Sensor and Simula-ion Notes Note November 1972 FIELD DISTRIBUTION FOR PARALLEL PLATE TRANSMISSIOC LINE OF FINITE WID)TH IN I{fLOXIMITY TO A CONDUCTING PLi NAE Soo) K. C, I);n Cn (i.Lo-I Ch ii:e' University of lMiciigan adil(iaion Laboratory Department of Electrical and Computer Engineering Ann Arbor, Michigan 48105 Abstract A parametric study of a parallel plate transmission line of finite width near a perfectly conducting plane ground is carried out by a numerical method for various ground proximities and the plate widths. In particular, the impedance factor f. of the transmission line system and relative electric field in some re-ion surrounding the transmission line system are computed. 11323-1-T = RL-2219

011323-1-T THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Radiation Laboratory Field Distribution for Parallel Plate Transmission Line of Finite Width By Soon K. Cho Chiao-Min Chu November 1972 Technical Report No. 011323-1-T Contract No. F29601-72-C-0087 Purchase Order No. DC-SC-72-09 Prepared for: The Dikewood Corporation 1009 Bradbury Drive, S. E. Albuquerque, New Mexico 87106 Ann Arbor, Michigan 48105

011323-1-T ABSTRACT A parametric study of a parallel plate transmission line of finite width near a perfectly conducting plane ground is carried out by a numerical method for various ground proximities and the plate widths. In particular, the impedance factor f of the transmission line system and relative g - electric field in some region surrounding the transmission line system are computed. i

0. 23 — T Table of Contents Abstract i List of Graphs iii Graphs of Relative Fields iv Graphs for Comparison of Relative Fields vii I Introduction 1 II Mathematical Formulation 4 III Impedance Factor and Electric Field Intensities 21 3.1 Relative Fields 23 3.2 Comparison of Relative Field Variations 91 IV Conclusion 109 References 110 Appendix: Approximate formulas for the complex 111 potential function and relative electric field intensity ii

011323-1-T List of Graphs List No. Captions Figure No. Page No. 1Geometry of a parallel plate trans- 1-1 1 mission line of finite width near a plane ground in (x, y) coordinate system. 8 Y8 to III Gcoeometry of parallel plate transmission 2-1 5 line near a plane ground in (~, r) coordinate system. 3 Variation of the impedance factor f as 2-2 12 d a g a function of - for different M. -= 1. 0. b b d 4 Charge density function for = 0. 5, 2-3-a 13 b a -= 1. 0 with various M. b 5 Charge density function for -= 0.5, 2-3-b 14 b a -=1.0 with M = 8. b d 6 Charge density function for = 0. 5, 2-3-c 15 =1. 0 with M = 10. b 7 Impedance factor, f, vs a/b for 3-0-a 23 d different /b 8 The difference of f as a function of 3-0-b 24 a d g d - between -=o and- = 10. b b b 9 Geometry of a parallel plate trans- 3-0-c 25 mission line of finite width near a plane ground in (x, y) coordinate system for the relative fields. iii

0 ' 12 2-1':- - T Graphs of Relative Fields Remarks: I in II in v x r Cepresents Y - b b represents = b b 0, 0.2, 0.4, 0.G6, 0.8, 1.2, 1.4; 1 and 1. List No. Range of - Parameters Field Components Figure No. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 I II II II I II I I I I I I I I I II a b 0.1 0.1 0. 1 0. 1 0.1 0. 1 0. 1 0.1 0. 1 0. 1 0.1 0. 1 0. 1 0. 1 0. 1 0. 1 d b 0.1 0.1 0.1 0. 1 0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 Eyrel E yrel E xrel E y xrel E yrel yrel Exrel E xrel yrel yrel xrel xrel E yrel yrel E xrel xrel 3-1-a 3-1-b 3-1-c 3-1-d 3-2-a 3-2-b 3-2-c 3-2-d 3-3-a 3-3-b 3-3-c 3-3-d 3-4-a 3-4-b 3-4-c 3-4-d Page No. 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 iv

011323-i -T Range of b List No. b 26 27 28 2,) 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 I II I II II I I II I II I II I I II I iI Parameters a d b b 0.2 0.1 0.2 0.1 0.2 0.1 0. 2 0. 1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.2 0.5 0.2 0.5 0.2 0.5 0.2 1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.5 0.1 0.5 0.1 0.5 0.1 0.5 0.1 Field Components Figure No. E yrel E yrel xrel xrcl xrel yrel E xrel E xrel yrel yrel E xrel E xrel E yrel yrel E xrel E xrel E yrel yrel E xrel E xrel E yrel yrel E xrel xrel 3-5-a 3-5-b 3-5-c 3-5-d 3-6-a 3-6-b 3-6-c 3-6-d 3-7-a 3-7-b 3-7-c 3-7-d 3-8-a 3-8-b 3-8-c 3-8-d 3-9-a 3-9-b 3-9-c 3-9-d Page No. 43 44 45 4(; 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 v

0113;23-.ID, -T List No. Rageo b 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 I II I II I la I II I II I II I II I TI1 I II I II Parameters a d b b 0. 5 0. 2 0. 5 0. 2 0.5 0. 2 0.5 0. 2 0. 5 0.5 0. 5 0.5 0. 5 0. 5 0. 5 0. 5 0. 5 1. 0 0. 5 1. 0 0. 5 1. 0 0. 5 1. 0 1. 0 0.1 1. 0 0. 1 1. 0 0.1 1. 0 0.1 1. 0 0. 2 1. 0 0. 2 1. 0 0. 2 1. 0 0. 2 Field Componnts Figure No. Page No._ Eyrei Eyrei Eyrei Eyrei Exe Ex vol1 Eyrci Eyrei Eyrei Eyrel Eyrei Eyrel Exrei 3-10-a 3-10-b 3-10-c 3 -10-d 3-11-a 3-11-b 3-11-c 3-1 1-d 3-12-a 3-12-b 3-12-c 3-1 2-d 3-13-a 3-1 3-b 3-13-c 3-13-d 3-14-a 3-14-b 3-14-c 3-14-d 63 64 65 66 67 68 69 70 71 72 73 714 75 76 77 78 79 80 81 82 vi

0i13'23 -iT Lit N T Range of I List No. Range of _ _ _ _ b1 66 67 68 69 70 71 72 73 I II I II I I II i~ Parameters a b 1.0 0. 1.0 0. 1.0 0. 1.0 0. 1.0 1. 1.0 1. 1.0 1. 1.0 1. 5 5 5 5 0 0 0 0 Field Components Figure No. d b yre yrel E yrel E xrel E xrel E yrel E yrel xrel E xrel 3-15-a 3-15-b 3-15-c 3-15-d 3-16-a 3-16-b 3-16-c 3-16-d Page No. 83 84 85 86 87 88 89 90 Graphs for Comparison of Relative Fields Y Range of - List No. b 74 75 76 77 78 79 80 81 0 0.2 0.4 0.6 0.8 1 1+ 1.2 Parameters d a b' b 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Field Components E yrel E yrel yrel yrel E yrel E yrel yrel E yrel Ficure No. 3-17-1 3-17-2 3-17-3 3-17-4 3-17-5 3-17-6 3-17-7 3-17-8 91 92 93 94 95 96 97 98 vii

011323-1-T List No. 82 83 84 85 86 87 88 89 90 91 Range of y b 1.4 1.5 0.2 0.4 0.6 0.8 1, 1 1.2 1.4 1.5 Parameters d a b-$ b' b 0.5 0.5 0.5 0. 5 0.5 0.5 0.5 0.5 0.5 0.5 Field Components Figure No. E yrel E yrel E - xrel E xrel Exrel Ex xrel Ex xrel E xrel E xrel E xrel xrel 3;17-9 3-17-10 3-18-1 3-18-2 3-18-3 3-18-4 3-18-5 3-18-6 3-18-7 3-18-8 Page No. 99 100 101 102 103 104 105 106 107 108 viii

01 I2 - -T I INTRODUCTION The ultimate goal we hope to achieve is the computation of propagation constant and field distribution for a parallel plate transmission line of finite width which is placed near a finitely conducting plane ground. See Figure 1-1 for the geometry of the transmission line system. y free space < -- 2a b parallel b/ plates d image plates \ b \ b a-j ---._ x FIG. 1-1: Geometry of a parallel plate transmission line of finite width near a plane ground in (x, y) coordinate system. SuIC1i a pioblei p)osrs a 1'Ws i(!S Of mtinlhcin:illitca lil1i1c8lii"ad rquircs ai Cxtensive analytical investigation. As a step leading to that problem, one can consider a simpler case where the plane ground is perfectly conducting. In that 1

011323-1-T case, the transmission line system supports dominantly an unattenuated TEM mode propagation and hence enables a formulation of electrostatic problems. In this report, we are concerned essentially with the field distribution in an immediate neighborhood of a parallel plate transmission line of finite width when a perfectly conducting plane ground is present to a varying degree of proximity. The cases of the ground proximity considered in terms of normalized form, d/b, are: 0.1, 0.2, 0. 5 and 1. 0; the cases of variation of the line width, a/b, are: 0. 1, 0. 2, 0. 5 and 1. 0; the range of the field points where dhe relative electric field intensities are computed are taken to be: 0 < bl< 3at+ d -b- b 0 < < 1. 5. - b - In Section II, the mathematical formulation of the problem is presented; a more detailed version will be found in Appendix A. Starting with an integral equation for the electric potential for the transmission line system of the geometry as shown in Fig. 2-1, the electric charge density function is found by (3) means of numerical method following the Kammler's procedure. Based on the charge density function computed, then the impedance factor, f, and electric g field intensity are derived. The latter is derived through the complex potential function. In Section III, we present in plotthe numerical results of relative electric field intensity for some representative cases of ground proximity and transmission line width, in a different cartesian coordinate system (cf Fig. 3-0-c) employed in (4) our previous work. In this coordinate system, the edges of the parallel plates facing the ground are on the y-axis for all different ground proximities. It is to be noted that this coordinate system is introduced solely for the graphical presentation of relative field strengths and is not to be confused with the coordinate system shown in Fig. 1-1. *See Section 3. 1 to see the meaning of absolute signs for Ixl and lal. 2

Based on the study of the numerical results shown in Section HI, conclusion is drawn in Section IV on the effect of the presence of a perfectly conducting plane ground upon the field intensity. 3

J1. 1,5 -.L-'1' MATHEMATICAL FORMULATION For a parallel plate transmission line of finite width in the proximity of a perfectly conducting plane ground (cf Fig. 1-1), the complex potential at any point (x,y) in the rectangular coordinate system may be expressed in the integral form 2a+d (x+iy - x +ib)(xiy+x'- ib)7 V (x, y) + i V (x, y) = 2 0 j dx' o(x') Ln (x+iy-x'-ib)(x+iy+x+ib) (1) where o(x') represents the electric charge density function of the upper plate. o(x') is an unknown function and we will find it by a numerical method with a known constant potential of the upper plate, V0. For that purpose, it is convenient to introduce the following change of variables: x' - a- d s =-, (2) a x-a-d (3) (3) a n =, (4) a and D 2 a + d (5) B =2 -. (6) a In the (C, r) coordinate system, the geometry of the transmission line system shown in Fig. 1-1 takes a form as shown in Fig. 2-1. 4

plane ground plate 1 1 r \ D 0)/ U) ( ---O0)-O.._,2'0 - B (lower plate 2 FIG. 2-1: Geometry of parallel plate transmission line near a plane ground in (g, n) coordinate system. The geometric parameters involved in the problem is then represented by D and B. For practical purposes, however, it is convenient to compute electric field intensity in a coordinate system normalized with respect to b, x v,(4) ( - ),,conforming to our previous work b' b 5

0112 3-1-T In this normalization a 2 b B (7) I and d D-2 b B (8) d a Conversion of - and a to D and B b b nd values of our interest for - and b b is presented in Table 1 for the set of TABLE 1 D and B Chart D=2a+d, B=2b a a a/b d/b 0. 1 0.2 0.5 1.0 0. 1 4 6 12 22 0.2 3 4 7 12 0.5 2.4 2.8 4 6 1.0 2.2 2.4 3 4 In the (5, r]) coordinate system, Eq. (1) becomes V(5,~) i~p~ r)= 2- r ds u(s) Ln 0 (g+ir-s+iB) (g+irts+D) (C+in-s) ( +irts+D+iB) (9) If we introduce, mainly for notational convenience, z=e+iin, (10) G

011323-1-T then d a s sLn(z-s+iB)(z+s+D) -a V(z)+i(z) = 2ds (s) Ln )(+I)+ ) (9-a) 2wE t (z-s) (z+s+D+iB) -1 where V(z) and V(z) are real functions of A,. Now, a point on the upper plate in ([, n) coordinate system is ([, 0), -1 < < 1, as seen in Fig. 2-1. Thus, by letting rp = 0 in Eq. (9) and denoting V(, O) by VO, we obtain 1 a V= i Ads (s)G (,s); -1 < <, (11) V 27o E where G(5, s)=Ln 1 +s+d s) +B2. (12) - s|y+ s+d) + B Solution of the integral equation of the type described by Eq. (11) has (3) been investigated by Kammler. In spite of singularities of a(s) at s =+ 1 due to edge effect, reasonably accurate results may be obtained by representing o(s) in terms of a series of piece-wise linear equations in s. Thus, let us divide the interval -1 s < 1 into 2M segments by a set of 2M+1 points: s, j = 1, 2, 3..., 2M+1. In each of the segments, (s., s ) j. j j+1 j = 1, 2,..., 2M, we approximate the charge variation by a linear equation, i. e., 7 - T. a(S)= j. + 1 (-); s. <s < j+ (13) 3 Sj+1 S. J sj+ - j- - 3+1 where T.'s are unknown constants yet to be determined. Substituting Eq. (13) into Eq. (11), one obtains 7

U1 12'j-1-T 2M - -V = ds.+1 l - (s- s.) G(, s). (14) a j = 1i J ji Next, let us evaluate the integral on the right-hand side of Eq. (14) at a set of (2N+1) discrete points of s: (k1) = 1, 2,..., 2N+1, where N M. Formally, then, we obtain a set of 2M+1 linear equations for T.: 2M+1 aV0 = Aj, k. a VO 2+1\ Aj (15) a jkj where =- ~., 2 M+1 j,k s j+lj j+1 k jk) k j+1, k j k s+-s (gj j-l j-1 (k -l (16) f S, = (E - s ) L-, k * 1 Jk kk j +(k + s' + D) Ln Ik + D (k ( + s. +D)2 +B2 +B an -l()3 + tan ( + D)] (17) \"k + s + 8

2 - 3 j 1z 2 -. -T and 2 2 2-. (P +D) - s. | +s.+D k j Ln j Jk 2 L 2/( s) + 2 2 Lu +3 tan K+D)tan -+sj+D>) +B k j k + k\j (18) For further details for derivation of f. and g. see Appendix A. To determine {(}in Eq. (15), we use the least square method with M j N. Thus, we set 2N+T1 /l 2M+1 7re 2 Aj = Minimum. (19) k = 1 j 1 Whence, one obtains for each I = 1, 2, 3,...., 2M+1, 2M+1 2N+1 2N+1 jk Ak = -- V0 Ak * (20) j =1 - 1 k = 1 Note that, for I = 1, 2,...., 2M+1, Eq. (20) can be written in the matrix form LA= [A] [I0] (21) where [A] [A ]; j = 1,2,..., 2M+1, k = 1.2,..,2N+1, [A T = Transpose of [A], 9

011323-I -.T [ ] = [ T = Column matrix of 1 x (2M+1) and [Icl = 1 x (2N+1) Column matrix of element 1. [A] [AT] is a (2M+1) x (2M+1) square matrix and Eq. (21) is our (2M+1) linear equation for ( j = 1, 2,..., 2M+1. Without loss of generality, we may set - V0 = 1 in Eq. (21) for computational convenience. It is pointed out by Kammler( that accurate results may be obtained if the interval -1 < s < 1 is sinusoidally divided into segments by s. sin (-M-1) M; j = 2,...,2M+l (22) and sin [(-N-); k = 12,...,2N+1. (23) We let N = 2 M. In principle, the greater the value of M one chooses, the higher the degree of accuracy one attains for charge density function represented by r in Eq. (21), but the greater the cost incurred in computation. Therefore, one would desire to find an optimum upper limit of M for a given degree of accuracy one is to maintain for charge density function. In our present work, we used the impedance factor, f, as a criterion in obtaining such an optimum upper limit of M: the numerical value of f for the case where the ground 0o is not present (i.e., the ground is removed infinitely far away from the transmission line) for a given value of a/b is compared with the value of f., for the same a/b and a sufficiently large d/b, by taking a series of monotonically increasing values for M. It turned out that, for a/b = 1 and d/b = 1, 000, the impedance factor, f, is 0.47244 for M = 10, which compares favorably with 10

e] 4_2- 1 (2) f = 0.46264( without the ground presence.* In Table 2 we present the comg puted numerical values of f for different d/b and M; in Fig. 2-2, the corresponding plots, including f for the case where there is no ground present. M = 10 g is chosen in our present work. Figures 2-3-a through 2-3-c show electric charge density for different M. TABLE 2 f for different d/b as a function of M with a/b = 1.0 g Variation of f g d/b = oo = 0.47264 for a/b = 1.0. d/b M 1 2 3 4 5 6 8 10 0. 1 0.30548 0.29669 0.29753 0.298455 0.299042 0.299414 0.299830 0.30041 0.2 0.3332 0.33442 0.33631 0.337284 0.337811 0.338122 0.338454 0.338617 1.0 0.41811 0.42642 0.42852 0.42936 0.429770 0.430022 0.430239 0.430352 The concept of the impedance factor, denoted by f, was introduced (1) g by Baum,) which is defined as the normalized impedance of the transAn extensive tabulation is compiled in Ref. (2) for f for various b/a for the case where the ground is not present. 11

011323-1-T 0.5 0.4 f 03 0.3 0.2 1 0.01 0.1 1.0 10 100 d b 1000 FIG. 2-2: Variation of the impedance factor f as a function of b for different M. - = 1.0. b 12

0112-I-, -) 3 2 T 1 0 M = 5 M = 3 M = 1 -1 0 1 S.1f = (sn -1-M) 7 d a= FIG. 2-3a: Charge density function for b = 0.5, 5 = 1.0 with various M. 13

011323-1-T 4 3 1 24 I 1 II 711 I - 1 1 -1 0 1 7f s. = sin (j-i-M) 2 2M d a FIG. 2-3b: Charge density function for = 0.5, = 1.0 with M = 8. 14

011323- 1-T 6 4 -FIG. 2-3c: Charore density function for Id. a b 'btA / 0 I - 1 0 s = sin (j-1-M) 7 n 2 15

011323-1-T mission line with respect to the wave impedance. In our present case, then, if L and C denote the inductance and the capacitance per unit length of the line, and IU and c the free space permeability and permitivity, Z =;, Z - representing the impedances of the line and the free-space respectively, we have Z o Zo C (24) Since C = - -, where q and V denote the total charge on the upper AV plate and the potential difference between the upper and the lower plates, respectively, and 2a+d 1 q = o(x) dx = a j o (s) ds -1 2M j=1 2M a = j = 1 ft d +l j (s-s S s j+1 j ( + j)( Sj+l-), ji j j+1 16

( -". - 2 - -'rl' A V = O, we obtain for the geometric factor 7rEO ' 2 f = I. g v 1 2M (T + T (S.+1-S) j=l (25) Equation (25) is used for numerical computation of f for various geometries g of the transmission line. We now derive the expression for the electric field intensity at a point in the z-plane. From Eqs. (9-a) and (13), after a little algebraic manipulation, we obtain the following: V (z) + VI (z)=a 2zc [(z, Sj+i) 2M r s - s j - T+1 Sj) j= j+l5 ( z j+l Q j P (z, s )] - %tt. ) [Q (z, s + )- Q(z S)]j (26) where P (z, s.) = (z + s. +D) Ln(z+s. + D) J J J + (z - s.) Ln (z - sj) - (z - s + i B) Ln (z - s.+ i B) - (z + s +D + i B) Ln(z+ s.+ D + i B) 3 J (27) 17

01 i32 03-1-T 2 2 (z+D) - s. and Q(z, s.) - -,n (z+s. + s )) j 2 j 3 2 (z + i B) -s. J Ln (z - s. + i B) 2 J 2 2 z -S. + + 2 — Ln (z- s) 2 2 (s + D + i B)2 - s.2 + Ln (z + s. + D + i B) 2 3 BD +i ( -s.B). (28) 2 J Now the electric field components E and E are related to the comx y plex potential functions V (z) and / (z) in V(z) + iip (z) as E V( V(z) (29) x ax a a E =- (z) a (z) S (z) (30) y ay ax a Hence, by Eqs. (26) through (30), we obtain the electric field intensity a for V = - 0 2rc0 2 M+ 1 - Ex (z) + i E (z) 2 (z) (31) x and y, j=1 E (z), E (z) being real functions of x and y, x y 18

011323-1-T where R. (z) Js [P j+ (z) - PI (z)] - [Qj(z) - QJ (z)jj + - sj 1 f2M Q! W - S. [P! (Z) Pj-l (z)]} P (32) jl if j - r 6. r (33),r t0 if r r. (z+s.+D) (z - s.) (34) and Q( (z) = Q( (z s.) = - (z + D) Ln (z + s. + D) J - (z + i B) Ln(z - S. + i B) J + z Ln (z - s.) +(z + D + i B) Ln(z+ s. + D + i B). (35) (4) For convenience and to conform to our previous work, we shall normalize the field intensity with respect to the constant field that exists between the upper and the lower plates of infinite extent of the line; i. e., 2a b. Thus, from Eq. (31), the relative field intensity is given by 0 19

011323-1-T 2M+l -E + iE = xrel yrel a j =1 B T. R'. (z) - 3 ] 2 2M+l I T. '. (z) j = j =1 (36) Equation (36) is used for computation of relative field intensity at a set of the field points in a surroundino area of the transmission line for different ground proximity and line width. 20

