Post on 18-Mar-2021
Field and Wave Electromagnetic
Chapter10
Waveguides and Cavity Resonators
Electromagnetic Theory 2 2
Introduction (1)
- TEM waves are not the only mode of guided waves - The three types of transmission lines (parallel-plate, two-wire, and coaxial) are not the only possible wave-guiding structur
* Waveguide
1( )
2
22( )s c
c
C LR G R
L C
R fR f
w w
α
α
π μσ
≅ + ∝
= = ∝
∴⇒
e. - Attenuation constant for loss line
Attenuation of TEM waves tends to increase monotonically with frequency prohibitively high in the microwave range.
0
Electromagnetic Theory 2 3
Introduction (2)
0
( 0, 0)z z
z z
E H
H E
= == ≠
- TEM waves :
- TM waves : transverse magnetic waves
- TE waves : transverse electric(TE) waves - single conductor wave guide : rectangular and cylindrical wave guide. - dielectric-slab waveguide : surface waves
Electromagnetic Theory 2 4
General Wave Behaviors along Uniform Guiding Structures (1)
* General wave behaviors along uniform g uiding structures- straight guiding structures with a uniform cross section.
- Assume that the waves propagate in the +z direction with a propagation constant
- For harmonic time dependence on z and t
for all field components :
jγ α β= +
( ) ( )z j t j t z z j t ze e e e eγ ω ω γ α ω β− − − −= =
Electromagnetic Theory 2 5
General Wave Behaviors along Uniform Guiding Structures (2)
0 ( )
0
( , , ; ) [ ( , ) ]
( , ) :
,
j t zE x y z t e E x y e
E x y
jwt z
ω γ
γ
−= ℜ
∂ ∂→ → − ,
∂ ∂
- For a cosine reference
where two-dimensional vector phasor
- in using a phasor representation
- In the charge-free dielectr
2
2
2 2
0
0,
( ) (xy z xy
E k E
H k H k
E E
ω με
∇ + =
∇ + = =
∇ = ∇ +∇ = ∇
2
2
2
ic region inside, Helmholtz's equations should be satisfied
where
- In Cartesian coordinates, rectangular wave guide
2
2 22) xyE E E
zγ∂
+ = ∇ +∂
2
Electromagnetic Theory 2 6
General Wave Behaviors along Uniform Guiding Structures (3)
2 2
2 2
2 2
2 2
2 2 2 2
( ) 0
( ) 0
( ) 0
( ) ( )( ) 0
( ) 0, ( ) 0
xy
xy
xy
xy x y z x y z
xy x x xy y y
E k E
H k H
E k E
xE yE zE k xE yE zE
E k E E k E
γ
γ
γ
γ
γ γ
∴ ∇ + + =
∇ + + =
∇ + + =
→ ∇ + + + + + + =
∇ + + = ∇ + + =
2
2
2
2
2 2
cf)
i.e
2 2
2 2
( ) 0xy z z
r xy
E k E
φ
γ∇ + + =
∇ ∇
2
The solution of above equations depends on the cross-sectional geometry and the boundary conditions
cf) instead of for waveguides with a circular cross section
Electromagnetic Theory 2 7
General Wave Behaviors along Uniform Guiding Structures (4)
0 00 0 0 0
0
z zy x y x
x
E j H H j E
E HE j H H j E
y y
E
ωμ ωε
γ ωμ γ ωε
γ
∇× = − ∇× =
∂ ∂+ = − + =
∂ ∂
∂− −
- Interrelationships among the six components in Cartesian coordinates
1 4
2 0 0
0 0 0
0 00 00 0
z zy x y
y yx xz z
yz
x z
y xx y z
E Hj H H j E
x x
E HE Hj H j E
x y x y
EE
x y z y z
