Chapter.5.fall09.ver3

Chapter.5.fall09.ver3 - Chapter 5 Cylindrical Cavities and...

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Chapter 5 Cylindrical Cavities and Waveguides We shall consider an electromagnetic field propagating inside a hollow (in the present case cylindrical) conductor. There are no sources inside the conductor, but we shall assume the material is isotropic with electric permittivity , and magnetic permeability, . The speed of the propagating wave is 1 /  The direction of propagation will be along the cylindrical axis which is the z ̂ direction We shall assume that E ( r , t )= E ( r ) e i t and B ( r , t B ( r ) e i t . Maxwell’s equations give: [∇ 2  2 t 2 ] E ( r . t 0 [∇ 2  2 t 2 ] B ( r . t 0 [∇ 2 +  2 ] B ( r 0 [∇ 2 +  2 ] E ( r 0 5.1 5.2 5.3 5.4 Since the wave is propagating along the z ̂ direction we shall further assume that: ∇ × E = i B ( r ) ; ∇ × B =− i  E ( r ) . 5.5 5.6 Since the wave is propagating along the z ̂ direction we shall further assume that: E ( r E ( x , y ) e ± ikz B ( r B ( x , y ) e ± ikz 5.7a 5.7b Thus Eq. (5.3 and 5.4) become [∇ t 2 +  2 k 2 ] E ( x , y 0 [∇ t 2 +  2 k 2 ] B ( x , y 0 5.8b 5.8a where
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t 2 = 2 x 2 + 2 y 2 t = x ̂ x + ŷ y The expressions in Eqs. 5.5 and 5.6 then become; ∇ × E =[ z ̂ z + ∇ t ]×[ z ̂ E z + E t ]= i B ( r ) where E t = E z ̂ E z =( z ̂ × E z ̂ B t = B z ̂ B z z ̂ × B z ̂ Then ∇ × E = z ̂ z × E t z ̂ × ∇ t E z + ∇ t × E t = i ( B t + z ̂ B z ) ∇ × B = z ̂ z × B t z ̂ × ∇ t B z + ∇ t × B t =− i  ( E t + z ̂ E z ) Thus, z ̂ ×∇ z E t z ̂ × ∇ t E z i z ̂ ×( z ̂ × B ) z E t + i ( z ̂ × B t )= t E z z ̂ (∇ t × E t i B z z B t i  ( z ̂ × E t t B z z ̂ (∇ t × B t )=− i  E z 5.9 5.10 5.11 5.12 Also, t E t +∇ z E z = 0 t B t z B z = 0 5.13a 5.13b Finally, one can solve for E t and B t if E z and B z are known (and not both are zero). ik E t = t E z i ( z ̂ × B t ) ik B t = t B z + i  ( z ̂ × E t ) ik ( z ̂ × B t )=( z ̂ × ∇ t B z )+ i  ( z ̂ ×( z ̂ × E t )) and ik E t = t E z −( i / ik )( z ̂ × ∇ t B z )+( 2  / ik )(− E t ) ( 2  k 2 ) E t = ik t E z i ( z ̂ × ∇ t B z ) E t = i ( 2  k 2 ) 1 [ k t E z ( z ̂ × ∇ t B z )] likewise B t = i ( 2  k 2 ) 1 [ k t B z +  ( z ̂ × ∇ t E z )] 5.14a 5.14b For waves in the opposite direction change k to k .
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Transverse electromagnetic wave (TEM): E z and B z are zero everywhere inside cylinder . For TEM waves E TEM = E t : t × E TEM = 0 t E TEM = 0 k = k 0 =  B TEM ( z ̂ × E TEM ) 5.15a 5.15b 5.15c 5.15d Unfortunately, the TEM wave is not supported by a single hollow cylindrical conductor (with infinite conductivity). The surface must be an equipotential surface and inside such a conductor, the electric field vanishes. One needs two or more cylindrical surfaces (such as a coaxial cable) to support a TEM wave.
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Chapter.5.fall09.ver3 - Chapter 5 Cylindrical Cavities and...

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