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Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 1 7222012
UNIFORM PLANE WAVES (PROPAGATION IN FREE SPACE) Starting with point form of Maxwells equations for time varying fields in free space
tto
HBE
tto
EDH
0 E
0 H Let
xx zE aE ˆ
Then
y
y
ooyx
t
zH
tz
Ea
HaE ˆˆ
And
xx
oox
y
t
E
tz
Ha
EaH ˆˆ
Collecting results
t
zH
z
E y
ox
t
E
z
Hx
o
y
Good reminder of telegraphist equations
To obtain the wave equations differentiate the first wrt z and the second wrt t and rearranging to get
E for equation wave
ldimensiona One2
2
2
2
t
E
z
E xoo
x
Or reversing differentiations to get
H for equation wave
ldimensiona One2
2
2
2
t
H
z
H y
oo
y
And a general solution is given by
EE
v
ztf
v
ztftzEx 21
From which the velocity of wave propagation may be deduced (by substituting 1f
in the wave equation performing the indicated diffrsquos)
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 2 7222012
cs
mvoo
81031
TEM waves Transverse ElectroMagnetic waves implies E is perpendicular to H and both lying in a transverse plane (a plane normal to the direction of propagation)
Uniform Plane Waves UPW E and H fields have constant magnitude and phase in the transverse plane (Constant phase and amplitude) For sinusoidal waves
2
1
2
1
coscos
coscos
zktEzktE
v
ztE
v
ztE
tzEtzEtzE
oxooxo
p
xo
p
xo
xxtotalx
timeunit per shift phases
rad
distanceunit per shift phasem
radko
constant phase implies
space) free(in
0constant
constant
1
1
cvkdt
zd
dt
dzkt
dt
d
zkt
p
o
o
o
The wave number in free space is defined as
m
radc
ko
The wavenumber is a property of a wave its spatial frequency that is proportional to
the reciprocal of the wavelength It is also the magnitude of the wave vector (to be seen
later) The wavenumber has dimensions of reciprocal length so its SI unit is m-1
Simply the number of wavelengths per 2π units of distance
Also the wave length is given by
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 3 7222012
m
f
c
ko
o
2
2πby shifts phase
spatial the over which
distance
Maxwells equations and the wave equations may be written in frequency domain with the help of the transformation
jt
soso jtt
HEHB
E
soso jtt
EHED
H
0 sE
0 sH
3D WAVE EQUATIONS (FREE SPACE) Taking the curl of the first equation namely
sos j HE
Using the identity
FFF2
Substituting
sosss j HEEE 2
Using the rest of Maxwellrsquos equations
sos j HE 2
soos jj EE 2
soos EE 22
) foreqation (Wave E
EE
ationmholtz equVector Hel
sos k 22
With
m
radv
k oo
p
o
Similar approach may be followed to obtain the wave equation for the magnetic field
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 4 7222012
)for eqation (Wave equation Helmholtz Vector H
HH sos k 22
RELATION BETWEEN AND
For the previous assumption xx zE aE ˆ the wave equation reduces to
zEk
dz
zEdxso
xs 2
2
2
With the frequency domain solution given by
zjk
x
zjk
xoxsoo eEeEzE
0
But from Maxwells equation
sos j HE
zHj
dz
zdEyso
xs
Substituting the solution
zHj
dz
eEeEdyso
zjkx
zjkxo
oo
0
Differentiating and solving for zH ys
zjk
x
o
ozjk
xo
o
oys
oo eEeEzH
0
Identifying
zjk
yoyfoeHzH
zjk
yyboeHzH
0
xo
o
oyo EH
0
x
o
oyo EH
And the intrinsic impedance of free space is
377120
o
o
yo
xoo
H
E
Or
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 5 7222012
o
o
yo
xoo
H
E
It can be shown that
HaEEa
H
no
o
n or ˆˆ
Or
HEn aaa ˆˆˆ Where
na unit vector in the direction of propagation
Ea unit vector in the direction of
Ha unit vector in the direction of
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 6 7222012
PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 7 7222012
k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 8 7222012
2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 9 7222012
Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 2 7222012
cs
mvoo
81031
TEM waves Transverse ElectroMagnetic waves implies E is perpendicular to H and both lying in a transverse plane (a plane normal to the direction of propagation)
Uniform Plane Waves UPW E and H fields have constant magnitude and phase in the transverse plane (Constant phase and amplitude) For sinusoidal waves
2
1
2
1
coscos
coscos
zktEzktE
v
ztE
v
ztE
tzEtzEtzE
oxooxo
p
xo
p
xo
xxtotalx
timeunit per shift phases
rad
distanceunit per shift phasem
radko
constant phase implies
space) free(in
0constant
constant
1
1
cvkdt
zd
dt
dzkt
dt
d
zkt
p
o
o
o
The wave number in free space is defined as
m
radc
ko
The wavenumber is a property of a wave its spatial frequency that is proportional to
the reciprocal of the wavelength It is also the magnitude of the wave vector (to be seen
later) The wavenumber has dimensions of reciprocal length so its SI unit is m-1
Simply the number of wavelengths per 2π units of distance
Also the wave length is given by
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Dr Naser Abu-Zaid Page 3 7222012
m
f
c
ko
o
2
2πby shifts phase
spatial the over which
distance
Maxwells equations and the wave equations may be written in frequency domain with the help of the transformation
jt
soso jtt
HEHB
E
soso jtt
EHED
H
0 sE
0 sH
3D WAVE EQUATIONS (FREE SPACE) Taking the curl of the first equation namely
sos j HE
Using the identity
FFF2
Substituting
sosss j HEEE 2
Using the rest of Maxwellrsquos equations
sos j HE 2
soos jj EE 2
soos EE 22
) foreqation (Wave E
