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Studies of dense plasmas in laser generated shock wave experiments Parfeniuk, Dean Allister 1982

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AN INVESTIGATION OF THE EFFECT OF CONDUCTING SHELLS ON ELECTRODELESS BREAKDOWN by DEAN ALLISTER PARFENIUK B. A. Sc., The U n i v e r s i t y  of B r i t i s h Columbia, 1981  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS)  We accept  this  t h e s i s as  to the r e q u i r e d  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA November ©  1982  Dean A l l i s t e r P a r f e n i u k ,  1982  In presenting  this  thesis i n partial  fulfilment of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y of B r i t i s h Columbia, I agree that it  freely  a v a i l a b l e f o r r e f e r e n c e and study.  agree that permission f o rextensive for  t h e L i b r a r y s h a l l make copying o f t h i s  understood that copying o r p u b l i c a t i o n o f t h i s financial  gain  Department  of  PHYSICS  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3  »E-6 C3/81)  0 . , .  It i s thesis  s h a l l n o t b e a l l o w e d w i t h o u t my  permission.  Date  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . for  I further  ICs/fi-Z.  Columbia  written  ii  ABSTRACT  Two effect low  t h e o r e t i c a l models were developed of  conducting  predict  the  s h e l l s on e l e c t r o d e l e s s breakdown at  f r e q u e n c i e s ( < 1 kHz). I f g l a s s s h e l l s c o n t a i n i n g gases  at low p r e s s u r e s are immersed field, of  to  in  an  alternating  of s u f f i c i e n t magnitude, these bulbs w i l l  light.  Together  electric  emit p u l s e s  the t h e o r e t i c a l models p r e d i c t  the form  of the c a l i b r a t i o n curves of these b u l b s , which are p l o t s of the pulse r a t e as a f u n c t i o n of the magnitude of the a p p l i e d field. Experimental v e r i f i c a t i o n Furthermore, neoprene  was  the  surface  measured  range 40 Hz to 1 kHz.  as  of these models was  conductivity a  of  observed.  paxolin  f u n c t i o n of frequency  and i n the  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF TABLES  vi  LIST OF FIGURES  •  vii  ACKNOWLEDGEMENTS CHAPTER  ix  1 INTRODUCTION  1  CHAPTER 2 THEORY  5  2.1 BASIC MECHANISM  5  2.2 GENERAL THEORY WITH CONDUCTIVITY INCLUDED  14  2.3 THEORY PARTICULARIZED SPHEROIDAL COORDINATES  22  TO  PROLATE  2.4 THE CYLINDRICAL LIMIT OF A SPHEROID  33  2.5'APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PERPENDICULAR TO THE AXIS OF THE SPHEROID  37  2.6 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PARALLEL TO THE AXIS OF THE SPHEROID  47  2.7 APPROXIMATE SOLUTION OF THE "SWITCH PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE X-DIRECTION  49  ON"  2.8 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE Z-DIRECTION  52  2.9 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR A SPHERE WITH THE APPLIED FIELD IN THE Z-DIRECTION  54  2.10 EXACT SOLUTION OF THE "SWITCH PROBLEM FOR A CYLINDER WITH THE APPLIED FIELD PERPENDICULAR TO ITS AXIS  57  2.11 RESPONSE OF A CONDUCTING ALTERNATING ELECTRIC FIELD 2.12 "RESET SCREENING" MODEL  SHELL TO AN  ON"  60 64  iv  2.13 LINKING OF THE "RESET SCREENING" AND "STATIC SCREENING" MODELS  72  CHAPTER 3 EXPERIMENTS  75  3.1 INTRODUCTION 3.2 VERIFICATION OF THE "STATIC SCREENING" MODEL  78  3.3 EXPERIMENTAL VERIFICATION OF FREQUENCY DEPENDENCE OF THE CONDUCTIVITY OF NEOPRENE  94  3.4 EXPERIMENTAL VERIFICATION OF WHICH EXPLAINS THE FORM OF THE CALIBRATION CURVES OF THE BULBS  THE  75  THE  THEORY  CHAPTER 4 CONCLUSIONS  105 116  4.1 INTRODUCTION  116  4.2 CONCLUSIONS  117  BIBLIOGRAPHY ' APPENDIX A  ..122 123  A.1 MOMENT OF EQUATION 2.3-19 FOR THE Z-DIRECTION PROBLEM  123  A. 2 MOMENT OF EQUATION 2.3-19 FOR THE X-DIRECTION PROBLEM  125  APPENDIX B  1 28  B. 1 SOLUTION OF THE X-DIRECTION PROBLEM FOR AN ASYMPTOTICALLY LONG SPHEROID  128  B.2 1-DIMENSIONAL EIGENVALUE PROBLEM  132  B. 3 3-DIMENSIONAL EIGENVALUE PROBLEM  133  APPENDIX C C. 1 SOLUTION OF THE Z-DIRECTION PROBLEM FOR  136  AN ASYMPTOTICALLY LONG SPHEROID  136  C.2 1-DIMENSIONAL EIGENVALUE PROBLEM  141  C.3 3-DIMENSIONAL EIGENVALUE PROBLEM  142  V  APPENDIX D D. 1 SOLUTION OF THE X-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT APPENDIX E E. 1 SOLUTION OF THE Z-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT  1 49  149 1 54  154  vi  LIST OF TABLES Page TABLE I,  STATIC SCREENING DATA  TABLE I I , CONDUCTIVITY OF PAXOLIN AND NEOPRENE  88 94  TABLE III CONDUCTIVITY OF NEOPRENE (RUN WITH 3 SHEETS) ..100 TABLE IV, CONDUCTIVITY OF NEOPRENE (RUN WITH 5 SHEETS) ..100 TABLE V,  BULB DATA  107  TABLE VI, EXPERIMENTALLY OBTAINED TIME CONSTANTS  112  TABLE VII EXPERIMENTALLY OBTAINED PHASE ANGLES  114  vii  LIST OF FIGURES Page FIGURE 1, TYPICAL BULB  8  FIGURE 2, FIELD INTERNAL TO THE BULB  8  FIGURE 3, CALIBRATION CURVE FOR A NON-CONDUCTING BULB  ....10  FIGURE 4, PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB  10  FIGURE 5, SAMPLE CALIBRATION CURVE  13  FIGURE 6, RESULTS OF THE PROPOSED THEORETICAL MODELS  13  FIGURE 7, ELLIPTICAL COORDINATES  23  FIGURE 8, PROLATE SPHEROIDAL COORDINATES  23  6  FIGURE 9, THEORETICAL AGAINST Qt/Re  1 X 0  (r*,t)/e E Cos* PLOTTED 0  .  ..  41  x  FIGURE 10,THEORETICAL  5  AGAINST flt/Re FIGURE 11,THEORETICAL AGAINST Gt/Re FIGURE 12,THEORETICAL  . 6 ( r , t ) / c E Cos t PLOTTED .. « ( r , t ) / t E Cos • PLOTTED  0  ( r , t ) / e E Cos • PLOTTED 0  3  X  5  X  42  0  0  43  0  AGAINST Qt/Re . 7  X  0  ;  44  FIGURE 13,DEMONSTRATION OF THE CONVERGENCE OF 6 ,6 AND 6 . . ? 3 X  X  45  7X  FIGURE 14,DEMONSTRATION OF THE CONVERGENCE OF 6 TO 6 . 7  x  46  FIGURE 15,SPHERICAL COORDINATES  55  FIGURE 16,CYLINDRICAL COORDINATES  55  FIGURE 17,DEFINITION OF THE RADIUS OF A SPHEROID  59  FIGURE 18,DEFINITION OF PHASE SHIFT  63  FIGURE 19,DEFINITION OF "a" FOR A TRIANGULAR  WAVE  70  FIGURE 20,DEFINITION OF "a" FOR A CLIPPED TRIANGULAR WAVE  70  FIGURE 21,RESULTS OF THE "STATIC" AND SCREENING" MODELS .  73  "RESET  FIGURE 22,DEVICE FOR PRODUCING ELECTRIC FIELDS  76  vi i i  FIGURE 23,DEVICE FOR MEASURING ELECTRIC FIELDS  77  FIGURE 24,TUBE ORIENTATION WITH RESPECT TO CARTESIAN COORDINATES  79 X /X  FIGURE 25,A PLOT OF THE THEORETICALLY PREDICTED FIGURE 26,CONFIGURATION  M  x  82  OF THE APPARATUS USED TO  MEASURE (EA/E, ) FIGURE 27,CONFIGURATION OF THE APPARATUS USED TO  85  MEASURE (EA/E, )„ FIGURE 28,A PLOT OF THE THEORETICAL AND EXPERIMENTAL X„ A i FIGURE 29,CONDUCTIVITY OF PAXOLIN PLOTTED FREQUENCY  86  x  89 AGAINST 92  FIGURE 30,CONDUCTIVITY OF NEOPRENE PLOTTED AGAINST FREQUENCY  93  FIGURE 31,NEOPRENE DIMENSIONS  '.  95  FIGURE 32,CONFIGURATION OF THE APPARATUS USED TO MEASURE fl/e FIGURE 3.3,EXPERIMENTAL e / c , FOR NEOPRENE, PLOTTED AGAINST FREQUENCY  102  FIGURE 34 /EXPERIMENTAL n, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY  103  98  0  FIGURE 35,EXPERIMENTAL  Q/c, FOR NEOPRENE, PLOTTED  AGAINST FREQUENCY  104  FIGURE 36,SODA-LIME  BULB CALIBRATION CURVE FOR 40 Hz  108  FIGURE 37,SODA-LIME  BULB CALIBRATION CURVE FOR 60 Hz  109  FIGURE 38,SODA-LIME  BULB CALIBRATION CURVE FOR 80 Hz  110  FIGURE 39,SODA-LIME  BULB CALIBRATION CURVE FOR  100 Hz ...111  FIGURE 40,PHASE SHIFT OF THE BULB'S PULSES AT THRESHOLD  115  FIGURE 41 ,RESULTS OF THE PROPOSED THEORETICAL MODELS  118  FIGURE 42,PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB FIGURE 43,APPROXIMATE  118  FORM OF  FIGURE 44,EMPIRICAL FIT OF  *  z  6,  145 147  ix  ACKNOWLEDGEMENTS  I  would  like  to thank my s u p e r v i s o r Dr. F. L. Curzon  f o r suggesting t h i s p r o j e c t and suggestions  were  most  useful  t h e o r e t i c a l aspects of t h i s  f o r h i s supervision. His i n both the experimental and  project.  I would a l s o l i k e to thank A. Cheuck the  experimental  apparatus,  the computer programming discussions.  and  for h i s help  with  L. Da S i l v a  f o r h i s h e l p with  M.  f o r many  Feeley  useful  -  1 -  CHAPTER 1 INTRODUCTION  In the past  few years,  there has been concern about the  p o s s i b l e e f f e c t s of s t r o n g , on  an  individual's  low frequency,  health.  Those  electric  who  may  i n c l u d e employees of u t i l i t y companies and l i v e near t r a n s m i s s i o n term  who  l i n e s s i n c e these people r e c e i v e  long  to  these  develop d e v i c e s  to  monitor  could  be  fields.  Hence, i t i s d e s i r e d t o  electric  used  in  field  studies  determining the h e a l t h hazards of these A l l currently based on c u r r e n t most  devices  required into the  existing  estimate  disadvantages.  field  For  and  of the e l e c t r i c  plates.  Clearly  example,  convert  field  these  are  the  result  i s l o c a t e d between devices  the s e n s i t i v i t y  c o n s i s t of metal p a r t s .  u s e r s of the device  electric  monitors  to  have  many  of the device  the  field.  The  i s a f f e c t e d by harmonic d i s t o r t i o n s i n the f i e l d .  These d e v i c e s  It  towards  p l a t e c a p a c i t o r . The e l e c t r o n i c s  depends on i t s o r i e n t a t i o n with r e s p e c t accuracy  directed  These  i n d u c t i o n . For example, the primary part of  i s a parallel  capacitor  doses.  fields.  electric  to measure the i n d u c t i o n  an  be concerned  individuals  exposures  devices  fields  has  been  receiving e l e c t r i c a l found  f i e l d s can be  electrical  This  breakdown  can  result  in  shocks.  that the magnitude of a l t e r n a t i n g  measured of .gas  with  a  device  based  inside . insulating  on  shells.  -  Harries  established  1  2 -  that e l e c t r i c a l  breakdown occurs when  g l a s s bulbs f i l l e d with gas at low p r e s s u r e s are exposed alternating electric known as an of  f i e l d s of s u f f i c i e n t  magnitude. T h i s i s  " e l e c t r o d e l e s s breakdown". I t t u r n s out the r a t e  breakdown occurrence  the e l e c t r i c  f i e l d and  i s p r o p o r t i o n a l to the magnitude of  the frequency.  These  bulbs  c a l i b r a t e d so that the magnitude of an e l e c t r i c determined An  to  can  f i e l d can  be be  from the breakdown r a t e of a bulb.  electric  c u r r e n t l y being  field  monitor  developed  at  based  the  on t h i s phenomena i s  University  of  British  5  Columbia.  This  sort  of  device has many advantages over a  d e v i c e based  on c a p a c i t i v e c o u p l i n g .  glass  i s s p h e r i c a l , the breakdown r a t e is. independent  of  bulb  For  example,  the o r i e n t a t i o n of the bulb with respect  Furthermore  the  breakdown  rate  of  to  the  if  the  field.  bulb  s u b s t a n t i a l l y a f f e c t e d by small harmonic d i s t o r t i o n s electric  f i e l d . Each breakdown r e s u l t s i n a p u l s e of  which can be t r a n s m i t t e d from the v i c i n i t y of the  pulse  counting  device  f i b e r . Thus the e l e c t r o n i c s without  a  nonconducting  (pulse counter) can be  noise  problem.  compatible  with  Furthermore,  this  1  not  in  the  light,  bulb  to  optical shielded  device  d i g i t a l processing. F i n a l l y ,  d e t e c t i o n d e v i c e c o n t a i n s no metal p a r t s , hence the of  is  a f f e c t i n g the f i e l d near the bulb. T h i s reduces  electronic directly  with  the  the  the is the  chances  a user r e c e i v i n g an e l e c t r i c a l shock are minimized. In this thesis, the results of research directed  Harries,w.L. and Von Sect. B64,915(1951)  Engel,A.,  Proc.  Phys.  Soc,  London,  -  towards  understanding  presented.  the  -  operation  of  the  bulbs  are  In p a r t i c u l a r , the e f f e c t of the c o n d u c t i v i t y of  the bulb's s h e l l was the  3  conductivity  cleanliness  of  studied. This effect  of the  the  shell  glass  and  can  i s important s i n c e  change  due  to  the  humidity. T h i s a f f e c t s  accuracy of the monitor. T h e o r e t i c a l models  were  the  developed  to p r e d i c t the e f f e c t of the conduction and experiments  were  d e v i s e d to t e s t these models. Chapter  2  c o n t a i n s the d e r i v a t i o n s of the  theoretical  models. I t t u r n s out two models were r e q u i r e d to e x p l a i n o p e r a t i o n of a bulb with a conducting are  called;  the  "static  s c r e e n i n g " model. These breakdown, E , A  and  f  B  shell.  screening"  models  A  predict  f . The  . The mathematical  The  are  A  r e s u l t s p r e d i c t the form  of  B  versus E  A  of at  the fixed  d e t a i l s appear i n the appendices at the  thesis.  1 the experimental apparatus s c r e e n i n g " model was  presented  measuring  in  tested  section  the e l e c t r i c  2.  field  i s d e s c r i b e d . Next the directly.  This  was  inside  consequence  of t h i s experiment  neoprene was  measured as a  range  frequency  experimental r e s u l t s are presented i n chapter 3. In  section "static  the  "reset  , as a f u n c t i o n of the a p p l i e d f i e l d s t r e n g t h ,  frequency,  end of the  models  model and the  c a l i b r a t i o n curves, which are p l o t s of f f  These  the  was  The  results  accomplished  conducting  tubes.  by A  that the c o n d u c t i v i t y of  function  of  frequency  in  the  0 Hz to 1 k H z . The  frequency  neoprene was  dependence  measured with a  of  different  the  conductivity  technique  and  of the  -  results  are  given  in  4 -  s e c t i o n 3. T h i s was accomplished  measuring the magnitudes and phases of neoprene  sheets  and  the  voltage  by  across  the c u r r e n t through the sheets. These  r e s u l t s agreed with those from the p r e v i o u s s e c t i o n . Finally,  the  "static  screening"  model  and  "reset  s c r e e n i n g " model were t e s t e d s i m u l t a n e o u s l y . The r e s u l t s are presented  in  section  4. The t h e o r e t i c a l  curve" and the t h e o r e t i c a l  "static  screening  " r e s e t s c r e e n i n g c u r v e " were f i t  to a t y p i c a l c a l i b r a t i o n curve. The parameters obtained this  f i t were  used  to  g l a s s i n two independent were  consistent.  Hence  from  c a l c u l a t e the c o n d u c t i v i t y of the  ways. I t turned  out  these  values  these models were determined  to be  valid. Chapter  4 i s a summary  t h e s i s , emphasizing  of  the  their practical  main  results  significance.  of  the  -  5 -  CHAPTER 2 THEORY 2.1  BASIC MECHANISM  I t has been e s t a b l i s h e d by H a r r i e s containing  gases  emit p u l s e s  of l i g h t when p l a c e d  field  of s u f f i c i e n t  pulse and  at  rate  pressures  in a specific  regime would  i n an a l t e r n a t i n g  shell  field.  An e x p l a n a t i o n  electric  of these  below with the assumption that the  bulb's  i s nonconducting. T h i s means the e l e c t r i c  i n s i d e the bulb i s e s s e n t i a l l y u n a f f e c t e d  by  the  field  shell.  A  bulb i s shown i n f i g u r e 1.  To  explain  initially As  bulbs  was approximately p r o p o r t i o n a l t o the magnitude  f i n d i n g s i s given  typical  glass  magnitude. Furthermore, i t was found the  frequency of the e l e c t r i c  glass  that  1  there  the  operation  of  a  bulb,  suppose that  i s no e x t e r n a l or i n t e r n a l e l e c t r i c  the e x t e r n a l  field  i s increased,  fields.  the i n t e r n a l f i e l d  also  i n c r e a s e s . A f r e e e l e c t r o n i n t.he bulb tends t o move i n the opposite  direction  t r a v e l s along, in  the  bulb.  denoted  The  field  average  inside  distance  an  the bulb. As i t with  X . The mean f r e e path i s much s m a l l e r T  electric  I f the e l e c t r i c  field  E  0  field  will  than the  i s below  , where the t h r e s h o l d  of l i g h t , the f r e e e l e c t r o n s  atoms  electron travels  d e f i n e d as the minimum f i e l d which causes the bulb pulses  gas  i s the thermal mean f r e e path which  dimensions of the bulb. threshold  the  i t i s a f f e c t e d by c o l l i s i o n s  between c o l l i s i o n s be  to  the  field is to  emit  i n the b u l b do not gain  -  enough energy dimension below  6 -  to i o n i z e the gas molecules on impact  of the b u l b . There are two  threshold  which  prevent  mechanisms  electrons  over the  at  from  becoming  e n e r g e t i c enough to i o n i z e gas m o l e c u l e s . F i r s t l y , molecules  absorb  a  T  to  the  electron  d i s t a n c e , to reach the molecule  on  ionization  to  One  of the bulb i s  gain  energy  collision.  gas  energy  which occur at an average d i s t a n c e  a p a r t . Secondly, the dimension allow  the  f r a c t i o n of the f r e e e l e c t r o n ' s  due t o thermal c o l l i s i o n s X  fields  too  small  enough energy, over  required  to  ionize  this  a  gas  can say the mean f r e e path f o r  i s g r e a t e r than the dimension  of the b u l b ,  where  the mean f r e e path f o r i o n i z a t i o n , which w i l l be denoted is  defined  as the average d i s t a n c e an e l e c t r o n must  travel  in order to gain enough energy to i o n i z e a gas molecule. the  magnitude  of  the e l e c t r i c  field  decrease s i n c e the thermal c o l l i s i o n s of  a  free  fields, bulb.  electron's  energy.  the  dimension  As will  fraction electric of  the  these c o n d i t i o n s , the e l e c t r o n p o p u l a t i o n w i l l  approximately distance  a  With l a r g e enough  X, w i l l become smaller than Under  i s i n c r e a s e d , X, only take  A,,  equal  double to  every X,.  time  the  electrons  move  a  Hence the e l e c t r o n p o p u l a t i o n w i l l  grow e x p o n e n t i a l l y . T h i s phenomenon i s known as an  electron  avalanche.  electron  The  number of e l e c t r o n s i n v o l v e d i n an  avalanche i s :  N-2  L / X  '  Here N i s the number of e l e c t r o n s i n v o l v e d i n the and  2.1-1 avalanche  L i s the l e n g t h of the avalanche which i s approximately  7  -  equal to the dimension of the b u l b . However, the of  these  avalanches  does  not  mean  threshold  reached, s i n c e i n order to s u s t a i n the  secondary  electron  will  has  ionization  each avalanche must produce at l e a s t one This  occurrence been  process,  secondary e l e c t r o n .  produce  another avalanche,  hence the i o n i z a t i o n process i s s u s t a i n e d . These  secondary e l e c t r o n s can be  mechanisms. positive  Firstly,  secondary e l e c t r o n s may  ions s t r i k i n g  electrons  involved  striking  the  produced  the  wall  of  the  i n the avalanche may  bulb's  wall  which  by  several  be produced  bulb.  by  Secondly,  produce photons upon  in  turn  could  produce  secondary e l e c t r o n s by i o n i z i n g gas molecules. I f  Y  probability  produce a  that  secondary  electron,  self-sustaining  1/Y  one  electron then  ion  the  pair  will  requirement  is  for  a  i o n i z a t i o n process i s :  . WX,  2.1-2  2  The  the  minimum  self-sustaining  electric  field  required  i o n i z a t i o n process i s  the  to  obtain  threshold  a  field  E . A, and Y are a f u n c t i o n of gas p r e s s u r e and the e l e c t r i c 0  field.  