```WAVEGUIDES AND SYSTEMS
Parallel-plate waveguide: TE case
Plane wave interference satisfies boundary conditions
x̂
kz
θi
ŷ
θi
ko
E
kz = kosin θi
λo
x
ẑ
ko
(
E
k o = ω μo εo
λx = 2π/kx = λo/cosθi
Null lines
λo
kx = kocos θi
x
ẑ
vp = vo/sinθi
vg = vosin θi
λz
= λo/sinθi
)
E = yˆ Eo e jk x x − jk z z − Eo e− jk x x − jk zz = yˆ 2jEo sink x x ⋅ e− jk z z ⇒ k x = m π
a
∂E y ⎞
∇ ×E
1 ⎛ ∂E y
ˆ
ˆ
H=−
=−
−x
⎜z
⎟
jωμ
jωμ ⎝ ∂x
∂z ⎠
=
x=a
⇒ m
E
2Eo
(xˆ ⋅ k z sink x x − zˆ ⋅ jk x cosk x x)e− jk zz
jωμ
x=0
λx
=a
2
m=3
L16-1
TE10 WAVEGUIDE MODE
Add Sidewalls to TE1 Parallel Plate Waveguide ⇒ TE10:
E
Surface charge σs
x̂
a
-
σs = n̂ ⋅ εE
+
+
-
Power flow
b
ŷ
ẑ
H
Surface current Js
Js = nˆ × H
Antennas can be
coupled through
holes or slots; slots at
x̂
a
b
ŷ
H
Sidewall slots that
don’t block current
L16-2
MODE PATTERNS - TE
TE modes:
( )
π x ⋅ e − jk zz
E = y2jE
sin
m
ˆ
o
a
()
2
− k 2x
() ( )
ω
c
=
2
− mπ
a
H(t, x,z)
ω3,0 = 3πc/a
E(t, x,z)
ŷ
S
⇒
a=λx/2
ω2,0 = 2πc/a
Slope = vg
ω1,0 = πc/a
Slope = vp
Slope = c
0
⇒
kz
v phase = ω
kz
TE2 mode:
S
2
= 0 for 'cutoff frequency' ωmn
ω
TE1 mode:
x̂
kz =
ω
c
a=λx
ẑ
v group = dω → 0 at ωco = m πc
dk z
a
ω < ωco → k z = jα
Evanescent
L16-3
RECTANGULAR WAVEGUIDE MODES
a x̂
ẑ
ẑ x̂
ŷ
b
TE11
b
TE10
a
a
E
H
ŷ
x̂
E
b
a
b
H
ŷ
x̂
H E
ŷ
E
H
ẑ
ẑ
ŷ
H
x̂
E
kz =
a
b
() ( ) ( )
ω
c
2
− mπ
a
ω
a
TM11
No TEoo,TMoo,
TMmo,TMon
x̂
ẑ
ŷ
b
E
TE20
H
TM11TE11
TE01
ω20
ω11
ω01
TE10 ω10
ẑ
2
− nπ
b
2
TE(M)mn : a < b
Dispersion Relations
slope=vg=dω/dkz
k z = ω με
slope=vp =ω/kz
kz
L16-4
RECTANGULAR WAVEGUIDE DESIGN
Modes and Dimensions:
TEm0 modes:
a = mλx/2
b
E
Em0 = yˆ ⋅ Eo 2jsink x x e − jk zz
kz =
(k o2 − k x 2 )
=
a
ωo2
2
− ( mπ )2 = 0 if ωo = mcπ
a
a
c
TEmn modes: kx = mπ/a, ky = nπ/b ⇒ k z = k − k − k =
2
o
Cutoff Frequencies:
Want b ≤ a/2 so that
f01 ≥ f20 = c/a (f01 = c/2b)
fm0 =
mc
2a
m 4
3
; f0n =
2
x
2
y
x
( ) ( )
ωo2
mπ
−
a
c2
2
− nπ
b
2
nc
2b
Evanescence
Single propagating mode
2
(if a ≥ 2b) [TE10]
1
(Two modes would interfere,
0
producing nulls in frequency)
