DTIC AD-A275 525 FE CONTROLLING C!{AOS IN I° PLASMA FILLED DIODES I0 4 PA.Lindsay J.Watkins X.Chen King's College 222 Strand London ',C2R 2LS UK !S _ FZ49620-92-J-0219 P 00001 1 15 May 1992 - 14 November 1993 I ° [ [ Best Available Copy H i0 REPORTForm Approved REPORT DOCUMENTATION PAGE oF No. o7o oU AigmaAdie ina nng endm amtamnfhonr t htheo d caotan inleecten do f ainnfo mcno(cid:127)amtiopnle tni go etntp (cid:127)itreedw ~ewtoin 8g .tah~ae(cid:127) c olIl ehcotuorn poefr urfimn onati(cid:127)o.., in1leinndgc 1 1Wtu neIer n.f,aom, r revlnwg tin, WbinnUudUamn Ma itunwata.wtec aa1wm "otnhe r daeete c iaoowf ethe,,. colleOt o fotmetion. icluding swufogrmgatoons foe reducng thu uden to Weshunonf e e l u O r in . .perios nd Repoes is *"Orion Dam Highway. Swt* 1204. Ar(cid:127)lngton. VA 22M0-4302. wn to the OflK. of #AnAgement and Sudgat Paperwork naduction Progect (07040 51). Wa0hington, OC J205 . 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 14 Jan 94 ]Final, 15 MAy 92 - 14 Nov 93 4L TITLE AND SUBTITLE S. FUNDING NUMBERS CONTROLLING CHAOS IN PLASMA FILLFD fIO,)1.S F9" •IRl F49620-92-J-0219 .AUTHOR(S) P.' .Lindsay P 00001 ,J.Watkins X.Chen 7. PERFORMING ORGANIZATION NAME(S) AND ADORESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER King's College London 222 Strpnc(cid:127) London YC2R 2LS UK 9. SPONSORING/ MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/ MONITORING AGENCY REPORT NUMBER EOARD 223/231 Old Marylebone Road 9 4- r London N'l 5TH UK 11. SUPPLEMENTARY NOTES 0 12a. DISTRIBUTION /AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Approved for )ublic release; distribution unlimited 13. ABSTRACT (Maximum 200 words) This report covers the first eighteer months of our investigations. Following a brief introduction the report discusses in some detail the mathematical model of the system, the methods used in solving the corresponding differential equations and the various ways of presentation of the results. The effect of an external load is then discussed in some detail. The report ends with suggestions for future work based on the results obtained so far. 14. SUBJECT TERMS 15. NUMBER OF PAGES 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT Of THIS PAGE OF ABSTRACT UNCLASSIFIED UNCLASSIFIED IJNCLASSIFIED NONE NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) PrtCribd by ANSI Std Z39-I8 2M102 -3- 0 CONTENTS Summary 4 1. Introduction [ 2. Mathematical model 6 0 3. Computational details 18 . 4. Presentation and discussion of the results 20 5. The effect of the load 26 [ 6. Chaos prolegomena 32 7. The influence of positive ions 38 0 444 Cclusions References 46 47 0 Captions AcU lo. FA " Cl- - T0 ~L (cid:127)9,-\1 0 9O4-0578 ,(cid:127) 'r9'4 2 09 112 1-- SUMMARY 0 This report covers the first eighteen months of our investigations. Following a brief introdactton the report discusses in some detail the mathematical model of the system, the -methods used in solving the corresponding 0 differential equations anid the various ways of presentation of the resilts. The effect of an external load is then discussed in some 0 detail. The report ends with suggestions for f-ture work based on the results obtained so far. 0 0 S.. o .-., . (cid:127) .. .._ . "(cid:127). .. . .. "i. ... i i _ . __ (cid:127) . (cid:127) i i i il l mi S 1. Introduction The main purpose of this work is to inveqtignte the control of chaos in a plasma-filled diode. A mathematical model of suich a system has been 0 considered before 1,2 its main feature being the replacement of a smooth, interelectrode electron cloud by a large number of infinitely thin electron sheets. 0 In the absence of ions similar, continijois systems have been considered in great detail , including the effects associated with the granular nature of the 0 electron stream . In the presence of positive ions the system was first considered by J.R.Pierce 5 and then in more detail by others 6,7 . When the electron cloud is discretized, the ions can be treated 1 as a smooth .3ositive bpckground , or they can be discretized as well. Drawing on the experience quoted above we are sugg-esting that the latter is the more appropriate approach to adopt for further investigations. 