Potential and field distribution inside an axially split...

9
Indian Journal of Radio & Space Ph ys ics Vol. 33, April2004, pp. 130- 138 Potential and field distribution inside an axially split cylinder B N Das & Arun Gayen Department of El ectro ni cs and El ec trical Communication Engin ee ring, Indian Institute of Tech nology, Kharagpur 721 302, India Received 25 March 2003; revised 12 November 2003; accepted 30 January 2004 The paper prese nts a method of eva lu a ti on of potential a nd field distribution in side an axially split cy linder, one part of whi ch is in s ul ated and oth er maintained at ground potential. The Lapl ace's eq uation expressed in cy li ndr ical polar co- ordinates is solved assuming no axial variation, and using boundary co ndition for the potential. Th e radial and ang ul ar co mponents of th e fi eld int ens ity are found us in g the appropriate de ri vati ve opera tor in the cylindri ca l co -ordinate system. The co nd iti on under which the radial fi e ld is maximum, is de ri ved in terms of ang ul ar widths of the co nductors. If the radial fi eld intens it y exceeds the breakdown limit, the resulting breakdown of the gas leads to pl asma generat io n in side the structure. Keywords: Split cy linde r, Laplace 's equa ti on, Potential distribution, Field intensity distribution. PACS No: 77.22 Jp, 84.40 Ua IPC Code: HOIS-3/00 1 Introduction Ga s la se rs are widely used the se days for scientific and industrial applications. Th ere are many possible electrode co nfigurations, which produce high field intensity required for electrical breakdown. On e possible electrode configuration is to split the wall of a conducting cylinder in two parts by two uniform slits along the axis of the cylinder. Wh en hi gh voltage is applied b etwee n the two electrodes, di sc harge may take place when the field intensity is above the breakdown field, which has a value of 30 kV/cm at nor mal atmospheric pressure (I torr). Th e re is di scharge in the region betwee n two electrodes at diff erent potentials. The disc har ge be tween the electrodes ge nerates the las er beam. Th e particular electrode configuration is chose n in order that there is sufficient space betwee n the electrodes for generat io n and emission of la ser beam. It is now necessa ry to find the potential and field distribution in order to determine the condition favourable for di sc har ge and ge neration of lase r bea m. Condition favourable for dischar ge can be found from the evaluation of potential and field distribution in the region between the two electrodes. Some studies on the evaluation of field distribution inside a split cylinder partially filled with a dielectric tube ha s been studied by Kajfez and Wu 1 The method is ba sed on use of a MATLAB for matrix inversion required for finding field distribution. Similar studies have also been presented in the literature 2 - 4 . We prefer to find a so lution in closed form employing the summation formula using contour integration technique 5 . In the pre se nt paper Laplace 's equat ion is so lved in circular cylindrical co-ordinates und er longitudinal invariance condition 6 to find the potential and fi e ld distribution in th e reg ion betwee n two halves of the cylinder for the general case in which the two boundaries form parts of sa me cylinder of different angular width. The Laplac e's equation is solved in terms of radial and angular functions in circ ul ar cylindrical co-ordinates 6 . The constants appearing in the expre ssion for the potential functions are evaluated from the bound a ry conditions a nd int eg ration of harmonic functions. Th e field distribution is found from the gradient of the sca lar potential function 7 The pote ntial for which breakdown for plas ma generation takes pla ce can be found from the results of the ana lysis, for an axia ll y split cylinder. The conditions under which the field intensity is derived with the objectives of achieving breakd ow n with minimum applied voltage is as follows: 2 Mathematical analysis Consid er the configuration of Fig. 1 which shows an axially split cylinder, two parts of which are maintained at potentials v, and v 2. The Laplace 's equation for the po tential function satisfies the equation where ¢is the pot ential function.

Transcript of Potential and field distribution inside an axially split...

  • Indian Journal of Radio & Space Physics Vol. 33, April2004, pp. 130- 138

    Potential and field distribution inside an axially split cylinder

    B N Das & Arun Gayen

    Department of Electronics and Electrical Communication Engineering, Indian Institute of Tech nology, Kharagpur 721 302, Indi a

    Received 25 March 2003; revised 12 November 2003; accepted 30 January 2004

    The paper presents a method of evaluation of potential and field distribution inside an ax ially split cy linder, o ne part of which is insul ated and other maintained at ground potential. The Laplace's equation expressed in cy li ndrical polar co-ordinates is solved assuming no ax ial variatio n, and using boundary conditio n for the potential. The radia l and ang ul ar components of the fi eld intensity are found using the appropriate deri vati ve operator in the cy lindrical co-ord inate sys tem. The conditio n under which the radia l fi e ld is maximum, is deri ved in terms of angul ar wid ths of the conducto rs. If the radial fi eld intensity exceeds the breakdown limit, the resulting breakdown of the gas leads to plasma generation inside the structure.

