MSU LIBRARIES -:—. RETURNING MATERIALS: PIace in book drop to remove this checkout from your record. FINES wiII be charged if book is returned after the date stamped below. MEASUREMENTS OF TRANSIENT ELECTROMAGNETIC FIELDS WITH SURFACE PROBES By Neila Gharsallah A THESIS Submitted to Michigan State University in partial fulfillement of the requirements for the degree of MASTER OF SCIENCE Departement of Electrical Engineering and Systems Science 1987 ABSTRACT MEASUREMENTS OF TRANSIENT ELECTROMAGNETIC FIELDS lflITfli SURFACE PROBES By Neila Gharsallah An alternative experimental technique for determining the natural resonant frequencies of a conducting target is developed in this research. This thesis presents a surface field probing scheme that can be used to measure the transient electromagnetic fields induced on conductors. This study was motivated by the fact that the surface charge and current responses have larger S/N ratio and shorter early-time than those of scattered-field responses, and the need for an efficent and accurate method of determining the natural frequencies of electromagnetic scatterers. This surface field probing scheme gives an additional validation of the singularity expansion method description of the late-time response of conducting bodies, and also a good verification of the aspect independence of the target natural modes. A variety of conducting scatterers, along with experimental data, are presented to illustrate the versatility and the accuracy of such technique. To my father and mother with all my love ii ACKNOWLEDGEMENTS I wish to express my sincere appreciations to Dr. K. M. Chen for his constant guidance, valuable suggestions, and interminable patience throughout the course of this study. Also, I wish to thank Dr. D. P. Nyquist for his helpful suggestions, and Dr. E. J. Rothwell for his encouragement, time and assistance concerning this research. Above all, I especially wish to thank my fiancee Lassaad A. Turki, for his endless encouragement and patience throughout this study. This research was supported in part by the Naval Air Systems Command under contract No. NOOOl9-85-c-O4ll and by the Tunisian government through the University Mission of Tunisia in Washington, DC. iii £11m: TABLE OF CONTENTS INTRODUCTION ............................................ SINGULARITY EXPANSION REPRESENTATION OF TRANSIENT SCATTERING .............................................. 2.1 Introduction ....................................... 2.2 SEM representation of scattered fields ............. 2.3 Conclusion ......................................... MEASUREMENTS OF SURFACE CHARGES AND CURRENTS INDUCED ON GOOD CONDUCTORS ..................................... 3.1 Introduction ...................................... 3.2 Transient performance of charge and current probes ............................................ 3.2.1 Charge probe ............................... 3.2.2 Current probe .............................. Construction of charge and current probes ......... Advantages of the surface probing scheme .......... Experimental results .............................. www Lnl-‘w 3.5.1 Thin wire cylinders ........................ 3.5.1.1 Numerical determination of the natural frequencies of thin wire cylinders ......................... 3.5.1.2 Surface field measurements ........ 3.5.2 Aspect independence of the natural frequencies extracted from surface charge and current transient responses ............ 3.5.2.1 Elliptical loop measurements ...... 3.5.2.2 Bent wire measurements ............ 3.5.2.3 Aspect independent target discrimination .................... 3.5.3 Application of the surface field probing scheme to complex and low-Q targets ........ 3.5.3.1 Surface transient response of an airplane model .................... iv 01> 10 10 11 11 13 15 17 18 18 18 20 21 30 3O 33 3A 34 TABLE OF CONTENTS continued Chapter 3.5.3.2 Experimental characterization of transient surface measurements of a sphere .......................... 4 THEORETICAL AND EXPERIMENTAL ANALYSIS OF A CIRCULAR LOOP SCATTERER ......................................... 4.1 Introduction ...................................... 4 2 SEM representation of a circular loop scatterer ......................................... 4.3 Numerical results ................................. 5 DETERMINATION OF THE NATURAL FREQUENCIES 0F THIN WIRE ELLIPTICAL LOOP USING HALLEN INTEGRAL EQUATION ......... 5.1 Introduction ...................................... 5.2 Hallen integral equation for the elliptical loop .............................................. 5.3 Loop symmetry ..................................... 5.4 Moment method solutions ........................... 6 EXPERIMENT ............................................. 6.1 Introduction ....................................... 6.2 Time-domain scattering range ....................... 7 CONCLUSION ........................; .................... BIBLIOGRAPHY ..................................................... ‘7 35 63 63 64 67 79 79 80 85 88 .1. .2. .l. 1 l 1 LIST OF TABLES Natural frequencies of a thin wire elliptical loop of major axis A - 16 cm, minor axis B - 7.6 cm, and wire radius a - .25cm ........................... Natural frequencies of a thin bent wire with dimensions 1. - 28.5 cm, 1..2 - 32.4 cm, extracted from surface charge responses at various aspects .... Natural resonant frequencies of a sphere of diameter 30 cm ( x 10 ) ........................... First six type I poles of circular loop scatterer of radius A and wire radius a - .016A .................. Extracted natural frequencies, relative amplitudes, and phases of natural resonant modes of a circular loop scatterer at various aspects ................... Entries in moment method matrix for the elliptical loop scatterer ( k - n, k-l - n ) ................... Entries in moment method matrix for the elliptical loop scatterer ( k n n, k-l # n ) ................... Boundary conditions in each symmetry case of the elliptical loop scatterer ........................... vi Page 31 32 39 7O 71 98 102 LIST OF FIGURES Figure Page 2.2.1 Scattering of a transient electromagnetic pulse from a conducting body .......................... 5 3.3.1 Arrangement for charge(a), and current(b) probe measurements on conducting surfaces ............. 16 3 5.1.2.1 Fourier spectra of (a) 6", (b) 12", and (c) 18" thin cylinders, obtained via FFT ................ 22 3.5.1.2.2 (a) Surface charge response of a 6" thin cylinder, (b) its corresponding Fourier spectrum .......... 23 3.5.1.2.3 (a) Surface current response of a 12" thin cylinder, and (b) its corresponding Fourier spectrum ........................................ 24 3.5.1.2.4 (a) Surface charge response of a 12" thin cylinder, and (b) its corresponding Fourier spectrum ...... 25 3.5.1.2.5 (a) Surface charge response of an 18" thin cylinder, and (b) its corresponding Fourier spectrum ...... 26 3.5.1.2.6 Natural frequencies extracted from different measured responses of a 6" thin cylinder ........ 27 3.5.1.2.7 Natural frequencies extracted from different measured responses of a 12" thin cylinder ....... 28 3.5 1.2.8 Natural frequencies extracted from different measured responses of an 18" thin cylinder ...... 29 3 5.2.1.1 Experimental arrangement for surface current measurements of an elliptical loop .............. 4O 3.5.2.1.2 (a) Surface current response of a thin wire elliptical loop measured at 0°, and (b) corresponding Fourier spectrum .................. 41 3.5 2.1.3 (a) Surface current response of a thin wire elliptical loop measured at 45°, and (b) corresponding Fourier spectrum .................. 42 vii Figure 3.5.2.1. J.‘ LIST OF FIGURES continued (a) Surface current response of a thin wire elliptical loop measured at 120°, and (b) corresponding Fourier spectrum .................. Natural frequencies of thin wire elliptical loop extracted from surface current responses measured at various aspects .............................. Surface charge response of a thin bent wire measured at 0° and (b) corresponding Fourier spectrum ........................................ Surface charge response of a thin bent wire measured at 45° and (b) corresponding Fourier spectrum ........................................ Surface charge response of a thin bent wire measured at 900 and (b) corresponding Fourier spectrum ........................................ Surface charge response of a thin bent wire measurements at 135° and (b) corresponding Fourier spectrum ......................................... Natural frequencies of a thin bent wire extracted from surface charge responses measured at various aspects .......................................... (a) Natural E-pulse constructed to kill the first six modes of long bent wire, (b) measured response of long bent wire, (c) convolution of long bent wire E-pulse with its measured response .......... (a) Natural E-pulse constructed to kill the first six modes of long bent wire, (b) measured response of small bent wire, (c) convolution of long bent wire E-pulse with small bent wire response ....... Measured (a) surface charge and (b) backscattered- field responses of an airplane model ............. Fourier spectra of (a) surface charge, and (b) backscattered-field responses of an airplane model ............................................ 43 44 45 46 47 48 49 50 51 52 S3 Figure 3.5.3.1. LIST OF FIGURES continued Natural frequencies of an airplane model extracted from surface charge and backscatered-field responses ........................................ (a) Surface charge response of a 30 cm diameter sphere, (b) its corresponding Fourier spectrum.... (a) Backscattered-field response of 30 cm diameter sphere, (b) its corresponding Fourier spectrum.... Comparison of the natural frequencies of a 30 cm diameter sphere extracted from different measured responses ........................................ Convolved output of the natural E-pulse of a sphere (30 cm diameter) with its backscattered-field response measured at distance of 5 cm from receiving probe ............................................ Convolved output of the natural E-pulse of a sphere (30 cm diameter) with its backscattered-field response measured at a distance of 10 cm from receiving probe .................................. Convolved output of the natural E-pulse of a sphere (30 cm diameter) with its surface charge response.. Backseattered-field response of a sphere of 12 cm diameter ......................................... Convolved output of the natural E-pulse of the 30 cm diameter sphere with (a) its backscattered- field response, (b) backscattered-field response of the 12 cm diameter sphere ........................ Geometry of a semi-circular loop scatterer mounted on a ground plane ........................ Late-time scattered field response of a thin wire circular loop scatterer (a - 0.016A, A — 15.2cm) measured at an aspect angle - 0o ................ ix 54 55 56 57 58 59 60 61 62 65 72 Figure 4. 3. 2 LIST OF FIGURES continued Page -Fourier spectrum of a thin wire circular loop ( a - 0, 016A, A - 15. 2cm ) scattered response measured at an aspect angle - 0o ................. 73 Late-time scattered field response of a thin wire circular loop ( a - 0.016A, A - 15 2cm ) measured at an aspect angle of 45° ........................ 74 Fourier spectrum of a thin wire circular loop (a - 0. 016A, A - 15. 2cm ) scattered response measured at an aspect angle of 45° ............... 75 Late-time scattered field response of a thin wire circular loop ( a - 0.016A, A - 15.2cm ) measured at an aspect angle of 90° ........................ 76 Fourier spectrum of a thin wire circular loop ( a - 0. 016A, A - 15. 2cm ) scattered field response measured at an aspect angle of 90° ............... 77 Natural frequencies of circular loop scatterer extracted from scattered field responses measured at various aspects ............................... 78 Geometry of the elliptical loop scatterer ........ 82 Symmetry cases for current distribution on an elliptical loop scatterer ........................ 86 Transient scattering range at Michigan State University ....................................... 108 Functional block diagram of the time- domain scattering range ................................. 109 Relative location of the elements on the ground plane for scattered field measurements (r - 1.4 cm, d - 6 cm), and corresponding incident pulse traces ..................................... 110 Relative location of the elements on the ground plane for surface field measurements ( r - 1.4 m), and corresponding incident pulse traces .......... lll CHLAPTTH! 1 INTRODUCTION Over the last few years, there has been an increasing interest in radar target identification and classification. A typical scheme consists of illuminating the target with a radar pulse and then identifying it with its natural modes, which are extracted from the late-time period of the return signal. One of the major problems associated with this scheme is the difficulty in obtaining accurate natural frequencies of the target from a noisy measured response. This thesis mainly presents an experimental surface probing scheme, that can be used to measure the target parameters with more accuracy, and therefore justify the SEM expansion of the late- time surface charge and current responses of various targets. The main advantages of the surface probing technique is that it provides larger S/N ratio and shorter early-time of the measured surface charge and current responses than those of the scattered field responses, thus allowing more accurate natural mode extraction. This thesis covers various topics most of which are related to the surface probing technique. Chapter 2 presents a brief review of the singularity expansion method (SEM) of a conducting target in terms of its parameters, namely the natural resonances. Chapter 3 presents a technique of applying surface field probes on, conducting targets to measure their induced transient charge and current responses, and therefore to determine their natural resonant frequencies. Experimental results of this technique are presented on various targets. The surface probing scheme is first implemented on thin wire cylindrical scatterers in order to assess its viability. A verification of the aspect independence of the target natural modes is also given using simple shaped targets such as loop or bent wire that can perform measurements above a conducting ground plane at multiple aspects. As an example of an application of this technique on complex targets, an airplane model is considered in this chapter. Using charge probe on the target surface, it appears feasible to measure the airplane induced surface charge response at different probe locations. A final application of this scheme is presented on a low-Q structure, where a conducting spherical target is considered. Such target is chosen because of its exact solution that can be used to verify the experiment. Affirmation of the natural resonance description of the late-time response of a conducting target is also accomplished in chapter 4. A semi- circular loop scatterer provides the opportunity to test both the usefulness of the singularity expansion method and the natural mode aspect independence. Comparison between the experimental and theoretical natural frequencies of the loop provides additional validation of the natural mode representation. Chapter 5 presents a theoretical analysis of a thin wire elliptical loop. The natural frequencies of such particular target are determined by solving the transform domain Hallen integral equation. This approach is based on the assumption of a natural resonance expansion of the late-time scattered field response. Finally, chapter 6 describes the experimental set-up and the basic operating principle of the time-domain scattering range that has been established in our laboratory. CHAPTER 2 SINGULARITY EXPANSION REPRESENTATION OF TRANSIENT SCATTERING 2-1 We}: The singularity expansion method (SEM) is a well known-technique, first introduced by C.E.Baum in 1971, for analysing the transient electromagnetic problems of complex scatterers. The conceptual basis of the SEM, namely natural mode representation, has been employed as the framework for electromagnetic system identification. Similarly to classical circuit theory, such technique is based upon the knowledge of the target's natural resonant frequencies from its late-time domain response which is obtained as a sum of exponentially damped sinusoids. These complex natural frequencies ( s-plane poles) of the scattering body are defined only by the structural geometry of the target and not by its particular form of excitation. The SEM representation of the scattered-field of a perfectly conducting scatterer is derived in the following section, followed by a time-domain scattering representation. 2-2 W The derivation of the integral equation for the current density on the surface of a perfectly conducting scatterer, leads to the determination 'of the electromagnetic field scattered by the target. Figure 2.2.1 shows / scattered wave 0 P,(r.t) .500 fi" * incident wave ' , / a &‘ {kg e» Fig 2.2.1 Scattering of a transient electromagnetic pulse from a conducting body an electromagnetic wave incident on a conducting target which will induce currents on the surface of the target in a way that the boundary condition on the surface is satisfied. These currents in turn produce a scattered field. Hence once the currents are determined, the field at any point in space can be computed from them. With the development of high-speed digital computers nowadays, solutions of such integral equation can be computed easily. The surface current can be represented as a sum of natural mode damped sinusoids, and consequently has the natural mode representation in the late-time period: N -v-e e-o-eant K(r,t) -n§1 AnKn(r)e cos(wnt + dn), t > TL (2.2.1) where An and ¢n are the amplitude and phase of the n'th mode, Sn - on + jwn is the complex natural frequencies (appearing in conjugate pairs), with on < 0 due to the radiation from the body, Kn(r) is the spatial current distribution of the n'th mode and TL is the begining of the late-time period. It will be shown in the following sections that these complex natural frequencies are aspect independent and depend only on the object's geometry, while the amplitude and phase depend upon the form of excitation. These natural frequencies of the radar target are commonly extracted from the late-time period scattered field to provide a unique characterization of the target. This leads to a target identification and discrimination thechnique. To evaluate these natural frequencies, we need to work in the frequency domain. The laplace transform of equation (2.2.1) is: 2N fi(¥;5) - 2 an(3)iin(¥)(s-sn)'1 (2.2.2) npl It is seen that the late-time surface current is a sum of pole singularities in the complex frequency plane, an(S) is the coupling coefficient which contains all of the amplitude and phase information of the n'th mode. From the SEM analysis such current form representation is valid in the late-time period only. Using Maxwell's equations or particularly the retarded potential theory, the scattered electric-field can be represented as: aA(¥,t) Es<¥.t) - -V¢S(r.t> - 6t s g #0 J K<¥'.t-R/c) where A (r,t) - ds' ... Vector potential 43 s R 9 S _. 1 J P3(r',t'R/C) and d (r,t) - ds' ... Scalar potential 4neo s R With the continuity equation, the Laplace transform of Es(;,t) can be expressed as: -7R e ds' [ [v'.i<'(i:’,3)v - 12136358” 5 4nR where R - I f - f’ , 7 - S/c, and c is the speed of light . e-flR Let -- - G(?,;';S), and introducing the green's dyadic notation: 4nR VV , 2 r > 6G,? ;s> 8(¥,?';5) - (‘1’ - The SEM representation (in the frequency domain ) of the scattered electric-field can be rewritten as: 2N Es(¥,5) - “3% 2 an(S)(S - sn)'1J8(¥,?';5).k’n(?')as' (2.2.3) n-1 5 After taking the inverse laplace transform, we have: 2N _. Es(¥,t) - -po 2 snan(sn)eSnc I G(r,r’;Sn).Kn(r')ds' (2.2.4) n-l s In order to determine the scattered field, we need to evaluate the surface current induced by the excitation field. From (2.2.2), such evaluation requires the calculation of the natural frequencies, which are dependent upon the target geometry, coupling coefficient and current distributions, which depend on the form of illumination with which the target was excited. The calculations of the surface current induced on the target was carried out in [14]. The time domain representation of the scattered electric-field in (2.2.4) becomes: N Es('r’,c) -nflAn(;)eantcos(wnt + ¢n(E’)) c > TL (2.2.5) It is shown then, that the scattered electric-field, similarly to the surface current, can be represented as a sum of damped sinusoids in its late-time period. 