THE UNIVERSITY OF MICHIGAN 5548-1-T THE MINIMIZATION OF THE RADAR CROSS SECTION OF A CYLINDER BY CENTRAL LOADING by Kun-MuhLx-and Valdis V. Liepa i April 1964 Scientific Report No. Contract AF 19(628)-2374 Project 5635 Task 563502.Prepared' for L.. Hanscom Field Bedford, Massachusetts,::.;. /.-.

fip" L THE TUNIVERSITY OF MICHIGAN.AN.. 5548-1-T ey ) U-MetJ' Requests for additional copies by of Defense, their contractors, and other be directed to: agencies of the Department government agencies should DEFENSE DOCUMENTATION CENTER CAMERON STATION ALEXANDRIA, VIRGINIA 22314 Department of Defense contractors must be established for DDC services or have their 'need-to-know' certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to: U.S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D.C. I I ii

THE UNIVERSITY OF MICHIGAN - 5548-1-T TABLE OF CONTENTS INTRODUCTION 1 I. THE MINIMIZATION OF THE CROSS SECTION OF A CYLINDER BY CENTRAL LOADING (BROADSIDE ASPECT) 4 1-1. Induced Current on a Center-Loaded Cylinder 4 1-1.1. Basic Equation and Solution 4 1-1.2. Induced Current on a Cylinder Without Central Loading 13 1-1.3. Induced Current on a Cylinder with an Infinite Midpoint Impedance 16 1-1.4. Induced Current on a Cylinder of Near-Resonant Length with Various Central Impedances 21 1-1.5. Induced Current on a Cylinder of Near AntiResonant Length with Various Central Impedances 26 1-2. Current Measurement on a Center-Loaded Cylinder 29 1-2.1. Experimental Setup 29 1-2.2. Experimental Results 35 1-2.3. Equivalent Circuit for Coaxial Cavity and Gap Capacitance 41 1-3. The Radar Cross Section of a Center-Loaded Cylinder 44 1-3. 1. Optimum Impedance for Zero Broadside Back Scattering from a Thin Cylinder 46 1-3.2. Scattered Fields of a Center-Loaded Cylinder 49 1-4. Summary 53 II. THE MINIMIZATION OF THE CROSS SECTION OF A CYLINDER BY CENTRAL LOADING (ARBITRARY ASPECT) 57 2-1. Induced Current on a Center-Loaded Cylinder Illuminated by an Obliquely Incident Plane Wave 57 2-1. 1. Integral Equation for the Induced Current on the Cylinder 58 2-1.2. Symmetrical Component of the Induced Current 62 2-1.3. Antisymmetrical Component of the Induced Current 69 2-1.4. Numerical Results 72 2-1.5. Comparison Between Theory and Experiment 78 iii

THE UNIVERSITY OF MICHIGAN 5548-1-T Table of Contents (Cont'd) 2-2. Back Scattering of a Center-Loaded Cylinder Illuminated by a Plane Wave at an Arbitrary Angle 80 2-2.1. Back Scattered Field of a Center- Loaded Cylinder 80 2-2.2. Measurements of the Back Scattered Field of a Center-Loaded Cylinder 83 2-2.3. Comparison Between Theory and Experiment 92 2-3. Summary 98 ACKNOWLEDGMENT 99 REFERENCES 99 I iv

THE UNIVERSITY OF MICHIGAN 5548-1-T INTRODUCTION Many investigations have been made concerning methods of reducing the radar cross section of metallic bodies, especially with regard to applications to radar camouflage techniques. Two methods have been widely used: the first utilizes radar absorbing materials, the second consists in reshaping the body to change the reflection pattern. A third method, known as the method of reactive loading, is the subject of investigation of this report. Only the case of back scattering is considered, and all references to cross sections are to be understood as such. The first known use of reactive loading to minimize the back scattering cross section was made by lams (1950) who applied the technique to metallic posts in a parallel plate pillbox structure. Shortly after this Sletten (1962) employed the method to decrease the radar cross section of objects in space. o Several authors (King, 1956; Hu, 1958; As and Schmitt, 1958) have studied the problem of cross sections of a cylinder with and without a central load. Although these investigations indicated that the cross section of a half-wavelength cylinder can be significantly reduced by the use of a high reactive impedance load at its center, the exact way in which the reactive loading behaves, as well as the optimum method (i.e. that loading which minimizes the cross section) of loading are still not well understood. This report has two purposes: (1) to develop a theory to explain the behavior of the cross section of a cylinder with loading; and (2) to determine the optimum loading. The problem is studied by considering the currents induced in a body illuminated by an electromagnetic wave. We consider the case of a plane wave which illuminates a perfectly conducting cylinder whose radius is small and whose length is less than two wavelengths. The plane wave induces a current on the cylinder; II I 1

THE UNIVERSITY OF MICHIGAN 5548-1-T this in turn produces a scattered electromagnetic field. If an impedance is added at the center of the cylinder, the induced current is modified, hence so is the scattered field. It should be noted that there are three ways in which an impedance change can reduce the scattered field: (i) by reducing the magnitude of the induced current; (ii) by reversing the phase of the induced current over some part of the cylinder; and (iii) by the combination of (i) and (ii). This third way is the most effective for reducing the back scattering cross section. In fact, we shall show that with central loading it is possible to reduce the broadside cross section to zero. For a center-loaded cylinder the induced current is first determined as a function of the cylinder dimensions, the midpoint impedance andthe incident electric field. Using this solution we obtain an optimum impedance, i.e. an impedance which gives a minimum back scattering. In order to verify this solution experimentally, the induced current on loaded cylinders and the return from a cylinder whose impedance is close to the calculated value are measured. The experimental data for induced current and cross section areas are found to be in excellent agreement with the theoretical values. Throughout the study a resonant cylinder whose total length is equal to 0.43X (X = wavelength) and an antiresonant cylinder of total length 0. 85X are used as typical examples. When a plane wave is obliquely incident on the cylinder the induced current can be divided into a symmetrical and an antisymmetrical component. The symmetrical component is predominant for a resonant cylinder while the antisymmetrical component is predominant for an antiresonant cylinder. Although the midpoint impedance can have a strong effect on the symmetrical component of the induced current it does not affect the antisymmetrical component. For this reason central loading cannot appreciably reduce the large cross section lobes occurring at off-normal aspects for the case of an antiresonant cylinder. I 2

THE UNIVERSITY OF MICHIGAN 5548-1-T This report is divided into two parts. In Part I we consider the case of a center-loaded cylinder illuminated by a plane wave at normal incidence, and develop the basic theory of central loading. In Part II the case is generalized to cover incidence at any arbitrary aspect angle. In the interests of simplicity, the analysis in both parts is limited to the case of a thin cylinder. The case of a thicker cylinder and the effect of multiple loadings will be investigated in the future. MKS rationalized units are used in the analysis and the time dependence factor e is omitted. 3

THE UNIVERSITY OF MICHIGAN 5548-1-T I THE MINIMIZATION OF THE CROSS SECTION OF A CYLINDER BY CENTRAL LOADING (BROADSIDE ASPECT) We consider here only the cross section of a center-loaded cylinder for broadside aspects. We first determine the induced current and then investigate the scattered field. 1-1 INDUCED CURRENT ON A CENTER-LOADED CYLINDER 1-1. 1 Basic Equation and Solution The geometry of the problem is as shown in Fig. 1-1. A cylinder of radius a and length 2h is assumed to be perfectly conducting and illuminated by a plane electromagnetic wave at normal incidence with the E field parallel to the axis. At the center of the cylinder is connected a lumped impedance ZL. The dimensions of interest are - X < 2h < 2 X 4 2 2 a << 1 0 where X is the wavelength and Io the wave number. The second condition implies that the cylinder is thin, and allows us to assume that only the axial current is induced. 1-1. la Integral Equation for the Induced Current on the Cylinder In order to determine the induced current on the cylinder we apply an integral equation method. The incident tangential electric field is assumed to be Ein E (1.1) z 0 where E is constant along the cylinder. The tangential electric field at the cylinder surface due to the current and charge on the cylinder is I 4

THE UNIVERSITY OF MICHIGAN 5548-1-T z = h z ELi 26 Incident - EM Wave z = 1/4X < 2h < 2X 2a 2~<< 1 0 z = -h FIG. 1-1: CYLINDER WITH CENTRAL LOADING ILLUMINATED BY AN ELECTROMAGNETIC WAVE 5

THE UNIVERSITY OF MICHIGAN 5548-1-T Ea = _ _ jA (1.2) z az z where f is the scalar potential maintained by the charge and A is the tangential z component of the vector component maintained by the current. By using the Lorentz condition ~ = j - V ' A (1.3) i30 (1.2) can be expressed as E — +. (1.4) 2 az 2 The electric field maintained across the gap at the center of the cylinder is related to the voltage drop across the center by the relation r6 Egdz = V = ZLI (z=O) = ZLI (1.5) z Lz L ( J-6 where V is the voltage drop across the center load ZL and I is the induced curL o rent at the center of the cylinder. From (1.5) Eg can be expressed as Z Eg = Z I 6(z) (1.6) z L o where 6(z) is the usual Dirac delta function. If the cylinder is perfectly conducting, the tangential electric field at the surface (excluding the gap) of the cylinder vanishes. That is Ea + Ein =0 for 6< z < h and -h < z < -6. (1.7) z z At the gap, the electric field is continuous. Hence I --- I ll ----I-I ---I-II-Ill-6 _____ --- —-- II

THE UNIVERSITY OF MICHIGAN 5548-1-T Ea+Ein = Eg = Z I 6(z) for -6 < z < 6. (1.8) z z z L ' By combining (1. 7) and (1. 8) and making use of (1. 4) it is possible to obtain a single equation valid for the entire length of the cylinder: 2 32 2A + A = -j - [E- -ZLI 6(z)] (1. 9) -z 2z - z - Eo L o for -h< z < h. Equation (1. 9) is an inhomogeneous differential equation for A. Consez quently the general solution is expressible as the sum of a complementary function and a particular integral. A = [ cos30 z + C2sin3 z + 0(z) (1.10) z v 1 ~ 2 So 0 where, V is 1/TfE, C and C are arbitrary constants, and 0(z) is a particu0 o ' 0 1 2 lar integral which can be written as Oz 0(z) = \ [E (s) - ZLI6(s) sin3(z -s)ds Jo E 1 = 0 (1 -coso3 z) - - Z I sin3 | (1.11) Ifo 2 Lo j 0 If E is assumed to be constant along the cylinder, the symmetry implies that C2 o0 must be zero. Equation (1.10) then becomes A (z) = Cl Ccos 3 z+ - (l -cos3 Z) - - ZLI sino 0z (1.12) 1 cOS~oz v l o p (l c oSfor 2-h L z o h for -h < z ~< h. 7

THE UNIVERSITY OF MICHIGAN 5548-1-T From (1.12), C1 can be expressed in terms of A (h) as 1 z E C, = sec 3 h jv oAz(h) - (1 - cos 0oh) + 2 ZLI sinf30h (1.13) Thus from (1.12) and (1.13) we obtain the following equation: zAh) 'A E A (z)- A (h) = sec h (jv A (h) - )(cosf3 z - cos 3 h) z z v o z o0 0 + 2ZLI sino(h-IIz) for - h< z<h. (1.14) On the other hand, the left side of (1.14) is also related to the induced current by /h A (z)-A (h) = A I (z')Kd(z,z')dz' (1.15) -h where Kd(z,z) K (z,z')-K (h,z'), (1.16) -jf z(z-z') +a2 K (z,z') = e 2i (1.17) a ((z-z') +a and I (z') is the induced current on the cylinder. z By equating (1.14) and (1.15) we obtain an integral equation for the induced current on the cylinder. 8

