Radar Cross Section (RCS)

Transcription

1 Chapter 11 Radar Cross Section (RCS) In this chapter, the phenomenon of target scattering and methods of RCS calculation are examined. Target RCS fluctuations due to aspect angle, frequency, and polarization are presented. Radar cross section characteristics of some simple and complex targets are also introduced RCS Definition Electromagnetic waves, with any specified polarization, are normally diffracted or scattered in all directions when incident on a target. These scattered waves are broken down into two parts. The first part is made of waves that have the same polarization as the receiving antenna. The other portion of the scattered waves will have a different polarization to which the receiving antenna does not respond. The two polarizations are orthogonal and are referred to as the Principal Polarization (PP) and Orthogonal Polarization (OP), respectively. The intensity of the backscattered energy that has the same polarization as the radar s receiving antenna is used to define the target RCS. When a target is illuminated by RF energy, it acts like an antenna, and will have near and far fields. Waves reflected and measured in the near field are, in general, spherical. Alternatively, in the far field the wavefronts are decomposed into a linear combination of plane waves. Assume the power density of a wave incident on a target located at range R away from the radar is P Di, as illustrated in Fig The amount of reflected power from the target is P r = σp Di (11.1)

2 R scattering object radar Radar Figure Scattering object located at range R. σ denotes the target cross section. Define P Dr as the power density of the scattered waves at the receiving antenna. It follows that P Dr = P r ( 4πR 2 ) Equating Eqs. (11.1) and (11.2) yields (11.2) σ 4πR 2 P = Dr P Di (11.3) and in order to ensure that the radar receiving antenna is in the far field (i.e., scattered waves received by the antenna are planar), Eq. (11.3) is modified σ 4πR 2 P = lim Dr R P Di (11.4) The RCS defined by Eq. (11.4) is often referred to as either the monostatic RCS, the backscattered RCS, or simply target RCS. The backscattered RCS is measured from all waves scattered in the direction of the radar and has the same polarization as the receiving antenna. It represents a portion of the total scattered target RCS σ t, where σ t > σ. Assuming a spherical coordinate system defined by ( ρ, θ, ϕ), then at range ρ the target scattered cross section is a function of ( θ, ϕ). Let the angles ( θ i, ϕ i ) define the direction of propagation of the incident waves. Also, let the angles ( θ s, ϕ s ) define the direction of propagation of the scattered waves. The special case,

3 θ s when = θ i and ϕ s = ϕ i, defines the monostatic RCS. The RCS measured by the radar at angles θ s θ i and ϕ s ϕ i is called the bistatic RCS. The total target scattered RCS is given by 2π 1 σ t = σθ ( 4π s, ϕ s ) sinθ s ϕ s = 0 π θ s = 0 (11.5) The amount of backscattered waves from a target is proportional to the ratio of the target extent (size) to the wavelength, λ, of the incident waves. In fact, a radar will not be able to detect targets much smaller than its operating wavelength. For example, if weather radars use L-band frequency, rain drops become nearly invisible to the radar since they are much smaller than the wavelength. RCS measurements in the frequency region, where the target extent and the wavelength are comparable, are referred to as the Rayleigh region. Alternatively, the frequency region where the target extent is much larger than the radar operating wavelength is referred to as the optical region. In practice, the majority of radar applications fall within the optical region. The analysis presented in this book mainly assumes far field monostatic RCS measurements in the optical region. Near field RCS, bistatic RCS, and RCS measurements in the Rayleigh region will not be considered since their treatment falls beyond this book s inted scope. Additionally, RCS treatment in this chapter is mainly concerned with Narrow Band (NB) cases. In other words, the extent of the target under consideration falls within a single range bin of the radar. Wide Band (WB) RCS measurements will be briefly addressed in a later section. Wide band radar range bins are small (typically cm); hence, the target under consideration may cover many range bins. The RCS value in an individual range bin corresponds to the portion of the target falling within that bin. dθ dϕ s RCS Prediction Methods Before presenting the different RCS calculation methods, it is important to understand the significance of RCS prediction. Most radar systems use RCS as a means of discrimination. Therefore, accurate prediction of target RCS is critical in order to design and develop robust discrimination algorithms. Additionally, measuring and identifying the scattering centers (sources) for a given target aid in developing RCS reduction techniques. Another reason of lesser importance is that RCS calculations require broad and extensive technical knowledge; thus, many scientists and scholars find the subject challenging and intellectually motivating. Two categories of RCS prediction methods are available: exact and approximate.

4 Exact methods of RCS prediction are very complex even for simple shape objects. This is because they require solving either differential or integral equations that describe the scattered waves from an object under the proper set of boundary conditions. Such boundary conditions are governed by Maxwell s equations. Even when exact solutions are achievable, they are often difficult to interpret and to program using digital computers. Due to the difficulties associated with the exact RCS prediction, approximate methods become the viable alternative. The majority of the approximate methods are valid in the optical region, and each has its own strengths and limitations. Most approximate methods can predict RCS within few dbs of the truth. In general, such a variation is quite acceptable by radar engineers and designers. Approximate methods are usually the main source for predicting RCS of complex and exted targets such as aircrafts, ships, and missiles. When experimental results are available, they can be used to validate and verify the approximations. Some of the most commonly used approximate methods are Geometrical Optics (GO), Physical Optics (PO), Geometrical Theory of Diffraction (GTD), Physical Theory of Diffraction (PTD), and Method of Equivalent Currents (MEC). Interested readers may consult Knott or Ruck (see bibliography) for more details on these and other approximate methods Depency on Aspect Angle and Frequency Radar cross section fluctuates as a function of radar aspect angle and frequency. For the purpose of illustration, isotropic point scatterers are considered. An isotropic scatterer is one that scatters incident waves equally in all directions. Consider the geometry shown in Fig In this case, two unity ( 1m 2 ) isotropic scatterers are aligned and placed along the radar line of sight (zero aspect angle) at a far field range R. The spacing between the two scatterers is 1 meter. The radar aspect angle is then changed from zero to 180 degrees, and the composite RCS of the two scatterers measured by the radar is computed. This composite RCS consists of the superposition of the two individual radar cross sections. At zero aspect angle, the composite RCS is 2m 2. Taking scatterer-1 as a phase reference, when the aspect angle is varied, the composite RCS is modified by the phase that corresponds to the electrical spacing between the two scatterers. For example, at aspect angle 10, the electrical spacing between the two scatterers is λ elec spacing is the radar operating wavelength. 2 ( 1.0 cos( 10 )) = λ (11.6)