23 mustang In developping these SEM forms, one observes that the singularity expansion method characterizes the late-time transient responses of the scattering object. It is important to recognize that the SEM representation cannot represent the early-time response because it is not a sum of natural resonant modes, it is rather a forced response by the excitation pulse. CRUKPTEEI 3 MEASUREMENTS OF SURFACE CHARGES AND CURRENTS INDUCED ON CONDUCTORS 3-1 1112123351211 As defined previously, the SEM as applied to electromagnetic scattering, is a method of characterizing the response of a target in terms of its parameters such as poles and residues. To this end, these poles are assumed to be invariant and can assist in determining the identity of the target. In practice, the estimates of the natural frequencies from the measured data of the scattered-field response tend to vary with target's orientation and illumination. If we can reduce the noise and the clutter for example by repeating the measurements several times and substructing each time the clutter from the total response, or by averaging the associated signals of the scattered-field response, this data may be improved sufficiently enough to garantee accurate mode extraction. However in practice obtaining a large number of measurements is somewhat impossible rand may be expensive or time consuming. For example, if measurements are made of a moving target such as an airplane, repeating many measurements -and subtructing the clutter each time, can be done for a limited time and most of the cases the energy associated with the clutter cannot be reduced. Consequently, the resulted scattered-field responses have a small S/N 10 11 ratio, and the natural mode extraction becomes inaccurate. In other words, natural mode extraction is critically dependent upon the S/N ratio of the measured response. Therefore, careful measurements should be made for accurate results. Because these natural frequencies are shared by both target's scattered-field and its induced surface charge and current distributions, it is suggested that mode extraction may be made from transient surface charge and current responses, rather than from scattered-field responses. Indeed, the purpose of this chapter is to develop an experimental surface probing scheme, that can be used to measure parameters necessary for the implementation of the SEM description of complex scatterers. The subsequent sections will look closely at how such probes can be implemented on the target surfaces, how they're related to the measurements of either the electric or the magnetic fields, and finally how results are improved using this surface field probing technique. 3.2 Transient performance of charge and cugrent probes 3.2.1 Charge pgobe Measurements of the transient surface charges on good conductors actually involve measurements of the normal electric-field at the surface. If a short monopole probe located normal to the conducting target surface is used for the measurements, it is shown that the output of the probe is proportional to the normal electric-field or the surface charge. The surface charge p(t) is related to the electric-field by: 12 p(t) - eon.E(t) (3.2.1.1) A where n is an outward unit vector normal to the surface and E(t) is the surface field to be measured. When E(t) is added to the electric field maintained by the induced charge on the probe Ei(t), the total electric field has zero tangential component on the surface: n.[ E‘(c) + €(c) 1 - 0 We can also write this boundary condition as: A 41 A 4 n.E (t) - - n.E(t) (3.2.1.2) for an electrically short probe E‘m .. - V¢i «— dt is proportional to the time rate the surface charge being a differentiator of the electric derivative of the SEM response is also a sum of natural modes, the probe output can provide sufficient information on the complex natural frequencies of the target. 3.2.2 Cuggppt pgpbp Measurements of the transient surface current on good conductors actually involve measurements of the magnetic-fields at the target surface. A small semi-circular loop normal to the target surface can be used to measure the induced currents on the surface. From Maxwell's equations, it is shown that the output voltage of the loop is proportional to the surface current as follow: 14 V(t) - E(t).dl - - . ds c at 5 (3.2.2.1) a g _, a - - —— ( B(t).ds) - - —— (B(t).A) 8t at where V(t) is the output voltage of the semi-circular loop accross its gap, E(t) is the electric-field associated with the magnetic-field B(t) being measured, and A is the area of the loop. In addition, we know that the magnetic-field at the surface is directly proportional to the surface current from the following boundary condition: n x Eu) - #0122“) (3.2.2.2) A Where n is the local normal vector to the surface, and R'(t) is the surface density current. Substituting equation (3.2.2.2) into (3.2.2.1), we have: 8Ks(t) V(t) - -“6A (3.2.2.3) at We conclude then, that this semi-circular loop is a time differentiating surface current sampler, and its output voltage is a sum of the natural resonant modes of the target. 15 3.3 Construction of charge and current probes Charge and current probes can be used either in free-standing arrangement or in conjunction with an image plane. Since our time-domain scattering range uses a conducting ground plane, the scattering of a target was measured on a half of a target which was mounted on the ground plane due to the image effect. Thus charge and current probes located on the target were also used in conjunction with the image plane. The probe was mounted at the end of a cable that served to move the probe and also as a lead wire which connected the probe to its receiving system as depicted in figure 3.3.1. The semi-circular loop depicted in figure 3.3.1 was mounted through a slot parallel to the direction of the current to avoid the interuption of current flow. At the middle point on the upper half of the loop, a small gap ( typical width was 1-2mm ) was left in the outer conductor of the coaxial line. Typical dimensions of the coaxial line was .032-in and a subminiature connector compatible with the cable was used to provide a 50 ohms connector for the feed line. In our experiments, a .020-in diameter semi-rigid 50 ohms coaxial cable and a loop diameter of a .l25-in semi- circular loop was found to be very efficient. For charge or electric-field measurements, an even simpler probe can be used. For many applications a short monopole over a conducting surface is an efficient electric-field sampler. Such probe was easily made by protruding the inner conductor of a micro-coax cable through a hole or slot on the target surface. sca“ gap coaxial line to receiver inner c n tor E t p 0 due conducting surface ._coaxial line to receiver 0' Figure 3.3.1 Arrangement for (a) current, and (b) charge probe measurements on conducting surfaces 17 3.4 Advantages of the surface probing scheme It appears feasible to implement an experimental technique to measure the transient induced current and charge on a scatterer using the probing technique discussed above. Indeed, this scheme has several advantages: * The probe early-time responses are shorter than those of scattered- field responses, thus allowing natural mode extraction from earlier measured responses of relatively large amplitude. * The surface probe responses have relatively large amplitude, thus providing improved S/N ratio of these measured responses. * Probes have satisfactory sensitivity, and can sense relatively small induced fields. * Probes can be moved along the target surface, therefore eliminating the need for multiple probes. * Probes are simple, and economical, since they can be built in the laboratory at low cost. In the following subsections, results are first presented on thin cylinders since their transient responses are well documented, and hence we can test the validity of this probing technique. The aspect independence of the natural frequencies has been confirmed using simple shaped targets such as bent wires and loops. Also, natural resonant frequencies of complex aircraft models were measured using optimally located surface probes. Finally, application of this probing technique on a low -Q target such as a sphere is shown. 18 3.5 KW 3.5-1 W The surface probing scheme has been first implemented on thin wire cylindrical scatterers in order to assess its viability. These cylindrical wires were chosen because of their well known and characterized transient responses, and also because of their simple geometry which allows probes to move along them. Three different length wires were studied in this section and for each cylinder all surface charge, current, and scattered field responses have been measured. Natural frequencies extracted from each measurement were compared to the theoretical values, which validates the above technique. As will be shown, the charge and current results are more accurate in terms of natural mode extraction, than those of the scattered field. In the first part of this section, a brief theoretical analysis is included to show how the theoretical natural frequencies of a target are numerically determined. Then experimental results are shown not only to validate the surface field probing technique but also to verify the singularity expansion method as applied to thin wire cylindrical electromagnetic scatterer. 3.5.1.1 MaWMWWn 2111mm: A thin wire analysis has been investigated by several researchers. Tesche [18] determined the natural frequencies of thin wire cylinders numerically by solving the Pocklington [l7] equation for a general complex frequency 8: 19 L d2 32 I(z',s)( - )K(z,z';s)dz' - 0 (3.5.1.1.1) 2 2 0 dz c where the kernel : 23 e-(s/c)R l ———d¢ 2x 0 4nR K(z,z';s) - with a - [(z - z')2 + a2 sin2(¢/2)]1/2 and where I(z',s) is the induced current along the thin wire cylinder, L and d are the length and diameter of the cylinder respectively, and c is the speed of light. It is assumed that there is no d variation for the current flowing in the z-direction. By determining the location of the poles of the current function I(z;s) in the complex plane, the natural frequencies can then be determined. Equation (3.5.1.1.1) may be cast into the matrix form as: [Z(s)][I(s)] - [V(s)] (3.5.1.1.2) where [Z(s)] is an an matrix and called the system impedance, [1(5)] and [V(s)] are the response and source vectors respectively, and with dimension 20 n. The natural frequencies of the cylinder are the roots of the homogenous equation: [Z(s)][I(s)] - 0 (3.5.1.1.3) For a nontrivial solution for [1(5)], the determinant of [2(5)] must vanish at s - sn (det [2(5)] - O ). Similarly to the circuit theory, the natural resonances of this thin wire, must occur in the left-hand side of the complex plane since the time behavior is as eSt which should be decaying as t goes to infinity (radiation process) and must not reside on jw axis. Also, since the time domain current is real, these poles must occur in conjugate pairs. Such poles are simple and of the form: 8 - a + jw From figure 3.5.1.2.1 which presents the FFT spectra of the backseattered- field of the three different cylinders, it is seen that the resonances appear to occur with wL/c z n«. Thus once an initial guess to the pole location is made, equation (3.5.1.1.3) may be used to iteratively find the poles. 