THE UNIVERSITY OF MICHIGAN 5548-1-T h I(z')K (zz)dz' = -jr sec 3 h (jv A (h) - — )(cosf3 z-cosf3 h) \ z d O z oZ J-h o o + 2 ZLI sin, (h- Izz (1.18) where ' = 1207r and (1. 18) is valid for -h< z < h. Both A (h) and I in the right side of (1.18) are functions of I (z) and are still unknown. z 1-1. lb Solution for the Induced Current on a Cylinder The kernel K d(z, z) in (1.18) has a sharp peak at z'=z. Moreover, it can be shown numerically that the left side of (1.18) is nearly proportional to I (z) for -h,< z < h. Hence we may assume I (z) = C (cosS 0z-coso3 h) + C sin/3 (h- IzI) (1.19) where C and C are constants to be determined. To obtain approximations to c s C and C it is reasonable to divide (1.18) into two parts: C C (cos Z,- z cosf oh)K (z z)dz1 = seco3 h(jv A (h)- - ) ' 0 0 * (cosl z-cos 3 h) (1.20) Ch \ sin 3((h- zKzz)dz = sec3hZIsin (h- [z[). (1.21) J-h Both (1. 20) and (1. 21) are valid for -h < z < h. They also agree at the end points, z = +h, since both sides of the equations become zero at these points. To find the constants C and C we can match both sides of (1.20) and (1.21) at the center of c s the cylinder, z = 0. 1 9

THE UNIVERSITY OF MICHIGAN 5548-1-T From (1.20) E C = -j4g sec3 h(jv A (h) - -2)(1 - cosf h) (1.22) c Tro o z 0 o cd o where h, cd = (cos/3 - cos3 h)Kd(0,z')dz' (1.23) -h From (1.21) c -sj2 sec hZIOsinl h (1.24) o sd where h T d \ sinp 3(h- Iz'I)Kd(0,z')dz (1.25) J-h It should be noted that if C and Cs had been evaluated using a value of z other than z = 0 the values for C and C would be relatively little affected. The c s substitution of (1.22) and (1.24) in (1. 19) gives 1 I~E I (z) - 7 T (jvoAz(h) - - )(secf3 h- l)(cos /3z - cos 3 h) o - cd 00 1 2T ZLIotanf3 hsind o(h- |z ]. sd -'(1.26) In order to obtain the final form of the solution we must still determine A (h) and I in (1. 26). By setting z = 0, I = I (z = 0) and we can express I z 0 ''0 Z 0 I 10

I --- THE UNIVERSITY OF MICHIGAN 5548-1-T in terms of A (h). A straightforward rearrangement of (1.26) then yields z I (z) = -ji4 z 0 (jv A (h)O z E 2) M cosI z-cosP h)+N'sinfo(h- [zi) o (1.27) where M' = - (secr3 h-l) cd -Z tan3 h(secf3 h+cosf3 h - 2) N' = Tcd ZLtan oh sin P0h - j60T cdT sd (1.28) (1.29) In (1.27) the only remaining unknown is A (h). To determine it, we use the z definition of the vector' potential A (h) = - Z 47r 1h -h I (z')K (h,z')dz' z a (1.30) where K (h, z') is defined in (1.17). Substituting (1.27) in (1.30) gives a jE A (h) = - z v 0 00 M'T +N'T ca sa 1- M'T - N'T ca sa (1.31) where T h T = \ ca \ J-I (cos3 z' - cos l h)K (h,z')dz' (c~So o a h sinf3 (h- |z'|)Ka(h, z')dz' -h (1.32) T - sa (1.33) 11

I THE UNIVERSITY OF MICHIGAN —, 5548-1-T A final form for the solution of I (z) is then obtained by substituting (1. 31) into (1. 27). After some rearrangement we have jE ] z) 30l0 (cos Ph-MT -NT S) [ (co z 3)(- )] (1.34) where 1 M = Td(1-cosf3 h) (1.35) cd 0 -Z sinf3 h(I-cos3 h) 2 N= (1.36) cdT Z sin 23oh-j 60T T cosf h dE o0 cd sd o and T cd T sd T, T are defined in equations (1.23), (1.25), (1. 32) and (1.33) cd sd' ca sa respectively. Equation (1. 34) gives a complete expression for the induced current on a cylinder with a central load ZL when illuminated by a broadside constant electric field E. Moreover, (1.34) is both simple in form and accurate. That it agrees well with experimental values will be shown in section 1-2. For completeness as well as convenience in computations, the integrals T c Tsd T and T are reformulated as cd sd ca sa Tcd = C a(h, 0)-C a(h, h) - cos h E (h, 0) - E (h, h) (1.37) Tsd = sin3oh [C(h, 0) - C (h, h) - cos 3Sh [S(h, 0) - S (h, h) (1.38) T = C (h, h) - cos/3oh Ea(h, h) (1.39) ca a 0 a T = sinf3 hC (h,h)-cos3 hS (h,hh) (1.40) sa o a o a I 12

THE UNIVERSITY OF MICHIGAN 5548-1-T where rh C (h, O) = \ a J-h rh C (h,h) = \ a ~-h h E (h, 0) = a J-h h E (h,h) = \ J-h rh S (h, 0) = -h rh S (hh) = \ J-h cos/3 z'K (0,z')dz' o a cos 3 z'K (h,z')dz' o a K (0, z')dz' a (1.41) (1.42) (1.43) (1.44) (1.45) (1.46) K (h, z')dz' a sin [z'1 K (0, z')dz' sin z Kh )d sin/o!z' K (hz')dzt o a The integrals of (1. 41) through (1.46) can be calculated on a digital computer. We now consider a number of examples. 1-1.2 Induced Current on a Cylinder without Central Loading The first and simplest case is that of a cylinder without loading. By setting Z = 0 the induced current can be found directly from (1.34): L jE / 1-cosO3 h \ I (z) 0 T = cos h-T (os z - cos j h) O K cd ca o ca Using (1.37) and (1.39) to express Tcd and T, I (z) can be written jE (l-cos13 h)(cosf3 z-cosf3 h) I (z) = 0 0 0 0 30 0LC (h,0)cos13 h-E (h,0)cos 23 h-C (h,h)+E (h,h)cos/3h: a o a o a a o (1.47) 13

THE UNIVERSITY OF MICHIGAN 5548-1-T In this case the distribution of the induced current is that of a shifted cosine curve. The maximum induced current occurs at z = 0 for h < 2 }2 l (1.48) z = X/2 for 2 < f0h < 27r and is given by jE (1 -cos h)2h I (0) 3 — h- (o)2 o C (h, O)cos/3 h-E (h, O)cos j h-C (h,h) + (h,h,h)cos3 h1 a o a o a a o (1.49) or 2 jE -sin 3 h I (x 2) -- z30o C (h, 0)cos3 h-E (h,O)cos P h-C (h,h)+E (h,h)cos30hJ a o a o a a o (1. 50) Theoretical and experimental results of I (0) as a function of h are comz pared in Fig. 1-2. The theoretical results were obtained from (1.49) by the use of a high speed computer. The experimental results were obtained by measuring the current induced at the center of a cylinder whose radius is 3/16" and whose length varied between X/4 and 2 X. The cylinder was illuminated by a plane wave with a frequency of 1.088 Gc. Except near h = 0.7 X, where a resonant peak occurs, the agreement between theory and experiment is good. The discrepancy near h = 0.7 X may be de due to a deficiency in the theoretical treatment or it may have its origin in the experiment, i.e. the incident electromagnetic wave may not be uniform along the cylinder for all cylinder lengths. 14

- Theory o o o Experiment E o I (0)" z T h -2a N 1-q o4-q 0 0).1 P^ H Fci z -4 C0 zo O >4 Z- 0.6, 0.4 0.5 h/X -- 0.6 0.7 0.8 FIG. 1-2: I (0) vs h/X WHEN 3 a = 0.11, f= 1.088 Kmc ZL=O Z 0

THE UNIVERSITY OF MICHIGAN 5548-1-T However, in view of the good general agreement the discrepancy does not appear serious. In Fig. 1-3 the current distributions for cylinders of various lengths are shown. Good agreement between theoretical predicution and experimental results is evident. For a cylinder whose length is 0.43 X or 1.4 X the induced current reaches a resonant peak for i3 a = 0.11. Since these current peaks imply large radar cross sections we propose to decrease the cross sections by eliminating these current peaks by using suitable impedance loading. This will be carried out in section 1-3.1. Before proceeding, we consider another example. 1-1.3 Induced Current on a Cylinder with an Infinite Midpoint Impedance The second case to be studied is that of a cylinder with an infinite midpoint impedance. Theoretically, the induced current can be obtained from (1.34) by setting Z = co. Experimentally, an infinite impedance is approximated by a coaxial L cavity tuned at its antiresonant position. This coaxial cavity is built inside the cylinder as described in section 1-2.1. A small probe then measures the induced current. When ZL= o, we have from (1.35) and (1.36) N -(1 - cos oh) M sin: h ' for 3 h ~ nor. (1.51) M sin h ~ Substituting (1.51) into (1.34) then gives -jE I (z) = z 30f3 (1-cos3oh) [sinl3 o z |- sin3 h+ sinf3 (h- z |)] sin3h [C (h, 0)-(2-cos3 h)C (h, h)-cos3 h E (h, 0) + E (h,h) -(1-cos h)2S (h,h) 0 oa o a o a a o a (1.52) 16

Theory h/X=0.258 H** 1.0 0.8 H )T1 z M N I' a) 0.6 0.4 z Cl) Ul " 0 O H0 PT 0.2 0 C) "-4 0 z 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 z /x 0.4 FIG. 1-3: CURRENT DISTRIBUTION ON UNLOADED CYLINDERS

THE UNIVERSITY OF MICHIGAN 5548-1-T In this case the induced current is zero at the center of the cylinder and its distribution along the cylinder is a combination of a sine and shifted sine curves. The maximum induced current occurs at z = h/2 for j h < 27 (1.53) and is given by -jE I(h/2) = 30 (1.54) 30 o 0 (3 h j3 h 2 sin - (1 - cos - )(1 - cos f h) Lin h [C(h, 0)-(2-cos3 h)C (h, h)-~osS h E (h, 0)+E (h, h) -(1-cosl h)2S (h, h) 0 oa a a a o a The theoretical value of I (h/2) as a function of h/X is compared with the experimental curve in Fig. 1-4. The agreement between the theoretical predictions and the experimental observations is only fair. This lack of agreement is probably due to the difficulty in experimentally obtaining an infinite impedance from a coaxial cavity structure. The fact that closer agreement between theory and experiment is obtained for ZL= j 2000 Q tends to support this explanation. For three cylinders of different lengths the induced current distribution for the case of infinite impedance loading is shown graphicall in Fig. 1-5. The agreement between theory and experiment is good. Moreover, with infinite impedance loading at the center, the induced currents at resonant lengths (namely 2h = 0.43X and 2h = 1.4X) are greatly reduced. However, the induced current appears to have a peak when 2h = 0.9 X, and if a small cross section over a wide frequency band is desired, this current peak must also be suppressed. Thus we conclude that an infinite (or very high) impedance is not optimum for minimizing the scattering over a wide range of frequencies. We shall take up this problem again in section 1-3.1. I I 18

1.4 1.2 Theory 1.2 --- -Theory o o o Experiment Iz(h N 0 (0 Q-A -4~ (with a cavity of er=4 1.0+ 2 =6.34 cm, 26=0.127 cm) E 0 H- z ci: ZL= o 0.8 0.6 0.4 ZL=j2000 Q 0 \ 0 / o 0 0 0 0 z 0.2 0 0 0.1 0.6 0.7 0.8 0.9 1.0 FIG. 1-4: Iz(h/2) vs h/X WHEN 3o=0.11, f=1.088 Kmc, ZL= LARGE.

mdmmmm Theory F '-I ' A - - I -I___ 0* * 6-lQ 1.0 0.8 0.6 A A A Experiment 0 00 2z = 6.34cm 26 = 0.127cm - wT )z O0 H h/X = 0.326 OI: OU ul cc H 0 A I 0 PTI 0.4 0.2 h/X = 0O.2M9 -n / / A Lu / p0~ 0 z / / / ~0 9 h/X = 0.213 4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 z /i FIG. 1-5: CURRENT DISTRIBUTION ON LOADED CYLINDERS WITH Z = oo L

