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General Field Expression inside a Waveguide
( ) zeyx = ,~EE
EE 22
2
=
z
Then
Helmholtzs equations:
022 =+ EE k
=k
022 =+ HH k
Transverse directions: (x,y) or (r, )
Longitudinal direction: z (propagation direction)
In general
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Method of Solution:
Express transverse field componentsEx,Ey in
terms of longitudinal field componentEz
Obtain solution for the longitudinal field
Ezfrom the wave equation
ObtainEx,Ey fromEz
Step 1
Step 2
Step 3
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In rectangular coordinates:
022
2
2
2
2
2
=
+
+
+
Ekzyx
022
22 =
+
+ Ek
z
xy
0222 =++ EE kxy
0222 =++ Ekxy
Similarly,
0222 =++ HH kxy
(1a)
(1b)
022 =+ EE k
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HE j=
H
zyx
j
EEE zyx zyx
=
xyz
x
yz HjE
y
EHj
z
E
y
E ~~~
=+
=
yz
xyzx Hj
x
EEHj
x
E
z
E ~~
~ =
=
zxy
zxy Hj
y
E
x
EHj
y
E
x
E ~~~
=
=
(2a)
(2b)
(2c)
Note that:( ) ( )
( ) ( )
zyxi
eyxHzyxH
eyxEzyxE
z
ii
z
ii
,,
,~
,,
,~,,
=
=
=
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Similarly from
EH j=
xyz EjH
y
H ~~~
=+
(3a)
yz
x Ejx
HH
~~~=
zxy Ej
yH
xH ~
~~
=
(3b)
(3c)
We have
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Finally from equation sets in (2) and (3), we have:
+
=y
Ej
x
H
kH zzx
~~1~
22 (4a)
+
+
=x
Ej
y
H
kH zzy
~~1~22
(4b)
+
+= yH
jx
E
kE zzx
~~1~
22 (4c)
+
=
x
Hj
y
E
k
E zzy
~~1~
22 (4d)
Hence, we can solve the scalar Helmholtzs equations forEzand
Hz, and use the above formulas to determine the other components.
~~
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Waveguide Mode Classification
It is convenient to first classify waveguide modes as to
whetherEz
orHz
exists according to:
TEM: Ez= 0 Hz= 0
TE: Ez= 0 Hz 0TM: Ez 0 Hz= 0
TEM = Transverse ElectroMagneticTE = Transverse Electric
TM = Transverse Magnetic
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(1) TEM Modes:
0~~
==== zzzz HEHE
From the equations in (4), for the existence of non-trivial solutions,the denominators must be zero also. That is,
022 =+ k
jjk ==
1==k
upPhase velocity:
Propagation constants:
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From the equations in (4), the field components take an indefinitemathematical form of 0/0, whose definite values have to be
determined by boundary conditions. In general, we can write:
jkz
xxxx eEEEE
== 00 ,~
jkz
yyyy eEEEE == 00 ,
~
jkzxxxx eHHHH
== 00 ,~
jkz
yyyy eHHHH == 00 ,
~
The relations between Ex, Ey, Hx, and Hy can be further obtained
from the equations in (2) and (3), as shown below.
2 0xy =E
2 0xy =H
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Wave impedance:
=====
j
H
E
H
EZ
y
x
y
xTEM ~
~
Wave impedance = Intrinsic impedance of the medium
(5a)
TEM
x
y
x
yZ
j
H
E
H
E====
~
~
(5b)
Combining (5a) & (5b),
yZ
Ex
Z
EyHxH
TEM
x
TEM
y
yx +=+
EzH = 1
TEMZ
Therefore
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(2) TE and TM Modes:
TE and TM modes in general exist in hollow waveguides such as
rectangular waveguides and circular waveguides. They will be
studied in the context of these waveguides.
TEM modes can only exist in two-conductor waveguides such as
two-wire transmission lines, co-axial lines, parallel-plate
waveguides, etc, but not in single-conductor waveguides such as
rectangular waveguides and circular waveguides. This is because
either longitudinal field components or longitudinal currents arerequired to support the transverse magnetic field components HxandHy which form close loops in the transverse plane. There are
no longitudinal currents (not longitudinal surface currents) inside
hollow waveguides and hence hollow waveguides cannot supportTEM modes. But they can support TE and TM modes.
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Rectangular Waveguide(A) TM Modes:
yzx
a
b
0~
== zz HH
( ) ( ) zzz eyxEzyxE,
~,, =
( ) 0~22
2
2
2
=
+
+
x,yEhyx
z
From (1a), the equation for theEzfield is:
222 kh +=
We first find the longitudinal fieldEz
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function ofx only function ofy only
( ) ( ) ( )yYxXyxEz =,~
Let
Then
The above equation can be satisfied for all values of x
andy inside the waveguide only when both terms on theleft-hand side being equal to a constant.
