Download - Propagation of Electromagnetic Waves
Propagation of
Electromagnetic Waves Dr. Sikder Sunbeam Islam
Dept. of EEE. IIUC
Waves: General Concept
• A wave is a function of both space and time. Waves are means of transporting energy or information.
• Examples: Light rays, TV signals, Radar beams etc.
• Wave motion occurs when a disturbance at point A at time , is related to what happens at point B at t> .
• The oscillating current creates oscillating electric (E) and magnetic (H) fields which in turn generate more electric and magnetic fields. Thus a outward propagating electromagnetic wave is created.
• Most often we are interested not so much in how an electromagnetic (EM) wave originated , but how it propagates.
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Waves: General Concept
• Waves may be polarized.
• For linearly polarized waves, the plane of oscillation is fixed.
For linearly polarized waves the components of the wave in each
direction are in phase.
• For circularly polarized light one component is 90 degrees out
of phase with the other, leading to a rotation of the plane of
oscillation.
• For a plane wave, the electric and magnetic field remain
perpendicular to each other to the direction of propagation.
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Fig.1a.Linearly polarized waves Fig.1b.Circularly polarized wave
Waves: Source Free Wave Equation
• In a source free region , ρ=0, J=0.
• We know from Maxwell’s equation,
𝛻 × 𝐸 = −𝜕𝐵
𝜕𝑡
Equation (1), (2),(3), (4) are first order differential equations with two variables E and H. They can be combined to give a second order equation.
------------------(1) B=𝜇0H
𝜵 × 𝑬 = −𝝁𝟎
𝝏𝑯
𝝏𝒕 ------------------(2)
𝜵 × 𝑯 = 𝐽 +𝜕𝐷
𝜕𝑡 =
𝜕𝐷
𝜕𝑡 =𝝐𝟎
𝝏𝑬𝝏𝒕
Also,
As, J=0
D=𝜖0E
------------------(3)
As, ρ=0,𝛻.D=𝛻.(𝜖E)= ρ=0; So, 𝜵.E=0
------------------(4)
As, 𝛻.B=𝛻.(𝜇0H)= 0; So, 𝜵.H=0 ------------------(5)
As,𝛻.B=0
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Source Free Wave Equation: continues.
• Taking curl in equ.(2),
𝜵 × 𝜵 × 𝑬 = −𝝁𝝏 𝜵 × 𝑯
𝝏𝒕
= −𝝁𝝐𝝏𝟐𝑬
𝝏𝒕𝟐
𝜵 × 𝑯 = 𝜖𝝏𝑬
𝝏𝒕 ------------------(6)
We know that for a vector field A,
So, from Equ.(6) and (4),
𝛻.E=0
𝜵 × 𝜵 × 𝑬 = −𝛁𝟐E=−𝝁𝝐𝝏𝟐𝑬
𝝏𝒕𝟐
therefore, 𝛁𝟐E−𝝁𝝐𝝏𝟐𝑬
𝝏𝒕𝟐 = 0
------------------(7) Or, 𝛁𝟐E−𝟏
𝒖𝟐
𝝏𝟐𝑬
𝝏𝒕𝟐 = 0 𝑠𝑖𝑛𝑐𝑒, 𝑢 =1
√(𝜇𝜖)
similarly can be obtained, 𝛁𝟐𝐻 −𝟏
𝒖𝟐
𝝏𝟐𝑯
𝝏𝒕𝟐= 0 ------------------(8)
Equation (7), (8) are homogeneous vector wave equation.
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One Dimension Wave Equation
• From Maxwell’s Equation, 𝜵 × 𝑯 =𝝐𝟎
𝝏𝑬𝝏𝒕
------------(3)
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Fig.1c
c
Time Harmonic Maxwell’s Equations
• Field vectors that vary with space coordinates and are sinusoidal
functions of time can similarly be represented by vector phasors that
depends on space coordinates but not on time. For example, we can
write time harmonic E field referring to cosωt as,
E(x,y,z,t)=Re[E(x,y,z)𝑒𝑗𝜔𝑡].
• Therefore, if E (x,y,z,t) is to be represented by vector phasors E (x,y,z),
then 𝝏E(x,y,z,t)/𝝏t and ⨜E(x,y,z,t)dtwould be represented by vector
phasors jωE(x,y,z) and E(x,y,z)/jωrespectively.
