Dept. of Communication Eng. - U.O.T. Radio Wave

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,

2. Electromagnetic Fundamentals

2.1. Electromagnetics Field Components

Electric field intensity [strength] �̅� (V 𝑚⁄ )

Magnetic field intensity �̅� (A m⁄ )

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 �̅� = ϵ�̅� C/m2 (coulombs/m2)

Current density]

Magnetic flux density [the magnetic �̅� = μ�̅� (Tesla = weber/m2)

induction]

Where:

𝜖 (epsilon) = 𝜖𝑜𝜖𝑟 F m⁄ (farad/m) is the permittivity of the medium.

𝜖𝑜 ≅ 1

36𝜋× 10−9 ≅ 8.854 × 10−12 (F m⁄ ) Permittivity [dielectric constant]

of a vacuum or free space.

𝜖𝑟 Is the relative permittivity of medium (dimensionless).

𝜇 (mu) = 𝜇𝑜𝜇𝑟 H/m (henry/m) is the permeability of the medium.

𝜇𝑜 = 4𝜋 × 10−7 (H m⁄ ) Permeability of a free space.

𝜇𝑟 The relative permeability of a material (dimensionless).

The units for 𝜖𝑜 and 𝜇𝑜 are the units of the ratios D/E and B/H, that is,

coulomb m2⁄

volt m⁄=

coulomb

volt∙m=

weber m2⁄

ampere m⁄=

ampere∙m=

2.2. Maxwell’s Equations

2.2.1. Gauss’s law

States that the total flux of �̅� = ϵ�̅� from a volume (𝑉) is equal to

the net charge contained within 𝑉.

The net charge contained within 𝑉 is 𝑄 = ∫ 𝜌 𝑑𝑣 𝑉

(coulomb).

Where 𝜌 (rho) is the volume charge density (coulombs/m3).

Then, Gauss’s law may be written as

∮ �̅� ∙ 𝑑�̅�S

= ∫ 𝜌 𝑑𝑣

∮ �̅� ∙ 𝑑�̅�S

= 𝑄1 + 𝑄2 = 𝑄 ∮ �̅� ∙ 𝑑�̅�S

Where 𝑄 in above equation represents the total charge contained in the closed

volume V (enclosed by a closed surface S).

By divergence theorem we get

∮ �̅� ∙ 𝑑�̅�S

= ∫ 𝛁 ∙ �̅�𝑑𝑣

= ∫ 𝜌 𝑑𝑣

= 𝑄

𝛁 ∙ �̅� = 𝜌 Gauss’s law

2.2.2. Ampere’s law

The circulation of magnetic field �̅� around a closed contour C is equal to the

sum of electric current and shift current passing through the surface (S).

∮ �̅� ∙ 𝑑�̅� = I + ∫∂�̅�

∂t∙ 𝑑𝒔 ̅

= ∫ 𝐉̅T ∙ 𝑑�̅�

𝑆+ ∫

∂�̅�

∂t∙ 𝑑𝒔 ̅

Application of stokes’ theorem, we get

∫ 𝛁 × �̅� ∙ 𝑑�̅�𝑆

= ∫ 𝐉̅T ∙ 𝑑�̅�

𝑆+ ∫

∂�̅�

∂t∙ 𝑑𝒔 ̅

𝛁 × �̅� =∂�̅�

∂t+ 𝐉̅

T Ampere’s law

Where I = ∫ 𝐉̅T ∙ 𝑑�̅�

And �̅� 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.

∮ �̅� ∙ 𝑑�̅� = 0𝑆

𝛁 ∙ �̅� = 0 Ampere’s law

2.2.3. Faraday’s law

A time-varying magnetic fields generates an electric field �̅�. The time rate of

change of total magnetic flux through the surfaces (S), ∂(∫ �̅�∙𝑑�̅�

∂t 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 Faraday’s law.

∂t(∫ �̅� ∙ 𝑑�̅�

𝑆) = − ∮ �̅� ∙ 𝑑�̅�

Application of stokes’s theorem:

∮ �̅� ∙ 𝑑�̅�C

= ∫ 𝛁 × �̅� ∙ 𝑑�̅�𝑆

= −∂

∂t(∫ �̅� ∙ 𝑑�̅�

𝛁 × �̅� = −∂�̅�

∂t Faraday’s law

2.2.4. Maxwell’s Equations [Conclusion]

Faraday’s law 𝛁 × �̅� = −∂�̅�

∂t (2.1)

Ampere’s law 𝛁 × �̅� =∂�̅�

∂t+ 𝐉̅

T (2.2)

Ampere’s law 𝛁 ∙ �̅� = 0 (2.3)

Gauss’s law 𝛁 ∙ �̅� = 𝜌𝑇(𝑡) (2.4)

Continuity equation:-

𝛁 ∙ 𝐉̅T(𝑡) = −

𝜕𝜌𝑇(𝑡)

𝜕𝑡 (2.5)

If the sources 𝜌𝑇(𝑡) and 𝐉̅T(𝑡) vary sinusoidally with time at radial (angular)

frequency 𝜔 (𝜔 = 2𝜋𝑓), the fields will also very sinusoidally and are

frequently called time-harmonic fields.

The qualitative mechanism by which Maxwell’s 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 Faraday’s law

generates a circulating electric field E, which through Ampere’s law

generates a magnetic field, and so on. The cross-linked electric and magnetic

fields propagate away from the current source.

If phasor fields are introduced as follows:-

�̅� = Re[𝐁ejωt],∂

∂t𝐁ejωt = jω𝐁ejωt = jω𝐁

�̅� = Re[𝐃ejωt],∂

∂t𝐃ejωt = jω𝐃ejωt = jω𝐃

Then, the equations (2.1) to (2.5) became

𝛁 × 𝐄 = −jω𝐁 (2.6)

𝛁 × 𝐇 = jω𝐃 + 𝐉𝑇 (2.7)

𝛁 ∙ 𝐃 = 𝜌𝑇 (2.8)

𝛁 ∙ 𝐁 = 0 (2.9)

𝛁 ∙ 𝐉𝑇 = −𝑗𝜔𝜌𝑇 (2.10)

The total current density ( 𝐉𝑇) is

𝐉T = σ𝐄 + 𝐉 (2.11)

Where σ𝐄 = a conducting current density which occurs in response.

𝜎 (sigma) A conductivity of the medium (℧ m⁄ ).

𝐉 an impressed, or source, current.

Also, 𝐃 = ϵ𝐄 (2.12) ; and 𝐁 = μ𝐇 (2.13)

Substituting equations (11 & 12) into (7) gives

𝛁 × 𝐇 = jω (ϵ +𝜎

𝑗𝜔) 𝐄 + 𝐉 = jωϵ∗𝐄 + 𝐉 (2.14)

𝜖∗ = 𝜖 +𝜎

𝑗𝜔= 𝜖′ − 𝑗𝜖′′ = 𝜖𝑜𝜖𝑟

∗ = 𝜖𝑜(𝜖𝑟′ − 𝑗𝜖𝑟

′′) (F/m) (2.15)

where 𝜖′ = 𝜖 (2.16)

and 𝜖′′ =𝜎

𝜔 (2.17)

The ratio 𝜖′′ 𝜖′ ⁄ 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:

𝑡𝑎𝑛𝛿 = 𝜖′′ 𝜖′ ⁄ =𝜎

𝜔𝜖 (2.18)

The 𝛿 may be called the loss angle.

Finally Maxwell’s Equations

𝛁 × 𝐄 = −jωμ𝐇 (2.19)

𝛁 × 𝐇 = jω𝜖∗𝐄 + 𝐉 (2.20)

𝛁 ∙ 𝐄 =𝜌

𝜖∗ (2.21)

𝛁 ∙ 𝐇 = 0 (2.22)

𝛁 ∙ 𝐉 = −jω𝜌 (2.23)

𝐉 = Electric source current density.

𝜌 = Source charge density.

Convenient equation (2.19) to introduce a fictitious magnetic

current density (M), is

𝛁 × 𝐄 = −jωμ𝐇 − 𝐌 (2.24)

2.3. Boundary conditions

Consider a plane interface between two media, as shown in Figure 2.3.

