uniform plane waves (1)
TRANSCRIPT
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EMG2016
ELECTROMAGNETIC THEORY
UNIFORM PLANE WAVES
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Learning Outcomes
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Outline Plane Wave Propagation
Time Harmonic Fields
Wave Equations Plane wave propagation in lossless media
Polarization of Waves
Plane wave propagation in lossy media
Electromagnetic Power density
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Plane wave propagation
a time-varying electric field E(t) produces a magnetic fieldH(t)and, conversely, a timevarying magnetic field
produces an electric field.
This cyclic pattern generates electromagnetic (EM) waves
capable of propagating through free space and in material
media.
When its propagation is guided by a material structure, the
EM wave is said to be traveling in a guided medium (e.g.
a transmission line).
EM waves also can travel in unbounded media. (e.g. lightwaves emitted by the sun and radio transmissions by
antennas).
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For a transmission-line circuit, we can model wave
propagation on such a transmission line either in terms ofthe voltages across the line and the currents through its
conductors or in terms of the electric and magnetic fields in
the dielectric medium between the conductors (bounded
case).
In this chapter we focus our attention on wave propagation
in unbounded media that can be categorized into two
classes lossless & lossy
The medium are assumed to be homogeneous &
isotropic
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When energy isemitted by a
source, such as an
antenna, it
expands outwardly
from the source inthe form of
spher ical waves.
Plane Wave
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To an observer very far way from the source, the
wavefront of the spherical wave appears
approximately planar, as if it were part of auni form planewave with uniform properties at all
points in the plane tangent to the wavefront
To analyse the unbounded EM wave, Maxwellsequations are used
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MAXWELL EQUATIONS. D = v
E= -B/t
. B= 0
H = J+ D/t
where E= electric field intensity D= electric flux density
H= magnetic field intensity B= magnetic flux density
v= electric charge volume density
J= conduction current density
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Time-Harmonic Fields D, E, B, H, Jand v depend on spatial coordinates (x,y,z)
and the time variable, t.
If their time variation is sinusoidalwith angular frequency, then these quantities can be represented by a phasor
that depends on (x,y,z) only.
Why phasor form?
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For a linear, isotropic, and homogeneous medium, the
MaxwellsEquations in phasor form is given as:
For time-harmonic quantities, differentiation in timedomain corresponds to multiplication by j in phasor
domain.
/v E
j E H
0 H
j H J E
Converted using
( , , , ) ( , , ) j tx y z t e x y z e E E
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j H J E
j j -j
H E E
By introducing the complex permittivity c which is
defined as
c= (-j/)
c= (-j/) = -j
with= , and = /
For a lossless medium (= 0), it follows that = 0 and c
= = .
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0 E
-j E H0 H
cj H E
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HjE ~~
EEjjE cc~~~ 2
EEE ~~~ 2
To the derive the wave equation from Maxwell
equation, we do a curl operation on both sides of the
second and fourth Maxwellsequations
Wave Equations
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2 2- 0 E E
2 2- 0 H H
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THE WAVE EQUATIONS FORLOSSLESS MEDIUM
0~~ 22 EE
0
~~ 22 HH
Homogeneous Vector Helmhotzs equations
0~
E~ 22 Ek
kSince
therefore,
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a plane wave has no electric-or magnetic-field components along its
direction of propagation
Wave equation:
For the phasor quantity x , the general solution of the ordinary
differential equation of the wave equation is
0~
E~
z zH
jkz
xo
jkz
xoxxx eEeEzEzEE )(
~)(
~~
Uniform Plane Wave
A uniform plane wave is characterized by electric and magnetic
fields that have uniform properties at all points across an infinite
plane and if this is the x-y plane so Eand Hdo not vary with x
andy
0~
~2
2
2
xx Ek
dz
Ed(x component ofE)
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Assume thatEhas only a component inxand that xconsists of a
wave travelling in the +z direction
( ) ( ) jkzx xoz E z E e E x x
0 0
x y z
x
j H H Hx y z
E
x y z
E x y z
0
~1~
~1~
0~
y
E
j
H
z
E
jH
H
xz
xy
x
jkz
yo
jkz
xoy eHeEkzH ~~)(~
yo xo
kH E
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I ntr insic impedance,h,of a lossless medium is defined as
h
k
We can summarize our results as
The electric and magnetic fields are perpendicular to each other, and
both are perpendicular to the direction of wave travel. These
directional properties characterize a transverse electromagnetic
(TEM) wave
jkz
xoeExz )(E
~x(z)E
~x
jkzxo eEyz hh //)(E~
y(z)H~
x
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TEM Wave [1]
[1] http://www.inchem.org/documents/ehc/ehc/ehc137.htm
E(z,t)andH(z,t) are in-phase
Phase velocity
wavelength
1pu
k
2 pu
k f
http://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htmhttp://www.inchem.org/documents/ehc/ehc/ehc137.htm -
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In lossless dielectric:
thus
rr 00 ,,0 then
0, k
1u 0
h
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Exercise 1
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General Relation between E and H
It can be shown that, for any uniform plane wave traveling
in an arbitrary direction denoted by the unit vector , themagnetic field phasor is interrelated to the electric field
phasor by
Right hand rule applies: when we rotate the four fingers of
the right hand from the direction of Etoward that of H, the
thumb points in the direction of wave travel,
1/h H k E
-h E k H
(7.39a)
k
k
E
Hk
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The wave may be considered the sum of two waves, one with (E+x,H+
y)
components and another with (E+y, H+
x) components.
