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    E l e c t r ic a l a n d M e c h a n i c a l P a s s iv e N e t w o r kS y n t h e s i sM i c h a e l Z . Q . C h e n l ' 2 a n d M a l c o l m C . S m i t h 11 Dep ar tm en t o f Enginee r ing , Unive rs i ty of Cam br idge , U .K.

    z c 2 1 4 O c a m , a c . u k / m z q c h e n 9 c o m , m c s 9 c a m . a c . u k2 D e p a r t m e n t o f E n g i n e e r i n g , U n i v e r s it y o f L e ic e st e r, U . K .

    S u m m a r y . T he c on t e x t o f t h is pa pe r is t he a pp l i c a ti on o f e l ec t ri c al c ir c u it s yn t he -s is to problems of mech anica l cont ro l. Th e use of the e l ec t ri ca l -mechan ica l an a logyand the iner ter mechanical e lement i s br ief ly reviewed. Class ical resul ts f rom pass ivene twork synthes i s a re surveyed inc lud ing Brune ' s syn thes i s , Bot t -Duf f in ' s p rocedure ,Da r l ing ton ' s syn thes i s, min im um reac tan ce ex t rac t ion and the synthes i s o f b iqu adra t i cfunc t ions . New resu l t s a re presen ted on the syn thes i s o f b iquadra t i c func t ions whichare rea l i s ab le us ing two reac t ive e l ement s and no t rans formers .

    1 I n t r o d u c t i o nP a s s i v e n e t w o r k s y n t h e s i s i s a c l a s s i c a l s u b j e c t i n e l e c t r i c a l c i r c u i t t h e o r y w h i c he x p e r i e n c e d a " g o l d en e ra " f r o m th e 1 9 30 s t h r o u g h t o t h e 1 9 60 s. R e n e w e d i n t e r e s ti n t h i s s u b j e c t h a s r e c e n t l y a r i s e n d u e t o t h e i n t r o d u c t i o n o f a n e w t w o - t e r m i n a l e l-e m e n t c a ll e d th e i n e r t e r a n d t h e p o s s i b i l i t y t o d i r e c t l y e x p l o i t e l e c t r i c a l s y n t h e s i sr e s u l t s f o r m e c h a n i c a l c o n t r o l [ 38 ]. A p p l i c a t i o n s o f t h i s a p p r o a c h t o v e h i c le su s -p e n s i o n [ 39 , 3 0], c o n t r o l o f m o t o r c y c l e s t e e r i n g i n s t a b i l i ti e s [ 1 9, 2 0] a n d v i b r a t i o na b s o r p t i o n [3 8 ] h a v e b e e n i d e n ti f ie d .

    D e s p i t e t h e r e l a ti v e m a t u r i t y o f t h e f ie ld , t h e r e a r e a s p e c t s o f p a s s iv e n e t w o r ks y n t h e s i s w h i c h c a n b e c o n s i d e r e d a s i n c o m p l e t e . F o r e x a m p l e , t h e q u e s t i o n o fm i n i m a l i t y o f r e a l i s a ti o n i n t e r m s o f t h e t o t a l n u m b e r o f e l e m e n t s u s e d i s f a r fr o ms o lv e d . F o r m e c h a n i c a l n e t w o r k s , e f f ic ie n c y o f r e a l i s a t i o n i s m u c h m o r e i m p o r t a n tt h a n f o r e l e c t r i c a l n e t w o r k s . A l s o , f o r m e c h a n i c a l n e t w o r k s i t i s o f t e n d e s i r a b l et h a t n o t r a n s f o r m e r s a r e e m p l o y e d , d u e t o t h e f a c t t h a t l e v e r s w i t h u n r e s t r i c t e dr a t i o s c a n b e a w k w a r d t o i m p l e m e n t . H o w e v e r , t h e o n l y g e n e r a l m e t h o d f o rt r a n s f o r m e r l e s s e l e c t ri c a l s y n t h e s i s - - t h e m e t h o d o f B o t t a n d D u f f in [7] a n d i tsv a r i a n t s [ 29 , 3 1 , 4 0 , 2 1 J - - a p p e a r s t o b e h i g h l y n o n - m i n i m a l .

    T h e p u r p o s e o f t h i s p a p e r is t o r e v i ew s o m e o f t h e b a c k g r o u n d e l e c tr i ca l c i rc u i ts y n t h e s i s t h e o r y a n d p r e s e n t s o m e n e w r e s u l t s o n t h e t r a n s f o r m e r l e s s s y n t h e s i so f a s u b - c l a s s o f b i q u a d r a t i c f u n c t i o n s .

    2 T h e E l e c t r i c a l a n d M e c h a n i c a l A n a l o g yT h e p r i n c i p a l m o t i v a t i o n f or th e i n t r o d u c t i o n o f t h e i n e r t e r i n [ 38 ] w a s t h e s y n -t h e si s o f p a s s iv e m e c h a n i c a l n e t w o r k s . I t w a s p o i n t e d o u t t h a t t h e s t a n d a r dV.D. Blondel et al. (Eds.) Recent Advances in Learning and Control, LNCIS 371, pp. 35-50, 2008.springerl ink.com (~) Springer-Verlag Berlin Heidelberg 2008

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    36 M.Z .Q . Chen and M.C. Sm i th