E Ej H
x y z z xE EE E Ex
ωμ γ ωε
ωμ ωε
ωμ
∂= − − − =
∂ ∂∂ ∂∂ ∂
− = − − =∂ ∂ ∂ ∂
∂∂−
∂ ∂∂ ∂∂ ∂ ∂
= − −∂ ∂ ∂ ∂ ∂
∂ ∂−
∂
5
3 6
cf)
x
y
z
z
j H
j H
j H
y
ez
γ
ωμωμ
ωμ
γ −
⎛ ⎞⎜ ⎟⎜ ⎟ ⎛ ⎞−⎜ ⎟ ⎜ ⎟
= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎜ ⎟⎜ ⎟∂⎝ ⎠
∂→ −
∂
where and is surpressed
Electromagnetic Theory 2 8
General Wave Behaviors along Uniform Guiding Structures (5)
00 0
00 0
zy x
zx y
EE j H
y
HH j E
x
γ ωμ
γ ωε
∂+ = −
∂
∂− − =
∂
- Transverse component can be expressed in terms of longitudinal components. Ex) Combining 1 and 5
1 :
5 :
eliminati 0
00 2 0
02 0 0
0 02 2 0
0 00 2 2
2
)
1( )
y
zy x
zx y
z zx
z zx
E
Ej j E H
y
HH j E
x
E Hk H j
y x
E HH j h k
h y x
ωε γ ωε ω με
γ γ γ ωε
γ ωε γ
ωε γ
∂+ =
∂
∂− − =
∂∂ ∂
+ = −∂ ∂
∂ ∂∴ = − − + =
∂ ∂
ng from 1,5
1' :
5' :
(
where 2γ+
Electromagnetic Theory 2 9
General Wave Behaviors along Uniform Guiding Structures (6)
0 00
2
0 00
2
0 00
2
0 00 2 2 2
2
1( )
1( )
1( )
1( )
z zx
z zy
z zx
z zy
x
E HH j
h y x
E
n
HH j
h x y
H EE j
h y x
H EE j h k
h
o
y
e
x
t
ωε γ
ωε γ
ωμ γ
ωμ γ γ
∂ ∂= − − +
∂ ∂
∂ ∂= − + +
∂ ∂
∂ ∂= − +
∂ ∂
∂ ∂= − − + = +
∂ ∂
∇
- i.e
where
First, so lve 2 2
2 2
0
0
, , ,
y
xy
x y x y
E h E
H h H
H H E E
+ =
∇ + = and for longitudinal components
then find using above equation
Electromagnetic Theory 2 10
TEM Waves (1)
2
2 2
2
0, 0
0 0
0
z z
x y x y
TEM
TEM
E H
E E H H h
k
jk j
γ
γ ω με
= =
∴ = = = = =
+ =
= =
- for TEM waves
unless
- TEM waves exist only
where
or : propagation constant of a uniform plan
* TEM wave
( )
1p TEM k
ωμμε
= =
e wave
on a lossless transmission line.
- Phase velocity
Electromagnetic Theory 2 11
TEM Waves (2)
0
0
0
0
xTEM
y
TEM
yTEM
x
n
E jZ
H j
Z
EZ
te
H
o
γωμγ ωε
μ ηε
με
ΤΕΜ
ΤΕΜ
= = =
= =
= − = −
- Wave impedance from 2,4
is the same as the intrinsic impedance of the dielectric medium
1
TEM
H z EZ
∴ = ×
Electromagnetic Theory 2 12
TEM Waves (3)
B
B H
Why?
1. lines always close upon themselves
2. For TEM waves to exist, and lines would form closed loops in a t
* Single-conductor waveguides cannot support TEM waves.
c d
c
d
H dl I I
I
I
= +∫ i
ransverse plane.
3. By the Ampere's circuital law.
transverse plane
: conductor current
: displacement current
0cI =
4. Without an inner conductor
Electromagnetic Theory 2 13
TEM Waves (4)
0
0
0
z
d s
z
E
DI ds
tE
= →
∂= =
∂=
∫ i
∵
5. For TEM wave, no longitudinal displacement current
cf) in the z direction
6. Therefore there can be no closed loops of magnetic field lines in any transverse plane
7. Assuming perfect conductors, a coaxial transmission line having an inner conductor can support TEM waves
8. When the conductors have losses, no longer TEM waves
Electromagnetic Theory 2 14
TM / TE Waves (1)
solvingsolving
For TE waves For TM waves
2 0 2 0
00
2
00
2
00
2
00
2
0
.