5 (a) radar line of sight scat1 scat2 radar 1m (b) radar line of sight 0.707m radar Figure RCS depency on aspect angle. (a) Zero aspect angle, zero electrical spacing. (b) aspect angle, electrical spacing λ Fig shows the composite RCS corresponding to this experiment. This plot can be reproduced using MATLAB function rcs_aspect.m given in Listing 11.1 in Section As clearly indicated by Fig. 11.3, RCS is depent on the radar aspect angle; thus, knowledge of this constructive and destructive interference between the individual scatterers can be very critical when a radar tries to extract the RCS of complex or maneuvering targets. This is true because of two reasons. First, the aspect angle may be continuously changing. Second, complex target RCS can be viewed to be made up from contributions of many individual scattering points distributed on the target surface. These scattering points are often called scattering centers. Many approximate RCS prediction methods generate a set of scattering centers that define the backscattering characteristics of such complex targets. MATLAB Function rcs_aspect.m The function rcs_aspect.m computes and plots the RCS depency on aspect angle. Its syntax is as follows: [rcs] = rcs_aspect (scat_spacing, freq) where Symbol Description Units Status scat_spacing scatterer spacing meters input freq radar frequency Hz input rcs array of RCS versus aspect angle dbsm output

6 Figure Illustration of RCS depency on aspect angle. Next, to demonstrate RCS depency on frequency, consider the experiment shown in Fig In this case, two far field unity isotropic scatterers are aligned with radar line of sight, and the composite RCS is measured by the radar as the frequency is varied from 8 GHz to 12.5 GHz (X-band). Figs and 11.6 show the composite RCS versus frequency for scatterer spacing of 0.25 and 0.75 meters. radar line of sight scat1 scat2 radar dist Figure Experiment setup which demonstrates RCS depency on frequency; dist = 0.1, or 0.7 m.

7 Figure Illustration of RCS depency on frequency. Figure Illustration of RCS depency on frequency.

8 The plots shown in Figs and 11.6 can be reproduced using MATLAB function rcs_frequency.m given in Listing 11.2 in Section From those two figures, RCS fluctuation as a function of frequency is evident. Little frequency change can cause serious RCS fluctuation when the scatterer spacing is large. Alternatively, when scattering centers are relatively close, it requires more frequency variation to produce significant RCS fluctuation. MATLAB Function rcs_frequency.m The function rcs_frequency.m computes and plots the RCS depency on frequency. Its syntax is as follows: where [rcs] = rcs_frequency (scat_spacing, frequ, freql) Symbol Description Units Status scat_spacing scatterer spacing meters input freql start of frequency band Hz input frequ of frequency band Hz input rcs array of RCS versus aspect angle dbsm output Referring to Fig. 11.2, assume that the two scatterers complete a full revolution about the radar line of sight in T rev = 3sec. Furthermore, assume that an X-band radar ( f 0 = 9GHz ) is used to detect (observe) those two scatterers using a PRF f r = 300Hz for a period of 3 seconds. Finally, assume a NB bandwidth B NB = 1MHz and a WB bandwidth B WB = 2GHz. It follows that the radar s NB and WB range resolutions are respectively equal to = 150m and R WB = 7.5cm. R NB Fig shows a plot of the detected range history for the two scatterers using NB detection. Clearly, the two scatterers are completely contained within one range bin. Fig shows the same; however, in this case WB detection is utilized. The two scatterers are now completely resolved as two distinct scatterers, except during the times where both point scatterers fall within the same range bin RCS Depency on Polarization The material in this section covers two topics. First, a review of polarization fundamentals is presented. Second, the concept of the target scattering matrix is introduced.

9 Figure NB detection of the two scatterers shown in Fig Figure WB detection of the two scatterers shown in Fig

10 Polarization The x and y electric field components for a wave traveling along the positive z direction are given by E x = E 1 sin( ωt kz) (11.7) E y = E 2 sin( ωt kz+ δ) (11.8) where k = 2π λ, ω is the wave frequency, the angle δ is the time phase angle which E y leads E x, and, finally, E 1 and E 2 are, respectively, the wave amplitudes along the x and y directions. When two or more electromagnetic waves combine, their electric fields are integrated vectorially at each point in space for any specified time. In general, the combined vector traces an ellipse when observed in the x-y plane. This is illustrated in Fig The ratio of the major to the minor axes of the polarization ellipse is called the Axial Ratio (AR). When AR is unity, the polarization ellipse becomes a circle, and the resultant wave is then called circularly polarized. Alternatively, when E 1 = 0 and AR = the wave becomes linearly polarized. Eqs. (11.7) and (11.8) can be combined to give the instantaneous total electric field, E = â x E 1 sin( ωt kz) + â y E 2 sin( ωt kz + δ) (11.9) Y E 2 E X Z E 1 Figure Electric field components along the x and y directions. The positive z direction is out of the page.

11 â x where and â y are unit vectors along the x and y directions, respectively. At z = 0, E x = E 1 sin( ωt) and E y = E 2 sin( ωt + δ), then by replacing sin( ωt) by the ratio E x E 1 and by using trigonometry properties Eq. (11.9) can be rewritten as 2E x E y cosδ = ( sinδ) 2 E 1 E 2 E x 2 E 1 2 E y 2 E 2 2 (11.10) Note that Eq. (11.10) has no depency on ωt. In the most general case, the polarization ellipse may have any orientation, as illustrated in Fig The angle ξ is called the tilt angle of the ellipse. In this case, AR is given by AR OA = ( 1 AR ) OB (11.11) When E 1 = 0, the wave is said to be linearly polarized in the y direction, while if E 2 = 0 the wave is said to be linearly polarized in the x direction. Polarization can also be linear at an angle of 45 when E 1 = E 2 and ξ = 45. When E 1 = E 2 and δ = 90, the wave is said to be Left Circularly Polarized (LCP), while if δ = 90 the wave is said to Right Circularly Polarized (RCP). It is a common notation to call the linear polarizations along the x and y directions by the names horizontal and vertical polarizations, respectively. Y B E 2 E y A E ξ E x Z O E 1 X Figure Polarization ellipse in the general case.

