Sensor and Simulation Notes Note January 1973 A PARAMETRIC STUDY OF A CIRCULAR CYLINDER WITHIN TWO PARALLEL PLATES OF FINITE WIDTH Soon K. Cho and Chiao-Min Chu The University of Michigan Radiation Laboratory Department of Electrical and Computer Engineering Ann Arbor, Michigan 48105 Abstract A parametric study of the presence of an infinitely long circular cylinder within two parallel plates of finite width is conducted. The numerical method is used for a special case where the axis of the cylinder is constrained on the center plane between the plates in terms of the impedance factor of the system and charge distribution on the cylinder. 11323-1-F = RL-2218

011323-1-F I INTRODUCTION In this report, for TEM wave propagation, we are concerned with the effect of the presence of an infinitely long circular cylinder placed within two parallel plates of finite width. The geometry of such a system is shown in Fig. 1. As seen in Fig. 1, the y-coordinate of the axis of the cylinder is '.. J. I cylinder k- a --- a - + V upper plate 0 I a —h -- - " (x y) I I b * I- i — ` I -- x j b -V: lower plate 0 Fig. 1: Geometry of two parallel plates of finite width with a circular cylinder within them. restricted to be symmetric between the plates. For a parametric study of the effect of the presence of a circular cylinder, we investigated the impedance factor f of the system and the g 1

011323-1-F charge distribution on the cylinder for the following cases of parameters: (a) for the impedance factor, f g -=0.1, 0.2, 0.5 and 1.0, b h= 0, 0.2, 0.5, 1.0 and 10.0, a -=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.99. (b) for the charge distribution on the cylinder, a=0.1, 0.2, 0.5 andl.0, b h 0. a c< 0.1, 0.5 and 0.9. bIn Section II, mathematical formulations for the impedance factor of the system shown in Fig. 1 and the charge distribution on the circular cylinder are presented. As pointed out at the outset, the y-coordinate of the axis of the cylinder is restricted to the center plane between the plates. The mathematical formulation of the impedance factor for our present system parallels closely that of our previous work(. Knowing the charge distribution function of the upper plate, the derivation of the charge on the cylinder is straightforward. In Section m, numerical results of the impedance factor of our system and the charge distribution on the cylinder are presented both in tabulations and graphs. The graphical presentations are limited for a few representative cases only. In Section IV, we present some qualitative conclusions for parametric effects of the circular cylinder on the impedance factor of the csystem and the charge distribution on the cylinder. 2

011323-1 —F II MATHEMATICAL FORMULATIONS 2. 1: Impedance Factor, f g For a system shown in Fig. 1, the complex potential of any point (x, y) outside the cylinder is (x, y)+iq(x,y) 1 I 0 0 h+a f dx'c(x')L n h-a (1) We introduce the coordinate transformation and the normalization of parameters as follows: x' -h = - a x-h y-b a a B 2b a H a - a. (2) (3) (4) (5) (6) 9 and c a Now, eq. (1) can be written in the following form 27rE F a O(9 7) +iP (go 7) a L v (z-s+iB)(s+X-iv) (s+H+i)2 = Ids or(s)Ln " J (z-s)(s++iB-iv)(s+H-i -) -1 2, (7) 3

011323 —1-F where =(A ) H- C (z+H) (8) i H [1 2 (8) (z+H) +( 2) V =- -— c, —) (9) (z+H) + 2 Note that the factor (s+H+iB )/(s+H-iB-) in the integrand of eq. (7) contributes 2 2 only an imaginary constant and hence may be ignored in the computation. The mathematical procedure for obtaining the impedance factor for our present system is similar to that in our previous work; therefore, we will simply list the results, omitting the detailed steps involved. Thus, f s)Ln -s jk k j1) 2+B2j k k + (\+ ) LnL.) +( P j +B tan[(k B ta) t'n ( (10)K and 2 2 2 9 i kS ] Ln, 2 +s.) +(B+vB) k Ik 2 2j + - -] 4

011323-1-F 2 vk +- Ln (7 +s)2 + Vk kj k (B -v)2 - ' Ln 2 ( +s)2 +(B-v )2 k j k. Bk1 '+B Bkta ( )+ Ek i v B-v Vk tan k - (BV )tan \k \k e ) (11) where C (k +H) A k kk^, (.k' 0) - -- 2 (k+H) + (') Ik 2 (12) (13) kA B vk- 0 k0) - k ks 2 The impedance factor f is evaluated by use of eq. (25) in our previous work() g 2.2: Charge distribution on the circular cylinder, (2) R. W. Latham investigated effects of the presence of a cylinder in two parallel plates of infinite width*. For our system of Fig. 1, we let x+iy reB and compute -(0+ i ) for eq. (1): 8 r (14) ^+1r ~1 Or Br 2lre 0 h+a I dx' o(x') h-a +- 1 2 ie c re -x' -ib x' -ib 1 1 ie i - e re -x'-ib iO c re - +ib t '+ ib (15) Our O differs from that in Ref. (2). For mathematical simplicity, it is convenient in our case to choose the O-coordinate as shown in Fig. 1. 5

011323-1-F We now let r — c. Noting that - - = 0 as it should, we obtain, after some ar r=c algebraic manipulation, the charge distribution on the cylinder, a(O): E Or r=c i - -- ds 1 (s) T -. -1 i=1 J~j sin n 0 (16) The charge function a(0) can be Fourier-expanded as follows: -w 0D a(0) X A sinn0 n=l (17) D where -1 An.- if w, d s a(s) F - B+H+i (s+H+i2 L 2 (18) For a piece-wise linear charge distribution on the plate, a(s), which we assumed to be the case, we obtain 2M =( ('rsT, T )I.P(s )- P (s)I n + sj j+ +1 j+l Pn j+l Pn J j..) j+l (~ +1- FQn (5~ nj1 ij (19) where P (s) — i n-1C ds - n 7 B n (s+H+i2) q6 - (20) Appendix A the algebraic sepsleading to Eq. (16) are outlined In Appendix A, the algebraic steps leading to Eq. (16) are outlined. 6

uiijZJ-I-F Q (8)w -i Cn1d n 7 S. (21) Carrying out the Integrations for eqs. (20) and (21),, one finds 2 for n= l P1(s) - aC(S) (22-a) for n >2, p GOs) n 7 ~no-1 dM~M I sin' Fn- a a n.i au a (22-b)' for nn 1, Q (S) =- 2 [Ha(s) +11 nR(sj (23-a) I for n =2, Q(11) a 2C "(-20 CF in a -B coca 2 T w- R(s) " (23-b) for n 3 Q (a)mn-l~ sin Ii-2) a(sj 2 1 (cY"' r -si-n* n-). a -- Cs nw)a( w' n-i R(s)J 0 (23-c) a (sh=taninl B4 2(s +H) (4 where R(s~s (S+H)2 (Bi)2 S (25) 7

011323-1-F m NUMERICAL RESULTS OF IMPEDANCE FACTOR OF THE SYSTEM AND CHARGE DISTRIBUTION ON THE CYLINDER In Section II, we introduced the normalized parameters B, H and C: B 2 b/a, H = h/a, C = c/a. In practice, it is more convenient to normalize the radius of the cylinder with respect to b, while it is physically more reasonable to normalize the position of the center of the cylinder (i. e., the x-coordinate of the center of the cylinder) with respect to a. For convenient reference, we present in Tables I, UI and III, the conversion charts for C, h/b, and B, respectively. Table I: Conversion Chart for C, "/b /, 0.1 0.2 0.5 1. 0 ' 0 0 0 0 0.1 1 0.5 0.2 0.1 0.2 2 1.0 0.4 0.2 0.3 3 1.5 0.6 0.3 0.4 4 2.0 0.8 0.4 0.5 5 2.5 1.0 0.5 0.6 6 3.0 1.2 0.6 0.7 7 3.5 1.4 0.7 0.8 8 4.0 1.6 0.8 0.9 9 4.5 1.8 0.9 0.99 9.9 4.95 1.98 0.99..........., 8

011323-1-F Table H: Conversion Chart for h/b 0.1 0.2 ^ 0.5 1.0 0 0 0 0 0 0.2 1 02 0 0.04 0.1 0.2 0.5 0.05 0.1 0.25 0.5 1.0 0.1 0.2 0.5 1.0 10.0 1.0 2.0 5.0 10.0 Table m: Conversion Chart for B a/b 0.1 0.2 0.5 1.0 B 20 10 4 2 As was the case in our previous work, it is necessary for us to know the charge distribution on the plate in order to compute the impedance factor of the system. The mathematical procedure for f of our present system is formally g quite similar to that of the imaged finite parallel plates and was briefly discussed in Section 2.1. As in our previous work, we divided the upper plate width into 20 segments and in each segment a linear approximation was made for the charge function. An examination of preliminary data of f indicated that, the the range of g h/a, 0 <h/a < 1. O, the dependence of f on c/b and a/b is more critical than g on h/a. As indicated in the Introduction, the numerical computation of the impedance factor, fg, of the system was carried out for the following cases: -a O. 1 0.2, 0.5 and 1.0, b h ho0, 0.2, 0.5, 1.0 and 10.0, a b 0 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9and0.99, and the results are presented in Table IV. / 9 t I

%,l- - i-&-Jr Table IV: Impedance factor, f. of two parallel plates - ~-c- -- - I -- - h/a c/b~ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 h/a c/b 0 0.2 1. 174366 1.168062 1. 149140 1. 117491 1.072685 1.013551 0.937401 0.838333 0.702627 0.491631' 0.097914 0 0.954908 0.948786 0.930410 0.899687 0.856256 1.174366 1.168067 1.149159 1.117532 1.072754 1.013656 0.937549 0.838541 0.704521 0.492144 0.098775, 0.2 0.954908 0.948804 0.930478 0.899835 0.856506;. I a/b = 0. 1 0.5 1.174366 1.168093 1. 149259 1. 117746 1.073118 1.014202 0.938323 0.839625 0.704521 0.494828 0.103422 a/b " 0.2 0.5 0.954908 0.948895 0.930833 0.900600 0.857803 1.0 10.0 1. 174366 1.168182 1. 149608 1.118502 1.074403 1. 016134 0.941059 0.843457 0.710134 0.504272 0.121392 1.0 0.954908 0.949204 0.932041 0.903219 0.862257 1. 174366 1.172758 1.167839 1.159313 1.146672 1.129170 1.105782 1.075137 1.035383 0.983910 0.924284 10.0 0.954908 0.954648 0.953858 0.952506 0.950539 0 0.1 0.2 0.3 0.4 10

011323-1-F a/b - 0.2 (continued) h/a c/b 0.5 v 0.6 0.7 0.8 0.9 0.99 0 0. 799146 0. 726178 0. 632813 0.509478 0.333668 0.066840 0.2 0.799518 0. 726696 0.633513 0.510425 0.334965 0.067626 0.5 0.801452 0.729390 0.637155 0. 515361 0.341766 0.072330 a/b a 0.5 1.0 0. 808120 0.738703 0.649779 0.532525 0.365780 0.097912 10.0 0.947875 0.944403 0. 939971 0.934378 0.927360 0.919543 h/a c/b 0 0.2 0.5 1.0 10.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.670940 0.665832 0.650491 0.624850 0.588725 0.541690 0.482869 0.410538 0.321262 0.207082 0.501227 0.670940 0.665889 0.650713 0.625329 0.589529 0. 542859 0.484409 0.412411 0.323338 0.208947 0.051561 0.670940 0.666179 0.651845 0.627780 0.593662 0.548902 0.492436 0.422295 0.334544 0.219582 0. 053861 0.670940 0.667086 0.655414 0.635593 0.607048 0.568886 0.519726 0.457257 0.377001 0.267424 0.085490 0.670940 0. 670931 0. 670901 0.670853 0.670783 0. 670694 0.670582 0.670448 0.670290 0.670106 0.669917 / 11

01132 -1-F a/b 1.0 h/a 0 0.2 0.5 1.0 10.0 o/b 0 0.472441 0.472441 0.472441 0.472441 0.472441 0.1 0.469161 0.469202 0.469439 0.470389 0.472441 0.2 0.459270 0.459432 0.460364 0.464133 0.472439 0.3 0.442615 0.442967 0.445006 0.453377 0.472436 0.4 0.418922 0.419519 0.422992 0.437594 0.472431 0.5 0.387767 0.388637 0.393738 0.415960 0.472425 0.6 0.348487 0.349616 0.356327 0.387235 0.472418 0. 7 0.299953 0.301264 0.309225 0. 349317 0.472410 0.8 0.239925 0. 241235 0.249491 0. 298196 0.472400 0.9 0.162440 0.163395 0.223328 0.228828 0.472389 0.99 0.046983 0.047522 0. 048281 0.079989 0.472377 In Table IV, c/b = 0 corresponds to the case where the cylinder is not present in the two parallel plates of finite width. Such cases were analyzed by Brown and Granzow(3) by a method different from ours* and some of their results are compared with our present results below. a f f f -f b B-G g C-C g B-G g C-C 0.2 0.95512 0.954908 - 0.000212 0.5 0.67116 0. 670940 -0.000220 1.0 0.47264 0.472441 -0. 000199 * B-C f B-Cd denote, respectively, the values of f by Brown Granzow ga gC and Chu-Cho..12

011323-1-F It appears that, by increasing the number of segments into which the upper plate was divided in our numerical scheme, a better agreement between the two sets of impedance factors could be brought about. This could be easily done, if desired, but at a considerably higher expense for running such a computer program. For a parametric study of the impedance factor, f, for the system shown in Fig. 1, we define Af: g Af f- f g g o where f denotes the impedance factor of two parallel plates of finite width oo without the presence of a circular cylinder. In Table V, Af are tabulated to see the g and h/a with a/b as a parameter. Table V: Dependence of Af on c/b. h/a dependence of A f on c/b with a/b as a Darameter a/b = 0. 1 h/a o/0.1 0.1 0 -0.006304 0.2 -0.025226 0.3 -0. 056875 0.4 -0. 101681 0.5 -0. 160815 0.6 -0.236965 0.7 - 0.336033 0.8 -0.471739 0.9 -0.682735 0.99 -1.086452 0.2 -0.006299 -0.025209 -0.056834 -0.101612 -0.160710 - 0. 236817 -0.335825 -0.471435 -0.682222 -1.075591 0.5 -0.006273 -0.025107 -0.056620 -0.101248 -0.160164 -0.236043 -0.334741 -0.469845 -0.679538 -1.070944 I 1.0 -0.006184 -0.024758 -0.055864 -0.099963 -0.158232 - 0.233307 '-0.330909 10.0 - 0.001608 -0.006527 - 0.015053 -0.027694 - 0.045196 - 0.068584 - 0.099229 -0.464232 -0.138983 -0.670094 -1.052974 - 0.190456 -0.250084 13

u11323- 1-F a/b w 0.2 h/a a/b 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 h/a c/b 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0 - 0. 006122 -0.624498 -0. 055221 -0.098652 -0. 155762 -0.228730 -0.322095 -0.445430 - 0. 621240 - 0.888068 0 -0.005108 - 0.020449 -0.046090 -0. 082215 -0.129250 - 0.188071 -0.260402 -0.349678 -0.463858 -0.619712 0.2 0.5 -0.006104 - 0. 006013 -0.024430 -0.024075 -0.055073 -0.054308 -0.098402 -0. 097105 - 0.155390 - 0.153456 -0.228312 - 0.225518 - 0.321395 -0.327753 - 0.444473 -0.439547 -0.619943 -0. 613142 - 0.887282, -0.882578 a/b a 0.5 0.2 0.5 -0.005051 -0.004761 - 0.020227 -0.019095 -0.045611 -0.043160 -0.081411 -0.077278 -0.128081 -0.122038 -0.186531 -0.178504 -0.258529 - 0.248645 -0.347602 -0.336396 -0.461993 -0.451358 -0.618379 -0.617079 1.0 -0.005704 - 0.022867 - 0.051689 - 0.092651 -0.146788 -0.216205 - 0.305129 -0.422383 -0.589128 -0.856996 1.0 -0.003854 - 0.015526 - 0.035347 - 0.063892 -0.112054 -0.151214 -0.213683 -0.293939 -0.403516 -0.595450 10.0 -0.000260 -0.001050 -0.002402 - 0.004369 -0.007033 -0.010505 -0.014937 -0.020530 - 0.027548 -0.035365 10.0 - 0. 000009 -0.000039 -0.000090 -0.000157 - 0.000246 -0.000358 -0.000492 - 0.000650 -0.000836 -0.001023 14

U lltZ0O-l-?r h/a c/b 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0 -0.003280 -0.01370 -0,029826 e 0.053519 -0.084674 - 0.124954 -0.172488 -0.232516 -0.310001 -0.425458 0.2 -0.003239 -0.013009 - 0.029474 -0.052922 -0.083804 - 0. 122825 -0.171177 - 0.231206 -0.309046 - 0.424919 a/b =1.0 0.5 - 0. 003002 -0.012077 - 0.027435 -0.049449 -0.078703 -0.116114 -0, 163216 -0.222950 0.302607 - 0.424160 1.0 - 0. 002052 -0.008308 - 0. 019064 - 0.034847 - 0.056481 -0.085206 -0.123124 - 0.174245 -0.249113 -0.392452 10.0 0 -0.000002 -0.000005 -0. 0000010 -0.000016 -0.000023 - 0. 000031 -0.000041 -0.000052 -0.000064 In order to exhibit more clearly the dependence of A f on a/b and g h/a with c/b as a parameter, we rearrange the data in Table V and the results are presented in Table VI. Table VI: Dependence of Af on a/b and h/a with c/b as a parameter c/b = 0.1 h/a 0 0.2 0.5 1.0 10.0 a al u 0.1 0.2 0.5 - 1.0 -0.006304 - 0.006122 - 0.005108 -0 003280 0.006299 -0.006104 - 0.005051 -0.003239 0. 006273 - 0.006013 -0. 004761 -0.003002 -0 006184 -0.005704 -0.003854 -0., 002052 -0. 001608 - 0.000260 -0.000009 0 15

011323-1-'F h/a ab 0.1 0.2 0.5 1.0 h/a a/b 0.1 0.2 0.5 1.0 0 - 0.025226 -0. 024498 -0.020449 -0.01370 0 -0.056875 - 0. 055221 -0.046090 -0.029826 0.2 -0.025209 -0.024430 - 0.020227 -0.013009 0.2 v - 0.056834 - 0.055073 - 0.045611 -0.029474 c/b = 0.2 0.5 -0. 024107 - 0.024075 -0.019095 - 0. 012077 c/b 0.3 0.5 -0.056620 -0.054308 -0.043160 - 0.027435 1.0 -0.024758 -0.022867 -0.015526 -0.008308 1.0 -0.055864 - 0.051689 -0.035347 -0.019064 10.0 - 0.006527 - 0.001050 -0.000039 -0.000002 10.0 -0. 015053 -0.002402 -0.000090 - 0.000005 h/a a/b 0.1 0.2 0.5 1.0 0 -0. 101681 -0.098652 - 0.102215 -0.053519 0.2 -0.010612 -0.098402 - 0.081411 -0.052922 c/b. 0.4 0.5 - 0.101248 - 0.097105 - 0.077278 - 0.049449 1.0 -0.099963 - 0.092651 -0.063892 -0.034847 10.0 - 0.027694 -0.004369 -0.000157 -0.000010 16

