Download Plane wave propagation in lossy medium

Microwave Engineering Course
4th Stage Electrical Engineering
Department
College of Engineering
Misan University
2016-2017
Prof. Dr. Ahmad H. Abood
Part I
Electromagnetic Theory
Outline course
 Maxwell’s
 Plane
 Filed
Equations
wave solution
component of plane wave
 General
development of the wave equation
 Plane
wave propagation in lossy medium
 Plane
wave propagation in conducting medium
 Plane
wave propagation in lossy dielectric medium
Text Book
D. M. Pozar, Microwave Engineering,
4th ed., Wiley, 2012.
 Solution
Manual
of
Microwave
Engineering, 4th ed., Wiley, 2012.

References
1.
D. M. Pozar, Microwave Engineering,
3rd ed., Wiley, 2005.
2.
S. C. Harsany, Principles of
Microwave Technology, Prentice-Hall,
1997. (undergraduate text)
3.
R. E. Collin, Foundations for
Microwave Engineering, 2nd ed., IEEE
Press, 2001. (advanced text)
Outline course
 Maxwell’s Equations
 Plane
 Filed
wave solution
component of plane wave
 General
development of the wave equation
 Plane
wave propagation in lossy medium
 Plane
wave propagation in conducting medium
 Plane
wave propagation in lossy dielectric medium
A BRIEF HISTORY OF ELECTROMAGNETICS
 Magnetism was known to the ancient Greeks (Plato and
Socrates).
 Hans Christian Oersted (1777-1851) discovers the relation
between current carrying wire and the magnetic field.
 André Ampère (1775-1836) discovers the force between two
current carrying wires.
 Jean-Baptiste Biot (1774-1862) and Félix Savart (1791-1841)
formulate their law quantifying the force between current
element.
 Benjamin Franklin (1706-1790) and Joseph Priestly (1733-1804)
postulate the inverse square law of electrostatics.
A BRIEF HISTORY OF ELECTROMAGNETICS, cont.
 Coulomb (in 1785) proves experimentally the inverse square
law for stationary electric charges.
 Alessandro Volta (1745-1827) investigated reactions between
dissimilar metals, and developed the first electric battery
(1800).
 Karl Friedrich Gauss (1777-1855) discovers the divergence
theorem of electricity.
A BRIEF HISTORY OF ELECTROMAGNETICS, cont.
A BRIEF HISTORY OF ELECTROMAGNETICS, cont.
Four Time-domain Maxwell’s Equations
Definitions:
Faraday's law states: The induced electromotive force in any closed
circuit is equal to the negative of the time rate of change of
the magnetic flux enclosed by the circuit.
 
 B   B  dS
S
 B   B  ds  cos 
S
 B   Wb
1Wb  1T  m 2
d m
Therefore, E 
dt
Definitions:
Ampere's Law states that for any closed loop path, the sum of the length
elements times the magnetic field in the direction of the length element
is equal to the permeability times the electric current enclosed in the
loop.
 Ampere’s
 Where
law,

 
B.ds   0ienc
the integral is a line integral.
 B.ds is integrated around a closed loop called an
Amperian loop.
 The current ienc is net current enclosed by the loop.
Definitions:
Gauss' Law of electric field is the first of Maxwell's Equations which
dictates how the Electric Field behaves around electric charges. Gauss'
Law can be written in terms of the Electric Flux Density and the Electric
Charge Density
Definitions:
Gauss' law of magnetic filed, the electric flux through any closed
surface is directly proportional to the net electric charge enclosed by
that surface.
Gauss’ Law in magnetism: the net magnetic flux
through a closed surface is zero
 