    Keywords: Split cylinder, Laplace's equation, Potential distribution, Field intensity di stributi on. PACS No: 77.22 Jp, 84.40 Ua IPC Code: HOIS-3/00

    1 Introduction Gas lasers are widely used these days for scientific

    and industrial applications. There are many poss ible electrode configurations, which produce high field intensity required for electrical breakdown. One possible electrode configuration is to split the wall of a conducting cylinder in two parts by two uniform slits along the axis of the cylinder. When high voltage is applied between the two electrodes, discharge may take place when the field intensity is above the breakdown field , which has a value of 30 kV/cm at normal atmospheric pressure ( I torr). There is di scharge in the reg ion between two electrodes at different potential s. The d ischarge between the electrodes generates the laser beam. The particular electrode configuration is chosen in order that there is sufficient space between the electrodes for generation and emission of laser beam. It is now necessary to find the potential and field distribution in order to determine the condition favourable for discharge and generation of laser beam. Condition favourable for discharge can be found from the evaluation of potential and field distribution in the region between the two electrodes. Some studies on the evaluation of field di stribution inside a split cylinder partially filled with a dielectric tube has been studied by Kajfez and Wu 1• The method is based on use of a MATLAB for matrix inversion required for finding field distribution. Similar studies have also been presented in the literature2-4. We prefer to find a solution in closed form employing the summation formula using contour integration technique5.

    In the present paper Laplace's equation is solved in circular cylindrical co-ordinates under longitudinal invariance condition6 to find the potenti al and fi e ld distributi on in the region between two halves of the cylinder for the general case in which the two boundaries form parts of same cylinder of different angular width. The Laplace's equ ation is solved in terms of radial and angular functions in circul ar cylindrical co-ordinates6. The constants appearing in the expression for the potenti al function s are evaluated from the boundary conditions and integration of harmonic functions.

    The field di stribution is found from the gradient of the scalar potential function 7 • The potential for wh ich breakdown for plasma generation takes place can be found from the results of the analys is , for an axia lly split cylinder.

    The conditions under which the field intensity is derived with the objectives of ach ieving breakdown with minimum applied voltage is as fo llows:

    2 Mathematical analysis Consider the configuration of Fig. 1 which shows

    an axially split cylinder, two parts of which are

    maintained at potential s v, and v2. The Laplace's equation for the potential function

    satisfies the equation

    where ¢is the potential function .

  • DAS & GA YEN: POTENTIAL AND FIELD DISTRII3UTION INSIDE AN AXIALLY SPLIT CYLINDER 131

    In circular cylindrical co-ordinates (r,

  • 132 INDIAN J RADIO & SPACE PHYS, APRIL 2004

    cos ne Jo, = Rll B' f1 - cos 2ne de a, II o 2

    Therefore,

    B' = 2V1 . cos ne1 -cos ne2 II nR 11 n

    Simple integration over 0 to n, leads to C0 = V1 Hence,

    ifJ(r,e)= 2V1 f(~ J"[sinne2 -sinne1 cosne n 11=1 R) n

    cos nel -cos ne? . fl] v + -Sill /l.o + 1 ll

    ... (11)

    for el..::; e ..::;ol For the conductor at potential V2 ( e3 ..::; e..::; e4 ) it

    can similarly be shown that

    sin 11.e4 -sin ne1 11 n V? · =R·A-- n 11 2

    Therefore,

    A = 2V2 sin ne4 -sin n03 ... (12) II n n

    B = 2V2 sin ne, - sin ne4 . .. (13) II n n

    ... (14)

    Hence, general expression for the potential at any point is given by

    [

    sin ne, -sin ne1 • n j II - COS/1!7

    r n 2V ~ - +V - 1 ~J R) cosHe1 -cosne2 . 1.) 1

    7r 11=1 + Slllllu II

    ifJ(r,e)= for el ..::;e ..::;e2

    r

    sinne4 -sinne3 a j 11 cosnu r n

    ?V ~ - + V, -

    2 ~J R) cosl!e3 -cosne4 • n -n 11=l + smnu ll

    for e3 ..::;e ..::;e4 ... (15)

    Angular field intensity £0 is given by

    2V ~ (,. )11 [-(sinne~ -sinne3 )sinn8 l n; ~~l R + (cos n83 -cos n84 )cos ne

    ... (15a)

    Equation (15) can be expressed as

    ifJ(e)j = 2V1 f[sinn(e2 -e) _ :5inn(e1 -e)]+v r=R 7r 11= l H H I

    for e l ~ e ..::; el ... (16)

    and

    ( )I 2V? ~ [sinn(e4 -e) sinn(e3 -e)] ifJ e = - - L, - + v, r=R 7r 11 =1 H H -

    From the series summation available in appendix3, it is found that

    ~ e ) llx . jn L, -=-ln2smx+-

    1.3 .s n 4

    Separation of left hand side into real and imaginary parts leads to

    f sinnx = n 11=l n 4

    ... ( 18)