3.5.1.2 W One advantage of using the surface probes is that they are movable along the surface of the target. Consequently, we can determine different surface distributions at different probe locations. We can select optimal probe locations to obtain maximum probe responses. In our experiments, measurements were made on thin cylinders mounted on a ground plane, which yielded to imaging effects. Consequently, 21 only odd modes of induced currents and charges were present. Different surface charge and current responses were measured on a 6" long cylinder at different probe locations. Figure 3.5.1.2.2 shows the surface charge response of the 6" thin cylinder and its corresponding Fourier spectrum. Figure 3.5.1.2.6 through figure 3.5.1.2.8 show the natural frequencies extracted from different measured responses of 6", 12", and 18" thin cylinders. Depending on where the probe was located, some modes tended to appear or disappear. When the probe was located at the top of the thin cylinder, where there was a maximum charge distribution, all the three modes of the 6" thin cylinder appeared as shown in figure 3.5.1.2.2 (b). Similar results occured for the 12" and 18” thin wire cylinders and were shown in figure 3.5.1.2.3 through figure 3.5.1.2.5. Natural mode extraction from the surface charge and current responses was more accurate than that from the scattered field responses which was due basically to the small S/N ratio of the latter responses. This mode extraction was based on the E-pulse [16] and the continuation method algorithms. It should be observed though that the scattered field Fourier spectra of these thin wire cylinders provided more higher order frequency components than those of the surface charge or current responses as expected. However, it was very difficult to extract all of these high frequency components from the scattered field responses. 3.5.2 W W To confirm aspect independence of the natural resonant frequencies of a target, a thin bent wire and an elliptical loop scatterers were considered. Such targets have simple geometries, and can be easy to analyse. II 0" O .i .4 1 3l :31 3 3 3 81 Relative amplitude r 1" t. t'. . Rodion frequency (G—rod/s) (C) 1:4“ L Figure 3.5.1.2.1 Fourier spectra of (a) 6”, (b) 12", and (c) 18' thin cylinders. obtained via FFT 404 N LJJ Q 4 3 p— 3 “é °‘ Lu 4 2 3 Lu "4°“ a: .- Y_ 7. fi f _r r !‘ soo “I 4 fie— E ‘0 TIME (ns) (0) scum: AMPLITUDE N § °r'1'x'5'1'nfla'nn'n' mouwm-m/S) (b) Figure 3.5.1.2.2 (a) Surface charge response of a 6" thin cylinder, (b) its corresponding Fourier spectrum . I .. I 0 £2 51 3 CL 22 st Lu 2 )— -5: 53 o: ‘1 r r r r V’ r so 2 4 K E 10 TIME (ns) (0) , T 12" ’47 3... 72%! S . g m c r {if V' ' fr f f I_T"—[ r r ' i e r 5 :0 ii to l. to Figure 3.5.1.2.3 meow ms (b)( I) (a) Surface current response of a 12' thin cylinder, and (b) its corresponding Fourier spectrum t“ 204 C3 ID I.— :3 Q l “g -20- S u: m 1 ‘60 r T I r I r I o 2 4 e a 10 TIME (ns) (0) ‘0004 RELAJAEEIAHHUUUDE 0f'ic; 5 f m l1 :4 to II ' sarcoma!) )(G-IAD /3) Figure 3.5.1.2.4 (a) Surface charge response of a 12' thin cylinder. and (b) its corresponding Fourier spectrum 26 J '53 10« 3 Z -J ‘I o. E 01 LU > c 4 “J - 4 0: IO 1 -2o 4 e - a - o 2 i 3 5 {CT TIME (ns) (0) RELAIWE AMPLITUDE 3;. I) 6' TEX r5' 5 't'ort'afite II II moumcvto-m/S) (b) Figure 3.5.1.2.5 (a) Surface charge response of an 18" thin cylinder, and (b) its corresponding Fourier spectrum radian frequency x 10‘9 27 18 15- Us): 12- 9.4 ‘0 m I 61 d a current probe reap. 3 } a charge probe reap. - .. I beckecottered field reap. o._z.m , p - f -::.o -1.e -1.2 -d.e 414 ofo damping coefficient x 10" Figure 3.5 1.2.6 Natural frequencies extracted from different measured responses of a 6" thin cylinder 28 d! ‘ a . ‘9. x 12- > o I I: ‘ ‘. I“ o 3- 3: "‘ ['ltifl ‘3 . .2 13 a current probe reap. . a charge probe reap. I bockeccttered field reap. '- 0 e meanticol reg. q r’ I_ ' I r I r I’ r’ I ' -4..O -l .6 -l .2 -O.8 -O.4 0.0 0.4 damping coefficient x 10" Figure 3.5.1.2.7 Natural frequencies extracted from different measured responses of a 12" thin cylinder '9 radian coefficient x 10 29 IS a O 12-1 I. y 9-1 .‘ "b 6-4 ' a eu‘ 3“ s- cumntprobe reap. i“ a charge probe reep. II backseanwrea fleklrsep. II ..g. (PM L , , . -::.o -1.e -1.2 -6.e 4: 05 0.? damping coefficient x 10" Figure 3.5.1.2.8 Natuarl frequencies extracted from different measured responses of an 18" thin cylinder 30 Unlike other simple targets, such as thin wire cylinders or a sphere, the bent wire and the elliptical loop scatterers can perform measurements above a ground plane at multiple aspects. Natural frequencies were extracted from the measured late-time transient waveforms of the induced surface charge and current. This mode extraction was done using algorithms based upon E-Pulse and continuation methods [16] subsequent to the FFT spectrum of the corresponding time domain responses, to provide an estimate of the frequency content in the measured data. 3.5.2-1 MW Measurements of the surface current responses of an elliptical loop scatterer mounted on a ground plane, were made at different aspect angles of the loop. Figure 3.5.2.1.2 through figure 3.5.2.1.4 show surface current responses of the loop measured at aspect angles of 0, 45, 120 degree, and their corresponding Fourier spectra. The natural frequencies extracted from different surface current responses each measured at different aspect angle were plotted in figure 3.5.2.1.5. They all seemed to be quite aspect- invariant. As shown in table 3.5.2.1.1, the natural frequencies showed close agreement with the theoretical values determined from the homogeneous transform domain electric-field integral equation ( EFIE ). This provides additional validation of the natural mode representation. 3-5.2-2 W Measurements of the surface charge responses were made on a bent wire, mounted on a ground plane and rotated at different aspect angles similarly to the elliptical loop case. A short monopole probe was mounted Table 3.5.2.1.1 Natural frequencies (x 1045 of a thin wire elliptical loop of major axis A - 16 cm, minor axis 8 - 7.6 cm, and wire radius a - .25 cm Mode l 2 3 Aspect angle 0° .354 + 32.396 .400 + 34.997 .528 + 37.801 45° .489 + 32.395 .392 + 35.246 .458 + 37.695 90° .508 + 32.392 .472 + 35.397 .253 + 37.525 135° .360 + 32.465 .396 + 35.217 .095 + 37.642 180° .470 + 32.410 .381 + 35.173 .301 + 37.558 Theoretical natural frequencies .375 + j2.365 .255 + j5.250 .625 + j7.852 9.. s ‘ pfi. (4., -r J -‘-z-,.e.¢e. O? ‘ ' Table 3.5.2.2.1 32 dimensions 1.1 - 28.5cm and L2 - 32.4 cm, extracted from surface charge responses at various aspects Natural frequencies of a thin bent wire with Aspect angle 00 45° 90° 135° Mode 1 -.017 + j .857 .091 + j .860 .143 + j .849 -.053 + j.881 2 -.042 + 32.500 .137 + 32.560 .158 + 32.491 -.067 + 32.580 3 -.040 + 33.948 .042 + 33.936 7 - 073 + 33.948 4 -.258 + 35.383 .253 + 35.383 .258 + 35.390 -.296 + 35.493 5 -.233 + 37.270 .230 + 37.271 .554 + 37.205 -.315 + 37.312 6 -.274 + 38.798 .185 + 38.801 .198 + 38.669 2 H I ..— 33 on the bent wire through a slot parralel to the wire and at a distance of about 6 cm from the ground plane. Figure 3.5.2.2.1 through figure 3.5.2.2.4 show surface charge responses of the bent wire measured at various aspects (0 ,45 ,90 ,135 degrees ), and below are their corresponding FFT spectra. The natural frequencies extracted from those responses were given in table 3.5.2.2.1 and plotted in figure 3.5.2.2.5. Again they all seemed to be aspect-invariant. 3.5.2.3 Agpgpp-ipgepppggpt taggpt digpgimipatiop As previously shown the bent wire resonant frequencies have been proven to be aspect-independent. Therefore, and on the basis of these frequencies, discriminant signals, called E-Pulse (Extintion pulse) for the bent wire can be synthesized for a radar target discrimination [2]. When the E-Pulse signal for an expected target (right bent wire ) was convolved with its late-time radar return, the convolved output was zero. On the other hand, when the E-Pulse signal for the expected target was convolved with a wrong target, the convolved output was significally different and large. Thus the different targets were discriminated. This E-Pulse technique had been experimentally proved on two different bent wires. The E-Pulse of the bigger bent wire was synthesized based upon the average frequencies extracted from the responses measured at different angles as shown in figure 3.5.2.3. This E-Pulse, when convolved with the response of the same bent wire measured at any aspect angle, gave small (nearly zero) convolved late-time output as shown in figure 3.5.2.3.1 On the other hand, when this same E-Pulse was convolved with the responses of the smaller bent wire measured at various aspect angles, the convolved 34 output responses had large late-time responses for all of the aspect angles as shown in figure 3.5.2.3.2 (c). 3.5.3 Applippgipp of thp gugfppe-fielg ppobipg gcheme to pppplex 5nd low-Q taggetg 3.5.3.1 Sugggpe tgppgient gpsponse of an pirplane mode; The previous sections described the surface-field probing scheme. The technique used and some results obtained from transient surface charge and current induced on simple targets such as thin wires or loops were presented. As an example of an application of this technique on complex targets, an airplane model was considered. Over the past few years, there has been an increasing interest in determining the transient electromagnetic responses of aircraft models. Such transient response is useful for target identification, since it contains the information about the target which is closely related to the target parameters and consequently the target geometry. Because an airplane does not have exact or theoretical natural frequencies that we can refer to or may compare the experimental results, careful measurements must be done. As previously mentioned, natural mode extraction is critically dependent upon the S/N of the measured response, it is suggested then that this mode extraction may be made from surface charge or current waveforms rather than from scattered field response which is subject to more noise problems. Backscattered-field responses of this airplane were first measured at different aspect angles. Natural frequencies were extracted from these responses using E-Pulse technique, and then compared to those extracted from surface charge responses of the airplane (table 3.5.3.1). 