( )
( )
( )
( ) 22
2
2
2 11
hdy
yYd
yYdx
xXd
xX =+
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Hence let
where222
hkk yx =+
Boundary conditions:( ) 00~ =,yEz( ) 0~ =a,yEz( ) 00~ =x,Ez( ) 0~ =x,bEz
( ) xkCxX xsin1= with am
kx= (m = 1, 2, 3, )
( ) ykCyY ysin2= with
b
nky= (n = 1, 2, 3, )
Solutioins:
C1 andC2 are
constants to be
determined bythe boundary
conditions along
thezdirection.
( )( ) 22
2
1xk
dxxXd
xX= ( ) ( )
22
2
1 ykdy
yYdyY
=
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( )
= yb
n
xa
m
Ex,yEz sinsin
~0
22
222
+
=+=b
n
a
mkkh yx
( )
= yb
nx
a
mE
a
m
h
x,yEx sincos
~02
( )
= yb
nx
a
mE
b
n
h
x,yEy cossin
~02
( )
= ybn
xa
m
Eb
n
h
j
x,yHx cossin
~02
( )
= yb
nx
a
mE
a
m
h
jx,yHy sincos
~02
(m = 1, 2, 3,)(n = 1, 2, 3, )
E0 is a contant
equal to C1C2and is to bedetermined by
the excitation
condition of the
waveguide.
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( )
222
2
22
22
2
:constantnPropagatio
fb
n
a
m
b
n
a
m
kh
+
=
+
=
=
Every combination of the integers m and n defines a possible TM
mode that may be designated as a TMmn mode. Hence there are
infinite number of TM mode that can exist inside the waveguide.
The frequency at which = 0 is called the cutoff frequencyfc.
( )22
2
1
+
=b
n
a
mf
mnc
( )22
21
+
==
b
n
a
mfcmnc
Note that the cutoff
frequency for a TEM
mode is zero (i.e., DC).
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(a) Whenf >fc, the propagation constant is an imaginary number and
the mode can travel inside the waveguide.
2
22
2
2
1
===
f
f
jkhkjj c
2
1
=
f
fjk c
2
1
=
ffk c
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Guided wavelength:
2
=g 21
2
=
f
fk c
2
1
=
f
fc
fk
12 ==
where is the wavelength of a plane wave with a frequencyf.
Note that . >g
2
2
2
1
=
c
g
2
22
1gc
=
222 111
cg +=
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Phase velocity: 2
1
==
f
fk
uc
p
2
1
=
f
fc
2
1
=
f
f
uu
c
p
The phase velocity is frequency dependent.
A rectangular waveguide is a dispersive device.
=
1u
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Group velocity:2
11
===
f
fu
ddd
du cg
uug
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Patslope=gu
linestraight
thisofslope=pu
Pc
Group velocity ug is the signalpropagation velocity if we assume the signal
composed of a narrow band of frequencies centered around f. Phasevelocity up is the speed of a constant-phase point of a particular mode.
Group velocity is also the speed of energy flow inside the waveguide. (See
Ref. 5, Section 8.5, for more details.)
-curve for waveguide TE and TM modes
-curve for TEM modes
Graphical Interpretation of up
and ug
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(b) When f < fc, the propagation constant is a real number and the
mode is non-propagating. The amplitude of the mode becomes
smaller (with the e-z) along the zdirection. This mode is called an
evanescent mode.
2
2
1constantnattenuatiohkh ===
2
1
=
cf
fh
Note that the energy of an evanescent mode is not lost but only
transferred back to the excitation source. That is, an evanescent mode
is constantly exchanging energy with the excitation source.
imaginary1
2
====
f
fj
j
jH
EZ c
y
xTM
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( )
= yb
x
a
E
b
h
jx,yHx cossin
~02
( )
= yb
x
a
E
a
h
jx,yHy sincos
~02
( ) 0~
=x,yHz
( ) ( ) ( ) zyxieyxEeyxEzyxE zjiz
ii ,,,,~,~,, ===
( ) ( ) ( ) zyxieyxHeyxHzyxH zjiz
ii ,,,,~
,~
,, ===
Instantaneous field expressions:( ) ( ){ } zyxiezyxEtzyxE tjii ,,,,,Re;,, ==
( ) ( ){ } zyxiezyxHtzyxH tjii ,,,,,Re;,, ==
For propagation
modes: = j
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( ) ( )ztyb
xa
Etzx,yEz
= cossinsin;, 0
( ) ( )ztyb
x
a
E
a
htzx,yEx
= sinsincos;, 02
( ) ( )ztyb
x
a
E
b
htzx,yEy
= sincossin;, 02
( ) ( )ztyb
x
a
E
b
h
tzx,yHx
= sincossin;, 02
( ) ( )ztyb
x
a
E
a
h
tzx,yHy
= sinsincos;, 02
( ) 0;, =tzx,yHz
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TM11
mode has the lowest cutoff frequency among
all the TM modes. Its field lines are shown below.
