Application of Metallic Strip Gratings for Enhancement of Electromagnetic Performance of A-sandwich Radome

Enhancement of the electromagnetic (EM) performance characteristics of A-sandwich radome wall over X-band using metallic strip gratings is presented in this work. Equivalent transmission line method in conjunction with equivalent circuit model (ECM) is used for modeling the A-sandwich radome panel with metallic strip gratings and the computation of radome performance parameters. Metallic strip grating embedded in the mid-plane of the core and those in the skin-core interface are the configurations considered in the present work. For a given thickness of metallic strip grating, its width and pitch are optimized at different angles of incidence such that the new radome wall configuration offers superior EM performance over the entire X-band as compared to the conventional A-sandwich wall. The EM analysis shows that the superior EM performance of A-sandwich with metallic strip gratings makes it suitable for the design of normal incidence and streamlined airborne radomes.

Keywords:  Radomesmetallic strip gratingsradome performance parameters

β                   Constant
εr                  Dielectric constant
Φ                  Electrical length
λ                    Wavelength
θ                    Incidence angle
A, B, C, D     Elements of transmission matrix
B                Shunt susceptance of the metallic strip gratings
Cn                  Coefficients of nth order
Ptr                  Power transmission coefficient
Prf                  Power reflection coefficient
P                     Pitch
tan δe             Electric loss tangent
tms                 Thickness of metallic strip
tc1                  Thickness of core for configuration 1
tc2                  Thickness of core for configuration 2
 tp                   Thickness of radome paint layer
ts                    Thickness of skin layer
T1, T2              Voltage transmission coefficients
w                    Width of the strip
Z0                   Impedance of free space
Zc                   Impedance of core layer
Zp                   Impedance of paint layer
Zs                   Impedance of skin layer     

The stringent electromagnetic (EM) performance requirements of modern airborne radar systems require novel radome designs. Many techniques based on metallic structures (wire grids, wire meshes, etc.), anisotropic inclusions, and resonant/ semi-resonant  structures were reported for modifying the radome wall configurations to enhance the EM performance of radome walls 1-4. Monolithic radome panels, centrally loaded by a periodic array of conducting inclusions, offered a wider frequency bandwidth compared to the conventional radome designs of the same material and thickness5. Frequency selective surfaces (FSS) find potential applications in the design of high performance radomes due to their inherent frequency selective characteristics6-8. Recently reported works show that metamaterial-based FSS structures are used in the design of novel radome wall configurations9-11. In airborne radome applications, A-sandwich radome wall configuration is generally preferred to monolithic wall due to high strength-to-weight ratio and bandwidth12. However the EM performance of conventional A-sandwich wall may not be sufficient to meet the requirements of modern radome applications. Hence in the present work, novel designs based on metallic strip gratings are used to enhance the EM performance of A-sandwich radome for airborne applications. Since metallic strips considered in the present work are thin (thickness = 0.1 mm), it is easy to load them in the layers of A-sandwich wall.


The A-sandwich radome wall considered here is composed of three layers namely, outer skin, core and inner skin. The outer and inner skin layers are made of glass-epoxy, while the core is made of polyurethane foam. The core thickness is optimized for maximum power transmission over the X-band. In order to enhance the EM performance, the metallic strip gratings consisting of planar array of strips are either fixed on the surfaces or embedded in the layers. The design configurations with metallic strip gratings studied are:


(i)    metallic strip gratings embedded in the mid-plane of the core layer (Configuration1) and
(ii)   metallic strip gratings embedded in each skin-core interface (Configuration2).


Figure 1. Schematic of A-sandwich radome panel with metallic strip gratings (b) embedded in each skin-core interface (Configuration 2).



Figure 1. Schematic of A-sandwich radome panel with metallic strip gratings (b) embedded in each skin-core interface (Configuration 2).


For each configuration mentioned above as shown in Fig. 1, the optimum design parameters of the metallic strip gratings and radome .EM performance parameters are computed based on the equivalent transmission line method in conjunction with equivalent circuit model.


Further EM performance parameters (power transmission, power reflection and insertion phase delay) of both novel radome wall configurations with metallic strip gratings and A-sandwich alone structure are evaluated for perpendicular polarization over a range of incidence angles from normal incidence to high incidence angle 80°. The EM analysis shows that the A-sandwichradome wall with metallic strip gratings have superior EM performance as compared to conventional A-sandwich wall. The results indicate that these configurations have potential applications in the design of both normal incidence and highly streamlined airborne radomes.


The A-sandwich radome wall is composed of three layers namely, outer skin, core and inner skin. The outer and inner skin layers considered are made of glass-epoxy (dielectric constant, εr = 4; and loss tangent, tan δe = 0.015) with constant thickness of 0.75 mm for structural rigidity. The core is made of polyurethane foam (εr = 1.15 and tan δe = 0.005) with optimized thickness of 5.84 mm for maximum power transmission over the X-band. The surface of the outer skin layer is coated with a radome paint (εr = 3.46 and tan δe = 0.068) of thickness 0.2 mm. The metallic strip grating consists of a planar array of thin parallel strips of uniform cross-section. The thickness of the metallic strip is assumed to be very small (of the order of 0.1 mm).