EE
ationmholtz equVector Hel
sos k 22
With
m
radv
k oo
p
o
Similar approach may be followed to obtain the wave equation for the magnetic field
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edition 2012
Dr Naser Abu-Zaid Page 4 7222012
)for eqation (Wave equation Helmholtz Vector H
HH sos k 22
RELATION BETWEEN AND
For the previous assumption xx zE aE ˆ the wave equation reduces to
zEk
dz
zEdxso
xs 2
2
2
With the frequency domain solution given by
zjk
x
zjk
xoxsoo eEeEzE
0
But from Maxwells equation
sos j HE
zHj
dz
zdEyso
xs
Substituting the solution
zHj
dz
eEeEdyso
zjkx
zjkxo
oo
0
Differentiating and solving for zH ys
zjk
x
o
ozjk
xo
o
oys
oo eEeEzH
0
Identifying
zjk
yoyfoeHzH
zjk
yyboeHzH
0
xo
o
oyo EH
0
x
o
oyo EH
And the intrinsic impedance of free space is
377120
o
o
yo
xoo
H
E
Or
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Dr Naser Abu-Zaid Page 5 7222012
o
o
yo
xoo
H
E
It can be shown that
HaEEa
H
no
o
n or ˆˆ
Or
HEn aaa ˆˆˆ Where
na unit vector in the direction of propagation
Ea unit vector in the direction of
Ha unit vector in the direction of
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PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
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k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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Dr Naser Abu-Zaid Page 8 7222012
2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
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Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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0amp
r
ro
xE and yH are in phase
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(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
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jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
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Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
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Illustration of skin depth
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POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
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Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
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oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
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Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
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Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
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Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
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m
f
c
ko
o
2
2πby shifts phase
spatial the over which
distance
Maxwells equations and the wave equations may be written in frequency domain with the help of the transformation
jt
soso jtt
HEHB
E
soso jtt
EHED
H
0 sE
0 sH
3D WAVE EQUATIONS (FREE SPACE) Taking the curl of the first equation namely
sos j HE
Using the identity
FFF2
Substituting
sosss j HEEE 2
Using the rest of Maxwellrsquos equations
sos j HE 2
soos jj EE 2
soos EE 22
) foreqation (Wave E
EE
ationmholtz equVector Hel
sos k 22
With
m
radv
k oo
p
o
Similar approach may be followed to obtain the wave equation for the magnetic field
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 4 7222012
)for eqation (Wave equation Helmholtz Vector H
HH sos k 22
RELATION BETWEEN AND
For the previous assumption xx zE aE ˆ the wave equation reduces to
zEk
dz
zEdxso
xs 2
2
2
With the frequency domain solution given by
zjk
x
zjk
xoxsoo eEeEzE
0
But from Maxwells equation
sos j HE
zHj
dz
zdEyso
xs
Substituting the solution
zHj
dz
eEeEdyso
zjkx
zjkxo
oo
0
Differentiating and solving for zH ys
zjk
x
o
ozjk
xo
o
oys
oo eEeEzH
0
Identifying
zjk
yoyfoeHzH
zjk
yyboeHzH
0
xo
o
oyo EH
0
x
o
oyo EH
And the intrinsic impedance of free space is
377120
o
o
yo
xoo
H
E
Or
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Dr Naser Abu-Zaid Page 5 7222012
o
o
yo
xoo
H
E
It can be shown that
HaEEa
H
no
o
n or ˆˆ
Or
HEn aaa ˆˆˆ Where
na unit vector in the direction of propagation
Ea unit vector in the direction of
Ha unit vector in the direction of
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Dr Naser Abu-Zaid Page 6 7222012
PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
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Dr Naser Abu-Zaid Page 7 7222012
k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
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Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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0amp
r
ro
xE and yH are in phase
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(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
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jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
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Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
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Illustration of skin depth
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POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
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Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