12 In general, an arbitrarily polarized electric field may be written as the sum of two circularly polarized fields. More precisely, E = E R + E L (11.12) where E R and E L are the RCP and LCP fields, respectively. Similarly, the RCP and LCP waves can be written as E R = E V + je H (11.13) E L = E V je H (11.14) where E V and E H are the fields with vertical and horizontal polarizations, respectively. Combining Eqs. (11.13) and (11.14) yields E R = E H je V 2 E L = E H + je V 2 Using matrix notation Eqs. (11.15) and (11.16) can be rewritten as (11.15) (11.16) E R E H E V = E L 12 1 j = 1 j [ T] E H E V (11.17) E H E R E L = E V = j j [ T] 1 E H E V (11.18) For many targets the scattered waves will have different polarization than the incident waves. This phenomenon is known as depolarization or cross-polarization. However, perfect reflectors reflect waves in such a fashion that an incident wave with horizontal polarization remains horizontal, and an incident wave with vertical polarization remains vertical but is phase shifted 180. Additionally, an incident wave which is RCP becomes LCP when reflected, and a wave which is LCP becomes RCP after reflection from a perfect reflector. Therefore, when a radar uses LCP waves for transmission, the receiving antenna needs to be RCP polarized in order to capture the PP RCS, and LCR to measure the OP RCS.

13 Example: Plot the locus of the electric field vector for the following cases: case1: Etz (, ) â 2πz x ω 0 t πz = cos + â λ y 3 cos ω 0 t λ case 2: case 3: Etz (, ) = â x cos ω 0 t + 2πz â λ sin y ω 0 t πz λ Etz (, ) â 2πz x ω 0 t πz π = cos + â λ cos y ω 0 t λ 6 Solution: case 4: Etz (, ) â 2πz x ω 0 t πz π = cos + â λ y 3 cos ω 0 t λ 3 The MATLAB program example11_1.m was developed to calculate and plot the loci of the electric fields. Figs through show the desired electric fields loci. See listing 11.3 in Section Figure Linearly polarized electric field.

14 Figure Circularly polarized electric field. Figure Elliptically polarized electric field.

15 Figure Elliptically polarized electric field Target Scattering Matrix Target backscattered RCS is commonly described by a matrix known as the scattering matrix, and is denoted by [ S]. When an arbitrarily linearly polarized wave is incident on a target, the backscattered field is then given by E 1 s E 2 s [ S] E 1 = = i E 2 i s 11 s 12 E 1 s 21 s 22 i E 2 i (11.19) The superscripts i and s denote incident and scattered fields. The quantities s ij are in general complex and the subscripts 1 and 2 represent any combination of orthogonal polarizations. More precisely, 1 = H, R, and 2 = V, L. From Eq. (11.3), the backscattered RCS is related to the scattering matrix components by the following relation: 2 s 12 2 σ 11 σ 12 4πR 2 s 11 = σ 21 σ 22 s s 22 (11.20)

16 It follows that once a scattering matrix is specified, the target backscattered RCS can be computed for any combination of transmitting and receiving polarizations. The reader is advised to see Ruck for ways to calculate the scattering matrix [ S]. Rewriting Eq. (11.20) in terms of the different possible orthogonal polarizations yields E H s E V s = s HH s HV s VH s VV i E H E V i (11.21) E R s E L s = s RR s RL s LR s LL i E R E L i (11.22) By using the transformation matrix [ T] in Eq. (11.17), the circular scattering elements can be computed from the linear scattering elements s RR s RL s LR s LL = [ T] s HH s HV 1 0 s VH s VV 0 1 [ T] 1 (11.23) and the individual components are = s VV + s HH js ( HV + s VH ) s RL = s VV + s HH + js ( HV s VH ) s LR = s VV + s HH js ( HV s VH ) = s VV + s HH + js ( HV + s VH ) s RR s LL Similarly, the linear scattering elements are given by (11.24) s HH s HV s VH s VV = [ T] 1 s RR s RL 1 0 s LR s LL 0 1 [ T] (11.25) and the individual components are

17 s HH = s RR s RL s LR s LL js ( s RR s LR + s RL s LL ) VH = js ( s RR + s LR s RL s LL ) HV = s s RR + s LL + js RL + s LR VV = (11.26) RCS of Simple Objects This section presents examples of backscattered radar cross section for a number of simple shape objects. In all cases, except for the perfectly conducting sphere, only optical region approximations are presented. Radar designers and RCS engineers consider the perfectly conducting sphere to be the simplest target to examine. Even in this case, the complexity of the exact solution, when compared to the optical region approximation, is overwhelming. Most formulas presented are Physical Optics (PO) approximation for the backscattered RCS measured by a far field radar in the direction ( θ, ϕ), as illustrated in Fig In this section, it is assumed that the radar is always illuminating an object from the positive z-direction. Z θ Direction to receiving radar sphere Y X ϕ Figure Direction of antenna receiving backscattered waves.

18 Sphere Due to symmetry, waves scattered from a perfectly conducting sphere are co-polarized (have the same polarization) with the incident waves. This means that the cross-polarized backscattered waves are practically zero. For example, if the incident waves were Left Circularly Polarized (LCP), then the backscattered waves will also be LCP. However, because of the opposite direction of propagation of the backscattered waves, they are considered to be Right Circularly Polarized (RCP) by the receiving antenna. Therefore, the PP backscattered waves from a sphere are LCP, while the OP backscattered waves are negligible. The normalized exact backscattered RCS for a perfectly conducting sphere is a Mie series given by σ πr 2 = j ---- ( 1) n ( 2n + 1) kr n = 1 J n ( kr) ( 1) H n ( kr) krj n 1 ( kr) nj n ( kr) ( 1) ( 1) krh n 1 ( kr) nh n ( kr) (11.27) where r is the radius of the sphere, k = 2π λ, λ is the wavelength, J n is the ( 1) spherical Bessel of the first kind of order n, and H n is the Hankel function of order n, and is given by Y n ( 1 H ) n ( kr) = J n ( kr) + jy n ( kr) (11.28) is the spherical Bessel function of the second kind of order n. Plots of the normalized perfectly conducting sphere RCS as a function of its circumference in wavelength units are shown in Figs a and 11.16b. These plots can be reproduced using the function rcs_sphere.m given in Listing 11.4 in Section In Fig , three regions are identified. First is the optical region (corresponds to a large sphere). In this case, σ = πr 2 r» λ Second is the Rayleigh region (small sphere). In this case, (11.29) σ 9πr 2 ( kr) 4 r «λ (11.30) The region between the optical and Rayleigh regions is oscillatory in nature and is called the Mie or resonance region.

19 σ πr πr λ Figure 11.16a. Normalized backscattered RCS for a perfectly conducting sphere. 5 0 Normalized sphere RCS - db Rayleigh region Mie region optical region Sphere circumference in wavelengths Figure 11.16b. Normalized backscattered RCS for a perfectly conducting sphere using semi-log scale.

