UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The calibration of a portable induction magnetometer system Zambresky, Liana 1977

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1977_A6_7 Z34.pdf [ 5.63MB ]
Metadata
JSON: 831-1.0052942.json
JSON-LD: 831-1.0052942-ld.json
RDF/XML (Pretty): 831-1.0052942-rdf.xml
RDF/JSON: 831-1.0052942-rdf.json
Turtle: 831-1.0052942-turtle.txt
N-Triples: 831-1.0052942-rdf-ntriples.txt
Original Record: 831-1.0052942-source.json
Full Text
831-1.0052942-fulltext.txt
Citation
831-1.0052942.ris

Full Text

THE CALIBRATION OF A PORTABLE  INDUCTION  MAGNETOMETER SYSTEM  by  L i a n a Zambresky B.Sc,  U n i v e r s i t y of Redlands, 1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Geophysics and Astronomy  We a c c e p t t h i s  t h e s i s as conforming t o the  required standard  The  U n i v e r s i t y of B r i t i s h June  ©  Columbia  1977  L i a n a Fran ces> Zamb r e s k y , 1977  In p r e s e n t i n g  this  an a d v a n c e d d e g r e e the L i b r a r y I  further  for  of  this  written  at  agree  for  British  by  for  gain  of Columbia  shall  the  requirements  Columbia, reference  copying of  I agree and this  that  not  copying  or  for that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  of  for extensive  p u r p o s e s may be g r a n t e d It  fulfilment of  available  permission.  Department  Date  freely  that permission  representatives. thesis  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t  scholarly  by h i s  thesis  or  publication  be a l l o w e d w i t h o u t  my  ABSTRACT  An i n v e s t i g a t i o n made c o n c e r n i n g the c h a r a c t e r i s t i c s o f a sensor c o i l  f o r an i n d u c t i o n magnetometer shows t h a t i t i s f e a s i b l e  to make the f i r s t stage 60 H e r t z r e j e c t i o n f i l t e r of the B u t t e r worth type.  T h i s i s an improvement i n d e s i g n over the Twinr-T  f i l t e r which i s sometimes used as a f i r s t stage f i l t e r as  the  number of e l e c t r i c a l components i s reduced and t h e r e i s no sibility  o f r i n g i n g between the c o i l  i n d u c t o r and the  pos-  filter  capacitors. Two  methods of r e l a t i v e  c a l i b r a t i o n f o r the i n d u c t i o n magne-  tometer system g i v e r e l i a b l e response  curves.  Wheatstone b r i d g e .  arm o f the b r i d g e and i t  The sensor i s one  i s shown t h a t the e f f e c t of if  the c o i l was  dary c o i l .  The  the s e n s o r by a f i e l d  ted  of  c r e a t e d by a s m a l l secongood.  to the a b s o l u t e c a l i b r a t i o n i s suc-  c a r r i e d out by comparing the output from the u n c a l i b r a -  system  brated.  a  The second method  agreement between the two methods i s  An e x p e r i m e n t a l approach cessfully  method uses  the s i g n a l g e n e r a t o r i s the same as  e x c i t e d by a n a t u r a l event.  involves exciting  One  to an a i r core system which has been p r e v i o u s l y c a l i -  A t h e o r e t i c a l approach  the s e n s i t i v i t y  i s used  of the sensor c o i l .  dent p r i m a r i l y upon the t u r n number and  to g i v e a good i n d i c a t i o n The s e n s i t i v i t y i s depenthe l e n g t h o f the  coil.  iii  TABLE OF CONTENTS  ABSTRACT  Ii  LIST OF FIGURES  v  LIST OF TABLES  viii  ACKNOWLEDGEMENTS  ix  CHAPTER I  GENERAL INTRODUCTION  1  CHAPTER I I - 1  THE 60 HERTZ REJECTION PROBLEM  6  II-l  Introduction  6  II-2  Theory o f Twin-T f i l t e r  7  II-2.1  Computer R e s u l t s  II-3  Theory o f B u t t e r w o r t h  I I - 3.1  Laboratory Results  22  THEORY OF THE BRIDGE METHOD  25  III- l  Introduction  25  III-2  An I n t u i t i v e Approach Using the  CHAPTER I I I  Norton  15 filter  Equivalent  17  26  III-2.1  Theory a t H i g h e r F r e q u e n c i e s  29  III-3  Theory o f the B r i d g e Method  33  III-3.1  Case of O b s e r v a t i o n  37  III-3.2  Case of C a l i b r a t i o n  37  III-3.3  Computer and L a b o r a t o r y R e s u l t s  40  III-4  The Case o f O b s e r v a t i o n w i t h and Without the B r i d g e  46  iv.  CHAPTER IV  THEORY OF THE SECONDARY COIL METHOD  52  IV-1  Introduction  52  IV-2  Theory  of Operation  53  IV- 3  Theory  of C a l i b r a t i o n  56  CHAPTER V  THE ABSOLUTE CALIBRATION  66  V- l  A L a b o r a t o r y Approach  66  V-2  A T h e o r e t i c a l Approach to the A b s o l u t e Sensitivity  74  CHAPTER VI  SUMMARY AND CONCLUDING REMARKS  81  APPENDIX 1  Methods o f Determining the Inductance and C a p a c i t a n c e of a C o i l w i t h F i n i t e Resistance  83  The T r a n s f e r F u n c t i o n o f the A m p l i f i e r System  88"  The T r a n s f e r F u n c t i o n f o r the T h i r d Order Low Pass F i l t e r  91  APPENDIX 2  APPENDIX 3  APPENMxRIiESEFCSS COT^eMbstolute C a l i b r a t i o n A c c o r d i n g to the B r i d g e Method  -- 'J 93  LIST OF REFERENCES CONSULTED  95.  V  List  of Figures  Chapter I Fig.  1-1  Equivalent c i r c u i t  o f sensor c o i l  Fig.  2.2-1  The symmetric Twin-T f i l t e r  Fig.  2.2-2  B l o c k diagram network  2  Chapter I I  Fig.  2.2.1-1  7  o f sensor c o i l / T w i n - T 12  Computer s i m u l a t i o n o f the frequency response o f the c o i l / T w i n - T s e n s i n g system  16 17  Fig.  2.3-1  An analog B u t t e r w o r t h  Fig.  2.3-2  The t r a n s f e r f u n c t i o n f o r the B u t t e r worth f i l t e r  20  L a b o r a t o r y set-up f o r the B u t t e r w o r t h filter  22  B u t t e r w o r t h f i l t e r c h a r a c t e r i s t i c s of the sensor c o i l system  24  Fig.  Fig.  2.3.1-1  2.3.1-2  filter  Chapter I I I Fig.  3.2-1  The Wheats tone b r i d g e  26  Fig.  3.2-2  Norton e q u i v a l e n t of the sensor c o i l  28  Fig.  3.2.1-1  Wheatstone b r i d g e a t h i g h e r f r e q u e n c i e s  29  Fig.  3.2.1-2  Norton e q u i v a l e n t o f the Wheatstone bridge  30  C i r c u i t used t o d e r i v e a new equivalent  30  Fig.  Fig.  3.2.1-3  3.3-1  Norton  The b r i d g e c i r c u i t w i t h parameters as they are d e f i n e d f o r c i r c u i t a n a l y s i s  34  VI  Fig.  3.3.3-1  Computer s i m u l a t i o n to determine the e f f e c t o f the c o n d i t i o n s R ^ » and R^>> R2 on the response o f the magnetometer system when u s i n g the b r i d g e method  41  Fig.  3.3.3-2  The l a b o r a t o r y b r i d g e c i r c u i t  43  Fig.  3.3.3-3  The n o r m a l i z e d amplitude response o f the magnetometer system a c c o r d i n g to the b r i d g e method  44  The phase response o f the magnetometer system a c c o r d i n g to the b r i d g e method  45  Diagram o f s e n s o r c o i l and l o a d impedance  47  The e s s e n t i a l elements o f the b r i d g e circuit  50  The e f f e c t of the r e s i s t a n c e R^ on the frequency response as o b t a i n e d from the b r i d g e method  51  The secondary c o i l method a t the time of o p e r a t i o n  53  The secondary c o i l method a t the time of c a l i b r a t i o n  56  The secondary c o i l method in>the oratory  60  Fig.  Fig.  Fig.  Fig.  3.3.3-4  3.4-1  3.4-^2  3.4-3  Chapter-jLV Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  4.2-1  4.3-1  4.3-2  4.3-3  4.3-4  4.3-5  lab-  The n o r m a l i z e d amplitude response o f the magnetometer system a c c o r d i n g to the secondary c o i l method  62  The phase response of the magnetometer system a c c o r d i n g to the secondary c o i l method  63  A comparison o f the frequency response curves o b t a i n e d i n the l a b o r a t o r y from the secondary c o i l method and the b r i d g e method  64  vii  Fig.  4.3-6  A comparison o f the phase response curves o b t a i n e d i n the l a b o r a t o r y from the secondary c o i l method and the b r i d g e method  65  Chapter V Fig.  5.1-1  C i r c u i t diagram f o r the a i r core system  69  Fig.  5.1-2  Computer s i m u l a t e d frequency of the a i r core c o i l system  71  Fig.  5.1-3  Method t o determine of a l a r g e c o i l  Fig.  5.1-4  The m i c r o p u l s a t i o n event used absolute c a l i b r a t i o n  Fig.  5.2-1  response  the i n n e r r a d i u s 72 f o r the 73  The dependence of the g e o m e t r i c p e r m e a b i l i t y M\ upon the r a t i o of the l e n g t h s of the s e m i p r i n c i p l e axes o f a p r o l a t e s p h e r o i d , a/b  80  Appendix 1 Fig.  A.1-1  Equivalent c i r c u i t of a sensor c o i l  83  Fig.  A.1-2  F i r s t method of d e t e r m i n i n g L  84  Fig.  A.1-3  Second method of d e t e r m i n i n g L  85  Fig.  A.1-4  The a n t i - r e s o n a n c e p o i n t  86  Fig.  A.2-1  Mu-metal core a m p l i f i e r system  89  Fig.  A.2-2  T r a n s f e r f u n c t i o n f o r f i l t e r s o f the Mu-metal core a m p l i f i e r system  90  The t h i r d order low pass  91  Appendix 2  Appendix 3 Fig.  A.3-1  filter  L i s t of Tables  Chapter  I  T a b l e 1-1  Coil  specifications  ix  ACKNOWLEDGEMENTS  I would l i k e  t o express my s i n c e r e g r a t i t u d e and a p p r e c i a t i o n  t o Dr. Tomiya Watanabe f o r h i s continuous h e l p and guidance as my research  a d v i s o r throughout the work p r e s e n t e d  i n this thesis.  I g r a t e f u l l y acknowledge Dr. R.D. R u s s e l l f o r many h e l p f u l d i s c u s s i o n s and i d e a s . I would l i k e T. Oguti  t o express many thanks t o Dr. K. Hayashi and Dr.  f o r t h e i r p a t i e n t guidance i n the e l e c t r o n i c s l a b o r a t o r y .  I would l i k e  t o acknowledge the V i c t o r i a G e o p h y s i c a l  Observa-  t o r y , Dept. of Energy, Mines and Resources f o r equipment which we borrowed. For  the o p e r a t i o n a t C h u r c h i l l , Manitoba, I would l i k e  Mr. C.R. B a r r e t t , s u p e r i n t e n d e n t  and h i s s t a f f t . a t the C h u r c h i l l Re-  s e a r c h Range o f NRC. T h i s r e s e a r c h r e c e i v e d f i n a n c i a l support grants  to thank  from the f o l l o w i n g  : NRC  A-3564  NRC  E-2923  DRB  9511-112  UBC  A r c t i c and A l p i n e Research Committee 65-0444  NRC UBC  D-6409 Summer Research  Scholarship  1  CH&PTEB I GENERAL INTRODUCTION  Recently, a p o r t a b l e i n d u c t i o n designed  by  Columbia.,  the  a  number  long of  could  be  term  intent  observing  Canada f o r m i c r o p u l s a t i o n system  system  was  aeronomy group at the U n i v e r s i t y of B r i t i s h  I t i s the  establish  magnetometer  at  commercial magnetometers.  a  this  stations  research.,  built  of  cost  It  group  a c r o s s northern  was  felt  that  significantly less  The reguirements  of t h i s  a r e such t h a t i t s h o u l d cover the freguency  to  a  than  instrument  band from  0.002  Hz  to  4 Hz and take the s i g n a l l e v e l from the order of milligamma  to  ten gammas.  core  The sensor was  constructed  with  a  Mu-metal  i n order t o reduce the p h y s i c a l s i z e from t h a t of an a i r  core sensor.  Mu-metal i s a high p e r m e a b i l i t y a l l o y of  and i r o n with a s m a l l amount o f chromium and (1975) demonstrated t h a t the  Mu-metal  nickel  molybdenum.  core  does  not  Ueda cause  s i g n i f i c a n t d i s t o r t i o n s or harmonics t o the s i g n a l c o n t r a r y to what some e a r l i e r i n v e s t i g a t o r s had The  e q u i v a l e n t c i r c u i t which i s used f o r the sensor  throughout capacitance  this  research  effect  of  is  the  shown  windings,  from g e n e r a l antenna use, Campbell capacitor of  the  current flux.  thought.  which  coil.,  The  in  Fig-  noted  (1969),  1-1.  i s p a r a l l e l to the r e s i s t a n c e and voltage  which i s induced  generator,  V,  The  as e a r l y as  gives r i s e  arises  i n the c o i l by the changing  coil  to  1910 the  inductance from  the  magnetic  2  It  is  difficult  capacitance  using  to  a  measure  bridge  the  coil  which i s designed  pure inductance and c a p a c i t a n c e . ,  For t h i s  and  f o r measuring  reason^  l a b o r i o u s methods must be put i n t o p r a c t i c e . such  inductance  somewhat  A c o m p i l a t i o n of A-1.,  methods used f o r t h i s r e s e a r c h i s given i n Appendix  These methods lend equivalent  a  circuit  great  amount  concept  of  because  credibility  the  coil  was  to  the  found  to  behave e x a c t l y as the c i r c u i t theory p r e d i c t s i n each method.  