2. .Mathematical model Let us first of all consider a system which is devoid of positive ions (pure vacuum). The equation of motion for a single sheet of electrons is then eiven by sI( (2.1) Fms'Kl - ,€here x is the position of the sheet, Fs and prs are the respective charge and mass surface densities (ps./ms - - e/m.) and F, is the averr.ge field acting on the sheet, 1 (cid:127)' " T TIT (. ~. 4ere FI and 7.TT Pre respectively the fields in the ,rid/electron-sheet Pnd electron-sheet/anode soaces, the Peneral Peometry beinp thrt shown in Fig.l. Since the field mtist change from ET to TI ai'-e pass through the sheet, the act jal rielt acting nni the shet is the av-,raie field "I" Stibstititinng (2.2) in (2.1) we find .h:t the -ingFle sheet is :lecelerited w'hen x 4I/ 2 and acceler.-ted when xI /2. --(cid:127) ' . . . . .. mmlm~m ~ wi m(cid:127), mm w m- --(cid:127) .--- ,m mmmmm-m l mI, m0 7 I Tn the case of ,ev'ril sheets the nver,-e field actine on qheet j is -iven by (cid:127)j . (all sheets to the left .)f j) + 9 + 1 E T (all sheets t,(cid:127) the right of j) -. (TIi>j) + E + F i -'IT- bag where the sheets are counted from ri?ht tq left, since sheet I ,¢n'ild be injected first and ,¢'olcd have travelled to the ri.ht hv the time sheet 2 hns been injected, and so on. t-ere M represents the tntal nm,.n,#r rf sheets momentarily ini the interelectrode space. In n.ace of (2.1) wf now have the f-llowin-, set of M coupled el-tations of motion, each representing one of the M electron sheets present in the interelectro-le pmslj Psj - reduced aibe: It is nowi convenient to introduce the following reduced variables I' = X/d. t' - t/t (2,5) S.... .. 0 1 - 8- i S where d Is the separatinn of the ,lectrodes ard to 0 is the transit time in tVe absence Of fields of an electron injected with initial velocity i v 0 i.e. ti d/v Thus !,e now have:; dx' t. dx d&xI _ x dt' d cit da oT IM d (2.6) In order to introdtice the .onne:jt if convecti')n carrent J we write C Ps - . "VeX(2.?) N v.N7 where is na-in the eharge surface-density of a single 0 sheet, p tot is the surface density of N sheets And PC is the smoothed-nlit volume density of N sheets. A typical equation (2.4) can now be written, substituting from (2.7), (2.6) and (2.7): dd-tt I = -fp s 'fa. va-A % I L9T i - + (2, ) Note that 4 may be different from N , since N is Sa constant and M may vary, bein.ý eqtt.4l to the numler of sheets which are present in the interelectrode space [ at any given time. In the final step of our derivation of the eqtrtions [ of motion ifec hoose to express the convection ciorrent -c as P fra[c tion of the convection current flowing in a similar • [' C: -9- space-char-e-limited diode- c, .T s~l giv"e• n b,y: 0 J2l 5 0 LL!2 V (2.9) c& 9 I*9-9 Sisbstittiting (2.9) in (2.8) we finally obtain daXt 2u M I d t 9 V- I E' &= I t -I- + (2.10) L * Jco/Jc (2.1l) c',scco'l as a typical equ;,titcn of -,otion for the jth electron sheet 1 . The .M equations (2.10) fully describe the motion of Nt electron sheets, each sheet injected with the same initial velocity dx'./dt' - i /v = 1 into the interelectrode space. Since the diode is assumed to be ;hort-circiiited, the only fields present are those due to the space charge of the sheets themselves. Thus (2.10) re)resent a computer model of the conditions inside a plane diode, where the cathode has been replaced by a grid, so that the electrons can he injected with a constant velocity v which is different from zero 0 0 So far we have only considered a short-circ;ited r diode. If a load is connected across the electrodes, we reqtire an additional equation which rplates the [ conditions inside and outside of the diode. Noting the fact that the total current in Fig. 1 must be continuous [(cid:1)we can frite: