Electromagnetic Fields
Rectangular Wave Guide
a
x
z
y
b
Assume perfectly conducting walls and perfect dielectric filling the
wave guide.
Convention :
a is always the wider side of the wave guide.
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240
Electromagnetic Fields
It is useful to consider the parallel plate wave guide as a starting
point. The rectangular wave guide has the same TE modes
corresponding to the two parallel plate wave guides obtained by
considering opposite metal walls
E
a
E
b
TEm0
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TE0n
241
Electromagnetic Fields
The TE modes of a parallel plate wave guide are preserved if
perfectly conducting walls are added perpendicularly to the electric
field.
E
H
The added metal plate does
not disturb normal electric
field and tangent magnetic
field.
On the other hand, TM modes of a parallel plate wave guide
disappear if perfectly conducting walls are added perpendicularly to
the magnetic field.
H
E
© Amanogawa, 2001 – Digital Maestro Series
The magnetic field cannot
be normal and the electric
field cannot be tangent to a
perfectly conducting plate.
242
Electromagnetic Fields
TEmn
TMmn
The remaining modes are TE and TM modes bouncing off each wall,
all with non-zero indices.
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243
Electromagnetic Fields
We have the following propagation vector components for the
modes in a rectangular waveguide
2 2 x2 2y z2
m
n
x ;
y a
b
2
2
2 2 2 z 2 x2 2y
z g 2
2
m
n
2 a b
At cut-off we have
2
2
m
n
2
z2 0 2 fc a b
© Amanogawa, 2001 – Digital Maestro Series
244
Electromagnetic Fields
The cut-off frequencies for all modes are
2
1
m
n
fc b
2 a
2
with cut-off wavelengths
c 2
2
2
m
n
a
b
with indices
TE modes m 0, 1, 2, 3,
n 0, 1, 2, 3,
(but m n 0 not allowed)
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TM modes m 1, 2, 3,
n 1, 2, 3,
245
Electromagnetic Fields
The guide wavelengths and guide phase velocities are
2
g z z
1 c v pz z
1
© Amanogawa, 2001 – Digital Maestro Series
2
2
2
m n a b
2
fc 1 f
2
1
1 c 2
1
2
1
fc 1 f
2
246
Electromagnetic Fields
The fundamental mode is the TE10 with cut-off frequency
m
fc TE10 2a The TE10 electric field has only the y-component. From Ampere’s
law
E j H
Ez E y j H x
y
z
iˆy
iˆz iˆx
E z j H y 0
Ex det
x
y
z x
z
E
=
0
E
E
=
0
y
z
x
x
© Amanogawa, 2001 – Digital Maestro Series
Ey y
E x j H z
247
Electromagnetic Fields
The complete field components for the TE10 mode are then
x j z z
E y Eo sin
e
a
z
1 E y j z
x j z z
Hx Ey Eo sin
e
a
j z
j 1 E x
j x j z z
Hz Eo cos
e
a
j z a
with
2
z a
© Amanogawa, 2001 – Digital Maestro Series
2
248
Electromagnetic Fields
The time-average power density is given by the Poynting vector
*
1
1
x j z z e
P ( t ) Re E H Re{ Eo sin
iy a
2
2
(
z
x
*
Eo sin
a
e
j z z
ix j a
E
x
*
Eo cos
a
e
j z z
iz )}
*
H
2
E 2 x
E
x
x
2
o z
o
Re iz j
sin
sin
cos
a
a
a
a
2
2
Eo z
2 x iz
sin
a
2 1
© Amanogawa, 2001 – Digital Maestro Series
"
#
$
249
Electromagnetic Fields
The resulting time-average power density is space-dependent
Eo2 z 2 x sin
P( t ) iz
a
2 The total transmitted power for the TE10 mode is obtained by
integrating over the cross-section of the rectangular wave guide
2
2
E
E
a
b
x
2
o
z
sin
Ptot ( t ) % dx% dy
o z ab
a 4 0
0
2 The rectangular waveguide has a high-pass behavior, since signals
can propagate only if they have frequency higher than the cut-off
for the TE10 mode.
© Amanogawa, 2001 – Digital Maestro Series
250
Electromagnetic Fields
For mono-mode (or single-mode) operation, only the fundamental
TE10 mode should be propagating over the frequency band of
interest.