R  .  V  L  pAAA/W © — W W — |  •  lie Fig.  1-1  E g u i v a l e n t C i r c u i t of Sensor C o i l  Further support observations  that  f o r the e g u i v a l e n t the  p a r a l l e l r e s i s t o r and characteristics Butterworth theory. support  of  filter  output  of  capacitor a  of  Butterworth  i s developed  60 Hz r e j e c t i o n f i l t e r  sensor  occurs coil  appropriate filter.  The  from with a  values  has  theory of the  i n d e t a i l according to c i r c u i t  Laboratory r e s u l t s concur to F i g . 1-1..  the  circuit  with theory and  give  more  A u s e f u l improvement i n the design of a r e s u l t s from  this  analysis  which  is  d i s c u s s e d i n d e t a i l i n Chapter I I . The  primary  practicality  of  i n t e n t of t h i s t h e s i s i s to i n v e s t i g a t e the two  methods  of  relative  calibration  for  3  finding  the  freguency  magnetometer.  One  method i n which one sensor  coil  of  generated  i n the  magnetic  field  of  methods  the  these  arm  of a Hheatstone b r i d g e c i r c u i t i s  sensor  coil  is  induction  of  the magnetometer.  called  the bridge the  I t w i l l be shown t h a t  by  an  external  time-varying  The  other method, c a l l e d the secondary  i n which a s m a l l secondary  coil  c o i l i s wound on  core c o a x i a l l y with the sensor c o i l and c r e a t e s magnetic to  be detected by the sensor The  secondary  coil  e x t r a cable i s needed amplifier  system.  method has the disadvantage  between  In  prevent  the  secondary  noise generated  being mixed with t h e n a t u r a l s i g n a l . to  have one c a b l e i n s t e a d of two,  of  c o s t and convenience,  problem  calibration  makes  bridge  curve. core  not only from the  an  air  order  method  standpoint  possibility  of  more d e s i r a b l e as a  procedure,  the  level  of  the  relative  of  t o another system which i s a l r e a d y  core  coil  order  freguency  T h i s second system i s an a i r core magnetometer. of  amplifier  i . e . "cross-^talk**.  T h i s i s done by comparing the output system  the  Thus, i t would be b e t t e r  An a b s o l u t e c a l i b r a t i o n must be c a r r i e d out i n determine  and  by the e l e c t r o n i c s from  but a l s o because the  the  flux  t h a t an  100 yards i n  u n d e s i r a b l e c a b l e e f f e c t s would be reduced, This  coil  the f i e l d , the c o i l s and the  spurious  the  coil.  e l e c t r o n i c s are separated by approximately to  emf  can be simulated by d r i v i n g the bridge with a  s i g n a l generator.. method, i s one  response  the  to  response Mu—metal  calibrated. Sensitivity  can be c a l c u l a t e d e x a c t l y knowing the  4  geometry  of  the  coil  and  its  turn  number.  The  two  magnetometers are s e t up i n the f i e l d , approximately 100 yards apart, that  and such  record a  distance.  global  Some  m i c r o p u l s a t i o n s . The event  will  not  micropulsation  assumption change  events  are  i s made  over recorded  s i n u s o i d a l s i g n a l s and i t i s t h i s type of event which i s for  the  comparison.  Readings  can  this as used  be made over a number of  c y c l e s so that the e r r o r o f t h i s measurement i s s m a l l . Finally, a theoretical sensitivity are  significant.  dimensions, are  of  the  approach  is  taken  towards  a c o i l i n order t o e l u c i d a t e which The parameters gauge  of  interest  are  parameters the  coil  o f the wire out o f which the windings  made, and the p e r m e a b i l i t y o f the c o r e . , Such an  would be h e l p f u l when d e s i g n i n g new The  the  specifications  of  r e f e r e n c e d throughout t h i s  sensors.  a l l of  r e s e a r c h are given i n Table 1-1. thesis.  approach  the c o i l s used f o r t h i s  These s p e c i f i c a t i o n s w i l l  be  Turn Number  Air  Core  Coil  Mu-metal Core C o i l (1976)  Secondary  Coil  Calibration Mu-metal Core C o i l  L (H)  R (fi)  C  Inner Diam. (cm)  Outer Diam. (cm)  5000  5130  120  .50 y f  149.02  151.40  50,000  1831  930  .188 y f  3.17  7.30  20  .011  Coil  1000  11  (19 75)  50,000  2230  T a b l e 1-1  Coil  4.76  45\72  r  3.17  241.3  17.14  1050  Length (cm)  .048 n f  Specifications  4.00  9.5  35.56  6  CHAPTER I I THE  II-J  60-HERTZ REJECTION PROBLEM  Introduction  The  r e j e c t i o n of 60 Hz n o i s e i s a problem which  careful  consideration  designed are  for  a  magnetometer  to detect micropulsations.  carefully  At low  the  60  Hz  noise  by at l e a s t  a l l o w i n g the s i g n a l t o be a m p l i f i e d . may to  be  overloaded  and  method  freguency  has  at  been  60 Hz,  a f a c t o r of 100  the Twin-T f i l t e r  Hz  amplifier due  used  for  directly  some  a notch  previous filter  with  between the sensor  coil  discussion  which  response of the s i g n a l output  from the  T h i s i s not a d e s i r a b l e e f f e c t .  rejection  approach t o the f i r s t  problem t h a t i s b e t t e r f o r two  reason  i s t h a t the frequency  range  is  ensured  r e s i s t o r and matched  before  i s dependent upon the input impedance of  There i s a d i f f e r e n t 60  to  ratio.  I t w i l l be shown i n the  t h a t the freguency  amplifier.  which  Otherwise the  s i g n a l to n o i s e  which  the a m p l i f i e r .  follows  noise s i t e s  I t i s necessary  magnetometers i s t o p l a c e a Twin-T f i l t e r , center  being  or the m i c r o p u l s a t i o n i n f o r m a t i o n l o s t  the extremely low One  is  chosen i n the f i e l d , the n o i s e i s c o n s i d e r a b l y  reduced from what i t would be i n a c i t y . reduce  that  reguires  to  be  response flat.  c a p a c i t o r are needed  components  for  a  The  in  of  the  reasons.  One  micropulsation  other i s t h a t only  instead  Twin-T  the  stage  of  filter.  six This  one  carefully approach  7  requires  making  the  sensor  coil  and  combination i n t o an analog Butterworth  I1-2 Theory o f Twin-T  The circuit  filter diagram  of  interest  is  filter.  the  symmetric  Twin-T.  A  R  -A/WW  l  Fig.  RC  i s shown i n P i g . 2.2-1.  <  R 2  2.2-1  Twin-T  X  v.  2C  The symmetric  To a i d e making a c i r c u i t to that of:  parallel  Filter  R  L  the  filter  a n a l y s i s , the c i r c u i t  is  generalized  8  This  is a  parallel  connection  of  two networks and may be  d e p i c t e d s c h e m a t i c a l l y as:  Each network may be d e s c r i b e d  by a matrix  equation  i n current,  admittance and v o l t a g e .  To d e s c r i b e the network as a whole, consider  the f i g u r e :  Then:  i * i> r v  - y  T h i s l e a d s t o the d e s i r e d  T h i s l a s t eguation admittance  matrix  = v~ result:  i s important of  a  number  because of  i t means  networks  that  the  connected i n  9  parallel  is  the  sum  individual  network.  the e n t i r e  network  of In  can  the  the be  admittance  matrix  of  each  case of the p a r a l l e l - T network, broken  down  into  two  simpler  networks o f t h e type:  The mesh eguations f o r t h i s network a r e :  V,  Z I , (2 2c)r b  +  b  +  z  The matrix r e p r e s e n t a t i o n which f o l l o w s i s :  It  is  I=¥V.  of  interest  t o have the matrix eguations i n t h e form  The I matrix, or t h e admittance  of the Z. matrix.  Thus, I=YV i m p l i e s :  matrix, i s t h e i n v e r s e  10  I  v,  III JZ|  is  the  determine networks.  the  determinant Y  matrix  of the %_ matrix. for  each  of  the  The f i r s t network t o c o n s i d e r i s : R  2jwC  The r e s u l t i n g  Y matrix i s :  2j ui C  i w  The second  network to c o n s i d e r i s : _JL_  jwC  JwC i  *  R 2  I t i s necessary to parallel-T  sub-  11  T h i s has the Y matrix:  R 2.  C  z  \utC  - JL z The  2.  Y matrix f o r the e n t i r e p a r a l l e l - T network as d e p i c t e d i n  F i g . 2.2-1 i s then  Y  T h i s matrix i s :  Y = Y ' + Y '  Y  Y 2.2-1  Y,  where:  V The u l t i m a t e aim freguency  of  1 » (ju>CR )* 2R0  +  this  discussion  i t i s terminated  f i n i t e impedance o f t h e a m p l i f i e r system. i s d e p i c t e d by F i g 2.2-2.  V  find  the  response o f the Twin-T network when i t has a s i g n a l  from t h e sensor as i n p u t and when  system i s :  i s to  by the  Schematically, t h i s  The matrix equation d e s c r i b i n g  this  12  A, 8  8,0,  + t  C, B , + D, Dj  /  A'  3  2  "1 4 A  l  B  C  l  D  1  Sensor  A  l  V  2  2  C  l  2  B  2  D  2  y  R.  Twin-T  Coil  Network  Fig.  Prom  the  matrix  2.2-2  equation, i t f o l l o w s that V, = A ' ? +B' I J 3  and  s i n c e 1 =? /R. , the r e s u l t i n g t r a n s f e r f u n c t i o n i s : %  3  t  V,  R; A' R; + B'  The m a t r i c e s d e p i c t e d above a r e not matrix.  In  the  terminology  in  the  The  reguired  is:  / --  where:  of  the  Y  o f e l e c t r i c a l e n g i n e e r i n g these  have the form of the F matrix.  F  form  i  *1X  m  V  Y»,  Yn Y  t  l  ~ Xi,.  * i Y, Z  transformation  13  To  compute  the F  following c i r c u i t  matrix  f o r t h e sensor c o i l , c o n s i d e r t h e  diagram:  From t h e RLC l o o p , an equation can be d e r i v e d f o r I,  From the p o t e n t i a l drop a c r o s s R and L:  V -V I, =  R + juiL  T h i s l e a d s to the e x p r e s s i o n f o r V, :  V,-- (l-"*LC + i u , C R ) v + (R + j u , L ) l . t  The sensor c o i l matrix i s then:  l-wHC + iwCR  V Sow a l l o f transfer final  l  w C  the g u a n t i t i e s  f u n c t i o n o f t h e system  result i s :  R*jwL  to  determine  the  complete  i n F i g . 2.2-2 a r e known.. The  14  V. where:  2.2-2  A' R- + B '  A  -  Bv'  A,--  8, -  A , A A, B  1-  x  +  l  B , C  l  + B,  u, LC + 3 1 0 C R l  R  +  1L Yx \ Y,x  Y,  t  -  Y„  Y.»  Y»  0.  »' 1  Y,  Ml  are defined by 2.2-1  :  15  11-2.1  Computer  Besnlts  A computer program was w r i t t e n to function  of  eguation  2.2-2-.  compute  the  The r e s u l t s of t h i s computing  may be seen i n P i g . 2,2.1-1.  The s t r i k i n g f e a t u r e  is  transfer  how  the  shape  of  the  f u n c t i o n of the t e r m i n a t i n g This  terminating  a m p l i f i e r system.  function  impedance of the  impedance  transfer  is  the  to  changes as a  Twin-T  input  notice  network.  impedance to the  I f the a m p l i f i e r i s an i n t e g r a t e d  circuit,  then i t s i n p u t impedance could e a s i l y be o f the order of 10Mix and a severe ringing and can  is  ringing the  effect  occur  near  5  Hz.  r e s u l t of c o u p l i n g between the c o i l  the Twin-T c a p a c i t o r s . be  will  reduced,  then  I f the a m p l i f i e r  the  input  ringing effect  D i l l  curve f o r an i n p u t impedance o f 7.5 Ksi. shows  no  a  response  curve  Butterworth f i l t e r . capacitor  in  six carefully case  of  matched  be  obtained  I t would r e g u i r e only  p a r a l l e l to the c o i l . . matched r e s i s t o r s and  balanced instead  troublesome  can  input, of  twelve  six.  requirement..  the t e r m i n a t i n g  In  by one  lessen.  i s reduced.  The  ringing  at  earlier,  exploiting resistor  a  and  The,Twin-T would r e g u i r e capacitors,  elements  practice,  I t should  and  in  the  would have to be this  may  be  a  a l s o be n o t i c e d t h a t as  impedance i s reduced, the s e n s i t i v i t y  lower f r e q u e n c i e s  inductor impedance  a l l and has a smooth drop o f f , . However, as d e s c r i b e d such  This  at  the  T h i s i s not a d e s i r a b l e e f f e c t .  16  FREQUENCY (HER^Z)  F i g . 2.2.1-1  Computer s i m u l a t i o n o f the frequency of the c o i l / T w i n - T s e n s i n g system  response  17  II~3  Theory, of the Butterworth  The  theory  will  now  Filter  be  developed  Butterworth  filter.  