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A comparison of a ferromagnetic core coil and an air core coil as the sensor head of the induction magnetometer Ueda, Hajime 1975

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A COMPARISON OF A FESROMAGNETIC CORE COIL AND  AN AIR TEE  CORE CCIL AS THE SEBSCB INDUCTION  K E A E OI  MAGNETOMETER  by Hajime Ueda B.Sc,  Tchoku U n i v e r s i t y ,  A THESIS SUBMITTED IN PARTIAL THE REQUIRE PINTS FOR MASTER OF  in  1969  FULFILMENT  TEE EEGBEE  OF  OF  SCIENCE  the Department of  G e o p h y s i c s and Astronomy  We  accept "this '»  thesis  required  The  University  as c c n f c r n i n g standard  Of B r i t i s h  April,  t c the  1975  Cclumtia  In p r e s e n t i n g t h i s  t h e s i s in p a r t i a l  an advanced degree at  further  agree  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  fulfilment  freely  available  for  this  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department  of  this  thesis for  It  financial  The  g a i n s h a l l not  of  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  or  i s understood that copying or p u b l i c a t i o n  written permission.  Department  that  r e f e r e n c e and study.  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 copying o f  by h i s r e p r e s e n t a t i v e s .  for  be allowed without my  ii  ABSTRACT An  experimental  output  characteristics  sensor  head  are mainly the  concerned  of  the  of  T h i s suggests is  linear  in  The linear  t h a t the  the  or  ene  air  as  obtained  here  of  that  coil.  taken  to  The  p o r t i o n of results  or  the p e r m a l l o y  the  by  Blackman-Tukey*s  distortions  by  a  acquired  core  values are  amplitude.  significant  providing  demagnetization  demagnetization geometry  of  suppressed reciprocal  does  coil,  peak-to-peak  response  used  permalloy  the a i r core  of  harmonics core  coil.  core  coil  coil  sensor  is  for  the  response.  response  factor:  to  p e a k s a r e compared.  data  input-  made  estimation i s applied tc a  spectral  in  coil  Analyses  respect  output  show t h a t no  recognized  linear  of the  power s p e c t r u m  analyses  sensor  with  corresponding  magnetograms, and  are  coil  core  the  o f magnetograms o b t a i n e d  by a l a r g e open l o o p  examine d i s t o r t i o n s method  a ferromagnetic  with comparison  core  simultaneously Ratios  of  elucidate  o f an i n d u c t i o n m a g n e t o m e t e r .  permalloy  these  a p p r o a c h i s made t o  of factor  the  is  depend  demagnetization suggests sensitivity  the c o r e  The  on  i f i t has  much  be.  an  explanation  core  coil  so l a r g e  the  than  longer  core  a the  of  core,  the  the  that  its  demagnetization  core  demagnetization the  The  material i s  value  sensitivity  permeability The  senscrs.  m a t e r i a l d e p e n d s o n l y on  smaller  the  factor.  be  p e r m e a b i l i t y of the  circumstances,  that . rhe will  may  permalloy  of  in i t s effect  under such  not  the  core.  value  effect  of  the  but  on  effect  higher  the  coil the also coil  iii  TABLE OF CONTENTS  ABSTRACT  ,  . . . i i  LIST OF FIGURES  V.  LIST OF TABLES  ,  vi  ACKNOWLEDGEMENTS 1.  INTRODUCTION  2.  CHARACTERISTICS  3.  4.  .vii 1 OF A FERROMAGNETIC CORE COIL  2.1  N o n l i n e a r Response  2.2  Demagnetization E f f e c t  2.3  Hysteresis  3  Of Ferromagnetic M a t e r i a l  .......3 4  Response  ...........9  EXPERIMENT  14  3.1  Instrumentation  14  3.2  Some C o n s i d e r a t i o n s On C o i l C h a r a c t e r i s t i c s  20  ANALYSIS  23  4.1  L e a s t Sguares F i t  23  4.2  Amplitude R a t i o  29  4.2.1  R e l a t i v e S e n s i t i v i t y And R a t i o Of Two Normal V a r i a b l e s  ....30  4.2.2  Noise A n a l y s i s on P r o b a b i l i t y Graph  4.2.3  P r o b a b i l i t y D e n s i t y F u n c t i o n Of A R a t i o Of Two Normal V a r i a b l e s  .  4.3  Effective  4.4  Comparison Of Power S p e c t r a  4.4.1  P e r m e a b i l i t y Of Permalloy Core  ......37 40 41  Confidence I n t e r v a l And Bandwidth Of The Tukey Window  4.4.2  34  Results  Of S p e c t r a l  42 Analysis  44  iv  5.  SUHMiiRY  AND  CONCLUDING REMARKS  49  APPENDICES  .51  REFERENCES  59  V  LIST OF FIGURES Figure 2.1  Page Apparent P e r m e a b i l i t y Of Rod With Respect To True P e r m e a b i l i t y And R a t i o Of Length To Diameter  8  3.1  Schematic Diagram Of The Experimental Setup .......15  3.2  Frequency Response  Of System A  .17  3.3  Frequency Response Of System B  .......18  3.4  Frequency Response  4.1  Magnetogram Of October 10, 1973  4.2  Least  B Output  26  4.3  Least Squares F i t To System C Output  27  4.4  R a t i o Of System  B Output To System  4.5  Ratio Of System  C Output To System A Output  4.6  Normal D i s t r i b u t i o n Of The  Of System C  Squares F i t To System  ...............19 ......24  A Output .......32  Difference  Between S i g n a l s By System A And B 4.7  Normal D i s t r i b u t i o n Of The  ..36  Difference  Between S i g n a l s By System A And C 4.8  Maximum L i k e l i h o o d Power S p e c t r a l Of System  4.9  Record By Blackman-Tukey 4.10  .37 Estimate  A Record Of October 10, 1973  D i f f e r e n c e i n Power S p e c t r a  Record By Blackman-Tukey  ..45  Of System B  Method  D i f f e r e n c e In Power S p e c t r a  33  Of System  Method  ..........46 C .....47  vi  LIST OF TABLES Table  Page 2.1  Demagnetization Factors Of Bod And E l l i p s o i d s Magnetized P a r a l l e l To Long Axis ....... 8  3.1  Coil Characteristics  4.1  Least Squares F i t C o e f f i c i e n t s  4.2  Comparison Of Power Spectra  ....16 .......28 ....48  vii  ACKNOWLEDGEMENTS I express advisor  my  appreciation  Dr. Tomiya Watanabe  great extent throughout to  analyze  data  and gratitude  who  directed  t h i s experiment  to my  and  thesis  helped me to a  and also  encouraged  me  and obtain the r e s u l t s which are presented i n  t h i s thesis. I g r a t e f u l l y acknowledge not only  encouragement  but  also  the h e l p f u l discussions and suggestions of Dr. R. D. Russell. I  appreciate  provision  of  the main  Dr. B . Caner  and  Observatory.  I  magnetometer  also  the generous equipment  Dr. L. Law, thank  system  with  of  as  well  as the  f o r the experiment, the V i c t o r i a  Dr. T. Oguchi permalloy  help  for  the  from  Geophysical induction  core c o i l sensor.  I also  thank Mr. J. Walter f o r the provision of an experimental s i t e i n the University of B r i t i s h Columbia Research Forest. This research received f i n a n c i a l support from the National Research  Council  Defence Research  of Canada  under  grant A-3564, and from the  Board under grant 9511-112.  1.  INTRODUCTION There a r e many types of magnetometers  the  activities  of  the e a r t h ' s  used  f o r observing  magnetic f i e l d .  They a r e the  p r o t o n p r e c e s s i o n magnetometer, f l u x g a t e magnetometer, i n d u c t i o n magnetometer,  etc.  From  reguirements  f o r these  should cover  the frequency  take  standpoint  cf  gammmas.  of  instrument  of  a coil  o b s e r v a t i o n systems a r e such band from UHz t o 0.002Hz  t h a t they  and  should  The i n d u c t i o n magnetometer t u r n s o u t t c be a which s a t i s f i e s these two c o n d i t i o n s .  type  I t consists  with many windings as a sensor f o l l o w e d by a low n o i s e  amplifier.  Output s i g n a l s a r e time d e r i v a t i v e s o f f l u c t u a t i o n s  the magnetic f i e l d  of the earth.  An open loop c o i l , or a i r core c o i l , has a to  aeronomy,  the s i g n a l l e v e l from the order o f milligamma t o gamma or  ten  of  the  large  diameter  g a i n as much e f f e c t i v e area as p o s s i b l e so t h a t i t can save a  great  number o f windings.  