In general, a TEM wave may have an electric field in any direction in the
plane orthogonal to the direction of wave travel, and the associated magnetic
field is also in the same plane and its direction is dictated by Eq.(7.39a,
Ulaby).
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Exercise 2
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POLARIZATION OF PLANE WAVES
EEy
Ex
cos cosx ya t kz a t kz E x y
The polarization of a uniform plane wave describes the locus traced by
the tip of the E vector (in the plane orthogonal to the direction ofpropagation) at a given point in space as a function of time
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SENSE OF POLARIZATION
Linear circularelliptical elliptical
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Answer:
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To examine wave propagation in a conducting medium we
return to the wave equation
with2= -2c= -
2(- j)
where = and = /. Since is complex, we express it
as
= +j,
where is the attenuation constantof the medium and is
its phase constant.
2 2E 0E
Plane Wave Propagation inLossy media (0)
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By replacing with (+j), we have
(+j)2= (2- 2) +j2= -2+j2.
Hence,
2- 2= -2,
2= 2
.Solving these two equations for and gives
2/12
1
'
"1
2
'
2/12
1
'
"1
2
'
or2/1
2
112
2/12
112
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zAssuming that the wave propagates along direction
and E
~
has only an x-component
x ( )E E zx
0~)-( 22 ESubstitute into
yields solution ' z0 0( )
z
x x xE z E e E e
Inserting the time factor
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( , ) Re[ ( ) ]j t
xz t E z e
E x ( )0
Re[ ]z j t zE e e
x
0( , ) cos( )zz t E e t z E x
For magnetic field,
( )
0( , ) Re[ ]z j t zz t H e e H y
where 00 c
EHh
y
nj| | | |ec c n cjj
h h h
and
intrinsic impedance
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thus0
( , ) cos( )
| |
z
c
Ez t e t z
h
h
H y
hc= intrinsic impedance for lossy medium= attenuation constant(factor)
= phase constant
h = phase angle of the intrinsic impedance for lossy medium
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1
/||
hc
h2tan
E(z,t) & H(z,t) not in-phase
Magnitude ofEx(z) decrease exponentially.
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For a lossy medium, the ratio /= /appears in
all these expressions and plays an important role in
determining how lossy a medium is.
When /1, the medium is characterized as a
good conductor. (/>102)
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Low-Loss Dielectric
The general expression for is given by
2'2
"
'
h c
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thus
then
In good conductors:
r 00 ,,
f2
2p
u
45c
h
1 (1 )cj f
j j
h
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0
( , ) cos( )zz t E e t z E x
thus
0( , ) cos( 45 )zE
z t e t z
H y
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Electromagnetic Power Density
The Poynting vector S is defined as
represents the power density (power per unit area) carried
by the wave. Its direction is along the propagation direction of
the wave
The instantaneous Poynting vector or power density vector is
given by
na
HE
P
P
tjtj etzHetzEetzHtzEtz
),(~
),(~
),(),(),(
P
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The total power that flows through or is intercepted by
the aperture is
(W)
Sd.
SaveP aveP
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In practice, the quantity of greater interest is the average
power density of the wave, Pave, which is the time-averaged value of
where and are in phasor formE~ H~
]~~
Re[)2/1( *ave HEP
aveP
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A plane wave is propagating in the +z direction in a lossy
medium
For a lossless medium
The attenuation raterepresents the rate of decrease of the
magnitude of Pave (z) as a function of propagation distance
A= 10 log10[Pave(z)/Pave(0) ] = -8.68 z (dB)
)cos(2
2
2
ave nzo e
Ez
hP
hh 2
2
22
ave
Ez
Ez
c
o P
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Decibel Scale for power ratios
G = P1 / P2
G(dB) = 10 log(G) = 10 log(P1/P2) (dB)
For voltage or current ratio, it can be written as
G(dB) = 20 log(V1/V2) (dB)
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