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Techniques applied to the design of the TRIUMF magnet poles Oraas, Sherman Roy 1970

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TECHNIQUES APPLIED TO THE DESIGN OF THE TRIUMF MAGNET POLES by SHERMAN ROY ORAAS B . A . S c , The U n i v e r s i t y of B r i t i s h Columbia (196?) A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF APPLIED SCIENCE in the Department of PHYSICS We accept t h i s t h e s i s as conforming to the requ i red s tandard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, Canada Date ApoU ' 7 , I?7Q i i ABSTRACT T h i s t h e s i s p r e s e n t s some o f t h e t e c h n i q u e s u s e d i n d e s i g n i n g t h e s e c t o r - f o c u s e d m a gnet f o r t h e TRIUMF c y c l o t r o n . An e m p i r i c a l m e t h o d i s g i v e n f o r c a l c u l a t i n g t h e magnet p o l e t i p s h a p e r e q u i r e d t o c o n t a i n a 500 MeV beam o f H" i o n s . The m e t h o d i s g o o d o n l y f o r s m a l l c h a n g e s i n t h e s h a p e . In t h e t e s t c a s e , t h e g e n e r a t e d p o l e t i p h a d a s p i r a l a n g l e c o r r e c t t o w i t h i n ±5 d e g r e e s , a n d a h i l l a n g l e c o r r e c t t o ±1 d e g r e e . T h e a v e r a g e f i e l d was f o u n d t o be i s o c h r o n o u s t o ±70 g a u s s . An e m p i r i c a l s o l u t i o n t o t h e p r o b l e m o f f i n d i n g t h e f i e l d i n s i d e t h e magnet a i r g a p i s a l s o g i v e n . The m a g n e t i c f i e l d r e s u l t i n g f r o m a g i v e n p o l e t i p c o n t o u r i s c a l c u l a t e d a t a p o i n t on t h e m e d i a n s u r f a c e by f i n d i n g t h e p e r p e n d i c u l a r d i s t a n c e f r o m t h e p o i n t t o t h e e d g e o f t h e p o l e a n d c o m p a r i n g t h i s t o an e x p e r i m e n t a l l y m e a s u r e d c u r v e o f f i e l d a g a i n s t d i s t a n c e . F i e l d s g e n e r a t e d by t h i s t e c h n i q u e h a v e t h e i r a v e r a g e s c o r r e c t t o w i t h i n 70 g a u s s a n d f l u t t e r t o w i t h i n 8%. A g a i n , p r e v i o u s k n o w l e d g e o f s i m i l a r p o l e t i p s i s a s s u m e d . The m e t h o d and r e s u l t s o f c a l c u l a t i n g t h e p o l e e d g e p o s i t i o n t o l e r a n c e s f o r t h e l a t e s t m odel magnet a r e g i v e n . The f i e l d s t r e n g t h s i n s i d e t h e s t e e l r e t u r n y o k e a s o b t a i n e d f r o m a s e r i e s o f f l u x m e a s u r e m e n t s a r e a l s o p r e s e n t e d . F i n a l l y , i t i s shown t h a t a s i m p l e a p p r o x i m a t i o n t o t h e m a g n e t i c c i r c u i t o f t h e m a g n e t p r e d i c t s t h e c o i l i n d u c t i o n r e q u i r e d t o an a c c u r a c y o f o n l y 25%. i v TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. POLE T IP DESIGN 3 2.1 I n t r o d u c t i o n 3 2.2 D e s i g n C r i t e r i a 3 2.2.1 I s o c h r o n i s m 3 2.2.2 F o c u s i n g 5 2.2.3 Ion L i f e t i m e 9 2.3 M a g n e t i c F i e l d D e s i g n 10 2.3-1 H i l l W i d t h 10 2.3.2 F l u t t e r 17 2.3.3 The D e s i g n P r o g r a m 21 2.4 Mk VI D e s i g n 22 2.4.1 R e s u l t s 22 2.4.2 Z e r o - o f f s e t i n e 27 3. AZIMUTHAL MAGNETIC FIELD PREDICTION 33 3-1 I n t r o d u c t i o n 33 3.2 F i t s t o V a r i o u s F o r m u l a e 33 3.3 E x p e r i m e n t a l F i e l d C u r v e s 34 4. TOLERANCES 42 4.1 T h e o r y 42 4.2 R e s u l t s 44 5. FLUX CALCULATIONS 46 5.1 F l u x Measurements 46 5.2 F l u x C a l c u l a t i o n s 51 5.3 C a l c u l a t i o n o f Magnet R e l u c t a n c e 54 R e f e r e n c e s 60 A p p e n d i x 1 61 A p p e n d i x 2 76 A p p e n d i x 3 88 A p p e n d i x 4 97 B i o g r a p h i c a l I n f o r m a t i o n 101 V LIST OF TABLES Page I. C o m p a r i s o n o f P r e d i c t e d B and F w i t h A c t u a l B and F f o r t h e Mk VI-1 Mod 10 Model Magnet 39 I I . T o l e r a n c e s f o r t h e Mk V l - l Mod 10 P o l e T i p 45 I I I . H o r i z o n t a l Yoke R a d i a l F l u x 50 IV. V e r t i c a l R e t u r n Yoke F l u x 50 V . C e n t r e R e g i o n F l u x 50 V I . H o r i z o n t a l Yoke T r a n s v e r s e F l u x 50 V I I . L e a k a g e F l u x 51 V I I I . C a l c u l a t e d F l u x D e n s i t y i n S t e e l f o r Mk V l - l Mod 9 Magnet 55 IX . C a l c u l a t e d F l u x D e n s i t i e s Compared w i t h A c t u a l F l u x D e n s i t i e s 55 V I LIST OF FIGURES Page 1. A p h o t o g r a p h o f t h e Mk V l - l model magnet 2 2. S p i r a l a n g l e vs r a d i u s f o r t h e Mk V l - l Mod 10 model magnet 8 3. " S q u a r e w a v e " a p p r o x i m a t i o n t o t h e a z i m u t h a l f i e l d c o n t o u r 12 k. H i l l - t o - h i l l d i s t a n c e and s p i r a l a n g l e o f p o l e t i p a r e d e f i n e d 12 5- H i l l - t o - h i l l d i s t a n c e D p l o t t e d a g a i n s t r a d i u s f o r t h e Mk V~3 model magnet 13 6. P a r a m e t e r s £ and <5 p l o t t e d a g a i n s t h i l l - t o - h i l l d i s t a n c e f o r t h e Mk V-3 model magnet 15 7. P a r a m e t e r a p l o t t e d a g a i n s t R n ' f o r t h e Mk V - l model magnet 16 8. Maximum h i l l f i e l d as used i n t h e d e s i g n o f t h e Mk VI magnet 16 9. Fudge f a c t o r p l o t t e d a g a i n s t h i l l - t o - h i l l d i s t a n c e f o r t h e Mk I I I model magnet 19 10. F l u t t e r p l o t t e d a g a i n s t <J>D f o r the Mk V l - l and Mk V-3 model magnets 20 11. D e v i a t i o n f r o m i s o c h r o n i s m f o r t h e Mk VI-1 Mod 0 model magnet 2k 12. V e r t i c a l b e t a t r o n f r e q u e n c y v p l o t t e d a g a i n s t e n e r g y f o r t h e Mk V l - l Mod 0 model magnet 2k 1 3 . D e s i g n f l u t t e r compared t o a c t u a l f l u t t e r f o r t h e Mk V l - l Mod 0 model magnet 25 1 4 . D e s i g n s p i r a l a n g l e compared t o the a c t u a l s p i r a l a n g l e f o r t h e Mk V l - l Mod 0 model magnet 25 15. S p i r a l a n g l e p r e d i c t i o n s f r o m e q u a t i o n s 8 and 19 compared t o t h e a c t u a l s p i r a l a n g l e o f the f i e l d 28 16. D e s i g n s p i r a l a n g l e p l o t t e d a g a i n s t a c t u a l s p i r a l a n g l e f o r t h e Mk V l - l Mod 10 model magnet 30 v i i 17- H i l l a n g l e h o f t h e d e s i g n and a c t u a l p o l e t i p s f o r t h e Mk V l - l Mod 10 model magnet 31 18. Saxon-Woods a p p r o x i m a t i o n t o f i e l d c u r v e 35 19- E x p e r i m e n t a l a r r a n g e m e n t t o d e t e r m i n e v a r i a t i o n o f f i e l d w i t h d i s t a n c e D 36 20. F i e l d v a r i a t i o n w i t h d i s t a n c e f o r t h e e x p e r i m e n t a l a r r a n g e m e n t o f F i g . 19 37 21. P r e d i c t e d and a c t u a l f i e l d c o n t o u r s f o r t h e Mk V l - l model magnet 22. R a d i a l f l u x m e a s u r i n g c o i l s 48 23. V e r t i c a l r e t u r n y o k e c o i l s hS 2k. T r a n s v e r s e c o i l s kS 25. F i e l d t h r o u g h c o i l s H and 16 as a f u n c t i o n o f t h e power s u p p l y p o t e n t i o m e t e r s e t t i n g 52 26. L e a k a g e f l u x c o i l s 53 27. M a g n e t i c c i r c u i t o f TRIUMF magnet 57 28. D i v i s i o n o f p o l e t i p used f o r t w o - s e c t i o n f l u x c a l c u l a t i o n s 57 v i i f ACKNOWLEDGEMENTS I s h o u l d l i k e t o t h a n k D r . E . G . A u l d f o r h i s many h e l p f u l s u g g e s t i o n s , and a l s o f o r h i s c o n t i n u a l o p t i m i s m and e n c o u r a g e m e n t , w i t h o u t w h i c h t h i s t h e s i s m i g h t n e v e r have been c o m p l e t e d . I s h o u l d a l s o l i k e t o t h a n k D r . G. M a c k e n z i e , D r . M. C r a d d o c k , and M r . A . O t t e r f o r t h e i r s u g g e s t i o n s and i m p r o v e m e n t s ; and Ada S t r a t h d e e , P a u l van R o o k , and t h e r e s t o f t h e TRIUMF s t a f f f o r t h e i r i n v a l u a b l e h e l p . I am g r a t e f u l t o TRIUMF f o r f i n a n c i a l s u p p o r t . 1 . INTRODUCTION The p r o b l e m o f d e s i g n i n g a magnet w h i c h w i l l p r o d u c e a mag-n e t i c f i e l d c a p a b l e o f c o n t a i n i n g a beam o f c h a r g e d p a r t i c l e s i s a v e r y d i f f i c u l t o n e . P r e s e n t - d a y n u m e r i c a l t e c h n i q u e s a r e n o t a d e q u a t e t o s o l v e M a x w e l l ' s e q u a t i o n s f o r t h e f i e l d i n s i d e a s t e e l shape as c o m p l e x as t h e TRIUMF magnet (see F i g u r e 1 ) , and hence d i f f e r e n t a p p r o a c h e s must be u s e d . S e c t i o n 2 o f t h i s t h e s i s i s an a t t e m p t t o c a l c u l a t e t h e p o l e g e o m e t r y r e q u i r e d t o p r o d u c e a c y c l o t r o n m a g n e t . S e c t i o n 3 i s t h e r e v e r s e a p p r o a c h - g i v e n a magnet s i m i l a r t o t h e TRIUMF m a g n e t , we can p r e d i c t w i t h some s u c c e s s t h e a z i m u t h a l f i e l d c o n t o u r s . S e c t i o n k g i v e s t o l e r a n c e s on t h e d i m e n s i o n s o f t h e p o l e t i p , as d e r i v e d f r o m t h e c o n d i t i o n s n e c e s s a r y t o p r o d u c e a c o n t a i n e d beam o f H - i o n s . S e c t i o n 5 g i v e s t h e r e s u l t s o f t h e f l u x m e a s u r e -ments on t h e TRIUMF model m a g n e t , and i t a l s o shows t h a t t h e s i m p l e e q u i v a l e n t m a g n e t i c c i r c u i t c a n n o t be e a s i l y a p p l i e d h e r e . P o l e P i e c e H o r i z o n t a l Yoke V e r t i c a l Yoke F i g u r e 1 . A p h o t o g r a p h o f t h e Mk V l - l model magnet - 3 2. POLE T I P DESIGN 2.1 In t r o d u c t i on The method used f o r c a l c u l a t i n g t h e s h a p e o f a p o l e t i p i s b a s i c a l l y t h a t o u t l i n e d i n " P r o g r e s s i n N u c l e a r T e c h n i q u e s and I n s t r u m e n t a t i o n " , by J . R . R i c h a r d s o n , 1 w i t h some m o d i f i c a t i o n s . The m o d i f i c a t i o n s a r e an a t t e m p t a t a p p l y i n g s e c o n d o r d e r c o r r e c t i o n s t o t h e s q u a r e wave a p p r o x i -m a t i o n t o t h e a z i m u t h a l f i e l d , t a k i n g i n t o a c c o u n t t h e g e o m e t r i c a l shape o f t h e s e c t o r e d magnet and t h e c o m p l e x i n t e r d e p e n d e n c e o f t h e v a r i o u s d e s i g n p a r a m e t e r s . From t h e measurements done on t h e model m a g n e t s , t h e s e c o r r e c t i o n s were found t o be r e l i a b l e o n l y f o r s m a l l changes i n t h e p o l e t i p s h a p e . 2 .2 D e s i g n C r i t e r i a T h e r e a r e t h r e e c o n d i t i o n s w h i c h must be met i f a 500 MeV, 100 pA, CW H~ beam s u i t a b l e f o r e x t r a c t i o n f r o m t h e c y c l o t r o n i s t o be p r o d u c e d . These a r e : 1) I s o c h r o n i s m 2) F o c u s i n g 3) Ion L i f e t i m e 2 .2 .1 I s o c h ron ? sm The i s o c h r o n o u s c o n d i t i o n r e q u i r e s t h a t the i o n must r o t a t e a t a c o n s t a n t a n g u l a r v e l o c i t y m, and hence i t d e f i n e s t h e a z i m u t h a l a v e r a g e f i e l d r e q u i r e d f o r any p a r t i c l e e n e r g y . We b e g i n w i t h t h e r e l a t i v i s t i c e q u a t i o n s E T Y E m c ^ " = 1 + ^ ( , ) and 3 ~= * = / T H ; ( 2 ) where E = t o t a l e n e r g y o f p a r t i c l e , T = k i n e t i c e n e r g y , m = r e s t m a s s , v = v e l o c i t y , and c = v e l o c i t y o f l i g h t . I f we assume t h a t t h e p a r t i c l e f o l l o w s a c i r c u l a r o r b i t , we can use t h e e q u a t i o n f o r t h e L o r e n t z f r e q u e n c y o f a p a r t i c l e w i t h c h a r g e q i n a m a g n e t i c f i e l d : a) = ^ = c o n s t a n t f o r any c y c l o t r o n B = ^ l (3) where B i s t h e f i e l d a v e r a g e d o v e r one r e v o l u t i o n . T h i s e q u a t i o n d e f i n e s t h e i s o c h r o n i s m c o n d i t i o n . The c e n t r a l f i e l d , o r t h e f i e l d r e -q u i r e d a t z e r o p a r t i c l e e n e r g y , i s B = ™ L . ' (k) c q f r o m eqn.(3) , l e t t i n g y = The r a d i u s a t w h i c h a p a r t i c l e o f e n e r g y E o r b i t s i s d e t e r m i n e d f r o m v _ gc w r r Thus r = 1^  . (5) to We can d e f i n e t h e r a d i u s a t w h i c h a p a r t i c l e o f i n f i n i t e e n e r g y w o u l d r o t a t e : c_ f r o m e q n . (5). From eqns.(3) and (k) we have B" 1 1 = Y B c / l - B z A - ( r / r ) 2 (6) The v a l u e o f t h e o r b i t i n g f r e q u e n c y to i s s e t by d e s i g n c r i t e r i a .1) and 3). 2.2.2 Focus i n g In o r d e r t o a c h i e v e an a d e q u a t e e x t r a c t i o n e f f i c i e n c y , t h e beam must be f o c u s e d r a d i a l l y and v e r t i c a l l y . The r a d i a l c o n d i t i o n i s s a t i s -f i e d a u t o m a t i c a l l y , s i n c e f o r any m a n y - s e c t o r e d i s o c h r o n o u s c y c l o t r o n s u c h as TRIUMF, t h e number o f r a d i a l b e t a t r o n o s c i l l a t i o n s o f the beam p e r t u r n v r % Y , and hence > 0.8 Thus t h e beam i s c o n t a i n e d r a d i a l l y . The o n l y r e a l p r o b l e m i s t o a v o i d r e s o n a n c e s s u c h as t h e = 1 r e s o n a n c e , b u t t h i s w i l l a f f e c t o n l y d e t a i l s o f t h e m a g n e t , no t t h e o v e r a l l g e n e r a l s h a p e . Hence i t w i l l n o t be c o n s i d e r e d h e r e . V e r t i c a l f o c u s i n g i s more d i f f i c u l t t o a c h i e v e . The r a d i a l l y i n -c r e a s i n g f i e l d r e q u i r e d f o r i s o c h r o n i s m p r o d u c e s a v e r t i c a l l y d e f o c u s i n g f o r c e w h i c h must be compensated f o r by o t h e r p r o p e r t i e s o f t h e m a g n e t . U s i n g t h e method o f S m i t h and G a r r e n , 2 we can e x p r e s s t h e r a d i u s o f c u r v a t u r e y ( r , e ) o f t h e o r b i t i n te rms o f a mixed F o u r i e r and power s e r i e s , as f o l l o w s : y(r,e) = y(r ,0 ) 1 + y'x + y " — + y' x 3 + y (a c o s n0 + b s i n n6) + x y ( a ' cos n0 + b ' s i n n0) 1 n n y ( a " cos n6 + b " s i n n0) + 2 k n n (7) where x i s d e f i n e d by r = r n ( l + x) , - 6 -i s t h e r a d i u s a t w h i c h t h e a z i m u t h a l a v e r a g e o f p i s a p p r o p r i a t e t o t h e p a r t i c l e e n e r g y , i . e . , r Q VU~BT - 1 , and a and b a r e t h e s t a n d a r d F o u r i e r c o e f f i c i e n t s , n n I f we use t h i s a p p r o x i m a t i o n t o s o l v e t h e e q u a t i o n s o f m o t i o n o f t h e p a r t i c l e , we o b t a i n t h e r a t h e r cumbersome e x p r e s s i o n f o r v 2 g i v e n i n a p p e n d i x 2 o f S m i t h and G a r r e n . T h i s e q u a t i o n c a n n o t be e a s i l y r e d u c e d t o a u s e f u l f o r m by a p p r o x i m a t i o n s w h i c h a r e v a l i d f o r TRIUMF. Hence i t was d e c i d e d t o t r y t h e r e v e r s e a p p r o a c h - c h o o s e an a p p r o x i m a t e f o r m u l a and see how a c c u r a t e i t i s . The f o r m u l a c h o s e n was t h e smooth a p p r o x i -m a t i o n f o r v 2 , as g i v e n by Symen e t a l , 1 1 v 2 = + F ^ l + 2 t a n 2 e ) (8) where F 2 = " f l u t t e r " o f f i e l d = ¥ 2 ) / B 2 , e = s p i r a l a n g l e o f f i e l d (see F i g u r e 4), and u ' = J l — = g 2 v 2 i f i s o c h r o n i s m h o l d s . B 9r T h i s f o r m u l a i s c o m p r i s e d o f a f o c u s i n g and a d e f o c u s i n g t e r m . The d e -f o c u s i n g t e r m , -\x", i s due t o t h e i n w a r d " b o w i n g " o f t h e f i e l d l i n e s i n t h e r a d i a l l y i n c r e a s i n g f i e l d r e q u i r e d i n a r e l a t i v i s t i c c y c l o t r o n . The f o c u s i n g t e r m , F ^ l + 2 t a n 2 e ) , i s due t o t h e t h r e e f o c u s i n g f o r c e s a r i s i n g f r o m t h e s p i r a l s e c t o r geomet ry o f t h e m a g n e t : t h e Thomas, a l t e r n a t i n g g r a d i e n t , and L a s l e t t f o r c e s . 8 A t h i g h e n e r g i e s , e a c h o f t h e s e two terms w i l l be q u i t e l a r g e , but t h e sum s h o u l d be s m a l l and f o c u s i n g . The l i m i t s on w i l l be d i s c u s s e d i n S e c t i o n 4.1. From e q u i l i b r i u m o r b i t p rograms s u c h as C Y C L O P S , 3 v 2 can be d e t e r m i n e d w i t h i n t h e n e c e s s a r y a c c u r a c y . S i n c e F 2 i s known f o r a g i v e n - 7 -f i e l d , we can use e q n . (8) t o c a l c u l a t e t h e s p i r a l a n g l e e. The a c t u a l s p i r a l a n g l e o f t h e f i e l d can be o b t a i n e d d i r e c t l y f r o m the F o u r i e r h a r m o n i c s o f t h e f i e l d . I f co B(e) = F + J A s i n ( n 6 + cb ) , u n T n ' n=l t h e n e = r £*n n d r where A = a m p l i t u d e o f n t n h a r m o n i c n r cb = phase " " " n r e = s p i r a l a n g l e " " n r 3 The s p i r a l a n g l e e o f t h e f i e l d B(e) can be f o u n d by a p p l y i n g t h e r e l a t i o n oo ^ T A 2 ( l + 2 t a n 2 e ) = F^ l + 2 t a n 2 e ) . 2 L n n n=o H e n c e , by c o m p a r i n g t h e s p i r a l a n g l e o b t a i n e d by e q n . (8) w i t h t h e a c t u a l s p i r a l a n g l e o f t h e f i e l d , we can e s t i m a t e t h e a c c u r a c y o f e q n . ( 8 ) . The r e s u l t s f r o m a t y p i c a l model measurement a r e shown i n F i g u r e 2 . I t can be seen t h a t f o r R > 250 i n . t h e maximum d e v i a t i o n Ae f r o m the two c u r v e s i s ± 2 ° , and f o r R < 150 i n . , Ae = 1 4 ° . These e r r o r s i n e can be t r a n s l a t e d i n t o e r r o r s i n v 2 by t h e use o f e q n . ( 8 ) . The w o r s t v a l u e s o f Av| o c c u r a t R = 150 i n . and R = 305 i n . , where Av 2 = ±0 .022 and ± 0 . 1 7 7 , r e s p e c t i v e l y . However , t h e s e numbers do not mean much by t h e m s e l v e s . S i n c e v 2 i s a d i f f e r e n c e o f two r e l a t i v e l y l a r g e n u m b e r s , we s h o u l d c o n -s i d e r Av 2 as a p e r c e n t a g e o f e i t h e r t h e f o c u s i n g t e r m o r t h e d e f o c u s i n g t e r m . We f i n d t h a t a t b o t h 150 i n . and 305 i n . r a d i u s Av 2 = ±\k%. An e r r o r was t o be e x p e c t e d b e c a u s e o f t h e i g n o r e d te rms i n t h e f u l l - 8 -R a d i u s ( i n c h e s ) F i g u r e 2 . S p i r a l a n g l e v s r a d i u s f o r t h e Mk V l - l Mod 10 model magnet - 9 -e x p r e s s i o n f o r v 2 . These r e s u l t s g i v e some c o n f i r m a t i o n t h a t t h e a p p r o x i m a t i o n s i m p l i c i t i n eqn .(8) a r e s u f f i c i e n t l y v a l i d t h a t t h e use o f s u c h an e q u a t i o n and the use o f model magnet measurements a r e a d e q u a t e t o d e s i g n a s e c t o r - f o c u s e d , i s o c h r o n o u s m a g n e t . 