    f o r m o f t h e e l e c t r i c a l -m e c h a n i c a l c o r r e s p o n d e n c e s ( w h e r e t h e s p ri n g , m a s s a n dd a m p e r a r e a n a l o g o u s t o t h e i n d u c t o r , c a p a c i t o r a n d r e s i s t o r ) w a s r e s t r i c t i v e f o rt h i s p u r p o s e , b e c a u s e t h e m a s s e l e m e n t e f f e ct iv e ly h a s o n e t e r m i n a l c o n n e c t e d t og r o u n d . T o a l lo w t h e f u l l p o w e r o f e l e c t r i c a l c i r c u i t s y n t h e s i s t h e o r y t o b e t r a n s -l a t e d o v e r t o m e c h a n i c a l n e t w o r k s , i t i s n e c e s s a r y t o r e p l a c e t h e m a s s e l e m e n tb y a g e n u i n e t w o - t e r m i n a l e l e m e n t w i t h t h e p r o p e r t y t h a t t h e ( e q u a l a n d o p p o -s i t e ) f o r c e a p p l i e d a t t h e t e r m i n a l s i s p r o p o r t i o n a l t o t h e r e l a t i v e a c c e l e r a t i o nb e t w e e n t h e m . I n t h e n o t a t i o n o f F i g. 1, t h e i n e r t e r o b e y s t h e f o r c e- v e lo c i ty la wF = b (~ )l - ~ )2 ), w h e r e t h e c o n s t a n t o f p r o p o r t i o n a l i t y b i s c a l l e d th e i n e r t a n c ea n d h a s t h e u n i t s o f k i l o g r a m s a n d v l , v2 a re t h e v e l o c it ie s o f t h e t w o t e r m i n a l sw i t h v = v l - v 2. F i g . 2 s h o w s t h e n e w t a b l e o f e l e m e n t c o r r e s p o n d e n c e s i nt h e f o r c e - c u r r e n t a n a l o g y w h e r e f or ce a n d c u r r e n t a r e th e " t h r o u g h " v a r i a b l e sa n d v e l o c i t y a n d v o l t a g e a r e t h e " a c r o s s " v a r i a b l e s . T h e a d m i t t a n c e Y ( s ) i s t her a t i o o f t h r o u g h t o a c r o s s q u a n t i t i e s , w h e r e s is t h e s t a n d a r d L a p l a c e t r a n s f o r mv a r i a b l e .

    T h e m e c h a n i c a l r e a l i s a t i o n o f a n i n e r t e r c a n b e a c h i e v e d u s i n g a fl y w h e e l t h a tis d r i v e n b y a r a c k a n d p i n i o n , a n d g e a r s ( s e e F i g . 3 ). T h e v a l u e o f t h e i n e r t a n c e bis e a s y t o c o m p u t e i n t e r m s o f t h e v a r i o u s g e a r r a t io s a n d t h e f l y w h e e l' s m o m e n t

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    s122 V la r = k ( v 2 - Vl) spring77

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    F i g . 2 . C i r c u i t sy m b o l s a n d c o r r e sp o n d e n c e s w i t h d e f i n i n g e q u a t i o n s a n d a d m i t t a n c e

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    38 M.Z.Q. Chen and M.C. Sm i thd r i v in g - p o i n t i m p e d a n c e o r a d m i t t a n c e o f a n y l in e a r t w o - te r m i n M ( o n e -p o r t)n e t w o r k is p o s i t i v e - r e a l, a n d c o n v e r s e ly , ( 2 ) g iv e n any p o s i t i v e - r e a l f u n c t i o n , at w o - t e r m i n a l n e t w o r k c o m p r i s i n g r e s i s t o r s , c a p a c i t o r s , i n d u c t o r s a n d t r a n s f o r m -e r s c a n b e f o u n d w h i c h h a s t h e g i v e n f u n c t i o n a s i t s d r i v i n g - p o i n t i m p e d a n c e o ra d m i t ta n c e . B r u n e ' s c o n s t r u c ti o n b e g i n s w i th t h e Foster preamble w h i c h r e d u c e st h e p o s i t i v e - r e a l f u n c t i o n t o a " m i n i m u m f u n c t i o n " , w h i c h is a p o s i t i v e - r e a l f u nc -t i o n t h a t h a s n o p o l e s o r z e r o s o n t h e i m a g i n a r y a x i s o r i n f i n i t y a n d h a s a r e a lp a r t t h a t v a n i s h e s f o r a t l e a s t o n e fi n it e r e a l fr e q u e n c y . T h e n e x t p a r t o f t h ec o n s t r u c t i o n is t h e " B r u n e c y cl e" w h i c h e x p r e ss e s t h e m i n i m u m f u n c t i o n a s al o ss le s s c o u p l i n g n e t w o r k c o n n e c t e d t o a p o s i t i v e - r e a l f u n c t i o n o f s t r i c t l y l o w e rd e g r e e . T h e w h o l e p r o c e s s is t h e n r e p e a t e d u n t i l a d e g r e e z e ro f u n c t i o n ( r e s i st o r )i s r e a c he d .

    F o r a n u m b e r o f y e a r s fo ll o w in g B r u n e ' s p a p e r , i t w a s th o u g h t t h a t t h e t r a n s -f o r m e r s a p p e a r i n g i n t h e s y n t h e s i s o f g e n e r a l p o s i t iv e - r e a l f u n c t io n s w e r e u n -a v o i d a b l e . I t w a s t h e r e f o r e a s u r p r i s e w h e n a r e a l i s a t i o n p r o c e d u r e w a s p u b -l is h e d b y B o t t a n d D u f f i n w h i c h d o e s n o t r e q u i r e t r a n s f o r m e r s [ 7 ]. S i m i l a r t oB r u n e ' s p r o c e d u r e , B o t t - D u f f i n ' s a p p r o a c h a l s o s t a r t s w i t h t h e F o s t e r p r e a m b l et o r e d u c e t h e p o s i t i v e - r e a l f u n c t i o n t o a m i n i m u m f u n c t i o n . I t t h e n m a k e s u s eo f t h e R i c h a r d ' s t h e o r e m [ 33 ], w h i c h i s a g e n e r a l i s a t i o n o f S c h w a r z ' s l e m m a [2 6],t o e x p r e s s t h e m i n i m u m f u n c t i o n a s a l o s s l e s s c o u p l i n g n e t w o r k c o n n e c t e d t otwo p o s i t i v e - r e a l f u n c t i o n s o f s t r i c t l y l o w e r d e g r e e . T h u s t h e p r o c e d u r e g i v es t h ea p p e a r a n c e o f b e i n g w a s t e f u l i n t e r m s o f t h e n u m b e r o f c o m p o n e n t s r e q u i r e d .H o w w a s t e f u l i t i s r e m a i n s a n o p e n q u e s t i o n .