Fig.  A r e s i s t o r and c a p a c i t o r a r e placed  2.3-1.  The c i r c u i t  for  the output of the sensor  an  analog  to be analyzed i s shown i n parallel  to  coil.  L - Sinnnnrr  v.  $  l  :R.  vwvww  v  R Fig.2.3-1  To  determine  three p a r a l l e l  V  0  An Analog Butterworth  , first  find  Filter  the e q u i v a l e n t impedance of the  q u a n t i t i e s C, C, and R, .  1 + iu»U + O R, The e q u i v a l e n t c i r c u i t  becomes: L  ©  R  -MMAAr  V  can now be found from the v o l t a q e d i v i d e r method.  R •* j L w  +H  18  A f t e r s u b s t i t u t i n g f o r Z, a l e a d s to the  considerable  amount  of  algebra  result:  2.3-1  ~1  This  equation  form.  The 1.  for  V„  can  be  put i n t o a non-dimensional  f o l l o w i n g s u b s t i t u t i o n s w i l l be made:  Let oc r e p r e s e n t the r a t i o of d.c.  r e s i s t a n c e of the  sensor c o i l t o the t e r m i n a t i n g r e s i s t a n c e . R  2.3-2  2. Let f? r e p r e s e n t the i n v e r s e Q f a c t o r of the series  circuit.  Q. 3. Let f series  S-L-C  r  2.3-3  4  r e p r e s e n t the resonance freguency  of the  B-L-C  circuit. 2.3-4  2-tr ] L (c Now  equation  v.  2.3-1  *0  can be re-expressed  V;  e  as:  2.3-5  19  where  In  -Q~ --  tan j  order f o r the c i r c u i t  filter,  the  term  Butterworth f i l t e r  2.3-1 t o  {f/f )  respond  as  a  Butterworth  must be made to approach z e r o .  2  r  o f any order i s present when  the  A  transfer  f u n c t i o n c o n t a i n s the f o l l o w i n g e x p r e s s i o n ;  A  p l o t of the t r a n s f e r f u n c t i o n f o r the Butterworth f i l t e r of  orders n=1,2,3 can be seen i n F i g 2.3-2- I t i s e v i d e n t t h a t i n the  case o f e q u a t i o n 2.3-5, t h i s i s a  Butterworth  filter  of  order 2 s i n c e i t i s o f the form: 1 1  i  2.3-6  i f the second term i n the denominator can be made n e g l i g a b l e . It  can  be  seen  from the c i r c u i t diagram 2.3-1 that i t  would be d e s i r a b l e t o have S, l a r g e would  i n c r e a s e the s e n s i t i v i t y .  compared  to  R  as  this  Thus i t can be expected t h a t  *<1.. , In t h e c a s e that oc»0, t h e second term vanishes f o r : 0 -  In the  ^  the case t h a t 0<«<1, the second term vanishes f o r e i t h e r of f o l l o w i n g two values o f p :  F i g . 2.3-2  The  transfer  function  f o r the B u t t e r w o r t h  filter  21  I f 0 « « 1 ., the b i n o m i a l s e r i e s expansion may be used to  show  t h a t t h e above two values are approximately equal t o :  P • «/nr In  t h e case that rt>1, t h e r e i s no p which can make the second  term v a n i s h .  T h i s i s another reason why * i s d e s i r e d  to be  small. In  practice,  i t i s best t o determine  make * equal t o t h a t term. it the  As  value  which  will  range  from . -002 Hz - 4 Hz .  c o n s i d e r a b l e amount necessary  of  man  made  freguency  The  Thus,  since  noise  at  over  there  is a  Hz,  i t is  60  of  noise  C, can  U  >  i s made  R, L and C a r e known.  L  -  by the  egual  by a to 6  (2irfvr  This leads t o :  r ^  P can now be found from eguation 2.3-3.  ,  up  be determined from eguation 2.3-4 provided  = J_  c  will nullify  picked  amplitude o f the 60 Hz n o i s e can be dropped  the c o i l parameters  P>JT  second  response  Also,  f a c t o r o f 100 i f t h e c u t - o f f freguency  If  the  to o p t i m i z e the g u a l i t i e s o f the Butterworth f i l t e r  i n order t o reduce t h e amount  Hz.  nullify  the sensor w i l l be used to d e t e c t m i c r o p u l s a t i o n s ,  i s d e s i r a b l e that i t have a f l a t  sensor.  P f i r s t and then  2.3-7 The value of o( which  t h e second term i n 2.3-5 i s :  no  «C  can make the second term v a n i s h .  type o f sensor i s t y p i c a l l y of the order o f 1 0  - 4  -10  0 - 2  f o r this ,  Oeda  22  and  watanabe  condition  (1975) ,  might  so  there  occur., Thus,  i s not any t h r e a t t h a t B  can  be  found  this  from  the  eguation:  R  2.3-8  R  =  I I - 3 . 1 Laboratory J e s u i t s  The to  circuit  determine  of F i g . , 2 . 3 . 1 - 1 was s e t up i n the l a b o r a t o r y  whether the t h e o r y of the Butterworth  filter  is  c o r r e c t f o r the c o i l s .  -©  wvvvv  I  1 ma rms  -VVVvVVV1  v out Fig.  A  2 . 3 . 1 - 1 Laboratory Set-up f o r the Butterworth  sinusoidal  magnetic  field  was  created  c a l i b r a t i o n c o i l , , The i n t e n s i t y o f the f i e l d  inside  Filter  a  at the c e n t e r of  the c o i l i s |^71: Gamma from the rms c u r r e n t o f 1 ma that through large  i t s windings. c o i l . . The  large  flows  The sensor c o i l was placed i n s i d e the  time-varying  magnetic  field  induced  a  23  s i n u s o i d a l emf i n the sensor c o i l was  and the output from the  measured a c r o s s a p a r a l l e l HC l o a d .  the c o r n e r frequency of 6 Hz and the the  flu-metal  core c o i l  (1975),  the p a r a l l e l c a p a c i t o r was  specifications  This  to  equation  =28KA  as determined from equation 2 . 3 - 8 .  2.3.1-2.  The  leads  to  noted  by n e a r l y a f a c t o r of 100 that  v  o u +  the  can be  value B,  seen  in  at  60  The amplitude Hz.  It  should  has be  i n F i g 2.3.1-1 has been d i v i d e d by freguency  because the emf induced i n a c o i l  by a changing  magnetic  i s p r o p o r t i o n a l t o freguency and the amplitude of the field.  Then,  frequency response i s f l a t at the lower  f r e q u e n c i e s and begins to drop o f f a t 4 Hz. dropped  .67 y u f .  be  The r e s u l t s of the l a b o r a t o r y experiment Fiq.  of  {see Table 1 - 1 ) , the value of  determined  0 =.0565.  From e g u a t i o n 2 . 3 - 7 ,  coil  from  2.3-3,  coil  flux  magnetic  24  F i g . 2.3.1-2  Butterworth f i l t e r c h a r a c t e r i s t i c s sensor c o i l system  of the  25  CHAPTER I I I THEORY OF THE BRIDGE METHOD  III-1  Introduction  The  o r i g i n a l i d e a t o c a l i b r a t e the i n d u c t i o n magnetometer  by using a Hheatstone b r i d g e was conceived at  the U n i v e r s i t y of B r i t i s h Columbia.  a  previously  successful  electromechanical and  T h i s i d e a stemmed from  undertaking  seismometer  to  calibrate  u s i n g a Maxwell b r i d g e ,  first  intuitively Norton's  p a r t o f t h i s chapter  analyzing  theorem,  model.. The  the  otherwise  remainder  will  Hheatstone known  be analyzed the  o f the chapter  f o r three configurations.  magnetometer  Kollar  remain  bridge the  current  to  source  w i l l present a d e t a i l e d The bridge system w i l l be  that  the  will when  bridge  can  a p a r t o f the magnetometer e l e c t r o n i c s so that  i s made.  The  the  second  system  every  configuration w i l l  proposed c a l i b r a t i o n procedure i n which the b r i d g e by an e l e c t r i c a l o s c i l l a t o r . case  with  according  The f i r s t  I t i s hoped  w i l l not need t o be wired i n t o  calibration  concerned  i s making observations and the b r i d g e i s a  part o f the e l e c t r o n i c s . , always  be  as  a n a l y s i s a c c o r d i n g t o K i r c h o f f * s laws.  is.  an  R u s s e l l (1966). The  it  by Dr. R.D. R u s s e l l  is  time  a  be the driven  The t h i r d w i l l be t o compare the  of o b s e r v a t i o n when the bridge i s not present to when i t T h i s l a s t step i s r e q u i r e d i n order t o determine how  presence of the b r i d g e could d i s t o r t  data.  the  26  I I I - 2 An I n t u i t i v e Approach Dsinq t h e Norton  It  is  the  show t h a t a simulated  intent  rate in  of  the  with an e l e c t r i c a l arm used  of  the  for  it  of the developement which f o l l o w s t o  change  sensor  of  has  and  magnetic  f l u x , <ij ,  can  be  c o i l by d r i v i n g a Hheatstone b r i d g e  oscillator.  bridge  Equivalent  The  sensor c o i l  the e q u i v a l e n t c i r c u i t  already  been  given  will  be  which w i l l  Fig.,, 1 - 1 . /  in  one be The  Hheatstone b r i d g e i s shown i n F i g . 3 . 2 - 1 .  F i g . , 3 . 2 - 1 The  As  the  sensor  is  to  be  used f o r m i c r o p u l s a t i o n  f r e q u e n c i e s of i n t e r e s t w i l l the is  c a p a c i t i v e reactance assumed  electrical  that  the  oscillator  which c o u l d be induced  Hheatstone b r i d g e  be between .002  of F i g . 1-1  amplitude will in  the  be  i s negligable.  of  the  much coil  - ,4 Hz,  by  signal  research, so  Also, i t from  l a r g e r than any a  natural  that  the signal  magnetic  27  event.  Therefore  made concerning  E»V.  the  Two  further  assumptions  which are  magnitudes of the b r i d g e components a r e :  3.2-1 Also,  the  bridge  is  balanced  c o n d i t i o n i s expressed  d.c..  This  balancing  as:  R R^  RR 3  In  at  3.2-2  Z  order to s i m p l i f y t h e a n a l y s i s , the Norton o r c u r r e n t  source e q u i v a l e n t i s to be found between the p o i n t s a and b o f 3.2-1.  Fig.  the two  T h i s i s done by removing L and  p o i n t s a and b.  p o i n t s would  The  short  circuiting  c u r r e n t which flows between  these  be:  3.2-3  Next,  consider  t e r m i n a l s a and and  R  3  »R  2  the b.  impedance The  when  l o o k i n g i n from the  impedance would be B*R *R L  t  if H  l t  two »R  Then, the e q u i v a l e n t c i r c u i t which f o l l o w s i s : R  -M/VVW  L  28  For flux  a  per one  c o i l i s N<|>.  coil  of  N t u r n s immersed i n an average magnetic  t u r n of the c o i l , <t> , the t o t a l f l u x Thus, the emf  induced  through  the  i n the c o i l toy the changing  flux i s : E  =  "-3t(N<J)  3.2-4  )  But, i t i s a l s o t r u e t h a t :  E =Applyiag  (^ L i;)  3.2-5  u i  Norton's theorem, the e g u i v a l e n t source  current i s : 3.2-6  Using the r e s u l t o f 3.2-6, the Norton e g u i v a l e n t of the  sensor  becomes: R  —'WWW—  L  F i g . 3.2-2  Norton e g u i v a l e n t of the sensor  coil  I f the c u r r e n t s through the i n d u c t o r i n the cases of 3.2-3 3.2-6  are  egual, then an important  amplitude of the d r i v i n g  voltage  and  result exists the  flux  and  between the through  the  coil:  A..  3.2-7  ~  I  29  I I I - 2 . 1 Theory a t Higher  Frequencies  I f the Hheatstone b r i d g e i s going t o be frequencies, i n t o account. to  used  at  higher  then the c a p a c i t a n c e of the sensor must be taken To compensate f o r t h i s , an i n d u c t o r L  the fij arm o f t h e b r i d q e .  The b r i d g e c i r c u i t  3  i s added  which f o l l o w s  i s shown i n F i g . , 3. 2. 1-r 1. ,  Fig..3.2.1-1 Hheatstone b r i d g e a t higher f r e g u e n c i e s  The  Norton e q u i v a l e n t c i r c u i t which  results  from  c o n d i t i o n 3.2-1 i s shown i n the f o l l o w i n g f i g u r e .  using  the  30  F i g . 3.2.1-2 Norton e g u i v a l e n t o f the Wheatstone  In two  order t o a r r i v e a t a new Norton e g u i v a l e n t , s h o r t the  p o i n t s a and b, , Then c a l c u l a t e the  through S.  Currents  Fig.  L e t Z,  which  flows  to F i g . , 3. 2. 1-3.  ±  R,  R  3.2.1-3 C i r c u i t  current  are defined according E  E  bridge  R +303L 3  used t o d e r i v e a new Norton  3  equivalent  be the p a r a l l e l composite impedance o f R and C. R 3.2-8  applying  Kirchoff's  law  to  the  R-Z -R L  Z  l o o p , the eguation  which r e s u l t s i s :  3.2-9  31  Using the c o n d i t i o n f o r the d.c. balanced R R  and another  3  R  =  z  bridge;  R^  c o n d i t i o n on L, t h a t :  CR eguation  3.2-10  3.2-9 s i m p l i f i e s t o : 3.2-11  R. Then, the c u r r e n t i  t  f l o w i n g through R i s given by:  T h e r e f o r e , the e g u i v a l e n t c i r c u i t becomes: R  R„ R  4  (1+JOJCR)  I f i , and i o f equation  3.2-6 a r e e q u a l , then the r e s u l t i s ;  Ru. Ci+^cR)  L  The importance o f the r e s u l t s 3.2-12 and 3.2-7  3.2-12  are  that  they p r e d i c t t h a t a r a t e o f change of magnetic f l u x , <l) , can be simulated  in  the  sensor c o i l by d r i v i n g a Hheatstone b r i d g e  32  with  an  electrical  oscillator.  