450fall09hw14_Sol - ECE 450 Fall 2009 1. Problem #1...

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ECE 450 Fall 2009 Homework 14 Solutions Due: Thursday, Dec 10 th , 2009 1. Problem #1 a. Check the f c relation derived from the characteristic equations: 1 ,0 , 1 , 2 2c o s m c c mv fm d θ == (Could also use 22 12 , 1 , 2 2 m c mc dn n ) Æ 0 0 c f = , OK, since 0 c f f > TE 0 mode propagation as indicated Æ 1 c f d 1 o s n 1 c c = , what is c ? 11 2 1 sin 18.44 10 cc n n θθ −− ⎛⎞ ⎜⎟ ⎝⎠ 1 sin = ° Thus, () 1 c f 8 3 10 2.5 10 3 2 c 18.44 × ×° 10 os 20 GHz and since 1 c f f NO PROPAGATION FOR TE 1 mode…and for TE 2 , TE 3 , etc. b. Essentially we want to determine min max ii i for which TIR will occur Æ For guidance to occur, ic , this sets the lower bound , a loose inequality. Æ Is there a max i ? That is, as i increases does it allow for higher order mode propagation? No, for a set f , increasing i does not allow for higher order mode propagation. In that case , physically 90 ci ≤° c. Use , 1 , 2 2 m c mc Æ 0 0 c f = : 0 c f f > YES 1
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Æ from before: 1 c 1 20 GHz c f = f f > YES Æ () 2 23 22 . 5 c f × 8 3 1 0 10 11 0 × = 40 GHz: 2 c = f f > NO, does not propagate (standing) Therefore only 2 modes propagate d. Intuition should tell you that if you were using a metallic parallel/ rectangular waveguide once can decrease the dimensions of the guide to make the higher order modes “more” evanescent. Recall, c f mn ab π Here utilize ⎛⎞ + ⎜⎟ ⎝⎠ = 1 mc 12 n 2c o s c mv d θ == 2 dn m c f , the idea is to make m c f larger so higher order modes will not propagate at 40 GHz (push them away from f ) * Aim for the TE 1 mode, the rest will follow suit. Define 1 c f be the cutoff frequency of the TE 1 mode in the modified waveguide. Thus compared to the unmodified guide: 1 1 20 GHz 1 2 40 GHz c c f f , need this ratio to be less than or equal to ½ i.e., 1 1 1 2 c c f f ** Proposed method #1: Vary the thickness of the dielectric slab, hold the medium parameters constant (i.e., 1.25 m 2 2 2 dd d c dn n c nn d ≤⇒ m) Æ Make the modified thickness less than or equal to one-half the original distance will only allow the TE 0 mode to propagate at 40 GHz. ** Proposed method #2: Vary the medium properties, hold the thickness of the dielectric 2
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slab constant. In particular vary the quantity 22 12 nn , better yet keep n 2 fixed (air) for convenience.
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450fall09hw14_Sol - ECE 450 Fall 2009 1. Problem #1...

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