chapter 5 waveguides and resonators - university of...
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Chapter 5 Waveguides and Resonators
ECE 3317
Dr. Stuart Long
5-1
Structure that transmits electromagnetic waves in such a way that the wave intensity is limited to a finite cross-
sectional area
In this chapter we will focus on three types of waveguides:
1. Parallel-Plate Waveguides
2. Rectangular Waveguides
3. Coaxial Lines
What is a “waveguide" (or
transmission line) ?
5-2
http://cal-crete.physics.uoc.gr/VLF-sprites/images/waveguide.jpg
5-3
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Parallel Plate Waveguide
5-4
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ˆ (w>> As
)sume both
plates to be perfect conductors Assume waveguide to be very large in direction
Fi 0
eld vectors have no -dependence,
Neglect any fringing fields at and Pr
y
y
y = 0 y = w
•
•
∂• =
∂
•
•
y
λ
opagation along the direct ˆ ionz+
Parallel Plate Waveguide
5-5
y x z z
y x z z
E H H TE E 0
From Maxwell Equations
( , , ) (Transverse Electric)
or
( , , ) (Transverse Magnetic)
H E E TM H 0
=
=
(Sect. 5.1)
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Parallel Plate Waveguide
5-6
( )y x z yB.C. at and
0 Remember:
, ,
The wave equation for the case derived from Maxwell's Equations is
E H H 0 E 0
TE
y
x x a ⇒
∂=
∂
= = =
2 22
2 2
E 0
yx zω µε
∂ ∂+ + = ∂ ∂
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TE Waves in Parallel Plate Waveguides
5-7
y
y 0
2 2 2 2
For ppg. in direction
(satisfies B.C a )
with
To satisfy B.C. ( at
( is any
ˆ+
t 0
E 0)
E sin( )
zjk zx
x z
x
x
x a
m
E k x e
k k k
k a m
ω µ ε
π
− =
= =
=
+ = =
=
z
exin ceteger pt 0)
(5.5)
(5.6)
(5.7)
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TE Waves in Parallel Plate Waveguides
5-8
y 0
1 12 22 2
2
The Electric field can be expressed in another form bysubstituting in for
with
E sin
12
z
x
jk z
z
k
mE x ea
m mka a
π
π λω µ ε ω µ ε
− =
= − = −
whe e 2
r
kπ
λ =
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TE Waves in Parallel Plate Waveguides
5-9
12 2
The
guided wave propagates with the phase velocity
is imaginary for
If bec
Note:
1 12
2
or or
z
z
pz
x
k
k
m vk a
m ak ka m
ω λµ ε
πω µ ε λ
− = = −
< < >
omes imaginary, the wave will attenuate exponentially,
and the velocity for the guided wave will also become undefined.
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TE Waves in Parallel Plate Waveguides
5-10
( )m
of modeTE
frequency at which
2 2z
c
cc
k
ma
mfa
πωµ ε
ωπ µ ε
=
= = cutoff frequency
( )m
exponential attenuation when
of modeTE
becomes imaginary
2
c
f fc
am
λ
<
= cutoff wavelength wavelength at which
becomes imaginary zk
TE Waves in Parallel Plate Waveguides
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5-11
Physical Interpretation TE mode
x
z
(5.8)
0
wave trav. in and dir.
coef
ˆ
ˆ
2f jE
+ x+ z
0
wave trav. in - and dir.
coe
ˆˆ
2
ff - jE
x+ z
5-12
Physical Interpretation of Guided Waves
xmkaπ
=
aπ
2aπ
3aπ
zk
xk
(fig 5.3)
z
For this case
so modes will ppg. but for
3.5
1,2,3 4 , is imaginary .
ka
m m k
π
= ≥
12 2
2z
mk kaπ = −
(5.9)
k
M=1
M=3
M=2
5-13
( ) -y x z y 0
z y z
z 0,
can find
For B.C. at ( )
H ,E ,E H cos
E ~ H E ~sin
E
0
zjk zx
x
x a
x
H k xe
k xx
x a
mkaπ
=
=
∂∂
= =
=
⇒
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TM Waves in Parallel Plate Waveguides
5-14
( )
z
y 0
2 2 2 2
For ppg. in direction
(satisfies B.C a )
with
To satisfy B.C.
ˆ+
t 0
E 0 ( ) at
(
H cos
zjk zx
x z
x
x
x a
m
H k x e
k k k
mka
ω µ ε
π
− =
= =
=
+ = =
=
z
includin is any integer 0 )
g
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TM Waves in Parallel Plate Waveguides
5-15
y 0
12 22
2
The field solutions can be expressed in another form bysubstituting in for
wi th
H cos
12
z
x
jk z
z
k
mH x ea
m mka a
π
π λω µ ε ω µ ε
− =
= − = −
12
where 2
kπ
λ =
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TM Waves in Parallel Plate Waveguides
5-16
z y
z 0
x 0
E ~ H
E sin
E cos
since
z
z
jk zx
jk zz
x
k mH x ej a
k mH x ea
πωε
πωε
−
−
∂∂
= −
=
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TM Waves in Parallel Plate Waveguides
5-17
0
Remember that for waves we can now have .