The wavelengths corresponding to low frequency edge (8 GHz) and high frequency edge (12 GHz) of X-band are 37.5 mm and 25 mm, respectively. It is obvious that the optimized core thickness considered in the present work is less than quarter wavelength corresponding to the above mentioned lower and upper frequency limits of X-band. The  susceptance of a dielectric layer with thickness less than quarter wavelength is purely capacitive4. If the capacitive susceptance of the core can be cancelled out by proper inductive loading within the core using suitable metallic structures, superior EM performance characteristics over broadband of frequencies can be achieved. In this paper, the inductive loading within the core is achieved by embedding the array of metallic strips in the mid-plane of the core (Configuration1) or at the core-skin interface (Configuration 2). The EM design strategy involved is that the design parameters (pitch and width, keeping constant thickness) of the metallic strip grating matches with the capacitive susceptance of the entire A-sandwich radome wall. Hence the resulting structure acts as spatial low-pass filter, providing superior EM performance over the entire X-band. The metallic strip gratings are very sensitive to the polarization and angle of incidence. For the configurations considered in the work, the design parameters of the metallic strip gratings (width and pitch) and radome EM performance parameters are computed based on the equivalent transmission line method13,14 in conjunction with equivalent circuit model15. It is observed that for a given radome wall configuration with identical materials and dimensions, the radome EM performance characteristics are generally superior for parallel polarization, as compared to perpendicular polarization at a given incidence angle. This is due to the fact that the wave impedance for parallel polarization is less than that of perpendicular polarization at a given incidence angle. In other words, the performance degradations for a given radome wall configuration may be more for the perpendicular polarization case as compared to the parallel polarization case.


Hence in the present work, the EM performance parameters are analyzed and reported only for perpendicular polarization, which is capable of catering to the worst-case scenarios.


The metallic strip grating incorporated A-sandwich radome wall is considered as an equivalent transmission line with different sections corresponding to slab and metallic strip grating structure (Fig. 1). The change in the characteristic impedance of the free space and A-sandwich wall is a major source of reflection of the wave incident on the structure. As compared to the free space, different layers of radome wall can be considered as low impedance lines connected end to end. Hence the whole configuration can be represented by a single matrix obtained by the multiplication of matrices corresponding to the individual layers.
For Configuration1 Fig. 1(a), the matrix representing each layer of A-sandwich wall are as follows.
The inner skin is represented by


[ A 1 B 1 C 1 D 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaIXaaaleqa aKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aadkealmaaBaaabaqcLbmacaaIXaaaleqaaaGcbaqcLbsacaaMc8Ua am4qaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWGebWcdaWgaaqaaKqzadGaaGym aaWcbeaaaaGccaGLBbGaayzxaaaaaa@5C9E@ = [ cos Φ 1 j z s z o sin Φ 1 j z o z s sin Φ 1 cos Φ 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiGacogacaGGVbGaai4CaiabfA6agTWaaSba aeaajugWaiaaigdaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOAaKqbaoaalaaakeaajugGbiaadQhalmaaBaaabaqcLb macaWGZbaaleqaaaGcbaqcLbyacaWG6bqcfa4aaSbaaSqaaKqzadGa am4BaaWcbeaaaaqcLbsacaaMc8UaaGPaVlGacohacaGGPbGaaiOBai abfA6agTWaaSbaaeaajugWaiaaigdaaSqabaaakeaajugibiaadQga juaGdaWcaaGcbaqcLbyacaWG6bWcdaWgaaqaaKqzadGaam4BaaWcbe aaaOqaaKqzagGaamOEaKqbaoaaBaaaleaajugWaiaadohaaSqabaaa aKqzGeGaci4CaiaacMgacaGGUbGaeuOPdy0cdaWgaaqaaKqzadGaaG ymaaWcbeaajugibiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8Uaci4yaiaac+gacaGGZbGaeuOPdy0cdaWgaaqaaK qzadGaaGymaaWcbeaaaaGccaGLBbGaayzxaaaaaa@8F41@ (1)

                          

In Configuration1, the metallic strip grating is located at the mid-plane of the core. Hence the core can be considered to be made up of two identical sections with strip grating in between them. Then the half-section of the core is represented by