20 The backscattered RCS for a perfectly conducting sphere is constant in the optical region. For this reason, radar designers typically use spheres of known cross sections to experimentally calibrate radar systems. For this purpose, spheres are flown attached to balloons. In order to obtain Doppler shift, spheres of known RCS are dropped out of an airplane and towed behind the airplane whose velocity is known to the radar Ellipsoid An ellipsoid centered at (0,0,0) is shown in Fig It is defined by the following equation: x -- a 2 y -- b 2 z c 2 = 1 (11.31) One widely accepted approximation for the ellipsoid backscattered RCS is given by σ πa 2 b 2 c 2 = ( a 2 ( sinθ) 2 ( cosϕ) 2 + b 2 ( sinθ) 2 ( sinϕ) 2 + c 2 ( cosθ) 2 ) 2 (11.32) When a = b, the ellipsoid becomes roll symmetric. Thus, the RCS is indepent of ϕ, and Eq. (11.32) is reduced to Z θ Direction to receiving radar c a b Y X ϕ Figure Ellipsoid.

21 σ = and for the case when a = b = c, πb 4 c ( a 2 ( sinθ) 2 + c 2 ( cosθ) 2 ) 2 (11.33) σ = πc 2 (11.34) Note that Eq. (11.34) defines the backscattered RCS of a sphere. This should be expected, since under the condition a = b = c the ellipsoid becomes a sphere. Fig a shows the backscattered RCS for an ellipsoid versus θ for ϕ = 45. This plot can be generated using MATLAB program fig11_18a.m given in Listing 11.5 in Section Note that at normal incidence ( θ = 90 ) the RCS corresponds to that of a sphere of radius c, and is often referred to as the broadside specular RCS value. Figure 11.18a. Ellipsoid backscattered RCS versus aspect angle. MATLAB Function rcs_ellipsoid.m The function rcs_ellipsoid.m computes and plots the RCS of an ellipsoid versus aspect angle. It is given in Listing 11.6 in Section 11.9, and its syntax is as follows: [rcs] = rcs_ellipsoid (a, b, c, phi)

22 where Symbol Description Units Status a ellipsoid a-radius meters input b ellipsoid b-radius meters input c ellipsoid c-radius meters input phi ellipsoid roll angle degrees input rcs array of RCS versus aspect angle dbsm output Fig b shows the GUI workspace associated with function. To execute this GUI type rcs_ellipsoid_gui from the MATLAB Command window. Figure 11.18b. GUI workspace associated with the function rcs_ellipsoid.m Circular Flat Plate Fig shows a circular flat plate of radius r, centered at the origin. Due to the circular symmetry, the backscattered RCS of a circular flat plate has no depency on ϕ. The RCS is only aspect angle depent. For normal incidence (i.e., zero aspect angle) the backscattered RCS for a circular flat plate is

23 σ 4π 3 r 4 = θ = 0 λ 2 (11.35) For non-normal incidence, two approximations for the circular flat plate backscattered RCS for any linearly polarized incident wave are σ = λr 8πsinθ( tan( θ) ) 2 (11.36) σ πk 2 r 4 2J 1 ( 2krsinθ) = ( cosθ) 2 2krsinθ (11.37) where k = 2π λ, and J 1 ( β) is the first order spherical Bessel function evaluated at β. The RCS corresponding to Eqs. (11.35) through (11.37) is shown in Fig These plots can be reproduced using MATLAB function rcs_circ_gui.m. Z θ Direction to receiving radar r Y X ϕ Figure Circular flat plate. MATLAB Function rcs_circ_plate.m The function rcs_circ_plate.m calculates and plots the backscattered RCS from a circular plate. It is given in Listing 11.7 in Section 11.9; its syntax is as follows: where [rcs] = rcs_circ_plate (r, freq) Symbol Description Units Status r radius of circular plate meters input freq frequency Hz input rcs array of RCS versus aspect angle dbsm output

24 Figure Backscattered RCS for a circular flat plate Truncated Cone (Frustum) Figs and show the geometry associated with a frustum. The half cone angle α is given by tanα ( r 2 r 1 ) = = H (11.38) Define the aspect angle at normal incidence with respect to the frustum s surface (broadside) as θ n. Thus, when a frustum is illuminated by a radar located at the same side as the cone s small, the angle is r L θ n θ n = 90 α (11.39) Alternatively, normal incidence occurs at θ n = 90 + α (11.40) At normal incidence, one approximation for the backscattered RCS of a truncated cone due to a linearly polarized incident wave is

25 Z r 2 θ z 2 z 1 H L r 1 Y X Figure Truncated cone (frustum). Z Z θ r 2 α H r 1 r 1 r 2 Figure Definition of half cone angle.

26 π( z σ 2 z 1 ) 2 θn = tanα( sinθ 9λsin n cosθ n tanα) 2 θ n (11.41) where λ is the wavelength, and z 1, z 2 are defined in Fig Using trigonometric identities, Eq. (11.41) can be reduced to π( z σ 2 z 1 ) 2 sinα θn = λ ( cosα) 4 (11.42) For non-normal incidence, the backscattered RCS due to a linearly polarized incident wave is σ λztanα sinθ cosθtanα = πsinθ sinθtanα+ cosθ (11.43) where z is equal to either z 1 or z 2 deping on whether the RCS contribution is from the small or the large of the cone. Again, using trigonometric identities Eq. (11.43) (assuming the radar illuminates the frustum starting from the large ) is reduced to (11.44) When the radar illuminates the frustum starting from the small (i.e., the radar is in the negative z direction in Fig ), Eq. (11.44) should be modified to (11.45) For example, consider a frustum defined by H = cm, r 1 = 2.057cm, r 2 = 5.753cm. It follows that the half cone angle is 10. Fig a shows a plot of its RCS when illuminated by a radar in the positive z direction. Fig b shows the same thing, except in this case, the radar is in the negative z direction. Note that for the first case, normal incidence occur at 100, while for the second case it occurs at 80. These plots can be reproduced using MATLAB function rcs_frustum_gui.m given in Listing 11.8 in Section MATLAB Function rcs_frustum.m σ σ λztanα = ( tan( 8πsinθ θ α )) 2 λztanα = ( tan( 8πsinθ θ + α )) 2 The function rcs_frustum.m computes and plots the backscattered RCS of a truncated conic section. The syntax is as follows: [rcs] = rcs_frustum (r1, r2, freq, indicator)

