⁰

M

ap ^o _V

R

+ L&

p ⁰

Mgp i-B M

-M

ur Ra +Ldp -Lqur Vi. M

aut

M

P Ldcjr Rq

⁺^-

Lqp

■V

The v a r io u s s e lf a n d m u tu a l in d u c ta n c e te r m s a re f ic t itio u s a n d a re r e la te d to th e o r ig in a l m a c h in e p a r a m e te r s b y th e r e la tio n s h ip s g iv e n in A p p e n d ix 1. A lte r n a tiv e ly th e y c a n a ls o be d ir e c t ly m e a s u re d i f th e tr a n s fo r m e d q u a n titie s a r e d ir e c t ly c o n tr o lla b le a n d m e a s u ra b le b y a p p r o p r ia t e c ir c u it s f r o m th e p h y s ic a l v a ria b le s

( v T,vy ,vb ,'i7., i y , i b) o f th e m a c h in e .

N ow th e a b o v e r e s u lt s m a y be a p p lie d to th e p a r t ic u la r case o f a th r e e p h a se s y n c h ro n o u s m o t o r d r iv e s h o w n in Fig. 5 .9. The r o t o r c a r r ie s b a la n c e d th r e e p h a s e w in d in g s (r , y , b) a n d e x c ite d b y

162

THREE

Dalam dokumen SLIDING MODE CONTROL OF PO"WER CONVERTERS (Halaman 153-168)