Page 396 - Electromagnetics

We also have
˜
2
2
˜
∇ ψ(ρ,ω) + k ψ(ρ,ω) = 0, (5.136)
t c
2
2
where k c = k −k is called the cutoff wavenumber. The solution to this equation depends
z
on the geometry of the waveguide cross-section and whether the field is TE or TM.
The fields may be computed from the Hertzian potentials using u = z in (5.122)–
(5.123) and (5.124)–(5.125). Because the fields all contain the common term e ∓ jk z z ,we
˜
define the field quantities ˜ e and h through
˜
˜
˜
E(r,ω) = ˜ e(ρ,ω)e ∓ jk z z , H(r,ω) = h(ρ,ω)e ∓ jk z z .
˜
˜
Then, substituting e = ψ e e ∓ jk z z , we have for TM fields
˜
2
˜
˜ e =∓ jk z ∇ t ψ e + ˆ zk ψ e ,
c
˜
˜
c
h =− jω˜ ˆ z ×∇ t ψ e .
˜
˜
Because we have a simple relationship between the transverse parts of E and H,wemay
also write the fields as
˜
2
˜ e z = k ψ e , (5.137)
c
˜
˜ e t =∓ jk z ∇ t ψ e , (5.138)
˜
h t =±Y e (ˆ z × ˜ e t ). (5.139)
Here
ω˜ c
Y e =
k z
˜
˜
is the complex TM wave admittance. For TE fields we have with h = ψ h e ∓ jk z z
˜
˜ e = jω ˜µˆ z ×∇ t ψ h ,
˜
˜
˜
2
h =∓ jk z ∇ t ψ h + ˆ zk ψ h ,
c
or
˜
2
˜
h z = k ψ h , (5.140)
c
˜
˜
h t =∓ jk z ∇ t ψ h , (5.141)
˜
˜ e t =∓Z h (ˆ z × h t ). (5.142)
Here
ω ˜µ
Z h =
k z
is the TM wave impedance.
Modal solutions for the transverse field dependence. Equation (5.136) describes
the transverse behavior of the waveguide fields. When coupled with an appropriate
boundary condition, this homogeneous equation has an infinite spectrum of discrete so-
lutions called eigenmodes or simply modes. Each mode has associated with it a real
eigenvalue k c that is dependent on the cross-sectional shape of the waveguide, but inde-
pendent of frequency and homogeneous material parameters. We number the modes so
that k c = k cn for the nth mode. The amplitude of each modal solution depends on the
excitation source within the waveguide.
© 2001 by CRC Press LLC