THE UNIVERSITY OF MICHIGAN 5548-4-T AFCRL-65-254 THE MINIMIZATION OF BACK SCATTERING OF A CYLINDER BY DOUBLE LOADING by Kun-Mu Chen April 1965 Scientific Report No. 4 Contract AF 19(628)-2374 Project 5635 Task 563502 Prepared for Air Force Cambridge Research Laboratories Office of Aerospace Research L.G. Hanscom Field Bedford, Massachusetts I

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THE UNIVERSITY OF MICHIGAN 5548-4-T TABLE OF CONTENTS ABSTRACT 1 I. INTRODUCTION 2 II. INTEGRAL EQUATION FOR INDUCED CURRENT 4 III. SYMMETRICAL COMPONENT OF INDUCED CURRENT 10 IV. ANTISYMMETRICAL COMPONENT OF INDUCED CURRENT 15 V. OPTIMUM LOADING FOR ZERO BROADSIDE BACK SCATTERING 18 VI. OPTIMUM LOADING FOR LOW OFF-BROADSIDE BACK SCATTERING 23 VII. OPTIMUM LOADING FOR LOW BACK SCATTERING OVER A WIDE ASPECT RANGE 26 VIII. CONCLUSION 29 ACKNOWLEDGMENT 30 REFERENCES 31 APPENDIX: Antisymmetrical Component of Induced Current With and Without Optimum Loading 32 iii

THE UNIVERSITY OF MICHIGAN 5548-4-T ABSTRACT A theory on the minimization of the back scattering of a cylinder by loading a cylinder at two points with lumped impedances is presented. The induced current on a doubly loaded cylinder when illuminated by a plane electromagnetic wave at an arbitrary angle is obtained. The optimum loading to eliminate the broadside back scattering which is caused by the symmetrical component of the induced current is established. A suitable loading to reduce the off-broadside back scattering, due to the antisymmetrical component of the induced current, is also determined. A proper choice of the impedance and the position of double loading can lead to the reduction of the back scattering over a wide aspect range. The useful formulas for optimum loading and some graphical illustrations are included. The advantages of double loading over central loading are discussed. 1

THE UNIVERSITY OF MICHIGAN 5548-4-T I INTRODUCTION The first use of reactive loading to reduce the back scattering of a metallic body was by Iams (1950). The idea of using the technique to decrease the radar cross section of objects in space was suggested and employed by Sletten (1962) in 1950. The method of modifying the back scattering cross section of a cylinder by loading techniques has been studied by others (Hu, 1958; As and Schmitt, 1958; Harrington, 1963). The method of minimizing the back scattering of a cylinder by central loading was studied by Chen and Liepa (1964b) from the viewpoint of the surface current. In their study, it was found that a cylinder center-loaded with an optimum impedance which has both reactive and resistive components can lead to zero broadside back scattering. The optimum impedance is passive for a cylinder shorter than a wavelength but an active impedance is required when a cylinder is longer than a wavelength. They also found that central loading can not reduce the off-broadside scattering which is due to the antisymmetrical component of the induced current and is very large in the case of an anti-resonant cylinder (Chen and Liepa, 1964a)(a cylinder with an anti-resonant length). The purpose of this study is to investigate the double loading method which overcomes the shortcomings of the central loading method. In general, when a conducting cylinder is illuminated by a plane wave the induced current on the cylinder can be divided into symmetrican and antisymmetrical components. In a resonant cylinder, the symmetrical component is predominant and it gives a large broadside return. In an anti-resonant cylinder, the antisymmetrical component can be very large resulting in a large off-broadside return. Central loading can modify greatly the symmetrical component of the induced current but it does not affect the antisymmetrical component. The double loading which load the cylinder at two symmetrical points with two identical lumped impedances is designed to modify both components of the induced current. In this report we show that I' 2

THE UNIVERSITY OF MICHIGAN 5548-4-T a properly designed double loading can eliminate the broadside back scatter or reduce the off-broadside back scatter. Furthermore, we show that with the freedom to choose the impedance and the position of double loading, we can reduce the back scattering over a wide aspect range. The scheme of the analysis is (1) to find the induced current on a doubly loaded cylinder as a function of cylinder dimensions, the impedance and the position of the double loading, and the parameters of the incident plane wave; (2) to determine an optimum loading for zero broadside back scattering; (3) to determine a suitable loading to reduce the off-broadside back scattering; and (4) to obtain an optimum loading which functions to reduce the back scattering over a wide aspect range. The accuracy of the results in this report should be quite adequate in practical applications since a similar analysis of the central loading case has been completely checked by experiment (Chen and Liepa, 1964a, b). I 3