Hence equation 2.1-2  o b t a i n an equation of the E= 0  may  be i n v e r t e d i n p r i n c i p l e to  form:  F(pL)/L  2 > 1  T h i s i s known as Paschen's from t a b l e s , hence E  0  may  Law.  Values of F may  be c a l c u l a t e d .  be  _  3  obtained  -  8 -  TYPICAL BULB  ^  GAS C O N T E N T IS ARGON A T ONE TORR  Iv /J FIGURE 1  FIELD INTERNAL TO THE BULB  FIGURE 2  GLASS  SHELL  FULL  SCALE  -  This  entire  Mechanism.  breakdown  9 -  process  is  known  as the Townsend  2  Many of the ions i n v o l v e d will  have  will  de-excite  in  been e x c i t e d to higher to  the  electron  energy s t a t e s . These  more s t a b l e energy s t a t e s thus  photons, hence a p u l s e of l i g h t  is  formed.  process  the  field  will  approximately  avalanche  continue  until  cancelled  by  the  field  The in  ions  emitting  ionization the  created  bulb i s  from  the  e l e c t r o n s and ions c o l l e c t i n g on the w a l l s of the bulb. The  time  required  f i e l d t o be reset period  f o r the avalanche t o occur and the  i s assumed t o be  compared  of the a p p l i e d f i e l d . As the e l e c t r i c  to the bulb i s f u r t h e r i n c r e a s e d , bulb  small  will  also  increase u n t i l  then another avalanche w i l l light  and again  This  process  the f i e l d threshold  field  external  i n t e r n a l t o the i s again  reached,  occur e m i t t i n g another p u l s e  r e s e t t i n g the bulb's i n t e r n a l continues  to the  indefinitely.  field  See  to zero.  figure 2 for a  graph of the i n t e r n a l f i e l d v e r s u s time. I t i s evident the  frequency  E . 0  B  two  the  applied  field  E  A  ,  which  i n f i g u r e 2, can be d i v i d e d by the t h r e s h o l d  Moreover, i t i s c l e a r that f  frequency  of  that  of the breakdown, f , i s p r o p o r t i o n a l -to the  i n t e g e r number of times defined  of  the a p p l i e d f i e l d .  B  i s proportional  is  field, t o the  Furthermore, there w i l l be  breakdowns per c y c l e at t h r e s h o l d ,  one  at  the  extreme  maximum of the f i e l d and one at the extreme minimum. Hence:  2  Meek,J.M. and Craggs,J.D., E l e c t r i c a l Breakdown of Gases John Wiley and Sons, New York(1978)  ,  - 10 -  CALIBRATION CURVE  FOR A NON-CONDUCTING  BULB  4  o  APPLIED  ELECTRIC  FIELD  FIGURE 3  PHASE OF THE PULSES AT THRESHOLD  FOR A NON-CONDUCTING  FIGURE 4  BULB  -11-  2.1-4 where  u  is  the  frequency of the a p p l i e d f i e l d and  i n d i c a t e s the q u a n t i t y i n the brackets i s the  down  to  nearest i n t e g e r . T h i s breakdown process would imply  the  c a l i b r a t i o n curve should be a figure  3.  The c a l i b r a t i o n  applied f i e l d ,  stepped  curve  rounded  int( )  curve  i s a plot  as  of f  B  shown  in  versus  the  E . In f a c t , not a l l curves are stepped, some A  are q u i t e l i n e a r . There are two mechanisms which cause t h i s . Firstly,  the s t a t i s t i c s of avalanche  curve somewhat. changing.  Secondly,  2  For example, the f i r s t  of  smooth  the  the path of the breakdowns may breakdown may  c e n t r e of the bulb. Thus the f i e l d centre  breakdowns  will  be  occur  reset  be  i n the in  the bulb, but only p a r t i a l l y reset away from  the  c e n t r e . Hence the next breakdown w i l l occur  away  centre  threshold. This  of the bulb at an a p p a r e n t l y reduced  apparent  changing  of the t h r e s h o l d f i e l d  in nature hence i t tends to smooth See  figure  3  for  an  curve. T h i s curve w i l l c u r v e . " Furthermore suggests minimum  the  is also  called  the  "zero  the  statistical  calibration  e x p e r i m e n t a l l y obtained be  from  the  curve.  calibration  conductivity  at t h r e s h o l d , where f = W/TT, t h i s B  theory  the p u l s e s should occur at the extreme maximum and electric  configuration  will  field be  as  shown  in  figure  4.  This  d e f i n e d to be a phase s h i f t of zero  degrees. For operation  some  bulbs,  the  above  adequately,  but  for  theory  explains  many bulbs the  their  calibration  - 12 -  curves are d i s p l a c e d from the o r i g i n as shown i n Furthermore,  the  origin  and  pulses  t h r e s h o l d . Therefore t h i s theory  have in i t s  allowed  problem  can  be  present  form  does  5.  to be s l i g h t l y conducting, where s l i g h t l y conducting  field  external  internal  field, E . A  charge d i s t r i b u t i o n Two  models  to  the  Current  bulb's  are  operation.  the s h e l l i n  can two  self-consistency  affect  bulb, E, , with respect to the  flow i n the g l a s s  required  Both  in  a  to  modifies  explain  different  models  the  the  observed  regime  of  the  c o n t a i n one parameter, the  c o n d u c t i v i t y of the s u r f a c e , so from one models  noticeably  on the bulb s u r f a c e .  r e s u l t s , each being v a l i d  these  goes  overcome i f the bulb's s h e l l i s  means the c o n d u c t i v i t y of the s h e l l w i l l the  curve  zero phase s h i f t at  not e x p l a i n the o b s e r v a t i o n s shown i n f i g u r e This  5.  at t h r e s h o l d , the p u l s e s have a non-zero phase  s h i f t . The above theory demands the c a l i b r a t i o n through  figure  calibration  curve  be a p p l i e d to o b t a i n the c o n d u c t i v i t y of independent of  the  c o n d u c t i v i t y can account  ways,  thereby  models.  testing  Furthermore,  f o r the phase s h i f t of  the  the this  pulses  at t h r e s h o l d . The below  f i r s t model, the " s t a t i c s c r e e n i n g " model, i s v a l i d  threshold  is certainly threshold  not v a l i d when f  value.  w i l l be reduced this  model  calibration  and p o s s i b l y s l i g h t l y above t h r e s h o l d , but  It  B  is  much  greater  than  its  turns out that t h i s model p r e d i c t s E,  and phase s h i f t e d with r e s p e c t to E . Hence,  predicts  A  a  reduction  in  the  slope  of  the  curve as w e l l as a phase s h i f t of the p u l s e s at  - 13 -  SAMPLE CALIBRATION CURVE  FOR A BULB  4  CD  % L U  1<  ZERO C O N D U C T I V I T Y CURVE  PULSE  or  OBSERVED ^ CURVE  APPLIED  ELECTRIC FIELD,  E  CALIBRATION  — • A  FIGURE 5  RESULTS OF THE PROPOSED THEORETICAL MODELS  OBSERVED  CALIBRATION  APPLIED  FIGURE 6  FIELD,  E  A  - 14 -  threshold. of  the  These e f f e c t s are a f u n c t i o n of the  shell.  Note,  p o s s i b l y s l i g h t l y past on  since  conductivity  t h i s model i s v a l i d up t o and  threshold,  a t l e a s t the  first  point  the c a l i b r a t i o n curve must l i e on the curve p r e d i c t e d by  t h i s model. T h i s curve i s shown i n c a l l e d the " s t a t i c The the  screening  6.  It  will  be  curve".  second model, the " r e s e t s c r e e n i n g "  form  model, p r e d i c t s  of the c a l i b r a t i o n curve at high count r a t e s . The  t h e o r e t i c a l curve p r e d i c t e d 6 and w i l l count  figure  by t h i s model i s shown i n f i g u r e  be c a l l e d the " r e s e t  screening  curve".  r a t e s , the t h e o r e t i c a l curve i s p a r a l l e l  conductivity  curve"  but  offset  from  it  Moreover, the o f f s e t of the " r e s e t screening  At  high  to the "zero  as  expected.  curve" from the  "zero c o n d u c t i v i t y curve" depends on the c o n d u c t i v i t y  of the  glass. There e x i s t two ways of c a l c u l a t i n g the c o n d u c t i v i t y of the  glass  with these models. F i r s t l y ,  be c a l c u l a t e d from the r a t i o of the screening  curve" to the " s t a t i c  the c o n d u c t i v i t y can  slopes  screening  of  screening  offset  curve" from the "zero c o n d u c t i v i t y  There i s one parameter i n  these  models  "reset  curve". Secondly,  the c o n d u c t i v i t y can be c a l c u l a t e d from the "reset  the  and  two  of  the  curve". ways  of  measuring i t . Thus s e l f - c o n s i s t e n c y can be demonstrated.  2.2 GENERAL THEORY WITH CONDUCTIVITY  Firstly derived.  INCLUDED  the theory f o r the " s t a t i c  The assumptions r e q u i r e d  screening"  f o r t h i s . model  model i s a r e ; the  - 15 -  frequency . of can  the  be n e g l e c t e d ,  avalanche  c u r r e n t can  breakdowns  conductivity  in  immersed  so magnetic e f f e c t s  flow  the  on  the  gas)  and  shell the  shell's  in  arbitrary  an  shape.  external  Let  electric  Assume  magnetic  the effects  potential  0  can  be  of  the  applied f i e l d  neglected.  at a point on the s h e l l due  d e n s i t y and  *<r >-  frequency  Then  R  6dA.  The  shell. the  E  2  to  the  0  0  and  the  infinitesimal  i n t e g r a l i s taken over the e n t i r e Furthermore,  since  surface  magnetic e f f e c t s are  >  2  „  1  external  0  d i s t a n c e between 5t  field  so  electric  <J>($) i s the t o t a l e l e c t r i c p o t e n t i a l at X*  D  the  the  +* (*;>  0  at X ,  is  i s low  to the s u r f a c e charge  Here <£ ()t ) i s the e l e c t r i c p o t e n t i a l due field  density  the e x t e r n a l f i e l d i s :  </> T § T - 5 -  E  the  field.  Furthermore, l e t the s h e l l possess a s u r f a c e charge 6.  (no  Ohmic.  a c l o s e d s h e l l of  be  i s low  only  i s i s o t r o p i c and  Consider shell  applied f i e l d  and  charge of  the  neglected,  i s written:  F ( X ) — V<t>(X) 0  2.2-2  0  Here l f ( x " ) i s the e l e c t r i c 0  dimensional  gradient  f i e l d at X* and  V  0  operator.  is  the  three  Therefore:  2.2-3  Next, surface.  assume This  current  assumption  can means  only there  flow are  on no  the  shell  avalanche  -  breakdowns  or  16 -  gas d i s c h a r g e s  i n s i d e the s h e l l .  Furthermore  assume the s h e l l has i s o t r o p i c and Ohmic conduction. c o n d u c t i v i t y be Q, where Q i s a s u r f a c e Ohms  per square).  The e l e c t r i c  field  Let the  conductivity ( i . e .  tangent to the s h e l l ' s  s u r f a c e w i l l d r i v e a s u r f a c e c u r r e n t on the s h e l l : Ji-OEj. where J  2.2-4 i s the s u r f a c e c u r r e n t d e n s i t y on the s h e l l and  ±  i s the e l e c t r i c is  current  f i e l d tangent to the bulb's  per  operator,  since  is  the  the  appropriate  current  requires  the  S u b s t i t u t e equation V • flE +|4 A  V  11  *" hi 71/  Here &  3  i n t o equation  further,  c u r v i l i n e a r coordinates  ... i i  two  gradient  + i*!I + h 3C 2  ^ , fc  2  2  2.2-5, t o o b t a i n :  on  the  properties  of  orthogonal  operator  i s written:  the three  3  h*l  22  _  S U  and £  depend  the  coordinates).  must be reviewed. F i r s t l y ,  3  3  2  ,  7 2  7  3  are generalized coordinates,  a r e u n i t v e c t o r s and h, , h  which  of  on  2.2-6  progress  dimensional  t o the s h e l l ' s  "°  A  To  dimensional  ( i . e . any p o s i t i o n  specification  2.2-4  two  i s constrained  s u r f a c e which i s two dimensional shell  (i.e. \  therefore:  operator  x  x  l e n g t h ) . The s u r f a c e c u r r e n t must obey the  c o n t i n u i t y equation,  Here the V  surface  t  the  2  and h  coordinate  3  are system  a,,  scaling used.  a  2  and  factors The  two  dimensional divergence operator i s w r i t t e n :  3  —L_  7 •  h,h  x  2.2-8  2  S u b s t i t u t e equation 2.2-7 i n t o equation 2.2-8 to o b t a i n :  V • Vx 1  x  1  [i_(h  i l \  2  +  i _ / h  1  2.2-9  Let:  Vi  v  i  v  v.h, h  2.2-10  :  S u b s t i t u t e equation 2.2-3 i n t o equation 2.2-6 t o o b t a i n :  2.2-11  S u b s t i t u t e equation 2.2-10 i n t o equation 2.2-11 to o b t a i n :  -nv <j>6dA 2  A  -ntf$ + | | -o  2.2-12  E  Equation 2.2-12 i s a g e n e r a l equation  which  motion of charges, under an e x t e r n a l f i e l d  describes  the  -V<$, on any s h e l l  which can be d e s c r i b e d by h o l d i n g one c o o r d i n a t e constant i n some orthogonal c u r v i l i n e a r c o o r d i n a t e system. Firstly  I  will  uniform s t a t i c e l e c t r i c  3  find  the  response  of  a s h e l l to a  f i e l d which i s suddenly a p p l i e d on a  Morse,P.M. and Feshbach,H., Methods of T h e o r e t i c a l P h y s i c s ,McGraw-Hill Book Company,New York(1953T  - 18 -  s h e l l which i n i t i a l l y w i l l be c a l l e d the problem  had no s u r f a c e  "switch  charge  density.  This  on" problem. Once the "switch on"  has been s o l v e d  i t i s easy t o use these r e s u l t s t o  o b t a i n the response of a s h e l l t o an  alternating  electric  field. The  static  field  induces a s u r f a c e charge d e n s i t y , 6 ,  on the s h e l l . E v e n t u a l l y charge the  will  the  f i e l d produced by  c a n c e l the f i e l d  surface  charge  monotonically.  This  the  induced  i n t e r n a l t o the s h e l l . Assume  density i s valid  approaches since  equilibrium  magnetic e f f e c t s  are  assumed t o be n e g l i g i b l e . The response of the s h e l l w i l l  be  s i m i l a r t o an RC c i r c u i t . Hence:  6(r,t)«]TcHr)(l-exp(-Xt:))  2.2-13  x  X  This i s v a l i d  s i n c e any monotonic f u n c t i o n can be w r i t t e n as  the  sum  exponentials.  t=0.  The X are the time constants  shell  of  and the ( r ) ^ 5  These are  This the  which  (t-*» ) equation  «(r,-)-^«(r) X  of  i s applied at  the response  of the  are the amplitudes of each response.  the q u a n t i t i e s  equilibrium  The e x t e r n a l f i e l d  must . be  determined.  At  2.2-13 becomes: 2.2-14  x  puts a c o n s t r a i n t on the 5 ( r ) ^ s , s i n c e a t e q u i l i b r i u m f i e l d c r e a t e d by the  induced  charge  must e x a c t l y c a n c e l the a p p l i e d f i e l d w i l l be c a l l e d the It  6(r,«»),  i n s i d e the s h e l l .  This  "equilibrium condition".  turns out the X's and the  of an eigenvalue  density,  problem, where the  6(r)j^s  are the s o l u t i o n s  X*s are eigenvalues  and  - 19 -  the  6(r) s  the  X's  are e i g e n s o l u t i o n s . I t a l s o becomes e v i d e n t  x  and  6 ( r ) ^ s depend on the geometry of the s h e l l  the o r i e n t a t i o n of the e l e c t r i c shell.  The  f i e l d with  respect  the  problem  must be c o n s i s t e n t with  the  "switch  in  the  "equilibrium condition".  sensitivity.  t h i s s o r t of s e n s i t i v i t y  the  i s posed. Furthermore they  There i s i n t e r e s t i n t u b u l a r bulbs axi-symmetric  equation  and  linear  x  combination of the s o l u t i o n s to Laplace's system  to  6 ( r ) w i l l be made up of a  components of  coordinate  that  Electric  f o r some  since  they  possess  f i e l d monitors r e q u i r e  applications.  Hence  the  on"  problem  w i l l be s o l v e d f o r a s p h e r o i d a l  described  by  prolate  spheroidal  s h e l l can  be taken to a t u b u l a r or s p h e r i c a l l i m i t . I t turns  out  this  problem  can  l i m i t s . Note, surfaces  be  solved  will  always  be  coordinate  system s i n c e i n t e g r a l s l i k e the one  constant  which  in any  are  chosen  appearing  in  2.2-12 must be s o l v e d . With s u r f a c e s chosen in t h i s 1/R  appropriate This  coordinate  this  f o r these  chosen  by  manner  since  analytically  described  equation  h o l d i n g one  coordinates,  shell  allows  appearing  in  Green's f u n c t i o n the  the  integrals will  for  that  simply  coordinate  i n t e g r a l s to be e v a l u a t e d .  be  the  system.  I f the s h e l l i s  allowed  to be of a r b i t r a r y shape i t w i l l be necessary to  sum  over an  i n f i n i t e number of Green's f u n c t i o n s to  the  appropriate  1/R.  This  makes  the  obtain  problem  hopelessly  complicated. Firstly  the spheroid  w i l l be long  (asymptotically)  that i t i s s i m i l a r t o a c y l i n d e r . A long  spheroid  will  such be  c a l l e d a c y l i n d e r i n the f o l l o w i n g a n a l y s i s although i t w i l l  - 20 -  become apparent l a t e r there a r e s l i g h t  differences  responses  If  subject  of  these  to the  problem  it  two  geometries.  conditions  i s clear  specified  that  depends on i t s o r i e n t a t i o n field.  However  one  the  in  the the  response  with  respect  cylinder i s "switch  to  the  knows from e l e c t r o s t a t i c s that i f t h i s  be s o l v e d with  the e l e c t r i c  the  and  asymptotic  electric  f i e l d p a r a l l e l to  perpendicular  w i l l be asymptotic an  the  easy  axis  to will of  to the a x i s . These r e s u l t s  s i n c e the spheroid w i l l be made  long  in  manner. The r e s u l t s f o r the problem with the  f i e l d perpendicular  w i l l be checked by an exact  to the  axis  of  the  cylinder  c a l c u l a t i o n , but the r e s u l t s f o r  the problem with the e l e c t r i c not  electric  these r e s u l t s t o a l l o r i e n t a t i o n s . T h i s problem  cylinder  on"  of the c y l i n d e r  problem i s solved f o r two o r i e n t a t i o n s , then i t i s extend  i n the  f i e l d p a r a l l e l t o the a x i s are  r e a d i l y s o l u b l e by any other method. Secondly  limit  the  spheroid  w i l l be taken to the s p h e r i c a l  ( a s y m p t o t i c a l l y ) and the "switch  Clearly  the  on"  problem  solved.  r e s u l t s w i l l be independent of the o r i e n t a t i o n  of the sphere with  respect  to the f i e l d . These r e s u l t s  will  be checked by an exact c a l c u l a t i o n . To  obtain  the eigenvalue  2.2-13 i n t o equation V  *  6  problem, s u b s t i t u t e  2.2-12 t o o b t a i n :  (l-exp(-Xt))dA  Air e R 0  equation  ^  n»^+L*« «PMt)-0  2.2-15  x  A  Assume the time c o n s t a n t s , X ,  are  independent  of  spatial  c o o r d i n a t e s . Hence the decay r a t e s can be equated to o b t a i n :  - 21 -  7l<h X j  rr + V.'cJ) -0 2.2-16  E  and: MA +  /  Equation  X 6  X-°  2.2-17  2.2-16  condition".  To  i n t o equation  is  2  <We R 0  Equation  of  the  "equilibrium  s u b s t i t u t e equation  2.2-14  2.2-16 t o o b t a i n :  2  * 9  statement  demonstrate t h i s  _ i 6 (r,»)dA . „ V  a  +  _  n  ^ E -0  V  2.2-18  2.2-18 may be w r i t t e n :  2.2-19  I n s i d e the s h e l l the q u a n t i t y between the b r a c k e t s  is  zero,,  by. the " e q u i l i b r i u m c o n d i t i o n " , hence no new information can be gained To specified  from equation find  the response of a s h e l l due t o the c o n d i t i o n s  i n the "switch  2.2-17. T h i s equation the  eigenvalues  equation the  2.2-16.  and  on" problem one must s o l v e  i s an eigenvalue a  r  e  t  n  e  equation  problem, where  X are  eigensolutions. Solving  2.2-17 w i l l determine the eigenvalues  exactly  e i g e n s o l u t i o n s t o w i t h i n undetermined c o e f f i c i e n t s .  of the " e q u i l i b r i u m c o n d i t i o n " w i l l  remove a l l  and Use  undetermined  - 22 -  coef f i c i e n t s .  2.3  THEORY PARTICULARIZED TO PROLATE SPHEROIDAL COORDINATES  To  obtain  the  eigenvalue  spheroidal coordinates V , 2  in  terms  of  these  1/R,  problem in terms of p r o l a t e (*)  and dA,  fi  A  coordinates.  