permeability  On the other hand, some high magnetic  m a t e r i a l s a r e used as a core of t h e sensor c o i l t o  reduce i t s s i z e and weight. It i s generally believed ferromagnetic  material  n o n l i n e a r i t y i n response Barkhausen of  a  sensor  devices  a r e not  due t o h y s t e r e s i s , eddy  of the f e r r o m a g n e t i c  on a core which has extremely  m a t e r i a l s commonly  permalloy, their to  for  magnetic  using free  current  material, etc.  these s p e c u l a t i o n s , many magnetic s t a t i o n s a r e using  wound The  noise  that  supermalloy,  s p e c t r a , amplitude  other  stations  used etc.  a  from less,  In s p i t e a  coil  high magnetic p e r m e a b i l i t y .  f o r these  sensors  are  As a n a l y s i s of output  a t t e n u a t i o n as w e l l as  mumetal,  wave forms,  phase  a r e a l l fundamental and important  relative part of  2  r e s e a r c h on geomagnetic m i c r o p u l s a t i o n s , i t  cannot  be  t h a t the sensor head i n t r o d u c e s n o n l i n e a r i t y a t the It  is  investigate  The  to  the  author  distortions  due  to  core  approach  magnetometers observation systems.  interest  the  ferromagnetic data.  of  using and  to  sensor made  through here  is  different compare  the  beginning.  this  thesis  nonlinearity  to of  the a n a l y s i s on magnetogram to  kinds the  of  allowed  simultaneously of  output  sensors from  operate  for  the  actual  different  R a t i o s of the peak-to-peak values i n the output of the  ferromagnetic  core  coil  system  r e l a t i v e t o the  peak-to-peak values by the a i r c o r e c o i l  system  corresponding  were  analyzed.  A l s o power s p e c t r a are compared between d i f f e r e n t systems.  3  CHAHACTERISTICS OF A FERROMAGNETIC CORE COIL  2.  There  are  ferromagnetic sensor  many  factors  which should be c o n s i d e r e d i f  core a c t s as a n o n l i n e a r  configuration  of  a  magnetometer.  s e c t i o n 2.1 shows the e f f e c t s nonlinear  response.  On  case,  the  which i s  response  2.1.  other  element  A list possible  sources  hand, one of the  becomes  coil  linear.  i n the s e c t i o n 2 . 2 . It  is  of  probable  may r e s u l t  response.  It requires Hysteresis  in a  given i n the  demagnetization e f f e c t ,  i n the s e c t i o n 2 . 3 .  to h y s t e r e s i s  are  which dominates the  discussed  discussed  which  the  source of n o n l i n e a r r e s p o n s e , i n a main f a c t o r  response  the  In  that  a condition response  is  shown t h a t the response due  produces odd number harmonics.  NONLINEAR RESPONSE OF FERROMAGNETIC MATERIAL Many  been  possible  speculated  problems with a ferromagnetic  and  discussed.  Campbell  core c o i l  (1967)  have  lists  the  following; Effective  1) coil  M a g n e t i z a t i o n caused magnetic  direction  to  the  'demagnetization  4)  through the s i d e  by the primary magnetic f i e l d  field  Barkhausen  which i s ,  original  wall  one.  produces a  i n most c a s e s , o p p o s i t e This  is  referred  to  in as  effect'.  Some l o s s may be caused  of elements 5)  leaks  core.  secondary  3)  depending upon the p o s i t i o n of  windings because magnetic f l u x  of the 2)  permeability varies  noise  by eddy c u r r e n t s .  may be caused  which form the core  by m i c r o s c o p i c m a g n e t i z a t i o n  material.  P e r m e a b i l i t y of a ferromagnetic  material i s  unstable  against  4  thermal s t r e s s as w e l l as  mechanical shocks.  6)  'B-H • curve i s  7)  Some l o s s may be caused i n a large  response.  by completing the h y s t e r e s i s b i a s such as  loop. of  the  e a r t h produces second harmonics and a l s o causes v a r i a t i o n of  the  8)  Operation  a hysteresis  the magnetic f i e l d  permeability. 9) Cross modulation may happen i n case of two applied  at  the  or  more  signals  same time because of the n o n l i n e a r response of  the ' B - H ' c u r v e . The items 3, discussion  in  4,  7,  terms  8  and  are  of freguency.  are the ones  which  originally  hysteresis.  Above  all,  important r o l e i n l a t e r  2.2.  9  the  the  ones  which  The problems 6,  stem  from  7,  need  8 and 9  characteristics  demagnetization  effect  plays  an  discussion.  DEMAGNETIZATION EFFECT When a f e r r o m a g n e t i c  field,  it  becomes  body i s  magnetized.  placed i n an ambient Further,  magnetization  magnetic f i e l d  case o f a s t r a i g h t  r o d , opposite  to the magnetization so t h a t  tends  the  magnetic f i e l d .  to  reduce  applied  magnetic f i e l d  is called  of  eguation d e s c r i b e s  Laplace's  whose  this  magnetic  produces secondary  'demagnetization  direction  field'.  field,  but i t  is  d i f f i c u l t to c a l c u l a t e  the m a g n e t i z a t i o n except one  such  case.  p o t e n t i a l through L a p l a c e ' s  in  the it  This  secondary  The  solution  parallel  exactly  f o r a few simple c a s e s .  Stratton  is,  the magnetization cf a body of  homogeneous m a t e r i a l placed i n a uniform and  is  of  (1941)  equation.  obtained  magnetic  the s t r e n g t h  of  An  ellipsoid  the  magnetic  5  The Laplace's equation becomes in e l l i p s o i d a l coordinates  where  where a, b and c are semiprincipal and  z  coordinates  axes of e l l i p s o i d along x,  respectively.  If  the  primary  directed along the x-coordinate axis which i s major  axis, a,  of  field  parallel  to  y is the  the e l l i p s o i d , the potential of the applied  magnetic f i e l d i s  i C^-d^X^-^)  (  ..  (2 2 3)  This i s also a solution of 2.2-1, and can be rewritten  such as  where 3T  (2.2-5)  m  4=-Et  {Cb-d^X^)}  The induced potential i s expected  to have the form such as  <h = C>4-,&)J^<y)fi(ZJ This i s substituted  Knowing  (s|J)=  (2.2-6)  into 2.2-1, and  \J^-f^(is  a  solution  of  2.2-7,  independent solution.  Then the potential of induced magnetization becomes  we  get  an  6  ^ where  C2  is  ~ ^  c  2  j ? y ^ h  ^  an undetermined c o n s t a n t .  O u t s i d e the  2  _  9  )  ellipsoid,  therefore,  goes i n f i n i t e at  4=_c2, the  potential  w i t h i n the  body  has the form, (2.2-11) The  boundary c o n d i t i o n s determine C2 and C3.  where m e t r i c a l  JJ.Q i s  the  ellipsoidal  These  are  coefficient,  p e r m e a b i l i t y of ambient m a t e r i a l and^C body.  The f i r s t  of these l e a d s  that  of  to  C^-hf-)^ The  is  (2.2-13)  second one g i v e s  -2 Rith 2.2-3,  ' /AX  the p o t e n t i a l  (2.2-14)  D  at any i n t e r i o r p o i n t of the  ellipsoid  is  =  n*  AJ—J*2— (2.2-15)  The  field  intensity  becomes  ^ " ^ ^ ^ y ^ ^  (2.2-16)  7  Consequently,  the  flux  density inside  the  / from  this  equation,  effective the  ^  i t i s clear  p e r m e a b i l i t y B/Ho  core  coil In  along  in this i s free  the  case,  that  reduces  p e r m a b i l i t y i s determined  Therefore,  by  overall  (2.2-18) i s very  to a value  cf  geometrical  response  of  i t s nonlinear  hysteresis  of  magnetization  of a s l i m  factor  Nd  large,  1/Nd.  response.  elliptic is  cylinder  calculated  diameter.  2.1  is  and  an  a given  shows  that  that,  its  length  to  the c r o s s - s e c t i o n a l  the  core  experiment  diameter,  sensitivity the  of  factor  oblate e l l i p s o i d .  reciprocal  the  f o r a longer  core,  by  case,  higher  Bozorth  longer  sensor,  value  length.  for a red, a  Data f o r the  the  of the  permeability in this  sguare  (2.2-20)  demagnetization  through  i s the  the  of  >>1,  shows t h e  higher  effective  ratio  (2.2-19,  Nd= -j£r{JtL2»<-l }  obtained For  a  If m  ellipsoid is  by  (1945) .  m  Table  only.  ferromagnetic  W-^{»^Afa-fifrT~)-l) where  the Thus  constant  a  from  case  becomes  ^  when  the  i t s length, demagnetization  Osborn  ellipsoid  of  given  here  length  the  (1951).  the c o r e  because  eguation  magnetization,  i s almost  F i g . 2.1  rod  2.2-18 or  proportional  f o r a rod  permeability  prolate  is  shows  the to  clearly  reguired  to,  8  Dimensional Ratio (length/diameter) 0 .  '  Table 2 . 1 .  