Solid lines: E field, dash lines: H field
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(B) TE Modes: 0~
== zz
EE
Using a similar analysis as for the TM modes, we can obtain field
expressions for TE modes as:
( ) = y
bnx
amHyxHz coscos,~ 0
( )
= yb
nx
a
mH
b
n
h
jyxEx
sincos,
~02
( )
= yb
nx
a
mH
a
m
h
jyxEy
cossin,
~02
( )
= ybn
xa
m
Ha
m
hyxHx
cossin,
~02
( )
= yb
nx
a
mH
b
n
hyxHy
sincos,
~02
(m = 0,1, 2, )
(n = 0,1, 2, )m & n cannot
be both equal
to zero
H0 is a constant
to be determined
by the excitation
condition of thewaveguide.
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( )22
2
1
+
=b
n
a
mf
mnc
( )22
21
+
==
bn
amfc
mnc
Cutoff frequency:
Cutoff wavelength:
2
1
=
f
fk c
Propagation constant:
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2
1
=
f
fc
g
Guided wavelength:
2
1
=
f
f
uu
c
p
Phased velocity:
2
1
=
f
fuu cg
Group velocity:
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2
1
=
f
f
Z
c
TE
Wave impedance:
2
1
==
cf
fh
Attenuation constant for evanescent modes:
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1 2 3
b/a=1/2
11
11
TM
TE
20
01
TETE
10TE
( )10TEcc
ff /
Note that in TE mode propagation, the lowest order mode is TE10 which
also has the lowest cutoff frequency among all the propation modes in arectangular waveguide. The cutoff frequencies of the different modes
are shown below for two cases of waveguide dimensions.
1 2
b/a=1
11
11
TM
TE
20
02
TE
TE
10
01
TE
TE
( )10TEcc
ff /
Case 1:
Case 2:
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TE10 Mode - Rectangular Waveguide
TE10 is the dominant mode in a rectangular waveguide with lowest
cutoff frequency (when a > b).
TE10
H field: dash lines
Surface current
E field: solid lines
(Picture form)
(Schematic form)
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Field expression of TE10 mode (m = 1 & n = 0):
( ) zjzjzz exa
HeyxHH 0 cos,
~
==
0=== yxz HEE
( ) zjzjyy exaHajeyxEE
0 sin2,~
==
( ) 0, sinj z j z
x x
aH H x y e j H x e
a
= =
( )a
fc2
110TE =
Cutoff frequency:
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Excitation of the Rectangular Waveguide
Excitation of a rectangular waveguide by a coaxial line.
Cross-section at x = a/2
Coaxial lineProbe
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A Note on the Propagating Modes inside
the Rectangular WaveguideNote that in a rectangular waveguide with an excitation source
frequency f= fi, all those TM and TE modes with a cutoff frequency
lower than fi
can propagate inside the waveguide. Whether they will
actually appear inside the waveguide depends on the excitation method.
The excitation method, for example the orientation of the coaxial
probe, can be chosen to excite certain modes while suppress other
modes. Those modes with a cutoff frequency higher than fi cannotpropagate inside the waveguide no matter what excitation method
chosen to excite them.
However, in the most general case, an EM wave inside the rectangularwaveguide is a linear combination of all those TE and TM modes
whose cutoff frequencies being lower than the excitation frequency.
Hence the rectangular waveguide is a high-pass filter.
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Example 2
A standard rectangular waveguide WG-16 is to be designed for the X-
band (8-12.4 GHz) radar application. The dimensions are a = 2.29 cm
and b = 1.02 cm. If only the lowest mode TE10 mode is to propagate
inside the waveguide and that the operating frequency be at least 25%above the cutoff frequency of the TE10 mode but no higher than 95% of
the next higher cutoff frequency, what is the allowable operating-
frequency range of this waveguide?
Solution
a = 2.29 cm b = 1.02 cm
( ) ( )Hz1055.60229.02
103
2
1 98TE10
=
==a
fc
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( ) ( )Hz1010.130229.01031 9
8
0,2TE20 =
==== af nm
c
( )22
TE2
1
+
=
bn
amf
mnc
( ) ( ) ( )2001 TE
98
1,0TE
Hz1071.140102.02
103
2
1c
nmc f
b
f >=
==
==
Hence the allowable operating-frequency range is:
( ) ( )2010 TETE
%95%125cc
fff
GHz45.12GHz19.8 f
That is:
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References:
1. David K. Cheng, Field and Wave Electromagnetic, Addison-
Wesley Pub. Co., New York, 1989.
2. David M. Pozar, Microwave Engineering, John Wiley & Sons,
Inc., New Jersey, 2005.3. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall, Inc.,
New Jersey, 2007.
4. Robert E. Collin, Field theory of guided waves, IEEE Press, New
York, 1991.
5. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons,
Inc., New York, 1975, Chapter 8, Section 8.5.
6. Joseph A. Edminister, Schaums Outline of Theory and Problemsof Electromagnetics, McGraw-Hill, Singapore, 1993.
7. Yung-kuo Lim (Editor), Problems and solutions on
electromagnetism, World Scientific, Singapore, 1993.
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