UBC Theses and Dissertations

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UBC Theses and Dissertations

An analysis of the Fraser River tidal control Kay, Donald Hughie James 1951

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AN ANALYSIS OF THE FRASER RIVER MODEL TIDAL CONTROL by DONALD HUG-HIE JAMES KAY A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLI-ED SCIENCE i n the Department of •ELECTRICAL ENGINEERING We accept t h i s t h e s i s as conforming to the standard required from candidates f o r the degree of MASTER OF APPLIED SCIENCE Members of the Department of ELECTRICAL ENGINEERING THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1951 A b s t r a c t The development of an a u t o m a t i c c o n t r o l system i n v o l v e s a c o n s i d e r a t i o n of t h e problems o f t h e s t a b i l i t y and response o f such a system. The purpose o f t h i s t h e s i s i s t o o u t l i n e t h e s e problems as t h e y appear i n t h e a u t o m a t i c c o n t r o l o f w a t e r l e v e l i n a t i d a l b a s i n model. A s y n o p s i s o f g e n e r a l servomechanism t h e o r y i s b r i e f l y o u t l i n e d s t r e s s i n g t h r e e p o i n t s (1) D e s i g n t h e system so t h a t o s c i l l a t o r y c o n d i t i o n s p r e v a i l . (2) D e s i g n t h e n a t u r a l f r e q u e n c y w e l l above the o p e r a t i n g f r e q u e n c y . (3) Where n e c e s s a r y i n t r o d u c e s t a b i l i z i n g n e t w o r k s . The r e s u l t s o b t a i n e d by t e s t s r e v e a l e d one t h i n g ; t h e i n t r o d u c t i o n o f e x t e r n a l c i r c u i t s had v e r y l i t t l e e f f e c t on the c o n t i n u o u s o p e r a t i o n o f t h e t i d a l model. An examin-a t i o n o f t h e t h e o r e t i c a l a n a l y s i s o f t h e c o n t r o l system however brought out t h r e e r e a s o n s f o r l i m i t i n g t h e v a l u e o f t h e con-s t a n t K 4. The c o n s t a n t K 4 c o r r e s p o n d s t o t h e moment o f i n e r t i a o f a m e c h a n i c a l system. The c o n s t a n t K 4 i s r e l a t e d t o the parameters o f the system, w h i c h a r e t h e a r e a o f t h e b a s i n , t h e l e n g t h o f t h e w e i r s , and t h e pump d i s c h a r g e . i T a b l e o f C o n t e n t s page I n t r o d u c t i o n 1 G e n e r a l Theory 2 D e s c r i p t i o n o f T i d a l C o n t r o l Equipment 8 T h e o r e t i c a l A n a l y s i s 11 T e s t s and R e s u l t s 24 D i s c u s s i o n and C o n c l u s i o n s 26 Acknowledgments 29 B i b l i o g r a p h y 30 L i s t o f Photographs f o l l o w i n g page Photo #1 West P o r t i o n o f R i v e r B a s i n 7 Photo #2 C e n t r e P o r t i o n o f R i v e r B a s i n 7 Photo #3 E a s t P o r t i o n o f R i v e r B a s i n 7 Photo #4 T i d a l B a s i n 7 Photo #5 E l e c t r o n i c C o n t r o l Equipment 8 Photo #6 H y d r a u l i c A m p l i f i e r 8 Photo #7 H y d r a u l i c J a c k 8 Photo #8 The Set o f W e i r s 8 L i s t o f I l l u s t r a t i o n s i i F i g . #1 B l o c k Diagram o f a S i m p l e Servomechanism F i g . #2 E r r o r - t i m e Curves f o r Viscous-clamped Servomechanism F i g . #3 Resonance Curves o f Servomechanism w i t h V i s c o u s Output Damping F i g . #4 B l o c k Diagram o f t h e C h a r t Reader F i g . #5 B l o c k Diagram o f T i d a l C o n t r o l System F i g . #6 Schematic Diagram o f T i d a l B a s i n page 2 f o l l o w i n g 6 f o l l o w i n g 6 f o l l o w i n g 9 13 13 F i g . #7 B l o c k Diagram o f M o d i f i e d T i d a l C o n t r o l System 16 L i s t o f Graphs f o l l o w i n g page Graph #1 Output V e c t o r Locus f o r C o n t r o l System 25 Graph #2 Servo Response w i t h I n t e r n a l Feedback C i r c u i t 25 Graph #3 Servo Response w i t h o u t I n t e r n a l Feedback C i r c u i t 25 Graph #4 Servo Response w i t h o u t D i f f e r e n t i a t i n g C i r c u i t 25 Graph #5 Servo Response w i t h D i f f e r e n t i a t i n g C i r c u i t 25 Graph #6 Step F u n c t i o n Response w i t h o u t D i f f e r e n t i a t i n g C i r c u i t 25 Graph #7 Step F u n c t i o n Response w i t h D i f f e r e n t i a t i n g C i r c u i t ' 25 I n t r o d u c t i o n The development o f an a u t o m a t i c c o n t r o l system i n v o l v e s a c o n s i d e r a t i o n o f t h e problems o f t h e s t a b i l i t y and response o f such a system. Under s t a b i l i t y , t h e system must n e i t h e r i n d u c e n o r s u p p o r t any harmonic o s c i l l a t i o n s . F u r t h e r , any o s c i l l a t i o n s t h a t a r e i n d u c e d by o u t s i d e d i s t u r b a n c e s must be a t t e n u a t e d t o a n e g l i g i b l e v a l u e i n a v e r y few c y c l e s . Under r e s p o n s e , t h e system must follow t h e i n p u t s i g n a l as a c c u r a t e l y as p o s s i b l e and w i t h a minimum t i m e d e l a y . The purpose o f t h i s t h e s i s i s t o o u t l i n e t h e s e problems as t h e y appear i n t h e a u t o m a t i c c o n t r o l o f w a t e r l e v e l i n a t i d a l b a s i n model. The servo-mechanism i t s e l f i s n o t u n l i k e t h a t used i n many o t h e r systems but t h e e q u a t i o n s i n t r o -duced by t h e h y d r a u l i c s o f the system make t h e problem u n i q u e . One o f t h e f a c t o r s i n v o l v e d i s t h e n o n - l i n e a r i t y o f t h e w e i r d i s c h a r g e e q u a t i o n . A n o t h e r i s t h e l a g i n t h e w a t e r l e v e l r e s p o nse t o w e i r movement caused by t h e f i n i t e v e l o c i t y o f a wave i n w a t e r . 