B

d
A

0

(1)
(2)
(3)
(4)
Outline course
 Maxwell’s
Equations
 Plane wave solution
 Filed
component of plane wave
 General
development of the wave equation
 Plane
wave propagation in lossy medium
 Plane
wave propagation in conducting medium
 Plane
wave propagation in lossy dielectric medium
A
B
Outline course
 Maxwell’s
 Plane
Equations
wave solution
 Filed component of plane wave
 General
development of the wave equation
 Plane
wave propagation in lossy medium
 Plane
wave propagation in conducting medium
 Plane
wave propagation in lossy dielectric medium
𝒙
𝒚
𝒛
𝝏
𝝏
𝝏
𝝏𝑬𝒚 𝝏𝑬𝒙
𝝏𝑬𝒛 𝝏𝑬𝒚
𝝏𝑬𝒙 𝝏𝑬𝒛
𝜵×𝑬=
=
−
𝒙+
−
𝒚+
−
𝒛
𝝏𝒙 𝝏𝒚 𝝏𝒛
𝝏𝒚
𝝏𝒛
𝝏𝒛
𝝏𝒙
𝝏𝒙
𝝏𝒚
𝑬𝒙 𝑬𝒚 𝑬𝒛
In the same manner
𝒙
𝝏
𝜵×𝑯=
𝝏𝒙
𝑯𝒙
𝒚
𝝏
𝝏𝒚
𝑯𝒚
𝒛
𝝏
𝝏𝑯𝒚 𝝏𝑯𝒙
𝝏𝑯𝒛 𝝏𝑯𝒚
𝝏𝑯𝒙 𝝏𝑯𝒛
=
−
𝒙+
−
𝒚+
−
𝒛
𝝏𝒛
𝝏𝒚
𝝏𝒛
𝝏𝒛
𝝏𝒙
𝝏𝒙
𝝏𝒚
𝑯𝒛
So,
𝜕𝐸𝑧 𝜕𝐸𝑦
𝜕𝐻𝑥
−
= −𝜇
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝐻𝑦
𝜕𝐸𝑧 𝜕𝐸𝑦
−
= −𝜇
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝐸𝑧 𝜕𝐸𝑦
𝜕𝐻𝑧
−
= −𝜇
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝜕𝐻𝑧 𝜕𝐻𝑦
𝜕𝐸𝑥
−
=𝜀
𝜕𝑦
𝜕𝑧
𝜕𝑥
and,
𝜕𝐸𝑦
𝜕𝐻𝑧 𝜕𝐻𝑦
−
=𝜀
𝜕𝑦
𝜕𝑧
𝜕𝑦
𝜕𝐻𝑧 𝜕𝐻𝑦
𝜕𝐸𝑧
−
=𝜀
𝜕𝑦
𝜕𝑧
𝜕𝑧
Outline course
 Maxwell’s
 Plane
 Filed
Equations
wave solution
component of plane wave
 General
development of the wave equation
 Plane wave propagation
in lossy medium
 Plane
wave propagation in conducting medium
 Plane
wave propagation in lossy dielectric medium
Now consider the effect of a lossy medium. If the
medium is conductive, with a conductivity σ,
Maxwell’s curl equations can be written, from (1) and
(2) as
.................. (1)
…………….. (2)
The resulting wave equation for 𝐸 then becomes
…………….. (3)
where we see a similarity with the wave equation for .E
in the lossless case. The difference is that the quantity
𝑘 2 = 𝜔2 𝜇𝜀 of lossless case is replaced by 𝑘 2
= 𝜔2 𝜇𝜀 1 − 𝑗 𝜎 𝜔𝜀
in (3). We then define a
complex propagation constant for the medium as
…………….. (4)
where α is the attenuation constant and β is the phase
constant. If we again assume an electric field with only
an 𝑥 component and uniform in x and y, the wave
equation of (3) reduces to
…………….. (5)
…………….. (6)
…………….. (7)
…………….. (8)
c
Outline course
 Maxwell’s
 Plane
 Filed
Equations
wave solution
component of plane wave
 General
 Plane
development of the wave equation
wave propagation in lossy medium
 Plane wave propagation
 Plane
in conducting medium
wave propagation in lossy dielectric medium
Many problems of practical interest involve loss or
attenuation due to good (but not perfect) conductors. A
good conductor is a special case of the preceding
analysis, where the conductive current is much greater
than the displacement current, which means that σ>>ω.
Most metals can be categorized as good conductors. In
terms of a complex, rather than conductivity, this
condition is equivalent to 𝜖 ′′ ≫ 𝜖 ′ . The propagation
constant of (4) can then be adequately approximated by
ignoring the displacement current term, to give
(1)
(2)
Thus the amplitude of the fields in the conductor will
decay by an amount 1/e, or 36.8%, after traveling a
distance of one skin depth, because
. At
microwave frequencies, for a good conductor, this
distance is very small. The practical importance of this
result is that only a thin plating of a good conductor
(e.g., silver or gold) is necessary for low-loss microwave
components.
Compute the skin depth of aluminum, copper, gold,
and silver at a frequency of 10 GHz?
Solution
The conductivities for these metals are listed in Appendix F.
Equation (2) gives the skin depths as
Outline course
 Maxwell’s
 Plane
 Filed
Equations
wave solution
component of plane wave
 General
development of the wave equation
 Plane
wave propagation in lossy medium
 Plane
wave propagation in conducting medium
 Plane wave propagation
in lossy dielectric medium