011323-1-T III IMPEDANCE FACTOR AND ELECTRIC FIELD INTENSITIES For the geometry of the system shown in Fig. 1-1, impedance factor d f is compared for various ground proximity d and is presented in Fig. d 3-0-a. In Fig. 3-0-b, the difference of f for the cases of -= 10 and oo g b is presented. For reference we also present in Table 3-1 the impedance a d factor f for - 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0 and 10.0 g b'b are presented. As mentioned in the Introduction, for the graphical presentation of relative fields, we adopt a cartesian coordinate system different from that shown in Fig. 1-1 (see Fig. 3-0-c). The coordinate system shown in Fig. 3-0-c (4) is the same as the one employed in our previous work) for graphical presentation of relative fields and the adaptation of the same coordinate system for our present case will enable us to compare field variations for finite and semiinfinite transmission lines. It should be noted that the coordinate system shown in Fig. 3-0-c is valid only in connection with the graphs of relative fields. In this new coordinate system, then, the edge of the upper plate facing the ground falls on the y-axis regardless of the ground proximity and the positive end of the x-axis in the graphs Figs. 3-1-a through 3-18-8 corresponds to the position of the ground plane. The introduction of this coordinate system for field variations is advantageous in that one is easily able to locate the x-coordin:lat of bo1()t cI of thle ulppelr plate: tho x-coordinate of the upper plate fla'cing tlue grounll is on tllo 0 () line (i.o., -:xis):and thl:t of thec olter t(led is on b b x 2a the -- line. b s In section 3. 1, variations of relative electric field intensity are presented graphically for 16 representative cases corresponding to permutation 21

(1 I i.2 - 1 ' of the cases i- d-0 1 0. 2, 0. 5 and 1. 0. The region of the field points for b b Li...b.Vd all cases is limited to 0 < < 3a+d 0 < b < 1 5. The mode of the presenb- b ' -b - tation is the same as that in our previous work where the width of the parallel (4) plates was infinite (c. f. section 3 in our previous work (). We mention that the numerical computation for relative electric field intensity was carried out for wider range of variations, but because of the time and also the b b fact that the examination of the data revealed no significant information which is not contained in the above cases, we chose to present in this report only 16 cases. In section 3. 2, we compare the field variations between the case of =0. 5 and co for the ground proximity =0. 5 3.1 Relative Electric Field Intensities, Ey E yrel' xrel As previously noted, for the field points, the b interval is taken as 0 < l<3+ and the y interval as 0 < b< 1.5 by the steps of = 0.2. - b a b - b- b The upper plate lies on the b = 1 line. The variation of relative fields on the b inner and the outer surfaces of the upper plate, which we denote by = 1 —, 1+, respectively, are also presented. The dotted and solid lines in the graphs in Figs. 3-1-a through 3-16-d represent the positive and the negative values, respectively. 22

r,..23-1-T 1 0'... '. F I Im 5-. - 1, -- i i, 0.79 - i l 0.36 * f, vs /b for different d/b i $'' 0.2 — x <P o% *1. 1 ~._ 0 2 8 10 23

011323-1-T.-3 2.Ox 103 1.5 - 1.0 Af g 0.5 0 2 4 6 8 10 a b FIG. 3-0-b: The difference of b 10. b a d f as a function of between =oo and g b b 24

, - -., ' ^. - y ---- 22a parallel plates - x plane ground Fig. 3-0-c: Geometry of a parallel plate transmission line of finite width near a plane ground in (x, y) coordinate system for the relative fields. 25

Table 3-1: Impedance Factor f Vb 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 10.0 d/b 0.01.6393.5257.4145.3531.3048.2510.1958.1542.1064.0686 0.02.78131.6391.4944.4136.3506.2822.2317.1785.1111.07007 0.05 1.009.8323.6380.5232.4324.3354.2671.1991.1187.0728 0.1 1.2017 1.0069.7781.6340.5156.3876.3000.2172.1250.0751 0.2 1.4031 1.1962.9406.7685.6190.4515.3386.2372.1316.0774 0.5 1.6557 1.4399 1.1618.9610.7726.5463.3940.2645.1399.0803 1.0 1.7982 1.5791 1.2919 1.07811.8698.6085.4304.2822.1451.0821 2.0 1.8718 1.6514 1.3607 1.1418.9249.6467.4543.2946.1490.08340 5.0 1.9007 1.6801 1.3885 1.1683.9491.6657.4680.3029.1521.0846 10.0 1.9053 1.6847 1.3931 1.1728.9534.6695.4711.3051.1532.0851

14 1, / 10.0 / E / -.24 0. 6 0. 4 0. 2 -0.2 -0.1 x b Fig,. 3-1-a: E on X O 0.2, 0. 4, 0. 6, 0. 8, 1. 2, 1. 4and1. 5 yrel b lines for the case ofdI = 0.1 d.1 b 'b 27

01 13 3-1 - 12. 0 10. 0 0 Eyrel 0 4. 0,2. 0 - 0. 3 - 0. 2 -0.1i 0 x b 0. 1 Fig. 3-1-b: E on 1-=l and 1 lines for the case yrcl, b - + -= 0. 1. b Of LI= 0. 1, b 28

01 13O23-1-T // xrel /111 '00.4 /1/7 -" -//"'ll/ 02 -0.3 -0.2.0. 1 00. 1 x b Figr. 3-1-c: E on 0, 0.2, 0. 4, 0.6, 0.8, 1.2, 1. 4 and xrel b. 0 1. 5 lines for the case of ' -0 1 0. 1 (E on b 'b xrel Y-= 0 line is zero). b 29

~12. 0 I ~ 0. 0 I. I E xrei 1 6. 0 ii2.. 1.30 0 1 4.01 I b Ii.31d onI ie o h aeo 'l=0 xre bI b 1d - 0.2.0 Ib 30

-III-,2-. -. 1. 2 '1. 0,8 Eyrel 6 0.4 0. 2 -02-0. 1 0 0.1I x b 0. 29 Fig. 3-2-a: E on Y- 0. 0.2,0 0.4, 0.6., 0. yrel b for the case of U 0. 1. = 0.2. b 'b 8,P 1. 2, 1. 4 and 1. 5 line s 31

011323-1-T 12. 0 -10. 0 -8.0 E yrel -6. 0 — 4. 0 -2. 0 0.2 -0.3 -0.2 -0.1 0 0.1 b Fig. 3-2-b: Eyr on b= 1 and 1 lines for the case of -= 0.1, = 0.2. yrel b - + b b 32

U "i. 1 43 2 1 -T 1. 2 1. 0 0o.8 Eixe -0. 6 '0.4.0. 2 -.0.:i x 0. I b Fig. 3-2-c: Exr on b for the case of u = 0. b 0.4, 0.6, 0.8, 1.2, 1. 4and1. 5 lines 1 = 0.2 (E on IL=0 is zero). # bxrel b 33

01132t- 1-T -12.0 -10. 0 -8.0 xrel -6.0 -4.0 -2.0 -0.3 -0.1 0 0.1 0.2 b Fig. 3-2-d: E on xrel b b = 1 and 1 lines for the case of bl= 0.1 - + = o.2. b 34

0 1 i'32 3 - 1 - T -)?-X CM T-< co CD C6 0 r-C?e-( O r-* "o I Xl. CM To * 0 /Y / /I / I 1 I I 1/ II I I! i II II II It 4 -4 0 C1) C.) 0 oo '-4 c~iJ (0 -0 0 C) / / I \.\ \ rt *i-i - - Co 1o 35

15 2 I 12.0 I I I I I4 I2 I 1'.2-.10 x 0.2030 4 8.05 I I Fig.3-31): onX 1 I ilis fo th Cas j. E I Ib-11 +o 0 iyrel

1. 2 ~1. 0 0. 8 1/00 1/..-''0.6 - - 4 0.42 0.42 /1/ 0. 0 2-0 10 0.21. 03_.40.25 0b Fig 33-: o <-I:0002 A.1...201 n.5lnsfrtecs f 0..5 0E Oi - ln-i er) xxcebl b