must be  Prolate  c o o r d i n a t e s are obtained by r o t a t i n g e l l i p t i c a l about  the  major  P and  f  coordinates  on the Z-axis as  shown  +a/2  on the Z-axis and -a/2  l e t r2 be the d i s t a n c e  in  on the Z-axis. Define £ and n to  between  be:  a  i  of  7. Let r1 be the d i s t a n c e between an a r b i t r a r y p o i n t  p o i n t P and  r. *  spheroidal  a x i s of the e l l i p s e s . Suppose the f o c i  the e l l i p s e s are l o c a t e d at ±a/2 figure  found  2.3-1  + 2 r  r, -r  2  2.3-2  a  where  "a"  is  the  distance  between the f o c i . Rotate  coordinate  system through an angle  shown  f i g u r e 8. •  in  by r1 and  r2 and  the  i  about  the  Thus,  the  position  T h i s i s the p r o l a t e s p h e r o i d a l c o o r d i n a t e system. definitions  and  of  £, n and  of  2*. Furthermore i t i s c l e a r £=£  0  From  any i. the  4 i t i s c l e a r that £ must be between  », n must be between 1 and  constant  as  formed  p o i n t P can be d e s c r i b e d by s p e c i f y i n g r,, n and  arbitrary  1 and  Z-axis  i s the angle between the plane X-axis.  this  -1, and  i must be between 0  from equation  2.3-1  that  a  w i l l d e s c r i b e a s p h e r o i d a l s h e l l . T h i s i s the  -  23 -  ELLIPTICAL  COORDINATES  —  z  FIGURE 7  PROLATE SPHEROIDAL COORDINATES  P  -—T^iN  Y  \  FIGURE 8  - 24 geometry d e s i r e d f o r t h i s problem. Any p o i n t on is  described  by  the  correct  specification  this of  n  shell and 4.  Moreover, r e c t a n g u l a r c o o r d i n a t e s x,y, and z can be obtained i n terms of £,n and 4. The r e s u l t s a r e :  3  x- -ja/( r / - l ) (1-n*) Cos$  2.3-3  y - j a / ( ^ - 1 ) (l-n«) S i n *  2.3-4  8  "i'  2.3-5  5 n  The s c a l i n g found  by  between  f a c t o r s which  equating  two c o o r d i n a t e s systems, as shown below:  2  2  2  2  2  2  2  2  2  are  3  '  If v a r i a b l e s n and 4 are not allowed 2.3-6  2.2-10  equation  the l e n g t h of an i n f i n i t e s i m a l a r c , dS,  dS «dx +dy +dz «h dS +h dn +h d* 2  enter  2.3-6 to vary  then  equation  becomes:  2.3-7 Substitute 2.3-7  h  r  equations  2.3-3, 2.3-4,  and  2.3-5  into  equation  to o b t a i n :  2  a  \T^T~  Similarly:  2.3-8  - 25 -  h  " '* "  n  2.3-9  h^a/(l-n')(?'-l)  2.3-10  The v a r i a b l e s which enter since  a  spheroidal  surface  the operator  V are n and 4 2  i s d e f i n e d by constant  Thus, s u b s t i t u t e equations 2.3-9 and  2.3-10  into  £=£ . 0  equation  2.2-10 to o b t a i n :  •'^t'-n*  The area  an/t'-n*  an  +  element, dA, i n equation  dA-  h  where  the s c a l e f a c t o r s h  order  t o conserve a r c lengths  a~F/  2  2  An function  1 1  2.3-12  Substitute  and h enter equation a  ( i . e . h^dn  equations  *  sa  2.3-12 i n  length  whereas  2.3-9 and 2.3-10  into  2.3-12 to o b t a i n :  dA^ (C -n ) (C -D dnd * 2  -3"  2.2-16 i s w r i t t e n :  dn d *  dn i s n o t ) . equation  <^-i)<i-n*)  2  J s  2  appropriate  representation  f o r an i n f i n i t e  spheroidal coordinates. of the equation  2.3-13  l s  domain  of 1/R i s the Green's i n terms  of  prolate  The Green's f u n c t i o n i s the s o l u t i o n  shown below s o l v e d  i n an i n f i n i t e  domain:  - 26 -  2.3-14  V G(X,X )--4T?6 (X-X ) 2  3  0  where  0  x  i s an a r b i t r a r y  0  f i x e d p o i n t i n three  dimensional  space and x i s a v a r i a b l e p o i n t . The Green's f u n c t i o n , G, i s a f u n c t i o n of both x  and x. The operator  0  dimensional The  Laplacian  V  is  2  the  three  only i n v o l v i n g the d e r i v a t i v e s of x.  s o l u t i o n of equation  2 . 3 - 1 4 can be o b t a i n e d with the use  of F o u r i e r transforms. The r e s u l t i s :  Ix-x-J " I  G =  2.3-15  where the v e r t i c a l bars mean the magnitude of  the  quantity  between them i s t o be taken. Hence R i s the d i s t a n c e between x  and  D  x.  The  Green's f u n c t i o n f o r an i n f i n i t e domain i n  terms of p r o l a t e s p h e r o i d a l c o o r d i n a t e s i s s o l v e d and Feshbach . The r e s u l t i s :  where and  e  m  equal t o 2 f o r m>0 . P j J ( n ) i s a Legendre f u n c t i o n of the kind and Q™(0  kind  and i i s the square  0  3  i s the Neumann f a c t o r which i s equal t o 1 f o r m=0  first  •  i n Morse  i s a Legendre f u n c t i o n  of  root of - 1 . Furthermore  the  second  £, n 0  0  and  are the c o o r d i n a t e s of the f i x e d p o i n t and £ , n and i a r e  the c o o r d i n a t e s of the v a r i a b l e p o i n t . equation  2.2-17  is  the  distance  The  quantity  R  in  between two p o i n t s on a  - 27 -  e  spheroidal shell  . Hence, set 0= £  0  0  in  equation  2.3-16  to o b t a i n :  i-|£(2o i)2. ij|5^1c..[-<«-dp;<, >F;(n)F;(t0)Q^t ) +  B  nsO  0  L  msO'  J  0  J  L  2.3-17  Up  to t h i s p o i n t a l l the q u a n t i t i e s necessary  the eigenvalue except  for selecting  insight  2.2-17, have been  an a p p r o p r i a t e form of  obtained  6 ( r ) . To o b t a i n x  i n t o the a p p r o p r i a t e form c o n s i d e r equation  6\ dA /  problem, equation  to solve  7nn-R  +  x  V  2.2-17:  0  2.2-17  Let: dA • • f t  e R  2.3-18  0  S u b s t i t u t e equation Vjl  + X«  x  Substitute  2.3-18  i n t o 2.2-17 to o b t a i n :  -0  2.3-19 equations  2.3-17 and 2.3-13 i n t o equation  2.3-18  to o b t a i n : 6, ( t 2 - n ) a ( C - l ) 2  ! 5  2  nsO  u  0  -  , i  m:o  L  *Cos • ( • - • ) P ; ( n , ) p ; ( n ) p ; ( C o ) Q ; ( C o ) 0  6  X  J  -1  must depend on  n  (  2  -  and 4 where n ranges from -1 t o 1 and  3  -  2  4  0  - 28  from  0  to  2*.  L ( - l , l ) and  However,  ^ ^  p  f°  n  r m s  complete b a s i s on  L (0,2ir), where 2  v  is  reasonable f u n c t i o n of n and  means any  complete b a s i s  a  an  integer.  • can  and 4 are -1  to 1 and  0 to  2ir  This  be w r i t t e n as a  combination of these f u n c t i o n s where the  function  on  a l i n e a r combination of Cosv <and Sinv <forms a  2  linear  -  respectively.  ranges of r\  A  reasonable  i s a bounded f u n c t i o n which i s continuous and  has  a  continuous f i r s t d e r i v a t i v e . Hence: 6  X o^ ) -2]R^P;(n)(co8v4 U  2  >S  +C^Sinv^  2  .3_  2 1  M v  where  R y, X  and  v  C y.X are undetermined c o e f f i c i e n t s . T h i s  convenient form of 6^ ( t^-n )" 2  was  substituted  into  4  because,  if  equation 2.3-20 the  could be done a n a l y t i c a l l y  s i n c e the  equation  2.3-21  can  symmetry  properties  be  simplified  and  problems that are to be  equation  would this  conditions  In t h i s case the  a x i s of the s p h e r o i d .  done  with  the  of  the  solve.  Consider the problem with the a p p l i e d e l e c t r i c Z-direction.  contain is  considerably  equilibrium  2.3-21  integrals in " I "  integrand  only orthogonal p o l y n o m i a l s . However, before  the  is a  v  At e q u i l i b r i u m  f i e l d in  f i e l d i s p a r a l l e l to 6 ( r ) . can  have  the  no  i  A  dependence  due  to  the  symmetry of the e q u i l i b r i u m charge  d e n s i t y d i s t r i b u t i o n , hence an exact expression  z<'-->'  6  eE  for  «pj(n)  0  2h yu mii-i> e Q  0  v=0 . Furthermore one  6 (r,»), 2  "V A  ^  ( r )  where:  ^  can  obtain  3  2  -" 3  22  - 29 -  6z ( r , « ° ) i s the e q u i l i b r i u m charge d e n s i t y  Here  resulting and  E  from  a  uniform a p p l i e d  i s the s t r e n g t h  field  of the f i e l d .  distribution  i n the Z - d i r e c t i o n  The second term i n t h i s  e q u a t i o n was obtained from e q u a t i o n 2.2-14.  6 ( r ) . are  the  A Z  e i g e n s o l u t ions  of the Z - d i r e c t i o n  ( i . e . *(r*)  problem  A2  s a t i s f i e s equation 2.2-17). The from  e q u i l i b r i u m charge d e n s i t y  a  been  uniform  calculated:  £ 6  x  applied  0  E  X  8 P  )(n)Cosi  ._  "2/  i s the e q u i l i b r i u m  x  resulting  i n the X - d i r e c t i o n has a l s o  ^ (r)  A  6 (r,«0  *x  2.3-23  charge  from a uniform e l e c t r i c  density  field,  equation 2.2-14. * ( r ) ^  X - d i r e c t i o n problem Hence, and  distribution  E , applied x  X - d i r e c t i o n . The second term i n t h i s equation from  resulting  3  ' - > " 2h Q { ( € 0 ) ( t J - D  ( r  field  distribution  was  obtained  a r e the e i g e n s o l u t i o n s  x  (i.e.6(r).  i n the  f o r the  s a t i s f i e s equation  2.2-17).  v=l i n t h i s case. Thus, from an i n s p e c t i o n  of 2.3-22  2.3-23 equation 2.3-21 becomes:  «(r) (C -n ) -^R ° P;(n) 2  A z  2  2.3-24  i $  0  p  x  and: 6(  ' Xx ?" )  U  The  n2)l5  "S Jx t R  exact  equilibrium  *  •  2.3-25  charge  density  distributions,  a r e not used i n t h i s a n a l y s i s , but  X  rather in  (n)Co8  M  6 ( r , » l and 6 ( r , « ) , Z  P  the expansions of S ( r )  terms  of orthogonal  ( £ -n ) 2  X z  2  polynomials  J i  and  6(r)  because,  Uj-n ) * 2  X z  1  the exact  - 30 -  equilibrium  d i s t r i b u t i o n s are not e i g e n s o l u t i o n s ,  do not s a t i s f y equation 2.2-17. S u b s t i t u t e  hence they  equation  2.3-24  i n t o equation 2.3-20 and e v a l u a t e " I " . The r e s u l t i s :  where  1^ i s the value of " I " r e s u l t i n g from an a p p l i e d  in  the  and  e v a l u a t e " I " . The r e s u l t i s :  where I  Z-direction.  Substitute  field  equation 2.3-25 i n t o 2.3-20  i s the value of " I " r e s u l t i n g from an a p p l i e d  x  field  in the X - d i r e c t i o n . To o b t a i n the applied  field  V?I + X «  Substitute  i n the Z - d i r e c t i o n  resulting  from  c o n s i d e r equation 2.3-19:  equations 2.3-26, 2.3-24 and 2.3-11 i n t o  equation  then take the moment of the r e s u l t i n g equation  respect  to  the f u n c t i o n of  a  function  variables  moment  of  an  equation  with with  means the equation i s m u l t i p l i e d by  and the e n t i r e  the  an  2.3-19  r e s p e c t t o P°(n). Taking the  range  problem  -0 •  x  2.3-19,  eigenvalue  equation  integrated  f o r which the f u n c t i o n  over  the  i s defined.  See appendix A f o r the d e t a i l e d c a l c u l a t i o n . The r e s u l t i s :  L  R° B°  u  where:  + X R°  -0  2 3-28  - 31 -  1  ac  2.3-29 Here X  i s the eigenvalue r e s u l t i n g  z  the Z - d i r e c t i o n . Let B form, then the X's d  e  t  [  * z ]  B  X  z  u  be the tensor B °  N  written in  matrix  can be found with the e q u a t i o n : 2.3-30  0  where  u  found,  to w i t h i n undetermined c o e f f i c i e n t s , by s o l v i n g :  [ z * B  X  is  =  z  from an a p p l i e d f i e l d i n  z K°X U  the  unit  field  can be  X  °°  2  To o b t a i n the applied  t e n s o r . The e i g e n v e c t o r s R °  eigenvalue  in  the  problem  X-direction  2.3-27, 2.3-25 and 2.3-11 i n t o equation  resulting substitute 2.3-19,  ' " 3  from  3 1  an  equations then  take  moment of the r e s u l t i n g equation with respect to P*("n).  the  See appendix A f o r the d e t a i l e d c a l c u l a t i o n . T h e . r e s u l t i s :  Z U  R yX . B 1  + X x R* -0 NX  1  M N  2.3-32  where: l . ^ - > ' < ^ ) p l ) o l MN a c p (y + l ) N ( N + l ) p 9 y i  B  0  1  l  2  W  f  P  g  (  Q  f  n (  0  / p l V / N P  f  n (  x  n  )  <  t  *  -1  (,}-n ) i _ _ 2  Here X  x  -LlEaL_V<n)dn  i s the eigenvalue r e s u l t i n g  2.3-33  from an a p p l i e d f i e l d i n  - 32 -  the X - d i r e c t i o n . L e t B  be the matrix form  of  tensor  B  .  1  Thus the A^s may be obtained by s o l v i n g :  d e t  [ x  * x ]  B  X  and  u  the  =  2.3-34  0  R*  may  x  be  obtained,  to  within  undetermined  c o e f f i c i e n t s , by s o l v i n g :  [ x B  +  X  X  U  It  ]  J  R  A  -  °  2  i s clear  matrices  B  are  from equations 2.3-29 and 2.3-33 infinite  suppose the dimension Furthermore  in  of the matrix  suppose  all x  2.3-30  and  2.3-34  that  3  "  3  5  the  However, f o r now  i s chosen  the  c o r r e s p o n d i n g e i g e n v e c t o r s , R^ equations  dimension.  '  t o be  eigenvalues,  , are c a l c u l a t e d  finite. A ,  and  ( i . e . solve  r e s p e c t i v e l y , then use these  r e s u l t s i n equations 2.3-31 and 2.3-35 t o o b t a i n the the R^  x  to  within  undetermined  eigensolutions, undetermined  c  a  n  ke  found  within and  the these 2.3-25  F i n a l l y the e i g e n s o l u t i o n s , 6 ^ ( r ) , are summed  so that they  add  distribution,  up  to  the  equilibrium  charge  density  6(r,«), where:  6(r . - ) - ^ * ( r ) A  2.2-14  x  undetermined  coefficients  2.2-14. I t t u r n s out  the  are  removed  "equilibrium  s a t i s f i e d r e g a r d l e s s of the dimension the response  to  Next  c o e f f i c i e n t s with equations 2.3-24  respectively.  All  >  coefficients).  of the s h e l l i s :  with equation  condition"  can  be  of the matrix B. Hence  - 33 -  6(r,t)-^6(r)  6  N X  solving  (l-exp(-Xt))  A  2.2-13  ( r ) i s used t o denote the e i g e n s o l u t i o n s o b t a i n e d by the N-dimensional  approximation  t o the i n f i n i t e  d i m e n s i o n a l m a t r i x . T h i s i s o b t a i n e d by s e t t i n g  a^=0 f o r i  or j > N where a^. i s an element of the m a t r i x . Furthermore, when t h i s approximation i s made the response of the s h e l l i s denoted  * ('»t)-  by  increases,  t h e dimension  i t t u r n s out that  6 (r,t)  that  As  N  i s essentially  N  of t h e matrix  6 ( r , t ) r a p i d l y converges indistinguishable  from 4  as expected. Furthermore, i t turns out that the i s so r a p i d ,  i n our examples, that  approximation t o  such  N  N + 1  (r,t),  convergence  6 ( r , t ) can be used as an 7  6 (.r,t). n  2.4 THE CYLINDRICAL LIMIT OF A SPHEROID  The  response of a s p h e r o i d a l s h e l l under the i n f l u e n c e  of an e l e c t r i c its  axis  f i e l d applied p a r a l l e l  can be  thus f a r . There equations  one problem,  and 2.3-33  the i n t e g r a l s i n  cannot be done a n a l y t i c a l l y ,  they must be done n u m e r i c a l l y . To a v o i d t h i s , limits  of a  spheroid w i l l  to  c a l c u l a t e d with the i n f o r m a t i o n o b t a i n e d  i s only  2.3-29  or p e r p e n d i c u l a r  spheroid  will  be made long  be  two asymptotic  considered.  F i r s t l y , the  ( a s y m p t o t i c a l l y ) so i t i s l i k e  a  c y l i n d e r . Secondly, the s p h e r o i d w i l l be allowed t o approach a sphere  ( a s y m p t o t i c a l l y ) . I t t u r n s out the i n t e g r a l s can be  done a n a l y t i c a l l y  i n these  limits.  - 34 -  In  the  first  limit  l e t the  ratio  of the l e n g t h t o  diameter of the spheroid become a r b i t r a r i l y  large.  equations  for a spheroidal  shell  2.3-3  and  2.3-4.  It i s clear,  Consider  £=e; , that X and Y take on t h e i r maximum values a t n-0 0  (the c e n t r e of the s p h e r o i d ) . Thus: M"  "J**TO-1  M"  J  X  Y  where  a /  So"  1  X  M  COB*  2.4-1  Sin* and  2.4-2 Y  M  are  the  maximum  values  of  X  and  Y  r e s p e c t i v e l y . The maximum r a d i u s of the s p h e r o i d i s :  ^-/Xg+Y* - y a / ^ F T As  2.4-3  the s p h e r o i d becomes long, i t w i l l become c y l i n d r i c a l i n  appearance. The diameter of will  the  cylinder  (long  spheroid)  be d e f i n e d to be:  D-2r  2.4-4  M  Moreover,  i t i s clear  and Y are zero at  from equations  2.3-3 and 2.3-4 that X  thus from equation  2.3-5  i t can be  seen that the maximum value of Z i s : 2  M-  -jaCo  Hence, the l e n g t h of the c y l i n d e r (long spheroid) i s :  2.4-5  - 35 -  L«2Z «ae M  2.4-6  0  Therefore:  £ * /#T  2.4-7  L/D i s c a l l e d the aspect from  equation  infinity ^o"  as  2.4-7,  ( i . e . the s p h e r o i d  0  approaches  one L/D approaches  becomes c y l i n d r i c a l ) . L e t : 2.4-8  where A i s small but g r e a t e r 2.4-8  £  A  +  1  r a t i o of the c y l i n d e r . Furthermore,  i n t o equation  than zero. S u b s t i t u t e  2 . 4 - 7 to obtain:  1+A  L  » " < 2 « * ( l * | >* Equation  5  equation  2  - " 4  9  2 . 4 - 9 can be s e r i e s expanded t o o b t a i n :  " 7 T T %  (  l  +  7  A  +  O  U  2  )  2.4-10  >  (2A) It  i s c l e a r from equation  L/D approaches only L  analysis  term of any such s e r i e s .  I  zero will  Thus:  1  ^  k  2.4-11  (2A>*  where left  that as A approaches  1/(2A)"* . In the f o l l o w i n g  keep the f i r s t  D D  2.4-10  the ^ hand  Furthermore:  means  quantity  the r i g h t hand q u a n t i t y approaches the asymptotically  as  A  becomes  small.  - 36 -  r *u al r  ''  1  A  is  2.4-12  the  parameter which must be small f o r the asymptotic  expansions Hence, of  to be v a l i d .  A< 0.1  will  be  small.  from equation 2.4-11, L/D>3 i s the range of v a l i d i t y  a l l the asymptotic  expansions.  To o b t a i n the asymptotic  e i g e n v a l u e equations  "switch on" problem w i t h the e l e c t r i c to  considered  the  a x i s of the c y l i n d e r  f i e l d applied  f o r the parallel  s u b s t i t u t e equation 2.4-8  equation 2.3-29 and keep only  zero  order  terms  in  into A  to  obtain: > , R°. B° + A R° -0 fcjj-' pA pN z NA  2.3-28 1  B M  The  " equilibrium  charge  2.4-13  1  density  distribution  f o r the  Z - d i r e c t i o n problem, equation 2.3-22, becomes: « E pO-<n>0  6 2  and  8  (r,-) ~ ' (l-n ) Q°(l+A)(2A) 2  l s  l s  2.4-14  -  the e i g e n s o l u t i o n s , equation 2.3-24, become:  Z  R° P ° ( n ) .,_„2x s  2.4-15  i  To o b t a i n the asymptotic "switch  on"  problem  with  e i g e n v a l u e equations the  electric  field  f o r the applied  - 37 -  perpendicular equation largest  Z  2.4-8  pX U N  into  axis  of  -  equation  2.3-33  and  substitute  keep  only the  O  2.3-32 2 4-16  -  a c y (u + l)N(N+l) ( 2 A )  J  4  0  equilibrium  charge  « (r,«) % e E C o s * x  0  x  / l  (l-n ) * 2  density  X - d i r e c t i o n problem, equation  The  cylinder  + )i R ' *0 x NX  —  W N  The  the  i n < 2 N + i ) Q j ( l + A ) p i ( l + A ) f?l<n)P*(n)dn  t  Li  the  terms to o b t a i n :  R'  U  to  1  distribution  for  2.3-23, becomes:  ^ ^  2.4-17  n  e i g e n s o l u t i o n s , equation  2.3-25, become:  *»» y > ..,^  --  (r>  (1  The  2  n  integrals  in  the  2.4-13  and  2.4-16  can  4  be  ,e  solved  analytically.  2.5 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PERPENDICULAR TO THE AXIS OF THE SPHEROID  Firstly with will  the  the "switch on" problem  applied  electric  field  be s o l v e d . T h i s i s done i n  briefly  long  perpendicular  appendix  B.  2.4-16  B i s chosen. Next, the  spheroid  to i t s a x i s  Here  o u t l i n e the procedure and give the r e s u l t s .  the dimension of matrix equation  for a  I  Firstly  integral  i s performed. Then the eigenvalue  will  in  problem,  - 38 -  equation 2.3-32, N-dimensional.  is  solved  f o r the  chosen  matrix,  say  The e i g e n v a l u e s are obtained by s o l v i n g :  detCB  1  • X u)=0  2.3-34  and  the e i g e n v e c t o r s , Rj^ , are found t o w i t h i n undetermined  c o e f f i c i e n t s by s o l v i n g :  (  B  J N  *  X  x  Hence,  u ) R  l x -°  the  2.3-35  eigensolutions  can  be  found  to  within  undetermined c o e f f i c i e n t s with equation 2.4-18: Pj (n)Cos4 j  "  Here  the  N in $  ( l  N A X  _  2.5-1  n l ) %  ( r ) denotes the dimension  F i n a l l y a l l undetermined c o e f f i c i e n t s can the  use  of  of the matrix.  