EE540_S07_HW_1_Solution

EE540_S07_HW_1_Solution - TE Modes TE Modes m p q Frequency...

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Page 1 of 2 EE540 (S’ 07) Homework #1 Solutions Problem 7.1 Total number of modes for the given cavity between 0 and 10GHz is nearly 9 3 6 8 3 8 (10 10 ) ( ) ( ) (99 10 ) 30 3(3 10 ) N cavityvolume ! " # $ $ $ = $ $ % $ modes. (1.2) Then, in order to count the exact number of allowed modes, we have to find all the modes for the given conditions; Only the m or p index of the q p m TE , , mode may be zero, but not both, and only the q index may be zero for the TM mode. Using the following equation, we find the appropriate frequencies listed in the table below ( a = 1 cm, b = 9 cm , and d = 11 cm ) 2 / 1 2 2 2 2 2 2 ] [ 2 d q b p a m n c + + = . (1.3) When the index of reflection n=1, allowed modes by are listed as follows
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Unformatted text preview: TE Modes TE Modes m p q Frequency (GHz) m p q Frequency (GHz) 0 1 1 2.1520 3 4 7.3944 0 1 2 3.1940 2 5 7.5842 0 2 1 3.5990 4 3 7.8164 0 2 2 4.3039 1 6 8.3441 0 1 3 4.4144 5 1 8.4384 0 3 1 5.1791 3 5 8.4492 0 2 3 5.2734 4 4 8.6078 0 3 2 5.6915 5 2 8.7623 0 1 4 5.6996 2 6 8.8287 0 2 4 6.3881 5 3 9.2710 0 3 3 6.4559 4 5 9.5293 0 4 1 6.8000 3 6 9.5821 0 1 5 7.0141 1 7 9.6832 0 4 2 7.1980 5 4 9.952925 Therefore, total 28 TE modes exist between 0 and 10 GHz and there is no TM mode in our given cavity. Page 2 of 2 The graphs by exact counting and approximated formula are given below....
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EE540_S07_HW_1_Solution - TE Modes TE Modes m p q Frequency...

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