In  practice,  i t i s much  s i m p l e r to simulate a magnetic f l u x i n t h i s manner than i t i s to  immerse  the  coil  in  a  uniform  calibrating  field.  A  freguent method which i s used t o c r e a t e an a r t i f i c i a l f i e l d i s to  put  coil.  the  Besides  investigator to  the  coil.  sensor  coil  being  inside  physically  a  larger  calibration  cumbersome  so  that  would n o t want t o b r i n g a c a l i b r a t i o n c o i l  an  along  f i e l d , i t c o s t s n e a r l y as much to b u i l d as the sensor  By u s i n g the Hheatstone bridge method, the  can be c a r r i e d  out a t any time and with very l i t t l e  calibration expense.  At t h i s p o i n t , t h e problem of the a b s o l u t e c a l i b r a t i o n i s not  worked out too w e l l .  external f i e l d  The t o t a l f l u x N<|> i s r e l a t e d  to the  B by the r e l a t i o n ;  s According t o Dr. B.D. sensors i s reduced and  for L  in  Bussell  to finding  terms  absolute c a l i b r a t i o n  of  at  DBC,  the  calibration  of  expressions f o r 4> i n terms of H,  the c o i l geometry.  The problem of  w i l l be d i s c u s s e d i n d e t a i l i n Chapter  V.  33  I I I - 3 Theory of the Bridge Method  A d e t a i l e d c i r c u i t diagram f o r the Fig.  3.3-1.  In p r a c t i c e Z  was  originally  hoped  e  ,Z  that  and  2  Z  3  Z^  bridge  shown  in  are pure r e s i s t o r s .  It  would  is  consist  of  the  SLC  c o n f i g u r a t i o n sketched below.  AVVWVTheoretical  In t h i s way, u s i n g the  both an a.c.  bridge.  configuration  would  work  sketch  to  render  except  balance c o u l d be  the  experimentally,  for  the f a c t t h a t a l a r g e found  Even a r e s i s t a n c e of 50XL was  a.c.  of the l a b o r a t o r y Z  achieved  and  of 10 Henry, c o u l d not be  negligable resistance.  enough  a d.c.  Theoretically  i n d u c t o r , of the order had  and  Z,  3  balancing  this  which large  condition useless.  A  i s shown below.  R  -MAAAAr  -AA/VVW—nfwrrExperimental  Z.  Hhat f o l l o w s i s a c i r c u i t a n a l y s i s of will  show  that  this  type  o f bridge  Fig.  3.3-1  arrangement w i l l  which truly  r e f l e c t the freguency response of the i n d u c t i o n magnetometer. Let fi, ,L, and  and  C,  be the d.c.  c a p a c i t y of the sensor.  resistance,  self-inductance  For convenience, i n t r o d u c e  the  35  quantities:  X, = R,+ ju,L,  3.3-1  3.3-3  X.+ Y, E  0  w i l l be the emf caused by the s i g n a l generator.  the emf generated by the time v a r i a t i o n s of  the  earth.  A  applying Kirchoff's  system  E, w i l l be  of the magnetic f i e l d  o f equations can now be defined by  law t o c u r r e n t l o o p s .  The  X,~Z -Z Z  S  loop  yields:  E, = X ^ , - 2 , 1 , • 2 , 1 , The Y,-Z -Zj. z  The Z -Z - Z 0  z  3  loop y i e l d s :  loop y i e l d s : eo=  The Zy-Z^-Zj  eguations  the independent is:  2  2  i  2  loop y i e l d s :  O  These  z.Cl-U-Virl,)-  -  2 , 1 , - 2 , ^ 1 - 1 , )  - 2 3 ( 1 , - 1 3 )  can be arranged, r e g a r d i n g I , I , I #  variables.  a  The corresponding matrix  3  and I as equation  36  X,  Z,  o o  -U„*z^z,} -z.  2, (2,  + 3.3-4  Let  D  be  the  coefficients.  determinant  which  is  defined  by  the  I t can be e v a l u a t e d as f o l l o w s :  3.3-5  where  + Z^(Z Z 2  The c u r r e n t I because  it  is  3  flows  +  3  Z  the  key  to  through  the  load  3  Z , - 7  the  the  sensor,  calibration,  =  C  o  n  s  t  then  response  impedance Z .  If this  s  J  sensor  by  a  3  signal  fluctuating  generator  '  be T  3  and  f o r the two  shown  that  observation  e  magnetic f i e l d and E =0. 3  E »E,. 0  cases.  The  o  i s caused by  the  emf  of  T h e r e f o r e , the system  simultaneous l i n e a r a l g e b r a i c eguations must r e s p e c t to I  h  response  i s the r e s u l t o f c u r r e n t s induced i n  c a l i b r a t i o n c o n d i t i o n means t h a t I the  freguency  must  * < 3 > observation  condition implies that I the  it  7 J  freguency  method i s going t o p r o p e r l y determine the of  5  be  solved  of  with  37  III-3.1  Case o f Observation  For t h i s case, E,#0 equation 3.3-4  and E =0. o  The s o l u t i o n of tae matrix  qives:  D The  n a t u r a l magnetic f i e l d  f o l l o w i n g manner, where  f l u c t u a t i o n s may  B  is  the  be d e f i n e d i n the  amplitude  of  the  changes and S i s the a b s o l u t e s e n s i t i v i t y o f the sensor  E = 1 ]u> S B e  >U>  field coil:  3.3.1-2  x  III-3.2 Case of C a l i b r a t i o n -  For  this  case, E » E , . 0  The s o l u t i o n o f 3.3-4  gives f o r  E =0 and E„ #0: 4  The numerator o f t h i s simplified  form  approximations.  last by  First,  e g u a t i o n 3.3-3., Then:  equation making  substitute  can  some for  be  reduced  substitutions Z, a c c o r d i n g  to  a and to  38  (X  Hext,  T  X.Y.Z,- (X,*Y,) 2 Z ,  Y,)( Z.Z.-Z^ZJ*  +  assume  resistors.  that  2  Z  t h e branches  That i s Z =R l  2  ,  Z  (y *Y,Hz,Z,-2 Zj* (  3  = a  3  T  a  B  d  a n d 2^. a r e a l l p u r e  3  Z^=R .  Then:  V  X,Y,R -  1  ,Z  3  U.+ Y J R ^  f o r X, a n d I, a c c o r d i n g t o e g u a t i o n s 3 . 3 - 1  How s u b s t i t u t e  and  3.3-3.  If  the bridge  i s balanced f o r t h e d.c. c a l i b r a t i o n  then t h e c o n d i t i o n  signal,  w h i c h i s met i s :  This leads t o the r e s u l t :  (y fY,Kz z -2 7^l  (  as  L,ai10 H 3  satisfied  3  C,«i10- F 7  r  l  and  (~-  3-L,R,R  B,«* 1 0 X L ,  3  the condition  3  i s  well  that: —  >>  R,  3.3.2-3  1  The e g u a t i o n r e d u c e s t o :  (x +Y Kz,VZJ,) t  (  T  R  B  "  > -  L  '  R  .  R  3  39  For is  micropulsation  r e s e a r c h , t h e frequency range o f i n t e r e s t  .002 Hz_i< f < 4 Hz-, The q u o t i e n t  1/CR*10*.  Then,  a  final  assumption can be made t h a t : I  CJ  It  should  higher  <<  3.3.2-4  be noted that t h i s l a s t c o n d i t i o n w i l l break down a t  f r e q u e n c i e s . , Whereas i t i s w e l l s a t i s f i e d  frequencies,  below  1  Hz,  at  a  frequency  c o n d i t i o n i s r e a l l y n o t too w e l l met. and  a t t h e lower  of  10  Substituting  Hz, t h e  C, =.2 /xF  R =2Ka:  3 Thus,  4-0  the f i n a l r e d u c t i o n o f t h e numerator o f equation  3.3.2-  1, keeping i n mind t h a t f < 5 Hz, i s :  (X,+ Y , K Z . l ^ H , Now t h e r a t i o  (I ) 3  C O l  i /{I ) \ tt  a%  ±- R  3.3.2-5  3  can be c a l c u l a t e d u s i n g 3.3.1-1,  3.3.2-1 and 3.3.2-5:  Osing 3.3.1-2, the f i n a l r e s u l t , keeping  i n mind  conditions  3.3.2-3 and 3.3.2-4, i s :  i l . R ,  --  {U) ^ 0  Thus,  the important  Z (R +R^^+ 0  3  i  R (R +R ) U f  1  3  o  _  3  3  2  _  6  SB  c o n c l u s i o n i s t h a t f o r the d.c. balanced  no  bridge: (i )c*l  "  3  The  ratio  freguency. ( I ) .i 3  /(X,') v>s  i  independent  s  0  Therefore,  faithfully  co  3.3.2-7  the  freguency  reflects  that  of  of  response the  of  induction  magnetometer.  I I I - 3 . 3 Computer and Laboratory R e s u l t s  The  success  of  the  b r i d g e method w i l l depend upon  w e l l two c o n d i t i o n s are s a t i s f i e d .  how  These c o n d i t i o n s a r e :  B^»R, 3.3.3-1  R »R 3  Z  I f these c o n d i t i o n s are p o o r l y met, bridge  may  differ  conditions  are  met,.  p r e c i s e l y determine would was  considerably  how  a  computer  then the output from  the  program  case was  the  when these written  to  the response curve f o r the sensor  coil  be a f f e c t e d by the i n e g u a l i t y . . The eguation programmed  V=(I ) . 3  results  ob<  3  % , where (I ) v, s  3  0  i s defined  t  be seen i n F i g 3 . 3 . 3 - 1 . ,  may  with R =R,.  by  The  3-3-1-1-  program was  The curve which r e s u l t s from R = .5R,  substantially  R^SOR^ .  There i s not much d i f f e r e n c e between the  R =.05R, 1  R^=50R-, . R >20R,  and  R =20R, IV  from  the  and  the  curve  curve  when  when  The c o n c l u s i o n i s t h a t the c o n d i t i o n R and R » R 3  1  i m p l i e s R <. 05R . a  3  These written  and R^ =2R ,  l  differs  lt  from  Hj_=.02R,  and  curve  for  a =.02H,  and  i  lfr  »R  l  implies  41  ,2  .3  .4  £  ,8  10  2  3  4  6  8  • 10  . 5 ^  V  2 R  .lR^  R  Frequency ( H e r t z )  KEY:  V V V V  Fig.  3.3.3-1  .051^ .02R!  Computer s i m u l a t i o n to determine the e f f e c t of the c o n d i t i o n s R ^ » R^ and R ^ » R on the response of the magnetometer system when usi n g the b r i d g e method 2  i  =10R. 4 1 R=20R 4 1 R =5QR., 4 1 1  42  A l a b o r a t o r y experiment was performed  to  determine  the  freguency response o f t h e magnetometer system using the b r i d g e method.. The  circuit  diagram  i s shown i n F i g . 3.3.3-2.  b r i d g e r e s i s t o r s were determined a c c o r d i n g 3.3.2- 2 R  3  and  3.3.3-1  .  To  satisfy  to  the  The  conditions  t h e c o n d i t i o n 3.3.2-2,  was made approximately egual t o R , R^-R,/20 and the b r i d g e (  was  adjusted  R^.  Also,  t o zero d.c. i t must  be  output noted  by that  the  variable  the f i r s t  resistor  stage o f t h e  a m p l i f i e r system i s a chopper a m p l i f i e r (see Appendix 2) which r e q u i r e s a balanced i n p u t using  the  inverting  signal.  amplifier  This  as  i s t h e reason > . for  p a r t of the i n p u t s i g n a l  e l e c t r o n i c s t o the b r i d g e . , The 3.3.3- 3., value  r e s u l t s of t h e l a b o r a t o r y t e s t can be  laboratory.  The d o t s r e p r e s e n t  The  smooth  The eguation  V T(jw) and  This  Fig.  which was programmed i s :  (X^)  e k s  *  2  f  *•  Ttju/)  3.3.3-2  electronics  As can be seen, t h e agreement  and t h e l a b o r a t o r y  i s excellent.  phase i s s h i f t e d by 90° i n t h e low freguency  i s because  i n the  i s the r e s u l t o f a computer  i s d e r i v e d i n Appendix 2.  between theory  law  curve  data p o i n t s obtained  i s the t r a n s f e r function f o r the a m p l i f i e r  The  in  For convenience, t h e data has been normalized t o t h e  a t 2 Hz.  analysis.  seen  the eiaf i s induced i n the c o i l  range.  by Faraday*s  o f i n d u c t i o n which s t a t e s t h a t t h e emf around a s t a t i o n a r y  l o o p i s p r o p o r t i o n a l t o t h e r a t e o f change of f l u x through t h e loop.  ^  .1  .2  .3  .5  .7 Frequency  Fig.  3.3.3-3  1.0  2.0  3.0  5.0  (Hertz)  The n o r m a l i z e d amplitude response of the magnetometer system a c c o r d i n g to the b r i d g e method  45  1 1  T T X 1 T i I - L l -11 1 1 i lu.1 T ij  1 180  1 1 I  -  1 ! Laboratory  •  _  i III II II T tt  1 1 11 n  1  i t 1 i  results  I  ! I  1  ill :i ill 1 it1' !il H II 11 i ll II U1 ii FI ' ! !' 1 ll1 III  I  j  i  T T i y rt I /i •J /' i 1 ' i! 1 1 1  /'•  1  il III m i l i n in I ! ! '1 III JLiiL i 1 il 1 |l lihi i i| l|i  1 1  90  l  i  r UJ  11  a) nl U 00 CD T3 v y  45 i  _  al CO ct)  I  I  1 fu  1  ]  i  1 : |1  0  11 11  i  1  1  j  i i j J l 1  ; II ll I i  '  '  +trrr i Ii i  1 11 Ii 1 i 1 1j I 1 1 i 1 1 1 1 1 j 1  1 I "i 1  !  1  1i 1 1 ! 1 1 i 1  / A \ i I  i  I  1  A• i1 1  i i i i  i\ »' l P  ii /  \V  I YJ '-,\ III i */ 1 III y 1 I 1 II 1 1 i i l l j II I 1 llII inn 1 1 II III | ..Iii i 1 ' 1 1 1 1 ! 1 1 i I 1 11 1 .11. 1 ill i I M I I TTllH I 1 ill i  1  1  1 1 1 I i 1 1 1 1 1 ' I 1  -90  t  i>  r  I i  1i  .1  .2  Fig.  M III TT HI I MI . iT  1 I'll  .3  3.3.3-4  i  Frequency  (Hertz)  1 ; j jT T1T i i i r j i ; i 1 ! i 1 1II 1ill' 1 1 1 I 1 I  /'-P^5  ii 1  .7  !  1.0  1  1  l JJ 1 I T Ijl I t TH 1  I I I T1i ' l t t i i i I 1 i 1  I 1 ii i  1 I  1  iii  1I i  I Till  1  I  f  •  ""tT" II Ji  -45  1  I ; i i 1I i  IT 1i  '  nil ill! ii i! III Illl I HI JlJiiil -I ill nil ii 1 i|i1 1 | 1111 i i i I' i 1i i I i I ill 1 ! II i 1 II i 1 ll j 1 I 1 i 1I 1 1 ""TI I 1 i  11 1 1 I i  Ji  11" T i  II  ll l|  t- - -i- 4- i iT j i jm 1 i i i I  f\ O  tt Tit  1  ill  1  I' i ii • Illl Ii '  1  i 11 !1 i 111 11 i 1i 11 I 11 1 11 i  '  i  ni 111  1 11 i i 1 1 l j in 1H1 1 Lj_ 1 IT ij 1 i 1 i ii i 1 in in  l lili i i 11 IIII iih i i i 1  i  1  11 11I  1  1  i II i II 11 II i i n i TT i H i ni l.Ji_L  i  i  11  I I i I i  -JJ.