be  removed  with  the " e q u i l i b r i u m c o n d i t i o n " , equations 2.2-14,  2.4-17 and 2.5-1:  ^  *  2.2-14  P  S u b s t i t u t e equations  l  ( n  >  2.4-17  2.5-1 and 2.4-17 i n t o  equation  2.2-14  to o b t a i n :  £  o  The  x  . - -  ^  XM  undetermined  equation  2.5-2  MA"y  2.5-2.  coefficients Hence  the  in R j  x  are  removed  eigensolutions, 6  with  f r ) , are NXX  V  '  - 39 -  known.  Finally  the  response of the s h e l l i s obtained  with  equation 2.2-13: 6  NX  ( ?  »^ -X>  N X x  (?)U-e*p(-Xt>)  2  2  _  1 3  X  where the N i n 6  ( r , t ) denotes the dimension of the matrix.  This c a l c u l a t i o n  i s done i n appendix B. The r e s u l t s are  given below. Note, i t turns out only matrix  in  the  elements  of the  which both y and N are odd need to be c o n s i d e r e d  i n the c a l c u l a t i o n , t h i s i s demonstrated i n appendix B.  The  r e s u l t s of the c a l c u l a t i o n a r e :  «  l  x  ( ? . t )  « 0 «,c.M(i-«,(- ^ f £ a „  «  3  x  ( ? . t )  e E 0  2  Co.•(. 8 3 4 9 - .  34 73 ( S n - 1 ) ) ( l - « * p < -  -"  2  8 Q t  4  ))  • " )> 8  2  ] t  2  2.5~5  0t  c„ E C o « •(.6 3 7 4 - . 4 1 4 0 ( 5 n - l ) + . 1 7 * 6 ( 2 1 n - 1 4 n + l ) ) l,  2  x  (l-«xp(-  '"*  0 t  ))  9 0 t  ))  + c E Co.»(.3182+.2910(5tl -l)-.3216(21n '-Ufi +l))(l-«xp<-  - *  + e E Co«4(.0444+.1230(5fl -l>+.1472(21ti -l4ti +l))(l-«xp(-  'j. * '))  2  ,  2  0  2  x  _  *  + t„E Co««(.U5H-.3473(Sn -D>a-«xp(-  0  5  , ,  2  6  2  to  0  2 . 5~6  - 40 -  « <r.t> + 7X  t 0 E x C o . » ( . 5 0 9 3 - . 3 7 53(5n 2 -J) + . J556(21n l, -l*n 2 +l)-.0209(429n s -*95n''+135n 2 -S)) ( l - « x p (  Rc  + c 0 E x C o « » (.3408+. 0878 (5n 2 -l)-.3602 (2In*-Un 2 +1) + . 0472 (429n*-495nl,+ l 3 5 n 2 - 5 ) ) ( l - « x p (  0  •liiOt))  + i 0 E x C o » •(.1337+.238l(Sn 2 -D+.029*(21n''-Un 2 +l)-.0413(*29n'-*95ri , , +135n*-5)) ( l - « x p (  + e 0 E x Co»f(.0162+.0495(5n 2 -D+.0476(2lTi ,, -I4D 2 +l)+.0151(429n*-495n''+135n 2 -S)) ( l - « x p (  2.5-7  These  functions  * /e E Cos< 1 x  6  3X  0  i s plotted  x  /e E Cos i o x 0  6  3 X  /  0  E  X  against  C  °  S  functions  *  tfl/Re  '  6  tO./Re  0  13. From t h i s  5 X  /  E  °  E  X  C  O  S  *  3  1  1  6  and o v e r l a y e d figure  7 X  6  /  E  0  exactly. This 6  of  e a X  (r,t)  X  To  as C  O  S  N  demonstrate increases, P  *  l o t t e d  f o r n = 0 and n=.75 i n f i g u r e  one can see that  these  functions  fi^C^t).  L a t e r an i n f i n i t e c y l i n d e r "switch field  3  n.  E  9,  i n f i g u r e 11, and '  a r e converging  converge r a p i d l y t o  applied  in figure  0  T  i n f i g u r e 12, f o r v a r i o u s  these e  against 0  x  that  on the f o l l o w i n g pages;  i n f i g u r e 10, 6 /e„E C o s * ^ sx' x  6 /e E Cos< 7 X  are p l o t t e d  perpendicular  to  on" problem with the  i t s axis  w i l l be s o l v e d  s o l u t i o n makes i t p o s s i b l e t o p r e d i c t the form f o r a long  spheroid  the geometries. I t turns  through the s i m i l a r i t y of  o u t , as  will  be  shown  later,  «. (r,t) i s : x  «. (r,t) * e E x  hence:  0  CoB*Yl-exp(  0  t  \\  2.5-8  - 41 -  FIGURE 9  - 42 -  FIGURE 10  - 43 -  THEORETICAL  6^(r,t)/e E^Cos j 0  FIGURE 11  PLOTTED AGAINST  nt/Re  0  - 44 -  THEORETICAL  g ^(r,t)/c E Cos 4 y  0  y  n . 75 s  FIGURE  12  PLOTTED AGAINST  flt/Re  0  -  45 -  DEMONSTRATION OF THE CONVERGENCE OF 6 •  —  (A  FIGURE 13  3  X  1  ,6  AND 6  5X  7X  FIGURE 14  - 47 -  A  x  ^  2e R(l- ) 2  0  The  l s  2.5-9  n  reason  solution  of  equations  t h i s r e s u l t was  equations  were  spatial  not  2.2-16  derived  coordinates.  obtained  and  2.2-17  assuming Hence,  by  A  was  these  is  an  exact  that  these  independent  equations  of  could  be  r e - d e r i v e d t a k i n g i n t o account the s p a t i a l d e r i v a t i v e s of and  the  problem  complicate  solved  /e„E «ex  However,  this  would  the problem c o n s i d e r a b l y . Moreover, t h i s  only example c o n s i d e r e d 6  exactly.  O  Cos 4 X  /e E °  the A.  Cos * are p l o t t e d i n f i g u r e 14  D  7X  for v a r i o u s n. The  is  which has a s p a t i a l l y dependent  and 6  A  X  agreement  J  between  these  functions  is  PROBLEM WITH  THE  excellent.  2.6  APPROXIMATE SOLUTION OF THE  APPLIED PARALLEL TO THE  The  spheroid  has  "SWITCH ON"  AXIS OF  been  THE  SPHEROID  defined  so  that i t s a x i s i s  p a r a l l e l to the Z-axis. Hence, i n t h i s s e c t i o n on"  problem  for  a  long  done  in  appendix  C.  Here  calculation  g i v e the r e s u l t s of  calculation. Firstly  all  field  I w i l l b r i e f l y o u t l i n e the  procedure used to s o l v e t h i s problem and the  "switch  s p h e r o i d with the e l e c t r i c  a p p l i e d i n the Z - d i r e c t i o n w i l l be s o l v e d . T h i s is  the  the  the dimension of the matrix  elements found. The  s o l v i n g equation  2.3-30:  eigenvalues  B°  is  chosen  , A , are found  and by  - 48 -  det(B°  and  2.3-30  • X u)=0 z  N  the e i g e n v e c t o r s ,  , are found t o w i t h i n undetermined  c o e f f i c i e n t s with equation  (B  2N  +  A  z  u)R  2x  =  2.3-31:  2.3-31  0  Hence the e i g e n s o l u t i o n s , 6  ( r ) , may be  found  these undetermined c o e f f i c i e n t s with equation  Z where  N  R°  within  2.4-15:  P°(n)  appearing  in *  N A Z  ( r ) denotes  the dimension  matrix. The undetermined c o e f f i c i e n t s may the  to  "equilibrium  condition",  be  of the  removed  with  equations 2.2-14, 2.4-14 and  2.6-1. S u b s t i t u t e equations 2.4-14 and 2.6-1  i n t o 2.2-14  to  obtain:  uX y  x  "  2.6-2  A l l p r e v i o u s l y undetermined c o e f f i c i e n t s may be removed with equation 2.6-2. T h e r e f o r e , the e i g e n s o l u t i o n s , * be  known  6  (r,t),  exactly.  Finally  the  i s found with equation  response  of  N A Z  ( r ) , will  the  shell,  2.2-13:  N Z 6  N (r.t) -^« Z  N A l  (?)(l-exp(-A t))  2.2-13  8  A  where N i n 6 , ( r , t ) denotes N  the dimension  of the matrix  used  Z  in the c a l c u l a t i o n . I would  like  to  obtain  the  quantity  - 49 -  ,(*»*)•  5  turns  I f c  (r,t)  out 6  6  o o z  converges very  r a p i d l y to  NX  ^ X  (r,t)  as the dimension of  reasonable  approximation  matrix  to  B  (r,t)  6  increases. is  obtained  Thus  a  with a  minimum amount of work. T h i s c a l c u l a t i o n i s done i n appendix C. The r e s u l t s a r e : % ——,  «. (r,t) 2  - _  DQjf 1+  —  (l-exp(-X  firW-n )* 2  t)) z  2  '  6  3  where:  X  z  *  -  - H 1 J  2  -  2  2.6-4  6  where D i s the maximum diameter of the s p h e r o i d and L i s the l e n g t h of the s p h e r o i d . Note, the time constant  i n t h i s case  i s an e m p i r i c a l f i t to  constant  the  theoretical  time  as  shown i n appendix C.  2.7  APPROXIMATE  SOLUTION OF THE "SWITCH ON" PROBLEM IN THE  SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE X-DIRECTION  A spheroidal s h e l l coordinates  by  holding  i s described £  constant.  the s p h e r i c a l l i m i t by a l l o w i n g  5  0  in . prolate This s h e l l  t o approach  spheroidal i s taken to  infinity  as  shown below. From show:  equations  2.3-3,  2.3-4  and 2.3-5 i t i s easy to  - 50 -  x +y +z ^ i 2  2  Let  a U -l+n > 2  2  2  2  2 > ?  £ -^» . n v a r i e s between -1 and 0  large,  the l a s t  two terms  1 hence,  i n equation 2.7-1  _  1  as  £  becomes  may  be ignored.  0  Thus: x +y +z ~ 2  2  2  ja £  2 2  where  t >>l  2.7-2  0  Hence the r a d i u s of the sphere i s : R * J*S>o  2.7-3  In t h i s s e c t i o n the response of a s p h e r o i d a l an  electric  f i e l d applied  shell  i n the X - d i r e c t i o n w i l l be  to  found.  The equations which govern t h i s response are; 2.3-32, 2.3-33 and  2.3-25  2.3-23.  Take  with  the  these  "equilibrium equations  a l l o w i n g f, to approach 0  E  RlyX B  infinity  to  condition", the  spherical limit  by  to o b t a i n :  + X xR , -0 NX  1  2.7-4  1  M N  equation  1  iO(2N+l)Pj;(E)Qj(Q .1 „ V o V o UN a c y (u + l)N(N+l) 0  2.7-5  o  2.7-6 M  The e q u i l i b r i u m charge d e n s i t y  d i s t r i b u t i o n becomes:  - 51 -  « (r,»)  * 3c E P}(n)Cos•  x  T h i s problem  2.7-7  x  0  i s s o l v e d i n appendix D. Here  I  will  briefly  o u t l i n e the procedure used t o s o l v e the problem and give the r e s u l t s of the c a l c u l a t i o n . Firstly hence the B  the i n t e g r a l  i n equation 2.7-5 i s performed,  are found. Next the e i g e n v a l u e s , x  1  , are found  with: det|B„ [ x • * X„u|=0 x ] B  and  X  U  2.3-34  = (  the e i g e n v e c t o r s ,  R  ,  1  are  found  to  within  undetermined c o e f f i c i e n t s w i t h : [ x B  The  +  X  x ] Jx u  "°  R  2.3-35  undetermined  2.7-6, 2.2-14.  coefficients  a r e removed with equations  2.7-7 and the " e q u i l i b r i u m Substitute  condition",  equation  equations 2.7-6 and 2.7-7 i n t o equation  2.2-14 t o o b t a i n :  3  'o  * ] £ i  R  i x  2.7-8  AM  Equation  2.7-8 i s used  coefficients.  Hence  to  remove  a l l undetermined  the response of the s h e l l can be found  with equation 2.2-13: «(r,t)-^6(r)  x  (l-exp(-Xt))  The r e s u l t s a r e :  2.2-13  - 52  « (r,t) x  *  -  3 e P j (n)Cos « (l-exp(-A  t))  0  2.7-9  where: A  2n  x  3e  2.8  0  R  2.7-10  APPROXIMATE  SOLUTION OF  SPHERICAL LIMIT WITH THE  If  limit  the  o r i e n t a t i o n of the section,  as  spherical  limit  results  last  problem with the  should  a  2.3-28,  2.3-29 and  equation  2.3-22.  to the  The  spheroidal  field  applied  shell in  the  limit,  then  in  the  depend  on  the  field.  which  f o r the  0  ,  CD  this  these  the  govern  "switch  Z-direction  "equilibrium C ^  In  £ * • < * > to o b t a i n  the  THE  Z-DIRECTION  equations  2.3-24 with the In  not  section, I l e t  of a s p h e r o i d . of  PROBLEM IN  is correct,  bulb with respect  in the  response  "SWITCH ON"  APPLIED FIELD IN THE  the mathematical a n a l y s i s  spherical  the  THE  on" are;  condition", equations  become:  n(2N+l)Q°U)P,?U) B°  ^  yi  o  u  P° (n)^(l-r, )^P ;(n)dn  o  N  2  )  2.8-1  -1  2.8-2  2.8-3 The  equilibrium  charge d e n s i t y  distribution  becomes:  - 53 -  6  * 3c E P0(n)  z  o  2.8-4  z  T h i s problem i s s o l v e d i n appendix E. Here outline  the  the r e s u l t s  procedure  used  hence  the  found  with:  in t h i s c a l c u l a t i o n  the i n t e g r a l B°  are  i n equation  known.  2.8-1  +  X  To  performed, X  , are  z  ,  ,  are  found  to  within  2.3-31  R  the  3 0  with:  z ] p°x "° u  _  2 > 3  eigenvectors  "equilibrium  undetermined  condition".  coefficients  Substitute  consider  equations  the  2.8-3  and  i n t o equation 2.2-14 to o b t a i n :  3e E P0 o  present  z  remove  2.8-4  is  3  undetermined c o e f f i c i e n t s B  and  • X iij = 0  z  the  [z  briefly  Next the e i g e n v a l u e s ,  JIN  and  will  obtained.  Firstly  det[B  I  2  ( n )  ^2\Kx  P  y °  (  n  )  2  '  8  ~  5  AM  All  undetermined  2.8-5. Thus 2.8-3.  the  Finally  coefficients  are  eigensolutions the  response  of  removed  are the  known  with with  equation equation  s h e l l i s found  with  equation 2.2-13: «(r.t)-^p«(r) (l-exp(-Xt)> A  The  results  are:  2.2-13  - 54 -  6 (r,t)  3e E pO(n)(l-exp(-X t))  %  z  0  z  2.8-6  z  where:  z  2.8-7  As  expected  the d i r e c t i o n of in  the s o l u t i o n  f o r 6 (r,t)  application  of  section  7).  These  compared with the exact s o l u t i o n  the  i s independent of applied  approximate  field  (see  results  are  f o r a s p h e r i c a l bulb, which  i s presented below. 2.9 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR  A  SPHERE  WITH THE APPLIED FIELD IN THE Z-DIRECTION Consider  a  conducting  sphere  c o o r d i n a t e s as shown i n f i g u r e shell  to  a uniform s t a t i c  the Z - d i r e c t i o n  will  15.  electric  d e s c r i b e d by s p h e r i c a l The field  response applied  of  this  at t=0 i n  be found. T h i s can be accomplished  with  the use of equation 2.2-17:  2.2-17 and  with  the  use  of  the  equilibrium  charge  density  d i s t r i b u t i o n : ** 6  k  z  "  3  E  O  E  C  O  8  0  Z  Jackson,J.D.,  -  3E E Pj(Co«e) 0  a  Classical  Sons, New York(l9T5~l  Electrodynamics  2.9-1  , John Wiley and  - 55 -  SPHERICAL COORDINATES  y /  r  Z'  e  FIGURE 15  CYLINDRICAL COORDINATES  z|  FIGURE 16  Y  - 56 -  Here  i s t h e magnitude  out,  i n this  hence  the eigensolution  density  6  z  (  To  ^»  -  distribution.  '  )  6  show  where  r  i sequal t othe  6  i s correct  equation  Furthermore  one e i g e n s o l u t i o n ,  equilibrium  2.2-17  t h e Green's  of spherical  one  and  function  coordinates,  demonstrate  i n an i n f i n i t e  with  r=r , i s :  ?  I  0  2 > g  0  i s the radius  of t h e sphere. Moreover  V  2  2  -  2.9-4  is: * 1  1  i-csinei^) + — i  Substitute equation  2.2-17 a n d p e r f o r m the spatial  as  terms  perform. equation 6 (r,t) z  where:  as  Equation 2.2-17  | i  the resulting  derivatives  possible.  z  resulting  This  2.9-2 t u r n s  . The r e s u l t  integration. from  calculation  V  2  Then  and cancel  i s easy  o u t t o be c o n s i s t e n t  to with  of the calculation i s :  - 3e E Cose(l-exp(-X t)) 0  2.9-5  2  e q u a t i o n s 2 . 9 - 2 , 2 . 9 - 3 , 2.9-4 a n d 2.9-5 i n t o  perform many  3  dA i s :  2  V  _  *  dA - r S i n 0 d e d $ and  3  Q  m=o  0  charge  can substitute  T^fTT ' : < c o . V ' ; ( c o . e ) c o . ( . ( - * ) )  *  turns  Thus:  assumption  i nterms  nso  there i sonly  I t  2.9-2  2.9-2 i n t o  " * " E E  field.  (  consistency.  G  that  electric  Xz *>  this  equation  domain  problem,  of the  2  2.9-6  - 57 -  A  * ' 3e r 0  The  2.9-7  o  exact r e s u l t s f o r the sphere, equations 2.9-6 and 2.9-7,  are c o n s i s t e n t with the approximate r e s u l t s  for a  sphere,  equations 2.8-6 and 2.8-7.  2.10  EXACT  SOLUTION  OF  THE  "SWITCH  ON"  PROBLEM  FOR A  CYLINDER WITH THE APPLIED FIELD PERPENDICULAR TO ITS AXIS  Consider a conducting polar  coordinates  this shell suddenly  to at  a  as  infinite  shown  uniform  cylinder  in figure  static  described  by  16. The response of  electric  field  applied  t=0 i n the X - d i r e c t i o n w i l l be found. T h i s can  be accomplished  with the use of equation 2.2-17:  J dA n  V  ^  and  +  X  6  2.2-17  A - °  the e q u i l i b r i u m charge  density  distribution,  6  Cr,"),  where: 6 ( r , « 0 x  It  -  c E Cos  6  out,  in  Q  turns  x  2.10-1  e i g e n s o l u t i o n , hence  this  problem,  the  that  eigensolution  there i s only one is  equal  to the  e q u i l i b r i u m charge d e n s i t y d i s t r i b u t i o n . Thus:  6(r,-> x  To  "  *  X X  <*>  demonstrate  2.10-2  this  assumption  i s correct  substitute  - 58  equation  2.10-2  consistent. domain  into  equation  Furthermore  2.2-17 a n d show t h e r e s u l t  the Green's  i n terms of c y l i n d r i c a l  function  coordinates,  i n an  with  i s  infinite  r=r ,  i  0  3 s  :  03  -21n[rJ+2y^Cos(n(»-»  G- £ -  where r  i s the radius  0  ))  0  2.10-3  of the c y l i n d e r . Moreover  dA i s :  dA « r d 6  2.10-4  0  and  V  i s :  2  1 ?  "  V  3 3~6 2  rT o  2.10-5  2  Substitute into  equation  Then p e r f o r m many t e r m s Equation 2.2-17  equations  2.2-17 a n d p e r f o r m  as p o s s i b l e . This  . The r e s u l t  -  v  be  and cancel  2  i s easy  consistent  as  t o perform.  with  equation  of the c a l c u l a t i o n i s :  t))  e. E C o s 6 ( l - e x p ( - X u  from V  calculation  out to  2.10-5  the resulting integration.  the derivatives resulting  2.10-2 t u r n s  « (r,t) X  2.10-2, 2.10-3, 2.10-4 a n d  X  2.10-6  *  where:  X  2 c  ° °  The on  l/r .  to  a  0  2.10-7  r  time  constant  A long  spheroid  c y l i n d e r , except  coordinates.  The  radius  of a i n f i n i t e could  be t h o u g h t  i t s radius of  c y l i n d e r , A ^ , depends  a  of  being  similar  i sa function of spatial  long  spheroid  defined  in  - 59 -  DEFINITION OF THE RADIUS OF A SPHEROID  r  r • R/l-n  z  \ 1  1  1  z =  7* to"  FIGURE 17  prolate spheroidal coordinates i s : r - R/l-n*. as  shown  2.10-8 in figure  17. Here R i s the maximum r a d i u s of the  s p h e r o i d . Hence the time constant would be expected  f o r a long s p h e r o i d ,  X  x  ,  to be:  ft x  2e R(l-n ) 2  2.10-9  l s  0  Thus, the response of the s h e l l would be: « (r,t) % e E Co89(l-ex (-X t)) 0  x  This  is  x  P  x  2.10-10  i n e x c e l l e n t agreement with the r e s u l t s of s e c t i o n  - 60 -  2.11  RESPONSE  OF  A  CONDUCTING  SHELL  TO  AN  ALTERNATING  ELECTRIC FIELD  Let the  the p o t e n t i a l which d e s c r i b e s the f i e l d e x t e r n a l to  b u l b be  $ (r,t) E  Here  -  *  u  $>. Furthermore l e t : E  2.11-1  (r)exp(iwt)  E  is  the frequency of the a l t e r n a t i n g e l e c t r i c  and <t (r) i s a p o t e n t i a l which  describes  E  Hence  as  u  approaches  zero  I  a  obtain  p o t e n t i a l as i n the "switch on" problem This the  problem  will  uniform  discussed  earlier.  be s e t up such that as u approaches zero  induced by the e x t e r n a l  field  2.11-2  E  induced  charge  density  i s assumed to f l u c t u a t e at the  same frequency as the e x t e r n a l density  field.  6 (r) E  is  a  charge  d i s t r i b u t i o n which i s a s o l u t i o n to the "switch on"  problem. Thus as u approaches zero a  must  this  "switch  problem  to  approach  the  Furthermore a i s expected to approach 0 as since  charge  be:  " «4 (r)exp(iut) The  field.  the same e x t e r n a l  "switch on" problem i s o b t a i n e d . Let the s u r f a c e  density  field  the  bulb  will  not  respond  approach  u  on"  1  problem.  approaches  to a f i e l d of  for  •  infinite  frequency. S u b s t i t u t e e q u a t i o n s 2.11-1 and 2.2-12. u  2.11-2  into  equation  i s assumed t o be small so magnetic e f f e c t s can be  n e g l e c t e d , hence e q u a t i o n 2.2-12 i s v a l i d . The r e s u l t i s :  - 61 -  6 dA E  - f l a V <j> 2  - 0 7 * * iau)6 »0  C w e  R  E  S u b s t i t u t e equations The  2.11-3  2  4  E  2.2-16 and 2.2-17 i n t o equation 2.11-3.  summation over A i n equation  since  the  2.2-16  has  been  dropped,  s o l u t i o n s t o the "switch on" problems, that have  been s o l v e d , c o n t a i n one e i g e n v a l u e , X. The  result  of  the  substitution i s : _  o  1_ , , <w  2.11-4  *X  1  Notice  that  a  approaches  0  approaches  as  u  1  as  approaches  t o approaches infinity  0  as  and a  expected.  S u b s t i t u t e equation 2.11-4 i n t o 2.11-2 to o b t a i n :  {  w  (  r  »  t  exp(lut)  )  1  +  i  2.11-5  r  6 ( r ) induces an e l e c t r i c  field  E  magnitude  to  i n s i d e the  the e x t e r n a l f i e l d  shell  equal  in.  but o p p o s i t e i n d i r e c t i o n .  Hence:  l + *r  Here  £  A  component due  is  -  2.11-6  the  of  applied  the  electric  electric  t o the s u r f a c e charges.  induced  field,  E* , N  is  field  and  t  u  i s the  f i e l d produced i n s i d e the s h e l l Note,  equal  as  u  tends  to  0  the  i n magnitude but o p p o s i t e i n  - 62 -  direction  to  the  applied  field  as  expected.  This  is  c o n s i s t e n t with the "switch on" problem. The e l e c t r i c E. + E  field  i n t e r n a l to the bulb, ?  ,  is:  k  2.11-7  S u b s t i t u t e equation  2.11-6 i n t o 2.11-7 to o b t a i n :  iL-  1 -  2.11-8  Hi  Thus the magnitude of the s h e l l ' s i n t e r n a l f i e l d i s : E.  E. =  2.11-9  where the  E,  is  magnitude  equation  the magnitude of the i n t e r n a l f i e l d and E i s A  of  2.11-8,  the the  applied  field.  Furthermore,  from  phase of the i n t e r n a l e l e c t r i c  field  with r e s p e c t to the e x t e r n a l e l e c t r i c  field i s :  6 - T a n —u  2.11-10  1  where e i s d e f i n e d i n f i g u r e 18. Now enough i n f o r m a t i o n has been obtained e f f e c t of " s t a t i c bulb.  As  a reminder, the " s t a t i c  below and up t o Consider  s c r e e n i n g " on a  equation  threshold 2.1-4:  when  calibration  to p r e d i c t the curve  of  a  s c r e e n i n g " model i s v a l i d applied  to  these  bulbs.  -  63 -  DEFINITION  E, ,  INTERNAL  OF  PHASE  SHIFT  FIELD  E , A  FIGURE  This  equation  The  " i n t " will  calibration f 6  In  i sv a l i d  curves  FIELD  18  f o rbulbs  be d r o p p e d  APPLIED  with  a nonconducting  since, as described  areactually  quite  linear,  earlier, the  hence:  - I * " EQ *  2.11-11  t h e d e r i v a t i o n o f 2.11-11  field  inside  account  the  equation E.  f  bulb  ui  i t was  was  unaffected  conductivity  applied  field,  2.11-11  assumed  of  by the  the  shell  E , by t h e i n t e r n a l A  A  2E f  the "static  To  simply  field,  E,.  becomes:  A  A  2.11-12  i st h e frequency  describes  electric  theshell.  1 +  «E„  Here  the  f o rthefinite  replace Hence  shell.  of the applied f i e l d .  screening  curve*.  The  This  equation  slope  of the  - 64 -  "static "zero  screening  curve"  conductivity  e x p l a i n s the curve The  curve"  depends  as  expected.  in  the  This  X/u. X  depends  conductivity  on  the  a  geometry,  only  size  and  sphere  and  s p h e r o i d , where two X's a r e r e q u i r e d to d e s c r i b e the  response of a s p h e r o i d ; one f o r the f i e l d axis  screening  curve"  c o n d u c t i v i t y of the s h e l l . I have found X f o r a for  equation  slope of the " s t a t i c  respect to the "zero  on  with respect to the  shown i n f i g u r e 6.  reduction  curve" with  i s reduced  parallel  of the spheroid and one f o r the f i e l d  to the  p e r p e n d i c u l a r to  the a x i s . Furthermore t h i s theory p r e d i c t s a phase s h i f t the  internal  field.  This  electric model  field  is valid  of  with respect to the e x t e r n a l at  threshold,  therefore i t  p r e d i c t s the phase s h i f t of the p u l s e s a t t h r e s h o l d .  2.12  "RESET SCREENING" MODEL  At fails the  high  f i e l d s the " s t a t i c  s c r e e n i n g " model  t o e x p l a i n the p p e r a t i o n of the bulbs. T h i s i s due t o avalanche  avalanche the  electric  bulb  breakdowns which occur a t high f i e l d s . These  breakdowns r e s e t the e l e c t r i c and w i t h i n the conducting  field  to  0  within  m a t e r i a l which makes up  the s u r f a c e of the bulb. The assumptions r e q u i r e d t o the " r e s e t s c r e e n i n g " model a r e i d e n t i c a l required account  f o r the " s t a t i c  derive  t o the assumptions  s c r e e n i n g " model except,  explicit  i s made f o r the " r e s e t t i n g " of the e l e c t r i c  field in  the bulb a f t e r each avalanche  breakdown.  are;  be  magnetic  effects  can  These  neglected  and  assumptions the  shell  - 65 -  possesses  i s o t r o p i c and Ohmic c o n d u c t i o n .  To account  f o r t h i s " r e s e t t i n g of the f i e l d " c o n s i d e r  uniform e x t e r n a l f i e l d which i s an i n f i n i t e  ramp:  (at)z  2.12-1  Here "a" i s the parameter which determines ramp, Let  the slope of  the  t i s time and z i n d i c a t e s the d i r e c t i o n of the f i e l d .  t h i s a p p l i e d f i e l d be c r e a t e d by the p o t e n t i a l :  * - atd.(r)  2.12-2  E  where tfr) c r e a t e s a uniform f i e l d . surface  charge  grows a charge  on  no  i t s s h e l l at t=0. As the e x t e r n a l f i e l d  density w i l l  bulb as determined  Consider a bulb with  be induced on the s u r f a c e of the  by equation 2.2-12:  2.2-12  T h i s charge reaches  density  the  will  threshold  grow  until  field  E . Q  avalanche  breakdown w i l l  bulb w i l l  be r e s e t to 0. The time  occurs field  will  be  the  Once  designated  at  t .  which  surface  s u r f a c e charge applied  charge.  It  will  produces a uniform  field  the  be  field  i n the  avalanche  Note, the bulb's  0  field  i s reached an.  Q  occur and the e l e c t r i c  i s the a p p l i e d f i e l d p l u s the f i e l d  induced  E  internal  internal  produced  by  the  assumed the induced which  opposes  the  field.  This  problem  will  constructed  "switch on" problem d i s c u s s e d e a r l i e r ,  so  i t reduces  when  "a"  t o the  tends  to  - 66 -  zero  and  transient  t  tends  s o l u t i o n s of  Furthermore, field  that  the  infinity. equation  consider  i n the bulb  slowly  to  "a"  Consider 2.2-12  t l a r g e so a l l  have  disappeared.  small and t l a r g e such that the  v a n i s h e s . The a p p l i e d  field  field  screened. Then the  i s completely  changes  so  induced s u r f a c e charge on the bulb i s : <5(r,t) - 6 ( r ) a t  2.12-3  where 6(r) i s the s o l u t i o n to the problem.  However,  I  am  appropriate  not i n t e r e s t e d  "switch  i n "a" small and-t  l a r g e . Hence I w i l l assume that the induced s u r f a c e 6(r,t),  has  a  transient  on"  charge,  which i s of the same form as the  t r a n s i e n t s which occur i n the -"switch on" problem, thus: 6 ( r , t ) - «(r)(at-S(l-exp(-At))) where 6 i s an undetermined by f o r c i n g  6 ( r ) to s a t i s f y  satisfies  the  "switch  2.12-4  c o e f f i c i e n t . B w i l l be determined the  "switch  on" problem  on"  problem. 6 ( r )  i f equations 2.2-16 and  2.2-17 a r e v a l i d . S u b s t i t u t e equations 2.12-4 and 2.2-12  and  equate  growth  and  2.12-2  into  equation  decay r a t e s . The r e s u l t i n g  equations a r e :  2.12-5  -  g " ^ ; : :  B  }  •o  A  +  67 -  -o  2.12-6  K  ^JAwcoR  2. 12-7  These equations 2.2-17 has  are equivalent  one eigenvalue.  induced «(r,t) The  Hence,  field, fi (r,t) L  -  is  2.12-8  first  term  in  equation  e x a c t l y c a n c e l s the a p p l i e d f i e l d . Thus the i n t e r n a l E, , i s e q u i v a l e n t -  t o a s u r f a c e charge  6 (r,t) L  -6(r) -(l-exp(-Xt)) f i e l d , E, ,  where: 2.12-9  a  is:  -(l-exp(-Xt))i  avalanche  2.12-10 field  breakdown  reaches the t h r e s h o l d v a l u e ,  occurs and the i n t e r n a l  to 0. T h i s occurs at time t , 0  0  there  6 ( r ) ( a t - f-( 1-exp (-X t ) ) )  When the i n t e r n a l  E -  interest  on the s h e l l due t o an i n f i n i t e ramp i s :  Thus the i n t e r n a l a  of  2.2-16  the s u r f a c e charge which i s  e l e c t r i c f i e l d c r e a t e d by the  2.12-8  2.2-16 and  i f B=a/\. Note t h a t the summation i n equation  been dropped s i n c e i n a l l cases  only  E, -  t o equations  J(l-exp(-Xtj >) )  Rearrange 2.12-11 t o o b t a i n :  field  E , D  an  i s reset  thus: 2.12-11  - 68 -  2.12-12  If one assumes the that the charge t r a n s p o r t mechanisms are unaffected  by  the  presence of the w a l l charges, then each  subsequent breakdown w i l l be l i k e the f i r s t frequency of breakdown i s given  by  one.  Hence  the  f = i / t . Thus: B  0  2.12-13  This  solution  Consider the amplitude and  £  of  wave  can  be  shown  applied  in  t o a t r i a n g u l a r wave.  figure  19.  Let  E  A  be  the  the t r i a n g u l a r wave, T the p e r i o d of the wave  the frequency of the wave. Then "a" i s :  A  a - 2f E A  2.12-14  A  S u b s t i t u t e equation 2.12-14 i n t o 2.12-13 t o o b t a i n :  2. 12-15  In equation 2.12-15 the corners been assumed not to a f f e c t , the bulb. T h i s assumption higher  then  the  of the t r i a n g u l a r wave  s u b s t a n t i a l l y , the pulse is  threshold  valid pulse  i n t e r e s t e d i n high a p p l i e d e l e c t r i c  at rate.  pulse  have  r a t e of  rates  Moreover,  f i e l d s such t h a t :  much I  am  -  AE  69  -  0  Equation 2.12-15 can be s e r i e s expanded The  results are:  f *  ^|^  B  11  - |  A  o  I  when  *  am  -Ala- << 2 f EA  for this  i  2.12-16  A  interested  in sinusoidal applied  f i e l d s . Hence,  parameter "a" i n equation 2.12-13 changes with "a"  i s an  like  to approximate the sine wave with a wave  similar  important  slope  the  Since I would  which  has  and has the same amplitude. The simplest shown  in  figure  20.  a  and This  form has the same slope as the sine wave at the  zero c r o s s i n g s has  time.  parameter i n the d e r i v a t i o n ,  most a c c u r a t e approximation i s approximate  extreme.  but i s c l i p p e d  o f f so  the  approximate  wave  same amplitude as the sine wave. I t i s c l e a r , from  f i g u r e 20, t h a t : a - wfA E  2.12-17  A  Furthermore, m u l t i p l y for  clipping  the  the pulse r a t e f  peaks  of  the  B  by  2/ir to  triangular  account  wave.  Hence  equation 2.12-13 becomes:  B  it  2.12-18 V  **AE /  I am i n t e r e s t e d  A  in  high  electric  fields,  2.12-18 can be s e r i e s expanded t o o b t a i n :  thus  equation  - 70 -  DEFINITION  A  Q _l  OF " a " FOR A T R I A N G U L A R  WAVE  A  s  2EA_ T  E / APPLIED FIELD , A  f = 7  UJ  A  U -  C_>  \  DC  HO UJ _l  /  \  /  a  s  2E f A  I -A  111  f  \l  1  I.  TIME  FIGURE  DEFINITION  19  OF " a " FOR A C L I P P E D  TRIANGULAR  WAVE  BEST APPROXIMATION TO A SINE WAVE  E 2  FIGURE  20  A  - 71 -  f *  2f | A  A  B  The  i  -i£a « x  when  "reset  screening"  model,  2.12-19  as  seen from  equation  2.12-19, p r e d i c t s the o f f s e t of the c a l i b r a t i o n curves the  "zero  conductivity  curve"  at  high  electric  fields.  F i g u r e 21 shows the t h e o r e t i c a l  " r e s e t screening curve"  "static  the  screening  curve"  t y p i c a l c a l i b r a t i o n curve For many bulbs "static  screening  on  same  screening"  the c a l i b r a t i o n curve curve"  model  p l o t as w e l l as a  at  low  will  "t " o  equation  derivation  of  the  w i l l e x p l a i n t h i s phenomenon.  The  2.2-12  the  transient  p l a y an important  r o l e . Let  > >  2.12-20  1  then the t r a n s i e n t s o l u t i o n s play no r o l e i n of  the bulb. Hence equation  the  "reset screening"  t «l/2f . A  0  7f > A  the  the  be the time between p u l s e s . I f :  0  A t  of  follow  count r a t e s as shown i n  heart of the " r e s e t s c r e e n i n g " model i s that solutions  and  of a bulb.  f i g u r e 21. A c a r e f u l examination of the "reset  from  model  the  operation  2.12-4 i s not v a l i d which means is  not  valid.  At  threshold  Thus i f :  1  calibration  2.12-21  curve  is  expected  to f o l l o w the " s t a t i c  s c r e e n i n g curve" at low count r a t e s . I f :  - 72 -  2.12-22 only the t h r e s h o l d p o i n t i s expected to be screening  curve".  Of  course  on  the  "static  there i s a t r a n s i t i o n zone i n  which n e i t h e r model a p p l i e s .  2.13 LINKING OF THE "RESET SCREENING" AND "STATIC SCREENING" MODELS  I  now  calibration  have  two  curve.  ways  of  Firstly,  calculating I  Y - i n t e r c e p t of the " r e s e t s c r e e n i n g x  * *  find  from X  one  from  the  curve": 2.13-1  Y S  where Y as  can  X  i s the Y - i n t e r c e p t of the " r e s e t  g  screening  shown i n f i g u r e 21. Secondly, I can f i n d  r a t i o of the slopes of the " r e s e t screening "static  curve"  X by talcing the curve"  to the  s c r e e n i n g curve". From equation 2.11-12:  2.13-2  where  M  s  i s the slope of the " s t a t i c  s c r e e n i n g curve". From  equation 2.12-19:  2.13-3  where M  R  i s the slope of the " r e s e t s c r e e n i n g curve",  hence:  -  RESULTS  OF T H E S T A T I C  73 -  AND  RESET  V ACTUAL CURVE  LU <  SCREENING  RESET  MODELS  SCREENING  CALIBRATION  2 f £E Eo A A  A  CURVE  A±  ir  LU CO I CL  STATIC  SCREENING  2E f A A  T  I T  / /  APPLIED  / / /  /  /  Y = A/ir 5  FIGURE  21  FIELD/  E,  CURVE  - 74 -  2.13-4  Finally, shift,  as shown e a r l i e r , A 0 ,  of  describes this X • wTanG  the  pulses  can be determined by the at  t h r e s h o l d . Equation  phase  2.11-10  relationship: 2.11-10  - 75 -  CHAPTER 3 EXPERIMENTS 3.1  INTRODUCTION  The  purpose  r e s e a r c h was the  of  the  experiments  performed  in  to e v a l u a t e the t h e o r e t i c a l models presented i n  previous  chapter.  Firstly  I w i l l b r i e f l y d e s c r i b e the  apparatus common to a l l the experiments.  The  d e s c r i b e d i n more d e t a i l than given below  i n a paper by  Friedmann The  apparatus  is D.E.  et. a l . apparatus  consisted  of  used  two  in  parts;  alternating electric magnitude  this  this a  experiment  device  f i e l d s , and a  to  device  essentially  create to  uniform  measure  the  of these f i e l d s . The f i e l d was c r e a t e d by a p p l y i n g  transformer  outputs  to  two  sets  of  parallel  plate  c a p a c i t o r s . For one s e t , the p l a t e dimensions were 3 f t . by 3  f t . and p l a t e s e p a r a t i o n ranged from 6 i n . to 1 f t . These  p l a t e s were d r i v e n by a 7 sinusoidal  and  kV  the frequency range was  other set of p l a t e s were 8 separation at 60  Hz  transformer  transformer.  f t . by  8  The  field  was  40 Hz to 1 kHz. f t . and  the  The  plate  ranged from 6 i n . to 3 f t . These p l a t e s were f e d by  a  30  output  kV was  transformer. varied  by  In  both  cases  the  using a v a r i a c f o r the  input supply. The p l a t e v o l t a g e was measured with the use of a 1000:1 v o l t a g e d i v i d e r  and  a  digital  volt  meter.  See  Friedmann,D.E.-, Curzon,F.L., Feeley,M., Young,J.F. Auchinleck,G. Rev. S c i . Instrum., V o l . 53, 1273 (1982)  and  DEVICE  120  V.A.C.  60 Hz  FOR PRODUCING ELECTRIC FIELDS  FREQUENCY GENERATOR  AMPLIFIER  PARALLEL PLATE CAPACITOR  TRANSFORMER  T VOLTAGE DIVIDER  FIGURE 22  DEVICE FOR MEASURING  OPTICAL  BULB  "  ELECTRIC  FIELDS  FIBER  PHOTOMULTIPLIER  Q -  I I l o FREQUENCY COUNTER  OSCILLOSCOPE  FIGURE 23  - 78 -  f i g u r e 22 f o r a diagram E  A  of the apparatus. The a p p l i e d  , can be c a l c u l a t e d from  the p l a t e v o l t a g e and  field,  the  plate  separation. Calibrated the e l e c t r i c plot  of  electric  f i e l d s . A c a l i b r a t e d bulb i s one f o r which  f i e l d has been determined. from  nonconducting was  the  region  The p u l s e s of l i g h t were  near  the  o p t i c a l f i b e r . T h i s ensured  not  connected  to  monitored  a  with  the f i e l d  a  near the  optical  fiber  p h o t o m u l t i p l i e r whose a m p l i f i e d output  by a frequency counter  so  as  to  determine  of the bulb, f . The p h o t o m u l t i p l i e r output  rate  B  a l s o monitored with an o s c i l l o s c o p e . See diagram  bulb  s u b s t a n t i a l l y a f f e c t e d by the d e v i c e used to  measure the p u l s e r a t e of the b u l b . The  pulse  the  the pulse r a t e versus the magnitude of the a p p l i e d  transmitted  bulb  bulbs were used t o measure the amplitude of  figure  23  was was the was  for  a  of the apparatus.  3.2 VERIFICATION OF THE "STATIC SCREENING" MODEL  The " s t a t i c the  screening  conducting  s c r e e n i n g " model can be t e s t e d by of  alternating  electric  on  i t . Furthermore,  that the a x i s of the tube in  inside  tube  with  d e f i n e a c o o r d i n a t e system  is parallel  no such  to the Z-axis as shown  f i g u r e 24. Let  Aj.  fields  tubes.  For example, c o n s i d e r a long conducting ends  measuring  when  the time constant f o r the response the  applied f i e l d  of the  tube  be  i s p e r p e n d i c u l a r t o the a x i s of  - 79 -  TUBE ORIENTATION WITH RESPECT TO CARTESIAN  COORDINATES  X  TUBE  L Y  FIGURE  the tube and A „ when the f i e l d These  time  chapter. \ of  constants  have  is been  perpendicular X  2.10-7,  parallel  to  derived  the  X  to  cylinder  i t s axis.  axis.  i n the p r e v i o u s  can be assumed to be equal t o the time  ±  the response of an i n f i n i t e  field  24  with  Hence,  constant  the  applied  from  equation  is:  Q  This  3.2-1  result  is  would expect modified cylinder finite  valid  the  near  f o r an i n f i n i t e  response  the  ends  of of  a  c y l i n d e r , hence one  finite  cylinder  to  the c y l i n d e r . However,  i f the  i s long enough the response near the c e n t r e of cylinder  will  c y l i n d e r . I t was found infinitely  long  when  be  similar  to  that s p h e r o i d s L/D>3;  i t will  that of an  could  be  be  the  infinite  considered  be assumed t h i s a l s o  - 80 -  applies  to  tubes.  Hence  equation  3.2-1  i s valid  for  conducting tubes with L/D>3. The d i s t o r t i o n s of the e l e c t r i c field,  near  the  ends  of the tube, w i l l extend a d i s t a n c e  approximately equal t o i t s diameter c o n s i s t e n t with the assumption like  i n t o the tube.  is  that tubes with L/D>3 respond  infinite  cylinders.  X„  be assumed to be equal to the time constant of  can  the response of a parallel  to  long  spheroid  with  the  applied  tube  field  i t s a x i s . As i n the p r e v i o u s case, the l e n g t h s  of the s p h e r o i d and c y l i n d e r a r e equal and the the  This  diameter  of  i s equal t o the maximum diameter of the s p h e r o i d .  Hence, from equation 2.6-4, X„ i s :  3.2-2  There i s some e r r o r differences  in  i n t h i s assumption  the  t h e i r ends. However, objects  becomes  near  the  to  the  drastic  geometries of the s p h e r o i d and tube at i t i s clear  larger  c o n s t a n t s i s reduced. field,  due  the  It  ends  that  differences  i s expected  of  as  the  tube,  L/D in  of  these  their  time  that  the  will  be  electric distorted  approximately one diameter i n l e n g t h i n from the ends of the tube. Near the c e n t r e of the tube the time c o n s t a n t response  of  the  electric  f i e l d w i l l be approximately the  same as the time constant i n the c e n t r e of a long A  spheroid  can  be  considered  long  spheroid.  when L/D>3. The time  constant f o r a tube w i l l be assumed to the same s p h e r o i d ' s when L/D>3.  of the  as  a  long  - 81 -  The  ratio  of the time c o n s t a n t s f o r the f i e l d a p p l i e d  p a r a l l e l t o the a x i s of the tube t o the the f i e l d  time  constant  for  a p p l i e d p e r p e n d i c u l a r t o the a x i s i s :  3.2-3  This  quantity  conductivity  is  from  convenient  the  analysis.  measured e x p e r i m e n t a l l y . figure  25.  Equation  since  A„L /X D 2  3.2-3  2  1  i t eliminates  Furthermore  the  i t can  i s plotted against  be  L/D  in  i s a t h e o r e t i c a l p r e d i c t i o n of  X|, / Xx. The  r a t i o of the time c o n s t a n t s can a l s o be  experimentally  by  electric  i n s i d e conducting tubes.  fields  measuring  the  determined  s c r e e n i n g of a l t e r n a t i n g Consider  equation  2.11-9:  2.11-9  Here  E,  and  E  A  are  the  magnitudes  of the i n t e r n a l and  e x t e r n a l f i e l d s r e s p e c t i v e l y and u i s the frequency field.  