    S i n c e 1 94 9 t h e o n l y g e n e r a l s i m p l i f i c a ti o n s o f B o t t - D u f f i n ' s m e t h o d a r e j u s tv a r i a n t s o f t h e p r o c e d u r e , e .g . P a n t e l l ' s p r o c e d u r e [ 2 9] , R e z a ' s p r o c e d u r e [3 1]a n d S t o r e r ' s p r o c e d u r e [ 40 ]. A l l t h r e e v a r i a n t s w o r k b y u n b a l a n c i n g t h e b r i d g ec o n f i g u r a t i o n w i t h i n t h e l o s s l e s s c o u p l i n g n e t w o r k i n B o t t - D u f f i n ' s r e a l i s a t i o n t or e d u c e t h e n u m b e r o f e l e m e n t s i n t h e n e t w o r k f r o m s i x t o f iv e i n e a c h c y c le .L a t e r , F i a l k o w a n d G e r s t i n d e p e n d e n t l y p r o v e d a s i m i l a r r e s u l t [ 2 1 ] .

    A n i m p o r t a n t a l t e rn a t i v e p r o o f o f B r u n e ' s t h e o r e m w a s o b t a i n e d in 1 9 39 b yD a r l i n g t o n [ 16 ]. T h e r e a l i s a t io n m e t h o d e x p r e s s e d t h e p o s i t iv e - r e a l f u n c t io n a sa l o s s l e s s t w o - p o r t t e r m i n a t e d i n a s i n g l e r e s i s t o r . T h e l o s s l e s s t w o - p o r t w a sr e a li s e d u si n g t r a n s f o r m e r s a s w e ll a s i n d u c t o r s a n d c a p a c i t o r s . T h e m e t h o d w a sa ls o c a ll e d " m i n i m u m r e s i st a n c e s y n t h e s is " . C o n n e c t i o n s o f t h e m e t h o d w i t hc l a s si c a l i n t e r p o l a t i o n w e r e l a t e r i d e n t i f ie d [ 1 8] w h i c h h a v e s e r v e d t o s e t t h em e t h o d i n a g e n e r a l c o n t e x t .

    A d i f f e re n t s e t o f t e c h n i q u e s f o r p a s s i v e n e t w o r k s y n t h e s i s w a s b a s e d o n a s t a t e -s p a c e f o r m u l a t i o n [ 1] . O n e o f t h e c e n t r a l i d e a s i s " r e a c t a n c e e x t r a c t i o n " i n w h i c ht h e i m p e d a n c e is r e p r e s e n t e d a s a m u l t i - p o r t w i t h n o f t h e p o r t s t e r m i n a t e d b yi n d u c t o r s o r c a p a c i t o r s , w h e r e n i s t h e M c M i l la n d e g r e e o f t h e t r a n s f e r -f u n c t i o n .C e n t r a l t o t h e a p p r o a c h is t h e " p o s i ti v e - re a l l e m m a " w h i c h g iv e s n e c e s s a r y a n ds u f f i c i e n t c o n d i t i o n s f o r a r a t i o n a l t r a n s f e r - f u n c t i o n t o b e p o s i t i v e - r e M a s a m a -t r i x c o n d i t i o n i n t e r m s o f t h e s t a t e - s p a c e r e a l i sa t io n . T h e r e a c t a n c e e x t r a c t i o nt e c h n i q u e a p p e a r s t o h a v e o r i g i n a t e d i n a p a p e r b y Y o u l a a n d T i s si [ 4 8 ], w h i c hd e a l s w i t h t h e r a t i o n a l b o u n d e d - r e a l s c a t t e r i n g m a t r i x s y n t h e s i s p r o b l e m .

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    Elec t r i ca l and Mechanica l Pass ive Ne two rk Synthes i s 39I n t h e r e s e a r c h w o r k o n e le c t r ic a l n e t w o r k s y n t h e si s , s p e c ia l a t t e n t i o n h a s b e e n

    p a i d t o t h e b i q u a d r a t i c f u n c t i o n s [ 2 4 , 2 5 , 2 3, 3 4 , 4 4 , 4 5 , 4 2] , w h e r e t h e i m p e d a n c ei s g i ve n by

    a282 + h i8 + aoZ( s) - d2s2 + d i s + d 0 '( h i _> 0 a n d d i _> 0 ) . F o r th e b i q u a d r a t i c i m p e d a n c e f u n c t i o n t o b e p o s i t i v e r e a l ,i t is ne c e s s a r y a n d s u f f ic i e n t t o h a ve ( a x / - ~ 2~ - v / a 0d2 ) 2 _< aid1 [ 23 ]. B i q u a d r a t i cf u n c t i o n s h a v e b e e n u s e d a s a n i m p o r t a n t t e s t c as e f or t h e q u e s t i o n o f m i n i m a lr e a l i s a t i o n .