35 It was observed that eventhough the FFT spectra of the former responses have more high frequency components than those of the surface charge responses, the natural mode extraction from the surface charge waveforms was easier and more consistant than that from the backscattered-field waveforms of the airplane model. That was basically due to the large S/N ratio and the short early-time of the measured surface charge response which allowed us to extract frequencies from earlier time of relatively large amplitude. In addition, charge probes can be placed easily at different locations on the airplane surface through small holes; thus allowing transient measurements of different charge distributions on the airplane. Figure 3.5.3.1.1(a) shows the surface charge response of an airplane model, its corresponding FFT spectrum is shown in figure 3.5.3.1.2 (a). The two sets of frequencies extracted from both the backscattered-field and the surface charge responses of the airplane were plotted in figure 3.5.3.1.3. The radian frequencies appeared very close, while the damping coefficients were not as close especially for the higher order modes. 3-5-3-2 W W As an application of the surface-field probing technique on a low-Q structure, a conducting spherical target has been considered in this last section. The study of the transient electromagnetic response of the sphere was mainly motivated by the need of understanding the transient response of a low-Q structure in connection with radar target identification and discrimination. Such target was also chosen because of 36 its exact solution that can be used to verify the experiment. Our time-domain scattering range was built on a ground plane, which acts as an electromagnetic mirror, and thus, we conducted the measurements on a hemisphere with mirror symmetry. Such hemisphere (of 30cm diameter ) was illuminated by a gaussian pulse generated by a picosecond pulse generator. Using charge probe on the target surface, it appears feasible to measure the sphere induced surface charge response at any desired probe location. Backseattered-field responses of the sphere have been also measured using a receiving monopole mounted over the same ground plane. Confirmation of the natural resonance description of the late-time response of the sphere can be accomplished by comparing the experimental natural frequencies extracted from both the surface charge and the backseattered-field responses and the theoretical values. Natural extraction was based on the E-pulse method [16]. Figure 3.5.3.2.1 shows the surface charge response of the sphere, and below its corresponding fourier spectrum obtained via FFT. More than five peaks clearly dominate the spectrum. Backseattered-field response of the same sphere measured at a distance of about 10 cm from the receiving probe is shown in figure 3.5.3.2.3, and below is its fourier spectrum. The arrows in the fourier spectra refered to the theoretical radian frequencies of the sphere. As seen, the backscattered-field FFT spectrum had more high frequency components than those of the surface charge response. However, these high components were very hard to extract, which was basically due to the small S/N ratio of the backscattered-field response. The first six natural modes of the sphere extracted from different measured responses were given in table 3.5.3.2.1 and also plotted in 37 figure 3.5.3.2.3. Comparison between the experimental and the theoretical natural frequencies show good agreement, especially for the radian frequencies. It appears that the extracted damping coefficients of the sphere from the backscattered-field response seemed closer to the theoretical values than those of the surface charge response. On the other hand, the radian frequencies extracted from the surface charge responses were more accurate and closer to the theoretical values. This was mainly due to the large S/N ratio and the short early-time of the surface charge response of the sphere. As previously shown, the sphere resonant frequencies have been experimentally extracted from different measured responses, and proven to be very consistent with the theoretiacal values. Therefore, and on the basis of these natural frequencies, discriminant signals, called E-pulse (extinction pulse) [4], for the sphere were synthesized for the radar target identification and discrimination in a similar way as for the bent wire. This E-pulse technique had been experimentally proven on two different spheres. The E-pulse of the sphere was synthesized based upon the average frequencies extracted from the surface charge and the backscattered-field responses of the sphere. This E-pulse, when convolved with the sphere response gave small (nearly zero) convolved late-time output. Figure 3.5.3.2.4 shows the convolved output of the natural E-pulse of the sphere (30 cm diameter) with its backscattered-field response measured at a distance of 5 cm from the receiving probe. The late-time output was very small. Another convolved output of the same E-pulse of the same sphere with its backscattered-field response measured at a distance of about 10 cm from the receiving probe antenna was shown in if V...- .- mlrflJu _ 38 figure 3.5.3.2.5. Figure 3.5.3.2.6 shows a convolved output of the same E-pulse of the 30 cm diameter sphere with its surface charge response. Backscattered-field response of a smaller sphere of 12 cm diameter was subsequently measured and shown in figure 3.5.3.2.7. Its convolved output with the previous E-pulse of the larger sphere is shown in figure 3.5.3.2.8 where the late-time output seemed significantly large compared to that of the larger sphere (right target); Therefore, the two spheres were discrimimated. 39 Table 3.5.3.1.1 Natural resonant frequencies of a sphere of diameter 30 cm.( x 10-9 ) mun in: arm I] Mode Theoretical Extracted natural frequencies frequencies from Surface charge Scattered field response response 1 -l.000 + 31.732 -.241 + 31.685 -.l35 + 32.208 2 -l.404 + 33.614 -.349 + 33.547 -.625 + 33.502 3 -l.686 + 35.516 -.537 + 35.386 -l.180 + 35.233 4 -1.908 + 37.430 -.575 + 37.365 -l.707 + 37.206 5 -2.096 + 39.352 -.691 + 39.251 -2.005 + 39.122 6 -2.258 + 311.284 -.796 + jll.277 -2.062 + jll.542 40 Transmitting antenna Elliptical / loop ’ Current / probe I Ground plane Figure 3.5.2.1.1 Experimental arrangements for surface current measurements of an elliptical loop scatterer mas 1 1‘1- 1 94 1 CD a 104 23 3' u 2: h 1 5 u c: I I Late-time -10-.~a..,-,_p'_3 O 2 A 6 8 10 TIME (ns) (6) 1&0qu g 1200-1 .5. 3 60041 4 0‘. 1— r ' f f I " rim 0 2 e O 0 IO 12 14 1| FREQUENCY (G-RAD/ S ) (b) Figure 3.5.2.1.2 (a) Surface current response of a thin wire elliptical loop measured at 0°, (b) its corresponding Fourier spectrum RELATIVE AMPLITUDE LdtlrEIMI -10 f h T F r ti f7 r— r a O 2 A 6 8 IO TIME (ns) (3) law!) REMIIVE AMPLITUDE 0" r1 #177 *1f r T'* F fi I v:_?——F'—-_'- o 2 s I 10 12 1s 10 1| FREQUENCY (G-RAO/ s ) (b) Figure 3.5.2.1.3 (a) Surface current response of a thin wire elliptical loop measured at 45°, (b) its corresponding Fourier spectrum RELATIVE AMPLITUDE 1 f\ “J ‘T \\\r\\\J//F\\./'\\~J///\\ffl“. C3 2 :1 . ‘5 h- .5 us a: I |___’ Lats-tin. -12 r I T T" r I ' r—— r fl T o 2 4 6 a 10 TIME (ns) (3) 1201M Mm 4 O v v r r I fi r—c—fi o z 4 F 5 10 11 16 1e 10 FREQUENCY (c-RAD/S ) (b) Figure 3.5.2.1.4 (a) Surface current response of a thin wire elliptical loop measured at 120°, (b) its corresponding Fourier spectrum L\ L\ 3'2 8 ‘ e X > k . O C: Q) g d f ‘0 .5 ‘7 13 E a angle-180 . O ‘1 e angle-1.35 a angle-90 I angle-45 0' 101r00g¥gflo r I’ r I—’ ' r' I” I ‘ -2.0 -I .6 - l .2 -0.8 -0.4 0.0 damping coefficient x 10" Figure 3.5 2.1.5 Natural frequencies of a thin wire elliptical loop extracted from surface current responses measured at various aspects as ....“ Q- L RELATIVE AMPLITUDE 2« fl ' I l-t /\ I r I On \\V//r\\\J// —l< -25 '____.,late-t1me 1-3 ... r’ e- r* . e4 ._, r4 ,1 e 0 2 A 0 0 10 TIME (ns) (5) 3 500-1 13 b g; 2001i 5 . a IW-II cfi—fi T'ir I—' T7 '— T’ * I ' 1— * If:—T 0 2 4 0 I 10 12 14 10 ' FREQUENCY (G-RAD/ S ) (b) Figure 3.5.2.2.1 (a) Surface charge response of a thin bent wire measured at 0°, (b) its corresponding Fourier spectrum -1 73*?“th puma... A u). . " . RELATIVE AMPLITUDE 2i 40 5. 0.1 \r O \ 2 1 I: a; 0d \\\J//A\\.K\J/ L4.) 4 2: 3 Lu ‘2‘ 0: 3..___—e Iatrtime “0' TTTFTETIB TIME (ns) (3) 300d zoo. ‘mdi o-r r f’fi— r V' T ' 'r' r 1 s e a to 12 to Is Figure 3.5.2.2.2 encounter (G-RAD/S ) (b) (a) Surface chage response of a thin bent wire measured at 45°, (b) its correponding Fourier spectrum _’T_-Tv -. warm-fl -r r ”1...!” L\ RELATIVE AMPLITUDE O ‘2 ‘T r '1 If ‘7 I’ ' r'— ,1 T—( I o 2 4- 6 a 10 TIME (ns) (8) 300 l 3 .... i . S § 10041 c—r' f fi'if *‘ r— * I— f IT ' A z e e I 10 12 10 1e FREQUENCY (o-RAD/ S ) (b) Figure 3.5.2.2 3 (a) Surface chgrge response of a thin bent wire measured at 90 , (b) its corresponding Fourier spectrum "war—.7 37732—7—v -K3E?l F Irmnlb~ 7; ; .f' 1 "1 2~ I A HJ / \\ a ‘/ f “ .3 \ ’3' °“ LIJ 4 Z 3' 1.1.1 '2‘ a: “a ' I I I I {73519 TIME (ns) (a) 600‘} g . :1 3 «soul 5 g zoo- emf r r——fi Tfi * 1* 'r r—* T—fi Tifir 1:“ 0 2 e I I 10 12 to 10 FREQUENCY (G-RAD/S) (b) Figure 3.5.2.2.4 (a) Surface charge response of a thin bent wire measured at 135°, (b) its corresponding Fourier spectrum A Mann“. ~ ..r, 11‘..-" ‘5‘. - .A.‘ 49 1.. w r 64 55 g I I c: I) a 3 g ~A .5. .. . ‘6 c. I. ‘ I angle-135 e angle-90 I angle-45 -0 e (31 ' T47 ' r’ ”5* I I r’ r’* r’ -2.0 -1.0 -l .2 -0.0 -0.4 0.0 0.4 damping coefficient ‘11)"9 Figure 3.5.2 2.5 Natural frequencies of a thin bent wire extracted from surface charge responses measured at various aspects u. (Ara-pal “mart (ii-1’ ‘- ..t‘stl .‘i 05.01; a 50 40.0 i Jilin 0 0 (I W relative amplitude 0 O C: (:2 relative amplitude 410.0 1__. lets-time ~60.0 0.0 4.0 ' 8.0 time (ns) time (ns) (a) (b) 40.0 relative amplitude 0.0 i 0 3 Late time ~40.0 o.0 410 I 8.0 time (115) (c) Figure 3.5.2.3.1 (a) Natural E-pulse constructed to kill the first six modes of long bent wire, (b) measured response of long bent wire, and (c) Convolution of the long bent wire E-pulse with its measured response ~n relative amplitude 51 to 3.4 0'1 .3 a" g ..:r 3 ‘ E H 0 JUL] .................. EC, =_ o [[1] 0 <3 V F > e ...-3 .- 1 +3 1 L o r1 0 01 o e H In a .3 ‘ <53 ‘ "T < 3 _. Late-time ~ 0.0 . 111.0 8.0 0.0 ‘ 430 T 8.0 10.0 time (ns) time (ns) I <3 o<5 ‘ 13.: 3 4.3 "-4 H 1 at 59. AIM/1317a,... 0o 1 so = UV: -0 > ca 4.5 d «4 ,4 20 Li I 0.0 420 ’ 8.0 7 10To time (as) (C) Figure 3.5.2.3.2 (a) Natural E-pulse constructed to kill the first six modes of long bent wire, (b) measured response of small bent wire, and (c) convolution of the long bent wire E-pulse with the small bent wire response 2.. 3: .9: ,. E- «"5 2 0" L. Q 825 a... I late-time -2 T1 '— l I f 0 2 4 0 0 10 time (ns) (6:) .’ i3, 10- I .t". 