of  the  T h i s equation can be rearranged to o b t a i n :  3.2-4  Thus, with the use of equation 3.2-4, X„/x i s : A  3.2-5  A PLOT OR THE THEORETICALLY PREDICTED X,, /X  x  - 83 -  Here  (E /E, )  field  to the  i s the r a t i o of the magnitudes of the a p p l i e d  A  parallel same  internal  to  the  ratio  axis  for  perpendicular  field  the  was  field  when  the  is  applied  (E /E, ) A  field  is  i s the  x  applied  to the a x i s . These q u a n t i t i e s can be measured  with equation 3.2-5  tube  the  of the c y l i n d e r and case  e x p e r i m e n t a l l y . Hence  Firstly  when  X /Xj. can be determined M  and  t h e o r e t i c a l l y with equation 3.2-3.  I w i l l d e s c r i b e how  held  experimentally  between  the  (E /E|) A  plates  was  ±  measured.  of the p a r a l l e l  The plate  c a p a c i t o r with p l e x i g l a s s h o l d e r s as shown i n f i g u r e 26. p l e x i g l a s s h o l d e r s d i d not a f f e c t the since  the  experimental  The  results  m a t e r i a l s which made up the tubes were much more  conducting than the p l e x i g l a s s . The p l a t e s e p a r a t i o n was l e a s t three times the diameter centred  between  applied e l e c t r i c distance ensure ends  field,  defined  of  the  a  plexiglass  equipment was  built  out of the p l a t e s important affected  because  in  done'to ensure  the  Furthermore,  the  uniform.  figure  26  was  r e s u l t s were not  g r e a t e r than 3D to influenced  p l a t e c a p a c i t o r . The  h e l d i n the c e n t r e holder  as  shown  of in  by  electric the  figure  the field  tube  with  26.  The  such that the tube c o u l d be moved i n and without  disturbing  the  bulb.  This  the c a l i b r a t i o n curve of the bulb can  the tube was  calibrated.  without  A  was  was  is be  i f the bulb i s d i s t u r b e d .  Firstly bulb  This  E , was  parallel  (a bulb) was  of the tube, and the tube  plates.  the experimental  monitor with  S  the  at  Next  removed from the the  tube  was  plates  and  the  placed into p o s i t i o n  d i s t u r b i n g the bulb. Some tubes had  seams which  ran  - 84 -  parallel  to  their  o r i e n t e d as shown  axes; in  these  figure  were p l a c e d with the seam  26.  The  applied  field  was  i n c r e a s e d u n t i l the bulb was o p e r a t i n g . From the c a l i b r a t i o n curve  the  applied  internal  field,  E, ,  f i e l d , E , was determined A  To measure configuration plates  (E /E,) A  M  determined;  and the  from the p l a t e v o l t a g e and  A  p l a t e s e p a r a t i o n . Hence ( E / E , )  was  was measured.  x  the apparatus was rearranged t o the  shown i n f i g u r e  27.  The  separation  of the  was such that the ends of the tube were a t l e a s t one  tube diameter away from the p l a t e s . T h i s was done t o the  applied  field  was r e l a t i v e l y  d i s t a n c e S, d e f i n e d ensure  the  was  A  (E /E, ) A  Tubes the  disturbing  ( E / E , ),,  measuring  2D+L  were  could the  to  from  the  The  procedure  for  the  procedure  for  earlier. since  c o n d u c t i v i t i e s of these m a t e r i a l s are i n the range  which  E / E , to A  conductivity  of  made e i t h e r  removed  bulb.  identical  as d e s c r i b e d  x  be  from p a x o l i n or neoprene  allows  be  measured  the  material  with  the  must  monitor  apparatus.  be  somewhere between 0.05 and 0.95 because  tubes,  The  such that E, / E  the  electric  A  is  field  i s , a t b e s t , a c c u r a t e t o w i t h i n 5%.  A diameter of approximately 3 inches was chosen since  this  dimension  i s l a r g e i n comparison  b u l b diameter and small compared t o The  to  experimental r e s u l t s were not i n f l u e n c e d by the  without  measuring  uniform. Furthermore, the  i n f i g u r e 27, was g r e a t e r than  edges of the p l a t e s . The tube plates  ensure  neoprene  neoprene which  tubes  were  the  constructed  ranged i n t h i c k n e s s from  plate  to  t o the  separation.  from f l a t 1/8  f o r the  sheets of  1/4  of  an  CONFIGURATION  OF  THE  APPARATUS  USED  TO  MEASURE  (EA/E, )  ±  L/2  SEAM OPTICAL  BULB  zzzz  FIBER  v  V*  D/2  T  t >>>>>>>>  \  RUBBER  3D  D  i. _\ > >  TUBE  7"  X  PLEXIGLASS  HOLDERS  SEAM S  >  3D  TUBE  FIGURE  26  CONFIGURATION OF THE APPARATUS USED TO MEASURE  T  ( E / H , )„ A  D -*»  D  1 RUBBER  H  TUBE  BULB OPTICAL  FIBER  D  CAPACITOR  7  "7 PLEXIGLASS  7*  X  HOLDERS  PLATE  S > L+2D  FIGURE 27  -  inch.  The  87 -  tubes were wrapped with black e l e c t r i c a l  order t o maintain the c y l i n d r i c a l turned out t o be much l e s s hence i t d i d npt a f f e c t  tape i n  shape. The e l e c t r i c a l  conducting  than  the experimental  the  neoprene,  results.  Each neoprene tube had a seam which ran p a r a l l e l  to the  a x i s of the tube. The seam was o r i e n t e d i n the manner in  figure  26,  tape  shown  s i n c e by symmetry there was no v o l t a g e drop  a c r o s s i t . In t h i s way the j o i n t d i d not a f f e c t the  current  flow. To check the theory as thoroughly as p o s s i b l e and  (E /E,) A  1  were  (40 Hz - 1 kHz), and f o r 3  Theory  L/D>3.  diameter accuracy are:  A  measured over a range of f r e q u e n c i e s of  the a p p l i e d f i e l d requires  (E /E, )„  The  r a t i o was s e t by the  upper need  <  L/D  <  11.  l i m i t on the l e n g t h to to  measure  E,  to an  of b e t t e r than 5%. The r e s u l t s of the measurements  -  88  -  TABLE I, STATIC SCREENING DATA  L/D  MATERIAL  3  paxolin  40  Hz  1.212  1.792  4.15  3.50  3  paxolin  60  Hz  1. 0 9 3  1.610  3. 14  3.50  3  paxolin  60  Hz  1 . 106  1.580  3.48  3.50  3  paxolin  120  Hz  1.111  1. 4 5 6  4.12  3.50  3  paxolin  160  Hz  1 . 104  1.585  3.42  3.50  3  paxolin  200  Hz  1.096  1. 5 9 0  3.27  3.50  3  neoprene  400  Hz  2.890  7.200  3.42  3.50  3  neoprene  700  Hz  2.440  5.800  3.51  3.50  3  neoprene  1000  2.270  5.000  3.74  3.50  5  neoprene  60  Hz  1. 5 6 3  5. 182  5.97  5.81  5  neoprene  60  Hz  1.742  5.464  6.64  5.81  11.3  neoprene  60  Hz  2.800  32.43  10.3  9.51  11.3  neoprene  60  Hz  1. 8 5 0  17.08  11.7  9.51  ( E / E , )„ A  Hz  (EA/E, )  L  (X L /Xi D )  E X  was  (X„ L /Xi  T H  was c a l c u l a t e d using equation  M  Z  z  2  D ) 2  calculated  using  Ui D^/EX  between the theory  reasonable.  Notice  that  D*/TH  3 . 2 - 5  equation  and  Both of  3.2-3.  2 8 . The  these q u a n t i t i e s a r e p l o t t e d a g a i n s t L/D i n f i g u r e agreement  Ui  and the experimental p o i n t s i s  X,,/Xj.  i s relatively  frequency  independent as expected. It  turned  difficult small  out the experimental p o i n t s were  relatively  t o o b t a i n . F i r s t l y , the tube m a t e r i a l had  conductivity  that merely handling  the c o n d u c t i v i t y enough t o a l t e r  such  a  the tubes a f f e c t e d  the r e s u l t s .  Hence  much  - 90 -  care  was  taken  to  ensure  the  tubes  Moreover, the humidity a f f e c t e d the f o r the tube with L/D The  quantity  = 11.3,  (E /E, ) A  two  For  clean. example,  changes  in  independent  = 3 and L/D  humidity.  rubber  was  = 5 were  Fortunately  of the c o n d u c t i v i t y of  experimental  results  are  washed  result  also  the the  relatively  changes i n the c o n d u c t i v i t y due  Hz.  changed d r a m a t i c a l l y between  t h i s changed the c o n d u c t i v i t y of the rubber. The  at 60 Hz with L/D  can  very  runs where taken at 60 A  between these runs because the runs;  results.  (E /E,)„  and  A  were  runs  influenced  quantity rubber,  by  Xi,/X i s ±  hence  the  u n a f f e c t e d . However,  to c l e a n l i n e s s and  humidity  i n some n o n - i s o t r o p i c c o n d u c t i o n . T h i s e f f e c t i s  l i k e l y not important  s i n c e the r e s u l t s agree reasonably w e l l  with the theory, but  i t is non-statistical  it  is  difficult  to  p r e d i c t the average  e f f e c t . Moreover, i t was tubes  in a c y l i n d r i c a l  For  tubes  even  difficult  as  little  Unfortunately a r i g i d  insulating  10%  affected  was  found s t a t i c charges tend to b u i l d  neoprene  of  as  rigid.  round  much  the  as  30%.  maintain the shape of the neoprene, because i t  the  insulating  therefore  problem. The e l e c t r i c  they  up  on  the  contact  s t r u c t u r e and the neoprene.  T h i s a f f e c t e d the experimental r e s u l t s . solid,  to t h i s  support s t r u c t u r e c o u l d not  used  were  out  by  be  between  hence  e r r o r due  to maintain the  as  were  areas  nature  shape s i n c e the tubes were not  experimental r e s u l t s  to  in  were  not  f i e l d monitor was  The  paxolin  affected another  tubes  by source  this of  e r r o r , s i n c e the bulbs are s l i g h t l y u n s t a b l e . The p u l s e r a t e of  a  bulb v a r i e d as much as 10% from run t o run at a given  -  applied f i e l d nature  of  strength  the  91  and  -  frequency.  experiment,  Thus,  due  to  the  the r e s u l t s should be taken as  qualitative. The assumptions  r e q u i r e d to d e r i v e the s t a t i c  model, which i s the theory t e s t e d i n t h i s the  frequency  of  the  applied  field  screening  experiment, is  are;  low so magnetic  e f f e c t s can be n e g l e c t e d , c u r r e n t can only flow on the s h e l l surface  and  conduction.  the The  shell first  possesses two  experiment. The remaining  two  isotropic  and  Ohmic  assumptions were v a l i d  in this  assumptions  depend  on  the  m a t e r i a l used to c o n s t r u c t the tubes. However, the m a t e r i a l s used  do  appear  to  have  isotropic  because the data agreed reasonably Furthermore,  and  well  Ohmic conduction with  the  theory.  i t seems t h a t the assumption t h a t a long, open  ended, tube can be t r e a t e d as a long s p h e r o i d ,  is  a  valid  assumption. It  i s i n t e r e s t i n g t o p l o t the c o n d u c t i v i t y of the tube  m a t e r i a l as a f u n c t i o n of frequency, s i n c e the seems t o be dependent  conductivity  on frequency. From e q u a t i o n 3 . 2 - 4 :  Furthermore, from equation 3 . 2 - 1 ,  Ai  is: 3.2-1  3.2-6  The diameter of the L/D = 3 neoprene  tube was 10 cm. and the  - 92 -  CONDUCTIVITY  3000f  OF PAXOLIN  PLOTTED AGAINST  FREQUENCY  © - EXPERIMENTAL POINT  e J7 I  2000  (0  I?  u  a  1000 +  8  50  150  100  «/2tr (Hz)  FIGURE 29  200  1  -  CONDUCTIVITY  4  OF  © -  NEOPRENE  93  -  PLOTTED  EXPERIMENTAL  AGAINST  POINT  60, 000 +  w a  40, 000 +  20,000 +  400  800 w/2« (Hz)  FIGURE  30  FREQUENCY  - 94 -  diameter of the p a x o l i n tube was 6 cm. The data from Table 1 was  used  to  obtain  fl/e  0  with equation 3.2-6. The r e s u l t s  are:  TABLE I I , CONDUCTIVITY OF PAXOLIN AND NEOPRENE M.K.S. u n i t s MATERIAL  n/e  fA  0  paxolin  40 Hz  23  paxolin  60 Hz  29  paxolin  60 Hz  28  paxolin  120 Hz  48  paxolin  160 Hz  74  paxolin  200 Hz  93  neoprene  400 Hz  1790  neoprene  700 Hz  2520  neoprene  1000  3080  Hz  T h i s data i s p l o t t e d i n f i g u r e s 29 and 30.  3.3 EXPERIMENTAL VERIFICATION OF THE FREQUENCY DEPENDENCE OF THE CONDUCTIVITY OF NEOPRENE  0/e  was  measured  experimentally e l e c t r i c current voltage  as  determining flowing  a p p l i e d across  a  function  the  in f l a t  of  frequency  by  magnitude and phase of the sheets of neoprene due t o a  the neoprene. For example, consider a  - 95  flat  rectangular  electric  piece  of  potential applied  neoprene  with  an  alternating  a c r o s s i t as shown i n f i g u r e 31.  NEOPRENE  DIMENSIONS  FLAT PIECE OF NEOPRENE  i  FIGURE  The f i e l d applied  31  i n t e r n a l to the neoprene, E , , i s  field  by equation  related  to the  2.11-8:  V.  E \  1-  2.11-8 0k  where the the  z  indicates  the d i r e c t i o n of the a p p l i e d  field,  V is  v o l t a g e drop a c r o s s the neoprene and L i s the length neoprene  field,  sheet  as  shown  Is, , w i l l d r i v e a surface  in  of  f i g u r e 31. The i n t e r n a l  current  i s r e l a t e d t o E \ through Ohms law:  A  J on the neoprene. 3  -  j -0E  -  96  3.3-1  A  where Q i s the s u r f a c e c o n d u c t i v i t y of one (i.e.  Ohms  per  square).  It  turns  sheet  of neoprene  the  experimental  out  r e s u l t s are c o n s i s t e n t with the assumption t h a t c u r r e n t flows on the s u r f a c e of the 3.3-1 *  i n t o equation J  •"  To  V  *  (i  =  »  2.11-8 to o b t a i n :  3  current  this  t o t a l current the  flow  material  through  were  the  neoprene  bundled together  flow through the neoprene i s  number of s h e e t s . The  "n" sheets  equation  4)L  -  of  Substitute  -  i n c r e a s e the  sheets  neoprene.  only  -  3  ?  2  many  since  the  proportional  to  t o t a l c u r r e n t , I, flowing  through  of neoprene i s :  I • Jn£  3.3-3  where J i s the magnitude of defined  in  figure  31  and  sheets. Thus, from equation I  V  the  surface  "n"  current  , £  is  i s the number of neoprene  3.3-2:  A  -  3.3-4  2  Let 0 be the phase d i f f e r e n c e between the a p p l i e d f i e l d the c u r r e n t , then from equation 6 - Tan -^  3.3-2: 3.3-5  1  Hence equation  and  3.3-4  becomes:  - 97 -  LI V £nCos9  3.3-6  A  C l e a r l y the time c o n s t a n t , X , of a f l a t  sheet i s :  The  the charge  time  constant  is  dependent  i n t e r a c t i o n of the e l e c t r o n s as they e  Thus was  was  used r a t h e r then e  0  on  flow  the  i n equation 3.3-7  q u i t e l a r g e , which means most of the  w i t h i n the d i e l e c t r i c .  in  Equation 3.3-5  to  current  charge rubber.  s i n c e "n" flow  was  becomes:  ^ = LuTane  3.3-8  S u b s t i t u t e equation 3.3-8  E  i n t o equation 3.3-6  to obtain:  I " V -twSine  3.3-9  A  Thus,  e  ,  3.3-9, 3.3-6 phase  ft and  and  ft/e  3.3-8  can  be c a l c u l a t e d w i t h  respectively  if  the  equations  magnitude  and  of the v o l t a g e a c r o s s the neoprene sheets i s known as  w e l l as the magnitude and the phase of the  current  through  the s h e e t s . The  apparatus  i n f i g u r e 32. The between  the  was  flat  arranged t o the c o n f i g u r a t i o n shown neoprene  capacitor  plates.  around the neoprene t o c r e a t e an allows  only  capacitively  the  circuit  due  to  A  mylar  were  held  sheet was  insulating  coupled  c i r c u i t . Of course t h e r e was in  sheets  current  a contribution to  barrier  firmly wrapped which  t o flow i n the the  current  the c a p a c i t a n c e of the p l a t e s . I t  - 98 -  CONFIGURATION OF THE APPARATUS  b H.V.  O OSCILLOSCOPE  FIGURE 32  USED TO MEASURE fl/e  -  turns out  t h i s current  The  magnitude and  an  oscilloscope  with  voltage  was  resistors voltage  -  is negligible. phase of the voltage as  shown  then c a l c u l a t e d with V which  form  the  in  voltage  H  was  figure  and  H  V  the  measured  32.  The  values  1000V  The  magnitude and  of  d i v i d e r . Thus the  the plate  3.3-10  H  phase of v o l t a g e  the o s c i l l o s c o p e . Thus the c u r r e n t  V,  was  a l s o measured with  through the  neoprene  1= V, /R  was:  3.3-11  1  I t t u r n s out  V,  the c o n d u c t i v i t y of  plate  was:  V= A  99  the  was  s i n u s o i d a l when V  of the neoprene was  resistances  in the c i r c u i t  experimental r e s u l t s were not The  dimensions  f i g u r e 31,  of  the  was  H  Ohmic.  s i n u s o i d a l , hence The  magnitudes  were chosen such that  the  a f f e c t e d by t h e i r presence. neoprene  sheets,  defined  in  were:  1=0.151m L = 0.1  25m  D = 0.0050m The  results  of  between the p l a t e s  a run with three sheets of neoprene are:  placed  -100 -  TABLE I I I , CONDUCTIVITY OF NEOPRENE (RUN WITH 3 SHEETS) M.K.S. u n i t s n  e/e  U/2T  0  OxlO"  O/e  n/e  8  0  3  95 Hz  1 1 .9  1 .8  171  2030  3  142 Hz  10.6  2.1  224  2370  3  191 Hz  9.8  2.4  276  2700  3  238 Hz  8.6  2.6  341  2930  3  286 Hz  8.0  3.0  423  3380  3  333 Hz  7.8  3.3  477  3720  3  1000 Hz  5.9  7.2  1377  81 20  The r e s u l t s of a run with f i v e  sheets  of  neoprene  between the p l a t e s a r e :  TABLE IV, CONDUCTIVITY OF NEOPRENE (RUN WITH 5 SHEETS) M.K.S. u n i t s n  <i)/2*  OxlO"  fl/e  n/e  8  0  5  95 Hz  11.5  2.0  196  2250  5  142 Hz  10.5  2.5  269  2820  5  191 Hz  9.6  2.8  329  3158  5  238 Hz  9.0  3.0  376  3380  5  286 Hz  8.6  3.2  420  3600  5  333 Hz  8.3  3.4  462  3830  5  1000 Hz  6.0  7.2  1354  8120  placed  -101  e/e ,  a  and  35  0  34  u>/2» i n f i g u r e s 33,  and ft/c a r e p l o t t e d a g a i n s t  measured,c/e sheets  -  or  used  respectively.  As  a and fl/e do not  expected, depend  on  the  quantities  the  number  i n the run. Any d i f f e r e n c e s were l i k e l y due to  the l a r g e e r r o r s i n v o l v e d i n t h i s experiment. the  conductivity  cleanliness handling  of  of  For  example,  of the neoprene was very s e n s i t i v e to the the  neoprene  the neoprene  An e f f o r t  was  made  possible,  however  and  the  humidity.  Merely  from run t o run a f f e c t e d the r e s u l t s . to  this  keep  the  was s t i l l  neoprene a major  Furthermore t h i s e r r o r was n o n - s t a t i s t i c a l  as  clean  as  source of e r r o r . in  nature ( i . e .  the c o n d u c t i v i t y of the neoprene e v o l v e s with time) hence i t was  inappropriate  to  estimate the magnitude  Thus, these r e s u l t s should be taken results,  tables  as  of the e r r o r .  qualitative.  I I I and IV, can be compared t o the r e s u l t s  from the p r e v i o u s s e c t i o n f o r the L/D=3 neoprene I I . As can be seen from these t a b l e s , 0/e be  the  same  order  of  magnitude  0  was  to be v a l i d ,  the " s t a t i c  the  frequency  of  magnetic e f f e c t s can be  which  was  a  s c r e e n i n g " model must be v a l i d . The  the  applied  neglected,  screening"  model  f i e l d must be low so  no  gaseous  breakdowns  i n or near the c o n d u c t i n g m a t e r i a l and the c o n d u c t i n g  m a t e r i a l must possess i s o t r o p i c and first  to  i n order f o r e q u a t i o n s 3.3-9, 3.3-6 and 3.3-8  assumptions r e q u i r e d t o d e r i v e the " s t a t i c  occur  measured  t h i c k n e s s and was from a d i f f e r e n t source.  Finally,  are;  tube, t a b l e  i n both experiments even  though the tube was c o n s t r u c t e d from neoprene different  These  two  assumptions  were  valid  Ohmic in  c o n d u c t i o n . The this  experiment.  -102 -  EXPERIMENTAL  12 +  e/e  0  , FOR NEOPRENE,  PLOTTED AGAINST  © - RUN  WITH 5 SHEETS OF NEOPRENE  + - RUN  WITH 3 SHEETS OF NEOPRENE  FREQUENCY  + €>  o  u  © +  8 +  © +  4 +  •+  100  4-  200 o)/2w (Hz) FIGURE 33  300  e +  -103 -  EXPERIMENTAL fl, FOR NEOPRENE,  PLOTTED AGAINST  ©  - RUN  WITH 5 SHEETS OF NEOPRENE  +  - RUN  WITH 3 SHEETS OF NEOPRENE  FREQUENCY  © +  3 +  6  XI O  co  i  o  7 a  2  +  1 +  +  +  100  200 w/2w  (Hz)  FIGURE 34  —t-* 300  e +  -104 -  EXPERIMENTAL  n/e, FOR NEOPRENE,  PLOTTED AGAINST  FREQUENCY  + © 400  +-  . RUN RUN  WITH 5 SHEETS OF NEOPRENE WITH 3 SHEETS OF NEOPRENE  3004-  e f  v  U  a 200 +  © +  lOOf  +  100 W  200 /2w  (Hz)  FIGURE 35  300  -105 -  Furthermore the neoprene possessed V,  was s i n u s o i d a l when V H  determine  whether  The important frequency  material  possessed  is likely a valid  c o n c l u s i o n i s that fl/e  since,  0  to  isotropic  assumption.  