For
For this case and are both perpendicular to the direction of propagation ( ).
TM
We refer to this case as
0
TM
ˆ
0.0
x
z
m
mk
k kω µ ε
=
=
= = =
E Hz
mode (Transverse Electroma T gneEM tic)
TM Waves in Parallel Plate Waveguides
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5-18
0
y 0
x 0
z
The field solutions for the or mode are given by
or equivalently
TM TEM
H
E
E 0
jk z
jk z
z
z
H e
H eη
−
−
=
=
=
x 0
0y
E
H
jk z
jk z
z
z
E e
E eη
−
−
=
=
TM0Mode
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(5.13a)
(5.13b)
5-19
Physical Interpretation TEM mode
x
z
0 0x=0
0
ˆ ˆ ˆ ˆ= = =
ˆ ˆ
(on bottom p
= =
late) jkz jkz
jkz
E Ee e
E e
η η
ρ ε ε
− −
−
× ×
=
s
s
J H
D E
n x y z
n x
(5.13c)
0
0
ˆ ˆ- = -
(on top plat
e )
jkz
jkzs
En e
E eη
ρ ε
−
−
=
= −
s Jx z
(5.11d)
n
n
s J
s J
for TE and TM modes has both and components.
5-20
dielectricsubstrate
ground plane
microstrip line
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Microstrip Lines
http://3.bp.blogspot.com/_QOpBRAPC2rY/TGW9OMdLRnI/AAAAAAAAAC8/2y2RGiyuRQ0/s1600/Picture2.jpg
substrate
5-21
- 00
20
1 = Re2
1 ˆ ˆ
The time-average Poynting power densit
= R
y can b
e ( )2
ˆ =2
e found b
y:
jkz jkzEE e e
E
η
η
∗
+
×
×
S E H
S
S
x y
z
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Microstrip Lines
5-22
Microstrip Lines
0
quasi TEM
-fields
k
ε ε
ω µε
=
0
ε ε>
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5-23
y
z
xa
b
Rectangular Waveguide
http://t1.gstatic.com/images?q=tbn:ANd9GcShyNLomTqQz3wl_U9PM7jxslFsPkjS4624DvinKOuUhHrSO37h
5-24
y
zxa
b
2 2
2 2
2 2 22z z z
z2 2 2
(wave equation)MAX equations HELMHOLTZ equation
6 equation like
0
0
H H 0 H Hx y z
ω µε
ω µε
ω µε
+ =
+ =
∂ ∂ ∂+ + + =
∂
⇒
∂ ∂
E E
H H
∇
∇
Rectangular Waveguide
5-25
y
zxa
b
z2 2 2
22 2 2
2 2 2 2
22
2
22
2
( , , ) ( ) ( ) ( )H X Y Z1 X 1 Y 1 ZX
For a se
Y Z
- - -
1 X -X
X X 0
parable solution assume
each term is a constant x y z
x
x
x y z x y z
x y z
k k k
kx
kx
ω µε
ω µε
=
∂ ∂ ∂+ + = −
∂ ∂ ∂
= −
∂=
∂∂
+
⇒
⇒ =∂
Rectangular Waveguide
5-26
y
zxa
b
x y x y
z z
Using Maxell's equations we can solve for the
transverse fields ( , , , )in terms of the
longtitudinal fields
E E H H
E ,H( )
Rectangular Waveguide
1 2
3 4
5 6
z
X sin cos
Y sin cos
Z
H X Y Z
x x
y y
jk z jk zz z
c k x c k x
c k y c k y
c e c e− +
= +
= +
= +
=
5-27
y
zxa
b
z z z zx y2 2 2 2
z z z zx y2 2 2 2
2 2 2 2 2
;
;
wher
E H E HE E
H E H EH H
e
z z
c c c c
z z
c c c c
c x y z
jk j jk jx y y xk k k k
jk j jk jx y y xk k k k
k k k k
ωµ ωµ
ωε ωε
ω µε
− ∂ ∂ − ∂ ∂= − = +
∂ ∂ ∂ ∂
− ∂ ∂ − ∂ ∂= + = −
∂ ∂ ∂ ∂
= + = −
Rectangular Waveguide
5-28
Rectangular Waveguide y
zxa
b
z
zz
z z
z when ; Transverse Electric mode ( )
when ; Transverse Magnetic mode ( )
Special cases
If all comp. Transverse
:
1)
E
E 0
2
l
H 0
E 0
ectric
0
E 0 0
) H
H
≠
≠
=
⇒=
= ⇒
=
=
⇒
TE
TM
Magnetic mode ( ) can not
TEMexist
5-29
Rectangular Waveguide TE:
Case (Ez=0)
y
zxa
b
zy
y 1 2 1
zx
x 3 4
tangential 0 0, y 0,
HE ~ 0 0, )
B.C.