Maxwell’s 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

“pillbox”- shaped surface shown in Figure 2.4, can be written as

∮ �̅� ∙ 𝑑�̅�S

= ∫ 𝛁 ∙ �̅�𝑑𝑣𝑉

= ∫ 𝜌 𝑑𝑣 𝑉

(2.25)

Figure 2.4 Closed surface S for equation (2.25).

In the limit as ℎ → 0 , the contribution of 𝐷𝑡𝑎𝑛 through the sidewalls goes

to zero, so equation (2.25) reduces to

∆𝑆𝐷2𝑛 − ∆𝑆𝐷1𝑛 = ∆𝑆𝜌𝑠

𝐷2𝑛 − 𝐷1𝑛 = 𝜌𝑠 (2.26)

Where 𝜌𝑠 is the surface charge density on the interface. In vector form, we

can write

�̂� ∙ (𝐃𝟐 − 𝐃𝟏) = 𝜌𝑠 (C m2⁄ ) (2.27)

A similar argument for 𝐁 leads to the result that

𝐧 ̂ ∙ 𝐁2 = �̂� ∙ 𝐁1 (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

∮ 𝐄 ∙ 𝑑𝒍C

== −jω ∫ 𝐁 ∙ 𝑑𝒔𝑆

− ∫ 𝐌 ∙ 𝑑𝒔𝑆

(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 ℎ → 0 , the surface integral of 𝐁 vanishes (because 𝑆 = ℎ∆𝑙 vanishes).

The contribution from the surface integral of 𝐌 , however, may be nonzero

if a magnetic surface current density 𝐌S exists on the surface. The Dirac delta

function can then be used to write

𝐌 = 𝐌S δ(h) (2.30)

Where ℎ is a coordinate measured normal from the interface. Equation

(2.29) then gives

∆𝑙𝐸𝑡𝑎𝑛2 − ∆𝑙𝐸𝑡𝑎𝑛1 = − ∆𝑙𝑀𝑠

𝐸𝑡𝑎𝑛2 − 𝐸𝑡𝑎𝑛1 = − 𝑀𝑠 (2.31)

Which can be generalized in vector form as

(𝐄2 − 𝐄1) × �̂� = 𝐌𝑠 (2.32)

A similar argument for the magnetic field leads to

�̂� × (𝐇2 − 𝐇1) = 𝐉𝑠 (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

�̂� ∙ (𝐃𝟐 − 𝐃𝟏) = 𝜌𝑠 (2.27)

𝐧 ̂ ∙ 𝐁2 = �̂� ∙ 𝐁1 (2.28)

(𝐄2 − 𝐄1) × �̂� = 𝐌s (2.32)

�̂� × (𝐇2 − 𝐇1) = 𝐉s (2.33)

Where 𝐉s electric surface current density

𝐌s Magnetic surface current density

𝐉s and 𝐌s flow on the boundary between two homogeneous media with

parameters 𝜖1 , 𝜇1, 𝜎1 and 𝜖2 , 𝜇2, 𝜎2.

𝐸𝑡𝑎𝑛2 = 𝐸𝑡𝑎𝑛1 + 𝑀𝑠 (2.34)

𝐻𝑡𝑎𝑛2 = 𝐻𝑡𝑎𝑛1 + 𝐽𝑠 (2.35)

Tangent component of E and H are continuous across the boundary.

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 (σ →∞). In

this case of a perfect conductor, all field components must be zero inside the

conducting region. The boundary conditions become

�̂� ∙ 𝐃 = 𝜌𝑠 (2.36)

𝐧 ̂ ∙ 𝐁 = 0 (2.37)

𝐄 × �̂� = 0 (2.38)

�̂� × 𝐇 = 𝐉s (2.39)

𝐇tan = 𝐉s (2.40)

𝐄tan = 0 (2.41)

Figure 2.6 Magnetic field intensity boundary condition.

(a) General case. (b) One medium a perfect conductor.