27 Figure 11.23a. Backscattered RCS for a frustum. Figure 11.23b. Backscattered RCS for a frustum.

28 where Symbol Description Units Status r1 small radius meters input r2 large radius meters input freq frequency Hz input indicator indicator = 1 when viewing from none input large indicator = 0 when viewing from small rcs array of RCS versus aspect angle dbsm output Cylinder Fig shows the geometry associated with a finite length conducting cylinder. Two cases are presented: first, the general case of an elliptical cross section cylinder; second, the case of a circular cross section cylinder. The normal and non-normal incidence backscattered RCS due to a linearly polarized incident wave from an elliptical cylinder with minor and major radii being r 1 and are, respectively, given by r 2 2πH 2 r 2 2 σ 2 r 1 θn = λ r 1 ( cosϕ) 2 2 [ + r 2 ( sinϕ) 2 ] 1.5 (11.46) σ = λr r 1 sinθ π ( cosθ) 2 2 r 1 ( cosϕ) 2 2 [ + r 2 ( sinϕ) 2 ] 1.5 (11.47) For a circular cylinder of radius r, then due to roll symmetry, Eqs. (11.46) and (11.47), respectively, reduce to σ θn = 2πH 2 r λ (11.48) σ λrsinθ = π( cosθ) 2 (11.49) Fig a shows a plot of the cylinder backscattered RCS for a symmetrical cylinder. Fig b shows the backscattered RCS for an elliptical cylinder. These plots can be reproduced using MATLAB function rcs_cylinder.m given in Listing 11.9 in Section 11.9.

29 Z Z r 2 θ r 1 r H H Y X ϕ (a) (b) Figure (a) Elliptical cylinder; (b) circular cylinder. Figure 11.25a. Backscattered RCS for a symmetrical cylinder, and H = 1m. r = 0.125m

30 Figure 11.25b. Backscattered RCS for an elliptical cylinder, r 1 = 0.125m, r 2 = 0.05m, and H = 1m. MATLAB Function rcs_cylinder.m The function rcs_cylinder.m computes and plots the backscattered RCS of a cylinder. The syntax is as follows: where [rcs] = rcs_cylinder(r1, r2, h, freq, phi, CylinderType) Symbol Description Units Status r1 radius r1 meters input r2 radius r2 meters input h length of cylinder meters input freq frequency Hz input phi roll viewing angle degrees input CylinderType Circular, i.e., r 1 = r 2 none input Elliptic, i.e., r 1 r 2 rcs array of RCS versus aspect angle dbsm output

31 Rectangular Flat Plate Consider a perfectly conducting rectangular thin flat plate in the x-y plane as shown in Fig The two sides of the plate are denoted by 2a and 2b. For a linearly polarized incident wave in the x-z plane, the horizontal and vertical backscattered RCS are, respectively, given by b 2 σ V σ σ π 1V σ 2V V 1 = ( σ cosθ 4 3V + σ 4V ) σ 5V 2 (11.50) b 2 1 σ H ---- σ π 1H σ 2H σ 2H 1 = ( σ cosθ 4 3H + σ 4H ) σ 5H 2 (11.51) where k = 2π λ and sin( kasinθ) σ 1V = cos( kasinθ) j = ( σ sinθ 1H ) (11.52) e jka ( π 4) σ 2V = π( ka) 3 2 (11.53) ( 1 + sinθ)e jkasinθ σ 3V = ( 1 sinθ) 2 (11.54) σ ( 1 sinθ)e jkasinθ 4V = ( 1 + sinθ) 2 (11.55) Z radar θ -a -b b Y X a Figure Rectangular flat plate.

32 e j( 2ka π 2) σ 5V = π( ka) 3 4e jka ( + π 4) σ 2H = π( ka) 1 2 e jkasinθ σ 3H = sinθ e jkasinθ σ 4H = sinθ (11.56) (11.57) (11.58) (11.59) e j( 2ka + ( π 2) ) σ 5H = π( ka) (11.60) Eqs. (11.50) and (11.51) are valid and quite accurate for aspect angles 0 θ 80. For aspect angles near 90, Ross 1 obtained by extensive fitting of measured data an empirical expression for the RCS. It is given by σ H 0 ab 2 σ V π π 3π λ cos 2ka a ( λ) 2 22a ( λ) 2 5 = (11.61) The backscattered RCS for a perfectly conducting thin rectangular plate for incident waves at any θ, ϕ can be approximated by σ 4πa 2 b 2 sin( aksinθcosϕ) sin( bksinθsinϕ) = ( cosθ) 2 aksinθcosϕ bksinθsinϕ λ 2 (11.62) Eq. (11.62) is indepent of the polarization, and is only valid for aspect angles θ 20. Fig shows an example for the backscattered RCS of a rectangular flat plate, for both vertical (Fig a) and horizontal (Fig b) polarizations, using Eqs. (11.50), (11.51), and (11.62). In this example, a = b = 10.16cm and wavelength λ = 3.33cm. This plot can be reproduced using MATLAB function rcs_rect_plate given in Listing MATLAB Function rcs_rect_plate.m The function rcs_rect_plate.m calculates and plots the backscattered RCS of a rectangular flat plate. Its syntax is as follows: [rcs] = rcs_rect_plate (a, b, freq) 1. Ross, R. A., Radar Cross Section of Rectangular Flat Plate as a Function of Aspect Angle, IEEE Trans., AP-14,320, 1966.

33 Figure 11.27a. Backscattered RCS for a rectangular flat plate. Figure 11.27b. Backscattered RCS for a rectangular flat plate.

34 where Symbol Description Units Status a short side of plate meters input b long side of plate meters input freq frequency Hz input rcs array of RCS versus aspect angle dbsm output Fig c shows the GUI workspace associated with this function. Figure 11.27c. GUI workspace associated with the function rcs_rect_plate.m Triangular Flat Plate Consider the triangular flat plate defined by the isosceles triangle as oriented in Fig The backscattered RCS can be approximated for small aspect angles ( θ 30 ) by σ = 4πA ( cosθ) 2 σ 0 λ 2 (11.63) σ 0 = [( sinα) 2 ( sin( β 2) ) 2 ] 2 + σ α 2 ( β 2) 2 (11.64) σ 01 = 0.25( sinϕ) 2 [( 2a b) cosϕsinβ sinϕsin2α] 2 (11.65) where α = kasinθcosϕ, β = kbsinθsinϕ, and A = ab 2. For waves incident in the plane ϕ = 0, the RCS reduces to

