A Study of Electromagnetic Wave Propagation. in the Foam Core Sandwich Structures

Transcription

1 ID-83 A Study of Electroagnetic Wave Propagation in the Foa Core Sandwich Structures H. J. Chun and H. S. Shin School of Electrical and Mechanical Engineering, Yonsei University 34, Shinchon-dong, Seodaeun-gu, Seoul, -749, Korea : hjchun@yonsei.ac.kr Departent of Mechanical Engineering, Yonsei University 34, Shinchon-dong, Seodaeun-gu, Seoul, -749, Korea : uujajuck@yonsei.ac.kr SUMMARY: In this study, efforts were ade to understand the propagation of electroagnetic wave through the foa core sandwich structure by the analytical odel. Foa core sandwich structure odel is coposed of glass/epoxy coposite skins and foa. Transittance and reflectance of the arbitrary linearly polarized incident TEM waves through unidirectional coposites and foa were calculated as functions of fiber orientation of coposites and incident angle by the analytical odel. Fro the results of the analysis, the general tendency of transittance and reflectance of electroagnetic wave through coposites and foa was obtained. While transittance of foa as isotropic aterial changed slowly, transittance of coposites as anisotropic aterials changed draatically since the attenuation coefficient of coposites increased in proportion to incident angle. Fro the results, it was observed that the propagation of electroagnetic wave through the foa core sandwich structure was ainly affected by the coposite skins since the effect of foa on the wave propagation was trivial due to translucent characteristics of foa to electroagnetic field. Fro the results of analysis of unidirectional coposites as functions of incident and polarization angles and fiber orientation, the relationship between transittance and polarization angle of the propagation of electroagnetic wave was ore coplex than that between transittance and fiber orientation. Fro the results of analysis of foa core sandwich structure, it was observed that the axiu transittances of all cases were ore or less siilar. However, the gradient of transittances in the case of [/3/6]-foa-[6/3/] and [-6//6]-foa-[6//-6] were ore onotonous than [ 3 ]-foa-[ 3 ] and [/9/]-foa-[/9/] since, in the forer case, electroagnetic properties becae ore quasi-isotropic. Fro the results, it was found the general tendency of the propagation of electroagnetic wave through foa core sandwich structures. KEYWORD : Anisotropic, Electroagnetic Wave, Transittance

2 INTRODUCTION There are any practical situations where electroagnetic wave s interact with coposite aterials. The knowledge of interaction of electroagnetic waves in coposite structures is iportant for designing the shielding structure for antenna such as radoe. Recently, radoes are constructed in the for of foa core sandwich structures that have any echanical advantages such as high strength, long fatigue life, low density and adaptability to the intended function of structure. However, the propagation of electroagnetic waves is affected by high anisotropic pereability and loss tangent of coposite skins. Even though any investigations were focused on the propagation of electroagnetic waves in the coposite aterials in last several decades, little investigations were carried out to understand adequately the propagation of the electroagnetic waves in the foa core sandwich structures[-4]. In this paper, nuerical analyses of unidirectional coposites, foa and foa core sandwich structures as functions of incident and polarization angles and fiber orientation were perfored and general tendency of the propagation of electroagnetic wave through foa core sandwich structures was observed. THEORY Foa core sandwich structure odel is coposed of coposite skins ade up of one or ore unidirectional plies stacked together at various orientations and foa. Coposite aterials have high anisotropic characteristics for echanical and electroagnetic properties. Incident angle of electroagnetic wave for surface of radoe is variable since radoe is heisphere shape. Therefore, incident and polarization angles of TEM wave and fiber orientation of coposites are the variables of ain concern in this paper. The scheatic drawings showing the incident TEM wave entering the foa core sandwich structure and skins ade up of several unidirectional plies stacked together at various orientations are shown in Figs. and, respectively. Fig. Scheatic drawing showing the incident TEM wave entering the sandwich structure

3 Fig. Scheatic drawing showing the skin ade up of several unidirectional plies stacked together at various orientations In these figures, φ, δ, θ, N and T are the incident and polarization angles of TEM wave, fiber orientation, noral and tangential direction of fiber in the coposite skins, respectively. The wave equation of electric field through anisotropic nonagnetic edia( µ µ can be expressed as follows: E E e [ α]u r e sin φ E e ( ω t[ β ]u r j e ( t j( t ω [ β ]u r [ α]u r ω [ β ]u r [ α ]u r j cos δ cosφ E e e i sin δ cos φ E e e j ( [ α]u r ( t ω [ β ]u r k j where, [ ] [ T ] [ α α ][ T ], [ β ] [ T ] [ β ][ T ] vector, cosθ sin θ T sin θ cosθ : rotating tensor, [ ] ωµ g ij α ij, βij and µ is pereability. [ ], u : propagation vector, r : position ( ωε ij ω ε ij g ij ωµ β ij ( i, j g and [ ] principal ply directions, respectively.,, 3 ε are conductivity and perittivity tensor along the In the case of a linearly polarization TEM wave traveling perpendicular to the -axis, phase velocity( V can be expressed as follows : ω sin φ V β sin φ β 33 cos φ V ( V 3 β ω cos φ sin φ β cos 33 φ For the interface between the and ply, Snell s law can be expressed as follows :