THE UNIVERSITY OF MICHIGAN 5548-1-T 1-1.4 Induced Current on a Cylinder of Near-Resonant Length with Various Central Impedances In this section we study the induced currents when the cylinder length is chosen to be a resonant length. For experimental convenience we select the following specific case: a - 0.173X 2h = 0.43X ZL= jXL This last condition restricts the central impedance to be purely reactive, and is chosen because only a reactive impedance can be obtained experimentally from a coaxial cavity. With these conditions the theoretical value of the induced current can be expressed as jE 0 (cosf3 z-0.216)+ sin(77. 50-/3z ) I (z) - o M L -M (1.55) Iz 30J 0 MjO.215- -(0.218-j0.25) where -0.765X -N -0.765_ L /(1.56) M 0.955X -24.6 Values of I (z) are calculated for values of ZL equal to 0, oo, -j 1600 Q, -j 8001Q, -j 600Qf, j 1600Q2, j 800Q, j 600 2 and j 400Q and shown in Figs. 1-6a and 1-6b. From the graphs we may observe the following: (1) When Z L= 0 (no loading) the induced current is very large and is distributed along the cylinder as a shifted cosine curve. (2) When ZL= oo the induced current is greatly reduced in magnitude. Its distribution curve becomes double-humped with a null at the center. (3) When ZL is capacitive and finite the induced current is smaller than the case of Z L= 0 but larger than for Z L= o. -L L 1 21

|ZL=O ZL= I 6 // \" /" \ I/ \ /ZL=-j8sOO2\ 5./ \ / \ / 1/ Nx4 \ I 3 4 ( I \ i I \ I N I ^ ' r t I 0 / - < a.) 4 \ \ v / I-.-I, o (pI / x / /\z '~3 X I /, S.. '* V 7 *ZL= j8000-. \ 7 I 6 4' * \0 ID.*. / \ 0*. /~~~~~~~, I:'. *~ 0. 0 0 ** *~~ ~ ~ ~ ~~~~~~~~ *.0 ** 77.50 600 40~ 20~ 0 200 400 600 77.50 FIG. 1-6a: CURRENT DISTRIBUTION, Iz(z), ON CYLINDER OF h=0. 215X, a=0.0173X FOR DIFFERENT CENTRAL LOADS, ZL (THEORETICAL) H < ci c-H 0I zH > Zlp

L=-j600 \ A / \ ZL=-jl6OOQ -N4/A, Z L= j 1600Q 12 / / Z A \ z./, \/ -=-4 / ' \ V / 'I\ < G / L ia y \j: \. Cm I / ' / \c,\H FIG. 1-6b: CURRENT DISTRIBUTION, /z(z) ON CYLINDER OF h=\215, a=0.0173 < / * \ * \ / N y2-: \ I' ~ \ I 2 0-. \ I /\. I / * * * o 1 o/ ' / \ A / \A '.**. \) ~400~ **\* *Z \:.!\ /...".. / I/ \.."'-.. \' o z \\. ** z

THE UNIVERSITY OF MICHIGAN 5548-1-T (4) When ZL is inductive and finite, the magnitude of the induced current is smaller than the case of Z = oo and is distributed in the form of three loops L along the cylinder. It is of interest to note that the phase of the current at the center loop is reversed. The most significant result is (4). From it we conclude that it is possible to reduce the broadside back scattering from a cylinder to zero by properly adjusting the value of ZL. As we shall see in a later section (1-3), the optimum impedance for zero broadside back scattering with a cylinder of this size is inductive and has a small resistive component. To compare the theoretical predictions of Figs. 1-6a and 1-6b the induced current along a cylinder of the specified dimension and with various cavity lengths was determined experimentally. The results are shown in Figs. 1-14a and 1-14b. The experimental curves closely resemble their theoretical counterparts. When the cavity length (total length) is longer than 6.2 cm, the impedance of the cavity is capacitive; when the length is less than 6.2 cm, the impedance is inductive. It should be noted that the effective cavity length is greater than these values since it is loaded with a dielectric material for which E = 4.0. The approximate value of r the cavity impedance is calculated by using a standard impedance formula for a transmission line and assuming that a capacitance of 0. 4 Mf is shunted across the gap at the center of the cylinder. The comparison between theory and experiment is made in Fig. 1-7, where theoretical curves for Z = -j 800 Q, oo, and j 800 Q are shown. These curves L are compared with experimental results for L = 3.32 cm, 3. 10 cm and 2.91 cm (where fj is the half-length of the coaxial cavity). The agreement between theory and experiment is very good, indicating that the calculated value of the cavity impedance is quite close to the corresponding theoretical impedance. 24

THE UNIVERSITY OF MICHIGAN 5548-1-T h = 0.215Xo -- Theory, ZL= oo 2a = 0.92 cm A Experiment, 1=3.10 cm Xo = 27.57cm --- Theory, ZL=j800Q2 26 = 0.127cm o o o Experiment,1 =2.91cm Theory, ZL=-j800Q2 * * * Experiment, -=3.32cm 226 V^ ---- h 16 14 z L: -jSoOQ o, - 28 20~ 0 20~ 40-0~ 80 Q___y 17 / __ I'' ' 80- 600 400 200 0 200 400 600 80 -Poi?PZ FIG. 1-7: CURRENT DISTRIBUTION ALONG A CYLINDER AS A FUNCTION OF CENTRAL LOAD FOR h=0.215Xo (THEORETICAL AND EXPERIMENTAL) 25

THE UNIVERSITY OF MICHIGAN 5548-1-T 1-1.5 Induced Current on a Cylinder of Near Anti-Resonant Length with Various Central Impedances In this section we study the induced current when the cylinder length is an antiresonant length. The following specific case is chosen: a= 0.173X 2h = 0.9X ZL = jXL where we again consider only the reactive loading case. For these conditions the theoretical value of the induced current is jE ~(cos3o z+0.951) + N sin(162~- \o zl) ~ I (z) -3 MLM1 (1.57) z N ' (1.57) o 0 -0.911 + j 0.217 - (0.111- j 0.128) where -0.604X M 0.096kL+ 71 - j52.2 (1.58) The magnitude of I (z) is calculated for values of ZL equal to 0, oo, -j 1600 2, -j 800 2, -j 600 Q, j 1600Q, j 800 Q2, j 600 Q and shown graphically in Figs. 1-8a and 1-8b. This family of curves is quite different from those of the preceeding section. Although a purely reactive impedance reduces the magnitude of the induce current and tends to reverse the phase of the induced current, it is not possible to reduce the radar cross section to zero because current nulls do not occur in this case. Actually, as will be shown in 1-3.1, the optimum impedance for zero broadside back scattering from a cylinder of this size should have a large resistive component. I J 26

7. N N 0 -e — "r-4 "-4iC3) /SL=-o \ / 'ZL= -j800 Q...\ / N -0* N. cC) O) 0 I z P-tl 0 A_4 / I / / / / 0 0 -q**..# 0 0 0 t 0 0 0 0 0 / 400 [loz IPozl FIG. 1-8a: CURRENT DISTRIBUTION, Iz(z), ON A CYLINDER OF h=0.45X, a=0.0173X FOR DIFFERENT CENTRAL LOAD (THEORETICAL)

'S3- - \ / / \ _ = 1 l / ' L -j 600Q -. \\ D /I a. //,* /. V v V....*-"*.\ /^. 2/ z=j600o2 '...-** * * ~ S 1- - 0/ L% ' -. 0 oz FIG. 1-8b: CURRENT DISTRIBUTION ON A CYLINDER FOR DIFFERENT CENTRAL LOAD h = 0.45X, a = 0.0173X (THEORETICAL)

THE UNIVERSITY OF MICHIGAN 5548-1-T In Figs. 1-15a and 1-15b the experimental results for the currents induced on this cylinder are summarized. The general shapes of the experimental curves are similar to the theoretical curves. A comparison between theory and experiment is made in Fig. 1-9 for three typical theoretical curves and their experimental coun terparts. The agreement between theory and experiment is again good, although not as good as in the case of a shorter cylinder. 1-2 CURRENT MEASUREMENT ON A CENTER-LOADED CYLINDER 1-2.1 Experimental Setup A block diagram of the equipment used for the current measurement is shown in Fig. 1-10. The cylinder was illuminated at broadside by a plane wave of 1.088 Gc from an L-band horn antenna with the electric field polarized in the direction parallel to the cylinder. A conventional probing method with a small current probe was employed to measure the induced current amplitude on the cylinder. The coaxial line leading from the probe was covered with radar absorbing material (RAM) and oriented perpendicular to the E field to minimize its interaction with the E field. The measurement area was also lined with RAM to reduce unwanted reflections. This is shown in Fig. 1-11. Figs. 1-12 and 1-13 show partially disassembled components of the loaded cylinder. The cylinder diameter is about 0.95cm and its length can be changed from 10Ocm (h = 0.182X) to 51.29cm (h = 0. 93X) by the combination of center and end pieces of different lengths. The center sections of the cylinder contain a symmetric coaxial cavity with an input gap at the center of the cylinder. The diameters of the center and outer conductors of the cavity are 0.32 cm and 0. 79 cm respectively. By varying the cavity length, various input reactances are obtained. The coaxial cavity is filled with a dielectric (Stycast HiK: c = 4, 6 - 0. 0001) in order to reduce the cavity length to fit within the cylinder. J 29

THE UNIVERSITY OF 5548-1-T MICHIGAN 16 14 N 0.14 12 10 8 6 4 2 0 40~ 0 40~ lUozI < I - K A FIG. 1-9: CURRENT DISTRIBUTION ALONG A CYLINDER AS A FUNCTION OF CENTRAL LOAD FOR h=0.45XO (THEORETICAL AND EXPERIMENTAL) 30

Horn Pr Cylinder Tuner obe M C (1 Coaxial u i Line A 4t '~I CO I-. Frequency Meter Meter HP 415B Detector z FIG. 1-10: BLOCK DIAGRAM OF THE EXPERIMENTAL SETUP

probe positioning r==F3? _3 - C carriage cylinder high performance pyramid type absorber current probe H an n z -* m e2v 0-4 01 0 Cf) 0 z 0-4 o z 2;4 3m E Y7 L-band horn antenna hairflex absorber Side View Top View FIG. 1-11: ROOM LAYOUT

Cavity Length, L slot for tighter contact 1/32'~- -gap width 26 1/8" H ffi z r TI ct t I. c0 01 0-^. ^H ft End Extension Pieces Pieces 2 13' 5" 2pcs: 116 I, 16 - 3" 2 -4 13 pcs: 0.177 to 3" center section loaded cavity gap Material: Brass Dielectric: Stycast HiK (e = 4) Minimum Dipole Length = 10.03 cm = 0. 362 X at Maximum Dipole Length = 51.19cm = 1.91X at 1.088Gc (6 = 0) 1.088Gc (6 = 0) C) 0 z FIG. 1-12: CYLINDER WITH VARIABLE OVERALL LENGTH AND CAVITY LENGTH

THE UNIVERSITY OF MICHIGAN 5548-1-T........ ~~~~~~~~..)A z ee.[:::: i ~~~~~~~~~~~~': ' i X:t: i; E;.:~ ~ ~ ~ ~ ~ ~ ~' - t:S: f:: iS~j: ff;F}XSft;;jj):; DSSE: W030 - fF: ffI::E; ILL;::; S~~~~~~~~i led'~~.. -qo X i 34

THE UNIVERSITY OF MICHIGAN 5548-1-T 1-2.2 Experimental Results Figs. 1-14a and 1-14b show the cylinder current distributions with h = 0.215 X for various central loadings. The curves are given in terms of the cavity length L. The central impedance ZL is not explicitly defined here, but its relation to the cavity length L is given in the next section (1-2.3). When L = 0, ZL= 0; ZL achieves its maximum ("infinite") when L is about 6.21cm. For 0 <L <6.21 cm ZL is inductive while for L > 6.21 cm but less than a certain critical length, ZL is capacitive. The maximum current on a cylinder with h = 0. 215X and Z L= 0 corresponds to a resonant current. The relative amplitude of the current is normalized to this maximum current. For the case of L = 5.83 cm the current distribution is close to optimum, i.e. the cross section is close to zero. For other cavity lengths the current distributions are also in good agreement with the theoretical predictions. In Figs. 1-15a and 1-15b we have the current distributions for a cylinder with various cavity lengths and h = 0.444X. This particular cylinder length corresponds to an antiresonant length. The manner in which this affects the induced current may be seen from the graphs: the introduction of a central impedance increases, rather than decreases, the induced current in general. This implies an increase in back scattering. Although the current distribution corresponding to L = 5. 58 cm is close to the optimum distribution, it is impossible to reduce the broadside cross section to zero by purely reactive loading. This is due to the fact that the phase of the current is not completely reversed at the center part of the cylinder and no current null appears with this loading. The asymmetries which exist in the measured currents are caused by room reflections. Fig. 1-16 shows the current distributions on a cylinder with fixed central impedance but whose length varies from 2h = 10.03 cm to 2h = 39.73 cm. The I __ 35

Relative Amplitude of II(Z) I 0 0 0 0 0 0 0 ^S" ~~~~0 S~~~~~~~~~~~~~~~~ 0t — 0 0] oooooo~ e OII. \ 0 Q s;,0-* / / "to. ~.. \ ' C.0 nO ^00 0 ' o H rtl;dI cr) U> 00 o Q4 FIG. 1-14a: CURRENT DISTRIBUTION ON A CYLINDJER UO' hl=U.zioA, a=u.ul 5iA WITH CAVITY LENGTHS, L (cm) (EXPERIMENTAL

THE UNIVER SITY OF MICHIGAN 5548-1-T 0.6 0.5 0.4 -N N 0.3-.-4 a < aC) i-. ct L = 4.32cm L = 6.72cm 5^ -~, 0o - / / 6 0 0 0 0 e \ ~ i \ 0 * \ / 0 L = 6.09cm \ 0. 0-h of400 U I I *. '.9 L4 ~., Ns * 0O /. 5. 58cm * I.*. a / 0 * 0o 0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.5 Position Along Cylinder (in.) 1.0 1.5 2.0 2.5 FIG. 14b: CURRENT DISTRIBUTION ON A CYLINDER WITH VARIOUS CAVITY LENGTHS h = 0.215)t, a = 0.0173X. (EXPERIMENTAL) 37

Relative Current Amplitude H z m 0-4.e CD+0:o CO E-D 00 OQ. CD Cr) 1-i. 0 ^= (D. Ca~ Ul ^ 00,_~ I 0 Pt1 0 z FIG. 1-15a: CURRENT DISTRIBUTION ON A CENTER-LOADED CYLINDER WITH VARIOUS CAVITY LENGTHS. h=0.444Xo, 26=0.050 in, cr=4. CURRENT SCALE IS RELATIVE TO MAXIMUM CURRENT (1.0) ON h=0.215X0, UNLOADED CYLINDER (EXPERIMENTAL)

Relative Current Amplitude oo ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~-. * 0 ~ CC Cl 01 CURRENT (1.0) ON h 25 UNLOADED CYLINDER (EXPERIMENTAL) O. I '~''.. ) ~ I. __ "-, — —.......-'CURRENT (1.0) ON h= 0. 215.o UNLOADED CYLINDER (EXPERMENTAL) - z - ci H0. m;o - c) cp " I-A ^-i0 PTl P-4 0 z

The current scale is relative to the maximum current (1.0) on an unloaded cylinder; h = 0.215 X Relative Current Amplitude 26 = 0.05in. = 4 L = 6.34 (Z Cylinder A0 A2 A4 A7 A12 BO B1 B3 B5 C3 Cylinder Length, 2h 10.03 cm 11.77 cm 14.29cm 18.00 cm 24.46 cm 22.90 cm 23. 50 cm 25.80 cm 28.36cm 39.73 cm H z ci?d CO) cn," 0 4 2O / -7 -6 -1 1 2 A 3 6 C3 B3 B1 A7 A4 AO A2 BO'A12 B5 Position along the cylinder, inches FIG. 1-16: CURRENT DISTRIBUTION ON CENTER LOADED CYLINDER FOR FIXED CAVITY LENGTH (EXPERIMENTAL)

I THE UNIVERSITY OF MICHIGAN 5548-1-T cavity length is kept at L = 6.34cm. The cavity length is rather arbitrary, but it appears to give a high central impedance. An accurate antiresonant length of the cavity was later found to be L = 6.21 cm. 1-2.3 Equivalent Circuit for Coaxial Cavity and Gap Capacitance An approximate value for the input impedance of the coaxial cavity can be obtained from the following observations. In Fig. 1-17a the dielectric loaded coaxial cavity is schematically represented. The input terminals are at the center of the outer conductor and a stray capacitance is assumed to be shunted across the input terminals. If a voltage V is applied at the input terminals the input current i is i = i + i (1. 59) t c where i is the transmission line current which flows inside the coaxial cavity and t i is the current which flows through the stray capacitance. c It is evident that i = jWC V (1.60) c S but the transmission line current i requires a more involved argument for its determination: As shown in Fig. 1-17b, the coaxial cavity can first be simplified to the case of a transmission line driven by a voltage V at the center of one wire of the line. This transmission line can then be considered as the superposition of two modes, as shown in Fig. 1-17c. Hence it = is+i a (1.61) where i is the current of the symmetrical mode and i is the current of the antisymmetrical mode. Since is can be assumed to be a usual TEM mode current I 41

THE UNIVERSITY OF MICHIGAN 5548-1-T V + Z,7/////// / /7 7 /A /1 /(//////// ( a). A 7, -.. —,-I l,/ X ', + v, (b) S v/2 + -07 V/2 + I 1+ 1. a 1 I I ->.- -o+ V/2 symmetrical mode + V/2 V/2 antisymmetrical mode t.s a 1 +1 FIG. 1-17: EQUIVALENT CIRCUITS FOR COAXIAL CAVITY 42

THE UNIVERSITY OF MICHIGAN 5548-1-T it can be obtained simply as i = V (1.62) jZ tan 3P 0 where Z is the characteristic impedance of the (assumed) lossless cavity, j3 is a propagation constant, and J, is the half-length. Furthermore, a coaxial cavity does not allow the existence of the antisymmetrical mode and so the antisymmetrical current is small. Thus, neglecting i the total input current i becomes i~i+i - + jWcV (1.63) t c jZ tan301 s The input impedance of the coaxial cavity is V j2Z tan f3. L i 1-2C C Z tanyL (1.64 so where Z is known to be 0 ~~~o and e is the dielectric constant of the dielectric in the cavity and r1 and r2 are the inner and outer cavity radii respectively. The value of the stray capacitance C in (1. 64) is difficult to determine either theoretically or experimentally and an s indirect method was used: We know that corresponding to Z L= oo the current distribution has a null at z = 0 and maxima at z = +th/2. From the current distributions of Figs. 1-14a and

i THE UNIVERSITY OF MICHIGAN 5548-1-T For the cavity used in the experiment, Z =27.2Q, = 27r x 1.088 x 10, e = 4, 0 r and we obtain from (1.6 6) C = 0.402,u/pf (1.67) S Using this value of C, other input impedances have been obtained as functions of L (= 21) L(cm) Z L (p) 5.07 j212 5.86 j 600 6.04 j 800 6.10 j 1600 6.21 joo 6.39 -j 1600 6.55 -j 800 6.64 -j 600 The effect of the gap width on the induced current is shown in Fig. 1-18. Since the amplitude of the induced current is greatly affected by the gap width, the stray capacitance at the cavity input cannot be neglected in the calculation of the input impedance. 1-3. THE RADAR CROSS SECTION OF A CENTER-LOADED CYLINDER In the preceding sections we found the induced current on a center-loaded cylinder as a function of cylinder dimension and the central impedance. We now proceed to study the scattering from such a cylinder and determine the optimum impedance for reducing the broadside cross section. I 44

U -i z Z ci -l1 C) 0 -3 -2 -1 0 1 2 Position along Cylinder in Inches vs Relative Current Amplitude FIG. 1-18: CURRENT DISTRIBUTION ON CENTER-LOADED CYLINDER FOR VARIOUS GAP THICKNESS, 26. CURRENT SCAL IS RELATIVE TO MAXIMUM CURRENT (1.0) ON UNLOADED CYLINDER h=0.215X, er=4, L=6.85cm.

THE UNIVERSITY OF MICHIGAN 5548-1-T 1-3.1 Optimum Impedance for Zero Broadside Back Scattering from a Thin Cylinder Since the induced currents give rise to a vector potential from which the scattered field may be computed, we use the expressions for the induced current in (1.34) through (1.36) to first determine the associated vector potential. For the far zone of the cylinder the vector potential is o jE \ -jf R -h 4 N 3013 h-MT -NT L (cos3~ o s 0 R dz ah -jhR + N \ ins3(h- IzI) e R dz -h - (1.68) where R = R - z cosO = distance between a point on the cylinder and the observation point. The scattered electric field in the far zone is then Es = -jwA = jwtA sinO. (1.69) 0 0 z and the corresponding Poynting vector is P=-W 1 EQ12 (1.70) The scattered field in the broadside direction is obtained when 0 = 90 and R = R. (1.71) 0 We then obtain __ o _ M(sin h - 30 h cos, h) + N(1 - cos 3 h) s o 2 e 0 0 0 E(0=90 ) - E R l cos h-MT -NT 2 (1.72) 0 o (o ca sa i 46

THE UNIVERSITY OF MICHIGAN 5548-1-T and - E2 M(sin/3 h- 3 hcos 3 h) + N(l - cos 3 h) 2 pso(0=90) 2 0 0 0 0 0 (1.73) P( = 2 22 cosl3 h-MT -NT o/ R o ca sa Equation (1.36) gives N as a function of ZL. Hence (1.73) expresses the Poynting power density as a function of the central impedance. Thus to reduce the back scattering (0 = 90 ) radar cross section to zero we simply make P equal to zero. This gives the condition N sinj3 h-(3 hcosf3 h M 1- cos/3 h (1.74) Using (1.35) and (1.36), (1.74) can be rewritten as ZLsin'/3 h(l-cosf3 h) sin:3 h —3 hcos3o h (1.75) Z sin /3 h-j60T cos/3 h 1-cos h L o sd o By solving for ZL in (1. 75) we obtain the optimum central impedance -j60T d (l-/30h cot 3 h) FZ(1 (1.76) [ZL 2cos/3 h-2+0 fhsinp h where T d is expressed in (1.38). sd Equation (1. 76) gives the complete expression for optimum central impedance as a function of the cylinder dimension. In view of its simplicity, the expression should prove useful in practical design. In Fig. 1-19 the calculated values of [ZL] for a cylinder with a = 0. 0173 X are plotted as a function of the cylinder length h/X. Certain observations are evident: (1) In general, the optimum central impedance for zero broadside back scattering should have both resistive and reactive components. 47

800 t 600 t I I I I I I I I I I I,J ~~~~~~~~~~I I Et ZL T h *2a H ffi z 400 RJo = RL+ jXL z co 200 + RL R o p 0 I I I I I -p _ p CO) 00 H0 PTJ 0 -200. I,I I I I I I I I I I I I [ZLo= -j60Ts(1- /oh cot:3h) 2 cos goh-2 + 3oh sinm oh 0 0o 0 0 z -400 - I 0.1 0.2 0.3 0.4 0.5 hA s 0.6 0.7 0.8 0.9 FIG. 1-19: OPTIMUM CENTRAL IMPEDANCE [Zi] OF RADIUS a=0.0173X AS FUNCTION OF FOR ZERO BACKSCATTERING FROM CYLINDER LENGTH CYLINDER

THE UNIVERSITY OF MICHIGAN 5548-1-T (2) For a cylinder shorter than one wavelength (h < 0. 5 X) the optimum impedance is inductive and requires a resistive component. (3) For a cylinder longer than one wavelength (h > 0. 5 X) the optimum impedance is inductive or capacitive but it requires a negative resistive component. These results indicate that for cylinders shorter than one wavelength a passive impedance can reduce the cross section to zero, but for cylinders longer than one wavelength an active impedance is required. However, in the latter case an active impedance may not be needed if the cylinder is loaded at two points. In Fig. 1-20 the cylinder parameter is changed to a = 0. 0517X and [Zo is again obtained as a function of cylinder length h/X. The similarity to Fig. 1-19 is evident. For the thicker cylinder the resistive component remains relatively constant but the reactive component is reduced-almost by a factor of two. Another property which can be studied from (1. 76) is the bandwidth characteristic of this technique. To do this we consider a cylinder with h = 4 cm, a = 0.476 cm and calculate the optimum impedance for a range of frequencies between 1 Gc and 3 Gc. The results are shown in Fig. 1-21. From the graph it is seen that within this frequency range the optimum impedance is inductive and requires a resistive component. For a wider range of frequencies an active impedance is needed. This impedance appears to be obtainable by a simple network synthesis. 1-3.2 Scattered Fields of a Center-Loaded Cylinder In this section we calculate the bistatic scattered field of a center loaded cylinder which is illuminated by a normally incident plane wave. When a center loaded cylinder is illuminated by a plane wave at normal incidence, the induced currents are given by equations (1.34) to (1.36). In turn, these induced currents give rise to a vector potential in the far zone of the cylinder. Using (1. 68) which expresses the vector potential and (1.69) which express the 49 --