E ~ cos sin 0
at and
Side walls (at
and
Top and bot
HE ~ 0 0, )
E ~ c
tom walls (
os si
at
n
x x x x x
y y y
x a b
x ax
mk c k x k c k x c ka
y by
k c k y k c
π
= =
∂= =
∂
− = =
∂= =
∂
−
→
⇒
E
3
5 2 4 0 z
z 0
0
H X Y Z
and
H cos
Let
cos z
y y
jk z
nk y c kb
c c c H
m x n yH ea b
π
π π −
= =
= =
=
⇒
⇒
5-30
y
zxa
b
2 2
2 22
2
2 22
-
=
Propagation constant
where
substituing in for
propagation ( real) for
ex
- -
Note
ponen
:
t
z c
c
c
z
z c
k k k
m nka b
k
m nka b
k k k
π π
π πω µε
=
+
=
>
ial attenuation ( imaginary) for z ck k k<
(5.19)
Rectangular Waveguide TE:
Case (Ez=0)
5-31
y
zxa
b
( )mn
12 2 2
of mT odE e
frequency at which
1 2 2
z
cc
k
m nfa b
ω π ππ π µ ε
= = +
cutoff frequency
2 2
exponential attenuation when
becomes imaginary
2
2 = -
c
gz c
f f
k k k
π πλ
<
= guide wav( )mn of modeTE
wavelength at which
zk
elength
mn
becomes imaginary
defines a doubly infinite set of modes TE
(5.21)
(5.20)
Rectangular Waveguide TE:
Case (Ez=0)
5-32
y
zxa
b
(5.18)
Rectangular Waveguide TM:
Case (Hz=0)
5-33
z 0
z zx y 02 2
z zx 0 y2 2
H cos
H HE = 0 E sin
H HH sin H 0
;
;
z
z
z
jk z
jk z
cc c
jk zz z z
cc c
xH ea
j j j xH ey x k ak k
jk jk jkxH ex k a yk k
π
ωµ ωµ ωµ π
π
−
−
−
=
∂ ∂= − = = −
∂ ∂
− ∂ − ∂= = = =
∂ ∂
Rectangular Waveguide TE10
mode (m=1 , n=0)
y
zxa
b
5-34
22
2
2
12 ; ; 2
2
12
z
c c c
gz
k ka
a f kaa
k
a
π
πλµε
π λλλ
= −
= = =
= = −
Note: these are the eqns. found in slide 5-32 and 5-33 with m=1 and n=0
y
z
xa
b
Rectangular Waveguide TE10
mode (m=1 , n=0)
5-35
Physical Interpretation TE10mode (m=1 , n=0)
y
zxa
b
5-36
81000000
0.275
k z
ω
10cω
11cω zk ω µε=TE10
TM11TE11
2 2z ck k k= −
(function of frequency )
where
ω versus kz graph
gz
vkω∂
=∂and
5-37
y
z
xa
b
Design of Practical Rectangular Waveguide
Example 5.4
TE10
TE01
TE20
1 12
1 12
1 22
c
c
c
fa
fb
fa
µε
µε
µε
=
=
=
1/ 22 2
c1 mπ nπ f = +
a b2π μ ε
5-38
y
zxa
b
(Cont. e.g 5.4)
for for 2 2a ab b> <
a
b ab
1 2 3
2for ab =
10TE
TE10
c
c
ff
10TE
01TE 01TE
10TE
Design of Practical Rectangular Waveguide
5-39
Time-average Poynting vector
y
zxa
b
( Example 5.3)
Total transmitted power
Design of Practical Rectangular Waveguide
5-40
20
Do not want for bandwith
Do not want small for power handling
Usual
No
case
te:
2
4
z
ab
E abkb Pωµ
>
=
ab =2
y
zxa
b
(Cont. e.g 5.3)
Design of Practical Rectangular Waveguide
5-41
TE10
TE20
TE01
6.
