Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 25 ECE 6340 Intermediate EM Waves 1

Image Theory Vertical electric dipole (VED) over an infinite ground plane PEC (E, H)(E, H) Dipole z E = 0 H = 0 h 2

Boundary Conditions (2) Boundary condition on S : E t = 0 everywhere on S. (1) Source condition inside S : A single VED exists inside S. PEC E t = 0 ( r =  ) E t = 0 (PEC) S S  S = S c + S  V S cS c h Any solution that satisfies these two conditions must be correct inside S. 3

Image Picture Hence, at z = 0, Also, z = 0 Original dipole Image dipole h h No ground plane is present in the image problem. 4

Image Picture (cont.) Hence, the image solution for z > 0 is the same as for the original problem. For z < 0 the image solution is not valid because the source in the image problem in this region is not the same as the original source in this region (which has zero sources). 5

Vertical Magnetic Dipole PEC For a single electric dipole: For a single magnetic dipole: Duality: 6

Vertical Magnetic Dipole (cont.) Therefore At z = 0 : For a single magnetic dipole: EoEo EiEi Image picture: so but Hence 7

All Possible Cases The PMC cases can be obtained from the PEC cases by duality. PECPMC 8

Example: Monopole Antenna Gap voltage model V M (0) I (z) h - V M (0) 9

Example: Monopole Antenna Magnetic frill model Gap voltage model V M (0) I (z) + -   V M (0)

Monopole Antenna (cont.) V D (0) The image picture is a physical dipole antenna: I (z) + -     The fields are symmetric about z = 0, and satisfy the correct boundary conditions on the entire wire. 11

Monopole Antenna (cont.) Coaxial frill model At z = 0: x y a b x y + - V M (0) I (z) 12 We close off the ground plane an out a magnetic surface current (“frill”) where the aperture used to be.

Monopole Antenna (cont.) Coaxial frill model This is an accurate model of the coax feed for modeling purposes (often used in antenna theory). + - V M (0)   I (z) 13

Monopole Antenna (cont.) After image theory:   I (z) 14 Electric Field Integral Equation (EFIE): The factor of 2 is from image theory.

Corner Reflector S Corner reflector: Image picture: 15

Corner Reflector (cont.) Also, we can have this orientation: S 16

Dipole in a Rectangular Waveguide a b x y 17