2.2.3 Ion L i f e t i m e C o n s i d e r a t i o n s s u c h as r a d i a t i o n l e v e l s i n t h e c y c l o t r o n v a u l t and r e s i d u a l a c t i v i t y i n t h e component p a r t s o f t h e c y c l o t r o n have l e d us t o l i m i t t h e a l l o w a b l e beam l o s s i n TRIUMF t o 10 kW. T h i s means t h a t t h e beam l o s s s h o u l d be l e s s t h a n 20% o f t h e e x t r a c t e d beam. T h e r e a r e two m a j o r c a u s e s o f beam l o s s , b o t h o f w h i c h r e s u l t f r o m t h e v e r y low b i n d i n g e n e r g y o f t h e e x t r a e l e c t r o n i n t h e H~ i o n . F i r s t , t h e e l e c t r o n can be s t r i p p e d by c o l l i s i o n w i t h gas m o l e c u l e s . T h i s removes t h e c h a r g e o f t h e i o n , and s i n c e i t w i l l no l o n g e r be a f f e c t e d by t h e m a g n e t i c f i e l d , i t w i l l f l y o u t o f t h e m a c h i n e . Hence we must have as low a gas p r e s s u r e as p o s s i b l e i n t h e vacuum t a n k . A p r e s s u r e o f 10" 7 T o r r was c h o s e n f o r TRIUMF, as t h i s i s f a i r l y e a s i l y a c h i e v e d w i t h p r e s e n t - d a y vacuum t e c h n o l o g y , and i t r e s u l t s i n a beam power l o s s . The e l e c t r o n s c a n a l s o be s t r i p p e d by t h e e l e c t r i c f i e l d w h i c h r e s u l t s f r o m t h e p a r t i c l e mov ing t h r o u g h a m a g n e t i c f i e l d . The e q u i v a -l e n t e l e c t r i c f i e l d i n t h e r e s t f rame o f t h e H~ i o n i s e = 0.3 ByB x 10 6 v o l t s / c m (9) where $ and y a r e as d e f i n e d p r e v i o u s l y , and B i s t h e m a g n e t i c f i e l d i n k i l o g a u s s . The maximum e l e c t r i c f i e l d i n t h e c a s e o f TRIUMF i s a b o u t 2.0 MV/cm. T h e o r e t i c a l c a l c u l a t i o n s o f t h e l i f e t i m e o f t h e H~ i o n i n t h i s f i e l d have been made by M u l l e n , 5 H i s k e s 1 2 a n d o t h e r s , b u t t h e y do n o t a g r e e w e l l w i t h t h e e x p e r i m e n t a l r e s u l t s o f S t i n s o n e t a l . 6 U s i n g - 10 -a f u n c t i o n o f t h e f o r m D exp - , . fc* (10) S t i n s o n e t a l f o u n d t h a t C = 7-96 x 10"11+ sec -MV/cm and D = 42.56 MV/cm. F o r e << D, x(e) i s a v e r y r a p i d l y d e c r e a s i n g f u n c t i o n o f e. In o u r c a s e , h a l v i n g t h e m a g n e t i c f i e l d r a i s e s x by a f a c t o r o f h x 10 3. Hence e q n . (9) s e t s a l i m i t on t h e maximum f i e l d . L e t t i n g t h e maximum h i l l f i e l d e q u a l 5.76 kG w i l l g i v e a beam power l o s s o f a b o u t 11.2% f o r T R I U M F . k M a g n e t i c F i e l d D e s i g n So f a r , f o r any g i v e n beam e n e r g y we can c a l c u l a t e y, g, B, and R. Two t a s k s r e m a i n . One i s t o f i n d t h e h i l l w i d t h n w h i c h i s c o n s i s -c o n d i t i o n . The o t h e r i s t o o b t a i n t h e f l u t t e r and s p i r a l a n g l e n e c e s s a r y t o s a t i s f y t h e v e r t i c a l f o c u s i n g c r i t e r i o n . The t e c h n i q u e s a p p l i e d t o s o l v i n g t h e s e two t a s k s a r e i n e x a c t , as t h e r e l a t i o n s h i p between t h e r e q u i r e d f i e l d shape and t h e shape o f t h e magnet p o l e p i e c e i s a c o m p l e x g e o m e t r i c a l f u n c t i o n . An e m p i r i c a l s o l u t i o n has been a p p l i e d u s i n g the a p p r o x i m a t e f o r m u l a e d i s c u s s e d p r e v i o u s l y . 2.3.1 H i l l W i d t h I f t h e f i e l d were a s q u a r e w a v e , we w o u l d have no p r o b l e m c a l c u -l a t i n g n , t h e h i l l w i d t h i n d e g r e e s . Tha t i s , i f t h e h i l l f i e l d and v a l l e y f i e l d were f l a t and e q u a l t o F^ and F^ r e s p e c t i v e l y , t h e n we can see f r o m F i g u r e 3 t h a t , f o r an N - s e c t o r m a c h i n e , t e n t w i t h B max b e i n g l e s s t h a n 5-76 kG and B s a t i s f y i n g t h e i s o c h r o n i s m n - 3 6 0 ( B ' ! v ) . (11) N (FH - By) F o r TRIUMF, N = 6 . The r e a s o n s a r e : 1) f o r N 4 , t h e beam w i l l p a s s t h r o u g h the r a d i a l s t o p band r e s o n a n c e s b e f o r e i t r e a c h e s 500 M e V ; 1 0 2) N > 8 w i l l n o t y i e l d enough f l u t t e r a t l a r g e r a d i i t o p e r m i t a d e q u a t e v e r t i c a l f o c u s i n g ; and 3) N = 5 and N = 1 a r e i n c o n v e n i e n t numbers t o work w i t h . Hence e q n . ( 1 1 ) becomes n - 6 0 _ ' ° - J v » (,2) B H - By So i f we know B, B u , and B.. , t h e n n i s d e t e r m i n e d . The a v e r a g e f i e l d B i s H V f i x e d by the i s o c h r o n i s m c o n d i t i o n and we assume we can o b t a i n any d e s i r e d B^ by a d d i n g o r r e m o v i n g s t e e l t o o r f r o m t h e r e t u r n y o k e . Hence t h e t a s k i s t o f i n d By . A r e a l f i e l d l o o k s more l i k e t h e " r o u n d e d " c u r v e i n F i g u r e 3-What i s needed i s some p a r a m e t e r w h i c h depends s o l e l y on t h e geomet ry o f the p o l e t i p and p r o v i d e s some measure o f t h e " r o u n d i n g " o f t h e c o r n e r s o f t h e f i e l d . Such a p a r a m e t e r i s t h e h i l l - t o - h i l l d i s t a n c e D . 9 T h i s i s d e f i n e d f o r a g i v e n r a d i u s R as t h e d i a m e t e r o f t h e l a r g e s t c i r c l e t h a t can be drawn c e n t r e d on R and j u s t t o u c h i n g t h e edges o f t h e a d j a c e n t p o l e t i p s (see F i g u r e k). F i g u r e 5 i s a g r a p h o f D p l o t t e d a g a i n s t R f o r t h e Mk V-3 model m a g n e t . The p r o b l e m o f f i n d i n g t h e h i l l w i d t h w o u l d be s o l v e d i f we c o u l d f i n d some way o f r e l a t i n g t h e minimum v a l l e y f i e l d B y , t h e a v e r a g e v a l l e y f i e l d B y , and t h e a v e r a g e h i l l f i e l d B^ t o t h e maximum h i l l f i e l d B^ v i a some p a r a m e t e r w h i c h depends o n l y on t h e g e o m e t r y o f t h e p o l e t i p . - 12 -P o l e tip P o l e tip K\NNN A c t u a l f i e l d S q u a r e wave a p p r o x i m a t i o n i g u r e 3. " S q u a r e w a v e " a p p r o x i m a t i o n t o t h e a z i m u t h a l f i e l d c o n t o u r - 13 -F i g u r e 5. Hi 11-to-hi11 d i s t a n c e D p l o t t e d a g a i n s t r a d i u s f o r t h e Mk V-3 model magnet - lit -D has p r o v e n t o be v e r y u s e f u l f o r t h i s . The r a t i o (13) i s a g o o d , s i n g l e - v a l u e d f u n c t i o n o f D, as can be s e e n i n F i g u r e 6,. A l t h o u g h D f i r s t i n c r e a s e s and t h e n d e c r e a s e s w i t h r a d i u s , E, r e m a i n s s i n g l e - v a l u e d i n D t o w i t h i n ±5% f o r r a d i i w i t h i n 0 t o 14.5 i n . on t h e m o d e l . A t t h e 15 i n . r a d i u s , t h e dependence o f 5 on D i s n o t s o g o o d , as t h e a b r u p t c u t - o f f o f the p o l e t i p a t 15.5 i n . b e g i n s t o have an e f f e c t . To a l e s s e r e x t e n t , t h e r a t i o i s a l s o s i n g l e - v a l u e d i n D. As shown i n F i g u r e 6, t h e r e i s a d e f i n i t e d i f f e r e n c e between 6 f o r i n c r e a s i n g D and f o r d e c r e a s i n g D. F i t t i n g a c u r v e between t h e two r e s u l t s i n a d e t e r m i n a t i o n o f 6 t o w i t h i n ± 3 % . The f i n a l p a r a m e t e r n e c e s s a r y t o p r o d u c e a s q u a r e wave f i e l d a p p r o x i m a t i o n i s We f i n d t h a t a i s a f u n c t i o n o f t h e h i l l w i d t h i n i n c h e s Rn' ( see F i g u r e 7 ) , where n' i s t h e h i l l w i d t h i n r a d i a n s . T h r e e d i f f e r e n t p r o -c e s s e s a r e i n e v i d e n c e h e r e . For Rn' << s , where s i s t h e gap l e n g t h , a i s a d e c r e a s i n g f u n c t i o n o f Rn', b e c a u s e t h e s t e e l i s e f f e c t i v e l y so f a r f r o m t h e median p l a n e t h a t t h e f i e l d c o n t o u r does not f o l l o w t h e s t e e l c o n t o u r . F o r Rn' > s , a i n c r e a s e s w i t h Rn' b e c a u s e t h e f i e l d i s now f l a t e x c e p t n e a r t h e edges o f t h e s t e e l . F o r l a r g e Rn', s a t u r a t i o n 6 s (14) (15) - 15 -1.0 0.9 0.8 0.7 0.6 > cn ii 0.5 0.4 0.3 0.2 + - 5 f o r i n c r e a s i n g D ° - £ f o r d e c r e a s i n g D • - 6 f o r i n c r e a s i n g D A - & f o r d e c r e a s i n g D £ (use LH s c a l e ) • S -~ ~~ A 6 (use RH s c a l e ) / / A/ , 9 « R = 311" / / + o 1.7 1.6 1.5 1.4 « n o 1.3 1.2 0.1 / • 1.1 / _L 1.0 2 3 D (model i n c h e s ) F i g u r e 6. P a r a m e t e r s £• and 6 p l o t t e d a g a i n s t h i l l - t o - h i l l d i s t a n c e f o r t h e Mk V-3 model magnet C O 0 50 100 150 200 R n " ( f u l l - s c a l e i n c h - r a d i a n s ) F i g u r e 7. P a r a m e t e r a p l o t t e d a g a i n s t R n ' f o r t h e Mk V - l model magnet J L J L JL J L J L F i g u r e 8. 100 200 R a d i u s ( f u l l s c a l e i n c h e s ) 300 Maximum h i l l f i e l d as used i n t h e d e s i g n o f t h e Mk VI magnet - 17 -i n t h e s t e e l c a u s e s t h e f i e l d t o d r o o p f u r t h e r i n f r o m t h e e d g e , and hence a a g a i n d e c r e a s e s . In any c a s e , s i n c e a v a r i e s o v e r s u c h a s m a l l r a n g e , i t can e a s i l y be d e t e r m i n e d t o w i t h i n ±\%. I t i s no t as c r i t i c a l a p a r a m e t e r as e i t h e r E, o r 6. By means o f t h e p a r a m e t e r s E,, 6, and a, we a r e now a b l e t o p r o -duce a s q u a r e - w a v e a p p r o x i m a t i o n t o t h e r e a l f i e l d , g i v e n j u s t B^ and D. The n e c e s s a r y r e l a t i o n s a r e : B y = ?6BH . We can f i n d n by s u b s t i t u t i n g t h e s e v a l u e s i n t o eqn.(12)to g e t „ _ 6o ( B - ( 1 6 ) (ct - £<5)B.H B^ i s an e x p e r i m e n t a l l y a d j u s t a b l e p a r a m e t e r . Tha t i s , we assume we can o b t a i n any d e s i r e d v a l u e j u s t by r e m o v i n g o r a d d i n g s t e e l above t h e p o l e t i p . The o n l y r e s t r i c t i o n i s t h a t B u must be l e s s t h a n 5.76 k G , b e c a u s e o f t h e e l e c t r i c d i s s o c i a t i o n o f t h e H~ i o n . H e n c e , any r e a s o n a b l e -l o o k i n g c u r v e o f B^ a g a i n s t r a d i u s can be used f o r d e s i g n i n g a p o l e t i p (some e x p e r i e n c e i s r e q u i r e d t o d e t e r m i n e what i s a r e a s o n a b l e c u r v e ) . The c u r v e used i n t h e Mk VI model magnet d e s i g n i s shown i n F i g u r e 8. 2.3-2 F l u t t e r Once a g a i n , i f t h e a z i m u t h a l f i e l d were s q u a r e , c o m p u t i n g t h e f l u t t e r w o u l d be t r i v i a l . However , i t i s n o t , and t h u s e m p i r i c a l t e c h n i q u e s were u s e d . Two methods were t r i e d . 1) Fudge F a c t o r The f u d g e f a c t o r FF i s d e f i n e d a s : - 18 -a c t u a l f l u t t e r (17) FF = f l u t t e r c a l c u l a t e d f o r " s q u a r e " f i e l d The f l u t t e r f o r a s q u a r e f i e l d i s F 2 = ( B H - B) 2n + (60 - n)(B - B y ) 2 60 Q2 (aBj_| - B) 2n + (60 - n) (B - e 6 B H ) 60 F Some a c t u a l v a l u e s o f FF w e r e o b t a i n e d f r o m t h e Mk I I I model m a g n e t . An a t t e m p t was made t o c o r r e l a t e t h e m w i t h some p a r a m e t e r o f t h e magnet g e o m e t r y . The b e s t o n e seemed t o be D , t h e h i l l - t o - h i l l d i s t a n c e , b u t e v e n t h a t d i d n o t r e s u l t i n FF b e i n g s i n g l e - v a l u e d , a s c a n be s e e n i n F i g u r e 9- T h e s o l i d l i n e i s a q u a d r a t i c l e a s t - s q u a r e s f i t t o t h e d a t a p o i n t s . An e r r o r o f up t o ±15% i n t h e f l u t t e r c a n be e x p e c t e d by t h e u s e o f t h i s t e c h n i q u e . I t was f o u n d t h a t t h e f l u t t e r i s a p p r o x i m a t e l y a s i n g l e - v a l u e d f u n c t i o n o f cbD. <j>D i s t h e p r o d u c t o f D , t h e h i l l - t o - h i l l d i s t a n c e , a n d cb, t h e v a l l e y w i d t h i n r a d i a n s . F i g u r e 10 shows a p l o t o f F v s cbD, u s i n g d a t a f r o m two d i f f e r e n t m o d e l s . T h e s o l i d c u r v e i s a q u a d r a t i c f i t t e d v i a a l e a s t - s q u a r e s t e c h n i q u e t o t h e Mk V-3 d a t a p o i n t s o n l y . T h i s c u r v e g i v e s v a l u e s o f F c o r r e c t t o w i t h i n ± 0 . 0 2 , o r ±15% a t t h e l a r g e s t r a d i u s , w h i c h i s t h e m o s t c r i t i c a l p o i n t . T h e e r r o r s i n e i t h e r t h e f u d g e - f a c t o r o r t h e cbD m e t h o d a r e r o u g h -l y t h e same. The p r o g r a m t o be d e s c r i b e d h a d an o p t i o n w h e r e b y e i t h e r t e c h n i q u e c o u l d be u s e d . 2) $D - 19 -0.8 r R = 3 1 1 " 0.7 o 0.6 u (0 cn •a 3 0.5 0.4 / / / + / / / / / / / / + +. 0.3 -L 2 3 4 H i l l - t o - h i l l D i s t a n c e D (model i n c h e s ) F i g u r e 9. F u d g e . f a c t o r p l o t t e d a g a i n s t h i l l - t o - h i l l d i s t a n c e f o r t h e Mk I I I model m a g n e t . The d a s h e d l i n e i s p o l y n o m i a l f i t t o t h e d a t a p o i n t s by a l e a s t s q u a r e s t e c h n i q u e . - 20 -<j>D F i g u r e 10. F l u t t e r p l o t t e d a g a i n s t <J>D f o r the Mk V l - l and Mk V-3 model magnets - 21 -2 . 3 - 3 The D e s i g n P r o g r a m A c o m p u t e r p r o g r a m c a l l e d "MAGNET", a l i s t i n g o f w h i c h c a n be f o u n d i n A p p e n d i x 1 , was w r i t t e n t o a s s i s t i n t h e d e s i g n o f a p o l e t i p . I t makes s a t i s f a c t o r y p r e d i c t i o n s o n l y f o r s m a l l changes i n a p o l e t i p . I t s o p e r a t i o n i s as f o l l o w s . The w h o l e e n e r g y range o f t h e m a c h i n e i s d i v i d e d i n t o many s m a l l i n c r e m e n t s . F o r each o f t h e s e , y> 3, r , and B a r e c a l c u l a t e d . N e x t , a c u r v e o f D vs R i s assumed f r o m t h e l a s t magnet p o l e p i e c e . From t h i s and f r o m the model m e a s u r e m e n t s , v a l u e s o f 5, 6, and a a r e f o u n d . S i n c e we have assumed we can o b t a i n any d e s i r e d v a l u e o f B u , a c u r v e o f B u vs R, w h i c h i s s i m i l a r t o t h e l a s t m a g n e t , i s used t o o b t a i n v a l u e s o f B u and hence n f o r e a c h e n e r g y i n c r e m e n t . The t e c h n i q u e s g i v e n i n S e c t i o n 2 . 3 . 2 a r e now employed t o y i e l d t h e f l u t t e r , and t h e n by e q n . (8) we can c a l c u l a t e t h e s p i r a l a n g l e e. The g e o m e t r i c r e l a t i o n s h i p h = t a " £ dg h = h i 11 a n g l e (18) a l l o w s us t o compute the a c t u a l a n g u l a r p o s i t i o n s o f t h e edges o f t h e p o l e t i p t o g i v e t h e d e s i r e d s p i r a l a n g l e e. F o l l o w i n g t h i s p r o c e d u r e t h r o u g h f o r a l l t h e e n e r g y i n t e r v a l s r e s u l t s i n a p o l e t i p . We can now i t e r a t e t h e w h o l e p r o c e s s by m e a s u r -i n g t h e h i l l - t o - h i l l d i s t a n c e o f t h i s p o l e t i p and p u t t i n g i t b a c k i n t o t h e p r o g r a m . When t h e h i l l - t o - h i l l d i s t a n c e o f the d e s i g n e d p o l e t i p i s s i m i l a r t o t h a t used i n the c a l c u l a t i o n s , t h e n the d e s i g n i s c o m p l e t e . - 22 -Mk VI D e s i g n 2.4.1 R e s u l t s I t was d e c i d e d t o use t h e p r o g r a m t o d e s i g n t h e Mk VI p o l e t i p . The Mk V p o l e t i p s had f a i l e d f o r v a r i o u s r e a s o n s . The Mk V - l d e s i g n , w h i c h had a h i l l w i d t h o f 24.5° f o r R < 150 i n . , was u n a c c e p t a b l e b e c a u s e t h e h i g h h i l l f i e l d r e q u i r e d f o r i s o c h r o n i s m a t low r a d i i was u n a t t a i n a b l e e x p e r i m e n t a l l y . The Mk V-3 p o l e p i e c e had a c e n t r a l h i l l w i d t h o f 26° and a l t h o u g h i s o c h r o n i s m c o u l d be a c h i e v e d , t h e p o l e t i p was s o w i d e t h a t beam e x t r a c t i o n became i m p o s s i b l e . A t l a r g e r a d i i t h e p o l e t i p o v e r l a p p e d t h e r e t u r n y o k e o f the a d j a c e n t s e c t o r , r e s u l t i n g i n a c o n t i n u o u s v e r t i c a l r e t u r n y o k e a r o u n d t h e c y c l o t r o n . A l s o , t h e r a t h e r l a r g e h i l l w i d t h r e s u l t e d i n a low f l u t t e r , w h i c h gave r i s e t o i n a d e q u a t e v e r t i c a l f o c u s i n g a t low r a d i i . On b o t h t h e Mk V - l and Mk V-3 models v e r t i c a l f o c u s i n g was s u f f i c i e n t l y h i g h f o r r a d i i g r e a t e r t h a n 150 i n . On t h e b a s i s o f t h e s e two m o d e l s , i t was d e c i d e d t h a t t h e Mk VI d e s i g n s h o u l d have a c e n t r a l h i l l w i d t h o f 2 5 ° . The g e n e r a l shape s h o u l d be somewhere between t h e Mk V - l and Mk V-3 d e s i g n s , and hence most o f t h e magnet p a r a m e t e r s , s u c h as 5, 6, and F 2 , s h o u l d l i e between t h o s e f o r t h e two Mk V m o d e l s . A t t h i s p o i n t , t h e p r o g r a m was a l t e r e d s l i g h t l y . As m e n t i o n e d i n S e c t i o n s 2.3-1 and 2.3-2, 6 can be p r e d i c t e d t o w i t h i n ± 3 % , and F 2 t o w i t h i n ±15%. S i n c e t h e s e p a r a m e t e r s s h o u l d l i e between t h o s e f o u n d f o r t h e Mk V m o d e l s , i t was f e l t t h a t more a c c u r a t e v a l u e s c o u l d be f o u n d by c h o o s i n g some v a l u e between t h e Mk V - l and Mk V-3 v a l u e s f o r t h a t p a r t i c u l a r r a d i u s . The f o r m u l a used t o f i n d 6 was - 23 -6 = n 0 - T l x 63 - 6 l where nQ = 2 5 ° , r\x = 2 4 . 5 ° , n 3 = 2 6 ° , Sl = Mk V - l , & 3 = Mk V-3. F 2 was f o u n d i n a s i m i l a r m a n n e r . I t was hoped t h a t t h i s i n t e r p o l a t i o n t e c h n i q u e w o u l d r e d u c e t h e e r r o r i n 6 t o ±1%, and F 2 t o l e s s t h a n ±5%. A p o l e t i p was b u i l t u s i n g t h i s d e s i g n . The r e s u l t s a r e shown i n F i g u r e s 11 and 1 2 . I s o c h r o n i s m was e x c e l l e n t ; t h e maximum d i f f e r e n c e was 75 g a u s s . F i g u r e 11 shows the d i f f e r e n c e between t h e i s o c h r o n o u s f i e l d and t h e a c t u a l a v e r a g e f i e l d w h i c h has been n o r m a l i z e d t o t h e d e s i g n h i l l f i e l d . The n o r m a l i z a t i o n was n e c e s s a r y b e c a u s e t h e s t e e l r e t u r n y o k e was n o t the c o r r e c t shape r e q u i r e d t o o b t a i n t h e d e s i g n B u . n U n f o r t u n a t e l y , was i n a d e q u a t e (see F i g u r e 12). The d e s i g n was 0.35, but t h e a c t u a l r e a c h e d 0 . 2 5 a t o n l y one r a d i u s . The r e a s o n f o r t h i s d i s c r e p a n c y i s i m m e d i a t e l y a p p a r e n t i f we p l o t F 2 vs E , as i n F i g u r e 13- The p r o g r a m b a d l y o v e r e s t i m a t e d t h e f l u t t e r , and hence i t d i d not a l l o w enough s p i r a l a n g l e t o m a i n t a i n v . A t 500 MeV, t h e e r r o r i n f l u t t e r A F 2 = 0 . 0 2 4 . By t h e use o f e q n .(8), we f i n d t h a t t h e e r r o r i n v c a u s e d by A F 2 i s z ' A ( v 2 ) = A F 2 ( 1 + 2 t a n 2e) = 0 . 2 4 . T h e r e i s a l s o a s m a l l c o n t r i b u t i o n t o A v 2 a r i s i n g f r o m t h e i n a b i l i t y o f t h e f i e l d t o f o l l o w t h e p o l e t i p s h a p e . T h i s a p p e a r s as an e r r o r i n t h e s p i r a l a n g l e . F i g u r e 14 shows t h e d e s i g n s p i r a l a n g l e o f the p o l e t i p p l o t t e d a g a i n s t e n e r g y , and t h e a c t u a l s p i r a l a n g l e o f t h e m a g n e t i c f i e l d , as computed f r o m t h e phase o f t h e 6 t h h a r m o n i c . The two c u r v e s - 2k, -F i g u r e 11. D e v i a t i o n f r o m " i s o c h r o n i s m f o r t h e Mk V l - l Mod 0 model m a g n e t . 