X-Band Waveguide (8-12 GHz)
GHz
GHz
G
55
H
13.1
1 . z4 7
c
c
c
f
f
f
=
=
=
y
zxa
b
0.4′′
0.9′′
0.4′′
a
Design of Practical Rectangular Waveguide
Historical standard design
5-42
y
z
xa
b
Rectangular Waveguide TE10
mode
z 0
y 0
x 0
H cos
E sin
H sin
z
z
z
jk z
jk z
jk zz
xH ea
j xH ea a
jk xH ea a
π
ωµ ππ
ππ
−
−
−
=
= −
=
HZ
HX
Ey
5-43
y
zxa
b
y 0
0 0
x 0
0z
or
whe e
r
E sin
H sin
H cos
z
z
z
jk z
jk zz
jk z
xE ea
jE Ha
k xE ea
E xa ej a
π
ωµπ
πωµ
ππ
ωµ
−
−
−
=
= −
−=
=−
(5.23)
Rectangular Waveguide TE10
mode
5-44
Example
Example 5.5 & 5.6
8 8
8 89
2
3 10 3
Design a Rectangul
10 <
ar waveguide with 10 GHz ( =3cm) at Mid-Band and
le
2
1 3 10 3 1010 10 2.252 2
t
1.1252
cm
c
BW
m
ab
fa a
aa a
ab b
λ =
× ×<
× ×× = + =
= =⇒
⇒
⇒
5-45
(Cont. e.g 5.5 & 5.6)
[ ]
( )[ ]
20
TE
TE
6
5max
10 7
TE 22
4
2 10
2 10
where
Vm
Vtake (safety fact
(2 10 )(4 10 )
or of 10)m
Max power handl
505
ing
.8
z
BDair
E abPZ
Zk
E
E
Zk a
ωµ
π π
π
−
=
=
= ×
= ×
× ×
⇒
Ω
=−
= Ω
Example 5-46
( ) ( )
[ ]
-1 -
2
5
max
1
2
2
2 1 2 2 1 3 4.53
1.561 156.1
note:
(2 10 ) (0.0225)(0.01125) 5.004 4(50
cm m
KW5.8)
z
z
ak
k
P
π λ πλ
− −= =
= =
×=
=
(Cont. e.g 5.5 & 5.6)
Example 5-47
Rectangular Waveguide Simulation
5-48
Coaxial Transmission Lines
b
aε
5-49
Cylindrical Coordinates
φ
z
y
x
ρ
z
ρ
φz
ˆˆ ˆ×ρ φ = z
( fig. 5.16)
5-50
Coaxial Transmission Lines
b
aε
Maxwell Eqns. V ˆ
V ˆj
z
kz
jke
eηρ
ρ
−
−⇒
=
=
0
0 H
E ρ
φ
k ω µε
µηε
=
=
( 5.48a)
( 5.48b)
( 5.49a)
( 5.49b)
5-51
The “Del” Operator in Cylindrical
Coordinates
φ
z
y
x
ρ
z
ρ
φz
ˆˆ ˆ×ρ φ = z
( fig. 5.16)
(5.44)
(5.45)
5-52
Currents on Coaxial Lines
b
aε
( 5.50a)
( 5.50b)
n sJ
H
5-53
Short-Circuited Coax
(Fig. 5.18)
( 5.52a)
0 1
0 1
1 0
for 0
both Incident Fields and Reflected Fields exist
B.C at
ˆ=
ˆ=
0 tangential 0 0
jkz jkz
jkz jkz
z
V Ve e
V Ve e
z E V Vρ
ρ ρ
ρη ρη
− +
− +
<
+
−
⇒ ⇒ ⇒= = = = −
E
H
E
ρ
φ
( 5.52b)
0z =
z
Negative sign comes from same place as in reflected H-field in Chapter 4
5-54
Short-Circuited Coax
(Fig. 5.18)
( 5.53a)( )0 0
0Similarly,
2ˆ ˆ= - sin
2ˆ= cos
jkz jkzV jVe e kz
V kz
ρ ρ
ηρ
− + − =
E
H
ρ ρ
φ ( 5.53b)
To measure standing wave pattern cut longitudinal slot in outer conductor for moveable probe.
: totally in direction so totally i
ˆˆ
Noten direction
sH Jz
φ
0z =
z
5-55
Example(Example 5.8)
20
2
Calculate the transmitted power along a coaxial line.
V ˆ
V ˆ
1 1 V V Vˆˆ ˆ= Re = R
= 2
e
2
jkz
jkz
jkz jkz
e
e
e e
ρ
ηρ
ρ ηρ ηρ
−
−
∗ −
=
=
× ×
0
0
0 0
E
H
S E H z
ρ
φ
ρ φ
20 V ln bP
aπ
η =
b
aε
TEM mode in a coaxial line
5-56
Transmission Lines
b
aε
Coaxial
two-wire (shielded
)
three-wire (shielded)
2 or more conductors TEM possible
along with TE and TM
5-57
Transmission Lines
2 or more conductors TEM possible
along with TE and TMParallel Plate
Microstrip Stripline
Conductor Dielectric
5-58
Waveguides
Elliptical
Ridged
Rectangular
Optical FiberDielectric Slab
Circular
http://optics.org
5-59
CircularElliptical
Rectangular Bends
5-60