Sadiku, M.N.O., Demarest, K. “Wave Propagation”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
37
Wave Propagation
37.1Space Propagation
Propagation in Simple Media?Propagation in the Atmosphere
37.2Waveguides
Waveguide Modes?Rectangular Waveguides?Circular
Waveguides?Commercially Available Waveguides?Waveguide
Losses?Mode Launching
37.1 Space Propagation
Matthew N. O. Sadiku
This section summarizes the basic principles of electromagnetic (EM) wave propagation in space. The principles
essentially state how the characteristics of the earth and the atmosphere affect the propagation of EM waves.
Understanding such principles is of practical interest to communication system engineers. Engineers cannot
competently apply formulas or models for communication system design without an adequate knowledge of
the propagation issue.
Propagation of an EM wave may be regarded as a means of transferring energy or information from one
point (a transmitter) to another (a receiver). EM wave propagation is achieved through guided structures such
as transmission lines and waveguides or through space. Wave propagation through waveguides and microstrip
lines will be treated in Section 37.2. In this section, our major focus is on EM wave propagation in space and
the power resident in the wave.
For a clear understanding of the phenomenon of EM wave propagation, it is expedient to break the discussion
of propagation effects into categories represented by four broad frequency intervals [Collin, 1985]:
?Very low frequencies (VLF), 3–30 kHz
?Low-frequency (LF) band, 30–300 kHz
?High-frequency (HF) band, 3–30 MHz
?Above 50 MHz
In the first range, wave propagates as in a waveguide, using the earth’s surface and the ionosphere as boundaries.
Attenuation is comparatively low, and hence VLF propagation is useful for long-distance worldwide telegraphy
and submarine communication. In the second frequency range, the availability of increased bandwidth makes
standard AM broadcasting possible. Propagation in this band is by means of surface wave due to the presence
of the ground. The third range is useful for long-range broadcasting services via sky wave reflection and
refraction by the ionosphere. Basic problems in this band include fluctuations in the ionosphere and a limited
usable frequency range. Frequencies above 50 MHz allow for line-of-sight space wave propagation, FM radio
and TV channels, radar and navigation systems, and so on. In this band, due consideration must be given to
reflection from the ground, refraction by the troposphere, scattering by atmospheric hydrometeors, and mul-
tipath effects of buildings, hills, trees, etc.
Matthew N.O. Sadiku
Temple University
Kenneth Demarest
University of Kansas
? 2000 by CRC Press LLC
EM wave propagation can be described by two complementary models. The physicist attempts a theoretical
model based on universal laws, which extends the field of application more widely than currently known. The
engineer prefers an empirical model based on measurements, which can be used immediately. This section
presents complementary standpoints by discussing theoretical factors affecting wave propagation and the
semiempirical rules allowing handy engineering calculations. First, we consider wave propagation in idealistic
simple media, with no obstacles. We later consider the more realistic case of wave propagation around the
earth, as influenced by its curvature and by atmospheric conditions.
Propagation in Simple Media
The conventional propagation models, on which the basic calculation of radio links is based, result directly
from Maxwell’s equations:
? × D = r
v
(37.1)
? × B = 0 (37.2)
(37.3)
(37.4)
In these equations, E is electric field strength in volts per meter, H is magnetic field strength in amperes per
meter, D is electric flux density in coulombs per square meter, B is magnetic flux density in webers per square
meter, J is conduction current density in amperes per square meter, and r
v
is electric charge density in coulombs
per cubic meter. These equations go hand in hand with the constitutive equations for the medium:
D = eE (37.5)
B = mH (37.6)
J = sE (37.7)
where e = e
o
e
r
, m = m
o
m
r
, and s are the permittivity, the permeability, and the conductivity of the medium,
respectively.
Consider the general case of a lossy medium which is charge-free (r
v
= 0). Assuming time-harmonic fields
and suppressing the time factor e
jwt
, Eqs. (37.1) to (37.7) can be manipulated to yield Helmholtz’s wave
equations
?
2
E – g
2
E = 0 (37.8)
?
2
H
– g
2
H = 0 (37.9)
where g = a + jb is the propagation constant, a is the attenuation constant in nepers per meter or decibels
per meter, and b is the phase constant in radians per meter. Constants a and b are given by
(37.10)
?′ =-E
B?
?t
?′ =+H
D
J
?
?t
aw
ms
w
=
+
?
è
?
?
?
÷
-
é
?
ê
ê
ê
ù
?
ú
ú
ú
e
e2
11
2
? 2000 by CRC Press LLC
(37.11)
where w = 2pf is the frequency of the wave. The wavelength l and wave velocity u are given in terms of b as
(37.12)
(37.13)
Without loss of generality, if we assume that wave propagates in the z-direction and the wave is polarized
in the x-direction, solving the wave equations (37.8) and (37.9) results in
E(z,t) = E
o
e
– az
cos(wt – bz)a
x
(37.14)
(37.15)
where h = ÷h÷Dq
h
is the intrinsic impedance of the medium and is given by
(37.16)
Equations (37.14) and (37.15) show that as the EM wave travels in the medium, its amplitude is attenuated
according to e
– az
, as illustrated in Fig. 37.1. The distance d through which the wave amplitude is reduced by a
factor of e
–1
(about 37%) is called the skin depth or penetration depth of the medium, i.e.,
(37.17)
FIGURE 37.1 The magnetic and electric field components of a plane wave in a lossy medium.
bw
ms
w
=
+
?
è
?
?
?
÷
+
é
?
ê
ê
ê
ù
?
ú
ú
ú
e
e2
11
2
l
p
b
=
2
uf==
w
b
l
Ha(,) cos( )zt
E
etz
o z
y
=-
-
**h
wbq
a
h
**h
m
s
w
q
s
w
q
hh
=
+
?
è
?
?
?
÷
é
?
ê
ù
?
ú
=
￡￡°
/
tan2
,0 45
e
e
e
41
14
,
d
a
=
1
? 2000 by CRC Press LLC
The power density of the EM wave is obtained from the Poynting vector
P = E 2 H (37.18)
with the time-average value of
(37.19)
It should be noted from Eqs. (37.14) and (37.15) that E and H are everywhere perpendicular to each other
and also to the direction of wave propagation. Thus, the wave described by Eqs. (37.14) and (37.15) is said to
be plane-polarized, implying that the electric field is always parallel to the same plane (the xz-plane in this case)
and is perpendicular to the direction of propagation. Also, as mentioned earlier, the wave decays as it travels
in the z-direction because of loss. This loss is expressed in the complex relative permittivity of the medium
(37.20)
and measured by the loss tangent, defined by
(37.21)
The imaginary part e
r
￠￠ = s/we
o
corresponds to the losses in the medium. The refractive index of the medium
n is given by
(37.22)
Having considered the general case of wave propagation through a lossy medium, we now consider wave
propagation in other types of media. A medium is said to be a good conductor if the loss tangent is large (s >>
we) or a lossless or good dielectric if the loss tangent is very small (s << we). Thus, the characteristics of wave
propagation through other types of media can be obtained as special cases of wave propagation in a lossy
medium as follows:
1.Good conductors: s >> we, e = e
o
, m = m
o
m
r
2.Good dielectric: s << we, e = e
o
e
r
, m = m
o
m
r
3.Free space: s = 0, e = e
o
, m = m
o
where e
o
= 8.854210
–12
F/m is the free-space permittivity, and m
o
= 4p210
–7
H/m is the free-space permeability.