0.5 0.25 N •> oi 0.25 i 0.50 i 0.75 i 100 200 E n e r g y (MeV) + 300 F i g u r e 12. V e r t i c a l b e t a t r o n f r e q u e n c y v z p l o t t e d a g a i n s t e n e r g y f o r t h e Mk V l - l Mod 6 model magnet . I m a g i n a r y v a l u e s o f v z i m p l y t h a t t h e beam i s not s t a b l e and w i l l b l o w up i n t h e v e r t i c a l d i r e c t i o n . - 26 -a g r e e w e l l e x c e p t a r o u n d R = 260 t o 290 i n . T h i s d i s c r e p a n c y i s due t o t h e p o s i t i o n i n g o f t h e r e t u r n y o k e s t e e l above t h e v a l l e y ( see F i g u r e 1 ) , c a u s i n g t h e f i e l d t o l a g . A t t h e w o r s t p o i n t (R = 280 i n . ) , Ae = 5 ° , w h i c h c o r r e s p o n d s t o a change in v 2 o f Av 2 = 4F 2 t a n e s e c 2 e Ae z = 0.16. However , Ae t h e n d e c r e a s e s , and a t 500 MeV Ae = 1 ° , w h i c h c o r r e s p o n d s t o Av 2 = 0.10. z W h i l e t h i s e f f e c t i s a n n o y i n g , i t i s not c a t a s t r o p h i c , and can e a s i l y be c o r r e c t e d f o r by t h e use o f s h i m s . However , t h e e r r o r i n f l u t t e r i s more s e r i o u s , as i t c a n n o t be shimmed o u t w i t h o u t r u i n i n g i s o c h r o n i s m . The r e a s o n s f o r the l a r g e AF 2 a r e n o t h a r d t o s e e i n h i n d s i g h t . The two c u r v e s o f F 2 vs R used i n p r e d i c t i n g t h e f l u t t e r were t a k e n f r o m two m o d e l s , n e i t h e r o f w h i c h a c t u a l l y had an i s o c h r o n o u s a v e r a g e f i e l d . I f t h e y h a d , t h e i r h i l l w i d t h s w o u l d have been d i f f e r e n t , r e s u l t i n g i n d i f f e r e n t f l u t t e r c u r v e s . A n o t h e r , p r o b a b l y more i m p o r t a n t , r e a s o n i s t h e z e r o e r r o r i n t h e s p i r a l a n g l e , w h i c h w i l l be e x p l a i n e d i n t h e n e x t s e c t i o n . A l t h o u g h was q u i t e b a d , t h i s p o l e t i p was s u f f i c i e n t l y good t h a t i t c o u l d be m o d i f i e d t o o b t a i n a p o l e t i p w h i c h f o c u s e d and was i s o c h r o n o u s o u t t o 500 MeV. A n o t h e r Mk VI d e s i g n was done a t t h e same t i m e , b u t w i t h o u t t h e use o f t h i s p r o g r a m . I t p r o v e d t o be u n s a t i s f a c t o r y b e c a u s e o f i s o c h r o n i s m p r o b l e m s . Hence i t i s f e l t t h a t t h e p r o g r a m , even i n t h i s s t a t e , s e r v e d a u s e f u l p u r p o s e i n t h e d e s i g n o f t h e m a g n e t . - 27 -2 . 4 . 2 Z e r o - o f f s e t i n e Once i t was d i s c o v e r e d t h a t t h e f l u t t e r had been p r e d i c t e d b a d l y by t h e i n t e r p o l a t i o n t e c h n i q u e , t h e p rog ram was r e r u n u s i n g the <f>D t e c h n i q u e d e s c r i b e d i n S e c t i o n 2 . 3 - 2 . T h i s p r o d u c e d a p o l e t i p w i t h a v e r y l a r g e s p i r a l . E x t r a c t i o n w o u l d be i m p o s s i b l e w i t h t h i s magnet w i t h o u t d r i l l i n g h o l e s i n t h e r e t u r n y o k e . A t t h i s p o i n t , t h e p rog ram was a g a i n m o d i f i e d . A z e r o - o f f s e t s p i r a l a n g l e was i n c l u d e d i n the f o r m u l a f o r v 2 ( e q n . 8). The r e a s o n f o r t h i s can be seen i n F i g u r e 2 . F o r R 150 i n . , t h e a c t u a l s p i r a l a n g l e o f t h e f i e l d was z e r o , b u t use o f e q n . (8) gave a s p i r a l a n g l e o f a b o u t 1 4 ° . C l e a r l y , t h e a s s u m p t i o n made e a r l i e r , t h a t t h i s e r r o r a t low r a d i i d i d n o t m a t t e r , was w r o n g . The h i l l a n g l e h i s e f f e c t i v e l y t h e i n t e g r a l o f t h e s p i r a l a n g l e and i s hence a c u m u l a t i v e p a r a m e t e r . Even though t h e s p i r a l a n g l e i s wrong by o n l y a l i t t l e b i t , i t s t o t a l e f f e c t on h i s q u i t e l a r g e . The m o d i f i e d f o r m c h o s e n f o r e q n . (8) i s v 2 = _ g 2 Y 2 + F 2 [ 1 + 2 ( t a n 2 e + t a n 2 ^ ) ] (19) where e Q = 14° . In t h i s f o r m , e Q has a l a r g e e f f e c t a t low e, but a n e g l i g i b l e e f f e c t a t l a r g e e . T h a t t h i s does i n d e e d a p p r o x i m a t e t h e f u l l v 2 e x p r e s s i o n somewhat b e t t e r can be seen i n F i g u r e 1 5 - Cu rve 2 i s t h e a c t u a l s p i r a l a n g l e o f t h e f i e l d , c u r v e 1 i s c a l c u l a t e d u s i n g e q n . (8), and c u r v e 3 uses e q n . ( 1 9 ) - The s p i r a l a n g l e i s s t i l l o v e r e s t i -mated by a b o u t 2° f o r 200 i n . < R < 250 i n . , and u n d e r e s t i m a t e d by t h e same amount a r o u n d R = 305 i n . H o w e v e r , t h i s i s a p p r o a c h i n g t h e l i m i t s o f o u r e x p e r i m e n t a l a c c u r a c y i n d e t e r m i n i n g e , and i s c e r t a i n l y much b e t t e r t h a n t h e 14° e r r o r p r e v i o u s l y f o r R ^ 150 i n . - 28 -+ £, - f r o m a c c u r a t e v a l u e s o f v 2 (CYCLOPS) a n d e q u a t i o n 8 O e 2 - F i e l d (d<()/dr) 250 300 R a d i u s ( i n c h e s ) F i g u r e 15- S p i r a l a n g l e p r e d i c t i o n s f r o m e q u a t i o n s 8 and 19 c o m p a r e d t o t h e a c t u a l s p i r a l a n g l e o f t h e f i e l d - 29 -A t t h e t i m e o f w r i t i n g t h i s r e p o r t , a w o r k i n g p o l e t i p d e s i g n had been a c h i e v e d . I t i s b a s i c a l l y t h e Mk V l - l d e s i g n as d e s c r i b e d i n t h e p r e v i o u s s e c t i o n , but w i t h s e v e r a l m o d i f i c a t i o n s . C a l l e d t h e Mk V l - l Mod 9 d e s i g n , i t i s i s o c h r o n o u s t o w i t h i n ±100 g a u s s , and f o c u s i n g i s g o o d ; = 0 . 3 ± 0 . 1 e x c e p t a t R = 305 i n . , where = 0 . 6 5 . Hence we have a p o l e t i p w h i c h w o r k s , and we can c h e c k t h e r e s u l t s o f p r o g r a m MAGNET by c o m p a r i n g them t o t h i s d e s i g n . The p r o g r a m was m o d i f i e d t o use e q n . (19) r a t h e r t h a n e q n . ( 8 ) . A run u s i n g the same i n p u t d a t a as used f o r t h e Mk V l - l d e s i g n p r o d u c e d a p o l e t i p f a i r l y s i m i l a r t o t h e Mk V l - l Mod 10 p o l e t i p . (The Mk V l - l Mod 10 p o l e t i p i s v e r y s i m i l a r t o t h e Mod 9 t i p . I t was used b e c a u s e d a t a on i t were more r e a d i l y a v a i l a b l e . ) F i g u r e s 16 and 17 show t h e c o m p a r i s o n . The s p i r a l a n g l e s a r e w i t h i n ±5% o f e a c h o t h e r . A l t h o u g h t h e Mod 10 p o l e t i p seems t o have a n o n - s m o o t h s p i r a l a n g l e c u r v e , most o f t h e bumps w i l l no t a p p e a r i n t h e f i e l d , as t h e f i e l d c o n t o u r w i l l no t f o l l o w i r r e g u l a r i t i e s i n t h e s t e e l s m a l l e r than , t h e gap w i d t h o f 20 i n . Hence t h e s m o o t h e r p o l e t i p p r e d i c t e d by t h e p r o g r a m s h o u l d p r o d u c e t h e same s p i r a l a n g l e i n t h e f i e l d as t h e Mod 10 p o l e t i p . The h i l l a n g l e s a r e w i t h i n ±1° o f e a c h o t h e r , w h i c h i s a l s o e n c o u r a g i n g . The h i l l w i d t h o f the c a l c u l a t e d magnet i s much h i g h e r t h a n t h a t f o r t h e Mk V l - l Mod 10 p o l e t i p . The r e a s o n i s t h a t t h i s i s n o t a f i n i s h e d d e s i g n - i . e . , t h e c u r v e o f D vs R has n o t been i t e r a t e d . The h i l l w i d t h n i s v e r y s t r o n g l y dependent on D. D i n t u r n i s s t r o n g l y d e p e n d e n t on t h e amount and shape o f the s t e e l beyond 312 i n . , w h i c h i s n o t s e t by t h e p r o g r a m . Hence c o n s i d e r a b l e work and a c e r t a i n amount o f j udgement must be used when i t e r a t i n g D. On e a c h i t e r a t i o n , t h e p r o -gram s e t s t h e new h i l l - t o - h i l l d i s t a n c e o u t t o R % 280 i n . , b u t f o r - 30 -O - Mk V l - l Mod 10 p o l e t i p A - MAG 1-31/12/69 d e s i g n A' o / o 150 200 2 5 0 R a d i u s R ( i n c h e s ) 300 F i g u r e 16. D e s i g n s p i r a l a n g l e p l o t t e d a g a i n s t a c t u a l s p i r a l a n g l e f o r t h e Mk V l - l Mod 10 model magnet - 31 -+ - Mk V l - l Mod 10 p o l e t i p O - MAG 1-31/12/69 d e s i g n 150 200 250 R a d i u s R ( i n c h e s ) F i g u r e 17. H i l l a n g l e h o f t h e d e s i g n and a c t u a l p o l e t i p s f o r t h e Mk V l - l Mod 10 model magnet - 32 -l a r g e r r a d i i t h e programmer must c h o o s e an a p p r o p r i a t e and p h y s i c a l l y -r e a l i z a b l e D. Two o r t h r e e i t e r a t i o n s a r e u s u a l l y s u f f i c i e n t . In t h e c a l c u l a t i o n a b o v e , o n l y one i t e r a t i o n was p e r f o r m e d , u s i n g t h e h i l l - t o -h i l l d i s t a n c e o f t h e Mk V-3 model m a g n e t . I t i s no t t o be e x p e c t e d t h a t t h e p rog ram w i l l g e n e r a t e a p e r f e c t p o l e t i p . As can be seen i n F i g u r e s 6, 9, and 10, t h e p a r a m e t e r s £ and F b o t h d e p a r t f r o m b e i n g s i n g l e - v a l u e d f u n c t i o n s o f t h e magnet geomet ry as t h e r a d i u s a p p r o a c h e s t h e " c o r n e r " o f t h e p o l e t i p . The s i t u a t i o n a t t h i s r a d i u s i s v e r y c r i t i c a l , as r e g a r d s beam e x t r a c t i o n . The main c r i t e r i o n f o r e a s e o f e x t r a c t i o n i s n + h 4 60°. O t h e r w i s e , t h e p o l e t i p w i l l bend o v e r i n t o t h e r e t u r n y o k e f o r the a d j a c e n t p o l e t i p . F o r t h e Mk V l - l Mod 10 m a g n e t , n + h = 61°, and n must s t i l l be i n c r e a s e d t o c o r r e c t f o r i s o c h r o n i s m . Hence t h e p o i n t a t w h i c h t h e p r o g r a m becomes l e a s t a c c u r a t e i s a l s o t h e p o i n t where t h e d e s i g n i s most c r i t i c a l . A model must be used t o c h e c k t h e d e s i g n . - 33 -3. AZIMUTHAL MAGNETIC FIELD PREDICTION 3.1 I n t r o d u c t i o n The p r o b l e m o f f i n d i n g t h e m a g n e t i c f i e l d s t r e n g t h i n s i d e a com-p l e x magnet s t r u c t u r e when h i g h f i e l d s a r e i n v o l v e d has n o t been s o l v e d . The r e a s o n i s t h a t t h e p e r m e a b i l i t y o f t h e s t e e l i s a f u n c t i o n o f t h e f i e l d . T h i s i s i n c o n t r a s t t o t h e e l e c t r i c f i e l d p r o b l e m , where t h e d i e l e c t r i c c o n s t a n t i s i n g e n e r a l no t dependent on t h e f i e l d s t r e n g t h , and hence M a x w e l l ' s e q u a t i o n s can be s o l v e d . In t h i s s e c t i o n , o t h e r l e s s g e n e r a l t e c h n i q u e s f o r f i n d i n g t h e f i e l d w i l l be i n v e s t i g a t e d . 3.2 F i t s t o V a r i o u s F o r m u l a e An a t t e m p t was made t o f i t t h e a z i m u t h a l f i e l d c o n t o u r s w i t h v a r i o u s a n a l y t i c a l e x p r e s s i o n s , and t h e n r e l a t e t h e e x p r e s s i o n s t o t h e m a g n e t . g e o m e t r y . The f i r s t f i t t h a t was t r i e d was o f t h e f o r m B = a6 n where B i s t h e f i e l d and 0 i s t h e a z i m u t h a l a n g l e . The v a l l e y f i e l d o n l y was f i t by t h i s f o r m u l a , and t h e h i l l f i e l d was assumed t o be f l a t . A good f i t was o b t a i n e d f o r k i n . < R < 10 i n . , but f o r l a r g e r a d i i , t h e s p i r a l a n g l e c a u s e d an a s y m m e t r i c v a l l e y f i e l d w h i c h c o u l d not be f i t by o u r s y m m e t r i c f o r m u l a , and f o r s m a l l r a d i i , t h e a s s u m p t i o n t h a t t h e h i l l f i e l d was f l a t d i d not h o l d . A l s o , no u s e f u l c o r r e l a t i o n c o u l d be f o u n d between t h e p a r a m e t e r s a and n and t h e magnet g e o m e t r y . A Saxon-Woods e x p r e s s i o n was a l s o t r i e d : Bn B = B 0 -1 + e < 8 - e 0 ) / d * ' The a d v a n t a g e o f t h i s f o r m u l a i s t h a t i t does have t h e r i g h t s o r t o f c u r v e . F o r o u r f i e l d c o n t o u r s , two i n d e p e n d e n t l y - v a r i e d Saxon-Woods - 3k -c u r v e s were added t o g e t h e r . F i g u r e 18 shows a l e a s t - s q u a r e s f i t t o a t y p i c a l f i e l d c u r v e t a k e n f r o m t h e model m a g n e t . C u r v e s II and III a r e t h e two c u r v e s , and c u r v e IV i s t h e i r sum. The f i t i s q u i t e b a d , e s p e -c i a l l y a t t h e " c o r n e r s " , where t h e e r r o r i s ±500 g a u s s . No a t t e m p t was made t o c o r r e l a t e t h e p a r a m e t e r s t o t h e magnet g e o m e t r y . O t h e r t y p e s o f f i t s were t r i e d , but t h e y were even l e s s s u c c e s s f u l . E x p e r i m e n t a l F i e l d C u r v e s An e x p e r i m e n t was p e r f o r m e d t o see how t h e f i e l d v a r i e s w i t h d i s t a n c e f r o m t h e p o l e t i p . F i g u r e 19 shows t h e e x p e r i m e n t a l a r r a n g e -m e n t . Two s t e e l p l a t e s were a t t a c h e d t o the p o l e s o f a V a r i a n V - 4 0 0 0 e l e c t r o m a g n e t . The gap was s e t t o 1 . 0 0 0 ± 0 . 0 0 1 i n . The s t e e l was t h e same as t h a t i n t h e model magnet p o l e t i p s , b o t h i n c o m p o s i t i o n and i n t h i c k n e s s . I t was hoped t h a t t h e f i e l d a l o n g t h e median p l a n e b o t h i n s i d e and o u t s i d e t h e gap w o u l d c l o s e l y r e s e m b l e t h e f i e l d p r o d u c e d by t h e model m a g n e t . R e a d i n g s were t a k e n w i t h a H a l l p r o b e , u s i n g the edge o f t h e s t e e l as t h e z e r o p o i n t ( i . e . D ' = 0 ) . T h r e e s e t s o f p o l e t i p s were u s e d , t h e r a d i i o f c u r v a t u r e o f t h e r e l e v a n t edges b e i n g 5 . 5 i n . c o n c a v e , 5 . 5 i n . c o n v e x , and i n f i n i t y (a s t r a i g h t e d g e ) . These c u r v a -t u r e s a r e r o u g h l y t h o s e e n c o u n t e r e d i n t h e model m a g n e t . T h r e e runs were t a k e n f o r e a c h s e t o f p o l e t i p s , w i t h t h e maximum f i e l d s e t a t a b o u t 3 , 6 , and 10 k G . F o r e a c h s e t - u p , was v a r i e d f r o m - 1 . 5 i n . t o + 4 . 0 i n . The f i e l d s o b t a i n e d a r e p l o t t e d i n F i g u r e 20 ( a , b , c ) . A c o m p u t e r p r o g r a m was t h e n w r i t t e n t o g e n e r a t e f i e l d c o n t o u r s o f a r e a l p o l e t i p , u s i n g t h i s e x p e r i m e n t a l d a t a . The f i e l d a t any p o i n t i s computed by c a l c u l a t i n g t h e p e r p e n d i c u l a r d i s t a n c e s t o t h e n e a r e s t edges o f t h e two c l o s e s t p o l e t i p s , l o o k i n g up i n F i g u r e 20 t h e f i e l d - 36 -i a n V4000 c t r o m a g n e t P o l e F a c e s : 1/2" t h i c k 1006 s t e e l F i g u r e 19. E x p e r i m e n t a l a r r a n g e m e n t t o d e t e r m i n e v a r i a t i o n o f f i e l d w i t h d i s t a n c e Q D i s t a n c e D ' ( i n c h e s ) . F i g u r e 20. F i e l d v a r i a t i o n w i t h d i s t a n c e f o r t h e e x p e r i m e n t a l a r r a n g e m e n t o f F i g u r e 19- R r e f e r s t o t h e r a d i u s of c u r v a t u r e o f the p o l e t i p b e i n g u s e d . - 38 -c o n t r i b u t i o n f r o m e a c h p o l e t i p , and a d d i n g t h e two t o g e t h e r . By f o l l o w i n g t h i s p r o c e d u r e f o r many p o i n t s a l o n g a r a d i u s , one can o b t a i n an a z i m u t h a l f i e l d c o n t o u r . In t h i s c a s e , f o r e a s e o f p rogramming j u s t one o f t h e B vs D c u r v e s i n F i g u r e 20 was used - t h e 6 k G , R = °° c u r v e . The d i f f e r e n c e s between t h i s c u r v e and any o f t h e o t h e r s a r e j u s t s e c o n d - o r d e r e f f e c t s , and we can a t l e a s t f i n d o u t how w e l l t h e method works by u s i n g o n l y one c u r v e . The f i e l d c o n t o u r s o b t a i n e d so f a r a r e now m u l t i p l i e d by the r a t i o o f t h e maximum h i l l f i e l d o f t h e r e a l m a g n e t , t o t h e maximum f i e l d o f o u r c a l c u l a t e d c o n t o u r . T h i s i s an a c c e p t a b l e p r o c e d u r e , as we can o b t a i n v i r t u a l l y any d e s i r e d h i l l f i e l d by a d d i n g o r r e m o v i n g s t e e l above t h e p o l e t i p . The f i r s t r e s u l t s were not e n c o u r a g i n g . A l t h o u g h t h e g e n e r a l s h a p e s o f t h e f i e l d c o n t o u r s were p r o d u c e d , t h e v a l l e y f i e l d was t o o h i g h . The r e a s o n i s t h a t t h e c l o s e p r o x i m i t y o f the c o i l s and t h e s u p p o r t i n g s t r u c t u r e i n t h e V a r i a n e l e c t r o m a g n e t c a u s e s t h e f r i n g i n g f i e l d a r o u n d i t t o be h i g h e r t h a n t h a t a r o u n d a p o l e t i p . T h i s p r o b l e m was overcome by t h e f o l l o w i n g t e c h n i q u e . We know what t h e minimum f i e l d s h o u l d be f o r a g i v e n p o l e t i p f r o m the p a r a m e t e r E, ( see S e c t i o n 2.3-1)• The g e n e r a t e d f i e l d c o n t o u r s can now be expanded t o g e t B^ - By c o r r e c t , and s h i f t e d u n t i l t h e a v e r a g e f i e l d i s c o r r e c t . The m o d i f i e d p rog ram was run u s i n g t h e a c c u r a t e l y measured d a t a p o i n t s f o r t h e Mk V l - l Mod 10 p o l e t i p . The l i s t i n g o f t h i s p rog ram i s g i v e n i n A p p e n d i x 2. R e s u l t s were q u i t e g o o d . T a b l e I compares the f l u t t e r and a v e r a g e f i e l d o f t h e g e n e r a t e d c o n t o u r w i t h t h a t o f t h e f i e l d a c t u a l l y o b t a i n e d f r o m t h i s p o l e t i p . N e g l e c t i n g r a d i i \h i n . o r g r e a t e r , t h e maximum e r r o r i n B was 66 g a u s s , o r 2%, and t h e maximum e r r o r i n f l u t t e r i s 7-3% a t R = 2 i n . The two r a d i i w i t h t h e g r e a t e s t - 39 -TABLE I COMPARISON OF PREDICTED B AND F WITH ACTUAL B AND F FOR THE MK V l - l MOD 10 MODEL MAGNET A c t u a l Mk V l - l Mod 10 R B F 2 B F 2 AB A F 2 / F 2 (model i n . ) (kG) (kG) (G) % 1 3.082 0.00123 3.061 0.00120 -21 -2.4 2 2.889 0.0215 2.920 0.0198 31 -7.9 3 2.932 0.0645 2.990 0.0613 58 -5.0 4 3.153 0.1203 3.219 0.1135 66 -5-7 5 3.215 0.1671 3.263 0.1612 48 -3.5 6 3.225 0.2089 3.241 0.2143 16 +2.6 7 3.269 0.2530 3.290 0.2572 21 + 1 .7 8 3.299 0.2871 3.299 0.2911 0 + 1.4 9 3-351 0.3074 3-335 0.3096 -16 +0.7 10 3.489 0.2969 3.479 0.2958 -10 -0.4 11 3.675 0.2712 3.645 0.2767 -30 +2.0 12 3.869 0.2306 3.831 0.2399 -38 +4.0 13 4.080 0.1825 4.067 0.1876 -13 +2.8 14 4.242 0.1324 4.373 0.1035 + 131 -21 .8 15 4.694 0.0635 4.547 0.0860 -147 +35.4 - ko -e r r o r i n B a r e p l o t t e d i n F i g u r e 21. In t h e k i n . r a d i u s c u r v e , t h e minimum v a l l e y f i e l d has been i n c o r r e c t l y p r e d i c t e d ; t h a t i s , 5 i s wrong by a b o u t 3%- In t h e 12 i n . r a d i u s c u r v e s , we s e e t h a t the h i l l f i e l d o f t h e a c t u a l model magnet i s not f l a t . T h u s , i n a s e n s e , t h e r e a l f i e l d i s no t c o r r e c t , s i n c e t h e f i e l d c o u l d have been made f l a t by c h a n g i n g t h e r e t u r n y o k e s t e e l . The p r o g r a m w i l l g e n e r a t e a f i e l d w i t h a n e a r l y f l a t h i l l . Even though we have t h e s e two e r r o r s o u r c e s , we have s t i l l p r e d i c t e d B w i t h i n ±70 g a u s s , and F 2 w i t h i n ±6% i n t h e range 3 i n . ^ R £ 13 i n . As w i t h t h e magnet p r o g r a m , t h i s p rog ram does n o t work w e l l f o r v e r y s m a l l o r l a r g e r a d i i . F o r s m a l l r a d i i , t h e w i d t h o f t h e p o l e t i p i s becoming l e s s t h a n t h e gap s i z e , and hence t h e c o n t o u r o f t h e f i e l d w i l l no t c o n f o r m t o the geomet ry o f the p o l e p i e c e . A t l a r g e r a d i i , t h e " c o r n e r " i n t h e s t e e l where t h e s p i r a l has been a b r u p t l y c u t o f f has an e f f e c t . O b v i o u s l y t h e f i e l d a d i s t a n c e away f r o m a " c o r n e r " w i l l n o t be t h e same as t h e f i e l d t h e same d i s t a n c e f r o m a s t r a i g h t e d g e . A t w o - d i m e n s i o n a l g r i d o f p o i n t s w o u l d have t o be measured a r o u n d an a p p r o p r i a t e p o l e f a c e , and t h e n the p r o g r a m w o u l d have t o be m o d i f i e d t o mesh t h e s e p o i n t s s m o o t h l y w i t h t h e f i e l d c o n t o u r s o b t a i n e d o v e r o t h e r p a r t s o f t h e p o l e t i p . The p r o b l e m was not a t t e m p t e d . = P r e d i c t e d f i e l d 3 A c t u a l f i e l d 1/1 3 m l Olil o .2 3 (b) R - 1 2 " F i g u r e 21 P r e d i c t e d and a c t u a l f i e l d c o n t o u r s f o r t h e Mk V l - l Mod 10 model magnet 10 20 30 40 50 A z i m u t h a l A n g l e ( d e g r e e s ) 60 - kl -k. TOLERANCES k.1 T h e o r y The t o l e r a n c e s on t h e d i m e n s i o n s o f t h e p o l e t i p a r e d e t e r m i n e d f r o m a) t h e i s o c h r o n i s m c o n d i t i o n on B, and b) t h e r e q u i r e m e n t t h a t v e r t i c a l f o c u s i n g be r e a l and not pass t h r o u g h any s e r i o u s r e s o n a n c e s . The method t o be used h e r e i s t h a t o f C r a d d o c k and R i c h a r d s o n . 