This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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) = Id 3 ~ r- h 2 z z ~ J 2 ( ~ r) = Id e j 3 ~ r + h 2 z z Assuming A = 1, = , and h = , find the radi- ation pattern in the z- x plane. Sketch p ( , = 0) and indicate the angular location of all the nulls in the radiation pattern. Hint: Make sure you do not miss the sin 2 term that is due to the an- gular dependence of the radiation patterns of the individual dipoles. x z 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 kI 1 d 4 r sin e jkr e jk r ~ h 1 + I 2 I 1 e jk r ~ h 2 Make sure the students understand where this comes from...probably you should go back to the vector potential for one Hertzian dipole: ~ A( ~ r ) = o Id 4 r e- jk ( r- r ~ h) noting that this is in the same direction as the current, and yields a magnetic field in the direction (via ~ ~ A) and an electric field in the direction ( to ~ H and ~ r. Remember that r = sin cos x + sin sin y + cos z and that r, , and form an orthogonal set. In the far-field: ~ E( ~ r ) = j kId 4 r sin e- jkr e jk r ~ h ~ H( ~ r ) = j kId 4 r sin e- jkr e jk r ~ h Be careful with the unit vector in the dot product r ~ h. Note | E | / | H | = , and that ~ H ~ E ~ r....
View Full Document