This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 34 Resonant modes and field fluctuations • Since in a rectangular cavity the resonant frequencies f mnl = c 2 ( m a ) 2 + ( n b ) 2 + ( l d ) 2 ⇒ 2 f mnl c = ( m a ) 2 + ( n b ) 2 + ( l d ) 2 , we can consider 2 f mnl /c to be the “length” of a “vector” ( m a , n b , l d ) point ing away from the origin of a “3D Cartesian space” where each lattice point , e.g., ( m a , n b , l d ) = ( 1 a , 2 b , 1 d ) , is associated with two resonant modes (TE and TM) of the cavity. – In this space, “volume” per lattice point is 1 abd , and thus volume per resonant mode is 1 / 2 abd . – Also, all the resonant modes with resonance frequencies f mnl less than a given frequency f can be associated with lattice points residing within one eight (an octant) of a sphere of “radius” 2 f/c centered about the origin of the same space — only an octant is involved since the indices m , n , l employed are all nonnegative. Thus, the number of resonant modes with frequencies less than f , to be denoted as the cumulative distribution C ( f ) , is found to be C ( f ) = 1 8 × ( sphere of radius 2 f/c ) 1 / 2 abd = 1 8 × 4 π 3 ( 2 f c ) 3 1 / 2 abd = 8 πf 3 3 c 3 V 1 where V = abd is the physical volume of the cavity. Consequently, the number density N ( f ) of the available resonant modes in a cavity of volume V is obtained as N ( f ) = dC df = 8 πf 2 c 3 V modes Hz which grows quadratically with frequency f . As illustrated later in this lecture, the distribution N ( f ) has deep theoretical implications. Example 1: Consider a rectangular cavity with dimensions a = b = d = 0 . 3 m. Determine N ( f ) for f = 50 GHz and the number of resonant modes to be found within a bandwidth of Δ f = 1 GHz centered about f = 50 GHz. Solution: Using the density function derived above, we find that N (50 × 10 9 ) = 8 π (50 × 10 9 ) 2 (3 × 10 8 ) 3 (3 × 10 1 ) 3 = 8 π × 25 × 10 7 = 2 π × 10 5 modes Hz ....
View
Full
Document
This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Staff
 Electromagnet

Click to edit the document details