[ A 2 B 2 C 2 D 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaIYaaaleqa aKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOqaK qbaoaaBaaaleaajugWaiaaikdaaSqabaaakeaajugibiaadoeajuaG daWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaamiraKqbaoaaBaaaleaajugWaiaaikda aSqabaaaaOGaay5waiaaw2faaaaa@5B36@ = [ cos Φ 2 j z c z o sin Φ 2 j z o z c sin Φ 2 cos Φ 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiGacogacaGGVbGaai4CaiabfA6agTWaaSba aeaajugWaiaaikdaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOAaKqbaoaalaaakeaajugGbiaadQhalmaaBaaabaqcLb macaWGJbaaleqaaaGcbaqcLbyacaWG6bqcfa4aaSbaaSqaaKqzadGa am4BaaWcbeaaaaqcLbsaciGGZbGaaiyAaiaac6gacqqHMoGrjuaGda WgaaWcbaqcLbmacaaIYaaaleqaaaGcbaqcLbsacaWGQbqcfa4aaSaa aOqaaKqzagGaamOEaSWaaSbaaeaajugWaiaad+gaaSqabaaakeaaju gGbiaadQhalmaaBaaabaqcLbmacaWGJbaaleqaaaaajugibiGacoha caGGPbGaaiOBaiabfA6agLqbaoaaBaaaleaajugWaiaaikdaaSqaba qcLbsacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua ci4yaiaac+gacaGGZbGaeuOPdy0cdaWgaaqaaKqzadGaaGOmaaWcbe aaaaGccaGLBbGaayzxaaaaaa@8B12@ (2)                    
          

Let AMS, BMS, CMS and DMS be the elements of the matrix representing metallic strip grating


  [ A MS B MS C MS D MS ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaWGnbGaam4u aaWcbeaajugibiaaykW7caaMc8UaaGPaVlaadkealmaaBaaabaqcLb macaWGnbGaam4uaaWcbeaaaOqaaKqzGeGaam4qaKqbaoaaBaaaleaa jugWaiaad2eacaWGtbaaleqaaKqzGeGaaGPaVlaaykW7caaMc8Uaam iraKqbaoaaBaaaleaajugWaiaad2eacaWGtbaaleqaaaaakiaawUfa caGLDbaaaaa@551E@ = [ 10 j B G 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7aO qaaKqzGeGaamOAaiaadkeajuaGdaWgaaWcbaqcLbmacaWGhbaaleqa aKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGymaa aakiaawUfacaGLDbaaaaa@5BB7@ (3)

Here BG represents the shunt susceptance of the metallic strip grating15. For perpendicular polarization, the shunt susceptance of the strip for perpendicular polarization is given by


  B G = 4psecθ λ [ (lncosec( πg 2p )+G ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWGhbaabeaakiabg2da9maalaaabaGaeyOeI0IaaGinaiaa dchaciGGZbGaaiyzaiaacogacqaH4oqCaeaacqaH7oaBaaWaamWaae aacaGGOaGaciiBaiaac6gacaaMe8Uaci4yaiaac+gacaGGZbGaamyz aiaadogacaaMe8UaaiikamaalaaabaGaeqiWdaNaam4zaaqaaiaaik dacaWGWbaaaiaacMcacqGHRaWkcaWGhbaacaGLBbGaayzxaaaaaa@555A@ (4)                                                                                                         

Here the correction term is given by


G= 0.5 (1 β 2 ) 2 [ (1 β 2 4 )( C 1+ + C 1 )+4 β 2 C 1+ C 1 ] (1 β 2 4 )+ β 2 (1+ β 2 2 β 4 8 )( C 1+ + C 1 )+2 β 6 C 1+ C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaaGimaiaac6cacaaI1aGaaiikaiaaigdacqGHsisl cqaHYoGydaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabeaaca aIYaaaaOWaamWaaeaacaGGOaGaaGymaiabgkHiTmaalaaabaGaeqOS di2aaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaaaacaGGPaGaaiikai aadoeadaWgaaWcbaGaaGymaiabgUcaRaqabaGccqGHRaWkcaWGdbWa aSbaaSqaaiaaigdacqGHsislaeqaaOGaaiykaiabgUcaRiaaisdacq aHYoGydaahaaWcbeqaaiaaikdaaaGccaWGdbWaaSbaaSqaaiaaigda cqGHRaWkaeqaaOGaam4qamaaBaaaleaacaaIXaGaeyOeI0cabeaaaO Gaay5waiaaw2faaaqaaiaacIcacaaIXaGaeyOeI0YaaSaaaeaacqaH YoGydaahaaWcbeqaaiaaikdaaaaakeaacaaI0aaaaiaacMcacqGHRa WkcqaHYoGydaahaaWcbeqaaiaaikdaaaGccaGGOaGaaGymaiabgUca RmaalaaabaGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaa aacqGHsisldaWcaaqaaiabek7aInaaCaaaleqabaGaaGinaaaaaOqa aiaaiIdaaaGaaiykaiaacIcacaWGdbWaaSbaaSqaaiaaigdacqGHRa WkaeqaaOGaey4kaSIaam4qamaaBaaaleaacaaIXaGaeyOeI0cabeaa kiaacMcacqGHRaWkcaaIYaGaeqOSdi2aaWbaaSqabeaacaaI2aaaaO Gaam4qamaaBaaaleaacaaIXaGaey4kaScabeaakiaadoeadaWgaaWc baGaaGymaiabgkHiTaqabaaaaaaa@7F3E@    (5)

