i v i e r L a b o r a t o i r e de P h y s i q u e e t M 6 t r o l o g i e des O s c i l l a t e u r s du CNRS associe a 1 ' U n i v e r s i t C de Franche-Comt6 Besancon 32, Avenue de 1 'Observatoi r e -25000 Besancon -France and F. L. Walls National Bureau o f Standards -Time and Frequency Division ' Boulder, Col orado 80303 Summary Frequency and phase noise i n q u a r t z c r ys t a l r e s o n a t o r s a r e s t u d i e d as a f u n c t i o n o f t h e d r i v i n g power. A t low power, where the crystal behaves l i n e a r l y , l / f f l u c t u a t i o n s o f t h e r e s o nance freuqency are observed. A t medium power the n o n l i n e a r i t i e s o f t h e c r y s t a l s i g n i f i c a n t l y 'increase the phase f l u c t u a t i o n s a t l o w F o u r i e r frequencies. A t high power, thermal instabili t i e s and c h a o t i c b e h a v i o r o c c u r c h a r a c t e r i z e d b y t h e g e n e r a t i o n o f h i g h l e v e l w h i t e n o i s e . I n t r o d u c t i o n The s h o r t and medium term frequency stabili t i e s o f q u a r t z c r y s t a l o s c i 1 l a t o r s a r e 1 i m i t e d b y a d d i t i v e and parametric noises o f t h e e l e ct r o n i c s and b y p a r a m e t r i c n o i s e o f t h e q u a r t z r e s o n a t o r . ' I n an o s c i l l a t o r it i s n o t always p o s s i b l e t o d i s t i n g u i s h b e t w e e n t h e s e d i f f e r e n t c o n t r i b u t i o n s and thereby determine the level o f ' f r e q u e n c y f l u c t u a t i o n s o f the resonator. A s i g n i f i c a n t advance i n s o l v i n g t h i s p r o b l e m came w i t h t h e i n t r o d u c t i o n o f a new technique t o t e s t q u a r t z c r y s t a l s i n d e p e n d e n t l y o f t h e o s c i l l a t o r e l e c t r o n i c s . T h i s e n a b l e d measurements o f f l i c k e r and random walk frequency noises o f v a r i o u s p a i r s o f c r y s t a l s a t d i f f e r e n t f r e q u e nc i e s and o f d i f f e r e n t rigi ins.''^ A second advance on t h e same problem came with t h e i n t r o -CH1957-0/83/0000 0218 $1.00 @ 1983 IEEE 218 d u c t i o n o f t h e c o n c e p t o f a dual crystal system, i n which a q u a r t z o s c i l l a t o r w i t h good s p e c t r a l p u r i t y i s phase locked t o a p a s s i v e q u a r t z r e s o n a t o r . 4 I n t h i s a p p r o a c h , t h e f r e q u e n c y s t a b i l i t y i n medium and long term i s dominated by the passive quartz resonator and hence i t s s t a b i l i t y becomes observable. Such measurement systems can be used f o r comparing quartz crystal i n s t a b i l i t i e s and f o r t r y i n g t o u n d e r s t a n d t h e o r i g i n s and t h e mechanisms of these frequency f l u c t u a t i o n s . The p r e v i o u s measurement^^'^ have shown that the resonance frequency o f quartz resonat o r s e x h i b i t s f l i c k e r ( l / f ) f l u c t u a t i o n s and random walk ( l / f ) f l u c t u a t i o n s . Random walk frequency noise has been shown t o be c o r r e l a t e d t o t ...
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