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Unformatted text preview: ECE 329 Spring 2010 Homework 9  Solution Due: Mar. 30, 2010 1. a) For plane waves propagating in free space, E and H are related by E = η H × ˆ β, where ˆ β is the unit vector in the wave propagation direction. Therefore, H 1 = 1 η ( ˆ y ) × E 1 = 1 η [ ˆ x 2cos( ω t + β y )] ( A/m ) . Wave #1 is a travelling wave propagating towards the ˆ y direction. The power per square meter is simply the Poynting vector:  S  =  E × H  = E × 1 η ˆ β × E = 1 η  E  2 ( W/m 2 ) . Therefore,  S 1  = 1 η  E 1  2 = 1 η 4cos 2 ( ω t + β y ) = 4 η cos 2 ( ω t + β y ) ( W/m 2 ) . The phasors are ˜ E 1 = ˆ z 2 e jβy ( V/m ) . ˜ H 1 = 1 η ˆ x 2 e jβy ( A/m ) . The timeaverage power cross a 1 m 2 area can be calculated as < P > = 1 T ˆ T  S  dt. By using  S 1  , we can obtain < P 1 > = 2 η ( W ) . To determine polarization, we notice that ˜ E 1 only has ˆ z component, so Wave #1 is linearly polarized and the unit vector in the polarization direction is ˆ z . b) H 2 = 1 η ˆ y × E 2 = 1 η [ ˆ z 10cos( ω t β y ) ˆ x 10sin( ω t β y )] ( A/m ) . Wave #2 is a travelling wave propagating towards the ˆ y direction.  S 2  = 1 η  E 2  2 = 1 η 100cos 2 ( ω t β y ) + 100sin 2 ( ω t β y ) = 100 η ( W/m 2 ) . ˜ E 2 = ˆ x 10 e jβy ˆ z 10 e jβy j π 2 ( V/m ) . 1 ECE 329 Spring 2010 ˜ H 2 = 1 η ˆ z 10 e jβy ˆ x 10 e jβy j π 2 ( A/m ) . < P 2 > = 100 η ( W ) . ˜ E z ˜ E x = 10 e j π 2 10 = j, Refer to Page 4 of Lecture 24, since the propagation direction is +ˆ y, we can see that the rotating direction of electric eld and the direction of propagation satisfy righthand rule. Thus, Wave #2 is righthand circularly polarized. c) For plane waves propagating in free space, E and H are also related by H = 1 η ˆ β × E . Then we have E 3 = η H 3 × ( ˆ z ) = η h ˆ y cos ω t + β z + π 3 ˆ x sin ω t + β z π 6 i ( V/m ) . Wave #3 is a travelling wave propagating towards the ˆ z direction. Since  S  =  E × H  = η H × ˆ β × H = η  H  2 ( W/m 2 ) , we have  S 3  = η  H 3  2 = η h cos 2 ω t + β z + π 3 + sin 2 ω t + β z π 6 i = 2 η cos 2 ω...
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This note was uploaded on 04/20/2010 for the course ECE ECE 329 taught by Professor Goddart during the Spring '10 term at University of Illinois at Urbana–Champaign.
 Spring '10
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