1 2  $ 10 20 50 100 200 500 1000 2000  Rod  Prolate Ellipsoid  Oblate Ellipsoid  1.0 .27 .14 .040 .0172 .00617 .00129 .00036 .000090 .000014 .0000036 .0000009  1.0 .3333 .1735 .0558 .0203 .00675 .00144 • .000430 .000125 .0000236 ' .0000066 .0000019  1.0 .3333 .2364 .1248 .0696 .0369 .01472 .00776 .00390 .001567 .000784 .000392  Demagnetization magnetized p a r a l l e l  factors of r o d and ellipsoids t o the long a x i s ( B o z o r t h , .1951) .  T R U E P E R M E A B I L I T Y , fX  Fig.  2.1.  Chart f o r c o n v e r t i n g t r u e to apparent p e r m e a b i l i t y f e r r o m a g n e t i c c y l i n d e r s of given r a t i o of length diameter (Bozorth, 1951).  of to  9  secure  the  dominates  2.3.  inherent  no  Considering  difficulty-  i n the o r i g i n a l  in  response  from  Peterson  hysteresis It  sinusoidal  field,  high  materials, nonlinear  magnetic permeability.  by  dominates c o i l  response  v i r t u e of the demagnetization  hysteresis  carried  rather  out  of a  the  field  mathematical  effect,  are expected  ferromagnetic  than  to  material.  analysis  of  the  response.  is  current  the  (1928)  sufficiently  suppressing  some h a r m o n i c s o f t h e a l t e r n a t i n g p r i m a r y result  effect  RESPONSE  t h e 'B-H' r e l a t i o n  linear  the demagnetization  a t t a i n a b l e f o r some f e r r o m a g n e t i c  be  HYSTERESIS If  the  to  i n which  response.  values  seems  response  region  overall  permeability there  linear  assumed  that  the a p p l i e d  magnetic  and i t s i n t e n s i t y i s n o t s t r o n g  loss.  When  H  i s t h e maximum  field  i s purely  enough t o c a u s e  amplitude,  eddy  the magnetic  h, becomes (2.3-1)  For  one c y c l e o f  assumed  to  function  of the a p p l i e d  sign  be  this  o f dh/dt.  excitation,  one c o m p l e t e field  loop,  each  hysteresis  and t h e f l u x  h, i t s maximum r>  curve  density  value  H,  is  E, i s a and  the  10  The  sign  of dh/dt s p l i t s  the h y s t e r e s i s loop i n t o two  top and bottom ones; B1 and B2, r e s p e c t i v e l y .  Each  branches, branch  is  expressed by the same form  (2.3-2)  M=0 A=0 where ffjMJtX,—  (2.3-3,  As B=0,  at  h=0 and H=0, (2.3-4)  Q.oo — 0 The  h y s t e r e s i s loop i s  branch has the  symmetric about  the o r i g i n , and i t s  each  relation,  5iC^H)=~-fia.C-A/H>  (2.3-5)  Then,  (2. 3-6)  •4  (2.3-7) These  branches  meet at the loop  tip,  PiCH/H^Bi^/H^ Substituting third  2.3-6 and -7 i n t o - 8 , we o b t a i n a  (2.3-8)  relation  up  to  order of H. O.0lH + (^  Since t h i s  relation  + *c*)tf'+<A2\+*c*}tt*=O  (2.3-9)  holds f o r any value of H, d-tz  =  , <2a3=-#2f  (2.3-10)  11  2.3-6  and  -7 become  (2.3-11)  (2. 3-12)  Let  (2.3-13) Then,  R> Gicasti&ti^-x+p cosofr+ txcas*zot + Tcog* cot From  these  remanence and  two  equations  Q  we  an a p p r o x i m a t e  see  that  CX  (2.3-15)  represents  the  permeability.  Letting  c—A,  A = - # , the equations  2.3-14 and  />—£  2.3-15 c a n be m o d i f i e d  3,<rrf^A/i,tf)«A^£^  ,2.3-16, further, (2.3-17)  (2.3-18) Both  b r a n c h e s o f h y s t e r e s i s c u r v e a r e combined  k=\  J  where  i n the form,  (2.3-19)  12  '70  Jo  (2.3-20) PL  k « r - ^ / { B 6 A , W [ I+ H J** 1} teS-hftdwt 1  (2.3-21) so  that  (2.3-22)  (2.3-23) The  coefficients  bc>r t>x, txt,' all  become z e r o ,  f u n d a m e n t a l and  - , b- *-' 2  because of the  third  the  symmetry  of  the  h a r m o n i c components have  loop.  The  coefficients,  (2.3-25) _  7 (2.3-26)  (2.3-27) The  voltage  across  the  output  terminals i s  13  where  N  sectional  i s the number of turns of the winding, A i s the crossarea,  E " — / 9 " ^ V / 4 (ivAi c&tvt-t-3toT^3 casStot •  -iobi?tii0it—3a)tbB'Sir\Sa)t}  (2.3-29)  Therefore i t i s more l i k e l y that the third harmonic rather the  second  response.  harmonic  than  s i g n a l w i l l be produced by the hysteresis  14  3.  EXPERIMENT  3.1.  INSTRUMENTATION Three  induction  University British  setup  3.1.  head and  noise  figure  output  of the  A consists  core c o i l  FET  as  3db.  A.  t h e two  with  0.1  intensive  than  system  together  output  and  recorder of  FM  range  carrier  constants  c o n s t r u c t e d and  ferromagnetic freguency is  signal.  of s y s t e m s and Coil  was  adjusted to  core  response  introduced  a  carrier be  System To  of the  are coil.  to the c o i l  as  with  core  C has  a  a  coil to  between  the  permalloy  amplifier  as  employs  Outputs  of  tape  these  recorder  per  Sensitivity 40%  obtained  through  this  deviation  the  dynamic  base.  B. C a n e r  The  of  freguency  a r e g i v e n i n T a b l e 3.1.  coil  sensor  r e c o r d e r determines  by  take  identical  magnetic  signal.  i t i s 40db i n rms  calibrated  sensor  obtain ncise levels  this  FM  on  f l u x g a t e magnetometer, c l o c k  2 volts  T h i s FM  Ridge,  the  a permalloy  ( F i g . 3.4).  of  as  i s possible  B.  a 7 channel  a r e f e r e n c e FM is  B uses  peak-to-peak,  on  the  shown  which c a n  k-ohm o f t h e  comparison  throughout  the  coil  amplifier  chopper a m p l i f i e r .  filtering  schematically  a i r core  200  s y s t e m s A and  at  F o r e s t , Maple  to p a r a m e t r i c a m p l i f i e r  systems are recorded  pulses  is  System  operated  1973.  o f an  the  Direct  micro-volt  with  10,  parametric  impedance  been  Research  experiment  i s connected  from  as  the  less  which  outputs  low  of  a semiconductor  an  that  Columbia  s i n c e September  System  high  sensor  British  Columbia  The Fig.  of  m a g n e t o m e t e r s have  by  signal  The  a i r core  (1970). T.  of  the  Watanabe from  from  a resistor  Data  coil  the  which  the  oscillator has  a  value  r System  A  45  I : 0.128  mV/ff-Hz  o .-..liconcluctor P;.rai.iotric Amplifier  db  7-channel  6'0 d b S y s t e m Li  v0.0189  iuV/y-Hz  System  •0.0139  C  mV/y-Hz  Fig. 3.1.  Schematic  FM.  Tapo Smenri;  diagram of experimental setup.  l/l6lr>a  Air  core  Weight  175 k g  Dimension  Diameter;  coil  P e r m a l l o y core  coil  12 k g 1.2? ni  Core l e n g t h ; 1.0 m Cross-section;  2.0x2.0 cm Number o f t u r n s  16,000 t u r n s  50,000 turns  DG r e s i s t a n c e  5 . ^5 k-ohm  158 ohm  Resonance  3*f Hz  400 Hz  550 H  170 II  6.7 nF  0.8 nF  200 k-ohm  10 k-ohm  18.5 nV(RMS)  5.17  0.128 mV/ -Hz  0.0189 mV/ - H z  Self  frequency  inductance  Stray capacitance  1  Damping r e s i s t o r Johnson n o i s e (DC Sensitivity  kllz)  Table 3.1. C o i l c h a r a c t e r i s t i c s  nV(Hl-iS)  A m p l i t u d e Ugnnortsg  0.1  J  Overall Response C o i l Response UJ  a  0.0J J  g  Si  S30 6.7 S.4S  COIL INDUCTPNCC CAPACITANCE RESISTANCE DAMPING RESISTANCE  H.  NANO-f K-OHM K-OHM  0.001 J  O'.l F i g . 3.2.  "i i i  FREflUENCr(HZ)  Attenuation and response includes preamplifier.  iI 1  1.0  phase response of system A. the input impedance of  Coil the  H  « . «  —I  Phase  O.I  J  Overall Response C o l l Response  a  S o.oi J  I  COIL INDUCTRNCE CAPACITANCE RESISTANCE  170 O.B CIS  DAMPING RESISTANCE  0.001  H.  NANO-F K-OHM K-OHM  J  1.0  Oil FREQUENCriHZ)  F i g . 3.3.  Attenuation and response includes preamplifier.  I  HIM  10*  phase response of system B. the input impedance of  Coil the  Amplitude  Oil  i'.o  FREQUENCY(HZ) Fig.  3.4.  Attenuation and response includes preamplifier.  phase response of system C. the input impedance of  Coil the H  20  much s m a l l e r t h a n current  through  frequency.  the r e s i s t a n c e  this  The  frequency  t h e i n p u t impedance of  coil  overall and  and  resistor  of the  amplifier  system  response  of the  is  coil  measured  response  as  a  of the c o i l  preamplifier. impedance  f o r each  windings.  is  The  Then  the  function  of  i s sensitive  combined  response  shown t o g e t h e r  sensor  coils  on  to  with  the  F i g . 3.2,  3.3  3.4.  3.2  SOME CONSIDERATIONS ON The  whose  sensor  radius  varying  where  is  component  field,  CHARACTERISTICS  approximated  the  coil  voltage induced  cere,  perpendicular  to  i s measured  as  t o be  Corresponding  cross-sectional  of the  permeability so  is  constant.  