12. 0 10. 0 0 E xrel 6. 0 Co 0 12. 0 - 0. 3 - 0. 2 -0.1I 0 x 0.1I 0. 2 0. 3 0.4 b 0. 5 Fig. 3-3-d: 13re onl 1-=l and 1+ xrol b ~ line's for the clase0 of Lji 0. 1.d 0. 5. b 'b

011323-1 -T '1. 2 '1. 0.0. 8 E yrei. 6. 4 ~.2 x Fig. 3-4-a: yrel b for 1.~' 0. 1 d 1. 0. bIt 0. 4, 0. 6, 0. 8, 1. 2, 1. 4and1. 5 lines 39

/,-, -1 -4 e-, - -,, -. I" li,LI - -" - -: - - 0 0 0."yrel ~.0:.0..0 x b FPi(g. 3-4-b: E on 1 and 1 linos for the case of o1 1. o. C>yrcl b -' + b P b 40

01i323- I-T -1.2 -1. 0 -0.8 E x 0 I. // // I/ 11. 1..- 0.2, // I / / _ *- -- _ f/!^!! rel "0.4 0.2.0 i i, -.2 -0.2 0 0.2 0.4 0. 0. 6 0. 8 1 x b Fig. 3-4-c: Eel on = 0, 0.2, 0.4, 0.6, 0.8, 1.2, 1.4 and 1.5 lines xrel b P. for the case of = 0 = 1.0 (E ' b b xrel on - = 0 line is zero). b 41

^ I (. (- t-.4,, i -, +I 4. II. 0 060 8. 0 xE IIb Fi.34d n l nd 1 lnsfrtecseoI.11c.0 xIlbI - 0. 2 42

0113to2 3-1-T / \ I yrel V-0. 4 x.0. 2 01. b / 01,0.8 Eye 0. 6 0.4 0. -0.6 -0.5 -04 -0.3 -0.2 -0.1 0 0. 1 b Figr. 3-5-a: E on Y- 0,. 0. 2, 0. 4, 0. 6, 0. 8, 1. 2, 1. 4and1. 5 lines for the yrel, b I'1.d =.1 case of 02=-00 1. b 'b 43

01132 3-1 -T 12.0 10.0 8.0 6.0 4.0 2.0 -0.G - o. 5 - o. 4 -. 2 -0.1 0 0. 1 x b Fig. 3-5-b: Ey on b= 1 and 1 lines for yrel b - + the case of fb= d 0 0. 2, 44

011 2 3- 1 -Tr 1.2 1.0 0.8 E xrel 0.6 0.4 0.2 -0.6 b b Fig. 3-5-c: Er on b = 0, xrel b case of I - 0. 2, b 0.2, 0.4, 0.6, 0.8, 1.2, 1.4 and 1.5 lines for the d = 0.1 (E on 0 line is zero). b xrel b 45

01 192-3-1-T.12. 0.10 0 Exrl - 0. 6 x d1 Fig-. 3-5 —d: E on = 1 1 lines for the case of If= 0.2 - = 0 1 C)xrel b -.0 + b '~b 46

co 0. I1u-~C) it It 4-4 C)) LCO LO co co C) 0 LO 6t 6 &%* 0 6Z 47

->. 0o o1 Q 0 C} '-; I. ----1 ---I I.I --- —---— o \.. I. + 0 ' -.. l0 0 + 0 -1' 7II >|,Q Lo1 I 48

Exrel (0 - 0.2 -o. 1 0 0.1i 0. 2 F ig. 3 -C) —c: K on X Y O' xr.e1 b 0. 2, 0. 4, 0. 6, 0. 8, 1. 2, 1. 4 and 1. 5lines for the case ofb - Itb- 0 1 -3 (E on Y-=0 line is zero). xrcl b

la) '-4 0' r —i ' —q 0 cis C' Cs C' 0'; i i iI I --- I I i -I -- I I I I A 0 S j I / / / / I 1' 0u - +T V-6 CN ) 0 'I 6 11 0 0 cJ; 0 0 0 44-4 f) 0 xj..o * + '-4 0 0 (0 * 1 i (-0 0 50

fjl-. ') ') - -,r I - — - -,J - - I, - yrel 0. 6 0.4 /J 1. 3 -0.4 -0.2 > 0 b -0. 6 - 0.4 -0. 2 0 0. 2 0.4 0.5 x b Fig. 3-7-a: E ye on bO, 0. 2, 0. 4, 0. 6, 0.8, 12.2, 1. 4and1. 5 lines for the case of L 0.2 0. 5. b.0'b 51

01I1323-1-Tr 8. 0 if ~I yrei 'I 6. 0 / 4. 0 2. 0 -0.6 60.4 -0.2 0 0. 2 0.4 0. 5 x b Figr. 3-7.-b: E on Y —1 and liefotecsofL-0. 2 - =0. 5. Oyrel b - + b ' b 52

(. j11323)-1i-Tf,1. 2 1. 0 I. 0.66 / 0.2 /1/ / - - - -0.6 -0.4 -0.2 0 0. 2 0.4 0. 5 b Fig. 3-7-c: E on Y=O 0. 2, 0. 4, 0. 6, 0. 8, 1. 2, 1. 4and1. 5 xrel b Ua 0 d v lines for the case of b02 = 0. 5 (EKre on b= line is zero). 53

I i ~. -.1 12. 0 10. 0 8. 0 E xrel 6. 0,4. 0 '2. 0 - 0. 6 - 0. 2 0 0. 2 x b 0.4 0. 5 Fig. 3-7-d: E on.- =1 -xrel b hil (I=.5 andi 1lines for the case of - = 0 2 -05 + b 'b 54

0 LO d co -* *- rh 1~ /001 co?I / lo *o; I - I 0: I 011323-1-T COa III '31 co 0 6; CI)C Q)Q LO Illy~ 0 (0 CCI // /l 7-/ 00, C, ~/ IC) ///3 / / /. -!- / cI 6 wco TL,6 // / / /0 /0 6/ // ^/ / ^ 6 y / c / / ^/0 ^ ^//</ -. 'I // //?/ ~6 ^ / //0 /y / ///-/ /^/ xi C) /^ -9 ^/^^ ":Co -^' / //I IC") k.* \ \ ~\ 4.4 Nz ~ %- \ I I I 55

0 0) 0o 0 0) 0 C-0 C\ C 0 0 C~ cIq ~0 - C) C co 0 1 + — +T 56

T-4 I II2-3)II liii' c C);I IIly C 01 C) ii~iI I I-/D1IV "IV (Z ti CD. f-4 44 C5 1 57

cco I 75 ~~~~1 I I 4'*16) +1-4 co &~ 58

1. 2 1. 0 0. 8 0. (3 0. 4 0. 2 C") CD 0-1.2- 1. 0 - 0. 8 - 0. 6 Fig. 3-9-a: E on Y 0 10. 2, 0. 4, 0. 6,0 0. 8,0 1. 2, 1. 4 yrel b the case of b. b ~0 1. and 1. 5 lines for

rqM1 -- I I I I I I I I I I I I I I I I I I I I I I I II I I I I + I I\ I I I I I I I II I I I I I I I I II I I I I1 / + I I I I I I I I I 1 i i rT I I I I I I I I I I I I I I I I I I I I, I I I I I I I.-I 12. 0 A 10o. 0 0 8. 0 6. 0 4. 0 2. 0 J yrel P 1I -1 _ — -I- - - -I I I I I I - -, I I I - - I - - - -1 I I -- I -~- - -1. 2 - 1.0 x -0. 8 b - 0. 6 0 0. 1 Yo 1 and 1 lines for the case -In 05 ci 0 Fig. 3-rG-b: E ofOfl b +5.b = 1..