The conditions for each medium type are merely substituted in Eqs. (37.10) to (37.21) to obtain the wave
properties for that medium. The formulas for calculating attenuation constant, phase constant, and intrinsic
impedance for different media are summarized in Table 37.1.
P
E
e
o z
z
ave
=′
=
*
-
1
2
2
2
2
Re( )
cos
EH
a
**h
q
a
h
eeee
e
crrr
jj=￠- ￠￠=-
?
è
?
?
?
÷
1
s
w
tand
s
w
=
￠￠
￠
=
e
ee
r
r
n
c
= e
? 2000 by CRC Press LLC
The classical model of a wave propagation presented in this subsection helps us understand some basic
concepts of EM wave propagation and the various parameters that play a part in determining the motion of a
wave from the transmitter to the receiver. We now apply the ideas to the particular case of wave propagation
in the atmosphere.
Propagation in the Atmosphere
Wave propagation hardly occurs under the idealized conditions assumed in the previous subsection. For most
communication links, the analysis must be modified to account for the presence of the earth, the ionosphere,
and atmospheric precipitates such as fog, raindrops, snow, and hail. This will be done in this subsection.
The major regions of the earth’s atmosphere that are of importance in radio wave propagation are the
troposphere and the ionosphere. At radar frequencies (approximately 100 MHz to 300 GHz), the troposphere
is by far the most important. It is the lower atmosphere consisting of a nonionized region extending from the
earth’s surface up to about 15 km. The ionosphere is the earth’s upper atmosphere in the altitude region from
50 km to one earth radius (6370 km). Sufficient ionization exists in this region to influence wave propagation.
Wave propagation over the surface of the earth may assume one of the following three principal modes:
?Surface wave propagation along the surface of the earth
?Space wave propagation through the lower atmosphere
?Sky wave propagation by reflection from the upper atmosphere
These modes are portrayed in Fig. 37.2. The sky wave is directed toward the ionosphere, which bends the
propagation path back toward the earth under certain conditions in a limited frequency range (0–50 MHz
approximately). The surface wave is directed along the surface over which the wave is propagated. The space
wave consists of the direct wave and the reflected wave. The direct wave travels from the transmitter to the
receiver in nearly a straight path, while the reflected wave is due to ground reflection. The space wave obeys
the optical laws in that direct and reflected wave components contribute to the total wave. Although the sky
and surface waves are important in many applications, we will only consider space waves in this section.
Figure 37.3 depicts the electromagnetic energy transmission between two antennas in space. As the wave
radiates from the transmitting antenna and propagates in space, its power density decreases, as expressed ideally
in Eq. (37.19). Assuming that the antennas are in free space, the power received by the receiving antenna is
given by the Friis transmission equation [Liu and Fang, 1988]:
(37.23)
TABLE 37.1Attenuation Constant, Phase Constant, and Intrinsic Impedance for Different Media
Good Good
Conductor Dielectric Free
Lossy Medium s/we >> 1 s/we << 1 Space
Attenuation constant a
. 00
Phase constant b
Intrinsic impedance h 377
w
ms
w
e
e2
11
2
+
?
è
?
?
?
÷
-
é
?
ê
ê
ù
?
ú
ú
wms
2
w
ms
w
e
e2
11
2
+
?
è
?
?
?
÷
+
é
?
ê
ê
ù
?
ú
ú
wms
2
wme wm
oo
e
j
j
wm
sw+ e
wm
s2
1()+j
m
e
PGG
r
P
rrt t
=
?
è
?
?
?
÷
l
p4
2
? 2000 by CRC Press LLC
where the subscripts t and r, respectively, refer to transmitting and receiving antennas. In Eq. (37.23), P is the
power in watts, G is the antenna gain (dimensionless), r is the distance between the antennas in meters, and l
is the wavelength in meters. The Friis equation relates the power received by one antenna to the power
transmitted by the other provided that the two antennas are separated by r > 2d
2
/l, where d is the largest
dimension of either antenna. Thus, the Friis equation applies only when the two antennas are in the far-field
of each other. In case the propagation path is not in free space, a correction factor F is included to account for
the effect of the medium. This factor, known as the propagation factor, is simply the ratio of the electric field
intensity E
m
in the medium to the electric field intensity E
o
in free space, i.e.,
(37.24)
The magnitude of F is always less than unity since E
m
is always less than E
o
. Thus, for a lossy medium, Eq. (37.23)
becomes
(37.25)
For practical reasons, Eqs. (37.23) and (37.25) are commonly expressed in the logarithmic form. If all terms
are expressed in decibels (dB), Eq. (37.25) can be written in the logarithmic form as
FIGURE 37.2Modes of wave propagation.
FIGURE 37.3Transmitting and receiving antennas in free space.
F
E
E
m
o
=
PGG
r
PF
rrt t
=
?
è
?
?
?
÷
l
p4
2
2
**
? 2000 by CRC Press LLC
P = P + G + G – L – L (37.26)
r t r t o m
where P is power in decibels referred to 1 W (or simply dBW), G is gain in decibels, L
o
is free-space loss in
decibels, and L
m
is loss in decibels due to the medium.
The free-space loss is obtained from standard monograph or directly from
(37.27)
while the loss due to the medium is given by
L
m
= –20 log *F* (37.28)
Our major concern in the rest of the section is to determine L
o
and L
m
for two important cases of space
propagation that differ considerably from the free-space conditions.
Effect of the Earth
The phenomenon of multipath propagation causes significant departures from free-space conditions. The term
multipath denotes the possibility of EM wave propagation along various paths from the transmitter to the
receiver. In multipath propagation of an EM wave over the earth’s surface, two such paths exist: a direct path
and a path via reflection and diffractions from the interface between the atmosphere and the earth. A simplified
geometry of the multipath situation is shown in Fig. 37.4. The reflected and diffracted component is commonly
separated into two parts, one specular (or coherent) and the other diffuse (or incoherent), that can be separately
analyzed. The specular component is well defined in terms of its amplitude, phase, and incident direction. Its
main characteristic is its conformance to Snell’s law for reflection, which requires that the angles of incidence
and reflection be equal and coplanar. It is a plane wave and, as such, is uniquely specified by its direction. The
diffuse component, however, arises out of the random nature of the scattering surface and, as such, is nonde-
terministic. It is not a plane wave and does not obey Snell’s law for reflection. It does not come from a given
direction but from a continuum.
