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. . . Sensor and Simulation Notes Note go. July 1969 Impedance and Fields of Two Parallel Plates of Unequal Breadths by G. W. Carlisle Northrop Corporate Laboratories Pasadena, California Abstract The method of conformal transformation is used to determine the impedance and fields of two parallel plates of unequal breadths. Parallel plates of this ‘#gg y%J#% ~.: :&““-z, F.,.=, -w.. type make up the transmission”lines that are used to transport electromagnetic energy along certain portions of some nuclear EMP simulators. Pertinent data are displayed graphically. .. ..._ ..=>..,,~.... ,,. ● Acknowledgment Thanks are due Mr. R. W. Sassman for his expert programming, Drs. R. W. Latham and K. S. H. Lee for their helpful advice, and Mrs. G. Peralta for preparing the manuscript. ✎ . , 6’ ● (-1. Introduction In simulating the phenomenon of a nuclear electromagnetic pulse (EM2) by artificial means, a parallel-plate transmission-line has proved to be a practical way of transporting the electromagnetic energy along certain portions of the simulator. In a previous notel the properties of two parallel plates of equal breadths were considered. This model also represents the electromagnetically equivalent configuration of a single plate parallel to an infinite ground plate --- a configuration that approximates an actual physical set-up of a plate parallel to a large, but finite, ground plate. In this note the actual geometry of two parallel plates of unequal breadths, as shown in figure 1, is considered, and the impedance of such a structure is computed. Curves are presented to show how the impedance depends on the size of the ground plate, which varies from a breadth equal to that (a = a) to a breadth of infinite extent (al=@) 1 In addition curves are presented to show the values of a/h and of the upper plate al/h (e where . corresponding to standard impedance values of 50, 75, 100 and 150 ohms a and a are the half-breadths of the plates and h is the total 1 spacing between the plates. The latter set of curves is particularly useful because it can be used to design a set of parallel plates of different breadths --- one plate perhaps much larger than the other --- whose impedance is exactly a prescribed value , without resorting to the infinite plate approximation. Finally the magnitude of the electric field beneath the finite ground plate is compared with the magnitude of the electric field between the parallel plates. This consideration is important if electronic equipment is placed beneath the ground plate. 1 . ._, . ..__., ,.... — —— . ● * 11. Exact Conformal Transformation A. E. H. Love gives the conformal transformation for two parallel plates ~ of unequal breadths.~ For the geometry depicted in figure 1 the transformation is iz .—= h~ K(m) Z(wlml) + — cn(wlml)dn(wlml) 2K(mY(ml) + v + sn(w ml) T ‘Zzm [ 1 “ (1) Associated with this transformation are the equations K(m) - v2E(m) C1C2 = (2) mlv2K(m) - E(m) (v’ - 1)(mlvL - l)K(m) . (3) In these equations Z(wlml) is Jacobi’s Zeta-function; K(m) is the complete (v + cl)(v + C*) = mlv2K(m) - E(m) elliptic integral; sn(wlml) s Cn(wlml) , and dn(wlrnl) are Jacobian elliptic functions. The notation used here is consistent with that used in reference 3. The parameter v the two places. As V+m, (1) relates to the difference in the breadths of reduces to the transformation for the case of two plates of equal breadth; as v + I/q , the breadth of the larger plate becomes infinite while the smaller plate remains finite. The range of and throughout this range azab ~ v , therefore, is v > l/~ Evaluating the real