4 V V sin φ sin φ Based on Snell s law, an angle of refraction is deterined as follows[5] : (3 3 3 n n 4 / 7 n n 4 / 7 φ 3 3 sin (4 where, ( β 33 ( β ( β33 sin φ and n {( β sin φ ( β33 cos φ } ( β ( 33 β It is assued that a linearly polarized TEM wave of known aplitude E and known frequency f ipinges on foa core sandwich structure. The scheatic drawing showing the electric field vectors (E entering the foa core sandwich structure is shown in Fig. 3. st ply θ nd ply θ M th ply θ M Mth ply θ M ( P ( P ( P M ( P, δ M φ φ φ ( E, δ ( P Foa φ M ( P M Fig. 3 Scheatic drawing showing electric field vectors at foa core sandwich structure At the interface, a portion transitted of the arriving wave through the interface is denoted by, and a portion reflected of it is denoted by ( E r. The reflected portion of the right traveling wave is denoted by ( E r, and the reflected portion of the left traveling wave is denoted by ( E r [7]. At the interface, the electroagnetic waves entering and leaving ust be equal. This condition requires that the following equalities be satisfied at the interface. ( E r E E ( E r (5 Two electric field vectors are defined as follows : P ( E r ( E r P (6 For the interface between the - and ply, the refraction coefficient tensor( r ij can be expressed as follows :

5 r ij ( Nij ( N ij ( Nij ( Nij (7 where, [ ] [ T ] [ N ][ T ] N and N ij µ /εij The reflected and incident electric field vectors are related by the expressions ( [ r]( E E r ( [ r]( E E r (8 The electric field traveling through the aterial is attenuated. Thus, attenuation coefficient of the electric field can be expressed as follows[6]: E ( ω t[ β ]u r j( ω t[ β]u r [ α]u r j E e E where, A e Ae (9 [ α ]u r e : attenuation coefficient The scheatic drawing showing electric field vectors at the -th ply is shown in Fig [( E ] t ( P A ( P ( E [( E ] r A ( P ( E ( P [( E ] r [( E ] t Fig.4 Scheatic drawing showing electric field vectors at the -th ply The electric field vectors in the -th ply can be expressed as follows : ( P [( E ] [( E ] t r ( E [( E ] [( ] r E r ( E [ r] ( ( ( E [ r] ( ( ( E ([ I ] [ r] ( ( [ A] ( P [ r] ( ( [ A] ( P ( P [( E ] [( ] t E r ( E [( ] [( ] E r E r ( E [ r] ( ( ( E [ r] ( ( ( E ([ I ] [ r] ( ( [ A] ( P [ r] ( ( [ A] ( P where, [ I ] : identity atrix and [ A ] : attenuation tensor For a given incident TEM wave, incident, reflected and transitted energy flux are obtained as follows [8]: (

6 F i ( E, Z F r ( P, Z F t ( P M ( Z where, Z : ipedence in free space ( π Solving Eq., ( P and ( M P can be obtained. Using Eq., transittance and reflectance of the incident linearly polarized TEM waves can be calculated as functions of fiber orientation of coposites, incident and polarization angles. RESULTS AND DISCUSSION Glass/epoxy coposites and foa are the aterials of ain concern in this study, because they are widely used in radoe. These aterials are nonagnetic( µ µ. Input constants used in this analysis are shown in Table. Table Input constants for analysis Foa Glass/Epoxy Coposites Longitudinal Transverse In- Transverse Out-of- Relative.3 Direction Plane Direction Plane Direction Pereability Conductivity[Ω - - ]. x x -.3 x -. x - Thickness [] 7.3 Frequency [GHz] In the case of relative pereability and conductivity, typical values were deterined within general range of aterial constants of foa and glass/epoxy coposites. Using above constants, nuerical analysis of unidirectional coposites and foa was perfored to obtain the general tendencies of transittance and reflectance of electroagnetic wave through coposites and foa. The plots of transittance and reflectance of TEM wave through glass/epoxy coposites and foa as a function of incident angle are shown in Fig. 5. (a (b Fig. 5 Transittance and reflectance of TEM wave through (a glass/epoxy coposites and (b foa as a function of incident angle (polarization angle θ