7 The e q u a t i o n f o r v e r t i c a l f o c u s i n g , as i n S e c t i o n 2 . 2 . 2 , i s : v 2 = _ y- + F 2 ( , + 2 t a n 2 e ) . (8) The l i m i t s on v 2 a r e s e t f r o m t h e f o l l o w i n g c o n s i d e r a t i o n s . The l o w e r l i m i t i s v 2 = 0 , b e c a u s e v , t h e number o f b e t a t r o n o s c i l l a t i o n s t h e z ' z p a r t i c l e makes p e r r e v o l u t i o n , must be r e a l . The u p p e r l i m i t i s s e t by t h e v a r i o u s r e s o n a n c e s between the v e r t i c a l o s c i l l a t i o n s and e i t h e r t h e magnet s t r u c t u r e o r t h e r a d i a 1 o s c i 1 1 a t i o n s . The c l o s e s t , most s e r i o u s v e r t i c a l r e s o n a n c e i s t h e v z = 1/2 r e s o n a n c e . Hence t h e upper l i m i t i s v 2 = 0 . 2 5 . The v a l u e c h o s e n f o r t h e TRIUMF c y c l o t r o n i s v 2 = 0 . 1 2 5 ± 0 . 0 5 . The t o l e r a n c e q u o t e d a l l o w s 2 . 5 s t a n d a r d d e v i a t i o n s i n v 2 b e f o r e we r e a c h e i t h e r l i m i t . Hence v 2 s h o u l d r e m a i n s u f f i c i e n t l y f a r away f r o m b o t h 0 and 1/2 t h a t t h e a m p l i t u d e o f t h e beam o s c i l l a t i o n s w i l l no t i n c r e a s e a p p r e c i a b l y . We w i s h t o f i n d the a l l o w a b l e e r r o r s i n F 2 and e by means o f e q n . ( 8 ) . The t o l e r a n c e on u ' comes f r o m the e q u a t i o n d e f i n i n g i t : - = j-dJL ^ F d r * D i f f e r e n t i a t i n g , we f i n d Ay' = A r \ _r_ .B. ^ + - 4 - A d r B dB d r dB IdrJ (20) - 43 -s i n c e t h e p e r m i s s i b l e p e r c e n t a g e e r r o r i n B i s n e g l i g i b l e compared t o t h a t i n d B / d r . The t o l e r a n c e on dB/dr i s s e t t o ±2 G / f t b e c a u s e o f t r i m c o i l c o n s i d e r a t i o n s . Hence f r o m e q n . (20) we can f i n d A y ' f o r any r and B, and t h e n f r o m e q n . (8) we can f i n d t h e a l l o w a b l e e r r o r i n F 2 ( l + 2 t a n 2 e ) . T h i s e r r o r i s now b r o k e n up a r b i t r a r i l y s u c h t h a t 35-5% o f i t a p p l i e s t o t h e ( 1 + 2 t a n 2 e ) t e r m , s i n c e t h e s p i r a l a n g l e e i s t h e most c r i t i c a l p a r a m e t e r a t l a r g e r a d i i . T h i s l e a v e s 30% o f t h e e r r o r t o t h e F 2 t e r m , f o l l o w i n g t h e r u l e s f o r a d d i t i o n o f s t a n d a r d d e v i a t i o n s . We a r e now a b l e t o f i n d t o l e r a n c e s f o r F 2 and e . We must now t r a n s l a t e t h e s e i n t o u s a b l e t o l e r a n c e s on t h e a z i m u t h a l p o s i t i o n s o f t h e f o c u s i n g and d e f o c u s i n g edges o f t h e p o l e t i p . From g e o m e t r y , we have 2 t a n e = t a n + t a n 2A t a n e = / ( A t a n e f ) 2 + (A tan e^)1 (21) where and a r e t h e s p i r a l a n g l e s o f t h e f o c u s i n g and d e f o c u s i n g e d g e s , r e s p e c t i v e l y . We now i n t r o d u c e a w e i g h t i n g f a c t o r X d e f i n e d by A t a n e , = Ae , / c o s 2 e , = 2 cos X • A t a n . e d d d A t a n = A e ^ / c o s 2 e ^ = 2 s i n A • A t a n e. T h i s d e f i n i t i o n o f X s a t i s f i e s e q n . ( 2 1 ) . I f we assume t h a t t h e work i n v o l v e d i n sh imming i s i n v e r s e l y p r o p o r t i o n a l t o t h e a n g u l a r t o l e r a n c e r a i s e d t o t h e power n , where n i s n o r m a l l y assumed t o be 1 , t h e n m i n i -m i z i n g t h e t o t a l work y i e l d s t a n X = _2n C O S £,-1 •2+n C O S EfJ From t h i s we can f i n d X, and hence Ae^ and A e ^ . (22) - kk -To f i n d a t o l e r a n c e on t h e a z i m u t h a l v a l u e o f the measured p o i n t s , we h a v e , f r o m g e o m e t r i c a l c o n s i d e r a t i o n s , /2~ c o s e , A y , - . J L J ± d _ ( 2 3 ) where h'= d i s t a n c e between t h e measured p o i n t s , and Ay^ = a l l o w a b l e a z i -mutha l e r r o r i n t h e measured p o i n t . The t o l e r a n c e s on t h e p o i n t s i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e t a n g e n t t o t h e s p i r a l can a g a i n be f o u n d f r o m geomet ry t o be A a d = 7 f A e d - ( 2 Z t ) S i m i l a r e q u a t i o n s h o l d f o r t h e f o c u s i n g edge o f t h e p o l e t i p . We now have t h e n e c e s s a r y f o r m u l a e t o d e t e r m i n e t h e r e q u i r e d t o l e r a n c e s on the m a n u f a c t u r e d p o l e t i p . R e s u l t s The Mk V l - l Mod 10 p o l e t i p was measured w i t h t h e a i d o f a m i l l i n g e d g e - f i n d e r t o an a c c u r a c y o f ±0.002 i n . on e a c h p o i n t . A c o m p u t e r p r o -gram was w r i t t e n w h i c h d i f f e r e n t i a t e d t h e smooth c u r v e a p p r o x i m a t i o n t o t h e s e p o i n t s t o f i n d e^ and e^.. The p o l e t i p t o l e r a n c e s were t h e n f o u n d by t h e a p p l i c a t i o n o f t h e f o r m u l a e i n t h e p r e v i o u s s e c t i o n . The w h o l e p r o g r a m i s l i s t e d i n A p p e n d i x 3. The c a l c u l a t e d t o l e r a n c e s a p p e a r i n T a b l e I I . These t o l e r a n c e s a r e beyond t h e c o n t o u r i n g a c c u r a c y a v a i l a b l e i n f l a m e c u t t i n g l a r g e 10 i n . t h i c k p l a t e s t e e l . T h e r e f o r e , t h e p o l e p i e c e w i l l be c u t u n d e r -s i z e , and s h i m s w i l l be added d u r i n g a s s e m b l y t o a c h i e v e t h e c o r r e c t c o n t o u r . TABLE I I TOLERANCES FOR THE MK V l - l MOD 1 0 POLE T I P R ( i n . ) A F 2 / F 2 {%) A ( ( B - B ) 2 ) ^ G (deg) £ d (deg) A e d (mrad) A a d ( i n . ) A ^ d ( i n . ) (deg) Ae.p (mrad) Aa.p ( i n . ) Ayf ( i n . ) 50 1 0 . 7 6 0 . 6 0 0 - - - 0 - - -100 7 . 9 5 3 . 3 0 0 - - - 0 - - -150 5 . 3 4 5 . 3 1 . 0 0 . 2 - - - 1 . 8 - - -2 0 0 3 . 4 3 4 . 7 1 8 . 3 2 4 . 5 1 1 8 . 9 0 . 8 4 0 0 . 9 2 3 1 1 . 6 1 3 1 . 1 0.927 0 . 9 4 6 2 5 0 2 . 0 1 9 . 8 4 3 . 0 4 6 . 9 3 3 . 0 0 . 2 3 3 0 . 3 4 1 3 8 . 4 3 9 . 6 0 . 2 8 0 0 . 3 5 7 2 6 0 1 . 8 1 6 . 3 5 0 . 0 5 3 . 3 2 5 . 0 0 . 1 7 7 0 . 2 9 5 4 6 . 1 3 0 . 4 0 . 2 1 5 0 . 3 1 0 2 7 0 1 . 6 1 3 . 2 5 6 . 4 5 8 . 7 1 9 . 1 0 . 1 3 5 0 . 2 6 0 5 3 . 7 2 2 . 8 0 . 1 6 1 0 . 2 7 2 2 8 0 1 . 5 1 0 . 6 6 2 . 0 6 1 . 2 1 6 . 2 0 . 1 1 5 0 . 2 3 7 6 2 . 7 1 5 . 1 0 . 1 0 7 0.233 290 1 . 3 1 0 . 0 6 2 . 7 6 0 . 3 1 5 . 0 0 . 1 0 6 0 . 2 1 4 6 4 . 7 1 2 . 3 0 . 0 8 7 0 . 2 0 3 3 0 0 1 . 2 9 - 5 6 3 . 4 6 2 . 5 1 2 . 2 0 . 0 8 6 0 . 1 8 6 6 4 . 3 1 1 . 2 0 . 0 7 9 0 . 1 8 2 3 0 5 1.1 8 . 4 6 6 . 1 6 7 . 6 9 . 0 6 0 . 0 6 4 0 . 1 6 8 6 4 . 4 1 0 . 7 0 . 0 7 6 0 . 1 7 6 3 1 0 1 . 0 6 . 7 7 0 . 7 7 4 . 7 5 . 4 8 0 . 0 3 9 0 . 1 4 7 6 3 - 9 1 0 . 9 0 . 0 7 7 0 . 1 7 5 3 1 1 1 . 0 6 . 2 72.0 7 6 . 4 4 . 7 5 0 . 0 3 4 0 . 1 4 2 6 3 . 8 1 1 . 0 0 . 0 7 8 0 . 1 7 6 - k6 -5. FLUX CALCULATIONS 5.1 F l u x Measurements A s u r v e y o f t h e f l u x i n t h e s t e e l r e t u r n y o k e o f t h e Mk V l - l Mod 9 model magnet was t a k e n . A s i n g l e l o o p o f f i n e w i r e was wrapped a r o u n d t h e s t e e l t h r o u g h w h i c h the f l u x was t o be m e a s u r e d . The ends o f t h i s l o o p were c o n n e c t e d t o a d i g i t a l v o l t m e t e r (DVM) w h i c h was s e t t o read v o l t s i n t e g r a t e d o v e r t i m e ( v o l t - s e c ) . The magnet power s u p p l y was run up t o i t s f u l l r a t i n g and back down a g a i n , w i t h t h e DVM b e i n g r e a d and r e s e t a f t e r b o t h t h e up c y c l e and t h e down c y c l e . The two r e a d i n g s were a v e r a g e d t o e l i m i n a t e z e r o d r i f t i n t h e DVM. The m a g n e t i c f l u x i n t h e s t e e l was t h e n c a l c u l a t e d by a p p l y i n g F a r a d a y ' s Law, w h i c h s t a t e s t h a t V - (25) where V i s t h e e l e c t r o m o t i v e f o r c e i n d u c e d i n t h e c o i l , and 0 i s t h e f l u x f l o w i n g t h r o u g h t h e a r e a e n c l o s e d by t h e c o i l . S o l v i n g f o r $ , we f i n d t Vdx _ (26) 0 The t e r m on t h e r i g h t i s j u s t t h e DVM r e a d i n g . I f we know <S, we can f i n d B s , t h e f i e l d s t r e n g t h i n s i d e t h e s t e e l , by u s i n g t h e d e f i n i t i o n o f $ : B • n da = B A (27) n s where B i s the f i e l d normal t o t h e c o i l , and A i s the a r e a o f t h e c o i l , n We t h e n have B = / B 2 + B 2 + B 2 (28) S 1 2 3 ' kl where B j , B 2 and B 3 a r e t h e f i e l d s t r e n g t h s a t one s p o t i n t h r e e m u t u a l l y p e r p e n d i c u l a r d i r e c t i o n s . A s e r i e s o f h o r i z o n t a l y o k e r a d i a l f l u x m e a s u r i n g c o i l s were s e t u p , as i n F i g u r e 2 2 . See F i g u r e 1 f o r a g e n e r a l v i e w o f the m a g n e t . The a v e r a g e f i e l d s i n s i d e e a c h c o i l a r e g i v e n i n T a b l e I I I . A l l a r e a s and r a d i i a r e g i v e n i n f u l l - s c a l e d i m e n s i o n s . C o i l s 1 8 and 1 9 show the e x i s t e n c e o f a v e r t i c a l B c o m p o n e n t . I t s c o n t r i b u t i o n t o t h e t o t a l f i e l d can be c a l c u l a t e d by e q n . ( 2 8 ) as f o l l o w s : C o i l 5 C o i l 6 C o i l 7 12. 7 12. 1 12, 1 Hence t h e i n c r e a s e i n | B | i s o f t h e o r d e r o f 1 0 % , and must be t a k e n i n t o a c c o u n t in any p e r m e a b i l i t y c a l c u l a t i o n s . S i m i l a r l y , a s e r i e s o f v e r t i c a l r e t u r n y o k e f l u x m e a s u r i n g c o i l s were s e t up as shown i n F i g u r e 2 3 , and t h e r e s u l t s a r e g i v e n i n T a b l e IV. T a b l e V g i v e s t h e r e s u l t s o f t h e runs done on t h e e i g h t h - s c a l e c e n t r a l r e g i o n m o d e l . The c o i l s f o r m e a s u r i n g t h e t r a n s v e r s e f l u x i n t h e h o r i z o n t a l y o k e a r e shown i n F i g u r e 2k, and t h e r e s u l t s i n T a b l e V I . As e x p e c t e d , c o i l s D and E show v e r y l i t t l e t r a n s v e r s e f l u x , b u t t h e l o w e r s e c t i o n o f c o i l F has a 6 . 1 kG a v e r a g e f i e l d t h r o u g h i t . C o i l H d e m o n s t r a t e s t h e e x t r e m e l y c o m p l e x n a t u r e o f t h e f l u x l i n e s i n t h e m a g n e t . F i g u r e 2 5 i s a g r a p h o f t h e f i e l d f l o w i n g t h r o u g h c o i l H p l o t t e d a g a i n s t t h e p o t e n t i o m e t e r s e t t i n g o f t h e magnet power s u p p l y w h i c h i s a l i n e a r - k8 -Figure 2^. Transverse Coi 1 s - 50 -TABLE 111 - HORIZONTAL YOKE RADIAL FLUX C o i l # A r e a ( i n . 2 ) A p p r o x . Rad i us ( i n . ) B (kG) C o i l # A r e a ( i n . 2 ) A p p r o x . Rad i us ( i n . ) B (kG) 1 2680 354 18.1 11 2380 114 14.7 2 1790 354 22.9 12 1220 83 17.6 3 2680 312 18.9 13 1860 260 12.0 4 1790 312 22.0 14 3030 312 11.5 5 1790 218 12.7 15 1770 338 22.1 6 1950 218 12.1 16 1930 338 28.1 7 2380 218 12.1 17 2920 312 10.0 8 780 187 9.7 18 2530 - 4.9 9 1570 187 13.8 19 2600 - 6.1 10 1570 187 15.9 TABLE IV - VERTICAL RETURN YOKE FLUX TABLE V - CENTRE REGION FLUX Coi 1 A r e a B Rad i us A r e a B ( i n . 2 ) (kG) ( i n . ) ( i n . 2 ) (kG) A 4970 1 7 . 7 16 2 5 - 1 2 0 . 3 B 4020 1 6 . 3 24 7 9 . 4 1 5 . 6 C 2620 2 0 . 3 32 1 0 5 . 6 1 9 - 7 36 1 7 3 - 4 16 .1 TABLE VI - HORIZONTAL YOKE TRANSVERSE FLUX Co i 1 A r e a B Coi 1 A r e a B ( i n . 2 ) (kG) ( i n . 2 ) (kG) D u p p e r 1280 - 1 . 0 8 F u p p e r 2080 0 . 6 0 l o w e r 805 0 . 1 4 ^1 owe r 2000 6 . 1 0 D t o t a l 2080 - 0 . 6 6 F t o t a l 4080 3 . 4 8 E u p p e r 2370 - 1 . 4 8 G 813 - 0 . 7 5 E, l o w e r 1560 3 . 1 5 H 1460 - 6 . 1 1 E t o t a l 3920 0 . 3 5 - 51 -f u n c t i o n o f the magnet c o i l e x c i t a t i o n (940 c o r r e s p o n d s t o f u l l e x c i t a -t i o n ) . T h i s b e h a v i o u r i s d i f f i c u l t t o e x p l a i n , b u t i t has t o do w i t h t h e v a r i a t i o n o f s t e e l p e r m e a b i l i t y w i t h c h a n g i n g c o i l i n d u c t i o n . F i g u r e 25 a l s o has a s i m i l a r p l o t f o r c o i l 1 6 . Note t h e change o f s c a l e f o r t h i s c u r v e . I t can be seen t h a t the s t e e l i n t h i s r e g i o n i s h i g h l y s a t u r a t e d , as t h e i n c r e m e n t a l p e r m e a b i l i t y i s t h e same as t h a t o f a i r . A s m a l l l e a k a g e f l u x e x i s t s a r o u n d t h e main magnet c o i l s . F i g u r e 26 and T a b l e VII show t h i s . The l e a k a g e f l u x i s due t o the f a c t t h a t t h e f l u x r e t u r n p a t h l e n g t h between the c o i l s i s c o m p a r a b l e t o t h e magnet g a p , and hence a s h o r t i n g e f f e c t o c c u r s . TABLE VI I LEAKAGE FLUX C o i l A r e a ( i n . 2 ) Bi:(kG) J 247 1 . 1 6 K 2310 0 . 9 2 F I u x C a l c u l a t i o n s L e t us assume t h a t a l l t h e f l u x c r o s s i n g t h e median p l a n e o f t h e magnet r e a c h e d t h a t p o i n t by t r a v e r s i n g a p a t h t h r o u g h t h e s t e e l r e t u r n y o k e . L e t us a l s o assume t h a t the f l u x c r o s s i n g the median p l a n e a t any r a d i u s l e f t the s t e e l a t t h a t same r a d i u s . We can t h e n f i n d t h e t o t a l f l u x $ c r o s s i n g t h e median p l a n e i n s i d e r a d i u s R f r o m rR $ = 2TT B r d r . (29) '0 By o u r f i r s t a s s u m p t i o n , $ i s a l s o the f l u x f l o w i n g t h r o u g h t h e s t e e l a t r a d i u s R. I f we know A g , t h e a r e a o f t h e s t e e l r e t u r n y o k e as a f u n c t i o n o f R, t h e n we can use r e l a t i o n 27 t o compute t h e f i e l d s t r e n g t h B s i n F i g u r e 25. F i e l d t h r o u g h c o i l s H and 16 as a f u n c t i o n o f t h e power s u p p l y p o t e n t i o m e t e r s e t t i n g , - w h i c h i s p r o p o r t i o n a l t o t h e m a i n c o i l e x c i t a t i o n - 53 -Co i 1 s V e r t i c a l r e t u r n y o k F l u x m e a s u r i n g c o i l J S K SECTION B-B S c a l e : 1/2" = 1" F i g u r e 26. L e a k a g e F l u x C o i l s - 5k -t h e s t e e l . The r e s u l t s o f t h i s c a l c u l a t i o n f o r the Mk V l - l Mod 9 model magnet a r e shown i n T a b l e V I I I . T a b l e IX compares t h e s e r e s u l t s w i t h the a c t u a l e x p e r i m e n t a l B s v a l u e s o b t a i n e d i n S e c t i o n k.]. The c a l c u l a t e d v a l u e s a r e t o o b i g by a f a c t o r o f a b o u t 1.2. S i n c e t h e c a l c u l a t e d f l u x c r o s s i n g t h e median p l a n e must be a c c u r a t e t o w i t h i n one o r two p e r c e n t , most o f t h i s d i s -c r e p a n c y i s due t o f l u x l e a k a g e - some o f t h e f l u x t r a v e l s t h r o u g h t h e a i r r a t h e r t h a n t h e s t e e l . The r a t i o o f t h e a c t u a l B t o t h e c a l c u l a t e d s B s i s a b o u t 0 . 8 5 ; hence a b o u t 1 5 % o f t h e f l u x t a k e s a p a t h t h r o u g h t h e a i r . C a l c u l a t i o n o f Magnet R e l u c t a n c e The c a l c u l a t i o n o f t h e m a g n e t i c r e l u c t a n c e o f t h e c y c l o t r o n magnet i s a v e r y c o m p l e x p r o b l e m r e q u i r i n g s e v e r a l s i m p l i f y i n g a s s u m p -t i o n s . F i r s t , l e t us assume a r a d i a l f l u x f l o w , w i t h t h e f i e l d c r o s s i n g t h e median p l a n e a t r a d i u s R c o m i n g f r o m t h e s t e e l a t R. To a f i r s t -o r d e r a p p r o x i m a t i o n we can d i v i d e t h e magnet i n t o N r a d i a l i n c r e m e n t s . Each i n c r e m e n t has two components o f r e l u c t a n c e - t h e s t e e l and t h e g a p . Z = Z + Z . ( 3 0 ) s 9 The r e l u c t a n c e o f t h e s t e e l i s j u s t Z s = (1 + l)as (3D where Jd g i s t h e e f f e c t i v e p a t h l e n g t h t h r o u g h t h e s t e e l ( t h e r a d i a l i n c r e m e n t i n t h i s c a s e ) , a i s t h e c r o s s - s e c t i o n a l a r e a o f the s t e e l , and s p i s t h e s t e e l p e r m e a b i l i t y . The t e r m 1 + u a c c o u n t s f o r t h e p o s s i b i l i t y o f the f l u x t a k i n g an a i r p a t h , a s s u m i n g t h a t t h e l e n g t h and a r e a o f t h e a i r p a t h a r e t h e same as t h o s e f o r t h e s t e e l f l u x p a t h . The gap - 55 -TABLE VI I 1 - CALCULATED FLUX DENSITY IN STEEL FOR MK VI - 1 MOD 9 MAGNET Rad i us $ As B s (model i n c h e s ) ( k G - i n . 2 ) ( i n .2) (kG) 2 4.92 • 0 . 