011323-1-F h/a a/b 0.1 0.2 0.5 1.0 h/a a/b 0.1 0.2 0.5 1.0 0 -0.160815 -0.155762 - 0.129850 -0.084674 0 - 0.236965 - 0.228730 -0.188071 -0.124954 0.2 - 0.160710 -0.155390 -0.128081 -0.083804 0.2 - 0.236817 - 0.228312 - 0.186531 -0.122825 c/b 0.5 0.5 -0. 160164 -0.153456 -0. 122038 - 0.078703 c/b = 0.6 0.5 - 0.236043 -0.225518 -0.178504 -0.116114 1.0 - 0.158232. 0.146788 -0.112054 -0.056481 1.0 - 0.233307 - 0.216205 -0.151214 -0.0825206 10.0 - 0. 045196 -0.007033 -0.000246 -0.000016 10.0 -0.068584 - 0.010505 -0.000358 -0.000023 h/a a/b 0.1 0.2 0.5 1.0 0 -0.336033 -0.322095 - 0.260402 -0.172488 0.2 -0.335825 -0.321395 - 0.258529 -0.171177 c/b = 0.7 0.5 -0.334741 -0.327753 - 0.248645 - 0.163216 1.0 - 0.330909 - 0.305129 - 0.213683 -0. 123124 10.0 -0.099229 - 0.014937 - 0.000492 -0.000031 17

011323-1-F h/a a/b\ 0.1 0.2 0.5 1.0 h/a a/b 0.1 0.2 0.5 1.0 h/a a/b 0.1 0.2 0.5 1.0 0 - 0.471739 - 0.445430 - 0.349678 - 0.232516 0 -0.682735 -0.621240 -0.463858 - 0,310001 0 -1.086452 - 0.888068 -0.619712 -0.425458 c/b = 0.8 0.2 0.5 - 0.471435 -0.469845 -0.444473 -0.439547 -0.347602 -0.336396 -0.231206 -0.222950 c/b 0.9 0.2 0.5 - 0.682222 - 0.679538 -0.619943 -0.613142 -0.461993 - 04513586 -0.309046 -0.302607 c/b = 0.99 0.2 0.5 -1.075591 -1.070944 -0.887282 -0.882578 - 0.618379 - 0.617079 -0.424919 -0. 424160 1.0 -0.464232 - 0.422383 -0.293939 -0.174245 1.0 - 0.670094 -0.589128 -0.463516 -0.249113 1.0 -1.052974 - 0.856996 -0.595450 -0.392452 10.0 -0. 138783 -0. 020530 -0.000650 -0.000041 10.0 - 0.190456 -0.027548 -0. 000836 -0.000062 10.0 -0.250084 -0.035365 -0. 001023 -0.000064 18

011323-1-F In Figs. 3-1 through 3-4, we present Af as functions of C/b with a/b and g b/a as parameters. In Figs. 3-5 through 3-8, Af are shown as functions of a/b g with C and h/a as parameters; in Figs. 3-9 through 3-12, Af are shown as functions of a/b for each given h/a The charge distribution a(0) on the circular cylinder is numerically obtained for cases of = 0.1, 0.2, 0.5 and 1. 0; b < A with - = 0 by Fourier expansion in 0 (see Fig. 1 for 0). o(0) is anti-symmetric about 0 = 0 in the 0-domain, 0 < 0 < 2f in a general case for H 4 0, and in the domain -7r < 0 < r in a special case for H=0. For the cases of H u 0, with which we are concerned in this report, it is easily seen that the terms of even integer in Eq. (17) vanish, so that a(0) can be written more explicitly in the form a(o) = s A in (2k-1) o] kul The Fourier coefficients for the cases mentioned above are listed in Table VIL The charge distribution on a circular cylinder within two parallel plates of (2) infinite width was treated by Latham. It is observed that, due to the employment of normalization different from ours, the Fourier coefficient in our work, An, is related to that in the work of Latham, Cn, by IA a/b) C A for n A for - 10, H = 0, - 0. 5 was computed and the result is as shown below: n b b A - -3.488, A3 =+0. 000206. In terms of C, when the difference in 0 is taken into account, we get C=1. 252, C 0.00006 indicating that, although the leading term agrees fairly closely, the convergence of Fourier expansion for the case of finite widths a 10 is much more rapid than that for the case of infinite width. 19

