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.

0t0t

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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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23

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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.

31

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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