seems  to  have  a  dependence, at low f r e q u e n c i e s ( i . e . 0 •*• 1000 Hz),  as measured i n the p r e v i o u s  3.4  conduction  was s i n u s o i d a l . I t i s d i f f i c u l t  the  c o n d u c t i o n , however, t h i s  Ohmic  EXPERIMENTAL  section.  VERIFICATION OF THE THEORY WHICH EXPLAINS  THE FORM OF THE CALIBRATION  In chapter  2, s e c t i o n  CURVES OF THE BULBS  13, the " s t a t i c  screening"  model  was combined with the " r e s e t s c r e e n i n g " model t o e x p l a i n the form  of  a t y p i c a l c a l i b r a t i o n curve of a bulb, as shown i n  f i g u r e 21. The " s t a t i c and  sometimes  s c r e e n i n g " model i s v a l i d slightly  r e q u i r e d to d e r i v e the applied no  field  avalanche  possesses  above model  below  threshold. are;  the  threshold  The assumptions  frequency  of  the  i s low so magnetic e f f e c t s can be n e g l e c t e d , breakdowns  occur  and  the  bulb's  surface  i s o t r o p i c and Ohmic c o n d u c t i o n . The r e s u l t s of the  d e r i v a t i o n are the p u l s e r a t e , f , B  in a sinusoidal  field i s : 2.11-12  Furthermore,  the phase s h i f t  of the p u l s e s a t t h r e s h o l d i s :  6 • Tan  where e i s d e f i n e d i n f i g u r e  2.11-10 18.  -106 -  The  "reset  screening"  model  i s v a l i d at high  pulse  r a t e s . The assumptions r e q u i r e d t o d e r i v e t h i s model are the same as the ones r e q u i r e d  t o d e r i v e the  "static  screening"  model, except, i t i s assumed many avalanche breakdowns occur during  one c y c l e of the a p p l i e d f i e l d .  v a l i d at high  f i e l d s . According  to  Hence, t h i s model i s  this  model  the  pulse  rate,f , i s : B  f  B^ " * t * - *  where  the  2.12-19  applied f i e l d  used to d e r i v e t h i s  approximation to a s i n u s o i d a l f i e l d were  combined  in  equation  as shown i n  These  models  obtain  three ways t o measure X. They a r e :  figure  chapter 2, s e c t i o n  X -  nY  2.13-4  2.13-1  s  wtanG  where M  s  2.11-10 i s the slope of the " s t a t i c  the slope of the " r e s e t Y-intercept  of  the  screening  "reset  f i g u r e 21. 9 i s d e f i n e d the  20.  13, to  1  X  i s an  screening curve"  screening  i n f i g u r e 18.X  curve", M„ i s  and  curve"  Y  i s the  s  as shown i n  depends d i r e c t l y  on  c o n d u c t i v i t y of the bulb's s u r f a c e . The c o n d u c t i v i t y of  many types of described construct relatively  glass  above. the bulbs  is  too  Hence, for  conductive  low  to  soda-lime  this  observe glass  experiment  the was  since  effect  used it  to  is a  g l a s s . One of these bulbs was placed  -107  -  between the c a p a c i t o r p l a t e s and c a l i b r a t e d . T h i s bulb was 4 cm. i n  diameter  experimentally E ,  against 37,  and  contained  obtained  are presented  A  argon  at  calibration  1.6  curves,  torr. f  i n g r a p h i c a l form i n  B  The  plotted  figures  36,  38 and 39 f o r v a r i o u s f r e q u e n c i e s of the a p p l i e d f i e l d .  M , M R and Y S  S  as measured from t h i s data  are  presented  in  t a b l e V.  TABLE V, BULB DATA  w/2*  X equation  may  M  R  ( T V )  "•(iv)  Y  S  (Hz)  40 Hz  11.7  3.5  280  60 Hz  21.7  6.5  460  80 Hz  25.5  9.7  440  100 Hz  31.4  12.7  520  be  2.13-1  calculated and  with  i n two ways with t h i s data, with 2.13-4.  Let  X  M  be  the  e x p e r i m e n t a l l y obtained X as c a l c u l a t e d with equation 2.13-4 and  let X  Y  be the e x p e r i m e n t a l l y o b t a i n e d X as c a l c u l a t e d  with equation 2.13-1. The r e s u l t s a r e presented  in table VI.  -108 -  SODA-LIME  BULB  CALIBRATION  FIGURE  36  CURVE  FOR  40Hz  -109 -  S O D A - L I M E BULB  CALIBRATION  k>/2w »  FIGURE  CURVE  60Hz  37  FOR  60Hz  — 110 —  S O D A - L I M E BULB  CALIBRATION  u/2«  =  FIGURE  80Hz  38  CURVE  FOR  80Hz  -111 -  SODA-LIME  BULB  CALIBRATION  FIGURE  CURVE  39  FOR  100Hz  -112-  TABLE VI, EXPERIMENTALLY OBTAINED TIME CONSTANTS  u/2*  f ,  X  Ay/A  Hz  800  880  1.10  60  Hz  1200  1450  1.21  80  Hz  1220  1 380  1.13  1420  1630  1.15  Hz  magnitude  obtained  (s" ) 1  y  40  100  The  A  of  the  is difficult  error  in  to determine  the  vary  approximately  calibration  curves  Furthermore,  this  necessarily slightly  a  10%  from  are  not  change  from  statistical  unstable  run  field,  to  to  fluctuation,  E  run.  entirely run  experimentally  s i n c e the pulse r a t e ,  at a given frequency, u , and a p p l i e d  B  M  A  ,  would  Hence,  the  reproducible. run  in X  these  i s not  bulbs  are  hence t h e i r p r o p e r t i e s evolve with time.  Thus, the experimental e r r o r s were not  necessarily  reduced  by a v e r a g i n g the r e s u l t s over many runs. From t a b l e VI, one can see the agreement between X  M  i s q u i t e reasonable. X  v  X  y  and  i s , on average, approximately  15%  higher than X . There are two p o s s i b l e e x p l a n a t i o n s f o r t h i s M  error.  The  distorted and  glass  makes  up  the  i n the r e g i o n where the bulb was  sealed  bulbs i s s l i g h t l y filled  with  gas  o f f . T h i s means the c o n d u c t i v i t y of the s u r f a c e  i s not i s o t r o p i c expected.  which  i n that r e g i o n . Hence some e r r o r  Fortunately  in  X  is  t h i s area i s a s m a l l f r a c t i o n of the  -113-  total  surface  derived  with  sinusoidal This to  area. an  Furthermore,  applied  field  equation  2.12-19  was  which only approximated a  f i e l d . This applied f i e l d  i s shown i n f i g u r e  20.  i s a l s o a source of e r r o r . A l l other assumptions used  d e r i v e the " r e s e t " and  considered As  "static  screening"  models  were  valid.  a  final  t e s t , the phase of the p u l s e s at t h r e s h o l d  was measured and compared t o the t h e o r e t i c a l  prediction:  0 = Tan -  2.11-10  The p l a t e v o l t a g e and the bulb's p u l s e s were observed  a t the  same time w i t h a storage o s c i l l o s c o p e as shown i n f i g u r e and  23.  The  applied  electric  f i e l d was i n phase with the  p l a t e v o l t a g e . I t was found that the fluctuated.  phase  of  the  screen  16 mode o v e r l a y s 16  signals  pulses  t o o b t a i n an average phase angle  these t r a c e s . Instead I w i l l X  M  and  AY  substitute  into  equation  the  The  position  of  the  from  experimentally  2.11-10 and o b t a i n the  v a l u e s of e which a r e c o n s i s t e n t with the c a l i b r a t i o n data.  groups  a r e wide i n comparison t o the width of a p e r i o d .  Hence, i t i s d i f f i c u l t  obtained  9  calculated  will  curve then be  i n d i c a t e d on the p i c t u r e s of the e x p e r i m e n t a l l y o b t a i n e d Let  9  M  results  be the r e s u l t s c a l c u l a t e d calculated  table VII.  on  and s t o r e s them. Thus, one c o u l d see the p u l s e s  bunching up i n t o groups as shown i n f i g u r e 40. These of  pulses  Hence, the o s c i l l o s c o p e was s e t to the envelope  16 mode, where the envelope the  22  from  from  X  M  and  9  Y  9.  be the  A . These r e s u l t s a r e presented i n Y  -114-  TABLE VII EXPERIMENTALLY OBTAINED PHASE ANGLES  «/2ir  eM  eY  40 Hz  73°  74°  73.5°  60 Hz  73°  75°  74.0°  80 Hz  68°  70°  69.0°  100 Hz  66°  69°  67.5°  The p o s i t i o n of 6 VG A  experimentally  i s indicated  obtained  6  in  on  QAVG  the  pictures  f i g u r e 40. As expected, the  groups of p u l s e s are more or l e s s centered around The " s t a t i c avalanche  s c r e e n i n g " model was  breakdowns  when 9 was measured. starts  to  fluctuations  breakdown. i n 6.  occur. Thus, This  of the  derived  QAVG  .  assuming  no  The f i e l d was above t h r e s h o l d the  "static  could  be  screening" the  cause  model of the  -1 15  PHASE  S H I F T OF  THE  SODA-LIME  BULB'S  P U L S E S AT  THE T R A C E S WERE O B T A I N E D BY S U P E R I M P O S I N G 16 SWEEPS AT THE THRESHOLD F I E L D  u/2v  « 40Hz  i- — i — — — THE B L A C K V E R T I C A L L I N E S ARE eAVG AS P R E D I C T E D BY A AND A  \ ... ii  M  Y  t  PULSES  APPLIED  u/2*  « 60Hz  u/2t  *  FIELD  eAVG  M/2IT  «  80Hz  lQQHz  TIME  FIGURE  40  THRESHOLD  -116-  CHAPTER 4 CONCLUSIONS 4.1 INTRODUCTION The  r e s u l t s presented i n t h i s t h e s i s  directed  towards  understanding  g a s - f i l l e d g l a s s b u l b s , which placed  in  an  alternating  emit electric  concern  the  research  operation  pulses  of  field  of the  light  of  when  sufficient  magnitude. In p a r t i c u l a r , the e f f e c t of the c o n d u c t i v i t y  of  the bulb's s h e l l was s t u d i e d . Two  theoretical  models  were developed  t o e x p l a i n the  i n f l u e n c e of conducting s h e l l s on the c a l i b r a t i o n curves f o r e l e c t r o d e l e s s breakdown at low f r e q u e n c i e s ( < 1kHz ). These r e s u l t s are of p a r t i c u l a r used It  s i g n i f i c a n c e s i n c e such bulbs  i n devices which monitor  environmental  electric  was found t h a t the c o n d u c t i v i t y of the g l a s s ,  the bulbs are made, can be conditions.  this w i l l  monitors.  by  the  fields.  from which  environmental  In p a r t i c u l a r , the humidity and the c l e a n l i n e s s  of the g l a s s a f f e c t predict  affected  are  the c o n d u c t i v i t y . The t h e o r e t i c a l models  lower the accuracy of the  Thus, i t i s important  electric  to design monitors  the bulbs are s h i e l d e d from the environment  such  field  i n which that  the  changes i n t h e i r c o n d u c t i v i t y are minimized. Experimental v e r i f i c a t i o n of the t h e o r e t i c a l models was observed. Furthermore of  some  materials  a method of measuring at  low  frequencies  developed. The t h e o r e t i c a l models and summarized i n more d e t a i l below.  the  the c o n d u c t i v i t y (  <  1 kHz ) was  experiments  are  -1 17 -  4.2 CONCLUSIONS It  turns  which forms  out  the  that  bulb,  i f the c o n d u c t i v i t y of the g l a s s , is  ignored  then  the t h e o r e t i c a l  c a l i b r a t i o n curve of the b u l b i s :  F  B  as  - E 7  2.1-4  0  shown  i n chapter 2, s e c t i o n 1. T h i s curve i s p l o t t e d i n  f i g u r e 41 and  i s called  Furthermore t h i s theory pulses  at  threshold  the  "zero  conductivity  p r e d i c t s that the phase s h i f t of the should  be as i n d i c a t e d i n f i g u r e 42.  However i t was found that many c a l i b r a t i o n curves the  curve".  are  like  one shown i n f i g u r e 41, which i s l a b e l l e d the "observed  calibration  curve".  Furthermore  shifted  at  threshold.  allowing  the bulb's s h e l l t o be s l i g h t l y  These  the  pulses  findings  were  were  phase  e x p l a i n e d by  conducting,  where  s l i g h t l y conducting means the c o n d u c t i v i t y of the s h e l l noticeably  affect  respect  the  to  completely. modifies  the  external  This  internal field,  models  results,  each  operation.  are  of not  flow  the  bulb with  screen  i t out  i n the g l a s s which  surface.  required  i s valid  Both  but  i s due to c u r r e n t  charges on the bulb  Two  field  will  to  explain  the  observed  i n a d i f f e r e n t regime of the bulb's  models  contain  one  parameter,  the  c o n d u c t i v i t y of the s u r f a c e . The below  first  model, the " s t a t i c  threshold  and  possibly  screening" slightly  However i t c e r t a i n l y i s not v a l i d when f  model, i s v a l i d  above B  threshold.  i s much  greater  than i t s t h r e s h o l d value. The assumptions r e q u i r e d t o d e r i v e  RESULTS OF THE PROPOSED THEORETICAL  OBSERVED  MODELS  CALIBRATION  APPLIED  FIELD/  E  A  *  FIGURE 41  PHASE  OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING  FIGURE 42  BULB  -119-  t h i s model are; that  the  magnetic  breakdowns  effects  occur  possesses  in  E  .  A  Hence  at  this  screening  the  of the  and  predicts  occur during  one  c a l l e d the  and  of  are  phase the  "static  41. model, p r e d i c t s  t h i s model are  to d e r i v e  the  the  the  "static  rates.  The  same  as  screening"  many  avalanche  breakdowns  c y c l e of the a p p l i e d  f i e l d . The  theoretical  by  t h i s model i s shown in f i g u r e 41 and curve." At high count r a t e s ,  parallel  to  the  "zero  is the  conductivity  o f f s e t from i t as expected. Moreover, the o f f s e t  "reset  screening  curve" only depends on conductivity  independent  ways  conductivity  can  of the  screening  curve".  this  respect  slope  pulses  "reset screening"  to d e r i v e  t h e o r e t i c a l curve i s  The  the  i s shown i n f i g u r e  "reset screening  but  surface  s h e l l . T h i s curve i s c a l l e d the  required  predicted  of the  bulb's  that the  so  avalanche  phase s h i f t e d with  model except i t i s assumed that  curve"  no  of the c a l i b r a t i o n curve at high pulse  assumptions  curve  the  i s low  These e f f e c t s are a f u n c t i o n of  second model, the  form  neglected,  bulb  reduced  curve", and  assumptions r e q u i r e d the  model  is  field  Ohmic c o n d u c t i o n . I t t u r n s out  threshold.  conductivity  The  the  be  i s reduced and  c a l i b r a t i o n curve shifted  can  i s o t r o p i c and  model p r e d i c t s E, to  frequency of the a p p l i e d  "reset  Secondly,  the o f f s e t of the  curve" from the  "zero  the c o n d u c t i v i t y  of the  of the g l a s s can  be c a l c u l a t e d in  with  these  models.  conductivity glass.  Firstly,  be c a l c u l a t e d from the r a t i o of the  the "reset  curve"  to  the  c o n d u c t i v i t y can screening  curve"  "static  two the  slopes  screening  be c a l c u l a t e d from from  the  "zero  -120  conductivity  curve."  measured i n two models c o u l d In  Thus,  this  ways. Hence,  2,  the  single  the  be demonstrated  chapter  screening"  -  parameter can  self-consistency  of  "static  screening"  and  "reset  These models were a p p l i e d  s e v e r a l geometries. They i n c l u d e ; s p h e r i c a l s h e l l s , s h e l l s and  long  spheroidal  In chapter 3, s e c t i o n 2, the was  t e s t e d d i r e c t l y . T h i s was  screening tubes.  measurements  measured when the the  tube and  axis.  when the  With  these  constants of the compared screening"  field  to  was  neoprene  was  frequency was  the  screening"  to the a x i s  of  of  the  by  the  model.  The  was  and  the  In t h i s  of neoprene  30. conductivity  of  another technique in chapter  accomplished by measuring the  calculated  time  "static  with  a p p l i e d across  r e s u l t i n g current  results  the were  use  of  the  consistent  3,  magnitude flat  through  With these q u a n t i t i e s , the c o n d u c t i v i t y  was  the  result  found that the c o n d u c t i v i t y  of  to  the  good as shown i n f i g u r e 28.  sheets of neoprene as w e l l as the  neoprene  was  ratio  the phase of an a l t e r n a t i n g v o l t a g e  sheets.  screening  c a l c u l a t e d . This  dependence  the  conducting  agreement between the theory  measured using  s e c t i o n 3. T h i s was  the  the  the t h e o r e t i c a l r a t i o p r e d i c t e d  it  model  applied perpendicular  frequency dependent as shown in f i g u r e The  and  inside  required;  was  responses was  experimental r e s u l t s was  was  fields  measurements,  model. The  experiment  screening"  applied p a r a l l e l  field  infinite  accomplished by measuring  were  to  shells.  "static  of a l t e r n a t i n g e l e c t r i c  Two  the  experimentally.  models were d e r i v e d .  cylindrical  be  of  the  "static with  the  -121  -  r e s u l t s from the p r e v i o u s s e c t i o n . Finally, screening" chapter bulb  and  the  found to resemble the one "static  screening  screening  curve"  and  the  these  values  "static  curve.  to c a l c u l a t e  independent  ways.  were c o n s i s t e n t . Furthermore  phase of the p u l s e s at t h r e s h o l d were  consistent  with  s c r e e n i n g " model. T h e r e f o r e the " s t a t i c " and  s c r e e n i n g " models are c o n s i s t e n t ,  hence  The  theoretical  curve" were f i t to the c a l i b r a t i o n  the c o n d u c t i v i t y of the g l a s s i n two out  filled  shown i n f i g u r e 41.  parameters obtained from t h i s f i t were used  turned  "reset  model were t e s t e d s i m u l t a n e o u s l y as d e s c r i b e d i n  theoretical  The  s c r e e n i n g " model  3, s e c t i o n 4. The c a l i b r a t i o n curve of a gas  was  "reset  the " s t a t i c  they  It the the  "reset  explain  the  form of the c a l i b r a t i o n curve of t h i s bulb. Thus account surface curves  the  models  developed  of the e f f e c t describe, obtained  frequencies  ( <  of  leakage  very for 1  such bulbs are used conductivity  of  in  well,  this  current  electrodeless  to take  the  bulb's  calibration at  low  kHz). These r e s u l t s are important  since  in e l e c t r i c  these  breakdown  f i e l d monitors.  bulbs i s dependent on devices  which,  threshold,  the  on  the form of the  conditions, portable testing measure  thesis  bulb's  should  Because  the  environmental be  obtain  developed the bulb's  c a l i b r a t i o n curve from the value of the t h r e s h o l d f i e l d  with  the t h e o r e t i c a l models and  into  the  field  monitor.  feeds t h i s i n f o r m a t i o n back  P e r i o d i c use of the t e s t i n g d e v i c e  i n c r e a s e the accuracy of the monitor changing  c o n d u c t i v i t y of the b u l b .  by c o r r e c t i n g  for  will the  -122 -  BIBLIOGRAPHY  1  H a r r i e s W . L . and Von Engel,A., S e c t . B64,915(1951) f  2  3  4  5  Proc.  Phys.  Soc,  Meek,J.M., and Craggs,J.D., E l e c t r i c a l Breakdown John Wiley and Sons, New York(1978)  Morse,P.M. and Feshback,H., Methods of T h e o r e t i c a l McGraw-Hill Book Company, New York ( 1 953")  Jackson,J.D., C l a s s i c a l E l e c t r o d y n a m i c s Sons, New York(1975)  ,  John  London,  of Gases ,  Physics  Wiley  and  Friedmann,D.E., Curzon,F.L., Feeley,M., Young,J.F. Auchinleck,G., Rev. S c i . Instrum. , V o l . 53, 1273(1982)  and  -123 -  APPENDIX A  In chapter 2, s e c t i o n 2.3-19  with  appendix  respect  these  will  the moment f o r  polynomials. be d e r i v e d . the  In  this  There are two  X-direction  "switch  problem and the moment f o r the Z - d i r e c t i o n problem. For  the X - d i r e c t i o n with  Legendre  quantities  cases to c o n s i d e r , on"  to  3, I gave the moments of equation  respect  Z-direction  problem the moment of the equation i s to  P/vOl)  and  with  respect  taken  t o ^(71) f o r the  problem.  A-1 - MOMENT OF EQUATION 2.3-19 FOR THE Z-DIRECTION PROBLEM  For  the  Z-direction  "switch  on"  problem,  equation  2.3-19 becomes: A. 1 - 1 The  quantities  , 5^2.  