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Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
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Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
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Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
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Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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)for eqation (Wave equation Helmholtz Vector H
HH sos k 22
RELATION BETWEEN AND
For the previous assumption xx zE aE ˆ the wave equation reduces to
zEk
dz
zEdxso
xs 2
2
2
With the frequency domain solution given by
zjk
x
zjk
xoxsoo eEeEzE
0
But from Maxwells equation
sos j HE
zHj
dz
zdEyso
xs
Substituting the solution
zHj
dz
eEeEdyso
zjkx
zjkxo
oo
0
Differentiating and solving for zH ys
zjk
x
o
ozjk
xo
o
oys
oo eEeEzH
0
Identifying
zjk
yoyfoeHzH
zjk
yyboeHzH
0
xo
o
oyo EH
0
x
o
oyo EH
And the intrinsic impedance of free space is
377120
o
o
yo
xoo
H
E
Or
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edition 2012
Dr Naser Abu-Zaid Page 5 7222012
o
o
yo
xoo
H
E
It can be shown that
HaEEa
H
no
o
n or ˆˆ
Or
HEn aaa ˆˆˆ Where
na unit vector in the direction of propagation
Ea unit vector in the direction of
Ha unit vector in the direction of
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PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 7 7222012
k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
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Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
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edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 5 7222012
o
o
yo
xoo
H
E
It can be shown that
HaEEa
H
no
o
n or ˆˆ
Or
HEn aaa ˆˆˆ Where
na unit vector in the direction of propagation
Ea unit vector in the direction of
Ha unit vector in the direction of
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 6 7222012
PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 7 7222012
k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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edition 2012
Dr Naser Abu-Zaid Page 8 7222012
2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 9 7222012
Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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edition 2012
Dr Naser Abu-Zaid Page 10 7222012
0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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PROPAGATION IN DIELECTRICS Assuming a simple dielectric the wave equation is written as
)for eqation (Wave22 EEE ss k
Where k is the wave number
Ex 121 Let m
Ae zj
y
o
x
o
s
070ˆ203ˆ402 aaH for a uniform plane
wave traveling in free space Find
1)
2) nstPH x 31321
3) origin the 0 tH
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k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
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Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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0amp
r
ro
xE and yH are in phase
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(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
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jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
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edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
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Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
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edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
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edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
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edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 7 7222012
k
Allowing the permittivity to be a complex constant (to be explained later) implies that the wave number may be complex and it is called the complex propagation constant
The propagation constant of an electromagnetic wave is a measure of the change
undergone by the amplitude of the wave as it propagates in a given direction The
propagation constant itself measures change per metre but is otherwise dimensionless
The quantity measured such as voltage or electric field intensity is expressed as a
sinusoidal phasor The phase of the sinusoid varies with distance which results in the
propagation constant being a complex number the imaginary part being caused by the phase change
For a One dimensional problem xxs zE aE ˆ the wave equation reduces to
zEkdz
zEdxs
xs 2
2
2
Define
jjk
So the solution is given by
zjzxo
zjzxo
zjkxo
zjkxo
zxo
zxoxs
eeEeeE
eEeE
eEeEzE
Transferring to time domain and considering only the forward part
zteEtzE zxox cos
Define the complex permittivity (dipole oscillations and conduction electrons and holes) as
rrororo jjj
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edition 2012
Dr Naser Abu-Zaid Page 8 7222012
2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
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edition 2012
Dr Naser Abu-Zaid Page 9 7222012
Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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edition 2012
Dr Naser Abu-Zaid Page 10 7222012
0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
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Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
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Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
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edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 8 7222012
2
1
1
1
j
jjk
With
2
1
2
1
1
2
ReRe
jk
2
1
2
1
1
2
ImIm
jk
Clearly from the time domain expression of tzE x the phase velocity is given
by
s
mv p
And the wave length is (distance required to change the phase by 2 )
m
2
And the magnetic field associated with the forward propagating part is (can be found through the use of Maxwellrsquos equations)
With the intrinsic impedance being a complex quantity given by
je
j
j
1
1
jzjzxozjzxo
zjkxozxoys
eeeE
eeE
eE
eE
zH
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 9 7222012
Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 10 7222012
0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
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edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
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Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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Since
zteEtzE zxox cos
zteE
tzH zxoy cos
Then xE leads yH by And you may do the same for the backward wave
Lossless medium (Perfect dielectric)
ro 0
01
1
2
ReRe
2
1
2
jk
1
1
2
ImIm
2
1
2
jk
s
mcv
rr
p
1
mf
c
rr
o
rr
22
0amp
1
1
r
ro
j
r
ro
ro
ro
e
j
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0amp
r
ro
xE and yH are in phase
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Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
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edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 10 7222012
0amp
r
ro
xE and yH are in phase
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
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Illustration of skin depth
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edition 2012
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POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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edition 2012
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Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
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Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
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Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
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Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 11 7222012
(Lossy Dielectrics)
ss
sss
j
jjj
EE
EEH
But on the other hand
sssss jj EEEJH
Comparing the two equations
Also from the second equation
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 12 7222012
jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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edition 2012
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2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
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Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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jjJ
J
d
c
cJ and dJ are o90 out of time phase and we identify the material as having
large losses or small losses depending on the magnitude of the loss tangent defined by
tantangnet Loss
is the angle by which dJ lead the total current density J
Good Dielectric Approximations
losses Small1
1
or
2
1
1
jjjjk
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Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
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Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
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edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 13 7222012
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
2
ReRe
jk
8
11ImIm
2
jk
21
j
Why Complex Permittivity Loss mechanism occurs in dielectrics even in the absence of free electrons
0 this is due to rotation of the dipoles to align with applied time varying
field or due to the net shift of the electron cloud with respect to the positive
nucleus At high frequencies the polarization P of the material is out of time
phase with applied field This loss mechanism is modeled by a complex permittivity as shown previously even with zero conductivity
j
So again from Ampers law
tyconductividielectricastreated
bemayormechanismLoss
s
mechanismstorageEnergy
sss
d
jjj
EEEH
J
And the loss tangent shown earlier is
tantangnet Loss
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
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Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
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edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
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edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
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edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