• We now write time-harmonic Maxwell’s equations in term of vector
phasors in a simple (linear, isotropic and homogeneous ) medium as
follows:
𝛻 × 𝐸 = −𝑗ωμ𝐻
𝛻 × 𝐻 = 𝐽 + 𝑗ω𝞊𝐸
------------------(14)
------------------(16) 𝛻.E=ρ/𝞊
𝛻.H=0
------------------(15)
------------------(17)
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Time Harmonic Maxwell’s Equations: Source
free nonconducting medium
• Source free nonconducting medium characterized by, J=0,
σ=0,ρ=0. So time-harmonic Maxwell’s equations will become:
• According to equ. (7), we can write
• Where, 𝑘 = β =𝜔
𝑢= ω√(𝜇𝜖) =
2𝜋𝑓
𝑢=
2𝜋
λ
𝛻 × 𝐸 = −𝑗ωμ𝐻
𝛻 × 𝐻 = 𝑗ω𝞊𝐸
𝛻.E=0
𝛻.H=0
------------------(18)
------------------(19)
------------------(20)
------------------(21)
𝛁𝟐E−𝟏
𝒖𝟐
𝝏𝟐𝑬
𝝏𝒕𝟐 = 0 𝛁𝟐E+𝒌𝟐𝑬 = 0 ------------------(22)
Similarly from (8), 𝛁𝟐𝐻 + 𝒌𝟐𝑯 = 0 ------------------(23)
Wavelength, λ=𝑢
𝑓
[ In some books k is denoted by β ] k=wave number
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Time Harmonic Maxwell’s Equations continues.
• In Cartesian coordinates Equ.(22) is equivalent to three scalar equations with components of 𝐸𝑥, 𝐸𝑦, 𝐸𝑧. Writing it for component
• With time factor the possible solution of Equ.(22) are,
𝛁𝟐E+𝒌𝟐𝑬 = 0 ------------------(22)
𝐸𝑥,
------------------(23)
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Time Harmonic Maxwell’s Equations continues.
(Can be showed/proved)
𝑒𝑖𝑥 = 𝑐𝑜𝑠𝑥 + 𝑖𝑠𝑖𝑛𝑥
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Time Harmonic Maxwell’s Equations continues.
• The equation below is similar to Equ.(26) where β=k and
A=𝐸0considering the imaginary part.
• The characteristics of the wave equ.(27),
------------------(27)
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Problem:1.
Solution:
T=1
𝑓=
2𝜋
2𝜋𝑓=
2𝜋
𝜔
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Fig.3
u
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Wave Propagation in Lossy Dielectrics
• A lossy dielectric is a medium in which an EM wave losses as it
propagates due to poor conduction.
• A lossy dielectric is partially conducting medium with σ ≠ 0.
• Considering a linear, isotropic , homogeneous, lossy dielectric that is
charge free (ρ=0). Considering time factor, from equ.(14) and (15),
𝛻 × 𝐸𝑠 = −𝑗ωμ𝐻𝑠
𝛻 × 𝐻𝑠 = 𝐽 + 𝑗ω𝞊𝐸𝑠=(σ+𝑗ω𝞊)𝐸𝑠 ------------------(28)
------------------(14)
------------------(29)
Now from equ.(29)
------------------(30)
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Wave Propagation in Lossy Dielectrics: continues.
------------------(31)
Equation (30), (31) are knows as homogeneous vector Helmholtz’s
wave equation.
From equ.(28)
𝛻 × 𝐻𝑠 = 𝐽 + 𝑗ω𝞊𝐸𝑠=(σ+𝑗ω𝞊)𝐸𝑠=jω 𝜖 +𝜎
𝑗𝜔𝐸𝑠=jω𝜖𝑐𝐸𝑠 ----------(32)
𝜖𝑐 = 𝜖 − 𝐽𝜎
𝜔 (F/m)
𝝐𝒄is complex permittivity for conducting media.
------------------(33)
Including ohmic losses in the imaginary part of complex permittivity,
𝜖"=𝜎
𝜔
------------------(34)
Now comparing, equ.(33) and (34);
𝜖𝑐 = 𝜖′ − 𝑗𝜖"
(Semense/m)
𝜖′ and 𝜖" are functions of frequency.
------------------(35)
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Wave Propagation in Lossy Dielectrics: continues
The ratio 𝜖"/𝜖 is called loss tangent because it is the measure of power loss
in the medium. The quantity 𝛿𝑐may be called loss angle.
tan𝛿𝑐 = 𝜖"/𝜖′=𝜎/(𝜔𝜖) ------------------(36)
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Wave Propagation in Lossy Dielectrics: contin.