1 z G e n e r a l Theory-C o n t r o l systems may be e i t h e r o f t h e open c y c l e o r of t h e c l o s e d c y c l e t y p e s . I n an open c y c l e system t h e s i g n a l t h a t o p e r a t e s t h e c o n t r o l l e r i s independent o f t h e o u t p u t . I n a c l o s e d c y c l e system a p e r c e n t a g e o f t h e o u t p u t i s f e d back and compared w i t h t h e i n p u t s i g n a l . The d i f f e r e n c e o r e r r o r t h e n o p e r a t e s t h e c o n t r o l l e r so as t o reduce t h e e r r o r . The open c y c l e system i s f a i r l y s i m p l e and w i l l not be c o n s i d e r e d i n t h i s t h e s i s . i n t o a u t o m a t i c c o n t r o l o r r e g u l a t o r systems and servomechanisms. The fundamental d i f f e r e n c e i n t h e two systems i s i n t h e i r a p p l i c a t i o n , r a t h e r t h a n the p r i n c i p l e s i n v o l v e d . The a u t o -m a t i c r e g u l a t o r i s d e s i g n e d t o m a i n t a i n t h e o u t p u t c l o s e t o some f i x e d i n p u t , as f o r example, i n a v o l t a g e r e g u l a t o r . The servomechanism i s d e s i g n e d t o m a i n t a i n t h e o u t p u t a r b i t r a r i l y c l o s e t o some i n p u t w h i c h v a r i e s w i t h t i m e , as f o r example, an au t o m a t i c p o s i t i o n c o n t r o l l e r . The i n p u t i n t h e l a t t e r c a s e may be c o n t i n u o u s o r d i s c o n t i n u o u s . The c l o s e d c y c l e system can be f u r t h e r s u b d i v i d e d K Controller Load F i g . 1 B l o c k Diagram o f a S i m p l e Servomechanism 3 F i g . 1 i s a diagram o f an e l e m e n t a r y c l o s e d c y c l e c o n t r o l system. r e p r e s e n t s t h e i n p u t s i g n a l , t h e o u t p u t and S t h e d i f f e r e n c e o r e r r o r . The d i f f e r e n t i a l compares t h e i n p u t and o u t p u t s i g n a l and i n j e c t s t h e d i f f e r e n c e i n t o t h e c o n t r o l l e r . The c o n t r o l l e r w i l l a m p l i f y t h e e r r o r and w i l l produce such a change i n t h e o u t p u t as w i l l t e n d t o reduce t h e e r r o r . C o n s i d e r f o r example a s i m p l e m e c h a n i c a l p o s i t i o n i n g system w i t h &? and So a n g u l a r p o s i t i o n s . The l o a d i n t h i s case w i l l be a moment o f i n e r t i a " J " and a f r i c t i o n component "F". L e t t h e t o r q u e a p p l i e d t o the l o a d be p r o p o r t i o n a l t o Then t h e b a s i c e q u a t i o n s w i l l be Gj - Go - e T « KS c7tx W Where J = moment o f i n e r t i a o f system F s v i s c o u s f r i c t i o n T - o u t p u t t o r q u e o f c o n t r o l l e r Combining t h e t h r e e e q u a t i o n s or 6>0 = K e 7 p , jL T h i s y i e l d s a C h a r a c t e r i s t i c l i n e a r e q u a t i o n t h a t i s a second o r d e r d i f f e r e n t i a l e q u a t i o n J p 2 4- Fp •+• K. The t h r e e c o n s t a n t s a r e i n g e n e r a l independent and can be i n d i v i d u a l l y a d j u s t e d i n t h e d e s i g n . 4 There i s one b a s i c f a u l t w i t h t h i s s i m p l e s e r v o system. An e r r o r must e x i s t b e f o r e any c o r r e c t i o n i s a p p l i e d . T h i s w i l l cause a v e l o c i t y e r r o r i n t h e system, o r t h e e r r o r w i l l be p r o p o r t i o n a l t o t h e r a t e o f change o f p o s i t i o n . T h i s v e l o c i t y e r r o r p l u s t h e i n e r t i a o f t h e system w i l l t e n d t o cause o s c i l l a t i o n . I n c r e a s i n g t h e f r i c t i o n f a c t o r w i l l d e c r e a s e the t i m e o f o s c i l l a t i o n but i t w i l l i n c r e a s e t h e magnitude o f the v e l o c i t y e r r o r . Where more r i g i d d e s i g n r e q u i r e m e n t s have t o be met, s t a b i l i z i n g c i r c u i t s must be i n t r o d u c e d t o reduce t h e magnitude o f t h e v e l o c i t y e r r o r and reduce t h e t i m e o f o s c i l l a t i o n . I n the c o n t r o l l e r o f the el e m e n t a r y system t h e o u t p u t i s p r o p o r -t i o n a l t o t h e e r r o r , t h a t i s t h e c o n t r o l l e r t r a n s f e r f u n c t i o n e q u a l s a c o n s t a n t . T.F. = K The s t a b i l i z i n g c i r c u i t s may be i n t r o d u c e d i n t o t h e c o n t r o l l e r so t h e t r a n s f e r f u n c t i o n may be a d i f f e r e n t i a l e q u a t i o n o f t h e t y p e T.F. « A+B+Cp-HDp 2+ ... P = d_ p d t The c i r c u i t s can be a d j u s t e d t o make any o f t h e c o n s t a n t s o f any d e s i r e d v a l u e . The d e s i g n r e q u i r e m e n t s w i l l d e t e r m i n e t h e v a r i o u s v a l u e s . The term A r e p r e s e n t s t h e i n t e g r a l f a c t o r o r r e s e t P component. I n c r e a s e i n t h e v a l u e o f t h i s t e r m d e c r e a s e s t h e a m p l i t u d e o f t h e v e l o c i t y e r r o r . The d i s a d v a n t a g e o f t h i s i s t h a t as the v e l o c i t y e r r o r approaches z e r o t h e system approaches i n s t a b i l i t y . I f t h e v e l o c i t y e r r o r e q u a l s z e r o any i n d u c e d o s c i l l a t i o n w i l l be s u s t a i n e d . I f t h e v e l o c i t y e r r o r becomes • n e g a t i v e t h e n t h e o s c i l l a t i o n s w i l l b u i l d up i n d e f i n i t e l y . The terms i n v o l v i n g p,p 2, e t c . r e p r e s e n t t h e v a r i o u s powers o f t h e d e r i v a t i v e o f t h e e r r o r . The p r i m a r y r e a s o n f o r i n t r o d u c i n g t h e s e terms i s t o improve t h e f r e q u e n c y r e s p o n s e . I n most p r a c t i c a l systems t h e reponse w i l l f a l l o f f as t h e f r e q u e n c y i n c r e a s e s . The d e r i v a t i v e f a c t o r s can be a d j u s t e d t o i n c r e a s e t h e c u t o f f f r e q u e n c y . I n most p r a c t i c a l a p p l i -c a t i o n s t h e f i r s t o r second o r d e r o f d e r i v a t i v e i s a l l t h a t i s used. R e f e r r i n g t o t h e s i m p l e s e r v o system a g a i n t h e c h a r a c t e r i s t i c e q u a t i o n has i t s r o o t s as P= -I_XFJSZ - K 2 J f\ZZj J T h i s w i l l p e r m i t t h r e e s o l u t i o n s depending on t h e magnitude o f t h e p a r a m e t e r s . (1) (^Tj) > "j- overdamped (2) /LEL^f = c r i t i c a l l y damped \ Z J J J (3) ^-E—J2 < under damped I n t h e f i r s t case p i s r e a l w i t h two s u r d r o o t s . The t r a n s i e n t s o l u t i o n o f d i f f e r e n t i a l e q u a t i o n s o f t h i s f o rm may be e x p r e s s e d i n h y p e r b o l i c f u n c t i o n s w i t h an