CN U) co ( LO If_ 4t 000 C;~ C \ V 0 T —i:-I I t -4;L4 6 1

12. 0 110. 0 I ~I\ 8. 0 16 \ I4 6. 0 1..21. 1.80..20 0 I d Fi.39d I n ie o h aco.D.1 Iv~ IIn I - 4 E xrel ALI' 1 D -T- L) E onlY 0 is zero. xrel b

OIt Ijj -.00*.000 0 1 6 Ue C)1 -b 0 r.1 XI I m i \ 1! \j. 1 - 63

C)21.8 x 0.GI.200 1 /1 Fig. 3-10-b: E O~~ilX 1 adI If sfo th cae f 05 d 0.2 Yro..b + 1 - 12. 0 * 10. 0 * 8. 0 E yreli * 6. 0 * 4. 0 2. 0 C(7) I. 0

0(' 2) ' ---T;-(4;4 G) 0.-I cO 0 0 C0 C 8'0 i \! \ 1 \! \ \ \ \ \I \I 0 Cl 0 In CT U I n 0 1, 0 - C '-4,- 2 o CC * 0 ~, — 0 Co. '-4\ ~1) *00 bb. 65

C) 0 C C3 r —4 -— 4 CD C) C:) x i- I 1.1.0 A-I — oo...OO..-,W --.O.- am- M- --- --- -- — O -", —.. _..W -. —. - - -ft. -- a — - -I-. -.- em- W- 4-0 ""oft- am"* 4-~ a — - 4 — 0-..- Sftow am 4-4 co It >i 6 6 + T

o-O rnOI33 -4;1 CZ C1C) C~j C\) \\ \ \\\ \x 67

C)C)0113023 - 1 -4 IfI CC3 CJ2 if C) 0 rf+ T T1 4 r —il I -H I - I i v..q I co I tB.,.4;Z4 llzl T" I I i i i I - 68

69 hi cJ C) frd C) 0 Ii 0 0 0 0 0 0 q rND I. I. C) CI) 0 0 C) C-) U) 0 0 11 0 I! 0 tQ 0 0 0 0 0 0 "IN I It I' I IF I? VI I I I 0 (ii - -T-'-" 71:T T 'I, L Cs N:~

-T I - 1 — -T -I I I I I I I I 112. 0 10. 0 II I I I I I. + II I I I I III II 0 0 '-4A 'I I I I I I I I \ e \ -% . e \ \ x 1%1.. %%% I-" "... —.- --!.0 1. 0 - 0. 8 x b - 0.0C 0 0. 2) 0.4 0. o Fig. 3-11-d: E onl 1-=l and 1 xrel b + lines for' the case of UlI= (15d=.5 b b xr1e%,l o -= 0 is zero. b

0 1 13'j2 3Qj-1- T — 1. 2 J.0. 8 E yre — 0. 6 0. 4 + 0. 2 ),l.f -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 x b 0 0. 2 0.4 0. 6 0. 8 1. 0 Figr. 3-12-a: E on Y 0 0. 2 0. 4, 0.6., t> ~ yrel b 1 0. 8, 1. 2, 1. 4 and 1. 5 lTa d for the case of L =0. 5 -1. 0. b $ b 71

eJ I I;-, j, - - I 12.0 10.0 8.0 yrel *6.0 '4.0 '2.0 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 x b 1.0 Fig. 3-12-b: Ey on b = 1 and 1 yrel b - + lines for the case of _Jl= 0 5 b = 1.0. b 72

u 1L'J2J-1-T -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 x b 0 0.2 0.4 0.6 0.8 1.0 Fig. 3-12-c: rel on = 0, 0.2, 0.4, 0.6, 0.8, 1. 2 1.4 and 1.5 lines xrel b for the case of 0. 5, =1 0. (E on 0 line is zero). b b xrel b 73

U. I 1`2).J1-I1- T 112. 0 10. 0 Ii 16. 0 I4 -1.2 -.1.0 -'0.8 -0.6 -0.4 -0.2 0 0. 2 0.4 0.6 0. 8 1. 0 b b - 1.4 74

co col C) 6 6 C _ _ _ _ _ i co C) I C )0 C7 C) coCl 6's IhB / / / / / K 75

0 0 E yrel 0)j 2. 8 - 2. 4 - 2. 0 - 1.6 - 1. 2 -0 b Fig. 3-13 —b: E on 1-= and 1 lines for the case ofLA 1. 0. d 0. 1. yrel b + b 'b 0 0. 1 C~k I)

co C.) 100 1-4 1 '-4 P.% y 0 0 CI 2~N 0 co 00 0l c 0Z \ \\\ \ - I;;I 77

.12.0 i -10. 0 I I I I I 8.0 IK E I Kxrc l b 6.0 IO I 44. I I..................1.. ----h-t-k —,-I ---! ---I ---1 — -2.8 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0 0.1 Fig. 3-13-cl: E on X = 1 and 1 lines for the case of 1 0, ~=0.1. xrel b - + b 'b 1 C> C') -1

C) — co C-0 (0 Nu If -Q 07 ft 00 79

.12. 0 0 3. 0 E yreil ~.0 co ~.0 - 2. 8 - 2. 4 - 2. 0 - 1. 6 - 1. 2 - 0. 8 x b 0 0.2' Fig. 3-14-1): Fi.3-4h on Y — and 1. yrol b - + lines for the case-, Of Li- 1. 0P! 0. 2. b 'b

0 U11 0 00 co 0 - '-40 0 Nii 81

0 C-' 10 I Y-= 1 I b. + ' v Ix I~ 0 co 0 AN T ' O XI~ 0-1

1. 2 -1. 0.8 EB1o -0. 6 co 4. 2 - 2. 4 - 1. 2 - 0. 8 - 0.4 0 0. 2 0. 4 0. 5 x b 0.4, 0.6, 0.8, 1.2, 1. 4 and 1. 5 lines for the case of ILI 1 0,.0 lp p 0b. Fig. 3-15-a: E ylOnl Y= 0. 2, b d = 0. 5. b

l l..X I.. \ o0 0 00 cco i o C \\ + ~ i I c O - I I C I -, — L1.2 I co C! CM cl 84

1. 2,>~' 4;i 11If~ 11 1 co Nv -2.8 -2.4 -2..0 -1. 6 -1. 2 -0.8 -0.4 0 0. 2 0. 4 x b Fig. 3-15-c: KOi 0. 2, 0. 4, 0. 6, 0. 8, 1. 2, 1. 4and1. 5lines for the casof = 1. 0 and Fig 315c:b 'b d 0. 5 (E on Y- 0 line is zero). b xr(el b.1. 0 0. 8 E xrci 0. 6.0.4 0O. 2 0. 5

.12. 0.10. 0 0o E xrel jI 6. 0 II 4. 0 I' IK 0 0.2'. 0. 4 0. 5 - 2. 4 - 2. 0 -1. 6 - 1. 2 x b La =1. d! Exe Oil Y- 1 and 1- lines for the case of 1 0 0. 5. xrcl b +b Figr. 3-15 —d:

C (CD CS C 0 0 -0 / // 01 1323-1-T 0:C CD cif '-44 C; I IO I I 'I I \ %\\ N.. I ft-fN ";QH C I r-,-, i — I C r —4 I co t.r-j PL4 co 87

