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Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 1/24 Part II
Division of Wireless Communication Engineering Systems
REFERENCES
1) Radio Wave Propagation for Telecommunication Applications; H.
Sizun, Springer, 2005.
2) Introduction to RF propagation; John S. Seybold, Ph.D., John Wiley
& sons, Inc., 2005.
3) Antennas and Radio wave Propagation; Robert E. Collin, International
Student Edition.
4) Antenna: Introductory Topics in Electronics and Telecommunication;
Frank Robert Connor
5) Radio wave propagation and antennas; an introduction; John
Griffiths, 621-3841'35.
6) Antenna and Wave propagation; A. K. Gautam, Published by S. K.
Kataria & Sons, Delhi 2003.
7) Electromagnetic Waves and Radiating systems; Edward C. Jordan,
Prentice-Hall, Inc., 1968.
8) Field and Wave Electromagnetics; David K. Cheng, Addison-Wesley
Publishing Company.
9) Mathematical Handbook of Formulas and Tables; Murray R. Spiegel,
Schaum's outlines series in mathematics, McGraw Hill Book Company,
1968.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 2/24 Part II
2. Electromagnetic Fundamentals
2.1. Electromagnetics Field Components
Electric field intensity [strength] ï¿œÌ ï¿œ (V ðâ )
Magnetic field intensity ï¿œÌ ï¿œ (A mâ )
The electric and magnetic flux densities D, B are related to the field intensities
E, H via the so-called constitutive relations, whose precise form depends on
the material in which the fields exist. The simplest form of the constitutive
relations is for simple homogeneous isotropic dielectric and for magnetic
materials:
Electric flux density [displacement ï¿œÌ ï¿œ = Ïµï¿œÌ ï¿œ C/m2 (coulombs/m2)
Current density]
Magnetic flux density [the magnetic ï¿œÌ ï¿œ = ÎŒï¿œÌ ï¿œ (Tesla = weber/m2)
induction]
Where:
ð (epsilon) = ðððð F mâ (farad/m) is the permittivity of the medium.
ðð â 1
36ðà 10â9 â 8.854 à 10â12 (F mâ ) Permittivity [dielectric constant]
of a vacuum or free space.
ðð Is the relative permittivity of medium (dimensionless).
ð (mu) = ðððð H/m (henry/m) is the permeability of the medium.
ðð = 4ð Ã 10â7 (H mâ ) Permeability of a free space.
ðð The relative permeability of a material (dimensionless).
The units for ðð and ðð are the units of the ratios D/E and B/H, that is,
coulomb m2â
volt mâ=
coulomb
voltâm=
farad
m ,
weber m2â
ampere mâ=
weber
ampereâm=
henry
m
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 3/24 Part II
2.2. Maxwellâs Equations
2.2.1. Gaussâs law
States that the total flux of ï¿œÌ ï¿œ = Ïµï¿œÌ ï¿œ from a volume (ð) is equal to
the net charge contained within ð.
The net charge contained within ð is ð = â« ð ðð£ ð
(coulomb).
Where ð (rho) is the volume charge density (coulombs/m3).
Then, Gaussâs law may be written as
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œS
= â« ð ðð£
ð
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œS
= ð1 + ð2 = ð â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œS
= 0
Where ð in above equation represents the total charge contained in the closed
volume V (enclosed by a closed surface S).
By divergence theorem we get
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œS
= â« ð â ï¿œÌ ï¿œðð£
ð
= â« ð ðð£
ð
= ð
ð â ï¿œÌ ï¿œ = ð Gaussâs law
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 4/24 Part II
2.2.2. Ampereâs law
The circulation of magnetic field ï¿œÌ ï¿œ around a closed contour C is equal to the
sum of electric current and shift current passing through the surface (S).
