Chapter 5 Waveguides and Resonators - University of...
Transcript of Chapter 5 Waveguides and Resonators - University of...
5-2
What is a “waveguide”(or transmission line)?
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:
Parallel-Plate Waveguides.
Rectangular Waveguides.
Coaxial Lines.
5-5
Assume both plates to be perfect conductors.
Assume waveguide to be very large in direction (w>>λ) .
Field vectors have no y-dependence .
Neglect any fringing fields at y=0 and y=w .
Propagation along the direction.
y
ˆ+z
0y
∂=
∂
Parallel Plate Waveguide
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5-6
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
=
=
Parallel Plate Waveguide
(Sect. 5.1)
5-7
TE Waves in Parallel Plate Waveguides
( )y x z yB.C. at and
The wave
Remem
equation for the case derived from Maxwell's Equations is
T
r: 0
E
be
E H H 0 E 0
, ,
y
x x a ⇒
∂=
∂
= = =
2 22
2 2
E 0
yx zω με
⎛ ⎞∂ ∂+ + =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
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5-8
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)
TE Waves in Parallel Plate Waveguides
(5.5)
(5.6)
(5.7)
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5-9
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 =⎛ ⎞⎜ ⎟⎝ ⎠
TE Waves in Parallel Plate Waveguides
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5-10
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.
TE Waves in Parallel Plate Waveguides
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5-11
( )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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5-12
Physical Interpretation TE mode
x
z
y 0
0y
E sin
E2
z
x z x z
jk z
jk x jk z jk x jk z
mE x ea
jE e e
π −
− − −
⎛ ⎞= ⎜ ⎟⎝ ⎠
⎡ ⎤= −⎣ ⎦(5.8)
0
wave trav. in and dir.
coef
ˆ
ˆ
2f jE⎛ ⎞
⎜ ⎟⎝ ⎠
+ x+ z
0
wave trav. in - and dir.
coe
ˆˆ
2ff - jE⎛ ⎞
⎜ ⎟⎝ ⎠
x+ z
5-13
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
5-14
( ) -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
=
=
∂∂
= =
=
⇒
TM Waves in Parallel Plate Waveguides
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5-15
( )
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
includinis any integer 0 )
g
TM Waves in Parallel Plate Waveguides
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5-16
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 =
⎥
⎛ ⎞⎜ ⎟⎝ ⎠
TM Waves in Parallel Plate Waveguides
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5-17
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
−
−
∂∂
⎛ ⎞= − ⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
TM Waves in Parallel Plate Waveguides
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5-18
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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5-19
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
−
−
=
=
TM0 mode
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(5.13a)
(5.13b)
5-20
Physical Interpretation TEM mode
x
z
0 0x =0
0
ˆ ˆ ˆ ˆ= = =
ˆ ˆ
(on bottom plate)
=
=
jkz jkz
jkz
E Ee e
E e
η η
ρ ε ε
− −
−
× ×
=
s
s
J H
D Ei i
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
5-21
Microstrip Lines
ground plane
dielectricsubstrate
microstrip line
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5-22
Microstrip Lines
- 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
5-23
Microstrip Lines
0
quasi TEM
-fields
k
ε ε
ω με
= 0
ε ε>
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5-25
Rectangular Waveguide
2 2
2 2
2 2 22z z z
z2 2 2
0
MAX EQN HELMHOLTZ EQN (wave equation)
6 EQNS like
0
H H 0 H H x y z
ω με
ω με
ω με
⇒
+ =
+ =
∂ ∂ ∂+ + + =
∂ ∂ ∂
E E
H H
∇
∇
5-26
Rectangular Waveguide
2 2 22
z 2 2 2
2 2 2 2
2 22 2
2 2
( , , ) ( ) ( ) ( )1 X 1 Y 1 ZH X Y ZX
For a se
Y Z
- - -
1 X X- X 0
parable solution assume
each term is a constant
X
x y z
x x
x y z x y zx y z
k k k
k kx x
ω με
ω με
∂ ∂ ∂= + + = −
∂ ∂ ∂
= −
∂ ∂= + =
∂
⇒
⇒∂
5-27
Rectangular Waveguide
1 2
3 4
5 6
z
x y
X sin cos
Y sin cos
Z
H X Y
Using Max. eqns. can solve for transverse fields (E ,
Z
E
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− +
= +
= +
= +
=
x y
z z
, , )
in terms of longtitudinal on
H H
(Ee ,Hs )
5-28
Rectangular Waveguide
