11 engineering electromagnetics essentials chapter 9 waveguides: solution of the wave equation for a...
TRANSCRIPT
11
Engineering Electromagnetics
EssentialsChapter 9
Waveguides: solution of the wave equation For a wave in a bounded medium
2
Objective
Topics dealt with
To solve wave equation and study propagation of electromagnetic wave through bounded media, namely hollow-pipe waveguides, and predict their characteristics
Solution of the wave equation of a rectangular waveguide in transverse electric (TE) mode for all the field components
Characteristic equation or dispersion relation of a rectangular waveguide excited in transverse electric (TE) mode in terms of wave frequency, wave phase propagation constant and waveguide cutoff frequency
Dependence of waveguide cutoff frequency on waveguide dimensions with reference to a waveguide excited in TE mode
Characteristic parameters namely phase velocity, group velocity, guide wavelength and wave impedance and their dependence on wave frequency relative to waveguide cutoff frequency
Evanescent mode in a waveguide
Dimension-wise and mode-wise operating frequency criteria
33
Excitation of a rectangular waveguide in transverse magnetic (TM) mode
Significance of mode numbers vis-à-vis field pattern
Cylindrical waveguide
Inability of a hollow-pipe waveguide to support a TEM mode
Power flow and power loss in a waveguide
Power loss per unit area, power loss per unit length and attenuation constant there from
Background
Maxwell’s equations (Chapter 5), electromagnetic boundary conditions at conductor-dielectric interface (Chapter 7), basic concepts of electromagnetic power flow (Chapter 8) and those of circuit theory
3
44
A ‘waveguide’ is a hollow metal pipe used to ‘guide’ the transmission of an electromagnetic ‘wave’ from one point to another.
It is a microwave-frequency counterpart of the lower-frequency connecting wire.
It is thus essentially a type of transmission line, some of the other types being the two-wire line, the coaxial cable, the parallel-plate line, the stripline, the microstrip line, etc.
We have seen that a transverse electromagnetic wave (TEM) mode of propagation is supported by a free-space medium with a phase velocity equal to the velocity of light c.
Can a TEM mode be supported by a hollow-pipe waveguide? Would the phase velocity of a wave propagating through such a waveguide be the same as or different from c?
We can answer to such question and many others concerning the characteristics of a waveguide with the help of electromagnetic analysis of the waveguide.
Such an analysis is based on setting up the wave equation in electric and magnetic fields in the waveguide and solving them with the help of relevant electromagnetic boundary conditions.
We take up here for analysis two types of waveguides, namely rectangular and cylindrical or circular waveguides which have rectangular and circular cross sections respectively perpendicular to the direction of wave propagation.
55
Rectangular waveguide
Rectangular waveguide can be treated in rectangular system of coordinates in view of the rectangular symmetry of its geometry. The waveguide will be considered for both TE and TM modes of excitation. (We will see later the inability of the hollow-pipe waveguides to support TEM mode of propagation).
The TE mode is also known as the H mode since it is associated with a non-zero value of the axial magnetic field:
The TM mode is also known as the E mode since it is associated with a non-zero value of the axial electric field:
.0,0 zz EH
.0,0 zz HE
a
b
Z
Y
X
Rectangular waveguide with its right side wall located at x = 0; left side at x = a; bottom wall at y = 0 and top wall at y = b.
mode) (TM )0,0(
mode) (TE )0,0(
2
2
002
2
2
2
2
2
2
2
002
2
2
2
2
2
zzzzzz
zzzzzz
HEt
E
z
E
y
E
x
E
EHt
H
z
H
y
H
x
H
Wave equations have already been introduced in Chapter 6.
Wave equations
66
(rewritten)
a
b
Z
Y
X
Rectangular waveguide with its right side wall located at x = 0; left side at x = a; bottom wall at y = 0 and top wall at y = b.
TE mode
mode) (TE )0,0(2
2
002
2
2
2
2
2
zzzzzz EH
t
H
z
H
y
H
x
H
)(exp ztj
jt /jz /
zzzz
zzzz
HHjHjjt
H
HHjHjjz
H
2222
2
2222
2
))((
))((
Field quantities varying as
zzzz HH
y
H
x
H 200
22
2
2
2
0)( 222
2
2
2
zzz Hk
y
H
x
H
constant)n propagatio
space-(free
)( 2/100k
(wave equation rewritten)
(wave equation)
77
x
XY
x
H z
XYH z
Field solutions
Let us use the method of separation of variables for solving wave equation
0)( 222
2
2
2
zzz Hk
y
H
x
H (wave equation rewritten)
X is a function of only x
2
2
2
2
x
XY
x
H z
0)( 222
2
2
2
XYky
YX
x
XY
2
2
2
2
y
YX
y
H z
Y is a function of only y
y
YX
y
H z
88
a
b
Z
Y
X
0)( 222
2
2
2
XYky
YX
x
XY (rewritten)
)(11 22
2
2
2
2
ky
Y
Yx
X
X
Dividing by XY and rearranging terms
Equates a function of x alone in its left-hand side with a function of y alone on its right-hand side. This can happen only when the individual functions of x and y are constants.
22
21A
x
X
X
222
2
2
)(1
Aky
Y
Y
22222
2
)(1
BkAy
Y
Y
A2 is a constant
B2 is a constant
99
)(exp)sincos)(sincos( 4321 ztjByCByCAxCAxCH z
ByCByCY sincos 43 AxCAxCX sincos 21
(rewritten) (rewritten)22
21A
x
X
X
2
2
21B
y
Y
Y
Solution
C1 and C2 are constants
C3 and C4 are constants
XYH z
Invoking the factor exp j(t - z) which was otherwise understood
(transverse components of the electric field and magnetic field components)
Maxwell’s equations
yxyx HHEE , and ,
1010
t
EH
t
HE
0
0
j
t
)(exp
as vary quantities Field
ztj
)(
)(
0
0
zzyyxx
zxy
yzx
xyz
zzyyxx
zxy
yzx
xyz
aEaEaEj
ay
H
x
Ha
x
H
z
Ha
z
H
y
H
aHaHaHj
ay
E
x
Ea
x
E
z
Ea
z
E
y
E
(Maxwell’s equation)
Expanding the curl in rectangular system
1111
)(
)(
0
0
yyxx
zxy
yzx
xyz
zzyyxx
zxy
yx
xy
aEaEj
ay
H
x
Ha
x
H
z
Ha
z
H
y
H
aHaHaHj
ay
E
x
Ea
z
Ea
z
E
)(
)(
0
0
zzyyxx
zxy
yzx
xyz
zzyyxx
zxy
yzx
xyz
aEaEaEj
ay
H
x
Ha
x
H
z
Ha
z
H
y
H
aHaHaHj
ay
E
x
Ea
x
E
z
Ea
z
E
y
E
0zE
(TE mode)
(rewritten)
1212
)(
)(
0
0
yyxx
zxy
yzx
xyz
zzyyxx
zxy
yx
xy
aEaEj
ay
H
x
Ha
x
H
z
Ha
z
H
y
H
aHaHaHj
ay
E
x
Ea
z
Ea
z
E
(rewritten)
x component
y component
yzx
xy
Ejx
H
z
H
Hjz
E
0
0
j
z
)(exp
as vary quantities Field
ztj
yz
x
xy
Ejx
HHj
HjEj
0
0
)/( 0 yx EH
yzy Ejx
HEj 0
0
2
Rearranging terms
00
0022
x
HEj zy
1313
)(
)(
0
0
yyxx
zxy
yzx
xyz
zzyyxx
zxy
yx
xy
aEaEj
ay
H
x
Ha
x
H
z
Ha
z
H
y
H
aHaHaHj
ay
E
x
Ea
z
Ea
z
E
(rewritten)
y component
x component
jz
)(exp
as vary quantities Field
ztj
)/( 0 xy EH
xzx Ejy
HEj 0
0
2
Rearranging terms
00
0022
y
HEj zx
Similarly
xyz
yx
Ejz
H
y
H
Hjz
E
0
0
xyz
yx
EjHjy
H
HjEj
0
0
1414
00
0022
y
HEj zx
0
0
0022
x
HEj zy
(recalled) (recalled)
x
H
k
jE zy
22
0
2/100 )( k
(recalled)
y
H
k
jE zx
220
y
HE
x
HE
zx
zy
We will recall the above relations of proportionality in the electromagnetic boundary conditions at the walls of the waveguide in the analysis to follow.
Electromagnetic boundary condition at the waveguide wall:
0)(2 zzyyxxnn aEaEaEaEa
0)( 12 EEan
(subscript 1 referring to the conducting waveguide wall region 1 and subscript 2 to the free-space region 2 inside the waveguide with unit vector directed from region 1 to 2)
mode) (TE 0zE
0)(2 yyxxnn aEaEaEa
1515
xn aa
ax
: wallsideLeft
0)( yyxxx aEaEa
0)( yyxxx aEaEa
(electromagnetic boundary condition at the left and right side walls of the rectangular waveguide)
0)(2 yyxxnn aEaEaEa
0zyaE
xn aa
x
0
: wallsideRight
0 zyaE
axbyEy at )0(0 0at )0(0 xbyEyx
HE zy
axbyx
H z
at )0(0 0at )0(0
xbyx
H z
(recalled)
(left side wall) (right side wall)
1616
)ofvaluesallfor(
0)sincos)()(( 432
y
ByCByCACx
H z
)cossin)(sincos(
)sincos)(cossin(
4321
4321
ByBCByBCAxCAxCy
H
ByCByCAxACAxACx
H
z
z
axbyx
H z
at )0(0 0at )0(0
xbyx
H z
)(exp)sincos)(sincos( 4321 ztjByCByCAxCAxCH z (recalled)
(left side wall) (right side wall)
ISituation
0,02 AC
IISituation
0,0 2 CA
This can happen only if
1717
0)sincos)(cossin)(( 4321
ByCByCAxACAxAC
x
H z
ISituation
0,02 AC axbyx
H z
at )0(0 (left side wall)
)ofvaluesallfor(
0)sincos(sin 431
y
ByCByCAxAC
)(exp)sincos)(sincos( 4321 ztjByCByCAxCAxCH z
0
I) (Situation 0,0
2
1
C
AC (left side wall at x = a)
0zH The finding contradicts with the TE-mode condition (Situation I):
(Situation I)
0zH
0,0 21 CC
18
0
I) (Situation 0,0
2
1
C
AC leads to the contradiction with the TE-mode condition
(left side wall at x = a)
0 besides 0,0 21 CCA
Therefore, in order to avoid this contradiction in Situation I let us put
(Situation I)
(left side wall at x = a)
)ofvaluesallfor(
0)sincos)(cossin)(( 4321
y
ByCByCAxACAxACx
H z
0sin Aa
(rewritten)(left side wall at x = a)
...),3,2,1( mmAa
(m = 0 that makes A = 0 is excluded to avoid contradiction with A ≠ 0 taken here)I)Situation(...),3,2,1( m
a
mA
I)Situation(...),3,2,1(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
19
Similarly we can proceed recalling Situation II: 0,0 2 CA
)(exp)sincos)(( 431 ztjByCByCCH z II)Situation()ofvaluesallfor( y
)(exp)sincos)(sincos( 4321 ztjByCByCAxCAxCH z
(recalled)
Tacitly expressed as
)(exp)sincos)(cos( 431 ztjByCByCAxCH z
II)Situation()0( nginterpreti ma
mA
II)Situation()0(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
19
20
I)Situation(...),3,2,1(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
II)Situation()0(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
(rewritten)
(recalled)Combining
...),3,2,1,0(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
)ofvaluesallfor( y
)ofvaluesallfor( y
20
We have so far considered electromagnetic boundary conditions at the left and right sidewalls of the rectangular waveguide. Let us next turn towards electromagnetic boundary conditions at the top and bottom walls of the rectangular waveguide.
212121
yn aa
y
0
: wallBottom
0)( yyxxy aEaEa
0)( yyxxy aEaEa
0)(2 yyxxnn aEaEaEa
0zxaE
yn aa
by
: wallTop
0 zxaE
0at )0(0 yaxEx byaxEx at )0(0y
HE zx
0at )0(0
yax
y
H z byaxy
H z
at )0(0
(recalled)
(bottom wall) (top wall)
(electromagnetic boundary condition at the bottom and top walls of the rectangular waveguide)
22
0at )0(0
yax
y
H z byaxy
H z
at )0(0
(bottom wall) (top wall)
)cossin)(sincos( 4321 ByBCByBCAxCAxCy
H z
...),3,2,1,0(with
)(exp)sincos)(cos( 213
nb
nB
ztjAxCAxCByCH z
Hence using the same procedure as followed earlier, which starts from the boundary conditions at the left and right side walls, we can obtain, now starting from the boundary conditions at the bottom and top walls, the following expression:
)ofvaluesallfor( x
23
...),3,2,1,0(with
)(exp)sincos)(cos( 213
nb
nB
ztjAxCAxCByCH z
)ofvaluesallfor( x
We have obtained the following:
...),3,2,1,0(with
)(exp)sincos)(cos( 431
ma
mA
ztjByCByCAxCH z
)ofvaluesallfor( y
When combined, these two expressions for Hz, while remembering that the former is valid for all values of y and that the latter for all values of x, prompt us to take the factor cosAx from the former and the factor cosBy from the latter in the field expression to write it as
...),3,2,1,0,(,with )(expcoscos0 nmb
nB
a
mAztjByAxHH zz
)(exp)cos)(cos( 31 ztjByCAxCHH zz
310 Putting CCH z
...),3,2,1,0,(, nmb
nB
a
mA
2424
...),3,2,1,0(...);,3,2,1,0(with
)(expcoscos0
nb
nBm
a
mA
ztjByAxHH zz
...),3,2,1,0,()(exp)cos()cos(0 nmztjyb
nx
a
mHH zz
...),3,2,1,0,()(exp)cos()sin(0
nmztjy
b
nx
a
mH
a
m
x
Hz
z
)(exp)sin()cos(0 ztjyb
nx
a
mH
b
n
y
Hz
z
...),3,2,1,0,(, nmb
nB
a
mA
2525
x
H
k
jE zy
22
0
y
H
k
jE zx
220
...),3,2,1,0,()(exp)cos()sin(0
nmztjy
b
nx
a
mH
a
m
x
Hz
z
)(exp)sin()cos(0 ztjyb
nx
a
mH
b
n
y
Hz
z
...),3,2,1,0,()(exp)cos()sin(0220
nmztjy
b
nx
a
mH
a
m
k
jE zy
...),3,2,1,0,()(exp)sin()cos(0220
nmztjy
b
nx
a
mH
b
n
k
jE zx
(rewritten)
(rewritten)
(rewritten)
(rewritten)
2626
...),3,2,1,0,()(exp)cos()sin(0220
nmztjy
b
nx
a
mH
a
m
k
jE zy
2222 )( BkA
(rewritten)
(rewritten)
(introduced earlier while solving the wave equation by the method of separation of variables)
2222 BAk
...),3,2,1,0,()(exp)cos()sin(0220
nmztjy
b
nx
a
mH
a
m
BA
jE zy
...),3,2,1,0,()(exp)sin()cos(0220
nmztjy
b
nx
a
mH
b
n
k
jE zx
...),3,2,1,0,()(exp)sin()cos(0220
nmztjy
b
nx
a
mH
b
n
BA
jE zx
2727
...),3,2,1,0,(, nmb
nB
a
mA
...),3,2,1,0,()(exp)cos()sin(0220
nmztjy
b
nx
a
mH
a
m
BA
jE zy
...),3,2,1,0,(
)(exp)cos()sin(0220
nm
ztjyb
nx
a
mH
a
m
bn
am
jE zy
...),3,2,1,0,(
)(exp)sin()cos(0220
nm
ztjyb
nx
a
mH
b
n
bn
am
jE zx
...),3,2,1,0,()(exp)sin()cos(0220
nmztjy
b
nx
a
mH
b
n
BA
jE zx
2828
yx EH0
xy EH0
...),3,2,1,0,()(exp)cos()sin(0220
nmztjyb
nx
a
mH
a
m
b
n
a
m
jE zy
(rewritten)
...),3,2,1,0,(
)(exp)cos()sin(022
nm
ztjyb
nx
a
mH
a
m
bn
am
jH zx
...),3,2,1,0,()(exp)sin()cos(0220
nmztjyb
nx
a
mH
b
n
bn
am
jE zx
...),3,2,1,0,(
)(exp)sin()cos(022
nm
ztjyb
nx
a
mH
b
n
bn
am
jH zy
(rewritten)
2929
Field expressions of a rectangular waveguide excited in the TEmn mode put together
...),3,2,1,0,()(exp)cos()cos(0 nmztjyb
nx
a
mHH zz
...),3,2,1,0,()(exp)sin()cos(0220
nmztjyb
nx
a
mH
b
n
bn
am
jE zx
...),3,2,1,0,()(exp)cos()sin(0220
nmztjyb
nx
a
mH
a
m
bn
am
jE zy
...),3,2,1,0,()(exp)cos()sin(022
nmztjyb
nx
a
mH
a
m
bn
am
jH zx
...),3,2,1,0,()(exp)sin()cos(022
nmztjyb
nx
a
mH
b
n
bn
am
jH zy
0zE
(m and n are the mode numbers of the rectangular waveguide excited in the TEmn mode)
3030
Characteristic equation or dispersion relation of a rectangular waveguide excited in the TE mode
...),3,2,1,0,(22
22
nm
b
n
a
mk
222ckk
...),3,2,1,0,( Putting
2/122
nm
b
n
a
mkc
(recalled)
222
2
ckc 022222 ckc c
frequency cutoff thebeing number, wavecutoff ,/ ccc ck
light of velocity thebeing )/(1
constant,n propagatio space-free ,/)(2/1
00
2/100
c
ck
02222 cc (dispersion relation)
(- relationship with reference to a rectangular waveguide excited in the TE mode)
31
Dispersion relation
2 2 2 2 0cc
cc -
line
cc
g
k
k
2
2
2
222ckk
222222
cg
222
111
gc
versus dispersion characteristics (dispersion curve) of the waveguide which meets the c-line at , the latter shown as a dotted line having positive and negative slopes of magnitude equal to the velocity of light c for the positive and the negative values of respectively. The intercept of the dispersion curve with the axis gives the cutoff frequency c.
(recalled)
(characteristic equation of the waveguide)
Dispersion curve: a hyperbola
32
cc f
c
...),3,2,1,0,(
2
2
2/122
2/122
2/122
nm
bn
amf
c
b
n
a
mcf
b
n
a
mc
cc
c
c
2/122
b
n
a
mkc
2c
cf
ck cc
(cutoff wave number)
(cutoff frequency)
(cutoff wavelength)
2
222
cc
02222 cc
cc
2/122 )(
2 2 2 2 0cc
cc -
line
In general, there will be two values of for a given value of one positive and the other negative.
33
Waveguide a (inch) b (inch) c (mm) fc(=c/2a) (GHz)
WR-2300 23 11.5 1168.4 0.257
WR-340 3.4 1.7 172.72 1.737
WR-284 2.84 1.34 144.27 2.08
WR-137 1.372 0.62 69.70 4.30
WR-112 1.122 0.497 57.0 5.26
WR-90 0.9 0.4 45.72 6.56
WR-3 0.034 0.017 1.73 173.7
Typical waveguide dimensions and corresponding cutoff frequencies (fc) and cutoff wavelengths (c) for the TE10 mode for typical commercial rectangular waveguide types
WaveguideWaveguide modes in order of increasing cutoff frequency (fc) shown in parenthesis against the
modes
WR-2300
TE10 (0.257 GHz)
TE20/ TE01 (0.51 GHz)
TE11 (0.61 GHz)
WR-284
TE10 (2.08 GHz)
TE20 (4.16 GHz)
TE01 (4.41 GHz)
TE11 (4.87 GHz)
WR-90
TE10 (6.56 GHz)
TE11 (9.88 GHz)
TE20 (13.1 GHz)
TE01 (14.76 GHz)
Modes in order of increasing cutoff frequency for typical commercial rectangular waveguide types
34
2/1222/122ph)()( cc
c
c
v
g
k
2
2 2
g
Characteristic parameters
cc
2/122 )(
2/1
2
22/1
2
22/122
ph
1
1
1
1
)(
f
fc
v
ccc
2/1
2
2
ph
1
1
ffc
vk
c
g
Wave phase velocity:
Guide wavelength:
ck
phv
2/1
2
2
ph
1
1
f
fc
v
c
g
(recalled)
(rewritten)
35
Wave group velocity
We have already defined the phase velocity of a wave vph that involves the phase propagation constant , which, in turn, occurs in the representation of a wave associated with a physical quantity, for instance, the electric field:
)(exp0 ztjEE
velocityphase /
/constantn propagatio phase
amplitude field
ph
ph
0
v
v
E
However, the phase velocity has a meaning only with reference to an infinite monochromatic wave train.
In order to convey information, we have to resort to a group of wave trains at different frequencies the form a finite wave train or a wave packet, say, in the form of a modulated wave such that the modulation envelope contains the information to be conveyed.
The velocity of the wave packet or group of waves is called the group velocity of the wave, which also represents the velocity with which the energy is transported.
We may then find the group velocity as the velocity of the amplitude of the group of waves in the wave packet. Through any medium, which, in general, is dispersive, the wave components of the group travel with different phase velocities.
In what will follow, we are going to show, considering two components of the wave packet, that the wave group velocity vg can be found from the relation:
gv
36
])()[(exp])()[()[exp(exp0 ztjztjztjEE
])()[([exp])()[([exp 00 ztjEztjEE
constant zt
Let us consider two components of the wave packet one at - and the other at + , which are separated in frequency by 2 with phase propagation constants - and + respectively.
Considering the magnitudes of the two electric field components to be the same for the sake of simplicity
sincosexprelation theof In view jj
)(exp)cos(2 0 ztjztEE
0dt
dz
The amplitude of the electric field of the combination of the wave components of the group has the wave-like variation with distance and time appearing in the cosine function.
Putting the argument of the cosine function as constant in order to find the wave group velocity vgDifferentiating
dt
dz
gvFor small values of in the limit (wave group velocity)
3737
2ph cvv g
ph
222
v
cccvg
02222 cc
Group velocity of the waveguide:
gv
2 2 2 2 0cc
cc -
line
The group velocity vg of the waveguide can be found from the relation
phv
022 2 c
The slope of the - dispersion curve of the waveguide at any point represents the group velocity of the waveguide at the point while the slope of the line joining the point and the origin of the curve represents the phase velocity of the waveguide.
Taking the partial derivative
2/1
2
2
ph
1
1
ffc
v
c
2/1
2
2
phph
11
f
f
c
vv
c
c
vcg
38
You will surely enjoy an illustrative example in which we will find the length of the waveguide of cutoff wavelength 69.70 mm that will ensure that a signal at 8.6 GHz emerging out of the guide is delayed by 1 μs with respect to the signal that propagates outside the waveguide.
s 10s 1 6delay
GHz 3.4Hz103.4107.69
103 93
8
cc
cf
m/s 103 8c
s 10)0516.0()103333.0103849.0(11 888
delay
llcv
lg
8
2/1
2
2
1031
f
fv cg
(time delay)
(given) m 107.69 3c
2/1
2
2
1
f
f
c
vcg
m/s 10598.210325.01 882/1 gv
m/s 103 (given); GHz 3.4 and GHz 6.8 8 cff c
(time delay) (given)
km 938.1m 1038.1910)0516.0(
10 28
6
ll = waveguide length
(required waveguide length)
3939
2/100
ph
)(
1
c
v
2/1002/1
0
0
0
0
0
0
)(
WZ
2/122
2/122
transverse
transverse
)(
)(
yx
yxW
HH
EE
H
EZ
Wave impedance:
xy
yx
EH
EH
0
0
(recalled)
0
2/122
20
2
2
2/122
transverse
transverse
)])([(
)(
xy
yxW
EE
EE
H
EZ
2/1
2
2
ph
01
1
ffc
vZ
c
W
2/1
2
2
ph
1
1
f
fc
v
c
4040
2/1
2
2
ph
01
1
ffc
vZ
c
W
2/1
2
2
1
f
f
c
vcg
2/1
2
2
ph
1
1
f
fc
v
c
2/1
2
2
1
1
f
fc
g
Waveguide characteristic parameters put together:
)(
0
ph
f
Z
cv
cv
W
g
g
)(
0
ph
c
W
g
gff
Z
v
v
gv c
pv c
g
0WZ
cf f
p gv c
0g Wv c Z
1 2
1
2
0WZ
41
Evanescent modecc
2/122 )( (recalled)
For frequencies above the cutoff frequency: >c, the phase propagation constant becomes positive real, which occurring in the phase factor exp j(t-z) and, in turn, in the expressions for the RF quantities, refers to a ‘propagating’ wave through the waveguide along positive z.
Taking the upper sign
cc
2/122 )( Positive real for c
Positive or negative real for c cc
jj
cj
cjj
cc
cc
2/1222/122
2/1222/122
)()()(
])(1[()(
)()((
Let us now take frequencies less than the cutoff frequency: <c and take the phase factor as exp j(t-z) :
)exp(exp
))(
(exp)exp()exp()exp()(exp2/122
ztj
zc
tjztjztj c
Phase factor occurring in field expressions taking the positive lower sign in :
cc
2/122 )(
Non-propagating evanescent mode )( c
42
)exp(exp
))(
(exp)exp()exp()exp()(exp2/122
ztj
zc
tjztjztj c
(Non-propagating evanescent mode)
Phase factor occurring in field expressions
Recalled
)( c
...),3,2,1,0,()(exp)sin()cos(0220
nmztjyb
nx
a
mH
b
n
bn
am
jE zx
(recalled)
cc
2/122 )(
...),3,2,1,0,(exp)exp()sin()cos(0220
nmtjzyb
nx
a
mH
b
n
bn
am
jE zx
)( c
)exp(expby )(exp Replacing ztjztj
)( c
The field amplitude decays exponentially with the axial distance z as exp(-z) for <c. is known as the attenuation constant in this non-propagating mode called the evanescent mode (corresponding to non-propagating mode vanishing).
Power consideration in this mode is taken up later.
4343
)exp()exp()sin()cos(0220 tjzy
b
nx
a
mH
b
n
bn
am
jE zx
)exp()exp()sin()cos(0220* tjzy
b
nx
a
mH
b
n
bn
am
jE zx
(rewritten)
Taking complex conjugate
Following the same procedure we can write
)exp()exp(cossin0220 tjzy
b
nx
a
mH
a
m
bn
am
jE zy
)exp()exp(cossin0220* tjzy
b
nx
a
mH
a
m
bn
am
jE zy
)( c
)( c
44
xy
yx
Ej
H
Ej
H
0
0
xy
yx
EH
EH
0
0
(recalled)
j
)Re(2
1
)Re(2
1)PRe()P(
**
*componentcomplexcomponentaverage
xyyx
zzz
HEHE
HE
2jj
*
0
*
*
0
*
xy
yx
Ej
H
Ej
H
(z-component of average Poynting vector)
)( c
)( c
45
)(Re2
1
)Re(2
1)P(
**
0
**componentaverage
yEEEEj
HEHE
yxx
xyyxz
)Re(2
1
)Re(2
1)PRe()P(
**
*componentcomplexcomponentaverage
xyyx
zzz
HEHE
HE
(z-component of average Poynting vector)
)( c (rewritten)
*
0
*
*
0
*
xy
yx
Ej
H
Ej
H
)( c
46
)2exp(cossin
)2exp(sincos
2220
2
220*
2220
2
220*
zyb
nx
a
mH
a
m
bn
am
EE
zyb
nx
a
mH
b
n
bn
am
EE
zyy
zxx
Real quantities
0
)(Re2
1)P( **
0componentaverage
yEEEEj yxxz
imaginarypurely is )( **
yEEEEj yxx
)( c
)( c
The average Poynting vector being thus zero, we infer that there would be no power flow in the waveguide in the evanescent mode below cutoff (<c). The waveguide in this mode acts as a reactive load such that the power oscillates back and forth between the source and the waveguide. .
4747
Let us take up a simple numerical example to show that a rectangular waveguide of cross-sectional dimensions 2.5 cm × 1 cm can support typically the propagating mode TE10 and the evanescent mode TE01 at 10 GHz operating frequency.
2/122
2
b
n
a
mcfc
GHz 6 Hz 106105.22
103 92
8
cf
mode TE
m/s103
m 100.1
m 105.2
0,1
10
8
2
2
c
b
a
nm
(given) GHz 10 fcff
mode) (TE
m/s103
m 100.1
m 105.2
1,0
01
8
2
2
c
b
a
nm
GHz 15 Hz 1015100.12
103 92
8
cf
cff
(Evanescent mode) (Propagating mode)
2/12
2
2
2
8
100.1105.22
103
nmfc
mode) (TE01mode) (TE10
Thus, the waveguide can support non-propagating evanescent mode TE01 and the propagating mode TE10
48
What will be the nature of the attenuation constant versus frequency plot of a waveguide in the evanescent mode?
2222cc
)()( 2/122
cc
c
Attenuation constant in evanescent mode
(recalled)
The plot of versus becomes an ellipse. The intercept of the ellipse on the axis (ordinate), corresponding to = 0, is found to be c//c. Similarly, the intercept of the ellipse on axis (abscissa), corresponding to = 0, is found to be c.
1)/( 2
2
2
2
ccc
1
2
2
2
2
b
y
a
x
/,
,
bca
yx
cc
(equation of an ellipse)
4949
TM mode
2
2
002
2
2
2
2
2
t
E
z
E
y
E
x
E zzzz
0)( 222
2
2
2
zzz Hk
y
H
x
H
)(exp)sincos(
)sincos(
43
21
ztjyBCyBC
xACxACEz
The method of analysis of the rectangular waveguide excited in the TM mode closely follows that of the waveguide excited in the TE mode the latter already presented. We can then obtain the TM-mode expressions using the same steps as that followed while obtaining the TE-mode expressions as follows.
)0,0( zz HE)0,0( zz EH
TE-mode expressions already developed in steps
Corresponding TM-mode expressions that follow
2
2
002
2
2
2
2
2
t
H
z
H
y
H
x
H zzzz
0)( 222
2
2
2
zzz Ek
y
E
x
E
)(exp)sincos(
)sincos(
43
21
ztjByCByC
AxCAxCH z
50
TE-mode expressions already developed
Corresponding TM-mode expressions that follow
xbyExyEyxEyaxE
z
z
z
z
allfor)walltop(0allfor)0wallbottom(0
allfor)0sidewallright(0allfor)sidewallleft(0
)(expsinsin0 ztjyBxAEE zz )(expcoscos0 ztjByAxHH zz
xbyE
xyE
yxE
yaxE
x
x
y
y
allfor)walltop(0
allfor)0wallbottom(0
allfor)0sidewallright(0
allfor)sidewallleft(0
...),3,2,1,0,(
nm
b
nB
a
mA
...),3,2,1,0,(
nm
b
nB
a
mA
)(exp
)cos()cos(0
ztj
yb
nx
a
mHH zz
)(exp
)cos()cos(0
ztj
yb
nx
a
mEE zz
51
TE-mode expressions already developed
Corresponding TM-mode expressions that follow
2222 )( BkA
2222
b
n
a
mk
222ckk
...),3,2,1,0,(
2/122
nm
b
n
a
mkc
02222 cc
2222 )( BkA
2222
b
n
a
mk
222ckk
...),3,2,1,0,(
2/122
nm
b
n
a
mkc
022222 ckc c 022222 ckc c
02222 cc
5252
TE-mode expressions already developed
Corresponding TM-mode expressions that follow
...),3,2,1,0,(
2
2
2/122
2/122
2/122
nm
bn
amf
c
b
n
a
mcf
b
n
a
mc
cc
c
c
...),3,2,1,0,(
2
2
2/122
2/122
2/122
nm
bn
amf
c
b
n
a
mcf
b
n
a
mc
cc
c
c
53
Dominant mode of a rectangular waveguide
• The cutoff frequency of the TE10 mode is the lowest of all other modes.
• If we choose the operating frequency of the waveguide above the cutoff frequency of the TE10 mode, then the TE10 mode will propagate through the waveguide. Also, any next higher order mode will co-propagate if its cutoff frequency is lower than the operating frequency.
• For instance, let the operating frequency be 4.5 GHz for waveguide WR-284. This is higher than the cutoff frequencies of TE10, TE20, and TE01 modes , which are respectively 2.08 GHz, 4.16 GHz, and 4.41 GHz, but lower than the cutoff frequency 4.87 GHz of TE11. ( See slide number 33 for dimensions and cutoff frequencies of waveguide WR-284.)
• Therefore, at the operating frequency of 4.5 GHz, waveguide WR-284 will allow propagation of TE10, TE20, and TE01 but makes mode TE11 evanescent, attenuating it. • Thus, TE10 mode (which has the lowest cutoff frequency of all the modes) will be excited in a rectangular waveguide if the operating frequency is higher than its cutoff frequency irrespective of whether any other modes are exited or not. Hence the TE10 mode is called the dominant mode of a rectangular waveguide.
• If the operating frequency is chosen at 3 GHz for waveguide WR-284, which is above the cutoff frequency 2.08 GHz of the dominant mode TE10 but below the cutoff frequency 4.16 GHz of TE20, then only the mode TE10, which is the dominant mode, will be excited in the rectangular waveguide.
• Moreover, the cutoff frequency of the TE10 mode is the lowest of the cutoff frequencies of all the TE and TM modes taken together and continues to be called the dominant mode with due consideration to the excitation of the waveguide in both the TE and TM mode types.
54
Starting from the expression for Ez and taking help from Maxwell’s equations, we can find the transverse electric and magnetic field components in the TM mode as we have obtained earlier in the TE mode. Thus, we obtain the field expressions for the TMmn mode as follows:
0zH
...),3,2,1,0,()(expsincos022
nmztjy
b
nx
a
mE
a
m
bn
am
jE zx
...),3,2,1,0,()(expsincos0220
nmztjyb
nx
a
mE
a
m
bn
am
jH zy
...),3,2,1,0,()(expcossin0220
nmztjyb
nx
a
mE
b
n
bn
am
jH zx
...),3,2,1,0,()(exp)cos()cos(0 nmztjyb
nx
a
mEE zz
...),3,2,1,0,()(expcossin022
nmztjy
b
nx
a
mE
b
n
bn
am
jE zy
(TMmn-mode field expressions of a rectangular waveguide)
55
Field pattern and significance of mode numbers of a rectangular waveguide
mode) TE(
)(expsin
)(expsin
)(expcos
10
0
00
0
ztjxa
Haj
H
ztjxa
Haj
E
ztjxa
HH
zx
zy
zz
mode) TE(
)(expsin
)(expsin
)(expcos
01
0
00
0
ztjyb
Hbj
H
ztjyb
Hbj
E
ztjyb
HH
zy
zx
zz
Expressions for non-zero field components of a rectangular waveguide for typical lower-order modes TE10, TE10, TE20 and TM11:
56
mode) TE(
)(exp2
sin2
)(exp2
sin2
)(exp2
cos
20
0
00
0
ztjxa
Haj
H
ztjxa
Haj
E
ztjxa
HH
zx
zy
zz
mode) TM(
)(expsincos
)(expcossin
)(expsinsin
)(expcossin
)(expsincos
11
0220
0220
0
022
022
ztjyb
xa
Ea
ba
jH
ztjyb
xa
Eb
ba
jH
ztjyb
xa
EE
ztjyb
xa
Eb
ba
jE
ztjyb
xa
Ea
ba
jE
zy
zx
zz
zy
zx
57
a
b
TE10 mode
a
b
TE20 mode
a
b
x
yz
TM11 mode
Field pattern of typical lower-order rectangular waveguide modes
The field pattern of the rectangular waveguide has been more elaborately explained in the book. However, in what follows next, let us discuss the significance of the mode number vis-à-vis the field pattern.
58
Significance of the TEmn-mode or TMmn-mode field patterns of a rectangular waveguide vis-à-vis the mode numbers m and n
It is easy to infer by examining field patterns that
• mode number m, the first suffix of TEmn or TMmn, indicates the number of maxima of any field component, electric or magnetic, along the broad dimension of the waveguide
• mode number n, the second suffix of TEmn or TMmn, indicates the number of maxima of
any field component, electric or magnetic, along the narrow dimension of the waveguide
Alternatively, m and n can be interpreted as the numbers of half-wave field patterns across the broad and narrow dimensions of the waveguide respectively.
59
Examples:
For TE10 mode (m = 1, n = 0), the first suffix being unity (m = 1), there is a single maximum of the field component along the waveguide broad dimension. The second suffix being zero (n = 0), there is no maximum of the field component along the waveguide narrow dimension. For TE20 mode (m =2, n = 0), the first suffix being 2 (m = 2), there exist two maxima of the field component along the waveguide broad dimension. The second suffix being zero (n = 0) there is no maximum of the field component along the waveguide narrow dimension. Similar observations can be made by examining the field pattern of the TMmn mode, for instance, the TM11 mode as well.
a
b
a
b
x
yz
TE10 mode
TE20 mode
TM11 mode
a
b
Later on we are going to make similar inferences on the significance of mode numbers vis-à-vis field pattern with respect to a cylindrical waveguide.
60
Cylindrical waveguide
Cylindrical or circular waveguide can be treated in cylindrical system of coordinates (r,,z) in view of the circular symmetry of this type of waveguide.
rθ
X
Y
Z
O
Z
For a cylindrical waveguide it turns out that the dominant mode is the TE11 mode characterized by having the lowest cutoff frequency. Let us emphasize here mainly the TE11 mode for analysis, although the analytical approach presented here is rather general and is easily applicable to the analysis of other higher order TEmn and TMmn modes as well.
02
2
002
t
HH zz
011
2
2
002
2
2
2
2
t
H
z
HH
rr
Hr
rrzzzz
)0,0( zz EHWave equation for the TE mode
Expanding Laplacian in cylindrical coordinates
61
011
2
2
002
2
2
2
2
t
H
z
HH
rr
Hr
rrzzzz
011
2
2
22
2
002
2
2
2
zzzzz H
rt
H
z
H
r
H
rr
H
01
)(1
2
2
22
002
2
2
z
zzz H
rH
r
H
rr
H
2
2
22
002
2
2 1)(
1
z
zzz H
rH
r
H
rr
H
(rewritten)
Carrying out the differentiation of the first term
In view of the field dependence exp j(t-z) which is understood, enabling us to put /t = j and /z = - j
Rearranging terms
62
2
2
22
002
2
2 1)(
1
z
zzz H
rH
r
H
rr
H
RH z
2
2
2
2
2
2
2
2
)(
R
H
r
R
r
H
r
R
r
R
r
H
z
z
z
2
2
22
002
2
2 1)(
1
Rr
Rr
R
rr
R
(wave equation) (rewritten)
Let us use the method of separation of variables for solving the wave equation and put
alone offunction a is
and alone offunction a is
rR
63
Rby Dividing
11
)(111
2
2
22
002
2
2
rr
R
Rrr
R
R
2/1222/12222/1200
2 )()/()( kc
1111
2
2
22
2
2
rr
R
Rrr
R
R
2
2
22
002
2
2 1)(
1
Rr
Rr
R
rr
R(rewritten)
1
2
222
2
22
r
r
R
R
r
r
R
R
r
2by gMultiplyin r
64
1
2
222
2
22
r
r
R
R
r
r
R
R
r(rewritten)
2m
22
2
2222
22
1m
mrr
R
R
r
r
R
R
r
Equates a function of r alone on its left hand side with a function of alone on its right hand side. This can hold good only when the individual functions each become equal to a constant.
Putting this constant as
65
2
22
2
2 11
r
m
r
R
Rrr
R
R
01
2
222
2
22
Rr
m
x
R
xx
R
2222
22
mrr
R
R
r
r
R
R
r
(rewritten)
0)(1
2
22
2
2
Rr
m
r
R
rr
R
2by Dividng r
x
r
rx
termsgRearrangin
In terms of a dimensionless quantity x
66
RH z
Rfor Solve
22
2 1m
)()( xQYxPJR mm
01
1
2
222
2
22
22
2
Rr
m
x
R
xx
R
m
(rewritten)
for Solve
We can obtain the expression for the axial component of magnetic field Hz from the solution of its components R and :
01
2
222
2
22
Rr
m
x
R
xx
R
mDmC sincos
022
2
m
Well known solution
Bessel equation
Well known solution
C and D are constants
P and Q are constants.
Jm(x) is the mth order ordinary Bessel function of the first kind.
Ym(x) is the mth order Bessel function of the second kind.
67
RH z
)sinsincos(cos mmM 0 as )( xxYm
mDmC sincos )()( xQYxPJR mm (rewritten)
(rewritten)
sin
cos
MD
MC
)cos( mM
]2)cos[(
])2(cos[)cos(
mm
mm
)( rPJR m
C
D
DCM
1
2/122
tan
)(
mM cos
The value of at angle would be the same as that at an angle + 2 (the radial coordinate r remaining unchanged)
a relation that shall hold good only for integral values of m, that is, for m = 0, 1, 2, 3, ….
)cos( mM (rewritten)
Choosing ψ = 0 that amounts to taking the reference of such that it corresponds to the maximum value = M at = 0
which in turn would make
0 as xR
which is an impossibility since the field cannot shoot up to infinity at x (= r) 0, that is at r 0 (axis of the waveguide).
In order to prevent this impossibility we must put the constant Q = 0 in the expression for R.
)()( xQYxPJR mm 0Q
)(xPJR m
rx
68
mM
rPJR m
cos
)(
0zHPM
RH z
)(expcos)(0 ztjmrJHH mzz
(recalled)
mrPMJH mz cos)(
mrJHH mzz cos)(0
Invoking the understood factor )(exp ztj
Next we can find the rest of the field expressions with the help of the above axial component of magnetic field and the two of the following Maxwell’s equations:
t
EH
t
HE
0
0
(Maxwell’s equations)
69
zz
rr
zr
zrr
z
at
Ha
t
Ha
t
H
aE
r
rE
r
ar
E
z
Ea
z
EE
r
0
)(1
1
t
H
z
EE
rrz
0
1
t
E
r
H
z
H zr
0
zz
rr
zr
zrr
z
at
Ea
t
Ea
t
E
aH
r
rH
r
ar
H
z
Ha
z
HH
r
0
)(1
1
t
H
r
E
z
E zr
0
t
E
z
HH
rrz
0
1
t
EH
0
t
HE
0 Maxwell’s equations
(-component)
(r-component)(r-component)
(-component)rHE 0 r
z EjHjH
r 0
1
Ejr
HHj zr 0
HEr 0
0zE
0zE (TE mode)
0zE
70
Ejr
HHj
HE
zr
r
0
0
HE
EjHjH
r
r
rz
0
0
1
(recalled)
r
HjH
r
HjE
zr
z
2
20
zr
z
H
r
jE
H
r
jH
20
2
(recalled)
Eliminating Hr and E respectively
Eliminating Er and H respectively
mrJHH mzz cos)(0
)(expcos)(
)(expsin)(
0
0
ztjmrJHr
H
ztjmrmJHH
mzz
mzz
Taking partial derivatives
(recalled)
)()( rJdr
drJ mm
2/122
2/1200
2
)(
)(
k
71
r
HjH
r
HjE
zr
z
2
20
zr
z
H
r
jE
H
r
jH
20
2
(rewritten) (rewritten)
)(expsin)(
)(expsin)(
020
02
ztjmrJHr
mjE
ztjmrmJHr
jH
mzr
mz
)(expcos)(
)(expcos)(
0
00
ztjmrJHj
H
ztjmrJHj
E
mzr
mz
)(expcos)(
)(expsin)(
0
0
ztjmrJHr
H
ztjmrmJHH
mzz
mzz
(rewritten)
72
Field expressions of a cylindrical waveguide excited in the TEmn mode put together
)(expsin)(020 ztjmrJHr
mjE mzr
)(expcos)(00 ztjmrJH
jE mz
0zE
)(expcos)(0 ztjmrJHj
H mzr
)(expsin)(02ztjmrmJH
r
jH mz
)(expcos)(0 ztjmrJHH mzz
(m and n are the mode numbers of the cylindrical waveguide excited in the TEmn mode)
73
arn Ea
02
arE
0
rn aa
zr
rr
aaa
aa
0
zzrr aEaEaEE
2
Electromagnetic boundary conditions at the waveguide wall of a cylindrical waveguide
Boundary condition at the interface between a conductor and a dielectric/free-space introduced in Chapter 7, here the interface being the conducting waveguide wall of radius r = a, the subscript 2 referring to the free-space region inside the waveguide
Unit vector directed from region 1 (here, conducting waveguide wall) to region 2 (here, free-space region inside the waveguide), thus being radially inward at the waveguide wall
arrrr aEaEa
0)(
0)( aJm
arzaE 0
mnXa
(electromagnetic boundary condition at the conducting wall of the waveguide)
)(expcos)(00 ztjmrJH
jE mz
has a number of zeros or roots corresponding to its multiple solutions
(corresponding to the nth zero or root, where Xmn is called the eigenvalue of the cylindrical waveguide excited in the mode TEmn)
74
mnXa (corresponding to the nth zero or root, where Xmn is called the eigenvalue of the cylindrical waveguide excited in the mode TEmn)
(rewritten)
970.9,706.6,054.3
536.8,31.5,841.1
174.10,016.7,832.3
232221
131211
030201
XXX
XXX
XXX
mnXak 2/122 )(
2/1222/12222/1200
2 )()/()( kc
0222 ckk
mnXa
(a few lower order eigenvalues of the roots taken from easily available text on Bessel functions, the eigenvalue X11 = 1.841 corresponding to m =1, n =1 for the TE11 mode being the lowest of them)
Dispersion relation and cutoff frequency of a cylindrical waveguide
a
Xk mnc
ck
02222 cc
a
cXmnc
222 )(a
Xk mn
ck cc
(dispersion relation)
ca
X cmn
(cutoff frequency)
75
02222 cc Dispersion relation of a cylindrical waveguide is the same as that of a rectangular waveguide with appropriate interpretation of the waveguide cutoff frequency.
(rewritten)
Obviously, the nature of the - dispersion curve of a a cylindrical waveguide is identical with that of a rectangular waveguide.
2 2 2 2 0cc
cc -
line
a
cXmnc
Dominant mode of a cylindrical waveguide
For a cylindrical waveguide, here considered as excited in the TE mode, we have seen that the TE11 mode has the lowest eigenvalue X11 = 1.841 and correspondingly the lowest cutoff frequency c = Xmnc/a = 1.841c/a.
In fact, the analysis in the TM mode (which, though presented earlier for a rectangular waveguide, has not been done here for a cylindrical waveguide) would reveal that, of all the modes of the TE and TM modes of a cylindrical waveguide taken together, the TE11 mode has the lowest cutoff frequency.
Therefore,
TE11 mode is the dominant mode in a cylindrical waveguide
TE10 is the dominant mode is in a rectangular waveguide.
76
a
cfc 2
2
22
rect822
)area(cf
caaaba
cf
ca
2
841.1
2
222
circular4
)841.1()area(
cf
ca
a
cXfc
112
cf
ca
2
a
c
a
cXfc
2
841.1
211
In an illustrative example you will find it of interest to compare the cross-sectional area of a circular or cylindrical waveguide with that of a rectangular waveguide that has its wider dimension twice its narrower dimension and that has its cutoff frequency the same as that of the rectangular waveguide, taking both the waveguides excited in the dominant mode.
2/122
2
b
n
a
mcfc
(a = wider dimension; b = narrower dimension)
Rectangular waveguide Cylindrical waveguide
Dominant mode: m =1, n = 0
(cross-sectional area of the given rectangular waveguide in terms of the cutoff frequency, the latter being the same as that of the given circular or cylindrical waveguide in the example)
a
cXmnc
(recalled)
Dominant mode: m =1, n = 1
(replacing the symbol a with a/ to avoid the confusion with the symbol a for the narrow dimension of a rectangular waveguide)
)841.1( 11 X
16.2143.3
)841.1(2
)8/(
)4/()841.1(
)area(
)area( 2
22
222
rect
circular
c
c
fc
fc
77
Field pattern and significance of mode numbers of a cylindrical waveguide
)()( 10 xJxJ
)(exp832.3
)(exp832.3
0
)(exp832.3
0
00
10
100
ztjra
JHH
ztjra
JHj
H
E
ztjra
JHj
E
E
zz
zr
z
z
r
Field expressions of a cylindrical waveguide for typical lower-order modes TE01, TE02 and TE11 (interpreting the TEmn-mode field expressions already deduced):
(recurrence relation taking help of)
(TE01 mode)
78
)(exp016.7
)(exp016.7
0
)(exp016.7
0
00
10
100
ztjra
JHH
ztjra
JHj
H
E
ztjra
JHj
E
E
zz
zr
z
z
r
(TE02 mode)
)(expcos841.1
)(expsin841.1
)(expcos841.1
0
)(expcos841.1
)(expsin841.1
10
102
10
100
1020
ztjra
JHH
ztjra
JHr
jH
ztjra
JHj
H
E
ztjra
JHj
E
ztjra
JHr
jE
zz
z
zr
z
z
zr
(TE11 mode) )/841.1(
)/841.1()841.1/(
)/841.1(
2
1
1
arJ
arJra
arJ
(relation to be made use of)
79
(TE01 mode) (TE02 mode)
(TE0n mode; n = 1, 2, 3)
D
C
B
A
(TE11 mode)
Field pattern of typical lower-order cylindrical waveguide modes
The field pattern of the cylindrical waveguide has been more elaborately explained in the book. However, in what follows next let us discuss the significance of the mode number vis-à-vis the field pattern of a cylindrical waveguide.
Significance of mode numbers of a cylindrical waveguide TE01 mode:mode number m = 0 no half-wave field patterns around the half circumferencemode number n =1 one field maximum across the waveguide radius approximately mid-way between the axis and the wall of the waveguide TE02 mode: mode number m = 0 no half-wave field patterns around the half circumferencemode number n = 2 two maxima across the waveguide radius, between the axis and the wall of the waveguide, such that if one of the maxima is positive the other is negative. TE11 mode: mode number m = 1 a single half-wave field pattern around the half circumference (or a single full-wave field pattern around the full circumference) of the waveguidemode number n =1 across the waveguide radius, a single maximum at the axis of the waveguide Based on the observation for the TE11 mode, TE01 and the TE02 modes above, the meaning of the mode numbers of a cylindrical waveguide excited in the TEmn mode is
• m (azimuthal mode number) is the number of half-wave field patterns around the half waveguide circumference; •n (radial mode number) is the number of maxima, positive and negative inclusive, across the waveguide radius.
Such interpretation is valid for the TMmn mode as well.
D
C
B
A
(TE01 mode)
(TE02 mode)
(TE11 mode)
81
The TEM mode is characterized by transverse electric and magnetic fields and no axial electric and magnetic field components (Ez = Hz = 0).
However, for continuous magnetic field lines to exist in a hollow-pipe waveguide, there should be an axial current present in the waveguide in the form of either conduction or displacement current in the axial direction.
The absence of a conductor does not allow such conduction current in the axial direction. Further, for the displacement current in the axial direction to exist in the waveguide, there should be an axial electric field, the time variation of which being responsible for such current.
However, the TEM mode does not permit the axial electric field. Therefore, a hollow-pipe waveguide cannot support the TEM mode.
On the other hand, a two-conductor structure comprising a hollow cylindrical waveguide with a conducting coaxial solid circular rod insert, known as a coaxial waveguide, can support the TEM mode.
Inability of a hollow-pipe waveguide to support a TEM mode:
82
Power Flow and power loss in a waveguide
We can find power transmitted in the axial direction z of a rectangular waveguide by integrating the axial z-component of the average complex Poynting vector (see Chapter 8):
*average Re2/1P HE
ax
x
by
y
zdxdyHEP0 0
*)Re(2
1
***)( xyyxz HEHEHE
over the waveguide cross-sectional area (= ab) transverse to the axis z of the waveguide as follows:
ax
x
by
y
xyyx dxdyHEHEP0 0
** )Re(2
1
83
)(expsin/
)(expsin/
)(expsin/
0
0*
0
00
ztjxa
Ha
jH
ztjxa
Ha
jH
ztjxa
Ha
jE
HE
zx
zx
zy
yx
ax
x
by
y
z dydxa
xH
aaP
0 0
220
0 sin2
1
ax
x
by
y
xyyx dxdyHEHEP0 0
** )Re(2
1
(power transmitted in the axial direction z of a rectangular waveguide)
(recalled)
Restricting the analysis to the dominant mode TE10
(TE10 mode)
)(22
1 20
0 ba
Haa
P z
bdya
dxa
xax
x
by
y
0 0
2 ;2
sin
20
3024
1zbHaP
(TE10 mode)
(power transmitted in the axial direction z of a rectangular waveguide)
84
20
3024
1zbHaP
(power handling capability of the rectangular waveguide in the dominant TE10 mode)
Power handling capability of a waveguide
(power transmitted in the axial direction z of a rectangular waveguide)
a
EH
y
z0
maximun0
)(expsin00 ztjx
aH
ajE zy
Let us take a rectangular waveguide excited in the dominant TE10 mode.