I I THE UNIVERSITY OF MICHIGAN 5548-4-T II INTEGRAL EQUATION FOR INDUCED CURRENT The geometry of the problem is as shown in Fig. 1. A cylinder with a radius a and length 2h is assumed to be perfectly conducting. Two identical impedances ZL are loaded at z = +d on the cylinder. A plane electromagnetic wave is incident upon the cylinder at an angle 0. The dimensions of interest are 1/4X < 2h < 2X and 2 2 3 a < 1, where X is the wavelength and /3 the wave number. We assume that the 0 0 cylinder is thin enough so that only the axial current is induced. The tangential component of the incident electric field on the surface of the cylinder is in -josinOz E = E cosOe (1) Z 0 The tangential component of the electric field maintained by the current and the charge on the cylinder at the surface of the cylinder is Ea = -j + A, (2) z - 22 olz where A is the tangential component of the vector potential maintained by the curz rent on the cylinder. The total tangential component of the electric field on the surface of the cylinder is zero except at z = +d where potential differences exist across the lumped impedances. The electric fields across the gaps at z = +d can be expressed as Eg = Z I(d)6(z-d)+Z I(-d)6(z+d),(3) z L L where I(d) is the induced current at z =d on the cylinder and 6(z) is a delta function. M.t

THE UNIVERSITY 5548-4-T OF MICHIGAN -— z= h ZL I. z Az=O Z z=-d I L 2a -_-z= —h FIG. 1: DOUBLY LOADED CYLINDER ILLUMINATED OBLIQUELY BY A PLANE WAVE 5

TH E UNIVERSITY OF MICHIGAN 5548-4-T The combination of (1) to (3) gives a differential equation for A as z ( O z 2 -j w Lr c -jf3 sin0z E cos0 e L-O - ZL [I(d) 6 (z-d) + I(-d)6 (z+d)] (4) The solution for A can be expressed as z A = [C cosoz + C2sin3 z +(z), z,v + (z O (5) where v = 1/\~F, C1 and C2 are arbitrary constants and O(z) is a particular integral. 0(z) can be found to be integral. 0(z) can be found to be E -jl3 sin0z Z O(Z) = co e 2 I(d)sin z-d[+I(-d)sin3oz+d|l 00) os0 2 Lz 0 (6) For convenience, A is divided into symmetrical and antisymmetrical components as follows: A (z) = -J ccos3 z z v 1 o a(z) aO 0 E ZL + 3 0 cos( sin0z) - - [I(d) + I(-d) cos os(O3sin~z) - 4 [sin olz-dl+ sin 3lz+d jE0 ZL _ os sin(o s in0z) - d 0 -". (7) [sin3 |z-d - sinf |z+d|]. (8) After some modification we obtain the following relations 6

THE UNIVERSITY OF MICHIGAN 5548-4-T AS(z)-AS(A (= -j sec pc h - sin (cos 3 z - cos 3 h) E cose h{- z z v /3oh o z (oOS o o + E cosoh cos(13 zsinO) - cos(I3 hsinO f~cosOo 0 + - [(d) + I(-d] [2 sin3 h cos odcos3oz - cos 3h(sin/3 |z-d + sin3 0z+di (9) A(z) = - csc3 hv Az(h)sinf3 z o E+ o sin(3 hsin) sin3 z- sinj hsin(L3 zsinO) p cos0 L o o o o 0 - 4 I(d) - I(-d)] L2 cos oh sinod sinp z+sinp h(sinp oIz-d -sinp0 |z+d[)-. (10) According to the definitions of the vector potential, A (z) and A (z) can be exz z pressed in terms of the symmetrical component of the induced current I (z) and the z antisymmetrical component of the induced current I (z) as follows: z AS(z) =(z')K (z )dz, (11) z 47 z a -h h Aa(z) = 4~ Iz)K (z, z')dz' (12) z 47 z a-h -h 7

THE UNIVERSITY OF MICHIGAN 5548-4-T where exp -j, (z-z') +a K (z, z') = 2. (13) a (z-z?' +a a An integral equation for IS(z) can be obtained from (9) and (11) as z h E h + co( sin - cos-~h sino z F 0 z -h a E^ -i right hand side of (14).(cos z-cos h) iecospe o 0 Kd(zz') = K (z,z')- K (h, z'), (15) and = 120 ohms. Notice that AS(h), I(d) and I(-d) are still unknown in the O Z right hand side of (14). An integral equation for Ia(z) can be obtained from (10) and (12) to be Z 8

THE UNIVERSITY OF MICHIGAN 5548-4-T h Ia(z')K (z = h jA (z'h) sin z \ o z o -h o jE r -' + Is [sin(3 h sinO) sinf z -sino hsin(f z sin) cos L o o o0 0 0 [I- I-d [2cos Ph sin od sin z + sin,oh(sin oz-d - sin3 |z+d]i)}. (16) Equations (14) and (16) are solved separately in the following sections. 9