The mono-mode bandwith depends on the cut-off frequency of the
second propagating mode. We have two possible modes to
consider, TE01 and TE20
1
fc TE01 2b 1
2 fc TE10 fc TE20 a © Amanogawa, 2001 – Digital Maestro Series
251
Electromagnetic Fields
a
b
2
If
1
fc TE01 fc TE20 2 fc TE10 a Mono-mode bandwidth
fc TE10 0
a
a b
2
If
fc TE20 f
fc TE01 fc TE10 fc TE01 fc TE20 Mono-mode bandwidth
0
© Amanogawa, 2001 – Digital Maestro Series
fc TE10 fc TE01 fc TE20 f
252
Electromagnetic Fields
If
0
a
b
2
fc TE20 fc TE01 Mono-mode bandwidth
fc TE10 f
fc TE20 fc TE01 In practice, a safety margin of about 20% is considered, so that the
useful bandwidth is less than the maximum mono-mode bandwidth.
This is necessary to make sure that the first mode (TE10) is well
above cut-off, and the second mode (TE01 or TE20) is strongly
evanescent.
Safety margin
Useful bandwidth
0
fc TE10 © Amanogawa, 2001 – Digital Maestro Series
f
fc TE20 fc TE01 253
Electromagnetic Fields
a b
If
(square wave guide)
0
fc TE10 fc TE01 fc TE10 fc TE01 fc TE20 f
fc TE02 In the case of perfectly square wave guide, TEm0 and TE0n modes
with m=n are degenerate with the same cut-off frequency.
Except for orthogonal field orientation, all other properties of
degenerate modes are the same.
© Amanogawa, 2001 – Digital Maestro Series
254
Electromagnetic Fields
Example - Design an air-filled rectangular waveguide for the
following operation conditions:
a) 10 GHz is the middle of the frequency band (single-mode
operation)
b) b = a/2
The fundamental mode is the TE10 with cut-off frequency
1
c 3 10 8
&
Hz
fc (TE10 ) 2a
2a o o 2a
For b=a/2, TE01 and TE20 have the same cut-off frequency.
1
c c 2 c 3 10 8
&
Hz
fc (TE01 ) a
2b o o 2b 2 a a
1
c 3 108
Hz
&
fc (TE20 ) a
a o o a
© Amanogawa, 2001 – Digital Maestro Series
255
Electromagnetic Fields
The operation frequency can be expressed in terms of the cut-off
frequencies
fc (TE10 ) fc (TE01 )
f fc (TE10 ) 2
fc (TE10 ) fc (TE01 )
10.0 GHz
2
8
8 1
3
10
3
10
10.0 109 2 2 a
a
a 2.25 10
© Amanogawa, 2001 – Digital Maestro Series
2
m
a
b 1.125 10 2 m
2
256
Electromagnetic Fields
Maxwell’s equations for TE modes
Since the electric field must be transverse to the direction of
propagation for a TE mode, we assume
Ez 0
In addition, we assume that the wave has the following behavior
along the direction of propagation
e
j z z
In the general case of TEmn modes it is more convenient to start
from an assumed intensity of the z-component of the magnetic field
H z Ho cos x x cos y y e
j z z
m n j z z
Ho cos
x cos
y e
a b © Amanogawa, 2001 – Digital Maestro Series
257
Electromagnetic Fields
Faraday’s law for a TE mode, under the previous assumptions, is
E j H
iˆx iˆy iˆz det
x y z
E x E y 0 © Amanogawa, 2001 – Digital Maestro Series
E y j z E y j H x (1)
z
E x j z E x j H y (2)
z
E y E x j H z (3)
x
y
258
Electromagnetic Fields
Ampere’s law for a TE mode, under the previous assumptions, is
H j E
iˆx
det
x
H x
iˆz y z
H y H z iˆy
© Amanogawa, 2001 – Digital Maestro Series
H z j zH y j E x (4)
y
j zH x H z j E y (5)
x
H y H x j E z 0 (6)
x
y
259
Electromagnetic Fields
From (1) and (2) we obtain the characteristic wave impedance for
the TE modes
Ey Ex
TE
Hy
Hx z
At cut-off
2
2
m
n
z 0 2 fc a
b
vp
1
2
c fc 2
2
c c m
n
a
b
© Amanogawa, 2001 – Digital Maestro Series