7 Though reflectance decreased in proportion to incident angle, transittance decreased because of the attenuation of both coposites and foa as shown in Fig. 5. While transittance of foa as isotropic aterials changed slowly, transittance of coposites as anisotropic aterials changed draatically since attenuation coefficient of coposites increased in proportion to incident angle. It was observed that the propagation of electroagnetic wave through the foa core sandwich structure was ainly affected by the coposite skins since the effect of foa on the wave propagation was trivial due to translucent characteristics of foa to electroagnetic field. Nuerical analysis of unidirectional coposites as functions of incident and polarization angles and fiber orientation was perfored to obtain the relationship aong transittance, polarization angle and fiber orientation. The plots of transittance of TEM wave through glass/epoxy coposites as functions of polarization and incident angles are shown in Fig. 6. (a (b Fig. 6 Transittance of glass/epoxy coposites as functions of (a polarization and incident angles and (b fiber orientation and incident angle The relationship between transittance and polarization angles of the propagation of electroagnetic wave was ore coplex than that between transittance and fiber orientation as shown in Fig. 6. In the case of foa core sandwich structure, nuerical analysis as functions of incident and polarization angles was perfored for various stacking sequences. The plots of transittance of TEM wave in foa core sandwich structures with various stacking sequences of coposite skins as functions of polarization and incident angles are shown in Fig. 7.

8 (a (b (c (d Fig. 7 Transittance of TEM wave in foa core sandwich structure (a [ 3 ]-foa-[ 3 ] (b [/3/6]-foa-[6/3/] (c [-6//6]-foa-[6//-6] (d [/9/]-foa-[/9/] as functions of polarization and incident angles It was observed that axiu transittances of all cases were ore or less siilar. However, the gradient of transittance in the case of Fig. 7 (b and (c was ore onotonous than that in the case of Fig. 7 (a and (d since, in the forer case, property becae ore quasiisotropic. Fro Fig. 7, general tendency of the propagation of electroagnetic wave through foa core sandwich structures with various stacking sequences of coposite skins can be obtained. To design spherical foa core sandwich structure radoe, transittance of electroagnetic wave is iportant. The tendency of electroagnetic wave propagation necessary for optiizing radoe ade of foa core sandwich structure with anisotropic coposite skins can be obtained fro nuerical odel proposed in this paper. CONCLUSIONS Recently, radoes are constructed in the for of foa core sandwich structures that have

9 any echanical advantages such as high strength, long fatigue life, low density and adaptability to the intended function of structure. However, the propagation of electroagnetic waves is affected by high anisotropic pereability and loss tangent of coposite skins. Nuerical analyses of unidirectional coposites, foa and foa core sandwich structure as functions of incident and polarization angles and incident angle and fiber orientation were perfored. Fro the results of nuerical analysis, it was observed that the propagation of electroagnetic wave through the foa core sandwich structure was ainly affected by the coposite skins since the effect of foa on the wave propagation was trivial due to translucent characteristics of foa to electroagnetic field. Fro the results, the general tendency of the propagation of electroagnetic wave through foa core sandwich structures was observed. In this paper, the analytical odel and ethod to predict the propagation of electroagnetic waves through the foa core sandwich structures were proposed. ACKNOWLEDGEMENTS The authors are grateful for the support provided by Brain Korea fro Korea Research Foundation (KRF. REFERENCES. Krohn, T.L. and Medgyesi-Mitschang, L.N., Scattering fro coposite aterials: a firstorder odel Antennas and Propagation, IEEE Transactions on, 989, Vol. 37, Issue :, pp O. Poncelet and M. Deschaps, Reflection and Refraction of an Inhoogeneous Plane Wave on Fluid/Anisotropic Solid Interface, ULTRASONICS SYMPOSIUM, 994. Proceeding., 994 IEEE, Vol., pp Sobrinho, C.L.d.S.S. and Giarola, A.J., Analysis of rectangular anisotropic dielectric waveguide structures, Antennas and Propagation Society International Syposiu, 99. AP-S. Digest, 99, Vol., pp Torres Lia, I., Jr. and Giarola, A.J., Electroagnetic wave propagation in two diensional anisotropic dielectric gratings, Antennas and Propagation Society International Syposiu, 997, IEEE., 997, Digest, Vol. 4, pp von Hippel, A. R., Dielectrics and Waves. New ed, Artech House, Cornbleet, S., Microwave Optics, Acadeic Press, Lee, W. I. and George S. Springer, Interaction of Electroagnetic Radiation with Organic Matrix Coposites, Journal of Coposite Materials, 984, Vol. 8, pp Willia H. Hayt, Jr., Engineering Electroagnetics, McGraw-Hill, 989