THE UNIVERSITY OF MICHIGAN 5548-4-T III SYMMETRICAL COMPONENT OF INDUCED CURRENT In this section, I (z) is determined approximately from (14). In view of the z complexity of the integral equation (14), it is very difficult to obtain a solution which is simple enough for further theoretical development and with the accuracy higher than the first order approximation. In general, this type of integral equation can be solved very accurately by using King-Middleton's iterative method (King, 1956) or Tai's variation method (Tai, 1950) but the solutions would be too complicated to make further theoretical development tractable. A compromise technique which is similar to King's recent method (1961) is used here. From the nature of the kernal Kd(z, z') which has a sharp peak at z' = z and from the form of the right hand side of (14), we assume IS(z) as Z Is(z) = Cc(cos 3oz-cos3 oh) + C oos(o z sin) - cos(3 hsin ) z C O 0 o 0 0 +C. sin Phcos30dco c-cos3ih(sin3 |z-d|+sin3o z+dj (17) This distribution has the advantage of satisfying the boundary conditions at z = h and z = -h. In other words, with this distribution (14) is satisfied at z =h and z = -h. Furthermore, the problem of finding I (z) is simplified due to the fortunate z relation of I(d) +I(-d) = I(d) +I(-d) +a(d) a(-d) = I(d)+ I(-d) (18) because (d) (-d) 0 by definition.Z Z Z Z because Ia(d)+Ia(-d) = 0 by definition. Z Z The constants, C, C and Ci can be determined by the substitution of (17) c' 0 1 in (14) and letting z = 0. By doing so the current in (17) has been made to satisfy the boundary conditions at three points, i.e. z =0, z =h and z = -h. It is noted that this method yields a quite satisfactory solution as discussed by King (1961) and Chen and Liepa (1964a, b). 10

THE UNIVERSITY OF MICHIGAN 5548-4-T After Cc, C and Ci are determined, I (z) can be expressed as C 0 1 z _ r E 2E I (z) -= j 4 seclP h jv A (h)- - 0 cos(3 hcosO) (1-cos1oh)(cos /3z-cos oh) z o T cd 0 (3 z Pcos 0 0 0o O 0 E 3,+ 1 o -cos(P h sin0) eos(/B zsin0)-cos( h sin0) + "27r sec3 hZL I (d)+IS(-d) sin/3 (h-d) 2sin 1 hcos 3 dcos 3 z o Tid o LL z z - oo o o o id - cos P h(sin/3 |z-d+ sin3o |z+d|, (19) where h Tcd = (cos 1oZ? - cos Poh)Kd(0, z')dz',(20) -h rh Tod = I cos(o z' sin0)-cos(P3ohsin0)Kd(0,z')dz' (21) y~d i L o o - d -h h Tid= h [2sin3 hcos3o dcos/3 z' -coso h(sin3o z'-d[ -h + sinp |z'+dK d(0, z ')dz. (22) Equation (19) is not the final form for I (z) because A (h) and (d)+(- in Z Z L. z Zd he right hand side of (19) are still unknown. IS (d)+Is(-d) can be determined directly from (19) as LZ z - 11

THE UNIVERSITY 5548-4-T OF MICHIGAN S(d)+I (-d= Dj T sec hv A(h) z z~1 D o cd E..... cos(3 hsin0) - cos 3 h)(cos d- cos h p cos& o o o 0 -j87r l o Od E0 l-cos( hsinO) EosC3 dsin)- cos(3 hsinO), (23) P cos@ 0 o 0 L o 0o - 0 where jZ D1 = 1+ LT sec3 hcosPf dsin 21 (h-d) 5Tid id (24) AS(h) can be determined from (11), (19) and (23) as follows: oj E T AS(h) = hsinO)(sec h- 1) ca z D21207rf3 cos j0 S 0 T cd 2 o LLc jZL 30D1 Tcd T. 7 Ta sin/3 (h-d)(secf3 hcos p3d-1) Tid o 0 -TO - - cos(3 hsin0)] Koa 0Od jL 1 dL 30D1TOd where T. -ia secp hsin/3 (h-d) cos(P3 dsino) - cos(/3 hsinO~)] T id oLo (25) jZL T.i +3D T sin 3 (h-d)(sec, h- 1) ~ 30DT T o id 1 cd id D = l-(sec3 h-l) D2 o T ca Td cd * (secp hcos3 d-1), o O (26) 12