the  permeability  one,  is  magnetic  S  coil  COIL  area  and the  h coil  N turn  to  the  i n the  of is  an  uniformly  ceil,  the the  Eo,  coil, magnetic  value  tc  is  JLL i s  cross-section.  a normalized  winding  field  Usually,  the  vacuum  that  S =4.70 • /d^fee, A/sJL  (3.2-2)  0  According  to data  f o r the  S=1.28 mz  and ^ ^ = 1 .  a i r core c o i l  The  magnetic  conventionally  i n milligammas  in  The  MKS  units.  i n Table  field,  which i s e q u a l  sensitivity  3.1,  N=16,000,  is  measured  h, to  of a i r core c o i l  1/^x  x  10  - 5  becomes  Bo^A/ZZ^r/^r-ttfr As  the  magnetic  total field  deflection  g a i n of the variation  o f 23.5  Campbell  mV  (1960)  on  amplifier of  (3.2-3)  i s calibrated  1 milligamma  AT/m  to  peak g i v e s  1.87x10 , a 5  rise  to  a  the r e c o r d e r s .  suggested  a  simple  form  of  equivalent  21  circuit  t o approximate  characteristics  o f sensor  coil.  AW— nrsip* —i—o—,—> -r- C  Where  r  i s  the  capacitance, resistor  which  resistor effort,  L  resistance  i s  the  the  the output  resonance  p r o v i d e s maximal f l a t  function  windings,  C  self-inductance.  i sshunting  damps  of  S  i s  i s an  terminals of  peak  of c o i l  amplitude  the  optional  coil.  This  and, with  response.  stray  The  minimal transfer  o f t h e above e q u i v a l e n t c i r c u i t i s  (3.2-4) So  the amplitude  response i s  <--  M"H Htfo)h -£/isi<w  32  where  ^  NcV= It  i s  r+fc  desirable  amplitude  for  u*/-Cg] + io\L -h r CRf)*  that  there  be  no  irregular  ( 3  peaks  i n the  That  i s ,  ^  { W/CzLC-r^A^, On  2 L > r C , which  the  highly  other  most  hand,  resistive  resistive  .2-6)  response,  a l l tO .  where  5)  wire  component  appear  h i g h e r than  R also  satisfies  sensors  for a coil and w i t h  (3.2-8) with  many  wound w i t h  windings  satisfy.  extremely  t h i n , or  small inductance, the l o s s  becomes s o l a r g e  that  the l e v e l  The maximum  the  a t DC.  condition  that  no  the  resonance limit  first  by t h e peaks  value of  and  second  22  derivatives response the  damped  higher As  of  N(w) by  be  R,  therefore,  freguency  the  positive,  damping  resistor  shunts the  components,  ^ r  representation,  are and  amplitude  monotonically  toward  output  of  the  coil,  it  noise.  O  L.  and  decreases  The  region.  produces a d d i t i o n a l thermal  where  f o r a l l OU .  rms  value  of  thermal  R,  respectively.  noise  of  resistive  Using  the  current  source  t o be  unccrrelated,  i t becomes  L  These two  two  thermal noises  current  sources are  are  considered  replaced  by  one  source such  that  •j f Similarly, magnetic  a current  field  source  variation  noise,  then,  to  the  voltage  (3.2-9) induced  r+j&L can  by  is  r-t-JioL Thermal  equivalent  and  be  variation.  r+JooL  estimated  (3.2-10) as  an  equivalent  magnetic  23  4.  ANALYSIS Outputs  by  an  FM  played  of the a m p l i f i e r s slow  speed  back a t a speed  Then  the  after  passing through  8  Hz  and  data  October  was  10,  1973.  UT  sporadic  noise  tape  times  digitized  faster  The  were a n a l y z e d inherent  filter  The  for  o f 29  tape  the  tape  was  recording.  p o i n t s per  whose c u t o f f  then  second  freguency  is  db/oct.  active  data  than  at a rate  a lowpass  of  recorder.  a magnetic  geomagnetic  o f a 282 mainly  to  sec d u r a t i o n s t a r t i n g  because  the  micrcpulsation  the data  is  on from  free  parametric amplifiers  of  used  in  experiment. As  a preliminary  squares  fits  amplitude  of  ratio  of the  coil  signal  three  were  compared.  4.1.  was  LEAST SQUARES A least  signal 21h  and 42m  the  each  coil system  Yb  and  systems, B has  signals  examined  the were  and  f i t was  data  tried  permalloy o f 2 min  October  be  B and  channels,  taken.  core c o i l  least  Secondly,  signal  to  the  the  finally,  power s p e c t r a  between  the  10,  core c o i l 21  1973.  activity  components l i e i n t h e Yc  three  air  of  the  FIT  micropulsation  frequency Let  UT,  of  permalloy  of the  r e c o r d of  geomagnetic major  squares  06s  comparison  these  core  from  15  a period  21:42'06"  on  analog  a t t e n u a t i o n s l o p e 24  There  this  was  were r e c o r d e d on  the C,  output  sec  There during range  a parametric amplifier  As  signals.  duration  voltage of  respectively.  air  is a this  core  coil  Commencing was  medium period  sampled level  of  and i t s  0.02-0.08HZ. the  permalloy  mentioned  immediately  already,  following  core the the  23H 42H 055 UT OCT. 10/1973  FLUXGRTE MAGNETOMETER  OflHHfl-HZ -o.oaJ 0  og  _, PERMALLOY CORE COIL + PflRRMETRlCJWPLlFIER  GRMMR-HZ -0.0SU Q  Qa  _, RIR CORE COIL • FRRflHETRIC RHPLIFJER  6HHMH-HZ -o.oaJ ~5a  F i g . 4.1.  i i  is  Hi  Magnetograns o f 21h 42m  !ul  5IB9  tas  -  06s UT on October 10.  bs4  l<m  S E C  fa  25  sensor used  coil  and  t o denote  (system  A).  the  system  the output  The  fittng  C has  a FET  chopper  v o l t a g e of the a i r  curve  i s given  amplifier.  core  X is  coil  system  by  fi-^O From  this,  squares  the  curve  (4.1-1) is  fitted  t o Yb  and  Yc  u s i n g the  least  fitting i.e.  (4.1-2) is  minimized  difference  over  sec d u r a t i o n of the r e c o r d .  r e p r e s e n t the  ferromagnetic  the  systems i s taken the s i g n a l , the  filters  amplifiers.  possible  amplitude  t o be  flat  differences results  characteristics coefficients  i s essentially  c o r e as m e n t i o n e d  Although  with  141  between c h a n n e l s  coefficients  affects  the  of  i f  sguares  is  of  have  are  least  response  the  of each  obtained.  If  2.1  order  the a m p l i f i e r s  not  will  be  relative  projected  responses  the  lag  in  the  smallest  is  ~C-  each  sec  of  This  f i t method.  realigning  and  the  content  are compensated,  by  to  negligible.  complementary  channel  due  already.  squares  i t i s done s i m p l y  data  higher  the frequency  thes.e p h a s e r e s p o n s e s  Here  of the  response  the  the c o i l  which  position  of  in section  phase  The  gain  nonlinear effects  enough o v e r  in  k1.  The  on  The the  ideally to  the  relative sum  of  between  records,  so t h a t  this  operation i s equivalent to using  which  gives  a phase s h i f t  result  i s obtained  proportional  when a l a g o f -0.55  a  phase  to f r e q u e n c y . sec  is  placed  shifter The  best  on  the  LERST SOURRE FIT  Ei  F i g . 4.2.  l  U~  S  Si  S  Least squares f i t . the one of system B.  7L  Til  5Ti  !l27  Output of system A i s f i t t e d to  !l  LERST SOUARE FIT DIFFERENCE YC-VF= YD  0.081 VOLTS  -o.oaJ  VOLTS  -0.42J 2  Q  ]  _ OUTPUT OF AIR CORE CDIL= X  VOLTS  -L.B7.  S  F i g . 4.3.  TA  ST  ^2  11  71  H5  5i  ~113  Least squares f i t . output of system A i s f i t t e d the one of system C.  127  S E C  to  \  •  •  n  System B Kn  System C Kn x 2  N  Kn x 2  Kn  1  0.14154  0.28508  2  (0.01532)  (0.05523)  3  (-0.00340)  (-0.02720)  0..01111  0.03838  k  -O.OO56I  -O.O8976  (0.00157)  (0.02512)  5  (O.C0195)  (0.. 06240)  0.36730  0.18365 (-0.00733)  (-0.00097)  N  (-0.02932)  (-0.05104) *  Table 4.1. C o e f f i c i e n t s of the l e a s t sguares f i t . A i r core coil s i g n a l i s f i t t e a to permalloy core c o i l s i g n a l . Bracketted values are not s i g n i f i c a n t .  29  record lag  of  system  i s equivalent  B with respect t o about  Hz.  I n case o f system  59.4  degree The  t o that  9.9 d e g r e e s l e a d  C, i t was 3.3 s e c  results  maximum  listed  are  shown  fit  amount  value  o f -^Cw2 . A  appears  t o the s i g n a l  system  C  o f system  signal.  one by s y s t e m in  recorder  l a g , which  These  This  o f 0.05  is  about  i s 0.02 v o l t s  still  a  substantial  part  T a b l e 4.1.  