-L  1 1 i T  i  2.0  i Ml i • ! i  3.0  1  111 1•  1  1  1nr f --4J444-1  "1  11 1 1 i11 i '' i11 illl  iiii if\  i  11  1  I  l  1  ML  1 j U. 1 1 1 11 \)1 ' !  1  |  1  mi i  l  III III j Ii! !|l ii  H nt  1  1(1  1  I  1  1 i  T  1  1 1'II  I l i i | HI I 1 I11 II  Computer r e s u l t s  1 135  iiI i 1 ll  •; 1 1 |  i  5.0  The phase response o f the magnetometer system a c c o r d i n g t o the b r i d g e method , ,  I  46  For a f i e l d which i s normal to the l o o p :  B-n Let  B=B e  dA -- BA  Then :  0  f » - ju>BA Thus, i t i s expected freguencies.  which  simple  there  The  are  seven  orders.  freguency  It  will  be  electronics,  i s no longer such a  v o l t a g e output  I t i s given by V  With and Without the  as measured a c r o s s Z  obt  =  (I ) 3  o l ) S  be  found  to  electronics. without  have  the b r i d g e  eg.  the  of  3.3.3-  3.4-1  bridge  so as  bridge , ., then  concur with the  that a  I f the sensor c o i l i s to the  using  0  i s d e s i r e d t h a t the r e s u l t 3.4-1  necessary  i n the case  5  ob<  o b s e r v a t i o n without  5  Bridge  *Z .  V = It  as  case.  o b s e r v a t i o n with the b r i d g e can  field  phase  pass f i l t e r s of the a m p l i f i e r  I l l — 4 The Case of Observation  of  the  problem t o p r e d i c t what the phase response w i l l be,  i n the low  "2,  w i l l l a g by 90° a t the lower  i t the higher f r e g u e n c i e s ,  a f f e c t e d by the low of  t h a t the emf  it  will  not  case be  permanent p a r t of the be  operating  in  the  a diagram r e p r e s e n t i n g the  47  i n p u t v o l t a g e t o the a m p l i f i e r system i s shown i n F i g . 3.4-1.  REAL ^  ^  REAL  E  (£)  I F i g . 3.4-1 Diagram o f sensor c o i l and load impedance  X, and Y, respectively.  a r e given  by  eguations  3.3-1  The composite impedance of I, and Z  and s  3.3-2  i s given  by:  Y.Zr  Z -Then the c u r r e n t I i s :  I  E/(X,*Z)  The p o t e n t i a l drop across Z V  RCAL  y  is:  2  r-  *,+ z  The r a t i o of the output voltage t o the input voltage i s :  V REAL  By making the s u b s t i t u t i o n f o r Z, t h i s l a s t r e s u l t  J_ E  2,Z  y  X,' (Z.+Z*)  becomes:  3.4-2  48  If  t h e b r i d g e method i s going t o be used f o r c a l i b r a t i o n  purposes,  then i t i s necessary t o show  eguation  for V  V  that  V  obs  » V^  . The  EAL  i s much more complicated than t h e one f o r  ots  , but by c a r e f u l l y c o n s i d e r i n g o r d e r s o f magnitude o f t h e  REAL  g u a n t i t i e s which a r e i n v o l v e d i n each term, i t can be that  Vfe«  i s essentially  0  identical  to V  J£flL  shown  . A s a starting  p o i n t t o e v a l u a t e these o r d e r s of magnitude, l e t Z =10fl., Z =R, 0  Z =R/20 and Z =20R, where R i s t h e r e s i s t a n c e 1  lf  coil  and  taken  Z =10000^t.  approximately  of t h e sensor  2000^.  Also l e t  By e v a l u a t i n g a l l of the terms o f D a c c o r d i n g  5  eguation  t o be  3  3.3-5,  to  and r e t a i n i n g o n l y t h e l a r g e s t terms, i t can  be shown t h a t :  D The  ~  2 ^ 2 , + 2,2^2, • Z , Z , Z  e r r o r o f t h i s approximation  evaluating  i s of  Z (Z +2j 0  T h i s approximation reductions 3  3  t h e order  of  6%. By  t h e numerator o f 3.4-1 with the same s u b s t i t u t i o n s  as were made f o r D, i t i s apparent  Z =R  3.4-3  S  3  +  that:  2^(2^2,) »  ZjZ^.  i s o f the order o f 5%., 8 i t h a l l  of terms  and n o t i n g  that  3.4-4 of  these  f o r pure r e s i s t a n c e s  and Z.=R,, 3.4-1 becomes: H  V  V.obs  .  Y, R , Z  g  3.4-5  49  Upon r e a r r a n g i n g terms:  VL  «  Z  ?  .  r 1 +~  If  can be made  large  enough,  (  then  )  the r a d i c a l  denominator c o n t a i n i n g R^ becomes s m a l l and  V o k s  ~ ~  J_ X,"  This i s i d e n t i c a l to 3 . 4 - 2 -  Z  ( Z  '  Z  l +  * Z ) f  3.4-6  i n the  reduces t o :  3.4-7-  E  *  I t must be noted t h a t R^ o f 3 - 4 - 6  cannot be i n c r e a s e d t o a very high value without r e c o n s i d e r i n g the  approximations  of  3 . 4 - 3 and 3 . 4 - 4 . ,  I f R^ i s t o be made  l a r g e r than 2 0 times t h e sensor c o i l r e s i s t a n c e , imperative and  t h a t Zz  preferably  approximations  of  the  leading  the v o l t a g e output observation  and Z  with  as and  Q  then  be s m a l l , c e r t a i n l y l e s s than order  to  of  10XL .  3 . 4 - 7 will  measured without  across  i ti s 100XL  Otherwise,  the  no longer be v a l i d and  Zs  in  the c a s e  of  the b r i d g e w i l l no longer be  comparable. The p h y s i c a l s i g n i f i c a n c e itself.  of  3 . 4 - 6 i s interesting  What i t means i s t h a t t h e b r i d g e c i r c u i t  1 with E = 0 e f f e c t i v e l y reduces t o the f o l l o w i n g : o  in  of F i g . 3 . 3 -  50  E  P i g . 3.4-2  The from  The  e s s e n t i a l elements o f the bridge  effect  computer  analysis  can  of  the term c o n t a i n i n g fi^ has  simulations. be  seen  programmed with and  The  in  without  results  F i g . 3.4-3.  of  It  can  s e p a r a t i o n o c c u r r i n g at 2  Hz.  reduced  comparing  by  oVj  F i g . 3.4-1 5s V  making  and  curves  bridge  as  without  the b r i d g e , may  permanent  a t t r i b u t e d p r i m a r i l y to the output  3.3.3-1  shown  a  high  of the sensor  part  with  in  the  3.4-2, the f i n a l  only i f the  R H L  it  two  effect  value.  comparison of o b s e r v a t i o n s between two a  computer was  Fig.  be seen t h a t the e f f e c t o f Z^. i s to cause a the  reached i s t h a t V  the  the 8^ term as given by 3.4-6. The  s m a l l s e p a r a t i o n between  By  been observed  Eguation  same values of bridge components were used as 3.3-3-2.  circuit  conclusion  of  can  systems, one  with  of the e l e c t r o n i c s and  a c t i n g as a  be  T h i s i n f e r s t h a t the  have a s l i g h t discrepancy  system.  maximum  parallel  another  which can load  the  be  across  Frequency  Fig.  3.4-3  (Hertz)  The e f f e c t of the r e s i s t a n c e R, on the frequency response as o b t a i n e d from the b r i d g e method  52  CHAPTER IV THEORY QF THE SEC-QNDA3Y- COIL fiETflOD-  IY--1  Introduction  When  using  response field  method  to  determine  coaxially  coil..  with  It  i s the  purpose  when  the  i s detected  is  by t h e  o f t h e d e r i v a t i o n s which  amplifier  be  accurately  system  will  be  the system i s being c a l i b r a t e d and a l s o when i t  i s i n i t s normal o b s e r v a t i o n a l mode.. will  coil  using t h i s approach.. As i n t h e case of the b r i d g e  method, the i n p u t v o l t a g e t o derived  magnetic  t h e sensor c o i l and at t h e c e n t e r of  f o l l o w t o show t h a t t h e freguency response can determined  freguency  The secondary  c o i l s i s a Mu-metal c o r e . . The f i e l d  sensor  the  the magnetometer system, a time v a r y i n g  i s c r e a t e d by a secondary c o i l .  aligned the  of  this  Finally,  a  comparison  be made between the freguency response which i s obtained  from the secondary c o i l b r i d g e method.  method t o t h e one  obtained  from  the  53  IV-2 Theory o f Operation  Secondary coil  Fig.,4.2-1  Sensor coil  The secondary c o i l  method at t h e time of o p e r a t i o n  The c i r c u i t which i s r e p r e s e n t a t i v e o f system Fig.  when  the  magnetometer  i t i s i n t h e o b s e r v a t i o n a l mode i s d e p i c t e d i n  4 . 2 - 1 . , The parameters C , 8 2  and  2  L  2  are  the  secondary  c o i l constants., I i s defined as: Y  --  jwM  4  where M i s the mutual inductance between the two c o i l s . the  load  on  the secondary  coil  (see F i g s . , 4 . 2 - 1  and  coils  and  the  fact  that  1  H is 0  for calibration  4.3-1)..  The sensor c o i l  c o n s t a n t s are R, , L, and C,.. Because of the geometry two  2  that i s adjusted t o remain  constant when t h e s i g n a l generator i s added purposes  - -  o f the  they have a Mu-metal core i n  common, there i s a p o s s i b i l i t y t h a t the response of the sensor c o i l i n the o b s e r v a t i o n a l mode could be inductance  between  itself  distorted  and the secondary c o i l .  purpose o f t h i s s e c t i o n t o e x p l o r e t h i s problem  by  mutual  I t i s the  in detail.,  Let Z, = R, + jio L,  4 > 2  _  2  54  4.2-4  J  Then,  the  equivalent  observation  4.2-5  R,  1  circuit  at  the time of operation f o r  becomes:.  (  E -0" Applyinq  Kirch o f f s  theorem,  t h e two  ( z. + z,)!, - E : - Y I  The  solution  input  make the  I,  2  ( zl  z5)(z^zj-  +  voltage to the a m p l i f i e r independent  following  o f any  upper  c o n d i t i o n must be  limit  bound on H.,  i s V =i z . g  (  3  In  and  i t i s assumed t h a t  by  the  coil  order  the secondary  to  coil,  satisfied:  lYl  1  4.2-7  t o t h e m a g n i t u d e o f I c a n be f o u n d by  I f the s e l f  first  4.2-6  Y*  e f f e c t s from  I (*.•*,)( Z , . Z . ) | » An  e q u a t i o n s are:  f o r I, i s :  I.The  circuit  i n d u c t a n c e o f t h e two  a l l the magnetic  coils  putting  a r e known  l i n e s of f o r c e s e t  c u t a l l t h e t u r n s of the second  a  coil,  up then  55  the  mutual i n d u c t a n c e  M i s given \l,Lj_  M =  It  i s not c e r t a i n  one to  coil  will  make t h i s  by: 4.2-8  what p e r c e n t a g e o f m a g n e t i c l i n e s  cut the turns of the other  last  eguation  into  an  coil,  s e t up  so i t i s s a f e r  ineguality.  M ^ JUT Then^  i t i s of interest  4  and  the c o i l s 8  =7.5  9  a  Ka., The c o i l  worst case  possible  freguency  2  9  1  u s e d , L,.-1000 H and L =.01 H. ,  reasonable  ' "  t o show t h a t :  >>  For  by  constants  w h i c h would be  g  =4.5 f  i n Table  1-1. A  M  highest  to  observe  m i c r o p u l s a t i o n s , would be a t 10 B z . S u b s t i t u t i n g t h e s e  numbers  that  one  720,000  3  u - L , L-j.  loo  1  T h u s , t h e c o n d i t i o n 4.2-7  by  Vow  expect  ineguality:  (Z,^ XV2j  result  might  at  C  the  into the l a s t  result,  are given  Also,  f o r the input  i s  easily  satisfied.,  The  final  voltage t o the a m p l i f i e r system i s given  =Z I,: 3  v  o k <  --  o b s  4.2-10  E.  Z + Z,  The  c o n c l u s i o n i s t h a t t h e i n d u c t i o n magnetometer a t t h e  of  operation  the secondary  f o r observation coil. ,  i s unaffected  time  by t h e p r e s e n c e o f  56  IV-3  Theory o f C a l i b r a t i o n  In  this  discussed., the  section  A time varying  calibration  magnetic  procedure  field will  will  natural  4.2-1  t h i s s i g n a l i s much l a r g e r  magnetic  fluctuations  so that  c a n be c o n s i d e r e d n e g l i g a b l e .  calibration  in  It sill  magnitude  £ as d e f i n e d  A circuit  be  be i n d u c e d i n t o  s e n s o r c o i l by a s i g n a l f r o m t h e s e c o n d a r y c o i l .  be assumed t h a t any  the  diagram  than  i nFig. f o r the  i s shown i n F i g . , 4 . 3 - 1 . ,  Secondary coil  Fig.,4.3-1  Sensor coil  T h e s e c o n d a r y c o i l method a t t h e t i m e o f calibration  In  order  to simplify  the  following  the picture  substitutions Z,  ^1  =  R , +•  will ju»L  f o r algebraic  be made:  ; j ;  K  9  calculations,  57  The e q u i v a l e n t c i r c u i t diagram  at  the time  of  calibration  becomes: 2  L  -4-  0  J  ©  L  There a r e t h r e e loop equations v h i c h  (Z.'Z.U.- - Y I Hc(l.-I,)-I R 5  result.  2  0  * - E.  S o l v i n g t h i s system o f eguations f o r I , :  Y  I. - ~  RoZ  where  c  *.*z  c  Z / ( Z •"B )=1/(1*jwC R ) i s approximately c  t  0  l  0  egual t o 1 i f  | w C , . R . | C< 1* The  laboratory  value  for R  0  c a p a c i t a n c e o f the secondary connecting  t h e secondary  c a b l e i s normally capacitance  i s 6 Kru coil  and  C  i n c l u d e s the  x  that  of  the cable  c o i l t o the s i g n a l generator.  