3- 3 O .2 z '1'qu 3 -30 ' I— r r f If r I— r r 0 2 4 0 0 10 time (ns) (b) Figure 3.5.3.1.1 Measured (a) surface charge and (b) backscattered- field responses of an airplane model “TI .P ‘- a-me-(l -0... Iu‘«u ”as. 5 1 1) '3 3 a—l E- 5 g 4 '3 '6 Q) a: o . I T T I - 1' - o 2 ‘ . a 10 11 I‘ " Radian frequencies (C—rad/S) (a) a- d) ‘U 3 an) ... i 3 ‘6 C) > 3 fl 0) a: a r I'L Ti '1 T— Y T r o z 4 e e to Is to 10 Radian frequencies (Grmd/S) (b) Figure 3 5.3.1 2 Fourier spectra of (a) surface charge and (b) backseattered-field responses of an airplane model 54 m 15. I 'e 3‘ 3 e A 5‘ S 12- :5 I A 0’ 3 o 6‘: r: a“ ‘ 0 .2 U 4 9- I 4.1 0a ‘ a charge probe reap. 0‘ e pgckscottm field [332. F ‘. r r r F a . -2.0 -l .6 -l .2 -0.8 -0.4 0.0 0.4 damping coefficient ‘10‘9 Figure 3.5.3.1.3 Natural frequencies of an airplane model extracted from measured surface charge and backscattered-field responses ‘J ]——Late-time The (79's) (a) Ile ‘3' 1.0-1 5 1) I- .2 L5 ’1) - “-1 3’. IL —3.3 A 'J 1 40.04 4) U 3 .3 1 C \- O Q) 3 20.04 2 I Q) 2: 1 0.0 0.0 Figure 3.5.3.2.1 (a) Surface charge response of a 30 cm diameter sphere, (b) its corresponding Fourier spectrum f 2.0 4.0 GB T are ' 10.0 12.0 1137 1 . Radian frequencies (C-Rcd/s) (b) I: IIIIliIIlIIHl' [(CIQITJG lhflohve onufiflude l l I l 1 ~20-H IJ I f—oLate-time ”308 2 — 6 a 3 12 ’kne (ns) (a) 300~ I 2004 100- i 0 J L I l ,L L ,l ' ' T r 1 E T ’ Iifi 0 2 4 6 a 1‘0 1'2 14 ”Sig. Radian frequencies (G-rad/S ) (b) Figure 3.5.3.2.2 (a) Backscattered-field response of a 30 cm diameter sphere, (b) its corresponding Fourier spectrum III—g ”Err momcoamot coczmooE Epcot? Eat uoaoobxm ocozam tmLoEofi Soon 9: Co mo_ocoaco.c 6:50: 9: U6 comtanoo 35.2 883.5 356580 9.3800 . o T N... n1 :1 L r p b IF 1— T 3 1P 0 boofi I one. potato course a I1 I .32 Ho: cocofioom I m a I e m. 1.0 U I17 4 I I a m a 9 4 II u m. C I 0 now INF Relative amplitude 58 (JI _l s: .4 Late-time ' E -155 ‘ T i I ' I l I ‘TT Time (NS) Figure 3.5-3.2.4- Convolved ouput of the natural E-pulse of the sphere (30 cm diameter) with its backseattered- field response measured at distance 5 cm from receiving probe Relative amplitude O - i Jw/WHWW’V-dv‘ q l +—-—-—1D Late-time «(a 004 o 2 4 6 Time (NS) Figure 3.5.3.2.5 Convolved output of the natural E-pulse of the sphere (30 cu diameter) with its backscattered- field response measured at distance 10 cm from receiving probe 'TFIH'EI? Relative amplitude 60 4 l l 2 -‘ ’3 l 2-1 —8 - I . Late-time -153 r T’ ‘T’ I ‘_‘ - V r T7 0 2 4 6 8 lO - Time (NS) Figure 3-5-3-2-5 Convolved output of the natural E-pulse of the sphere (30 cm diameter) with its surface charge response ' Relative amplitude 70.0- 30.0% —l0.0 —50.0-* 61 \ if l—° Late-time -90.0' 2 4 6 8 10 Time (Ms) F1Sure 3 . 5 . 3 . 2 . 7 Backseattered- field response of a sphere of 12 cm diameter IV (”I ‘. A - ' . Jul—f Relative arriplitude Relative amplitude Figure {1| LI] 1 i l r. l 62 Hk\be/\JpNflVXfN’\/\J’\/-v~v~§_n\/—_UFF I Late-time 1 f—‘Late-time _. _ i ' l V T ' 30 o i 34 6 a 10 Time (ns) (b) 3 5.3.2.3 Convolved output of the natural E-pulse of the sphere (30 cm diameter) with (3) its backscattered- field response, (1)) backseattered-field response of a 12 cm diameter sphere :- val "Er CHflAPTTHl 4 THEORETICAL AND EXPERIMENTAL ANALYSIS OF A CIRCULAR LOOP SCATTERER «(CAL-6’ '1. A- bet.- ‘ I,” i 4-1 W The basic idea behind the development of the Singularity Expansion isn't-.5 a. -7 filth " Method, is the need to provide a general analysis of the natural mode representation of a conducting target transient response. As an attempt to determine the validity of the SEM to electromagnetic problems, an experimental verification has been made on a conducting target. An appropriate target for this experimental investigation is a thin wire circular loop scatterer. This loop has a simple geometry, and therefore, involves a simple theoretical analysis. In our experiments, a semi-circular loop mounted on a ground plane as shown in figure 4.2.1. was used. Scattered-field responses of the loop at different aspect angles were measured, and thus, natural mode aspect independence was demonstrated. Affirmation of the natural resonance description of the late-time response of the circular loop was accomplished by comparing the natural frequencies extracted from the measured scattered-field responses of the loop with those obtained from the theoretical SEM analysis. Close agreement between the two sets of frequencies appeared to validate this SEM representation. 63 6A 4.2 S e en tio 0 Ci cular loo A circular loop antenna has been analyzed in depth by Blackburn and Wilton [1] using the Singularity Expansion Method. Their analysis was more concerned about an impedance-loaded antenna in free space arrangement. Similar approach can be used to a semi-circular loop mounted on a ground plane. The characteristic equation describing the natural frequencies of the loop can be obtained using the following integral equation: n J K(¢ - ¢’)I(¢')d¢' - O for all -« S d S « (4.2.1) -K Using the "thin wire approximation" of antenna theory, it is assumed that the surface current induced on the loop surface has only an axial component I(¢) along the loop. ¢ is the angular variable in the polar coordinates, and K(¢ - ¢') is the kernel of the integral equation which can be expanded in fourier series as: K(¢ - 4") - z an(s.)i=.'j“(Q6 ‘ 45') (4.2.2) n--0 where . . 2 -JSA -Jnc [K (S) + K _ (5)] 2c n+1 n 1 SA an(s) - a_n(s) - Kn(S) 65 ¢ - é' 2Asin( )' 2a I(¢) ground plane l Figure 4.2.1 Geometry of a semi-circular loop scatterer mounted on a ground plane 66 d0 A « ejn0 e-(s/c)R(€) and Kn(s) - n R(0) where A is the outer radius of the loop," a " is the wire radius, c is the speed of light, and R is the approximated distance between the axial location of the field point and the source point on the surface and is given as: ¢ - ¢' R<¢.¢'> - R<¢ - w - <4A2s1n2<—> + a2) 2 1/2 In addition, we can expand the current in fourier series as: - °° -3... I(d) m_2}mIm(s)e (4.2.3) where the amplitude In(s) depends on the form of excitation. Substituting equations (4.2.3) and (4.2.2) into equation (4.2.1), the latter becomes: n. I 2 Z an(s)Im(s)e'jn¢ ej(n-m)¢ dd' - O -« 5 ¢ 5 n m n -n (4.2.4) 67 Since nm x I J ej(n - m)¢ dd' - 6 ’1! equation (4.2.4) can be reduced to: -jn¢ - 0 g an(S)In(S)e (4.2.5) ii . Using the orthogonality property of the exponential function, equation (4.2.5) leads to: an(5) — 0 (4.2.6) where the roots of equation (4.2.6) are the natural frequencies 3 - sn to be determined. from the expression of an(s), equation (4.2.6) leads to the following characteristic equation: A n2 -;-(S/C)[Kn+1(5) - Kn-1(s)] + A (s/c)Kn(s) - 0 (4.2.7) 4.3 Numerical_rssnlts The theoretical natural frequencies of the circular loop scatterer were obtained by solving equation (4.2.6). Moreover, Blackburn and Wilton [1], have treated these natural frequencies in more detail. After numerical calculations, they have been able to find three seperate 68 categories of pole locations in the s-plane for each mode number n. There is one type I pole which is located near w a n in the s-plane. This pole has the smallest damping coefficient and gives the major contribution to the late-time period of the sacttered-field response of the loop. Then, there are n+1 type II poles for each n, which lie on the left hand side of the type I poles. Finally, there are the type III poles which lie in a layer roughly parallel to the s - jw axis. Since the type I poles give the principle contribution to the time- domain response of the loop, they are those of the greatest interest to us. The first six type I poles have been computed and shown in table 4.3.1 and also plotted in figure 4.3.7. These poles correspond to a circular loop of outer radius A - 15.2 cm and wire radius " a " - .25 cm. The behavior of the type I poles as the loop is deformed into an ellipse is investigated in the next chapter. Confirmation of the natural resonance description of the late- time response of the circular loop can be accomplished by comparing the natural frequencies extracted from the measured data with those determined theoretically. Figure 4.3.1 shows the measured scattered field response of a semi-circular loop mounted normal to the ground plane. This response is apparently represented by a pure natural resonance series. These modes are shown in figure 4.3.2 which is the Fourier spectrum of the scattered field response obtained via the FFT. More than five peaks clearly dominate the spectrum. These modes were used as initial guesses for the E-Pulse method for natural frequency extraction. The extracted natural frequencies corresponding to these modes and the theoretical frequencies predicted from the SEM analysis are given in table 4.3.1 The imaginary parts of these 69 frequencies seem to match very well. The damping coefficients are not as close, which is not surprising because from our previous experience, it has been shown that extracting the damping coefficients is not accurate most of the time. A verification of the aspect independence of the natural frequencies of the loop scatterer can be obtained by rotating the loop over the ground screen and making measurements at different aspect angles. Table 4.3.2 shows the resulting natural frequencies, relative amplitudes and phases extracted from three different aspect angles. As seen, the amplitudes and phases change dramatically, while the natural frequencies remain constant. These results for the circular loop scatterer provide a considerable amount of confidence in the natural resonance expansion of the late-time response of a conducting target. Table 4.3.1 First six type I poles of a circular loop of radius A and wire radius a -.016A n S x 10-9 n Theoretical poles Experimental poles 1 -.216 + 32.006 -.235 + 32.185 2 -.327 + 34.072 -.267 + 34.250 3 -.417 + 36.065 -.487 + 36.154 4 -.498 + 38.055 -.512 + 38.046 5 -.S73 + 310.041 -.546 + 310.098 6 -.642 + 312.027 - 645 + 312.300 T .3 mks-Tii'm fi-‘. .021: 3 EAW'I— E-I Table 4.3.2 Extracted natural frequencies, relative amplitudes and phases of natural resonant modes of a circular loop scatterer at various aspects .' “VI“ Aspect Mode Damping _9 Radian _9 Relative Relative angle coeff.x 10 freq.x 10 ampli. phase 1 -.123 2.345 .238 4.306 2 -.267 4.252 .412 -.050 0° 3 -.278 6.154 .359 1.016 4 -.512 8.046 1.027 -.775 5 -.534 9.878 1.726 -.379 6 -.645 12.300 .105 -1.115 1 -.118 2.213 .396 -1.518 2 -.266 4.098 .589 1.185 45° 3 -.323 6.122 .220 3.444 4 -.545 8.081 .695 -1.036 5 -.S46 10.098 .817 ..993 6 -.435 12.101 4.127 4 127 1 -.233 2.039 .448 -.765 2 -.323 4.044 . .720 1.459 90° 3 -.400 5.989 3.481 3.916 4 -.318 8.200 .394 -1.000 5 -.436 9.879 .531 .302 6 -.544 12.340 .298 3.920 Relative amplitude 20- __L I N O L 6 8 10 #m Time (ns) Figure 4.3.1 Late-time scattered field response of a thin wire circular loop scatterer (a - 0.016A, A - 15.2 cm) measured at an aspect angle - 0° Relative amplitude 2000‘ fig . ll Radian frequency (G-rad/S) (Tl—1h. 11,1 i T . r 10 12 14 Figure 4.3.2 Fourier spectrum of a thin wire circular loop (a - 0.016A, A - 15.2 cm) scattered field measured at an aspect angle - 0° ‘6'... ‘ I...“ Relative amplitude 74 20‘“ A 0‘ NWW . -20... “'40 T f . T T T ' f T r O 2 4 6 8 10 Time (ns) Figure 4.3.3 Late-time scattered field response of a thin wire circular loop (a - 0.016A, A - 15.2 cm) measured at an aspect angle - 45o 'vl‘ n.