order c o e f f i c i e n t order  The  i n the to the  o f the magnitude o f l e a s t B and t h e  by t h e n o i s e o f t h e t a p e  recorder  The  less  contribute  by s y s t e m  noise  rms, s o t h a t  with p r o b a b i l i t y  4.1, s o t h a t  -£CM- on  between t h e r e s u l t s  experiment.  k  As t h e  maximum c o n t r i b u t i o n s a r e  B, and f o r t h e t h i r d  C may be c a u s e d  this  -fc*x  terms  The i n c o n s i s t e n c y  occurs  Table  the  f o r the fourth  squares f i tc o e f f i c i e n t s  used  f o r a signal  on F i g . 4.2 and F i g . 4.3.  t o g e t h e r with t h e c o e f f i c i e n t s  largest  A.  l a g f o r 0.05 Hz.  maximum a m p l i t u d e o f X i s 2 v o l t s , the  of t h e system  level  noise  of  t h e FM  h i g h e r t h a n 0.06  t h a n 0.12 %.  But  such  tape volts  noise  is  o f many v a l u e s i n c o l u m n s 3 and 5 o f  many o f t h e c o e f f i c i e n t s  a r e masked  by  tape  noise.  4.2.  AMPLITUDE A  straightforward  made b y t a k i n g the  comparison  the r a t i o  a i r core c o i l  readings. they  RATIO  signal.  They a r e f r e e  are less  o f each  affected  ratios.  Data  disturbed  day r e c o r d ,  of  the s i g n a l  permalloy core c o i l  I t i s appropriate o f dc o f f s e t  signal  to  t o use peak-to-peak  and phase  by n o i s e b e c a u s e  commencing  amplitude i s  shift.  of r e l a t i v e l y  Besides, h i g h e r S/N  on 21h 42m 06s UT was t a k e n f r o m t h e  October  10, 1973.  30  To o b t a i n peak-to-peak r e a d i n g s , of  maximum and minimum values  the s i g n a l waveform were c o l l e c t e d .  Then,  d i f f e r e n c e between  a s u c c e s i v e p a i r s o f v a l u e s was o b t a i n e d which g i v e s a peak  reading.  One  maximum/minimum  calculating  two  Conseguently,  they are o p p o s i t e i n p o l a r i t y .  Readings  of n o i s e ,  are then  in  used  the  for  series.  mostly s p o r a d i c p u l s e s caused by a m p l i f i e r s ,  4.4 and F i g . 4.5 show the r a t i o s of  permalloy  core c o i l  affects  small  the  readings  systems to the peak-to-peak  the a i r core c o i l system s i g n a l . system  values  is  delated.  Fig. the  peak-to-peak  reading  peak-to-  reading of  Apparently the n o i s e of  peak-to-peak  values.  every  P l o t s f c r smaller  v a l u e s are more s c a t t e r e d .  S t u d i e s of noise c h a r a c t e r i s t i c s  well  of the r a t i o of two normal  as  the  probability  enable us to draw the d i s t r i b u t i o n  range of  by  ratio  as  variables  values  with  r e s p e c t to the value af the output of system A.  4.2.1.  RELATIVE SENSITIVITY AND RATIO OF TWO NORMAL VARIAELES  Let signal,  n1 Y1.  where x i s  be the n o i s e i n a peak-to-peak r e a d i n g on system A Then,  the  corresponding  we have  J/» =  Kl  signal,  the  peak-to-peak  (4.2.1-1) true reading  peak-to-peak  reading.  The  on system B c h a n n e l ,  Y2 i s  given by (4.2.1-2)  31  where S i s the g a i n o f system B r e l a t i v e t o system A.  The r a t i o  is •X  (4.2.1-3)  I f x i s much l a r g e r than n1,  (4. 2.1-4) so  that  XCR-S^ «  n-i- H.iS  I f n o i s e o f each c h a n n e l distribution,  the g u a n t i t y  normal d i s t r i b u t i o n ,  (4.2.1-5)  i s assumed x (R-S),  follows  2  standard  a  normal  approximately a  N (0 ,07* • ( C\ S) 2 ) ; the average v a l u e i s zero  and the v a r i a n c e i s equal t o 0~i + ( rr^ S) 2 the  to f o l l o w  deviations  of  n1  where and  CJj" and r/^ n2,  are  respectively.  Consequently, the mean v a l u e becomes  <X(RS)>  V>0  (4.2.1-6)  or  <XR>  *.SX>S  (4.2.1-7)  That i s ,  <X>  (4.2.1-8)  Thus, an approximate v a l u e o f the r e l a t i v e obtained.  The s e n s i t i v i t y  check the  knowledge  of  this  For system C,  i t i s 0.1838.  s e n s i t i v i t y makes i t p o s s i b l e t o  the assumption on the c h a r a c t e r i s t i c s  of  ncise,  because  quantity,  ^-5^=^-5^11 must  can be  of system B r e l a t i v e t o the one o f  system A was found t o be 0.1438. Moreover,  sensitivity  follow  a  normal  distribution,  (4.2.1-9) N (0 , <fe 2+ (<7T S) 2 ) , i f t h e  Fig.  4.4.  R a t i o of s y s t e m B o u t p u t t o s y s t e m A output. Ratio i s p l o t t e d v e r s u s system A o u t p u t . The i n s i d e p a i r of distribution limits correspond to one standard d e v i a t i o n , 68.3%. The outside ones are twice the standard deviation, 95.5%,  2.00 _ STRN0RRO DEV1RTION OF NOISE SYSTEM Ri SYSTEM C;  1.75  0.27 3.81  MILtIGRMMR-HZ HILLIGRMKR-HZ  1.50  1.25 J  a  1.00 J  0.75 J  0.50  25 Fig.  PERK TO PERK REflDlNGCMILLIGRMHR-HZ)  4.5.  130  145  160  ITS  Ratio of system C output to system A output. Ratio i s p l o t t e d versus system A output. The i n s i d e p a i r of distribution limits correspond to one standard' deviation, 68.3X. The o u t s i d e ones are twice the s t a n d a r d d e v i a t i o n , 95.5?.  VM  34  noise  on  4.2.2  each channel  follows  NOISE ANALYSIS ON As  defined  differences  by  the  between  the  two  channels  The  p r o b a b i l i t y , y,  normal d i s t r i b u t i o n .  PROBABILITY  left the  are  a  GRAPH  hand s i d e  of  corresponding  taken that  and  plotted  a difference  the  eguation  peak-to-peak on  be  a  4.2.1-9,  readings  probability  l e s s than  on  graph.  x i s given  as  follows.  % Providing  that  distribution is,  )/27C the  with  Ntm,^^).  V  (4.2.2-1)  differences  mean, m,  The  and  form  a sample  a standard  p r o b a b i l i t y , y,  ensemble  deviation,  i s rewritten  as  of  CT  normal ;  that  follows.  where  t^CZ-^)/<T The of  range of classes,  each  class.  probability  sample  and  the  (4.2.2-2)  v a l u e s x,  i s divided  cumulative  For  a  should  be  large close  into regular  probability, y i , is number  to  that  of of  samples,  intervals  obtained the  on  cumulative  a normal d i s t r i b u t i o n .  i h ' - T ^ r /  (  , ,  2  .  3  )  —  As  the  p r o b a b i l i t i e s , y i ' s , are  where t i s r e l a t e d tc The  plots  follows  a  should  to =  x  plotted  along  the  axis  t  linearly;  CXt-r*L)/<T be  of  a  (4.2.2-4)  straight  normal d i s t r i b u t i o n .  The  line  i f the  reading  at  sample t=0  ensemble or  y=0.5  35  gives is  t h e mean g i v e s t h e mean v a l u e ,  o b t a i n e d by  the f o l l o w i n g  m.  The  standard  deviation  relation,  ti-%J^CX.C-Xj)/q-  (4.2.2-5)  CZC -X-p/CVt—lj)  (4.2.2-6)  or <T= The  standard deviation  and In  system  of noise  B i s f o u n d t o be  the case of comparison  be 3.81  milligamma-Hz  4.2.3.  f o r comparison  13.59  between s y s t e m  milligamma-Hz  between s y s t e m  A and  from C,  Fig .  A 4.6.  i t i s found  to  as shown i n F i g . 4.7.  PROBABILITY DENSITY FUNCTION OF  A RATIO OF  TWO  NORMAL  VARIABLES The normal  probability  variables  Let variables standard  a and  b  which  was  density  be  constant  follow  normal  p  deviations  and  or  the  given  (X  and  also  the  ratio  by G e a r y y  distributions , respectively.  and  _  two  (1935).  x  be  random  w i t h z e r o mean and The  ratio  A + X  value  (1.2.3-1)  y a r e i n d e p e n d e n t of each probability  of  other,  the  joint  probability,  s i m u l t a n e o u s o c c u r r e n c e o f x and  by  y, i s  .  ' Changing  of  , X  x and  for  o b t a i n e d , f o r example,  becomes  if  function  the v a r i a b l e s  (x,y) t o  (x,z) by 4.2.3-1, we  (4.2.3-2)  get  T=  21.BB 19.69 37.50 15.31 13.12 JO.93 8.74 •  6.55 4.36 2.17  £  p  -0.01 -2.20  g-4.39  _j  d -6.58 sr. -8.77  J  -10.96  J  -13.15 -15.34 -17.53  J  -19.72  J  -21.91  001  F i g . 4.6.  0.01  O'.OS O'.l 0'.2 0.3 0.4 0.5 0.6 0'.7 O'.B CUMULATIVE PROBABILITY  0.95  0.99  Normal d i s t r i b u t i o n of the difference between signals of system A and B. Mean value i s found at t=0. Standard deviation appears as the slope of the l i n e .  0.999  F i g . 4.7.  Normal d i s t r i b u t i o n of the difference between signals of system A and C. Mean value i s found at t=0. Standard deviation appears as the slope of the l i n e .  