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œ = I + â«âï¿œÌ ï¿œ
âtâ ðð Ì
ðC
= â« ðÌ T â ðï¿œÌ ï¿œ
ð+ â«
âï¿œÌ ï¿œ
âtâ ðð Ì
ð
Application of stokesâ theorem, we get
â« ð Ã ï¿œÌ ï¿œ â ðï¿œÌ ï¿œð
= â« ðÌ T â ðï¿œÌ ï¿œ
ð+ â«
âï¿œÌ ï¿œ
âtâ ðð Ì
ð
ð Ã ï¿œÌ ï¿œ =âï¿œÌ ï¿œ
ât+ ðÌ
T Ampereâs law
Where I = â« ðÌ T â ðï¿œÌ ï¿œ
S
And ï¿œÌ ï¿œ is the current density, (Ampere/m2).
Since magnetic charge does not exist in nature. Thus the flux of B through any
closed surface (S) is always zero.
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œ = 0ð
ð â ï¿œÌ ï¿œ = 0 Ampereâs law
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 5/24 Part II
2.2.3. Faradayâs law
A time-varying magnetic fields generates an electric field ï¿œÌ ï¿œ. The time rate of
change of total magnetic flux through the surfaces (S), â(â« ï¿œÌ ï¿œâðï¿œÌ ï¿œ
ð)
ât is equal to
the negative value of the total voltage measured around, Figure 2.1.
Figure 2.1 The closed contour C and surface S associated
with Faradayâs law.
â
ât(â« ï¿œÌ ï¿œ â ðï¿œÌ ï¿œ
ð) = â â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œ
C
Application of stokesâs theorem:
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œC
= â« ð Ã ï¿œÌ ï¿œ â ðï¿œÌ ï¿œð
= ââ
ât(â« ï¿œÌ ï¿œ â ðï¿œÌ ï¿œ
ð)
ð Ã ï¿œÌ ï¿œ = ââï¿œÌ ï¿œ
ât Faradayâs law
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 6/24 Part II
2.2.4. Maxwellâs Equations [Conclusion]
Faradayâs law ð Ã ï¿œÌ ï¿œ = ââï¿œÌ ï¿œ
ât (2.1)
Ampereâs law ð Ã ï¿œÌ ï¿œ =âï¿œÌ ï¿œ
ât+ ðÌ
T (2.2)
Ampereâs law ð â ï¿œÌ ï¿œ = 0 (2.3)
Gaussâs law ð â ï¿œÌ ï¿œ = ðð(ð¡) (2.4)
Continuity equation:-
ð â ðÌ T(ð¡) = â
ððð(ð¡)
ðð¡ (2.5)
If the sources ðð(ð¡) and ðÌ T(ð¡) vary sinusoidally with time at radial (angular)
frequency ð (ð = 2ðð), the fields will also very sinusoidally and are
frequently called time-harmonic fields.
The qualitative mechanism by which Maxwellâs equations give rise to
propagating electromagnetic fields is shown in the Figure 2.2.
Figure 2.2 The qualitative mechanism.
For example, a time-varying current J on a linear antenna generates a
circulating and time-varying magnetic field H, which through Faradayâs law
generates a circulating electric field E, which through Ampereâs law
generates a magnetic field, and so on. The cross-linked electric and magnetic
fields propagate away from the current source.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 7/24 Part II
If phasor fields are introduced as follows:-
ï¿œÌ ï¿œ = Re[ðejÏt],â
âtðejÏt = jÏðejÏt = jÏð
ï¿œÌ ï¿œ = Re[ðejÏt],â
âtðejÏt = jÏðejÏt = jÏð
Then, the equations (2.1) to (2.5) became
ð Ã ð = âjÏð (2.6)
ð Ã ð = jÏð + ðð (2.7)
ð â ð = ðð (2.8)
ð â ð = 0 (2.9)
ð â ðð = âðððð (2.10)
The total current density ( ðð) is
ðT = Ïð + ð (2.11)
Where Ïð = a conducting current density which occurs in response.
ð (sigma) A conductivity of the medium (⧠mâ ).
ð an impressed, or source, current.