The coefficients are given by  C 1± = 1 ( psinθ λ ±1) 2 p 2 λ 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaDa aaleaacaaIXaGaeyySaelabaaaaOGaeyypa0ZaaSaaaeaacaaIXaaa baWaaOaaaeaacaGGOaWaaSaaaeaacaWGWbGaci4CaiaacMgacaGGUb GaeqiUdehabaGaeq4UdWgaaiabgglaXkaaigdacaGGPaWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0YaaSaaaeaacaWGWbWaaWbaaSqabeaaca aIYaaaaaGcbaGaeq4UdW2aaWbaaSqabeaacaaIYaaaaaaaaeqaaaaa kiabgkHiTiaaigdaaaa@4EEF@ (6)   Here   β=sin( πw 2p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey ypa0Jaci4CaiaacMgacaGGUbWaaeWaaeaadaWcaaqaaiabec8aWjaa dEhaaeaacaaIYaGaamiCaaaaaiaawIcacaGLPaaaaaa@4178@ and n =   ±1, ±2, ±3,……..


The other half-section of the core is represented by


[ A 3 B 3 C 3 D 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzaeabaeqakeaajugabiaadgeajuaGdaWgaaWcbaqcLbmacaaIZaaa leqaaKqzaeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam OqaKqbaoaaBaaaleaajugWaiaaiodaaSqabaaakeaajugabiaadoea juaGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzaeGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGebWcdaWgaaqaaKqzadGaaG4maaWcbeaa aaGccaGLBbGaayzxaaaaaa@590F@ = [ cos Φ 3 j z c z o sin Φ 3 j z o z c sin Φ 3 cos Φ 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiGacogacaGGVbGaai4CaiabfA6agTWaaSba aeaajugWaiaaiodaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOAaKqbaoaalaaakeaajugGbiaadQhajuaGdaWgaaWcba qcLbmacaWGJbaaleqaaaGcbaqcLbyacaWG6bqcfa4aaSbaaSqaaKqz adGaam4BaaWcbeaaaaqcLbsaciGGZbGaaiyAaiaac6gacqqHMoGrju aGdaWgaaWcbaqcLbmacaaIZaaaleqaaaGcbaqcLbsacaWGQbqcfa4a aSaaaOqaaKqzagGaamOEaSWaaSbaaeaajugWaiaad+gaaSqabaaake aajugGbiaadQhajuaGdaWgaaWcbaqcLbmacaWGJbaaleqaaaaajugi biGacohacaGGPbGaaiOBaiabfA6agTWaaSbaaeaajugWaiaaiodaaS qabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8Uaci4yaiaac+gacaGGZbGaeuOPdyucfa4aaSbaaSqaaKqzadGaaG 4maaWcbeaaaaGccaGLBbGaayzxaaaaaa@8C32@ (7)                             

The outer skin is represented by

 

[ A 4 B 4 C 4 D 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzaeabaeqakeaajugabiaadgeajuaGdaWgaaWcbaqcLbmacaaI0aaa leqaaKqzaeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGcbWcda WgaaqaaKqzadGaaGinaaWcbeaaaOqaaKqzaeGaam4qaKqbaoaaBaaa leaajugWaiaaisdaaSqabaqcLbqacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaaaakiaa wUfacaGLDbaaaaa@5788@ = [ cos Φ 4 j z s z o sin Φ 4 j z o z s sin Φ 4 cos Φ 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiGacogacaGGVbGaai4CaiabfA6agTWaaSba aeaajugWaiaaisdaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaamOAaKqbaoaalaaakeaajugGbiaadQhajuaGdaWgaaWcba qcLbmacaWGZbaaleqaaaGcbaqcLbyacaWG6bqcfa4aaSbaaSqaaKqz adGaam4BaaWcbeaaaaqcLbsaciGGZbGaaiyAaiaac6gacqqHMoGrju aGdaWgaaWcbaqcLbmacaaI0aaaleqaaaGcbaqcLbsacaWGQbqcfa4a aSaaaOqaaKqzagGaamOEaSWaaSbaaeaajugWaiaad+gaaSqabaaake aajugGbiaadQhajuaGdaWgaaWcbaqcLbmacaWGZbaaleqaaaaajugi biGacohacaGGPbGaaiOBaiabfA6agTWaaSbaaeaajugWaiaaisdaaS qabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlGacogacaGGVbGaai4CaiabfA6agTWaaSbaaeaajugWai aaisdaaSqabaaaaOGaay5waiaaw2faaaaa@8D53@ (8)      

 

The radome paint is represented by

 

[ A 5 B 5 C 5 D 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaI1aaaleqa aKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGcbWcdaWgaa qaaKqzadGaaGynaaWcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugW aiaaiwdaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aadsealmaaBaaabaqcLbmacaaI1aaaleqaaaaakiaawUfacaGLDbaa aaa@5682@ = [ cos Φ 5 j z p z o sin Φ 5 j z o z p sin Φ 5 cos Φ 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiGacogacaGGVbGaai4CaiabfA6agLqbaoaa BaaaleaajugWaiaaiwdaaSqabaqcLbsacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaamOAaKqbaoaalaaakeaajugGbiaadQhajuaGdaWgaa WcbaqcLbmacaWGWbaaleqaaaGcbaqcLbyacaWG6bWcdaWgaaqaaKqz adGaam4BaaWcbeaaaaqcLbsaciGGZbGaaiyAaiaac6gacqqHMoGrlm aaBaaabaqcLbmacaaI1aaaleqaaaGcbaqcLbsacaWGQbqcfa4aaSaa aOqaaKqzagGaamOEaKqbaoaaBaaaleaajugWaiaad+gaaSqabaaake aajugGbiaadQhajuaGdaWgaaWcbaqcLbmacaWGWbaaleqaaaaajugi biGacohacaGGPbGaaiOBaiabfA6agLqbaoaaBaaaleaajugWaiaaiw daaSqabaqcLbsacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlGacogacaGGVbGaai4CaiabfA6agTWaaSbaaeaaju gWaiaaiwdaaSqabaaaaOGaay5waiaaw2faaaaa@8DDF@ (9)                            