38  2.7CC*£ with  y=(a+x)z-b and  probability  ^  where  function  r  |a+x|  is  0*"*  a  (4.2.3-3)  positive  value.  The  P ( z ) becomes  (4.2.3-4) with  y=(a+x)z-b.  second  integrals  Let  on t h e r i g h t =  The i n t e g r a t i o n  so  Q (z)  and  B(z) r e p r e s e n t  hand s i d e  of t h i s  the f i r s t  equation.  +  (4.2.3-5)  i n Q(z) c a n be p e r f o r m e d  through  a transform  (4.2.3-7)  P  Then  ^7  C\<ltht* the  of  that  \4d  Q(z)  and  becomes a n o r m a l d i s t r i b u t i o n  VpW/  function  (4.2.3-8)  by t h e t r a n s f o r m  of  variable  V^TD Practically, probability  Q (z)  density  i^i-Ct^  ' becomes  function,  a  P (z).  good From  (4.2.3-9) approximation 4.2.3-5,  of  the  39  -tf The  left  -V  (4.2.3-10)  1  hand s i d e o f t h e e q u a t i o n  i s unity.  (4. 2.3-1 1) That  i s the error,  , by e q u a t i n g  Q(z) t o P ( z )  is  ,p0 (4.2.3-12)  s hd In  other  from It  words,  , i s the p r o b a b i l i t y  the average i n a b s o l u t e  will  be n o t e d  negative it  £  value  that  value  ^-/z i  o f a + x.  s  of f i n d i n g  g r e a t e r than  the  a deviation  or egual to  probability  cf  S i n c e R (z) i s a p o s i t i v e  finding  a  f u n c t i o n o f z,  follows that  (4.2.3-13)  z  f o r a l l v a l u e s o f z1 and z 2 . finding  a value  equation  I f P (z) d e n o t e s t h e p r o b a b i l i t y o f  o f z between z1 and  z2,  i t follows  from  the  4.2.3-10 t h a t  (4.2.3-14) Even  i f  greater as  Ct/cZ  than  equal  ,  1/3,  to  the £  coefficient  i s very  Q(z1,z2).  of  variation  m i n u t e and P ( z 1 , z 2 )  In the particular  case  c f a+x, i s n o t  can of  be  taken  ^#=1/3,  £=0.0027. The  expression  4.2.3-8  shows  with  zero  normally  distributed  Therefore,  | t |<1 g i v e s t h e r a n g e  that  the  mean . and  variable, t , i s unit  variance.  o f 68.3% f o r t h e r a t i o  value  40  distribution.  | t  |<2  g i v e s t h e r a n g e o f 95.5%.  distribution  limits,  the standard  d e v i a t i o n of noise of system  from  records  of  the r e s u l t s  o f 4.2.1  geomagnetically  To draw  these  a r e used  together  A which  is  guiet  days.  with  estimated  It  is  0.27  of  the  milligamma-Hz.  4.3.  E F F E C T I V E PERMEABILITY In  the  permalloy system  section  core  were  4.2.1  coil  of  relative  systems with  obtained.  permeability  OF PERMALLOY  each  With of  CORE  sensitivities  respect tp the a i r core  these  values,  the permalloy  the  coil  effective  cores i s calculated  as  follows. From s e c t i o n be  0.0235  3.2, o u t p u t  volts  per  sinusoidal  milligamma-Hz.  Then, output  with  relative  sensitivity,  total  gain of the  estimate  The  relation  The  amplifier  of the output  3.2-6  constants  permeability  of the a i r core  geomagnetic  of a  permalloy  S, i s 0.0235xS of  this  voltage of the sensor  in  of system  Table  3.1  B i s obtained  system  core  coil  volts. G,  coil  used.  a s 152 w h i l e  cf 1  system  Knowing t h e we  get  an  as  p e r m e a b i l i t y such  are  will  variations  system,  g i v e s the e f f e c t i v e  given  coil  The  as  effective  t h a t of system  C as 146. According expected is  50.  true  to  to  F i g . 2.1,  the  be 800 f o r t h e c o r e  The a p p a r e n t permeability  apparent whose  length to diameter  p e r m e a b i l i t y o f 150 c a n is  150,  but  this  permeability  value  be  obtained  is ratio when  i s extraordinarily  41  small  f o r a ferromagnetic  this  low a p p a r e n t  sensor It  response  i s  biased  in  fluctuation  amplitude  this  of the earth's  of magnetic  records  starting their  at  power  magnetic  field  field  the  where core.  response f o r a i s subjected to  linear  permeability within  a small  fluctuation.  SPECTRA  by t h e t h r e e  systems,  282  21h 42m 06s UT on O c t o b e r spectra  region  incremental  may a p p e a r  Besides,  of  t h e dynamic  namely t h e  I t s response  COMPARISON OF POWER The  i s i n the nonlinear  case t h a t  permeability,  (Bozorth,1951).  4.4  permeability  such as permalloy.  d e p e n d s more on t h e p e r m e a b i l i t y  possible  much s m a l l e r  material  were  obtained  sec  long  duration  10, 1973, were t a k e n and  f o r another  interchannel  comparison. The  conventional  maximum  likelihood  spectrum taken  o f each.  first  Blackman-Tukey  method  were employed  The maximum  with  spectral  out  estimators  points  (Lacoss  1971).  power  estimate  is  its  spectral  maximum spectral the  given  estimate  estimator  window s h a p e  of  relatively  i s  f o r the  reproducing  Various  likelihood lengths of  method  until  t o t h e one g i v e n  by t h e  Then a c o n f i d e n c e  as w e l l as the bandwidth  and t h e l e n g t h  t h e power  number o f d a t a  maximum  t h e Blackman-Tukey  becomes c l o s e s t  method.  the  as the  estimator  for  small  i n the appendix. with  well  a reference  excellent  A d e r i v a t i o n of  were t r i e d  likelihood  power  an i n t e n t i o n o f o b t a i n i n g because i t i s  as  f o r estimating  likelihood  B l a c k m a n - T u k e y method  autocorrelation  method  interval  f o r the  was c a l c u l a t e d from  o f t h e a u t o c o r r e l a t i c n used.  42  4.4.1.  CONFIDENCE INTERVAL  According estimated  to the  by  taking This  from  length  sampled  truncated  and/or  window,  or  where M i s t h e weighting direct  defined  of  weighted  i s given  I  O  length  of  Fourier  with  in lag  The  THE  TUKEY WINDOW  a  is  series window  the  calculated  directly  window  form  to smoothing of  for  bandwidth  of  the  then  i t is  The  Tukey  weighting  the  as  (4.4.  > M  autocorrelation.  the  data  and  function.  here  raw  of  this  are  of  data  used  truncated  spectra  transform  IUI  transform  filter.  Fourier  time  bell,  i s equivalent  bandpass  the  autocorrelation  cosine  autocorrelation  BANDWIDTH OF  B l a c k m a n - T u k e y method, power  autocorrelation. a  AND  spectra,  1-1)  Sthis  which i s a  time domain, through s p e c t r a l window,  b,  a is  as /  /,  = Iw  (4.4.1-2)  H v ) A » = ^ Q + ^ i c v f  M E A T S '  2-3)  'A  That  i s , the  bandwidth  i s r e l a t e d to  £>~C^7£~/~/) The  smoothed  windowing  sample  K  by  ~'  spectra  (4.4.1-4) C (f) e s t i m a t e d  through  the  atove  is  (4.4.1-5) where R (u)  i s the  a u t o c o r r e l a t i o n and  T is  the  length  of  the  43  time s e r i e s  data.  In the frequency  ,  where  WCft If  P (f)  varies  Using  may  theorem,  be  the  variance  (4.4.1-8)  o f smoothed  spectral  ZiO  (4.4.1-9)  lV*CH)dU= /fr  (4.4.1-10)  ^ 1 = /  this  l  mean and v a r i a n c e ,  v a r i a b l e , CK ^  Z  ,  which  degrees of freedom, l > = -2  4 other  V  follows  chi-squared  distribution  \  of  i n the  with  (4. 4.1-11)  E[cCpJ/\j the  as a  random  .  _  random  variable",  range of  chi-square  \J  distribution  (^.4.1-12)  \J  with d e g r e e s of freedom,  100(1-CV)% d i s t r i b u t i o n table  C (f) c a n be a p p r o x i m a t e d t o  (B ICCpVforiCCp\  words,  distributed  ^\)(^")  then  described.  That i s ,  the  t o t h e b a n d w i d t h , b,  Kfif-wtpdz * Pep  '  In  (4.4.1-7)  smoothly r e l a t i v e  Parseval's  estimator  r  &  sleep]-  With  domain,  is  V ,  . we  which g i v e s  Tc f i n d  the  refer  to  —  and  relation. CX f^f)  where  ft  { ^  Then c o n f i d e n c e  < ^  (J*->  interval  (4.4.1-13)  _ CX  ] =  f "  f o r P (f)  becomes  uu  In  logarithmic scale,  (a.a.1-15) Therefore, limit  and  which  from log  the  value  V^v£"§f")  true  probability, confidence  the  100  and  of l o g C ( f ) ,  i  value  st  higher  e  of  (l-cx/)  the  h  the  of  limit  of the  spectrum  These  degrees  b^cfvO-^r)  log  limits  freedom  i s the interval  P (f),  could  depend  only  which a r e  same  M  on  the  to M  by  (a.a.1-16)  c o n t r o l s the  RESULTS OF  Table Spectrum  a.2  bandwidth  the  SPECTRAL  show t h e  patterns are  These spectrum  and  confidence  level  at  the  ones  ANALYSIS  numerical  peaks o c c u r  obtained  by  the  their  peak. coil within  The  low  difference  system 96%  level,  and  each  confidence  lower  results  F i g . a.8,  g i v e n on  p e a k s a t 0.110Hz, 0.138Hz and of  with  time.  4.4.2.  