Also, ð = ϵð (2.12) ; and ð = ÎŒð (2.13)
Substituting equations (11 & 12) into (7) gives
ð à ð = jÏ (ϵ +ð
ðð) ð + ð = jÏϵâð + ð (2.14)
ðâ = ð +ð
ðð= ðâ² â ððâ²â² = ðððð
â = ðð(ððâ² â ððð
â²â²) (F/m) (2.15)
where ðâ² = ð (2.16)
and ðâ²â² =ð
ð (2.17)
The ratio ðâ²â² ðâ² â measures the magnitude of the conduction current relative to
that of the displacement current. It is called a loss tangent because it is a
measure of the ohmic loss in the medium:
ð¡ððð¿ = ðâ²â² ðâ² â =ð
ðð (2.18)
The ð¿ may be called the loss angle.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 8/24 Part II
Finally Maxwellâs Equations
ð à ð = âjÏÎŒð (2.19)
ð Ã ð = jÏðâð + ð (2.20)
ð â ð =ð
ðâ (2.21)
ð â ð = 0 (2.22)
ð â ð = âjÏð (2.23)
ð = Electric source current density.
ð = Source charge density.
Convenient equation (2.19) to introduce a fictitious magnetic
current density (M), is
ð à ð = âjÏÎŒð â ð (2.24)
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 9/24 Part II
2.3. Boundary conditions
Consider a plane interface between two media, as shown in Figure 2.3.
Maxwellâs equations in integral form can be used to deduce conditions
involving the normal and tangential fields at this interface.
Figure 2.3 Fields, currents, and surface charge at a general
interface between two media.
The time-harmonic version of equation (2.25), where S is the closed
âpillboxâ- shaped surface shown in Figure 2.4, can be written as
â® ï¿œÌ ï¿œ â ðï¿œÌ ï¿œS
= â« ð â ï¿œÌ ï¿œðð£ð
= â« ð ðð£ ð
(2.25)
Figure 2.4 Closed surface S for equation (2.25).
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 10/24 Part II
In the limit as â â 0 , the contribution of ð·ð¡ðð through the sidewalls goes
to zero, so equation (2.25) reduces to
âðð·2ð â âðð·1ð = âððð
Or
ð·2ð â ð·1ð = ðð (2.26)
Where ðð is the surface charge density on the interface. In vector form, we
can write
ï¿œÌï¿œ â (ðð â ðð) = ðð (C m2â ) (2.27)
A similar argument for ð leads to the result that
ð§ Ì â ð2 = ï¿œÌï¿œ â ð1 (T) (2.28)
Because there is no free magnetic charge.
For the tangential components of the electric field we use the phasor form of
equation below
â® ð â ððC
== âjÏ â« ð â ððð
â â« ð â ððð
(2.29)
Figure 2.5 Closed contour C for equation (2.29).
In connection with the closed contour C shown in Figure 2.5. In the limit
as â â 0 , the surface integral of ð vanishes (because ð = ââð vanishes).
The contribution from the surface integral of ð , however, may be nonzero
if a magnetic surface current density ðS exists on the surface. The Dirac delta
function can then be used to write
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 11/24 Part II
ð = ðS ÎŽ(h) (2.30)
Where â is a coordinate measured normal from the interface. Equation
(2.29) then gives
âððžð¡ðð2 â âððžð¡ðð1 = â âððð
Or
ðžð¡ðð2 â ðžð¡ðð1 = â ðð (2.31)
Which can be generalized in vector form as
(ð2 â ð1) à ᅵÌï¿œ = ðð (2.32)
A similar argument for the magnetic field leads to
ï¿œÌï¿œ à (ð2 â ð1) = ðð (2.33)
Summary
A sufficient set of boundary conditions (in the time-harmonic form) at an
arbitrary interface of materials and/or surface currents are
ï¿œÌï¿œ â (ðð â ðð) = ðð (2.27)
ð§ Ì â ð2 = ï¿œÌï¿œ â ð1 (2.28)
(ð2 â ð1) à ᅵÌï¿œ = ðs (2.32)
ï¿œÌï¿œ à (ð2 â ð1) = ðs (2.33)
Where ðs electric surface current density
ðs Magnetic surface current density
ðs and ðs flow on the boundary between two homogeneous media with
parameters ð1 , ð1, ð1 and ð2 , ð2, ð2.