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edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
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edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
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edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
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edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
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edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
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edition 2012
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Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 14 7222012
Even if the conductivity is nonzero 0 but quite small still we may write
Ampers law as
)(
tyconductividielectricJJ
EEE
EEH
astreatedbemayor
mechanismLossDielectric
s
mechanismstorageEnergy
s
mechanismloss
Conductor
s
sss
dc
j
jj
So we may write
d
mechanismstorageEnergy
s
typermittiviEffective
mechanismslossdielectricandConductor
s
sss
j
jj
J
tyconductiviEffective
EE
EEH
And the loss tangent is defined as
tantangnet Loss
Depth of penetration (Skin Depth) Consider a forward traveling wave in a lossy dielectric
zteEtzE zxox cos
At
1z the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
The quantity
1
is called the skin depth or depth of penetration
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
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edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 15 7222012
Illustration of skin depth
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 16 7222012
POYNTINGrsquoS THEOREM (POWER THEOREM)
t
DJH
Left dot both sides with E then using the identity
HEEHHE
And with some vector manipulations one can obtain (Follow text book)
Theorem sPoynting the of form alDifferenti
2
1
2
1
HBEDJEHE
tt
Integrating over a volume v enclosed by a surface s
vv v vt
dvt
dvdv HBEDJEHE2
1
2
1
Upon using the divergence theorem for LHS
v v vs
dvt
dvt
dvds HBEDJEHE2
1
2
1
s
v v v
d
dvt
dvt
dv
sHE
HBEDJE2
1
2
1 volumeinto flowingpower Total
s
dsHE volumeofout flowingpower Total
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 17 7222012
Illustration of power balance for EM fields
And the instantaneous Poynting vector S (or instantaneous power density) is defined as
2m
WHES
For UPW with
xx tzEtz aE ˆ
yy tzHtz aH ˆ
Then
zzyyxx StzHtzE aaaHES ˆˆˆ
1) Perfect Dielectric (forward part only)
ztEtzE xox cos
ztE
tzH xoy
cos
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 18 7222012
ztE
tzS xoz
2
2
cos
2) Lossy Dielectric
zteEtzE zxox cos
zteE
tzH zxoy cos
ztzte
EtzS zxo
z coscos 22
The average power density (time averaged Poyntingrsquos vector) is (for time harmonic case)
T
zxo
T
dtzteE
T
dtT
cos22cos
2
11
1
22
SS
cos2
11 22
zxo
T
eE
dtT
SS
The above expression is easily evaluated using phasors by defining
Re2
1ss HES
Doing it for lossy dielectric (Sinusoidal wave)
Re2
1
jzjzxozjzxo eee
EeeES
cos2
1 22
zxo eE S
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 19 7222012
Try to fill the rest
Good conductors Approximations (Skin effect)
losses High1
1
or
2
1
1
jjjjk
2
1
1
j
The above two exact expressions may be approximated using the binomial expansion
1
2
111 2
xx
nnnxx
n
Hence
f
Solution
)(MHz
f
r
S
z=0
S
z=10m
1 415 012 2717 2582
100 345 0035
3000 32 00009 238
Example 125 At frequencies of 1 100 and 3000MHz the dielectric constant
of ice made from pure water has values 0f 415 345 and 32 respectively
while the loss tangent is 012 0035 and 00009 also respectively If a UPW
with amplitude of 100(Vm) z=0 is propagating through the ice fine the time
average power density z=0 and z=10m for each frequency
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 20 7222012
oj45
Consider a forward traveling wave
zteEtzE zxox cos
fzteEtzEfz
xox
cos
fzteEEtzJfz
xoxx
cos
At
f
z11
the wave is attenuated by a factor of
amplitudemaximum sit Of
1
1
370
eee z
f1
Which is the skin depth again or depth of penetration
For copper m
S71085 o
fcopper
0660
f copper
Hz60 mm538
GHz10 mm41066
After a few skin depths within the conductor all fields are almost zero
2
2
pv
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 21 7222012
o
jj 452
111
If
fzteEtzEfz
xox
cos
Then
o
z
xoy
zteEtzH 45cos
2
yH lags xE by o45
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 22 7222012
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 23 7222012
Attacking the power problem (in good conductors)
zteEtzE
z
xox cos
o
z
xoy
zteEtzH 45cos
2
zteJ
zteEtzJ
z
xo
z
xox
cos
cos
z
z
xoss eE aHES ˆ4
Re2
12
2
The total average power (loss) crossing the conductor surface at 0z
22
0 0 0
2
2
44
ˆˆ4
xoxo
z
b
y
L
x
z
z
z
xo
S
L
JbL
EbL
dxdyeEdP
aasS
What result would be obtained for the power loss if it is assumed that the total current is distributed uniformly in one skin depth
To calculate the total current crossing the surface at 0x rewrite in phasor form
zj
xoxs eJzJ
1
So
y
x
z
by
0y
Lx
UPW
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 24 7222012
j
bJdydzeJdI xo
z
b
y
zj
xo
S
s
10 0
1 sJ
and
4cos
2
t
bJtI xo
Assuming this current is distributed uniformly with current density S
IJ uniform
through the cross section bS then
4cos
2
t
J
S
IJ xo
uniform
Then the total instantaneous power dissipated in volume of one skin depth thickness is