to  in  fall  related  x  Therefore  lower  in a  the  F i g . a.9  position  maximum  than  -20db,  permalloy  intervals.  to  with  be  F i g . a.  method. noise  coil  the  10. well  Three because  r e s p e c t t c the  between core  and  analysis.  corresponding  likelihood  0.166Hz seem  i n power s p e c t r u m of  of s p e c t r a l  air  main core  systems i s w e l l  F i g . 4.8.  Maximum l i k e l i h o o d power estimate of system A record of October 10, 1973. (see F i g . 4.1.)  o'.OO  Fig.  0\02  4.9.  OTM  G\06 o!u8 FREQUENCY(HZ)  O'.IO  JA2  OYU  GLIB  CUB  Difference in power s p e c t r a of system B r e c o r d by Blackman-Tukey method. The h o r i z o n t a l cross line shows the bandwidth, and the v e r t i c a l ene i s the 96% confidence interval. Also, the center pcint corresponds to zero d i f f e r e n c e of power s p e c t r a by the scale on the right hand s i d e . D i f f e r e n c e of pewer s p e c t r a i s taken a f t e r each spectrum i s normalized to i t s peak.  5.20  CLOD  CL02  CUM  o'.06  CUOB  (TH)  cTTlZ  OLH  O'.16  cTTlB  0.20  FREQUENCY IHZ)  Fig.  4.10.  Difference in power s p e c t r a of system C r e c o r d by Blackman-Tukey method. The h o r i z o n t a l cross line shows the bandwidth, and the v e r t i c a l one i s the 96% confidence interval. Also, the center point corresponds to zero d i f f e r e n c e of power s p e c t r a by the scale on the right hand s i d e . D i f f e r e n c e of power s p e c t r a i s taken a f t e r each spectrum i s normalized to i t s peak.  D i f f e r e n c e from P e r m a l l o y  Peaks i n the power spectrum of  the  a i r core c o i l  Maximum l i k e l i h o o d db  Hz  • 0.0106  -0.2053  0.0106  0.0566  0.191  -0.^235 0.0*  C.0566  -2.856  —  0.110  -  System C  db  db  -0.9582  -0.2095  db  0.0518  0.0*  System B  Blackman- -Tukey  Hz  O.OJ518  core c o i l s i £nal  signal  0.0*  0.0*  -0.9299  0.1953  -22.9825  1.2^36  0.6705  0.138  -25.9197  2.286*f  0.571^  —  0.166  -27.3723  -O.O956  0.765^  -28.697  0.191  -29.3180  0.0206  0.66^1  -2.7992  Table 4.2.  Comparison o f power spectra. Each spectra normalized to i t s peak. That i s , peaks with a r b i t r a r i l y s e t to O.Odb.  are are  49  5.  SUMMARY AND In  the  CONCLUDING REMARKS  region  correlations  stations  with  data  to  study  the  magnetic  high  to  between g e o m a g n e t i c  various the  of  provide  along  the  behavior  mid  important  covers  micropulsation ferromagnetic  the  plasma  the  For  core  coil  this  the  the  large  ferromagnetic Three  open core  geomagnetic  systems,  one  using  permalloy  core  coil  September  20,  10,  1973  1973.  The  coil  system.  be  a  permanent  The  specifically, core  coil coil  system,  was  found  plots  are  it  is  still  w i t h i n the  The  the  more s c a t t e r e d a s  on  the  coil  coil  the a  sensor  theories  the  was  of  two  by  each  i n the the  the  data  sensors compared  sensors  coil  role  other  of  of  designed  since October no  to the a i r designed  f o r use  at  signals,  or  of  the  permalloy  value  of the a i r  c f the  becomes p o o r e r , based  estimated  two  produce  were  vicinity  S/N  predicted limits was  of  to  and  corresponding  to f a l l  power s p e c t r u m  of  simultaneously  signals  core  a peak-to-peak r e a d i n g  The  noise.  output  ratio  s y s t e m s d i v i d e d by  choice  coil,  sensor  core  a i r core  The  1.  of  permalloy  p o r t a b l e whereas t h e station.  coil  analysis carried  permalloy  band  detection/recording  have been r u n  i n the  induction  from s p e c u l a t i o n s .  a i r core  sensors,  distortion  to  core  an  The  electro  observation,  or a i r core  free  interesting  contrary  micropulsation  r e v e a l s t h a t the  significant core  i s not  field  at  example,  w e l l as t h e  i s a practical  loop c o i l ,  is  frequency  But,  obtained For  boundary.  the  because of i t s p h y s i c a l c o n s t a n t s . about  i t  pause as  entire  signal. iron  information.  geomagnetic meridian  of  latitude,  m i c r o p u l s a t i o n data  phenomena a s s o c i a t e d w i t h  magnetometer  geomagnetic  on an  for  more  value but  estimation  each  of  the  50  three  by B l a c k m a n - T u k e y method  method.  No  significant  a s v w e l l a s t h e maximum  d i f f e r e n c e was r e c o g n i z e d  likelihood among  these  spectra. The e x p l a n a t i o n response the  inherent  also  sensitive  effect.  suggests  i f  permeability  i t of  is the  suppress the nonlinear Although, effect, account  other  a  longer. core  As  of  the  the  core  core  of  considerable  reduction 2 ) .  by  demagnetization coil  becomes  i s required  placed  on  i s more longer,  t c be h i g h e r  ferromagnetism  the should  f o r the i n t e r p r e t a t i o n of the a c t u a l core  appendix  the nonlinear  material i s suppressed  ferromagnetic  material  the emphasis i s effects  be t h a t  to  response.  permeability,  records  may  The t h e o r y  that  incremental  (see  results  t o any f e r r o m a g n e t i c  demagnetization  effect  f o r these  for  example,  o f the e f f e c t i v e  be t a k e n  into  response.  The  seems  to  permeability  Also, further analyses  of b e t t e r q u a l i t y , e s p e c i a l l y  demagnetization  should  better S/N.  explain  of the core  be done on t h e  51  APPENDICES  1.  MAXIMUM The  LIKELIHOOD POWER  Blackman-Tukey  procedure  which  approximation data.  maximum  power  through  adaptive  electrical  above  input  concept,  this  infinite  applied  to the  a  different t o pass  the residual or  concept. the s i g n a l  power a t t h e of the  power e s t i m a t o r , i t i s  regarded  filter  process  i sa filter.  i s designed  Based  as f o l l o w s .  F o r an  and z e r o mean n o i s e p r o d u c e d  k i s a time index filter  A-1, this  retains  -ft*  AM  by a  and ^  i s the sampling  ,1-1, interval.  Output  the signal.  i s reduced to  VU=\ In  on  process.  this  With  i s  f o r an  optimization  of t h i s  of signal  c  of  on  minimizing,  analog  x^M ^ where  a  method.  which c o n s i s t s  random  which  i s designed  i s done f o r i n d i v i d u a l  as a d a t a  i s  length of data a better  calculated  i s based  process  this  estimation  artificial.  likelihood  estimation  power  a finite  function,  i s rather  Because  process,  the  from  of  b u t t o a t t e n u a t e and m i n i m i z e  output.  An  infers  weight  autocorrelation,  The  method  of the autocorrelation  The  The  ESTIMATOR  matrix notation,  CL=  column  CJ&C/-Aa./  (A- 3) vectors are defined as follows.  (A-4)  52  y * * ™ - )  m - a t a . e ' * * , A-3  becomes  The o t h e r  ( s  I= E*T^  .  5 )  (A-6)  c o n d i t i o n i s t h a t the variance o f the noise  is  t o be  minimized. The v a r i a n c e o f o u t p u t ,  y,  i s  <r*=<ty±-<^>f>  (A-7)  (A-9)  (A-10)  (A-11)  > (A-12) The c o r r e l a t i o n  i s defined as (A-13)  A-10  becomes  where  R i s N by N c o r r e l a t i o n  A-13  i s to  Lagrange's  be  minimized  multiplier  f-B Then,  with  method,  \  -I  matrix. under  the constraint  we f i r s t  A-U.  Taking  rewrite A - U .  --=o  <*-i6)  =• CK^ -f  JOijl  y  Sfl  (A-17)  53  where  In  &  i s an u n d e t e r m i n e d  matrix  multiplier.  notation,  =  (A-19)  The f i l t e r c o e f f i c i e n t  This  i s substituted  vector  becomes  i n t o a complementary e q u a t i o n  t o A-4, (A-21)  which  describes  the negative  c9  region.  So  that,  R~' E*  I= Eliminating  frequency  with  A-17,  '(»:»>  we g e t  (A-23) The power  estimator  by t h e maximum l i k e l i h o o d  Finally,  method i s  ^  (A-25)  2.  