THE m UNIVERSITY 5548-4-T OF MICHIGAN T = \ (cos z' - cos h)K (h, z')dz' ca 0o o a -h (27) (28) h TOa = \ cos(3 z'sin0)-cos( hsin)Ka(h, z')dz' -h h Ti = 2 sin3 h c dos z dcos '-cos3h(sin 3 z'-d +sin3 |z'+d) K (h,z')dz' -h (29) With (19), (23) and (25), the final solution for I (z) can be summarized as z follows: jE (z) =30 0 F(cos 0 z - os h)+ F os(z sina)- cos(3 h sinO F si cos d os + F. 2sin3 hcosp dcosp3 z-cosp3 h(sinP Iz-dI+sinO |z+dI)i (30) where 1 F = ---c Tcd (sec 3h h-1) cos(,3 h sine)+ 0 0 D Fcs, rn)sc l T - Los(3 hsino)(secp h-1) ca 22 ~s" ocTd j~~ZL Tia L( hi ) 10 5 0-0 T J 7z- T. r-cos(,hsin T - 30D1 D - sin / (h-d) cos(3h sinO)(sec3oh -l)(sec3 hcos d-1) 1 30D1D2 Tid 0 o Tcd - sec oh [ - cos(3 h sin) [cos(3od sin0)- cos(P3 hsin0) T-] } oL 0 J 0 T 0 (31) 13

THE UNIVERSITY OF MICHIGAN 5548-4-T F = T 1 -cos(P h sino) (32) TOdo F = (hd seinc (h-cd) (sec/3 hos d-l)sinO 30D1Tid 0 0 0 0 0 - lid 0e0 cos(3 dsinO)-cos(3 hsin0) (33) L o ' We see that the symmetrical component of the induced current IS(z) is a function of z the cylinder dimensions (h, a), the impedance (ZL) and the position (d) of the loading, and the incidence angle (0) and the magnitude (E ) of the incident plane wave. 14

I THE UNIVERSITY OF MICHIGAN 5548-4-T IV ANTISYMMETRICAL COMPONENT OF INDUCED CURRENT In this section, I (z) is determined from (16). By the same reasoning as in z he preceding section we can assume I (z) as z Iz(z) = C sin(po hsinn) sino z-sinB3 hsin(3 zsinO) z m 0 o z+ ~ +C 2cos3: hsin3, dsin31 z+sin3 h(sinp jz-dl-sin3 l|z+d 1. (34) An important relation, I(d)- I(-d) = I(d) - Is(-d)+ Ia(d)- Ia(-d) = Ia(d)-Ia(-d) (35) z z z z z z is also needed in the analysis. The constants, C and Cn, can be determined approximately by substituting (34) in (16) and letting z = h/2. It is also assumed that the first term of the right hand side of (16) is negligible compared with the other terms. This assumption is not rigorously justified but it gives a reasonable answer (King, 1956). The final results for C and C are: m n E /3 h /h h C =303 ~-0 sin( sin) cos(- sinO sec( -1) -- T, (36) o O / k~ \ /.J ma h sin/3 d, for h/2>d - c = Iz [a(d)_ ia(_d)s s a _ (37) n 120 L z z J 2 L2 sinP (h-d), for d > h/22 na where h Tma = s( hin(/ hsinO)sinp oz'-isin hsin(3 z'sinO) Ka(h/2,z')dz, (38) -h I 15

w THE UNIVERSITY OF MICHIGAN 5548-4-T hT T = \ [2cos/] hsinp]d sin 3z' + sin h(sin3o z'-di| na 0 0 - J-h -sin 3 | z '+d K (h/2, z')dz' 0' I Z (39) After (36) and (37) are substituted in (34), [I(d) - Ia(-d) can be obtained as Z Z [Ia(d)- Ia(-d)j z z E O 303 cos0 0 sin( snh Eos( ) sin(-2sin) cos(2 s in0) \2 " - eoh - 2 sec - 1 T D 2 ma 3 * sin(P hsinO)sinf3 d-sin/3 hsin(,3 dsin0), (40) where jZ,3h D = 1- 30 sec -2sin3odsin o(h-d) na sin,3d, for h/2 > d L sin 0(h-d), for d > h/2I sin3 (h-d), for d > h/2 (41) With (34), (36), (37) and (40), the antisymmetrical component of the induced current Ia(z) can be expressed as follows: z Ia(z) = 0 o F [sin( ohsinO)sin z-sin hsin( i) iaz 30(3zcos- sin(3 hsinn(3 z sinOil z 308 cos0 smn o o + Fn[ cos h sin3odsinz+siini iz + sinh(sin z-d-sin 3 z+d|] (42) where F = sin 2 sin0/ m 2 cos _ (Bh sin) (-2 '8 sec - 1 sec 2 _ 1 ma ma (43) I______I 16

THE UNIVERSITY OF MICHIGAN 5548-4-T F F (6DZL)Tin( [ sin(sin)sino d \ 30D na P h - sin3 d, for h/2 >d-sin/3 hsin(B3 dsin) sec 2-. (44) L sinP (h-d), for d > h/2 Ia () is also determined as a function of the cylinder dimensions, the impedance and z location of the loading, and the incident angle and magnitude of the plane wave. 17