ðžð¡ðð2 = ðžð¡ðð1 + ðð (2.34)
ð»ð¡ðð2 = ð»ð¡ðð1 + ðœð (2.35)
Tangent component of E and H are continuous across the boundary.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 12/24 Part II
2.3.1. One Side Perfect Conductor (Electric Wall)
If one side is a perfect electrical conductor, shown in Figure 2.6, Many
problems in microwave engineering involve boundaries with good
conductors (e.g. metals), which can often be assumed as lossless (Ï ââ). In
this case of a perfect conductor, all field components must be zero inside the
conducting region. The boundary conditions become
ï¿œÌï¿œ â ð = ðð (2.36)
ð§ Ì â ð = 0 (2.37)
ð à ᅵÌï¿œ = 0 (2.38)
ï¿œÌï¿œ à ð = ðs (2.39)
Or
ðtan = ðs (2.40)
ðtan = 0 (2.41)
Figure 2.6 Magnetic field intensity boundary condition.
(a) General case. (b) One medium a perfect conductor.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 13/24 Part II
2.4. Plane Wave Propagation in Conducting Media
In a source-free conducting medium, the homogeneous vector Helmholtzâs
equation to be solve is
â2ð + ð2ð = 0 (2.42)
Define a complex propagation constant ðŸ (gamma), a complex wave
number ð, an attenuation constant ðŒ (Alpha), a phase constant ðœ (beta) and
intrinsic impedance ð (eta) for the medium as
ðŸ = ðŒ + ððœ = ðð = ððâððâ1 â ðð
ðð (mâ1) (2.43)
ð = ðâððâ1 â ðð
ðð (2.44)
Where ðŒ and ðœ are, respectively, the real and imaginary parts of ðŸ, and both of
them positive quantities.
ðŒ = ðâðð
2{â1 + (
ð
ðð)2 â 1} (neper mâ = Np mâ ) (2.45)
ðœ = ðâðð
2{â1 + (
ð
ðð)
2+ 1} (ððð ðâ ) (2.46)
ð =ððð
ðŸ (Ω) (2.47)
Helmholtzâs equation, Eq. (2.53), becomes
â2ð â ðŸ2ð = 0 (2.48)
The solution of Eq. (2.48), which corresponding to a uniform plane wave
propagating in the +ð§ direction, is
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 14/24 Part II
ð = ðð¥ðžð¥ = ðð¥ðžððâðŸð§ = ðð¥ðžððââð§ðâððœð§ (2.49)
Where we have assumed that the wave is linearly polarized in the ð¥ direction.
2.4.1. A lossless medium
For lossless medium, ð = 0, then from Eq. (2.45), attenuation constant
become â= 0 (2.50)
And from Eq. (2.46), a phase constant equal to real wave number, and
become ðœ = ð = ðâðð (rad mâ ) (2.51)
While, the intrinsic impedance is
ð = âð ðâ (Ω) (2.52)
2.4.2. Low-Loss Dielectric
A low-loss dielectric is a good but imperfect insulator with nonzero
conductivity, such that ðâ²â² ⪠ðâ² or ð
ðð⪠1. Under this condition ðŸ in Eq.
(2.43) can be approximated by using the binomial expansion.
ðŸ = ðŒ + ððœ â ððâðð[1 +ð
ð2ðð+
1
8 (
ð
ðð)
2] (mâ1) (2.53)
From which we obtain the attenuation constant
ðŒ â ð
2â
ð
ð (Np mâ ) (2.54)
And the phase constant
ðœ â ðâðð[1 +1
8 (
ð
ðð)
2] (rad mâ ) (2.55)
The phase constant in Eq. (2.55) deviates only very slightly from its value for
a perfect (lossless) dielectric.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 15/24 Part II
The intrinsic impedance of a low-loss dielectric is a complex quantity.
ð = âð
â[1 +
ð
ððð]â1 2â
ð â âð
â[1 + ð
ð
2ðð] (Ω) (2.56)
Since the intrinsic impedance is the ratio of ðžð¥ and ð»ðŠ for a uniform plane
wave, the electric and magnetic field intensity in a lossy dielectric are, thus,
not in time phase, as they would be in a lossless medium.