and Ij.  are d e f i n e d  i n equations  2.3-11, 2.3-24 and 2.3-26 r e s p e c t i v e l y . As a reminder equations are l i s t e d  these  below: 2.3-11  2.3-24  -124  Note, 97  has r e p l a c e d 7?  0  -  i n these equations s i n c e t h i s p o i n t  i s no longer f i x e d . S u b s t i t u t e these equations  into  equation  A.1 - 1 to o b t a i n :  2 R>^c^«^j^^ ^ ^ ^  ZnS&=iP  + 7  A. 1-2 Multiply  The  t h i s equation by  fjft??) and  integrate over7):  second term i n t h i s equation can be e v a l u a t e d  use of the i d e n t i t y  shown below:  Thus equation A.1-3  becomes:  the  3  + j I^WX Equation A.1-5  with  A.1-5 can be w r i t t e n i n the form:  -125  -  A. 1-6  J  -»  A.1-7  A-2 MOMENT OF EQUATION 2.3-19 FOR THE X-DIRECTION PROBLEM  For  the  X-direction  "switch  on"  problem,  equation  2.3-19 becomes:  Vflx+'Xxk'XX-O The  quantities  2.3-11,  2.3-25  A.2-1 b-\y.  and  and I  2.3-27  are  x  defined  respectively.  in As  equations  a reminder,  equations 2.3-25 and 2.3-27 a r e :  M € -1)"*- £ RJAKMCOS  4 >  Note 7l„ and <£> are r e p l a c e d by D and ty s i n c e t h i s 0  no  longer  fixed.  2.3-27 i n t o equation  Substitute  2.3-25  point  is  equations 2.3-11, 2.3-25 and  A.2-1 t o o b t a i n :  -126 -  2ja(tNift-  M u l t i p l y t h i s equation by  PJtfi)  and i n t e g r a t e over?? :  ZCdtt-  i y^'^ The  second  term  V  V  ° ' ~  A.2-3  i n t h i s equation can be c a l c u l a t e d with  use of the i d e n t i t y  shown below:  the  3  I  n j  A.2-4  —I  Hence equation A.2-4 becomes:  n.&-ifHzH+\)iK  Rudolf*) f p U j L ^ 1  Equation A.2-5 may be w r i t t e n :  _ ayfft "Ip'mj]  -127 -  -I  A.2-7  -128 -  APPENDIX B  B-1  SOLUTION  OF THE X-DIRECTION "SWITCH ON" PROBLEM FOR AN  ASYMPTOTICALLY LONG SPHEROID  In chapter 2, s e c t i o n 5, I o b t a i n e d the equations which govern  the response of a  electric appendix  field  applied  long  conducting  spheroid  to  p e r p e n d i c u l a r to i t s a x i s . In t h i s  these equations w i l l be s o l v e d . As a reminder  equations are l i s t e d  an  these  below:  2.3-32  2.4-16 -1  2.5-2  2.5-1  2.2-13  The procedure this chosen  used t o s o l v e these equations i s  paragraph.  Firstly  , say N-dimensional,  the and  dimension the  outlined  of matrix  elements  of  in is  E^itf  are  -129  found.  -  Next the e i g e n v a l u e s , ^ , can be found with equati  ion  2.3-32:  ^ [ B ^ A , + T^u]  The  =  o  B. 1 -1  eigenvectors, l^yX, can be  found  to  within  undetermined  c o e f f i c i e n t s with equation 2.3-32:  B. 1-2  The  undetermined  coefficients  are  removed  with  the  " e q u i l i b r i u m c o n d i t i o n " , equation 2.5-2. The e i g e n s o l u t i o n s , S/v"X(i*) , are then found with  equation  2.5-1.  Finally  the  response of the bulb i s found with equation 2.2-13. This  problem  may  symmetry of the problem function  be to  simplified be  r^u(7l) i s even w h e n i s  Thus the matrix ByiN i s of the  x  where  x  o  O  x  X o  O  x  is  a  x  o  x  o X o  x  o  o X o  x  o x  x  non-zero  in  the  sign  form:  considered.  odd and odd  The  w h e n ^ i s even.  form:  of 7)  SffkiY*) ,  must  be  even  to  by the symmetry of the problem.  Hence, from an i n s p e c t i o n of 2.5-1, the must be of the  is  q u a n t i t y . However, the e q u i l i b r i u m  charge d e n s i t y d i s t r i b u t i o n , change  solved  c o n s i d e r a b l y i f the  e i g e n v e c t o r s , R^X ,  a  -130  From  an  -  of o^H and ry*> i t can be seen that the  inspection  elements of B^utf i n which bothyiA and A/ are even p l a y no in determining the e i g e n v e c t o r s , R/A . Hence they w i l l included To  role  not be  i n the c a l c u l a t i o n . start,  e v a l u a t e B^KN .  The  limit  A —* O  of  Qil^W) i s : B. 1-3  Thus equation 2.4-6 becomes:  B. 1-4 -I Let:  J|-71-  U  "  B.1-5  -I S u b s t i t u t e equation B.1-5 i n t o equation B.1-4 t o o b t a i n :  7*^  Za6.(Nti)^AVL  B  - -e 1  S u b s t i t u t e equation 2.4-12 i n t o equation B.1-6 t o o b t a i n :  -131  D'  XUZN-t-l)  -  T  N  S u b s t i t u t e t h i s equation i n t o equations B . 1 - 1 and  ,  N  B . 1 - 2 to  obtain:  The Jyjiw have been c a l c u l a t e d f o r y u = 1 , 3 , 7 and N = 1 , 3 , 7 where the Legendre polynomials a r e :  3  P '=/F^(ai n -I4-77N-|)£ ,  +  5  P,»/rrrF(4Z? n *- *fi 5  The  I„  13 srt-5 ) ^  I^ are: w  =1.5707  I „  =0.5890  I,5  =0.3682  I  Ij3  =5.7432  I,5  = 2.5541  \  I55 =  10.656  =0.2684  n  n  =1.7315  IS7 =5. 1366  -132 -  I  where  7 7  =1 6.035  = Iw/u. . The e i g e n v a l u e problem has been solved f o r  1, 3, 5  and  presented  7  dimensional  matrices.  These  r e s u l t s are  below.  B.2 1-DIMENSIONAL EIGENVALUE PROBLEM  With  equations  B.1-8  and  B.1-9  and the [pm the one  dimensional eigenvalue problem i s :  B.2-1  B.2-2  Hence:  -\ _ 0.589-TL Ax  ~  R£  B.2-3  0  R' = X  B.2-4  /?s  where x i s an undetermined c o e f f i c i e n t . S u b s t i t u t e  equation  B.2-4  2.5-2,to  obtain:  Hence:  into  the  " e q u i l i b r i u m c o n d i t i o n " , equation  -133 -  B.2-6  S u b s t i t u t e equation B.2-6 i n t o equation 2.5-1 to o b t a i n :  B.2-7  S u b s t i t u t e equation B.2-7 i n t o equation 2.2-13 t o o b t a i n :  B.2-8  B.3 3-DIMENSIONAL EIGENVALUE PROBLEM  Substitute  the  i n t o equations B.1-8  and  B.1-9  to  obtain:  [0,5890B.3-1  0.08590  o.8376 -  0.5890 - f j k *  O.0 2 . Z 0 9  E  R  ^**  B.3-2  0.8376-  0.0859O  The s o l u t i o n s of t h i s e i g e n v a l u e problem a r e :  Xx" °'^^ ' _ n  w i  th  eigenvector  R'^TMX**  B.3-3  -134 -  -O.GI78 with e i g e n v e c t o r  R^»AiX y =  -o.8fo6f where  x  and  y  are  B.3-4  undetermined c o e f f i c i e n t s . S u b s t i t u t e  equations B.3-3 and B.3-4 i n t o equation 2.5-2 to o b t a i n : [-0.<*636 R(77)  - 0 - 6 1 7 8 R^7)| I" B.3-5  o  The  .  solution  of these equations i s :  B.3-6 -0.267 3€jix  B.3-7  S u b s t i t u t e these equations i n t o equations B.3-3 and B.3-4 to obtain: 0.834<J B.3-8  I-O.Z3I5 0.1651 £  c.2.316  °E  X  B.3-9  S u b s t i t u t e equations B.3-8 and B.3-9 i n t o equation 2.5-1  to  obtain:  o^'X.xVT)~  ^0^4(0.8^9 -0.3473 C57> - 1 ) )  B.3-10  -135 -  B.3-11  Finally  substitute  o b t a i n the response  these equations  R f e  °  J  (0.I&5I t o . 3 4 7 3 ( 5 ? ? - l 1 ) ( l - ^  required  to  i s identical  +  B.3-12  J  x  problems  equation 2.2-13 t o  of the s h e l l :  (0.S34^-0.3473C5^-'))(l-e  The method  into  solve  the  5  and  7  dimensional  to the method used to s o l v e the 1 and  3 dimensional problems. Since nothing new s o l v i n g these problems i n d e t a i l ,  i s demonstrated  I w i l l simply present  by the  results.  (.6*74--  .4140( 577*-1) t. j74-4Ul?7*-|1  +(.Jl ex - .2.9 10(577**-/) -.32.1 b (2,1T)- \47th 4  R  £  e  * J "^£ ] P  B.3-13  (.0444- ,|iJO ( 577 - iH.(472-fej?7 -t477N-/|(| - e ^ R f e ? ^ 1  +  )V| - e  & 7 XIV*, •£)/€. E* Co s £  (.5093-  +  ^  .3753(577-l)4-.2554(2J  Tr*-I477V 1) -.c£o«K^7j-4*>S??* iiS7v-5))(i4  C  +(35o5r.087«S(^--i)-.%o2{£ I 77*-|4?7 +•)+-0472(42.9?»-4<?9fa\%stf-fy-of&^\ ,  +(. 1 1 - 2 3  6  o 1 (S^- )-.o2.9M 2l77*- l47fH I) -.0413 (4Z7 ?7-4 9S?7 + 6  Hztf^jfy.  B.3-14  -136  -  APPENDIX C  C.1  SOLUTION  OF THE Z-DIRECTION "SWITCH ON" PROBLEM FOR AN  ASYMPTOTICALLY LONG SPHEROID  In chapter 2, s e c t i o n 6, I obtained the equations which govern  the response of a  electric appendix  field  applied  long  conducting  parallel  to  spheroid  i t s axis.  to In  this  these equations w i l l be s o l v e d . As a reminder  equations a r e l i s t e d  2RUXB£W  •+•  an  these  below:  AzR X=°  2-3-28  W  2.4-13  Q,°Ci +A)(£A^  ^  >  K  '  U^l-yW^-t^) The p r o c e d u r e this  used t o s o l v e  paragraph.  Firstly  these equations the  dimension  2.2-,3  is of  outlined matrix  in  E&yis  -137 -  chosen, say N-dimensional, found.  and  the  elements  BJLw are  of  Next the e i g e n v a l u e s , Az , can be found with equation  2.3-28:  +'X uJ = 0  c. 1-1  z  The e i g e n v e c t o r s , ry,x, can be found to  within  undetermined  c o e f f i c i e n t s with equation 2.3-28:  The  undetermined  coefficients  are  removed  with  the  " e q u i l i b r i u m c o n d i t i o n " , equation 2.6-2. The e i g e n s o l u t i o n s , 8*A£(r), are then found response of the s h e l l Firstly equation  I  2.4-13  with  2.6-1.  Finally  the  i s found with equation 2.2-13.  will is  equation  demonstrate equivalent  to  that  the  integral  ~J^XN d e f i n e d  in  i n the  p r e v i o u s appendix. L e t :  S u b s t i t u t e the f o l l o w i n g i d e n t i t y  3  i n t o equation C.1-3:  C. 1-4  The  r e s u l t of the s u b s t i t u t i o n i s :  -138  -  C. 1-5  j-  w »  —I Integrate  this  v  equation  by  parts.  Note, the  polynomials  are equal to zero when 7\ = ± | . Hence:  ^uCfl)  -I S u b s t i t u t e equation C.1-4  i n t o equation C.1-6  to o b t a i n :  C. 1-7 —I where the T^N  have been c a l c u l a t e d  S u b s t i t u t e equation C.1-7  T h i s problem may symmetry ^.(T>) matrix  properties  be of  appendix.  i n t o equation 2.4-13 to o b t a i n :  simplified the  i s even whenyu. i s odd and OJJ&I  i n the p r e v i o u s  considerably  if  the  problem are c o n s i d e r e d . Since  f^'C^)  i s odd w h e n ^ i s even,  i s of the form:  " x  o  x.  o  x  o  x  o  X  O  X  O  X  O  X  O  X  o  x  o  x  o  x  where x i s a  o x  non-zero  quantity.  However  the  equilibrium  -139  -  charge d e n s i t y d i s t r i b u t i o n , S ( . r i ) , must be odd t o a change 2  in  ;  s i g n of 77 by the symmetry of the problem. Hence, from an  i n s p e c t i o n of 2.6-1, the e i g e n v e c t o r s , ryCX , must be  of the  form: X o  X  o  From  an  inspection  i n which b o t h a n d N are even play no  elements of in determining included  of B^, and iyuCk i t can be seen that the role  the e i g e n v e c t o r s , fyuC< • Hence they w i l l not be  i n the c a l c u l a t i o n .  To  start,  evaluate  the B^.A/ .  The  limits  A—>0 of  are:  u  _L 2  Q;(I+A)R°(I t A )  C.1-9  S u b s t i t u t e equations C.1-9 i n t o equation B,° ~ *(2.35 6  _ 4. ., 2_) 7  B j , — If(o.8g30|n^) - 3 . 2 . 3 9 ) Bs, — y(o.5 5 2 i l « ( - Z.5 a 2.) B7<~y(o.4-DZ6  -2.088)  C.1-8 to o b t a i n :  -140 -  Y(83*>7 |n(2^ -4-0.82.) By?^(6,060  m)-3l.4-$)  g ^ ' f r U . O Z S Uj^j-4:050)  B ^ - ^ g . ^ l In^-Zfo7.7) Bys—*(zo.Z6 1^-146.5) B,°7^y(2 oi3 ln(^- 4.0 t  B j ~ Y O 2.99 1*^-47.62) 7  857 - V(^8.5Z  where V  - 175.7)  is:  S u b s t i t u t e equations 2.4-11 and 2.4-12 i n t o equation  C.1-10  to o b t a i n :  €.L>  In  C.1-1 1  this  problem  the A dependence of the  f a c t o r e d out. Hence the eigenvalue problem for from  must  cannot be by  solved  each chosen A . T h i s problem w i l l be s o l v e d f o r A . ranging IO  i n t o the  to B > * N  lO  . Firstly,  to obtain:  l e t A=.o2_. S u b s t i t u t e A=o.o2,  -141  -  B°j 5 ^ ' V 13.183  0.8 2 7 0  B° 7 /~ tfb.2.34  B ^ - Y 18.86  Bs-r^-y I2..I90  -YO.344-  Bg-,A/  -yiA8^  B77^*69.8l8  C.2 1-DIMENSIONAL EIGENVALUE PROBLEM Substitute  the ^uw i n t o equations C.1-1 and C.1-2 t o o b t a i n :  cJet[fe.l36- ^ " ] = 0  C.2-1  [6.138 -  C.2-2  ^ ] R ^ 7 C °  Hence:  ^ = V 6.1383  C.2-3  C.2-4  where x i s an undetermined C.2-4 obtain:  into  c o e f f i c i e n t . Substitute  equation  the " e q u i l i b r i u m c o n d i t i o n " , equation 2.6-2, t o  -142 -  /"V  XR°(?7)  C.2-5  Hence:  S u b s t i t u t e equation C.2-6 i n t o equation 2.6-1 t o o b t a i n  the  eigensolution, S/^C^) :  S u b s t i t u t e equation C.2-7 i n t o equation 2.2-13 t o o b t a i n :  C.3 3-DIMENSIONAL EIGENVALUE PROBLEM Substitute  4..I38 -  the  i n t o equations C.1-1 and C.1-2 t o o b t a i n :  5.3*72-  Jet  =0 0.62.^3  '6.138 - ?f-  18.87-  ^  5372. | ^ / A X - 0  0.6 Z 7 3  c.3-1  I8.B7-^  C.3-2  -143 -  The s o l u t i o n s of t h i s eigenvalue problem a r e :  with e i g e n v e c t o r  Q.997  R^,* X  C.3-3  -O.063  with e i g e n v e c t o r  where  x  and  y  are  Ryu7v y =  0.380Z C.3-4  zl  undetermined c o e f f i c i e n t s . S u b s t i t u t e  equations C.3-3 and C.3-4 i n t o the " e q u i l i b r i u m  condition",  equation 2.6-2, to o b t a i n :  'oSftl Q,°(I+A)(2-AY4.  O  R°(7>)  -0.O63P°0?)  0.3 802. 0.94-9  P>)'  X  Lyj  c.3-5  The s o l u t i o n of these equations i s :  C.3-6  y  QJC 1 + A ) C 2 A ^  Substitute  these equations  C.3-7  i n t o equations C.3-3 and C.3-4 t o  obtain:  O.T75 -0.062.  C.3-8  -144 -  O.OT5  O.ObO Substitute  £*Ez  C.3-9  equations C.3-8 and C.3-9 i n t o equation 2.6-1 t o  obtain:  C.3-10  C.3-11  Thus, from equation 2.2-13, the response of the s h e l l i s :  ^W^^  75  -0.oS.C577^-3)Xl-e^ ) t  _ t9.2|JCLD\  ^zQX'+^KaA^ypTry^Ea.??  i s plotted  against  C. 3-1 1  n.Dt/L £o  for  l  v a r i o u s 71 i n f i g u r e 43. Furthermore: _ 5.79.n.Dt I -  Q  is  plotted  graph  C.3-12  on  the  same  i t i s apparent  essentially  independent  graph. From an i n s p e c t i o n of the t h a t 8 Q,°(| + ^ R ^ V V l - ? V^Eiz.7i 1  5Z  of 7) .  Moreover  i t may  i  s  be  approximated by equation C.3-12 s i n c e the experiments I w i l l perform c o u l d not p o s s i b l y d e t e c t the e r r o r  resulting  from  -145  FIGURE  -  43  -146  this  approximation.  to be independent  -  Note, I would expect  of 71 from the  geometry  the time  constant  of  problem.  the  Thus, assume:  The method used to s o l v e the 5 and 7 dimensional problems i s identical  to  the  method  used  t o s o l v e the 3 dimensional  problem. Since nothing new i s demonstrated by s o l v i n g  these  problems i n d e t a i l , I w i l l simply present the r e s u l t s . It S^Cv^t-)  turns  out  that S^Cr^-t-)  and S zO^ t)  f o r the number of s i g n i f i c a n t  7  figures  are equal to considered.  Hence:  C.3-14  Thus:  This  problem  was  solved  for various  A  technique o u t l i n e d above. The r e s u l t s a r e :  using  the same  -147 -  EMPIRICAL  F I T OF X  <  A c L /DA 2  2  FIGURE  44  0  -148 -  'Xa£oL"/DJL  i s plotted  l i e on a s t r a i g h t  a g a i n s t A i n f i g u r e 44.  These  l i n e which may be approximated by:  - f j ^ - 2 . - 2 6 ln(3.82vA)  as  demonstrated  results  in  figure  C.3-16  44. S u b s t i t u t e  equation 2.4-11  i n t o equation C.3-16 t o o b t a i n :  C.3-17  Hence the response of the s h e l l may be  written:  where:  U ~ I h ( W I ^ )  C.3-19  -149 -  APPENDIX D  D.I  SOLUTION OF THE X-DIRECTION "SWITCH ON"  PROBLEM  FOR  A  SPHEROID TAKEN TO THE SPHERICAL LIMIT  In govern  chapter 2, s e c t i o n 7, I o b t a i n e d the equations which the  response  of  a  s p h e r i c a l l i m i t , to an a p p l i e d this  appendix  spheroidal s h e l l , field  taken to the  i n the x - d i r e c t i o n .  In  these equations w i l l be s o l v e d . As a reminder  these equations a r e :  2.7-5  36.ExP/(nw££R^P (7i) ,  S/\ '"~^Cr y /A ^ i \(i7 i flt i),C . io / i5 n <ots> u; )X^ ~  2.7-6  >  2.2-13  The procedure used t o s o l v e t h i s problem paragraph. F i r s t l y  i s outlined  the elements of B^n are found.  in this  It  turns  -150 -  out  the  matrix  i s d i a g o n a l . Thus, from equation 2.7-4, the  eigenvalues are:  D. 1-1  and  the e i g e n v e c t o r s a r e :  I  o  O  I  O  o  O O =2 I  O  o  where x, y, z, e t c e t e r a are undetermined eigenvalue  -&',,  coefficient  x, ~B>ZJL  etcetera.  An  to  The  eigenvector  with  the  t o the e i g e n v e c t o r with  inspection  equation 2.7-8, undetermined  corresponds  coefficients.  indicates  coefficients  of  the  "equilibrium condition",  that are  coefficient  and equal  to  a l l other  zero.  Hence the  eigenvector i s :  o o o  D. 1-2  Thus, from equation 2.7-6, the e i g e n s o l u t i o n i s :  D. 1-3  From equation 2.2-13, the response  of the s h e l l i s :  y,  -151  -  D. 1-4  All  that i s l e f t  to do i s to f i n d the elements of B^»/  Let:  D. 1-5  This  integral  can be e v a l u a t e d a n a l y t i c a l l y  with the use of  the f o l l o w i n g i d e n t i t i e s : ^  C2yx+i)7lf>'(rj)'/^.(^)  H/^-0^,(7))  D. 1-6  D. 1-7  S u b s t i t u t e equation D.1-7  i n t o equation D.1-5  to o b t a i n :  D. 1-8  S u b s t i t u t e equation D.1-7  i n t o equation D.1-8  d?7  S u b s t i t u t e equation D.1-6  = 0  i n t o equation D.1-9  to o b t a i n :  D. 1-9  to o b t a i n :  -152 -  P>)^'Cr))(^f.^c)77  D.  1-10  D.  1 -11  -I Thus:  T h i s i n t e g r a l can be e v a l u a t e d with the use of the f o l l o w i n g identity:^  I R.'m^nlcJ'n = ^ - ^ ( ^ + 0 ^  D.1-12  —i  Hence:  ^dak2j&mi  D  K  The l i m i t  of  ? —•• oo  Hence B,'  R'(!JQ,'Cf„)  is:  J Sa  becomes:  The r a d i u s of the sphere,  from equation 2.7-3, i s :  .,-,  3  -153 -  R  ^  % ^  D.1-16  S u b s t i t u t e equation  D.1-16 i n t o equation  D.1-15 to obtain:  ^ ' ^ ^ f t "  From equation  D. 1 - 1 7  D.1-4 the response of the s h e l l i s :  6 Cf\t) ^3£„R'(7))Co4(| x  where:  -  D.i-18  -154 -  APPENDIX E  E.1  SOLUTION OF THE Z-DIRECTION "SWITCH ON"  PROBLEM  FOR  A  SPHEROID TAKEN TO THE SPHERICAL LIMIT  In chapter 2, s e c t i o n 8, I o b t a i n e d the equations which govern  the  response  of  a  s p h e r o i d a l s h e l l , taken t o the  s p h e r i c a l l i m i t , to an a p p l i e d f i e l d this  appendix  i n the Z - d i r e c t i o n .  In  these equations w i l l be s o l v e d . As a reminder  these equations a r e :  2.8-2 .  2.8-1  2.8-5  2.8-3  2.2-13  7\ The procedure used to s o l v e t h i s problem  i s outlined  paragraph. F i r s t l y the  It  matrix  is  diagonal.  ' are Thus,  found. from  turns  equation  in this out  the  2.8-2,  the  -155 -  eigenvalues are:  E. 1-1  and  the e i g e n v e c t o r s a r e :  i  o  o  o  I 0  o  o  o  o  where x, y, z, e t c e t e r a are undetermined  coefficients.  The  eigenvector  with  eigenvalue  -B t l  corresponds  coefficient  x, " B ^  t o the e i g e n v e c t o r with  etcetera. equation  An  inspection  2.8-5,  undetermined  indicates  coefficients  of  to  the  coefficient  y,  "equilibrium condition",,  x= Se.o£-£%  that are  the  equal  to  Q  and  a l l other  zero.  Hence the  eigenvector i s :  o  f^xi ^  3e.E f £  o!  o o  E. 1-2  Thus, from equation 2.8-3, the e i g e n s o l u t i o n i s :  E. 1-3  From equation  2.2-13, the response  of the s h e l l i s :  -156  -  E. 1-4  All  that  is left  to do  is  calculate  the  elements  of  Bp* • L e t :  E. 1-5  This  i n t e g r a l can be e v a l u a t e d a n a l y t i c a l l y with the use of  the f o l l o w i n g  identities:  3  E. 1-6  E. 1-7  S u b s t i t u t e equation E.1-7  i n t o equation E.1-5  to obtain:  I  E. 1-8  S u b s t i t u t e equation E.1-7  i n t o equation E.1-8  to obtain:  I  E. 1-9  S u b s t i t u t e equation E.1-6  i n t o equation E.1-9  to obtain:  -157 -  E. 1-10 -1  Thus:  -l  T h i s i n t e g r a l may be e v a l u a t e d with the use of the f o l l o w i n g identity:  E. 1-12  -1 Hence:  E. 1-13  The l i m i t  U  of  ^iS.^tfJ)  is:  PfltoQ^.)~-^  Hence B,',  E.1-14  becomes:  E. 1-15  The r a d i u s of the sphere,  from equation 2.7-3, i s :  -158 -  Hence:  From equation E.1-4 the response of the s h e l l  where:  

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