FIGURE 37.4Multipath geometry.
L
r
o
=
?
è
?
?
?
÷
20
4
log
p
l
? 2000 by CRC Press LLC
The loss factor F that accounts for the departures from free-space conditions is given by
F = 1 + G r
s
D S(q)e
–jD
(37.29)
where G is the Fresnel reflection coefficient, r
s
is the roughness coefficient, D is the divergence factor, S(q) is
the shadowing function, and D is the phase angle corresponding to the path difference. We now account for
each of these terms.
The Fresnel reflection coefficient G accounts for the electrical properties of the earth’s surface. Because the
earth is a lossy medium, the value of the reflection coefficient depends on the complex relative permittivity e
c
of the surface, the grazing angle y, and the wave polarization. It is given by
(37.30)
where
(37.31)
(37.32)
(37.33)
e
r
and s are the dielectric constant and conductivity of the surface; w and l are the frequency and wavelength
of the incident wave; and y is the grazing angle. It is apparent that 0 <÷G÷ < 1.
To account for the spreading (or divergence) of the reflected rays because of the earth’s curvature, we
introduce the divergence factor D. The curvature has a tendency to spread out the reflected energy more than
a corresponding flat surface. The divergence factor is defined as the ratio of the reflected field from curved
surface to the reflected field from flat surface [Kerr, 1951]. Using the geometry of Fig. 37.5, D is given by
(37.34)
where G = G
1
+ G
2
is the total ground range and a
e
= 6370 km is the effective earth radius. Given the transmitter
height h
1
, the receiver height h
2
, and the total ground range G, we can determine G
1
, G
2
, and y. If we define
(37.35)
(37.36)
G=
-
+
sin
sin
y
y
z
z
z
c
=-e cos
2
y for horizontal polarization
z
c
c
=
-e
e
cos
2
y
for vertical polarization
ee
e
e
cr
o
r
jj=- =-
s
w
sl60
D
GG
aG
e
.1
2
12
12
+
?
è
?
?
?
÷
-
siny
/
pahh
G
e
=++
é
?
ê
ê
ù
?
ú
ú
2
3
4
12
2
12
()
/
a=
-
é
?
ê
ê
ù
?
ú
ú
-
cos
()
1 12
3
2ahhG
p
e
? 2000 by CRC Press LLC
and assume h
1
￡ h
2
, G
1
￡ G
2
, using small angle approximation yields [Blake, 1986]
(37.37)
G
2
= G – G
1
(37.38)
(37.39)
(37.40)
The grazing angle is given by
(37.41)
or
(37.42)
FIGURE 37.5 Geometry of spherical earth reflection.
G
G
p
1
23
=+
+
?
è
?
?
?
÷
cos
pa
f
i
i
e
G
a
i==,,12
Rh aah i
ii eei i
=+ + =[ ( ) sin ( )] ,
/2212
4212f /,
y=
+-
é
?
ê
ê
ù
?
ú
ú
-
sin
1 11
2
1
2
1
2
2
ah h R
aR
e
e
yf=
++
+
é
?
ê
ê
ù
?
ú
ú
-
-
sin
()
1 11
2
1
2
11
1
2
2
ah h R
ahR
e
e
? 2000 by CRC Press LLC
Although D varies from 0 to 1, in practice D is a significant factor at low grazing angle y.
The phase angle corresponding to the path difference between direct and reflected waves is given by
(37.43)
The roughness coefficient r
s
takes care of the fact that the earth’s surface is not sufficiently smooth to produce
specular (mirrorlike) reflection except at a very low grazing angle. The earth’s surface has a height distribution
that is random in nature. The randomness arises out of the hills, structures, vegetation, and ocean waves. It is
found that the distribution of the heights of the earth’s surface is usually the Gaussian or normal distribution
of probability theory. If s
h
is the standard deviation of the normal distribution of heights, we define the
roughness parameters
(37.44)
If g < 1/8, specular reflection is dominant; if g > 1/8, diffuse scattering results. This criterion, known as Rayleigh
criterion, should only be used as a guideline since the dividing line between a specular and diffuse reflection
or between a smooth and a rough surface is not well defined [Beckman and Spizzichino, 1963]. The roughness
is taken into account by the roughness coefficient (0 < r
s
< 1), which is the ratio of the field strength after
reflection with roughness taken into account to that which would be received if the surface were smooth. The
roughness coefficient is given by
r
s
= exp[–2(2pg)
2
] (37.45)
The shadowing function S(q) is important at a low grazing angle. It considers the effect of geometric
shadowing—the fact that the incident wave cannot illuminate parts of the earth’s surface shadowed by higher
parts. In a geometric approach, where diffraction and multiple scattering effects are neglected, the reflecting
surface will consist of well-defined zones of illumination and shadow. As there will be no field on a shadowed
portion of the surface, the analysis should include only the illuminated portions of the surface. The phenomenon
of shadowing of a stationary surface was first investigated by Beckman in 1965 and subsequently refined by
Smith [1967] and others. A pictorial representation of rough surfaces illuminated at angle of incidence q (= 90° –
y) is shown in Fig. 37.6. It is evident from the figure that the shadowing function S(q) is equal to unity when
q = 0 and zero when q = p /2. According to Smith [1967],
(37.46)
where erfc(x) is the complementary error function,
(37.47)
and
D= + -
2
12
p
l
()RRR
d
g
h
=
sy
l
sin
S
a
B
()
(
q.
1
1
2
12
-
é
?
ê
ê
ù
?
ú
ú
+
erfc)
erfc) erf)((xxedt
t
x
=- =
-
￥
ò
1
2
2
p
? 2000 by CRC Press LLC
(37.48)
(37.49)
(37.50)
In Eq. (37.50) s
h
is the rms roughness height and s
l
is the correlation length. Alternative models for S(q) are
available in the literature. Using Eqs. (37.30) to (37.50), the loss factor in Eq. (37.29) can be calculated. Thus
(37.51)
(37.52)
Effect of Atmospheric Hydrometeors
The effect of atmospheric hydrometeors on satellite–earth propagation is of major concern at microwave
frequencies. The problem of scattering of electromagnetic waves by atmospheric hydrometeors has attracted
much interest since the late 1940s. The main hydrometeors that exist for long duration and have the greatest
interaction with microwaves are rain and snow. At frequencies above 10 GHz, rain has been recognized as the
most fundamental obstacle on the earth–space path. Rain has been known to cause attenuation, phase difference,
and depolarization of radio waves. For analog signals, the effect of rain is more significant above 10 GHz, while
for digital signals, rain effects can be significant down to 3 GHz. Attenuation of microwaves because of
precipitation becomes severe owing to increased scattering and beam energy absorption by raindrops, thus
impairing terrestrial as well as earth–satellite communication links. Cross-polarization distortion due to rain
has also engaged the attention of researchers. This is of particular interest when frequency reuse employing
signals with orthogonal polarizations is used for doubling the capacity of a communication system. A thorough
review on the interaction of microwaves with hydrometeors has been given by Oguchi [1983].