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
− ∂ ∂ − ∂ ∂= − = +
∂ ∂ ∂ ∂
− ∂ ∂ − ∂ ∂= + = −
∂ ∂ ∂ ∂
= + = −
5-29
Rectangular Waveguide
z z
z z
z z
Special cases
1) when ; Transverse Electric mode ( )
2) when ; Transverse Magnetic mode ( )
If all comp. Transverse Electric
E 0 H 0 TE
H 0 E 0 TM
E H 0M
0
= ≠
= ≠
= = =
⇒
⇒
⇒agnetic mode ( )
can not TEM
exist
5-30
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 −
= =
= =
=
⇒
⇒
5-31y
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<
Rectangular Waveguide TE-Case (Ez=0)
(5.19)
5-32y
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( )mnof modeTE
wavelength at which
zk
elength
mn
becomes imaginary
defines a doubly infinite set of modes TE
Rectangular Waveguide TE-Case (Ez=0)
(5.21)
(5.20)
5-33y
zxa
b
z 0
11
10
Same , , , as
For ; and lowest order (smallest ) is
The lowest ord
E sin sin
TE
TM m 0 n 0 TM
er of all is called for rectangular wavegui
Ed
Tes
zjk z
z c c g
c
m x n yE ea b
k k f
f
π π
λ
−=
≠ ≠
DOMINANT MODE ( possible)
(there exist a frequency range were only it can propaTE
go M n
ate)
Rectangular Waveguide TM-Case (Hz=0)
(5.18)
5-34
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 x jkH ex k a yk k
π
ωμ ωμ ωμ π
π
−
−
−
=
∂ ∂= − = = −
∂ ∂
− ∂ − ∂= = = =
∂ ∂
Rectangular Waveguide TE10 mode (m=1 , n=0)
y
zxa
b
5-35
Rectangular Waveguide TE10 mode (m=1 , n=0)
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
zxa
b
5-37
8 1 0 0 0 0 0 0
0 .2 7 5
k z
ω
10cω
11cω
θ
zk ω με=TE10
TM11TE11
2 2z ck k k= −
tanpz
vkω θ= = (function of frequency )
where
ω versus kz graph
gz
vkω∂
=∂
and
5-38y
zxa
b
Design of Practical Rectangular Waveguide
(Example 5.4)
TE10
TE01
TE20
12 2 21
2
1 12
1 12
1 22
c
c
c
c
m nfa b
fa
fb
fa
π ππ μ ε
με
με
με
⎡ ⎤⎛ ⎞ ⎛ ⎞= +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
⎛ ⎞= ⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
⎛ ⎞= ⎜ ⎟⎝ ⎠
5-39y
zxa
b
Design of Practical Rectangular Waveguide
(Cont. e.g 5.4)
for for 2 2a ab b> <
a
b ab
1 2 3
2for ab =
10TE11
11
TETM
01
20
TETM
TE10
c
c
ff
10TE
01TE
20TM
01TE
10TE 20TM
5-40
[ ]10
220
0 0
20
TE
1 = Re2
ˆ
Transmitting Power in wavegui
= sin
de
2
ˆ
mode
4
z
a b
z
E k xa
P dxdy
E abkP
πωμ
ωμ
∗⎡ ⎤×⎣ ⎦
⎛ ⎞⎜ ⎟⎝ ⎠
=
=
∫ ∫ i
S E H
S
S
z
z
Time-average Poynting vector
y
zxa
b
Design of Practical Rectangular Waveguide
( Example 5.3)
Total transmitted power
5-41
204
zE abkPωμ
=
: Do not want for bandwith
Do not want small for power handling
Usual case
Not
2
2
e ab
b
ab
>
y
zxa
b
Design of Practical Rectangular Waveguide
(Cont. e.g 5.3)
5-42
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
0.9a ′′=
0.4b ′′=
a
Design of Practical Rectangular Waveguide
5-43y
zxa
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
−
−
−
=
= −
=
5-44y
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
5-45
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 =
× ×<
⎡ ⎤× ×× = + =⎢
⎣
= =⇒
⎦
⇒
⎥ ⇒
5-46
(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
5-47
( ) ( )
[ ]
-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
5-50
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
( ) ( )1 1 zA A A
zρ φρ ρ
ρ ρ ρ φ
∂ ∂ ∂= + +
∂ ∂ ∂Ai∇
( )1 1ˆˆ ˆ- z zAA A AA A
z zφφ ρ ρρ
ρ φ ρ ρ ρ φ
⎛ ⎞∂∂ ∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂⎜ ⎟× − + + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠
A = z∇ ρ φ
φ
z
y
x
ρ
z
ρ
ˆφz
ˆˆ ˆ×ρ φ = z
( f i g . 5 .1 6 )
(5.44)
(5.45)
5-52
Currents on Coaxial Lines
b
aε
( 5.50a)
( 5.50b)
=a0 0
=a
20
0 =a
ˆˆ ˆ = =
2
ˆ
I =
= jkz jkz
jkz
V Ve ea
Va e
ρρ
π
ρ
ηρ η
πφη
− −
−
×
∂ =
×
∫
s
sJ
HJ zn ρ φ
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
5-54
Short-Circuited Coax
(Fig. 5.18)( 5.53a) ( )0 0
0
2ˆ ˆ= - sin
Similarl
2ˆ= c
y
os
jkz jkzV jVe e kz
V kz
ρ ρ
ηρ
− + ⎛ ⎞− = ⎜ ⎟
⎝ ⎠ E
H
ρ ρ
φ( 5.53b)
0z =
z
To measure standing wave pattern cut longitudinal slot in outer conductor for moveable proble.
: totally in direction so totally i
ˆ
ˆNoten direction
sH Jz
φ
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
2 or more conductors , (TEM possible ) (along with TE and TM) Coaxial
two-wire (shielded)
5-57
Transmission Lines
2 or more conductors , (TEM possible ) (along with TE and TM) Parallel
Plate
microstrip
www.amanogawa.com/transmission.html
stripline
Conductor
Dielectric