    I n [ 34 ], S e s h u p r o v e d t h a t a t l e a s t tw o r e s i s t o r s a r e r e q u i r e d f o r a t r a n s f o r m e r -l e s s r e a l i s a t i on o f a b iq u a d ra t i c m in imu m f u n c t i o n , i . e . a b i q u a d r a t i c f u n c t i o nt h a t i s m i n i m u m . ( T h i s r e s u l t w a s al s o g i v e n b y R e z a i n [ 32 ].) S e s h u a l so p r o v e dt h a t a t r a n s f o r m e r l e s s r e a l is a t i o n o f a n y m i n i m u m f u n c t i o n r e q u i r e s a t le a s tt h r e e r e a c t i v e e l e m e n t s . T h e a u t h o r w e n t o n t o p r o v e t h a t , f o r a biquadraticm i n i m u m f u n c t i o n , s e v e n e l e m e n t s a r e g e n e r a l l y r e q u i r e d , e x c e p t f o r t h e s p e c i a lc a s es Z ( 0 ) = 4 Z ( e c ) a n d Z ( e c ) = 4 Z ( 0 ) , w h i c h a r e r e a l is a b l e w i t h a f iv e - e le m e n tb r i d g e s t r u c t u r e . I n f a c t , t h e s e v e n - e l e m e n t r e a l i s a t i o n s t u r n e d o u t t o b e t h em o d i f i e d B o t t - D u m n r e a l i s a t i on s [ 29 , 40 ]. F o l l ow i ng [ 3 4 ] , i t is s u f f i c i e n t t o r e a l i s ea g e n e r a l b i q u a d r a t i c f u n c t i o n u s i n g e i g h t e l e m e n t s ( w i t h o n e r e s i s t o r t o r e d u c ea p o s i t iv e - r e a l fu n c t i o n t o a m i n i m u m f u n c t io n ) . W h e t h e r i t is n e c e s s a r y t o u s ee i g h t e l e m e n t s i s s t i l l a n o p e n q u e s t i o n .A t p r e s e n t , t h e r e e x i s t s n o g e n e r a l p r o c e d u r e f o r r e a l i s i n g b i q u a d r a t i c f u n c -t i o n s w i t h t h e l e a s t n u m b e r o f e l e m e n t s w i t h o u t t r a n s f o r m e r s . G i v e n t h e lo w e ro r d e r , i t is v e r y o f t e n t h e c a s e t h a t a c e n s u s a p p r o a c h i s u s e d t o c o v e r a ll t h e p o s -s ib le c o m b i n a t i o n s w h e n t h e n e t w o r k s t r u c t u r e o r t h e n u m b e r o f e l e m e n t s is f ix e d( e. g. a f i v e - el e m e n t b r i d g e n e t w o r k w i t h 3 r e a c t i v e e l e m e n t s ) . O n e a t t e m p t t og e n e r a li s e a ll b i q u a d r a t i c i m p e d a n c e f u n c t i o n s r e a l is a b l e w i t h o n e i n d u c t o r a n do n e c a p a c i t o r ( m i n i m u m r e a c t i v e ) w i t h o u t u s i n g a c e n s u s a p p r o a c h w a s m a d e b yA u t h [2, 3]. H e f o r m u l a t e d t h e p r o b l e m a s a t h r e e - p o r t n e t w o r k s y n t h e s i s p r o b -l e m a n d p r o v i d e d c e r t a i n c o n d i t io n s o n t h e p h y s i c a l re a l i sa b i l it y o f t h e t h r e e -p o r t r e s is t iv e n e t w o r k t h a t is t e r m i n a t e d b y o n e i n d u c t o r a n d o n e c a p a c i t o r . H i sa p p r o a c h c o m b i n es e l e m e n t s f ro m r e a c t a n c e e x t r a c t i o n a n d t r a n s f o r m e r l e s s s y n -t h e si s. H o w e v e r, it se e m s t h a t t h e r e i s n o g e n e r a l m e t h o d t o s y s t e m a t i c a l l y c h e c kt h e c o n d i t i o n s o n t h e p h y s i c a l r e a l i s a b i l i t y t h a t A u t h d e r i v e d . A l s o h i s d i r e c t u s eo f T e l l e g e n 's f o r m m e a n s t h a t s ix r e s is t o r s a r e n e e d e d [ 4 1 ] ( s e e S e c t i on 4 . 2 ) . I nS e c t i o n 4 , w e r e - c o n s i d e r A u t h ' s p r o b l e m a n d d e r i v e a m o r e e x p l i c i t r e s u l t . I np a r t i c u l a r , w e s h o w t h a t o n l y f o ur d i s si p a t iv e e l e m e n t s ( r e si s to r s o r d a m p e r s )a r e n e e d e d .

    T r a n s fo r m e r le s s S e c o n d - o r d e r M i n i m u m R e a c t a n c eS y n t h e s i s

    T h i s s e c t io n c o n s i d e r s t h e s u b - c l a s s o f b i q u a d r a t i c f u n c t i o n s r e a l i s a b l e w i t h o n es p r in g , o n e i n e r te r , a n a r b i t r a r y n u m b e r o f d a m p e r s w i t h n o l ev e r s ( t r a n s f o r m e r s ) ,

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    40 M.Z.Q. Chen and M.C. Sm ithw h i c h i s e x a c t l y t h e p r o b l e m c o n s i d e r e d b y A u t h [2, 3] u n d e r t h e f o r c e - c u r re n tan a l o g y . H e re , w e p ro v i d e a mo r e ex p l i c i t ch a r ac t e r i s a t i o n o f t h i s c l as s .

    4 .1 P r o b l e m F o r m u l a t i o nW e c o n s i d e r a m e c h a n i c a l o n e - p o r t n e t w o r k c o n s i st i n g o f a n a r b i t r a r y n u m b e ro f d a m p e r s , o n e s p r in g a n d o n e i n e rt e r . W e c a n a r r a n g e t h e n e t w o r k i n t h e f o r mo f F i g . 4 w h e r e Q i s a t h r e e - p o r t n e t w o r k c o n t a i n i n g a l l t h e d a m p e r s . W e b r i n gi n a m i l d a s s u m p t i o n t h a t t h e o n e - p o r t h a s a w e l l - d e f i n e d a d m i t t a n c e a n d t h en e t w o r k Q h a s a w e l l -d e f in e d i m p e d a n c e . A s i n th e p r o o f o f [ 36 , T h e o r e m 8 . 1 /2 ]w e can d e r i v e an ex p l i c i t f o rm fo r t h e i mp ed an ce ma t r i x . T h i s i s d e f i n ed b y

    [ X l x 4 x 5/~2 - - X 4 X 2 X 6?)3 X 5 X 6 X 3

    - . x ( I )

    w h e r e X is a n o n - n e g a t i v e d e f i n it e m a t r i x ( ^ d e n o t e s t h e L a p l a c e t r a n s f o r m ) ._ k ~2, an d el i m ina t in g ~2 an d ~?3 gives the fo l lowinge t t i n g / ~ 3 - - b s ~ 3 an d F2 -

    e x p r e s s i o n f o r t h e a d m i t t a n c eY ( s ) - F 1 b X 3 s 2 + [1 + k b ( X 2 X 3 - X 2 ) ] s + k X 2

    v l = b ( X l X 3 - X 2 ) s 2 + (X1 -t- k bd et X ) s + ~ ( X l X 2 - - 2 2 ) (2 )w h e r e d e t X - X I X 2 X 3 - X 1 X 2 - X 2 X ~ - X 3 X 2 -+ -2 X 4X 5X 6. N o t e t h a t X 1 - 0r e q u i r e s t h a t X 4 - X 5 - 0 f o r n o n - n e g a t i v e d e fi n it e n e ss w h i c h m e a n s t h a t t h ea d m i t t a n c e d o e s n o t e x is t. T h u s t h e a s s u m p t i o n o f e x i st e n c e o f t h e a d m i t t a n c ereq u i r e s t h a t X 1 > 0 .