part of (1) at (-a,h) and (-al,O) gives (4) -. . *“ . {/-- I ‘- al E (m) K(m) —h = — ‘lT ‘2 K(m) where v 1 and v~ E(v21m) - #M3Fm “) satisfy (6) and dn(v2\m) + ~ = O C2 r’ The constants c1 and ‘2 9 0~v2~K(m) . (7) are obtained from (2) and (3) such that 9 (8) where ~ c~ is the positive value of (8) and quantities a and 8 C2 is the negative value. The are given by ‘a= V2E(m) - K(m) (9) E (m) - mlv2K(m) = $V(l - m/2)K(m) - E(m) E (m) - mlv2K(m) ‘. (lo) . ● ● The impedance of a parallel-plate transmission-lineimmersed in free1 space is Cz 00 (11) ‘L=-T-where in this particular case C , the capacitance per unit length, is given by coK(m) c = K(m,) (12) “ It is convenient to define a normalized impedance z. (13) which, if expressed explicitly, is K(ml) z;=-— (14) . h K(m) In figures 2 through 4 the normalized impedance Z: a/h with d/h = (al - a)/h is plotted versus as a parameter. The data in figures 2 and 3 were obtained by choosing a set of values for ml values to compute the corresponding values of and v a/h , al/h and by using these and Z; . To compute these three quantities, (9) and (10) are used to obtain a which are substituted into (8) to compute c1 and and C2 . The values of $ c1 -“ 4 ● ‘1 /-- [ are substituted into (6) and (7) and a root-finding technique is ‘2 used to determine and v With all of these constants known, (4) ‘1 2“ and (5) are evaluated to find a/h and al/h . The impedance Z~ is and computed from (14). A different method was used to obtain the data in figure 4. The method, as well as the reason for using it, is discussed in the next section. The data presented in figures 5 through 7 were obtained in the same way as the data in figures 2 and 3 were obtained exeept that a root-finding technique was used first to determine from (14) what values of ml correspond In these figures A = a/afi is plotted versus ‘L “ = 50, 75, 100 and 150 ohms as a parameter. The quantity to specific values of ‘1 aQ =a/a ~ ~ with Z L is measured in meters and is assigned the subscript $2 to emphasize that it takes on a different value for each ohmic value of listed a table of the values of values of 2ao/h ZL . Below is that correspond to each of the four Z L“ /--- Table 1. Values of ZL (ohms) and an 2afi ‘L T 50 5.998 75 3.618 100 2.457 150 1.337 ... Physically is the half-breadth of two parallel plates of equal breadth an Multiplying the values that are one meter apart and have an impedance ‘L “ by the appropriate of A and Al that correspond to a particular value of ‘L value of 2aQ/h listed in Table 1 gives 2a/h and 2al/h , respectively, which can be used to design an actual parallel plate structure, 5 ._—--- ..-— .-— ._. . III. Approximate Transformation for Plates Close Together For the case of two parallel plates that are very close together (a/h >> 1) , the parameter ml ml w 1~14 ; when is very small. For example when 2a/h s 20, ml s 1~24 2a/hss 10 , . Numerical computation becomes difficult for such small values of ml; hence, it is convenient to use an approximate conformal transformation for 4 transformation is TT(Z - a) h where the parameter T 2 =*+ 2a/h> 10 . The approximate (1-T)w-Tlllw++ -’c (15) satisfies (16) The real and imaginary parts of (15) are 2 7rT(x- a) =>cos28+(l-T)r h cos~-~lnr++-~ (17) 2 y Setting x =Oande=n => sin 2f3+(l-T)r sin 6--r6 (18) in (17) gives ?rTa s;-(1-T)r T 1 - T hi r +—- 2 -r (19) -0’ J ,- ( For a set of values of a/h and T , two values of r < 1 and and such that ‘A ‘B ‘A are obtained from (16), which depends on d/h . radii are denoted T In terms of ‘A and ‘B satisfy (19); these rB> 1 . Values of the transmission line impedance ‘n ‘L .4 ‘s Z. ‘L = Y ln(rB/rA) (20) “ The curves in figure 4 were obtained by using this approximate transformation. In figure 4 the normalized impedance Z: a/h with d/h defined by (13) is plotted versus as a parameter. Note that the values of d/h are different from those in figures 2 and 3. /’-- .. ..