400 - 300' 200 100 - XL I / / / I I,1 I I I I I I I I I I I I t I E 0 u~ H z M / / / L a = 0.0517X [z o= RL+iX L J QO C),.0 z - -100 --200 [Z -j 60T (1-3 hcot/ oh) 2 cos 3 h-2+f3 hsinf h 0 0 0 I I I I I / / /L / / I h/X - g 0.6 0.7 0.8 0.9 FIG. 1-20: OPTIMUM CENTRAL IMPEDANCE, [zLI FOR ZERO BACK SCATTERING FROM A CYLINDER o

1200 - 1000 \ E 6\~~~~~~~~~~~~~~~~~0 - h 4cm a= 0.476cm L ZL] = R +jX 600 X 400 - - 200 _ _ - I I I 0 1 2 Frequency (Gc) 3 FIG. 1-21: OPTIMUM CENTRAL IMPEDANCE, [ZLj, FOR ZERO BACK SCATTERING FROM A CYLINDER H z (i < CO 0) cfr z I H Z 0

THE UNIVERSITY OF MICHIGAN 5548-1-T scattered electric field we perform the indicated integration and obtain s -jRo e O E Mh si h( +cosE 0 47 R 3013 kcosp h-MT -NT ins /L3 h(l+cos0) o0 0o ca sa 0o + sin [13 h( - cosO)l sin(P3 hcosO) + -h(1 cos 0) -2cosi h cos0h j 0 0 N 2 cos(,3 hcos0)-cos 3 h) M h sin0 f ~~~o zJ (1.77) By plotting E0 as a function of 0 we obtain the bistatic scattered field. An example will be considered. For the case of a cylinder of resonant length we choose the following parameters: h = 0. 213X, T = 3.17-jO0.327 T = 0. 696-j 1.071 T = 0.62-j0.928 ca T d = 3.63-jO.377 sd a = 0.0173X where the T's are calculated with the aid of a computer. We choose several different central loads: ZL = 0, o, j 800 2, -j 800 Q, -j 600 Q2, j 626 Q and the optimum central load Z = 65+j 626Q (which can be obtained from Fig. 1-19 or from (1.76)). 52

I THE UNIVERSITY OF MICHIGAN 5548-1-T For these seven cases of Z the back scattered field (0 = 90 ) are calculaL ted and listed in Table I. In the table the field strengths are normalized to the value of E(0O= 900) when Z= 0. 0 L TABLE I Field Impedance te e nPower L Es(0=900) L db 0 0 1.0 1.0 0 co.0841.00707 -21.5 j800.022.000484 -33.2 -j800.149.0222 -16.5 -j650.144.0207 -16.8 j626.00943.0000889 -40.5 65+j 626 0 0 -oo In Fig. 1-22 is shown the bistatic field patterns for the cylinder with ZL= 0, j 800, -j 800 Q and co Q. Although the scattered field is greatly reduced by the introduction of the center loading, since none of these is the optimum loading, the scattered field can be further reduced. In Fig. 1-23 where the bistatic field patterns for ZL - 65 + j 626 and for ZL = j 626 are shown, the scale is magnified by a factor of 100 with respect to the one used in Fig. 1-22. Even when ZL is optimum the scattered field is not zero except in the back scatter direction. Instead, the pattern is seen to consist of four loops of very small field intensity with maxima at 0 = 35 and 0 = 145. However, the maximum field intensity is 56 db below its value for Z = 0. I-4 SUMMARY The important results obtained in Part I are summarized here. The induced current on a center-loaded cylinder which is illuminated normally by a plane wave is obtained as a function of the cylinder dimension and the central impedance in (1.34) through (1.36). II I 53

THE UNIVERSITY OF MICHIGAN 5548-1-T 0 Cf rz q HS 1.0 54

THE UNIVERSITY OF MICHIGAN 5548-1-T z ^^^\\^ A x i,^^Axis of Cylinder 2 80~ e/\S 0~ Z() t S ZL=-65+j626(optimum) c'q co jh-l"^~~~~~~~~~~~~~~~~~~P /i^~~~~~~~~xsofClne 800\ 0 \PQ~~~~~~ C 0.01 55

THE UNIVERSITY OF MICHIGAN 5548-1-T The effect of a central impedance on the induced current on a resonant cylinder is investigated in Section 1-1.4 It is found that there exists an optimum impedance which will reduce the magnitude of the induced current greatly and in addition, reverse the phase of the induced current over the central part of the cylinder. This optimum impedance will give zero broadside back scattering and very low return in other aspects. The effect of a central impedance on the induced current on an antiresonant cylinder is investigated in section 1-1.5. It is found that the induced current is, in general, not reduced by a central impedance. Although we can make the broadside back scattering vanish the scattering in other aspects may be enhanced. An experimental study on the induced current has been carried out in parallel with the development of the theory. The theory was carefully checked by experiment at every step and the agreement between theory and experiment is excellent. The scattering nature of a center-loaded cylinder is studied in section I-3. The optimum impedance which makes the broadside back scattering vanish is obtained in (1. 76). The expression for this optimum impedance is very simple and should prove useful in practical design situations. The optimum impedance as a function of the cylinder dimension is shown graphically in Figs. 1-19 and 1-20. The frequency characteristic of an optimum impedance for broadband effect is shown graphically in Fig. 1-21. 56

THE UNIVERSITY OF MICHIGAN 5548-1-T II THE MINIMIZATION OF THE CROSS SECTION OF A CYLINDER BY CENTRAL LOADING (ARBITRARY ASPECT) In Part I we considered the case of broadside illumination and established optimum loading which results in zero back scatter for the broadside direction. It is natural to ask whether that same optimum loading will minimize the back scattering for off-broadside illumination angles. This question is taken up in part II where we seek to find the induced current on a center-loaded cylinder illuminated by an obliquely incident plane wave. After the induced current is obtained, both the theoretical back scattered field in an arbitrary direction and the radar cross section of the cylinder can be obtained and compared with experimental observations. Good agreement between experiment and theory is obtained. 2-1 INDUCED CURRENT ON A CENTER-LOADED CYLINDER ILLUMINATED BY AN OBLIQUELY INCIDENT PLANE WAVE When a conducting finite cylinder is illuminated by a plane EM wave at oblique incidence, the induced current does not have symmetry with respect to the center of the cylinder. Nonetheless the induced current along the cylinder can be divided into a symmetrical and an antisymmetrical component. Although a central load will greatly change the symmetrical component of the induced current, such a load has no effect on the antisymmetrical component. On a cylinder with a resonant length, the symmetrical component of the induced current dominates the antisymmetrical component; therefore, back scattering can be modified greatly by a single load. On the other hand, on a cylinder with an antiresonant length, the antisymmetrical component of the induced current dominates its symmetrical component; hence a single center load can only modify the back scattering very slightly. The same integral equation method of Part I will be used. It is perhaps worthwhile to note that King (1956) solved a similar problem of a center-loaded receiving antenna but he ignored the antisymmetrical component of the antenna I 57

THE UNIVERSITY OF MICHIGAN 5548-1-T current. His results are also rather complicated for our purpose. We use a somewhat different method in order to obtain a reasonably simple solution with an accuracy which is satisfactory for our purpose. 2-1.1 Integral Equation for the Induced Current on the Cylinder The geometry of the problem is as shown in Fig. 2-1. A cylinder with a radius a and length 2h is assumed to be perfectly conducting. A plane EM wave is incident to the cylinder at an angle 0. A lumped impedance ZL is connected at the center of the cylinder. The dimensions of interest are 1X <2h < 2X 4 2 2 a3 a << 0 where X is the wavelength and /3 is the wave number. The second condition implies that the cylinder is thin and only the axial current is induced. The tangential component of the incident EM wave along the cylinder is assumed to be.in -j3 sinOz E = E cos0e e (2.1) z 0 where E is a constant and the time-dependent factor ej is omitted. o The current and the charge on the cylinder maintain a tangential electric field at the surface which can be expressed as E = -_ _-jLA (2.2) z az z where 0 is the scalar potential maintained by the charge and A is the tangential z component of the vector potential maintained by the current. By using the Lorentz condition, 58

THE UNIVERSITY 5548-1-T gap = 26 z \E > 90 z OF MICHIGAN z z=h z=0 z =-h FIG. 2-1: A CENTRALLY LOADED CYLINDER ILLUMINATED OBLIQUELY BY AN EM WAVE 59

THE UNIVERSITY OF MICHIGAN 5548-1-T 0 = j —2 V-A (2.3) o3 (2.2) can be expressed as Ba =+-j A2) (2.4) 0 The electric field maintained across the gap at the center of the cylinder can be expressed as Eg = Z I 6(z) (2.5) z Lo where ZL is the center load, I is the induced current at the center of the cylinder -L o and 6(z) is a delta function. Since the tangential electric field should be continuous at the boundary we obtain the following equation E +E = ZLI 6(z) (2.6) z z Lo for -h < z < h. Equation (2.6) implies that the total tangential electric field vanishes on the surface of the cylinder and maintains a voltage drop of Z LI across the gap at the center of Lo the cylinder. The substitution of (2.1) and (2.4) in (2.6) gives 2 22 -j sin0z - a2 A +3 A = -jA S E cosOe 0 -ZI6(z) (2.7) 2 z ozLfor -hz Oh. az for -h x< z.< h. i 60

THE UNIVERSITY OF MICHIGAN 5548-1-T 1 The general solution for A is the sum of the complementary function and a particz ular integral A = - C cos /3z + C2 sin3 z + e(z) z v 1 o 2 o o O (2.8) where v is 1/Jo e, C and C2 are arbitrary constants, and where the particular integral O(z) can be obtained as E -jf sinOz O(z) = --- e ~ cos0 e 0 1 _ - Z I sin3 IzI 2 Lo 0 (2.9) In (2.8) A can be divided into a symmetrical and an antisymmetrical component z A (z) = A (z) + A (z) z z z (2.10) thus A (z) = Z Ccos3 z z v L-1 o o E + -0 3 cos0 0 cos (G3z sin0) - 2 ZLIOsin 3oIz T (2. 11) a F A (z) = - sin z z v 2 o 0 L E 0o - J /3 cosO 0 sin (, z sin0) *o _ (2.12) We also divide the induced current on the cylinder into a symmetrical and an antisymmetrical component: I (z) = IS(z) + Ia(z) z z z and, by the assumed symmetries, IS(z) = IS(-z), z z Ia(z) = -Ia(-z) z z (2.13) From the definition of the vector potential, we can write A in terms of I as z z follows: I, I,, 61

THE UNIVERSITY OF MICHIGAN 5548-1-T h AS(z) = ~ IS(z')K (z,z')dz' z 47r z z -h E 1 v= L Clcospoz + 3 co cos(W3 zsinO) - - ZLI sin3 [z (2.14) h a _ 0 a Az(z) = 4 Iz (z') K(z, z') dz' -h = SC2 z sin (2.15) v 2 0 -o 2 where -j3~ (Z-z')2 +a K (z,z') - 2e (2.16) a ( S Equations (2.14) and (2.15) are integral equations for the induced currents, I (z) z and I a(z). We will determine I (z) and I a(z) in sections 2-1.2 and 2-1.3 respecz z z tively. 2-1.2 Symmetrical Component of the Induced Current Instead of solving (2.14) directly for I (z), for convenience we will start z from (2.11), from which C1 can be expressed as F E_ 1 C = sec h jv A (h)- cos(P3 hsin0) + - Z Isino3h. (2.17) Wi 1 thh [ z n cos. 12 Lwo h With (2.11) and (2.17), we obtain I j 62