The loss due to a rain-filled medium is given by
L
m
= g(R) l
e
(R) p(R) (37.53)
where g is attenuation per unit length at rain rate R, l is the equivalent path length at rain rate R, and p(R) is
the probability in percentage of rainfall rate R.
FIGURE 37.6 Rough surface illuminated at an angle of incidence q.
B
a
eaa
a
=-
é
?
ê
ê
ù
?
ú
ú
1
4
1
2
p
erfc )(
a
s
=
cot q
2
s
h
l
==
s
s
rms surface slope
L
R
o
d
=
?
è
?
?
?
÷
20
4
log
p
l
L log
ms
j
DS e=- +
( )
[]
-
20 1 G
D
rq
? 2000 by CRC Press LLC
Attenuation is a function of the cumulative rain-rate distribution, drop-size distribution, refractive index of
water, temperature, and other variables. A rigorous calculation of g(R) incorporating raindrop-size distribution,
velocity of raindrops, and refractive index of water can be found in Sadiku [1992]. For practical engineering
purposes, what is needed is a simple formula relating attenuation to rain parameters. Such is found in the aR
b
empirical relationship, which has been used to calculate rain attenuation directly [Collin, 1985], i.e.,
g(R) = aR
b
dB/km (37.54)
where R is the rain rate and a and b are constants. At 0°C, the values of a and b are related to frequency f in
gigahertz as follows:
a = G
a
f
Ea
(37.55)
where G
a
= 6.39 2 10
–5
, E
a
= 2.03, for f < 2.9 GHz; G
a
= 4.21 2 10
–5
, E
a
= 2.42, for 2.9 GHz ￡ f ￡ 54 GHz;
G
a
= 4.09 2 10
–2
, E
a
= 0.699, for 54 GHz ￡ f < 100 GHz; G
a
= 3.38, E
a
= –0.151, for 180 GHz < f; and
b = G
b
f
Eb
(37.56)
where G
b
= 0.851, E
b
= 0.158, for f < 8.5 GHz; G
b
= 1.41, E
b
= –0.0779, for 8.5 GHz ￡ f < 25 GHz; G
b
= 2.63,
E
b
= –0.272, for 25 GHz ￡ f < 164 GHz; G
b
= 0.616, E
b
= 0.0126, for 164 GHz ￡ f.
The effective length l
e
(R) through the medium is needed since rain intensity is not uniform over the path.
Its actual value depends on the particular area of interest and therefore has a number of representations [Liu
and Fang, 1988]. Based on data collected in western Europe and eastern North America, the effective path
length has been approximated as [Hyde, 1984]
l
e
(R) = [0.00741R
0.766
+ (0.232 - 0.00018R) sin q]
–1
(37.57)
where q is the elevation angle.
The cumulative probability in percentage of rainfall rate R is given by [Hyde, 1984]
(37.58)
where M is the mean annual rainfall accumulation in milli-
meters and b is the Rice–Holmberg thunderstorm ratio.
The effect of other hydrometeors such as water vapor, fog,
hail, snow, and ice is governed by similar fundamental prin-
ciples as the effect of rain [Collin, 1985]. In most cases, how-
ever, their effects are at least an order of magnitude less than
the effect of rain.
Other Effects
Besides hydrometeors, the atmosphere has the composition
given in Table 37.2. While attenuation of EM waves by
hydrometeors may result from both absorption and scatter-
ing, gases act only as absorbers. Although some of these gases
do not absorb microwaves, some possess permanent electric
and/or magnetic dipole moment and play some part in
pR
M
eee
RRR
()
.
[. .( )( . )]
..
=+-+
---
8766
003 021 186
003 0258 163
bb
TABLE 37.2Composition of Dry Atmosphere
from Sea Level to about 90 km
Percent Percent
Constituent by Volume by Weight
Nitrogen 78.088 75.527
Oxygen 20.949 23.143
Argon 0.93 1.282
Carbon dioxide 0.03 0.0456
Neon 1.8 2 10
–3
1.25 2 10
–3
Helium 5.24 2 10
–4
7.24 2 10
–5
Methane 1.4 2 10
–4
7.75 2 10
–5
Krypton 1.14 2 10
–4
3.30 2 10
–4
Nitrous oxide 5 2 10
–5
7.60 2 10
–5
Xenon 8.6 2 10
–6
3.90 2 10
–5
Hydrogen 5 2 10
–5
3.48 2 10
–6
Source: D.C. Livingston, The Physics of Microwave
Propagation, Englewood Cliffs, N.J.: Prentice-Hall,
1970, p. 11. With permission.
? 2000 by CRC Press LLC
microwave absorption. For example, nitrogen molecules do not possess permanent electric or magnetic dipole
moment and therefore play no part in microwave absorption. Oxygen has a small magnetic moment, which
enables it to display weak absorption lines in the centimeter and millimeter wave regions. Water vapor is a
molecular gas with a permanent electric dipole moment. It is more responsive to excitation by an EM field
than is oxygen.
Defining Terms
Multipath: Propagation of electromagnetic waves along various paths from the transmitter to the receiver.
Propagation constant: The negative of the partial logarithmic derivative, with respect to the distance in the
direction of the wave normal, of the phasor quantity describing a traveling wave in a homogeneous
medium.
Propagation factor: The ratio of the electric field intensity in a medium to its value if the propagation took
place in free space.
Wave propagation: The transfer of energy by electromagnetic radiation.
Related Topic
35.1 Maxwell Equations
References
P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Random Surfaces, New York:
Macmillan, 1963.
L.V. Blake, Radar Range-Performance Analysis, Norwood, Mass.: Artech House, 1986, pp. 253–271.
R.E. Collin, Antennas and Radiowave Propagation, New York: McGraw-Hill, 1985, pp. 339–456.
G. Hyde, “Microwave propagation,” in Antenna Engineering Handbook, 2nd ed., R.C. Johnson and H. Jasik,
Eds., New York: McGraw-Hill, 1984, pp. 45.1–45.17.
D.E. Kerr, Propagation of Short Radio Waves, New York: McGraw-Hill (republished by Peter Peregrinus, London,
1987), 1951, pp. 396–444.
C.H. Liu and D.J. Fang, “Propagation,” in Antenna Handbook: Theory, Applications, and Design, Y.T. Lo and
S.W. Lee, Eds., New York: Van Nostrand Reinhold, 1988, pp. 29.1–29.56.
T. Oguchi, “Electromagnetic wave propagation and scattering in rain and other hydrometeors,” Proc. IEEE, vol.
71, pp. 1029–1078, 1983.
M.N.O. Sadiku, Numerical Techniques in Electromagnetics, Boca Raton, Fla.: CRC Press, 1992, pp. 96–116.