THE UNIVERSITY OF MICHIGAN 5548-4-T V OPTIMUM LOADING FOR ZERO BROADSIDE BACK SCATTERING The total induced current on the cylinder is I (z) = IS(z) + Ia(z) (45) z z z This total current maintains the scattered field. As-far as the back scattered field is concerned, I (z) maintains a large broadside back scatter and I (z) maintains Z z a large off-broadside back scatter at 0 " 400 when the cylinder has an anti-resonani length. In this section, we shall determine an optimum loading to achieve zero broadside back scattering. With this optimum loading, I (z) has a minimum ampz litude and an appropriate phase distribution along the cylinder. When the plane wave is incident broadside (0 =0 00) on the cylinder, the induced current on the cylinder is jE 0 (cos h F h cos I (z) = Lc fz (c - os z h) + F. sin3o hcos3 dcosp z z 300o 0 cs o o i o o - cos3 h(sin 3Jz-d +sin3oz+d), (46) where F (0= 0) = T (sec3 h-1) (47) ec Tcd D2 -jZ 0iZL F.( 30DDT T sinp3 (h-d)(sec 3 h-1)(sec o hcos3 d-l), (48) 1 30D1D 2 cd id 0 0 0 0 F (0=0~) = 0.(49) The antisymmetrical component of the induced current I (z) = 0 when 0 = 0 z 18

I THE UNIVERSITY OF MICHIGAN 5548-4-T The back scattered field in the broadside direction can be found as jW/t -j3oR h E (0= ~) =We Iz)dz scat 47rR z 0 -h 2E -j3 R e ~ F (0=00)(sin h-3 hcos/3 h) R 0 c 0oh o o o + F.(0= 0)(2cosf3 d-2cos3o h). 1 0 0 (50) To make E (0- 0 ) = 0, we let scat F (0=0 ) sin3 h-3 hcosf3 h c 0 0 0 Fi(0=0 ) 2(cosl3 d-cos/3 h) 1 0 0 The optimum impedance for zero broadside back scattering ZL can be determined from (51), (47) and (48) as follows: [ s L l-jJl5T idcos 3 h(sin 3oh-3oh cosj3 h) r -i5 - id 0 0 0 0 ~ sino3 (h-d)(cosp d-cosh3 h) - cosl3 dsin (h-d)(sinf3 h-fo hcos3 h) 0 0 0 0 0 0 0 0 (52) |[zJs is thus determined as a function of the cylinder dimensions (h, d) and the position (d) of the loading. The form for [ZL is quite simple and should prove useful in pracitcal design. The numerical value of [Z is readily obtained once Tid as defined in (22) is calculated by a digital computer. It is noted that [ZL] reduces to the corresponding value for central loading (Chen and Leipa, 1964b) when d=0. Two numerical examples for [z ] are given in Figs. 2 and 3. In Fig. 2 we show [ZL] for the case of a =0.0173X and d =h/2 as a function of the cylinder length h/X. In this case two impedances are loaded at the centers of two halves of I j 19 I

-Z h I E I ~\ I m/I 2a XL 600 t L t! A a=0.0173X \ 400 d = h/2 0 L L ~ R L RL a v (a) _ ---- l\ ) O f _____ I ' ~I.'. i\ ' i\ 0 -200 - '1 0 z 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 h/X — W FIG. 2: OPTIMUM IMPEDANCE FOR ZERO BROADSIDE BACK SCATTERING VS CYLINDER LENGTH WITH d= h/2.

3 r*l ct z rl CO) ^0.0 c.n cl cn 00 I 1 tj 0 T) c) 0 5 0.6 h/X- FIG. 3: OPTIMUM IMPEDANCE FOR ZERO VS CYLINDER LENGTH WITH h-d BROADSIDE BACK SCATTERING = X/4.

THE UNIVERSITY OF MICHIGAN 5548-4-T the cylinder and [zL] behaves very smoothly up to h=0.7X. In Fig. 3, [ZL for the case of a = 0.0173X and h- d = X/4 is shown graphically as a function of the cylinder length h/X. In this case two impedances are loaded at the points a quarterwavelength from the ends of the cylinder and [ZL] requires a negative resistance for h > 0. 7X. From these two examples we see that [ZL] is dependent on the position of the loading and it appears possible to have a passive impedance for [ZL]s up to h = X by a proper choice of d. In this way we can avoid using an active impedance which is necessary for central loading (Chen and Liepa, 1964b). 22

