Demo11Hertzian

# Demo11Hertzian - ECE 3030 Electromagnetic Fields and Waves...

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Unformatted text preview: ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 11 INSTRUCTOR NOTES: Hertzian Dipole Antennas Reminder: Date: Tuesday 11/10 1. Problem 17.2 The two Hertzian dipoles shown have current densities specified by the phasors: ~ J 1 ( ~ r) = Idδ 3 ~ r- h 2 ˆ z ¶ ˆ z ~ J 2 ( ~ r) = Id e jα δ 3 ~ r + h 2 ˆ z ¶ ˆ z Assuming A = 1, α = π , and h = λ , find the radi- ation pattern in the z- x plane. Sketch p ( θ,φ = 0) and indicate the angular location of all the nulls in the radiation pattern. Hint: Make sure you do not miss the sin 2 θ term that is due to the an- gular dependence of the radiation patterns of the individual dipoles. x z θ h J 1 J 2 Figure 1: Another two Hertzian dipole array. Solution: The general far-field formula for 2 Hertzian z-directed dipoles (from Slide 9 of Lecture 32) is ~ E ff ( ~ r ) = j η kI 1 d 4 πr sin θ e jkr • e jk ˆ r · ~ h 1 + I 2 I 1 e jk ˆ r · ~ h 2 ‚ ˆ θ Make sure the students understand where this comes from...probably you should go back to the vector potential for one Hertzian dipole: ~ A( ~ r ) = μ o Id 4 πr e- jk ( r- ˆ r · ~ h) noting that this is in the same direction as the current, and yields a magnetic field in the ˆ φ direction (via ~ ∇ × ~ A) and an electric field in the ˆ θ direction ( ⊥ to ~ H and ~ r. Remember that ˆ r = sin θ cos φ ˆx + sin θ sin φ ˆy + cos θ ˆ z and that ˆ r, ˆ θ , and ˆ φ form an orthogonal set. In the far-field: ~ E( ~ r ) = j η kId 4 πr sin θ e- jkr e jk ˆ r · ~ h ˆ θ ~ H( ~ r ) = j kId 4 πr sin θ e- jkr e jk ˆ r · ~ h ˆ φ • Be careful with the unit vector in the dot product ˆ r · ~ h. • Note | E | / | H | = η , and that ~ H ⊥ ~ E ⊥ ~ r....
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## This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell.

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Demo11Hertzian - ECE 3030 Electromagnetic Fields and Waves...

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