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ElectromagnetismINEL 4151
Ch 10 Waves
Sandra Cruz-Pol, Ph. D.ECE UPRM
Mayagüez, PR
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Electromagnetic Spectrum
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Maxwell Equations Maxwell Equations in General Form in General Form
Differential formDifferential form Integral FormIntegral FormGaussGauss’’ss Law for Law for EE field.field.
GaussGauss’’ss Law for Law for HH field. Nonexistence field. Nonexistence of monopole of monopole
FaradayFaraday’’ss Law Law
AmpereAmpere’’ss Circuit Circuit LawLaw
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vD
0 B
t
BE
t
DJH
v
v
s
dvdSD
0s
dSB
sL
dSBt
dlE
sL
dSt
DJdlH
Who was NikolaTesla?
• Find out what inventions he made• His relation to Thomas Edison• Why is he not well know?
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Special case• Consider the case of a lossless medium
• with no charges, i.e. .
The wave equation can be derived from Maxwell equations as
What is the solution for this differential equation? • The equation of a wave!
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00v
022 EE c
Phasors & complex #’s
Working with harmonic fields is easier, but requires knowledge of phasor, let’s review
• complex numbers and• phasors
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COMPLEX NUMBERS:
• Given a complex number z
where
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sincos jrrrrejyxz j
magnitude theis || 22 yxzr
angle theis tan 1
x
y
Review:
• Addition, • Subtraction, • Multiplication, • Division, • Square Root, • Complex Conjugate
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For a time varying phase
• Real and imaginary parts are:
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t
)cos(}Re{ trre j
)sin(}Im{ trre j
PHASORS
• For a sinusoidal current equals the real part of
• The complex term which results from dropping the time factor is called the phasor current, denoted by (s comes from sinusoidal)
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)cos()( tItI otjj
o eeI
joeI
tje
sI
To change back to time domain
• The phasor is multiplied by the time factor, ejt, and taken the real part.
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}Re{ tjseAA
Advantages of phasors
• Time derivative is equivalent to multiplying its phasor by jj
• Time integral is equivalent to dividing by the same term.
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sAjt
A
jA
tA s
Time-Harmonic fields (sines and cosines)
• The wave equation can be derived from Maxwell equations, indicating that the changes in the fields behave as a wave, called an electromagnetic field.
• Since any periodic wave can be represented as a sum of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify the equations.
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Maxwell Equations for Harmonic fields
Differential form* Differential form*
GaussGauss’’ss Law for E field. Law for E field.
GaussGauss’’ss Law for H field. Law for H field. No monopoleNo monopole
FaradayFaraday’’ss Law Law
AmpereAmpere’’ss Circuit Law Circuit Law
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t
DJH
t
BE
0 B
vD vE
0 H
HjE
EjJH
* (substituting and )ED BH
A wave
• Start taking the curl of Faraday’s law
• Then apply the vectorial identity
• And you’re left withAAA 2)(
s
sss
E
EjjEE2
2
)()(
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ss HjE
A Wave
022 EE
zo
zo eEe EE(z)
tzE
'
issolution general whose
),(
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LetLet’’s look at a special case for simplicity s look at a special case for simplicity without loosing generality:without loosing generality:
•The electric field has only an The electric field has only an xx-component-component•The field travels in The field travels in zz direction directionThen we haveThen we have
To change back to time domain
• From phasor
• …to time domain
)()( jzo
zoxs eEeEzE
xzteEtzE zo
)cos(),(
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Several Cases of Media
1. Free space 2. Lossless dielectric3. Lossy dielectric4. Good Conductor )or ,,(
),,0(
)or ,,0(
),,0(
oro
oror
oror
oo
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o=8.854 x 10-12[ F/m]
o= 4 x 10-7 [H/m]
1. Free space
There are no losses, e.g.
Let’s define• The phase of the wave• The angular frequency• Phase constant• The phase velocity of the wave• The period and wavelength• How does it moves?
xztAtzE
)sin(),(
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3. Lossy Dielectrics(General Case)
• In general, we had
• From this we obtain
• So , for a known material and frequency, we can find j
11
2 and 11
2
22
)(2 jj
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xzteEtzE zo
)cos(),(
j
222222
2222Re
Intrinsic Impedance, • If we divide E by H, we get units of ohms and the
definition of the intrinsic impedance of a medium at a given frequency.