The phase velocity (ðð) is obtained from the ratio ð ðœâ in a manner, using
Eq. (2.55), we found
ðð =ð
ðœâ
1
âðð[1 â
1
8 (
ð
ðð)
2] (m sâ ) (2.57)
2.4.3. Good Conductor
A good conductor is a medium for which ðâ²â² â« ðâ² or ð
ððâ« 1. Under this
condition we can neglect 1 in comparison with the term ð ðððâ in Eq. (2.43)
and write
ðŸ â ððâððâð
ððð= âðâððð =
1+ð
â2 âððð
Or ðŸ = ðŒ + ððœ â (1 + ð)âðððð (mâ1) (2.58)
Where we have used the relations âð = (ððð 2â )1 2â = ððð 4â =1+ð
â2 and ð =
2ðð. Eq. (2.58) indicates that ðŒ and ðœ for a good conductor are approximately
equal and both increase as âð ððð âð. For a good conductor
ðŒ = ðœ = âðððð (2.59)
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 16/24 Part II
The wave impedance (or intrinsic impedance) inside a good conductor is
ð =ððð
ðŸâ â
ððð
ð= (1 + ð)â
ðð
2ð= (1 + ð)
ðŒ
ð (Ω) (2.60)
Which has a phase angle of ððð. Hence the magnetic field intensity lags
behind the electric field intensity by 45ð.
Notice that
The phase angle of the impedance for a lossless material is ððš, and
The phase angle of the impedance of an arbitrary lossy medium is
somewhere between ðð and ððð.
The phase velocity (ðð) in a good conductor is
ðð =ð
ðœâ â
2ð
ðð (m sâ ) (2.61)
Which is proportional to âð ððð 1 âðâ .
The wavelength (ð) of a plane wave in a good conductor is
ð =2ð
ðœ=
ðð
ð= 2â
ð
ððð (m) (2.62)
The attenuation factor is ðâðŒð§, amplitude of a wave will be attenuated by a
factor of ðâ1 = 0.368 when it travels a distance (skin depth) ð¿ð = 1 ðŒâ . The
skin depth given by
ð¿ð = 1 ðŒâ = ( 2
ðððð )1 2â (m) (2.62)
Since ðŒ = ðœ for a good conductor, ð¿ð can also be written as
ð¿ð = 1 ðœâ = ð 2ð â (m) (2.63)
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 17/24 Part II
Table 2.1 Summary of results for plane wave propagation in various media
Ex: Consider copper as ð = 5.8 Ã 107 S m,â ð = 4ð Ã 10â7 H mâ .
Solution: the phase velocity in a good conductor media at ð = 3 MHz are
ðð = â4ðÃ3Ã106
4ðÃ10â7Ã5.8Ã107= 720 m sâ
Which is about twice the velocity of sound in air and is many orders of
magnitude slower than the velocity of light in air.
The wavelength in copper is ð =ðð
ð=
720
3Ã106= 0.24 mm
As comparison, a 3MHz electromagnetic wave in air has ð = 100 m.
The attenuation in copper is
ðŒ = âðððð = âð à 3 à 106 à 4ð à 10â7 à 5.8 à 107 = 2.62 à 104 Np mâ
The skin depth ð¿ð = 1 ðŒâ = 0.038 mm
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 18/24 Part II
2.5. Group Velocity
If the phase velocity is different for different frequencies, then the individual
frequency components will not maintain their original phase relationships as
they propagate down the transmission line or waveguide, and signal
distortion will occur. Such an effect is called dispersion since different phase
velocities allow the âfasterâ waves to lead in phase relative to the âslowerâ
waves, and the original phase relationships will gradually be dispersed as the
signal propagates down the line. In such a case, there is no single phase
velocity that can be attributed to the signal as a whole. However, if the
bandwidth of the signal is relatively small or if the dispersion is not too
severe, a group velocity can be defined in a meaningful way. This velocity
can be used to describe the speed at which the signal propagates.
The physical interpretation of group velocity (ðð) is the velocity at which a
narrowband signal propagates, Figure 2.7.
Figure 2.7 Sum of two time-harmonic traveling waves of equal amplitude
and slightly frequencies at a given t.