THE UNIVERSITY 5548-1-T OF MICHIGAN A (z) - A (h) z z = -J sec 3 h V 0 0 v AS(h) LLO z E - 7~osO cos(f hsinO) (cos3 z-cosf3 h) 0 cosO 0o ( o o 0 + 1 ZIo sin3o(h-lzl) 2 Lo o' E cos3 h ci + COS' - cos(W z sinO) - cos(3 h sinO) ] (2.18) With the help of (2.14), another integral equation for I (z) z is obtained as follows: h J-h IS(z) Kd(, z') dz' -j47r 0 s ~ eAS(h secff h - A (h 'o [J z E ) - cos cos (3 h sin0) 0 E cos/3 h ~ c os (30z sinO) 0 (cos 80z - cos Oh) 0 0 1 + I ZIsin3 (h-zI)+ 2 Lo 0 - (/3h sin0) (2.19) where K d(z z) = K (z,z) - K (h,z') d ' a z (2.20) in the and ~ is 1207r. Equation (2.19) is valid for -h<z <h but A (h) and I o z 0 right hand side of (2.19) are still unknown. However, the right hand side of (2.19) suggests a form for the solution of I (z) as Is(z) = C (cos/3 z-cos/ h) + C sin (h - z ) + C 0cos(3 z sinO) - cos(3h sinO) Z C O O0 0 0 - 0 (2.21) It is then reasonable to divide (2.19) into three parts as follows: 63

THE UNIVERSITY OF MICHIGAN 5548-1-T h C (cos 3 z' - cos oh)K (z z')dz' J-h -j47r E j= <4rsecf h Lv AS(h) - ~ ( hsin)] (cos z-cos h), (2.22) D 0 0 Z cose (P~ (COS-z - COS 0oh) (2 22) h C \ sin o(h- Iz'l)Kd(z,z')dz = -j2rsec hZLI sin, (h-|z|) (2.23) -h and Co \ [cos( oz' sino) - cos(3 hsinO)] Kd(z,z')dz' -h E 0= ' Lcos(1 zsinO) - cos(3 hsinO). (2.24) o f cos L~ o o Equations (2.22) through (2.24) are well matched at the end points, z = +h. Furthermore, the constants C, C and Co can be determined by matching these equations at the center of the cylinder, z = 0. This matching yields E B C = 4 secl3 h v A (h)- cos(3 hsin0 (1-cos h) (2.25) c T o oz fcos0 o o o cd L o where h T cd (cos0 z' - cos h)Kd(0, z)dzt (2.26) cd 0 o d -h and -jr sec ohZLIo sin3~ h (2.27) s o sd I I m 64

THE UNIVERSITY OF MICHIGAN 5548-1-T where h Tsd \ -h and sinf3 (h- Iz')K d(0, z')dz' E r - 1 - cos(L 3 h sin0) d f cos0 0 -J (2.28) C _ -j47T 0 =T o 0( (2.29) m I where h T Od = -h cos(3 z sin0) - cos(/h sin0)] K (0, z')dz'. (2.30) The substitution of (2.25), (2.27) and (2.29) in (2.21) gives IS(z) =-j4 z 0 {T d jv AS(h) Lcd L E - cos(3 h sinO) (sec h- 1)(cos 3 z- cos3 h) 3~ cos0 o J 0o o o I I + 2T z LI tan3ohsinf (h-jzj) sd + T ~cos l- cos(3 h sino) cos(O z sinO) - cos(O3 hsin0. T cos s0 -0 c o o Od 0 (2.31) In (2.31), A (h) and I are still unknown, but I can be determined from z o o By definition, (2.31).! I I (0) = IS() + a(0), 0 z z z but I (0) = 0 z hence I I(0) 0 z (2.32) 65

THE UNIVERSITY OF MICHIGAN 5548-1-T I can then be expressed in terms of A S(h) by letting z = 0 in (2.31), and after o z some algebraic manipulation (2.31) itself can be rearranged to give I(z) = {v A(h)- cos(-3 hsin0) M'l(cos3 z-cosf h),o - Cos(3 hsin + N' sin/3 (h-Izj + j M CO zsino) - cos(o hsin) + N2sin/ (h- ztz (2.33) where M' = — (sec/3 h-1) (2.34) 1 T o cd -z tanf3 h(sec h+cos/3 h-2) Nt = L (2.35) 1 T dZLtan/ hsin3 h - j 60 T dT d cd L o o cd sd M2 =To- fl-cos(o3 hsin0)j (2.36) -ZLsin3oh L- cos(3 h sin0)t2 N = -i- (2.37) ZOd Zsin 2h-j60TdTsd cosf h In (2.33) the remaining unknown is A (h). To determine it we use the definition z of the vector potential h AS(h) = I (z')K (h z')dz" (2.38) -h After substituting (2.33) in (2.38), A (h) becomes z 66

THE UNIVERSITY OF 5548-1-T MICHIGAN jE F[cos(3 hsinO)(M' T +N T )- (M2T +NT T) L o 1 cea 1 sa 2 0a 2sa A (h) = z (2.39) v f COSO 1-M' T -N' T 1 ca 1 sa where h T = \ (cos z' - cos 3 h)K (h, z') dz' ca I o o a -h (2.40) (2.41) h T = sin z (h - Iz' )K (h, z')dz' sa J z -h h T a = TOa - J-h Lcos(3 0z' sinO) - cos(/3h sin3O K (h,z')dz' z (2.42) I If (2. 39) is substituted in (2.33) and the result rearranged, the final form of the solution for IS(z) becomes z 15z = jE cos(O3hsin0) - M2TO- N2Tsa I ( 3z 03) = -- --- -^_ ---- -- - - -- -- Ml(cos z - cos h) z 30/3o cos 3oh-M T -NT 1 s '3o os o0 LL o 1 ca i sa] FNIcos(O hsin0) - N1M2 T +M1N2T -N2cos/ h + 0oin 1 2 a ca o n/(h-jzJ) L+ os-cos 3h- M T - N T sin(h-lz|) -o 1 ca 1 sa - M2 cos(3 z sin0) - cos(3O hsinG0) M1 (1 - cos 3 h) cd where (2.43) (2.44) (2.45) -ZLsin/ h(l - cos/3 h) -Lsi _o o N1 T Z sin 3 h- j60 TT cos h cd L o cd sd o 67

- THE UNIVERSITY OF MICHIGAN 5548-1-T and M2 and N2 are expressed in (2.36) and (2.37). Equation (2.43) expresses I (z) as a function of the cylinder size, z load ZL and the incidence angle 0 of an EM wave. As a matter of completeness and convenience, the integrals T d' T, T and T are expressed in terms of better known integrals: ca sa Oa T = C (h, 0) - C (h, h)- cos 3oh rE (h, 0) - E (h, h)] T = C (h, 0)-C (h,h) - cos(3 hsin0) E (h, 0)-E (h,h) Od a a o La a T = C (hh) - cosf3 hE (h,h) ca a o a T = sin 3 hC (h,h)-cosf3 hS (h, h) sa o a o a T = C (h,h)-cos(3 hsin0)E (h, h) O0a a o a the center T T sd' Od' (2.46) (2.47) (2.48) (2.49) (2.50) (2.51) where h C (h, 0) = -h cos 3 z'K (0, z')dz' o a h Ca(h,h) = cos f z'K (h, z') dz' -h E (h, 0) = K (0, z')dz' a a -h (2.52) (2.53) (2.54) 68

- THE UNIVERSITY OF MICHIGAN 5548-1-T 1 E (h,h) = a h K (h,z')dz' a -h (2.55) (2.56) h S (h, 0) = sin3O |z'K (0, z')dzt a -h -h S (h,h) = a C (h, 0) = a C (h,h) = a h sin3 Iz'lK (h, z') dz' -h (2.57) (2.58) h \ cos(!3 z'sin0)K (O,z') dz 0~s f~0 a J-h h \ cos( z' sin0) K (h, z')dz': O a -h (2.59) 1 The integrals of (2. 52) through (2. 59) can be calculated on a digital computer. 2-1.3 Antisymmetrical Component of the Induced Current We can determine C2 from (2.12) as r 2=c hh+i E B C2 =csc oh vAa(h) +J Cos0 sin( h sinO] (2.60) Substituting (2.60) in (2.15), we obtain 69

I ----- THE UNIVERSITY OF MICHIGAN 5548-1-T h I(z')K (z, z') dz' z a h = 47 c h 0 0 rE 3 cos0 0 [sin( 0h sinO) sin/3 z - sin/3h sin(3 z sin0)] + v Aa(h)sinf3 z o z o (2.61) If the solution for I (z) is assumed to be z Ia(z) = C sin(h3 hsin(0) zsin sin(3 zsin0) z a - o o 0o o (2.62) equation (2.61) is matched at z = 0. We will match (2.61) at two more points. If we set z = h/2 in (2.61) and use the substitution of (2.62), the constant C can be expressed as a 1 a T (h/2) a r 2irE ke icosO Lo oo 1.n^ h sin( h sinO) sec -2 L 2 s h )] 9 * 2ir + - sec 0 f3h 2 z (2.63) where h T (h/2) = [sin(ph sin)sin 30 z' - sin) sin( 'sin K (h/2, z') dz'. -h (2.64) By definition, Aa(h) is z 70

THE UNIVERSITY OF MICHIGAN 5548-1-T h A a(h) 0 Ia(z')K (h,z')dz' -h = - C T (h2.65) 4?r a a where h T (h) = \ zsin( hsin)s z'sinhsin K (h,z')dz'. a in Lo o o 0 0 a -h (2.66) From (2.63) and (2.65), A (h) is determined as z E T (h) sin(3 hsin0)sec - 2 sin ( —sino A(h) = (2.67) wcos0 2Ta(h/2) - sec -- T (h) After substituting (2. 63) and (2. 67) into (2. 62) we obtain the final form of the solution for Iza(z) z f h 0Ph - E - sin(3 h sin) (sec- sin sin)I a o 2 o 2 2 / z 30/ Ocos0 T (h/2) - sec hT (h) * sin(P hsinO)sin oz - sin o hsin(P zsin)i (2.68) Equation (2.68) gives the complete solution for the antisymmetrical component of a the induced current on a cylinder. It is noted that I (z) is a function of the cylinder z size and the incidence angle 0 only, and is entirely independent of the center load I ZL I L (1 I

THE UNIVERSITY OF MICHIGAN 5548-1-T For convenience the integrals T (h/2) and T (h) are expressed alternatively as follows: T (h/2) = sin(3 h sin)S9 0(h, h/2) - sin:3 hS (h, h/2) (2.69) a 0 Z o a T (h) = sin(36 hsinO)S9 (h, h) - sinl3 hS (h,h) (2.70) a o a o a where h S (h, h/2) = sin(13 z'sin0)K (h/2, z')dz' (2.71) a o a h S (h,h) = sin(3 z' sin0)K (h,z')dz' (2.72) a o a -h h 9h S90(h,h) = sin3 zK (hz')dz ' (2.74) -h The integrals in (2.71) through (2.74) can be readily calculated on a computer. 2-1.4 Numerical Results To demonstrate the solutions we have obtained in the preceeding sections, numerical calculations are made for two typical cases. The first case is that of a resonant cylinder for which h = 0.215X and a = 0.0173 X and with a central load ZL =/j 800 2, as found in Part I. This value of Z is close to the optimum value for minimizing the broadside back scattering. Using it, the symmetrical component of the induced current is reduced more than I J - -- 72

THE UNIVERSITY OF MICHIGAN 5548-1-T 20 db from the value when ZL = 0. The distribution of the symmetrical component of the induced current, I (z), is shown in Fig. 2-2 for different incident angles. z It is observed that the general behavior of I (z) is essentially independent of inciz dent angle. This means that an optimum load for reducing the broadside back scattering is also effective in reducing the off-broadside back scattering. As shown in Fig. 2-3, the antisymmetrical component of the induced current, I (z), on this z cylinder is quite small. As already noted, I (z) is entirely independent of ZL but z is strongly dependent on the cylinder dimension and on the incidence angle. For a cylinder with a resonant length, I (z) is usually very small compared to I (z). z z Hence the fact that the magnitudes of I (z) and I (z) are comparable in Figs. 2-2 z z and 2-3 is a consequence of the large reduction in the symmetrical component produced by the nearly optimum load ZL= j 600 72. The second case is that of an antiresonant cylinder whose dimensions are h = 0.425X and a = 0.0173X and with a central load ZL= j 600 2. This Z is L L reather arbitrarily selected for actual numerical calculation. The distribution of the symmetrical component of the induced current, I (z), for different incidence z angles is shown graphically in Fig. 2.4. We observe that the magnitude of I (z) is z only slightly affected by the incidence angle. This also assures that the optimum loading for reducing the broadside back scattering will remain effective in reducing the off-broadside back scattering. The antisymmetrical component of the induced current, I (z), is very large for this cylinder and its dependence upon the angle of z incidence is shown graphically in Fig. 2-5. Comparing Figs. 2-4 and 2-5, we observe that the antisymmetrical component of the induced current dominates the symmetrical component. Since a central load can not change the antisymmetrical component of the induced current, central loading will not be an effective method for reducing the overall back scattering from a cylinder with an antiresonant length. 73

h = 0.215X _ _ \ 1 r7o '\ 8 — a = u. I (Symmetrical component of the induced current) Z =j 800 2 7 6 I - - 0= 0~ 0-IZ \ I \/ 0F - T I AN Y R A OL0= 01t^c-0-50 -30 -10~ 0 1 30 50~ 70~ F:FO A FIG. 2-2: DISTRIBUTION OF IIS(z) ALONG A CYLINDER WITH A RESONANT LENGTH Z - Ca Cal O tl ni -- C) 0 1z P>

h = 0.215X a = 0.0173X (Antisymmetrical component of the induced current) Hz 6 r N _ N 0 "o An g a) '._ X * 3 4 0 = 45~ N%. _ - _ 3 0 = UJ1 Ul 00 I H-A I m 2 CH) i-q 0 - - doool / lopo;o 2 0 P*tJ — 0= i 1 PC) 0 z -3 z --- I-o z FIG. 2-3: DISTRIBUTION OF I Ia(Z) ALONG A CYLINDER WITH A RESONANT LENGTH z