B.G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Ant. Prog., vol. 15, pp. 668–671,
1967.
Further Information
There are several sources of information dealing with the theory and practice of wave propagation in space.
Some of these are in the reference section. Journals such as Radio Science, IEE Proceedings Part H, and IEEE
Transactions on Antennas and Propagation are devoted to EM wave propagation. Radio Science is available from
the American Geophysical Union, 2000 Florida Avenue NW, Washington DC 20009; IEE Proceedings Part H
from IEE Publishing Department, Michael Faraday House, 6 Hills Way, Stevenage, Herts, SG1 2AY, U.K.; and
IEEE Transactions on Antennas and Propagation from IEEE, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ
08855-1331.
Other mechanisms that can affect EM wave propagation in space, not discussed in this section, include
clouds, dust, and the ionosphere. The effect of the ionosphere is discussed in detail in standard texts.
? 2000 by CRC Press LLC
37.2 Waveguides
Kenneth Demarest
Waveguide Modes
Any structure that guides electromagnetic waves can be considered a waveguide. Most often, however, this
term refers to closed metal cylinders that maintain the same cross-sectional dimensions over long distances.
Such a structure is shown in Fig. 37.7, which consists of a metal cylinder filled with a dielectric. When filled
with low-loss dielectrics (such as air), waveguides typically exhibit lower losses than transmission lines, which
makes them useful for transporting RF energy over relatively long distances. They are most often used for
frequencies ranging from 1 to 150 GHz.
Every type of waveguide has an infinite number of distinct electromagnetic field configurations that can exist
inside it. Each of these configurations is called a waveguide mode. The characteristics of these modes depend
upon the cross-sectional dimensions of the conducting cylinder, the type of dielectric material inside the
waveguide, and the frequency of operation.
Waveguide modes are typically classed according to the nature of the electric and magnetic field components
E
z
and H
z
. These components are called the longitudinal components of the fields. Several types of modes are
possible in waveguides:
TE modes: Transverse-electric modes, sometimes called H modes. These modes have E
z
= 0 at all points
within the waveguide, which means that the electric field vector is always perpendicular (i.e.,
transverse) to the waveguide axis. These modes are always possible in waveguides with
uniform dielectrics.
TM modes: Transverse-magnetic modes, sometimes called E modes. These modes have H
z
= 0 at all
points within the waveguide, which means that the magnetic field vector is perpendicular
to the waveguide axis. Like TE modes, they are always possible in waveguides with uniform
dielectrics.
EH modes: EH modes are hybrid modes in which neither E
z
nor H
z
are zero, but the characteristics of
the transverse fields are controlled more by E
z
than H
z
. These modes are often possible in
waveguides with inhomogeneous dielectrics.
HE modes: HE modes are hybrid modes in which neither E
z
nor H
z
are zero, but the characteristics of
the transverse fields are controlled more by H
z
than E
z
. Like EH modes, these modes are
often possible in waveguides with inhomogeneous dielectrics.
TEM modes: Transverse-electromagnetic modes, often called transmission line modes. These modes can
exist only when a second conductor exists within the waveguide, such as a center conductor
on a coaxial cable. Because these modes cannot exist in single, closed conductor structures,
they are not waveguide modes.
FIGURE 37.7A uniform waveguide with arbitrary cross section.
? 2000 by CRC Press LLC
Waveguide modes are most easily determined by first computing the longitudinal field components, E
z
and
H
z
, that can be supported by the waveguide. From these, the transverse components (such as E
x
and E
y
) can
easily be found simply by taking spatial derivatives of the longitudinal fields [Collin, 1992].
When the waveguide properties are constant along the z axis, E
z
and H
z
vary in the longitudinal direction
as E
z
, H
z
μ exp(wt – gz), where w = 2pf is the radian frequency of operation and g is a complex number of
the form
g = a + jb (37.59)
The parameters g, a, and b are called the propagation, attenuation, and phase constants, respectively, and j =
. When there are no metal or dielectric losses, g is always either purely real or imaginary. When g is real,
E
z
and H
z
have constant phase and decay exponentially with increasing z. When g is imaginary, E
z
and H
z
vary
in phase with increasing z but do not decay in amplitude. When this occurs, the fields are said to be propagating.
When the dielectric is uniform (i.e., homogeneous), E
z
and H
z
satisfy the scalar wave equation at all points
within the waveguide:
?
t
2
E
z
+ h
2
E
z
= 0 (37.60)
and
?
t
2
H
z
+ h
2
H
z
= 0 (37.61)
where
h
2
= (2p f)
2
me + g
2
= k
2
+ g
2
(37.62)
Here, m and e are the permeability and permittivity of the dielectric media, respectively, and k = 2pf is the
wavenumber of the dielectric. The operator ?
t
2
is called the transverse Laplacian operator. In Cartesian
coordinates,
Most of the properties of the allowed modes in real waveguides can usually be found by assuming that the
walls are perfectly conducting. Under this condition, E
z
= 0 and ?H
z
/?p = 0 at the waveguide walls, where p
is the direction perpendicular to the waveguide wall. When these conditions are imposed upon the general
solutions of Eqs. (37.60) and (37.61), it is found that only certain values of h are allowed. These values are
called the modal eigenvalues and are determined by the cross-sectional shape of the waveguide. Using Eq. (37.62),
the propagation constant g for each mode varies with frequency according to
(37.63)
where
(37.64)
1–
me
?= +
t
xy
2
2
2
2
2
?
?
?
?
ga b=+ = -
?
è
?
?
?
÷
jh
f
f
c
1
2
f
h
c
=
2pme
? 2000 by CRC Press LLC
The modal parameter f
c
has units hertz and is called the cut-off frequency of the mode it is associated with.
According to Eq. (37.63), when f > f
c
, the propagation constant g is imaginary and thus the mode is propagating.
On the other hand, when f < f
c
, g is real, which means that the fields decay exponentially with increasing values
of z. Modes operated at frequencies below their cut-off frequency are not able to propagate energy over long
distances and are called evanescent modes.
The dominant mode of a waveguide is the one with the lowest cut-off frequency. Although higher-order
modes are often useful for a variety of specialized uses of waveguides, signal distortion is usually minimized
when a waveguide is operated in the frequency range where only the dominant mode exists. This range of
frequencies is called the dominant range of the waveguide.
The distance over which the fields of propagating modes repeat themselves is called the guide wavelength
l
g
. From Eq. (37.63), it can be shown that l
g
always varies with frequency according to
(37.65)
where l
o
= 1/(f ) is the wavelength of a plane wave of the same frequency in an infinite sample of the
waveguide dielectric. For f >> f
c
, l
g
?
l
o
. Also, l
g
? ￥ as f ? f
c
, which is one reason why it is usually undesirable
to operate a waveguide mode near modal cut-off frequencies.