    1

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    Fig . 4 . Three-por t damper ne twork t e rmina ted wi th one iner t e r and one sp r ing

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    Elec t r ica l and Mechan ica l Pass ive Netw ork Syn thes i s 41T h e v a l u e s o f b a n d k c a n b e s e t t o 1 a n d t h e f o l lo w i n g sc a l i n g s a r e c a r r i e d o u t :

    X l ~ R1, kX 2 --+ R2, bX3 ---* R 3 , v / k X 4 ~ R 4 , v /b X 5 ~ R 5 a n d x / ~ X 6 ~ R 6 .T h e r e s u l t i n g a d m i t t a n c e i s

    Y ( s ) - ( R I R 3 - R2)s 2 4 - (R1 4- d e t R)s 4 - ( R I R 2 - R 2 )a n d

    w h e r e

    R : = [R 1R 4R s ] [ X lx 4x s ]R 4 R 2 R 6 - T X 4 X2 X6 T ,R 5 R 6 R a X 5 X 6 X a

    (3)

    1 0 0 ]T - 0 v / k 0 ( 4)o o v~

    a n d R i s n o n - n e g a t i v e d e fi n it e. F r o m t h e e x p r e s s i o n d e t ( R ) - R I R 2 R 3 - R I R 2 -R2R~ - RaR24 + 2R4 RsR 6 , w e n o t e t h a t (3 ) d e p e n d s o n s i g n ( R 4 R s R 6 ) b u t n o to n t h e i n d i v i d u a l s i g n s o f R 4 , R 5 a n d R 6 .

    T h e r e a c t a n c e e x t r a c t i o n a p p r o a c h t o n e t w o r k s y n t h e s i s [ 1,4 8] a l lo w s t h e f ol-l o w i n g c o n c l u s io n t o be d r a w n : a n y p o s i ti v e - r e a l b i q u a d r a t i c ( i m m i t a n c e ) f u nc -t i o n s h o u l d b e r e a l i s a b l e i n t h e f o r m o f F i g . 4 f o r s o m e n o n - n e g a t i v e d e f i n i teX . I t is a ls o k n o w n t h a t a n y n o n - n e g a t i v e d e fi n it e m a t r i x X c a n b e r ea l is e da s t h e d r i v i n g - p o i n t i m p e d a n c e o f a n e t w o r k c o n s i s t i n g o f d a m p e r s a n d l e v e rs( a n a l o g o u s l y , r e s i s t o r s a n d t r a n s f o r m e r s ) [ 1 0 , C h a p t e r 4 , p a g e s 1 7 3 - 1 7 9] . W e n o we x a m i n e t h e q u e s t i o n o f t h e a d d i t i o n a l r e s t r i c ti o n s t h a t a r e i m p o s e d w h e n n ot r a n s f o r m e r s a r e a l l o w e d i n Q .4 .2 T r a n s f o r m e r l e s s R e a l i s a t i o n a n d P a r a m o u n t c yT h i s s e c t io n r e v ie w s t h e c o n c e p t o f p a r a m o u n t c y a n d i ts ro l e i n t r a n s f o r m e r l e s ss y n t h e s i s . W e a l s o s t a t e s o m e r e l e v a n t r e s u l t s f r o m [ 1 3 , 1 4 ] w h i c h w i l l b e n e e d e df o r o u r l a t e r r e s u l t s .

    A m a t r i x i s d e f i n e d t o b e p a ra m o u n t i f i t s p r i n c i p a l m i n o r s , o f a l l o r d e r s ,a r e g r e a t e r t h a n o r e q u a l to t h e a b s o l u t e v a l u e o f a n y m i n o r b u i l t f r o m t h es a m e r o w s [ 1 1 , 3 5 ] . I t h a s b e e n s h o w n t h a t p a r a m o u n t c y i s a n e c e s s a r y c o n d i t i o nf or t h e r e a l i s a b il i ty o f a n n - p o r t r e s is t iv e n e t w o r k w i t h o u t t r a n s f o r m e r s [ 11 ,3 5 ] . I n g e n e r a l , p a r a m o u n t c y i s n o t a s u f f i c i e n t c o n d i t i o n f o r t h e r e a l i s a b i l i t yo f a t r a n s f o r m e r l e s s r e s is t iv e n e t w o r k a n d a c o u n t e r - e x a m p l e f or n = 4 w a sg i v e n i n [ 1 2, 4 6] . H o w e v e r , i n [ 4 1 , p p . 1 6 6 - 1 6 8 ] , i t w a s p r o v e n t h a t p a r a m o u n t c yis n e c e s s a r y a n d s u f fi c ie n t f o r t h e r e a l i s a b i l i t y o f a r e s i st i v e n e t w o r k w i t h o u tt r a n s f o r m e r s w i t h o r d e r le s s t h a n o r e q u a l t o t h r e e ( n _< 3 ). T h e c o n s t r u c t i o no f [4 1] f o r t h e n = 3 c a s e m a k e s u s e o f t h e n e t w o r k c o n t a i n i n g s ix r e s is t o r s s h o w ni n F i g . 5 . I t i s s h o w n t h a t t h i s c i r c u i t i s s u ff ic i e nt t o r e a li s e a n y p a r a m o u n t m a t r i xs u b j e c t t o j u d i c i o u s r e l a b e l l i n g o f t e r m i n a l s a n d c h a n g e s o f p o l a r i t y . A r e w o r k i n g( i n En g l i s h ) o f Te l l eg en ' s p ro o f i s g i v en i n [1 3 ] .