H IT1 zf m -1 0n N UI N 0 0 *-a 0 a4) j * - l P4 a).1-4 4-+ (a) x 24 20 16 12 / h = 0.425X a = 0.0173X Z = j600Q L 0= 45~0 \ Symmetrical Component of the Induced Current 7 P-4 C, -4 0 z 8 4 -80~ I0o 0 z E I0oz FIG. 2-4: DISTRIBUTION OF I (z) ALONG A CYLINDER WITH AN ANTIRESONANT LENGTH z

h = 0.425X (Antisymmetrical component of the induced current) 56. a = 0.0173A 0 =45 F 5IOLN LW48' G/ I - 300 40 2 /0=15~ 0 0/ FIG. 2-5: DISTRIBUTION OF jIa(Z) ALONG A CYLINDER WITH AN ANTIRESONANT LENGTH Z; H,.q z,=.Q1.-4 cn i.j c. 0 I C) 0 z >4

THE UNIVERSITY OF MICHIGAN 5548-1-T It should be remembered that the total induced current is the vector sum of its symmetrical and antisymmetrical components so that a simple addition of I (z).y ~z and I (z) in Figs. 2-2 through 2-5 does not produce the total induced current, I (z). z z 2-1. 5 Comparison Between Theory and Experiment In order to check the theory, the current distribution was measured on a cylinder with dimensions h = 0. 425X and a = 0.0173 X. A coaxial cavity, built in at the center, was adjusted to simulate the central impedance of about j 600 Q. The actual cavity was filled with a dielectric (er= 4) and the length of this cavity was set equal to 5.07 cm. The induced current on the cylinder was measured by a small current loop and the cylinder was illuminated by an EM wave radiated from a horn antenna. The corresponding theoretical current distribution was calculated on a digital computer and a desk calculator. The theoretical and the experimental results are compared in Fig. 2-6, and the agreement is quite good. For this particular cylinder whose length is in the antiresonant region, a large antisymmetrical current is predicted by the theory when the incidence angle is other than zero degrees. This was confirmed by experiment, as was the prediction that the antisymmetrical component of the induced current should not be affected by the central load. The main disagreement between theory and experiment is at the center of the cylinder. One explanation may be that an ideal delta function impedance is assumed in the theory but a finite gap exists on the experimental cylinder. Another may be the difficulty encountered in obtaining a specified impedance by a coaxial cavity. However, the general behavior of the induced current predicted by theory is confirmed by experiment, and we will assume that our theory is adequate for our purpose. 78

60 50 40 h = 0.425X a = 0.0173X theoretical impedance ZL= j600 Q experimental impedance cavity length L = 5.07 cm (with e = 4) r H z m 0 -co C2 ~-e H 0f) 034 l-q 0-} 0?'II N N i-t 0 a) *-> 1) 04 30 20 nl oj I Q P-4 0 z P-4 0 >4 10 -10 z dw -W 0 0 FIG. 2-6: THEORETICAL AND EXPERIMENTAL CURRENT DISTRIBUTION ALONG A CYLINDER

THE UNIVERSITY OF MICHIGAN 5548-1-T 2-2 BACK SCATTERING OF A CENTER-LOADED CYLINDER ILLUMINATED BY A PLANE WAVE AT AN ARBITRARY ANGLE The induced current on a center-loaded cylinder illuminated by a plane wave at an arbitrary angle was obtained in section 2-1. Using the solution obtained there we can calculate the back scattered field. 2-2.1 Back Scattered Field of a Center-Loaded Cylinder With the same geometry as in Fig. 2-1 and the solution of the induced current as expressed in (2.43) and (2.68) we proceed as follows: The symmetrical component of the induced current I (z) maintains a vector z potential at a point in the far zone of the cylinder in the direction of 0: P joR o h jo13 zsin0 As = o e IS(z) e dz (2.75) z 47r R z 0 -h where I (z) can be obtained from (2.43) and where R is the distance between the z o center of the cylinder and an observation point. It is important to note that in this part of the study 0 is defined as shown in Fig. 2-1 and differs by 90 from the 0 defined in Part I. Performing the indicated integration in (2.75), the final expression for A z becomes E j60 cR-oo hsin0)-M T - NT 2M As _ 0 e o e 2 Oa 2 sa __ 1 z 1,- 2 R f cos3 h-M T -NT 3 1207r13 o L2 R o 1 ca 1 sa cos 0sin0 0 Lsinl hsinocosGohsinc)-cos3 hsin(3 hsinO) N cos(3 h sin0) - N1M T MLN2T -N 2cos(o h 2[cos( hsin0)-cosf h] l11 1 oa _a _2 ca o o_____ +cosR h-M T -N T _ 3 L So 1 ca 1 sa -i cos 0 (cont'd) I 80

--- THE UNIVERSITY OF MICHIGAN - 5548-1-T 1 2h sin - o.....^22 h sine - sin(2 6 h sin). 2 sin0cos0 f o o (2.76) Similarly, the antisymmetrical component of the induced current I (z) mainz tains a vector potential at a point in the far zone of the cylinder in the direction of 0: -Jfo R a =o e A a z 4wT R 0 h -h jii z sin0 I (z) e dz z (2.77) where I can be obtained from (2.68). After the integration in (2. 77) is performed, z j/u E Aa(0) = - - 1207r9 2 O -J3oRo e R 0 T (h/2) - sec3 hT (h) a 2 o a 1 — - s sinL o h 1+sin2 0)sin(20 hsin0)-2 hsin0cos 2 cos3 0 sin 0 - 4cos B hsin0 sin (j3 hsin0) 0 o -J (2.78) I To obtain an expression for the back scattered field, the following argument is employed: The total vector potential maintained by the induced current on the cylinder is A (0) = A(0) + Aa(0) z z z (2.79) The scattered electric field in the far zone of the cylinder due to the induced current is 81

I THE UNIVERSITY OF MICHIGAN 5548-1-T E= -jwA = jA cos0 u u ~Z (2.80) and the Poynting power density of the scattered field is 1 12 0 (2.81) Hence, using the values of As(0) from (2.76) and Aa(0) from (2.78), the final exz z pression for the back scattered field becomes E _j~R0 rTcos(3o hsin0)- M2T0a-NT 2M 0(0) R cos 3h-MT -NT 2 * i hsin0cos( hsin0) - cos a hsin(G hsin0 Nos(3o hsin0)-N.M2T +M1N2T -N2cos o h cos h-MT -NT e~Sio 1 ca I sa 2 [cos(/3 h sin0) - cos 3oh] 2 cos 0 2 sin0- 2 3oh sin(2 h sin(2 hsin0 2 sin0 o o - + T (h/2)- sec3 hT (h) a o2 a 1 2 2 cos 0sin0 r~r 2 2 * sin, h (1+sin 0)sin(23 hsin0)-2 0 hsin0cos 02 2o n - 4cos o hsin0sin 2(3 hsino). 0o 0 j (2.82) 82 --

THE UNIVERSITY OF MICHIGAN 5548-1-T Equation (2.82) gives the complete expression for the back scattered electric field of a center-loaded cylinder when illuminated by a plane wave with an electric field E at an angle 0 with respect to the normal to the cylinder. When typical values are calculated from (2.82) and compared with the experimental results, the agreement is excellent, as will be seen in section 2-2.3. It is noted that the radar cross section is usually defined as E (0) 2 lim 2 R2 o o 0 0 The experimental procedure will now be described. 2-2.2 Measurements of the Back Scattered Field of a Center-Loaded Cylinder The radar cross section measurements were made at a range of 10 feet at a frequency of 1.088 Gc, the same as used for the current distribution measurements. The center-loaded cylinders were illuminated by a plane wave whose electric field vector was in the plane containing the cylinders. The back scattered fields from the cylinders were recorded as a function of aspect angle, where the zero degree aspect was chosen to represent the broadside direction. Fig. 2-7 shows three scattering patterns for a resonant length cylinder with h = 0.2134X for three different coaxial cavities of L = 0 (Z L= 0), L = 6.22 cm (Z L- oo) and L = 5.83 cm (ZL ji 600 2). We observe that the introduction of an L L impedance of Z j 600 2 reduces the cross section of a resonant cylinder by more than 30 db. This impedance is only approximately optimum and it is believed that an optimum impedance, which is about Z L= 65+j 626 Q, would reduce the cross section even more. Actually, in the experiment a maximum reduction of 35 db was achieved by a purely reactive loading using a dielectric loaded coaxial cavity. Figs. 2.8 through 2. 11 show the scattering patterns for an anti-resonant length cylinder with h = 0.4435 X for four different coaxial cavities: L = 0 (ZL= 0) L I 83

-v I 1- E I, _ __ J-2-~ ' '-Z....... r _;Calibration.' —.:... L v | r!g |:. i *t'\L= 0 |L'(Z.0);;- 'p-.................. o -- -o -.- -....... -6-..... / ~. '.-.......... -.-....... ~ '.:. 1',L= 6.22cm(Z c'oo) i i; t — H[l 61DTl '" i, r'| i. i i.2;,:i;; i -! ji ~iii ii;t!ji.1 F. ',:!. l rT i.: X['fi |!t.-t i l |U ~ -l.,l.', ][ l,:-|!'.!iX ^ +, T.,!! 1!'-.'.:!!, ~ -T i 7-:":' 1 ^!::- i''i- L= 5.83cm (Z6 j600 0)j.:.Htf:-:::: ~.:,172:..,,,1:,,..~e; lr} 1i 1 Ht tI, 1.l~l:l,3. TI ~M HIM,zz L J -~ — c#,.72^.,, -.;ALE;; 40CMB ir *=ANGLE -i --- 00 41 - H 24 p-4 00 q < ril 4 FIG. 2-7: SCATTERING PATTERN FOR h = 0.2134X CYLINDER

THE UNIVERSITY OF MICHIGAN 5548-1-T -1-. I —t'V 4 — — 4 — A - ----.-I — I *^-5 eE 7 — 4 —. —+ —.... 4- - - - _ _ -_4 - _+ -- I - _ — - t- -- It.7b - -+ -4 --- t. m- i I i+-4 — ^ ---........... ~.,- I... L —.... I.................... ";-........ tl.... 4 'o _, -' -I._..._._..-; —1- -t- - t ---/ ---- _. --- ---- = ~ ~~~~~~~~.: ->4 -- ==-i - -t= t + t sI - > '- ~..... '" --.....,.._.......... '...... _-.... -' ---'"+'..___..___.___*- - - -,-,,4 ---. -, _. - +v._;- — 4 - '-! — 4- 4t - t4 — t - 1 - - t —t' t -: - -:.=_. ___...... -'- -'* —_- 1 - -.....::,l-. -. _-. - -- -. - - -. -._ — -- - - - 1- - _ -.- - -. - t 4- 1 —;- - - 4 --- W - *; - - 5. --. -. j 4~ - t - ~- ~ =- - -, - =-.. I... I:: -'-::-'':1....._,.... ~ — 1 1 s — '-1 { 'I' '' --- * -4- 1' —1it I-0-70 1 —1 ---10;-:- -. -- —...-, i........ - -.... -.-...- * - _, 4.4- | [.. s - - - - - *_ |__ _ _g 4 - _ 4 _ _ | _ _* _ +_ _ - - -, ~-.; ~. _ - -- |- 8 1 — -:t:.I --- —|L. - | —. * — t 4 -S1 L- -1- + A f4 — ' 1 -. --..0-1-+-t —: ---- I- - - -T ---l --- —] - - 1z uO 0 LO o HE CO I C" C\] C-D'-I 1 | -— 8 5 -:- 85