e x p o n e n t i a l decay term. T h i s case i s termed as the overdamped c a s e . There i s no tendency f o r o s c i l l a t i o n a t a l l but t h e t i m e o f response 6 i s t o o . l o n g t o be p r a c t i c a l . I n t h e second case t h e e q u a t i o n has two i d e n t i c a l r e a l r o o t s . T h i s e q u a t i o n has a s o l u t i o n o f an e x p o n e n t i a l decay term m u l t i p l y i n g a c o n s t a n t term p l u s a c o n s t a n t t i m e s t i m e . T h i s case i s r e f e r r e d t o as t h e c r i t i c a l l y damped c a s e . The damping f a c t o r F c i s known as t h e c r i t i c a l damping and e q u a l s E / K J . A g a i n t h e r e i s no tendency f o r o s c i l l -a t i o n but t h e t i m e o f r e s p o n s e , a l t h o u g h s h o r t e r t h a n t h e f i r s t c a s e , i s s t i l l t oo l o n g t o be p r a c t i c a l . I n t h e t h i r d and most i m p o r t a n t case t h e e q u a t i o n has two r o o t s t h a t a r e complex c o n j u g a t e s . T h i s y i e l d s t h e o s c i l l -a t o r y s o l u t i o n , an e x p o n e n t i a l f a c t o r m u l t i p l y i n g a s i n e and c o s i n e term. T h i s case i s c a l l e d t h e underdamped ca s e . The s i n e and c o s i n e terms w i l l have a f r e q u e n c y r e f e r r e d t o as t h e n a t u r a l f r e q u e n c y o f t h e system * J - / £ - ( £ . ) * Jl> zn 271 v J U / / F i g . 2 i s a graph o f servomechanism response to a s t e p f u n c t i o n i n p u t . The v a r i o u s c u r v e s c o r r e s p o n d t o d i f f e r e n t damping r a t i o s "G". The damping r a t i o "C" i s d e f i n e d as t h e r a t i o o f t h e a c t u a l damping t o t h e c r i t i c a l damping o f t h e system. CJ^ i s t h e n a t u r a l f r e q u e n c y o f t h e system w i t h o u t damping and COj i s t h e a p p l i e d f r e q u e n c y . The c u r v e s a r e made d i m e n s i o n l e s s to p e r m i t comparison o f d i f f e r e n t systems. I f i n s t e a d o f a s t e p f u n c t i o n i n p u t a s i n u s o i d a l i n p u t i s a p p l i e d , t h e o u t p u t w i l l a l s o be s i n u s o i d a l w i t h e i t h e r p o s i t i v e o r n e g a t i v e a m p l i f i c a t i o n and some ti m e phase d i s p l a c e m e n t . The degree o f a m p l i f i c a t i o n and phase d i s p l a c e -ment w i l l depend on t h e a p p l i e d f r e q u e n c y . I f t h e a p p l i e d 0 5 10 15 30 F I G . 2 E r r o r - t i m e C u r v e s f o r V i s c o u s - d a m n e d S e r v o m e c h a n i s m 0 1 2 3 R e l a t i v e F r e q u e n c y FIG. 3 Resonance C u r v e s o f S e r v o m e c h a n i s m W i t h V i s c o u s Output Damping 7 f r e q u e n c y approaches t h e n a t u r a l f r e q u e n c y a c o n d i t i o n e x i s t s v e r y s i m i l a r t o t h a t encountered i n an o s c i l l a t o r t a n k c i r c u i t . A v e r y s m a l l i n p u t s i g n a l w i l l r e s u l t i n a l a r g e o u t p u t s i g n a l , t h e o n l y l i m i t i n g f a c t o r i s t h e damping o r f r i c t i o n c o e f f i c i e n t . T h i s i s v e r y u n d e s i r a b l e i n most se r v o systems. The i d e a l s i t u a t i o n e x i s t s when t h e o u t p u t e q u a l s -the i n p u t and t h e phase d i s p l a c e m e n t i s a minimum. F i g . 3 i s a graph o f t h e response o f a s i m p l e s e r v o s u b j e c t t o v a r i o u s f r e q u e n c y i n p u t s . T h i s r e v i e w o f t h e servomechanism t h e o r y has r e v e a l e d t h r e e p o i n t s . (1) D e s i g n t h e system so t h a t o s c i l l a t o r y c o n d i t i o n s w i l l p r e v a i l , t h a t i s have i t s a t i s f y case I I I , t h e underdamped c a s e . (2) D e s i g n t h e n a t u r a l f r e q u e n c y w e l l above t h e o p e r a t i n g f r e q u e n c y e s p e c i a l l y i f t h e i n p u t s h o u l d c o n t a i n predominant harmonics h i g h e r t h a n t h e f u n d a m e n t a l . (3) Where more r i g i d d e s i g n r e q u i r e m e n t s have t o be met, i n t r o -duce networks so as t o reduce t h e v e l o c i t y e r r o r , o r t o i n c r e a s e t h e f r e q u e n c y r e s p o n s e . Photo #2 Centre Port-ion of R i v e r Basin Photo #4 T i d a l Basin w i t h Recorder i n Foreground 8 D e s c r i p t i o n o f t h e T i d a l C o n t r o l Equipment The model c o v e r s an a r e a o f t h r e e a c r e s . I t i n c l u d e s t h e E r a s e r R i v e r from M i s s i o n t o t h e mouth, P i t t R i v e r and P i t t Lake, and a p o r t i o n o f t h e G-ulf o f G e o r g i a i n t h e v i c i n i t y o f t h e t i d e f l a t s . The h o r i z o n t a l s c a l e i s one i n s i x hundred, t h e t i m e and v e r t i c a l s c a l e i s one i n s e v e n t y , and t h e v e l o c i t y s c a l e i s one i n e i g h t p o i n t f i v e . V a r i o u s views o f t h e model a r e i l l u s t r a t e d i n photographs 1,2,3 and 4. Photographs 1,2 and 3, a r e views o f t h e west, c e n t r e , and e a s t p o r t i o n s o f t h e ; r i v e r b a s i n . Photograph 4 i s a v i e w o f t h e t i d a l b a s i n . The w a t e r l e v e l c o n t r o l equipment c o n s i s t s o f s i x components i l l u s t r a t e d i n photographs 5,6,7 and 8. (1) A photo c e l l r e a d e r t h a t p r o v i d e s a v o l t a g e p r o p o r t i o n a l t o t h e d e s i r e d t i d e l e v e l . (2) a p a i r o f l o a d r e s i s t o r s t h a t a c t s as t h e d i f f e r e n t i a l . (3) An a m p l i f i e r t h a t c o n s i s t s o f a D.C. e l e c t r o n i c a m p l i f i e r and a h y d r a u l i c a m p l i f i e r . (4) The h y d r a u l i c j a c k t h a t r a i s e s and l o w e r s t h e w e i r s . (5) The s e t o f w e i r s t h a t c o n t r o l t h e w a t e r l e v e l . (6) The t i d e f l o a t p o t e n t i o m e t e r t h a t produces a v o l t a g e p r o p o r t i o n a l t o t h e a c t u a l t i d e l e v e l . There a r e o t h e r p i e c e s o f equipment, such as t h e 20 c u b i c f o o t p e r second pump t h a t s u p p l i e s w a t e r t o t h e b a s i n ; The power s u p p l i e s f o r t h e f l o a t p o t e n t i o m e t e r , t h e photo c e l l t o f o l l o w page 8 Photo - # 5 E l e c t r o n i c C o n t r o l Equipment I n c l u d i n g (1) Photo C e l l Reader (2) D.C. A m p l i f i e r (3) J u n c t i o n box c o n t a i n i n g power s u p p l i e s and d i f f e r e n t i a l Photo #6 H y d r a u l i c A m p l i f i e r Photo #8 The Set o f Weirs r e a d e r , and the D.C. a m p l i f i e r ; and t h e pump and r e g u l a t i n g system t h a t s u p p l i e s t h e p r e s s u r i z e d o i l f o r t h e h y d r a u l i c a m p l i f i e r , t h a t a r e not a c t u a l l y p a r t o f t h e c l o s e d c y c l e c o n t r o l system. The photo c e l l r e a d e r , i w h i c h i s encased i n a l i g h t t i g h t box, i s shown s c h e m a t i c a l l y i n F i g . 