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    42 M.Z.Q. Chen and M.C. Sm i thI i

    I l

    R5

    /3R2

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    R6

    h .._ h2 ' 2

    Fi g. 5 . Tel legen 's circui t for the co nstruc t ion of resist ive 3-por ts w itho ut t ransfo rm ers

    I n t h e n e x t t w o l e m m a s w e e s t a b l i s h a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o ra t h i r d - o r d e r n o n - n e g a t i v e d e f i n i t e m a t r i x[ 11 R4 R5R - R4R2R6 ( 5 )R5 R6 Rat o b e r e d u c i b l e t o a p a r a m o u n t m a t r i x u s i n g a d i a g o n a l t r a n s f o r m a t i o n . S e e[13 ,14] for the proofs .L e m m a 1 . L e t R be n o n - n e g a t i v e d e f in i te . I f a n y f i r st - o r s e c o n d - o r d e r m i n o ro f R i s z e ro , t h e n t h e r e e x i s t s a n in v e r t i b l e D - d i a g { 1 , x , y } s u c h t h a t D R D i sa p a r a m o u n t m a t r i x .L e m m a 2 . L e t R b e n o n - n e g a t i v e d e f i n i t e a n d s u p p o s e t h a t a ll f i r s t - a n d s e c o n d -o r d e r m i n o r s a r e n o n - z e r o . T h e n t h e r e e x is t s a n i n v e r t i b l e D - d i a g { 1 , x , y } s u c ht h a t D R D is a p a r a m o u n t m a t r i x i f a n d o n l y i f o n e o f t h e f o l l o w i n g h o l d s:( i ) R a R s R 6 < O ;

    R 4 R 6 R 5 R 6R4R5 R 2 > a n d R 3 > ;( i i ) R 4 R 5 R 6 > O , R 1 > R 6 , R 5 R 4R 5 R 6 a n d R I R 2 R 3 + R 4 R 5 R 6 - R I R 2 - R 2 R 2 > O;i i i ) R 4 R 5 R 6 > O , R 3 < ~R 4 R 6 a n d R 1 R 2 R 3 + R 4 R 5 R 6 - R 1 R ~ - R 3 R ~ > O;i v ) R 4 R 5 R 6 > O, R 2 < R 5R 4 R5 a n d R 1 R 2 R 3 + R 4 R 5 R 6 - R 3R 24 - R 2 R 2 > O.v ) R 4 R s R 6 > O , R ~ < R ~

    4 .3 S y n t h e s i s o f B i q u a d r a t i c F u n c t i o n s w i t h R e s t r i c t e d C o m p l e x i t yT h i s s e c t i o n d e r i v e s a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r t h e r e a l i s a b i l i t yo f a n a d m i t t a n c e f u n c t io n u s i n g o n e sp r in g , o n e i n e r te r , a n a r b i t r a r y n u m b e r

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    Elec t rica l and Mechan ica l Pass ive Netw ork Syn thes i s 43o f d a m p e r s a n d n o le v e rs ( t r a n s f o r m e r s ) ( T h e o r e m 1 ). T h e p r o o f re li es o n t h er e s u l t s o f S e c t i o n 4 .2 a n d t h e c o n s t r u c t i o n o f [ 41 ] . A s t r o n g e r v e r s i o n o f t h es u ff ic ie n c y p a r t o f t h i s r e su l t, w h i c h s h o w s t h a t a t m o s t f o u r d a m p e r s a r e n e e d e d ,i s g i v e n i n T h e o r e m 2 w i t h e x p l i c it c i r c u i t c o n s t r u c t i o n s . S i n g u l a r c a s e s a r et r e a t e d i n T h e o r e m 3.L e m m a 3. A pos i t i ve-real fun ct i on Y (s ) can be reali sed as the dr iv ing-po in t ad-m i t t ance o f a ne tw ork i n t he f o r m o f F ig. ~ , where Q has a we ll -de fi ned impedanceand i s rea l isable wi th d am pers only and b , k ~ O, i f and only i f Y (s ) can be wr i t t eni n t he f o r m o f

    R 3s 2 + [1 + (R2 R3 - R~)] s + R2Y ( s ) - ( R I R 3 - R ~ ) s 2 + (R 1 + d e t R ) s + (R IR 2 - R 2 ) ' ( 6 )

    where J 1 R 4 R 5 ]R 4 R 2 R 6R 5 R 6 R 3i s non-negat ive def in i te , and there exi s t s an inver t ib le d iagonal matr ix D =d ia g {1 , x , y } s u c h t h a t D R D i s p a r a m o u n t .Proof: ( O n l y if .) A s i n S e c t i o n 4, w e c a n w r i t e t h e i m p e d a n c e o f Q i n t h e f o r m o f( 1 ). S in c e Q is r e a l i s e d u s i n g d a m p e r s o n l y ( n o t r a n s f o r m e r s ) , w e c l a i m t h a t t h em a t r i x X i n (3 ) is p a r a m o u n t . T h e t r a n s f o r m a t i o n t o ( 3 ), as i n S e c t i o n 4 , n o wp r o v id e s t h e r e q u ir e d m a t r i x R w i t h t h e p r o p e r t y t h a t X - D R D is p a r a m o u n twh ere x = 1 / x /~ an d y = 1 /v /b .

    ( I f . ) I f we def ine k = 1 / x 2 a n d b = 1 / y 2 , t h e n X = D R D is p a r a m o u n t . U s i ngt h e c o n s t r u c t i o n o f T e l l e g e n ( s ee S e c t i o n 4 .2 , F i g . 5 ), w e c a n f i n d a n e t w o r kc o n s is t in g o f 6 d a m p e r s a n d n o t r a n s f o r m e r s w i t h i m p e d a n c e m a t r i x e q u a l to X .U s i n g t h is n e t w o r k i n p l ac e o f Q i n F i g . 4 p r o v i d e s a d r i v i n g - p o i n t a d m i t t a n c eg i v e n b y ( 2) w h i c h i s e q u a l t o ( 6) a f t e r t h e s a m e t r a n s f o r m a t i o n o f S e c t i o n 4 . mW e n o w c o m b i n e L e m m a s 1, 2 a n d 3 t o o b t a i n t h e f o l lo w i n g t h e o r e m .T h e o r e m 1 . A pos i ti ve - rea l f un c t i o n Y (s ) can be r ea li sed as the dr i v ing -po in tadm i t t ance o f a ne twork i n t he f o r m o f F ig. ~ , where Q has a we l l -de f inedimped ance and i s rea li sable wi th d am pers only an d b, k ~ O, i f and only i f Y (s )can be wr i t t en i n the f o rm o f (6 ) and R sa t i sf i e s the cond i t i ons o f e i t her L e m m a 1o r L e m m a 2.I n T h e o r e m 2 , w e p r o v i d e s p e c if ic r e a l i s a t i o n s f o r t h e Y ( s ) i n T h e o r e m 1 f o r a llc a s e s w h e r e R s a t i s fi e s t h e c o n d i t i o n s o f L e m m a 2 . T h e r e a l i s a t i o n s a r e m o r ee f fi ci en t t h a n t h e c o n s t r u c t i o n o f T e l l e g en ( se e S e c t i o n 4 .2 , F i g . 5 ) i n t h a t o n l yf o u r d a m p e r s a r e n e ed e d . T h e s i n g u l a r c a se s s a t is f y i n g t h e c o n d i t i o n s o f L e m m a 1a r e a l s o t r e a t e d i n T h e o r e m 3 .