‘fl . Relative amplitude 2000« \ \ \Ak i 1,1,1. - r . T i 8 10 12 i4 #71,... O) Radian frequency (G-rad/S) Figure 4.3.4 Fourier spectrum of a thin wire circular loop (a - 0.016A, A - 15.2 cm) scattered field measured at an aspect angle - 45° ‘1. Ami-mats- ...---st .a. ‘0-1 Relative amplitude 76 WT 20-” . ii ‘51 u- r‘ v. ' . ‘40 T T T T F O 2 4- 5 8 10 T r f Time (ns) Figure 4.3.5 Late- time scattered field response of a thin wire circular loop (a - 0. 016A, A - 15. 2 cm) measured at an aspect angle - 90° 4000+ L 2000‘ Relative amplitude 6.4—:5141-1_.L-.4, 0 2 '4 6 8101214 Radian frequency (G—rad/S) Figure 4.3.6 Fourier spectrum of a thin wire circular loop (a - 0.016A, A - 15.2 cm) scattered field measured at an aspect angle - 90° 78 ‘i‘ 3 12« 3 ‘4 O >- (.29 i ‘5 a ‘5‘ 84 an ' 8 E J g A O 3 44 a I O W n. . ‘ a M e ends-45 ‘3-F_£L_JIIII=;I_1 ' T' ' r’ r TT T ‘r -1l.2 -1.o -o.e -o.e 41.4 -0.2 0.0 DAMPING COEFFICIENT '10'9 Figure 4.3.7 Natural frequencies of a circular loop scatterer extracted from scattered field responses measured at various aspects w are-3. urn-5.: ..-. ”Le CHAIHHHR 5 DETERMINATION OF THE NATURAL FREQUENCIES OF THIN WIRE ELLIPTICAL LOOP USING HALLEN INTEGRAL EQUATION 5.1 W In the recent years, the natural frequencies of a conducting body have been theoretically determined through solving homogeneous transform domain electric-field integral equations ( EFIE ), which involve a simple mathematical description. Because of the effects of its derivative terms, the EFIE is not stable, consequently, it does not always provide accurate values of the natural resonant frequencies. An alternative method for obtaining these natural frequencies is presented in this chapter. It is found that the Hallen integral equation is the most adequate and useful equation in the numerical determination of the target natural resonant frequencies. Mei [9] has utilized the Hallen integral equation, and has extended it to describe thin wire antenna of arbitrary geometry. A similar derivation of the Hallen type equation for an elliptical loop scatterer is accomplished in the following sections. This approach is based on the SEM assumption of a natural resonance expansion of the late-time scattered field. Thus, agreement between the theoretical and the experimental natural frequencies provides additional validation of the natural mode representation. 79 “LR §'— I‘m 80 5.2 fiallen integral equation for the elliptical loop Using Mei's integral equation [10 ] which describes thin wire antennas of arbitrary shape, it is feasable to write the homogeneous transform domain Hallen integral equation as: 2 a g A A J I(u';s)[ --—- (u,u';s) + 12(u.u')g(u,u';s)] du' - 0 (5.2.1) P auau' where I(u';s) is the Laplace transform of the current distribution which is assumed to be axially directed and azimuthally invariant, u is the arc length measured along the wire, 0 is the unit tangent vector at u, 7 - s/c, where s is the transform variable and c is the speed of light. After using the " thin wire approximation " the green's function can be represented by its average around the periphery of the wire [14]: e"7R g(u.u';5) - 4rR where R - ( d2 + a2 )l/2 and d is the distance between the axial points u and u' as shown in figure 5.2.1, "a" is the wire radius, which is assumed to be sufficiently small. For the special case of an elliptical loop, equation (5.2.1) can be 81 rearranged using the geometry shown in figure 5.2.1. Using the standard form of elliptical coordinates [14], any point (x,y) that lies on the outer periphery of the loop can be written as: x - Acosn y - Bsinn where A and B are the major and minor axis of the ellipse, respectively, and n is the angular variable of the elliptical loop. Thus equation (5.2.1) can be rewritten as: where and where the 62; ~2— ~ [ (n.n';S) + 7 S(n.n')g(n.n';8) ] dn' - 0 (5.2.2) anan' S(n,n') - sinnsinn' + 2 cosncosn', A ~ e-vAR g(n.n';S) - ——:—-. GA)?» i - ( 32 + (a/A)2 )“2. ~ 2 32 2 12 d - ( (cosn - cosn') + 2 (sinn - sinn') )/ A " refers to normalized parameters (dimenssionless). Equation (5.2.2) can be manipulated and simplified using the relation: 82 Figure 5.2.1 Geometry of the elliptical loop scatterer 83 62 ~2 In sinh7(0-n) - 7 ) <———— 8(9) d6 - -g(n) (5.2.3) 6n2 0 1 where 0 is any arbitrary angular parameter. 2.. a g(n.n';5) ~2~ ~ Let g(n;S) - + 7 S(n.n')g(n.n';5) anan' Substituting g(n;s) into equation (5.2.3) gives: 62 ~2 n azg ~2~ ~ sinh7(0-n) -( 2 - 7 ) [ + 7 S(n.n')g(n.n';5)] .. d9 an 0 398n' 7 628(n.n';s) ~2~ ~ - + 7 S(n.n')g(n.n';5) (5.2.4) 6060' After substituting equation (5.2.4) into equation (5.2.2), this later becomes: 84 2 ... fl 8 ~2 n 328(9.n';s) ~2~ ~ I(n';5)( 2 - 7 ) I + 7 3(9.n')g(6.n';5)] -« an' 0 8986' sinh7(0-n) X d0dn - O 7 or, 82 ~2 « ( 2 - 7 ) I(n';S)K1(n.n';S)dn' - 0 (5 2.5) an -« where 2~ , - n 6 8(0.n :s) ~2~ ~ sinh1(0-n) K1(n.n';5) - [ + 7 S(n.n')g(0.n';5)] - d0 0 aoan' 7 Equation (5.2.5) is an integro-differential form equation [10], the solution of which is a more stable Hallen integral equation, and is represented as: w J I(n';s)K(n,n';s)dn' - Csinh;n + Dcosh7n (5.2.6) -n 85 where n 62(0.n';5) ~ ~~ - r K(n.n';S) - [ cosh7(9-n) - 75(9.n')g(6.n':5)sinh7(9-n)id9 0’ 80' which is a simplified form of the kernel K1(n,n';s). 5.3 W Because of its symmetrical geometry, the elliptical loop spatial current distribution can be decomposed into four possible ways [14]. As shown in figure 5.3.1, there are four symmetry cases. In each case the ellipse is divided into four quadrants, each corresponds to the relative sign of the current in that quadrant. The arrows indicate the current direction for the lowest mode. The current signs in each symmetry case and each quadrant are represented as: Case (J) Qlj sz Q3j Q4] 1 + + - - 2 + - - + 3 + - + - 4 + + + + where Q13 indicates the sign of the current in each quadrant i for symmetry case 3. Thus, equation (5.2.3) can be divided into four integrals: 86 Figure 5.3.1 Symmetry cases for current distribution on an elliptical loop scatterer 87 1I’ J I(n';s)K(n,n';s)dn' - II + 12 + 13 + I4 “K n/2 where I - I(n';S)K (u,u';s)dfl'. 1 0 1 n/2 12 - 0 I(n';S)K2(n.n';S)dn'. «/2 I - I(n';S)K (n.n';8)dfl'. 3 0 3 n/2 and I4 - 0 I(n ;S)K4(n.n ;S)dn Equation (5.2.3) then becomes: 4 «/2 ~ ~ 2 Q I(n':S)K (u,u';s)dn' - Csinh7n + Dcosh7n 1-1 13 0 1 where 831 (0.0';S)cosh7(0-n) - V - Ki(n.n';5) - J [(-1)1 1 0 an' - 7§i(0.n')§i(0.n':8)sinh7(0-n)ld9 (5.3.1) 'PJZ’L'I34 1m. :34»: 88 with ~ 82 Si(0,n ) - $5151n651nn + SCi A2 cosfis1nn and _ 2 32 2 uz di(0,n ) - ( (c056 - Ssicosn ) + A2 (Slnfl - Scisinn ) ) where Ssi and SC are the signs on the sine and cosine terms, respectively. i 1 Sci 831 l + + 2 + - 3 - - 4 - + The main advantages of using the above symmetry, is to simplify the numerical determination of the natural frequencies of the elliptical loop, and also to save a significant amount of time and efforts. 5.4 e me o o ut one The natural frequencies of the elliptical loop are determined by solving the inhomogenous equation (5.3.1). These frequencies are the values of the transform complex variables 5 which lead to a nontrivial solution to equation (5.3.1). Such equation can be solved numerically using the well known moment method [17]. The current along the loop can be expanded into a set of rectangular 89 pulse basis functions: N 1(0) - 2 anPn(n) n-l where an is the amplitude of the current, and 1 "n-l S 0 S "n Pn(n) - O ... else with an angular width of "n - nn_1. After substituting this expansion into equation (5.3.1), and multiplying it by 6(n - gm) where (m is the matching point ( l < m < N ), equation (5.3.1) yields to: 4 N n n ,. , _ ~ - .2 Qij 2 an Ki(§m,n ,s)dn Csinhygm + Dcosh7§m (5.4.1) l-l n—1 "n-1 1 S m S N where f a; (€,n';s) i-l i ~ Ki(§'m,n';s) - 1m[ (-1) —— cosh1(0-§'m) - O 60' - 7§i(0.n')§i(6.n':5)sinh7(6-§m) ldfi Equation (5.4.1) may be cast into the form: N nflan LInn - Cs1nh~y§In + Dcoshyg'm m - 1,2,...N (5.4.2) 0. -o-\ -...I’ lb 1in m. L! 5 f1" 3; _ 90 where gm 4 1-1 ~ ~ ~ LInn - 0 ifloiji <1) [ gi(0,nn;5) - gi(6.nn_l;5) ] cosh7 Qij [ 81(9,fln;5) - gi -[ (9-nn_1)201n + (a/A)2 1 where 2 2 2 2 C - sin qn_1 + (B/A) cos nn_ 1n 1 Also ~ +~ - eiyfi a e_7nn1 + .~ i771n-1 - 7e (0 nn_1) Hence, and after using the relation in equation (5.4.10), the integral in equation (5.4.4) is approximated as: U— 94 Q . Fm’n - -—ll—- 2 { - cosh1(nn_1-§m) x 2 2 ]1/2 (Cn‘fln_1) + [ (fn’fln_1) + (a/Acln) x log 2 11/2 2 (§n_1'fln_1) + [ (§n_1'fln_1) + (a/Acln) h ‘as til 7 2 ) + (a/AC 2 11/2 - i 1n) + 7sinh7(nn_l-§m) T T (Tn-nn_l - T (§n_1-nn_1)2 + (mom)2 11/2 1 5 "l. . (§n+1-nn) + T (§n+1-nn)2 + (at/401,1)2 11/2 2 1/2 + cosh;(nn-§m)log 2 (tn-fin) + T (§n~nn) + (a/Acln) ] + 7sinh7(nn-§m) T T (§n+1-nn)2 + (a/Acln)2 11/2 - T (tn-fin)2 + (ea/40m)2 11/2 1 } (5.4.16) Similar procedure can be made for equation (5.4.11) when i - 1. Because of the double integrals carried by equation (5.4.11), careful approximations and calculations must be made. Expanding in a Taylor series about 0' = (n, gives: 95 81(0)") 3 C1110 + c2n0(’7"§n) where . . 2 clnfl - $1n051n§n + (B/A) cosacosg'n and . 2 C2n0 - Sinficosg'n - (B/A) cosacosg'n and let 1/2 C3n0 - (Clnfi) Also ~ 1 81(9.'7';S) ‘-* ~ - 1 (7A)R1 where 81 a [ (n'-rn)2c23no + (a/A)2 11/2 for R << (7A) With these approximations, the integral in equation (5.4.13) becomes: Iin(0) - 11(0) + 12(0) + 13(0) + 14(6) where: 96 11(0) - - ClnaAn (5.4.17) I a - 1 2>c [ < - >2 - < - )2 l (5 4 18) 2( ) - ( / 2n9 "n gin "n-l gin ' ' (an-(n) + [Tun-rn)2 + (a/Ac3np211/2 13(9) ' (Clue/C3noIA)1°5 (nn_1-§n) + [(nn_1-§n)2 + (a/Ac3np211/2 2 2 2 2 1/2 -[ (nn_1-§n) + (a/Ac3n,> 1 } (5.4.20) A - - where n n n Moreover, it has been shown that the integrand of equation (5.4.12) is varying linearily. Therefore, we can use the rectangular rule integration to evaluate this integral rather than using the numerical analysis. For i - 1, equation (5.4.12) becomes: n _ ;nn 2 Q1311n(‘n)e 4k (5.4.21) 97 where Ak - Ck — (k-l’ and Kn 15 any arbitrary paint in the interval A k Equation (5.4.11) then becomes: m ,n _ E k-l m C Qljlin(nn)sinh7(nn-§m)ak (5.4.22) Finally, and after substituting equations (5.4.16) and (5.4.22) into equation (5.4.3), the entries in the moment method matrix for the case when k - n or k-l - n will be filled as shown in table 5.4.1. TI 98 Table 5.4.1 Entries in moment method matrix for the elliptical loop scatterer (k - n, k-l - n) m 4 Fm'“ - 2 2 013.01)1 { 21(nk_1.nn_1;8) + §i(nk_l.nn;5)}cosh7(nk_l-§m)4k k-l i-2 Q11 m ~ + 2 - cosh7(nn_1-§m) x (1A)C1n k-l 2 2 11/2 x lo (tn-an-1) + [ (tn-nn_1) + (a/Acln) 3 2 2 11/2 (§n_1'fln_1) + [ (§n_1’fln_1) + (a/Acln) + 7sinh7(nn_1-§m) T T (in-nn_1)2 + (amen)2 11/2 - T (§n_1-nn_1)2 + (a/Ac1n>2 11/2 ] -nn)2 + (a/Acln)2 11/2 (tn-an) + T (Tn-an)2 + (a/Acln)2 11/2 (§n+1'"n) + [ (§n+1 + cosh;(nn-§m)log 2 1/2 -nn)2 + (a/Acln) T + 7sinh7(nn-§m) T T 2 + (a/Acln)2 11/2 1 } 99 Table 5.4.1 (continued) m 4 ~ ~ ~ m,n G 2 E Qijsi("k-1’§n)51("k-1’Cn‘5)51“h7("n 1- Wm)A k-l i-2 m + k2 JQij in (Kn )Sinh7(~n -§ m)Ak ... “LA—.n ‘1. ‘. .