D I S P E R S I O N O N T H E P H A S E F L U C T U A T I O N S O F

M U T U A L L Y S Y N C H R O N I Z E D O S C I L L A T O R S T H A T

A R E C O U P L E D V I A A R E S I S T A N C E

V . I . K a n a v e t s a n d A . Y u . S t a b i n i s UDC 621.373

It fo l lows f r o m a g e n e r a l c o n s i d e r a t i o n of mu tua l s y n c h r o n i z a t i o n of o s c i l l a t o r s in the p r e s e n c e of no i se tha t o p t i m a l coup l ings e x i s t which l e a d to a n a r r o w i n g of the s p e c t r a l l ine of the o s c i l l a t i o n s [1, 2]: Le t u s show tha t such o p t i m a l cond i t ions a r e r e a l i z e d in a s y s t e m of o s c i l l a t o r s tha t a r e coup led v ia r e s i s - t a n c e s . We s h a l l a s s u m e tha t the o s c i l l a t o r s a r e s u b j e c t e d to n a t u r a l no i s e . Le t us c o n s i d e r a s y s t e m of N i d e n t i c a l coup led o s c i l l a t o r s tha t a r e d e s i g n e d a c c o r d i n g to the p a r a l l e l - t u n e d - c i r c u i t con f igu ra t i on . The g e n e r a l i z e d equ iva l en t c i r c u i t i s d i s p l a y e d in F i g . 1, w h e r e G n i s a n o n l i n e a r conduc tance ; Gn = G( / ) + G(al) U2n; r i s the coupl ing r e s i s t a n c e ; U n i s the vo l t age a c r o s s an i nd iv idua l o s c i l l a t o r . The o p e r a t i o n of the s e l f - e x c i t e d o s c i l l a t o r s can be d e s c r i b e d by the s y s t e m of equa t ions

N

r C ~). + ~'o 0. + ~ U~. U. + ,~ U,, = ~ Z ~)" + ~e~(t), (i) m=1 (m~n)

where s 0 = G(/)/C, ~2 = 3G(Ul)/C, c~ c = i/Cr, o)2 = I/LC, En(t ) is the "white noise" acting on the n-th oscil- lator, n = 1, 2, 3, . . . , N.

Le t us f ind the s t a t i o n a r y o p e r a t i n g mode wi thout a l l o w a n c e fo r the no i s e . Le t us a s s u m e tha t a p h a s e d o p e r a t i n g m o d e at the f r e q u e n c y w 0 h o l d s . We s e e k the so lu t i on in the f o r m U n = a n cos (w~t + ~n), w h e r e a n and go n a r e quan t i t i e s which v a r y s l o w l y d u r i n g a p e r i o d of o s c i l l a t i o n i f I~0[ << w0, 1~21 a2a << COo, ~ c << w0, ]Enl << an . We ob ta in the fo l lowing a b r i d g e d equa t ions :

m§ (2) ~C

~" - 2 ann ~ am sin ( ~ - ~,), r~

The g e n e r a t i o n of o s c i l l a t i o n s i s p o s s i b l e when the cond i t ions s 0 < 0, a2 > 0 a r e fu l f i l l ed . Le t us u se the subs t i t u t i on a 2 = 4 Ic~ 0 l / a 2 , and l e t us i n t r o d u c e the d i m e n s i o n l e s s qua n t i t i e s t ' = coot , fll = I s 0 W2co0, fl0 = ~c/2W0, Yn = an /ao . H e r e a0 i s the s t a t i o n a r y a m p l i t u d e of the o s c i l l a t i o n of an i s o l a t e d o s c i l l a t o r ; fit and fl0 a r e the l o s s and coupl ing coe f f i c i en t s ; fit << 1, fl0 << 1. Dropp ing the p r i m e s , we ob ta in

m §

~o (3)

m

In the s y s t e m t h e r e e x i s t s a s t a t i o n a r y o p e r a t i n g mode wi th i d e n t i c a l Yn and ~o n fo r a l l o s c i l l a t o r s . I n v e s t i g a t i o n s showed tha t the o p e r a t i n g mode i s s t a b l e , wh i l e the s t a t i o n a r y va lue of the a m p l i t u d e i s equa l to Yst = ~/1 + fl0 ( N - 1 ) / f l i N . The s y n c h r o n o u s p h a s e d o p e r a t i n g m o d e of the o s c i l l a t i o n i s the r e s u l t of coupl ing the o s c i l l a t o r s v ia r e s i s t o r s .