Eyrol II - I cs co 11 C;. C) 4. C) 2. 0 v-4f I' 1. -29.4 -2. -1. 6 - 1. 2 0 0. 4 0. 8 1. 0 x b E,,,on I= 1 and 1 1inec." for mce Case of -1. 0 ~-= 1. 0. yrl b + IL b - Fig. 3-16 —b: (L -) .4 C- ) 1\.) I %-, -A,I A

.1. 2 xrel OV - 1. (j 0. C C() ' 0.2r F 0. 6 I / 0.4 --- - 0. 2 1. O) - 2. 8 - 2. 0 -1. 6 -1. 2 - 0.4 0 0. 4 0. 8 x b Fig. 3-16-c: E on = O 0.2, 0.4, 0.6, 0.8, 1.2, 1. 4and1. 5 lines for the caseoof-1t= 1. 0 -I. 0 xrcdl b.0 p. 0b b (E on Y-=0 line is zero). xrcl b

fN! i 0 Cs 0 011323-1-T - I I I --- I i i I 1 — 1-% x I I I I / / / / / 1-, 1+_ ~,1- _ ---— ~ 1 I - o iI i -IC 1-k 1:1 -A I I C co - 6 1 C113 - 1 I 0 -4 To t 0 r-I - 0 ~-4 4 - 0 - )-q * if??q c - I 1- P 1 + c CS 4 '-4 -i T-(i I;L4 C3 I co CO c'...... ~.. i

2 -- -,-, -— b I' lv O 0, b() - (+) E yrel *I 0 A.J, C) *.j. I J. 1 a:) C 1. 0 - 0. 6 - 0. 4 - 0. 2 0 0. 2 0. 4 0.5I x b Fig. 3-17.-i: Comparison of 0ln o h aeo n yrel b = 0 line for the case of d 0. 5 between LI" = 0. 5 and co. b b

C)C0 / C\] 0 CO C)co 'C) I? 92

2 al y= lb I= b = 0.4 (qb - '# - - I,..,. 0,8 0 yrel C(D C0 0. -J 0.2 -1.4 -1.0 -0. 8 -0.6 - 0.2 0 2 0.5 x b Fig. 3-17-3: d lal 0. 5 nc. Comparison of Eyr, on = 0. 4 line for the case of = 0. 5 between = 0. 5 and Co. yrel b b b b —" -, co I!-A H*i

I I. C C C C C II 0 C C C o C C 0 C I' Q1 S 0 0 I 0 4-1

'-4 0 G:; ' —4 0D I --- I 1-5p - - -,-.- - I 0 — If) 0O 8 0QI 0 C)) co 6 11 IAP z lk e,\;:L I 0u I I 0 '-.4 C'I 0 0t0 I.4 0 ':-4 -'-4 '-4 95

1 2. C -1I0. C 6 I). 0 C.D 9. ( ca o 0. 5,0;:= 1 -1. 2.-0. 8 '-0. 6 0 0. 2 0. 4 0. 5 x b Fig. 3'-17 —6: Com-parison of ~ onI = 19 line for the case of -1 0. 5 betwee-n UIa 0. 5 andI co. yrel b b b

E yrel C-0 -11 I.I2). C.10. ( 18. 0 u" 0 "I. (.L 9. 0 (-I II I c') I I tat = 0. 5 Y- - 1 " b - + - 1. 2 - 1. 0 0 0. 2 0. 4 (). 5 x b Y-:4 1 lal - v. 5 and oc). Fig. 3-17-7: Comparison of r-, Oil 11110 for the, case of ' = 0. 5 betwUL1,11 IL yrel b + b b I I

cI' C L"' 0 6 o 0 Th, CMl O 8 L-J 0 L'"~ 0 0 6 6. C),-4 0 C) 0 0 d Cs e co-0 0 l Xl ',, 0 III.C; 0 0 ~ C) XI..-e 0,,-4 C: ^ 0 II 11 CO C) 0 >~ 4-1 *i-s? -( 0 Ot C.* in * —l '-4 98

11. 2 1%1- 0 Y- =. 4 (~ bb * 1. U 0. 8' 0. C Eyrci 4 Vi)f Vx, I0. 2) - 1. 2 - 1. 0 - 0. 8 - 0.4 - 0. 2 0. 2) 0. 4 0. 5 x b Fig. 3-17-9: Comparison of E yrel onl b- 0. 4 line, for the case Fig 3-7-: Cmp~rion f Ere 0. an b of -. 0. 5 between bi. bb 0. 5 and oo.

GOT I C3 -I o C 0 2 o * 0 C o 0 C-, 0 ,iZ II o 0 C-" o o 0o oI 0 0II / o / C 2 0 / / / / - 0 / - 0 / / 1 / /I 47 21. 0

lal 0. o 2 (-t) b 1: oll 0. `0, E0 O. 2 line for lc 0.2 'Is o 0 5 bet~ve.,1~ b f i, 1. 2 E: xrel - It 0. 9 i 11 1i ~; "I'

0 i i 1323-1-T, LO ' ~ '' C; 6 —C iit C) C) It C: i u 1II o -d 0 ~ '\ ~r xi.., % ~f,o I ""'~ o /i "'; ~ I -~t,.. o!X o...~ I'( I I 102

.1. 2 1. 0_.r _- -0. C) - I - A E xrel..0. 6 0..1, -0. 2 i-s.(O \ cP' I 3 - 1. 4 - 0. 6 0 0. 2 0. 4 0. 5 x b Fig. 3-18 —3: Comparison of E OnlY 0. 6 line for the case of -! 0. 5 between l= 0. 5 and co. xrcl b b, b

co; — ' o:(l<2 — C0 D -W ^011323-1-T #-4! fr( X^ - - K O 8 C) 5= I CM C. C: fI L'II! e- O 0 2 \ O o X I o ^VY ' —4* Go e\S'D, ~ i1./ \ C) / \ OX ^^I ~ ^r \ ' o ^"^ 1!,^^ ^ ~ I \ T I 104

C)) TT C) ~0 a~0 C) 6 C) ClC) 11:11 I 105

coQ l011323-1.-T I 6 '-4) 0C Ii ~0I co 11 - r —i I ti I r-4 44 1 - -I.H -14 1 1 106

I.2 (-0) -1.4 -1.2 -1.0 -0.8 -0..6 -0. 4.202 x b Fig. 3-18-7: Compari-son of E xel. onX 1. 4 line for the case of d-0. 5 between Ii0. 5 and co. xrl b b b I r(1 - 0 ). (2

cNq (I1.32)) - i-T I -fs Q; x 'N -P /:\ \ I %P. \.-I. \\ P IPA (6 \ V, IA 11\, '.N 0PJ 6 it C) 0 0 C) co CC t 711 I I 108

,'J 1 9-1 -'^U.Z ^l ^ IV CONCLUSION From the numerical parametric study of field distribution for imaged parallel-plate transmission line of finite width, the following qualitative observations are made: (a) The presence of the ground tends to decrease f compared to the a. f without the ground. As - increases, however, for all ground proximities, g b the effect of the presence of the ground on f increasingly diminishes. In a db particular, for all - it appears that, for > 1, the decrease of f from the d b b- g value of f for b = o is less than 10 percent. g b (b) The ground causes an asymmetry in the charge distribution on the plates; this asymmetry appears to be rather pronounced for - < 0. 5 and d. b increasingly less pronounced as - increases. b (c) The division of the charge between the inner and outer surface a d of the upper plate is effected both by b and b. In general, the ratio (innerd a surface charge/outer surface charge) appears to increase with both - and b b b (d) In the region 0 < < al+ 0 <b < 1. 0, the deivation of the relative b- b ' - - - field strength of finite-width case from that of a semi-infinite case tends to be d not significant for > 0. 5. b The conclusions drawn above are qualitative in nature. Based on those, however, it seems to be indicated that some approximate expansion of charge a d distribution appropriate for various range of the parameters - and - could be derived based on the solutions of simple problems such as a finite parallel plate without ground and a semi-infinite parallel plate in the presence of a ground. Such an attempt, however, has not been made. 109