THE UNIVERSITY OF MICHIGAN 5548-4-T VI OPTIMUM LOADING FOR LOW OFF-BROADSIDE BACK-SCATTERING When a cylinder has a resonant length (2h - (2n+l)X/2), the symmetrical component of the induced current I (z) is the predominant component and [ZLl] is adequate to minimize the scattered field in the broadside direction and other directions as well (Chen and Liepa, 1964a). But when a Cylinder has an anti-resonant length (2h - nX), the antisymmetrical component of the induced current I (z) z can be very large and it gives a large off-broadside return. In this section we seek an optimum impedance which keeps I (z) small along the cylinder, resulting in a z small off-broadside back scatter. For a cylinder shorter than two wavelengths and with two impedances loaded at z = +d, the distribution of I (z) along one half of the cylinder can be shown from z (42) to be the sum of two shifted sinusoidal curves. After the distribution of Ia(z) z is studied graphically, one would write the simplest criterion for small I (z) and z low off-broadside back scattering as Ia(z) dz = 0 (at a particular 0). (53a) z Notice that the lower limit of the integral is 0 instead of -h. The condition imposed in (53a) usually requires I (z) to be small and causes cancellation between two z shifted sinusoidal curves. This criterion is rather arbitrary but quite sufficient, as we can see in the Appendix that I (z) under (53a) becomes very small indeed. An z alternative and perhaps more accurate criterion, such as demanding that j3 z sinO Ia(z) e dz = 0 (at a particular 0) (53b) | reight zmiu o-rddbcsted tI q J-h or requiring that the maximum off-broadside back scatter due to I (z) equal zero z l 23

THE UNIVERSITY OF MICHIGAN 5548-4-T in the 0 directionwould be more rigorous, but the algebraic evaluation is more involved. In case more rigorous treatment is desired, (53b) may be used. For the sake of simplicity, (53a) is used as the criterion in this report and the corresponding optimum impedance is obtained as follows. The substitution of (42) in (53a) gives Fn _ inf3 hcos(I3 hsinO)-sing lh+sin(IB hsin0)sin0-sin(IB hsin0)sinOcost3 h Fm 2sin0 sinh- sin3 hd-sino3 (h-d) (54) An optimum impedance Lo which gives small I (z) and low off-broadside back scattering can be solved from (54) after the substitution of (43) and (44). The final expression for [z La is L O -jl5Ta sin P hcos(P hsin0)- sin P h +sin(f3h sinO)sin0- sin(f3 hsin0)sin0cos PhIq Lz sLin( h sin)sin dsin(0 1- cos (h-d] rsin3dfor >d 0 f3 h r o 0 in F d,for->d sin 2- +sin(o dsin0)sinsin (h-d)-sinf h+sin3~ d] 2 Sio! ~sin(10inI3o(h-d),for - L+sin3p dsin 3(h-d)[cos(i hsin0) -1] L in(h-d),for d>h 2 (55) where Ta was defined in (39). [ZLa is a function of the cylinder dimensions na L 0LJ (h, a), the location d of the loading and the incident angle 0 of the plane wave. In the practical application, [ZLJ1 is calculated from (55) by assigning 0 as the angle where the maximum off-broadside back scatter occurs. Usually this angle is about 40 off the broadside direction. A numerical example is shown in Fig. 4. We consider a cylinder with a=0.0173X, d=h/2 and 0=450. [Zda for this case is passive and well behaved up to h = 0. 95 as shown in Fig. 4. 24

1200 - 1000 800 I [ZL]o I I I I Ii H L a = 0.0173X d = h/2 = RL+jXL 1000 800 1200 d z (-1 c3 o / / I 1N Cin / / 600 t XL RL (Q) / I XL RL (,) -600 Cn cn I I^ 0 400, _ 400 200 RL _.00 z 200 0 s / 0 L I at_ _ t _l_ I tX_ JO.0 0.1 0.2 0.3 0.4 0'5 0.6 h/X — 0.7 0.8 0.9 I FIG. 4: OPTIMUM IMPEDANCE FOR MINIMUM ANTISYMMETRICAL CURRENT WITH 450 INCIDENCE ANGLE, d = h/2.

THE UNIVERSITY OF MICHIGAN 5548-4-T VII OPTIMUM LOADING FOR LOW BACK SCATTERING OVER A WIDE ASPECT RANGE If the back scattering cross section of a cylinder with 2h < 2 X is plotted as a function of the aspect angle 0, there are three possible peaks. The first peak occurs at the broadside direction (0=00) for a cylinder with any length between X/4 < 2h< 2X. The two other peaks occur at each side of the first peak and about 400 off the broadside direction. The off-broadside back scatter is insignificant for a resonant cylinder but it is very large for an anti-resonant cylinder. In the preceding sections, [ZL] is designed to eliminate the broadside back scatter and dnga - 0 [ a is designed to reduce the off-broadside scatter separately. It is then desirable to design an omptimum impedance [zL]- to function as [ZL] and zL simultaneously. Actually [ZL] and [Za are both functions of d and it is possible to adjust the value of d to make [ZL and [ZLl equal or nearly equal. If L L such an empedance [ZJand the position d are obtainable, the back scattering of a cylinder can be reduced greatly over a wide aspect range. To seek an optimum impedance [ZL] such as O [ZLu [z L- [ ZLa (56) it is necessary to find an optimum value of d from the following equation.0 it is necessary to find an optimum value of d from the following equation. 26