MOscow State U n i v e r s i t y . T r a n s l a t e d f r o m I z v e s t i y a V y s s h i k h Uchebnykh Z a v e d e n i i , R a d i o f i z i k a , Vol . 15, No. 8, pp . 1264-1267, Augus t , 1972. O r i g i n a l a r t i c l e s u b m i t t e d A p r i l 19, 1971.

o 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g:est 17th Street, New York, N. Y. 10011. :Vo part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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[ ]r @r [

~.c, i uNL! Fig. I

F

Let us in t roduce the subs t i tu t ion of va r i ab l e s Yn =Ys t (1 + Xn). We r e s t r i c t our cons ide ra t ion to f a i r l y sma l l coupl ing tic~ill << I and s m a l l r e l a t i ve devia t ions of the ampl i tude x n << 1. Reta in ing the t e r m s having an ident ica l o r d e r of s m a l l n e s s , we obtain

�9 6o N - - i x,, = - 2 6: ~. + ~- Y, ~o~ ('e,,, - +,,) - ~o ~ ,

m#n (4)

�9 60 ~. = ~ y , ~ i . (+,,, - +.) m

ins tead of (3). In t roduc ing the noise Ea ( t ) , which is such that , a s p r e v i o u s l y , the condit ion x a << 1 is fu l - f i l led, into cons ide ra t i on and us ing the p r o c e d u r e developed in [3], we obtain the ab r idged equa t ions :

~o - - 1 1

m#n �9 ~o _ 1__ ~ ( 5 ) %t = ~ " m~ s in ( ~ m - - ~ n ) 2no [ In '

w h e r e x n has the mean ing of r e l a t i ve f luc tua t ions of the ampl i tudes ; E ,n( t ) = e, n cos ~v n + e• a s in ~vn; E • = e • ~ o n - e t n s i n ~on; eft n and e • a r e r e l a t e d to E n ( t ) [3].

Le t us l imi t our c o n s i d e r a t i o n to phase f luc tua t ions while neglec t ing ampl i tude f luc tua t ions . Apply ing the me thod of s t a t i s t i c a l equiva lence of the equat ions fo r the p h a s e s , the s y s t e m (5) m a y be r e p l a c e d by the s y s t e m of equa t ions

,e,, = ~ - ~ ~., (~,. - +,,) + ~,, (,~ = L 2 ..... N). (6) //l

where ~ n = -(1/2ao)eun a r e n o r m a l l y d i s t r ibu ted r a n d o m quant i t ies with the c o r r e l a t i o n funct ions ~ = D6 (T).

<<1 . Le t us c o n s i d e r the s t rong synch ron iza t ion mode in which the m e a n - s q u a r e devia t ion is ( ( ~ m - ~n) 2) Equa t ions (6) can be s impl i f i ed :

~. = to

m

~m -- in = -- $0 (,~m -- %) + ~,n - ~n. (7)

In t eg ra t ing the l a t t e r equat ions , it m a y be shown that the condi t ions for s m a l l n e s s of ((~0 m - qn) 2) << 1 is given by the inequal i ty D / • << 1. F o r the phase of an i nd i~dua l o sc i l l a to r ~o n we obtain the fol lowing r e - sul t on the a s s u m p t i o n that ~ ~m(0) = 0:

tn t

6o

m 0 t

We use the subs t i tu t ion ~?(t) = ~n(t) + (ri0/N) ~ S ~mdt" The s t a t i s t i ca l s t r u c t u r a l funct ion is equal to m 0

d ~ [ O ; , l =D~(O)- - ID~(r - O) :1 { 1 o (N < IO[)"