 

Thus, the entire A-sandwich configuration with metallic strip grating embedded at the mid-plane of the core may be expressed as

 

[ AB CD ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGcbaakeaajugibiaadoeacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadseaaaGccaGLBbGaayzxaaaaaa@4E5F@ = [ A 1 B 1 C 1 D 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIXaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbWcdaWgaaqaaKqzadGaaGymaa WcbeaaaOqaaKqzGeGaam4qaKqbaoaaBaaaleaajugWaiaaigdaaSqa baqcLbsacaaMc8UaaGPaVlaadsealmaaBaaabaqcLbmacaaIXaaale qaaaaakiaawUfacaGLDbaaaaa@4E4C@ [ A 2 B 2 C 2 D 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbWcdaWgaaqaaKqzadGaaGOmaa WcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqc LbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaIYaaale qaaaaakiaawUfacaGLDbaaaaa@4E50@ [ 10 j B G 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMe8UaaGPaVlaaicdaaOqaaKqzGeGaamOAaiaadk eajuaGdaWgaaWcbaqcLbmacaWGhbaaleqaaKqzGeGaaGPaVlaaykW7 caaMc8UaaGymaaaakiaawUfacaGLDbaaaaa@5277@ [ A 3 B 3 C 3 D 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIZaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbqcfa4aaSbaaSqaaKqzadGaaG 4maaWcbeaajugibiaaykW7aOqaaKqzGeGaam4qaSWaaSbaaeaajugW aiaaiodaaSqabaqcLbsacaaMc8UaaGPaVlaadsealmaaBaaabaqcLb macaaIZaaaleqaaaaakiaawUfacaGLDbaaaaa@506E@ [ A 4 B 4 C 4 D 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaI0aaaleqa aKqzGeGaaGPaVlaaykW7caWGcbqcfa4aaSbaaSqaaKqzadGaaGinaa WcbeaaaOqaaKqzGeGaam4qaKqbaoaaBaaaleaajugWaiaaisdaaSqa baqcLbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaI0a aaleqaaaaakiaawUfacaGLDbaaaaa@4EE6@ [ A 5 B 5 C 5 D 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaI1aaaleqa aKqzGeGaaGPaVlaaykW7caWGcbqcfa4aaSbaaSqaaKqzadGaaGynaa WcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugWaiaaiwdaaSqabaqc LbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaI1aaale qaaaaakiaawUfacaGLDbaaaaa@4E5C@    (10)
                                                                                        

In Configuration2,metallic strip grating is embedded in each skin-core interface Fig. 1(b). Then the entire A-sandwich radome wall with metallic strip gratings is represented by



  [ AB CD ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGcbaakeaajugibiaadoeacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadseaaaGccaGLBbGaayzxaaaaaa@4E5F@ = [ A 1 B 1 C 1 D 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIXaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbWcdaWgaaqaaKqzadGaaGymaa WcbeaaaOqaaKqzGeGaam4qaKqbaoaaBaaaleaajugWaiaaigdaaSqa baqcLbsacaaMc8UaaGPaVlaadsealmaaBaaabaqcLbmacaaIXaaale qaaaaakiaawUfacaGLDbaaaaa@4E4C@ [ 10 j B G 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMe8UaaGPaVlaaicdaaOqaaKqzGeGaamOAaiaadk eajuaGdaWgaaWcbaqcLbmacaWGhbaaleqaaKqzGeGaaGPaVlaaykW7 caaMc8UaaGymaaaakiaawUfacaGLDbaaaaa@5277@ [ A 2 B 2 C 2 D 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbWcdaWgaaqaaKqzadGaaGOmaa WcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqc LbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaIYaaale qaaaaakiaawUfacaGLDbaaaaa@4E50@ [ 10 j B G 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMe8UaaGPaVlaaicdaaOqaaKqzGeGaamOAaiaadk eajuaGdaWgaaWcbaqcLbmacaWGhbaaleqaaKqzGeGaaGPaVlaaykW7 caaMc8UaaGymaaaakiaawUfacaGLDbaaaaa@5277@ [ A 4 B 4 C 4 D 4 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaI0aaaleqa aKqzGeGaaGPaVlaaykW7caWGcbqcfa4aaSbaaSqaaKqzadGaaGinaa WcbeaaaOqaaKqzGeGaam4qaKqbaoaaBaaaleaajugWaiaaisdaaSqa baqcLbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaI0a aaleqaaaaakiaawUfacaGLDbaaaaa@4EE6@ [ A 5 B 5 C 5 D 5 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgealmaaBaaabaqcLbmacaaI1aaaleqa aKqzGeGaaGPaVlaaykW7caWGcbqcfa4aaSbaaSqaaKqzadGaaGynaa WcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugWaiaaiwdaaSqabaqc LbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaI1aaale qaaaaakiaawUfacaGLDbaaaaa@4E5C@    (11)