4 12.3 4 25-3 1.2 21 .1 6 57.8 3.8 15.2 8 106 6.5 16.3 10 169 12.0 14.1 12 254 16.5 15.4 14 366 21.4 17.1 15 432 23.5 18.4 16.25 528 25.9 20.4 TABLE IX - CALCULATED FLUX DENSITIES COMPARED WITH ACTUAL FLUX DENSITIES R ( i n c h e s ) B (measured) (kG) B ( c a l c ) kG R a t i o ' 15 14.8 18.4 .80 10.5 12.3 14.3 .86 9 13-8 15.9 .87 5.5 14.7 17.5 .84 - 56 -r e l u c t a n c e can be b r o k e n i n t o two p a r a l l e l components - t h e h i l l gap and the v a l l e y gap r e l u c t a n c e s : zn • r V r - (32) g L, + L 3 h v it, 20 I Z = J L = _ _ > Z = — h a h a h v a v . (33) For a g i v e n m a g n e t , we know the h i l l and v a l l e y a r e a s a^ and a v > but we do not know t h e e f f e c t i v e v a l l e y p a t h l e n g t h l^. We can e s t i m a t e f r o m the r e l a t i o n s h i p a, Z, B = a Z B h h h v v v w h i c h comes f r o m t h e f a c t t h a t the two r e l u c t a n c e s a r e i n p a r a l l e l . Thus we have a K K K K A = a Z = a Z, — — = £ , — = 20 — V V V V h a B H B B . (34) d V V v V The m a g n e t i c c i r c u i t i s shown i n F i g u r e 27. Z^ i s t h e v e r t i c a l r e t u r n y o k e r e l u c t a n c e , and Nl i s t h e m a g n e t o m o t i v e f o r c e o f t h e main e x c i t a t i o n c o i l s . The t o t a l r e l u c t a n c e o f t h e magnet can be c a l c u l a t e d as f o l l o w s : Z l = 2Z • h l V 2 K + z K 1 z h 2 v 2 i Z , = 2Z + h 2 v 2 i n 2 v 2 T h i s p r o c e s s i s r e p e a t e d u n t i l 1^ has been c o m p u t e d . Then t h e t o t a l r e l u c t a n c e i s Z t = Z n + Z y . (36) - 57 -F i g u r e 28. D i v i s i o n o f P o l e T w o - s e c t i o n F l u x T i p u s e d f o r Ca 1 c u l a t io r is - 58 -z t c a n be used t o c a l c u l a t e N l , v i a t h e r e l a t i o n s h i p Nl = T T T - $ Z„ (37) The c a l c u l a t i o n was c a r r i e d o u t on t h e Mk V l - l Mod 9 model magnet . The p r o g r a m used f o r t h i s i s l i s t e d i n A p p e n d i x k. The r e s u l t s were not e n c o u r a g i n g . The c a l c u l a t e d number o f a m p e r e - t u r n s r e q u i r e d was 15,800, but t h e a c t u a l number i s 33,650. One o f the m a j o r c a u s e s o f t h i s d i s c r e p a n c y was t h e a s s u m p t i o n i m p l i c i t i n t h e c a l c u l a t i o n t h a t a l l t h e s t e e l a t any p a r t i c u l a r r a d i u s has the same p e r m e a b i l i t y . The f l u x measurements i n S e c t i o n 5-1 show how bad an a s s u m p t i o n t h i s i s . However , a c a l c u l a t i o n w h i c h t a k e s t h i s f a c t o r i n t o a c c o u n t w o u l d be v e r y d i f f i -c u l t . A p a r t i a l c o r r e c t i o n c a n be done by d i v i d i n g t h e magnet a z i m u t h a l l y i n t o two s e c t i o n s , c a l c u l a t i n g the r e l u c t a n c e f o r e a c h s e c t i o n , and a d d i n g them i n p a r a l l e l . The n a t u r a l d i v i s i o n i s a l o n g t h e l i n e shown i n F i g u r e 28, as t h e h e i g h t o f the s t e e l r e t u r n y o k e s u d d e n l y changes h e r e . A l s o , s i n c e t h e r e t u r n yoke p l a t e s a r e a l o n g a r a d i u s l i n e , t h e a i r gaps between t h e p l a t e s w i l l t e n d t o p r e v e n t f l u x g o i n g f r o m one s e c t i o n t o the o t h e r . The f l u x measurements do t e n d t o v i n d i c a t e t h i s a p p r o x i m a -t i o n . T h i s c a l c u l a t i o n was a l s o p e r f o r m e d u s i n g t h e Mk V l - l Mod 9 d a t a , and r e s u l t e d i n an e x c i t a t i o n o f 25,500 a m p e r e - t u r n s . T h e r e a r e many m a j o r s o u r c e s o f e r r o r i n a c a l c u l a t i o n o f t h i s t y p e . F o r one t h i n g , the s t e e l p e r m e a b i l i t y i s v e r y s t r o n g l y dependent on t h e f i e l d s t r e n g t h - a s m a l l A B s r e s u l t s i n a l a r g e A y . The c a l c u l a -t i o n s assume t h a t y i s c o n s t a n t o v e r t h e w h o l e s t e e l c r o s s - s e c t i o n , but t h i s f a r f r o m r e a l i t y . The computed p a t h l e n g t h s i n t h e s t e e l a r e a l s o w r o n g , as t h e f l u x does not n e c e s s a r i l y t r a v e l r a d i a l l y . A i r l e a k a g e f l u x must p r o d u c e some d i s c r e p a n c y . The t w o - s e c t i o n c a l c u l a t i o n does - 59 -n o t a l l o w any f l u x l e a k a g e f r o m one s e c t i o n t o t h e o t h e r , w h i c h i s a l s o n o t t r u e . In g e n e r a l , t h e n , t h e v e r y c o m p l e x g e o m e t r i c a l shape o f t h e TRIUMF magnet w i l l make any f a i r l y s i m p l e r e l u c t a n c e c a l c u l a t i o n s a c c u r a t e t o o n l y ±25% o r s o . The method d e s c r i b e d h e r e i s a c c u r a t e enough f o r t o l e r a n c e c a l c u -l a t i o n s , h o w e v e r . When t h e t w o - s e c t i o n c a l c u l a t i o n above was redone w i t h t h e s t e e l p e r m e a b i l i t y changed by 10%, i t was f o u n d t h a t t h e magnet r e l u c t a n c e changed by h%. A d d i n g on 25% f o r the e r r o r i n the c a l c u l a -t i o n g i v e s a maximum r e l u c t a n c e change o f 5%- So we have A H - ^ 2 ^ - . (38) We can compare t h i s w i t h t h e r e s u l t s o b t a i n e d by C r a d d o c k and R i c h a r d s o n . 7 S i n c e ^ - = r^jo" f o r TRIUMF, t h e n by e q n . (38) , Ay y 100 ' T h i s c o r r e s p o n d s c l o s e l y w i t h C r a d d o c k and R i c h a r d s o n ' s r e s u l t o f Ay „ 1 (39) y 150 • - 60 -REFERENCES 1. J . R . R i c h a r d s o n , P r o g r e s s i n N u c l e a r T e c h n i q u e s and I n s t r u m e n t a t i o n , I, 1-101 (1965) 2. L . S m i t h and A . A . G a r r e n , O r b i t Dynamics i n t h e S p i r a l - R i d g e d C y c l o t r o n , UCRL 8598 (1959) 3. T . I . A r n e t t e , H . G . B l o s s e r , M.M. Gordon and D .A . J o h n s o n , C y c l o t r o n P rograms f o r a S m a l l C o m p u t e r , NIM j_8_,_[9_, 3^ 3 (1962) k. M . K . C r a d d o c k and V . J . H a r w o o d , Beam L o s s by Gas S t r i p p i n g i n TRIUMF, p r i v a t e c o m m u n i c a t i o n (1968) 5. B . C . M u l l e n , Toward an A c c u r a t e C a l c u l a t i o n o f t h e Mean L i f e t i m e o f t h e H" Ion i n an E l e c t r i c F i e l d , M . S c . T h e s i s , Depar tment o f P h y s i c s , U n i v e r s i t y o f B r i t i s h C o l u m b i a (1968), u n p u b l i s h e d 6. G . M . S t i n s o n , W . C . O l s e n , W . J . M c D o n a l d , P . F o r d , D. A x e n , and E.W. B l a c k m o r e , E l e c t r i c D i s s o c i a t i o n o f H" Ions by M a g n e t i c F i e l d s , TR I-69-1 0969) 7. M .K . C r a d d o c k and J . R. R i c h a r d s o n , M a g n e t i c F i e l d T o l e r a n c e s f o r a S i x - S e c t o r 500 MeV H - C y c l o t r o n , TRI-67-2 (1968) 8. J . J . L i v i n g o o d , P r i n c i p l e s o f C y c l i c P a r t i c l e A c c e l e r a t o r s (D. van N o s t r a n d Company, I n c . , P r i n c e t o n , New J e r s e y , 1961) 9. E . G . A u l d , J . J . B u r g e r j o n and M . K . C r a d d o c k , I n i t i a l Model Magnet S t u d y f o r a S i x - S e c t o r 500 MeV FT C y c l o t r o n , TRI-67-1 (1967) 10. W. W a l k i n s h a w and N . M . K i n g , L i n e a r Dynamics i n S p i r a l R i d g e C y c l o t r o n D e s i g n , AERE GP/R 2050 11. K . R . Symen, D.W. K e r s t , L .W. J o n e s , L . J . L a s l e t t , and K . M . T e r w i l l i g e r , F i x e d F i e l d A l t e r n a t i n g G r a d i e n t P a r t i c l e A c c e l e r a t o r s , P h y s . R e v . 103, 1837 (1956) 12. J . R . H i s k e s , u n p u b l i s h e d , q u o t e d i n D . L . J u d d , NIM 18,19, 70 (1962) - 61 -A p p e n d i x 1 F o l l o w i n g i s a b r i e f d e s c r i p t i o n o f p r o g r a m MAGNET, w h i c h was used t o c a l c u l a t e p o l e t i p c o n t o u r s . A l i s t i n g o f t h e p r o g r a m , a t y p i c a l s e t o f i n p u t d a t a , and a t y p i c a l o u t p u t a r e a l s o g i v e n . The d e s c r i p t i o n b e l o w a p p l i e s o n l y t o t h e main r o u t i n e . The p r o g r a m was run on an IBM 36O/67. S t e p D e s c r i p t i on L i nes 1. Read i n p u t d a t a : a) No . o f r a d i i , i n i t i a l e n e r g y , e n e r g y i n c r e m e n t , f i n a l r a d i u s , i o n m a s s , maximum e n e r g y 1 - 6 b) Curve o f D vs R 7 - 8 c) Cu rve o f B^ vs R 9 - 10 d) m , n3, no 11 e) Cu rve o f a vs Rn 12 - 13 f ) P o l y n o m i a l c o e f f i c i e n t s f o r c u r v e £ vs D 14 - 15 g) C u r v e s - 6 , 6 3 vs R 16 - 19 h) P o l y n o m i a l c o e f f i c i e n t s f o r c u r v e o f F vs <f>D 20 - 21 i ) Z e r o o f f s e t s p i r a l a n g l e 22 2. I n i t i a l i z e , w r i t e m i s c e l l a n e o u s s t u f f 23 - 39 3. C a l c u l a t e y, 3, F, R 40 - 49 4. C a l c u l a t e D, £ , 6 1 , 6 3 , B f r o m i n p u t d a t a 50 - 55 5. F i n d a f r o m i n p u t d a t a , u s i n g i t e r a t i v e t e c h n i q u e 56 - 62 6. C a l c u l a t e <j>D and hence f i n d F f r o m i n p u t d a t a 63 - 64 7. Use e q u a t i o n 19 ( see S e c t i o n 2.4.2) t o f i n d e 65 - 70 8. F i n d h i l l a n g l e h by n u m e r i c a l i n t e g r a t i o n 71 - 82 9. C a l c u l a t e p o s i t i o n s o f p o l e t i p edges 83 - 84 10. O u t p u t r e s u l t s 85 - 86 11. I f n o t f i n i s h e d a l l e n e r g i e s , go t o s t e p 3 87 - 88 12. C a l c u l a t e h i l l - t o - h i l l d i s t a n c e s f o r new p o l e t i p 89 - 99 13- O u t p u t them 100 - 101 14. S t o p 102 FORTRAN IV G COMPILER MAIN 0 4 - 0 6 - 7 0 TTT57T5T- PAGE 0001 0001 0002 0003 000'. C — THIS VERSION ACCEPTS A BH VS R C U R V E , F VS P H I O E E , AND 2 C - - CURVES OF DELTA VS R. DATA FOR THIS IS STORED IN ( 3 0 0 0 ) . DIMENSION A N G ( 2 , 1 0 0 ) , Z X P ( 1 0 ) , R ( 5 0 ) , R E T A ( 5 0 ) , B R ( 5 0 ) , B H ( 5 0 > , 1 E ( 1 0 1 ) , X X ( 5 0 ) , Y Y ( 5 0 ) , R H O ( 1 0 0 ) , A L P H A ! 5 0 ) , E P S L N ( 1 0 ) , 967 101 2~~BA V n W ) , F L0T"( 100T7E T A~(T0 0) TGA'MM'A"(TO'0")TSPRANGT10 0)~ 3 D E L T 1 ( 5 0 ) , D E L T R 1 ( 5 0 ) , F 1 ( 5 0 ) , D E L T 3 I 5 0 ) , 4 0 E L T R 3 ( 5 0 ) , X A 1 ( 1 0 0 ) , Y A l ( 100) ,XA2( 100) , Y A 2 ( 100) READ ( 5 , 1 0 1 ) K l READ (5 ,102 ) E S T A R T , EDELTA FORMAT (10 15) 0005 102 FORMAT ( 8 F 1 0 . 5 ) 0006 READ (5 ,102) RHOF, AMASS, EMAX 0007 READ (5 ,101) NO 0008 READ (5 ,102) ( X X ( J ) , Y Y ( J ) , J=1 ,N0) 0009 READ (5 ,101) NB 00 10 READ (5 ,102) ( B R ( I ) , B H ( I ) , 1 = 1 , N B ) 00 11 READ (5 ,102 ) E T 1 , E T 3 , E T 0 0012 READ (5 ,1011NALPHA 0013 READ (5 ,102) ( R E T A I I ) , A L P H A ! I ) , I = 1 , N A L P H A ) 0014 READ (5 ,101) NEPS 0015 READ (5 ,102) ( E P S L N ( J ) , J = 1 , N E P S ) 0016 READ (5 ,101) NDELT1 0017 REATTT5" ,1TI2 )~I'O'E'LTRl ( J ) , D b L T l ( J ) , J = l , N D f c L T l ) 0018 READ (5 ,101) NDELT3 0019 READ (5 ,102) ( D E I T R 3 U ) , D E L T 3 ( J ) , J = 1 , N D E L T 3 I 0020 READ (5 , 101 )NF1 00 21 READ (5 ,102) ( F l ( J ) , J = 1 , N F 1 ) 0022 READ (5 ,102) EPSO ( ^ "t-0023 TN2E0=2 • * T A N ( E P S 0 / 5 7 . 2 9 5 8 ) * * 2 { 0024 B l = - 0 . 1 0025 B2=0.0 0026 Y1=0.0 0027 Y2=0.0 0028 H = 0 .0 0029 R A T 1 0 = ( E T 0 - E T 1 ) / ( E T 3 - E T 1 ) 0030 GAMAX = 1.0 + EMAX/AMASS 0031 BETMX = S Q R T d . O - 1 . 0 / G A M A X * * 2 ) 0032 BC ENT = A M A S S * 1 . 7 8 2 5 3 4 * 2 . 9 9 7 9 2 * B E T M X / ( R H O F * 1 . 6 0 2 0 7 * 2 . 5 4 ) 0033 WRITE ( 6 , 1 0 5 ) RHOF 0034 105 FORMAT (1H1 ,13HFINAL R A D I U S = , F 1 0 . 3 . 7 H INCHES ) 00 3 5 WRITE ( 6 , 1 0 6 ) BCENT.EPSO 0036 106 FORMAT * 'ZERO (15H CENTRAL F I E L D = , F 1 0 . 5 , 10H K I L O G A U S S , / , SPIRAL ANGLE = ' , F 6 . 1 , ' D E G R E E S ' , / ) 0037 WRITE (6 ,110 ) 0038 110 FORMAT (103H E R B AV B HILL FLUT 1 SP ANG ETA H ANG 1 ANG 2) ro 0040 0041 0042 0043 0044 ~ W 4 5 ~ 0046 0047 33 ~ r r r 34 DO 12 J = 1 , K I G A M M A ( J ) - I , 0 + E ( J ) / A M A S S B R 2 = 1 . 0 - 1 . O / G A M M A I J ) * * 2 IF ( B R 2 ) 3 3 , 3 4 , 3 4 WRITE ( 6 , 1 1 1 ) "FORMAT ("5'2'H-5ET7P BR2 = 0 . 0 BETA = SORT(BR2) SQUARE R"001 OF NEGATIVE NTJTOTTK^ D~EF I NED ZERO) ( FORTRAN IV G COMPILER RATFJ 04-06-70 H:b/ibl MAUL 0UU2 \ 0048 BAV(J)=GAMMA(J)*BCENT Q 0049 RHO (J) = RHOF*BETA/BETMX 0050 22 CALL F IT(XX , YY,RHOU),N0,D) 0051 EPS = AUX(EPSLN,ZXP,0,0,NEPS) (~ O0~5Z : CALL h I 1 rffECTTa"7DTL"TlTRH0TT)1 ,NDfcL T1,DEL IA 1 I < 0053 CALL FIT(DELTR3,DELT3,RHO(J),NDELT3,DELTA3 I 0054 DELTA=DELTA1+(0ELTA3-DELTA1)*RATI0 0055 CALL FIT(BR,BH,RHO(J),NB,BHILL) 0056 ALPH=0.965 0057 47 ETA(J)=(BAV(J)-EPS*DELTA*BHILL)*60./(BHILL«(ALPH-EPS*DELTA )) 0DT8 FrET=RH0T"J")"*"ETArj)TJ7T2 95"8 • — — 0059 CALL FITIRETA,ALPHA,RET,NALPHA,ALPH1) 0060 IF (ABS(ALPHl-ALPH).LE.0.001) GO TO 41 0061 ALPH=ALPH1 0062 GO TO 47 0063 41 PHIDEE=II60.-ETAIJ)>/57.29578)*D OTTCT FL"TJTT3T=Air!rt"Fl, L X P , PFTD bb,U,Nhl ) 006 5 Y21 = ((.12 50+1 BETA*GAMMA(J))**2)/FLDT(J)-1.0-TN2E0)/2 .0 0066 IF (Y21) 31,31,32 0067 31 Y21=0.0 0068 32 TANY21 = S0RTIY21) 0069 IF(E(J>.LT.50.0)TANY21=0.0 0TT7D 9U3 S"PRANG"m~sTA"T AN IT AN Y2 T T T*5 1. 2y 57 8 0071 B 3 =B E TA I 0072 Y3=TANY21*57.29578/B£TA ^ 0073 B3B23=B3=*3-B2**3 " s 0074 B3B"22 = B3-B3-B2*B2 | 0075 AIl=Yl*(B3B23/3.-(B2+B3)*B3B22/2.+82*B3*(B3-B2)) 1 j r r r r B T^B2 r™T8T*83-n j 0076 AI2=Y2*(B3B23/3.-(B1+B3)*B3B22/2.+B1*B3*(B3-B2)) ' 1/((B2-B1)*(B2-B3)) ; 0077 AI3=Y3*(B3B23/3.-(Bl+B2)*B3B22/2.+Bl*B2*(B3-B2)) 1/((B3-B1)*(B3-B2)) 0078 H=AI1+AI2+AI3+H 0079^  Y1"=Y2 • 0080 Y2=Y3 0081 B1=B2 0082 B2=B3 0083 ANGIl,J)=H+ETA(J)/2.0 0084 ANG(2,J)=H-ETAU)/2.0 0CTF5 HRTTE-r6"7T04-TFrUTTRH0"( J ) , BAV ( J I , BH I LL , FLUT ( J ) , SPRANG 1 J ) , E I A (J ) , 1 H,ANGI1,J),ANG(2,J) 0086 104 FORMAT (5X.10F10.4) 0087 11 E(J+1)=E(J)+EDELTA 0088 12 CONTINUE 0089 WR1TE(6,108) otr70 nrs FORMAT-(THl-) : 0091 DO 117 1=1,Kl 0092 ANG(1,I)=ANG(1,I)-60. 0093 XA1 (I)=RHO(II*COS((ANG(1,1 ) 1/57.29578) 0094 YA1( I )=RHO(I 1*SIN((ANG(1 ,1)1/57.29578) 0095 XA2( I ) = R H ( J ( I )*C0S(ANG(2, I )/57.29578) VGSb tr? rA-2-nT=RH0Tn»S"INrANGC2Trr/57T2957Tn • 0097 DO 118 1=1,Kl 0098 CALL DHILL(RHO(I),D,XA1,YA1,XA2,YA2,RHO,RHO,ANG.KI,KI) ( FORTRAN iV G COMPILER HSTN 04-06-70 ITT57T5T. 01707 0099 0 = 0 / 2 0 . 8 0100 118 W R I T E ( 6 , 1 0 9 ) R H O ( I ) , D 0101 109 F 0 R M A T ( F 1 0 . 3 , F 1 5 . ^ ) 0102 STOP  0103 END TOTAL MEMORY REQUIREMENTS 002CF6 BYTES I ON I F0R'TR7CN~rV~Tr~C0M PICE R~ ~AUX~ 0001 0002 0003 0004 ~W05~" 0006 0007 0008 0009 FUNCTION AUX(P,D,X,L,M) OIMENS 11IN P ( 10),D(10) 0(I)=I.0 AUX = P(l) o DT) 10 J=2,M D(J)=DU-1)*X 10 AUX=AUX+P(J)*D(J) RETURN END " T O T A L MEMORY RE0UTRTM F/NTS 000"1EC bYlfcS ON v_n "FORTRAN IV G COMPTCF/R - T T 4 - 0 6 W 0 11:58:46 PAGb OUOl 0001 0002 SUBROUTINE L O F ( X , Y , Y F , W , E 1 , E 2 , P , W Z , N , M , N I , N D , E P , A U X ) 01 MENS ION X( 100) , Y ( 100) , Y F ( 100) ,P( 1 0 ) , E l ( 10 ) , E 2 ( 10) ,W(100) , 1 C ( 1 0 , 1 0 ) , V ( 1 0 ) , 0 1 1 0 ) ^ 0003 ND = 1 J f 0004 NT = 1 0005 IV=0 0006 5 DO 10 1=1,M 0007 1 V ( I ) = 0 . 0 0008 DO 10 J=1,M 0009 10 C ( I , J )=0.0 00 10 TT=0 0011 DO 20 1=1,N 0012 IF(WZ) 6 , 7 , 6 0013 6 WT=W(L) 0014 22 GO TO 3 0015 7 WT = 1 . 0016 8 U = A U X ( P , D , X ( L ) , 1 , M ) 0017 DO 30 1=1,M 0018 DO 30 J = l , I 0019 30 C ( I , J ) = C ( I , J ) + W T * D ( I ) * D ( J ) 0020 DO 40 1=1,M 0021 40 V ( I ) = V ( I ) + W T * ( Y ( L ) - U ) * D ( I ) 0022 20 LUNIINUb 0023 DO 50 1=1,M 0024 JK=1+1 0025 I F ( J K . G T . M ) GO TO 50 0026 DO 49 J = JK , M 0027 C ( I , J ) = C ( J , I ) 0028 <tSJ LUN1INUb 0029 50 CONTINUE 0030 I F ( I V . E O . l ) GO TO 45 0031 I F ( N T - N I ) 3 5 , 4 5 , 5 5 00 32 35 CALL S 0 L T N ( C , V , M , 5 0 , D E T ) 0033 I F ( A B S I D E T ) . L T . 1 . 0 E - 1 9 ) GO TO 65 0034 DO 75 1=1,M 0035 P ( I ) = P ( I ) + V ( I ) 0036 TC = ABS (V(I ) / P ( I ) ) 0037 I F ( T C . G T . T T ) TT=TC 0038 75 CONT INUE 0039 NT =NT +1 0040 1b( 1 1 .L1 . b P ) IV = i 0041 . GO TO 5 0042 45 CALL I N V E R T ( C , M , 1 0 , D E T , C 0 N D ) 0043 I F ( A B S ( D E T ) . L T . 1 . 0 E - 1 9 ) GO TO 65 0044 DO 85 1=1,M 0045 DO 85 J=1,M 0046 85 P ( 1 ) =P ( i ) +C 1 1 , J ) *V ( J I 0047 55 DO 95 1=1,M 0048 95 E l ( I ) = S O R T ( C ( I , I ) ) 0049 3 F 0 R M A T ( 1 X , 8 G 1 5 . 5 ) 0050 S=0.0 0051 DO 105 L=1,N 0052 lb(WZ) 1 6 , W , 1 6 0053 16 WT=W(L) 0054 GO TO 18 OS FUR I KAN IV U LUMP 1 L b R - L0> U 4 - 0 6 - / U 1 1 : 5 8 : 4 6 PAGE 0002 0055 0056 0057 0058 W 0060 0061 0062 0063 0064 ~ W 6 5 ~ 0066 0067 0068 17 18 105 115 65 ~2 WT = 1 . Y F ( L ) = AUX < P , D , X ( L >,L,M> S = l Y ( L J - Y F ( L ) ) * * 2 * W T + S . CONTINUE o "TP -ST^M F I = S Q R T ( S / P P ) DO 115 1=1,M E 2 ( I ) = F I * E 1 ( I ) RETURN W R I T E ( 6 , 2 ) "FORMATT72H LINEAR EUUAIIUNS F A I L ) ND=0 RETURN END TOTAL MEMORY REQUIREMENTS OOOADO BYTES I I ( F O R T R A N iv G C O M P I L E R "FTT" 0 4 - 0 6 - f O 1 1 : 5 9 : 1 0 PAbt OUOl 0001 0002 . 0003 ? 0004 S U B R O U T I N E F I T ( X X , Y Y , R H 0 , N O , D > D I M E N S I O N X X ( 100) , Y Y ( 1 0 0 ) T H I S I S A L I N E A R I N T E R P O L A T I O N S U B R O U T I N E DO 21 1=2,NO I F r i R H o - x x i m " * T R H O - x x n " = i _ n . L b . o . o i GO Trr-72 ~ 21 C O N T I N U E I = N O I F ( ( X X ( 2 l - X X ( 1 ) ) * ( R H O - X X ( 1 ) ) . L E . 0 . 0 ) 1=2 22 0 = YY ( I - l ).+ ( Y Y ( I ) - Y Y < I - l ) ) * ( R H O - X X ( I -1 ) ) / ( XX ( I ) - X X ( I -1 ) ) RETURN 0005 0006 0007 0008 0009 0010 E N D T O T A L M E M O R Y R E Q U I R E M E N T S 000238 B Y T E S CTS Of FORTRAN IV U COMPILER SLOPE U * t - 0 6 - / 0 11 : 5 9 : l<t I ' Ab t U U U l 0001 0002 000 3 0004 SUBROUTINE S L O P E ( X , Y , X X , S , N ) OIMENS ION XI (7) ,Y1 (7) , Y F (7) ,W(5) , E 1 ( 7) , E 2 ( 7 ) , P ( 7 ) , X ( 100) ,Y< 100) W ( l ) = . 2 5 W(2)=.5 o 000b 0006 0007 0008 0009 0010 ~wT3~r=r; " ~ W<4)=.5 W<5>=.25 NN=N-2 DO 21 L=3,NN IF ( ( X X - X ( L ) ) * < X X - X ( L + 1 > ) ( 2 2 , 2 2 , 2 1 0012 0013 0014 0015 0016 "'21 C'ONTTNU r L=N-2 IF ( X X . L T . X I 3 ) > L=3 22 DO 27 K = l , 5 KL=L+K-3 P(K)=0 .0 ' 001 / 0018 0019 0020 0021 0022 UU23 XTTKT=XTK L ) -XX 27 Y 1 ( K ) = Y ( K L ) EXTERNAL AUX CALL L Q F ( X 1 , Y 1 , Y F , W , E 1 , E 2 , P , 1 . 0 , 5 , 3 , 1 , N D , 0 . 0 1 , A U X ) S=P (2 ) RETURN E N D TOTAL MEMORY REQUIREMENTS 0003CA BYTES C FORTRAN IV G COMPILER 0010 AX=R*C0S(ANG/57 .2958> 0011 A Y = R * S I N ( A N G / 5 7 . 2 9 5 8 ) 0012 CALL D I S T ( A X , A Y , D 1 , D 2 , X 1 , Y 1 X2 Y 2 , K I 1 , K I 2 ) 0013 IF (A8S <m-D2 ) . L E . 0 . 0 0 1 ) GO TO 20 0014 IF ( D 1 . G T . D 2 ) ANG2=ANG 0015 IF ( D 1 . L T . 0 2 ) ANG1=ANG D H I L l 0 4 - 0 6 - 7 0 TTT59TT9 - H A b b U U U 1 0001 0002 0003 0004 0005 0006 0007 0008 0009 T 5 ~ 16 10 SUBROUTINE D H I L L ( R , D , X 1 , Y 1 , X 2 , Y 2 , R 1 , R 2 , A N , K I 1 , K I 2 ) DIMENSION X I ( 1 0 0 ) , Y 1 ( 1 0 0 ) , X 2 ( 1 0 0 ) , Y 2 ( 1 0 0 ) , R 1 ( 1 0 0 ) , R 2 ( 1 0 0 ) , * T H E T A 1 ( 1 0 0 ) , T H E T A 2 ( 1 0 0 ) , A N ( 2 , 1 0 0 > DO 15 1=1,KI 1 "THE Txn n~=A _N"rrrn :  DO 16 1=1,KI2 T H E T A 2 I 1 ) = A N ( 2 , I ) CALL F I T ( R I , T H E T A 1 , R , K I 1 , A N G 1 ) CALL F I T ( R 2 , T H E T A 2 , R , K I 2 , A N G 2 ) ANG=(ANG1+ANG2)/2 .0 0016 0017 0018 0019 20 D=Dl+02 RETURN END TOTAL MEMORY REQUIREMENTS 000716 BYTES O D i. 0 FURIRAN IV G CUMPILfcR DIS1 0 4 - 0 6 - / 0 1 1 : 5 9 : 4 0 PAGE 0001 0001 0002 0003 ~o"oo~zr 0005 0006 0007 0008 "TJOTjg-0010 0011 0012 0013 0014 12 1 4 SUBROUTINE D 1 S T ( A X , A Y , D 1 , D 2 , X 1 , Y 1 , X 2 , Y 2 , K I 1 , K I 2 ) THIS SUBROUTINE CALCS DISTANCE FROM POINT TO TWO POLE TIP EDGES DIMENSION X 1 ( 1 0 0 ) , Y 1 ( 1 0 0 ) , X 2 ( 1 0 0 ) , Y 2 ( 1 0 0 ) 1=2 DT^S0T<riTAX^XTrr)T*^2T("AY=Yrrr)T*"*?T DO 11 1=2,KI1 CALL DAUXIXl ( I ) ,Y1 ( I ) ,X1 ( I - l ) , Y H I - l ) , AX , AY , DL I NE , XN , YN) I F ( ( X N - X 1 ( I ) ) * ( X N - X 1 ( I - l ) ) . L T . O . O . O R . ( Y N - Y 1 ( I ) ) * ( Y M - Y 1 ( I - l ) ) . * L T . O . O ) GO. TO 13 D P N T = S Q R T ( ( A X - X I ( I ) ) * * 2 + ( A Y - Y 1 ( I ) ) * * 2 ) rFTWNTTLT7DT)TJT=TTPNT o 13 11 ~0TJT5-0016 0017 0018 0019 GO TO 11 IF ( D L I N E . L T . D 1 ) D 1 = D L I N E GO TO 14 CONTINUE D2=S0RTI(AX-X211) )*=2+<AY-Y2(1)1**2> "DTJ-2T^1^2TKT2 " CALL D A U X ( X 2 ( I ) , Y 2 ( I ) , X 2 ( I - l ) , Y 2 ( I - l ) , A X , A Y , O L I N E , X N , Y N ) I F ( ( X N - X 2 ( I ) >*< X N - X 2 I I - l ) ) . L T . O . O . O R . ( Y N - Y 2 I I ) ) * ( Y N - Y 2 ( I -* L T . O . O ) GO TO 23 D P N T = S 0 R T ( ( A X - X 2 ( I ) ) * * 2 + ( A Y - Y 2 ( 1 ) ) * * 2 ) I F ( D P N T . L T . D 2 ) D 2 = D P N T 22 24 1 ) ) . O O ^ U GU 1U 21 0021 23 I F I D L I N E . L T . D 2 ) D 2 = D L I N E 0022 GO TO 24 0023 21 CONTINUE 0024 20 RETURN 0025 END I TOTAL MEMORY REQUIREMENTS 000536 BYTES C F O R T R A N IV G COHFTLTfT - 0 A W 11 i y v i ^ y 0001 0002 0003 0004 000 5 0006 0007 0008 SUBROUTINE D A U X < X X 1 , Y Y 1 , X X 2 , Y Y 2 , P X , P Y , D , X N E W , Y N E W ) C THIS SUBROUTINE FINDS THE D I S T . BETWEEN A POINT AND A LINE C AND GIVES THE POINT OF INTERSECTION OF THE P E R P . AND THE LINE A = S O R T ( < P X - X X 2 ) * * 2 + ( P Y - Y Y 2 ) * * 2 ) B^yORT11T"X=XXl"r*"-2+TPY-YYlT**21 C = S O R T ( ( X X I - X X 2 ) * * 2 + < Y Y l - Y Y 2 1 * * 2 ) S = ( A + B + C ) / 2 . 