1 Chapter 6 Equivalence theorem EMLAB Aperture type

1 Chapter 6. Equivalence theorem EMLAB

Aperture type antennas 2 EMLAB

Green’s functions for physical laws 4 Poisson’s equation 1. G는 원래 미분 방정식을 만족. 2. G는 경계조건도 만족시켜야 한다. Free space scalar Green function Helmholtz’ equation Vector wave equation EMLAB

Example - Free space Green function 6 Source Boundary condition : radiation condition EMLAB

Example – Green function of rectangular cavity 7 Boundary condition : h PEC b a 똑같은 미분방정식이어도 경계 조건에 따라 다른 형태의 solution이 얻어짐. EMLAB

Derivation of Equivalence theorem for E-field 11 (a는 임의의 상수 벡터. ) (Vector Green’s identity) EMLAB

Equivalence theorem with free space kernel : G 0 13 ; for free space Free space Green function을 이용하여 만든 Green’s theorem식. EMLAB

Dyadic Green’s function for PEC half space : G 1 14 ; for half space 전기장의 접선 성분을 0으로 만 들기 위한 Green function들. t는 지면과 평행한 성분, n은 지 면에 수직인 성분. ; image current PEC EMLAB

Half space Green function with sources on PEC 15 Non-zero Perfect electric conductor Equivalence theorem 에서는 tangential current만 중요함. EMLAB

Dyadic Green’s function for half space – PMC : G 2 16 ; for half space 전기장의 접선 성분을 0으로 만 들기 위한 Green function들. t는 지면과 평행한 성분, n은 지 면에 수직인 성분. ; image current Perfect magnetic conductor Non-zero EMLAB

Image current for magnetic current 17 전기장은 자기장을 1번 미분한 양이므 로 대칭성이 반대임. Perfect electric conductor Perfect magnetic conductor EMLAB

Summary – image current Perfect electric conductor Perfect magnetic conductor The equivalent currents contain only tangential components. 18 Perfect electric conductor Perfect magnetic conductor EMLAB

Slit 문제 19 (1) Free space dyadic Green function 인 경우 Je는 Screen의 빨간 영역에 대해 적분, PEC Jm은 Screen의 파란 영역에 대해 적분 (2) Half space dyadic Green function 인 경우 Jm만 Screen의 파란 영역에 대해 적분하면 됨. PEC EMLAB

Schelkunoff’s field equivalence principles 21 image current EMLAB

Fresnel zone, Fraunhofer zone 22 EMLAB

Kirchhoff approximation, Fresnel diffraction 23 : Fresnel diffraction EMLAB

Diffraction of half plane 24 PEC EMLAB

25 EMLAB

26 EMLAB

Fraunhofer diffraction 27 : Fourier transform EMLAB

Radiation from open-ended coaxial cable PEC 28 PMC Free space EMLAB

Babinet principle – slot antenna 29 PEC EMLAB

Complementary antenna 30 EMLAB

Aperture antenna – horn antennas 32 EMLAB

Waveguide aperture antenna 33 Waveguide slot array antenna EMLAB

34 EMLAB

TEM mode (Transverse electromagnetic mode) 35 EMLAB

Eigen-function expansions in waveguides TM modes 38 TE modes TEM modes EMLAB

Rectangular waveguide (TE mode) 39 경계 조건 EMLAB

40 Field lines for some of the lower order modes of a rectangular waveguide. Reprinted from Fields and Waves in Communication Electronics, Ramo et al, © Wiley, 1965) EMLAB

41 EMLAB

Cylindrical waveguide 42 By separation of variables EMLAB

Cylindrical waveguide mode 43 EMLAB

44 Cutoff frequencies of the first few TE and TM modes of a circular waveguide, relative to the cutoff frequency of the dominant TE 11 mode. EMLAB

Micro-strip patch antenna 45 EMLAB

Resonant modes of micro-strip patch antenna 46 EMLAB

Equivalent current distributions of resonant patch 47 EMLAB