:
:
:
:
xy z z
z
zx
zy
zx
zy
E h E
for E
EjH
h y
EjH
h x
EE
h x
EE
h y
ωε
ωε
γ
γ
∇ + =
∂=
∂
∂= −
∂∂
= −∂∂
= −∂
①
②
③
④
w ith proper
boundary conditions
0zH = 0zE =
2 2
00
2
00
2
00
2
00
2
0
.
:
:
:
:
xy z z
z
zx
zy
zx
zy
H h H
for H
HH
h x
HH
h y
HjE
h y
HjE
h x
γ
γ
ωμ
ωμ
∇ + =
∂= −
∂∂
= −∂
∂= −
∂
∂=
∂
⑤
⑥
⑦
⑧
w ith proper
boundary conditions
Electromagnetic Theory 2 15
TM / TE Waves (2)
0 0 0 02
0 0 0 0
00
0 0
ˆ ˆ( )
, , ,
)
,
T TM x y T z
x y x y
yxTM
y x
TM
E xE yE Eh
E E H H
EEZ
H H j
jcf Z
j
γ
γωε
ωμγ
γ
ω με
= + = − ∇
= = − =
≠
∵
② ③ ① ④
are given can be
determ ined from the wave impedance for the TM mode
from , and ,
for TM is not equal to
whic
1ˆ( )
TEM
TM
H z EZ
γ
∴ = ×
h is
0 0 0 02
0 0
0 0
00
0 0
ˆ ˆ( )
,
,
)
ˆ( )
ˆ( )
T TE x y T z
x y
x y
yxTE
y x
TE TM
TE
TE
H xH yH Hh
E E
H H
EE jZ
H H
cf Z Zj
E Z z H
Z H z
γ
ωμγ
γωε
= + = − ∇
= = − =
≠ =
∴ = − ×
= ×
⑥ ⑦ ⑤ ⑧
sim ilar way, can be
obtained from
from , and ,
Electromagnetic Theory 2 16
TM / TE Waves (3)
2 0 2 0 0xy z zE h E
h
h
h
∇ + =
⇒⇒
⇒
solution of
for a given boundary conditionare possible only for discretevalues of infinity of 's but solutions are not possible for all values of eigenvalues or characteri
2 2 2
2 2
2 2
h k
h k
h
γ
γ
ω με
= +
= −
= −
stic values
2 0 2 0 0xy z zH h H∇ + =
e igen va lues
Electromagnetic Theory 2 17
TM / TE Waves (4)
2 2
2
0,
:2
1 ( )
c
c
c
for
h
hf
f
fh
f
γ
ω με
π με
γ
=
=
=
= −
cutoff frequency
cf) The value of for a particular
mode in a waveguide depends on the eigenvalue of this mode
2 0 2 0 0xy z zH h H∇ + =
e igen va lues
Electromagnetic Theory 2 18
TM / TE Waves (5)
2
2
2 2
2
2
1 ( )
( ) 1
1 ( )
1 ( )
1
c
cc
c
fh
f
ff f
f
h
hj jk
k
fjk
f
k
γ
ω με γ
γ β
β
β
= −
> >
⇒ >
= = −
= −
⇒
= −
(a) or
in this range, and is imaginary
propagation mode with a phase constant
2( ) (rad/m)cf
f
Electromagnetic Theory 2 19
TM / TE Waves (6)
2
22 2
22
2 2 2 2 2
2 2 2
2 2 1
1 ( )
1 1(1 ( ) )
1 1 1 1( )
1 1 1
g
c
cc
c
g
c
g c
g c
u
k ff f
f
u
f
f
f
ff
u f
π λ πλ λ λβ με
λ
λ λ
λ λ λ λ
λ λ λ
= = > = = =−
=
= −
= − = −
∴ + =
- Guided wavelength
where
let cutoff wavelength,
then
Electromagnetic Theory 2 20
TM / TE Waves (7)
21 ( )
gp
c
uu u u
f
f
λωβ λ
= = = >−
- Phase velocity
cf) 1. Phase velocity of guided wave is faster than that of unbounded medium. 2. Phase velocity depends on frequency so that single conductor waveguides are dispersive transmission systems
Electromagnetic Theory 2 21
TM / TE Waves (8)
2
2
2 2
11 ( )
( 1 ( ) [2 1 ( ) ]
(2 )
1 (
cg
g
g p
c c
c