2.4. Plane Wave Propagation in Conducting Media

In a source-free conducting medium, the homogeneous vector Helmholtz’s

equation to be solve is

∇2𝐄 + 𝑘2𝐄 = 0 (2.42)

Define a complex propagation constant 𝛾 (gamma), a complex wave

number 𝑘, an attenuation constant 𝛼 (Alpha), a phase constant 𝛽 (beta) and

intrinsic impedance 𝜂 (eta) for the medium as

𝛾 = 𝛼 + 𝑗𝛽 = 𝑗𝑘 = 𝑗𝜔√𝜇𝜖√1 − 𝑗𝜎

𝜔𝜖 (m−1) (2.43)

𝑘 = 𝜔√𝜇𝜖√1 − 𝑗𝜎

𝜔𝜖 (2.44)

Where 𝛼 and 𝛽 are, respectively, the real and imaginary parts of 𝛾, and both of

them positive quantities.

𝛼 = 𝜔√𝜇𝜖

2{√1 + (

𝜔𝜖)2 − 1} (neper m⁄ = Np m⁄ ) (2.45)

𝛽 = 𝜔√𝜇𝜖

2{√1 + (

𝜔𝜖)

2+ 1} (𝑟𝑎𝑑 𝑚⁄ ) (2.46)

𝜂 =𝑗𝜔𝜇

𝛾 (Ω) (2.47)

Helmholtz’s equation, Eq. (2.53), becomes

∇2𝐄 − 𝛾2𝐄 = 0 (2.48)

The solution of Eq. (2.48), which corresponding to a uniform plane wave

propagating in the +𝑧 direction, is

𝐄 = 𝒂𝑥𝐸𝑥 = 𝒂𝑥𝐸𝑜𝑒−𝛾𝑧 = 𝒂𝑥𝐸𝑜𝑒−∝𝑧𝑒−𝑗𝛽𝑧 (2.49)

Where we have assumed that the wave is linearly polarized in the 𝑥 direction.

2.4.1. A lossless medium

For lossless medium, 𝜎 = 0, then from Eq. (2.45), attenuation constant

become ∝= 0 (2.50)

And from Eq. (2.46), a phase constant equal to real wave number, and

become 𝛽 = 𝑘 = 𝜔√𝜇𝜖 (rad m⁄ ) (2.51)

While, the intrinsic impedance is

𝜂 = √𝜇 𝜖⁄ (Ω) (2.52)

2.4.2. Low-Loss Dielectric

A low-loss dielectric is a good but imperfect insulator with nonzero

conductivity, such that 𝜖′′ ≪ 𝜖′ or 𝜎

𝜔𝜖≪ 1. Under this condition 𝛾 in Eq.

(2.43) can be approximated by using the binomial expansion.

𝛾 = 𝛼 + 𝑗𝛽 ≅ 𝑗𝜔√𝜇𝜖[1 +𝜎

𝑗2𝜔𝜖+

𝜔𝜖)

2] (m−1) (2.53)

From which we obtain the attenuation constant

𝛼 ≅𝜎

𝜖 (Np m⁄ ) (2.54)

And the phase constant

𝛽 ≅ 𝜔√𝜇𝜖[1 +1

𝜔𝜖)

2] (rad m⁄ ) (2.55)

The phase constant in Eq. (2.55) deviates only very slightly from its value for

a perfect (lossless) dielectric.

The intrinsic impedance of a low-loss dielectric is a complex quantity.

𝜂 = √𝜇

∈[1 +

𝑗𝜔𝜖]−1 2⁄

𝜂 ≅ √𝜇

∈[1 + 𝑗

2𝜔𝜖] (Ω) (2.56)

Since the intrinsic impedance is the ratio of 𝐸𝑥 and 𝐻𝑦 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 (𝜐𝑝) is obtained from the ratio 𝜔 𝛽⁄ in a manner, using

Eq. (2.55), we found

𝜐𝑝 =𝜔

𝛽≅

√𝜇𝜖[1 −

𝜔𝜖)

2] (m s⁄ ) (2.57)