Consider the simplest case of a wave packet that consists of two traveling
waves having equal amplitude and slightly different angular frequencies
ðð + âð and ðð â âð (âð ⪠ðð). The phase constant, being functions of
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 19/24 Part II
frequency, will also be slightly different. Let the phase constants
corresponding to the two frequencies be ðœð + âðœ and ðœð â âðœ. We have
ðž(ð§, ð¡) = ðžð cos[( ðð + âð)ð¡ â (ðœð + âðœ)ð§]
+ðžð cos[( ðð â âð)ð¡ â (ðœð â âðœ)ð§]
= 2ðžð cos(ð¡âð â ð§âðœ)cos ( ððð¡ + ðœðð§) (2.64)
Since âð ⪠ðð, the expression in Eq. (2.64) represents a rapidly oscillating
wave an angular frequency ððand an amplitude that varies slowly with an
angular frequency âð, as shown in Figure 2.7.
The wave inside the envelope propagates with a phase velocity (ðð)
discused above.
The velocity of the original modulation envelope (the group velocity ðð) can
be determined by setting the argument of the first cosine factor in Eq. (2.64)
equal to a constant:
(ð¡âð â ð§âðœ = Constant) (2.65)
From which we obtain
ðð =ðð§
ðð¡=
âð
âðœ=
1
âðœ âðâ
In the limit that âð â 0, we have the formula for computing the group
velocity in a dispersive medium.
ðð =1
ððœ ððâ (m sâ ) (2.66)
This is the velocity of a point on the envelope of the wave packet, as shown
in Figure 2.7, and is identified as the velocity of the narrow-band signal.
A relation between the group and phase velocities may be obtained by
combining Eqs. (2.61) and (2.66). From Eq. (2.61), we have
ððœ
ðð=
ð
ðð(
ð
ðð) =
1
ððâ
ð
ðð2
ððð
ðð
Substitution of the above in Eq. (2.66) yields
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 20/24 Part II
ðð =ðð
1â ð
ðð
ððð
ðð
(2.67)
From Eq. (2.67) we see three possible cases:
a) No dispersion:
ððð
ðð= 0 (ðð independent of ð, ðœ linear function of ð),
ðð = ðð
b) Normal dispersion: ððð
ðð< 0 (ðð decreasing with ð),
ðð < ðð
c) Anomalous dispersion: ððð
ðð> 0 (ðð increasing with ð),
ðð > ðð
2.6. Negative Index Media
Maxwellâs equations do not preclude the possibility that one or both of the
quantities ð, ð be negative. For example, plasmas below their plasma
frequency, and metals up to optical frequencies, have ð < 0 and ð > 0,
with interesting applications such as surface Plasmon.
Negative-index media, also known as left-handed media, have ð, ð that are
simultaneously negative, ð < 0 and ð < 0 . Veselago was the first to study
their unusual electromagnetic properties, such as having a negative index of
refraction and the reversal of Snellâs law.
When, ðð < 0 and ðð < 0, the refractive index, ð2 = ðððð , must be
defined by the negative square root ð = ââðððð . Because then ð < 0
and ðð < 0 , will imply that the characteristic impedance of the medium
ð = ðððð ðâ will be positive, that the energy flux of a wave is in the same
direction as the direction of propagation.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 21/24 Part II
2.7. Poyntingâs Theorem
Poyntingâs theorem or power equation. Consider a volume ( ð ) bounded
by a closed surface (S). The complex power ( ðð ) delivered by the sources in
ð is:
ðð = ðð + ðððð£+ ð2ð(ðððð£
â ðððð£) (2.68)
Where
ðð Power flowing out of a closed surface (s),
ðððð£ Time-averaging power dissipated in a volume (ð) ,
ð2ð(ðððð£â ðððð£
) Time-averaging stored power in a volume ( ð ).
And ðð =1
2⯠(ð à ðâ)
Sâ ðï¿œÌ ï¿œ (2.69)
Where
ðï¿œÌ ï¿œ = ðð ï¿œÌï¿œ
ï¿œÌï¿œ is the unit normal to the surface directed out from the surface.