Here  [ A 2 B 2 C 2 D 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaK qzGeabaeqakeaajugibiaadgeajuaGdaWgaaWcbaqcLbmacaaIYaaa leqaaKqzGeGaaGPaVlaaykW7caWGcbWcdaWgaaqaaKqzadGaaGOmaa WcbeaaaOqaaKqzGeGaam4qaSWaaSbaaeaajugWaiaaikdaaSqabaqc LbsacaaMc8UaaGPaVlaadseajuaGdaWgaaWcbaqcLbmacaaIYaaale qaaaaakiaawUfacaGLDbaaaaa@4E50@

represents the core of the radome wall, which is different from that of Configuration1. 
Using Eqns (10) and (11), the A, B, C, a nd D parameters of the final matrix are computed for each Configuration. The power transmission coefficient is given by


   Ptr= [ 4 (A+B+C+D) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqbaoaalaaakeaajugibiaaisdaaOqaaKqzGeGaaiikaiaadgea cqGHRaWkcaaMc8UaamOqaiaaykW7cqGHRaWkcaaMc8Uaam4qaiaayk W7cqGHRaWkcaaMc8UaamiraiaacMcajuaGdaahaaWcbeqaaKqzGeGa aGOmaaaaaaaakiaawUfacaGLDbaaaaa@4BF2@                                     (12)

The power reflection coefficient is given by


Prf= [ A+BCD A+B+C+D ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqbaoaalaaakeaajugibiaadgeacaaMc8Uaey4kaSIaaGPaVlaa dkeacaaMc8UaeyOeI0IaaGPaVlaadoeacaaMc8UaeyOeI0IaaGPaVl aadseaaOqaaKqzGeGaamyqaiaaykW7cqGHRaWkcaWGcbGaaGPaVlab gUcaRiaaykW7caWGdbGaaGPaVlabgUcaRiaaykW7caWGebaaaaGcca GLBbGaayzxaaqcfa4aaWbaaSqabeaajugibiaaikdaaaaaaa@58F7@                                              (13)

The phase distortions are determined by the insertion phase delay (IPD) of the radome wall. For the Configuration 1, two skin layers, cores sections, metallic strip grating and radome paint are cascaded. Hence the insertion phase delays for Configuration1 is given by
IPD1= T 1 2π λ ( 2 t s +2 t c1 + t ms + t p )cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey iiIaDdcaWGubWaaSbaa4qaaiaaigdaaeqaa0GaaGPaVRGaeyOeI0Ia aGPaVlaaykW7daWcaaqaaiaaikdacqaHapaCaeaacqaH7oaBaaGaaG PaVpaabmaabaGaaGOmaiaadshadaWgaaWcbaGaam4CaaqabaGccqGH RaWkcaaMc8UaaGjbVlaaikdacaWG0bWaaSbaaSqaaiaadogacaaIXa aabeaakiabgUcaRiaadshadaWgaaWcbaGaamyBaiaadohaaeqaaOGa ey4kaSIaamiDamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaai GacogacaGGVbGaai4CaiabeI7aXjaaykW7caaMc8oaaa@6052@ 14


  Similarly, insertion phase delay for Configuration2 is given by
 IPD2 = T 2 2π λ ( 2 t s + t c2 +2 t ms + t p )cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey iiIaDdcaWGubWaaSbaa4qaaiaaikdaaeqaa0GaaGPaVRGaeyOeI0Ia aGPaVlaaykW7daWcaaqaaiaaikdacqaHapaCaeaacqaH7oaBaaGaaG PaVpaabmaabaGaaGOmaiaadshadaWgaaWcbaGaam4CaaqabaGccqGH RaWkcaaMc8UaaGjbVlaadshadaWgaaWcbaGaam4yaiaaikdaaeqaaO Gaey4kaSIaaGOmaiaadshadaWgaaWcbaGaamyBaiaadohaaeqaaOGa ey4kaSIaamiDamaaBaaaleaacaWGWbaabeaaaOGaayjkaiaawMcaai GacogacaGGVbGaai4CaiabeI7aXjaaykW7caaMc8oaaa@6054@   (15)


Here  and  are the phase angles associated with the voltage transmission coefficients corresponding to Configuration 1 and Configuration 2 respectively. The thicknesses of skin layers, metallic strip grating and radome paint are given by ts, tms, and tp respectively. Let tc1 be the thickness of each section of the core for Configuration 1, while tc2 be the core thickness for Configuration 2.