260
Electromagnetic Fields
In general,
2
2
2
2
4
m
n
2
z 1
2 2
a b
2
c
2
1 z c 2
and we obtain an alternative expression for the characteristic wave
impedance of TE modes as
2 1 2
o 1 TE c z
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261
Electromagnetic Fields
From (4) and (5) we obtain
H z j zH y j E x j TE H y
y
H z
H z
1
1
Hy j TE j z y
j j z y
z
2
H z
c H z
Hy 2
j
z
2 y
2 y
z
j z
j zH x H z j E y j TEH x
x
2
H z
c H z
Hx 2
j z
2 x
2 x
z
j z
© Amanogawa, 2001 – Digital Maestro Series
262
Electromagnetic Fields
We have used
2
c
2 z2 x2 2y m 2 n 2 2 a b
1
1
1
The final expressions for the magnetic field components of TE
modes in rectangular waveguide are
2
m c m n j z z
H x j z
Ho sin
x cos
y e
a b a 2 n c 2
m n j z z
H y j z
Ho cos
x sin
y e
a b b 2 m n j z z
H z Ho cos
x cos
y e
a b © Amanogawa, 2001 – Digital Maestro Series
263
Electromagnetic Fields
The final electric field components for TE modes in rectangular
wave guide are
E x TE H y
n c 2
m n j z z
j
TE z
Ho cos
x sin
y e
a b b 2 E y TE H x
m c 2
m n j z z
j
TE z
Ho sin
x cos
y e
a b a 2 Ez 0
© Amanogawa, 2001 – Digital Maestro Series
264
Electromagnetic Fields
Maxwell’s equations for TM modes
Since the magnetic field must be transverse to the direction of
propagation for a TM mode, we assume
Hz 0
In addition, we assume that the wave has the following behavior
along the direction of propagation
e
j z z
In the general case of TMmn modes it is more convenient to start
from an assumed intensity of the z-component of the electric field
E z Eo cos x x cos y y e
j z z
m n j z z
Eo cos
x cos
y e
a b © Amanogawa, 2001 – Digital Maestro Series
265
Electromagnetic Fields
Faraday’s law for a TM mode, under the previous assumptions, is
E j H
iˆx iˆy iˆz det
x y z
E x E y E z © Amanogawa, 2001 – Digital Maestro Series
E z j z E y j H x (1)
y
E z j H y (2)
j z E x x
E y E x j H z (3)
x
y
266
Electromagnetic Fields
Ampere’s law for a TM mode, under the previous assumptions, is
H j E
iˆx
det
x
H x
iˆy
y
Hy
iˆz z
0 © Amanogawa, 2001 – Digital Maestro Series
j zH y j E x
(4)
j zH x j E y (5)
H y H x j E z
x
y
(6)
267
Electromagnetic Fields
From (4) and (5) we obtain the characteristic wave impedance for
the TM modes
Ey z
Ex
TM
Hy
Hx The same cut-off conditions found earlier for TE modes also apply
for TM modes.
We obtain a different expression for the characteristic wave
impedance
z
TM o 1 c © Amanogawa, 2001 – Digital Maestro Series
2
268
Electromagnetic Fields
From (1) and (2) we obtain
Ey
E z j z E y j H x j TM
y
E z
E z
1
1
Ey j / TM j z y
y
j j z
z
2
E z
c E z
Ey 2
j z
2 y
2 y
z
j z
Ex
j z E x E z j H y j TM
x
2
E z
c E z
Ex 2
j z
2 x
2 x
z
j z
© Amanogawa, 2001 – Digital Maestro Series
269
Electromagnetic Fields
The final expressions for the electric field components of TM modes
in rectangular waveguide are
m c 2
m n j z z
E x j z
Eo cos
x sin
y e
a b a 2 2
n c m n j z z
E y j z
Eo sin
x cos
y e
a b b 2 m n j z z
E z Eo sin
x sin
y e
a b © Amanogawa, 2001 – Digital Maestro Series
270
Electromagnetic Fields
The final magnetic field components for TM modes in rectangular
wave guide are
H x E y / TM
z n c 2
m n j z z
j
Eo sin
x cos
y e
a b TM b 2 H y E x / TM
z m c 2
m n j z z
j
Eo cos
x sin
y e
a b TM a 2 Hz 0
Note: all the TM field components are zero if either βx=0 or βy=0.