4. The l i g h t s o u r c e i s r e f l e c t e d o f f the galvanometer m i r r o r t h r o u g h a p i e c e o f t r a n s p a r e n t paper and a p a i r o f convex l e n s e s onto a photo m u l t i p l i e r t u b e . The o u t p u t o f t h e tube i s connected i n s e r i e s w i t h t h e galvanometer and one o f t h e l o a d r e s i s t o r s i n t h e d i f f e r e n t i a l . The galvanometer w i l l d e f l e c t u n t i l t h e l i g h t i s i n t e r r u p t e d by the t i d e c y c l e , w h i c h i s a b l a c k l i n e p a i n t e d on t h e t r a n s p a r e n t paper. Time s c a l e extends a l o n g t h e paper and w a t e r l e v e l s c a l e a c r o s s . The t i d e c h a r t i s d r i v e n p a s t the l i g h t by synchronous motor. Thus t h e o u t p u t v o l t a g e o f t h e photo c e l l r e a d e r i s p r o p o r t i o n a l t o t h e d e s i r e d t i d e l e v e l . The d i f f e r e n t i a l r e c e i v e s t h e v o l t a g e p r o p o r t i o n a l t o t h e d e s i r e d t i d e l e v e l f r om t h e photo c e l l r e a d e r and t h a t p r o p o r t i o n a l t o a c t u a l t i d e l e v e l from t h e t i d e f l o a t p o t e n t i o m e t e r and has an o u t p u t v o l t a g e e q u a l t o t h e d i f f e r e n c e . The c i r c u i t c o n s i s t s o f two l o a d r e s i s t o r s connected i n s e r i e s w i t h t h e i n p u t v o l t a g e s i m p ressed a c r o s s the r e s i s t o r s i n o p p o s i t e p o l a r i t y . The a m p l i f i e r system c o n s i s t s o f a D.C. e l e c t r o n i c a m p l i f i e r t h a t a c t u a t e s a b a l a n c e d o i l v a l v e i n t h e h y d r a u l i c system. The d i s p l a c e m e n t o f t h e b a l a n c e d o i l v a l v e i s p r o -p o r t i o n a l t o t h e magnitude and p o l a r i t y o f t h e e r r o r o r d i f f e r e n c e v o l t a g e . The o u t p u t o f t h e h y d r a u l i c a m p l i f i e r i s n o . \ -LOG:-: D I A G R A M OF TH:, CHART R E A D E R 10 i n t u r n p r o p o r t i o n a l t o the d i s p l a c e m e n t o f t h e b a l a n c e d o i l v a l v e . The'output o f t h e h y d r a u l i c a m p l i f i e r moves t h e h y d r a u l i c j a c k w h i c h i s m e c h a n i c a l l y connected t o t h e w e i r s . The h y d r a u l i c j a c k speed and hence t h e w e i r speed i s p r o -p o r t i o n a l t o t h e h y d r a u l i c a m p l i f i e r o u t p u t . Water i s pumped c o n t i n u o u s l y a t a r a t e o f 20 c u b i c f e e t p e r second from a sump i n t o t h e t i d a l b a s i n . The s u r p l u s w a t e r i s s p i l l e d o v e r t h e w e i r s back i n t o t h e sump. Any v a r i a t i o n i n w e i r l e v e l w i l l t h u s r e s u l t i n a s i m i l a r v a r i a t i o n i n t h e w a t e r l e v e l . A v o l t a g e p r o p o r t i o n a l t o t h e a c t u a l t i d e l e v e l i s o b t a i n e d by means o f t h e t i d e f l o a t p o t e n t i o m e t e r . The p o t e n t -i o m e t e r i s m e c h a n i c a l l y connected t o a f l o a t . The o u t p u t o f t h e p o t e n t i o m e t e r i s f e d i n t o t h e o t h e r l o a d r e s i s t o r i n the d i f f e r e n t i a l . D u r i n g o p e r a t i o n t h e t i d e c h a r t , w h i c h has t h e sequence o f t i d e s f o r an e n t i r e y e a r p l o t t e d on i t , i s d r i v e n c o n t i n u o u s l y p a s t t h e l i g h t s o u r c e . T h i s c a l l s f o r a c o n t i n u o u s sequence o f o p e r a t i o n s f o r f i v e days. I t i s h i g h l y d e s i r a b l e t o have t h e system o p e r a t e w i t h o u t any break and t o have t h e e r r o r i n t h e o u t p u t l e s s t h a n 5%, p r e f e r a b l y l e s s t h a n 3 % . T h i s e r r o r o n l y r e f e r s t o the a m p l i t u d e , t h e phase d i s p l a c e m e n t i s r e l a t i v e l y u n i m p o r t a n t as l o n g as i t i s not u n r e a s o n a b l y l a r g e and remains f a i r l y c o n s t a n t . 11 T h e o r e t i c a l A n a l y s i s I n d e s i g n i n g a se r v o system more t h a n one approach can he t a k e n . The i n p u t can be assumed t o be a s e r i e s o f s t e p s . T h i s approach imposes v e r y s e v e r e r e q u i r e m e n t s on t h e system and i n c e r t a i n c a ses may not be j u s t i f i e d . A n o t h e r approach can be t o assume t h e i n p u t i s a s i n u s o i d a l f u n c t i o n . The system can t h e n be d e s i g n e d t o have t h e a m p l i f i c a t i o n l e s s t h a n some p r e d e t e r m i n e d v a l u e and t h e phase d i s p l a c e m e n t a t t h e o p e r a t i n g f r e q u e n c i e s l e s s t h a n t h e a l l o w a b l e v a l u e . I f the f u n c t i o n i s more complex a F o u r i e r a n a l y s i s o f t h e f u n c t i o n can be made and t h e h i g h e s t a p p r e c i a b l e harmonic can be made t o comply w i t h the a m p l i t u d e and phase d i s p l a c e m e n t r e q u i r e m e n t s . I n a n a l y z i n g t h e t i d a l c o n t r o l system i t i s o b v i o u s t h a t r e q u i r i n g a good s t e p f u n c t i o n r e s p o n s e , i s i m p o s i n g c o n d i t i o n s t h a t a r e more s e v e r e t h a n n e c e s s a r y . The non r e p e t e t i v e n