l o n g , f r e q u e n t l y 100 meters  or  more..  The The  o f the c a b l e i s o f t h e order of 10 n f . , & t y p i c a l  c a b l e c a p a c i t a n c e i s 50 p f / f t . Then, t h e t o t a l c a p a c i t a n c e f o r 100m  i s approximately  * This i s n o t a necessary  15 n f .  The c a p a c i t a n c e o f the secondary  r e s t r i c t i o n , but a  convenience.  58  c o i l should  be much s m a l l e r than t h i s . , T h e r e f o r e , a t  w C R « (20-IT) (6 K) { 1 . 5 x 1 0 - 8 ) = 5 . 6 5 x 1 0 - 3 « 1 . z  10  Hz,  I t has a l r e a d y been  0  shown t h a t : Kz.+z.Kz^+zjl » T h e r e f o r e , the s i m p l i f i e d  Y  2  r e s u l t f o r the c u r r e n t I, i s :  i, -  Y  The r e s u l t i n g v o l t a g e drop a c r o s s the i n p u t to  the  amplifier  system when the magnetometer i s being c a l i b r a t e d i s V  Y Zs  V  The  ratio  and  4.3-1.  V  obs  /V | C0l  can  cal  V*L , - _Ii*l±._L  The emf induced  i n t h e sensor  3  :  4.3-1  r  now be found using eguations  Y  =1 z  4.2-10  4.3-2  E,  c o i l by n a t u r a l  magnetic  field  f l u c t u a t i o n s can be d e s c r i b e d by:  S  D  E = i - wBe  4.3-3  2.TT  where  S i s the s e n s i t i v i t y c o e f f i c i e n t  of the o s c i l l a t i n g magnetic  field.  4 . 2 - 1 i n t o 4.3-2, the r e s u l t i s :  and B i s the amplitude  Substituting  4.3-3  and  59  The  f i n a l observation  signal  to make i s t h a t t h e a.c. s i g n a l from the  generator i s given by:  E - E e >"* e  s  Then :  -  —  —  (R^ju-L^R.)  As shown p r e v i o u s l y , w C B « 1 l  order  to  make  4.3-6  and t h i s i m p l i e s t h a t Z^=B . , In  t >  0  4.3-6 independent o f freguency, the f o l l o w i n g  c o n d i t i o n must be met: (R  x  +  R ) >> a  u u L  4.3-7  z  Again, the worst c a s e would be a t t h e high freguency 10  Hz. , S u b s t i t u t e t h e values  B = l l x j . , B, =6 U t  end, at  and L =.01 H.  Then :  R+ x  R„  & * lo'  The c o n d i t i o n 4.3-7 i s e a s i l y s a t i s f i e d , . The f i n a l r e s u l t i s :  60  -  _L.il  ME  21T  Thus,  i t i s seen  that  U  l  4.3-8  * R . )  S  the voltage  output a t the time o f  c a l i b r a t i o n i s d i r e c t l y p r o p o r t i o n a l t o the output a t t h e time of  observation.  calibration the  will  I t i s expected  that  this  method  of  c o r r e c t l y produce the freguency response of  magnetometer system., The l a b o r a t o r y c i r c u i t  diagram  f o r the secondary  coil  method i s shown i n P i g . ,4.3-2. 6K - A W W V  1 Mu-metal core M-+.3 K  10K  Magnetometer amplifiers and f i l t e r s  W.4- pi  0  Fig.  The 4.3-3  4.3-2 The secondary c o i l  results  method i n the l a b o r a t o r y  from t h e l a b o r a t o r y a n a l y s i s can be seen i n F i g .  which shows the normalized  freguency  response  and i n  Fig.,4.3-4 which shows t h e phase response.. In both cases, the laboratory  data  i s plotted  as sguares and t h e smooth curve  r e p r e s e n t s the r e s u l t s o f a computer a n a l y s i s . ,  The  eguation  programmed i s :  V  -  T(iu,)  where X, and Z, a r e d e f i n e d by 3.3-1 and 3.3-3 and :  4.3-9  61  £  This  follows  -  R  from  +.  '  I  - ('OKX^^  eg., 3.4-7 and from the d e r i v a t i o n of the  a m p l i f i e r system t r a n s f e r f u n c t i o n given i n the appendix. The agreement  is  experimentally  between  r .'good  and t h e r e s u l t s  the  data  obtained  p r e d i c t e d from t h e computer  analysis., a p l o t with l a b o r a t o r y data can analogous  t o t h e computer  done simply the  now  be made  plot of Fig.,3.4-3.  T h i s can be  by comparing the l a b o r a t o r y r e s u l t s obtained  bridge method and the secondary c o i l method.  the  This  computer  i s shown i n F i g . , 4 . 3 - 5 . analysis  o f F i g . 3.4-3  from  On t h e same  graph appears the l a b o r a t o r y data of F i g . , 3.3.3-3 4.3-3.  which i s  and  The agreement  Fig.  between  and the l a b o r a t o r y  a n a l y s i s o f F i g . „JI.3-5 i s e x c e l l e n t . a  comparison  o f t h e r e s u l t s of the phase a n a l y s i s from  the bridge method and t h e secondary c o i l method Fig., good.  4.3-6.  The agreement  i s shown i n  between the two methods i s q u i t e  .1  .2  .3  .5  .7  1.0  2.0  3.0  5.0  Frequency (Hertz)  Fig. 4.3-3  The normalized amplitude response of the magnetometer system according to the secondary c o i l method ON ho  63  Fig.  4.3-4  The phase response of the magnetometer system a c c o r d i n g to the secondary c o i l method  Frequency  F i g . 4.3-5  (Hertz)  A comparison of the frequency response curves o b t a i n e d i n the l a b o r a t o r y from the secondary c o i l method and the b r i d g e method (see F i g . 3.4-3 and d i s c u s s i o n on page 50 about the s e p a r a t i o n between the, curves) .  65  ui  hi!  in  iii  180  i  II!  Ill in ill  TTT  135-  Secondary  •  B r i d g e method  i  UU  in  i ill  111 to cu cu  nu  ii U  CU CD n)  Mi! !i I i I ! I  I I I  III!  !  X  hil  i-U-L Tl~[  iii  III  i  m illl  ilii  I  11  ill  III!  iii Ui Illl  Mil  i  I I I !  i i i mi i i i Mil i 11  iu  m  Mi. iltr i  45  rmiT  iii  TTT Iii  90  60 01  M  4444  Ill III  ! i  u  hi MI-  I  i !  TT  c o i l method  I I !  TT  TTiTt  _!___  Wi  +  •II;  i  M  i. i  Mi  i ! I1  M  i Tl Tl Tl MIL  111 111  -45  i  i  I  Ml  III!! in  rn"  ! I  Ui!  II! II!  mua. TTT  iiii UU  ill!  Ill  ilii  iiii  iii  _LL i ! I  m TTTfi  UM  ill.  Ttffl Tl  ! I I I I  mm  TTtti Ul  III  -90  I ! I  .1  .3  III!  Illl  .5  1.0  2.0  445 llllll 3.0  Frequency ( H e r t z )  Fig.  4.3-6  A comparison of the phase response curves o b t a i n e d i n the l a b o r a t o r y from the secondary c o i l method and the b r i d g e method  66  CHAP-TE1 V1SI  V-1  A Laboratory  Now system  ABSOLUTE-CAL-IBRATIOH--  Approach  that the r e l a t i v e is  well  known,  s e n s i t i v i t y of it  is  the  necessary  Mu-metal  to  determine  a b s o l u t e s e n s i t i v i t y . , C o n c e p t u a l l y , the s i m p l e s t this  would  be  to  put  In  artificial  practice,  field  way  the  to  do  the sensor c o i l i n a known, uniform,  s i n u s o i d a l l y v a r y i n g magnetic the system.,  core  which  field it  and r e c o r d the response of  is  not  easy  to  create  an  would he uniform over a volume l a r g e  enough t o accomodate a t h r e e f o o t long sensor c o i l . , A l o g i c a l s o l u t i o n i s t o use the e a r t h * s n a t u r a l magnetic s i n u s o i d a l m i c r o p u l s a t i o n event i s o c c u r i n g . magnetometer  to  An a i r c o r e c o i l  a b s o l u t e amplitude of t h e m i c r o p u l s a t i o n event., The a i r  core  the  used  a  the  and  is  when  determine  coil  system  field  Mu-metal c o r e c o i l  precisely  are l o c a t e d about  100  yards  apart and i t i s assumed that the n a t u r a l event i s uniform over t h i s d i s t a n c e . , The response o f the Mu-metal core then  be  compared  to  that  determining the a b s o l u t e magnetometer.,  It  is  of  the  sensitivity necessary  to  air  core  of  the  do  this  system  can  system,  thus  Mu-metal at  only  frequency as the r e l a t i v e freguency response i s a l r e a d y  core one known  and l i n e a r i t y i s assumed., The f l u x , <$ ,through :a c i r c u l a r where  A=irR *N 2  is  coil i s ;  AB  5.1-1  the average c r o s s - s e c t i o n a l area times the  67  number o f t u r n s electromotive  and B i s t h e  f o r c e induced  V The  output  voltage  .  by  5.1-3. .. H{f) i s t h e t r a n s f e r  is  for  i s i n volts,  and  i s i n milli-gammas.,  „ , f  V^f  8' Substituting  these  g  m  the  yields  Ve f W  or S =2ira*10  - 6  to  express  vLf i s  two  strength eguations  B i s i n tesla  - I 0 V. f  5.1-4  -  5.1-5  b  m  to"" B into  5. 1-2  leads to the r e s u l t :  (^IO") ii'  B' i s a s i n u s o i d a l f i e l d , 5.1-6  following  i s i n micro-volts,  f  -  f  a  V  to the  eguations  'vL  into  research  f o r c e i n m i c r o - v o l t s and m a g n e t i c f i e l d  V  this  to  5.1-3  micropulsation  where  If  normalized  ew  milli-gammas. / T h i s l e a d s  B'  function  i s given  - V , * G - HlfW  ottt  convenient  e  a d.c. gain  and R i s t h e d . c . r e s i s t a n c e o f t h e c o i l . ,  electromotive in  5.1-2  i m p e d a n c e B- , which would be o b s e r v e d  V It  The  flux i s :  from t h e a m p l i f i e r system, with  a n d an i n p u t  value  strength...  A i§. dt  G  d.c.  field  by t h e c h a n g i n g  i£ dt  t M f  magnetic  then  B'=B sin2irft. 0  5 > 1  _  6  Substituting  :  = =  called  ( 21r A * \0' ) • f • B„ cos 2lr-f+ b  S^'f-  8 o>s lirft  5.1-7  e  the absolute  sensitivity  when  V e m f  is  68  measured i n m i c r o - v o l t s , B Hertz.,  i t a frequency of  Gauss)  will  sensitivy  induce  will  be  a  i n m i l l i - g a m m a s and  0  1 Hz,  potential  two  e g u a t i o n s can  system  and  the  substituting air  c o r e and  V  0  (  2  n  be  other  5.1-7 n  of  of  1 milli-gamma  1  micro-volt  in  C10—  and  a  the  1 Caner.,  1 Caner =  How  a field  freguency  1,uV/(my*Hz)  written, for  into  f o r the  air  core  t h e Mu-metal c o r e c o i l  5.1-3.  i s f o r the  one  The  superscript "1"  Mu-metal  coil  system  by  i s f o r the  core.,  °-  •  H 'V) l  5.1-8  i  V. * S i " - f • B. « *** • 6". w>  An  important  is  taken..  as  the  analysis  the  circuit this  o c c u r s when t h e r a t i o  T h i s i s g i v e n by  agreement has  method and of  result  system  on  these  computer  analysis  and l a b o r a t o r y  coil  system  method, t h e  freguency  i s determined  the computer.  The  o f two  transfer  bridge  response  by programming  circuit  diagram  i s shown i n F i g . , 5. 1-r 1..,, H<*>{f) c a n be  as t h e product  eguations  good f o r t h e work c o n c e r n i n g t h e  the secondary  a i r core c o i l  of  :  between  been v e r y  parameters  • H~(«)  -ggf^  f u n c t i o n s . , The  first  the for  expressed one  is  F i g . 5.1-1  Circuit  diagram f o r the a i r core system  70  H,  < l J  the  (f)»  approximately of a Butterworth type.  transfer  function  of  the  The second  t h i r d order low pass  is  filter,  H^Uf). In  order to determine  H,<*>(f),-  consider  the  following  figure :  R  From t h i s  diagram  where:  Z =  R* I + ju> C  x  Normalizing 7 /E t o one at d.c., X  H <*5(f) a  is  derived  R  x  H,* *^) becomes 1  i n Appendix 3.  T h e r e f o r e , the t r a n s f e r  f u n c t i o n f o r the a i r core c o i l system, normalized d. c.  H  result  was  =  H?»m.  programmed  response curve which was  on the computer.  determined  the  Mu-metal J  one.  The  t 2 ,  freguency  i s shown i n Fig.,5.1-2.. the  core systems are f l a t i n the low  range, the terms H< *(f) and H to  at  Href)  As the freguency response curves f o r both and  one  is : H (f)  This  to  air  core  freguency  { f ) i n 5 . 1 - 1 0 are both  egual  71  F i g . 5.1-2  Computer s i m u l a t e d frequency a i r core c o i l system  response  of  the  72  The absolute s e n s i t i v i t y determined  from  of the a i r c o i l , S<->^ i s e a s i l y 4  i t s geometry  and t u r n number.  f i n d t h e i n n e r r a d i u s of the c o i l , the  following  In order to method  was  used. A  Fig.  5 . 1 - 3 Method t o determine the i n n e r r a d i u s of a l a r g e coil  Three measurements were taken along the i n n e r circumference the c o i l as shown i n P i g . ,5. 1-3.  of  From t h e law o f c o s i n e s , the  angle C i s given by : C  =  co.-'/'-'+b'-c ) ^ 2cb 1  1  Then,  an  eguation  was  used f o r a c i r c u m s c r i b e d  order t o f i n d t h e r a d i u s . o-  K After  finding  determined  by  t h i s approach, *••*  *  T h i s eguation  2-mA  the  inner  adding fi.  "  c  2-<nB  2 «.»vC  radius  was  the t h i c k n e s s of the c o i l t o i t .  