a = λ10/2 ⇐ c/2a = f10
Propagation
TE20 TM20
TE10
f
2c/2a = f20 ⇒ a = λ10
c/2b = f01
L16-5
CAVITY RESONATOR DESIGN
Derivation of
resonant frequencies fmnp
z
b
x
y a
d≥a≥b
TEmn waveguide modes:
λy
ωo2
λx
mπ
2
2
2
a=m , b=n
⇒ k z = ko − k x − k y =
−
2
2
a
c2
( ) ( )
2
− nπ
b
2
Shortest side Longest side
TEmnp resonator modes: a = mλx/2, b = nλy/2, d = pλz/2
pπ
⇒ k z = k o2 − k 2x − k 2y = 2π =
⇒
λz
d
⇒ ωmnp = c
( maπ ) + ( nbπ )
2
2
2
2
ωmnp
c
2
⎛ pπ ⎞
+ ⎜ ⎟ ⇒ fmnp
⎝ d ⎠
( ) ( )
⎛ pc ⎞
= ( mc ) + ( nc ) + ⎜ ⎟
2a
2b
⎝ 2d ⎠
− mπ
a
2
− nπ
b
2
2
2
⎛ pπ ⎞
−⎜ ⎟ = 0
⎝ d ⎠
2
2
L16-6
CAVITY RESONATOR PERTURBATION
Work = force • distance = Δw
Fe
[N/m2]
<⎯fe >
= ρsE/2 (attractive)
= εE2/2; <Fe> = εοE2/4 = <We>[J/m3]
⎯E
Fm [N/m2] =⎯Js ×⎯Hμo/2(repulsive)
Cross-section
= μH2/2; <Fm> = μoH2/4 = <Wm>[J/m3]
At resonance <we> = <wm> = wT/2
wT = n hf, so ΔwT = nhΔf
Δf [Hz] = ΔwT/nh = f (ΔwT)/wT
Δf [Hz] = f •(Δwm – Δwe)/wT
Thus pushing in the wall where Wm
dominates does work and raises
fmnp; where We dominates fmnp drops
<⎯fm >
⎯H
<⎯fe >
<We>, <Wm>
L16-7
WAVEGUIDE SYSTEMS
Physics of Matching:
Choose waveguide that propagates only one mode at f.
Structural discontinuities cause reflections; tuning cancels them.
Given some reflecting obstacle:
Cancel reactance
with
[= f(freq.)] near obstacle
Cancel reflection with offset
[magnitude and phase (Δ)]
Δ
Example:
Matches coupler
Waveguide coupler
Horn transition
Matches transition
Horn aperture
L16-8
WAVEGUIDES AND SYSTEMS
Parallel-plate waveguide: TE case
Plane wave interference satisfies boundary conditions
kz
θi
θi
(
kz = kosin θi
λo
ko
x
ẑ
ko
x̂
E
E
k o = ω μo εo
λx = 2π/kx = λo/cosθi
Null lines
λo
kx = kocos θi
x
ẑ
vp = vo/sinθi
vg = vosin θi
λz
= λo/sinθi
)
E = yˆ Eo e jk x x − jk z z − Eo e− jk x x − jk zz = yˆ 2jEo sink x x ⋅ e− jk z z ⇒ k x = m π
a
∂E y ⎞
∇ ×E
1 ⎛ ∂E y
ˆ
ˆ
H=−
=−
−x
⎜z
⎟
jωμ
jωμ ⎝ ∂x
∂z ⎠
=
x=a
⇒ m
E
2Eo
(xˆ ⋅ k z sink x x − zˆ ⋅ jk x cosk x x)e− jk zz
jωμ
x=0
λx
=a
2
m=3
L16-9
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
```