A comparative  study  of the EM performance of A-sandwich wall with metallic strip gratings and A-sandwich wall alone is carried out for Configuration1. The EM performance parameters are computed at normal incidence, 45°, 60°, and 80° for perpendicular polarization over X-band frequency range. The optimized design parameters of metallic strip gratings for Configuration 1 are given in Table 1. Figures 2-4 show the EM performance characteristics of Configuration 1. It may be observed that the A-sandwich wall with metallic strip grating shows superior power transmission characteristics as compared to A-sandwich wall alone Fig. 2. The power transmission of the A-sandwich wall embedded with strip grating is well above 90 per cent at normal incidence, 45°, and 60°. But there is degradation of power transmission efficiency at high incidence angle 80°. The power reflection is very low (a desirable characteristic) for the A-sandwich wall with strip grating as compared to the A-sandwich alone wall (Figs. 3(a) and 3(d)). It is observed that the power reflection of the conventional A-sandwich structure increases with the increase in the incidence angle. TThe insertion phase delay (IPD) characteristics are shown in Figs. 4(a) and 4(d). It is observed that the IPD of the A-sandwich wall embedded with strip gratings is same as that of the conventional structure at normal incidence. However, the IPD of the strip embedded structure increases with incidence angle.




Figure 2.Power transmission characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).

Figure 2.Power transmission characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Figure 2.Power transmission characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Figure 3.Power transmission characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Figure 3.Power transmission characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Table 1. Design parameters of metallic strip gratings .



The EM performance parameters of A-sandwich wall with metallic strip gratings embedded in each skin-core interface (Configuration 2) and A-sandwich wall alone are shown in Figs. 5 and 7. The optimized design parameters of metallic strip gratings corresponding to Configuration 2 are given in Table 1. Figure 5 show the power transmission characteristics of Configuration 2 at normal incidence, 45°, 60°, and 80°. It is noted that the inclusion of metallic strip gratings at each skin-core interface improves the power transmission efficiency. The A-sandwich wall with strip gratings shows excellent power transmission characteristics (above 90%) at normal incidence. There is degradation in the power transmission efficiency of the A-sandwich wall with strip gratings at other incidence angle. However, the power transmission characteristics of A-sandwich wall with strip gratings are better than that of A-sandwich alone at all incidence angles over the X-band. The power reflection characteristics of Configuration 2 are shown in Fig. 6. The A-sandwich wall embedded with metallic strip gratings shows very low power reflection as compared to that of the conventional structure. The power reflection for Configuration 2 is very low at normal incidence, 45°, and 60°. However, the power reflection of the conventional structure increases drastically at high incidence angle 80°. The insertion phase delay characteristics of Configuration 2 are shown in Fig. 7. It is observed that the IPD of the A-sandwich wall embedded with metallic strip gratings for Configuration 2 is higher than that of A-sandwich alone at all incidence angles. It is desirable for reducing phase distortions and hence boresight error.

Figure 4. Power reflection characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60° and 80°. (Polarization: Perpendicular).


Figure 4. Power reflection characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 1) at normal incidence, 45°, 60° and 80°. (Polarization: Perpendicular).


Figure 5.Figure 5. Power reflection characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 2) at normal incidence, 45°, 60° and 80°. (Polarization: Perpendicular).


Figure 5.Figure 5. Power reflection characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 2) at normal incidence, 45°, 60° and 80°. (Polarization: Perpendicular).



Figure 6. Power reflection characteristics of A-sandwich radome with metallic strip gratings embedded in the mid-plane of the core and A-sandwich alone (Configuration 2) at normal incidence, 45°, 60° and 80°. (Polarization: Perpendicular).


Figure 7. Insertion phase delay characteristics of A-sandwich radome with metallic strip gratings embedded in each skin-core interface and A-sandwich alone (Configuration 2) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Figure 7. Insertion phase delay characteristics of A-sandwich radome with metallic strip gratings embedded in each skin-core interface and A-sandwich alone (Configuration 2) at normal incidence, 45°, 60°, and 80°. (Polarization: Perpendicular).


Among the two configurations considered, Configuration 1 has better EM performance characteristics as compared to Configuration 2. The average power transmission efficiency (around 85 per cent) for Configuration 1, while that of Configuration 2 is around 82 per cent. Further, the average  power reflection for Configuration 1 is nearly zero at high incidence angle 80°, while it is around 5 per cent for Configuration 2. At normal incidence, IPD of Configuration 2 is much higher than that of Configuration 1, indicating more phase distortions.  

The application of metallic strip gratings for improving the EM performance characteristics of A-sandwich radome over X-Band is established in this work.  It is observed that the novel A-sandwich wall configurations (Configuration 1 and Configuration2) offer better EM performance characteristics as compared to the conventional A-sandwich wall with optimized core thickness. Considering the EM performance characteristics, Configuration1 is preferable to Configuration2.  Further regarding fabrication aspects, Configuration1 is desirable as only one set of strip grating has to be embedded in the structure.  The present work also shows that A-sandwich wall with metallic strip gratings is a better choice for both the normal-incidence (i.e., cylindrical or spherical) radomes, and the highly streamlined (e.g. conical, ogival) nosecone radomes. 