Idi i idl k I i~~~ T ~ I - I ~~~~~~~~~~~~~~I Ijil II~~~~gI~~III ~ ~Li I 4 -I V H! - t4Tt I1~ it~~~~~~~~~~~~~~~~~~ ~~~~F'l I 0 1 1* 20 1 K 3 6. ' V; i I9 II_ _ _ _ _ _ _ CATt 0olA,1, mH cTon ci z Q-4 0 0 0 z FIG. 2-9: SCATTERING PATTERN FOR h=0.4435X CYLINDER L= 5.58cm (ZL j415Q)

4!-.I,I;;:! _, ii:.- -ji' iil.l............;.....-..i-t.i i 0 Vt',calibration i:12 --- 1- _ —............:.:.11ill l: ll -6 i -- -,.. t I^_ _ _ _ll_, I;r:::::-: iiiL-i::li1. __ __ -..:::::i -:i:ii ti:::li' ' ' 1 - 2 - - _______ 41111111111111111 11111 11111111111 1. ----11111 I Ij;_, 72 3.6' 3it l l 6 i r __..... C..HANT NO ho 121 A ' - H rtz ml 0 It 0 0 z ZT J'RLV. i FIG. 2-10: SCATTERING PATTERN FOR h= 0.4435X CYLINDER L= 6.22 cm (ZL oo)

-7 T7, 11111 Hill H I JI IWi;I11 - ' - '; / ';;:."^I. / i x! 1;1 i -t-4: 2Kt I1 t7t | ii I -- K'.2 IL' f2KTtII ' ^-|'.- - *) * ^ ii |i;;.! K 7 ',. ~,......- - - - - -.T. - T -e l ^ - ^,-t' A-+ -* \ l;. I IIU,.;..::^..^:::^::..;.^.^}-jm-:-:^ -^^~~~~~~~~~~~~~~~~~~~~~~~~~~~ ff~ ~ ~ ~ ~ ~ ~ ~~f +4 - ^ - -- - - - - -* * * -V;*-i-; g~, p,. -\I i,,,.!j z 0 --e4 00 - z ANGLE FIG. 2-11: SCATTERING PATTERN FOR h= 0.4435X CYLINDER L= 5.07 cm (ZL= j 212Q) L

THE UNIVERSITY OF MICHIGAN 5548-1-T L = 5.58 cm (Zm (Z- C oo) and L = 5.07cm (Z- j212Q). L L L In these figures, we see that the back scatter lobes for some off-broadside aspects are not reduced by central loading. In particular, the large cross sections at about 40 off-broadside are due to a large antisymmetrical current induced on the cylinder at this angle of incidence. Since the antisymmetrical current is not affected by a change in the central impedance, the only advantage with an antiresonant cylinder is the large reduction of the cross section at broadside incidence which results from the modification of the symmetrical component of the induced current. Some additional measurements of the maximum reduction of the back scattered cross section as a function of the cavity length are summarized in Table II. In Table II, e is the dielectric constant of the dielectric inside the coaxial cavity, a is the maximum back scattered cross section of the loaded cylinder and a max o is the maximum broadside back scattered cross section of a particular non-loaded cylinder for which h = 0.2134X. The variation of the maximum back scattered cross section as a function of the cavity length is shown in Fig. 2-12 where three curves show the information summarized in Table II. It is interesting to compare curve 1 and curve 3. Since the dielectric constant of the dielectric of the cavity changes from 4 to 16, one woul expect the required physical length of the cavity to decrease by a factor of 2. In the experiment the required physical length of the cavity was decreased by a factor of 1. 5. This tends to indicate that the shunt capacitance across the input of the cavity may change if the cavity is filled with different dielectric. Fig. 2-13 shows the maximum back scattering cross section as a function of cavity length for a cylinder with h = 0.215 X. The three curves in the figure are obtained by three different methods. Curve 1 is directly obtained from a back scattering measurement; curve 2 is obtained from the measured current distribution by means of a graphical integration; and curve 3 is the theoretical curve with an I I 89

0 -10/ -20 - 0 - I I l l l ) 0 1 2 3x 4 5 6 7 8 c Cavity Length, L (cm) FIG. 2-12: MAXIMUM BACK SCATTERING CROSS SECTION VS CAVITY LENGTH (1) h=0.2134X, e =4; (2) h=0.4435X, e =4; (3) h=0.2134X, e =16 H tT V4 -~ 00C)0 z

04. 0 0000010,.000000 A6 -1ot (2) -20-. 000, o a (1). OA A -r Theoretical Current Distributions I I~ Scattering Measurements --- Experimental Current Distributions - - Theoretical Current Distributions I I (C =0.4A/f) 5 H z, q I '-3 0 C)f z 00 -30+ -40 T A I I -50. 5 FIG. 2-13: MAXIMUM BACK SCATTERING CROSS SECTION VS CAVITY LENGTH h = 0.215X, e = 4 r

I i THE UNIVERSITY OF MICHIGAN 5548-1-T assumption of a shunt capacitance C = 0. 4p//f which was deduced in section 1-2.3. However, if C = 0.46 pf had been chosen, curve (3) would shift to the left and coincide with (1) and (2). TABLE II Cylinder Length e h r 0.2134X 0.4435X 0.2134X 4 4 16 Cavity Length L (cm) 0 5.07 5.58 5.71 5.83 6.10 6.22 6.35 7.12 0 5.07 5.58 5.71 5.83 6.10 6.22 6.35 7.12 2.92 3.18 3.29 3.81 4.07 (7 max,, db 0 0 -14.3 -25.7 -34.9 -32.9 -22.5 -19.1 -17.4 -10.0 0.1 0.1 0.0 -0.2 -0.7 3.0 3.6 3.0 2.5 -4.8 -7.5 -8.9 -25.6 -19.3 2-2.3 Comparison Between Theory and Experiment In this section we compare the theoretical predictions and experimental observations of the back scattering cross section dependence on aspect angle. Fig. 2-14 shows the back scattering cross section of a resonant cylinder (h = 0.215X) 92

- tw It g v.. r-l a) P4 z c! H m COD h-q 0 ftl.n 00 I 3 0 z J -40 90O 900 70~ 50~ 30~ 10~ 0 10~ 30~ 50~ 70~ "e I '- 0 FIG. 2-14: BACK SCATTERING CROSS SECTIONS OF LOADED CYLINDERS VS ASPECT ANGLE (0)

THE UNIVERSITY OF MICHIGAN 5548- 1-T as a function of the aspect angle for three different loadings. When Z = 0 (unloaded cylinder) the cross section is very large. When the same cylinder is loaded with an infinite impedance (ZL= oo), the back scattering cross section is reduced about 20 db. If the loading is adjusted close to the optimum value (ZL= j 600 0), a reduction of more than 30 db is obtained. The agreement between theory and experiment is excellent. Figs. 2-15 through 2-17 show the radar cross sections of an antiresonant cylinder as a function of the aspect angle for three different loadings. When the cylinder is not loaded (Z L= 0), the back scattering is approximately constant over the aspect angle range of 0 4 0 500 with a slight maximum at 0 = 40~. The theoretical and the experimental results for this case are compared in Fig. 2-15 in which the zerodb -level is chosen to have the same absolute scale as in Fig. 2-14. Fig. 2-16 shows the theoretical curve for Z L= j 300 Q compared with an experimental curve for ZL= j 212 2. The point of interest is that for this loading the back scattering in the broadside direction is reduced considerably. These two curves, though with different loadings, agree quite well over most of the aspect range except for small 0. The maximum back scattering occurs at 0 = 420 and its amplitude is not reduced by the loading. Fig. 2-17 shows the theoretical curve for Z j 600 2 and a comparison with an experimental curve for ZL= j 415 2. The general behavior of these curves agrees very well. The maximum back scattering occurs at 0 = 42 and again its amplitude is not reduced by loading. In these three figures we find that the maximum cross section for an antiresonant cylinder at 0 ^' 40 is not modified at all by a central loading. As mentioned before, this is due to the fact that this maximum back scattering is produced by the antisymmetrical component of the induced current which is not affected by a central impedance. 94

10 10 5 0 -5 -10 -15 H )-i z m -1 10 I-I $- - 11 01 <U co P., r"a) z ci) H-4 CO) >-q 0-4 Cn 00 I. -3 0 PT1 -20 -25 -30 0 -z 9 I — — 0 FIG. 2-15: BACK SCATTERING CROSS SECTION OF LOADED CYLINDER VS ASPECT ANGLE (0)

10 10 Lt: ' / A!\ -20 -20 -51 - \ -25 I I^Vj \ -205 -20 -30 -30 30 0 0o..0 o0 0,0 H z oi! cm) -4 0 z u - -An o- - FIG. 2-16: BACK SCATTERING CROSS SECTION OF LOADED CYLINDER VS ASPECT ANGLE (0)

10 5 0 -5 10 -i rT1 z m sq la) v r3-I?-q (a) CD I< 0 P*, a). r-4 -Cd z C2 H 0-1 eIt cr) P-4 0 PTl -10 -15 Cn1 00 I on -20 -25 -30 P-( 0 z 0 e - I - e FIG. 2-17: BACK SCATTERING CROSS SECTION OF LOADED CYLINDER VS ASPECT ANGLE (0)

THE UNIVERSITY OF MICHIGAN 5548-1-T The agreement between theory and experiment is found to be excellent. This confirms the accuracy of the theory and the experiment and also the feasibility of the reactive loading technique for the reduction of the radar cross section of a metallic body. 2-3. SUMMARY The induced current on a center-loaded cylinder illuminated by a plane wave at an arbitrary angle is obtained in Section 2-1. This is essentially the generalized version of the case studied in Section 1-1. It has been found that when a cylinder is illuminated by an obliquely incident plane wave the induced current has a symmetrical and an antisymmetrical component. The symmetrical component can be modified greatly by central loading but the antisymmetrical component is not affected. In a resonant cylinder the symmetrical component of the induced current dominates the antisymmetrical component. In an antiresonant cylinder the antisymmetrical component is dominant. Therefore, the scattering cross section of a resonant cylinder can be greatly reduced by central loading while the cross section of an antiresonant cylinder can be modified only slightly. The back scattered field of a center-loaded cylinder illuminated by a plane wave at an arbitrary angle is obtained in Section 2-2. The effect of central loading on the cross section of a resonant and an antiresonant cylinder is carefully studied theoretically and experimentally. The scattering cross section of a resonant cylinder can be reduced more than 30 db by an optimum loading. The scattering cross section of an antiresonant cylinder can be reduced only in the broadside direction but the large cross section in the off-broadside direction can not be reduced by central loading. To reduce the overall cross section of an antiresonant cylinder or to increase the bandwidth of the reactive loading technique a double or a multiple loading may prove to be more effective. a I.1 I 98 J

THE UNIVERSITY OF MICHIGAN -- 5548-1-T ACKNOWLE DGME NT The authors are grateful to Mr. R.E. Hiatt and Dr. T.B.A. Senior for many helpful discussions and acknowledge the assistance of Mr. L. Zukowski in the preparation of this report. The authors also wish to thank Mr. H. Hunter and Mr. J.A. Ducmanis for their numerical calculations and Mr. E. F. Knott and Mr. V. M. Powers for their contributions on the experimental side. RE FERENCES o As, B.O., and H.J. Schmitt (1958) "Backscattering Cross Section of Reactively Loaded Cylindrical Antennas", Harvard University Cruft Laboratory, Scientific Report No. 18. Hu, Yueh-Ying (1958) T"Backscattering Cross Section of a Center-Loaded Cylinder", IRE Trans. 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) The Theory of Linear Antennas, Harvard University Press, Cambridge. pp. 506-511. Sletten, C.J. (1962) Air Force Cambridge Research Laboratories, private communication. i: 99

UNIVERSITY OF MICHIGAN 3 9015 02829 62111 11 3 9015 02829 6211