0" ' i 32-i-T REFERENCES (1) Baul, CLarl E. (June, 1iMG(;), Sensor and(l Simulation Note 21, "Impe(1dance and Field Distribution for Parallel Plate Transmission Line Simulators". (2) Brown, T. and Granzow, K. (April, 19G8), Electromagnetic Pulse Sensor and Simulation Note 52, "A Parameter Study of Two Parallel Plate Transmission Line Simulators of EMP Sensor and Simulation Note 21". (3) Kammler, D. W., "Calculation of Characteristic Impedance and Coupling Coefficients for Strip Tira nsmission Lines," IEEE Transactions, MTT-16, 1968. pp. 925-937. (4) Chu, C-M., Cho, S. K., Tai, C-T. (August, 1971), Sensor and Simulation Note 137, "Proximity Effects of Semi-infinite Parallel Plate Transmission Line in the Presence of a Perfectly Conducting Ground". 110

311 323-1-T APPENDLX A: Approximate formulas for the complex potential function and relative electric field intensity The complex potential function in the z-plane z = 4 + irt, V( z) + i( z), is related to the charge density function u( s ) by the integral equation ( Eq. (10)): r(lo))... \ (L\z)+i((z)ij [ I (z)+i(z) ds c(s)Ln (z-siB) (z+s+D. (A-) a z-s) (z+s+D+iB) (A) -1 If o((s) for s in s.<s<sj+ for all j =, 2, 3,..., 2M+1 is approximated by a piece-wise linear equation of the form ( (s) = Tr.+ +1 - (s-s.), (A-2) J Si -S J j+1 j then the Eq. (A-l) becomes 2re r - o V(z)+i (z) 2M Sj+ l ( s - st)) Ln c(z-s)(z4s+D+i B)J Ln (z -in11 ) ( z.+ S -- i ) ) (Ar In carryina- out the integral on the right-hand side of Eq. (A-3), we recog-nize the 0 0 0 0 111

01 1:3J23... 1 -T following, two types of integrals invo> t P (z, s)= ds Ln- - (A-4) and Q (z s) =fas sLn (z-s+iB) (z+s+D)7l.L (z-s) (z+s+D+iB)j We know that (A-5) fs Ln (a+ib~s) = +(a+ibjs) Ln (a+ib~s) -s (A-6) and ds sLn(a+ib+s) ( a- ib) Ln (a +s +ib ) Jo 2 — 1(a+s+ib) 2 (A-7) 4 By use of the formulas (A-6) and (A.-7), we can write down P C z, s') and Q ( z, s). They are P(z, s) =(z+s+D) Ln (z+s+D) - (z-s+iB) Ln (z-s+iB) + (z-s) Ln (z-s) - (z+s+D+iB) Ln (z+s+D+iB) (A-8) and Q(z, s) = (zD)- Ln (z+s+D) 112

0 113%j23'- 1 -T (z~B) sLn (z-s+iB) 2 2 2 + Z - s Ln (z-s) 2 + (zDiY- Ln (z+s+D+iB) 2 BD + 2 Bs. We can now express Eq. (A-3) in terms of P and Q:. a Lv )+ -() (A-9) 2 M j=1j+ j 'r.sji-T-5s) I j P(, j+ ) - P( Z* s.)]j - (7T-7T 1) j+1 -11 Q(z S ) - Q(z ''S.) p j + 1 A i 0 (A-lO) Eq. (A-10) can be rearranged to yield 2M+l where V(z)+ iv'(z) = I 0 R. (z) =.61M+] 3 5j 1- S. ji X Tr.R. (z) j=1 (A-li) L. P(z, s j+1)-PZ 113

0-.j2o- -T1 Q(zj Sj j + 1 ZJ S. ); i- 6.,, P + 1 ( Z. S. ) - n ( z. Si s v-Pij-s.r- 1. -S- 1 [P(ZS)-P(Z's.i1)J 6. and (A-12) (A-13) 1 if j=r 6. = 0 if j^ r In Eq. (A-12), if we let z approach to a point on the upper plate, i.e., z-> k (note that -1 <k < 1 for all k= 1, 2, 3,..., 2N+1), then V(z) =V( k+iO) = V, which is the potential of the upper plate. The complex potential there assumes the form k 27c 0 2M+l i,. IRj(k+i0) J=1 j=l (A-14) If we denote the real part of Rj (k+iO) by Aj k' then we obtain a system of 2 M+ 1 linear equations for T.: J V = a o 2r E o 2M+1 J j, k j=l (A-15) To obtain the explicit form of Aj, k we first denote real parts of P( k+i 0, s.) j, k' k, and Q(k +iO, s ) by fj k and gj k respectively. Then, by Eq. (A-12), we obtain the explicit expression for Aj in the following form: obt j, k i 114

A -Lt2M ] 1 j, k s -s sj- sji1 j,k where f. and gj are, by I J, k ~j,k f. k k-s)Ln J' j + ( s.+D) Ln + B tan-1 B tl- (fi+lk- fjk) - (gj+l, k- gj,k) i-. 9A-8 ) a-d(fj.k f(-9), 0 Eqs. (A-8) and (A-9), (A-16) t l n __B__ + tan +-1+ Jk+ sj+ (A-17) and 2 2 -S. 2k ':'j, k 2 Ln ( +D)2 2 2 B2 r + 2 Ln -I 2 w, i +s.+D Ln 2 2 ( +S.+D) +B 2 k J 115

i 1 3 2 -1 -i j -L t^. Z *j -L JI + B tan ' - B ( k+D) tan ks k- s k (A-18) In Eqs. (A-17) and (A-18), the principal branch of the arctangent functions must be chosen. Since B is a positive number, it follows, for any A, that < tan -<. A A i.e., -lB 7T 0 <tan A < 2 for 0<A A -2 < tan < 7r for A< 0 2- A - 0 (A-19) (A-20) and In Section 2, the solution of Eq. (A-15) for T is discussed for the case of V ~ = 2e. \ r. is calculated by a numerical method. Once 7. is mkown, 00 J then we can obtain the expressions for component electric field intensities by differentiating the complex potential function Eq. (A-ll). i.e., av av E- V x ax aa3 (A-21) av _, _ E 3 y 3y ax aa& Therefore, from Eq. (A-ll), for V = we have o 27r 0 (A-22) -E (z)+iE (z)= x y 27r 0 2M+l '. R.' (z ) J J j= (A-23) 116

The explicit form of R.' (z) is the same as Eq. (A-12), except that now P( z, s.) and Q(z, s.) are replaced by 3 and Q.'(z)= Q'(.+irn)- - Q(z,s.) -(z+D) Ln(z+s.+D) - (z+iB) Ln (z-s.+iB) + (z+D+iB) Ln (z+s.+D+iB) + z Ln (z-s.) (A-25) 33 - (z);(z+iB) Ln Ezs.i) J For convenience, we normalize the field intensity with respect to the field intensity between the plates of infinite extent, which is 2 ab ' 0 Hence, by Eq. (A-23), we obtain the relative field intensity in the following form: 2M+1 -E +iE - b xrel yrel a JRj 3 j j=1 2 M + 1 2M+1. t.R(z). (A-26) j=l 117

Im1 -1-7 Eq. (A-26) is used for numerical computation of the relative field intensities. 118