------------------(31.1)
The attenuation constant defines the rate at which the fields of the wave are
attenuated as the wave propagates. The phase constant defines the rate at which
the phase changes as the wave propagates. In loss less media , α=0. From (31.1),
------------------(31.2)
------------------(31.3)
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Wave Propagation in Lossy Dielectrics: contin.
In Fig4, the uniform plane wave for this example has
only a z-component of electric field and an x-
component of magnetic field which are both functions
of only y.
An electromagnetic wave which has no electric or
magnetic field components in the direction of
propagation (all components of E and H are
perpendicular to the direction of propagation) is called a
transverse electromagnetic (TEM) wave. All plane
waves are TEM waves.
For this uniform plane wave, the only two field components (Ezs, Hxs) can be
simplified significantly given the field dependence on y only.
------------------(31.4)
------------------(31.5)
------------------(31.6)
------------------(31.7)
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Wave Propagation: TEM waves
The remaining single partial derivative from equ. (31.6),
------------------(31.8)
The general solutions to the reduced waves equations (will be similar for H-field),
where (E1, E2) are constants (electric field amplitudes). Inserting the time factor,
------------------(31.9)
as, ------------------(31.10)
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Wave Propagation: TEM waves
Assuming a +ay traveling uniform plane wave (see equ.31.4),
and
Intrinsic Impedance (η): The intrinsic impedance of the wave is defined as
the ratio of the electric field and magnetic field phasors (complex amplitudes)
------------------(31.11)
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Wave Propagation: in Lossy vs. Lossless media
Wave Propagation in
Lossy Media
Wave Propagation in
Lossless Media
• The electric field and magnetic field
in a lossless medium are in phase. • The electric field and magnetic field
in a lossy medium are out of phase.
(i.e. No attenuation )
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Wave Propagation: in Free Space vs. Good conductor media
Free Space Good conductor
Low loss (negligible attenuation)
[Note that attenuation in a good conductor increases with frequency. The attenuation rate in a good conductor can be characterized by the skin depth.]
Skin depth (δ) or Penetration depth is a distance over which a plane wave is attenuated by a factor of 𝑒−1 in a good conductor. It is a measure of depth to which EM wave can penetrate in the medium.
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• The group velocity (𝒖𝒈) is the speed at which
information (usually by modulating the frequency
or amplitude of the wave) can travel (Fig.5). 𝒖𝒈=𝝎
𝜷.
• In some cases the waves of different frequencies
will propagate with different phase velocities-
causing a distortion in the signal wave shape. The
phenomenon of signal distortion caused by the
dependence of the phase velocity on frequencies is
called dispersion.
Fig.5. Sum of two travelling waves with
slight different frequencies
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EM Wave Propagation in Ionized Media
• In the earth’s upper atmosphere (50-500km altitude) there exist
an ionized gas layers called Ionosphere, that consists of free
electrons and positive ions.
• Ionized gases with equal electron and ion densities are called
plasmas.
• As electrons are much lighter than positive ions, they are more
accelerated by the electric fields of EM waves passing through
the ionosphere and affects telecommunication.
or, ------------------(31.12)
Such displacement give rise to dipole moment, p = -ex ------------------(31.13)
------------------(31.14)
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EM Wave Propagation in Ionized Media contin.
------------------(31.16)
------------------(31.15)
------------------(31.17)
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Power and Poynting Vector
• Energy can be transported from one point (where a transmitter is located)
to another point (with a receiver) by means of EM waves. The rate of such
energy transportation can be obtained from Maxwell's equations:
• Poynting vector is a measurement of intensity of electromagnetic
Radiation.
• We know, for any vector A and B,
------------------(2)
-----------------(3)
------------------(37)
-----------------(38)
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Power and Poynting Vector (cont.)
Integrating both sides (divergence theorem) of equ.(39)
---------------(39)
Therefore, from equ.(38)
---------------(40)
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Power and Poynting Vector (cont.)
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Prob.4.
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32
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Reference
• Engineering Electromagnetics; William Hayt & John Buck, 7th & 8th editions; 2012
• Electromagnetics with Applications, Kraus and Fleisch, 5th edition, 2010
• Elements of Electromagnetics ; Matthew
N.O. Sadiku
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