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HW2Solution

# HW2Solution - ECE 201A Homework 2 solution 1"" ˆ...

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Unformatted text preview: ECE 201A Homework 2 solution 1. ! " ! "- ˆ ˆ ˆ ˆ + 2 j y j y x z x z E a ja e a ja e # # \$ % & ! (a) ! " ! "- 1 ˆ ˆ ˆ ˆ + 2 j y j y z x z x H a ja e a ja e # # ’ ( ) \$ & % % * + ! (b) E ! and H ! fields propagating in +y direction are ! "- ˆ ˆ j y x z E a ja e # \$ % ! and ! "- 1 ˆ ˆ j y z x H a ja e # ’ ( ) \$ & % * + ! . Hence the time average power density is ! " ! " ! " ! " * 2 ˆ 1 1 1 1 1 ˆ ˆ ˆ ˆ ˆ Re Re Re 1 1 W/m 2 2 2 y j y j y y av x z z x y a S E H a ja e a ja e a # # ’ ’ ’ & % % ,- ,- ( ) \$ . \$ % . & & \$ % \$ / / * + 1 2 1 2 ! ! ! (c) E ! and H ! fields propagating in -y direction are ! " ˆ ˆ 2 j y x z E a ja e # \$ & ! and ! " 1 ˆ ˆ 2 j y z x H a ja e # ’ ( ) \$ % * + ! . Hence the time average power density is ! " ! " ! " ! " * 2 1 1 1 1 1 5 ˆ ˆ ˆ ˆ ˆ ˆ Re Re 2 2 Re 4 1 W/m 2 2 2 2 j y j y y av x z z x y y S E H a ja e a ja e a a # # ’ ’ ’ & & ,- ,- ( ) \$ . \$ & . & \$ & % \$ & / / * + 1 2 1 2 ! ! ! (d) Net time average power density is 2 1 5 3 ˆ ˆ 1 =- W/m 2 2 y y av av av y y S S S a a ’ ’ % & ,- \$ % \$ & / 1 2 ! ! ! . (e) The part propagating in +y direction has two mutually orthogonal components with equal amplitude and 90 out of phase. So by definition this part is circularly polarized. The part propagating in - y direction has two mutually orthogonal components with unequal amplitude and 90 out of phase. So this part is elliptically polarized. 2. ! " ! " 3 ˆ ˆ 10 sin x y E a ja z # & \$ % ! (a) This is a perfect standing wave. We can express the same electric field as ! " 3 3 3 ˆ ˆ ˆ ˆ ˆ ˆ 10 10 10 2 2 2 j z j z x y x y j z j z x y a ja a ja e e E a ja e e j j j # # # # & & & & & % % ,- ,- ,- & \$ % \$ & / / / 1 2 1 2 1 2 ! . As is seen this is the superposition of two waves of equal amplitude traveling in opposite directions. (b) The electric field of the wave traveling in +z direction is 3 ˆ ˆ 10 2 x y j z a ja E e j # & & % ,- \$ & / 1 2 ! and the electric field of the wave traveling in –z directions is 3 ˆ ˆ 10 2 x y j z a ja E e j # & % ,- \$ / 1 2 ! . (c) The magnetic field of the wave traveling in +z direction is 3 ˆ ˆ 10 2 y x j z a ja H e j # ’ & & & ,- \$ & / 1 2 ! and the magnetic field of the wave traveling in –z directions is 3 ˆ ˆ 10 2 y x j z a ja H e j # ’ & & % ,- \$ / 1 2 !...
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HW2Solution - ECE 201A Homework 2 solution 1"" ˆ...

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