2.4.3. Good Conductor

A good conductor is a medium for which 𝜖′′ ≫ 𝜖′ or 𝜎

𝜔𝜖≫ 1. Under this

condition we can neglect 1 in comparison with the term 𝜎 𝑗𝜔𝜖⁄ in Eq. (2.43)

and write

𝛾 ≅ 𝑗𝜔√𝜇𝜖√𝜎

𝑗𝜔𝜖= √𝑗√𝜔𝜇𝜎 =

1+𝑗

√2 √𝜔𝜇𝜎

Or 𝛾 = 𝛼 + 𝑗𝛽 ≅ (1 + 𝑗)√𝜋𝑓𝜇𝜎 (m−1) (2.58)

Where we have used the relations √𝑗 = (𝑒𝑗𝜋 2⁄ )1 2⁄ = 𝑒𝑗𝜋 4⁄ =1+𝑗

√2 and 𝜔 =

2𝜋𝑓. Eq. (2.58) indicates that 𝛼 and 𝛽 for a good conductor are approximately

equal and both increase as √𝑓 𝑎𝑛𝑑 √𝜎. For a good conductor

𝛼 = 𝛽 = √𝜋𝑓𝜇𝜎 (2.59)

The wave impedance (or intrinsic impedance) inside a good conductor is

𝜂 =𝑗𝜔𝜇

𝛾≅ √

𝑗𝜔𝜇

𝜎= (1 + 𝑗)√

𝜔𝜇

2𝜎= (1 + 𝑗)

𝜎 (Ω) (2.60)

Which has a phase angle of 𝟒𝟓𝒐. Hence the magnetic field intensity lags

behind the electric field intensity by 45𝑜.

Notice that

The phase angle of the impedance for a lossless material is 𝟎𝐨, and

The phase angle of the impedance of an arbitrary lossy medium is

somewhere between 𝟎𝒐 and 𝟒𝟓𝒐.

The phase velocity (𝜐𝑝) in a good conductor is

𝜐𝑝 =𝜔

𝛽≅ √

𝜇𝜎 (m s⁄ ) (2.61)

Which is proportional to √𝑓 𝑎𝑛𝑑 1 √𝜎⁄ .

The wavelength (𝜆) of a plane wave in a good conductor is

𝜆 =2𝜋

𝜐𝑝

𝑓= 2√

𝑓𝜇𝜎 (m) (2.62)

The attenuation factor is 𝑒−𝛼𝑧, amplitude of a wave will be attenuated by a

factor of 𝑒−1 = 0.368 when it travels a distance (skin depth) 𝛿𝑠 = 1 𝛼⁄ . The

skin depth given by

𝛿𝑠 = 1 𝛼⁄ = ( 2

𝜔𝜇𝑜𝜎 )1 2⁄ (m) (2.62)

Since 𝛼 = 𝛽 for a good conductor, 𝛿𝑠 can also be written as

𝛿𝑠 = 1 𝛽⁄ = 𝜆 2𝜋 ⁄ (m) (2.63)

Table 2.1 Summary of results for plane wave propagation in various media

Ex: Consider copper as 𝜎 = 5.8 × 107 S m,⁄ 𝜇 = 4𝜋 × 10−7 H m⁄ .

Solution: the phase velocity in a good conductor media at 𝑓 = 3 MHz are

𝜐𝑝 = √4𝜋×3×106

4𝜋×10−7×5.8×107= 720 m s⁄

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 𝜆 =𝜐𝑝

3×106= 0.24 mm

As comparison, a 3MHz electromagnetic wave in air has 𝜆 = 100 m.

The attenuation in copper is

𝛼 = √𝜋𝑓𝜇𝜎 = √𝜋 × 3 × 106 × 4𝜋 × 10−7 × 5.8 × 107 = 2.62 × 104 Np m⁄

The skin depth 𝛿𝑠 = 1 𝛼⁄ = 0.038 mm

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 “faster” waves to lead in phase relative to the “slower”

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 (𝜐𝑔) 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