 

1.     Robinson, L.A. Electrical properties of metal loaded radomes. WADD Technical Report. Report No. 60-84, 1960.

2.     Walton, Jr. J.D. Techniques for airborne radome design. AFAL Report. Report No. 45433, 1966, 105-108.

3.     Crone, G.A.E.; Rudge, A.W. & Taylor, G.N. Design and performance of airborne radomes: A review. IEE Proc., 1981, 128(7), 451-464. [Full text via CrossRef]

4.     Cary, R.H.J. Radomes, In The Handbook of Antenna Design. Peter Peregrinus, London, 1982.

5.     Frenkel, A. Thick metal dielectric window. Electronics Letters, 2001, 37(23), 1374-1375. [Full text via CrossRef]

6.     Munk, B.A. Frequency selective surfaces - Theory and design. Wiley, New York, 2005.

7.     Lin, B.-Q.; Li, F.; Zheng, Q.-R. & Zen, Y.-S. Design and simulation of a miniature thick-screen frequency selective surface radome. Progress Electromagnetics Research, 2009, 138, 537-553. [Full text via CrossRef]

8.     Nair, R.U.; Madhumitha, J. & Jha, R.M. Broadband EM performance characteristics of single square loop FSS embedded monolithic radome. Int. J. Antenn. Propag., 2013, 1-8. [Full text via CrossRef]

9.     Latrach, M.; Rmili, H.; Sabatier, C.; Seguenot, E. & Toutain, S. Design of a new type of metamaterial radome at low frequencies. Microwave Optical Technol. Letters, 2010, 52(5), 1119-1123. [Full text via CrossRef]

10.   Basiry, R.; Abiri, H. & Yahaghi, A. Electromagnetic performance analysis of omega type metamaterial radome. Int. J. RF Microw. Comput. Aided Eng., 2011, 21(6), 665-673. [Full text via CrossRef]

11.   Narayan, S.; Shamala, J.B.; Nair, R.U. & Jha, R.M. Electromagnetic performance analysis of novel multiband metamaterial FSS for millimeter wave radome applications. Computers, Materials  Continua, 2012, 31(1), 1-16.

12.   Kozakoff, D.J. Analysis of radome enclosed antennas. Artech House, Norwood, 2010.

13.   Chen, F.; Shen, Q. & Zhang, L. Electromagnetic optimal design and preparation of broadband ceramic radome material with graded porous structure. Progress Electromagnetics Research, 2010, 105, 445-461. [Full text via CrossRef]

14.   Pei, Y.M.; Zeng, A.M.; Zhou, L.C.; Zhang, R.B. & Xu, K.X. Electromagnetic optimal design for dual-band radome wall with alternating layers of staggered composite and Kagome lattice structure. Progress Electromagnetics Research, 2012, 122, 437-452. [Full text via CrossRef]

15.          Lee, C.K. & Langley, R.J. Equivalent-circuit models for frequency-selective surfaces at oblique angles of incidence. IEE Proc., Pt. H, 1985,18(6), 395-399. [Full text via CrossRef]

Dr Raveendranath U. Nair received his MSc and PhD in Physics (Microwave Electronics) from the School of Pure and applied Physics, Mahatma Gandhi University, Kerala, India, in 1989 and 1997, respectively. Since September 1999 he has been working as Scientist at the Centre for Electromagnetics in CSIR-NAL, Bangalore, India where currently he is a principal scientist. He has authored/co-authored over 100 research publications including journal papers, symposium papers and technical reports. His research interests include computational electromagnetics, EM design, analysis and performance measurements of radomes, metamaterials, frequency selective surfaces, EM material characterization techniques.

*, Ms J. Madhumitha obtained the BE (Electronics and Communication) from Anna University, Chennai, in 2005. She is currently working as a Technical Manager in Sterling Electronics at Tiruchirappalli, Tamilnadu, India. She was earlier working as a Project Engineer at the Centre for Electromagnetics, CSIR-National Aerospace Laboratories (CSIR-NAL), Bangalore, India. Her research interests area are Radomes and EM material characterization.

, and Dr Rakesh Mohan Jha obtained a dual degree in BE (Hons.) EEE and MSc Physics from BITS Pilani (Raj.), in 1982 and PhD from Department of Aerospace Engineering, Indian Institute of Science, Bangalore, in 1989. He is currently a Chief Scientist and Head of Centre for Electromagnetics at CSIR-National Aerospace Laboratories, Bangalore. He is also concurrently Professor and Associate Dean of the Academy of Scientific and Innovative Research, New Delhi. His active areas of research are in the domain of computational electromagnetics for aerospace applications; these include GTD/UTD, 3-D ray tracing and surface modelling, aerospace antennas and radomes, radar cross section (RCS) studies including active RCS reduction, radar absorbing materials and radar absorbing structures, and metamaterials for aerospace applications. He has published more than 400 scientific research papers and technical reports.