(dominant-mode TE10 field expressions) (recalled)
Let us find an expression for the maximum permissible Pmaximum, that is, the power handling capability of the waveguide for a known magnitude of the maximum electric field amplitude
which the atmosphere of the inside of the waveguide can withstand before it breaks down.
2
maximun0
maximum )(4
1yEabP
maximunyE
(axial magnetic field amplitude in terms of the breakdown electric field)
85
m.1054.27.17.1 and m 1054.24.34.3 22 ba
2
maximun0
2/122
maximum )(4
)/1(y
c Ec
ffabP
In an illustrative example let us calculate the maximum power handling capability of WR-340 rectangular waveguide, excited in the dominant mode, operating at 3 GHz frequency, taking the breakdown limit of air as 29 kV/cm and taking the waveguide dimensions as:
2
maximun0
maximum )(4
1yEabP
(rewritten)
2/122
ph
ph
)/1/(1/
/2/
ffcv
fv
c
(recalled)
GHz 737.1Hz10737.1 6 cf
a
cfc 2
Recalling the expression for the cutoff frequency of a rectangular waveguide in the dominant mode, we can calculate:
m 1054.24.34.3 2a (dominant mode: m =1, n = 0)
86
m/s 103
H/m 1048
70
c
2252
maximun
2
24
4
)V/m()1029()(
335.0
m1029.37
1054.27.154.24.3
y
c
E
f
f
ab
MW 952.16 W10952.16
W10103143.344
29298155.029.37
6
94maximum
P
GHz 737.1Hz10737.1 6 cf(rewritten)
2
maximun0
2/122
maximum )(4
)/1(y
c Ec
ffabP
V/m1029kV/cm29 5
maximunyE
(given)
Hz103GHz3 6f
m1054.27.17.1
m 1054.24.34.32
2
b
a
(given)
(given)
87
*LA 2
1ssS JJRP
sn JHa 2
Power loss per unit area and power loss per unit length of a rectangular waveguide
Power loss per unit area at a conducting surface here the interface between the conducting wall and the inside of the rectangular waveguide is given by (see Chapter 8)
density current surface
resistance surface
areaunit per losspower LA
s
S
J
R
P
Surface current density at any of the four waveguide walls can be found from the following electromagnetic boundary condition at the concerned waveguide wall:
Recalling (from Chapter 7) the electromagnetic boundary condition at the interface between region 2, here the free-space region inside the waveguide wall, and region 1, here the conducting wall region
Let us next find the surface current densities developed at the right side wall (x = 0), left side wall (x = a), bottom wall (y = 0) and top wall (y = b) respectively.
a
b
Z
Y
X
inside) space-(free 2region to wall)g(conductin
1region waveguide thefrom directed
runit vecto theis na
88
zzxz aztjxa
Haztjxa
Haj
H
)(expcos)(expsin 002
zzyyxx aHaHaHH
2
sn JHa 2
mode) TE(
)(expcos
0
)(expsin
10
0
0
ztjxa
HH
H
ztjxa
Haj
H
zz
y
zx
xzxs aztjx
aH
ajaJ
)(expsin][ 0 sidewallright
(recalled)
zz aztjx
aH
)(expcos0
wall)side(right xn aa
(boundary condition) (recalled)
yzs aztjHJ
)(exp0 sidewallright
10cos
wall)side(right 0
0
x
aaa
aa
yzx
xx
89
zzxz aztjxa
Haztjxa
Haj
H
)(expcos)(expsin 002
sn JHa 2
xzxs aztjx
aH
ajaJ
)(expsin][ 0 sidewallleft
wall)side(left xn aa
(boundary condition) (recalled)
yzs aztjHJ
)(exp0 sidewallleft
zz aztjx
aH
)(expcos0
1cos
wall)side(left
0
ax
aaa
aa
yzx
xx
Similarly, we can derive next an expression for the surface current density developed at the left side wall.
90
zzxz aztjxa
Haztjxa
Haj
H
)(expcos)(expsin 002
sn JHa 2
xzys aztjx
aH
ajaJ
)(expsin][ 0 wallbottom
wall)(bottom yn aa
(boundary condition) (recalled)
xzzzs aztjxa
Haztjxa
Haj
J
)(expcos)(expsin 00 wallbottom
zz aztjx
aH
)(expcos0
wall)(bottom 0y
aaa
aaa
xzy
zxy
Next, let us derive next an expression for the surface current density developed at the bottom wall.
(TE10 mode)
91
zzxz aztjxa
Haztjxa
Haj
H
)(expcos)(expsin 002
sn JHa 2
xzys aztjx
aH
ajaJ
)(expsin][ 0 walltop
wall)(top yn aa
(boundary condition) (recalled)
xzzzs aztjxa
Haztjxa
Haj
J
)(expcos)(expsin 00 walltop
zz aztjx
aH
)(expcos0
wall)(top by
aaa
aaa
xzy
zxy
Next, let us derive next an expression for the surface current density developed at the top wall.
92
Expressions for surface current densities at the waveguide walls are required to find power loss per unit area PLA in terms of their surface resistance RS for each of these walls with the help of the expression:
xzzzs
xzzzs
yzs
yzs
aztjxa
Haztjxa
Haj
J
aztjxa
Haztjxa
Haj
J
aztjHJ
aztjHJ
)(expcos)(expsin
)(expcos)(expsin
)(exp
)(exp
00 walltop
00 wallbottom
0 sidewallleft
0 sidewallright
xzzzs
xzzzs
yzs
yzs
aztjxa
Haztjxa
Haj
J
aztjxa
Haztjxa
Haj
J
aztjHJ
aztjHJ
)(expcos)(expsin
)(expcos)(expsin
)(exp
)(exp
00 walltop*
00 wallbottom*
0 sidewallleft *
0 sidewallright *
Taking complex conjugate
(rewritten)
*LA 2
1ssS JJRP
(deduced in Chapter 8)
(TE10 mode)
93
])(exp[])(exp[ 00*
yzyzss aztjHaztjHJJ
])(exp[])(exp[ 00*
yzyzss aztjHaztjHJJ
(right side wall)
(left side wall)
])(expcos)(expsin[
])(expcos)(expsin[
00
00*
xzzz
xzzzss
aztjxa
Haztjxa
Haj
aztjxa
Haztjxa
Haj
JJ
(bottom wall)
])(expcos)(expsin[
])(exp)(expcos)(expsin[
00
00*
xzzz
xxzzzss
aztjxa
Haztjxa
Haj
aztjaztjxa
Haztjxa
Haj
JJ
(top wall)
Using the expressions for the surface current densities and those of their complex conjugates already presented, which are involved in the expression for power loss per unit area PLA, we can write the following expressions with respect to all the four sides of the waveguide wall:
We can next simplify the above expressions before they can be used in the expression for PLA.
94
xa
xa
aHR
aztjxa
Haztjxa
Haj
aztjxa
Haztjxa
Haj
RJJRP
zS
xzzz
xzzzSssS
222
222
0
00
00*
LA
cossin2
1
])(expcos)(expsin[
])(expcos)(expsin[2
1
2
1
20
00*
LA
2
1
])(exp[])(exp[2
1
2
1
zS
yzyzssS
HR
aztjHaztjHJJRP
(right side wall)
20
00*
LA
2
1
])(exp[])(exp[2
1
2
1
zS
yzyzssS
HR
aztjHaztjHJJRP
(left side wall)
(bottom wall)
xa
xa
aHR
aztjxa
Haztjxa
Haj
aztjxa
Haztjxa
Haj
RJJRP
zS
xzzz
xzzzSssS
222
222
0
00
00*
LA
cossin2
1
])(expcos)(expsin[
])(expcos)(expsin[2
1
2
1
(top wall)
95
xa
xa
aHRPP
HRPP
zS
zS
222
222
0 walltopLA, wallbottomLA,
20sidewallleft LA,sidewallrightLA,
cossin2
1
2
1
We can then put together the expressions for power loss per unit area on all the four walls of the waveguide:
dydzHRdP zS2
0LL 2
1
From the expression for power loss per unit area, we can also find power loss per unit length of the waveguide as follows.