THE UNIVERSITY OF MICHIGAN 5548-4-T T (d) T (d) na rsin (h-d)(cosB3 d-cos P h) -cos,3 dsin B3 (h-d)(sinl3 h-J3 hcos3 h) E[inoh cos(oo hsin) - sin 3 h+sin(Bo hsin0)sin- sin((3 hsine)sin0cos oh i h rsin3 d for h/2 > d *o ~ cosf h sin (sin/3 h - h cos 3 h) *coshsin 2 (sin oh- ohos o Lsin3 (h-d) for d >h/2_ o r sin(3oh sinO)sin3 d sin& 01 - cos l3(h-d +sin(I3 dsinO)sino[sin3 (h-d)-sin3 h+sin3 d]j + sinSod sin (h-d) [cos(13h sin0) - (57) Actually, Tid and Tna are both complex numbers and it is only possible to find an optimum d which makes both sides of (57) nearly equal. After the optimum d is determined from (57), [ZJ can be obtained easily from (52) or (55). A numerical example is given in Fig. 5. [ZL] and d for a cylinder of a = 0.0173 X is shown graphically as functions of the cylinder length h/X. We observe that d varies almost linearly with h and [ZJ is passive and well-behaved 0 in the entire range of X/4 < 2h < 2X. This [zL appears to be obtainable by a simjple network synthesis. One remaining question is whether one can design an appropriate double loading for a fixed cylinder to minimize its back scattering cross section over a wide frequency range. The answer is self evident. We can calculate [ZL] and d for a fixed cylinder as a function of frequency by the same technique employed in this section. Of course, the calculation will be somewhat complicated because 13 h and Poa both vary with frequency in this case and we can expect similar sets of curves for [ZL~ and d as those shown in Fig. 5. I 27

1200 1000 800 600 F I I [ZJ -0.6 'L XL RL (A) a = 0.0173X '1/' /./0 / f 7 0.5 0.4 t d (X) = RL+ jXL z C-. vil,3,< 0 I, Hr t 400 / 0.3 0.2 / / / 200 I RL _.SN.a ~ m ~ m. M ~ ~. W WO 0 z / 0.1 i - -7' 0.I 0:2 0.3 0:4 0:5 0:6 h/X -- 0.7 0.8 0.9 1.0 FIG. 5: OPTIMUM IMPEDANCE AND OPTIMUM POSITION FOR DOUBLE LOADING TO MINIMIZE BACK SCATTERING OVER A WIDE ASPECT RANGE.

THE UNIVERSITY OF MICHIGAN 5548-4-T VIII CONCLUSION The induced current on a doubly-loaded cylinder when illuminated by a plane wave at an arbitrary angle is obtained as a function of the cylinder dimensions, the impedance and the position of the double loading and the amplitude and the incidence angle of the plane wave. The optimum loading to eliminate the broadside back scattering is established. A suitable loading to reduce the off-broadside back scattering is also determined. The combination of these two techniques enables one to design an optimum loading to reduce the back scattering over a wide aspect range. Some reasonalby simple formulas for optimum loading derived in this paper should prove useful in practical design. The accuracy of this theory has been justified in a central loading case (Chen and Liepa, 1964a) by experiment and we feel that no additional experimental check is needed for the double loading case. 29

THE UNIVERSITY OF MICHIGAN 5548-4-T ACKNOWLEDGMENT The author gratefully acknowledges the assistance and helpful discussions of Mr. R.E. Hiatt and Dr. T.B.A. Senior. The author also expresses his appreciation to Mr. J.A. Ducmanis for the numerical computations. 30

I THE UNIVERSITY OF MICHIGAN 5548-4-T REFERENCES As, B.O. and H.J. Schmitt (1958) "Backscattering Cross Section of Reactively Loaded Cylindrical Antennas, " Harvard University Cruft Laboratory Scientific Report No. 18. Chen, K-M and V.V. Liepa (1964a) "Minimization of the Radar Cross Section of a Cylinder by Central Loading, " The University of Michigan Radiation Laboratory Report No. 5548-1-T. Chen, K-M and V.V. Liepa (1964b) IEEE Trans. Antennas and Propagation, AP-12 pp. 576-582. Harrington, R.F. (1963) "Some Bounds to the Behavior of Loaded Scatterers," AFCRL Symposium on Modification of Electromagnetic Scattering Cross Sections in the Resonant Region; Cambridge, Massachusetts. Hu, Y-Y (1958) IRE Trans. Antennas and Propagation, AP-6, pp. 140-148. Iams, H.A. (1950) "Radio Wave Conducting Device, " U.S. Patent No. 2, 528,367. King, R. W.P. (1956) Theory of Linear Antennas, Harvard University Press, Cambridge, Massachusetts. King, R. W. P. (1961) "Dipoles in Dissipative Media, " Harvard University Cruft Laboratory Technical Report No. 336. Sletten, C.J. (1962) Private Communication. Tai, C-T (1950) "A Variational Solution to the Problem of Cylindrical Antennas, Stanford Research Institute Technical Report No. 12, Project No. 188. 31