‘fl-L" -_ .‘ A. 0 100 Table 5.4.2 Entries in moment method matrix for the elliptical loop scatterer ( k i n, k-l # n ) m 4 ,n 1 ~ . ~ . Fm -k§1 151 Qij(-1) [ 81("k-1’"n-1’S) + 81("k-1’"n’s) ] X 0°5h7(’7k_1'§m) A'k m 4 m,n ~ - . ~ _ c kEl 131 Qij51(nk_1.(n)gi(nk_l.Cn.S)sinh7(nk_1 rm)4k4n 101 The partitioned Hallen equation (5.4.2) represents a system of N simultaneous linear equations in the N unknown current expansion coefficients a. Since equation (5.4.2) is a nonhomogenous equation, it is necessary to have two boundary conditions for the two unknown coefficients C and D. From the symmetry of the loop, it is seen that: I(fl - 0 or n - n/2) - O which implies (5.4.23) a - 0 or a - 0 Also, from the continuity distribution of the current along the loop: dI -———(n - 0 or n = n/2) - 0 dn (5.4.24) thus, a ora '0 ' “ N-l N For each symmetry case, the boundary conditions are specified and given in table 5.4.3. 102 Table 5.4.3 Boundary conditions in each symmetry case of the elliptical loop scatterer Case Boundary condition Boundary condition No. 1 No. 2 1 a1 - 0 aN - aN-l 2 a1 - a2 aN - O 3 a1 - 0 aN - 0 4 a1 - a2 aN - aN-l -0 -. ‘Il-~€p'! I’l' IE’J' '\ 103 Consequently, the partitioned Hallen equation (5.4.2) can be split into four equations, each corresponds to one symmetry case. For each case, the equation can be written in a matrix form as: A.a - 0 for i - 1,2,3,4. 1 i Case 1 i l -cosh(7A§l) L1,2 ..... L1,N-2 L1,N-1 + L1,N -sinh(1A§1) : . . . . . . 2 -cosh(7A§m) Lm,2 ..... Lm,N-2 Lm,N-l + Lm,N -sinh(1A§m) . - O . . . . . . QB'I -cosh(7A§N) LN,2 ..... LN,N-2 LN,N-1 + LN,M -51nh(7A§N) L .. .. .7 (5.4.25) Case 2 7 7 -cosh(7A§l) L1,1 + L1,2 L1,3 ..... L1,N-1 -sinh(7A§1) C . . . . . a2 -cosh(1A§m) Lm,1 + Lm’2 Lm,3 ..... Lm,N-l -sinh(1A§m) . - 0 . . . . . 03-1 -cosh(1A§N) LN,1 + LN,2 LN,3 ..... LN,N-l -51nh(1A§N) j (5.4.26) Case 3 104 -cosh(7A§1) L1 2 ..... L1 N 1 -sinh(7A§l) C -cosh(7A§m) Lm 2 ..... Lm N 1 -sinh(1A§m) . - O . . . . a _1 -cosh(7A§N) LN 2 ..... LN N-l -sinh(7A§N) B L _ _ _ (5.4.27) Case 4 -cosh(7A§1) L1’1+L1,2 L1,3 °"L1,N-2 L1,N-1+L1,N -51nh(7A§1) C ~cosh(7A§m) Lm,1+Lm,2 Lm,3 °"Lm,N-2 Lm,N-1+Lm,N -sinh(7A§m) . . . . . . . . aB_1 'C°Sh(7A§N) LN,1+LN,2 LN,3 "'LN,N-2 LN,N-1+LN,N 'Sinh(7A§N)j (5.4.28) Since equations (5.4.25) through (5.4.28) are all homogenous systems, each will have a solution only when the determinant of the coefficient matrix goes to zero. The poles which cause this to occur are the natural frequencies of the thin wire elliptical loop scatterer at which there can exist a nontrivial solution. ‘7“ , CHAIHHHR 6 EXPERIMENT 6.1 Introduction This chapter will describe the basic operating principle of the time-domain scattering range of the surface current and charge responses as well as the scattered field transient responses of radar targets, that has eF—em -mhm- h—un-‘mv ...a '. ‘ ' . u been established in Michigan State University. Figure 6.2.1 shows a rough sketch of the experimental facilities used in our laboratory, followed by a functional block diagram of the time-domain scattering range. Finally, a representation of the relative location of the elements on the ground plane is shown for both scattered-field and surface charge measurements. 6.2 Time-domain scgttgring rang; All of the time-domain scattering ranges have been built on a ground plane as shown in figure 6.2.1. The conducting ground screen consists of nine individual 4 x 8 ft aluminum sheathed modules, and has total dimensions of 16 x 20 ft. This screen acts as an electromagnetic mirror, and thus limits application to the measurements of targets with mirror symmetry. Also, it provides the isolation from equipments and cabling necessary for its application. As shown in figure 6.2.2, the system signal source is a picosecond pulse lab model 10008-01, which generates a gaussian pulse of 1 picosec rise-time. This signal is radiated from a monocone antenna suspended over the ground screen. The antenna has an axial height of 105 106 2.4 m, a polar angle of 8° and a characteristic impedance of 160 ohms. It transmits TEM spherical waves polarized perpendicular to the ground plane. Scattered-field of the target is measured by a short monopole probe situated on the ground plane at a distance of about 1.4 meters from the transmitting antenna. The transient induced charge and current on the target surface, are measured respectively by charge (short monopole) and current (semi-circular loop) probes located on the target surface (construction and implementation of these surface field probes were described in chapter 3). These surface field probe responses are then measured using a picosecond-rise time sampling and wave processing oscilloscope (model Tektronix 7854 ) which is triggered by the initial pulse. Processed data is displayed on the oscilloscope CRT, which is in turn connected to a micro-computer. Once the measured data has been acquired in the computer memory, it can be either stored in magnetic floppy disks for further operations, or transmitted to the VAX/VMS mainframe computer system for more sophisticated and/or more rapid graphic plotting. The main characteristics of the above range are the speed and the simplicity with which transient responses can be obtained. The sketch on figure 6.2.3 shows the relative location of the elements on the ground plane. As mentioned earlier, the SEM representation is valid only in the late-time period of the target response. Thus it is very important to determine the exact value of the begining of the late-time or the length of the early-time period of the measured response. The transmitted signal travels outward from the monoconical transmitting antenna and is received at r at time to - r/c, where c is the speed of light. This time is marked by the pulse at the left end of the trace in figure 6.2.3. The transmitted 107 wave reaches the target, then gets reflected, and arrives at the receiver at time t1 - TL - 2(d +L)/c + Tp where IL is the begining of the late-time period of the scattered-field response of the target, Tp is the pulse width (about 1 ns), L is the length of the target, d is the distance between the target and the receiving probe, and c is the speed of light. The duration of the early-time period of the scattered-field response is longer than that of the surface charge or current response. Indeed, the target response in the latter case is received by the surface field probe, which is mounted on the target surface as shown in figure 6.2.4. Thus the begining of the late-time period of the surface charge and current responses is determined as: T - 2D/c + T L p where D is the distance between the charge probe and the edge of the target (D < L), Tp is the width of the pulse, and c is the speed of light. Indeed, the surface field measurements provide shorter early-time period than the scattered-field measurements. Thus permitting natural mode extraction from earlier response of relatively large amplitude, which is one of the main advantages of the surface field probing scheme. 108 Transmittina antenna I l I, Ground plane I ’C---- -’ - - ‘ -/ ----- / —- - I / / / ,’ / lReceiving ,/ Target // I b Ill.‘ / me lii’ & /’ Figure 6.2.1 Transient scattering range at Michigan State University 109 Dulse trans. generator antenna target recelv. antenna Tcottered l FGSDODSE sampling 8 process. scope r’i 1 micro- —1 computer, vax/vms disk visual (DIOttln;1*_ Storage —4 insoect trigge Figure 6.2.2 Functional block diagram of the time-domain scattering range 110 incident 3 pulse Trans. antenna (d+L) c 1 1 ‘ .— to -0 Tp Receiv. Target probe L 3, ' ,I,r/'/g/,/’/V/r}’/'7rl l5.(./ /.4"7’ I /”7’r”7 | ' I 2.; r 4—4mt— d-4;‘F-' L -‘L Figure 6 2.3 Relative location of the elements on the ground plane for scattered field measurements (r - 1.4 m, d - 6 cm) and corresponding incident pulse traces lll incident pulse Trans. 0/ ,__., C .... antenna . L L . .1 0 -o- Tp‘t- Charge Target probe ////| f// (fr/fl . I l i ' 0 ‘t r 4—4IIh-D «-m- Figure 6.2.4 Relative location of the elements on the ground plane for surface field measurements (r - 1.4 m) and corresponding incident pulse traces CHAPTER 7 CONCLUSION This thesis has introduced a surface field probing scheme, that has been used to measure conducting target parameters necessary for the implementation of the SEM representation. Such technique has been shown to be very effective and accurate in terms of natural mode extraction. Measurements of surface charge and current reponses of various conducting targets have been accomplished. Comparison between experimental and theoretical results has provided a considerable amount of confidence in this new surface probing technique and has also validated the natural resonance expansion of the late-time response of a conducting target. A verification of the aspect independence of the natural modes has been also made using simple targets such as thin wire loop or bent wire, which have performed measurements above ground screen at various aspects. Surface field probes have been also used on complex targets such as aircraft models, and on low-Q structure targets where a conducting sphere has been considered. The success of this technique on such targets has provided an additional validation of the SEM description of the late-time response of these targets and a good justification of the surface field scheme applied to any complex scatterer. Several advantages of this surface probing scheme have been deduced. Mainly, the large S/N ratio and the short early-time of the surface charge and current waveforms seemed to be the keys to success and usefulness of 112 113 this technique over the scattered field measurements. The simple structure and behavior of the surface field probes have provided easy and practical way of measuring the induced surface charge and current distributions on different conductors. Other topics have been included in this thesis as well. Backscattered-field measurements of a semi-circular loop mounted on a ground plane have been presented. Such measurements have meant to validate more the usefulness of the SEM description and also the natural mode aspect independence. Lastly, a theoretical analysis of a thin wire elliptical loop scatterer has been conducted. A Hallen integral equation has been derived and subsequently solved in the frequency domain to determine the natural resonant frequencies of such particular target. BIBLIOGRAPHY 10. 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