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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Studies of dense plasmas in laser generated shock wave experiments Parfeniuk, Dean Allister 1982
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Title | Studies of dense plasmas in laser generated shock wave experiments |
Creator |
Parfeniuk, Dean Allister |
Date Issued | 1982 |
Description | Two theoretical models were developed to predict the effect of conducting shells on electrodeless breakdown at low frequencies (<1 kHz). If glass shells containing gases at low pressures are immersed in an alternating electric field, of sufficient magnitude, these bulbs will emit pulses of light. Together the theoretical models predict the form of the calibration curves of these bulbs, which are plots of the pulse rate as a function of the magnitude of the applied field. Experimental verification of these models was observed. Furthermore, the surface conductivity of paxolin and neoprene was measured as a function of frequency in the range 40 Hz to 1 kHz. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-04-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085786 |
URI | http://hdl.handle.net/2429/24086 |
Degree |
Master of Applied Science - MASc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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