35 Z radar θ X -b/2 a ϕ b/2 Y Figure Coordinates for a perfectly conducting isosceles triangular plate. σ = 4πA ( cosθ) 2 ( sinα)4 ( sin2α 2α) λ 2 α 4 4α 4 (11.66) and for incidence in the plane ϕ = π 2 σ = 4πA ( cosθ) 2 ( sin( β 2) )4 λ 2 ( β 2) 4 (11.67) Fig shows a plot for the normalized backscattered RCS from a perfectly conducting isosceles triangular flat plate. In this example a = 0.2m, b = 0.75m. This plot can be reproduced using MATLAB function rcs_isosceles.m given in Listing in Section MATLAB Function rcs_isosceles.m The function rcs_isosceles.m calculates and plots the backscattered RCS of a triangular flat plate. Its syntax is as follows: where [rcs] = rcs_isosceles (a, b, freq, phi) Symbol Description Units Status a height of plate meters input b base of plate meters input freq frequency Hz input phi roll angle degrees input rcs array of RCS versus aspect angle dbsm output

36 Figure Backscattered RCS for a perfectly conducting triangular a = 20cm b = 75cm flat plate, and Scattering From a Dielectric-Capped Wedge The geometry of a dielectric-capped wedge is shown in Fig It is required to find to the field expressions for the problem of scattering by a 2-D perfect electric conducting (PEC) wedge capped with a dielectric cylinder. Using the cylindrical coordinates system, the excitation due to an electric line current of complex amplitude I located at results in TM z 0 ( ρ 0, ϕ 0 ) incident field with the electric field expression given by ωµ = 4 i 0 Ez Ie H k ( 2 ) 0 ( ρ ρ0 ) (11.68) The problem is divided into three regions, I, II, and III shown in Fig The field expressions may be assumed to take the following forms:

37 y (ρ 0, ϕ 0 ) III I e II ε 0, µ 0 a ε, µ 0 I α β PEC x Figure Scattering from dielectric-capped wedge. where I z n v n= 0 ( ) sin ( ) sin ( ) E = a J k ρ v φ α v φ α 1 0 ( 2 ( ( ) ) ( )) sin ( ) sin ( 0 ) E = b J kρ + ch kρ v φ α v φ α II z n v n ν n= 0 III z n ν n= 0 ( 2 ) ( ) sin ( ) sin ( ) E = d H kρ v φ α v φ α 0 (11.69) nπ v = 2π α β (11.70) ( 2) while J v ( x) is the Bessel function of order v and argument x and H v is the Hankel function of the second kind of order v and argument x. From Maxwell's equations, the magnetic field component H ϕ is related to the electric field component for a TM z wave by E z H φ = 1 Ez jωµ ρ (11.71)

38 Thus, the magnetic field component as H ϕ in the various regions may be written k H = a J k ρ v φ α v φ α I 1 φ n v jωµ 0 ( 1 ) sin ( ) sin ( 0 ) n= 0 ( 2) ( ( ) ( )) sin ( ) sin ( 0 ) k H = b J kρ + ch kρ v φ α v φ α II φ n v n ν jωµ 0 n= 0 III k ( 2) φ n ν sin jωµ 0 n= 0 ( ) ( ) sin ( ) H = d H kρ v φ α v φ α (11.72) Where the prime indicated derivatives with respect to the full argument of the function. The boundary conditions require that the tangential electric field components vanish at the PEC surface. Also, the tangential field components should be continuous across the air-dielectric interface and the virtual boundary between region II and III, except for the discontinuity of the magnetic field at the source point. Thus, 0 E = 0 at φ= α,2π β z (11.73) E = E I II z z at I II φ Hφ H = ρ = a (11.74) E = E H H = J II III z z at II III φ φ e ρ = ρ 0 (11.75) The current density J e may be given in Fourier series expansion as I 2 I Je = δ φ φ = ν φ α ν φ α ρ π α β ρ e n= 0 e ( ) sin ( ) sin ( ) (11.76) The boundary condition on the PEC surface is automatically satisfied by the ϕ depence of the electric field Eq. (11.72). From the boundary conditions in Eq. (11.73) n= 0 n v ( ) sin ( ) sin ( ) aj ka vφ α v φ α = 1 0 ( 2 ( bj ( ) ) n v ka + ch n ν ( ka) ) sin v( φ α) sin v( φ0 α) n= 0 (11.77)

39 k1 jωµ 0 n= 0 n v ( ) sin ( ) sin ( ) aj ka vφ α v φ α = k jωµ 1 0 ( 2) ( bnj v( ka) + cnh ν ( ka) ) sin v ( φ α) sin v( φ0 α) 0 n= 0 From the boundary conditions in Eq. (11.75), we have ( 2 ( bj ( ) ) n v kρ ch ( )) sin ( ) sin ( ) 0 n ν kρ v φ α v φ α 0 0 n= 0 n= 0 n + = ( 2 ) ( ) sin ( ) sin ( ) dh kρ v φ α v φ α ν 0 0 (11.78) (11.79) k jωµ ( 2) ( bj n v( kρ ) ch ( )) sin ( ) sin ( ) 0 n ν kρ v φ α v φ α n= 0 + = k jωµ 0 n= 0 n ( 2) ν ( ) sin ( ) sin ( ) dh kρ v φ α v φ α Ie sin ν φ α ν φ0 α 2π α β ρ 0 n= 0 ( ) sin ( ) (11.80) Since Eqs. (11.77) and (11.80) hold for all ϕ, the series on the left and right hand sides should be equal term by term. More precisely, ( ) ( ) n v 1 n v n ν From Eqs. (11.81) and (11.83), we have ( 2 ) ( ) a J k a = b J ka + c H ka 1 k ( 2) anj v( k1a ) = ( bnj v( ka) + cnh ν ( ka) ) 0 0 k µ µ ( ) ( ) ( ) ( ) ( ) ( ) bj kρ + ch kρ = dh kρ 2 2 n v 0 n ν 0 n ν 0 ( ) ( ) bj kρ ch kρ dh kρ 2 jη Ie 2π α β ρ ( 2) ( 2) 0 n v 0 + n ν 0 = n ν 0 1 ( 2 a = b J ( ka) c H ) ( ka) J k a + ( ) n n v n v v 1 J v ( kρ0 ) ( 2 ) ( ) dn = cn + bn H v kρ 0 0 (11.81) (11.82) (11.83) (11.84) (11.85) (11.86)