Although waveguide modes are not plane waves, the ratio of their transverse electric and magnetic field
magnitudes is constant throughout the cross section of the waveguide, just as for plane waves. This ratio is
called the modal wave impedance and has the following values for TE and TM modes:
(37.66)
and
(37.67)
where E
T
and H
T
are the magnitudes of the transverse electric and magnetic fields, respectively. In the limit as
f ? ￥, both Z
TE
and Z
TM
approach , which is the intrinsic impedance of the dielectric medium. On the
other hand, as f ? f
c
, Z
TE
? ￥ and Z
TM
? 0, which means that the transverse electric fields are dominant in
TE modes near cut-off and vice versa for TM modes.
Rectangular Waveguides
A rectangular waveguide is shown in Fig. 37.8. The conducting walls are formed such that the inner surfaces
form a rectangular cross section, with dimensions a and b along the x and y coordinate axes, respectively.
If the walls are perfectly conducting and the dielectric material is lossless, the field components for the TE
mn
modes are given by
(37.68)
l
l
g
o
c
f
f
=
-
?
è
?
?
?
÷
1
2
me
Z
E
H
j
TE
T
T
==
wm
g
Z
E
Hj
TM
T
T
==
g
we
me
EH
j
h
n
b
m
a
x
n
b
yjtz
x
mn
mn
=
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
wm p p p
wgcos sin exp( )
? 2000 by CRC Press LLC
(37.69)
E
z
= 0
(37.70)
(37.71)
(37.72)
where
(37.73)
For the TM
mn
modes, m and n can be any positive integer value, including zero, as long as both are not zero.
The field components for the TM
mn
modes are
(37.74)
(37.75)
(37.76)
FIGURE 37.8 A rectangular waveguide.
EH
j
h
m
a
m
a
x
n
b
yjtz
y
mn
mn
=-
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
wm p p p
wgsin cos exp( )
HH
h
m
a
m
a
x
n
b
yjtz
x
mn
mn
mn
=
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
gp p p
wgsin cos exp( )
HH
h
n
b
m
a
x
n
b
yjtz
y
mn
mn
mn
=
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
gp p p
wgcos sin exp( )
HH
m
a
x
n
b
yjtz
zm
=
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
cos cos exp( )
pp
wg
h
m
a
n
b
f
mn c
mn
=
?
è
?
?
?
÷
+
?
è
?
?
?
÷
=
pp
pm
22
2 e
EE
h
m
a
m
a
x
n
b
yjtz
x
mn
mn
mn
=-
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
gp p p
wgcos sin exp( )
EE
h
n
b
m
a
x
n
b
yjtz
y
mn
mn
mn
=-
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
gp p p
wgsin cos exp( )
EE
m
a
x
n
b
yjtz
zm
=
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
sin sin exp( )
pp
wg
? 2000 by CRC Press LLC
(37.77)
(37.78)
H
z
= 0 (37.79)
where the values of h
mn
and f
cmn
are given by Eq. (37.73). For the TM
mn
modes, m and n can be any positive
integer value except zero.
The dominant mode in a rectangular waveguide is the TE
10
mode, which has a cut-off frequency
(37.80)
where c is the speed of light in the dielectric media. The modal field patterns for this mode are shown in Fig. 37.9.
Table 37.3 shows the cut-off frequencies of the lowest-order rectangular waveguide modes (as referenced to
the cut-off frequency of the dominant mode) when a/b = 2.1. The modal field patterns for several lower-order
modes are shown in Fig. 37.10.
Circular Waveguides
A circular waveguide with inner radius a is shown in Fig. 37.11. Here the axis of the waveguide is aligned with
the z axis of a circular-cylindrical coordinate system, where r and f are the radial and azimuthal coordinates,
FIGURE 37.9Field configurations for the TE
10
(dominant) mode of a rectangular waveguide. Solid lines, E; dashed lines, H.
(Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGraw-
Hill, 1986, p. 63. With permission.)
TABLE 37.3Cut-off Frequencies of the
Lowest-Order Rectangular Waveguide
Modes (Referenced to the Cut-off
Frequency of the Dominant Mode) for a
Rectangular Waveguide with a/b = 2.1
f
c
/f
c
10
Modes
1.0 TE
10
2.0 TE
20
2.1 TE
01
2.326 TE
11
,
TM
11
2.9 TE
21
,
TM
21
3.0 TE
30
3.662 TE
31
,
TM
31
4.0 TE
40
HE
j
h
n
b
m
a
x
n
b
yjtz
x
mn
mn
=
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
wp p p
wg
e
sin cos exp( )
HE
j
h
m
a
m
a
x
n
b
yjtz
y
mn
mn
=-
?
è
?
?
?
÷
?
è
?
?
?
÷
?
è
?
?
?
÷
-
0
2
wp p p
wg
e
cos sin exp( )
f
a
c
a
c
10
1
2
2
==
me
? 2000 by CRC Press LLC
respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the
TE
nm
modes are
(37.81)
(37.82)
E
z
= 0 (37.83)
(37.84)
(37.85)
(37.86)
where n is any positive valued integer, including zero, and J
n
(x) and J ￠
n
(x) are the regular Bessel function of
order n and its first derivative, respectively. The allowed values of the modal eigenvalues h
nm
satisfy
J ￠
n
(h
nm
a) = 0 (37.87)
FIGURE 37.10 Field configurations for the TE
11
, TM
11
, and the TE
21
modes. Solid lines, E; dashed lines, H. (Source: Adapted
from N. Marcuvitz, Waveguide Handbook, 2nd. ed., London: Peter Peregrinus Ltd., and New York: McGraw-Hill, 1986, p. 59.
With permission.)
FIGURE 37.11 A circular waveguide.
EH
jn
h
Jh n jt z
nm
nnm nmr
wm
r
rf wg=-
0
2
( ) sin( ) exp( )
EH
j
h
Jh n jt z
nm
nnm nmf
wm
rfwg= ￠ -
0
( ) cos ( ) exp( )
HH
h
Jh n jt z
nm
nm
nnm nmr
g
rfwg=- ￠ -
0
( ) cos ( ) exp( )
HH
h
Jh n jt z
nm
nm
nnm nmf
g
r
rfwg
0
2
( ) sin ( ) exp( )
HHJh n jt z
z n nm nm
=-
0
( ) cos ( ) exp( )rfwg
? 2000 by CRC Press LLC
where m signifies the root number of Eq. (37.87). By convention, 1 < m < ￥, where m = 1 indicates the smallest
root.
The equations that define the TM
nm
modes in circular waveguides are
(37.88)
(37.89)
(37.90)
(37.91)
(37.92)
H
z
= 0 (37.93)
where n is any positive valued integer, including zero. For the TM
nm
modes, the values of the modal eigenvalues
are solutions of
J
n
(h
nm
a) = 0 (37.94)
where m signifies the root number of Eq. (37.94). As in the case of the TE modes, 1 < m < ￥.
