FUNDAMENTAL ELECTROMAGNETIC CONCEPTS FOR RAM

2 FUNDAMENTAL ELECTROMAGNETIC CONCEPTS FOR RAM III Chapler I the concept of RAM was introduced as a means for reducing the ReS of aerospace vchicles Numerous RAM were also introduced in that chapter; many of which still find extensive use in stealth , antenna engineering and microwave related technologies It was a lso mentioned that these RAM vary considerably in their absorption characteristics An enhancement in the absorption due to th e RAM coating on It target r(sulls in lower scattered EM field s and hence radar cross sect ion red uct ion (RCSR) or the target II is possib le in principle 10 predict the electromagnet ic fi elds al an observation point i.e., the receiving radar, by the application of EM fi eld theOl)' In th is chapler, we first introduce Maxwell' s equations in their most genera l form These equations constit{Jtc the starting point for EM wave propagation Ilna lysis The wave analys is involves not on ly free space propagation and interaction at the interface of two medi a, but also Ihrough bounded material medium EM wave propagat ion equations are sel up in this chapter for the three most common cases o f free spoce, homogeneous, and inhomogeneo J/s propagation Wave propagation thro ugh a maleria lm edium is governed by the intrin sic ph ysica l parameters of the medium , viz ils penllillivity permeabi lity and conductivity The well known c lasses of dielectric and magnetic RAM are essentially a manifestation of these intrin sic H l parameters of the mediulll It is a lso possible to explain the various propen ies, such as isotropy linearity and rec iproc ity of the med ium in Ic ml S of the nat ure o f these parameters In the optical reg ion one frequentl y encou nters the phenomena of optical act ivity and c ircular dichroism Drawing upon a microwave ana log of these, one may visualize chiml materials, which are in fact highly effective absorbers The electromagnetic parameters corresponding to chiral ity arc de fined in Sect ion 2.4 Fwu/amell laf E leclroll/ agllelic Concepts/or RAM 20 2.1 l\lAXWELL'S EQUATIONS The most fund amental Jaws of e!cclromagnclics arc Maxwe\l' s equations which originate from Faraday's law, Ampere ' s law, and Gauss's law These a rc expressed in differenti al form as 8B 'V' x E= - - at ao (2 1) '\1 x H = - + J (22) V · D =p (2.3) '\7- = (2.4) at In the equations above, E and H are the electromagllelic fie ld vec/ors The wave propagation analysis is oftell carried oul in terms of these E and H refer to the electric fie ld slrellg lll and the magneric field strength , respectively is known as the electric displacemellt density whereas its analog, B, is called the magnetic flrl x density The partial derivative in cqs (2 1) and (2.2) is with respect to Ihe time I Fina lly, the symbol s J and p a ppearing in the right hand side of eqs (2.2) and (2.]) refer to the electric current densily and the volume charge density, respectively Equation (2 1) follows from the Faraday induction Iml', while eq (2 2) is a generaliz.1tio n of the Ampere circuital loll' by Maxwell and is al so referred to as the Maxwe ll.A mpere law Equation (2.]) is the differential form of Gauss '5 law for the electric displacement density Finally, eq (2 4) merely stales Iha l magnetic monopoles arc non cxislent It is in fa ct the magnetic analog of Gauss's law, and ca n be readily derived fromlhe Biot-Savortlml' Although one is accustomed to fou r Maxwell equations, two of Ihese, namely, the dive rgence relations (2.3) and (2.4) can be derived from the curl equations (2 1) and (2.2) (Corson & Lorrain, 1962) The alternate representation oflhe Ma well 's equations is in Ihe integral form and can be obtained by suitably integrating eqs (2 1) through (2 4) The integration in the case of cqs (2.1) and (2 2) is with respect 10 the area clement da, whereas in the case of eqs_ (2 ]) and (2 4) it is \\~th respect to the vo lume clemenl dv The Maxwell equations in the integral form arc : (2.5) Radar Absorbing ,\/meria/s f E dl = f J.da + J -o-t ·da " " 00 fO.da = f PdV " 21 (2 6) (2 7) , fB.da = O (2 8) " The derivation of cqs (2.5) and (2 6) is carried OUI by resorting to Stokes ' theorem Stokes ' theore m is a well known resu lt of veclor calculu s which establi shes an equivalence between the surface integmls and the line integrals: the curl ofa vector A over a surface area is thu s related to the \'ector on the curve enclosing tIlat area, f 'V' x A.da = f A dl (2 9) Likewise ill the case of volume integral s the area contour integra l appe.1Ting on the len hand side of (2 7) and (2 8) follow from Gallss's divergence (heorem which relates the divergence of a vcclor A from a volume, to thc vcctor o\'er the surface area enclosi ng that volul1Ie, (2 10) , " Bolh the differential and integral forills of Maxwell's equations arc extensively uscd ill the EM wave propagation analysis For example, the differential form of these equalions fonns the starting point for the finite difference time domain (FDTD) analysis, and EM wave propagation in free space and 01 her continuous media On the other hand, as shall be shown in the next section , the integral form of Maxwel l's equations arc used to derive the boundar)' conditions at the interface between two media 2.2 SURFACE BOUNDARY CONDITIONS Equations (2 1) throug h (2 4) describe the spalial value of EM vectors E, n, D and B These vectors are continuous along the di rection of propagation within a medium It is of interest to examine whether these vectors remain cOlll inuous across the interface of Iwo different media 22 F'llldwllell/(/{ flec[l'oll/agnelic COIICepfS!Or f?A ,\f We beg in the discu ss ion with cqs (2 1) and (2 ) 1·lenec for the corresponding illlcgral fOnll S (2.5 ) and (2.6), it possible 10 visualize a surface art'a enclosed by a curve Boulldary CmldiliOlI I Let an area completely enclosed by a curve be intersected by an interface oflhe two media as shown in Fig By appl ying Farad~" s law of eq (2.5), one call visualize the electric fields £-' and £ "2' \ '0 )' close 10 the media interface along the x-direction Similarly, £'1' £ )1 ' £" and /:,~~ are along the y -direction The exact values fo r these veclOrs arc not known a priori Integrat ing along the closed curve we gel, ~ oil (2.11 ) ' ilxfiy 0' The assumption that E" I and tends to zero Thus Ex] are vel)' close to the interrace, requires that 11)' (2.12) I.e , (2.1 3) Since E~ l and E r! arc tangential to the interrace, eq (2 13) is generalized as: (2 14) The first bOlll/dmy condition thererore requires that the tangential component or the electric field be continuous at the interface or two arbitrary media Without loss or generalization, let LI S assume that Medium is a perrect conductor For such a case, it can be shown Ihat E,m,! is zero, so that E',,,,I = (2 15) Hence in those cases where one of the media is a conductor, the electric fi eld can onl y be normal to the interrace Radar Ah.wrbillg Materials 'J , Medium J- Fi j:ure 2.1 Thc bou ndaf) co ndi tion fo r the e lcClr ic fi eld tl\O me dia \"C~ tors at the inlcrfa c( ~t\\ccn BOlIl/ dary Com/ilioll As ment ioned