40 ( 2)' H v ( 2) H v Multiplying Eq. (11.83) by and Eq. (11.84) by, and by subtraction and using the Wronskian of the Bessel and Hankel functions, we get πωµ I Substituting in Eqs. (11.81) and (11.82) and solving for yield b n From Eqs. (11.86) through (11.88), ( 2 ) ( ) 0 e bn = Hν kρ0 2π α β may be given by c n ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) πωµ I ( ) kj ka J k a k J ka J k a = ( ) π α β kh ka J k a k H ka J k a 0 e 2 v v 1 1 v v 1 cn Hν kρ0 2 ( 2) 2 ν v 1 1 ν v 1 d n (11.87) (11.88) πωµ I kj ( ka) J ( k a) k J ( ka) J ( k a) d H k J kρ ( ) ( I ) 0 e 2 v v 1 1 v v 1 n = v 0 v 0 2π α β ( 2) ( 2 kh ( ) ( ) ) v ka J v k1a k1h v ( ka) J v( k1a ) which can be written as ( ) (11.89) ( 2 ( ) ( ) ) ( 2) kjv k1a J v ka Hν ( kρ0) H ν ( ka) Jv ( kρ ) 0 + K ( 2 ( ) ) ( 2 ( ) ( ) ( ) ) πωµ 1 v 1 ν v 0 v ν ( 0) 0I kj ka H ka J kρ J ka H kρ e dn = 2π α β ( 2) ( 2 kh ( ) ( ) ) ν ka Jv k1a k1h ν ( ka) J v ( k1a ) (11.90) Substituting for the Hankel function in terms of Bessel and Neumann functions, Eq. (11.90) reduces to d ( 1 ) ( ) ( 0) ( ) ( 0) + K kj 1 v ( ka 1 ) Yν ( ka) Jv( kρ0) Jv( ka) Yν ( kρ0) ( 2 ( 2 )( ) ( ) ) ( ) ( ) kjv k a Jv ka Yν kρ Yν ka Jv kρ πωµ I 2π α β kh ν ka Jv k1a k1h ν ka J v k1a 0 e n = j (11.91) With these closed form expressions for the expansion coeffiecients a n, b n, c n and d n, the field components E z and H ϕ can be determined from Eq. (11.69) and Eq. (11.72), respectively. Alternatively, the magnetic field component can be computed from H ρ H ρ = 1 1 Ez jωµ ρ φ (11.92)

41 Thus, the H ρ expressions for the three regions defined in Fig become 1 ρ = n v( ρ)cos ( ϕ α)sin ( ϕ α) I H a vj k1 v v 0 jωµρ n= 0 1 II (2) Hρ = vbj ( n v( kρ) ch n v ( kρ) ) cos v( ϕ α)sin v( ϕ0 α) jωµρ + n= 0 (11.93) 1 H vh k v v III (2) ρ = dn v ( ρ)cos ( ϕ α)sin ( ϕ0 α) jωµρ n= Far Scattered Field In region III, the scattered field may be found as the difference between the total and incident fields. Thus, using Eqs. (11.68) and (11.69) and considering the far field condition ( ρ ) we get Note that 2 j E = E + E = e d j sin v( φ α)sin v( φ α) III i s jkρ v z z z n πkρ n= 0 ωµ 2 j 4 πkρ H i 0 jkρ + jkρ0cos( φ φ0) Ez = Ie e e ρ d n can be written as 1 1 Ez = jωµ ρ φ 0 (11.94) where d ωµ I 4 0 e n = d% n (11.95) kjv( k1a ) J v( ka) Yν( kρ0) Y ν( ka) Jv ( kρ0) + K 4π kj 1 v ( ka 1 ) Yν ( ka) Jv( kρ0) Jv( ka) Yν ( kρ0) d% n = j 2π α β ( 2) ( 2 kh ( ) ( ) ) ν ka Jv k1a k1h ν ( ka) J v ( k1a ) Substituting Eq. (11.95) into Eq. (11.94), the scattered field f( ϕ) is (11.96)

42 E z s = ωµ 0 I e 4 2j e jkρ πkρ (11.97) n = 0 d n j ν sinνϕ ( α) sinνϕ ( 0 α) e jkρ 0 cos( ϕ ϕ 0 ) Plane Wave Excitation For plane wave excitation ( (11.88) reduce to ρ 0 ), the expression in Eqs. (11.87) and πωµ I 2 j 0 e ν jkρ0 bn = j e 2π α β πkρ0 ( ) ( 1 ) 1 ( ) ( 1 ) ( ( ) ( ) ) ( ) ( ) πωµ I 2 j kj ka J k a k J ka J k a π α β πkρ kh ka J k a k H ka J k a 0 e ν jkρ0 v v v v cn = j e 2 ( 2) 0 2 ν v 1 1 ν v 1 where the complex amplitude of the incident plane wave, E 0 (11.98), can be given by ωµ 0 jkρ0 E0 = Ie e 4 πkρ0 In this case, the field components can be evaluated in regions I and II only Special Cases Case I: α = β (reference at bisector); The definition of ν reduces to ν = nπ (11.99) (11.100) and the same expression will hold for the coefficients (with α = β). Case II: α = 0 (reference at face); the definition of ν takes on the form (11.101) and the same expression will hold for the coefficients (with α = 0 ). Case III: k 1 (PEC cap); Fields at region I will vanish, and the coefficients will be given by 2 j 2( π β) nπ ν = 2π β

43 πωµ I = 2π α β ( ka) ( 2 ) ( ) 0 e bn Hν kρ0 0 e cn H kρ d n n n v n ν Jv 1 ( 2 ) Jv ( ka) ν ( 0 ) ( 2 ) ν ( ) Yν ( ka) Jv( kρ0) Jv( ka) Yν ( kρ0) ( 2 ) ( ) πωµ I = 2π α β H ka πωµ 0Ie = j 2π α β H ka 1 ( 2 a = b J ( ka) c H ) ( ka) + = 0 (11.102) Note that the expressions of b n and c n will yield zero tangential electric field at ρ = a when substituted in Eq.(11.69). Case IV: a 0 (no cap); The expressions of the coefficients in this case may be obtained by setting k 1 = k, or by taking the limit as a approaches zero. Thus, (11.103) E Case V: a 0 and α = β = 0 (semi-infinite PEC plane); In this case, the coefficients in Eq. (11.103) become valid with the exception that the values of v reduce to n 2. Once, the electric field component in the different z regions is computed, the corresponding magnetic field component H ϕ can be computed using Eq. (11.71) and the magnetic field component H ρ may be computed as ν ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) πωµ 0I ( 2 ) kj v ka Jv ka kjv ka J v ka e cn = Hν ( kρ0 ) = 0 2π α β ( 2) 2 kh ν ka Jv ka khν ka Jv ka πωµ 0Ie ( 2 b ) n = Hν ( kρ0 ) 2π α β 1 ( 2 a = b J ( ka) + c H ) ( ka) = b J ( ka) n n v n ν n v ( 2 ( ) ( ) ) ( 2 kjv k1a J v ka Hν ( kρ0) H )( ν ka) Jv ( kρ0) + K ( 2 ( ) ) πωµ kj 1 v ka 1 Hν ( ka) Jv( kρ0) Jv( ka) H 0I e dn = ν 2π α β ( 2) ( 2 ( ) ( ) ) kh ν ka Jv k1a k1h ν ( ka) J v ( k1a ) πωµ 0Ie = Jv ( kρ0 ) 2π α β ( 2 ) ( kρ ) 0 H ρ = 1 1 Ez jωµ ρ φ (11.104)