-0. 1 am 0. 3 - 0.4 Af g -o0. 5 -00.6 - 0.7 -00.81 0.2 0.~v4 0.6 0.8 1.0 c/b 20

.10. 1.0.2 600~ 3 -.0.4 -0.5 1.0O 0 00 9 O* 2 0.4 b Oe 6 21

0 m 0. 1 so 0. 2 fm 0. 3 so 0. 4 Af go 0. 7 -60.8 - 40.9 0 0.2 0.4 0.6 c/b 22 0.8 1.0

0 gf 1.0 c/b 23

-0.01 Af 1o0. 02 -0.03 0 0.2 0.4 0.6 0.8 1.0 a/b 24

-0.05 0-0.1 g, -0.15j mw 0. 0 0.2 0.4 ab 0.6 0.8 1.0 25

0.I -0. 1 - 0. 2 -0.3 -0.4 Af g -o0. 5 so 0. 6 - 0. 7 - 0. 8 -0.9 L 0 0.2 -0.4 0.6 a/b 0.8 1. 0 26

0.0( -101. 0 - 0.2 %00. 3 -. 0. 4i gf -o0. 5 df 0.6 -00. 7 Ono0.8 - 0. 9S I 0 0.2 0. 4 0.6 0.8 1.0 a/b 27

0.0 Af 8 0 0.2 0.4 A 0.6 0.8 1.0 28

0. -0.1 - 0.3 -0. Afg g a -0. -0. 7 Fig. 3-10: Af vs. a/b for various c/b for the case of H = 0. -0. 0 0.2 0.4 0.6 0.8 1.0 a/b 29

0.0 oft Oll -0.2 so 0. 3 am 0. 4 gf so0. 5 -w0.6 - 0. 7 -0. 8 -.0.9 1 L 0 0.2 0.4 a 0.6 0.8 1.0 30

0. 0 *0 ol 02 j "m 0, 04 a* 0* I - 0, no Wu ,4f 9 -w 0. 12 - 1,06 Pt& 31-12- 4 f Va. 10 a/b for 9 various, O/b fOr the case of 11 a Io, 0 me ol.. 0. m 0) 0 d we 0. 0 Oo 2 0A 0. 6 a/b 0.8 1,6 o 31

1I 1.4O -&-x Figures 3-13 through 3-15 show the variation of charge densities on circular cylinders for the cases mentioned previously. 0 indicated in each figure corresponds to O in the work of Latham. It is to be noted that, due to the normalization V in the computation, the actual charge density is equal to -- times the values 2 given in these figures. 0 Table VII: Fourier coefficients H=O.0 A e/~ A1 0. 1 - 0. 0172 * A for charge density on the circular cylinder with a/bx 0.1 A3 A5 A7 a/b a 0.2 A c/b 0.1 0.2 A c/b\ 0.2 0.5 0.99 A1 - 0.0418 -0.0427 A1 -0. 140 -0.168 -0.452 A3 A5 A7 +0.0016 a/b u 0.5 A3 +0.006 +0. 077 a/b 1.0 A7 A5 A7 -0.01 A c/b0.1 0.2 0.4 0.6 0.8 0.9 A1 -0.31 -0.318 -0.354 -0.438 -0.667 -1.03 A3 +0.0018 +0. 015 +0.1 +0.32 A5 -0.01 -0.085 A9 A11 A13 A15 +0. 022 +0. 029 -0.011 +0. 003 32

