Microwave theory, April 7, 2015
Anders Karlsson, [email protected]
Electrical and information technology
Microwave theory, April 7, 2015
Hollow waveguides
Γ
z
n
S
Ω
Three type of waves:
I
TE-waves⇒ Ez = 0, Hz = w(ρ)eikz z
I
TM-waves⇒ Hz = 0, Ez = v(ρ)eikz z
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TEM-waves⇒ Ez = 0 and Hz = 0, E = −∇φ(ρ)eikz z
(TEM only in waveguides with at least two conductors)
Anders Karlsson, Electrical and information technology
Hollow waveguides
Eigenvalue problem TE-waves
∇2 w(ρ) + kt2 w(ρ) = 0, ρ ∈ Ω
∂w(ρ)
= 0, ρ ∈ Γ
∂n
2 and eigenfunctions w (ρ)
Eigenvalues ktn
n
Anders Karlsson, Electrical and information technology
Hollow waveguides
Eigenvalue problem TM-waves
∇2 v(ρ) + kt2 v(ρ) = 0, ρ ∈ Ω
v(ρ) = 0, ρ ∈ Γ
2 and eigenfunctions v (ρ)
Eigenvalues ktn
n
Anders Karlsson, Electrical and information technology
Rectangular waveguide TE-modes
y
b
x
a
TE-waves ⇒
Eigenvalue problem
∇2 w(ρ) + kt2 w(ρ) = 0, ρ ∈ Ω
∂w(ρ)
= 0, on all four sides
∂n
Anders Karlsson, Electrical and information technology
Rectangular waveguide TE-modes
y
b
x
a
2
Eigenvalues: ktmn
=
mπ 2
+
nπ 2
a
b
nπy mπx Eigenfunctions: wmn (ρ) = Amn cos
cos
a
b
m = 0, 1, 2 . . ., n = 0, 1, 2 . . .
but (m, n) 6= (0, 0)
Anders Karlsson, Electrical and information technology
Rectangular waveguide
The TE10 mode. Fundamental mode!
Anders Karlsson, Electrical and information technology
Rectangular waveguide
The TE20 mode
Anders Karlsson, Electrical and information technology
Rectangular waveguide
The TE01 mode
Anders Karlsson, Electrical and information technology
Rectangular waveguide
The TE21 mode
Anders Karlsson, Electrical and information technology
Cut-off frequencies
eikz z p
2 .
kz = k 2 − ktmn
1. k < ktmn =⇒ kz imaginary =⇒ non-propagating mode
2. k = ktmn =⇒ kz = 0 =⇒ standing wave, cut-off frequency fc
3. k > ktmn =⇒ kz real =⇒ propagating mode
Anders Karlsson, Electrical and information technology
Today
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TM-wave in rectangular waveguides
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COMSOL
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Dispersion, phase- and group speeds
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The fundamental mode TE10 .
Anders Karlsson, Electrical and information technology