—— in figure 4 ● Iv. Electric Field below the Lower Plate In an actual simulator it may be that electronic equipment is located beneath the lower plate, i.e. the plate of greater breadth. In placing such equipment in this position it might be assumed that the field of a simulated EMP is contained wholly within the transmission line and that none of it leaks around the ends of the lower plate into the region where it could interfere with the electronic equipment. In the case of a finite lower plate a portion of the field does of course penetrate into the region beneath the lower plate. In this section of the note the magnitude of the electric field just beneath the center of the lower plate (ZA = O + iO-) is compared with that just above the center of the lower plate (ZB = O + iO+) . The magnitude of the electric field is given by IE[ = $ (21) ‘1 o which, in this case, is (snw - cl)(sn w - c2) IEl = (22) A (snw+ where the constant A ;)2 relates to the specific geometry of the parallel plates and does not depend on w . Under the conformal transformation (1) correspond respectively to ‘A = - K(ml) and and the points ‘B ‘A = - K(ml) -tiK(m) . Upon substituting these values of w into (22) ‘B the ratio between the magnitude of the electric field at ‘A and z~ is ]EA\ (l+c1)(l+c2)Gp 1)2 ● ~= (l+qcl) (l+~c2) (23) (v-1)2 0 8 9- Plots of \EA[/ lEB\ x 100% versus a/h with d/h as a parameter are presented in figure 8. 9 ..—. Y d 1 +,-x Figure 1. Two Parallel Plates of Unequal Breadths. a -J e . .25 d/h = O .20 z’ L .“1: hz ‘; = zL/(~) .1( h .o! 1 ( .01 .02 I .03 1 .04 I I .05 .06 I 1 .08 I 1 1 .2 .1 t .3 t .4 a/h Figure 2. Z; versus a/h with d/h as a parameter (.Ol<a/hs 1.0). I .5 I .6 I I .8 [ 1.0 .95 hZ z; = ZLI(=) 2a .85 k-2a+’l .75 h 4/ dfh = O .65 . z; .55 . .45 . .35 . .25 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 2 3 456 78910 a/h Figure 3. ‘i versus a/h with d/h as a parameter (.1< a/h~ 10). . I 1.00 Cl/h= o .98 5.0 .96 2.0 1.0 .94 0.25 z’ hz z; = zLf(~) L Ku .92 .90 t+ .88 .86 10 20 30 40 50 60 80 200 100 300 400 500 a/h Figure 4. Z; versus a/h with d/h as a parameter (10 < a/h< 1000). 600 800 1000 . 1.00 I I I I 1 i I I t I 1 I i . @ 1 . s .9e . * . .96 . . .94 . . .92 . A *90 .88 .86 .84 .82 150Q .80 1.0 I 1.1 Figure 5. A 1*3 1.2 versus ‘1 with 1.4 1.5 ‘1 as a parameter ‘L 1.7 1.6 -’ (1.OsA@ 1.7). . -3 .86 I I ,. (- 0 .84 ZL = 50$2 .82 .80 .78 A .76 .74 .72 .70 150sl .68 .66 1.3 1.4 1.7 1.6 1.5 1.8 1.9 ‘1 Figure 6. A versus ‘1 with ZL as a parameter 15 (1.3 <Al ~ 2.0). 2.0 .85 .80 .75 i’xl .70 A \ .65 100$2 ‘ .60 \ .55 1 1500 .45 1 1 I 1 I 1.5 2 3 456 1 I I I I I t 15 . 78910 1 20 I 30 I 40 % Figure 7. A versus ‘1 with ZL 1 50 I I I I 1 60 70 80 93100 ● as a parameter (l<AIS 100). . & 80 70 60 50 a/h Figure 8. ‘Ai/lEB~x 100% versus a/h with ZL as a parameter. * ● ✍✍✍✍✍ b References 1. Carl E. Baurn,Sensor and Simulation Note 21, “Impedances and Field Distributions for Parallel-Plate Transmission-Line Simulators,” June 1966. 2. A. E. H. Love, “Some Electrostatic Distributions in Two Dimensions,’r Proc. London Math. Sot. ~, 3. pp. 337-369, 1923. Milton Abromowitz and Irene A. Stegun, Editors, Handbook of Mathematical Functions, National Bureau of Standards, AIM-55, June 1964. 4. F, Assadourian and E. Rimai, “Simplified Theory of Microstrip Transmission Systerns,rl Proc. of IRE, December 1952, pp. 1651-1657. a 18