TEST BANK FOR Antennas for all Applications By John D. Kraus Converted

Show that the directivity D of an antenna may be written ( ) ( ) ( ) ( ) ∫∫ Ω = ∗ ∗ π θ φ θ φ π θ φ θ φ 4 2 max max 2 , , 4 1 , , r d Z E E r Z E E D Solution: av U D U max (θ ,φ ) = , 2 max max U(θ ,φ ) = S(θ ,φ ) r , = ∫∫ Ω π θ φ π 4 ( , ) 4 U 1 U d av U(θ ,φ ) = S(θ ,φ )r2 , ( ) ( ) Z S E E θ φ θ φ (θ ,φ ) , , ∗ = Therefore ( ) ( ) ( ) ( ) ∫∫ Ω = ∗ ∗ π θ φ θ φ π θ φ θ φ 4 2 m...