][
j
j
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yzteE
tzH
xzteEtzE
zo
zo
ˆ)cos(),(
)cos(),(
*Not in-phase for a lossy medium
Note…
• E and H are perpendicular to one another• Travel is perpendicular to the direction of
propagation• The amplitude is related to the impedance• And so is the phase
yzteE
tzH
xzteEtzE
zo
zo
ˆ)cos(),(
)cos(),(
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Loss Tangent
• If we divide the conduction current by the displacement current
tangentosstan lEj
E
J
J
s
s
ds
cs
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http://fipsgold.physik.uni-kl.de/software/java/polarisation
Relation between tan and c
EjjEjEH
1
Ej c
'''1
isty permittivicomplex The
jjc
'
"tanas also defined becan tangent loss The
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2. Lossless dielectric)or ,,0( oror
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• Substituting in the general equations:
o
u
0
21
,0
Review: 1. Free Space
• Substituting in the general equations:
mAyztE
tzH
mVxztEtzE
o
o
o
/ˆ)cos(),(
/)cos(),(
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) ,,0( oo
3771200
21
/,0
o
o
o
oo
cu
c
4. Good Conductors
• Substituting in the general equations:
]/[ˆ)45cos(),(
]/[)cos(),(
mAyzteE
tzH
mVxzteEtzE
oz
o
o
zo
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) ,,( oro
o
u
45
22
2
Is water a good conductor???
SummaryAny medium Lossless
medium (=0)
Low-loss medium
(”/’<.01)
Good conductor
(”/’>100)Units
0 [Np/m]
[rad/m]
[ohm]
uucc
up/f
[m/s]
[m]
**In free space; **In free space; o =8.85 x 10-12 F/m o=4 x 10-7 H/m
j
j
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f
u p
1
2
f
f
f
u p
1
f
u
f
p
4
)1( j
11
2
2
Skin depth, • Is defined as the depth
at which the electric amplitude is decreased to 37%
/1at
%)37(37.01
1
zee
ez
[m] /1
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Short Cut …
• You can use Maxwell’s or use
where k is the direction of propagation of the wave, i.e., the direction in which the EM wave is traveling (a unitary vector).
HkE
EkH
ˆ
ˆ1
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Waves
• Static charges > static electric field, E
• Steady current > static magnetic field, H
• Static magnet > static magnetic field, H
• Time-varying current > time varying E(t) & H(t) that are interdependent > electromagnetic wave
• Time-varying magnet > time varying E(t) & H(t) that are interdependent > electromagnetic wave
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EM waves don’t need a medium to propagate
• Sound waves need a medium like air or water to propagate
• EM wave don’t. They can travel in free space in the complete absence of matter.
• Look at a “wind wave”; the energy moves, the plants stay at the same place.
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Exercises: Wave Propagation in Lossless materials
• A wave in a nonmagnetic material is given by
Find:
(a) direction of wave propagation,
(b) wavelength in the material
(c) phase velocity
(d) Relative permittivity of material
(e) Electric field phasor
Answer: +y, up= 2x108 m/s, 1.26m, 2.25,2.25,
[mA/m])510cos(50ˆ 9 ytzH
[V/m]57.12ˆ 5 yjexE
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Power in a wave
• A wave carries power and transmits it wherever it goes
Cruz-Pol, Electromagnetics UPRMSee Applet by Daniel Roth at
http://fipsgold.physik.uni-kl.de/software/java/polarisation
The power density per area carried by a wave is given by the Poynting vector.
Poynting Vector Derivation
• Start with E dot Ampere
• Apply vectorial identity
• And end up with
EHHEEH
BAABBA
:case in thisor
t
EEEEHE
t
EEHE
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t
EEEHEH
2
2
2
1
Poynting Vector Derivation…
• Substitute Faraday in 1rst term
t
EEEH
t
HH
2
2
2
1
t
HH
t
HH
2
:function square of derivativein As
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t
EEHE
t
H
2
22
22
HEEH
(-) sit' order, invert the if and
222
22E
t
H
t
EHE
Rearrange
Poynting Vector Derivation…
• Taking the integral wrt volume
• Applying theory of divergence
• Which simply means that the total power coming out of a volume is either due to the electric or magnetic field energy variations or is lost in ohmic losses.
dvEdvHEt
dvHEvvv
222
22
dvEdvHEt
dSHEvvS
222
22
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Total power across surface of volume
Rate of change of stored energy in E or H
Ohmic losses due to conduction current
Power: Poynting Vector
• Waves carry energy and information• Poynting says that the net power flowing out of a
given volume is = to the decrease in time in energy stored minus the conduction losses.