fu u u u
d d f
u u u
f fd k d f
f fd
d d d f
fd f
df u f
λβ ω λ
π μεβω ω π
= = − = </
∴ =
− −= =
= −
- Group velocity
cf)
2
2
)
1 1
1 ( )cu f
f
=−
Electromagnetic Theory 2 22
TM / TE Waves (9)2
2 2
2
1 ( )
1 ( ) 1 ( )
1 ( )
c
TM
c c
TE
c
fjk
fZ
j j
f f
f f
j jZ
fjk
f
γωε ωε
μ ηε
ωμ ωμγ
−= =
= − = −
= =−
-
; purely resistive and less than the intrinsic impedance of the dielectric medium
-
2 2
1
1 ( ) 1 ( )c cf f
f f
μ ηε
= =− −
; purely resistive larger than the intrinsic impedance of the dielectric medium
Electromagnetic Theory 2 23
TM / TE Waves (10)2
2
( ) 1
1 ( )
z
cc
c
z z
ff f
f
fh
f
e eγ α
γ α
− −
< <
= = −
∴ = ⇒
⇒
(b) or
: real number
wave diminishes rapidly with and is said to be evanescent waveguide : high-pass filt
2
2
1 ( )
1 ( ) ,cTM c
c
TE
fh
f h fZ j f f
j j f
jZ
γωε ωε ωε
ωμγ
−= = = − − <
⇒ ⇒
= =
er
purely reactive no power flow associated with evanescent mode
21 ( )
c
jf
hf
ωμ
− : purely reactive. no power flow.
Electromagnetic Theory 2 24
TM / TE Waves (11)
21 ( )c
u
ω β
ωωβω
−
= −
- diagram
Electromagnetic Theory 2 25
Parallel-Plate Waveguide (1)
( )
-
( 0)z
j t z
x
H
e ω γ
ε μ
−
=
1. Assuming and 2. Infinite in extent in the direction3. TM waves
4.
- Parallel plate waveguide can support TM and TE waves
as well as TEM waves
* TM waves between parallel plates
Electromagnetic Theory 2 26
Parallel-Plate Waveguide (2)0
2 02 0
2
2 2 2
0
0
( , ) ( ) -
( )( ) 0
( ) 0 0
( ) sin( )
zz z
zz
z
z n
n
E y z E y e x
d E yh E y
dy
h k
E y y y b
n y nE y A h
b bA
γ
γ
π π
−=
+ =
= +
= = =
∴ = =
(no variation along direction)
whereB.C.
at and
from
where depends on the strength of excitation of
the particular TM wave
Electromagnetic Theory 2 27
Parallel-Plate Waveguide (3)
00
2
00
2
00
2
00
2
2 2
( ) cos( )
( ) 0
( ) 0
( ) cos( )
( )
2
zx n
zy
zx
zy n
c
Ej j n yH y A
h y h b
EjH y
h x
EE y
h x
E n yE y A
h y h b
n
bn
fb
ωε ωε π
ωε
γ
γ γ π
πγ ω με
γμε
∂∴ = =
∂
∂= − =
∂∂
= − =∂∂
= − = −∂
= −
= 0 ∴ =Cutoff frequency that makes
Electromagnetic Theory 2 28
Parallel-Plate Waveguide (4)
1 1
2 2
0
1
2
2
2
0
0
c
c
c
z
f TMb
f TMb
TM f
E
με
με
=
=
==∵
cf) for mode with n=1
for mode with n=2
cf) mode is the TEM mode with
- Dominant mode of the waveguide = the mode having the lowest cutoff frequency - For parallel plate waveguides, the dominant mode is the TEM mode
Electromagnetic Theory 2 29
Ex. 10-3(1)
ˆ ˆ ˆ ˆˆ ˆ( )x y z
x y z
dl xdx ydy zdz kE k xE yE zE
dx dy dzk
E E E
= + + = = + +
= = = ⇒
cf) Field line : the direction of the field in space
i.e.