- 0.,06 -0.05.0.04 -0.03 -0.02 M0. 01 0 0 00 300 60 90 6 900 600 300 00 Fig. 3-13: aQ(G) for'.0 * 1. -0.1 and0. 2 -.0 1 0.2, -0o. bb b b a 33

too. 6;.-0.4 c~ -0.2 -0.12 00 3Q0 600 900 6 900 600 300 00 Fig. 34t4: 'a(9) for O,* 05 -a, 2.02, 0. 5,v 0. 99,0 0. b ~ba 34

we1L so1. 4 -0.8 -e0.4 o -0.22 /b 00 3Q0 600 90 09 900 600 300 000O a h. Fig. 3-15: o(O) for -1. 0.0. 2, 0. 4. 0. 6, 0. 8, aO b a 35

, j-.L-r IV CONCLUSIONS Based on the numerical results presented in Section Im, the following qualitative observations can be made for the effect of the presence of a circular cylinder within two parallel plates of finite width on the impedance of the system and charge density of the circular cylinder. It is to be noted that the charge density on the cylinder is computed only for the case of - = H = 0. a (a) The presence of a circular cylinder, for all - < oD, decreases the a impedance of the system from that of the system without the cylinder. (b) With the position of the cylinder fixed, as its radius increases, the impedance of the system decreases somewhat in an exponential fashion. (c) For all radii of the cylinders within the limit 0 <( < 1 and for all -a considered in this report the lowering of impedance of the system due to the presence of the cylinder is maximum at h = 0. a (d) For the fixed radius of the cylinder, the lowering effect of impedance by a the cylinder becomes less as a increases. (e) For all - (0 < - < -) considered, a(O) increases with -. b b - b b (f) a () is always maximum at 0 = 90. (g) With a fixed, as - increases to 1- 6, where 6> 0 are small, a(0) increases sharply as 0 approaches 90~. 36

011311-1-F V REFERENCES (1) C-M. Chu, S. K. Cho, Sensor and Simulation Note 161, "Field Distribution for Parallel Plate Transmission Line of Finite Width in Proximity to a Conducting Plane, " November 1972. (2) R. W. Latham. Sensor and Simulation Note 55, "Interaction Between a Cylindrical Test Body and a Parallel Plate Simulator," May 1968. (3) T. L. Brown and K. D. Granzow, Sensor and Simulation Note 52, "A parameter study of two Parallel Plate Transmission Line Simulators of EMP Sensor and Simulation Note 21, 1 April 1968. 37

011323-1-F APPENDIX A Charge Distribution on the Cylinder In this Appendix, the algebraic steps leading from Eq. (15) to Eq. (16), concerning the charge density on the cylinder, are outlined. In Eq. (15) 8r a +i a r R dx'a(x') 1 + I red ( -xI+ib ie c re " If we let r-)c, each term in the integrand of (A-l) may be written as: ie, (A-2) r —c io re r-x' +ib x'-ib-ce1 ie ie Lim -ei e 19 (A-3) re -x'-ib x'+ib-ce i - (x' ib) -io Lim e, c, 1 e_____ (A re + (A-4) r —+ o 2 -io (A-to rio c (x' -ib)-ce18 c (x'-ib)-ce1 XI —ib t e i E (xib) -ib Lim - e c e (A-5) i8o 2 x' +ib) -woe (x' + ib)-oe - re x + ib Introducing these into Eq. (A-l) yields 7 38

21E F ie h-a0 LXI+lb -eoe xi..lb - ce- x' -lb-ce _Ox'+ ib - c ij (A-6) Equation (A-6) reveals that 8r N rxc Introducin the normalized quantities,, X'=as+h= a(s+H) B lI b a (s+H+1-i) and C a into (As.6), we have:rue cr(s) do B 9B i 2ir Ls+H+l —_Cei S+H#*i BCe i L 2 2 ie ei Be 1 B L(A-7) s+H*-iliCe s+H+iB".CeJ Using the relation B qU s C B'e (A-B) s+H+i ZieCe nul H - 2 39

we may reduce Eq. (A-7) to ifnl 1 1 L B() a cn-1 " sinn90 (A-9) n (s+H+i))n This is Eq. (16) in the text. 40