The dominant mode in a circular waveguide is the TE
11
mode, which has a cut-off frequency given by
(37.95)
The configuration of the electric and magnetic fields of this mode is shown in Fig. 37.12.
FIGURE 37.12 Field configuration for the TE
11
(dominant) mode of a circular waveguide. Solid lines, E; dashed lines, H.
(Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGraw-
Hill, 1986, p. 68. With permission.)
EE
h
Jh n jt z
nm
nm
nnm nmr
g
rfwg=- ￠ -
0
( )cos( )exp( )
EE
h
Jh n jt z
nm
nm
nnm nmf
g
r
rfwg
0
2
( )sin( )exp( )
EEJh n jtz
z n nm nm
=-
0
( )cos( )exp( )rfwg
HE
jn
h
Jh n jt z
nm
nnm nmr
w
r
rfwg=- -
0
2
e
( )sin( )exp( )
HE
j
h
Jh n jt z
nm
nnm nmf
w
rfwg=- ￠ -
0
e
( )cos( )exp( )
f
a
c
11
0293
=
.
me
? 2000 by CRC Press LLC
Table 37.4 shows the cut-off frequencies of the lowest-order modes for circular waveguides, referenced to
the cut-off frequency of the dominant mode. The modal field patterns for several lower-order modes are shown
in Fig. 37.13.
Commercially Available Waveguides
The dimensions of standard rectangular waveguides are given in Table 37.5.
In addition to rectangular and circular waveguides, several other waveguide types are commonly used in
microwave applications. Among these are ridge waveguides and elliptical waveguides. The modes of elliptical
waveguides can be expressed in terms of Mathieu functions [Kretzschmar, 1970] and are similar to those of
circular waveguides but are less perturbed by minor twists and bends of the waveguide. This property makes
them attractive for coupling to antennas.
Single-ridge and double-ridge waveguides are shown in Fig. 37.14. The modes of these waveguides bear
similarities to those of rectangular guides, but can only be derived numerically [Montgomery, 1971]. Ridge
waveguides are useful because their dominant ranges exceed those of rectangular waveguides. However, this
range increase is obtained at the expense of higher losses.
Waveguides are also available in a number of constructions, including rigid, semirigid, and flexible. In
applications where it is not necessary for the waveguide to bend, rigid construction is always the best since it
exhibits the lowest loss. In general, the more flexible the waveguide construction, the higher the loss.
Waveguide Losses
There are two mechanisms that cause losses in waveguides: dielectric losses and metal losses. In both cases,
these losses cause the amplitudes of the propagating modes to decay as exp(- az), where a is the attenuation
constant, measured in units of nepers/meter. Typically, the attenuation constant is considered as the sum of
TABLE 37.4Cut-off Frequencies of
the Lowest-Order Circular Waveguide
Modes, Referenced to the Cut-off
Frequency of the Dominant Mode
f
c
/f
c
11
Modes
1.0 TE
11
1.307 TM
01
1.66 TE
21
2.083 TE
01
,
TM
11
2.283 TE
31
2.791 TM
21
2.89 TE
41
3.0 TE
12
FIGURE 37.13Field configurations for the TM
01
, TE
21
, and TE
01
circular waveguide modes. Solid lines, E; dashed lines, H.
(Source: Adapted from N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., and New York: McGraw-
Hill, 1986, p. 71. With permission.)
? 2000 by CRC Press LLC
TABLE 37.5 Standard Rectangular Waveguides
Cut-off Recommended
Frequency for Frequency
EIA
a
Designation
Physical Dimensions
Air-filled Range for
Inside, cm (in.) Outside, cm (in.)
Waveguide, TE
10
Mode,
WR
b
( ) Width Height Width Height GHz GHZ
2300 58.420 29.210 59.055 29.845 0.257 0.32–0.49
(23.000) (11.500) (23.250) (11.750)
2100 53.340 26.670 53.973 27.305 0.281 0.35–0.53
(21.000) (10.500) (21.250) (10.750)
1800 45.720 22.860 46.350 23.495 0.328 0.41–0.62
(18.000) (9.000) (18.250) (9.250)
1500 38.100 19.050 38.735 19.685 0.394 0.49–0.75
(15.000) (7.500) (15.250) (7.750)
1150 29.210 14.605 29.845 15.240 0.514 0.64–0.98
(11.500) (5.750) (11.750) (6.000)
975 24.765 12.383 25.400 13.018 0.606 0.76–1.15
(9.750) (4.875) (10.000) (5.125)
770 19.550 9.779 20.244 10.414 0.767 0.96–1.46
(7.700) (3.850) (7.970) (4.100)
650 16.510 8.255 16.916 8.661 0.909 1.14–1.73
(6.500) (3.250) (6.660) (3.410)
510 12.954 6.477 13.360 6.883 1.158 1.45–2.20
(5.100) (2.500) (5.260) (2.710)
430 10.922 5.461 11.328 5.867 1.373 1.72–2.61
(4.300) (2.150) (4.460) (2.310)
340 8.636 4.318 9.042 4.724 1.737 2.17–3.30
(3.400) (1.700) (3.560) (1.860)
284 7.214 3.404 7.620 3.810 2.079 2.60–3.95
(2.840) (1.340) (3.000) (1.500)
229 5.817 2.908 6.142 3.233 2.579 3.22–4.90
(2.290) (1.145) (2.418) (1.273)
187 4.755 2.215 5.080 2.540 3.155 3.94–5.99
(1.872) (0.872) (2.000) (1.000)
159 4.039 2.019 4.364 2.344 3.714 4.64–7.05
(1.590) (0.795) (1.718) (0.923)
137 3.485 1.580 3.810 1.905 4.304 5.38–8.17
(1.372) (0.622) (1.500) (0.750)
112 2.850 1.262 3.175 1.588 5.263 6.57–9.99
(1.122) (0.497) (1.250) (0.625)
90 2.286 1.016 2.540 1.270 6.562 8.20–12.50
(0.900) (0.400) (1.000) (0.500)
75 1.905 0.953 2.159 1.207 7.874 9.84–15.00
(0.750) (0.375) (0.850) (0.475)
62 1.580 0.790 1.783 0.993 9.494 11.90–18.00
(0.622) (0.311) (0.702) (0.391)
51 1.295 0.648 1.499 0.851 11.583 14.50–22.00
(0.510) (0.255) (0.590) (0.335)
42 1.067 0.432 1.270 0.635 14.058 17.60–26.70
(0.420) (0.170) (0.500) (0.250)
34 0.864 0.432 1.067 0.635 17.361 21.70–33.00
(0.340) (0.170) (0.420) (0.250)
28 0.711 0.356 0.914 0.559 21.097 26.40–40.00
(0.280) (0.140) (0.360) (0.220)
22 0.569 0.284 0.772 0.488 26.362 32.90–50.10
(0.224) (0.112) (0.304) (0.192)
19 0.478 0.239 0.681 0.442 31.381 39.20–59.60
(0.188) (0.094) (0.268) (0.174)
15 0.376 0.188 0.579 0.391 39.894 49.80–75.80
(0.148) (0.074) (0.228) (0.154)
? 2000 by CRC Press LLC
two components: a = a
die
+ a
met
, where a
die
and a
met
are the dielectric and metal attenuation constants,
respectively.