field line
Electromagnetic Theory 2 30
Ex. 10-3(2)
1
( , ; 0)
( , ; 0)
( , ;0) cos( )sin
0
y
z
x
E y z tdyy z
dz E y x t
b yH y z A z
by y b
ωε π βπ
=∴ − =
=−
=
− = =
1
in the plane
For TM mode at t=0,
At and - There are surface currents because of a discontinuity in the tangential magnetic field. - There are surface charges because of the presence of a normal electric field
Electromagnetic Theory 2 31
Ex. 10-4 (1)
/ /11( , ) sin( ) ( )
2j z j y b j y b
z
AyE y z A e e e
b jβ π ππ − −= = −
1(a) A propagating TM wave = the superposition of two plane
waves bouncing back and forth obliquely between the two conducting platesproof>
( / ) ( / )1 [ ]2
j z
j z y b j z y b
e
Ae e
j
β
β π β π
−
− − − += −1 2
Electromagnetic Theory 2 32
Ex. 10-4 (2)
z y
bz y
πβ
+ −
+ +
1 Term : A plane wave propagating obliquely in the and
directions with phase constants and
2 Term : A plane wave propagating obliquely in the and directions w
0 0 0 0
0 0
ˆ
ˆ ˆˆ ˆsin cos sin cos
ˆ ˆsin cos
ˆ ˆˆ ˆcos sin cos sin
x
i i i i i r r i r i
i i i i
i i i r i i
H xH
E yE zE E yE zE
yE zE
y z y z
θ θ θ θθ θ
β β θ β θ β β θ β θ1 1 1 1
= −
= − = − −= +
= + = − +
ith the same phase constants
Electromagnetic Theory 2 33
Ex. 10-4 (3)1 1 1cos cos sin
0
2 2 2 2
( , ) cos ( )
sin , cos
( ) ( )
cos 22
11,
2 2
0
i i ij y j y j zz i i
i i
i
i
E y z E e e e
b
b b
bb b
uf
b b
β θ β θ β θθπβ θ β β θ
π πβ β ω με
π λθ λβ
λλ με
θ
− −
1 1
1
1
= −
∴ = =
= − = −
= = ⇒ ≤
= = = ⇒
= ⇒
solution exists only for
at cutoff frequency
then waves bounce ba
2
-
-
2 .
cos sin 1 ( )
c c
c ci i
c g p
y
z
b f f
f fu
f u f
λ λ
λ λθ θλ λ
⇒ < = >
= = = = = −
1
ck and forth in the direction
and no propagation in the direction TM mode propagates only when or
Electromagnetic Theory 2 34
TE Waves between Parallel Plates (1)
2 02 0
2
0
00
2
0
0
0, 0
( )( ) 0
( , ) ( )
0
( )0 0
( ) cos( )
z
zz
zz z
zx
z
z n
Ex
d H yh H y
dy
H y z H y e
HjE
h y
dH yy y b
dy
n yH y B
b
γ
ωμ
π
−
∂= =
∂
∴ + =
=
∂− = − =
∂
= = =
∴ =
* TE waves
We note that
B.C.