][W/m 2HE
P
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Represents the instantaneous power vector associated to the electromagnetic wave.
Time Average Power
• The Poynting vector averaged in time is
• For the general case wave:
*
00
Re2
111ss
TT
ave HEtdHET
tdT
PP
]/[ˆ
]/[ˆ
mAyeeE
H
mVxeeEE
zjzos
zjzos
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][W/m ˆcos2
222
zeE zo
ave
P
Total Power in W
The total power through a surface S is
• Note that the units now are in Watts• Note that power nomenclature, P is not cursive.• Note that the dot product indicates that the surface area
needs to be perpendicular to the Poynting vector so that all the power will go thru. (give example of receiver antenna)
][WdSPS
aveave P
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Exercises: Power
1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cm2. A radar radiates a wave with an electric field amplitude E that decays with distance as E(R)=3000/R [V/m], where R is the distance in meters. What is the radius of the unsafe region?
• Answer: 34.64 m
2. A 5GHz wave traveling In a nonmagnetic medium with r=9 is characterized by Determine the direction of wave travel and the average power density carried by the wave
• Answer: ][W/m 05.0ˆ 2xave P
[V/m])cos(2ˆ)cos(3ˆ xtzxtyE
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TEM wave
Transverse ElectroMagnetic = plane wave• There are no fields parallel to the direction of
propagation,• only perpendicular (transverse).• If have an electric field Ex(z)
– …then must have a corresponding magnetic field Hx(z)
• The direction of propagation is – aE x aH = ak
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z
x
y
z
x
PE 10.7
In free space, H=0.2 cos (t-x) z A/m. Find the total power passing through a
• square plate of side 10cm on plane x+z=1
• square plate at x=1, 0
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Answer; Ptot = 53mWHz
Ey
x
Answer; Ptot = 0mW!
Polarization of a waveIEEE Definition:
The trace of the tip of the E-field vector as a function of time seen from behind.
Simple cases• Vertical, Ex
• Horizontal, Ey
xztEzE
eEzE
ox
zjoxs
ˆ)cos()(
)(
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x
x
y
y
z
x
x
y
y
Dual-Pol in Weather Radars• Dual polarization radars can
estimate several return signal properties beyond those available from conventional, single polarization Doppler systems.
• Hydrometeors: Shape, Direction, Behavior, Type, etc…
• Events: Development, identification, extinction
ZHH ZVV
ZHV ZVH
Lineal Typical•Horizontal•Vertical
Dra. Leyda León
Polarization:Why do we care?? • Antenna applications –
– Antenna can only TX or RX a polarization it is designed to support. Straight wires, square waveguides, and similar rectangular systems support linear waves (polarized in one direction, often) Round waveguides, helical or flat spiral antennas produce circular or elliptical waves.
• Remote Sensing and Radar Applications – – Many targets will reflect or absorb EM waves differently for different
polarizations. Using multiple polarizations can give different information and improve results.
• Absorption applications – – Human body, for instance, will absorb waves with E oriented from head to
toe better than side-to-side, esp. in grounded cases. Also, the frequency at which maximum absorption occurs is different for these two polarizations. This has ramifications in safety guidelines and studies.
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Polarization• In general, plane wave has 2 components; in x & y
• And y-component might be out of phase wrt to x-component, is the phase difference between x and y.
Ey ExzE yx ˆˆ)(
zj
oy
zjox
e E E
e E E
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x
yEy
Ex
y
x
Front View
Several Cases
• Linear polarization: y-x =0o or ±180on
• Circular polarization: y-x =±90o & Eox=Eoy
• Elliptical polarization: y-x=±90o & Eox≠Eoy, or ≠0o or ≠180on even if Eox=Eoy
• Unpolarized- natural radiation
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Linear polarization
• =0
• @z=0 in time domain
t)cos(
t)cos(
yoy
xox
E E
E E
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zjoy
zjox
e E E
e E E
x
yEy
Ex
Front View
y
x
Back View:
Circular polarization• Both components have
same amplitude Eox=Eoy,
• = y- x= -90o = Right circular polarized (RCP)
• =+90o = LCP ˆˆˆˆ
:phasorin
)90tcos(
t)cos(
90
o
yjEExe EyExE
E E
E E
yoxoj
yoxo
yoy
xox
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Elliptical polarization
• X and Y components have different amplitudes Eox≠Eoy,
and =±90o
• Or ≠±90o and Eox=Eoy,
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Polarization example
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Polarizing glasses
Unpolarizedradiation enters
Nothing comes out this time.