From  =74.511  the  outer  cm and R_„ „_ = 7 5 . 7 0 2 cm.  i n n e r  absolute s e n s i t i v i t y  is :  b  radius,  triangle in  +  o u T e r  can be found..  Now the  73  where r. and r  o  a r e the i n n e r and outer  radii,  respectively. .  S u b s t i t u t i n g i n the numbers l e a d s t o S^< >=.0557 Caner ± .2%. 1  The  ratio V  < 2 > 0  /V <»> 0  can be determined by measuring the  amplitude of the s i g n a l from t h e a i r core core  systems  when  and  the  Hu-metal  a s i n u s o i d a l m i c r o p u l s a t i o n event o c c u r s . ,  The output s i g n a l s from the event used i s shown i n F i g . 5.1-4.  Air  core  Feb. 17, 19 77' 21  h  27  m  Mu-metal core Z comp.  F i g . 5.1-4 The m i c r o p u l s a t i o n event used f o r the a b s o l u t e calibration  Peak-to-peak measurements were made from m i c r o f i l m by using travelling  a  microscope. .. The r a t i o s were taken with the r e s u l t  of : — ° —  =  1.4-1 + . 7 4 7 .  F i n a l l y , the remaining parameters values :  .24-5- H  2  t l._7„  of 5.1-10 have the f o l l o w i n g  74  G < > = 5:09 7'* 10* R.C O  that  another  G  R; «2>  = i|.0 K „ t . 135  RO. = 5130  Note  < z> = 4.01*10* ±.%  tU  1  si  R t > 2  t  R<2)  t . 1$  is  = 7.21  Kxi ±4%  = 1831 -v •. 1%  d i f f e r e n t from R, =7.5  a m p l i f i e r u n i t was  K__  used f o r the f i e l d  (p. 55)  because  observation.  The  value of S,.<z> a c c o r d i n g to 5.1-10 i s .0548 Caner ±6.4%.  V-2  A T h e o r e t i c a l Approach t o the Absolute  The  absolute  sensitivity  t h e o r e t i c a l p o i n t of view. semiprinciple strength  The  axes  a,b  Consider immersed  dealt  a prolate in  a  with  from  spheroid  magnetic  field  a  with of  0  approximating  i s i n reality a cylinder,  but  the c y l i n d e r as a p r o l a t e spheroid i s reasonably  and saves a great amount o f mathematical d i f f i c u l t y .  The r e s u l t s of an a n a l y s i s (1940)  be  H.  Mu-metal core of the sensor  accurate  can  Sensitivity-  indicates  that  shown  the  i n s i d e the c a v i t y i s given by  in  the  magnetic f i e l d :  book  by  Stratton  s t r e n g t h anywhere  75  A -  where:  The magnetic f i e l d permeability  ~~T~7 ( ~ 2 6 H n —  cv  f o l l o w s from  and  B=/u/u„H  where /* i s t h e r e l a t i v e  the p e r m e a b i l i t y of f r e e space.  u n i t s , /*„=4 rr*10- Henry/meter, ,  )  i-e '  /u- i s non-dimensional  7  In - HKS and i s  the p e r m e a b i l i t y o f the metal from which the core i s made.  By  l e t t i n g B = /A,H ., i t f o l l o w s that : a  0  B  -  rr  2  —  B  '  The t o t a l f l u x $ through the c o i l f o l l o w s from $  TT  •B•  Or, s u b s t i t u t i n g f o r B according  5.2-2.  c  5.1-1.  N t o 5.2-2 :  5.2-3 |+ 2^  Let  Cyu-l) A , i  S  ^  M - ^ b - N  , 5.2-4  7. Then $=SB„. f l u x would  The e l e c t r o m o t i v e f o r c e generated by the changing be :  76  Comparing  this  to  5.1-7,  i t is  seen  that S represents a  theoretical  s e n s i t i v i t y . , Expressing S i n u n i t s of Caner,  sensitivity  becomes :  For  a  Mu-metal  restriction i s  core, put  /x i s  on  the  of  the  ratio  of  order the  of  10 .  lengths  s  of  the  If a the  s e m i p r i n c i p l e axes: a  then  the  reduced  200  denominator  of  the  e x p r e s s i o n 5.2-4 f o r S can be  t o a s i m p l e r form,,  | + f___L ( u. ,) A « Z  The  error  of  this  approximation  expression f o r s e n s i t i v i t y  is  5.2-6  A  less  than  8%.  The  reduces t o :  mi  S The  o-b V 2  '  5.2-7  OLA  e x p r e s s i o n 5,2-1 f o r A can a l s o be s i m p l i f i e d .  If  ? ~,  then e-1 and A becomes ;  A- ^ By t h e b i n o m i a l expansion  0  '  [ " 2*  In 2 - In  5.2-8  :  Z  a-  a  a  Then 5.2-8 reduces t o :  A -where  e =2.71828. a  a'  The  In  e. b  final  5.2-9 a  simplified  expression  for  77  s e n s i t i v i t y f o l l o w s from 5 . 2 - 9 ,  S  The  a  P  P  -.„  importance  and  5.2-5  *  =  5.2-6. ,  Caner  10  5.2-10  o f t h i s e x p r e s s i o n i s t h a t the s e n s i t i v i t y of  the sensor  c o i l i s primarily  windings,  ti,  and  the  dependent  length  of  upon  the  the c o i l , a.  weight of the sensor c o u l d be reduced by using wire  and  adjusted  a t h i n n e r core.  S ,<2> = S  =-0583  +K»,r  Sapp-**  These  results  expression  Caner +  .0548  0  =  the  agreement  of  this  i n c r e a s e d without take p l a c e . . Hhen about  larger.,  2%  and  and  values are shown below :  6.4%  Caner  -0587  that  the  approximate  theoretical  a b s o l u t e s e n s i t i v i t y , eg. 5 . 2 - 1 0 ,  Another i n t e r e s t i n g  by  be  amongst 5.2-5  r e l i a b l y used to a i d e i n the design of f u t u r e sensor  result  gauge  Caner  imply  for  the  r e s u l t s of 5 . 2 - 1 0 ,  and experimental  of 5 . 1 - 1 0 i s g u i t e good,. The  2  lighter  overall  to o b t a i n the d e s i r e d l e v e l of s e n s i t i v i t y .  theoretical  S^ *  The  of  The l e n g t h and t u r n number could  In s p i t e of many approximations, the  number  observation  analysis  coils.  which has been made as  bound u n l e s s an u n d e s i r a b l e e f f e c t begins  to  +Ka(  ,  r  and  the  S  a f p r o  r a t i o a/b  a be  S  that  be  cannot  ^>100,  is  can  »  s t a r t to  the agreement becomes worse as a/b  There i s an important  reason  behind  diverge becomes  t h i s which can  be  * F o r a b s o l u t e c a l i b r a t i o n a c c o r d i n g to the b r i d g e method, see Appendix IV.  78  unfolded expressed  by onee a g a i n  t a k i n g a look  a t 5-2-2.  i n terms o f a d e m a g n e t i z a t i o n c o e f f i c i e n t  B  --  ^ l + N  d  be  coefficient given  seen  that  the  a  5.2-11  A  inverse  i sthe permeability  of  the demagnetization  which l e a d s t o t h e s e n s i t i v i t y  by eq- ,5-.2-10.'„- W r i t i n g r •—  yU, = — Na it  N . ,.  B.  N - Of can  be r e -  (/*-•)  d  It  B can  i s seen with  the help yU,  5.2-12  ab^A  o f 5.2-9 :  -  !  5.2-13  e„b Note t h a t t h i s not  the  ratio is  i s d e p e n d e n t upon t h e g e o m e t r y o f t h e c o r e  properties  of  t h e metal.,  but  yu, i s d e p e n d e n t upon t h e  a/b*, By making t h e s u b s t i t u t i o n 5-2-12 i n t o  5.2-13, i t  seen t h a t :  B  where:  For: the research,  most  * M&f$ 8  M ~r r -  recent  Mu-metal  a=:>45-.:7-2cm and b<?i,/Q~ r  5.2-14  0  cm  core  coil  {see T a b l e  used f o r  this  1-1).  This  79  leads  to  fx  i s s a t i s f i e d . , Then  »/JLX  desirable core  i s  yu&IO , t h e i n e g u a l i t y 5  5.2-14, yu-  the  eff  effective  i t s geometry.  a yu., . ,  This  i s a  permeability  of the  When  a/b*100,  the  y u » y u , b e g i n s t o weaken., When a/b=1000, //, = 1.5*10  €  of  the  dependent m e t a l , /x,  i s not a d e s i r a b l e r e s u l t  not always constant.,  to  environmental  upon t h e m e t a l . in  upon  / ~ f f i s no l o n g e r  This  by  because  dependent  permeability  is  f o r /u., o f 5 9 5 ' - S i n c e  result  ineguality and  a value  principally  upon yu, .  becomes e g u a l l y  The  as important.,  as the permeability  of a  I t may change s i g n i f i c a n t l y  conditions  s  such, as temperature  metal  according and s t r e s s  & p l o t o f yu, a s a f u n c t i o n o f a/b c a n b e s e e n  F i g . 5.2-1. / The  give  conclusion  a  shaped  good  of this  a n a l y s i s i s that  indication of the sensitivity  sensor c o i l  provided  s;> I i L ^  eg. „ 5.2-10  will  of a c y l i n d r i c a l l y  ISO .  b  Note: The  v a l u e o f b a t the bottom o f p. 78 i s n o t simply  e t e r o f the c o i l w i n d i n g s . ing  I t a c t u a l l y denotes an e f f e c t i v e r a d i u s  from the c r o s s - s e c t i o n a l area o f the Mu-metal core i t s e l f .  c o n s i s t s o f approximately of  the i n n e r diam-  one s t r i p  48 r e c t a n g u l a r s t r i p s .  The  resultcore  The c r o s s - s e c t i o n a l area  i s (3/4")(.014")=(1.905 cm)(.03556 cm)=.06774 cm . 2  2 are 48 s t r i p s , then the t o t a l c r o s s - s e c t i o n a l area i s 3.252 cm . fective, radius f e c t i v e r a d i u s f o r t h i s area i s ^3.252/lT = 1.02 cm.  I f there The e f -  80  1,000,000  500,000 300,000  100,000  50,000 30,000 4-1 •rl rH •rl  •8 '10,000 p. o  •rl S-l  4J  OJ  5000  e  o  0J 60  3000  1000'  500 300  100 50  10  500  100  1000  a. b. F i g . 5.2-1  The dependence o f the g e o m e t r i c permeab i l i t y /x, upon the r a t i o o f the l e n g t h s of the s e m i p r i n c i p l e axes o f a p r o l a t e spheroid,a/b  81  CHAPTER-VISUMMARY AND  Many  useful  results  investigations carried i n t o two  CONCLODINS HIARK-S-  have  obtained  out i n t h i s t h e s i s .  from  the  These r e s u l t s  v  fall  categories.  One category i s concerned of  been  the  magnetometer system.  with improvements i n the design As d e s c r i b e d i n Chapter I I , use  can be made of a Butterworth f i l t e r r a t h e r than a notch f i l t e r to reduce 60 Hz n o i s e . . T h i s guarantees a f l a t response i n the low freguency range and r e g u i r e s o n l y two components  rather  than s i x c a r e f u l l y  Chapter V, which was it  was  shown  that  concerned the  matched  electrical  matched components.  with the absolute  sensitivity  of  In  calibration,  the sensor c o i l i s  p r i m a r i l y dependent upon the number of windings and the l e n g t h of the c o i l . , The reduced  by  using  overall lighter  weight  of  gauge  However, f o r optimum performance,  the  wire  sensor  and  could  be  a thinner core. length  to  t h e diameter of the core must be w i t h i n a s p e c i f i c range..  One  of  the  reasons f o r not being a b l e t o i n c r e a s e s e n s i t i v i t y ad  infinitum induce  the r a t i o of the  i s that  small  thermal  currents  in  agitations the  at  sensor  the  atomic  level  and other e l e c t r i c a l  components.. T h i s i s known as Johnson n o i s e . The second category concerns the Two  methods  were  investigated  in  method and the secondary c o i l method. both  calibration  procedure.  great d e t a i l , the bridge I t was  determined  that  methods can produce r e l i a b l e r e l a t i v e freguency response  82  curves. coil  An advantage of the b r i d g e method over the  method  is  that  only  one  c a b l e i s needed between the  sensor c o i l and t h e remaining e l e c t r o n i c s , For  this  secondary  reason, the b r i d g e method w i l l  rather  than  two.  probably be used f o r  future calibrations. The a b s o l u t e c a l i b r a t i o n was comparing  calibration  was  A  theoretical  approach  to  the  calibrated absolute  the approximations made, with the l a b o r a t o r y r e s u l t s .  the  by  d i s c u s s e d which agreed w e l l , c o n s i d e r i n g a l l  t h e o r e t i c a l approach may of  performed  the Mu-metal core system to a p r e v i o u s l y  a i r core system.,  of  successfully  sensitivity  of  be used to o b t a i n a  good  This  indication  a c y l i n d r i c a l l y shaped c o i l before a  l a b o r a t o r y a n a l y s i s i s performed.  83  APPENDIX 1 METHODS OF DETERMINING THE  INDOCTANCE AND  CAPACITANCE OF A  COIL WITH FINITE RESISTANCE  It  is  not a t r i v i a l problem  t o determine  and c a p a c i t a n c e of a c o i l which has a f i n i t e sensor  coils  used  eguivalent c i r c u i t  the inductance  resistance.  The  with the magnetometer system a l l have the of F i g . A . 1 - 1 . R  V  L  —nM/wl—©—Tomr—i— lie Fig.  A.1-1  Eguivalent c i r c u i t  of a sensor  coil  A meter which i s normally used t o measure an inductance capacitance  operates  sensor c o i l s , Because  of  R i s of  on the  the  principle  order  of  few  R=0.  a  For the  thousand  ohms.  t h i s , o t h e r l e s s s t r a i g h t forward t e c h n i q u e s must  be employed  to  determine  compilation  of  those  laboratory.  I t i s s a f e to measure c o i l  ohm  a  that  or  L  and  technigues  C.  