0 D = 2 . 0 * S O R T ( A B S I S * ( S - A ) * ( S - B ) * ( S - C ) ) ) / C E = ( B * B + C * C - A * A ) / ( 2 . 0 * C ) XNEW=XX1+(XX2—XX1)*E/C 0009 0010 0011 "YN"Ew=YYl + ( T T 2 - Y Y T ) *"E / 'C~ RETURN END TOTAL MEMORY REQUIREMENTS 000328 BYTES ho EXECUTION TERMINATED . S L I S A f 3TJOTJ-3001 3002 3003 3004 3005 ~ 3 0 0 6 " 3007 3008 3009 3010 3011 (3000,3400) 5"cr~ 2 0 . 0 311 . 15 0 . 190 . "240. 2 9 5 . 13 0 . 130 . 170 . Id 1 0 . 0 E S T A R T , EDELTA 939 .278 5 0 0 . R F I N A L , MASS OF H - , E MAX D VS R FROM OUTPUT OF MAG 1 - 3 1 / 1 2 / 6 9 0 . 1 0 0 . 2 . 9 4 170. 4 . 8 9 5 . 0 9 2 0 0 . 5 .06 2 1 0 . 4 .97 "4.32 2 6 0 . 3.76" — '280; 2 . 46 311 . MK V I - l MOD 0 DESIGN 4 . 6 4 110. 5 .14 150 . 5 .42 190. o ~3~0T~Z~ 3013 3014 3015 3016 3017 . 0 T .48 300 . BHILL VS R .0052 .04 .37 100. 1 4 0 . 180 . 97" 56 79 23 46 180. 220 . ""290"; 120. 160. 2 0 0 . 5.06 4 . 8 1 "2T5T" 4 . 9 2 5 . 3 0 5 . 5 0 ~~JTT~.— 2 4 . 5 25 0 . 2 0 . 4 0 . 5 . lb 2 6 . 1 . . 9605 .9658 ""31)18 3019 3020 3021 3022 3023 2 5 . ALPHA VS R*ETA MK V - l 5 . 0 .9868 10. 2 5 . .9601 3 0 . 5 0 . .9702 6 0 . ETA 1, E T A 3 , ETAO .9734 15 . .9613 3 5 . .9741 7 0 . .9641 .9634 .9776 120 . 1 6 0 . 2 0 0 . 4 1 .20124 .yaob . 9856 .9610 .9718 EPS VS - . 5 6 2 8 9 5 DELIA 1.0 1 .135 1.640 1 .781 1 .59 9"0~; 130. 1 7 0 . D CUBIC 0 .108713 -"MK" r9879~ .9852 .9790 "TOUT" 140. 180 . .9845 .9842 .9767 V - l Tixr. 150 . 190 . r985~4~ . 9828 .9744 ~3TJ2"4~ 3025 3026 3027 3028 3029 "3D3TJ" 3031 3032 3033 3034 3035 C O E F F I C I E N T S MK 00723508 Z2~ 0 . 0 41 .6 124 .8 2 0 8 . 2 8 0 . 8 VS R 1 0 . 4 62 .4 145 .6 228 .8 287 .04 -I . 004 .263 .723 .742 .57 2 0 . 8 83 .2 166 .4 2 4 9 . 6 2 9 1 . 2 1.024 1.395 1.77 1.69 1.565 31 .2 104. 187.2 2 7 0 . 4 2 9 5 . 3 6 1 .07 1 .518 1 .790 1 .626 1 .57 "301.6 27 0 . 41 .6 83 .2 166 .4 "TT5"97 DELTA 1 . 1 .108 1 . 367 1 .633 3T2~. VS R 10 .4 5 2 . 104 . 187 .2 F5"B"" -3 .00 3 .17 .468 .643 2 0 . 8 6 2 . 4 124 .8 2 0 8 . 1.02 1.235 1.547 1.633 31 .2 72 .8 145.6 2 2 8 . 8 ~2"4"9T5 2 9 3 . 2 8 3 0 1 . 6 3 - . 0 7 1 7 5 5 1 4 . 0 10 .0 1 . 5bb 1 .398 1 .394 "ZT0T4" r 295 .36 1 3 0 5 . 7 6 1 ".510 • .393 .400 V - 3 1 . 4 / 0 1.391 1.42 ~2?rT2 299.52 1.058 1.302 1 .604 1 .608 1 .410 1.392 "28TJT8 2 9 7 . 4 4 3 1 2 . 0 F VS PHIOEE MK .157608 "TTO-- . 0 1 1 8 1 4 9 ZERO SPIRAL ANGLE ~Crrr6"6-B"E-67 F I L E SRUN —LOAD# 5=A(3000 ,3400) 6=*SINK* tTxrrctrnorr-B Etrm s :  ( FINAL RAOIUS= 311 .000 INCHES CENTRAL F I E L 0 = 3 .00520 K1L0GAUSS ZERO SPIRAL ANGLE = 1 4 . 0 DEGREES 20.0000 83.3760 B AV 3.0692 B HILL 4.3682 30. 40. 50, 60 . 70. 80. 0000 OOOO 0000 0000 0000 0000 " T O T 116. 128. 140, 150 159. 90 , 100 , 110. 120. 1 30 . 140 . 0000 0000 0000 0000 0000 0000 ~T5T 1 75 182 189 196 202 T 2 T 9 -1103 8383 0840 1952 3980 8509 6730 9523 7584 1489 1682 ~3TTDTr 3.1332 3 .1652 3 .1972 3 .2292 3 .2612 r&5W" .8694 .0261 . 1408 .2314 .2958 3 . 2 9 3 T~ 3 .3251 3 .3571 3 .3891 3 .4211 3 .4531 4 . 5 . 5 . 5 . 5. ~5TT5"5"0~ 5 .3984 5 .4318 5 .4590 5 .4846 5 .5051 FLUT 0.1358 0.17X3""" 0.2031 0 .2275 0 .2472 0 . 2 6 3 9 0 .2771 —TTTZSKiT 0.2952 0 .2977 0 .2970 0 .2942 0 .2904 SP ANG 0.0 ETA 25.3681 H 0.0 ANG 1 12.6840 ANG 2 -12.6840 ~0T0""" 0 . 0 0 . 0 0 .0 0 . 0 0 . 0 — " 2 . 4 / 2 1 9 .7155 14.2162 18 .0334 21 .3853 24 .3328 " 7 5 T 4 T T 8 -25 .1282 24 .9021 24 .8611 24 .8673 2 5 . 0 5 8 0 "75T2T5T"" 2 5 . 4 6 1 3 25 .7000 25 .9353 26 .1813 26 .4596 '0.0 0 .0 0 . 0 0 . 0 0 . 0 0 .0 12 .7089 12 .5641 1 2 . 4 5 1 0 12 .4306 12 .4336 12 .5290 - 1 2 . / 0 8 9 -12 .5641 - 1 2 . 4 5 1 0 - 1 2 . 4 3 0 6 -12 .4336 - 1 2 . 5 2 9 0 0 .0523 0 .3110 0 . 8 1 1 7 1 .4179 2 .0984 2 .8290 12.65^3" 13 .0417 13 .6617 14 .3855 1 5 . 1 8 9 0 16 .0588 -12.55"4"B~ - 1 2 . 4 1 9 6 -12 .0383 -11 .5498 -10 .9923 -10 .4008 213 . 218. 223 . 227. 2 32. 2440 3587 2248 8625 2892 150. 160. 170. 180. 190. 200. 0000 0000 0000 0000 0000 0000 3.48"5T" 3 .5171 3 .5491 3 .5811 3 .6131 3 .6451 5 .5184 5 .5310 5 .5430 5 .5544 5 .5653 5 .5756 TTCT, 220. 2 30. 240. 250, 260. 0000 0000 0000 0000 0000 0000 "735" 240 244 248 251 255 r5~2T5~ ,5733 .4565 .1833 .7633 ,2055 "T75T7T" 3.7091 3.7411 3.7731 3.8051 3.8371 "T75"8~55-5 .5950 5 .6041 5 . 6 1 2 9 5 .6212 5.6293. "0. 2853" 0 .2788 0 .2717 0 .2637 0 .2553 0 .2469 ""0T2"3"8T-0 .2307 0 .2225 0 .2146 0 .2070 0 .1995 2 /. 0747"" 29 .6927 32 .1222 34 .4690 36 .6922 38 .7733 2 6 . / T T B ~ 2 7 . 1 4 4 4 27 .4955 27 .8110 28 .1091 2 8 . 4 2 8 3 T0T724TT 42.5592 44.3249 45.9906 47.5730 49.0749 ~2 8 .15 T 4 ~ 2 9 . 0 6 8 9 29 .3418 29 .6060 29 .8706 30 .1286 3 . 5 9 W 4 . 3 8 5 4 5 .1983 6 . 0 2 9 9 6 . 8 7 2 9 7 .7261 8 .5"B"6"3~ 9 .4510 10 .3215 11 .1905 12 .0677 12 .9444 16 .9833 17 .9576 1 8 . 9 4 6 0 19 .9354 2 0 . 9 2 7 4 2 1 . 9 4 0 2 - 9 . / 9 4 5 - 9 . 1 8 6 8 - 8 . 5 4 9 4 - 7 . 8 7 5 6 - 7 . 1 8 1 7 - 6 . 4 8 8 1 -72T9"6"3"5~ 2 3 . 9 8 5 5 2 4 . 9 9 2 4 2 5 . 9 9 3 5 27 .0031 2 8 . 0 0 8 7 - 5 . / 9 0 9 - 5 . 0 8 3 4 - 4 . 3 4 9 4 - 3 . 6 1 2 6 - 2 . 8 6 7 6 - 2 . 1 1 9 8 ~ZTiT, 280. 290, 300 . 310. 320, "0000 0000 0000 0000 0000 0000 "75-5: 261 , 264. 267, 270 , 273, T 3 7 X 340, 350. 360. 370. 380 , TJOTJTT 0000 0000 0000 0000 0000 "776" 278 281 283 285 288 rsrw" 7102 7874 7561 6221 3916 "05BSr 6587 1658 5935 9458 2266 T 7 W 3 T " 3 .9010 "T97T. 400 . 410, 420 . 430, 440 . 0000 0000 0000 0000 0000 0000 zvrr, 292 . 294 , 2 96. 298 . 300 . ?3TT5~ 5R50 6689 6926 6592 5706 .9330 .9650 .9970 .0290 r0"6T0~ . 0930 . 1250 .1570 , 1890 4 .2210 -4-775"3"Cr 4 . 2 8 5 0 4 . 3 1 7 0 4 .3490 4 . 3 8 1 0 4 .4130 "5TU3TT 5 .6445 5 .6518 5 .6587 5 .6654 5 .6719 ~5T5787~ 5 .6842 5 .6901 5 .6958 5 .7013 5 .7067 0 .1851 0 . 1778 0 . 1710 0 . 1646 0 .1586 -0T1"530 _ 0.1478 0 .1430 0 .1386 0 .1345 0 .1310 460 . 470. 480 . 490 500. 73TJ07r 0000 0000 0000 0000 0000 T10.00W" ~3TJ7^ 304. 30 5. 307 . 30 9. 310. "3T2"; "47"97r 2368 9961 7083 3757 9998 "587T" ^T4"T5"[T 4 .4770 5090 5409 5729 6049 "6"35"9~~ " 5 T T I T 8 " 5 .7169 5 .7217 5 .7265 5.7311 5 .7356 T 3 W 7442 7483 7523 7562 7600 "TJ. 1279 0 .1244 0 .1214 0 .1186 0'. 1156 0. 1124 "577537" "TJTT0TJ7" 0 .1051 0 . 1013 0 .0976 0 .0932 0 .0896 ""070"8"4"9~ "snT^gff?" 51 .9005 53 .2589 54 .5319 55 .7258 56 .8608 "57792 5T~ 58 .9213 59 .8542 6 0 . 7 3 5 9 6 1 . 5 5 6 9 6 2 . 2 9 6 8 ~67T9^8T2~ 6 3 . 6 9 1 7 64 .3416 6 4 . 9 4 5 8 65 .5631 6 6 . 1 9 6 0 ~65T86"6"0~ 6 7 . 5 1 1 2 6 8 . 1 6 3 3 6 8 . 8 0 9 4 69 .5141 70 .1265 "3"0T373"8~ 30 .5171 30 .5455 30 .5271 30 .4623 30 .3726 ~70723"50~ 30 .0461 29 .8001 2 9 . 5 0 8 4 29 .1562 2 8 . 7 0 1 2 "13 .818T" 14 .6938 15 .5660 16 .4429 17 .3153 18 .1848 ~T9TD5T2~ 19 .9265 20 .7989 21 .6662 22 .5395 2 3 . 3 9 2 3 "79TTJTJ52" 2 9 . 9 5 2 4 30 .8388 31 .7065 32 .5465 33 .3711 - 1 . 3 6 8 6 - 0 . 5 6 4 7 0 . 2 9 3 3 1 .1794 2 .0842 2 .9985 3^TT7-ZT-3 4 . 9 4 9 5 35 .6989 36 . 4204 37 .1176 3 7 . 7 4 2 9 3 .9377 4 . 9 0 3 4 5 .8988 6 . 9 1 2 0 7 . 9 6 1 3 9 . 0 4 1 7 "7B"72"97T" 2 8 . 4 4 9 3 28 .5360 28 .8809 29 .3337 29 .9299 " 7 0 T 8 W T 31 .6428 32 .4843 33 .3111 34 .2246 34 .9837 "7ZT72T8"9_ 25 .0895 25 .9083 2 6 . 7 2 0 6 27 .5436 28 .3495 29 .9546 30 .7580 31 .5866 32 .3652 33 .1987 3 8 . 3 8 / 4 39 .3141 4 0 . 1 7 6 3 4 1 . 1 6 1 0 4 2 . 2 1 0 4 4 3 . 3 1 4 5 10 .0903 1 0 . 8 6 4 9 11 .6403 12 .2801 12 .8767 13 .3846 "T0TB5"8"4~ "35T9T87" "3"4"7"070~8~ 44.53"9T" 45 .7760 47.0001 48 .2421 49.4775 50.6905 ""5T797W" "T3T73TJ0 -14 .1332 14 .5158 14 .9311 15 .2528 15 .7068 "T570617 c — , 8 3 . 3 7 6 2 . 4137 1 0 1 . 3 2 6 2 . 8967 116 .110 3 . 3 3 5 9 1 2 8 . 8 3 8 3 .7243 140 .084 4 . 0 6 3 3 ( 150 .195 4 .359^ < 159 .398 4 . 6 1 8 9 167 .851 4 . 8 4 1 0 175 .673 5 .0038 1 8 2 . 9 5 2 5 .0786 189 .758 5 .0868 1 9 6 . 1 4 9 5 .0H08 202 .168 5 . 0 4 4 9 2 0 7 . 8 5 6 4.9922 213 .244 4 . 9 3 1 5 2 1 8 . 3 5 9 4 . 8 5 5 9 2 2 3 . 2 2 5 4 .7812 22 1 .862 4 . ab9 2 3 2 . 2 8 9 4 . 6 4 9 9 2 3 6 . 5 2 2 4 . 5 8 2 9 2 4 0 . 5 7 3 4 . 5 2 1 7 2 4 4 . 4 5 7 4 .4642 2 4 8 . 1 8 3 4 . 4 1 1 7 2 5 1 . I6i 4 . i 1 u 1 2 5 5 . 2 0 6 4 . 2 7 0 6 2 5 8 . 5 1 9 4 . 1 4 9 0 261 .710 4 .0160 2 6 4 . 7 8 7 3 .8786 2 6 7 . 7 5 6 3 .7450 2 /0.622 3 .6151 273 .392 3 .4882 2 7 6 . 0 6 9 3 . 3668 2 7 8 . 6 5 9 3 . 2 4 7 7 2 8 1 . 1 6 6 3.1313 283 .594 3 .0572 2 8 5 . 9 4 6 3 .0312 2 8 8 . 2 2 7 3 . 0 4 0 2 2 9 0 . 4 3 8 3 .0716 292 .585 3 .1220 2 9 4 . 6 6 9 3 .1840 2 9 6 . 6 9 3 3 .2552 2 9 8 . 6 5 9 3 .3351 300 .571 3 . 4 1 7 9 3 0 2 . 4 2 9 3 . 5 0 6 3 3 0 4 . 2 3 7 3 .5977 3 0 5 . 9 9 6 3 . 6 8 9 7 3 0 7 . 7 0 8 3 . 7 8 2 9 3 0 9 . 3 / 6 3 .8798 311 .000 3 .9763 312 .582 4 . 0 7 1 9 STOP 0 EXECUTION TERM I NAT EO SSIG - 76 -A p p e n d i x 2 F o l l o w i n g i s a b r i e f d e s c r i p t i o n o f p r o g r a m F I E L D , w h i c h was used t o c a l c u l a t e t h e a z i m u t h a l f i e l d c o n t o u r s p r o d u c e d by a g i v e n p o l e t i p . A l i s t i n g o f t h e p r o g r a m , a t y p i c a l s e t o f d a t a , and a t y p i c a l o u t p u t a r e a l s o g i v e n . The p r o g r a m was run on an IBM 36O/67. S t e p D e s c r i p t i o n L i n e s 1. Read i n p u t d a t a : a) X,Y c o o r d i n a t e s o f b o t h edges o f t h e p o l e t i p 1 - 8 b) Curve o f B vs D' 9 - 10 c) Curve o f £ vs D 11 - 12 d) R a d i u s , B i n i t i a l , and B f i n a l f o r each t o be g e n e r a t e d f i e l d 13 " 14 2. C o n v e r t X , Y c o o r d i n a t e s t o R,8 c o o r d i n a t e s 15 " 25 3. i :a C a l c u l a t e h i l l - t o - h i l l d i s t a n c e D ( s u b r o u t i n e DHILL) 27 h. Use t h i s t o f i n d £ f r o m c u r v e o f £ vs D 28 5. G e n e r a t e f i e l d c o n t o u r ( s u b r o u t i n e CURVE) 29 6. C a l c u l a t e B and F ( s u b r o u t i n e FLUT) 30 7- O u t p u t r e s u l t s 32 - 37 8. I f n o t f i n i s h e d a l l r a d i i , go t o s t e p 3 38 9. S top 39 Sub r o u t i nes DIST (AX , AY , D 1 , D2) - c a l c u l a t e s d i s t a n c e s D1 and D2 o f t h e p o i n t A X , Ay f r o m t h e two p o l e t i p e d g e s . FLUT ( B , N, BAV, F) - c a l c u l a t e s t h e a v e r a g e BAV and f l u t t e r F o f t h e N p o i n t f i e l d B. DAUX ( X X I , Y Y 1 , X X 2 , Y Y 2 , P X , P Y , D, XNEW, YNEW) - f i n d s t h e d i s t a n c e D between p o i n t P X , P Y and t h e l i n e d e f i n e d by t h e two p o i n t s XX1,YY1 and X X 2 , Y Y 2 , and g i v e s t h e p o i n t o f i n t e r s e c t i o n XNEW,YNEW between t h e l i n e and the p e r p e n d i c u l a r . - 77 -CURVE (R, B , BSTART, BEND, EPS) - c o n s t r u c t s a 61 p o i n t f i e l d c o n t o u r B w h i c h has i n i t i a l h i l l f i e l d BSTART and f i n a l h i l l f i e l d BEND. The r a t i o between h i l l and v a l l e y f i e l d s w i l l be E P S , and R i s the r a d i u s . F IT (XX , Y Y , RHO, NO, D) - g i v e n a c u r v e w i t h NO d a t a p o i n t s X X , Y Y , a l i n e a r i n t e r p o l a t i o n t e c h n i q u e i s used t o f i n d t h e Y v a l u e D c o r r e s p o n d i n g t o t h e X v a l u e RHO. DHILL (R,D) - f i n d s t h e h i l l - t o - h i l l d i s t a n c e D a t r a d i u s R. RFS NO. 034254 ; UNIVERSITY OF B C COMPUTING CENTRE MTS(AN120) 1 4 : 2 7 : 2 3 0 4 - 0 6 - 7 0 F I L E S PERMITTED IN THE OLD SYSTEM MUST BE PERMITTED AGAIN. 1S1G TRSU PLEASE RETURN TO TRIUMF * * * * * * * * * * * * * * * * * * * * * * * * * * * t $SIG TRSO o •**LAST SIGNON WAS: 1 1 : 5 7 : 3 9 USER "TRSO" SIGNED ON AT SCOP ' S O U R C E * - A S L I S *SOURCE*+-A 0 4 - 0 6 - 7 0 1 4 : 2 7 : 2 4 ON 0 4 - 0 6 - 7 0 1 1 1 DIMENSION X I ( 1 0 0 ) , Y 1 ( 1 0 0 ) , X 2 ( 1 0 0 ) , Y 2 ( 1 0 0 ) , R 1 ( 1 0 0 ) , R 2 < 1 0 0 ) , 2 * T H E T A 1 ( 1 0 0 ) , T H E T A 2 ( 1 0 0 ) , B E X P ( 1 0 0 ) , D E X P ( 1 0 0 ) 3 CUMMON XI , Y 1 , X 2 , Y 2 , R 1 , R 2 , T H E T A 1 , T H E T A 2 , K I 1, K I 2 , NE XP , B EXP ,DE XP 4 D l MENS ION B I ( 5 O ) , B F ( 5 0 ) , E P S ( 5 0 ) , R ( 5 0 ) , B ( 6 1 ) , D H ( 5 0 ) 5 READ (5 ,101 ) KI1 6 READ ( 5 , 1 0 7 ) ( X 1 ( I ) , Y 1 ( I ) , I = 1 , K I 1 ) 7 READ ( 5 , 1 0 1 ) KI2 8 R E A D ( 5 , 1 0 7 ) ( X 2 ( I ) , Y 2 ( I ) , I = 1 , K 1 2 ) 9 READ ( 5 , 1 0 1 ) NEXP 10 READ ( 5 , 1 0 2 ) ( D E X P I I ) , B E X P ( I ) , 1 = 1 , N E X P ) 11 REAO (5 ,101 ) NEPS 12 READ (-5,102) (DH( I ) , E P S ( I ) , 1 = 1 ,NEPS) 13 READ ( 5 , 1 0 1 ) NR 14 READ ( 5 , 1 0 2 ) (R(I ) ,81 ( I ) , 8 F ( I ) ,11111,1 = 1,NR) 15 107 FORMAT ( 1 0 F 7 . 3 ) 16 101 FORMAT (15) 17 102 FORMAT ( R F 1 0 . 5 ) 18 WRITE (6 ,103 ) 19 103 FORMAT (1H1) 20 DO 20 1=1,K11 21 R I ( I ! = S O R T ( X I ( I ) * * 2 + Y l ( I ) * * 2 ) 22 20 T H E T A 1 ( I ) = A T A N ( Y 1 ( I ) / X l ( I ) 1 * 5 7 . 2 9 5 7 8 23 DO 21 1=1,KI2 24 R 2 ( I ) = S O R T ( X 2 ( I ) * * 2 + Y 2 ( I ) * * 2 ) 25 21 T H E T A 2 ( I ) = A T A N ( Y 2 ( I ) / X 2 ( I ) 1 * 5 7 . 2 9 5 7 8 26 00 10 1=1,NR 27 CALL D H I L L ( R ( I ) , D ) 28 CALL F I T I D H , E P S , D , N E P S , E P ) 29 CALL C U R V E ( R ( I ) , B , B I ( I ) , B F ( 1 ) , E P ) 30 CALL F L U T ( B , 6 1 , B A V , F ) 31 WRITE (6 ,104 ) R ( I ) , D 32 104 FORMAT (' RADIUS = • , F 6 . 0 2 , ' H I L L - T O - H I L L DISTANCE = 33 * F6 .03 , / ) 34 WRITE (6 ,105 ) ( B ( J ) , J = 1 , 6 1 ) -35 105 FORMAT ( 1 0 F 9 . 3 ) 36 WRITE (6 ,106 ) B A V , F 37 106 FORMAT ( / , ' AVERAGE F I E L D = ' , F 7 . 4 , ' FLUTTER = S F 7 . 5 , / / / ) 38 10 CONTINUE 39 STOP 40 END 41 SUBROUTINE 0 I S T ( A X , A Y , D 1 , D 2 ) 42 C THIS SUBROUTINE CALCS DISTANCE FROM POINT TO TWO POLE TIP EDGES 43 DIMENSION X I ( 1 0 0 ) , Y 1 ( 1 0 0 ) , X 2 ( 1 0 0 ) , Y 2 I 1 0 0 ) , R 1 ( 1 0 0 ) , R 2 ( 1 0 0 ) , 44 * T H E T A 1 ( 1 0 0 ) , T H E T A 2 ( 1 0 0 ) , B E X P ( 1 0 0 ) , D E X P ( 1 0 0 ) 45 COMMON X 1 , Y 1 , X 2 , Y 2 , R 1 , R 2 , T H E T A 1 , T H E T A 2 , K I 1 , K I 2 , N E X P , B E X P , D E X P 46 1 = 2 47 D 1 = S 0 R T ( ( A X - X 1 ( 1 ) ) * * 2 + ( A Y - Y 1 ( 1 ) ) * * 2 ) 48 DO 11 1=2,KI1 49 12 CALL D A U X I X l ( I ) , Y 1 ( I ) , X 1 ( I - 1 ) , Y 1 ( I - 1 ) , A X , A Y , D L I NE , XN, YN) ) oo f 50 51 IF( ( X N - X l <I > >*(XN-X1I I - l ) ) . L T . O . O . O R . ( Y N - Y 1 ( I ) )*(YN-Y1< I - l ) ) . * L T . O . O ) GO TO 13 52 14 O P N T = S O R T ( ( A X - X 1 ( I ) ) * * 2 + ( A Y - Y 1 ( 1 ) 1 * * 2 ) 5 3 IF ini 'NT.I .T.01 ) n i = ni'NT 54 GO TO 11 \ 55 13 IF < D L I N E . L T . D 1 ) D 1 = D L I N E / / 56 GO TO 14 \ 57 11 CONTINUE 58 0 2 = S 0 R T ( ( A X - X 2 ( 1 ) ) * * 2 + ( A Y - Y 2 ( 1 1 ) * * 2 ) 59 00 21 1=2,KI2 60 22 CALL D A U X U 2 (I ) , Y 2 ( I > ,X2< I - l ) ,Y2( I - l ) , A X , A Y , D L I N E , X N , Y N ) 61 I F ( ( X N - X 2 ( I ) ) * ( X N - X 2 ( I - l 1 ) . L T . 0 . 0 . O R . ( Y N - Y 2 ( I ) ) * ( Y N - Y 2 ( I - l ) ) . 62 * L T . O . O ) GO TO 23 63 24 D P N T = S O R T ( ( A X - X 2 ( I ) 1 * * 2 + ( A Y - Y 2 ( I ) ) * * 2 ) 64 I F ( O P N T . L T , D 2 ) D 2 = D P N T 65 GO T0 21 66 23 I F I D L I N E . L T . D 2 ) D 2 = D L I N E 67 GO TO 24 68 21 CONTINUE 69 20 RETURN 70 ENO 71 SUBROUTINE F L U T ( B , N , B A V , F ) 72 C THIS SUBROUTINE CALCS AVERAGES AND FLUTTER OF 6 1 - P T . F I E L D DATA 73 DI MENS I ON B ( 6 1 ) , D ( 6 1 ) 74 AN = N 75 K = 0 76 15 M=(N-11 /2 77 K = K + 1 78 SUM4=0.0 79 00 11 J=1,M 80 JA=2*J 81 11 SUM4=SUM4+B(JA) 32 M=M-1 83 SUM2=0.0 84 00 12 J=1 ,M 85 JA=2*J + 1 86 12 SUM2=SUM2+B(JA) 87 M = 2 * ( ( N - 1 ) / 2 ) + l 88 SUM=(4 .*SUM4+2 ,*SUM2+B(1 )+B(M)) /3 . 89 IF ( M . N E . N ) SUM=SUM+B(N) 90 SUM=SUM/(AN-1.1 91 I F ( K . E 0 . 2 1 GO TO 17 92 BAV=SUM 93 DO 18 J=1,N 94 D (J )=B(J 1 95 18 B ( J ) = ( B ( J ) - B A V ) * * 2 96 GO TO 15 97 17 F=SUM/(BAV*BAV) 98 DO 19 J=1,N 99 19 B ( J ) = D ( J ) 100 RETURN 101 END 102 SUB ROUT INE D A U X ( X X 1 , Y Y 1 , X X 2 , Y Y 2 , P X , P Y , 0 , X N E W , Y N E W ) 103 C THIS SUBROUTINE FINDS THE D I S T . BETWEEN A POINT AND A LINE 104 C AND GIVES THE POINT OF INTERSECTION OF THE P E R P . AND THE LINE 105 A = S O R T ( ( P X - X X 2 1 * * 2 + ( P Y - Y Y 2 1 * * 2 ) 106 B = S 0 R T ( ( P X - X X 1 ) * * 2 + ( P Y - Y Y 1 1 * * 2 ) 107 C = S O R T ( ( X X 1 - X X 2 ) * * 2 + ( Y Y l - Y Y 2 ) * * 2 ) 108 S=(A + B+C 1 /2 .0 \ 109 D = 2 . 0 * S O R T ( A 8 S ( S * ( S - A ) * ( S - B 1 * ( S - C 1 I ) / C J o - 80 -O O O CO tt "V • O UJ UJ CD • tt tt + (\j — — a: - x > uu • x >- > : I I cc .. rsj (NJ D <I X >- o X > O —' UJ tt + + z X > h-) X >- Z ID II II CC C ) 3 ^ ^ cc » LU UJ I— O CO i Z Z UJ Z D J X >- OC UJ to O OJ fO vt m I — <: o : CC o > - . cc — — > D o n : o o x O L O O — ' > < Q . — X I CD fNJ LU I X CC LO O O i I— o o O —t <-H z> — — , CC (NJ , >- *3 - X • - i C .-H — «-x o >- CO •—li 1 (\j • *• o < r\j - t\j' I— \ • i LU — il r-I (NJ ' U N I— C O LU <-• : _J o o : o *• z CO o H LfN fNJ • 0 s •*-* C I oo o r- >- oj , in < O i »• z ; O x < • z <r • < — CD CD CO i X OL. UJ X • CD LU o o 2! LO >-i CD O f-1 o CD .-t — O I —• — D to s: CO CO I — I O O —i •c « <r cc to i -z LU LO LU X L O ' Z <-0 LL 1 o z 2 : UJ _j . 2 1 —I O ~ < O O O o • < z < o < : >- lO O 1 • O O i C tt . CC i II ( O X . o < • I CO • < i • co —' i— • I LO | < o CQ . cc z — . II < II i ~ M a • i —i o >- : ~ z s: ( I CD < CQ i O o ' s: z II ^ o II t-0 O s CQ O * O I L II — O o Q M ^ £fl (J Q 1 O — o CO — u II CO c I — II —I — ' o —• —> ' - i II O LL II M UJ w • z ^ cc —« o II h H o z ~ o o ( D U O — CO I — OJ II — < CO < r-< CO >0 tt . _ < I CO I - ' I I -£ o a: ; w z < a u j i -I CO to - — CO II II _ - CO o - CO CO O - CD O II _ O — • O < CD sO r- CO f> O —* I -I H r-t (\J N I co 0 s o »-« fNJ ro . i oo oj ro ro ro ro i o oj ro *t in >r *t -4- -j- vj- iO t* O O1 O H ( • J • ! >f -t I T M T I • co fJ* o •-* fM ro m J5 r- oo >£) 45 Ji -O >0 ' 170 102 CONTINUE 171 RETURN 172 END 173 SUBROUTINE F I T ( X X , Y Y , R H O , N O , D ) 174 DIMENSION X X ( 1 0 0 ) , Y Y ( 1 0 0 ) k 175 C THIS IS A LINEAR INTERPOLATION SUBROUTINE ( 176 DO 21 1=2,NO 177 IF ( ( R H O - X X I I ) > * ( R H O - X X < I - l ) ) . L E . 0 . 0 ) GO TO 22 178 . 21 CONTINUE 179 I=NO 180 IF ( (XX (2 )-XX-( 1 ) )* (RHO-XX ( 1) ) . L E . 0 . 0 ) 1 = 2 181 22 D = Y Y ( I - l ) + ( Y Y ( I J - Y Y I I - l ) ) * ( R H O - X X ( I - 1 ) ) / ( X X I I ) - X X ( I - 1 ) ) 182 RETURN 183 ENO 184 SUBROUTINE D H I L L ( R , D ) 185 DIMENSION X 1 ( 1 0 0 ) , Y 1 ( 1 0 0 ) , X 2 ( 1 0 0 ) , Y 2 ( 1 0 0 ) , R 1 ( 1 0 0 ) , R 2 ( 1 0 0 ) , 186 * T H E T A 1 ( 1 0 0 ) , T H E T A 2 ( 1 0 0 ) , B E X P ( 1 0 0 ) , D E X P ( 1 0 0 ) 187 COMMON X I , Y 1 , X 2 , Y 2 , R 1 , R 2 , T H E T A 1 , T H E T A 2 , K I 1 ,KI 2 , N E X P , B E X P , D E X P 188 CALL F I T ( R 1 , T H E T A 1 , R , K I 1 , A N G 1 ) 189 CALL F I T ( R 2 , T H E T A 2 , R , K I 2 , A N G 2 ) 190 10 ANG=(ANGl+ANG2) /2 .0 191 A X = R * C O S ( A N G / 5 7 . 2 9 5 8 ) 192 A Y = R * S I N ( A N G / 5 7 . 