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    4 4 M . Z . Q . C h e n a n d M . C . S m i t h

    T h e o r e m 2 . L e ty ( ~ ) - R 3 s 2 + [1 + ( R 2 R 3 - R 2 ) ] s + R 2

    ( R 1 R 3 - R ~ ) s 2 + ( R 1 + d e t R ) s + ( R 1 R 2 - R 2 ) ( 7 )where

    R ~R ] R 4 R 5 ]R4 R2 t:16R 5 R 6 R 3

    i s non- negat i ve de f in i t e and sa t i s f i e s t he cond i t ions in Lemma 2 . Then Y ( s )can be rea li sed wi th one spr ing , one ine r t e r an d fou r dam pers in t he for m o fF ig . 6 ( a ) - 6 ( e ).Proof: F i g . 6 ( a ) - 6 ( e ) c o r r e s p o n d t o C a s e s ( i ) - ( v ) i n L e m m a 2 , r e sp e c t iv e l y .E x p l i c i t f o r m u l a e c a n b e g i v e n f or t h e c o n s t a n t s i n e a c h c i r c u i t a r r a n g e m e n t .H e r e w e c o n s i d e r o n l y t h e c a s e o f F i g . 6 ( a ) .

    U LI

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    F i g . 6 . F i v e c i rc u i t a rr a n g e m e n t s o f T h e o r e m 2

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    Elec t r i ca l and Me chan ica l Pass ive Ne twork Syn the s i s 45

    I f R 4 R 5 R 6 < O, Y ( s ) c a n b e r e a l i s e d i n t h e f o r m o f F i g . 6 ( a ) w i t h1C l - - R 1 - R 4R 5 'R6R 2 ( R 2 _ R 4 R 6 )R 5

    C 3 - - ( R 1 R 6 - R 4 R 5 1 2 'b - ( R 3R 4 - R 5 R 6 )

    ( R 1 R 6 - R 4 R s ) 2 '

    ( R 3 R 4 - R 5 R 6 ) ( R 4 R 6 - R 2 R 5 )C 2 - - - d e t R ( R 1 R 6 - R n R 5 )R n ( R 3 - R 4C4 - - ( R I R 6 - R 4 R 5 ) 2 '

    k - ( R 4 R 6 - R 2 R 5 ) 2( R I R 6 - R 4 R s ) 2"

    T h e s e f o r m u l a e w e r e d e r i v e d d i r e c t l y i n [ 1 5] . T h e y c a n a l so b e c h e c k e d b y d i r e c ts u b s t i t u t i o n . S e e [ 15 ] f o r t h e p r o c e d u r e s a n d t h e e x p r e s s i o n s o f o t h e r c a se s . ( As i m i l a r p r o c e d u r e h a s a p p e a r e d i n [ 13 ,1 4 ]. ) mT h e o r e m 3 . L e t

    - R 3 s 2 + [1 + (R2 R3 - R~) ] s + R2( R I R 3 - R 2 ) s 2 + ( R 1 + d e t R ) s + ( R 1 R 2 - R 2 )w h er e R a s d e f i n ed in (5 ) i s n o n -n eg a t i ve d e f i n i te . I f o n e o r mo r e o f t h e f i r s t -o r s eco n d -o r d er m in o r s o f R i s z e r o, t h en Y ( s ) ca n be r ea li s ed w i th a t m o s t o n es p r in g , o n e i n e r t e r a n d t h r ee d a mp er s .Proo f : T h e p r o o f is o m i t t e d f o r b r e v i t y . S e e [ 1 5] f o r d e t a i l s . m4 .4 E x a m p l e o f N o n - r e a l i s a b i li t yW e n o w p r o v i d e a n e x p li c it e x a m p l e o f a b i q u a d r a t i c f u n c t i o n w h i c h c a n n o t b er e a li s e d w i t h t w o r e a c t i v e e l e m e n t s a n d n o t r a n s f o r m e r s . F i r s t o f a ll, w e n e e d t oe s t a b l i s h t h e f o l l o w i n g r e s u l t .T h e o r e m 4 . Th e p o s i t i v e - r ea l b iq u a d r a t i c f u n c t i o n

    1 a o s 2 - 4 - a l s ~ - 1Y ( s ) - - ~ . d o s2 + d i s + 1 ( s )

    ca n b e r ea li s ed i n t h e f o r m o f (3 ) , eq u i va l en t l y F ig . ~ , f o r a g i ven n o n - n eg a t i ved e f i n i t e R i f a n d o n l y i f R 2 s a t i s f i e s

    R2 >_ m a x { a 1 1 , d 1 1 , d o / ( a o d 1 ) } , (9)0

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    4 6 M .Z .Q . C h e n a n d M .C . S m i t ha n d s a t i s f y i n g

    h R 2 ( d ~ - 1 )

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    E l e c t r ic a l a n d M e c h a n i c a l P a s s i v e N e t w o r k S y n t h e s i s 4 7S u b s t i t u t i n g ( 20 ) i n to (2 4) a n d r e a r r a n g i n g t h e t e r m s , w e h a v e R 2 d e t R -h ( d l R 2 - 1 ) R 2 - R 2 , a n d t h e re f o re

    d e t R > 0 r h ( d l R 2 - 1 ) R 2 > R 2 ( 2 7 )w h i c h s h o w s t h e n e c e s s i t y o f t h e u p p e r i n e q u a l i t y i n ( 14 ) . F o r ( 2 7 ) t o h a v e as o l u t i o n i t is n e c e s s a r y t h a t R2 _> d l 1. (28)F o r t h e r a n g e d e f i n e d i n (1 4 ) t o b e n o n - e m p t y , i t is n e c e s s a r y t h a t