All light comes out
Example
• Determine the polarization state of a plane wave with electric field:
a. b.
c.
d.
)45z-t4sin(y-)30z-tcos(3ˆ),( oo xtzE
)45z-t8sin(y)45z-tcos(3ˆ),( oo xtzE
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)45z-t4sin(y-)45z-tcos(4ˆ),( oo xtzEa. Elliptic
b. -90, RHEP
c. +90, LHCP
d. -90, RHCP
)(cos)180(c
)sin()180sin(o
o
os )(s)90(c
)cos()90sin(o
o
inos
Cell phone & brain
• Computer model for Cell phone Radiation inside the Human Brain
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Radar bandsBand Name
Nominal FreqRange
Specific Bands Application
HF, VHF, UHF 3-30 MHz0, 30-300 MHz, 300-1000MHz
138-144 MHz216-225, 420-450 MHz890-942
TV, Radio,
L 1-2 GHz (15-30 cm) 1.215-1.4 GHz Clear air, soil moist
S 2-4 GHz (8-15 cm) 2.3-2.5 GHz2.7-3.7>
Weather observationsCellular phones
C 4-8 GHz (4-8 cm) 5.25-5.925 GHzTV stations, short range
Weather
X 8-12 GHz (2.5–4 cm) 8.5-10.68 GHzCloud, light rain, airplane
weather. Police radar.
Ku 12-18 GHz 13.4-14.0 GHz, 15.7-17.7 Weather studies
K 18-27 GHz 24.05-24.25 GHz Water vapor content
Ka 27-40 GHz 33.4-36.0 GHz Cloud, rain
V 40-75 GHz 59-64 GHz Intra-building comm.
W 75-110 GHz 76-81 GH, 92-100 GHz Rain, tornadoes
millimeter 110-300 GHz Tornado chasersCruz-Pol, Electromagnetics UPRM
Microwave Oven
Most food is lossy media at microwave frequencies, therefore EM power is lost in the food as heat.
• Find depth of penetration if chicken which at 2.45 GHz has the complex permittivity given.
The power reaches the inside as soon as the oven in turned on! [/m] 2817.4
)30(2
j
jc
f
jj co
cm 3.21/1 Cruz-Pol, Electromagnetics UPRM
)130( joc
Decibel Scale
• In many applications need comparison of two powers, a power ratio, e.g. reflected power, attenuated power, gain,…
• The decibel (dB) scale is logarithmic
• Note that for voltages, the log is multiplied by 20 instead of 10.
2
12
2
21
2
1
2
1
log20log10log10][V
V
/RV
/RV
P
PdBG
P
P G
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Attenuation rate, A
• Represents the rate of decrease of the magnitude of Pave(z) as a function of propagation distance
]Np/m[68.8]dB/m[
where
[dB] -z8.68- log20
log100
log10
dB
dB
2
zez
e)(P
(z)PA z
ave
ave
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Submarine antennaA submarine at a depth of 200m uses a wire
antenna to receive signal transmissions at 1kHz. • Determine the power density incident upon the
submarine antenna due to the EM wave with |Eo|= 10V/m.
• [At 1kHz, sea water has r=81, =4].
• At what depth the amplitude of E has decreased to 1% its initial value at z=0 (sea surface)?
][W/m ˆcos2
222
zeE zo
ave
P
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Exercise: Lossy media propagation
For each of the following determine if the material is low-loss dielectric, good conductor, etc.
(a) Glass with r=1, r=5 and =10-12 S/m at 10 GHZ
(b) Animal tissue with r=1, r=12 and =0.3 S/m at 100 MHZ
(c) Wood with r=1, r=3 and =10-4 S/m at 1 kHZ
Answer:(a) low-loss,
xNp/mr/mcmupxc
(b) general, cmupxm/scj31.7
(c) Good conductor, xxkm upxcj
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