    Substituting Eq. (18) in Eqs (16) and (17), it is found that

    I 2V [n n] ifJ(e) =-1 X --- +V1 =V1 r=R 7r 4 4 .. . (19) for the conductor at potential V, and

    qy(e)l = 2V2 x[n _ n]+v =V r=R 7r 4 4 2 2

    ... (20)

    for the conductor at potential v2 . Substituting Eqs (18)-(20) in Eq. (15), and

    differentiating with respect tor, it is found that

  • DAS & GA YEN: POTENTIAL AND FIELD DISTRIBUTION INSIDE AN AXIALLY SPLIT CYLINDER 133

    2V1 f [sinn(e2 -e)-sinn(e1 -e)] ()t~> nR n =I.3... ... for e

    1 $; e $; e,_

    f;(R ,e)= 2V

    2 ~ [sinn(e4 -e)-sinn(e3 -e)] -L

    R n=I.3...... for e3

    $; e $; e4

    . .. (21)

    From Eq. (21), it is found that the radial component of the field intensity E, is a function of e, e, and fh.

    For e = e 1 and r = R , the field intensity is of the form

    .. . (22a)

    and fore =e2

    .. . (22b)

    Similarly, for e = e3 ,e4 , the expressions are:

    l/J(r,e)= 2V0 f(!...)" sinne + V0 n n =l R 1z 2

    At r=R

    S. ~ sinne n mce L..J --=-

    11=1 ll 4 2V n V

    Hence, l/J(R,e)=-0 x -+__Q_=V0 n 4 2

    . .. (24)

    ... (25)

    which satisfies the boundary condition for all values of e and radial field intensity given by

    ol/J = 2V0 f ~(!_)"- I sin ne = 2V0 f(!_)" sin ne dr n n=l R R n n R n= l R

    .. . (26) At r = R, i.e. on the conductor surface

    E, =- =-0 LSIO/le acpl 2v ~ . dr r=R nR n=l

    · · ·

  • 134 INDIAN J RADIO & SPACE PHYS, APRIL 2004

    Differentiating right hand side with respect to 8

    sin(82 -8)+sin(8-81)=0

    sin (82 -8) = sin(8 -81 ) 82 -8=rc+8-81

    which gives

    e = - rc + 82 +f) I 2 2

    .. . (29a)

    Hence, the angle for which field intensity is maximum for 81 ::; e::; 82 is given by

    Q =- rc + e, + 84 f Q < Q < Q u or u 3 _v _u4 2 2 ... (29b)

    Substituting Eqs (29a) and (29b) in the two parts of Eq. (28)

    E (R)I =-1 sin-2--1 -sin-1--2 v [ e -e e -e J r max rcR 2 2

    = 2VI . ( 82 -f) I ) f a

  • DAS & GA YEN: POTENTIAL AND FIELD DISTRIBUTION INSIDE AN AXIALLY SPLIT CYLINDER 135

    . (~Jsin8{1+(1-~ cose} NormahzedE, = I+(~ J _2( ~ }osO

    ... (32e)

    The variation of angular field intensity ( E0 ) with r

    and 8 can be found as

    E = ..!_ d

  • 136 INDIAN J RADIO & SPACE PHYS, APRIL 2004

    axially split cylinder of Fig. 1 as a function of 0 with

    (~)as a parameter are presented in Fig. 5.

    4 Conclusions Radial field intensity ( E,) inside the tube with

    increase in ( ~) has a non-linear variation. The non-linearlity is less for ( 82 -81) ranging from 90° to

    180° and it can be approximated as linear function . But for ( 82 -81 ) less than 90°, the non-linearity as a

    function of ( ~ ) is noticeable and for ( 82 -81 ) between 30° and 60°, non-linearity is appreciable.

    Value of Er inside the tube with increase in ()has a

    non-linear variation for the values of ( ~) ranging from 0.25 to 1.