4cos
2
4cos
2
1
22
0 0 0
2
tbLJ
dxdydztJ
dvP
xo
z
b
y
L
x
xo
v
uniformLinsEJ
And the time average power loss within this volume is
bLJ
dtPP xo
T
LL ins 4
2
This is exactly the same formula obtained before Conclusion The average power loss in a conductor with skin effect may be calculated assuming that the total current is distributed uniformly in one skin depth Or the resistance of width b and length L of an infinitely thick slab with skin effect is the same as the resistance of a rectangular slab of width b length
L and thickness without skin effect
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 25 7222012
S
LRdc
b
LRac
For a circular cross-section wire with radius a and at high frequencies
a
L
S
LR
2
More often the surface or skin resistance is defined as the
real part of the intrinsic impedance for a good conductor Thus
fRs
1
since
o
jj 452
111
b
L
With Skin effect Without Skin effect
b
L
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 26 7222012
WAVE POLARIZATION
Polarization is defined as The locus that the tip of the E field traces
as time varies for a fixed point in space Or the time-varying behavior of E at a given point in space
Consider a wave propagating in the negative z-direction
tztztz yyxx ˆˆ EΕ aa E
Assume each component have a sinusoidal time dependence and Since
tjet Recos then each component maybe written as a real
part of some complex quantity (complex phasor)
phasor Complex
with
xjjkzx
jkzxx
tjxx
eeEeEzE
ezEtz
ˆ
ˆReE
phasor Complex
with
yjjkzy
jkzyy
tjyy
eeEeEzE
ezEtz
ˆ
ˆReE
So the time domain representation is obtained as
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 27 7222012
xxkztj
xx kztEeEtz x
cosReE
yy
kztj
yy kztEeEtz y
cosReE
Three cases are to be considered Case1 Linear polarization
210 nnxy with
Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
kztkzttz yx cos5ˆcos10ˆ aaE
Solution
Take 0z
ttt yx cos5ˆcos10ˆ0 aaE
And since tt coscos
ttt yx cos5ˆcos10ˆ0 aa E
2
1tan
cos10
cos5tan0
cos125cos125
cos5cos100
11
2
2222
t
tt
tt
ttt
E
E
x
y
0t 125
2
t
t
125
50tan 1
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 28 7222012
Linearly polarized with an angle of o562650tan 1
Note that y 0x and 0xy So from the beginning we may state
that the polarization is linear but with what angle
Case2 Circular polarization
210
LHCPor CCW for 22
1
RHCPor CW for 22
1
n
n
n
EE
xy
yx
If the direction of propagation is in the positive z-direction then the phases for CW and CCW must be reversed Example Find the polarization (linear circular elliptical) and sense of rotation for the uniform plane wave whose electric field is given by
2
5cos10ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
2
5cos10ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
5cos
ttt yx sin10ˆcos10ˆ0 aa E
ttt
tt
tt
ttt
tantancos10
sin10tan0
10100sincos100
sin10cos100
11
22
2222
E
E
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 29 7222012
Since the wave is propagating in negative z-direction this is a RHCP circular polarization or CW polarization Note that
10 yx EE 2
5 y 0x
and
12
2
1
2
50
2
5xy
So from the beginning we may state that the polarization is CW circular
Case3 Elliptical polarization
210
LHEPor CCW for 0
RHEPor CW for 0
2
Or
LHEPor CCW for 22
1
RHEPor CW for 22
1
n
n
n
n
EE
xy
xy
yx
x
y
0t
10
2
t
t
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 30 7222012
The curve traced is a tilted ellipse
2cos22
1
2cos22
1
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
OB
OAAR
axisminor
axismajor oAxial rati
And the tilt angle wrt the y-axis is
cos
2tan
2
1
2 22
1
yx
yx
EE
EE
Example Find the polarization (linear circular elliptical) sense of rotation axial ratio AR and the tilt angle for the uniform plane wave whose electric field is given by
2cos50ˆcos10ˆ
kztkzttz yx aaE
Solution Take 0z
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 31 7222012
2cos5ˆcos10ˆ0
ttt yx aaE
And since tt
sin2
cos
ttt yx sin5ˆcos10ˆ0 aa E
tt
tt
ttt
tan50tancos10
sin5tan0
sin5cos100
11
2222
E
E
This is an elliptical polarization with
52
2cos51025105102
1
2cos22
1
102
2cos51025105102
1
2cos22
1
224422
224422
224422
224422
yxyxyx
yxyxyx
EEEEEEOB
EEEEEEOA
25
10
axisminor
axismajor ratio Axial
OB
OAAR
And the tilt angle is
22cos
510
5102tan
2
1
2
cos2
tan2
1
2
22
1
22
1
yx
yx
EE
EE
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 32 7222012
Since the wave is propagating in negative z-direction this is a RH elliptical polarization or CW elliptical polarization Note that
yx EE 2
y 0x
and
02
2
1
20
2xy
So from the beginning we may state that the polarization is CW elliptical
copyFrom Wikipedia the free encyclopedia
x
y
0t
10
2
t
t
5
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 33 7222012
THE ELECTROMAGNETIC SPECTRUM
wherec = 299792458 ms is the speed of light in vacuum and
h = 662606896(33)times10minus34 J s = 413566733(10)times10minus15 eV s is Plancks constant[5]
copyFrom Wikipedia the free encyclopedia
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
Dr Naser Abu-Zaid Lecture notes in electromagnetic theory 1 Referenced to Engineering electromagnetics by Hayt 8th
edition 2012
Dr Naser Abu-Zaid Page 34 7222012
copyFrom Microwave Engineering by David M Pozar copy2005
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