INCREMENTAL  PERMEABILITY  The o r d i n a r y increasing  or  h y s t e r e s i s curve  decreasisng  between  the s a t u r a t i o n  occurs  when m a g n e t i c  a  relatively  large  magnetization,  limits. field  bias  describes  The  the  monotonically  o r EC  magnetization  hysteresis  response  c o n s i s t s o f an a l t e r n a t i n g ' f i e l d field.  B o z o r t h (1951)  reviewd  also and this  phenomena. In  the  presense  of  a  relatively  large  bias  field,  the  54  alternating  field  produces loops  magnetic  minor  hystersis  tips.  In most cases the s l o p e of  induction  bounded to the major  which  loop at  each branch of  i s l e s s steep than the main l o o p .  fellows  ene c f  these  its  sublcops  These are shown i n  Fig.A-1.  10 XI6 3  (b)  (a) •4  i  •^SLOPE-AX  1  12  1 i  -Bn  2  10  io  «0 „v S L O P E ,  AH  «  /I -LB/c\H L  B ^,  -B  < O Z  8  ID  t  "B  r ffi 6  r  ®%  /l  Is  6  p  IRON  /l  IRON  0  u-  0^2  6  IO 3 X 6  0 IO X 16 3  4 - 7 9  (c)  MO  (d)  PERMALLOY  •  'p. *\Z,000  ML  Ba  1260  7000  T- 725  7600  *-  7850 6 000  r  *» 5  <fl 10  __|  IU  1 10 0  3<  5 « z O  3  200 SO  z  //  ID  s  IRON  / x / •  -0.04  0  Is  0.04  0-G3  0.12  0.18  2 3 * H IN OE.RSTEDS  0.20  H IN OERSTEDS  Fig.A-1.  the  field  magnetization.  permeability becomes  the  for  much  the e x i t a t i o n  less  than  asymptotic  value  of a  variational  the one t r a c e d fcy CC  The i n c r e m e n t a l p s r m e a b i l i t y , j J J  the average of each b r a n c h . is  e  Minor h y s t e r e s i s l o o p s taken under v a r i o u s c o n d i t i o n s i n i r o n ( a , b , d ) and 4-79 Permalloy ( c ) . (from E o z o r t h 1951)  Briefly, magnetic  s  The r e v e r s i b l e when the amplitude  , is  defined  as  p e r m e a b i l i t y , Jj,y , cf  the  alternating  55  field,  , is  increased  in  increased,  the i n c r e m e n t a l p e r m e a b i l i t y  a  relation  linear  F i g . A - 2 shows t h a t of 2.4x about  103, 5x 10 3  permeability iron,  Its  the  such as J^t~ J^f*  45 Permalloy has the i n i t i a l  nominal p e r m e a b i l i t y  for  a  field  becomes 1x 1 0 3 difference  of at  is  0.5  this  by  CC  oersted.  bias.  also  til'  ^  Hn  re  permeability  magnetization  is  The i n c r e m e n t a l  In the case  of  pure  between the i n c r e m e n t a l p e r m e a b i l i t y and  the one by DC m a g n e t i z a t i o n  appears even  more  pronounced  (see  F i g . A - 1 (d)).  BIASING FIELD S T R E N G T H , H , IN O E R S T E D S b  Fig.A-2.  It  Incremental and reversible permeabilities of Permalloy measured with v a r i o u s biasing f i e l d s incremental f i e l d strength. (from Bozorth 1951)  was  permeability  suggested is  by  determined by  GansH^IO) the  bias  that  the  induction,  45 and  reversible  By  •  The  56  departure  from  materials.  this  Fig.A-3  simplification shows t h e  i s not  plot  large  f o r 45  i n case  of  some  Permalloy.  o -  \  \  •  \  *\ \\ \\  )  i  5  X  •  «  2 Ol  A.  s  A  Fig.A-2.  But,  Reversible permeability (form B o z o r t h 1951)  genarally,  permeability  there  curves  characteristics The with  depend the  the  field. on  the  magnetic  that  for shown  in  they  magnetic The  ii)  always  vs  the  induction  for  in  45  Permalloy,  the  materials.  reversible  Four d i f f e r e n t  Fig.A-4. is a  measure  domains  of t h e  has  X  differences  m a g n e t i c domain  domain  IS  GAUSSES  different  composition  magnetization, so  wide  reversible permeability  which  biasing  are  are  IN  4  2  10  BIASING INOUCTION, Bo,  no  are  material. principal  or  the  the  in  i s discussed  d i r e c t i o n of  l i e parallel  held  of  p o s i t i o n by fcy  These  models  the  which  models a r e :  direction domains are  antiparallel  firmness  to  cf  i)  easy  restricted the  field,  57  (a) M O L Y B D E N U M  -Ac  PERMALLOY  '  , -f  r  \ a,  H=0  10  I2110  3  O  IS  2 0  2 * X I 0 »  (d) 65 PERMALLOY, HARD f ^ k  H  =  0  10  BIASING INDUCTION, Bfc,, IN  12«K>>  CAUSSES  Fig„A-4.  Reversible permeability as dependent on biasing induction in (a) Molybdenum P e r m a l l o y , (b) Vanadium P e r m e n d u r , (c) P e r m a l l o y c o n t a i n i n g 74% n i c k e l and (d) hard P e r m a l l o y c o n t a i n i n g 65% n i c k e l . (from Eozorth 1951)  Fig.A-5.  Reduced reversible permeability vs r e d u c e d b i a s i n g induction f o r several materials, i n comparison with t h e t h e o r y : (a) G a n s ' r e l a t i o n , (b) i s o t r o p i c d o m a i n s , (c) anisotropic d o m a i n s (Drown). B r o k e n c u r v e shows t h e a p p r o x i m a t e c h a n g e i n d u / d l l ( f o r weak fields) as d e p e n d e n t on b i a s i n g i n d u c t i o n , o b s e r v e d f o r a v a r i e t y of m a t e r i a l s . ( f r o m B o z o r t h 1951)  58  iii)  they  lie  parallel  or  anti  parallel  crystallographic direction i n crystals random  i n the material.  by Brown(1938). reversible  that  a  single  oriented  at  The t h i r d model i s designed for cobalt  With these assumptions,  permeability  are  to  of  coincided by p l o t t i n g jULy/jJL  0  the d i f f e r e n c e s i n the  different  materials  -are further  Bj>/ B 5 , where y,0  against  i s the  i n i t i a l permeability and B$\ i s the induction at saturation, i n Fig.A-5.  An i n t e r e s t i n g case of superposition of f i e l d occurs when a weak  ^alternating f i e l d  field.  Fig.A-6  Molybdenum  shows  Permalloy  i s applied at r i g h t angles to a constant that  the  reversible  permeability  of  i s larger than the ordinary one except a t  the end points where these values must coincide.  e  **-JL T O B  b  j 6 ffi <  •Sf * a UJ  vii  UJ _J  TO  e  b  a  s  3  \  UJ •  >  2  t O O  Fig.A-6.  \ \ \  UJ _ OC  \  \  I 2 3 4 5 e 7 8 PRINCIPAL C O M P O N E N T C P INDUCTION, B b , IN G A U S S E S  9 x  E f f e c t o f b i a s i n g i n d u c t i o n on r e v e r s i b l e p e r m e a b i l i t y when i t i s measured p a r a l l e l o r p e r p e n d i c u l a r to the d i r e c t i o n o f uieasurement. (from E o z o r t h 1951)  59  REFERENCES  Bozorth,  R. H.  (1951) .  £-112 ma^ ne t i sra, New  York,  Van  Nostrand  Co. , pp. 968. Brown,  W.  F.  (1 9 3 8 ) .  stress  Domain  III.  theory  Reversible  of  ferromagnetics  susceptibility,  under  Physical  R a y i s w , 54, 279-299. Campbell,  W.  H.  (1967).  field  I n d u c t i o n loop antennas f o r geomagnetic  variation  measurement,  ESSA T e c h n i c a l  Report,  ERL 123-ESL 6. Caner,  B.  ( 1970) .  Design  considerations,  micropulsation  Gans,  R.  Observatory,  Unpublished  Observatory,  R.  (1910) .  R.  C. of  (1930).  J e n k i n s , G . M.  corresponding  variables,  states,  Magnetic  Geophysical  Fhvskalisch <  distribution J o u r n a l of  of the q u o t i e n t  Royal  Statistics  442-444.  and D. G. W a t t s ,  Its  for  B. C.  11 , 988-99 1.  two n o r m a l  coils  Victoria  report, Victoria  The f r e q u e n c y  S o c i e t y , 93,  at  R. 7 V i c t o r i a ,  Magnetic  Zei t s c h r i f t. Geary,  measurement  induction  A ogl i c a t i o n  t  5 an  (1 9 6 8 ) .  Spectral  Francisco,  Holden  Ana l y s i s Day  and  Inc.,  pp. 525. Kollar,  F.  and using  R.  D. R u s s e l l ,  an e l e c t r i c  (1966).  current  analog.  §i.i§!S2l23.i£S.£ S o c i e t y o f America, Lacoss,  R. T.  (197 1).  £§2Eiiy.§i£§» O s b o r n , J . A.  (1945).  Data  Seismometer  56,  adaptive s p e c t r a l  Bulletin  analysis of  the  1 193-1205. analysis  methods,  3 6 , 66 1-675. Demagnetizing  f a c t o r s . of  the  general  60  ellipsoid, Peterson,  E.  (1928).  materials Bell Stratton,  Physical  J . A.  Harmonic a t low  System T e c h n i c a l (1941).  67,  351-357.  production  frequency  and  Journal,  low 7,  Ilectrcmacjne t i c  London, M c G r a w - H i l l  i  Review,  Inc.,  pp.615.  in  ferromagnetic  flux  densities,  762-796. The or j ,  New  York  and  

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