a t u r e o f t h e t i d e c y c l e does not p e r m i t the u s u a l F o u r i e r a n a l y s i s o f t h e c y c l e . Harmonic a n a l y s e s o f t i d e c y c l e s have been made t o f a c i l i t a t e t i d e p r e d i c t i o n . T h i s a n a l y s i s r e l a t e s t h e v a r i o u s components o f t h e t i d e c y c l e t o t h e b e h a v i o u r o f d i f f e r e n t c e l e s t i a l b o d i e s . Some 170 components e x i s t i n a d e t a i l e d a n a l y s i s but t h e f o u r major components a r e t h e p u l l o f t h e moon, t h e p u l l o f t h e sun, and two components due t o moons d e c l i n a t i o n . T a b l e #1 g i v e s t h e r e l a t i v e magnitude and p e r i o d o f t h e v a r i o u s components. The magnitude o f the moons component i s t a k e n as u n i t y . These f o u r major components a r e near o r l e s s t h a n t h e fundamental f r e q u e n c y , w h i c h i s t a k e n as t h e f r e q u e n c y 12 \ Component R e l a t i v e A m p l i t u d e P e r i o d Moons p u l l 1.0 12.4 h r s . Suns p u l l 0.280 12,0 h r s . Moons d e c l i n a t i o n #1 0.415 24 h r s . Moons d e c l i n a t i o n #2 0.258 25.9 h r s . T a b l e #1 T a b l e o f p e r i o d s and r e l a t i v e a m p l i t u d e s o f major components o f t h e harmonic c o n s t i t u e n t s o f t h e t i d a l o c y c l e o f t h e moons a t t r a c t i v e component. There are components o f h i g h e r f r e q u e n c y , i n s h a l l o w w a t e r t i d e s p e r i o d s may be as low as t h r e e h o u r s , but t h e r e l a t i v e magnitudes o f t h e s e components are l e s s t h a n 0.01. W i t h t h e s e f a c t o r s i n mind t h e d e s i g n f r e q u e n c y was t a k e n t o be the f r e q u e n c y o f t h e moons a t t r a c t i v e component. The s t e p f u n c t i o n response a l t h o u g h o f secondary i m p o r t a n c e can not be c o m p l e t e l y i g n o r e d . A r e a s o n a b l e r e s p o n s e t o a s t e p f u n c t i o n i s d e s i r a b l e f o r two r e a s o n s . One, when s t a r t i n g t h e system i t i s d e s i r a b l e t o have t h e w a t e r l e v e l s e t t l e t o t h e s t a r t i n g t i d e l e v e l i n a r e a s o n a b l e l e n g t h o f t i m e . Two, i f i n t h e m i d d l e o f the r u n t h e wat e r l e v e l i s e i t h e r i n t e n t i o n a l l y o r a c c i d e n t a l l y s h i f t e d o f f t h e c u r v e i t w i l l be d e s i r a b l e t o have i t r e t u r n as q u i c k l y as p o s s i b l e and w i t h t h e minimum amount o f o s c i l l a t i o n . A f t e r c o n s i d e r i n g t h e s e p r o p e r t i e s o f t h e t i d a l c y c l e , i t was d e c i d e d t o a n a l y z e the system u s i n g a d e s i g n f r e q u e n c y e q u a l t o t h e f r e q u e n c y o f t h e moons a t t r a c t i v e com-ponent. The s t e p f u n c t i o n a n a l y s i s w i l l be used as a check. Amplifier TM Basin F i g . 5 B l o c k Diagram o f T i d a l C o n t r o l System F i g . 6 Schematic Diagram o f T i d a l B a s i n D e f i n i t i o n s Oj - i n p u t s i g n a l 0o =. o u t p u t s i g n a l £^ = e r r o r between and OA £, - e r r o r between €, and Kg dy_ * d t K]_ =. a m p l i f i e r c o n s t a n t y s. w e i r l e v e l A = a r e a o f t i d a l b a s i n = 12,000 s q . f t . 14 H = head over w e i r Kg - feed hack constant Q, =pump discharge=20 c . f . s . b = weir length = 40 f t . A block diagram of the system as i t o r i g i n a l l y e x i s t e d i s i l l u s t r a t e d i n F i g . 5 and a diagram of the t i d a l b a s i n at the weirs i s shown i n F i g . 6. The basic r e l a t i o n s h i p s taken o f f these diagrams are € - f j f - £ * ( 2 ) ea = H + y (*) The water f l o w i n g i n t o the basin minus the water f l o w i n g out over the w e i r s w i l l equal the change i n volume. The water f l o w i n g over the weir can be c a l c u l a t e d by the weir discharge formula. Weir discharge = J.J 3 h " K3 w i t h ft - 3.33 b .". Q - K,H% = dXz' ° cit o r KSH* = Q - Ad& ( 5 ) dt Equation f i v e i s a non- l i n e a r equation but i t can be expressed by a MacLaurens S e r i e s . Therefore by power expansion »• ®fo - i iff * /ia )YJ -*4& - ±:f& & f - -0 I f £d$g\s s m a l l , a l l but the f i r s t two terms may be Q at neglected. Here the maximum value i s of the order of 0.5 so i reduces the v a l i d i t y of the approximation but the t h i r d term i l e s s than one t e n t h of the second term, so (6) From equations 3,4 and 6 d t 1 7 ? c > where 1/ = J? a l s o from equations 2 and 3 £ X = - € j (8) from equations 1,7 and 8 l-rKtK2Cl dt TV-K + + K,£&-d& + K f d ^ ( 1 0 ) J+KtKz dt 7 dt* dt 7 cit-Equation (9.) and (10) r e l a t e the e r r o r , input, and output of a servo system. The equations could be solved f o r any given input c o n d i t i o n s . I n i t i a l l y the i n v e s t i g a t i o n w i l l be c a r r i e d out without the i n t e r n a l feedback loop, see F i g . 7. 16 F ig . 7 Block Diagram of Modified Tidal Control System The basic equations now are Combining these equations give ( 6 ) Kre>+-H?+Ki&-#+«*1k 1131 Comparing equations (9) and (12) one can see that the two equations are identical i f K 5 = .:ffl . i n other words 1 K X K 2 the only effect of this internal feedback loop is the reduction of the apparent amplification of the amplifier. This effect can be more simply accomplished by the use of an amplifier of 17 l o w e r g a i n . S i n c e i t i s d e s i r a b l e t o have t h e system work as a c c u r a t e l y as p o s s i b l e and y e t be as s i m p l e and t r o u b l e f r e e , so the f i r s t m o d i f i c a t i o n suggested i s removal o f t h e i n t e r n a l feedback l o o p . The s o l u t i o n o f e q u a t i o n s (12) and (13) w i l l now be c o n s i d e r e d f o r a f r e q u e n c y i n p u t and f o r a s t e p i n p u t . F i r s t a t r a n s f e r f u n c t i o n a n a l y s i s o f e q u a t i o n (12) w i l l be c o n s i d e r e d . I f the e r r o r i s a p p l i e d a r b i t r a r i l y as a c o s i n e f u n c t i o n o