This  which  appendix  were  used  is in  a the  r e s i s t a n c e by using an  meters  Methods of Determining L  The most r e l i a b l e method wiring  a  signal  generator,  of  determing  the c o i l  L  follows  from  and a l o a d r e s i s t o r i n  84  series. r~  C J  F i g * a.1-2 F i r s t  method of determining L  The s i g n a l generator must produce a freguency  sinusoidal  signal  of  which i s low enough such t h a t :  >> |R+i<-L| Then  the  circuit  a  presence  of  a n a l y s i s . „. For  the  A.  capacitor  the  R-L -R  1-1  can be ignored i n the loop,  the  following  eguation i s t r u e :  Rearranging  t h i s i n t o a d i f f e r e n t form :  A.  Thus, slope  a  plot  of  2  versus w  2  i s a straight line  with  (L/R,) . 2  Another c i r c u i t shown  (V/Vjj)  1-2  in  which can be used  F i g . A.1-3.  This  c i r c u i t except t h a t t h e s i g n a l  is  for  determining  identical  generator  L  is  t o the previous  produces  a  sguare  85  wave i n s t e a d o f a s i n u s o i d . ,  1  c  i i i I  !  Fig.,A.1-3 Second method o f determining L  In order f o r t h i s method to produce the d e s i r e d r e s u l t s , are two c o n d i t i o n s which must be met.  there  These are :  A. 1-3 -!=- >> CRq  Then the scope p i c t u r e a c r o s s R  A  decreasing e x p o n e n t i a l . for  VJJ  w i l l show an i n c r e a s i n g and a  By measuring t h e time which i t  takes  t o o b t a i n one h a l f o f the f i n a l v a l u e , the inductance  can be found  from : A. 1-4  T h i s method of determining is  difficult  accuracy  L i s r e l i a b l e , but i n  practice i t  t o o b t a i n b e t t e r than one s i g n i f i c a n t f i g u r e of  because the time i n t e r v a l  of  r,,  t  i s not  measure using a v a i l a b l e scopes or c h a r t r e c o r d e r s .  easy  to  86  Methods of Determining  The  C-  most r e l i a b l e method to determine  using the c i r c u i t freguencies.  The  of  Fig.  a.1-2,  but  C f o l l o w s by again  this  time  at  high  c o n d i t i o n which needs to be met i s ; a. 1-5  A l s o , another  necessary c o n d i t i o n i s : a. 1-6  Bhen the freguency that V  4  becomes high enough, IjwLJ becomes so l a r g e  n e a r l y a l l the c u r r e n t passes through the c a p a c i t o r . , I f  i s the v o l t a g e drop a c r o s s R , the  from  t  capacitance  C  follows  A. 1-'7. , a. 1-7  V R On  the p l o t of l o g w versus l o g l , t h i s eguation i s v a l i d t  the 45°  s l o p i n g l i n e t o the r i g h t of the resonance  on  point.  o  loq  U  F i g . A.1-i+ The anti-resonance p o i n t  w  r  is  called  the  anti-resonance  freguency.„  The  coil  87  inductance and c a p a c i t a n c e are r e l a t e d t o w  r  8.  a c c o r d i n g t o A. 1-  T h i s e x p r e s s i o n a c t u a l l y d e f i n e s t h e resonance  but the resonance  point and the a n t i - r e s o n a n c e  t h i s case e s s e n t i a l l y  i f w  point  are i n  identical. u  Thus,  freguency,  r  I  --  / JIT  A.  and L a r e known, then C can be determined  r  1-8  using  A. 1-$. , I f a c a p a c i t o r i s wired p a r a l l e l t o the  coil  circuit  p o i n t w i l l be  of  F i g . , A.1-2,  then  s h i f t e d t o a lower freguency  the  resonance  i n the  according t o A. 1-IT" where  C  a  is  the a d d i t i o n a l c a p a c i t o r .  Both L and C can be determined as  A.1-8  and  by s h i f t i n g t h e resonance  point  A.1-90 are two eguations with two unknowns. A  practical  difficulty  resonance  point  is  with  this  shifted  trough o f F i g . A. 1 - 4 ; becomes  to  approach the  shallower  is  that  as  the  lower f r e g u e n c i e s , the and  w  r  more  poorly  defined. A  more  complete  review  of  these  techniques  determining L and C may be found i n Oeda and Hatanabe  for  (1975).  88  APPENDIX 2 THE TRANSFER FUNCTION OF THE AMPLIFIER SYSTEM-  Both methods o f c a l i b r a t i o n have i n common the system coil.  amplifier  which f i l t e r s and a m p l i f i e s the s i g n a l from the sensor In order t o  response  curves,  make the  computer  transfer  simulations function  of  freguency  f o r the a m p l i f i e r  e l e c t r o n i c s must be determined. A schematic f o r the a m p l i f i e r i s shown i n F i g . A. 2-1- The transfer function transfer  f o r each stage i s given i n F i g .  A.2-2.  The  f u n c t i o n f o r a l l o f t h e e l e c t r o n i c s of the a m p l i f i e r  system i s then :  A. 2-1  where :  w  t  -  l i r r  -AAAAAA*—i I—WWV—I  I.8J K A = input r output  i m p i J U m c * i HH-.3 in>p«donce<  K  \0-fT-  5K AAAAAr  2.5SK  o—o—wwv (00  si  200 ja O u t » & t  so X L  1 •\5V Fig.  A.2-1  Mu-metal  core  amplifier  system  Voltage Divider VS - IS  Fig.  A.2-2  Transfer functions for f i l t e r s of the Mu-metal core amplifier s y s tem  An com amplifier  ° i  1 °  (jw)  + w  *  N  RC  u» « 1 2 T T ft  * Butterworth f i l t e r  • st of 1 — degree.  Corner frequency = 6 Hz.  91  APPENDIX 3 THE TRANSFER FONCTION FOR THE THIRD ORDER LOW PASS FILTER  The c i r c u i t diagram which  is  used  f o r t h e t h i r d order low  pass  as a p a r t o f the a i r core magnetometer  filter system  e l e c t r o n i c s i s shown i n F i g . A . 3 - 1 .  R  _L_ . C  Fig.  A.3-1 The t h i r d order low pass  filter  Four fundamental e g u a t i o n s which can be w r i t t e n a r e e g u a t i o n s A.3-1 through  given ~. by  A.3-4.  A. 3-1 A. 3-2 A. 3-3 V.--  For  Kc", Z ,  these e g u a t i o n s , Z,, Z  of C,, C. and C,.  K - 2  ^  t  and Z  A. 3-4  3  a r e t h e complex impedances  Using the top t h r e e e g u a t i o n s ,  e g u a t i o n that r e s u l t s i s :  the  matrix  92  -2,  o  2. -2.  A. 3-5  \  I  The s o l u t i o n f o r I , i s :  Z R V . Z , -  I.Baking  the s u b s t i t u t i o n f o r V  o  s o l v i n g f o r t h e r a t i o V„ /V  s  V  VeR  +  Vs?.2_  A. 3-6  from A.3-* i n t o A.3-6 and then  leads to :  (R*Z,K 2 * ^ + 2 , 2 , + R + 2 , R ) - R Z . K U Z . + R^+R^Z^R+Z,)  "  s  l  l  A. 3-7  The f i n a l s u b s t i t u t i o n s t o make a r e f o r Z ,  Z  (  A.3-7  and  x  Z^  into  where :  A f t e r rearrangement o f terms, t h i s l e a d s t o the r e s u l t o f A . 3 8 where s=jw.  K \J  %  '  J V C A C J R * * sX{2R*CFCCt4. R^Cl-^C.C^ +  Let  H  2  be  S£3RC,* 2 R ( l - K ) C  the t r a n s f e r  function  2  + RC,  2R*C,C}  ] + I*]  S  A. 3-8  o f t h e t h i r d order low pass  f i l t e r normalized to one a t d.c. Then :  u  _ J L Vo  A. 3-9  93  APPENDIX 4121  ABSOLUTE CALIBRATION ACCORDING TO- THE- BRIDGE ~METHOD  Using in  the developement of the b r i d g e method as  III-2,  a  relationship  can  v a r y i n g e x t e r n a l magnetic f i e l d , the  sensor  coil,,  This  is  be  3.2-7  the emf  calculation  s e n s i t i v i t y as d i s c u s s e d i n Chapter Eg.  d e r i v e d between the time  B , and  a  generated  of the  absolute  driving  voltage  the f l u x through the c o i l .  This r e s u l t i s :  E  A.4-1  L  It  is  in  V.  i s a r e l a t i o n s h i p between the  of the bridge and  discussed  necessary t o express <fr i n terms of H, and  L i n terms of  c o i l geometry. , For a s o l e n o i d o f l e n g t h 'J. and  cross-section  area  c l o s e l y wound  with  A such t h a t end N  turns  c u r r e n t sheet,  of  e f f e c t s are n e g l i g a b l e , and thin  the two  wire so that the winding resembles a  expressions  f o r 4 and  L are : A. 4-2 A. 4-3  where ju. i s the p e r m e a b i l i t y 0  relative  permeability.  A.4-1, the r e s u l t i s :  of  free  space  S u b s t i t u t i n g these two  and  k  is  the  eguations i n t o  APPENDIX  THE  in  ABSOLUTE CALIBRATION  Using  the  III-2,  a  varying the  sensor  sensitivity Eg. of the  ACCORDING TO  d e v e l o p e m e n t of relationship  external  magnetic  coil. as  discussed  3.2-7  bridge  This  the  can  is  method  the  and  the  necessary  geometry.  area  A such t h a t N  turns  of  current  sheet,  the  two  generated  of  the  driving  coil.  voltage  This result i s : A.4-1  L  of  length  H, and  JL and  negligable,  wire so  that  expressions  the  for 4  L i n terms  and  c l o s e l y wound  winding  and  where  ju i s t h e 0  relative A.4-1,  the  ~- M. &  permeability  permeability. result  is :  of  L are  resembles a : A  M'A/i  free  of  cross-section  4> ~- ^ i A H L  in  absolute  -  e f f e c t s are thin  time  V.  f l u x through the  a solenoid  end  discussed  between t h e  calculation  t o e x p r e s s <fr i n t e r m s of  For  as  emf  i s a r e l a t i o n s h i p between t h e  coil  with  BRIDGE METHOD  derived  B , and  a  %  is  be  field,  THE  bridge  i n Chapter  —  It  4  -~ 4  2  A.4-3  space  Substituting these  two  and  k  is  the  eguations into  94  The sensor used f o r t h e f i e l d o b s e r v a t i o n (see F i g - 5.14)  was  not  available  d i r e c t comparison obtained from sensors  to  test  i n t h e l a b o r a t o r y so t h a t a  cannot be made between  the s e n s i t i v i t y  A.4-4 and t h e r e s u l t s on page 77.  as  The d i f f e r e n t  a r e n e a r l y i d e n t i c a l , s o t h e best t h a t can be done a t  t h i s point  i s to  make  the s e n s i t i v i t y  calculation  using  another s e n s o r . At of  a freguency o f 1 Hertz, an i n p u t s i g n a l t o the b r i d g e  . 7 5 mV p-p caused an output from the a m p l i f i e r o f 1 3 V p-p.  For N = 5 0 , 0 0 0 , l=.9144m B=/* H, 0  where  and  /*„ = 4 i r x 1 0 * ,  R =38787a,  A.4-4  indicates  that  i s 13.3x10* m!f. , The output from the  b r i d g e or the i n p u t t o t h e a m p l i f i e r i s 13.0V/2x10s=65/«.V. , To f i n d t h e emf i n t h e sensor c o i l , use can be made o f e g . , 3 . 4 - 6 . The f o l l o w i n g v a l u e s a r e used: R, = 1.83 Ko. L,=930 H C, =4.4juf Z = 44.3 -Ka II 10 Kfl.= 8.158 Kn. f  R = 38.8 K*a %  T h i s i n d i c a t e s t h a t t h e r a t i o of the output from t h e b r i d g e t o t h e emf i n the sensor c o i l i s . 7 7 1 . sensor c o i l i s 65/uV/. 771=84.3/*.V.  B  T h e r e f o r e , the emf i n the  The s e n s i t i v i t y i s then :  IS.BMcfm*  T h i s i s a r e a s o n a b l e r e s u l t when i t i s compared t o the r e s u l t s on page 77.  LIST OF REFERENCES CONSULTED  Campbell, W.H. (1967) I n d u c t i o n Loop Antennas f o r Geomagnetic F i e l d V a r i a t i o n Measurement, ESSA T e c h n i c a l Report, ERL 123-ESL 6. Kanasewich, E.R. (1973) Time Sequence A n a l y s i s i n Geophysics, Edmonton, The U n i v e r s i t y o f A l b e r t a P r e s s , pp 170-186. K o l l a r , F. and R u s s e l l , R.D. (1966) Seismometer A n a l y s i s U s i n g an E l e c t r i c C u r r e n t A n a l o g , BSSA, 56, 1193-1205. Lewis, W.E. and P r y c e , D.G. (1965) The A p p l i c a t i o n o f M a t r i x Theory to E l e c t r i c a l E n g i n e e r i n g , London, E. & F.N. Spon L t d , pp 141-171. Schwartz, M. (19 72) P r i n c i p l e s o f E l e c t r o d y n a m i c s , San F r a n c i s c o , McGrawH i l l Inc. S l u r z b e r z , M. and O s t e r h e l d , W. (1944) E l e c t r i c a l E s s e n t i a l s of Radio, McGraw-Hill I n c . , pp. 269-303. S t r a t t o n , J.A. (1941) E l e c t r o m a g n e t i c Theory, New York and London, McGraw-Hill Inc. Ueda, H. (1975) U n i v e r s i t y of B r i t i s h M.Sc. T h e s i s .  Columbia, G e o p h y s i c s ,  Ueda, H. and Watanabe, T. (1975) Comments on the Anti-Resonance Method t o Measure the C i r c u i t Constants o f a C o i l Used as a Sensor of an Induct i o n Magnetometer, The S c i e n c e Reports o f the Tohoku U n i v e r s i t y , S e r i e s 5, G e o p h y s i c s , V o l . 22, No. 3-4, pp 129-135. Ueda, H. and Watanabe, T. (1975) S e v e r a l Problems about S e n s i t i v i t y and Frequency Response of an I n d u c t i o n Magnetometer, The S c i e n c e Reports o f the Tohoku U n i v e r s i t y , S e r i e s 5, Geophysics, V o l . 22, No. 3-4, pp. 107-127.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052942/manifest

Comment

Related Items