    Table 2- Normalized radial field intensity ( E,) inside an axially

    split cylinder as a function of 0 with (!....I as parameter Rj

    0 Normalized E, (de g)

    r!R=0.25 r!R=0.5 r!R=l

    30 0.081888 0.466506 1.866025

    45 0.095397 0.440744 1.207107

    60 0.09 1599 0.360844 0.866025

    90 0.058824 0.2 0.5

    120 0.025775 0.092788 0.288675

    150 0.007324 0.033494 0.133975

    180 -4.10E-12 -2.28E-Il - 1.03E- l0

    2 - -f =30 --- " =45

    1.5 ·· ·· ·- If =60

    :[ -·- · d' =90 - - . -If =120

    ~ --~' =1 50 ~;q'- -If =1 80 "C Gl .!!! 0.5 iii - · E ... 0 z 0

    0.2 0.4 0.6 0.8 12

    - 0.5 ......._ _ _ _ ______ -------'

    (i) Fig. 2-Normalized radial fie ld intensity (E )' for an axially split

    ' r

    I 2:. ~ .. 'C .. .:a -; a .. 0 z

    cylinder as a func tion of !.__with () as parameter R

    2 .-------------.------~

    1.5

    0.5 - ·-- '._ · · ··· · ·-·-·-----0

    50 100

    · · · · - · (r/R)=0.25

    -- · - · (r/R)=O.S --- (r/R)=1

    150 2 0

    - 0.5 ...__ ______________ __.J

    O(deg)

    Fig. 3- Normalized radial fie ld intensity ( E, ) inside an axially

    split cylinder as a function of(} with !.... as parameter R

    Table 3-Normalized angular fie ld intensity ( E0 ) inside an axially split cylinder as a func tion of ( ~) with 0 as parameter

    (~) Normalized ( E0 )

    () =30° () =45° ()=60° ()=90° () =120° ()= 150° ()= 180°

    0.1 0.09 1543 0.069897 0.043956 - 0.0099 -0.05405 -0.08165 -0.09091

    0.2 0.192052 0.13395 0.071429 -0.03846 - 0.1129 -0.15378 -0.16667

    0.3 0.297707 0. 183454 0.075949 -0.08257 -0.17266 -0.21732 -0.23077

    0.4 0.399012 0.206696 0.052632 - 0.13793 -0.23077 -0.27332 -0.2857 1

    0.5 0.476627 0. 190744 -7.89E- I I -0.2 - 0.28571 - 0.32278 -0.33333

    0.6 0.497601 0.125645 -0.07895 -0.26471 -0.33674 -0.36662 -0.375

    0.7 0.418706 0.009948 -0.17722 -0.32886 - 0.38356 - 0.40564 -0.41177

    0.8 0.20766 - 0.14611 - 0.28571 - 0.39024 -0.42623 -0.44051 --0.44444 0.9 -0.12175 -0.32316 -0.3956 -0.44751 -0.46495 -0.4718 -0.47368

    -0.5 -0.5 - 0.5 -0.5 - 0.5 -0.5 -0.5

  • DAS & GAYEN: POTENTIAL AND FIELD DISTRIBUTION INSIDE AN AXIALLY SPLIT CYLINDER 137

    Table 4-Normalized angular field intensity ( E0

    ) inside an

    axially split cylinder as a function of() with (~)as parameter

    e Normalized (E0 ) (deg)

    r!R=0.25 r!R=0.5 r!R= l

    30 0.244654 0.476627 -0.5 45 0.161192 0.190744 -0.5

    60 0.076923 - 7.89E- Il -0.5

    90 -0.05882 -0.2 -0.5

    120 -0.14286 -0.28571 -0.5

    150 -0.18656 -0.32278 - 0.5

    180 -0.2 - 0.33333 - 0.5

    210 -0.1 8656 - 0.32278 - 0.5 240 -0.14286 - 0.28571 -0.5

    300 0.076923 -1.20E-07 - 0.5

    360 0.333333 -0.53468

    0.6 -- (/'_=30 - - - (f =45 ------ d' =60

    0.4 - --- d' =90 -- -- if'=120 --+- (1. =150

    I -o- If =180 0.2 ?:. ' ' / -- '

  • 138 INDIAN 1 RADIO & SPACE PHYS, APRIL 2004

    The studies reveal the characteristics of a new electrode configuration , which can be used for the generation of plasma and hence a laser beam.

    Acknowledgements The authors thank Mr Anirban Dhar and Miss

    Sarbari Ray for useful di scussion .

    References I Kajfez D & Wu Y, /££ Proc H. Micro Antennas & Propag

    (UK), 140 ( 1993) 29.

    2 Jackson 1 D, Classical electrodynamics. 2"d edn, (John Wiley. New York), 1975 .

    3 PC-MATLAB user's guide (The Mathworks Inc. , Soputh Natick, MA), 1989.

    4 Kajfez D, Mahadevan K & Gerald 1 A, Microwave J (USA). 32 ( 1989) 267.

    5 Ramo S, Whinnery 1 R & Van Duzer T, Fields and waves in commu11ication electronics (Prentice Hall, New Delhi ), 1970.

    6 Pipes L A, Applied Mathematics for Engineers and Physicists (McGraw Hill , New York and London), 1946.

    7 Collin R E, Field Theory of Guided Waves, Second Editio n. (IEEE Press, New York), 1994.