Element of power loss dPL over an element of strip of elemental thickness dz and elemental area dydz across the narrow dimension of the waveguide on its right sidewall (x = 0)
(right sidewall)
Power loss per unit length PLL across the narrow dimension of the waveguide on its right sidewall bHR
dzdyHRdPP
zS
z
z
by
y
zS
20
1
00
20LLsidewallrightLL,
2
1
2
1
(right sidewall)
dydzHRdP zS2
0LL 2
1
Element of power loss dPL over an element of strip of elemental thickness dz and elemental area dydz across the narrow dimension of the waveguide on its left sidewall (x = a)
(left sidewall)
bHR
dzdyHRdPP
zS
z
z
by
yzS
20
1
00
20LLsidewallleftLL,
2
12
1
(left sidewall) Power loss per unit length PLL across the narrow dimension of the waveguide on its left sidewall
96
dxdzxa
xa
aHRdP zS
222
222
0LL cossin2
1Element of power loss dPL over an element of strip of elemental thickness dz and elemental area dxdz across the broad dimension of the waveguide on its bottom wall (y = 0)
(bottom wall)
Power loss per unit length PLL across the broad dimension of the waveguide on its bottom wall
1
00
222
222
0LLwallbottomLL, )cossin(2
1 z
z
ay
y
zS dzdxxa
xa
aHRdPP
x
ax
a
aHRP zS
222
222
0 wallbottomLA, cossin2
1
(bottom wall)
2
2cos1
cos
2
2cos1
sin
2
2
xax
a
xax
a
1
22
12
222
0wallbottomLL, aa
HRP zS
(using the relation
and evaluating the integral)
97
dxdzxa
xa
aHRdP zS
222
222
0LL cossin2
1Element of power loss dPL over an element of strip of elemental thickness dz and elemental area dxdz across the broad dimension of the waveguide on its top wall (y = b)
(top wall)
Power loss per unit length PLL across the broad dimension of the waveguide on its top wall
1
00
222
222
0LLwalltopLL, )cossin(2
1 z
z
ay
yzS dzdxx
ax
a
aHRdPP
x
ax
a
aHRP zS
222
222
0 walltopLA, cossin2
1
(top wall)
2
2cos1
cos
2
2cos1
sin
2
2
xax
a
xax
a
1
22
12
222
0walltopLL, aa
HRP zS
(using the relation
and evaluating the integral)
98
Power loss per unit length of the rectangular waveguide adding contributions of all the four walls
2
222
0
2
222
02
222
02
02
0LL
12
122
11
22
1
2
1
2
1
aabHR
aaHR
aaHRbHRbHRP
zS
zSzSzSzS
)(exp)exp()exp()exp(exp)exp( ztjzzjztjztj
)2exp()0()( zPzP
The above expression is useful in finding the expression for the attenuation constant of the waveguide.
j
Power loss caused by the finite resistivity of the waveguide wall may be accounted for by generalizing the dependence of field components as
propagation constant
*average Re2/1P HE
phase propagation constant attenuation constant
The amplitude of the field component decreases exponentially with z, with the factor exp (-z)
Power P (z) transmitted through the waveguide decreases exponentially with z, with the factor exp (-2z)
See for instance average complex Poynting or power density vector involving the electric and magnetic field components each decreasing with the factor exp (-z) making their product depending on the factor exp (-2z)
(power transmitted through the waveguide)
(TE10 mode)
99
)2exp()0()( zPzP (rewritten)
)(2)2exp()0(2)(
zPzPdz
zdP
)(2
)(
zPdzzdP
LL
)(P
dz
zdP
Differentiating
)(2LL
zP
P
P
P
2LL
dropping the parenthesis from P(z)
100
)(
1
2
12/1
2
2
2
2
0 a
cab
b c
c
cc
ba
aabRS
30
2
222 1
22
P
P
2LL (rewritten)
2
222
0LL 12
aabHRP zS
ba
aab
30
2
222 1
22
(recalled)
20
3024
1zbHaP
(recalled)
(TE10 mode)
(TE10 mode)
(TE10 mode)
1
sR
phv
(in terms of the conductivity and skin depth of the material of the waveguide wall; see Chapter 6)
(recalled)2/1
2
2
ph
1
1
c
c
v
(attenuation constant of the rectangular waveguide in the dominant TE10 mode)
101
In a numerical example, let us calculate the attenuation constant of a rectangular waveguide (a = 0.9// and b = 0.4//; cutoff frequency = 6.56 GHz) made of aluminum ( = 35.4x106 mho/m) at the operating frequency 9 GHz in the dominant mode.
mho/m104.35
Hz1056.6
Hz109
m1054.24.04.0
m1054.29.09.0
6
9
9
2
2
cf
f
b
a
0
1
f
2/1
2
2
2
2
0
01
21
1
ff
ff
ab
f
bc
c
)(
1
2
12/1
2
2
2
2
0 a
cab
b c
c
cc
After a simple algebra and recalling
(skin depth)
(TE10 dominant mode) (recalled)
(given)
154.00177.068.80177.0 Np/m dB/m0177.0
ohm 3770 H/m 104 7
0
102
2/1
2
2
2
2
01
2
1
c
cc
f
f
f
fab
f
f
b
01 f
2
f
fc
2/1
4/34/1
)1(
2
ab
G
cf
bG 0
0
1
In another interesting problem, let us find the value of the wave frequency relative to the cutoff frequency of a rectangular waveguide excited in the dominant mode that results in the minimum attenuation due to the finite conductivity of the material of the waveguide, in terms of the waveguide dimensions. Numerically appreciate the problem taking a = 1.8b and fc = 6.56 GHz.
)// nginterpreti (recalled ffcc
103
2/1
4/34/1
)1(
2
ab
G
0df
d
d
d
df
d
0)1(2
2)1(
32
1
)1(
24/94/54/54/1
2/1
a
bab
f
G
df
d
c
(rewritten)
2
f
fc (rewritten)
At the frequency where the minimum attenuation of the waveguide takes place
0)1(2
2)1(
32
1 4/94/54/54/1
ab
ab
Quantity outside the parenthesis being non-zero for practical values of
)/( 22 c
104
4/1by Dividing
0)63(2 2 abab 2
12)1(
3 2
a
b
a
b
abbaba
b
f
f
c 8)63()63(
42
b
abbaba
f
fc4
8)63()63( 22
4/14/94/54/5
2
12)1(
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106106
Waveguidea hollow pipe made of a conducting materialis extensively used for the transmission of power in the microwave frequency range.
Waveguide can support transverse electric (TE) mode, which is characterized by non-zero axial magnetic field and zero axial electric field
Waveguide can also support transverse magnetic TM mode, which is characterized by non-zero axial electric field and zero axial magnetic field.
Waveguide behaves as a high-pass filter supporting propagating waves above a cutoff frequency that is related to waveguide dimensions.
TE-mode and TM-mode field solutions for both the rectangular and cylindrical waveguides have been obtained.
Characteristic equation or dispersion relation of a waveguide can be found with the help of the field solutions and electromagnetic boundary condition that the tangential component of the electric field is nil at the conducting surface of the waveguide wall.
One and the same dispersion relation is obtained between the wave angular frequency and phase propagation constant of a waveguide for the TE and the TM modes involving their respective cutoff frequencies c.
- dispersion plots of Identical nature are generated for rectangular and cylindrical waveguides excited in TE or TM mode.
Summarising Notes
107107
Cutoff frequency of the waveguide is the frequency corresponding to zero value of phase propagation constant (which can be identified as the point of intersection between the - dispersion plot and the abscissa, that is, -axis of the plot).
Cutoff frequency of the waveguide depends on the waveguide dimensions and waveguide mode chosen.
Characteristic parameters of a waveguide are guide wavelength, phase propagation constant, phase velocity, group velocity and wave impedance, each of them depending on the operating frequency relative to the cutoff frequency of the waveguide.
Evanescent mode is supported by a waveguide below its cutoff frequency associated with no component of the average Poynting vector in the direction of wave propagation, corresponding to no power flow in the waveguide.
Mode numbers m and n are subscripted in the nomenclatures TEmn and the TMmn representing respectively the transverse electric and transverse magnetic modes of the waveguide.
Dominant mode of a waveguide is characterized by the lowest value of the cutoff frequencies of all the TEmn and the TMmn modes of the waveguide.
Dominant mode of a rectangular waveguide is the mode TE10.
Dominant mode of a cylindrical waveguide is the mode TE11.
108108
Mode numbers m and n of the TEmn and TMmn modes of a rectangular or cylindrical waveguide can be correlated with their respective field patterns across the waveguide cross section.
For a rectangular waveguide, the mode number m indicates the number of maxima (of any field component) along the broad dimension of the waveguide, while the mode number n indicates the number of maxima (of any field component) along the narrow dimension of the waveguide. Alternatively, you may interpret m and n as the numbers of half-wave field patterns across the broad and the narrow dimensions of the waveguide respectively.
For a cylindrical waveguide, the mode number m indicates the number of half-wave field pattern around the half circumference and n indicates the number of positive or negative maxima across the waveguide radius
Why a hollow-pipe waveguide cannot support transverse electromagnetic (TEM) mode, for which the axial electric field and the axial magnetic field are each nil, has been explained.
Expression for the power propagating through a rectangular waveguide above its cutoff frequency has been developed and hence the power handling capability of the waveguide has been found in terms of the breakdown voltage of the medium filling the hollow region of the waveguide for dominant-mode excitation of the waveguide.
109109
Expression for the power loss per unit length of the walls of a rectangular waveguide due to the finite resistivity of the material making the walls has been developed for dominant-mode excitation of the waveguide.
Expression for the attenuation constant of a dominant-mode-excited rectangular waveguide has been developed using the expressions for propagating power and power loss per unit length of the waveguide.
Attenuation constant of a waveguide depends on the operating frequency and the waveguide dimensions which should be taken into consideration while choosing the waveguide mode and frequency for lower waveguide attenuation.
Readers are encouraged to go through Chapter 9 of the book for more topics and more worked-out examples and review questions.