f time o f c o n s t a n t a m p l i t u d e , say u n i t y , and v a r y i n g f r e q u e n c y t h e n t h e o u t p u t w i l l a l s o be a c o s i n e f u n c t i o n o f time o f t h e same f r e q u e n c y d i s p l a c e d by some phase a n g l e o f a d i f f e r e n t a m p l i t u d e . The i n p u t and out p u t can th e n be r e p r e s e n t e d by jCOt Si =e j(cot +X) eQ - C e Then e q u a t i o n (12) becomes (14) CeJ = *> w h i c h may be r e p r e s e n t e d as a v e c t o r o f magnitude and o f phase d i s p l a c e m e n t K ~ /80° + ore tan—1-7? (16) Cu A 4 The l o c u s o r N y q u i s t p l o t o f t h e s e • v e c t o r s f o r e x i s t i n g c o n d i t i o n s i s shown i n graph #1. A l l t h e n e c e s s a r y i n f o r m a t i o n 18 f o r a f r e q u e n c y s t u d y i s c o n t a i n e d i n t h i s g r aph. A v e c t o r from t h e o r i g i n t o t h e c u r v e i s t h e o u t p u t o f the system, a v e c t o r f r o m t h e o r i g i n t o (-1,0) i s the e r r o r , and from (-1,0) t o t h e c u r v e i s t h e i n p u t s i g n a l . The a n g l e between t h e i n p u t and ou t p u t v e c t o r s i s t h e phase d i s p l a c e m e n t . i n p u t and o u t p u t v e c t o r s o f e q u a l magnitude. T h i s c o u l d be a c c o m p l i s h e d i f t h e l o c u s c o u l d be swung c o u n t e r - c l o c k w i s e u n t i l t h e p o i n t on t h e l o c u s c o r r e s p o n d i n g t o t h e o p e r a t i n g f r e q u e n c y i s on t h e l i n e , r e a l s e q u a l minus one h a l f . To a c c o m p l i s h t h i s by a d j u s t i n g K 5 would not be p r a c t i c a l . Adjustment o f K 5 w i l l o n l y change t h e l e n g t h o f the ou t p u t v e c t o r f o r a g i v e n e r r o r v e c t o r , i t w i l l not a f f e c t t h e phase a n g l e . R e f e r t o , g r a p h #1. I f t h e v e c t o r ( 0 ; ^ ) were reduced t o t h e p o i n t where i t i n t e r -s e c t s t h e l i n e r e a l s e q u a l minus one h a l f , t h e n i t would be s h o r t e r t h a n t h e e r r o r v e c t o r and the a n g l e between ( 0 ; ^ ) and (-1,0;C*^) would be g r e a t e r t h a n 90 degre e s . m u l t i p l y i n g c o n s t a n t g r e a t e r t h a n u n i t y , the c u r v e would be moved c l o s e r t o t h e n e g a t i v e r e a l a x i s . T h i s j term o r i g i n a t e s from t h e f i r s t d e r i v a t i v e o f the o u t p u t . Here i t can be seen how d e s i r a b l e i t i s t o have a f l e x i b l e c o n s t a n t m u l t i p l y i n g t h i s t e r m. Now a s t e p f u n c t i o n a n a l y s i s o f e q u a t i o n (13) w i l l be c a r r i e d o u t . The c o n d i t i o n s w i l l be As has a l r e a d y been mentioned t h e aim i s t o have t h e C o n s i d e r now e q u a t i o n ( 1 4 ) , i f t h e j t e r m had a t < o 19 Pei = f > * e t = o t >o E q u a t i o n (13) becomes \Kf PZ -I-? -I- Ks) £ = (kff2 +-f>) Qt =o e = f-—i )foj £ = e e H r , Cos/% f I") E q u a t i o n (17) i s an e x p r e s s i o n f o r t h e e r r o r o f t h e s e r v o system s u b j e c t t o a s t e p i n p u t . T h i s i s a t r a n s i e n t term t h a t approaches z e r o as time becomes l a r g e . I t i s d e s i r a b l e t o keep t h i s t e r m s m a l l and have i t approach z e r o as soon as p o s s i b l e . T h i s may be a c c o m p l i s h e d by making K 4 s m a l l o r K 5 l a r g e . Making K 4 s m a l l w i l l cause t h e e x p o n e n t i a l term t o d i e away f a s t e r and i n the range o f v a l u e s c o n s i d e r e d tend t o i n c r e a s e t h e v a l u e o f the r a d i c a l . U n f o r t u n a t e l y K 4 i s f o r a l l p r a c t i c a l purposes f i x e d , i t r e p r e s e n t s the c o n f i g u r a t i o n o f t h e model a l r e a d y c o n s t r u c t e d . T h i s l e a v e s K 5 w h i c h i s the a m p l i f i e r g a i n . Any m a n i p u l a t i o n o f t h i s q u a n t i t y does not reduce t h e t i m e o f d i e away. S i n c e t h e t i m e s c a l e i s 1:70, t h e p e r i o d o f t h e compon-ent o f t h e moon i s - 10.63 m i n u t e s . T h i s component th e n has an a n g u l a r v e l o c i t y ; • T 6 0 0 ' The n a t u r a l p e r i o d o f t h e system; 20 Kj- = Amplifier Constant = 0.2 of Q foot /sec /foot of 7 ^ (3.33 ky% T a k i n g e r r o r CO a - / — =2>98xi0~*rds/sec Cun = / Q* A * (18) / A E q u a t i o n (18) i n d i c a t e s t h e r e l a t i o n s h i p between t h e parameters and t h e n a t u r a l f r e q u e n c y o f t h e system. The n e x t p o i n t o f i n t e r e s t on t h e s e e q u a t i o n s i s t h e c o e f f i c i e n t o f t h e f i r s t d e r i v a t i v e o f t h e o u t p u t . T h i s c o e f f i c -i e n t o f t h e f i r s t d e r i v a t i v e i s t h e f r i c t i o n o r damping c o e f f i c i e n t , I f i t were z e r o any in d u c e d o s c i l l a t i o n would be s u s t a i n e d u n t i l some e x t e r n a l f o r c e appeared t o a l t e r i t . The system o p e r a t i o n may be improved by i n c r e a s i n g t h i s f a c t o r but t h i s i s i m p o s s i b l e because o f h y d r a u l i c r e l a t i o n s h i p s . One a l t e r n a t i v e i s t o r e -duce t h e magnitude o f K 4 and K 5 . A l t h o u g h K 5 i s f l e x i b l e any adju s t m e n t o f K 4 would i n v o l v e e x t e n s i v e m o d i f i c a t i o n s o f t h e system. A n o t h e r a l t e r n a t i v e would be t o l o o k f o r some s t a b i l -i z i n g c i r c u i t t h a t would improve t h e o p e r a t i o n . From t h e c o n s i d e r a t i o n o f e q u a t i o n s ( 1 4 ) , ( 1 5 ) , ( 1 6 ) and (17) i t has been seen- how d e s i r a b l e i t i s t o have something e l s e t o i n c r e a s e t h e f l e x i b i l i t y o f t h e system and improve t h e res p o n s e . To t h i s end a c o n s i d e r a t i o n o f d e r i v a t i v e c o n t r o l 21 w i l l be s t u d i e d . The only, d i f f e r e n c e i n t h e e q u a t i o n s o f t h e system w i l l be i n t h e t r a n s f e r f u n c t i o n o f th e a m p l i f i e r . I t w i l l now be fa + K < i i ) The b a s i c e q u a t i o n s now become 6i - e0 = £ ( i ) u-/Q^/f _ 2 . _ d (e) n 1—1(1 3 Q c/tj (19) (4) (20) Combining t h e s e e q u a t i o n s Ks£+ 0 + - $ + ( 8 1 ) A s t u d y o f t h i s system w i t h a s i n u s o i d a l i n p u t w i l l now be c a r r i e d o u t . E q u a t i o n (20) becomes Ks e -tj^r^ - J C c o G -cujftyc Ce. = n* J — (22) wh i c h a g a i n may be r e p r e s e n t e d as a v e c t o r o f magnitude 22 and o f phase d i s p l a c e m e n t A « / 8 0 ° + Arcfan^-r, + A r c t o n ^ - 6 (24) The e x t r a t e r m a r c t a n r e s u l t s i n an i n c r e a s e i n t h e a n g l e A . Curve (2) i n graph #1 i s a p l o t o f t h i s r e s u l t . I n c r e a s i n g K 5 improves t h e a m p l i f i c a t i o n r a t i o . Now c o n s i d e r a s t e p f u n c t i o n a n a l y s i s o f e q u a t i o n (21) £ = + < 2 5 ) w h i c h has as a s o l u t i o n E q u a t i o n (26) i s t h e s o l u t i o n o f t h e e q u a t i o n t o a s t e p f u n c t i o n i n p u t . Here t h e advantage o f t h e i n t r o d u c t i o n o f t h e Kg f a c t o r i s r e v e a l e d . I t i n c r e a s e s t h e a t t e n u a t i o n con-s t a n t o f t h e c i r c u i t . The i n t r o d u c t i o n o f t h i s f a c t o r i s q u i t e p r a c t i c a l and does not i n v o l v e any e x t e n s i v e m o d i f i c a t i o n o f t h e equipment. However t h e i n t r o d u c t i o n o f t h e Kg f a c t o r reduces t h e n a t u r a l f r e q u e n c y o f t h e system. E i t h e r t h i s c o n d i t i o n must be a c c e p t e d o r one o f t h e o t h e r parameters must be. a d j u s t e d t o o f f s e t t h e change. A p r o b a b l e a d j ustment would be t o i n c r e a s e K 5 . The p o s s i b i l i t y o f i n t r o d u c i n g an i n t e g r a l component was c o n s i d e r e d and d i s c a r d e d . I n t e g r a l c o n t r o l reduces th e v e l o c i t y e r r o r but does not improve t h e a m p l i f i c a t i o n r a t i o . Summarizing t h e t h e o r y t h e r e a r e two t h i n g s t o be done. One, o b t a i n t h e b e s t r e s u l t s by m a n i p u l a t i o n o f and K2 and t h e n do t h e same by m a n i p u l a t i o n o f K 5 and show e x p e r i m e n t a l l y t h a t the re s p o n s e o f t h e l a t t e r w i l l e q u a l o r b e t t e r t h e f o r m e r . Two, d e t e r m i n e t h e v a l u e o f K 6 t h a t w i l l improve t h e r e s p o n s e w i t h o u t c a u s i n g overdamping. 24 T e s t s and R e s u l t s To impose t h e most s e v e r e c o n d i t i o n s p o s s i b l e a c u r v e was p l o t t e d t h a t i n c l u d e d t h e h i g h e s t w a t e r on r e c o r d , t h e l o w e s t w a t e r on r e c o r d , and t h e g r e a t e s t r a t e o f change o f w a t e r l e v e l . T h i s was a c t u a l l y more s e v e r e t h a n any r e c o r d e d t i d e because t h e p e r i o d s o f h i g h e s t and l o w e s t w a t e r s do not o c c u r on t h e same day. The e a r l y t e s t s were r u n u s i n g t h i s h y p o t h e t i c a l c u r v e as a s t a n d a r d f o r comparing t h e response t o th e v a r i o u s servomechanism c o n d i t i o n s . L a t e r t e s t s were r u n u s i n g t h e r e c o r d o f J a n u a r y 4 t o 12 1947 w h i c h i s t h e week o f most s e v e r e c o n d i t i o n s . Graph #2 i s a graph o f one o f t h e b e t t e r c u r v e s o b t a i n e d w i t h t h e i n t e r n a l feedback c i r c u i t . Graph #3 i s a p l o t o f one o f t h e c u r v e s o b t a i n e d w i t h o u t t h e c i r c u i t . The -c u r v e s a r e compared w i t h t h e i n p u t s i g n a l . A l t h o u g h graph §3 has b e t t e r a m p l i t u d e c h a r a c t e r i s t i c s t h a n graph #2, an o s c i l l -a t i o n has appeared a t t h e low t i d e . The c u r v e drawn up from most s e v e r e c o n d i t i o n s i s made up o f two major components. One i s a t the fundamental f r e q u e n c y and the o t h e r i s a t the second harmonic f r e q u e n c y . The o s c i l l -a t i o n a p p e a r i n g a t t h e low t i d e i s caused by t h e pr e s e n c e o f the second harmonic. The f r e q u e n c y o f t h e second harmonic i s c l o s e t o t h e n a t u r a l f r e q u e n c y o f t h e system. The second h a r -monic i s not as predominant i n the t i d e c y c l e o f the-most s e v e r e r e c o r d e d week nor was i t c o n s i d e r e d i n t h e t h e o r e t i c a l a n a l y s i s . 25 Graphs #4 and #5 compare t h e r e s u l t s o f t e s t s w i t h and w i t h o u t t h e d i f f e r e n t i a t i n g c i r c u i t . The week o f J a n u a r y 4 t o 12 was used f o r t h e s e t e s t s w i t h t h e day o f J a n u a r y 5 used h e r e f o r comparison. Some tendency f o r o s c i l l a t i o n i s s t i l l a p p arent but not n e a r l y as s e v e r e as f o r t h e h y p o t h e t i c a l c u r v e . I n comparing t h e response w i t h and w i t h o u t t h e d i f f e r e n t i a t i n g c i r c u i t v e r y l i t t l e improvement i s o b t a i n e d by t h e a d d i t i o n . Graphs #6 and #7 show th e response o f t h e system t o a one and one h a l f i n c h s t e p f u n c t i o n w i t h and w i t h o u t t h e d i f f e r -e n t i a t i n g c i r c u i t . Here, as i n t h e a n a l y s i s o f the system, t h e advantage o f t h e d i f f e r e n t i a t i n g c i r c u i t i s brought o u t . Both t h e a m p l i t u d e o f t h e o v e r s h o o t and t h e t i m e f o r t h e o s c i l l a t i o n s t o d i e away a r e reduced c o n s i d e r a b l y . I t i s on t h e s t r e n g t h o f t h i s l a t t e r r e s u l t t h a t t h e d i f f e r e n t i a t i n g c i r c u i t i s suggested as a m o d i f i c a t i o n t o t h e c o n t r o l system. JLO. follow page. Zpi-.. D i s c u s s i o n and C o n c l u s i o n s The r e s u l t s o b t a i n e d by t e s t s have r e v e a l e d one t h i n g ; t h e i n t r o d u c t i o n o f e x t e r n a l c i r c u i t s has v e r y l i t t l e e f f e c t on th e c o n t i n u o u s o p e r a t i o n o f t h e t i d a l model. T