44 MATLAB Program Capped_WedgeTM.m The MATLAB program "Capped_WedgeTM.m" given in listing 11.12, along with the following associated functions "DielCappedWedgeTMFields_Ls.m", DielCappedWedgeTMFields_PW", "polardb.m", "dbesselj.m", "dbesselh.m", and "dbessely.m" given in the following listings, calculates and plots the far field of a capped wedge in the presence of an electric line source field. The near field distribution is also computed for both line source or plane wave excitation. All near field components are computed and displayed, in separate windows, using 3-D output format. The program is also capable of analyzing the field variations due to the cap parameters. The user can execute this MATLAB program from the MATLAB command window and manually change the input parameters in the designated section in the program in order to perform the desired analysis. Alternatively, the "Capped_Wedge_GUI.m" function along with the "Capped_Wedge_GUI.fig" file can be used to simplify the data entry procedure. A sample of the data entry screen of the "Capped_Wedge_GUI" program is shown in Fig for the case of a line source exciting a sharp conducting wedge. The corresponding far field pattern is shown in Fig When keeping all the parameters in Fig the same except that selecting a dielectric or conducting cap, one obtains the far field patterns in Figs and 11.34, respectively. It is clear from these figures how the cap parameters affect the direction of the maximum radiation of the line source in the presence of the wedge. The distribution of the components of the fields in the near field for these three cases (sharp edge, dielectric capped edge, and conducting capped edge) is computed and shown in Figs to The near field distribution for an incident plane wave field on these three types of wedges is also computed and shown in Figs to These near field distributions clearly demonstrated the effect or cap parameters in altering the sharp edge singular behavior. To further illustrate this effect, the following set of figures (Figs. (11.53) to (11.55)) presents the near field of the electric component of plane wave incident on a half plane with a sharp edge, dielectric capped edge, and conducting capped edge. The user is encouraged to experiment with this program as there are many parameters that can be altered to change the near and far field characteristic due to the scattering from a wedge structure.

45 Figure The parameters for computing the far field pattern of a 60 degrees wedge excited by a line source Figure The far field pattern of a line source near a conducting wedge with sharp edge characterized by the parameters in Fig

46 Figure The far field pattern of a line source near a conducting wedge with a dielectric capped edge characterized by the parameters in Fig Figure The far field pattern of a line source near a conducting wedge with a conducting capped edge characterized by the parameters in Fig

47 E z Figure The near field pattern of a line source near a conducting wedge with a sharp edge characterized by the parameters in Fig H ρ Figure The near field pattern of a line source near a conducting wedge with a sharp edge characterized by the parameters in Fig

48 H ϕ Figure The near field pattern of a line source near a conducting wedge with a sharp edge characterized by the parameters in Fig E z Figure The near field pattern of a line source near a conducting wedge with a dielectric cap edge characterized by Fig

49 H ρ Figure The near field pattern of a line source near a conducting wedge with a dielectric cap edge characterized by Fig H φ Figure The near field pattern of a line source near a conducting wedge with a dielectric cap edge characterized by Fig

50 E z Figure The near field pattern of a line source near a conducting wedge with a conducting capped edge characterized by Fig H ρ Figure The near field pattern of a line source near a conducting wedge with a conducting capped edge characterized by Fig

51 H ϕ Figure The near field pattern of a line source near a conducting wedge with a conducting capped edge characterized by Fig E z Figure The near field pattern of a plane wave incident on a conducting wedge with a sharp edge characterized by Fig

52 H ρ Figure The near field pattern of a plane wave incident on a conducting wedge with a sharp edge characterized by Fig H ϕ Figure The near field pattern of a plane wave incident on a conducting wedge with a sharp edge characterized by Fig

53 E z Figure The near field pattern of a plane wave incident on a conducting wedge with a dielectric edge characterized by Fig H ρ Figure The near field pattern of a plane wave incident on a conducting wedge with a dielectric edge characterized by Fig

54 H ϕ Figure The near field pattern of a plane wave incident on a conducting wedge with dielectric capped edge characterized by Fig E z Figure The near field pattern of a plane wave incident on a conducting wedge with a conducting capped edge characterized by Fig

55 H ρ Figure The near field pattern of a plane wave incident on a conducting wedge with a conducting capped edge characterized by Fig H ϕ Figure The near field pattern of a plane wave incident on a conducting wedge with a conducting capped edge characterized by Fig

56 E z Figure The near field pattern of a plane wave incident on a half plane with sharp edge. All other parameters are as in Fig E z Figure near field pattern of a plane wave incident on a half plane with a dielectric capped edge. All other parameters are as in Fig

57 E z Figure near field pattern of a plane wave incident on a half plane with a conducting capped edge. All other parameters are as in Fig RCS of Complex Objects A complex target RCS is normally computed by coherently combining the cross sections of the simple shapes that make that target. In general, a complex target RCS can be modeled as a group of individual scattering centers distributed over the target. The scattering centers can be modeled as isotropic point scatterers (N-point model) or as simple shape scatterers (N-shape model). In any case, knowledge of the scattering centers locations and strengths is critical in determining complex target RCS. This is true, because as seen in Section 11.3, relative spacing and aspect angles of the individual scattering centers drastically influence the overall target RCS. Complex targets that can be modeled by many equal scattering centers are often called Swerling 1 or 2 targets. Alternatively, targets that have one dominant scattering center and many other smaller scattering centers are known as Swerling 3 or 4 targets. In NB radar applications, contributions from all scattering centers combine coherently to produce a single value for the target RCS at every aspect angle. However, in WB applications, a target may straddle many range bins. For each range bin, the average RCS extracted by the radar represents the contributions from all scattering centers that fall within that bin.