Page 397 - Electromagnetics

The appropriate boundary conditions can be found by employing the condition that
˜
for both TM and TE fields the tangential component of E must be zero on the waveguide
˜
walls: ˆ n × E = 0, where ˆ n is the unit inward normal to the waveguide wall. For TM
˜
fields we have E z = 0 and thus
˜
ψ e (ρ,ω) = 0, ρ ∈ , (5.143)
˜
where is the contour describing the waveguide boundary. For TE fields we have ˆ n×E t =
0,or
˜
ˆ n × (ˆ z ×∇ t ψ h ) = 0.
Using
˜ ˜ ˜
ˆ n × (ˆ z ×∇ t ψ h ) = ˆ z(ˆ n ·∇ t ψ h ) − (ˆ n · ˆ z)∇ t ψ h
and noting that ˆ n · ˆ z = 0, we have the boundary condition
˜
∂ψ h (ρ,ω)
˜
ˆ n ·∇ t ψ h (ρ,ω) = = 0, ρ ∈ . (5.144)
∂n
The wave nature of the waveguide fields. We have seen that all waveguide field
2
components, for both TE and TM modes, vary as e ∓ jk zn z . Here k 2 = k − k 2 is the
zn cn
propagation constant of the nth mode. Letting
k z = β − jα

we thus have
˜ ˜
E, H ∼ e ∓ jβz ∓αz .
e
For z > d we choose the minus sign so that we have a wave propagating away from the
source; for z < −d we choose the plus sign.
When the guide is filled with a good dielectric we may assume ˜µ = µ is real and
independent of frequency and use (4.254) to showthat
!
2 2 2

k z = β − jα = ω µ − k − jω µ tan δ c
c

!
tan δ c
2
= µ ω − ω 2 1 − j
c 2
1 − (ω c /ω)
where δ c is the loss tangent (4.253) and where
k c
ω c = √
µ
is called the cutoff frequency. Under the condition
tan δ c
1 (5.145)
1 − (ω c /ω) 2
we may approximate the square root using the first two terms of the binomial series to
showthat

!
1 tan δ c
2
β − jα ≈ µ ω − ω 2 1 − j . (5.146)
c 2
2 1 − (ω c /ω)
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