U s i n g t h e re la t ionsh ip d~v[0; T ] = (1 / 2 ) (~o2 )= S S exp [-fl0 (u + v ) ] d ~ [ u - v , T]dudv p r e s e n t e d in [3], we u 0

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obtain the following express ion for the condition ri0[T[ >> 1 and when quantit ies of o rder D/rio a re neglected:

<~,~)_"i:l N (9)

Thus, the intensi ty of the phase f luctuations of an individuaI osc i l la tor is dec reased by a fac tor of N. In this express ion the quantity D/N is the coefficient of phase diffusion. For a f ree osc i l la tor it is equal to the quantity D. When amplitude fluctuations a r e neglected, the width of the spec t ra l line of the osc i l la tor is propor t ional to this coefficient [3]. Thus, the width of the spec t r a l line is l ikewise dec rea sed by a fac tor of N.

Let us consider the t r ans fo rma t ion of the spec t rum using the example of two osc i l la tors without l imi t - ing ourse lves to the condition ( ( 9 m - 9n) 2 ) << 1. The phase equations have the f o r m

~1 = ~t + "~ sin ( ~ t - - ,%), (lO)

~2 = ~2 - - ~'~ sin (,~ - - ~1 ) .

t ]

Let us introduce the functions 9 = 92 - 91 and r = 92 + 91. They a re de te rmined by the equations

= ~ , - - ~ l - - ~o sin V,

+ = h + ~,. (11)

Since ( 9 r = 0, we obtain ( 9 ] ) = (922 ) = 1/4{ (r + (9~)}, whence

Dl~l 1 < ~ > = ~ + ~ - < Y' >. (12)

In the case ri0/D >> 1 we have the following r e su l t s on the bas i s of using the express ion for the diffusion coeff icients der ived in [3] (equation 9.2.43):

< ~ > = 4 *t~ o exp ('-- 2 ~o/n) [ ~ I (13) and

( ~ ) = ~l '~I, (14)

where ~ = 1 + (2~ri0/D) exp (-2ri0/D). The mul t ip l ie r ~ de te rmines the inc rease of the phase diffusion coef- f icient 9t for a la rge but finite value of the ra t io ri0/D.

In the other ex t r em e case rio/D << 1 we have the following re su l t s using the cor responding express ion for the diffusion coefficient in [3] (equation 9.2.23):

D,/ I'~ ! (i5)

and

< ~ > = DI*]'~I, (16)

where # = 1 - (1/4)(fl~/D 2 ). Here the mul t ip l ie r /* shows how the phase diffusion coefficient of a f r ee osc i l - l a tor d e c r e a s e s when weak coupling is introduced.

The narrowing of the line in a s y s t e m of coupled osc i l l a to rs which a re subjected to noise may be t r ea ted as the resu l t of nonlinear f requency modulation. In this sense it is analogous to spec t rum conve r - sion in a s y s t e m of coupled osc i l l a to r s , which was cons idered by Wiener [4].

L I T E R A T U R E C I T E D

1. I . M . Klibanova, A. N. Malakhov, and A. A. Mal ' t sev , Izv. VUZ., Radiofizika, 14, No. 2, 173 (1971). 2. A . N . Malakhov and A. A. Mal ' t s ev , Dokl. Akad. Nauk SSSR, 196, No. 5, 1065 (1971). 3. A . N . Malakhov, Fluctuations in Se l f -Osc i l la tory Sys tems [in Russian] , Nauka, Moscow (1968). 4. N. Wiener , Nonlinear P rob l ems in the Theory of Random P r o c e s s e s [Russian t rans la t ion] , IL, Mos-

cow (1969).

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