Lecture Slides on Equivalence Principle - Intermediate Electromagnetic Waves | ECE 6340, Study notes of Electrical and Electronics Engineering

Download Lecture Slides on Equivalence Principle - Intermediate Electromagnetic Waves | ECE 6340 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Prof. David R. Jackson Dept. of ECE Fall 2008 Notes 26 ECE 6340 Intermediate EM Waves 1 Equivalence Principle Basic idea: We can replace the actual sources in a region by equivalent sources at the boundary. Keep original fields E , H outside S. Put zero fields (and no sources) inside S. E , H a b ANT S E , H 2 Outside S, these sources radiate the same fields as the original antenna, and produce zero fields inside S. This is justified by the uniqueness theorem: Hence e s e s J n H M n E = × = − × Equivalent sources: zero fields s sJ e sMzero sources S Equivalence Principle (cont.) Maxwell's equations are satisfied along with boundary conditions at the interface. 5 Note about materials: If there are zero fields throughout a region, it doesn’t matter what material is placed there (or removed). zero fields s sJ e sM rε (E , H) Equivalence Principle (cont.) 6 Scattering by a PEC sE (E , H) source iE 0tE = PEC S sJ i sE E E= + i sH H H= + 7 Example (cont.) source 0 0,ε μ sJ (E , H) (0 ,0) [ ] 0it tsE J E+ = [ ] it tsE J E= − This integral equation has to be solved numerically. Integral equation for the unknown current so 0 t s i tE E+ = “Electric Field Integral Equation (EFIE)” 10 Example: Scattering by Dielectric Body sE (E , H) source ,r rε μiE i sE E E= + i s E E = = incident field scattered field (E , H) 11 Exterior Equivalence Replace body by free space (since material doesn’t matter in zero-field region). S source ,r rε μ Ea = E Ha = H Eb = 0 Hb = 0 ( ) ( ) 0 0 e a s e a s J n H M n E + + = × − = − × − S source 0 0,ε μ e sJ e sM (E , H) (0 ,0) n 12 Summary for Interior ,r rε μ S Original problem: S e sJ +− e sM +− (0 ,0) n Eb, Hb ,r rε μ ,r rε μ Homogeneous- medium problem: 15 Integral Equation ,r rε μ S , , , , e e i e e t s t t ss s e e i e e t s t t ss s E J M E E J M H J M H H J M + + + − − − + + + − − − ⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤+ =⎣ ⎦ ⎣ ⎦ , , , , e e e e i t s t s ts s e e e e i t s t s ts s E J M E J M E H J M H J M H + + + − + + + + + − + + ⎡ ⎤ ⎡ ⎤+ = −⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤+ = −⎣ ⎦ ⎣ ⎦ Boundary conditions: Hence: The – means calculate the fields inside the surface, assuming an infinite dielectric region. Note: The + means calculate the fields outside the surface, radiated by the sources in free space. “PMCHWT" Integral Equation* Poggio-Miller-Chang-Harrington-Wu-Tsai * Poggio-Miller-Chang-Harrington-Wu-Tsai 16 Fields in a Half Space Equivalent sources: e s e s J z H M z E = × = − × (0 ,0) e s e s J M (E , H) sources z (E , H) region of interest (z > 0) 17 Fields in a Half Space: Summary sources z (E , H) region of interest (z > 0) incorrect fields (E , H) sM correct fields 0 0( , )ε μ 2sM z E= − × 20 Alternative (better when H is known on the interface) PMC e sJ Mse does not radiate on PMC, and is therefore not included. image theory: Incorrect fields (E ,H) sJ correct fields0 0( , )ε μ 2 2es sJ J z H= = × Fields in a Half Space (cont.) 21 Example: Radiation from Waveguide z b y 0( , ,0) cos xE x y y E a π⎛ ⎞= ⎜ ⎟ ⎝ ⎠ b a x y 22