i.e at and
Electromagnetic Theory 2 35
TE Waves between Parallel Plates (2)
00
2
00
2
00
2
00
2
2 2 2 2
( ) 0 ( 0)
( ) sin( )
( ) sin( )
( ) 0( 0)
( )
z Zx
zy n
zx n
z Zy
H HH y
h x x
H n yH y B
h y h b
Hj j n yE y B
h y h b
H HjE y
h x x
nh k
b
γ
γ γ π
ωμ ωμ π
ωμ
πγ ω με
∂ ∂∴ = − = =
∂ ∂∂
= − =∂
∂= − =
∂
∂ ∂= = =
∂ ∂
⇒ = − = − ⇒
⇒
∵
∵
the same as that for TM waves
The cutoff frequency is0, 0 0y xn H E⇒ = = =
the sameFor and
Electromagnetic Theory 2 36
TE Waves between Parallel Plates (3)
0
01 10
01 10
)
)
)
cf TM TEM
cf TM TM
cf TM TM
=
0i.e, TE mode doesn't exist
or does not exist
or does exist
for the rectangular waveguide
00( , ) sin( )sin( )z
m x m yE x y E
a b
π π=
00( , ) cos( )cos( )z
m x m yH x y H
a b
π π=
Electromagnetic Theory 2 37
Energy-transport Velocity (1)
⇒⇒
* Energy-transport velocity - Wave guide high pass filter - Broadband signal 1. low frequency components may be below cutoff 2. high frequency components will travel widely different velocity - Energy-transport velocity : veloc
( )(m/s)
( )
z aven
av
z av avs
Pu
W
P P ds
=′
= ∫ i
ity at which energy propagates along a waveguide
: the time average power
Electromagnetic Theory 2 38
Energy-transport Velocity (2)
2
[( ) ( ) ]
1 ( )
[
(
av e av m avs
cen
W w w ds
fu u
f
w
′ = +
= −
∫ : the time average stored
energy per unit length
H.W] prove that
*
*
) ( )4
( ) ( )4
e av
m av
e E E
w e H H
ε
μ
= ℜ
= ℜ
i
i
Electromagnetic Theory 2 39
Energy-transport Velocity (3)2
2 2 22
* 2
22 2 2
2 20
2 22 2
2
2 2 2 2 22 20
0
2 220
( ) [sin ( ) cos ( )]4
( ) ( )
( ) [1 ]8 8
( ) ( ) cos ( )4
( ) ( )8 8
ˆ( )
cos (
e av n
b
e av n n
m av n
b
m av n n
b
z av av
b
n
n y n yw A
b h b
E E j j
b bw dy A k A
h h
n yw A
h bb b
w dy A k Ah h
P P zdy
Ah
ε π β π
β β β
ε β ε
μ ω ε π
μ εω ε
ωεβ
= +
⇒ − =
= + =
=
= =
=
=2
∫
∫
∫
∫
i i
i
cf)
22
)4 n
n y bdy A
b h
π ωεβ=
Electromagnetic Theory 2 40
Energy-transport Velocity (4)*
0 0* 0 0*
0 0*
2 22
22
1) ( )
21
ˆˆ( )2
1( ) ( )
2
cos ( )
( ) 1 ( )
av
y x z x
av y x
n
cen
cf P e E H
e zE H yE H
P z e E H
n yA
h b
fu u
k k k f
ωεβ π
ωβ ω β
= ℜ ×
= ℜ − +
= − ℜ
=2
= = = −
i
Electromagnetic Theory 2 41
Attenuation in Parallel-plate Waveguides (1)
( )
d c
LP z
α α α= +
=
* Attenuation in parallel-plate waveguide - Losses are very small -
For TEM mode cf) For a lossy transmission line the time-average power loss per unit length
22 2 2 20
02
0
2* 20
02
0
1[ ( ) ( ) ] ( )