The attenuation constant a
die
can be found directly from Eq. (37.63) simply by generalizing the dielectric
wavenumber k to include the effect of the dielectric conductivity s. For a lossy dielectric, the wavenumber is
given by k
2
= w
2
me[1 + (s/jwe)]. Thus, from Eqs. (37.62) and (37.63) the attenuation constant a
die
due to
dielectric losses is given by
(37.96)
where the allowed values of h are given by Eq. (37.73) for rectangular modes and Eqs. (37.87) and (37.94) for
circular modes.
12 0.310 0.155 0.513 0.358 48.387 60.50–91.90
(0.122) (0.061) (0.202) (0.141)
10 0.254 0.127 0.457 0.330 59.055 73.80–112.00
(0.100) (0.050) (0.180) (0.130)
8 0.203 0.102 0.406 0.305 73.892 92.20–140.00
(0.080) (0.040) (0.160) (0.120)
7 0.165 0.084 0.343 0.262 90.909 114.00–173.00
(0.065) (0.033) (0.135) (0.103)
5 0.130 0.066 0.257 0.193 115.385 145.00–220.00
(0.051) (0.026) (0.101) (0.076)
4 0.109 0.056 0.211 0.157 137.615 172.00–261.00
(0.043) (0.022) (0.083) (0.062)
3 0.086 0.043 0.163 0.119 174.419 217.00–333.00
(0.034) (0.017) (0.064) (0.047)
a
Electronic Industry Association.
b
Rectangular waveguide.
FIGURE 37.14Single- and double-ridged waveguides.
TABLE 37.5 (continued) Standard Rectangular Waveguides
Cut-off Recommended
Frequency for Frequency
EIA
a
Designation
Physical Dimensions
Air-filled Range for
Inside, cm (in.) Outside, cm (in.)
Waveguide, TE
10
Mode,
WR
b
( ) Width Height Width Height GHz GHZ
a wm
s
w
die
2
real= -+
?
è
?
?
?
÷
é
?
ê
ê
ù
?
ú
ú
h
j
2
1e
e
? 2000 by CRC Press LLC
The metal loss constant a
met
is usually obtained by assuming that the wall conductivity is high enough to
have only a negligible effect on the transverse properties of the modal field patterns. Using this assumption,
the power loss in the walls per unit distance along the waveguide can then be calculated to obtain a
met
[Marcuvitz, 1986]. Figure 37.15 shows the metal attenuation constants for several circular waveguide modes,
each normalized to the resistivity R
s
of the walls, where R
s
= and where m and s are the permeability
and conductivity of the metal walls, respectively. As can be seen from this figure, the TE
0m
modes exhibit
particularly low loss at frequencies significantly above their cut-off frequencies, making them useful for trans-
porting microwave energy over large distances.
Mode Launching
When coupling electromagnetic energy into a waveguide, it is important to ensure that the desired modes are
excited and that reflections back to the source are minimized. Similar concerns must be considered when
FIGURE 37.15Values of metallic attenuation constant a for the first few waveguide modes in a circular waveguide of
diameter d, plotted against normalized wavelength. (Source: A.J. Baden Fuller, Microwaves, 2nd ed., New York: Pergamon
Press, 1979, p. 138. With permission.)
pfms¤()
? 2000 by CRC Press LLC
coupling energy from a waveguide to a transmission line or circuit element. This is achieved by using launching
(or coupling) structures that allow strong coupling between the desired modes on both structures.
Figure 37.16 shows a mode launching structure for coaxial cable to rectangular waveguide transitions.
This structure provides good coupling between the TEM (transmission line) mode on a coaxial cable and
the TE
10
mode in the waveguide because the antenna probe excites a strong transverse electric field in the center
of the waveguide, directed between the broad walls. The distance between the probe and the short circuit back
wall is chosen to be approximately l/4, which allows the TE
10
mode launched in this direction to reflect off the
short circuit and arrive in phase with the mode launched towards the right.
Launching structures can also be devised to launch higher-order modes. Mode launchers that couple the
transmission line mode on a coaxial cable to the TM
11
and TM
21
waveguide mode are shown in Fig. 37.17.
Defining Terms
Cut-off frequency: The minimum frequency at which a waveguide mode will propagate energy with little or
no attenuation.
Guide wavelength: The distance over which the fields of propagating modes repeat themselves in a waveguide.
Waveguide: A closed metal cylinder, filled with a dielectric, used to transport electromagnetic energy over
short or long distances.
Waveguide modes:Unique electromagnetic field configurations supported by a waveguide that have distinct
electrical characteristics.
Wave impedance:The ratio of the transverse electric and magnetic fields inside a waveguide.
Related Topics
35.1 Maxwell Equations?39.1 Passive Microwave Devices?42.1 Lightwave Waveguides
FIGURE 37.16Coaxial to rectangular waveguide transition that couples the transmission line mode to the dominant
waveguide mode
FIGURE 37.17Coaxial to rectangular waveguide transitions that couple the transmission line mode to the TM
11
and TM
21
waveguide modes.
Antenna
probe
TE
10
mode
2
l
Antenna
probe
Short-circuited
end
TM
11
mode TM
21
mode
2
l
? 2000 by CRC Press LLC
References
A. J. Baden Fuller, Microwaves, 2nd ed., New York: Pergamon Press, 1979.
R. E. Collin, Foundations for Microwave Engineering, 2nd ed., New York: McGraw-Hill, 1992.
J. Kretzschmar, “Wave propagation in hollow conducting elliptical waveguides,” IEEE Transactions on Microwave
Theory and Techniques, vol. MTT-18, no. 9, pp. 547–554, Sept. 1970.
S. Y. Liao, Microwave Devices and Circuits, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1990.
N. Marcuvitz, Waveguide Handbook, 2nd ed., London: Peter Peregrinus Ltd., 1986.
J. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” IEEE Transactions on Microwave
Theory and Techniques, vol. MTT-19, no. 6, pp. 457–555, June 1971.
Further Information
There are many textbooks and handbooks that cover the subject of waveguides in great detail. In addition to
the references cited above, others include
L. Lewin, Theory of Waveguides, New York: John Wiley, 1975.
Reference Data for Radio Engineers, Howard W. Sams Co., 1975.
R. E. Collin, Field Theory of Guided Waves, 2nd ed., Piscataway, N.J.: IEEE Press, 1991.
F. Gardiol, Introduction to Microwaves, Dedham, Mass.: Artech House, 1984.
S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, New York: John Wiley,
1965.
? 2000 by CRC Press LLC