ð = ð Ã ð (ð ð2)â Instantaneous poynting vector (2.70)
ð =1
2ð Ã ðâ (ð ð2)â Complex poynting vector (2.71)
ðððð£=
1
2â Ï|E|2
Vðð (2.72)
Time-average stored magnetic energy is
ðððð£=
1
2â
1
2 Ό|H|2
Vðð (2.73)
Time-average stored electric energy is
ðððð£=
1
2â
1
2 ε|E|2
Vðð (2.74)
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 22/24 Part II
If the source power is not known explicitly, it may calculated from the
volume current density
ðð = â1
2â (ð â ðâ)
Vðð (2.75)
Or
ðð = â1
2â (ðâ â ð)
Vðð (2.76)
The real power flowing through surface (S) is
ððð£ =1
2ð ð[⯠(ð à ðâ)
Sâ ðï¿œÌ ï¿œ] (2.77)
2.8. Solution of Maxwellâs Equations for Radiation Problems
Summarize the procedure for finding the fields generated by an electric
source current density distribution (J).
1) The auxiliary magnetic vector potential A is found from
ð = â ð eâjβR
4ÏRðâ² ððâ² (2.78)
2) H field is found from
ð = ð Ã ð (2.79)
3) E field is simpler to find from
a) If we are in the source region, or from just
ð =1
ððð(ð Ã ð â ð) (2.80)
b) If the field point is removed in distance from the source, ð = 0 at point
p.
ð =1
ðððð Ã ð (2.81)
Note that: term ðððð¡ is eliminating. In free space case
Phase constant (beta) ðœ = ð âðððð =ð
ð=
2ð
ðð (2.82)
And ð =1
âÎŒoϵoâ 3 à 108
ð
ð ðð (2.83)
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 23/24 Part II
2.9. Field Regions
The space surrounding an antenna is usually subdivided into three regions
as shown in Figure 2.8:
(a) Reactive near-field,
(b) Radiating near-field (Fresnel) and
(c) Far-field (Fraunhofer) regions.
Figure 2.8 Field regions of an antenna.
The boundaries separating these regions are not unique, although various
criteria have been established and are commonly used to identify the regions.
Reactive near-field region is defined as âthat portion of the near-field region
immediately surrounding the antenna wherein the reactive field
predominatesâ.
1) For most antennas, the outer boundary of this region is commonly
taken to exist at a distance ð < 0.62âð·3/ð from the antenna surface,
where λ is the wavelength and D is the largest dimension of the antenna.
Dept. of Communication Eng. - U.O.T. Radio Wave Propagation [CEM 2206] Assist. Prof. R.T. Hussein (Ph.D.)
2017 Page 24/24 Part II
2) For a very short dipole, or equivalent radiator, the outer boundary is
commonly taken to exist at a distance ð 1 = ð/2ð from the antenna
surface.
Radiating near-field (Fresnel) region is defined as âthat region of the
field of an antenna between the reactive near-field region and the far-field
region wherein radiation fields predominate and wherein the angular field
distribution is dependent upon the distance from the antenna. If the antenna
has a maximum dimension that is not large compared to the wavelength,
this region may not exist. For an antenna focused at infinity, the radiating
near-field region is sometimes referred to as the Fresnel region on the basis
of analogy to optical terminology. If the antenna has a maximum overall
dimension which is very small compared to the wavelength, this field
region may not exist.â The inner boundary is taken to be the distance ð â¥
0.62âð·3/ð and the outer boundary the distance ð < 2ð·2/ð where D is
the largestâ dimension of the antenna. This criterion is based on a
maximum phase error of Ï/8. In this region the field pattern is, in general,
a function of the radial distance and the radial field component may be
appreciable.
* To be valid, D must also be large compared to the wavelength (D > λ).
Far-field (Fraunhofer) region is defined as âthat region of the field of an
antenna where the angular field distribution is essentially independent of
the distance from the antenna. If the antenna has a maximumâ overall
dimension D, the far-field region is commonly taken to exist at distances
greater than 2ð·2/ð from the antenna, λ being the wavelength.
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