2 9 5 8 ) 193 CALL D I S T ( A X , A Y , 0 1 , D 2 ) 194 IF ( A B S ( D l - 0 2 ) . L E . 0 . 0 0 1 ) GO TO 20 195 . IF ( 0 1 . G T . D 2 ) ANG2 =ANG 196 IF ( D 1 . L T . D 2 ) ANG1=ANG 197 GO TO 10 198 20 D=D1+D2 199 RETURN 200 END END OF F ILE $SIG • - - - •- ->. J R FS NO. 033083 UNIVERSITY OF B C COMPUTING CENTRE MTSIAN059) JOB S T A R T : 1 1 : 4 6 : 2 4 01 -15-70 #*#a«*#*9*»#Wft*a<is PLEASE RETURN TO TRIUMF a*************  o $SIG TRSO THE DISK SPACE ALLOTTED THIS USER * * L A S T . S I G N O N WAS: 11:23:22 01 USER "TRSU" SIGNED ON AT 11:46 SCOP C I 4 , 3 6 ) + « S 0 U R C E * D SLIS D  ID HAS BEEN E X C E E D E D . 15-70 30 ON 0 1 - 1 5 - 7 0 1 72 CURVE 1 OF MK V l - l MOD 10 TIP AS MEASURED BY D . S . 2 .000001 .01 .000002 2. .000003 4 . .000004 6 . .000005 6 . 5 3 .001 6 .916 .010 7 .306 .020 7 .498 .040 7 .806 .060 8 . 0 6 7 4 . 089 8 .350 . 1 1 9 8 .575 .149 8 .783 .199 9 .093 .248 9 .352 5 .298 9 .578 .347 9 .771 .396 9 .946 .446 10 .104 .494 10 .245 6 .593 10.498 .691 10 .716 .790 10 .918 .889 11 . 113 .987 11 .287 9 10 11 12 1 .085 1 .570 2 . 5 4 6 3 .670 4 . 9 0 9 6 .152 11 .442 11.978 12 .647 13.002 13 .167 13.233 1.178 1 .766 2 .742 3 . 9 1 8 5 .158 6 .400 11. 12. 12. 13. 13. 13 . 555 151 738 049 186 233 277 961 938 166 408 649 11 .674 12 .298 12 .817 13 .088 13 .206 13 .230 1.374 11.782 2 .156 12 .429 3 . 1 8 0 12 .894 4 .412 13 .119 5 .655 13 .219 6 .898 13 .224 1 .472 2 .351 3 .423 4 .661 5 .904 7 .146 884 545 951 146 228 215 13 14 15 16 17 18 7 .395 8 .642 9 . 7 3 9 9 .800 78 .001 13 .203 13.122 13 .012 13.004 002137 7 .644 8 .892 9 .750 9 . 8 1 0 CURVE 1 .000 13. 13 . 13. 13. 2 OF 2. 188 7. 101 9. 010 9. 003 MK V I -137 2. 895 140 760 13 .175 13 .076 13 .009 8 .144 13 .159 9 .390 13 .050 9 .770 13.008 1 MOD 10 TIP AS MEASURED OOP 4 . 2 7 3 2 .500 5.342 8 .393 9 .639 9 .790 BY D . S . 2 .750 141 023 005 C O 5 .876 19 20 21 22 23 24 2 . 8 0 9 3 .310 3 . 8 1 8 4 . 4 3 7 5 .457 6 .473 6 .002 7 .057 7 .935 8 .563 9.141 9 .507 2 .909 3 .411 3 .921 4 . 6 4 1 5 .660 6 . 6 7 5 215 256 068 712 230 560 010 6 .426 511 7 . 4 4 9 025 8 .188 849 8 . 8 3 7 863 9 .311 8B1 9 .600 109 6 .640 614 7 .620 127 8 .297 054 8 .945 066 9 .384 135 9 .640 3 .210 3 .714 4 .232 5 .255 6 . 2 7 0 7.391 850 785 395 045 449 663 25 26 27 28 29 30 7 .647 8 .920 10 .181 1 1 . 3 3 9 1 2 . 3 5 9 13 .178 9 .670 9 .534 9 .149 8 .707 8.122 7 .303 7 .902 9 .176 10 .434 11 .540 12 .563 13 .281 666 8 464 9 057 10 619 11 956 12 164 13 .157 .428 .682 . 745 .768 .382 649 391 8 .410 9 .682 8 .974 10 .936 8 . 5 1 5 11 .949 7 .766 12 .972 7 .017 13 .484 9 .622 9 .305 8 .877 8 .400 7 .552 6 .858 8 .664 9 .928 11 .139 12 .154 13 .076 13.5 86 .587 .232 .791 .270 .431 .687 31 32 33 34 35 36 13 .687 13 .941 14 .190 23 - 1 0 . - 1 . 6.50 9 6 .009 5 .416 6 .05 6 .04 13 .738 13 .992 14 .240 6 . 5 . 5 . B VS 1.75 0.75 414 13 891 14 312 14 D AS 6 6 .788 6 . .043 5 . .290 5 . MEASURED .05 .02 6 .219 5 .635 320 13 .839 766 14 .093 206 EXPERIMENTALLY - 1 . 5 0 6 .05 - 0 . 5 0 5 .96 13 .889 14.142 . 119 .516 - 1 . 2 5 - 0 . 2 5 6 . 0 5 5 .72 37 38 39 40 41 42 0 . 0 1 .00 2 .00 5 .00 14 0 . 5 .00 1 .60 0 .73 0 .13 1. .25 .25 .50 .00 EPS 5 VS D 3.82 1.27 ) . 53 3.08 .945 0 .50 1.50 3 .00 10 .0 1. 2 .78 1.02 0. 39 0 .05 .743 0 .75 1.75 4 .00 1.5 2 . 0 7 0 . 8 6 0 .22 .575 43 44 45 46 47 48 15 .45 .238 .199 3 .230 3 .913 .5 .5 ,B I , .217 .860 B F . 368 3 .219 5 . 199 FOR MK VI .31 .206 •1 MOD 10 MAGNET 2 . 4 . 3 . 448 4 .500 3 .5 5 .5 3 .441 4 .4 50 .267 .2 f O O rn ro co 0> «£> (\j <T o-co m m %o %O ,t m m in in m co o co co o fv cc r\j -o o m m r— r-• • • • • m m in m m co r- -4" m -J" nj --t cc .3- » r-r- -H ^- m -o -o m m in m in -4  ^ o m o m - H to OC r- ^  -O co •—i .0 1 r— •4- in in in m m o o —1 nj rn <r vj- m m in m m f ANGLE = RAHIllS 4 7 . 1 8 6 3 .230 3 . 1 5 0 1 .00 HILL - T n - H I L L OISTANCE = 0 .591 3 .221 3 .212 3 .203 3 . 195 3 . 187 3. 179 3 .171 3 . 1 6 4 3 .157 3 .144 3 .138 3 .128 3 .103 3 . 0 7 9 3 .054 3 .031 3 . 0 1 7 3 . 0 0 6 2 .985 2 .975 2 .966 2. 957 2 .949 2. 941 2 .93 3 2 .930 2 .930 2 .929 2 .930 2 .932 2 .939 2 .947 2 .955 2 .9 64 2 . 9 7 3 2 .982 3 . 0 0 2 3 .013 3 . 0 2 6 . 3 . 0 5 0 . 3 . 074 . . 3 .098 3 .123 3 . 1 3 2 . 3 . 1 3 7 3 . 1 5 0 3 .157 3 . 164 3 . 171 3 . 178 3 .186 3 .194 3 .203 3 .211 o 2 .995 2 .929 2 .992 3.1'-3 3 . 2 2 0 AVERAGE F I E L D 3 .0700 FLUTTER 0 .00108 ANGLE = RADIUS 4 7 . 1 8 5 = 2 . 0 0 H l L L - T P - H I l t -DISTANCE' 1. 182 3 . 4 4 8 3 . 4 4 3 3 .438 3 .433 3 . 4 2 9 3 . 4 2 4 3 .406 3 .364 3 .322 3 . 2 8 2 3 .246 - . 3 . 2 0 9 . 3 . 173 3 . 130 3 .058 - 2 .986 . . 2 . 914 2 .842 2 . 7 7 9 . 2 . 718 2 .658 2 . 6 0 9 2 .560 2.5 12 2 .46 5 2 .41R 2 .387 2.3 59 2 . 3 5 0 2 . 3 4 9 2 . 349 2 . 349 2 .350 2 . 3 5 9 2. 386 2 .417 2 .464 2 .511 2 . 5 5 9 2 .607 2 . 6 5 6 2 . 7 1 6 2 .777 2 .840 2 . 9 1 1 2 . a n ? 3 .054 3 .176 '• .205 3 .241 3 .278 3 .317 3 . 3 5 9 3 .400 3 .418 3 .422 3 .427 3 .431 3 . 4 3 6 3 .441 AVERAGE F I E L D = 2 .9370 FLUTTER 0 .01882 00 -c-ANGLE = 4 7 . 1 8 3 RAOIUS = 3.00. H I L L - T O - H I L L DISTANCE 1.772 3 . 9 1 3 3 .6 4 7 3 .912 3 .911 3 . 910 3 .893 870 3 .849 3 .830 3 . 810 3 .730 3 .565 3.4 34 3 .393 3 .254 3 , 1 1 4 2 .975 2 .336 ? . 713 2 .6'04 2 .3 88 2 .281 2 .204 2. 151 2. 099 2 .047 1.996 1 .972 1 .971 1 .970 1 .970 1 .993 2 .04 3 2 . 093 2 .145 2 .197 2 .272 2 .378 2 .591 2 .698 2 .3 18 2 .955 3. 092 3 . 229 3 .36 7 3 .455 3 . 534 3 .694 3 .772 3 .790 3 .808 3 . 826 3 .847 3 .863 3 .862 3 .861 2 .496 1.970 2 .484 3 .614 3 .860 AVERAGE F I E L D 3 .0163 FLUTTER = 0 .05873 ANGLE = 47 .182 RADIUS = 4 . 0 0 H I L L - T O - H I L - L - DISTANCE = 2 .363 3 .504 3 . 503 3 .502 3 .500 3 • 498 3 . 4 9 6 3 .483 3 .453 3 . 4 2 3 3 . 3 9 4 3 . 309 3 .207 3 .104 2 . 989 2 . 814 2 .639 2. 464 . 2 . 303 . 2 .155 2 .008 1 .862 1 .768 1.678 1 .590 1. 516 1 . 46 6 1.427 1 .392 1 .372 1 .359 1 .353 1 .358 1.370 1 .389 1. 4 24 1.462 1. 511 1 . 584 1 .671 1.760 1 .852 1 .997 2 .142 2 .288 2. 44 7 2 .619 7.791 ? .9 64 3 .077 3 . 1 7 7 3 .277 3 .360 3 .386 3 .414 3 . 44 3 3 .453 • 3. 454 3 . 45 4 3 .455 3 . 4 5 4 3 . 4 5 3 AVERAGE F I E L D = 2 .5260 ' FLUTTER = 0 .10953 ANGLE = 4 7 . 1 8 3 RADIUS = 5 . 0 0 H I L L - T O - H I L L DISTANCE = 2 . 9 5 4 4 . 8 1 6 /. . 814 4 .813 4 .811 4 .810 4 .802 4 .793 4 . 7 84 4 . 7 3 1 4 . 6 7 4 4 . 6 1 0 4 .42 8 4 .248 4 .045 3 .74 2 3 . 439 3. 149 2 . 884 7 .627 2 .410 2 .244 2 . 078 1.948 1 .848 1 .751 1.683 1.628 1. 576 1 .545 1.521 1 . 50 3 1 . 5 ?. 0 1 . 54.3 l .5 7 3 l . 6 2 3 1 .677 1. 743 1. «30 ] . 9 3 f i 2 .065 2 . 2 2 8 2 .392 2 .605 2 .859 ' 3 .119 3 . 404 3 .701 3 . 9 99 4 . 1 9 6 4 . 3 7 1 4 . 5 4 8 4 .607 4 .661 4 .710 4 .7 16 4 .722 4 .725 4 . 724 4 . 7 2 2 4 .721 4 . 7 1 9 AVERAGE F I E L D 3 .2993 FLUTTER 0 .15611 ANGLE = 4 7 . 1 8 3 ' RADIUS = 6 .00 H I L L - T O - H I L L DISTANCE 5 .003 5 .001 4 .999 4 . 9 9 7 4 .995 4 .992 4 .987 4 . 9 7 6 4 . 9 6 5 4 . 8 9 6 4 . 8 2 5 4 .661 4 .428 4 . 167 3 . 7 7 9 . 3 .392 . 3 .044 2 .705 2 . 4 1 9 2 .201 1 .984 ' 1 .347 1.712 1 .608 1.529 1.455 1. 404 1.354 1 .377 1 .316 1 .308 1.315 1.325 1.350 1.400 1.450 1.522 1.599 1.701 1 .835 1 .96 9 2 .182 7 .397 7 .678 3 .011 3. 3 54 3. 733 4 .113 4 . 3 6 8 4 . 5 9 4 4 .752 4 . 8 1 8 4 . 8 8 3 4 .890 4 .897 4 .898 4 .897 4 .8 96 4 . 8 94 4 .892 4 .890 AVERAGE F I E L D 3 .2807 FLUTTER = 0 .20737 oo ANGLE = 4 7 . 1 9 1 RADIUS = 7 .00 H I L L - T O - H I L L DISTANCE = .138 5 .185 5 .031 5 .185 5.185 5 . 185 5 . 185 5. 185 5 . 184 5 .177 5 .160 5 .121 4 . 8 9 3 4 .608 4 .290 3 .8 76 3. 370 2. 963 2 .570 2 .298 2 .031 1 .683 1.568 1 .461 1 .385 1. 312 * 1. 268 1.240 1 . 2 2 4 1.212. 1 .212 1 .224 1 .241 1 . 269 1. 312 1. 385 1 .462 1 .568 1 .683 2 .031 2 .298 2 .570 2 . 963 3 . 370 . 3 . 826 4 .291 . . 608 4 . 8 9 3 5 .121 5 .160 5 . 177 5 . 184 5. 1S5 5. 185 5 .185 5 .185 5 .185 1 .85 7 1 .207 1 .857 5 .031 5 .185 AVERAGE F I E L D = 3 .3283 FLUTTER = 0 .25028 ANGLE = 4 6 . 9 1 9 RADIUS = 8 .00 H I L L - T O - H I L L DISTANCE = 4 . 7 6 0 5 . 329 5 .330 5 .330 5 .331 5 .331 5. 329 5. 325 5 .318 5 . 297 5 .276 5 . 174 5 .027 . 4 .700 4 .285 3 .742 3 . 238 2 .760 2 .409 2 . 098 1.896 1 .70 9 l .573 1 .453 1 .355 1. 283 1. 227 1. 194 1 . 165 1 .149 1.135 1 . 134 1 .137 1.150 1. 167 1.195 1. 233 1.287 1.363 1 .459 1.580 1 .714 1 .903 2 . 106 2 .422 2 .775 7 50 3 .757 4 . 7 9 5 u .711 5 .040 5 .190 5 .294 5 .319 5 .341 5 .349 5. 355 5 .358 5 .358 5 .359 5 . 3 5 9 5 .360 AVERAGE F I E L D = 3 .3303 FLUTTER = 0 .28566 ANGLE = 4 5 . 9 6 7 RADIUS = 9 .00 H I L L - T 0 - H I L L DISTANCE 5.212 5 .485 5 .340 5 .484 5 .218 5 .483 4 .863 5 .482 4 . 4 3 7 5 .481 3 . 8 3 6 5 .479 3 .7R5 5 .476 7 .760 5 .46 7 7 .405 5 . 4 5 3 7 . 0 9 3 5 .424 1.863 1 .684 1 .536 1 .426 1 .337 1.275 1. 225 1.131 1.151 1 .131 1.115 1 .109 1 .113 1.12 7 1. 146 1.173 1.215 1.261 1.319 1 .401 1.506 1 .654 1 .325 . .. 2 .054 . 2 . 3 5 5 - 2 .710 . . 3 .228 .. 3 .780 4 . 390 4 . 8 3 3 5 .182 5 .305 5 .381 5 .409 5 .418 5 .424 5.42 5 5 .424 5.42 3 5 . 4 2 2 5 .421 5 .42 0 o AVERAGE F I E L D = 3 .3724 ANGLE = 4 3 . 9 8 3 RADIUS = 10 .00 FLUTTER 0 .30109 H I L L - T O - H I L L DISTANCE = 5 .208 5 .595 5 .593 5.591 5 .588 5 .586 5 .584 5 . 5 7 9 5 .573 5 . 5 6 4 5 .541 5 .510 5 .391 5.195 4 . 6 2 6 4 . 3 1 7 . 3 .714 3 .193 2 .712 2 . 3 5 3 2 .072 1 .339 1 .672 1.525 1.416 1.335 1.274 1.228 1.188 1 .158 1.141 1 .127 1 .129 1 .142 1. 162 1. 191 1.235 . 1.288 1 .357 1 .451 1 .573 1 .737 1 .937 2 .188 2 .562 3 . 0 2 3 3 .602 4 .253 4 .813 5 .212 5 .367 5 -4so 5 .478 5 .489 5 .495 5 .498 5. 49 6 5. 4<?4 5 .492 5 . 4 8 9 5 .487 5 .485 AVERAGE F I E L D 3 .5167 FLUTTER = 0 .23593 OO ANGLE = 4 1 . 0 8 0 . RADIUS = 11 .00 H I L L - T O - H I L L DISTANCE = 4 . S 2 6 5 .690 5 .686 682 5 . 6 7 8 5 . 6 7 4 5.671 5. 667 3 . 5 .661 3 . 5 . 6 5 3 5 .642 2 .005 1 .777 1.620 1 .491 1.392 1. 321 1.269 1.230 1 . 196 1.186 1.1 84 1 .183 1.199 1.224 1.256 1. 30 2 1 .356 1.423 1 .550 1.703 1 .882 2 .142 2 .483 2 .914 3 .520 4. 190 4. 82 3 5 .253 5 .44 1 5 .535 5 .566 5 .573 5 .577 5 .578 5 .574 5 .570 5 .566 5 .562 • 5 . 5 5 8 5 .554 5 .550 AVERAGE F I E L D 3 .6833 FLUTTER 0 .26637 ANGLE = 37 .215 RADIUS = 1 2 . 0 0 H I L L - T O - H I L L DISTANCE = 4 .262 5 . 7 3 6 5 .730 5 .725 5 . 7 1 9 5 .713 5 .703 5 . 702 5 .696 5 . 689 5 .676 5 . 659 5 .624 . 5 .539 5 .410 5 .078 . 4 .6 97 4. 3 . 0 7 6 . ...2 .657 . 2 .342 2 .074 1.884 1.728 1.617 1.515 1. 439 1.369 1 .333 1 .300 1 .293 1 .293 1.306 1.323 1. 353 1. 395 1. 456 1 .576 1 . 7 7 4 1 .392 2 .1 44 2 .479 7 .880 3 .476 4 .100 4 .754 5 . ! 71 S.461 5 . 5 8 0 5 . 6 ~ ? 5 .653 5 .652 5 .647 5 .641 5 .635 5.629 5. 624 5 .618 5 . 6 1 2 5 .606 5 . 600 AVERAGE F I E L D = 3 .8616 FLUTTER = 0 . 2 3 0 3 4 ANGLE = RADIUS 3 1 . 4 0 8 = 1 5 .764 5 . 679 . 0 0 H I L L - T O - H I L L - DISTANCE = 3 .7 44 5 .758 5 . 649 5 .752 5 .565 5 .747 5 .456 5 .741 5 .222 5 .735 4 .920 5 .729 4. 549 5 .723 4 . 0 7 4 5 .713 3 . 6 2 4 5 .701 3 .215 2 .497 1 .451 3 . 2 4 5 5 .716 2 .238 1 .485 3 .757 5 .710 2 .046 1 .530 4 .273 5 .704 1.857 1.593 4 .305 5 .698 1.720 1.712 5 .195 5.692 1.609 1. 857 5 .458 . 5. 686 1.532 2 .028 5 .633 5 .680 1.462 2 . 2 50 5 .681 5 . 6 7 4 1 .447 2 .491 5 .728 5 .668 2 . 6 7 0 1 .444 2 . 8 4 9 5 .722 5 .662 AVERAGE F I E L D FLUTTER = 0 .18081 ANGLE = 2 5 . 9 5 6 RADIUS = 1^.00 H I L L - T O - H I L L DISTANCE 2 .358 5 .76v6 5 .731 3 .706 2 .258 3 .078 5 .764 5 .762 5 .759 5 .757 5 .755 5. 753 5 .750 5 .748 5 .746 5 .686 . . . 5 . 5 0 3 . 4 .855 4 .237 4 . 3 7 9 . 4 . 1 2 3 . 4 . 6 5 6 4 . 6 4 0 ... 4 . 1 5 1 . . . . . .. 3 .311 2 .984 2 .754 2 .544 2 .429 2. 340 2 .263 2 . 2 3 6 2 .240 2 . 2 9 6 2 .345 2 .404 2 .463 2 .57 1 2 .6 76 2 .799 2 .911 3 .009 3 . 1 9 8 3 .355 3 .5 86 •3.872 4. 165 4. 514 4 . 8 8 0 5 .153 5 .385 5 .620 5 .673 5 .673 5 .676 5 .673 5 .671 5 .669 5 .667 5 . 6 6 4 5.5 67 5 .662 AVERAGE F I E L D = 4 . 2 7 6 0 FLUTTER = 0 .10142 co ANGLE = 2 1 . 8 9 6 RADIUS = 1 5 . 0 0 H I L L - T O - H I L L DISTANCE = 2 .854 5 .760 5 .758 5 .757 5 .755 5 .754 5. 752 5 . 746 5 .730 5 .676 '•3 .744 4 . 7 7 0 4 . 1 0 7 4 .775 3 .293 7.339 7. 714 2. 521 2 . 347 3 .243 3 .618 3 .994 4 . 3 5 7 4 .687 4 .954 4 .782 4 . 6 57 ' 4 . 586 4 .549 4 .531 4 .507 4 .497 4 . 4 97 4 .492 4 .478 3 .975 3 . 176 2. 612 2 .258 2 .031 1 .917 1.863 1 .874 1.905 1.987 2 .109 2. 27 2 . 2 . 522 2 .770 3 . 163 3 .5 67 4 . 0 2 4 4 . 5 0 8 4 . 894 5 .209 5.431 5. 535 5. 60 5 5 . 6 2 7 5 .638 5 . 640 5 .640 AVERAGE F I E L D = 4 .0659 FLUTTER = 0 .10364 STOP EXECUTION TERMINATED S S I G SHORT - 88 -A p p e n d i x 3 F o l l o w i n g i s a b r i e f d e s c r i p t i o n o f t h e p r o g r a m used f o r c a l c u l a t i n g t h e a l l o w a b l e t o l e r a n c e s on a g i v e n p o l e t i p . A l i s t i n g o f t h e p r o g r a m , a t y p i c a l d a t a s e t , and a t y p i c a l o u t p u t a r e a l s o g i v e n . The p r o g r a m was run on an IBM 3 6 0 / 6 7 . S t e p D e s c r i p t i o n L i nes 1 . Input d a t a : a) F i n a l r a d i u s , number o f r a d i i , i n i t i a l r a d i u s , r a d i u s i n c r e m e n t 1 - 8 b) Data p o i n t s i n C a r t e s i a n c o o r d i n a t e s d e s c r i b i n g t h e f o c u s s i n g and d e f o c u s s i n g edges o f t h e p o l e t i p 9 - 1 2 c) Maximum p e r m i s s i b l e e r r o r i n 2 B / 2 r , p a r a m e t e r n , f i and the d i s t a n c e h a l o n g t h e edge o f t h e p o l e t i p a t w h i c h measurements w i l l be t a k e n 13 2 . C o n v e r t c a r t e s i a n c o o r d i n a t e s t o c y l i n d r i c a l c o o r d i n a t e s 14 - 25 3 . G e n e r a t e p o i n t s a l o n g c e n t r e l i n e o f p o l e t i p . F i n d a n g u l a r w i d t h o f p o l e t i p a t t h e same t i m e 26 - 32 4 . C a l c u l a t e s p i r a l a n g l e o f p o l e t i p a t r a d i i s p e c i -f i e d i n t h e i n p u t d a t a 33 - 38 5 . C a l c u l a t e m i s c e l l a n e o u s i n f o r m a t i o n , s u c h as 8 , y » E , B , f l u t t e r , and a n g u l a r p o s i t i o n s o f t h e p o l e t i p e d g e s . O u t p u t t h e s e r e s u l t s . 39 - 58 6 . C a l c u l a t e maximum p e r m i s s i b l e e r r o r s on t h e p o l e t i p p a r a m e t e r s and o u t p u t them. The p r o c e d u r e i s s t r a i g h t -f o r w a r d and j u s t a m a t t e r o f a p p l y i n g t h e e q u a t i o n s g i v e n i n S e c t i o n 4 . 59 - 89 7- S t o p 90 - 93 Sub r o u t i nes SLOPE (X , Y , X X , S , N) - f i n d s s l o p e S o f t h e l i n e d e s c r i b e d by N d a t a p o i n t s X , Y a t the p o i n t X X . F IT (XX , YY , RHO, NO, D) - g i v e n a c u r v e d e s c r i b e d by t h e NO d a t a p o i n t s X X , Y Y , t h e y - v a l u e - 89 -D c o r r e s p o n d i n g t o t h e x - v a l u e RHO i s c a l c u l a t e d by a L a g r a n g i a n i n t e r p o l a t i o n t e c h n i q u e . LQF ( S , Y, Y F , W, E 1 , E 2 , P , WZ, N , M, N l , ND, E P , AUX) - a l i b r a r y r o u t i n e f o r p e r f o r m i n g a l e a s t - s q u a r e s f i t . AUX ( P , D, X , L, M) - a r o u t i n e r e q u i r e d by LQF t o d e s c r i b e t h e f u n c t i o n t o be f i t . ^JLF.S NO. Q3itZ13 UNIVERSITY OF 6 C COMPUTING CENTRE MTSIAN12Q) 1 5 : 5 0 : 2 0 Q4-Q7-70 L I S *IHC ( . X X X , . X X X ) FOR FORTRAN ERROR IHCXXX SSIG TRSO * * * * * * * * * * * * * * * * * * * * PLEASE RETURN TO TRIUMF * I S S I G TRSO  * * L A S T SIGNON USER "TRSO" $ L I S A ( l , 2 4 2 1 1 2 2 WAS: 1 2 : 2 6 : 3 2 SIGNED ON AT 0 4 - 0 7 - 7 0 1 5 : 5 0 : 2 5 ON 0 4 - 0 7 - 7 0 C ; I r : DIMENSION R H O ( 1 0 0 ) , B A V ( 1 0 0 ) , F L U T ( 1 0 0 ) , G A M M A ( 1 0 0 ) , X 1 ( 1 0 0 ) , Y 1 ( 1 0 0 ) , * X 2 ( 1 0 0 ) , Y 2 ( 1 0 0 ) , R 1 ( 1 0 0 ) , A N G 1 ( 1 0 0 ) , R 2 ( 1 0 0 ) , A N G 2 ( 1 0 0 ) , ..* AC(100 ) ,ETA(100 ) .SPRANG( 100) ,XC( 100) ,YC1 1 0 0 ) , E ( 100 )  101 102 FORMAT (1015) FORMAT ( B F 1 0 . READ ( 5 , 1 0 2 ) READ ( 5 , 1 0 1 ) READ (5 ,102) READ ( 5 , 1 0 1 ) 5 ) RFINAL K l RO,RDEL TA K l C c o 10 11 12 13 14 15 22 READ ( 5 , 1 0 2 ) READ ( 5 , 1 0 1 ) READ ( 5 , 1 0 2 ) READ ( 5 , 1 0 2 ) DO 22 1=1,KI RHO(I)=R0+(I I X 1 ( I ) , Y 1 ( I ) , I = 1, K1 ) K2 ( X 2 ( I ) , Y 2 ( I ) , I = 1,K2) DGRADB,AN,H -1 ) *RDELTA 16 17 18 19 20 21 20 DO 20 1=1 ,Kl R l ( I ) = S O R T ( X I ( I ) * * 2 + Y l ( I ) * * 2 ) * 2 0 . 8 I F ( X I ( 1 ) . L T . Y 1 ( I ) )ANG1(I ) = 3 . 1 4 1 5 9 / 2 . - A T A N ( X 1 { I ) / Y 1 ( I ) ) IF ( X l ( I ) . G E . Y l ( I ) ) A N G 1 ( 1 ) = A T A N ( Y 1 ( I ) / X 1 ( I ) ) CONTINUE DO 21 1 = 1,K2  22 23 24 25 26 27 21 R 2 ( I ) = S O R T ( X 2 ( I )**2+Y2<I )**2)*20 .8 IF ( X 2 ( I ) . L T . Y 2 ( I ) ) A N G 2 ( I ) = 3 . 1 4 1 5 9 / 2 . - A T A N ( X 2 ( I ) / Y 2 ( I ) ) IF ( X 2 ( I ) . G E . Y 2 ( I ) ) A N G 2 ( I ) = A T A N ( Y 2 ( I ) / X 2 ( I ) ) CONTINUE DO 30 I=1,K2 CALL F I T ( R 1 , A N G 1 , R 2 ( I ) , K 1 , A I )  28 AC(I)=IA1+ANG2(I)112. 29 30 E T A ( I ) = A B S ( A 1 - A N G 2 ( I ) ) * 5 7 . 2 9 5 8 30 DO 32 1=1,K2 31 X C ( I ) = R 2 ( I ) * C O S ( A C ( I ) ) 32 32 Y C ( I ) = R 2 ( I l * S I N ( A C ( I ) ) 33 DO 31 1=1 ,Kl 34 35 36 37 38 39 40 41 42 43 44 45 31 CALL F I T ( R 2 , X C , R H O ( I ) , K 2 , X C E N T ) CALL S L O P E ( X C , Y C , X C E N T , S , K 2 ) IF ( A B S ( X C E N T ) . G E . A B S ( R H O ( 1 ) 1 ) A N G 0 = 0 . 0 IF ( A B S ( X C E N T 1 . L T . A B S ( R H O ( I 1)1ANG0 = ARC0S(XCENT/RHO(I 11 SPRANG!I 1=ABS(ANGO-ATAN(S)1*5 7 .2958 DO 23 1=1 ,Kl 23 B E T A = R H O ( I ) * . 7 5 7 7 / R F I N A L GAMMA(I ) = S O R T ( 1 . / ( 1 • - B E TA**2) ) E( I )=939.278*(GAMMA(I 1-1.0) B A V ( I ) = 9 3 4 . 6 2 3 * G A M M A ( I ) / R F I N A L F L U T ( I 1 = (GAMMA!I 1**2-.875 ) / ( 2 . * WRITE (6 ,104) (TAN(SPRANG(I 1 /57 .295 811**2+1 . ) 46 104 FORMAT ( 1 H 1 , 1 R E GAMMA 47 *DTH SP A N G ' / ) 48 00 24 1=1,Kl 49 CALL F I T ( R l , A N G 1 , R H O ( I 1 ,K1,ANGF1 50 CALL F I T ( R 2 , A N G 2 , R H O ( I ) , K 2 , A N G D ) 51 ANGF=ANGF*57.2958 B AV ANGF ANGD WI 52 53 ANGD=ANG0*57.2958 WIDTH=ANGF-ANGD 54 24 WRITE(6 , 1 0 3 ) R H O ( I ) , E ( I ) , G A M M A ( I ) , B A V ( I ) , A N G F , A N G D , W I D T H , S P R A N G ( I ) 55 103 FORMAT ( F 6 . 1 , F 9 . 2 T F 9 . 4 , F 8 . 3 , 4 F 8 . 2 ) 56 WRITE ( 6 , 6 0 0 ) \ 57 600 FORMAT (1H1,105H R X . 3*X 0 B .955*X 0 TAN E 0 E 58 1 ED D ED D AD D YD EF D EF D AF D YF , / ) ( 59 DO 500 1=1,Kl 60 DMU=DGRADB*RHO<I) / (BAV(I)*1000. ) 61 X=100.0*SQRT(.05*.05-DMU*DMU)/(GAMMA(I) **2 + 0 . 1 2 5 - 1 . 0 ) 62 X3=.3*X 63 X9=.955*X 64 DBBB=BAV(I ) * X 3 * ( S 0 R T ( F L U T < I ) ) ) / 2 0 0 . 0 65 TANSP=TAN(SPRANG(11/57 .29578) 66 D T N E = X 9 * ( 1 . 0 + 2 , 0 * T A N S P * * 2 ) / ( 4 . 0 * T A N S P * 100 . ) 67 DE=DTNE/(1 .0+TANSP-*2) 68 CALL S L O P E ( R 2 , E T A , R H O ( I ) , S , K 2 ) 69 TNUO=RHO(1)*S*3.1415 9 2 6 5 / 3 6 0 . 70 TNED=TANSP+TNUO 71 TNEr=TANSP-TNUO 72 ED=ATAN(TNED1*57.29 57 8 73 EF=ATAN.(TNEF 1*57.29578 74 COSED=COS(ATAN(TNEO11 75 COSEF=COS(ATAN(TNEFI) 76 ALAMDA=ATAN 1 ((COS E D / C O S E F ) * * 2 ) * * ( A N / ( 2 . 0 + A N ) ) ) 77 DED=2.0*COS(ALAMDAJ*(COSED*COSED)*DTNE*1000. 78 DAD=H*[)E0/ ( 1000.*S0RT ( 2 . 0 ) ) 79 DYD=DAD/CaSED 80 DEF=2 .0*SIN(ALAMDA)*(COSEF*COSEF)*DTNE*1000 . 81 DA F =H*DE F / ( 1 0 0 0 . * S 0 R T ( 2 . 0 ) ) 82 DYF=OAF/COSEF 83 DBBB=DBBB*1000. 84 DE=DE*1000. 85 W R I T E ( 6 , 6 0 1 ) R H 0 ( I 1 , X , X 3 , D B B B , X 9 , D T N E , D E , E D , D E D , 0 A D , D Y D , E F , 86 1 D E F , D A F , D Y F 87 601 F O R M A T ( 1 X , F 7 . 2 , 4 F 7 . 3 , F 7 . 4 , F 8 . 3 , F 7 . 3 , F 8 . 3 , F 7 . 4 , 2 F 7 . 3 , F 8 . 3 , 88 1 F 7 . 4 , F 7 . 