𝜔𝑜 + ∆𝜔 and 𝜔𝑜 − ∆𝜔 (∆𝜔 ≪ 𝜔𝑜). The phase constant, being functions of

frequency, will also be slightly different. Let the phase constants

corresponding to the two frequencies be 𝛽𝑜 + ∆𝛽 and 𝛽𝑜 − ∆𝛽. We have

𝐸(𝑧, 𝑡) = 𝐸𝑜 cos[( 𝜔𝑜 + ∆𝜔)𝑡 − (𝛽𝑜 + ∆𝛽)𝑧]

+𝐸𝑜 cos[( 𝜔𝑜 − ∆𝜔)𝑡 − (𝛽𝑜 − ∆𝛽)𝑧]

= 2𝐸𝑜 cos(𝑡∆𝜔 − 𝑧∆𝛽)cos ( 𝜔𝑜𝑡 + 𝛽𝑜𝑧) (2.64)

Since ∆𝜔 ≪ 𝜔𝑜, the expression in Eq. (2.64) represents a rapidly oscillating

wave an angular frequency 𝜔𝑜and an amplitude that varies slowly with an

angular frequency ∆𝜔, as shown in Figure 2.7.

The wave inside the envelope propagates with a phase velocity (𝜐𝑝)

discused above.

The velocity of the original modulation envelope (the group velocity 𝜐𝑔) can

be determined by setting the argument of the first cosine factor in Eq. (2.64)

equal to a constant:

(𝑡∆𝜔 − 𝑧∆𝛽 = Constant) (2.65)

From which we obtain

𝜐𝑔 =𝑑𝑧

𝑑𝑡=

∆𝜔

∆𝛽=

∆𝛽 ∆𝜔⁄

In the limit that ∆𝜔 → 0, we have the formula for computing the group

velocity in a dispersive medium.

𝜐𝑔 =1

𝑑𝛽 𝑑𝜔⁄ (m s⁄ ) (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

𝑑𝛽

𝑑𝜔=

𝑑𝜔(

𝜐𝑝) =

𝜐𝑝−

𝜐𝑝2

𝑑𝜐𝑝

𝑑𝜔

Substitution of the above in Eq. (2.66) yields

𝜐𝑔 =𝜐𝑝

1− 𝜔

𝜐𝑝

𝑑𝜐𝑝

𝑑𝜔

(2.67)

From Eq. (2.67) we see three possible cases:

a) No dispersion:

𝑑𝜐𝑝

𝑑𝜔= 0 (𝜐𝑝 independent of 𝜔, 𝛽 linear function of 𝜔),

𝜐𝑔 = 𝜐𝑝

b) Normal dispersion: 𝑑𝜐𝑝

𝑑𝜔< 0 (𝜐𝑝 decreasing with 𝜔),

𝜐𝑔 < 𝜐𝑝

c) Anomalous dispersion: 𝑑𝜐𝑝

𝑑𝜔> 0 (𝜐𝑝 increasing with 𝜔),

𝜐𝑔 > 𝜐𝑝

2.6. Negative Index Media

Maxwell’s equations do not preclude the possibility that one or both of the

quantities 𝜖, 𝜇 be negative. For example, plasmas below their plasma

frequency, and metals up to optical frequencies, have 𝜖 < 0 and 𝜇 > 0,

with interesting applications such as surface Plasmon.

Negative-index media, also known as left-handed media, have 𝜖, 𝜇 that are

simultaneously negative, 𝜖 < 0 and 𝜇 < 0 . Veselago was the first to study

their unusual electromagnetic properties, such as having a negative index of

refraction and the reversal of Snell’s law.

When, 𝜖𝑟 < 0 and 𝜇𝑟 < 0, the refractive index, 𝑛2 = 𝜖𝑟𝜇𝑟 , must be

defined by the negative square root 𝑛 = −√𝜖𝑟𝜇𝑟 . Because then 𝑛 < 0

and 𝜇𝑟 < 0 , will imply that the characteristic impedance of the medium

𝜂 = 𝜂𝑜𝜇𝑟 𝑛⁄ will be positive, that the energy flux of a wave is in the same

direction as the direction of propagation.

