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# Class Note for ECE 6341 with Professor Jackson at UH 2

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This 23 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 44 Summary cont In this set of notes we use the SDI theory from the last set of notes to solve the classical quotSommerfeld problemquot of a vertical dipole over an semiinfinite earth qg Summary cont Planar vertical electric current J fltxygt6ltzgt For a vertical electric dipole of amplitude 11 we have fxy5x5y Example Sommerfeld Problem Vertical dipole over the earth Find E2 on the surface of the earth in the air region Z l 1 220 90 91 Z 805 Sommerfeld Problem cont TEN The vertical electric dipole excites TMZ waves only Sommerfeld Problem cont From Notes 40 x y 9 080 We use the quotMichalskiquot TM Z 20 VTM Z Zo VSTM normalized current function k TM The v subscript indicates a 1V source Iv Z ZOHw 0 We need to calculate the Michalski normalized current function at z 0 Sommerfeld Problem cont 2 1V 21W k 25M 20 km leg k 080 Z1TM kZI kzl 12 1 Sommerfeld Problem cont Z Z0 ll ZfM 20quot 1V ll 25M gtZ This figure shows how to calculate the Michalski normalized current function it will be calculated later Sommerfeld Problem cont Return to the calculation of the field Ezkxky0 1 k k TM 0 II 080 I v 3Z0 080 Hencewehave Ezxy0 1 T j 1 1915M 0zo11 kt ejkxxkyydkxdky 080 10 Sommerfeld Problem cont J Note VTM 020 is only a function of kt 1 I 2 802 2 j j 13M 020 e jquotxquotquotyy k3 dkxdky OO 00 E2 xy0 Change to polar coordinates kx kt c035 ky kt sin xpcos dkx dky gt kt dkt d5 y psm 5 Sommerfeld Problem cont EZ xjyjo I 1 j J J IVTM 020 e jkxXkyy k dkxdky my weer E x 1 of m k3 2f jktcos 7pcos ktsin 7psin z 9y 2 2 mgby 0 v 9Z0 t 0 6 EL EZ X 1 O Z0 J ejkp cos 7 0 80 2 0 Sommerfeld Problem cont 0 27 Ezxy0 1 I13MOzokfj eJrkzmwsw mw 0 0 use 27 27r 27 J jktp cos J jktp cosor d J jktp cosor d 0 0 We see that the vertical field of the vertical electric dipole should not vary with angle Integral identity 1 I lm 50 da 2 27rJ0 kip 0 12 Sommer eld Problem cont Hence we have Sommerfeld Problem cont We now return to the calculation of the Michalski normalized current function gt Z A rm 2 OltZltZ0 VVTM Z3Z02Ae jkzOZ Bejkzoz B I I zTM ZTM Bekzoz FTMe jkzoz 2 Z 25M 14 Sommerfeld Problem cont TM 39k ZgtZo Vv ZZOCeJZOZ Sommerfeld Problem cont D 1 am 20W lt12 IV 1 25M Z Boundary conditions VVTM Z32Z0VVTM ZSaZo1 VTM 2320 2 VTM 2620 Sommerfeld Problem cont Hence we have VVTM Z32Z0VVTM Z62Z021 Ce szozo Beszozo FTMeJkZOZO 1 TM TM I ZOZO I 2020 6 1 2on eszozo FTMe J39kzozo 0 0 Sommerfeld Problem cont Substitute the first of these into the second one Ce szozo 1 BeijOZO FTMeJkZOZO Substitute into the equation below C e J39kzozo B TM TM Z0 Z0 eszozo FTMe szozo This gives us TM Z0 18 Sommenfeld Problem cont 1 TM Z0 B ZOTM i1 1 B 6jkzozo B ekzozo cancels B Ze kzozo 1 B 64172020 FTMeJkZOZO eijOZO FTMe szozo H Hence we have Sommerfeld Problem cont For the current we then have B 13M 220 ekzoz FTMe szOZ TM Z0 Hence Forz0 20 Sommerfeld Problem cont We thus have 1 080 2 j 10 kt p 15M 020 k3 dkt 0 Sommerfeld Problem cont E2 p0 wgo2 10 k p 2 km 1FTM6 E E2 p90 wlg0 J0 kipi1 FTM eJ39kzozo Sommerfeldl Problem cont Final Result I 80 23

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