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Demo12Dishes - ECE 3030 Electromagnetic Fields and Waves...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 12 INSTRUCTOR NOTES: Reflectors and 2- and 3-Dimensional Arrays Reminder: Prelim 3 on Thursday night. Date: Tuesday 11/17 1. Prelim 3 Thursday night, November 19...no Workshop 12. 2. Problem 17.8 A short dipole of physical length d , pointing in the z -direction, and carrying a current pha- sor I , is located at ( h, h, 0) in the ( x, y, z ) coordinate system, as shown in the figure to the right. The space for which x < 0 or y < 0 is filled with a perfect con- ductor. a. Find the expression for the far-field electric field vector of the radiation and explain your result. b. Find an exact expression for the gain G ( θ, φ ) of the antenna. What value of h (in terms of the wavelength λ ) will give the maximum value for the gain and in which directions ( θ, φ ) does this maximum value of Gain occur? c. Sketch the radiation pattern p ( θ, φ ) in the x - y plane assuming that the distance h equals λ . y x h h = σ Figure 1: A short dipole with a corner reflector. Solution: Pattern from a corner reflector. a. Note that there are three image dipoles in this problem. (Draw the figure.) The far field only exits for x > 0 and y > 0 with ~ E ff ( ~ r ) = { Element Factor } × { Array Factor } The two factors are { Element Factor } = j kId eff 4 πr e - jkr sin θ { Array Factor } = e jk ˆ r · ~ h 1 - e jk ˆ r · ~ h 2 + e jk ˆ r · ~ h 3 - e jk ˆ r · ~ h 4 where the effective length of the dipole and its images is d eff = d 2 , and the signs account for the current reversals of the images. We need the dot products, but first make sure the class understands the pieces. The unit vector ˆ r = ~ r /r is ˆ
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