e s t s were c a r r i e d out on s e v e r a l s t a b i l i z i n g c i r c u i t s t h a t have n ot been mentioned i n t h i s t h e s i s , but i n a l l c a s e s t h e r e s u l t s were p o o r e r t h a n any obta-i-ned by methods d i s c u s s e d h e r e i n . Some improvement i n t h e s t e p f u n c t i o n response c o u l d be o b t a i n e d i n some cases but t h e b e s t o v e r a l l r e s u l t s were o b t a i n e d u s i n g t h e d i f f e r -e n t i a t i n g c i r c u i t t h a t has a l r e a d y been d i s c u s s e d . R e p e a t i n g e q u a t i o n (12) t h e t r a n s f e r f u n c t i o n o f t h e system i s (12) K 5 i s t h e o r e t i c a l l y a r b i t r a r y but p r a c t i c a l l y i t i s l i m i t e d t o one t e n t h o f a f o o t p e r second p e r f o o t o f e r r o r . I f K5 i s l a r g e r t h a n 0.1 t h e w e i r s would r i s e f a s t e r t h a n t h e pump can b r i n g up t h e w a t e r l e v e l w h i c h means the w e i r s w i l l b r eak t h r o u g h t h e w a t e r s u r f a c e and the w a t e r l e v e l w i l l no l o n g e r f o l l o w t h e w e i r movement. The c o n s t a n t K 4 i s r e p r e s e n t a t i v e o f the time c o n s t a n t o f t h e system. The s m a l l e r t h i s t ime c o n s t a n t i s t h e b e t t e r w i l l be t h e res p o n s e o f the system. R e p e a t i n g e q u a t i o n (17) T s 2K4 • 224 seconds T h i s e q u a t i o n r e v e a l s t h e importance o f the ti m e con-s t a n t . Any r e d u c t i o n i n K 4 w i l l r educe t h e ti m e f o r t h e expon-e n t i a l t e r m to approach z e r o . I f t h e c o e f f i c i e n t o f t h e f i r s t d e r i v a t i v e had a c o n s t a n t g r e a t e r t h a n u n i t y i t a l s o would de-c r e a s e t h e t i m e c o n s t a n t o f t h e system. R e f e r now t o F i g . (3) i n t h e g e n e r a l t h e o r y . T h i s f i g u r e i s a graph o f i n p u t , o u t p u t a m p l i f i c a t i o n t o r e l a t i v e f r e q u e n c y . I t i s h i g h l y d e s i r a b l e t o have t h e a m p l i f i c a t i o n r a t i o e q u a l t o u n i t y . The g e n e r a l shape o f t h e cur v e i s -depend-ent on t h e damping r a t i o "G". I n t h i s case t h e c r i t i c a l damping C - 1 = 0.149 6.7 The resonance c u r v e when G = 0.149 r i s e s v e r y s h a r p l y as t h e a p p l i e d f r e q u e n c y a p p r o a c h e s t h e n a t u r a l f r e q u e n c y . The a c t u a l v a l u e o f "F" i s f i x e d a t u n i t y , t h e r e f o r e nCV. can o n l y be i n c r e a s e d by r e d u c i n g F c w h i c h a g a i n c a l l s f o r a r e d u c t i o n i n K 4. A good v a l u e f o r "C" Would be 0.6. To keep away from t h e r e s o n a n t p o i n t t h e n a t u r a l f r e q u e n c y s h o u l d be w e l l above the a p p l i e d f r e q u e n c y . S i n c e t h e a p p l i e d f r e q u e n c y i s f i x e d by d e s i g n r e q u i r e m e n t s t h e n a t u r a l f r e q u e n c y s h o u l d then be made as l a r g e as p o s s i b l e . Here a g a i n t h e c o n s t a n t K4 s h o u l d be made as s m a l l as p o s s i b l e . T h i s i n v e s t i g a t i o n has brought out t h r e e r e a s o n s why 28 the constant K 4 should be kept as low as p o s s i b l e . K 4 = "j(J.33til's Q* <28) E q u a t i o n (28) shows the r e l a t i o n s h i p between K 4 and the components o f t h e t i d a l b a s i n . "A" i s the area o f the b a s i n which i s approximately 12,TOO square f e e t , b i s the l e n g t h of the weirs which i s 40 f e e t , and Q, i s the pump di s c h a r g e which i s 20 cubic f e e t per second. In t h i s p a r t i c u l a r problem an i n c r e a s e i n b and Q, i s p r a c t i c a l . In f a c t f a c i l i t i e s are a v a i l a b l e to double both components. K 4 w i l l then be halv e d . T h i s m o d i f i c a t i o n w i l l i n c r e a s e the n a t u r a l frequency from 2 . 9 6 x 10- 2 r a d i a n s per second to 4.2 x 10- 2 r a d i a n s per second, i t w i l l i n c r e a s e " C " from 0.149 to 0.21 and i t w i l l reduce the time constant from 224 seconds to 112 seconds. In the d e s i g n of any f u t u r e model ba s i n s of t h i s type formula (28) may w e l l be used as a guide. The area should be made as small as p o s s i b l e and the weir l e n g t h and pump discharge should be as larp:e as p o s s i b l e . When "C" i s 0.3 or l e s s , then COn should be at l e a s t f i v e times 29 Acknowledgment The a u t h o r w i s h e s t o e x p r e s s h i s i n d e b t e d n e s s t o t h o s e who have a s s i s t e d h i m t h r o u g h o u t t h e c o u r s e o f t h i s r e s e a r c h ; e s p e c i a l l y t o Mr. W.B. C o u l t h a r d f o r h i s g u i d a n c e and e n c o u r a g e -ment . Acknowledgment i s a l s o made t o t h e N a t i o n a l R e s e a r c h C o u n c i l whose g r a n t made t h i s i n v e s t i g a t i o n p o s s i b l e . D . H .J . K a y The s e r v o - c o n t r o l mechanism o f t h e F r a s e r R i v e r M o d e l d i s c u s s e d i n t h i s t h e s i s was d e s i g n e d , t e s t e d and f a b r i c a t e d , i n t h e N a t i o n a l R e s e a r c h C o u n c i l l a b o r a t o r i e s , Ottawa. Inserted by request. Library, March 10th., 1952 30 B i b l i o g r a p h y L a v e r H e n r y , L e s n i c k R o b e r t , and M a t s o n L e s l i e E;, S e r v o m e c h a n i s m F u n d a m e n t a l s . New Y o r k and London, M c G r a w - H i l l , 1947. Brown G.S. and C a m p b e l l D.P., P r i n c i p l e s o f S e r v o m e c h a n i s m s , New Y o r k , W i l e y , 1948. H a l l A . C , " A p p l i c a t i o n o f C i r c u i t T h e o r y t o t h e D e s i g n o f S e r v o m e c h a n i s m s , " J o u r n a l o f t h e F r a n k l i n I n s t i t u t e , v o l . 242, pp.279-307, O c t o b e r , 1 9 4 6 . Schureman P a u l , Manual o f Harmonic A n a l y s i s and P r e d i c t i o n o f T i d e s , W a s h i n g t o n , U n i t e d S t a t e s Government P r i n t i n g O f f i c e , 1941. 

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