3 1 89 500 CONTINUE 90 WRITE ( 6 , 6 0 2 ) 91 602 FORMAT (1H1) 92 15 STOP 93 END 94 FUNCTION A U X ( P , D , X , L , M ) 95 DIMENSION P ( 1 0 ) , D ( 1 0 ) 96 D ( l 1 = 1 . 0 97 AUX = P ( 1 ) 98 DO 10 J=2,M 99 D ( J ) = D ( J - 1 ) * X 100 10 AUX=AUX+P(J)*D(J1 101 RETURN 102 END 103 SUBROUTINE L 0 F ( X , Y , Y F , W , E 1 , E 2 , P , W Z , N , M , N I , N D , E P , A U X ) 104 DIMENSION X ( 1 0 0 ) , Y ( 1 0 0 ) , Y F ( 1 0 0 ) , P ( 1 0 ) , E l ( 1 0 ) , E 2 ( 1 0 ) , W ( 1 0 0 ) , 105 1 C(10 , 1 0 ) , V ( 1 0 ) , D ( 1 0 ) 106 ND=1 107 NT = 1 108 IV = 0 109 5 DO 10 1=1,M no v(i) =0.0 111 DO 10 J = l ,M J c 112 10 C U , J ) = 0 . 0 \ 113 TT = 0 114 DO 20 L = 1,N . 115 IFIWZ ) 6 , 7 , 6 116 6 WT=W(L) V 117 22 GO TO 8 / f 118 7 WT = 1. \ 119 8 U = AUXIP , D , X ( L ) , L , M ) 120 DO 30 1=1,M 121 DO 30 J=1 ,1 122 30 C( I , J ) =C I I , J ) + W T * 0 ( I )*DIJ) 123 DO 40 1=1 ,M 124 40 V( I ) = V ( I ) + W T * ( Y ( L ) - U ) * D ( I > 125 20 CONTINUE 126 DO 50 1 = 1, M 127 JK=I+1 128 I F ( J K . G T . M ) GO TO 50 129 DO 49 J = JK , M 130 C I I , J ) =C I J , I ) 131 49 CONTINUE 132 50 CONTINUE 133 I F ! I V . E O . l ) GO TO 45 134 I F ( N T - N I ) 3 5 , 4 5 , 5 5 135 35 CALL S 0 L T N ( C , V , H , 1 0 , D E T ) 136 I F ( A B S ( G E T ) . I T . l . O E - 1 9 ) GO TO 65 137 DO 75 1=1,M 138 P ( I )=PI I)+V 11 ! 139 TC = A B S ( V ( I ) / P ( I ) ) 140 I F ( T C . G T . T T ) TT=TC 141 75 CONTINUE 142 NT =NT+1 143 I F ( T T . L T . E P ) IV=1 144 GO TO 5 145 45 CALL I N V E R T ( C , M , 1 0 , D E T , C O N D ) 146 I F ( A B S ( D E T 1 . L T . l . O E - 1 9 ) GO TO 65 147 DO 85 I=1,M 148 DO 85 J=1 ,M 149 85 P11 )=PI I)+CI I , J ) * V ( J ) 150 55 . DO 95 1 = 1,H 151 95 E l ( I ) = S O R T ( C ( 1 , 1 ) ) 152 3 F 0 R M A T ( 1 X , 8 G 1 5 . 5 ) 153 S = 0 .0 154 DO 105 L=1,N 1 55 IFIWZ) 1 6 , 1 7 , 1 6 156 16 WT =W(L) 157 GO TO 18 158 17 WT = 1 . 159 18 Y F ( L ) = A U X ( P , D , X ( L ) , L , H ) 160 S=IY(L)—YFIL ) )**2*WT+S 161 105 CONTINUE 162 PP=N-M 163 FI = SOP.T ( S / P P ) 164 DO 115 1=1,M 165 115 E 2 ( I ) = F I * E 1 ( I ) 166 RETURN 167 65 WRITE ( 6 , 2 ) 168 2 FORMAT(22H LINEAR EQUATIONS F A I L ) 169 ND = 0 170 RETURN s 171 END J ( 172 SUBROUTINE F IT(XX , Y Y , R H O , N O , 0 ) 173 DI MENS ION X X ( 1 0 0 ) , Y Y ( 1 0 0 ) , X ( 4 ) , Y ( 4 ) 174 NN=N0-1 175 C FINDS 4 POINTS AROUND RHO 176 DO 21 L=1,NN \ 177 IF ( ( R H O - X X I L ) ) * ( R H 0 - X X ( L + 1 ) ) ) 2 2 , 2 2 , 2 1 / ( 178 21 CONTINUE " 1 179 IF { ( ( R H O - X X ( 1 ) ) * ( X X ( 2 1 - X X ( 1 ) ) ) . L T . 0 . 0 1 G O TO 24 180 L = N0-1 131 GO TO 26 182 24 L = 3 183 GO TO 26 184 22 I F ( L . L T . 3 ) L = 3 185 26 DO 27 K = l , 4 186 KL=L-3+K 187 X(K)= XX (KL) 188 27 Y ( K ) =Y Y ( K L ) 189 C PERFORMS LAGRANGI AN INTERPOLATION 190 A1=RH0-X(1) 191 A2=RH0-X(2) 192 A3=RHO-X(3) 193 A4=RH0-X(4) 194 B 1 = A 2 * A 3 * A 4 / ( ( X ( 1 ) - X ( 2 ) ) * ( X ( 1 ) - X { 3 ) ) * ( X ( 1 ) - X ( 4 ) ) ) 195 B 2 = A 1 * A 3 * A 4 / ( ( X ( 2 ) - X ( 1 ) ) * ( X ( 2 ) - X ( 3 ) ) * ( X ( 2 ) - X ( 4 ) ) ) 196 B 3 = A 1 * A 2 * A 4 / ( ( X ( 3 ) - X < 1 ) ) * ( X ( 3 ) - X ( 2 ) ) * ( X ( 3 ) - X ( 4 ) ) ) 197 B 4 = A 1 * A 2 * A 3 / ( ( X ( 4 ) - X ( 1 ) ) * ( X ( 4 ) - X ( 2 ) ) * ( X ( 4 ) - X ( 3 ) ) ) 198 D=B1*Y(1 )+B2*Y(2)+83*Y(3)+B4*Y(4) 199 RETURN 200 END 201 SUBROUTINE S L O P E ( X , Y , X X , S , N ) 202 C FINOS SLOPE OF LINE BY F I T T I N G WEIGHTED QUADRATIC TO 5 NEIGHBOURING P T S . 203 DI MENS I ON X I ( 5 ) , Y 1 ( 5 ) , Y F ( 5 ) , W ( 5 ) , E l ( 5 ) , E 2 ( 5 ) , P ( 5 ) , X ( 1 0 0 ) , Y ( 1 0 0 ) , 204 * WW(9 ) 205 DATA W W / . 0 6 2 5 , . 1 2 5 , . 2 5 , . 5 , 1 . , . 5 , . 2 5 , . 1 2 5 , . 0 6 2 5 / 206 C FINDS 5 PTS SURROUNDING XX AND SETS UP APPROPRIATE WEIGHTS 207 IF ( X X . E O . X I 1 ) ) GO TO 25 208 DO 21 L=2,N 209 IF ( ( ( X X - X I L - 1 ) > * ( X X - X ( I ) ) ) . L E . 0 . 0 ) GO TO 22 210 21 CONTINUE 211 IF ( ( ( X X - X I 1 ) ) * ( X ( N ) - X ( 1 ) ) ) . L T . O . O ) GO TO 25 212 IF (( ( X X - X I N ! ) * ( X ( 1 ) - X ( N ) )) . L T . O . O ) GO TO 27 213 22 IF ( L . E 0 . 2 ) GO TO 24 214 IF ( L . E Q . ( N - 1 ) ) GO TO 26 215 IF ( L . E Q . N ) GO TO 27 216 JW = 2 217 GO TO 23 218 24 L = 3 219 JW=3 220 GO TO 23 221 25 L = 3 222 JW = 4 223 GO TO 23 224 26 L = N - Z 225 JW = 1 226 GO TO 23 227 27 L = N-2 228 JW = 0 229 GO TO 23 230 23 DO 28 K = l , 5 231 KL=L+K-3 J 2 32 P ( K ) =0.0 N 233 X I ( K ) = X ( K L ) - XX 234 Y I ( K ) = Y ( K L ) 235 2 8 W(K) =WW(K+JW) 236 C F I T S WEIGHTED QUADRATIC TO CURVE ^ 237 EXTERNAL AUX / 238 CALL L Q F I X l , YI , Y F , W , E 1 , E 2 , P , 1 . 0 5 , 3 , 1 , N D 0 . 0 1 , A U X ) 1 < 239 C DEFINES SLOPE 240 S=P(2) 241 RETURN 242 END END OF F I L E $L 1 S A(IOOO) . „ . _ 1000 312 . 1001 32 1002 10 .0 10 .0 1003 63 CURVE 1 1004 0 . 0 0 .01 0 . 0 3 . 0 0 . 0 4 . 0 0 . 0 5 . 0 1005 0 . 0 6 . 0 0 . 0 6 . 5 0 . 0 7 . 0 0 . 0 7 .15 1006 0 . 0 7 .25 0 .02 7 .5 0 . 0 3 7 .75 0 .045 8 . 100 7 0 . 065 8 .25 0 . 0 9 8 . 5 .13 8 .75 .165 9 . 1008 .215 9 .25 .26 9 . 5 .315 9 . 7 5 .395 10 . 1009 .47 10 .25 .565 10 .5 .675 10 .75 .3 1 1 . 1010 . 8 9 11 .2 1.02 1 1 . 4 1. 17 11 .6 1 .335 11 .8 1011 1.515 12 . 1.71 12 .165 1.915 12 .32 2 .105 12 .475 1012 2 . 3 1 0 12 .610 2 .540 12 .72 2 .76 12 .82 3 . 12 .9 1013 3 . 2 5 12 .985 3 .5 13 .035 3 .75 13 .085 4 . 13 .125 1014 4 . 2 5 13 .16 4 . 5 13 .185 4 . 7 5 13.22 5 . 13 .24 1015 5 .25 13 .27 5 .5 13.28 5 .75 13 .285 6 . 13 .29 1016 6 . 2 5 13 .295 6 . 5 13 .29 6 .75 13 .285 7 . 13 .28 1017 7 .25 13 .275 7 .5 13 .26 7 .75 1 3 . 2 4 8 . 13 .225 1018 8 .25 13 .21 8 .5 13 .19 8 .75 13 .17 9 . 13 .15 1019 9 .25 13.12 9 .5 13.1 9 .75 13 .07 1020 75 1021 0 .00212 0 .00453 0 .212 0.^53 0 .424 0 .906 0 .636 1.359 1022 0 . 8 4 8 1.811 1.060 2 . 2 6 4 1.27 2 2 .717 1 .484 3 . 1 7 1023 1 .696 3 .623 1 .907 4 .076 2 .119 4 . 5 2 9 2 .225 4 . 7 5 5 1024 2 . 3 4 0 5 .000 2 .560 5 .500 2 .790 6 .000 2 .915 6 . 2 5 1025 3 .035 6 .500 3 .155 6 .750 3 .280 7 . 0 0 0 3 .400 7 . 2 5 1026 3 . 5 3 0 7 .500 3 .685 7 .750 3 .870 8. 000 4 .025 8.2 1027 4 . 1 9 5 8 .375 4 . 3 8 5 8 .565 4 . 575 8 . 7 1 5 4 . 7 8 0 8 .855 1028 5 . 0 0 0 8 .980 5 .220 9 .085 5 .450 9 . 180 5 .680 9 . 2 8 5 1029 5 .920 9 .380 6 . 165 9 .460 6 .375 9 .520 6 .625 9 . 5 8 5 1030 6 . 8 0 0 9 .625 7 .000 9 .655 7 .250 9 .690 . 7 .500 9 . 7 1 5 1031 7 .750 9 .720 8 .000 9 .715 8 .250 9 .700 8 .500 9 .660 1032 8 . 7 5 0 9 .615 9 .000 9 .570 9.2 50 9 .490 9 .500 9 .41 1033 9 .750 9 .335 1 0 . 9 .26 10 .25 9 . 1 7 10 .5 9 . 0 8 5 1034 1 0 . 7 5 8 . 985 11 . 8 . 8 9 0 11 .235 8 .8 11 .46 8 . 6 8 5 1035 11 .68 fi .56 11 .88 8 .43 12 .095 8 .28 12 .29 8 .135 1036 12 .485 7 .980 12 .655 7 .81 12 .835 7 .625 13 . 7 . 445 1037 1 3 . 1 4 5 7 .25 1 3 . 2 9 7 .055 13 .43 6 .84 13 .565 6 . 6 3 1038 1 3 . 6 9 6 .42 13 .815 6 .19 13 .935 5 .98 14 .025 5 .7 5 1039 14 .135 5 .53 14 .25 5.31 14 .305 5 .19 1040 0 . 1 6 6 6 6 7 1.0 1 0 . END OF F I L E R 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 E 0 . 2 8 1.11 2 .50 4 . 4 6 7 .00 GAMMA 1 .0003 1 .0012 1 .0027 1.0048 1.0075 8 AV 2 .996 2 . 9 9 9 3 .004 3 .010 3 .018 ANGF 9 0 . 00 9 0 . 0 0 9 0. 00 9 0 . 0 0 9 0 . 0 0 ANGD 64 .92 64 .92 64 .92 6 4 . 9 1 64 .91 WIDTH 2 5 . 0 8 2 5 . 0 8 2 5 . 0 8 2 5 . 0 9 2 5 . 0 9 SP ANG 0 . 0 0 0 .00 0 .01 0 .00 0 .00 6 0 . 0 7 0 . 0 8 0 . 0 9 0 . 0 100 .0 110 .0 10 .13 13 .87 18 .24 2 3 . 2 7 28 .99 35 .42 1 .0108 1 .0148 1.0194 1 .0248 1 .0309 1 .0377 3 .028 3 .040 3 .054 3 .070 3 .088 3 .109 9 0 . 0 0 9 0. 00 9 0 . 0 0 9 0 . 00 90 .00 9 0 . 00 6 4 . 9 1 64 .91 64 .91 64 .92 6 4 . 9 3 64 .92 2 5 .09 2 5 . 0 9 2 5 . 0 9 2 5 . 0 8 25 .07 2 5 . 0 8 0 .00 0 .01 0 .02 0 .02 0 . 00 0 .21 1 2 0 . 0 1 3 0 i 0 140 .0 150 .0 1 6 0 . 0 1 7 0 . 0 4 2 . 6 2 50 .63 5 9 . 5 0 6 9 . 3 0 8 0 . 1 0 91 .99 1 .0454 1 .0539 1 .0633 1 .0738 1.0853 1 .0979 3 .132 3 .157 3 .185 3 .217 3 .251 3 .289 9 0 . 0 0 9 0 . 00 9 0 . 0 0 9 0 . 00 8 9 . 7 7 8 9 . 5 9 64 .94 6 5 . 0 7 65 .04 6 4 . 9 7 6 4 . 9 0 64 .84 2 5 . 0 6 2 4 . 9 3 2 4 . 9 6 2 5 . 0 3 2 4 . 8 6 24 .75 0 .47 0 . 1 5 0 .42 1.20 2. 17 3. 50 180 .0 190 .0 2 0 0 . 0 2 1 0 . 0 2 2 0 . 0 2 3 0 . 0 105 .06 119 .44 135 .26 152 .68 171 .88 193.11 1 .1119 1.1272 1.1440 1 .1625 1.1830 1 .2056 3 .331 3 .377 3 .427 3 .483 3 .544 3 .611 89 .26 88 .82 88 .32 8 7 . 6 0 86 .80 85 .78 6 4 . 4 9 6 3 . 8 6 6 2 . 9 0 6 1 . 5 3 59 .72 57 .88 2 4 . 7 7 • 2 4 . 9 6 25 .42 26 .07 2 7 . 0 8 27 .90 8 . 19 11 .84 16 .25 2 3 . 1 4 2 8 . 4 7 3 0 . 4 8 2 4 0 . 0 2 5 0 . 0 2 6 0 . 0 2 7 0 . 0 2 8 0 . 0 2 9 0 . 0 2 1 6 . 6 4 2 4 2 . 8 0 2 7 2 . 0 0 304 .77 3 4 1 . 7 5 3 8 3 . 7 8 1 .2306 1 .2585 1 .2896 1 .3245 1 .3638 1 .4086 3 .686 3 .770 3 .863 3 .968 4 .085 4 .220 84 .61 8 3 . 06 81 .02 78 .65 75 .31 7 1.06 55 .84 5 3 . 5 9 5 1 . 0 3 47 .83 4 4 . 0 2 4 0 . 4 4 28 .77 2 9 . 4 8 2 9 . 9 9 30 .82 31 .29 30 .62 36 . 61 4 1 . 7 0 4 8 . 2 4 5 6 . 5 4 6 1 . 7 8 62 .92 3 0 0 . 0 3 1 0 . 0 3 2 0 . 0 4 3 1 . 9 6 4 8 7 . 7 6 553 .20 1 .4599 1 .5193 1.5890 4 .373 4 .551 4 . 7 6 0 6 7 . 0 6 6 3 . 05 59 .33 36 .76 31 .12 -72 .21 30 . 30 31 .93 131 .54 6 6 . 0 6 7 3 . 19 8 0 . 54 R X . 3 * X 0 B .955*X D TAN E D E ED 1 0 . 0 0 39 .810 11 .943 63 .412 38 .019*************** 0 . 2 0 . 0 0 3 9 . 2 4 8 11 .775 63 .013 37 .482 * * * * * * * * * * * * * * * 0. 3 0 . O U 38 .342 1 1 .503 62 .363 36 .616 * * * * * * * * * * * * * * * 0 . 4 0 . 0 0 3 7 . 1 3 1 11 .139 6 1 . 4 8 6 35.461*************** 0 . 5 0 . 0 0 3 5 . 6 6 9 10 .701 60 .408 34.064********>s****** 0 . D ED D AD D YD 001 * * * * * * * * * * * * * * * * * * * * * * 0 0 5 * * * * * * * * * * * * * * * * * * * * * * n 11****************** * * * * 003********************** E F D E F D A F D Y F 0 .001 * * * * * * * * * * * * * * * * * * * * * * 0 . 0 0 2 * * * * * * * * * * * * * * * * * * * * * * 0 . 0 0 3 * * * * * * * * * * * * * * * * * * * * * * - 0 . 0 0 2 * * * * * * * * * * * * * * * * * * * * * * » 0 . 0 0 2 * * * * * * * * * * * * * * * * * * * * * * 120 .00 22 .76 8 6 .830 49 .909 21 .743 6 . 5 9 4 8 6 5 9 4 . 3 4 8 0 . 0 1 5 9 3 2 6 . 0 3 9 6 5 . 9 4 5 1 6 5 . 9 4 5 0 .9309324 . 3 9 5 6 5 . 9 3 3 4 6 5 . 942 130 .00 21 .012 6 . 304 4 8 . 3 0 9 20 . 06719 .3754******** - 0 . 0 2 4 * * * * * * * * * * * * * * * 19 3 . 7 5 2 0 . 3 2 0 * * * * *********** 193. 751 140 .00 19 .343 5 .80 3 46 .732 18 .473 6 . 3 2 8 9 6 3 2 8 . 6 0 2 0 . 5 7 0 8 9 4 9 . 6 9 1 6 3 . 2 8 3 9 63 .287 0 .2668950 . 1 5 6 6 3 . 2 8 7 2 6 3 . 288 150 .00 17 . 767 5 .3 30 4 5 . 1 7 9 16 .967 2 . 0 2 7 6 2 0 2 6 . 6 9 3 0 . 7 4 6 2 8 6 6 . 6 2 9 2 0 . 2 7 0 1 20 .272 1 .6532865 . 3 5 8 2 0 . 2 6 1 1 2 0 . 270 160 .00 l b .287 4 .886 43 . 6 4 5 15 .554 1 .02961028 .103 1 . 1 2 0 1 4 5 4 . 8 1 3 1 0 . 2 8 7 1 10 .289 3 .2171452 . 1 2 6 1 0 . 2 6 8 1 10 . 284 170 . 0 0 14 .904 4 .471 42 .111 14 .233 0 .5867 584 .523 2 .904 827 .247 5.8495 5 .857 A .088 825 .855 5 .8397 5 . 855 60 .00 7 0 . 0 0 8 0 . 0 0 90 .00 100 .00 1 1 0 . 0 0 3 4 . 0 10 3 2 . 2 1 3 3 0 . 3 3 0 2 0 . 4 0 8 2 6 . 4 8 7 2 4 . 5 9 9 10 .20 3 9 .664 9 . 0 9 9 8 .522 7 .946 7 .380 5 9 .163 57 .783 56 .301 54 .747 53 .149 51 .530 32 . 30, 28, 27. 25. 23 . 4H0**< 764*** 965*** 129**s 2 9 5 * * « 49215. >;= * if if if if s,1: >;• * >;< if # if if if if if if # if if. if if £ if. if if if. if if if if & if. ; if if i.: )Jt * if # if if if i,i J|; # sjt J,* if if J;: ;|t if i,t 7261******** - 0 . 0. 0. -0 . o. 0. 001********** 001********** 002********** 00 l********s 001********* 016********=: * * * * * * * * ******** ******** ******** *****157 .260 * * * * **** * * * * * 0 .009*"** 0 . 0 1 6 * * * * 0 . 0 3 1 * * * 0 . 0 4 0 * * * * 0 . 0 0 4 * * * * 0 . 4 1 2 * * * * * * * * * * >: * * * * * * $ ******* ^  I' * * * * * * * * * * * ****** 5**157 .258 * * * * * *** * * * 180 .00 190 .00 2 0 0 . 0 0 2 1 0 . 0 0 220 .00 2 3 0 . 0 0 13 .615 12 .418 11 .307 10 .279 9 . 3 2 7 3 .447 4 . 0 8 5 3 .725 3 .392 3 .0 84 2 .798 2. 534 40 .062 3 7.922 35 .391 31 .719 28 .492 26 .746 003 0. 859 0, 799 0, 816 0. 907 0. 066 0. 2353 1539 1083 0784 0652 0580 2 3 0 . 4 9 2 1 4 7 . 3 R 6 9 9 . 8 5 8 6 6 . 2 8 0 5 0 . 3 9 6 4 3 . 0 7 5 157 324. 684 205. ,259 135. 276 85 . 917 63. ,363 54. 813 032 1. 365 0. 686 0, 432 0. 348 0. 2968 4498 9572 6059 44 8 5 3843 326 499 027 702 554 484 7 . 2 1 4 8 .934 10 .978 15 .169 IS .809 2 2 . 4 7 8 326, 210. 145, 99. 77 , 66, 933 8 55 1 084 1 377 0 464 0 437 0 3118 4910 0259 7027 5478 4698 330 509 045 728 582 508 240 .00 7 .632 2 .290 23 .270 7 .289 0 . 05 16 33 .254 4 1 . 293 42 . 927 0 .3035 0 .404 31 .273 50 .976 0 . 3605 0 . 422 2 5 0 . 0 0 6 .880 2 .064 20 .360 6 .570 0 . 0477 26 .592 46 . 023 34 . 046 0 .2407 0 . 3 4 7 36 .711 41 .234 0 . 2916 0. 364 2 6 0 . 0 0 6 .183 1 .855 16 .979 5 .905 0. 0462 20 .514 52 . 161 2 5 . 932 0 .1834 0 .299 43 .613 32 .345 0 . 2287 0 . 316 2 7 0 . 0 0 5 .539 1 .662 13 .084 5 .289 0 . 0488 14.820 58 . 735 19. 319 0 .1366 0 .263 54 .066 22 .758 0 . 1609 0. 274 2 8 0 . 0 0 u .942 1 .482 10.662 4 .7 19 0 . 0503 11 .246 6 1 . 283 16. 2 49 0 .1149 0 . 2 3 9 62 .270 15 .568 0. 1101 0 . 237 2 9 0 . 0 0 4 .388 1 .316 9 .943 4 . 191 0 . 0463 9 .602 6 1 . 052 14. 728 0 .1041 0 .215 64 .584 12 .547 0 . 0887 0 . 207 300 .00 3 1 0 . 0 0 320 .00 3 . 8 7 5 3 .397 2 . 9 5 4 1 .162 1 .019 0 . 8 8 6 8.531 5 .802 3 .170 ,700 0. .245 0. ,821 0. 0458 0561 0858 7 .537 4 .697 2 .319 522 10. .646 4. .964 4. 402 0. 367 0. 008 0. 0736 0. 0309 0. 0283 0, 185 144 148 65 . 5 8 9 6 4 . 0 2 5 8 1 . 7 2 1 10, 11 . 2 , 923 0. 352 0. 741 0. 0772 0, 0803 0. 0194 0. 187 183 135 - 97 -A p p e n d i x 4 A b r i e f d e s c r i p t i o n o f t h e p r o g r a m used t o c a l c u l a t e t h e m a g n e t i c r e l u c t a n c e o f t h e TRIUMF model magnet i s g i v e n h e r e . A l i s t i n g o f t h e p r o -g r a m , a t y p i c a l d a t a s e t , and a t y p i c a l o u t p u t f o l l o w . The p r o g r a m was run on t h e HP2116B c o m p u t e r . S t e p D e s c r i p t i o n L i nes 1 . Input d a t a : a) number o f r a d i a l i n c r e m e n t s 4 b) r a d i u s , c r o s s - s e c t i o n a l s t e e l a r e a , r e l a t i v e p e r m e a b i l i t y o f s t e e l , a n g u l a r w i d t h s o f h i l l and v a l l e y , a v e r a g e f i e l d , and h i l l f i e l d 5 - 6 2. Compute v a l l e y f i e l d 7 3. Input r e l u c t a n c e o f v e r t i c a l y o k e and t o t a l f l u x 8 4. I n i t i a l i z e t o t a l r e l u c t a n c e and w r i t e h e a d i n g 9 -12 5. Set up p a r a m e t e r s t o be used i n c a l c u l a t i o n s w h i c h i n v o l v e t r a p e z o i d a l i n t e g r a t i o n s 13 -26 6. F i n d r e l u c t a n c e o f h i l l 27 -29 7. F i n d r e l u c t a n c e o f v a l l e y by a 1 0 - s t e p n u m e r i c a l i n t e g r a t i o n 30 -37 8. F i n d r e l u c t a n c e o f s t e e l 38 -45 9. F i n d t o t a l r e l u c t a n c e 46 -47 10. Output and r e p e a t above f o r r e m a i n i n g r a d i i 48 11. Compute t o t a l magnet r e l u c t a n c e 49 12. Compute n o . o f a m p - t u r n s r e q u i r e d 50 13- O u t p u t 51 14. S t o p 57 - 98 -1 FTN,B 2 DIMENSION RC 40) * ASC 40) / AMUC 40>t WHC 40)> WVC 40 > t BAVC 40 ) * BHC 40>* 3 * BVC40) " " ' ' " 4 99 READC5>*)N 5 DO 10 I= 1 * N 6 READC 5**>'RCn»AS<I>>AMLKI)*WHCI>* WV CI )> BAVC I),BHC I ) 7 10 BVC I )'='( C WHCI) + WVCI) )*BAVC I7-WHCI)*BHCI))/WVCI) " 8 READC5-*)ZYOKE*FLUX 9 WRITEC 2* 102) 10 WRI TEC 2t 103>CRCI),BAVCI),BHCI),BVCI) J>WHCI) J>WVCI),AMUCr)>ASCI) J> 11 * i= i,N)' • ' 12 ZTOT=1.0E+10 13 DO 20~I=2>N IA C - - S E T UP PARAMETERS 15 AS1 = A S C I - 1 ) - ( A S C I ) - A S C I - 1 ) ) * R C I - l ) / C R C I )-RCI- 1 ) ) 16 AS2=CASCT)-ASC I- 1)7/CRCT>-RCI- 17) * . " 17 AMU 1 = AMUC I - 1)-'( AMUC I )-AMOCI- 17)*RCI- D/CRCI )-RC I - 1) ) 18 AMU2=CAMUCT)-AMUCI-1)7/CRCI7-RCI-1)7 " 19 WH1=WH< I - O-CWHCI )'-WHC I - 1) ) *RC I - l )V C RC I ) - RC I- 1 ) ) 20 WH 2= C WH CI5 -WH CI-157/C R C15 -R CI- 17J" " 21 W V1 = W V CI-T)-< W V CI)-WV CI- 17)* R C I - i ) / C R CI)-R CI- 1)) 22 WV2=CWVCI>~-V)VCI- 1 57/C RC 15 - RC I-17) 23 BV1=BVCI- 17-CBVCI)-6VCI- 17)*R'Ci- 1)/C RC I ) - RC I - 1) ) 24 . BV2=CBVCl5-BVCI-l)7/CRCT>-RCI-l7)~~ 25 BH1=BHCI-17-CBHCi)-BHCI-i7)*RCI-1)/CRCI)-RCI-1)) 26 BH2=CBHC'I5-BHCI-1>7/CRCI)-RU-175 :" ' 27 C -- FIND RELUCTANCE OF H l L L 28 ZH=WH1*CRCI)**2-RCI-1)**2)/2«+WH2*CRCI)**3-RCI-1>**3>/3. 29 Z H =180./C3.14l6 * Z H > • • - -30 C -- FIND"RELUCTANCE OF VALLEY C10 STEP NUMERICAL INTEGRATION) 31 . H=CRCI)-Rcl-1))/10. " 32 ZV=0. " " 33 RZ=R'C'I- 1 )-H/2. 34 DO"30 J=It 10 " 35 RZ=RZ+H"" " 36 30 ZV=H*RZ*CWV1+WV2*RZ)*CBV1+BV2*RZ)/CBH1+BH2*RZ) 37 Z V = 1 8 0 . / C 3 . i 4 1 6 * Z V ) ' 38 C -- FIND"RELUCTANCE OF STEEL 39 " " ZS= CAS1 + AS2*RC I ) )*( AMU1 + AMU2*RC I-1 ) )/C CAMU1 + AMU2*RC I ) ) 40 * *C AS 1 + AS2*RCI- 1))) " 41 TEMP=AMU1*AS2-'AMU2*AS1 42 IF CTEMP) 40>'4l> 40 ' " . ' • 43 41 ZS=CRCI ) -Ri.I-l>7/CAMUCI)*ASCI)) 44 GO TO 42 "~ ' ~" 45 40 ' ZS=ALOGCZS>/TEMP 46 C -- FIND TOTAL RELUCTANCE 47 42""' ZT0T=2.*ZS+L/C l./ZH+l./ZV+l./ZTOT) 48 20 WRI TEt'2* 101 )ZTOT" ' 49 Z=ZTOT+ZYOKE 50 A M = FLUX* Z * 2020. 51 WRlTEi.2*104)ANI" 52 101 FORMATcEi 5. 5) 53 102 FORM ATC ///" ~R ; i6X"B AV B HILL B VALLEY H WIDTH V WIDTH 54 * P£RM"6X"AREA"/) 55 103 FORMATCF5.2j.3F9.3;2F9.2*F9.0,F9.2) 56 104 FORMATC///"iMI= "HI 5. 5> /// ) ' " 57 " PAUSE 58 GO TO 99 59 END 60 ENDS - 99 -0 .0001 2000 25 35 1. 954 2. 592 1 .V0002 2000 25 35 1. 954 2. 59 2 2 0. 400 2000 25 35 2. 921 3.668 3 0.650 400 25 35 3.' 095 4.230 4 1. 150 165 25 35 3. 169 4. 544 5 1 .9 50 300 25 35 3. 185 4. 7 34 6 3.800 1900 25 35 3. 203 4.909 7 5'. 550 2000 25 35 3'. 277 5.240 8 6'. 500 1600 25 35 3'. 309 5.392 9 8.'500 i800 25 • 3 34 .7 3. 34S 5. 492 10 12.00 2000 26 • 4 33 .6 3. 48 3 5. 619 11 14. 50 2000 28 • 3 31 • 7 3. 671 5. 723 12 16. 50 1850 29 • 6 30 .4 3. 865 5. 755 13 19. 50 i800 31 .2 28 .8 4. 078 5. 766 14 21.4 i300 30 .7 29 • 3 4. 242 5. 768 15 23. 50 750 30 .8 29 .2 4. 637 5. 761 15. 5 26.45 1 100 38.6 '21 • 4"' 4.786 ' 5. 746 i6. 25 25.9 270 33. 5 26 '•'5 4.786 5". 7 46 ."00 351 529.56 - 100 -R B AV B HILL B VALLEY • 00 1 . 9 5 4 2 . 592 1. 498 1. 00 1. 9 5 4 2 . 592 1. 498 2 . 00 2 . 921 3 . 668 2 . 38 7 3 . 00 3 . 0 9 5 4 . 2 3 0 2 . 28 4 4 . 00 3 . 169 4 . 544 2 . 187 5". 00 3 . 185 4 . 7 3 4 2 . 0 7 9 6 . 00 3 . 20 3 4 . 9 0 9 1. 9 8 4 7 « 00 3 . 277 5 . 2 4 0 i". 8 7 5 8 . 00 3 . 309 5 . 392 i . 821 9 . 00 3 . 348 5 . 4 9 2 1. 7 8 5 10. 00 3 . 48 3 5 . 6 1 9 1. 8 0 5 ir. 00 3 . 671 5 . 7 2 3 1. 839 12. 00 3 . 8 6 5 5 . 7 5 5 2 . 0 2 5 13'. 00 4 . 0 7 S 5 . 7 6 6 2 . 249 14. 00 4 . 2 4 2 5 . 768 2 . 6 4 3 "15. 00 4 . 637 5 . 761 3 . 451 15'. 50 4 . 7 8 6 5'. 746 3 . 0 5 4 16 ; 25 4 . 7 8 6 5 . 7 46 3 . 572 • 1 0 9 0 4 E + 0 2 "." 1 2 3 3 4 E + 0 1 '. 5 0 3 7 4 E + 0 0 28 3 8 0 E + 0 0 1 8 3 9 2 E + 0 0 1 2 6 1 6 E + 0 0 9 18 3 7 E - 0 1 6 9 9 8 9 E - ' 0 i 5 5 1 1 5 E - 0 1 4 4 3 1 4 E - 0 1 3 6 0 7 8 E - 0 1 2 9 6 9 4 E - 0 1 2 4 7 1 3 E - 0 1 2 0 8 6 2 E - 0 1 1 7 8 9 3 E - 0 1 1 6 5 2 7 E - 0 1 1 4 7 3 1 E - 0 1 Nl= 1 9 5 1 2 . 3 2 0 0 9 H WIDTH V WIDTH PERM AREA 2 5 . 0 0 3 5 . 00 2 0 0 0 . • 00 2 5 . 0 0 3 5 . 00 2000." 00 2 5 . 0 0 3 5 . 00 2000." 40 2 5 . 0 0 3 5 . 00 4 0 0 . 65 2 5 . 0 0 3 5 . 0 0 1 6 5 . 1'• 15 2 5 . 0 0 3 5 . 0 0 300." i". 9 5 2 5 . 0 0 3 5 . 00 1 9 0 0 . 3". 8 0 2 5 . 0 0 3 5 . 00 2000". 5". 55 2 5 . 0 0 35". 00 1600". 6". 50 2 5 . 3 0 3 4 . 70 1 8 0 0 . 8. 50 2 6 . ' 4 0 3 3 . 60 2000." 1 2 . 00 2 8 . 3 0 3 1 . 70 2 0 0 0 . 1 4 . 50 2 9 . 6 0 3 0 . 40 18 5 0 . 16- 50 3 1 . 2 0 2 8 . 80 1800." 19'. 50 3 0 . 7 0 2 9 . 30 1300". 2 1 . 40 3 0 . 8 0 2 9 . 20 "750." 2 3 . 50 3 8 . 6 0 2 1 . 40 1100." 26". 45 3 3 . 5 0 2 6 . 50 "270". 25". 9 0 P A U S E 

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