This proves that TMmo or TMon modes cannot exist in the
rectangular wave guide.
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271
Electromagnetic Fields
Field patterns for the TE10 mode in rectangular wave guide
z
x
Cross-section
E
y
y
z
x
E
Side view
H
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Top view
H
272
Electromagnetic Fields
The simple arrangement below can be used to excite the TE10 in a
rectangular waveguide.
The inner conductor of the coaxial cable behaves like a dipole
antenna and it creates a maximum electric field in the middle of the
cross-section.
Closed end
TE10
© Amanogawa, 2001 – Digital Maestro Series
273
Electromagnetic Fields
Waveguide Cavity Resonator
m
x a
n
y b
l
z d
d
a
x z
y
b
The cavity resonator is obtained from a section of rectangular wave
guide, closed by two additional metal plates. We assume again
perfectly conducting walls and loss-less dielectric.
© Amanogawa, 2001 – Digital Maestro Series
274
Electromagnetic Fields
The addition of a new set of plates introduces a condition for
standing waves in the z−direction which leads to the definition of
oscillation frequencies
2
2
1
m
n
l
fc b
d
2 a
2
The high-pass behavior of the rectangular wave guide is modified
into a very narrow pass-band behavior, since cut−off frequencies of
the wave guide are transformed into oscillation frequencies of the
resonator.
0
fc1
fc 2
f
In the wave guide, each mode is
associated with a band of frequencies
larger than the cut-off frequency.
© Amanogawa, 2001 – Digital Maestro Series
0
fr 1
fr 2
f
In the resonator, resonant modes can
only exist in correspondence of
discrete resonance frequencies.
275
Electromagnetic Fields
The cavity resonator will have modes indicated as
TEmnl
TMmnl
The value of the index corresponds to periodicity (number of half
sine or cosine waves) in the three directions. Using z-direction as
the reference for the definition of transverse electric or magnetic
fields, the allowed indices are
TE m 0, 1, 2, 3
n 0, 1, 2, 3
l 1, 2, 3
with only one zero index
m or n allowed
m 1, 2, 3
TM n 1, 2, 3
l 0, 1, 2, 3
The mode with lowest resonance frequency is called dominant
mode. In the case a ≥ d > b the dominant mode is the TE101.
© Amanogawa, 2001 – Digital Maestro Series
276
Electromagnetic Fields
Note that for a TM cavity mode, with magnetic field transverse to
the z-direction, it is possible to have the third index equal to zero.
This is because the magnetic field is going to be parallel to the third
set of plates, and it can therefore be uniform in the third direction,
with no periodicity.
The electric field components will have the following form that
satisfies the boundary conditions for perfectly conducting walls
m n l E x Ex cos
x sin
y sin
z
a b d m n l E y E y sin
x cos
y sin
z
a b d m n l E z Ez sin
x sin
y cos
z
a b d © Amanogawa, 2001 – Digital Maestro Series
277
Electromagnetic Fields
The amplitudes of the electric field components also must satisfy
the divergence condition which, in absence of charge, is
m n l E 0 Ex Ey Ez 0
a b
d
The magnetic field intensities are obtained from Ampere’s law
Hx z E y y Ez
j m n l sin
x cos
y cos
z
a b d x Ez z Ex
m n l Hy cos
x sin
y cos
z
a b d j Hz y Ex x E y
j © Amanogawa, 2001 – Digital Maestro Series
m n l cos
x cos
y sin
z
a b d 278
Electromagnetic Fields
Similar considerations for modes and indices can be made if the
other axes are used as reference for transverse fields, leading to
analogous resonant field configurations.
Movable piston changes
the resonance frequencies
INPUT
OUTPUT
A cavity resonator can be coupled to a wave guide through a small
opening. When the input frequency resonates with the cavity,
electromagnetic radiation enters the resonator and a lowering in the
output is detected. By using carefully tuned cavities, this scheme
can be used for frequency measurements.
© Amanogawa, 2001 – Digital Maestro Series
279
Electromagnetic Fields
Example of resonant cavity excited by using coaxial cables.
The termination of the inner conductor of the cable acts like an
elementary dipole (left) or an elementary loop (right) antenna.
E
H
© Amanogawa, 2001 – Digital Maestro Series
E
H
280