2.7. Poynting’s Theorem

Poynting’s theorem or power equation. Consider a volume ( 𝑉 ) bounded

by a closed surface (S). The complex power ( 𝑃𝑠) delivered by the sources in

𝑉 is:

𝑃𝑠 = 𝑃𝑓 + 𝑃𝑑𝑎𝑣+ 𝑗2𝜔(𝑊𝑚𝑎𝑣

− 𝑊𝑒𝑎𝑣) (2.68)

𝑃𝑓 Power flowing out of a closed surface (s),

𝑃𝑑𝑎𝑣 Time-averaging power dissipated in a volume (𝑉) ,

𝑗2𝜔(𝑊𝑚𝑎𝑣− 𝑊𝑒𝑎𝑣

) Time-averaging stored power in a volume ( 𝑉 ).

And 𝑃𝑓 =1

2∯ (𝐄 × 𝐇∗)

S∙ 𝑑�̅� (2.69)

𝑑�̅� = 𝑑𝑠�̂�

�̂� is the unit normal to the surface directed out from the surface.

𝐒 = 𝐄 × 𝐇 (𝑊 𝑚2)⁄ Instantaneous poynting vector (2.70)

𝐒 =1

2𝐄 × 𝐇∗ (𝑊 𝑚2)⁄ Complex poynting vector (2.71)

𝑃𝑑𝑎𝑣=

2∭ σ|E|2

V𝑑𝜈 (2.72)

Time-average stored magnetic energy is

𝑊𝑚𝑎𝑣=

2 μ|H|2

V𝑑𝜈 (2.73)

Time-average stored electric energy is

𝑊𝑒𝑎𝑣=

2 ε|E|2

V𝑑𝜈 (2.74)

If the source power is not known explicitly, it may calculated from the

volume current density

𝑃𝑆 = −1

2∭ (𝐄 ∙ 𝐉∗)

V𝑑𝜈 (2.75)

𝑃𝑆 = −1

2∭ (𝐇∗ ∙ 𝐌)

V𝑑𝜈 (2.76)

The real power flowing through surface (S) is

𝑃𝑎𝑣 =1

2𝑅𝑒[∯ (𝐄 × 𝐇∗)

S∙ 𝑑�̅�] (2.77)

2.8. Solution of Maxwell’s 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

𝐀 = ∭ 𝐉 e−jβR

4πR𝐕′ 𝑑𝜈′ (2.78)

2) H field is found from

𝐇 = 𝛁 × 𝐀 (2.79)

3) E field is simpler to find from

a) If we are in the source region, or from just

𝐄 =1

𝑗𝜔𝜖(𝛁 × 𝐇 − 𝐉) (2.80)

b) If the field point is removed in distance from the source, 𝐉 = 0 at point

𝐄 =1

𝑗𝜔𝜖𝛁 × 𝐇 (2.81)

Note that: term 𝑒𝑗𝜔𝑡 is eliminating. In free space case

Phase constant (beta) 𝛽 = 𝜔 √𝜇𝑜𝜖𝑜 =𝜔

𝜆𝑜 (2.82)

And 𝑐 =1

√μoϵo≅ 3 × 108

𝑠𝑒𝑐 (2.83)

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 “that portion of the near-field region

immediately surrounding the antenna wherein the reactive field

predominates”.

1) For most antennas, the outer boundary of this region is commonly

taken to exist at a distance 𝑅 < 0.62√𝐷3/𝜆 from the antenna surface,

where λ is the wavelength and D is the largest dimension of the antenna.

2) For a very short dipole, or equivalent radiator, the outer boundary is

commonly taken to exist at a distance 𝑅1 = 𝜆/2𝜋 from the antenna

surface.

Radiating near-field (Fresnel) region is defined as “that 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.” The inner boundary is taken to be the distance 𝑅 ≥

0.62√𝐷3/𝜆 and the outer boundary the distance 𝑅 < 2𝐷2/𝜆 where D is

the largest∗ dimension of the antenna. This criterion is based on a

maximum phase error of π/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 “that 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 maximum∗ overall

dimension D, the far-field region is commonly taken to exist at distances

greater than 2𝐷2/𝜆 from the antenna, λ being the wavelength.

Dept. of Communication Eng. - U.O.T. Radio Wave

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