2 2
1( ) [ ( ) ( )]
2 2
z
z
VI z R V z G R G Z e
Z
VP z e V z I z R e
Z
α
α
−
−
+ = +
= ℜ =
Dielectric losses
Ohmic losses
Electromagnetic Theory 2 42
Attenuation in Parallel-plate Waveguides (2)
2
00
0 0 0
0
( )( ) 2 ( )
( ) 1( )
2 ( ) 2
(2 2 2
L
L
d
P zP z P z
zP z
R G ZP z R
GR Z R
Gbwhere
bR
α
α
σ μ σα ηεωσ
ηω
∂− = =
∂
∴ = = +
∴ = = =
⎛ =⎜⎜⎜ =⎜⎝
∵ for low loss conductor)
independent of frequency
Electromagnetic Theory 2 43
Attenuation in Parallel-plate Waveguides (3)
0
0
1
2
2
( )
cc
c
c
d c
d
R ff
R b
b bR
fR
f f
j
π εασ
μηω ω ε
π μω σ
ασε εω
∴ = = ∝
= =
=
>
= +
cf)
For TM mode to find dielectric losses, at
-
Electromagnetic Theory 2 44
Attenuation in Parallel-plate Waveguides (4)
2 2 1/ 2
2 2 2 2 1 1/ 2
2 2 2 2 1
2 2
[ (1 ) ( ) ]
( ) {1 ( ) ] }
( ) {1 ( ) ] }2
( )
j nj
b
n nj j
b b
n j nj
b bn
b
σ πγ ω μεωεπ πω με ωμσ ω με
π ωμσ πω με ω με
πωμσ ω με
−
−
= − −
= − − [ −
≅ − − [ −
− Assumption that
Electromagnetic Theory 2 45
Attenuation in Parallel-plate Waveguides (5)
2 2 2
2
2
2
2
( ) 1 ( )
1 ( )
11 ( )
21 ( )
c
c
c
cd
c
nf
b
n
b
f
f
fj j
ff
f
π π με
ωπω με ω μεω
ω με
σ μγ α β ω μεε
=
⇒ − = −
= −
∴ = + = + −−
For cutoff frequency
Electromagnetic Theory 2 46
Attenuation in Parallel-plate Waveguides (6)
2
2
0 0 *
0
2 2 2
0
2 1 ( )
1 ( )
( )
2 ( )
1( ) ( )( )
2
( ) cos ( ) ( )2
d
c
c
c
Lc
b
y x
bn n
f
f
f
f
P z
P z
P z w E H dy
bA bAw n ydy w b
n b n
σηα
β ω με
α
α
ωεβ π ωεβπ π
= ⇒−
= −
=
= −
= =2
∫
∫
We obtain decreases when frequency increases
and
To find
Electromagnetic Theory 2 47
Attenuation in Parallel-plate Waveguides (7)20
2 0 0
2
2
1( ) 2 ( )
2
( ) ( 0)
( ) 22
2 ( )1 ( )
2 1
( )[1 ( ) ]
L
L
sz s
n ns sz x
sc s
c
cs
c
c cc
c c c
P z w J R
bA bAw R J H y
n nP z R
RP z b f
bf
fR
f
b f f
f f
ωε ωεπ π
ωεαβ
η
π μσ
πμαη σ
=
= = = =
∴ = = =−
=
∴ =−
where
Electromagnetic Theory 2 48
Attenuation in Parallel-plate Waveguides (8)
0 0 *
0
2 2 2
0
20
20 2
22
2 2
:
1: ( ) ( )( )
2
( ) sin ( ) ( )2
1( ) 2 ( )
2
( 0)
2 2( )( )
2 ( )1 ( )
L
d
b
c x y
bn n
sx s
z z n s
s s cLc
c
P z w E H dy
bB bBw n ydy w b
n b n
P z w J R
w H y R wB R
R R fP z n
P z b b fbf
f
α
α
ωμβ π ωμβπ π
παωμβ
η
=
= =2
=
= = =
∴ = = =−
⇒
∫
∫
TE modes the same as TM
decreases monotonically as frequency increases
Electromagnetic Theory 2 49
Attenuation in Parallel-plate Waveguides (9)
Electromagnetic Theory 2 50
Homework
H.W10-2, 10-4, 10-5, 10-8, 10-9, 10-11, 10-14