THE UNIVERSITY OF MICHIGAN 5548-4-T APPENDIX ANTISYMMETRICAL COMPONENT OF INDUCED CURRENT WITH AND WITHOUT OPTIMUM LOADING To show the effectiveness of [Zja which is derived under the condition of (53a) in reducing I (z) and off-broadside back scattering, I (z) on some cylinders z z are calculated for the cases with and without ZL]a. We consider the cylinders of o o various lengths and with A= 0.0173 X, d = h/2 and 0= 45. The numerical results are shown graphically in Fig. 6. The solid'lines represent I (z) without [ZJ and the dashed lines represent Iz(z) with [ZL. From Fig. 6, it is evident that with LZLT the antisymmetrical component of the induced current is greatly reduced in amplitude and the phase is reversed so that the off-broadside back scattering is reduced to a very low value. An interesting point in Fig. 6 is also worth mentioning. For a cylinder with h = 0. 5X, Ia(z) can be eliminated theoretically with Z [ZLT while without [ZLj, Ia(z) would be very large on this cylinder. It is note that I (z) without [ for various cylinders are drawn in comparable scale for z the sake of clarity in Fig. 6. Actually, Ia(z) on a cylinder of h=0.5X would be much larger than that on a cylinder of h= 25 much larger than that on a cylinder of h = 0. 25X. 32

r THE UNIVERSITY OF MICHIGAN 5548-4-T - Iwithout [ZLa zwith [zLJ t,,, o d = h/2 0 = 45~ 0 V (h = 0.25X) h Ia with [ZL]a is zero z L LO z=h z=h z=0 z=d (h = 0.5X) h (h = 0.625X) h (h = 0.75X) FIG. 6: ANTISYMMETRICAL COMPONENT OF THE INDUCED WITH AND WITHOUT OPTIMUM LOADING [Za CURRENT 33

Unclassified Security Classification DOCUMENT CONTROL DATA R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified Radiation Laboratory 2b GROUP 3. REPORT TITLE The Minimization of Back Scattering of a Cylinder by Double Loading 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Technical Report 5. AUTHOR(S) (Last name, first name, initial) Chen, Kun-Mu 6. REPO RT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS April 1965 32 10 8a. CONTRACT OR GRANT NO. 9a, ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-2374 5548-4-T b. PROJECT NO. 5635. C. 9 b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) 563502 AFCRL-65-254 d. 10. AVAIL ABILITY/LIMITATION NOTICES U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified users shall request through CFSTI 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Laboratories Bedford, Massachusetts 13. ABSTRACT A theory on the minimization of the back scattering of a cylinder by loading a cylinder at two points with lumped impedances is presented. The induced current on a doubly loaded cylinder when illuminated by a plane electromagnetic wave at an arbitrary angle is obtained. The optimum loading to eliminate the broadside back scattering which is caused by the symmetrical component of the induced current is established. A suitable loading to reduce the off-broadside back scattering, due to the antisymmetrical component of the induced current, is also determined. A proper choice of the impedance and the position of double loading can lead to the reduction of the back scattering over a wide aspect range. The useful formulas for optimum loading and some graphical illustrations are included. The advantages of double loading over central loading are discussed. _ i.. I J _ ~. DD JAN 64 1473 DDOR Unclassified Security Classification

Unclassified I Security Classification,~1~~~~~4 ~LINK A LINK B LINK C 14. KEY WORDS ROLE WT ROLE WT ROLE WT Back Scattering Cylinder Double Loading __________________________________________________________________________I_______ _______ h _,.___. I INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal;author is an absolute minimum requirement. 6. REPORT DAT_: Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NUMBER(S): If the report has been assigned any other report numbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) (2) "Qualified requesters may obtain copies of this report from DDC." "Foreign announcement and dissemination of this report by DDC is not authorized. " (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through (4) "U. S. military agencies may obtain copies of this report directly from DDC Other qualified users shall request through,, (5) "All distribution of this report is controlled. Qualified DDC users shall request through,f If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11. SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. L Unclassified Security Classification

UNIVERSITY OF MICHIGAN 3 9015 02829 6203i 3 9015 02829 6203