    R 2 > d o / ( a o d l ) . (2 9 )C o m b i n i n g ( 2 6 ), ( 2 8 ) a n d ( 2 9) g i v e s ( 9 ). mN o w w e w il l s h o w i n t h e e x a m p l e b e l o w t h a t i t is n o t a l w a y s p o s s i b le t o r e a l is ea b i q u a d r a t i c i n t h e f o r m o f F i g . 4 w i t h o u t t r a n s f o r m e r s ( le v e rs ) .E x a m p l e ( n o n -r e a l i s a b il i t y ) . C o n s i d er t h e a d m i t t a n c e f u n c t i o n

    2 s 2 + s + 1Y ( s ) - s 2 + s + l ,

    w h i c h t a k e s t h e f o r m ( 8 ) w i t h a 0 - 2 , a l - d o - d l - h - 1 . S i n c e aid1 >( x / - ~ - v ~ 0 ) 2 , Y ( s ) is p o s i ti v e - re a l . N o w w e a p p l y t h e p r o c e d u r e i n T h e o r e m 4 .B y ( 9) a n d ( 1 4 ), i t is n e c e s s a r y t o h a v e R 2 > 1 a n d

    0 < R ~ < R 2 ( R 2 - 1 ). ( 3 0 )N o t e t h a t ( 10 ) is r e d u n d a n t i n t h i s c a s e . F o r a p a r t i c u l a r R 2 , R 2 is s o l v e d b y( 11 ). T h e n R 1 , R 3 , R 5 a n d R 6 a r e d e t e r m i n e d b y ( 1 5 ) - ( 1 8 ) . T h e s o l u t i o n o f R 2f r o m ( 11 ) a n d t h e u p p e r b o u n d i n (1 4 ) a r e p l o t t e d i n F i g . 7. T h e r e f o r e , w e c a ns e e t h a t a n y R 2 s u f f i c i e n t l y l a r g e ( i n f a c t R 2 >_ 1 .5 ) g iv e s a n o n - n e g a t i v e d e f i n i t eR s a t i s f y i n g T h e o r e m 4 .

    N o w w e w o u l d l ik e t o s h o w t h a t i t is n o t p o s s i b l e t o r e a l is e t h i s a d m i t t a n c ef u n c t i o n i n t h e f o r m o f F i g . 4 w i t h o u t t r a n s f o r m e r s ( l ev e r s ). F i r s t , w e n o t e f r o m( 1 9 ) t h a t P - R 2 + ( 4 R 2 - 1 )R 4 2 > 0 fo r a ll R 2 _> 0 . T h e r e f o r e R n R h R 6 > 0 fo ra n y a d m i s s i b l e R 2 . B y s u b s t i t u t i n g f r o m ( 17 ) a n d ( 1 8) , i t is e a s y t o s h o w t h a t

    2 2R h R 6 - + 1 ) + + - 1 ) - n 4 )) > _ O ,w h i c h i m p l i e s t h a t R 3 < R h R 6 / R 4 f o r a n y a d m i s s i b l e R 2 . H o w e v e r ,

    R ~ R 2 R a + R 4 R 5 R 6 - R 1 R ~ - R 2 R ~1= 2 R 2 ( ( R 2 - 2 ) R 2 - R 2 ( R 2 - 2 R 2 + 2 ) )1< ( ( R 2 - 2 ) R 2 ( R 2 - 1 ) - R 2 ( R 2 - 2 R 2 + 2 ) )2 R 2R 2= < 0 ,2

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    w h e r e t h e f i r s t i n e q u a l i t y m a d e u s e o f ( 3 0 ) . T h e r e f o r e t h e s e c o n d c o n d i t i o n o fC a s e ( i i i ) i n L e m m a 2 f a i l s f o r a n y a d m i s s i b l e R 2 . T h e r e f o r e , Y(s) c a n n o t b er e a l i s e d in t h e f o r m o f F i g . 4 w i t h o u t t r a n s f o r m e r s ( l e v e r s ).

    5 C o n c l u s i o n sT h e t h e m e o f t h i s p a p e r i s t h e a p p l i c a t i o n o f e l e c t r ic a l c i rc u i t s y n t h e s i s t o m e -c h a n i c a l n e t w o r k s . R e l e v a n t r e s u l t s f ro m t h e f i e ld o f p a s s i v e n e t w o r k s h a v e b e e ns u r v e y e d . I t w a s p o i n t e d o u t t h a t t h e p r o b l e m o f m i n i m a l r e a l is a t i o n ( i n t e r m so f t h e n u m b e r o f e l e m e n t s u s e d ) i s s t il l u n s o l v e d , a n d t h a t t h i s is an i m p o r t a n tq u e s t i o n f or m e c h a n i c a l i m p l e m e n t a t i o n . T h e c l a s s o f b i q u a d r a t i c p o s i t i v e - r e a lf u n c t i o n s w a s h i g h l i g h t e d a s a n in t e r e s t i n g t e s t c a s e . F o r t h i s c l a s s , a n e x p l i c i tp r o c e d u r e w a s p r o v i d e d t o d e t e r m i n e i f a g i v e n f u n c t i o n c a n b e r e a l i se d w i t h t w or e a c t i v e e l e m e n t s a n d n o t r a n s f o r m e r s .

    R e f e r e n c e s1 . A n d erso n , B .D .O . , V o n g p a n i t l erd , S . : N etw o rk A n a ly s i s a n d S y n th es i s : A Mo d ern

    S y stem s T h eo ry A p p ro a ch . P ren t i ce H a l l , E n g lew o o d C l i f f s (1 9 7 3 )2 . A u th , L .V . : S y n th es i s o f a S u b c la ss o f B iq u a d ra t i c Im m it ta n ce F u n ct io n s , P h D

    thesis , U nive rsi ty o f I l l ino is , Urba na , I l l. (1962)3 . A u th , L .V . : R L C B iq u a d ra t i c D r iv in g -P o in t S y n th es i s u s in g th e R es i s t iv e T h ree-

    p o rt . IE E E T ra n s . o n C ircu i t T h eo ry , 8 2 -8 8 (1 9 6 4 )4 . B a h er , H . : S y n th es i s o f E lec tr i ca l N etw o rk s . W i ley , C h ich es ter (1 9 8 4 )

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    Electr ica l and M echanical Passive Netw ork Synthesis 49.

    6.

    10 .11.12.13.

    14 .

    15.16.17.

    18.

    19.

    20 .

    21 .22 .23 .24 .

    25 .26 .27 .

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